LMFDB - Embedded newform 3640.2.a.y.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3640,2,Mod(1,3640)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3640, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("3640.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3640.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:	$29.0655463357$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.1194649.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 8x^{3} - 3x^{2} + 10x + 4 $ x^5 - 8x^3 - 3x^2 + 10*x + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.75291$ of defining polynomial
Character	$\chi$	$=$	3640.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.75291 q^{3} -1.00000 q^{5} +1.00000 q^{7} +4.57851 q^{9} +O(q^{10})$	$q-2.75291 q^{3} -1.00000 q^{5} +1.00000 q^{7} +4.57851 q^{9} -0.480110 q^{11} -1.00000 q^{13} +2.75291 q^{15} -3.04559 q^{17} +4.05862 q^{19} -2.75291 q^{21} -5.17109 q^{23} +1.00000 q^{25} -4.34549 q^{27} -5.48402 q^{29} +7.19289 q^{31} +1.32170 q^{33} -1.00000 q^{35} -1.29990 q^{37} +2.75291 q^{39} +4.61829 q^{41} +8.76985 q^{43} -4.57851 q^{45} +4.03874 q^{47} +1.00000 q^{49} +8.38423 q^{51} -11.9458 q^{53} +0.480110 q^{55} -11.1730 q^{57} +1.43121 q^{59} -3.32170 q^{61} +4.57851 q^{63} +1.00000 q^{65} -12.6443 q^{67} +14.2355 q^{69} -7.70401 q^{71} +13.0785 q^{73} -2.75291 q^{75} -0.480110 q^{77} +15.8014 q^{79} -1.77279 q^{81} +15.7026 q^{83} +3.04559 q^{85} +15.0970 q^{87} +12.9356 q^{89} -1.00000 q^{91} -19.8014 q^{93} -4.05862 q^{95} +1.11828 q^{97} -2.19819 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{5} + 5 q^{7} + q^{9}+O(q^{10}) $ 5 * q - 5 * q^5 + 5 * q^7 + q^9	$ 5 q - 5 q^{5} + 5 q^{7} + q^{9} - 7 q^{11} - 5 q^{13} + q^{17} + 3 q^{19} - 5 q^{23} + 5 q^{25} - 9 q^{27} - 9 q^{29} + 6 q^{31} + q^{33} - 5 q^{35} - 10 q^{37} - 8 q^{41} + 6 q^{43} - q^{45} - 13 q^{47} + 5 q^{49} - 4 q^{51} - 16 q^{53} + 7 q^{55} - 10 q^{57} - q^{59} - 11 q^{61} + q^{63} + 5 q^{65} - 30 q^{67} - 15 q^{69} - 12 q^{71} + 23 q^{73} - 7 q^{77} + 2 q^{79} - 11 q^{81} - 2 q^{83} - q^{85} - 5 q^{87} - 25 q^{89} - 5 q^{91} - 22 q^{93} - 3 q^{95} - 5 q^{97} - 12 q^{99}+O(q^{100}) $ 5 * q - 5 * q^5 + 5 * q^7 + q^9 - 7 * q^11 - 5 * q^13 + q^17 + 3 * q^19 - 5 * q^23 + 5 * q^25 - 9 * q^27 - 9 * q^29 + 6 * q^31 + q^33 - 5 * q^35 - 10 * q^37 - 8 * q^41 + 6 * q^43 - q^45 - 13 * q^47 + 5 * q^49 - 4 * q^51 - 16 * q^53 + 7 * q^55 - 10 * q^57 - q^59 - 11 * q^61 + q^63 + 5 * q^65 - 30 * q^67 - 15 * q^69 - 12 * q^71 + 23 * q^73 - 7 * q^77 + 2 * q^79 - 11 * q^81 - 2 * q^83 - q^85 - 5 * q^87 - 25 * q^89 - 5 * q^91 - 22 * q^93 - 3 * q^95 - 5 * q^97 - 12 * q^99

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	0
$2$	0	0
$3$	−2.75291	−1.58939	−0.794696	−	0.607007i	$-0.792370\pi$
$3$	−2.75291	−1.58939	−0.794696	+	0.607007i	$0.792370\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	4.57851	1.52617
$10$	0	0
$11$	−0.480110	−0.144759	−0.0723794	−	0.997377i	$-0.523059\pi$
$11$	−0.480110	−0.144759	−0.0723794	+	0.997377i	$0.523059\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	2.75291	0.710798
$16$	0	0
$17$	−3.04559	−0.738664	−0.369332	−	0.929298i	$-0.620413\pi$
$17$	−3.04559	−0.738664	−0.369332	+	0.929298i	$0.620413\pi$
$18$	0	0
$19$	4.05862	0.931111	0.465555	−	0.885019i	$-0.345855\pi$
$19$	4.05862	0.931111	0.465555	+	0.885019i	$0.345855\pi$
$20$	0	0
$21$	−2.75291	−0.600734
$22$	0	0
$23$	−5.17109	−1.07825	−0.539123	−	0.842227i	$-0.681244\pi$
$23$	−5.17109	−1.07825	−0.539123	+	0.842227i	$0.681244\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−4.34549	−0.836290
$28$	0	0
$29$	−5.48402	−1.01836	−0.509178	−	0.860661i	$-0.670051\pi$
$29$	−5.48402	−1.01836	−0.509178	+	0.860661i	$0.670051\pi$
$30$	0	0
$31$	7.19289	1.29188	0.645940	−	0.763388i	$-0.276465\pi$
$31$	7.19289	1.29188	0.645940	+	0.763388i	$0.276465\pi$
$32$	0	0
$33$	1.32170	0.230078
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−1.29990	−0.213702	−0.106851	−	0.994275i	$-0.534077\pi$
$37$	−1.29990	−0.213702	−0.106851	+	0.994275i	$0.534077\pi$
$38$	0	0
$39$	2.75291	0.440818
$40$	0	0
$41$	4.61829	0.721255	0.360628	−	0.932710i	$-0.382563\pi$
$41$	4.61829	0.721255	0.360628	+	0.932710i	$0.382563\pi$
$42$	0	0
$43$	8.76985	1.33739	0.668695	−	0.743537i	$-0.266853\pi$
$43$	8.76985	1.33739	0.668695	+	0.743537i	$0.266853\pi$
$44$	0	0
$45$	−4.57851	−0.682524
$46$	0	0
$47$	4.03874	0.589110	0.294555	−	0.955634i	$-0.404829\pi$
$47$	4.03874	0.589110	0.294555	+	0.955634i	$0.404829\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	8.38423	1.17403
$52$	0	0
$53$	−11.9458	−1.64088	−0.820441	−	0.571732i	$-0.806272\pi$
$53$	−11.9458	−1.64088	−0.820441	+	0.571732i	$0.806272\pi$
$54$	0	0
$55$	0.480110	0.0647381
$56$	0	0
$57$	−11.1730	−1.47990
$58$	0	0
$59$	1.43121	0.186328	0.0931638	−	0.995651i	$-0.470302\pi$
$59$	1.43121	0.186328	0.0931638	+	0.995651i	$0.470302\pi$
$60$	0	0
$61$	−3.32170	−0.425300	−0.212650	−	0.977128i	$-0.568209\pi$
$61$	−3.32170	−0.425300	−0.212650	+	0.977128i	$0.568209\pi$
$62$	0	0
$63$	4.57851	0.576838
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	−12.6443	−1.54475	−0.772376	−	0.635165i	$-0.780932\pi$
$67$	−12.6443	−1.54475	−0.772376	+	0.635165i	$0.780932\pi$
$68$	0	0
$69$	14.2355	1.71376
$70$	0	0
$71$	−7.70401	−0.914297	−0.457149	−	0.889390i	$-0.651129\pi$
$71$	−7.70401	−0.914297	−0.457149	+	0.889390i	$0.651129\pi$
$72$	0	0
$73$	13.0785	1.53072	0.765362	−	0.643600i	$-0.222560\pi$
$73$	13.0785	1.53072	0.765362	+	0.643600i	$0.222560\pi$
$74$	0	0
$75$	−2.75291	−0.317879
$76$	0	0
$77$	−0.480110	−0.0547137
$78$	0	0
$79$	15.8014	1.77779	0.888896	−	0.458109i	$-0.151473\pi$
$79$	15.8014	1.77779	0.888896	+	0.458109i	$0.151473\pi$
$80$	0	0
$81$	−1.77279	−0.196977
$82$	0	0
$83$	15.7026	1.72359	0.861793	−	0.507260i	$-0.169342\pi$
$83$	15.7026	1.72359	0.861793	+	0.507260i	$0.169342\pi$
$84$	0	0
$85$	3.04559	0.330340
$86$	0	0
$87$	15.0970	1.61857
$88$	0	0
$89$	12.9356	1.37117	0.685587	−	0.727991i	$-0.259546\pi$
$89$	12.9356	1.37117	0.685587	+	0.727991i	$0.259546\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−19.8014	−2.05331
$94$	0	0
$95$	−4.05862	−0.416405
$96$	0	0
$97$	1.11828	0.113544	0.0567720	−	0.998387i	$-0.481919\pi$
$97$	1.11828	0.113544	0.0567720	+	0.998387i	$0.481919\pi$
$98$	0	0
$99$	−2.19819	−0.220926
$100$	0	0
$101$	−17.3411	−1.72551	−0.862754	−	0.505625i	$-0.831262\pi$
$101$	−17.3411	−1.72551	−0.862754	+	0.505625i	$0.831262\pi$
$102$	0	0
$103$	5.63618	0.555349	0.277675	−	0.960675i	$-0.410436\pi$
$103$	5.63618	0.555349	0.277675	+	0.960675i	$0.410436\pi$
$104$	0	0
$105$	2.75291	0.268656
$106$	0	0
$107$	3.89299	0.376349	0.188175	−	0.982136i	$-0.439743\pi$
$107$	3.89299	0.376349	0.188175	+	0.982136i	$0.439743\pi$
$108$	0	0
$109$	16.3140	1.56260	0.781300	−	0.624155i	$-0.214557\pi$
$109$	16.3140	1.56260	0.781300	+	0.624155i	$0.214557\pi$
$110$	0	0
$111$	3.57851	0.339657
$112$	0	0
$113$	−8.82752	−0.830423	−0.415212	−	0.909725i	$-0.636292\pi$
$113$	−8.82752	−0.830423	−0.415212	+	0.909725i	$0.636292\pi$
$114$	0	0
$115$	5.17109	0.482206
$116$	0	0
$117$	−4.57851	−0.423283
$118$	0	0
$119$	−3.04559	−0.279189
$120$	0	0
$121$	−10.7695	−0.979045
$122$	0	0
$123$	−12.7137	−1.14636
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−19.7863	−1.75575	−0.877877	−	0.478886i	$-0.841041\pi$
$127$	−19.7863	−1.75575	−0.877877	+	0.478886i	$0.841041\pi$
$128$	0	0
$129$	−24.1426	−2.12564
$130$	0	0
$131$	1.54558	0.135038	0.0675189	−	0.997718i	$-0.478492\pi$
$131$	1.54558	0.135038	0.0675189	+	0.997718i	$0.478492\pi$
$132$	0	0
$133$	4.05862	0.351927
$134$	0	0
$135$	4.34549	0.374000
$136$	0	0
$137$	−11.1672	−0.954078	−0.477039	−	0.878882i	$-0.658290\pi$
$137$	−11.1672	−0.954078	−0.477039	+	0.878882i	$0.658290\pi$
$138$	0	0
$139$	−12.6192	−1.07035	−0.535175	−	0.844741i	$-0.679754\pi$
$139$	−12.6192	−1.07035	−0.535175	+	0.844741i	$0.679754\pi$
$140$	0	0
$141$	−11.1183	−0.936328
$142$	0	0
$143$	0.480110	0.0401488
$144$	0	0
$145$	5.48402	0.455423
$146$	0	0
$147$	−2.75291	−0.227056
$148$	0	0
$149$	1.66387	0.136310	0.0681550	−	0.997675i	$-0.478289\pi$
$149$	1.66387	0.136310	0.0681550	+	0.997675i	$0.478289\pi$
$150$	0	0
$151$	1.46745	0.119420	0.0597098	−	0.998216i	$-0.480982\pi$
$151$	1.46745	0.119420	0.0597098	+	0.998216i	$0.480982\pi$
$152$	0	0
$153$	−13.9442	−1.12733
$154$	0	0
$155$	−7.19289	−0.577747
$156$	0	0
$157$	−23.7196	−1.89303	−0.946513	−	0.322665i	$-0.895421\pi$
$157$	−23.7196	−1.89303	−0.946513	+	0.322665i	$0.895421\pi$
$158$	0	0
$159$	32.8857	2.60800
$160$	0	0
$161$	−5.17109	−0.407539
$162$	0	0
$163$	−6.60716	−0.517513	−0.258756	−	0.965943i	$-0.583313\pi$
$163$	−6.60716	−0.517513	−0.258756	+	0.965943i	$0.583313\pi$
$164$	0	0
$165$	−1.32170	−0.102894
$166$	0	0
$167$	−4.33436	−0.335403	−0.167701	−	0.985838i	$-0.553634\pi$
$167$	−4.33436	−0.335403	−0.167701	+	0.985838i	$0.553634\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	18.5824	1.42103
$172$	0	0
$173$	−18.4696	−1.40422	−0.702109	−	0.712070i	$-0.747758\pi$
$173$	−18.4696	−1.40422	−0.702109	+	0.712070i	$0.747758\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	−3.93999	−0.296148
$178$	0	0
$179$	−13.9777	−1.04474	−0.522370	−	0.852719i	$-0.674952\pi$
$179$	−13.9777	−1.04474	−0.522370	+	0.852719i	$0.674952\pi$
$180$	0	0
$181$	−12.7219	−0.945611	−0.472806	−	0.881167i	$-0.656759\pi$
$181$	−12.7219	−0.945611	−0.472806	+	0.881167i	$0.656759\pi$
$182$	0	0
$183$	9.14434	0.675969
$184$	0	0
$185$	1.29990	0.0955706
$186$	0	0
$187$	1.46222	0.106928
$188$	0	0
$189$	−4.34549	−0.316088
$190$	0	0
$191$	4.29886	0.311055	0.155527	−	0.987832i	$-0.450292\pi$
$191$	4.29886	0.311055	0.155527	+	0.987832i	$0.450292\pi$
$192$	0	0
$193$	6.05722	0.436009	0.218004	−	0.975948i	$-0.430045\pi$
$193$	6.05722	0.436009	0.218004	+	0.975948i	$0.430045\pi$
$194$	0	0
$195$	−2.75291	−0.197140
$196$	0	0
$197$	−7.66032	−0.545775	−0.272888	−	0.962046i	$-0.587979\pi$
$197$	−7.66032	−0.545775	−0.272888	+	0.962046i	$0.587979\pi$
$198$	0	0
$199$	−4.91383	−0.348332	−0.174166	−	0.984716i	$-0.555723\pi$
$199$	−4.91383	−0.348332	−0.174166	+	0.984716i	$0.555723\pi$
$200$	0	0
$201$	34.8087	2.45522
$202$	0	0
$203$	−5.48402	−0.384903
$204$	0	0
$205$	−4.61829	−0.322555
$206$	0	0
$207$	−23.6759	−1.64559
$208$	0	0
$209$	−1.94858	−0.134786
$210$	0	0
$211$	−7.03189	−0.484095	−0.242048	−	0.970264i	$-0.577819\pi$
$211$	−7.03189	−0.484095	−0.242048	+	0.970264i	$0.577819\pi$
$212$	0	0
$213$	21.2084	1.45318
$214$	0	0
$215$	−8.76985	−0.598099
$216$	0	0
$217$	7.19289	0.488285
$218$	0	0
$219$	−36.0040	−2.43292
$220$	0	0
$221$	3.04559	0.204868
$222$	0	0
$223$	−16.5843	−1.11057	−0.555285	−	0.831660i	$-0.687391\pi$
$223$	−16.5843	−1.11057	−0.555285	+	0.831660i	$0.687391\pi$
$224$	0	0
$225$	4.57851	0.305234
$226$	0	0
$227$	−19.3062	−1.28140	−0.640700	−	0.767791i	$-0.721356\pi$
$227$	−19.3062	−1.28140	−0.640700	+	0.767791i	$0.721356\pi$
$228$	0	0
$229$	−0.0339520	−0.00224361	−0.00112181	−	0.999999i	$-0.500357\pi$
$229$	−0.0339520	−0.00224361	−0.00112181	+	0.999999i	$0.500357\pi$
$230$	0	0
$231$	1.32170	0.0869615
$232$	0	0
$233$	18.2211	1.19370	0.596852	−	0.802352i	$-0.296418\pi$
$233$	18.2211	1.19370	0.596852	+	0.802352i	$0.296418\pi$
$234$	0	0
$235$	−4.03874	−0.263458
$236$	0	0
$237$	−43.4997	−2.82561
$238$	0	0
$239$	−15.6642	−1.01324	−0.506618	−	0.862171i	$-0.669105\pi$
$239$	−15.6642	−1.01324	−0.506618	+	0.862171i	$0.669105\pi$
$240$	0	0
$241$	−24.3596	−1.56914	−0.784571	−	0.620039i	$-0.787117\pi$
$241$	−24.3596	−1.56914	−0.784571	+	0.620039i	$0.787117\pi$
$242$	0	0
$243$	17.9168	1.14936
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−4.05862	−0.258244
$248$	0	0
$249$	−43.2279	−2.73945
$250$	0	0
$251$	25.1697	1.58870	0.794348	−	0.607462i	$-0.207812\pi$
$251$	25.1697	1.58870	0.794348	+	0.607462i	$0.207812\pi$
$252$	0	0
$253$	2.48269	0.156086
$254$	0	0
$255$	−8.38423	−0.525041
$256$	0	0
$257$	5.43616	0.339098	0.169549	−	0.985522i	$-0.445769\pi$
$257$	5.43616	0.339098	0.169549	+	0.985522i	$0.445769\pi$
$258$	0	0
$259$	−1.29990	−0.0807719
$260$	0	0
$261$	−25.1086	−1.55418
$262$	0	0
$263$	2.43929	0.150413	0.0752065	−	0.997168i	$-0.476038\pi$
$263$	2.43929	0.150413	0.0752065	+	0.997168i	$0.476038\pi$
$264$	0	0
$265$	11.9458	0.733824
$266$	0	0
$267$	−35.6106	−2.17933
$268$	0	0
$269$	5.34113	0.325655	0.162827	−	0.986655i	$-0.447939\pi$
$269$	5.34113	0.325655	0.162827	+	0.986655i	$0.447939\pi$
$270$	0	0
$271$	29.5432	1.79462	0.897310	−	0.441402i	$-0.145519\pi$
$271$	29.5432	1.79462	0.897310	+	0.441402i	$0.145519\pi$
$272$	0	0
$273$	2.75291	0.166614
$274$	0	0
$275$	−0.480110	−0.0289517
$276$	0	0
$277$	−0.233460	−0.0140273	−0.00701363	−	0.999975i	$-0.502233\pi$
$277$	−0.233460	−0.0140273	−0.00701363	+	0.999975i	$0.502233\pi$
$278$	0	0
$279$	32.9327	1.97163
$280$	0	0
$281$	−17.9321	−1.06974	−0.534870	−	0.844935i	$-0.679639\pi$
$281$	−17.9321	−1.06974	−0.534870	+	0.844935i	$0.679639\pi$
$282$	0	0
$283$	−25.6401	−1.52415	−0.762074	−	0.647490i	$-0.775819\pi$
$283$	−25.6401	−1.52415	−0.762074	+	0.647490i	$0.775819\pi$
$284$	0	0
$285$	11.1730	0.661832
$286$	0	0
$287$	4.61829	0.272609
$288$	0	0
$289$	−7.72439	−0.454376
$290$	0	0
$291$	−3.07852	−0.180466
$292$	0	0
$293$	−7.23936	−0.422928	−0.211464	−	0.977386i	$-0.567823\pi$
$293$	−7.23936	−0.422928	−0.211464	+	0.977386i	$0.567823\pi$
$294$	0	0
$295$	−1.43121	−0.0833282
$296$	0	0
$297$	2.08631	0.121060
$298$	0	0
$299$	5.17109	0.299052
$300$	0	0
$301$	8.76985	0.505486
$302$	0	0
$303$	47.7386	2.74251
$304$	0	0
$305$	3.32170	0.190200
$306$	0	0
$307$	−2.04758	−0.116861	−0.0584307	−	0.998291i	$-0.518610\pi$
$307$	−2.04758	−0.116861	−0.0584307	+	0.998291i	$0.518610\pi$
$308$	0	0
$309$	−15.5159	−0.882668
$310$	0	0
$311$	−3.77436	−0.214024	−0.107012	−	0.994258i	$-0.534128\pi$
$311$	−3.77436	−0.214024	−0.107012	+	0.994258i	$0.534128\pi$
$312$	0	0
$313$	−13.6332	−0.770596	−0.385298	−	0.922792i	$-0.625901\pi$
$313$	−13.6332	−0.770596	−0.385298	+	0.922792i	$0.625901\pi$
$314$	0	0
$315$	−4.57851	−0.257970
$316$	0	0
$317$	−7.18794	−0.403715	−0.201857	−	0.979415i	$-0.564698\pi$
$317$	−7.18794	−0.403715	−0.201857	+	0.979415i	$0.564698\pi$
$318$	0	0
$319$	2.63293	0.147416
$320$	0	0
$321$	−10.7170	−0.598167
$322$	0	0
$323$	−12.3609	−0.687778
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	−44.9110	−2.48359
$328$	0	0
$329$	4.03874	0.222663
$330$	0	0
$331$	10.2436	0.563037	0.281518	−	0.959556i	$-0.409162\pi$
$331$	10.2436	0.563037	0.281518	+	0.959556i	$0.409162\pi$
$332$	0	0
$333$	−5.95160	−0.326146
$334$	0	0
$335$	12.6443	0.690835
$336$	0	0
$337$	−17.7160	−0.965052	−0.482526	−	0.875882i	$-0.660281\pi$
$337$	−17.7160	−0.965052	−0.482526	+	0.875882i	$0.660281\pi$
$338$	0	0
$339$	24.3014	1.31987
$340$	0	0
$341$	−3.45338	−0.187011
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−14.2355	−0.766415
$346$	0	0
$347$	−0.662334	−0.0355559	−0.0177780	−	0.999842i	$-0.505659\pi$
$347$	−0.662334	−0.0355559	−0.0177780	+	0.999842i	$0.505659\pi$
$348$	0	0
$349$	−26.7604	−1.43245	−0.716224	−	0.697870i	$-0.754131\pi$
$349$	−26.7604	−1.43245	−0.716224	+	0.697870i	$0.754131\pi$
$350$	0	0
$351$	4.34549	0.231945
$352$	0	0
$353$	5.69202	0.302956	0.151478	−	0.988461i	$-0.451597\pi$
$353$	5.69202	0.302956	0.151478	+	0.988461i	$0.451597\pi$
$354$	0	0
$355$	7.70401	0.408886
$356$	0	0
$357$	8.38423	0.443740
$358$	0	0
$359$	−6.13307	−0.323691	−0.161846	−	0.986816i	$-0.551745\pi$
$359$	−6.13307	−0.323691	−0.161846	+	0.986816i	$0.551745\pi$
$360$	0	0
$361$	−2.52762	−0.133033
$362$	0	0
$363$	29.6474	1.55609
$364$	0	0
$365$	−13.0785	−0.684561
$366$	0	0
$367$	25.4203	1.32693	0.663465	−	0.748207i	$-0.269085\pi$
$367$	25.4203	1.32693	0.663465	+	0.748207i	$0.269085\pi$
$368$	0	0
$369$	21.1449	1.10076
$370$	0	0
$371$	−11.9458	−0.620195
$372$	0	0
$373$	−24.6484	−1.27625	−0.638123	−	0.769935i	$-0.720289\pi$
$373$	−24.6484	−1.27625	−0.638123	+	0.769935i	$0.720289\pi$
$374$	0	0
$375$	2.75291	0.142160
$376$	0	0
$377$	5.48402	0.282441
$378$	0	0
$379$	−13.3988	−0.688250	−0.344125	−	0.938924i	$-0.611824\pi$
$379$	−13.3988	−0.688250	−0.344125	+	0.938924i	$0.611824\pi$
$380$	0	0
$381$	54.4700	2.79058
$382$	0	0
$383$	−23.5438	−1.20303	−0.601517	−	0.798860i	$-0.705437\pi$
$383$	−23.5438	−1.20303	−0.601517	+	0.798860i	$0.705437\pi$
$384$	0	0
$385$	0.480110	0.0244687
$386$	0	0
$387$	40.1528	2.04108
$388$	0	0
$389$	22.7937	1.15569	0.577843	−	0.816148i	$-0.303895\pi$
$389$	22.7937	1.15569	0.577843	+	0.816148i	$0.303895\pi$
$390$	0	0
$391$	15.7490	0.796461
$392$	0	0
$393$	−4.25484	−0.214628
$394$	0	0
$395$	−15.8014	−0.795053
$396$	0	0
$397$	−11.2880	−0.566526	−0.283263	−	0.959042i	$-0.591417\pi$
$397$	−11.2880	−0.566526	−0.283263	+	0.959042i	$0.591417\pi$
$398$	0	0
$399$	−11.1730	−0.559350
$400$	0	0
$401$	17.7462	0.886204	0.443102	−	0.896471i	$-0.353878\pi$
$401$	17.7462	0.886204	0.443102	+	0.896471i	$0.353878\pi$
$402$	0	0
$403$	−7.19289	−0.358303
$404$	0	0
$405$	1.77279	0.0880906
$406$	0	0
$407$	0.624096	0.0309353
$408$	0	0
$409$	7.57756	0.374686	0.187343	−	0.982295i	$-0.440012\pi$
$409$	7.57756	0.374686	0.187343	+	0.982295i	$0.440012\pi$
$410$	0	0
$411$	30.7423	1.51640
$412$	0	0
$413$	1.43121	0.0704252
$414$	0	0
$415$	−15.7026	−0.770811
$416$	0	0
$417$	34.7396	1.70121
$418$	0	0
$419$	−9.35867	−0.457201	−0.228601	−	0.973520i	$-0.573415\pi$
$419$	−9.35867	−0.457201	−0.228601	+	0.973520i	$0.573415\pi$
$420$	0	0
$421$	14.2277	0.693417	0.346708	−	0.937973i	$-0.387299\pi$
$421$	14.2277	0.693417	0.346708	+	0.937973i	$0.387299\pi$
$422$	0	0
$423$	18.4914	0.899082
$424$	0	0
$425$	−3.04559	−0.147733
$426$	0	0
$427$	−3.32170	−0.160748
$428$	0	0
$429$	−1.32170	−0.0638123
$430$	0	0
$431$	20.1556	0.970861	0.485431	−	0.874275i	$-0.338663\pi$
$431$	20.1556	0.970861	0.485431	+	0.874275i	$0.338663\pi$
$432$	0	0
$433$	22.2801	1.07071	0.535356	−	0.844626i	$-0.320177\pi$
$433$	22.2801	1.07071	0.535356	+	0.844626i	$0.320177\pi$
$434$	0	0
$435$	−15.0970	−0.723846
$436$	0	0
$437$	−20.9875	−1.00397
$438$	0	0
$439$	−14.6823	−0.700746	−0.350373	−	0.936610i	$-0.613945\pi$
$439$	−14.6823	−0.700746	−0.350373	+	0.936610i	$0.613945\pi$
$440$	0	0
$441$	4.57851	0.218024
$442$	0	0
$443$	−5.28346	−0.251025	−0.125512	−	0.992092i	$-0.540057\pi$
$443$	−5.28346	−0.251025	−0.125512	+	0.992092i	$0.540057\pi$
$444$	0	0
$445$	−12.9356	−0.613208
$446$	0	0
$447$	−4.58050	−0.216650
$448$	0	0
$449$	11.8227	0.557947	0.278973	−	0.960299i	$-0.410006\pi$
$449$	11.8227	0.557947	0.278973	+	0.960299i	$0.410006\pi$
$450$	0	0
$451$	−2.21729	−0.104408
$452$	0	0
$453$	−4.03976	−0.189805
$454$	0	0
$455$	1.00000	0.0468807
$456$	0	0
$457$	17.0962	0.799725	0.399862	−	0.916575i	$-0.369058\pi$
$457$	17.0962	0.799725	0.399862	+	0.916575i	$0.369058\pi$
$458$	0	0
$459$	13.2346	0.617737
$460$	0	0
$461$	−31.3834	−1.46167	−0.730836	−	0.682553i	$-0.760870\pi$
$461$	−31.3834	−1.46167	−0.730836	+	0.682553i	$0.760870\pi$
$462$	0	0
$463$	27.1224	1.26048	0.630241	−	0.776399i	$-0.282956\pi$
$463$	27.1224	1.26048	0.630241	+	0.776399i	$0.282956\pi$
$464$	0	0
$465$	19.8014	0.918266
$466$	0	0
$467$	−30.5064	−1.41167	−0.705834	−	0.708377i	$-0.749428\pi$
$467$	−30.5064	−1.41167	−0.705834	+	0.708377i	$0.749428\pi$
$468$	0	0
$469$	−12.6443	−0.583862
$470$	0	0
$471$	65.2978	3.00876
$472$	0	0
$473$	−4.21049	−0.193599
$474$	0	0
$475$	4.05862	0.186222
$476$	0	0
$477$	−54.6939	−2.50426
$478$	0	0
$479$	8.54273	0.390327	0.195164	−	0.980771i	$-0.437476\pi$
$479$	8.54273	0.390327	0.195164	+	0.980771i	$0.437476\pi$
$480$	0	0
$481$	1.29990	0.0592704
$482$	0	0
$483$	14.2355	0.647739
$484$	0	0
$485$	−1.11828	−0.0507784
$486$	0	0
$487$	4.64312	0.210400	0.105200	−	0.994451i	$-0.466452\pi$
$487$	4.64312	0.210400	0.105200	+	0.994451i	$0.466452\pi$
$488$	0	0
$489$	18.1889	0.822531
$490$	0	0
$491$	−15.8381	−0.714763	−0.357382	−	0.933958i	$-0.616330\pi$
$491$	−15.8381	−0.714763	−0.357382	+	0.933958i	$0.616330\pi$
$492$	0	0
$493$	16.7021	0.752223
$494$	0	0
$495$	2.19819	0.0988013
$496$	0	0
$497$	−7.70401	−0.345572
$498$	0	0
$499$	36.2817	1.62419	0.812096	−	0.583523i	$-0.198326\pi$
$499$	36.2817	1.62419	0.812096	+	0.583523i	$0.198326\pi$
$500$	0	0
$501$	11.9321	0.533087
$502$	0	0
$503$	31.3669	1.39858	0.699290	−	0.714839i	$-0.253500\pi$
$503$	31.3669	1.39858	0.699290	+	0.714839i	$0.253500\pi$
$504$	0	0
$505$	17.3411	0.771670
$506$	0	0
$507$	−2.75291	−0.122261
$508$	0	0
$509$	27.7239	1.22884	0.614420	−	0.788979i	$-0.289390\pi$
$509$	27.7239	1.22884	0.614420	+	0.788979i	$0.289390\pi$
$510$	0	0
$511$	13.0785	0.578560
$512$	0	0
$513$	−17.6367	−0.778678
$514$	0	0
$515$	−5.63618	−0.248360
$516$	0	0
$517$	−1.93904	−0.0852789
$518$	0	0
$519$	50.8451	2.23185
$520$	0	0
$521$	2.71912	0.119127	0.0595634	−	0.998225i	$-0.481029\pi$
$521$	2.71912	0.119127	0.0595634	+	0.998225i	$0.481029\pi$
$522$	0	0
$523$	−28.7118	−1.25548	−0.627740	−	0.778423i	$-0.716020\pi$
$523$	−28.7118	−1.25548	−0.627740	+	0.778423i	$0.716020\pi$
$524$	0	0
$525$	−2.75291	−0.120147
$526$	0	0
$527$	−21.9066	−0.954265
$528$	0	0
$529$	3.74014	0.162615
$530$	0	0
$531$	6.55280	0.284367
$532$	0	0
$533$	−4.61829	−0.200040
$534$	0	0
$535$	−3.89299	−0.168308
$536$	0	0
$537$	38.4792	1.66050
$538$	0	0
$539$	−0.480110	−0.0206798
$540$	0	0
$541$	29.4731	1.26715	0.633574	−	0.773682i	$-0.281587\pi$
$541$	29.4731	1.26715	0.633574	+	0.773682i	$0.281587\pi$
$542$	0	0
$543$	35.0222	1.50295
$544$	0	0
$545$	−16.3140	−0.698816
$546$	0	0
$547$	8.21762	0.351360	0.175680	−	0.984447i	$-0.443788\pi$
$547$	8.21762	0.351360	0.175680	+	0.984447i	$0.443788\pi$
$548$	0	0
$549$	−15.2084	−0.649080
$550$	0	0
$551$	−22.2575	−0.948203
$552$	0	0
$553$	15.8014	0.671942
$554$	0	0
$555$	−3.57851	−0.151899
$556$	0	0
$557$	1.10376	0.0467678	0.0233839	−	0.999727i	$-0.492556\pi$
$557$	1.10376	0.0467678	0.0233839	+	0.999727i	$0.492556\pi$
$558$	0	0
$559$	−8.76985	−0.370925
$560$	0	0
$561$	−4.02535	−0.169951
$562$	0	0
$563$	9.77670	0.412039	0.206019	−	0.978548i	$-0.433949\pi$
$563$	9.77670	0.412039	0.206019	+	0.978548i	$0.433949\pi$
$564$	0	0
$565$	8.82752	0.371376
$566$	0	0
$567$	−1.77279	−0.0744502
$568$	0	0
$569$	−26.4686	−1.10962	−0.554810	−	0.831977i	$-0.687209\pi$
$569$	−26.4686	−1.10962	−0.554810	+	0.831977i	$0.687209\pi$
$570$	0	0
$571$	−22.6664	−0.948559	−0.474279	−	0.880374i	$-0.657291\pi$
$571$	−22.6664	−0.948559	−0.474279	+	0.880374i	$0.657291\pi$
$572$	0	0
$573$	−11.8344	−0.494388
$574$	0	0
$575$	−5.17109	−0.215649
$576$	0	0
$577$	−26.8973	−1.11975	−0.559875	−	0.828577i	$-0.689151\pi$
$577$	−26.8973	−1.11975	−0.559875	+	0.828577i	$0.689151\pi$
$578$	0	0
$579$	−16.6750	−0.692989
$580$	0	0
$581$	15.7026	0.651454
$582$	0	0
$583$	5.73530	0.237532
$584$	0	0
$585$	4.57851	0.189298
$586$	0	0
$587$	−26.4855	−1.09317	−0.546587	−	0.837402i	$-0.684073\pi$
$587$	−26.4855	−1.09317	−0.546587	+	0.837402i	$0.684073\pi$
$588$	0	0
$589$	29.1932	1.20288
$590$	0	0
$591$	21.0882	0.867451
$592$	0	0
$593$	−3.93917	−0.161762	−0.0808811	−	0.996724i	$-0.525773\pi$
$593$	−3.93917	−0.161762	−0.0808811	+	0.996724i	$0.525773\pi$
$594$	0	0
$595$	3.04559	0.124857
$596$	0	0
$597$	13.5273	0.553637
$598$	0	0
$599$	−47.6048	−1.94508	−0.972540	−	0.232737i	$-0.925232\pi$
$599$	−47.6048	−1.94508	−0.972540	+	0.232737i	$0.925232\pi$
$600$	0	0
$601$	−15.3439	−0.625891	−0.312946	−	0.949771i	$-0.601316\pi$
$601$	−15.3439	−0.625891	−0.312946	+	0.949771i	$0.601316\pi$
$602$	0	0
$603$	−57.8922	−2.35755
$604$	0	0
$605$	10.7695	0.437842
$606$	0	0
$607$	−7.06089	−0.286593	−0.143296	−	0.989680i	$-0.545770\pi$
$607$	−7.06089	−0.286593	−0.143296	+	0.989680i	$0.545770\pi$
$608$	0	0
$609$	15.0970	0.611761
$610$	0	0
$611$	−4.03874	−0.163390
$612$	0	0
$613$	−27.7702	−1.12163	−0.560814	−	0.827942i	$-0.689512\pi$
$613$	−27.7702	−1.12163	−0.560814	+	0.827942i	$0.689512\pi$
$614$	0	0
$615$	12.7137	0.512667
$616$	0	0
$617$	−11.3617	−0.457405	−0.228702	−	0.973496i	$-0.573448\pi$
$617$	−11.3617	−0.457405	−0.228702	+	0.973496i	$0.573448\pi$
$618$	0	0
$619$	14.1288	0.567885	0.283942	−	0.958841i	$-0.408358\pi$
$619$	14.1288	0.567885	0.283942	+	0.958841i	$0.408358\pi$
$620$	0	0
$621$	22.4709	0.901726
$622$	0	0
$623$	12.9356	0.518255
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	5.36428	0.214229
$628$	0	0
$629$	3.95896	0.157854
$630$	0	0
$631$	−44.1768	−1.75865	−0.879325	−	0.476222i	$-0.842006\pi$
$631$	−44.1768	−1.75865	−0.879325	+	0.476222i	$0.842006\pi$
$632$	0	0
$633$	19.3581	0.769417
$634$	0	0
$635$	19.7863	0.785197
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−35.2729	−1.39537
$640$	0	0
$641$	29.1197	1.15016	0.575080	−	0.818097i	$-0.304971\pi$
$641$	29.1197	1.15016	0.575080	+	0.818097i	$0.304971\pi$
$642$	0	0
$643$	46.6058	1.83795	0.918977	−	0.394310i	$-0.129016\pi$
$643$	46.6058	1.83795	0.918977	+	0.394310i	$0.129016\pi$
$644$	0	0
$645$	24.1426	0.950614
$646$	0	0
$647$	−42.2035	−1.65919	−0.829595	−	0.558365i	$-0.811429\pi$
$647$	−42.2035	−1.65919	−0.829595	+	0.558365i	$0.811429\pi$
$648$	0	0
$649$	−0.687138	−0.0269725
$650$	0	0
$651$	−19.8014	−0.776077
$652$	0	0
$653$	43.4964	1.70215	0.851073	−	0.525048i	$-0.175952\pi$
$653$	43.4964	1.70215	0.851073	+	0.525048i	$0.175952\pi$
$654$	0	0
$655$	−1.54558	−0.0603908
$656$	0	0
$657$	59.8801	2.33614
$658$	0	0
$659$	−11.7287	−0.456887	−0.228444	−	0.973557i	$-0.573364\pi$
$659$	−11.7287	−0.456887	−0.228444	+	0.973557i	$0.573364\pi$
$660$	0	0
$661$	29.9391	1.16450	0.582248	−	0.813011i	$-0.302173\pi$
$661$	29.9391	1.16450	0.582248	+	0.813011i	$0.302173\pi$
$662$	0	0
$663$	−8.38423	−0.325616
$664$	0	0
$665$	−4.05862	−0.157386
$666$	0	0
$667$	28.3583	1.09804
$668$	0	0
$669$	45.6552	1.76513
$670$	0	0
$671$	1.59478	0.0615659
$672$	0	0
$673$	−7.10458	−0.273861	−0.136931	−	0.990581i	$-0.543724\pi$
$673$	−7.10458	−0.273861	−0.136931	+	0.990581i	$0.543724\pi$
$674$	0	0
$675$	−4.34549	−0.167258
$676$	0	0
$677$	39.3092	1.51077	0.755387	−	0.655278i	$-0.227449\pi$
$677$	39.3092	1.51077	0.755387	+	0.655278i	$0.227449\pi$
$678$	0	0
$679$	1.11828	0.0429156
$680$	0	0
$681$	53.1483	2.03665
$682$	0	0
$683$	−11.7644	−0.450152	−0.225076	−	0.974341i	$-0.572263\pi$
$683$	−11.7644	−0.450152	−0.225076	+	0.974341i	$0.572263\pi$
$684$	0	0
$685$	11.1672	0.426677
$686$	0	0
$687$	0.0934668	0.00356598
$688$	0	0
$689$	11.9458	0.455099
$690$	0	0
$691$	−15.3777	−0.584997	−0.292499	−	0.956266i	$-0.594487\pi$
$691$	−15.3777	−0.584997	−0.292499	+	0.956266i	$0.594487\pi$
$692$	0	0
$693$	−2.19819	−0.0835023
$694$	0	0
$695$	12.6192	0.478675
$696$	0	0
$697$	−14.0654	−0.532765
$698$	0	0
$699$	−50.1610	−1.89726
$700$	0	0
$701$	−9.31103	−0.351673	−0.175836	−	0.984419i	$-0.556263\pi$
$701$	−9.31103	−0.351673	−0.175836	+	0.984419i	$0.556263\pi$
$702$	0	0
$703$	−5.27580	−0.198981
$704$	0	0
$705$	11.1183	0.418739
$706$	0	0
$707$	−17.3411	−0.652180
$708$	0	0
$709$	7.05246	0.264861	0.132430	−	0.991192i	$-0.457722\pi$
$709$	7.05246	0.264861	0.132430	+	0.991192i	$0.457722\pi$
$710$	0	0
$711$	72.3467	2.71321
$712$	0	0
$713$	−37.1950	−1.39297
$714$	0	0
$715$	−0.480110	−0.0179551
$716$	0	0
$717$	43.1222	1.61043
$718$	0	0
$719$	−24.1940	−0.902283	−0.451141	−	0.892452i	$-0.648983\pi$
$719$	−24.1940	−0.902283	−0.451141	+	0.892452i	$0.648983\pi$
$720$	0	0
$721$	5.63618	0.209902
$722$	0	0
$723$	67.0599	2.49398
$724$	0	0
$725$	−5.48402	−0.203671
$726$	0	0
$727$	−15.6579	−0.580718	−0.290359	−	0.956918i	$-0.593775\pi$
$727$	−15.6579	−0.580718	−0.290359	+	0.956918i	$0.593775\pi$
$728$	0	0
$729$	−44.0049	−1.62981
$730$	0	0
$731$	−26.7093	−0.987881
$732$	0	0
$733$	−37.4320	−1.38258	−0.691291	−	0.722576i	$-0.742958\pi$
$733$	−37.4320	−1.38258	−0.691291	+	0.722576i	$0.742958\pi$
$734$	0	0
$735$	2.75291	0.101543
$736$	0	0
$737$	6.07068	0.223616
$738$	0	0
$739$	−19.6276	−0.722012	−0.361006	−	0.932563i	$-0.617567\pi$
$739$	−19.6276	−0.722012	−0.361006	+	0.932563i	$0.617567\pi$
$740$	0	0
$741$	11.1730	0.410451
$742$	0	0
$743$	−5.88857	−0.216031	−0.108015	−	0.994149i	$-0.534450\pi$
$743$	−5.88857	−0.216031	−0.108015	+	0.994149i	$0.534450\pi$
$744$	0	0
$745$	−1.66387	−0.0609597
$746$	0	0
$747$	71.8945	2.63048
$748$	0	0
$749$	3.89299	0.142247
$750$	0	0
$751$	14.0324	0.512049	0.256025	−	0.966670i	$-0.417587\pi$
$751$	14.0324	0.512049	0.256025	+	0.966670i	$0.417587\pi$
$752$	0	0
$753$	−69.2899	−2.52506
$754$	0	0
$755$	−1.46745	−0.0534060
$756$	0	0
$757$	−12.2366	−0.444745	−0.222373	−	0.974962i	$-0.571380\pi$
$757$	−12.2366	−0.444745	−0.222373	+	0.974962i	$0.571380\pi$
$758$	0	0
$759$	−6.83463	−0.248081
$760$	0	0
$761$	−17.7089	−0.641949	−0.320974	−	0.947088i	$-0.604010\pi$
$761$	−17.7089	−0.641949	−0.320974	+	0.947088i	$0.604010\pi$
$762$	0	0
$763$	16.3140	0.590608
$764$	0	0
$765$	13.9442	0.504155
$766$	0	0
$767$	−1.43121	−0.0516780
$768$	0	0
$769$	34.6116	1.24813	0.624063	−	0.781374i	$-0.285481\pi$
$769$	34.6116	1.24813	0.624063	+	0.781374i	$0.285481\pi$
$770$	0	0
$771$	−14.9652	−0.538960
$772$	0	0
$773$	28.1810	1.01360	0.506800	−	0.862063i	$-0.330828\pi$
$773$	28.1810	1.01360	0.506800	+	0.862063i	$0.330828\pi$
$774$	0	0
$775$	7.19289	0.258376
$776$	0	0
$777$	3.57851	0.128378
$778$	0	0
$779$	18.7439	0.671569
$780$	0	0
$781$	3.69877	0.132353
$782$	0	0
$783$	23.8307	0.851641
$784$	0	0
$785$	23.7196	0.846587
$786$	0	0
$787$	21.4594	0.764946	0.382473	−	0.923967i	$-0.375072\pi$
$787$	21.4594	0.764946	0.382473	+	0.923967i	$0.375072\pi$
$788$	0	0
$789$	−6.71514	−0.239065
$790$	0	0
$791$	−8.82752	−0.313870
$792$	0	0
$793$	3.32170	0.117957
$794$	0	0
$795$	−32.8857	−1.16634
$796$	0	0
$797$	15.4647	0.547788	0.273894	−	0.961760i	$-0.411688\pi$
$797$	15.4647	0.547788	0.273894	+	0.961760i	$0.411688\pi$
$798$	0	0
$799$	−12.3003	−0.435154
$800$	0	0
$801$	59.2259	2.09264
$802$	0	0
$803$	−6.27913	−0.221586
$804$	0	0
$805$	5.17109	0.182257
$806$	0	0
$807$	−14.7037	−0.517593
$808$	0	0
$809$	−36.2656	−1.27503	−0.637516	−	0.770437i	$-0.720038\pi$
$809$	−36.2656	−1.27503	−0.637516	+	0.770437i	$0.720038\pi$
$810$	0	0
$811$	52.1033	1.82960	0.914798	−	0.403912i	$-0.132350\pi$
$811$	52.1033	1.82960	0.914798	+	0.403912i	$0.132350\pi$
$812$	0	0
$813$	−81.3296	−2.85235
$814$	0	0
$815$	6.60716	0.231439
$816$	0	0
$817$	35.5935	1.24526
$818$	0	0
$819$	−4.57851	−0.159986
$820$	0	0
$821$	10.7092	0.373754	0.186877	−	0.982383i	$-0.440163\pi$
$821$	10.7092	0.373754	0.186877	+	0.982383i	$0.440163\pi$
$822$	0	0
$823$	−32.2825	−1.12530	−0.562648	−	0.826697i	$-0.690217\pi$
$823$	−32.2825	−1.12530	−0.562648	+	0.826697i	$0.690217\pi$
$824$	0	0
$825$	1.32170	0.0460157
$826$	0	0
$827$	−10.6743	−0.371183	−0.185591	−	0.982627i	$-0.559420\pi$
$827$	−10.6743	−0.371183	−0.185591	+	0.982627i	$0.559420\pi$
$828$	0	0
$829$	47.6105	1.65358	0.826791	−	0.562509i	$-0.190164\pi$
$829$	47.6105	1.65358	0.826791	+	0.562509i	$0.190164\pi$
$830$	0	0
$831$	0.642694	0.0222948
$832$	0	0
$833$	−3.04559	−0.105523
$834$	0	0
$835$	4.33436	0.149997
$836$	0	0
$837$	−31.2566	−1.08039
$838$	0	0
$839$	13.6762	0.472154	0.236077	−	0.971734i	$-0.424138\pi$
$839$	13.6762	0.472154	0.236077	+	0.971734i	$0.424138\pi$
$840$	0	0
$841$	1.07446	0.0370502
$842$	0	0
$843$	49.3654	1.70024
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	−10.7695	−0.370044
$848$	0	0
$849$	70.5850	2.42247
$850$	0	0
$851$	6.72190	0.230424
$852$	0	0
$853$	−36.8199	−1.26069	−0.630344	−	0.776316i	$-0.717086\pi$
$853$	−36.8199	−1.26069	−0.630344	+	0.776316i	$0.717086\pi$
$854$	0	0
$855$	−18.5824	−0.635505
$856$	0	0
$857$	−32.3429	−1.10481	−0.552406	−	0.833575i	$-0.686290\pi$
$857$	−32.3429	−1.10481	−0.552406	+	0.833575i	$0.686290\pi$
$858$	0	0
$859$	20.6611	0.704947	0.352473	−	0.935822i	$-0.385341\pi$
$859$	20.6611	0.704947	0.352473	+	0.935822i	$0.385341\pi$
$860$	0	0
$861$	−12.7137	−0.433283
$862$	0	0
$863$	31.9507	1.08761	0.543807	−	0.839210i	$-0.316982\pi$
$863$	31.9507	1.08761	0.543807	+	0.839210i	$0.316982\pi$
$864$	0	0
$865$	18.4696	0.627985
$866$	0	0
$867$	21.2646	0.722182
$868$	0	0
$869$	−7.58640	−0.257351
$870$	0	0
$871$	12.6443	0.428437
$872$	0	0
$873$	5.12004	0.173287
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	11.5911	0.391402	0.195701	−	0.980664i	$-0.437302\pi$
$877$	11.5911	0.391402	0.195701	+	0.980664i	$0.437302\pi$
$878$	0	0
$879$	19.9293	0.672199
$880$	0	0
$881$	−12.7019	−0.427937	−0.213969	−	0.976841i	$-0.568639\pi$
$881$	−12.7019	−0.427937	−0.213969	+	0.976841i	$0.568639\pi$
$882$	0	0
$883$	29.0479	0.977540	0.488770	−	0.872413i	$-0.337446\pi$
$883$	29.0479	0.977540	0.488770	+	0.872413i	$0.337446\pi$
$884$	0	0
$885$	3.93999	0.132441
$886$	0	0
$887$	−13.7994	−0.463340	−0.231670	−	0.972794i	$-0.574419\pi$
$887$	−13.7994	−0.463340	−0.231670	+	0.972794i	$0.574419\pi$
$888$	0	0
$889$	−19.7863	−0.663613
$890$	0	0
$891$	0.851135	0.0285141
$892$	0	0
$893$	16.3917	0.548527
$894$	0	0
$895$	13.9777	0.467222
$896$	0	0
$897$	−14.2355	−0.475311
$898$	0	0
$899$	−39.4459	−1.31560
$900$	0	0
$901$	36.3820	1.21206
$902$	0	0
$903$	−24.1426	−0.803415
$904$	0	0
$905$	12.7219	0.422890
$906$	0	0
$907$	−2.52477	−0.0838336	−0.0419168	−	0.999121i	$-0.513346\pi$
$907$	−2.52477	−0.0838336	−0.0419168	+	0.999121i	$0.513346\pi$
$908$	0	0
$909$	−79.3965	−2.63342
$910$	0	0
$911$	−18.5669	−0.615148	−0.307574	−	0.951524i	$-0.599517\pi$
$911$	−18.5669	−0.615148	−0.307574	+	0.951524i	$0.599517\pi$
$912$	0	0
$913$	−7.53899	−0.249504
$914$	0	0
$915$	−9.14434	−0.302303
$916$	0	0
$917$	1.54558	0.0510395
$918$	0	0
$919$	−2.12755	−0.0701813	−0.0350907	−	0.999384i	$-0.511172\pi$
$919$	−2.12755	−0.0701813	−0.0350907	+	0.999384i	$0.511172\pi$
$920$	0	0
$921$	5.63679	0.185739
$922$	0	0
$923$	7.70401	0.253580
$924$	0	0
$925$	−1.29990	−0.0427405
$926$	0	0
$927$	25.8053	0.847557
$928$	0	0
$929$	36.9945	1.21375	0.606875	−	0.794797i	$-0.292423\pi$
$929$	36.9945	1.21375	0.606875	+	0.794797i	$0.292423\pi$
$930$	0	0
$931$	4.05862	0.133016
$932$	0	0
$933$	10.3905	0.340168
$934$	0	0
$935$	−1.46222	−0.0478197
$936$	0	0
$937$	15.6718	0.511974	0.255987	−	0.966680i	$-0.417600\pi$
$937$	15.6718	0.511974	0.255987	+	0.966680i	$0.417600\pi$
$938$	0	0
$939$	37.5311	1.22478
$940$	0	0
$941$	−41.6448	−1.35758	−0.678791	−	0.734332i	$-0.737496\pi$
$941$	−41.6448	−1.35758	−0.678791	+	0.734332i	$0.737496\pi$
$942$	0	0
$943$	−23.8816	−0.777691
$944$	0	0
$945$	4.34549	0.141359
$946$	0	0
$947$	−18.8055	−0.611097	−0.305548	−	0.952177i	$-0.598840\pi$
$947$	−18.8055	−0.611097	−0.305548	+	0.952177i	$0.598840\pi$
$948$	0	0
$949$	−13.0785	−0.424547
$950$	0	0
$951$	19.7877	0.641661
$952$	0	0
$953$	18.0320	0.584113	0.292057	−	0.956401i	$-0.405660\pi$
$953$	18.0320	0.584113	0.292057	+	0.956401i	$0.405660\pi$
$954$	0	0
$955$	−4.29886	−0.139108
$956$	0	0
$957$	−7.24823	−0.234302
$958$	0	0
$959$	−11.1672	−0.360608
$960$	0	0
$961$	20.7376	0.668956
$962$	0	0
$963$	17.8241	0.574373
$964$	0	0
$965$	−6.05722	−0.194989
$966$	0	0
$967$	−21.7343	−0.698928	−0.349464	−	0.936950i	$-0.613636\pi$
$967$	−21.7343	−0.698928	−0.349464	+	0.936950i	$0.613636\pi$
$968$	0	0
$969$	34.0284	1.09315
$970$	0	0
$971$	14.4643	0.464181	0.232090	−	0.972694i	$-0.425444\pi$
$971$	14.4643	0.464181	0.232090	+	0.972694i	$0.425444\pi$
$972$	0	0
$973$	−12.6192	−0.404554
$974$	0	0
$975$	2.75291	0.0881636
$976$	0	0
$977$	−12.4145	−0.397177	−0.198588	−	0.980083i	$-0.563636\pi$
$977$	−12.4145	−0.397177	−0.198588	+	0.980083i	$0.563636\pi$
$978$	0	0
$979$	−6.21053	−0.198489
$980$	0	0
$981$	74.6939	2.38479
$982$	0	0
$983$	17.0057	0.542398	0.271199	−	0.962523i	$-0.412580\pi$
$983$	17.0057	0.542398	0.271199	+	0.962523i	$0.412580\pi$
$984$	0	0
$985$	7.66032	0.244078
$986$	0	0
$987$	−11.1183	−0.353899
$988$	0	0
$989$	−45.3496	−1.44203
$990$	0	0
$991$	−15.3159	−0.486525	−0.243263	−	0.969960i	$-0.578218\pi$
$991$	−15.3159	−0.486525	−0.243263	+	0.969960i	$0.578218\pi$
$992$	0	0
$993$	−28.1996	−0.894886
$994$	0	0
$995$	4.91383	0.155779
$996$	0	0
$997$	25.0386	0.792982	0.396491	−	0.918039i	$-0.370228\pi$
$997$	25.0386	0.792982	0.396491	+	0.918039i	$0.370228\pi$
$998$	0	0
$999$	5.64870	0.178717

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3640.2.a.y.1.1	✓	5
4.3	odd	2		7280.2.a.cc.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3640.2.a.y.1.1	✓	5	1.1	even	1	trivial
7280.2.a.cc.1.5		5	4.3	odd	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:	\( N \)	\(=\)	\( 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3640.a (trivial)

\(f(q)\)	\(=\)	\(q-2.75291 q^{3} -1.00000 q^{5} +1.00000 q^{7} +4.57851 q^{9} +O(q^{10})\)	\(q-2.75291 q^{3} -1.00000 q^{5} +1.00000 q^{7} +4.57851 q^{9} -0.480110 q^{11} -1.00000 q^{13} +2.75291 q^{15} -3.04559 q^{17} +4.05862 q^{19} -2.75291 q^{21} -5.17109 q^{23} +1.00000 q^{25} -4.34549 q^{27} -5.48402 q^{29} +7.19289 q^{31} +1.32170 q^{33} -1.00000 q^{35} -1.29990 q^{37} +2.75291 q^{39} +4.61829 q^{41} +8.76985 q^{43} -4.57851 q^{45} +4.03874 q^{47} +1.00000 q^{49} +8.38423 q^{51} -11.9458 q^{53} +0.480110 q^{55} -11.1730 q^{57} +1.43121 q^{59} -3.32170 q^{61} +4.57851 q^{63} +1.00000 q^{65} -12.6443 q^{67} +14.2355 q^{69} -7.70401 q^{71} +13.0785 q^{73} -2.75291 q^{75} -0.480110 q^{77} +15.8014 q^{79} -1.77279 q^{81} +15.7026 q^{83} +3.04559 q^{85} +15.0970 q^{87} +12.9356 q^{89} -1.00000 q^{91} -19.8014 q^{93} -4.05862 q^{95} +1.11828 q^{97} -2.19819 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{5} + 5 q^{7} + q^{9}+O(q^{10}) \) 5 * q - 5 * q^5 + 5 * q^7 + q^9	\( 5 q - 5 q^{5} + 5 q^{7} + q^{9} - 7 q^{11} - 5 q^{13} + q^{17} + 3 q^{19} - 5 q^{23} + 5 q^{25} - 9 q^{27} - 9 q^{29} + 6 q^{31} + q^{33} - 5 q^{35} - 10 q^{37} - 8 q^{41} + 6 q^{43} - q^{45} - 13 q^{47} + 5 q^{49} - 4 q^{51} - 16 q^{53} + 7 q^{55} - 10 q^{57} - q^{59} - 11 q^{61} + q^{63} + 5 q^{65} - 30 q^{67} - 15 q^{69} - 12 q^{71} + 23 q^{73} - 7 q^{77} + 2 q^{79} - 11 q^{81} - 2 q^{83} - q^{85} - 5 q^{87} - 25 q^{89} - 5 q^{91} - 22 q^{93} - 3 q^{95} - 5 q^{97} - 12 q^{99}+O(q^{100}) \) 5 * q - 5 * q^5 + 5 * q^7 + q^9 - 7 * q^11 - 5 * q^13 + q^17 + 3 * q^19 - 5 * q^23 + 5 * q^25 - 9 * q^27 - 9 * q^29 + 6 * q^31 + q^33 - 5 * q^35 - 10 * q^37 - 8 * q^41 + 6 * q^43 - q^45 - 13 * q^47 + 5 * q^49 - 4 * q^51 - 16 * q^53 + 7 * q^55 - 10 * q^57 - q^59 - 11 * q^61 + q^63 + 5 * q^65 - 30 * q^67 - 15 * q^69 - 12 * q^71 + 23 * q^73 - 7 * q^77 + 2 * q^79 - 11 * q^81 - 2 * q^83 - q^85 - 5 * q^87 - 25 * q^89 - 5 * q^91 - 22 * q^93 - 3 * q^95 - 5 * q^97 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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