LMFDB - Embedded newform 5070.2.b.z.1351.2

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			1351.2
Root			$-0.445042i$ of defining polynomial
Character	$\chi$	$=$	5070.1351
Dual form			5070.2.b.z.1351.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} +3.35690i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} +3.35690i q^{7} +1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} +1.69202i q^{11} -1.00000 q^{12} +3.35690 q^{14} +1.00000i q^{15} +1.00000 q^{16} -0.939001 q^{17} -1.00000i q^{18} +4.85086i q^{19} -1.00000i q^{20} +3.35690i q^{21} +1.69202 q^{22} -4.04892 q^{23} +1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} -3.35690i q^{28} -8.12498 q^{29} +1.00000 q^{30} -4.08815i q^{31} -1.00000i q^{32} +1.69202i q^{33} +0.939001i q^{34} -3.35690 q^{35} -1.00000 q^{36} +11.6310i q^{37} +4.85086 q^{38} -1.00000 q^{40} -3.86294i q^{41} +3.35690 q^{42} -4.02177 q^{43} -1.69202i q^{44} +1.00000i q^{45} +4.04892i q^{46} -1.27413i q^{47} +1.00000 q^{48} -4.26875 q^{49} +1.00000i q^{50} -0.939001 q^{51} +5.74094 q^{53} -1.00000i q^{54} -1.69202 q^{55} -3.35690 q^{56} +4.85086i q^{57} +8.12498i q^{58} +0.417895i q^{59} -1.00000i q^{60} +0.198062 q^{61} -4.08815 q^{62} +3.35690i q^{63} -1.00000 q^{64} +1.69202 q^{66} -8.93900i q^{67} +0.939001 q^{68} -4.04892 q^{69} +3.35690i q^{70} +5.15883i q^{71} +1.00000i q^{72} -11.5308i q^{73} +11.6310 q^{74} -1.00000 q^{75} -4.85086i q^{76} -5.67994 q^{77} -4.94869 q^{79} +1.00000i q^{80} +1.00000 q^{81} -3.86294 q^{82} -13.1739i q^{83} -3.35690i q^{84} -0.939001i q^{85} +4.02177i q^{86} -8.12498 q^{87} -1.69202 q^{88} -9.47650i q^{89} +1.00000 q^{90} +4.04892 q^{92} -4.08815i q^{93} -1.27413 q^{94} -4.85086 q^{95} -1.00000i q^{96} -15.4547i q^{97} +4.26875i q^{98} +1.69202i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 12 q^{14} + 6 q^{16} + 14 q^{17} - 6 q^{23} - 6 q^{25} + 6 q^{27} + 6 q^{30} - 12 q^{35} - 6 q^{36} + 2 q^{38} - 6 q^{40} + 12 q^{42} - 18 q^{43} + 6 q^{48} - 10 q^{49} + 14 q^{51} + 6 q^{53} - 12 q^{56} + 10 q^{61} - 32 q^{62} - 6 q^{64} - 14 q^{68} - 6 q^{69} + 40 q^{74} - 6 q^{75} + 14 q^{77} + 34 q^{79} + 6 q^{81} - 34 q^{82} + 6 q^{90} + 6 q^{92} + 14 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 + 12 * q^14 + 6 * q^16 + 14 * q^17 - 6 * q^23 - 6 * q^25 + 6 * q^27 + 6 * q^30 - 12 * q^35 - 6 * q^36 + 2 * q^38 - 6 * q^40 + 12 * q^42 - 18 * q^43 + 6 * q^48 - 10 * q^49 + 14 * q^51 + 6 * q^53 - 12 * q^56 + 10 * q^61 - 32 * q^62 - 6 * q^64 - 14 * q^68 - 6 * q^69 + 40 * q^74 - 6 * q^75 + 14 * q^77 + 34 * q^79 + 6 * q^81 - 34 * q^82 + 6 * q^90 + 6 * q^92 + 14 * 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
$7$	3.35690i	1.26879i	0.773010	+	0.634394i	$0.218750\pi$
$7$	3.35690i	1.26879i	−0.773010	+	0.634394i	$0.781250\pi$
$8$	1.00000i	0.353553i
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	1.69202i	0.510164i	0.966919	+	0.255082i	$0.0821024\pi$
$11$	1.69202i	0.510164i	−0.966919	+	0.255082i	$0.917898\pi$
$12$	−1.00000	−0.288675
$13$	0	0
$13$	0	0
$14$	3.35690	0.897168
$15$	1.00000i	0.258199i
$16$	1.00000	0.250000
$17$	−0.939001	−0.227741	−0.113871	−	0.993496i	$-0.536325\pi$
$17$	−0.939001	−0.227741	−0.113871	+	0.993496i	$0.536325\pi$
$18$	− 1.00000i	− 0.235702i
$19$	4.85086i	1.11286i	0.830894	+	0.556431i	$0.187830\pi$
$19$	4.85086i	1.11286i	−0.830894	+	0.556431i	$0.812170\pi$
$20$	− 1.00000i	− 0.223607i
$21$	3.35690i	0.732535i
$22$	1.69202	0.360740
$23$	−4.04892	−0.844258	−0.422129	−	0.906536i	$-0.638717\pi$
$23$	−4.04892	−0.844258	−0.422129	+	0.906536i	$0.638717\pi$
$24$	1.00000i	0.204124i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	− 3.35690i	− 0.634394i
$29$	−8.12498	−1.50877	−0.754386	−	0.656432i	$-0.772065\pi$
$29$	−8.12498	−1.50877	−0.754386	+	0.656432i	$0.772065\pi$
$30$	1.00000	0.182574
$31$	− 4.08815i	− 0.734253i	−0.930171	−	0.367126i	$-0.880342\pi$
$31$	− 4.08815i	− 0.734253i	0.930171	−	0.367126i	$-0.119658\pi$
$32$	− 1.00000i	− 0.176777i
$33$	1.69202i	0.294543i
$34$	0.939001i	0.161037i
$35$	−3.35690	−0.567419
$36$	−1.00000	−0.166667
$37$	11.6310i	1.91213i	0.293157	+	0.956064i	$0.405294\pi$
$37$	11.6310i	1.91213i	−0.293157	+	0.956064i	$0.594706\pi$
$38$	4.85086	0.786913
$39$	0	0
$40$	−1.00000	−0.158114
$41$	− 3.86294i	− 0.603289i	−0.953420	−	0.301645i	$-0.902464\pi$
$41$	− 3.86294i	− 0.603289i	0.953420	−	0.301645i	$-0.0975356\pi$
$42$	3.35690	0.517980
$43$	−4.02177	−0.613314	−0.306657	−	0.951820i	$-0.599210\pi$
$43$	−4.02177	−0.613314	−0.306657	+	0.951820i	$0.599210\pi$
$44$	− 1.69202i	− 0.255082i
$45$	1.00000i	0.149071i
$46$	4.04892i	0.596980i
$47$	− 1.27413i	− 0.185850i	−0.995673	−	0.0929252i	$-0.970378\pi$
$47$	− 1.27413i	− 0.185850i	0.995673	−	0.0929252i	$-0.0296218\pi$
$48$	1.00000	0.144338
$49$	−4.26875	−0.609821
$50$	1.00000i	0.141421i
$51$	−0.939001	−0.131486
$52$	0	0
$53$	5.74094	0.788579	0.394289	−	0.918986i	$-0.370991\pi$
$53$	5.74094	0.788579	0.394289	+	0.918986i	$0.370991\pi$
$54$	− 1.00000i	− 0.136083i
$55$	−1.69202	−0.228152
$56$	−3.35690	−0.448584
$57$	4.85086i	0.642511i
$58$	8.12498i	1.06686i
$59$	0.417895i	0.0544053i	0.999630	+	0.0272026i	$0.00865994\pi$
$59$	0.417895i	0.0544053i	−0.999630	+	0.0272026i	$0.991340\pi$
$60$	− 1.00000i	− 0.129099i
$61$	0.198062	0.0253593	0.0126796	−	0.999920i	$-0.495964\pi$
$61$	0.198062	0.0253593	0.0126796	+	0.999920i	$0.495964\pi$
$62$	−4.08815	−0.519195
$63$	3.35690i	0.422929i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	1.69202	0.208273
$67$	− 8.93900i	− 1.09207i	−0.837761	−	0.546036i	$-0.816136\pi$
$67$	− 8.93900i	− 1.09207i	0.837761	−	0.546036i	$-0.183864\pi$
$68$	0.939001	0.113871
$69$	−4.04892	−0.487432
$70$	3.35690i	0.401226i
$71$	5.15883i	0.612241i	0.951993	+	0.306120i	$0.0990310\pi$
$71$	5.15883i	0.612241i	−0.951993	+	0.306120i	$0.900969\pi$
$72$	1.00000i	0.117851i
$73$	− 11.5308i	− 1.34958i	−0.738011	−	0.674789i	$-0.764235\pi$
$73$	− 11.5308i	− 1.34958i	0.738011	−	0.674789i	$-0.235765\pi$
$74$	11.6310	1.35208
$75$	−1.00000	−0.115470
$76$	− 4.85086i	− 0.556431i
$77$	−5.67994	−0.647289
$78$	0	0
$79$	−4.94869	−0.556771	−0.278386	−	0.960469i	$-0.589799\pi$
$79$	−4.94869	−0.556771	−0.278386	+	0.960469i	$0.589799\pi$
$80$	1.00000i	0.111803i
$81$	1.00000	0.111111
$82$	−3.86294	−0.426590
$83$	− 13.1739i	− 1.44602i	−0.690836	−	0.723012i	$-0.742757\pi$
$83$	− 13.1739i	− 1.44602i	0.690836	−	0.723012i	$-0.257243\pi$
$84$	− 3.35690i	− 0.366267i
$85$	− 0.939001i	− 0.101849i
$86$	4.02177i	0.433679i
$87$	−8.12498	−0.871089
$88$	−1.69202	−0.180370
$89$	− 9.47650i	− 1.00451i	−0.864720	−	0.502254i	$-0.832504\pi$
$89$	− 9.47650i	− 1.00451i	0.864720	−	0.502254i	$-0.167496\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	4.04892	0.422129
$93$	− 4.08815i	− 0.423921i
$94$	−1.27413	−0.131416
$95$	−4.85086	−0.497687
$96$	− 1.00000i	− 0.102062i
$97$	− 15.4547i	− 1.56919i	−0.620009	−	0.784595i	$-0.712871\pi$
$97$	− 15.4547i	− 1.56919i	0.620009	−	0.784595i	$-0.287129\pi$
$98$	4.26875i	0.431209i
$99$	1.69202i	0.170055i
$100$	1.00000	0.100000
$101$	−5.38404	−0.535732	−0.267866	−	0.963456i	$-0.586318\pi$
$101$	−5.38404	−0.535732	−0.267866	+	0.963456i	$0.586318\pi$
$102$	0.939001i	0.0929750i
$103$	−4.08575	−0.402581	−0.201291	−	0.979532i	$-0.564514\pi$
$103$	−4.08575	−0.402581	−0.201291	+	0.979532i	$0.564514\pi$
$104$	0	0
$105$	−3.35690	−0.327599
$106$	− 5.74094i	− 0.557609i
$107$	−16.1196	−1.55834	−0.779171	−	0.626812i	$-0.784359\pi$
$107$	−16.1196	−1.55834	−0.779171	+	0.626812i	$0.784359\pi$
$108$	−1.00000	−0.0962250
$109$	14.2325i	1.36323i	0.731713	+	0.681613i	$0.238721\pi$
$109$	14.2325i	1.36323i	−0.731713	+	0.681613i	$0.761279\pi$
$110$	1.69202i	0.161328i
$111$	11.6310i	1.10397i
$112$	3.35690i	0.317197i
$113$	−13.3274	−1.25373	−0.626866	−	0.779127i	$-0.715663\pi$
$113$	−13.3274	−1.25373	−0.626866	+	0.779127i	$0.715663\pi$
$114$	4.85086	0.454324
$115$	− 4.04892i	− 0.377563i
$116$	8.12498	0.754386
$117$	0	0
$118$	0.417895	0.0384703
$119$	− 3.15213i	− 0.288955i
$120$	−1.00000	−0.0912871
$121$	8.13706	0.739733
$122$	− 0.198062i	− 0.0179317i
$123$	− 3.86294i	− 0.348309i
$124$	4.08815i	0.367126i
$125$	− 1.00000i	− 0.0894427i
$126$	3.35690	0.299056
$127$	12.8509	1.14033	0.570164	−	0.821531i	$-0.306879\pi$
$127$	12.8509	1.14033	0.570164	+	0.821531i	$0.306879\pi$
$128$	1.00000i	0.0883883i
$129$	−4.02177	−0.354097
$130$	0	0
$131$	−0.570024	−0.0498032	−0.0249016	−	0.999690i	$-0.507927\pi$
$131$	−0.570024	−0.0498032	−0.0249016	+	0.999690i	$0.507927\pi$
$132$	− 1.69202i	− 0.147272i
$133$	−16.2838	−1.41199
$134$	−8.93900	−0.772212
$135$	1.00000i	0.0860663i
$136$	− 0.939001i	− 0.0805187i
$137$	21.5405i	1.84033i	0.391533	+	0.920164i	$0.371945\pi$
$137$	21.5405i	1.84033i	−0.391533	+	0.920164i	$0.628055\pi$
$138$	4.04892i	0.344667i
$139$	7.09783	0.602030	0.301015	−	0.953619i	$-0.402674\pi$
$139$	7.09783	0.602030	0.301015	+	0.953619i	$0.402674\pi$
$140$	3.35690	0.283709
$141$	− 1.27413i	− 0.107301i
$142$	5.15883	0.432920
$143$	0	0
$144$	1.00000	0.0833333
$145$	− 8.12498i	− 0.674743i
$146$	−11.5308	−0.954295
$147$	−4.26875	−0.352081
$148$	− 11.6310i	− 0.956064i
$149$	1.14914i	0.0941416i	0.998892	+	0.0470708i	$0.0149886\pi$
$149$	1.14914i	0.0941416i	−0.998892	+	0.0470708i	$0.985011\pi$
$150$	1.00000i	0.0816497i
$151$	0.0217703i	0.00177164i	1.00000		0.000885819i	$0.000281965\pi$
$151$	0.0217703i	0.00177164i	−1.00000		0.000885819i	$0.999718\pi$
$152$	−4.85086	−0.393456
$153$	−0.939001	−0.0759137
$154$	5.67994i	0.457703i
$155$	4.08815	0.328368
$156$	0	0
$157$	−10.1588	−0.810763	−0.405382	−	0.914148i	$-0.632861\pi$
$157$	−10.1588	−0.810763	−0.405382	+	0.914148i	$0.632861\pi$
$158$	4.94869i	0.393697i
$159$	5.74094	0.455286
$160$	1.00000	0.0790569
$161$	− 13.5918i	− 1.07118i
$162$	− 1.00000i	− 0.0785674i
$163$	19.1511i	1.50003i	0.661422	+	0.750014i	$0.269953\pi$
$163$	19.1511i	1.50003i	−0.661422	+	0.750014i	$0.730047\pi$
$164$	3.86294i	0.301645i
$165$	−1.69202	−0.131724
$166$	−13.1739	−1.02249
$167$	− 15.2174i	− 1.17756i	−0.808293	−	0.588780i	$-0.799608\pi$
$167$	− 15.2174i	− 1.17756i	0.808293	−	0.588780i	$-0.200392\pi$
$168$	−3.35690	−0.258990
$169$	0	0
$170$	−0.939001	−0.0720181
$171$	4.85086i	0.370954i
$172$	4.02177	0.306657
$173$	−21.8213	−1.65904	−0.829522	−	0.558474i	$-0.811387\pi$
$173$	−21.8213	−1.65904	−0.829522	+	0.558474i	$0.811387\pi$
$174$	8.12498i	0.615953i
$175$	− 3.35690i	− 0.253757i
$176$	1.69202i	0.127541i
$177$	0.417895i	0.0314109i
$178$	−9.47650	−0.710294
$179$	20.2513	1.51365	0.756826	−	0.653616i	$-0.226749\pi$
$179$	20.2513	1.51365	0.756826	+	0.653616i	$0.226749\pi$
$180$	− 1.00000i	− 0.0745356i
$181$	−14.4969	−1.07755	−0.538775	−	0.842450i	$-0.681113\pi$
$181$	−14.4969	−1.07755	−0.538775	+	0.842450i	$0.681113\pi$
$182$	0	0
$183$	0.198062	0.0146412
$184$	− 4.04892i	− 0.298490i
$185$	−11.6310	−0.855130
$186$	−4.08815	−0.299757
$187$	− 1.58881i	− 0.116185i
$188$	1.27413i	0.0929252i
$189$	3.35690i	0.244178i
$190$	4.85086i	0.351918i
$191$	−10.1806	−0.736643	−0.368321	−	0.929699i	$-0.620067\pi$
$191$	−10.1806	−0.736643	−0.368321	+	0.929699i	$0.620067\pi$
$192$	−1.00000	−0.0721688
$193$	27.6407i	1.98962i	0.101741	+	0.994811i	$0.467559\pi$
$193$	27.6407i	1.98962i	−0.101741	+	0.994811i	$0.532441\pi$
$194$	−15.4547	−1.10958
$195$	0	0
$196$	4.26875	0.304911
$197$	13.0261i	0.928070i	0.885817	+	0.464035i	$0.153599\pi$
$197$	13.0261i	0.928070i	−0.885817	+	0.464035i	$0.846401\pi$
$198$	1.69202	0.120247
$199$	−6.67563	−0.473223	−0.236611	−	0.971604i	$-0.576037\pi$
$199$	−6.67563	−0.473223	−0.236611	+	0.971604i	$0.576037\pi$
$200$	− 1.00000i	− 0.0707107i
$201$	− 8.93900i	− 0.630509i
$202$	5.38404i	0.378820i
$203$	− 27.2747i	− 1.91431i
$204$	0.939001	0.0657432
$205$	3.86294	0.269799
$206$	4.08575i	0.284668i
$207$	−4.04892	−0.281419
$208$	0	0
$209$	−8.20775	−0.567742
$210$	3.35690i	0.231648i
$211$	−11.3056	−0.778309	−0.389154	−	0.921173i	$-0.627233\pi$
$211$	−11.3056	−0.778309	−0.389154	+	0.921173i	$0.627233\pi$
$212$	−5.74094	−0.394289
$213$	5.15883i	0.353477i
$214$	16.1196i	1.10191i
$215$	− 4.02177i	− 0.274282i
$216$	1.00000i	0.0680414i
$217$	13.7235	0.931611
$218$	14.2325	0.963947
$219$	− 11.5308i	− 0.779179i
$220$	1.69202	0.114076
$221$	0	0
$222$	11.6310	0.780623
$223$	− 18.0911i	− 1.21147i	−0.795666	−	0.605736i	$-0.792879\pi$
$223$	− 18.0911i	− 1.21147i	0.795666	−	0.605736i	$-0.207121\pi$
$224$	3.35690	0.224292
$225$	−1.00000	−0.0666667
$226$	13.3274i	0.886523i
$227$	9.37196i	0.622039i	0.950404	+	0.311019i	$0.100670\pi$
$227$	9.37196i	0.622039i	−0.950404	+	0.311019i	$0.899330\pi$
$228$	− 4.85086i	− 0.321256i
$229$	21.2664i	1.40532i	0.711526	+	0.702660i	$0.248005\pi$
$229$	21.2664i	1.40532i	−0.711526	+	0.702660i	$0.751995\pi$
$230$	−4.04892	−0.266978
$231$	−5.67994	−0.373713
$232$	− 8.12498i	− 0.533431i
$233$	−2.03146	−0.133085	−0.0665427	−	0.997784i	$-0.521197\pi$
$233$	−2.03146	−0.133085	−0.0665427	+	0.997784i	$0.521197\pi$
$234$	0	0
$235$	1.27413	0.0831149
$236$	− 0.417895i	− 0.0272026i
$237$	−4.94869	−0.321452
$238$	−3.15213	−0.204322
$239$	11.9903i	0.775589i	0.921746	+	0.387794i	$0.126763\pi$
$239$	11.9903i	0.775589i	−0.921746	+	0.387794i	$0.873237\pi$
$240$	1.00000i	0.0645497i
$241$	21.7627i	1.40186i	0.713230	+	0.700930i	$0.247231\pi$
$241$	21.7627i	1.40186i	−0.713230	+	0.700930i	$0.752769\pi$
$242$	− 8.13706i	− 0.523070i
$243$	1.00000	0.0641500
$244$	−0.198062	−0.0126796
$245$	− 4.26875i	− 0.272720i
$246$	−3.86294	−0.246292
$247$	0	0
$248$	4.08815	0.259598
$249$	− 13.1739i	− 0.834862i
$250$	−1.00000	−0.0632456
$251$	−13.7506	−0.867932	−0.433966	−	0.900929i	$-0.642886\pi$
$251$	−13.7506	−0.867932	−0.433966	+	0.900929i	$0.642886\pi$
$252$	− 3.35690i	− 0.211465i
$253$	− 6.85086i	− 0.430710i
$254$	− 12.8509i	− 0.806334i
$255$	− 0.939001i	− 0.0588025i
$256$	1.00000	0.0625000
$257$	24.1782	1.50820	0.754098	−	0.656762i	$-0.228075\pi$
$257$	24.1782	1.50820	0.754098	+	0.656762i	$0.228075\pi$
$258$	4.02177i	0.250384i
$259$	−39.0441	−2.42608
$260$	0	0
$261$	−8.12498	−0.502924
$262$	0.570024i	0.0352162i
$263$	0.960771	0.0592437	0.0296218	−	0.999561i	$-0.490570\pi$
$263$	0.960771	0.0592437	0.0296218	+	0.999561i	$0.490570\pi$
$264$	−1.69202	−0.104137
$265$	5.74094i	0.352663i
$266$	16.2838i	0.998425i
$267$	− 9.47650i	− 0.579952i
$268$	8.93900i	0.546036i
$269$	10.0761	0.614348	0.307174	−	0.951653i	$-0.400617\pi$
$269$	10.0761	0.614348	0.307174	+	0.951653i	$0.400617\pi$
$270$	1.00000	0.0608581
$271$	− 14.9584i	− 0.908657i	−0.890834	−	0.454328i	$-0.849879\pi$
$271$	− 14.9584i	− 0.908657i	0.890834	−	0.454328i	$-0.150121\pi$
$272$	−0.939001	−0.0569353
$273$	0	0
$274$	21.5405	1.30131
$275$	− 1.69202i	− 0.102033i
$276$	4.04892	0.243716
$277$	13.0465	0.783890	0.391945	−	0.919989i	$-0.371802\pi$
$277$	13.0465	0.783890	0.391945	+	0.919989i	$0.371802\pi$
$278$	− 7.09783i	− 0.425700i
$279$	− 4.08815i	− 0.244751i
$280$	− 3.35690i	− 0.200613i
$281$	16.2881i	0.971668i	0.874051	+	0.485834i	$0.161484\pi$
$281$	16.2881i	0.971668i	−0.874051	+	0.485834i	$0.838516\pi$
$282$	−1.27413	−0.0758731
$283$	12.3773	0.735756	0.367878	−	0.929874i	$-0.380084\pi$
$283$	12.3773	0.735756	0.367878	+	0.929874i	$0.380084\pi$
$284$	− 5.15883i	− 0.306120i
$285$	−4.85086	−0.287340
$286$	0	0
$287$	12.9675	0.765446
$288$	− 1.00000i	− 0.0589256i
$289$	−16.1183	−0.948134
$290$	−8.12498	−0.477115
$291$	− 15.4547i	− 0.905972i
$292$	11.5308i	0.674789i
$293$	− 1.73663i	− 0.101455i	−0.998713	−	0.0507274i	$-0.983846\pi$
$293$	− 1.73663i	− 0.101455i	0.998713	−	0.0507274i	$-0.0161540\pi$
$294$	4.26875i	0.248959i
$295$	−0.417895	−0.0243308
$296$	−11.6310	−0.676039
$297$	1.69202i	0.0981810i
$298$	1.14914	0.0665682
$299$	0	0
$300$	1.00000	0.0577350
$301$	− 13.5007i	− 0.778165i
$302$	0.0217703	0.00125274
$303$	−5.38404	−0.309305
$304$	4.85086i	0.278216i
$305$	0.198062i	0.0113410i
$306$	0.939001i	0.0536791i
$307$	25.4403i	1.45195i	0.687720	+	0.725976i	$0.258612\pi$
$307$	25.4403i	1.45195i	−0.687720	+	0.725976i	$0.741388\pi$
$308$	5.67994	0.323645
$309$	−4.08575	−0.232430
$310$	− 4.08815i	− 0.232191i
$311$	10.0325	0.568892	0.284446	−	0.958692i	$-0.408190\pi$
$311$	10.0325	0.568892	0.284446	+	0.958692i	$0.408190\pi$
$312$	0	0
$313$	−22.3274	−1.26202	−0.631008	−	0.775776i	$-0.717359\pi$
$313$	−22.3274	−1.26202	−0.631008	+	0.775776i	$0.717359\pi$
$314$	10.1588i	0.573296i
$315$	−3.35690	−0.189140
$316$	4.94869	0.278386
$317$	5.33811i	0.299818i	0.988700	+	0.149909i	$0.0478981\pi$
$317$	5.33811i	0.299818i	−0.988700	+	0.149909i	$0.952102\pi$
$318$	− 5.74094i	− 0.321936i
$319$	− 13.7476i	− 0.769720i
$320$	− 1.00000i	− 0.0559017i
$321$	−16.1196	−0.899709
$322$	−13.5918	−0.757441
$323$	− 4.55496i	− 0.253445i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	19.1511	1.06068
$327$	14.2325i	0.787059i
$328$	3.86294	0.213295
$329$	4.27711	0.235805
$330$	1.69202i	0.0931427i
$331$	4.49827i	0.247247i	0.992329	+	0.123624i	$0.0394516\pi$
$331$	4.49827i	0.247247i	−0.992329	+	0.123624i	$0.960548\pi$
$332$	13.1739i	0.723012i
$333$	11.6310i	0.637376i
$334$	−15.2174	−0.832661
$335$	8.93900	0.488390
$336$	3.35690i	0.183134i
$337$	11.5676	0.630129	0.315064	−	0.949070i	$-0.397974\pi$
$337$	11.5676	0.630129	0.315064	+	0.949070i	$0.397974\pi$
$338$	0	0
$339$	−13.3274	−0.723843
$340$	0.939001i	0.0509245i
$341$	6.91723	0.374589
$342$	4.85086	0.262304
$343$	9.16852i	0.495054i
$344$	− 4.02177i	− 0.216839i
$345$	− 4.04892i	− 0.217986i
$346$	21.8213i	1.17312i
$347$	27.1226	1.45602	0.728008	−	0.685568i	$-0.240446\pi$
$347$	27.1226	1.45602	0.728008	+	0.685568i	$0.240446\pi$
$348$	8.12498	0.435545
$349$	12.8562i	0.688178i	0.938937	+	0.344089i	$0.111812\pi$
$349$	12.8562i	0.688178i	−0.938937	+	0.344089i	$0.888188\pi$
$350$	−3.35690	−0.179434
$351$	0	0
$352$	1.69202	0.0901850
$353$	− 24.6262i	− 1.31072i	−0.755316	−	0.655361i	$-0.772516\pi$
$353$	− 24.6262i	− 1.31072i	0.755316	−	0.655361i	$-0.227484\pi$
$354$	0.417895	0.0222109
$355$	−5.15883	−0.273802
$356$	9.47650i	0.502254i
$357$	− 3.15213i	− 0.166828i
$358$	− 20.2513i	− 1.07031i
$359$	− 23.8629i	− 1.25944i	−0.776823	−	0.629719i	$-0.783170\pi$
$359$	− 23.8629i	− 1.25944i	0.776823	−	0.629719i	$-0.216830\pi$
$360$	−1.00000	−0.0527046
$361$	−4.53079	−0.238463
$362$	14.4969i	0.761942i
$363$	8.13706	0.427085
$364$	0	0
$365$	11.5308	0.603549
$366$	− 0.198062i	− 0.0103529i
$367$	33.1933	1.73267	0.866337	−	0.499459i	$-0.166468\pi$
$367$	33.1933	1.73267	0.866337	+	0.499459i	$0.166468\pi$
$368$	−4.04892	−0.211064
$369$	− 3.86294i	− 0.201096i
$370$	11.6310i	0.604668i
$371$	19.2717i	1.00054i
$372$	4.08815i	0.211960i
$373$	−10.8465	−0.561613	−0.280806	−	0.959764i	$-0.590602\pi$
$373$	−10.8465	−0.561613	−0.280806	+	0.959764i	$0.590602\pi$
$374$	−1.58881	−0.0821554
$375$	− 1.00000i	− 0.0516398i
$376$	1.27413	0.0657081
$377$	0	0
$378$	3.35690	0.172660
$379$	6.09485i	0.313071i	0.987672	+	0.156536i	$0.0500326\pi$
$379$	6.09485i	0.313071i	−0.987672	+	0.156536i	$0.949967\pi$
$380$	4.85086	0.248844
$381$	12.8509	0.658369
$382$	10.1806i	0.520885i
$383$	7.95407i	0.406434i	0.979134	+	0.203217i	$0.0651397\pi$
$383$	7.95407i	0.406434i	−0.979134	+	0.203217i	$0.934860\pi$
$384$	1.00000i	0.0510310i
$385$	− 5.67994i	− 0.289477i
$386$	27.6407	1.40688
$387$	−4.02177	−0.204438
$388$	15.4547i	0.784595i
$389$	−1.50365	−0.0762380	−0.0381190	−	0.999273i	$-0.512137\pi$
$389$	−1.50365	−0.0762380	−0.0381190	+	0.999273i	$0.512137\pi$
$390$	0	0
$391$	3.80194	0.192272
$392$	− 4.26875i	− 0.215604i
$393$	−0.570024	−0.0287539
$394$	13.0261	0.656245
$395$	− 4.94869i	− 0.248996i
$396$	− 1.69202i	− 0.0850273i
$397$	34.9627i	1.75473i	0.479826	+	0.877364i	$0.340700\pi$
$397$	34.9627i	1.75473i	−0.479826	+	0.877364i	$0.659300\pi$
$398$	6.67563i	0.334619i
$399$	−16.2838	−0.815210
$400$	−1.00000	−0.0500000
$401$	− 31.1148i	− 1.55380i	−0.629624	−	0.776900i	$-0.716791\pi$
$401$	− 31.1148i	− 1.55380i	0.629624	−	0.776900i	$-0.283209\pi$
$402$	−8.93900	−0.445837
$403$	0	0
$404$	5.38404	0.267866
$405$	1.00000i	0.0496904i
$406$	−27.2747	−1.35362
$407$	−19.6799	−0.975498
$408$	− 0.939001i	− 0.0464875i
$409$	32.7318i	1.61849i	0.587474	+	0.809243i	$0.300122\pi$
$409$	32.7318i	1.61849i	−0.587474	+	0.809243i	$0.699878\pi$
$410$	− 3.86294i	− 0.190777i
$411$	21.5405i	1.06251i
$412$	4.08575	0.201291
$413$	−1.40283	−0.0690287
$414$	4.04892i	0.198993i
$415$	13.1739	0.646681
$416$	0	0
$417$	7.09783	0.347582
$418$	8.20775i	0.401454i
$419$	13.0694	0.638480	0.319240	−	0.947674i	$-0.396572\pi$
$419$	13.0694	0.638480	0.319240	+	0.947674i	$0.396572\pi$
$420$	3.35690	0.163800
$421$	− 16.4789i	− 0.803132i	−0.915830	−	0.401566i	$-0.868466\pi$
$421$	− 16.4789i	− 0.803132i	0.915830	−	0.401566i	$-0.131534\pi$
$422$	11.3056i	0.550347i
$423$	− 1.27413i	− 0.0619502i
$424$	5.74094i	0.278805i
$425$	0.939001	0.0455482
$426$	5.15883	0.249946
$427$	0.664874i	0.0321755i
$428$	16.1196	0.779171
$429$	0	0
$430$	−4.02177	−0.193947
$431$	41.0640i	1.97798i	0.147974	+	0.988991i	$0.452725\pi$
$431$	41.0640i	1.97798i	−0.147974	+	0.988991i	$0.547275\pi$
$432$	1.00000	0.0481125
$433$	6.04593	0.290549	0.145275	−	0.989391i	$-0.453593\pi$
$433$	6.04593	0.290549	0.145275	+	0.989391i	$0.453593\pi$
$434$	− 13.7235i	− 0.658748i
$435$	− 8.12498i	− 0.389563i
$436$	− 14.2325i	− 0.681613i
$437$	− 19.6407i	− 0.939543i
$438$	−11.5308	−0.550963
$439$	26.2543	1.25305	0.626524	−	0.779402i	$-0.284477\pi$
$439$	26.2543	1.25305	0.626524	+	0.779402i	$0.284477\pi$
$440$	− 1.69202i	− 0.0806640i
$441$	−4.26875	−0.203274
$442$	0	0
$443$	−39.3846	−1.87122	−0.935610	−	0.353035i	$-0.885150\pi$
$443$	−39.3846	−1.87122	−0.935610	+	0.353035i	$0.885150\pi$
$444$	− 11.6310i	− 0.551984i
$445$	9.47650	0.449229
$446$	−18.0911	−0.856640
$447$	1.14914i	0.0543527i
$448$	− 3.35690i	− 0.158598i
$449$	− 9.39075i	− 0.443177i	−0.975140	−	0.221588i	$-0.928876\pi$
$449$	− 9.39075i	− 0.443177i	0.975140	−	0.221588i	$-0.0711241\pi$
$450$	1.00000i	0.0471405i
$451$	6.53617	0.307776
$452$	13.3274	0.626866
$453$	0.0217703i	0.00102286i
$454$	9.37196	0.439848
$455$	0	0
$456$	−4.85086	−0.227162
$457$	2.66487i	0.124658i	0.998056	+	0.0623288i	$0.0198527\pi$
$457$	2.66487i	0.124658i	−0.998056	+	0.0623288i	$0.980147\pi$
$458$	21.2664	0.993712
$459$	−0.939001	−0.0438288
$460$	4.04892i	0.188782i
$461$	24.5284i	1.14240i	0.820810	+	0.571201i	$0.193522\pi$
$461$	24.5284i	1.14240i	−0.820810	+	0.571201i	$0.806478\pi$
$462$	5.67994i	0.264255i
$463$	− 4.50498i	− 0.209364i	−0.994506	−	0.104682i	$-0.966618\pi$
$463$	− 4.50498i	− 0.209364i	0.994506	−	0.104682i	$-0.0333825\pi$
$464$	−8.12498	−0.377193
$465$	4.08815	0.189583
$466$	2.03146i	0.0941055i
$467$	−21.3317	−0.987112	−0.493556	−	0.869714i	$-0.664303\pi$
$467$	−21.3317	−0.987112	−0.493556	+	0.869714i	$0.664303\pi$
$468$	0	0
$469$	30.0073	1.38561
$470$	− 1.27413i	− 0.0587711i
$471$	−10.1588	−0.468094
$472$	−0.417895	−0.0192352
$473$	− 6.80492i	− 0.312891i
$474$	4.94869i	0.227301i
$475$	− 4.85086i	− 0.222572i
$476$	3.15213i	0.144478i
$477$	5.74094	0.262860
$478$	11.9903	0.548424
$479$	− 32.9517i	− 1.50560i	−0.658249	−	0.752800i	$-0.728703\pi$
$479$	− 32.9517i	− 1.50560i	0.658249	−	0.752800i	$-0.271297\pi$
$480$	1.00000	0.0456435
$481$	0	0
$482$	21.7627	0.991264
$483$	− 13.5918i	− 0.618448i
$484$	−8.13706	−0.369867
$485$	15.4547	0.701763
$486$	− 1.00000i	− 0.0453609i
$487$	9.33034i	0.422798i	0.977400	+	0.211399i	$0.0678020\pi$
$487$	9.33034i	0.422798i	−0.977400	+	0.211399i	$0.932198\pi$
$488$	0.198062i	0.00896586i
$489$	19.1511i	0.866041i
$490$	−4.26875	−0.192842
$491$	20.4349	0.922213	0.461107	−	0.887345i	$-0.347453\pi$
$491$	20.4349	0.922213	0.461107	+	0.887345i	$0.347453\pi$
$492$	3.86294i	0.174155i
$493$	7.62937	0.343609
$494$	0	0
$495$	−1.69202	−0.0760507
$496$	− 4.08815i	− 0.183563i
$497$	−17.3177	−0.776804
$498$	−13.1739	−0.590337
$499$	− 3.54958i	− 0.158901i	−0.996839	−	0.0794505i	$-0.974683\pi$
$499$	− 3.54958i	− 0.158901i	0.996839	−	0.0794505i	$-0.0253166\pi$
$500$	1.00000i	0.0447214i
$501$	− 15.2174i	− 0.679865i
$502$	13.7506i	0.613721i
$503$	−19.8146	−0.883490	−0.441745	−	0.897141i	$-0.645640\pi$
$503$	−19.8146	−0.883490	−0.441745	+	0.897141i	$0.645640\pi$
$504$	−3.35690	−0.149528
$505$	− 5.38404i	− 0.239587i
$506$	−6.85086	−0.304558
$507$	0	0
$508$	−12.8509	−0.570164
$509$	− 32.4325i	− 1.43754i	−0.695246	−	0.718772i	$-0.744704\pi$
$509$	− 32.4325i	− 1.43754i	0.695246	−	0.718772i	$-0.255296\pi$
$510$	−0.939001	−0.0415797
$511$	38.7077	1.71233
$512$	− 1.00000i	− 0.0441942i
$513$	4.85086i	0.214170i
$514$	− 24.1782i	− 1.06646i
$515$	− 4.08575i	− 0.180040i
$516$	4.02177	0.177049
$517$	2.15585	0.0948142
$518$	39.0441i	1.71550i
$519$	−21.8213	−0.957849
$520$	0	0
$521$	7.87694	0.345095	0.172547	−	0.985001i	$-0.444800\pi$
$521$	7.87694	0.345095	0.172547	+	0.985001i	$0.444800\pi$
$522$	8.12498i	0.355621i
$523$	−23.4276	−1.02442	−0.512208	−	0.858861i	$-0.671172\pi$
$523$	−23.4276	−1.02442	−0.512208	+	0.858861i	$0.671172\pi$
$524$	0.570024	0.0249016
$525$	− 3.35690i	− 0.146507i
$526$	− 0.960771i	− 0.0418916i
$527$	3.83877i	0.167220i
$528$	1.69202i	0.0736358i
$529$	−6.60627	−0.287229
$530$	5.74094	0.249370
$531$	0.417895i	0.0181351i
$532$	16.2838	0.705993
$533$	0	0
$534$	−9.47650	−0.410088
$535$	− 16.1196i	− 0.696911i
$536$	8.93900	0.386106
$537$	20.2513	0.873908
$538$	− 10.0761i	− 0.434410i
$539$	− 7.22282i	− 0.311109i
$540$	− 1.00000i	− 0.0430331i
$541$	− 1.21552i	− 0.0522593i	−0.999659	−	0.0261297i	$-0.991682\pi$
$541$	− 1.21552i	− 0.0522593i	0.999659	−	0.0261297i	$-0.00831828\pi$
$542$	−14.9584	−0.642517
$543$	−14.4969	−0.622123
$544$	0.939001i	0.0402593i
$545$	−14.2325	−0.609654
$546$	0	0
$547$	−15.1371	−0.647214	−0.323607	−	0.946192i	$-0.604896\pi$
$547$	−15.1371	−0.647214	−0.323607	+	0.946192i	$0.604896\pi$
$548$	− 21.5405i	− 0.920164i
$549$	0.198062	0.00845309
$550$	−1.69202	−0.0721480
$551$	− 39.4131i	− 1.67905i
$552$	− 4.04892i	− 0.172333i
$553$	− 16.6122i	− 0.706424i
$554$	− 13.0465i	− 0.554294i
$555$	−11.6310	−0.493709
$556$	−7.09783	−0.301015
$557$	0.147817i	0.00626321i	0.999995	+	0.00313160i	$0.000996822\pi$
$557$	0.147817i	0.00626321i	−0.999995	+	0.00313160i	$0.999003\pi$
$558$	−4.08815	−0.173065
$559$	0	0
$560$	−3.35690	−0.141855
$561$	− 1.58881i	− 0.0670796i
$562$	16.2881	0.687073
$563$	−40.6945	−1.71507	−0.857535	−	0.514426i	$-0.828005\pi$
$563$	−40.6945	−1.71507	−0.857535	+	0.514426i	$0.828005\pi$
$564$	1.27413i	0.0536504i
$565$	− 13.3274i	− 0.560686i
$566$	− 12.3773i	− 0.520258i
$567$	3.35690i	0.140976i
$568$	−5.15883	−0.216460
$569$	26.2433	1.10017	0.550087	−	0.835107i	$-0.314594\pi$
$569$	26.2433	1.10017	0.550087	+	0.835107i	$0.314594\pi$
$570$	4.85086i	0.203180i
$571$	−2.31575	−0.0969110	−0.0484555	−	0.998825i	$-0.515430\pi$
$571$	−2.31575	−0.0969110	−0.0484555	+	0.998825i	$0.515430\pi$
$572$	0	0
$573$	−10.1806	−0.425301
$574$	− 12.9675i	− 0.541252i
$575$	4.04892	0.168852
$576$	−1.00000	−0.0416667
$577$	14.5241i	0.604646i	0.953206	+	0.302323i	$0.0977621\pi$
$577$	14.5241i	0.604646i	−0.953206	+	0.302323i	$0.902238\pi$
$578$	16.1183i	0.670432i
$579$	27.6407i	1.14871i
$580$	8.12498i	0.337372i
$581$	44.2234	1.83470
$582$	−15.4547	−0.640619
$583$	9.71379i	0.402304i
$584$	11.5308	0.477148
$585$	0	0
$586$	−1.73663	−0.0717394
$587$	32.9028i	1.35804i	0.734119	+	0.679021i	$0.237596\pi$
$587$	32.9028i	1.35804i	−0.734119	+	0.679021i	$0.762404\pi$
$588$	4.26875	0.176040
$589$	19.8310	0.817122
$590$	0.417895i	0.0172045i
$591$	13.0261i	0.535821i
$592$	11.6310i	0.478032i
$593$	− 36.0146i	− 1.47894i	−0.673188	−	0.739471i	$-0.735076\pi$
$593$	− 36.0146i	− 1.47894i	0.673188	−	0.739471i	$-0.264924\pi$
$594$	1.69202	0.0694245
$595$	3.15213	0.129225
$596$	− 1.14914i	− 0.0470708i
$597$	−6.67563	−0.273215
$598$	0	0
$599$	−29.4644	−1.20388	−0.601942	−	0.798540i	$-0.705606\pi$
$599$	−29.4644	−1.20388	−0.601942	+	0.798540i	$0.705606\pi$
$600$	− 1.00000i	− 0.0408248i
$601$	−28.4004	−1.15848	−0.579239	−	0.815158i	$-0.696650\pi$
$601$	−28.4004	−1.15848	−0.579239	+	0.815158i	$0.696650\pi$
$602$	−13.5007	−0.550246
$603$	− 8.93900i	− 0.364024i
$604$	− 0.0217703i	0 0.000885819i
$605$	8.13706i	0.330819i
$606$	5.38404i	0.218712i
$607$	15.6450	0.635012	0.317506	−	0.948256i	$-0.397155\pi$
$607$	15.6450	0.635012	0.317506	+	0.948256i	$0.397155\pi$
$608$	4.85086	0.196728
$609$	− 27.2747i	− 1.10523i
$610$	0.198062	0.00801931
$611$	0	0
$612$	0.939001	0.0379569
$613$	17.6329i	0.712188i	0.934450	+	0.356094i	$0.115892\pi$
$613$	17.6329i	0.712188i	−0.934450	+	0.356094i	$0.884108\pi$
$614$	25.4403	1.02669
$615$	3.86294	0.155769
$616$	− 5.67994i	− 0.228851i
$617$	− 10.8582i	− 0.437133i	−0.975822	−	0.218566i	$-0.929862\pi$
$617$	− 10.8582i	− 0.437133i	0.975822	−	0.218566i	$-0.0701380\pi$
$618$	4.08575i	0.164353i
$619$	− 34.2083i	− 1.37495i	−0.726208	−	0.687475i	$-0.758719\pi$
$619$	− 34.2083i	− 1.37495i	0.726208	−	0.687475i	$-0.241281\pi$
$620$	−4.08815	−0.164184
$621$	−4.04892	−0.162477
$622$	− 10.0325i	− 0.402268i
$623$	31.8116	1.27451
$624$	0	0
$625$	1.00000	0.0400000
$626$	22.3274i	0.892381i
$627$	−8.20775	−0.327786
$628$	10.1588	0.405382
$629$	− 10.9215i	− 0.435470i
$630$	3.35690i	0.133742i
$631$	31.9571i	1.27219i	0.771611	+	0.636095i	$0.219451\pi$
$631$	31.9571i	1.27219i	−0.771611	+	0.636095i	$0.780549\pi$
$632$	− 4.94869i	− 0.196848i
$633$	−11.3056	−0.449357
$634$	5.33811	0.212003
$635$	12.8509i	0.509971i
$636$	−5.74094	−0.227643
$637$	0	0
$638$	−13.7476	−0.544274
$639$	5.15883i	0.204080i
$640$	−1.00000	−0.0395285
$641$	45.8297	1.81016	0.905082	−	0.425238i	$-0.139810\pi$
$641$	45.8297	1.81016	0.905082	+	0.425238i	$0.139810\pi$
$642$	16.1196i	0.636190i
$643$	− 20.6418i	− 0.814032i	−0.913421	−	0.407016i	$-0.866569\pi$
$643$	− 20.6418i	− 0.814032i	0.913421	−	0.407016i	$-0.133431\pi$
$644$	13.5918i	0.535592i
$645$	− 4.02177i	− 0.158357i
$646$	−4.55496	−0.179212
$647$	−1.40389	−0.0551928	−0.0275964	−	0.999619i	$-0.508785\pi$
$647$	−1.40389	−0.0551928	−0.0275964	+	0.999619i	$0.508785\pi$
$648$	1.00000i	0.0392837i
$649$	−0.707087	−0.0277556
$650$	0	0
$651$	13.7235	0.537866
$652$	− 19.1511i	− 0.750014i
$653$	−4.54229	−0.177753	−0.0888767	−	0.996043i	$-0.528328\pi$
$653$	−4.54229	−0.177753	−0.0888767	+	0.996043i	$0.528328\pi$
$654$	14.2325	0.556535
$655$	− 0.570024i	− 0.0222727i
$656$	− 3.86294i	− 0.150822i
$657$	− 11.5308i	− 0.449859i
$658$	− 4.27711i	− 0.166739i
$659$	17.4161	0.678435	0.339217	−	0.940708i	$-0.389838\pi$
$659$	17.4161	0.678435	0.339217	+	0.940708i	$0.389838\pi$
$660$	1.69202	0.0658618
$661$	43.3682i	1.68683i	0.537263	+	0.843415i	$0.319458\pi$
$661$	43.3682i	1.68683i	−0.537263	+	0.843415i	$0.680542\pi$
$662$	4.49827	0.174830
$663$	0	0
$664$	13.1739	0.511246
$665$	− 16.2838i	− 0.631459i
$666$	11.6310	0.450693
$667$	32.8974	1.27379
$668$	15.2174i	0.588780i
$669$	− 18.0911i	− 0.699443i
$670$	− 8.93900i	− 0.345344i
$671$	0.335126i	0.0129374i
$672$	3.35690	0.129495
$673$	3.49875	0.134867	0.0674334	−	0.997724i	$-0.478519\pi$
$673$	3.49875	0.134867	0.0674334	+	0.997724i	$0.478519\pi$
$674$	− 11.5676i	− 0.445568i
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−13.3284	−0.512253	−0.256126	−	0.966643i	$-0.582446\pi$
$677$	−13.3284	−0.512253	−0.256126	+	0.966643i	$0.582446\pi$
$678$	13.3274i	0.511834i
$679$	51.8799	1.99097
$680$	0.939001	0.0360090
$681$	9.37196i	0.359134i
$682$	− 6.91723i	− 0.264874i
$683$	− 24.1739i	− 0.924989i	−0.886622	−	0.462494i	$-0.846955\pi$
$683$	− 24.1739i	− 0.924989i	0.886622	−	0.462494i	$-0.153045\pi$
$684$	− 4.85086i	− 0.185477i
$685$	−21.5405	−0.823020
$686$	9.16852	0.350056
$687$	21.2664i	0.811362i
$688$	−4.02177	−0.153329
$689$	0	0
$690$	−4.04892	−0.154140
$691$	37.8810i	1.44106i	0.693423	+	0.720530i	$0.256102\pi$
$691$	37.8810i	1.44106i	−0.693423	+	0.720530i	$0.743898\pi$
$692$	21.8213	0.829522
$693$	−5.67994	−0.215763
$694$	− 27.1226i	− 1.02956i
$695$	7.09783i	0.269236i
$696$	− 8.12498i	− 0.307977i
$697$	3.62730i	0.137394i
$698$	12.8562	0.486616
$699$	−2.03146	−0.0768368
$700$	3.35690i	0.126879i
$701$	21.5967	0.815696	0.407848	−	0.913050i	$-0.366279\pi$
$701$	21.5967	0.815696	0.407848	+	0.913050i	$0.366279\pi$
$702$	0	0
$703$	−56.4204	−2.12794
$704$	− 1.69202i	− 0.0637705i
$705$	1.27413	0.0479864
$706$	−24.6262	−0.926821
$707$	− 18.0737i	− 0.679730i
$708$	− 0.417895i	− 0.0157054i
$709$	41.3062i	1.55129i	0.631172	+	0.775643i	$0.282574\pi$
$709$	41.3062i	1.55129i	−0.631172	+	0.775643i	$0.717426\pi$
$710$	5.15883i	0.193608i
$711$	−4.94869	−0.185590
$712$	9.47650	0.355147
$713$	16.5526i	0.619898i
$714$	−3.15213	−0.117965
$715$	0	0
$716$	−20.2513	−0.756826
$717$	11.9903i	0.447786i
$718$	−23.8629	−0.890557
$719$	35.5338	1.32519	0.662593	−	0.748980i	$-0.269456\pi$
$719$	35.5338	1.32519	0.662593	+	0.748980i	$0.269456\pi$
$720$	1.00000i	0.0372678i
$721$	− 13.7154i	− 0.510790i
$722$	4.53079i	0.168619i
$723$	21.7627i	0.809364i
$724$	14.4969	0.538775
$725$	8.12498	0.301754
$726$	− 8.13706i	− 0.301995i
$727$	−19.6635	−0.729281	−0.364640	−	0.931148i	$-0.618808\pi$
$727$	−19.6635	−0.729281	−0.364640	+	0.931148i	$0.618808\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	− 11.5308i	− 0.426774i
$731$	3.77645	0.139677
$732$	−0.198062	−0.00732059
$733$	46.8702i	1.73119i	0.500743	+	0.865596i	$0.333060\pi$
$733$	46.8702i	1.73119i	−0.500743	+	0.865596i	$0.666940\pi$
$734$	− 33.1933i	− 1.22519i
$735$	− 4.26875i	− 0.157455i
$736$	4.04892i	0.149245i
$737$	15.1250	0.557136
$738$	−3.86294	−0.142197
$739$	− 8.57779i	− 0.315539i	−0.987476	−	0.157770i	$-0.949570\pi$
$739$	− 8.57779i	− 0.315539i	0.987476	−	0.157770i	$-0.0504303\pi$
$740$	11.6310	0.427565
$741$	0	0
$742$	19.2717	0.707488
$743$	44.9506i	1.64908i	0.565805	+	0.824539i	$0.308565\pi$
$743$	44.9506i	1.64908i	−0.565805	+	0.824539i	$0.691435\pi$
$744$	4.08815	0.149879
$745$	−1.14914	−0.0421014
$746$	10.8465i	0.397120i
$747$	− 13.1739i	− 0.482008i
$748$	1.58881i	0.0580926i
$749$	− 54.1118i	− 1.97720i
$750$	−1.00000	−0.0365148
$751$	−44.0374	−1.60695	−0.803474	−	0.595339i	$-0.797018\pi$
$751$	−44.0374	−1.60695	−0.803474	+	0.595339i	$0.797018\pi$
$752$	− 1.27413i	− 0.0464626i
$753$	−13.7506	−0.501101
$754$	0	0
$755$	−0.0217703	−0.000792301 0
$756$	− 3.35690i	− 0.122089i
$757$	−13.2798	−0.482661	−0.241331	−	0.970443i	$-0.577584\pi$
$757$	−13.2798	−0.482661	−0.241331	+	0.970443i	$0.577584\pi$
$758$	6.09485	0.221375
$759$	− 6.85086i	− 0.248670i
$760$	− 4.85086i	− 0.175959i
$761$	9.11231i	0.330321i	0.986267	+	0.165160i	$0.0528142\pi$
$761$	9.11231i	0.330321i	−0.986267	+	0.165160i	$0.947186\pi$
$762$	− 12.8509i	− 0.465537i
$763$	−47.7770	−1.72964
$764$	10.1806	0.368321
$765$	− 0.939001i	− 0.0339497i
$766$	7.95407	0.287392
$767$	0	0
$768$	1.00000	0.0360844
$769$	31.7784i	1.14596i	0.819570	+	0.572979i	$0.194212\pi$
$769$	31.7784i	1.14596i	−0.819570	+	0.572979i	$0.805788\pi$
$770$	−5.67994	−0.204691
$771$	24.1782	0.870757
$772$	− 27.6407i	− 0.994811i
$773$	− 7.11960i	− 0.256074i	−0.991769	−	0.128037i	$-0.959132\pi$
$773$	− 7.11960i	− 0.256074i	0.991769	−	0.128037i	$-0.0408677\pi$
$774$	4.02177i	0.144560i
$775$	4.08815i	0.146851i
$776$	15.4547	0.554792
$777$	−39.0441	−1.40070
$778$	1.50365i	0.0539084i
$779$	18.7385	0.671378
$780$	0	0
$781$	−8.72886	−0.312343
$782$	− 3.80194i	− 0.135957i
$783$	−8.12498	−0.290363
$784$	−4.26875	−0.152455
$785$	− 10.1588i	− 0.362584i
$786$	0.570024i	0.0203321i
$787$	− 44.2121i	− 1.57599i	−0.615682	−	0.787995i	$-0.711119\pi$
$787$	− 44.2121i	− 1.57599i	0.615682	−	0.787995i	$-0.288881\pi$
$788$	− 13.0261i	− 0.464035i
$789$	0.960771	0.0342044
$790$	−4.94869	−0.176066
$791$	− 44.7385i	− 1.59072i
$792$	−1.69202	−0.0601234
$793$	0	0
$794$	34.9627	1.24078
$795$	5.74094i	0.203610i
$796$	6.67563	0.236611
$797$	−19.2241	−0.680954	−0.340477	−	0.940253i	$-0.610589\pi$
$797$	−19.2241	−0.680954	−0.340477	+	0.940253i	$0.610589\pi$
$798$	16.2838i	0.576441i
$799$	1.19641i	0.0423258i
$800$	1.00000i	0.0353553i
$801$	− 9.47650i	− 0.334836i
$802$	−31.1148	−1.09870
$803$	19.5104	0.688505
$804$	8.93900i	0.315254i
$805$	13.5918	0.479048
$806$	0	0
$807$	10.0761	0.354694
$808$	− 5.38404i	− 0.189410i
$809$	0.0988996	0.00347713	0.00173856	−	0.999998i	$-0.499447\pi$
$809$	0.0988996	0.00347713	0.00173856	+	0.999998i	$0.499447\pi$
$810$	1.00000	0.0351364
$811$	− 6.70304i	− 0.235376i	−0.993051	−	0.117688i	$-0.962452\pi$
$811$	− 6.70304i	− 0.235376i	0.993051	−	0.117688i	$-0.0375482\pi$
$812$	27.2747i	0.957155i
$813$	− 14.9584i	− 0.524613i
$814$	19.6799i	0.689782i
$815$	−19.1511	−0.670833
$816$	−0.939001	−0.0328716
$817$	− 19.5090i	− 0.682534i
$818$	32.7318	1.14444
$819$	0	0
$820$	−3.86294	−0.134900
$821$	23.7450i	0.828706i	0.910116	+	0.414353i	$0.135992\pi$
$821$	23.7450i	0.828706i	−0.910116	+	0.414353i	$0.864008\pi$
$822$	21.5405	0.751311
$823$	−18.0049	−0.627611	−0.313806	−	0.949487i	$-0.601604\pi$
$823$	−18.0049	−0.627611	−0.313806	+	0.949487i	$0.601604\pi$
$824$	− 4.08575i	− 0.142334i
$825$	− 1.69202i	− 0.0589086i
$826$	1.40283i	0.0488107i
$827$	40.4566i	1.40682i	0.710787	+	0.703408i	$0.248339\pi$
$827$	40.4566i	1.40682i	−0.710787	+	0.703408i	$0.751661\pi$
$828$	4.04892	0.140710
$829$	−8.56704	−0.297546	−0.148773	−	0.988871i	$-0.547532\pi$
$829$	−8.56704	−0.297546	−0.148773	+	0.988871i	$0.547532\pi$
$830$	− 13.1739i	− 0.457273i
$831$	13.0465	0.452579
$832$	0	0
$833$	4.00836	0.138881
$834$	− 7.09783i	− 0.245778i
$835$	15.2174	0.526621
$836$	8.20775	0.283871
$837$	− 4.08815i	− 0.141307i
$838$	− 13.0694i	− 0.451474i
$839$	− 6.75600i	− 0.233243i	−0.993176	−	0.116622i	$-0.962794\pi$
$839$	− 6.75600i	− 0.233243i	0.993176	−	0.116622i	$-0.0372065\pi$
$840$	− 3.35690i	− 0.115824i
$841$	37.0153	1.27639
$842$	−16.4789	−0.567900
$843$	16.2881i	0.560993i
$844$	11.3056	0.389154
$845$	0	0
$846$	−1.27413	−0.0438054
$847$	27.3153i	0.938564i
$848$	5.74094	0.197145
$849$	12.3773	0.424789
$850$	− 0.939001i	− 0.0322075i
$851$	− 47.0930i	− 1.61433i
$852$	− 5.15883i	− 0.176739i
$853$	− 37.1269i	− 1.27120i	−0.772018	−	0.635600i	$-0.780753\pi$
$853$	− 37.1269i	− 1.27120i	0.772018	−	0.635600i	$-0.219247\pi$
$854$	0.664874	0.0227515
$855$	−4.85086	−0.165896
$856$	− 16.1196i	− 0.550957i
$857$	24.3937	0.833274	0.416637	−	0.909073i	$-0.363209\pi$
$857$	24.3937	0.833274	0.416637	+	0.909073i	$0.363209\pi$
$858$	0	0
$859$	43.9366	1.49910	0.749549	−	0.661949i	$-0.230270\pi$
$859$	43.9366	1.49910	0.749549	+	0.661949i	$0.230270\pi$
$860$	4.02177i	0.137141i
$861$	12.9675	0.441930
$862$	41.0640	1.39864
$863$	− 16.3846i	− 0.557739i	−0.960329	−	0.278870i	$-0.910040\pi$
$863$	− 16.3846i	− 0.557739i	0.960329	−	0.278870i	$-0.0899598\pi$
$864$	− 1.00000i	− 0.0340207i
$865$	− 21.8213i	− 0.741947i
$866$	− 6.04593i	− 0.205449i
$867$	−16.1183	−0.547405
$868$	−13.7235	−0.465805
$869$	− 8.37329i	− 0.284044i
$870$	−8.12498	−0.275463
$871$	0	0
$872$	−14.2325	−0.481973
$873$	− 15.4547i	− 0.523063i
$874$	−19.6407	−0.664357
$875$	3.35690	0.113484
$876$	11.5308i	0.389589i
$877$	− 28.1347i	− 0.950040i	−0.879975	−	0.475020i	$-0.842441\pi$
$877$	− 28.1347i	− 0.950040i	0.879975	−	0.475020i	$-0.157559\pi$
$878$	− 26.2543i	− 0.886039i
$879$	− 1.73663i	− 0.0585750i
$880$	−1.69202	−0.0570380
$881$	−39.1454	−1.31884	−0.659421	−	0.751773i	$-0.729199\pi$
$881$	−39.1454	−1.31884	−0.659421	+	0.751773i	$0.729199\pi$
$882$	4.26875i	0.143736i
$883$	−46.2833	−1.55756	−0.778779	−	0.627298i	$-0.784161\pi$
$883$	−46.2833	−1.55756	−0.778779	+	0.627298i	$0.784161\pi$
$884$	0	0
$885$	−0.417895	−0.0140474
$886$	39.3846i	1.32315i
$887$	−3.08097	−0.103449	−0.0517244	−	0.998661i	$-0.516472\pi$
$887$	−3.08097	−0.103449	−0.0517244	+	0.998661i	$0.516472\pi$
$888$	−11.6310	−0.390312
$889$	43.1390i	1.44684i
$890$	− 9.47650i	− 0.317653i
$891$	1.69202i	0.0566849i
$892$	18.0911i	0.605736i
$893$	6.18060	0.206826
$894$	1.14914	0.0384332
$895$	20.2513i	0.676926i
$896$	−3.35690	−0.112146
$897$	0	0
$898$	−9.39075	−0.313373
$899$	33.2161i	1.10782i
$900$	1.00000	0.0333333
$901$	−5.39075	−0.179592
$902$	− 6.53617i	− 0.217631i
$903$	− 13.5007i	− 0.449274i
$904$	− 13.3274i	− 0.443261i
$905$	− 14.4969i	− 0.481895i
$906$	0.0217703	0.000723269 0
$907$	2.61224	0.0867379	0.0433689	−	0.999059i	$-0.486191\pi$
$907$	2.61224	0.0867379	0.0433689	+	0.999059i	$0.486191\pi$
$908$	− 9.37196i	− 0.311019i
$909$	−5.38404	−0.178577
$910$	0	0
$911$	24.9302	0.825973	0.412987	−	0.910737i	$-0.364486\pi$
$911$	24.9302	0.825973	0.412987	+	0.910737i	$0.364486\pi$
$912$	4.85086i	0.160628i
$913$	22.2905	0.737709
$914$	2.66487	0.0881462
$915$	0.198062i	0.00654774i
$916$	− 21.2664i	− 0.702660i
$917$	− 1.91351i	− 0.0631897i
$918$	0.939001i	0.0309917i
$919$	−54.4462	−1.79602	−0.898008	−	0.439980i	$-0.854986\pi$
$919$	−54.4462	−1.79602	−0.898008	+	0.439980i	$0.854986\pi$
$920$	4.04892	0.133489
$921$	25.4403i	0.838285i
$922$	24.5284	0.807800
$923$	0	0
$924$	5.67994	0.186856
$925$	− 11.6310i	− 0.382426i
$926$	−4.50498	−0.148043
$927$	−4.08575	−0.134194
$928$	8.12498i	0.266716i
$929$	40.8267i	1.33948i	0.742596	+	0.669740i	$0.233595\pi$
$929$	40.8267i	1.33948i	−0.742596	+	0.669740i	$0.766405\pi$
$930$	− 4.08815i	− 0.134056i
$931$	− 20.7071i	− 0.678647i
$932$	2.03146	0.0665427
$933$	10.0325	0.328450
$934$	21.3317i	0.697993i
$935$	1.58881	0.0519596
$936$	0	0
$937$	−46.9571	−1.53402	−0.767010	−	0.641635i	$-0.778256\pi$
$937$	−46.9571	−1.53402	−0.767010	+	0.641635i	$0.778256\pi$
$938$	− 30.0073i	− 0.979773i
$939$	−22.3274	−0.728626
$940$	−1.27413	−0.0415574
$941$	− 35.8525i	− 1.16876i	−0.811481	−	0.584379i	$-0.801338\pi$
$941$	− 35.8525i	− 1.16876i	0.811481	−	0.584379i	$-0.198662\pi$
$942$	10.1588i	0.330993i
$943$	15.6407i	0.509332i
$944$	0.417895i	0.0136013i
$945$	−3.35690	−0.109200
$946$	−6.80492	−0.221247
$947$	− 8.73019i	− 0.283693i	−0.989889	−	0.141846i	$-0.954696\pi$
$947$	− 8.73019i	− 0.283693i	0.989889	−	0.141846i	$-0.0453039\pi$
$948$	4.94869	0.160726
$949$	0	0
$950$	−4.85086	−0.157383
$951$	5.33811i	0.173100i
$952$	3.15213	0.102161
$953$	−10.2107	−0.330758	−0.165379	−	0.986230i	$-0.552885\pi$
$953$	−10.2107	−0.330758	−0.165379	+	0.986230i	$0.552885\pi$
$954$	− 5.74094i	− 0.185870i
$955$	− 10.1806i	− 0.329437i
$956$	− 11.9903i	− 0.387794i
$957$	− 13.7476i	− 0.444398i
$958$	−32.9517	−1.06462
$959$	−72.3092	−2.33498
$960$	− 1.00000i	− 0.0322749i
$961$	14.2871	0.460873
$962$	0	0
$963$	−16.1196	−0.519447
$964$	− 21.7627i	− 0.700930i
$965$	−27.6407	−0.889786
$966$	−13.5918	−0.437309
$967$	34.6907i	1.11558i	0.829983	+	0.557789i	$0.188350\pi$
$967$	34.6907i	1.11558i	−0.829983	+	0.557789i	$0.811650\pi$
$968$	8.13706i	0.261535i
$969$	− 4.55496i	− 0.146326i
$970$	− 15.4547i	− 0.496221i
$971$	34.8039	1.11691	0.558454	−	0.829535i	$-0.311395\pi$
$971$	34.8039	1.11691	0.558454	+	0.829535i	$0.311395\pi$
$972$	−1.00000	−0.0320750
$973$	23.8267i	0.763849i
$974$	9.33034	0.298963
$975$	0	0
$976$	0.198062	0.00633982
$977$	− 23.3086i	− 0.745707i	−0.927890	−	0.372854i	$-0.878379\pi$
$977$	− 23.3086i	− 0.745707i	0.927890	−	0.372854i	$-0.121621\pi$
$978$	19.1511	0.612383
$979$	16.0344	0.512463
$980$	4.26875i	0.136360i
$981$	14.2325i	0.454409i
$982$	− 20.4349i	− 0.652103i
$983$	41.3639i	1.31930i	0.751571	+	0.659652i	$0.229296\pi$
$983$	41.3639i	1.31930i	−0.751571	+	0.659652i	$0.770704\pi$
$984$	3.86294	0.123146
$985$	−13.0261	−0.415045
$986$	− 7.62937i	− 0.242969i
$987$	4.27711	0.136142
$988$	0	0
$989$	16.2838	0.517795
$990$	1.69202i	0.0537760i
$991$	49.4209	1.56991	0.784953	−	0.619555i	$-0.212687\pi$
$991$	49.4209	1.56991	0.784953	+	0.619555i	$0.212687\pi$
$992$	−4.08815	−0.129799
$993$	4.49827i	0.142748i
$994$	17.3177i	0.549283i
$995$	− 6.67563i	− 0.211632i
$996$	13.1739i	0.417431i
$997$	44.1879	1.39944	0.699722	−	0.714415i	$-0.253307\pi$
$997$	44.1879	1.39944	0.699722	+	0.714415i	$0.253307\pi$
$998$	−3.54958	−0.112360
$999$	11.6310i	0.367989i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.b.z.1351.2		6
13.5	odd	4		5070.2.a.bp.1.2	✓	3
13.8	odd	4		5070.2.a.bw.1.2	yes	3
13.12	even	2	inner	5070.2.b.z.1351.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5070.2.a.bp.1.2	✓	3	13.5	odd	4
5070.2.a.bw.1.2	yes	3	13.8	odd	4
5070.2.b.z.1351.2		6	1.1	even	1	trivial
5070.2.b.z.1351.5		6	13.12	even	2	inner

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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} +3.35690i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} +3.35690i q^{7} +1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} +1.69202i q^{11} -1.00000 q^{12} +3.35690 q^{14} +1.00000i q^{15} +1.00000 q^{16} -0.939001 q^{17} -1.00000i q^{18} +4.85086i q^{19} -1.00000i q^{20} +3.35690i q^{21} +1.69202 q^{22} -4.04892 q^{23} +1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} -3.35690i q^{28} -8.12498 q^{29} +1.00000 q^{30} -4.08815i q^{31} -1.00000i q^{32} +1.69202i q^{33} +0.939001i q^{34} -3.35690 q^{35} -1.00000 q^{36} +11.6310i q^{37} +4.85086 q^{38} -1.00000 q^{40} -3.86294i q^{41} +3.35690 q^{42} -4.02177 q^{43} -1.69202i q^{44} +1.00000i q^{45} +4.04892i q^{46} -1.27413i q^{47} +1.00000 q^{48} -4.26875 q^{49} +1.00000i q^{50} -0.939001 q^{51} +5.74094 q^{53} -1.00000i q^{54} -1.69202 q^{55} -3.35690 q^{56} +4.85086i q^{57} +8.12498i q^{58} +0.417895i q^{59} -1.00000i q^{60} +0.198062 q^{61} -4.08815 q^{62} +3.35690i q^{63} -1.00000 q^{64} +1.69202 q^{66} -8.93900i q^{67} +0.939001 q^{68} -4.04892 q^{69} +3.35690i q^{70} +5.15883i q^{71} +1.00000i q^{72} -11.5308i q^{73} +11.6310 q^{74} -1.00000 q^{75} -4.85086i q^{76} -5.67994 q^{77} -4.94869 q^{79} +1.00000i q^{80} +1.00000 q^{81} -3.86294 q^{82} -13.1739i q^{83} -3.35690i q^{84} -0.939001i q^{85} +4.02177i q^{86} -8.12498 q^{87} -1.69202 q^{88} -9.47650i q^{89} +1.00000 q^{90} +4.04892 q^{92} -4.08815i q^{93} -1.27413 q^{94} -4.85086 q^{95} -1.00000i q^{96} -15.4547i q^{97} +4.26875i q^{98} +1.69202i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{3} - 6 q^{4} + 6 q^{9}+O(q^{10}) \) 6 * q + 6 * q^3 - 6 * q^4 + 6 * q^9	\( 6 q + 6 q^{3} - 6 q^{4} + 6 q^{9} + 6 q^{10} - 6 q^{12} + 12 q^{14} + 6 q^{16} + 14 q^{17} - 6 q^{23} - 6 q^{25} + 6 q^{27} + 6 q^{30} - 12 q^{35} - 6 q^{36} + 2 q^{38} - 6 q^{40} + 12 q^{42} - 18 q^{43} + 6 q^{48} - 10 q^{49} + 14 q^{51} + 6 q^{53} - 12 q^{56} + 10 q^{61} - 32 q^{62} - 6 q^{64} - 14 q^{68} - 6 q^{69} + 40 q^{74} - 6 q^{75} + 14 q^{77} + 34 q^{79} + 6 q^{81} - 34 q^{82} + 6 q^{90} + 6 q^{92} + 14 q^{94} - 2 q^{95}+O(q^{100}) \) 6 * q + 6 * q^3 - 6 * q^4 + 6 * q^9 + 6 * q^10 - 6 * q^12 + 12 * q^14 + 6 * q^16 + 14 * q^17 - 6 * q^23 - 6 * q^25 + 6 * q^27 + 6 * q^30 - 12 * q^35 - 6 * q^36 + 2 * q^38 - 6 * q^40 + 12 * q^42 - 18 * q^43 + 6 * q^48 - 10 * q^49 + 14 * q^51 + 6 * q^53 - 12 * q^56 + 10 * q^61 - 32 * q^62 - 6 * q^64 - 14 * q^68 - 6 * q^69 + 40 * q^74 - 6 * q^75 + 14 * q^77 + 34 * q^79 + 6 * q^81 - 34 * q^82 + 6 * q^90 + 6 * q^92 + 14 * q^94 - 2 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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