LMFDB - Embedded newform 8470.2.a.cr.1.3

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:	$4$
Coefficient field:	$\Q(\zeta_{24})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 1 $ x^4 - 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.517638$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +0.414214 q^{3} +1.00000 q^{4} +1.00000 q^{5} +0.414214 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.82843 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +0.414214 q^{3} +1.00000 q^{4} +1.00000 q^{5} +0.414214 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.82843 q^{9} +1.00000 q^{10} +0.414214 q^{12} +0.696775 q^{13} -1.00000 q^{14} +0.414214 q^{15} +1.00000 q^{16} -0.449490 q^{17} -2.82843 q^{18} -3.37894 q^{19} +1.00000 q^{20} -0.414214 q^{21} -9.00997 q^{23} +0.414214 q^{24} +1.00000 q^{25} +0.696775 q^{26} -2.41421 q^{27} -1.00000 q^{28} -1.60040 q^{29} +0.414214 q^{30} +10.1769 q^{31} +1.00000 q^{32} -0.449490 q^{34} -1.00000 q^{35} -2.82843 q^{36} +3.32780 q^{37} -3.37894 q^{38} +0.288614 q^{39} +1.00000 q^{40} +6.12044 q^{41} -0.414214 q^{42} +7.14162 q^{43} -2.82843 q^{45} -9.00997 q^{46} -9.51399 q^{47} +0.414214 q^{48} +1.00000 q^{49} +1.00000 q^{50} -0.186185 q^{51} +0.696775 q^{52} -2.84909 q^{53} -2.41421 q^{54} -1.00000 q^{56} -1.39960 q^{57} -1.60040 q^{58} -7.97469 q^{59} +0.414214 q^{60} -8.29253 q^{61} +10.1769 q^{62} +2.82843 q^{63} +1.00000 q^{64} +0.696775 q^{65} +11.1708 q^{67} -0.449490 q^{68} -3.73205 q^{69} -1.00000 q^{70} -11.9202 q^{71} -2.82843 q^{72} +0.585786 q^{73} +3.32780 q^{74} +0.414214 q^{75} -3.37894 q^{76} +0.288614 q^{78} -10.5811 q^{79} +1.00000 q^{80} +7.48528 q^{81} +6.12044 q^{82} -17.5950 q^{83} -0.414214 q^{84} -0.449490 q^{85} +7.14162 q^{86} -0.662907 q^{87} -11.9700 q^{89} -2.82843 q^{90} -0.696775 q^{91} -9.00997 q^{92} +4.21541 q^{93} -9.51399 q^{94} -3.37894 q^{95} +0.414214 q^{96} -8.56993 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{5} - 4 q^{6} - 4 q^{7} + 4 q^{8}+O(q^{10}) $ 4 * q + 4 * q^2 - 4 * q^3 + 4 * q^4 + 4 * q^5 - 4 * q^6 - 4 * q^7 + 4 * q^8	$ 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{5} - 4 q^{6} - 4 q^{7} + 4 q^{8} + 4 q^{10} - 4 q^{12} - 4 q^{14} - 4 q^{15} + 4 q^{16} + 8 q^{17} - 12 q^{19} + 4 q^{20} + 4 q^{21} - 8 q^{23} - 4 q^{24} + 4 q^{25} - 4 q^{27} - 4 q^{28} - 8 q^{29} - 4 q^{30} + 4 q^{32} + 8 q^{34} - 4 q^{35} - 16 q^{37} - 12 q^{38} + 8 q^{39} + 4 q^{40} + 8 q^{41} + 4 q^{42} - 8 q^{43} - 8 q^{46} - 16 q^{47} - 4 q^{48} + 4 q^{49} + 4 q^{50} - 8 q^{51} - 4 q^{54} - 4 q^{56} - 4 q^{57} - 8 q^{58} - 8 q^{59} - 4 q^{60} - 8 q^{61} + 4 q^{64} + 8 q^{68} - 8 q^{69} - 4 q^{70} - 8 q^{71} + 8 q^{73} - 16 q^{74} - 4 q^{75} - 12 q^{76} + 8 q^{78} - 24 q^{79} + 4 q^{80} - 4 q^{81} + 8 q^{82} + 8 q^{83} + 4 q^{84} + 8 q^{85} - 8 q^{86} + 16 q^{87} - 8 q^{92} + 16 q^{93} - 16 q^{94} - 12 q^{95} - 4 q^{96} - 8 q^{97} + 4 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 - 4 * q^3 + 4 * q^4 + 4 * q^5 - 4 * q^6 - 4 * q^7 + 4 * q^8 + 4 * q^10 - 4 * q^12 - 4 * q^14 - 4 * q^15 + 4 * q^16 + 8 * q^17 - 12 * q^19 + 4 * q^20 + 4 * q^21 - 8 * q^23 - 4 * q^24 + 4 * q^25 - 4 * q^27 - 4 * q^28 - 8 * q^29 - 4 * q^30 + 4 * q^32 + 8 * q^34 - 4 * q^35 - 16 * q^37 - 12 * q^38 + 8 * q^39 + 4 * q^40 + 8 * q^41 + 4 * q^42 - 8 * q^43 - 8 * q^46 - 16 * q^47 - 4 * q^48 + 4 * q^49 + 4 * q^50 - 8 * q^51 - 4 * q^54 - 4 * q^56 - 4 * q^57 - 8 * q^58 - 8 * q^59 - 4 * q^60 - 8 * q^61 + 4 * q^64 + 8 * q^68 - 8 * q^69 - 4 * q^70 - 8 * q^71 + 8 * q^73 - 16 * q^74 - 4 * q^75 - 12 * q^76 + 8 * q^78 - 24 * q^79 + 4 * q^80 - 4 * q^81 + 8 * q^82 + 8 * q^83 + 4 * q^84 + 8 * q^85 - 8 * q^86 + 16 * q^87 - 8 * q^92 + 16 * q^93 - 16 * q^94 - 12 * q^95 - 4 * q^96 - 8 * q^97 + 4 * 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
$3$	0.414214	0.239146	0.119573	−	0.992825i	$-0.461847\pi$
$3$	0.414214	0.239146	0.119573	+	0.992825i	$0.461847\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0.414214	0.169102
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	−2.82843	−0.942809
$10$	1.00000	0.316228
$11$	0	0
$11$	0	0
$12$	0.414214	0.119573
$13$	0.696775	0.193251	0.0966253	−	0.995321i	$-0.469195\pi$
$13$	0.696775	0.193251	0.0966253	+	0.995321i	$0.469195\pi$
$14$	−1.00000	−0.267261
$15$	0.414214	0.106949
$16$	1.00000	0.250000
$17$	−0.449490	−0.109017	−0.0545086	−	0.998513i	$-0.517359\pi$
$17$	−0.449490	−0.109017	−0.0545086	+	0.998513i	$0.517359\pi$
$18$	−2.82843	−0.666667
$19$	−3.37894	−0.775181	−0.387591	−	0.921832i	$-0.626693\pi$
$19$	−3.37894	−0.775181	−0.387591	+	0.921832i	$0.626693\pi$
$20$	1.00000	0.223607
$21$	−0.414214	−0.0903888
$22$	0	0
$23$	−9.00997	−1.87871	−0.939354	−	0.342949i	$-0.888574\pi$
$23$	−9.00997	−1.87871	−0.939354	+	0.342949i	$0.888574\pi$
$24$	0.414214	0.0845510
$25$	1.00000	0.200000
$26$	0.696775	0.136649
$27$	−2.41421	−0.464616
$28$	−1.00000	−0.188982
$29$	−1.60040	−0.297187	−0.148593	−	0.988898i	$-0.547475\pi$
$29$	−1.60040	−0.297187	−0.148593	+	0.988898i	$0.547475\pi$
$30$	0.414214	0.0756247
$31$	10.1769	1.82782	0.913912	−	0.405912i	$-0.133046\pi$
$31$	10.1769	1.82782	0.913912	+	0.405912i	$0.133046\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−0.449490	−0.0770869
$35$	−1.00000	−0.169031
$36$	−2.82843	−0.471405
$37$	3.32780	0.547088	0.273544	−	0.961860i	$-0.411804\pi$
$37$	3.32780	0.547088	0.273544	+	0.961860i	$0.411804\pi$
$38$	−3.37894	−0.548136
$39$	0.288614	0.0462151
$40$	1.00000	0.158114
$41$	6.12044	0.955852	0.477926	−	0.878400i	$-0.341389\pi$
$41$	6.12044	0.955852	0.477926	+	0.878400i	$0.341389\pi$
$42$	−0.414214	−0.0639145
$43$	7.14162	1.08909	0.544543	−	0.838733i	$-0.316703\pi$
$43$	7.14162	1.08909	0.544543	+	0.838733i	$0.316703\pi$
$44$	0	0
$45$	−2.82843	−0.421637
$46$	−9.00997	−1.32845
$47$	−9.51399	−1.38776	−0.693879	−	0.720092i	$-0.744100\pi$
$47$	−9.51399	−1.38776	−0.693879	+	0.720092i	$0.744100\pi$
$48$	0.414214	0.0597866
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	−0.186185	−0.0260711
$52$	0.696775	0.0966253
$53$	−2.84909	−0.391353	−0.195676	−	0.980669i	$-0.562690\pi$
$53$	−2.84909	−0.391353	−0.195676	+	0.980669i	$0.562690\pi$
$54$	−2.41421	−0.328533
$55$	0	0
$56$	−1.00000	−0.133631
$57$	−1.39960	−0.185382
$58$	−1.60040	−0.210143
$59$	−7.97469	−1.03822	−0.519108	−	0.854709i	$-0.673736\pi$
$59$	−7.97469	−1.03822	−0.519108	+	0.854709i	$0.673736\pi$
$60$	0.414214	0.0534747
$61$	−8.29253	−1.06175	−0.530875	−	0.847450i	$-0.678137\pi$
$61$	−8.29253	−1.06175	−0.530875	+	0.847450i	$0.678137\pi$
$62$	10.1769	1.29247
$63$	2.82843	0.356348
$64$	1.00000	0.125000
$65$	0.696775	0.0864243
$66$	0	0
$67$	11.1708	1.36474	0.682368	−	0.731009i	$-0.260950\pi$
$67$	11.1708	1.36474	0.682368	+	0.731009i	$0.260950\pi$
$68$	−0.449490	−0.0545086
$69$	−3.73205	−0.449286
$70$	−1.00000	−0.119523
$71$	−11.9202	−1.41466	−0.707331	−	0.706882i	$-0.750101\pi$
$71$	−11.9202	−1.41466	−0.707331	+	0.706882i	$0.750101\pi$
$72$	−2.82843	−0.333333
$73$	0.585786	0.0685611	0.0342806	−	0.999412i	$-0.489086\pi$
$73$	0.585786	0.0685611	0.0342806	+	0.999412i	$0.489086\pi$
$74$	3.32780	0.386849
$75$	0.414214	0.0478293
$76$	−3.37894	−0.387591
$77$	0	0
$78$	0.288614	0.0326790
$79$	−10.5811	−1.19047	−0.595236	−	0.803551i	$-0.702941\pi$
$79$	−10.5811	−1.19047	−0.595236	+	0.803551i	$0.702941\pi$
$80$	1.00000	0.111803
$81$	7.48528	0.831698
$82$	6.12044	0.675890
$83$	−17.5950	−1.93130	−0.965652	−	0.259837i	$-0.916331\pi$
$83$	−17.5950	−1.93130	−0.965652	+	0.259837i	$0.916331\pi$
$84$	−0.414214	−0.0451944
$85$	−0.449490	−0.0487540
$86$	7.14162	0.770101
$87$	−0.662907	−0.0710711
$88$	0	0
$89$	−11.9700	−1.26882	−0.634411	−	0.772996i	$-0.718757\pi$
$89$	−11.9700	−1.26882	−0.634411	+	0.772996i	$0.718757\pi$
$90$	−2.82843	−0.298142
$91$	−0.696775	−0.0730418
$92$	−9.00997	−0.939354
$93$	4.21541	0.437117
$94$	−9.51399	−0.981293
$95$	−3.37894	−0.346672
$96$	0.414214	0.0422755
$97$	−8.56993	−0.870145	−0.435072	−	0.900396i	$-0.643277\pi$
$97$	−8.56993	−0.870145	−0.435072	+	0.900396i	$0.643277\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	−0.464102	−0.0461798	−0.0230899	−	0.999733i	$-0.507350\pi$
$101$	−0.464102	−0.0461798	−0.0230899	+	0.999733i	$0.507350\pi$
$102$	−0.186185	−0.0184350
$103$	−7.55051	−0.743974	−0.371987	−	0.928238i	$-0.621323\pi$
$103$	−7.55051	−0.743974	−0.371987	+	0.928238i	$0.621323\pi$
$104$	0.696775	0.0683244
$105$	−0.414214	−0.0404231
$106$	−2.84909	−0.276728
$107$	−14.4901	−1.40081	−0.700405	−	0.713746i	$-0.746997\pi$
$107$	−14.4901	−1.40081	−0.700405	+	0.713746i	$0.746997\pi$
$108$	−2.41421	−0.232308
$109$	5.91359	0.566419	0.283210	−	0.959058i	$-0.408601\pi$
$109$	5.91359	0.566419	0.283210	+	0.959058i	$0.408601\pi$
$110$	0	0
$111$	1.37842	0.130834
$112$	−1.00000	−0.0944911
$113$	−19.0599	−1.79300	−0.896500	−	0.443043i	$-0.853899\pi$
$113$	−19.0599	−1.79300	−0.896500	+	0.443043i	$0.853899\pi$
$114$	−1.39960	−0.131085
$115$	−9.00997	−0.840184
$116$	−1.60040	−0.148593
$117$	−1.97078	−0.182198
$118$	−7.97469	−0.734130
$119$	0.449490	0.0412047
$120$	0.414214	0.0378124
$121$	0	0
$122$	−8.29253	−0.750770
$123$	2.53517	0.228589
$124$	10.1769	0.913912
$125$	1.00000	0.0894427
$126$	2.82843	0.251976
$127$	21.2334	1.88416	0.942078	−	0.335395i	$-0.108870\pi$
$127$	21.2334	1.88416	0.942078	+	0.335395i	$0.108870\pi$
$128$	1.00000	0.0883883
$129$	2.95816	0.260451
$130$	0.696775	0.0611112
$131$	7.67147	0.670259	0.335130	−	0.942172i	$-0.391220\pi$
$131$	7.67147	0.670259	0.335130	+	0.942172i	$0.391220\pi$
$132$	0	0
$133$	3.37894	0.292991
$134$	11.1708	0.965014
$135$	−2.41421	−0.207782
$136$	−0.449490	−0.0385434
$137$	−0.211493	−0.0180691	−0.00903454	−	0.999959i	$-0.502876\pi$
$137$	−0.211493	−0.0180691	−0.00903454	+	0.999959i	$0.502876\pi$
$138$	−3.73205	−0.317693
$139$	−7.34971	−0.623395	−0.311697	−	0.950181i	$-0.600897\pi$
$139$	−7.34971	−0.623395	−0.311697	+	0.950181i	$0.600897\pi$
$140$	−1.00000	−0.0845154
$141$	−3.94082	−0.331877
$142$	−11.9202	−1.00032
$143$	0	0
$144$	−2.82843	−0.235702
$145$	−1.60040	−0.132906
$146$	0.585786	0.0484800
$147$	0.414214	0.0341638
$148$	3.32780	0.273544
$149$	3.64173	0.298342	0.149171	−	0.988811i	$-0.452340\pi$
$149$	3.64173	0.298342	0.149171	+	0.988811i	$0.452340\pi$
$150$	0.414214	0.0338204
$151$	5.56099	0.452547	0.226274	−	0.974064i	$-0.427346\pi$
$151$	5.56099	0.452547	0.226274	+	0.974064i	$0.427346\pi$
$152$	−3.37894	−0.274068
$153$	1.27135	0.102782
$154$	0	0
$155$	10.1769	0.817428
$156$	0.288614	0.0231076
$157$	−13.8332	−1.10401	−0.552006	−	0.833840i	$-0.686138\pi$
$157$	−13.8332	−1.10401	−0.552006	+	0.833840i	$0.686138\pi$
$158$	−10.5811	−0.841790
$159$	−1.18013	−0.0935906
$160$	1.00000	0.0790569
$161$	9.00997	0.710085
$162$	7.48528	0.588099
$163$	−16.2220	−1.27060	−0.635302	−	0.772264i	$-0.719124\pi$
$163$	−16.2220	−1.27060	−0.635302	+	0.772264i	$0.719124\pi$
$164$	6.12044	0.477926
$165$	0	0
$166$	−17.5950	−1.36564
$167$	−18.0547	−1.39711	−0.698557	−	0.715554i	$-0.746174\pi$
$167$	−18.0547	−1.39711	−0.698557	+	0.715554i	$0.746174\pi$
$168$	−0.414214	−0.0319573
$169$	−12.5145	−0.962654
$170$	−0.449490	−0.0344743
$171$	9.55708	0.730848
$172$	7.14162	0.544543
$173$	−16.2419	−1.23485	−0.617425	−	0.786630i	$-0.711824\pi$
$173$	−16.2419	−1.23485	−0.617425	+	0.786630i	$0.711824\pi$
$174$	−0.662907	−0.0502548
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−3.30323	−0.248286
$178$	−11.9700	−0.897193
$179$	−13.4848	−1.00790	−0.503949	−	0.863733i	$-0.668120\pi$
$179$	−13.4848	−1.00790	−0.503949	+	0.863733i	$0.668120\pi$
$180$	−2.82843	−0.210819
$181$	13.5300	1.00568	0.502839	−	0.864380i	$-0.332289\pi$
$181$	13.5300	1.00568	0.502839	+	0.864380i	$0.332289\pi$
$182$	−0.696775	−0.0516484
$183$	−3.43488	−0.253913
$184$	−9.00997	−0.664224
$185$	3.32780	0.244665
$186$	4.21541	0.309089
$187$	0	0
$188$	−9.51399	−0.693879
$189$	2.41421	0.175608
$190$	−3.37894	−0.245134
$191$	−18.2779	−1.32254	−0.661272	−	0.750146i	$-0.729983\pi$
$191$	−18.2779	−1.32254	−0.661272	+	0.750146i	$0.729983\pi$
$192$	0.414214	0.0298933
$193$	14.2986	1.02923	0.514617	−	0.857420i	$-0.327934\pi$
$193$	14.2986	1.02923	0.514617	+	0.857420i	$0.327934\pi$
$194$	−8.56993	−0.615285
$195$	0.288614	0.0206680
$196$	1.00000	0.0714286
$197$	7.35023	0.523682	0.261841	−	0.965111i	$-0.415670\pi$
$197$	7.35023	0.523682	0.261841	+	0.965111i	$0.415670\pi$
$198$	0	0
$199$	13.7201	0.972593	0.486296	−	0.873794i	$-0.338348\pi$
$199$	13.7201	0.972593	0.486296	+	0.873794i	$0.338348\pi$
$200$	1.00000	0.0707107
$201$	4.62712	0.326372
$202$	−0.464102	−0.0326541
$203$	1.60040	0.112326
$204$	−0.186185	−0.0130355
$205$	6.12044	0.427470
$206$	−7.55051	−0.526069
$207$	25.4840	1.77126
$208$	0.696775	0.0483126
$209$	0	0
$210$	−0.414214	−0.0285835
$211$	−2.06574	−0.142212	−0.0711059	−	0.997469i	$-0.522653\pi$
$211$	−2.06574	−0.142212	−0.0711059	+	0.997469i	$0.522653\pi$
$212$	−2.84909	−0.195676
$213$	−4.93749	−0.338311
$214$	−14.4901	−0.990522
$215$	7.14162	0.487054
$216$	−2.41421	−0.164266
$217$	−10.1769	−0.690853
$218$	5.91359	0.400519
$219$	0.242641	0.0163961
$220$	0	0
$221$	−0.313193	−0.0210676
$222$	1.37842	0.0925136
$223$	−2.30839	−0.154581	−0.0772904	−	0.997009i	$-0.524627\pi$
$223$	−2.30839	−0.154581	−0.0772904	+	0.997009i	$0.524627\pi$
$224$	−1.00000	−0.0668153
$225$	−2.82843	−0.188562
$226$	−19.0599	−1.26784
$227$	8.40441	0.557820	0.278910	−	0.960317i	$-0.410027\pi$
$227$	8.40441	0.557820	0.278910	+	0.960317i	$0.410027\pi$
$228$	−1.39960	−0.0926909
$229$	15.4830	1.02315	0.511573	−	0.859240i	$-0.329063\pi$
$229$	15.4830	1.02315	0.511573	+	0.859240i	$0.329063\pi$
$230$	−9.00997	−0.594100
$231$	0	0
$232$	−1.60040	−0.105071
$233$	12.9927	0.851180	0.425590	−	0.904916i	$-0.360067\pi$
$233$	12.9927	0.851180	0.425590	+	0.904916i	$0.360067\pi$
$234$	−1.97078	−0.128834
$235$	−9.51399	−0.620624
$236$	−7.97469	−0.519108
$237$	−4.38285	−0.284697
$238$	0.449490	0.0291361
$239$	24.4799	1.58347	0.791737	−	0.610862i	$-0.209177\pi$
$239$	24.4799	1.58347	0.791737	+	0.610862i	$0.209177\pi$
$240$	0.414214	0.0267374
$241$	13.6203	0.877363	0.438681	−	0.898643i	$-0.355446\pi$
$241$	13.6203	0.877363	0.438681	+	0.898643i	$0.355446\pi$
$242$	0	0
$243$	10.3431	0.663513
$244$	−8.29253	−0.530875
$245$	1.00000	0.0638877
$246$	2.53517	0.161636
$247$	−2.35436	−0.149804
$248$	10.1769	0.646234
$249$	−7.28810	−0.461864
$250$	1.00000	0.0632456
$251$	3.27135	0.206486	0.103243	−	0.994656i	$-0.467078\pi$
$251$	3.27135	0.206486	0.103243	+	0.994656i	$0.467078\pi$
$252$	2.82843	0.178174
$253$	0	0
$254$	21.2334	1.33230
$255$	−0.186185	−0.0116593
$256$	1.00000	0.0625000
$257$	24.2767	1.51434	0.757169	−	0.653219i	$-0.226582\pi$
$257$	24.2767	1.51434	0.757169	+	0.653219i	$0.226582\pi$
$258$	2.95816	0.184167
$259$	−3.32780	−0.206780
$260$	0.696775	0.0432121
$261$	4.52661	0.280190
$262$	7.67147	0.473945
$263$	−3.94887	−0.243498	−0.121749	−	0.992561i	$-0.538850\pi$
$263$	−3.94887	−0.243498	−0.121749	+	0.992561i	$0.538850\pi$
$264$	0	0
$265$	−2.84909	−0.175018
$266$	3.37894	0.207176
$267$	−4.95816	−0.303434
$268$	11.1708	0.682368
$269$	2.25866	0.137713	0.0688565	−	0.997627i	$-0.478065\pi$
$269$	2.25866	0.137713	0.0688565	+	0.997627i	$0.478065\pi$
$270$	−2.41421	−0.146924
$271$	15.0479	0.914096	0.457048	−	0.889442i	$-0.348907\pi$
$271$	15.0479	0.914096	0.457048	+	0.889442i	$0.348907\pi$
$272$	−0.449490	−0.0272543
$273$	−0.288614	−0.0174677
$274$	−0.211493	−0.0127768
$275$	0	0
$276$	−3.73205	−0.224643
$277$	−3.85714	−0.231753	−0.115876	−	0.993264i	$-0.536968\pi$
$277$	−3.85714	−0.231753	−0.115876	+	0.993264i	$0.536968\pi$
$278$	−7.34971	−0.440807
$279$	−28.7846	−1.72329
$280$	−1.00000	−0.0597614
$281$	−22.2655	−1.32825	−0.664123	−	0.747623i	$-0.731195\pi$
$281$	−22.2655	−1.32825	−0.664123	+	0.747623i	$0.731195\pi$
$282$	−3.94082	−0.234673
$283$	5.73862	0.341125	0.170563	−	0.985347i	$-0.445441\pi$
$283$	5.73862	0.341125	0.170563	+	0.985347i	$0.445441\pi$
$284$	−11.9202	−0.707331
$285$	−1.39960	−0.0829052
$286$	0	0
$287$	−6.12044	−0.361278
$288$	−2.82843	−0.166667
$289$	−16.7980	−0.988115
$290$	−1.60040	−0.0939786
$291$	−3.54978	−0.208092
$292$	0.585786	0.0342806
$293$	8.40425	0.490981	0.245491	−	0.969399i	$-0.421051\pi$
$293$	8.40425	0.490981	0.245491	+	0.969399i	$0.421051\pi$
$294$	0.414214	0.0241574
$295$	−7.97469	−0.464304
$296$	3.32780	0.193425
$297$	0	0
$298$	3.64173	0.210960
$299$	−6.27792	−0.363061
$300$	0.414214	0.0239146
$301$	−7.14162	−0.411636
$302$	5.56099	0.319999
$303$	−0.192237	−0.0110437
$304$	−3.37894	−0.193795
$305$	−8.29253	−0.474829
$306$	1.27135	0.0726782
$307$	20.8683	1.19102	0.595508	−	0.803349i	$-0.296951\pi$
$307$	20.8683	1.19102	0.595508	+	0.803349i	$0.296951\pi$
$308$	0	0
$309$	−3.12752	−0.177919
$310$	10.1769	0.578009
$311$	−20.1336	−1.14167	−0.570835	−	0.821065i	$-0.693380\pi$
$311$	−20.1336	−1.14167	−0.570835	+	0.821065i	$0.693380\pi$
$312$	0.288614	0.0163395
$313$	−11.3326	−0.640557	−0.320279	−	0.947323i	$-0.603777\pi$
$313$	−11.3326	−0.640557	−0.320279	+	0.947323i	$0.603777\pi$
$314$	−13.8332	−0.780655
$315$	2.82843	0.159364
$316$	−10.5811	−0.595236
$317$	−9.67951	−0.543655	−0.271828	−	0.962346i	$-0.587628\pi$
$317$	−9.67951	−0.543655	−0.271828	+	0.962346i	$0.587628\pi$
$318$	−1.18013	−0.0661785
$319$	0	0
$320$	1.00000	0.0559017
$321$	−6.00199	−0.334999
$322$	9.00997	0.502106
$323$	1.51880	0.0845082
$324$	7.48528	0.415849
$325$	0.696775	0.0386501
$326$	−16.2220	−0.898452
$327$	2.44949	0.135457
$328$	6.12044	0.337945
$329$	9.51399	0.524523
$330$	0	0
$331$	−8.60572	−0.473013	−0.236507	−	0.971630i	$-0.576002\pi$
$331$	−8.60572	−0.473013	−0.236507	+	0.971630i	$0.576002\pi$
$332$	−17.5950	−0.965652
$333$	−9.41245	−0.515799
$334$	−18.0547	−0.987909
$335$	11.1708	0.610328
$336$	−0.414214	−0.0225972
$337$	24.7814	1.34993	0.674964	−	0.737851i	$-0.264159\pi$
$337$	24.7814	1.34993	0.674964	+	0.737851i	$0.264159\pi$
$338$	−12.5145	−0.680699
$339$	−7.89485	−0.428789
$340$	−0.449490	−0.0243770
$341$	0	0
$342$	9.55708	0.516788
$343$	−1.00000	−0.0539949
$344$	7.14162	0.385050
$345$	−3.73205	−0.200927
$346$	−16.2419	−0.873171
$347$	−28.6040	−1.53554	−0.767771	−	0.640725i	$-0.778634\pi$
$347$	−28.6040	−1.53554	−0.767771	+	0.640725i	$0.778634\pi$
$348$	−0.662907	−0.0355355
$349$	−13.7967	−0.738521	−0.369261	−	0.929326i	$-0.620389\pi$
$349$	−13.7967	−0.738521	−0.369261	+	0.929326i	$0.620389\pi$
$350$	−1.00000	−0.0534522
$351$	−1.68216	−0.0897872
$352$	0	0
$353$	18.0265	0.959454	0.479727	−	0.877418i	$-0.340736\pi$
$353$	18.0265	0.959454	0.479727	+	0.877418i	$0.340736\pi$
$354$	−3.30323	−0.175564
$355$	−11.9202	−0.632656
$356$	−11.9700	−0.634411
$357$	0.186185	0.00985394
$358$	−13.4848	−0.712692
$359$	8.39127	0.442875	0.221437	−	0.975175i	$-0.428925\pi$
$359$	8.39127	0.442875	0.221437	+	0.975175i	$0.428925\pi$
$360$	−2.82843	−0.149071
$361$	−7.58278	−0.399094
$362$	13.5300	0.711122
$363$	0	0
$364$	−0.696775	−0.0365209
$365$	0.585786	0.0306615
$366$	−3.43488	−0.179544
$367$	−20.7882	−1.08513	−0.542566	−	0.840013i	$-0.682547\pi$
$367$	−20.7882	−1.08513	−0.542566	+	0.840013i	$0.682547\pi$
$368$	−9.00997	−0.469677
$369$	−17.3112	−0.901186
$370$	3.32780	0.173004
$371$	2.84909	0.147917
$372$	4.21541	0.218559
$373$	17.8705	0.925300	0.462650	−	0.886541i	$-0.346899\pi$
$373$	17.8705	0.925300	0.462650	+	0.886541i	$0.346899\pi$
$374$	0	0
$375$	0.414214	0.0213899
$376$	−9.51399	−0.490647
$377$	−1.11512	−0.0574314
$378$	2.41421	0.124174
$379$	8.83324	0.453733	0.226866	−	0.973926i	$-0.427152\pi$
$379$	8.83324	0.453733	0.226866	+	0.973926i	$0.427152\pi$
$380$	−3.37894	−0.173336
$381$	8.79514	0.450589
$382$	−18.2779	−0.935180
$383$	9.77249	0.499351	0.249675	−	0.968330i	$-0.419676\pi$
$383$	9.77249	0.499351	0.249675	+	0.968330i	$0.419676\pi$
$384$	0.414214	0.0211377
$385$	0	0
$386$	14.2986	0.727779
$387$	−20.1996	−1.02680
$388$	−8.56993	−0.435072
$389$	−2.04508	−0.103690	−0.0518448	−	0.998655i	$-0.516510\pi$
$389$	−2.04508	−0.103690	−0.0518448	+	0.998655i	$0.516510\pi$
$390$	0.288614	0.0146145
$391$	4.04989	0.204812
$392$	1.00000	0.0505076
$393$	3.17763	0.160290
$394$	7.35023	0.370299
$395$	−10.5811	−0.532395
$396$	0	0
$397$	35.7259	1.79303	0.896517	−	0.443009i	$-0.146089\pi$
$397$	35.7259	1.79303	0.896517	+	0.443009i	$0.146089\pi$
$398$	13.7201	0.687727
$399$	1.39960	0.0700677
$400$	1.00000	0.0500000
$401$	12.5334	0.625889	0.312944	−	0.949771i	$-0.398685\pi$
$401$	12.5334	0.625889	0.312944	+	0.949771i	$0.398685\pi$
$402$	4.62712	0.230780
$403$	7.09100	0.353228
$404$	−0.464102	−0.0230899
$405$	7.48528	0.371947
$406$	1.60040	0.0794264
$407$	0	0
$408$	−0.186185	−0.00921752
$409$	−35.7625	−1.76834	−0.884170	−	0.467165i	$-0.845275\pi$
$409$	−35.7625	−1.76834	−0.884170	+	0.467165i	$0.845275\pi$
$410$	6.12044	0.302267
$411$	−0.0876034	−0.00432116
$412$	−7.55051	−0.371987
$413$	7.97469	0.392409
$414$	25.4840	1.25247
$415$	−17.5950	−0.863706
$416$	0.696775	0.0341622
$417$	−3.04435	−0.149083
$418$	0	0
$419$	9.13416	0.446233	0.223116	−	0.974792i	$-0.428377\pi$
$419$	9.13416	0.446233	0.223116	+	0.974792i	$0.428377\pi$
$420$	−0.414214	−0.0202116
$421$	−31.3404	−1.52744	−0.763720	−	0.645548i	$-0.776629\pi$
$421$	−31.3404	−1.52744	−0.763720	+	0.645548i	$0.776629\pi$
$422$	−2.06574	−0.100559
$423$	26.9096	1.30839
$424$	−2.84909	−0.138364
$425$	−0.449490	−0.0218035
$426$	−4.93749	−0.239222
$427$	8.29253	0.401304
$428$	−14.4901	−0.700405
$429$	0	0
$430$	7.14162	0.344400
$431$	1.68216	0.0810269	0.0405135	−	0.999179i	$-0.487101\pi$
$431$	1.68216	0.0810269	0.0405135	+	0.999179i	$0.487101\pi$
$432$	−2.41421	−0.116154
$433$	−5.91608	−0.284309	−0.142154	−	0.989845i	$-0.545403\pi$
$433$	−5.91608	−0.284309	−0.142154	+	0.989845i	$0.545403\pi$
$434$	−10.1769	−0.488507
$435$	−0.662907	−0.0317839
$436$	5.91359	0.283210
$437$	30.4441	1.45634
$438$	0.242641	0.0115938
$439$	16.6904	0.796588	0.398294	−	0.917258i	$-0.369602\pi$
$439$	16.6904	0.796588	0.398294	+	0.917258i	$0.369602\pi$
$440$	0	0
$441$	−2.82843	−0.134687
$442$	−0.313193	−0.0148971
$443$	−27.4203	−1.30278	−0.651389	−	0.758744i	$-0.725813\pi$
$443$	−27.4203	−1.30278	−0.651389	+	0.758744i	$0.725813\pi$
$444$	1.37842	0.0654170
$445$	−11.9700	−0.567435
$446$	−2.30839	−0.109305
$447$	1.50845	0.0713474
$448$	−1.00000	−0.0472456
$449$	0.608726	0.0287276	0.0143638	−	0.999897i	$-0.495428\pi$
$449$	0.608726	0.0287276	0.0143638	+	0.999897i	$0.495428\pi$
$450$	−2.82843	−0.133333
$451$	0	0
$452$	−19.0599	−0.896500
$453$	2.30344	0.108225
$454$	8.40441	0.394438
$455$	−0.696775	−0.0326653
$456$	−1.39960	−0.0655424
$457$	−13.4794	−0.630542	−0.315271	−	0.949002i	$-0.602095\pi$
$457$	−13.4794	−0.630542	−0.315271	+	0.949002i	$0.602095\pi$
$458$	15.4830	0.723473
$459$	1.08516	0.0506511
$460$	−9.00997	−0.420092
$461$	−19.5891	−0.912356	−0.456178	−	0.889888i	$-0.650782\pi$
$461$	−19.5891	−0.912356	−0.456178	+	0.889888i	$0.650782\pi$
$462$	0	0
$463$	−24.3330	−1.13085	−0.565424	−	0.824800i	$-0.691288\pi$
$463$	−24.3330	−1.13085	−0.565424	+	0.824800i	$0.691288\pi$
$464$	−1.60040	−0.0742966
$465$	4.21541	0.195485
$466$	12.9927	0.601875
$467$	−28.3412	−1.31147	−0.655737	−	0.754990i	$-0.727642\pi$
$467$	−28.3412	−1.31147	−0.655737	+	0.754990i	$0.727642\pi$
$468$	−1.97078	−0.0910992
$469$	−11.1708	−0.515822
$470$	−9.51399	−0.438848
$471$	−5.72991	−0.264020
$472$	−7.97469	−0.367065
$473$	0	0
$474$	−4.38285	−0.201311
$475$	−3.37894	−0.155036
$476$	0.449490	0.0206023
$477$	8.05845	0.368971
$478$	24.4799	1.11968
$479$	26.9207	1.23004	0.615019	−	0.788513i	$-0.289149\pi$
$479$	26.9207	1.23004	0.615019	+	0.788513i	$0.289149\pi$
$480$	0.414214	0.0189062
$481$	2.31873	0.105725
$482$	13.6203	0.620389
$483$	3.73205	0.169814
$484$	0	0
$485$	−8.56993	−0.389140
$486$	10.3431	0.469175
$487$	10.5353	0.477402	0.238701	−	0.971093i	$-0.423278\pi$
$487$	10.5353	0.477402	0.238701	+	0.971093i	$0.423278\pi$
$488$	−8.29253	−0.375385
$489$	−6.71936	−0.303860
$490$	1.00000	0.0451754
$491$	−12.3321	−0.556540	−0.278270	−	0.960503i	$-0.589761\pi$
$491$	−12.3321	−0.556540	−0.278270	+	0.960503i	$0.589761\pi$
$492$	2.53517	0.114294
$493$	0.719363	0.0323985
$494$	−2.35436	−0.105928
$495$	0	0
$496$	10.1769	0.456956
$497$	11.9202	0.534692
$498$	−7.28810	−0.326587
$499$	37.4485	1.67643	0.838214	−	0.545342i	$-0.183600\pi$
$499$	37.4485	1.67643	0.838214	+	0.545342i	$0.183600\pi$
$500$	1.00000	0.0447214
$501$	−7.47850	−0.334115
$502$	3.27135	0.146007
$503$	−7.53393	−0.335921	−0.167961	−	0.985794i	$-0.553718\pi$
$503$	−7.53393	−0.335921	−0.167961	+	0.985794i	$0.553718\pi$
$504$	2.82843	0.125988
$505$	−0.464102	−0.0206523
$506$	0	0
$507$	−5.18368	−0.230215
$508$	21.2334	0.942078
$509$	33.7306	1.49508	0.747541	−	0.664216i	$-0.231234\pi$
$509$	33.7306	1.49508	0.747541	+	0.664216i	$0.231234\pi$
$510$	−0.186185	−0.00824440
$511$	−0.585786	−0.0259137
$512$	1.00000	0.0441942
$513$	8.15748	0.360161
$514$	24.2767	1.07080
$515$	−7.55051	−0.332715
$516$	2.95816	0.130226
$517$	0	0
$518$	−3.32780	−0.146215
$519$	−6.72762	−0.295310
$520$	0.696775	0.0305556
$521$	17.4407	0.764092	0.382046	−	0.924143i	$-0.375220\pi$
$521$	17.4407	0.764092	0.382046	+	0.924143i	$0.375220\pi$
$522$	4.52661	0.198124
$523$	−11.7069	−0.511907	−0.255954	−	0.966689i	$-0.582389\pi$
$523$	−11.7069	−0.511907	−0.255954	+	0.966689i	$0.582389\pi$
$524$	7.67147	0.335130
$525$	−0.414214	−0.0180778
$526$	−3.94887	−0.172179
$527$	−4.57441	−0.199264
$528$	0	0
$529$	58.1795	2.52954
$530$	−2.84909	−0.123757
$531$	22.5558	0.978840
$532$	3.37894	0.146496
$533$	4.26457	0.184719
$534$	−4.95816	−0.214560
$535$	−14.4901	−0.626461
$536$	11.1708	0.482507
$537$	−5.58557	−0.241035
$538$	2.25866	0.0973778
$539$	0	0
$540$	−2.41421	−0.103891
$541$	−24.2686	−1.04339	−0.521695	−	0.853132i	$-0.674700\pi$
$541$	−24.2686	−1.04339	−0.521695	+	0.853132i	$0.674700\pi$
$542$	15.0479	0.646363
$543$	5.60431	0.240504
$544$	−0.449490	−0.0192717
$545$	5.91359	0.253310
$546$	−0.288614	−0.0123515
$547$	21.6629	0.926240	0.463120	−	0.886296i	$-0.346730\pi$
$547$	21.6629	0.926240	0.463120	+	0.886296i	$0.346730\pi$
$548$	−0.211493	−0.00903454
$549$	23.4548	1.00103
$550$	0	0
$551$	5.40765	0.230373
$552$	−3.73205	−0.158847
$553$	10.5811	0.449956
$554$	−3.85714	−0.163874
$555$	1.37842	0.0585108
$556$	−7.34971	−0.311697
$557$	31.4409	1.33220	0.666098	−	0.745864i	$-0.267963\pi$
$557$	31.4409	1.33220	0.666098	+	0.745864i	$0.267963\pi$
$558$	−28.7846	−1.21855
$559$	4.97610	0.210467
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	−22.2655	−0.939212
$563$	44.4010	1.87128	0.935640	−	0.352955i	$-0.114823\pi$
$563$	44.4010	1.87128	0.935640	+	0.352955i	$0.114823\pi$
$564$	−3.94082	−0.165939
$565$	−19.0599	−0.801854
$566$	5.73862	0.241212
$567$	−7.48528	−0.314352
$568$	−11.9202	−0.500159
$569$	2.44077	0.102322	0.0511611	−	0.998690i	$-0.483708\pi$
$569$	2.44077	0.102322	0.0511611	+	0.998690i	$0.483708\pi$
$570$	−1.39960	−0.0586229
$571$	4.79639	0.200723	0.100361	−	0.994951i	$-0.468000\pi$
$571$	4.79639	0.200723	0.100361	+	0.994951i	$0.468000\pi$
$572$	0	0
$573$	−7.57096	−0.316281
$574$	−6.12044	−0.255462
$575$	−9.00997	−0.375742
$576$	−2.82843	−0.117851
$577$	−23.9982	−0.999060	−0.499530	−	0.866297i	$-0.666494\pi$
$577$	−23.9982	−0.999060	−0.499530	+	0.866297i	$0.666494\pi$
$578$	−16.7980	−0.698703
$579$	5.92267	0.246138
$580$	−1.60040	−0.0664529
$581$	17.5950	0.729965
$582$	−3.54978	−0.147143
$583$	0	0
$584$	0.585786	0.0242400
$585$	−1.97078	−0.0814816
$586$	8.40425	0.347176
$587$	13.2344	0.546241	0.273121	−	0.961980i	$-0.411944\pi$
$587$	13.2344	0.546241	0.273121	+	0.961980i	$0.411944\pi$
$588$	0.414214	0.0170819
$589$	−34.3871	−1.41690
$590$	−7.97469	−0.328313
$591$	3.04456	0.125237
$592$	3.32780	0.136772
$593$	36.6395	1.50460	0.752302	−	0.658818i	$-0.228943\pi$
$593$	36.6395	1.50460	0.752302	+	0.658818i	$0.228943\pi$
$594$	0	0
$595$	0.449490	0.0184273
$596$	3.64173	0.149171
$597$	5.68306	0.232592
$598$	−6.27792	−0.256723
$599$	2.81485	0.115011	0.0575057	−	0.998345i	$-0.481685\pi$
$599$	2.81485	0.115011	0.0575057	+	0.998345i	$0.481685\pi$
$600$	0.414214	0.0169102
$601$	27.0224	1.10227	0.551134	−	0.834417i	$-0.314195\pi$
$601$	27.0224	1.10227	0.551134	+	0.834417i	$0.314195\pi$
$602$	−7.14162	−0.291071
$603$	−31.5959	−1.28669
$604$	5.56099	0.226274
$605$	0	0
$606$	−0.192237	−0.00780910
$607$	−18.7130	−0.759538	−0.379769	−	0.925081i	$-0.623997\pi$
$607$	−18.7130	−0.759538	−0.379769	+	0.925081i	$0.623997\pi$
$608$	−3.37894	−0.137034
$609$	0.662907	0.0268623
$610$	−8.29253	−0.335755
$611$	−6.62911	−0.268185
$612$	1.27135	0.0513912
$613$	−13.1333	−0.530449	−0.265224	−	0.964187i	$-0.585446\pi$
$613$	−13.1333	−0.530449	−0.265224	+	0.964187i	$0.585446\pi$
$614$	20.8683	0.842176
$615$	2.53517	0.102228
$616$	0	0
$617$	−0.553017	−0.0222636	−0.0111318	−	0.999938i	$-0.503543\pi$
$617$	−0.553017	−0.0222636	−0.0111318	+	0.999938i	$0.503543\pi$
$618$	−3.12752	−0.125807
$619$	−17.8938	−0.719213	−0.359607	−	0.933104i	$-0.617089\pi$
$619$	−17.8938	−0.719213	−0.359607	+	0.933104i	$0.617089\pi$
$620$	10.1769	0.408714
$621$	21.7520	0.872877
$622$	−20.1336	−0.807283
$623$	11.9700	0.479570
$624$	0.288614	0.0115538
$625$	1.00000	0.0400000
$626$	−11.3326	−0.452942
$627$	0	0
$628$	−13.8332	−0.552006
$629$	−1.49581	−0.0596420
$630$	2.82843	0.112687
$631$	45.6251	1.81631	0.908154	−	0.418636i	$-0.137492\pi$
$631$	45.6251	1.81631	0.908154	+	0.418636i	$0.137492\pi$
$632$	−10.5811	−0.420895
$633$	−0.855659	−0.0340094
$634$	−9.67951	−0.384422
$635$	21.2334	0.842620
$636$	−1.18013	−0.0467953
$637$	0.696775	0.0276072
$638$	0	0
$639$	33.7153	1.33376
$640$	1.00000	0.0395285
$641$	−23.2522	−0.918408	−0.459204	−	0.888331i	$-0.651865\pi$
$641$	−23.2522	−0.918408	−0.459204	+	0.888331i	$0.651865\pi$
$642$	−6.00199	−0.236880
$643$	−10.3407	−0.407796	−0.203898	−	0.978992i	$-0.565361\pi$
$643$	−10.3407	−0.407796	−0.203898	+	0.978992i	$0.565361\pi$
$644$	9.00997	0.355042
$645$	2.95816	0.116477
$646$	1.51880	0.0597563
$647$	14.5229	0.570952	0.285476	−	0.958386i	$-0.407848\pi$
$647$	14.5229	0.570952	0.285476	+	0.958386i	$0.407848\pi$
$648$	7.48528	0.294050
$649$	0	0
$650$	0.696775	0.0273297
$651$	−4.21541	−0.165215
$652$	−16.2220	−0.635302
$653$	−15.2419	−0.596461	−0.298231	−	0.954494i	$-0.596396\pi$
$653$	−15.2419	−0.596461	−0.298231	+	0.954494i	$0.596396\pi$
$654$	2.44949	0.0957826
$655$	7.67147	0.299749
$656$	6.12044	0.238963
$657$	−1.65685	−0.0646400
$658$	9.51399	0.370894
$659$	−45.7851	−1.78354	−0.891768	−	0.452494i	$-0.850535\pi$
$659$	−45.7851	−1.78354	−0.891768	+	0.452494i	$0.850535\pi$
$660$	0	0
$661$	−4.64993	−0.180861	−0.0904307	−	0.995903i	$-0.528824\pi$
$661$	−4.64993	−0.180861	−0.0904307	+	0.995903i	$0.528824\pi$
$662$	−8.60572	−0.334471
$663$	−0.129729	−0.00503825
$664$	−17.5950	−0.682819
$665$	3.37894	0.131030
$666$	−9.41245	−0.364725
$667$	14.4195	0.558327
$668$	−18.0547	−0.698557
$669$	−0.956164	−0.0369675
$670$	11.1708	0.431567
$671$	0	0
$672$	−0.414214	−0.0159786
$673$	21.5428	0.830413	0.415206	−	0.909727i	$-0.363709\pi$
$673$	21.5428	0.830413	0.415206	+	0.909727i	$0.363709\pi$
$674$	24.7814	0.954543
$675$	−2.41421	−0.0929231
$676$	−12.5145	−0.481327
$677$	44.5318	1.71150	0.855748	−	0.517393i	$-0.173098\pi$
$677$	44.5318	1.71150	0.855748	+	0.517393i	$0.173098\pi$
$678$	−7.89485	−0.303200
$679$	8.56993	0.328884
$680$	−0.449490	−0.0172371
$681$	3.48122	0.133401
$682$	0	0
$683$	24.3276	0.930869	0.465435	−	0.885082i	$-0.345898\pi$
$683$	24.3276	0.930869	0.465435	+	0.885082i	$0.345898\pi$
$684$	9.55708	0.365424
$685$	−0.211493	−0.00808074
$686$	−1.00000	−0.0381802
$687$	6.41327	0.244682
$688$	7.14162	0.272272
$689$	−1.98517	−0.0756291
$690$	−3.73205	−0.142077
$691$	28.8708	1.09830	0.549148	−	0.835725i	$-0.314952\pi$
$691$	28.8708	1.09830	0.549148	+	0.835725i	$0.314952\pi$
$692$	−16.2419	−0.617425
$693$	0	0
$694$	−28.6040	−1.08579
$695$	−7.34971	−0.278791
$696$	−0.662907	−0.0251274
$697$	−2.75108	−0.104204
$698$	−13.7967	−0.522213
$699$	5.38175	0.203557
$700$	−1.00000	−0.0377964
$701$	40.0109	1.51119	0.755595	−	0.655039i	$-0.227348\pi$
$701$	40.0109	1.51119	0.755595	+	0.655039i	$0.227348\pi$
$702$	−1.68216	−0.0634891
$703$	−11.2444	−0.424092
$704$	0	0
$705$	−3.94082	−0.148420
$706$	18.0265	0.678436
$707$	0.464102	0.0174543
$708$	−3.30323	−0.124143
$709$	−16.3266	−0.613157	−0.306578	−	0.951845i	$-0.599184\pi$
$709$	−16.3266	−0.613157	−0.306578	+	0.951845i	$0.599184\pi$
$710$	−11.9202	−0.447356
$711$	29.9280	1.12239
$712$	−11.9700	−0.448596
$713$	−91.6935	−3.43395
$714$	0.186185	0.00696779
$715$	0	0
$716$	−13.4848	−0.503949
$717$	10.1399	0.378682
$718$	8.39127	0.313160
$719$	−29.1213	−1.08604	−0.543020	−	0.839720i	$-0.682719\pi$
$719$	−29.1213	−1.08604	−0.543020	+	0.839720i	$0.682719\pi$
$720$	−2.82843	−0.105409
$721$	7.55051	0.281196
$722$	−7.58278	−0.282202
$723$	5.64173	0.209818
$724$	13.5300	0.502839
$725$	−1.60040	−0.0594373
$726$	0	0
$727$	−0.229273	−0.00850327	−0.00425164	−	0.999991i	$-0.501353\pi$
$727$	−0.229273	−0.00850327	−0.00425164	+	0.999991i	$0.501353\pi$
$728$	−0.696775	−0.0258242
$729$	−18.1716	−0.673021
$730$	0.585786	0.0216809
$731$	−3.21009	−0.118729
$732$	−3.43488	−0.126957
$733$	−25.0740	−0.926130	−0.463065	−	0.886324i	$-0.653250\pi$
$733$	−25.0740	−0.926130	−0.463065	+	0.886324i	$0.653250\pi$
$734$	−20.7882	−0.767305
$735$	0.414214	0.0152785
$736$	−9.00997	−0.332112
$737$	0	0
$738$	−17.3112	−0.637235
$739$	7.95973	0.292803	0.146402	−	0.989225i	$-0.453231\pi$
$739$	7.95973	0.292803	0.146402	+	0.989225i	$0.453231\pi$
$740$	3.32780	0.122333
$741$	−0.975207	−0.0358251
$742$	2.84909	0.104593
$743$	−11.9597	−0.438759	−0.219379	−	0.975640i	$-0.570403\pi$
$743$	−11.9597	−0.438759	−0.219379	+	0.975640i	$0.570403\pi$
$744$	4.21541	0.154544
$745$	3.64173	0.133423
$746$	17.8705	0.654286
$747$	49.7662	1.82085
$748$	0	0
$749$	14.4901	0.529456
$750$	0.414214	0.0151249
$751$	−26.7728	−0.976954	−0.488477	−	0.872577i	$-0.662447\pi$
$751$	−26.7728	−0.976954	−0.488477	+	0.872577i	$0.662447\pi$
$752$	−9.51399	−0.346940
$753$	1.35504	0.0493803
$754$	−1.11512	−0.0406102
$755$	5.56099	0.202385
$756$	2.41421	0.0878041
$757$	−15.7539	−0.572586	−0.286293	−	0.958142i	$-0.592423\pi$
$757$	−15.7539	−0.572586	−0.286293	+	0.958142i	$0.592423\pi$
$758$	8.83324	0.320838
$759$	0	0
$760$	−3.37894	−0.122567
$761$	27.5271	0.997858	0.498929	−	0.866643i	$-0.333727\pi$
$761$	27.5271	0.997858	0.498929	+	0.866643i	$0.333727\pi$
$762$	8.79514	0.318614
$763$	−5.91359	−0.214086
$764$	−18.2779	−0.661272
$765$	1.27135	0.0459657
$766$	9.77249	0.353094
$767$	−5.55656	−0.200636
$768$	0.414214	0.0149466
$769$	−27.3409	−0.985940	−0.492970	−	0.870046i	$-0.664089\pi$
$769$	−27.3409	−0.985940	−0.492970	+	0.870046i	$0.664089\pi$
$770$	0	0
$771$	10.0557	0.362148
$772$	14.2986	0.514617
$773$	−37.4833	−1.34818	−0.674090	−	0.738649i	$-0.735464\pi$
$773$	−37.4833	−1.34818	−0.674090	+	0.738649i	$0.735464\pi$
$774$	−20.1996	−0.726058
$775$	10.1769	0.365565
$776$	−8.56993	−0.307643
$777$	−1.37842	−0.0494506
$778$	−2.04508	−0.0733197
$779$	−20.6806	−0.740959
$780$	0.288614	0.0103340
$781$	0	0
$782$	4.04989	0.144824
$783$	3.86370	0.138077
$784$	1.00000	0.0357143
$785$	−13.8332	−0.493729
$786$	3.17763	0.113342
$787$	−3.34973	−0.119405	−0.0597025	−	0.998216i	$-0.519015\pi$
$787$	−3.34973	−0.119405	−0.0597025	+	0.998216i	$0.519015\pi$
$788$	7.35023	0.261841
$789$	−1.63567	−0.0582316
$790$	−10.5811	−0.376460
$791$	19.0599	0.677690
$792$	0	0
$793$	−5.77802	−0.205184
$794$	35.7259	1.26787
$795$	−1.18013	−0.0418550
$796$	13.7201	0.486296
$797$	22.1429	0.784341	0.392171	−	0.919893i	$-0.371724\pi$
$797$	22.1429	0.784341	0.392171	+	0.919893i	$0.371724\pi$
$798$	1.39960	0.0495454
$799$	4.27644	0.151290
$800$	1.00000	0.0353553
$801$	33.8564	1.19626
$802$	12.5334	0.442570
$803$	0	0
$804$	4.62712	0.163186
$805$	9.00997	0.317560
$806$	7.09100	0.249770
$807$	0.935568	0.0329336
$808$	−0.464102	−0.0163270
$809$	2.22095	0.0780843	0.0390421	−	0.999238i	$-0.487569\pi$
$809$	2.22095	0.0780843	0.0390421	+	0.999238i	$0.487569\pi$
$810$	7.48528	0.263006
$811$	46.3962	1.62919	0.814595	−	0.580031i	$-0.196959\pi$
$811$	46.3962	1.62919	0.814595	+	0.580031i	$0.196959\pi$
$812$	1.60040	0.0561630
$813$	6.23305	0.218603
$814$	0	0
$815$	−16.2220	−0.568231
$816$	−0.186185	−0.00651777
$817$	−24.1311	−0.844240
$818$	−35.7625	−1.25041
$819$	1.97078	0.0688645
$820$	6.12044	0.213735
$821$	35.1399	1.22639	0.613195	−	0.789932i	$-0.289884\pi$
$821$	35.1399	1.22639	0.613195	+	0.789932i	$0.289884\pi$
$822$	−0.0876034	−0.00305552
$823$	−8.90147	−0.310286	−0.155143	−	0.987892i	$-0.549584\pi$
$823$	−8.90147	−0.310286	−0.155143	+	0.987892i	$0.549584\pi$
$824$	−7.55051	−0.263034
$825$	0	0
$826$	7.97469	0.277475
$827$	22.4976	0.782319	0.391159	−	0.920323i	$-0.372074\pi$
$827$	22.4976	0.782319	0.391159	+	0.920323i	$0.372074\pi$
$828$	25.4840	0.885632
$829$	−5.11350	−0.177599	−0.0887995	−	0.996050i	$-0.528303\pi$
$829$	−5.11350	−0.177599	−0.0887995	+	0.996050i	$0.528303\pi$
$830$	−17.5950	−0.610732
$831$	−1.59768	−0.0554228
$832$	0.696775	0.0241563
$833$	−0.449490	−0.0155739
$834$	−3.04435	−0.105417
$835$	−18.0547	−0.624809
$836$	0	0
$837$	−24.5692	−0.849236
$838$	9.13416	0.315534
$839$	4.06199	0.140236	0.0701178	−	0.997539i	$-0.477662\pi$
$839$	4.06199	0.140236	0.0701178	+	0.997539i	$0.477662\pi$
$840$	−0.414214	−0.0142917
$841$	−26.4387	−0.911680
$842$	−31.3404	−1.08006
$843$	−9.22266	−0.317645
$844$	−2.06574	−0.0711059
$845$	−12.5145	−0.430512
$846$	26.9096	0.925172
$847$	0	0
$848$	−2.84909	−0.0978382
$849$	2.37701	0.0815789
$850$	−0.449490	−0.0154174
$851$	−29.9834	−1.02782
$852$	−4.93749	−0.169156
$853$	40.3033	1.37996	0.689980	−	0.723828i	$-0.257619\pi$
$853$	40.3033	1.37996	0.689980	+	0.723828i	$0.257619\pi$
$854$	8.29253	0.283764
$855$	9.55708	0.326845
$856$	−14.4901	−0.495261
$857$	25.7912	0.881009	0.440505	−	0.897750i	$-0.354800\pi$
$857$	25.7912	0.881009	0.440505	+	0.897750i	$0.354800\pi$
$858$	0	0
$859$	30.2497	1.03211	0.516054	−	0.856556i	$-0.327400\pi$
$859$	30.2497	1.03211	0.516054	+	0.856556i	$0.327400\pi$
$860$	7.14162	0.243527
$861$	−2.53517	−0.0863983
$862$	1.68216	0.0572947
$863$	16.2691	0.553806	0.276903	−	0.960898i	$-0.410692\pi$
$863$	16.2691	0.553806	0.276903	+	0.960898i	$0.410692\pi$
$864$	−2.41421	−0.0821332
$865$	−16.2419	−0.552242
$866$	−5.91608	−0.201037
$867$	−6.95794	−0.236304
$868$	−10.1769	−0.345426
$869$	0	0
$870$	−0.662907	−0.0224746
$871$	7.78356	0.263736
$872$	5.91359	0.200259
$873$	24.2394	0.820380
$874$	30.4441	1.02979
$875$	−1.00000	−0.0338062
$876$	0.242641	0.00819807
$877$	−34.6670	−1.17062	−0.585310	−	0.810809i	$-0.699027\pi$
$877$	−34.6670	−1.17062	−0.585310	+	0.810809i	$0.699027\pi$
$878$	16.6904	0.563273
$879$	3.48115	0.117416
$880$	0	0
$881$	−15.0738	−0.507849	−0.253924	−	0.967224i	$-0.581721\pi$
$881$	−15.0738	−0.507849	−0.253924	+	0.967224i	$0.581721\pi$
$882$	−2.82843	−0.0952381
$883$	−1.63994	−0.0551884	−0.0275942	−	0.999619i	$-0.508785\pi$
$883$	−1.63994	−0.0551884	−0.0275942	+	0.999619i	$0.508785\pi$
$884$	−0.313193	−0.0105338
$885$	−3.30323	−0.111037
$886$	−27.4203	−0.921202
$887$	−14.7479	−0.495185	−0.247592	−	0.968864i	$-0.579639\pi$
$887$	−14.7479	−0.495185	−0.247592	+	0.968864i	$0.579639\pi$
$888$	1.37842	0.0462568
$889$	−21.2334	−0.712144
$890$	−11.9700	−0.401237
$891$	0	0
$892$	−2.30839	−0.0772904
$893$	32.1472	1.07576
$894$	1.50845	0.0504502
$895$	−13.4848	−0.450746
$896$	−1.00000	−0.0334077
$897$	−2.60040	−0.0868248
$898$	0.608726	0.0203135
$899$	−16.2871	−0.543205
$900$	−2.82843	−0.0942809
$901$	1.28064	0.0426642
$902$	0	0
$903$	−2.95816	−0.0984413
$904$	−19.0599	−0.633921
$905$	13.5300	0.449753
$906$	2.30344	0.0765266
$907$	−55.5813	−1.84555	−0.922774	−	0.385342i	$-0.874083\pi$
$907$	−55.5813	−1.84555	−0.922774	+	0.385342i	$0.874083\pi$
$908$	8.40441	0.278910
$909$	1.31268	0.0435388
$910$	−0.696775	−0.0230979
$911$	14.8844	0.493141	0.246571	−	0.969125i	$-0.420696\pi$
$911$	14.8844	0.493141	0.246571	+	0.969125i	$0.420696\pi$
$912$	−1.39960	−0.0463454
$913$	0	0
$914$	−13.4794	−0.445860
$915$	−3.43488	−0.113554
$916$	15.4830	0.511573
$917$	−7.67147	−0.253334
$918$	1.08516	0.0358158
$919$	−39.1462	−1.29131	−0.645657	−	0.763628i	$-0.723416\pi$
$919$	−39.1462	−1.29131	−0.645657	+	0.763628i	$0.723416\pi$
$920$	−9.00997	−0.297050
$921$	8.64393	0.284827
$922$	−19.5891	−0.645133
$923$	−8.30566	−0.273384
$924$	0	0
$925$	3.32780	0.109418
$926$	−24.3330	−0.799631
$927$	21.3561	0.701425
$928$	−1.60040	−0.0525356
$929$	6.76998	0.222116	0.111058	−	0.993814i	$-0.464576\pi$
$929$	6.76998	0.222116	0.111058	+	0.993814i	$0.464576\pi$
$930$	4.21541	0.138229
$931$	−3.37894	−0.110740
$932$	12.9927	0.425590
$933$	−8.33960	−0.273026
$934$	−28.3412	−0.927351
$935$	0	0
$936$	−1.97078	−0.0644168
$937$	−9.93426	−0.324538	−0.162269	−	0.986747i	$-0.551881\pi$
$937$	−9.93426	−0.324538	−0.162269	+	0.986747i	$0.551881\pi$
$938$	−11.1708	−0.364741
$939$	−4.69412	−0.153187
$940$	−9.51399	−0.310312
$941$	−55.3511	−1.80439	−0.902197	−	0.431324i	$-0.858047\pi$
$941$	−55.3511	−1.80439	−0.902197	+	0.431324i	$0.858047\pi$
$942$	−5.72991	−0.186691
$943$	−55.1450	−1.79577
$944$	−7.97469	−0.259554
$945$	2.41421	0.0785344
$946$	0	0
$947$	11.9252	0.387517	0.193758	−	0.981049i	$-0.437932\pi$
$947$	11.9252	0.387517	0.193758	+	0.981049i	$0.437932\pi$
$948$	−4.38285	−0.142348
$949$	0.408161	0.0132495
$950$	−3.37894	−0.109627
$951$	−4.00938	−0.130013
$952$	0.449490	0.0145680
$953$	31.9262	1.03419	0.517096	−	0.855928i	$-0.327013\pi$
$953$	31.9262	1.03419	0.517096	+	0.855928i	$0.327013\pi$
$954$	8.05845	0.260902
$955$	−18.2779	−0.591460
$956$	24.4799	0.791737
$957$	0	0
$958$	26.9207	0.869768
$959$	0.211493	0.00682947
$960$	0.414214	0.0133687
$961$	72.5692	2.34094
$962$	2.31873	0.0747589
$963$	40.9842	1.32070
$964$	13.6203	0.438681
$965$	14.2986	0.460288
$966$	3.73205	0.120077
$967$	−38.6060	−1.24149	−0.620743	−	0.784014i	$-0.713169\pi$
$967$	−38.6060	−1.24149	−0.620743	+	0.784014i	$0.713169\pi$
$968$	0	0
$969$	0.629107	0.0202098
$970$	−8.56993	−0.275164
$971$	22.2352	0.713561	0.356780	−	0.934188i	$-0.383874\pi$
$971$	22.2352	0.713561	0.356780	+	0.934188i	$0.383874\pi$
$972$	10.3431	0.331757
$973$	7.34971	0.235621
$974$	10.5353	0.337574
$975$	0.288614	0.00924303
$976$	−8.29253	−0.265437
$977$	−17.0281	−0.544779	−0.272389	−	0.962187i	$-0.587814\pi$
$977$	−17.0281	−0.544779	−0.272389	+	0.962187i	$0.587814\pi$
$978$	−6.71936	−0.214862
$979$	0	0
$980$	1.00000	0.0319438
$981$	−16.7262	−0.534025
$982$	−12.3321	−0.393533
$983$	−33.4596	−1.06720	−0.533598	−	0.845738i	$-0.679160\pi$
$983$	−33.4596	−1.06720	−0.533598	+	0.845738i	$0.679160\pi$
$984$	2.53517	0.0808182
$985$	7.35023	0.234198
$986$	0.719363	0.0229092
$987$	3.94082	0.125438
$988$	−2.35436	−0.0749021
$989$	−64.3458	−2.04608
$990$	0	0
$991$	18.3129	0.581729	0.290864	−	0.956764i	$-0.406057\pi$
$991$	18.3129	0.581729	0.290864	+	0.956764i	$0.406057\pi$
$992$	10.1769	0.323117
$993$	−3.56461	−0.113119
$994$	11.9202	0.378084
$995$	13.7201	0.434957
$996$	−7.28810	−0.230932
$997$	−48.5331	−1.53706	−0.768530	−	0.639814i	$-0.779012\pi$
$997$	−48.5331	−1.53706	−0.768530	+	0.639814i	$0.779012\pi$
$998$	37.4485	1.18541
$999$	−8.03403	−0.254186

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8470.2.a.cr.1.3	yes	4
11.10	odd	2		8470.2.a.cp.1.4	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8470.2.a.cp.1.4	✓	4	11.10	odd	2
8470.2.a.cr.1.3	yes	4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +0.414214 q^{3} +1.00000 q^{4} +1.00000 q^{5} +0.414214 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.82843 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +0.414214 q^{3} +1.00000 q^{4} +1.00000 q^{5} +0.414214 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.82843 q^{9} +1.00000 q^{10} +0.414214 q^{12} +0.696775 q^{13} -1.00000 q^{14} +0.414214 q^{15} +1.00000 q^{16} -0.449490 q^{17} -2.82843 q^{18} -3.37894 q^{19} +1.00000 q^{20} -0.414214 q^{21} -9.00997 q^{23} +0.414214 q^{24} +1.00000 q^{25} +0.696775 q^{26} -2.41421 q^{27} -1.00000 q^{28} -1.60040 q^{29} +0.414214 q^{30} +10.1769 q^{31} +1.00000 q^{32} -0.449490 q^{34} -1.00000 q^{35} -2.82843 q^{36} +3.32780 q^{37} -3.37894 q^{38} +0.288614 q^{39} +1.00000 q^{40} +6.12044 q^{41} -0.414214 q^{42} +7.14162 q^{43} -2.82843 q^{45} -9.00997 q^{46} -9.51399 q^{47} +0.414214 q^{48} +1.00000 q^{49} +1.00000 q^{50} -0.186185 q^{51} +0.696775 q^{52} -2.84909 q^{53} -2.41421 q^{54} -1.00000 q^{56} -1.39960 q^{57} -1.60040 q^{58} -7.97469 q^{59} +0.414214 q^{60} -8.29253 q^{61} +10.1769 q^{62} +2.82843 q^{63} +1.00000 q^{64} +0.696775 q^{65} +11.1708 q^{67} -0.449490 q^{68} -3.73205 q^{69} -1.00000 q^{70} -11.9202 q^{71} -2.82843 q^{72} +0.585786 q^{73} +3.32780 q^{74} +0.414214 q^{75} -3.37894 q^{76} +0.288614 q^{78} -10.5811 q^{79} +1.00000 q^{80} +7.48528 q^{81} +6.12044 q^{82} -17.5950 q^{83} -0.414214 q^{84} -0.449490 q^{85} +7.14162 q^{86} -0.662907 q^{87} -11.9700 q^{89} -2.82843 q^{90} -0.696775 q^{91} -9.00997 q^{92} +4.21541 q^{93} -9.51399 q^{94} -3.37894 q^{95} +0.414214 q^{96} -8.56993 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{5} - 4 q^{6} - 4 q^{7} + 4 q^{8}+O(q^{10}) \) 4 * q + 4 * q^2 - 4 * q^3 + 4 * q^4 + 4 * q^5 - 4 * q^6 - 4 * q^7 + 4 * q^8	\( 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{5} - 4 q^{6} - 4 q^{7} + 4 q^{8} + 4 q^{10} - 4 q^{12} - 4 q^{14} - 4 q^{15} + 4 q^{16} + 8 q^{17} - 12 q^{19} + 4 q^{20} + 4 q^{21} - 8 q^{23} - 4 q^{24} + 4 q^{25} - 4 q^{27} - 4 q^{28} - 8 q^{29} - 4 q^{30} + 4 q^{32} + 8 q^{34} - 4 q^{35} - 16 q^{37} - 12 q^{38} + 8 q^{39} + 4 q^{40} + 8 q^{41} + 4 q^{42} - 8 q^{43} - 8 q^{46} - 16 q^{47} - 4 q^{48} + 4 q^{49} + 4 q^{50} - 8 q^{51} - 4 q^{54} - 4 q^{56} - 4 q^{57} - 8 q^{58} - 8 q^{59} - 4 q^{60} - 8 q^{61} + 4 q^{64} + 8 q^{68} - 8 q^{69} - 4 q^{70} - 8 q^{71} + 8 q^{73} - 16 q^{74} - 4 q^{75} - 12 q^{76} + 8 q^{78} - 24 q^{79} + 4 q^{80} - 4 q^{81} + 8 q^{82} + 8 q^{83} + 4 q^{84} + 8 q^{85} - 8 q^{86} + 16 q^{87} - 8 q^{92} + 16 q^{93} - 16 q^{94} - 12 q^{95} - 4 q^{96} - 8 q^{97} + 4 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 - 4 * q^3 + 4 * q^4 + 4 * q^5 - 4 * q^6 - 4 * q^7 + 4 * q^8 + 4 * q^10 - 4 * q^12 - 4 * q^14 - 4 * q^15 + 4 * q^16 + 8 * q^17 - 12 * q^19 + 4 * q^20 + 4 * q^21 - 8 * q^23 - 4 * q^24 + 4 * q^25 - 4 * q^27 - 4 * q^28 - 8 * q^29 - 4 * q^30 + 4 * q^32 + 8 * q^34 - 4 * q^35 - 16 * q^37 - 12 * q^38 + 8 * q^39 + 4 * q^40 + 8 * q^41 + 4 * q^42 - 8 * q^43 - 8 * q^46 - 16 * q^47 - 4 * q^48 + 4 * q^49 + 4 * q^50 - 8 * q^51 - 4 * q^54 - 4 * q^56 - 4 * q^57 - 8 * q^58 - 8 * q^59 - 4 * q^60 - 8 * q^61 + 4 * q^64 + 8 * q^68 - 8 * q^69 - 4 * q^70 - 8 * q^71 + 8 * q^73 - 16 * q^74 - 4 * q^75 - 12 * q^76 + 8 * q^78 - 24 * q^79 + 4 * q^80 - 4 * q^81 + 8 * q^82 + 8 * q^83 + 4 * q^84 + 8 * q^85 - 8 * q^86 + 16 * q^87 - 8 * q^92 + 16 * q^93 - 16 * q^94 - 12 * q^95 - 4 * q^96 - 8 * q^97 + 4 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more