LMFDB - Embedded newform 3822.2.a.bx.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3822,2,Mod(1,3822)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3822, 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("3822.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3822.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:	$30.5188236525$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.10304.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 7x^{2} + 8x + 8 $ x^4 - 2x^3 - 7x^2 + 8*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.69230$ of defining polynomial
Character	$\chi$	$=$	3822.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.307703 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.307703 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +0.307703 q^{10} +1.25714 q^{11} -1.00000 q^{12} +1.00000 q^{13} +0.307703 q^{15} +1.00000 q^{16} +2.64173 q^{17} -1.00000 q^{18} +7.86343 q^{19} -0.307703 q^{20} -1.25714 q^{22} -6.47884 q^{23} +1.00000 q^{24} -4.90532 q^{25} -1.00000 q^{26} -1.00000 q^{27} +8.88438 q^{29} -0.307703 q^{30} -5.22170 q^{31} -1.00000 q^{32} -1.25714 q^{33} -2.64173 q^{34} +1.00000 q^{36} +8.44922 q^{37} -7.86343 q^{38} -1.00000 q^{39} +0.307703 q^{40} -0.192517 q^{41} +10.4788 q^{43} +1.25714 q^{44} -0.307703 q^{45} +6.47884 q^{46} -9.74825 q^{47} -1.00000 q^{48} +4.90532 q^{50} -2.64173 q^{51} +1.00000 q^{52} +7.74825 q^{53} +1.00000 q^{54} -0.386825 q^{55} -7.86343 q^{57} -8.88438 q^{58} -10.0920 q^{59} +0.307703 q^{60} -9.38639 q^{61} +5.22170 q^{62} +1.00000 q^{64} -0.307703 q^{65} +1.25714 q^{66} +9.62724 q^{67} +2.64173 q^{68} +6.47884 q^{69} -1.56484 q^{71} -1.00000 q^{72} -6.83487 q^{73} -8.44922 q^{74} +4.90532 q^{75} +7.86343 q^{76} +1.00000 q^{78} -16.0082 q^{79} -0.307703 q^{80} +1.00000 q^{81} +0.192517 q^{82} +1.04145 q^{83} -0.812869 q^{85} -10.4788 q^{86} -8.88438 q^{87} -1.25714 q^{88} -1.24308 q^{89} +0.307703 q^{90} -6.47884 q^{92} +5.22170 q^{93} +9.74825 q^{94} -2.41960 q^{95} +1.00000 q^{96} +15.7688 q^{97} +1.25714 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} - 6 q^{5} + 4 q^{6} - 4 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 - 6 * q^5 + 4 * q^6 - 4 * q^8 + 4 * q^9	$ 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} - 6 q^{5} + 4 q^{6} - 4 q^{8} + 4 q^{9} + 6 q^{10} + 2 q^{11} - 4 q^{12} + 4 q^{13} + 6 q^{15} + 4 q^{16} - 2 q^{17} - 4 q^{18} - 2 q^{19} - 6 q^{20} - 2 q^{22} - 2 q^{23} + 4 q^{24} + 6 q^{25} - 4 q^{26} - 4 q^{27} + 6 q^{29} - 6 q^{30} - 4 q^{32} - 2 q^{33} + 2 q^{34} + 4 q^{36} + 6 q^{37} + 2 q^{38} - 4 q^{39} + 6 q^{40} - 16 q^{41} + 18 q^{43} + 2 q^{44} - 6 q^{45} + 2 q^{46} - 16 q^{47} - 4 q^{48} - 6 q^{50} + 2 q^{51} + 4 q^{52} + 8 q^{53} + 4 q^{54} - 2 q^{55} + 2 q^{57} - 6 q^{58} - 16 q^{59} + 6 q^{60} + 2 q^{61} + 4 q^{64} - 6 q^{65} + 2 q^{66} + 12 q^{67} - 2 q^{68} + 2 q^{69} - 8 q^{71} - 4 q^{72} - 6 q^{73} - 6 q^{74} - 6 q^{75} - 2 q^{76} + 4 q^{78} - 24 q^{79} - 6 q^{80} + 4 q^{81} + 16 q^{82} - 28 q^{83} + 38 q^{85} - 18 q^{86} - 6 q^{87} - 2 q^{88} - 28 q^{89} + 6 q^{90} - 2 q^{92} + 16 q^{94} + 22 q^{95} + 4 q^{96} + 4 q^{97} + 2 q^{99}+O(q^{100}) $ 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 - 6 * q^5 + 4 * q^6 - 4 * q^8 + 4 * q^9 + 6 * q^10 + 2 * q^11 - 4 * q^12 + 4 * q^13 + 6 * q^15 + 4 * q^16 - 2 * q^17 - 4 * q^18 - 2 * q^19 - 6 * q^20 - 2 * q^22 - 2 * q^23 + 4 * q^24 + 6 * q^25 - 4 * q^26 - 4 * q^27 + 6 * q^29 - 6 * q^30 - 4 * q^32 - 2 * q^33 + 2 * q^34 + 4 * q^36 + 6 * q^37 + 2 * q^38 - 4 * q^39 + 6 * q^40 - 16 * q^41 + 18 * q^43 + 2 * q^44 - 6 * q^45 + 2 * q^46 - 16 * q^47 - 4 * q^48 - 6 * q^50 + 2 * q^51 + 4 * q^52 + 8 * q^53 + 4 * q^54 - 2 * q^55 + 2 * q^57 - 6 * q^58 - 16 * q^59 + 6 * q^60 + 2 * q^61 + 4 * q^64 - 6 * q^65 + 2 * q^66 + 12 * q^67 - 2 * q^68 + 2 * q^69 - 8 * q^71 - 4 * q^72 - 6 * q^73 - 6 * q^74 - 6 * q^75 - 2 * q^76 + 4 * q^78 - 24 * q^79 - 6 * q^80 + 4 * q^81 + 16 * q^82 - 28 * q^83 + 38 * q^85 - 18 * q^86 - 6 * q^87 - 2 * q^88 - 28 * q^89 + 6 * q^90 - 2 * q^92 + 16 * q^94 + 22 * q^95 + 4 * q^96 + 4 * q^97 + 2 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−0.307703	−0.137609	−0.0688044	−	0.997630i	$-0.521918\pi$
$5$	−0.307703	−0.137609	−0.0688044	+	0.997630i	$0.521918\pi$
$6$	1.00000	0.408248
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0.307703	0.0973041
$11$	1.25714	0.379042	0.189521	−	0.981877i	$-0.439306\pi$
$11$	1.25714	0.379042	0.189521	+	0.981877i	$0.439306\pi$
$12$	−1.00000	−0.288675
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0.307703	0.0794485
$16$	1.00000	0.250000
$17$	2.64173	0.640715	0.320357	−	0.947297i	$-0.396197\pi$
$17$	2.64173	0.640715	0.320357	+	0.947297i	$0.396197\pi$
$18$	−1.00000	−0.235702
$19$	7.86343	1.80399	0.901997	−	0.431741i	$-0.142101\pi$
$19$	7.86343	1.80399	0.901997	+	0.431741i	$0.142101\pi$
$20$	−0.307703	−0.0688044
$21$	0	0
$22$	−1.25714	−0.268023
$23$	−6.47884	−1.35093	−0.675465	−	0.737392i	$-0.736057\pi$
$23$	−6.47884	−1.35093	−0.675465	+	0.737392i	$0.736057\pi$
$24$	1.00000	0.204124
$25$	−4.90532	−0.981064
$26$	−1.00000	−0.196116
$27$	−1.00000	−0.192450
$28$	0	0
$29$	8.88438	1.64979	0.824894	−	0.565288i	$-0.191235\pi$
$29$	8.88438	1.64979	0.824894	+	0.565288i	$0.191235\pi$
$30$	−0.307703	−0.0561786
$31$	−5.22170	−0.937844	−0.468922	−	0.883239i	$-0.655358\pi$
$31$	−5.22170	−0.937844	−0.468922	+	0.883239i	$0.655358\pi$
$32$	−1.00000	−0.176777
$33$	−1.25714	−0.218840
$34$	−2.64173	−0.453054
$35$	0	0
$36$	1.00000	0.166667
$37$	8.44922	1.38904	0.694521	−	0.719472i	$-0.255616\pi$
$37$	8.44922	1.38904	0.694521	+	0.719472i	$0.255616\pi$
$38$	−7.86343	−1.27562
$39$	−1.00000	−0.160128
$40$	0.307703	0.0486521
$41$	−0.192517	−0.0300660	−0.0150330	−	0.999887i	$-0.504785\pi$
$41$	−0.192517	−0.0300660	−0.0150330	+	0.999887i	$0.504785\pi$
$42$	0	0
$43$	10.4788	1.59801	0.799004	−	0.601326i	$-0.205361\pi$
$43$	10.4788	1.59801	0.799004	+	0.601326i	$0.205361\pi$
$44$	1.25714	0.189521
$45$	−0.307703	−0.0458696
$46$	6.47884	0.955252
$47$	−9.74825	−1.42193	−0.710964	−	0.703229i	$-0.751741\pi$
$47$	−9.74825	−1.42193	−0.710964	+	0.703229i	$0.751741\pi$
$48$	−1.00000	−0.144338
$49$	0	0
$50$	4.90532	0.693717
$51$	−2.64173	−0.369917
$52$	1.00000	0.138675
$53$	7.74825	1.06430	0.532152	−	0.846649i	$-0.321384\pi$
$53$	7.74825	1.06430	0.532152	+	0.846649i	$0.321384\pi$
$54$	1.00000	0.136083
$55$	−0.386825	−0.0521595
$56$	0	0
$57$	−7.86343	−1.04154
$58$	−8.88438	−1.16658
$59$	−10.0920	−1.31387	−0.656934	−	0.753948i	$-0.728147\pi$
$59$	−10.0920	−1.31387	−0.656934	+	0.753948i	$0.728147\pi$
$60$	0.307703	0.0397242
$61$	−9.38639	−1.20180	−0.600902	−	0.799323i	$-0.705192\pi$
$61$	−9.38639	−1.20180	−0.600902	+	0.799323i	$0.705192\pi$
$62$	5.22170	0.663156
$63$	0	0
$64$	1.00000	0.125000
$65$	−0.307703	−0.0381658
$66$	1.25714	0.154743
$67$	9.62724	1.17615	0.588077	−	0.808805i	$-0.299885\pi$
$67$	9.62724	1.17615	0.588077	+	0.808805i	$0.299885\pi$
$68$	2.64173	0.320357
$69$	6.47884	0.779960
$70$	0	0
$71$	−1.56484	−0.185713	−0.0928563	−	0.995680i	$-0.529600\pi$
$71$	−1.56484	−0.185713	−0.0928563	+	0.995680i	$0.529600\pi$
$72$	−1.00000	−0.117851
$73$	−6.83487	−0.799961	−0.399981	−	0.916524i	$-0.630983\pi$
$73$	−6.83487	−0.799961	−0.399981	+	0.916524i	$0.630983\pi$
$74$	−8.44922	−0.982202
$75$	4.90532	0.566417
$76$	7.86343	0.901997
$77$	0	0
$78$	1.00000	0.113228
$79$	−16.0082	−1.80107	−0.900534	−	0.434786i	$-0.856824\pi$
$79$	−16.0082	−1.80107	−0.900534	+	0.434786i	$0.856824\pi$
$80$	−0.307703	−0.0344022
$81$	1.00000	0.111111
$82$	0.192517	0.0212599
$83$	1.04145	0.114314	0.0571569	−	0.998365i	$-0.481796\pi$
$83$	1.04145	0.114314	0.0571569	+	0.998365i	$0.481796\pi$
$84$	0	0
$85$	−0.812869	−0.0881680
$86$	−10.4788	−1.12996
$87$	−8.88438	−0.952505
$88$	−1.25714	−0.134012
$89$	−1.24308	−0.131766	−0.0658831	−	0.997827i	$-0.520986\pi$
$89$	−1.24308	−0.131766	−0.0658831	+	0.997827i	$0.520986\pi$
$90$	0.307703	0.0324347
$91$	0	0
$92$	−6.47884	−0.675465
$93$	5.22170	0.541465
$94$	9.74825	1.00545
$95$	−2.41960	−0.248246
$96$	1.00000	0.102062
$97$	15.7688	1.60107	0.800537	−	0.599283i	$-0.204548\pi$
$97$	15.7688	1.60107	0.800537	+	0.599283i	$0.204548\pi$
$98$	0	0
$99$	1.25714	0.126347
$100$	−4.90532	−0.490532
$101$	−19.3509	−1.92549	−0.962745	−	0.270409i	$-0.912841\pi$
$101$	−19.3509	−1.92549	−0.962745	+	0.270409i	$0.912841\pi$
$102$	2.64173	0.261571
$103$	18.0642	1.77992	0.889959	−	0.456042i	$-0.150733\pi$
$103$	18.0642	1.77992	0.889959	+	0.456042i	$0.150733\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	−7.74825	−0.752576
$107$	16.0577	1.55236	0.776180	−	0.630511i	$-0.217155\pi$
$107$	16.0577	1.55236	0.776180	+	0.630511i	$0.217155\pi$
$108$	−1.00000	−0.0962250
$109$	10.8053	1.03496	0.517478	−	0.855697i	$-0.326871\pi$
$109$	10.8053	1.03496	0.517478	+	0.855697i	$0.326871\pi$
$110$	0.386825	0.0368824
$111$	−8.44922	−0.801964
$112$	0	0
$113$	14.8367	1.39572	0.697858	−	0.716236i	$-0.254137\pi$
$113$	14.8367	1.39572	0.697858	+	0.716236i	$0.254137\pi$
$114$	7.86343	0.736478
$115$	1.99356	0.185900
$116$	8.88438	0.824894
$117$	1.00000	0.0924500
$118$	10.0920	0.929045
$119$	0	0
$120$	−0.307703	−0.0280893
$121$	−9.41960	−0.856327
$122$	9.38639	0.849804
$123$	0.192517	0.0173586
$124$	−5.22170	−0.468922
$125$	3.04789	0.272612
$126$	0	0
$127$	−1.53584	−0.136284	−0.0681421	−	0.997676i	$-0.521707\pi$
$127$	−1.53584	−0.136284	−0.0681421	+	0.997676i	$0.521707\pi$
$128$	−1.00000	−0.0883883
$129$	−10.4788	−0.922610
$130$	0.307703	0.0269873
$131$	−2.16934	−0.189536	−0.0947682	−	0.995499i	$-0.530211\pi$
$131$	−2.16934	−0.189536	−0.0947682	+	0.995499i	$0.530211\pi$
$132$	−1.25714	−0.109420
$133$	0	0
$134$	−9.62724	−0.831666
$135$	0.307703	0.0264828
$136$	−2.64173	−0.226527
$137$	−8.46972	−0.723617	−0.361809	−	0.932252i	$-0.617841\pi$
$137$	−8.46972	−0.723617	−0.361809	+	0.932252i	$0.617841\pi$
$138$	−6.47884	−0.551515
$139$	12.4015	1.05188	0.525941	−	0.850521i	$-0.323713\pi$
$139$	12.4015	1.05188	0.525941	+	0.850521i	$0.323713\pi$
$140$	0	0
$141$	9.74825	0.820950
$142$	1.56484	0.131319
$143$	1.25714	0.105127
$144$	1.00000	0.0833333
$145$	−2.73375	−0.227025
$146$	6.83487	0.565658
$147$	0	0
$148$	8.44922	0.694521
$149$	0.213022	0.0174514	0.00872571	−	0.999962i	$-0.497222\pi$
$149$	0.213022	0.0174514	0.00872571	+	0.999962i	$0.497222\pi$
$150$	−4.90532	−0.400518
$151$	12.5116	1.01818	0.509090	−	0.860713i	$-0.329982\pi$
$151$	12.5116	1.01818	0.509090	+	0.860713i	$0.329982\pi$
$152$	−7.86343	−0.637809
$153$	2.64173	0.213572
$154$	0	0
$155$	1.60673	0.129056
$156$	−1.00000	−0.0800641
$157$	2.21569	0.176831	0.0884157	−	0.996084i	$-0.471820\pi$
$157$	2.21569	0.176831	0.0884157	+	0.996084i	$0.471820\pi$
$158$	16.0082	1.27355
$159$	−7.74825	−0.614476
$160$	0.307703	0.0243260
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	16.4337	1.28718	0.643592	−	0.765369i	$-0.277443\pi$
$163$	16.4337	1.28718	0.643592	+	0.765369i	$0.277443\pi$
$164$	−0.192517	−0.0150330
$165$	0.386825	0.0301143
$166$	−1.04145	−0.0808321
$167$	−12.9544	−1.00244	−0.501220	−	0.865320i	$-0.667115\pi$
$167$	−12.9544	−1.00244	−0.501220	+	0.865320i	$0.667115\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0.812869	0.0623442
$171$	7.86343	0.601332
$172$	10.4788	0.799004
$173$	−2.74930	−0.209026	−0.104513	−	0.994524i	$-0.533328\pi$
$173$	−2.74930	−0.209026	−0.104513	+	0.994524i	$0.533328\pi$
$174$	8.88438	0.673523
$175$	0	0
$176$	1.25714	0.0947605
$177$	10.0920	0.758562
$178$	1.24308	0.0931727
$179$	−3.10007	−0.231710	−0.115855	−	0.993266i	$-0.536961\pi$
$179$	−3.10007	−0.231710	−0.115855	+	0.993266i	$0.536961\pi$
$180$	−0.307703	−0.0229348
$181$	−4.94032	−0.367211	−0.183606	−	0.983000i	$-0.558777\pi$
$181$	−4.94032	−0.367211	−0.183606	+	0.983000i	$0.558777\pi$
$182$	0	0
$183$	9.38639	0.693862
$184$	6.47884	0.477626
$185$	−2.59985	−0.191145
$186$	−5.22170	−0.382873
$187$	3.32103	0.242858
$188$	−9.74825	−0.710964
$189$	0	0
$190$	2.41960	0.175536
$191$	−24.4187	−1.76688	−0.883438	−	0.468547i	$-0.844778\pi$
$191$	−24.4187	−1.76688	−0.883438	+	0.468547i	$0.844778\pi$
$192$	−1.00000	−0.0721688
$193$	−8.43428	−0.607113	−0.303556	−	0.952813i	$-0.598174\pi$
$193$	−8.43428	−0.607113	−0.303556	+	0.952813i	$0.598174\pi$
$194$	−15.7688	−1.13213
$195$	0.307703	0.0220350
$196$	0	0
$197$	12.4770	0.888953	0.444476	−	0.895791i	$-0.353390\pi$
$197$	12.4770	0.888953	0.444476	+	0.895791i	$0.353390\pi$
$198$	−1.25714	−0.0893410
$199$	−5.44878	−0.386254	−0.193127	−	0.981174i	$-0.561863\pi$
$199$	−5.44878	−0.386254	−0.193127	+	0.981174i	$0.561863\pi$
$200$	4.90532	0.346858
$201$	−9.62724	−0.679053
$202$	19.3509	1.36153
$203$	0	0
$204$	−2.64173	−0.184958
$205$	0.0592379	0.00413735
$206$	−18.0642	−1.25859
$207$	−6.47884	−0.450310
$208$	1.00000	0.0693375
$209$	9.88543	0.683790
$210$	0	0
$211$	−13.8763	−0.955285	−0.477643	−	0.878554i	$-0.658509\pi$
$211$	−13.8763	−0.955285	−0.477643	+	0.878554i	$0.658509\pi$
$212$	7.74825	0.532152
$213$	1.56484	0.107221
$214$	−16.0577	−1.09768
$215$	−3.22437	−0.219900
$216$	1.00000	0.0680414
$217$	0	0
$218$	−10.8053	−0.731824
$219$	6.83487	0.461858
$220$	−0.386825	−0.0260798
$221$	2.64173	0.177702
$222$	8.44922	0.567074
$223$	13.2589	0.887884	0.443942	−	0.896056i	$-0.353580\pi$
$223$	13.2589	0.887884	0.443942	+	0.896056i	$0.353580\pi$
$224$	0	0
$225$	−4.90532	−0.327021
$226$	−14.8367	−0.986920
$227$	−18.4459	−1.22430	−0.612150	−	0.790742i	$-0.709695\pi$
$227$	−18.4459	−1.22430	−0.612150	+	0.790742i	$0.709695\pi$
$228$	−7.86343	−0.520768
$229$	12.5471	0.829133	0.414566	−	0.910019i	$-0.363933\pi$
$229$	12.5471	0.829133	0.414566	+	0.910019i	$0.363933\pi$
$230$	−1.99356	−0.131451
$231$	0	0
$232$	−8.88438	−0.583288
$233$	9.52761	0.624174	0.312087	−	0.950053i	$-0.398972\pi$
$233$	9.52761	0.624174	0.312087	+	0.950053i	$0.398972\pi$
$234$	−1.00000	−0.0653720
$235$	2.99956	0.195670
$236$	−10.0920	−0.656934
$237$	16.0082	1.03985
$238$	0	0
$239$	17.6651	1.14266	0.571330	−	0.820721i	$-0.306428\pi$
$239$	17.6651	1.14266	0.571330	+	0.820721i	$0.306428\pi$
$240$	0.307703	0.0198621
$241$	0.986672	0.0635572	0.0317786	−	0.999495i	$-0.489883\pi$
$241$	0.986672	0.0635572	0.0317786	+	0.999495i	$0.489883\pi$
$242$	9.41960	0.605515
$243$	−1.00000	−0.0641500
$244$	−9.38639	−0.600902
$245$	0	0
$246$	−0.192517	−0.0122744
$247$	7.86343	0.500338
$248$	5.22170	0.331578
$249$	−1.04145	−0.0659991
$250$	−3.04789	−0.192766
$251$	14.7117	0.928597	0.464299	−	0.885679i	$-0.346306\pi$
$251$	14.7117	0.928597	0.464299	+	0.885679i	$0.346306\pi$
$252$	0	0
$253$	−8.14481	−0.512060
$254$	1.53584	0.0963674
$255$	0.812869	0.0509038
$256$	1.00000	0.0625000
$257$	19.8280	1.23684	0.618418	−	0.785849i	$-0.287774\pi$
$257$	19.8280	1.23684	0.618418	+	0.785849i	$0.287774\pi$
$258$	10.4788	0.652384
$259$	0	0
$260$	−0.307703	−0.0190829
$261$	8.88438	0.549929
$262$	2.16934	0.134022
$263$	20.4260	1.25952	0.629762	−	0.776788i	$-0.283152\pi$
$263$	20.4260	1.25952	0.629762	+	0.776788i	$0.283152\pi$
$264$	1.25714	0.0773716
$265$	−2.38416	−0.146458
$266$	0	0
$267$	1.24308	0.0760752
$268$	9.62724	0.588077
$269$	20.6344	1.25810	0.629051	−	0.777364i	$-0.283444\pi$
$269$	20.6344	1.25810	0.629051	+	0.777364i	$0.283444\pi$
$270$	−0.307703	−0.0187262
$271$	2.28137	0.138584	0.0692918	−	0.997596i	$-0.477926\pi$
$271$	2.28137	0.138584	0.0692918	+	0.997596i	$0.477926\pi$
$272$	2.64173	0.160179
$273$	0	0
$274$	8.46972	0.511675
$275$	−6.16667	−0.371864
$276$	6.47884	0.389980
$277$	23.6232	1.41938	0.709690	−	0.704514i	$-0.248835\pi$
$277$	23.6232	1.41938	0.709690	+	0.704514i	$0.248835\pi$
$278$	−12.4015	−0.743793
$279$	−5.22170	−0.312615
$280$	0	0
$281$	−3.92911	−0.234391	−0.117196	−	0.993109i	$-0.537390\pi$
$281$	−3.92911	−0.234391	−0.117196	+	0.993109i	$0.537390\pi$
$282$	−9.74825	−0.580500
$283$	20.1002	1.19484	0.597418	−	0.801930i	$-0.296193\pi$
$283$	20.1002	1.19484	0.597418	+	0.801930i	$0.296193\pi$
$284$	−1.56484	−0.0928563
$285$	2.41960	0.143325
$286$	−1.25714	−0.0743362
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	−10.0212	−0.589485
$290$	2.73375	0.160531
$291$	−15.7688	−0.924381
$292$	−6.83487	−0.399981
$293$	−2.75842	−0.161149	−0.0805743	−	0.996749i	$-0.525675\pi$
$293$	−2.75842	−0.161149	−0.0805743	+	0.996749i	$0.525675\pi$
$294$	0	0
$295$	3.10534	0.180800
$296$	−8.44922	−0.491101
$297$	−1.25714	−0.0729467
$298$	−0.213022	−0.0123400
$299$	−6.47884	−0.374681
$300$	4.90532	0.283209
$301$	0	0
$302$	−12.5116	−0.719962
$303$	19.3509	1.11168
$304$	7.86343	0.450999
$305$	2.88822	0.165379
$306$	−2.64173	−0.151018
$307$	−5.64397	−0.322118	−0.161059	−	0.986945i	$-0.551491\pi$
$307$	−5.64397	−0.322118	−0.161059	+	0.986945i	$0.551491\pi$
$308$	0	0
$309$	−18.0642	−1.02764
$310$	−1.60673	−0.0912561
$311$	24.9822	1.41661	0.708306	−	0.705906i	$-0.249460\pi$
$311$	24.9822	1.41661	0.708306	+	0.705906i	$0.249460\pi$
$312$	1.00000	0.0566139
$313$	27.5680	1.55824	0.779118	−	0.626878i	$-0.215667\pi$
$313$	27.5680	1.55824	0.779118	+	0.626878i	$0.215667\pi$
$314$	−2.21569	−0.125039
$315$	0	0
$316$	−16.0082	−0.900534
$317$	0.274794	0.0154340	0.00771699	−	0.999970i	$-0.497544\pi$
$317$	0.274794	0.0154340	0.00771699	+	0.999970i	$0.497544\pi$
$318$	7.74825	0.434500
$319$	11.1689	0.625339
$320$	−0.307703	−0.0172011
$321$	−16.0577	−0.896256
$322$	0	0
$323$	20.7731	1.15585
$324$	1.00000	0.0555556
$325$	−4.90532	−0.272098
$326$	−16.4337	−0.910176
$327$	−10.8053	−0.597532
$328$	0.192517	0.0106300
$329$	0	0
$330$	−0.386825	−0.0212940
$331$	21.4660	1.17988	0.589939	−	0.807448i	$-0.299152\pi$
$331$	21.4660	1.17988	0.589939	+	0.807448i	$0.299152\pi$
$332$	1.04145	0.0571569
$333$	8.44922	0.463014
$334$	12.9544	0.708832
$335$	−2.96233	−0.161849
$336$	0	0
$337$	23.0350	1.25480	0.627398	−	0.778698i	$-0.284120\pi$
$337$	23.0350	1.25480	0.627398	+	0.778698i	$0.284120\pi$
$338$	−1.00000	−0.0543928
$339$	−14.8367	−0.805817
$340$	−0.812869	−0.0440840
$341$	−6.56440	−0.355482
$342$	−7.86343	−0.425206
$343$	0	0
$344$	−10.4788	−0.564981
$345$	−1.99356	−0.107329
$346$	2.74930	0.147804
$347$	−1.68647	−0.0905346	−0.0452673	−	0.998975i	$-0.514414\pi$
$347$	−1.68647	−0.0905346	−0.0452673	+	0.998975i	$0.514414\pi$
$348$	−8.88438	−0.476253
$349$	22.2877	1.19303	0.596516	−	0.802601i	$-0.296551\pi$
$349$	22.2877	1.19303	0.596516	+	0.802601i	$0.296551\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	−1.25714	−0.0670058
$353$	−13.0664	−0.695455	−0.347728	−	0.937596i	$-0.613047\pi$
$353$	−13.0664	−0.695455	−0.347728	+	0.937596i	$0.613047\pi$
$354$	−10.0920	−0.536384
$355$	0.481506	0.0255557
$356$	−1.24308	−0.0658831
$357$	0	0
$358$	3.10007	0.163844
$359$	3.70320	0.195448	0.0977238	−	0.995214i	$-0.468844\pi$
$359$	3.70320	0.195448	0.0977238	+	0.995214i	$0.468844\pi$
$360$	0.307703	0.0162174
$361$	42.8336	2.25440
$362$	4.94032	0.259658
$363$	9.41960	0.494401
$364$	0	0
$365$	2.10311	0.110082
$366$	−9.38639	−0.490634
$367$	−4.32176	−0.225594	−0.112797	−	0.993618i	$-0.535981\pi$
$367$	−4.32176	−0.225594	−0.112797	+	0.993618i	$0.535981\pi$
$368$	−6.47884	−0.337733
$369$	−0.192517	−0.0100220
$370$	2.59985	0.135160
$371$	0	0
$372$	5.22170	0.270732
$373$	23.6349	1.22377	0.611883	−	0.790948i	$-0.290412\pi$
$373$	23.6349	1.22377	0.611883	+	0.790948i	$0.290412\pi$
$374$	−3.32103	−0.171726
$375$	−3.04789	−0.157393
$376$	9.74825	0.502727
$377$	8.88438	0.457569
$378$	0	0
$379$	3.21196	0.164987	0.0824937	−	0.996592i	$-0.473712\pi$
$379$	3.21196	0.164987	0.0824937	+	0.996592i	$0.473712\pi$
$380$	−2.41960	−0.124123
$381$	1.53584	0.0786837
$382$	24.4187	1.24937
$383$	−37.3559	−1.90880	−0.954398	−	0.298536i	$-0.903502\pi$
$383$	−37.3559	−1.90880	−0.954398	+	0.298536i	$0.903502\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	8.43428	0.429294
$387$	10.4788	0.532669
$388$	15.7688	0.800537
$389$	−10.0334	−0.508713	−0.254357	−	0.967110i	$-0.581864\pi$
$389$	−10.0334	−0.508713	−0.254357	+	0.967110i	$0.581864\pi$
$390$	−0.307703	−0.0155811
$391$	−17.1154	−0.865561
$392$	0	0
$393$	2.16934	0.109429
$394$	−12.4770	−0.628584
$395$	4.92578	0.247843
$396$	1.25714	0.0631737
$397$	24.3552	1.22235	0.611175	−	0.791495i	$-0.290697\pi$
$397$	24.3552	1.22235	0.611175	+	0.791495i	$0.290697\pi$
$398$	5.44878	0.273123
$399$	0	0
$400$	−4.90532	−0.245266
$401$	22.4434	1.12077	0.560385	−	0.828232i	$-0.310653\pi$
$401$	22.4434	1.12077	0.560385	+	0.828232i	$0.310653\pi$
$402$	9.62724	0.480163
$403$	−5.22170	−0.260111
$404$	−19.3509	−0.962745
$405$	−0.307703	−0.0152899
$406$	0	0
$407$	10.6218	0.526506
$408$	2.64173	0.130785
$409$	17.5729	0.868926	0.434463	−	0.900690i	$-0.356938\pi$
$409$	17.5729	0.868926	0.434463	+	0.900690i	$0.356938\pi$
$410$	−0.0592379	−0.00292555
$411$	8.46972	0.417781
$412$	18.0642	0.889959
$413$	0	0
$414$	6.47884	0.318417
$415$	−0.320457	−0.0157306
$416$	−1.00000	−0.0490290
$417$	−12.4015	−0.607304
$418$	−9.88543	−0.483512
$419$	−8.69651	−0.424852	−0.212426	−	0.977177i	$-0.568137\pi$
$419$	−8.69651	−0.424852	−0.212426	+	0.977177i	$0.568137\pi$
$420$	0	0
$421$	9.83648	0.479401	0.239700	−	0.970847i	$-0.422951\pi$
$421$	9.83648	0.479401	0.239700	+	0.970847i	$0.422951\pi$
$422$	13.8763	0.675489
$423$	−9.74825	−0.473976
$424$	−7.74825	−0.376288
$425$	−12.9586	−0.628582
$426$	−1.56484	−0.0758169
$427$	0	0
$428$	16.0577	0.776180
$429$	−1.25714	−0.0606953
$430$	3.22437	0.155493
$431$	−30.0518	−1.44754	−0.723772	−	0.690040i	$-0.757593\pi$
$431$	−30.0518	−1.44754	−0.723772	+	0.690040i	$0.757593\pi$
$432$	−1.00000	−0.0481125
$433$	6.83816	0.328621	0.164311	−	0.986409i	$-0.447460\pi$
$433$	6.83816	0.328621	0.164311	+	0.986409i	$0.447460\pi$
$434$	0	0
$435$	2.73375	0.131073
$436$	10.8053	0.517478
$437$	−50.9459	−2.43707
$438$	−6.83487	−0.326583
$439$	−1.19387	−0.0569803	−0.0284902	−	0.999594i	$-0.509070\pi$
$439$	−1.19387	−0.0569803	−0.0284902	+	0.999594i	$0.509070\pi$
$440$	0.386825	0.0184412
$441$	0	0
$442$	−2.64173	−0.125655
$443$	2.07404	0.0985407	0.0492703	−	0.998785i	$-0.484310\pi$
$443$	2.07404	0.0985407	0.0492703	+	0.998785i	$0.484310\pi$
$444$	−8.44922	−0.400982
$445$	0.382499	0.0181322
$446$	−13.2589	−0.627829
$447$	−0.213022	−0.0100756
$448$	0	0
$449$	31.2736	1.47589	0.737947	−	0.674859i	$-0.235796\pi$
$449$	31.2736	1.47589	0.737947	+	0.674859i	$0.235796\pi$
$450$	4.90532	0.231239
$451$	−0.242020	−0.0113963
$452$	14.8367	0.697858
$453$	−12.5116	−0.587847
$454$	18.4459	0.865710
$455$	0	0
$456$	7.86343	0.368239
$457$	1.30635	0.0611084	0.0305542	−	0.999533i	$-0.490273\pi$
$457$	1.30635	0.0611084	0.0305542	+	0.999533i	$0.490273\pi$
$458$	−12.5471	−0.586285
$459$	−2.64173	−0.123306
$460$	1.99356	0.0929500
$461$	−12.4723	−0.580891	−0.290445	−	0.956892i	$-0.593803\pi$
$461$	−12.4723	−0.580891	−0.290445	+	0.956892i	$0.593803\pi$
$462$	0	0
$463$	−31.7415	−1.47515	−0.737577	−	0.675263i	$-0.764030\pi$
$463$	−31.7415	−1.47515	−0.737577	+	0.675263i	$0.764030\pi$
$464$	8.88438	0.412447
$465$	−1.60673	−0.0745103
$466$	−9.52761	−0.441358
$467$	31.6006	1.46230	0.731152	−	0.682215i	$-0.238983\pi$
$467$	31.6006	1.46230	0.731152	+	0.682215i	$0.238983\pi$
$468$	1.00000	0.0462250
$469$	0	0
$470$	−2.99956	−0.138359
$471$	−2.21569	−0.102094
$472$	10.0920	0.464522
$473$	13.1734	0.605712
$474$	−16.0082	−0.735283
$475$	−38.5726	−1.76983
$476$	0	0
$477$	7.74825	0.354768
$478$	−17.6651	−0.807982
$479$	19.1429	0.874660	0.437330	−	0.899301i	$-0.355924\pi$
$479$	19.1429	0.874660	0.437330	+	0.899301i	$0.355924\pi$
$480$	−0.307703	−0.0140446
$481$	8.44922	0.385251
$482$	−0.986672	−0.0449417
$483$	0	0
$484$	−9.41960	−0.428164
$485$	−4.85209	−0.220322
$486$	1.00000	0.0453609
$487$	−18.7563	−0.849929	−0.424965	−	0.905210i	$-0.639713\pi$
$487$	−18.7563	−0.849929	−0.424965	+	0.905210i	$0.639713\pi$
$488$	9.38639	0.424902
$489$	−16.4337	−0.743156
$490$	0	0
$491$	−21.2050	−0.956967	−0.478483	−	0.878097i	$-0.658813\pi$
$491$	−21.2050	−0.956967	−0.478483	+	0.878097i	$0.658813\pi$
$492$	0.192517	0.00867932
$493$	23.4702	1.05704
$494$	−7.86343	−0.353793
$495$	−0.386825	−0.0173865
$496$	−5.22170	−0.234461
$497$	0	0
$498$	1.04145	0.0466684
$499$	−38.9864	−1.74527	−0.872636	−	0.488371i	$-0.837591\pi$
$499$	−38.9864	−1.74527	−0.872636	+	0.488371i	$0.837591\pi$
$500$	3.04789	0.136306
$501$	12.9544	0.578759
$502$	−14.7117	−0.656617
$503$	−36.1321	−1.61105	−0.805526	−	0.592560i	$-0.798117\pi$
$503$	−36.1321	−1.61105	−0.805526	+	0.592560i	$0.798117\pi$
$504$	0	0
$505$	5.95434	0.264965
$506$	8.14481	0.362081
$507$	−1.00000	−0.0444116
$508$	−1.53584	−0.0681421
$509$	11.2053	0.496664	0.248332	−	0.968675i	$-0.420118\pi$
$509$	11.2053	0.496664	0.248332	+	0.968675i	$0.420118\pi$
$510$	−0.812869	−0.0359944
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	−7.86343	−0.347179
$514$	−19.8280	−0.874575
$515$	−5.55840	−0.244932
$516$	−10.4788	−0.461305
$517$	−12.2549	−0.538970
$518$	0	0
$519$	2.74930	0.120681
$520$	0.307703	0.0134937
$521$	24.2741	1.06347	0.531733	−	0.846912i	$-0.321541\pi$
$521$	24.2741	1.06347	0.531733	+	0.846912i	$0.321541\pi$
$522$	−8.88438	−0.388859
$523$	10.9494	0.478785	0.239393	−	0.970923i	$-0.423052\pi$
$523$	10.9494	0.478785	0.239393	+	0.970923i	$0.423052\pi$
$524$	−2.16934	−0.0947682
$525$	0	0
$526$	−20.4260	−0.890618
$527$	−13.7943	−0.600891
$528$	−1.25714	−0.0547100
$529$	18.9753	0.825014
$530$	2.38416	0.103561
$531$	−10.0920	−0.437956
$532$	0	0
$533$	−0.192517	−0.00833882
$534$	−1.24308	−0.0537933
$535$	−4.94101	−0.213618
$536$	−9.62724	−0.415833
$537$	3.10007	0.133778
$538$	−20.6344	−0.889613
$539$	0	0
$540$	0.307703	0.0132414
$541$	−8.63858	−0.371402	−0.185701	−	0.982606i	$-0.559456\pi$
$541$	−8.63858	−0.371402	−0.185701	+	0.982606i	$0.559456\pi$
$542$	−2.28137	−0.0979934
$543$	4.94032	0.212010
$544$	−2.64173	−0.113263
$545$	−3.32481	−0.142419
$546$	0	0
$547$	3.32536	0.142182	0.0710910	−	0.997470i	$-0.477352\pi$
$547$	3.32536	0.142182	0.0710910	+	0.997470i	$0.477352\pi$
$548$	−8.46972	−0.361809
$549$	−9.38639	−0.400601
$550$	6.16667	0.262948
$551$	69.8617	2.97621
$552$	−6.47884	−0.275758
$553$	0	0
$554$	−23.6232	−1.00365
$555$	2.59985	0.110357
$556$	12.4015	0.525941
$557$	−26.5544	−1.12515	−0.562573	−	0.826747i	$-0.690189\pi$
$557$	−26.5544	−1.12515	−0.562573	+	0.826747i	$0.690189\pi$
$558$	5.22170	0.221052
$559$	10.4788	0.443208
$560$	0	0
$561$	−3.32103	−0.140214
$562$	3.92911	0.165740
$563$	30.5664	1.28822	0.644110	−	0.764933i	$-0.277228\pi$
$563$	30.5664	1.28822	0.644110	+	0.764933i	$0.277228\pi$
$564$	9.74825	0.410475
$565$	−4.56528	−0.192063
$566$	−20.1002	−0.844876
$567$	0	0
$568$	1.56484	0.0656593
$569$	40.6092	1.70243	0.851213	−	0.524821i	$-0.175868\pi$
$569$	40.6092	1.70243	0.851213	+	0.524821i	$0.175868\pi$
$570$	−2.41960	−0.101346
$571$	12.3552	0.517047	0.258524	−	0.966005i	$-0.416764\pi$
$571$	12.3552	0.517047	0.258524	+	0.966005i	$0.416764\pi$
$572$	1.25714	0.0525637
$573$	24.4187	1.02011
$574$	0	0
$575$	31.7808	1.32535
$576$	1.00000	0.0416667
$577$	13.2006	0.549547	0.274773	−	0.961509i	$-0.411397\pi$
$577$	13.2006	0.549547	0.274773	+	0.961509i	$0.411397\pi$
$578$	10.0212	0.416829
$579$	8.43428	0.350517
$580$	−2.73375	−0.113513
$581$	0	0
$582$	15.7688	0.653636
$583$	9.74063	0.403416
$584$	6.83487	0.282829
$585$	−0.307703	−0.0127219
$586$	2.75842	0.113949
$587$	−29.6452	−1.22359	−0.611794	−	0.791017i	$-0.709552\pi$
$587$	−29.6452	−1.22359	−0.611794	+	0.791017i	$0.709552\pi$
$588$	0	0
$589$	−41.0605	−1.69187
$590$	−3.10534	−0.127845
$591$	−12.4770	−0.513237
$592$	8.44922	0.347261
$593$	−28.7560	−1.18087	−0.590435	−	0.807085i	$-0.701044\pi$
$593$	−28.7560	−1.18087	−0.590435	+	0.807085i	$0.701044\pi$
$594$	1.25714	0.0515811
$595$	0	0
$596$	0.213022	0.00872571
$597$	5.44878	0.223004
$598$	6.47884	0.264939
$599$	1.11790	0.0456762	0.0228381	−	0.999739i	$-0.492730\pi$
$599$	1.11790	0.0456762	0.0228381	+	0.999739i	$0.492730\pi$
$600$	−4.90532	−0.200259
$601$	18.4091	0.750924	0.375462	−	0.926838i	$-0.377484\pi$
$601$	18.4091	0.750924	0.375462	+	0.926838i	$0.377484\pi$
$602$	0	0
$603$	9.62724	0.392051
$604$	12.5116	0.509090
$605$	2.89844	0.117838
$606$	−19.3509	−0.786078
$607$	−2.75978	−0.112016	−0.0560080	−	0.998430i	$-0.517837\pi$
$607$	−2.75978	−0.112016	−0.0560080	+	0.998430i	$0.517837\pi$
$608$	−7.86343	−0.318904
$609$	0	0
$610$	−2.88822	−0.116940
$611$	−9.74825	−0.394372
$612$	2.64173	0.106786
$613$	37.3489	1.50851	0.754254	−	0.656583i	$-0.227999\pi$
$613$	37.3489	1.50851	0.754254	+	0.656583i	$0.227999\pi$
$614$	5.64397	0.227772
$615$	−0.0592379	−0.00238870
$616$	0	0
$617$	32.4033	1.30451	0.652254	−	0.758001i	$-0.273824\pi$
$617$	32.4033	1.30451	0.652254	+	0.758001i	$0.273824\pi$
$618$	18.0642	0.726648
$619$	−31.5448	−1.26789	−0.633947	−	0.773377i	$-0.718566\pi$
$619$	−31.5448	−1.26789	−0.633947	+	0.773377i	$0.718566\pi$
$620$	1.60673	0.0645278
$621$	6.47884	0.259987
$622$	−24.9822	−1.00170
$623$	0	0
$624$	−1.00000	−0.0400320
$625$	23.5888	0.943550
$626$	−27.5680	−1.10184
$627$	−9.88543	−0.394786
$628$	2.21569	0.0884157
$629$	22.3206	0.889980
$630$	0	0
$631$	−21.7900	−0.867447	−0.433723	−	0.901046i	$-0.642800\pi$
$631$	−21.7900	−0.867447	−0.433723	+	0.901046i	$0.642800\pi$
$632$	16.0082	0.636773
$633$	13.8763	0.551534
$634$	−0.274794	−0.0109135
$635$	0.472583	0.0187539
$636$	−7.74825	−0.307238
$637$	0	0
$638$	−11.1689	−0.442181
$639$	−1.56484	−0.0619042
$640$	0.307703	0.0121630
$641$	32.3750	1.27874	0.639369	−	0.768900i	$-0.279196\pi$
$641$	32.3750	1.27874	0.639369	+	0.768900i	$0.279196\pi$
$642$	16.0577	0.633748
$643$	−47.4249	−1.87025	−0.935127	−	0.354313i	$-0.884715\pi$
$643$	−47.4249	−1.87025	−0.935127	+	0.354313i	$0.884715\pi$
$644$	0	0
$645$	3.22437	0.126959
$646$	−20.7731	−0.817307
$647$	33.7131	1.32540	0.662699	−	0.748886i	$-0.269411\pi$
$647$	33.7131	1.32540	0.662699	+	0.748886i	$0.269411\pi$
$648$	−1.00000	−0.0392837
$649$	−12.6871	−0.498011
$650$	4.90532	0.192402
$651$	0	0
$652$	16.4337	0.643592
$653$	−39.3341	−1.53926	−0.769631	−	0.638489i	$-0.779560\pi$
$653$	−39.3341	−1.53926	−0.769631	+	0.638489i	$0.779560\pi$
$654$	10.8053	0.422519
$655$	0.667512	0.0260819
$656$	−0.192517	−0.00751651
$657$	−6.83487	−0.266654
$658$	0	0
$659$	−8.61944	−0.335766	−0.167883	−	0.985807i	$-0.553693\pi$
$659$	−8.61944	−0.335766	−0.167883	+	0.985807i	$0.553693\pi$
$660$	0.386825	0.0150572
$661$	−37.4537	−1.45678	−0.728391	−	0.685162i	$-0.759731\pi$
$661$	−37.4537	−1.45678	−0.728391	+	0.685162i	$0.759731\pi$
$662$	−21.4660	−0.834300
$663$	−2.64173	−0.102596
$664$	−1.04145	−0.0404161
$665$	0	0
$666$	−8.44922	−0.327401
$667$	−57.5604	−2.22875
$668$	−12.9544	−0.501220
$669$	−13.2589	−0.512620
$670$	2.96233	0.114445
$671$	−11.8000	−0.455534
$672$	0	0
$673$	−44.4792	−1.71455	−0.857274	−	0.514861i	$-0.827843\pi$
$673$	−44.4792	−1.71455	−0.857274	+	0.514861i	$0.827843\pi$
$674$	−23.0350	−0.887275
$675$	4.90532	0.188806
$676$	1.00000	0.0384615
$677$	−21.6604	−0.832478	−0.416239	−	0.909255i	$-0.636652\pi$
$677$	−21.6604	−0.832478	−0.416239	+	0.909255i	$0.636652\pi$
$678$	14.8367	0.569799
$679$	0	0
$680$	0.812869	0.0311721
$681$	18.4459	0.706850
$682$	6.56440	0.251364
$683$	3.19702	0.122331	0.0611654	−	0.998128i	$-0.480518\pi$
$683$	3.19702	0.122331	0.0611654	+	0.998128i	$0.480518\pi$
$684$	7.86343	0.300666
$685$	2.60616	0.0995761
$686$	0	0
$687$	−12.5471	−0.478700
$688$	10.4788	0.399502
$689$	7.74825	0.295185
$690$	1.99356	0.0758934
$691$	6.51428	0.247815	0.123907	−	0.992294i	$-0.460457\pi$
$691$	6.51428	0.247815	0.123907	+	0.992294i	$0.460457\pi$
$692$	−2.74930	−0.104513
$693$	0	0
$694$	1.68647	0.0640176
$695$	−3.81598	−0.144748
$696$	8.88438	0.336761
$697$	−0.508578	−0.0192638
$698$	−22.2877	−0.843601
$699$	−9.52761	−0.360367
$700$	0	0
$701$	24.4705	0.924237	0.462119	−	0.886818i	$-0.347089\pi$
$701$	24.4705	0.924237	0.462119	+	0.886818i	$0.347089\pi$
$702$	1.00000	0.0377426
$703$	66.4398	2.50583
$704$	1.25714	0.0473802
$705$	−2.99956	−0.112970
$706$	13.0664	0.491761
$707$	0	0
$708$	10.0920	0.379281
$709$	−24.9327	−0.936367	−0.468184	−	0.883631i	$-0.655091\pi$
$709$	−24.9327	−0.936367	−0.468184	+	0.883631i	$0.655091\pi$
$710$	−0.481506	−0.0180706
$711$	−16.0082	−0.600356
$712$	1.24308	0.0465864
$713$	33.8305	1.26696
$714$	0	0
$715$	−0.386825	−0.0144664
$716$	−3.10007	−0.115855
$717$	−17.6651	−0.659715
$718$	−3.70320	−0.138202
$719$	24.2084	0.902820	0.451410	−	0.892317i	$-0.350921\pi$
$719$	24.2084	0.902820	0.451410	+	0.892317i	$0.350921\pi$
$720$	−0.307703	−0.0114674
$721$	0	0
$722$	−42.8336	−1.59410
$723$	−0.986672	−0.0366947
$724$	−4.94032	−0.183606
$725$	−43.5807	−1.61855
$726$	−9.41960	−0.349594
$727$	−30.5312	−1.13234	−0.566170	−	0.824288i	$-0.691576\pi$
$727$	−30.5312	−1.13234	−0.566170	+	0.824288i	$0.691576\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−2.10311	−0.0778395
$731$	27.6823	1.02387
$732$	9.38639	0.346931
$733$	15.1098	0.558093	0.279047	−	0.960278i	$-0.409982\pi$
$733$	15.1098	0.558093	0.279047	+	0.960278i	$0.409982\pi$
$734$	4.32176	0.159519
$735$	0	0
$736$	6.47884	0.238813
$737$	12.1028	0.445812
$738$	0.192517	0.00708664
$739$	12.9782	0.477410	0.238705	−	0.971092i	$-0.423277\pi$
$739$	12.9782	0.477410	0.238705	+	0.971092i	$0.423277\pi$
$740$	−2.59985	−0.0955723
$741$	−7.86343	−0.288870
$742$	0	0
$743$	−5.44171	−0.199637	−0.0998186	−	0.995006i	$-0.531826\pi$
$743$	−5.44171	−0.199637	−0.0998186	+	0.995006i	$0.531826\pi$
$744$	−5.22170	−0.191437
$745$	−0.0655474	−0.00240147
$746$	−23.6349	−0.865333
$747$	1.04145	0.0381046
$748$	3.32103	0.121429
$749$	0	0
$750$	3.04789	0.111293
$751$	32.3086	1.17896	0.589479	−	0.807784i	$-0.299333\pi$
$751$	32.3086	1.17896	0.589479	+	0.807784i	$0.299333\pi$
$752$	−9.74825	−0.355482
$753$	−14.7117	−0.536126
$754$	−8.88438	−0.323550
$755$	−3.84986	−0.140111
$756$	0	0
$757$	−15.8745	−0.576970	−0.288485	−	0.957484i	$-0.593151\pi$
$757$	−15.8745	−0.576970	−0.288485	+	0.957484i	$0.593151\pi$
$758$	−3.21196	−0.116664
$759$	8.14481	0.295638
$760$	2.41960	0.0877681
$761$	−41.2621	−1.49575	−0.747875	−	0.663840i	$-0.768926\pi$
$761$	−41.2621	−1.49575	−0.747875	+	0.663840i	$0.768926\pi$
$762$	−1.53584	−0.0556378
$763$	0	0
$764$	−24.4187	−0.883438
$765$	−0.812869	−0.0293893
$766$	37.3559	1.34972
$767$	−10.0920	−0.364401
$768$	−1.00000	−0.0360844
$769$	−48.1001	−1.73453	−0.867267	−	0.497843i	$-0.834126\pi$
$769$	−48.1001	−1.73453	−0.867267	+	0.497843i	$0.834126\pi$
$770$	0	0
$771$	−19.8280	−0.714088
$772$	−8.43428	−0.303556
$773$	9.45474	0.340063	0.170032	−	0.985439i	$-0.445613\pi$
$773$	9.45474	0.340063	0.170032	+	0.985439i	$0.445613\pi$
$774$	−10.4788	−0.376654
$775$	25.6141	0.920085
$776$	−15.7688	−0.566065
$777$	0	0
$778$	10.0334	0.359715
$779$	−1.51384	−0.0542390
$780$	0.307703	0.0110175
$781$	−1.96723	−0.0703929
$782$	17.1154	0.612044
$783$	−8.88438	−0.317502
$784$	0	0
$785$	−0.681774	−0.0243336
$786$	−2.16934	−0.0773779
$787$	−39.2061	−1.39755	−0.698774	−	0.715343i	$-0.746271\pi$
$787$	−39.2061	−1.39755	−0.698774	+	0.715343i	$0.746271\pi$
$788$	12.4770	0.444476
$789$	−20.4260	−0.727186
$790$	−4.92578	−0.175251
$791$	0	0
$792$	−1.25714	−0.0446705
$793$	−9.38639	−0.333320
$794$	−24.3552	−0.864332
$795$	2.38416	0.0845573
$796$	−5.44878	−0.193127
$797$	−18.1246	−0.642006	−0.321003	−	0.947078i	$-0.604020\pi$
$797$	−18.1246	−0.642006	−0.321003	+	0.947078i	$0.604020\pi$
$798$	0	0
$799$	−25.7523	−0.911050
$800$	4.90532	0.173429
$801$	−1.24308	−0.0439220
$802$	−22.4434	−0.792504
$803$	−8.59239	−0.303219
$804$	−9.62724	−0.339526
$805$	0	0
$806$	5.22170	0.183926
$807$	−20.6344	−0.726366
$808$	19.3509	0.680764
$809$	−47.5457	−1.67162	−0.835810	−	0.549019i	$-0.815001\pi$
$809$	−47.5457	−1.67162	−0.835810	+	0.549019i	$0.815001\pi$
$810$	0.307703	0.0108116
$811$	5.91697	0.207773	0.103886	−	0.994589i	$-0.466872\pi$
$811$	5.91697	0.207773	0.103886	+	0.994589i	$0.466872\pi$
$812$	0	0
$813$	−2.28137	−0.0800113
$814$	−10.6218	−0.372296
$815$	−5.05668	−0.177128
$816$	−2.64173	−0.0924792
$817$	82.3996	2.88280
$818$	−17.5729	−0.614424
$819$	0	0
$820$	0.0592379	0.00206868
$821$	23.2754	0.812317	0.406159	−	0.913803i	$-0.366868\pi$
$821$	23.2754	0.812317	0.406159	+	0.913803i	$0.366868\pi$
$822$	−8.46972	−0.295416
$823$	−33.2570	−1.15927	−0.579633	−	0.814878i	$-0.696804\pi$
$823$	−33.2570	−1.15927	−0.579633	+	0.814878i	$0.696804\pi$
$824$	−18.0642	−0.629296
$825$	6.16667	0.214696
$826$	0	0
$827$	−28.6956	−0.997845	−0.498922	−	0.866647i	$-0.666271\pi$
$827$	−28.6956	−0.997845	−0.498922	+	0.866647i	$0.666271\pi$
$828$	−6.47884	−0.225155
$829$	−33.3026	−1.15665	−0.578324	−	0.815807i	$-0.696293\pi$
$829$	−33.3026	−1.15665	−0.578324	+	0.815807i	$0.696293\pi$
$830$	0.320457	0.0111232
$831$	−23.6232	−0.819480
$832$	1.00000	0.0346688
$833$	0	0
$834$	12.4015	0.429429
$835$	3.98610	0.137945
$836$	9.88543	0.341895
$837$	5.22170	0.180488
$838$	8.69651	0.300416
$839$	−33.1610	−1.14484	−0.572422	−	0.819959i	$-0.693996\pi$
$839$	−33.1610	−1.14484	−0.572422	+	0.819959i	$0.693996\pi$
$840$	0	0
$841$	49.9321	1.72180
$842$	−9.83648	−0.338988
$843$	3.92911	0.135326
$844$	−13.8763	−0.477643
$845$	−0.307703	−0.0105853
$846$	9.74825	0.335152
$847$	0	0
$848$	7.74825	0.266076
$849$	−20.1002	−0.689839
$850$	12.9586	0.444475
$851$	−54.7411	−1.87650
$852$	1.56484	0.0536106
$853$	17.3008	0.592369	0.296184	−	0.955131i	$-0.404286\pi$
$853$	17.3008	0.592369	0.296184	+	0.955131i	$0.404286\pi$
$854$	0	0
$855$	−2.41960	−0.0827485
$856$	−16.0577	−0.548842
$857$	−45.6377	−1.55896	−0.779478	−	0.626430i	$-0.784516\pi$
$857$	−45.6377	−1.55896	−0.779478	+	0.626430i	$0.784516\pi$
$858$	1.25714	0.0429181
$859$	−10.2247	−0.348861	−0.174431	−	0.984669i	$-0.555808\pi$
$859$	−10.2247	−0.348861	−0.174431	+	0.984669i	$0.555808\pi$
$860$	−3.22437	−0.109950
$861$	0	0
$862$	30.0518	1.02357
$863$	−49.9830	−1.70144	−0.850721	−	0.525618i	$-0.823834\pi$
$863$	−49.9830	−1.70144	−0.850721	+	0.525618i	$0.823834\pi$
$864$	1.00000	0.0340207
$865$	0.845968	0.0287638
$866$	−6.83816	−0.232370
$867$	10.0212	0.340339
$868$	0	0
$869$	−20.1246	−0.682680
$870$	−2.73375	−0.0926827
$871$	9.62724	0.326206
$872$	−10.8053	−0.365912
$873$	15.7688	0.533691
$874$	50.9459	1.72327
$875$	0	0
$876$	6.83487	0.230929
$877$	−13.2274	−0.446657	−0.223329	−	0.974743i	$-0.571692\pi$
$877$	−13.2274	−0.446657	−0.223329	+	0.974743i	$0.571692\pi$
$878$	1.19387	0.0402912
$879$	2.75842	0.0930391
$880$	−0.386825	−0.0130399
$881$	4.45983	0.150255	0.0751277	−	0.997174i	$-0.476064\pi$
$881$	4.45983	0.150255	0.0751277	+	0.997174i	$0.476064\pi$
$882$	0	0
$883$	−9.38980	−0.315992	−0.157996	−	0.987440i	$-0.550503\pi$
$883$	−9.38980	−0.315992	−0.157996	+	0.987440i	$0.550503\pi$
$884$	2.64173	0.0888512
$885$	−3.10534	−0.104385
$886$	−2.07404	−0.0696788
$887$	7.68051	0.257886	0.128943	−	0.991652i	$-0.458842\pi$
$887$	7.68051	0.257886	0.128943	+	0.991652i	$0.458842\pi$
$888$	8.44922	0.283537
$889$	0	0
$890$	−0.382499	−0.0128214
$891$	1.25714	0.0421158
$892$	13.2589	0.443942
$893$	−76.6547	−2.56515
$894$	0.213022	0.00712451
$895$	0.953899	0.0318853
$896$	0	0
$897$	6.47884	0.216322
$898$	−31.2736	−1.04361
$899$	−46.3915	−1.54724
$900$	−4.90532	−0.163511
$901$	20.4688	0.681915
$902$	0.242020	0.00805840
$903$	0	0
$904$	−14.8367	−0.493460
$905$	1.52015	0.0505315
$906$	12.5116	0.415670
$907$	52.6792	1.74918	0.874592	−	0.484860i	$-0.161129\pi$
$907$	52.6792	1.74918	0.874592	+	0.484860i	$0.161129\pi$
$908$	−18.4459	−0.612150
$909$	−19.3509	−0.641830
$910$	0	0
$911$	−4.37413	−0.144921	−0.0724607	−	0.997371i	$-0.523085\pi$
$911$	−4.37413	−0.144921	−0.0724607	+	0.997371i	$0.523085\pi$
$912$	−7.86343	−0.260384
$913$	1.30925	0.0433298
$914$	−1.30635	−0.0432102
$915$	−2.88822	−0.0954815
$916$	12.5471	0.414566
$917$	0	0
$918$	2.64173	0.0871902
$919$	−22.0592	−0.727667	−0.363834	−	0.931464i	$-0.618532\pi$
$919$	−22.0592	−0.727667	−0.363834	+	0.931464i	$0.618532\pi$
$920$	−1.99356	−0.0657256
$921$	5.64397	0.185975
$922$	12.4723	0.410752
$923$	−1.56484	−0.0515074
$924$	0	0
$925$	−41.4461	−1.36274
$926$	31.7415	1.04309
$927$	18.0642	0.593306
$928$	−8.88438	−0.291644
$929$	−7.47152	−0.245133	−0.122566	−	0.992460i	$-0.539112\pi$
$929$	−7.47152	−0.245133	−0.122566	+	0.992460i	$0.539112\pi$
$930$	1.60673	0.0526868
$931$	0	0
$932$	9.52761	0.312087
$933$	−24.9822	−0.817881
$934$	−31.6006	−1.03400
$935$	−1.02189	−0.0334194
$936$	−1.00000	−0.0326860
$937$	−10.5523	−0.344729	−0.172365	−	0.985033i	$-0.555141\pi$
$937$	−10.5523	−0.344729	−0.172365	+	0.985033i	$0.555141\pi$
$938$	0	0
$939$	−27.5680	−0.899648
$940$	2.99956	0.0978349
$941$	40.0419	1.30533	0.652664	−	0.757647i	$-0.273651\pi$
$941$	40.0419	1.30533	0.652664	+	0.757647i	$0.273651\pi$
$942$	2.21569	0.0721911
$943$	1.24728	0.0406172
$944$	−10.0920	−0.328467
$945$	0	0
$946$	−13.1734	−0.428303
$947$	26.8315	0.871905	0.435953	−	0.899970i	$-0.356412\pi$
$947$	26.8315	0.871905	0.435953	+	0.899970i	$0.356412\pi$
$948$	16.0082	0.519923
$949$	−6.83487	−0.221869
$950$	38.5726	1.25146
$951$	−0.274794	−0.00891081
$952$	0	0
$953$	14.9987	0.485855	0.242928	−	0.970044i	$-0.421892\pi$
$953$	14.9987	0.485855	0.242928	+	0.970044i	$0.421892\pi$
$954$	−7.74825	−0.250859
$955$	7.51371	0.243138
$956$	17.6651	0.571330
$957$	−11.1689	−0.361039
$958$	−19.1429	−0.618478
$959$	0	0
$960$	0.307703	0.00993106
$961$	−3.73388	−0.120448
$962$	−8.44922	−0.272414
$963$	16.0577	0.517453
$964$	0.986672	0.0317786
$965$	2.59525	0.0835441
$966$	0	0
$967$	24.4407	0.785961	0.392980	−	0.919547i	$-0.371444\pi$
$967$	24.4407	0.785961	0.392980	+	0.919547i	$0.371444\pi$
$968$	9.41960	0.302757
$969$	−20.7731	−0.667328
$970$	4.85209	0.155791
$971$	6.27014	0.201218	0.100609	−	0.994926i	$-0.467921\pi$
$971$	6.27014	0.201218	0.100609	+	0.994926i	$0.467921\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	18.7563	0.600991
$975$	4.90532	0.157096
$976$	−9.38639	−0.300451
$977$	53.5154	1.71211	0.856054	−	0.516886i	$-0.172909\pi$
$977$	53.5154	1.71211	0.856054	+	0.516886i	$0.172909\pi$
$978$	16.4337	0.525490
$979$	−1.56272	−0.0499449
$980$	0	0
$981$	10.8053	0.344985
$982$	21.2050	0.676678
$983$	41.7110	1.33038	0.665188	−	0.746676i	$-0.268351\pi$
$983$	41.7110	1.33038	0.665188	+	0.746676i	$0.268351\pi$
$984$	−0.192517	−0.00613721
$985$	−3.83922	−0.122328
$986$	−23.4702	−0.747442
$987$	0	0
$988$	7.86343	0.250169
$989$	−67.8907	−2.15880
$990$	0.386825	0.0122941
$991$	−4.84665	−0.153959	−0.0769795	−	0.997033i	$-0.524528\pi$
$991$	−4.84665	−0.153959	−0.0769795	+	0.997033i	$0.524528\pi$
$992$	5.22170	0.165789
$993$	−21.4660	−0.681203
$994$	0	0
$995$	1.67660	0.0531519
$996$	−1.04145	−0.0329996
$997$	10.0643	0.318740	0.159370	−	0.987219i	$-0.449054\pi$
$997$	10.0643	0.318740	0.159370	+	0.987219i	$0.449054\pi$
$998$	38.9864	1.23409
$999$	−8.44922	−0.267321

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3822.2.a.bx.1.3	✓	4
7.6	odd	2		3822.2.a.by.1.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3822.2.a.bx.1.3	✓	4	1.1	even	1	trivial
3822.2.a.by.1.2	yes	4	7.6	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:	\( N \)	\(=\)	\( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3822.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.307703 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.307703 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +0.307703 q^{10} +1.25714 q^{11} -1.00000 q^{12} +1.00000 q^{13} +0.307703 q^{15} +1.00000 q^{16} +2.64173 q^{17} -1.00000 q^{18} +7.86343 q^{19} -0.307703 q^{20} -1.25714 q^{22} -6.47884 q^{23} +1.00000 q^{24} -4.90532 q^{25} -1.00000 q^{26} -1.00000 q^{27} +8.88438 q^{29} -0.307703 q^{30} -5.22170 q^{31} -1.00000 q^{32} -1.25714 q^{33} -2.64173 q^{34} +1.00000 q^{36} +8.44922 q^{37} -7.86343 q^{38} -1.00000 q^{39} +0.307703 q^{40} -0.192517 q^{41} +10.4788 q^{43} +1.25714 q^{44} -0.307703 q^{45} +6.47884 q^{46} -9.74825 q^{47} -1.00000 q^{48} +4.90532 q^{50} -2.64173 q^{51} +1.00000 q^{52} +7.74825 q^{53} +1.00000 q^{54} -0.386825 q^{55} -7.86343 q^{57} -8.88438 q^{58} -10.0920 q^{59} +0.307703 q^{60} -9.38639 q^{61} +5.22170 q^{62} +1.00000 q^{64} -0.307703 q^{65} +1.25714 q^{66} +9.62724 q^{67} +2.64173 q^{68} +6.47884 q^{69} -1.56484 q^{71} -1.00000 q^{72} -6.83487 q^{73} -8.44922 q^{74} +4.90532 q^{75} +7.86343 q^{76} +1.00000 q^{78} -16.0082 q^{79} -0.307703 q^{80} +1.00000 q^{81} +0.192517 q^{82} +1.04145 q^{83} -0.812869 q^{85} -10.4788 q^{86} -8.88438 q^{87} -1.25714 q^{88} -1.24308 q^{89} +0.307703 q^{90} -6.47884 q^{92} +5.22170 q^{93} +9.74825 q^{94} -2.41960 q^{95} +1.00000 q^{96} +15.7688 q^{97} +1.25714 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} - 6 q^{5} + 4 q^{6} - 4 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 - 6 * q^5 + 4 * q^6 - 4 * q^8 + 4 * q^9	\( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} - 6 q^{5} + 4 q^{6} - 4 q^{8} + 4 q^{9} + 6 q^{10} + 2 q^{11} - 4 q^{12} + 4 q^{13} + 6 q^{15} + 4 q^{16} - 2 q^{17} - 4 q^{18} - 2 q^{19} - 6 q^{20} - 2 q^{22} - 2 q^{23} + 4 q^{24} + 6 q^{25} - 4 q^{26} - 4 q^{27} + 6 q^{29} - 6 q^{30} - 4 q^{32} - 2 q^{33} + 2 q^{34} + 4 q^{36} + 6 q^{37} + 2 q^{38} - 4 q^{39} + 6 q^{40} - 16 q^{41} + 18 q^{43} + 2 q^{44} - 6 q^{45} + 2 q^{46} - 16 q^{47} - 4 q^{48} - 6 q^{50} + 2 q^{51} + 4 q^{52} + 8 q^{53} + 4 q^{54} - 2 q^{55} + 2 q^{57} - 6 q^{58} - 16 q^{59} + 6 q^{60} + 2 q^{61} + 4 q^{64} - 6 q^{65} + 2 q^{66} + 12 q^{67} - 2 q^{68} + 2 q^{69} - 8 q^{71} - 4 q^{72} - 6 q^{73} - 6 q^{74} - 6 q^{75} - 2 q^{76} + 4 q^{78} - 24 q^{79} - 6 q^{80} + 4 q^{81} + 16 q^{82} - 28 q^{83} + 38 q^{85} - 18 q^{86} - 6 q^{87} - 2 q^{88} - 28 q^{89} + 6 q^{90} - 2 q^{92} + 16 q^{94} + 22 q^{95} + 4 q^{96} + 4 q^{97} + 2 q^{99}+O(q^{100}) \) 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 - 6 * q^5 + 4 * q^6 - 4 * q^8 + 4 * q^9 + 6 * q^10 + 2 * q^11 - 4 * q^12 + 4 * q^13 + 6 * q^15 + 4 * q^16 - 2 * q^17 - 4 * q^18 - 2 * q^19 - 6 * q^20 - 2 * q^22 - 2 * q^23 + 4 * q^24 + 6 * q^25 - 4 * q^26 - 4 * q^27 + 6 * q^29 - 6 * q^30 - 4 * q^32 - 2 * q^33 + 2 * q^34 + 4 * q^36 + 6 * q^37 + 2 * q^38 - 4 * q^39 + 6 * q^40 - 16 * q^41 + 18 * q^43 + 2 * q^44 - 6 * q^45 + 2 * q^46 - 16 * q^47 - 4 * q^48 - 6 * q^50 + 2 * q^51 + 4 * q^52 + 8 * q^53 + 4 * q^54 - 2 * q^55 + 2 * q^57 - 6 * q^58 - 16 * q^59 + 6 * q^60 + 2 * q^61 + 4 * q^64 - 6 * q^65 + 2 * q^66 + 12 * q^67 - 2 * q^68 + 2 * q^69 - 8 * q^71 - 4 * q^72 - 6 * q^73 - 6 * q^74 - 6 * q^75 - 2 * q^76 + 4 * q^78 - 24 * q^79 - 6 * q^80 + 4 * q^81 + 16 * q^82 - 28 * q^83 + 38 * q^85 - 18 * q^86 - 6 * q^87 - 2 * q^88 - 28 * q^89 + 6 * q^90 - 2 * q^92 + 16 * q^94 + 22 * q^95 + 4 * q^96 + 4 * q^97 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more