LMFDB - Embedded newform 4009.2.a.b.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4009,2,Mod(1,4009)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4009, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4009 = 19 \cdot 211 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4009.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:	$32.0120261703$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.24698$ of defining polynomial
Character	$\chi$	$=$	4009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.24698 q^{2} -2.24698 q^{3} +3.04892 q^{4} -2.35690 q^{5} -5.04892 q^{6} +2.69202 q^{7} +2.35690 q^{8} +2.04892 q^{9} +O(q^{10})$	\(q+2.24698 q^{2} -2.24698 q^{3} +3.04892 q^{4} -2.35690 q^{5} -5.04892 q^{6} +2.69202 q^{7} +2.35690 q^{8} +2.04892 q^{9} -5.29590 q^{10} +0.109916 q^{11} -6.85086 q^{12} +6.04892 q^{14} +5.29590 q^{15} -0.801938 q^{16} -0.554958 q^{17} +4.60388 q^{18} +1.00000 q^{19} -7.18598 q^{20} -6.04892 q^{21} +0.246980 q^{22} -0.801938 q^{23} -5.29590 q^{24} +0.554958 q^{25} +2.13706 q^{27} +8.20775 q^{28} +3.35690 q^{29} +11.8998 q^{30} +4.26875 q^{31} -6.51573 q^{32} -0.246980 q^{33} -1.24698 q^{34} -6.34481 q^{35} +6.24698 q^{36} -7.50365 q^{37} +2.24698 q^{38} -5.55496 q^{40} +0.268750 q^{41} -13.5918 q^{42} -3.49396 q^{43} +0.335126 q^{44} -4.82908 q^{45} -1.80194 q^{46} -11.8388 q^{47} +1.80194 q^{48} +0.246980 q^{49} +1.24698 q^{50} +1.24698 q^{51} -7.64310 q^{53} +4.80194 q^{54} -0.259061 q^{55} +6.34481 q^{56} -2.24698 q^{57} +7.54288 q^{58} +4.18598 q^{59} +16.1468 q^{60} +4.49396 q^{61} +9.59179 q^{62} +5.51573 q^{63} -13.0368 q^{64} -0.554958 q^{66} +11.0315 q^{67} -1.69202 q^{68} +1.80194 q^{69} -14.2567 q^{70} -5.52111 q^{71} +4.82908 q^{72} +8.74094 q^{73} -16.8605 q^{74} -1.24698 q^{75} +3.04892 q^{76} +0.295897 q^{77} -9.64310 q^{79} +1.89008 q^{80} -10.9487 q^{81} +0.603875 q^{82} +5.34481 q^{83} -18.4426 q^{84} +1.30798 q^{85} -7.85086 q^{86} -7.54288 q^{87} +0.259061 q^{88} -18.5405 q^{89} -10.8509 q^{90} -2.44504 q^{92} -9.59179 q^{93} -26.6015 q^{94} -2.35690 q^{95} +14.6407 q^{96} +2.29590 q^{97} +0.554958 q^{98} +0.225209 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{2} - 2 q^{3} - 3 q^{5} - 6 q^{6} + 3 q^{7} + 3 q^{8} - 3 q^{9}+O(q^{10}) $ 3 * q + 2 * q^2 - 2 * q^3 - 3 * q^5 - 6 * q^6 + 3 * q^7 + 3 * q^8 - 3 * q^9	$ 3 q + 2 q^{2} - 2 q^{3} - 3 q^{5} - 6 q^{6} + 3 q^{7} + 3 q^{8} - 3 q^{9} - 2 q^{10} + q^{11} - 7 q^{12} + 9 q^{14} + 2 q^{15} + 2 q^{16} - 2 q^{17} + 5 q^{18} + 3 q^{19} - 7 q^{20} - 9 q^{21} - 4 q^{22} + 2 q^{23} - 2 q^{24} + 2 q^{25} + q^{27} + 7 q^{28} + 6 q^{29} + 13 q^{30} + 5 q^{31} - 7 q^{32} + 4 q^{33} + q^{34} + 4 q^{35} + 14 q^{36} + 9 q^{37} + 2 q^{38} - 17 q^{40} - 7 q^{41} - 13 q^{42} - q^{43} - 4 q^{45} - q^{46} - 3 q^{47} + q^{48} - 4 q^{49} - q^{50} - q^{51} - 27 q^{53} + 10 q^{54} - 15 q^{55} - 4 q^{56} - 2 q^{57} + 4 q^{58} - 2 q^{59} + 21 q^{60} + 4 q^{61} + q^{62} + 4 q^{63} - 11 q^{64} - 2 q^{66} + 8 q^{67} + q^{69} - 16 q^{70} - q^{71} + 4 q^{72} + 12 q^{73} - 15 q^{74} + q^{75} - 13 q^{77} - 33 q^{79} + 5 q^{80} - q^{81} - 7 q^{82} - 7 q^{83} - 14 q^{84} + 9 q^{85} - 10 q^{86} - 4 q^{87} + 15 q^{88} + 4 q^{89} - 19 q^{90} - 7 q^{92} - q^{93} - 30 q^{94} - 3 q^{95} + 7 q^{96} - 7 q^{97} + 2 q^{98} - q^{99}+O(q^{100}) $ 3 * q + 2 * q^2 - 2 * q^3 - 3 * q^5 - 6 * q^6 + 3 * q^7 + 3 * q^8 - 3 * q^9 - 2 * q^10 + q^11 - 7 * q^12 + 9 * q^14 + 2 * q^15 + 2 * q^16 - 2 * q^17 + 5 * q^18 + 3 * q^19 - 7 * q^20 - 9 * q^21 - 4 * q^22 + 2 * q^23 - 2 * q^24 + 2 * q^25 + q^27 + 7 * q^28 + 6 * q^29 + 13 * q^30 + 5 * q^31 - 7 * q^32 + 4 * q^33 + q^34 + 4 * q^35 + 14 * q^36 + 9 * q^37 + 2 * q^38 - 17 * q^40 - 7 * q^41 - 13 * q^42 - q^43 - 4 * q^45 - q^46 - 3 * q^47 + q^48 - 4 * q^49 - q^50 - q^51 - 27 * q^53 + 10 * q^54 - 15 * q^55 - 4 * q^56 - 2 * q^57 + 4 * q^58 - 2 * q^59 + 21 * q^60 + 4 * q^61 + q^62 + 4 * q^63 - 11 * q^64 - 2 * q^66 + 8 * q^67 + q^69 - 16 * q^70 - q^71 + 4 * q^72 + 12 * q^73 - 15 * q^74 + q^75 - 13 * q^77 - 33 * q^79 + 5 * q^80 - q^81 - 7 * q^82 - 7 * q^83 - 14 * q^84 + 9 * q^85 - 10 * q^86 - 4 * q^87 + 15 * q^88 + 4 * q^89 - 19 * q^90 - 7 * q^92 - q^93 - 30 * q^94 - 3 * q^95 + 7 * q^96 - 7 * q^97 + 2 * q^98 - 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 $

$2$	2.24698	1.58885	0.794427	−	0.607359i	$-0.207771\pi$
$2$	2.24698	1.58885	0.794427	+	0.607359i	$0.207771\pi$
$3$	−2.24698	−1.29729	−0.648647	−	0.761089i	$-0.724665\pi$
$3$	−2.24698	−1.29729	−0.648647	+	0.761089i	$0.724665\pi$
$4$	3.04892	1.52446
$5$	−2.35690	−1.05404	−0.527018	−	0.849854i	$-0.676690\pi$
$5$	−2.35690	−1.05404	−0.527018	+	0.849854i	$0.676690\pi$
$6$	−5.04892	−2.06121
$7$	2.69202	1.01749	0.508744	−	0.860918i	$-0.330110\pi$
$7$	2.69202	1.01749	0.508744	+	0.860918i	$0.330110\pi$
$8$	2.35690	0.833289
$9$	2.04892	0.682972
$10$	−5.29590	−1.67471
$11$	0.109916	0.0331410	0.0165705	−	0.999863i	$-0.494725\pi$
$11$	0.109916	0.0331410	0.0165705	+	0.999863i	$0.494725\pi$
$12$	−6.85086	−1.97767
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	6.04892	1.61664
$15$	5.29590	1.36739
$16$	−0.801938	−0.200484
$17$	−0.554958	−0.134597	−0.0672986	−	0.997733i	$-0.521438\pi$
$17$	−0.554958	−0.134597	−0.0672986	+	0.997733i	$0.521438\pi$
$18$	4.60388	1.08514
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−7.18598	−1.60683
$21$	−6.04892	−1.31998
$22$	0.246980	0.0526562
$23$	−0.801938	−0.167216	−0.0836078	−	0.996499i	$-0.526644\pi$
$23$	−0.801938	−0.167216	−0.0836078	+	0.996499i	$0.526644\pi$
$24$	−5.29590	−1.08102
$25$	0.554958	0.110992
$26$	0	0
$27$	2.13706	0.411278
$28$	8.20775	1.55112
$29$	3.35690	0.623360	0.311680	−	0.950187i	$-0.399108\pi$
$29$	3.35690	0.623360	0.311680	+	0.950187i	$0.399108\pi$
$30$	11.8998	2.17259
$31$	4.26875	0.766690	0.383345	−	0.923605i	$-0.374772\pi$
$31$	4.26875	0.766690	0.383345	+	0.923605i	$0.374772\pi$
$32$	−6.51573	−1.15183
$33$	−0.246980	−0.0429936
$34$	−1.24698	−0.213855
$35$	−6.34481	−1.07247
$36$	6.24698	1.04116
$37$	−7.50365	−1.23359	−0.616796	−	0.787123i	$-0.711570\pi$
$37$	−7.50365	−1.23359	−0.616796	+	0.787123i	$0.711570\pi$
$38$	2.24698	0.364508
$39$	0	0
$40$	−5.55496	−0.878316
$41$	0.268750	0.0419717	0.0209858	−	0.999780i	$-0.493320\pi$
$41$	0.268750	0.0419717	0.0209858	+	0.999780i	$0.493320\pi$
$42$	−13.5918	−2.09726
$43$	−3.49396	−0.532824	−0.266412	−	0.963859i	$-0.585838\pi$
$43$	−3.49396	−0.532824	−0.266412	+	0.963859i	$0.585838\pi$
$44$	0.335126	0.0505221
$45$	−4.82908	−0.719877
$46$	−1.80194	−0.265681
$47$	−11.8388	−1.72686	−0.863431	−	0.504466i	$-0.831689\pi$
$47$	−11.8388	−1.72686	−0.863431	+	0.504466i	$0.831689\pi$
$48$	1.80194	0.260087
$49$	0.246980	0.0352828
$50$	1.24698	0.176350
$51$	1.24698	0.174612
$52$	0	0
$53$	−7.64310	−1.04986	−0.524931	−	0.851145i	$-0.675909\pi$
$53$	−7.64310	−1.04986	−0.524931	+	0.851145i	$0.675909\pi$
$54$	4.80194	0.653461
$55$	−0.259061	−0.0349318
$56$	6.34481	0.847861
$57$	−2.24698	−0.297620
$58$	7.54288	0.990428
$59$	4.18598	0.544968	0.272484	−	0.962160i	$-0.412155\pi$
$59$	4.18598	0.544968	0.272484	+	0.962160i	$0.412155\pi$
$60$	16.1468	2.08454
$61$	4.49396	0.575393	0.287696	−	0.957722i	$-0.407111\pi$
$61$	4.49396	0.575393	0.287696	+	0.957722i	$0.407111\pi$
$62$	9.59179	1.21816
$63$	5.51573	0.694917
$64$	−13.0368	−1.62960
$65$	0	0
$66$	−0.554958	−0.0683106
$67$	11.0315	1.34771	0.673854	−	0.738865i	$-0.264638\pi$
$67$	11.0315	1.34771	0.673854	+	0.738865i	$0.264638\pi$
$68$	−1.69202	−0.205188
$69$	1.80194	0.216928
$70$	−14.2567	−1.70400
$71$	−5.52111	−0.655235	−0.327617	−	0.944810i	$-0.606246\pi$
$71$	−5.52111	−0.655235	−0.327617	+	0.944810i	$0.606246\pi$
$72$	4.82908	0.569113
$73$	8.74094	1.02305	0.511525	−	0.859269i	$-0.329081\pi$
$73$	8.74094	1.02305	0.511525	+	0.859269i	$0.329081\pi$
$74$	−16.8605	−1.96000
$75$	−1.24698	−0.143989
$76$	3.04892	0.349735
$77$	0.295897	0.0337206
$78$	0	0
$79$	−9.64310	−1.08493	−0.542467	−	0.840077i	$-0.682510\pi$
$79$	−9.64310	−1.08493	−0.542467	+	0.840077i	$0.682510\pi$
$80$	1.89008	0.211318
$81$	−10.9487	−1.21652
$82$	0.603875	0.0666869
$83$	5.34481	0.586670	0.293335	−	0.956010i	$-0.405235\pi$
$83$	5.34481	0.586670	0.293335	+	0.956010i	$0.405235\pi$
$84$	−18.4426	−2.01226
$85$	1.30798	0.141870
$86$	−7.85086	−0.846579
$87$	−7.54288	−0.808681
$88$	0.259061	0.0276160
$89$	−18.5405	−1.96529	−0.982644	−	0.185503i	$-0.940609\pi$
$89$	−18.5405	−1.96529	−0.982644	+	0.185503i	$0.940609\pi$
$90$	−10.8509	−1.14378
$91$	0	0
$92$	−2.44504	−0.254913
$93$	−9.59179	−0.994623
$94$	−26.6015	−2.74373
$95$	−2.35690	−0.241812
$96$	14.6407	1.49426
$97$	2.29590	0.233113	0.116557	−	0.993184i	$-0.462814\pi$
$97$	2.29590	0.233113	0.116557	+	0.993184i	$0.462814\pi$
$98$	0.554958	0.0560592
$99$	0.225209	0.0226344
$100$	1.69202	0.169202
$101$	−14.8605	−1.47868	−0.739340	−	0.673333i	$-0.764862\pi$
$101$	−14.8605	−1.47868	−0.739340	+	0.673333i	$0.764862\pi$
$102$	2.80194	0.277433
$103$	−5.74094	−0.565672	−0.282836	−	0.959168i	$-0.591275\pi$
$103$	−5.74094	−0.565672	−0.282836	+	0.959168i	$0.591275\pi$
$104$	0	0
$105$	14.2567	1.39131
$106$	−17.1739	−1.66808
$107$	−9.86592	−0.953775	−0.476887	−	0.878964i	$-0.658235\pi$
$107$	−9.86592	−0.953775	−0.476887	+	0.878964i	$0.658235\pi$
$108$	6.51573	0.626976
$109$	−16.7235	−1.60182	−0.800909	−	0.598785i	$-0.795650\pi$
$109$	−16.7235	−1.60182	−0.800909	+	0.598785i	$0.795650\pi$
$110$	−0.582105	−0.0555016
$111$	16.8605	1.60033
$112$	−2.15883	−0.203991
$113$	−19.6039	−1.84418	−0.922089	−	0.386979i	$-0.873519\pi$
$113$	−19.6039	−1.84418	−0.922089	+	0.386979i	$0.873519\pi$
$114$	−5.04892	−0.472874
$115$	1.89008	0.176251
$116$	10.2349	0.950286
$117$	0	0
$118$	9.40581	0.865875
$119$	−1.49396	−0.136951
$120$	12.4819	1.13943
$121$	−10.9879	−0.998902
$122$	10.0978	0.914215
$123$	−0.603875	−0.0544496
$124$	13.0151	1.16879
$125$	10.4765	0.937047
$126$	12.3937	1.10412
$127$	0.411190	0.0364872	0.0182436	−	0.999834i	$-0.494193\pi$
$127$	0.411190	0.0364872	0.0182436	+	0.999834i	$0.494193\pi$
$128$	−16.2620	−1.43738
$129$	7.85086	0.691229
$130$	0	0
$131$	0.542877	0.0474313	0.0237157	−	0.999719i	$-0.492450\pi$
$131$	0.542877	0.0474313	0.0237157	+	0.999719i	$0.492450\pi$
$132$	−0.753020	−0.0655420
$133$	2.69202	0.233428
$134$	24.7875	2.14131
$135$	−5.03684	−0.433502
$136$	−1.30798	−0.112158
$137$	9.13706	0.780632	0.390316	−	0.920681i	$-0.372366\pi$
$137$	9.13706	0.780632	0.390316	+	0.920681i	$0.372366\pi$
$138$	4.04892	0.344667
$139$	−7.72348	−0.655097	−0.327549	−	0.944834i	$-0.606222\pi$
$139$	−7.72348	−0.655097	−0.327549	+	0.944834i	$0.606222\pi$
$140$	−19.3448	−1.63494
$141$	26.6015	2.24025
$142$	−12.4058	−1.04107
$143$	0	0
$144$	−1.64310	−0.136925
$145$	−7.91185	−0.657044
$146$	19.6407	1.62548
$147$	−0.554958	−0.0457722
$148$	−22.8780	−1.88056
$149$	−8.28621	−0.678833	−0.339416	−	0.940636i	$-0.610230\pi$
$149$	−8.28621	−0.678833	−0.339416	+	0.940636i	$0.610230\pi$
$150$	−2.80194	−0.228777
$151$	17.5308	1.42664	0.713318	−	0.700841i	$-0.247192\pi$
$151$	17.5308	1.42664	0.713318	+	0.700841i	$0.247192\pi$
$152$	2.35690	0.191169
$153$	−1.13706	−0.0919261
$154$	0.664874	0.0535771
$155$	−10.0610	−0.808119
$156$	0	0
$157$	−2.54527	−0.203135	−0.101567	−	0.994829i	$-0.532386\pi$
$157$	−2.54527	−0.203135	−0.101567	+	0.994829i	$0.532386\pi$
$158$	−21.6679	−1.72380
$159$	17.1739	1.36198
$160$	15.3569	1.21407
$161$	−2.15883	−0.170140
$162$	−24.6015	−1.93288
$163$	11.4397	0.896024	0.448012	−	0.894028i	$-0.352132\pi$
$163$	11.4397	0.896024	0.448012	+	0.894028i	$0.352132\pi$
$164$	0.819396	0.0639841
$165$	0.582105	0.0453168
$166$	12.0097	0.932133
$167$	−22.2446	−1.72134	−0.860669	−	0.509165i	$-0.829954\pi$
$167$	−22.2446	−1.72134	−0.860669	+	0.509165i	$0.829954\pi$
$168$	−14.2567	−1.09993
$169$	−13.0000	−1.00000
$170$	2.93900	0.225411
$171$	2.04892	0.156685
$172$	−10.6528	−0.812268
$173$	5.21744	0.396675	0.198337	−	0.980134i	$-0.436446\pi$
$173$	5.21744	0.396675	0.198337	+	0.980134i	$0.436446\pi$
$174$	−16.9487	−1.28488
$175$	1.49396	0.112933
$176$	−0.0881460	−0.00664425
$177$	−9.40581	−0.706984
$178$	−41.6601	−3.12256
$179$	−3.32006	−0.248153	−0.124076	−	0.992273i	$-0.539597\pi$
$179$	−3.32006	−0.248153	−0.124076	+	0.992273i	$0.539597\pi$
$180$	−14.7235	−1.09742
$181$	21.6775	1.61128	0.805640	−	0.592406i	$-0.201822\pi$
$181$	21.6775	1.61128	0.805640	+	0.592406i	$0.201822\pi$
$182$	0	0
$183$	−10.0978	−0.746453
$184$	−1.89008	−0.139339
$185$	17.6853	1.30025
$186$	−21.5526	−1.58031
$187$	−0.0609989	−0.00446068
$188$	−36.0954	−2.63253
$189$	5.75302	0.418471
$190$	−5.29590	−0.384205
$191$	8.76271	0.634047	0.317024	−	0.948418i	$-0.397317\pi$
$191$	8.76271	0.634047	0.317024	+	0.948418i	$0.397317\pi$
$192$	29.2935	2.11408
$193$	12.9758	0.934021	0.467011	−	0.884252i	$-0.345331\pi$
$193$	12.9758	0.934021	0.467011	+	0.884252i	$0.345331\pi$
$194$	5.15883	0.370383
$195$	0	0
$196$	0.753020	0.0537872
$197$	0.158834	0.0113164	0.00565821	−	0.999984i	$-0.498199\pi$
$197$	0.158834	0.0113164	0.00565821	+	0.999984i	$0.498199\pi$
$198$	0.506041	0.0359628
$199$	−23.9705	−1.69922	−0.849610	−	0.527411i	$-0.823163\pi$
$199$	−23.9705	−1.69922	−0.849610	+	0.527411i	$0.823163\pi$
$200$	1.30798	0.0924880
$201$	−24.7875	−1.74837
$202$	−33.3913	−2.34941
$203$	9.03684	0.634262
$204$	3.80194	0.266189
$205$	−0.633415	−0.0442397
$206$	−12.8998	−0.898770
$207$	−1.64310	−0.114204
$208$	0	0
$209$	0.109916	0.00760307
$210$	32.0344	2.21059
$211$	1.00000	0.0688428
$211$	1.00000	0.0688428
$212$	−23.3032	−1.60047
$213$	12.4058	0.850032
$214$	−22.1685	−1.51541
$215$	8.23490	0.561615
$216$	5.03684	0.342713
$217$	11.4916	0.780098
$218$	−37.5773	−2.54506
$219$	−19.6407	−1.32720
$220$	−0.789856	−0.0532521
$221$	0	0
$222$	37.8853	2.54269
$223$	−8.16182	−0.546556	−0.273278	−	0.961935i	$-0.588108\pi$
$223$	−8.16182	−0.546556	−0.273278	+	0.961935i	$0.588108\pi$
$224$	−17.5405	−1.17197
$225$	1.13706	0.0758042
$226$	−44.0495	−2.93013
$227$	20.3491	1.35062	0.675309	−	0.737534i	$-0.264010\pi$
$227$	20.3491	1.35062	0.675309	+	0.737534i	$0.264010\pi$
$228$	−6.85086	−0.453709
$229$	−0.210144	−0.0138867	−0.00694335	−	0.999976i	$-0.502210\pi$
$229$	−0.210144	−0.0138867	−0.00694335	+	0.999976i	$0.502210\pi$
$230$	4.24698	0.280038
$231$	−0.664874	−0.0437455
$232$	7.91185	0.519439
$233$	−14.2808	−0.935568	−0.467784	−	0.883843i	$-0.654948\pi$
$233$	−14.2808	−0.935568	−0.467784	+	0.883843i	$0.654948\pi$
$234$	0	0
$235$	27.9028	1.82018
$236$	12.7627	0.830782
$237$	21.6679	1.40748
$238$	−3.35690	−0.217595
$239$	19.1521	1.23885	0.619424	−	0.785057i	$-0.287366\pi$
$239$	19.1521	1.23885	0.619424	+	0.785057i	$0.287366\pi$
$240$	−4.24698	−0.274141
$241$	13.1763	0.848760	0.424380	−	0.905484i	$-0.360492\pi$
$241$	13.1763	0.848760	0.424380	+	0.905484i	$0.360492\pi$
$242$	−24.6896	−1.58711
$243$	18.1903	1.16691
$244$	13.7017	0.877162
$245$	−0.582105	−0.0371893
$246$	−1.35690	−0.0865125
$247$	0	0
$248$	10.0610	0.638874
$249$	−12.0097	−0.761083
$250$	23.5405	1.48883
$251$	−5.10560	−0.322263	−0.161131	−	0.986933i	$-0.551514\pi$
$251$	−5.10560	−0.322263	−0.161131	+	0.986933i	$0.551514\pi$
$252$	16.8170	1.05937
$253$	−0.0881460	−0.00554169
$254$	0.923936	0.0579729
$255$	−2.93900	−0.184047
$256$	−10.4668	−0.654176
$257$	24.7482	1.54375	0.771876	−	0.635773i	$-0.219318\pi$
$257$	24.7482	1.54375	0.771876	+	0.635773i	$0.219318\pi$
$258$	17.6407	1.09826
$259$	−20.2000	−1.25517
$260$	0	0
$261$	6.87800	0.425738
$262$	1.21983	0.0753615
$263$	−4.19269	−0.258532	−0.129266	−	0.991610i	$-0.541262\pi$
$263$	−4.19269	−0.258532	−0.129266	+	0.991610i	$0.541262\pi$
$264$	−0.582105	−0.0358261
$265$	18.0140	1.10659
$266$	6.04892	0.370883
$267$	41.6601	2.54956
$268$	33.6340	2.05452
$269$	−18.1782	−1.10835	−0.554173	−	0.832402i	$-0.686965\pi$
$269$	−18.1782	−1.10835	−0.554173	+	0.832402i	$0.686965\pi$
$270$	−11.3177	−0.688771
$271$	−24.8170	−1.50753	−0.753763	−	0.657147i	$-0.771763\pi$
$271$	−24.8170	−1.50753	−0.753763	+	0.657147i	$0.771763\pi$
$272$	0.445042	0.0269846
$273$	0	0
$274$	20.5308	1.24031
$275$	0.0609989	0.00367837
$276$	5.49396	0.330697
$277$	−4.71379	−0.283224	−0.141612	−	0.989922i	$-0.545229\pi$
$277$	−4.71379	−0.283224	−0.141612	+	0.989922i	$0.545229\pi$
$278$	−17.3545	−1.04085
$279$	8.74632	0.523628
$280$	−14.9541	−0.893676
$281$	13.5386	0.807643	0.403822	−	0.914838i	$-0.367682\pi$
$281$	13.5386	0.807643	0.403822	+	0.914838i	$0.367682\pi$
$282$	59.7730	3.55943
$283$	20.6679	1.22858	0.614288	−	0.789082i	$-0.289443\pi$
$283$	20.6679	1.22858	0.614288	+	0.789082i	$0.289443\pi$
$284$	−16.8334	−0.998878
$285$	5.29590	0.313702
$286$	0	0
$287$	0.723480	0.0427057
$288$	−13.3502	−0.786668
$289$	−16.6920	−0.981884
$290$	−17.7778	−1.04395
$291$	−5.15883	−0.302416
$292$	26.6504	1.55960
$293$	4.79417	0.280078	0.140039	−	0.990146i	$-0.455277\pi$
$293$	4.79417	0.280078	0.140039	+	0.990146i	$0.455277\pi$
$294$	−1.24698	−0.0727253
$295$	−9.86592	−0.574416
$296$	−17.6853	−1.02794
$297$	0.234898	0.0136302
$298$	−18.6189	−1.07857
$299$	0	0
$300$	−3.80194	−0.219505
$301$	−9.40581	−0.542142
$302$	39.3913	2.26672
$303$	33.3913	1.91828
$304$	−0.801938	−0.0459943
$305$	−10.5918	−0.606484
$306$	−2.55496	−0.146057
$307$	16.7506	0.956009	0.478004	−	0.878357i	$-0.341360\pi$
$307$	16.7506	0.956009	0.478004	+	0.878357i	$0.341360\pi$
$308$	0.902165	0.0514056
$309$	12.8998	0.733842
$310$	−22.6069	−1.28398
$311$	11.4916	0.651627	0.325813	−	0.945434i	$-0.394362\pi$
$311$	11.4916	0.651627	0.325813	+	0.945434i	$0.394362\pi$
$312$	0	0
$313$	1.07308	0.0606541	0.0303270	−	0.999540i	$-0.490345\pi$
$313$	1.07308	0.0606541	0.0303270	+	0.999540i	$0.490345\pi$
$314$	−5.71917	−0.322751
$315$	−13.0000	−0.732467
$316$	−29.4010	−1.65394
$317$	−20.9530	−1.17684	−0.588419	−	0.808556i	$-0.700249\pi$
$317$	−20.9530	−1.17684	−0.588419	+	0.808556i	$0.700249\pi$
$318$	38.5894	2.16399
$319$	0.368977	0.0206588
$320$	30.7265	1.71766
$321$	22.1685	1.23733
$322$	−4.85086	−0.270328
$323$	−0.554958	−0.0308787
$324$	−33.3817	−1.85454
$325$	0	0
$326$	25.7047	1.42365
$327$	37.5773	2.07803
$328$	0.633415	0.0349745
$329$	−31.8702	−1.75706
$330$	1.30798	0.0720018
$331$	21.5418	1.18405	0.592023	−	0.805921i	$-0.298330\pi$
$331$	21.5418	1.18405	0.592023	+	0.805921i	$0.298330\pi$
$332$	16.2959	0.894354
$333$	−15.3744	−0.842509
$334$	−49.9831	−2.73496
$335$	−26.0000	−1.42053
$336$	4.85086	0.264636
$337$	−31.3653	−1.70857	−0.854287	−	0.519801i	$-0.826006\pi$
$337$	−31.3653	−1.70857	−0.854287	+	0.519801i	$0.826006\pi$
$338$	−29.2107	−1.58885
$339$	44.0495	2.39244
$340$	3.98792	0.216275
$341$	0.469205	0.0254089
$342$	4.60388	0.248949
$343$	−18.1793	−0.981589
$344$	−8.23490	−0.443996
$345$	−4.24698	−0.228650
$346$	11.7235	0.630258
$347$	5.78986	0.310816	0.155408	−	0.987850i	$-0.450331\pi$
$347$	5.78986	0.310816	0.155408	+	0.987850i	$0.450331\pi$
$348$	−22.9976	−1.23280
$349$	−22.4470	−1.20156	−0.600779	−	0.799415i	$-0.705143\pi$
$349$	−22.4470	−1.20156	−0.600779	+	0.799415i	$0.705143\pi$
$350$	3.35690	0.179434
$351$	0	0
$352$	−0.716185	−0.0381728
$353$	28.4741	1.51552	0.757762	−	0.652531i	$-0.226293\pi$
$353$	28.4741	1.51552	0.757762	+	0.652531i	$0.226293\pi$
$354$	−21.1347	−1.12330
$355$	13.0127	0.690641
$356$	−56.5284	−2.99600
$357$	3.35690	0.177666
$358$	−7.46011	−0.394279
$359$	−23.8159	−1.25696	−0.628479	−	0.777827i	$-0.716322\pi$
$359$	−23.8159	−1.25696	−0.628479	+	0.777827i	$0.716322\pi$
$360$	−11.3817	−0.599866
$361$	1.00000	0.0526316
$362$	48.7090	2.56009
$363$	24.6896	1.29587
$364$	0	0
$365$	−20.6015	−1.07833
$366$	−22.6896	−1.18601
$367$	3.00777	0.157004	0.0785022	−	0.996914i	$-0.474986\pi$
$367$	3.00777	0.157004	0.0785022	+	0.996914i	$0.474986\pi$
$368$	0.643104	0.0335241
$369$	0.550646	0.0286655
$370$	39.7385	2.06591
$371$	−20.5754	−1.06822
$372$	−29.2446	−1.51626
$373$	6.88471	0.356477	0.178238	−	0.983987i	$-0.442960\pi$
$373$	6.88471	0.356477	0.178238	+	0.983987i	$0.442960\pi$
$374$	−0.137063	−0.00708738
$375$	−23.5405	−1.21563
$376$	−27.9028	−1.43897
$377$	0	0
$378$	12.9269	0.664889
$379$	−34.9517	−1.79535	−0.897673	−	0.440661i	$-0.854744\pi$
$379$	−34.9517	−1.79535	−0.897673	+	0.440661i	$0.854744\pi$
$380$	−7.18598	−0.368633
$381$	−0.923936	−0.0473347
$382$	19.6896	1.00741
$383$	4.33752	0.221637	0.110818	−	0.993841i	$-0.464653\pi$
$383$	4.33752	0.221637	0.110818	+	0.993841i	$0.464653\pi$
$384$	36.5405	1.86470
$385$	−0.697398	−0.0355427
$386$	29.1564	1.48402
$387$	−7.15883	−0.363904
$388$	7.00000	0.355371
$389$	23.0616	1.16927	0.584635	−	0.811297i	$-0.301238\pi$
$389$	23.0616	1.16927	0.584635	+	0.811297i	$0.301238\pi$
$390$	0	0
$391$	0.445042	0.0225067
$392$	0.582105	0.0294008
$393$	−1.21983	−0.0615324
$394$	0.356896	0.0179802
$395$	22.7278	1.14356
$396$	0.686645	0.0345052
$397$	6.89307	0.345953	0.172977	−	0.984926i	$-0.444662\pi$
$397$	6.89307	0.345953	0.172977	+	0.984926i	$0.444662\pi$
$398$	−53.8611	−2.69981
$399$	−6.04892	−0.302825
$400$	−0.445042	−0.0222521
$401$	6.42998	0.321098	0.160549	−	0.987028i	$-0.448674\pi$
$401$	6.42998	0.321098	0.160549	+	0.987028i	$0.448674\pi$
$402$	−55.6969	−2.77791
$403$	0	0
$404$	−45.3086	−2.25419
$405$	25.8049	1.28226
$406$	20.3056	1.00775
$407$	−0.824773	−0.0408825
$408$	2.93900	0.145502
$409$	16.7463	0.828052	0.414026	−	0.910265i	$-0.364122\pi$
$409$	16.7463	0.828052	0.414026	+	0.910265i	$0.364122\pi$
$410$	−1.42327	−0.0702904
$411$	−20.5308	−1.01271
$412$	−17.5036	−0.862343
$413$	11.2687	0.554499
$414$	−3.69202	−0.181453
$415$	−12.5972	−0.618371
$416$	0	0
$417$	17.3545	0.849854
$418$	0.246980	0.0120802
$419$	24.4741	1.19564	0.597819	−	0.801631i	$-0.296034\pi$
$419$	24.4741	1.19564	0.597819	+	0.801631i	$0.296034\pi$
$420$	43.4674	2.12099
$421$	34.1739	1.66553	0.832767	−	0.553624i	$-0.186755\pi$
$421$	34.1739	1.66553	0.832767	+	0.553624i	$0.186755\pi$
$422$	2.24698	0.109381
$423$	−24.2567	−1.17940
$424$	−18.0140	−0.874837
$425$	−0.307979	−0.0149392
$426$	27.8756	1.35058
$427$	12.0978	0.585455
$428$	−30.0804	−1.45399
$429$	0	0
$430$	18.5036	0.892325
$431$	−6.28621	−0.302796	−0.151398	−	0.988473i	$-0.548378\pi$
$431$	−6.28621	−0.302796	−0.151398	+	0.988473i	$0.548378\pi$
$432$	−1.71379	−0.0824548
$433$	18.3599	0.882320	0.441160	−	0.897429i	$-0.354567\pi$
$433$	18.3599	0.882320	0.441160	+	0.897429i	$0.354567\pi$
$434$	25.8213	1.23946
$435$	17.7778	0.852379
$436$	−50.9885	−2.44191
$437$	−0.801938	−0.0383619
$438$	−44.1323	−2.10872
$439$	−1.31873	−0.0629397	−0.0314698	−	0.999505i	$-0.510019\pi$
$439$	−1.31873	−0.0629397	−0.0314698	+	0.999505i	$0.510019\pi$
$440$	−0.610580	−0.0291083
$441$	0.506041	0.0240972
$442$	0	0
$443$	−8.21073	−0.390104	−0.195052	−	0.980793i	$-0.562488\pi$
$443$	−8.21073	−0.390104	−0.195052	+	0.980793i	$0.562488\pi$
$444$	51.4064	2.43964
$445$	43.6980	2.07148
$446$	−18.3394	−0.868398
$447$	18.6189	0.880646
$448$	−35.0954	−1.65810
$449$	9.41789	0.444458	0.222229	−	0.974995i	$-0.428667\pi$
$449$	9.41789	0.444458	0.222229	+	0.974995i	$0.428667\pi$
$450$	2.55496	0.120442
$451$	0.0295400	0.00139098
$452$	−59.7706	−2.81137
$453$	−39.3913	−1.85077
$454$	45.7241	2.14594
$455$	0	0
$456$	−5.29590	−0.248003
$457$	4.52648	0.211740	0.105870	−	0.994380i	$-0.466237\pi$
$457$	4.52648	0.211740	0.105870	+	0.994380i	$0.466237\pi$
$458$	−0.472189	−0.0220639
$459$	−1.18598	−0.0553568
$460$	5.76271	0.268688
$461$	−24.1685	−1.12564	−0.562820	−	0.826579i	$-0.690284\pi$
$461$	−24.1685	−1.12564	−0.562820	+	0.826579i	$0.690284\pi$
$462$	−1.49396	−0.0695053
$463$	3.62565	0.168498	0.0842490	−	0.996445i	$-0.473151\pi$
$463$	3.62565	0.168498	0.0842490	+	0.996445i	$0.473151\pi$
$464$	−2.69202	−0.124974
$465$	22.6069	1.04837
$466$	−32.0887	−1.48648
$467$	27.3129	1.26389	0.631945	−	0.775013i	$-0.282257\pi$
$467$	27.3129	1.26389	0.631945	+	0.775013i	$0.282257\pi$
$468$	0	0
$469$	29.6969	1.37128
$470$	62.6969	2.89199
$471$	5.71917	0.263525
$472$	9.86592	0.454116
$473$	−0.384043	−0.0176583
$474$	48.6872	2.23628
$475$	0.554958	0.0254632
$476$	−4.55496	−0.208776
$477$	−15.6601	−0.717026
$478$	43.0344	1.96835
$479$	24.3575	1.11292	0.556461	−	0.830874i	$-0.312159\pi$
$479$	24.3575	1.11292	0.556461	+	0.830874i	$0.312159\pi$
$480$	−34.5066	−1.57501
$481$	0	0
$482$	29.6069	1.34856
$483$	4.85086	0.220722
$484$	−33.5013	−1.52278
$485$	−5.41119	−0.245709
$486$	40.8732	1.85405
$487$	−32.4432	−1.47014	−0.735072	−	0.677989i	$-0.762852\pi$
$487$	−32.4432	−1.47014	−0.735072	+	0.677989i	$0.762852\pi$
$488$	10.5918	0.479468
$489$	−25.7047	−1.16241
$490$	−1.30798	−0.0590884
$491$	−7.22952	−0.326264	−0.163132	−	0.986604i	$-0.552160\pi$
$491$	−7.22952	−0.326264	−0.163132	+	0.986604i	$0.552160\pi$
$492$	−1.84117	−0.0830062
$493$	−1.86294	−0.0839024
$494$	0	0
$495$	−0.530795	−0.0238575
$496$	−3.42327	−0.153709
$497$	−14.8629	−0.666694
$498$	−26.9855	−1.20925
$499$	4.59611	0.205750	0.102875	−	0.994694i	$-0.467196\pi$
$499$	4.59611	0.205750	0.102875	+	0.994694i	$0.467196\pi$
$500$	31.9420	1.42849
$501$	49.9831	2.23308
$502$	−11.4722	−0.512029
$503$	7.08277	0.315805	0.157903	−	0.987455i	$-0.449527\pi$
$503$	7.08277	0.315805	0.157903	+	0.987455i	$0.449527\pi$
$504$	13.0000	0.579066
$505$	35.0248	1.55858
$506$	−0.198062	−0.00880494
$507$	29.2107	1.29729
$508$	1.25368	0.0556232
$509$	13.1328	0.582099	0.291049	−	0.956708i	$-0.405996\pi$
$509$	13.1328	0.582099	0.291049	+	0.956708i	$0.405996\pi$
$510$	−6.60388	−0.292425
$511$	23.5308	1.04094
$512$	9.00538	0.397985
$513$	2.13706	0.0943537
$514$	55.6088	2.45280
$515$	13.5308	0.596238
$516$	23.9366	1.05375
$517$	−1.30127	−0.0572300
$518$	−45.3889	−1.99428
$519$	−11.7235	−0.514604
$520$	0	0
$521$	−0.107523	−0.00471068	−0.00235534	−	0.999997i	$-0.500750\pi$
$521$	−0.107523	−0.00471068	−0.00235534	+	0.999997i	$0.500750\pi$
$522$	15.4547	0.676435
$523$	18.9782	0.829860	0.414930	−	0.909853i	$-0.363806\pi$
$523$	18.9782	0.829860	0.414930	+	0.909853i	$0.363806\pi$
$524$	1.65519	0.0723071
$525$	−3.35690	−0.146507
$526$	−9.42088	−0.410770
$527$	−2.36898	−0.103194
$528$	0.198062	0.00861955
$529$	−22.3569	−0.972039
$530$	40.4771	1.75821
$531$	8.57673	0.372198
$532$	8.20775	0.355851
$533$	0	0
$534$	93.6094	4.05087
$535$	23.2529	1.00531
$536$	26.0000	1.12303
$537$	7.46011	0.321927
$538$	−40.8461	−1.76100
$539$	0.0271471	0.00116931
$540$	−15.3569	−0.660856
$541$	27.3381	1.17536	0.587679	−	0.809094i	$-0.300042\pi$
$541$	27.3381	1.17536	0.587679	+	0.809094i	$0.300042\pi$
$542$	−55.7633	−2.39524
$543$	−48.7090	−2.09030
$544$	3.61596	0.155033
$545$	39.4155	1.68837
$546$	0	0
$547$	−35.7547	−1.52876	−0.764380	−	0.644766i	$-0.776955\pi$
$547$	−35.7547	−1.52876	−0.764380	+	0.644766i	$0.776955\pi$
$548$	27.8582	1.19004
$549$	9.20775	0.392977
$550$	0.137063	0.00584440
$551$	3.35690	0.143009
$552$	4.24698	0.180763
$553$	−25.9594	−1.10391
$554$	−10.5918	−0.450002
$555$	−39.7385	−1.68681
$556$	−23.5483	−0.998668
$557$	−16.4343	−0.696343	−0.348172	−	0.937431i	$-0.613197\pi$
$557$	−16.4343	−0.696343	−0.348172	+	0.937431i	$0.613197\pi$
$558$	19.6528	0.831969
$559$	0	0
$560$	5.08815	0.215013
$561$	0.137063	0.00578682
$562$	30.4209	1.28323
$563$	−9.71081	−0.409262	−0.204631	−	0.978839i	$-0.565599\pi$
$563$	−9.71081	−0.409262	−0.204631	+	0.978839i	$0.565599\pi$
$564$	81.1057	3.41517
$565$	46.2043	1.94383
$566$	46.4403	1.95203
$567$	−29.4741	−1.23780
$568$	−13.0127	−0.546000
$569$	−30.5013	−1.27868	−0.639340	−	0.768925i	$-0.720792\pi$
$569$	−30.5013	−1.27868	−0.639340	+	0.768925i	$0.720792\pi$
$570$	11.8998	0.498427
$571$	27.8321	1.16474	0.582368	−	0.812925i	$-0.302126\pi$
$571$	27.8321	1.16474	0.582368	+	0.812925i	$0.302126\pi$
$572$	0	0
$573$	−19.6896	−0.822546
$574$	1.62565	0.0678531
$575$	−0.445042	−0.0185595
$576$	−26.7114	−1.11297
$577$	39.1105	1.62819	0.814096	−	0.580731i	$-0.197233\pi$
$577$	39.1105	1.62819	0.814096	+	0.580731i	$0.197233\pi$
$578$	−37.5066	−1.56007
$579$	−29.1564	−1.21170
$580$	−24.1226	−1.00164
$581$	14.3884	0.596930
$582$	−11.5918	−0.480495
$583$	−0.840101	−0.0347934
$584$	20.6015	0.852495
$585$	0	0
$586$	10.7724	0.445004
$587$	−6.71081	−0.276985	−0.138492	−	0.990364i	$-0.544226\pi$
$587$	−6.71081	−0.276985	−0.138492	+	0.990364i	$0.544226\pi$
$588$	−1.69202	−0.0697778
$589$	4.26875	0.175891
$590$	−22.1685	−0.912664
$591$	−0.356896	−0.0146807
$592$	6.01746	0.247316
$593$	15.5778	0.639703	0.319852	−	0.947468i	$-0.396367\pi$
$593$	15.5778	0.639703	0.319852	+	0.947468i	$0.396367\pi$
$594$	0.527811	0.0216563
$595$	3.52111	0.144351
$596$	−25.2640	−1.03485
$597$	53.8611	2.20439
$598$	0	0
$599$	23.7614	0.970864	0.485432	−	0.874275i	$-0.338662\pi$
$599$	23.7614	0.970864	0.485432	+	0.874275i	$0.338662\pi$
$600$	−2.93900	−0.119984
$601$	−8.97392	−0.366054	−0.183027	−	0.983108i	$-0.558590\pi$
$601$	−8.97392	−0.366054	−0.183027	+	0.983108i	$0.558590\pi$
$602$	−21.1347	−0.861385
$603$	22.6025	0.920447
$604$	53.4499	2.17485
$605$	25.8974	1.05288
$606$	75.0297	3.04787
$607$	44.8789	1.82158	0.910788	−	0.412874i	$-0.135475\pi$
$607$	44.8789	1.82158	0.910788	+	0.412874i	$0.135475\pi$
$608$	−6.51573	−0.264248
$609$	−20.3056	−0.822824
$610$	−23.7995	−0.963615
$611$	0	0
$612$	−3.46681	−0.140138
$613$	35.3163	1.42641	0.713207	−	0.700954i	$-0.247242\pi$
$613$	35.3163	1.42641	0.713207	+	0.700954i	$0.247242\pi$
$614$	37.6383	1.51896
$615$	1.42327	0.0573919
$616$	0.697398	0.0280990
$617$	−12.3220	−0.496064	−0.248032	−	0.968752i	$-0.579784\pi$
$617$	−12.3220	−0.496064	−0.248032	+	0.968752i	$0.579784\pi$
$618$	28.9855	1.16597
$619$	18.5670	0.746272	0.373136	−	0.927777i	$-0.378282\pi$
$619$	18.5670	0.746272	0.373136	+	0.927777i	$0.378282\pi$
$620$	−30.6752	−1.23194
$621$	−1.71379	−0.0687721
$622$	25.8213	1.03534
$623$	−49.9114	−1.99966
$624$	0	0
$625$	−27.4668	−1.09867
$626$	2.41119	0.0963705
$627$	−0.246980	−0.00986342
$628$	−7.76032	−0.309670
$629$	4.16421	0.166038
$630$	−29.2107	−1.16378
$631$	24.4142	0.971913	0.485956	−	0.873983i	$-0.338471\pi$
$631$	24.4142	0.971913	0.485956	+	0.873983i	$0.338471\pi$
$632$	−22.7278	−0.904063
$633$	−2.24698	−0.0893094
$634$	−47.0810	−1.86982
$635$	−0.969132	−0.0384588
$636$	52.3618	2.07628
$637$	0	0
$638$	0.829085	0.0328238
$639$	−11.3123	−0.447507
$640$	38.3279	1.51505
$641$	−45.5502	−1.79912	−0.899562	−	0.436793i	$-0.856114\pi$
$641$	−45.5502	−1.79912	−0.899562	+	0.436793i	$0.856114\pi$
$642$	49.8122	1.96593
$643$	−4.23623	−0.167060	−0.0835302	−	0.996505i	$-0.526620\pi$
$643$	−4.23623	−0.167060	−0.0835302	+	0.996505i	$0.526620\pi$
$644$	−6.58211	−0.259371
$645$	−18.5036	−0.728580
$646$	−1.24698	−0.0490618
$647$	−43.6708	−1.71688	−0.858439	−	0.512916i	$-0.828565\pi$
$647$	−43.6708	−1.71688	−0.858439	+	0.512916i	$0.828565\pi$
$648$	−25.8049	−1.01371
$649$	0.460107	0.0180608
$650$	0	0
$651$	−25.8213	−1.01202
$652$	34.8786	1.36595
$653$	−15.6246	−0.611437	−0.305719	−	0.952122i	$-0.598897\pi$
$653$	−15.6246	−0.611437	−0.305719	+	0.952122i	$0.598897\pi$
$654$	84.4355	3.30169
$655$	−1.27950	−0.0499943
$656$	−0.215521	−0.00841467
$657$	17.9095	0.698715
$658$	−71.6118	−2.79172
$659$	−29.4741	−1.14815	−0.574074	−	0.818803i	$-0.694638\pi$
$659$	−29.4741	−1.14815	−0.574074	+	0.818803i	$0.694638\pi$
$660$	1.77479	0.0690836
$661$	−9.04461	−0.351794	−0.175897	−	0.984409i	$-0.556283\pi$
$661$	−9.04461	−0.351794	−0.175897	+	0.984409i	$0.556283\pi$
$662$	48.4040	1.88128
$663$	0	0
$664$	12.5972	0.488865
$665$	−6.34481	−0.246041
$666$	−34.5459	−1.33862
$667$	−2.69202	−0.104235
$668$	−67.8219	−2.62411
$669$	18.3394	0.709044
$670$	−58.4215	−2.25702
$671$	0.493959	0.0190691
$672$	39.4131	1.52039
$673$	−5.90946	−0.227793	−0.113896	−	0.993493i	$-0.536333\pi$
$673$	−5.90946	−0.227793	−0.113896	+	0.993493i	$0.536333\pi$
$674$	−70.4771	−2.71468
$675$	1.18598	0.0456484
$676$	−39.6359	−1.52446
$677$	−32.5733	−1.25189	−0.625946	−	0.779866i	$-0.715287\pi$
$677$	−32.5733	−1.25189	−0.625946	+	0.779866i	$0.715287\pi$
$678$	98.9783	3.80124
$679$	6.18060	0.237190
$680$	3.08277	0.118219
$681$	−45.7241	−1.75215
$682$	1.05429	0.0403710
$683$	−21.1752	−0.810248	−0.405124	−	0.914262i	$-0.632772\pi$
$683$	−21.1752	−0.810248	−0.405124	+	0.914262i	$0.632772\pi$
$684$	6.24698	0.238859
$685$	−21.5351	−0.822814
$686$	−40.8485	−1.55960
$687$	0.472189	0.0180151
$688$	2.80194	0.106823
$689$	0	0
$690$	−9.54288	−0.363291
$691$	6.13275	0.233301	0.116650	−	0.993173i	$-0.462784\pi$
$691$	6.13275	0.233301	0.116650	+	0.993173i	$0.462784\pi$
$692$	15.9075	0.604714
$693$	0.606268	0.0230302
$694$	13.0097	0.493841
$695$	18.2034	0.690496
$696$	−17.7778	−0.673865
$697$	−0.149145	−0.00564927
$698$	−50.4379	−1.90910
$699$	32.0887	1.21371
$700$	4.55496	0.172161
$701$	−11.7345	−0.443206	−0.221603	−	0.975137i	$-0.571129\pi$
$701$	−11.7345	−0.443206	−0.221603	+	0.975137i	$0.571129\pi$
$702$	0	0
$703$	−7.50365	−0.283005
$704$	−1.43296	−0.0540067
$705$	−62.6969	−2.36130
$706$	63.9807	2.40795
$707$	−40.0049	−1.50454
$708$	−28.6775	−1.07777
$709$	5.20583	0.195509	0.0977546	−	0.995211i	$-0.468834\pi$
$709$	5.20583	0.195509	0.0977546	+	0.995211i	$0.468834\pi$
$710$	29.2392	1.09733
$711$	−19.7579	−0.740980
$712$	−43.6980	−1.63765
$713$	−3.42327	−0.128203
$714$	7.54288	0.282285
$715$	0	0
$716$	−10.1226	−0.378299
$717$	−43.0344	−1.60715
$718$	−53.5139	−1.99712
$719$	−47.8055	−1.78284	−0.891422	−	0.453173i	$-0.850292\pi$
$719$	−47.8055	−1.78284	−0.891422	+	0.453173i	$0.850292\pi$
$720$	3.87263	0.144324
$721$	−15.4547	−0.575564
$722$	2.24698	0.0836239
$723$	−29.6069	−1.10109
$724$	66.0930	2.45633
$725$	1.86294	0.0691877
$726$	55.4771	2.05895
$727$	−46.5368	−1.72595	−0.862976	−	0.505244i	$-0.831402\pi$
$727$	−46.5368	−1.72595	−0.862976	+	0.505244i	$0.831402\pi$
$728$	0	0
$729$	−8.02715	−0.297302
$730$	−46.2911	−1.71331
$731$	1.93900	0.0717165
$732$	−30.7875	−1.13794
$733$	−11.7028	−0.432252	−0.216126	−	0.976365i	$-0.569342\pi$
$733$	−11.7028	−0.432252	−0.216126	+	0.976365i	$0.569342\pi$
$734$	6.75840	0.249457
$735$	1.30798	0.0482455
$736$	5.22521	0.192604
$737$	1.21254	0.0446644
$738$	1.23729	0.0455453
$739$	2.54766	0.0937173	0.0468586	−	0.998902i	$-0.485079\pi$
$739$	2.54766	0.0937173	0.0468586	+	0.998902i	$0.485079\pi$
$740$	53.9211	1.98218
$741$	0	0
$742$	−46.2325	−1.69725
$743$	11.2000	0.410887	0.205444	−	0.978669i	$-0.434136\pi$
$743$	11.2000	0.410887	0.205444	+	0.978669i	$0.434136\pi$
$744$	−22.6069	−0.828808
$745$	19.5297	0.715514
$746$	15.4698	0.566389
$747$	10.9511	0.400679
$748$	−0.185981	−0.00680013
$749$	−26.5593	−0.970455
$750$	−52.8950	−1.93145
$751$	24.6112	0.898074	0.449037	−	0.893513i	$-0.351767\pi$
$751$	24.6112	0.898074	0.449037	+	0.893513i	$0.351767\pi$
$752$	9.49396	0.346209
$753$	11.4722	0.418070
$754$	0	0
$755$	−41.3183	−1.50373
$756$	17.5405	0.637941
$757$	8.50258	0.309032	0.154516	−	0.987990i	$-0.450618\pi$
$757$	8.50258	0.309032	0.154516	+	0.987990i	$0.450618\pi$
$758$	−78.5357	−2.85254
$759$	0.198062	0.00718920
$760$	−5.55496	−0.201500
$761$	16.8019	0.609070	0.304535	−	0.952501i	$-0.401499\pi$
$761$	16.8019	0.609070	0.304535	+	0.952501i	$0.401499\pi$
$762$	−2.07606	−0.0752079
$763$	−45.0200	−1.62983
$764$	26.7168	0.966579
$765$	2.67994	0.0968934
$766$	9.74632	0.352149
$767$	0	0
$768$	23.5187	0.848658
$769$	20.1142	0.725338	0.362669	−	0.931918i	$-0.381866\pi$
$769$	20.1142	0.725338	0.362669	+	0.931918i	$0.381866\pi$
$770$	−1.56704	−0.0564722
$771$	−55.6088	−2.00270
$772$	39.5623	1.42388
$773$	17.9336	0.645028	0.322514	−	0.946565i	$-0.395472\pi$
$773$	17.9336	0.645028	0.322514	+	0.946565i	$0.395472\pi$
$774$	−16.0858	−0.578190
$775$	2.36898	0.0850962
$776$	5.41119	0.194250
$777$	45.3889	1.62832
$778$	51.8189	1.85780
$779$	0.268750	0.00962896
$780$	0	0
$781$	−0.606859	−0.0217151
$782$	1.00000	0.0357599
$783$	7.17390	0.256374
$784$	−0.198062	−0.00707365
$785$	5.99894	0.214111
$786$	−2.74094	−0.0977660
$787$	11.5579	0.411996	0.205998	−	0.978552i	$-0.433956\pi$
$787$	11.5579	0.411996	0.205998	+	0.978552i	$0.433956\pi$
$788$	0.484271	0.0172514
$789$	9.42088	0.335392
$790$	51.0689	1.81695
$791$	−52.7741	−1.87643
$792$	0.530795	0.0188610
$793$	0	0
$794$	15.4886	0.549669
$795$	−40.4771	−1.43557
$796$	−73.0840	−2.59039
$797$	25.4336	0.900903	0.450451	−	0.892801i	$-0.351263\pi$
$797$	25.4336	0.900903	0.450451	+	0.892801i	$0.351263\pi$
$798$	−13.5918	−0.481144
$799$	6.57002	0.232431
$800$	−3.61596	−0.127843
$801$	−37.9879	−1.34224
$802$	14.4480	0.510178
$803$	0.960771	0.0339049
$804$	−75.5749	−2.66532
$805$	5.08815	0.179334
$806$	0	0
$807$	40.8461	1.43785
$808$	−35.0248	−1.23217
$809$	4.32975	0.152226	0.0761129	−	0.997099i	$-0.475749\pi$
$809$	4.32975	0.152226	0.0761129	+	0.997099i	$0.475749\pi$
$810$	57.9831	2.03732
$811$	4.73184	0.166157	0.0830787	−	0.996543i	$-0.473525\pi$
$811$	4.73184	0.166157	0.0830787	+	0.996543i	$0.473525\pi$
$812$	27.5526	0.966905
$813$	55.7633	1.95570
$814$	−1.85325	−0.0649563
$815$	−26.9621	−0.944441
$816$	−1.00000	−0.0350070
$817$	−3.49396	−0.122238
$818$	37.6286	1.31565
$819$	0	0
$820$	−1.93123	−0.0674415
$821$	45.9667	1.60425	0.802125	−	0.597156i	$-0.203703\pi$
$821$	45.9667	1.60425	0.802125	+	0.597156i	$0.203703\pi$
$822$	−46.1323	−1.60905
$823$	−29.7918	−1.03848	−0.519238	−	0.854630i	$-0.673784\pi$
$823$	−29.7918	−1.03848	−0.519238	+	0.854630i	$0.673784\pi$
$824$	−13.5308	−0.471368
$825$	−0.137063	−0.00477193
$826$	25.3207	0.881018
$827$	−20.1250	−0.699814	−0.349907	−	0.936784i	$-0.613787\pi$
$827$	−20.1250	−0.699814	−0.349907	+	0.936784i	$0.613787\pi$
$828$	−5.00969	−0.174099
$829$	33.7125	1.17088	0.585441	−	0.810715i	$-0.300921\pi$
$829$	33.7125	1.17088	0.585441	+	0.810715i	$0.300921\pi$
$830$	−28.3056	−0.982501
$831$	10.5918	0.367425
$832$	0	0
$833$	−0.137063	−0.00474896
$834$	38.9952	1.35029
$835$	52.4282	1.81435
$836$	0.335126	0.0115906
$837$	9.12259	0.315323
$838$	54.9928	1.89970
$839$	−41.3666	−1.42813	−0.714067	−	0.700078i	$-0.753149\pi$
$839$	−41.3666	−1.42813	−0.714067	+	0.700078i	$0.753149\pi$
$840$	33.6015	1.15936
$841$	−17.7313	−0.611422
$842$	76.7881	2.64629
$843$	−30.4209	−1.04775
$844$	3.04892	0.104948
$845$	30.6396	1.05404
$846$	−54.5042	−1.87389
$847$	−29.5797	−1.01637
$848$	6.12929	0.210481
$849$	−46.4403	−1.59383
$850$	−0.692021	−0.0237361
$851$	6.01746	0.206276
$852$	37.8243	1.29584
$853$	42.3957	1.45160	0.725800	−	0.687906i	$-0.241470\pi$
$853$	42.3957	1.45160	0.725800	+	0.687906i	$0.241470\pi$
$854$	27.1836	0.930203
$855$	−4.82908	−0.165151
$856$	−23.2529	−0.794769
$857$	−37.1183	−1.26794	−0.633968	−	0.773359i	$-0.718575\pi$
$857$	−37.1183	−1.26794	−0.633968	+	0.773359i	$0.718575\pi$
$858$	0	0
$859$	−40.5295	−1.38285	−0.691424	−	0.722449i	$-0.743016\pi$
$859$	−40.5295	−1.38285	−0.691424	+	0.722449i	$0.743016\pi$
$860$	25.1075	0.856159
$861$	−1.62565	−0.0554019
$862$	−14.1250	−0.481099
$863$	−36.0157	−1.22599	−0.612994	−	0.790088i	$-0.710035\pi$
$863$	−36.0157	−1.22599	−0.612994	+	0.790088i	$0.710035\pi$
$864$	−13.9245	−0.473722
$865$	−12.2970	−0.418109
$866$	41.2543	1.40188
$867$	37.5066	1.27379
$868$	35.0368	1.18923
$869$	−1.05993	−0.0359558
$870$	39.9463	1.35431
$871$	0	0
$872$	−39.4155	−1.33478
$873$	4.70410	0.159210
$874$	−1.80194	−0.0609515
$875$	28.2030	0.953434
$876$	−59.8829	−2.02326
$877$	8.50711	0.287265	0.143632	−	0.989631i	$-0.454122\pi$
$877$	8.50711	0.287265	0.143632	+	0.989631i	$0.454122\pi$
$878$	−2.96316	−0.100002
$879$	−10.7724	−0.363344
$880$	0.207751	0.00700328
$881$	21.3056	0.717803	0.358902	−	0.933375i	$-0.383151\pi$
$881$	21.3056	0.717803	0.358902	+	0.933375i	$0.383151\pi$
$882$	1.13706	0.0382869
$883$	−36.5314	−1.22938	−0.614690	−	0.788769i	$-0.710719\pi$
$883$	−36.5314	−1.22938	−0.614690	+	0.788769i	$0.710719\pi$
$884$	0	0
$885$	22.1685	0.745187
$886$	−18.4494	−0.619818
$887$	−30.9312	−1.03857	−0.519285	−	0.854601i	$-0.673802\pi$
$887$	−30.9312	−1.03857	−0.519285	+	0.854601i	$0.673802\pi$
$888$	39.7385	1.33354
$889$	1.10693	0.0371253
$890$	98.1885	3.29129
$891$	−1.20344	−0.0403167
$892$	−24.8847	−0.833202
$893$	−11.8388	−0.396169
$894$	41.8364	1.39922
$895$	7.82504	0.261562
$896$	−43.7778	−1.46251
$897$	0	0
$898$	21.1618	0.706179
$899$	14.3297	0.477924
$900$	3.46681	0.115560
$901$	4.24160	0.141308
$902$	0.0663757	0.00221007
$903$	21.1347	0.703318
$904$	−46.2043	−1.53673
$905$	−51.0917	−1.69835
$906$	−88.5115	−2.94060
$907$	13.4069	0.445168	0.222584	−	0.974914i	$-0.428551\pi$
$907$	13.4069	0.445168	0.222584	+	0.974914i	$0.428551\pi$
$908$	62.0428	2.05896
$909$	−30.4480	−1.00990
$910$	0	0
$911$	31.2519	1.03542	0.517711	−	0.855556i	$-0.326784\pi$
$911$	31.2519	1.03542	0.517711	+	0.855556i	$0.326784\pi$
$912$	1.80194	0.0596681
$913$	0.587482	0.0194428
$914$	10.1709	0.336424
$915$	23.7995	0.786789
$916$	−0.640711	−0.0211697
$917$	1.46144	0.0482608
$918$	−2.66487	−0.0879540
$919$	39.3978	1.29961	0.649806	−	0.760100i	$-0.274850\pi$
$919$	39.3978	1.29961	0.649806	+	0.760100i	$0.274850\pi$
$920$	4.45473	0.146868
$921$	−37.6383	−1.24022
$922$	−54.3062	−1.78848
$923$	0	0
$924$	−2.02715	−0.0666882
$925$	−4.16421	−0.136918
$926$	8.14675	0.267719
$927$	−11.7627	−0.386338
$928$	−21.8726	−0.718004
$929$	15.2295	0.499664	0.249832	−	0.968289i	$-0.419625\pi$
$929$	15.2295	0.499664	0.249832	+	0.968289i	$0.419625\pi$
$930$	50.7972	1.66570
$931$	0.246980	0.00809443
$932$	−43.5411	−1.42624
$933$	−25.8213	−0.845352
$934$	61.3715	2.00814
$935$	0.143768	0.00470172
$936$	0	0
$937$	33.3357	1.08903	0.544515	−	0.838751i	$-0.316714\pi$
$937$	33.3357	1.08903	0.544515	+	0.838751i	$0.316714\pi$
$938$	66.7284	2.17876
$939$	−2.41119	−0.0786862
$940$	85.0732	2.77478
$941$	4.51706	0.147252	0.0736259	−	0.997286i	$-0.476543\pi$
$941$	4.51706	0.147252	0.0736259	+	0.997286i	$0.476543\pi$
$942$	12.8509	0.418703
$943$	−0.215521	−0.00701832
$944$	−3.35690	−0.109258
$945$	−13.5593	−0.441083
$946$	−0.862937	−0.0280565
$947$	36.9633	1.20115	0.600573	−	0.799570i	$-0.294939\pi$
$947$	36.9633	1.20115	0.600573	+	0.799570i	$0.294939\pi$
$948$	66.0635	2.14564
$949$	0	0
$950$	1.24698	0.0404574
$951$	47.0810	1.52670
$952$	−3.52111	−0.114120
$953$	−21.0344	−0.681372	−0.340686	−	0.940177i	$-0.610659\pi$
$953$	−21.0344	−0.681372	−0.340686	+	0.940177i	$0.610659\pi$
$954$	−35.1879	−1.13925
$955$	−20.6528	−0.668309
$956$	58.3933	1.88857
$957$	−0.829085	−0.0268005
$958$	54.7308	1.76827
$959$	24.5972	0.794284
$960$	−69.0417	−2.22831
$961$	−12.7778	−0.412186
$962$	0	0
$963$	−20.2145	−0.651402
$964$	40.1734	1.29390
$965$	−30.5827	−0.984492
$966$	10.8998	0.350694
$967$	34.4058	1.10642	0.553208	−	0.833043i	$-0.313403\pi$
$967$	34.4058	1.10642	0.553208	+	0.833043i	$0.313403\pi$
$968$	−25.8974	−0.832373
$969$	1.24698	0.0400588
$970$	−12.1588	−0.390397
$971$	−42.4075	−1.36092	−0.680460	−	0.732785i	$-0.738220\pi$
$971$	−42.4075	−1.36092	−0.680460	+	0.732785i	$0.738220\pi$
$972$	55.4607	1.77890
$973$	−20.7918	−0.666554
$974$	−72.8993	−2.33584
$975$	0	0
$976$	−3.60388	−0.115357
$977$	26.2892	0.841066	0.420533	−	0.907277i	$-0.361843\pi$
$977$	26.2892	0.841066	0.420533	+	0.907277i	$0.361843\pi$
$978$	−57.7579	−1.84689
$979$	−2.03790	−0.0651316
$980$	−1.77479	−0.0566936
$981$	−34.2650	−1.09400
$982$	−16.2446	−0.518386
$983$	−36.3317	−1.15880	−0.579400	−	0.815043i	$-0.696713\pi$
$983$	−36.3317	−1.15880	−0.579400	+	0.815043i	$0.696713\pi$
$984$	−1.42327	−0.0453722
$985$	−0.374354	−0.0119279
$986$	−4.18598	−0.133309
$987$	71.6118	2.27943
$988$	0	0
$989$	2.80194	0.0890964
$990$	−1.19269	−0.0379060
$991$	−12.3308	−0.391701	−0.195851	−	0.980634i	$-0.562747\pi$
$991$	−12.3308	−0.391701	−0.195851	+	0.980634i	$0.562747\pi$
$992$	−27.8140	−0.883096
$993$	−48.4040	−1.53605
$994$	−33.3967	−1.05928
$995$	56.4959	1.79104
$996$	−36.6165	−1.16024
$997$	−26.5555	−0.841023	−0.420511	−	0.907287i	$-0.638149\pi$
$997$	−26.5555	−0.841023	−0.420511	+	0.907287i	$0.638149\pi$
$998$	10.3274	0.326907
$999$	−16.0358	−0.507349

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.b.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.b.1.3	✓	3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 4009 = 19 \cdot 211 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4009.a (trivial)

\(f(q)\)	\(=\)	\(q+2.24698 q^{2} -2.24698 q^{3} +3.04892 q^{4} -2.35690 q^{5} -5.04892 q^{6} +2.69202 q^{7} +2.35690 q^{8} +2.04892 q^{9} +O(q^{10})\)	\(q+2.24698 q^{2} -2.24698 q^{3} +3.04892 q^{4} -2.35690 q^{5} -5.04892 q^{6} +2.69202 q^{7} +2.35690 q^{8} +2.04892 q^{9} -5.29590 q^{10} +0.109916 q^{11} -6.85086 q^{12} +6.04892 q^{14} +5.29590 q^{15} -0.801938 q^{16} -0.554958 q^{17} +4.60388 q^{18} +1.00000 q^{19} -7.18598 q^{20} -6.04892 q^{21} +0.246980 q^{22} -0.801938 q^{23} -5.29590 q^{24} +0.554958 q^{25} +2.13706 q^{27} +8.20775 q^{28} +3.35690 q^{29} +11.8998 q^{30} +4.26875 q^{31} -6.51573 q^{32} -0.246980 q^{33} -1.24698 q^{34} -6.34481 q^{35} +6.24698 q^{36} -7.50365 q^{37} +2.24698 q^{38} -5.55496 q^{40} +0.268750 q^{41} -13.5918 q^{42} -3.49396 q^{43} +0.335126 q^{44} -4.82908 q^{45} -1.80194 q^{46} -11.8388 q^{47} +1.80194 q^{48} +0.246980 q^{49} +1.24698 q^{50} +1.24698 q^{51} -7.64310 q^{53} +4.80194 q^{54} -0.259061 q^{55} +6.34481 q^{56} -2.24698 q^{57} +7.54288 q^{58} +4.18598 q^{59} +16.1468 q^{60} +4.49396 q^{61} +9.59179 q^{62} +5.51573 q^{63} -13.0368 q^{64} -0.554958 q^{66} +11.0315 q^{67} -1.69202 q^{68} +1.80194 q^{69} -14.2567 q^{70} -5.52111 q^{71} +4.82908 q^{72} +8.74094 q^{73} -16.8605 q^{74} -1.24698 q^{75} +3.04892 q^{76} +0.295897 q^{77} -9.64310 q^{79} +1.89008 q^{80} -10.9487 q^{81} +0.603875 q^{82} +5.34481 q^{83} -18.4426 q^{84} +1.30798 q^{85} -7.85086 q^{86} -7.54288 q^{87} +0.259061 q^{88} -18.5405 q^{89} -10.8509 q^{90} -2.44504 q^{92} -9.59179 q^{93} -26.6015 q^{94} -2.35690 q^{95} +14.6407 q^{96} +2.29590 q^{97} +0.554958 q^{98} +0.225209 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{2} - 2 q^{3} - 3 q^{5} - 6 q^{6} + 3 q^{7} + 3 q^{8} - 3 q^{9}+O(q^{10}) \) 3 * q + 2 * q^2 - 2 * q^3 - 3 * q^5 - 6 * q^6 + 3 * q^7 + 3 * q^8 - 3 * q^9	\( 3 q + 2 q^{2} - 2 q^{3} - 3 q^{5} - 6 q^{6} + 3 q^{7} + 3 q^{8} - 3 q^{9} - 2 q^{10} + q^{11} - 7 q^{12} + 9 q^{14} + 2 q^{15} + 2 q^{16} - 2 q^{17} + 5 q^{18} + 3 q^{19} - 7 q^{20} - 9 q^{21} - 4 q^{22} + 2 q^{23} - 2 q^{24} + 2 q^{25} + q^{27} + 7 q^{28} + 6 q^{29} + 13 q^{30} + 5 q^{31} - 7 q^{32} + 4 q^{33} + q^{34} + 4 q^{35} + 14 q^{36} + 9 q^{37} + 2 q^{38} - 17 q^{40} - 7 q^{41} - 13 q^{42} - q^{43} - 4 q^{45} - q^{46} - 3 q^{47} + q^{48} - 4 q^{49} - q^{50} - q^{51} - 27 q^{53} + 10 q^{54} - 15 q^{55} - 4 q^{56} - 2 q^{57} + 4 q^{58} - 2 q^{59} + 21 q^{60} + 4 q^{61} + q^{62} + 4 q^{63} - 11 q^{64} - 2 q^{66} + 8 q^{67} + q^{69} - 16 q^{70} - q^{71} + 4 q^{72} + 12 q^{73} - 15 q^{74} + q^{75} - 13 q^{77} - 33 q^{79} + 5 q^{80} - q^{81} - 7 q^{82} - 7 q^{83} - 14 q^{84} + 9 q^{85} - 10 q^{86} - 4 q^{87} + 15 q^{88} + 4 q^{89} - 19 q^{90} - 7 q^{92} - q^{93} - 30 q^{94} - 3 q^{95} + 7 q^{96} - 7 q^{97} + 2 q^{98} - q^{99}+O(q^{100}) \) 3 * q + 2 * q^2 - 2 * q^3 - 3 * q^5 - 6 * q^6 + 3 * q^7 + 3 * q^8 - 3 * q^9 - 2 * q^10 + q^11 - 7 * q^12 + 9 * q^14 + 2 * q^15 + 2 * q^16 - 2 * q^17 + 5 * q^18 + 3 * q^19 - 7 * q^20 - 9 * q^21 - 4 * q^22 + 2 * q^23 - 2 * q^24 + 2 * q^25 + q^27 + 7 * q^28 + 6 * q^29 + 13 * q^30 + 5 * q^31 - 7 * q^32 + 4 * q^33 + q^34 + 4 * q^35 + 14 * q^36 + 9 * q^37 + 2 * q^38 - 17 * q^40 - 7 * q^41 - 13 * q^42 - q^43 - 4 * q^45 - q^46 - 3 * q^47 + q^48 - 4 * q^49 - q^50 - q^51 - 27 * q^53 + 10 * q^54 - 15 * q^55 - 4 * q^56 - 2 * q^57 + 4 * q^58 - 2 * q^59 + 21 * q^60 + 4 * q^61 + q^62 + 4 * q^63 - 11 * q^64 - 2 * q^66 + 8 * q^67 + q^69 - 16 * q^70 - q^71 + 4 * q^72 + 12 * q^73 - 15 * q^74 + q^75 - 13 * q^77 - 33 * q^79 + 5 * q^80 - q^81 - 7 * q^82 - 7 * q^83 - 14 * q^84 + 9 * q^85 - 10 * q^86 - 4 * q^87 + 15 * q^88 + 4 * q^89 - 19 * q^90 - 7 * q^92 - q^93 - 30 * q^94 - 3 * q^95 + 7 * q^96 - 7 * q^97 + 2 * q^98 - q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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