LMFDB - Embedded newform 1425.2.c.k.799.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1425,2,Mod(799,1425)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1425, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1425.799");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1425 = 3 \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1425.c (of order $2$, degree $1$, not 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:	$11.3786822880$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 285)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			799.4
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1425.799
Dual form			1425.2.c.k.799.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.73205i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.73205 q^{6} +2.73205i q^{7} +1.73205i q^{8} -1.00000 q^{9} +4.73205 q^{11} -1.00000i q^{12} +0.732051i q^{13} -4.73205 q^{14} -5.00000 q^{16} -1.73205i q^{18} -1.00000 q^{19} -2.73205 q^{21} +8.19615i q^{22} +3.46410i q^{23} -1.73205 q^{24} -1.26795 q^{26} -1.00000i q^{27} -2.73205i q^{28} -8.19615 q^{29} +8.92820 q^{31} -5.19615i q^{32} +4.73205i q^{33} +1.00000 q^{36} +6.19615i q^{37} -1.73205i q^{38} -0.732051 q^{39} +1.26795 q^{41} -4.73205i q^{42} +4.19615i q^{43} -4.73205 q^{44} -6.00000 q^{46} +3.46410i q^{47} -5.00000i q^{48} -0.464102 q^{49} -0.732051i q^{52} -9.46410i q^{53} +1.73205 q^{54} -4.73205 q^{56} -1.00000i q^{57} -14.1962i q^{58} -2.53590 q^{59} -6.53590 q^{61} +15.4641i q^{62} -2.73205i q^{63} -1.00000 q^{64} -8.19615 q^{66} -8.00000i q^{67} -3.46410 q^{69} -4.39230 q^{71} -1.73205i q^{72} -16.9282i q^{73} -10.7321 q^{74} +1.00000 q^{76} +12.9282i q^{77} -1.26795i q^{78} +10.9282 q^{79} +1.00000 q^{81} +2.19615i q^{82} -12.9282i q^{83} +2.73205 q^{84} -7.26795 q^{86} -8.19615i q^{87} +8.19615i q^{88} -10.7321 q^{89} -2.00000 q^{91} -3.46410i q^{92} +8.92820i q^{93} -6.00000 q^{94} +5.19615 q^{96} +6.19615i q^{97} -0.803848i q^{98} -4.73205 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} - 4 q^{9} + 12 q^{11} - 12 q^{14} - 20 q^{16} - 4 q^{19} - 4 q^{21} - 12 q^{26} - 12 q^{29} + 8 q^{31} + 4 q^{36} + 4 q^{39} + 12 q^{41} - 12 q^{44} - 24 q^{46} + 12 q^{49} - 12 q^{56} - 24 q^{59}+ \cdots - 12 q^{99}+O(q^{100}) $ 4 * q - 4 * q^4 - 4 * q^9 + 12 * q^11 - 12 * q^14 - 20 * q^16 - 4 * q^19 - 4 * q^21 - 12 * q^26 - 12 * q^29 + 8 * q^31 + 4 * q^36 + 4 * q^39 + 12 * q^41 - 12 * q^44 - 24 * q^46 + 12 * q^49 - 12 * q^56 - 24 * q^59 - 40 * q^61 - 4 * q^64 - 12 * q^66 + 24 * q^71 - 36 * q^74 + 4 * q^76 + 16 * q^79 + 4 * q^81 + 4 * q^84 - 36 * q^86 - 36 * q^89 - 8 * q^91 - 24 * q^94 - 12 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1425\mathbb{Z}\right)^\times$.

$n$	$476$	$1027$	$1351$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$	1.73205i	1.22474i	0.790569	+	0.612372i	$0.209785\pi$
$2$	1.73205i	1.22474i	−0.790569	+	0.612372i	$0.790215\pi$
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−1.73205	−0.707107
$7$	2.73205i	1.03262i	0.856402	+	0.516309i	$0.172694\pi$
$7$	2.73205i	1.03262i	−0.856402	+	0.516309i	$0.827306\pi$
$8$	1.73205i	0.612372i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	4.73205	1.42677	0.713384	−	0.700774i	$-0.247162\pi$
$11$	4.73205	1.42677	0.713384	+	0.700774i	$0.247162\pi$
$12$	− 1.00000i	− 0.288675i
$13$	0.732051i	0.203034i	0.994834	+	0.101517i	$0.0323697\pi$
$13$	0.732051i	0.203034i	−0.994834	+	0.101517i	$0.967630\pi$
$14$	−4.73205	−1.26469
$15$	0	0
$16$	−5.00000	−1.25000
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	− 1.73205i	− 0.408248i
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−2.73205	−0.596182
$22$	8.19615i	1.74743i
$23$	3.46410i	0.722315i	0.932505	+	0.361158i	$0.117618\pi$
$23$	3.46410i	0.722315i	−0.932505	+	0.361158i	$0.882382\pi$
$24$	−1.73205	−0.353553
$25$	0	0
$26$	−1.26795	−0.248665
$27$	− 1.00000i	− 0.192450i
$28$	− 2.73205i	− 0.516309i
$29$	−8.19615	−1.52199	−0.760994	−	0.648759i	$-0.775288\pi$
$29$	−8.19615	−1.52199	−0.760994	+	0.648759i	$0.775288\pi$
$30$	0	0
$31$	8.92820	1.60355	0.801776	−	0.597624i	$-0.203889\pi$
$31$	8.92820	1.60355	0.801776	+	0.597624i	$0.203889\pi$
$32$	− 5.19615i	− 0.918559i
$33$	4.73205i	0.823744i
$34$	0	0
$35$	0	0
$36$	1.00000	0.166667
$37$	6.19615i	1.01864i	0.860577	+	0.509321i	$0.170103\pi$
$37$	6.19615i	1.01864i	−0.860577	+	0.509321i	$0.829897\pi$
$38$	− 1.73205i	− 0.280976i
$39$	−0.732051	−0.117222
$40$	0	0
$41$	1.26795	0.198020	0.0990102	−	0.995086i	$-0.468432\pi$
$41$	1.26795	0.198020	0.0990102	+	0.995086i	$0.468432\pi$
$42$	− 4.73205i	− 0.730171i
$43$	4.19615i	0.639907i	0.947433	+	0.319954i	$0.103667\pi$
$43$	4.19615i	0.639907i	−0.947433	+	0.319954i	$0.896333\pi$
$44$	−4.73205	−0.713384
$45$	0	0
$46$	−6.00000	−0.884652
$47$	3.46410i	0.505291i	0.967559	+	0.252646i	$0.0813007\pi$
$47$	3.46410i	0.505291i	−0.967559	+	0.252646i	$0.918699\pi$
$48$	− 5.00000i	− 0.721688i
$49$	−0.464102	−0.0663002
$50$	0	0
$51$	0	0
$52$	− 0.732051i	− 0.101517i
$53$	− 9.46410i	− 1.29999i	−0.759937	−	0.649997i	$-0.774770\pi$
$53$	− 9.46410i	− 1.29999i	0.759937	−	0.649997i	$-0.225230\pi$
$54$	1.73205	0.235702
$55$	0	0
$56$	−4.73205	−0.632347
$57$	− 1.00000i	− 0.132453i
$58$	− 14.1962i	− 1.86405i
$59$	−2.53590	−0.330146	−0.165073	−	0.986281i	$-0.552786\pi$
$59$	−2.53590	−0.330146	−0.165073	+	0.986281i	$0.552786\pi$
$60$	0	0
$61$	−6.53590	−0.836836	−0.418418	−	0.908255i	$-0.637415\pi$
$61$	−6.53590	−0.836836	−0.418418	+	0.908255i	$0.637415\pi$
$62$	15.4641i	1.96394i
$63$	− 2.73205i	− 0.344206i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−8.19615	−1.00888
$67$	− 8.00000i	− 0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$	− 8.00000i	− 0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$	0	0
$69$	−3.46410	−0.417029
$70$	0	0
$71$	−4.39230	−0.521271	−0.260635	−	0.965437i	$-0.583932\pi$
$71$	−4.39230	−0.521271	−0.260635	+	0.965437i	$0.583932\pi$
$72$	− 1.73205i	− 0.204124i
$73$	− 16.9282i	− 1.98130i	−0.136441	−	0.990648i	$-0.543566\pi$
$73$	− 16.9282i	− 1.98130i	0.136441	−	0.990648i	$-0.456434\pi$
$74$	−10.7321	−1.24758
$75$	0	0
$76$	1.00000	0.114708
$77$	12.9282i	1.47331i
$78$	− 1.26795i	− 0.143567i
$79$	10.9282	1.22952	0.614759	−	0.788715i	$-0.289253\pi$
$79$	10.9282	1.22952	0.614759	+	0.788715i	$0.289253\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	2.19615i	0.242524i
$83$	− 12.9282i	− 1.41905i	−0.704678	−	0.709527i	$-0.748908\pi$
$83$	− 12.9282i	− 1.41905i	0.704678	−	0.709527i	$-0.251092\pi$
$84$	2.73205	0.298091
$85$	0	0
$86$	−7.26795	−0.783723
$87$	− 8.19615i	− 0.878720i
$88$	8.19615i	0.873713i
$89$	−10.7321	−1.13760	−0.568798	−	0.822478i	$-0.692591\pi$
$89$	−10.7321	−1.13760	−0.568798	+	0.822478i	$0.692591\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	− 3.46410i	− 0.361158i
$93$	8.92820i	0.925812i
$94$	−6.00000	−0.618853
$95$	0	0
$96$	5.19615	0.530330
$97$	6.19615i	0.629124i	0.949237	+	0.314562i	$0.101858\pi$
$97$	6.19615i	0.629124i	−0.949237	+	0.314562i	$0.898142\pi$
$98$	− 0.803848i	− 0.0812009i
$99$	−4.73205	−0.475589
$100$	0	0
$101$	−10.3923	−1.03407	−0.517036	−	0.855963i	$-0.672965\pi$
$101$	−10.3923	−1.03407	−0.517036	+	0.855963i	$0.672965\pi$
$102$	0	0
$103$	9.85641i	0.971181i	0.874187	+	0.485590i	$0.161395\pi$
$103$	9.85641i	0.971181i	−0.874187	+	0.485590i	$0.838605\pi$
$104$	−1.26795	−0.124333
$105$	0	0
$106$	16.3923	1.59216
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	1.00000i	0.0962250i
$109$	14.3923	1.37853	0.689266	−	0.724508i	$-0.257933\pi$
$109$	14.3923	1.37853	0.689266	+	0.724508i	$0.257933\pi$
$110$	0	0
$111$	−6.19615	−0.588113
$112$	− 13.6603i	− 1.29077i
$113$	18.9282i	1.78062i	0.455359	+	0.890308i	$0.349511\pi$
$113$	18.9282i	1.78062i	−0.455359	+	0.890308i	$0.650489\pi$
$114$	1.73205	0.162221
$115$	0	0
$116$	8.19615	0.760994
$117$	− 0.732051i	− 0.0676781i
$118$	− 4.39230i	− 0.404344i
$119$	0	0
$120$	0	0
$121$	11.3923	1.03566
$122$	− 11.3205i	− 1.02491i
$123$	1.26795i	0.114327i
$124$	−8.92820	−0.801776
$125$	0	0
$126$	4.73205	0.421565
$127$	4.00000i	0.354943i	0.984126	+	0.177471i	$0.0567917\pi$
$127$	4.00000i	0.354943i	−0.984126	+	0.177471i	$0.943208\pi$
$128$	− 12.1244i	− 1.07165i
$129$	−4.19615	−0.369451
$130$	0	0
$131$	−9.12436	−0.797199	−0.398599	−	0.917125i	$-0.630504\pi$
$131$	−9.12436	−0.797199	−0.398599	+	0.917125i	$0.630504\pi$
$132$	− 4.73205i	− 0.411872i
$133$	− 2.73205i	− 0.236899i
$134$	13.8564	1.19701
$135$	0	0
$136$	0	0
$137$	− 19.8564i	− 1.69645i	−0.529638	−	0.848224i	$-0.677672\pi$
$137$	− 19.8564i	− 1.69645i	0.529638	−	0.848224i	$-0.322328\pi$
$138$	− 6.00000i	− 0.510754i
$139$	8.39230	0.711826	0.355913	−	0.934519i	$-0.384170\pi$
$139$	8.39230	0.711826	0.355913	+	0.934519i	$0.384170\pi$
$140$	0	0
$141$	−3.46410	−0.291730
$142$	− 7.60770i	− 0.638424i
$143$	3.46410i	0.289683i
$144$	5.00000	0.416667
$145$	0	0
$146$	29.3205	2.42658
$147$	− 0.464102i	− 0.0382785i
$148$	− 6.19615i	− 0.509321i
$149$	19.8564	1.62670	0.813350	−	0.581775i	$-0.197641\pi$
$149$	19.8564	1.62670	0.813350	+	0.581775i	$0.197641\pi$
$150$	0	0
$151$	14.0000	1.13930	0.569652	−	0.821886i	$-0.307078\pi$
$151$	14.0000	1.13930	0.569652	+	0.821886i	$0.307078\pi$
$152$	− 1.73205i	− 0.140488i
$153$	0	0
$154$	−22.3923	−1.80442
$155$	0	0
$156$	0.732051	0.0586110
$157$	− 6.39230i	− 0.510161i	−0.966920	−	0.255081i	$-0.917898\pi$
$157$	− 6.39230i	− 0.510161i	0.966920	−	0.255081i	$-0.0821021\pi$
$158$	18.9282i	1.50585i
$159$	9.46410	0.750552
$160$	0	0
$161$	−9.46410	−0.745876
$162$	1.73205i	0.136083i
$163$	9.26795i	0.725922i	0.931804	+	0.362961i	$0.118234\pi$
$163$	9.26795i	0.725922i	−0.931804	+	0.362961i	$0.881766\pi$
$164$	−1.26795	−0.0990102
$165$	0	0
$166$	22.3923	1.73798
$167$	− 3.46410i	− 0.268060i	−0.990977	−	0.134030i	$-0.957208\pi$
$167$	− 3.46410i	− 0.268060i	0.990977	−	0.134030i	$-0.0427919\pi$
$168$	− 4.73205i	− 0.365086i
$169$	12.4641	0.958777
$170$	0	0
$171$	1.00000	0.0764719
$172$	− 4.19615i	− 0.319954i
$173$	6.92820i	0.526742i	0.964695	+	0.263371i	$0.0848343\pi$
$173$	6.92820i	0.526742i	−0.964695	+	0.263371i	$0.915166\pi$
$174$	14.1962	1.07621
$175$	0	0
$176$	−23.6603	−1.78346
$177$	− 2.53590i	− 0.190610i
$178$	− 18.5885i	− 1.39326i
$179$	−23.3205	−1.74306	−0.871528	−	0.490345i	$-0.836871\pi$
$179$	−23.3205	−1.74306	−0.871528	+	0.490345i	$0.836871\pi$
$180$	0	0
$181$	−2.39230	−0.177819	−0.0889093	−	0.996040i	$-0.528338\pi$
$181$	−2.39230	−0.177819	−0.0889093	+	0.996040i	$0.528338\pi$
$182$	− 3.46410i	− 0.256776i
$183$	− 6.53590i	− 0.483148i
$184$	−6.00000	−0.442326
$185$	0	0
$186$	−15.4641	−1.13388
$187$	0	0
$188$	− 3.46410i	− 0.252646i
$189$	2.73205	0.198727
$190$	0	0
$191$	0.339746	0.0245832	0.0122916	−	0.999924i	$-0.496087\pi$
$191$	0.339746	0.0245832	0.0122916	+	0.999924i	$0.496087\pi$
$192$	− 1.00000i	− 0.0721688i
$193$	17.1244i	1.23264i	0.787497	+	0.616319i	$0.211377\pi$
$193$	17.1244i	1.23264i	−0.787497	+	0.616319i	$0.788623\pi$
$194$	−10.7321	−0.770516
$195$	0	0
$196$	0.464102	0.0331501
$197$	24.0000i	1.70993i	0.518686	+	0.854965i	$0.326421\pi$
$197$	24.0000i	1.70993i	−0.518686	+	0.854965i	$0.673579\pi$
$198$	− 8.19615i	− 0.582475i
$199$	15.3205	1.08604	0.543021	−	0.839719i	$-0.317280\pi$
$199$	15.3205	1.08604	0.543021	+	0.839719i	$0.317280\pi$
$200$	0	0
$201$	8.00000	0.564276
$202$	− 18.0000i	− 1.26648i
$203$	− 22.3923i	− 1.57163i
$204$	0	0
$205$	0	0
$206$	−17.0718	−1.18945
$207$	− 3.46410i	− 0.240772i
$208$	− 3.66025i	− 0.253793i
$209$	−4.73205	−0.327323
$210$	0	0
$211$	1.07180	0.0737855	0.0368928	−	0.999319i	$-0.488254\pi$
$211$	1.07180	0.0737855	0.0368928	+	0.999319i	$0.488254\pi$
$212$	9.46410i	0.649997i
$213$	− 4.39230i	− 0.300956i
$214$	0	0
$215$	0	0
$216$	1.73205	0.117851
$217$	24.3923i	1.65586i
$218$	24.9282i	1.68835i
$219$	16.9282	1.14390
$220$	0	0
$221$	0	0
$222$	− 10.7321i	− 0.720288i
$223$	− 17.8564i	− 1.19575i	−0.801588	−	0.597877i	$-0.796011\pi$
$223$	− 17.8564i	− 1.19575i	0.801588	−	0.597877i	$-0.203989\pi$
$224$	14.1962	0.948520
$225$	0	0
$226$	−32.7846	−2.18080
$227$	− 10.3923i	− 0.689761i	−0.938647	−	0.344881i	$-0.887919\pi$
$227$	− 10.3923i	− 0.689761i	0.938647	−	0.344881i	$-0.112081\pi$
$228$	1.00000i	0.0662266i
$229$	18.5359	1.22489	0.612443	−	0.790515i	$-0.290187\pi$
$229$	18.5359	1.22489	0.612443	+	0.790515i	$0.290187\pi$
$230$	0	0
$231$	−12.9282	−0.850613
$232$	− 14.1962i	− 0.932023i
$233$	− 7.85641i	− 0.514690i	−0.966320	−	0.257345i	$-0.917152\pi$
$233$	− 7.85641i	− 0.514690i	0.966320	−	0.257345i	$-0.0828477\pi$
$234$	1.26795	0.0828884
$235$	0	0
$236$	2.53590	0.165073
$237$	10.9282i	0.709863i
$238$	0	0
$239$	9.80385	0.634158	0.317079	−	0.948399i	$-0.397298\pi$
$239$	9.80385	0.634158	0.317079	+	0.948399i	$0.397298\pi$
$240$	0	0
$241$	−3.07180	−0.197872	−0.0989359	−	0.995094i	$-0.531544\pi$
$241$	−3.07180	−0.197872	−0.0989359	+	0.995094i	$0.531544\pi$
$242$	19.7321i	1.26842i
$243$	1.00000i	0.0641500i
$244$	6.53590	0.418418
$245$	0	0
$246$	−2.19615	−0.140022
$247$	− 0.732051i	− 0.0465793i
$248$	15.4641i	0.981971i
$249$	12.9282	0.819292
$250$	0	0
$251$	28.0526	1.77066	0.885331	−	0.464961i	$-0.153932\pi$
$251$	28.0526	1.77066	0.885331	+	0.464961i	$0.153932\pi$
$252$	2.73205i	0.172103i
$253$	16.3923i	1.03058i
$254$	−6.92820	−0.434714
$255$	0	0
$256$	19.0000	1.18750
$257$	24.0000i	1.49708i	0.663090	+	0.748539i	$0.269245\pi$
$257$	24.0000i	1.49708i	−0.663090	+	0.748539i	$0.730755\pi$
$258$	− 7.26795i	− 0.452483i
$259$	−16.9282	−1.05187
$260$	0	0
$261$	8.19615	0.507329
$262$	− 15.8038i	− 0.976365i
$263$	6.00000i	0.369976i	0.982741	+	0.184988i	$0.0592246\pi$
$263$	6.00000i	0.369976i	−0.982741	+	0.184988i	$0.940775\pi$
$264$	−8.19615	−0.504438
$265$	0	0
$266$	4.73205	0.290141
$267$	− 10.7321i	− 0.656791i
$268$	8.00000i	0.488678i
$269$	−0.588457	−0.0358789	−0.0179394	−	0.999839i	$-0.505711\pi$
$269$	−0.588457	−0.0358789	−0.0179394	+	0.999839i	$0.505711\pi$
$270$	0	0
$271$	0.392305	0.0238308	0.0119154	−	0.999929i	$-0.496207\pi$
$271$	0.392305	0.0238308	0.0119154	+	0.999929i	$0.496207\pi$
$272$	0	0
$273$	− 2.00000i	− 0.121046i
$274$	34.3923	2.07772
$275$	0	0
$276$	3.46410	0.208514
$277$	− 2.00000i	− 0.120168i	−0.998193	−	0.0600842i	$-0.980863\pi$
$277$	− 2.00000i	− 0.120168i	0.998193	−	0.0600842i	$-0.0191369\pi$
$278$	14.5359i	0.871805i
$279$	−8.92820	−0.534518
$280$	0	0
$281$	1.26795	0.0756395	0.0378198	−	0.999285i	$-0.487959\pi$
$281$	1.26795	0.0756395	0.0378198	+	0.999285i	$0.487959\pi$
$282$	− 6.00000i	− 0.357295i
$283$	24.9808i	1.48495i	0.669873	+	0.742476i	$0.266349\pi$
$283$	24.9808i	1.48495i	−0.669873	+	0.742476i	$0.733651\pi$
$284$	4.39230	0.260635
$285$	0	0
$286$	−6.00000	−0.354787
$287$	3.46410i	0.204479i
$288$	5.19615i	0.306186i
$289$	17.0000	1.00000
$290$	0	0
$291$	−6.19615	−0.363225
$292$	16.9282i	0.990648i
$293$	27.7128i	1.61900i	0.587120	+	0.809500i	$0.300262\pi$
$293$	27.7128i	1.61900i	−0.587120	+	0.809500i	$0.699738\pi$
$294$	0.803848	0.0468813
$295$	0	0
$296$	−10.7321	−0.623788
$297$	− 4.73205i	− 0.274581i
$298$	34.3923i	1.99229i
$299$	−2.53590	−0.146655
$300$	0	0
$301$	−11.4641	−0.660780
$302$	24.2487i	1.39536i
$303$	− 10.3923i	− 0.597022i
$304$	5.00000	0.286770
$305$	0	0
$306$	0	0
$307$	32.3923i	1.84873i	0.381514	+	0.924363i	$0.375403\pi$
$307$	32.3923i	1.84873i	−0.381514	+	0.924363i	$0.624597\pi$
$308$	− 12.9282i	− 0.736653i
$309$	−9.85641	−0.560711
$310$	0	0
$311$	32.4449	1.83978	0.919890	−	0.392177i	$-0.128278\pi$
$311$	32.4449	1.83978	0.919890	+	0.392177i	$0.128278\pi$
$312$	− 1.26795i	− 0.0717835i
$313$	6.39230i	0.361314i	0.983546	+	0.180657i	$0.0578225\pi$
$313$	6.39230i	0.361314i	−0.983546	+	0.180657i	$0.942178\pi$
$314$	11.0718	0.624818
$315$	0	0
$316$	−10.9282	−0.614759
$317$	11.3205i	0.635823i	0.948120	+	0.317912i	$0.102982\pi$
$317$	11.3205i	0.635823i	−0.948120	+	0.317912i	$0.897018\pi$
$318$	16.3923i	0.919235i
$319$	−38.7846	−2.17152
$320$	0	0
$321$	0	0
$322$	− 16.3923i	− 0.913507i
$323$	0	0
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−16.0526	−0.889069
$327$	14.3923i	0.795896i
$328$	2.19615i	0.121262i
$329$	−9.46410	−0.521773
$330$	0	0
$331$	−25.7128	−1.41330	−0.706652	−	0.707561i	$-0.749795\pi$
$331$	−25.7128	−1.41330	−0.706652	+	0.707561i	$0.749795\pi$
$332$	12.9282i	0.709527i
$333$	− 6.19615i	− 0.339547i
$334$	6.00000	0.328305
$335$	0	0
$336$	13.6603	0.745228
$337$	− 5.12436i	− 0.279141i	−0.990212	−	0.139571i	$-0.955428\pi$
$337$	− 5.12436i	− 0.279141i	0.990212	−	0.139571i	$-0.0445723\pi$
$338$	21.5885i	1.17426i
$339$	−18.9282	−1.02804
$340$	0	0
$341$	42.2487	2.28790
$342$	1.73205i	0.0936586i
$343$	17.8564i	0.964155i
$344$	−7.26795	−0.391862
$345$	0	0
$346$	−12.0000	−0.645124
$347$	0.928203i	0.0498286i	0.999690	+	0.0249143i	$0.00793128\pi$
$347$	0.928203i	0.0498286i	−0.999690	+	0.0249143i	$0.992069\pi$
$348$	8.19615i	0.439360i
$349$	22.0000	1.17763	0.588817	−	0.808267i	$-0.299594\pi$
$349$	22.0000	1.17763	0.588817	+	0.808267i	$0.299594\pi$
$350$	0	0
$351$	0.732051	0.0390740
$352$	− 24.5885i	− 1.31057i
$353$	− 14.7846i	− 0.786905i	−0.919345	−	0.393453i	$-0.871281\pi$
$353$	− 14.7846i	− 0.786905i	0.919345	−	0.393453i	$-0.128719\pi$
$354$	4.39230	0.233448
$355$	0	0
$356$	10.7321	0.568798
$357$	0	0
$358$	− 40.3923i	− 2.13480i
$359$	−0.339746	−0.0179311	−0.00896555	−	0.999960i	$-0.502854\pi$
$359$	−0.339746	−0.0179311	−0.00896555	+	0.999960i	$0.502854\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	− 4.14359i	− 0.217782i
$363$	11.3923i	0.597941i
$364$	2.00000	0.104828
$365$	0	0
$366$	11.3205	0.591732
$367$	− 16.1962i	− 0.845432i	−0.906262	−	0.422716i	$-0.861077\pi$
$367$	− 16.1962i	− 0.845432i	0.906262	−	0.422716i	$-0.138923\pi$
$368$	− 17.3205i	− 0.902894i
$369$	−1.26795	−0.0660068
$370$	0	0
$371$	25.8564	1.34240
$372$	− 8.92820i	− 0.462906i
$373$	− 6.19615i	− 0.320825i	−0.987050	−	0.160412i	$-0.948718\pi$
$373$	− 6.19615i	− 0.320825i	0.987050	−	0.160412i	$-0.0512824\pi$
$374$	0	0
$375$	0	0
$376$	−6.00000	−0.309426
$377$	− 6.00000i	− 0.309016i
$378$	4.73205i	0.243390i
$379$	−20.9282	−1.07501	−0.537505	−	0.843261i	$-0.680633\pi$
$379$	−20.9282	−1.07501	−0.537505	+	0.843261i	$0.680633\pi$
$380$	0	0
$381$	−4.00000	−0.204926
$382$	0.588457i	0.0301081i
$383$	− 17.0718i	− 0.872328i	−0.899867	−	0.436164i	$-0.856337\pi$
$383$	− 17.0718i	− 0.872328i	0.899867	−	0.436164i	$-0.143663\pi$
$384$	12.1244	0.618718
$385$	0	0
$386$	−29.6603	−1.50967
$387$	− 4.19615i	− 0.213302i
$388$	− 6.19615i	− 0.314562i
$389$	−7.85641	−0.398336	−0.199168	−	0.979965i	$-0.563824\pi$
$389$	−7.85641	−0.398336	−0.199168	+	0.979965i	$0.563824\pi$
$390$	0	0
$391$	0	0
$392$	− 0.803848i	− 0.0406004i
$393$	− 9.12436i	− 0.460263i
$394$	−41.5692	−2.09423
$395$	0	0
$396$	4.73205	0.237795
$397$	− 8.92820i	− 0.448094i	−0.974578	−	0.224047i	$-0.928073\pi$
$397$	− 8.92820i	− 0.448094i	0.974578	−	0.224047i	$-0.0719269\pi$
$398$	26.5359i	1.33012i
$399$	2.73205	0.136774
$400$	0	0
$401$	−34.0526	−1.70050	−0.850252	−	0.526376i	$-0.823550\pi$
$401$	−34.0526	−1.70050	−0.850252	+	0.526376i	$0.823550\pi$
$402$	13.8564i	0.691095i
$403$	6.53590i	0.325576i
$404$	10.3923	0.517036
$405$	0	0
$406$	38.7846	1.92485
$407$	29.3205i	1.45336i
$408$	0	0
$409$	26.3923	1.30502	0.652508	−	0.757782i	$-0.273717\pi$
$409$	26.3923	1.30502	0.652508	+	0.757782i	$0.273717\pi$
$410$	0	0
$411$	19.8564	0.979444
$412$	− 9.85641i	− 0.485590i
$413$	− 6.92820i	− 0.340915i
$414$	6.00000	0.294884
$415$	0	0
$416$	3.80385	0.186499
$417$	8.39230i	0.410973i
$418$	− 8.19615i	− 0.400887i
$419$	28.0526	1.37046	0.685229	−	0.728328i	$-0.259702\pi$
$419$	28.0526	1.37046	0.685229	+	0.728328i	$0.259702\pi$
$420$	0	0
$421$	−18.7846	−0.915506	−0.457753	−	0.889079i	$-0.651346\pi$
$421$	−18.7846	−0.915506	−0.457753	+	0.889079i	$0.651346\pi$
$422$	1.85641i	0.0903685i
$423$	− 3.46410i	− 0.168430i
$424$	16.3923	0.796081
$425$	0	0
$426$	7.60770	0.368594
$427$	− 17.8564i	− 0.864132i
$428$	0	0
$429$	−3.46410	−0.167248
$430$	0	0
$431$	−11.3205	−0.545290	−0.272645	−	0.962115i	$-0.587898\pi$
$431$	−11.3205	−0.545290	−0.272645	+	0.962115i	$0.587898\pi$
$432$	5.00000i	0.240563i
$433$	− 10.5885i	− 0.508849i	−0.967093	−	0.254424i	$-0.918114\pi$
$433$	− 10.5885i	− 0.508849i	0.967093	−	0.254424i	$-0.0818860\pi$
$434$	−42.2487	−2.02800
$435$	0	0
$436$	−14.3923	−0.689266
$437$	− 3.46410i	− 0.165710i
$438$	29.3205i	1.40099i
$439$	−26.9282	−1.28521	−0.642607	−	0.766196i	$-0.722147\pi$
$439$	−26.9282	−1.28521	−0.642607	+	0.766196i	$0.722147\pi$
$440$	0	0
$441$	0.464102	0.0221001
$442$	0	0
$443$	− 5.32051i	− 0.252785i	−0.991980	−	0.126392i	$-0.959660\pi$
$443$	− 5.32051i	− 0.252785i	0.991980	−	0.126392i	$-0.0403399\pi$
$444$	6.19615	0.294056
$445$	0	0
$446$	30.9282	1.46449
$447$	19.8564i	0.939176i
$448$	− 2.73205i	− 0.129077i
$449$	5.66025	0.267124	0.133562	−	0.991040i	$-0.457358\pi$
$449$	5.66025	0.267124	0.133562	+	0.991040i	$0.457358\pi$
$450$	0	0
$451$	6.00000	0.282529
$452$	− 18.9282i	− 0.890308i
$453$	14.0000i	0.657777i
$454$	18.0000	0.844782
$455$	0	0
$456$	1.73205	0.0811107
$457$	− 4.53590i	− 0.212180i	−0.994357	−	0.106090i	$-0.966167\pi$
$457$	− 4.53590i	− 0.212180i	0.994357	−	0.106090i	$-0.0338332\pi$
$458$	32.1051i	1.50017i
$459$	0	0
$460$	0	0
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	− 22.3923i	− 1.04178i
$463$	− 35.5167i	− 1.65060i	−0.564695	−	0.825300i	$-0.691006\pi$
$463$	− 35.5167i	− 1.65060i	0.564695	−	0.825300i	$-0.308994\pi$
$464$	40.9808	1.90248
$465$	0	0
$466$	13.6077	0.630364
$467$	− 20.5359i	− 0.950288i	−0.879908	−	0.475144i	$-0.842396\pi$
$467$	− 20.5359i	− 0.950288i	0.879908	−	0.475144i	$-0.157604\pi$
$468$	0.732051i	0.0338391i
$469$	21.8564	1.00924
$470$	0	0
$471$	6.39230	0.294542
$472$	− 4.39230i	− 0.202172i
$473$	19.8564i	0.912999i
$474$	−18.9282	−0.869401
$475$	0	0
$476$	0	0
$477$	9.46410i	0.433331i
$478$	16.9808i	0.776682i
$479$	−25.5167	−1.16589	−0.582943	−	0.812513i	$-0.698099\pi$
$479$	−25.5167	−1.16589	−0.582943	+	0.812513i	$0.698099\pi$
$480$	0	0
$481$	−4.53590	−0.206819
$482$	− 5.32051i	− 0.242343i
$483$	− 9.46410i	− 0.430632i
$484$	−11.3923	−0.517832
$485$	0	0
$486$	−1.73205	−0.0785674
$487$	32.3923i	1.46784i	0.679238	+	0.733918i	$0.262310\pi$
$487$	32.3923i	1.46784i	−0.679238	+	0.733918i	$0.737690\pi$
$488$	− 11.3205i	− 0.512455i
$489$	−9.26795	−0.419111
$490$	0	0
$491$	16.0526	0.724442	0.362221	−	0.932092i	$-0.382019\pi$
$491$	16.0526	0.724442	0.362221	+	0.932092i	$0.382019\pi$
$492$	− 1.26795i	− 0.0571636i
$493$	0	0
$494$	1.26795	0.0570477
$495$	0	0
$496$	−44.6410	−2.00444
$497$	− 12.0000i	− 0.538274i
$498$	22.3923i	1.00342i
$499$	−10.5359	−0.471652	−0.235826	−	0.971795i	$-0.575779\pi$
$499$	−10.5359	−0.471652	−0.235826	+	0.971795i	$0.575779\pi$
$500$	0	0
$501$	3.46410	0.154765
$502$	48.5885i	2.16861i
$503$	− 23.0718i	− 1.02872i	−0.857574	−	0.514360i	$-0.828029\pi$
$503$	− 23.0718i	− 1.02872i	0.857574	−	0.514360i	$-0.171971\pi$
$504$	4.73205	0.210782
$505$	0	0
$506$	−28.3923	−1.26219
$507$	12.4641i	0.553550i
$508$	− 4.00000i	− 0.177471i
$509$	10.0526	0.445572	0.222786	−	0.974867i	$-0.428485\pi$
$509$	10.0526	0.445572	0.222786	+	0.974867i	$0.428485\pi$
$510$	0	0
$511$	46.2487	2.04592
$512$	8.66025i	0.382733i
$513$	1.00000i	0.0441511i
$514$	−41.5692	−1.83354
$515$	0	0
$516$	4.19615	0.184725
$517$	16.3923i	0.720933i
$518$	− 29.3205i	− 1.28827i
$519$	−6.92820	−0.304114
$520$	0	0
$521$	37.2679	1.63274	0.816369	−	0.577530i	$-0.195983\pi$
$521$	37.2679	1.63274	0.816369	+	0.577530i	$0.195983\pi$
$522$	14.1962i	0.621349i
$523$	8.67949i	0.379528i	0.981830	+	0.189764i	$0.0607722\pi$
$523$	8.67949i	0.379528i	−0.981830	+	0.189764i	$0.939228\pi$
$524$	9.12436	0.398599
$525$	0	0
$526$	−10.3923	−0.453126
$527$	0	0
$528$	− 23.6603i	− 1.02968i
$529$	11.0000	0.478261
$530$	0	0
$531$	2.53590	0.110049
$532$	2.73205i	0.118449i
$533$	0.928203i	0.0402049i
$534$	18.5885	0.804401
$535$	0	0
$536$	13.8564	0.598506
$537$	− 23.3205i	− 1.00635i
$538$	− 1.01924i	− 0.0439425i
$539$	−2.19615	−0.0945950
$540$	0	0
$541$	41.7128	1.79337	0.896687	−	0.442665i	$-0.145967\pi$
$541$	41.7128	1.79337	0.896687	+	0.442665i	$0.145967\pi$
$542$	0.679492i	0.0291867i
$543$	− 2.39230i	− 0.102664i
$544$	0	0
$545$	0	0
$546$	3.46410	0.148250
$547$	− 43.3205i	− 1.85225i	−0.377215	−	0.926126i	$-0.623118\pi$
$547$	− 43.3205i	− 1.85225i	0.377215	−	0.926126i	$-0.376882\pi$
$548$	19.8564i	0.848224i
$549$	6.53590	0.278945
$550$	0	0
$551$	8.19615	0.349168
$552$	− 6.00000i	− 0.255377i
$553$	29.8564i	1.26962i
$554$	3.46410	0.147176
$555$	0	0
$556$	−8.39230	−0.355913
$557$	− 0.928203i	− 0.0393292i	−0.999807	−	0.0196646i	$-0.993740\pi$
$557$	− 0.928203i	− 0.0393292i	0.999807	−	0.0196646i	$-0.00625985\pi$
$558$	− 15.4641i	− 0.654648i
$559$	−3.07180	−0.129923
$560$	0	0
$561$	0	0
$562$	2.19615i	0.0926391i
$563$	− 27.4641i	− 1.15747i	−0.815514	−	0.578737i	$-0.803546\pi$
$563$	− 27.4641i	− 1.15747i	0.815514	−	0.578737i	$-0.196454\pi$
$564$	3.46410	0.145865
$565$	0	0
$566$	−43.2679	−1.81869
$567$	2.73205i	0.114735i
$568$	− 7.60770i	− 0.319212i
$569$	−22.0526	−0.924491	−0.462246	−	0.886752i	$-0.652956\pi$
$569$	−22.0526	−0.924491	−0.462246	+	0.886752i	$0.652956\pi$
$570$	0	0
$571$	−34.2487	−1.43326	−0.716632	−	0.697452i	$-0.754317\pi$
$571$	−34.2487	−1.43326	−0.716632	+	0.697452i	$0.754317\pi$
$572$	− 3.46410i	− 0.144841i
$573$	0.339746i	0.0141931i
$574$	−6.00000	−0.250435
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 15.1769i	− 0.631823i	−0.948789	−	0.315912i	$-0.897690\pi$
$577$	− 15.1769i	− 0.631823i	0.948789	−	0.315912i	$-0.102310\pi$
$578$	29.4449i	1.22474i
$579$	−17.1244	−0.711664
$580$	0	0
$581$	35.3205	1.46534
$582$	− 10.7321i	− 0.444858i
$583$	− 44.7846i	− 1.85479i
$584$	29.3205	1.21329
$585$	0	0
$586$	−48.0000	−1.98286
$587$	3.46410i	0.142979i	0.997441	+	0.0714894i	$0.0227752\pi$
$587$	3.46410i	0.142979i	−0.997441	+	0.0714894i	$0.977225\pi$
$588$	0.464102i	0.0191392i
$589$	−8.92820	−0.367880
$590$	0	0
$591$	−24.0000	−0.987228
$592$	− 30.9808i	− 1.27330i
$593$	− 38.7846i	− 1.59269i	−0.604841	−	0.796347i	$-0.706763\pi$
$593$	− 38.7846i	− 1.59269i	0.604841	−	0.796347i	$-0.293237\pi$
$594$	8.19615	0.336292
$595$	0	0
$596$	−19.8564	−0.813350
$597$	15.3205i	0.627027i
$598$	− 4.39230i	− 0.179615i
$599$	13.8564	0.566157	0.283079	−	0.959097i	$-0.408644\pi$
$599$	13.8564	0.566157	0.283079	+	0.959097i	$0.408644\pi$
$600$	0	0
$601$	−47.1769	−1.92439	−0.962193	−	0.272368i	$-0.912193\pi$
$601$	−47.1769	−1.92439	−0.962193	+	0.272368i	$0.912193\pi$
$602$	− 19.8564i	− 0.809287i
$603$	8.00000i	0.325785i
$604$	−14.0000	−0.569652
$605$	0	0
$606$	18.0000	0.731200
$607$	11.6077i	0.471142i	0.971857	+	0.235571i	$0.0756960\pi$
$607$	11.6077i	0.471142i	−0.971857	+	0.235571i	$0.924304\pi$
$608$	5.19615i	0.210732i
$609$	22.3923	0.907382
$610$	0	0
$611$	−2.53590	−0.102591
$612$	0	0
$613$	42.3923i	1.71221i	0.516803	+	0.856105i	$0.327122\pi$
$613$	42.3923i	1.71221i	−0.516803	+	0.856105i	$0.672878\pi$
$614$	−56.1051	−2.26422
$615$	0	0
$616$	−22.3923	−0.902212
$617$	− 27.7128i	− 1.11568i	−0.829950	−	0.557838i	$-0.811631\pi$
$617$	− 27.7128i	− 1.11568i	0.829950	−	0.557838i	$-0.188369\pi$
$618$	− 17.0718i	− 0.686728i
$619$	15.3205	0.615783	0.307892	−	0.951421i	$-0.400377\pi$
$619$	15.3205	0.615783	0.307892	+	0.951421i	$0.400377\pi$
$620$	0	0
$621$	3.46410	0.139010
$622$	56.1962i	2.25326i
$623$	− 29.3205i	− 1.17470i
$624$	3.66025	0.146527
$625$	0	0
$626$	−11.0718	−0.442518
$627$	− 4.73205i	− 0.188980i
$628$	6.39230i	0.255081i
$629$	0	0
$630$	0	0
$631$	−34.9282	−1.39047	−0.695235	−	0.718783i	$-0.744700\pi$
$631$	−34.9282	−1.39047	−0.695235	+	0.718783i	$0.744700\pi$
$632$	18.9282i	0.752923i
$633$	1.07180i	0.0426001i
$634$	−19.6077	−0.778721
$635$	0	0
$636$	−9.46410	−0.375276
$637$	− 0.339746i	− 0.0134612i
$638$	− 67.1769i	− 2.65956i
$639$	4.39230	0.173757
$640$	0	0
$641$	48.5885	1.91913	0.959564	−	0.281489i	$-0.0908284\pi$
$641$	48.5885	1.91913	0.959564	+	0.281489i	$0.0908284\pi$
$642$	0	0
$643$	− 12.1962i	− 0.480969i	−0.970653	−	0.240485i	$-0.922694\pi$
$643$	− 12.1962i	− 0.480969i	0.970653	−	0.240485i	$-0.0773064\pi$
$644$	9.46410	0.372938
$645$	0	0
$646$	0	0
$647$	4.14359i	0.162901i	0.996677	+	0.0814507i	$0.0259553\pi$
$647$	4.14359i	0.162901i	−0.996677	+	0.0814507i	$0.974045\pi$
$648$	1.73205i	0.0680414i
$649$	−12.0000	−0.471041
$650$	0	0
$651$	−24.3923	−0.956010
$652$	− 9.26795i	− 0.362961i
$653$	17.0718i	0.668071i	0.942560	+	0.334036i	$0.108411\pi$
$653$	17.0718i	0.668071i	−0.942560	+	0.334036i	$0.891589\pi$
$654$	−24.9282	−0.974770
$655$	0	0
$656$	−6.33975	−0.247525
$657$	16.9282i	0.660432i
$658$	− 16.3923i	− 0.639039i
$659$	−5.07180	−0.197569	−0.0987846	−	0.995109i	$-0.531495\pi$
$659$	−5.07180	−0.197569	−0.0987846	+	0.995109i	$0.531495\pi$
$660$	0	0
$661$	39.1769	1.52381	0.761903	−	0.647692i	$-0.224265\pi$
$661$	39.1769	1.52381	0.761903	+	0.647692i	$0.224265\pi$
$662$	− 44.5359i	− 1.73094i
$663$	0	0
$664$	22.3923	0.868990
$665$	0	0
$666$	10.7321	0.415859
$667$	− 28.3923i	− 1.09935i
$668$	3.46410i	0.134030i
$669$	17.8564	0.690369
$670$	0	0
$671$	−30.9282	−1.19397
$672$	14.1962i	0.547628i
$673$	17.1244i	0.660095i	0.943964	+	0.330048i	$0.107065\pi$
$673$	17.1244i	0.660095i	−0.943964	+	0.330048i	$0.892935\pi$
$674$	8.87564	0.341877
$675$	0	0
$676$	−12.4641	−0.479389
$677$	− 0.679492i	− 0.0261150i	−0.999915	−	0.0130575i	$-0.995844\pi$
$677$	− 0.679492i	− 0.0261150i	0.999915	−	0.0130575i	$-0.00415645\pi$
$678$	− 32.7846i	− 1.25909i
$679$	−16.9282	−0.649645
$680$	0	0
$681$	10.3923	0.398234
$682$	73.1769i	2.80209i
$683$	− 5.07180i	− 0.194067i	−0.995281	−	0.0970335i	$-0.969065\pi$
$683$	− 5.07180i	− 0.194067i	0.995281	−	0.0970335i	$-0.0309354\pi$
$684$	−1.00000	−0.0382360
$685$	0	0
$686$	−30.9282	−1.18084
$687$	18.5359i	0.707189i
$688$	− 20.9808i	− 0.799884i
$689$	6.92820	0.263944
$690$	0	0
$691$	12.3923	0.471425	0.235713	−	0.971823i	$-0.424258\pi$
$691$	12.3923	0.471425	0.235713	+	0.971823i	$0.424258\pi$
$692$	− 6.92820i	− 0.263371i
$693$	− 12.9282i	− 0.491102i
$694$	−1.60770	−0.0610273
$695$	0	0
$696$	14.1962	0.538104
$697$	0	0
$698$	38.1051i	1.44230i
$699$	7.85641	0.297157
$700$	0	0
$701$	−33.7128	−1.27332	−0.636658	−	0.771147i	$-0.719684\pi$
$701$	−33.7128	−1.27332	−0.636658	+	0.771147i	$0.719684\pi$
$702$	1.26795i	0.0478557i
$703$	− 6.19615i	− 0.233692i
$704$	−4.73205	−0.178346
$705$	0	0
$706$	25.6077	0.963758
$707$	− 28.3923i	− 1.06780i
$708$	2.53590i	0.0953049i
$709$	29.1769	1.09576	0.547881	−	0.836556i	$-0.315435\pi$
$709$	29.1769	1.09576	0.547881	+	0.836556i	$0.315435\pi$
$710$	0	0
$711$	−10.9282	−0.409840
$712$	− 18.5885i	− 0.696632i
$713$	30.9282i	1.15827i
$714$	0	0
$715$	0	0
$716$	23.3205	0.871528
$717$	9.80385i	0.366131i
$718$	− 0.588457i	− 0.0219610i
$719$	11.6603	0.434854	0.217427	−	0.976077i	$-0.430234\pi$
$719$	11.6603	0.434854	0.217427	+	0.976077i	$0.430234\pi$
$720$	0	0
$721$	−26.9282	−1.00286
$722$	1.73205i	0.0644603i
$723$	− 3.07180i	− 0.114241i
$724$	2.39230	0.0889093
$725$	0	0
$726$	−19.7321	−0.732325
$727$	− 25.6603i	− 0.951686i	−0.879530	−	0.475843i	$-0.842143\pi$
$727$	− 25.6603i	− 0.951686i	0.879530	−	0.475843i	$-0.157857\pi$
$728$	− 3.46410i	− 0.128388i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	0	0
$732$	6.53590i	0.241574i
$733$	− 18.7846i	− 0.693825i	−0.937897	−	0.346913i	$-0.887230\pi$
$733$	− 18.7846i	− 0.693825i	0.937897	−	0.346913i	$-0.112770\pi$
$734$	28.0526	1.03544
$735$	0	0
$736$	18.0000	0.663489
$737$	− 37.8564i	− 1.39446i
$738$	− 2.19615i	− 0.0808415i
$739$	−6.14359	−0.225996	−0.112998	−	0.993595i	$-0.536045\pi$
$739$	−6.14359	−0.225996	−0.112998	+	0.993595i	$0.536045\pi$
$740$	0	0
$741$	0.732051	0.0268926
$742$	44.7846i	1.64409i
$743$	− 3.21539i	− 0.117961i	−0.998259	−	0.0589806i	$-0.981215\pi$
$743$	− 3.21539i	− 0.117961i	0.998259	−	0.0589806i	$-0.0187850\pi$
$744$	−15.4641	−0.566941
$745$	0	0
$746$	10.7321	0.392928
$747$	12.9282i	0.473018i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	26.0000	0.948753	0.474377	−	0.880322i	$-0.342673\pi$
$751$	26.0000	0.948753	0.474377	+	0.880322i	$0.342673\pi$
$752$	− 17.3205i	− 0.631614i
$753$	28.0526i	1.02229i
$754$	10.3923	0.378465
$755$	0	0
$756$	−2.73205	−0.0993637
$757$	− 32.2487i	− 1.17210i	−0.810275	−	0.586050i	$-0.800682\pi$
$757$	− 32.2487i	− 1.17210i	0.810275	−	0.586050i	$-0.199318\pi$
$758$	− 36.2487i	− 1.31661i
$759$	−16.3923	−0.595003
$760$	0	0
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	− 6.92820i	− 0.250982i
$763$	39.3205i	1.42350i
$764$	−0.339746	−0.0122916
$765$	0	0
$766$	29.5692	1.06838
$767$	− 1.85641i	− 0.0670310i
$768$	19.0000i	0.685603i
$769$	20.6410	0.744334	0.372167	−	0.928166i	$-0.378615\pi$
$769$	20.6410	0.744334	0.372167	+	0.928166i	$0.378615\pi$
$770$	0	0
$771$	−24.0000	−0.864339
$772$	− 17.1244i	− 0.616319i
$773$	− 25.1769i	− 0.905551i	−0.891625	−	0.452775i	$-0.850434\pi$
$773$	− 25.1769i	− 0.905551i	0.891625	−	0.452775i	$-0.149566\pi$
$774$	7.26795	0.261241
$775$	0	0
$776$	−10.7321	−0.385258
$777$	− 16.9282i	− 0.607296i
$778$	− 13.6077i	− 0.487860i
$779$	−1.26795	−0.0454290
$780$	0	0
$781$	−20.7846	−0.743732
$782$	0	0
$783$	8.19615i	0.292907i
$784$	2.32051	0.0828753
$785$	0	0
$786$	15.8038	0.563705
$787$	− 8.67949i	− 0.309390i	−0.987962	−	0.154695i	$-0.950560\pi$
$787$	− 8.67949i	− 0.309390i	0.987962	−	0.154695i	$-0.0494396\pi$
$788$	− 24.0000i	− 0.854965i
$789$	−6.00000	−0.213606
$790$	0	0
$791$	−51.7128	−1.83870
$792$	− 8.19615i	− 0.291238i
$793$	− 4.78461i	− 0.169906i
$794$	15.4641	0.548800
$795$	0	0
$796$	−15.3205	−0.543021
$797$	− 44.7846i	− 1.58635i	−0.608992	−	0.793176i	$-0.708426\pi$
$797$	− 44.7846i	− 1.58635i	0.608992	−	0.793176i	$-0.291574\pi$
$798$	4.73205i	0.167513i
$799$	0	0
$800$	0	0
$801$	10.7321	0.379198
$802$	− 58.9808i	− 2.08268i
$803$	− 80.1051i	− 2.82685i
$804$	−8.00000	−0.282138
$805$	0	0
$806$	−11.3205	−0.398748
$807$	− 0.588457i	− 0.0207147i
$808$	− 18.0000i	− 0.633238i
$809$	−14.7846	−0.519799	−0.259900	−	0.965636i	$-0.583689\pi$
$809$	−14.7846	−0.519799	−0.259900	+	0.965636i	$0.583689\pi$
$810$	0	0
$811$	37.5692	1.31923	0.659617	−	0.751602i	$-0.270719\pi$
$811$	37.5692	1.31923	0.659617	+	0.751602i	$0.270719\pi$
$812$	22.3923i	0.785816i
$813$	0.392305i	0.0137587i
$814$	−50.7846	−1.78000
$815$	0	0
$816$	0	0
$817$	− 4.19615i	− 0.146805i
$818$	45.7128i	1.59831i
$819$	2.00000	0.0698857
$820$	0	0
$821$	−32.5359	−1.13551	−0.567755	−	0.823197i	$-0.692188\pi$
$821$	−32.5359	−1.13551	−0.567755	+	0.823197i	$0.692188\pi$
$822$	34.3923i	1.19957i
$823$	12.9808i	0.452481i	0.974071	+	0.226240i	$0.0726435\pi$
$823$	12.9808i	0.452481i	−0.974071	+	0.226240i	$0.927356\pi$
$824$	−17.0718	−0.594724
$825$	0	0
$826$	12.0000	0.417533
$827$	5.32051i	0.185012i	0.995712	+	0.0925061i	$0.0294878\pi$
$827$	5.32051i	0.185012i	−0.995712	+	0.0925061i	$0.970512\pi$
$828$	3.46410i	0.120386i
$829$	−34.1051	−1.18452	−0.592260	−	0.805747i	$-0.701764\pi$
$829$	−34.1051	−1.18452	−0.592260	+	0.805747i	$0.701764\pi$
$830$	0	0
$831$	2.00000	0.0693792
$832$	− 0.732051i	− 0.0253793i
$833$	0	0
$834$	−14.5359	−0.503337
$835$	0	0
$836$	4.73205	0.163661
$837$	− 8.92820i	− 0.308604i
$838$	48.5885i	1.67846i
$839$	19.6077	0.676933	0.338466	−	0.940978i	$-0.390092\pi$
$839$	19.6077	0.676933	0.338466	+	0.940978i	$0.390092\pi$
$840$	0	0
$841$	38.1769	1.31645
$842$	− 32.5359i	− 1.12126i
$843$	1.26795i	0.0436705i
$844$	−1.07180	−0.0368928
$845$	0	0
$846$	6.00000	0.206284
$847$	31.1244i	1.06945i
$848$	47.3205i	1.62499i
$849$	−24.9808	−0.857338
$850$	0	0
$851$	−21.4641	−0.735780
$852$	4.39230i	0.150478i
$853$	27.1769i	0.930520i	0.885174	+	0.465260i	$0.154039\pi$
$853$	27.1769i	0.930520i	−0.885174	+	0.465260i	$0.845961\pi$
$854$	30.9282	1.05834
$855$	0	0
$856$	0	0
$857$	42.2487i	1.44319i	0.692316	+	0.721594i	$0.256590\pi$
$857$	42.2487i	1.44319i	−0.692316	+	0.721594i	$0.743410\pi$
$858$	− 6.00000i	− 0.204837i
$859$	−32.0000	−1.09183	−0.545913	−	0.837842i	$-0.683817\pi$
$859$	−32.0000	−1.09183	−0.545913	+	0.837842i	$0.683817\pi$
$860$	0	0
$861$	−3.46410	−0.118056
$862$	− 19.6077i	− 0.667841i
$863$	− 12.0000i	− 0.408485i	−0.978920	−	0.204242i	$-0.934527\pi$
$863$	− 12.0000i	− 0.408485i	0.978920	−	0.204242i	$-0.0654731\pi$
$864$	−5.19615	−0.176777
$865$	0	0
$866$	18.3397	0.623210
$867$	17.0000i	0.577350i
$868$	− 24.3923i	− 0.827929i
$869$	51.7128	1.75424
$870$	0	0
$871$	5.85641	0.198437
$872$	24.9282i	0.844175i
$873$	− 6.19615i	− 0.209708i
$874$	6.00000	0.202953
$875$	0	0
$876$	−16.9282	−0.571951
$877$	− 53.1244i	− 1.79388i	−0.442150	−	0.896941i	$-0.645784\pi$
$877$	− 53.1244i	− 1.79388i	0.442150	−	0.896941i	$-0.354216\pi$
$878$	− 46.6410i	− 1.57406i
$879$	−27.7128	−0.934730
$880$	0	0
$881$	8.53590	0.287582	0.143791	−	0.989608i	$-0.454071\pi$
$881$	8.53590	0.287582	0.143791	+	0.989608i	$0.454071\pi$
$882$	0.803848i	0.0270670i
$883$	36.9808i	1.24450i	0.782818	+	0.622251i	$0.213782\pi$
$883$	36.9808i	1.24450i	−0.782818	+	0.622251i	$0.786218\pi$
$884$	0	0
$885$	0	0
$886$	9.21539	0.309597
$887$	24.0000i	0.805841i	0.915235	+	0.402921i	$0.132005\pi$
$887$	24.0000i	0.805841i	−0.915235	+	0.402921i	$0.867995\pi$
$888$	− 10.7321i	− 0.360144i
$889$	−10.9282	−0.366520
$890$	0	0
$891$	4.73205	0.158530
$892$	17.8564i	0.597877i
$893$	− 3.46410i	− 0.115922i
$894$	−34.3923	−1.15025
$895$	0	0
$896$	33.1244	1.10661
$897$	− 2.53590i	− 0.0846712i
$898$	9.80385i	0.327159i
$899$	−73.1769	−2.44059
$900$	0	0
$901$	0	0
$902$	10.3923i	0.346026i
$903$	− 11.4641i	− 0.381501i
$904$	−32.7846	−1.09040
$905$	0	0
$906$	−24.2487	−0.805609
$907$	32.3923i	1.07557i	0.843082	+	0.537784i	$0.180739\pi$
$907$	32.3923i	1.07557i	−0.843082	+	0.537784i	$0.819261\pi$
$908$	10.3923i	0.344881i
$909$	10.3923	0.344691
$910$	0	0
$911$	−54.9282	−1.81985	−0.909926	−	0.414770i	$-0.863862\pi$
$911$	−54.9282	−1.81985	−0.909926	+	0.414770i	$0.863862\pi$
$912$	5.00000i	0.165567i
$913$	− 61.1769i	− 2.02466i
$914$	7.85641	0.259867
$915$	0	0
$916$	−18.5359	−0.612443
$917$	− 24.9282i	− 0.823202i
$918$	0	0
$919$	−51.4256	−1.69637	−0.848187	−	0.529696i	$-0.822306\pi$
$919$	−51.4256	−1.69637	−0.848187	+	0.529696i	$0.822306\pi$
$920$	0	0
$921$	−32.3923	−1.06736
$922$	10.3923i	0.342252i
$923$	− 3.21539i	− 0.105836i
$924$	12.9282	0.425307
$925$	0	0
$926$	61.5167	2.02156
$927$	− 9.85641i	− 0.323727i
$928$	42.5885i	1.39803i
$929$	1.60770	0.0527468	0.0263734	−	0.999652i	$-0.491604\pi$
$929$	1.60770	0.0527468	0.0263734	+	0.999652i	$0.491604\pi$
$930$	0	0
$931$	0.464102	0.0152103
$932$	7.85641i	0.257345i
$933$	32.4449i	1.06220i
$934$	35.5692	1.16386
$935$	0	0
$936$	1.26795	0.0414442
$937$	16.2487i	0.530822i	0.964135	+	0.265411i	$0.0855077\pi$
$937$	16.2487i	0.530822i	−0.964135	+	0.265411i	$0.914492\pi$
$938$	37.8564i	1.23606i
$939$	−6.39230	−0.208605
$940$	0	0
$941$	0.588457	0.0191832	0.00959158	−	0.999954i	$-0.496947\pi$
$941$	0.588457	0.0191832	0.00959158	+	0.999954i	$0.496947\pi$
$942$	11.0718i	0.360739i
$943$	4.39230i	0.143033i
$944$	12.6795	0.412682
$945$	0	0
$946$	−34.3923	−1.11819
$947$	− 28.1436i	− 0.914544i	−0.889327	−	0.457272i	$-0.848827\pi$
$947$	− 28.1436i	− 0.914544i	0.889327	−	0.457272i	$-0.151173\pi$
$948$	− 10.9282i	− 0.354932i
$949$	12.3923	0.402271
$950$	0	0
$951$	−11.3205	−0.367093
$952$	0	0
$953$	37.8564i	1.22629i	0.789971	+	0.613145i	$0.210096\pi$
$953$	37.8564i	1.22629i	−0.789971	+	0.613145i	$0.789904\pi$
$954$	−16.3923	−0.530720
$955$	0	0
$956$	−9.80385	−0.317079
$957$	− 38.7846i	− 1.25373i
$958$	− 44.1962i	− 1.42791i
$959$	54.2487	1.75178
$960$	0	0
$961$	48.7128	1.57138
$962$	− 7.85641i	− 0.253301i
$963$	0	0
$964$	3.07180	0.0989359
$965$	0	0
$966$	16.3923	0.527414
$967$	− 4.87564i	− 0.156790i	−0.996922	−	0.0783951i	$-0.975020\pi$
$967$	− 4.87564i	− 0.156790i	0.996922	−	0.0783951i	$-0.0249796\pi$
$968$	19.7321i	0.634212i
$969$	0	0
$970$	0	0
$971$	−27.7128	−0.889346	−0.444673	−	0.895693i	$-0.646680\pi$
$971$	−27.7128	−0.889346	−0.444673	+	0.895693i	$0.646680\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	22.9282i	0.735044i
$974$	−56.1051	−1.79772
$975$	0	0
$976$	32.6795	1.04605
$977$	39.0333i	1.24879i	0.781110	+	0.624393i	$0.214654\pi$
$977$	39.0333i	1.24879i	−0.781110	+	0.624393i	$0.785346\pi$
$978$	− 16.0526i	− 0.513304i
$979$	−50.7846	−1.62308
$980$	0	0
$981$	−14.3923	−0.459511
$982$	27.8038i	0.887256i
$983$	− 41.3205i	− 1.31792i	−0.752178	−	0.658960i	$-0.770997\pi$
$983$	− 41.3205i	− 1.31792i	0.752178	−	0.658960i	$-0.229003\pi$
$984$	−2.19615	−0.0700108
$985$	0	0
$986$	0	0
$987$	− 9.46410i	− 0.301246i
$988$	0.732051i	0.0232896i
$989$	−14.5359	−0.462215
$990$	0	0
$991$	13.0718	0.415239	0.207620	−	0.978210i	$-0.433428\pi$
$991$	13.0718	0.415239	0.207620	+	0.978210i	$0.433428\pi$
$992$	− 46.3923i	− 1.47296i
$993$	− 25.7128i	− 0.815971i
$994$	20.7846	0.659248
$995$	0	0
$996$	−12.9282	−0.409646
$997$	17.6077i	0.557641i	0.960343	+	0.278821i	$0.0899435\pi$
$997$	17.6077i	0.557641i	−0.960343	+	0.278821i	$0.910056\pi$
$998$	− 18.2487i	− 0.577653i
$999$	6.19615	0.196038

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1425.2.c.k.799.4		4
5.2	odd	4		285.2.a.e.1.1	✓	2
5.3	odd	4		1425.2.a.o.1.2		2
5.4	even	2	inner	1425.2.c.k.799.1		4
15.2	even	4		855.2.a.f.1.2		2
15.8	even	4		4275.2.a.t.1.1		2
20.7	even	4		4560.2.a.bh.1.2		2
95.37	even	4		5415.2.a.r.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
285.2.a.e.1.1	✓	2	5.2	odd	4
855.2.a.f.1.2		2	15.2	even	4
1425.2.a.o.1.2		2	5.3	odd	4
1425.2.c.k.799.1		4	5.4	even	2	inner
1425.2.c.k.799.4		4	1.1	even	1	trivial
4275.2.a.t.1.1		2	15.8	even	4
4560.2.a.bh.1.2		2	20.7	even	4
5415.2.a.r.1.2		2	95.37	even	4

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:	\( N \)	\(=\)	\( 1425 = 3 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1425.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.73205 q^{6} +2.73205i q^{7} +1.73205i q^{8} -1.00000 q^{9} +4.73205 q^{11} -1.00000i q^{12} +0.732051i q^{13} -4.73205 q^{14} -5.00000 q^{16} -1.73205i q^{18} -1.00000 q^{19} -2.73205 q^{21} +8.19615i q^{22} +3.46410i q^{23} -1.73205 q^{24} -1.26795 q^{26} -1.00000i q^{27} -2.73205i q^{28} -8.19615 q^{29} +8.92820 q^{31} -5.19615i q^{32} +4.73205i q^{33} +1.00000 q^{36} +6.19615i q^{37} -1.73205i q^{38} -0.732051 q^{39} +1.26795 q^{41} -4.73205i q^{42} +4.19615i q^{43} -4.73205 q^{44} -6.00000 q^{46} +3.46410i q^{47} -5.00000i q^{48} -0.464102 q^{49} -0.732051i q^{52} -9.46410i q^{53} +1.73205 q^{54} -4.73205 q^{56} -1.00000i q^{57} -14.1962i q^{58} -2.53590 q^{59} -6.53590 q^{61} +15.4641i q^{62} -2.73205i q^{63} -1.00000 q^{64} -8.19615 q^{66} -8.00000i q^{67} -3.46410 q^{69} -4.39230 q^{71} -1.73205i q^{72} -16.9282i q^{73} -10.7321 q^{74} +1.00000 q^{76} +12.9282i q^{77} -1.26795i q^{78} +10.9282 q^{79} +1.00000 q^{81} +2.19615i q^{82} -12.9282i q^{83} +2.73205 q^{84} -7.26795 q^{86} -8.19615i q^{87} +8.19615i q^{88} -10.7321 q^{89} -2.00000 q^{91} -3.46410i q^{92} +8.92820i q^{93} -6.00000 q^{94} +5.19615 q^{96} +6.19615i q^{97} -0.803848i q^{98} -4.73205 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} - 4 q^{9} + 12 q^{11} - 12 q^{14} - 20 q^{16} - 4 q^{19} - 4 q^{21} - 12 q^{26} - 12 q^{29} + 8 q^{31} + 4 q^{36} + 4 q^{39} + 12 q^{41} - 12 q^{44} - 24 q^{46} + 12 q^{49} - 12 q^{56} - 24 q^{59}+ \cdots - 12 q^{99}+O(q^{100}) \) 4 * q - 4 * q^4 - 4 * q^9 + 12 * q^11 - 12 * q^14 - 20 * q^16 - 4 * q^19 - 4 * q^21 - 12 * q^26 - 12 * q^29 + 8 * q^31 + 4 * q^36 + 4 * q^39 + 12 * q^41 - 12 * q^44 - 24 * q^46 + 12 * q^49 - 12 * q^56 - 24 * q^59 - 40 * q^61 - 4 * q^64 - 12 * q^66 + 24 * q^71 - 36 * q^74 + 4 * q^76 + 16 * q^79 + 4 * q^81 + 4 * q^84 - 36 * q^86 - 36 * q^89 - 8 * q^91 - 24 * q^94 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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