LMFDB - Embedded newform 5070.2.b.v.1351.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5070,2,Mod(1351,5070)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5070, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

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

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

chi := DirichletCharacter("5070.1351");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5070.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$40.4841538248$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.153664.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 5x^{4} + 6x^{2} + 1 $ x^6 + 5x^4 + 6x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.6
Root			$1.24698i$ of defining polynomial
Character	$\chi$	$=$	5070.1351
Dual form			5070.2.b.v.1351.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} +4.24698i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} +4.24698i q^{7} -1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.91185i q^{11} +1.00000 q^{12} -4.24698 q^{14} +1.00000i q^{15} +1.00000 q^{16} +3.33513 q^{17} +1.00000i q^{18} -4.85086i q^{19} +1.00000i q^{20} -4.24698i q^{21} +1.91185 q^{22} +0.445042 q^{23} +1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} -4.24698i q^{28} -8.56465 q^{29} -1.00000 q^{30} +5.29590i q^{31} +1.00000i q^{32} +1.91185i q^{33} +3.33513i q^{34} +4.24698 q^{35} -1.00000 q^{36} -1.75302i q^{37} +4.85086 q^{38} -1.00000 q^{40} +3.24698i q^{41} +4.24698 q^{42} +1.97823 q^{43} +1.91185i q^{44} -1.00000i q^{45} +0.445042i q^{46} -10.3840i q^{47} -1.00000 q^{48} -11.0368 q^{49} -1.00000i q^{50} -3.33513 q^{51} -1.14914 q^{53} -1.00000i q^{54} -1.91185 q^{55} +4.24698 q^{56} +4.85086i q^{57} -8.56465i q^{58} -13.4601i q^{59} -1.00000i q^{60} +1.13169 q^{61} -5.29590 q^{62} +4.24698i q^{63} -1.00000 q^{64} -1.91185 q^{66} +13.5308i q^{67} -3.33513 q^{68} -0.445042 q^{69} +4.24698i q^{70} -2.14675i q^{71} -1.00000i q^{72} -5.15883i q^{73} +1.75302 q^{74} +1.00000 q^{75} +4.85086i q^{76} +8.11960 q^{77} -14.5526 q^{79} -1.00000i q^{80} +1.00000 q^{81} -3.24698 q^{82} -9.49157i q^{83} +4.24698i q^{84} -3.33513i q^{85} +1.97823i q^{86} +8.56465 q^{87} -1.91185 q^{88} +1.25667i q^{89} +1.00000 q^{90} -0.445042 q^{92} -5.29590i q^{93} +10.3840 q^{94} -4.85086 q^{95} -1.00000i q^{96} -5.01507i q^{97} -11.0368i q^{98} -1.91185i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{3} - 6 q^{4} + 6 q^{9}+O(q^{10}) $ 6 * q - 6 * q^3 - 6 * q^4 + 6 * q^9	$ 6 q - 6 q^{3} - 6 q^{4} + 6 q^{9} + 6 q^{10} + 6 q^{12} - 16 q^{14} + 6 q^{16} + 18 q^{17} + 4 q^{22} + 2 q^{23} - 6 q^{25} - 6 q^{27} - 8 q^{29} - 6 q^{30} + 16 q^{35} - 6 q^{36} + 2 q^{38} - 6 q^{40} + 16 q^{42} + 18 q^{43} - 6 q^{48} - 10 q^{49} - 18 q^{51} - 34 q^{53} - 4 q^{55} + 16 q^{56} + 2 q^{61} - 4 q^{62} - 6 q^{64} - 4 q^{66} - 18 q^{68} - 2 q^{69} + 20 q^{74} + 6 q^{75} + 6 q^{77} - 6 q^{79} + 6 q^{81} - 10 q^{82} + 8 q^{87} - 4 q^{88} + 6 q^{90} - 2 q^{92} + 42 q^{94} - 2 q^{95}+O(q^{100}) $ 6 * q - 6 * q^3 - 6 * q^4 + 6 * q^9 + 6 * q^10 + 6 * q^12 - 16 * q^14 + 6 * q^16 + 18 * q^17 + 4 * q^22 + 2 * q^23 - 6 * q^25 - 6 * q^27 - 8 * q^29 - 6 * q^30 + 16 * q^35 - 6 * q^36 + 2 * q^38 - 6 * q^40 + 16 * q^42 + 18 * q^43 - 6 * q^48 - 10 * q^49 - 18 * q^51 - 34 * q^53 - 4 * q^55 + 16 * q^56 + 2 * q^61 - 4 * q^62 - 6 * q^64 - 4 * q^66 - 18 * q^68 - 2 * q^69 + 20 * q^74 + 6 * q^75 + 6 * q^77 - 6 * q^79 + 6 * q^81 - 10 * q^82 + 8 * q^87 - 4 * q^88 + 6 * q^90 - 2 * q^92 + 42 * q^94 - 2 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5070\mathbb{Z}\right)^\times$.

$n$	$1691$	$1861$	$4057$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.00000	−0.500000
$5$	− 1.00000i	− 0.447214i
$5$	− 1.00000i	− 0.447214i
$6$	− 1.00000i	− 0.408248i
$7$	4.24698i	1.60521i	0.596513	+	0.802604i	$0.296553\pi$
$7$	4.24698i	1.60521i	−0.596513	+	0.802604i	$0.703447\pi$
$8$	− 1.00000i	− 0.353553i
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	− 1.91185i	− 0.576446i	−0.957563	−	0.288223i	$-0.906936\pi$
$11$	− 1.91185i	− 0.576446i	0.957563	−	0.288223i	$-0.0930644\pi$
$12$	1.00000	0.288675
$13$	0	0
$13$	0	0
$14$	−4.24698	−1.13505
$15$	1.00000i	0.258199i
$16$	1.00000	0.250000
$17$	3.33513	0.808887	0.404443	−	0.914563i	$-0.367465\pi$
$17$	3.33513	0.808887	0.404443	+	0.914563i	$0.367465\pi$
$18$	1.00000i	0.235702i
$19$	− 4.85086i	− 1.11286i	−0.830894	−	0.556431i	$-0.812170\pi$
$19$	− 4.85086i	− 1.11286i	0.830894	−	0.556431i	$-0.187830\pi$
$20$	1.00000i	0.223607i
$21$	− 4.24698i	− 0.926767i
$22$	1.91185	0.407609
$23$	0.445042	0.0927976	0.0463988	−	0.998923i	$-0.485225\pi$
$23$	0.445042	0.0927976	0.0463988	+	0.998923i	$0.485225\pi$
$24$	1.00000i	0.204124i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	− 4.24698i	− 0.802604i
$29$	−8.56465	−1.59041	−0.795207	−	0.606337i	$-0.792638\pi$
$29$	−8.56465	−1.59041	−0.795207	+	0.606337i	$0.792638\pi$
$30$	−1.00000	−0.182574
$31$	5.29590i	0.951171i	0.879669	+	0.475586i	$0.157764\pi$
$31$	5.29590i	0.951171i	−0.879669	+	0.475586i	$0.842236\pi$
$32$	1.00000i	0.176777i
$33$	1.91185i	0.332811i
$34$	3.33513i	0.571969i
$35$	4.24698	0.717871
$36$	−1.00000	−0.166667
$37$	− 1.75302i	− 0.288195i	−0.989564	−	0.144097i	$-0.953972\pi$
$37$	− 1.75302i	− 0.288195i	0.989564	−	0.144097i	$-0.0460279\pi$
$38$	4.85086	0.786913
$39$	0	0
$40$	−1.00000	−0.158114
$41$	3.24698i	0.507093i	0.967323	+	0.253547i	$0.0815971\pi$
$41$	3.24698i	0.507093i	−0.967323	+	0.253547i	$0.918403\pi$
$42$	4.24698	0.655323
$43$	1.97823	0.301677	0.150839	−	0.988558i	$-0.451803\pi$
$43$	1.97823	0.301677	0.150839	+	0.988558i	$0.451803\pi$
$44$	1.91185i	0.288223i
$45$	− 1.00000i	− 0.149071i
$46$	0.445042i	0.0656178i
$47$	− 10.3840i	− 1.51467i	−0.653028	−	0.757334i	$-0.726501\pi$
$47$	− 10.3840i	− 1.51467i	0.653028	−	0.757334i	$-0.273499\pi$
$48$	−1.00000	−0.144338
$49$	−11.0368	−1.57669
$50$	− 1.00000i	− 0.141421i
$51$	−3.33513	−0.467011
$52$	0	0
$53$	−1.14914	−0.157847	−0.0789236	−	0.996881i	$-0.525148\pi$
$53$	−1.14914	−0.157847	−0.0789236	+	0.996881i	$0.525148\pi$
$54$	− 1.00000i	− 0.136083i
$55$	−1.91185	−0.257794
$56$	4.24698	0.567527
$57$	4.85086i	0.642511i
$58$	− 8.56465i	− 1.12459i
$59$	− 13.4601i	− 1.75236i	−0.481987	−	0.876178i	$-0.660085\pi$
$59$	− 13.4601i	− 1.75236i	0.481987	−	0.876178i	$-0.339915\pi$
$60$	− 1.00000i	− 0.129099i
$61$	1.13169	0.144898	0.0724488	−	0.997372i	$-0.476919\pi$
$61$	1.13169	0.144898	0.0724488	+	0.997372i	$0.476919\pi$
$62$	−5.29590	−0.672580
$63$	4.24698i	0.535069i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−1.91185	−0.235333
$67$	13.5308i	1.65305i	0.562900	+	0.826525i	$0.309686\pi$
$67$	13.5308i	1.65305i	−0.562900	+	0.826525i	$0.690314\pi$
$68$	−3.33513	−0.404443
$69$	−0.445042	−0.0535767
$70$	4.24698i	0.507611i
$71$	− 2.14675i	− 0.254773i	−0.991853	−	0.127386i	$-0.959341\pi$
$71$	− 2.14675i	− 0.254773i	0.991853	−	0.127386i	$-0.0406588\pi$
$72$	− 1.00000i	− 0.117851i
$73$	− 5.15883i	− 0.603796i	−0.953340	−	0.301898i	$-0.902380\pi$
$73$	− 5.15883i	− 0.603796i	0.953340	−	0.301898i	$-0.0976202\pi$
$74$	1.75302	0.203784
$75$	1.00000	0.115470
$76$	4.85086i	0.556431i
$77$	8.11960	0.925315
$78$	0	0
$79$	−14.5526	−1.63729	−0.818646	−	0.574299i	$-0.805275\pi$
$79$	−14.5526	−1.63729	−0.818646	+	0.574299i	$0.805275\pi$
$80$	− 1.00000i	− 0.111803i
$81$	1.00000	0.111111
$82$	−3.24698	−0.358569
$83$	− 9.49157i	− 1.04183i	−0.853607	−	0.520917i	$-0.825590\pi$
$83$	− 9.49157i	− 1.04183i	0.853607	−	0.520917i	$-0.174410\pi$
$84$	4.24698i	0.463383i
$85$	− 3.33513i	− 0.361745i
$86$	1.97823i	0.213318i
$87$	8.56465	0.918227
$88$	−1.91185	−0.203804
$89$	1.25667i	0.133207i	0.997780	+	0.0666033i	$0.0212162\pi$
$89$	1.25667i	0.133207i	−0.997780	+	0.0666033i	$0.978784\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	−0.445042	−0.0463988
$93$	− 5.29590i	− 0.549159i
$94$	10.3840	1.07103
$95$	−4.85086	−0.497687
$96$	− 1.00000i	− 0.102062i
$97$	− 5.01507i	− 0.509203i	−0.967046	−	0.254601i	$-0.918056\pi$
$97$	− 5.01507i	− 0.509203i	0.967046	−	0.254601i	$-0.0819443\pi$
$98$	− 11.0368i	− 1.11489i
$99$	− 1.91185i	− 0.192149i
$100$	1.00000	0.100000
$101$	−14.8116	−1.47381	−0.736906	−	0.675995i	$-0.763714\pi$
$101$	−14.8116	−1.47381	−0.736906	+	0.675995i	$0.763714\pi$
$102$	− 3.33513i	− 0.330227i
$103$	13.6896	1.34888	0.674440	−	0.738330i	$-0.264385\pi$
$103$	13.6896	1.34888	0.674440	+	0.738330i	$0.264385\pi$
$104$	0	0
$105$	−4.24698	−0.414463
$106$	− 1.14914i	− 0.111615i
$107$	−12.8334	−1.24065	−0.620326	−	0.784344i	$-0.713000\pi$
$107$	−12.8334	−1.24065	−0.620326	+	0.784344i	$0.713000\pi$
$108$	1.00000	0.0962250
$109$	10.8726i	1.04141i	0.853737	+	0.520704i	$0.174331\pi$
$109$	10.8726i	1.04141i	−0.853737	+	0.520704i	$0.825669\pi$
$110$	− 1.91185i	− 0.182288i
$111$	1.75302i	0.166389i
$112$	4.24698i	0.401302i
$113$	−11.2295	−1.05638	−0.528192	−	0.849125i	$-0.677130\pi$
$113$	−11.2295	−1.05638	−0.528192	+	0.849125i	$0.677130\pi$
$114$	−4.85086	−0.454324
$115$	− 0.445042i	− 0.0415004i
$116$	8.56465	0.795207
$117$	0	0
$118$	13.4601	1.23910
$119$	14.1642i	1.29843i
$120$	1.00000	0.0912871
$121$	7.34481	0.667710
$122$	1.13169i	0.102458i
$123$	− 3.24698i	− 0.292770i
$124$	− 5.29590i	− 0.475586i
$125$	1.00000i	0.0894427i
$126$	−4.24698	−0.378351
$127$	−20.8267	−1.84807	−0.924035	−	0.382308i	$-0.875129\pi$
$127$	−20.8267	−1.84807	−0.924035	+	0.382308i	$0.875129\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	−1.97823	−0.174173
$130$	0	0
$131$	17.5036	1.52930	0.764650	−	0.644445i	$-0.222912\pi$
$131$	17.5036	1.52930	0.764650	+	0.644445i	$0.222912\pi$
$132$	− 1.91185i	− 0.166406i
$133$	20.6015	1.78638
$134$	−13.5308	−1.16888
$135$	1.00000i	0.0860663i
$136$	− 3.33513i	− 0.285985i
$137$	0.929312i	0.0793965i	0.999212	+	0.0396983i	$0.0126397\pi$
$137$	0.929312i	0.0793965i	−0.999212	+	0.0396983i	$0.987360\pi$
$138$	− 0.445042i	− 0.0378845i
$139$	3.12200	0.264804	0.132402	−	0.991196i	$-0.457731\pi$
$139$	3.12200	0.264804	0.132402	+	0.991196i	$0.457731\pi$
$140$	−4.24698	−0.358935
$141$	10.3840i	0.874494i
$142$	2.14675	0.180151
$143$	0	0
$144$	1.00000	0.0833333
$145$	8.56465i	0.711255i
$146$	5.15883	0.426948
$147$	11.0368	0.910303
$148$	1.75302i	0.144097i
$149$	6.57673i	0.538787i	0.963030	+	0.269393i	$0.0868232\pi$
$149$	6.57673i	0.538787i	−0.963030	+	0.269393i	$0.913177\pi$
$150$	1.00000i	0.0816497i
$151$	− 6.29590i	− 0.512353i	−0.966630	−	0.256176i	$-0.917537\pi$
$151$	− 6.29590i	− 0.512353i	0.966630	−	0.256176i	$-0.0824628\pi$
$152$	−4.85086	−0.393456
$153$	3.33513	0.269629
$154$	8.11960i	0.654296i
$155$	5.29590	0.425377
$156$	0	0
$157$	19.9148	1.58938	0.794689	−	0.607017i	$-0.207634\pi$
$157$	19.9148	1.58938	0.794689	+	0.607017i	$0.207634\pi$
$158$	− 14.5526i	− 1.15774i
$159$	1.14914	0.0911331
$160$	1.00000	0.0790569
$161$	1.89008i	0.148959i
$162$	1.00000i	0.0785674i
$163$	− 13.1075i	− 1.02666i	−0.858191	−	0.513330i	$-0.828412\pi$
$163$	− 13.1075i	− 1.02666i	0.858191	−	0.513330i	$-0.171588\pi$
$164$	− 3.24698i	− 0.253547i
$165$	1.91185	0.148838
$166$	9.49157	0.736688
$167$	− 21.5351i	− 1.66644i	−0.552944	−	0.833218i	$-0.686496\pi$
$167$	− 21.5351i	− 1.66644i	0.552944	−	0.833218i	$-0.313504\pi$
$168$	−4.24698	−0.327662
$169$	0	0
$170$	3.33513	0.255792
$171$	− 4.85086i	− 0.370954i
$172$	−1.97823	−0.150839
$173$	13.9541	1.06091	0.530454	−	0.847714i	$-0.322021\pi$
$173$	13.9541	1.06091	0.530454	+	0.847714i	$0.322021\pi$
$174$	8.56465i	0.649284i
$175$	− 4.24698i	− 0.321041i
$176$	− 1.91185i	− 0.144111i
$177$	13.4601i	1.01172i
$178$	−1.25667	−0.0941913
$179$	−8.32975	−0.622595	−0.311297	−	0.950313i	$-0.600764\pi$
$179$	−8.32975	−0.622595	−0.311297	+	0.950313i	$0.600764\pi$
$180$	1.00000i	0.0745356i
$181$	−17.1371	−1.27379	−0.636894	−	0.770951i	$-0.719781\pi$
$181$	−17.1371	−1.27379	−0.636894	+	0.770951i	$0.719781\pi$
$182$	0	0
$183$	−1.13169	−0.0836567
$184$	− 0.445042i	− 0.0328089i
$185$	−1.75302	−0.128885
$186$	5.29590	0.388314
$187$	− 6.37627i	− 0.466279i
$188$	10.3840i	0.757334i
$189$	− 4.24698i	− 0.308922i
$190$	− 4.85086i	− 0.351918i
$191$	23.1685	1.67642	0.838208	−	0.545351i	$-0.183604\pi$
$191$	23.1685	1.67642	0.838208	+	0.545351i	$0.183604\pi$
$192$	1.00000	0.0721688
$193$	− 6.38942i	− 0.459920i	−0.973200	−	0.229960i	$-0.926140\pi$
$193$	− 6.38942i	− 0.459920i	0.973200	−	0.229960i	$-0.0738596\pi$
$194$	5.01507	0.360061
$195$	0	0
$196$	11.0368	0.788345
$197$	− 14.8605i	− 1.05877i	−0.848382	−	0.529385i	$-0.822423\pi$
$197$	− 14.8605i	− 1.05877i	0.848382	−	0.529385i	$-0.177577\pi$
$198$	1.91185	0.135870
$199$	−26.3860	−1.87045	−0.935226	−	0.354052i	$-0.884803\pi$
$199$	−26.3860	−1.87045	−0.935226	+	0.354052i	$0.884803\pi$
$200$	1.00000i	0.0707107i
$201$	− 13.5308i	− 0.954389i
$202$	− 14.8116i	− 1.04214i
$203$	− 36.3739i	− 2.55295i
$204$	3.33513	0.233505
$205$	3.24698	0.226779
$206$	13.6896i	0.953802i
$207$	0.445042	0.0309325
$208$	0	0
$209$	−9.27413	−0.641505
$210$	− 4.24698i	− 0.293069i
$211$	17.4383	1.20050	0.600252	−	0.799811i	$-0.295067\pi$
$211$	17.4383	1.20050	0.600252	+	0.799811i	$0.295067\pi$
$212$	1.14914	0.0789236
$213$	2.14675i	0.147093i
$214$	− 12.8334i	− 0.877273i
$215$	− 1.97823i	− 0.134914i
$216$	1.00000i	0.0680414i
$217$	−22.4916	−1.52683
$218$	−10.8726	−0.736387
$219$	5.15883i	0.348602i
$220$	1.91185	0.128897
$221$	0	0
$222$	−1.75302	−0.117655
$223$	− 9.95838i	− 0.666862i	−0.942774	−	0.333431i	$-0.891794\pi$
$223$	− 9.95838i	− 0.666862i	0.942774	−	0.333431i	$-0.108206\pi$
$224$	−4.24698	−0.283763
$225$	−1.00000	−0.0666667
$226$	− 11.2295i	− 0.746977i
$227$	− 19.6112i	− 1.30164i	−0.759232	−	0.650820i	$-0.774425\pi$
$227$	− 19.6112i	− 1.30164i	0.759232	−	0.650820i	$-0.225575\pi$
$228$	− 4.85086i	− 0.321256i
$229$	8.72886i	0.576819i	0.957507	+	0.288410i	$0.0931265\pi$
$229$	8.72886i	0.576819i	−0.957507	+	0.288410i	$0.906874\pi$
$230$	0.445042	0.0293452
$231$	−8.11960	−0.534231
$232$	8.56465i	0.562297i
$233$	−0.975837	−0.0639292	−0.0319646	−	0.999489i	$-0.510176\pi$
$233$	−0.975837	−0.0639292	−0.0319646	+	0.999489i	$0.510176\pi$
$234$	0	0
$235$	−10.3840	−0.677380
$236$	13.4601i	0.876178i
$237$	14.5526	0.945291
$238$	−14.1642	−0.918129
$239$	− 7.39373i	− 0.478261i	−0.970987	−	0.239130i	$-0.923138\pi$
$239$	− 7.39373i	− 0.478261i	0.970987	−	0.239130i	$-0.0768623\pi$
$240$	1.00000i	0.0645497i
$241$	2.11529i	0.136258i	0.997677	+	0.0681290i	$0.0217029\pi$
$241$	2.11529i	0.136258i	−0.997677	+	0.0681290i	$0.978297\pi$
$242$	7.34481i	0.472143i
$243$	−1.00000	−0.0641500
$244$	−1.13169	−0.0724488
$245$	11.0368i	0.705118i
$246$	3.24698	0.207020
$247$	0	0
$248$	5.29590	0.336290
$249$	9.49157i	0.601504i
$250$	−1.00000	−0.0632456
$251$	1.62863	0.102798	0.0513991	−	0.998678i	$-0.483632\pi$
$251$	1.62863	0.102798	0.0513991	+	0.998678i	$0.483632\pi$
$252$	− 4.24698i	− 0.267535i
$253$	− 0.850855i	− 0.0534928i
$254$	− 20.8267i	− 1.30678i
$255$	3.33513i	0.208854i
$256$	1.00000	0.0625000
$257$	8.87263	0.553459	0.276730	−	0.960948i	$-0.410749\pi$
$257$	8.87263	0.553459	0.276730	+	0.960948i	$0.410749\pi$
$258$	− 1.97823i	− 0.123159i
$259$	7.44504	0.462612
$260$	0	0
$261$	−8.56465	−0.530138
$262$	17.5036i	1.08138i
$263$	22.4185	1.38238	0.691192	−	0.722672i	$-0.257086\pi$
$263$	22.4185	1.38238	0.691192	+	0.722672i	$0.257086\pi$
$264$	1.91185	0.117666
$265$	1.14914i	0.0705914i
$266$	20.6015i	1.26316i
$267$	− 1.25667i	− 0.0769068i
$268$	− 13.5308i	− 0.826525i
$269$	26.9071	1.64055	0.820276	−	0.571967i	$-0.193820\pi$
$269$	26.9071	1.64055	0.820276	+	0.571967i	$0.193820\pi$
$270$	−1.00000	−0.0608581
$271$	− 31.7265i	− 1.92725i	−0.267266	−	0.963623i	$-0.586120\pi$
$271$	− 31.7265i	− 1.92725i	0.267266	−	0.963623i	$-0.413880\pi$
$272$	3.33513	0.202222
$273$	0	0
$274$	−0.929312	−0.0561418
$275$	1.91185i	0.115289i
$276$	0.445042	0.0267884
$277$	30.6069	1.83899	0.919494	−	0.393104i	$-0.128599\pi$
$277$	30.6069	1.83899	0.919494	+	0.393104i	$0.128599\pi$
$278$	3.12200i	0.187245i
$279$	5.29590i	0.317057i
$280$	− 4.24698i	− 0.253806i
$281$	− 16.6649i	− 0.994143i	−0.867710	−	0.497072i	$-0.834409\pi$
$281$	− 16.6649i	− 0.994143i	0.867710	−	0.497072i	$-0.165591\pi$
$282$	−10.3840	−0.618361
$283$	4.32304	0.256978	0.128489	−	0.991711i	$-0.458987\pi$
$283$	4.32304	0.256978	0.128489	+	0.991711i	$0.458987\pi$
$284$	2.14675i	0.127386i
$285$	4.85086	0.287340
$286$	0	0
$287$	−13.7899	−0.813989
$288$	1.00000i	0.0589256i
$289$	−5.87694	−0.345702
$290$	−8.56465	−0.502933
$291$	5.01507i	0.293988i
$292$	5.15883i	0.301898i
$293$	− 5.62804i	− 0.328794i	−0.986394	−	0.164397i	$-0.947432\pi$
$293$	− 5.62804i	− 0.328794i	0.986394	−	0.164397i	$-0.0525677\pi$
$294$	11.0368i	0.643681i
$295$	−13.4601	−0.783678
$296$	−1.75302	−0.101892
$297$	1.91185i	0.110937i
$298$	−6.57673	−0.380980
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	8.40150i	0.484254i
$302$	6.29590	0.362288
$303$	14.8116	0.850906
$304$	− 4.85086i	− 0.278216i
$305$	− 1.13169i	− 0.0648002i
$306$	3.33513i	0.190656i
$307$	− 12.6635i	− 0.722747i	−0.932421	−	0.361373i	$-0.882308\pi$
$307$	− 12.6635i	− 0.722747i	0.932421	−	0.361373i	$-0.117692\pi$
$308$	−8.11960	−0.462657
$309$	−13.6896	−0.778776
$310$	5.29590i	0.300787i
$311$	−8.49635	−0.481784	−0.240892	−	0.970552i	$-0.577440\pi$
$311$	−8.49635	−0.481784	−0.240892	+	0.970552i	$0.577440\pi$
$312$	0	0
$313$	−25.2610	−1.42784	−0.713918	−	0.700230i	$-0.753081\pi$
$313$	−25.2610	−1.42784	−0.713918	+	0.700230i	$0.753081\pi$
$314$	19.9148i	1.12386i
$315$	4.24698	0.239290
$316$	14.5526	0.818646
$317$	4.96615i	0.278927i	0.990227	+	0.139463i	$0.0445377\pi$
$317$	4.96615i	0.278927i	−0.990227	+	0.139463i	$0.955462\pi$
$318$	1.14914i	0.0644408i
$319$	16.3744i	0.916788i
$320$	1.00000i	0.0559017i
$321$	12.8334	0.716290
$322$	−1.89008	−0.105330
$323$	− 16.1782i	− 0.900180i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	13.1075	0.725959
$327$	− 10.8726i	− 0.601258i
$328$	3.24698	0.179284
$329$	44.1008	2.43136
$330$	1.91185i	0.105244i
$331$	− 35.3749i	− 1.94438i	−0.234188	−	0.972191i	$-0.575243\pi$
$331$	− 35.3749i	− 1.94438i	0.234188	−	0.972191i	$-0.424757\pi$
$332$	9.49157i	0.520917i
$333$	− 1.75302i	− 0.0960649i
$334$	21.5351	1.17835
$335$	13.5308	0.739266
$336$	− 4.24698i	− 0.231692i
$337$	4.70171	0.256118	0.128059	−	0.991767i	$-0.459125\pi$
$337$	4.70171	0.256118	0.128059	+	0.991767i	$0.459125\pi$
$338$	0	0
$339$	11.2295	0.609904
$340$	3.33513i	0.180873i
$341$	10.1250	0.548299
$342$	4.85086	0.262304
$343$	− 17.1444i	− 0.925708i
$344$	− 1.97823i	− 0.106659i
$345$	0.445042i	0.0239602i
$346$	13.9541i	0.750175i
$347$	14.9028	0.800022	0.400011	−	0.916510i	$-0.369006\pi$
$347$	14.9028	0.800022	0.400011	+	0.916510i	$0.369006\pi$
$348$	−8.56465	−0.459113
$349$	− 35.2669i	− 1.88780i	−0.330237	−	0.943898i	$-0.607129\pi$
$349$	− 35.2669i	− 1.88780i	0.330237	−	0.943898i	$-0.392871\pi$
$350$	4.24698	0.227011
$351$	0	0
$352$	1.91185	0.101902
$353$	10.5574i	0.561911i	0.959721	+	0.280956i	$0.0906514\pi$
$353$	10.5574i	0.561911i	−0.959721	+	0.280956i	$0.909349\pi$
$354$	−13.4601	−0.715397
$355$	−2.14675	−0.113938
$356$	− 1.25667i	− 0.0666033i
$357$	− 14.1642i	− 0.749650i
$358$	− 8.32975i	− 0.440241i
$359$	− 28.7724i	− 1.51855i	−0.650770	−	0.759275i	$-0.725554\pi$
$359$	− 28.7724i	− 1.51855i	0.650770	−	0.759275i	$-0.274446\pi$
$360$	−1.00000	−0.0527046
$361$	−4.53079	−0.238463
$362$	− 17.1371i	− 0.900704i
$363$	−7.34481	−0.385503
$364$	0	0
$365$	−5.15883	−0.270026
$366$	− 1.13169i	− 0.0591542i
$367$	−24.1511	−1.26068	−0.630338	−	0.776321i	$-0.717084\pi$
$367$	−24.1511	−1.26068	−0.630338	+	0.776321i	$0.717084\pi$
$368$	0.445042	0.0231994
$369$	3.24698i	0.169031i
$370$	− 1.75302i	− 0.0911352i
$371$	− 4.88040i	− 0.253377i
$372$	5.29590i	0.274579i
$373$	−21.0664	−1.09078	−0.545388	−	0.838184i	$-0.683618\pi$
$373$	−21.0664	−1.09078	−0.545388	+	0.838184i	$0.683618\pi$
$374$	6.37627	0.329709
$375$	− 1.00000i	− 0.0516398i
$376$	−10.3840	−0.535516
$377$	0	0
$378$	4.24698	0.218441
$379$	− 8.42327i	− 0.432674i	−0.976319	−	0.216337i	$-0.930589\pi$
$379$	− 8.42327i	− 0.432674i	0.976319	−	0.216337i	$-0.0694111\pi$
$380$	4.85086	0.248844
$381$	20.8267	1.06698
$382$	23.1685i	1.18540i
$383$	10.6377i	0.543562i	0.962359	+	0.271781i	$0.0876127\pi$
$383$	10.6377i	0.543562i	−0.962359	+	0.271781i	$0.912387\pi$
$384$	1.00000i	0.0510310i
$385$	− 8.11960i	− 0.413813i
$386$	6.38942	0.325213
$387$	1.97823	0.100559
$388$	5.01507i	0.254601i
$389$	25.7429	1.30521	0.652607	−	0.757696i	$-0.273675\pi$
$389$	25.7429	1.30521	0.652607	+	0.757696i	$0.273675\pi$
$390$	0	0
$391$	1.48427	0.0750628
$392$	11.0368i	0.557444i
$393$	−17.5036	−0.882942
$394$	14.8605	0.748663
$395$	14.5526i	0.732219i
$396$	1.91185i	0.0960743i
$397$	− 11.4373i	− 0.574020i	−0.957928	−	0.287010i	$-0.907339\pi$
$397$	− 11.4373i	− 0.574020i	0.957928	−	0.287010i	$-0.0926613\pi$
$398$	− 26.3860i	− 1.32261i
$399$	−20.6015	−1.03136
$400$	−1.00000	−0.0500000
$401$	− 22.1511i	− 1.10617i	−0.833124	−	0.553086i	$-0.813450\pi$
$401$	− 22.1511i	− 1.10617i	0.833124	−	0.553086i	$-0.186550\pi$
$402$	13.5308	0.674855
$403$	0	0
$404$	14.8116	0.736906
$405$	− 1.00000i	− 0.0496904i
$406$	36.3739	1.80521
$407$	−3.35152	−0.166129
$408$	3.33513i	0.165113i
$409$	− 22.0965i	− 1.09260i	−0.837589	−	0.546301i	$-0.816035\pi$
$409$	− 22.0965i	− 1.09260i	0.837589	−	0.546301i	$-0.183965\pi$
$410$	3.24698i	0.160357i
$411$	− 0.929312i	− 0.0458396i
$412$	−13.6896	−0.674440
$413$	57.1648	2.81290
$414$	0.445042i	0.0218726i
$415$	−9.49157	−0.465923
$416$	0	0
$417$	−3.12200	−0.152885
$418$	− 9.27413i	− 0.453612i
$419$	9.03444	0.441361	0.220681	−	0.975346i	$-0.429172\pi$
$419$	9.03444	0.441361	0.220681	+	0.975346i	$0.429172\pi$
$420$	4.24698	0.207231
$421$	7.17928i	0.349896i	0.984578	+	0.174948i	$0.0559758\pi$
$421$	7.17928i	0.349896i	−0.984578	+	0.174948i	$0.944024\pi$
$422$	17.4383i	0.848885i
$423$	− 10.3840i	− 0.504889i
$424$	1.14914i	0.0558074i
$425$	−3.33513	−0.161777
$426$	−2.14675	−0.104010
$427$	4.80625i	0.232591i
$428$	12.8334	0.620326
$429$	0	0
$430$	1.97823	0.0953987
$431$	8.36360i	0.402860i	0.979503	+	0.201430i	$0.0645589\pi$
$431$	8.36360i	0.402860i	−0.979503	+	0.201430i	$0.935441\pi$
$432$	−1.00000	−0.0481125
$433$	12.2851	0.590386	0.295193	−	0.955438i	$-0.404616\pi$
$433$	12.2851	0.590386	0.295193	+	0.955438i	$0.404616\pi$
$434$	− 22.4916i	− 1.07963i
$435$	− 8.56465i	− 0.410643i
$436$	− 10.8726i	− 0.520704i
$437$	− 2.15883i	− 0.103271i
$438$	−5.15883	−0.246499
$439$	−20.4849	−0.977689	−0.488845	−	0.872371i	$-0.662581\pi$
$439$	−20.4849	−0.977689	−0.488845	+	0.872371i	$0.662581\pi$
$440$	1.91185i	0.0911441i
$441$	−11.0368	−0.525564
$442$	0	0
$443$	−0.928247	−0.0441024	−0.0220512	−	0.999757i	$-0.507020\pi$
$443$	−0.928247	−0.0441024	−0.0220512	+	0.999757i	$0.507020\pi$
$444$	− 1.75302i	− 0.0831947i
$445$	1.25667	0.0595718
$446$	9.95838	0.471543
$447$	− 6.57673i	− 0.311069i
$448$	− 4.24698i	− 0.200651i
$449$	22.9355i	1.08240i	0.840895	+	0.541198i	$0.182029\pi$
$449$	22.9355i	1.08240i	−0.840895	+	0.541198i	$0.817971\pi$
$450$	− 1.00000i	− 0.0471405i
$451$	6.20775	0.292312
$452$	11.2295	0.528192
$453$	6.29590i	0.295807i
$454$	19.6112	0.920398
$455$	0	0
$456$	4.85086	0.227162
$457$	10.3773i	0.485431i	0.970097	+	0.242716i	$0.0780382\pi$
$457$	10.3773i	0.485431i	−0.970097	+	0.242716i	$0.921962\pi$
$458$	−8.72886	−0.407873
$459$	−3.33513	−0.155670
$460$	0.445042i	0.0207502i
$461$	7.69740i	0.358504i	0.983803	+	0.179252i	$0.0573677\pi$
$461$	7.69740i	0.358504i	−0.983803	+	0.179252i	$0.942632\pi$
$462$	− 8.11960i	− 0.377758i
$463$	39.0640i	1.81546i	0.419558	+	0.907729i	$0.362185\pi$
$463$	39.0640i	1.81546i	−0.419558	+	0.907729i	$0.637815\pi$
$464$	−8.56465	−0.397604
$465$	−5.29590	−0.245591
$466$	− 0.975837i	− 0.0452048i
$467$	−0.916166	−0.0423951	−0.0211976	−	0.999775i	$-0.506748\pi$
$467$	−0.916166	−0.0423951	−0.0211976	+	0.999775i	$0.506748\pi$
$468$	0	0
$469$	−57.4650	−2.65349
$470$	− 10.3840i	− 0.478980i
$471$	−19.9148	−0.917627
$472$	−13.4601	−0.619552
$473$	− 3.78209i	− 0.173901i
$474$	14.5526i	0.668421i
$475$	4.85086i	0.222572i
$476$	− 14.1642i	− 0.649216i
$477$	−1.14914	−0.0526157
$478$	7.39373	0.338181
$479$	23.1715i	1.05873i	0.848393	+	0.529367i	$0.177570\pi$
$479$	23.1715i	1.05873i	−0.848393	+	0.529367i	$0.822430\pi$
$480$	−1.00000	−0.0456435
$481$	0	0
$482$	−2.11529	−0.0963490
$483$	− 1.89008i	− 0.0860018i
$484$	−7.34481	−0.333855
$485$	−5.01507	−0.227722
$486$	− 1.00000i	− 0.0453609i
$487$	− 11.9812i	− 0.542921i	−0.962450	−	0.271460i	$-0.912493\pi$
$487$	− 11.9812i	− 0.542921i	0.962450	−	0.271460i	$-0.0875066\pi$
$488$	− 1.13169i	− 0.0512290i
$489$	13.1075i	0.592743i
$490$	−11.0368	−0.498593
$491$	24.8745	1.12257	0.561286	−	0.827622i	$-0.310307\pi$
$491$	24.8745	1.12257	0.561286	+	0.827622i	$0.310307\pi$
$492$	3.24698i	0.146385i
$493$	−28.5642	−1.28647
$494$	0	0
$495$	−1.91185	−0.0859314
$496$	5.29590i	0.237793i
$497$	9.11721	0.408963
$498$	−9.49157	−0.425327
$499$	− 42.0737i	− 1.88348i	−0.336347	−	0.941738i	$-0.609192\pi$
$499$	− 42.0737i	− 1.88348i	0.336347	−	0.941738i	$-0.390808\pi$
$500$	− 1.00000i	− 0.0447214i
$501$	21.5351i	0.962118i
$502$	1.62863i	0.0726893i
$503$	−20.0151	−0.892428	−0.446214	−	0.894926i	$-0.647228\pi$
$503$	−20.0151	−0.892428	−0.446214	+	0.894926i	$0.647228\pi$
$504$	4.24698	0.189176
$505$	14.8116i	0.659109i
$506$	0.850855	0.0378251
$507$	0	0
$508$	20.8267	0.924035
$509$	25.6907i	1.13872i	0.822088	+	0.569360i	$0.192809\pi$
$509$	25.6907i	1.13872i	−0.822088	+	0.569360i	$0.807191\pi$
$510$	−3.33513	−0.147682
$511$	21.9095	0.969217
$512$	1.00000i	0.0441942i
$513$	4.85086i	0.214170i
$514$	8.87263i	0.391355i
$515$	− 13.6896i	− 0.603237i
$516$	1.97823	0.0870867
$517$	−19.8528	−0.873124
$518$	7.44504i	0.327116i
$519$	−13.9541	−0.612516
$520$	0	0
$521$	−24.7342	−1.08363	−0.541813	−	0.840499i	$-0.682262\pi$
$521$	−24.7342	−1.08363	−0.541813	+	0.840499i	$0.682262\pi$
$522$	− 8.56465i	− 0.374864i
$523$	−18.7439	−0.819615	−0.409807	−	0.912172i	$-0.634404\pi$
$523$	−18.7439	−0.819615	−0.409807	+	0.912172i	$0.634404\pi$
$524$	−17.5036	−0.764650
$525$	4.24698i	0.185353i
$526$	22.4185i	0.977492i
$527$	17.6625i	0.769390i
$528$	1.91185i	0.0832028i
$529$	−22.8019	−0.991389
$530$	−1.14914	−0.0499157
$531$	− 13.4601i	− 0.584119i
$532$	−20.6015	−0.893188
$533$	0	0
$534$	1.25667	0.0543814
$535$	12.8334i	0.554836i
$536$	13.5308	0.584441
$537$	8.32975	0.359455
$538$	26.9071i	1.16005i
$539$	21.1008i	0.908877i
$540$	− 1.00000i	− 0.0430331i
$541$	43.7434i	1.88068i	0.340239	+	0.940339i	$0.389492\pi$
$541$	43.7434i	1.88068i	−0.340239	+	0.940339i	$0.610508\pi$
$542$	31.7265	1.36277
$543$	17.1371	0.735422
$544$	3.33513i	0.142992i
$545$	10.8726	0.465732
$546$	0	0
$547$	17.9681	0.768259	0.384130	−	0.923279i	$-0.374502\pi$
$547$	17.9681	0.768259	0.384130	+	0.923279i	$0.374502\pi$
$548$	− 0.929312i	− 0.0396983i
$549$	1.13169	0.0482992
$550$	−1.91185	−0.0815217
$551$	41.5459i	1.76991i
$552$	0.445042i	0.0189422i
$553$	− 61.8044i	− 2.62819i
$554$	30.6069i	1.30036i
$555$	1.75302	0.0744116
$556$	−3.12200	−0.132402
$557$	8.26981i	0.350403i	0.984533	+	0.175202i	$0.0560577\pi$
$557$	8.26981i	0.350403i	−0.984533	+	0.175202i	$0.943942\pi$
$558$	−5.29590	−0.224193
$559$	0	0
$560$	4.24698	0.179468
$561$	6.37627i	0.269206i
$562$	16.6649	0.702965
$563$	−19.8984	−0.838619	−0.419310	−	0.907843i	$-0.637728\pi$
$563$	−19.8984	−0.838619	−0.419310	+	0.907843i	$0.637728\pi$
$564$	− 10.3840i	− 0.437247i
$565$	11.2295i	0.472430i
$566$	4.32304i	0.181711i
$567$	4.24698i	0.178356i
$568$	−2.14675	−0.0900757
$569$	−14.6528	−0.614277	−0.307139	−	0.951665i	$-0.599371\pi$
$569$	−14.6528	−0.614277	−0.307139	+	0.951665i	$0.599371\pi$
$570$	4.85086i	0.203180i
$571$	9.47517	0.396524	0.198262	−	0.980149i	$-0.436470\pi$
$571$	9.47517	0.396524	0.198262	+	0.980149i	$0.436470\pi$
$572$	0	0
$573$	−23.1685	−0.967879
$574$	− 13.7899i	− 0.575577i
$575$	−0.445042	−0.0185595
$576$	−1.00000	−0.0416667
$577$	− 8.79225i	− 0.366026i	−0.983110	−	0.183013i	$-0.941415\pi$
$577$	− 8.79225i	− 0.366026i	0.983110	−	0.183013i	$-0.0585851\pi$
$578$	− 5.87694i	− 0.244448i
$579$	6.38942i	0.265535i
$580$	− 8.56465i	− 0.355628i
$581$	40.3105	1.67236
$582$	−5.01507	−0.207881
$583$	2.19700i	0.0909903i
$584$	−5.15883	−0.213474
$585$	0	0
$586$	5.62804	0.232492
$587$	− 2.38942i	− 0.0986219i	−0.998783	−	0.0493110i	$-0.984297\pi$
$587$	− 2.38942i	− 0.0986219i	0.998783	−	0.0493110i	$-0.0157025\pi$
$588$	−11.0368	−0.455151
$589$	25.6896	1.05852
$590$	− 13.4601i	− 0.554144i
$591$	14.8605i	0.611281i
$592$	− 1.75302i	− 0.0720487i
$593$	− 21.4034i	− 0.878933i	−0.898259	−	0.439467i	$-0.855167\pi$
$593$	− 21.4034i	− 0.878933i	0.898259	−	0.439467i	$-0.144833\pi$
$594$	−1.91185	−0.0784443
$595$	14.1642	0.580676
$596$	− 6.57673i	− 0.269393i
$597$	26.3860	1.07991
$598$	0	0
$599$	12.4765	0.509776	0.254888	−	0.966971i	$-0.417961\pi$
$599$	12.4765	0.509776	0.254888	+	0.966971i	$0.417961\pi$
$600$	− 1.00000i	− 0.0408248i
$601$	−12.9487	−0.528188	−0.264094	−	0.964497i	$-0.585073\pi$
$601$	−12.9487	−0.528188	−0.264094	+	0.964497i	$0.585073\pi$
$602$	−8.40150	−0.342420
$603$	13.5308i	0.551017i
$604$	6.29590i	0.256176i
$605$	− 7.34481i	− 0.298609i
$606$	14.8116i	0.601681i
$607$	−11.5230	−0.467705	−0.233853	−	0.972272i	$-0.575133\pi$
$607$	−11.5230	−0.467705	−0.233853	+	0.972272i	$0.575133\pi$
$608$	4.85086	0.196728
$609$	36.3739i	1.47394i
$610$	1.13169	0.0458206
$611$	0	0
$612$	−3.33513	−0.134814
$613$	− 44.9842i	− 1.81689i	−0.417999	−	0.908447i	$-0.637269\pi$
$613$	− 44.9842i	− 1.81689i	0.417999	−	0.908447i	$-0.362731\pi$
$614$	12.6635	0.511059
$615$	−3.24698	−0.130931
$616$	− 8.11960i	− 0.327148i
$617$	24.9318i	1.00372i	0.864950	+	0.501859i	$0.167350\pi$
$617$	24.9318i	1.00372i	−0.864950	+	0.501859i	$0.832650\pi$
$618$	− 13.6896i	− 0.550678i
$619$	11.2929i	0.453900i	0.973906	+	0.226950i	$0.0728755\pi$
$619$	11.2929i	0.453900i	−0.973906	+	0.226950i	$0.927125\pi$
$620$	−5.29590	−0.212688
$621$	−0.445042	−0.0178589
$622$	− 8.49635i	− 0.340673i
$623$	−5.33704	−0.213824
$624$	0	0
$625$	1.00000	0.0400000
$626$	− 25.2610i	− 1.00963i
$627$	9.27413	0.370373
$628$	−19.9148	−0.794689
$629$	− 5.84654i	− 0.233117i
$630$	4.24698i	0.169204i
$631$	20.6165i	0.820732i	0.911921	+	0.410366i	$0.134599\pi$
$631$	20.6165i	0.820732i	−0.911921	+	0.410366i	$0.865401\pi$
$632$	14.5526i	0.578870i
$633$	−17.4383	−0.693112
$634$	−4.96615	−0.197231
$635$	20.8267i	0.826482i
$636$	−1.14914	−0.0455666
$637$	0	0
$638$	−16.3744	−0.648267
$639$	− 2.14675i	− 0.0849242i
$640$	−1.00000	−0.0395285
$641$	−47.1487	−1.86226	−0.931130	−	0.364687i	$-0.881176\pi$
$641$	−47.1487	−1.86226	−0.931130	+	0.364687i	$0.881176\pi$
$642$	12.8334i	0.506494i
$643$	− 8.41981i	− 0.332045i	−0.986122	−	0.166023i	$-0.946908\pi$
$643$	− 8.41981i	− 0.332045i	0.986122	−	0.166023i	$-0.0530924\pi$
$644$	− 1.89008i	− 0.0744797i
$645$	1.97823i	0.0778927i
$646$	16.1782	0.636523
$647$	43.3836	1.70558	0.852792	−	0.522251i	$-0.174907\pi$
$647$	43.3836	1.70558	0.852792	+	0.522251i	$0.174907\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	−25.7338	−1.01014
$650$	0	0
$651$	22.4916	0.881514
$652$	13.1075i	0.513330i
$653$	−21.2862	−0.832994	−0.416497	−	0.909137i	$-0.636742\pi$
$653$	−21.2862	−0.832994	−0.416497	+	0.909137i	$0.636742\pi$
$654$	10.8726	0.425153
$655$	− 17.5036i	− 0.683924i
$656$	3.24698i	0.126773i
$657$	− 5.15883i	− 0.201265i
$658$	44.1008i	1.71923i
$659$	24.9075	0.970260	0.485130	−	0.874442i	$-0.338772\pi$
$659$	24.9075	0.970260	0.485130	+	0.874442i	$0.338772\pi$
$660$	−1.91185	−0.0744188
$661$	17.6209i	0.685372i	0.939450	+	0.342686i	$0.111337\pi$
$661$	17.6209i	0.685372i	−0.939450	+	0.342686i	$0.888663\pi$
$662$	35.3749	1.37489
$663$	0	0
$664$	−9.49157	−0.368344
$665$	− 20.6015i	− 0.798891i
$666$	1.75302	0.0679282
$667$	−3.81163	−0.147587
$668$	21.5351i	0.833218i
$669$	9.95838i	0.385013i
$670$	13.5308i	0.522740i
$671$	− 2.16362i	− 0.0835256i
$672$	4.24698	0.163831
$673$	28.8888	1.11358	0.556790	−	0.830653i	$-0.312033\pi$
$673$	28.8888	1.11358	0.556790	+	0.830653i	$0.312033\pi$
$674$	4.70171i	0.181103i
$675$	1.00000	0.0384900
$676$	0	0
$677$	24.3927	0.937486	0.468743	−	0.883335i	$-0.344707\pi$
$677$	24.3927	0.937486	0.468743	+	0.883335i	$0.344707\pi$
$678$	11.2295i	0.431267i
$679$	21.2989	0.817376
$680$	−3.33513	−0.127896
$681$	19.6112i	0.751502i
$682$	10.1250i	0.387706i
$683$	− 47.6558i	− 1.82350i	−0.410748	−	0.911749i	$-0.634732\pi$
$683$	− 47.6558i	− 1.82350i	0.410748	−	0.911749i	$-0.365268\pi$
$684$	4.85086i	0.185477i
$685$	0.929312	0.0355072
$686$	17.1444	0.654575
$687$	− 8.72886i	− 0.333027i
$688$	1.97823	0.0754193
$689$	0	0
$690$	−0.445042	−0.0169425
$691$	50.2180i	1.91038i	0.295987	+	0.955192i	$0.404351\pi$
$691$	50.2180i	1.91038i	−0.295987	+	0.955192i	$0.595649\pi$
$692$	−13.9541	−0.530454
$693$	8.11960	0.308438
$694$	14.9028i	0.565701i
$695$	− 3.12200i	− 0.118424i
$696$	− 8.56465i	− 0.324642i
$697$	10.8291i	0.410181i
$698$	35.2669	1.33487
$699$	0.975837	0.0369095
$700$	4.24698i	0.160521i
$701$	12.8140	0.483979	0.241989	−	0.970279i	$-0.422200\pi$
$701$	12.8140	0.483979	0.241989	+	0.970279i	$0.422200\pi$
$702$	0	0
$703$	−8.50365	−0.320721
$704$	1.91185i	0.0720557i
$705$	10.3840	0.391086
$706$	−10.5574	−0.397331
$707$	− 62.9047i	− 2.36577i
$708$	− 13.4601i	− 0.505862i
$709$	34.5515i	1.29761i	0.760955	+	0.648804i	$0.224731\pi$
$709$	34.5515i	1.29761i	−0.760955	+	0.648804i	$0.775269\pi$
$710$	− 2.14675i	− 0.0805662i
$711$	−14.5526	−0.545764
$712$	1.25667	0.0470956
$713$	2.35690i	0.0882664i
$714$	14.1642	0.530082
$715$	0	0
$716$	8.32975	0.311297
$717$	7.39373i	0.276124i
$718$	28.7724	1.07378
$719$	−20.8528	−0.777677	−0.388839	−	0.921306i	$-0.627124\pi$
$719$	−20.8528	−0.777677	−0.388839	+	0.921306i	$0.627124\pi$
$720$	− 1.00000i	− 0.0372678i
$721$	58.1396i	2.16523i
$722$	− 4.53079i	− 0.168619i
$723$	− 2.11529i	− 0.0786686i
$724$	17.1371	0.636894
$725$	8.56465	0.318083
$726$	− 7.34481i	− 0.272592i
$727$	−32.7192	−1.21349	−0.606743	−	0.794898i	$-0.707524\pi$
$727$	−32.7192	−1.21349	−0.606743	+	0.794898i	$0.707524\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	− 5.15883i	− 0.190937i
$731$	6.59764	0.244023
$732$	1.13169	0.0418283
$733$	52.2737i	1.93077i	0.260826	+	0.965386i	$0.416005\pi$
$733$	52.2737i	1.93077i	−0.260826	+	0.965386i	$0.583995\pi$
$734$	− 24.1511i	− 0.891432i
$735$	− 11.0368i	− 0.407100i
$736$	0.445042i	0.0164045i
$737$	25.8689	0.952893
$738$	−3.24698	−0.119523
$739$	13.4571i	0.495028i	0.968884	+	0.247514i	$0.0796137\pi$
$739$	13.4571i	0.495028i	−0.968884	+	0.247514i	$0.920386\pi$
$740$	1.75302	0.0644423
$741$	0	0
$742$	4.88040	0.179165
$743$	22.9989i	0.843749i	0.906654	+	0.421875i	$0.138628\pi$
$743$	22.9989i	0.843749i	−0.906654	+	0.421875i	$0.861372\pi$
$744$	−5.29590	−0.194157
$745$	6.57673	0.240953
$746$	− 21.0664i	− 0.771295i
$747$	− 9.49157i	− 0.347278i
$748$	6.37627i	0.233140i
$749$	− 54.5032i	− 1.99150i
$750$	1.00000	0.0365148
$751$	−34.1159	−1.24491	−0.622453	−	0.782657i	$-0.713864\pi$
$751$	−34.1159	−1.24491	−0.622453	+	0.782657i	$0.713864\pi$
$752$	− 10.3840i	− 0.378667i
$753$	−1.62863	−0.0593506
$754$	0	0
$755$	−6.29590	−0.229131
$756$	4.24698i	0.154461i
$757$	−38.9004	−1.41386	−0.706929	−	0.707285i	$-0.749920\pi$
$757$	−38.9004	−1.41386	−0.706929	+	0.707285i	$0.749920\pi$
$758$	8.42327	0.305947
$759$	0.850855i	0.0308841i
$760$	4.85086i	0.175959i
$761$	36.8528i	1.33591i	0.744201	+	0.667956i	$0.232831\pi$
$761$	36.8528i	1.33591i	−0.744201	+	0.667956i	$0.767169\pi$
$762$	20.8267i	0.754471i
$763$	−46.1758	−1.67168
$764$	−23.1685	−0.838208
$765$	− 3.33513i	− 0.120582i
$766$	−10.6377	−0.384357
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	− 23.4198i	− 0.844540i	−0.906470	−	0.422270i	$-0.861233\pi$
$769$	− 23.4198i	− 0.844540i	0.906470	−	0.422270i	$-0.138767\pi$
$770$	8.11960	0.292610
$771$	−8.87263	−0.319540
$772$	6.38942i	0.229960i
$773$	0.275192i	0.00989795i	0.999988	+	0.00494898i	$0.00157531\pi$
$773$	0.275192i	0.00989795i	−0.999988	+	0.00494898i	$0.998425\pi$
$774$	1.97823i	0.0711060i
$775$	− 5.29590i	− 0.190234i
$776$	−5.01507	−0.180030
$777$	−7.44504	−0.267089
$778$	25.7429i	0.922926i
$779$	15.7506	0.564325
$780$	0	0
$781$	−4.10428	−0.146863
$782$	1.48427i	0.0530774i
$783$	8.56465	0.306076
$784$	−11.0368	−0.394173
$785$	− 19.9148i	− 0.710791i
$786$	− 17.5036i	− 0.624334i
$787$	− 27.2543i	− 0.971510i	−0.874095	−	0.485755i	$-0.838545\pi$
$787$	− 27.2543i	− 0.971510i	0.874095	−	0.485755i	$-0.161455\pi$
$788$	14.8605i	0.529385i
$789$	−22.4185	−0.798119
$790$	−14.5526	−0.517757
$791$	− 47.6915i	− 1.69572i
$792$	−1.91185	−0.0679348
$793$	0	0
$794$	11.4373	0.405894
$795$	− 1.14914i	− 0.0407560i
$796$	26.3860	0.935226
$797$	−14.2819	−0.505891	−0.252945	−	0.967481i	$-0.581399\pi$
$797$	−14.2819	−0.505891	−0.252945	+	0.967481i	$0.581399\pi$
$798$	− 20.6015i	− 0.729285i
$799$	− 34.6321i	− 1.22520i
$800$	− 1.00000i	− 0.0353553i
$801$	1.25667i	0.0444022i
$802$	22.1511	0.782181
$803$	−9.86294	−0.348055
$804$	13.5308i	0.477194i
$805$	1.89008	0.0666167
$806$	0	0
$807$	−26.9071	−0.947174
$808$	14.8116i	0.521071i
$809$	−8.82610	−0.310309	−0.155155	−	0.987890i	$-0.549588\pi$
$809$	−8.82610	−0.310309	−0.155155	+	0.987890i	$0.549588\pi$
$810$	1.00000	0.0351364
$811$	6.37196i	0.223750i	0.993722	+	0.111875i	$0.0356856\pi$
$811$	6.37196i	0.223750i	−0.993722	+	0.111875i	$0.964314\pi$
$812$	36.3739i	1.27647i
$813$	31.7265i	1.11270i
$814$	− 3.35152i	− 0.117471i
$815$	−13.1075	−0.459137
$816$	−3.33513	−0.116753
$817$	− 9.59611i	− 0.335725i
$818$	22.0965	0.772586
$819$	0	0
$820$	−3.24698	−0.113389
$821$	13.7476i	0.479796i	0.970798	+	0.239898i	$0.0771140\pi$
$821$	13.7476i	0.479796i	−0.970798	+	0.239898i	$0.922886\pi$
$822$	0.929312	0.0324135
$823$	−47.8587	−1.66825	−0.834125	−	0.551575i	$-0.814027\pi$
$823$	−47.8587	−1.66825	−0.834125	+	0.551575i	$0.814027\pi$
$824$	− 13.6896i	− 0.476901i
$825$	− 1.91185i	− 0.0665622i
$826$	57.1648i	1.98902i
$827$	29.1317i	1.01301i	0.862238	+	0.506504i	$0.169062\pi$
$827$	29.1317i	1.01301i	−0.862238	+	0.506504i	$0.830938\pi$
$828$	−0.445042	−0.0154663
$829$	25.7748	0.895195	0.447598	−	0.894235i	$-0.352280\pi$
$829$	25.7748	0.895195	0.447598	+	0.894235i	$0.352280\pi$
$830$	− 9.49157i	− 0.329457i
$831$	−30.6069	−1.06174
$832$	0	0
$833$	−36.8092	−1.27536
$834$	− 3.12200i	− 0.108106i
$835$	−21.5351	−0.745253
$836$	9.27413	0.320752
$837$	− 5.29590i	− 0.183053i
$838$	9.03444i	0.312090i
$839$	2.91292i	0.100565i	0.998735	+	0.0502826i	$0.0160122\pi$
$839$	2.91292i	0.100565i	−0.998735	+	0.0502826i	$0.983988\pi$
$840$	4.24698i	0.146535i
$841$	44.3532	1.52942
$842$	−7.17928	−0.247414
$843$	16.6649i	0.573969i
$844$	−17.4383	−0.600252
$845$	0	0
$846$	10.3840	0.357011
$847$	31.1933i	1.07181i
$848$	−1.14914	−0.0394618
$849$	−4.32304	−0.148366
$850$	− 3.33513i	− 0.114394i
$851$	− 0.780167i	− 0.0267438i
$852$	− 2.14675i	− 0.0735465i
$853$	− 12.9661i	− 0.443952i	−0.975052	−	0.221976i	$-0.928749\pi$
$853$	− 12.9661i	− 0.443952i	0.975052	−	0.221976i	$-0.0712507\pi$
$854$	−4.80625	−0.164466
$855$	−4.85086	−0.165896
$856$	12.8334i	0.438636i
$857$	9.85623	0.336682	0.168341	−	0.985729i	$-0.446159\pi$
$857$	9.85623	0.336682	0.168341	+	0.985729i	$0.446159\pi$
$858$	0	0
$859$	30.6746	1.04660	0.523301	−	0.852148i	$-0.324700\pi$
$859$	30.6746	1.04660	0.523301	+	0.852148i	$0.324700\pi$
$860$	1.97823i	0.0674571i
$861$	13.7899	0.469957
$862$	−8.36360	−0.284865
$863$	34.1588i	1.16278i	0.813625	+	0.581390i	$0.197491\pi$
$863$	34.1588i	1.16278i	−0.813625	+	0.581390i	$0.802509\pi$
$864$	− 1.00000i	− 0.0340207i
$865$	− 13.9541i	− 0.474452i
$866$	12.2851i	0.417466i
$867$	5.87694	0.199591
$868$	22.4916	0.763414
$869$	27.8224i	0.943810i
$870$	8.56465	0.290369
$871$	0	0
$872$	10.8726	0.368194
$873$	− 5.01507i	− 0.169734i
$874$	2.15883	0.0730236
$875$	−4.24698	−0.143574
$876$	− 5.15883i	− 0.174301i
$877$	20.1588i	0.680715i	0.940296	+	0.340358i	$0.110548\pi$
$877$	20.1588i	0.680715i	−0.940296	+	0.340358i	$0.889452\pi$
$878$	− 20.4849i	− 0.691331i
$879$	5.62804i	0.189829i
$880$	−1.91185	−0.0644486
$881$	2.79550	0.0941827	0.0470913	−	0.998891i	$-0.485005\pi$
$881$	2.79550	0.0941827	0.0470913	+	0.998891i	$0.485005\pi$
$882$	− 11.0368i	− 0.371630i
$883$	16.8605	0.567402	0.283701	−	0.958913i	$-0.408438\pi$
$883$	16.8605	0.567402	0.283701	+	0.958913i	$0.408438\pi$
$884$	0	0
$885$	13.4601	0.452457
$886$	− 0.928247i	− 0.0311851i
$887$	−8.71379	−0.292580	−0.146290	−	0.989242i	$-0.546733\pi$
$887$	−8.71379	−0.292580	−0.146290	+	0.989242i	$0.546733\pi$
$888$	1.75302	0.0588275
$889$	− 88.4505i	− 2.96654i
$890$	1.25667i	0.0421236i
$891$	− 1.91185i	− 0.0640495i
$892$	9.95838i	0.333431i
$893$	−50.3715	−1.68562
$894$	6.57673	0.219959
$895$	8.32975i	0.278433i
$896$	4.24698	0.141882
$897$	0	0
$898$	−22.9355	−0.765369
$899$	− 45.3575i	− 1.51276i
$900$	1.00000	0.0333333
$901$	−3.83254	−0.127681
$902$	6.20775i	0.206695i
$903$	− 8.40150i	− 0.279584i
$904$	11.2295i	0.373488i
$905$	17.1371i	0.569655i
$906$	−6.29590	−0.209167
$907$	−14.2295	−0.472483	−0.236242	−	0.971694i	$-0.575916\pi$
$907$	−14.2295	−0.472483	−0.236242	+	0.971694i	$0.575916\pi$
$908$	19.6112i	0.650820i
$909$	−14.8116	−0.491271
$910$	0	0
$911$	−19.3672	−0.641663	−0.320832	−	0.947136i	$-0.603962\pi$
$911$	−19.3672	−0.641663	−0.320832	+	0.947136i	$0.603962\pi$
$912$	4.85086i	0.160628i
$913$	−18.1465	−0.600561
$914$	−10.3773	−0.343252
$915$	1.13169i	0.0374124i
$916$	− 8.72886i	− 0.288410i
$917$	74.3376i	2.45484i
$918$	− 3.33513i	− 0.110076i
$919$	−44.1420	−1.45611	−0.728055	−	0.685519i	$-0.759575\pi$
$919$	−44.1420	−1.45611	−0.728055	+	0.685519i	$0.759575\pi$
$920$	−0.445042	−0.0146726
$921$	12.6635i	0.417278i
$922$	−7.69740	−0.253500
$923$	0	0
$924$	8.11960	0.267115
$925$	1.75302i	0.0576390i
$926$	−39.0640	−1.28372
$927$	13.6896	0.449626
$928$	− 8.56465i	− 0.281148i
$929$	14.8616i	0.487594i	0.969826	+	0.243797i	$0.0783930\pi$
$929$	14.8616i	0.487594i	−0.969826	+	0.243797i	$0.921607\pi$
$930$	− 5.29590i	− 0.173659i
$931$	53.5381i	1.75464i
$932$	0.975837	0.0319646
$933$	8.49635	0.278158
$934$	− 0.916166i	− 0.0299779i
$935$	−6.37627	−0.208526
$936$	0	0
$937$	54.3196	1.77454	0.887272	−	0.461247i	$-0.152598\pi$
$937$	54.3196	1.77454	0.887272	+	0.461247i	$0.152598\pi$
$938$	− 57.4650i	− 1.87630i
$939$	25.2610	0.824361
$940$	10.3840	0.338690
$941$	− 14.0121i	− 0.456781i	−0.973570	−	0.228390i	$-0.926654\pi$
$941$	− 14.0121i	− 0.456781i	0.973570	−	0.228390i	$-0.0733463\pi$
$942$	− 19.9148i	− 0.648860i
$943$	1.44504i	0.0470570i
$944$	− 13.4601i	− 0.438089i
$945$	−4.24698	−0.138154
$946$	3.78209	0.122966
$947$	− 56.1831i	− 1.82571i	−0.408289	−	0.912853i	$-0.633874\pi$
$947$	− 56.1831i	− 1.82571i	0.408289	−	0.912853i	$-0.366126\pi$
$948$	−14.5526	−0.472645
$949$	0	0
$950$	−4.85086	−0.157383
$951$	− 4.96615i	− 0.161038i
$952$	14.1642	0.459065
$953$	−42.9090	−1.38996	−0.694979	−	0.719030i	$-0.744586\pi$
$953$	−42.9090	−1.38996	−0.694979	+	0.719030i	$0.744586\pi$
$954$	− 1.14914i	− 0.0372049i
$955$	− 23.1685i	− 0.749716i
$956$	7.39373i	0.239130i
$957$	− 16.3744i	− 0.529308i
$958$	−23.1715	−0.748637
$959$	−3.94677	−0.127448
$960$	− 1.00000i	− 0.0322749i
$961$	2.95348	0.0952734
$962$	0	0
$963$	−12.8334	−0.413550
$964$	− 2.11529i	− 0.0681290i
$965$	−6.38942	−0.205683
$966$	1.89008	0.0608124
$967$	2.89977i	0.0932504i	0.998912	+	0.0466252i	$0.0148466\pi$
$967$	2.89977i	0.0932504i	−0.998912	+	0.0466252i	$0.985153\pi$
$968$	− 7.34481i	− 0.236071i
$969$	16.1782i	0.519719i
$970$	− 5.01507i	− 0.161024i
$971$	32.8146	1.05307	0.526535	−	0.850153i	$-0.323491\pi$
$971$	32.8146	1.05307	0.526535	+	0.850153i	$0.323491\pi$
$972$	1.00000	0.0320750
$973$	13.2591i	0.425066i
$974$	11.9812	0.383903
$975$	0	0
$976$	1.13169	0.0362244
$977$	− 13.1806i	− 0.421685i	−0.977520	−	0.210842i	$-0.932379\pi$
$977$	− 13.1806i	− 0.421685i	0.977520	−	0.210842i	$-0.0676207\pi$
$978$	−13.1075	−0.419132
$979$	2.40257	0.0767864
$980$	− 11.0368i	− 0.352559i
$981$	10.8726i	0.347136i
$982$	24.8745i	0.793779i
$983$	− 31.3726i	− 1.00063i	−0.865844	−	0.500315i	$-0.833218\pi$
$983$	− 31.3726i	− 1.00063i	0.865844	−	0.500315i	$-0.166782\pi$
$984$	−3.24698	−0.103510
$985$	−14.8605	−0.473496
$986$	− 28.5642i	− 0.909669i
$987$	−44.1008	−1.40374
$988$	0	0
$989$	0.880395	0.0279949
$990$	− 1.91185i	− 0.0607627i
$991$	−44.1842	−1.40356	−0.701778	−	0.712395i	$-0.747610\pi$
$991$	−44.1842	−1.40356	−0.701778	+	0.712395i	$0.747610\pi$
$992$	−5.29590	−0.168145
$993$	35.3749i	1.12259i
$994$	9.11721i	0.289180i
$995$	26.3860i	0.836491i
$996$	− 9.49157i	− 0.300752i
$997$	50.8254	1.60966	0.804828	−	0.593509i	$-0.202258\pi$
$997$	50.8254	1.60966	0.804828	+	0.593509i	$0.202258\pi$
$998$	42.0737	1.33182
$999$	1.75302i	0.0554631i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.b.v.1351.6		6
13.5	odd	4		5070.2.a.bs.1.1	yes	3
13.8	odd	4		5070.2.a.bl.1.3	✓	3
13.12	even	2	inner	5070.2.b.v.1351.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5070.2.a.bl.1.3	✓	3	13.8	odd	4
5070.2.a.bs.1.1	yes	3	13.5	odd	4
5070.2.b.v.1351.1		6	13.12	even	2	inner
5070.2.b.v.1351.6		6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5070.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} +4.24698i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} +4.24698i q^{7} -1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.91185i q^{11} +1.00000 q^{12} -4.24698 q^{14} +1.00000i q^{15} +1.00000 q^{16} +3.33513 q^{17} +1.00000i q^{18} -4.85086i q^{19} +1.00000i q^{20} -4.24698i q^{21} +1.91185 q^{22} +0.445042 q^{23} +1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} -4.24698i q^{28} -8.56465 q^{29} -1.00000 q^{30} +5.29590i q^{31} +1.00000i q^{32} +1.91185i q^{33} +3.33513i q^{34} +4.24698 q^{35} -1.00000 q^{36} -1.75302i q^{37} +4.85086 q^{38} -1.00000 q^{40} +3.24698i q^{41} +4.24698 q^{42} +1.97823 q^{43} +1.91185i q^{44} -1.00000i q^{45} +0.445042i q^{46} -10.3840i q^{47} -1.00000 q^{48} -11.0368 q^{49} -1.00000i q^{50} -3.33513 q^{51} -1.14914 q^{53} -1.00000i q^{54} -1.91185 q^{55} +4.24698 q^{56} +4.85086i q^{57} -8.56465i q^{58} -13.4601i q^{59} -1.00000i q^{60} +1.13169 q^{61} -5.29590 q^{62} +4.24698i q^{63} -1.00000 q^{64} -1.91185 q^{66} +13.5308i q^{67} -3.33513 q^{68} -0.445042 q^{69} +4.24698i q^{70} -2.14675i q^{71} -1.00000i q^{72} -5.15883i q^{73} +1.75302 q^{74} +1.00000 q^{75} +4.85086i q^{76} +8.11960 q^{77} -14.5526 q^{79} -1.00000i q^{80} +1.00000 q^{81} -3.24698 q^{82} -9.49157i q^{83} +4.24698i q^{84} -3.33513i q^{85} +1.97823i q^{86} +8.56465 q^{87} -1.91185 q^{88} +1.25667i q^{89} +1.00000 q^{90} -0.445042 q^{92} -5.29590i q^{93} +10.3840 q^{94} -4.85086 q^{95} -1.00000i q^{96} -5.01507i q^{97} -11.0368i q^{98} -1.91185i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} - 6 q^{4} + 6 q^{9}+O(q^{10}) \) 6 * q - 6 * q^3 - 6 * q^4 + 6 * q^9	\( 6 q - 6 q^{3} - 6 q^{4} + 6 q^{9} + 6 q^{10} + 6 q^{12} - 16 q^{14} + 6 q^{16} + 18 q^{17} + 4 q^{22} + 2 q^{23} - 6 q^{25} - 6 q^{27} - 8 q^{29} - 6 q^{30} + 16 q^{35} - 6 q^{36} + 2 q^{38} - 6 q^{40} + 16 q^{42} + 18 q^{43} - 6 q^{48} - 10 q^{49} - 18 q^{51} - 34 q^{53} - 4 q^{55} + 16 q^{56} + 2 q^{61} - 4 q^{62} - 6 q^{64} - 4 q^{66} - 18 q^{68} - 2 q^{69} + 20 q^{74} + 6 q^{75} + 6 q^{77} - 6 q^{79} + 6 q^{81} - 10 q^{82} + 8 q^{87} - 4 q^{88} + 6 q^{90} - 2 q^{92} + 42 q^{94} - 2 q^{95}+O(q^{100}) \) 6 * q - 6 * q^3 - 6 * q^4 + 6 * q^9 + 6 * q^10 + 6 * q^12 - 16 * q^14 + 6 * q^16 + 18 * q^17 + 4 * q^22 + 2 * q^23 - 6 * q^25 - 6 * q^27 - 8 * q^29 - 6 * q^30 + 16 * q^35 - 6 * q^36 + 2 * q^38 - 6 * q^40 + 16 * q^42 + 18 * q^43 - 6 * q^48 - 10 * q^49 - 18 * q^51 - 34 * q^53 - 4 * q^55 + 16 * q^56 + 2 * q^61 - 4 * q^62 - 6 * q^64 - 4 * q^66 - 18 * q^68 - 2 * q^69 + 20 * q^74 + 6 * q^75 + 6 * q^77 - 6 * q^79 + 6 * q^81 - 10 * q^82 + 8 * q^87 - 4 * q^88 + 6 * q^90 - 2 * q^92 + 42 * q^94 - 2 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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