LMFDB - Embedded newform 3640.2.a.p.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:	$0$
Dimension:	$3$
Coefficient field:	3.3.1101.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 9x + 12 $ x^3 - x^2 - 9*x + 12
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.43163$ 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)

$f(q)$	$=$	$q-1.43163 q^{3} -1.00000 q^{5} +1.00000 q^{7} -0.950444 q^{9} +O(q^{10})$	$q-1.43163 q^{3} -1.00000 q^{5} +1.00000 q^{7} -0.950444 q^{9} -1.00000 q^{13} +1.43163 q^{15} -3.95044 q^{17} -5.95044 q^{19} -1.43163 q^{21} -5.03763 q^{23} +1.00000 q^{25} +5.65556 q^{27} +4.46926 q^{29} +2.56837 q^{31} -1.00000 q^{35} +5.60601 q^{37} +1.43163 q^{39} -3.95044 q^{41} -1.13675 q^{43} +0.950444 q^{45} +9.03763 q^{47} +1.00000 q^{49} +5.65556 q^{51} +2.00000 q^{53} +8.51882 q^{57} -0.294881 q^{59} -10.0000 q^{61} -0.950444 q^{63} +1.00000 q^{65} -8.12482 q^{67} +7.21201 q^{69} +2.86325 q^{71} -8.76414 q^{73} -1.43163 q^{75} +5.95044 q^{79} -5.24533 q^{81} +5.03763 q^{83} +3.95044 q^{85} -6.39831 q^{87} +2.74275 q^{89} -1.00000 q^{91} -3.67695 q^{93} +5.95044 q^{95} -9.21201 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{3} - 3 q^{5} + 3 q^{7} + 10 q^{9}+O(q^{10}) $ 3 * q - q^3 - 3 * q^5 + 3 * q^7 + 10 * q^9	$ 3 q - q^{3} - 3 q^{5} + 3 q^{7} + 10 q^{9} - 3 q^{13} + q^{15} + q^{17} - 5 q^{19} - q^{21} + 4 q^{23} + 3 q^{25} + 14 q^{27} - 9 q^{29} + 11 q^{31} - 3 q^{35} + q^{37} + q^{39} + q^{41} - 10 q^{43} - 10 q^{45} + 8 q^{47} + 3 q^{49} + 14 q^{51} + 6 q^{53} + 16 q^{57} + 9 q^{59} - 30 q^{61} + 10 q^{63} + 3 q^{65} + q^{67} - 10 q^{69} + 2 q^{71} + 6 q^{73} - q^{75} + 5 q^{79} + 7 q^{81} - 4 q^{83} - q^{85} - 7 q^{87} - q^{89} - 3 q^{91} + 15 q^{93} + 5 q^{95} + 4 q^{97}+O(q^{100}) $ 3 * q - q^3 - 3 * q^5 + 3 * q^7 + 10 * q^9 - 3 * q^13 + q^15 + q^17 - 5 * q^19 - q^21 + 4 * q^23 + 3 * q^25 + 14 * q^27 - 9 * q^29 + 11 * q^31 - 3 * q^35 + q^37 + q^39 + q^41 - 10 * q^43 - 10 * q^45 + 8 * q^47 + 3 * q^49 + 14 * q^51 + 6 * q^53 + 16 * q^57 + 9 * q^59 - 30 * q^61 + 10 * q^63 + 3 * q^65 + q^67 - 10 * q^69 + 2 * q^71 + 6 * q^73 - q^75 + 5 * q^79 + 7 * q^81 - 4 * q^83 - q^85 - 7 * q^87 - q^89 - 3 * q^91 + 15 * q^93 + 5 * q^95 + 4 * 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	0
$2$	0	0
$3$	−1.43163	−0.826550	−0.413275	−	0.910606i	$-0.635615\pi$
$3$	−1.43163	−0.826550	−0.413275	+	0.910606i	$0.635615\pi$
$4$	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
$9$	−0.950444	−0.316815
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	1.43163	0.369645
$16$	0	0
$17$	−3.95044	−0.958123	−0.479062	−	0.877781i	$-0.659023\pi$
$17$	−3.95044	−0.958123	−0.479062	+	0.877781i	$0.659023\pi$
$18$	0	0
$19$	−5.95044	−1.36513	−0.682563	−	0.730827i	$-0.739135\pi$
$19$	−5.95044	−1.36513	−0.682563	+	0.730827i	$0.739135\pi$
$20$	0	0
$21$	−1.43163	−0.312407
$22$	0	0
$23$	−5.03763	−1.05042	−0.525210	−	0.850973i	$-0.676013\pi$
$23$	−5.03763	−1.05042	−0.525210	+	0.850973i	$0.676013\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.65556	1.08841
$28$	0	0
$29$	4.46926	0.829921	0.414960	−	0.909839i	$-0.363795\pi$
$29$	4.46926	0.829921	0.414960	+	0.909839i	$0.363795\pi$
$30$	0	0
$31$	2.56837	0.461293	0.230647	−	0.973038i	$-0.425916\pi$
$31$	2.56837	0.461293	0.230647	+	0.973038i	$0.425916\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	5.60601	0.921622	0.460811	−	0.887498i	$-0.347559\pi$
$37$	5.60601	0.921622	0.460811	+	0.887498i	$0.347559\pi$
$38$	0	0
$39$	1.43163	0.229244
$40$	0	0
$41$	−3.95044	−0.616956	−0.308478	−	0.951232i	$-0.599820\pi$
$41$	−3.95044	−0.616956	−0.308478	+	0.951232i	$0.599820\pi$
$42$	0	0
$43$	−1.13675	−0.173352	−0.0866761	−	0.996237i	$-0.527625\pi$
$43$	−1.13675	−0.173352	−0.0866761	+	0.996237i	$0.527625\pi$
$44$	0	0
$45$	0.950444	0.141684
$46$	0	0
$47$	9.03763	1.31827	0.659137	−	0.752023i	$-0.270922\pi$
$47$	9.03763	1.31827	0.659137	+	0.752023i	$0.270922\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	5.65556	0.791937
$52$	0	0
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	8.51882	1.12834
$58$	0	0
$59$	−0.294881	−0.0383903	−0.0191951	−	0.999816i	$-0.506110\pi$
$59$	−0.294881	−0.0383903	−0.0191951	+	0.999816i	$0.506110\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	0	0
$63$	−0.950444	−0.119745
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	−8.12482	−0.992605	−0.496303	−	0.868150i	$-0.665309\pi$
$67$	−8.12482	−0.992605	−0.496303	+	0.868150i	$0.665309\pi$
$68$	0	0
$69$	7.21201	0.868224
$70$	0	0
$71$	2.86325	0.339806	0.169903	−	0.985461i	$-0.445655\pi$
$71$	2.86325	0.339806	0.169903	+	0.985461i	$0.445655\pi$
$72$	0	0
$73$	−8.76414	−1.02577	−0.512883	−	0.858459i	$-0.671422\pi$
$73$	−8.76414	−1.02577	−0.512883	+	0.858459i	$0.671422\pi$
$74$	0	0
$75$	−1.43163	−0.165310
$76$	0	0
$77$	0	0
$78$	0	0
$79$	5.95044	0.669477	0.334739	−	0.942311i	$-0.391352\pi$
$79$	5.95044	0.669477	0.334739	+	0.942311i	$0.391352\pi$
$80$	0	0
$81$	−5.24533	−0.582814
$82$	0	0
$83$	5.03763	0.552952	0.276476	−	0.961021i	$-0.410833\pi$
$83$	5.03763	0.552952	0.276476	+	0.961021i	$0.410833\pi$
$84$	0	0
$85$	3.95044	0.428486
$86$	0	0
$87$	−6.39831	−0.685971
$88$	0	0
$89$	2.74275	0.290731	0.145366	−	0.989378i	$-0.453564\pi$
$89$	2.74275	0.290731	0.145366	+	0.989378i	$0.453564\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−3.67695	−0.381282
$94$	0	0
$95$	5.95044	0.610503
$96$	0	0
$97$	−9.21201	−0.935338	−0.467669	−	0.883904i	$-0.654906\pi$
$97$	−9.21201	−0.935338	−0.467669	+	0.883904i	$0.654906\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−3.03763	−0.302256	−0.151128	−	0.988514i	$-0.548291\pi$
$101$	−3.03763	−0.302256	−0.151128	+	0.988514i	$0.548291\pi$
$102$	0	0
$103$	8.12482	0.800563	0.400281	−	0.916392i	$-0.368912\pi$
$103$	8.12482	0.800563	0.400281	+	0.916392i	$0.368912\pi$
$104$	0	0
$105$	1.43163	0.139713
$106$	0	0
$107$	−2.17438	−0.210205	−0.105103	−	0.994461i	$-0.533517\pi$
$107$	−2.17438	−0.210205	−0.105103	+	0.994461i	$0.533517\pi$
$108$	0	0
$109$	−16.0753	−1.53973	−0.769866	−	0.638206i	$-0.779677\pi$
$109$	−16.0753	−1.53973	−0.769866	+	0.638206i	$0.779677\pi$
$110$	0	0
$111$	−8.02571	−0.761767
$112$	0	0
$113$	17.2120	1.61917	0.809585	−	0.587003i	$-0.199692\pi$
$113$	17.2120	1.61917	0.809585	+	0.587003i	$0.199692\pi$
$114$	0	0
$115$	5.03763	0.469762
$116$	0	0
$117$	0.950444	0.0878686
$118$	0	0
$119$	−3.95044	−0.362137
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	5.65556	0.509945
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	18.7641	1.66505	0.832524	−	0.553989i	$-0.186895\pi$
$127$	18.7641	1.66505	0.832524	+	0.553989i	$0.186895\pi$
$128$	0	0
$129$	1.62740	0.143284
$130$	0	0
$131$	10.7641	0.940467	0.470234	−	0.882542i	$-0.344170\pi$
$131$	10.7641	0.940467	0.470234	+	0.882542i	$0.344170\pi$
$132$	0	0
$133$	−5.95044	−0.515969
$134$	0	0
$135$	−5.65556	−0.486753
$136$	0	0
$137$	20.4693	1.74881	0.874403	−	0.485200i	$-0.161253\pi$
$137$	20.4693	1.74881	0.874403	+	0.485200i	$0.161253\pi$
$138$	0	0
$139$	7.90089	0.670145	0.335072	−	0.942192i	$-0.391239\pi$
$139$	7.90089	0.670145	0.335072	+	0.942192i	$0.391239\pi$
$140$	0	0
$141$	−12.9385	−1.08962
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−4.46926	−0.371152
$146$	0	0
$147$	−1.43163	−0.118079
$148$	0	0
$149$	2.00000	0.163846	0.0819232	−	0.996639i	$-0.473894\pi$
$149$	2.00000	0.163846	0.0819232	+	0.996639i	$0.473894\pi$
$150$	0	0
$151$	20.9385	1.70395	0.851976	−	0.523580i	$-0.175404\pi$
$151$	20.9385	1.70395	0.851976	+	0.523580i	$0.175404\pi$
$152$	0	0
$153$	3.75467	0.303547
$154$	0	0
$155$	−2.56837	−0.206297
$156$	0	0
$157$	3.33251	0.265964	0.132982	−	0.991118i	$-0.457545\pi$
$157$	3.33251	0.265964	0.132982	+	0.991118i	$0.457545\pi$
$158$	0	0
$159$	−2.86325	−0.227071
$160$	0	0
$161$	−5.03763	−0.397021
$162$	0	0
$163$	9.43163	0.738742	0.369371	−	0.929282i	$-0.379573\pi$
$163$	9.43163	0.738742	0.369371	+	0.929282i	$0.379573\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	18.6650	1.44434	0.722172	−	0.691714i	$-0.243144\pi$
$167$	18.6650	1.44434	0.722172	+	0.691714i	$0.243144\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	5.65556	0.432492
$172$	0	0
$173$	−25.4078	−1.93172	−0.965859	−	0.259069i	$-0.916584\pi$
$173$	−25.4078	−1.93172	−0.965859	+	0.259069i	$0.916584\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	0.422160	0.0317315
$178$	0	0
$179$	6.04956	0.452165	0.226083	−	0.974108i	$-0.427408\pi$
$179$	6.04956	0.452165	0.226083	+	0.974108i	$0.427408\pi$
$180$	0	0
$181$	13.2120	0.982041	0.491021	−	0.871148i	$-0.336624\pi$
$181$	13.2120	0.982041	0.491021	+	0.871148i	$0.336624\pi$
$182$	0	0
$183$	14.3163	1.05829
$184$	0	0
$185$	−5.60601	−0.412162
$186$	0	0
$187$	0	0
$188$	0	0
$189$	5.65556	0.411382
$190$	0	0
$191$	13.4316	0.971878	0.485939	−	0.873993i	$-0.338478\pi$
$191$	13.4316	0.971878	0.485939	+	0.873993i	$0.338478\pi$
$192$	0	0
$193$	4.14867	0.298628	0.149314	−	0.988790i	$-0.452294\pi$
$193$	4.14867	0.298628	0.149314	+	0.988790i	$0.452294\pi$
$194$	0	0
$195$	−1.43163	−0.102521
$196$	0	0
$197$	27.9223	1.98938	0.994690	−	0.102917i	$-0.0328176\pi$
$197$	27.9223	1.98938	0.994690	+	0.102917i	$0.0328176\pi$
$198$	0	0
$199$	15.2120	1.07835	0.539175	−	0.842193i	$-0.318736\pi$
$199$	15.2120	1.07835	0.539175	+	0.842193i	$0.318736\pi$
$200$	0	0
$201$	11.6317	0.820438
$202$	0	0
$203$	4.46926	0.313681
$204$	0	0
$205$	3.95044	0.275911
$206$	0	0
$207$	4.78799	0.332788
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−2.39831	−0.165107	−0.0825534	−	0.996587i	$-0.526307\pi$
$211$	−2.39831	−0.165107	−0.0825534	+	0.996587i	$0.526307\pi$
$212$	0	0
$213$	−4.09911	−0.280867
$214$	0	0
$215$	1.13675	0.0775254
$216$	0	0
$217$	2.56837	0.174353
$218$	0	0
$219$	12.5470	0.847847
$220$	0	0
$221$	3.95044	0.265736
$222$	0	0
$223$	1.82562	0.122253	0.0611263	−	0.998130i	$-0.480531\pi$
$223$	1.82562	0.122253	0.0611263	+	0.998130i	$0.480531\pi$
$224$	0	0
$225$	−0.950444	−0.0633629
$226$	0	0
$227$	2.96237	0.196619	0.0983096	−	0.995156i	$-0.468656\pi$
$227$	2.96237	0.196619	0.0983096	+	0.995156i	$0.468656\pi$
$228$	0	0
$229$	−11.9504	−0.789708	−0.394854	−	0.918744i	$-0.629205\pi$
$229$	−11.9504	−0.789708	−0.394854	+	0.918744i	$0.629205\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−15.0376	−0.985148	−0.492574	−	0.870271i	$-0.663944\pi$
$233$	−15.0376	−0.985148	−0.492574	+	0.870271i	$0.663944\pi$
$234$	0	0
$235$	−9.03763	−0.589550
$236$	0	0
$237$	−8.51882	−0.553357
$238$	0	0
$239$	9.03763	0.584596	0.292298	−	0.956327i	$-0.405580\pi$
$239$	9.03763	0.584596	0.292298	+	0.956327i	$0.405580\pi$
$240$	0	0
$241$	23.7522	1.53001	0.765007	−	0.644021i	$-0.222735\pi$
$241$	23.7522	1.53001	0.765007	+	0.644021i	$0.222735\pi$
$242$	0	0
$243$	−9.45734	−0.606688
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	5.95044	0.378618
$248$	0	0
$249$	−7.21201	−0.457043
$250$	0	0
$251$	−14.6650	−0.925648	−0.462824	−	0.886450i	$-0.653164\pi$
$251$	−14.6650	−0.925648	−0.462824	+	0.886450i	$0.653164\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−5.65556	−0.354165
$256$	0	0
$257$	21.4530	1.33820	0.669101	−	0.743171i	$-0.266679\pi$
$257$	21.4530	1.33820	0.669101	+	0.743171i	$0.266679\pi$
$258$	0	0
$259$	5.60601	0.348340
$260$	0	0
$261$	−4.24778	−0.262931
$262$	0	0
$263$	21.0376	1.29724	0.648618	−	0.761114i	$-0.275347\pi$
$263$	21.0376	1.29724	0.648618	+	0.761114i	$0.275347\pi$
$264$	0	0
$265$	−2.00000	−0.122859
$266$	0	0
$267$	−3.92660	−0.240304
$268$	0	0
$269$	−5.31112	−0.323825	−0.161913	−	0.986805i	$-0.551766\pi$
$269$	−5.31112	−0.323825	−0.161913	+	0.986805i	$0.551766\pi$
$270$	0	0
$271$	−18.0753	−1.09799	−0.548997	−	0.835824i	$-0.684990\pi$
$271$	−18.0753	−1.09799	−0.548997	+	0.835824i	$0.684990\pi$
$272$	0	0
$273$	1.43163	0.0866460
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−9.31112	−0.559451	−0.279726	−	0.960080i	$-0.590244\pi$
$277$	−9.31112	−0.559451	−0.279726	+	0.960080i	$0.590244\pi$
$278$	0	0
$279$	−2.44109	−0.146144
$280$	0	0
$281$	−30.0514	−1.79272	−0.896359	−	0.443329i	$-0.853797\pi$
$281$	−30.0514	−1.79272	−0.896359	+	0.443329i	$0.853797\pi$
$282$	0	0
$283$	−18.3983	−1.09367	−0.546833	−	0.837242i	$-0.684167\pi$
$283$	−18.3983	−1.09367	−0.546833	+	0.837242i	$0.684167\pi$
$284$	0	0
$285$	−8.51882	−0.504611
$286$	0	0
$287$	−3.95044	−0.233187
$288$	0	0
$289$	−1.39399	−0.0819996
$290$	0	0
$291$	13.1882	0.773104
$292$	0	0
$293$	−12.1744	−0.711235	−0.355617	−	0.934632i	$-0.615729\pi$
$293$	−12.1744	−0.711235	−0.355617	+	0.934632i	$0.615729\pi$
$294$	0	0
$295$	0.294881	0.0171687
$296$	0	0
$297$	0	0
$298$	0	0
$299$	5.03763	0.291334
$300$	0	0
$301$	−1.13675	−0.0655209
$302$	0	0
$303$	4.34876	0.249830
$304$	0	0
$305$	10.0000	0.572598
$306$	0	0
$307$	2.96237	0.169071	0.0845356	−	0.996420i	$-0.473059\pi$
$307$	2.96237	0.169071	0.0845356	+	0.996420i	$0.473059\pi$
$308$	0	0
$309$	−11.6317	−0.661705
$310$	0	0
$311$	−0.447871	−0.0253964	−0.0126982	−	0.999919i	$-0.504042\pi$
$311$	−0.447871	−0.0253964	−0.0126982	+	0.999919i	$0.504042\pi$
$312$	0	0
$313$	29.4078	1.66223	0.831113	−	0.556104i	$-0.187704\pi$
$313$	29.4078	1.66223	0.831113	+	0.556104i	$0.187704\pi$
$314$	0	0
$315$	0.950444	0.0535514
$316$	0	0
$317$	9.80178	0.550523	0.275261	−	0.961369i	$-0.411236\pi$
$317$	9.80178	0.550523	0.275261	+	0.961369i	$0.411236\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	3.11290	0.173745
$322$	0	0
$323$	23.5069	1.30796
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	23.0138	1.27267
$328$	0	0
$329$	9.03763	0.498261
$330$	0	0
$331$	−18.0753	−0.993507	−0.496753	−	0.867892i	$-0.665475\pi$
$331$	−18.0753	−0.993507	−0.496753	+	0.867892i	$0.665475\pi$
$332$	0	0
$333$	−5.32819	−0.291983
$334$	0	0
$335$	8.12482	0.443906
$336$	0	0
$337$	7.72651	0.420890	0.210445	−	0.977606i	$-0.432509\pi$
$337$	7.72651	0.420890	0.210445	+	0.977606i	$0.432509\pi$
$338$	0	0
$339$	−24.6412	−1.33833
$340$	0	0
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−7.21201	−0.388282
$346$	0	0
$347$	30.9147	1.65959	0.829793	−	0.558071i	$-0.188458\pi$
$347$	30.9147	1.65959	0.829793	+	0.558071i	$0.188458\pi$
$348$	0	0
$349$	−23.1343	−1.23835	−0.619175	−	0.785253i	$-0.712533\pi$
$349$	−23.1343	−1.23835	−0.619175	+	0.785253i	$0.712533\pi$
$350$	0	0
$351$	−5.65556	−0.301872
$352$	0	0
$353$	11.5283	0.613589	0.306794	−	0.951776i	$-0.400744\pi$
$353$	11.5283	0.613589	0.306794	+	0.951776i	$0.400744\pi$
$354$	0	0
$355$	−2.86325	−0.151966
$356$	0	0
$357$	5.65556	0.299324
$358$	0	0
$359$	−32.0514	−1.69161	−0.845805	−	0.533493i	$-0.820879\pi$
$359$	−32.0514	−1.69161	−0.845805	+	0.533493i	$0.820879\pi$
$360$	0	0
$361$	16.4078	0.863567
$362$	0	0
$363$	15.7479	0.826550
$364$	0	0
$365$	8.76414	0.458736
$366$	0	0
$367$	30.0753	1.56992	0.784958	−	0.619549i	$-0.212684\pi$
$367$	30.0753	1.56992	0.784958	+	0.619549i	$0.212684\pi$
$368$	0	0
$369$	3.75467	0.195461
$370$	0	0
$371$	2.00000	0.103835
$372$	0	0
$373$	−12.3163	−0.637712	−0.318856	−	0.947803i	$-0.603299\pi$
$373$	−12.3163	−0.637712	−0.318856	+	0.947803i	$0.603299\pi$
$374$	0	0
$375$	1.43163	0.0739289
$376$	0	0
$377$	−4.46926	−0.230179
$378$	0	0
$379$	19.1129	0.981764	0.490882	−	0.871226i	$-0.336675\pi$
$379$	19.1129	0.981764	0.490882	+	0.871226i	$0.336675\pi$
$380$	0	0
$381$	−26.8633	−1.37625
$382$	0	0
$383$	1.62740	0.0831561	0.0415780	−	0.999135i	$-0.486761\pi$
$383$	1.62740	0.0831561	0.0415780	+	0.999135i	$0.486761\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.08041	0.0549205
$388$	0	0
$389$	33.4787	1.69744	0.848719	−	0.528843i	$-0.177374\pi$
$389$	33.4787	1.69744	0.848719	+	0.528843i	$0.177374\pi$
$390$	0	0
$391$	19.9009	1.00643
$392$	0	0
$393$	−15.4102	−0.777344
$394$	0	0
$395$	−5.95044	−0.299399
$396$	0	0
$397$	3.62740	0.182054	0.0910269	−	0.995848i	$-0.470985\pi$
$397$	3.62740	0.182054	0.0910269	+	0.995848i	$0.470985\pi$
$398$	0	0
$399$	8.51882	0.426474
$400$	0	0
$401$	12.5231	0.625376	0.312688	−	0.949856i	$-0.398771\pi$
$401$	12.5231	0.625376	0.312688	+	0.949856i	$0.398771\pi$
$402$	0	0
$403$	−2.56837	−0.127940
$404$	0	0
$405$	5.24533	0.260642
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−6.19822	−0.306482	−0.153241	−	0.988189i	$-0.548971\pi$
$409$	−6.19822	−0.306482	−0.153241	+	0.988189i	$0.548971\pi$
$410$	0	0
$411$	−29.3043	−1.44548
$412$	0	0
$413$	−0.294881	−0.0145102
$414$	0	0
$415$	−5.03763	−0.247288
$416$	0	0
$417$	−11.3111	−0.553908
$418$	0	0
$419$	15.9009	0.776809	0.388405	−	0.921489i	$-0.373026\pi$
$419$	15.9009	0.776809	0.388405	+	0.921489i	$0.373026\pi$
$420$	0	0
$421$	−0.273492	−0.0133292	−0.00666458	−	0.999978i	$-0.502121\pi$
$421$	−0.273492	−0.0133292	−0.00666458	+	0.999978i	$0.502121\pi$
$422$	0	0
$423$	−8.58976	−0.417649
$424$	0	0
$425$	−3.95044	−0.191625
$426$	0	0
$427$	−10.0000	−0.483934
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−24.2496	−1.16806	−0.584032	−	0.811731i	$-0.698526\pi$
$431$	−24.2496	−1.16806	−0.584032	+	0.811731i	$0.698526\pi$
$432$	0	0
$433$	−9.87518	−0.474571	−0.237285	−	0.971440i	$-0.576258\pi$
$433$	−9.87518	−0.474571	−0.237285	+	0.971440i	$0.576258\pi$
$434$	0	0
$435$	6.39831	0.306776
$436$	0	0
$437$	29.9762	1.43395
$438$	0	0
$439$	−23.8018	−1.13600	−0.567998	−	0.823030i	$-0.692282\pi$
$439$	−23.8018	−1.13600	−0.567998	+	0.823030i	$0.692282\pi$
$440$	0	0
$441$	−0.950444	−0.0452592
$442$	0	0
$443$	5.03763	0.239345	0.119673	−	0.992813i	$-0.461816\pi$
$443$	5.03763	0.239345	0.119673	+	0.992813i	$0.461816\pi$
$444$	0	0
$445$	−2.74275	−0.130019
$446$	0	0
$447$	−2.86325	−0.135427
$448$	0	0
$449$	2.78799	0.131573	0.0657866	−	0.997834i	$-0.479044\pi$
$449$	2.78799	0.131573	0.0657866	+	0.997834i	$0.479044\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−29.9762	−1.40840
$454$	0	0
$455$	1.00000	0.0468807
$456$	0	0
$457$	12.4693	0.583287	0.291644	−	0.956527i	$-0.405798\pi$
$457$	12.4693	0.583287	0.291644	+	0.956527i	$0.405798\pi$
$458$	0	0
$459$	−22.3420	−1.04283
$460$	0	0
$461$	−15.9223	−0.741574	−0.370787	−	0.928718i	$-0.620912\pi$
$461$	−15.9223	−0.741574	−0.370787	+	0.928718i	$0.620912\pi$
$462$	0	0
$463$	21.7522	1.01091	0.505456	−	0.862853i	$-0.331324\pi$
$463$	21.7522	1.01091	0.505456	+	0.862853i	$0.331324\pi$
$464$	0	0
$465$	3.67695	0.170515
$466$	0	0
$467$	−21.6531	−1.00199	−0.500993	−	0.865451i	$-0.667032\pi$
$467$	−21.6531	−1.00199	−0.500993	+	0.865451i	$0.667032\pi$
$468$	0	0
$469$	−8.12482	−0.375169
$470$	0	0
$471$	−4.77092	−0.219832
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−5.95044	−0.273025
$476$	0	0
$477$	−1.90089	−0.0870357
$478$	0	0
$479$	0.493106	0.0225306	0.0112653	−	0.999937i	$-0.496414\pi$
$479$	0.493106	0.0225306	0.0112653	+	0.999937i	$0.496414\pi$
$480$	0	0
$481$	−5.60601	−0.255612
$482$	0	0
$483$	7.21201	0.328158
$484$	0	0
$485$	9.21201	0.418296
$486$	0	0
$487$	−40.0967	−1.81695	−0.908476	−	0.417936i	$-0.862754\pi$
$487$	−40.0967	−1.81695	−0.908476	+	0.417936i	$0.862754\pi$
$488$	0	0
$489$	−13.5026	−0.610607
$490$	0	0
$491$	8.34876	0.376774	0.188387	−	0.982095i	$-0.439674\pi$
$491$	8.34876	0.376774	0.188387	+	0.982095i	$0.439674\pi$
$492$	0	0
$493$	−17.6556	−0.795167
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2.86325	0.128435
$498$	0	0
$499$	9.87704	0.442157	0.221079	−	0.975256i	$-0.429042\pi$
$499$	9.87704	0.442157	0.221079	+	0.975256i	$0.429042\pi$
$500$	0	0
$501$	−26.7214	−1.19382
$502$	0	0
$503$	30.0753	1.34099	0.670495	−	0.741914i	$-0.266082\pi$
$503$	30.0753	1.34099	0.670495	+	0.741914i	$0.266082\pi$
$504$	0	0
$505$	3.03763	0.135173
$506$	0	0
$507$	−1.43163	−0.0635808
$508$	0	0
$509$	30.1248	1.33526	0.667630	−	0.744494i	$-0.267309\pi$
$509$	30.1248	1.33526	0.667630	+	0.744494i	$0.267309\pi$
$510$	0	0
$511$	−8.76414	−0.387703
$512$	0	0
$513$	−33.6531	−1.48582
$514$	0	0
$515$	−8.12482	−0.358022
$516$	0	0
$517$	0	0
$518$	0	0
$519$	36.3745	1.59666
$520$	0	0
$521$	−43.1882	−1.89211	−0.946054	−	0.324009i	$-0.894969\pi$
$521$	−43.1882	−1.89211	−0.946054	+	0.324009i	$0.894969\pi$
$522$	0	0
$523$	−9.52828	−0.416643	−0.208321	−	0.978060i	$-0.566800\pi$
$523$	−9.52828	−0.416643	−0.208321	+	0.978060i	$0.566800\pi$
$524$	0	0
$525$	−1.43163	−0.0624813
$526$	0	0
$527$	−10.1462	−0.441976
$528$	0	0
$529$	2.37775	0.103380
$530$	0	0
$531$	0.280268	0.0121626
$532$	0	0
$533$	3.95044	0.171113
$534$	0	0
$535$	2.17438	0.0940066
$536$	0	0
$537$	−8.66071	−0.373737
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−15.8256	−0.680397	−0.340198	−	0.940354i	$-0.610494\pi$
$541$	−15.8256	−0.680397	−0.340198	+	0.940354i	$0.610494\pi$
$542$	0	0
$543$	−18.9147	−0.811706
$544$	0	0
$545$	16.0753	0.688589
$546$	0	0
$547$	−2.76414	−0.118186	−0.0590931	−	0.998252i	$-0.518821\pi$
$547$	−2.76414	−0.118186	−0.0590931	+	0.998252i	$0.518821\pi$
$548$	0	0
$549$	9.50444	0.405640
$550$	0	0
$551$	−26.5941	−1.13295
$552$	0	0
$553$	5.95044	0.253039
$554$	0	0
$555$	8.02571	0.340672
$556$	0	0
$557$	23.7522	1.00641	0.503207	−	0.864166i	$-0.332153\pi$
$557$	23.7522	1.00641	0.503207	+	0.864166i	$0.332153\pi$
$558$	0	0
$559$	1.13675	0.0480792
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−31.4787	−1.32667	−0.663335	−	0.748322i	$-0.730860\pi$
$563$	−31.4787	−1.32667	−0.663335	+	0.748322i	$0.730860\pi$
$564$	0	0
$565$	−17.2120	−0.724115
$566$	0	0
$567$	−5.24533	−0.220283
$568$	0	0
$569$	−20.2710	−0.849806	−0.424903	−	0.905239i	$-0.639692\pi$
$569$	−20.2710	−0.849806	−0.424903	+	0.905239i	$0.639692\pi$
$570$	0	0
$571$	−17.4316	−0.729491	−0.364745	−	0.931107i	$-0.618844\pi$
$571$	−17.4316	−0.729491	−0.364745	+	0.931107i	$0.618844\pi$
$572$	0	0
$573$	−19.2291	−0.803306
$574$	0	0
$575$	−5.03763	−0.210084
$576$	0	0
$577$	−25.4617	−1.05998	−0.529991	−	0.848003i	$-0.677805\pi$
$577$	−25.4617	−1.05998	−0.529991	+	0.848003i	$0.677805\pi$
$578$	0	0
$579$	−5.93935	−0.246831
$580$	0	0
$581$	5.03763	0.208996
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−0.950444	−0.0392960
$586$	0	0
$587$	23.7027	0.978314	0.489157	−	0.872196i	$-0.337305\pi$
$587$	23.7027	0.978314	0.489157	+	0.872196i	$0.337305\pi$
$588$	0	0
$589$	−15.2830	−0.629723
$590$	0	0
$591$	−39.9743	−1.64432
$592$	0	0
$593$	33.1129	1.35978	0.679892	−	0.733312i	$-0.262027\pi$
$593$	33.1129	1.35978	0.679892	+	0.733312i	$0.262027\pi$
$594$	0	0
$595$	3.95044	0.161952
$596$	0	0
$597$	−21.7779	−0.891311
$598$	0	0
$599$	26.0753	1.06541	0.532703	−	0.846302i	$-0.321176\pi$
$599$	26.0753	1.06541	0.532703	+	0.846302i	$0.321176\pi$
$600$	0	0
$601$	0.316271	0.0129010	0.00645048	−	0.999979i	$-0.497947\pi$
$601$	0.316271	0.0129010	0.00645048	+	0.999979i	$0.497947\pi$
$602$	0	0
$603$	7.72219	0.314472
$604$	0	0
$605$	11.0000	0.447214
$606$	0	0
$607$	−17.4316	−0.707528	−0.353764	−	0.935335i	$-0.615098\pi$
$607$	−17.4316	−0.707528	−0.353764	+	0.935335i	$0.615098\pi$
$608$	0	0
$609$	−6.39831	−0.259273
$610$	0	0
$611$	−9.03763	−0.365624
$612$	0	0
$613$	−2.34876	−0.0948655	−0.0474327	−	0.998874i	$-0.515104\pi$
$613$	−2.34876	−0.0948655	−0.0474327	+	0.998874i	$0.515104\pi$
$614$	0	0
$615$	−5.65556	−0.228054
$616$	0	0
$617$	47.8513	1.92642	0.963211	−	0.268746i	$-0.0866093\pi$
$617$	47.8513	1.92642	0.963211	+	0.268746i	$0.0866093\pi$
$618$	0	0
$619$	−14.2196	−0.571535	−0.285767	−	0.958299i	$-0.592248\pi$
$619$	−14.2196	−0.571535	−0.285767	+	0.958299i	$0.592248\pi$
$620$	0	0
$621$	−28.4907	−1.14329
$622$	0	0
$623$	2.74275	0.109886
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−22.1462	−0.883027
$630$	0	0
$631$	27.7027	1.10283	0.551413	−	0.834233i	$-0.314089\pi$
$631$	27.7027	1.10283	0.551413	+	0.834233i	$0.314089\pi$
$632$	0	0
$633$	3.43349	0.136469
$634$	0	0
$635$	−18.7641	−0.744632
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−2.72136	−0.107655
$640$	0	0
$641$	9.05902	0.357810	0.178905	−	0.983866i	$-0.442745\pi$
$641$	9.05902	0.357810	0.178905	+	0.983866i	$0.442745\pi$
$642$	0	0
$643$	41.3864	1.63212	0.816060	−	0.577967i	$-0.196154\pi$
$643$	41.3864	1.63212	0.816060	+	0.577967i	$0.196154\pi$
$644$	0	0
$645$	−1.62740	−0.0640787
$646$	0	0
$647$	6.56837	0.258229	0.129115	−	0.991630i	$-0.458786\pi$
$647$	6.56837	0.258229	0.129115	+	0.991630i	$0.458786\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−3.67695	−0.144111
$652$	0	0
$653$	−3.27864	−0.128303	−0.0641515	−	0.997940i	$-0.520434\pi$
$653$	−3.27864	−0.128303	−0.0641515	+	0.997940i	$0.520434\pi$
$654$	0	0
$655$	−10.7641	−0.420590
$656$	0	0
$657$	8.32982	0.324977
$658$	0	0
$659$	−26.0019	−1.01289	−0.506444	−	0.862273i	$-0.669040\pi$
$659$	−26.0019	−1.01289	−0.506444	+	0.862273i	$0.669040\pi$
$660$	0	0
$661$	−44.0019	−1.71147	−0.855737	−	0.517411i	$-0.826896\pi$
$661$	−44.0019	−1.71147	−0.855737	+	0.517411i	$0.826896\pi$
$662$	0	0
$663$	−5.65556	−0.219644
$664$	0	0
$665$	5.95044	0.230748
$666$	0	0
$667$	−22.5145	−0.871765
$668$	0	0
$669$	−2.61361	−0.101048
$670$	0	0
$671$	0	0
$672$	0	0
$673$	6.49065	0.250196	0.125098	−	0.992144i	$-0.460075\pi$
$673$	6.49065	0.250196	0.125098	+	0.992144i	$0.460075\pi$
$674$	0	0
$675$	5.65556	0.217683
$676$	0	0
$677$	−2.34876	−0.0902701	−0.0451351	−	0.998981i	$-0.514372\pi$
$677$	−2.34876	−0.0902701	−0.0451351	+	0.998981i	$0.514372\pi$
$678$	0	0
$679$	−9.21201	−0.353525
$680$	0	0
$681$	−4.24100	−0.162516
$682$	0	0
$683$	22.2992	0.853255	0.426628	−	0.904427i	$-0.359701\pi$
$683$	22.2992	0.853255	0.426628	+	0.904427i	$0.359701\pi$
$684$	0	0
$685$	−20.4693	−0.782090
$686$	0	0
$687$	17.1086	0.652733
$688$	0	0
$689$	−2.00000	−0.0761939
$690$	0	0
$691$	−3.15814	−0.120141	−0.0600706	−	0.998194i	$-0.519133\pi$
$691$	−3.15814	−0.120141	−0.0600706	+	0.998194i	$0.519133\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−7.90089	−0.299698
$696$	0	0
$697$	15.6060	0.591120
$698$	0	0
$699$	21.5283	0.814274
$700$	0	0
$701$	18.3230	0.692052	0.346026	−	0.938225i	$-0.387531\pi$
$701$	18.3230	0.692052	0.346026	+	0.938225i	$0.387531\pi$
$702$	0	0
$703$	−33.3582	−1.25813
$704$	0	0
$705$	12.9385	0.487293
$706$	0	0
$707$	−3.03763	−0.114242
$708$	0	0
$709$	30.4907	1.14510	0.572550	−	0.819870i	$-0.305954\pi$
$709$	30.4907	1.14510	0.572550	+	0.819870i	$0.305954\pi$
$710$	0	0
$711$	−5.65556	−0.212100
$712$	0	0
$713$	−12.9385	−0.484551
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−12.9385	−0.483198
$718$	0	0
$719$	−37.0804	−1.38287	−0.691433	−	0.722441i	$-0.743020\pi$
$719$	−37.0804	−1.38287	−0.691433	+	0.722441i	$0.743020\pi$
$720$	0	0
$721$	8.12482	0.302584
$722$	0	0
$723$	−34.0043	−1.26463
$724$	0	0
$725$	4.46926	0.165984
$726$	0	0
$727$	1.62985	0.0604479	0.0302239	−	0.999543i	$-0.490378\pi$
$727$	1.62985	0.0604479	0.0302239	+	0.999543i	$0.490378\pi$
$728$	0	0
$729$	29.2754	1.08427
$730$	0	0
$731$	4.49065	0.166093
$732$	0	0
$733$	43.8770	1.62064	0.810318	−	0.585991i	$-0.199295\pi$
$733$	43.8770	1.62064	0.810318	+	0.585991i	$0.199295\pi$
$734$	0	0
$735$	1.43163	0.0528064
$736$	0	0
$737$	0	0
$738$	0	0
$739$	22.1744	0.815698	0.407849	−	0.913049i	$-0.366279\pi$
$739$	22.1744	0.815698	0.407849	+	0.913049i	$0.366279\pi$
$740$	0	0
$741$	−8.51882	−0.312947
$742$	0	0
$743$	−26.5488	−0.973983	−0.486991	−	0.873407i	$-0.661906\pi$
$743$	−26.5488	−0.973983	−0.486991	+	0.873407i	$0.661906\pi$
$744$	0	0
$745$	−2.00000	−0.0732743
$746$	0	0
$747$	−4.78799	−0.175183
$748$	0	0
$749$	−2.17438	−0.0794501
$750$	0	0
$751$	10.1719	0.371179	0.185589	−	0.982627i	$-0.440581\pi$
$751$	10.1719	0.371179	0.185589	+	0.982627i	$0.440581\pi$
$752$	0	0
$753$	20.9949	0.765095
$754$	0	0
$755$	−20.9385	−0.762031
$756$	0	0
$757$	20.0753	0.729648	0.364824	−	0.931077i	$-0.381129\pi$
$757$	20.0753	0.729648	0.364824	+	0.931077i	$0.381129\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	7.57352	0.274540	0.137270	−	0.990534i	$-0.456167\pi$
$761$	7.57352	0.274540	0.137270	+	0.990534i	$0.456167\pi$
$762$	0	0
$763$	−16.0753	−0.581964
$764$	0	0
$765$	−3.75467	−0.135751
$766$	0	0
$767$	0.294881	0.0106475
$768$	0	0
$769$	19.1795	0.691631	0.345816	−	0.938302i	$-0.387602\pi$
$769$	19.1795	0.691631	0.345816	+	0.938302i	$0.387602\pi$
$770$	0	0
$771$	−30.7127	−1.10609
$772$	0	0
$773$	−6.89574	−0.248023	−0.124011	−	0.992281i	$-0.539576\pi$
$773$	−6.89574	−0.248023	−0.124011	+	0.992281i	$0.539576\pi$
$774$	0	0
$775$	2.56837	0.0922587
$776$	0	0
$777$	−8.02571	−0.287921
$778$	0	0
$779$	23.5069	0.842222
$780$	0	0
$781$	0	0
$782$	0	0
$783$	25.2762	0.903297
$784$	0	0
$785$	−3.33251	−0.118943
$786$	0	0
$787$	−16.5470	−0.589836	−0.294918	−	0.955523i	$-0.595292\pi$
$787$	−16.5470	−0.589836	−0.294918	+	0.955523i	$0.595292\pi$
$788$	0	0
$789$	−30.1180	−1.07223
$790$	0	0
$791$	17.2120	0.611989
$792$	0	0
$793$	10.0000	0.355110
$794$	0	0
$795$	2.86325	0.101549
$796$	0	0
$797$	−30.5445	−1.08194	−0.540971	−	0.841041i	$-0.681943\pi$
$797$	−30.5445	−1.08194	−0.540971	+	0.841041i	$0.681943\pi$
$798$	0	0
$799$	−35.7027	−1.26307
$800$	0	0
$801$	−2.60683	−0.0921079
$802$	0	0
$803$	0	0
$804$	0	0
$805$	5.03763	0.177553
$806$	0	0
$807$	7.60355	0.267658
$808$	0	0
$809$	−8.49743	−0.298754	−0.149377	−	0.988780i	$-0.547727\pi$
$809$	−8.49743	−0.298754	−0.149377	+	0.988780i	$0.547727\pi$
$810$	0	0
$811$	−46.2258	−1.62321	−0.811604	−	0.584208i	$-0.801405\pi$
$811$	−46.2258	−1.62321	−0.811604	+	0.584208i	$0.801405\pi$
$812$	0	0
$813$	25.8770	0.907547
$814$	0	0
$815$	−9.43163	−0.330375
$816$	0	0
$817$	6.76414	0.236647
$818$	0	0
$819$	0.950444	0.0332112
$820$	0	0
$821$	−41.7027	−1.45543	−0.727716	−	0.685878i	$-0.759418\pi$
$821$	−41.7027	−1.45543	−0.727716	+	0.685878i	$0.759418\pi$
$822$	0	0
$823$	16.5470	0.576792	0.288396	−	0.957511i	$-0.406878\pi$
$823$	16.5470	0.576792	0.288396	+	0.957511i	$0.406878\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−2.62225	−0.0911846	−0.0455923	−	0.998960i	$-0.514518\pi$
$827$	−2.62225	−0.0911846	−0.0455923	+	0.998960i	$0.514518\pi$
$828$	0	0
$829$	−17.6599	−0.613353	−0.306677	−	0.951814i	$-0.599217\pi$
$829$	−17.6599	−0.613353	−0.306677	+	0.951814i	$0.599217\pi$
$830$	0	0
$831$	13.3301	0.462415
$832$	0	0
$833$	−3.95044	−0.136875
$834$	0	0
$835$	−18.6650	−0.645930
$836$	0	0
$837$	14.5256	0.502078
$838$	0	0
$839$	−5.92473	−0.204545	−0.102272	−	0.994756i	$-0.532611\pi$
$839$	−5.92473	−0.204545	−0.102272	+	0.994756i	$0.532611\pi$
$840$	0	0
$841$	−9.02571	−0.311231
$842$	0	0
$843$	43.0224	1.48177
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	−11.0000	−0.377964
$848$	0	0
$849$	26.3395	0.903970
$850$	0	0
$851$	−28.2410	−0.968089
$852$	0	0
$853$	−44.7641	−1.53270	−0.766348	−	0.642426i	$-0.777928\pi$
$853$	−44.7641	−1.53270	−0.766348	+	0.642426i	$0.777928\pi$
$854$	0	0
$855$	−5.65556	−0.193416
$856$	0	0
$857$	28.2283	0.964259	0.482129	−	0.876100i	$-0.339863\pi$
$857$	28.2283	0.964259	0.482129	+	0.876100i	$0.339863\pi$
$858$	0	0
$859$	−9.97615	−0.340382	−0.170191	−	0.985411i	$-0.554438\pi$
$859$	−9.97615	−0.340382	−0.170191	+	0.985411i	$0.554438\pi$
$860$	0	0
$861$	5.65556	0.192741
$862$	0	0
$863$	8.47358	0.288444	0.144222	−	0.989545i	$-0.453932\pi$
$863$	8.47358	0.288444	0.144222	+	0.989545i	$0.453932\pi$
$864$	0	0
$865$	25.4078	0.863890
$866$	0	0
$867$	1.99568	0.0677768
$868$	0	0
$869$	0	0
$870$	0	0
$871$	8.12482	0.275299
$872$	0	0
$873$	8.75550	0.296329
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−43.5540	−1.47071	−0.735357	−	0.677680i	$-0.762985\pi$
$877$	−43.5540	−1.47071	−0.735357	+	0.677680i	$0.762985\pi$
$878$	0	0
$879$	17.4292	0.587871
$880$	0	0
$881$	−51.6360	−1.73966	−0.869831	−	0.493350i	$-0.835772\pi$
$881$	−51.6360	−1.73966	−0.869831	+	0.493350i	$0.835772\pi$
$882$	0	0
$883$	−13.8770	−0.467000	−0.233500	−	0.972357i	$-0.575018\pi$
$883$	−13.8770	−0.467000	−0.233500	+	0.972357i	$0.575018\pi$
$884$	0	0
$885$	−0.422160	−0.0141908
$886$	0	0
$887$	−43.5069	−1.46082	−0.730409	−	0.683010i	$-0.760671\pi$
$887$	−43.5069	−1.46082	−0.730409	+	0.683010i	$0.760671\pi$
$888$	0	0
$889$	18.7641	0.629329
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−53.7779	−1.79961
$894$	0	0
$895$	−6.04956	−0.202214
$896$	0	0
$897$	−7.21201	−0.240802
$898$	0	0
$899$	11.4787	0.382837
$900$	0	0
$901$	−7.90089	−0.263217
$902$	0	0
$903$	1.62740	0.0541564
$904$	0	0
$905$	−13.2120	−0.439182
$906$	0	0
$907$	49.5283	1.64456	0.822280	−	0.569083i	$-0.192702\pi$
$907$	49.5283	1.64456	0.822280	+	0.569083i	$0.192702\pi$
$908$	0	0
$909$	2.88710	0.0957591
$910$	0	0
$911$	59.7284	1.97889	0.989445	−	0.144911i	$-0.0462897\pi$
$911$	59.7284	1.97889	0.989445	+	0.144911i	$0.0462897\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−14.3163	−0.473281
$916$	0	0
$917$	10.7641	0.355463
$918$	0	0
$919$	−0.493106	−0.0162661	−0.00813303	−	0.999967i	$-0.502589\pi$
$919$	−0.493106	−0.0162661	−0.00813303	+	0.999967i	$0.502589\pi$
$920$	0	0
$921$	−4.24100	−0.139746
$922$	0	0
$923$	−2.86325	−0.0942452
$924$	0	0
$925$	5.60601	0.184324
$926$	0	0
$927$	−7.72219	−0.253630
$928$	0	0
$929$	8.27104	0.271364	0.135682	−	0.990752i	$-0.456677\pi$
$929$	8.27104	0.271364	0.135682	+	0.990752i	$0.456677\pi$
$930$	0	0
$931$	−5.95044	−0.195018
$932$	0	0
$933$	0.641184	0.0209914
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−12.1487	−0.396880	−0.198440	−	0.980113i	$-0.563587\pi$
$937$	−12.1487	−0.396880	−0.198440	+	0.980113i	$0.563587\pi$
$938$	0	0
$939$	−42.1010	−1.37391
$940$	0	0
$941$	26.4736	0.863014	0.431507	−	0.902110i	$-0.357982\pi$
$941$	26.4736	0.863014	0.431507	+	0.902110i	$0.357982\pi$
$942$	0	0
$943$	19.9009	0.648062
$944$	0	0
$945$	−5.65556	−0.183975
$946$	0	0
$947$	18.2001	0.591423	0.295712	−	0.955277i	$-0.404443\pi$
$947$	18.2001	0.591423	0.295712	+	0.955277i	$0.404443\pi$
$948$	0	0
$949$	8.76414	0.284496
$950$	0	0
$951$	−14.0325	−0.455035
$952$	0	0
$953$	6.20687	0.201060	0.100530	−	0.994934i	$-0.467946\pi$
$953$	6.20687	0.201060	0.100530	+	0.994934i	$0.467946\pi$
$954$	0	0
$955$	−13.4316	−0.434637
$956$	0	0
$957$	0	0
$958$	0	0
$959$	20.4693	0.660987
$960$	0	0
$961$	−24.4035	−0.787208
$962$	0	0
$963$	2.06663	0.0665961
$964$	0	0
$965$	−4.14867	−0.133550
$966$	0	0
$967$	10.7666	0.346230	0.173115	−	0.984902i	$-0.444617\pi$
$967$	10.7666	0.346230	0.173115	+	0.984902i	$0.444617\pi$
$968$	0	0
$969$	−33.6531	−1.08109
$970$	0	0
$971$	33.5283	1.07597	0.537987	−	0.842953i	$-0.319185\pi$
$971$	33.5283	1.07597	0.537987	+	0.842953i	$0.319185\pi$
$972$	0	0
$973$	7.90089	0.253291
$974$	0	0
$975$	1.43163	0.0458488
$976$	0	0
$977$	57.2805	1.83257	0.916283	−	0.400532i	$-0.131175\pi$
$977$	57.2805	1.83257	0.916283	+	0.400532i	$0.131175\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	15.2786	0.487809
$982$	0	0
$983$	−46.7641	−1.49154	−0.745772	−	0.666201i	$-0.767919\pi$
$983$	−46.7641	−1.49154	−0.745772	+	0.666201i	$0.767919\pi$
$984$	0	0
$985$	−27.9223	−0.889678
$986$	0	0
$987$	−12.9385	−0.411838
$988$	0	0
$989$	5.72651	0.182092
$990$	0	0
$991$	23.3282	0.741045	0.370522	−	0.928824i	$-0.379179\pi$
$991$	23.3282	0.741045	0.370522	+	0.928824i	$0.379179\pi$
$992$	0	0
$993$	25.8770	0.821183
$994$	0	0
$995$	−15.2120	−0.482253
$996$	0	0
$997$	−15.7999	−0.500388	−0.250194	−	0.968196i	$-0.580494\pi$
$997$	−15.7999	−0.500388	−0.250194	+	0.968196i	$0.580494\pi$
$998$	0	0
$999$	31.7051	1.00311

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3640.2.a.p.1.2	✓	3
4.3	odd	2		7280.2.a.bn.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3640.2.a.p.1.2	✓	3	1.1	even	1	trivial
7280.2.a.bn.1.2		3	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)

\(f(q)\)	\(=\)	\(q-1.43163 q^{3} -1.00000 q^{5} +1.00000 q^{7} -0.950444 q^{9} +O(q^{10})\)	\(q-1.43163 q^{3} -1.00000 q^{5} +1.00000 q^{7} -0.950444 q^{9} -1.00000 q^{13} +1.43163 q^{15} -3.95044 q^{17} -5.95044 q^{19} -1.43163 q^{21} -5.03763 q^{23} +1.00000 q^{25} +5.65556 q^{27} +4.46926 q^{29} +2.56837 q^{31} -1.00000 q^{35} +5.60601 q^{37} +1.43163 q^{39} -3.95044 q^{41} -1.13675 q^{43} +0.950444 q^{45} +9.03763 q^{47} +1.00000 q^{49} +5.65556 q^{51} +2.00000 q^{53} +8.51882 q^{57} -0.294881 q^{59} -10.0000 q^{61} -0.950444 q^{63} +1.00000 q^{65} -8.12482 q^{67} +7.21201 q^{69} +2.86325 q^{71} -8.76414 q^{73} -1.43163 q^{75} +5.95044 q^{79} -5.24533 q^{81} +5.03763 q^{83} +3.95044 q^{85} -6.39831 q^{87} +2.74275 q^{89} -1.00000 q^{91} -3.67695 q^{93} +5.95044 q^{95} -9.21201 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{3} - 3 q^{5} + 3 q^{7} + 10 q^{9}+O(q^{10}) \) 3 * q - q^3 - 3 * q^5 + 3 * q^7 + 10 * q^9	\( 3 q - q^{3} - 3 q^{5} + 3 q^{7} + 10 q^{9} - 3 q^{13} + q^{15} + q^{17} - 5 q^{19} - q^{21} + 4 q^{23} + 3 q^{25} + 14 q^{27} - 9 q^{29} + 11 q^{31} - 3 q^{35} + q^{37} + q^{39} + q^{41} - 10 q^{43} - 10 q^{45} + 8 q^{47} + 3 q^{49} + 14 q^{51} + 6 q^{53} + 16 q^{57} + 9 q^{59} - 30 q^{61} + 10 q^{63} + 3 q^{65} + q^{67} - 10 q^{69} + 2 q^{71} + 6 q^{73} - q^{75} + 5 q^{79} + 7 q^{81} - 4 q^{83} - q^{85} - 7 q^{87} - q^{89} - 3 q^{91} + 15 q^{93} + 5 q^{95} + 4 q^{97}+O(q^{100}) \) 3 * q - q^3 - 3 * q^5 + 3 * q^7 + 10 * q^9 - 3 * q^13 + q^15 + q^17 - 5 * q^19 - q^21 + 4 * q^23 + 3 * q^25 + 14 * q^27 - 9 * q^29 + 11 * q^31 - 3 * q^35 + q^37 + q^39 + q^41 - 10 * q^43 - 10 * q^45 + 8 * q^47 + 3 * q^49 + 14 * q^51 + 6 * q^53 + 16 * q^57 + 9 * q^59 - 30 * q^61 + 10 * q^63 + 3 * q^65 + q^67 - 10 * q^69 + 2 * q^71 + 6 * q^73 - q^75 + 5 * q^79 + 7 * q^81 - 4 * q^83 - q^85 - 7 * q^87 - q^89 - 3 * q^91 + 15 * q^93 + 5 * q^95 + 4 * q^97

Introduction

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