LMFDB - Embedded newform 8470.2.a.cu.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:	$0$
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			1.4
Root			$0.752765$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} -0.247235 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.247235 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.93888 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -0.247235 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.247235 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.93888 q^{9} +1.00000 q^{10} -0.247235 q^{12} -2.83439 q^{13} +1.00000 q^{14} +0.247235 q^{15} +1.00000 q^{16} +5.76549 q^{17} +2.93888 q^{18} -3.66941 q^{19} -1.00000 q^{20} +0.247235 q^{21} -7.17432 q^{23} +0.247235 q^{24} +1.00000 q^{25} +2.83439 q^{26} +1.46830 q^{27} -1.00000 q^{28} -1.41528 q^{29} -0.247235 q^{30} -0.356435 q^{31} -1.00000 q^{32} -5.76549 q^{34} +1.00000 q^{35} -2.93888 q^{36} +2.94752 q^{37} +3.66941 q^{38} +0.700760 q^{39} +1.00000 q^{40} -2.22578 q^{41} -0.247235 q^{42} -9.64018 q^{43} +2.93888 q^{45} +7.17432 q^{46} +3.13615 q^{47} -0.247235 q^{48} +1.00000 q^{49} -1.00000 q^{50} -1.42543 q^{51} -2.83439 q^{52} +0.874146 q^{53} -1.46830 q^{54} +1.00000 q^{56} +0.907206 q^{57} +1.41528 q^{58} -10.4962 q^{59} +0.247235 q^{60} +9.20135 q^{61} +0.356435 q^{62} +2.93888 q^{63} +1.00000 q^{64} +2.83439 q^{65} -13.5466 q^{67} +5.76549 q^{68} +1.77374 q^{69} -1.00000 q^{70} -12.5139 q^{71} +2.93888 q^{72} +7.19836 q^{73} -2.94752 q^{74} -0.247235 q^{75} -3.66941 q^{76} -0.700760 q^{78} -6.14508 q^{79} -1.00000 q^{80} +8.45361 q^{81} +2.22578 q^{82} -3.08217 q^{83} +0.247235 q^{84} -5.76549 q^{85} +9.64018 q^{86} +0.349905 q^{87} -5.75443 q^{89} -2.93888 q^{90} +2.83439 q^{91} -7.17432 q^{92} +0.0881230 q^{93} -3.13615 q^{94} +3.66941 q^{95} +0.247235 q^{96} +12.5278 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.247235	−0.142741	−0.0713705	−	0.997450i	$-0.522737\pi$
$3$	−0.247235	−0.142741	−0.0713705	+	0.997450i	$0.522737\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0.247235	0.100933
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	−2.93888	−0.979625
$10$	1.00000	0.316228
$11$	0	0
$11$	0	0
$12$	−0.247235	−0.0713705
$13$	−2.83439	−0.786119	−0.393060	−	0.919513i	$-0.628583\pi$
$13$	−2.83439	−0.786119	−0.393060	+	0.919513i	$0.628583\pi$
$14$	1.00000	0.267261
$15$	0.247235	0.0638357
$16$	1.00000	0.250000
$17$	5.76549	1.39834	0.699168	−	0.714957i	$-0.253554\pi$
$17$	5.76549	1.39834	0.699168	+	0.714957i	$0.253554\pi$
$18$	2.93888	0.692699
$19$	−3.66941	−0.841821	−0.420910	−	0.907102i	$-0.638289\pi$
$19$	−3.66941	−0.841821	−0.420910	+	0.907102i	$0.638289\pi$
$20$	−1.00000	−0.223607
$21$	0.247235	0.0539510
$22$	0	0
$23$	−7.17432	−1.49595	−0.747974	−	0.663728i	$-0.768973\pi$
$23$	−7.17432	−1.49595	−0.747974	+	0.663728i	$0.768973\pi$
$24$	0.247235	0.0504666
$25$	1.00000	0.200000
$26$	2.83439	0.555870
$27$	1.46830	0.282574
$28$	−1.00000	−0.188982
$29$	−1.41528	−0.262810	−0.131405	−	0.991329i	$-0.541949\pi$
$29$	−1.41528	−0.262810	−0.131405	+	0.991329i	$0.541949\pi$
$30$	−0.247235	−0.0451387
$31$	−0.356435	−0.0640176	−0.0320088	−	0.999488i	$-0.510190\pi$
$31$	−0.356435	−0.0640176	−0.0320088	+	0.999488i	$0.510190\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−5.76549	−0.988773
$35$	1.00000	0.169031
$36$	−2.93888	−0.489813
$37$	2.94752	0.484570	0.242285	−	0.970205i	$-0.422103\pi$
$37$	2.94752	0.484570	0.242285	+	0.970205i	$0.422103\pi$
$38$	3.66941	0.595257
$39$	0.700760	0.112211
$40$	1.00000	0.158114
$41$	−2.22578	−0.347608	−0.173804	−	0.984780i	$-0.555606\pi$
$41$	−2.22578	−0.347608	−0.173804	+	0.984780i	$0.555606\pi$
$42$	−0.247235	−0.0381491
$43$	−9.64018	−1.47011	−0.735057	−	0.678006i	$-0.762844\pi$
$43$	−9.64018	−1.47011	−0.735057	+	0.678006i	$0.762844\pi$
$44$	0	0
$45$	2.93888	0.438102
$46$	7.17432	1.05780
$47$	3.13615	0.457455	0.228727	−	0.973491i	$-0.426544\pi$
$47$	3.13615	0.457455	0.228727	+	0.973491i	$0.426544\pi$
$48$	−0.247235	−0.0356852
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	−1.42543	−0.199600
$52$	−2.83439	−0.393060
$53$	0.874146	0.120073	0.0600366	−	0.998196i	$-0.480878\pi$
$53$	0.874146	0.120073	0.0600366	+	0.998196i	$0.480878\pi$
$54$	−1.46830	−0.199810
$55$	0	0
$56$	1.00000	0.133631
$57$	0.907206	0.120162
$58$	1.41528	0.185835
$59$	−10.4962	−1.36649	−0.683243	−	0.730191i	$-0.739431\pi$
$59$	−10.4962	−1.36649	−0.683243	+	0.730191i	$0.739431\pi$
$60$	0.247235	0.0319179
$61$	9.20135	1.17811	0.589056	−	0.808092i	$-0.299500\pi$
$61$	9.20135	1.17811	0.589056	+	0.808092i	$0.299500\pi$
$62$	0.356435	0.0452673
$63$	2.93888	0.370263
$64$	1.00000	0.125000
$65$	2.83439	0.351563
$66$	0	0
$67$	−13.5466	−1.65498	−0.827492	−	0.561478i	$-0.810233\pi$
$67$	−13.5466	−1.65498	−0.827492	+	0.561478i	$0.810233\pi$
$68$	5.76549	0.699168
$69$	1.77374	0.213533
$70$	−1.00000	−0.119523
$71$	−12.5139	−1.48512	−0.742561	−	0.669779i	$-0.766389\pi$
$71$	−12.5139	−1.48512	−0.742561	+	0.669779i	$0.766389\pi$
$72$	2.93888	0.346350
$73$	7.19836	0.842504	0.421252	−	0.906944i	$-0.361591\pi$
$73$	7.19836	0.842504	0.421252	+	0.906944i	$0.361591\pi$
$74$	−2.94752	−0.342643
$75$	−0.247235	−0.0285482
$76$	−3.66941	−0.420910
$77$	0	0
$78$	−0.700760	−0.0793455
$79$	−6.14508	−0.691376	−0.345688	−	0.938350i	$-0.612354\pi$
$79$	−6.14508	−0.691376	−0.345688	+	0.938350i	$0.612354\pi$
$80$	−1.00000	−0.111803
$81$	8.45361	0.939290
$82$	2.22578	0.245796
$83$	−3.08217	−0.338312	−0.169156	−	0.985589i	$-0.554104\pi$
$83$	−3.08217	−0.338312	−0.169156	+	0.985589i	$0.554104\pi$
$84$	0.247235	0.0269755
$85$	−5.76549	−0.625355
$86$	9.64018	1.03953
$87$	0.349905	0.0375138
$88$	0	0
$89$	−5.75443	−0.609968	−0.304984	−	0.952357i	$-0.598651\pi$
$89$	−5.75443	−0.609968	−0.304984	+	0.952357i	$0.598651\pi$
$90$	−2.93888	−0.309785
$91$	2.83439	0.297125
$92$	−7.17432	−0.747974
$93$	0.0881230	0.00913793
$94$	−3.13615	−0.323469
$95$	3.66941	0.376474
$96$	0.247235	0.0252333
$97$	12.5278	1.27200	0.636001	−	0.771689i	$-0.280588\pi$
$97$	12.5278	1.27200	0.636001	+	0.771689i	$0.280588\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	5.90123	0.587195	0.293597	−	0.955929i	$-0.405148\pi$
$101$	5.90123	0.587195	0.293597	+	0.955929i	$0.405148\pi$
$102$	1.42543	0.141138
$103$	−11.9671	−1.17915	−0.589576	−	0.807713i	$-0.700705\pi$
$103$	−11.9671	−1.17915	−0.589576	+	0.807713i	$0.700705\pi$
$104$	2.83439	0.277935
$105$	−0.247235	−0.0241276
$106$	−0.874146	−0.0849046
$107$	6.28895	0.607976	0.303988	−	0.952676i	$-0.401682\pi$
$107$	6.28895	0.607976	0.303988	+	0.952676i	$0.401682\pi$
$108$	1.46830	0.141287
$109$	3.42125	0.327696	0.163848	−	0.986486i	$-0.447609\pi$
$109$	3.42125	0.327696	0.163848	+	0.986486i	$0.447609\pi$
$110$	0	0
$111$	−0.728730	−0.0691680
$112$	−1.00000	−0.0944911
$113$	−16.8223	−1.58251	−0.791253	−	0.611489i	$-0.790571\pi$
$113$	−16.8223	−1.58251	−0.791253	+	0.611489i	$0.790571\pi$
$114$	−0.907206	−0.0849676
$115$	7.17432	0.669008
$116$	−1.41528	−0.131405
$117$	8.32993	0.770102
$118$	10.4962	0.966251
$119$	−5.76549	−0.528521
$120$	−0.247235	−0.0225693
$121$	0	0
$122$	−9.20135	−0.833051
$123$	0.550290	0.0496180
$124$	−0.356435	−0.0320088
$125$	−1.00000	−0.0894427
$126$	−2.93888	−0.261816
$127$	−0.175760	−0.0155961	−0.00779807	−	0.999970i	$-0.502482\pi$
$127$	−0.175760	−0.0155961	−0.00779807	+	0.999970i	$0.502482\pi$
$128$	−1.00000	−0.0883883
$129$	2.38339	0.209845
$130$	−2.83439	−0.248593
$131$	11.4391	0.999438	0.499719	−	0.866187i	$-0.333437\pi$
$131$	11.4391	0.999438	0.499719	+	0.866187i	$0.333437\pi$
$132$	0	0
$133$	3.66941	0.318178
$134$	13.5466	1.17025
$135$	−1.46830	−0.126371
$136$	−5.76549	−0.494387
$137$	−9.95341	−0.850377	−0.425189	−	0.905105i	$-0.639792\pi$
$137$	−9.95341	−0.850377	−0.425189	+	0.905105i	$0.639792\pi$
$138$	−1.77374	−0.150991
$139$	0.627224	0.0532005	0.0266002	−	0.999646i	$-0.491532\pi$
$139$	0.627224	0.0532005	0.0266002	+	0.999646i	$0.491532\pi$
$140$	1.00000	0.0845154
$141$	−0.775365	−0.0652975
$142$	12.5139	1.05014
$143$	0	0
$144$	−2.93888	−0.244906
$145$	1.41528	0.117532
$146$	−7.19836	−0.595740
$147$	−0.247235	−0.0203916
$148$	2.94752	0.242285
$149$	15.0315	1.23143	0.615713	−	0.787970i	$-0.288868\pi$
$149$	15.0315	1.23143	0.615713	+	0.787970i	$0.288868\pi$
$150$	0.247235	0.0201866
$151$	−0.516467	−0.0420295	−0.0210148	−	0.999779i	$-0.506690\pi$
$151$	−0.516467	−0.0420295	−0.0210148	+	0.999779i	$0.506690\pi$
$152$	3.66941	0.297629
$153$	−16.9441	−1.36985
$154$	0	0
$155$	0.356435	0.0286295
$156$	0.700760	0.0561057
$157$	−22.7782	−1.81790	−0.908949	−	0.416907i	$-0.863114\pi$
$157$	−22.7782	−1.81790	−0.908949	+	0.416907i	$0.863114\pi$
$158$	6.14508	0.488876
$159$	−0.216119	−0.0171394
$160$	1.00000	0.0790569
$161$	7.17432	0.565415
$162$	−8.45361	−0.664178
$163$	3.72353	0.291649	0.145825	−	0.989310i	$-0.453416\pi$
$163$	3.72353	0.291649	0.145825	+	0.989310i	$0.453416\pi$
$164$	−2.22578	−0.173804
$165$	0	0
$166$	3.08217	0.239223
$167$	−15.5825	−1.20581	−0.602906	−	0.797812i	$-0.705991\pi$
$167$	−15.5825	−1.20581	−0.602906	+	0.797812i	$0.705991\pi$
$168$	−0.247235	−0.0190746
$169$	−4.96622	−0.382017
$170$	5.76549	0.442193
$171$	10.7839	0.824669
$172$	−9.64018	−0.735057
$173$	−5.89952	−0.448532	−0.224266	−	0.974528i	$-0.571998\pi$
$173$	−5.89952	−0.448532	−0.224266	+	0.974528i	$0.571998\pi$
$174$	−0.349905	−0.0265263
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	2.59502	0.195053
$178$	5.75443	0.431313
$179$	16.3323	1.22073	0.610367	−	0.792119i	$-0.291022\pi$
$179$	16.3323	1.22073	0.610367	+	0.792119i	$0.291022\pi$
$180$	2.93888	0.219051
$181$	−3.48610	−0.259120	−0.129560	−	0.991572i	$-0.541356\pi$
$181$	−3.48610	−0.259120	−0.129560	+	0.991572i	$0.541356\pi$
$182$	−2.83439	−0.210099
$183$	−2.27489	−0.168165
$184$	7.17432	0.528898
$185$	−2.94752	−0.216706
$186$	−0.0881230	−0.00646149
$187$	0	0
$188$	3.13615	0.228727
$189$	−1.46830	−0.106803
$190$	−3.66941	−0.266207
$191$	25.9703	1.87914	0.939572	−	0.342352i	$-0.111223\pi$
$191$	25.9703	1.87914	0.939572	+	0.342352i	$0.111223\pi$
$192$	−0.247235	−0.0178426
$193$	0.677502	0.0487677	0.0243838	−	0.999703i	$-0.492238\pi$
$193$	0.677502	0.0487677	0.0243838	+	0.999703i	$0.492238\pi$
$194$	−12.5278	−0.899441
$195$	−0.700760	−0.0501825
$196$	1.00000	0.0714286
$197$	−5.80812	−0.413811	−0.206906	−	0.978361i	$-0.566339\pi$
$197$	−5.80812	−0.413811	−0.206906	+	0.978361i	$0.566339\pi$
$198$	0	0
$199$	10.9475	0.776046	0.388023	−	0.921650i	$-0.373158\pi$
$199$	10.9475	0.776046	0.388023	+	0.921650i	$0.373158\pi$
$200$	−1.00000	−0.0707107
$201$	3.34919	0.236234
$202$	−5.90123	−0.415209
$203$	1.41528	0.0993329
$204$	−1.42543	−0.0998000
$205$	2.22578	0.155455
$206$	11.9671	0.833787
$207$	21.0844	1.46547
$208$	−2.83439	−0.196530
$209$	0	0
$210$	0.247235	0.0170608
$211$	6.42786	0.442512	0.221256	−	0.975216i	$-0.428984\pi$
$211$	6.42786	0.442512	0.221256	+	0.975216i	$0.428984\pi$
$212$	0.874146	0.0600366
$213$	3.09386	0.211988
$214$	−6.28895	−0.429904
$215$	9.64018	0.657455
$216$	−1.46830	−0.0999049
$217$	0.356435	0.0241964
$218$	−3.42125	−0.231716
$219$	−1.77968	−0.120260
$220$	0	0
$221$	−16.3417	−1.09926
$222$	0.728730	0.0489091
$223$	23.2048	1.55391	0.776955	−	0.629557i	$-0.216763\pi$
$223$	23.2048	1.55391	0.776955	+	0.629557i	$0.216763\pi$
$224$	1.00000	0.0668153
$225$	−2.93888	−0.195925
$226$	16.8223	1.11900
$227$	25.5715	1.69724	0.848621	−	0.529001i	$-0.177433\pi$
$227$	25.5715	1.69724	0.848621	+	0.529001i	$0.177433\pi$
$228$	0.907206	0.0600812
$229$	4.02501	0.265980	0.132990	−	0.991117i	$-0.457542\pi$
$229$	4.02501	0.265980	0.132990	+	0.991117i	$0.457542\pi$
$230$	−7.17432	−0.473060
$231$	0	0
$232$	1.41528	0.0929174
$233$	3.67721	0.240902	0.120451	−	0.992719i	$-0.461566\pi$
$233$	3.67721	0.240902	0.120451	+	0.992719i	$0.461566\pi$
$234$	−8.32993	−0.544544
$235$	−3.13615	−0.204580
$236$	−10.4962	−0.683243
$237$	1.51928	0.0986877
$238$	5.76549	0.373721
$239$	1.09444	0.0707931	0.0353966	−	0.999373i	$-0.488731\pi$
$239$	1.09444	0.0707931	0.0353966	+	0.999373i	$0.488731\pi$
$240$	0.247235	0.0159589
$241$	−6.24902	−0.402535	−0.201267	−	0.979536i	$-0.564506\pi$
$241$	−6.24902	−0.402535	−0.201267	+	0.979536i	$0.564506\pi$
$242$	0	0
$243$	−6.49491	−0.416649
$244$	9.20135	0.589056
$245$	−1.00000	−0.0638877
$246$	−0.550290	−0.0350852
$247$	10.4006	0.661772
$248$	0.356435	0.0226336
$249$	0.762019	0.0482910
$250$	1.00000	0.0632456
$251$	−19.2916	−1.21767	−0.608836	−	0.793296i	$-0.708363\pi$
$251$	−19.2916	−1.21767	−0.608836	+	0.793296i	$0.708363\pi$
$252$	2.93888	0.185132
$253$	0	0
$254$	0.175760	0.0110281
$255$	1.42543	0.0892638
$256$	1.00000	0.0625000
$257$	25.2663	1.57607	0.788035	−	0.615630i	$-0.211099\pi$
$257$	25.2663	1.57607	0.788035	+	0.615630i	$0.211099\pi$
$258$	−2.38339	−0.148383
$259$	−2.94752	−0.183150
$260$	2.83439	0.175782
$261$	4.15932	0.257455
$262$	−11.4391	−0.706710
$263$	−2.99352	−0.184589	−0.0922943	−	0.995732i	$-0.529420\pi$
$263$	−2.99352	−0.184589	−0.0922943	+	0.995732i	$0.529420\pi$
$264$	0	0
$265$	−0.874146	−0.0536984
$266$	−3.66941	−0.224986
$267$	1.42269	0.0870675
$268$	−13.5466	−0.827492
$269$	−22.5013	−1.37193	−0.685964	−	0.727635i	$-0.740619\pi$
$269$	−22.5013	−1.37193	−0.685964	+	0.727635i	$0.740619\pi$
$270$	1.46830	0.0893576
$271$	13.6286	0.827881	0.413940	−	0.910304i	$-0.364152\pi$
$271$	13.6286	0.827881	0.413940	+	0.910304i	$0.364152\pi$
$272$	5.76549	0.349584
$273$	−0.700760	−0.0424119
$274$	9.95341	0.601307
$275$	0	0
$276$	1.77374	0.106767
$277$	23.5465	1.41477	0.707387	−	0.706827i	$-0.249874\pi$
$277$	23.5465	1.41477	0.707387	+	0.706827i	$0.249874\pi$
$278$	−0.627224	−0.0376184
$279$	1.04752	0.0627132
$280$	−1.00000	−0.0597614
$281$	19.1837	1.14440	0.572201	−	0.820113i	$-0.306090\pi$
$281$	19.1837	1.14440	0.572201	+	0.820113i	$0.306090\pi$
$282$	0.775365	0.0461723
$283$	−17.2722	−1.02673	−0.513363	−	0.858172i	$-0.671601\pi$
$283$	−17.2722	−1.02673	−0.513363	+	0.858172i	$0.671601\pi$
$284$	−12.5139	−0.742561
$285$	−0.907206	−0.0537382
$286$	0	0
$287$	2.22578	0.131384
$288$	2.93888	0.173175
$289$	16.2409	0.955345
$290$	−1.41528	−0.0831079
$291$	−3.09730	−0.181567
$292$	7.19836	0.421252
$293$	32.9318	1.92389	0.961947	−	0.273235i	$-0.0880938\pi$
$293$	32.9318	1.92389	0.961947	+	0.273235i	$0.0880938\pi$
$294$	0.247235	0.0144190
$295$	10.4962	0.611111
$296$	−2.94752	−0.171321
$297$	0	0
$298$	−15.0315	−0.870750
$299$	20.3348	1.17599
$300$	−0.247235	−0.0142741
$301$	9.64018	0.555651
$302$	0.516467	0.0297194
$303$	−1.45899	−0.0838168
$304$	−3.66941	−0.210455
$305$	−9.20135	−0.526868
$306$	16.9441	0.968627
$307$	7.97195	0.454983	0.227492	−	0.973780i	$-0.426948\pi$
$307$	7.97195	0.454983	0.227492	+	0.973780i	$0.426948\pi$
$308$	0	0
$309$	2.95868	0.168313
$310$	−0.356435	−0.0202441
$311$	−4.00836	−0.227293	−0.113647	−	0.993521i	$-0.536253\pi$
$311$	−4.00836	−0.227293	−0.113647	+	0.993521i	$0.536253\pi$
$312$	−0.700760	−0.0396727
$313$	5.27260	0.298025	0.149013	−	0.988835i	$-0.452390\pi$
$313$	5.27260	0.298025	0.149013	+	0.988835i	$0.452390\pi$
$314$	22.7782	1.28545
$315$	−2.93888	−0.165587
$316$	−6.14508	−0.345688
$317$	−4.65978	−0.261720	−0.130860	−	0.991401i	$-0.541774\pi$
$317$	−4.65978	−0.261720	−0.130860	+	0.991401i	$0.541774\pi$
$318$	0.216119	0.0121194
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	−1.55485	−0.0867831
$322$	−7.17432	−0.399809
$323$	−21.1560	−1.17715
$324$	8.45361	0.469645
$325$	−2.83439	−0.157224
$326$	−3.72353	−0.206227
$327$	−0.845851	−0.0467757
$328$	2.22578	0.122898
$329$	−3.13615	−0.172902
$330$	0	0
$331$	−25.0199	−1.37522	−0.687608	−	0.726082i	$-0.741339\pi$
$331$	−25.0199	−1.37522	−0.687608	+	0.726082i	$0.741339\pi$
$332$	−3.08217	−0.169156
$333$	−8.66240	−0.474697
$334$	15.5825	0.852638
$335$	13.5466	0.740131
$336$	0.247235	0.0134878
$337$	−15.9367	−0.868130	−0.434065	−	0.900882i	$-0.642921\pi$
$337$	−15.9367	−0.868130	−0.434065	+	0.900882i	$0.642921\pi$
$338$	4.96622	0.270127
$339$	4.15905	0.225889
$340$	−5.76549	−0.312678
$341$	0	0
$342$	−10.7839	−0.583129
$343$	−1.00000	−0.0539949
$344$	9.64018	0.519764
$345$	−1.77374	−0.0954949
$346$	5.89952	0.317160
$347$	12.4275	0.667142	0.333571	−	0.942725i	$-0.391746\pi$
$347$	12.4275	0.667142	0.333571	+	0.942725i	$0.391746\pi$
$348$	0.349905	0.0187569
$349$	16.2129	0.867854	0.433927	−	0.900948i	$-0.357128\pi$
$349$	16.2129	0.867854	0.433927	+	0.900948i	$0.357128\pi$
$350$	1.00000	0.0534522
$351$	−4.16173	−0.222137
$352$	0	0
$353$	−21.8320	−1.16200	−0.580999	−	0.813904i	$-0.697338\pi$
$353$	−21.8320	−1.16200	−0.580999	+	0.813904i	$0.697338\pi$
$354$	−2.59502	−0.137924
$355$	12.5139	0.664166
$356$	−5.75443	−0.304984
$357$	1.42543	0.0754417
$358$	−16.3323	−0.863189
$359$	18.2106	0.961116	0.480558	−	0.876963i	$-0.340434\pi$
$359$	18.2106	0.961116	0.480558	+	0.876963i	$0.340434\pi$
$360$	−2.93888	−0.154892
$361$	−5.53541	−0.291338
$362$	3.48610	0.183226
$363$	0	0
$364$	2.83439	0.148563
$365$	−7.19836	−0.376779
$366$	2.27489	0.118911
$367$	−5.27857	−0.275539	−0.137770	−	0.990464i	$-0.543993\pi$
$367$	−5.27857	−0.275539	−0.137770	+	0.990464i	$0.543993\pi$
$368$	−7.17432	−0.373987
$369$	6.54129	0.340526
$370$	2.94752	0.153234
$371$	−0.874146	−0.0453834
$372$	0.0881230	0.00456897
$373$	−18.5799	−0.962031	−0.481016	−	0.876712i	$-0.659732\pi$
$373$	−18.5799	−0.962031	−0.481016	+	0.876712i	$0.659732\pi$
$374$	0	0
$375$	0.247235	0.0127671
$376$	−3.13615	−0.161735
$377$	4.01145	0.206600
$378$	1.46830	0.0755210
$379$	22.8406	1.17324	0.586621	−	0.809862i	$-0.300458\pi$
$379$	22.8406	1.17324	0.586621	+	0.809862i	$0.300458\pi$
$380$	3.66941	0.188237
$381$	0.0434539	0.00222621
$382$	−25.9703	−1.32876
$383$	−22.4735	−1.14834	−0.574171	−	0.818736i	$-0.694675\pi$
$383$	−22.4735	−1.14834	−0.574171	+	0.818736i	$0.694675\pi$
$384$	0.247235	0.0126166
$385$	0	0
$386$	−0.677502	−0.0344840
$387$	28.3313	1.44016
$388$	12.5278	0.636001
$389$	36.2498	1.83794	0.918968	−	0.394333i	$-0.129024\pi$
$389$	36.2498	1.83794	0.918968	+	0.394333i	$0.129024\pi$
$390$	0.700760	0.0354844
$391$	−41.3634	−2.09184
$392$	−1.00000	−0.0505076
$393$	−2.82814	−0.142661
$394$	5.80812	0.292609
$395$	6.14508	0.309193
$396$	0	0
$397$	−8.02992	−0.403010	−0.201505	−	0.979487i	$-0.564583\pi$
$397$	−8.02992	−0.403010	−0.201505	+	0.979487i	$0.564583\pi$
$398$	−10.9475	−0.548748
$399$	−0.907206	−0.0454171
$400$	1.00000	0.0500000
$401$	0.672421	0.0335791	0.0167896	−	0.999859i	$-0.494655\pi$
$401$	0.672421	0.0335791	0.0167896	+	0.999859i	$0.494655\pi$
$402$	−3.34919	−0.167043
$403$	1.01028	0.0503254
$404$	5.90123	0.293597
$405$	−8.45361	−0.420063
$406$	−1.41528	−0.0702390
$407$	0	0
$408$	1.42543	0.0705692
$409$	21.7385	1.07490	0.537450	−	0.843296i	$-0.319388\pi$
$409$	21.7385	1.07490	0.537450	+	0.843296i	$0.319388\pi$
$410$	−2.22578	−0.109923
$411$	2.46083	0.121384
$412$	−11.9671	−0.589576
$413$	10.4962	0.516483
$414$	−21.0844	−1.03624
$415$	3.08217	0.151298
$416$	2.83439	0.138968
$417$	−0.155072	−0.00759389
$418$	0	0
$419$	−7.38003	−0.360538	−0.180269	−	0.983617i	$-0.557697\pi$
$419$	−7.38003	−0.360538	−0.180269	+	0.983617i	$0.557697\pi$
$420$	−0.247235	−0.0120638
$421$	29.4497	1.43529	0.717646	−	0.696408i	$-0.245220\pi$
$421$	29.4497	1.43529	0.717646	+	0.696408i	$0.245220\pi$
$422$	−6.42786	−0.312903
$423$	−9.21676	−0.448134
$424$	−0.874146	−0.0424523
$425$	5.76549	0.279667
$426$	−3.09386	−0.149898
$427$	−9.20135	−0.445285
$428$	6.28895	0.303988
$429$	0	0
$430$	−9.64018	−0.464891
$431$	19.6701	0.947477	0.473739	−	0.880665i	$-0.342904\pi$
$431$	19.6701	0.947477	0.473739	+	0.880665i	$0.342904\pi$
$432$	1.46830	0.0706434
$433$	−17.0504	−0.819390	−0.409695	−	0.912222i	$-0.634365\pi$
$433$	−17.0504	−0.819390	−0.409695	+	0.912222i	$0.634365\pi$
$434$	−0.356435	−0.0171094
$435$	−0.349905	−0.0167767
$436$	3.42125	0.163848
$437$	26.3255	1.25932
$438$	1.77968	0.0850365
$439$	−29.9589	−1.42986	−0.714929	−	0.699197i	$-0.753541\pi$
$439$	−29.9589	−1.42986	−0.714929	+	0.699197i	$0.753541\pi$
$440$	0	0
$441$	−2.93888	−0.139946
$442$	16.3417	0.777293
$443$	41.2954	1.96201	0.981003	−	0.193994i	$-0.0621442\pi$
$443$	41.2954	1.96201	0.981003	+	0.193994i	$0.0621442\pi$
$444$	−0.728730	−0.0345840
$445$	5.75443	0.272786
$446$	−23.2048	−1.09878
$447$	−3.71630	−0.175775
$448$	−1.00000	−0.0472456
$449$	33.9263	1.60108	0.800541	−	0.599278i	$-0.204546\pi$
$449$	33.9263	1.60108	0.800541	+	0.599278i	$0.204546\pi$
$450$	2.93888	0.138540
$451$	0	0
$452$	−16.8223	−0.791253
$453$	0.127689	0.00599934
$454$	−25.5715	−1.20013
$455$	−2.83439	−0.132878
$456$	−0.907206	−0.0424838
$457$	−19.9928	−0.935225	−0.467613	−	0.883934i	$-0.654886\pi$
$457$	−19.9928	−0.935225	−0.467613	+	0.883934i	$0.654886\pi$
$458$	−4.02501	−0.188076
$459$	8.46544	0.395133
$460$	7.17432	0.334504
$461$	17.4739	0.813840	0.406920	−	0.913464i	$-0.366603\pi$
$461$	17.4739	0.813840	0.406920	+	0.913464i	$0.366603\pi$
$462$	0	0
$463$	−41.7730	−1.94135	−0.970677	−	0.240386i	$-0.922726\pi$
$463$	−41.7730	−1.94135	−0.970677	+	0.240386i	$0.922726\pi$
$464$	−1.41528	−0.0657026
$465$	−0.0881230	−0.00408661
$466$	−3.67721	−0.170343
$467$	−13.7476	−0.636164	−0.318082	−	0.948063i	$-0.603039\pi$
$467$	−13.7476	−0.636164	−0.318082	+	0.948063i	$0.603039\pi$
$468$	8.32993	0.385051
$469$	13.5466	0.625525
$470$	3.13615	0.144660
$471$	5.63156	0.259489
$472$	10.4962	0.483125
$473$	0	0
$474$	−1.51928	−0.0697827
$475$	−3.66941	−0.168364
$476$	−5.76549	−0.264261
$477$	−2.56901	−0.117627
$478$	−1.09444	−0.0500583
$479$	20.6636	0.944145	0.472072	−	0.881560i	$-0.343506\pi$
$479$	20.6636	0.944145	0.472072	+	0.881560i	$0.343506\pi$
$480$	−0.247235	−0.0112847
$481$	−8.35444	−0.380930
$482$	6.24902	0.284635
$483$	−1.77374	−0.0807079
$484$	0	0
$485$	−12.5278	−0.568856
$486$	6.49491	0.294615
$487$	−22.1049	−1.00167	−0.500835	−	0.865543i	$-0.666974\pi$
$487$	−22.1049	−1.00167	−0.500835	+	0.865543i	$0.666974\pi$
$488$	−9.20135	−0.416526
$489$	−0.920586	−0.0416303
$490$	1.00000	0.0451754
$491$	−10.6974	−0.482768	−0.241384	−	0.970430i	$-0.577601\pi$
$491$	−10.6974	−0.482768	−0.241384	+	0.970430i	$0.577601\pi$
$492$	0.550290	0.0248090
$493$	−8.15976	−0.367497
$494$	−10.4006	−0.467943
$495$	0	0
$496$	−0.356435	−0.0160044
$497$	12.5139	0.561323
$498$	−0.762019	−0.0341469
$499$	1.85809	0.0831794	0.0415897	−	0.999135i	$-0.486758\pi$
$499$	1.85809	0.0831794	0.0415897	+	0.999135i	$0.486758\pi$
$500$	−1.00000	−0.0447214
$501$	3.85254	0.172119
$502$	19.2916	0.861024
$503$	13.5487	0.604108	0.302054	−	0.953291i	$-0.402328\pi$
$503$	13.5487	0.604108	0.302054	+	0.953291i	$0.402328\pi$
$504$	−2.93888	−0.130908
$505$	−5.90123	−0.262601
$506$	0	0
$507$	1.22782	0.0545295
$508$	−0.175760	−0.00779807
$509$	−6.36772	−0.282244	−0.141122	−	0.989992i	$-0.545071\pi$
$509$	−6.36772	−0.282244	−0.141122	+	0.989992i	$0.545071\pi$
$510$	−1.42543	−0.0631190
$511$	−7.19836	−0.318437
$512$	−1.00000	−0.0441942
$513$	−5.38778	−0.237876
$514$	−25.2663	−1.11445
$515$	11.9671	0.527333
$516$	2.38339	0.104923
$517$	0	0
$518$	2.94752	0.129507
$519$	1.45857	0.0640240
$520$	−2.83439	−0.124296
$521$	−24.1693	−1.05888	−0.529439	−	0.848348i	$-0.677598\pi$
$521$	−24.1693	−1.05888	−0.529439	+	0.848348i	$0.677598\pi$
$522$	−4.15932	−0.182048
$523$	30.5849	1.33738	0.668692	−	0.743540i	$-0.266855\pi$
$523$	30.5849	1.33738	0.668692	+	0.743540i	$0.266855\pi$
$524$	11.4391	0.499719
$525$	0.247235	0.0107902
$526$	2.99352	0.130524
$527$	−2.05502	−0.0895181
$528$	0	0
$529$	28.4708	1.23786
$530$	0.874146	0.0379705
$531$	30.8469	1.33864
$532$	3.66941	0.159089
$533$	6.30873	0.273261
$534$	−1.42269	−0.0615660
$535$	−6.28895	−0.271895
$536$	13.5466	0.585125
$537$	−4.03791	−0.174249
$538$	22.5013	0.970100
$539$	0	0
$540$	−1.46830	−0.0631854
$541$	−22.1914	−0.954082	−0.477041	−	0.878881i	$-0.658291\pi$
$541$	−22.1914	−0.954082	−0.477041	+	0.878881i	$0.658291\pi$
$542$	−13.6286	−0.585400
$543$	0.861886	0.0369871
$544$	−5.76549	−0.247193
$545$	−3.42125	−0.146550
$546$	0.700760	0.0299898
$547$	−17.1879	−0.734903	−0.367452	−	0.930043i	$-0.619770\pi$
$547$	−17.1879	−0.734903	−0.367452	+	0.930043i	$0.619770\pi$
$548$	−9.95341	−0.425189
$549$	−27.0416	−1.15411
$550$	0	0
$551$	5.19323	0.221239
$552$	−1.77374	−0.0754954
$553$	6.14508	0.261315
$554$	−23.5465	−1.00040
$555$	0.728730	0.0309329
$556$	0.627224	0.0266002
$557$	6.02990	0.255495	0.127747	−	0.991807i	$-0.459225\pi$
$557$	6.02990	0.255495	0.127747	+	0.991807i	$0.459225\pi$
$558$	−1.04752	−0.0443449
$559$	27.3240	1.15568
$560$	1.00000	0.0422577
$561$	0	0
$562$	−19.1837	−0.809215
$563$	−27.1986	−1.14628	−0.573142	−	0.819456i	$-0.694276\pi$
$563$	−27.1986	−1.14628	−0.573142	+	0.819456i	$0.694276\pi$
$564$	−0.775365	−0.0326488
$565$	16.8223	0.707718
$566$	17.2722	0.726005
$567$	−8.45361	−0.355018
$568$	12.5139	0.525070
$569$	−31.8782	−1.33641	−0.668203	−	0.743979i	$-0.732936\pi$
$569$	−31.8782	−1.33641	−0.668203	+	0.743979i	$0.732936\pi$
$570$	0.907206	0.0379987
$571$	26.4539	1.10706	0.553531	−	0.832829i	$-0.313280\pi$
$571$	26.4539	1.10706	0.553531	+	0.832829i	$0.313280\pi$
$572$	0	0
$573$	−6.42075	−0.268231
$574$	−2.22578	−0.0929022
$575$	−7.17432	−0.299190
$576$	−2.93888	−0.122453
$577$	−30.0526	−1.25111	−0.625554	−	0.780181i	$-0.715127\pi$
$577$	−30.0526	−1.25111	−0.625554	+	0.780181i	$0.715127\pi$
$578$	−16.2409	−0.675531
$579$	−0.167502	−0.00696115
$580$	1.41528	0.0587661
$581$	3.08217	0.127870
$582$	3.09730	0.128387
$583$	0	0
$584$	−7.19836	−0.297870
$585$	−8.32993	−0.344400
$586$	−32.9318	−1.36040
$587$	26.7574	1.10440	0.552198	−	0.833713i	$-0.313789\pi$
$587$	26.7574	1.10440	0.552198	+	0.833713i	$0.313789\pi$
$588$	−0.247235	−0.0101958
$589$	1.30791	0.0538913
$590$	−10.4962	−0.432121
$591$	1.43597	0.0590679
$592$	2.94752	0.121142
$593$	3.33504	0.136954	0.0684768	−	0.997653i	$-0.478186\pi$
$593$	3.33504	0.136954	0.0684768	+	0.997653i	$0.478186\pi$
$594$	0	0
$595$	5.76549	0.236362
$596$	15.0315	0.615713
$597$	−2.70660	−0.110774
$598$	−20.3348	−0.831553
$599$	−3.33155	−0.136123	−0.0680617	−	0.997681i	$-0.521681\pi$
$599$	−3.33155	−0.136123	−0.0680617	+	0.997681i	$0.521681\pi$
$600$	0.247235	0.0100933
$601$	41.5433	1.69459	0.847293	−	0.531126i	$-0.178231\pi$
$601$	41.5433	1.69459	0.847293	+	0.531126i	$0.178231\pi$
$602$	−9.64018	−0.392904
$603$	39.8118	1.62126
$604$	−0.516467	−0.0210148
$605$	0	0
$606$	1.45899	0.0592674
$607$	−20.9676	−0.851049	−0.425525	−	0.904947i	$-0.639910\pi$
$607$	−20.9676	−0.851049	−0.425525	+	0.904947i	$0.639910\pi$
$608$	3.66941	0.148814
$609$	−0.349905	−0.0141789
$610$	9.20135	0.372552
$611$	−8.88908	−0.359614
$612$	−16.9441	−0.684923
$613$	2.77191	0.111956	0.0559782	−	0.998432i	$-0.482172\pi$
$613$	2.77191	0.111956	0.0559782	+	0.998432i	$0.482172\pi$
$614$	−7.97195	−0.321722
$615$	−0.550290	−0.0221898
$616$	0	0
$617$	48.3567	1.94677	0.973384	−	0.229180i	$-0.0736043\pi$
$617$	48.3567	1.94677	0.973384	+	0.229180i	$0.0736043\pi$
$618$	−2.95868	−0.119016
$619$	46.8795	1.88425	0.942123	−	0.335268i	$-0.108827\pi$
$619$	46.8795	1.88425	0.942123	+	0.335268i	$0.108827\pi$
$620$	0.356435	0.0143148
$621$	−10.5340	−0.422716
$622$	4.00836	0.160721
$623$	5.75443	0.230546
$624$	0.700760	0.0280529
$625$	1.00000	0.0400000
$626$	−5.27260	−0.210736
$627$	0	0
$628$	−22.7782	−0.908949
$629$	16.9939	0.677592
$630$	2.93888	0.117088
$631$	−21.5815	−0.859145	−0.429573	−	0.903032i	$-0.641336\pi$
$631$	−21.5815	−0.859145	−0.429573	+	0.903032i	$0.641336\pi$
$632$	6.14508	0.244438
$633$	−1.58919	−0.0631647
$634$	4.65978	0.185064
$635$	0.175760	0.00697481
$636$	−0.216119	−0.00856969
$637$	−2.83439	−0.112303
$638$	0	0
$639$	36.7766	1.45486
$640$	1.00000	0.0395285
$641$	22.9849	0.907850	0.453925	−	0.891040i	$-0.350023\pi$
$641$	22.9849	0.907850	0.453925	+	0.891040i	$0.350023\pi$
$642$	1.55485	0.0613649
$643$	−32.9082	−1.29777	−0.648886	−	0.760886i	$-0.724765\pi$
$643$	−32.9082	−1.29777	−0.648886	+	0.760886i	$0.724765\pi$
$644$	7.17432	0.282708
$645$	−2.38339	−0.0938457
$646$	21.1560	0.832370
$647$	19.9201	0.783140	0.391570	−	0.920148i	$-0.371932\pi$
$647$	19.9201	0.783140	0.391570	+	0.920148i	$0.371932\pi$
$648$	−8.45361	−0.332089
$649$	0	0
$650$	2.83439	0.111174
$651$	−0.0881230	−0.00345381
$652$	3.72353	0.145825
$653$	34.0859	1.33389	0.666943	−	0.745109i	$-0.267602\pi$
$653$	34.0859	1.33389	0.666943	+	0.745109i	$0.267602\pi$
$654$	0.845851	0.0330754
$655$	−11.4391	−0.446962
$656$	−2.22578	−0.0869021
$657$	−21.1551	−0.825338
$658$	3.13615	0.122260
$659$	−7.54952	−0.294088	−0.147044	−	0.989130i	$-0.546976\pi$
$659$	−7.54952	−0.294088	−0.147044	+	0.989130i	$0.546976\pi$
$660$	0	0
$661$	41.0094	1.59508	0.797541	−	0.603265i	$-0.206134\pi$
$661$	41.0094	1.59508	0.797541	+	0.603265i	$0.206134\pi$
$662$	25.0199	0.972424
$663$	4.04022	0.156909
$664$	3.08217	0.119611
$665$	−3.66941	−0.142294
$666$	8.66240	0.335661
$667$	10.1536	0.393150
$668$	−15.5825	−0.602906
$669$	−5.73703	−0.221807
$670$	−13.5466	−0.523352
$671$	0	0
$672$	−0.247235	−0.00953728
$673$	−48.3741	−1.86469	−0.932343	−	0.361576i	$-0.882239\pi$
$673$	−48.3741	−1.86469	−0.932343	+	0.361576i	$0.882239\pi$
$674$	15.9367	0.613860
$675$	1.46830	0.0565147
$676$	−4.96622	−0.191008
$677$	−5.38363	−0.206910	−0.103455	−	0.994634i	$-0.532990\pi$
$677$	−5.38363	−0.206910	−0.103455	+	0.994634i	$0.532990\pi$
$678$	−4.15905	−0.159727
$679$	−12.5278	−0.480771
$680$	5.76549	0.221096
$681$	−6.32217	−0.242266
$682$	0	0
$683$	12.5734	0.481106	0.240553	−	0.970636i	$-0.422671\pi$
$683$	12.5734	0.481106	0.240553	+	0.970636i	$0.422671\pi$
$684$	10.7839	0.412334
$685$	9.95341	0.380300
$686$	1.00000	0.0381802
$687$	−0.995121	−0.0379662
$688$	−9.64018	−0.367528
$689$	−2.47767	−0.0943919
$690$	1.77374	0.0675251
$691$	−26.0427	−0.990712	−0.495356	−	0.868690i	$-0.664962\pi$
$691$	−26.0427	−0.990712	−0.495356	+	0.868690i	$0.664962\pi$
$692$	−5.89952	−0.224266
$693$	0	0
$694$	−12.4275	−0.471741
$695$	−0.627224	−0.0237920
$696$	−0.349905	−0.0132631
$697$	−12.8327	−0.486073
$698$	−16.2129	−0.613666
$699$	−0.909134	−0.0343866
$700$	−1.00000	−0.0377964
$701$	46.1549	1.74325	0.871624	−	0.490176i	$-0.163067\pi$
$701$	46.1549	1.74325	0.871624	+	0.490176i	$0.163067\pi$
$702$	4.16173	0.157074
$703$	−10.8157	−0.407921
$704$	0	0
$705$	0.775365	0.0292019
$706$	21.8320	0.821656
$707$	−5.90123	−0.221939
$708$	2.59502	0.0975267
$709$	44.5132	1.67173	0.835865	−	0.548935i	$-0.184967\pi$
$709$	44.5132	1.67173	0.835865	+	0.548935i	$0.184967\pi$
$710$	−12.5139	−0.469637
$711$	18.0596	0.677289
$712$	5.75443	0.215656
$713$	2.55718	0.0957670
$714$	−1.42543	−0.0533453
$715$	0	0
$716$	16.3323	0.610367
$717$	−0.270582	−0.0101051
$718$	−18.2106	−0.679612
$719$	24.2008	0.902536	0.451268	−	0.892389i	$-0.350972\pi$
$719$	24.2008	0.902536	0.451268	+	0.892389i	$0.350972\pi$
$720$	2.93888	0.109525
$721$	11.9671	0.445678
$722$	5.53541	0.206007
$723$	1.54498	0.0574582
$724$	−3.48610	−0.129560
$725$	−1.41528	−0.0525620
$726$	0	0
$727$	−5.25083	−0.194742	−0.0973712	−	0.995248i	$-0.531043\pi$
$727$	−5.25083	−0.194742	−0.0973712	+	0.995248i	$0.531043\pi$
$728$	−2.83439	−0.105050
$729$	−23.7551	−0.879817
$730$	7.19836	0.266423
$731$	−55.5803	−2.05571
$732$	−2.27489	−0.0840825
$733$	47.6047	1.75832	0.879160	−	0.476526i	$-0.158104\pi$
$733$	47.6047	1.75832	0.879160	+	0.476526i	$0.158104\pi$
$734$	5.27857	0.194836
$735$	0.247235	0.00911939
$736$	7.17432	0.264449
$737$	0	0
$738$	−6.54129	−0.240788
$739$	−27.2113	−1.00098	−0.500491	−	0.865742i	$-0.666847\pi$
$739$	−27.2113	−1.00098	−0.500491	+	0.865742i	$0.666847\pi$
$740$	−2.94752	−0.108353
$741$	−2.57138	−0.0944619
$742$	0.874146	0.0320909
$743$	5.29833	0.194377	0.0971885	−	0.995266i	$-0.469015\pi$
$743$	5.29833	0.194377	0.0971885	+	0.995266i	$0.469015\pi$
$744$	−0.0881230	−0.00323075
$745$	−15.0315	−0.550710
$746$	18.5799	0.680259
$747$	9.05811	0.331419
$748$	0	0
$749$	−6.28895	−0.229793
$750$	−0.247235	−0.00902773
$751$	−20.0625	−0.732089	−0.366045	−	0.930597i	$-0.619288\pi$
$751$	−20.0625	−0.732089	−0.366045	+	0.930597i	$0.619288\pi$
$752$	3.13615	0.114364
$753$	4.76954	0.173812
$754$	−4.01145	−0.146088
$755$	0.516467	0.0187962
$756$	−1.46830	−0.0534014
$757$	−4.36865	−0.158781	−0.0793907	−	0.996844i	$-0.525297\pi$
$757$	−4.36865	−0.158781	−0.0793907	+	0.996844i	$0.525297\pi$
$758$	−22.8406	−0.829607
$759$	0	0
$760$	−3.66941	−0.133104
$761$	−2.16435	−0.0784576	−0.0392288	−	0.999230i	$-0.512490\pi$
$761$	−2.16435	−0.0784576	−0.0392288	+	0.999230i	$0.512490\pi$
$762$	−0.0434539	−0.00157417
$763$	−3.42125	−0.123857
$764$	25.9703	0.939572
$765$	16.9441	0.612613
$766$	22.4735	0.812000
$767$	29.7503	1.07422
$768$	−0.247235	−0.00892131
$769$	32.7883	1.18237	0.591187	−	0.806534i	$-0.298659\pi$
$769$	32.7883	1.18237	0.591187	+	0.806534i	$0.298659\pi$
$770$	0	0
$771$	−6.24671	−0.224970
$772$	0.677502	0.0243838
$773$	−7.29845	−0.262507	−0.131253	−	0.991349i	$-0.541900\pi$
$773$	−7.29845	−0.262507	−0.131253	+	0.991349i	$0.541900\pi$
$774$	−28.3313	−1.01835
$775$	−0.356435	−0.0128035
$776$	−12.5278	−0.449720
$777$	0.728730	0.0261430
$778$	−36.2498	−1.29962
$779$	8.16730	0.292624
$780$	−0.700760	−0.0250912
$781$	0	0
$782$	41.3634	1.47915
$783$	−2.07804	−0.0742632
$784$	1.00000	0.0357143
$785$	22.7782	0.812989
$786$	2.82814	0.100876
$787$	0.836360	0.0298130	0.0149065	−	0.999889i	$-0.495255\pi$
$787$	0.836360	0.0298130	0.0149065	+	0.999889i	$0.495255\pi$
$788$	−5.80812	−0.206906
$789$	0.740103	0.0263484
$790$	−6.14508	−0.218632
$791$	16.8223	0.598131
$792$	0	0
$793$	−26.0802	−0.926137
$794$	8.02992	0.284971
$795$	0.216119	0.00766496
$796$	10.9475	0.388023
$797$	−2.03251	−0.0719952	−0.0359976	−	0.999352i	$-0.511461\pi$
$797$	−2.03251	−0.0719952	−0.0359976	+	0.999352i	$0.511461\pi$
$798$	0.907206	0.0321147
$799$	18.0814	0.639676
$800$	−1.00000	−0.0353553
$801$	16.9115	0.597540
$802$	−0.672421	−0.0237440
$803$	0	0
$804$	3.34919	0.118117
$805$	−7.17432	−0.252861
$806$	−1.01028	−0.0355855
$807$	5.56310	0.195830
$808$	−5.90123	−0.207605
$809$	42.1998	1.48367	0.741833	−	0.670585i	$-0.233957\pi$
$809$	42.1998	1.48367	0.741833	+	0.670585i	$0.233957\pi$
$810$	8.45361	0.297030
$811$	3.03390	0.106535	0.0532673	−	0.998580i	$-0.483036\pi$
$811$	3.03390	0.106535	0.0532673	+	0.998580i	$0.483036\pi$
$812$	1.41528	0.0496665
$813$	−3.36947	−0.118173
$814$	0	0
$815$	−3.72353	−0.130430
$816$	−1.42543	−0.0499000
$817$	35.3738	1.23757
$818$	−21.7385	−0.760069
$819$	−8.32993	−0.291071
$820$	2.22578	0.0777276
$821$	−34.5900	−1.20720	−0.603599	−	0.797288i	$-0.706267\pi$
$821$	−34.5900	−1.20720	−0.603599	+	0.797288i	$0.706267\pi$
$822$	−2.46083	−0.0858312
$823$	50.7373	1.76859	0.884295	−	0.466928i	$-0.154639\pi$
$823$	50.7373	1.76859	0.884295	+	0.466928i	$0.154639\pi$
$824$	11.9671	0.416894
$825$	0	0
$826$	−10.4962	−0.365209
$827$	12.9190	0.449239	0.224619	−	0.974447i	$-0.427886\pi$
$827$	12.9190	0.449239	0.224619	+	0.974447i	$0.427886\pi$
$828$	21.0844	0.732734
$829$	12.8251	0.445434	0.222717	−	0.974883i	$-0.428507\pi$
$829$	12.8251	0.445434	0.222717	+	0.974883i	$0.428507\pi$
$830$	−3.08217	−0.106984
$831$	−5.82151	−0.201946
$832$	−2.83439	−0.0982649
$833$	5.76549	0.199762
$834$	0.155072	0.00536969
$835$	15.5825	0.539255
$836$	0	0
$837$	−0.523352	−0.0180897
$838$	7.38003	0.254939
$839$	−8.02823	−0.277165	−0.138583	−	0.990351i	$-0.544255\pi$
$839$	−8.02823	−0.277165	−0.138583	+	0.990351i	$0.544255\pi$
$840$	0.247235	0.00853041
$841$	−26.9970	−0.930931
$842$	−29.4497	−1.01490
$843$	−4.74287	−0.163353
$844$	6.42786	0.221256
$845$	4.96622	0.170843
$846$	9.21676	0.316879
$847$	0	0
$848$	0.874146	0.0300183
$849$	4.27029	0.146556
$850$	−5.76549	−0.197755
$851$	−21.1465	−0.724891
$852$	3.09386	0.105994
$853$	30.1638	1.03279	0.516395	−	0.856351i	$-0.327274\pi$
$853$	30.1638	1.03279	0.516395	+	0.856351i	$0.327274\pi$
$854$	9.20135	0.314864
$855$	−10.7839	−0.368803
$856$	−6.28895	−0.214952
$857$	−32.8279	−1.12138	−0.560689	−	0.828026i	$-0.689464\pi$
$857$	−32.8279	−1.12138	−0.560689	+	0.828026i	$0.689464\pi$
$858$	0	0
$859$	40.7882	1.39168	0.695838	−	0.718199i	$-0.255033\pi$
$859$	40.7882	1.39168	0.695838	+	0.718199i	$0.255033\pi$
$860$	9.64018	0.328727
$861$	−0.550290	−0.0187538
$862$	−19.6701	−0.669968
$863$	42.0371	1.43096	0.715479	−	0.698634i	$-0.246208\pi$
$863$	42.0371	1.43096	0.715479	+	0.698634i	$0.246208\pi$
$864$	−1.46830	−0.0499524
$865$	5.89952	0.200590
$866$	17.0504	0.579397
$867$	−4.01530	−0.136367
$868$	0.356435	0.0120982
$869$	0	0
$870$	0.349905	0.0118629
$871$	38.3964	1.30101
$872$	−3.42125	−0.115858
$873$	−36.8175	−1.24608
$874$	−26.3255	−0.890474
$875$	1.00000	0.0338062
$876$	−1.77968	−0.0601299
$877$	2.16379	0.0730659	0.0365330	−	0.999332i	$-0.488369\pi$
$877$	2.16379	0.0730659	0.0365330	+	0.999332i	$0.488369\pi$
$878$	29.9589	1.01106
$879$	−8.14187	−0.274619
$880$	0	0
$881$	9.74157	0.328202	0.164101	−	0.986444i	$-0.447528\pi$
$881$	9.74157	0.328202	0.164101	+	0.986444i	$0.447528\pi$
$882$	2.93888	0.0989571
$883$	25.2808	0.850768	0.425384	−	0.905013i	$-0.360139\pi$
$883$	25.2808	0.850768	0.425384	+	0.905013i	$0.360139\pi$
$884$	−16.3417	−0.549629
$885$	−2.59502	−0.0872306
$886$	−41.2954	−1.38735
$887$	45.8358	1.53901	0.769507	−	0.638638i	$-0.220502\pi$
$887$	45.8358	1.53901	0.769507	+	0.638638i	$0.220502\pi$
$888$	0.728730	0.0244546
$889$	0.175760	0.00589479
$890$	−5.75443	−0.192889
$891$	0	0
$892$	23.2048	0.776955
$893$	−11.5078	−0.385095
$894$	3.71630	0.124292
$895$	−16.3323	−0.545929
$896$	1.00000	0.0334077
$897$	−5.02747	−0.167862
$898$	−33.9263	−1.13214
$899$	0.504454	0.0168245
$900$	−2.93888	−0.0979625
$901$	5.03988	0.167903
$902$	0	0
$903$	−2.38339	−0.0793141
$904$	16.8223	0.559500
$905$	3.48610	0.115882
$906$	−0.127689	−0.00424217
$907$	−15.1299	−0.502381	−0.251190	−	0.967938i	$-0.580822\pi$
$907$	−15.1299	−0.502381	−0.251190	+	0.967938i	$0.580822\pi$
$908$	25.5715	0.848621
$909$	−17.3430	−0.575231
$910$	2.83439	0.0939592
$911$	14.7067	0.487255	0.243627	−	0.969869i	$-0.421663\pi$
$911$	14.7067	0.487255	0.243627	+	0.969869i	$0.421663\pi$
$912$	0.907206	0.0300406
$913$	0	0
$914$	19.9928	0.661304
$915$	2.27489	0.0752056
$916$	4.02501	0.132990
$917$	−11.4391	−0.377752
$918$	−8.46544	−0.279401
$919$	−1.78904	−0.0590151	−0.0295075	−	0.999565i	$-0.509394\pi$
$919$	−1.78904	−0.0590151	−0.0295075	+	0.999565i	$0.509394\pi$
$920$	−7.17432	−0.236530
$921$	−1.97094	−0.0649448
$922$	−17.4739	−0.575472
$923$	35.4692	1.16748
$924$	0	0
$925$	2.94752	0.0969140
$926$	41.7730	1.37275
$927$	35.1698	1.15513
$928$	1.41528	0.0464587
$929$	50.8053	1.66687	0.833433	−	0.552620i	$-0.186372\pi$
$929$	50.8053	1.66687	0.833433	+	0.552620i	$0.186372\pi$
$930$	0.0881230	0.00288967
$931$	−3.66941	−0.120260
$932$	3.67721	0.120451
$933$	0.991006	0.0324441
$934$	13.7476	0.449836
$935$	0	0
$936$	−8.32993	−0.272272
$937$	−56.0500	−1.83107	−0.915537	−	0.402233i	$-0.868234\pi$
$937$	−56.0500	−1.83107	−0.915537	+	0.402233i	$0.868234\pi$
$938$	−13.5466	−0.442313
$939$	−1.30357	−0.0425404
$940$	−3.13615	−0.102290
$941$	−31.8530	−1.03838	−0.519188	−	0.854660i	$-0.673766\pi$
$941$	−31.8530	−1.03838	−0.519188	+	0.854660i	$0.673766\pi$
$942$	−5.63156	−0.183486
$943$	15.9684	0.520004
$944$	−10.4962	−0.341621
$945$	1.46830	0.0477637
$946$	0	0
$947$	16.7087	0.542959	0.271479	−	0.962444i	$-0.412487\pi$
$947$	16.7087	0.542959	0.271479	+	0.962444i	$0.412487\pi$
$948$	1.51928	0.0493438
$949$	−20.4030	−0.662308
$950$	3.66941	0.119051
$951$	1.15206	0.0373581
$952$	5.76549	0.186861
$953$	−2.96987	−0.0962036	−0.0481018	−	0.998842i	$-0.515317\pi$
$953$	−2.96987	−0.0962036	−0.0481018	+	0.998842i	$0.515317\pi$
$954$	2.56901	0.0831747
$955$	−25.9703	−0.840379
$956$	1.09444	0.0353966
$957$	0	0
$958$	−20.6636	−0.667611
$959$	9.95341	0.321412
$960$	0.247235	0.00797946
$961$	−30.8730	−0.995902
$962$	8.35444	0.269358
$963$	−18.4824	−0.595588
$964$	−6.24902	−0.201267
$965$	−0.677502	−0.0218096
$966$	1.77374	0.0570691
$967$	20.4576	0.657873	0.328936	−	0.944352i	$-0.393310\pi$
$967$	20.4576	0.657873	0.328936	+	0.944352i	$0.393310\pi$
$968$	0	0
$969$	5.23049	0.168027
$970$	12.5278	0.402242
$971$	−60.0213	−1.92618	−0.963088	−	0.269187i	$-0.913245\pi$
$971$	−60.0213	−1.92618	−0.963088	+	0.269187i	$0.913245\pi$
$972$	−6.49491	−0.208324
$973$	−0.627224	−0.0201079
$974$	22.1049	0.708287
$975$	0.700760	0.0224423
$976$	9.20135	0.294528
$977$	16.7790	0.536808	0.268404	−	0.963306i	$-0.413504\pi$
$977$	16.7790	0.536808	0.268404	+	0.963306i	$0.413504\pi$
$978$	0.920586	0.0294371
$979$	0	0
$980$	−1.00000	−0.0319438
$981$	−10.0546	−0.321019
$982$	10.6974	0.341369
$983$	48.6058	1.55028	0.775141	−	0.631788i	$-0.217679\pi$
$983$	48.6058	1.55028	0.775141	+	0.631788i	$0.217679\pi$
$984$	−0.550290	−0.0175426
$985$	5.80812	0.185062
$986$	8.15976	0.259860
$987$	0.775365	0.0246802
$988$	10.4006	0.330886
$989$	69.1617	2.19921
$990$	0	0
$991$	55.7217	1.77006	0.885028	−	0.465537i	$-0.154139\pi$
$991$	55.7217	1.77006	0.885028	+	0.465537i	$0.154139\pi$
$992$	0.356435	0.0113168
$993$	6.18578	0.196300
$994$	−12.5139	−0.396915
$995$	−10.9475	−0.347058
$996$	0.762019	0.0241455
$997$	−38.8069	−1.22903	−0.614513	−	0.788907i	$-0.710647\pi$
$997$	−38.8069	−1.22903	−0.614513	+	0.788907i	$0.710647\pi$
$998$	−1.85809	−0.0588167
$999$	4.32784	0.136927

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8470.2.a.cu.1.4		6
11.7	odd	10		770.2.n.h.71.2	✓	12
11.8	odd	10		770.2.n.h.141.2	yes	12
11.10	odd	2		8470.2.a.da.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.n.h.71.2	✓	12	11.7	odd	10
770.2.n.h.141.2	yes	12	11.8	odd	10
8470.2.a.cu.1.4		6	1.1	even	1	trivial
8470.2.a.da.1.4		6	11.10	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -0.247235 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.247235 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.93888 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -0.247235 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.247235 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.93888 q^{9} +1.00000 q^{10} -0.247235 q^{12} -2.83439 q^{13} +1.00000 q^{14} +0.247235 q^{15} +1.00000 q^{16} +5.76549 q^{17} +2.93888 q^{18} -3.66941 q^{19} -1.00000 q^{20} +0.247235 q^{21} -7.17432 q^{23} +0.247235 q^{24} +1.00000 q^{25} +2.83439 q^{26} +1.46830 q^{27} -1.00000 q^{28} -1.41528 q^{29} -0.247235 q^{30} -0.356435 q^{31} -1.00000 q^{32} -5.76549 q^{34} +1.00000 q^{35} -2.93888 q^{36} +2.94752 q^{37} +3.66941 q^{38} +0.700760 q^{39} +1.00000 q^{40} -2.22578 q^{41} -0.247235 q^{42} -9.64018 q^{43} +2.93888 q^{45} +7.17432 q^{46} +3.13615 q^{47} -0.247235 q^{48} +1.00000 q^{49} -1.00000 q^{50} -1.42543 q^{51} -2.83439 q^{52} +0.874146 q^{53} -1.46830 q^{54} +1.00000 q^{56} +0.907206 q^{57} +1.41528 q^{58} -10.4962 q^{59} +0.247235 q^{60} +9.20135 q^{61} +0.356435 q^{62} +2.93888 q^{63} +1.00000 q^{64} +2.83439 q^{65} -13.5466 q^{67} +5.76549 q^{68} +1.77374 q^{69} -1.00000 q^{70} -12.5139 q^{71} +2.93888 q^{72} +7.19836 q^{73} -2.94752 q^{74} -0.247235 q^{75} -3.66941 q^{76} -0.700760 q^{78} -6.14508 q^{79} -1.00000 q^{80} +8.45361 q^{81} +2.22578 q^{82} -3.08217 q^{83} +0.247235 q^{84} -5.76549 q^{85} +9.64018 q^{86} +0.349905 q^{87} -5.75443 q^{89} -2.93888 q^{90} +2.83439 q^{91} -7.17432 q^{92} +0.0881230 q^{93} -3.13615 q^{94} +3.66941 q^{95} +0.247235 q^{96} +12.5278 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{2} - 5 q^{3} + 6 q^{4} - 6 q^{5} + 5 q^{6} - 6 q^{7} - 6 q^{8} + 11 q^{9}+O(q^{10}) \) 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

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Database

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