LMFDB - Embedded newform 6027.2.a.bm.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6027,2,Mod(1,6027)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6027, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6027.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6027 = 3 \cdot 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6027.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:	$48.1258372982$
Analytic rank:	$1$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 14 x^{14} + 68 x^{13} + 64 x^{12} - 456 x^{11} - 54 x^{10} + 1532 x^{9} - 400 x^{8} + \cdots - 49 $ x^16 - 4x^15 - 14x^14 + 68x^13 + 64x^12 - 456x^11 - 54x^10 + 1532x^9 - 400x^8 - 2708x^7 + 1218x^6 + 2424x^5 - 1276x^4 - 960x^3 + 500x^2 + 112*x - 49
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.64639$ of defining polynomial
Character	$\chi$	$=$	6027.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.64639 q^{2} +1.00000 q^{3} +0.710595 q^{4} +0.0457395 q^{5} -1.64639 q^{6} +2.12286 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.64639 q^{2} +1.00000 q^{3} +0.710595 q^{4} +0.0457395 q^{5} -1.64639 q^{6} +2.12286 q^{8} +1.00000 q^{9} -0.0753050 q^{10} +0.153516 q^{11} +0.710595 q^{12} +3.46056 q^{13} +0.0457395 q^{15} -4.91624 q^{16} -0.713284 q^{17} -1.64639 q^{18} -6.38590 q^{19} +0.0325023 q^{20} -0.252747 q^{22} -1.26532 q^{23} +2.12286 q^{24} -4.99791 q^{25} -5.69742 q^{26} +1.00000 q^{27} +1.30954 q^{29} -0.0753050 q^{30} +0.735747 q^{31} +3.84832 q^{32} +0.153516 q^{33} +1.17434 q^{34} +0.710595 q^{36} +9.12498 q^{37} +10.5137 q^{38} +3.46056 q^{39} +0.0970987 q^{40} +1.00000 q^{41} +2.39572 q^{43} +0.109088 q^{44} +0.0457395 q^{45} +2.08321 q^{46} +9.00229 q^{47} -4.91624 q^{48} +8.22850 q^{50} -0.713284 q^{51} +2.45905 q^{52} -14.1311 q^{53} -1.64639 q^{54} +0.00702175 q^{55} -6.38590 q^{57} -2.15600 q^{58} -11.3136 q^{59} +0.0325023 q^{60} -14.2763 q^{61} -1.21133 q^{62} +3.49665 q^{64} +0.158284 q^{65} -0.252747 q^{66} -13.3603 q^{67} -0.506856 q^{68} -1.26532 q^{69} -6.60480 q^{71} +2.12286 q^{72} +15.8710 q^{73} -15.0233 q^{74} -4.99791 q^{75} -4.53778 q^{76} -5.69742 q^{78} -6.18509 q^{79} -0.224867 q^{80} +1.00000 q^{81} -1.64639 q^{82} -2.20561 q^{83} -0.0326253 q^{85} -3.94428 q^{86} +1.30954 q^{87} +0.325893 q^{88} -14.3331 q^{89} -0.0753050 q^{90} -0.899130 q^{92} +0.735747 q^{93} -14.8213 q^{94} -0.292088 q^{95} +3.84832 q^{96} -8.42652 q^{97} +0.153516 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{2} + 16 q^{3} + 12 q^{4} - 12 q^{5} - 4 q^{6} - 12 q^{8} + 16 q^{9}+O(q^{10}) $ 16 * q - 4 * q^2 + 16 * q^3 + 12 * q^4 - 12 * q^5 - 4 * q^6 - 12 * q^8 + 16 * q^9	$ 16 q - 4 q^{2} + 16 q^{3} + 12 q^{4} - 12 q^{5} - 4 q^{6} - 12 q^{8} + 16 q^{9} - 4 q^{10} - 4 q^{11} + 12 q^{12} - 12 q^{15} - 8 q^{17} - 4 q^{18} + 4 q^{19} - 20 q^{20} - 16 q^{22} - 12 q^{23} - 12 q^{24} - 8 q^{25} - 8 q^{26} + 16 q^{27} - 16 q^{29} - 4 q^{30} - 4 q^{31} - 48 q^{32} - 4 q^{33} + 16 q^{34} + 12 q^{36} - 48 q^{37} - 4 q^{38} + 56 q^{40} + 16 q^{41} - 16 q^{43} - 12 q^{45} - 4 q^{46} - 36 q^{47} - 8 q^{50} - 8 q^{51} - 60 q^{53} - 4 q^{54} + 8 q^{55} + 4 q^{57} - 36 q^{58} - 36 q^{59} - 20 q^{60} - 4 q^{61} - 12 q^{62} + 52 q^{64} - 36 q^{65} - 16 q^{66} - 52 q^{67} - 8 q^{68} - 12 q^{69} - 12 q^{71} - 12 q^{72} - 16 q^{73} + 4 q^{74} - 8 q^{75} + 16 q^{76} - 8 q^{78} - 36 q^{79} - 68 q^{80} + 16 q^{81} - 4 q^{82} - 32 q^{83} - 28 q^{85} - 8 q^{86} - 16 q^{87} - 36 q^{88} - 12 q^{89} - 4 q^{90} - 36 q^{92} - 4 q^{93} + 24 q^{94} - 20 q^{95} - 48 q^{96} + 48 q^{97} - 4 q^{99}+O(q^{100}) $ 16 * q - 4 * q^2 + 16 * q^3 + 12 * q^4 - 12 * q^5 - 4 * q^6 - 12 * q^8 + 16 * q^9 - 4 * q^10 - 4 * q^11 + 12 * q^12 - 12 * q^15 - 8 * q^17 - 4 * q^18 + 4 * q^19 - 20 * q^20 - 16 * q^22 - 12 * q^23 - 12 * q^24 - 8 * q^25 - 8 * q^26 + 16 * q^27 - 16 * q^29 - 4 * q^30 - 4 * q^31 - 48 * q^32 - 4 * q^33 + 16 * q^34 + 12 * q^36 - 48 * q^37 - 4 * q^38 + 56 * q^40 + 16 * q^41 - 16 * q^43 - 12 * q^45 - 4 * q^46 - 36 * q^47 - 8 * q^50 - 8 * q^51 - 60 * q^53 - 4 * q^54 + 8 * q^55 + 4 * q^57 - 36 * q^58 - 36 * q^59 - 20 * q^60 - 4 * q^61 - 12 * q^62 + 52 * q^64 - 36 * q^65 - 16 * q^66 - 52 * q^67 - 8 * q^68 - 12 * q^69 - 12 * q^71 - 12 * q^72 - 16 * q^73 + 4 * q^74 - 8 * q^75 + 16 * q^76 - 8 * q^78 - 36 * q^79 - 68 * q^80 + 16 * q^81 - 4 * q^82 - 32 * q^83 - 28 * q^85 - 8 * q^86 - 16 * q^87 - 36 * q^88 - 12 * q^89 - 4 * q^90 - 36 * q^92 - 4 * q^93 + 24 * q^94 - 20 * q^95 - 48 * q^96 + 48 * q^97 - 4 * 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 $

$2$	−1.64639	−1.16417	−0.582086	−	0.813127i	$-0.697763\pi$
$2$	−1.64639	−1.16417	−0.582086	+	0.813127i	$0.697763\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0.710595	0.355297
$5$	0.0457395	0.0204553	0.0102277	−	0.999948i	$-0.496744\pi$
$5$	0.0457395	0.0204553	0.0102277	+	0.999948i	$0.496744\pi$
$6$	−1.64639	−0.672135
$7$	0	0
$7$	0	0
$8$	2.12286	0.750545
$9$	1.00000	0.333333
$10$	−0.0753050	−0.0238135
$11$	0.153516	0.0462868	0.0231434	−	0.999732i	$-0.492633\pi$
$11$	0.153516	0.0462868	0.0231434	+	0.999732i	$0.492633\pi$
$12$	0.710595	0.205131
$13$	3.46056	0.959786	0.479893	−	0.877327i	$-0.340675\pi$
$13$	3.46056	0.959786	0.479893	+	0.877327i	$0.340675\pi$
$14$	0	0
$15$	0.0457395	0.0118099
$16$	−4.91624	−1.22906
$17$	−0.713284	−0.172997	−0.0864984	−	0.996252i	$-0.527568\pi$
$17$	−0.713284	−0.172997	−0.0864984	+	0.996252i	$0.527568\pi$
$18$	−1.64639	−0.388057
$19$	−6.38590	−1.46503	−0.732513	−	0.680754i	$-0.761652\pi$
$19$	−6.38590	−1.46503	−0.732513	+	0.680754i	$0.761652\pi$
$20$	0.0325023	0.00726773
$21$	0	0
$22$	−0.252747	−0.0538858
$23$	−1.26532	−0.263838	−0.131919	−	0.991261i	$-0.542114\pi$
$23$	−1.26532	−0.263838	−0.131919	+	0.991261i	$0.542114\pi$
$24$	2.12286	0.433327
$25$	−4.99791	−0.999582
$26$	−5.69742	−1.11736
$27$	1.00000	0.192450
$28$	0	0
$29$	1.30954	0.243175	0.121587	−	0.992581i	$-0.461202\pi$
$29$	1.30954	0.243175	0.121587	+	0.992581i	$0.461202\pi$
$30$	−0.0753050	−0.0137487
$31$	0.735747	0.132144	0.0660721	−	0.997815i	$-0.478953\pi$
$31$	0.735747	0.132144	0.0660721	+	0.997815i	$0.478953\pi$
$32$	3.84832	0.680294
$33$	0.153516	0.0267237
$34$	1.17434	0.201398
$35$	0	0
$36$	0.710595	0.118432
$37$	9.12498	1.50014	0.750069	−	0.661360i	$-0.230020\pi$
$37$	9.12498	1.50014	0.750069	+	0.661360i	$0.230020\pi$
$38$	10.5137	1.70554
$39$	3.46056	0.554133
$40$	0.0970987	0.0153526
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	2.39572	0.365344	0.182672	−	0.983174i	$-0.441525\pi$
$43$	2.39572	0.365344	0.182672	+	0.983174i	$0.441525\pi$
$44$	0.109088	0.0164456
$45$	0.0457395	0.00681844
$46$	2.08321	0.307152
$47$	9.00229	1.31312	0.656560	−	0.754274i	$-0.272011\pi$
$47$	9.00229	1.31312	0.656560	+	0.754274i	$0.272011\pi$
$48$	−4.91624	−0.709599
$49$	0	0
$50$	8.22850	1.16369
$51$	−0.713284	−0.0998798
$52$	2.45905	0.341009
$53$	−14.1311	−1.94106	−0.970529	−	0.240985i	$-0.922529\pi$
$53$	−14.1311	−1.94106	−0.970529	+	0.240985i	$0.922529\pi$
$54$	−1.64639	−0.224045
$55$	0.00702175	0.000946812 0
$56$	0	0
$57$	−6.38590	−0.845833
$58$	−2.15600	−0.283097
$59$	−11.3136	−1.47291	−0.736454	−	0.676488i	$-0.763501\pi$
$59$	−11.3136	−1.47291	−0.736454	+	0.676488i	$0.763501\pi$
$60$	0.0325023	0.00419602
$61$	−14.2763	−1.82789	−0.913946	−	0.405836i	$-0.866980\pi$
$61$	−14.2763	−1.82789	−0.913946	+	0.405836i	$0.866980\pi$
$62$	−1.21133	−0.153839
$63$	0	0
$64$	3.49665	0.437081
$65$	0.158284	0.0196327
$66$	−0.252747	−0.0311110
$67$	−13.3603	−1.63222	−0.816109	−	0.577898i	$-0.803873\pi$
$67$	−13.3603	−1.63222	−0.816109	+	0.577898i	$0.803873\pi$
$68$	−0.506856	−0.0614653
$69$	−1.26532	−0.152327
$70$	0	0
$71$	−6.60480	−0.783846	−0.391923	−	0.919998i	$-0.628190\pi$
$71$	−6.60480	−0.783846	−0.391923	+	0.919998i	$0.628190\pi$
$72$	2.12286	0.250182
$73$	15.8710	1.85756	0.928778	−	0.370637i	$-0.120861\pi$
$73$	15.8710	1.85756	0.928778	+	0.370637i	$0.120861\pi$
$74$	−15.0233	−1.74642
$75$	−4.99791	−0.577109
$76$	−4.53778	−0.520520
$77$	0	0
$78$	−5.69742	−0.645106
$79$	−6.18509	−0.695877	−0.347938	−	0.937517i	$-0.613118\pi$
$79$	−6.18509	−0.695877	−0.347938	+	0.937517i	$0.613118\pi$
$80$	−0.224867	−0.0251409
$81$	1.00000	0.111111
$82$	−1.64639	−0.181813
$83$	−2.20561	−0.242097	−0.121048	−	0.992647i	$-0.538626\pi$
$83$	−2.20561	−0.242097	−0.121048	+	0.992647i	$0.538626\pi$
$84$	0	0
$85$	−0.0326253	−0.00353871
$86$	−3.94428	−0.425323
$87$	1.30954	0.140397
$88$	0.325893	0.0347403
$89$	−14.3331	−1.51930	−0.759651	−	0.650332i	$-0.774630\pi$
$89$	−14.3331	−1.51930	−0.759651	+	0.650332i	$0.774630\pi$
$90$	−0.0753050	−0.00793784
$91$	0	0
$92$	−0.899130	−0.0937408
$93$	0.735747	0.0762935
$94$	−14.8213	−1.52870
$95$	−0.292088	−0.0299676
$96$	3.84832	0.392768
$97$	−8.42652	−0.855583	−0.427792	−	0.903877i	$-0.640708\pi$
$97$	−8.42652	−0.855583	−0.427792	+	0.903877i	$0.640708\pi$
$98$	0	0
$99$	0.153516	0.0154289
$100$	−3.55149	−0.355149
$101$	−4.27869	−0.425746	−0.212873	−	0.977080i	$-0.568282\pi$
$101$	−4.27869	−0.425746	−0.212873	+	0.977080i	$0.568282\pi$
$102$	1.17434	0.116277
$103$	8.86287	0.873284	0.436642	−	0.899635i	$-0.356168\pi$
$103$	8.86287	0.873284	0.436642	+	0.899635i	$0.356168\pi$
$104$	7.34629	0.720363
$105$	0	0
$106$	23.2653	2.25973
$107$	2.93535	0.283771	0.141886	−	0.989883i	$-0.454683\pi$
$107$	2.93535	0.283771	0.141886	+	0.989883i	$0.454683\pi$
$108$	0.710595	0.0683770
$109$	−8.56169	−0.820061	−0.410030	−	0.912072i	$-0.634482\pi$
$109$	−8.56169	−0.820061	−0.410030	+	0.912072i	$0.634482\pi$
$110$	−0.0115605	−0.00110225
$111$	9.12498	0.866105
$112$	0	0
$113$	8.56761	0.805973	0.402986	−	0.915206i	$-0.367972\pi$
$113$	8.56761	0.805973	0.402986	+	0.915206i	$0.367972\pi$
$114$	10.5137	0.984695
$115$	−0.0578751	−0.00539688
$116$	0.930549	0.0863993
$117$	3.46056	0.319929
$118$	18.6266	1.71472
$119$	0	0
$120$	0.0970987	0.00886385
$121$	−10.9764	−0.997858
$122$	23.5043	2.12798
$123$	1.00000	0.0901670
$124$	0.522818	0.0469505
$125$	−0.457299	−0.0409021
$126$	0	0
$127$	−0.852507	−0.0756478	−0.0378239	−	0.999284i	$-0.512043\pi$
$127$	−0.852507	−0.0756478	−0.0378239	+	0.999284i	$0.512043\pi$
$128$	−13.4535	−1.18913
$129$	2.39572	0.210931
$130$	−0.260597	−0.0228559
$131$	−10.6329	−0.928999	−0.464499	−	0.885573i	$-0.653766\pi$
$131$	−10.6329	−0.928999	−0.464499	+	0.885573i	$0.653766\pi$
$132$	0.109088	0.00949486
$133$	0	0
$134$	21.9962	1.90018
$135$	0.0457395	0.00393663
$136$	−1.51420	−0.129842
$137$	21.2240	1.81329	0.906645	−	0.421895i	$-0.138635\pi$
$137$	21.2240	1.81329	0.906645	+	0.421895i	$0.138635\pi$
$138$	2.08321	0.177334
$139$	12.0384	1.02108	0.510541	−	0.859854i	$-0.329445\pi$
$139$	12.0384	1.02108	0.510541	+	0.859854i	$0.329445\pi$
$140$	0	0
$141$	9.00229	0.758130
$142$	10.8741	0.912531
$143$	0.531251	0.0444254
$144$	−4.91624	−0.409687
$145$	0.0598975	0.00497422
$146$	−26.1298	−2.16252
$147$	0	0
$148$	6.48416	0.532995
$149$	−15.5561	−1.27441	−0.637203	−	0.770696i	$-0.719909\pi$
$149$	−15.5561	−1.27441	−0.637203	+	0.770696i	$0.719909\pi$
$150$	8.22850	0.671854
$151$	23.1467	1.88365	0.941826	−	0.336100i	$-0.109108\pi$
$151$	23.1467	1.88365	0.941826	+	0.336100i	$0.109108\pi$
$152$	−13.5564	−1.09957
$153$	−0.713284	−0.0576656
$154$	0	0
$155$	0.0336527	0.00270305
$156$	2.45905	0.196882
$157$	13.5629	1.08244	0.541219	−	0.840882i	$-0.317963\pi$
$157$	13.5629	1.08244	0.541219	+	0.840882i	$0.317963\pi$
$158$	10.1831	0.810120
$159$	−14.1311	−1.12067
$160$	0.176020	0.0139156
$161$	0	0
$162$	−1.64639	−0.129352
$163$	18.8746	1.47837	0.739184	−	0.673503i	$-0.235211\pi$
$163$	18.8746	1.47837	0.739184	+	0.673503i	$0.235211\pi$
$164$	0.710595	0.0554881
$165$	0.00702175	0.000546642 0
$166$	3.63128	0.281842
$167$	1.10994	0.0858894	0.0429447	−	0.999077i	$-0.486326\pi$
$167$	1.10994	0.0858894	0.0429447	+	0.999077i	$0.486326\pi$
$168$	0	0
$169$	−1.02454	−0.0788109
$170$	0.0537139	0.00411967
$171$	−6.38590	−0.488342
$172$	1.70239	0.129806
$173$	−21.6857	−1.64874	−0.824368	−	0.566054i	$-0.808469\pi$
$173$	−21.6857	−1.64874	−0.824368	+	0.566054i	$0.808469\pi$
$174$	−2.15600	−0.163446
$175$	0	0
$176$	−0.754722	−0.0568893
$177$	−11.3136	−0.850383
$178$	23.5978	1.76873
$179$	−7.05942	−0.527646	−0.263823	−	0.964571i	$-0.584983\pi$
$179$	−7.05942	−0.527646	−0.263823	+	0.964571i	$0.584983\pi$
$180$	0.0325023	0.00242258
$181$	13.1070	0.974234	0.487117	−	0.873337i	$-0.338049\pi$
$181$	13.1070	0.974234	0.487117	+	0.873337i	$0.338049\pi$
$182$	0	0
$183$	−14.2763	−1.05533
$184$	−2.68610	−0.198022
$185$	0.417372	0.0306858
$186$	−1.21133	−0.0888187
$187$	−0.109501	−0.00800747
$188$	6.39698	0.466548
$189$	0	0
$190$	0.480890	0.0348874
$191$	9.51467	0.688457	0.344229	−	0.938886i	$-0.388140\pi$
$191$	9.51467	0.688457	0.344229	+	0.938886i	$0.388140\pi$
$192$	3.49665	0.252349
$193$	−10.1608	−0.731389	−0.365695	−	0.930735i	$-0.619169\pi$
$193$	−10.1608	−0.731389	−0.365695	+	0.930735i	$0.619169\pi$
$194$	13.8733	0.996047
$195$	0.158284	0.0113350
$196$	0	0
$197$	7.54126	0.537293	0.268646	−	0.963239i	$-0.413424\pi$
$197$	7.54126	0.537293	0.268646	+	0.963239i	$0.413424\pi$
$198$	−0.252747	−0.0179619
$199$	25.9232	1.83765	0.918823	−	0.394671i	$-0.129141\pi$
$199$	25.9232	1.83765	0.918823	+	0.394671i	$0.129141\pi$
$200$	−10.6099	−0.750231
$201$	−13.3603	−0.942361
$202$	7.04439	0.495641
$203$	0	0
$204$	−0.506856	−0.0354870
$205$	0.0457395	0.00319459
$206$	−14.5917	−1.01665
$207$	−1.26532	−0.0879458
$208$	−17.0129	−1.17964
$209$	−0.980337	−0.0678113
$210$	0	0
$211$	−14.2724	−0.982555	−0.491277	−	0.871003i	$-0.663470\pi$
$211$	−14.2724	−0.982555	−0.491277	+	0.871003i	$0.663470\pi$
$212$	−10.0415	−0.689653
$213$	−6.60480	−0.452553
$214$	−4.83273	−0.330359
$215$	0.109579	0.00747322
$216$	2.12286	0.144442
$217$	0	0
$218$	14.0959	0.954692
$219$	15.8710	1.07246
$220$	0.00498962	0.000336400 0
$221$	−2.46836	−0.166040
$222$	−15.0233	−1.00830
$223$	−15.5265	−1.03973	−0.519866	−	0.854248i	$-0.674018\pi$
$223$	−15.5265	−1.03973	−0.519866	+	0.854248i	$0.674018\pi$
$224$	0	0
$225$	−4.99791	−0.333194
$226$	−14.1056	−0.938291
$227$	−22.0233	−1.46174	−0.730870	−	0.682517i	$-0.760886\pi$
$227$	−22.0233	−1.46174	−0.730870	+	0.682517i	$0.760886\pi$
$228$	−4.53778	−0.300522
$229$	−14.6238	−0.966369	−0.483184	−	0.875519i	$-0.660520\pi$
$229$	−14.6238	−0.966369	−0.483184	+	0.875519i	$0.660520\pi$
$230$	0.0952849	0.00628290
$231$	0	0
$232$	2.77996	0.182514
$233$	4.56710	0.299200	0.149600	−	0.988747i	$-0.452201\pi$
$233$	4.56710	0.299200	0.149600	+	0.988747i	$0.452201\pi$
$234$	−5.69742	−0.372452
$235$	0.411760	0.0268603
$236$	−8.03940	−0.523320
$237$	−6.18509	−0.401765
$238$	0	0
$239$	15.1362	0.979079	0.489540	−	0.871981i	$-0.337165\pi$
$239$	15.1362	0.979079	0.489540	+	0.871981i	$0.337165\pi$
$240$	−0.224867	−0.0145151
$241$	−12.9806	−0.836153	−0.418077	−	0.908412i	$-0.637296\pi$
$241$	−12.9806	−0.836153	−0.418077	+	0.908412i	$0.637296\pi$
$242$	18.0715	1.16168
$243$	1.00000	0.0641500
$244$	−10.1447	−0.649445
$245$	0	0
$246$	−1.64639	−0.104970
$247$	−22.0988	−1.40611
$248$	1.56189	0.0991801
$249$	−2.20561	−0.139775
$250$	0.752892	0.0476171
$251$	−21.8010	−1.37607	−0.688033	−	0.725679i	$-0.741526\pi$
$251$	−21.8010	−1.37607	−0.688033	+	0.725679i	$0.741526\pi$
$252$	0	0
$253$	−0.194247	−0.0122122
$254$	1.40356	0.0880671
$255$	−0.0326253	−0.00204307
$256$	15.1564	0.947274
$257$	−8.80386	−0.549170	−0.274585	−	0.961563i	$-0.588540\pi$
$257$	−8.80386	−0.549170	−0.274585	+	0.961563i	$0.588540\pi$
$258$	−3.94428	−0.245560
$259$	0	0
$260$	0.112476	0.00697546
$261$	1.30954	0.0810582
$262$	17.5058	1.08151
$263$	−24.6440	−1.51962	−0.759808	−	0.650147i	$-0.774707\pi$
$263$	−24.6440	−1.51962	−0.759808	+	0.650147i	$0.774707\pi$
$264$	0.325893	0.0200573
$265$	−0.646350	−0.0397050
$266$	0	0
$267$	−14.3331	−0.877169
$268$	−9.49375	−0.579923
$269$	−13.5580	−0.826648	−0.413324	−	0.910584i	$-0.635632\pi$
$269$	−13.5580	−0.826648	−0.413324	+	0.910584i	$0.635632\pi$
$270$	−0.0753050	−0.00458292
$271$	19.5011	1.18461	0.592305	−	0.805714i	$-0.298218\pi$
$271$	19.5011	1.18461	0.592305	+	0.805714i	$0.298218\pi$
$272$	3.50668	0.212624
$273$	0	0
$274$	−34.9430	−2.11098
$275$	−0.767259	−0.0462674
$276$	−0.899130	−0.0541213
$277$	9.02931	0.542519	0.271259	−	0.962506i	$-0.412560\pi$
$277$	9.02931	0.542519	0.271259	+	0.962506i	$0.412560\pi$
$278$	−19.8198	−1.18871
$279$	0.735747	0.0440480
$280$	0	0
$281$	0.893294	0.0532895	0.0266447	−	0.999645i	$-0.491518\pi$
$281$	0.893294	0.0532895	0.0266447	+	0.999645i	$0.491518\pi$
$282$	−14.8213	−0.882594
$283$	20.6014	1.22463	0.612313	−	0.790615i	$-0.290239\pi$
$283$	20.6014	1.22463	0.612313	+	0.790615i	$0.290239\pi$
$284$	−4.69334	−0.278498
$285$	−0.292088	−0.0173018
$286$	−0.874645	−0.0517189
$287$	0	0
$288$	3.84832	0.226765
$289$	−16.4912	−0.970072
$290$	−0.0986146	−0.00579085
$291$	−8.42652	−0.493971
$292$	11.2778	0.659985
$293$	17.0744	0.997495	0.498747	−	0.866747i	$-0.333794\pi$
$293$	17.0744	0.997495	0.498747	+	0.866747i	$0.333794\pi$
$294$	0	0
$295$	−0.517479	−0.0301288
$296$	19.3711	1.12592
$297$	0.153516	0.00890790
$298$	25.6114	1.48363
$299$	−4.37871	−0.253228
$300$	−3.55149	−0.205045
$301$	0	0
$302$	−38.1085	−2.19290
$303$	−4.27869	−0.245804
$304$	31.3946	1.80061
$305$	−0.652991	−0.0373901
$306$	1.17434	0.0671327
$307$	6.13820	0.350325	0.175163	−	0.984539i	$-0.443955\pi$
$307$	6.13820	0.350325	0.175163	+	0.984539i	$0.443955\pi$
$308$	0	0
$309$	8.86287	0.504191
$310$	−0.0554055	−0.00314682
$311$	15.6468	0.887247	0.443624	−	0.896213i	$-0.353693\pi$
$311$	15.6468	0.887247	0.443624	+	0.896213i	$0.353693\pi$
$312$	7.34629	0.415901
$313$	16.9514	0.958150	0.479075	−	0.877774i	$-0.340972\pi$
$313$	16.9514	0.958150	0.479075	+	0.877774i	$0.340972\pi$
$314$	−22.3298	−1.26015
$315$	0	0
$316$	−4.39509	−0.247243
$317$	−8.16987	−0.458866	−0.229433	−	0.973324i	$-0.573687\pi$
$317$	−8.16987	−0.458866	−0.229433	+	0.973324i	$0.573687\pi$
$318$	23.2653	1.30465
$319$	0.201035	0.0112558
$320$	0.159935	0.00894065
$321$	2.93535	0.163836
$322$	0	0
$323$	4.55496	0.253445
$324$	0.710595	0.0394775
$325$	−17.2955	−0.959384
$326$	−31.0748	−1.72108
$327$	−8.56169	−0.473462
$328$	2.12286	0.117215
$329$	0	0
$330$	−0.0115605	−0.000636386 0
$331$	7.85298	0.431639	0.215820	−	0.976433i	$-0.430758\pi$
$331$	7.85298	0.431639	0.215820	+	0.976433i	$0.430758\pi$
$332$	−1.56729	−0.0860163
$333$	9.12498	0.500046
$334$	−1.82738	−0.0999900
$335$	−0.611093	−0.0333876
$336$	0	0
$337$	−10.3559	−0.564124	−0.282062	−	0.959396i	$-0.591018\pi$
$337$	−10.3559	−0.564124	−0.282062	+	0.959396i	$0.591018\pi$
$338$	1.68679	0.0917495
$339$	8.56761	0.465328
$340$	−0.0231834	−0.00125729
$341$	0.112949	0.00611653
$342$	10.5137	0.568514
$343$	0	0
$344$	5.08578	0.274207
$345$	−0.0578751	−0.00311589
$346$	35.7032	1.91941
$347$	−28.8340	−1.54789	−0.773944	−	0.633254i	$-0.781719\pi$
$347$	−28.8340	−1.54789	−0.773944	+	0.633254i	$0.781719\pi$
$348$	0.930549	0.0498827
$349$	−2.39890	−0.128410	−0.0642051	−	0.997937i	$-0.520451\pi$
$349$	−2.39890	−0.128410	−0.0642051	+	0.997937i	$0.520451\pi$
$350$	0	0
$351$	3.46056	0.184711
$352$	0.590779	0.0314886
$353$	−0.720190	−0.0383318	−0.0191659	−	0.999816i	$-0.506101\pi$
$353$	−0.720190	−0.0383318	−0.0191659	+	0.999816i	$0.506101\pi$
$354$	18.6266	0.989993
$355$	−0.302100	−0.0160338
$356$	−10.1850	−0.539804
$357$	0	0
$358$	11.6225	0.614271
$359$	−18.3701	−0.969535	−0.484767	−	0.874643i	$-0.661096\pi$
$359$	−18.3701	−0.969535	−0.484767	+	0.874643i	$0.661096\pi$
$360$	0.0970987	0.00511755
$361$	21.7797	1.14630
$362$	−21.5792	−1.13418
$363$	−10.9764	−0.576113
$364$	0	0
$365$	0.725930	0.0379969
$366$	23.5043	1.22859
$367$	−7.81632	−0.408009	−0.204004	−	0.978970i	$-0.565396\pi$
$367$	−7.81632	−0.408009	−0.204004	+	0.978970i	$0.565396\pi$
$368$	6.22062	0.324272
$369$	1.00000	0.0520579
$370$	−0.687157	−0.0357236
$371$	0	0
$372$	0.522818	0.0271069
$373$	−24.2133	−1.25372	−0.626859	−	0.779133i	$-0.715660\pi$
$373$	−24.2133	−1.25372	−0.626859	+	0.779133i	$0.715660\pi$
$374$	0.180280	0.00932208
$375$	−0.457299	−0.0236148
$376$	19.1106	0.985555
$377$	4.53172	0.233396
$378$	0	0
$379$	−15.6238	−0.802538	−0.401269	−	0.915960i	$-0.631431\pi$
$379$	−15.6238	−0.802538	−0.401269	+	0.915960i	$0.631431\pi$
$380$	−0.207556	−0.0106474
$381$	−0.852507	−0.0436753
$382$	−15.6648	−0.801483
$383$	32.9203	1.68215	0.841073	−	0.540921i	$-0.181924\pi$
$383$	32.9203	1.68215	0.841073	+	0.540921i	$0.181924\pi$
$384$	−13.4535	−0.686546
$385$	0	0
$386$	16.7286	0.851463
$387$	2.39572	0.121781
$388$	−5.98784	−0.303987
$389$	−22.7504	−1.15349	−0.576745	−	0.816924i	$-0.695678\pi$
$389$	−22.7504	−1.15349	−0.576745	+	0.816924i	$0.695678\pi$
$390$	−0.260597	−0.0131959
$391$	0.902533	0.0456431
$392$	0	0
$393$	−10.6329	−0.536358
$394$	−12.4158	−0.625501
$395$	−0.282903	−0.0142344
$396$	0.109088	0.00548186
$397$	−5.59969	−0.281040	−0.140520	−	0.990078i	$-0.544877\pi$
$397$	−5.59969	−0.281040	−0.140520	+	0.990078i	$0.544877\pi$
$398$	−42.6796	−2.13934
$399$	0	0
$400$	24.5709	1.22855
$401$	−24.1918	−1.20808	−0.604041	−	0.796953i	$-0.706444\pi$
$401$	−24.1918	−1.20808	−0.604041	+	0.796953i	$0.706444\pi$
$402$	21.9962	1.09707
$403$	2.54610	0.126830
$404$	−3.04042	−0.151266
$405$	0.0457395	0.00227281
$406$	0	0
$407$	1.40083	0.0694366
$408$	−1.51420	−0.0749643
$409$	−18.2159	−0.900716	−0.450358	−	0.892848i	$-0.648704\pi$
$409$	−18.2159	−0.900716	−0.450358	+	0.892848i	$0.648704\pi$
$410$	−0.0753050	−0.00371905
$411$	21.2240	1.04690
$412$	6.29791	0.310276
$413$	0	0
$414$	2.08321	0.102384
$415$	−0.100883	−0.00495217
$416$	13.3173	0.652937
$417$	12.0384	0.589521
$418$	1.61402	0.0789441
$419$	−39.5625	−1.93275	−0.966377	−	0.257128i	$-0.917224\pi$
$419$	−39.5625	−1.93275	−0.966377	+	0.257128i	$0.917224\pi$
$420$	0	0
$421$	−33.8389	−1.64921	−0.824603	−	0.565712i	$-0.808601\pi$
$421$	−33.8389	−1.64921	−0.824603	+	0.565712i	$0.808601\pi$
$422$	23.4980	1.14386
$423$	9.00229	0.437706
$424$	−29.9984	−1.45685
$425$	3.56493	0.172925
$426$	10.8741	0.526850
$427$	0	0
$428$	2.08585	0.100823
$429$	0.531251	0.0256490
$430$	−0.180410	−0.00870012
$431$	−16.8579	−0.812016	−0.406008	−	0.913869i	$-0.633080\pi$
$431$	−16.8579	−0.812016	−0.406008	+	0.913869i	$0.633080\pi$
$432$	−4.91624	−0.236533
$433$	−2.77987	−0.133592	−0.0667961	−	0.997767i	$-0.521278\pi$
$433$	−2.77987	−0.133592	−0.0667961	+	0.997767i	$0.521278\pi$
$434$	0	0
$435$	0.0598975	0.00287187
$436$	−6.08389	−0.291365
$437$	8.08020	0.386529
$438$	−26.1298	−1.24853
$439$	−15.5531	−0.742309	−0.371154	−	0.928571i	$-0.621038\pi$
$439$	−15.5531	−0.742309	−0.371154	+	0.928571i	$0.621038\pi$
$440$	0.0149062	0.000710625 0
$441$	0	0
$442$	4.06388	0.193299
$443$	−19.3228	−0.918056	−0.459028	−	0.888422i	$-0.651802\pi$
$443$	−19.3228	−0.918056	−0.459028	+	0.888422i	$0.651802\pi$
$444$	6.48416	0.307725
$445$	−0.655587	−0.0310778
$446$	25.5627	1.21043
$447$	−15.5561	−0.735779
$448$	0	0
$449$	−29.7895	−1.40585	−0.702926	−	0.711263i	$-0.748124\pi$
$449$	−29.7895	−1.40585	−0.702926	+	0.711263i	$0.748124\pi$
$450$	8.22850	0.387895
$451$	0.153516	0.00722879
$452$	6.08810	0.286360
$453$	23.1467	1.08753
$454$	36.2590	1.70172
$455$	0	0
$456$	−13.5564	−0.634835
$457$	−19.4583	−0.910223	−0.455111	−	0.890434i	$-0.650401\pi$
$457$	−19.4583	−0.910223	−0.455111	+	0.890434i	$0.650401\pi$
$458$	24.0765	1.12502
$459$	−0.713284	−0.0332933
$460$	−0.0411258	−0.00191750
$461$	−8.86653	−0.412955	−0.206478	−	0.978451i	$-0.566200\pi$
$461$	−8.86653	−0.412955	−0.206478	+	0.978451i	$0.566200\pi$
$462$	0	0
$463$	23.0442	1.07096	0.535478	−	0.844549i	$-0.320131\pi$
$463$	23.0442	1.07096	0.535478	+	0.844549i	$0.320131\pi$
$464$	−6.43800	−0.298877
$465$	0.0336527	0.00156061
$466$	−7.51921	−0.348321
$467$	−6.62006	−0.306340	−0.153170	−	0.988200i	$-0.548948\pi$
$467$	−6.62006	−0.306340	−0.153170	+	0.988200i	$0.548948\pi$
$468$	2.45905	0.113670
$469$	0	0
$470$	−0.677918	−0.0312700
$471$	13.5629	0.624946
$472$	−24.0172	−1.10548
$473$	0.367781	0.0169106
$474$	10.1831	0.467723
$475$	31.9161	1.46441
$476$	0	0
$477$	−14.1311	−0.647019
$478$	−24.9201	−1.13982
$479$	16.9788	0.775782	0.387891	−	0.921705i	$-0.373204\pi$
$479$	16.9788	0.775782	0.387891	+	0.921705i	$0.373204\pi$
$480$	0.176020	0.00803420
$481$	31.5775	1.43981
$482$	21.3711	0.973426
$483$	0	0
$484$	−7.79980	−0.354536
$485$	−0.385425	−0.0175012
$486$	−1.64639	−0.0746817
$487$	18.3314	0.830677	0.415338	−	0.909667i	$-0.363663\pi$
$487$	18.3314	0.830677	0.415338	+	0.909667i	$0.363663\pi$
$488$	−30.3066	−1.37192
$489$	18.8746	0.853537
$490$	0	0
$491$	−6.45168	−0.291160	−0.145580	−	0.989346i	$-0.546505\pi$
$491$	−6.45168	−0.291160	−0.145580	+	0.989346i	$0.546505\pi$
$492$	0.710595	0.0320361
$493$	−0.934072	−0.0420685
$494$	36.3831	1.63696
$495$	0.00702175	0.000315604 0
$496$	−3.61711	−0.162413
$497$	0	0
$498$	3.63128	0.162722
$499$	9.71812	0.435043	0.217521	−	0.976056i	$-0.430203\pi$
$499$	9.71812	0.435043	0.217521	+	0.976056i	$0.430203\pi$
$500$	−0.324955	−0.0145324
$501$	1.10994	0.0495882
$502$	35.8929	1.60198
$503$	19.3611	0.863269	0.431634	−	0.902049i	$-0.357937\pi$
$503$	19.3611	0.863269	0.431634	+	0.902049i	$0.357937\pi$
$504$	0	0
$505$	−0.195705	−0.00870877
$506$	0.319806	0.0142171
$507$	−1.02454	−0.0455015
$508$	−0.605787	−0.0268775
$509$	4.52576	0.200601	0.100300	−	0.994957i	$-0.468020\pi$
$509$	4.52576	0.200601	0.100300	+	0.994957i	$0.468020\pi$
$510$	0.0537139	0.00237849
$511$	0	0
$512$	1.95371	0.0863426
$513$	−6.38590	−0.281944
$514$	14.4946	0.639328
$515$	0.405383	0.0178633
$516$	1.70239	0.0749433
$517$	1.38200	0.0607801
$518$	0	0
$519$	−21.6857	−0.951899
$520$	0.336015	0.0147353
$521$	−6.12296	−0.268252	−0.134126	−	0.990964i	$-0.542823\pi$
$521$	−6.12296	−0.268252	−0.134126	+	0.990964i	$0.542823\pi$
$522$	−2.15600	−0.0943658
$523$	12.0382	0.526394	0.263197	−	0.964742i	$-0.415223\pi$
$523$	12.0382	0.526394	0.263197	+	0.964742i	$0.415223\pi$
$524$	−7.55567	−0.330071
$525$	0	0
$526$	40.5737	1.76910
$527$	−0.524797	−0.0228605
$528$	−0.754722	−0.0328451
$529$	−21.3990	−0.930390
$530$	1.06414	0.0462234
$531$	−11.3136	−0.490969
$532$	0	0
$533$	3.46056	0.149893
$534$	23.5978	1.02118
$535$	0.134262	0.00580464
$536$	−28.3620	−1.22505
$537$	−7.05942	−0.304636
$538$	22.3218	0.962361
$539$	0	0
$540$	0.0325023	0.00139867
$541$	−43.0826	−1.85226	−0.926132	−	0.377199i	$-0.876887\pi$
$541$	−43.0826	−1.85226	−0.926132	+	0.377199i	$0.876887\pi$
$542$	−32.1064	−1.37909
$543$	13.1070	0.562474
$544$	−2.74495	−0.117689
$545$	−0.391607	−0.0167746
$546$	0	0
$547$	−20.8040	−0.889514	−0.444757	−	0.895651i	$-0.646710\pi$
$547$	−20.8040	−0.889514	−0.444757	+	0.895651i	$0.646710\pi$
$548$	15.0817	0.644257
$549$	−14.2763	−0.609297
$550$	1.26321	0.0538633
$551$	−8.36256	−0.356257
$552$	−2.68610	−0.114328
$553$	0	0
$554$	−14.8657	−0.631585
$555$	0.417372	0.0177165
$556$	8.55440	0.362787
$557$	−17.5841	−0.745060	−0.372530	−	0.928020i	$-0.621510\pi$
$557$	−17.5841	−0.745060	−0.372530	+	0.928020i	$0.621510\pi$
$558$	−1.21133	−0.0512795
$559$	8.29052	0.350652
$560$	0	0
$561$	−0.109501	−0.00462312
$562$	−1.47071	−0.0620381
$563$	−17.7830	−0.749463	−0.374731	−	0.927133i	$-0.622265\pi$
$563$	−17.7830	−0.749463	−0.374731	+	0.927133i	$0.622265\pi$
$564$	6.39698	0.269362
$565$	0.391878	0.0164864
$566$	−33.9179	−1.42568
$567$	0	0
$568$	−14.0211	−0.588311
$569$	−2.63154	−0.110320	−0.0551599	−	0.998478i	$-0.517567\pi$
$569$	−2.63154	−0.110320	−0.0551599	+	0.998478i	$0.517567\pi$
$570$	0.480890	0.0201423
$571$	−5.24279	−0.219404	−0.109702	−	0.993965i	$-0.534990\pi$
$571$	−5.24279	−0.219404	−0.109702	+	0.993965i	$0.534990\pi$
$572$	0.377504	0.0157842
$573$	9.51467	0.397481
$574$	0	0
$575$	6.32395	0.263727
$576$	3.49665	0.145694
$577$	30.1299	1.25432	0.627162	−	0.778888i	$-0.284216\pi$
$577$	30.1299	1.25432	0.627162	+	0.778888i	$0.284216\pi$
$578$	27.1510	1.12933
$579$	−10.1608	−0.422268
$580$	0.0425629	0.00176733
$581$	0	0
$582$	13.8733	0.575068
$583$	−2.16935	−0.0898454
$584$	33.6919	1.39418
$585$	0.158284	0.00654425
$586$	−28.1110	−1.16126
$587$	−27.5676	−1.13784	−0.568918	−	0.822394i	$-0.692638\pi$
$587$	−27.5676	−1.13784	−0.568918	+	0.822394i	$0.692638\pi$
$588$	0	0
$589$	−4.69841	−0.193594
$590$	0.851972	0.0350751
$591$	7.54126	0.310206
$592$	−44.8606	−1.84376
$593$	18.6723	0.766780	0.383390	−	0.923586i	$-0.374756\pi$
$593$	18.6723	0.766780	0.383390	+	0.923586i	$0.374756\pi$
$594$	−0.252747	−0.0103703
$595$	0	0
$596$	−11.0541	−0.452793
$597$	25.9232	1.06096
$598$	7.20906	0.294801
$599$	16.7957	0.686253	0.343127	−	0.939289i	$-0.388514\pi$
$599$	16.7957	0.686253	0.343127	+	0.939289i	$0.388514\pi$
$600$	−10.6099	−0.433146
$601$	−7.92057	−0.323087	−0.161543	−	0.986866i	$-0.551647\pi$
$601$	−7.92057	−0.323087	−0.161543	+	0.986866i	$0.551647\pi$
$602$	0	0
$603$	−13.3603	−0.544073
$604$	16.4479	0.669257
$605$	−0.502057	−0.0204115
$606$	7.04439	0.286159
$607$	29.7076	1.20579	0.602897	−	0.797819i	$-0.294013\pi$
$607$	29.7076	1.20579	0.602897	+	0.797819i	$0.294013\pi$
$608$	−24.5750	−0.996648
$609$	0	0
$610$	1.07508	0.0435286
$611$	31.1529	1.26031
$612$	−0.506856	−0.0204884
$613$	21.3366	0.861776	0.430888	−	0.902406i	$-0.358200\pi$
$613$	21.3366	0.861776	0.430888	+	0.902406i	$0.358200\pi$
$614$	−10.1059	−0.407839
$615$	0.0457395	0.00184440
$616$	0	0
$617$	5.34012	0.214985	0.107493	−	0.994206i	$-0.465718\pi$
$617$	5.34012	0.214985	0.107493	+	0.994206i	$0.465718\pi$
$618$	−14.5917	−0.586965
$619$	8.18591	0.329019	0.164510	−	0.986375i	$-0.447396\pi$
$619$	8.18591	0.329019	0.164510	+	0.986375i	$0.447396\pi$
$620$	0.0239135	0.000960387 0
$621$	−1.26532	−0.0507756
$622$	−25.7607	−1.03291
$623$	0	0
$624$	−17.0129	−0.681063
$625$	24.9686	0.998745
$626$	−27.9086	−1.11545
$627$	−0.980337	−0.0391509
$628$	9.63774	0.384588
$629$	−6.50871	−0.259519
$630$	0	0
$631$	3.64661	0.145169	0.0725845	−	0.997362i	$-0.476875\pi$
$631$	3.64661	0.145169	0.0725845	+	0.997362i	$0.476875\pi$
$632$	−13.1301	−0.522287
$633$	−14.2724	−0.567278
$634$	13.4508	0.534199
$635$	−0.0389933	−0.00154740
$636$	−10.0415	−0.398171
$637$	0	0
$638$	−0.330981	−0.0131037
$639$	−6.60480	−0.261282
$640$	−0.615356	−0.0243241
$641$	−33.2611	−1.31373	−0.656867	−	0.754007i	$-0.728119\pi$
$641$	−33.2611	−1.31373	−0.656867	+	0.754007i	$0.728119\pi$
$642$	−4.83273	−0.190733
$643$	31.6609	1.24858	0.624291	−	0.781192i	$-0.285388\pi$
$643$	31.6609	1.24858	0.624291	+	0.781192i	$0.285388\pi$
$644$	0	0
$645$	0.109579	0.00431467
$646$	−7.49923	−0.295053
$647$	−19.7459	−0.776291	−0.388146	−	0.921598i	$-0.626884\pi$
$647$	−19.7459	−0.776291	−0.388146	+	0.921598i	$0.626884\pi$
$648$	2.12286	0.0833939
$649$	−1.73682	−0.0681762
$650$	28.4752	1.11689
$651$	0	0
$652$	13.4122	0.525261
$653$	12.9661	0.507403	0.253702	−	0.967283i	$-0.418352\pi$
$653$	12.9661	0.507403	0.253702	+	0.967283i	$0.418352\pi$
$654$	14.0959	0.551192
$655$	−0.486343	−0.0190030
$656$	−4.91624	−0.191947
$657$	15.8710	0.619185
$658$	0	0
$659$	4.26699	0.166218	0.0831091	−	0.996540i	$-0.473515\pi$
$659$	4.26699	0.166218	0.0831091	+	0.996540i	$0.473515\pi$
$660$	0.00498962	0.000194221 0
$661$	6.81866	0.265215	0.132608	−	0.991169i	$-0.457665\pi$
$661$	6.81866	0.265215	0.132608	+	0.991169i	$0.457665\pi$
$662$	−12.9291	−0.502502
$663$	−2.46836	−0.0958632
$664$	−4.68220	−0.181704
$665$	0	0
$666$	−15.0233	−0.582140
$667$	−1.65698	−0.0641586
$668$	0.788714	0.0305163
$669$	−15.5265	−0.600289
$670$	1.00610	0.0388689
$671$	−2.19164	−0.0846073
$672$	0	0
$673$	−35.9850	−1.38712	−0.693559	−	0.720400i	$-0.743958\pi$
$673$	−35.9850	−1.38712	−0.693559	+	0.720400i	$0.743958\pi$
$674$	17.0499	0.656738
$675$	−4.99791	−0.192370
$676$	−0.728034	−0.0280013
$677$	33.0463	1.27007	0.635036	−	0.772482i	$-0.280985\pi$
$677$	33.0463	1.27007	0.635036	+	0.772482i	$0.280985\pi$
$678$	−14.1056	−0.541723
$679$	0	0
$680$	−0.0692590	−0.00265596
$681$	−22.0233	−0.843936
$682$	−0.185958	−0.00712070
$683$	40.0689	1.53319	0.766596	−	0.642130i	$-0.221949\pi$
$683$	40.0689	1.53319	0.766596	+	0.642130i	$0.221949\pi$
$684$	−4.53778	−0.173507
$685$	0.970776	0.0370914
$686$	0	0
$687$	−14.6238	−0.557933
$688$	−11.7779	−0.449030
$689$	−48.9015	−1.86300
$690$	0.0952849	0.00362744
$691$	−13.5466	−0.515335	−0.257668	−	0.966234i	$-0.582954\pi$
$691$	−13.5466	−0.515335	−0.257668	+	0.966234i	$0.582954\pi$
$692$	−15.4098	−0.585792
$693$	0	0
$694$	47.4719	1.80201
$695$	0.550629	0.0208866
$696$	2.77996	0.105374
$697$	−0.713284	−0.0270176
$698$	3.94952	0.149492
$699$	4.56710	0.172743
$700$	0	0
$701$	14.1232	0.533428	0.266714	−	0.963776i	$-0.414062\pi$
$701$	14.1232	0.533428	0.266714	+	0.963776i	$0.414062\pi$
$702$	−5.69742	−0.215035
$703$	−58.2712	−2.19774
$704$	0.536792	0.0202311
$705$	0.411760	0.0155078
$706$	1.18571	0.0446249
$707$	0	0
$708$	−8.03940	−0.302139
$709$	25.2039	0.946554	0.473277	−	0.880914i	$-0.343071\pi$
$709$	25.2039	0.946554	0.473277	+	0.880914i	$0.343071\pi$
$710$	0.497374	0.0186661
$711$	−6.18509	−0.231959
$712$	−30.4271	−1.14030
$713$	−0.930956	−0.0348646
$714$	0	0
$715$	0.0242992	0.000908737 0
$716$	−5.01639	−0.187471
$717$	15.1362	0.565272
$718$	30.2443	1.12871
$719$	3.91519	0.146012	0.0730059	−	0.997332i	$-0.476741\pi$
$719$	3.91519	0.146012	0.0730059	+	0.997332i	$0.476741\pi$
$720$	−0.224867	−0.00838028
$721$	0	0
$722$	−35.8578	−1.33449
$723$	−12.9806	−0.482753
$724$	9.31375	0.346143
$725$	−6.54494	−0.243073
$726$	18.0715	0.670695
$727$	44.3489	1.64481	0.822406	−	0.568902i	$-0.192632\pi$
$727$	44.3489	1.64481	0.822406	+	0.568902i	$0.192632\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−1.19516	−0.0442350
$731$	−1.70883	−0.0632033
$732$	−10.1447	−0.374957
$733$	48.1470	1.77835	0.889175	−	0.457568i	$-0.151279\pi$
$733$	48.1470	1.77835	0.889175	+	0.457568i	$0.151279\pi$
$734$	12.8687	0.474993
$735$	0	0
$736$	−4.86936	−0.179487
$737$	−2.05102	−0.0755502
$738$	−1.64639	−0.0606044
$739$	34.4634	1.26776	0.633878	−	0.773433i	$-0.281462\pi$
$739$	34.4634	1.26776	0.633878	+	0.773433i	$0.281462\pi$
$740$	0.296583	0.0109026
$741$	−22.0988	−0.811818
$742$	0	0
$743$	−28.7819	−1.05591	−0.527953	−	0.849273i	$-0.677040\pi$
$743$	−28.7819	−1.05591	−0.527953	+	0.849273i	$0.677040\pi$
$744$	1.56189	0.0572617
$745$	−0.711529	−0.0260684
$746$	39.8645	1.45954
$747$	−2.20561	−0.0806989
$748$	−0.0778105	−0.00284503
$749$	0	0
$750$	0.752892	0.0274917
$751$	−45.4779	−1.65951	−0.829756	−	0.558127i	$-0.811520\pi$
$751$	−45.4779	−1.65951	−0.829756	+	0.558127i	$0.811520\pi$
$752$	−44.2575	−1.61390
$753$	−21.8010	−0.794472
$754$	−7.46098	−0.271713
$755$	1.05872	0.0385307
$756$	0	0
$757$	−32.6583	−1.18699	−0.593493	−	0.804839i	$-0.702252\pi$
$757$	−32.6583	−1.18699	−0.593493	+	0.804839i	$0.702252\pi$
$758$	25.7228	0.934293
$759$	−0.194247	−0.00705071
$760$	−0.620062	−0.0224920
$761$	22.5014	0.815676	0.407838	−	0.913054i	$-0.366283\pi$
$761$	22.5014	0.815676	0.407838	+	0.913054i	$0.366283\pi$
$762$	1.40356	0.0508456
$763$	0	0
$764$	6.76108	0.244607
$765$	−0.0326253	−0.00117957
$766$	−54.1995	−1.95831
$767$	−39.1514	−1.41368
$768$	15.1564	0.546909
$769$	6.24290	0.225125	0.112562	−	0.993645i	$-0.464094\pi$
$769$	6.24290	0.225125	0.112562	+	0.993645i	$0.464094\pi$
$770$	0	0
$771$	−8.80386	−0.317063
$772$	−7.22020	−0.259861
$773$	−21.8999	−0.787684	−0.393842	−	0.919178i	$-0.628854\pi$
$773$	−21.8999	−0.787684	−0.393842	+	0.919178i	$0.628854\pi$
$774$	−3.94428	−0.141774
$775$	−3.67720	−0.132089
$776$	−17.8883	−0.642154
$777$	0	0
$778$	37.4560	1.34286
$779$	−6.38590	−0.228798
$780$	0.112476	0.00402728
$781$	−1.01394	−0.0362817
$782$	−1.48592	−0.0531364
$783$	1.30954	0.0467990
$784$	0	0
$785$	0.620361	0.0221416
$786$	17.5058	0.624413
$787$	37.6171	1.34090	0.670452	−	0.741953i	$-0.266100\pi$
$787$	37.6171	1.34090	0.670452	+	0.741953i	$0.266100\pi$
$788$	5.35878	0.190899
$789$	−24.6440	−0.877351
$790$	0.465768	0.0165713
$791$	0	0
$792$	0.325893	0.0115801
$793$	−49.4039	−1.75438
$794$	9.21926	0.327179
$795$	−0.646350	−0.0229237
$796$	18.4209	0.652911
$797$	−20.5650	−0.728449	−0.364224	−	0.931311i	$-0.618666\pi$
$797$	−20.5650	−0.728449	−0.364224	+	0.931311i	$0.618666\pi$
$798$	0	0
$799$	−6.42119	−0.227166
$800$	−19.2336	−0.680009
$801$	−14.3331	−0.506434
$802$	39.8292	1.40642
$803$	2.43645	0.0859803
$804$	−9.49375	−0.334819
$805$	0	0
$806$	−4.19186	−0.147652
$807$	−13.5580	−0.477265
$808$	−9.08307	−0.319541
$809$	−43.8207	−1.54065	−0.770327	−	0.637649i	$-0.779907\pi$
$809$	−43.8207	−1.54065	−0.770327	+	0.637649i	$0.779907\pi$
$810$	−0.0753050	−0.00264595
$811$	−28.5288	−1.00178	−0.500891	−	0.865511i	$-0.666994\pi$
$811$	−28.5288	−1.00178	−0.500891	+	0.865511i	$0.666994\pi$
$812$	0	0
$813$	19.5011	0.683935
$814$	−2.30631	−0.0808362
$815$	0.863313	0.0302405
$816$	3.50668	0.122758
$817$	−15.2988	−0.535238
$818$	29.9904	1.04859
$819$	0	0
$820$	0.0325023	0.00113503
$821$	36.1062	1.26012	0.630058	−	0.776548i	$-0.283031\pi$
$821$	36.1062	1.26012	0.630058	+	0.776548i	$0.283031\pi$
$822$	−34.9430	−1.21878
$823$	42.3799	1.47727	0.738635	−	0.674106i	$-0.235471\pi$
$823$	42.3799	1.47727	0.738635	+	0.674106i	$0.235471\pi$
$824$	18.8146	0.655439
$825$	−0.767259	−0.0267125
$826$	0	0
$827$	−52.9489	−1.84121	−0.920606	−	0.390492i	$-0.872305\pi$
$827$	−52.9489	−1.84121	−0.920606	+	0.390492i	$0.872305\pi$
$828$	−0.899130	−0.0312469
$829$	−9.16919	−0.318459	−0.159230	−	0.987242i	$-0.550901\pi$
$829$	−9.16919	−0.318459	−0.159230	+	0.987242i	$0.550901\pi$
$830$	0.166093	0.00576518
$831$	9.02931	0.313223
$832$	12.1004	0.419505
$833$	0	0
$834$	−19.8198	−0.686305
$835$	0.0507679	0.00175690
$836$	−0.696622	−0.0240932
$837$	0.735747	0.0254312
$838$	65.1352	2.25006
$839$	−42.6983	−1.47411	−0.737055	−	0.675833i	$-0.763784\pi$
$839$	−42.6983	−1.47411	−0.737055	+	0.675833i	$0.763784\pi$
$840$	0	0
$841$	−27.2851	−0.940866
$842$	55.7119	1.91996
$843$	0.893294	0.0307667
$844$	−10.1419	−0.349099
$845$	−0.0468620	−0.00161210
$846$	−14.8213	−0.509566
$847$	0	0
$848$	69.4720	2.38568
$849$	20.6014	0.707039
$850$	−5.86926	−0.201314
$851$	−11.5460	−0.395793
$852$	−4.69334	−0.160791
$853$	12.6282	0.432381	0.216190	−	0.976351i	$-0.430637\pi$
$853$	12.6282	0.432381	0.216190	+	0.976351i	$0.430637\pi$
$854$	0	0
$855$	−0.292088	−0.00998919
$856$	6.23135	0.212983
$857$	−55.6169	−1.89984	−0.949919	−	0.312497i	$-0.898835\pi$
$857$	−55.6169	−1.89984	−0.949919	+	0.312497i	$0.898835\pi$
$858$	−0.874645	−0.0298599
$859$	−6.63053	−0.226231	−0.113115	−	0.993582i	$-0.536083\pi$
$859$	−6.63053	−0.226231	−0.113115	+	0.993582i	$0.536083\pi$
$860$	0.0778663	0.00265522
$861$	0	0
$862$	27.7546	0.945327
$863$	−35.9010	−1.22208	−0.611042	−	0.791598i	$-0.709250\pi$
$863$	−35.9010	−1.22208	−0.611042	+	0.791598i	$0.709250\pi$
$864$	3.84832	0.130923
$865$	−0.991895	−0.0337255
$866$	4.57675	0.155524
$867$	−16.4912	−0.560071
$868$	0	0
$869$	−0.949510	−0.0322099
$870$	−0.0986146	−0.00334335
$871$	−46.2340	−1.56658
$872$	−18.1753	−0.615492
$873$	−8.42652	−0.285194
$874$	−13.3032	−0.449986
$875$	0	0
$876$	11.2778	0.381042
$877$	−34.1017	−1.15153	−0.575766	−	0.817615i	$-0.695296\pi$
$877$	−34.1017	−1.15153	−0.575766	+	0.817615i	$0.695296\pi$
$878$	25.6064	0.864175
$879$	17.0744	0.575904
$880$	−0.0345206	−0.00116369
$881$	−4.69337	−0.158124	−0.0790619	−	0.996870i	$-0.525192\pi$
$881$	−4.69337	−0.158124	−0.0790619	+	0.996870i	$0.525192\pi$
$882$	0	0
$883$	20.6223	0.693995	0.346998	−	0.937866i	$-0.387201\pi$
$883$	20.6223	0.693995	0.346998	+	0.937866i	$0.387201\pi$
$884$	−1.75401	−0.0589936
$885$	−0.517479	−0.0173949
$886$	31.8129	1.06878
$887$	−4.96347	−0.166657	−0.0833285	−	0.996522i	$-0.526555\pi$
$887$	−4.96347	−0.166657	−0.0833285	+	0.996522i	$0.526555\pi$
$888$	19.3711	0.650051
$889$	0	0
$890$	1.07935	0.0361799
$891$	0.153516	0.00514298
$892$	−11.0331	−0.369414
$893$	−57.4877	−1.92375
$894$	25.6114	0.856573
$895$	−0.322894	−0.0107932
$896$	0	0
$897$	−4.37871	−0.146201
$898$	49.0451	1.63665
$899$	0.963488	0.0321341
$900$	−3.55149	−0.118383
$901$	10.0795	0.335797
$902$	−0.252747	−0.00841555
$903$	0	0
$904$	18.1878	0.604919
$905$	0.599507	0.0199283
$906$	−38.1085	−1.26607
$907$	−46.7190	−1.55128	−0.775640	−	0.631176i	$-0.782573\pi$
$907$	−46.7190	−1.55128	−0.775640	+	0.631176i	$0.782573\pi$
$908$	−15.6497	−0.519353
$909$	−4.27869	−0.141915
$910$	0	0
$911$	−7.07687	−0.234467	−0.117234	−	0.993104i	$-0.537403\pi$
$911$	−7.07687	−0.234467	−0.117234	+	0.993104i	$0.537403\pi$
$912$	31.3946	1.03958
$913$	−0.338596	−0.0112059
$914$	32.0360	1.05966
$915$	−0.652991	−0.0215872
$916$	−10.3916	−0.343348
$917$	0	0
$918$	1.17434	0.0387591
$919$	12.9722	0.427914	0.213957	−	0.976843i	$-0.431365\pi$
$919$	12.9722	0.427914	0.213957	+	0.976843i	$0.431365\pi$
$920$	−0.122861	−0.00405060
$921$	6.13820	0.202261
$922$	14.5977	0.480751
$923$	−22.8563	−0.752324
$924$	0	0
$925$	−45.6058	−1.49951
$926$	−37.9398	−1.24678
$927$	8.86287	0.291095
$928$	5.03952	0.165430
$929$	17.1271	0.561921	0.280961	−	0.959719i	$-0.409347\pi$
$929$	17.1271	0.561921	0.280961	+	0.959719i	$0.409347\pi$
$930$	−0.0554055	−0.00181682
$931$	0	0
$932$	3.24536	0.106305
$933$	15.6468	0.512253
$934$	10.8992	0.356633
$935$	−0.00500850	−0.000163796 0
$936$	7.34629	0.240121
$937$	51.2625	1.67467	0.837337	−	0.546687i	$-0.184111\pi$
$937$	51.2625	1.67467	0.837337	+	0.546687i	$0.184111\pi$
$938$	0	0
$939$	16.9514	0.553188
$940$	0.292595	0.00954339
$941$	35.2503	1.14913	0.574564	−	0.818460i	$-0.305172\pi$
$941$	35.2503	1.14913	0.574564	+	0.818460i	$0.305172\pi$
$942$	−22.3298	−0.727545
$943$	−1.26532	−0.0412045
$944$	55.6205	1.81029
$945$	0	0
$946$	−0.605510	−0.0196868
$947$	−36.1653	−1.17521	−0.587607	−	0.809146i	$-0.699930\pi$
$947$	−36.1653	−1.17521	−0.587607	+	0.809146i	$0.699930\pi$
$948$	−4.39509	−0.142746
$949$	54.9224	1.78286
$950$	−52.5463	−1.70483
$951$	−8.16987	−0.264926
$952$	0	0
$953$	−19.7601	−0.640093	−0.320046	−	0.947402i	$-0.603699\pi$
$953$	−19.7601	−0.640093	−0.320046	+	0.947402i	$0.603699\pi$
$954$	23.2653	0.753242
$955$	0.435196	0.0140826
$956$	10.7557	0.347864
$957$	0.201035	0.00649853
$958$	−27.9537	−0.903144
$959$	0	0
$960$	0.159935	0.00516188
$961$	−30.4587	−0.982538
$962$	−51.9889	−1.67619
$963$	2.93535	0.0945905
$964$	−9.22394	−0.297083
$965$	−0.464749	−0.0149608
$966$	0	0
$967$	−16.4406	−0.528695	−0.264347	−	0.964428i	$-0.585157\pi$
$967$	−16.4406	−0.528695	−0.264347	+	0.964428i	$0.585157\pi$
$968$	−23.3014	−0.748937
$969$	4.55496	0.146326
$970$	0.634559	0.0203745
$971$	41.5017	1.33185	0.665926	−	0.746018i	$-0.268037\pi$
$971$	41.5017	1.33185	0.665926	+	0.746018i	$0.268037\pi$
$972$	0.710595	0.0227923
$973$	0	0
$974$	−30.1807	−0.967051
$975$	−17.2955	−0.553901
$976$	70.1857	2.24659
$977$	−10.0675	−0.322089	−0.161044	−	0.986947i	$-0.551486\pi$
$977$	−10.0675	−0.322089	−0.161044	+	0.986947i	$0.551486\pi$
$978$	−31.0748	−0.993664
$979$	−2.20035	−0.0703236
$980$	0	0
$981$	−8.56169	−0.273354
$982$	10.6220	0.338961
$983$	49.9216	1.59225	0.796126	−	0.605132i	$-0.206879\pi$
$983$	49.9216	1.59225	0.796126	+	0.605132i	$0.206879\pi$
$984$	2.12286	0.0676744
$985$	0.344934	0.0109905
$986$	1.53784	0.0489750
$987$	0	0
$988$	−15.7033	−0.499587
$989$	−3.03135	−0.0963913
$990$	−0.0115605	−0.000367417 0
$991$	−27.4632	−0.872397	−0.436199	−	0.899850i	$-0.643675\pi$
$991$	−27.4632	−0.872397	−0.436199	+	0.899850i	$0.643675\pi$
$992$	2.83140	0.0898969
$993$	7.85298	0.249207
$994$	0	0
$995$	1.18571	0.0375896
$996$	−1.56729	−0.0496615
$997$	15.3608	0.486482	0.243241	−	0.969966i	$-0.421789\pi$
$997$	15.3608	0.486482	0.243241	+	0.969966i	$0.421789\pi$
$998$	−15.9998	−0.506465
$999$	9.12498	0.288702

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6027.2.a.bm.1.4	yes	16
7.6	odd	2		6027.2.a.bl.1.4	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6027.2.a.bl.1.4	✓	16	7.6	odd	2
6027.2.a.bm.1.4	yes	16	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:	\( N \)	\(=\)	\( 6027 = 3 \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6027.a (trivial)

\(f(q)\)	\(=\)	\(q-1.64639 q^{2} +1.00000 q^{3} +0.710595 q^{4} +0.0457395 q^{5} -1.64639 q^{6} +2.12286 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.64639 q^{2} +1.00000 q^{3} +0.710595 q^{4} +0.0457395 q^{5} -1.64639 q^{6} +2.12286 q^{8} +1.00000 q^{9} -0.0753050 q^{10} +0.153516 q^{11} +0.710595 q^{12} +3.46056 q^{13} +0.0457395 q^{15} -4.91624 q^{16} -0.713284 q^{17} -1.64639 q^{18} -6.38590 q^{19} +0.0325023 q^{20} -0.252747 q^{22} -1.26532 q^{23} +2.12286 q^{24} -4.99791 q^{25} -5.69742 q^{26} +1.00000 q^{27} +1.30954 q^{29} -0.0753050 q^{30} +0.735747 q^{31} +3.84832 q^{32} +0.153516 q^{33} +1.17434 q^{34} +0.710595 q^{36} +9.12498 q^{37} +10.5137 q^{38} +3.46056 q^{39} +0.0970987 q^{40} +1.00000 q^{41} +2.39572 q^{43} +0.109088 q^{44} +0.0457395 q^{45} +2.08321 q^{46} +9.00229 q^{47} -4.91624 q^{48} +8.22850 q^{50} -0.713284 q^{51} +2.45905 q^{52} -14.1311 q^{53} -1.64639 q^{54} +0.00702175 q^{55} -6.38590 q^{57} -2.15600 q^{58} -11.3136 q^{59} +0.0325023 q^{60} -14.2763 q^{61} -1.21133 q^{62} +3.49665 q^{64} +0.158284 q^{65} -0.252747 q^{66} -13.3603 q^{67} -0.506856 q^{68} -1.26532 q^{69} -6.60480 q^{71} +2.12286 q^{72} +15.8710 q^{73} -15.0233 q^{74} -4.99791 q^{75} -4.53778 q^{76} -5.69742 q^{78} -6.18509 q^{79} -0.224867 q^{80} +1.00000 q^{81} -1.64639 q^{82} -2.20561 q^{83} -0.0326253 q^{85} -3.94428 q^{86} +1.30954 q^{87} +0.325893 q^{88} -14.3331 q^{89} -0.0753050 q^{90} -0.899130 q^{92} +0.735747 q^{93} -14.8213 q^{94} -0.292088 q^{95} +3.84832 q^{96} -8.42652 q^{97} +0.153516 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{2} + 16 q^{3} + 12 q^{4} - 12 q^{5} - 4 q^{6} - 12 q^{8} + 16 q^{9}+O(q^{10}) \) 16 * q - 4 * q^2 + 16 * q^3 + 12 * q^4 - 12 * q^5 - 4 * q^6 - 12 * q^8 + 16 * q^9	\( 16 q - 4 q^{2} + 16 q^{3} + 12 q^{4} - 12 q^{5} - 4 q^{6} - 12 q^{8} + 16 q^{9} - 4 q^{10} - 4 q^{11} + 12 q^{12} - 12 q^{15} - 8 q^{17} - 4 q^{18} + 4 q^{19} - 20 q^{20} - 16 q^{22} - 12 q^{23} - 12 q^{24} - 8 q^{25} - 8 q^{26} + 16 q^{27} - 16 q^{29} - 4 q^{30} - 4 q^{31} - 48 q^{32} - 4 q^{33} + 16 q^{34} + 12 q^{36} - 48 q^{37} - 4 q^{38} + 56 q^{40} + 16 q^{41} - 16 q^{43} - 12 q^{45} - 4 q^{46} - 36 q^{47} - 8 q^{50} - 8 q^{51} - 60 q^{53} - 4 q^{54} + 8 q^{55} + 4 q^{57} - 36 q^{58} - 36 q^{59} - 20 q^{60} - 4 q^{61} - 12 q^{62} + 52 q^{64} - 36 q^{65} - 16 q^{66} - 52 q^{67} - 8 q^{68} - 12 q^{69} - 12 q^{71} - 12 q^{72} - 16 q^{73} + 4 q^{74} - 8 q^{75} + 16 q^{76} - 8 q^{78} - 36 q^{79} - 68 q^{80} + 16 q^{81} - 4 q^{82} - 32 q^{83} - 28 q^{85} - 8 q^{86} - 16 q^{87} - 36 q^{88} - 12 q^{89} - 4 q^{90} - 36 q^{92} - 4 q^{93} + 24 q^{94} - 20 q^{95} - 48 q^{96} + 48 q^{97} - 4 q^{99}+O(q^{100}) \) 16 * q - 4 * q^2 + 16 * q^3 + 12 * q^4 - 12 * q^5 - 4 * q^6 - 12 * q^8 + 16 * q^9 - 4 * q^10 - 4 * q^11 + 12 * q^12 - 12 * q^15 - 8 * q^17 - 4 * q^18 + 4 * q^19 - 20 * q^20 - 16 * q^22 - 12 * q^23 - 12 * q^24 - 8 * q^25 - 8 * q^26 + 16 * q^27 - 16 * q^29 - 4 * q^30 - 4 * q^31 - 48 * q^32 - 4 * q^33 + 16 * q^34 + 12 * q^36 - 48 * q^37 - 4 * q^38 + 56 * q^40 + 16 * q^41 - 16 * q^43 - 12 * q^45 - 4 * q^46 - 36 * q^47 - 8 * q^50 - 8 * q^51 - 60 * q^53 - 4 * q^54 + 8 * q^55 + 4 * q^57 - 36 * q^58 - 36 * q^59 - 20 * q^60 - 4 * q^61 - 12 * q^62 + 52 * q^64 - 36 * q^65 - 16 * q^66 - 52 * q^67 - 8 * q^68 - 12 * q^69 - 12 * q^71 - 12 * q^72 - 16 * q^73 + 4 * q^74 - 8 * q^75 + 16 * q^76 - 8 * q^78 - 36 * q^79 - 68 * q^80 + 16 * q^81 - 4 * q^82 - 32 * q^83 - 28 * q^85 - 8 * q^86 - 16 * q^87 - 36 * q^88 - 12 * q^89 - 4 * q^90 - 36 * q^92 - 4 * q^93 + 24 * q^94 - 20 * q^95 - 48 * q^96 + 48 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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