LMFDB - Embedded newform 8470.2.a.da.1.5

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:	6.6.38498000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 12x^{4} + 5x^{3} + 38x^{2} - x - 19 $ x^6 - x^5 - 12x^4 + 5x^3 + 38*x^2 - x - 19
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
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			1.5
Root			$2.15804$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.15804 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.15804 q^{6} +1.00000 q^{7} +1.00000 q^{8} -1.65894 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.15804 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.15804 q^{6} +1.00000 q^{7} +1.00000 q^{8} -1.65894 q^{9} -1.00000 q^{10} +1.15804 q^{12} -3.58654 q^{13} +1.00000 q^{14} -1.15804 q^{15} +1.00000 q^{16} +4.29025 q^{17} -1.65894 q^{18} -4.89322 q^{19} -1.00000 q^{20} +1.15804 q^{21} +2.85699 q^{23} +1.15804 q^{24} +1.00000 q^{25} -3.58654 q^{26} -5.39525 q^{27} +1.00000 q^{28} -0.754826 q^{29} -1.15804 q^{30} -9.35354 q^{31} +1.00000 q^{32} +4.29025 q^{34} -1.00000 q^{35} -1.65894 q^{36} -4.24080 q^{37} -4.89322 q^{38} -4.15337 q^{39} -1.00000 q^{40} +2.02891 q^{41} +1.15804 q^{42} -0.0689189 q^{43} +1.65894 q^{45} +2.85699 q^{46} +2.23785 q^{47} +1.15804 q^{48} +1.00000 q^{49} +1.00000 q^{50} +4.96830 q^{51} -3.58654 q^{52} +13.1026 q^{53} -5.39525 q^{54} +1.00000 q^{56} -5.66656 q^{57} -0.754826 q^{58} +1.22512 q^{59} -1.15804 q^{60} -4.03036 q^{61} -9.35354 q^{62} -1.65894 q^{63} +1.00000 q^{64} +3.58654 q^{65} -14.3059 q^{67} +4.29025 q^{68} +3.30852 q^{69} -1.00000 q^{70} -5.79751 q^{71} -1.65894 q^{72} -9.06652 q^{73} -4.24080 q^{74} +1.15804 q^{75} -4.89322 q^{76} -4.15337 q^{78} -5.03269 q^{79} -1.00000 q^{80} -1.27112 q^{81} +2.02891 q^{82} -16.6258 q^{83} +1.15804 q^{84} -4.29025 q^{85} -0.0689189 q^{86} -0.874121 q^{87} +9.92243 q^{89} +1.65894 q^{90} -3.58654 q^{91} +2.85699 q^{92} -10.8318 q^{93} +2.23785 q^{94} +4.89322 q^{95} +1.15804 q^{96} -6.71733 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{2} - 5 q^{3} + 6 q^{4} - 6 q^{5} - 5 q^{6} + 6 q^{7} + 6 q^{8} + 11 q^{9}+O(q^{10}) $ 6 * q + 6 * q^2 - 5 * q^3 + 6 * q^4 - 6 * q^5 - 5 * q^6 + 6 * q^7 + 6 * q^8 + 11 * q^9	$ 6 q + 6 q^{2} - 5 q^{3} + 6 q^{4} - 6 q^{5} - 5 q^{6} + 6 q^{7} + 6 q^{8} + 11 q^{9} - 6 q^{10} - 5 q^{12} + 6 q^{14} + 5 q^{15} + 6 q^{16} - 15 q^{17} + 11 q^{18} - 13 q^{19} - 6 q^{20} - 5 q^{21} - 2 q^{23} - 5 q^{24} + 6 q^{25} - 26 q^{27} + 6 q^{28} - 4 q^{29} + 5 q^{30} - 2 q^{31} + 6 q^{32} - 15 q^{34} - 6 q^{35} + 11 q^{36} - 13 q^{38} - 30 q^{39} - 6 q^{40} - 17 q^{41} - 5 q^{42} + 15 q^{43} - 11 q^{45} - 2 q^{46} - 18 q^{47} - 5 q^{48} + 6 q^{49} + 6 q^{50} + 6 q^{51} + 22 q^{53} - 26 q^{54} + 6 q^{56} - 2 q^{57} - 4 q^{58} - 7 q^{59} + 5 q^{60} - 20 q^{61} - 2 q^{62} + 11 q^{63} + 6 q^{64} - 29 q^{67} - 15 q^{68} + 12 q^{69} - 6 q^{70} - 2 q^{71} + 11 q^{72} + q^{73} - 5 q^{75} - 13 q^{76} - 30 q^{78} - 12 q^{79} - 6 q^{80} + 22 q^{81} - 17 q^{82} - 35 q^{83} - 5 q^{84} + 15 q^{85} + 15 q^{86} - 29 q^{89} - 11 q^{90} - 2 q^{92} + 8 q^{93} - 18 q^{94} + 13 q^{95} - 5 q^{96} - 19 q^{97} + 6 q^{98}+O(q^{100}) $ 6 * q + 6 * q^2 - 5 * q^3 + 6 * q^4 - 6 * q^5 - 5 * q^6 + 6 * q^7 + 6 * q^8 + 11 * q^9 - 6 * q^10 - 5 * q^12 + 6 * q^14 + 5 * q^15 + 6 * q^16 - 15 * q^17 + 11 * q^18 - 13 * q^19 - 6 * q^20 - 5 * q^21 - 2 * q^23 - 5 * q^24 + 6 * q^25 - 26 * q^27 + 6 * q^28 - 4 * q^29 + 5 * q^30 - 2 * q^31 + 6 * q^32 - 15 * q^34 - 6 * q^35 + 11 * q^36 - 13 * q^38 - 30 * q^39 - 6 * q^40 - 17 * q^41 - 5 * q^42 + 15 * q^43 - 11 * q^45 - 2 * q^46 - 18 * q^47 - 5 * q^48 + 6 * q^49 + 6 * q^50 + 6 * q^51 + 22 * q^53 - 26 * q^54 + 6 * q^56 - 2 * q^57 - 4 * q^58 - 7 * q^59 + 5 * q^60 - 20 * q^61 - 2 * q^62 + 11 * q^63 + 6 * q^64 - 29 * q^67 - 15 * q^68 + 12 * q^69 - 6 * q^70 - 2 * q^71 + 11 * q^72 + q^73 - 5 * q^75 - 13 * q^76 - 30 * q^78 - 12 * q^79 - 6 * q^80 + 22 * q^81 - 17 * q^82 - 35 * q^83 - 5 * q^84 + 15 * q^85 + 15 * q^86 - 29 * q^89 - 11 * q^90 - 2 * q^92 + 8 * q^93 - 18 * q^94 + 13 * q^95 - 5 * q^96 - 19 * 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
$3$	1.15804	0.668596	0.334298	−	0.942467i	$-0.391501\pi$
$3$	1.15804	0.668596	0.334298	+	0.942467i	$0.391501\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	1.15804	0.472769
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	−1.65894	−0.552979
$10$	−1.00000	−0.316228
$11$	0	0
$11$	0	0
$12$	1.15804	0.334298
$13$	−3.58654	−0.994727	−0.497363	−	0.867542i	$-0.665698\pi$
$13$	−3.58654	−0.994727	−0.497363	+	0.867542i	$0.665698\pi$
$14$	1.00000	0.267261
$15$	−1.15804	−0.299005
$16$	1.00000	0.250000
$17$	4.29025	1.04054	0.520270	−	0.854002i	$-0.325831\pi$
$17$	4.29025	1.04054	0.520270	+	0.854002i	$0.325831\pi$
$18$	−1.65894	−0.391015
$19$	−4.89322	−1.12258	−0.561291	−	0.827619i	$-0.689695\pi$
$19$	−4.89322	−1.12258	−0.561291	+	0.827619i	$0.689695\pi$
$20$	−1.00000	−0.223607
$21$	1.15804	0.252706
$22$	0	0
$23$	2.85699	0.595724	0.297862	−	0.954609i	$-0.403726\pi$
$23$	2.85699	0.595724	0.297862	+	0.954609i	$0.403726\pi$
$24$	1.15804	0.236385
$25$	1.00000	0.200000
$26$	−3.58654	−0.703378
$27$	−5.39525	−1.03832
$28$	1.00000	0.188982
$29$	−0.754826	−0.140168	−0.0700838	−	0.997541i	$-0.522327\pi$
$29$	−0.754826	−0.140168	−0.0700838	+	0.997541i	$0.522327\pi$
$30$	−1.15804	−0.211429
$31$	−9.35354	−1.67995	−0.839973	−	0.542628i	$-0.817429\pi$
$31$	−9.35354	−1.67995	−0.839973	+	0.542628i	$0.817429\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	4.29025	0.735772
$35$	−1.00000	−0.169031
$36$	−1.65894	−0.276489
$37$	−4.24080	−0.697183	−0.348592	−	0.937275i	$-0.613340\pi$
$37$	−4.24080	−0.697183	−0.348592	+	0.937275i	$0.613340\pi$
$38$	−4.89322	−0.793785
$39$	−4.15337	−0.665071
$40$	−1.00000	−0.158114
$41$	2.02891	0.316862	0.158431	−	0.987370i	$-0.449356\pi$
$41$	2.02891	0.316862	0.158431	+	0.987370i	$0.449356\pi$
$42$	1.15804	0.178690
$43$	−0.0689189	−0.0105100	−0.00525502	−	0.999986i	$-0.501673\pi$
$43$	−0.0689189	−0.0105100	−0.00525502	+	0.999986i	$0.501673\pi$
$44$	0	0
$45$	1.65894	0.247300
$46$	2.85699	0.421240
$47$	2.23785	0.326425	0.163212	−	0.986591i	$-0.447814\pi$
$47$	2.23785	0.326425	0.163212	+	0.986591i	$0.447814\pi$
$48$	1.15804	0.167149
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	4.96830	0.695701
$52$	−3.58654	−0.497363
$53$	13.1026	1.79977	0.899887	−	0.436123i	$-0.143649\pi$
$53$	13.1026	1.79977	0.899887	+	0.436123i	$0.143649\pi$
$54$	−5.39525	−0.734200
$55$	0	0
$56$	1.00000	0.133631
$57$	−5.66656	−0.750554
$58$	−0.754826	−0.0991135
$59$	1.22512	0.159497	0.0797486	−	0.996815i	$-0.474588\pi$
$59$	1.22512	0.159497	0.0797486	+	0.996815i	$0.474588\pi$
$60$	−1.15804	−0.149503
$61$	−4.03036	−0.516034	−0.258017	−	0.966140i	$-0.583069\pi$
$61$	−4.03036	−0.516034	−0.258017	+	0.966140i	$0.583069\pi$
$62$	−9.35354	−1.18790
$63$	−1.65894	−0.209006
$64$	1.00000	0.125000
$65$	3.58654	0.444855
$66$	0	0
$67$	−14.3059	−1.74774	−0.873870	−	0.486159i	$-0.838398\pi$
$67$	−14.3059	−1.74774	−0.873870	+	0.486159i	$0.838398\pi$
$68$	4.29025	0.520270
$69$	3.30852	0.398299
$70$	−1.00000	−0.119523
$71$	−5.79751	−0.688038	−0.344019	−	0.938963i	$-0.611789\pi$
$71$	−5.79751	−0.688038	−0.344019	+	0.938963i	$0.611789\pi$
$72$	−1.65894	−0.195508
$73$	−9.06652	−1.06116	−0.530578	−	0.847636i	$-0.678025\pi$
$73$	−9.06652	−1.06116	−0.530578	+	0.847636i	$0.678025\pi$
$74$	−4.24080	−0.492983
$75$	1.15804	0.133719
$76$	−4.89322	−0.561291
$77$	0	0
$78$	−4.15337	−0.470276
$79$	−5.03269	−0.566222	−0.283111	−	0.959087i	$-0.591367\pi$
$79$	−5.03269	−0.566222	−0.283111	+	0.959087i	$0.591367\pi$
$80$	−1.00000	−0.111803
$81$	−1.27112	−0.141236
$82$	2.02891	0.224055
$83$	−16.6258	−1.82492	−0.912460	−	0.409167i	$-0.865819\pi$
$83$	−16.6258	−1.82492	−0.912460	+	0.409167i	$0.865819\pi$
$84$	1.15804	0.126353
$85$	−4.29025	−0.465343
$86$	−0.0689189	−0.00743171
$87$	−0.874121	−0.0937156
$88$	0	0
$89$	9.92243	1.05177	0.525887	−	0.850554i	$-0.323733\pi$
$89$	9.92243	1.05177	0.525887	+	0.850554i	$0.323733\pi$
$90$	1.65894	0.174867
$91$	−3.58654	−0.375971
$92$	2.85699	0.297862
$93$	−10.8318	−1.12321
$94$	2.23785	0.230817
$95$	4.89322	0.502034
$96$	1.15804	0.118192
$97$	−6.71733	−0.682042	−0.341021	−	0.940056i	$-0.610773\pi$
$97$	−6.71733	−0.682042	−0.341021	+	0.940056i	$0.610773\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	3.15928	0.314360	0.157180	−	0.987570i	$-0.449760\pi$
$101$	3.15928	0.314360	0.157180	+	0.987570i	$0.449760\pi$
$102$	4.96830	0.491935
$103$	−14.4797	−1.42673	−0.713365	−	0.700792i	$-0.752830\pi$
$103$	−14.4797	−1.42673	−0.713365	+	0.700792i	$0.752830\pi$
$104$	−3.58654	−0.351689
$105$	−1.15804	−0.113013
$106$	13.1026	1.27263
$107$	−7.11161	−0.687505	−0.343753	−	0.939060i	$-0.611698\pi$
$107$	−7.11161	−0.687505	−0.343753	+	0.939060i	$0.611698\pi$
$108$	−5.39525	−0.519158
$109$	−15.0710	−1.44354	−0.721770	−	0.692133i	$-0.756671\pi$
$109$	−15.0710	−1.44354	−0.721770	+	0.692133i	$0.756671\pi$
$110$	0	0
$111$	−4.91103	−0.466134
$112$	1.00000	0.0944911
$113$	0.563261	0.0529871	0.0264936	−	0.999649i	$-0.491566\pi$
$113$	0.563261	0.0529871	0.0264936	+	0.999649i	$0.491566\pi$
$114$	−5.66656	−0.530722
$115$	−2.85699	−0.266416
$116$	−0.754826	−0.0700838
$117$	5.94984	0.550063
$118$	1.22512	0.112782
$119$	4.29025	0.393287
$120$	−1.15804	−0.105714
$121$	0	0
$122$	−4.03036	−0.364891
$123$	2.34956	0.211853
$124$	−9.35354	−0.839973
$125$	−1.00000	−0.0894427
$126$	−1.65894	−0.147790
$127$	12.3097	1.09231	0.546153	−	0.837685i	$-0.316092\pi$
$127$	12.3097	1.09231	0.546153	+	0.837685i	$0.316092\pi$
$128$	1.00000	0.0883883
$129$	−0.0798110	−0.00702697
$130$	3.58654	0.314560
$131$	−14.9957	−1.31018	−0.655090	−	0.755551i	$-0.727369\pi$
$131$	−14.9957	−1.31018	−0.655090	+	0.755551i	$0.727369\pi$
$132$	0	0
$133$	−4.89322	−0.424296
$134$	−14.3059	−1.23584
$135$	5.39525	0.464349
$136$	4.29025	0.367886
$137$	−0.423951	−0.0362206	−0.0181103	−	0.999836i	$-0.505765\pi$
$137$	−0.423951	−0.0362206	−0.0181103	+	0.999836i	$0.505765\pi$
$138$	3.30852	0.281640
$139$	19.3990	1.64540	0.822700	−	0.568475i	$-0.192466\pi$
$139$	19.3990	1.64540	0.822700	+	0.568475i	$0.192466\pi$
$140$	−1.00000	−0.0845154
$141$	2.59153	0.218246
$142$	−5.79751	−0.486517
$143$	0	0
$144$	−1.65894	−0.138245
$145$	0.754826	0.0626849
$146$	−9.06652	−0.750350
$147$	1.15804	0.0955138
$148$	−4.24080	−0.348592
$149$	20.2199	1.65648	0.828241	−	0.560372i	$-0.189342\pi$
$149$	20.2199	1.65648	0.828241	+	0.560372i	$0.189342\pi$
$150$	1.15804	0.0945538
$151$	−15.3843	−1.25195	−0.625977	−	0.779842i	$-0.715299\pi$
$151$	−15.3843	−1.25195	−0.625977	+	0.779842i	$0.715299\pi$
$152$	−4.89322	−0.396892
$153$	−7.11726	−0.575396
$154$	0	0
$155$	9.35354	0.751295
$156$	−4.15337	−0.332535
$157$	−4.83845	−0.386150	−0.193075	−	0.981184i	$-0.561846\pi$
$157$	−4.83845	−0.386150	−0.193075	+	0.981184i	$0.561846\pi$
$158$	−5.03269	−0.400380
$159$	15.1733	1.20332
$160$	−1.00000	−0.0790569
$161$	2.85699	0.225163
$162$	−1.27112	−0.0998688
$163$	3.98234	0.311921	0.155961	−	0.987763i	$-0.450153\pi$
$163$	3.98234	0.311921	0.155961	+	0.987763i	$0.450153\pi$
$164$	2.02891	0.158431
$165$	0	0
$166$	−16.6258	−1.29041
$167$	−23.9536	−1.85359	−0.926794	−	0.375570i	$-0.877447\pi$
$167$	−23.9536	−1.85359	−0.926794	+	0.375570i	$0.877447\pi$
$168$	1.15804	0.0893450
$169$	−0.136737	−0.0105182
$170$	−4.29025	−0.329047
$171$	8.11754	0.620764
$172$	−0.0689189	−0.00525502
$173$	9.32676	0.709100	0.354550	−	0.935037i	$-0.384634\pi$
$173$	9.32676	0.709100	0.354550	+	0.935037i	$0.384634\pi$
$174$	−0.874121	−0.0662669
$175$	1.00000	0.0755929
$176$	0	0
$177$	1.41874	0.106639
$178$	9.92243	0.743717
$179$	21.0160	1.57081	0.785405	−	0.618982i	$-0.212454\pi$
$179$	21.0160	1.57081	0.785405	+	0.618982i	$0.212454\pi$
$180$	1.65894	0.123650
$181$	−2.21284	−0.164479	−0.0822394	−	0.996613i	$-0.526207\pi$
$181$	−2.21284	−0.164479	−0.0822394	+	0.996613i	$0.526207\pi$
$182$	−3.58654	−0.265852
$183$	−4.66733	−0.345019
$184$	2.85699	0.210620
$185$	4.24080	0.311790
$186$	−10.8318	−0.794227
$187$	0	0
$188$	2.23785	0.163212
$189$	−5.39525	−0.392447
$190$	4.89322	0.354991
$191$	19.1652	1.38674	0.693372	−	0.720580i	$-0.256124\pi$
$191$	19.1652	1.38674	0.693372	+	0.720580i	$0.256124\pi$
$192$	1.15804	0.0835746
$193$	8.44654	0.607996	0.303998	−	0.952673i	$-0.401678\pi$
$193$	8.44654	0.607996	0.303998	+	0.952673i	$0.401678\pi$
$194$	−6.71733	−0.482276
$195$	4.15337	0.297429
$196$	1.00000	0.0714286
$197$	−11.1821	−0.796693	−0.398347	−	0.917235i	$-0.630416\pi$
$197$	−11.1821	−0.796693	−0.398347	+	0.917235i	$0.630416\pi$
$198$	0	0
$199$	−4.23045	−0.299888	−0.149944	−	0.988694i	$-0.547909\pi$
$199$	−4.23045	−0.299888	−0.149944	+	0.988694i	$0.547909\pi$
$200$	1.00000	0.0707107
$201$	−16.5668	−1.16853
$202$	3.15928	0.222286
$203$	−0.754826	−0.0529784
$204$	4.96830	0.347850
$205$	−2.02891	−0.141705
$206$	−14.4797	−1.00885
$207$	−4.73957	−0.329423
$208$	−3.58654	−0.248682
$209$	0	0
$210$	−1.15804	−0.0799126
$211$	−2.78498	−0.191726	−0.0958628	−	0.995395i	$-0.530561\pi$
$211$	−2.78498	−0.191726	−0.0958628	+	0.995395i	$0.530561\pi$
$212$	13.1026	0.899887
$213$	−6.71377	−0.460020
$214$	−7.11161	−0.486139
$215$	0.0689189	0.00470023
$216$	−5.39525	−0.367100
$217$	−9.35354	−0.634960
$218$	−15.0710	−1.02074
$219$	−10.4994	−0.709485
$220$	0	0
$221$	−15.3872	−1.03505
$222$	−4.91103	−0.329607
$223$	−11.2604	−0.754052	−0.377026	−	0.926203i	$-0.623053\pi$
$223$	−11.2604	−0.754052	−0.377026	+	0.926203i	$0.623053\pi$
$224$	1.00000	0.0668153
$225$	−1.65894	−0.110596
$226$	0.563261	0.0374676
$227$	4.17183	0.276894	0.138447	−	0.990370i	$-0.455789\pi$
$227$	4.17183	0.276894	0.138447	+	0.990370i	$0.455789\pi$
$228$	−5.66656	−0.375277
$229$	14.1775	0.936879	0.468439	−	0.883496i	$-0.344816\pi$
$229$	14.1775	0.936879	0.468439	+	0.883496i	$0.344816\pi$
$230$	−2.85699	−0.188384
$231$	0	0
$232$	−0.754826	−0.0495568
$233$	−9.49782	−0.622223	−0.311111	−	0.950373i	$-0.600701\pi$
$233$	−9.49782	−0.622223	−0.311111	+	0.950373i	$0.600701\pi$
$234$	5.94984	0.388953
$235$	−2.23785	−0.145982
$236$	1.22512	0.0797486
$237$	−5.82808	−0.378574
$238$	4.29025	0.278096
$239$	2.03180	0.131426	0.0657130	−	0.997839i	$-0.479068\pi$
$239$	2.03180	0.131426	0.0657130	+	0.997839i	$0.479068\pi$
$240$	−1.15804	−0.0747514
$241$	−18.6496	−1.20133	−0.600663	−	0.799502i	$-0.705097\pi$
$241$	−18.6496	−1.20133	−0.600663	+	0.799502i	$0.705097\pi$
$242$	0	0
$243$	14.7137	0.943886
$244$	−4.03036	−0.258017
$245$	−1.00000	−0.0638877
$246$	2.34956	0.149803
$247$	17.5497	1.11666
$248$	−9.35354	−0.593951
$249$	−19.2534	−1.22013
$250$	−1.00000	−0.0632456
$251$	−14.5068	−0.915663	−0.457832	−	0.889039i	$-0.651374\pi$
$251$	−14.5068	−0.915663	−0.457832	+	0.889039i	$0.651374\pi$
$252$	−1.65894	−0.104503
$253$	0	0
$254$	12.3097	0.772377
$255$	−4.96830	−0.311127
$256$	1.00000	0.0625000
$257$	−14.8443	−0.925962	−0.462981	−	0.886368i	$-0.653220\pi$
$257$	−14.8443	−0.925962	−0.462981	+	0.886368i	$0.653220\pi$
$258$	−0.0798110	−0.00496882
$259$	−4.24080	−0.263510
$260$	3.58654	0.222428
$261$	1.25221	0.0775097
$262$	−14.9957	−0.926437
$263$	−31.6286	−1.95030	−0.975152	−	0.221539i	$-0.928892\pi$
$263$	−31.6286	−1.95030	−0.975152	+	0.221539i	$0.928892\pi$
$264$	0	0
$265$	−13.1026	−0.804884
$266$	−4.89322	−0.300022
$267$	11.4906	0.703213
$268$	−14.3059	−0.873870
$269$	11.1022	0.676912	0.338456	−	0.940982i	$-0.390095\pi$
$269$	11.1022	0.676912	0.338456	+	0.940982i	$0.390095\pi$
$270$	5.39525	0.328344
$271$	−14.3128	−0.869442	−0.434721	−	0.900565i	$-0.643153\pi$
$271$	−14.3128	−0.869442	−0.434721	+	0.900565i	$0.643153\pi$
$272$	4.29025	0.260135
$273$	−4.15337	−0.251373
$274$	−0.423951	−0.0256118
$275$	0	0
$276$	3.30852	0.199149
$277$	3.88359	0.233342	0.116671	−	0.993171i	$-0.462778\pi$
$277$	3.88359	0.233342	0.116671	+	0.993171i	$0.462778\pi$
$278$	19.3990	1.16347
$279$	15.5169	0.928974
$280$	−1.00000	−0.0597614
$281$	−23.6842	−1.41288	−0.706441	−	0.707772i	$-0.749700\pi$
$281$	−23.6842	−1.41288	−0.706441	+	0.707772i	$0.749700\pi$
$282$	2.59153	0.154323
$283$	8.95928	0.532574	0.266287	−	0.963894i	$-0.414203\pi$
$283$	8.95928	0.532574	0.266287	+	0.963894i	$0.414203\pi$
$284$	−5.79751	−0.344019
$285$	5.66656	0.335658
$286$	0	0
$287$	2.02891	0.119763
$288$	−1.65894	−0.0977538
$289$	1.40627	0.0827218
$290$	0.754826	0.0443249
$291$	−7.77896	−0.456011
$292$	−9.06652	−0.530578
$293$	12.7431	0.744460	0.372230	−	0.928141i	$-0.378593\pi$
$293$	12.7431	0.744460	0.372230	+	0.928141i	$0.378593\pi$
$294$	1.15804	0.0675384
$295$	−1.22512	−0.0713293
$296$	−4.24080	−0.246491
$297$	0	0
$298$	20.2199	1.17131
$299$	−10.2467	−0.592583
$300$	1.15804	0.0668596
$301$	−0.0689189	−0.00397242
$302$	−15.3843	−0.885265
$303$	3.65858	0.210180
$304$	−4.89322	−0.280645
$305$	4.03036	0.230778
$306$	−7.11726	−0.406866
$307$	−5.75775	−0.328612	−0.164306	−	0.986409i	$-0.552538\pi$
$307$	−5.75775	−0.328612	−0.164306	+	0.986409i	$0.552538\pi$
$308$	0	0
$309$	−16.7682	−0.953907
$310$	9.35354	0.531246
$311$	12.2054	0.692106	0.346053	−	0.938215i	$-0.387522\pi$
$311$	12.2054	0.692106	0.346053	+	0.938215i	$0.387522\pi$
$312$	−4.15337	−0.235138
$313$	20.3162	1.14834	0.574170	−	0.818736i	$-0.305325\pi$
$313$	20.3162	1.14834	0.574170	+	0.818736i	$0.305325\pi$
$314$	−4.83845	−0.273050
$315$	1.65894	0.0934705
$316$	−5.03269	−0.283111
$317$	26.5294	1.49004	0.745020	−	0.667042i	$-0.232440\pi$
$317$	26.5294	1.49004	0.745020	+	0.667042i	$0.232440\pi$
$318$	15.1733	0.850878
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	−8.23555	−0.459663
$322$	2.85699	0.159214
$323$	−20.9931	−1.16809
$324$	−1.27112	−0.0706179
$325$	−3.58654	−0.198945
$326$	3.98234	0.220562
$327$	−17.4529	−0.965146
$328$	2.02891	0.112028
$329$	2.23785	0.123377
$330$	0	0
$331$	0.393891	0.0216502	0.0108251	−	0.999941i	$-0.496554\pi$
$331$	0.393891	0.0216502	0.0108251	+	0.999941i	$0.496554\pi$
$332$	−16.6258	−0.912460
$333$	7.03522	0.385527
$334$	−23.9536	−1.31068
$335$	14.3059	0.781614
$336$	1.15804	0.0631764
$337$	−1.24378	−0.0677532	−0.0338766	−	0.999426i	$-0.510785\pi$
$337$	−1.24378	−0.0677532	−0.0338766	+	0.999426i	$0.510785\pi$
$338$	−0.136737	−0.00743752
$339$	0.652280	0.0354270
$340$	−4.29025	−0.232672
$341$	0	0
$342$	8.11754	0.438946
$343$	1.00000	0.0539949
$344$	−0.0689189	−0.00371586
$345$	−3.30852	−0.178125
$346$	9.32676	0.501410
$347$	−15.7099	−0.843354	−0.421677	−	0.906746i	$-0.638558\pi$
$347$	−15.7099	−0.843354	−0.421677	+	0.906746i	$0.638558\pi$
$348$	−0.874121	−0.0468578
$349$	16.4116	0.878493	0.439247	−	0.898366i	$-0.355245\pi$
$349$	16.4116	0.878493	0.439247	+	0.898366i	$0.355245\pi$
$350$	1.00000	0.0534522
$351$	19.3503	1.03284
$352$	0	0
$353$	22.8233	1.21476	0.607382	−	0.794410i	$-0.292220\pi$
$353$	22.8233	1.21476	0.607382	+	0.794410i	$0.292220\pi$
$354$	1.41874	0.0754053
$355$	5.79751	0.307700
$356$	9.92243	0.525887
$357$	4.96830	0.262950
$358$	21.0160	1.11073
$359$	−10.8548	−0.572894	−0.286447	−	0.958096i	$-0.592474\pi$
$359$	−10.8548	−0.572894	−0.286447	+	0.958096i	$0.592474\pi$
$360$	1.65894	0.0874336
$361$	4.94358	0.260189
$362$	−2.21284	−0.116304
$363$	0	0
$364$	−3.58654	−0.187986
$365$	9.06652	0.474563
$366$	−4.66733	−0.243965
$367$	21.3899	1.11654	0.558271	−	0.829658i	$-0.311465\pi$
$367$	21.3899	1.11654	0.558271	+	0.829658i	$0.311465\pi$
$368$	2.85699	0.148931
$369$	−3.36583	−0.175218
$370$	4.24080	0.220469
$371$	13.1026	0.680251
$372$	−10.8318	−0.561603
$373$	2.90463	0.150396	0.0751981	−	0.997169i	$-0.476041\pi$
$373$	2.90463	0.150396	0.0751981	+	0.997169i	$0.476041\pi$
$374$	0	0
$375$	−1.15804	−0.0598011
$376$	2.23785	0.115409
$377$	2.70721	0.139429
$378$	−5.39525	−0.277502
$379$	0.0627414	0.00322281	0.00161140	−	0.999999i	$-0.499487\pi$
$379$	0.0627414	0.00322281	0.00161140	+	0.999999i	$0.499487\pi$
$380$	4.89322	0.251017
$381$	14.2551	0.730312
$382$	19.1652	0.980576
$383$	−8.43394	−0.430954	−0.215477	−	0.976509i	$-0.569131\pi$
$383$	−8.43394	−0.430954	−0.215477	+	0.976509i	$0.569131\pi$
$384$	1.15804	0.0590961
$385$	0	0
$386$	8.44654	0.429918
$387$	0.114332	0.00581182
$388$	−6.71733	−0.341021
$389$	−36.6681	−1.85915	−0.929573	−	0.368637i	$-0.879825\pi$
$389$	−36.6681	−1.85915	−0.929573	+	0.368637i	$0.879825\pi$
$390$	4.15337	0.210314
$391$	12.2572	0.619874
$392$	1.00000	0.0505076
$393$	−17.3657	−0.875982
$394$	−11.1821	−0.563347
$395$	5.03269	0.253222
$396$	0	0
$397$	−2.91177	−0.146137	−0.0730687	−	0.997327i	$-0.523279\pi$
$397$	−2.91177	−0.146137	−0.0730687	+	0.997327i	$0.523279\pi$
$398$	−4.23045	−0.212053
$399$	−5.66656	−0.283683
$400$	1.00000	0.0500000
$401$	15.3525	0.766667	0.383334	−	0.923610i	$-0.374776\pi$
$401$	15.3525	0.766667	0.383334	+	0.923610i	$0.374776\pi$
$402$	−16.5668	−0.826278
$403$	33.5469	1.67109
$404$	3.15928	0.157180
$405$	1.27112	0.0631626
$406$	−0.754826	−0.0374614
$407$	0	0
$408$	4.96830	0.245967
$409$	−14.9938	−0.741397	−0.370699	−	0.928753i	$-0.620882\pi$
$409$	−14.9938	−0.741397	−0.370699	+	0.928753i	$0.620882\pi$
$410$	−2.02891	−0.100201
$411$	−0.490954	−0.0242170
$412$	−14.4797	−0.713365
$413$	1.22512	0.0602843
$414$	−4.73957	−0.232937
$415$	16.6258	0.816129
$416$	−3.58654	−0.175845
$417$	22.4649	1.10011
$418$	0	0
$419$	26.8166	1.31008	0.655038	−	0.755596i	$-0.272653\pi$
$419$	26.8166	1.31008	0.655038	+	0.755596i	$0.272653\pi$
$420$	−1.15804	−0.0565067
$421$	39.0662	1.90397	0.951984	−	0.306148i	$-0.0990401\pi$
$421$	39.0662	1.90397	0.951984	+	0.306148i	$0.0990401\pi$
$422$	−2.78498	−0.135571
$423$	−3.71246	−0.180506
$424$	13.1026	0.636316
$425$	4.29025	0.208108
$426$	−6.71377	−0.325283
$427$	−4.03036	−0.195043
$428$	−7.11161	−0.343753
$429$	0	0
$430$	0.0689189	0.00332356
$431$	36.9117	1.77797	0.888987	−	0.457932i	$-0.151410\pi$
$431$	36.9117	1.77797	0.888987	+	0.457932i	$0.151410\pi$
$432$	−5.39525	−0.259579
$433$	−12.4708	−0.599307	−0.299654	−	0.954048i	$-0.596871\pi$
$433$	−12.4708	−0.599307	−0.299654	+	0.954048i	$0.596871\pi$
$434$	−9.35354	−0.448984
$435$	0.874121	0.0419109
$436$	−15.0710	−0.721770
$437$	−13.9799	−0.668749
$438$	−10.4994	−0.501682
$439$	−5.79416	−0.276540	−0.138270	−	0.990395i	$-0.544154\pi$
$439$	−5.79416	−0.276540	−0.138270	+	0.990395i	$0.544154\pi$
$440$	0	0
$441$	−1.65894	−0.0789970
$442$	−15.3872	−0.731893
$443$	11.8578	0.563380	0.281690	−	0.959506i	$-0.409105\pi$
$443$	11.8578	0.563380	0.281690	+	0.959506i	$0.409105\pi$
$444$	−4.91103	−0.233067
$445$	−9.92243	−0.470368
$446$	−11.2604	−0.533195
$447$	23.4156	1.10752
$448$	1.00000	0.0472456
$449$	−6.26914	−0.295859	−0.147929	−	0.988998i	$-0.547261\pi$
$449$	−6.26914	−0.295859	−0.147929	+	0.988998i	$0.547261\pi$
$450$	−1.65894	−0.0782030
$451$	0	0
$452$	0.563261	0.0264936
$453$	−17.8156	−0.837051
$454$	4.17183	0.195794
$455$	3.58654	0.168140
$456$	−5.66656	−0.265361
$457$	−30.0377	−1.40510	−0.702552	−	0.711632i	$-0.747956\pi$
$457$	−30.0377	−1.40510	−0.702552	+	0.711632i	$0.747956\pi$
$458$	14.1775	0.662473
$459$	−23.1470	−1.08041
$460$	−2.85699	−0.133208
$461$	−37.8904	−1.76473	−0.882365	−	0.470566i	$-0.844050\pi$
$461$	−37.8904	−1.76473	−0.882365	+	0.470566i	$0.844050\pi$
$462$	0	0
$463$	2.23043	0.103657	0.0518285	−	0.998656i	$-0.483495\pi$
$463$	2.23043	0.103657	0.0518285	+	0.998656i	$0.483495\pi$
$464$	−0.754826	−0.0350419
$465$	10.8318	0.502313
$466$	−9.49782	−0.439978
$467$	40.1699	1.85884	0.929420	−	0.369024i	$-0.120308\pi$
$467$	40.1699	1.85884	0.929420	+	0.369024i	$0.120308\pi$
$468$	5.94984	0.275031
$469$	−14.3059	−0.660584
$470$	−2.23785	−0.103225
$471$	−5.60313	−0.258179
$472$	1.22512	0.0563908
$473$	0	0
$474$	−5.82808	−0.267692
$475$	−4.89322	−0.224516
$476$	4.29025	0.196643
$477$	−21.7363	−0.995237
$478$	2.03180	0.0929322
$479$	−15.4649	−0.706609	−0.353304	−	0.935508i	$-0.614942\pi$
$479$	−15.4649	−0.706609	−0.353304	+	0.935508i	$0.614942\pi$
$480$	−1.15804	−0.0528572
$481$	15.2098	0.693507
$482$	−18.6496	−0.849466
$483$	3.30852	0.150543
$484$	0	0
$485$	6.71733	0.305018
$486$	14.7137	0.667428
$487$	−17.9184	−0.811960	−0.405980	−	0.913882i	$-0.633070\pi$
$487$	−17.9184	−0.811960	−0.405980	+	0.913882i	$0.633070\pi$
$488$	−4.03036	−0.182446
$489$	4.61172	0.208549
$490$	−1.00000	−0.0451754
$491$	26.6413	1.20230	0.601152	−	0.799135i	$-0.294709\pi$
$491$	26.6413	1.20230	0.601152	+	0.799135i	$0.294709\pi$
$492$	2.34956	0.105926
$493$	−3.23839	−0.145850
$494$	17.5497	0.789599
$495$	0	0
$496$	−9.35354	−0.419987
$497$	−5.79751	−0.260054
$498$	−19.2534	−0.862765
$499$	29.4299	1.31746	0.658732	−	0.752378i	$-0.271093\pi$
$499$	29.4299	1.31746	0.658732	+	0.752378i	$0.271093\pi$
$500$	−1.00000	−0.0447214
$501$	−27.7393	−1.23930
$502$	−14.5068	−0.647472
$503$	−25.1768	−1.12258	−0.561288	−	0.827620i	$-0.689694\pi$
$503$	−25.1768	−1.12258	−0.561288	+	0.827620i	$0.689694\pi$
$504$	−1.65894	−0.0738949
$505$	−3.15928	−0.140586
$506$	0	0
$507$	−0.158348	−0.00703246
$508$	12.3097	0.546153
$509$	28.8350	1.27809	0.639043	−	0.769171i	$-0.279330\pi$
$509$	28.8350	1.27809	0.639043	+	0.769171i	$0.279330\pi$
$510$	−4.96830	−0.220000
$511$	−9.06652	−0.401079
$512$	1.00000	0.0441942
$513$	26.4001	1.16559
$514$	−14.8443	−0.654754
$515$	14.4797	0.638053
$516$	−0.0798110	−0.00351348
$517$	0	0
$518$	−4.24080	−0.186330
$519$	10.8008	0.474102
$520$	3.58654	0.157280
$521$	8.86274	0.388284	0.194142	−	0.980973i	$-0.437808\pi$
$521$	8.86274	0.388284	0.194142	+	0.980973i	$0.437808\pi$
$522$	1.25221	0.0548077
$523$	−23.2339	−1.01595	−0.507973	−	0.861373i	$-0.669605\pi$
$523$	−23.2339	−1.01595	−0.507973	+	0.861373i	$0.669605\pi$
$524$	−14.9957	−0.655090
$525$	1.15804	0.0505411
$526$	−31.6286	−1.37907
$527$	−40.1291	−1.74805
$528$	0	0
$529$	−14.8376	−0.645113
$530$	−13.1026	−0.569139
$531$	−2.03240	−0.0881985
$532$	−4.89322	−0.212148
$533$	−7.27675	−0.315191
$534$	11.4906	0.497247
$535$	7.11161	0.307462
$536$	−14.3059	−0.617920
$537$	24.3375	1.05024
$538$	11.1022	0.478649
$539$	0	0
$540$	5.39525	0.232175
$541$	−21.2530	−0.913740	−0.456870	−	0.889534i	$-0.651030\pi$
$541$	−21.2530	−0.913740	−0.456870	+	0.889534i	$0.651030\pi$
$542$	−14.3128	−0.614789
$543$	−2.56256	−0.109970
$544$	4.29025	0.183943
$545$	15.0710	0.645571
$546$	−4.15337	−0.177748
$547$	39.8544	1.70405	0.852026	−	0.523500i	$-0.175374\pi$
$547$	39.8544	1.70405	0.852026	+	0.523500i	$0.175374\pi$
$548$	−0.423951	−0.0181103
$549$	6.68611	0.285356
$550$	0	0
$551$	3.69353	0.157350
$552$	3.30852	0.140820
$553$	−5.03269	−0.214012
$554$	3.88359	0.164998
$555$	4.91103	0.208462
$556$	19.3990	0.822700
$557$	19.7255	0.835794	0.417897	−	0.908494i	$-0.362767\pi$
$557$	19.7255	0.835794	0.417897	+	0.908494i	$0.362767\pi$
$558$	15.5169	0.656884
$559$	0.247180	0.0104546
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	−23.6842	−0.999058
$563$	20.5630	0.866627	0.433314	−	0.901243i	$-0.357344\pi$
$563$	20.5630	0.866627	0.433314	+	0.901243i	$0.357344\pi$
$564$	2.59153	0.109123
$565$	−0.563261	−0.0236966
$566$	8.95928	0.376587
$567$	−1.27112	−0.0533821
$568$	−5.79751	−0.243258
$569$	21.9128	0.918632	0.459316	−	0.888273i	$-0.348095\pi$
$569$	21.9128	0.918632	0.459316	+	0.888273i	$0.348095\pi$
$570$	5.66656	0.237346
$571$	−17.3899	−0.727745	−0.363872	−	0.931449i	$-0.618546\pi$
$571$	−17.3899	−0.727745	−0.363872	+	0.931449i	$0.618546\pi$
$572$	0	0
$573$	22.1941	0.927172
$574$	2.02891	0.0846849
$575$	2.85699	0.119145
$576$	−1.65894	−0.0691223
$577$	25.6014	1.06580	0.532899	−	0.846179i	$-0.321102\pi$
$577$	25.6014	1.06580	0.532899	+	0.846179i	$0.321102\pi$
$578$	1.40627	0.0584932
$579$	9.78146	0.406504
$580$	0.754826	0.0313424
$581$	−16.6258	−0.689755
$582$	−7.77896	−0.322448
$583$	0	0
$584$	−9.06652	−0.375175
$585$	−5.94984	−0.245996
$586$	12.7431	0.526413
$587$	−38.2688	−1.57952	−0.789762	−	0.613413i	$-0.789796\pi$
$587$	−38.2688	−1.57952	−0.789762	+	0.613413i	$0.789796\pi$
$588$	1.15804	0.0477569
$589$	45.7689	1.88588
$590$	−1.22512	−0.0504374
$591$	−12.9494	−0.532666
$592$	−4.24080	−0.174296
$593$	−11.8855	−0.488078	−0.244039	−	0.969765i	$-0.578473\pi$
$593$	−11.8855	−0.488078	−0.244039	+	0.969765i	$0.578473\pi$
$594$	0	0
$595$	−4.29025	−0.175883
$596$	20.2199	0.828241
$597$	−4.89904	−0.200504
$598$	−10.2467	−0.419019
$599$	−4.65368	−0.190144	−0.0950722	−	0.995470i	$-0.530308\pi$
$599$	−4.65368	−0.190144	−0.0950722	+	0.995470i	$0.530308\pi$
$600$	1.15804	0.0472769
$601$	−13.2306	−0.539688	−0.269844	−	0.962904i	$-0.586972\pi$
$601$	−13.2306	−0.539688	−0.269844	+	0.962904i	$0.586972\pi$
$602$	−0.0689189	−0.00280892
$603$	23.7325	0.966464
$604$	−15.3843	−0.625977
$605$	0	0
$606$	3.65858	0.148620
$607$	30.3137	1.23040	0.615198	−	0.788372i	$-0.289076\pi$
$607$	30.3137	1.23040	0.615198	+	0.788372i	$0.289076\pi$
$608$	−4.89322	−0.198446
$609$	−0.874121	−0.0354212
$610$	4.03036	0.163184
$611$	−8.02615	−0.324703
$612$	−7.11726	−0.287698
$613$	−12.6321	−0.510206	−0.255103	−	0.966914i	$-0.582109\pi$
$613$	−12.6321	−0.510206	−0.255103	+	0.966914i	$0.582109\pi$
$614$	−5.75775	−0.232364
$615$	−2.34956	−0.0947434
$616$	0	0
$617$	−48.4310	−1.94976	−0.974880	−	0.222731i	$-0.928503\pi$
$617$	−48.4310	−1.94976	−0.974880	+	0.222731i	$0.928503\pi$
$618$	−16.7682	−0.674514
$619$	−2.76873	−0.111285	−0.0556423	−	0.998451i	$-0.517721\pi$
$619$	−2.76873	−0.111285	−0.0556423	+	0.998451i	$0.517721\pi$
$620$	9.35354	0.375647
$621$	−15.4142	−0.618550
$622$	12.2054	0.489393
$623$	9.92243	0.397534
$624$	−4.15337	−0.166268
$625$	1.00000	0.0400000
$626$	20.3162	0.811999
$627$	0	0
$628$	−4.83845	−0.193075
$629$	−18.1941	−0.725446
$630$	1.65894	0.0660936
$631$	21.0875	0.839479	0.419740	−	0.907645i	$-0.362121\pi$
$631$	21.0875	0.839479	0.419740	+	0.907645i	$0.362121\pi$
$632$	−5.03269	−0.200190
$633$	−3.22512	−0.128187
$634$	26.5294	1.05362
$635$	−12.3097	−0.488494
$636$	15.1733	0.601661
$637$	−3.58654	−0.142104
$638$	0	0
$639$	9.61771	0.380471
$640$	−1.00000	−0.0395285
$641$	10.8696	0.429324	0.214662	−	0.976688i	$-0.431135\pi$
$641$	10.8696	0.429324	0.214662	+	0.976688i	$0.431135\pi$
$642$	−8.23555	−0.325031
$643$	3.71180	0.146379	0.0731896	−	0.997318i	$-0.476682\pi$
$643$	3.71180	0.146379	0.0731896	+	0.997318i	$0.476682\pi$
$644$	2.85699	0.112581
$645$	0.0798110	0.00314256
$646$	−20.9931	−0.825964
$647$	−25.1297	−0.987951	−0.493976	−	0.869476i	$-0.664457\pi$
$647$	−25.1297	−0.987951	−0.493976	+	0.869476i	$0.664457\pi$
$648$	−1.27112	−0.0499344
$649$	0	0
$650$	−3.58654	−0.140676
$651$	−10.8318	−0.424532
$652$	3.98234	0.155961
$653$	27.3470	1.07017	0.535085	−	0.844798i	$-0.320279\pi$
$653$	27.3470	1.07017	0.535085	+	0.844798i	$0.320279\pi$
$654$	−17.4529	−0.682461
$655$	14.9957	0.585930
$656$	2.02891	0.0792155
$657$	15.0408	0.586796
$658$	2.23785	0.0872406
$659$	−6.74968	−0.262930	−0.131465	−	0.991321i	$-0.541968\pi$
$659$	−6.74968	−0.262930	−0.131465	+	0.991321i	$0.541968\pi$
$660$	0	0
$661$	−34.5615	−1.34429	−0.672144	−	0.740421i	$-0.734626\pi$
$661$	−34.5615	−1.34429	−0.672144	+	0.740421i	$0.734626\pi$
$662$	0.393891	0.0153090
$663$	−17.8190	−0.692032
$664$	−16.6258	−0.645206
$665$	4.89322	0.189751
$666$	7.03522	0.272609
$667$	−2.15653	−0.0835013
$668$	−23.9536	−0.926794
$669$	−13.0400	−0.504157
$670$	14.3059	0.552684
$671$	0	0
$672$	1.15804	0.0446725
$673$	2.98731	0.115152	0.0575761	−	0.998341i	$-0.481663\pi$
$673$	2.98731	0.115152	0.0575761	+	0.998341i	$0.481663\pi$
$674$	−1.24378	−0.0479087
$675$	−5.39525	−0.207663
$676$	−0.136737	−0.00525912
$677$	5.39584	0.207379	0.103690	−	0.994610i	$-0.466935\pi$
$677$	5.39584	0.207379	0.103690	+	0.994610i	$0.466935\pi$
$678$	0.652280	0.0250507
$679$	−6.71733	−0.257787
$680$	−4.29025	−0.164524
$681$	4.83116	0.185130
$682$	0	0
$683$	2.39397	0.0916028	0.0458014	−	0.998951i	$-0.485416\pi$
$683$	2.39397	0.0916028	0.0458014	+	0.998951i	$0.485416\pi$
$684$	8.11754	0.310382
$685$	0.423951	0.0161983
$686$	1.00000	0.0381802
$687$	16.4182	0.626394
$688$	−0.0689189	−0.00262751
$689$	−46.9928	−1.79028
$690$	−3.30852	−0.125953
$691$	−28.2727	−1.07554	−0.537771	−	0.843091i	$-0.680734\pi$
$691$	−28.2727	−1.07554	−0.537771	+	0.843091i	$0.680734\pi$
$692$	9.32676	0.354550
$693$	0	0
$694$	−15.7099	−0.596341
$695$	−19.3990	−0.735846
$696$	−0.874121	−0.0331335
$697$	8.70452	0.329707
$698$	16.4116	0.621189
$699$	−10.9989	−0.416016
$700$	1.00000	0.0377964
$701$	42.6691	1.61159	0.805795	−	0.592194i	$-0.201738\pi$
$701$	42.6691	1.61159	0.805795	+	0.592194i	$0.201738\pi$
$702$	19.3503	0.730329
$703$	20.7512	0.782645
$704$	0	0
$705$	−2.59153	−0.0976027
$706$	22.8233	0.858967
$707$	3.15928	0.118817
$708$	1.41874	0.0533196
$709$	49.1230	1.84485	0.922427	−	0.386171i	$-0.126203\pi$
$709$	49.1230	1.84485	0.922427	+	0.386171i	$0.126203\pi$
$710$	5.79751	0.217577
$711$	8.34892	0.313109
$712$	9.92243	0.371859
$713$	−26.7230	−1.00078
$714$	4.96830	0.185934
$715$	0	0
$716$	21.0160	0.785405
$717$	2.35291	0.0878709
$718$	−10.8548	−0.405097
$719$	31.0636	1.15848	0.579239	−	0.815158i	$-0.303350\pi$
$719$	31.0636	1.15848	0.579239	+	0.815158i	$0.303350\pi$
$720$	1.65894	0.0618249
$721$	−14.4797	−0.539254
$722$	4.94358	0.183981
$723$	−21.5971	−0.803203
$724$	−2.21284	−0.0822394
$725$	−0.754826	−0.0280335
$726$	0	0
$727$	41.2246	1.52894	0.764468	−	0.644662i	$-0.223002\pi$
$727$	41.2246	1.52894	0.764468	+	0.644662i	$0.223002\pi$
$728$	−3.58654	−0.132926
$729$	20.8525	0.772315
$730$	9.06652	0.335567
$731$	−0.295679	−0.0109361
$732$	−4.66733	−0.172509
$733$	45.0160	1.66271	0.831353	−	0.555745i	$-0.187567\pi$
$733$	45.0160	1.66271	0.831353	+	0.555745i	$0.187567\pi$
$734$	21.3899	0.789515
$735$	−1.15804	−0.0427151
$736$	2.85699	0.105310
$737$	0	0
$738$	−3.36583	−0.123898
$739$	−39.4450	−1.45101	−0.725504	−	0.688218i	$-0.758393\pi$
$739$	−39.4450	−1.45101	−0.725504	+	0.688218i	$0.758393\pi$
$740$	4.24080	0.155895
$741$	20.3233	0.746596
$742$	13.1026	0.481010
$743$	37.9890	1.39368	0.696840	−	0.717227i	$-0.254589\pi$
$743$	37.9890	1.39368	0.696840	+	0.717227i	$0.254589\pi$
$744$	−10.8318	−0.397113
$745$	−20.2199	−0.740801
$746$	2.90463	0.106346
$747$	27.5811	1.00914
$748$	0	0
$749$	−7.11161	−0.259852
$750$	−1.15804	−0.0422858
$751$	44.7783	1.63399	0.816993	−	0.576648i	$-0.195640\pi$
$751$	44.7783	1.63399	0.816993	+	0.576648i	$0.195640\pi$
$752$	2.23785	0.0816061
$753$	−16.7995	−0.612209
$754$	2.70721	0.0985909
$755$	15.3843	0.559890
$756$	−5.39525	−0.196223
$757$	−45.7050	−1.66118	−0.830588	−	0.556888i	$-0.811995\pi$
$757$	−45.7050	−1.66118	−0.830588	+	0.556888i	$0.811995\pi$
$758$	0.0627414	0.00227887
$759$	0	0
$760$	4.89322	0.177496
$761$	−21.6462	−0.784675	−0.392338	−	0.919821i	$-0.628333\pi$
$761$	−21.6462	−0.784675	−0.392338	+	0.919821i	$0.628333\pi$
$762$	14.2551	0.516409
$763$	−15.0710	−0.545607
$764$	19.1652	0.693372
$765$	7.11726	0.257325
$766$	−8.43394	−0.304731
$767$	−4.39394	−0.158656
$768$	1.15804	0.0417873
$769$	−38.7211	−1.39632	−0.698159	−	0.715942i	$-0.745997\pi$
$769$	−38.7211	−1.39632	−0.698159	+	0.715942i	$0.745997\pi$
$770$	0	0
$771$	−17.1903	−0.619095
$772$	8.44654	0.303998
$773$	−31.6653	−1.13892	−0.569461	−	0.822018i	$-0.692848\pi$
$773$	−31.6653	−1.13892	−0.569461	+	0.822018i	$0.692848\pi$
$774$	0.114332	0.00410958
$775$	−9.35354	−0.335989
$776$	−6.71733	−0.241138
$777$	−4.91103	−0.176182
$778$	−36.6681	−1.31462
$779$	−9.92788	−0.355703
$780$	4.15337	0.148714
$781$	0	0
$782$	12.2572	0.438317
$783$	4.07247	0.145538
$784$	1.00000	0.0357143
$785$	4.83845	0.172692
$786$	−17.3657	−0.619413
$787$	−42.4147	−1.51192	−0.755961	−	0.654617i	$-0.772830\pi$
$787$	−42.4147	−1.51192	−0.755961	+	0.654617i	$0.772830\pi$
$788$	−11.1821	−0.398347
$789$	−36.6273	−1.30397
$790$	5.03269	0.179055
$791$	0.563261	0.0200272
$792$	0	0
$793$	14.4550	0.513313
$794$	−2.91177	−0.103335
$795$	−15.1733	−0.538142
$796$	−4.23045	−0.149944
$797$	13.7248	0.486159	0.243079	−	0.970006i	$-0.421842\pi$
$797$	13.7248	0.486159	0.243079	+	0.970006i	$0.421842\pi$
$798$	−5.66656	−0.200594
$799$	9.60096	0.339658
$800$	1.00000	0.0353553
$801$	−16.4607	−0.581609
$802$	15.3525	0.542116
$803$	0	0
$804$	−16.5668	−0.584267
$805$	−2.85699	−0.100696
$806$	33.5469	1.18164
$807$	12.8568	0.452581
$808$	3.15928	0.111143
$809$	−32.0943	−1.12838	−0.564189	−	0.825646i	$-0.690811\pi$
$809$	−32.0943	−1.12838	−0.564189	+	0.825646i	$0.690811\pi$
$810$	1.27112	0.0446627
$811$	44.8968	1.57654	0.788271	−	0.615329i	$-0.210977\pi$
$811$	44.8968	1.57654	0.788271	+	0.615329i	$0.210977\pi$
$812$	−0.754826	−0.0264892
$813$	−16.5749	−0.581306
$814$	0	0
$815$	−3.98234	−0.139495
$816$	4.96830	0.173925
$817$	0.337235	0.0117984
$818$	−14.9938	−0.524247
$819$	5.94984	0.207904
$820$	−2.02891	−0.0708525
$821$	−41.2775	−1.44060	−0.720298	−	0.693665i	$-0.755995\pi$
$821$	−41.2775	−1.44060	−0.720298	+	0.693665i	$0.755995\pi$
$822$	−0.490954	−0.0171240
$823$	14.7542	0.514298	0.257149	−	0.966372i	$-0.417217\pi$
$823$	14.7542	0.514298	0.257149	+	0.966372i	$0.417217\pi$
$824$	−14.4797	−0.504426
$825$	0	0
$826$	1.22512	0.0426274
$827$	51.5767	1.79350	0.896748	−	0.442541i	$-0.145923\pi$
$827$	51.5767	1.79350	0.896748	+	0.442541i	$0.145923\pi$
$828$	−4.73957	−0.164711
$829$	10.6814	0.370982	0.185491	−	0.982646i	$-0.440612\pi$
$829$	10.6814	0.370982	0.185491	+	0.982646i	$0.440612\pi$
$830$	16.6258	0.577090
$831$	4.49737	0.156012
$832$	−3.58654	−0.124341
$833$	4.29025	0.148648
$834$	22.4649	0.777895
$835$	23.9536	0.828950
$836$	0	0
$837$	50.4647	1.74432
$838$	26.8166	0.926363
$839$	−27.4262	−0.946857	−0.473429	−	0.880832i	$-0.656984\pi$
$839$	−27.4262	−0.946857	−0.473429	+	0.880832i	$0.656984\pi$
$840$	−1.15804	−0.0399563
$841$	−28.4302	−0.980353
$842$	39.0662	1.34631
$843$	−27.4273	−0.944648
$844$	−2.78498	−0.0958628
$845$	0.136737	0.00470390
$846$	−3.71246	−0.127637
$847$	0	0
$848$	13.1026	0.449944
$849$	10.3752	0.356077
$850$	4.29025	0.147154
$851$	−12.1159	−0.415329
$852$	−6.71377	−0.230010
$853$	50.2244	1.71965	0.859826	−	0.510587i	$-0.170572\pi$
$853$	50.2244	1.71965	0.859826	+	0.510587i	$0.170572\pi$
$854$	−4.03036	−0.137916
$855$	−8.11754	−0.277614
$856$	−7.11161	−0.243070
$857$	−2.69776	−0.0921539	−0.0460769	−	0.998938i	$-0.514672\pi$
$857$	−2.69776	−0.0921539	−0.0460769	+	0.998938i	$0.514672\pi$
$858$	0	0
$859$	19.0873	0.651252	0.325626	−	0.945499i	$-0.394425\pi$
$859$	19.0873	0.651252	0.325626	+	0.945499i	$0.394425\pi$
$860$	0.0689189	0.00235011
$861$	2.34956	0.0800728
$862$	36.9117	1.25722
$863$	−21.2845	−0.724532	−0.362266	−	0.932075i	$-0.617997\pi$
$863$	−21.2845	−0.724532	−0.362266	+	0.932075i	$0.617997\pi$
$864$	−5.39525	−0.183550
$865$	−9.32676	−0.317119
$866$	−12.4708	−0.423774
$867$	1.62852	0.0553075
$868$	−9.35354	−0.317480
$869$	0	0
$870$	0.874121	0.0296355
$871$	51.3086	1.73853
$872$	−15.0710	−0.510369
$873$	11.1436	0.377155
$874$	−13.9799	−0.472877
$875$	−1.00000	−0.0338062
$876$	−10.4994	−0.354742
$877$	14.7968	0.499654	0.249827	−	0.968291i	$-0.419626\pi$
$877$	14.7968	0.499654	0.249827	+	0.968291i	$0.419626\pi$
$878$	−5.79416	−0.195543
$879$	14.7571	0.497743
$880$	0	0
$881$	−38.9767	−1.31316	−0.656579	−	0.754257i	$-0.727997\pi$
$881$	−38.9767	−1.31316	−0.656579	+	0.754257i	$0.727997\pi$
$882$	−1.65894	−0.0558593
$883$	−40.8954	−1.37624	−0.688119	−	0.725597i	$-0.741563\pi$
$883$	−40.8954	−1.37624	−0.688119	+	0.725597i	$0.741563\pi$
$884$	−15.3872	−0.517526
$885$	−1.41874	−0.0476905
$886$	11.8578	0.398369
$887$	−30.9885	−1.04049	−0.520247	−	0.854016i	$-0.674160\pi$
$887$	−30.9885	−1.04049	−0.520247	+	0.854016i	$0.674160\pi$
$888$	−4.91103	−0.164803
$889$	12.3097	0.412853
$890$	−9.92243	−0.332600
$891$	0	0
$892$	−11.2604	−0.377026
$893$	−10.9503	−0.366438
$894$	23.4156	0.783134
$895$	−21.0160	−0.702488
$896$	1.00000	0.0334077
$897$	−11.8661	−0.396199
$898$	−6.26914	−0.209204
$899$	7.06030	0.235474
$900$	−1.65894	−0.0552979
$901$	56.2133	1.87274
$902$	0	0
$903$	−0.0798110	−0.00265594
$904$	0.563261	0.0187338
$905$	2.21284	0.0735572
$906$	−17.8156	−0.591885
$907$	−7.92243	−0.263060	−0.131530	−	0.991312i	$-0.541989\pi$
$907$	−7.92243	−0.263060	−0.131530	+	0.991312i	$0.541989\pi$
$908$	4.17183	0.138447
$909$	−5.24104	−0.173834
$910$	3.58654	0.118893
$911$	30.5244	1.01132	0.505659	−	0.862734i	$-0.331250\pi$
$911$	30.5244	1.01132	0.505659	+	0.862734i	$0.331250\pi$
$912$	−5.66656	−0.187638
$913$	0	0
$914$	−30.0377	−0.993559
$915$	4.66733	0.154297
$916$	14.1775	0.468439
$917$	−14.9957	−0.495202
$918$	−23.1470	−0.763964
$919$	−5.14113	−0.169590	−0.0847951	−	0.996398i	$-0.527024\pi$
$919$	−5.14113	−0.169590	−0.0847951	+	0.996398i	$0.527024\pi$
$920$	−2.85699	−0.0941922
$921$	−6.66772	−0.219709
$922$	−37.8904	−1.24785
$923$	20.7930	0.684410
$924$	0	0
$925$	−4.24080	−0.139437
$926$	2.23043	0.0732966
$927$	24.0210	0.788952
$928$	−0.754826	−0.0247784
$929$	9.36591	0.307286	0.153643	−	0.988126i	$-0.450899\pi$
$929$	9.36591	0.307286	0.153643	+	0.988126i	$0.450899\pi$
$930$	10.8318	0.355189
$931$	−4.89322	−0.160369
$932$	−9.49782	−0.311111
$933$	14.1344	0.462739
$934$	40.1699	1.31440
$935$	0	0
$936$	5.94984	0.194477
$937$	−32.4445	−1.05991	−0.529957	−	0.848024i	$-0.677792\pi$
$937$	−32.4445	−1.05991	−0.529957	+	0.848024i	$0.677792\pi$
$938$	−14.3059	−0.467103
$939$	23.5270	0.767776
$940$	−2.23785	−0.0729908
$941$	−11.9651	−0.390052	−0.195026	−	0.980798i	$-0.562479\pi$
$941$	−11.9651	−0.390052	−0.195026	+	0.980798i	$0.562479\pi$
$942$	−5.60313	−0.182560
$943$	5.79657	0.188762
$944$	1.22512	0.0398743
$945$	5.39525	0.175507
$946$	0	0
$947$	5.99311	0.194750	0.0973749	−	0.995248i	$-0.468955\pi$
$947$	5.99311	0.194750	0.0973749	+	0.995248i	$0.468955\pi$
$948$	−5.82808	−0.189287
$949$	32.5174	1.05556
$950$	−4.89322	−0.158757
$951$	30.7222	0.996235
$952$	4.29025	0.139048
$953$	37.6907	1.22092	0.610461	−	0.792046i	$-0.290984\pi$
$953$	37.6907	1.22092	0.610461	+	0.792046i	$0.290984\pi$
$954$	−21.7363	−0.703739
$955$	−19.1652	−0.620171
$956$	2.03180	0.0657130
$957$	0	0
$958$	−15.4649	−0.499648
$959$	−0.423951	−0.0136901
$960$	−1.15804	−0.0373757
$961$	56.4888	1.82222
$962$	15.2098	0.490383
$963$	11.7977	0.380176
$964$	−18.6496	−0.600663
$965$	−8.44654	−0.271904
$966$	3.30852	0.106450
$967$	−49.5830	−1.59448	−0.797241	−	0.603662i	$-0.793708\pi$
$967$	−49.5830	−1.59448	−0.797241	+	0.603662i	$0.793708\pi$
$968$	0	0
$969$	−24.3110	−0.780981
$970$	6.71733	0.215680
$971$	−38.2393	−1.22716	−0.613579	−	0.789633i	$-0.710271\pi$
$971$	−38.2393	−1.22716	−0.613579	+	0.789633i	$0.710271\pi$
$972$	14.7137	0.471943
$973$	19.3990	0.621903
$974$	−17.9184	−0.574143
$975$	−4.15337	−0.133014
$976$	−4.03036	−0.129009
$977$	−7.02182	−0.224648	−0.112324	−	0.993672i	$-0.535829\pi$
$977$	−7.02182	−0.224648	−0.112324	+	0.993672i	$0.535829\pi$
$978$	4.61172	0.147467
$979$	0	0
$980$	−1.00000	−0.0319438
$981$	25.0018	0.798247
$982$	26.6413	0.850157
$983$	−31.0557	−0.990525	−0.495262	−	0.868744i	$-0.664928\pi$
$983$	−31.0557	−0.990525	−0.495262	+	0.868744i	$0.664928\pi$
$984$	2.34956	0.0749013
$985$	11.1821	0.356292
$986$	−3.23839	−0.103131
$987$	2.59153	0.0824894
$988$	17.5497	0.558331
$989$	−0.196901	−0.00626108
$990$	0	0
$991$	20.2609	0.643608	0.321804	−	0.946806i	$-0.395711\pi$
$991$	20.2609	0.643608	0.321804	+	0.946806i	$0.395711\pi$
$992$	−9.35354	−0.296975
$993$	0.456143	0.0144753
$994$	−5.79751	−0.183886
$995$	4.23045	0.134114
$996$	−19.2534	−0.610067
$997$	16.0793	0.509236	0.254618	−	0.967042i	$-0.418050\pi$
$997$	16.0793	0.509236	0.254618	+	0.967042i	$0.418050\pi$
$998$	29.4299	0.931588
$999$	22.8802	0.723896

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8470.2.a.da.1.5		6
11.5	even	5		770.2.n.h.421.3	✓	12
11.9	even	5		770.2.n.h.631.3	yes	12
11.10	odd	2		8470.2.a.cu.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.n.h.421.3	✓	12	11.5	even	5
770.2.n.h.631.3	yes	12	11.9	even	5
8470.2.a.cu.1.5		6	11.10	odd	2
8470.2.a.da.1.5		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.15804 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.15804 q^{6} +1.00000 q^{7} +1.00000 q^{8} -1.65894 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.15804 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.15804 q^{6} +1.00000 q^{7} +1.00000 q^{8} -1.65894 q^{9} -1.00000 q^{10} +1.15804 q^{12} -3.58654 q^{13} +1.00000 q^{14} -1.15804 q^{15} +1.00000 q^{16} +4.29025 q^{17} -1.65894 q^{18} -4.89322 q^{19} -1.00000 q^{20} +1.15804 q^{21} +2.85699 q^{23} +1.15804 q^{24} +1.00000 q^{25} -3.58654 q^{26} -5.39525 q^{27} +1.00000 q^{28} -0.754826 q^{29} -1.15804 q^{30} -9.35354 q^{31} +1.00000 q^{32} +4.29025 q^{34} -1.00000 q^{35} -1.65894 q^{36} -4.24080 q^{37} -4.89322 q^{38} -4.15337 q^{39} -1.00000 q^{40} +2.02891 q^{41} +1.15804 q^{42} -0.0689189 q^{43} +1.65894 q^{45} +2.85699 q^{46} +2.23785 q^{47} +1.15804 q^{48} +1.00000 q^{49} +1.00000 q^{50} +4.96830 q^{51} -3.58654 q^{52} +13.1026 q^{53} -5.39525 q^{54} +1.00000 q^{56} -5.66656 q^{57} -0.754826 q^{58} +1.22512 q^{59} -1.15804 q^{60} -4.03036 q^{61} -9.35354 q^{62} -1.65894 q^{63} +1.00000 q^{64} +3.58654 q^{65} -14.3059 q^{67} +4.29025 q^{68} +3.30852 q^{69} -1.00000 q^{70} -5.79751 q^{71} -1.65894 q^{72} -9.06652 q^{73} -4.24080 q^{74} +1.15804 q^{75} -4.89322 q^{76} -4.15337 q^{78} -5.03269 q^{79} -1.00000 q^{80} -1.27112 q^{81} +2.02891 q^{82} -16.6258 q^{83} +1.15804 q^{84} -4.29025 q^{85} -0.0689189 q^{86} -0.874121 q^{87} +9.92243 q^{89} +1.65894 q^{90} -3.58654 q^{91} +2.85699 q^{92} -10.8318 q^{93} +2.23785 q^{94} +4.89322 q^{95} +1.15804 q^{96} -6.71733 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{2} - 5 q^{3} + 6 q^{4} - 6 q^{5} - 5 q^{6} + 6 q^{7} + 6 q^{8} + 11 q^{9}+O(q^{10}) \) 6 * q + 6 * q^2 - 5 * q^3 + 6 * q^4 - 6 * q^5 - 5 * q^6 + 6 * q^7 + 6 * q^8 + 11 * q^9	\( 6 q + 6 q^{2} - 5 q^{3} + 6 q^{4} - 6 q^{5} - 5 q^{6} + 6 q^{7} + 6 q^{8} + 11 q^{9} - 6 q^{10} - 5 q^{12} + 6 q^{14} + 5 q^{15} + 6 q^{16} - 15 q^{17} + 11 q^{18} - 13 q^{19} - 6 q^{20} - 5 q^{21} - 2 q^{23} - 5 q^{24} + 6 q^{25} - 26 q^{27} + 6 q^{28} - 4 q^{29} + 5 q^{30} - 2 q^{31} + 6 q^{32} - 15 q^{34} - 6 q^{35} + 11 q^{36} - 13 q^{38} - 30 q^{39} - 6 q^{40} - 17 q^{41} - 5 q^{42} + 15 q^{43} - 11 q^{45} - 2 q^{46} - 18 q^{47} - 5 q^{48} + 6 q^{49} + 6 q^{50} + 6 q^{51} + 22 q^{53} - 26 q^{54} + 6 q^{56} - 2 q^{57} - 4 q^{58} - 7 q^{59} + 5 q^{60} - 20 q^{61} - 2 q^{62} + 11 q^{63} + 6 q^{64} - 29 q^{67} - 15 q^{68} + 12 q^{69} - 6 q^{70} - 2 q^{71} + 11 q^{72} + q^{73} - 5 q^{75} - 13 q^{76} - 30 q^{78} - 12 q^{79} - 6 q^{80} + 22 q^{81} - 17 q^{82} - 35 q^{83} - 5 q^{84} + 15 q^{85} + 15 q^{86} - 29 q^{89} - 11 q^{90} - 2 q^{92} + 8 q^{93} - 18 q^{94} + 13 q^{95} - 5 q^{96} - 19 q^{97} + 6 q^{98}+O(q^{100}) \) 6 * q + 6 * q^2 - 5 * q^3 + 6 * q^4 - 6 * q^5 - 5 * q^6 + 6 * q^7 + 6 * q^8 + 11 * q^9 - 6 * q^10 - 5 * q^12 + 6 * q^14 + 5 * q^15 + 6 * q^16 - 15 * q^17 + 11 * q^18 - 13 * q^19 - 6 * q^20 - 5 * q^21 - 2 * q^23 - 5 * q^24 + 6 * q^25 - 26 * q^27 + 6 * q^28 - 4 * q^29 + 5 * q^30 - 2 * q^31 + 6 * q^32 - 15 * q^34 - 6 * q^35 + 11 * q^36 - 13 * q^38 - 30 * q^39 - 6 * q^40 - 17 * q^41 - 5 * q^42 + 15 * q^43 - 11 * q^45 - 2 * q^46 - 18 * q^47 - 5 * q^48 + 6 * q^49 + 6 * q^50 + 6 * q^51 + 22 * q^53 - 26 * q^54 + 6 * q^56 - 2 * q^57 - 4 * q^58 - 7 * q^59 + 5 * q^60 - 20 * q^61 - 2 * q^62 + 11 * q^63 + 6 * q^64 - 29 * q^67 - 15 * q^68 + 12 * q^69 - 6 * q^70 - 2 * q^71 + 11 * q^72 + q^73 - 5 * q^75 - 13 * q^76 - 30 * q^78 - 12 * q^79 - 6 * q^80 + 22 * q^81 - 17 * q^82 - 35 * q^83 - 5 * q^84 + 15 * q^85 + 15 * q^86 - 29 * q^89 - 11 * q^90 - 2 * q^92 + 8 * q^93 - 18 * q^94 + 13 * q^95 - 5 * q^96 - 19 * q^97 + 6 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more