LMFDB - Embedded newform 1815.2.a.s.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1815,2,Mod(1,1815)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1815, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1815.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1815 = 3 \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1815.a (trivial)

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:	yes
Analytic conductor:	$14.4928479669$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.8112.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 3 $ x^4 - 5*x^2 + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.07431$ of defining polynomial
Character	$\chi$	$=$	1815.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.628052 q^{2} -1.00000 q^{3} -1.60555 q^{4} +1.00000 q^{5} -0.628052 q^{6} -4.14863 q^{7} -2.26447 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.628052 q^{2} -1.00000 q^{3} -1.60555 q^{4} +1.00000 q^{5} -0.628052 q^{6} -4.14863 q^{7} -2.26447 q^{8} +1.00000 q^{9} +0.628052 q^{10} +1.60555 q^{12} -5.40473 q^{13} -2.60555 q^{14} -1.00000 q^{15} +1.78890 q^{16} -4.14863 q^{17} +0.628052 q^{18} +1.25610 q^{19} -1.60555 q^{20} +4.14863 q^{21} +5.21110 q^{23} +2.26447 q^{24} +1.00000 q^{25} -3.39445 q^{26} -1.00000 q^{27} +6.66083 q^{28} +7.04115 q^{29} -0.628052 q^{30} +4.00000 q^{31} +5.65246 q^{32} -2.60555 q^{34} -4.14863 q^{35} -1.60555 q^{36} -7.21110 q^{37} +0.788897 q^{38} +5.40473 q^{39} -2.26447 q^{40} +9.55336 q^{41} +2.60555 q^{42} -6.66083 q^{43} +1.00000 q^{45} +3.27284 q^{46} -1.78890 q^{48} +10.2111 q^{49} +0.628052 q^{50} +4.14863 q^{51} +8.67757 q^{52} +11.2111 q^{53} -0.628052 q^{54} +9.39445 q^{56} -1.25610 q^{57} +4.42221 q^{58} -5.21110 q^{59} +1.60555 q^{60} -8.29725 q^{61} +2.51221 q^{62} -4.14863 q^{63} -0.0277564 q^{64} -5.40473 q^{65} -13.2111 q^{67} +6.66083 q^{68} -5.21110 q^{69} -2.60555 q^{70} +5.21110 q^{71} -2.26447 q^{72} +2.89252 q^{73} -4.52894 q^{74} -1.00000 q^{75} -2.01674 q^{76} +3.39445 q^{78} +1.25610 q^{79} +1.78890 q^{80} +1.00000 q^{81} +6.00000 q^{82} +13.7020 q^{83} -6.66083 q^{84} -4.14863 q^{85} -4.18335 q^{86} -7.04115 q^{87} +6.00000 q^{89} +0.628052 q^{90} +22.4222 q^{91} -8.36669 q^{92} -4.00000 q^{93} +1.25610 q^{95} -5.65246 q^{96} +19.2111 q^{97} +6.41310 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + 8 q^{4} + 4 q^{5} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 + 8 * q^4 + 4 * q^5 + 4 * q^9	$ 4 q - 4 q^{3} + 8 q^{4} + 4 q^{5} + 4 q^{9} - 8 q^{12} + 4 q^{14} - 4 q^{15} + 36 q^{16} + 8 q^{20} - 8 q^{23} + 4 q^{25} - 28 q^{26} - 4 q^{27} + 16 q^{31} + 4 q^{34} + 8 q^{36} + 32 q^{38} - 4 q^{42} + 4 q^{45} - 36 q^{48} + 12 q^{49} + 16 q^{53} + 52 q^{56} - 40 q^{58} + 8 q^{59} - 8 q^{60} + 72 q^{64} - 24 q^{67} + 8 q^{69} + 4 q^{70} - 8 q^{71} - 4 q^{75} + 28 q^{78} + 36 q^{80} + 4 q^{81} + 24 q^{82} - 60 q^{86} + 24 q^{89} + 32 q^{91} - 120 q^{92} - 16 q^{93} + 48 q^{97}+O(q^{100}) $ 4 * q - 4 * q^3 + 8 * q^4 + 4 * q^5 + 4 * q^9 - 8 * q^12 + 4 * q^14 - 4 * q^15 + 36 * q^16 + 8 * q^20 - 8 * q^23 + 4 * q^25 - 28 * q^26 - 4 * q^27 + 16 * q^31 + 4 * q^34 + 8 * q^36 + 32 * q^38 - 4 * q^42 + 4 * q^45 - 36 * q^48 + 12 * q^49 + 16 * q^53 + 52 * q^56 - 40 * q^58 + 8 * q^59 - 8 * q^60 + 72 * q^64 - 24 * q^67 + 8 * q^69 + 4 * q^70 - 8 * q^71 - 4 * q^75 + 28 * q^78 + 36 * q^80 + 4 * q^81 + 24 * q^82 - 60 * q^86 + 24 * q^89 + 32 * q^91 - 120 * q^92 - 16 * q^93 + 48 * q^97

Display 100 coefficients 10 coefficients

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 $

$2$	0.628052	0.444099	0.222050	−	0.975035i	$-0.428725\pi$
$2$	0.628052	0.444099	0.222050	+	0.975035i	$0.428725\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.60555	−0.802776
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−0.628052	−0.256401
$7$	−4.14863	−1.56803	−0.784017	−	0.620740i	$-0.786832\pi$
$7$	−4.14863	−1.56803	−0.784017	+	0.620740i	$0.786832\pi$
$8$	−2.26447	−0.800612
$9$	1.00000	0.333333
$10$	0.628052	0.198607
$11$	0	0
$11$	0	0
$12$	1.60555	0.463483
$13$	−5.40473	−1.49900	−0.749501	−	0.662003i	$-0.769707\pi$
$13$	−5.40473	−1.49900	−0.749501	+	0.662003i	$0.769707\pi$
$14$	−2.60555	−0.696363
$15$	−1.00000	−0.258199
$16$	1.78890	0.447224
$17$	−4.14863	−1.00619	−0.503095	−	0.864231i	$-0.667805\pi$
$17$	−4.14863	−1.00619	−0.503095	+	0.864231i	$0.667805\pi$
$18$	0.628052	0.148033
$19$	1.25610	0.288170	0.144085	−	0.989565i	$-0.453976\pi$
$19$	1.25610	0.288170	0.144085	+	0.989565i	$0.453976\pi$
$20$	−1.60555	−0.359012
$21$	4.14863	0.905305
$22$	0	0
$23$	5.21110	1.08659	0.543295	−	0.839542i	$-0.317177\pi$
$23$	5.21110	1.08659	0.543295	+	0.839542i	$0.317177\pi$
$24$	2.26447	0.462233
$25$	1.00000	0.200000
$26$	−3.39445	−0.665706
$27$	−1.00000	−0.192450
$28$	6.66083	1.25878
$29$	7.04115	1.30751	0.653754	−	0.756707i	$-0.273193\pi$
$29$	7.04115	1.30751	0.653754	+	0.756707i	$0.273193\pi$
$30$	−0.628052	−0.114666
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	5.65246	0.999224
$33$	0	0
$34$	−2.60555	−0.446848
$35$	−4.14863	−0.701246
$36$	−1.60555	−0.267592
$37$	−7.21110	−1.18550	−0.592749	−	0.805387i	$-0.701957\pi$
$37$	−7.21110	−1.18550	−0.592749	+	0.805387i	$0.701957\pi$
$38$	0.788897	0.127976
$39$	5.40473	0.865449
$40$	−2.26447	−0.358044
$41$	9.55336	1.49198	0.745992	−	0.665955i	$-0.231976\pi$
$41$	9.55336	1.49198	0.745992	+	0.665955i	$0.231976\pi$
$42$	2.60555	0.402045
$43$	−6.66083	−1.01577	−0.507884	−	0.861426i	$-0.669572\pi$
$43$	−6.66083	−1.01577	−0.507884	+	0.861426i	$0.669572\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	3.27284	0.482554
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	−1.78890	−0.258205
$49$	10.2111	1.45873
$50$	0.628052	0.0888199
$51$	4.14863	0.580924
$52$	8.67757	1.20336
$53$	11.2111	1.53996	0.769982	−	0.638066i	$-0.220265\pi$
$53$	11.2111	1.53996	0.769982	+	0.638066i	$0.220265\pi$
$54$	−0.628052	−0.0854670
$55$	0	0
$56$	9.39445	1.25539
$57$	−1.25610	−0.166375
$58$	4.42221	0.580664
$59$	−5.21110	−0.678428	−0.339214	−	0.940709i	$-0.610161\pi$
$59$	−5.21110	−0.678428	−0.339214	+	0.940709i	$0.610161\pi$
$60$	1.60555	0.207276
$61$	−8.29725	−1.06235	−0.531177	−	0.847261i	$-0.678250\pi$
$61$	−8.29725	−1.06235	−0.531177	+	0.847261i	$0.678250\pi$
$62$	2.51221	0.319050
$63$	−4.14863	−0.522678
$64$	−0.0277564	−0.00346955
$65$	−5.40473	−0.670374
$66$	0	0
$67$	−13.2111	−1.61399	−0.806997	−	0.590556i	$-0.798908\pi$
$67$	−13.2111	−1.61399	−0.806997	+	0.590556i	$0.798908\pi$
$68$	6.66083	0.807745
$69$	−5.21110	−0.627343
$70$	−2.60555	−0.311423
$71$	5.21110	0.618444	0.309222	−	0.950990i	$-0.399931\pi$
$71$	5.21110	0.618444	0.309222	+	0.950990i	$0.399931\pi$
$72$	−2.26447	−0.266871
$73$	2.89252	0.338544	0.169272	−	0.985569i	$-0.445858\pi$
$73$	2.89252	0.338544	0.169272	+	0.985569i	$0.445858\pi$
$74$	−4.52894	−0.526479
$75$	−1.00000	−0.115470
$76$	−2.01674	−0.231336
$77$	0	0
$78$	3.39445	0.384346
$79$	1.25610	0.141323	0.0706613	−	0.997500i	$-0.477489\pi$
$79$	1.25610	0.141323	0.0706613	+	0.997500i	$0.477489\pi$
$80$	1.78890	0.200005
$81$	1.00000	0.111111
$82$	6.00000	0.662589
$83$	13.7020	1.50399	0.751994	−	0.659170i	$-0.229092\pi$
$83$	13.7020	1.50399	0.751994	+	0.659170i	$0.229092\pi$
$84$	−6.66083	−0.726756
$85$	−4.14863	−0.449982
$86$	−4.18335	−0.451102
$87$	−7.04115	−0.754891
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0.628052	0.0662024
$91$	22.4222	2.35049
$92$	−8.36669	−0.872288
$93$	−4.00000	−0.414781
$94$	0	0
$95$	1.25610	0.128873
$96$	−5.65246	−0.576902
$97$	19.2111	1.95059	0.975296	−	0.220902i	$-0.0709002\pi$
$97$	19.2111	1.95059	0.975296	+	0.220902i	$0.0709002\pi$
$98$	6.41310	0.647821
$99$	0	0
$100$	−1.60555	−0.160555
$101$	1.25610	0.124987	0.0624935	−	0.998045i	$-0.480095\pi$
$101$	1.25610	0.124987	0.0624935	+	0.998045i	$0.480095\pi$
$102$	2.60555	0.257988
$103$	1.21110	0.119333	0.0596667	−	0.998218i	$-0.480996\pi$
$103$	1.21110	0.119333	0.0596667	+	0.998218i	$0.480996\pi$
$104$	12.2389	1.20012
$105$	4.14863	0.404864
$106$	7.04115	0.683897
$107$	−2.89252	−0.279631	−0.139815	−	0.990178i	$-0.544651\pi$
$107$	−2.89252	−0.279631	−0.139815	+	0.990178i	$0.544651\pi$
$108$	1.60555	0.154494
$109$	16.5945	1.58947	0.794733	−	0.606960i	$-0.207611\pi$
$109$	16.5945	1.58947	0.794733	+	0.606960i	$0.207611\pi$
$110$	0	0
$111$	7.21110	0.684448
$112$	−7.42147	−0.701263
$113$	11.2111	1.05465	0.527326	−	0.849663i	$-0.323195\pi$
$113$	11.2111	1.05465	0.527326	+	0.849663i	$0.323195\pi$
$114$	−0.788897	−0.0738870
$115$	5.21110	0.485938
$116$	−11.3049	−1.04964
$117$	−5.40473	−0.499667
$118$	−3.27284	−0.301289
$119$	17.2111	1.57774
$120$	2.26447	0.206717
$121$	0	0
$122$	−5.21110	−0.471791
$123$	−9.55336	−0.861397
$124$	−6.42221	−0.576731
$125$	1.00000	0.0894427
$126$	−2.60555	−0.232121
$127$	1.63642	0.145209	0.0726044	−	0.997361i	$-0.476869\pi$
$127$	1.63642	0.145209	0.0726044	+	0.997361i	$0.476869\pi$
$128$	−11.3224	−1.00076
$129$	6.66083	0.586454
$130$	−3.39445	−0.297713
$131$	5.78505	0.505442	0.252721	−	0.967539i	$-0.418675\pi$
$131$	5.78505	0.505442	0.252721	+	0.967539i	$0.418675\pi$
$132$	0	0
$133$	−5.21110	−0.451860
$134$	−8.29725	−0.716774
$135$	−1.00000	−0.0860663
$136$	9.39445	0.805567
$137$	−16.4222	−1.40304	−0.701522	−	0.712648i	$-0.747496\pi$
$137$	−16.4222	−1.40304	−0.701522	+	0.712648i	$0.747496\pi$
$138$	−3.27284	−0.278603
$139$	−9.55336	−0.810305	−0.405153	−	0.914249i	$-0.632782\pi$
$139$	−9.55336	−0.810305	−0.405153	+	0.914249i	$0.632782\pi$
$140$	6.66083	0.562943
$141$	0	0
$142$	3.27284	0.274651
$143$	0	0
$144$	1.78890	0.149075
$145$	7.04115	0.584736
$146$	1.81665	0.150347
$147$	−10.2111	−0.842198
$148$	11.5778	0.951689
$149$	−22.8750	−1.87399	−0.936997	−	0.349336i	$-0.886407\pi$
$149$	−22.8750	−1.87399	−0.936997	+	0.349336i	$0.886407\pi$
$150$	−0.628052	−0.0512802
$151$	−15.3384	−1.24822	−0.624111	−	0.781336i	$-0.714539\pi$
$151$	−15.3384	−1.24822	−0.624111	+	0.781336i	$0.714539\pi$
$152$	−2.84441	−0.230712
$153$	−4.14863	−0.335397
$154$	0	0
$155$	4.00000	0.321288
$156$	−8.67757	−0.694762
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	0.788897	0.0627613
$159$	−11.2111	−0.889098
$160$	5.65246	0.446866
$161$	−21.6189	−1.70381
$162$	0.628052	0.0493444
$163$	13.2111	1.03477	0.517387	−	0.855752i	$-0.326905\pi$
$163$	13.2111	1.03477	0.517387	+	0.855752i	$0.326905\pi$
$164$	−15.3384	−1.19773
$165$	0	0
$166$	8.60555	0.667920
$167$	19.4870	1.50795	0.753976	−	0.656902i	$-0.228134\pi$
$167$	19.4870	1.50795	0.753976	+	0.656902i	$0.228134\pi$
$168$	−9.39445	−0.724797
$169$	16.2111	1.24701
$170$	−2.60555	−0.199837
$171$	1.25610	0.0960566
$172$	10.6943	0.815433
$173$	12.4459	0.946243	0.473121	−	0.880997i	$-0.343127\pi$
$173$	12.4459	0.946243	0.473121	+	0.880997i	$0.343127\pi$
$174$	−4.42221	−0.335247
$175$	−4.14863	−0.313607
$176$	0	0
$177$	5.21110	0.391690
$178$	3.76831	0.282447
$179$	−22.4222	−1.67591	−0.837957	−	0.545736i	$-0.816250\pi$
$179$	−22.4222	−1.67591	−0.837957	+	0.545736i	$0.816250\pi$
$180$	−1.60555	−0.119671
$181$	19.2111	1.42795	0.713975	−	0.700171i	$-0.246893\pi$
$181$	19.2111	1.42795	0.713975	+	0.700171i	$0.246893\pi$
$182$	14.0823	1.04385
$183$	8.29725	0.613351
$184$	−11.8004	−0.869937
$185$	−7.21110	−0.530171
$186$	−2.51221	−0.184204
$187$	0	0
$188$	0	0
$189$	4.14863	0.301768
$190$	0.788897	0.0572326
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0.0277564	0.00200314
$193$	−11.1898	−0.805458	−0.402729	−	0.915319i	$-0.631938\pi$
$193$	−11.1898	−0.805458	−0.402729	+	0.915319i	$0.631938\pi$
$194$	12.0656	0.866257
$195$	5.40473	0.387041
$196$	−16.3944	−1.17103
$197$	−9.17304	−0.653552	−0.326776	−	0.945102i	$-0.605962\pi$
$197$	−9.17304	−0.653552	−0.326776	+	0.945102i	$0.605962\pi$
$198$	0	0
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	−2.26447	−0.160122
$201$	13.2111	0.931839
$202$	0.788897	0.0555066
$203$	−29.2111	−2.05022
$204$	−6.66083	−0.466352
$205$	9.55336	0.667235
$206$	0.760635	0.0529959
$207$	5.21110	0.362197
$208$	−9.66851	−0.670390
$209$	0	0
$210$	2.60555	0.179800
$211$	−3.76831	−0.259421	−0.129711	−	0.991552i	$-0.541405\pi$
$211$	−3.76831	−0.259421	−0.129711	+	0.991552i	$0.541405\pi$
$212$	−18.0000	−1.23625
$213$	−5.21110	−0.357059
$214$	−1.81665	−0.124184
$215$	−6.66083	−0.454265
$216$	2.26447	0.154078
$217$	−16.5945	−1.12651
$218$	10.4222	0.705881
$219$	−2.89252	−0.195459
$220$	0	0
$221$	22.4222	1.50828
$222$	4.52894	0.303963
$223$	−2.42221	−0.162203	−0.0811014	−	0.996706i	$-0.525844\pi$
$223$	−2.42221	−0.162203	−0.0811014	+	0.996706i	$0.525844\pi$
$224$	−23.4500	−1.56682
$225$	1.00000	0.0666667
$226$	7.04115	0.468370
$227$	2.89252	0.191984	0.0959918	−	0.995382i	$-0.469398\pi$
$227$	2.89252	0.191984	0.0959918	+	0.995382i	$0.469398\pi$
$228$	2.01674	0.133562
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	3.27284	0.215805
$231$	0	0
$232$	−15.9445	−1.04681
$233$	25.7675	1.68809	0.844044	−	0.536274i	$-0.180169\pi$
$233$	25.7675	1.68809	0.844044	+	0.536274i	$0.180169\pi$
$234$	−3.39445	−0.221902
$235$	0	0
$236$	8.36669	0.544625
$237$	−1.25610	−0.0815927
$238$	10.8095	0.700673
$239$	8.29725	0.536705	0.268352	−	0.963321i	$-0.413521\pi$
$239$	8.29725	0.536705	0.268352	+	0.963321i	$0.413521\pi$
$240$	−1.78890	−0.115473
$241$	−3.27284	−0.210822	−0.105411	−	0.994429i	$-0.533616\pi$
$241$	−3.27284	−0.210822	−0.105411	+	0.994429i	$0.533616\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	13.3217	0.852832
$245$	10.2111	0.652363
$246$	−6.00000	−0.382546
$247$	−6.78890	−0.431967
$248$	−9.05789	−0.575176
$249$	−13.7020	−0.868328
$250$	0.628052	0.0397215
$251$	−27.6333	−1.74420	−0.872099	−	0.489329i	$-0.837242\pi$
$251$	−27.6333	−1.74420	−0.872099	+	0.489329i	$0.837242\pi$
$252$	6.66083	0.419593
$253$	0	0
$254$	1.02776	0.0644872
$255$	4.14863	0.259797
$256$	−7.05551	−0.440970
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	4.18335	0.260444
$259$	29.9162	1.85890
$260$	8.67757	0.538160
$261$	7.04115	0.435836
$262$	3.63331	0.224466
$263$	−7.91694	−0.488179	−0.244090	−	0.969753i	$-0.578489\pi$
$263$	−7.91694	−0.488179	−0.244090	+	0.969753i	$0.578489\pi$
$264$	0	0
$265$	11.2111	0.688693
$266$	−3.27284	−0.200671
$267$	−6.00000	−0.367194
$268$	21.2111	1.29567
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	−0.628052	−0.0382220
$271$	−22.8750	−1.38956	−0.694779	−	0.719223i	$-0.744498\pi$
$271$	−22.8750	−1.38956	−0.694779	+	0.719223i	$0.744498\pi$
$272$	−7.42147	−0.449993
$273$	−22.4222	−1.35705
$274$	−10.3140	−0.623091
$275$	0	0
$276$	8.36669	0.503616
$277$	−10.4291	−0.626626	−0.313313	−	0.949650i	$-0.601439\pi$
$277$	−10.4291	−0.626626	−0.313313	+	0.949650i	$0.601439\pi$
$278$	−6.00000	−0.359856
$279$	4.00000	0.239474
$280$	9.39445	0.561426
$281$	26.1479	1.55985	0.779925	−	0.625873i	$-0.215257\pi$
$281$	26.1479	1.55985	0.779925	+	0.625873i	$0.215257\pi$
$282$	0	0
$283$	31.5526	1.87561	0.937803	−	0.347167i	$-0.112856\pi$
$283$	31.5526	1.87561	0.937803	+	0.347167i	$0.112856\pi$
$284$	−8.36669	−0.496472
$285$	−1.25610	−0.0744051
$286$	0	0
$287$	−39.6333	−2.33948
$288$	5.65246	0.333075
$289$	0.211103	0.0124178
$290$	4.42221	0.259681
$291$	−19.2111	−1.12617
$292$	−4.64409	−0.271775
$293$	20.7431	1.21183	0.605913	−	0.795531i	$-0.292808\pi$
$293$	20.7431	1.21183	0.605913	+	0.795531i	$0.292808\pi$
$294$	−6.41310	−0.374020
$295$	−5.21110	−0.303402
$296$	16.3293	0.949124
$297$	0	0
$298$	−14.3667	−0.832240
$299$	−28.1646	−1.62880
$300$	1.60555	0.0926965
$301$	27.6333	1.59276
$302$	−9.63331	−0.554335
$303$	−1.25610	−0.0721612
$304$	2.24704	0.128877
$305$	−8.29725	−0.475099
$306$	−2.60555	−0.148949
$307$	7.42147	0.423566	0.211783	−	0.977317i	$-0.432073\pi$
$307$	7.42147	0.423566	0.211783	+	0.977317i	$0.432073\pi$
$308$	0	0
$309$	−1.21110	−0.0688972
$310$	2.51221	0.142684
$311$	6.78890	0.384963	0.192482	−	0.981301i	$-0.438346\pi$
$311$	6.78890	0.384963	0.192482	+	0.981301i	$0.438346\pi$
$312$	−12.2389	−0.692889
$313$	15.2111	0.859782	0.429891	−	0.902881i	$-0.358552\pi$
$313$	15.2111	0.859782	0.429891	+	0.902881i	$0.358552\pi$
$314$	6.28052	0.354430
$315$	−4.14863	−0.233749
$316$	−2.01674	−0.113450
$317$	−11.2111	−0.629678	−0.314839	−	0.949145i	$-0.601951\pi$
$317$	−11.2111	−0.629678	−0.314839	+	0.949145i	$0.601951\pi$
$318$	−7.04115	−0.394848
$319$	0	0
$320$	−0.0277564	−0.00155163
$321$	2.89252	0.161445
$322$	−13.5778	−0.756661
$323$	−5.21110	−0.289954
$324$	−1.60555	−0.0891973
$325$	−5.40473	−0.299800
$326$	8.29725	0.459542
$327$	−16.5945	−0.917678
$328$	−21.6333	−1.19450
$329$	0	0
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	−21.9992	−1.20736
$333$	−7.21110	−0.395166
$334$	12.2389	0.669681
$335$	−13.2111	−0.721800
$336$	7.42147	0.404874
$337$	−8.67757	−0.472697	−0.236349	−	0.971668i	$-0.575951\pi$
$337$	−8.67757	−0.472697	−0.236349	+	0.971668i	$0.575951\pi$
$338$	10.1814	0.553796
$339$	−11.2111	−0.608904
$340$	6.66083	0.361234
$341$	0	0
$342$	0.788897	0.0426587
$343$	−13.3217	−0.719302
$344$	15.0833	0.813235
$345$	−5.21110	−0.280556
$346$	7.81665	0.420226
$347$	−29.5359	−1.58557	−0.792784	−	0.609503i	$-0.791369\pi$
$347$	−29.5359	−1.58557	−0.792784	+	0.609503i	$0.791369\pi$
$348$	11.3049	0.606008
$349$	13.3217	0.713092	0.356546	−	0.934278i	$-0.383954\pi$
$349$	13.3217	0.713092	0.356546	+	0.934278i	$0.383954\pi$
$350$	−2.60555	−0.139273
$351$	5.40473	0.288483
$352$	0	0
$353$	12.7889	0.680684	0.340342	−	0.940302i	$-0.389457\pi$
$353$	12.7889	0.680684	0.340342	+	0.940302i	$0.389457\pi$
$354$	3.27284	0.173950
$355$	5.21110	0.276577
$356$	−9.63331	−0.510564
$357$	−17.2111	−0.910908
$358$	−14.0823	−0.744273
$359$	−15.8339	−0.835680	−0.417840	−	0.908521i	$-0.637213\pi$
$359$	−15.8339	−0.835680	−0.417840	+	0.908521i	$0.637213\pi$
$360$	−2.26447	−0.119348
$361$	−17.4222	−0.916958
$362$	12.0656	0.634152
$363$	0	0
$364$	−36.0000	−1.88691
$365$	2.89252	0.151402
$366$	5.21110	0.272389
$367$	18.4222	0.961631	0.480816	−	0.876822i	$-0.340341\pi$
$367$	18.4222	0.961631	0.480816	+	0.876822i	$0.340341\pi$
$368$	9.32213	0.485950
$369$	9.55336	0.497328
$370$	−4.52894	−0.235449
$371$	−46.5107	−2.41471
$372$	6.42221	0.332976
$373$	7.91694	0.409923	0.204962	−	0.978770i	$-0.434293\pi$
$373$	7.91694	0.409923	0.204962	+	0.978770i	$0.434293\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−38.0555	−1.95996
$378$	2.60555	0.134015
$379$	−14.4222	−0.740819	−0.370409	−	0.928869i	$-0.620783\pi$
$379$	−14.4222	−0.740819	−0.370409	+	0.928869i	$0.620783\pi$
$380$	−2.01674	−0.103456
$381$	−1.63642	−0.0838364
$382$	0	0
$383$	−5.21110	−0.266275	−0.133137	−	0.991098i	$-0.542505\pi$
$383$	−5.21110	−0.266275	−0.133137	+	0.991098i	$0.542505\pi$
$384$	11.3224	0.577792
$385$	0	0
$386$	−7.02776	−0.357703
$387$	−6.66083	−0.338589
$388$	−30.8444	−1.56589
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	3.39445	0.171885
$391$	−21.6189	−1.09332
$392$	−23.1228	−1.16788
$393$	−5.78505	−0.291817
$394$	−5.76114	−0.290242
$395$	1.25610	0.0632014
$396$	0	0
$397$	−13.6333	−0.684236	−0.342118	−	0.939657i	$-0.611144\pi$
$397$	−13.6333	−0.684236	−0.342118	+	0.939657i	$0.611144\pi$
$398$	−5.02441	−0.251851
$399$	5.21110	0.260881
$400$	1.78890	0.0894449
$401$	28.4222	1.41934	0.709669	−	0.704536i	$-0.248845\pi$
$401$	28.4222	1.41934	0.709669	+	0.704536i	$0.248845\pi$
$402$	8.29725	0.413829
$403$	−21.6189	−1.07692
$404$	−2.01674	−0.100336
$405$	1.00000	0.0496904
$406$	−18.3461	−0.910501
$407$	0	0
$408$	−9.39445	−0.465095
$409$	7.53662	0.372662	0.186331	−	0.982487i	$-0.440340\pi$
$409$	7.53662	0.372662	0.186331	+	0.982487i	$0.440340\pi$
$410$	6.00000	0.296319
$411$	16.4222	0.810048
$412$	−1.94449	−0.0957980
$413$	21.6189	1.06380
$414$	3.27284	0.160851
$415$	13.7020	0.672604
$416$	−30.5500	−1.49784
$417$	9.55336	0.467830
$418$	0	0
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	−6.66083	−0.325015
$421$	39.2111	1.91103	0.955516	−	0.294939i	$-0.0952993\pi$
$421$	39.2111	1.91103	0.955516	+	0.294939i	$0.0952993\pi$
$422$	−2.36669	−0.115209
$423$	0	0
$424$	−25.3872	−1.23291
$425$	−4.14863	−0.201238
$426$	−3.27284	−0.158570
$427$	34.4222	1.66581
$428$	4.64409	0.224481
$429$	0	0
$430$	−4.18335	−0.201739
$431$	2.51221	0.121009	0.0605044	−	0.998168i	$-0.480729\pi$
$431$	2.51221	0.121009	0.0605044	+	0.998168i	$0.480729\pi$
$432$	−1.78890	−0.0860684
$433$	−4.78890	−0.230140	−0.115070	−	0.993357i	$-0.536709\pi$
$433$	−4.78890	−0.230140	−0.115070	+	0.993357i	$0.536709\pi$
$434$	−10.4222	−0.500282
$435$	−7.04115	−0.337597
$436$	−26.6433	−1.27598
$437$	6.54568	0.313122
$438$	−1.81665	−0.0868031
$439$	−2.01674	−0.0962536	−0.0481268	−	0.998841i	$-0.515325\pi$
$439$	−2.01674	−0.0962536	−0.0481268	+	0.998841i	$0.515325\pi$
$440$	0	0
$441$	10.2111	0.486243
$442$	14.0823	0.669827
$443$	27.6333	1.31290	0.656449	−	0.754370i	$-0.272058\pi$
$443$	27.6333	1.31290	0.656449	+	0.754370i	$0.272058\pi$
$444$	−11.5778	−0.549458
$445$	6.00000	0.284427
$446$	−1.52127	−0.0720342
$447$	22.8750	1.08195
$448$	0.115151	0.00544037
$449$	16.4222	0.775012	0.387506	−	0.921867i	$-0.373337\pi$
$449$	16.4222	0.775012	0.387506	+	0.921867i	$0.373337\pi$
$450$	0.628052	0.0296066
$451$	0	0
$452$	−18.0000	−0.846649
$453$	15.3384	0.720661
$454$	1.81665	0.0852598
$455$	22.4222	1.05117
$456$	2.84441	0.133202
$457$	2.13189	0.0997255	0.0498628	−	0.998756i	$-0.484122\pi$
$457$	2.13189	0.0997255	0.0498628	+	0.998756i	$0.484122\pi$
$458$	6.28052	0.293469
$459$	4.14863	0.193641
$460$	−8.36669	−0.390099
$461$	−6.28052	−0.292513	−0.146256	−	0.989247i	$-0.546722\pi$
$461$	−6.28052	−0.292513	−0.146256	+	0.989247i	$0.546722\pi$
$462$	0	0
$463$	−26.4222	−1.22794	−0.613972	−	0.789328i	$-0.710429\pi$
$463$	−26.4222	−1.22794	−0.613972	+	0.789328i	$0.710429\pi$
$464$	12.5959	0.584750
$465$	−4.00000	−0.185496
$466$	16.1833	0.749679
$467$	−17.2111	−0.796435	−0.398217	−	0.917291i	$-0.630371\pi$
$467$	−17.2111	−0.796435	−0.398217	+	0.917291i	$0.630371\pi$
$468$	8.67757	0.401121
$469$	54.8079	2.53080
$470$	0	0
$471$	−10.0000	−0.460776
$472$	11.8004	0.543157
$473$	0	0
$474$	−0.788897	−0.0362353
$475$	1.25610	0.0576340
$476$	−27.6333	−1.26657
$477$	11.2111	0.513321
$478$	5.21110	0.238350
$479$	−3.27284	−0.149540	−0.0747700	−	0.997201i	$-0.523822\pi$
$479$	−3.27284	−0.149540	−0.0747700	+	0.997201i	$0.523822\pi$
$480$	−5.65246	−0.257998
$481$	38.9741	1.77706
$482$	−2.05551	−0.0936260
$483$	21.6189	0.983695
$484$	0	0
$485$	19.2111	0.872331
$486$	−0.628052	−0.0284890
$487$	−1.21110	−0.0548803	−0.0274401	−	0.999623i	$-0.508736\pi$
$487$	−1.21110	−0.0548803	−0.0274401	+	0.999623i	$0.508736\pi$
$488$	18.7889	0.850533
$489$	−13.2111	−0.597427
$490$	6.41310	0.289714
$491$	40.7256	1.83792	0.918961	−	0.394348i	$-0.129030\pi$
$491$	40.7256	1.83792	0.918961	+	0.394348i	$0.129030\pi$
$492$	15.3384	0.691509
$493$	−29.2111	−1.31560
$494$	−4.26378	−0.191836
$495$	0	0
$496$	7.15559	0.321295
$497$	−21.6189	−0.969741
$498$	−8.60555	−0.385624
$499$	14.4222	0.645627	0.322813	−	0.946463i	$-0.395371\pi$
$499$	14.4222	0.645627	0.322813	+	0.946463i	$0.395371\pi$
$500$	−1.60555	−0.0718024
$501$	−19.4870	−0.870616
$502$	−17.3551	−0.774598
$503$	−18.7264	−0.834969	−0.417484	−	0.908684i	$-0.637088\pi$
$503$	−18.7264	−0.834969	−0.417484	+	0.908684i	$0.637088\pi$
$504$	9.39445	0.418462
$505$	1.25610	0.0558959
$506$	0	0
$507$	−16.2111	−0.719960
$508$	−2.62736	−0.116570
$509$	16.4222	0.727901	0.363951	−	0.931418i	$-0.381428\pi$
$509$	16.4222	0.727901	0.363951	+	0.931418i	$0.381428\pi$
$510$	2.60555	0.115376
$511$	−12.0000	−0.530849
$512$	18.2135	0.804930
$513$	−1.25610	−0.0554583
$514$	3.76831	0.166213
$515$	1.21110	0.0533676
$516$	−10.6943	−0.470791
$517$	0	0
$518$	18.7889	0.825537
$519$	−12.4459	−0.546313
$520$	12.2389	0.536709
$521$	7.57779	0.331989	0.165995	−	0.986127i	$-0.446917\pi$
$521$	7.57779	0.331989	0.165995	+	0.986127i	$0.446917\pi$
$522$	4.42221	0.193555
$523$	25.7675	1.12674	0.563368	−	0.826206i	$-0.309505\pi$
$523$	25.7675	1.12674	0.563368	+	0.826206i	$0.309505\pi$
$524$	−9.28819	−0.405756
$525$	4.14863	0.181061
$526$	−4.97224	−0.216800
$527$	−16.5945	−0.722868
$528$	0	0
$529$	4.15559	0.180678
$530$	7.04115	0.305848
$531$	−5.21110	−0.226143
$532$	8.36669	0.362742
$533$	−51.6333	−2.23649
$534$	−3.76831	−0.163071
$535$	−2.89252	−0.125055
$536$	29.9162	1.29218
$537$	22.4222	0.967590
$538$	−11.3049	−0.487390
$539$	0	0
$540$	1.60555	0.0690919
$541$	−33.1890	−1.42691	−0.713454	−	0.700703i	$-0.752870\pi$
$541$	−33.1890	−1.42691	−0.713454	+	0.700703i	$0.752870\pi$
$542$	−14.3667	−0.617102
$543$	−19.2111	−0.824427
$544$	−23.4500	−1.00541
$545$	16.5945	0.710831
$546$	−14.0823	−0.602667
$547$	25.7675	1.10174	0.550870	−	0.834591i	$-0.314296\pi$
$547$	25.7675	1.10174	0.550870	+	0.834591i	$0.314296\pi$
$548$	26.3667	1.12633
$549$	−8.29725	−0.354118
$550$	0	0
$551$	8.84441	0.376785
$552$	11.8004	0.502258
$553$	−5.21110	−0.221599
$554$	−6.55004	−0.278284
$555$	7.21110	0.306094
$556$	15.3384	0.650493
$557$	3.38799	0.143554	0.0717769	−	0.997421i	$-0.477133\pi$
$557$	3.38799	0.143554	0.0717769	+	0.997421i	$0.477133\pi$
$558$	2.51221	0.106350
$559$	36.0000	1.52264
$560$	−7.42147	−0.313614
$561$	0	0
$562$	16.4222	0.692729
$563$	−22.7599	−0.959214	−0.479607	−	0.877483i	$-0.659221\pi$
$563$	−22.7599	−0.959214	−0.479607	+	0.877483i	$0.659221\pi$
$564$	0	0
$565$	11.2111	0.471655
$566$	19.8167	0.832956
$567$	−4.14863	−0.174226
$568$	−11.8004	−0.495134
$569$	1.25610	0.0526586	0.0263293	−	0.999653i	$-0.491618\pi$
$569$	1.25610	0.0526586	0.0263293	+	0.999653i	$0.491618\pi$
$570$	−0.788897	−0.0330433
$571$	31.1723	1.30452	0.652260	−	0.757996i	$-0.273821\pi$
$571$	31.1723	1.30452	0.652260	+	0.757996i	$0.273821\pi$
$572$	0	0
$573$	0	0
$574$	−24.8918	−1.03896
$575$	5.21110	0.217318
$576$	−0.0277564	−0.00115652
$577$	11.5778	0.481990	0.240995	−	0.970526i	$-0.422526\pi$
$577$	11.5778	0.481990	0.240995	+	0.970526i	$0.422526\pi$
$578$	0.132583	0.00551474
$579$	11.1898	0.465031
$580$	−11.3049	−0.469412
$581$	−56.8444	−2.35830
$582$	−12.0656	−0.500134
$583$	0	0
$584$	−6.55004	−0.271043
$585$	−5.40473	−0.223458
$586$	13.0278	0.538172
$587$	17.2111	0.710378	0.355189	−	0.934794i	$-0.384416\pi$
$587$	17.2111	0.710378	0.355189	+	0.934794i	$0.384416\pi$
$588$	16.3944	0.676096
$589$	5.02441	0.207027
$590$	−3.27284	−0.134741
$591$	9.17304	0.377328
$592$	−12.8999	−0.530184
$593$	9.93367	0.407927	0.203964	−	0.978978i	$-0.434618\pi$
$593$	9.93367	0.407927	0.203964	+	0.978978i	$0.434618\pi$
$594$	0	0
$595$	17.2111	0.705586
$596$	36.7270	1.50440
$597$	8.00000	0.327418
$598$	−17.6888	−0.723350
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	2.26447	0.0924467
$601$	43.9985	1.79474	0.897368	−	0.441284i	$-0.145477\pi$
$601$	43.9985	1.79474	0.897368	+	0.441284i	$0.145477\pi$
$602$	17.3551	0.707343
$603$	−13.2111	−0.537998
$604$	24.6266	1.00204
$605$	0	0
$606$	−0.788897	−0.0320468
$607$	18.2309	0.739970	0.369985	−	0.929038i	$-0.379363\pi$
$607$	18.2309	0.739970	0.369985	+	0.929038i	$0.379363\pi$
$608$	7.10008	0.287946
$609$	29.2111	1.18369
$610$	−5.21110	−0.210991
$611$	0	0
$612$	6.66083	0.269248
$613$	−41.8666	−1.69098	−0.845488	−	0.533995i	$-0.820690\pi$
$613$	−41.8666	−1.69098	−0.845488	+	0.533995i	$0.820690\pi$
$614$	4.66106	0.188105
$615$	−9.55336	−0.385229
$616$	0	0
$617$	24.7889	0.997963	0.498982	−	0.866613i	$-0.333707\pi$
$617$	24.7889	0.997963	0.498982	+	0.866613i	$0.333707\pi$
$618$	−0.760635	−0.0305972
$619$	8.00000	0.321547	0.160774	−	0.986991i	$-0.448601\pi$
$619$	8.00000	0.321547	0.160774	+	0.986991i	$0.448601\pi$
$620$	−6.42221	−0.257922
$621$	−5.21110	−0.209114
$622$	4.26378	0.170962
$623$	−24.8918	−0.997267
$624$	9.66851	0.387050
$625$	1.00000	0.0400000
$626$	9.55336	0.381829
$627$	0	0
$628$	−16.0555	−0.640685
$629$	29.9162	1.19284
$630$	−2.60555	−0.103808
$631$	−28.8444	−1.14828	−0.574139	−	0.818758i	$-0.694663\pi$
$631$	−28.8444	−1.14828	−0.574139	+	0.818758i	$0.694663\pi$
$632$	−2.84441	−0.113145
$633$	3.76831	0.149777
$634$	−7.04115	−0.279640
$635$	1.63642	0.0649394
$636$	18.0000	0.713746
$637$	−55.1882	−2.18664
$638$	0	0
$639$	5.21110	0.206148
$640$	−11.3224	−0.447556
$641$	6.00000	0.236986	0.118493	−	0.992955i	$-0.462194\pi$
$641$	6.00000	0.236986	0.118493	+	0.992955i	$0.462194\pi$
$642$	1.81665	0.0716976
$643$	−0.366692	−0.0144609	−0.00723047	−	0.999974i	$-0.502302\pi$
$643$	−0.366692	−0.0144609	−0.00723047	+	0.999974i	$0.502302\pi$
$644$	34.7103	1.36778
$645$	6.66083	0.262270
$646$	−3.27284	−0.128768
$647$	5.21110	0.204870	0.102435	−	0.994740i	$-0.467337\pi$
$647$	5.21110	0.204870	0.102435	+	0.994740i	$0.467337\pi$
$648$	−2.26447	−0.0889569
$649$	0	0
$650$	−3.39445	−0.133141
$651$	16.5945	0.650390
$652$	−21.2111	−0.830691
$653$	4.42221	0.173054	0.0865271	−	0.996249i	$-0.472423\pi$
$653$	4.42221	0.173054	0.0865271	+	0.996249i	$0.472423\pi$
$654$	−10.4222	−0.407540
$655$	5.78505	0.226040
$656$	17.0900	0.667251
$657$	2.89252	0.112848
$658$	0	0
$659$	33.1890	1.29286	0.646430	−	0.762973i	$-0.276261\pi$
$659$	33.1890	1.29286	0.646430	+	0.762973i	$0.276261\pi$
$660$	0	0
$661$	30.8444	1.19971	0.599854	−	0.800109i	$-0.295225\pi$
$661$	30.8444	1.19971	0.599854	+	0.800109i	$0.295225\pi$
$662$	5.02441	0.195279
$663$	−22.4222	−0.870806
$664$	−31.0278	−1.20411
$665$	−5.21110	−0.202078
$666$	−4.52894	−0.175493
$667$	36.6922	1.42073
$668$	−31.2874	−1.21055
$669$	2.42221	0.0936479
$670$	−8.29725	−0.320551
$671$	0	0
$672$	23.4500	0.904602
$673$	15.4536	0.595691	0.297845	−	0.954614i	$-0.403732\pi$
$673$	15.4536	0.595691	0.297845	+	0.954614i	$0.403732\pi$
$674$	−5.44996	−0.209925
$675$	−1.00000	−0.0384900
$676$	−26.0278	−1.00107
$677$	−3.38799	−0.130211	−0.0651056	−	0.997878i	$-0.520738\pi$
$677$	−3.38799	−0.130211	−0.0651056	+	0.997878i	$0.520738\pi$
$678$	−7.04115	−0.270414
$679$	−79.6997	−3.05859
$680$	9.39445	0.360261
$681$	−2.89252	−0.110842
$682$	0	0
$683$	−36.0000	−1.37750	−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000	−1.37750	−0.688751	+	0.724998i	$0.741841\pi$
$684$	−2.01674	−0.0771119
$685$	−16.4222	−0.627460
$686$	−8.36669	−0.319442
$687$	−10.0000	−0.381524
$688$	−11.9155	−0.454276
$689$	−60.5930	−2.30841
$690$	−3.27284	−0.124595
$691$	9.57779	0.364356	0.182178	−	0.983266i	$-0.441685\pi$
$691$	9.57779	0.364356	0.182178	+	0.983266i	$0.441685\pi$
$692$	−19.9825	−0.759621
$693$	0	0
$694$	−18.5500	−0.704150
$695$	−9.55336	−0.362379
$696$	15.9445	0.604374
$697$	−39.6333	−1.50122
$698$	8.36669	0.316684
$699$	−25.7675	−0.974618
$700$	6.66083	0.251756
$701$	−7.04115	−0.265941	−0.132970	−	0.991120i	$-0.542451\pi$
$701$	−7.04115	−0.265941	−0.132970	+	0.991120i	$0.542451\pi$
$702$	3.39445	0.128115
$703$	−9.05789	−0.341625
$704$	0	0
$705$	0	0
$706$	8.03209	0.302292
$707$	−5.21110	−0.195984
$708$	−8.36669	−0.314440
$709$	−5.63331	−0.211563	−0.105782	−	0.994389i	$-0.533734\pi$
$709$	−5.63331	−0.211563	−0.105782	+	0.994389i	$0.533734\pi$
$710$	3.27284	0.122828
$711$	1.25610	0.0471075
$712$	−13.5868	−0.509188
$713$	20.8444	0.780629
$714$	−10.8095	−0.404534
$715$	0	0
$716$	36.0000	1.34538
$717$	−8.29725	−0.309867
$718$	−9.94449	−0.371125
$719$	−10.4222	−0.388683	−0.194341	−	0.980934i	$-0.562257\pi$
$719$	−10.4222	−0.388683	−0.194341	+	0.980934i	$0.562257\pi$
$720$	1.78890	0.0666683
$721$	−5.02441	−0.187119
$722$	−10.9420	−0.407221
$723$	3.27284	0.121718
$724$	−30.8444	−1.14632
$725$	7.04115	0.261502
$726$	0	0
$727$	−21.5778	−0.800276	−0.400138	−	0.916455i	$-0.631038\pi$
$727$	−21.5778	−0.800276	−0.400138	+	0.916455i	$0.631038\pi$
$728$	−50.7745	−1.88183
$729$	1.00000	0.0370370
$730$	1.81665	0.0672374
$731$	27.6333	1.02205
$732$	−13.3217	−0.492383
$733$	−16.2142	−0.598885	−0.299442	−	0.954114i	$-0.596801\pi$
$733$	−16.2142	−0.598885	−0.299442	+	0.954114i	$0.596801\pi$
$734$	11.5701	0.427060
$735$	−10.2111	−0.376642
$736$	29.4556	1.08575
$737$	0	0
$738$	6.00000	0.220863
$739$	42.7424	1.57230	0.786152	−	0.618034i	$-0.212070\pi$
$739$	42.7424	1.57230	0.786152	+	0.618034i	$0.212070\pi$
$740$	11.5778	0.425608
$741$	6.78890	0.249396
$742$	−29.2111	−1.07237
$743$	−33.5693	−1.23154	−0.615770	−	0.787926i	$-0.711155\pi$
$743$	−33.5693	−1.23154	−0.615770	+	0.787926i	$0.711155\pi$
$744$	9.05789	0.332078
$745$	−22.8750	−0.838076
$746$	4.97224	0.182047
$747$	13.7020	0.501329
$748$	0	0
$749$	12.0000	0.438470
$750$	−0.628052	−0.0229332
$751$	36.8444	1.34447	0.672236	−	0.740337i	$-0.265334\pi$
$751$	36.8444	1.34447	0.672236	+	0.740337i	$0.265334\pi$
$752$	0	0
$753$	27.6333	1.00701
$754$	−23.9008	−0.870417
$755$	−15.3384	−0.558222
$756$	−6.66083	−0.242252
$757$	20.4222	0.742258	0.371129	−	0.928581i	$-0.378971\pi$
$757$	20.4222	0.742258	0.371129	+	0.928581i	$0.378971\pi$
$758$	−9.05789	−0.328997
$759$	0	0
$760$	−2.84441	−0.103178
$761$	−26.9085	−0.975432	−0.487716	−	0.873002i	$-0.662170\pi$
$761$	−26.9085	−0.975432	−0.487716	+	0.873002i	$0.662170\pi$
$762$	−1.02776	−0.0372317
$763$	−68.8444	−2.49233
$764$	0	0
$765$	−4.14863	−0.149994
$766$	−3.27284	−0.118253
$767$	28.1646	1.01696
$768$	7.05551	0.254594
$769$	10.8095	0.389799	0.194900	−	0.980823i	$-0.437562\pi$
$769$	10.8095	0.389799	0.194900	+	0.980823i	$0.437562\pi$
$770$	0	0
$771$	−6.00000	−0.216085
$772$	17.9658	0.646602
$773$	42.0000	1.51064	0.755318	−	0.655359i	$-0.227483\pi$
$773$	42.0000	1.51064	0.755318	+	0.655359i	$0.227483\pi$
$774$	−4.18335	−0.150367
$775$	4.00000	0.143684
$776$	−43.5030	−1.56167
$777$	−29.9162	−1.07324
$778$	11.3049	0.405301
$779$	12.0000	0.429945
$780$	−8.67757	−0.310707
$781$	0	0
$782$	−13.5778	−0.485541
$783$	−7.04115	−0.251630
$784$	18.2666	0.652379
$785$	10.0000	0.356915
$786$	−3.63331	−0.129596
$787$	−6.66083	−0.237433	−0.118717	−	0.992928i	$-0.537878\pi$
$787$	−6.66083	−0.237433	−0.118717	+	0.992928i	$0.537878\pi$
$788$	14.7278	0.524656
$789$	7.91694	0.281850
$790$	0.788897	0.0280677
$791$	−46.5107	−1.65373
$792$	0	0
$793$	44.8444	1.59247
$794$	−8.56242	−0.303869
$795$	−11.2111	−0.397617
$796$	12.8444	0.455258
$797$	54.4777	1.92970	0.964850	−	0.262802i	$-0.0846465\pi$
$797$	54.4777	1.92970	0.964850	+	0.262802i	$0.0846465\pi$
$798$	3.27284	0.115857
$799$	0	0
$800$	5.65246	0.199845
$801$	6.00000	0.212000
$802$	17.8506	0.630327
$803$	0	0
$804$	−21.2111	−0.748058
$805$	−21.6189	−0.761967
$806$	−13.5778	−0.478257
$807$	18.0000	0.633630
$808$	−2.84441	−0.100066
$809$	12.0656	0.424203	0.212101	−	0.977248i	$-0.431969\pi$
$809$	12.0656	0.424203	0.212101	+	0.977248i	$0.431969\pi$
$810$	0.628052	0.0220675
$811$	51.8003	1.81895	0.909477	−	0.415755i	$-0.136483\pi$
$811$	51.8003	1.81895	0.909477	+	0.415755i	$0.136483\pi$
$812$	46.8999	1.64586
$813$	22.8750	0.802262
$814$	0	0
$815$	13.2111	0.462765
$816$	7.42147	0.259803
$817$	−8.36669	−0.292714
$818$	4.73338	0.165499
$819$	22.4222	0.783495
$820$	−15.3384	−0.535640
$821$	−37.7180	−1.31637	−0.658183	−	0.752858i	$-0.728675\pi$
$821$	−37.7180	−1.31637	−0.658183	+	0.752858i	$0.728675\pi$
$822$	10.3140	0.359742
$823$	4.36669	0.152213	0.0761067	−	0.997100i	$-0.475751\pi$
$823$	4.36669	0.152213	0.0761067	+	0.997100i	$0.475751\pi$
$824$	−2.74251	−0.0955398
$825$	0	0
$826$	13.5778	0.472432
$827$	13.7020	0.476465	0.238232	−	0.971208i	$-0.423432\pi$
$827$	13.7020	0.476465	0.238232	+	0.971208i	$0.423432\pi$
$828$	−8.36669	−0.290763
$829$	−36.0555	−1.25226	−0.626130	−	0.779719i	$-0.715362\pi$
$829$	−36.0555	−1.25226	−0.626130	+	0.779719i	$0.715362\pi$
$830$	8.60555	0.298703
$831$	10.4291	0.361783
$832$	0.150016	0.00520086
$833$	−42.3621	−1.46776
$834$	6.00000	0.207763
$835$	19.4870	0.674376
$836$	0	0
$837$	−4.00000	−0.138260
$838$	−7.53662	−0.260348
$839$	−30.7889	−1.06295	−0.531475	−	0.847074i	$-0.678362\pi$
$839$	−30.7889	−1.06295	−0.531475	+	0.847074i	$0.678362\pi$
$840$	−9.39445	−0.324139
$841$	20.5778	0.709579
$842$	24.6266	0.848688
$843$	−26.1479	−0.900580
$844$	6.05021	0.208257
$845$	16.2111	0.557679
$846$	0	0
$847$	0	0
$848$	20.0555	0.688709
$849$	−31.5526	−1.08288
$850$	−2.60555	−0.0893697
$851$	−37.5778	−1.28815
$852$	8.36669	0.286638
$853$	19.4870	0.667223	0.333612	−	0.942711i	$-0.391733\pi$
$853$	19.4870	0.667223	0.333612	+	0.942711i	$0.391733\pi$
$854$	21.6189	0.739784
$855$	1.25610	0.0429578
$856$	6.55004	0.223876
$857$	40.6105	1.38723	0.693614	−	0.720347i	$-0.256017\pi$
$857$	40.6105	1.38723	0.693614	+	0.720347i	$0.256017\pi$
$858$	0	0
$859$	32.0000	1.09183	0.545913	−	0.837842i	$-0.316183\pi$
$859$	32.0000	1.09183	0.545913	+	0.837842i	$0.316183\pi$
$860$	10.6943	0.364673
$861$	39.6333	1.35070
$862$	1.57779	0.0537399
$863$	−15.6333	−0.532164	−0.266082	−	0.963950i	$-0.585729\pi$
$863$	−15.6333	−0.532164	−0.266082	+	0.963950i	$0.585729\pi$
$864$	−5.65246	−0.192301
$865$	12.4459	0.423173
$866$	−3.00767	−0.102205
$867$	−0.211103	−0.00716942
$868$	26.6433	0.904334
$869$	0	0
$870$	−4.42221	−0.149927
$871$	71.4024	2.41938
$872$	−37.5778	−1.27254
$873$	19.2111	0.650197
$874$	4.11103	0.139058
$875$	−4.14863	−0.140249
$876$	4.64409	0.156909
$877$	−13.7020	−0.462683	−0.231342	−	0.972873i	$-0.574311\pi$
$877$	−13.7020	−0.462683	−0.231342	+	0.972873i	$0.574311\pi$
$878$	−1.26662	−0.0427462
$879$	−20.7431	−0.699649
$880$	0	0
$881$	−49.2666	−1.65983	−0.829917	−	0.557887i	$-0.811612\pi$
$881$	−49.2666	−1.65983	−0.829917	+	0.557887i	$0.811612\pi$
$882$	6.41310	0.215940
$883$	−6.42221	−0.216124	−0.108062	−	0.994144i	$-0.534465\pi$
$883$	−6.42221	−0.216124	−0.108062	+	0.994144i	$0.534465\pi$
$884$	−36.0000	−1.21081
$885$	5.21110	0.175169
$886$	17.3551	0.583057
$887$	38.5937	1.29585	0.647926	−	0.761704i	$-0.275637\pi$
$887$	38.5937	1.29585	0.647926	+	0.761704i	$0.275637\pi$
$888$	−16.3293	−0.547977
$889$	−6.78890	−0.227692
$890$	3.76831	0.126314
$891$	0	0
$892$	3.88897	0.130212
$893$	0	0
$894$	14.3667	0.480494
$895$	−22.4222	−0.749492
$896$	46.9722	1.56923
$897$	28.1646	0.940389
$898$	10.3140	0.344182
$899$	28.1646	0.939342
$900$	−1.60555	−0.0535184
$901$	−46.5107	−1.54950
$902$	0	0
$903$	−27.6333	−0.919579
$904$	−25.3872	−0.844367
$905$	19.2111	0.638599
$906$	9.63331	0.320045
$907$	−14.4222	−0.478881	−0.239441	−	0.970911i	$-0.576964\pi$
$907$	−14.4222	−0.478881	−0.239441	+	0.970911i	$0.576964\pi$
$908$	−4.64409	−0.154120
$909$	1.25610	0.0416623
$910$	14.0823	0.466824
$911$	10.4222	0.345303	0.172652	−	0.984983i	$-0.444767\pi$
$911$	10.4222	0.345303	0.172652	+	0.984983i	$0.444767\pi$
$912$	−2.24704	−0.0744069
$913$	0	0
$914$	1.33894	0.0442881
$915$	8.29725	0.274299
$916$	−16.0555	−0.530489
$917$	−24.0000	−0.792550
$918$	2.60555	0.0859960
$919$	−18.6112	−0.613928	−0.306964	−	0.951721i	$-0.599313\pi$
$919$	−18.6112	−0.613928	−0.306964	+	0.951721i	$0.599313\pi$
$920$	−11.8004	−0.389048
$921$	−7.42147	−0.244546
$922$	−3.94449	−0.129905
$923$	−28.1646	−0.927049
$924$	0	0
$925$	−7.21110	−0.237100
$926$	−16.5945	−0.545329
$927$	1.21110	0.0397778
$928$	39.7998	1.30649
$929$	16.4222	0.538795	0.269398	−	0.963029i	$-0.413175\pi$
$929$	16.4222	0.538795	0.269398	+	0.963029i	$0.413175\pi$
$930$	−2.51221	−0.0823785
$931$	12.8262	0.420362
$932$	−41.3711	−1.35516
$933$	−6.78890	−0.222259
$934$	−10.8095	−0.353696
$935$	0	0
$936$	12.2389	0.400040
$937$	−10.4291	−0.340705	−0.170353	−	0.985383i	$-0.554491\pi$
$937$	−10.4291	−0.340705	−0.170353	+	0.985383i	$0.554491\pi$
$938$	34.4222	1.12392
$939$	−15.2111	−0.496396
$940$	0	0
$941$	6.28052	0.204739	0.102369	−	0.994746i	$-0.467358\pi$
$941$	6.28052	0.204739	0.102369	+	0.994746i	$0.467358\pi$
$942$	−6.28052	−0.204630
$943$	49.7835	1.62117
$944$	−9.32213	−0.303409
$945$	4.14863	0.134955
$946$	0	0
$947$	−1.57779	−0.0512714	−0.0256357	−	0.999671i	$-0.508161\pi$
$947$	−1.57779	−0.0512714	−0.0256357	+	0.999671i	$0.508161\pi$
$948$	2.01674	0.0655006
$949$	−15.6333	−0.507479
$950$	0.788897	0.0255952
$951$	11.2111	0.363545
$952$	−38.9741	−1.26316
$953$	34.8254	1.12811	0.564053	−	0.825738i	$-0.309241\pi$
$953$	34.8254	1.12811	0.564053	+	0.825738i	$0.309241\pi$
$954$	7.04115	0.227966
$955$	0	0
$956$	−13.3217	−0.430853
$957$	0	0
$958$	−2.05551	−0.0664106
$959$	68.1296	2.20002
$960$	0.0277564	0.000895833 0
$961$	−15.0000	−0.483871
$962$	24.4777	0.789193
$963$	−2.89252	−0.0932103
$964$	5.25471	0.169243
$965$	−11.1898	−0.360212
$966$	13.5778	0.436858
$967$	13.2065	0.424693	0.212346	−	0.977194i	$-0.431889\pi$
$967$	13.2065	0.424693	0.212346	+	0.977194i	$0.431889\pi$
$968$	0	0
$969$	5.21110	0.167405
$970$	12.0656	0.387402
$971$	−58.4222	−1.87486	−0.937429	−	0.348177i	$-0.886801\pi$
$971$	−58.4222	−1.87486	−0.937429	+	0.348177i	$0.886801\pi$
$972$	1.60555	0.0514981
$973$	39.6333	1.27059
$974$	−0.760635	−0.0243723
$975$	5.40473	0.173090
$976$	−14.8429	−0.475111
$977$	−30.0000	−0.959785	−0.479893	−	0.877327i	$-0.659324\pi$
$977$	−30.0000	−0.959785	−0.479893	+	0.877327i	$0.659324\pi$
$978$	−8.29725	−0.265317
$979$	0	0
$980$	−16.3944	−0.523701
$981$	16.5945	0.529822
$982$	25.5778	0.816220
$983$	26.0555	0.831042	0.415521	−	0.909584i	$-0.363599\pi$
$983$	26.0555	0.831042	0.415521	+	0.909584i	$0.363599\pi$
$984$	21.6333	0.689645
$985$	−9.17304	−0.292277
$986$	−18.3461	−0.584258
$987$	0	0
$988$	10.8999	0.346773
$989$	−34.7103	−1.10372
$990$	0	0
$991$	4.00000	0.127064	0.0635321	−	0.997980i	$-0.479763\pi$
$991$	4.00000	0.127064	0.0635321	+	0.997980i	$0.479763\pi$
$992$	22.6099	0.717864
$993$	−8.00000	−0.253872
$994$	−13.5778	−0.430662
$995$	−8.00000	−0.253617
$996$	21.9992	0.697072
$997$	−30.2965	−0.959499	−0.479750	−	0.877405i	$-0.659273\pi$
$997$	−30.2965	−0.959499	−0.479750	+	0.877405i	$0.659273\pi$
$998$	9.05789	0.286722
$999$	7.21110	0.228149

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1815.2.a.s.1.3	yes	4
3.2	odd	2		5445.2.a.bo.1.2		4
5.4	even	2		9075.2.a.dc.1.2		4
11.10	odd	2	inner	1815.2.a.s.1.2	✓	4
33.32	even	2		5445.2.a.bo.1.3		4
55.54	odd	2		9075.2.a.dc.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1815.2.a.s.1.2	✓	4	11.10	odd	2	inner
1815.2.a.s.1.3	yes	4	1.1	even	1	trivial
5445.2.a.bo.1.2		4	3.2	odd	2
5445.2.a.bo.1.3		4	33.32	even	2
9075.2.a.dc.1.2		4	5.4	even	2
9075.2.a.dc.1.3		4	55.54	odd	2

Overview	Random
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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:	\( N \)	\(=\)	\( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1815.a (trivial)

\(f(q)\)	\(=\)	\(q+0.628052 q^{2} -1.00000 q^{3} -1.60555 q^{4} +1.00000 q^{5} -0.628052 q^{6} -4.14863 q^{7} -2.26447 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.628052 q^{2} -1.00000 q^{3} -1.60555 q^{4} +1.00000 q^{5} -0.628052 q^{6} -4.14863 q^{7} -2.26447 q^{8} +1.00000 q^{9} +0.628052 q^{10} +1.60555 q^{12} -5.40473 q^{13} -2.60555 q^{14} -1.00000 q^{15} +1.78890 q^{16} -4.14863 q^{17} +0.628052 q^{18} +1.25610 q^{19} -1.60555 q^{20} +4.14863 q^{21} +5.21110 q^{23} +2.26447 q^{24} +1.00000 q^{25} -3.39445 q^{26} -1.00000 q^{27} +6.66083 q^{28} +7.04115 q^{29} -0.628052 q^{30} +4.00000 q^{31} +5.65246 q^{32} -2.60555 q^{34} -4.14863 q^{35} -1.60555 q^{36} -7.21110 q^{37} +0.788897 q^{38} +5.40473 q^{39} -2.26447 q^{40} +9.55336 q^{41} +2.60555 q^{42} -6.66083 q^{43} +1.00000 q^{45} +3.27284 q^{46} -1.78890 q^{48} +10.2111 q^{49} +0.628052 q^{50} +4.14863 q^{51} +8.67757 q^{52} +11.2111 q^{53} -0.628052 q^{54} +9.39445 q^{56} -1.25610 q^{57} +4.42221 q^{58} -5.21110 q^{59} +1.60555 q^{60} -8.29725 q^{61} +2.51221 q^{62} -4.14863 q^{63} -0.0277564 q^{64} -5.40473 q^{65} -13.2111 q^{67} +6.66083 q^{68} -5.21110 q^{69} -2.60555 q^{70} +5.21110 q^{71} -2.26447 q^{72} +2.89252 q^{73} -4.52894 q^{74} -1.00000 q^{75} -2.01674 q^{76} +3.39445 q^{78} +1.25610 q^{79} +1.78890 q^{80} +1.00000 q^{81} +6.00000 q^{82} +13.7020 q^{83} -6.66083 q^{84} -4.14863 q^{85} -4.18335 q^{86} -7.04115 q^{87} +6.00000 q^{89} +0.628052 q^{90} +22.4222 q^{91} -8.36669 q^{92} -4.00000 q^{93} +1.25610 q^{95} -5.65246 q^{96} +19.2111 q^{97} +6.41310 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 8 q^{4} + 4 q^{5} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 + 8 * q^4 + 4 * q^5 + 4 * q^9	\( 4 q - 4 q^{3} + 8 q^{4} + 4 q^{5} + 4 q^{9} - 8 q^{12} + 4 q^{14} - 4 q^{15} + 36 q^{16} + 8 q^{20} - 8 q^{23} + 4 q^{25} - 28 q^{26} - 4 q^{27} + 16 q^{31} + 4 q^{34} + 8 q^{36} + 32 q^{38} - 4 q^{42} + 4 q^{45} - 36 q^{48} + 12 q^{49} + 16 q^{53} + 52 q^{56} - 40 q^{58} + 8 q^{59} - 8 q^{60} + 72 q^{64} - 24 q^{67} + 8 q^{69} + 4 q^{70} - 8 q^{71} - 4 q^{75} + 28 q^{78} + 36 q^{80} + 4 q^{81} + 24 q^{82} - 60 q^{86} + 24 q^{89} + 32 q^{91} - 120 q^{92} - 16 q^{93} + 48 q^{97}+O(q^{100}) \) 4 * q - 4 * q^3 + 8 * q^4 + 4 * q^5 + 4 * q^9 - 8 * q^12 + 4 * q^14 - 4 * q^15 + 36 * q^16 + 8 * q^20 - 8 * q^23 + 4 * q^25 - 28 * q^26 - 4 * q^27 + 16 * q^31 + 4 * q^34 + 8 * q^36 + 32 * q^38 - 4 * q^42 + 4 * q^45 - 36 * q^48 + 12 * q^49 + 16 * q^53 + 52 * q^56 - 40 * q^58 + 8 * q^59 - 8 * q^60 + 72 * q^64 - 24 * q^67 + 8 * q^69 + 4 * q^70 - 8 * q^71 - 4 * q^75 + 28 * q^78 + 36 * q^80 + 4 * q^81 + 24 * q^82 - 60 * q^86 + 24 * q^89 + 32 * q^91 - 120 * q^92 - 16 * q^93 + 48 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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