LMFDB - Embedded newform 841.2.b.c.840.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [841,2,Mod(840,841)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(841, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("841.840"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 841 = 29^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	841.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$6.71541880999$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 29)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			840.3
Root			$-0.445042i$ of defining polynomial
Character	$\chi$	$=$	841.840
Dual form			841.2.b.c.840.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.445042i q^{2} -1.24698i q^{3} +1.80194 q^{4} -0.692021 q^{5} -0.554958 q^{6} +0.356896 q^{7} -1.69202i q^{8} +1.44504 q^{9} +0.307979i q^{10} +4.93900i q^{11} -2.24698i q^{12} +5.65279 q^{13} -0.158834i q^{14} +0.862937i q^{15} +2.85086 q^{16} +4.49396i q^{17} -0.643104i q^{18} -2.35690i q^{19} -1.24698 q^{20} -0.445042i q^{21} +2.19806 q^{22} +2.29590 q^{23} -2.10992 q^{24} -4.52111 q^{25} -2.51573i q^{26} -5.54288i q^{27} +0.643104 q^{28} +0.384043 q^{30} -6.69202i q^{31} -4.65279i q^{32} +6.15883 q^{33} +2.00000 q^{34} -0.246980 q^{35} +2.60388 q^{36} +4.93900i q^{37} -1.04892 q^{38} -7.04892i q^{39} +1.17092i q^{40} -3.10992i q^{41} -0.198062 q^{42} -3.40581i q^{43} +8.89977i q^{44} -1.00000 q^{45} -1.02177i q^{46} +6.44504i q^{47} -3.55496i q^{48} -6.87263 q^{49} +2.01208i q^{50} +5.60388 q^{51} +10.1860 q^{52} -4.69202 q^{53} -2.46681 q^{54} -3.41789i q^{55} -0.603875i q^{56} -2.93900 q^{57} -12.4940 q^{59} +1.55496i q^{60} -1.64310i q^{61} -2.97823 q^{62} +0.515729 q^{63} +3.63102 q^{64} -3.91185 q^{65} -2.74094i q^{66} +2.32304 q^{67} +8.09783i q^{68} -2.86294i q^{69} +0.109916i q^{70} +7.33513 q^{71} -2.44504i q^{72} -5.62565i q^{73} +2.19806 q^{74} +5.63773i q^{75} -4.24698i q^{76} +1.76271i q^{77} -3.13706 q^{78} -4.66487i q^{79} -1.97285 q^{80} -2.57673 q^{81} -1.38404 q^{82} +4.45473 q^{83} -0.801938i q^{84} -3.10992i q^{85} -1.51573 q^{86} +8.35690 q^{88} -5.67994i q^{89} +0.445042i q^{90} +2.01746 q^{91} +4.13706 q^{92} -8.34481 q^{93} +2.86831 q^{94} +1.63102i q^{95} -5.80194 q^{96} +0.180604i q^{97} +3.05861i q^{98} +7.13706i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{4} + 6 q^{5} - 4 q^{6} - 6 q^{7} + 8 q^{9} - 2 q^{13} - 10 q^{16} + 2 q^{20} + 22 q^{22} - 14 q^{23} - 14 q^{24} + 4 q^{25} + 12 q^{28} - 18 q^{30} + 20 q^{33} + 12 q^{34} + 8 q^{35} - 2 q^{36}+ \cdots - 26 q^{96}+O(q^{100}) $ 6 * q + 2 * q^4 + 6 * q^5 - 4 * q^6 - 6 * q^7 + 8 * q^9 - 2 * q^13 - 10 * q^16 + 2 * q^20 + 22 * q^22 - 14 * q^23 - 14 * q^24 + 4 * q^25 + 12 * q^28 - 18 * q^30 + 20 * q^33 + 12 * q^34 + 8 * q^35 - 2 * q^36 + 12 * q^38 - 10 * q^42 - 6 * q^45 - 8 * q^49 + 16 * q^51 + 32 * q^52 - 18 * q^53 - 8 * q^54 + 2 * q^57 - 56 * q^59 - 24 * q^62 - 22 * q^63 - 8 * q^64 - 16 * q^65 - 26 * q^67 + 42 * q^71 + 22 * q^74 - 8 * q^78 - 24 * q^80 - 10 * q^81 + 12 * q^82 - 18 * q^83 + 16 * q^86 + 42 * q^88 + 44 * q^91 + 14 * q^92 - 4 * q^93 + 22 * q^94 - 26 * q^96

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/841\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

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 $

$2$	− 0.445042i	− 0.314692i	−0.987544	−	0.157346i	$-0.949706\pi$
$2$	− 0.445042i	− 0.314692i	0.987544	−	0.157346i	$-0.0502938\pi$
$3$	− 1.24698i	− 0.719944i	−0.932963	−	0.359972i	$-0.882786\pi$
$3$	− 1.24698i	− 0.719944i	0.932963	−	0.359972i	$-0.117214\pi$
$4$	1.80194	0.900969
$5$	−0.692021	−0.309481	−0.154741	−	0.987955i	$-0.549454\pi$
$5$	−0.692021	−0.309481	−0.154741	+	0.987955i	$0.549454\pi$
$6$	−0.554958	−0.226561
$7$	0.356896	0.134894	0.0674470	−	0.997723i	$-0.478515\pi$
$7$	0.356896	0.134894	0.0674470	+	0.997723i	$0.478515\pi$
$8$	− 1.69202i	− 0.598220i
$9$	1.44504	0.481681
$10$	0.307979i	0.0973914i
$11$	4.93900i	1.48916i	0.667531	+	0.744582i	$0.267351\pi$
$11$	4.93900i	1.48916i	−0.667531	+	0.744582i	$0.732649\pi$
$12$	− 2.24698i	− 0.648647i
$13$	5.65279	1.56780	0.783901	−	0.620885i	$-0.213227\pi$
$13$	5.65279	1.56780	0.783901	+	0.620885i	$0.213227\pi$
$14$	− 0.158834i	− 0.0424501i
$15$	0.862937i	0.222809i
$16$	2.85086	0.712714
$17$	4.49396i	1.08995i	0.838454	+	0.544973i	$0.183460\pi$
$17$	4.49396i	1.08995i	−0.838454	+	0.544973i	$0.816540\pi$
$18$	− 0.643104i	− 0.151581i
$19$	− 2.35690i	− 0.540709i	−0.962761	−	0.270354i	$-0.912859\pi$
$19$	− 2.35690i	− 0.540709i	0.962761	−	0.270354i	$-0.0871409\pi$
$20$	−1.24698	−0.278833
$21$	− 0.445042i	− 0.0971161i
$22$	2.19806	0.468628
$23$	2.29590	0.478728	0.239364	−	0.970930i	$-0.423061\pi$
$23$	2.29590	0.478728	0.239364	+	0.970930i	$0.423061\pi$
$24$	−2.10992	−0.430685
$25$	−4.52111	−0.904221
$26$	− 2.51573i	− 0.493375i
$27$	− 5.54288i	− 1.06673i
$28$	0.643104	0.121535
$29$	0	0
$29$	0	0
$30$	0.384043	0.0701163
$31$	− 6.69202i	− 1.20192i	−0.799278	−	0.600961i	$-0.794785\pi$
$31$	− 6.69202i	− 1.20192i	0.799278	−	0.600961i	$-0.205215\pi$
$32$	− 4.65279i	− 0.822505i
$33$	6.15883	1.07212
$34$	2.00000	0.342997
$35$	−0.246980	−0.0417472
$36$	2.60388	0.433979
$37$	4.93900i	0.811967i	0.913880	+	0.405983i	$0.133071\pi$
$37$	4.93900i	0.811967i	−0.913880	+	0.405983i	$0.866929\pi$
$38$	−1.04892	−0.170157
$39$	− 7.04892i	− 1.12873i
$40$	1.17092i	0.185138i
$41$	− 3.10992i	− 0.485687i	−0.970065	−	0.242844i	$-0.921920\pi$
$41$	− 3.10992i	− 0.485687i	0.970065	−	0.242844i	$-0.0780802\pi$
$42$	−0.198062	−0.0305617
$43$	− 3.40581i	− 0.519382i	−0.965692	−	0.259691i	$-0.916379\pi$
$43$	− 3.40581i	− 0.519382i	0.965692	−	0.259691i	$-0.0836207\pi$
$44$	8.89977i	1.34169i
$45$	−1.00000	−0.149071
$46$	− 1.02177i	− 0.150652i
$47$	6.44504i	0.940106i	0.882638	+	0.470053i	$0.155765\pi$
$47$	6.44504i	0.940106i	−0.882638	+	0.470053i	$0.844235\pi$
$48$	− 3.55496i	− 0.513114i
$49$	−6.87263	−0.981804
$50$	2.01208i	0.284551i
$51$	5.60388	0.784700
$52$	10.1860	1.41254
$53$	−4.69202	−0.644499	−0.322249	−	0.946655i	$-0.604439\pi$
$53$	−4.69202	−0.644499	−0.322249	+	0.946655i	$0.604439\pi$
$54$	−2.46681	−0.335691
$55$	− 3.41789i	− 0.460869i
$56$	− 0.603875i	− 0.0806963i
$57$	−2.93900	−0.389280
$58$	0	0
$59$	−12.4940	−1.62657	−0.813287	−	0.581862i	$-0.802324\pi$
$59$	−12.4940	−1.62657	−0.813287	+	0.581862i	$0.802324\pi$
$60$	1.55496i	0.200744i
$61$	− 1.64310i	− 0.210378i	−0.994452	−	0.105189i	$-0.966455\pi$
$61$	− 1.64310i	− 0.210378i	0.994452	−	0.105189i	$-0.0335447\pi$
$62$	−2.97823	−0.378236
$63$	0.515729	0.0649758
$64$	3.63102	0.453878
$65$	−3.91185	−0.485206
$66$	− 2.74094i	− 0.337386i
$67$	2.32304	0.283805	0.141902	−	0.989881i	$-0.454678\pi$
$67$	2.32304	0.283805	0.141902	+	0.989881i	$0.454678\pi$
$68$	8.09783i	0.982007i
$69$	− 2.86294i	− 0.344657i
$70$	0.109916i	0.0131375i
$71$	7.33513	0.870519	0.435260	−	0.900305i	$-0.356657\pi$
$71$	7.33513	0.870519	0.435260	+	0.900305i	$0.356657\pi$
$72$	− 2.44504i	− 0.288151i
$73$	− 5.62565i	− 0.658432i	−0.944255	−	0.329216i	$-0.893216\pi$
$73$	− 5.62565i	− 0.658432i	0.944255	−	0.329216i	$-0.106784\pi$
$74$	2.19806	0.255520
$75$	5.63773i	0.650989i
$76$	− 4.24698i	− 0.487162i
$77$	1.76271i	0.200879i
$78$	−3.13706	−0.355202
$79$	− 4.66487i	− 0.524839i	−0.964954	−	0.262420i	$-0.915480\pi$
$79$	− 4.66487i	− 0.524839i	0.964954	−	0.262420i	$-0.0845205\pi$
$80$	−1.97285	−0.220572
$81$	−2.57673	−0.286303
$82$	−1.38404	−0.152842
$83$	4.45473	0.488970	0.244485	−	0.969653i	$-0.421381\pi$
$83$	4.45473	0.488970	0.244485	+	0.969653i	$0.421381\pi$
$84$	− 0.801938i	− 0.0874986i
$85$	− 3.10992i	− 0.337318i
$86$	−1.51573	−0.163445
$87$	0	0
$88$	8.35690	0.890848
$89$	− 5.67994i	− 0.602072i	−0.953613	−	0.301036i	$-0.902667\pi$
$89$	− 5.67994i	− 0.602072i	0.953613	−	0.301036i	$-0.0973325\pi$
$90$	0.445042i	0.0469115i
$91$	2.01746	0.211487
$92$	4.13706	0.431319
$93$	−8.34481	−0.865317
$94$	2.86831	0.295844
$95$	1.63102i	0.167339i
$96$	−5.80194	−0.592158
$97$	0.180604i	0.0183375i	0.999958	+	0.00916877i	$0.00291855\pi$
$97$	0.180604i	0.0183375i	−0.999958	+	0.00916877i	$0.997081\pi$
$98$	3.05861i	0.308966i
$99$	7.13706i	0.717302i
$100$	−8.14675	−0.814675
$101$	3.22952i	0.321349i	0.987007	+	0.160675i	$0.0513670\pi$
$101$	3.22952i	0.321349i	−0.987007	+	0.160675i	$0.948633\pi$
$102$	− 2.49396i	− 0.246939i
$103$	−13.6528	−1.34525	−0.672625	−	0.739984i	$-0.734833\pi$
$103$	−13.6528	−1.34525	−0.672625	+	0.739984i	$0.734833\pi$
$104$	− 9.56465i	− 0.937891i
$105$	0.307979i	0.0300556i
$106$	2.08815i	0.202819i
$107$	16.2446	1.57042	0.785212	−	0.619227i	$-0.212554\pi$
$107$	16.2446	1.57042	0.785212	+	0.619227i	$0.212554\pi$
$108$	− 9.98792i	− 0.961088i
$109$	−5.46681	−0.523626	−0.261813	−	0.965119i	$-0.584320\pi$
$109$	−5.46681	−0.523626	−0.261813	+	0.965119i	$0.584320\pi$
$110$	−1.52111	−0.145032
$111$	6.15883	0.584571
$112$	1.01746	0.0961408
$113$	10.6746i	1.00418i	0.864816	+	0.502089i	$0.167435\pi$
$113$	10.6746i	1.00418i	−0.864816	+	0.502089i	$0.832565\pi$
$114$	1.30798i	0.122503i
$115$	−1.58881	−0.148157
$116$	0	0
$117$	8.16852	0.755180
$118$	5.56033i	0.511870i
$119$	1.60388i	0.147027i
$120$	1.46011	0.133289
$121$	−13.3937	−1.21761
$122$	−0.731250	−0.0662043
$123$	−3.87800	−0.349668
$124$	− 12.0586i	− 1.08289i
$125$	6.58881	0.589321
$126$	− 0.229521i	− 0.0204474i
$127$	− 1.03684i	− 0.0920043i	−0.998941	−	0.0460021i	$-0.985352\pi$
$127$	− 1.03684i	− 0.0920043i	0.998941	−	0.0460021i	$-0.0146481\pi$
$128$	− 10.9215i	− 0.965337i
$129$	−4.24698	−0.373926
$130$	1.74094i	0.152690i
$131$	− 8.02177i	− 0.700865i	−0.936588	−	0.350433i	$-0.886035\pi$
$131$	− 8.02177i	− 0.700865i	0.936588	−	0.350433i	$-0.113965\pi$
$132$	11.0978	0.965943
$133$	− 0.841166i	− 0.0729384i
$134$	− 1.03385i	− 0.0893112i
$135$	3.83579i	0.330132i
$136$	7.60388	0.652027
$137$	17.2325i	1.47227i	0.676834	+	0.736136i	$0.263352\pi$
$137$	17.2325i	1.47227i	−0.676834	+	0.736136i	$0.736648\pi$
$138$	−1.27413	−0.108461
$139$	−16.6679	−1.41375	−0.706875	−	0.707339i	$-0.749896\pi$
$139$	−16.6679	−1.41375	−0.706875	+	0.707339i	$0.749896\pi$
$140$	−0.445042	−0.0376129
$141$	8.03684	0.676824
$142$	− 3.26444i	− 0.273946i
$143$	27.9191i	2.33472i
$144$	4.11960	0.343300
$145$	0	0
$146$	−2.50365	−0.207203
$147$	8.57002i	0.706844i
$148$	8.89977i	0.731557i
$149$	−18.6069	−1.52433	−0.762167	−	0.647381i	$-0.775864\pi$
$149$	−18.6069	−1.52433	−0.762167	+	0.647381i	$0.775864\pi$
$150$	2.50902	0.204861
$151$	7.52781	0.612605	0.306302	−	0.951934i	$-0.400908\pi$
$151$	7.52781	0.612605	0.306302	+	0.951934i	$0.400908\pi$
$152$	−3.98792	−0.323463
$153$	6.49396i	0.525005i
$154$	0.784479	0.0632151
$155$	4.63102i	0.371973i
$156$	− 12.7017i	− 1.01695i
$157$	18.2392i	1.45565i	0.685764	+	0.727824i	$0.259468\pi$
$157$	18.2392i	1.45565i	−0.685764	+	0.727824i	$0.740532\pi$
$158$	−2.07606	−0.165163
$159$	5.85086i	0.464003i
$160$	3.21983i	0.254550i
$161$	0.819396	0.0645775
$162$	1.14675i	0.0900973i
$163$	− 12.6407i	− 0.990097i	−0.868865	−	0.495048i	$-0.835150\pi$
$163$	− 12.6407i	− 0.990097i	0.868865	−	0.495048i	$-0.164850\pi$
$164$	− 5.60388i	− 0.437589i
$165$	−4.26205	−0.331800
$166$	− 1.98254i	− 0.153875i
$167$	0.796561	0.0616397	0.0308199	−	0.999525i	$-0.490188\pi$
$167$	0.796561	0.0616397	0.0308199	+	0.999525i	$0.490188\pi$
$168$	−0.753020	−0.0580968
$169$	18.9541	1.45801
$170$	−1.38404	−0.106151
$171$	− 3.40581i	− 0.260449i
$172$	− 6.13706i	− 0.467947i
$173$	9.15346	0.695924	0.347962	−	0.937509i	$-0.386874\pi$
$173$	9.15346	0.695924	0.347962	+	0.937509i	$0.386874\pi$
$174$	0	0
$175$	−1.61356	−0.121974
$176$	14.0804i	1.06135i
$177$	15.5797i	1.17104i
$178$	−2.52781	−0.189467
$179$	−3.40581	−0.254562	−0.127281	−	0.991867i	$-0.540625\pi$
$179$	−3.40581	−0.254562	−0.127281	+	0.991867i	$0.540625\pi$
$180$	−1.80194	−0.134309
$181$	12.6746	0.942093	0.471046	−	0.882108i	$-0.343876\pi$
$181$	12.6746	0.942093	0.471046	+	0.882108i	$0.343876\pi$
$182$	− 0.897853i	− 0.0665533i
$183$	−2.04892	−0.151460
$184$	− 3.88471i	− 0.286384i
$185$	− 3.41789i	− 0.251289i
$186$	3.71379i	0.272308i
$187$	−22.1957	−1.62311
$188$	11.6136i	0.847006i
$189$	− 1.97823i	− 0.143895i
$190$	0.725873	0.0526604
$191$	− 10.6703i	− 0.772072i	−0.922484	−	0.386036i	$-0.873844\pi$
$191$	− 10.6703i	− 0.772072i	0.922484	−	0.386036i	$-0.126156\pi$
$192$	− 4.52781i	− 0.326767i
$193$	22.7289i	1.63606i	0.575176	+	0.818029i	$0.304933\pi$
$193$	22.7289i	1.63606i	−0.575176	+	0.818029i	$0.695067\pi$
$194$	0.0803763	0.00577068
$195$	4.87800i	0.349321i
$196$	−12.3840	−0.884574
$197$	−19.5579	−1.39345	−0.696723	−	0.717340i	$-0.745359\pi$
$197$	−19.5579	−1.39345	−0.696723	+	0.717340i	$0.745359\pi$
$198$	3.17629	0.225729
$199$	−0.875018	−0.0620284	−0.0310142	−	0.999519i	$-0.509874\pi$
$199$	−0.875018	−0.0620284	−0.0310142	+	0.999519i	$0.509874\pi$
$200$	7.64981i	0.540923i
$201$	− 2.89679i	− 0.204324i
$202$	1.43727	0.101126
$203$	0	0
$204$	10.0978	0.706990
$205$	2.15213i	0.150311i
$206$	6.07606i	0.423339i
$207$	3.31767	0.230594
$208$	16.1153	1.11739
$209$	11.6407	0.805205
$210$	0.137063	0.00945827
$211$	18.2687i	1.25767i	0.777538	+	0.628836i	$0.216468\pi$
$211$	18.2687i	1.25767i	−0.777538	+	0.628836i	$0.783532\pi$
$212$	−8.45473	−0.580673
$213$	− 9.14675i	− 0.626725i
$214$	− 7.22952i	− 0.494200i
$215$	2.35690i	0.160739i
$216$	−9.37867	−0.638137
$217$	− 2.38835i	− 0.162132i
$218$	2.43296i	0.164781i
$219$	−7.01507	−0.474034
$220$	− 6.15883i	− 0.415228i
$221$	25.4034i	1.70882i
$222$	− 2.74094i	− 0.183960i
$223$	−1.81833	−0.121764	−0.0608822	−	0.998145i	$-0.519391\pi$
$223$	−1.81833	−0.121764	−0.0608822	+	0.998145i	$0.519391\pi$
$224$	− 1.66056i	− 0.110951i
$225$	−6.53319	−0.435546
$226$	4.75063	0.316007
$227$	−13.8562	−0.919670	−0.459835	−	0.888004i	$-0.652092\pi$
$227$	−13.8562	−0.919670	−0.459835	+	0.888004i	$0.652092\pi$
$228$	−5.29590	−0.350729
$229$	12.7724i	0.844024i	0.906590	+	0.422012i	$0.138676\pi$
$229$	12.7724i	0.844024i	−0.906590	+	0.422012i	$0.861324\pi$
$230$	0.707087i	0.0466239i
$231$	2.19806	0.144622
$232$	0	0
$233$	−8.86592	−0.580826	−0.290413	−	0.956901i	$-0.593793\pi$
$233$	−8.86592	−0.580826	−0.290413	+	0.956901i	$0.593793\pi$
$234$	− 3.63533i	− 0.237649i
$235$	− 4.46011i	− 0.290945i
$236$	−22.5133	−1.46549
$237$	−5.81700	−0.377855
$238$	0.713792	0.0462682
$239$	−25.5646	−1.65364	−0.826820	−	0.562467i	$-0.809852\pi$
$239$	−25.5646	−1.65364	−0.826820	+	0.562467i	$0.809852\pi$
$240$	2.46011i	0.158799i
$241$	−9.73125	−0.626845	−0.313422	−	0.949614i	$-0.601476\pi$
$241$	−9.73125	−0.626845	−0.313422	+	0.949614i	$0.601476\pi$
$242$	5.96077i	0.383173i
$243$	− 13.4155i	− 0.860605i
$244$	− 2.96077i	− 0.189544i
$245$	4.75600	0.303850
$246$	1.72587i	0.110038i
$247$	− 13.3230i	− 0.847725i
$248$	−11.3230	−0.719014
$249$	− 5.55496i	− 0.352031i
$250$	− 2.93230i	− 0.185455i
$251$	− 9.76032i	− 0.616066i	−0.951376	−	0.308033i	$-0.900329\pi$
$251$	− 9.76032i	− 0.616066i	0.951376	−	0.308033i	$-0.0996707\pi$
$252$	0.929312	0.0585412
$253$	11.3394i	0.712904i
$254$	−0.461435	−0.0289530
$255$	−3.87800	−0.242850
$256$	2.40150	0.150094
$257$	16.3569	1.02032	0.510158	−	0.860081i	$-0.329587\pi$
$257$	16.3569	1.02032	0.510158	+	0.860081i	$0.329587\pi$
$258$	1.89008i	0.117671i
$259$	1.76271i	0.109529i
$260$	−7.04892	−0.437155
$261$	0	0
$262$	−3.57002	−0.220557
$263$	23.7235i	1.46285i	0.681921	+	0.731426i	$0.261145\pi$
$263$	23.7235i	1.46285i	−0.681921	+	0.731426i	$0.738855\pi$
$264$	− 10.4209i	− 0.641361i
$265$	3.24698	0.199460
$266$	−0.374354	−0.0229531
$267$	−7.08277	−0.433458
$268$	4.18598	0.255699
$269$	− 25.3545i	− 1.54589i	−0.634472	−	0.772946i	$-0.718783\pi$
$269$	− 25.3545i	− 1.54589i	0.634472	−	0.772946i	$-0.281217\pi$
$270$	1.70709	0.103890
$271$	1.20344i	0.0731037i	0.999332	+	0.0365519i	$0.0116374\pi$
$271$	1.20344i	0.0731037i	−0.999332	+	0.0365519i	$0.988363\pi$
$272$	12.8116i	0.776819i
$273$	− 2.51573i	− 0.152259i
$274$	7.66919	0.463312
$275$	− 22.3297i	− 1.34653i
$276$	− 5.15883i	− 0.310525i
$277$	10.6649	0.640790	0.320395	−	0.947284i	$-0.396184\pi$
$277$	10.6649	0.640790	0.320395	+	0.947284i	$0.396184\pi$
$278$	7.41789i	0.444896i
$279$	− 9.67025i	− 0.578943i
$280$	0.417895i	0.0249740i
$281$	−16.2795	−0.971154	−0.485577	−	0.874194i	$-0.661390\pi$
$281$	−16.2795	−0.971154	−0.485577	+	0.874194i	$0.661390\pi$
$282$	− 3.57673i	− 0.212991i
$283$	5.24698	0.311901	0.155950	−	0.987765i	$-0.450156\pi$
$283$	5.24698	0.311901	0.155950	+	0.987765i	$0.450156\pi$
$284$	13.2174	0.784311
$285$	2.03385	0.120475
$286$	12.4252	0.734717
$287$	− 1.10992i	− 0.0655163i
$288$	− 6.72348i	− 0.396185i
$289$	−3.19567	−0.187981
$290$	0	0
$291$	0.225209	0.0132020
$292$	− 10.1371i	− 0.593227i
$293$	6.76377i	0.395144i	0.980288	+	0.197572i	$0.0633056\pi$
$293$	6.76377i	0.395144i	−0.980288	+	0.197572i	$0.936694\pi$
$294$	3.81402	0.222438
$295$	8.64609	0.503395
$296$	8.35690	0.485735
$297$	27.3763	1.58853
$298$	8.28083i	0.479696i
$299$	12.9782	0.750550
$300$	10.1588i	0.586521i
$301$	− 1.21552i	− 0.0700614i
$302$	− 3.35019i	− 0.192782i
$303$	4.02715	0.231354
$304$	− 6.71917i	− 0.385371i
$305$	1.13706i	0.0651081i
$306$	2.89008	0.165215
$307$	4.51812i	0.257863i	0.991654	+	0.128931i	$0.0411547\pi$
$307$	4.51812i	0.257863i	−0.991654	+	0.128931i	$0.958845\pi$
$308$	3.17629i	0.180986i
$309$	17.0248i	0.968504i
$310$	2.06100	0.117057
$311$	− 12.6165i	− 0.715419i	−0.933833	−	0.357709i	$-0.883558\pi$
$311$	− 12.6165i	− 0.715419i	0.933833	−	0.357709i	$-0.116442\pi$
$312$	−11.9269	−0.675229
$313$	19.2403	1.08752	0.543762	−	0.839239i	$-0.316999\pi$
$313$	19.2403	1.08752	0.543762	+	0.839239i	$0.316999\pi$
$314$	8.11721	0.458081
$315$	−0.356896	−0.0201088
$316$	− 8.40581i	− 0.472864i
$317$	− 2.86831i	− 0.161101i	−0.996751	−	0.0805503i	$-0.974332\pi$
$317$	− 2.86831i	− 0.161101i	0.996751	−	0.0805503i	$-0.0256678\pi$
$318$	2.60388	0.146018
$319$	0	0
$320$	−2.51275	−0.140467
$321$	− 20.2567i	− 1.13062i
$322$	− 0.364666i	− 0.0203220i
$323$	10.5918	0.589343
$324$	−4.64310	−0.257950
$325$	−25.5569	−1.41764
$326$	−5.62565	−0.311576
$327$	6.81700i	0.376981i
$328$	−5.26205	−0.290548
$329$	2.30021i	0.126815i
$330$	1.89679i	0.104415i
$331$	− 3.13408i	− 0.172265i	−0.996284	−	0.0861323i	$-0.972549\pi$
$331$	− 3.13408i	− 0.172265i	0.996284	−	0.0861323i	$-0.0274508\pi$
$332$	8.02715	0.440547
$333$	7.13706i	0.391109i
$334$	− 0.354503i	− 0.0193975i
$335$	−1.60760	−0.0878324
$336$	− 1.26875i	− 0.0692160i
$337$	− 4.98254i	− 0.271416i	−0.990749	−	0.135708i	$-0.956669\pi$
$337$	− 4.98254i	− 0.271416i	0.990749	−	0.135708i	$-0.0433310\pi$
$338$	− 8.43535i	− 0.458823i
$339$	13.3110	0.722952
$340$	− 5.60388i	− 0.303913i
$341$	33.0519	1.78986
$342$	−1.51573	−0.0819613
$343$	−4.95108	−0.267333
$344$	−5.76271	−0.310704
$345$	1.98121i	0.106665i
$346$	− 4.07367i	− 0.219002i
$347$	−20.1172	−1.07995	−0.539974	−	0.841682i	$-0.681566\pi$
$347$	−20.1172	−1.07995	−0.539974	+	0.841682i	$0.681566\pi$
$348$	0	0
$349$	20.4892	1.09676	0.548380	−	0.836229i	$-0.315245\pi$
$349$	20.4892	1.09676	0.548380	+	0.836229i	$0.315245\pi$
$350$	0.718104i	0.0383843i
$351$	− 31.3327i	− 1.67242i
$352$	22.9801	1.22485
$353$	−17.9976	−0.957916	−0.478958	−	0.877838i	$-0.658985\pi$
$353$	−17.9976	−0.957916	−0.478958	+	0.877838i	$0.658985\pi$
$354$	6.93362	0.368518
$355$	−5.07606	−0.269410
$356$	− 10.2349i	− 0.542449i
$357$	2.00000	0.105851
$358$	1.51573i	0.0801088i
$359$	23.6286i	1.24707i	0.781795	+	0.623536i	$0.214304\pi$
$359$	23.6286i	1.24707i	−0.781795	+	0.623536i	$0.785696\pi$
$360$	1.69202i	0.0891774i
$361$	13.4450	0.707634
$362$	− 5.64071i	− 0.296469i
$363$	16.7017i	0.876612i
$364$	3.63533	0.190543
$365$	3.89307i	0.203772i
$366$	0.911854i	0.0476634i
$367$	− 29.6431i	− 1.54736i	−0.633578	−	0.773679i	$-0.718414\pi$
$367$	− 29.6431i	− 1.54736i	0.633578	−	0.773679i	$-0.281586\pi$
$368$	6.54527	0.341196
$369$	− 4.49396i	− 0.233946i
$370$	−1.52111	−0.0790786
$371$	−1.67456	−0.0869390
$372$	−15.0368	−0.779624
$373$	−25.1879	−1.30418	−0.652090	−	0.758142i	$-0.726108\pi$
$373$	−25.1879	−1.30418	−0.652090	+	0.758142i	$0.726108\pi$
$374$	9.87800i	0.510779i
$375$	− 8.21611i	− 0.424278i
$376$	10.9051	0.562390
$377$	0	0
$378$	−0.880395	−0.0452826
$379$	− 26.9071i	− 1.38212i	−0.722796	−	0.691062i	$-0.757143\pi$
$379$	− 26.9071i	− 1.38212i	0.722796	−	0.691062i	$-0.242857\pi$
$380$	2.93900i	0.150768i
$381$	−1.29291	−0.0662379
$382$	−4.74871	−0.242965
$383$	19.6692	1.00505	0.502524	−	0.864563i	$-0.332405\pi$
$383$	19.6692	1.00505	0.502524	+	0.864563i	$0.332405\pi$
$384$	−13.6189	−0.694989
$385$	− 1.21983i	− 0.0621684i
$386$	10.1153	0.514855
$387$	− 4.92154i	− 0.250176i
$388$	0.325437i	0.0165216i
$389$	− 24.8552i	− 1.26021i	−0.776511	−	0.630103i	$-0.783012\pi$
$389$	− 24.8552i	− 1.26021i	0.776511	−	0.630103i	$-0.216988\pi$
$390$	2.17092	0.109929
$391$	10.3177i	0.521787i
$392$	11.6286i	0.587334i
$393$	−10.0030	−0.504584
$394$	8.70410i	0.438506i
$395$	3.22819i	0.162428i
$396$	12.8605i	0.646267i
$397$	−3.99462	−0.200484	−0.100242	−	0.994963i	$-0.531962\pi$
$397$	−3.99462	−0.200484	−0.100242	+	0.994963i	$0.531962\pi$
$398$	0.389420i	0.0195198i
$399$	−1.04892	−0.0525115
$400$	−12.8890	−0.644451
$401$	−24.9071	−1.24380	−0.621900	−	0.783097i	$-0.713639\pi$
$401$	−24.9071	−1.24380	−0.621900	+	0.783097i	$0.713639\pi$
$402$	−1.28919	−0.0642991
$403$	− 37.8286i	− 1.88438i
$404$	5.81940i	0.289526i
$405$	1.78315	0.0886055
$406$	0	0
$407$	−24.3937	−1.20915
$408$	− 9.48188i	− 0.469423i
$409$	0.283224i	0.0140045i	0.999975	+	0.00700227i	$0.00222891\pi$
$409$	0.283224i	0.0140045i	−0.999975	+	0.00700227i	$0.997771\pi$
$410$	0.957787	0.0473017
$411$	21.4886	1.05995
$412$	−24.6015	−1.21203
$413$	−4.45904	−0.219415
$414$	− 1.47650i	− 0.0725661i
$415$	−3.08277	−0.151327
$416$	− 26.3013i	− 1.28953i
$417$	20.7845i	1.01782i
$418$	− 5.18060i	− 0.253392i
$419$	26.4886	1.29405	0.647026	−	0.762468i	$-0.276013\pi$
$419$	26.4886	1.29405	0.647026	+	0.762468i	$0.276013\pi$
$420$	0.554958i	0.0270792i
$421$	17.5821i	0.856899i	0.903566	+	0.428450i	$0.140940\pi$
$421$	17.5821i	0.856899i	−0.903566	+	0.428450i	$0.859060\pi$
$422$	8.13036	0.395780
$423$	9.31336i	0.452831i
$424$	7.93900i	0.385552i
$425$	− 20.3177i	− 0.985552i
$426$	−4.07069	−0.197225
$427$	− 0.586417i	− 0.0283787i
$428$	29.2717	1.41490
$429$	34.8146	1.68087
$430$	1.04892	0.0505833
$431$	−27.7952	−1.33885	−0.669425	−	0.742880i	$-0.733459\pi$
$431$	−27.7952	−1.33885	−0.669425	+	0.742880i	$0.733459\pi$
$432$	− 15.8019i	− 0.760271i
$433$	− 5.87561i	− 0.282364i	−0.989984	−	0.141182i	$-0.954910\pi$
$433$	− 5.87561i	− 0.282364i	0.989984	−	0.141182i	$-0.0450902\pi$
$434$	−1.06292	−0.0510217
$435$	0	0
$436$	−9.85086	−0.471770
$437$	− 5.41119i	− 0.258852i
$438$	3.12200i	0.149175i
$439$	15.7409	0.751274	0.375637	−	0.926767i	$-0.377424\pi$
$439$	15.7409	0.751274	0.375637	+	0.926767i	$0.377424\pi$
$440$	−5.78315	−0.275701
$441$	−9.93123	−0.472916
$442$	11.3056	0.537752
$443$	− 6.73855i	− 0.320158i	−0.987104	−	0.160079i	$-0.948825\pi$
$443$	− 6.73855i	− 0.320158i	0.987104	−	0.160079i	$-0.0511749\pi$
$444$	11.0978	0.526680
$445$	3.93064i	0.186330i
$446$	0.809234i	0.0383183i
$447$	23.2024i	1.09743i
$448$	1.29590	0.0612254
$449$	12.3201i	0.581420i	0.956811	+	0.290710i	$0.0938914\pi$
$449$	12.3201i	0.581420i	−0.956811	+	0.290710i	$0.906109\pi$
$450$	2.90754i	0.137063i
$451$	15.3599	0.723268
$452$	19.2349i	0.904733i
$453$	− 9.38703i	− 0.441041i
$454$	6.16660i	0.289413i
$455$	−1.39612	−0.0654513
$456$	4.97285i	0.232875i
$457$	−13.6625	−0.639104	−0.319552	−	0.947569i	$-0.603532\pi$
$457$	−13.6625	−0.639104	−0.319552	+	0.947569i	$0.603532\pi$
$458$	5.68425	0.265608
$459$	24.9095	1.16267
$460$	−2.86294	−0.133485
$461$	− 11.6082i	− 0.540647i	−0.962770	−	0.270324i	$-0.912869\pi$
$461$	− 11.6082i	− 0.540647i	0.962770	−	0.270324i	$-0.0871307\pi$
$462$	− 0.978230i	− 0.0455114i
$463$	7.24267	0.336595	0.168298	−	0.985736i	$-0.446173\pi$
$463$	7.24267	0.336595	0.168298	+	0.985736i	$0.446173\pi$
$464$	0	0
$465$	5.77479	0.267800
$466$	3.94571i	0.182781i
$467$	− 2.06339i	− 0.0954824i	−0.998860	−	0.0477412i	$-0.984798\pi$
$467$	− 2.06339i	− 0.0954824i	0.998860	−	0.0477412i	$-0.0152023\pi$
$468$	14.7192	0.680394
$469$	0.829085	0.0382836
$470$	−1.98493	−0.0915582
$471$	22.7439	1.04798
$472$	21.1400i	0.973050i
$473$	16.8213	0.773445
$474$	2.58881i	0.118908i
$475$	10.6558i	0.488921i
$476$	2.89008i	0.132467i
$477$	−6.78017	−0.310443
$478$	11.3773i	0.520387i
$479$	3.88902i	0.177694i	0.996045	+	0.0888469i	$0.0283182\pi$
$479$	3.88902i	0.177694i	−0.996045	+	0.0888469i	$0.971682\pi$
$480$	4.01507	0.183262
$481$	27.9191i	1.27300i
$482$	4.33081i	0.197263i
$483$	− 1.02177i	− 0.0464922i
$484$	−24.1347	−1.09703
$485$	− 0.124982i	− 0.00567513i
$486$	−5.97046	−0.270826
$487$	9.84548	0.446141	0.223071	−	0.974802i	$-0.428392\pi$
$487$	9.84548	0.446141	0.223071	+	0.974802i	$0.428392\pi$
$488$	−2.78017	−0.125852
$489$	−15.7627	−0.712814
$490$	− 2.11662i	− 0.0956192i
$491$	− 7.79118i	− 0.351611i	−0.984425	−	0.175806i	$-0.943747\pi$
$491$	− 7.79118i	− 0.351611i	0.984425	−	0.175806i	$-0.0562530\pi$
$492$	−6.98792	−0.315040
$493$	0	0
$494$	−5.92931	−0.266772
$495$	− 4.93900i	− 0.221992i
$496$	− 19.0780i	− 0.856627i
$497$	2.61788	0.117428
$498$	−2.47219	−0.110781
$499$	20.5652	0.920626	0.460313	−	0.887757i	$-0.347737\pi$
$499$	20.5652	0.920626	0.460313	+	0.887757i	$0.347737\pi$
$500$	11.8726	0.530960
$501$	− 0.993295i	− 0.0443772i
$502$	−4.34375	−0.193871
$503$	− 8.23490i	− 0.367176i	−0.983003	−	0.183588i	$-0.941229\pi$
$503$	− 8.23490i	− 0.367176i	0.983003	−	0.183588i	$-0.0587712\pi$
$504$	− 0.872625i	− 0.0388698i
$505$	− 2.23490i	− 0.0994517i
$506$	5.04652	0.224345
$507$	− 23.6353i	− 1.04968i
$508$	− 1.86831i	− 0.0828930i
$509$	7.91617	0.350878	0.175439	−	0.984490i	$-0.443866\pi$
$509$	7.91617	0.350878	0.175439	+	0.984490i	$0.443866\pi$
$510$	1.72587i	0.0764230i
$511$	− 2.00777i	− 0.0888185i
$512$	− 22.9119i	− 1.01257i
$513$	−13.0640	−0.576789
$514$	− 7.27950i	− 0.321085i
$515$	9.44803	0.416330
$516$	−7.65279	−0.336895
$517$	−31.8321	−1.39997
$518$	0.784479	0.0344680
$519$	− 11.4142i	− 0.501027i
$520$	6.61894i	0.290260i
$521$	3.52542	0.154451	0.0772257	−	0.997014i	$-0.475394\pi$
$521$	3.52542	0.154451	0.0772257	+	0.997014i	$0.475394\pi$
$522$	0	0
$523$	10.0301	0.438587	0.219294	−	0.975659i	$-0.429625\pi$
$523$	10.0301	0.438587	0.219294	+	0.975659i	$0.429625\pi$
$524$	− 14.4547i	− 0.631458i
$525$	2.01208i	0.0878144i
$526$	10.5579	0.460348
$527$	30.0737	1.31003
$528$	17.5579	0.764111
$529$	−17.7289	−0.770820
$530$	− 1.44504i	− 0.0627686i
$531$	−18.0543	−0.783490
$532$	− 1.51573i	− 0.0657152i
$533$	− 17.5797i	− 0.761462i
$534$	3.15213i	0.136406i
$535$	−11.2416	−0.486017
$536$	− 3.93064i	− 0.169778i
$537$	4.24698i	0.183271i
$538$	−11.2838	−0.486480
$539$	− 33.9439i	− 1.46207i
$540$	6.91185i	0.297439i
$541$	8.22414i	0.353584i	0.984248	+	0.176792i	$0.0565719\pi$
$541$	8.22414i	0.353584i	−0.984248	+	0.176792i	$0.943428\pi$
$542$	0.535581	0.0230052
$543$	− 15.8049i	− 0.678254i
$544$	20.9095	0.896486
$545$	3.78315	0.162052
$546$	−1.11960	−0.0479147
$547$	25.8629	1.10582	0.552910	−	0.833241i	$-0.313517\pi$
$547$	25.8629	1.10582	0.552910	+	0.833241i	$0.313517\pi$
$548$	31.0519i	1.32647i
$549$	− 2.37435i	− 0.101335i
$550$	−9.93767	−0.423744
$551$	0	0
$552$	−4.84415	−0.206181
$553$	− 1.66487i	− 0.0707977i
$554$	− 4.74632i	− 0.201652i
$555$	−4.26205	−0.180914
$556$	−30.0344	−1.27374
$557$	−23.0097	−0.974952	−0.487476	−	0.873136i	$-0.662082\pi$
$557$	−23.0097	−0.974952	−0.487476	+	0.873136i	$0.662082\pi$
$558$	−4.30367	−0.182189
$559$	− 19.2524i	− 0.814288i
$560$	−0.704103	−0.0297538
$561$	27.6775i	1.16855i
$562$	7.24506i	0.305614i
$563$	− 43.1159i	− 1.81712i	−0.417757	−	0.908559i	$-0.637184\pi$
$563$	− 43.1159i	− 1.81712i	0.417757	−	0.908559i	$-0.362816\pi$
$564$	14.4819	0.609797
$565$	− 7.38703i	− 0.310775i
$566$	− 2.33513i	− 0.0981527i
$567$	−0.919624	−0.0386206
$568$	− 12.4112i	− 0.520762i
$569$	24.3123i	1.01922i	0.860404	+	0.509612i	$0.170211\pi$
$569$	24.3123i	1.01922i	−0.860404	+	0.509612i	$0.829789\pi$
$570$	− 0.905149i	− 0.0379125i
$571$	18.4886	0.773723	0.386862	−	0.922138i	$-0.373559\pi$
$571$	18.4886	0.773723	0.386862	+	0.922138i	$0.373559\pi$
$572$	50.3086i	2.10351i
$573$	−13.3056	−0.555849
$574$	−0.493959	−0.0206175
$575$	−10.3800	−0.432876
$576$	5.24698	0.218624
$577$	37.8582i	1.57606i	0.615640	+	0.788028i	$0.288898\pi$
$577$	37.8582i	1.57606i	−0.615640	+	0.788028i	$0.711102\pi$
$578$	1.42221i	0.0591560i
$579$	28.3424	1.17787
$580$	0	0
$581$	1.58987	0.0659591
$582$	− 0.100228i	− 0.00415457i
$583$	− 23.1739i	− 0.959765i
$584$	−9.51871	−0.393887
$585$	−5.65279	−0.233714
$586$	3.01016	0.124349
$587$	−14.4354	−0.595811	−0.297905	−	0.954595i	$-0.596288\pi$
$587$	−14.4354	−0.595811	−0.297905	+	0.954595i	$0.596288\pi$
$588$	15.4426i	0.636844i
$589$	−15.7724	−0.649890
$590$	− 3.84787i	− 0.158414i
$591$	24.3884i	1.00320i
$592$	14.0804i	0.578700i
$593$	13.0019	0.533925	0.266962	−	0.963707i	$-0.413980\pi$
$593$	13.0019	0.533925	0.266962	+	0.963707i	$0.413980\pi$
$594$	− 12.1836i	− 0.499899i
$595$	− 1.10992i	− 0.0455021i
$596$	−33.5284	−1.37338
$597$	1.09113i	0.0446570i
$598$	− 5.77586i	− 0.236192i
$599$	12.0935i	0.494128i	0.968999	+	0.247064i	$0.0794658\pi$
$599$	12.0935i	0.494128i	−0.968999	+	0.247064i	$0.920534\pi$
$600$	9.53916	0.389434
$601$	− 22.3472i	− 0.911562i	−0.890092	−	0.455781i	$-0.849360\pi$
$601$	− 22.3472i	− 0.911562i	0.890092	−	0.455781i	$-0.150640\pi$
$602$	−0.540958	−0.0220478
$603$	3.35690	0.136703
$604$	13.5646	0.551938
$605$	9.26875	0.376828
$606$	− 1.79225i	− 0.0728051i
$607$	− 41.4687i	− 1.68316i	−0.540129	−	0.841582i	$-0.681625\pi$
$607$	− 41.4687i	− 1.68316i	0.540129	−	0.841582i	$-0.318375\pi$
$608$	−10.9661	−0.444736
$609$	0	0
$610$	0.506041	0.0204890
$611$	36.4325i	1.47390i
$612$	11.7017i	0.473014i
$613$	−25.7845	−1.04143	−0.520713	−	0.853732i	$-0.674334\pi$
$613$	−25.7845	−1.04143	−0.520713	+	0.853732i	$0.674334\pi$
$614$	2.01075	0.0811474
$615$	2.68366	0.108216
$616$	2.98254	0.120170
$617$	− 8.86533i	− 0.356905i	−0.983949	−	0.178452i	$-0.942891\pi$
$617$	− 8.86533i	− 0.356905i	0.983949	−	0.178452i	$-0.0571090\pi$
$618$	7.57673	0.304781
$619$	− 29.4513i	− 1.18375i	−0.806031	−	0.591873i	$-0.798389\pi$
$619$	− 29.4513i	− 1.18375i	0.806031	−	0.591873i	$-0.201611\pi$
$620$	8.34481i	0.335136i
$621$	− 12.7259i	− 0.510672i
$622$	−5.61489	−0.225137
$623$	− 2.02715i	− 0.0812159i
$624$	− 20.0954i	− 0.804461i
$625$	18.0459	0.721837
$626$	− 8.56273i	− 0.342235i
$627$	− 14.5157i	− 0.579702i
$628$	32.8659i	1.31149i
$629$	−22.1957	−0.884999
$630$	0.158834i	0.00632808i
$631$	−23.7711	−0.946311	−0.473156	−	0.880979i	$-0.656885\pi$
$631$	−23.7711	−0.946311	−0.473156	+	0.880979i	$0.656885\pi$
$632$	−7.89307	−0.313969
$633$	22.7808	0.905454
$634$	−1.27652	−0.0506971
$635$	0.717513i	0.0284736i
$636$	10.5429i	0.418052i
$637$	−38.8495	−1.53927
$638$	0	0
$639$	10.5996	0.419312
$640$	7.55794i	0.298754i
$641$	28.4655i	1.12432i	0.827029	+	0.562160i	$0.190029\pi$
$641$	28.4655i	1.12432i	−0.827029	+	0.562160i	$0.809971\pi$
$642$	−9.01507	−0.355796
$643$	19.3467	0.762961	0.381480	−	0.924377i	$-0.375414\pi$
$643$	19.3467	0.762961	0.381480	+	0.924377i	$0.375414\pi$
$644$	1.47650	0.0581823
$645$	2.93900	0.115723
$646$	− 4.71379i	− 0.185462i
$647$	−22.7144	−0.892995	−0.446497	−	0.894785i	$-0.647329\pi$
$647$	−22.7144	−0.892995	−0.446497	+	0.894785i	$0.647329\pi$
$648$	4.35988i	0.171272i
$649$	− 61.7077i	− 2.42224i
$650$	11.3739i	0.446120i
$651$	−2.97823	−0.116726
$652$	− 22.7778i	− 0.892046i
$653$	− 22.9312i	− 0.897368i	−0.893690	−	0.448684i	$-0.851893\pi$
$653$	− 22.9312i	− 0.897368i	0.893690	−	0.448684i	$-0.148107\pi$
$654$	3.03385	0.118633
$655$	5.55124i	0.216905i
$656$	− 8.86592i	− 0.346156i
$657$	− 8.12929i	− 0.317154i
$658$	1.02369	0.0399076
$659$	− 19.1782i	− 0.747077i	−0.927615	−	0.373539i	$-0.878144\pi$
$659$	− 19.1782i	− 0.747077i	0.927615	−	0.373539i	$-0.121856\pi$
$660$	−7.67994	−0.298941
$661$	−3.38644	−0.131717	−0.0658585	−	0.997829i	$-0.520979\pi$
$661$	−3.38644	−0.131717	−0.0658585	+	0.997829i	$0.520979\pi$
$662$	−1.39480	−0.0542103
$663$	31.6775	1.23025
$664$	− 7.53750i	− 0.292512i
$665$	0.582105i	0.0225731i
$666$	3.17629	0.123079
$667$	0	0
$668$	1.43535	0.0555355
$669$	2.26742i	0.0876636i
$670$	0.715448i	0.0276402i
$671$	8.11529	0.313287
$672$	−2.07069	−0.0798785
$673$	−4.77048	−0.183888	−0.0919442	−	0.995764i	$-0.529308\pi$
$673$	−4.77048	−0.183888	−0.0919442	+	0.995764i	$0.529308\pi$
$674$	−2.21744	−0.0854126
$675$	25.0599i	0.964557i
$676$	34.1540	1.31362
$677$	43.0901i	1.65609i	0.560665	+	0.828043i	$0.310546\pi$
$677$	43.0901i	1.65609i	−0.560665	+	0.828043i	$0.689454\pi$
$678$	− 5.92394i	− 0.227507i
$679$	0.0644568i	0.00247362i
$680$	−5.26205	−0.201790
$681$	17.2784i	0.662111i
$682$	− 14.7095i	− 0.563255i
$683$	23.4330	0.896637	0.448319	−	0.893874i	$-0.352023\pi$
$683$	23.4330	0.896637	0.448319	+	0.893874i	$0.352023\pi$
$684$	− 6.13706i	− 0.234656i
$685$	− 11.9253i	− 0.455641i
$686$	2.20344i	0.0841277i
$687$	15.9269	0.607650
$688$	− 9.70948i	− 0.370170i
$689$	−26.5230	−1.01045
$690$	0.881723	0.0335666
$691$	43.1377	1.64103	0.820517	−	0.571622i	$-0.193686\pi$
$691$	43.1377	1.64103	0.820517	+	0.571622i	$0.193686\pi$
$692$	16.4940	0.627006
$693$	2.54719i	0.0967597i
$694$	8.95300i	0.339851i
$695$	11.5345	0.437529
$696$	0	0
$697$	13.9758	0.529373
$698$	− 9.11854i	− 0.345142i
$699$	11.0556i	0.418162i
$700$	−2.90754	−0.109895
$701$	−4.23085	−0.159797	−0.0798985	−	0.996803i	$-0.525460\pi$
$701$	−4.23085	−0.159797	−0.0798985	+	0.996803i	$0.525460\pi$
$702$	−13.9444	−0.526297
$703$	11.6407	0.439038
$704$	17.9336i	0.675899i
$705$	−5.56166	−0.209464
$706$	8.00969i	0.301449i
$707$	1.15260i	0.0433481i
$708$	28.0737i	1.05507i
$709$	−14.3037	−0.537185	−0.268593	−	0.963254i	$-0.586559\pi$
$709$	−14.3037	−0.537185	−0.268593	+	0.963254i	$0.586559\pi$
$710$	2.25906i	0.0847811i
$711$	− 6.74094i	− 0.252805i
$712$	−9.61058	−0.360172
$713$	− 15.3642i	− 0.575393i
$714$	− 0.890084i	− 0.0333105i
$715$	− 19.3207i	− 0.722551i
$716$	−6.13706	−0.229353
$717$	31.8786i	1.19053i
$718$	10.5157	0.392444
$719$	−23.4407	−0.874192	−0.437096	−	0.899415i	$-0.643993\pi$
$719$	−23.4407	−0.874192	−0.437096	+	0.899415i	$0.643993\pi$
$720$	−2.85086	−0.106245
$721$	−4.87263	−0.181466
$722$	− 5.98361i	− 0.222687i
$723$	12.1347i	0.451293i
$724$	22.8388	0.848796
$725$	0	0
$726$	7.43296	0.275863
$727$	51.9976i	1.92848i	0.265022	+	0.964242i	$0.414621\pi$
$727$	51.9976i	1.92848i	−0.265022	+	0.964242i	$0.585379\pi$
$728$	− 3.41358i	− 0.126516i
$729$	−24.4590	−0.905890
$730$	1.73258	0.0641256
$731$	15.3056	0.566098
$732$	−3.69202	−0.136461
$733$	− 34.1903i	− 1.26285i	−0.775438	−	0.631424i	$-0.782471\pi$
$733$	− 34.1903i	− 1.26285i	0.775438	−	0.631424i	$-0.217529\pi$
$734$	−13.1924	−0.486941
$735$	− 5.93064i	− 0.218755i
$736$	− 10.6823i	− 0.393756i
$737$	11.4735i	0.422632i
$738$	−2.00000	−0.0736210
$739$	39.7730i	1.46307i	0.681802	+	0.731537i	$0.261196\pi$
$739$	39.7730i	1.46307i	−0.681802	+	0.731537i	$0.738804\pi$
$740$	− 6.15883i	− 0.226403i
$741$	−16.6136	−0.610315
$742$	0.745251i	0.0273590i
$743$	− 7.17821i	− 0.263343i	−0.991293	−	0.131672i	$-0.957966\pi$
$743$	− 7.17821i	− 0.263343i	0.991293	−	0.131672i	$-0.0420344\pi$
$744$	14.1196i	0.517650i
$745$	12.8763	0.471753
$746$	11.2097i	0.410415i
$747$	6.43727	0.235527
$748$	−39.9952	−1.46237
$749$	5.79763	0.211841
$750$	−3.65651	−0.133517
$751$	27.1685i	0.991393i	0.868496	+	0.495697i	$0.165087\pi$
$751$	27.1685i	0.991393i	−0.868496	+	0.495697i	$0.834913\pi$
$752$	18.3739i	0.670026i
$753$	−12.1709	−0.443533
$754$	0	0
$755$	−5.20941	−0.189590
$756$	− 3.56465i	− 0.129645i
$757$	23.6926i	0.861123i	0.902561	+	0.430561i	$0.141684\pi$
$757$	23.6926i	0.861123i	−0.902561	+	0.430561i	$0.858316\pi$
$758$	−11.9748	−0.434943
$759$	14.1400	0.513251
$760$	2.75973	0.100106
$761$	−14.2543	−0.516717	−0.258359	−	0.966049i	$-0.583182\pi$
$761$	−14.2543	−0.516717	−0.258359	+	0.966049i	$0.583182\pi$
$762$	0.575400i	0.0208446i
$763$	−1.95108	−0.0706339
$764$	− 19.2271i	− 0.695613i
$765$	− 4.49396i	− 0.162479i
$766$	− 8.75361i	− 0.316281i
$767$	−70.6258	−2.55015
$768$	− 2.99462i	− 0.108059i
$769$	44.7066i	1.61216i	0.591805	+	0.806081i	$0.298415\pi$
$769$	44.7066i	1.61216i	−0.591805	+	0.806081i	$0.701585\pi$
$770$	−0.542877	−0.0195639
$771$	− 20.3967i	− 0.734570i
$772$	40.9560i	1.47404i
$773$	22.9487i	0.825407i	0.910865	+	0.412704i	$0.135415\pi$
$773$	22.9487i	0.825407i	−0.910865	+	0.412704i	$0.864585\pi$
$774$	−2.19029	−0.0787284
$775$	30.2553i	1.08680i
$776$	0.305586	0.0109699
$777$	2.19806	0.0788550
$778$	−11.0616	−0.396577
$779$	−7.32975	−0.262616
$780$	8.78986i	0.314727i
$781$	36.2282i	1.29635i
$782$	4.59179	0.164202
$783$	0	0
$784$	−19.5929	−0.699745
$785$	− 12.6219i	− 0.450496i
$786$	4.45175i	0.158789i
$787$	−14.3327	−0.510907	−0.255453	−	0.966821i	$-0.582225\pi$
$787$	−14.3327	−0.510907	−0.255453	+	0.966821i	$0.582225\pi$
$788$	−35.2422	−1.25545
$789$	29.5827	1.05317
$790$	1.43668	0.0511148
$791$	3.80971i	0.135458i
$792$	12.0761	0.429104
$793$	− 9.28813i	− 0.329831i
$794$	1.77777i	0.0630909i
$795$	− 4.04892i	− 0.143600i
$796$	−1.57673	−0.0558857
$797$	10.1933i	0.361064i	0.983569	+	0.180532i	$0.0577820\pi$
$797$	10.1933i	0.361064i	−0.983569	+	0.180532i	$0.942218\pi$
$798$	0.466812i	0.0165250i
$799$	−28.9638	−1.02466
$800$	21.0358i	0.743727i
$801$	− 8.20775i	− 0.290007i
$802$	11.0847i	0.391414i
$803$	27.7851	0.980514
$804$	− 5.21983i	− 0.184089i
$805$	−0.567040	−0.0199855
$806$	−16.8353	−0.592999
$807$	−31.6165	−1.11296
$808$	5.46442	0.192238
$809$	− 8.97285i	− 0.315469i	−0.987482	−	0.157734i	$-0.949581\pi$
$809$	− 8.97285i	− 0.315469i	0.987482	−	0.157734i	$-0.0504190\pi$
$810$	− 0.793577i	− 0.0278835i
$811$	28.5628	1.00298	0.501489	−	0.865164i	$-0.332786\pi$
$811$	28.5628	1.00298	0.501489	+	0.865164i	$0.332786\pi$
$812$	0	0
$813$	1.50066	0.0526306
$814$	10.8562i	0.380511i
$815$	8.74764i	0.306417i
$816$	15.9758	0.559266
$817$	−8.02715	−0.280834
$818$	0.126047	0.00440712
$819$	2.91531	0.101869
$820$	3.87800i	0.135426i
$821$	−11.3351	−0.395599	−0.197799	−	0.980243i	$-0.563379\pi$
$821$	−11.3351	−0.395599	−0.197799	+	0.980243i	$0.563379\pi$
$822$	− 9.56332i	− 0.333559i
$823$	5.67217i	0.197719i	0.995101	+	0.0988597i	$0.0315195\pi$
$823$	5.67217i	0.197719i	−0.995101	+	0.0988597i	$0.968480\pi$
$824$	23.1008i	0.804755i
$825$	−27.8447	−0.969429
$826$	1.98446i	0.0690482i
$827$	2.90323i	0.100955i	0.998725	+	0.0504776i	$0.0160744\pi$
$827$	2.90323i	0.100955i	−0.998725	+	0.0504776i	$0.983926\pi$
$828$	5.97823	0.207758
$829$	− 45.2137i	− 1.57034i	−0.619282	−	0.785169i	$-0.712576\pi$
$829$	− 45.2137i	− 1.57034i	0.619282	−	0.785169i	$-0.287424\pi$
$830$	1.37196i	0.0476215i
$831$	− 13.2989i	− 0.461333i
$832$	20.5254	0.711591
$833$	− 30.8853i	− 1.07011i
$834$	9.24996	0.320300
$835$	−0.551237	−0.0190764
$836$	20.9758	0.725464
$837$	−37.0930	−1.28212
$838$	− 11.7885i	− 0.407228i
$839$	45.6256i	1.57517i	0.616205	+	0.787586i	$0.288669\pi$
$839$	45.6256i	1.57517i	−0.616205	+	0.787586i	$0.711331\pi$
$840$	0.521106	0.0179799
$841$	0	0
$842$	7.82477	0.269659
$843$	20.3002i	0.699176i
$844$	32.9191i	1.13312i
$845$	−13.1166	−0.451225
$846$	4.14483	0.142502
$847$	−4.78017	−0.164248
$848$	−13.3763	−0.459343
$849$	− 6.54288i	− 0.224551i
$850$	−9.04221	−0.310145
$851$	11.3394i	0.388711i
$852$	− 16.4819i	− 0.564660i
$853$	36.9288i	1.26442i	0.774797	+	0.632210i	$0.217852\pi$
$853$	36.9288i	1.26442i	−0.774797	+	0.632210i	$0.782148\pi$
$854$	−0.260980	−0.00893056
$855$	2.35690i	0.0806041i
$856$	− 27.4862i	− 0.939459i
$857$	35.5502	1.21437	0.607185	−	0.794560i	$-0.292299\pi$
$857$	35.5502	1.21437	0.607185	+	0.794560i	$0.292299\pi$
$858$	− 15.4940i	− 0.528955i
$859$	− 42.3812i	− 1.44603i	−0.690834	−	0.723014i	$-0.742756\pi$
$859$	− 42.3812i	− 1.44603i	0.690834	−	0.723014i	$-0.257244\pi$
$860$	4.24698i	0.144821i
$861$	−1.38404	−0.0471681
$862$	12.3700i	0.421325i
$863$	−49.7797	−1.69452	−0.847260	−	0.531178i	$-0.821750\pi$
$863$	−49.7797	−1.69452	−0.847260	+	0.531178i	$0.821750\pi$
$864$	−25.7899	−0.877389
$865$	−6.33439	−0.215376
$866$	−2.61489	−0.0888577
$867$	3.98493i	0.135335i
$868$	− 4.30367i	− 0.146076i
$869$	23.0398	0.781572
$870$	0	0
$871$	13.1317	0.444950
$872$	9.24996i	0.313243i
$873$	0.260980i	0.00883284i
$874$	−2.40821	−0.0814588
$875$	2.35152	0.0794959
$876$	−12.6407	−0.427090
$877$	21.8931	0.739276	0.369638	−	0.929176i	$-0.379482\pi$
$877$	21.8931	0.739276	0.369638	+	0.929176i	$0.379482\pi$
$878$	− 7.00538i	− 0.236420i
$879$	8.43429	0.284481
$880$	− 9.74392i	− 0.328468i
$881$	− 33.5308i	− 1.12968i	−0.825200	−	0.564841i	$-0.808938\pi$
$881$	− 33.5308i	− 1.12968i	0.825200	−	0.564841i	$-0.191062\pi$
$882$	4.41981i	0.148823i
$883$	16.1274	0.542729	0.271365	−	0.962477i	$-0.412525\pi$
$883$	16.1274	0.542729	0.271365	+	0.962477i	$0.412525\pi$
$884$	45.7754i	1.53959i
$885$	− 10.7815i	− 0.362416i
$886$	−2.99894	−0.100751
$887$	52.7391i	1.77081i	0.464823	+	0.885403i	$0.346118\pi$
$887$	52.7391i	1.77081i	−0.464823	+	0.885403i	$0.653882\pi$
$888$	− 10.4209i	− 0.349702i
$889$	− 0.370042i	− 0.0124108i
$890$	1.74930	0.0586367
$891$	− 12.7265i	− 0.426353i
$892$	−3.27652	−0.109706
$893$	15.1903	0.508324
$894$	10.3260	0.345354
$895$	2.35690	0.0787823
$896$	− 3.89785i	− 0.130218i
$897$	− 16.1836i	− 0.540354i
$898$	5.48294	0.182968
$899$	0	0
$900$	−11.7724	−0.392413
$901$	− 21.0858i	− 0.702468i
$902$	− 6.83579i	− 0.227607i
$903$	−1.51573	−0.0504403
$904$	18.0616	0.600720
$905$	−8.77107	−0.291560
$906$	−4.17762	−0.138792
$907$	29.8437i	0.990943i	0.868624	+	0.495472i	$0.165005\pi$
$907$	29.8437i	0.990943i	−0.868624	+	0.495472i	$0.834995\pi$
$908$	−24.9681	−0.828594
$909$	4.66679i	0.154788i
$910$	0.621334i	0.0205970i
$911$	9.34050i	0.309465i	0.987956	+	0.154732i	$0.0494515\pi$
$911$	9.34050i	0.309465i	−0.987956	+	0.154732i	$0.950548\pi$
$912$	−8.37867	−0.277445
$913$	22.0019i	0.728157i
$914$	6.08038i	0.201121i
$915$	1.41789	0.0468742
$916$	23.0151i	0.760439i
$917$	− 2.86294i	− 0.0945425i
$918$	− 11.0858i	− 0.365884i
$919$	−18.4209	−0.607649	−0.303824	−	0.952728i	$-0.598264\pi$
$919$	−18.4209	−0.607649	−0.303824	+	0.952728i	$0.598264\pi$
$920$	2.68830i	0.0886306i
$921$	5.63401	0.185647
$922$	−5.16613	−0.170137
$923$	41.4639	1.36480
$924$	3.96077	0.130300
$925$	− 22.3297i	− 0.734198i
$926$	− 3.22329i	− 0.105924i
$927$	−19.7289	−0.647981
$928$	0	0
$929$	−4.84654	−0.159010	−0.0795050	−	0.996834i	$-0.525334\pi$
$929$	−4.84654	−0.159010	−0.0795050	+	0.996834i	$0.525334\pi$
$930$	− 2.57002i	− 0.0842744i
$931$	16.1981i	0.530870i
$932$	−15.9758	−0.523306
$933$	−15.7326	−0.515061
$934$	−0.918296	−0.0300476
$935$	15.3599	0.502322
$936$	− 13.8213i	− 0.451764i
$937$	44.7144	1.46076	0.730378	−	0.683044i	$-0.239344\pi$
$937$	44.7144	1.46076	0.730378	+	0.683044i	$0.239344\pi$
$938$	− 0.368977i	− 0.0120475i
$939$	− 23.9922i	− 0.782957i
$940$	− 8.03684i	− 0.262133i
$941$	−13.5851	−0.442861	−0.221431	−	0.975176i	$-0.571073\pi$
$941$	−13.5851	−0.442861	−0.221431	+	0.975176i	$0.571073\pi$
$942$	− 10.1220i	− 0.329793i
$943$	− 7.14005i	− 0.232512i
$944$	−35.6185	−1.15928
$945$	1.36898i	0.0445328i
$946$	− 7.48619i	− 0.243397i
$947$	15.0116i	0.487812i	0.969799	+	0.243906i	$0.0784288\pi$
$947$	15.0116i	0.487812i	−0.969799	+	0.243906i	$0.921571\pi$
$948$	−10.4819	−0.340436
$949$	− 31.8006i	− 1.03229i
$950$	4.74227	0.153859
$951$	−3.57673	−0.115983
$952$	2.71379	0.0879545
$953$	51.8447	1.67942	0.839708	−	0.543038i	$-0.182726\pi$
$953$	51.8447	1.67942	0.839708	+	0.543038i	$0.182726\pi$
$954$	3.01746i	0.0976938i
$955$	7.38404i	0.238942i
$956$	−46.0659	−1.48988
$957$	0	0
$958$	1.73078	0.0559188
$959$	6.15021i	0.198601i
$960$	3.13334i	0.101128i
$961$	−13.7832	−0.444618
$962$	12.4252	0.400604
$963$	23.4741	0.756443
$964$	−17.5351	−0.564768
$965$	− 15.7289i	− 0.506330i
$966$	−0.454731	−0.0146307
$967$	41.6249i	1.33857i	0.743007	+	0.669283i	$0.233399\pi$
$967$	41.6249i	1.33857i	−0.743007	+	0.669283i	$0.766601\pi$
$968$	22.6625i	0.728400i
$969$	− 13.2078i	− 0.424294i
$970$	−0.0556221	−0.00178592
$971$	− 55.7730i	− 1.78984i	−0.446226	−	0.894920i	$-0.647232\pi$
$971$	− 55.7730i	− 1.78984i	0.446226	−	0.894920i	$-0.352768\pi$
$972$	− 24.1739i	− 0.775378i
$973$	−5.94869	−0.190706
$974$	− 4.38165i	− 0.140397i
$975$	31.8689i	1.02062i
$976$	− 4.68425i	− 0.149939i
$977$	48.9584	1.56632	0.783159	−	0.621822i	$-0.213607\pi$
$977$	48.9584	1.56632	0.783159	+	0.621822i	$0.213607\pi$
$978$	7.01507i	0.224317i
$979$	28.0532	0.896585
$980$	8.57002	0.273759
$981$	−7.89977	−0.252220
$982$	−3.46740	−0.110649
$983$	− 5.17331i	− 0.165003i	−0.996591	−	0.0825015i	$-0.973709\pi$
$983$	− 5.17331i	− 0.165003i	0.996591	−	0.0825015i	$-0.0262909\pi$
$984$	6.56166i	0.209178i
$985$	13.5345	0.431246
$986$	0	0
$987$	2.86831	0.0912994
$988$	− 24.0073i	− 0.763774i
$989$	− 7.81940i	− 0.248642i
$990$	−2.19806	−0.0698590
$991$	16.8528	0.535346	0.267673	−	0.963510i	$-0.413745\pi$
$991$	16.8528	0.535346	0.267673	+	0.963510i	$0.413745\pi$
$992$	−31.1366	−0.988588
$993$	−3.90813	−0.124021
$994$	− 1.16506i	− 0.0369536i
$995$	0.605531	0.0191966
$996$	− 10.0097i	− 0.317169i
$997$	38.2097i	1.21011i	0.796183	+	0.605056i	$0.206849\pi$
$997$	38.2097i	1.21011i	−0.796183	+	0.605056i	$0.793151\pi$
$998$	− 9.15239i	− 0.289714i
$999$	27.3763	0.866147

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	841.2.b.c.840.3		6
29.2	odd	28		841.2.d.e.605.1		6
29.3	odd	28		29.2.d.a.20.1	yes	6
29.4	even	14		841.2.e.d.651.2		12
29.5	even	14		841.2.e.b.236.1		12
29.6	even	14		841.2.e.b.196.2		12
29.7	even	7		841.2.e.d.270.2		12
29.8	odd	28		841.2.d.c.574.1		6
29.9	even	14		841.2.e.c.267.2		12
29.10	odd	28		841.2.d.d.190.1		6
29.11	odd	28		841.2.d.b.778.1		6
29.12	odd	4		841.2.a.f.1.2		3
29.13	even	14		841.2.e.c.63.1		12
29.14	odd	28		841.2.d.e.645.1		6
29.15	odd	28		841.2.d.a.645.1		6
29.16	even	7		841.2.e.c.63.2		12
29.17	odd	4		841.2.a.e.1.2		3
29.18	odd	28		841.2.d.c.778.1		6
29.19	odd	28		29.2.d.a.16.1	✓	6
29.20	even	7		841.2.e.c.267.1		12
29.21	odd	28		841.2.d.b.574.1		6
29.22	even	14		841.2.e.d.270.1		12
29.23	even	7		841.2.e.b.196.1		12
29.24	even	7		841.2.e.b.236.2		12
29.25	even	7		841.2.e.d.651.1		12
29.26	odd	28		841.2.d.d.571.1		6
29.27	odd	28		841.2.d.a.605.1		6
29.28	even	2	inner	841.2.b.c.840.4		6
87.17	even	4		7569.2.a.r.1.2		3
87.32	even	28		261.2.k.a.136.1		6
87.41	even	4		7569.2.a.p.1.2		3
87.77	even	28		261.2.k.a.190.1		6
116.3	even	28		464.2.u.f.49.1		6
116.19	even	28		464.2.u.f.161.1		6
145.3	even	28		725.2.r.b.49.1		12
145.19	odd	28		725.2.l.b.451.1		6
145.32	even	28		725.2.r.b.49.2		12
145.48	even	28		725.2.r.b.74.2		12
145.77	even	28		725.2.r.b.74.1		12
145.119	odd	28		725.2.l.b.426.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.2.d.a.16.1	✓	6	29.19	odd	28
29.2.d.a.20.1	yes	6	29.3	odd	28
261.2.k.a.136.1		6	87.32	even	28
261.2.k.a.190.1		6	87.77	even	28
464.2.u.f.49.1		6	116.3	even	28
464.2.u.f.161.1		6	116.19	even	28
725.2.l.b.426.1		6	145.119	odd	28
725.2.l.b.451.1		6	145.19	odd	28
725.2.r.b.49.1		12	145.3	even	28
725.2.r.b.49.2		12	145.32	even	28
725.2.r.b.74.1		12	145.77	even	28
725.2.r.b.74.2		12	145.48	even	28
841.2.a.e.1.2		3	29.17	odd	4
841.2.a.f.1.2		3	29.12	odd	4
841.2.b.c.840.3		6	1.1	even	1	trivial
841.2.b.c.840.4		6	29.28	even	2	inner
841.2.d.a.605.1		6	29.27	odd	28
841.2.d.a.645.1		6	29.15	odd	28
841.2.d.b.574.1		6	29.21	odd	28
841.2.d.b.778.1		6	29.11	odd	28
841.2.d.c.574.1		6	29.8	odd	28
841.2.d.c.778.1		6	29.18	odd	28
841.2.d.d.190.1		6	29.10	odd	28
841.2.d.d.571.1		6	29.26	odd	28
841.2.d.e.605.1		6	29.2	odd	28
841.2.d.e.645.1		6	29.14	odd	28
841.2.e.b.196.1		12	29.23	even	7
841.2.e.b.196.2		12	29.6	even	14
841.2.e.b.236.1		12	29.5	even	14
841.2.e.b.236.2		12	29.24	even	7
841.2.e.c.63.1		12	29.13	even	14
841.2.e.c.63.2		12	29.16	even	7
841.2.e.c.267.1		12	29.20	even	7
841.2.e.c.267.2		12	29.9	even	14
841.2.e.d.270.1		12	29.22	even	14
841.2.e.d.270.2		12	29.7	even	7
841.2.e.d.651.1		12	29.25	even	7
841.2.e.d.651.2		12	29.4	even	14
7569.2.a.p.1.2		3	87.41	even	4
7569.2.a.r.1.2		3	87.17	even	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 841 = 29^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	841.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.445042i q^{2} -1.24698i q^{3} +1.80194 q^{4} -0.692021 q^{5} -0.554958 q^{6} +0.356896 q^{7} -1.69202i q^{8} +1.44504 q^{9} +0.307979i q^{10} +4.93900i q^{11} -2.24698i q^{12} +5.65279 q^{13} -0.158834i q^{14} +0.862937i q^{15} +2.85086 q^{16} +4.49396i q^{17} -0.643104i q^{18} -2.35690i q^{19} -1.24698 q^{20} -0.445042i q^{21} +2.19806 q^{22} +2.29590 q^{23} -2.10992 q^{24} -4.52111 q^{25} -2.51573i q^{26} -5.54288i q^{27} +0.643104 q^{28} +0.384043 q^{30} -6.69202i q^{31} -4.65279i q^{32} +6.15883 q^{33} +2.00000 q^{34} -0.246980 q^{35} +2.60388 q^{36} +4.93900i q^{37} -1.04892 q^{38} -7.04892i q^{39} +1.17092i q^{40} -3.10992i q^{41} -0.198062 q^{42} -3.40581i q^{43} +8.89977i q^{44} -1.00000 q^{45} -1.02177i q^{46} +6.44504i q^{47} -3.55496i q^{48} -6.87263 q^{49} +2.01208i q^{50} +5.60388 q^{51} +10.1860 q^{52} -4.69202 q^{53} -2.46681 q^{54} -3.41789i q^{55} -0.603875i q^{56} -2.93900 q^{57} -12.4940 q^{59} +1.55496i q^{60} -1.64310i q^{61} -2.97823 q^{62} +0.515729 q^{63} +3.63102 q^{64} -3.91185 q^{65} -2.74094i q^{66} +2.32304 q^{67} +8.09783i q^{68} -2.86294i q^{69} +0.109916i q^{70} +7.33513 q^{71} -2.44504i q^{72} -5.62565i q^{73} +2.19806 q^{74} +5.63773i q^{75} -4.24698i q^{76} +1.76271i q^{77} -3.13706 q^{78} -4.66487i q^{79} -1.97285 q^{80} -2.57673 q^{81} -1.38404 q^{82} +4.45473 q^{83} -0.801938i q^{84} -3.10992i q^{85} -1.51573 q^{86} +8.35690 q^{88} -5.67994i q^{89} +0.445042i q^{90} +2.01746 q^{91} +4.13706 q^{92} -8.34481 q^{93} +2.86831 q^{94} +1.63102i q^{95} -5.80194 q^{96} +0.180604i q^{97} +3.05861i q^{98} +7.13706i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{4} + 6 q^{5} - 4 q^{6} - 6 q^{7} + 8 q^{9} - 2 q^{13} - 10 q^{16} + 2 q^{20} + 22 q^{22} - 14 q^{23} - 14 q^{24} + 4 q^{25} + 12 q^{28} - 18 q^{30} + 20 q^{33} + 12 q^{34} + 8 q^{35} - 2 q^{36}+ \cdots - 26 q^{96}+O(q^{100}) \) 6 * q + 2 * q^4 + 6 * q^5 - 4 * q^6 - 6 * q^7 + 8 * q^9 - 2 * q^13 - 10 * q^16 + 2 * q^20 + 22 * q^22 - 14 * q^23 - 14 * q^24 + 4 * q^25 + 12 * q^28 - 18 * q^30 + 20 * q^33 + 12 * q^34 + 8 * q^35 - 2 * q^36 + 12 * q^38 - 10 * q^42 - 6 * q^45 - 8 * q^49 + 16 * q^51 + 32 * q^52 - 18 * q^53 - 8 * q^54 + 2 * q^57 - 56 * q^59 - 24 * q^62 - 22 * q^63 - 8 * q^64 - 16 * q^65 - 26 * q^67 + 42 * q^71 + 22 * q^74 - 8 * q^78 - 24 * q^80 - 10 * q^81 + 12 * q^82 - 18 * q^83 + 16 * q^86 + 42 * q^88 + 44 * q^91 + 14 * q^92 - 4 * q^93 + 22 * q^94 - 26 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more