LMFDB - Embedded newform 3640.2.a.s.1.2

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:	$4$
Coefficient field:	4.4.2225.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} + 2x + 4 $ x^4 - x^3 - 5x^2 + 2x + 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			1.2
Root			$1.13856$ 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)

$f(q)$	$=$	$q-1.13856 q^{3} +1.00000 q^{5} -1.00000 q^{7} -1.70367 q^{9} +O(q^{10})$	$q-1.13856 q^{3} +1.00000 q^{5} -1.00000 q^{7} -1.70367 q^{9} -3.23607 q^{11} +1.00000 q^{13} -1.13856 q^{15} +2.98080 q^{17} +1.93974 q^{19} +1.13856 q^{21} -2.17127 q^{23} +1.00000 q^{25} +5.35543 q^{27} +3.58697 q^{29} +7.29517 q^{31} +3.68447 q^{33} -1.00000 q^{35} +2.37463 q^{37} -1.13856 q^{39} +1.76846 q^{41} -5.51320 q^{43} -1.70367 q^{45} -7.53693 q^{47} +1.00000 q^{49} -3.39383 q^{51} -3.40734 q^{53} -3.23607 q^{55} -2.20852 q^{57} -11.6377 q^{59} -5.68447 q^{61} +1.70367 q^{63} +1.00000 q^{65} +2.04559 q^{67} +2.47214 q^{69} -10.9205 q^{71} -7.09181 q^{73} -1.13856 q^{75} +3.23607 q^{77} -14.7949 q^{79} -0.986489 q^{81} +2.61968 q^{83} +2.98080 q^{85} -4.08399 q^{87} +3.92952 q^{89} -1.00000 q^{91} -8.30602 q^{93} +1.93974 q^{95} -1.34192 q^{97} +5.51320 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{3} + 4 q^{5} - 4 q^{7} - q^{9}+O(q^{10}) $ 4 * q - q^3 + 4 * q^5 - 4 * q^7 - q^9	$ 4 q - q^{3} + 4 q^{5} - 4 q^{7} - q^{9} - 4 q^{11} + 4 q^{13} - q^{15} - q^{17} - 7 q^{19} + q^{21} - 6 q^{23} + 4 q^{25} - 4 q^{27} + q^{29} - 11 q^{31} - 4 q^{33} - 4 q^{35} - 3 q^{37} - q^{39} - 5 q^{41} - 6 q^{43} - q^{45} - 6 q^{47} + 4 q^{49} - 14 q^{51} - 2 q^{53} - 4 q^{55} + 14 q^{57} - q^{59} - 4 q^{61} + q^{63} + 4 q^{65} - 11 q^{67} - 8 q^{69} - 16 q^{71} + 2 q^{73} - q^{75} + 4 q^{77} - 15 q^{79} - 16 q^{81} - 2 q^{83} - q^{85} - 23 q^{87} - 3 q^{89} - 4 q^{91} - 5 q^{93} - 7 q^{95} + 8 q^{97} + 6 q^{99}+O(q^{100}) $ 4 * q - q^3 + 4 * q^5 - 4 * q^7 - q^9 - 4 * q^11 + 4 * q^13 - q^15 - q^17 - 7 * q^19 + q^21 - 6 * q^23 + 4 * q^25 - 4 * q^27 + q^29 - 11 * q^31 - 4 * q^33 - 4 * q^35 - 3 * q^37 - q^39 - 5 * q^41 - 6 * q^43 - q^45 - 6 * q^47 + 4 * q^49 - 14 * q^51 - 2 * q^53 - 4 * q^55 + 14 * q^57 - q^59 - 4 * q^61 + q^63 + 4 * q^65 - 11 * q^67 - 8 * q^69 - 16 * q^71 + 2 * q^73 - q^75 + 4 * q^77 - 15 * q^79 - 16 * q^81 - 2 * q^83 - q^85 - 23 * q^87 - 3 * q^89 - 4 * q^91 - 5 * q^93 - 7 * q^95 + 8 * q^97 + 6 * 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
$3$	−1.13856	−0.657350	−0.328675	−	0.944443i	$-0.606602\pi$
$3$	−1.13856	−0.657350	−0.328675	+	0.944443i	$0.606602\pi$
$4$	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
$9$	−1.70367	−0.567890
$10$	0	0
$11$	−3.23607	−0.975711	−0.487856	−	0.872924i	$-0.662221\pi$
$11$	−3.23607	−0.975711	−0.487856	+	0.872924i	$0.662221\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−1.13856	−0.293976
$16$	0	0
$17$	2.98080	0.722950	0.361475	−	0.932382i	$-0.382273\pi$
$17$	2.98080	0.722950	0.361475	+	0.932382i	$0.382273\pi$
$18$	0	0
$19$	1.93974	0.445007	0.222503	−	0.974932i	$-0.428577\pi$
$19$	1.93974	0.445007	0.222503	+	0.974932i	$0.428577\pi$
$20$	0	0
$21$	1.13856	0.248455
$22$	0	0
$23$	−2.17127	−0.452742	−0.226371	−	0.974041i	$-0.572686\pi$
$23$	−2.17127	−0.452742	−0.226371	+	0.974041i	$0.572686\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.35543	1.03065
$28$	0	0
$29$	3.58697	0.666083	0.333042	−	0.942912i	$-0.391925\pi$
$29$	3.58697	0.666083	0.333042	+	0.942912i	$0.391925\pi$
$30$	0	0
$31$	7.29517	1.31025	0.655126	−	0.755520i	$-0.272616\pi$
$31$	7.29517	1.31025	0.655126	+	0.755520i	$0.272616\pi$
$32$	0	0
$33$	3.68447	0.641384
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	2.37463	0.390387	0.195194	−	0.980765i	$-0.437466\pi$
$37$	2.37463	0.390387	0.195194	+	0.980765i	$0.437466\pi$
$38$	0	0
$39$	−1.13856	−0.182316
$40$	0	0
$41$	1.76846	0.276188	0.138094	−	0.990419i	$-0.455902\pi$
$41$	1.76846	0.276188	0.138094	+	0.990419i	$0.455902\pi$
$42$	0	0
$43$	−5.51320	−0.840755	−0.420377	−	0.907349i	$-0.638102\pi$
$43$	−5.51320	−0.840755	−0.420377	+	0.907349i	$0.638102\pi$
$44$	0	0
$45$	−1.70367	−0.253968
$46$	0	0
$47$	−7.53693	−1.09937	−0.549687	−	0.835371i	$-0.685253\pi$
$47$	−7.53693	−1.09937	−0.549687	+	0.835371i	$0.685253\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−3.39383	−0.475232
$52$	0	0
$53$	−3.40734	−0.468035	−0.234017	−	0.972232i	$-0.575187\pi$
$53$	−3.40734	−0.468035	−0.234017	+	0.972232i	$0.575187\pi$
$54$	0	0
$55$	−3.23607	−0.436351
$56$	0	0
$57$	−2.20852	−0.292525
$58$	0	0
$59$	−11.6377	−1.51510	−0.757551	−	0.652776i	$-0.773604\pi$
$59$	−11.6377	−1.51510	−0.757551	+	0.652776i	$0.773604\pi$
$60$	0	0
$61$	−5.68447	−0.727822	−0.363911	−	0.931434i	$-0.618559\pi$
$61$	−5.68447	−0.727822	−0.363911	+	0.931434i	$0.618559\pi$
$62$	0	0
$63$	1.70367	0.214642
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	2.04559	0.249909	0.124954	−	0.992162i	$-0.460122\pi$
$67$	2.04559	0.249909	0.124954	+	0.992162i	$0.460122\pi$
$68$	0	0
$69$	2.47214	0.297610
$70$	0	0
$71$	−10.9205	−1.29603	−0.648015	−	0.761628i	$-0.724401\pi$
$71$	−10.9205	−1.29603	−0.648015	+	0.761628i	$0.724401\pi$
$72$	0	0
$73$	−7.09181	−0.830034	−0.415017	−	0.909814i	$-0.636224\pi$
$73$	−7.09181	−0.830034	−0.415017	+	0.909814i	$0.636224\pi$
$74$	0	0
$75$	−1.13856	−0.131470
$76$	0	0
$77$	3.23607	0.368784
$78$	0	0
$79$	−14.7949	−1.66455	−0.832276	−	0.554362i	$-0.812962\pi$
$79$	−14.7949	−1.66455	−0.832276	+	0.554362i	$0.812962\pi$
$80$	0	0
$81$	−0.986489	−0.109610
$82$	0	0
$83$	2.61968	0.287547	0.143774	−	0.989611i	$-0.454076\pi$
$83$	2.61968	0.287547	0.143774	+	0.989611i	$0.454076\pi$
$84$	0	0
$85$	2.98080	0.323313
$86$	0	0
$87$	−4.08399	−0.437850
$88$	0	0
$89$	3.92952	0.416528	0.208264	−	0.978073i	$-0.433219\pi$
$89$	3.92952	0.416528	0.208264	+	0.978073i	$0.433219\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−8.30602	−0.861294
$94$	0	0
$95$	1.93974	0.199013
$96$	0	0
$97$	−1.34192	−0.136252	−0.0681258	−	0.997677i	$-0.521702\pi$
$97$	−1.34192	−0.136252	−0.0681258	+	0.997677i	$0.521702\pi$
$98$	0	0
$99$	5.51320	0.554097
$100$	0	0
$101$	7.49853	0.746132	0.373066	−	0.927805i	$-0.378307\pi$
$101$	7.49853	0.746132	0.373066	+	0.927805i	$0.378307\pi$
$102$	0	0
$103$	14.5685	1.43548	0.717738	−	0.696314i	$-0.245178\pi$
$103$	14.5685	1.43548	0.717738	+	0.696314i	$0.245178\pi$
$104$	0	0
$105$	1.13856	0.111112
$106$	0	0
$107$	−11.7736	−1.13820	−0.569100	−	0.822269i	$-0.692708\pi$
$107$	−11.7736	−1.13820	−0.569100	+	0.822269i	$0.692708\pi$
$108$	0	0
$109$	0.787665	0.0754446	0.0377223	−	0.999288i	$-0.487990\pi$
$109$	0.787665	0.0754446	0.0377223	+	0.999288i	$0.487990\pi$
$110$	0	0
$111$	−2.70367	−0.256621
$112$	0	0
$113$	−7.96160	−0.748964	−0.374482	−	0.927234i	$-0.622180\pi$
$113$	−7.96160	−0.748964	−0.374482	+	0.927234i	$0.622180\pi$
$114$	0	0
$115$	−2.17127	−0.202472
$116$	0	0
$117$	−1.70367	−0.157504
$118$	0	0
$119$	−2.98080	−0.273249
$120$	0	0
$121$	−0.527864	−0.0479876
$122$	0	0
$123$	−2.01351	−0.181552
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−3.23607	−0.287155	−0.143577	−	0.989639i	$-0.545861\pi$
$127$	−3.23607	−0.287155	−0.143577	+	0.989639i	$0.545861\pi$
$128$	0	0
$129$	6.27713	0.552670
$130$	0	0
$131$	−6.42467	−0.561326	−0.280663	−	0.959806i	$-0.590554\pi$
$131$	−6.42467	−0.561326	−0.280663	+	0.959806i	$0.590554\pi$
$132$	0	0
$133$	−1.93974	−0.168197
$134$	0	0
$135$	5.35543	0.460922
$136$	0	0
$137$	1.83708	0.156952	0.0784760	−	0.996916i	$-0.474995\pi$
$137$	1.83708	0.156952	0.0784760	+	0.996916i	$0.474995\pi$
$138$	0	0
$139$	8.62874	0.731880	0.365940	−	0.930638i	$-0.380747\pi$
$139$	8.62874	0.731880	0.365940	+	0.930638i	$0.380747\pi$
$140$	0	0
$141$	8.58128	0.722674
$142$	0	0
$143$	−3.23607	−0.270614
$144$	0	0
$145$	3.58697	0.297881
$146$	0	0
$147$	−1.13856	−0.0939072
$148$	0	0
$149$	−18.8405	−1.54347	−0.771735	−	0.635944i	$-0.780611\pi$
$149$	−18.8405	−1.54347	−0.771735	+	0.635944i	$0.780611\pi$
$150$	0	0
$151$	−12.6704	−1.03111	−0.515553	−	0.856858i	$-0.672413\pi$
$151$	−12.6704	−1.03111	−0.515553	+	0.856858i	$0.672413\pi$
$152$	0	0
$153$	−5.07830	−0.410557
$154$	0	0
$155$	7.29517	0.585962
$156$	0	0
$157$	2.32717	0.185728	0.0928641	−	0.995679i	$-0.470398\pi$
$157$	2.32717	0.185728	0.0928641	+	0.995679i	$0.470398\pi$
$158$	0	0
$159$	3.87948	0.307663
$160$	0	0
$161$	2.17127	0.171120
$162$	0	0
$163$	−11.1623	−0.874299	−0.437149	−	0.899389i	$-0.644012\pi$
$163$	−11.1623	−0.874299	−0.437149	+	0.899389i	$0.644012\pi$
$164$	0	0
$165$	3.68447	0.286836
$166$	0	0
$167$	−8.58128	−0.664039	−0.332020	−	0.943272i	$-0.607730\pi$
$167$	−8.58128	−0.664039	−0.332020	+	0.943272i	$0.607730\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−3.30468	−0.252715
$172$	0	0
$173$	−13.4010	−1.01886	−0.509431	−	0.860512i	$-0.670144\pi$
$173$	−13.4010	−1.01886	−0.509431	+	0.860512i	$0.670144\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	13.2503	0.995953
$178$	0	0
$179$	8.45356	0.631849	0.315925	−	0.948784i	$-0.397685\pi$
$179$	8.45356	0.631849	0.315925	+	0.948784i	$0.397685\pi$
$180$	0	0
$181$	−21.6989	−1.61286	−0.806432	−	0.591327i	$-0.798604\pi$
$181$	−21.6989	−1.61286	−0.806432	+	0.591327i	$0.798604\pi$
$182$	0	0
$183$	6.47214	0.478434
$184$	0	0
$185$	2.37463	0.174586
$186$	0	0
$187$	−9.64607	−0.705391
$188$	0	0
$189$	−5.35543	−0.389550
$190$	0	0
$191$	−14.0975	−1.02006	−0.510030	−	0.860157i	$-0.670366\pi$
$191$	−14.0975	−1.02006	−0.510030	+	0.860157i	$0.670366\pi$
$192$	0	0
$193$	14.9077	1.07308	0.536541	−	0.843874i	$-0.319731\pi$
$193$	14.9077	1.07308	0.536541	+	0.843874i	$0.319731\pi$
$194$	0	0
$195$	−1.13856	−0.0815343
$196$	0	0
$197$	18.1503	1.29315	0.646577	−	0.762848i	$-0.276200\pi$
$197$	18.1503	1.29315	0.646577	+	0.762848i	$0.276200\pi$
$198$	0	0
$199$	−26.6070	−1.88612	−0.943062	−	0.332618i	$-0.892068\pi$
$199$	−26.6070	−1.88612	−0.943062	+	0.332618i	$0.892068\pi$
$200$	0	0
$201$	−2.32904	−0.164278
$202$	0	0
$203$	−3.58697	−0.251756
$204$	0	0
$205$	1.76846	0.123515
$206$	0	0
$207$	3.69914	0.257108
$208$	0	0
$209$	−6.27713	−0.434198
$210$	0	0
$211$	6.28432	0.432631	0.216315	−	0.976324i	$-0.430596\pi$
$211$	6.28432	0.432631	0.216315	+	0.976324i	$0.430596\pi$
$212$	0	0
$213$	12.4337	0.851946
$214$	0	0
$215$	−5.51320	−0.375997
$216$	0	0
$217$	−7.29517	−0.495229
$218$	0	0
$219$	8.07449	0.545623
$220$	0	0
$221$	2.98080	0.200510
$222$	0	0
$223$	17.8885	1.19791	0.598953	−	0.800784i	$-0.295584\pi$
$223$	17.8885	1.19791	0.598953	+	0.800784i	$0.295584\pi$
$224$	0	0
$225$	−1.70367	−0.113578
$226$	0	0
$227$	7.77566	0.516089	0.258044	−	0.966133i	$-0.416922\pi$
$227$	7.77566	0.516089	0.258044	+	0.966133i	$0.416922\pi$
$228$	0	0
$229$	−5.60048	−0.370090	−0.185045	−	0.982730i	$-0.559243\pi$
$229$	−5.60048	−0.370090	−0.185045	+	0.982730i	$0.559243\pi$
$230$	0	0
$231$	−3.68447	−0.242420
$232$	0	0
$233$	6.88011	0.450731	0.225365	−	0.974274i	$-0.427642\pi$
$233$	6.88011	0.450731	0.225365	+	0.974274i	$0.427642\pi$
$234$	0	0
$235$	−7.53693	−0.491655
$236$	0	0
$237$	16.8449	1.09419
$238$	0	0
$239$	21.2722	1.37598	0.687991	−	0.725720i	$-0.258493\pi$
$239$	21.2722	1.37598	0.687991	+	0.725720i	$0.258493\pi$
$240$	0	0
$241$	−7.89805	−0.508758	−0.254379	−	0.967105i	$-0.581871\pi$
$241$	−7.89805	−0.508758	−0.254379	+	0.967105i	$0.581871\pi$
$242$	0	0
$243$	−14.9431	−0.958601
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	1.93974	0.123423
$248$	0	0
$249$	−2.98267	−0.189019
$250$	0	0
$251$	−29.0792	−1.83546	−0.917731	−	0.397203i	$-0.869981\pi$
$251$	−29.0792	−1.83546	−0.917731	+	0.397203i	$0.869981\pi$
$252$	0	0
$253$	7.02639	0.441746
$254$	0	0
$255$	−3.39383	−0.212530
$256$	0	0
$257$	8.12959	0.507110	0.253555	−	0.967321i	$-0.418400\pi$
$257$	8.12959	0.507110	0.253555	+	0.967321i	$0.418400\pi$
$258$	0	0
$259$	−2.37463	−0.147552
$260$	0	0
$261$	−6.11101	−0.378262
$262$	0	0
$263$	0.0416886	0.00257063	0.00128531	−	0.999999i	$-0.499591\pi$
$263$	0.0416886	0.00257063	0.00128531	+	0.999999i	$0.499591\pi$
$264$	0	0
$265$	−3.40734	−0.209311
$266$	0	0
$267$	−4.47401	−0.273805
$268$	0	0
$269$	24.0068	1.46372	0.731859	−	0.681456i	$-0.238653\pi$
$269$	24.0068	1.46372	0.731859	+	0.681456i	$0.238653\pi$
$270$	0	0
$271$	−23.8815	−1.45070	−0.725349	−	0.688381i	$-0.758322\pi$
$271$	−23.8815	−1.45070	−0.725349	+	0.688381i	$0.758322\pi$
$272$	0	0
$273$	1.13856	0.0689090
$274$	0	0
$275$	−3.23607	−0.195142
$276$	0	0
$277$	23.5256	1.41351	0.706757	−	0.707457i	$-0.250158\pi$
$277$	23.5256	1.41351	0.706757	+	0.707457i	$0.250158\pi$
$278$	0	0
$279$	−12.4286	−0.744079
$280$	0	0
$281$	−18.8405	−1.12393	−0.561964	−	0.827162i	$-0.689954\pi$
$281$	−18.8405	−1.12393	−0.561964	+	0.827162i	$0.689954\pi$
$282$	0	0
$283$	−9.11742	−0.541974	−0.270987	−	0.962583i	$-0.587350\pi$
$283$	−9.11742	−0.541974	−0.270987	+	0.962583i	$0.587350\pi$
$284$	0	0
$285$	−2.20852	−0.130821
$286$	0	0
$287$	−1.76846	−0.104389
$288$	0	0
$289$	−8.11483	−0.477343
$290$	0	0
$291$	1.52786	0.0895650
$292$	0	0
$293$	−10.2387	−0.598153	−0.299076	−	0.954229i	$-0.596679\pi$
$293$	−10.2387	−0.598153	−0.299076	+	0.954229i	$0.596679\pi$
$294$	0	0
$295$	−11.6377	−0.677574
$296$	0	0
$297$	−17.3305	−1.00562
$298$	0	0
$299$	−2.17127	−0.125568
$300$	0	0
$301$	5.51320	0.317775
$302$	0	0
$303$	−8.53756	−0.490470
$304$	0	0
$305$	−5.68447	−0.325492
$306$	0	0
$307$	18.9059	1.07902	0.539508	−	0.841981i	$-0.318610\pi$
$307$	18.9059	1.07902	0.539508	+	0.841981i	$0.318610\pi$
$308$	0	0
$309$	−16.5872	−0.943610
$310$	0	0
$311$	−17.7499	−1.00650	−0.503252	−	0.864140i	$-0.667863\pi$
$311$	−17.7499	−1.00650	−0.503252	+	0.864140i	$0.667863\pi$
$312$	0	0
$313$	5.53187	0.312680	0.156340	−	0.987703i	$-0.450031\pi$
$313$	5.53187	0.312680	0.156340	+	0.987703i	$0.450031\pi$
$314$	0	0
$315$	1.70367	0.0959910
$316$	0	0
$317$	−2.13491	−0.119908	−0.0599542	−	0.998201i	$-0.519095\pi$
$317$	−2.13491	−0.119908	−0.0599542	+	0.998201i	$0.519095\pi$
$318$	0	0
$319$	−11.6077	−0.649905
$320$	0	0
$321$	13.4050	0.748196
$322$	0	0
$323$	5.78198	0.321718
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	−0.896807	−0.0495935
$328$	0	0
$329$	7.53693	0.415524
$330$	0	0
$331$	12.8528	0.706454	0.353227	−	0.935538i	$-0.385084\pi$
$331$	12.8528	0.706454	0.353227	+	0.935538i	$0.385084\pi$
$332$	0	0
$333$	−4.04559	−0.221697
$334$	0	0
$335$	2.04559	0.111763
$336$	0	0
$337$	20.8147	1.13385	0.566924	−	0.823770i	$-0.308133\pi$
$337$	20.8147	1.13385	0.566924	+	0.823770i	$0.308133\pi$
$338$	0	0
$339$	9.06479	0.492332
$340$	0	0
$341$	−23.6077	−1.27843
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	2.47214	0.133095
$346$	0	0
$347$	17.2889	0.928114	0.464057	−	0.885805i	$-0.346393\pi$
$347$	17.2889	0.928114	0.464057	+	0.885805i	$0.346393\pi$
$348$	0	0
$349$	−28.9920	−1.55191	−0.775953	−	0.630791i	$-0.782731\pi$
$349$	−28.9920	−1.55191	−0.775953	+	0.630791i	$0.782731\pi$
$350$	0	0
$351$	5.35543	0.285852
$352$	0	0
$353$	−30.9149	−1.64544	−0.822718	−	0.568450i	$-0.807543\pi$
$353$	−30.9149	−1.64544	−0.822718	+	0.568450i	$0.807543\pi$
$354$	0	0
$355$	−10.9205	−0.579602
$356$	0	0
$357$	3.39383	0.179621
$358$	0	0
$359$	−25.6584	−1.35420	−0.677100	−	0.735891i	$-0.736764\pi$
$359$	−25.6584	−1.35420	−0.677100	+	0.735891i	$0.736764\pi$
$360$	0	0
$361$	−15.2374	−0.801969
$362$	0	0
$363$	0.601007	0.0315447
$364$	0	0
$365$	−7.09181	−0.371203
$366$	0	0
$367$	−12.4317	−0.648930	−0.324465	−	0.945898i	$-0.605184\pi$
$367$	−12.4317	−0.648930	−0.324465	+	0.945898i	$0.605184\pi$
$368$	0	0
$369$	−3.01288	−0.156844
$370$	0	0
$371$	3.40734	0.176900
$372$	0	0
$373$	−29.1613	−1.50991	−0.754957	−	0.655774i	$-0.772343\pi$
$373$	−29.1613	−1.50991	−0.754957	+	0.655774i	$0.772343\pi$
$374$	0	0
$375$	−1.13856	−0.0587952
$376$	0	0
$377$	3.58697	0.184738
$378$	0	0
$379$	12.4921	0.641677	0.320839	−	0.947134i	$-0.396035\pi$
$379$	12.4921	0.641677	0.320839	+	0.947134i	$0.396035\pi$
$380$	0	0
$381$	3.68447	0.188761
$382$	0	0
$383$	−11.7012	−0.597902	−0.298951	−	0.954268i	$-0.596637\pi$
$383$	−11.7012	−0.597902	−0.298951	+	0.954268i	$0.596637\pi$
$384$	0	0
$385$	3.23607	0.164925
$386$	0	0
$387$	9.39268	0.477457
$388$	0	0
$389$	15.1758	0.769444	0.384722	−	0.923032i	$-0.374297\pi$
$389$	15.1758	0.769444	0.384722	+	0.923032i	$0.374297\pi$
$390$	0	0
$391$	−6.47214	−0.327310
$392$	0	0
$393$	7.31490	0.368988
$394$	0	0
$395$	−14.7949	−0.744410
$396$	0	0
$397$	35.0150	1.75735	0.878677	−	0.477417i	$-0.158427\pi$
$397$	35.0150	1.75735	0.878677	+	0.477417i	$0.158427\pi$
$398$	0	0
$399$	2.20852	0.110564
$400$	0	0
$401$	−17.0059	−0.849237	−0.424618	−	0.905372i	$-0.639592\pi$
$401$	−17.0059	−0.849237	−0.424618	+	0.905372i	$0.639592\pi$
$402$	0	0
$403$	7.29517	0.363398
$404$	0	0
$405$	−0.986489	−0.0490191
$406$	0	0
$407$	−7.68447	−0.380905
$408$	0	0
$409$	4.05279	0.200397	0.100199	−	0.994967i	$-0.468052\pi$
$409$	4.05279	0.200397	0.100199	+	0.994967i	$0.468052\pi$
$410$	0	0
$411$	−2.09163	−0.103172
$412$	0	0
$413$	11.6377	0.572655
$414$	0	0
$415$	2.61968	0.128595
$416$	0	0
$417$	−9.82438	−0.481102
$418$	0	0
$419$	−8.73256	−0.426614	−0.213307	−	0.976985i	$-0.568423\pi$
$419$	−8.73256	−0.426614	−0.213307	+	0.976985i	$0.568423\pi$
$420$	0	0
$421$	18.0000	0.877266	0.438633	−	0.898666i	$-0.355463\pi$
$421$	18.0000	0.877266	0.438633	+	0.898666i	$0.355463\pi$
$422$	0	0
$423$	12.8405	0.624324
$424$	0	0
$425$	2.98080	0.144590
$426$	0	0
$427$	5.68447	0.275091
$428$	0	0
$429$	3.68447	0.177888
$430$	0	0
$431$	3.20530	0.154394	0.0771970	−	0.997016i	$-0.475403\pi$
$431$	3.20530	0.154394	0.0771970	+	0.997016i	$0.475403\pi$
$432$	0	0
$433$	−33.4199	−1.60606	−0.803028	−	0.595941i	$-0.796779\pi$
$433$	−33.4199	−1.60606	−0.803028	+	0.595941i	$0.796779\pi$
$434$	0	0
$435$	−4.08399	−0.195812
$436$	0	0
$437$	−4.21171	−0.201473
$438$	0	0
$439$	23.6551	1.12900	0.564499	−	0.825434i	$-0.309069\pi$
$439$	23.6551	1.12900	0.564499	+	0.825434i	$0.309069\pi$
$440$	0	0
$441$	−1.70367	−0.0811272
$442$	0	0
$443$	6.15864	0.292606	0.146303	−	0.989240i	$-0.453263\pi$
$443$	6.15864	0.292606	0.146303	+	0.989240i	$0.453263\pi$
$444$	0	0
$445$	3.92952	0.186277
$446$	0	0
$447$	21.4511	1.01460
$448$	0	0
$449$	15.4073	0.727117	0.363559	−	0.931571i	$-0.381562\pi$
$449$	15.4073	0.727117	0.363559	+	0.931571i	$0.381562\pi$
$450$	0	0
$451$	−5.72287	−0.269479
$452$	0	0
$453$	14.4261	0.677797
$454$	0	0
$455$	−1.00000	−0.0468807
$456$	0	0
$457$	12.6518	0.591824	0.295912	−	0.955215i	$-0.404376\pi$
$457$	12.6518	0.591824	0.295912	+	0.955215i	$0.404376\pi$
$458$	0	0
$459$	15.9635	0.745111
$460$	0	0
$461$	7.19838	0.335262	0.167631	−	0.985850i	$-0.446388\pi$
$461$	7.19838	0.335262	0.167631	+	0.985850i	$0.446388\pi$
$462$	0	0
$463$	19.2849	0.896248	0.448124	−	0.893972i	$-0.352092\pi$
$463$	19.2849	0.896248	0.448124	+	0.893972i	$0.352092\pi$
$464$	0	0
$465$	−8.30602	−0.385183
$466$	0	0
$467$	2.59407	0.120039	0.0600197	−	0.998197i	$-0.480884\pi$
$467$	2.59407	0.120039	0.0600197	+	0.998197i	$0.480884\pi$
$468$	0	0
$469$	−2.04559	−0.0944567
$470$	0	0
$471$	−2.64963	−0.122088
$472$	0	0
$473$	17.8411	0.820334
$474$	0	0
$475$	1.93974	0.0890013
$476$	0	0
$477$	5.80499	0.265792
$478$	0	0
$479$	−18.6808	−0.853548	−0.426774	−	0.904358i	$-0.640350\pi$
$479$	−18.6808	−0.853548	−0.426774	+	0.904358i	$0.640350\pi$
$480$	0	0
$481$	2.37463	0.108274
$482$	0	0
$483$	−2.47214	−0.112486
$484$	0	0
$485$	−1.34192	−0.0609335
$486$	0	0
$487$	−8.27081	−0.374786	−0.187393	−	0.982285i	$-0.560004\pi$
$487$	−8.27081	−0.374786	−0.187393	+	0.982285i	$0.560004\pi$
$488$	0	0
$489$	12.7090	0.574720
$490$	0	0
$491$	1.87448	0.0845940	0.0422970	−	0.999105i	$-0.486532\pi$
$491$	1.87448	0.0845940	0.0422970	+	0.999105i	$0.486532\pi$
$492$	0	0
$493$	10.6920	0.481545
$494$	0	0
$495$	5.51320	0.247800
$496$	0	0
$497$	10.9205	0.489853
$498$	0	0
$499$	−36.2217	−1.62151	−0.810754	−	0.585387i	$-0.800943\pi$
$499$	−36.2217	−1.62151	−0.810754	+	0.585387i	$0.800943\pi$
$500$	0	0
$501$	9.77034	0.436506
$502$	0	0
$503$	−1.53364	−0.0683817	−0.0341908	−	0.999415i	$-0.510885\pi$
$503$	−1.53364	−0.0683817	−0.0341908	+	0.999415i	$0.510885\pi$
$504$	0	0
$505$	7.49853	0.333680
$506$	0	0
$507$	−1.13856	−0.0505654
$508$	0	0
$509$	8.06229	0.357355	0.178677	−	0.983908i	$-0.442818\pi$
$509$	8.06229	0.357355	0.178677	+	0.983908i	$0.442818\pi$
$510$	0	0
$511$	7.09181	0.313723
$512$	0	0
$513$	10.3881	0.458648
$514$	0	0
$515$	14.5685	0.641964
$516$	0	0
$517$	24.3900	1.07267
$518$	0	0
$519$	15.2579	0.669749
$520$	0	0
$521$	−10.4835	−0.459291	−0.229646	−	0.973274i	$-0.573757\pi$
$521$	−10.4835	−0.459291	−0.229646	+	0.973274i	$0.573757\pi$
$522$	0	0
$523$	−25.2517	−1.10418	−0.552090	−	0.833784i	$-0.686170\pi$
$523$	−25.2517	−1.10418	−0.552090	+	0.833784i	$0.686170\pi$
$524$	0	0
$525$	1.13856	0.0496910
$526$	0	0
$527$	21.7454	0.947247
$528$	0	0
$529$	−18.2856	−0.795025
$530$	0	0
$531$	19.8269	0.860412
$532$	0	0
$533$	1.76846	0.0766007
$534$	0	0
$535$	−11.7736	−0.509018
$536$	0	0
$537$	−9.62493	−0.415346
$538$	0	0
$539$	−3.23607	−0.139387
$540$	0	0
$541$	−15.3947	−0.661870	−0.330935	−	0.943654i	$-0.607364\pi$
$541$	−15.3947	−0.661870	−0.330935	+	0.943654i	$0.607364\pi$
$542$	0	0
$543$	24.7055	1.06022
$544$	0	0
$545$	0.787665	0.0337398
$546$	0	0
$547$	37.9613	1.62311	0.811554	−	0.584277i	$-0.198622\pi$
$547$	37.9613	1.62311	0.811554	+	0.584277i	$0.198622\pi$
$548$	0	0
$549$	9.68447	0.413323
$550$	0	0
$551$	6.95778	0.296411
$552$	0	0
$553$	14.7949	0.629141
$554$	0	0
$555$	−2.70367	−0.114764
$556$	0	0
$557$	−43.5779	−1.84645	−0.923227	−	0.384254i	$-0.874459\pi$
$557$	−43.5779	−1.84645	−0.923227	+	0.384254i	$0.874459\pi$
$558$	0	0
$559$	−5.51320	−0.233183
$560$	0	0
$561$	10.9827	0.463689
$562$	0	0
$563$	21.6423	0.912116	0.456058	−	0.889950i	$-0.349261\pi$
$563$	21.6423	0.912116	0.456058	+	0.889950i	$0.349261\pi$
$564$	0	0
$565$	−7.96160	−0.334947
$566$	0	0
$567$	0.986489	0.0414287
$568$	0	0
$569$	−2.47845	−0.103902	−0.0519511	−	0.998650i	$-0.516544\pi$
$569$	−2.47845	−0.103902	−0.0519511	+	0.998650i	$0.516544\pi$
$570$	0	0
$571$	−10.0040	−0.418655	−0.209327	−	0.977846i	$-0.567127\pi$
$571$	−10.0040	−0.418655	−0.209327	+	0.977846i	$0.567127\pi$
$572$	0	0
$573$	16.0509	0.670537
$574$	0	0
$575$	−2.17127	−0.0905484
$576$	0	0
$577$	28.2953	1.17795	0.588974	−	0.808152i	$-0.299532\pi$
$577$	28.2953	1.17795	0.588974	+	0.808152i	$0.299532\pi$
$578$	0	0
$579$	−16.9734	−0.705391
$580$	0	0
$581$	−2.61968	−0.108683
$582$	0	0
$583$	11.0264	0.456667
$584$	0	0
$585$	−1.70367	−0.0704381
$586$	0	0
$587$	0.614357	0.0253572	0.0126786	−	0.999920i	$-0.495964\pi$
$587$	0.614357	0.0253572	0.0126786	+	0.999920i	$0.495964\pi$
$588$	0	0
$589$	14.1507	0.583071
$590$	0	0
$591$	−20.6653	−0.850056
$592$	0	0
$593$	−34.6227	−1.42178	−0.710892	−	0.703302i	$-0.751708\pi$
$593$	−34.6227	−1.42178	−0.710892	+	0.703302i	$0.751708\pi$
$594$	0	0
$595$	−2.98080	−0.122201
$596$	0	0
$597$	30.2938	1.23984
$598$	0	0
$599$	38.2311	1.56208	0.781040	−	0.624481i	$-0.214689\pi$
$599$	38.2311	1.56208	0.781040	+	0.624481i	$0.214689\pi$
$600$	0	0
$601$	−36.3390	−1.48230	−0.741149	−	0.671341i	$-0.765719\pi$
$601$	−36.3390	−1.48230	−0.741149	+	0.671341i	$0.765719\pi$
$602$	0	0
$603$	−3.48502	−0.141921
$604$	0	0
$605$	−0.527864	−0.0214607
$606$	0	0
$607$	−2.03537	−0.0826132	−0.0413066	−	0.999147i	$-0.513152\pi$
$607$	−2.03537	−0.0826132	−0.0413066	+	0.999147i	$0.513152\pi$
$608$	0	0
$609$	4.08399	0.165492
$610$	0	0
$611$	−7.53693	−0.304912
$612$	0	0
$613$	29.8464	1.20548	0.602742	−	0.797936i	$-0.294075\pi$
$613$	29.8464	1.20548	0.602742	+	0.797936i	$0.294075\pi$
$614$	0	0
$615$	−2.01351	−0.0811926
$616$	0	0
$617$	2.72163	0.109569	0.0547843	−	0.998498i	$-0.482553\pi$
$617$	2.72163	0.109569	0.0547843	+	0.998498i	$0.482553\pi$
$618$	0	0
$619$	−8.52795	−0.342767	−0.171384	−	0.985204i	$-0.554824\pi$
$619$	−8.52795	−0.342767	−0.171384	+	0.985204i	$0.554824\pi$
$620$	0	0
$621$	−11.6281	−0.466620
$622$	0	0
$623$	−3.92952	−0.157433
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	7.14691	0.285420
$628$	0	0
$629$	7.07830	0.282230
$630$	0	0
$631$	−1.39268	−0.0554415	−0.0277208	−	0.999616i	$-0.508825\pi$
$631$	−1.39268	−0.0554415	−0.0277208	+	0.999616i	$0.508825\pi$
$632$	0	0
$633$	−7.15510	−0.284390
$634$	0	0
$635$	−3.23607	−0.128419
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	18.6050	0.736003
$640$	0	0
$641$	3.19838	0.126329	0.0631643	−	0.998003i	$-0.479881\pi$
$641$	3.19838	0.126329	0.0631643	+	0.998003i	$0.479881\pi$
$642$	0	0
$643$	−25.2845	−0.997124	−0.498562	−	0.866854i	$-0.666138\pi$
$643$	−25.2845	−0.997124	−0.498562	+	0.866854i	$0.666138\pi$
$644$	0	0
$645$	6.27713	0.247162
$646$	0	0
$647$	0.414007	0.0162763	0.00813814	−	0.999967i	$-0.497410\pi$
$647$	0.414007	0.0162763	0.00813814	+	0.999967i	$0.497410\pi$
$648$	0	0
$649$	37.6605	1.47830
$650$	0	0
$651$	8.30602	0.325539
$652$	0	0
$653$	22.4170	0.877246	0.438623	−	0.898671i	$-0.355466\pi$
$653$	22.4170	0.877246	0.438623	+	0.898671i	$0.355466\pi$
$654$	0	0
$655$	−6.42467	−0.251033
$656$	0	0
$657$	12.0821	0.471368
$658$	0	0
$659$	18.8679	0.734990	0.367495	−	0.930026i	$-0.380216\pi$
$659$	18.8679	0.734990	0.367495	+	0.930026i	$0.380216\pi$
$660$	0	0
$661$	14.8398	0.577203	0.288601	−	0.957449i	$-0.406810\pi$
$661$	14.8398	0.577203	0.288601	+	0.957449i	$0.406810\pi$
$662$	0	0
$663$	−3.39383	−0.131806
$664$	0	0
$665$	−1.93974	−0.0752199
$666$	0	0
$667$	−7.78829	−0.301564
$668$	0	0
$669$	−20.3673	−0.787444
$670$	0	0
$671$	18.3953	0.710144
$672$	0	0
$673$	−0.488836	−0.0188432	−0.00942162	−	0.999956i	$-0.502999\pi$
$673$	−0.488836	−0.0188432	−0.00942162	+	0.999956i	$0.502999\pi$
$674$	0	0
$675$	5.35543	0.206131
$676$	0	0
$677$	24.5736	0.944442	0.472221	−	0.881480i	$-0.343452\pi$
$677$	24.5736	0.944442	0.472221	+	0.881480i	$0.343452\pi$
$678$	0	0
$679$	1.34192	0.0514982
$680$	0	0
$681$	−8.85309	−0.339251
$682$	0	0
$683$	19.8392	0.759126	0.379563	−	0.925166i	$-0.376074\pi$
$683$	19.8392	0.759126	0.379563	+	0.925166i	$0.376074\pi$
$684$	0	0
$685$	1.83708	0.0701910
$686$	0	0
$687$	6.37650	0.243279
$688$	0	0
$689$	−3.40734	−0.129809
$690$	0	0
$691$	25.4788	0.969259	0.484630	−	0.874719i	$-0.338954\pi$
$691$	25.4788	0.969259	0.484630	+	0.874719i	$0.338954\pi$
$692$	0	0
$693$	−5.51320	−0.209429
$694$	0	0
$695$	8.62874	0.327307
$696$	0	0
$697$	5.27144	0.199670
$698$	0	0
$699$	−7.83344	−0.296288
$700$	0	0
$701$	27.8981	1.05369	0.526847	−	0.849960i	$-0.323374\pi$
$701$	27.8981	1.05369	0.526847	+	0.849960i	$0.323374\pi$
$702$	0	0
$703$	4.60617	0.173725
$704$	0	0
$705$	8.58128	0.323190
$706$	0	0
$707$	−7.49853	−0.282011
$708$	0	0
$709$	21.7860	0.818190	0.409095	−	0.912492i	$-0.365845\pi$
$709$	21.7860	0.818190	0.409095	+	0.912492i	$0.365845\pi$
$710$	0	0
$711$	25.2056	0.945283
$712$	0	0
$713$	−15.8398	−0.593206
$714$	0	0
$715$	−3.23607	−0.121022
$716$	0	0
$717$	−24.2197	−0.904502
$718$	0	0
$719$	−2.76064	−0.102955	−0.0514773	−	0.998674i	$-0.516393\pi$
$719$	−2.76064	−0.102955	−0.0514773	+	0.998674i	$0.516393\pi$
$720$	0	0
$721$	−14.5685	−0.542559
$722$	0	0
$723$	8.99244	0.334432
$724$	0	0
$725$	3.58697	0.133217
$726$	0	0
$727$	5.00366	0.185575	0.0927877	−	0.995686i	$-0.470422\pi$
$727$	5.00366	0.185575	0.0927877	+	0.995686i	$0.470422\pi$
$728$	0	0
$729$	19.9732	0.739747
$730$	0	0
$731$	−16.4337	−0.607824
$732$	0	0
$733$	−1.26512	−0.0467283	−0.0233642	−	0.999727i	$-0.507438\pi$
$733$	−1.26512	−0.0467283	−0.0233642	+	0.999727i	$0.507438\pi$
$734$	0	0
$735$	−1.13856	−0.0419966
$736$	0	0
$737$	−6.61968	−0.243839
$738$	0	0
$739$	28.1432	1.03526	0.517632	−	0.855603i	$-0.326814\pi$
$739$	28.1432	1.03526	0.517632	+	0.855603i	$0.326814\pi$
$740$	0	0
$741$	−2.20852	−0.0811319
$742$	0	0
$743$	−13.1298	−0.481685	−0.240842	−	0.970564i	$-0.577424\pi$
$743$	−13.1298	−0.481685	−0.240842	+	0.970564i	$0.577424\pi$
$744$	0	0
$745$	−18.8405	−0.690261
$746$	0	0
$747$	−4.46307	−0.163295
$748$	0	0
$749$	11.7736	0.430199
$750$	0	0
$751$	27.3409	0.997685	0.498842	−	0.866693i	$-0.333759\pi$
$751$	27.3409	0.997685	0.498842	+	0.866693i	$0.333759\pi$
$752$	0	0
$753$	33.1085	1.20654
$754$	0	0
$755$	−12.6704	−0.461124
$756$	0	0
$757$	29.5640	1.07452	0.537260	−	0.843417i	$-0.319459\pi$
$757$	29.5640	1.07452	0.537260	+	0.843417i	$0.319459\pi$
$758$	0	0
$759$	−8.00000	−0.290382
$760$	0	0
$761$	−17.3639	−0.629440	−0.314720	−	0.949185i	$-0.601911\pi$
$761$	−17.3639	−0.629440	−0.314720	+	0.949185i	$0.601911\pi$
$762$	0	0
$763$	−0.787665	−0.0285154
$764$	0	0
$765$	−5.07830	−0.183606
$766$	0	0
$767$	−11.6377	−0.420214
$768$	0	0
$769$	10.9443	0.394661	0.197330	−	0.980337i	$-0.436773\pi$
$769$	10.9443	0.394661	0.197330	+	0.980337i	$0.436773\pi$
$770$	0	0
$771$	−9.25606	−0.333349
$772$	0	0
$773$	−38.4003	−1.38116	−0.690582	−	0.723254i	$-0.742646\pi$
$773$	−38.4003	−1.38116	−0.690582	+	0.723254i	$0.742646\pi$
$774$	0	0
$775$	7.29517	0.262050
$776$	0	0
$777$	2.70367	0.0969937
$778$	0	0
$779$	3.43036	0.122905
$780$	0	0
$781$	35.3396	1.26455
$782$	0	0
$783$	19.2098	0.686501
$784$	0	0
$785$	2.32717	0.0830602
$786$	0	0
$787$	22.1836	0.790761	0.395380	−	0.918517i	$-0.370613\pi$
$787$	22.1836	0.790761	0.395380	+	0.918517i	$0.370613\pi$
$788$	0	0
$789$	−0.0474651	−0.00168980
$790$	0	0
$791$	7.96160	0.283082
$792$	0	0
$793$	−5.68447	−0.201861
$794$	0	0
$795$	3.87948	0.137591
$796$	0	0
$797$	21.2368	0.752245	0.376123	−	0.926570i	$-0.377257\pi$
$797$	21.2368	0.752245	0.376123	+	0.926570i	$0.377257\pi$
$798$	0	0
$799$	−22.4661	−0.794793
$800$	0	0
$801$	−6.69461	−0.236542
$802$	0	0
$803$	22.9496	0.809874
$804$	0	0
$805$	2.17127	0.0765274
$806$	0	0
$807$	−27.3332	−0.962175
$808$	0	0
$809$	−20.3677	−0.716090	−0.358045	−	0.933704i	$-0.616557\pi$
$809$	−20.3677	−0.716090	−0.358045	+	0.933704i	$0.616557\pi$
$810$	0	0
$811$	41.3800	1.45305	0.726525	−	0.687140i	$-0.241134\pi$
$811$	41.3800	1.45305	0.726525	+	0.687140i	$0.241134\pi$
$812$	0	0
$813$	27.1906	0.953617
$814$	0	0
$815$	−11.1623	−0.390998
$816$	0	0
$817$	−10.6942	−0.374141
$818$	0	0
$819$	1.70367	0.0595311
$820$	0	0
$821$	−28.8328	−1.00627	−0.503136	−	0.864207i	$-0.667821\pi$
$821$	−28.8328	−1.00627	−0.503136	+	0.864207i	$0.667821\pi$
$822$	0	0
$823$	−11.1527	−0.388758	−0.194379	−	0.980926i	$-0.562269\pi$
$823$	−11.1527	−0.388758	−0.194379	+	0.980926i	$0.562269\pi$
$824$	0	0
$825$	3.68447	0.128277
$826$	0	0
$827$	−27.0330	−0.940028	−0.470014	−	0.882659i	$-0.655751\pi$
$827$	−27.0330	−0.940028	−0.470014	+	0.882659i	$0.655751\pi$
$828$	0	0
$829$	22.2928	0.774260	0.387130	−	0.922025i	$-0.373466\pi$
$829$	22.2928	0.774260	0.387130	+	0.922025i	$0.373466\pi$
$830$	0	0
$831$	−26.7854	−0.929174
$832$	0	0
$833$	2.98080	0.103279
$834$	0	0
$835$	−8.58128	−0.296967
$836$	0	0
$837$	39.0688	1.35042
$838$	0	0
$839$	50.3602	1.73863	0.869314	−	0.494260i	$-0.164561\pi$
$839$	50.3602	1.73863	0.869314	+	0.494260i	$0.164561\pi$
$840$	0	0
$841$	−16.1337	−0.556333
$842$	0	0
$843$	21.4511	0.738814
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	0.527864	0.0181376
$848$	0	0
$849$	10.3808	0.356267
$850$	0	0
$851$	−5.15598	−0.176745
$852$	0	0
$853$	35.8989	1.22915	0.614577	−	0.788857i	$-0.289327\pi$
$853$	35.8989	1.22915	0.614577	+	0.788857i	$0.289327\pi$
$854$	0	0
$855$	−3.30468	−0.113018
$856$	0	0
$857$	−8.48752	−0.289928	−0.144964	−	0.989437i	$-0.546307\pi$
$857$	−8.48752	−0.289928	−0.144964	+	0.989437i	$0.546307\pi$
$858$	0	0
$859$	16.8200	0.573891	0.286946	−	0.957947i	$-0.407360\pi$
$859$	16.8200	0.573891	0.286946	+	0.957947i	$0.407360\pi$
$860$	0	0
$861$	2.01351	0.0686203
$862$	0	0
$863$	−28.5718	−0.972594	−0.486297	−	0.873793i	$-0.661653\pi$
$863$	−28.5718	−0.972594	−0.486297	+	0.873793i	$0.661653\pi$
$864$	0	0
$865$	−13.4010	−0.455649
$866$	0	0
$867$	9.23926	0.313782
$868$	0	0
$869$	47.8772	1.62412
$870$	0	0
$871$	2.04559	0.0693123
$872$	0	0
$873$	2.28619	0.0773759
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	16.7638	0.566072	0.283036	−	0.959109i	$-0.408658\pi$
$877$	16.7638	0.566072	0.283036	+	0.959109i	$0.408658\pi$
$878$	0	0
$879$	11.6575	0.393196
$880$	0	0
$881$	−29.5933	−0.997023	−0.498512	−	0.866883i	$-0.666120\pi$
$881$	−29.5933	−0.997023	−0.498512	+	0.866883i	$0.666120\pi$
$882$	0	0
$883$	3.18203	0.107084	0.0535418	−	0.998566i	$-0.482949\pi$
$883$	3.18203	0.107084	0.0535418	+	0.998566i	$0.482949\pi$
$884$	0	0
$885$	13.2503	0.445404
$886$	0	0
$887$	−4.31856	−0.145003	−0.0725015	−	0.997368i	$-0.523098\pi$
$887$	−4.31856	−0.145003	−0.0725015	+	0.997368i	$0.523098\pi$
$888$	0	0
$889$	3.23607	0.108534
$890$	0	0
$891$	3.19235	0.106948
$892$	0	0
$893$	−14.6197	−0.489229
$894$	0	0
$895$	8.45356	0.282571
$896$	0	0
$897$	2.47214	0.0825422
$898$	0	0
$899$	26.1675	0.872736
$900$	0	0
$901$	−10.1566	−0.338366
$902$	0	0
$903$	−6.27713	−0.208890
$904$	0	0
$905$	−21.6989	−0.721294
$906$	0	0
$907$	−8.46653	−0.281127	−0.140563	−	0.990072i	$-0.544891\pi$
$907$	−8.46653	−0.281127	−0.140563	+	0.990072i	$0.544891\pi$
$908$	0	0
$909$	−12.7750	−0.423721
$910$	0	0
$911$	4.45356	0.147553	0.0737766	−	0.997275i	$-0.476495\pi$
$911$	4.45356	0.147553	0.0737766	+	0.997275i	$0.476495\pi$
$912$	0	0
$913$	−8.47746	−0.280563
$914$	0	0
$915$	6.47214	0.213962
$916$	0	0
$917$	6.42467	0.212161
$918$	0	0
$919$	−0.310467	−0.0102414	−0.00512068	−	0.999987i	$-0.501630\pi$
$919$	−0.310467	−0.0102414	−0.00512068	+	0.999987i	$0.501630\pi$
$920$	0	0
$921$	−21.5256	−0.709291
$922$	0	0
$923$	−10.9205	−0.359454
$924$	0	0
$925$	2.37463	0.0780774
$926$	0	0
$927$	−24.8199	−0.815193
$928$	0	0
$929$	−0.886424	−0.0290826	−0.0145413	−	0.999894i	$-0.504629\pi$
$929$	−0.886424	−0.0290826	−0.0145413	+	0.999894i	$0.504629\pi$
$930$	0	0
$931$	1.93974	0.0635724
$932$	0	0
$933$	20.2094	0.661626
$934$	0	0
$935$	−9.64607	−0.315460
$936$	0	0
$937$	−22.3307	−0.729513	−0.364757	−	0.931103i	$-0.618848\pi$
$937$	−22.3307	−0.729513	−0.364757	+	0.931103i	$0.618848\pi$
$938$	0	0
$939$	−6.29839	−0.205540
$940$	0	0
$941$	−10.0673	−0.328184	−0.164092	−	0.986445i	$-0.552469\pi$
$941$	−10.0673	−0.328184	−0.164092	+	0.986445i	$0.552469\pi$
$942$	0	0
$943$	−3.83982	−0.125042
$944$	0	0
$945$	−5.35543	−0.174212
$946$	0	0
$947$	26.0559	0.846703	0.423352	−	0.905965i	$-0.360853\pi$
$947$	26.0559	0.846703	0.423352	+	0.905965i	$0.360853\pi$
$948$	0	0
$949$	−7.09181	−0.230210
$950$	0	0
$951$	2.43073	0.0788218
$952$	0	0
$953$	11.7206	0.379666	0.189833	−	0.981816i	$-0.439205\pi$
$953$	11.7206	0.379666	0.189833	+	0.981816i	$0.439205\pi$
$954$	0	0
$955$	−14.0975	−0.456185
$956$	0	0
$957$	13.2161	0.427215
$958$	0	0
$959$	−1.83708	−0.0593222
$960$	0	0
$961$	22.2195	0.716759
$962$	0	0
$963$	20.0584	0.646373
$964$	0	0
$965$	14.9077	0.479897
$966$	0	0
$967$	−12.0758	−0.388332	−0.194166	−	0.980969i	$-0.562200\pi$
$967$	−12.0758	−0.388332	−0.194166	+	0.980969i	$0.562200\pi$
$968$	0	0
$969$	−6.58315	−0.211481
$970$	0	0
$971$	−60.2174	−1.93247	−0.966234	−	0.257666i	$-0.917047\pi$
$971$	−60.2174	−1.93247	−0.966234	+	0.257666i	$0.917047\pi$
$972$	0	0
$973$	−8.62874	−0.276625
$974$	0	0
$975$	−1.13856	−0.0364632
$976$	0	0
$977$	−27.1157	−0.867508	−0.433754	−	0.901031i	$-0.642811\pi$
$977$	−27.1157	−0.867508	−0.433754	+	0.901031i	$0.642811\pi$
$978$	0	0
$979$	−12.7162	−0.406411
$980$	0	0
$981$	−1.34192	−0.0428443
$982$	0	0
$983$	−28.4274	−0.906692	−0.453346	−	0.891335i	$-0.649770\pi$
$983$	−28.4274	−0.906692	−0.453346	+	0.891335i	$0.649770\pi$
$984$	0	0
$985$	18.1503	0.578316
$986$	0	0
$987$	−8.58128	−0.273145
$988$	0	0
$989$	11.9707	0.380645
$990$	0	0
$991$	25.3582	0.805529	0.402765	−	0.915304i	$-0.368049\pi$
$991$	25.3582	0.805529	0.402765	+	0.915304i	$0.368049\pi$
$992$	0	0
$993$	−14.6337	−0.464388
$994$	0	0
$995$	−26.6070	−0.843500
$996$	0	0
$997$	−40.7554	−1.29074	−0.645368	−	0.763872i	$-0.723296\pi$
$997$	−40.7554	−1.29074	−0.645368	+	0.763872i	$0.723296\pi$
$998$	0	0
$999$	12.7172	0.402354

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3640.2.a.s.1.2	✓	4
4.3	odd	2		7280.2.a.ca.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3640.2.a.s.1.2	✓	4	1.1	even	1	trivial
7280.2.a.ca.1.3		4	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)

\(f(q)\)	\(=\)	\(q-1.13856 q^{3} +1.00000 q^{5} -1.00000 q^{7} -1.70367 q^{9} +O(q^{10})\)	\(q-1.13856 q^{3} +1.00000 q^{5} -1.00000 q^{7} -1.70367 q^{9} -3.23607 q^{11} +1.00000 q^{13} -1.13856 q^{15} +2.98080 q^{17} +1.93974 q^{19} +1.13856 q^{21} -2.17127 q^{23} +1.00000 q^{25} +5.35543 q^{27} +3.58697 q^{29} +7.29517 q^{31} +3.68447 q^{33} -1.00000 q^{35} +2.37463 q^{37} -1.13856 q^{39} +1.76846 q^{41} -5.51320 q^{43} -1.70367 q^{45} -7.53693 q^{47} +1.00000 q^{49} -3.39383 q^{51} -3.40734 q^{53} -3.23607 q^{55} -2.20852 q^{57} -11.6377 q^{59} -5.68447 q^{61} +1.70367 q^{63} +1.00000 q^{65} +2.04559 q^{67} +2.47214 q^{69} -10.9205 q^{71} -7.09181 q^{73} -1.13856 q^{75} +3.23607 q^{77} -14.7949 q^{79} -0.986489 q^{81} +2.61968 q^{83} +2.98080 q^{85} -4.08399 q^{87} +3.92952 q^{89} -1.00000 q^{91} -8.30602 q^{93} +1.93974 q^{95} -1.34192 q^{97} +5.51320 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{3} + 4 q^{5} - 4 q^{7} - q^{9}+O(q^{10}) \) 4 * q - q^3 + 4 * q^5 - 4 * q^7 - q^9	\( 4 q - q^{3} + 4 q^{5} - 4 q^{7} - q^{9} - 4 q^{11} + 4 q^{13} - q^{15} - q^{17} - 7 q^{19} + q^{21} - 6 q^{23} + 4 q^{25} - 4 q^{27} + q^{29} - 11 q^{31} - 4 q^{33} - 4 q^{35} - 3 q^{37} - q^{39} - 5 q^{41} - 6 q^{43} - q^{45} - 6 q^{47} + 4 q^{49} - 14 q^{51} - 2 q^{53} - 4 q^{55} + 14 q^{57} - q^{59} - 4 q^{61} + q^{63} + 4 q^{65} - 11 q^{67} - 8 q^{69} - 16 q^{71} + 2 q^{73} - q^{75} + 4 q^{77} - 15 q^{79} - 16 q^{81} - 2 q^{83} - q^{85} - 23 q^{87} - 3 q^{89} - 4 q^{91} - 5 q^{93} - 7 q^{95} + 8 q^{97} + 6 q^{99}+O(q^{100}) \) 4 * q - q^3 + 4 * q^5 - 4 * q^7 - q^9 - 4 * q^11 + 4 * q^13 - q^15 - q^17 - 7 * q^19 + q^21 - 6 * q^23 + 4 * q^25 - 4 * q^27 + q^29 - 11 * q^31 - 4 * q^33 - 4 * q^35 - 3 * q^37 - q^39 - 5 * q^41 - 6 * q^43 - q^45 - 6 * q^47 + 4 * q^49 - 14 * q^51 - 2 * q^53 - 4 * q^55 + 14 * q^57 - q^59 - 4 * q^61 + q^63 + 4 * q^65 - 11 * q^67 - 8 * q^69 - 16 * q^71 + 2 * q^73 - q^75 + 4 * q^77 - 15 * q^79 - 16 * q^81 - 2 * q^83 - q^85 - 23 * q^87 - 3 * q^89 - 4 * q^91 - 5 * q^93 - 7 * q^95 + 8 * q^97 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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