LMFDB - Embedded newform 946.2.a.k.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [946,2,Mod(1,946)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(946, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("946.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 946 = 2 \cdot 11 \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	946.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:	$7.55384803121$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 11x^{5} + 31x^{4} + 39x^{3} - 91x^{2} - 48x + 64 $ x^7 - 3x^6 - 11x^5 + 31x^4 + 39x^3 - 91x^2 - 48x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.11676$ of defining polynomial
Character	$\chi$	$=$	946.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +2.11676 q^{3} +1.00000 q^{4} +2.64074 q^{5} +2.11676 q^{6} +0.778817 q^{7} +1.00000 q^{8} +1.48068 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +2.11676 q^{3} +1.00000 q^{4} +2.64074 q^{5} +2.11676 q^{6} +0.778817 q^{7} +1.00000 q^{8} +1.48068 q^{9} +2.64074 q^{10} +1.00000 q^{11} +2.11676 q^{12} -4.70137 q^{13} +0.778817 q^{14} +5.58981 q^{15} +1.00000 q^{16} -6.97819 q^{17} +1.48068 q^{18} +2.32047 q^{19} +2.64074 q^{20} +1.64857 q^{21} +1.00000 q^{22} +7.27978 q^{23} +2.11676 q^{24} +1.97349 q^{25} -4.70137 q^{26} -3.21605 q^{27} +0.778817 q^{28} -7.57509 q^{29} +5.58981 q^{30} -1.75230 q^{31} +1.00000 q^{32} +2.11676 q^{33} -6.97819 q^{34} +2.05665 q^{35} +1.48068 q^{36} -1.48870 q^{37} +2.32047 q^{38} -9.95168 q^{39} +2.64074 q^{40} -4.84309 q^{41} +1.64857 q^{42} +1.00000 q^{43} +1.00000 q^{44} +3.91007 q^{45} +7.27978 q^{46} -0.194669 q^{47} +2.11676 q^{48} -6.39344 q^{49} +1.97349 q^{50} -14.7712 q^{51} -4.70137 q^{52} +4.42475 q^{53} -3.21605 q^{54} +2.64074 q^{55} +0.778817 q^{56} +4.91188 q^{57} -7.57509 q^{58} +4.11453 q^{59} +5.58981 q^{60} +9.48065 q^{61} -1.75230 q^{62} +1.15317 q^{63} +1.00000 q^{64} -12.4151 q^{65} +2.11676 q^{66} +1.45470 q^{67} -6.97819 q^{68} +15.4095 q^{69} +2.05665 q^{70} +3.99986 q^{71} +1.48068 q^{72} +16.9297 q^{73} -1.48870 q^{74} +4.17740 q^{75} +2.32047 q^{76} +0.778817 q^{77} -9.95168 q^{78} -8.62636 q^{79} +2.64074 q^{80} -11.2496 q^{81} -4.84309 q^{82} -6.33315 q^{83} +1.64857 q^{84} -18.4276 q^{85} +1.00000 q^{86} -16.0346 q^{87} +1.00000 q^{88} +6.50010 q^{89} +3.91007 q^{90} -3.66151 q^{91} +7.27978 q^{92} -3.70921 q^{93} -0.194669 q^{94} +6.12775 q^{95} +2.11676 q^{96} +14.6459 q^{97} -6.39344 q^{98} +1.48068 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 7 q^{2} + 3 q^{3} + 7 q^{4} + 8 q^{5} + 3 q^{6} - 2 q^{7} + 7 q^{8} + 10 q^{9}+O(q^{10}) $ 7 * q + 7 * q^2 + 3 * q^3 + 7 * q^4 + 8 * q^5 + 3 * q^6 - 2 * q^7 + 7 * q^8 + 10 * q^9	\( 7 q + 7 q^{2} + 3 q^{3} + 7 q^{4} + 8 q^{5} + 3 q^{6} - 2 q^{7} + 7 q^{8} + 10 q^{9} + 8 q^{10} + 7 q^{11} + 3 q^{12} + 4 q^{13} - 2 q^{14} + 2 q^{15} + 7 q^{16} + 10 q^{17} + 10 q^{18} - 2 q^{19} + 8 q^{20} + 7 q^{22} - 4 q^{23} + 3 q^{24} + 11 q^{25} + 4 q^{26} + 15 q^{27} - 2 q^{28} + 5 q^{29} + 2 q^{30} - 2 q^{31} + 7 q^{32} + 3 q^{33} + 10 q^{34} - 10 q^{35} + 10 q^{36} + 6 q^{37} - 2 q^{38} - 8 q^{39} + 8 q^{40} + 4 q^{41} + 7 q^{43} + 7 q^{44} - 2 q^{45} - 4 q^{46} - 6 q^{47} + 3 q^{48} + 31 q^{49} + 11 q^{50} - 14 q^{51} + 4 q^{52} + q^{53} + 15 q^{54} + 8 q^{55} - 2 q^{56} - 30 q^{57} + 5 q^{58} - 4 q^{59} + 2 q^{60} + 9 q^{61} - 2 q^{62} - 24 q^{63} + 7 q^{64} + 18 q^{65} + 3 q^{66} - 6 q^{67} + 10 q^{68} + 2 q^{69} - 10 q^{70} + 10 q^{72} + 13 q^{73} + 6 q^{74} - 9 q^{75} - 2 q^{76} - 2 q^{77} - 8 q^{78} - 31 q^{79} + 8 q^{80} - 25 q^{81} + 4 q^{82} - 11 q^{83} - 24 q^{85} + 7 q^{86} - 13 q^{87} + 7 q^{88} + 10 q^{89} - 2 q^{90} - 12 q^{91} - 4 q^{92} + 12 q^{93} - 6 q^{94} - 18 q^{95} + 3 q^{96} + 23 q^{97} + 31 q^{98} + 10 q^{99}+O(q^{100}) \) 7 * q + 7 * q^2 + 3 * q^3 + 7 * q^4 + 8 * q^5 + 3 * q^6 - 2 * q^7 + 7 * q^8 + 10 * q^9 + 8 * q^10 + 7 * q^11 + 3 * q^12 + 4 * q^13 - 2 * q^14 + 2 * q^15 + 7 * q^16 + 10 * q^17 + 10 * q^18 - 2 * q^19 + 8 * q^20 + 7 * q^22 - 4 * q^23 + 3 * q^24 + 11 * q^25 + 4 * q^26 + 15 * q^27 - 2 * q^28 + 5 * q^29 + 2 * q^30 - 2 * q^31 + 7 * q^32 + 3 * q^33 + 10 * q^34 - 10 * q^35 + 10 * q^36 + 6 * q^37 - 2 * q^38 - 8 * q^39 + 8 * q^40 + 4 * q^41 + 7 * q^43 + 7 * q^44 - 2 * q^45 - 4 * q^46 - 6 * q^47 + 3 * q^48 + 31 * q^49 + 11 * q^50 - 14 * q^51 + 4 * q^52 + q^53 + 15 * q^54 + 8 * q^55 - 2 * q^56 - 30 * q^57 + 5 * q^58 - 4 * q^59 + 2 * q^60 + 9 * q^61 - 2 * q^62 - 24 * q^63 + 7 * q^64 + 18 * q^65 + 3 * q^66 - 6 * q^67 + 10 * q^68 + 2 * q^69 - 10 * q^70 + 10 * q^72 + 13 * q^73 + 6 * q^74 - 9 * q^75 - 2 * q^76 - 2 * q^77 - 8 * q^78 - 31 * q^79 + 8 * q^80 - 25 * q^81 + 4 * q^82 - 11 * q^83 - 24 * q^85 + 7 * q^86 - 13 * q^87 + 7 * q^88 + 10 * q^89 - 2 * q^90 - 12 * q^91 - 4 * q^92 + 12 * q^93 - 6 * q^94 - 18 * q^95 + 3 * q^96 + 23 * q^97 + 31 * q^98 + 10 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	2.11676	1.22211	0.611056	−	0.791587i	$-0.290745\pi$
$3$	2.11676	1.22211	0.611056	+	0.791587i	$0.290745\pi$
$4$	1.00000	0.500000
$5$	2.64074	1.18097	0.590486	−	0.807048i	$-0.298936\pi$
$5$	2.64074	1.18097	0.590486	+	0.807048i	$0.298936\pi$
$6$	2.11676	0.864164
$7$	0.778817	0.294365	0.147183	−	0.989109i	$-0.452980\pi$
$7$	0.778817	0.294365	0.147183	+	0.989109i	$0.452980\pi$
$8$	1.00000	0.353553
$9$	1.48068	0.493558
$10$	2.64074	0.835074
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	2.11676	0.611056
$13$	−4.70137	−1.30393	−0.651963	−	0.758251i	$-0.726054\pi$
$13$	−4.70137	−1.30393	−0.651963	+	0.758251i	$0.726054\pi$
$14$	0.778817	0.208148
$15$	5.58981	1.44328
$16$	1.00000	0.250000
$17$	−6.97819	−1.69246	−0.846230	−	0.532817i	$-0.821133\pi$
$17$	−6.97819	−1.69246	−0.846230	+	0.532817i	$0.821133\pi$
$18$	1.48068	0.348998
$19$	2.32047	0.532352	0.266176	−	0.963924i	$-0.414240\pi$
$19$	2.32047	0.532352	0.266176	+	0.963924i	$0.414240\pi$
$20$	2.64074	0.590486
$21$	1.64857	0.359747
$22$	1.00000	0.213201
$23$	7.27978	1.51794	0.758969	−	0.651126i	$-0.225703\pi$
$23$	7.27978	1.51794	0.758969	+	0.651126i	$0.225703\pi$
$24$	2.11676	0.432082
$25$	1.97349	0.394697
$26$	−4.70137	−0.922015
$27$	−3.21605	−0.618929
$28$	0.778817	0.147183
$29$	−7.57509	−1.40666	−0.703329	−	0.710864i	$-0.748304\pi$
$29$	−7.57509	−1.40666	−0.703329	+	0.710864i	$0.748304\pi$
$30$	5.58981	1.02055
$31$	−1.75230	−0.314723	−0.157361	−	0.987541i	$-0.550299\pi$
$31$	−1.75230	−0.314723	−0.157361	+	0.987541i	$0.550299\pi$
$32$	1.00000	0.176777
$33$	2.11676	0.368481
$34$	−6.97819	−1.19675
$35$	2.05665	0.347637
$36$	1.48068	0.246779
$37$	−1.48870	−0.244741	−0.122371	−	0.992484i	$-0.539050\pi$
$37$	−1.48870	−0.244741	−0.122371	+	0.992484i	$0.539050\pi$
$38$	2.32047	0.376430
$39$	−9.95168	−1.59354
$40$	2.64074	0.417537
$41$	−4.84309	−0.756364	−0.378182	−	0.925731i	$-0.623451\pi$
$41$	−4.84309	−0.756364	−0.378182	+	0.925731i	$0.623451\pi$
$42$	1.64857	0.254380
$43$	1.00000	0.152499
$43$	1.00000	0.152499
$44$	1.00000	0.150756
$45$	3.91007	0.582879
$46$	7.27978	1.07334
$47$	−0.194669	−0.0283954	−0.0141977	−	0.999899i	$-0.504519\pi$
$47$	−0.194669	−0.0283954	−0.0141977	+	0.999899i	$0.504519\pi$
$48$	2.11676	0.305528
$49$	−6.39344	−0.913349
$50$	1.97349	0.279093
$51$	−14.7712	−2.06838
$52$	−4.70137	−0.651963
$53$	4.42475	0.607786	0.303893	−	0.952706i	$-0.401714\pi$
$53$	4.42475	0.607786	0.303893	+	0.952706i	$0.401714\pi$
$54$	−3.21605	−0.437649
$55$	2.64074	0.356077
$56$	0.778817	0.104074
$57$	4.91188	0.650594
$58$	−7.57509	−0.994658
$59$	4.11453	0.535666	0.267833	−	0.963465i	$-0.413693\pi$
$59$	4.11453	0.535666	0.267833	+	0.963465i	$0.413693\pi$
$60$	5.58981	0.721641
$61$	9.48065	1.21387	0.606937	−	0.794750i	$-0.292398\pi$
$61$	9.48065	1.21387	0.606937	+	0.794750i	$0.292398\pi$
$62$	−1.75230	−0.222543
$63$	1.15317	0.145286
$64$	1.00000	0.125000
$65$	−12.4151	−1.53990
$66$	2.11676	0.260555
$67$	1.45470	0.177720	0.0888602	−	0.996044i	$-0.471678\pi$
$67$	1.45470	0.177720	0.0888602	+	0.996044i	$0.471678\pi$
$68$	−6.97819	−0.846230
$69$	15.4095	1.85509
$70$	2.05665	0.245817
$71$	3.99986	0.474695	0.237348	−	0.971425i	$-0.423722\pi$
$71$	3.99986	0.474695	0.237348	+	0.971425i	$0.423722\pi$
$72$	1.48068	0.174499
$73$	16.9297	1.98147	0.990734	−	0.135813i	$-0.0433646\pi$
$73$	16.9297	1.98147	0.990734	+	0.135813i	$0.0433646\pi$
$74$	−1.48870	−0.173058
$75$	4.17740	0.482364
$76$	2.32047	0.266176
$77$	0.778817	0.0887544
$78$	−9.95168	−1.12681
$79$	−8.62636	−0.970542	−0.485271	−	0.874364i	$-0.661279\pi$
$79$	−8.62636	−0.970542	−0.485271	+	0.874364i	$0.661279\pi$
$80$	2.64074	0.295243
$81$	−11.2496	−1.24996
$82$	−4.84309	−0.534830
$83$	−6.33315	−0.695153	−0.347577	−	0.937652i	$-0.612995\pi$
$83$	−6.33315	−0.695153	−0.347577	+	0.937652i	$0.612995\pi$
$84$	1.64857	0.179874
$85$	−18.4276	−1.99875
$86$	1.00000	0.107833
$87$	−16.0346	−1.71909
$88$	1.00000	0.106600
$89$	6.50010	0.689009	0.344505	−	0.938785i	$-0.388047\pi$
$89$	6.50010	0.689009	0.344505	+	0.938785i	$0.388047\pi$
$90$	3.91007	0.412158
$91$	−3.66151	−0.383830
$92$	7.27978	0.758969
$93$	−3.70921	−0.384627
$94$	−0.194669	−0.0200786
$95$	6.12775	0.628694
$96$	2.11676	0.216041
$97$	14.6459	1.48707	0.743533	−	0.668699i	$-0.233148\pi$
$97$	14.6459	1.48707	0.743533	+	0.668699i	$0.233148\pi$
$98$	−6.39344	−0.645835
$99$	1.48068	0.148813
$100$	1.97349	0.197349
$101$	13.4412	1.33745	0.668723	−	0.743511i	$-0.266841\pi$
$101$	13.4412	1.33745	0.668723	+	0.743511i	$0.266841\pi$
$102$	−14.7712	−1.46256
$103$	−10.9090	−1.07490	−0.537450	−	0.843296i	$-0.680612\pi$
$103$	−10.9090	−1.07490	−0.537450	+	0.843296i	$0.680612\pi$
$104$	−4.70137	−0.461008
$105$	4.35344	0.424852
$106$	4.42475	0.429769
$107$	−18.5276	−1.79114	−0.895568	−	0.444925i	$-0.853230\pi$
$107$	−18.5276	−1.79114	−0.895568	+	0.444925i	$0.853230\pi$
$108$	−3.21605	−0.309464
$109$	15.9309	1.52590	0.762951	−	0.646456i	$-0.223750\pi$
$109$	15.9309	1.52590	0.762951	+	0.646456i	$0.223750\pi$
$110$	2.64074	0.251784
$111$	−3.15123	−0.299101
$112$	0.778817	0.0735913
$113$	−4.84706	−0.455973	−0.227987	−	0.973664i	$-0.573214\pi$
$113$	−4.84706	−0.455973	−0.227987	+	0.973664i	$0.573214\pi$
$114$	4.91188	0.460040
$115$	19.2240	1.79264
$116$	−7.57509	−0.703329
$117$	−6.96121	−0.643564
$118$	4.11453	0.378773
$119$	−5.43474	−0.498201
$120$	5.58981	0.510277
$121$	1.00000	0.0909091
$122$	9.48065	0.858338
$123$	−10.2517	−0.924362
$124$	−1.75230	−0.157361
$125$	−7.99222	−0.714846
$126$	1.15317	0.102733
$127$	−10.6955	−0.949074	−0.474537	−	0.880236i	$-0.657384\pi$
$127$	−10.6955	−0.949074	−0.474537	+	0.880236i	$0.657384\pi$
$128$	1.00000	0.0883883
$129$	2.11676	0.186370
$130$	−12.4151	−1.08887
$131$	19.2913	1.68549	0.842744	−	0.538314i	$-0.180939\pi$
$131$	19.2913	1.68549	0.842744	+	0.538314i	$0.180939\pi$
$132$	2.11676	0.184240
$133$	1.80722	0.156706
$134$	1.45470	0.125667
$135$	−8.49273	−0.730938
$136$	−6.97819	−0.598375
$137$	15.8027	1.35012	0.675059	−	0.737763i	$-0.264118\pi$
$137$	15.8027	1.35012	0.675059	+	0.737763i	$0.264118\pi$
$138$	15.4095	1.31175
$139$	15.5620	1.31995	0.659976	−	0.751287i	$-0.270567\pi$
$139$	15.5620	1.31995	0.659976	+	0.751287i	$0.270567\pi$
$140$	2.05665	0.173819
$141$	−0.412068	−0.0347024
$142$	3.99986	0.335660
$143$	−4.70137	−0.393149
$144$	1.48068	0.123390
$145$	−20.0038	−1.66123
$146$	16.9297	1.40111
$147$	−13.5334	−1.11622
$148$	−1.48870	−0.122371
$149$	−8.30292	−0.680201	−0.340101	−	0.940389i	$-0.610461\pi$
$149$	−8.30292	−0.680201	−0.340101	+	0.940389i	$0.610461\pi$
$150$	4.17740	0.341083
$151$	2.33738	0.190213	0.0951066	−	0.995467i	$-0.469681\pi$
$151$	2.33738	0.190213	0.0951066	+	0.995467i	$0.469681\pi$
$152$	2.32047	0.188215
$153$	−10.3324	−0.835328
$154$	0.778817	0.0627589
$155$	−4.62737	−0.371679
$156$	−9.95168	−0.796772
$157$	−3.17011	−0.253002	−0.126501	−	0.991966i	$-0.540375\pi$
$157$	−3.17011	−0.253002	−0.126501	+	0.991966i	$0.540375\pi$
$158$	−8.62636	−0.686277
$159$	9.36613	0.742782
$160$	2.64074	0.208769
$161$	5.66962	0.446828
$162$	−11.2496	−0.883854
$163$	−13.0050	−1.01863	−0.509316	−	0.860579i	$-0.670102\pi$
$163$	−13.0050	−1.01863	−0.509316	+	0.860579i	$0.670102\pi$
$164$	−4.84309	−0.378182
$165$	5.58981	0.435166
$166$	−6.33315	−0.491548
$167$	11.7433	0.908724	0.454362	−	0.890817i	$-0.349867\pi$
$167$	11.7433	0.908724	0.454362	+	0.890817i	$0.349867\pi$
$168$	1.64857	0.127190
$169$	9.10290	0.700223
$170$	−18.4276	−1.41333
$171$	3.43586	0.262747
$172$	1.00000	0.0762493
$173$	−21.3985	−1.62690	−0.813450	−	0.581635i	$-0.802413\pi$
$173$	−21.3985	−1.62690	−0.813450	+	0.581635i	$0.802413\pi$
$174$	−16.0346	−1.21558
$175$	1.53698	0.116185
$176$	1.00000	0.0753778
$177$	8.70946	0.654643
$178$	6.50010	0.487203
$179$	12.5716	0.939646	0.469823	−	0.882761i	$-0.344318\pi$
$179$	12.5716	0.939646	0.469823	+	0.882761i	$0.344318\pi$
$180$	3.91007	0.291440
$181$	−11.8654	−0.881948	−0.440974	−	0.897520i	$-0.645367\pi$
$181$	−11.8654	−0.881948	−0.440974	+	0.897520i	$0.645367\pi$
$182$	−3.66151	−0.271409
$183$	20.0683	1.48349
$184$	7.27978	0.536672
$185$	−3.93127	−0.289033
$186$	−3.70921	−0.271972
$187$	−6.97819	−0.510296
$188$	−0.194669	−0.0141977
$189$	−2.50471	−0.182191
$190$	6.12775	0.444553
$191$	−14.3550	−1.03869	−0.519345	−	0.854564i	$-0.673824\pi$
$191$	−14.3550	−1.03869	−0.519345	+	0.854564i	$0.673824\pi$
$192$	2.11676	0.152764
$193$	−0.376050	−0.0270687	−0.0135343	−	0.999908i	$-0.504308\pi$
$193$	−0.376050	−0.0270687	−0.0135343	+	0.999908i	$0.504308\pi$
$194$	14.6459	1.05151
$195$	−26.2798	−1.88193
$196$	−6.39344	−0.456675
$197$	1.38133	0.0984157	0.0492079	−	0.998789i	$-0.484330\pi$
$197$	1.38133	0.0984157	0.0492079	+	0.998789i	$0.484330\pi$
$198$	1.48068	0.105227
$199$	−13.9603	−0.989621	−0.494811	−	0.869001i	$-0.664763\pi$
$199$	−13.9603	−0.989621	−0.494811	+	0.869001i	$0.664763\pi$
$200$	1.97349	0.139547
$201$	3.07926	0.217194
$202$	13.4412	0.945717
$203$	−5.89961	−0.414071
$204$	−14.7712	−1.03419
$205$	−12.7893	−0.893245
$206$	−10.9090	−0.760069
$207$	10.7790	0.749191
$208$	−4.70137	−0.325982
$209$	2.32047	0.160510
$210$	4.35344	0.300416
$211$	−8.52787	−0.587083	−0.293541	−	0.955946i	$-0.594834\pi$
$211$	−8.52787	−0.587083	−0.293541	+	0.955946i	$0.594834\pi$
$212$	4.42475	0.303893
$213$	8.46674	0.580131
$214$	−18.5276	−1.26652
$215$	2.64074	0.180097
$216$	−3.21605	−0.218824
$217$	−1.36472	−0.0926435
$218$	15.9309	1.07898
$219$	35.8361	2.42158
$220$	2.64074	0.178038
$221$	32.8071	2.20684
$222$	−3.15123	−0.211497
$223$	16.9526	1.13523	0.567616	−	0.823293i	$-0.307866\pi$
$223$	16.9526	1.13523	0.567616	+	0.823293i	$0.307866\pi$
$224$	0.778817	0.0520369
$225$	2.92209	0.194806
$226$	−4.84706	−0.322422
$227$	2.81586	0.186895	0.0934477	−	0.995624i	$-0.470211\pi$
$227$	2.81586	0.186895	0.0934477	+	0.995624i	$0.470211\pi$
$228$	4.91188	0.325297
$229$	−8.80344	−0.581748	−0.290874	−	0.956761i	$-0.593946\pi$
$229$	−8.80344	−0.581748	−0.290874	+	0.956761i	$0.593946\pi$
$230$	19.2240	1.26759
$231$	1.64857	0.108468
$232$	−7.57509	−0.497329
$233$	7.14193	0.467883	0.233942	−	0.972251i	$-0.424838\pi$
$233$	7.14193	0.467883	0.233942	+	0.972251i	$0.424838\pi$
$234$	−6.96121	−0.455068
$235$	−0.514069	−0.0335342
$236$	4.11453	0.267833
$237$	−18.2599	−1.18611
$238$	−5.43474	−0.352282
$239$	11.7651	0.761022	0.380511	−	0.924776i	$-0.375748\pi$
$239$	11.7651	0.761022	0.380511	+	0.924776i	$0.375748\pi$
$240$	5.58981	0.360820
$241$	−28.9037	−1.86185	−0.930924	−	0.365212i	$-0.880997\pi$
$241$	−28.9037	−1.86185	−0.930924	+	0.365212i	$0.880997\pi$
$242$	1.00000	0.0642824
$243$	−14.1646	−0.908661
$244$	9.48065	0.606937
$245$	−16.8834	−1.07864
$246$	−10.2517	−0.653622
$247$	−10.9094	−0.694148
$248$	−1.75230	−0.111271
$249$	−13.4058	−0.849555
$250$	−7.99222	−0.505473
$251$	−9.75093	−0.615473	−0.307737	−	0.951472i	$-0.599572\pi$
$251$	−9.75093	−0.615473	−0.307737	+	0.951472i	$0.599572\pi$
$252$	1.15317	0.0726432
$253$	7.27978	0.457676
$254$	−10.6955	−0.671096
$255$	−39.0067	−2.44270
$256$	1.00000	0.0625000
$257$	−18.3994	−1.14773	−0.573863	−	0.818951i	$-0.694556\pi$
$257$	−18.3994	−1.14773	−0.573863	+	0.818951i	$0.694556\pi$
$258$	2.11676	0.131784
$259$	−1.15943	−0.0720433
$260$	−12.4151	−0.769951
$261$	−11.2162	−0.694268
$262$	19.2913	1.19182
$263$	3.95265	0.243731	0.121865	−	0.992547i	$-0.461112\pi$
$263$	3.95265	0.243731	0.121865	+	0.992547i	$0.461112\pi$
$264$	2.11676	0.130278
$265$	11.6846	0.717778
$266$	1.80722	0.110808
$267$	13.7592	0.842047
$268$	1.45470	0.0888602
$269$	10.6325	0.648275	0.324138	−	0.946010i	$-0.394926\pi$
$269$	10.6325	0.648275	0.324138	+	0.946010i	$0.394926\pi$
$270$	−8.49273	−0.516851
$271$	4.70777	0.285977	0.142988	−	0.989724i	$-0.454329\pi$
$271$	4.70777	0.285977	0.142988	+	0.989724i	$0.454329\pi$
$272$	−6.97819	−0.423115
$273$	−7.75054	−0.469084
$274$	15.8027	0.954678
$275$	1.97349	0.119006
$276$	15.4095	0.927546
$277$	−10.8591	−0.652463	−0.326232	−	0.945290i	$-0.605779\pi$
$277$	−10.8591	−0.652463	−0.326232	+	0.945290i	$0.605779\pi$
$278$	15.5620	0.933347
$279$	−2.59459	−0.155334
$280$	2.05665	0.122908
$281$	−9.18273	−0.547796	−0.273898	−	0.961759i	$-0.588313\pi$
$281$	−9.18273	−0.547796	−0.273898	+	0.961759i	$0.588313\pi$
$282$	−0.412068	−0.0245383
$283$	9.70850	0.577110	0.288555	−	0.957463i	$-0.406825\pi$
$283$	9.70850	0.577110	0.288555	+	0.957463i	$0.406825\pi$
$284$	3.99986	0.237348
$285$	12.9710	0.768334
$286$	−4.70137	−0.277998
$287$	−3.77188	−0.222647
$288$	1.48068	0.0872496
$289$	31.6952	1.86442
$290$	−20.0038	−1.17466
$291$	31.0019	1.81736
$292$	16.9297	0.990734
$293$	11.5405	0.674202	0.337101	−	0.941469i	$-0.390554\pi$
$293$	11.5405	0.674202	0.337101	+	0.941469i	$0.390554\pi$
$294$	−13.5334	−0.789283
$295$	10.8654	0.632607
$296$	−1.48870	−0.0865291
$297$	−3.21605	−0.186614
$298$	−8.30292	−0.480975
$299$	−34.2250	−1.97928
$300$	4.17740	0.241182
$301$	0.778817	0.0448903
$302$	2.33738	0.134501
$303$	28.4517	1.63451
$304$	2.32047	0.133088
$305$	25.0359	1.43355
$306$	−10.3324	−0.590666
$307$	−6.12348	−0.349486	−0.174743	−	0.984614i	$-0.555909\pi$
$307$	−6.12348	−0.349486	−0.174743	+	0.984614i	$0.555909\pi$
$308$	0.778817	0.0443772
$309$	−23.0918	−1.31365
$310$	−4.62737	−0.262817
$311$	25.6979	1.45719	0.728597	−	0.684943i	$-0.240173\pi$
$311$	25.6979	1.45719	0.728597	+	0.684943i	$0.240173\pi$
$312$	−9.95168	−0.563403
$313$	30.0966	1.70116	0.850579	−	0.525847i	$-0.176252\pi$
$313$	30.0966	1.70116	0.850579	+	0.525847i	$0.176252\pi$
$314$	−3.17011	−0.178899
$315$	3.04523	0.171579
$316$	−8.62636	−0.485271
$317$	7.31933	0.411094	0.205547	−	0.978647i	$-0.434103\pi$
$317$	7.31933	0.411094	0.205547	+	0.978647i	$0.434103\pi$
$318$	9.36613	0.525226
$319$	−7.57509	−0.424123
$320$	2.64074	0.147622
$321$	−39.2186	−2.18897
$322$	5.66962	0.315955
$323$	−16.1927	−0.900985
$324$	−11.2496	−0.624979
$325$	−9.27809	−0.514656
$326$	−13.0050	−0.720282
$327$	33.7219	1.86482
$328$	−4.84309	−0.267415
$329$	−0.151612	−0.00835861
$330$	5.58981	0.307709
$331$	−16.2061	−0.890768	−0.445384	−	0.895340i	$-0.646933\pi$
$331$	−16.2061	−0.890768	−0.445384	+	0.895340i	$0.646933\pi$
$332$	−6.33315	−0.347577
$333$	−2.20429	−0.120794
$334$	11.7433	0.642565
$335$	3.84149	0.209883
$336$	1.64857	0.0899368
$337$	−10.0611	−0.548063	−0.274031	−	0.961721i	$-0.588357\pi$
$337$	−10.0611	−0.548063	−0.274031	+	0.961721i	$0.588357\pi$
$338$	9.10290	0.495133
$339$	−10.2601	−0.557251
$340$	−18.4276	−0.999375
$341$	−1.75230	−0.0948925
$342$	3.43586	0.185790
$343$	−10.4310	−0.563223
$344$	1.00000	0.0539164
$345$	40.6926	2.19081
$346$	−21.3985	−1.15039
$347$	−25.1456	−1.34989	−0.674945	−	0.737868i	$-0.735832\pi$
$347$	−25.1456	−1.34989	−0.674945	+	0.737868i	$0.735832\pi$
$348$	−16.0346	−0.859547
$349$	−28.0141	−1.49956	−0.749780	−	0.661687i	$-0.769841\pi$
$349$	−28.0141	−1.49956	−0.749780	+	0.661687i	$0.769841\pi$
$350$	1.53698	0.0821553
$351$	15.1198	0.807037
$352$	1.00000	0.0533002
$353$	36.2072	1.92712	0.963558	−	0.267500i	$-0.0861974\pi$
$353$	36.2072	1.92712	0.963558	+	0.267500i	$0.0861974\pi$
$354$	8.70946	0.462903
$355$	10.5626	0.560603
$356$	6.50010	0.344505
$357$	−11.5040	−0.608858
$358$	12.5716	0.664430
$359$	−18.1510	−0.957973	−0.478987	−	0.877822i	$-0.658996\pi$
$359$	−18.1510	−0.957973	−0.478987	+	0.877822i	$0.658996\pi$
$360$	3.91007	0.206079
$361$	−13.6154	−0.716601
$362$	−11.8654	−0.623631
$363$	2.11676	0.111101
$364$	−3.66151	−0.191915
$365$	44.7068	2.34006
$366$	20.0683	1.04899
$367$	−31.1395	−1.62547	−0.812734	−	0.582635i	$-0.802022\pi$
$367$	−31.1395	−1.62547	−0.812734	+	0.582635i	$0.802022\pi$
$368$	7.27978	0.379485
$369$	−7.17105	−0.373310
$370$	−3.93127	−0.204377
$371$	3.44607	0.178911
$372$	−3.70921	−0.192313
$373$	−20.1432	−1.04298	−0.521488	−	0.853259i	$-0.674623\pi$
$373$	−20.1432	−1.04298	−0.521488	+	0.853259i	$0.674623\pi$
$374$	−6.97819	−0.360834
$375$	−16.9176	−0.873622
$376$	−0.194669	−0.0100393
$377$	35.6133	1.83418
$378$	−2.50471	−0.128828
$379$	4.88175	0.250758	0.125379	−	0.992109i	$-0.459985\pi$
$379$	4.88175	0.250758	0.125379	+	0.992109i	$0.459985\pi$
$380$	6.12775	0.314347
$381$	−22.6398	−1.15987
$382$	−14.3550	−0.734465
$383$	2.10839	0.107734	0.0538669	−	0.998548i	$-0.482845\pi$
$383$	2.10839	0.107734	0.0538669	+	0.998548i	$0.482845\pi$
$384$	2.11676	0.108020
$385$	2.05665	0.104817
$386$	−0.376050	−0.0191405
$387$	1.48068	0.0752669
$388$	14.6459	0.743533
$389$	−19.2770	−0.977382	−0.488691	−	0.872457i	$-0.662525\pi$
$389$	−19.2770	−0.977382	−0.488691	+	0.872457i	$0.662525\pi$
$390$	−26.2798	−1.33073
$391$	−50.7997	−2.56905
$392$	−6.39344	−0.322918
$393$	40.8351	2.05986
$394$	1.38133	0.0695904
$395$	−22.7800	−1.14618
$396$	1.48068	0.0744067
$397$	28.0066	1.40561	0.702807	−	0.711381i	$-0.251930\pi$
$397$	28.0066	1.40561	0.702807	+	0.711381i	$0.251930\pi$
$398$	−13.9603	−0.699768
$399$	3.82545	0.191512
$400$	1.97349	0.0986743
$401$	14.9737	0.747750	0.373875	−	0.927479i	$-0.378029\pi$
$401$	14.9737	0.747750	0.373875	+	0.927479i	$0.378029\pi$
$402$	3.07926	0.153580
$403$	8.23823	0.410375
$404$	13.4412	0.668723
$405$	−29.7073	−1.47617
$406$	−5.89961	−0.292793
$407$	−1.48870	−0.0737923
$408$	−14.7712	−0.731282
$409$	0.508131	0.0251255	0.0125627	−	0.999921i	$-0.496001\pi$
$409$	0.508131	0.0251255	0.0125627	+	0.999921i	$0.496001\pi$
$410$	−12.7893	−0.631620
$411$	33.4506	1.65000
$412$	−10.9090	−0.537450
$413$	3.20446	0.157681
$414$	10.7790	0.529758
$415$	−16.7242	−0.820957
$416$	−4.70137	−0.230504
$417$	32.9410	1.61313
$418$	2.32047	0.113498
$419$	17.0240	0.831677	0.415838	−	0.909439i	$-0.363488\pi$
$419$	17.0240	0.831677	0.415838	+	0.909439i	$0.363488\pi$
$420$	4.35344	0.212426
$421$	−2.59012	−0.126235	−0.0631174	−	0.998006i	$-0.520104\pi$
$421$	−2.59012	−0.126235	−0.0631174	+	0.998006i	$0.520104\pi$
$422$	−8.52787	−0.415130
$423$	−0.288242	−0.0140148
$424$	4.42475	0.214885
$425$	−13.7714	−0.668009
$426$	8.46674	0.410215
$427$	7.38369	0.357322
$428$	−18.5276	−0.895568
$429$	−9.95168	−0.480472
$430$	2.64074	0.127348
$431$	29.4435	1.41824	0.709121	−	0.705087i	$-0.249092\pi$
$431$	29.4435	1.41824	0.709121	+	0.705087i	$0.249092\pi$
$432$	−3.21605	−0.154732
$433$	−0.274100	−0.0131724	−0.00658620	−	0.999978i	$-0.502096\pi$
$433$	−0.274100	−0.0131724	−0.00658620	+	0.999978i	$0.502096\pi$
$434$	−1.36472	−0.0655088
$435$	−42.3433	−2.03020
$436$	15.9309	0.762951
$437$	16.8925	0.808078
$438$	35.8361	1.71231
$439$	18.3500	0.875800	0.437900	−	0.899024i	$-0.355722\pi$
$439$	18.3500	0.875800	0.437900	+	0.899024i	$0.355722\pi$
$440$	2.64074	0.125892
$441$	−9.46661	−0.450791
$442$	32.8071	1.56047
$443$	23.2359	1.10397	0.551985	−	0.833854i	$-0.313871\pi$
$443$	23.2359	1.10397	0.551985	+	0.833854i	$0.313871\pi$
$444$	−3.15123	−0.149551
$445$	17.1651	0.813702
$446$	16.9526	0.802730
$447$	−17.5753	−0.831282
$448$	0.778817	0.0367956
$449$	10.1233	0.477747	0.238874	−	0.971051i	$-0.423222\pi$
$449$	10.1233	0.477747	0.238874	+	0.971051i	$0.423222\pi$
$450$	2.92209	0.137749
$451$	−4.84309	−0.228052
$452$	−4.84706	−0.227987
$453$	4.94767	0.232462
$454$	2.81586	0.132155
$455$	−9.66908	−0.453293
$456$	4.91188	0.230020
$457$	35.6134	1.66593	0.832963	−	0.553329i	$-0.186643\pi$
$457$	35.6134	1.66593	0.832963	+	0.553329i	$0.186643\pi$
$458$	−8.80344	−0.411358
$459$	22.4422	1.04751
$460$	19.2240	0.896322
$461$	23.3777	1.08881	0.544403	−	0.838824i	$-0.316756\pi$
$461$	23.3777	1.08881	0.544403	+	0.838824i	$0.316756\pi$
$462$	1.64857	0.0766984
$463$	−10.6802	−0.496351	−0.248175	−	0.968715i	$-0.579831\pi$
$463$	−10.6802	−0.496351	−0.248175	+	0.968715i	$0.579831\pi$
$464$	−7.57509	−0.351665
$465$	−9.79503	−0.454234
$466$	7.14193	0.330844
$467$	−24.4626	−1.13199	−0.565996	−	0.824408i	$-0.691508\pi$
$467$	−24.4626	−1.13199	−0.565996	+	0.824408i	$0.691508\pi$
$468$	−6.96121	−0.321782
$469$	1.13295	0.0523147
$470$	−0.514069	−0.0237123
$471$	−6.71036	−0.309197
$472$	4.11453	0.189386
$473$	1.00000	0.0459800
$474$	−18.2599	−0.838707
$475$	4.57941	0.210118
$476$	−5.43474	−0.249101
$477$	6.55161	0.299978
$478$	11.7651	0.538124
$479$	21.3377	0.974945	0.487472	−	0.873138i	$-0.337919\pi$
$479$	21.3377	0.974945	0.487472	+	0.873138i	$0.337919\pi$
$480$	5.58981	0.255139
$481$	6.99895	0.319125
$482$	−28.9037	−1.31653
$483$	12.0012	0.546074
$484$	1.00000	0.0454545
$485$	38.6760	1.75619
$486$	−14.1646	−0.642520
$487$	−5.98402	−0.271162	−0.135581	−	0.990766i	$-0.543290\pi$
$487$	−5.98402	−0.271162	−0.135581	+	0.990766i	$0.543290\pi$
$488$	9.48065	0.429169
$489$	−27.5285	−1.24488
$490$	−16.8834	−0.762714
$491$	37.4263	1.68903	0.844513	−	0.535534i	$-0.179890\pi$
$491$	37.4263	1.68903	0.844513	+	0.535534i	$0.179890\pi$
$492$	−10.2517	−0.462181
$493$	52.8604	2.38071
$494$	−10.9094	−0.490837
$495$	3.91007	0.175745
$496$	−1.75230	−0.0786807
$497$	3.11516	0.139734
$498$	−13.4058	−0.600726
$499$	19.0955	0.854832	0.427416	−	0.904055i	$-0.359424\pi$
$499$	19.0955	0.854832	0.427416	+	0.904055i	$0.359424\pi$
$500$	−7.99222	−0.357423
$501$	24.8578	1.11056
$502$	−9.75093	−0.435205
$503$	−9.76589	−0.435439	−0.217720	−	0.976011i	$-0.569862\pi$
$503$	−9.76589	−0.435439	−0.217720	+	0.976011i	$0.569862\pi$
$504$	1.15317	0.0513665
$505$	35.4946	1.57949
$506$	7.27978	0.323626
$507$	19.2687	0.855752
$508$	−10.6955	−0.474537
$509$	−11.5653	−0.512621	−0.256310	−	0.966595i	$-0.582507\pi$
$509$	−11.5653	−0.512621	−0.256310	+	0.966595i	$0.582507\pi$
$510$	−39.0067	−1.72725
$511$	13.1851	0.583275
$512$	1.00000	0.0441942
$513$	−7.46274	−0.329488
$514$	−18.3994	−0.811565
$515$	−28.8079	−1.26943
$516$	2.11676	0.0931852
$517$	−0.194669	−0.00856153
$518$	−1.15943	−0.0509423
$519$	−45.2956	−1.98825
$520$	−12.4151	−0.544437
$521$	10.0626	0.440849	0.220425	−	0.975404i	$-0.429256\pi$
$521$	10.0626	0.440849	0.220425	+	0.975404i	$0.429256\pi$
$522$	−11.2162	−0.490922
$523$	36.9531	1.61585	0.807923	−	0.589288i	$-0.200592\pi$
$523$	36.9531	1.61585	0.807923	+	0.589288i	$0.200592\pi$
$524$	19.2913	0.842744
$525$	3.25343	0.141991
$526$	3.95265	0.172344
$527$	12.2279	0.532656
$528$	2.11676	0.0921202
$529$	29.9952	1.30414
$530$	11.6846	0.507546
$531$	6.09228	0.264382
$532$	1.80722	0.0783530
$533$	22.7692	0.986243
$534$	13.7592	0.595417
$535$	−48.9266	−2.11528
$536$	1.45470	0.0628336
$537$	26.6111	1.14835
$538$	10.6325	0.458400
$539$	−6.39344	−0.275385
$540$	−8.49273	−0.365469
$541$	−40.8563	−1.75655	−0.878275	−	0.478156i	$-0.841305\pi$
$541$	−40.8563	−1.75655	−0.878275	+	0.478156i	$0.841305\pi$
$542$	4.70777	0.202216
$543$	−25.1162	−1.07784
$544$	−6.97819	−0.299188
$545$	42.0693	1.80205
$546$	−7.75054	−0.331692
$547$	−25.8937	−1.10714	−0.553568	−	0.832804i	$-0.686734\pi$
$547$	−25.8937	−1.10714	−0.553568	+	0.832804i	$0.686734\pi$
$548$	15.8027	0.675059
$549$	14.0378	0.599117
$550$	1.97349	0.0841497
$551$	−17.5778	−0.748838
$552$	15.4095	0.655874
$553$	−6.71836	−0.285694
$554$	−10.8591	−0.461361
$555$	−8.32156	−0.353231
$556$	15.5620	0.659976
$557$	−4.85334	−0.205642	−0.102821	−	0.994700i	$-0.532787\pi$
$557$	−4.85334	−0.205642	−0.102821	+	0.994700i	$0.532787\pi$
$558$	−2.59459	−0.109838
$559$	−4.70137	−0.198847
$560$	2.05665	0.0869093
$561$	−14.7712	−0.623639
$562$	−9.18273	−0.387350
$563$	−18.2459	−0.768973	−0.384487	−	0.923131i	$-0.625622\pi$
$563$	−18.2459	−0.768973	−0.384487	+	0.923131i	$0.625622\pi$
$564$	−0.412068	−0.0173512
$565$	−12.7998	−0.538492
$566$	9.70850	0.408079
$567$	−8.76140	−0.367944
$568$	3.99986	0.167830
$569$	44.7481	1.87594	0.937970	−	0.346717i	$-0.112704\pi$
$569$	44.7481	1.87594	0.937970	+	0.346717i	$0.112704\pi$
$570$	12.9710	0.543294
$571$	31.2368	1.30722	0.653609	−	0.756832i	$-0.273254\pi$
$571$	31.2368	1.30722	0.653609	+	0.756832i	$0.273254\pi$
$572$	−4.70137	−0.196574
$573$	−30.3861	−1.26940
$574$	−3.77188	−0.157435
$575$	14.3665	0.599126
$576$	1.48068	0.0616948
$577$	−36.7163	−1.52852	−0.764259	−	0.644909i	$-0.776895\pi$
$577$	−36.7163	−1.52852	−0.764259	+	0.644909i	$0.776895\pi$
$578$	31.6952	1.31835
$579$	−0.796008	−0.0330810
$580$	−20.0038	−0.830613
$581$	−4.93236	−0.204629
$582$	31.0019	1.28507
$583$	4.42475	0.183254
$584$	16.9297	0.700555
$585$	−18.3827	−0.760031
$586$	11.5405	0.476733
$587$	36.6481	1.51263	0.756315	−	0.654207i	$-0.226998\pi$
$587$	36.6481	1.51263	0.756315	+	0.654207i	$0.226998\pi$
$588$	−13.5334	−0.558108
$589$	−4.06617	−0.167543
$590$	10.8654	0.447320
$591$	2.92395	0.120275
$592$	−1.48870	−0.0611853
$593$	36.1741	1.48549	0.742746	−	0.669574i	$-0.233523\pi$
$593$	36.1741	1.48549	0.742746	+	0.669574i	$0.233523\pi$
$594$	−3.21605	−0.131956
$595$	−14.3517	−0.588362
$596$	−8.30292	−0.340101
$597$	−29.5507	−1.20943
$598$	−34.2250	−1.39956
$599$	−24.3251	−0.993895	−0.496948	−	0.867781i	$-0.665546\pi$
$599$	−24.3251	−0.993895	−0.496948	+	0.867781i	$0.665546\pi$
$600$	4.17740	0.170542
$601$	−6.75224	−0.275430	−0.137715	−	0.990472i	$-0.543976\pi$
$601$	−6.75224	−0.275430	−0.137715	+	0.990472i	$0.543976\pi$
$602$	0.778817	0.0317422
$603$	2.15394	0.0877154
$604$	2.33738	0.0951066
$605$	2.64074	0.107361
$606$	28.4517	1.15577
$607$	22.9946	0.933322	0.466661	−	0.884436i	$-0.345457\pi$
$607$	22.9946	0.933322	0.466661	+	0.884436i	$0.345457\pi$
$608$	2.32047	0.0941075
$609$	−12.4881	−0.506041
$610$	25.0359	1.01367
$611$	0.915211	0.0370255
$612$	−10.3324	−0.417664
$613$	−15.5832	−0.629400	−0.314700	−	0.949191i	$-0.601904\pi$
$613$	−15.5832	−0.629400	−0.314700	+	0.949191i	$0.601904\pi$
$614$	−6.12348	−0.247124
$615$	−27.0719	−1.09165
$616$	0.778817	0.0313794
$617$	24.0054	0.966423	0.483211	−	0.875504i	$-0.339470\pi$
$617$	24.0054	0.966423	0.483211	+	0.875504i	$0.339470\pi$
$618$	−23.0918	−0.928890
$619$	−36.2699	−1.45781	−0.728905	−	0.684615i	$-0.759970\pi$
$619$	−36.2699	−1.45781	−0.728905	+	0.684615i	$0.759970\pi$
$620$	−4.62737	−0.185840
$621$	−23.4121	−0.939496
$622$	25.6979	1.03039
$623$	5.06239	0.202820
$624$	−9.95168	−0.398386
$625$	−30.9728	−1.23891
$626$	30.0966	1.20290
$627$	4.91188	0.196162
$628$	−3.17011	−0.126501
$629$	10.3885	0.414215
$630$	3.04523	0.121325
$631$	−14.8666	−0.591828	−0.295914	−	0.955215i	$-0.595624\pi$
$631$	−14.8666	−0.591828	−0.295914	+	0.955215i	$0.595624\pi$
$632$	−8.62636	−0.343138
$633$	−18.0515	−0.717481
$634$	7.31933	0.290688
$635$	−28.2440	−1.12083
$636$	9.36613	0.371391
$637$	30.0580	1.19094
$638$	−7.57509	−0.299901
$639$	5.92249	0.234290
$640$	2.64074	0.104384
$641$	25.3741	1.00222	0.501108	−	0.865385i	$-0.332926\pi$
$641$	25.3741	1.00222	0.501108	+	0.865385i	$0.332926\pi$
$642$	−39.2186	−1.54783
$643$	24.8652	0.980588	0.490294	−	0.871557i	$-0.336889\pi$
$643$	24.8652	0.980588	0.490294	+	0.871557i	$0.336889\pi$
$644$	5.66962	0.223414
$645$	5.58981	0.220098
$646$	−16.1927	−0.637093
$647$	−29.4014	−1.15589	−0.577945	−	0.816076i	$-0.696145\pi$
$647$	−29.4014	−1.15589	−0.577945	+	0.816076i	$0.696145\pi$
$648$	−11.2496	−0.441927
$649$	4.11453	0.161509
$650$	−9.27809	−0.363917
$651$	−2.88879	−0.113221
$652$	−13.0050	−0.509316
$653$	−28.7249	−1.12409	−0.562046	−	0.827106i	$-0.689986\pi$
$653$	−28.7249	−1.12409	−0.562046	+	0.827106i	$0.689986\pi$
$654$	33.7219	1.31863
$655$	50.9432	1.99052
$656$	−4.84309	−0.189091
$657$	25.0674	0.977971
$658$	−0.151612	−0.00591043
$659$	26.2422	1.02225	0.511125	−	0.859507i	$-0.329229\pi$
$659$	26.2422	1.02225	0.511125	+	0.859507i	$0.329229\pi$
$660$	5.58981	0.217583
$661$	38.5979	1.50128	0.750642	−	0.660709i	$-0.229744\pi$
$661$	38.5979	1.50128	0.750642	+	0.660709i	$0.229744\pi$
$662$	−16.2061	−0.629868
$663$	69.4448	2.69701
$664$	−6.33315	−0.245774
$665$	4.77239	0.185065
$666$	−2.20429	−0.0854143
$667$	−55.1450	−2.13522
$668$	11.7433	0.454362
$669$	35.8847	1.38738
$670$	3.84149	0.148410
$671$	9.48065	0.365997
$672$	1.64857	0.0635949
$673$	−48.2633	−1.86041	−0.930206	−	0.367037i	$-0.880372\pi$
$673$	−48.2633	−1.86041	−0.930206	+	0.367037i	$0.880372\pi$
$674$	−10.0611	−0.387539
$675$	−6.34682	−0.244289
$676$	9.10290	0.350112
$677$	−23.0907	−0.887447	−0.443723	−	0.896164i	$-0.646343\pi$
$677$	−23.0907	−0.887447	−0.443723	+	0.896164i	$0.646343\pi$
$678$	−10.2601	−0.394036
$679$	11.4065	0.437741
$680$	−18.4276	−0.706665
$681$	5.96051	0.228407
$682$	−1.75230	−0.0670991
$683$	20.1118	0.769557	0.384778	−	0.923009i	$-0.374278\pi$
$683$	20.1118	0.769557	0.384778	+	0.923009i	$0.374278\pi$
$684$	3.43586	0.131373
$685$	41.7308	1.59445
$686$	−10.4310	−0.398259
$687$	−18.6348	−0.710961
$688$	1.00000	0.0381246
$689$	−20.8024	−0.792508
$690$	40.6926	1.54914
$691$	−5.46895	−0.208049	−0.104024	−	0.994575i	$-0.533172\pi$
$691$	−5.46895	−0.208049	−0.104024	+	0.994575i	$0.533172\pi$
$692$	−21.3985	−0.813450
$693$	1.15317	0.0438055
$694$	−25.1456	−0.954516
$695$	41.0951	1.55883
$696$	−16.0346	−0.607792
$697$	33.7960	1.28012
$698$	−28.0141	−1.06035
$699$	15.1178	0.571806
$700$	1.53698	0.0580925
$701$	6.10103	0.230433	0.115216	−	0.993340i	$-0.463244\pi$
$701$	6.10103	0.230433	0.115216	+	0.993340i	$0.463244\pi$
$702$	15.1198	0.570661
$703$	−3.45449	−0.130289
$704$	1.00000	0.0376889
$705$	−1.08816	−0.0409825
$706$	36.2072	1.36268
$707$	10.4682	0.393698
$708$	8.70946	0.327322
$709$	−4.76039	−0.178780	−0.0893900	−	0.995997i	$-0.528492\pi$
$709$	−4.76039	−0.178780	−0.0893900	+	0.995997i	$0.528492\pi$
$710$	10.5626	0.396406
$711$	−12.7728	−0.479019
$712$	6.50010	0.243602
$713$	−12.7564	−0.477730
$714$	−11.5040	−0.430528
$715$	−12.4151	−0.464298
$716$	12.5716	0.469823
$717$	24.9039	0.930054
$718$	−18.1510	−0.677389
$719$	17.2673	0.643962	0.321981	−	0.946746i	$-0.395651\pi$
$719$	17.2673	0.643962	0.321981	+	0.946746i	$0.395651\pi$
$720$	3.91007	0.145720
$721$	−8.49615	−0.316413
$722$	−13.6154	−0.506714
$723$	−61.1821	−2.27539
$724$	−11.8654	−0.440974
$725$	−14.9493	−0.555204
$726$	2.11676	0.0785604
$727$	−33.7460	−1.25157	−0.625784	−	0.779996i	$-0.715221\pi$
$727$	−33.7460	−1.25157	−0.625784	+	0.779996i	$0.715221\pi$
$728$	−3.66151	−0.135705
$729$	3.76576	0.139473
$730$	44.7068	1.65467
$731$	−6.97819	−0.258098
$732$	20.0683	0.741745
$733$	−12.7814	−0.472092	−0.236046	−	0.971742i	$-0.575852\pi$
$733$	−12.7814	−0.472092	−0.236046	+	0.971742i	$0.575852\pi$
$734$	−31.1395	−1.14938
$735$	−35.7381	−1.31822
$736$	7.27978	0.268336
$737$	1.45470	0.0535847
$738$	−7.17105	−0.263970
$739$	33.4197	1.22936	0.614681	−	0.788776i	$-0.289285\pi$
$739$	33.4197	1.22936	0.614681	+	0.788776i	$0.289285\pi$
$740$	−3.93127	−0.144516
$741$	−23.0926	−0.848327
$742$	3.44607	0.126509
$743$	−2.09991	−0.0770384	−0.0385192	−	0.999258i	$-0.512264\pi$
$743$	−2.09991	−0.0770384	−0.0385192	+	0.999258i	$0.512264\pi$
$744$	−3.70921	−0.135986
$745$	−21.9258	−0.803299
$746$	−20.1432	−0.737495
$747$	−9.37733	−0.343099
$748$	−6.97819	−0.255148
$749$	−14.4296	−0.527248
$750$	−16.9176	−0.617744
$751$	−51.5506	−1.88111	−0.940554	−	0.339644i	$-0.889693\pi$
$751$	−51.5506	−1.88111	−0.940554	+	0.339644i	$0.889693\pi$
$752$	−0.194669	−0.00709885
$753$	−20.6404	−0.752177
$754$	35.6133	1.29696
$755$	6.17240	0.224637
$756$	−2.50471	−0.0910955
$757$	−21.6663	−0.787477	−0.393738	−	0.919223i	$-0.628818\pi$
$757$	−21.6663	−0.787477	−0.393738	+	0.919223i	$0.628818\pi$
$758$	4.88175	0.177313
$759$	15.4095	0.559331
$760$	6.12775	0.222277
$761$	−18.9574	−0.687207	−0.343603	−	0.939115i	$-0.611648\pi$
$761$	−18.9574	−0.687207	−0.343603	+	0.939115i	$0.611648\pi$
$762$	−22.6398	−0.820155
$763$	12.4072	0.449172
$764$	−14.3550	−0.519345
$765$	−27.2852	−0.986500
$766$	2.10839	0.0761793
$767$	−19.3439	−0.698468
$768$	2.11676	0.0763820
$769$	2.47447	0.0892317	0.0446158	−	0.999004i	$-0.485794\pi$
$769$	2.47447	0.0892317	0.0446158	+	0.999004i	$0.485794\pi$
$770$	2.05665	0.0741165
$771$	−38.9472	−1.40265
$772$	−0.376050	−0.0135343
$773$	49.7433	1.78914	0.894571	−	0.446925i	$-0.147481\pi$
$773$	49.7433	1.78914	0.894571	+	0.446925i	$0.147481\pi$
$774$	1.48068	0.0532218
$775$	−3.45815	−0.124220
$776$	14.6459	0.525757
$777$	−2.45423	−0.0880450
$778$	−19.2770	−0.691113
$779$	−11.2382	−0.402652
$780$	−26.2798	−0.940966
$781$	3.99986	0.143126
$782$	−50.7997	−1.81659
$783$	24.3618	0.870621
$784$	−6.39344	−0.228337
$785$	−8.37142	−0.298789
$786$	40.8351	1.45654
$787$	−2.71152	−0.0966553	−0.0483277	−	0.998832i	$-0.515389\pi$
$787$	−2.71152	−0.0966553	−0.0483277	+	0.998832i	$0.515389\pi$
$788$	1.38133	0.0492079
$789$	8.36682	0.297867
$790$	−22.7800	−0.810474
$791$	−3.77497	−0.134223
$792$	1.48068	0.0526135
$793$	−44.5721	−1.58280
$794$	28.0066	0.993919
$795$	24.7335	0.877206
$796$	−13.9603	−0.494811
$797$	−1.95631	−0.0692959	−0.0346480	−	0.999400i	$-0.511031\pi$
$797$	−1.95631	−0.0692959	−0.0346480	+	0.999400i	$0.511031\pi$
$798$	3.82545	0.135420
$799$	1.35844	0.0480581
$800$	1.97349	0.0697733
$801$	9.62454	0.340066
$802$	14.9737	0.528739
$803$	16.9297	0.597435
$804$	3.07926	0.108597
$805$	14.9720	0.527692
$806$	8.23823	0.290179
$807$	22.5065	0.792265
$808$	13.4412	0.472859
$809$	5.76542	0.202701	0.101351	−	0.994851i	$-0.467684\pi$
$809$	5.76542	0.202701	0.101351	+	0.994851i	$0.467684\pi$
$810$	−29.7073	−1.04381
$811$	−7.03877	−0.247164	−0.123582	−	0.992334i	$-0.539438\pi$
$811$	−7.03877	−0.247164	−0.123582	+	0.992334i	$0.539438\pi$
$812$	−5.89961	−0.207036
$813$	9.96523	0.349496
$814$	−1.48870	−0.0521790
$815$	−34.3429	−1.20298
$816$	−14.7712	−0.517094
$817$	2.32047	0.0811829
$818$	0.508131	0.0177664
$819$	−5.42150	−0.189443
$820$	−12.7893	−0.446623
$821$	−4.35095	−0.151849	−0.0759245	−	0.997114i	$-0.524191\pi$
$821$	−4.35095	−0.151849	−0.0759245	+	0.997114i	$0.524191\pi$
$822$	33.4506	1.16672
$823$	17.1668	0.598396	0.299198	−	0.954191i	$-0.403281\pi$
$823$	17.1668	0.598396	0.299198	+	0.954191i	$0.403281\pi$
$824$	−10.9090	−0.380035
$825$	4.17740	0.145438
$826$	3.20446	0.111497
$827$	−52.9164	−1.84008	−0.920041	−	0.391822i	$-0.871845\pi$
$827$	−52.9164	−1.84008	−0.920041	+	0.391822i	$0.871845\pi$
$828$	10.7790	0.374596
$829$	−45.7035	−1.58735	−0.793674	−	0.608343i	$-0.791834\pi$
$829$	−45.7035	−1.58735	−0.793674	+	0.608343i	$0.791834\pi$
$830$	−16.7242	−0.580504
$831$	−22.9862	−0.797383
$832$	−4.70137	−0.162991
$833$	44.6147	1.54581
$834$	32.9410	1.14065
$835$	31.0110	1.07318
$836$	2.32047	0.0802551
$837$	5.63549	0.194791
$838$	17.0240	0.588084
$839$	16.2585	0.561307	0.280653	−	0.959809i	$-0.409449\pi$
$839$	16.2585	0.561307	0.280653	+	0.959809i	$0.409449\pi$
$840$	4.35344	0.150208
$841$	28.3819	0.978687
$842$	−2.59012	−0.0892615
$843$	−19.4376	−0.669468
$844$	−8.52787	−0.293541
$845$	24.0384	0.826945
$846$	−0.288242	−0.00990995
$847$	0.778817	0.0267605
$848$	4.42475	0.151946
$849$	20.5506	0.705293
$850$	−13.7714	−0.472354
$851$	−10.8374	−0.371502
$852$	8.46674	0.290066
$853$	1.94777	0.0666904	0.0333452	−	0.999444i	$-0.489384\pi$
$853$	1.94777	0.0666904	0.0333452	+	0.999444i	$0.489384\pi$
$854$	7.38369	0.252665
$855$	9.07320	0.310297
$856$	−18.5276	−0.633262
$857$	−31.0874	−1.06192	−0.530962	−	0.847396i	$-0.678169\pi$
$857$	−31.0874	−1.06192	−0.530962	+	0.847396i	$0.678169\pi$
$858$	−9.95168	−0.339745
$859$	−21.6102	−0.737330	−0.368665	−	0.929562i	$-0.620185\pi$
$859$	−21.6102	−0.737330	−0.368665	+	0.929562i	$0.620185\pi$
$860$	2.64074	0.0900483
$861$	−7.98417	−0.272100
$862$	29.4435	1.00285
$863$	29.4861	1.00372	0.501860	−	0.864949i	$-0.332649\pi$
$863$	29.4861	1.00372	0.501860	+	0.864949i	$0.332649\pi$
$864$	−3.21605	−0.109412
$865$	−56.5079	−1.92132
$866$	−0.274100	−0.00931430
$867$	67.0911	2.27853
$868$	−1.36472	−0.0463217
$869$	−8.62636	−0.292629
$870$	−42.3433	−1.43557
$871$	−6.83911	−0.231734
$872$	15.9309	0.539488
$873$	21.6858	0.733954
$874$	16.8925	0.571398
$875$	−6.22448	−0.210426
$876$	35.8361	1.21079
$877$	−3.45730	−0.116745	−0.0583723	−	0.998295i	$-0.518591\pi$
$877$	−3.45730	−0.116745	−0.0583723	+	0.998295i	$0.518591\pi$
$878$	18.3500	0.619284
$879$	24.4284	0.823950
$880$	2.64074	0.0890192
$881$	8.78200	0.295873	0.147937	−	0.988997i	$-0.452737\pi$
$881$	8.78200	0.295873	0.147937	+	0.988997i	$0.452737\pi$
$882$	−9.46661	−0.318757
$883$	−21.6560	−0.728781	−0.364391	−	0.931246i	$-0.618723\pi$
$883$	−21.6560	−0.728781	−0.364391	+	0.931246i	$0.618723\pi$
$884$	32.8071	1.10342
$885$	22.9994	0.773116
$886$	23.2359	0.780625
$887$	−3.95617	−0.132835	−0.0664177	−	0.997792i	$-0.521157\pi$
$887$	−3.95617	−0.132835	−0.0664177	+	0.997792i	$0.521157\pi$
$888$	−3.15123	−0.105748
$889$	−8.32985	−0.279374
$890$	17.1651	0.575374
$891$	−11.2496	−0.376877
$892$	16.9526	0.567616
$893$	−0.451723	−0.0151163
$894$	−17.5753	−0.587805
$895$	33.1983	1.10970
$896$	0.778817	0.0260184
$897$	−72.4460	−2.41890
$898$	10.1233	0.337818
$899$	13.2738	0.442708
$900$	2.92209	0.0974031
$901$	−30.8767	−1.02865
$902$	−4.84309	−0.161257
$903$	1.64857	0.0548609
$904$	−4.84706	−0.161211
$905$	−31.3334	−1.04156
$906$	4.94767	0.164375
$907$	24.1594	0.802199	0.401100	−	0.916034i	$-0.368628\pi$
$907$	24.1594	0.802199	0.401100	+	0.916034i	$0.368628\pi$
$908$	2.81586	0.0934477
$909$	19.9020	0.660108
$910$	−9.66908	−0.320527
$911$	−30.9629	−1.02585	−0.512923	−	0.858435i	$-0.671437\pi$
$911$	−30.9629	−1.02585	−0.512923	+	0.858435i	$0.671437\pi$
$912$	4.91188	0.162649
$913$	−6.33315	−0.209597
$914$	35.6134	1.17799
$915$	52.9950	1.75196
$916$	−8.80344	−0.290874
$917$	15.0244	0.496149
$918$	22.4422	0.740703
$919$	47.4869	1.56645	0.783224	−	0.621739i	$-0.213574\pi$
$919$	47.4869	1.56645	0.783224	+	0.621739i	$0.213574\pi$
$920$	19.2240	0.633796
$921$	−12.9619	−0.427111
$922$	23.3777	0.769902
$923$	−18.8048	−0.618968
$924$	1.64857	0.0542339
$925$	−2.93794	−0.0965987
$926$	−10.6802	−0.350973
$927$	−16.1527	−0.530526
$928$	−7.57509	−0.248664
$929$	−15.9714	−0.524003	−0.262002	−	0.965067i	$-0.584383\pi$
$929$	−15.9714	−0.524003	−0.262002	+	0.965067i	$0.584383\pi$
$930$	−9.79503	−0.321192
$931$	−14.8358	−0.486223
$932$	7.14193	0.233942
$933$	54.3963	1.78085
$934$	−24.4626	−0.800440
$935$	−18.4276	−0.602646
$936$	−6.96121	−0.227534
$937$	−6.74865	−0.220469	−0.110234	−	0.993906i	$-0.535160\pi$
$937$	−6.74865	−0.220469	−0.110234	+	0.993906i	$0.535160\pi$
$938$	1.13295	0.0369921
$939$	63.7072	2.07901
$940$	−0.514069	−0.0167671
$941$	−14.2990	−0.466133	−0.233066	−	0.972461i	$-0.574876\pi$
$941$	−14.2990	−0.466133	−0.233066	+	0.972461i	$0.574876\pi$
$942$	−6.71036	−0.218635
$943$	−35.2566	−1.14811
$944$	4.11453	0.133916
$945$	−6.61428	−0.215163
$946$	1.00000	0.0325128
$947$	−44.0603	−1.43177	−0.715883	−	0.698220i	$-0.753976\pi$
$947$	−44.0603	−1.43177	−0.715883	+	0.698220i	$0.753976\pi$
$948$	−18.2599	−0.593056
$949$	−79.5927	−2.58369
$950$	4.57941	0.148576
$951$	15.4933	0.502403
$952$	−5.43474	−0.176141
$953$	−24.2737	−0.786302	−0.393151	−	0.919474i	$-0.628615\pi$
$953$	−24.2737	−0.786302	−0.393151	+	0.919474i	$0.628615\pi$
$954$	6.55161	0.212116
$955$	−37.9077	−1.22667
$956$	11.7651	0.380511
$957$	−16.0346	−0.518326
$958$	21.3377	0.689390
$959$	12.3074	0.397428
$960$	5.58981	0.180410
$961$	−27.9294	−0.900949
$962$	6.99895	0.225655
$963$	−27.4334	−0.884030
$964$	−28.9037	−0.930924
$965$	−0.993050	−0.0319674
$966$	12.0012	0.386133
$967$	−48.7871	−1.56889	−0.784444	−	0.620200i	$-0.787051\pi$
$967$	−48.7871	−1.56889	−0.784444	+	0.620200i	$0.787051\pi$
$968$	1.00000	0.0321412
$969$	−34.2760	−1.10110
$970$	38.6760	1.24181
$971$	−15.4841	−0.496907	−0.248453	−	0.968644i	$-0.579922\pi$
$971$	−15.4841	−0.496907	−0.248453	+	0.968644i	$0.579922\pi$
$972$	−14.1646	−0.454331
$973$	12.1199	0.388548
$974$	−5.98402	−0.191740
$975$	−19.6395	−0.628967
$976$	9.48065	0.303468
$977$	−1.89591	−0.0606555	−0.0303277	−	0.999540i	$-0.509655\pi$
$977$	−1.89591	−0.0606555	−0.0303277	+	0.999540i	$0.509655\pi$
$978$	−27.5285	−0.880266
$979$	6.50010	0.207744
$980$	−16.8834	−0.539320
$981$	23.5885	0.753122
$982$	37.4263	1.19432
$983$	−49.7272	−1.58605	−0.793026	−	0.609188i	$-0.791495\pi$
$983$	−49.7272	−1.58605	−0.793026	+	0.609188i	$0.791495\pi$
$984$	−10.2517	−0.326811
$985$	3.64773	0.116226
$986$	52.8604	1.68342
$987$	−0.320925	−0.0102152
$988$	−10.9094	−0.347074
$989$	7.27978	0.231484
$990$	3.91007	0.124270
$991$	−45.1966	−1.43572	−0.717859	−	0.696189i	$-0.754878\pi$
$991$	−45.1966	−1.43572	−0.717859	+	0.696189i	$0.754878\pi$
$992$	−1.75230	−0.0556357
$993$	−34.3044	−1.08862
$994$	3.11516	0.0988067
$995$	−36.8655	−1.16872
$996$	−13.4058	−0.424778
$997$	4.13658	0.131007	0.0655034	−	0.997852i	$-0.479135\pi$
$997$	4.13658	0.131007	0.0655034	+	0.997852i	$0.479135\pi$
$998$	19.0955	0.604457
$999$	4.78774	0.151477

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	946.2.a.k.1.5	✓	7
3.2	odd	2		8514.2.a.bl.1.3		7
4.3	odd	2		7568.2.a.bg.1.3		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
946.2.a.k.1.5	✓	7	1.1	even	1	trivial
7568.2.a.bg.1.3		7	4.3	odd	2
8514.2.a.bl.1.3		7	3.2	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:	\( N \)	\(=\)	\( 946 = 2 \cdot 11 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	946.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +2.11676 q^{3} +1.00000 q^{4} +2.64074 q^{5} +2.11676 q^{6} +0.778817 q^{7} +1.00000 q^{8} +1.48068 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +2.11676 q^{3} +1.00000 q^{4} +2.64074 q^{5} +2.11676 q^{6} +0.778817 q^{7} +1.00000 q^{8} +1.48068 q^{9} +2.64074 q^{10} +1.00000 q^{11} +2.11676 q^{12} -4.70137 q^{13} +0.778817 q^{14} +5.58981 q^{15} +1.00000 q^{16} -6.97819 q^{17} +1.48068 q^{18} +2.32047 q^{19} +2.64074 q^{20} +1.64857 q^{21} +1.00000 q^{22} +7.27978 q^{23} +2.11676 q^{24} +1.97349 q^{25} -4.70137 q^{26} -3.21605 q^{27} +0.778817 q^{28} -7.57509 q^{29} +5.58981 q^{30} -1.75230 q^{31} +1.00000 q^{32} +2.11676 q^{33} -6.97819 q^{34} +2.05665 q^{35} +1.48068 q^{36} -1.48870 q^{37} +2.32047 q^{38} -9.95168 q^{39} +2.64074 q^{40} -4.84309 q^{41} +1.64857 q^{42} +1.00000 q^{43} +1.00000 q^{44} +3.91007 q^{45} +7.27978 q^{46} -0.194669 q^{47} +2.11676 q^{48} -6.39344 q^{49} +1.97349 q^{50} -14.7712 q^{51} -4.70137 q^{52} +4.42475 q^{53} -3.21605 q^{54} +2.64074 q^{55} +0.778817 q^{56} +4.91188 q^{57} -7.57509 q^{58} +4.11453 q^{59} +5.58981 q^{60} +9.48065 q^{61} -1.75230 q^{62} +1.15317 q^{63} +1.00000 q^{64} -12.4151 q^{65} +2.11676 q^{66} +1.45470 q^{67} -6.97819 q^{68} +15.4095 q^{69} +2.05665 q^{70} +3.99986 q^{71} +1.48068 q^{72} +16.9297 q^{73} -1.48870 q^{74} +4.17740 q^{75} +2.32047 q^{76} +0.778817 q^{77} -9.95168 q^{78} -8.62636 q^{79} +2.64074 q^{80} -11.2496 q^{81} -4.84309 q^{82} -6.33315 q^{83} +1.64857 q^{84} -18.4276 q^{85} +1.00000 q^{86} -16.0346 q^{87} +1.00000 q^{88} +6.50010 q^{89} +3.91007 q^{90} -3.66151 q^{91} +7.27978 q^{92} -3.70921 q^{93} -0.194669 q^{94} +6.12775 q^{95} +2.11676 q^{96} +14.6459 q^{97} -6.39344 q^{98} +1.48068 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 7 q^{2} + 3 q^{3} + 7 q^{4} + 8 q^{5} + 3 q^{6} - 2 q^{7} + 7 q^{8} + 10 q^{9}+O(q^{10}) \) 7 * q + 7 * q^2 + 3 * q^3 + 7 * q^4 + 8 * q^5 + 3 * q^6 - 2 * q^7 + 7 * q^8 + 10 * q^9	\( 7 q + 7 q^{2} + 3 q^{3} + 7 q^{4} + 8 q^{5} + 3 q^{6} - 2 q^{7} + 7 q^{8} + 10 q^{9} + 8 q^{10} + 7 q^{11} + 3 q^{12} + 4 q^{13} - 2 q^{14} + 2 q^{15} + 7 q^{16} + 10 q^{17} + 10 q^{18} - 2 q^{19} + 8 q^{20} + 7 q^{22} - 4 q^{23} + 3 q^{24} + 11 q^{25} + 4 q^{26} + 15 q^{27} - 2 q^{28} + 5 q^{29} + 2 q^{30} - 2 q^{31} + 7 q^{32} + 3 q^{33} + 10 q^{34} - 10 q^{35} + 10 q^{36} + 6 q^{37} - 2 q^{38} - 8 q^{39} + 8 q^{40} + 4 q^{41} + 7 q^{43} + 7 q^{44} - 2 q^{45} - 4 q^{46} - 6 q^{47} + 3 q^{48} + 31 q^{49} + 11 q^{50} - 14 q^{51} + 4 q^{52} + q^{53} + 15 q^{54} + 8 q^{55} - 2 q^{56} - 30 q^{57} + 5 q^{58} - 4 q^{59} + 2 q^{60} + 9 q^{61} - 2 q^{62} - 24 q^{63} + 7 q^{64} + 18 q^{65} + 3 q^{66} - 6 q^{67} + 10 q^{68} + 2 q^{69} - 10 q^{70} + 10 q^{72} + 13 q^{73} + 6 q^{74} - 9 q^{75} - 2 q^{76} - 2 q^{77} - 8 q^{78} - 31 q^{79} + 8 q^{80} - 25 q^{81} + 4 q^{82} - 11 q^{83} - 24 q^{85} + 7 q^{86} - 13 q^{87} + 7 q^{88} + 10 q^{89} - 2 q^{90} - 12 q^{91} - 4 q^{92} + 12 q^{93} - 6 q^{94} - 18 q^{95} + 3 q^{96} + 23 q^{97} + 31 q^{98} + 10 q^{99}+O(q^{100}) \) 7 * q + 7 * q^2 + 3 * q^3 + 7 * q^4 + 8 * q^5 + 3 * q^6 - 2 * q^7 + 7 * q^8 + 10 * q^9 + 8 * q^10 + 7 * q^11 + 3 * q^12 + 4 * q^13 - 2 * q^14 + 2 * q^15 + 7 * q^16 + 10 * q^17 + 10 * q^18 - 2 * q^19 + 8 * q^20 + 7 * q^22 - 4 * q^23 + 3 * q^24 + 11 * q^25 + 4 * q^26 + 15 * q^27 - 2 * q^28 + 5 * q^29 + 2 * q^30 - 2 * q^31 + 7 * q^32 + 3 * q^33 + 10 * q^34 - 10 * q^35 + 10 * q^36 + 6 * q^37 - 2 * q^38 - 8 * q^39 + 8 * q^40 + 4 * q^41 + 7 * q^43 + 7 * q^44 - 2 * q^45 - 4 * q^46 - 6 * q^47 + 3 * q^48 + 31 * q^49 + 11 * q^50 - 14 * q^51 + 4 * q^52 + q^53 + 15 * q^54 + 8 * q^55 - 2 * q^56 - 30 * q^57 + 5 * q^58 - 4 * q^59 + 2 * q^60 + 9 * q^61 - 2 * q^62 - 24 * q^63 + 7 * q^64 + 18 * q^65 + 3 * q^66 - 6 * q^67 + 10 * q^68 + 2 * q^69 - 10 * q^70 + 10 * q^72 + 13 * q^73 + 6 * q^74 - 9 * q^75 - 2 * q^76 - 2 * q^77 - 8 * q^78 - 31 * q^79 + 8 * q^80 - 25 * q^81 + 4 * q^82 - 11 * q^83 - 24 * q^85 + 7 * q^86 - 13 * q^87 + 7 * q^88 + 10 * q^89 - 2 * q^90 - 12 * q^91 - 4 * q^92 + 12 * q^93 - 6 * q^94 - 18 * q^95 + 3 * q^96 + 23 * q^97 + 31 * q^98 + 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more