LMFDB - Embedded newform 1547.2.a.i.1.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1547,2,Mod(1,1547)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1547.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1547, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1547 = 7 \cdot 13 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1547.a (trivial)

Newform invariants

comment:select newform

sage:traces = [14,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$12.3528571927$
Analytic rank:	$1$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - x^{13} - 20 x^{12} + 19 x^{11} + 151 x^{10} - 133 x^{9} - 536 x^{8} + 404 x^{7} + 924 x^{6} + \cdots + 4 $ x^14 - x^13 - 20x^12 + 19x^11 + 151x^10 - 133x^9 - 536x^8 + 404x^7 + 924x^6 - 467x^5 - 778x^4 + 97x^3 + 278x^2 + 73x + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-0.0757200$ of defining polynomial
Character	$\chi$	$=$	1547.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.0757200 q^{2} +2.32101 q^{3} -1.99427 q^{4} -2.99129 q^{5} -0.175747 q^{6} -1.00000 q^{7} +0.302446 q^{8} +2.38707 q^{9} +0.226501 q^{10} +5.73437 q^{11} -4.62871 q^{12} -1.00000 q^{13} +0.0757200 q^{14} -6.94281 q^{15} +3.96563 q^{16} -1.00000 q^{17} -0.180749 q^{18} +1.36989 q^{19} +5.96543 q^{20} -2.32101 q^{21} -0.434207 q^{22} -5.46282 q^{23} +0.701979 q^{24} +3.94783 q^{25} +0.0757200 q^{26} -1.42261 q^{27} +1.99427 q^{28} -4.59566 q^{29} +0.525710 q^{30} -10.0303 q^{31} -0.905170 q^{32} +13.3095 q^{33} +0.0757200 q^{34} +2.99129 q^{35} -4.76046 q^{36} -8.50525 q^{37} -0.103728 q^{38} -2.32101 q^{39} -0.904704 q^{40} -10.0541 q^{41} +0.175747 q^{42} +7.65461 q^{43} -11.4359 q^{44} -7.14043 q^{45} +0.413645 q^{46} -1.53575 q^{47} +9.20426 q^{48} +1.00000 q^{49} -0.298930 q^{50} -2.32101 q^{51} +1.99427 q^{52} +7.05436 q^{53} +0.107720 q^{54} -17.1532 q^{55} -0.302446 q^{56} +3.17952 q^{57} +0.347983 q^{58} +1.73896 q^{59} +13.8458 q^{60} -11.0594 q^{61} +0.759494 q^{62} -2.38707 q^{63} -7.86272 q^{64} +2.99129 q^{65} -1.00780 q^{66} +9.80006 q^{67} +1.99427 q^{68} -12.6792 q^{69} -0.226501 q^{70} -12.7742 q^{71} +0.721960 q^{72} -11.6203 q^{73} +0.644018 q^{74} +9.16293 q^{75} -2.73192 q^{76} -5.73437 q^{77} +0.175747 q^{78} +8.10592 q^{79} -11.8624 q^{80} -10.4631 q^{81} +0.761293 q^{82} -6.79580 q^{83} +4.62871 q^{84} +2.99129 q^{85} -0.579607 q^{86} -10.6665 q^{87} +1.73434 q^{88} +9.94229 q^{89} +0.540673 q^{90} +1.00000 q^{91} +10.8943 q^{92} -23.2804 q^{93} +0.116287 q^{94} -4.09774 q^{95} -2.10091 q^{96} -9.21971 q^{97} -0.0757200 q^{98} +13.6884 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + q^{2} - 4 q^{3} + 13 q^{4} - 9 q^{5} - 10 q^{6} - 14 q^{7} + 18 q^{9} - 9 q^{10} + 3 q^{11} - 16 q^{12} - 14 q^{13} - q^{14} - q^{15} + 11 q^{16} - 14 q^{17} + 5 q^{18} - 17 q^{19} - 24 q^{20}+ \cdots + 33 q^{99}+O(q^{100}) $ 14 * q + q^2 - 4 * q^3 + 13 * q^4 - 9 * q^5 - 10 * q^6 - 14 * q^7 + 18 * q^9 - 9 * q^10 + 3 * q^11 - 16 * q^12 - 14 * q^13 - q^14 - q^15 + 11 * q^16 - 14 * q^17 + 5 * q^18 - 17 * q^19 - 24 * q^20 + 4 * q^21 - 22 * q^22 - 6 * q^23 - 18 * q^24 + 11 * q^25 - q^26 - 19 * q^27 - 13 * q^28 + 3 * q^29 + 8 * q^30 - 19 * q^31 - 4 * q^32 - 34 * q^33 - q^34 + 9 * q^35 + 25 * q^36 - 45 * q^37 - 7 * q^38 + 4 * q^39 - 24 * q^40 - 29 * q^41 + 10 * q^42 + 7 * q^43 + 10 * q^44 - 48 * q^45 + q^46 - 10 * q^47 - 38 * q^48 + 14 * q^49 + 12 * q^50 + 4 * q^51 - 13 * q^52 - 15 * q^53 - 44 * q^54 - 2 * q^55 - 21 * q^57 - 42 * q^58 + 3 * q^59 + 8 * q^60 - 34 * q^61 - 16 * q^62 - 18 * q^63 - 20 * q^64 + 9 * q^65 - 16 * q^66 - 14 * q^67 - 13 * q^68 + 4 * q^69 + 9 * q^70 + 4 * q^71 - 6 * q^72 - 45 * q^73 - 22 * q^74 - 3 * q^75 - 29 * q^76 - 3 * q^77 + 10 * q^78 + 23 * q^79 - 37 * q^80 + 6 * q^81 + 15 * q^82 - 25 * q^83 + 16 * q^84 + 9 * q^85 + 40 * q^86 - 7 * q^87 - 36 * q^88 + 9 * q^89 - 57 * q^90 + 14 * q^91 - 38 * q^92 - 27 * q^93 + 13 * q^94 + 13 * q^95 + 3 * q^96 - 76 * q^97 + q^98 + 33 * q^99

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 $

$2$	−0.0757200	−0.0535422	−0.0267711	−	0.999642i	$-0.508523\pi$
$2$	−0.0757200	−0.0535422	−0.0267711	+	0.999642i	$0.508523\pi$
$3$	2.32101	1.34003	0.670017	−	0.742346i	$-0.266287\pi$
$3$	2.32101	1.34003	0.670017	+	0.742346i	$0.266287\pi$
$4$	−1.99427	−0.997133
$5$	−2.99129	−1.33775	−0.668873	−	0.743377i	$-0.733223\pi$
$5$	−2.99129	−1.33775	−0.668873	+	0.743377i	$0.733223\pi$
$6$	−0.175747	−0.0717483
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0.302446	0.106931
$9$	2.38707	0.795691
$10$	0.226501	0.0716258
$11$	5.73437	1.72898	0.864489	−	0.502651i	$-0.167642\pi$
$11$	5.73437	1.72898	0.864489	+	0.502651i	$0.167642\pi$
$12$	−4.62871	−1.33619
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0.0757200	0.0202370
$15$	−6.94281	−1.79263
$16$	3.96563	0.991408
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−0.180749	−0.0426030
$19$	1.36989	0.314274	0.157137	−	0.987577i	$-0.449774\pi$
$19$	1.36989	0.314274	0.157137	+	0.987577i	$0.449774\pi$
$20$	5.96543	1.33391
$21$	−2.32101	−0.506485
$22$	−0.434207	−0.0925732
$23$	−5.46282	−1.13908	−0.569538	−	0.821965i	$-0.692878\pi$
$23$	−5.46282	−1.13908	−0.569538	+	0.821965i	$0.692878\pi$
$24$	0.701979	0.143291
$25$	3.94783	0.789565
$26$	0.0757200	0.0148499
$27$	−1.42261	−0.273782
$28$	1.99427	0.376881
$29$	−4.59566	−0.853392	−0.426696	−	0.904395i	$-0.640323\pi$
$29$	−4.59566	−0.853392	−0.426696	+	0.904395i	$0.640323\pi$
$30$	0.525710	0.0959810
$31$	−10.0303	−1.80149	−0.900747	−	0.434344i	$-0.856980\pi$
$31$	−10.0303	−1.80149	−0.900747	+	0.434344i	$0.856980\pi$
$32$	−0.905170	−0.160013
$33$	13.3095	2.31689
$34$	0.0757200	0.0129859
$35$	2.99129	0.505621
$36$	−4.76046	−0.793410
$37$	−8.50525	−1.39825	−0.699127	−	0.714997i	$-0.746428\pi$
$37$	−8.50525	−1.39825	−0.699127	+	0.714997i	$0.746428\pi$
$38$	−0.103728	−0.0168269
$39$	−2.32101	−0.371659
$40$	−0.904704	−0.143046
$41$	−10.0541	−1.57018	−0.785089	−	0.619382i	$-0.787383\pi$
$41$	−10.0541	−1.57018	−0.785089	+	0.619382i	$0.787383\pi$
$42$	0.175747	0.0271183
$43$	7.65461	1.16732	0.583658	−	0.811999i	$-0.301621\pi$
$43$	7.65461	1.16732	0.583658	+	0.811999i	$0.301621\pi$
$44$	−11.4359	−1.72402
$45$	−7.14043	−1.06443
$46$	0.413645	0.0609886
$47$	−1.53575	−0.224012	−0.112006	−	0.993708i	$-0.535728\pi$
$47$	−1.53575	−0.224012	−0.112006	+	0.993708i	$0.535728\pi$
$48$	9.20426	1.32852
$49$	1.00000	0.142857
$50$	−0.298930	−0.0422750
$51$	−2.32101	−0.325006
$52$	1.99427	0.276555
$53$	7.05436	0.968991	0.484495	−	0.874794i	$-0.339003\pi$
$53$	7.05436	0.968991	0.484495	+	0.874794i	$0.339003\pi$
$54$	0.107720	0.0146589
$55$	−17.1532	−2.31293
$56$	−0.302446	−0.0404160
$57$	3.17952	0.421138
$58$	0.347983	0.0456924
$59$	1.73896	0.226394	0.113197	−	0.993573i	$-0.463891\pi$
$59$	1.73896	0.226394	0.113197	+	0.993573i	$0.463891\pi$
$60$	13.8458	1.78749
$61$	−11.0594	−1.41600	−0.708002	−	0.706210i	$-0.750403\pi$
$61$	−11.0594	−1.41600	−0.708002	+	0.706210i	$0.750403\pi$
$62$	0.759494	0.0964559
$63$	−2.38707	−0.300743
$64$	−7.86272	−0.982840
$65$	2.99129	0.371024
$66$	−1.00780	−0.124051
$67$	9.80006	1.19727	0.598634	−	0.801023i	$-0.295711\pi$
$67$	9.80006	1.19727	0.598634	+	0.801023i	$0.295711\pi$
$68$	1.99427	0.241840
$69$	−12.6792	−1.52640
$70$	−0.226501	−0.0270720
$71$	−12.7742	−1.51601	−0.758007	−	0.652246i	$-0.773827\pi$
$71$	−12.7742	−1.51601	−0.758007	+	0.652246i	$0.773827\pi$
$72$	0.721960	0.0850838
$73$	−11.6203	−1.36006	−0.680028	−	0.733186i	$-0.738032\pi$
$73$	−11.6203	−1.36006	−0.680028	+	0.733186i	$0.738032\pi$
$74$	0.644018	0.0748655
$75$	9.16293	1.05804
$76$	−2.73192	−0.313373
$77$	−5.73437	−0.653492
$78$	0.175747	0.0198994
$79$	8.10592	0.911987	0.455993	−	0.889983i	$-0.349284\pi$
$79$	8.10592	0.911987	0.455993	+	0.889983i	$0.349284\pi$
$80$	−11.8624	−1.32625
$81$	−10.4631	−1.16257
$82$	0.761293	0.0840708
$83$	−6.79580	−0.745936	−0.372968	−	0.927844i	$-0.621660\pi$
$83$	−6.79580	−0.745936	−0.372968	+	0.927844i	$0.621660\pi$
$84$	4.62871	0.505033
$85$	2.99129	0.324451
$86$	−0.579607	−0.0625007
$87$	−10.6665	−1.14357
$88$	1.73434	0.184881
$89$	9.94229	1.05388	0.526940	−	0.849902i	$-0.323339\pi$
$89$	9.94229	1.05388	0.526940	+	0.849902i	$0.323339\pi$
$90$	0.540673	0.0569920
$91$	1.00000	0.104828
$92$	10.8943	1.13581
$93$	−23.2804	−2.41406
$94$	0.116287	0.0119941
$95$	−4.09774	−0.420419
$96$	−2.10091	−0.214423
$97$	−9.21971	−0.936119	−0.468060	−	0.883697i	$-0.655047\pi$
$97$	−9.21971	−0.936119	−0.468060	+	0.883697i	$0.655047\pi$
$98$	−0.0757200	−0.00764888
$99$	13.6884	1.37573
$100$	−7.87302	−0.787302
$101$	2.90274	0.288833	0.144417	−	0.989517i	$-0.453869\pi$
$101$	2.90274	0.288833	0.144417	+	0.989517i	$0.453869\pi$
$102$	0.175747	0.0174015
$103$	−0.911312	−0.0897942	−0.0448971	−	0.998992i	$-0.514296\pi$
$103$	−0.911312	−0.0897942	−0.0448971	+	0.998992i	$0.514296\pi$
$104$	−0.302446	−0.0296573
$105$	6.94281	0.677549
$106$	−0.534156	−0.0518818
$107$	−12.0437	−1.16431	−0.582154	−	0.813078i	$-0.697790\pi$
$107$	−12.0437	−1.16431	−0.582154	+	0.813078i	$0.697790\pi$
$108$	2.83706	0.272997
$109$	19.3367	1.85212	0.926060	−	0.377376i	$-0.123174\pi$
$109$	19.3367	1.85212	0.926060	+	0.377376i	$0.123174\pi$
$110$	1.29884	0.123840
$111$	−19.7407	−1.87371
$112$	−3.96563	−0.374717
$113$	4.08481	0.384267	0.192133	−	0.981369i	$-0.438459\pi$
$113$	4.08481	0.384267	0.192133	+	0.981369i	$0.438459\pi$
$114$	−0.240754	−0.0225486
$115$	16.3409	1.52379
$116$	9.16496	0.850946
$117$	−2.38707	−0.220685
$118$	−0.131674	−0.0121216
$119$	1.00000	0.0916698
$120$	−2.09982	−0.191687
$121$	21.8830	1.98937
$122$	0.837414	0.0758159
$123$	−23.3355	−2.10409
$124$	20.0031	1.79633
$125$	3.14736	0.281508
$126$	0.180749	0.0161024
$127$	−14.6296	−1.29817	−0.649083	−	0.760717i	$-0.724847\pi$
$127$	−14.6296	−1.29817	−0.649083	+	0.760717i	$0.724847\pi$
$128$	2.40571	0.212636
$129$	17.7664	1.56424
$130$	−0.226501	−0.0198654
$131$	11.4907	1.00395	0.501975	−	0.864882i	$-0.332607\pi$
$131$	11.4907	1.00395	0.501975	+	0.864882i	$0.332607\pi$
$132$	−26.5427	−2.31025
$133$	−1.36989	−0.118784
$134$	−0.742061	−0.0641043
$135$	4.25544	0.366250
$136$	−0.302446	−0.0259345
$137$	−5.20377	−0.444588	−0.222294	−	0.974980i	$-0.571355\pi$
$137$	−5.20377	−0.444588	−0.222294	+	0.974980i	$0.571355\pi$
$138$	0.960072	0.0817268
$139$	2.88701	0.244873	0.122437	−	0.992476i	$-0.460929\pi$
$139$	2.88701	0.244873	0.122437	+	0.992476i	$0.460929\pi$
$140$	−5.96543	−0.504171
$141$	−3.56449	−0.300184
$142$	0.967261	0.0811707
$143$	−5.73437	−0.479532
$144$	9.46625	0.788854
$145$	13.7469	1.14162
$146$	0.879892	0.0728203
$147$	2.32101	0.191433
$148$	16.9617	1.39425
$149$	−5.38643	−0.441274	−0.220637	−	0.975356i	$-0.570814\pi$
$149$	−5.38643	−0.441274	−0.220637	+	0.975356i	$0.570814\pi$
$150$	−0.693818	−0.0566500
$151$	−4.55246	−0.370474	−0.185237	−	0.982694i	$-0.559305\pi$
$151$	−4.55246	−0.370474	−0.185237	+	0.982694i	$0.559305\pi$
$152$	0.414318	0.0336056
$153$	−2.38707	−0.192983
$154$	0.434207	0.0349894
$155$	30.0035	2.40994
$156$	4.62871	0.370593
$157$	10.7817	0.860475	0.430237	−	0.902716i	$-0.358430\pi$
$157$	10.7817	0.860475	0.430237	+	0.902716i	$0.358430\pi$
$158$	−0.613780	−0.0488297
$159$	16.3732	1.29848
$160$	2.70763	0.214057
$161$	5.46282	0.430530
$162$	0.792267	0.0622463
$163$	5.09298	0.398913	0.199457	−	0.979907i	$-0.436082\pi$
$163$	5.09298	0.398913	0.199457	+	0.979907i	$0.436082\pi$
$164$	20.0505	1.56568
$165$	−39.8127	−3.09941
$166$	0.514578	0.0399390
$167$	9.28546	0.718531	0.359265	−	0.933235i	$-0.383027\pi$
$167$	9.28546	0.718531	0.359265	+	0.933235i	$0.383027\pi$
$168$	−0.701979	−0.0541589
$169$	1.00000	0.0769231
$170$	−0.226501	−0.0173718
$171$	3.27002	0.250065
$172$	−15.2653	−1.16397
$173$	24.0491	1.82842	0.914209	−	0.405243i	$-0.132813\pi$
$173$	24.0491	1.82842	0.914209	+	0.405243i	$0.132813\pi$
$174$	0.807671	0.0612294
$175$	−3.94783	−0.298428
$176$	22.7404	1.71412
$177$	4.03615	0.303375
$178$	−0.752830	−0.0564270
$179$	−3.07531	−0.229860	−0.114930	−	0.993374i	$-0.536664\pi$
$179$	−3.07531	−0.229860	−0.114930	+	0.993374i	$0.536664\pi$
$180$	14.2399	1.06138
$181$	0.176809	0.0131421	0.00657105	−	0.999978i	$-0.497908\pi$
$181$	0.176809	0.0131421	0.00657105	+	0.999978i	$0.497908\pi$
$182$	−0.0757200	−0.00561274
$183$	−25.6688	−1.89749
$184$	−1.65221	−0.121802
$185$	25.4417	1.87051
$186$	1.76279	0.129254
$187$	−5.73437	−0.419339
$188$	3.06270	0.223370
$189$	1.42261	0.103480
$190$	0.310281	0.0225101
$191$	9.32070	0.674422	0.337211	−	0.941429i	$-0.390516\pi$
$191$	9.32070	0.674422	0.337211	+	0.941429i	$0.390516\pi$
$192$	−18.2494	−1.31704
$193$	−21.3127	−1.53412	−0.767059	−	0.641576i	$-0.778281\pi$
$193$	−21.3127	−1.53412	−0.767059	+	0.641576i	$0.778281\pi$
$194$	0.698117	0.0501218
$195$	6.94281	0.497185
$196$	−1.99427	−0.142448
$197$	13.0175	0.927457	0.463729	−	0.885977i	$-0.346511\pi$
$197$	13.0175	0.927457	0.463729	+	0.885977i	$0.346511\pi$
$198$	−1.03648	−0.0736596
$199$	−7.70115	−0.545920	−0.272960	−	0.962025i	$-0.588003\pi$
$199$	−7.70115	−0.545920	−0.272960	+	0.962025i	$0.588003\pi$
$200$	1.19400	0.0844289
$201$	22.7460	1.60438
$202$	−0.219795	−0.0154648
$203$	4.59566	0.322552
$204$	4.62871	0.324074
$205$	30.0746	2.10050
$206$	0.0690046	0.00480778
$207$	−13.0401	−0.906352
$208$	−3.96563	−0.274967
$209$	7.85546	0.543373
$210$	−0.525710	−0.0362774
$211$	−8.55109	−0.588681	−0.294341	−	0.955701i	$-0.595100\pi$
$211$	−8.55109	−0.588681	−0.294341	+	0.955701i	$0.595100\pi$
$212$	−14.0683	−0.966213
$213$	−29.6489	−2.03151
$214$	0.911949	0.0623396
$215$	−22.8972	−1.56157
$216$	−0.430263	−0.0292757
$217$	10.0303	0.680901
$218$	−1.46418	−0.0991665
$219$	−26.9709	−1.82252
$220$	34.2080	2.30630
$221$	1.00000	0.0672673
$222$	1.49477	0.100322
$223$	−22.1333	−1.48215	−0.741077	−	0.671420i	$-0.765685\pi$
$223$	−22.1333	−1.48215	−0.741077	+	0.671420i	$0.765685\pi$
$224$	0.905170	0.0604792
$225$	9.42375	0.628250
$226$	−0.309302	−0.0205745
$227$	−23.0070	−1.52703	−0.763515	−	0.645790i	$-0.776528\pi$
$227$	−23.0070	−1.52703	−0.763515	+	0.645790i	$0.776528\pi$
$228$	−6.34082	−0.419931
$229$	2.47261	0.163395	0.0816974	−	0.996657i	$-0.473966\pi$
$229$	2.47261	0.163395	0.0816974	+	0.996657i	$0.473966\pi$
$230$	−1.23733	−0.0815873
$231$	−13.3095	−0.875702
$232$	−1.38994	−0.0912539
$233$	−6.48271	−0.424696	−0.212348	−	0.977194i	$-0.568111\pi$
$233$	−6.48271	−0.424696	−0.212348	+	0.977194i	$0.568111\pi$
$234$	0.180749	0.0118159
$235$	4.59388	0.299672
$236$	−3.46796	−0.225745
$237$	18.8139	1.22209
$238$	−0.0757200	−0.00490820
$239$	−8.28300	−0.535783	−0.267892	−	0.963449i	$-0.586327\pi$
$239$	−8.28300	−0.535783	−0.267892	+	0.963449i	$0.586327\pi$
$240$	−27.5326	−1.77722
$241$	−11.8887	−0.765820	−0.382910	−	0.923786i	$-0.625078\pi$
$241$	−11.8887	−0.765820	−0.382910	+	0.923786i	$0.625078\pi$
$242$	−1.65698	−0.106515
$243$	−20.0171	−1.28410
$244$	22.0553	1.41195
$245$	−2.99129	−0.191107
$246$	1.76697	0.112658
$247$	−1.36989	−0.0871640
$248$	−3.03362	−0.192635
$249$	−15.7731	−0.999580
$250$	−0.238318	−0.0150726
$251$	−3.58705	−0.226412	−0.113206	−	0.993572i	$-0.536112\pi$
$251$	−3.58705	−0.226412	−0.113206	+	0.993572i	$0.536112\pi$
$252$	4.76046	0.299881
$253$	−31.3258	−1.96944
$254$	1.10775	0.0695066
$255$	6.94281	0.434776
$256$	15.5433	0.971455
$257$	7.95989	0.496524	0.248262	−	0.968693i	$-0.420141\pi$
$257$	7.95989	0.496524	0.248262	+	0.968693i	$0.420141\pi$
$258$	−1.34527	−0.0837530
$259$	8.50525	0.528490
$260$	−5.96543	−0.369960
$261$	−10.9702	−0.679036
$262$	−0.870078	−0.0537536
$263$	26.0672	1.60737	0.803686	−	0.595054i	$-0.202869\pi$
$263$	26.0672	1.60737	0.803686	+	0.595054i	$0.202869\pi$
$264$	4.02541	0.247747
$265$	−21.1016	−1.29626
$266$	0.103728	0.00635998
$267$	23.0761	1.41223
$268$	−19.5439	−1.19384
$269$	−4.01367	−0.244718	−0.122359	−	0.992486i	$-0.539046\pi$
$269$	−4.01367	−0.244718	−0.122359	+	0.992486i	$0.539046\pi$
$270$	−0.322222	−0.0196098
$271$	−27.8789	−1.69352	−0.846762	−	0.531972i	$-0.821451\pi$
$271$	−27.8789	−1.69352	−0.846762	+	0.531972i	$0.821451\pi$
$272$	−3.96563	−0.240452
$273$	2.32101	0.140474
$274$	0.394030	0.0238042
$275$	22.6383	1.36514
$276$	25.2858	1.52202
$277$	9.76388	0.586655	0.293327	−	0.956012i	$-0.405237\pi$
$277$	9.76388	0.586655	0.293327	+	0.956012i	$0.405237\pi$
$278$	−0.218605	−0.0131110
$279$	−23.9430	−1.43343
$280$	0.904704	0.0540664
$281$	−12.0792	−0.720586	−0.360293	−	0.932839i	$-0.617323\pi$
$281$	−12.0792	−0.720586	−0.360293	+	0.932839i	$0.617323\pi$
$282$	0.269903	0.0160725
$283$	−7.61619	−0.452736	−0.226368	−	0.974042i	$-0.572685\pi$
$283$	−7.61619	−0.452736	−0.226368	+	0.974042i	$0.572685\pi$
$284$	25.4751	1.51167
$285$	−9.51088	−0.563376
$286$	0.434207	0.0256752
$287$	10.0541	0.593472
$288$	−2.16071	−0.127321
$289$	1.00000	0.0588235
$290$	−1.04092	−0.0611249
$291$	−21.3990	−1.25443
$292$	23.1740	1.35616
$293$	28.6461	1.67352	0.836762	−	0.547567i	$-0.184446\pi$
$293$	28.6461	1.67352	0.836762	+	0.547567i	$0.184446\pi$
$294$	−0.175747	−0.0102498
$295$	−5.20175	−0.302857
$296$	−2.57238	−0.149516
$297$	−8.15778	−0.473362
$298$	0.407861	0.0236267
$299$	5.46282	0.315923
$300$	−18.2733	−1.05501
$301$	−7.65461	−0.441204
$302$	0.344713	0.0198360
$303$	6.73727	0.387046
$304$	5.43248	0.311574
$305$	33.0817	1.89426
$306$	0.180749	0.0103327
$307$	−20.7760	−1.18575	−0.592874	−	0.805295i	$-0.702007\pi$
$307$	−20.7760	−1.18575	−0.592874	+	0.805295i	$0.702007\pi$
$308$	11.4359	0.651619
$309$	−2.11516	−0.120327
$310$	−2.27187	−0.129033
$311$	15.2296	0.863591	0.431796	−	0.901971i	$-0.357880\pi$
$311$	15.2296	0.863591	0.431796	+	0.901971i	$0.357880\pi$
$312$	−0.701979	−0.0397417
$313$	25.2651	1.42807	0.714033	−	0.700112i	$-0.246867\pi$
$313$	25.2651	1.42807	0.714033	+	0.700112i	$0.246867\pi$
$314$	−0.816392	−0.0460717
$315$	7.14043	0.402318
$316$	−16.1654	−0.909373
$317$	−23.4025	−1.31441	−0.657206	−	0.753711i	$-0.728262\pi$
$317$	−23.4025	−1.31441	−0.657206	+	0.753711i	$0.728262\pi$
$318$	−1.23978	−0.0695234
$319$	−26.3532	−1.47550
$320$	23.5197	1.31479
$321$	−27.9535	−1.56021
$322$	−0.413645	−0.0230515
$323$	−1.36989	−0.0762227
$324$	20.8662	1.15923
$325$	−3.94783	−0.218986
$326$	−0.385641	−0.0213587
$327$	44.8806	2.48190
$328$	−3.04081	−0.167901
$329$	1.53575	0.0846687
$330$	3.01462	0.165949
$331$	30.7142	1.68821	0.844104	−	0.536180i	$-0.180133\pi$
$331$	30.7142	1.68821	0.844104	+	0.536180i	$0.180133\pi$
$332$	13.5526	0.743798
$333$	−20.3026	−1.11258
$334$	−0.703095	−0.0384717
$335$	−29.3148	−1.60164
$336$	−9.20426	−0.502133
$337$	−12.8963	−0.702506	−0.351253	−	0.936281i	$-0.614244\pi$
$337$	−12.8963	−0.702506	−0.351253	+	0.936281i	$0.614244\pi$
$338$	−0.0757200	−0.00411863
$339$	9.48087	0.514930
$340$	−5.96543	−0.323521
$341$	−57.5174	−3.11474
$342$	−0.247606	−0.0133890
$343$	−1.00000	−0.0539949
$344$	2.31511	0.124822
$345$	37.9273	2.04194
$346$	−1.82100	−0.0978974
$347$	−14.4989	−0.778340	−0.389170	−	0.921166i	$-0.627238\pi$
$347$	−14.4989	−0.778340	−0.389170	+	0.921166i	$0.627238\pi$
$348$	21.2719	1.14030
$349$	−9.18303	−0.491556	−0.245778	−	0.969326i	$-0.579043\pi$
$349$	−9.18303	−0.491556	−0.245778	+	0.969326i	$0.579043\pi$
$350$	0.298930	0.0159785
$351$	1.42261	0.0759333
$352$	−5.19058	−0.276659
$353$	13.6724	0.727706	0.363853	−	0.931456i	$-0.381461\pi$
$353$	13.6724	0.727706	0.363853	+	0.931456i	$0.381461\pi$
$354$	−0.305617	−0.0162434
$355$	38.2113	2.02804
$356$	−19.8276	−1.05086
$357$	2.32101	0.122841
$358$	0.232863	0.0123072
$359$	−0.0660933	−0.00348827	−0.00174414	−	0.999998i	$-0.500555\pi$
$359$	−0.0660933	−0.00348827	−0.00174414	+	0.999998i	$0.500555\pi$
$360$	−2.15959	−0.113821
$361$	−17.1234	−0.901232
$362$	−0.0133880	−0.000703656 0
$363$	50.7907	2.66582
$364$	−1.99427	−0.104528
$365$	34.7598	1.81941
$366$	1.94364	0.101596
$367$	9.16434	0.478375	0.239187	−	0.970973i	$-0.423119\pi$
$367$	9.16434	0.478375	0.239187	+	0.970973i	$0.423119\pi$
$368$	−21.6635	−1.12929
$369$	−23.9997	−1.24938
$370$	−1.92645	−0.100151
$371$	−7.05436	−0.366244
$372$	46.4273	2.40714
$373$	13.4621	0.697041	0.348520	−	0.937301i	$-0.386684\pi$
$373$	13.4621	0.697041	0.348520	+	0.937301i	$0.386684\pi$
$374$	0.434207	0.0224523
$375$	7.30504	0.377230
$376$	−0.464482	−0.0239538
$377$	4.59566	0.236688
$378$	−0.107720	−0.00554053
$379$	−28.9867	−1.48895	−0.744473	−	0.667652i	$-0.767299\pi$
$379$	−28.9867	−1.48895	−0.744473	+	0.667652i	$0.767299\pi$
$380$	8.17198	0.419214
$381$	−33.9554	−1.73959
$382$	−0.705764	−0.0361100
$383$	1.78626	0.0912737	0.0456368	−	0.998958i	$-0.485468\pi$
$383$	1.78626	0.0912737	0.0456368	+	0.998958i	$0.485468\pi$
$384$	5.58366	0.284940
$385$	17.1532	0.874207
$386$	1.61380	0.0821400
$387$	18.2721	0.928823
$388$	18.3866	0.933436
$389$	35.8791	1.81914	0.909571	−	0.415548i	$-0.136410\pi$
$389$	35.8791	1.81914	0.909571	+	0.415548i	$0.136410\pi$
$390$	−0.525710	−0.0266203
$391$	5.46282	0.276267
$392$	0.302446	0.0152758
$393$	26.6701	1.34533
$394$	−0.985684	−0.0496581
$395$	−24.2472	−1.22001
$396$	−27.2982	−1.37179
$397$	−21.3999	−1.07403	−0.537016	−	0.843572i	$-0.680449\pi$
$397$	−21.3999	−1.07403	−0.537016	+	0.843572i	$0.680449\pi$
$398$	0.583131	0.0292297
$399$	−3.17952	−0.159175
$400$	15.6556	0.782781
$401$	−6.61412	−0.330293	−0.165147	−	0.986269i	$-0.552810\pi$
$401$	−6.61412	−0.330293	−0.165147	+	0.986269i	$0.552810\pi$
$402$	−1.72233	−0.0859019
$403$	10.0303	0.499645
$404$	−5.78883	−0.288005
$405$	31.2982	1.55522
$406$	−0.347983	−0.0172701
$407$	−48.7723	−2.41755
$408$	−0.701979	−0.0347531
$409$	−9.57071	−0.473241	−0.236620	−	0.971602i	$-0.576040\pi$
$409$	−9.57071	−0.473241	−0.236620	+	0.971602i	$0.576040\pi$
$410$	−2.27725	−0.112465
$411$	−12.0780	−0.595763
$412$	1.81740	0.0895368
$413$	−1.73896	−0.0855688
$414$	0.987400	0.0485280
$415$	20.3282	0.997873
$416$	0.905170	0.0443796
$417$	6.70077	0.328138
$418$	−0.594816	−0.0290934
$419$	15.1697	0.741086	0.370543	−	0.928815i	$-0.379172\pi$
$419$	15.1697	0.741086	0.370543	+	0.928815i	$0.379172\pi$
$420$	−13.8458	−0.675606
$421$	10.7852	0.525637	0.262818	−	0.964845i	$-0.415348\pi$
$421$	10.7852	0.525637	0.262818	+	0.964845i	$0.415348\pi$
$422$	0.647489	0.0315193
$423$	−3.66595	−0.178245
$424$	2.13356	0.103615
$425$	−3.94783	−0.191498
$426$	2.24502	0.108771
$427$	11.0594	0.535199
$428$	24.0183	1.16097
$429$	−13.3095	−0.642590
$430$	1.73377	0.0836100
$431$	21.5974	1.04031	0.520155	−	0.854072i	$-0.325874\pi$
$431$	21.5974	1.04031	0.520155	+	0.854072i	$0.325874\pi$
$432$	−5.64155	−0.271429
$433$	24.3800	1.17163	0.585815	−	0.810445i	$-0.300774\pi$
$433$	24.3800	1.17163	0.585815	+	0.810445i	$0.300774\pi$
$434$	−0.759494	−0.0364569
$435$	31.9068	1.52981
$436$	−38.5625	−1.84681
$437$	−7.48346	−0.357982
$438$	2.04223	0.0975817
$439$	−14.2310	−0.679206	−0.339603	−	0.940569i	$-0.610293\pi$
$439$	−14.2310	−0.679206	−0.339603	+	0.940569i	$0.610293\pi$
$440$	−5.18791	−0.247324
$441$	2.38707	0.113670
$442$	−0.0757200	−0.00360163
$443$	−18.0860	−0.859290	−0.429645	−	0.902998i	$-0.641361\pi$
$443$	−18.0860	−0.859290	−0.429645	+	0.902998i	$0.641361\pi$
$444$	39.3683	1.86834
$445$	−29.7403	−1.40982
$446$	1.67593	0.0793577
$447$	−12.5019	−0.591321
$448$	7.86272	0.371479
$449$	24.0201	1.13358	0.566788	−	0.823864i	$-0.308186\pi$
$449$	24.0201	1.13358	0.566788	+	0.823864i	$0.308186\pi$
$450$	−0.713566	−0.0336378
$451$	−57.6537	−2.71481
$452$	−8.14620	−0.383165
$453$	−10.5663	−0.496448
$454$	1.74209	0.0817604
$455$	−2.99129	−0.140234
$456$	0.961634	0.0450326
$457$	−33.5221	−1.56810	−0.784050	−	0.620698i	$-0.786849\pi$
$457$	−33.5221	−1.56810	−0.784050	+	0.620698i	$0.786849\pi$
$458$	−0.187226	−0.00874851
$459$	1.42261	0.0664018
$460$	−32.5881	−1.51943
$461$	31.4992	1.46706	0.733531	−	0.679656i	$-0.237871\pi$
$461$	31.4992	1.46706	0.733531	+	0.679656i	$0.237871\pi$
$462$	1.00780	0.0468870
$463$	−22.9898	−1.06842	−0.534212	−	0.845350i	$-0.679392\pi$
$463$	−22.9898	−1.06842	−0.534212	+	0.845350i	$0.679392\pi$
$464$	−18.2247	−0.846060
$465$	69.6384	3.22940
$466$	0.490871	0.0227392
$467$	−14.7676	−0.683364	−0.341682	−	0.939816i	$-0.610997\pi$
$467$	−14.7676	−0.683364	−0.341682	+	0.939816i	$0.610997\pi$
$468$	4.76046	0.220052
$469$	−9.80006	−0.452525
$470$	−0.347849	−0.0160451
$471$	25.0244	1.15306
$472$	0.525943	0.0242085
$473$	43.8944	2.01827
$474$	−1.42459	−0.0654335
$475$	5.40809	0.248140
$476$	−1.99427	−0.0914071
$477$	16.8393	0.771017
$478$	0.627189	0.0286870
$479$	−4.52156	−0.206595	−0.103298	−	0.994650i	$-0.532939\pi$
$479$	−4.52156	−0.206595	−0.103298	+	0.994650i	$0.532939\pi$
$480$	6.28442	0.286843
$481$	8.50525	0.387806
$482$	0.900214	0.0410036
$483$	12.6792	0.576925
$484$	−43.6406	−1.98366
$485$	27.5788	1.25229
$486$	1.51570	0.0687534
$487$	−10.9106	−0.494404	−0.247202	−	0.968964i	$-0.579511\pi$
$487$	−10.9106	−0.494404	−0.247202	+	0.968964i	$0.579511\pi$
$488$	−3.34486	−0.151415
$489$	11.8208	0.534557
$490$	0.226501	0.0102323
$491$	10.6414	0.480240	0.240120	−	0.970743i	$-0.422813\pi$
$491$	10.6414	0.480240	0.240120	+	0.970743i	$0.422813\pi$
$492$	46.5372	2.09806
$493$	4.59566	0.206978
$494$	0.103728	0.00466695
$495$	−40.9459	−1.84038
$496$	−39.7765	−1.78602
$497$	12.7742	0.573000
$498$	1.19434	0.0535196
$499$	36.6406	1.64026	0.820130	−	0.572177i	$-0.193901\pi$
$499$	36.6406	1.64026	0.820130	+	0.572177i	$0.193901\pi$
$500$	−6.27667	−0.280701
$501$	21.5516	0.962855
$502$	0.271611	0.0121226
$503$	−4.06832	−0.181397	−0.0906987	−	0.995878i	$-0.528910\pi$
$503$	−4.06832	−0.181397	−0.0906987	+	0.995878i	$0.528910\pi$
$504$	−0.721960	−0.0321587
$505$	−8.68294	−0.386386
$506$	2.37199	0.105448
$507$	2.32101	0.103080
$508$	29.1753	1.29445
$509$	25.9423	1.14987	0.574936	−	0.818198i	$-0.305027\pi$
$509$	25.9423	1.14987	0.574936	+	0.818198i	$0.305027\pi$
$510$	−0.525710	−0.0232788
$511$	11.6203	0.514053
$512$	−5.98835	−0.264650
$513$	−1.94882	−0.0860425
$514$	−0.602723	−0.0265850
$515$	2.72600	0.120122
$516$	−35.4309	−1.55976
$517$	−8.80657	−0.387313
$518$	−0.644018	−0.0282965
$519$	55.8181	2.45014
$520$	0.904704	0.0396739
$521$	−22.6821	−0.993723	−0.496861	−	0.867830i	$-0.665514\pi$
$521$	−22.6821	−0.993723	−0.496861	+	0.867830i	$0.665514\pi$
$522$	0.830661	0.0363570
$523$	12.6502	0.553157	0.276578	−	0.960991i	$-0.410799\pi$
$523$	12.6502	0.553157	0.276578	+	0.960991i	$0.410799\pi$
$524$	−22.9156	−1.00107
$525$	−9.16293	−0.399903
$526$	−1.97381	−0.0860621
$527$	10.0303	0.436926
$528$	52.7806	2.29698
$529$	6.84237	0.297494
$530$	1.59782	0.0694047
$531$	4.15103	0.180139
$532$	2.73192	0.118444
$533$	10.0541	0.435489
$534$	−1.74732	−0.0756141
$535$	36.0262	1.55755
$536$	2.96399	0.128025
$537$	−7.13782	−0.308020
$538$	0.303915	0.0131027
$539$	5.73437	0.246997
$540$	−8.48649	−0.365200
$541$	32.3156	1.38936	0.694679	−	0.719320i	$-0.255547\pi$
$541$	32.3156	1.38936	0.694679	+	0.719320i	$0.255547\pi$
$542$	2.11099	0.0906749
$543$	0.410374	0.0176108
$544$	0.905170	0.0388088
$545$	−57.8417	−2.47767
$546$	−0.175747	−0.00752126
$547$	39.6861	1.69685	0.848427	−	0.529313i	$-0.177550\pi$
$547$	39.6861	1.69685	0.848427	+	0.529313i	$0.177550\pi$
$548$	10.3777	0.443314
$549$	−26.3995	−1.12670
$550$	−1.71417	−0.0730926
$551$	−6.29554	−0.268199
$552$	−3.83478	−0.163219
$553$	−8.10592	−0.344699
$554$	−0.739322	−0.0314108
$555$	59.0503	2.50655
$556$	−5.75747	−0.244171
$557$	26.3554	1.11671	0.558356	−	0.829601i	$-0.311432\pi$
$557$	26.3554	1.11671	0.558356	+	0.829601i	$0.311432\pi$
$558$	1.81297	0.0767490
$559$	−7.65461	−0.323755
$560$	11.8624	0.501276
$561$	−13.3095	−0.561928
$562$	0.914639	0.0385817
$563$	18.8598	0.794847	0.397423	−	0.917635i	$-0.369904\pi$
$563$	18.8598	0.794847	0.397423	+	0.917635i	$0.369904\pi$
$564$	7.10854	0.299324
$565$	−12.2189	−0.514051
$566$	0.576698	0.0242404
$567$	10.4631	0.439409
$568$	−3.86350	−0.162109
$569$	13.2685	0.556247	0.278123	−	0.960545i	$-0.410288\pi$
$569$	13.2685	0.556247	0.278123	+	0.960545i	$0.410288\pi$
$570$	0.720164	0.0301644
$571$	−26.4268	−1.10593	−0.552964	−	0.833205i	$-0.686503\pi$
$571$	−26.4268	−1.10593	−0.552964	+	0.833205i	$0.686503\pi$
$572$	11.4359	0.478158
$573$	21.6334	0.903748
$574$	−0.761293	−0.0317758
$575$	−21.5663	−0.899375
$576$	−18.7689	−0.782037
$577$	−41.5319	−1.72900	−0.864498	−	0.502636i	$-0.832364\pi$
$577$	−41.5319	−1.72900	−0.864498	+	0.502636i	$0.832364\pi$
$578$	−0.0757200	−0.00314954
$579$	−49.4668	−2.05577
$580$	−27.4151	−1.13835
$581$	6.79580	0.281937
$582$	1.62033	0.0671650
$583$	40.4523	1.67536
$584$	−3.51452	−0.145432
$585$	7.14043	0.295220
$586$	−2.16909	−0.0896041
$587$	−26.6959	−1.10186	−0.550929	−	0.834552i	$-0.685727\pi$
$587$	−26.6959	−1.10186	−0.550929	+	0.834552i	$0.685727\pi$
$588$	−4.62871	−0.190885
$589$	−13.7404	−0.566163
$590$	0.393876	0.0162156
$591$	30.2137	1.24282
$592$	−33.7287	−1.38624
$593$	−31.6804	−1.30096	−0.650478	−	0.759525i	$-0.725431\pi$
$593$	−31.6804	−1.30096	−0.650478	+	0.759525i	$0.725431\pi$
$594$	0.617707	0.0253448
$595$	−2.99129	−0.122631
$596$	10.7420	0.440008
$597$	−17.8744	−0.731551
$598$	−0.413645	−0.0169152
$599$	26.4182	1.07942	0.539709	−	0.841852i	$-0.318534\pi$
$599$	26.4182	1.07942	0.539709	+	0.841852i	$0.318534\pi$
$600$	2.77129	0.113138
$601$	22.7366	0.927443	0.463722	−	0.885981i	$-0.346514\pi$
$601$	22.7366	0.927443	0.463722	+	0.885981i	$0.346514\pi$
$602$	0.579607	0.0236230
$603$	23.3934	0.952654
$604$	9.07883	0.369412
$605$	−65.4585	−2.66127
$606$	−0.510147	−0.0207233
$607$	−3.03701	−0.123268	−0.0616341	−	0.998099i	$-0.519631\pi$
$607$	−3.03701	−0.123268	−0.0616341	+	0.998099i	$0.519631\pi$
$608$	−1.23998	−0.0502879
$609$	10.6665	0.432230
$610$	−2.50495	−0.101422
$611$	1.53575	0.0621299
$612$	4.76046	0.192430
$613$	−5.17686	−0.209091	−0.104546	−	0.994520i	$-0.533339\pi$
$613$	−5.17686	−0.209091	−0.104546	+	0.994520i	$0.533339\pi$
$614$	1.57316	0.0634875
$615$	69.8033	2.81474
$616$	−1.73434	−0.0698785
$617$	6.93458	0.279176	0.139588	−	0.990210i	$-0.455422\pi$
$617$	6.93458	0.279176	0.139588	+	0.990210i	$0.455422\pi$
$618$	0.160160	0.00644258
$619$	39.1133	1.57210	0.786049	−	0.618165i	$-0.212123\pi$
$619$	39.1133	1.57210	0.786049	+	0.618165i	$0.212123\pi$
$620$	−59.8350	−2.40303
$621$	7.77146	0.311858
$622$	−1.15319	−0.0462385
$623$	−9.94229	−0.398329
$624$	−9.20426	−0.368465
$625$	−29.1538	−1.16615
$626$	−1.91307	−0.0764617
$627$	18.2326	0.728139
$628$	−21.5016	−0.858008
$629$	8.50525	0.339126
$630$	−0.540673	−0.0215409
$631$	35.1958	1.40112	0.700561	−	0.713592i	$-0.252933\pi$
$631$	35.1958	1.40112	0.700561	+	0.713592i	$0.252933\pi$
$632$	2.45160	0.0975195
$633$	−19.8471	−0.788853
$634$	1.77203	0.0703765
$635$	43.7614	1.73662
$636$	−32.6525	−1.29476
$637$	−1.00000	−0.0396214
$638$	1.99547	0.0790013
$639$	−30.4929	−1.20628
$640$	−7.19617	−0.284453
$641$	−17.1347	−0.676778	−0.338389	−	0.941006i	$-0.609882\pi$
$641$	−17.1347	−0.676778	−0.338389	+	0.941006i	$0.609882\pi$
$642$	2.11664	0.0835371
$643$	−7.90036	−0.311560	−0.155780	−	0.987792i	$-0.549789\pi$
$643$	−7.90036	−0.311560	−0.155780	+	0.987792i	$0.549789\pi$
$644$	−10.8943	−0.429296
$645$	−53.1445	−2.09256
$646$	0.103728	0.00408113
$647$	−21.3512	−0.839401	−0.419701	−	0.907663i	$-0.637865\pi$
$647$	−21.3512	−0.839401	−0.419701	+	0.907663i	$0.637865\pi$
$648$	−3.16452	−0.124314
$649$	9.97186	0.391430
$650$	0.298930	0.0117250
$651$	23.2804	0.912430
$652$	−10.1568	−0.397770
$653$	−3.23913	−0.126757	−0.0633785	−	0.997990i	$-0.520188\pi$
$653$	−3.23913	−0.126757	−0.0633785	+	0.997990i	$0.520188\pi$
$654$	−3.39836	−0.132886
$655$	−34.3721	−1.34303
$656$	−39.8707	−1.55669
$657$	−27.7386	−1.08218
$658$	−0.116287	−0.00453335
$659$	−25.8976	−1.00883	−0.504413	−	0.863463i	$-0.668291\pi$
$659$	−25.8976	−1.00883	−0.504413	+	0.863463i	$0.668291\pi$
$660$	79.3970	3.09053
$661$	−13.7035	−0.533003	−0.266502	−	0.963834i	$-0.585868\pi$
$661$	−13.7035	−0.533003	−0.266502	+	0.963834i	$0.585868\pi$
$662$	−2.32568	−0.0903903
$663$	2.32101	0.0901404
$664$	−2.05536	−0.0797636
$665$	4.09774	0.158904
$666$	1.53732	0.0595698
$667$	25.1052	0.972078
$668$	−18.5177	−0.716471
$669$	−51.3715	−1.98614
$670$	2.21972	0.0857553
$671$	−63.4184	−2.44824
$672$	2.10091	0.0810442
$673$	8.37425	0.322804	0.161402	−	0.986889i	$-0.448399\pi$
$673$	8.37425	0.322804	0.161402	+	0.986889i	$0.448399\pi$
$674$	0.976507	0.0376137
$675$	−5.61622	−0.216168
$676$	−1.99427	−0.0767026
$677$	50.3790	1.93622	0.968111	−	0.250520i	$-0.0806015\pi$
$677$	50.3790	1.93622	0.968111	+	0.250520i	$0.0806015\pi$
$678$	−0.717892	−0.0275705
$679$	9.21971	0.353820
$680$	0.904704	0.0346938
$681$	−53.3994	−2.04627
$682$	4.35522	0.166770
$683$	−0.0471964	−0.00180592	−0.000902960	−	1.00000i	$-0.500287\pi$
$683$	−0.0471964	−0.00180592	−0.000902960		1.00000i	$0.500287\pi$
$684$	−6.52130	−0.249348
$685$	15.5660	0.594746
$686$	0.0757200	0.00289100
$687$	5.73895	0.218955
$688$	30.3554	1.15729
$689$	−7.05436	−0.268750
$690$	−2.87186	−0.109330
$691$	−13.9710	−0.531482	−0.265741	−	0.964044i	$-0.585617\pi$
$691$	−13.9710	−0.531482	−0.265741	+	0.964044i	$0.585617\pi$
$692$	−47.9603	−1.82318
$693$	−13.6884	−0.519978
$694$	1.09786	0.0416740
$695$	−8.63590	−0.327578
$696$	−3.22606	−0.122283
$697$	10.0541	0.380824
$698$	0.695339	0.0263190
$699$	−15.0464	−0.569107
$700$	7.87302	0.297572
$701$	42.5199	1.60595	0.802977	−	0.596010i	$-0.203248\pi$
$701$	42.5199	1.60595	0.802977	+	0.596010i	$0.203248\pi$
$702$	−0.107720	−0.00406563
$703$	−11.6513	−0.439435
$704$	−45.0878	−1.69931
$705$	10.6624	0.401570
$706$	−1.03527	−0.0389630
$707$	−2.90274	−0.109169
$708$	−8.04915	−0.302506
$709$	−1.01010	−0.0379351	−0.0189675	−	0.999820i	$-0.506038\pi$
$709$	−1.01010	−0.0379351	−0.0189675	+	0.999820i	$0.506038\pi$
$710$	−2.89336	−0.108586
$711$	19.3494	0.725659
$712$	3.00700	0.112692
$713$	54.7937	2.05204
$714$	−0.175747	−0.00657716
$715$	17.1532	0.641493
$716$	6.13299	0.229201
$717$	−19.2249	−0.717967
$718$	0.00500459	0.000186770 0
$719$	0.820627	0.0306042	0.0153021	−	0.999883i	$-0.495129\pi$
$719$	0.820627	0.0306042	0.0153021	+	0.999883i	$0.495129\pi$
$720$	−28.3163	−1.05529
$721$	0.911312	0.0339390
$722$	1.29658	0.0482539
$723$	−27.5938	−1.02622
$724$	−0.352604	−0.0131044
$725$	−18.1429	−0.673809
$726$	−3.84587	−0.142734
$727$	−27.4611	−1.01848	−0.509238	−	0.860626i	$-0.670073\pi$
$727$	−27.4611	−1.01848	−0.509238	+	0.860626i	$0.670073\pi$
$728$	0.302446	0.0112094
$729$	−15.0705	−0.558167
$730$	−2.63201	−0.0974152
$731$	−7.65461	−0.283116
$732$	51.1905	1.89205
$733$	5.58661	0.206346	0.103173	−	0.994663i	$-0.467100\pi$
$733$	5.58661	0.206346	0.103173	+	0.994663i	$0.467100\pi$
$734$	−0.693925	−0.0256132
$735$	−6.94281	−0.256089
$736$	4.94478	0.182267
$737$	56.1972	2.07005
$738$	1.81726	0.0668943
$739$	24.4714	0.900194	0.450097	−	0.892980i	$-0.351389\pi$
$739$	24.4714	0.900194	0.450097	+	0.892980i	$0.351389\pi$
$740$	−50.7375	−1.86515
$741$	−3.17952	−0.116803
$742$	0.534156	0.0196095
$743$	−13.4500	−0.493433	−0.246717	−	0.969088i	$-0.579352\pi$
$743$	−13.4500	−0.493433	−0.246717	+	0.969088i	$0.579352\pi$
$744$	−7.04106	−0.258138
$745$	16.1124	0.590312
$746$	−1.01935	−0.0373211
$747$	−16.2221	−0.593534
$748$	11.4359	0.418137
$749$	12.0437	0.440067
$750$	−0.553138	−0.0201977
$751$	33.0213	1.20497	0.602483	−	0.798132i	$-0.294178\pi$
$751$	33.0213	1.20497	0.602483	+	0.798132i	$0.294178\pi$
$752$	−6.09023	−0.222088
$753$	−8.32556	−0.303400
$754$	−0.347983	−0.0126728
$755$	13.6177	0.495601
$756$	−2.83706	−0.103183
$757$	−39.8309	−1.44768	−0.723839	−	0.689969i	$-0.757624\pi$
$757$	−39.8309	−1.44768	−0.723839	+	0.689969i	$0.757624\pi$
$758$	2.19487	0.0797214
$759$	−72.7075	−2.63911
$760$	−1.23934	−0.0449558
$761$	−24.7185	−0.896046	−0.448023	−	0.894022i	$-0.647872\pi$
$761$	−24.7185	−0.896046	−0.448023	+	0.894022i	$0.647872\pi$
$762$	2.57110	0.0931413
$763$	−19.3367	−0.700036
$764$	−18.5880	−0.672489
$765$	7.14043	0.258163
$766$	−0.135256	−0.00488699
$767$	−1.73896	−0.0627903
$768$	36.0761	1.30178
$769$	−48.0567	−1.73297	−0.866484	−	0.499205i	$-0.833626\pi$
$769$	−48.0567	−1.73297	−0.866484	+	0.499205i	$0.833626\pi$
$770$	−1.29884	−0.0468069
$771$	18.4750	0.665359
$772$	42.5031	1.52972
$773$	31.8597	1.14591	0.572956	−	0.819586i	$-0.305797\pi$
$773$	31.8597	1.14591	0.572956	+	0.819586i	$0.305797\pi$
$774$	−1.38356	−0.0497312
$775$	−39.5979	−1.42240
$776$	−2.78846	−0.100100
$777$	19.7407	0.708195
$778$	−2.71677	−0.0974008
$779$	−13.7729	−0.493467
$780$	−13.8458	−0.495760
$781$	−73.2519	−2.62116
$782$	−0.413645	−0.0147919
$783$	6.53783	0.233643
$784$	3.96563	0.141630
$785$	−32.2513	−1.15110
$786$	−2.01946	−0.0720317
$787$	45.8356	1.63386	0.816932	−	0.576734i	$-0.195673\pi$
$787$	45.8356	1.63386	0.816932	+	0.576734i	$0.195673\pi$
$788$	−25.9603	−0.924798
$789$	60.5021	2.15393
$790$	1.83600	0.0653218
$791$	−4.08481	−0.145239
$792$	4.13999	0.147108
$793$	11.0594	0.392729
$794$	1.62040	0.0575060
$795$	−48.9771	−1.73704
$796$	15.3581	0.544355
$797$	−20.5434	−0.727686	−0.363843	−	0.931460i	$-0.618536\pi$
$797$	−20.5434	−0.727686	−0.363843	+	0.931460i	$0.618536\pi$
$798$	0.240754	0.00852258
$799$	1.53575	0.0543310
$800$	−3.57345	−0.126341
$801$	23.7329	0.838562
$802$	0.500821	0.0176846
$803$	−66.6353	−2.35151
$804$	−45.3616	−1.59978
$805$	−16.3409	−0.575940
$806$	−0.759494	−0.0267520
$807$	−9.31575	−0.327930
$808$	0.877921	0.0308852
$809$	−32.0530	−1.12692	−0.563462	−	0.826142i	$-0.690531\pi$
$809$	−32.0530	−1.12692	−0.563462	+	0.826142i	$0.690531\pi$
$810$	−2.36990	−0.0832698
$811$	−46.3431	−1.62733	−0.813663	−	0.581337i	$-0.802530\pi$
$811$	−46.3431	−1.62733	−0.813663	+	0.581337i	$0.802530\pi$
$812$	−9.16496	−0.321627
$813$	−64.7071	−2.26938
$814$	3.69304	0.129441
$815$	−15.2346	−0.533645
$816$	−9.20426	−0.322213
$817$	10.4860	0.366858
$818$	0.724694	0.0253383
$819$	2.38707	0.0834110
$820$	−59.9768	−2.09448
$821$	−12.4633	−0.434972	−0.217486	−	0.976063i	$-0.569786\pi$
$821$	−12.4633	−0.434972	−0.217486	+	0.976063i	$0.569786\pi$
$822$	0.914546	0.0318985
$823$	−21.6700	−0.755370	−0.377685	−	0.925934i	$-0.623280\pi$
$823$	−21.6700	−0.755370	−0.377685	+	0.925934i	$0.623280\pi$
$824$	−0.275623	−0.00960177
$825$	52.5437	1.82934
$826$	0.131674	0.00458154
$827$	37.9405	1.31932	0.659660	−	0.751564i	$-0.270700\pi$
$827$	37.9405	1.31932	0.659660	+	0.751564i	$0.270700\pi$
$828$	26.0055	0.903754
$829$	0.654152	0.0227196	0.0113598	−	0.999935i	$-0.496384\pi$
$829$	0.654152	0.0227196	0.0113598	+	0.999935i	$0.496384\pi$
$830$	−1.53925	−0.0534283
$831$	22.6620	0.786137
$832$	7.86272	0.272591
$833$	−1.00000	−0.0346479
$834$	−0.507383	−0.0175692
$835$	−27.7755	−0.961212
$836$	−15.6659	−0.541816
$837$	14.2692	0.493216
$838$	−1.14865	−0.0396793
$839$	−41.2676	−1.42471	−0.712357	−	0.701817i	$-0.752372\pi$
$839$	−41.2676	−1.42471	−0.712357	+	0.701817i	$0.752372\pi$
$840$	2.09982	0.0724508
$841$	−7.87994	−0.271722
$842$	−0.816653	−0.0281437
$843$	−28.0360	−0.965610
$844$	17.0531	0.586994
$845$	−2.99129	−0.102904
$846$	0.277586	0.00954360
$847$	−21.8830	−0.751910
$848$	27.9750	0.960665
$849$	−17.6772	−0.606681
$850$	0.298930	0.0102532
$851$	46.4626	1.59272
$852$	59.1279	2.02569
$853$	−6.65736	−0.227944	−0.113972	−	0.993484i	$-0.536357\pi$
$853$	−6.65736	−0.227944	−0.113972	+	0.993484i	$0.536357\pi$
$854$	−0.837414	−0.0286557
$855$	−9.78160	−0.334524
$856$	−3.64257	−0.124500
$857$	−56.2615	−1.92186	−0.960928	−	0.276798i	$-0.910727\pi$
$857$	−56.2615	−1.92186	−0.960928	+	0.276798i	$0.910727\pi$
$858$	1.00780	0.0344056
$859$	−15.9034	−0.542617	−0.271309	−	0.962492i	$-0.587456\pi$
$859$	−15.9034	−0.542617	−0.271309	+	0.962492i	$0.587456\pi$
$860$	45.6631	1.55710
$861$	23.3355	0.795272
$862$	−1.63536	−0.0557005
$863$	4.58670	0.156133	0.0780666	−	0.996948i	$-0.475125\pi$
$863$	4.58670	0.156133	0.0780666	+	0.996948i	$0.475125\pi$
$864$	1.28770	0.0438086
$865$	−71.9378	−2.44596
$866$	−1.84606	−0.0627316
$867$	2.32101	0.0788255
$868$	−20.0031	−0.678949
$869$	46.4824	1.57681
$870$	−2.41598	−0.0819094
$871$	−9.80006	−0.332062
$872$	5.84831	0.198049
$873$	−22.0081	−0.744861
$874$	0.566648	0.0191671
$875$	−3.14736	−0.106400
$876$	53.7871	1.81730
$877$	−57.0868	−1.92768	−0.963842	−	0.266475i	$-0.914141\pi$
$877$	−57.0868	−1.92768	−0.963842	+	0.266475i	$0.914141\pi$
$878$	1.07757	0.0363662
$879$	66.4878	2.24258
$880$	−68.0232	−2.29306
$881$	41.0451	1.38285	0.691423	−	0.722450i	$-0.256984\pi$
$881$	41.0451	1.38285	0.691423	+	0.722450i	$0.256984\pi$
$882$	−0.180749	−0.00608614
$883$	−32.4847	−1.09320	−0.546599	−	0.837395i	$-0.684078\pi$
$883$	−32.4847	−1.09320	−0.546599	+	0.837395i	$0.684078\pi$
$884$	−1.99427	−0.0670744
$885$	−12.0733	−0.405839
$886$	1.36947	0.0460082
$887$	−0.597957	−0.0200774	−0.0100387	−	0.999950i	$-0.503195\pi$
$887$	−0.597957	−0.0200774	−0.0100387	+	0.999950i	$0.503195\pi$
$888$	−5.97051	−0.200357
$889$	14.6296	0.490661
$890$	2.25193	0.0754850
$891$	−59.9993	−2.01005
$892$	44.1397	1.47790
$893$	−2.10381	−0.0704013
$894$	0.946647	0.0316606
$895$	9.19916	0.307494
$896$	−2.40571	−0.0803690
$897$	12.6792	0.423347
$898$	−1.81880	−0.0606941
$899$	46.0958	1.53738
$900$	−18.7935	−0.626449
$901$	−7.05436	−0.235015
$902$	4.36554	0.145357
$903$	−17.7664	−0.591229
$904$	1.23543	0.0410900
$905$	−0.528887	−0.0175808
$906$	0.800080	0.0265809
$907$	−32.9138	−1.09288	−0.546442	−	0.837497i	$-0.684018\pi$
$907$	−32.9138	−1.09288	−0.546442	+	0.837497i	$0.684018\pi$
$908$	45.8821	1.52265
$909$	6.92904	0.229822
$910$	0.226501	0.00750843
$911$	33.8544	1.12165	0.560823	−	0.827935i	$-0.310485\pi$
$911$	33.8544	1.12165	0.560823	+	0.827935i	$0.310485\pi$
$912$	12.6088	0.417520
$913$	−38.9697	−1.28971
$914$	2.53830	0.0839594
$915$	76.7829	2.53837
$916$	−4.93105	−0.162926
$917$	−11.4907	−0.379457
$918$	−0.107720	−0.00355529
$919$	−16.6489	−0.549195	−0.274598	−	0.961559i	$-0.588545\pi$
$919$	−16.6489	−0.549195	−0.274598	+	0.961559i	$0.588545\pi$
$920$	4.94223	0.162941
$921$	−48.2212	−1.58894
$922$	−2.38512	−0.0785497
$923$	12.7742	0.420467
$924$	26.5427	0.873192
$925$	−33.5773	−1.10401
$926$	1.74079	0.0572058
$927$	−2.17537	−0.0714484
$928$	4.15985	0.136554
$929$	48.6803	1.59715	0.798574	−	0.601896i	$-0.205588\pi$
$929$	48.6803	1.59715	0.798574	+	0.601896i	$0.205588\pi$
$930$	−5.27302	−0.172909
$931$	1.36989	0.0448963
$932$	12.9282	0.423479
$933$	35.3480	1.15724
$934$	1.11820	0.0365888
$935$	17.1532	0.560969
$936$	−0.721960	−0.0235980
$937$	27.0573	0.883923	0.441961	−	0.897034i	$-0.354283\pi$
$937$	27.0573	0.883923	0.441961	+	0.897034i	$0.354283\pi$
$938$	0.742061	0.0242291
$939$	58.6404	1.91366
$940$	−9.16142	−0.298813
$941$	27.3172	0.890514	0.445257	−	0.895403i	$-0.353112\pi$
$941$	27.3172	0.890514	0.445257	+	0.895403i	$0.353112\pi$
$942$	−1.89485	−0.0617376
$943$	54.9234	1.78855
$944$	6.89609	0.224449
$945$	−4.25544	−0.138430
$946$	−3.32368	−0.108062
$947$	21.7110	0.705514	0.352757	−	0.935715i	$-0.385244\pi$
$947$	21.7110	0.705514	0.352757	+	0.935715i	$0.385244\pi$
$948$	−37.5199	−1.21859
$949$	11.6203	0.377212
$950$	−0.409501	−0.0132860
$951$	−54.3173	−1.76136
$952$	0.302446	0.00980233
$953$	19.7773	0.640648	0.320324	−	0.947308i	$-0.396208\pi$
$953$	19.7773	0.640648	0.320324	+	0.947308i	$0.396208\pi$
$954$	−1.27507	−0.0412819
$955$	−27.8809	−0.902206
$956$	16.5185	0.534247
$957$	−61.1660	−1.97722
$958$	0.342373	0.0110616
$959$	5.20377	0.168039
$960$	54.5894	1.76186
$961$	69.6068	2.24538
$962$	−0.644018	−0.0207640
$963$	−28.7492	−0.926429
$964$	23.7093	0.763624
$965$	63.7524	2.05226
$966$	−0.960072	−0.0308898
$967$	−22.0061	−0.707670	−0.353835	−	0.935308i	$-0.615123\pi$
$967$	−22.0061	−0.707670	−0.353835	+	0.935308i	$0.615123\pi$
$968$	6.61844	0.212725
$969$	−3.17952	−0.102141
$970$	−2.08827	−0.0670503
$971$	−39.0756	−1.25399	−0.626997	−	0.779021i	$-0.715716\pi$
$971$	−39.0756	−1.25399	−0.626997	+	0.779021i	$0.715716\pi$
$972$	39.9194	1.28042
$973$	−2.88701	−0.0925534
$974$	0.826147	0.0264715
$975$	−9.16293	−0.293449
$976$	−43.8573	−1.40384
$977$	11.6510	0.372750	0.186375	−	0.982479i	$-0.440326\pi$
$977$	11.6510	0.372750	0.186375	+	0.982479i	$0.440326\pi$
$978$	−0.895075	−0.0286213
$979$	57.0128	1.82214
$980$	5.96543	0.190559
$981$	46.1581	1.47371
$982$	−0.805768	−0.0257131
$983$	−22.5560	−0.719426	−0.359713	−	0.933063i	$-0.617125\pi$
$983$	−22.5560	−0.719426	−0.359713	+	0.933063i	$0.617125\pi$
$984$	−7.05773	−0.224992
$985$	−38.9391	−1.24070
$986$	−0.347983	−0.0110820
$987$	3.56449	0.113459
$988$	2.73192	0.0869141
$989$	−41.8157	−1.32966
$990$	3.10042	0.0985379
$991$	46.7348	1.48458	0.742290	−	0.670079i	$-0.233740\pi$
$991$	46.7348	1.48458	0.742290	+	0.670079i	$0.233740\pi$
$992$	9.07912	0.288262
$993$	71.2880	2.26226
$994$	−0.967261	−0.0306796
$995$	23.0364	0.730302
$996$	31.4558	0.996714
$997$	−7.46495	−0.236417	−0.118209	−	0.992989i	$-0.537715\pi$
$997$	−7.46495	−0.236417	−0.118209	+	0.992989i	$0.537715\pi$
$998$	−2.77443	−0.0878231
$999$	12.0997	0.382816

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1547.2.a.i.1.8	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1547.2.a.i.1.8	✓	14	1.1	even	1	trivial

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1547 = 7 \cdot 13 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1547.a (trivial)

\(f(q)\)	\(=\)	\(q-0.0757200 q^{2} +2.32101 q^{3} -1.99427 q^{4} -2.99129 q^{5} -0.175747 q^{6} -1.00000 q^{7} +0.302446 q^{8} +2.38707 q^{9} +0.226501 q^{10} +5.73437 q^{11} -4.62871 q^{12} -1.00000 q^{13} +0.0757200 q^{14} -6.94281 q^{15} +3.96563 q^{16} -1.00000 q^{17} -0.180749 q^{18} +1.36989 q^{19} +5.96543 q^{20} -2.32101 q^{21} -0.434207 q^{22} -5.46282 q^{23} +0.701979 q^{24} +3.94783 q^{25} +0.0757200 q^{26} -1.42261 q^{27} +1.99427 q^{28} -4.59566 q^{29} +0.525710 q^{30} -10.0303 q^{31} -0.905170 q^{32} +13.3095 q^{33} +0.0757200 q^{34} +2.99129 q^{35} -4.76046 q^{36} -8.50525 q^{37} -0.103728 q^{38} -2.32101 q^{39} -0.904704 q^{40} -10.0541 q^{41} +0.175747 q^{42} +7.65461 q^{43} -11.4359 q^{44} -7.14043 q^{45} +0.413645 q^{46} -1.53575 q^{47} +9.20426 q^{48} +1.00000 q^{49} -0.298930 q^{50} -2.32101 q^{51} +1.99427 q^{52} +7.05436 q^{53} +0.107720 q^{54} -17.1532 q^{55} -0.302446 q^{56} +3.17952 q^{57} +0.347983 q^{58} +1.73896 q^{59} +13.8458 q^{60} -11.0594 q^{61} +0.759494 q^{62} -2.38707 q^{63} -7.86272 q^{64} +2.99129 q^{65} -1.00780 q^{66} +9.80006 q^{67} +1.99427 q^{68} -12.6792 q^{69} -0.226501 q^{70} -12.7742 q^{71} +0.721960 q^{72} -11.6203 q^{73} +0.644018 q^{74} +9.16293 q^{75} -2.73192 q^{76} -5.73437 q^{77} +0.175747 q^{78} +8.10592 q^{79} -11.8624 q^{80} -10.4631 q^{81} +0.761293 q^{82} -6.79580 q^{83} +4.62871 q^{84} +2.99129 q^{85} -0.579607 q^{86} -10.6665 q^{87} +1.73434 q^{88} +9.94229 q^{89} +0.540673 q^{90} +1.00000 q^{91} +10.8943 q^{92} -23.2804 q^{93} +0.116287 q^{94} -4.09774 q^{95} -2.10091 q^{96} -9.21971 q^{97} -0.0757200 q^{98} +13.6884 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + q^{2} - 4 q^{3} + 13 q^{4} - 9 q^{5} - 10 q^{6} - 14 q^{7} + 18 q^{9} - 9 q^{10} + 3 q^{11} - 16 q^{12} - 14 q^{13} - q^{14} - q^{15} + 11 q^{16} - 14 q^{17} + 5 q^{18} - 17 q^{19} - 24 q^{20}+ \cdots + 33 q^{99}+O(q^{100}) \) 14 * q + q^2 - 4 * q^3 + 13 * q^4 - 9 * q^5 - 10 * q^6 - 14 * q^7 + 18 * q^9 - 9 * q^10 + 3 * q^11 - 16 * q^12 - 14 * q^13 - q^14 - q^15 + 11 * q^16 - 14 * q^17 + 5 * q^18 - 17 * q^19 - 24 * q^20 + 4 * q^21 - 22 * q^22 - 6 * q^23 - 18 * q^24 + 11 * q^25 - q^26 - 19 * q^27 - 13 * q^28 + 3 * q^29 + 8 * q^30 - 19 * q^31 - 4 * q^32 - 34 * q^33 - q^34 + 9 * q^35 + 25 * q^36 - 45 * q^37 - 7 * q^38 + 4 * q^39 - 24 * q^40 - 29 * q^41 + 10 * q^42 + 7 * q^43 + 10 * q^44 - 48 * q^45 + q^46 - 10 * q^47 - 38 * q^48 + 14 * q^49 + 12 * q^50 + 4 * q^51 - 13 * q^52 - 15 * q^53 - 44 * q^54 - 2 * q^55 - 21 * q^57 - 42 * q^58 + 3 * q^59 + 8 * q^60 - 34 * q^61 - 16 * q^62 - 18 * q^63 - 20 * q^64 + 9 * q^65 - 16 * q^66 - 14 * q^67 - 13 * q^68 + 4 * q^69 + 9 * q^70 + 4 * q^71 - 6 * q^72 - 45 * q^73 - 22 * q^74 - 3 * q^75 - 29 * q^76 - 3 * q^77 + 10 * q^78 + 23 * q^79 - 37 * q^80 + 6 * q^81 + 15 * q^82 - 25 * q^83 + 16 * q^84 + 9 * q^85 + 40 * q^86 - 7 * q^87 - 36 * q^88 + 9 * q^89 - 57 * q^90 + 14 * q^91 - 38 * q^92 - 27 * q^93 + 13 * q^94 + 13 * q^95 + 3 * q^96 - 76 * q^97 + q^98 + 33 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more