LMFDB - Embedded newform 45.5.g.d.28.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [45,5,Mod(28,45)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(45, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3]))

N = Newforms(chi, 5, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("45.28");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 45 = 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	45.g (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$4.65164833877$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{10})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 25 $ x^4 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			28.1
Root			$-1.58114 + 1.58114i$ of defining polynomial
Character	$\chi$	$=$	45.28
Dual form			45.5.g.d.37.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.58114 + 1.58114i) q^{2} +11.0000i q^{4} +(-20.5548 - 14.2302i) q^{5} +(-5.00000 + 5.00000i) q^{7} +(-42.6907 - 42.6907i) q^{8} +(55.0000 - 10.0000i) q^{10} -173.925 q^{11} +(-110.000 - 110.000i) q^{13} -15.8114i q^{14} -41.0000 q^{16} +(3.16228 - 3.16228i) q^{17} +198.000i q^{19} +(156.533 - 226.103i) q^{20} +(275.000 - 275.000i) q^{22} +(534.425 + 534.425i) q^{23} +(220.000 + 585.000i) q^{25} +347.851 q^{26} +(-55.0000 - 55.0000i) q^{28} +996.117i q^{29} -1192.00 q^{31} +(747.879 - 747.879i) q^{32} +10.0000i q^{34} +(173.925 - 31.6228i) q^{35} +(1810.00 - 1810.00i) q^{37} +(-313.065 - 313.065i) q^{38} +(270.000 + 1485.00i) q^{40} -1549.52 q^{41} +(-1310.00 - 1310.00i) q^{43} -1913.18i q^{44} -1690.00 q^{46} +(-2235.73 + 2235.73i) q^{47} +2351.00i q^{49} +(-1272.82 - 577.116i) q^{50} +(1210.00 - 1210.00i) q^{52} +(-1619.09 - 1619.09i) q^{53} +(3575.00 + 2475.00i) q^{55} +426.907 q^{56} +(-1575.00 - 1575.00i) q^{58} -996.117i q^{59} -1378.00 q^{61} +(1884.72 - 1884.72i) q^{62} +1709.00i q^{64} +(695.701 + 3826.36i) q^{65} +(1870.00 - 1870.00i) q^{67} +(34.7851 + 34.7851i) q^{68} +(-225.000 + 325.000i) q^{70} +6893.77 q^{71} +(-3905.00 - 3905.00i) q^{73} +5723.72i q^{74} -2178.00 q^{76} +(869.626 - 869.626i) q^{77} +612.000i q^{79} +(842.747 + 583.440i) q^{80} +(2450.00 - 2450.00i) q^{82} +(4765.55 + 4765.55i) q^{83} +(-110.000 + 20.0000i) q^{85} +4142.58 q^{86} +(7425.00 + 7425.00i) q^{88} -11384.2i q^{89} +1100.00 q^{91} +(-5878.67 + 5878.67i) q^{92} -7070.00i q^{94} +(2817.59 - 4069.85i) q^{95} +(-7205.00 + 7205.00i) q^{97} +(-3717.26 - 3717.26i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 20 q^{7} + 220 q^{10} - 440 q^{13} - 164 q^{16} + 1100 q^{22} + 880 q^{25} - 220 q^{28} - 4768 q^{31} + 7240 q^{37} + 1080 q^{40} - 5240 q^{43} - 6760 q^{46} + 4840 q^{52} + 14300 q^{55} - 6300 q^{58}+ \cdots - 28820 q^{97}+O(q^{100}) $ 4 * q - 20 * q^7 + 220 * q^10 - 440 * q^13 - 164 * q^16 + 1100 * q^22 + 880 * q^25 - 220 * q^28 - 4768 * q^31 + 7240 * q^37 + 1080 * q^40 - 5240 * q^43 - 6760 * q^46 + 4840 * q^52 + 14300 * q^55 - 6300 * q^58 - 5512 * q^61 + 7480 * q^67 - 900 * q^70 - 15620 * q^73 - 8712 * q^76 + 9800 * q^82 - 440 * q^85 + 29700 * q^88 + 4400 * q^91 - 28820 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/45\mathbb{Z}\right)^\times$.

$n$	$11$	$37$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−1.58114	+	1.58114i	−0.395285	+	0.395285i	−0.876566	−	0.481281i	$-0.840172\pi$
$2$	−1.58114	+	1.58114i	−0.395285	+	0.395285i	0.481281	+	0.876566i	$0.340172\pi$
$3$		0			0
$3$		0			0
$4$			11.0000i			0.687500i
$5$	−20.5548	−	14.2302i	−0.822192	−	0.569210i
$5$	−20.5548	−	14.2302i	−0.822192	−	0.569210i
$6$		0			0
$7$	−5.00000	+	5.00000i	−0.102041	+	0.102041i	−0.756284	−	0.654243i	$-0.772987\pi$
$7$	−5.00000	+	5.00000i	−0.102041	+	0.102041i	0.654243	+	0.756284i	$0.272987\pi$
$8$	−42.6907	−	42.6907i	−0.667043	−	0.667043i
$9$		0			0
$10$	55.0000	−	10.0000i	0.550000	−	0.100000i
$11$	−173.925			−1.43740			−0.718699	−	0.695321i	$-0.755262\pi$
$11$	−173.925			−1.43740			−0.718699	+	0.695321i	$0.755262\pi$
$12$		0			0
$13$	−110.000	−	110.000i	−0.650888	−	0.650888i	0.302319	−	0.953207i	$-0.402239\pi$
$13$	−110.000	−	110.000i	−0.650888	−	0.650888i	−0.953207	+	0.302319i	$0.902239\pi$
$14$		−	15.8114i		−	0.0806703i
$15$		0			0
$16$	−41.0000			−0.160156
$17$	3.16228	−	3.16228i	0.0109421	−	0.0109421i	−0.701615	−	0.712557i	$-0.747537\pi$
$17$	3.16228	−	3.16228i	0.0109421	−	0.0109421i	0.712557	+	0.701615i	$0.247537\pi$
$18$		0			0
$19$			198.000i			0.548476i	0.961662	+	0.274238i	$0.0884257\pi$
$19$			198.000i			0.548476i	−0.961662	+	0.274238i	$0.911574\pi$
$20$	156.533	−	226.103i	0.391332	−	0.565257i
$21$		0			0
$22$	275.000	−	275.000i	0.568182	−	0.568182i
$23$	534.425	+	534.425i	1.01026	+	1.01026i	0.999947	+	0.0103082i	$0.00328126\pi$
$23$	534.425	+	534.425i	1.01026	+	1.01026i	0.0103082	+	0.999947i	$0.496719\pi$
$24$		0			0
$25$	220.000	+	585.000i	0.352000	+	0.936000i
$26$	347.851			0.514572
$27$		0			0
$28$	−55.0000	−	55.0000i	−0.0701531	−	0.0701531i
$29$			996.117i			1.18444i	0.805775	+	0.592222i	$0.201749\pi$
$29$			996.117i			1.18444i	−0.805775	+	0.592222i	$0.798251\pi$
$30$		0			0
$31$	−1192.00			−1.24037			−0.620187	−	0.784454i	$-0.712943\pi$
$31$	−1192.00			−1.24037			−0.620187	+	0.784454i	$0.712943\pi$
$32$	747.879	−	747.879i	0.730350	−	0.730350i
$33$		0			0
$34$			10.0000i			0.00865052i
$35$	173.925	−	31.6228i	0.141980	−	0.0258145i
$36$		0			0
$37$	1810.00	−	1810.00i	1.32213	−	1.32213i	0.410086	−	0.912047i	$-0.365499\pi$
$37$	1810.00	−	1810.00i	1.32213	−	1.32213i	0.912047	−	0.410086i	$-0.134501\pi$
$38$	−313.065	−	313.065i	−0.216804	−	0.216804i
$39$		0			0
$40$	270.000	+	1485.00i	0.168750	+	0.928125i
$41$	−1549.52			−0.921782			−0.460891	−	0.887457i	$-0.652470\pi$
$41$	−1549.52			−0.921782			−0.460891	+	0.887457i	$0.652470\pi$
$42$		0			0
$43$	−1310.00	−	1310.00i	−0.708491	−	0.708491i	0.257727	−	0.966218i	$-0.417027\pi$
$43$	−1310.00	−	1310.00i	−0.708491	−	0.708491i	−0.966218	+	0.257727i	$0.917027\pi$
$44$		−	1913.18i		−	0.988212i
$45$		0			0
$46$	−1690.00			−0.798677
$47$	−2235.73	+	2235.73i	−1.01210	+	1.01210i	−0.0121748	+	0.999926i	$0.503875\pi$
$47$	−2235.73	+	2235.73i	−1.01210	+	1.01210i	−0.999926	+	0.0121748i	$0.996125\pi$
$48$		0			0
$49$			2351.00i			0.979175i
$50$	−1272.82	−	577.116i	−0.509127	−	0.230846i
$51$		0			0
$52$	1210.00	−	1210.00i	0.447485	−	0.447485i
$53$	−1619.09	−	1619.09i	−0.576392	−	0.576392i	0.357515	−	0.933907i	$-0.383624\pi$
$53$	−1619.09	−	1619.09i	−0.576392	−	0.576392i	−0.933907	+	0.357515i	$0.883624\pi$
$54$		0			0
$55$	3575.00	+	2475.00i	1.18182	+	0.818182i
$56$	426.907			0.136131
$57$		0			0
$58$	−1575.00	−	1575.00i	−0.468193	−	0.468193i
$59$		−	996.117i		−	0.286158i	−0.989711	−	0.143079i	$-0.954300\pi$
$59$		−	996.117i		−	0.286158i	0.989711	−	0.143079i	$-0.0457004\pi$
$60$		0			0
$61$	−1378.00			−0.370331			−0.185165	−	0.982707i	$-0.559282\pi$
$61$	−1378.00			−0.370331			−0.185165	+	0.982707i	$0.559282\pi$
$62$	1884.72	−	1884.72i	0.490301	−	0.490301i
$63$		0			0
$64$			1709.00i			0.417236i
$65$	695.701	+	3826.36i	0.164663	+	0.905646i
$66$		0			0
$67$	1870.00	−	1870.00i	0.416574	−	0.416574i	−0.467447	−	0.884021i	$-0.654826\pi$
$67$	1870.00	−	1870.00i	0.416574	−	0.416574i	0.884021	+	0.467447i	$0.154826\pi$
$68$	34.7851	+	34.7851i	0.00752272	+	0.00752272i
$69$		0			0
$70$	−225.000	+	325.000i	−0.0459184	+	0.0663265i
$71$	6893.77			1.36754			0.683770	−	0.729698i	$-0.260339\pi$
$71$	6893.77			1.36754			0.683770	+	0.729698i	$0.260339\pi$
$72$		0			0
$73$	−3905.00	−	3905.00i	−0.732783	−	0.732783i	0.238387	−	0.971170i	$-0.423381\pi$
$73$	−3905.00	−	3905.00i	−0.732783	−	0.732783i	−0.971170	+	0.238387i	$0.923381\pi$
$74$			5723.72i			1.04524i
$75$		0			0
$76$	−2178.00			−0.377078
$77$	869.626	−	869.626i	0.146673	−	0.146673i
$78$		0			0
$79$			612.000i			0.0980612i	0.998797	+	0.0490306i	$0.0156132\pi$
$79$			612.000i			0.0980612i	−0.998797	+	0.0490306i	$0.984387\pi$
$80$	842.747	+	583.440i	0.131679	+	0.0911625i
$81$		0			0
$82$	2450.00	−	2450.00i	0.364366	−	0.364366i
$83$	4765.55	+	4765.55i	0.691763	+	0.691763i	0.962620	−	0.270857i	$-0.0873071\pi$
$83$	4765.55	+	4765.55i	0.691763	+	0.691763i	−0.270857	+	0.962620i	$0.587307\pi$
$84$		0			0
$85$	−110.000	+	20.0000i	−0.0152249	+	0.00276817i
$86$	4142.58			0.560111
$87$		0			0
$88$	7425.00	+	7425.00i	0.958807	+	0.958807i
$89$		−	11384.2i		−	1.43722i	−0.695415	−	0.718609i	$-0.744779\pi$
$89$		−	11384.2i		−	1.43722i	0.695415	−	0.718609i	$-0.255221\pi$
$90$		0			0
$91$	1100.00			0.132834
$92$	−5878.67	+	5878.67i	−0.694550	+	0.694550i
$93$		0			0
$94$		−	7070.00i		−	0.800136i
$95$	2817.59	−	4069.85i	0.312198	−	0.450953i
$96$		0			0
$97$	−7205.00	+	7205.00i	−0.765756	+	0.765756i	−0.977356	−	0.211600i	$-0.932133\pi$
$97$	−7205.00	+	7205.00i	−0.765756	+	0.765756i	0.211600	+	0.977356i	$0.432133\pi$
$98$	−3717.26	−	3717.26i	−0.387053	−	0.387053i
$99$		0			0
$100$	−6435.00	+	2420.00i	−0.643500	+	0.242000i
$101$	−17250.2			−1.69103			−0.845516	−	0.533950i	$-0.820707\pi$
$101$	−17250.2			−1.69103			−0.845516	+	0.533950i	$0.820707\pi$
$102$		0			0
$103$	3085.00	+	3085.00i	0.290791	+	0.290791i	0.837393	−	0.546602i	$-0.184079\pi$
$103$	3085.00	+	3085.00i	0.290791	+	0.290791i	−0.546602	+	0.837393i	$0.684079\pi$
$104$			9391.96i			0.868340i
$105$		0			0
$106$	5120.00			0.455678
$107$	6321.39	−	6321.39i	0.552135	−	0.552135i	−0.374922	−	0.927057i	$-0.622330\pi$
$107$	6321.39	−	6321.39i	0.552135	−	0.552135i	0.927057	+	0.374922i	$0.122330\pi$
$108$		0			0
$109$			1782.00i			0.149987i	0.997184	+	0.0749937i	$0.0238937\pi$
$109$			1782.00i			0.149987i	−0.997184	+	0.0749937i	$0.976106\pi$
$110$	−9565.89	+	1739.25i	−0.790569	+	0.143740i
$111$		0			0
$112$	205.000	−	205.000i	0.0163425	−	0.0163425i
$113$	−8686.78	−	8686.78i	−0.680302	−	0.680302i	0.279766	−	0.960068i	$-0.409743\pi$
$113$	−8686.78	−	8686.78i	−0.680302	−	0.680302i	−0.960068	+	0.279766i	$0.909743\pi$
$114$		0			0
$115$	−3380.00	−	18590.0i	−0.255577	−	1.40567i
$116$	−10957.3			−0.814305
$117$		0			0
$118$	1575.00	+	1575.00i	0.113114	+	0.113114i
$119$			31.6228i			0.00223309i
$120$		0			0
$121$	15609.0			1.06612
$122$	2178.81	−	2178.81i	0.146386	−	0.146386i
$123$		0			0
$124$		−	13112.0i		−	0.852758i
$125$	3802.64	−	15155.2i	0.243369	−	0.969934i
$126$		0			0
$127$	−19745.0	+	19745.0i	−1.22419	+	1.22419i	−0.258065	+	0.966128i	$0.583085\pi$
$127$	−19745.0	+	19745.0i	−1.22419	+	1.22419i	−0.966128	+	0.258065i	$0.916915\pi$
$128$	9263.89	+	9263.89i	0.565423	+	0.565423i
$129$		0			0
$130$	−7150.00	−	4950.00i	−0.423077	−	0.292899i
$131$	11400.0			0.664298			0.332149	−	0.943227i	$-0.392226\pi$
$131$	11400.0			0.664298			0.332149	+	0.943227i	$0.392226\pi$
$132$		0			0
$133$	−990.000	−	990.000i	−0.0559670	−	0.0559670i
$134$			5913.46i			0.329331i
$135$		0			0
$136$	−270.000			−0.0145978
$137$	−5707.91	+	5707.91i	−0.304114	+	0.304114i	−0.842621	−	0.538507i	$-0.818988\pi$
$137$	−5707.91	+	5707.91i	−0.304114	+	0.304114i	0.538507	+	0.842621i	$0.318988\pi$
$138$		0			0
$139$			3438.00i			0.177941i	0.996034	+	0.0889706i	$0.0283577\pi$
$139$			3438.00i			0.177941i	−0.996034	+	0.0889706i	$0.971642\pi$
$140$	347.851	+	1913.18i	0.0177475	+	0.0976111i
$141$		0			0
$142$	−10900.0	+	10900.0i	−0.540567	+	0.540567i
$143$	19131.8	+	19131.8i	0.935585	+	0.935585i
$144$		0			0
$145$	14175.0	−	20475.0i	0.674197	−	0.973841i
$146$	12348.7			0.579316
$147$		0			0
$148$	19910.0	+	19910.0i	0.908966	+	0.908966i
$149$			4695.98i			0.211521i	0.994392	+	0.105761i	$0.0337277\pi$
$149$			4695.98i			0.211521i	−0.994392	+	0.105761i	$0.966272\pi$
$150$		0			0
$151$	31418.0			1.37792			0.688961	−	0.724798i	$-0.258067\pi$
$151$	31418.0			1.37792			0.688961	+	0.724798i	$0.258067\pi$
$152$	8452.77	−	8452.77i	0.365857	−	0.365857i
$153$		0			0
$154$			2750.00i			0.115955i
$155$	24501.3	+	16962.5i	1.01983	+	0.706034i
$156$		0			0
$157$	−11000.0	+	11000.0i	−0.446266	+	0.446266i	−0.894111	−	0.447845i	$-0.852191\pi$
$157$	−11000.0	+	11000.0i	−0.446266	+	0.446266i	0.447845	+	0.894111i	$0.352191\pi$
$158$	−967.657	−	967.657i	−0.0387621	−	0.0387621i
$159$		0			0
$160$	−26015.0	+	4730.00i	−1.01621	+	0.184766i
$161$	−5344.25			−0.206175
$162$		0			0
$163$	−14300.0	−	14300.0i	−0.538221	−	0.538221i	0.384785	−	0.923006i	$-0.374276\pi$
$163$	−14300.0	−	14300.0i	−0.538221	−	0.538221i	−0.923006	+	0.384785i	$0.874276\pi$
$164$		−	17044.7i		−	0.633725i
$165$		0			0
$166$	−15070.0			−0.546886
$167$	989.793	−	989.793i	0.0354904	−	0.0354904i	−0.689139	−	0.724629i	$-0.742011\pi$
$167$	989.793	−	989.793i	0.0354904	−	0.0354904i	0.724629	+	0.689139i	$0.242011\pi$
$168$		0			0
$169$		−	4361.00i		−	0.152691i
$170$	142.302	−	205.548i	0.00492396	−	0.00711239i
$171$		0			0
$172$	14410.0	−	14410.0i	0.487088	−	0.487088i
$173$	−19653.6	−	19653.6i	−0.656673	−	0.656673i	0.297919	−	0.954591i	$-0.403708\pi$
$173$	−19653.6	−	19653.6i	−0.656673	−	0.656673i	−0.954591	+	0.297919i	$0.903708\pi$
$174$		0			0
$175$	−4025.00	−	1825.00i	−0.131429	−	0.0595918i
$176$	7130.94			0.230208
$177$		0			0
$178$	18000.0	+	18000.0i	0.568110	+	0.568110i
$179$			14087.9i			0.439685i	0.975535	+	0.219842i	$0.0705543\pi$
$179$			14087.9i			0.439685i	−0.975535	+	0.219842i	$0.929446\pi$
$180$		0			0
$181$	1298.00			0.0396203			0.0198101	−	0.999804i	$-0.493694\pi$
$181$	1298.00			0.0396203			0.0198101	+	0.999804i	$0.493694\pi$
$182$	−1739.25	+	1739.25i	−0.0525073	+	0.0525073i
$183$		0			0
$184$		−	45630.0i		−	1.34777i
$185$	−62960.9	+	11447.4i	−1.83962	+	0.334476i
$186$		0			0
$187$	−550.000	+	550.000i	−0.0157282	+	0.0157282i
$188$	−24593.0	−	24593.0i	−0.695819	−	0.695819i
$189$		0			0
$190$	1980.00	+	10890.0i	0.0548476	+	0.301662i
$191$	3952.85			0.108354			0.0541768	−	0.998531i	$-0.482747\pi$
$191$	3952.85			0.108354			0.0541768	+	0.998531i	$0.482747\pi$
$192$		0			0
$193$	−8525.00	−	8525.00i	−0.228865	−	0.228865i	0.583353	−	0.812219i	$-0.301740\pi$
$193$	−8525.00	−	8525.00i	−0.228865	−	0.228865i	−0.812219	+	0.583353i	$0.801740\pi$
$194$		−	22784.2i		−	0.605383i
$195$		0			0
$196$	−25861.0			−0.673183
$197$	−40837.7	+	40837.7i	−1.05227	+	1.05227i	−0.0537165	+	0.998556i	$0.517107\pi$
$197$	−40837.7	+	40837.7i	−1.05227	+	1.05227i	−0.998556	+	0.0537165i	$0.982893\pi$
$198$		0			0
$199$			41922.0i			1.05861i	0.848432	+	0.529305i	$0.177547\pi$
$199$			41922.0i			1.05861i	−0.848432	+	0.529305i	$0.822453\pi$
$200$	15582.1	−	34366.1i	0.389553	−	0.859151i
$201$		0			0
$202$	27275.0	−	27275.0i	0.668439	−	0.668439i
$203$	−4980.59	−	4980.59i	−0.120862	−	0.120862i
$204$		0			0
$205$	31850.0	+	22050.0i	0.757882	+	0.524688i
$206$	−9755.63			−0.229890
$207$		0			0
$208$	4510.00	+	4510.00i	0.104244	+	0.104244i
$209$		−	34437.2i		−	0.788379i
$210$		0			0
$211$	−29938.0			−0.672447			−0.336223	−	0.941782i	$-0.609150\pi$
$211$	−29938.0			−0.672447			−0.336223	+	0.941782i	$0.609150\pi$
$212$	17809.9	−	17809.9i	0.396270	−	0.396270i
$213$		0			0
$214$			19990.0i			0.436501i
$215$	8285.17	+	45568.4i	0.179236	+	0.985796i
$216$		0			0
$217$	5960.00	−	5960.00i	0.126569	−	0.126569i
$218$	−2817.59	−	2817.59i	−0.0592877	−	0.0592877i
$219$		0			0
$220$	−27225.0	+	39325.0i	−0.562500	+	0.812500i
$221$	−695.701			−0.0142442
$222$		0			0
$223$	−44735.0	−	44735.0i	−0.899576	−	0.899576i	0.0958227	−	0.995398i	$-0.469452\pi$
$223$	−44735.0	−	44735.0i	−0.899576	−	0.899576i	−0.995398	+	0.0958227i	$0.969452\pi$
$224$			7478.79i			0.149051i
$225$		0			0
$226$	27470.0			0.537826
$227$	30465.4	−	30465.4i	0.591228	−	0.591228i	−0.346735	−	0.937963i	$-0.612710\pi$
$227$	30465.4	−	30465.4i	0.591228	−	0.591228i	0.937963	+	0.346735i	$0.112710\pi$
$228$		0			0
$229$			51498.0i			0.982018i	0.871155	+	0.491009i	$0.163372\pi$
$229$			51498.0i			0.982018i	−0.871155	+	0.491009i	$0.836628\pi$
$230$	34737.6	+	24049.1i	0.656666	+	0.454615i
$231$		0			0
$232$	42525.0	−	42525.0i	0.790075	−	0.790075i
$233$	30626.7	+	30626.7i	0.564141	+	0.564141i	0.930481	−	0.366340i	$-0.119389\pi$
$233$	30626.7	+	30626.7i	0.564141	+	0.564141i	−0.366340	+	0.930481i	$0.619389\pi$
$234$		0			0
$235$	77770.0	−	14140.0i	1.40824	−	0.256043i
$236$	10957.3			0.196734
$237$		0			0
$238$	−50.0000	−	50.0000i	−0.000882706	−	0.000882706i
$239$			87658.3i			1.53461i	0.641284	+	0.767304i	$0.278402\pi$
$239$			87658.3i			1.53461i	−0.641284	+	0.767304i	$0.721598\pi$
$240$		0			0
$241$	−62668.0			−1.07898			−0.539488	−	0.841993i	$-0.681382\pi$
$241$	−62668.0			−1.07898			−0.539488	+	0.841993i	$0.681382\pi$
$242$	−24680.0	+	24680.0i	−0.421419	+	0.421419i
$243$		0			0
$244$		−	15158.0i		−	0.254602i
$245$	33455.3	−	48324.3i	0.557356	−	0.805070i
$246$		0			0
$247$	21780.0	−	21780.0i	0.356997	−	0.356997i
$248$	50887.4	+	50887.4i	0.827383	+	0.827383i
$249$		0			0
$250$	17950.0	+	29975.0i	0.287200	+	0.479600i
$251$	45742.3			0.726057			0.363029	−	0.931778i	$-0.381743\pi$
$251$	45742.3			0.726057			0.363029	+	0.931778i	$0.381743\pi$
$252$		0			0
$253$	−92950.0	−	92950.0i	−1.45214	−	1.45214i
$254$		−	62439.2i		−	0.967809i
$255$		0			0
$256$	−56639.0			−0.864243
$257$	32182.5	−	32182.5i	0.487252	−	0.487252i	−0.420186	−	0.907438i	$-0.638035\pi$
$257$	32182.5	−	32182.5i	0.487252	−	0.487252i	0.907438	+	0.420186i	$0.138035\pi$
$258$		0			0
$259$			18100.0i			0.269823i
$260$	−42089.9	+	7652.71i	−0.622632	+	0.113206i
$261$		0			0
$262$	−18025.0	+	18025.0i	−0.262587	+	0.262587i
$263$	1577.98	+	1577.98i	0.0228133	+	0.0228133i	0.718421	−	0.695608i	$-0.244865\pi$
$263$	1577.98	+	1577.98i	0.0228133	+	0.0228133i	−0.695608	+	0.718421i	$0.744865\pi$
$264$		0			0
$265$	10240.0	+	56320.0i	0.145817	+	0.801994i
$266$	3130.65			0.0442458
$267$		0			0
$268$	20570.0	+	20570.0i	0.286395	+	0.286395i
$269$			48525.2i			0.670598i	0.942112	+	0.335299i	$0.108837\pi$
$269$			48525.2i			0.670598i	−0.942112	+	0.335299i	$0.891163\pi$
$270$		0			0
$271$	53102.0			0.723057			0.361528	−	0.932361i	$-0.382255\pi$
$271$	53102.0			0.723057			0.361528	+	0.932361i	$0.382255\pi$
$272$	−129.653	+	129.653i	−0.00175245	+	0.00175245i
$273$		0			0
$274$		−	18050.0i		−	0.240423i
$275$	−38263.6	−	101746.i	−0.505964	−	1.34541i
$276$		0			0
$277$	66370.0	−	66370.0i	0.864992	−	0.864992i	−0.126921	−	0.991913i	$-0.540509\pi$
$277$	66370.0	−	66370.0i	0.864992	−	0.864992i	0.991913	+	0.126921i	$0.0405093\pi$
$278$	−5435.96	−	5435.96i	−0.0703374	−	0.0703374i
$279$		0			0
$280$	−8775.00	−	6075.00i	−0.111926	−	0.0774872i
$281$	−64352.4			−0.814989			−0.407495	−	0.913208i	$-0.633597\pi$
$281$	−64352.4			−0.814989			−0.407495	+	0.913208i	$0.633597\pi$
$282$		0			0
$283$	54250.0	+	54250.0i	0.677371	+	0.677371i	0.959405	−	0.282033i	$-0.0910088\pi$
$283$	54250.0	+	54250.0i	0.677371	+	0.677371i	−0.282033	+	0.959405i	$0.591009\pi$
$284$			75831.4i			0.940183i
$285$		0			0
$286$	−60500.0			−0.739645
$287$	7747.58	−	7747.58i	0.0940594	−	0.0940594i
$288$		0			0
$289$			83501.0i			0.999761i
$290$	9961.17	+	54786.5i	0.118444	+	0.651444i
$291$		0			0
$292$	42955.0	−	42955.0i	0.503788	−	0.503788i
$293$	−80777.2	−	80777.2i	−0.940922	−	0.940922i	0.0574276	−	0.998350i	$-0.481710\pi$
$293$	−80777.2	−	80777.2i	−0.940922	−	0.940922i	−0.998350	+	0.0574276i	$0.981710\pi$
$294$		0			0
$295$	−14175.0	+	20475.0i	−0.162884	+	0.235277i
$296$	−154541.			−1.76384
$297$		0			0
$298$	−7425.00	−	7425.00i	−0.0836111	−	0.0836111i
$299$		−	117573.i		−	1.31512i
$300$		0			0
$301$	13100.0			0.144590
$302$	−49676.2	+	49676.2i	−0.544672	+	0.544672i
$303$		0			0
$304$		−	8118.00i		−	0.0878419i
$305$	28324.5	+	19609.3i	0.304483	+	0.210796i
$306$		0			0
$307$	−88010.0	+	88010.0i	−0.933803	+	0.933803i	−0.997941	−	0.0641380i	$-0.979570\pi$
$307$	−88010.0	+	88010.0i	−0.933803	+	0.933803i	0.0641380	+	0.997941i	$0.479570\pi$
$308$	9565.89	+	9565.89i	0.100838	+	0.100838i
$309$		0			0
$310$	−65560.0	+	11920.0i	−0.682206	+	0.124037i
$311$	−44524.9			−0.460343			−0.230172	−	0.973150i	$-0.573929\pi$
$311$	−44524.9			−0.460343			−0.230172	+	0.973150i	$0.573929\pi$
$312$		0			0
$313$	5305.00	+	5305.00i	0.0541498	+	0.0541498i	0.733663	−	0.679513i	$-0.237809\pi$
$313$	5305.00	+	5305.00i	0.0541498	+	0.0541498i	−0.679513	+	0.733663i	$0.737809\pi$
$314$		−	34785.1i		−	0.352804i
$315$		0			0
$316$	−6732.00			−0.0674171
$317$	48367.0	−	48367.0i	0.481317	−	0.481317i	−0.424235	−	0.905552i	$-0.639457\pi$
$317$	48367.0	−	48367.0i	0.481317	−	0.481317i	0.905552	+	0.424235i	$0.139457\pi$
$318$		0			0
$319$		−	173250.i		−	1.70252i
$320$	24319.5	−	35128.2i	0.237495	−	0.343048i
$321$		0			0
$322$	8450.00	−	8450.00i	0.0814976	−	0.0814976i
$323$	626.131	+	626.131i	0.00600150	+	0.00600150i
$324$		0			0
$325$	40150.0	−	88550.0i	0.380118	−	0.838343i
$326$	45220.6			0.425501
$327$		0			0
$328$	66150.0	+	66150.0i	0.614868	+	0.614868i
$329$		−	22357.3i		−	0.206551i
$330$		0			0
$331$	29402.0			0.268362			0.134181	−	0.990957i	$-0.457160\pi$
$331$	29402.0			0.268362			0.134181	+	0.990957i	$0.457160\pi$
$332$	−52421.1	+	52421.1i	−0.475587	+	0.475587i
$333$		0			0
$334$			3130.00i			0.0280577i
$335$	−65048.1	+	11826.9i	−0.579622	+	0.105386i
$336$		0			0
$337$	14245.0	−	14245.0i	0.125430	−	0.125430i	−0.641605	−	0.767035i	$-0.721731\pi$
$337$	14245.0	−	14245.0i	0.125430	−	0.125430i	0.767035	+	0.641605i	$0.221731\pi$
$338$	6895.35	+	6895.35i	0.0603563	+	0.0603563i
$339$		0			0
$340$	−220.000	−	1210.00i	−0.00190311	−	0.0104671i
$341$	207319.			1.78291
$342$		0			0
$343$	−23760.0	−	23760.0i	−0.201957	−	0.201957i
$344$			111850.i			0.945188i
$345$		0			0
$346$	62150.0			0.519145
$347$	−8942.92	+	8942.92i	−0.0742712	+	0.0742712i	−0.743267	−	0.668995i	$-0.766725\pi$
$347$	−8942.92	+	8942.92i	−0.0742712	+	0.0742712i	0.668995	+	0.743267i	$0.266725\pi$
$348$		0			0
$349$			11178.0i			0.0917726i	0.998947	+	0.0458863i	$0.0146112\pi$
$349$			11178.0i			0.0917726i	−0.998947	+	0.0458863i	$0.985389\pi$
$350$	9249.66	−	3478.51i	0.0755074	−	0.0283960i
$351$		0			0
$352$	−130075.	+	130075.i	−1.04980	+	1.04980i
$353$	107394.	+	107394.i	0.861849	+	0.861849i	0.991553	−	0.129704i	$-0.0414027\pi$
$353$	107394.	+	107394.i	0.861849	+	0.861849i	−0.129704	+	0.991553i	$0.541403\pi$
$354$		0			0
$355$	−141700.	−	98100.0i	−1.12438	−	0.778417i
$356$	125226.			0.988087
$357$		0			0
$358$	−22275.0	−	22275.0i	−0.173801	−	0.173801i
$359$		−	33868.0i		−	0.262785i	−0.991330	−	0.131393i	$-0.958055\pi$
$359$		−	33868.0i		−	0.262785i	0.991330	−	0.131393i	$-0.0419448\pi$
$360$		0			0
$361$	91117.0			0.699174
$362$	−2052.32	+	2052.32i	−0.0156613	+	0.0156613i
$363$		0			0
$364$			12100.0i			0.0913235i
$365$	24697.4	+	135836.i	0.185381	+	1.01960i
$366$		0			0
$367$	−127985.	+	127985.i	−0.950226	+	0.950226i	−0.998819	−	0.0485926i	$-0.984526\pi$
$367$	−127985.	+	127985.i	−0.950226	+	0.950226i	0.0485926	+	0.998819i	$0.484526\pi$
$368$	−21911.4	−	21911.4i	−0.161799	−	0.161799i
$369$		0			0
$370$	81450.0	−	117650.i	0.594960	−	0.859386i
$371$	16190.9			0.117631
$372$		0			0
$373$	117700.	+	117700.i	0.845977	+	0.845977i	0.989628	−	0.143651i	$-0.0458843\pi$
$373$	117700.	+	117700.i	0.845977	+	0.845977i	−0.143651	+	0.989628i	$0.545884\pi$
$374$		−	1739.25i		−	0.0124342i
$375$		0			0
$376$	190890.			1.35023
$377$	109573.	−	109573.i	0.770940	−	0.770940i
$378$		0			0
$379$			113418.i			0.789594i	0.918768	+	0.394797i	$0.129185\pi$
$379$			113418.i			0.789594i	−0.918768	+	0.394797i	$0.870815\pi$
$380$	44768.4	+	30993.5i	0.310030	+	0.214636i
$381$		0			0
$382$	−6250.00	+	6250.00i	−0.0428305	+	0.0428305i
$383$	−78301.2	−	78301.2i	−0.533790	−	0.533790i	0.387908	−	0.921698i	$-0.373198\pi$
$383$	−78301.2	−	78301.2i	−0.533790	−	0.533790i	−0.921698	+	0.387908i	$0.873198\pi$
$384$		0			0
$385$	−30250.0	+	5500.00i	−0.204082	+	0.0371058i
$386$	26958.4			0.180934
$387$		0			0
$388$	−79255.0	−	79255.0i	−0.526457	−	0.526457i
$389$			119961.i			0.792758i	0.918087	+	0.396379i	$0.129733\pi$
$389$			119961.i			0.792758i	−0.918087	+	0.396379i	$0.870267\pi$
$390$		0			0
$391$	3380.00			0.0221087
$392$	100366.	−	100366.i	0.653152	−	0.653152i
$393$		0			0
$394$		−	129140.i		−	0.831895i
$395$	8708.91	−	12579.5i	0.0558174	−	0.0806252i
$396$		0			0
$397$	−97460.0	+	97460.0i	−0.618366	+	0.618366i	−0.945112	−	0.326746i	$-0.894048\pi$
$397$	−97460.0	+	97460.0i	−0.618366	+	0.618366i	0.326746	+	0.945112i	$0.394048\pi$
$398$	−66284.5	−	66284.5i	−0.418452	−	0.418452i
$399$		0			0
$400$	−9020.00	−	23985.0i	−0.0563750	−	0.149906i
$401$	−44524.9			−0.276894			−0.138447	−	0.990370i	$-0.544211\pi$
$401$	−44524.9			−0.276894			−0.138447	+	0.990370i	$0.544211\pi$
$402$		0			0
$403$	131120.	+	131120.i	0.807344	+	0.807344i
$404$		−	189752.i		−	1.16258i
$405$		0			0
$406$	15750.0			0.0955495
$407$	−314805.	+	314805.i	−1.90043	+	1.90043i
$408$		0			0
$409$		−	210492.i		−	1.25831i	−0.777278	−	0.629157i	$-0.783400\pi$
$409$		−	210492.i		−	1.25831i	0.777278	−	0.629157i	$-0.216600\pi$
$410$	−85223.4	+	15495.2i	−0.506980	+	0.0921782i
$411$		0			0
$412$	−33935.0	+	33935.0i	−0.199919	+	0.199919i
$413$	4980.59	+	4980.59i	0.0291998	+	0.0291998i
$414$		0			0
$415$	−30140.0	−	165770.i	−0.175004	−	0.962520i
$416$	−164533.			−0.950752
$417$		0			0
$418$	54450.0	+	54450.0i	0.311634	+	0.311634i
$419$		−	189405.i		−	1.07885i	−0.842032	−	0.539427i	$-0.818641\pi$
$419$		−	189405.i		−	1.07885i	0.842032	−	0.539427i	$-0.181359\pi$
$420$		0			0
$421$	−97462.0			−0.549884			−0.274942	−	0.961461i	$-0.588659\pi$
$421$	−97462.0			−0.549884			−0.274942	+	0.961461i	$0.588659\pi$
$422$	47336.1	−	47336.1i	0.265808	−	0.265808i
$423$		0			0
$424$			138240.i			0.768957i
$425$	2545.63	+	1154.23i	0.0140935	+	0.00639021i
$426$		0			0
$427$	6890.00	−	6890.00i	0.0377888	−	0.0377888i
$428$	69535.3	+	69535.3i	0.379593	+	0.379593i
$429$		0			0
$430$	−85150.0	−	58950.0i	−0.460519	−	0.318821i
$431$	−223352.			−1.20236			−0.601180	−	0.799113i	$-0.705303\pi$
$431$	−223352.			−1.20236			−0.601180	+	0.799113i	$0.705303\pi$
$432$		0			0
$433$	−91295.0	−	91295.0i	−0.486935	−	0.486935i	0.420402	−	0.907338i	$-0.361889\pi$
$433$	−91295.0	−	91295.0i	−0.486935	−	0.486935i	−0.907338	+	0.420402i	$0.861889\pi$
$434$			18847.2i			0.100061i
$435$		0			0
$436$	−19602.0			−0.103116
$437$	−105816.	+	105816.i	−0.554101	+	0.554101i
$438$		0			0
$439$		−	177858.i		−	0.922878i	−0.887172	−	0.461439i	$-0.847333\pi$
$439$		−	177858.i		−	0.922878i	0.887172	−	0.461439i	$-0.152667\pi$
$440$	−46959.8	−	258279.i	−0.242561	−	1.33409i
$441$		0			0
$442$	1100.00	−	1100.00i	0.00563052	−	0.00563052i
$443$	−131418.	−	131418.i	−0.669649	−	0.669649i	0.287986	−	0.957635i	$-0.407014\pi$
$443$	−131418.	−	131418.i	−0.669649	−	0.669649i	−0.957635	+	0.287986i	$0.907014\pi$
$444$		0			0
$445$	−162000.	+	234000.i	−0.818079	+	1.18167i
$446$	141464.			0.711177
$447$		0			0
$448$	−8545.00	−	8545.00i	−0.0425751	−	0.0425751i
$449$			159663.i			0.791977i	0.918255	+	0.395989i	$0.129598\pi$
$449$			159663.i			0.791977i	−0.918255	+	0.395989i	$0.870402\pi$
$450$		0			0
$451$	269500.			1.32497
$452$	95554.5	−	95554.5i	0.467708	−	0.467708i
$453$		0			0
$454$			96340.0i			0.467407i
$455$	−22610.3	−	15653.3i	−0.109215	−	0.0756106i
$456$		0			0
$457$	170635.	−	170635.i	0.817026	−	0.817026i	−0.168650	−	0.985676i	$-0.553941\pi$
$457$	170635.	−	170635.i	0.817026	−	0.817026i	0.985676	+	0.168650i	$0.0539408\pi$
$458$	−81425.5	−	81425.5i	−0.388177	−	0.388177i
$459$		0			0
$460$	204490.	−	37180.0i	0.966399	−	0.175709i
$461$	9218.04			0.0433747			0.0216874	−	0.999765i	$-0.493096\pi$
$461$	9218.04			0.0433747			0.0216874	+	0.999765i	$0.493096\pi$
$462$		0			0
$463$	242485.	+	242485.i	1.13116	+	1.13116i	0.989985	+	0.141172i	$0.0450870\pi$
$463$	242485.	+	242485.i	1.13116	+	1.13116i	0.141172	+	0.989985i	$0.454913\pi$
$464$		−	40840.8i		−	0.189696i
$465$		0			0
$466$	−96850.0			−0.445993
$467$	195356.	−	195356.i	0.895763	−	0.895763i	−0.0992952	−	0.995058i	$-0.531659\pi$
$467$	195356.	−	195356.i	0.895763	−	0.895763i	0.995058	+	0.0992952i	$0.0316588\pi$
$468$		0			0
$469$			18700.0i			0.0850151i
$470$	−100608.	+	145322.i	−0.455445	+	0.657865i
$471$		0			0
$472$	−42525.0	+	42525.0i	−0.190880	+	0.190880i
$473$	227842.	+	227842.i	1.01838	+	1.01838i
$474$		0			0
$475$	−115830.	+	43560.0i	−0.513374	+	0.193064i
$476$	−347.851			−0.00153525
$477$		0			0
$478$	−138600.	−	138600.i	−0.606607	−	0.606607i
$479$			413246.i			1.80110i	0.434751	+	0.900551i	$0.356836\pi$
$479$			413246.i			1.80110i	−0.434751	+	0.900551i	$0.643164\pi$
$480$		0			0
$481$	−398200.			−1.72112
$482$	99086.8	−	99086.8i	0.426503	−	0.426503i
$483$		0			0
$484$			171699.i			0.732955i
$485$	250626.	−	45568.4i	1.06547	−	0.193723i
$486$		0			0
$487$	46675.0	−	46675.0i	0.196801	−	0.196801i	−0.601826	−	0.798627i	$-0.705560\pi$
$487$	46675.0	−	46675.0i	0.196801	−	0.196801i	0.798627	+	0.601826i	$0.205560\pi$
$488$	58827.9	+	58827.9i	0.247026	+	0.247026i
$489$		0			0
$490$	23510.0	+	129305.i	0.0979175	+	0.538546i
$491$	−218845.			−0.907767			−0.453884	−	0.891061i	$-0.649962\pi$
$491$	−218845.			−0.907767			−0.453884	+	0.891061i	$0.649962\pi$
$492$		0			0
$493$	3150.00	+	3150.00i	0.0129603	+	0.0129603i
$494$			68874.4i			0.282231i
$495$		0			0
$496$	48872.0			0.198654
$497$	−34468.8	+	34468.8i	−0.139545	+	0.139545i
$498$		0			0
$499$		−	166518.i		−	0.668744i	−0.942441	−	0.334372i	$-0.891476\pi$
$499$		−	166518.i		−	0.668744i	0.942441	−	0.334372i	$-0.108524\pi$
$500$	166707.	+	41829.0i	0.666829	+	0.167316i
$501$		0			0
$502$	−72325.0	+	72325.0i	−0.286999	+	0.286999i
$503$	183744.	+	183744.i	0.726236	+	0.726236i	0.969868	−	0.243632i	$-0.0783389\pi$
$503$	183744.	+	183744.i	0.726236	+	0.726236i	−0.243632	+	0.969868i	$0.578339\pi$
$504$		0			0
$505$	354575.	+	245475.i	1.39035	+	0.962553i
$506$	293934.			1.14802
$507$		0			0
$508$	−217195.	−	217195.i	−0.841632	−	0.841632i
$509$		−	234088.i		−	0.903531i	−0.892137	−	0.451765i	$-0.850794\pi$
$509$		−	234088.i		−	0.903531i	0.892137	−	0.451765i	$-0.149206\pi$
$510$		0			0
$511$	39050.0			0.149548
$512$	−58668.2	+	58668.2i	−0.223801	+	0.223801i
$513$		0			0
$514$			101770.i			0.385206i
$515$	−19511.3	−	107312.i	−0.0735649	−	0.404607i
$516$		0			0
$517$	388850.	−	388850.i	1.45479	−	1.45479i
$518$	−28618.6	−	28618.6i	−0.106657	−	0.106657i
$519$		0			0
$520$	133650.	−	193050.i	0.494268	−	0.713942i
$521$	238278.			0.877825			0.438912	−	0.898530i	$-0.355364\pi$
$521$	238278.			0.877825			0.438912	+	0.898530i	$0.355364\pi$
$522$		0			0
$523$	−215270.	−	215270.i	−0.787010	−	0.787010i	0.193993	−	0.981003i	$-0.437856\pi$
$523$	−215270.	−	215270.i	−0.787010	−	0.787010i	−0.981003	+	0.193993i	$0.937856\pi$
$524$			125400.i			0.456705i
$525$		0			0
$526$	−4990.00			−0.0180355
$527$	−3769.43	+	3769.43i	−0.0135723	+	0.0135723i
$528$		0			0
$529$			291379.i			1.04123i
$530$	−105241.	−	72858.9i	−0.374655	−	0.259377i
$531$		0			0
$532$	10890.0	−	10890.0i	0.0384773	−	0.0384773i
$533$	170447.	+	170447.i	0.599977	+	0.599977i
$534$		0			0
$535$	−219890.	+	39980.0i	−0.768242	+	0.139680i
$536$	−159663.			−0.555745
$537$		0			0
$538$	−76725.0	−	76725.0i	−0.265077	−	0.265077i
$539$		−	408898.i		−	1.40747i
$540$		0			0
$541$	−353518.			−1.20786			−0.603931	−	0.797037i	$-0.706400\pi$
$541$	−353518.			−1.20786			−0.603931	+	0.797037i	$0.706400\pi$
$542$	−83961.6	+	83961.6i	−0.285813	+	0.285813i
$543$		0			0
$544$		−	4730.00i		−	0.0159832i
$545$	25358.3	−	36628.7i	0.0853743	−	0.123318i
$546$		0			0
$547$	52690.0	−	52690.0i	0.176098	−	0.176098i	−0.613555	−	0.789652i	$-0.710261\pi$
$547$	52690.0	−	52690.0i	0.176098	−	0.176098i	0.789652	+	0.613555i	$0.210261\pi$
$548$	−62787.0	−	62787.0i	−0.209078	−	0.209078i
$549$		0			0
$550$	221375.	+	100375.i	0.731818	+	0.331818i
$551$	−197231.			−0.649640
$552$		0			0
$553$	−3060.00	−	3060.00i	−0.0100062	−	0.0100062i
$554$			209880.i			0.683837i
$555$		0			0
$556$	−37818.0			−0.122335
$557$	−85634.5	+	85634.5i	−0.276019	+	0.276019i	−0.831517	−	0.555499i	$-0.812527\pi$
$557$	−85634.5	+	85634.5i	−0.276019	+	0.276019i	0.555499	+	0.831517i	$0.312527\pi$
$558$		0			0
$559$			288200.i			0.922296i
$560$	−7130.94	+	1296.53i	−0.0227390	+	0.00413436i
$561$		0			0
$562$	101750.	−	101750.i	0.322153	−	0.322153i
$563$	−377905.	−	377905.i	−1.19225	−	1.19225i	−0.976435	−	0.215810i	$-0.930761\pi$
$563$	−377905.	−	377905.i	−1.19225	−	1.19225i	−0.215810	−	0.976435i	$-0.569239\pi$
$564$		0			0
$565$	54940.0	+	302170.i	0.172104	+	0.946574i
$566$	−171554.			−0.535509
$567$		0			0
$568$	−294300.	−	294300.i	−0.912207	−	0.912207i
$569$		−	292005.i		−	0.901914i	−0.892546	−	0.450957i	$-0.851083\pi$
$569$		−	292005.i		−	0.901914i	0.892546	−	0.450957i	$-0.148917\pi$
$570$		0			0
$571$	−232618.			−0.713462			−0.356731	−	0.934207i	$-0.616109\pi$
$571$	−232618.			−0.713462			−0.356731	+	0.934207i	$0.616109\pi$
$572$	−210450.	+	210450.i	−0.643215	+	0.643215i
$573$		0			0
$574$			24500.0i			0.0743605i
$575$	−195065.	+	430212.i	−0.589989	+	1.30121i
$576$		0			0
$577$	226765.	−	226765.i	0.681121	−	0.681121i	−0.279132	−	0.960253i	$-0.590047\pi$
$577$	226765.	−	226765.i	0.681121	−	0.681121i	0.960253	+	0.279132i	$0.0900466\pi$
$578$	−132027.	−	132027.i	−0.395190	−	0.395190i
$579$		0			0
$580$	225225.	+	155925.i	0.669515	+	0.463511i
$581$	−47655.5			−0.141176
$582$		0			0
$583$	281600.	+	281600.i	0.828506	+	0.828506i
$584$			333415.i			0.977595i
$585$		0			0
$586$	255440.			0.743864
$587$	451538.	−	451538.i	1.31044	−	1.31044i	0.389358	−	0.921087i	$-0.372697\pi$
$587$	451538.	−	451538.i	1.31044	−	1.31044i	0.921087	−	0.389358i	$-0.127303\pi$
$588$		0			0
$589$		−	236016.i		−	0.680316i
$590$	−9961.17	−	54786.5i	−0.0286158	−	0.157387i
$591$		0			0
$592$	−74210.0	+	74210.0i	−0.211748	+	0.211748i
$593$	−396793.	−	396793.i	−1.12838	−	1.12838i	−0.990441	−	0.137937i	$-0.955953\pi$
$593$	−396793.	−	396793.i	−1.12838	−	1.12838i	−0.137937	−	0.990441i	$-0.544047\pi$
$594$		0			0
$595$	450.000	−	650.000i	0.00127110	−	0.00183603i
$596$	−51655.8			−0.145421
$597$		0			0
$598$	185900.	+	185900.i	0.519849	+	0.519849i
$599$		−	50375.1i		−	0.140398i	−0.997533	−	0.0701992i	$-0.977637\pi$
$599$		−	50375.1i		−	0.140398i	0.997533	−	0.0701992i	$-0.0223635\pi$
$600$		0			0
$601$	−151822.			−0.420326			−0.210163	−	0.977666i	$-0.567399\pi$
$601$	−151822.			−0.420326			−0.210163	+	0.977666i	$0.567399\pi$
$602$	−20712.9	+	20712.9i	−0.0571542	+	0.0571542i
$603$		0			0
$604$			345598.i			0.947321i
$605$	−320840.	−	222120.i	−0.876552	−	0.606844i
$606$		0			0
$607$	−65945.0	+	65945.0i	−0.178980	+	0.178980i	−0.790911	−	0.611931i	$-0.790393\pi$
$607$	−65945.0	+	65945.0i	−0.178980	+	0.178980i	0.611931	+	0.790911i	$0.290393\pi$
$608$	148080.	+	148080.i	0.400580	+	0.400580i
$609$		0			0
$610$	−75790.0	+	13780.0i	−0.203682	+	0.0370331i
$611$	491861.			1.31753
$612$		0			0
$613$	−204380.	−	204380.i	−0.543898	−	0.543898i	0.380771	−	0.924669i	$-0.375659\pi$
$613$	−204380.	−	204380.i	−0.543898	−	0.543898i	−0.924669	+	0.380771i	$0.875659\pi$
$614$		−	278312.i		−	0.738236i
$615$		0			0
$616$	−74250.0			−0.195675
$617$	−187874.	+	187874.i	−0.493511	+	0.493511i	−0.909410	−	0.415900i	$-0.863467\pi$
$617$	−187874.	+	187874.i	−0.493511	+	0.493511i	0.415900	+	0.909410i	$0.363467\pi$
$618$		0			0
$619$		−	617958.i		−	1.61279i	−0.591378	−	0.806395i	$-0.701416\pi$
$619$		−	617958.i		−	1.61279i	0.591378	−	0.806395i	$-0.298584\pi$
$620$	−186587.	+	269515.i	−0.485398	+	0.701131i
$621$		0			0
$622$	70400.0	−	70400.0i	0.181967	−	0.181967i
$623$	56921.0	+	56921.0i	0.146655	+	0.146655i
$624$		0			0
$625$	−293825.	+	257400.i	−0.752192	+	0.658944i
$626$	−16775.9			−0.0428092
$627$		0			0
$628$	−121000.	−	121000.i	−0.306808	−	0.306808i
$629$		−	11447.4i		−	0.0289339i
$630$		0			0
$631$	−256228.			−0.643529			−0.321764	−	0.946820i	$-0.604276\pi$
$631$	−256228.			−0.643529			−0.321764	+	0.946820i	$0.604276\pi$
$632$	26126.7	−	26126.7i	0.0654110	−	0.0654110i
$633$		0			0
$634$			152950.i			0.380514i
$635$	686831.	−	124878.i	1.70334	−	0.309699i
$636$		0			0
$637$	258610.	−	258610.i	0.637333	−	0.637333i
$638$	273932.	+	273932.i	0.672980	+	0.672980i
$639$		0			0
$640$	−58590.0	−	322245.i	−0.143042	−	0.786731i
$641$	370809.			0.902472			0.451236	−	0.892405i	$-0.350983\pi$
$641$	370809.			0.902472			0.451236	+	0.892405i	$0.350983\pi$
$642$		0			0
$643$	311380.	+	311380.i	0.753128	+	0.753128i	0.975062	−	0.221934i	$-0.0712369\pi$
$643$	311380.	+	311380.i	0.753128	+	0.753128i	−0.221934	+	0.975062i	$0.571237\pi$
$644$		−	58786.7i		−	0.141745i
$645$		0			0
$646$	−1980.00			−0.00474461
$647$	−438972.	+	438972.i	−1.04864	+	1.04864i	−0.0498886	+	0.998755i	$0.515887\pi$
$647$	−438972.	+	438972.i	−1.04864	+	1.04864i	−0.998755	+	0.0498886i	$0.984113\pi$
$648$		0			0
$649$			173250.i			0.411324i
$650$	76527.1	+	203493.i	0.181129	+	0.481639i
$651$		0			0
$652$	157300.	−	157300.i	0.370027	−	0.370027i
$653$	250475.	+	250475.i	0.587404	+	0.587404i	0.936928	−	0.349523i	$-0.113656\pi$
$653$	250475.	+	250475.i	0.587404	+	0.587404i	−0.349523	+	0.936928i	$0.613656\pi$
$654$		0			0
$655$	−234325.	−	162225.i	−0.546180	−	0.378125i
$656$	63530.2			0.147629
$657$		0			0
$658$	35350.0	+	35350.0i	0.0816465	+	0.0816465i
$659$			24618.3i			0.0566876i	0.999598	+	0.0283438i	$0.00902331\pi$
$659$			24618.3i			0.0566876i	−0.999598	+	0.0283438i	$0.990977\pi$
$660$		0			0
$661$	508838.			1.16460			0.582300	−	0.812974i	$-0.302153\pi$
$661$	508838.			1.16460			0.582300	+	0.812974i	$0.302153\pi$
$662$	−46488.6	+	46488.6i	−0.106079	+	0.106079i
$663$		0			0
$664$		−	406890.i		−	0.922871i
$665$	6261.31	+	34437.2i	0.0141587	+	0.0778726i
$666$		0			0
$667$	−532350.	+	532350.i	−1.19659	+	1.19659i
$668$	10887.7	+	10887.7i	0.0243997	+	0.0243997i
$669$		0			0
$670$	84150.0	−	121550.i	0.187458	−	0.270773i
$671$	239669.			0.532313
$672$		0			0
$673$	−66425.0	−	66425.0i	−0.146657	−	0.146657i	0.629966	−	0.776623i	$-0.283069\pi$
$673$	−66425.0	−	66425.0i	−0.146657	−	0.146657i	−0.776623	+	0.629966i	$0.783069\pi$
$674$			45046.6i			0.0991614i
$675$		0			0
$676$	47971.0			0.104975
$677$	583937.	−	583937.i	1.27406	−	1.27406i	0.330115	−	0.943941i	$-0.392912\pi$
$677$	583937.	−	583937.i	1.27406	−	1.27406i	0.943941	−	0.330115i	$-0.107088\pi$
$678$		0			0
$679$		−	72050.0i		−	0.156277i
$680$	5549.80	+	3842.17i	0.0120022	+	0.00830919i
$681$		0			0
$682$	−327800.	+	327800.i	−0.704758	+	0.704758i
$683$	258140.	+	258140.i	0.553368	+	0.553368i	0.927411	−	0.374044i	$-0.122029\pi$
$683$	258140.	+	258140.i	0.553368	+	0.553368i	−0.374044	+	0.927411i	$0.622029\pi$
$684$		0			0
$685$	198550.	−	36100.0i	0.423145	−	0.0769354i
$686$	75135.7			0.159661
$687$		0			0
$688$	53710.0	+	53710.0i	0.113469	+	0.113469i
$689$			356199.i			0.750333i
$690$		0			0
$691$	509318.			1.06668			0.533338	−	0.845902i	$-0.320937\pi$
$691$	509318.			1.06668			0.533338	+	0.845902i	$0.320937\pi$
$692$	216189.	−	216189.i	0.451462	−	0.451462i
$693$		0			0
$694$		−	28280.0i		−	0.0587165i
$695$	48923.6	−	70667.4i	0.101286	−	0.146302i
$696$		0			0
$697$	−4900.00	+	4900.00i	−0.0100863	+	0.0100863i
$698$	−17674.0	−	17674.0i	−0.0362763	−	0.0362763i
$699$		0			0
$700$	20075.0	−	44275.0i	0.0409694	−	0.0903571i
$701$	−705615.			−1.43592			−0.717962	−	0.696082i	$-0.754925\pi$
$701$	−705615.			−1.43592			−0.717962	+	0.696082i	$0.754925\pi$
$702$		0			0
$703$	358380.	+	358380.i	0.725159	+	0.725159i
$704$		−	297238.i		−	0.599735i
$705$		0			0
$706$	−339610.			−0.681351
$707$	86251.1	−	86251.1i	0.172554	−	0.172554i
$708$		0			0
$709$			463158.i			0.921376i	0.887562	+	0.460688i	$0.152397\pi$
$709$			463158.i			0.921376i	−0.887562	+	0.460688i	$0.847603\pi$
$710$	379157.	−	68937.7i	0.752147	−	0.136754i
$711$		0			0
$712$	−486000.	+	486000.i	−0.958686	+	0.958686i
$713$	−637035.	−	637035.i	−1.25309	−	1.25309i
$714$		0			0
$715$	−121000.	−	665500.i	−0.236686	−	1.30178i
$716$	−154967.			−0.302283
$717$		0			0
$718$	53550.0	+	53550.0i	0.103875	+	0.103875i
$719$			751926.i			1.45451i	0.686366	+	0.727256i	$0.259205\pi$
$719$			751926.i			1.45451i	−0.686366	+	0.727256i	$0.740795\pi$
$720$		0			0
$721$	−30850.0			−0.0593451
$722$	−144069.	+	144069.i	−0.276373	+	0.276373i
$723$		0			0
$724$			14278.0i			0.0272389i
$725$	−582729.	+	219146.i	−1.10864	+	0.416924i
$726$		0			0
$727$	−353705.	+	353705.i	−0.669225	+	0.669225i	−0.957537	−	0.288311i	$-0.906906\pi$
$727$	−353705.	+	353705.i	−0.669225	+	0.669225i	0.288311	+	0.957537i	$0.406906\pi$
$728$	−46959.8	−	46959.8i	−0.0886061	−	0.0886061i
$729$		0			0
$730$	−253825.	−	175725.i	−0.476309	−	0.329752i
$731$	−8285.17			−0.0155048
$732$		0			0
$733$	−45650.0	−	45650.0i	−0.0849636	−	0.0849636i	0.663348	−	0.748311i	$-0.269135\pi$
$733$	−45650.0	−	45650.0i	−0.0849636	−	0.0849636i	−0.748311	+	0.663348i	$0.769135\pi$
$734$		−	404724.i		−	0.751220i
$735$		0			0
$736$	799370.			1.47568
$737$	−325240.	+	325240.i	−0.598783	+	0.598783i
$738$		0			0
$739$		−	827262.i		−	1.51480i	−0.652953	−	0.757398i	$-0.726470\pi$
$739$		−	827262.i		−	1.51480i	0.652953	−	0.757398i	$-0.273530\pi$
$740$	−125922.	−	692570.i	−0.229952	−	1.26474i
$741$		0			0
$742$	−25600.0	+	25600.0i	−0.0464978	+	0.0464978i
$743$	463131.	+	463131.i	0.838932	+	0.838932i	0.988718	−	0.149787i	$-0.0478587\pi$
$743$	463131.	+	463131.i	0.838932	+	0.838932i	−0.149787	+	0.988718i	$0.547859\pi$
$744$		0			0
$745$	66825.0	−	96525.0i	0.120400	−	0.173911i
$746$	−372200.			−0.668804
$747$		0			0
$748$	−6050.00	−	6050.00i	−0.0108131	−	0.0108131i
$749$			63213.9i			0.112681i
$750$		0			0
$751$	922568.			1.63576			0.817878	−	0.575392i	$-0.195150\pi$
$751$	922568.			1.63576			0.817878	+	0.575392i	$0.195150\pi$
$752$	91664.9	−	91664.9i	0.162094	−	0.162094i
$753$		0			0
$754$			346500.i			0.609482i
$755$	−645791.	−	447086.i	−1.13292	−	0.784327i
$756$		0			0
$757$	34030.0	−	34030.0i	0.0593841	−	0.0593841i	−0.676791	−	0.736175i	$-0.736630\pi$
$757$	34030.0	−	34030.0i	0.0593841	−	0.0593841i	0.736175	+	0.676791i	$0.236630\pi$
$758$	−179330.	−	179330.i	−0.312114	−	0.312114i
$759$		0			0
$760$	−294030.	+	53460.0i	−0.509055	+	0.0925554i
$761$	−855364.			−1.47700			−0.738502	−	0.674251i	$-0.764467\pi$
$761$	−855364.			−1.47700			−0.738502	+	0.674251i	$0.764467\pi$
$762$		0			0
$763$	−8910.00	−	8910.00i	−0.0153048	−	0.0153048i
$764$			43481.3i			0.0744931i
$765$		0			0
$766$	247610.			0.421998
$767$	−109573.	+	109573.i	−0.186257	+	0.186257i
$768$		0			0
$769$		−	272898.i		−	0.461474i	−0.973016	−	0.230737i	$-0.925886\pi$
$769$		−	272898.i		−	0.461474i	0.973016	−	0.230737i	$-0.0741138\pi$
$770$	39133.2	−	56525.7i	0.0660030	−	0.0953377i
$771$		0			0
$772$	93775.0	−	93775.0i	0.157345	−	0.157345i
$773$	−148333.	−	148333.i	−0.248244	−	0.248244i	0.572006	−	0.820250i	$-0.306166\pi$
$773$	−148333.	−	148333.i	−0.248244	−	0.248244i	−0.820250	+	0.572006i	$0.806166\pi$
$774$		0			0
$775$	−262240.	−	697320.i	−0.436612	−	1.16099i
$776$	615174.			1.02158
$777$		0			0
$778$	−189675.	−	189675.i	−0.313365	−	0.313365i
$779$		−	306804.i		−	0.505576i
$780$		0			0
$781$	−1.19900e6			−1.96570
$782$	−5344.25	+	5344.25i	−0.00873923	+	0.00873923i
$783$		0			0
$784$		−	96391.0i		−	0.156821i
$785$	382636.	−	69570.1i	0.620935	−	0.112897i
$786$		0			0
$787$	−771950.	+	771950.i	−1.24635	+	1.24635i	−0.289029	+	0.957320i	$0.593332\pi$
$787$	−771950.	+	771950.i	−1.24635	+	1.24635i	−0.957320	+	0.289029i	$0.906668\pi$
$788$	−449214.	−	449214.i	−0.723438	−	0.723438i
$789$		0			0
$790$	6120.00	+	33660.0i	0.00980612	+	0.0539337i
$791$	86867.8			0.138837
$792$		0			0
$793$	151580.	+	151580.i	0.241044	+	0.241044i
$794$		−	308196.i		−	0.488861i
$795$		0			0
$796$	−461142.			−0.727794
$797$	−59517.2	+	59517.2i	−0.0936971	+	0.0936971i	−0.752402	−	0.658705i	$-0.771105\pi$
$797$	−59517.2	+	59517.2i	−0.0936971	+	0.0936971i	0.658705	+	0.752402i	$0.271105\pi$
$798$		0			0
$799$			14140.0i			0.0221491i
$800$	602042.	+	272976.i	0.940691	+	0.426525i
$801$		0			0
$802$	70400.0	−	70400.0i	0.109452	−	0.109452i
$803$	679178.	+	679178.i	1.05330	+	1.05330i
$804$		0			0
$805$	109850.	+	76050.0i	0.169515	+	0.117357i
$806$	−414638.			−0.638262
$807$		0			0
$808$	736425.	+	736425.i	1.12799	+	1.12799i
$809$			181578.i			0.277438i	0.990332	+	0.138719i	$0.0442985\pi$
$809$			181578.i			0.277438i	−0.990332	+	0.138719i	$0.955701\pi$
$810$		0			0
$811$	1.26958e6			1.93027			0.965134	−	0.261756i	$-0.0843015\pi$
$811$	1.26958e6			1.93027			0.965134	+	0.261756i	$0.0843015\pi$
$812$	54786.5	−	54786.5i	0.0830924	−	0.0830924i
$813$		0			0
$814$		−	995500.i		−	1.50242i
$815$	90441.1	+	497426.i	0.136160	+	0.748882i
$816$		0			0
$817$	259380.	−	259380.i	0.388591	−	0.388591i
$818$	332817.	+	332817.i	0.497392	+	0.497392i
$819$		0			0
$820$	−242550.	+	350350.i	−0.360723	+	0.521044i
$821$	−504779.			−0.748884			−0.374442	−	0.927250i	$-0.622166\pi$
$821$	−504779.			−0.748884			−0.374442	+	0.927250i	$0.622166\pi$
$822$		0			0
$823$	299365.	+	299365.i	0.441979	+	0.441979i	0.892677	−	0.450698i	$-0.148825\pi$
$823$	299365.	+	299365.i	0.441979	+	0.441979i	−0.450698	+	0.892677i	$0.648825\pi$
$824$		−	263402.i		−	0.387940i
$825$		0			0
$826$	−15750.0			−0.0230845
$827$	364269.	−	364269.i	0.532612	−	0.532612i	−0.388737	−	0.921349i	$-0.627088\pi$
$827$	364269.	−	364269.i	0.532612	−	0.532612i	0.921349	+	0.388737i	$0.127088\pi$
$828$		0			0
$829$			597438.i			0.869328i	0.900593	+	0.434664i	$0.143133\pi$
$829$			597438.i			0.869328i	−0.900593	+	0.434664i	$0.856867\pi$
$830$	309761.	+	214450.i	0.449646	+	0.311293i
$831$		0			0
$832$	187990.	−	187990.i	0.271574	−	0.271574i
$833$	7434.51	+	7434.51i	0.0107143	+	0.0107143i
$834$		0			0
$835$	−34430.0	+	6260.00i	−0.0493815	+	0.00897845i
$836$	378809.			0.542011
$837$		0			0
$838$	299475.	+	299475.i	0.426454	+	0.426454i
$839$			1.11992e6i			1.59097i	0.605970	+	0.795487i	$0.292785\pi$
$839$			1.11992e6i			1.59097i	−0.605970	+	0.795487i	$0.707215\pi$
$840$		0			0
$841$	−284969.			−0.402908
$842$	154101.	−	154101.i	0.217361	−	0.217361i
$843$		0			0
$844$		−	329318.i		−	0.462307i
$845$	−62058.1	+	89639.5i	−0.0869131	+	0.125541i
$846$		0			0
$847$	−78045.0	+	78045.0i	−0.108787	+	0.108787i
$848$	66382.5	+	66382.5i	0.0923128	+	0.0923128i
$849$		0			0
$850$	−5850.00	+	2200.00i	−0.00809689	+	0.00304498i
$851$	1.93462e6			2.67138
$852$		0			0
$853$	−695120.	−	695120.i	−0.955348	−	0.955348i	0.0436966	−	0.999045i	$-0.486087\pi$
$853$	−695120.	−	695120.i	−0.955348	−	0.955348i	−0.999045	+	0.0436966i	$0.986087\pi$
$854$			21788.1i			0.0298747i
$855$		0			0
$856$	−539730.			−0.736595
$857$	−106117.	+	106117.i	−0.144485	+	0.144485i	−0.775649	−	0.631164i	$-0.782577\pi$
$857$	−106117.	+	106117.i	−0.144485	+	0.144485i	0.631164	+	0.775649i	$0.282577\pi$
$858$		0			0
$859$			1.25206e6i			1.69683i	0.529328	+	0.848417i	$0.322444\pi$
$859$			1.25206e6i			1.69683i	−0.529328	+	0.848417i	$0.677556\pi$
$860$	−501253.	+	91136.8i	−0.677735	+	0.123225i
$861$		0			0
$862$	353150.	−	353150.i	0.475275	−	0.475275i
$863$	−221818.	−	221818.i	−0.297835	−	0.297835i	0.542331	−	0.840165i	$-0.317542\pi$
$863$	−221818.	−	221818.i	−0.297835	−	0.297835i	−0.840165	+	0.542331i	$0.817542\pi$
$864$		0			0
$865$	124300.	+	683650.i	0.166126	+	0.913696i
$866$	288700.			0.384956
$867$		0			0
$868$	65560.0	+	65560.0i	0.0870161	+	0.0870161i
$869$		−	106442.i		−	0.140953i
$870$		0			0
$871$	−411400.			−0.542285
$872$	76074.9	−	76074.9i	0.100048	−	0.100048i
$873$		0			0
$874$		−	334620.i		−	0.438055i
$875$	56762.9	+	94789.3i	0.0741393	+	0.123806i
$876$		0			0
$877$	−20900.0	+	20900.0i	−0.0271736	+	0.0271736i	−0.720563	−	0.693389i	$-0.756117\pi$
$877$	−20900.0	+	20900.0i	−0.0271736	+	0.0271736i	0.693389	+	0.720563i	$0.256117\pi$
$878$	281218.	+	281218.i	0.364800	+	0.364800i
$879$		0			0
$880$	−146575.	−	101475.i	−0.189276	−	0.131037i
$881$	518044.			0.667444			0.333722	−	0.942671i	$-0.391695\pi$
$881$	518044.			0.667444			0.333722	+	0.942671i	$0.391695\pi$
$882$		0			0
$883$	−416240.	−	416240.i	−0.533854	−	0.533854i	0.387863	−	0.921717i	$-0.373213\pi$
$883$	−416240.	−	416240.i	−0.533854	−	0.533854i	−0.921717	+	0.387863i	$0.873213\pi$
$884$		−	7652.71i		−	0.00979289i
$885$		0			0
$886$	415580.			0.529404
$887$	−92076.0	+	92076.0i	−0.117031	+	0.117031i	−0.763197	−	0.646166i	$-0.776371\pi$
$887$	−92076.0	+	92076.0i	−0.117031	+	0.117031i	0.646166	+	0.763197i	$0.276371\pi$
$888$		0			0
$889$		−	197450.i		−	0.249835i
$890$	−113842.	−	626131.i	−0.143722	−	0.790470i
$891$		0			0
$892$	492085.	−	492085.i	0.618458	−	0.618458i
$893$	−442675.	−	442675.i	−0.555113	−	0.555113i
$894$		0			0
$895$	200475.	−	289575.i	0.250273	−	0.361506i
$896$	−92638.9			−0.115392
$897$		0			0
$898$	−252450.	−	252450.i	−0.313056	−	0.313056i
$899$		−	1.18737e6i		−	1.46915i
$900$		0			0
$901$	−10240.0			−0.0126139
$902$	−426117.	+	426117.i	−0.523740	+	0.523740i
$903$		0			0
$904$			741690.i			0.907581i
$905$	−26680.1	−	18470.9i	−0.0325755	−	0.0225523i
$906$		0			0
$907$	216040.	−	216040.i	0.262615	−	0.262615i	−0.563501	−	0.826116i	$-0.690546\pi$
$907$	216040.	−	216040.i	0.262615	−	0.262615i	0.826116	+	0.563501i	$0.190546\pi$
$908$	335119.	+	335119.i	0.406469	+	0.406469i
$909$		0			0
$910$	60500.0	−	11000.0i	0.0730588	−	0.0132834i
$911$	−354080.			−0.426643			−0.213322	−	0.976982i	$-0.568428\pi$
$911$	−354080.			−0.426643			−0.213322	+	0.976982i	$0.568428\pi$
$912$		0			0
$913$	−828850.	−	828850.i	−0.994339	−	0.994339i
$914$			539595.i			0.645916i
$915$		0			0
$916$	−566478.			−0.675137
$917$	−57000.1	+	57000.1i	−0.0677855	+	0.0677855i
$918$		0			0
$919$		−	951192.i		−	1.12626i	−0.826370	−	0.563128i	$-0.809598\pi$
$919$		−	951192.i		−	1.12626i	0.826370	−	0.563128i	$-0.190402\pi$
$920$	−649326.	+	937916.i	−0.767162	+	1.10812i
$921$		0			0
$922$	−14575.0	+	14575.0i	−0.0171454	+	0.0171454i
$923$	−758314.	−	758314.i	−0.890114	−	0.890114i
$924$		0			0
$925$	1.45705e6	+	660650.i	1.70291	+	0.772126i
$926$	−766805.			−0.894258
$927$		0			0
$928$	744975.	+	744975.i	0.865059	+	0.865059i
$929$		−	600801.i		−	0.696144i	−0.937468	−	0.348072i	$-0.886836\pi$
$929$		−	600801.i		−	0.696144i	0.937468	−	0.348072i	$-0.113164\pi$
$930$		0			0
$931$	−465498.			−0.537055
$932$	−336893.	+	336893.i	−0.387847	+	0.387847i
$933$		0			0
$934$			617770.i			0.708163i
$935$	19131.8	−	3478.51i	0.0218843	−	0.00397896i
$936$		0			0
$937$	672265.	−	672265.i	0.765705	−	0.765705i	−0.211642	−	0.977347i	$-0.567881\pi$
$937$	672265.	−	672265.i	0.765705	−	0.765705i	0.977347	+	0.211642i	$0.0678812\pi$
$938$	−29567.3	−	29567.3i	−0.0336052	−	0.0336052i
$939$		0			0
$940$	155540.	+	855470.i	0.176030	+	0.968164i
$941$	198101.			0.223721			0.111861	−	0.993724i	$-0.464319\pi$
$941$	198101.			0.223721			0.111861	+	0.993724i	$0.464319\pi$
$942$		0			0
$943$	−828100.	−	828100.i	−0.931235	−	0.931235i
$944$			40840.8i			0.0458301i
$945$		0			0
$946$	−720500.			−0.805103
$947$	−329845.	+	329845.i	−0.367798	+	0.367798i	−0.866674	−	0.498876i	$-0.833746\pi$
$947$	−329845.	+	329845.i	−0.367798	+	0.367798i	0.498876	+	0.866674i	$0.333746\pi$
$948$		0			0
$949$			859100.i			0.953919i
$950$	114269.	−	252018.i	0.126614	−	0.279244i
$951$		0			0
$952$	1350.00	−	1350.00i	0.00148957	−	0.00148957i
$953$	−811918.	−	811918.i	−0.893977	−	0.893977i	0.100918	−	0.994895i	$-0.467822\pi$
$953$	−811918.	−	811918.i	−0.893977	−	0.893977i	−0.994895	+	0.100918i	$0.967822\pi$
$954$		0			0
$955$	−81250.0	−	56250.0i	−0.0890875	−	0.0616759i
$956$	−964242.			−1.05504
$957$		0			0
$958$	−653400.	−	653400.i	−0.711948	−	0.711948i
$959$		−	57079.1i		−	0.0620640i
$960$		0			0
$961$	497343.			0.538529
$962$	629609.	−	629609.i	0.680332	−	0.680332i
$963$		0			0
$964$		−	689348.i		−	0.741796i
$965$	53916.8	+	296543.i	0.0578988	+	0.318444i
$966$		0			0
$967$	−422345.	+	422345.i	−0.451663	+	0.451663i	−0.895906	−	0.444243i	$-0.853473\pi$
$967$	−422345.	+	422345.i	−0.451663	+	0.451663i	0.444243	+	0.895906i	$0.353473\pi$
$968$	−666360.	−	666360.i	−0.711145	−	0.711145i
$969$		0			0
$970$	−324225.	+	468325.i	−0.344590	+	0.497742i
$971$	−852756.			−0.904453			−0.452227	−	0.891903i	$-0.649370\pi$
$971$	−852756.			−0.904453			−0.452227	+	0.891903i	$0.649370\pi$
$972$		0			0
$973$	−17190.0	−	17190.0i	−0.0181573	−	0.0181573i
$974$			147599.i			0.155585i
$975$		0			0
$976$	56498.0			0.0593108
$977$	499390.	−	499390.i	0.523180	−	0.523180i	−0.395351	−	0.918530i	$-0.629377\pi$
$977$	499390.	−	499390.i	0.523180	−	0.523180i	0.918530	+	0.395351i	$0.129377\pi$
$978$		0			0
$979$			1.98000e6i			2.06585i
$980$	531568.	+	368008.i	0.553486	+	0.383183i
$981$		0			0
$982$	346025.	−	346025.i	0.358826	−	0.358826i
$983$	516599.	+	516599.i	0.534622	+	0.534622i	0.921944	−	0.387322i	$-0.126600\pi$
$983$	516599.	+	516599.i	0.534622	+	0.534622i	−0.387322	+	0.921944i	$0.626600\pi$
$984$		0			0
$985$	1.42054e6	−	258280.i	1.46413	−	0.266206i
$986$	−9961.17			−0.0102461
$987$		0			0
$988$	239580.	+	239580.i	0.245435	+	0.245435i
$989$		−	1.40019e6i		−	1.43151i
$990$		0			0
$991$	−552958.			−0.563047			−0.281524	−	0.959554i	$-0.590840\pi$
$991$	−552958.			−0.563047			−0.281524	+	0.959554i	$0.590840\pi$
$992$	−891471.	+	891471.i	−0.905908	+	0.905908i
$993$		0			0
$994$		−	109000.i		−	0.110320i
$995$	596561.	−	861699.i	0.602571	−	0.870381i
$996$		0			0
$997$	830110.	−	830110.i	0.835113	−	0.835113i	−0.153098	−	0.988211i	$-0.548925\pi$
$997$	830110.	−	830110.i	0.835113	−	0.835113i	0.988211	+	0.153098i	$0.0489250\pi$
$998$	263288.	+	263288.i	0.264344	+	0.264344i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.5.g.d.28.1	✓	4
3.2	odd	2	inner	45.5.g.d.28.2	yes	4
5.2	odd	4	inner	45.5.g.d.37.1	yes	4
5.3	odd	4		225.5.g.k.82.2		4
5.4	even	2		225.5.g.k.118.2		4
15.2	even	4	inner	45.5.g.d.37.2	yes	4
15.8	even	4		225.5.g.k.82.1		4
15.14	odd	2		225.5.g.k.118.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.5.g.d.28.1	✓	4	1.1	even	1	trivial
45.5.g.d.28.2	yes	4	3.2	odd	2	inner
45.5.g.d.37.1	yes	4	5.2	odd	4	inner
45.5.g.d.37.2	yes	4	15.2	even	4	inner
225.5.g.k.82.1		4	15.8	even	4
225.5.g.k.82.2		4	5.3	odd	4
225.5.g.k.118.1		4	15.14	odd	2
225.5.g.k.118.2		4	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	45.g (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.58114 + 1.58114i) q^{2} +11.0000i q^{4} +(-20.5548 - 14.2302i) q^{5} +(-5.00000 + 5.00000i) q^{7} +(-42.6907 - 42.6907i) q^{8} +(55.0000 - 10.0000i) q^{10} -173.925 q^{11} +(-110.000 - 110.000i) q^{13} -15.8114i q^{14} -41.0000 q^{16} +(3.16228 - 3.16228i) q^{17} +198.000i q^{19} +(156.533 - 226.103i) q^{20} +(275.000 - 275.000i) q^{22} +(534.425 + 534.425i) q^{23} +(220.000 + 585.000i) q^{25} +347.851 q^{26} +(-55.0000 - 55.0000i) q^{28} +996.117i q^{29} -1192.00 q^{31} +(747.879 - 747.879i) q^{32} +10.0000i q^{34} +(173.925 - 31.6228i) q^{35} +(1810.00 - 1810.00i) q^{37} +(-313.065 - 313.065i) q^{38} +(270.000 + 1485.00i) q^{40} -1549.52 q^{41} +(-1310.00 - 1310.00i) q^{43} -1913.18i q^{44} -1690.00 q^{46} +(-2235.73 + 2235.73i) q^{47} +2351.00i q^{49} +(-1272.82 - 577.116i) q^{50} +(1210.00 - 1210.00i) q^{52} +(-1619.09 - 1619.09i) q^{53} +(3575.00 + 2475.00i) q^{55} +426.907 q^{56} +(-1575.00 - 1575.00i) q^{58} -996.117i q^{59} -1378.00 q^{61} +(1884.72 - 1884.72i) q^{62} +1709.00i q^{64} +(695.701 + 3826.36i) q^{65} +(1870.00 - 1870.00i) q^{67} +(34.7851 + 34.7851i) q^{68} +(-225.000 + 325.000i) q^{70} +6893.77 q^{71} +(-3905.00 - 3905.00i) q^{73} +5723.72i q^{74} -2178.00 q^{76} +(869.626 - 869.626i) q^{77} +612.000i q^{79} +(842.747 + 583.440i) q^{80} +(2450.00 - 2450.00i) q^{82} +(4765.55 + 4765.55i) q^{83} +(-110.000 + 20.0000i) q^{85} +4142.58 q^{86} +(7425.00 + 7425.00i) q^{88} -11384.2i q^{89} +1100.00 q^{91} +(-5878.67 + 5878.67i) q^{92} -7070.00i q^{94} +(2817.59 - 4069.85i) q^{95} +(-7205.00 + 7205.00i) q^{97} +(-3717.26 - 3717.26i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 20 q^{7} + 220 q^{10} - 440 q^{13} - 164 q^{16} + 1100 q^{22} + 880 q^{25} - 220 q^{28} - 4768 q^{31} + 7240 q^{37} + 1080 q^{40} - 5240 q^{43} - 6760 q^{46} + 4840 q^{52} + 14300 q^{55} - 6300 q^{58}+ \cdots - 28820 q^{97}+O(q^{100}) \) 4 * q - 20 * q^7 + 220 * q^10 - 440 * q^13 - 164 * q^16 + 1100 * q^22 + 880 * q^25 - 220 * q^28 - 4768 * q^31 + 7240 * q^37 + 1080 * q^40 - 5240 * q^43 - 6760 * q^46 + 4840 * q^52 + 14300 * q^55 - 6300 * q^58 - 5512 * q^61 + 7480 * q^67 - 900 * q^70 - 15620 * q^73 - 8712 * q^76 + 9800 * q^82 - 440 * q^85 + 29700 * q^88 + 4400 * q^91 - 28820 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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