LMFDB - Embedded newform 2401.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2401,2,Mod(1,2401)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2401, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2401.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2401 = 7^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2401.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:	$19.1720815253$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\Q(\zeta_{21})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 6x^{4} + 6x^{3} + 8x^{2} - 8x + 1 $ x^6 - x^5 - 6x^4 + 6x^3 + 8x^2 - 8x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 49)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.65248$ of defining polynomial
Character	$\chi$	$=$	2401.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.91115 q^{2} -1.73068 q^{3} +1.65248 q^{4} +2.12712 q^{5} +3.30759 q^{6} +0.664166 q^{8} -0.00473965 q^{9} +O(q^{10})$	\(q-1.91115 q^{2} -1.73068 q^{3} +1.65248 q^{4} +2.12712 q^{5} +3.30759 q^{6} +0.664166 q^{8} -0.00473965 q^{9} -4.06524 q^{10} +4.67354 q^{11} -2.85991 q^{12} +0.977662 q^{13} -3.68137 q^{15} -4.57427 q^{16} +2.08424 q^{17} +0.00905815 q^{18} -5.29116 q^{19} +3.51502 q^{20} -8.93181 q^{22} -3.77960 q^{23} -1.14946 q^{24} -0.475353 q^{25} -1.86845 q^{26} +5.20025 q^{27} +5.54823 q^{29} +7.03564 q^{30} -10.1646 q^{31} +7.41377 q^{32} -8.08841 q^{33} -3.98328 q^{34} -0.00783216 q^{36} -1.72678 q^{37} +10.1122 q^{38} -1.69202 q^{39} +1.41276 q^{40} -3.55006 q^{41} -10.0503 q^{43} +7.72292 q^{44} -0.0100818 q^{45} +7.22336 q^{46} -4.32864 q^{47} +7.91661 q^{48} +0.908468 q^{50} -3.60716 q^{51} +1.61556 q^{52} -5.17192 q^{53} -9.93843 q^{54} +9.94119 q^{55} +9.15731 q^{57} -10.6035 q^{58} -5.20025 q^{59} -6.08338 q^{60} +4.46701 q^{61} +19.4261 q^{62} -5.02025 q^{64} +2.07961 q^{65} +15.4581 q^{66} +0.482407 q^{67} +3.44416 q^{68} +6.54128 q^{69} +13.3901 q^{71} -0.00314791 q^{72} +5.96686 q^{73} +3.30012 q^{74} +0.822684 q^{75} -8.74352 q^{76} +3.23370 q^{78} +5.75814 q^{79} -9.73004 q^{80} -8.98576 q^{81} +6.78469 q^{82} +2.95977 q^{83} +4.43343 q^{85} +19.2076 q^{86} -9.60223 q^{87} +3.10401 q^{88} -11.2841 q^{89} +0.0192678 q^{90} -6.24570 q^{92} +17.5917 q^{93} +8.27266 q^{94} -11.2549 q^{95} -12.8309 q^{96} -1.76250 q^{97} -0.0221509 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{2} - 7 q^{3} + q^{4} + 3 q^{8} + 3 q^{9}+O(q^{10}) $ 6 * q - q^2 - 7 * q^3 + q^4 + 3 * q^8 + 3 * q^9	$ 6 q - q^{2} - 7 q^{3} + q^{4} + 3 q^{8} + 3 q^{9} + 10 q^{11} - 7 q^{13} - 7 q^{15} - 13 q^{16} + 10 q^{18} - 7 q^{19} - 7 q^{20} - 4 q^{22} - q^{23} - 7 q^{24} - 2 q^{25} - 7 q^{27} + 12 q^{29} + 21 q^{30} - 7 q^{31} - 5 q^{32} - 14 q^{33} - 3 q^{36} + 13 q^{37} - 7 q^{38} - 7 q^{40} - 14 q^{41} + 19 q^{43} + 18 q^{44} + 14 q^{45} - q^{46} - 21 q^{47} - 23 q^{50} - 14 q^{51} - 18 q^{53} - 21 q^{54} - 7 q^{55} + 21 q^{57} - 16 q^{58} + 7 q^{59} + 14 q^{60} - 28 q^{61} + 21 q^{62} - 7 q^{64} + 14 q^{65} - 7 q^{66} + 24 q^{67} + 28 q^{68} + 7 q^{69} + 33 q^{71} - 2 q^{72} + 7 q^{73} - 15 q^{74} + 14 q^{75} - 7 q^{76} + 7 q^{78} - 8 q^{79} + 21 q^{80} - 2 q^{81} - 21 q^{82} - 21 q^{83} - 28 q^{85} + 12 q^{86} - 49 q^{87} + 12 q^{88} - 21 q^{89} - 35 q^{90} - 6 q^{92} + 35 q^{93} + 7 q^{94} - 35 q^{95} + 21 q^{96} - 14 q^{97} - 9 q^{99}+O(q^{100}) $ 6 * q - q^2 - 7 * q^3 + q^4 + 3 * q^8 + 3 * q^9 + 10 * q^11 - 7 * q^13 - 7 * q^15 - 13 * q^16 + 10 * q^18 - 7 * q^19 - 7 * q^20 - 4 * q^22 - q^23 - 7 * q^24 - 2 * q^25 - 7 * q^27 + 12 * q^29 + 21 * q^30 - 7 * q^31 - 5 * q^32 - 14 * q^33 - 3 * q^36 + 13 * q^37 - 7 * q^38 - 7 * q^40 - 14 * q^41 + 19 * q^43 + 18 * q^44 + 14 * q^45 - q^46 - 21 * q^47 - 23 * q^50 - 14 * q^51 - 18 * q^53 - 21 * q^54 - 7 * q^55 + 21 * q^57 - 16 * q^58 + 7 * q^59 + 14 * q^60 - 28 * q^61 + 21 * q^62 - 7 * q^64 + 14 * q^65 - 7 * q^66 + 24 * q^67 + 28 * q^68 + 7 * q^69 + 33 * q^71 - 2 * q^72 + 7 * q^73 - 15 * q^74 + 14 * q^75 - 7 * q^76 + 7 * q^78 - 8 * q^79 + 21 * q^80 - 2 * q^81 - 21 * q^82 - 21 * q^83 - 28 * q^85 + 12 * q^86 - 49 * q^87 + 12 * q^88 - 21 * q^89 - 35 * q^90 - 6 * q^92 + 35 * q^93 + 7 * q^94 - 35 * q^95 + 21 * q^96 - 14 * q^97 - 9 * 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 $

$2$	−1.91115	−1.35138	−0.675692	−	0.737184i	$-0.736155\pi$
$2$	−1.91115	−1.35138	−0.675692	+	0.737184i	$0.736155\pi$
$3$	−1.73068	−0.999210	−0.499605	−	0.866253i	$-0.666521\pi$
$3$	−1.73068	−0.999210	−0.499605	+	0.866253i	$0.666521\pi$
$4$	1.65248	0.826239
$5$	2.12712	0.951278	0.475639	−	0.879641i	$-0.342217\pi$
$5$	2.12712	0.951278	0.475639	+	0.879641i	$0.342217\pi$
$6$	3.30759	1.35032
$7$	0	0
$7$	0	0
$8$	0.664166	0.234818
$9$	−0.00473965	−0.00157988
$10$	−4.06524	−1.28554
$11$	4.67354	1.40913	0.704563	−	0.709642i	$-0.251143\pi$
$11$	4.67354	1.40913	0.704563	+	0.709642i	$0.251143\pi$
$12$	−2.85991	−0.825586
$13$	0.977662	0.271155	0.135577	−	0.990767i	$-0.456711\pi$
$13$	0.977662	0.271155	0.135577	+	0.990767i	$0.456711\pi$
$14$	0	0
$15$	−3.68137	−0.950526
$16$	−4.57427	−1.14357
$17$	2.08424	0.505502	0.252751	−	0.967531i	$-0.418665\pi$
$17$	2.08424	0.505502	0.252751	+	0.967531i	$0.418665\pi$
$18$	0.00905815	0.00213503
$19$	−5.29116	−1.21387	−0.606937	−	0.794750i	$-0.707602\pi$
$19$	−5.29116	−1.21387	−0.606937	+	0.794750i	$0.707602\pi$
$20$	3.51502	0.785983
$21$	0	0
$22$	−8.93181	−1.90427
$23$	−3.77960	−0.788101	−0.394050	−	0.919089i	$-0.628926\pi$
$23$	−3.77960	−0.788101	−0.394050	+	0.919089i	$0.628926\pi$
$24$	−1.14946	−0.234633
$25$	−0.475353	−0.0950705
$26$	−1.86845	−0.366434
$27$	5.20025	1.00079
$28$	0	0
$29$	5.54823	1.03028	0.515141	−	0.857106i	$-0.327740\pi$
$29$	5.54823	1.03028	0.515141	+	0.857106i	$0.327740\pi$
$30$	7.03564	1.28453
$31$	−10.1646	−1.82562	−0.912811	−	0.408383i	$-0.866093\pi$
$31$	−10.1646	−1.82562	−0.912811	+	0.408383i	$0.866093\pi$
$32$	7.41377	1.31058
$33$	−8.08841	−1.40801
$34$	−3.98328	−0.683128
$35$	0	0
$36$	−0.00783216	−0.00130536
$37$	−1.72678	−0.283880	−0.141940	−	0.989875i	$-0.545334\pi$
$37$	−1.72678	−0.283880	−0.141940	+	0.989875i	$0.545334\pi$
$38$	10.1122	1.64041
$39$	−1.69202	−0.270940
$40$	1.41276	0.223377
$41$	−3.55006	−0.554427	−0.277213	−	0.960808i	$-0.589411\pi$
$41$	−3.55006	−0.554427	−0.277213	+	0.960808i	$0.589411\pi$
$42$	0	0
$43$	−10.0503	−1.53265	−0.766327	−	0.642450i	$-0.777918\pi$
$43$	−10.0503	−1.53265	−0.766327	+	0.642450i	$0.777918\pi$
$44$	7.72292	1.16427
$45$	−0.0100818	−0.00150291
$46$	7.22336	1.06503
$47$	−4.32864	−0.631397	−0.315698	−	0.948860i	$-0.602239\pi$
$47$	−4.32864	−0.631397	−0.315698	+	0.948860i	$0.602239\pi$
$48$	7.91661	1.14266
$49$	0	0
$50$	0.908468	0.128477
$51$	−3.60716	−0.505103
$52$	1.61556	0.224038
$53$	−5.17192	−0.710418	−0.355209	−	0.934787i	$-0.615590\pi$
$53$	−5.17192	−0.710418	−0.355209	+	0.934787i	$0.615590\pi$
$54$	−9.93843	−1.35245
$55$	9.94119	1.34047
$56$	0	0
$57$	9.15731	1.21292
$58$	−10.6035	−1.39231
$59$	−5.20025	−0.677015	−0.338507	−	0.940964i	$-0.609922\pi$
$59$	−5.20025	−0.677015	−0.338507	+	0.940964i	$0.609922\pi$
$60$	−6.08338	−0.785361
$61$	4.46701	0.571941	0.285971	−	0.958238i	$-0.407684\pi$
$61$	4.46701	0.571941	0.285971	+	0.958238i	$0.407684\pi$
$62$	19.4261	2.46712
$63$	0	0
$64$	−5.02025	−0.627531
$65$	2.07961	0.257943
$66$	15.4581	1.90276
$67$	0.482407	0.0589354	0.0294677	−	0.999566i	$-0.490619\pi$
$67$	0.482407	0.0589354	0.0294677	+	0.999566i	$0.490619\pi$
$68$	3.44416	0.417666
$69$	6.54128	0.787478
$70$	0	0
$71$	13.3901	1.58912	0.794558	−	0.607188i	$-0.207702\pi$
$71$	13.3901	1.58912	0.794558	+	0.607188i	$0.207702\pi$
$72$	−0.00314791	−0.000370985 0
$73$	5.96686	0.698368	0.349184	−	0.937054i	$-0.386459\pi$
$73$	5.96686	0.698368	0.349184	+	0.937054i	$0.386459\pi$
$74$	3.30012	0.383631
$75$	0.822684	0.0949954
$76$	−8.74352	−1.00295
$77$	0	0
$78$	3.23370	0.366144
$79$	5.75814	0.647842	0.323921	−	0.946084i	$-0.394999\pi$
$79$	5.75814	0.647842	0.323921	+	0.946084i	$0.394999\pi$
$80$	−9.73004	−1.08785
$81$	−8.98576	−0.998418
$82$	6.78469	0.749243
$83$	2.95977	0.324877	0.162438	−	0.986719i	$-0.448064\pi$
$83$	2.95977	0.324877	0.162438	+	0.986719i	$0.448064\pi$
$84$	0	0
$85$	4.43343	0.480873
$86$	19.2076	2.07121
$87$	−9.60223	−1.02947
$88$	3.10401	0.330888
$89$	−11.2841	−1.19611	−0.598056	−	0.801454i	$-0.704060\pi$
$89$	−11.2841	−1.19611	−0.598056	+	0.801454i	$0.704060\pi$
$90$	0.0192678	0.00203100
$91$	0	0
$92$	−6.24570	−0.651160
$93$	17.5917	1.82418
$94$	8.27266	0.853260
$95$	−11.2549	−1.15473
$96$	−12.8309	−1.30955
$97$	−1.76250	−0.178955	−0.0894774	−	0.995989i	$-0.528520\pi$
$97$	−1.76250	−0.178955	−0.0894774	+	0.995989i	$0.528520\pi$
$98$	0	0
$99$	−0.0221509	−0.00222625
$100$	−0.785510	−0.0785510
$101$	−9.64601	−0.959814	−0.479907	−	0.877319i	$-0.659330\pi$
$101$	−9.64601	−0.959814	−0.479907	+	0.877319i	$0.659330\pi$
$102$	6.89380	0.682588
$103$	−3.63655	−0.358320	−0.179160	−	0.983820i	$-0.557338\pi$
$103$	−3.63655	−0.358320	−0.179160	+	0.983820i	$0.557338\pi$
$104$	0.649330	0.0636720
$105$	0	0
$106$	9.88430	0.960048
$107$	−4.34427	−0.419976	−0.209988	−	0.977704i	$-0.567343\pi$
$107$	−4.34427	−0.419976	−0.209988	+	0.977704i	$0.567343\pi$
$108$	8.59329	0.826890
$109$	0.963919	0.0923267	0.0461634	−	0.998934i	$-0.485301\pi$
$109$	0.963919	0.0923267	0.0461634	+	0.998934i	$0.485301\pi$
$110$	−18.9991	−1.81149
$111$	2.98850	0.283656
$112$	0	0
$113$	3.76018	0.353728	0.176864	−	0.984235i	$-0.443405\pi$
$113$	3.76018	0.353728	0.176864	+	0.984235i	$0.443405\pi$
$114$	−17.5010	−1.63911
$115$	−8.03967	−0.749703
$116$	9.16833	0.851258
$117$	−0.00463377	−0.000428392 0
$118$	9.93843	0.914907
$119$	0	0
$120$	−2.44504	−0.223201
$121$	10.8420	0.985634
$122$	−8.53710	−0.772913
$123$	6.14403	0.553989
$124$	−16.7968	−1.50840
$125$	−11.6467	−1.04172
$126$	0	0
$127$	9.88589	0.877231	0.438615	−	0.898675i	$-0.355469\pi$
$127$	9.88589	0.877231	0.438615	+	0.898675i	$0.355469\pi$
$128$	−5.23312	−0.462546
$129$	17.3939	1.53144
$130$	−3.97443	−0.348580
$131$	−5.23104	−0.457038	−0.228519	−	0.973539i	$-0.573388\pi$
$131$	−5.23104	−0.457038	−0.228519	+	0.973539i	$0.573388\pi$
$132$	−13.3659	−1.16335
$133$	0	0
$134$	−0.921951	−0.0796444
$135$	11.0616	0.952028
$136$	1.38428	0.118701
$137$	18.3955	1.57164	0.785818	−	0.618458i	$-0.212242\pi$
$137$	18.3955	1.57164	0.785818	+	0.618458i	$0.212242\pi$
$138$	−12.5013	−1.06419
$139$	−7.09536	−0.601821	−0.300910	−	0.953652i	$-0.597291\pi$
$139$	−7.09536	−0.601821	−0.300910	+	0.953652i	$0.597291\pi$
$140$	0	0
$141$	7.49150	0.630898
$142$	−25.5905	−2.14751
$143$	4.56914	0.382091
$144$	0.0216804	0.00180670
$145$	11.8018	0.980084
$146$	−11.4035	−0.943763
$147$	0	0
$148$	−2.85346	−0.234553
$149$	7.02556	0.575556	0.287778	−	0.957697i	$-0.407083\pi$
$149$	7.02556	0.575556	0.287778	+	0.957697i	$0.407083\pi$
$150$	−1.57227	−0.128375
$151$	−1.05748	−0.0860566	−0.0430283	−	0.999074i	$-0.513701\pi$
$151$	−1.05748	−0.0860566	−0.0430283	+	0.999074i	$0.513701\pi$
$152$	−3.51421	−0.285040
$153$	−0.00987856	−0.000798634 0
$154$	0	0
$155$	−21.6214	−1.73667
$156$	−2.79603	−0.223861
$157$	4.76515	0.380300	0.190150	−	0.981755i	$-0.439103\pi$
$157$	4.76515	0.380300	0.190150	+	0.981755i	$0.439103\pi$
$158$	−11.0047	−0.875483
$159$	8.95095	0.709857
$160$	15.7700	1.24673
$161$	0	0
$162$	17.1731	1.34925
$163$	−1.42110	−0.111309	−0.0556546	−	0.998450i	$-0.517725\pi$
$163$	−1.42110	−0.111309	−0.0556546	+	0.998450i	$0.517725\pi$
$164$	−5.86640	−0.458089
$165$	−17.2050	−1.33941
$166$	−5.65654	−0.439033
$167$	−12.2212	−0.945703	−0.472851	−	0.881142i	$-0.656775\pi$
$167$	−12.2212	−0.945703	−0.472851	+	0.881142i	$0.656775\pi$
$168$	0	0
$169$	−12.0442	−0.926475
$170$	−8.47293	−0.649844
$171$	0.0250782	0.00191778
$172$	−16.6079	−1.26634
$173$	3.68519	0.280180	0.140090	−	0.990139i	$-0.455261\pi$
$173$	3.68519	0.280180	0.140090	+	0.990139i	$0.455261\pi$
$174$	18.3513	1.39121
$175$	0	0
$176$	−21.3780	−1.61143
$177$	8.99998	0.676480
$178$	21.5656	1.61641
$179$	−14.3463	−1.07229	−0.536146	−	0.844125i	$-0.680120\pi$
$179$	−14.3463	−1.07229	−0.536146	+	0.844125i	$0.680120\pi$
$180$	−0.0166600	−0.00124176
$181$	−26.4901	−1.96899	−0.984495	−	0.175410i	$-0.943875\pi$
$181$	−26.4901	−1.96899	−0.984495	+	0.175410i	$0.943875\pi$
$182$	0	0
$183$	−7.73097	−0.571489
$184$	−2.51028	−0.185060
$185$	−3.67306	−0.270049
$186$	−33.6204	−2.46517
$187$	9.74077	0.712316
$188$	−7.15298	−0.521685
$189$	0	0
$190$	21.5098	1.56049
$191$	−9.00474	−0.651560	−0.325780	−	0.945446i	$-0.605627\pi$
$191$	−9.00474	−0.651560	−0.325780	+	0.945446i	$0.605627\pi$
$192$	8.68845	0.627035
$193$	6.86330	0.494031	0.247016	−	0.969012i	$-0.420550\pi$
$193$	6.86330	0.494031	0.247016	+	0.969012i	$0.420550\pi$
$194$	3.36839	0.241837
$195$	−3.59914	−0.257739
$196$	0	0
$197$	11.3025	0.805273	0.402636	−	0.915360i	$-0.368094\pi$
$197$	11.3025	0.805273	0.402636	+	0.915360i	$0.368094\pi$
$198$	0.0423336	0.00300852
$199$	14.1121	1.00038	0.500191	−	0.865915i	$-0.333263\pi$
$199$	14.1121	1.00038	0.500191	+	0.865915i	$0.333263\pi$
$200$	−0.315713	−0.0223243
$201$	−0.834894	−0.0588889
$202$	18.4349	1.29708
$203$	0	0
$204$	−5.96074	−0.417336
$205$	−7.55142	−0.527414
$206$	6.94997	0.484228
$207$	0.0179140	0.00124511
$208$	−4.47209	−0.310084
$209$	−24.7284	−1.71050
$210$	0	0
$211$	−16.8556	−1.16039	−0.580195	−	0.814478i	$-0.697024\pi$
$211$	−16.8556	−1.16039	−0.580195	+	0.814478i	$0.697024\pi$
$212$	−8.54649	−0.586975
$213$	−23.1741	−1.58786
$214$	8.30253	0.567549
$215$	−21.3782	−1.45798
$216$	3.45383	0.235003
$217$	0	0
$218$	−1.84219	−0.124769
$219$	−10.3267	−0.697816
$220$	16.4276	1.10755
$221$	2.03768	0.137069
$222$	−5.71146	−0.383328
$223$	−16.4985	−1.10482	−0.552411	−	0.833572i	$-0.686292\pi$
$223$	−16.4985	−1.10482	−0.552411	+	0.833572i	$0.686292\pi$
$224$	0	0
$225$	0.00225300	0.000150200 0
$226$	−7.18625	−0.478022
$227$	−3.54867	−0.235533	−0.117767	−	0.993041i	$-0.537573\pi$
$227$	−3.54867	−0.235533	−0.117767	+	0.993041i	$0.537573\pi$
$228$	15.1323	1.00216
$229$	−14.2886	−0.944215	−0.472107	−	0.881541i	$-0.656507\pi$
$229$	−14.2886	−0.944215	−0.472107	+	0.881541i	$0.656507\pi$
$230$	15.3650	1.01314
$231$	0	0
$232$	3.68495	0.241929
$233$	14.2085	0.930827	0.465414	−	0.885093i	$-0.345906\pi$
$233$	14.2085	0.930827	0.465414	+	0.885093i	$0.345906\pi$
$234$	0.00885581	0.000578922 0
$235$	−9.20754	−0.600634
$236$	−8.59329	−0.559376
$237$	−9.96552	−0.647330
$238$	0	0
$239$	10.1700	0.657844	0.328922	−	0.944357i	$-0.393315\pi$
$239$	10.1700	0.657844	0.328922	+	0.944357i	$0.393315\pi$
$240$	16.8396	1.08699
$241$	−26.4375	−1.70299	−0.851494	−	0.524364i	$-0.824303\pi$
$241$	−26.4375	−1.70299	−0.851494	+	0.524364i	$0.824303\pi$
$242$	−20.7206	−1.33197
$243$	−0.0492558	−0.00315976
$244$	7.38163	0.472560
$245$	0	0
$246$	−11.7421	−0.748651
$247$	−5.17296	−0.329148
$248$	−6.75100	−0.428689
$249$	−5.12241	−0.324620
$250$	22.2586	1.40776
$251$	−24.3910	−1.53954	−0.769772	−	0.638319i	$-0.779630\pi$
$251$	−24.3910	−1.53954	−0.769772	+	0.638319i	$0.779630\pi$
$252$	0	0
$253$	−17.6641	−1.11053
$254$	−18.8934	−1.18548
$255$	−7.67286	−0.480493
$256$	20.0417	1.25261
$257$	−1.63981	−0.102289	−0.0511443	−	0.998691i	$-0.516287\pi$
$257$	−1.63981	−0.102289	−0.0511443	+	0.998691i	$0.516287\pi$
$258$	−33.2422	−2.06957
$259$	0	0
$260$	3.43650	0.213123
$261$	−0.0262967	−0.00162772
$262$	9.99727	0.617634
$263$	−15.0058	−0.925298	−0.462649	−	0.886541i	$-0.653101\pi$
$263$	−15.0058	−0.925298	−0.462649	+	0.886541i	$0.653101\pi$
$264$	−5.37205	−0.330627
$265$	−11.0013	−0.675805
$266$	0	0
$267$	19.5292	1.19517
$268$	0.797167	0.0486947
$269$	−7.37245	−0.449506	−0.224753	−	0.974416i	$-0.572158\pi$
$269$	−7.37245	−0.449506	−0.224753	+	0.974416i	$0.572158\pi$
$270$	−21.1403	−1.28656
$271$	18.7955	1.14175	0.570874	−	0.821038i	$-0.306604\pi$
$271$	18.7955	1.14175	0.570874	+	0.821038i	$0.306604\pi$
$272$	−9.53388	−0.578076
$273$	0	0
$274$	−35.1565	−2.12388
$275$	−2.22158	−0.133966
$276$	10.8093	0.650645
$277$	12.6086	0.757577	0.378788	−	0.925483i	$-0.376341\pi$
$277$	12.6086	0.757577	0.378788	+	0.925483i	$0.376341\pi$
$278$	13.5603	0.813291
$279$	0.0481768	0.00288427
$280$	0	0
$281$	−15.0498	−0.897795	−0.448897	−	0.893583i	$-0.648183\pi$
$281$	−15.0498	−0.897795	−0.448897	+	0.893583i	$0.648183\pi$
$282$	−14.3173	−0.852585
$283$	−23.7275	−1.41045	−0.705227	−	0.708981i	$-0.749155\pi$
$283$	−23.7275	−1.41045	−0.705227	+	0.708981i	$0.749155\pi$
$284$	22.1269	1.31299
$285$	19.4787	1.15382
$286$	−8.73229	−0.516351
$287$	0	0
$288$	−0.0351386	−0.00207056
$289$	−12.6559	−0.744467
$290$	−22.5549	−1.32447
$291$	3.05033	0.178813
$292$	9.86010	0.577019
$293$	−5.68397	−0.332061	−0.166031	−	0.986121i	$-0.553095\pi$
$293$	−5.68397	−0.332061	−0.166031	+	0.986121i	$0.553095\pi$
$294$	0	0
$295$	−11.0616	−0.644029
$296$	−1.14687	−0.0666602
$297$	24.3036	1.41024
$298$	−13.4269	−0.777797
$299$	−3.69517	−0.213697
$300$	1.35947	0.0784889
$301$	0	0
$302$	2.02100	0.116296
$303$	16.6942	0.959056
$304$	24.2032	1.38815
$305$	9.50187	0.544075
$306$	0.0188794	0.00107926
$307$	34.1390	1.94842	0.974209	−	0.225647i	$-0.0724497\pi$
$307$	34.1390	1.94842	0.974209	+	0.225647i	$0.0724497\pi$
$308$	0	0
$309$	6.29371	0.358037
$310$	41.3217	2.34691
$311$	−28.5311	−1.61785	−0.808924	−	0.587913i	$-0.799950\pi$
$311$	−28.5311	−1.61785	−0.808924	+	0.587913i	$0.799950\pi$
$312$	−1.12378	−0.0636217
$313$	27.1423	1.53417	0.767086	−	0.641544i	$-0.221706\pi$
$313$	27.1423	1.53417	0.767086	+	0.641544i	$0.221706\pi$
$314$	−9.10689	−0.513932
$315$	0	0
$316$	9.51520	0.535272
$317$	12.2846	0.689969	0.344985	−	0.938608i	$-0.387884\pi$
$317$	12.2846	0.689969	0.344985	+	0.938608i	$0.387884\pi$
$318$	−17.1066	−0.959289
$319$	25.9299	1.45180
$320$	−10.6787	−0.596956
$321$	7.51855	0.419644
$322$	0	0
$323$	−11.0280	−0.613616
$324$	−14.8488	−0.824931
$325$	−0.464734	−0.0257788
$326$	2.71593	0.150421
$327$	−1.66824	−0.0922538
$328$	−2.35783	−0.130189
$329$	0	0
$330$	32.8813	1.81006
$331$	−17.7755	−0.977027	−0.488514	−	0.872556i	$-0.662461\pi$
$331$	−17.7755	−0.977027	−0.488514	+	0.872556i	$0.662461\pi$
$332$	4.89095	0.268426
$333$	0.00818430	0.000448497 0
$334$	23.3564	1.27801
$335$	1.02614	0.0560640
$336$	0	0
$337$	6.10994	0.332830	0.166415	−	0.986056i	$-0.446781\pi$
$337$	6.10994	0.332830	0.166415	+	0.986056i	$0.446781\pi$
$338$	23.0182	1.25202
$339$	−6.50767	−0.353448
$340$	7.32614	0.397316
$341$	−47.5048	−2.57253
$342$	−0.0479281	−0.00259166
$343$	0	0
$344$	−6.67506	−0.359895
$345$	13.9141	0.749110
$346$	−7.04294	−0.378631
$347$	19.9468	1.07080	0.535399	−	0.844599i	$-0.320161\pi$
$347$	19.9468	1.07080	0.535399	+	0.844599i	$0.320161\pi$
$348$	−15.8675	−0.850585
$349$	−20.1484	−1.07852	−0.539260	−	0.842139i	$-0.681296\pi$
$349$	−20.1484	−1.07852	−0.539260	+	0.842139i	$0.681296\pi$
$350$	0	0
$351$	5.08408	0.271368
$352$	34.6485	1.84677
$353$	−18.5224	−0.985848	−0.492924	−	0.870072i	$-0.664072\pi$
$353$	−18.5224	−0.985848	−0.492924	+	0.870072i	$0.664072\pi$
$354$	−17.2003	−0.914184
$355$	28.4825	1.51169
$356$	−18.6467	−0.988275
$357$	0	0
$358$	27.4179	1.44908
$359$	−15.7687	−0.832240	−0.416120	−	0.909310i	$-0.636610\pi$
$359$	−15.7687	−0.832240	−0.416120	+	0.909310i	$0.636610\pi$
$360$	−0.00669599	−0.000352910 0
$361$	8.99635	0.473492
$362$	50.6264	2.66086
$363$	−18.7640	−0.984855
$364$	0	0
$365$	12.6922	0.664342
$366$	14.7750	0.772302
$367$	−14.3857	−0.750927	−0.375463	−	0.926837i	$-0.622516\pi$
$367$	−14.3857	−0.750927	−0.375463	+	0.926837i	$0.622516\pi$
$368$	17.2889	0.901247
$369$	0.0168260	0.000875929 0
$370$	7.01976	0.364940
$371$	0	0
$372$	29.0700	1.50721
$373$	−3.94528	−0.204279	−0.102139	−	0.994770i	$-0.532569\pi$
$373$	−3.94528	−0.204279	−0.102139	+	0.994770i	$0.532569\pi$
$374$	−18.6160	−0.962612
$375$	20.1568	1.04089
$376$	−2.87493	−0.148263
$377$	5.42430	0.279365
$378$	0	0
$379$	−26.7134	−1.37217	−0.686086	−	0.727520i	$-0.740673\pi$
$379$	−26.7134	−1.37217	−0.686086	+	0.727520i	$0.740673\pi$
$380$	−18.5985	−0.954084
$381$	−17.1093	−0.876538
$382$	17.2094	0.880508
$383$	−0.682570	−0.0348777	−0.0174388	−	0.999848i	$-0.505551\pi$
$383$	−0.682570	−0.0348777	−0.0174388	+	0.999848i	$0.505551\pi$
$384$	9.05686	0.462181
$385$	0	0
$386$	−13.1168	−0.667626
$387$	0.0476348	0.00242141
$388$	−2.91249	−0.147859
$389$	31.8613	1.61543	0.807715	−	0.589573i	$-0.200704\pi$
$389$	31.8613	1.61543	0.807715	+	0.589573i	$0.200704\pi$
$390$	6.87847	0.348305
$391$	−7.87759	−0.398387
$392$	0	0
$393$	9.05326	0.456677
$394$	−21.6008	−1.08823
$395$	12.2483	0.616278
$396$	−0.0366039	−0.00183942
$397$	−7.84339	−0.393649	−0.196824	−	0.980439i	$-0.563063\pi$
$397$	−7.84339	−0.393649	−0.196824	+	0.980439i	$0.563063\pi$
$398$	−26.9703	−1.35190
$399$	0	0
$400$	2.17439	0.108720
$401$	19.7771	0.987624	0.493812	−	0.869569i	$-0.335603\pi$
$401$	19.7771	0.987624	0.493812	+	0.869569i	$0.335603\pi$
$402$	1.59560	0.0795815
$403$	−9.93757	−0.495026
$404$	−15.9398	−0.793036
$405$	−19.1138	−0.949773
$406$	0	0
$407$	−8.07015	−0.400023
$408$	−2.39575	−0.118607
$409$	9.08319	0.449135	0.224567	−	0.974459i	$-0.427903\pi$
$409$	9.08319	0.449135	0.224567	+	0.974459i	$0.427903\pi$
$410$	14.4319	0.712739
$411$	−31.8368	−1.57039
$412$	−6.00931	−0.296058
$413$	0	0
$414$	−0.0342362	−0.00168262
$415$	6.29578	0.309048
$416$	7.24816	0.355370
$417$	12.2798	0.601345
$418$	47.2596	2.31154
$419$	6.15974	0.300923	0.150462	−	0.988616i	$-0.451924\pi$
$419$	6.15974	0.300923	0.150462	+	0.988616i	$0.451924\pi$
$420$	0	0
$421$	9.82795	0.478985	0.239493	−	0.970898i	$-0.423019\pi$
$421$	9.82795	0.478985	0.239493	+	0.970898i	$0.423019\pi$
$422$	32.2136	1.56813
$423$	0.0205162	0.000997533 0
$424$	−3.43502	−0.166819
$425$	−0.990749	−0.0480584
$426$	44.2890	2.14581
$427$	0	0
$428$	−7.17881	−0.347001
$429$	−7.90773	−0.381789
$430$	40.8568	1.97029
$431$	−2.35791	−0.113576	−0.0567882	−	0.998386i	$-0.518086\pi$
$431$	−2.35791	−0.113576	−0.0567882	+	0.998386i	$0.518086\pi$
$432$	−23.7874	−1.14447
$433$	25.4693	1.22398	0.611989	−	0.790866i	$-0.290370\pi$
$433$	25.4693	1.22398	0.611989	+	0.790866i	$0.290370\pi$
$434$	0	0
$435$	−20.4251	−0.979309
$436$	1.59285	0.0762839
$437$	19.9985	0.956656
$438$	19.7359	0.943017
$439$	38.2280	1.82452	0.912262	−	0.409607i	$-0.134334\pi$
$439$	38.2280	1.82452	0.912262	+	0.409607i	$0.134334\pi$
$440$	6.60260	0.314767
$441$	0	0
$442$	−3.89430	−0.185233
$443$	13.7242	0.652055	0.326027	−	0.945360i	$-0.394290\pi$
$443$	13.7242	0.652055	0.326027	+	0.945360i	$0.394290\pi$
$444$	4.93843	0.234367
$445$	−24.0027	−1.13784
$446$	31.5310	1.49304
$447$	−12.1590	−0.575101
$448$	0	0
$449$	−5.44848	−0.257130	−0.128565	−	0.991701i	$-0.541037\pi$
$449$	−5.44848	−0.257130	−0.128565	+	0.991701i	$0.541037\pi$
$450$	−0.00430582	−0.000202978 0
$451$	−16.5914	−0.781257
$452$	6.21361	0.292264
$453$	1.83016	0.0859886
$454$	6.78202	0.318296
$455$	0	0
$456$	6.08197	0.284815
$457$	23.0445	1.07798	0.538989	−	0.842313i	$-0.318806\pi$
$457$	23.0445	1.07798	0.538989	+	0.842313i	$0.318806\pi$
$458$	27.3075	1.27600
$459$	10.8386	0.505901
$460$	−13.2854	−0.619434
$461$	33.1294	1.54299	0.771495	−	0.636236i	$-0.219510\pi$
$461$	33.1294	1.54299	0.771495	+	0.636236i	$0.219510\pi$
$462$	0	0
$463$	1.68543	0.0783284	0.0391642	−	0.999233i	$-0.487530\pi$
$463$	1.68543	0.0783284	0.0391642	+	0.999233i	$0.487530\pi$
$464$	−25.3791	−1.17820
$465$	37.4198	1.73530
$466$	−27.1544	−1.25791
$467$	−19.9515	−0.923246	−0.461623	−	0.887076i	$-0.652733\pi$
$467$	−19.9515	−0.923246	−0.461623	+	0.887076i	$0.652733\pi$
$468$	−0.00765720	−0.000353954 0
$469$	0	0
$470$	17.5970	0.811687
$471$	−8.24696	−0.380000
$472$	−3.45383	−0.158975
$473$	−46.9704	−2.15970
$474$	19.0456	0.874791
$475$	2.51517	0.115404
$476$	0	0
$477$	0.0245131	0.00112238
$478$	−19.4364	−0.888999
$479$	2.93148	0.133943	0.0669713	−	0.997755i	$-0.478666\pi$
$479$	2.93148	0.133943	0.0669713	+	0.997755i	$0.478666\pi$
$480$	−27.2928	−1.24574
$481$	−1.68820	−0.0769754
$482$	50.5259	2.30139
$483$	0	0
$484$	17.9161	0.814369
$485$	−3.74905	−0.170236
$486$	0.0941350	0.00427005
$487$	−29.8588	−1.35303	−0.676515	−	0.736429i	$-0.736511\pi$
$487$	−29.8588	−1.35303	−0.676515	+	0.736429i	$0.736511\pi$
$488$	2.96683	0.134302
$489$	2.45947	0.111221
$490$	0	0
$491$	−0.0655631	−0.00295882	−0.00147941	−	0.999999i	$-0.500471\pi$
$491$	−0.0655631	−0.00295882	−0.00147941	+	0.999999i	$0.500471\pi$
$492$	10.1529	0.457727
$493$	11.5638	0.520809
$494$	9.88628	0.444805
$495$	−0.0471177	−0.00211778
$496$	46.4958	2.08772
$497$	0	0
$498$	9.78968	0.438686
$499$	4.64508	0.207942	0.103971	−	0.994580i	$-0.466845\pi$
$499$	4.64508	0.207942	0.103971	+	0.994580i	$0.466845\pi$
$500$	−19.2460	−0.860706
$501$	21.1510	0.944955
$502$	46.6147	2.08051
$503$	−22.1144	−0.986034	−0.493017	−	0.870020i	$-0.664106\pi$
$503$	−22.1144	−0.986034	−0.493017	+	0.870020i	$0.664106\pi$
$504$	0	0
$505$	−20.5182	−0.913050
$506$	33.7587	1.50076
$507$	20.8446	0.925743
$508$	16.3362	0.724802
$509$	10.7184	0.475083	0.237542	−	0.971377i	$-0.423658\pi$
$509$	10.7184	0.475083	0.237542	+	0.971377i	$0.423658\pi$
$510$	14.6640	0.649331
$511$	0	0
$512$	−27.8365	−1.23021
$513$	−27.5153	−1.21483
$514$	3.13392	0.138231
$515$	−7.73538	−0.340862
$516$	28.7430	1.26534
$517$	−20.2301	−0.889717
$518$	0	0
$519$	−6.37789	−0.279958
$520$	1.38120	0.0605698
$521$	3.90956	0.171281	0.0856405	−	0.996326i	$-0.472706\pi$
$521$	3.90956	0.171281	0.0856405	+	0.996326i	$0.472706\pi$
$522$	0.0502568	0.00219968
$523$	36.6457	1.60241	0.801203	−	0.598392i	$-0.204194\pi$
$523$	36.6457	1.60241	0.801203	+	0.598392i	$0.204194\pi$
$524$	−8.64417	−0.377622
$525$	0	0
$526$	28.6783	1.25043
$527$	−21.1855	−0.922856
$528$	36.9986	1.61016
$529$	−8.71463	−0.378897
$530$	21.0251	0.913272
$531$	0.0246473	0.00106960
$532$	0	0
$533$	−3.47076	−0.150335
$534$	−37.3231	−1.61513
$535$	−9.24079	−0.399514
$536$	0.320399	0.0138391
$537$	24.8289	1.07144
$538$	14.0898	0.607456
$539$	0	0
$540$	18.2790	0.786602
$541$	−34.0145	−1.46240	−0.731199	−	0.682164i	$-0.761039\pi$
$541$	−34.0145	−1.46240	−0.731199	+	0.682164i	$0.761039\pi$
$542$	−35.9210	−1.54294
$543$	45.8459	1.96743
$544$	15.4521	0.662502
$545$	2.05037	0.0878284
$546$	0	0
$547$	26.2909	1.12412	0.562059	−	0.827097i	$-0.310009\pi$
$547$	26.2909	1.12412	0.562059	+	0.827097i	$0.310009\pi$
$548$	30.3982	1.29855
$549$	−0.0211720	−0.000903600 0
$550$	4.24576	0.181040
$551$	−29.3566	−1.25063
$552$	4.34450	0.184914
$553$	0	0
$554$	−24.0969	−1.02378
$555$	6.35690	0.269835
$556$	−11.7249	−0.497248
$557$	−3.01222	−0.127632	−0.0638159	−	0.997962i	$-0.520327\pi$
$557$	−3.01222	−0.127632	−0.0638159	+	0.997962i	$0.520327\pi$
$558$	−0.0920728	−0.00389775
$559$	−9.82578	−0.415586
$560$	0	0
$561$	−16.8582	−0.711753
$562$	28.7623	1.21327
$563$	0.421258	0.0177539	0.00887696	−	0.999961i	$-0.497174\pi$
$563$	0.421258	0.0177539	0.00887696	+	0.999961i	$0.497174\pi$
$564$	12.3795	0.521272
$565$	7.99836	0.336493
$566$	45.3467	1.90607
$567$	0	0
$568$	8.89327	0.373153
$569$	−33.5878	−1.40807	−0.704037	−	0.710163i	$-0.748621\pi$
$569$	−33.5878	−1.40807	−0.704037	+	0.710163i	$0.748621\pi$
$570$	−37.2267	−1.55925
$571$	13.9589	0.584161	0.292080	−	0.956394i	$-0.405653\pi$
$571$	13.9589	0.584161	0.292080	+	0.956394i	$0.405653\pi$
$572$	7.55040	0.315698
$573$	15.5843	0.651045
$574$	0	0
$575$	1.79664	0.0749252
$576$	0.0237942	0.000991425 0
$577$	37.2005	1.54868	0.774338	−	0.632772i	$-0.218083\pi$
$577$	37.2005	1.54868	0.774338	+	0.632772i	$0.218083\pi$
$578$	24.1874	1.00606
$579$	−11.8782	−0.493641
$580$	19.5022	0.809783
$581$	0	0
$582$	−5.82962	−0.241645
$583$	−24.1712	−1.00107
$584$	3.96298	0.163989
$585$	−0.00985659	−0.000407520 0
$586$	10.8629	0.448742
$587$	−35.8868	−1.48120	−0.740602	−	0.671943i	$-0.765460\pi$
$587$	−35.8868	−1.48120	−0.740602	+	0.671943i	$0.765460\pi$
$588$	0	0
$589$	53.7827	2.21608
$590$	21.1403	0.870331
$591$	−19.5611	−0.804637
$592$	7.89874	0.324636
$593$	26.0179	1.06843	0.534214	−	0.845349i	$-0.320608\pi$
$593$	26.0179	1.06843	0.534214	+	0.845349i	$0.320608\pi$
$594$	−46.4477	−1.90577
$595$	0	0
$596$	11.6096	0.475547
$597$	−24.4236	−0.999592
$598$	7.06201	0.288787
$599$	−29.3004	−1.19718	−0.598591	−	0.801055i	$-0.704273\pi$
$599$	−29.3004	−1.19718	−0.598591	+	0.801055i	$0.704273\pi$
$600$	0.546399	0.0223066
$601$	−0.249287	−0.0101686	−0.00508432	−	0.999987i	$-0.501618\pi$
$601$	−0.249287	−0.0101686	−0.00508432	+	0.999987i	$0.501618\pi$
$602$	0	0
$603$	−0.00228644	−9.31110e−5 0
$604$	−1.74746	−0.0711033
$605$	23.0622	0.937611
$606$	−31.9050	−1.29605
$607$	4.31353	0.175081	0.0875404	−	0.996161i	$-0.472099\pi$
$607$	4.31353	0.175081	0.0875404	+	0.996161i	$0.472099\pi$
$608$	−39.2274	−1.59088
$609$	0	0
$610$	−18.1594	−0.735255
$611$	−4.23194	−0.171206
$612$	−0.0163241	−0.000659862 0
$613$	−20.4005	−0.823967	−0.411984	−	0.911191i	$-0.635164\pi$
$613$	−20.4005	−0.823967	−0.411984	+	0.911191i	$0.635164\pi$
$614$	−65.2447	−2.63306
$615$	13.0691	0.526997
$616$	0	0
$617$	31.4656	1.26676	0.633379	−	0.773841i	$-0.281667\pi$
$617$	31.4656	1.26676	0.633379	+	0.773841i	$0.281667\pi$
$618$	−12.0282	−0.483845
$619$	16.1171	0.647802	0.323901	−	0.946091i	$-0.395006\pi$
$619$	16.1171	0.647802	0.323901	+	0.946091i	$0.395006\pi$
$620$	−35.7289	−1.43491
$621$	−19.6549	−0.788722
$622$	54.5270	2.18633
$623$	0	0
$624$	7.73977	0.309839
$625$	−22.3973	−0.895891
$626$	−51.8729	−2.07326
$627$	42.7971	1.70915
$628$	7.87430	0.314219
$629$	−3.59901	−0.143502
$630$	0	0
$631$	−40.5989	−1.61622	−0.808108	−	0.589035i	$-0.799508\pi$
$631$	−40.5989	−1.61622	−0.808108	+	0.589035i	$0.799508\pi$
$632$	3.82436	0.152125
$633$	29.1717	1.15947
$634$	−23.4776	−0.932414
$635$	21.0285	0.834490
$636$	14.7913	0.586511
$637$	0	0
$638$	−49.5558	−1.96193
$639$	−0.0634645	−0.00251062
$640$	−11.1315	−0.440010
$641$	−26.0614	−1.02936	−0.514681	−	0.857381i	$-0.672090\pi$
$641$	−26.0614	−1.02936	−0.514681	+	0.857381i	$0.672090\pi$
$642$	−14.3690	−0.567101
$643$	18.1699	0.716550	0.358275	−	0.933616i	$-0.383365\pi$
$643$	18.1699	0.716550	0.358275	+	0.933616i	$0.383365\pi$
$644$	0	0
$645$	36.9989	1.45683
$646$	21.0762	0.829231
$647$	5.46792	0.214966	0.107483	−	0.994207i	$-0.465721\pi$
$647$	5.46792	0.214966	0.107483	+	0.994207i	$0.465721\pi$
$648$	−5.96804	−0.234447
$649$	−24.3036	−0.953999
$650$	0.888175	0.0348371
$651$	0	0
$652$	−2.34834	−0.0919680
$653$	−32.0505	−1.25423	−0.627117	−	0.778925i	$-0.715765\pi$
$653$	−32.0505	−1.25423	−0.627117	+	0.778925i	$0.715765\pi$
$654$	3.18825	0.124670
$655$	−11.1271	−0.434770
$656$	16.2390	0.634025
$657$	−0.0282808	−0.00110334
$658$	0	0
$659$	−42.5021	−1.65565	−0.827823	−	0.560989i	$-0.810421\pi$
$659$	−42.5021	−1.65565	−0.827823	+	0.560989i	$0.810421\pi$
$660$	−28.4309	−1.10667
$661$	−11.6801	−0.454302	−0.227151	−	0.973860i	$-0.572941\pi$
$661$	−11.6801	−0.454302	−0.227151	+	0.973860i	$0.572941\pi$
$662$	33.9715	1.32034
$663$	−3.52658	−0.136961
$664$	1.96578	0.0762869
$665$	0	0
$666$	−0.0156414	−0.000606092 0
$667$	−20.9701	−0.811966
$668$	−20.1952	−0.781376
$669$	28.5537	1.10395
$670$	−1.96110	−0.0757640
$671$	20.8767	0.805937
$672$	0	0
$673$	−6.30201	−0.242924	−0.121462	−	0.992596i	$-0.538758\pi$
$673$	−6.30201	−0.242924	−0.121462	+	0.992596i	$0.538758\pi$
$674$	−11.6770	−0.449781
$675$	−2.47195	−0.0951455
$676$	−19.9027	−0.765490
$677$	0.459540	0.0176615	0.00883077	−	0.999961i	$-0.497189\pi$
$677$	0.459540	0.0176615	0.00883077	+	0.999961i	$0.497189\pi$
$678$	12.4371	0.477644
$679$	0	0
$680$	2.94453	0.112918
$681$	6.14162	0.235347
$682$	90.7886	3.47647
$683$	−45.2098	−1.72990	−0.864952	−	0.501854i	$-0.832652\pi$
$683$	−45.2098	−1.72990	−0.864952	+	0.501854i	$0.832652\pi$
$684$	0.0414412	0.00158454
$685$	39.1295	1.49506
$686$	0	0
$687$	24.7290	0.943469
$688$	45.9728	1.75270
$689$	−5.05639	−0.192633
$690$	−26.5919	−1.01234
$691$	−12.3206	−0.468696	−0.234348	−	0.972153i	$-0.575296\pi$
$691$	−12.3206	−0.468696	−0.234348	+	0.972153i	$0.575296\pi$
$692$	6.08970	0.231495
$693$	0	0
$694$	−38.1212	−1.44706
$695$	−15.0927	−0.572499
$696$	−6.37747	−0.241738
$697$	−7.39918	−0.280264
$698$	38.5066	1.45750
$699$	−24.5903	−0.930092
$700$	0	0
$701$	−17.8501	−0.674189	−0.337095	−	0.941471i	$-0.609444\pi$
$701$	−17.8501	−0.674189	−0.337095	+	0.941471i	$0.609444\pi$
$702$	−9.71642	−0.366723
$703$	9.13664	0.344595
$704$	−23.4623	−0.884270
$705$	15.9353	0.600159
$706$	35.3990	1.33226
$707$	0	0
$708$	14.8723	0.558934
$709$	36.8226	1.38290	0.691450	−	0.722424i	$-0.256972\pi$
$709$	36.8226	1.38290	0.691450	+	0.722424i	$0.256972\pi$
$710$	−54.4341	−2.04288
$711$	−0.0272916	−0.00102351
$712$	−7.49452	−0.280869
$713$	38.4182	1.43877
$714$	0	0
$715$	9.71912	0.363474
$716$	−23.7069	−0.885969
$717$	−17.6011	−0.657324
$718$	30.1363	1.12468
$719$	19.3019	0.719841	0.359920	−	0.932983i	$-0.382804\pi$
$719$	19.3019	0.719841	0.359920	+	0.932983i	$0.382804\pi$
$720$	0.0461169	0.00171868
$721$	0	0
$722$	−17.1933	−0.639869
$723$	45.7549	1.70164
$724$	−43.7742	−1.62686
$725$	−2.63737	−0.0979494
$726$	35.8607	1.33092
$727$	−24.7011	−0.916113	−0.458057	−	0.888923i	$-0.651454\pi$
$727$	−24.7011	−0.916113	−0.458057	+	0.888923i	$0.651454\pi$
$728$	0	0
$729$	27.0425	1.00157
$730$	−24.2567	−0.897781
$731$	−20.9472	−0.774760
$732$	−12.7752	−0.472187
$733$	−29.4370	−1.08728	−0.543640	−	0.839318i	$-0.682954\pi$
$733$	−29.4370	−1.08728	−0.543640	+	0.839318i	$0.682954\pi$
$734$	27.4931	1.01479
$735$	0	0
$736$	−28.0211	−1.03287
$737$	2.25455	0.0830474
$738$	−0.0321570	−0.00118372
$739$	15.2693	0.561692	0.280846	−	0.959753i	$-0.409385\pi$
$739$	15.2693	0.561692	0.280846	+	0.959753i	$0.409385\pi$
$740$	−6.06965	−0.223125
$741$	8.95275	0.328888
$742$	0	0
$743$	−0.704363	−0.0258406	−0.0129203	−	0.999917i	$-0.504113\pi$
$743$	−0.704363	−0.0258406	−0.0129203	+	0.999917i	$0.504113\pi$
$744$	11.6838	0.428350
$745$	14.9442	0.547514
$746$	7.54000	0.276059
$747$	−0.0140282	−0.000513267 0
$748$	16.0964	0.588543
$749$	0	0
$750$	−38.5226	−1.40665
$751$	29.8809	1.09037	0.545185	−	0.838316i	$-0.316459\pi$
$751$	29.8809	1.09037	0.545185	+	0.838316i	$0.316459\pi$
$752$	19.8004	0.722045
$753$	42.2130	1.53833
$754$	−10.3666	−0.377530
$755$	−2.24939	−0.0818637
$756$	0	0
$757$	33.9540	1.23408	0.617039	−	0.786932i	$-0.288332\pi$
$757$	33.9540	1.23408	0.617039	+	0.786932i	$0.288332\pi$
$758$	51.0531	1.85433
$759$	30.5710	1.10966
$760$	−7.47515	−0.271152
$761$	16.9446	0.614241	0.307120	−	0.951671i	$-0.400635\pi$
$761$	16.9446	0.614241	0.307120	+	0.951671i	$0.400635\pi$
$762$	32.6984	1.18454
$763$	0	0
$764$	−14.8801	−0.538344
$765$	−0.0210129	−0.000759723 0
$766$	1.30449	0.0471331
$767$	−5.08408	−0.183576
$768$	−34.6859	−1.25162
$769$	−36.7623	−1.32568	−0.662841	−	0.748760i	$-0.730649\pi$
$769$	−36.7623	−1.32568	−0.662841	+	0.748760i	$0.730649\pi$
$770$	0	0
$771$	2.83799	0.102208
$772$	11.3414	0.408188
$773$	−48.3465	−1.73890	−0.869451	−	0.494019i	$-0.835527\pi$
$773$	−48.3465	−1.73890	−0.869451	+	0.494019i	$0.835527\pi$
$774$	−0.0910371	−0.00327226
$775$	4.83178	0.173563
$776$	−1.17059	−0.0420218
$777$	0	0
$778$	−60.8915	−2.18307
$779$	18.7839	0.673005
$780$	−5.94749	−0.212954
$781$	62.5793	2.23926
$782$	15.0552	0.538374
$783$	28.8522	1.03109
$784$	0	0
$785$	10.1361	0.361771
$786$	−17.3021	−0.617146
$787$	−24.8263	−0.884961	−0.442480	−	0.896778i	$-0.645901\pi$
$787$	−24.8263	−0.884961	−0.442480	+	0.896778i	$0.645901\pi$
$788$	18.6772	0.665348
$789$	25.9703	0.924567
$790$	−23.4082	−0.832828
$791$	0	0
$792$	−0.0147119	−0.000522764 0
$793$	4.36722	0.155085
$794$	14.9899	0.531970
$795$	19.0398	0.675271
$796$	23.3200	0.826555
$797$	10.9194	0.386787	0.193393	−	0.981121i	$-0.438051\pi$
$797$	10.9194	0.386787	0.193393	+	0.981121i	$0.438051\pi$
$798$	0	0
$799$	−9.02192	−0.319173
$800$	−3.52416	−0.124598
$801$	0.0534827	0.00188972
$802$	−37.7970	−1.33466
$803$	27.8863	0.984088
$804$	−1.37964	−0.0486563
$805$	0	0
$806$	18.9921	0.668970
$807$	12.7594	0.449151
$808$	−6.40655	−0.225382
$809$	−38.8530	−1.36600	−0.683000	−	0.730419i	$-0.739325\pi$
$809$	−38.8530	−1.36600	−0.683000	+	0.730419i	$0.739325\pi$
$810$	36.5293	1.28351
$811$	−54.3623	−1.90892	−0.954460	−	0.298339i	$-0.903568\pi$
$811$	−54.3623	−1.90892	−0.954460	+	0.298339i	$0.903568\pi$
$812$	0	0
$813$	−32.5291	−1.14085
$814$	15.4232	0.540584
$815$	−3.02285	−0.105886
$816$	16.5001	0.577620
$817$	53.1777	1.86045
$818$	−17.3593	−0.606954
$819$	0	0
$820$	−12.4785	−0.435770
$821$	7.98945	0.278834	0.139417	−	0.990234i	$-0.455477\pi$
$821$	7.98945	0.278834	0.139417	+	0.990234i	$0.455477\pi$
$822$	60.8448	2.12221
$823$	−41.5369	−1.44788	−0.723942	−	0.689861i	$-0.757672\pi$
$823$	−41.5369	−1.44788	−0.723942	+	0.689861i	$0.757672\pi$
$824$	−2.41527	−0.0841400
$825$	3.84485	0.133860
$826$	0	0
$827$	−6.55907	−0.228081	−0.114041	−	0.993476i	$-0.536379\pi$
$827$	−6.55907	−0.228081	−0.114041	+	0.993476i	$0.536379\pi$
$828$	0.0296024	0.00102876
$829$	25.1203	0.872465	0.436232	−	0.899834i	$-0.356313\pi$
$829$	25.1203	0.872465	0.436232	+	0.899834i	$0.356313\pi$
$830$	−12.0322	−0.417642
$831$	−21.8215	−0.756978
$832$	−4.90810	−0.170158
$833$	0	0
$834$	−23.4685	−0.812648
$835$	−25.9959	−0.899626
$836$	−40.8632	−1.41328
$837$	−52.8586	−1.82706
$838$	−11.7722	−0.406663
$839$	39.6930	1.37036	0.685178	−	0.728376i	$-0.259725\pi$
$839$	39.6930	1.37036	0.685178	+	0.728376i	$0.259725\pi$
$840$	0	0
$841$	1.78290	0.0614792
$842$	−18.7826	−0.647293
$843$	26.0464	0.897085
$844$	−27.8535	−0.958759
$845$	−25.6194	−0.881335
$846$	−0.0392095	−0.00134805
$847$	0	0
$848$	23.6578	0.812412
$849$	41.0648	1.40934
$850$	1.89347	0.0649453
$851$	6.52652	0.223726
$852$	−38.2946	−1.31195
$853$	−12.9489	−0.443363	−0.221681	−	0.975119i	$-0.571155\pi$
$853$	−12.9489	−0.443363	−0.221681	+	0.975119i	$0.571155\pi$
$854$	0	0
$855$	0.0533444	0.00182434
$856$	−2.88532	−0.0986181
$857$	49.0913	1.67693	0.838463	−	0.544959i	$-0.183455\pi$
$857$	49.0913	1.67693	0.838463	+	0.544959i	$0.183455\pi$
$858$	15.1128	0.515943
$859$	51.1512	1.74526	0.872629	−	0.488384i	$-0.162414\pi$
$859$	51.1512	1.74526	0.872629	+	0.488384i	$0.162414\pi$
$860$	−35.3270	−1.20464
$861$	0	0
$862$	4.50630	0.153485
$863$	7.24161	0.246507	0.123254	−	0.992375i	$-0.460667\pi$
$863$	7.24161	0.246507	0.123254	+	0.992375i	$0.460667\pi$
$864$	38.5534	1.31161
$865$	7.83885	0.266529
$866$	−48.6756	−1.65406
$867$	21.9034	0.743879
$868$	0	0
$869$	26.9109	0.912890
$870$	39.0354	1.32342
$871$	0.471631	0.0159806
$872$	0.640202	0.0216800
$873$	0.00835362	0.000282727 0
$874$	−38.2200	−1.29281
$875$	0	0
$876$	−17.0647	−0.576563
$877$	8.76137	0.295851	0.147925	−	0.988999i	$-0.452740\pi$
$877$	8.76137	0.295851	0.147925	+	0.988999i	$0.452740\pi$
$878$	−73.0593	−2.46563
$879$	9.83715	0.331799
$880$	−45.4737	−1.53292
$881$	−12.2501	−0.412715	−0.206358	−	0.978477i	$-0.566161\pi$
$881$	−12.2501	−0.412715	−0.206358	+	0.978477i	$0.566161\pi$
$882$	0	0
$883$	−26.9923	−0.908362	−0.454181	−	0.890910i	$-0.650068\pi$
$883$	−26.9923	−0.908362	−0.454181	+	0.890910i	$0.650068\pi$
$884$	3.36722	0.113252
$885$	19.1440	0.643520
$886$	−26.2289	−0.881177
$887$	−56.9720	−1.91293	−0.956466	−	0.291842i	$-0.905732\pi$
$887$	−56.9720	−1.91293	−0.956466	+	0.291842i	$0.905732\pi$
$888$	1.98486	0.0666075
$889$	0	0
$890$	45.8726	1.53765
$891$	−41.9953	−1.40690
$892$	−27.2634	−0.912846
$893$	22.9035	0.766437
$894$	23.2376	0.777183
$895$	−30.5163	−1.02005
$896$	0	0
$897$	6.39516	0.213528
$898$	10.4128	0.347481
$899$	−56.3957	−1.88090
$900$	0.00372304	0.000124101 0
$901$	−10.7795	−0.359118
$902$	31.7085	1.05578
$903$	0	0
$904$	2.49738	0.0830617
$905$	−56.3476	−1.87306
$906$	−3.49771	−0.116204
$907$	−36.9326	−1.22633	−0.613163	−	0.789956i	$-0.710103\pi$
$907$	−36.9326	−1.22633	−0.613163	+	0.789956i	$0.710103\pi$
$908$	−5.86409	−0.194607
$909$	0.0457187	0.00151639
$910$	0	0
$911$	0.553010	0.0183220	0.00916102	−	0.999958i	$-0.497084\pi$
$911$	0.553010	0.0183220	0.00916102	+	0.999958i	$0.497084\pi$
$912$	−41.8880	−1.38705
$913$	13.8326	0.457792
$914$	−44.0415	−1.45676
$915$	−16.4447	−0.543645
$916$	−23.6115	−0.780147
$917$	0	0
$918$	−20.7141	−0.683666
$919$	−10.7772	−0.355506	−0.177753	−	0.984075i	$-0.556883\pi$
$919$	−10.7772	−0.355506	−0.177753	+	0.984075i	$0.556883\pi$
$920$	−5.33967	−0.176044
$921$	−59.0838	−1.94688
$922$	−63.3151	−2.08517
$923$	13.0910	0.430896
$924$	0	0
$925$	0.820827	0.0269886
$926$	−3.22109	−0.105852
$927$	0.0172360	0.000566103 0
$928$	41.1333	1.35027
$929$	−12.6554	−0.415211	−0.207605	−	0.978213i	$-0.566567\pi$
$929$	−12.6554	−0.415211	−0.207605	+	0.978213i	$0.566567\pi$
$930$	−71.5147	−2.34506
$931$	0	0
$932$	23.4792	0.769086
$933$	49.3782	1.61657
$934$	38.1303	1.24766
$935$	20.7198	0.677610
$936$	−0.00307759	−0.000100594 0
$937$	−12.0477	−0.393580	−0.196790	−	0.980446i	$-0.563052\pi$
$937$	−12.0477	−0.393580	−0.196790	+	0.980446i	$0.563052\pi$
$938$	0	0
$939$	−46.9747	−1.53296
$940$	−15.2153	−0.496267
$941$	−21.0200	−0.685234	−0.342617	−	0.939475i	$-0.611313\pi$
$941$	−21.0200	−0.685234	−0.342617	+	0.939475i	$0.611313\pi$
$942$	15.7611	0.513526
$943$	13.4178	0.436944
$944$	23.7874	0.774213
$945$	0	0
$946$	89.7673	2.91859
$947$	−36.3745	−1.18201	−0.591006	−	0.806667i	$-0.701269\pi$
$947$	−36.3745	−1.18201	−0.591006	+	0.806667i	$0.701269\pi$
$948$	−16.4678	−0.534849
$949$	5.83357	0.189366
$950$	−4.80685	−0.155955
$951$	−21.2607	−0.689424
$952$	0	0
$953$	47.7840	1.54787	0.773937	−	0.633262i	$-0.218284\pi$
$953$	47.7840	1.54787	0.773937	+	0.633262i	$0.218284\pi$
$954$	−0.0468481	−0.00151676
$955$	−19.1542	−0.619815
$956$	16.8057	0.543536
$957$	−44.8764	−1.45065
$958$	−5.60248	−0.181008
$959$	0	0
$960$	18.4814	0.596485
$961$	72.3197	2.33289
$962$	3.22640	0.104023
$963$	0.0205903	0.000663513 0
$964$	−43.6873	−1.40707
$965$	14.5991	0.469961
$966$	0	0
$967$	13.1751	0.423682	0.211841	−	0.977304i	$-0.432054\pi$
$967$	13.1751	0.423682	0.211841	+	0.977304i	$0.432054\pi$
$968$	7.20087	0.231445
$969$	19.0860	0.613132
$970$	7.16498	0.230054
$971$	12.5598	0.403062	0.201531	−	0.979482i	$-0.435408\pi$
$971$	12.5598	0.403062	0.201531	+	0.979482i	$0.435408\pi$
$972$	−0.0813941	−0.00261072
$973$	0	0
$974$	57.0644	1.82846
$975$	0.804307	0.0257584
$976$	−20.4333	−0.654054
$977$	−58.9430	−1.88575	−0.942877	−	0.333142i	$-0.891891\pi$
$977$	−58.9430	−1.88575	−0.942877	+	0.333142i	$0.891891\pi$
$978$	−4.70041	−0.150303
$979$	−52.7367	−1.68547
$980$	0	0
$981$	−0.00456864	−0.000145865 0
$982$	0.125301	0.00399850
$983$	30.2675	0.965385	0.482692	−	0.875790i	$-0.339659\pi$
$983$	30.2675	0.965385	0.482692	+	0.875790i	$0.339659\pi$
$984$	4.08066	0.130087
$985$	24.0419	0.766038
$986$	−22.1002	−0.703814
$987$	0	0
$988$	−8.54820	−0.271955
$989$	37.9861	1.20789
$990$	0.0900488	0.00286194
$991$	−21.8922	−0.695427	−0.347714	−	0.937601i	$-0.613042\pi$
$991$	−21.8922	−0.695427	−0.347714	+	0.937601i	$0.613042\pi$
$992$	−75.3582	−2.39263
$993$	30.7637	0.976255
$994$	0	0
$995$	30.0182	0.951642
$996$	−8.46467	−0.268213
$997$	21.0576	0.666901	0.333451	−	0.942768i	$-0.391787\pi$
$997$	21.0576	0.666901	0.333451	+	0.942768i	$0.391787\pi$
$998$	−8.87742	−0.281010
$999$	−8.97966	−0.284104

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2401.2.a.c.1.1		6
7.6	odd	2		2401.2.a.d.1.1		6
49.4	even	21		343.2.g.b.226.1		12
49.12	odd	42		343.2.g.c.214.1		12
49.17	odd	42		343.2.g.a.79.1		12
49.20	odd	14		343.2.e.b.50.1		12
49.22	even	7		49.2.e.b.43.1	yes	12
49.23	even	21		343.2.g.d.165.1		12
49.26	odd	42		343.2.g.a.165.1		12
49.27	odd	14		343.2.e.b.295.1		12
49.29	even	7		49.2.e.b.8.1	✓	12
49.32	even	21		343.2.g.d.79.1		12
49.37	even	21		343.2.g.b.214.1		12
49.45	odd	42		343.2.g.c.226.1		12
147.29	odd	14		441.2.u.b.253.2		12
147.71	odd	14		441.2.u.b.190.2		12
196.71	odd	14		784.2.u.b.337.1		12
196.127	odd	14		784.2.u.b.449.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.2.e.b.8.1	✓	12	49.29	even	7
49.2.e.b.43.1	yes	12	49.22	even	7
343.2.e.b.50.1		12	49.20	odd	14
343.2.e.b.295.1		12	49.27	odd	14
343.2.g.a.79.1		12	49.17	odd	42
343.2.g.a.165.1		12	49.26	odd	42
343.2.g.b.214.1		12	49.37	even	21
343.2.g.b.226.1		12	49.4	even	21
343.2.g.c.214.1		12	49.12	odd	42
343.2.g.c.226.1		12	49.45	odd	42
343.2.g.d.79.1		12	49.32	even	21
343.2.g.d.165.1		12	49.23	even	21
441.2.u.b.190.2		12	147.71	odd	14
441.2.u.b.253.2		12	147.29	odd	14
784.2.u.b.337.1		12	196.71	odd	14
784.2.u.b.449.1		12	196.127	odd	14
2401.2.a.c.1.1		6	1.1	even	1	trivial
2401.2.a.d.1.1		6	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2401 = 7^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2401.a (trivial)

\(f(q)\)	\(=\)	\(q-1.91115 q^{2} -1.73068 q^{3} +1.65248 q^{4} +2.12712 q^{5} +3.30759 q^{6} +0.664166 q^{8} -0.00473965 q^{9} +O(q^{10})\)	\(q-1.91115 q^{2} -1.73068 q^{3} +1.65248 q^{4} +2.12712 q^{5} +3.30759 q^{6} +0.664166 q^{8} -0.00473965 q^{9} -4.06524 q^{10} +4.67354 q^{11} -2.85991 q^{12} +0.977662 q^{13} -3.68137 q^{15} -4.57427 q^{16} +2.08424 q^{17} +0.00905815 q^{18} -5.29116 q^{19} +3.51502 q^{20} -8.93181 q^{22} -3.77960 q^{23} -1.14946 q^{24} -0.475353 q^{25} -1.86845 q^{26} +5.20025 q^{27} +5.54823 q^{29} +7.03564 q^{30} -10.1646 q^{31} +7.41377 q^{32} -8.08841 q^{33} -3.98328 q^{34} -0.00783216 q^{36} -1.72678 q^{37} +10.1122 q^{38} -1.69202 q^{39} +1.41276 q^{40} -3.55006 q^{41} -10.0503 q^{43} +7.72292 q^{44} -0.0100818 q^{45} +7.22336 q^{46} -4.32864 q^{47} +7.91661 q^{48} +0.908468 q^{50} -3.60716 q^{51} +1.61556 q^{52} -5.17192 q^{53} -9.93843 q^{54} +9.94119 q^{55} +9.15731 q^{57} -10.6035 q^{58} -5.20025 q^{59} -6.08338 q^{60} +4.46701 q^{61} +19.4261 q^{62} -5.02025 q^{64} +2.07961 q^{65} +15.4581 q^{66} +0.482407 q^{67} +3.44416 q^{68} +6.54128 q^{69} +13.3901 q^{71} -0.00314791 q^{72} +5.96686 q^{73} +3.30012 q^{74} +0.822684 q^{75} -8.74352 q^{76} +3.23370 q^{78} +5.75814 q^{79} -9.73004 q^{80} -8.98576 q^{81} +6.78469 q^{82} +2.95977 q^{83} +4.43343 q^{85} +19.2076 q^{86} -9.60223 q^{87} +3.10401 q^{88} -11.2841 q^{89} +0.0192678 q^{90} -6.24570 q^{92} +17.5917 q^{93} +8.27266 q^{94} -11.2549 q^{95} -12.8309 q^{96} -1.76250 q^{97} -0.0221509 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{2} - 7 q^{3} + q^{4} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) 6 * q - q^2 - 7 * q^3 + q^4 + 3 * q^8 + 3 * q^9	\( 6 q - q^{2} - 7 q^{3} + q^{4} + 3 q^{8} + 3 q^{9} + 10 q^{11} - 7 q^{13} - 7 q^{15} - 13 q^{16} + 10 q^{18} - 7 q^{19} - 7 q^{20} - 4 q^{22} - q^{23} - 7 q^{24} - 2 q^{25} - 7 q^{27} + 12 q^{29} + 21 q^{30} - 7 q^{31} - 5 q^{32} - 14 q^{33} - 3 q^{36} + 13 q^{37} - 7 q^{38} - 7 q^{40} - 14 q^{41} + 19 q^{43} + 18 q^{44} + 14 q^{45} - q^{46} - 21 q^{47} - 23 q^{50} - 14 q^{51} - 18 q^{53} - 21 q^{54} - 7 q^{55} + 21 q^{57} - 16 q^{58} + 7 q^{59} + 14 q^{60} - 28 q^{61} + 21 q^{62} - 7 q^{64} + 14 q^{65} - 7 q^{66} + 24 q^{67} + 28 q^{68} + 7 q^{69} + 33 q^{71} - 2 q^{72} + 7 q^{73} - 15 q^{74} + 14 q^{75} - 7 q^{76} + 7 q^{78} - 8 q^{79} + 21 q^{80} - 2 q^{81} - 21 q^{82} - 21 q^{83} - 28 q^{85} + 12 q^{86} - 49 q^{87} + 12 q^{88} - 21 q^{89} - 35 q^{90} - 6 q^{92} + 35 q^{93} + 7 q^{94} - 35 q^{95} + 21 q^{96} - 14 q^{97} - 9 q^{99}+O(q^{100}) \) 6 * q - q^2 - 7 * q^3 + q^4 + 3 * q^8 + 3 * q^9 + 10 * q^11 - 7 * q^13 - 7 * q^15 - 13 * q^16 + 10 * q^18 - 7 * q^19 - 7 * q^20 - 4 * q^22 - q^23 - 7 * q^24 - 2 * q^25 - 7 * q^27 + 12 * q^29 + 21 * q^30 - 7 * q^31 - 5 * q^32 - 14 * q^33 - 3 * q^36 + 13 * q^37 - 7 * q^38 - 7 * q^40 - 14 * q^41 + 19 * q^43 + 18 * q^44 + 14 * q^45 - q^46 - 21 * q^47 - 23 * q^50 - 14 * q^51 - 18 * q^53 - 21 * q^54 - 7 * q^55 + 21 * q^57 - 16 * q^58 + 7 * q^59 + 14 * q^60 - 28 * q^61 + 21 * q^62 - 7 * q^64 + 14 * q^65 - 7 * q^66 + 24 * q^67 + 28 * q^68 + 7 * q^69 + 33 * q^71 - 2 * q^72 + 7 * q^73 - 15 * q^74 + 14 * q^75 - 7 * q^76 + 7 * q^78 - 8 * q^79 + 21 * q^80 - 2 * q^81 - 21 * q^82 - 21 * q^83 - 28 * q^85 + 12 * q^86 - 49 * q^87 + 12 * q^88 - 21 * q^89 - 35 * q^90 - 6 * q^92 + 35 * q^93 + 7 * q^94 - 35 * q^95 + 21 * q^96 - 14 * q^97 - 9 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more