LMFDB - Embedded newform 8470.2.a.db.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8470,2,Mod(1,8470)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8470, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8470.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8470.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:	$67.6332905120$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.4642000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 8x^{4} + 5x^{3} + 14x^{2} - 9x - 1 $ x^6 - x^5 - 8x^4 + 5x^3 + 14x^2 - 9x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 770)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.0970464$ of defining polynomial
Character	$\chi$	$=$	8470.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +0.0970464 q^{3} +1.00000 q^{4} +1.00000 q^{5} +0.0970464 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.99058 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +0.0970464 q^{3} +1.00000 q^{4} +1.00000 q^{5} +0.0970464 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.99058 q^{9} +1.00000 q^{10} +0.0970464 q^{12} -4.32646 q^{13} -1.00000 q^{14} +0.0970464 q^{15} +1.00000 q^{16} +2.12295 q^{17} -2.99058 q^{18} +5.51022 q^{19} +1.00000 q^{20} -0.0970464 q^{21} +0.922019 q^{23} +0.0970464 q^{24} +1.00000 q^{25} -4.32646 q^{26} -0.581365 q^{27} -1.00000 q^{28} -8.02820 q^{29} +0.0970464 q^{30} +1.21552 q^{31} +1.00000 q^{32} +2.12295 q^{34} -1.00000 q^{35} -2.99058 q^{36} -5.01306 q^{37} +5.51022 q^{38} -0.419868 q^{39} +1.00000 q^{40} -1.92806 q^{41} -0.0970464 q^{42} +3.75423 q^{43} -2.99058 q^{45} +0.922019 q^{46} -0.865052 q^{47} +0.0970464 q^{48} +1.00000 q^{49} +1.00000 q^{50} +0.206025 q^{51} -4.32646 q^{52} -0.729158 q^{53} -0.581365 q^{54} -1.00000 q^{56} +0.534747 q^{57} -8.02820 q^{58} -0.178208 q^{59} +0.0970464 q^{60} -13.0127 q^{61} +1.21552 q^{62} +2.99058 q^{63} +1.00000 q^{64} -4.32646 q^{65} -3.55383 q^{67} +2.12295 q^{68} +0.0894787 q^{69} -1.00000 q^{70} +5.96038 q^{71} -2.99058 q^{72} -14.8292 q^{73} -5.01306 q^{74} +0.0970464 q^{75} +5.51022 q^{76} -0.419868 q^{78} +13.1971 q^{79} +1.00000 q^{80} +8.91533 q^{81} -1.92806 q^{82} +2.63016 q^{83} -0.0970464 q^{84} +2.12295 q^{85} +3.75423 q^{86} -0.779108 q^{87} -18.4678 q^{89} -2.99058 q^{90} +4.32646 q^{91} +0.922019 q^{92} +0.117962 q^{93} -0.865052 q^{94} +5.51022 q^{95} +0.0970464 q^{96} +0.0855815 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{2} - q^{3} + 6 q^{4} + 6 q^{5} - q^{6} - 6 q^{7} + 6 q^{8} - q^{9}+O(q^{10}) $ 6 * q + 6 * q^2 - q^3 + 6 * q^4 + 6 * q^5 - q^6 - 6 * q^7 + 6 * q^8 - q^9	$ 6 q + 6 q^{2} - q^{3} + 6 q^{4} + 6 q^{5} - q^{6} - 6 q^{7} + 6 q^{8} - q^{9} + 6 q^{10} - q^{12} - 6 q^{13} - 6 q^{14} - q^{15} + 6 q^{16} - 21 q^{17} - q^{18} + 3 q^{19} + 6 q^{20} + q^{21} - 10 q^{23} - q^{24} + 6 q^{25} - 6 q^{26} - 4 q^{27} - 6 q^{28} - 10 q^{29} - q^{30} - 4 q^{31} + 6 q^{32} - 21 q^{34} - 6 q^{35} - q^{36} - 2 q^{37} + 3 q^{38} - 26 q^{39} + 6 q^{40} - 7 q^{41} + q^{42} - 19 q^{43} - q^{45} - 10 q^{46} + 10 q^{47} - q^{48} + 6 q^{49} + 6 q^{50} + 4 q^{51} - 6 q^{52} - 16 q^{53} - 4 q^{54} - 6 q^{56} - 16 q^{57} - 10 q^{58} - 3 q^{59} - q^{60} + 8 q^{61} - 4 q^{62} + q^{63} + 6 q^{64} - 6 q^{65} - 27 q^{67} - 21 q^{68} + 4 q^{69} - 6 q^{70} + 4 q^{71} - q^{72} - 13 q^{73} - 2 q^{74} - q^{75} + 3 q^{76} - 26 q^{78} - 14 q^{79} + 6 q^{80} - 14 q^{81} - 7 q^{82} - 51 q^{83} + q^{84} - 21 q^{85} - 19 q^{86} - 8 q^{87} + q^{89} - q^{90} + 6 q^{91} - 10 q^{92} + 4 q^{93} + 10 q^{94} + 3 q^{95} - q^{96} + 7 q^{97} + 6 q^{98}+O(q^{100}) $ 6 * q + 6 * q^2 - q^3 + 6 * q^4 + 6 * q^5 - q^6 - 6 * q^7 + 6 * q^8 - q^9 + 6 * q^10 - q^12 - 6 * q^13 - 6 * q^14 - q^15 + 6 * q^16 - 21 * q^17 - q^18 + 3 * q^19 + 6 * q^20 + q^21 - 10 * q^23 - q^24 + 6 * q^25 - 6 * q^26 - 4 * q^27 - 6 * q^28 - 10 * q^29 - q^30 - 4 * q^31 + 6 * q^32 - 21 * q^34 - 6 * q^35 - q^36 - 2 * q^37 + 3 * q^38 - 26 * q^39 + 6 * q^40 - 7 * q^41 + q^42 - 19 * q^43 - q^45 - 10 * q^46 + 10 * q^47 - q^48 + 6 * q^49 + 6 * q^50 + 4 * q^51 - 6 * q^52 - 16 * q^53 - 4 * q^54 - 6 * q^56 - 16 * q^57 - 10 * q^58 - 3 * q^59 - q^60 + 8 * q^61 - 4 * q^62 + q^63 + 6 * q^64 - 6 * q^65 - 27 * q^67 - 21 * q^68 + 4 * q^69 - 6 * q^70 + 4 * q^71 - q^72 - 13 * q^73 - 2 * q^74 - q^75 + 3 * q^76 - 26 * q^78 - 14 * q^79 + 6 * q^80 - 14 * q^81 - 7 * q^82 - 51 * q^83 + q^84 - 21 * q^85 - 19 * q^86 - 8 * q^87 + q^89 - q^90 + 6 * q^91 - 10 * q^92 + 4 * q^93 + 10 * q^94 + 3 * q^95 - q^96 + 7 * q^97 + 6 * q^98

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
$3$	0.0970464	0.0560298	0.0280149	−	0.999608i	$-0.491081\pi$
$3$	0.0970464	0.0560298	0.0280149	+	0.999608i	$0.491081\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0.0970464	0.0396190
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	−2.99058	−0.996861
$10$	1.00000	0.316228
$11$	0	0
$11$	0	0
$12$	0.0970464	0.0280149
$13$	−4.32646	−1.19994	−0.599972	−	0.800021i	$-0.704822\pi$
$13$	−4.32646	−1.19994	−0.599972	+	0.800021i	$0.704822\pi$
$14$	−1.00000	−0.267261
$15$	0.0970464	0.0250573
$16$	1.00000	0.250000
$17$	2.12295	0.514891	0.257445	−	0.966293i	$-0.417119\pi$
$17$	2.12295	0.514891	0.257445	+	0.966293i	$0.417119\pi$
$18$	−2.99058	−0.704887
$19$	5.51022	1.26413	0.632066	−	0.774915i	$-0.282207\pi$
$19$	5.51022	1.26413	0.632066	+	0.774915i	$0.282207\pi$
$20$	1.00000	0.223607
$21$	−0.0970464	−0.0211773
$22$	0	0
$23$	0.922019	0.192254	0.0961271	−	0.995369i	$-0.469354\pi$
$23$	0.922019	0.192254	0.0961271	+	0.995369i	$0.469354\pi$
$24$	0.0970464	0.0198095
$25$	1.00000	0.200000
$26$	−4.32646	−0.848489
$27$	−0.581365	−0.111884
$28$	−1.00000	−0.188982
$29$	−8.02820	−1.49080	−0.745399	−	0.666618i	$-0.767741\pi$
$29$	−8.02820	−1.49080	−0.745399	+	0.666618i	$0.767741\pi$
$30$	0.0970464	0.0177182
$31$	1.21552	0.218314	0.109157	−	0.994024i	$-0.465185\pi$
$31$	1.21552	0.218314	0.109157	+	0.994024i	$0.465185\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	2.12295	0.364083
$35$	−1.00000	−0.169031
$36$	−2.99058	−0.498430
$37$	−5.01306	−0.824142	−0.412071	−	0.911152i	$-0.635194\pi$
$37$	−5.01306	−0.824142	−0.412071	+	0.911152i	$0.635194\pi$
$38$	5.51022	0.893876
$39$	−0.419868	−0.0672326
$40$	1.00000	0.158114
$41$	−1.92806	−0.301113	−0.150556	−	0.988601i	$-0.548107\pi$
$41$	−1.92806	−0.301113	−0.150556	+	0.988601i	$0.548107\pi$
$42$	−0.0970464	−0.0149746
$43$	3.75423	0.572514	0.286257	−	0.958153i	$-0.407589\pi$
$43$	3.75423	0.572514	0.286257	+	0.958153i	$0.407589\pi$
$44$	0	0
$45$	−2.99058	−0.445810
$46$	0.922019	0.135944
$47$	−0.865052	−0.126181	−0.0630904	−	0.998008i	$-0.520096\pi$
$47$	−0.865052	−0.126181	−0.0630904	+	0.998008i	$0.520096\pi$
$48$	0.0970464	0.0140074
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	0.206025	0.0288492
$52$	−4.32646	−0.599972
$53$	−0.729158	−0.100158	−0.0500788	−	0.998745i	$-0.515947\pi$
$53$	−0.729158	−0.100158	−0.0500788	+	0.998745i	$0.515947\pi$
$54$	−0.581365	−0.0791137
$55$	0	0
$56$	−1.00000	−0.133631
$57$	0.534747	0.0708290
$58$	−8.02820	−1.05415
$59$	−0.178208	−0.0232008	−0.0116004	−	0.999933i	$-0.503693\pi$
$59$	−0.178208	−0.0232008	−0.0116004	+	0.999933i	$0.503693\pi$
$60$	0.0970464	0.0125286
$61$	−13.0127	−1.66610	−0.833051	−	0.553195i	$-0.813408\pi$
$61$	−13.0127	−1.66610	−0.833051	+	0.553195i	$0.813408\pi$
$62$	1.21552	0.154372
$63$	2.99058	0.376778
$64$	1.00000	0.125000
$65$	−4.32646	−0.536631
$66$	0	0
$67$	−3.55383	−0.434169	−0.217085	−	0.976153i	$-0.569655\pi$
$67$	−3.55383	−0.434169	−0.217085	+	0.976153i	$0.569655\pi$
$68$	2.12295	0.257445
$69$	0.0894787	0.0107720
$70$	−1.00000	−0.119523
$71$	5.96038	0.707367	0.353683	−	0.935365i	$-0.384929\pi$
$71$	5.96038	0.707367	0.353683	+	0.935365i	$0.384929\pi$
$72$	−2.99058	−0.352443
$73$	−14.8292	−1.73562	−0.867811	−	0.496895i	$-0.834474\pi$
$73$	−14.8292	−1.73562	−0.867811	+	0.496895i	$0.834474\pi$
$74$	−5.01306	−0.582756
$75$	0.0970464	0.0112060
$76$	5.51022	0.632066
$77$	0	0
$78$	−0.419868	−0.0475406
$79$	13.1971	1.48479	0.742394	−	0.669964i	$-0.233690\pi$
$79$	13.1971	1.48479	0.742394	+	0.669964i	$0.233690\pi$
$80$	1.00000	0.111803
$81$	8.91533	0.990592
$82$	−1.92806	−0.212919
$83$	2.63016	0.288698	0.144349	−	0.989527i	$-0.453891\pi$
$83$	2.63016	0.288698	0.144349	+	0.989527i	$0.453891\pi$
$84$	−0.0970464	−0.0105886
$85$	2.12295	0.230266
$86$	3.75423	0.404829
$87$	−0.779108	−0.0835291
$88$	0	0
$89$	−18.4678	−1.95758	−0.978791	−	0.204861i	$-0.934326\pi$
$89$	−18.4678	−1.95758	−0.978791	+	0.204861i	$0.934326\pi$
$90$	−2.99058	−0.315235
$91$	4.32646	0.453536
$92$	0.922019	0.0961271
$93$	0.117962	0.0122321
$94$	−0.865052	−0.0892233
$95$	5.51022	0.565337
$96$	0.0970464	0.00990476
$97$	0.0855815	0.00868949	0.00434474	−	0.999991i	$-0.498617\pi$
$97$	0.0855815	0.00868949	0.00434474	+	0.999991i	$0.498617\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	−7.57523	−0.753763	−0.376882	−	0.926261i	$-0.623004\pi$
$101$	−7.57523	−0.753763	−0.376882	+	0.926261i	$0.623004\pi$
$102$	0.206025	0.0203995
$103$	−3.76181	−0.370663	−0.185331	−	0.982676i	$-0.559336\pi$
$103$	−3.76181	−0.370663	−0.185331	+	0.982676i	$0.559336\pi$
$104$	−4.32646	−0.424244
$105$	−0.0970464	−0.00947076
$106$	−0.729158	−0.0708221
$107$	−13.3165	−1.28735	−0.643677	−	0.765297i	$-0.722592\pi$
$107$	−13.3165	−1.28735	−0.643677	+	0.765297i	$0.722592\pi$
$108$	−0.581365	−0.0559418
$109$	−8.04563	−0.770631	−0.385316	−	0.922785i	$-0.625907\pi$
$109$	−8.04563	−0.770631	−0.385316	+	0.922785i	$0.625907\pi$
$110$	0	0
$111$	−0.486499	−0.0461765
$112$	−1.00000	−0.0944911
$113$	9.17493	0.863104	0.431552	−	0.902088i	$-0.357966\pi$
$113$	9.17493	0.863104	0.431552	+	0.902088i	$0.357966\pi$
$114$	0.534747	0.0500837
$115$	0.922019	0.0859787
$116$	−8.02820	−0.745399
$117$	12.9386	1.19618
$118$	−0.178208	−0.0164054
$119$	−2.12295	−0.194610
$120$	0.0970464	0.00885909
$121$	0	0
$122$	−13.0127	−1.17811
$123$	−0.187112	−0.0168713
$124$	1.21552	0.109157
$125$	1.00000	0.0894427
$126$	2.99058	0.266422
$127$	−8.63204	−0.765970	−0.382985	−	0.923755i	$-0.625104\pi$
$127$	−8.63204	−0.765970	−0.382985	+	0.923755i	$0.625104\pi$
$128$	1.00000	0.0883883
$129$	0.364335	0.0320779
$130$	−4.32646	−0.379456
$131$	5.01748	0.438379	0.219189	−	0.975682i	$-0.429659\pi$
$131$	5.01748	0.438379	0.219189	+	0.975682i	$0.429659\pi$
$132$	0	0
$133$	−5.51022	−0.477797
$134$	−3.55383	−0.307004
$135$	−0.581365	−0.0500359
$136$	2.12295	0.182041
$137$	−9.38739	−0.802018	−0.401009	−	0.916074i	$-0.631340\pi$
$137$	−9.38739	−0.802018	−0.401009	+	0.916074i	$0.631340\pi$
$138$	0.0894787	0.00761693
$139$	−6.96159	−0.590474	−0.295237	−	0.955424i	$-0.595399\pi$
$139$	−6.96159	−0.590474	−0.295237	+	0.955424i	$0.595399\pi$
$140$	−1.00000	−0.0845154
$141$	−0.0839502	−0.00706988
$142$	5.96038	0.500184
$143$	0	0
$144$	−2.99058	−0.249215
$145$	−8.02820	−0.666706
$146$	−14.8292	−1.22727
$147$	0.0970464	0.00800426
$148$	−5.01306	−0.412071
$149$	−19.8855	−1.62908	−0.814542	−	0.580105i	$-0.803011\pi$
$149$	−19.8855	−1.62908	−0.814542	+	0.580105i	$0.803011\pi$
$150$	0.0970464	0.00792381
$151$	−6.91156	−0.562455	−0.281227	−	0.959641i	$-0.590742\pi$
$151$	−6.91156	−0.562455	−0.281227	+	0.959641i	$0.590742\pi$
$152$	5.51022	0.446938
$153$	−6.34886	−0.513275
$154$	0	0
$155$	1.21552	0.0976332
$156$	−0.419868	−0.0336163
$157$	9.00126	0.718378	0.359189	−	0.933265i	$-0.383053\pi$
$157$	9.00126	0.718378	0.359189	+	0.933265i	$0.383053\pi$
$158$	13.1971	1.04990
$159$	−0.0707622	−0.00561181
$160$	1.00000	0.0790569
$161$	−0.922019	−0.0726653
$162$	8.91533	0.700454
$163$	−7.07107	−0.553849	−0.276924	−	0.960892i	$-0.589315\pi$
$163$	−7.07107	−0.553849	−0.276924	+	0.960892i	$0.589315\pi$
$164$	−1.92806	−0.150556
$165$	0	0
$166$	2.63016	0.204140
$167$	−19.7421	−1.52769	−0.763846	−	0.645398i	$-0.776691\pi$
$167$	−19.7421	−1.52769	−0.763846	+	0.645398i	$0.776691\pi$
$168$	−0.0970464	−0.00748730
$169$	5.71826	0.439866
$170$	2.12295	0.162823
$171$	−16.4788	−1.26016
$172$	3.75423	0.286257
$173$	13.9142	1.05787	0.528937	−	0.848661i	$-0.322591\pi$
$173$	13.9142	1.05787	0.528937	+	0.848661i	$0.322591\pi$
$174$	−0.779108	−0.0590640
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−0.0172945	−0.00129993
$178$	−18.4678	−1.38422
$179$	−3.61007	−0.269829	−0.134915	−	0.990857i	$-0.543076\pi$
$179$	−3.61007	−0.269829	−0.134915	+	0.990857i	$0.543076\pi$
$180$	−2.99058	−0.222905
$181$	−1.95017	−0.144955	−0.0724776	−	0.997370i	$-0.523091\pi$
$181$	−1.95017	−0.144955	−0.0724776	+	0.997370i	$0.523091\pi$
$182$	4.32646	0.320699
$183$	−1.26283	−0.0933514
$184$	0.922019	0.0679722
$185$	−5.01306	−0.368567
$186$	0.117962	0.00864941
$187$	0	0
$188$	−0.865052	−0.0630904
$189$	0.581365	0.0422881
$190$	5.51022	0.399753
$191$	12.7463	0.922288	0.461144	−	0.887325i	$-0.347439\pi$
$191$	12.7463	0.922288	0.461144	+	0.887325i	$0.347439\pi$
$192$	0.0970464	0.00700372
$193$	−14.5920	−1.05035	−0.525176	−	0.850994i	$-0.676001\pi$
$193$	−14.5920	−1.05035	−0.525176	+	0.850994i	$0.676001\pi$
$194$	0.0855815	0.00614439
$195$	−0.419868	−0.0300673
$196$	1.00000	0.0714286
$197$	24.4598	1.74269	0.871346	−	0.490669i	$-0.163248\pi$
$197$	24.4598	1.74269	0.871346	+	0.490669i	$0.163248\pi$
$198$	0	0
$199$	−10.4998	−0.744315	−0.372157	−	0.928170i	$-0.621382\pi$
$199$	−10.4998	−0.744315	−0.372157	+	0.928170i	$0.621382\pi$
$200$	1.00000	0.0707107
$201$	−0.344887	−0.0243264
$202$	−7.57523	−0.532991
$203$	8.02820	0.563469
$204$	0.206025	0.0144246
$205$	−1.92806	−0.134662
$206$	−3.76181	−0.262098
$207$	−2.75737	−0.191651
$208$	−4.32646	−0.299986
$209$	0	0
$210$	−0.0970464	−0.00669684
$211$	14.9790	1.03120	0.515600	−	0.856830i	$-0.327569\pi$
$211$	14.9790	1.03120	0.515600	+	0.856830i	$0.327569\pi$
$212$	−0.729158	−0.0500788
$213$	0.578434	0.0396336
$214$	−13.3165	−0.910296
$215$	3.75423	0.256036
$216$	−0.581365	−0.0395569
$217$	−1.21552	−0.0825151
$218$	−8.04563	−0.544919
$219$	−1.43912	−0.0972465
$220$	0	0
$221$	−9.18486	−0.617840
$222$	−0.486499	−0.0326517
$223$	−6.98647	−0.467848	−0.233924	−	0.972255i	$-0.575157\pi$
$223$	−6.98647	−0.467848	−0.233924	+	0.972255i	$0.575157\pi$
$224$	−1.00000	−0.0668153
$225$	−2.99058	−0.199372
$226$	9.17493	0.610307
$227$	−23.6112	−1.56713	−0.783565	−	0.621309i	$-0.786601\pi$
$227$	−23.6112	−1.56713	−0.783565	+	0.621309i	$0.786601\pi$
$228$	0.534747	0.0354145
$229$	21.0574	1.39151	0.695755	−	0.718279i	$-0.255070\pi$
$229$	21.0574	1.39151	0.695755	+	0.718279i	$0.255070\pi$
$230$	0.922019	0.0607961
$231$	0	0
$232$	−8.02820	−0.527077
$233$	−29.7941	−1.95187	−0.975937	−	0.218052i	$-0.930030\pi$
$233$	−29.7941	−1.95187	−0.975937	+	0.218052i	$0.930030\pi$
$234$	12.9386	0.845825
$235$	−0.865052	−0.0564298
$236$	−0.178208	−0.0116004
$237$	1.28073	0.0831923
$238$	−2.12295	−0.137610
$239$	−18.4084	−1.19074	−0.595370	−	0.803452i	$-0.702995\pi$
$239$	−18.4084	−1.19074	−0.595370	+	0.803452i	$0.702995\pi$
$240$	0.0970464	0.00626432
$241$	−2.15635	−0.138903	−0.0694513	−	0.997585i	$-0.522125\pi$
$241$	−2.15635	−0.138903	−0.0694513	+	0.997585i	$0.522125\pi$
$242$	0	0
$243$	2.60929	0.167386
$244$	−13.0127	−0.833051
$245$	1.00000	0.0638877
$246$	−0.187112	−0.0119298
$247$	−23.8398	−1.51689
$248$	1.21552	0.0771858
$249$	0.255248	0.0161757
$250$	1.00000	0.0632456
$251$	19.6172	1.23823	0.619114	−	0.785301i	$-0.287492\pi$
$251$	19.6172	1.23823	0.619114	+	0.785301i	$0.287492\pi$
$252$	2.99058	0.188389
$253$	0	0
$254$	−8.63204	−0.541622
$255$	0.206025	0.0129018
$256$	1.00000	0.0625000
$257$	−5.06003	−0.315636	−0.157818	−	0.987468i	$-0.550446\pi$
$257$	−5.06003	−0.315636	−0.157818	+	0.987468i	$0.550446\pi$
$258$	0.364335	0.0226825
$259$	5.01306	0.311496
$260$	−4.32646	−0.268316
$261$	24.0090	1.48612
$262$	5.01748	0.309981
$263$	−16.0394	−0.989031	−0.494516	−	0.869169i	$-0.664655\pi$
$263$	−16.0394	−0.989031	−0.494516	+	0.869169i	$0.664655\pi$
$264$	0	0
$265$	−0.729158	−0.0447918
$266$	−5.51022	−0.337853
$267$	−1.79223	−0.109683
$268$	−3.55383	−0.217085
$269$	7.45344	0.454444	0.227222	−	0.973843i	$-0.427036\pi$
$269$	7.45344	0.454444	0.227222	+	0.973843i	$0.427036\pi$
$270$	−0.581365	−0.0353807
$271$	19.3298	1.17420	0.587102	−	0.809513i	$-0.300269\pi$
$271$	19.3298	1.17420	0.587102	+	0.809513i	$0.300269\pi$
$272$	2.12295	0.128723
$273$	0.419868	0.0254115
$274$	−9.38739	−0.567113
$275$	0	0
$276$	0.0894787	0.00538598
$277$	21.9563	1.31923	0.659614	−	0.751605i	$-0.270720\pi$
$277$	21.9563	1.31923	0.659614	+	0.751605i	$0.270720\pi$
$278$	−6.96159	−0.417528
$279$	−3.63512	−0.217629
$280$	−1.00000	−0.0597614
$281$	0.172162	0.0102703	0.00513515	−	0.999987i	$-0.498365\pi$
$281$	0.172162	0.0102703	0.00513515	+	0.999987i	$0.498365\pi$
$282$	−0.0839502	−0.00499916
$283$	−14.5989	−0.867817	−0.433909	−	0.900957i	$-0.642866\pi$
$283$	−14.5989	−0.867817	−0.433909	+	0.900957i	$0.642866\pi$
$284$	5.96038	0.353683
$285$	0.534747	0.0316757
$286$	0	0
$287$	1.92806	0.113810
$288$	−2.99058	−0.176222
$289$	−12.4931	−0.734887
$290$	−8.02820	−0.471432
$291$	0.00830538	0.000486870 0
$292$	−14.8292	−0.867811
$293$	−4.99491	−0.291806	−0.145903	−	0.989299i	$-0.546609\pi$
$293$	−4.99491	−0.291806	−0.145903	+	0.989299i	$0.546609\pi$
$294$	0.0970464	0.00565986
$295$	−0.178208	−0.0103757
$296$	−5.01306	−0.291378
$297$	0	0
$298$	−19.8855	−1.15194
$299$	−3.98908	−0.230694
$300$	0.0970464	0.00560298
$301$	−3.75423	−0.216390
$302$	−6.91156	−0.397716
$303$	−0.735149	−0.0422332
$304$	5.51022	0.316033
$305$	−13.0127	−0.745104
$306$	−6.34886	−0.362940
$307$	12.1296	0.692270	0.346135	−	0.938185i	$-0.387494\pi$
$307$	12.1296	0.692270	0.346135	+	0.938185i	$0.387494\pi$
$308$	0	0
$309$	−0.365071	−0.0207681
$310$	1.21552	0.0690371
$311$	−5.79026	−0.328335	−0.164168	−	0.986432i	$-0.552494\pi$
$311$	−5.79026	−0.328335	−0.164168	+	0.986432i	$0.552494\pi$
$312$	−0.419868	−0.0237703
$313$	−8.21454	−0.464313	−0.232157	−	0.972678i	$-0.574578\pi$
$313$	−8.21454	−0.464313	−0.232157	+	0.972678i	$0.574578\pi$
$314$	9.00126	0.507970
$315$	2.99058	0.168500
$316$	13.1971	0.742394
$317$	−9.85272	−0.553384	−0.276692	−	0.960959i	$-0.589238\pi$
$317$	−9.85272	−0.553384	−0.276692	+	0.960959i	$0.589238\pi$
$318$	−0.0707622	−0.00396815
$319$	0	0
$320$	1.00000	0.0559017
$321$	−1.29232	−0.0721301
$322$	−0.922019	−0.0513821
$323$	11.6979	0.650890
$324$	8.91533	0.495296
$325$	−4.32646	−0.239989
$326$	−7.07107	−0.391630
$327$	−0.780799	−0.0431783
$328$	−1.92806	−0.106459
$329$	0.865052	0.0476919
$330$	0	0
$331$	−5.71671	−0.314219	−0.157109	−	0.987581i	$-0.550218\pi$
$331$	−5.71671	−0.314219	−0.157109	+	0.987581i	$0.550218\pi$
$332$	2.63016	0.144349
$333$	14.9920	0.821554
$334$	−19.7421	−1.08024
$335$	−3.55383	−0.194166
$336$	−0.0970464	−0.00529432
$337$	−29.1366	−1.58717	−0.793585	−	0.608459i	$-0.791788\pi$
$337$	−29.1366	−1.58717	−0.793585	+	0.608459i	$0.791788\pi$
$338$	5.71826	0.311032
$339$	0.890394	0.0483596
$340$	2.12295	0.115133
$341$	0	0
$342$	−16.4788	−0.891070
$343$	−1.00000	−0.0539949
$344$	3.75423	0.202414
$345$	0.0894787	0.00481737
$346$	13.9142	0.748030
$347$	9.51730	0.510915	0.255458	−	0.966820i	$-0.417774\pi$
$347$	9.51730	0.510915	0.255458	+	0.966820i	$0.417774\pi$
$348$	−0.779108	−0.0417646
$349$	−8.24510	−0.441350	−0.220675	−	0.975347i	$-0.570826\pi$
$349$	−8.24510	−0.441350	−0.220675	+	0.975347i	$0.570826\pi$
$350$	−1.00000	−0.0534522
$351$	2.51525	0.134254
$352$	0	0
$353$	29.6347	1.57730	0.788649	−	0.614843i	$-0.210781\pi$
$353$	29.6347	1.57730	0.788649	+	0.614843i	$0.210781\pi$
$354$	−0.0172945	−0.000919192 0
$355$	5.96038	0.316344
$356$	−18.4678	−0.978791
$357$	−0.206025	−0.0109040
$358$	−3.61007	−0.190798
$359$	−24.7795	−1.30781	−0.653907	−	0.756575i	$-0.726871\pi$
$359$	−24.7795	−1.30781	−0.653907	+	0.756575i	$0.726871\pi$
$360$	−2.99058	−0.157618
$361$	11.3625	0.598028
$362$	−1.95017	−0.102499
$363$	0	0
$364$	4.32646	0.226768
$365$	−14.8292	−0.776194
$366$	−1.26283	−0.0660094
$367$	12.8613	0.671352	0.335676	−	0.941977i	$-0.391035\pi$
$367$	12.8613	0.671352	0.335676	+	0.941977i	$0.391035\pi$
$368$	0.922019	0.0480636
$369$	5.76603	0.300167
$370$	−5.01306	−0.260616
$371$	0.729158	0.0378560
$372$	0.117962	0.00611606
$373$	22.7708	1.17903	0.589514	−	0.807758i	$-0.299319\pi$
$373$	22.7708	1.17903	0.589514	+	0.807758i	$0.299319\pi$
$374$	0	0
$375$	0.0970464	0.00501146
$376$	−0.865052	−0.0446117
$377$	34.7337	1.78888
$378$	0.581365	0.0299022
$379$	11.0712	0.568691	0.284346	−	0.958722i	$-0.408224\pi$
$379$	11.0712	0.568691	0.284346	+	0.958722i	$0.408224\pi$
$380$	5.51022	0.282668
$381$	−0.837709	−0.0429171
$382$	12.7463	0.652156
$383$	28.6974	1.46637	0.733184	−	0.680030i	$-0.238033\pi$
$383$	28.6974	1.46637	0.733184	+	0.680030i	$0.238033\pi$
$384$	0.0970464	0.00495238
$385$	0	0
$386$	−14.5920	−0.742711
$387$	−11.2273	−0.570717
$388$	0.0855815	0.00434474
$389$	1.68387	0.0853754	0.0426877	−	0.999088i	$-0.486408\pi$
$389$	1.68387	0.0853754	0.0426877	+	0.999088i	$0.486408\pi$
$390$	−0.419868	−0.0212608
$391$	1.95740	0.0989900
$392$	1.00000	0.0505076
$393$	0.486928	0.0245623
$394$	24.4598	1.23227
$395$	13.1971	0.664017
$396$	0	0
$397$	16.9399	0.850189	0.425095	−	0.905149i	$-0.360241\pi$
$397$	16.9399	0.850189	0.425095	+	0.905149i	$0.360241\pi$
$398$	−10.4998	−0.526310
$399$	−0.534747	−0.0267709
$400$	1.00000	0.0500000
$401$	−15.0165	−0.749889	−0.374944	−	0.927047i	$-0.622338\pi$
$401$	−15.0165	−0.749889	−0.374944	+	0.927047i	$0.622338\pi$
$402$	−0.344887	−0.0172014
$403$	−5.25891	−0.261965
$404$	−7.57523	−0.376882
$405$	8.91533	0.443006
$406$	8.02820	0.398433
$407$	0	0
$408$	0.206025	0.0101997
$409$	33.7143	1.66707	0.833533	−	0.552469i	$-0.186314\pi$
$409$	33.7143	1.66707	0.833533	+	0.552469i	$0.186314\pi$
$410$	−1.92806	−0.0952202
$411$	−0.911012	−0.0449369
$412$	−3.76181	−0.185331
$413$	0.178208	0.00876907
$414$	−2.75737	−0.135518
$415$	2.63016	0.129109
$416$	−4.32646	−0.212122
$417$	−0.675598	−0.0330842
$418$	0	0
$419$	−17.7859	−0.868900	−0.434450	−	0.900696i	$-0.643057\pi$
$419$	−17.7859	−0.868900	−0.434450	+	0.900696i	$0.643057\pi$
$420$	−0.0970464	−0.00473538
$421$	18.2534	0.889619	0.444809	−	0.895625i	$-0.353271\pi$
$421$	18.2534	0.889619	0.444809	+	0.895625i	$0.353271\pi$
$422$	14.9790	0.729168
$423$	2.58701	0.125785
$424$	−0.729158	−0.0354111
$425$	2.12295	0.102978
$426$	0.578434	0.0280252
$427$	13.0127	0.629728
$428$	−13.3165	−0.643677
$429$	0	0
$430$	3.75423	0.181045
$431$	27.5547	1.32726	0.663632	−	0.748059i	$-0.269014\pi$
$431$	27.5547	1.32726	0.663632	+	0.748059i	$0.269014\pi$
$432$	−0.581365	−0.0279709
$433$	−3.18070	−0.152855	−0.0764273	−	0.997075i	$-0.524351\pi$
$433$	−3.18070	−0.152855	−0.0764273	+	0.997075i	$0.524351\pi$
$434$	−1.21552	−0.0583470
$435$	−0.779108	−0.0373554
$436$	−8.04563	−0.385316
$437$	5.08053	0.243035
$438$	−1.43912	−0.0687637
$439$	39.0123	1.86196	0.930978	−	0.365076i	$-0.118957\pi$
$439$	39.0123	1.86196	0.930978	+	0.365076i	$0.118957\pi$
$440$	0	0
$441$	−2.99058	−0.142409
$442$	−9.18486	−0.436879
$443$	5.00748	0.237912	0.118956	−	0.992900i	$-0.462045\pi$
$443$	5.00748	0.237912	0.118956	+	0.992900i	$0.462045\pi$
$444$	−0.486499	−0.0230882
$445$	−18.4678	−0.875457
$446$	−6.98647	−0.330819
$447$	−1.92982	−0.0912772
$448$	−1.00000	−0.0472456
$449$	12.5770	0.593546	0.296773	−	0.954948i	$-0.404090\pi$
$449$	12.5770	0.593546	0.296773	+	0.954948i	$0.404090\pi$
$450$	−2.99058	−0.140977
$451$	0	0
$452$	9.17493	0.431552
$453$	−0.670742	−0.0315142
$454$	−23.6112	−1.10813
$455$	4.32646	0.202828
$456$	0.534747	0.0250418
$457$	−36.1549	−1.69125	−0.845627	−	0.533775i	$-0.820773\pi$
$457$	−36.1549	−1.69125	−0.845627	+	0.533775i	$0.820773\pi$
$458$	21.0574	0.983946
$459$	−1.23421	−0.0576079
$460$	0.922019	0.0429894
$461$	26.0248	1.21209	0.606047	−	0.795429i	$-0.292754\pi$
$461$	26.0248	1.21209	0.606047	+	0.795429i	$0.292754\pi$
$462$	0	0
$463$	2.55526	0.118753	0.0593766	−	0.998236i	$-0.481089\pi$
$463$	2.55526	0.118753	0.0593766	+	0.998236i	$0.481089\pi$
$464$	−8.02820	−0.372700
$465$	0.117962	0.00547037
$466$	−29.7941	−1.38018
$467$	14.4176	0.667169	0.333585	−	0.942720i	$-0.391742\pi$
$467$	14.4176	0.667169	0.333585	+	0.942720i	$0.391742\pi$
$468$	12.9386	0.598089
$469$	3.55383	0.164101
$470$	−0.865052	−0.0399019
$471$	0.873540	0.0402506
$472$	−0.178208	−0.00820271
$473$	0	0
$474$	1.28073	0.0588259
$475$	5.51022	0.252826
$476$	−2.12295	−0.0973052
$477$	2.18061	0.0998432
$478$	−18.4084	−0.841981
$479$	32.6066	1.48984	0.744918	−	0.667156i	$-0.232489\pi$
$479$	32.6066	1.48984	0.744918	+	0.667156i	$0.232489\pi$
$480$	0.0970464	0.00442954
$481$	21.6888	0.988924
$482$	−2.15635	−0.0982190
$483$	−0.0894787	−0.00407142
$484$	0	0
$485$	0.0855815	0.00388606
$486$	2.60929	0.118360
$487$	16.9041	0.765997	0.382999	−	0.923749i	$-0.374891\pi$
$487$	16.9041	0.765997	0.382999	+	0.923749i	$0.374891\pi$
$488$	−13.0127	−0.589056
$489$	−0.686222	−0.0310320
$490$	1.00000	0.0451754
$491$	−38.7664	−1.74950	−0.874752	−	0.484571i	$-0.838976\pi$
$491$	−38.7664	−1.74950	−0.874752	+	0.484571i	$0.838976\pi$
$492$	−0.187112	−0.00843564
$493$	−17.0435	−0.767599
$494$	−23.8398	−1.07260
$495$	0	0
$496$	1.21552	0.0545786
$497$	−5.96038	−0.267360
$498$	0.255248	0.0114379
$499$	−37.6575	−1.68578	−0.842890	−	0.538086i	$-0.819148\pi$
$499$	−37.6575	−1.68578	−0.842890	+	0.538086i	$0.819148\pi$
$500$	1.00000	0.0447214
$501$	−1.91590	−0.0855963
$502$	19.6172	0.875559
$503$	28.4963	1.27059	0.635294	−	0.772270i	$-0.280879\pi$
$503$	28.4963	1.27059	0.635294	+	0.772270i	$0.280879\pi$
$504$	2.99058	0.133211
$505$	−7.57523	−0.337093
$506$	0	0
$507$	0.554937	0.0246456
$508$	−8.63204	−0.382985
$509$	21.2554	0.942130	0.471065	−	0.882099i	$-0.343870\pi$
$509$	21.2554	0.942130	0.471065	+	0.882099i	$0.343870\pi$
$510$	0.206025	0.00912293
$511$	14.8292	0.656003
$512$	1.00000	0.0441942
$513$	−3.20345	−0.141436
$514$	−5.06003	−0.223189
$515$	−3.76181	−0.165765
$516$	0.364335	0.0160389
$517$	0	0
$518$	5.01306	0.220261
$519$	1.35032	0.0592724
$520$	−4.32646	−0.189728
$521$	9.47858	0.415264	0.207632	−	0.978207i	$-0.433424\pi$
$521$	9.47858	0.415264	0.207632	+	0.978207i	$0.433424\pi$
$522$	24.0090	1.05084
$523$	−21.3021	−0.931475	−0.465737	−	0.884923i	$-0.654211\pi$
$523$	−21.3021	−0.931475	−0.465737	+	0.884923i	$0.654211\pi$
$524$	5.01748	0.219189
$525$	−0.0970464	−0.00423545
$526$	−16.0394	−0.699351
$527$	2.58050	0.112408
$528$	0	0
$529$	−22.1499	−0.963038
$530$	−0.729158	−0.0316726
$531$	0.532947	0.0231279
$532$	−5.51022	−0.238898
$533$	8.34169	0.361319
$534$	−1.79223	−0.0775575
$535$	−13.3165	−0.575722
$536$	−3.55383	−0.153502
$537$	−0.350344	−0.0151185
$538$	7.45344	0.321341
$539$	0	0
$540$	−0.581365	−0.0250180
$541$	−43.2284	−1.85853	−0.929267	−	0.369408i	$-0.879560\pi$
$541$	−43.2284	−1.85853	−0.929267	+	0.369408i	$0.879560\pi$
$542$	19.3298	0.830287
$543$	−0.189257	−0.00812181
$544$	2.12295	0.0910207
$545$	−8.04563	−0.344637
$546$	0.419868	0.0179687
$547$	−22.1833	−0.948491	−0.474245	−	0.880393i	$-0.657279\pi$
$547$	−22.1833	−0.948491	−0.474245	+	0.880393i	$0.657279\pi$
$548$	−9.38739	−0.401009
$549$	38.9155	1.66087
$550$	0	0
$551$	−44.2372	−1.88457
$552$	0.0894787	0.00380847
$553$	−13.1971	−0.561197
$554$	21.9563	0.932835
$555$	−0.486499	−0.0206507
$556$	−6.96159	−0.295237
$557$	3.41806	0.144828	0.0724140	−	0.997375i	$-0.476930\pi$
$557$	3.41806	0.144828	0.0724140	+	0.997375i	$0.476930\pi$
$558$	−3.63512	−0.153887
$559$	−16.2425	−0.686985
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	0.172162	0.00726220
$563$	−16.7614	−0.706411	−0.353205	−	0.935546i	$-0.614908\pi$
$563$	−16.7614	−0.706411	−0.353205	+	0.935546i	$0.614908\pi$
$564$	−0.0839502	−0.00353494
$565$	9.17493	0.385992
$566$	−14.5989	−0.613640
$567$	−8.91533	−0.374409
$568$	5.96038	0.250092
$569$	30.7237	1.28800	0.644001	−	0.765024i	$-0.277273\pi$
$569$	30.7237	1.28800	0.644001	+	0.765024i	$0.277273\pi$
$570$	0.534747	0.0223981
$571$	26.7892	1.12109	0.560546	−	0.828123i	$-0.310591\pi$
$571$	26.7892	1.12109	0.560546	+	0.828123i	$0.310591\pi$
$572$	0	0
$573$	1.23698	0.0516756
$574$	1.92806	0.0804758
$575$	0.922019	0.0384509
$576$	−2.99058	−0.124608
$577$	8.18950	0.340933	0.170467	−	0.985363i	$-0.445472\pi$
$577$	8.18950	0.340933	0.170467	+	0.985363i	$0.445472\pi$
$578$	−12.4931	−0.519644
$579$	−1.41610	−0.0588510
$580$	−8.02820	−0.333353
$581$	−2.63016	−0.109117
$582$	0.00830538	0.000344269 0
$583$	0	0
$584$	−14.8292	−0.613635
$585$	12.9386	0.534947
$586$	−4.99491	−0.206338
$587$	−19.3228	−0.797537	−0.398768	−	0.917052i	$-0.630562\pi$
$587$	−19.3228	−0.797537	−0.398768	+	0.917052i	$0.630562\pi$
$588$	0.0970464	0.00400213
$589$	6.69780	0.275978
$590$	−0.178208	−0.00733673
$591$	2.37374	0.0976427
$592$	−5.01306	−0.206035
$593$	−46.6962	−1.91758	−0.958792	−	0.284109i	$-0.908302\pi$
$593$	−46.6962	−1.91758	−0.958792	+	0.284109i	$0.908302\pi$
$594$	0	0
$595$	−2.12295	−0.0870325
$596$	−19.8855	−0.814542
$597$	−1.01897	−0.0417038
$598$	−3.98908	−0.163126
$599$	−20.0821	−0.820533	−0.410267	−	0.911966i	$-0.634564\pi$
$599$	−20.0821	−0.820533	−0.410267	+	0.911966i	$0.634564\pi$
$600$	0.0970464	0.00396190
$601$	−19.0105	−0.775456	−0.387728	−	0.921774i	$-0.626740\pi$
$601$	−19.0105	−0.775456	−0.387728	+	0.921774i	$0.626740\pi$
$602$	−3.75423	−0.153011
$603$	10.6280	0.432806
$604$	−6.91156	−0.281227
$605$	0	0
$606$	−0.735149	−0.0298634
$607$	27.4450	1.11396	0.556979	−	0.830526i	$-0.311960\pi$
$607$	27.4450	1.11396	0.556979	+	0.830526i	$0.311960\pi$
$608$	5.51022	0.223469
$609$	0.779108	0.0315711
$610$	−13.0127	−0.526868
$611$	3.74261	0.151410
$612$	−6.34886	−0.256637
$613$	9.50074	0.383731	0.191866	−	0.981421i	$-0.438546\pi$
$613$	9.50074	0.383731	0.191866	+	0.981421i	$0.438546\pi$
$614$	12.1296	0.489509
$615$	−0.187112	−0.00754507
$616$	0	0
$617$	33.6314	1.35395	0.676974	−	0.736007i	$-0.263291\pi$
$617$	33.6314	1.35395	0.676974	+	0.736007i	$0.263291\pi$
$618$	−0.365071	−0.0146853
$619$	27.2183	1.09400	0.546998	−	0.837134i	$-0.315770\pi$
$619$	27.2183	1.09400	0.546998	+	0.837134i	$0.315770\pi$
$620$	1.21552	0.0488166
$621$	−0.536029	−0.0215101
$622$	−5.79026	−0.232168
$623$	18.4678	0.739896
$624$	−0.419868	−0.0168082
$625$	1.00000	0.0400000
$626$	−8.21454	−0.328319
$627$	0	0
$628$	9.00126	0.359189
$629$	−10.6425	−0.424343
$630$	2.99058	0.119148
$631$	28.1766	1.12169	0.560846	−	0.827920i	$-0.310476\pi$
$631$	28.1766	1.12169	0.560846	+	0.827920i	$0.310476\pi$
$632$	13.1971	0.524952
$633$	1.45366	0.0577779
$634$	−9.85272	−0.391301
$635$	−8.63204	−0.342552
$636$	−0.0707622	−0.00280590
$637$	−4.32646	−0.171421
$638$	0	0
$639$	−17.8250	−0.705146
$640$	1.00000	0.0395285
$641$	9.40813	0.371599	0.185799	−	0.982588i	$-0.440513\pi$
$641$	9.40813	0.371599	0.185799	+	0.982588i	$0.440513\pi$
$642$	−1.29232	−0.0510037
$643$	−41.8999	−1.65237	−0.826185	−	0.563398i	$-0.809494\pi$
$643$	−41.8999	−1.65237	−0.826185	+	0.563398i	$0.809494\pi$
$644$	−0.922019	−0.0363326
$645$	0.364335	0.0143457
$646$	11.6979	0.460249
$647$	17.5282	0.689105	0.344553	−	0.938767i	$-0.388031\pi$
$647$	17.5282	0.689105	0.344553	+	0.938767i	$0.388031\pi$
$648$	8.91533	0.350227
$649$	0	0
$650$	−4.32646	−0.169698
$651$	−0.117962	−0.00462330
$652$	−7.07107	−0.276924
$653$	47.2120	1.84755	0.923775	−	0.382937i	$-0.125087\pi$
$653$	47.2120	1.84755	0.923775	+	0.382937i	$0.125087\pi$
$654$	−0.780799	−0.0305317
$655$	5.01748	0.196049
$656$	−1.92806	−0.0752782
$657$	44.3478	1.73017
$658$	0.865052	0.0337232
$659$	−4.53445	−0.176637	−0.0883185	−	0.996092i	$-0.528149\pi$
$659$	−4.53445	−0.176637	−0.0883185	+	0.996092i	$0.528149\pi$
$660$	0	0
$661$	−33.2581	−1.29359	−0.646794	−	0.762664i	$-0.723891\pi$
$661$	−33.2581	−1.29359	−0.646794	+	0.762664i	$0.723891\pi$
$662$	−5.71671	−0.222186
$663$	−0.891358	−0.0346175
$664$	2.63016	0.102070
$665$	−5.51022	−0.213677
$666$	14.9920	0.580927
$667$	−7.40215	−0.286612
$668$	−19.7421	−0.763846
$669$	−0.678012	−0.0262134
$670$	−3.55383	−0.137296
$671$	0	0
$672$	−0.0970464	−0.00374365
$673$	3.32610	0.128212	0.0641059	−	0.997943i	$-0.479580\pi$
$673$	3.32610	0.128212	0.0641059	+	0.997943i	$0.479580\pi$
$674$	−29.1366	−1.12230
$675$	−0.581365	−0.0223767
$676$	5.71826	0.219933
$677$	8.90346	0.342188	0.171094	−	0.985255i	$-0.445270\pi$
$677$	8.90346	0.342188	0.171094	+	0.985255i	$0.445270\pi$
$678$	0.890394	0.0341954
$679$	−0.0855815	−0.00328432
$680$	2.12295	0.0814114
$681$	−2.29138	−0.0878060
$682$	0	0
$683$	26.5881	1.01736	0.508682	−	0.860954i	$-0.330133\pi$
$683$	26.5881	1.01736	0.508682	+	0.860954i	$0.330133\pi$
$684$	−16.4788	−0.630081
$685$	−9.38739	−0.358674
$686$	−1.00000	−0.0381802
$687$	2.04354	0.0779660
$688$	3.75423	0.143129
$689$	3.15468	0.120184
$690$	0.0894787	0.00340639
$691$	−28.8459	−1.09735	−0.548674	−	0.836036i	$-0.684867\pi$
$691$	−28.8459	−1.09735	−0.548674	+	0.836036i	$0.684867\pi$
$692$	13.9142	0.528937
$693$	0	0
$694$	9.51730	0.361272
$695$	−6.96159	−0.264068
$696$	−0.779108	−0.0295320
$697$	−4.09318	−0.155040
$698$	−8.24510	−0.312082
$699$	−2.89141	−0.109363
$700$	−1.00000	−0.0377964
$701$	10.9136	0.412203	0.206101	−	0.978531i	$-0.433922\pi$
$701$	10.9136	0.412203	0.206101	+	0.978531i	$0.433922\pi$
$702$	2.51525	0.0949320
$703$	−27.6231	−1.04182
$704$	0	0
$705$	−0.0839502	−0.00316175
$706$	29.6347	1.11532
$707$	7.57523	0.284896
$708$	−0.0172945	−0.000649967 0
$709$	−35.0191	−1.31517	−0.657584	−	0.753381i	$-0.728422\pi$
$709$	−35.0191	−1.31517	−0.657584	+	0.753381i	$0.728422\pi$
$710$	5.96038	0.223689
$711$	−39.4670	−1.48013
$712$	−18.4678	−0.692110
$713$	1.12074	0.0419719
$714$	−0.206025	−0.00771028
$715$	0	0
$716$	−3.61007	−0.134915
$717$	−1.78647	−0.0667169
$718$	−24.7795	−0.924765
$719$	43.4663	1.62102	0.810510	−	0.585725i	$-0.199190\pi$
$719$	43.4663	1.62102	0.810510	+	0.585725i	$0.199190\pi$
$720$	−2.99058	−0.111452
$721$	3.76181	0.140097
$722$	11.3625	0.422870
$723$	−0.209266	−0.00778269
$724$	−1.95017	−0.0724776
$725$	−8.02820	−0.298160
$726$	0	0
$727$	−52.4475	−1.94517	−0.972585	−	0.232548i	$-0.925294\pi$
$727$	−52.4475	−1.94517	−0.972585	+	0.232548i	$0.925294\pi$
$728$	4.32646	0.160349
$729$	−26.4928	−0.981213
$730$	−14.8292	−0.548852
$731$	7.97004	0.294783
$732$	−1.26283	−0.0466757
$733$	−26.2363	−0.969059	−0.484529	−	0.874775i	$-0.661009\pi$
$733$	−26.2363	−0.969059	−0.484529	+	0.874775i	$0.661009\pi$
$734$	12.8613	0.474718
$735$	0.0970464	0.00357961
$736$	0.922019	0.0339861
$737$	0	0
$738$	5.76603	0.212250
$739$	−29.8531	−1.09816	−0.549081	−	0.835769i	$-0.685022\pi$
$739$	−29.8531	−1.09816	−0.549081	+	0.835769i	$0.685022\pi$
$740$	−5.01306	−0.184284
$741$	−2.31356	−0.0849909
$742$	0.729158	0.0267682
$743$	−39.8331	−1.46133	−0.730667	−	0.682734i	$-0.760791\pi$
$743$	−39.8331	−1.46133	−0.730667	+	0.682734i	$0.760791\pi$
$744$	0.117962	0.00432471
$745$	−19.8855	−0.728548
$746$	22.7708	0.833698
$747$	−7.86571	−0.287791
$748$	0	0
$749$	13.3165	0.486574
$750$	0.0970464	0.00354363
$751$	48.2801	1.76177	0.880884	−	0.473333i	$-0.156949\pi$
$751$	48.2801	1.76177	0.880884	+	0.473333i	$0.156949\pi$
$752$	−0.865052	−0.0315452
$753$	1.90378	0.0693776
$754$	34.7337	1.26493
$755$	−6.91156	−0.251537
$756$	0.581365	0.0211440
$757$	−8.61332	−0.313056	−0.156528	−	0.987673i	$-0.550030\pi$
$757$	−8.61332	−0.313056	−0.156528	+	0.987673i	$0.550030\pi$
$758$	11.0712	0.402125
$759$	0	0
$760$	5.51022	0.199877
$761$	−0.634416	−0.0229975	−0.0114988	−	0.999934i	$-0.503660\pi$
$761$	−0.634416	−0.0229975	−0.0114988	+	0.999934i	$0.503660\pi$
$762$	−0.837709	−0.0303470
$763$	8.04563	0.291271
$764$	12.7463	0.461144
$765$	−6.34886	−0.229543
$766$	28.6974	1.03688
$767$	0.771012	0.0278396
$768$	0.0970464	0.00350186
$769$	−34.4997	−1.24409	−0.622046	−	0.782981i	$-0.713698\pi$
$769$	−34.4997	−1.24409	−0.622046	+	0.782981i	$0.713698\pi$
$770$	0	0
$771$	−0.491058	−0.0176850
$772$	−14.5920	−0.525176
$773$	−32.9878	−1.18649	−0.593244	−	0.805023i	$-0.702153\pi$
$773$	−32.9878	−1.18649	−0.593244	+	0.805023i	$0.702153\pi$
$774$	−11.2273	−0.403558
$775$	1.21552	0.0436629
$776$	0.0855815	0.00307220
$777$	0.486499	0.0174531
$778$	1.68387	0.0603695
$779$	−10.6241	−0.380646
$780$	−0.419868	−0.0150337
$781$	0	0
$782$	1.95740	0.0699965
$783$	4.66731	0.166796
$784$	1.00000	0.0357143
$785$	9.00126	0.321269
$786$	0.486928	0.0173682
$787$	−33.6231	−1.19853	−0.599266	−	0.800550i	$-0.704541\pi$
$787$	−33.6231	−1.19853	−0.599266	+	0.800550i	$0.704541\pi$
$788$	24.4598	0.871346
$789$	−1.55657	−0.0554152
$790$	13.1971	0.469531
$791$	−9.17493	−0.326223
$792$	0	0
$793$	56.2988	1.99923
$794$	16.9399	0.601175
$795$	−0.0707622	−0.00250968
$796$	−10.4998	−0.372157
$797$	45.0594	1.59609	0.798043	−	0.602600i	$-0.205869\pi$
$797$	45.0594	1.59609	0.798043	+	0.602600i	$0.205869\pi$
$798$	−0.534747	−0.0189299
$799$	−1.83646	−0.0649694
$800$	1.00000	0.0353553
$801$	55.2294	1.95144
$802$	−15.0165	−0.530252
$803$	0	0
$804$	−0.344887	−0.0121632
$805$	−0.922019	−0.0324969
$806$	−5.25891	−0.185237
$807$	0.723330	0.0254624
$808$	−7.57523	−0.266496
$809$	−12.1005	−0.425432	−0.212716	−	0.977114i	$-0.568231\pi$
$809$	−12.1005	−0.425432	−0.212716	+	0.977114i	$0.568231\pi$
$810$	8.91533	0.313253
$811$	−19.6540	−0.690147	−0.345073	−	0.938576i	$-0.612146\pi$
$811$	−19.6540	−0.690147	−0.345073	+	0.938576i	$0.612146\pi$
$812$	8.02820	0.281735
$813$	1.87589	0.0657904
$814$	0	0
$815$	−7.07107	−0.247689
$816$	0.206025	0.00721231
$817$	20.6866	0.723734
$818$	33.7143	1.17879
$819$	−12.9386	−0.452113
$820$	−1.92806	−0.0673309
$821$	5.45071	0.190231	0.0951156	−	0.995466i	$-0.469678\pi$
$821$	5.45071	0.190231	0.0951156	+	0.995466i	$0.469678\pi$
$822$	−0.911012	−0.0317752
$823$	34.8782	1.21578	0.607889	−	0.794022i	$-0.292016\pi$
$823$	34.8782	1.21578	0.607889	+	0.794022i	$0.292016\pi$
$824$	−3.76181	−0.131049
$825$	0	0
$826$	0.178208	0.00620067
$827$	−16.7849	−0.583670	−0.291835	−	0.956469i	$-0.594266\pi$
$827$	−16.7849	−0.583670	−0.291835	+	0.956469i	$0.594266\pi$
$828$	−2.75737	−0.0958254
$829$	−7.98157	−0.277211	−0.138606	−	0.990348i	$-0.544262\pi$
$829$	−7.98157	−0.277211	−0.138606	+	0.990348i	$0.544262\pi$
$830$	2.63016	0.0912942
$831$	2.13078	0.0739160
$832$	−4.32646	−0.149993
$833$	2.12295	0.0735558
$834$	−0.675598	−0.0233940
$835$	−19.7421	−0.683205
$836$	0	0
$837$	−0.706662	−0.0244258
$838$	−17.7859	−0.614405
$839$	−6.47669	−0.223600	−0.111800	−	0.993731i	$-0.535662\pi$
$839$	−6.47669	−0.223600	−0.111800	+	0.993731i	$0.535662\pi$
$840$	−0.0970464	−0.00334842
$841$	35.4520	1.22248
$842$	18.2534	0.629055
$843$	0.0167077	0.000575443 0
$844$	14.9790	0.515600
$845$	5.71826	0.196714
$846$	2.58701	0.0889432
$847$	0	0
$848$	−0.729158	−0.0250394
$849$	−1.41678	−0.0486236
$850$	2.12295	0.0728166
$851$	−4.62213	−0.158445
$852$	0.578434	0.0198168
$853$	−11.1007	−0.380081	−0.190040	−	0.981776i	$-0.560862\pi$
$853$	−11.1007	−0.380081	−0.190040	+	0.981776i	$0.560862\pi$
$854$	13.0127	0.445285
$855$	−16.4788	−0.563562
$856$	−13.3165	−0.455148
$857$	−21.2871	−0.727154	−0.363577	−	0.931564i	$-0.618445\pi$
$857$	−21.2871	−0.727154	−0.363577	+	0.931564i	$0.618445\pi$
$858$	0	0
$859$	14.8904	0.508053	0.254027	−	0.967197i	$-0.418245\pi$
$859$	14.8904	0.508053	0.254027	+	0.967197i	$0.418245\pi$
$860$	3.75423	0.128018
$861$	0.187112	0.00637675
$862$	27.5547	0.938518
$863$	−47.6345	−1.62150	−0.810748	−	0.585395i	$-0.800939\pi$
$863$	−47.6345	−1.62150	−0.810748	+	0.585395i	$0.800939\pi$
$864$	−0.581365	−0.0197784
$865$	13.9142	0.473095
$866$	−3.18070	−0.108084
$867$	−1.21241	−0.0411756
$868$	−1.21552	−0.0412576
$869$	0	0
$870$	−0.779108	−0.0264142
$871$	15.3755	0.520979
$872$	−8.04563	−0.272459
$873$	−0.255939	−0.00866221
$874$	5.08053	0.171851
$875$	−1.00000	−0.0338062
$876$	−1.43912	−0.0486233
$877$	36.4415	1.23054	0.615270	−	0.788316i	$-0.289047\pi$
$877$	36.4415	1.23054	0.615270	+	0.788316i	$0.289047\pi$
$878$	39.0123	1.31660
$879$	−0.484738	−0.0163498
$880$	0	0
$881$	52.4724	1.76784	0.883920	−	0.467639i	$-0.154895\pi$
$881$	52.4724	1.76784	0.883920	+	0.467639i	$0.154895\pi$
$882$	−2.99058	−0.100698
$883$	−6.85399	−0.230655	−0.115328	−	0.993328i	$-0.536792\pi$
$883$	−6.85399	−0.230655	−0.115328	+	0.993328i	$0.536792\pi$
$884$	−9.18486	−0.308920
$885$	−0.0172945	−0.000581348 0
$886$	5.00748	0.168229
$887$	−23.4947	−0.788876	−0.394438	−	0.918922i	$-0.629061\pi$
$887$	−23.4947	−0.788876	−0.394438	+	0.918922i	$0.629061\pi$
$888$	−0.486499	−0.0163258
$889$	8.63204	0.289509
$890$	−18.4678	−0.619042
$891$	0	0
$892$	−6.98647	−0.233924
$893$	−4.76663	−0.159509
$894$	−1.92982	−0.0645427
$895$	−3.61007	−0.120671
$896$	−1.00000	−0.0334077
$897$	−0.387126	−0.0129258
$898$	12.5770	0.419700
$899$	−9.75846	−0.325463
$900$	−2.99058	−0.0996861
$901$	−1.54797	−0.0515702
$902$	0	0
$903$	−0.364335	−0.0121243
$904$	9.17493	0.305153
$905$	−1.95017	−0.0648260
$906$	−0.670742	−0.0222839
$907$	33.9046	1.12578	0.562892	−	0.826530i	$-0.309689\pi$
$907$	33.9046	1.12578	0.562892	+	0.826530i	$0.309689\pi$
$908$	−23.6112	−0.783565
$909$	22.6543	0.751397
$910$	4.32646	0.143421
$911$	−48.8133	−1.61726	−0.808628	−	0.588320i	$-0.799790\pi$
$911$	−48.8133	−1.61726	−0.808628	+	0.588320i	$0.799790\pi$
$912$	0.534747	0.0177073
$913$	0	0
$914$	−36.1549	−1.19590
$915$	−1.26283	−0.0417480
$916$	21.0574	0.695755
$917$	−5.01748	−0.165692
$918$	−1.23421	−0.0407349
$919$	16.4073	0.541228	0.270614	−	0.962688i	$-0.412773\pi$
$919$	16.4073	0.541228	0.270614	+	0.962688i	$0.412773\pi$
$920$	0.922019	0.0303981
$921$	1.17713	0.0387878
$922$	26.0248	0.857080
$923$	−25.7873	−0.848801
$924$	0	0
$925$	−5.01306	−0.164828
$926$	2.55526	0.0839711
$927$	11.2500	0.369499
$928$	−8.02820	−0.263539
$929$	−18.9935	−0.623157	−0.311579	−	0.950220i	$-0.600858\pi$
$929$	−18.9935	−0.623157	−0.311579	+	0.950220i	$0.600858\pi$
$930$	0.117962	0.00386813
$931$	5.51022	0.180590
$932$	−29.7941	−0.975937
$933$	−0.561924	−0.0183966
$934$	14.4176	0.471760
$935$	0	0
$936$	12.9386	0.422913
$937$	5.26363	0.171955	0.0859777	−	0.996297i	$-0.472599\pi$
$937$	5.26363	0.171955	0.0859777	+	0.996297i	$0.472599\pi$
$938$	3.55383	0.116037
$939$	−0.797192	−0.0260154
$940$	−0.865052	−0.0282149
$941$	36.6208	1.19380	0.596901	−	0.802315i	$-0.296398\pi$
$941$	36.6208	1.19380	0.596901	+	0.802315i	$0.296398\pi$
$942$	0.873540	0.0284615
$943$	−1.77771	−0.0578902
$944$	−0.178208	−0.00580019
$945$	0.581365	0.0189118
$946$	0	0
$947$	39.3320	1.27812	0.639060	−	0.769157i	$-0.279324\pi$
$947$	39.3320	1.27812	0.639060	+	0.769157i	$0.279324\pi$
$948$	1.28073	0.0415962
$949$	64.1578	2.08265
$950$	5.51022	0.178775
$951$	−0.956171	−0.0310060
$952$	−2.12295	−0.0688052
$953$	−30.1849	−0.977784	−0.488892	−	0.872344i	$-0.662599\pi$
$953$	−30.1849	−0.977784	−0.488892	+	0.872344i	$0.662599\pi$
$954$	2.18061	0.0705998
$955$	12.7463	0.412460
$956$	−18.4084	−0.595370
$957$	0	0
$958$	32.6066	1.05347
$959$	9.38739	0.303134
$960$	0.0970464	0.00313216
$961$	−29.5225	−0.952339
$962$	21.6888	0.699275
$963$	39.8241	1.28331
$964$	−2.15635	−0.0694513
$965$	−14.5920	−0.469732
$966$	−0.0894787	−0.00287893
$967$	−52.3303	−1.68283	−0.841414	−	0.540391i	$-0.818276\pi$
$967$	−52.3303	−1.68283	−0.841414	+	0.540391i	$0.818276\pi$
$968$	0	0
$969$	1.13524	0.0364692
$970$	0.0855815	0.00274786
$971$	34.1707	1.09659	0.548294	−	0.836285i	$-0.315277\pi$
$971$	34.1707	1.09659	0.548294	+	0.836285i	$0.315277\pi$
$972$	2.60929	0.0836932
$973$	6.96159	0.223178
$974$	16.9041	0.541642
$975$	−0.419868	−0.0134465
$976$	−13.0127	−0.416526
$977$	−43.2069	−1.38231	−0.691156	−	0.722706i	$-0.742898\pi$
$977$	−43.2069	−1.38231	−0.691156	+	0.722706i	$0.742898\pi$
$978$	−0.686222	−0.0219430
$979$	0	0
$980$	1.00000	0.0319438
$981$	24.0611	0.768212
$982$	−38.7664	−1.23709
$983$	43.9672	1.40233	0.701167	−	0.712997i	$-0.252663\pi$
$983$	43.9672	1.40233	0.701167	+	0.712997i	$0.252663\pi$
$984$	−0.187112	−0.00596490
$985$	24.4598	0.779356
$986$	−17.0435	−0.542774
$987$	0.0839502	0.00267216
$988$	−23.8398	−0.758444
$989$	3.46147	0.110068
$990$	0	0
$991$	11.0881	0.352224	0.176112	−	0.984370i	$-0.443648\pi$
$991$	11.0881	0.352224	0.176112	+	0.984370i	$0.443648\pi$
$992$	1.21552	0.0385929
$993$	−0.554786	−0.0176056
$994$	−5.96038	−0.189052
$995$	−10.4998	−0.332868
$996$	0.255248	0.00808783
$997$	14.3365	0.454042	0.227021	−	0.973890i	$-0.427101\pi$
$997$	14.3365	0.454042	0.227021	+	0.973890i	$0.427101\pi$
$998$	−37.6575	−1.19203
$999$	2.91441	0.0922080

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8470.2.a.db.1.4		6
11.3	even	5		770.2.n.g.141.2	yes	12
11.4	even	5		770.2.n.g.71.2	✓	12
11.10	odd	2		8470.2.a.cv.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.n.g.71.2	✓	12	11.4	even	5
770.2.n.g.141.2	yes	12	11.3	even	5
8470.2.a.cv.1.4		6	11.10	odd	2
8470.2.a.db.1.4		6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8470.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +0.0970464 q^{3} +1.00000 q^{4} +1.00000 q^{5} +0.0970464 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.99058 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +0.0970464 q^{3} +1.00000 q^{4} +1.00000 q^{5} +0.0970464 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.99058 q^{9} +1.00000 q^{10} +0.0970464 q^{12} -4.32646 q^{13} -1.00000 q^{14} +0.0970464 q^{15} +1.00000 q^{16} +2.12295 q^{17} -2.99058 q^{18} +5.51022 q^{19} +1.00000 q^{20} -0.0970464 q^{21} +0.922019 q^{23} +0.0970464 q^{24} +1.00000 q^{25} -4.32646 q^{26} -0.581365 q^{27} -1.00000 q^{28} -8.02820 q^{29} +0.0970464 q^{30} +1.21552 q^{31} +1.00000 q^{32} +2.12295 q^{34} -1.00000 q^{35} -2.99058 q^{36} -5.01306 q^{37} +5.51022 q^{38} -0.419868 q^{39} +1.00000 q^{40} -1.92806 q^{41} -0.0970464 q^{42} +3.75423 q^{43} -2.99058 q^{45} +0.922019 q^{46} -0.865052 q^{47} +0.0970464 q^{48} +1.00000 q^{49} +1.00000 q^{50} +0.206025 q^{51} -4.32646 q^{52} -0.729158 q^{53} -0.581365 q^{54} -1.00000 q^{56} +0.534747 q^{57} -8.02820 q^{58} -0.178208 q^{59} +0.0970464 q^{60} -13.0127 q^{61} +1.21552 q^{62} +2.99058 q^{63} +1.00000 q^{64} -4.32646 q^{65} -3.55383 q^{67} +2.12295 q^{68} +0.0894787 q^{69} -1.00000 q^{70} +5.96038 q^{71} -2.99058 q^{72} -14.8292 q^{73} -5.01306 q^{74} +0.0970464 q^{75} +5.51022 q^{76} -0.419868 q^{78} +13.1971 q^{79} +1.00000 q^{80} +8.91533 q^{81} -1.92806 q^{82} +2.63016 q^{83} -0.0970464 q^{84} +2.12295 q^{85} +3.75423 q^{86} -0.779108 q^{87} -18.4678 q^{89} -2.99058 q^{90} +4.32646 q^{91} +0.922019 q^{92} +0.117962 q^{93} -0.865052 q^{94} +5.51022 q^{95} +0.0970464 q^{96} +0.0855815 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{2} - q^{3} + 6 q^{4} + 6 q^{5} - q^{6} - 6 q^{7} + 6 q^{8} - q^{9}+O(q^{10}) \) 6 * q + 6 * q^2 - q^3 + 6 * q^4 + 6 * q^5 - q^6 - 6 * q^7 + 6 * q^8 - q^9	\( 6 q + 6 q^{2} - q^{3} + 6 q^{4} + 6 q^{5} - q^{6} - 6 q^{7} + 6 q^{8} - q^{9} + 6 q^{10} - q^{12} - 6 q^{13} - 6 q^{14} - q^{15} + 6 q^{16} - 21 q^{17} - q^{18} + 3 q^{19} + 6 q^{20} + q^{21} - 10 q^{23} - q^{24} + 6 q^{25} - 6 q^{26} - 4 q^{27} - 6 q^{28} - 10 q^{29} - q^{30} - 4 q^{31} + 6 q^{32} - 21 q^{34} - 6 q^{35} - q^{36} - 2 q^{37} + 3 q^{38} - 26 q^{39} + 6 q^{40} - 7 q^{41} + q^{42} - 19 q^{43} - q^{45} - 10 q^{46} + 10 q^{47} - q^{48} + 6 q^{49} + 6 q^{50} + 4 q^{51} - 6 q^{52} - 16 q^{53} - 4 q^{54} - 6 q^{56} - 16 q^{57} - 10 q^{58} - 3 q^{59} - q^{60} + 8 q^{61} - 4 q^{62} + q^{63} + 6 q^{64} - 6 q^{65} - 27 q^{67} - 21 q^{68} + 4 q^{69} - 6 q^{70} + 4 q^{71} - q^{72} - 13 q^{73} - 2 q^{74} - q^{75} + 3 q^{76} - 26 q^{78} - 14 q^{79} + 6 q^{80} - 14 q^{81} - 7 q^{82} - 51 q^{83} + q^{84} - 21 q^{85} - 19 q^{86} - 8 q^{87} + q^{89} - q^{90} + 6 q^{91} - 10 q^{92} + 4 q^{93} + 10 q^{94} + 3 q^{95} - q^{96} + 7 q^{97} + 6 q^{98}+O(q^{100}) \) 6 * q + 6 * q^2 - q^3 + 6 * q^4 + 6 * q^5 - q^6 - 6 * q^7 + 6 * q^8 - q^9 + 6 * q^10 - q^12 - 6 * q^13 - 6 * q^14 - q^15 + 6 * q^16 - 21 * q^17 - q^18 + 3 * q^19 + 6 * q^20 + q^21 - 10 * q^23 - q^24 + 6 * q^25 - 6 * q^26 - 4 * q^27 - 6 * q^28 - 10 * q^29 - q^30 - 4 * q^31 + 6 * q^32 - 21 * q^34 - 6 * q^35 - q^36 - 2 * q^37 + 3 * q^38 - 26 * q^39 + 6 * q^40 - 7 * q^41 + q^42 - 19 * q^43 - q^45 - 10 * q^46 + 10 * q^47 - q^48 + 6 * q^49 + 6 * q^50 + 4 * q^51 - 6 * q^52 - 16 * q^53 - 4 * q^54 - 6 * q^56 - 16 * q^57 - 10 * q^58 - 3 * q^59 - q^60 + 8 * q^61 - 4 * q^62 + q^63 + 6 * q^64 - 6 * q^65 - 27 * q^67 - 21 * q^68 + 4 * q^69 - 6 * q^70 + 4 * q^71 - q^72 - 13 * q^73 - 2 * q^74 - q^75 + 3 * q^76 - 26 * q^78 - 14 * q^79 + 6 * q^80 - 14 * q^81 - 7 * q^82 - 51 * q^83 + q^84 - 21 * q^85 - 19 * q^86 - 8 * q^87 + q^89 - q^90 + 6 * q^91 - 10 * q^92 + 4 * q^93 + 10 * q^94 + 3 * q^95 - q^96 + 7 * q^97 + 6 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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