LMFDB - Embedded newform 169.2.b.b.168.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [169,2,Mod(168,169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(169, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("169.168");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 169 = 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	169.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:	$1.34947179416$
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			168.2
Root			$-1.24698i$ of defining polynomial
Character	$\chi$	$=$	169.168
Dual form			169.2.b.b.168.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.801938i q^{2} -2.24698 q^{3} +1.35690 q^{4} -0.246980i q^{5} +1.80194i q^{6} -2.35690i q^{7} -2.69202i q^{8} +2.04892 q^{9} +O(q^{10})$	\(q-0.801938i q^{2} -2.24698 q^{3} +1.35690 q^{4} -0.246980i q^{5} +1.80194i q^{6} -2.35690i q^{7} -2.69202i q^{8} +2.04892 q^{9} -0.198062 q^{10} -4.24698i q^{11} -3.04892 q^{12} -1.89008 q^{14} +0.554958i q^{15} +0.554958 q^{16} -2.15883 q^{17} -1.64310i q^{18} +0.0881460i q^{19} -0.335126i q^{20} +5.29590i q^{21} -3.40581 q^{22} -1.49396 q^{23} +6.04892i q^{24} +4.93900 q^{25} +2.13706 q^{27} -3.19806i q^{28} +4.63102 q^{29} +0.445042 q^{30} +6.63102i q^{31} -5.82908i q^{32} +9.54288i q^{33} +1.73125i q^{34} -0.582105 q^{35} +2.78017 q^{36} +5.69202i q^{37} +0.0706876 q^{38} -0.664874 q^{40} +11.5918i q^{41} +4.24698 q^{42} +0.295897 q^{43} -5.76271i q^{44} -0.506041i q^{45} +1.19806i q^{46} -7.35690i q^{47} -1.24698 q^{48} +1.44504 q^{49} -3.96077i q^{50} +4.85086 q^{51} -10.3937 q^{53} -1.71379i q^{54} -1.04892 q^{55} -6.34481 q^{56} -0.198062i q^{57} -3.71379i q^{58} -6.78017i q^{59} +0.753020i q^{60} +3.47219 q^{61} +5.31767 q^{62} -4.82908i q^{63} -3.56465 q^{64} +7.65279 q^{66} -7.67994i q^{67} -2.92931 q^{68} +3.35690 q^{69} +0.466812i q^{70} +8.66487i q^{71} -5.51573i q^{72} +6.73556i q^{73} +4.56465 q^{74} -11.0978 q^{75} +0.119605i q^{76} -10.0097 q^{77} +9.97046 q^{79} -0.137063i q^{80} -10.9487 q^{81} +9.29590 q^{82} -1.60925i q^{83} +7.18598i q^{84} +0.533188i q^{85} -0.237291i q^{86} -10.4058 q^{87} -11.4330 q^{88} -2.88471i q^{89} -0.405813 q^{90} -2.02715 q^{92} -14.8998i q^{93} -5.89977 q^{94} +0.0217703 q^{95} +13.0978i q^{96} +8.05861i q^{97} -1.15883i q^{98} -8.70171i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{3} - 6 q^{9}+O(q^{10}) $ 6 * q - 4 * q^3 - 6 * q^9	$ 6 q - 4 q^{3} - 6 q^{9} - 10 q^{10} - 10 q^{14} + 4 q^{16} + 4 q^{17} + 6 q^{22} + 10 q^{23} + 10 q^{25} + 2 q^{27} - 2 q^{29} + 2 q^{30} + 8 q^{35} + 14 q^{36} - 24 q^{38} - 6 q^{40} + 16 q^{42} - 26 q^{43} + 2 q^{48} + 8 q^{49} + 2 q^{51} + 2 q^{53} + 12 q^{55} + 8 q^{56} + 8 q^{61} - 2 q^{62} + 22 q^{64} + 10 q^{66} - 42 q^{68} + 12 q^{69} - 16 q^{74} - 30 q^{75} - 16 q^{77} - 10 q^{79} - 2 q^{81} + 28 q^{82} - 36 q^{87} - 30 q^{88} + 24 q^{90} + 10 q^{94} - 6 q^{95}+O(q^{100}) $ 6 * q - 4 * q^3 - 6 * q^9 - 10 * q^10 - 10 * q^14 + 4 * q^16 + 4 * q^17 + 6 * q^22 + 10 * q^23 + 10 * q^25 + 2 * q^27 - 2 * q^29 + 2 * q^30 + 8 * q^35 + 14 * q^36 - 24 * q^38 - 6 * q^40 + 16 * q^42 - 26 * q^43 + 2 * q^48 + 8 * q^49 + 2 * q^51 + 2 * q^53 + 12 * q^55 + 8 * q^56 + 8 * q^61 - 2 * q^62 + 22 * q^64 + 10 * q^66 - 42 * q^68 + 12 * q^69 - 16 * q^74 - 30 * q^75 - 16 * q^77 - 10 * q^79 - 2 * q^81 + 28 * q^82 - 36 * q^87 - 30 * q^88 + 24 * q^90 + 10 * q^94 - 6 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/169\mathbb{Z}\right)^\times$.

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

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 $

$2$	− 0.801938i	− 0.567056i	−0.958964	−	0.283528i	$-0.908495\pi$
$2$	− 0.801938i	− 0.567056i	0.958964	−	0.283528i	$-0.0915048\pi$
$3$	−2.24698	−1.29729	−0.648647	−	0.761089i	$-0.724665\pi$
$3$	−2.24698	−1.29729	−0.648647	+	0.761089i	$0.724665\pi$
$4$	1.35690	0.678448
$5$	− 0.246980i	− 0.110453i	−0.998474	−	0.0552263i	$-0.982412\pi$
$5$	− 0.246980i	− 0.110453i	0.998474	−	0.0552263i	$-0.0175880\pi$
$6$	1.80194i	0.735638i
$7$	− 2.35690i	− 0.890823i	−0.895326	−	0.445411i	$-0.853057\pi$
$7$	− 2.35690i	− 0.890823i	0.895326	−	0.445411i	$-0.146943\pi$
$8$	− 2.69202i	− 0.951773i
$9$	2.04892	0.682972
$10$	−0.198062	−0.0626328
$11$	− 4.24698i	− 1.28051i	−0.768161	−	0.640256i	$-0.778828\pi$
$11$	− 4.24698i	− 1.28051i	0.768161	−	0.640256i	$-0.221172\pi$
$12$	−3.04892	−0.880147
$13$	0	0
$13$	0	0
$14$	−1.89008	−0.505146
$15$	0.554958i	0.143290i
$16$	0.554958	0.138740
$17$	−2.15883	−0.523594	−0.261797	−	0.965123i	$-0.584315\pi$
$17$	−2.15883	−0.523594	−0.261797	+	0.965123i	$0.584315\pi$
$18$	− 1.64310i	− 0.387283i
$19$	0.0881460i	0.0202221i	0.999949	+	0.0101110i	$0.00321850\pi$
$19$	0.0881460i	0.0202221i	−0.999949	+	0.0101110i	$0.996782\pi$
$20$	− 0.335126i	− 0.0749364i
$21$	5.29590i	1.15566i
$22$	−3.40581	−0.726122
$23$	−1.49396	−0.311512	−0.155756	−	0.987796i	$-0.549781\pi$
$23$	−1.49396	−0.311512	−0.155756	+	0.987796i	$0.549781\pi$
$24$	6.04892i	1.23473i
$25$	4.93900	0.987800
$26$	0	0
$27$	2.13706	0.411278
$28$	− 3.19806i	− 0.604377i
$29$	4.63102	0.859959	0.429980	−	0.902839i	$-0.358521\pi$
$29$	4.63102	0.859959	0.429980	+	0.902839i	$0.358521\pi$
$30$	0.445042	0.0812532
$31$	6.63102i	1.19097i	0.803368	+	0.595483i	$0.203039\pi$
$31$	6.63102i	1.19097i	−0.803368	+	0.595483i	$0.796961\pi$
$32$	− 5.82908i	− 1.03045i
$33$	9.54288i	1.66120i
$34$	1.73125i	0.296907i
$35$	−0.582105	−0.0983937
$36$	2.78017	0.463361
$37$	5.69202i	0.935763i	0.883791	+	0.467881i	$0.154983\pi$
$37$	5.69202i	0.935763i	−0.883791	+	0.467881i	$0.845017\pi$
$38$	0.0706876	0.0114670
$39$	0	0
$40$	−0.664874	−0.105126
$41$	11.5918i	1.81033i	0.425056	+	0.905167i	$0.360254\pi$
$41$	11.5918i	1.81033i	−0.425056	+	0.905167i	$0.639746\pi$
$42$	4.24698	0.655323
$43$	0.295897	0.0451239	0.0225619	−	0.999745i	$-0.492818\pi$
$43$	0.295897	0.0451239	0.0225619	+	0.999745i	$0.492818\pi$
$44$	− 5.76271i	− 0.868761i
$45$	− 0.506041i	− 0.0754361i
$46$	1.19806i	0.176645i
$47$	− 7.35690i	− 1.07311i	−0.843864	−	0.536557i	$-0.819725\pi$
$47$	− 7.35690i	− 1.07311i	0.843864	−	0.536557i	$-0.180275\pi$
$48$	−1.24698	−0.179986
$49$	1.44504	0.206435
$50$	− 3.96077i	− 0.560138i
$51$	4.85086	0.679256
$52$	0	0
$53$	−10.3937	−1.42769	−0.713844	−	0.700304i	$-0.753048\pi$
$53$	−10.3937	−1.42769	−0.713844	+	0.700304i	$0.753048\pi$
$54$	− 1.71379i	− 0.233218i
$55$	−1.04892	−0.141436
$56$	−6.34481	−0.847861
$57$	− 0.198062i	− 0.0262340i
$58$	− 3.71379i	− 0.487645i
$59$	− 6.78017i	− 0.882703i	−0.897334	−	0.441351i	$-0.854499\pi$
$59$	− 6.78017i	− 0.882703i	0.897334	−	0.441351i	$-0.145501\pi$
$60$	0.753020i	0.0972145i
$61$	3.47219	0.444568	0.222284	−	0.974982i	$-0.428649\pi$
$61$	3.47219	0.444568	0.222284	+	0.974982i	$0.428649\pi$
$62$	5.31767	0.675344
$63$	− 4.82908i	− 0.608407i
$64$	−3.56465	−0.445581
$65$	0	0
$66$	7.65279	0.941994
$67$	− 7.67994i	− 0.938254i	−0.883131	−	0.469127i	$-0.844569\pi$
$67$	− 7.67994i	− 0.938254i	0.883131	−	0.469127i	$-0.155431\pi$
$68$	−2.92931	−0.355231
$69$	3.35690	0.404123
$70$	0.466812i	0.0557947i
$71$	8.66487i	1.02833i	0.857691	+	0.514166i	$0.171898\pi$
$71$	8.66487i	1.02833i	−0.857691	+	0.514166i	$0.828102\pi$
$72$	− 5.51573i	− 0.650035i
$73$	6.73556i	0.788338i	0.919038	+	0.394169i	$0.128968\pi$
$73$	6.73556i	0.788338i	−0.919038	+	0.394169i	$0.871032\pi$
$74$	4.56465	0.530629
$75$	−11.0978	−1.28147
$76$	0.119605i	0.0137196i
$77$	−10.0097	−1.14071
$78$	0	0
$79$	9.97046	1.12176	0.560882	−	0.827896i	$-0.310462\pi$
$79$	9.97046	1.12176	0.560882	+	0.827896i	$0.310462\pi$
$80$	− 0.137063i	− 0.0153241i
$81$	−10.9487	−1.21652
$82$	9.29590	1.02656
$83$	− 1.60925i	− 0.176638i	−0.996092	−	0.0883192i	$-0.971850\pi$
$83$	− 1.60925i	− 0.176638i	0.996092	−	0.0883192i	$-0.0281495\pi$
$84$	7.18598i	0.784055i
$85$	0.533188i	0.0578323i
$86$	− 0.237291i	− 0.0255877i
$87$	−10.4058	−1.11562
$88$	−11.4330	−1.21876
$89$	− 2.88471i	− 0.305778i	−0.988243	−	0.152889i	$-0.951142\pi$
$89$	− 2.88471i	− 0.305778i	0.988243	−	0.152889i	$-0.0488577\pi$
$90$	−0.405813	−0.0427765
$91$	0	0
$92$	−2.02715	−0.211345
$93$	− 14.8998i	− 1.54503i
$94$	−5.89977	−0.608515
$95$	0.0217703	0.00223358
$96$	13.0978i	1.33679i
$97$	8.05861i	0.818227i	0.912483	+	0.409114i	$0.134162\pi$
$97$	8.05861i	0.818227i	−0.912483	+	0.409114i	$0.865838\pi$
$98$	− 1.15883i	− 0.117060i
$99$	− 8.70171i	− 0.874555i
$100$	6.70171	0.670171
$101$	13.3545	1.32882	0.664411	−	0.747367i	$-0.268682\pi$
$101$	13.3545	1.32882	0.664411	+	0.747367i	$0.268682\pi$
$102$	− 3.89008i	− 0.385176i
$103$	−1.36227	−0.134229	−0.0671144	−	0.997745i	$-0.521379\pi$
$103$	−1.36227	−0.134229	−0.0671144	+	0.997745i	$0.521379\pi$
$104$	0	0
$105$	1.30798	0.127646
$106$	8.33513i	0.809579i
$107$	3.26875	0.316002	0.158001	−	0.987439i	$-0.449495\pi$
$107$	3.26875	0.316002	0.158001	+	0.987439i	$0.449495\pi$
$108$	2.89977	0.279031
$109$	− 15.7017i	− 1.50395i	−0.659191	−	0.751976i	$-0.729101\pi$
$109$	− 15.7017i	− 1.50395i	0.659191	−	0.751976i	$-0.270899\pi$
$110$	0.841166i	0.0802021i
$111$	− 12.7899i	− 1.21396i
$112$	− 1.30798i	− 0.123592i
$113$	12.0489	1.13347	0.566733	−	0.823901i	$-0.308207\pi$
$113$	12.0489	1.13347	0.566733	+	0.823901i	$0.308207\pi$
$114$	−0.158834	−0.0148761
$115$	0.368977i	0.0344073i
$116$	6.28382	0.583438
$117$	0	0
$118$	−5.43727	−0.500541
$119$	5.08815i	0.466430i
$120$	1.49396	0.136379
$121$	−7.03684	−0.639712
$122$	− 2.78448i	− 0.252095i
$123$	− 26.0465i	− 2.34854i
$124$	8.99761i	0.808009i
$125$	− 2.45473i	− 0.219558i
$126$	−3.87263	−0.345001
$127$	9.80731	0.870258	0.435129	−	0.900368i	$-0.356703\pi$
$127$	9.80731	0.870258	0.435129	+	0.900368i	$0.356703\pi$
$128$	− 8.79954i	− 0.777777i
$129$	−0.664874	−0.0585389
$130$	0	0
$131$	−6.57673	−0.574611	−0.287306	−	0.957839i	$-0.592760\pi$
$131$	−6.57673	−0.574611	−0.287306	+	0.957839i	$0.592760\pi$
$132$	12.9487i	1.12704i
$133$	0.207751	0.0180143
$134$	−6.15883	−0.532042
$135$	− 0.527811i	− 0.0454267i
$136$	5.81163i	0.498343i
$137$	6.21983i	0.531396i	0.964056	+	0.265698i	$0.0856024\pi$
$137$	6.21983i	0.531396i	−0.964056	+	0.265698i	$0.914398\pi$
$138$	− 2.69202i	− 0.229160i
$139$	−14.7071	−1.24744	−0.623719	−	0.781648i	$-0.714379\pi$
$139$	−14.7071	−1.24744	−0.623719	+	0.781648i	$0.714379\pi$
$140$	−0.789856	−0.0667550
$141$	16.5308i	1.39214i
$142$	6.94869	0.583121
$143$	0	0
$144$	1.13706	0.0947553
$145$	− 1.14377i	− 0.0949848i
$146$	5.40150	0.447031
$147$	−3.24698	−0.267806
$148$	7.72348i	0.634866i
$149$	− 4.33513i	− 0.355147i	−0.984108	−	0.177574i	$-0.943175\pi$
$149$	− 4.33513i	− 0.355147i	0.984108	−	0.177574i	$-0.0568248\pi$
$150$	8.89977i	0.726663i
$151$	3.94438i	0.320989i	0.987037	+	0.160494i	$0.0513089\pi$
$151$	3.94438i	0.320989i	−0.987037	+	0.160494i	$0.948691\pi$
$152$	0.237291	0.0192468
$153$	−4.42327	−0.357600
$154$	8.02715i	0.646846i
$155$	1.63773	0.131545
$156$	0	0
$157$	4.45473	0.355526	0.177763	−	0.984073i	$-0.443114\pi$
$157$	4.45473	0.355526	0.177763	+	0.984073i	$0.443114\pi$
$158$	− 7.99569i	− 0.636103i
$159$	23.3545	1.85213
$160$	−1.43967	−0.113816
$161$	3.52111i	0.277502i
$162$	8.78017i	0.689835i
$163$	16.1588i	1.26566i	0.774292	+	0.632829i	$0.218106\pi$
$163$	16.1588i	1.26566i	−0.774292	+	0.632829i	$0.781894\pi$
$164$	15.7289i	1.22822i
$165$	2.35690	0.183484
$166$	−1.29052	−0.100164
$167$	16.1172i	1.24719i	0.781749	+	0.623594i	$0.214328\pi$
$167$	16.1172i	1.24719i	−0.781749	+	0.623594i	$0.785672\pi$
$168$	14.2567	1.09993
$169$	0	0
$170$	0.427583	0.0327942
$171$	0.180604i	0.0138111i
$172$	0.401501	0.0306142
$173$	21.5362	1.63736	0.818682	−	0.574247i	$-0.194705\pi$
$173$	21.5362	1.63736	0.818682	+	0.574247i	$0.194705\pi$
$174$	8.34481i	0.632619i
$175$	− 11.6407i	− 0.879955i
$176$	− 2.35690i	− 0.177658i
$177$	15.2349i	1.14513i
$178$	−2.31336	−0.173393
$179$	−11.4330	−0.854540	−0.427270	−	0.904124i	$-0.640525\pi$
$179$	−11.4330	−0.854540	−0.427270	+	0.904124i	$0.640525\pi$
$180$	− 0.686645i	− 0.0511795i
$181$	−20.9705	−1.55872	−0.779361	−	0.626575i	$-0.784456\pi$
$181$	−20.9705	−1.55872	−0.779361	+	0.626575i	$0.784456\pi$
$182$	0	0
$183$	−7.80194	−0.576736
$184$	4.02177i	0.296489i
$185$	1.40581	0.103357
$186$	−11.9487	−0.876120
$187$	9.16852i	0.670469i
$188$	− 9.98254i	− 0.728052i
$189$	− 5.03684i	− 0.366376i
$190$	− 0.0174584i	− 0.00126657i
$191$	−14.4373	−1.04464	−0.522322	−	0.852748i	$-0.674934\pi$
$191$	−14.4373	−1.04464	−0.522322	+	0.852748i	$0.674934\pi$
$192$	8.00969	0.578049
$193$	− 13.5797i	− 0.977489i	−0.872427	−	0.488745i	$-0.837455\pi$
$193$	− 13.5797i	− 0.977489i	0.872427	−	0.488745i	$-0.162545\pi$
$194$	6.46250	0.463980
$195$	0	0
$196$	1.96077	0.140055
$197$	0.560335i	0.0399222i	0.999801	+	0.0199611i	$0.00635424\pi$
$197$	0.560335i	0.0399222i	−0.999801	+	0.0199611i	$0.993646\pi$
$198$	−6.97823	−0.495921
$199$	−11.4916	−0.814616	−0.407308	−	0.913291i	$-0.633532\pi$
$199$	−11.4916	−0.814616	−0.407308	+	0.913291i	$0.633532\pi$
$200$	− 13.2959i	− 0.940162i
$201$	17.2567i	1.21719i
$202$	− 10.7095i	− 0.753516i
$203$	− 10.9148i	− 0.766071i
$204$	6.58211	0.460840
$205$	2.86294	0.199956
$206$	1.09246i	0.0761151i
$207$	−3.06100	−0.212754
$208$	0	0
$209$	0.374354	0.0258946
$210$	− 1.04892i	− 0.0723822i
$211$	8.78448	0.604748	0.302374	−	0.953189i	$-0.402221\pi$
$211$	8.78448	0.604748	0.302374	+	0.953189i	$0.402221\pi$
$212$	−14.1032	−0.968613
$213$	− 19.4698i	− 1.33405i
$214$	− 2.62133i	− 0.179191i
$215$	− 0.0730805i	− 0.00498405i
$216$	− 5.75302i	− 0.391443i
$217$	15.6286	1.06094
$218$	−12.5918	−0.852824
$219$	− 15.1347i	− 1.02271i
$220$	−1.42327	−0.0959570
$221$	0	0
$222$	−10.2567	−0.688383
$223$	2.25906i	0.151278i	0.997135	+	0.0756390i	$0.0240996\pi$
$223$	2.25906i	0.151278i	−0.997135	+	0.0756390i	$0.975900\pi$
$224$	−13.7385	−0.917945
$225$	10.1196	0.674640
$226$	− 9.66248i	− 0.642739i
$227$	6.96615i	0.462359i	0.972911	+	0.231180i	$0.0742585\pi$
$227$	6.96615i	0.462359i	−0.972911	+	0.231180i	$0.925741\pi$
$228$	− 0.268750i	− 0.0177984i
$229$	− 24.1739i	− 1.59746i	−0.601692	−	0.798728i	$-0.705507\pi$
$229$	− 24.1739i	− 1.59746i	0.601692	−	0.798728i	$-0.294493\pi$
$230$	0.295897	0.0195109
$231$	22.4916	1.47984
$232$	− 12.4668i	− 0.818486i
$233$	3.06100	0.200533	0.100266	−	0.994961i	$-0.468031\pi$
$233$	3.06100	0.200533	0.100266	+	0.994961i	$0.468031\pi$
$234$	0	0
$235$	−1.81700	−0.118528
$236$	− 9.19998i	− 0.598868i
$237$	−22.4034	−1.45526
$238$	4.08038	0.264492
$239$	25.1468i	1.62661i	0.581839	+	0.813304i	$0.302333\pi$
$239$	25.1468i	1.62661i	−0.581839	+	0.813304i	$0.697667\pi$
$240$	0.307979i	0.0198799i
$241$	− 20.2664i	− 1.30547i	−0.757586	−	0.652735i	$-0.773621\pi$
$241$	− 20.2664i	− 1.30547i	0.757586	−	0.652735i	$-0.226379\pi$
$242$	5.64310i	0.362752i
$243$	18.1903	1.16691
$244$	4.71140	0.301616
$245$	− 0.356896i	− 0.0228012i
$246$	−20.8877	−1.33175
$247$	0	0
$248$	17.8509	1.13353
$249$	3.61596i	0.229152i
$250$	−1.96854	−0.124501
$251$	23.7211	1.49726	0.748631	−	0.662987i	$-0.230712\pi$
$251$	23.7211	1.49726	0.748631	+	0.662987i	$0.230712\pi$
$252$	− 6.55257i	− 0.412773i
$253$	6.34481i	0.398895i
$254$	− 7.86486i	− 0.493485i
$255$	− 1.19806i	− 0.0750256i
$256$	−14.1860	−0.886624
$257$	−14.2241	−0.887278	−0.443639	−	0.896206i	$-0.646313\pi$
$257$	−14.2241	−0.887278	−0.443639	+	0.896206i	$0.646313\pi$
$258$	0.533188i	0.0331948i
$259$	13.4155	0.833599
$260$	0	0
$261$	9.48858	0.587329
$262$	5.27413i	0.325837i
$263$	−17.0954	−1.05415	−0.527075	−	0.849819i	$-0.676711\pi$
$263$	−17.0954	−1.05415	−0.527075	+	0.849819i	$0.676711\pi$
$264$	25.6896	1.58109
$265$	2.56704i	0.157692i
$266$	− 0.166603i	− 0.0102151i
$267$	6.48188i	0.396684i
$268$	− 10.4209i	− 0.636556i
$269$	−6.46681	−0.394288	−0.197144	−	0.980374i	$-0.563167\pi$
$269$	−6.46681	−0.394288	−0.197144	+	0.980374i	$0.563167\pi$
$270$	−0.423272	−0.0257595
$271$	6.44803i	0.391690i	0.980635	+	0.195845i	$0.0627449\pi$
$271$	6.44803i	0.391690i	−0.980635	+	0.195845i	$0.937255\pi$
$272$	−1.19806	−0.0726432
$273$	0	0
$274$	4.98792	0.301331
$275$	− 20.9758i	− 1.26489i
$276$	4.55496	0.274176
$277$	−13.4601	−0.808739	−0.404370	−	0.914596i	$-0.632509\pi$
$277$	−13.4601	−0.808739	−0.404370	+	0.914596i	$0.632509\pi$
$278$	11.7942i	0.707367i
$279$	13.5864i	0.813398i
$280$	1.56704i	0.0936485i
$281$	5.03684i	0.300472i	0.988650	+	0.150236i	$0.0480034\pi$
$281$	5.03684i	0.300472i	−0.988650	+	0.150236i	$0.951997\pi$
$282$	13.2567	0.789423
$283$	−22.1280	−1.31537	−0.657686	−	0.753293i	$-0.728464\pi$
$283$	−22.1280	−1.31537	−0.657686	+	0.753293i	$0.728464\pi$
$284$	11.7573i	0.697669i
$285$	−0.0489173	−0.00289761
$286$	0	0
$287$	27.3207	1.61269
$288$	− 11.9433i	− 0.703766i
$289$	−12.3394	−0.725849
$290$	−0.917231	−0.0538616
$291$	− 18.1075i	− 1.06148i
$292$	9.13946i	0.534846i
$293$	14.9463i	0.873172i	0.899663	+	0.436586i	$0.143813\pi$
$293$	14.9463i	0.873172i	−0.899663	+	0.436586i	$0.856187\pi$
$294$	2.60388i	0.151861i
$295$	−1.67456	−0.0974968
$296$	15.3230	0.890634
$297$	− 9.07606i	− 0.526647i
$298$	−3.47650	−0.201388
$299$	0	0
$300$	−15.0586	−0.869409
$301$	− 0.697398i	− 0.0401974i
$302$	3.16315	0.182019
$303$	−30.0073	−1.72387
$304$	0.0489173i	0.00280560i
$305$	− 0.857560i	− 0.0491037i
$306$	3.54719i	0.202779i
$307$	19.1293i	1.09177i	0.837861	+	0.545883i	$0.183806\pi$
$307$	19.1293i	1.09177i	−0.837861	+	0.545883i	$0.816194\pi$
$308$	−13.5821	−0.773912
$309$	3.06100	0.174134
$310$	− 1.31336i	− 0.0745936i
$311$	0.269815	0.0152998	0.00764990	−	0.999971i	$-0.497565\pi$
$311$	0.269815	0.0152998	0.00764990	+	0.999971i	$0.497565\pi$
$312$	0	0
$313$	−23.3937	−1.32229	−0.661146	−	0.750257i	$-0.729930\pi$
$313$	−23.3937	−1.32229	−0.661146	+	0.750257i	$0.729930\pi$
$314$	− 3.57242i	− 0.201603i
$315$	−1.19269	−0.0672002
$316$	13.5289	0.761059
$317$	13.9952i	0.786050i	0.919528	+	0.393025i	$0.128571\pi$
$317$	13.9952i	0.786050i	−0.919528	+	0.393025i	$0.871429\pi$
$318$	− 18.7289i	− 1.05026i
$319$	− 19.6679i	− 1.10119i
$320$	0.880395i	0.0492156i
$321$	−7.34481	−0.409948
$322$	2.82371	0.157359
$323$	− 0.190293i	− 0.0105882i
$324$	−14.8562	−0.825346
$325$	0	0
$326$	12.9584	0.717698
$327$	35.2814i	1.95107i
$328$	31.2054	1.72303
$329$	−17.3394	−0.955954
$330$	− 1.89008i	− 0.104046i
$331$	− 17.8213i	− 0.979548i	−0.871849	−	0.489774i	$-0.837079\pi$
$331$	− 17.8213i	− 0.979548i	0.871849	−	0.489774i	$-0.162921\pi$
$332$	− 2.18359i	− 0.119840i
$333$	11.6625i	0.639100i
$334$	12.9250	0.707225
$335$	−1.89679	−0.103633
$336$	2.93900i	0.160336i
$337$	27.8485	1.51700	0.758501	−	0.651672i	$-0.225932\pi$
$337$	27.8485	1.51700	0.758501	+	0.651672i	$0.225932\pi$
$338$	0	0
$339$	−27.0737	−1.47044
$340$	0.723480i	0.0392362i
$341$	28.1618	1.52505
$342$	0.144833	0.00783167
$343$	− 19.9041i	− 1.07472i
$344$	− 0.796561i	− 0.0429477i
$345$	− 0.829085i	− 0.0446364i
$346$	− 17.2707i	− 0.928477i
$347$	1.50365	0.0807200	0.0403600	−	0.999185i	$-0.487150\pi$
$347$	1.50365	0.0807200	0.0403600	+	0.999185i	$0.487150\pi$
$348$	−14.1196	−0.756890
$349$	− 14.1860i	− 0.759358i	−0.925118	−	0.379679i	$-0.876034\pi$
$349$	− 14.1860i	− 0.759358i	0.925118	−	0.379679i	$-0.123966\pi$
$350$	−9.33513	−0.498983
$351$	0	0
$352$	−24.7560	−1.31950
$353$	7.16852i	0.381542i	0.981635	+	0.190771i	$0.0610988\pi$
$353$	7.16852i	0.381542i	−0.981635	+	0.190771i	$0.938901\pi$
$354$	12.2174	0.649350
$355$	2.14005	0.113582
$356$	− 3.91425i	− 0.207455i
$357$	− 11.4330i	− 0.605096i
$358$	9.16852i	0.484571i
$359$	19.8853i	1.04951i	0.851255	+	0.524753i	$0.175842\pi$
$359$	19.8853i	1.04951i	−0.851255	+	0.524753i	$0.824158\pi$
$360$	−1.36227	−0.0717981
$361$	18.9922	0.999591
$362$	16.8170i	0.883882i
$363$	15.8116	0.829895
$364$	0	0
$365$	1.66355	0.0870740
$366$	6.25667i	0.327041i
$367$	1.08383	0.0565757	0.0282878	−	0.999600i	$-0.490994\pi$
$367$	1.08383	0.0565757	0.0282878	+	0.999600i	$0.490994\pi$
$368$	−0.829085	−0.0432190
$369$	23.7506i	1.23641i
$370$	− 1.12737i	− 0.0586094i
$371$	24.4969i	1.27182i
$372$	− 20.2174i	− 1.04823i
$373$	−6.13036	−0.317418	−0.158709	−	0.987325i	$-0.550733\pi$
$373$	−6.13036	−0.317418	−0.158709	+	0.987325i	$0.550733\pi$
$374$	7.35258	0.380193
$375$	5.51573i	0.284831i
$376$	−19.8049	−1.02136
$377$	0	0
$378$	−4.03923	−0.207756
$379$	2.40880i	0.123732i	0.998084	+	0.0618658i	$0.0197051\pi$
$379$	2.40880i	0.123732i	−0.998084	+	0.0618658i	$0.980295\pi$
$380$	0.0295400	0.00151537
$381$	−22.0368	−1.12898
$382$	11.5778i	0.592371i
$383$	30.3913i	1.55292i	0.630164	+	0.776462i	$0.282988\pi$
$383$	30.3913i	1.55292i	−0.630164	+	0.776462i	$0.717012\pi$
$384$	19.7724i	1.00901i
$385$	2.47219i	0.125994i
$386$	−10.8901	−0.554291
$387$	0.606268	0.0308184
$388$	10.9347i	0.555125i
$389$	15.9409	0.808237	0.404118	−	0.914707i	$-0.367578\pi$
$389$	15.9409	0.808237	0.404118	+	0.914707i	$0.367578\pi$
$390$	0	0
$391$	3.22521	0.163106
$392$	− 3.89008i	− 0.196479i
$393$	14.7778	0.745440
$394$	0.449354	0.0226381
$395$	− 2.46250i	− 0.123902i
$396$	− 11.8073i	− 0.593340i
$397$	16.9148i	0.848931i	0.905444	+	0.424466i	$0.139538\pi$
$397$	16.9148i	0.848931i	−0.905444	+	0.424466i	$0.860462\pi$
$398$	9.21552i	0.461932i
$399$	−0.466812	−0.0233698
$400$	2.74094	0.137047
$401$	26.6625i	1.33146i	0.746192	+	0.665730i	$0.231880\pi$
$401$	26.6625i	1.33146i	−0.746192	+	0.665730i	$0.768120\pi$
$402$	13.8388	0.690215
$403$	0	0
$404$	18.1207	0.901537
$405$	2.70410i	0.134368i
$406$	−8.75302	−0.434405
$407$	24.1739	1.19826
$408$	− 13.0586i	− 0.646497i
$409$	− 28.5163i	− 1.41004i	−0.709187	−	0.705021i	$-0.750938\pi$
$409$	− 28.5163i	− 1.41004i	0.709187	−	0.705021i	$-0.249062\pi$
$410$	− 2.29590i	− 0.113386i
$411$	− 13.9758i	− 0.689377i
$412$	−1.84846	−0.0910672
$413$	−15.9801	−0.786332
$414$	2.45473i	0.120643i
$415$	−0.397452	−0.0195102
$416$	0	0
$417$	33.0465	1.61830
$418$	− 0.300209i	− 0.0146837i
$419$	−29.6093	−1.44651	−0.723253	−	0.690583i	$-0.757354\pi$
$419$	−29.6093	−1.44651	−0.723253	+	0.690583i	$0.757354\pi$
$420$	1.77479	0.0866009
$421$	11.6606i	0.568301i	0.958780	+	0.284151i	$0.0917115\pi$
$421$	11.6606i	0.568301i	−0.958780	+	0.284151i	$0.908288\pi$
$422$	− 7.04461i	− 0.342926i
$423$	− 15.0737i	− 0.732907i
$424$	27.9801i	1.35884i
$425$	−10.6625	−0.517206
$426$	−15.6136	−0.756480
$427$	− 8.18359i	− 0.396032i
$428$	4.43535	0.214391
$429$	0	0
$430$	−0.0586060	−0.00282623
$431$	− 4.34913i	− 0.209490i	−0.994499	−	0.104745i	$-0.966597\pi$
$431$	− 4.34913i	− 0.209490i	0.994499	−	0.104745i	$-0.0334026\pi$
$432$	1.18598	0.0570605
$433$	14.3884	0.691460	0.345730	−	0.938334i	$-0.387631\pi$
$433$	14.3884	0.691460	0.345730	+	0.938334i	$0.387631\pi$
$434$	− 12.5332i	− 0.601612i
$435$	2.57002i	0.123223i
$436$	− 21.3056i	− 1.02035i
$437$	− 0.131687i	− 0.00629942i
$438$	−12.1371	−0.579931
$439$	20.2325	0.965645	0.482822	−	0.875718i	$-0.339612\pi$
$439$	20.2325	0.965645	0.482822	+	0.875718i	$0.339612\pi$
$440$	2.82371i	0.134615i
$441$	2.96077	0.140989
$442$	0	0
$443$	8.12200	0.385888	0.192944	−	0.981210i	$-0.438196\pi$
$443$	8.12200	0.385888	0.192944	+	0.981210i	$0.438196\pi$
$444$	− 17.3545i	− 0.823608i
$445$	−0.712464	−0.0337740
$446$	1.81163	0.0857830
$447$	9.74094i	0.460731i
$448$	8.40150i	0.396934i
$449$	12.4916i	0.589513i	0.955572	+	0.294757i	$0.0952386\pi$
$449$	12.4916i	0.589513i	−0.955572	+	0.294757i	$0.904761\pi$
$450$	− 8.11529i	− 0.382559i
$451$	49.2301	2.31816
$452$	16.3491	0.768998
$453$	− 8.86294i	− 0.416417i
$454$	5.58642	0.262184
$455$	0	0
$456$	−0.533188	−0.0249688
$457$	− 5.98121i	− 0.279789i	−0.990166	−	0.139895i	$-0.955324\pi$
$457$	− 5.98121i	− 0.279789i	0.990166	−	0.139895i	$-0.0446764\pi$
$458$	−19.3860	−0.905847
$459$	−4.61356	−0.215343
$460$	0.500664i	0.0233436i
$461$	2.05669i	0.0957895i	0.998852	+	0.0478947i	$0.0152512\pi$
$461$	2.05669i	0.0957895i	−0.998852	+	0.0478947i	$0.984749\pi$
$462$	− 18.0368i	− 0.839150i
$463$	− 8.44935i	− 0.392675i	−0.980536	−	0.196337i	$-0.937095\pi$
$463$	− 8.44935i	− 0.392675i	0.980536	−	0.196337i	$-0.0629048\pi$
$464$	2.57002	0.119310
$465$	−3.67994	−0.170653
$466$	− 2.45473i	− 0.113713i
$467$	−33.5139	−1.55084	−0.775420	−	0.631446i	$-0.782462\pi$
$467$	−33.5139	−1.55084	−0.775420	+	0.631446i	$0.782462\pi$
$468$	0	0
$469$	−18.1008	−0.835818
$470$	1.45712i	0.0672121i
$471$	−10.0097	−0.461222
$472$	−18.2524	−0.840133
$473$	− 1.25667i	− 0.0577817i
$474$	17.9661i	0.825213i
$475$	0.435353i	0.0199754i
$476$	6.90408i	0.316448i
$477$	−21.2959	−0.975072
$478$	20.1661	0.922377
$479$	− 24.7313i	− 1.13000i	−0.825091	−	0.565000i	$-0.808876\pi$
$479$	− 24.7313i	− 1.13000i	0.825091	−	0.565000i	$-0.191124\pi$
$480$	3.23490	0.147652
$481$	0	0
$482$	−16.2524	−0.740275
$483$	− 7.91185i	− 0.360002i
$484$	−9.54825	−0.434012
$485$	1.99031	0.0903754
$486$	− 14.5875i	− 0.661702i
$487$	37.7555i	1.71087i	0.517913	+	0.855433i	$0.326709\pi$
$487$	37.7555i	1.71087i	−0.517913	+	0.855433i	$0.673291\pi$
$488$	− 9.34721i	− 0.423128i
$489$	− 36.3086i	− 1.64193i
$490$	−0.286208	−0.0129296
$491$	−31.3110	−1.41304	−0.706522	−	0.707691i	$-0.749737\pi$
$491$	−31.3110	−1.41304	−0.706522	+	0.707691i	$0.749737\pi$
$492$	− 35.3424i	− 1.59336i
$493$	−9.99761	−0.450270
$494$	0	0
$495$	−2.14914	−0.0965969
$496$	3.67994i	0.165234i
$497$	20.4222	0.916061
$498$	2.89977	0.129942
$499$	− 21.4873i	− 0.961902i	−0.876748	−	0.480951i	$-0.840292\pi$
$499$	− 21.4873i	− 0.961902i	0.876748	−	0.480951i	$-0.159708\pi$
$500$	− 3.33081i	− 0.148959i
$501$	− 36.2150i	− 1.61797i
$502$	− 19.0228i	− 0.849031i
$503$	37.5924	1.67616	0.838081	−	0.545546i	$-0.183678\pi$
$503$	37.5924	1.67616	0.838081	+	0.545546i	$0.183678\pi$
$504$	−13.0000	−0.579066
$505$	− 3.29829i	− 0.146772i
$506$	5.08815	0.226196
$507$	0	0
$508$	13.3075	0.590425
$509$	17.1075i	0.758278i	0.925340	+	0.379139i	$0.123780\pi$
$509$	17.1075i	0.758278i	−0.925340	+	0.379139i	$0.876220\pi$
$510$	−0.960771	−0.0425437
$511$	15.8750	0.702269
$512$	− 6.22282i	− 0.275012i
$513$	0.188374i	0.00831690i
$514$	11.4069i	0.503136i
$515$	0.336454i	0.0148259i
$516$	−0.902165	−0.0397156
$517$	−31.2446	−1.37414
$518$	− 10.7584i	− 0.472697i
$519$	−48.3913	−2.12414
$520$	0	0
$521$	−19.8465	−0.869493	−0.434746	−	0.900553i	$-0.643162\pi$
$521$	−19.8465	−0.869493	−0.434746	+	0.900553i	$0.643162\pi$
$522$	− 7.60925i	− 0.333048i
$523$	−11.4300	−0.499798	−0.249899	−	0.968272i	$-0.580397\pi$
$523$	−11.4300	−0.499798	−0.249899	+	0.968272i	$0.580397\pi$
$524$	−8.92394	−0.389844
$525$	26.1564i	1.14156i
$526$	13.7095i	0.597762i
$527$	− 14.3153i	− 0.623583i
$528$	5.29590i	0.230474i
$529$	−20.7681	−0.902960
$530$	2.05861	0.0894201
$531$	− 13.8920i	− 0.602862i
$532$	0.281896	0.0122218
$533$	0	0
$534$	5.19806	0.224942
$535$	− 0.807315i	− 0.0349033i
$536$	−20.6746	−0.893005
$537$	25.6896	1.10859
$538$	5.18598i	0.223584i
$539$	− 6.13706i	− 0.264342i
$540$	− 0.716185i	− 0.0308197i
$541$	16.1884i	0.695993i	0.937496	+	0.347996i	$0.113138\pi$
$541$	16.1884i	0.695993i	−0.937496	+	0.347996i	$0.886862\pi$
$542$	5.17092	0.222110
$543$	47.1202	2.02212
$544$	12.5840i	0.539536i
$545$	−3.87800	−0.166115
$546$	0	0
$547$	5.33081	0.227929	0.113965	−	0.993485i	$-0.463645\pi$
$547$	5.33081	0.227929	0.113965	+	0.993485i	$0.463645\pi$
$548$	8.43967i	0.360525i
$549$	7.11423	0.303628
$550$	−16.8213	−0.717263
$551$	0.408206i	0.0173902i
$552$	− 9.03684i	− 0.384633i
$553$	− 23.4993i	− 0.999293i
$554$	10.7942i	0.458600i
$555$	−3.15883	−0.134085
$556$	−19.9560	−0.846322
$557$	7.39075i	0.313156i	0.987666	+	0.156578i	$0.0500463\pi$
$557$	7.39075i	0.313156i	−0.987666	+	0.156578i	$0.949954\pi$
$558$	10.8955	0.461242
$559$	0	0
$560$	−0.323044	−0.0136511
$561$	− 20.6015i	− 0.869795i
$562$	4.03923	0.170385
$563$	9.47889	0.399488	0.199744	−	0.979848i	$-0.435989\pi$
$563$	9.47889	0.399488	0.199744	+	0.979848i	$0.435989\pi$
$564$	22.4306i	0.944497i
$565$	− 2.97584i	− 0.125194i
$566$	17.7453i	0.745889i
$567$	25.8049i	1.08370i
$568$	23.3260	0.978738
$569$	10.1438	0.425249	0.212624	−	0.977134i	$-0.431799\pi$
$569$	10.1438	0.425249	0.212624	+	0.977134i	$0.431799\pi$
$570$	0.0392287i	0.00164311i
$571$	14.0925	0.589751	0.294876	−	0.955536i	$-0.404722\pi$
$571$	14.0925	0.589751	0.294876	+	0.955536i	$0.404722\pi$
$572$	0	0
$573$	32.4403	1.35521
$574$	− 21.9095i	− 0.914483i
$575$	−7.37867	−0.307712
$576$	−7.30367	−0.304319
$577$	− 25.1545i	− 1.04720i	−0.851965	−	0.523598i	$-0.824589\pi$
$577$	− 25.1545i	− 1.04720i	0.851965	−	0.523598i	$-0.175411\pi$
$578$	9.89546i	0.411597i
$579$	30.5133i	1.26809i
$580$	− 1.55197i	− 0.0644422i
$581$	−3.79284	−0.157354
$582$	−14.5211	−0.601919
$583$	44.1420i	1.82817i
$584$	18.1323	0.750319
$585$	0	0
$586$	11.9860	0.495137
$587$	− 43.8353i	− 1.80928i	−0.426180	−	0.904639i	$-0.640141\pi$
$587$	− 43.8353i	− 1.80928i	0.426180	−	0.904639i	$-0.359859\pi$
$588$	−4.40581	−0.181693
$589$	−0.584498	−0.0240838
$590$	1.34290i	0.0552861i
$591$	− 1.25906i	− 0.0517909i
$592$	3.15883i	0.129827i
$593$	− 24.9965i	− 1.02648i	−0.858244	−	0.513242i	$-0.828444\pi$
$593$	− 24.9965i	− 1.02648i	0.858244	−	0.513242i	$-0.171556\pi$
$594$	−7.27844	−0.298638
$595$	1.25667	0.0515184
$596$	− 5.88231i	− 0.240949i
$597$	25.8213	1.05680
$598$	0	0
$599$	−6.24027	−0.254971	−0.127485	−	0.991840i	$-0.540691\pi$
$599$	−6.24027	−0.254971	−0.127485	+	0.991840i	$0.540691\pi$
$600$	29.8756i	1.21967i
$601$	6.32975	0.258196	0.129098	−	0.991632i	$-0.458792\pi$
$601$	6.32975	0.258196	0.129098	+	0.991632i	$0.458792\pi$
$602$	−0.559270	−0.0227941
$603$	− 15.7356i	− 0.640802i
$604$	5.35211i	0.217774i
$605$	1.73795i	0.0706579i
$606$	24.0640i	0.977532i
$607$	−43.6480	−1.77162	−0.885809	−	0.464050i	$-0.846396\pi$
$607$	−43.6480	−1.77162	−0.885809	+	0.464050i	$0.846396\pi$
$608$	0.513811	0.0208378
$609$	24.5254i	0.993820i
$610$	−0.687710	−0.0278445
$611$	0	0
$612$	−6.00192	−0.242613
$613$	25.9541i	1.04827i	0.851634	+	0.524137i	$0.175612\pi$
$613$	25.9541i	1.04827i	−0.851634	+	0.524137i	$0.824388\pi$
$614$	15.3405	0.619092
$615$	−6.43296	−0.259402
$616$	26.9463i	1.08570i
$617$	− 45.9396i	− 1.84946i	−0.380626	−	0.924729i	$-0.624291\pi$
$617$	− 45.9396i	− 1.84946i	0.380626	−	0.924729i	$-0.375709\pi$
$618$	− 2.45473i	− 0.0987437i
$619$	6.73556i	0.270725i	0.990796	+	0.135363i	$0.0432199\pi$
$619$	6.73556i	0.270725i	−0.990796	+	0.135363i	$0.956780\pi$
$620$	2.22223	0.0892467
$621$	−3.19269	−0.128118
$622$	− 0.216375i	− 0.00867583i
$623$	−6.79895	−0.272394
$624$	0	0
$625$	24.0887	0.963549
$626$	18.7603i	0.749813i
$627$	−0.841166	−0.0335930
$628$	6.04461	0.241206
$629$	− 12.2881i	− 0.489960i
$630$	0.956459i	0.0381063i
$631$	45.0998i	1.79539i	0.440614	+	0.897696i	$0.354761\pi$
$631$	45.0998i	1.79539i	−0.440614	+	0.897696i	$0.645239\pi$
$632$	− 26.8407i	− 1.06767i
$633$	−19.7385	−0.784537
$634$	11.2233	0.445734
$635$	− 2.42221i	− 0.0961223i
$636$	31.6896	1.25658
$637$	0	0
$638$	−15.7724	−0.624435
$639$	17.7536i	0.702322i
$640$	−2.17331	−0.0859075
$641$	−32.5821	−1.28692	−0.643458	−	0.765482i	$-0.722501\pi$
$641$	−32.5821	−1.28692	−0.643458	+	0.765482i	$0.722501\pi$
$642$	5.89008i	0.232463i
$643$	− 25.5754i	− 1.00860i	−0.863530	−	0.504298i	$-0.831751\pi$
$643$	− 25.5754i	− 1.00860i	0.863530	−	0.504298i	$-0.168249\pi$
$644$	4.77777i	0.188271i
$645$	0.164210i	0.00646578i
$646$	−0.152603	−0.00600408
$647$	30.1715	1.18616	0.593082	−	0.805142i	$-0.297911\pi$
$647$	30.1715	1.18616	0.593082	+	0.805142i	$0.297911\pi$
$648$	29.4741i	1.15785i
$649$	−28.7952	−1.13031
$650$	0	0
$651$	−35.1172	−1.37635
$652$	21.9259i	0.858683i
$653$	36.9028	1.44412	0.722058	−	0.691832i	$-0.243196\pi$
$653$	36.9028	1.44412	0.722058	+	0.691832i	$0.243196\pi$
$654$	28.2935	1.10636
$655$	1.62432i	0.0634673i
$656$	6.43296i	0.251165i
$657$	13.8006i	0.538413i
$658$	13.9051i	0.542079i
$659$	23.6866	0.922701	0.461350	−	0.887218i	$-0.347365\pi$
$659$	23.6866	0.922701	0.461350	+	0.887218i	$0.347365\pi$
$660$	3.19806	0.124484
$661$	− 31.7590i	− 1.23528i	−0.786460	−	0.617641i	$-0.788089\pi$
$661$	− 31.7590i	− 1.23528i	0.786460	−	0.617641i	$-0.211911\pi$
$662$	−14.2916	−0.555458
$663$	0	0
$664$	−4.33214	−0.168120
$665$	− 0.0513102i	− 0.00198973i
$666$	9.35258	0.362405
$667$	−6.91856	−0.267888
$668$	21.8694i	0.846152i
$669$	− 5.07606i	− 0.196252i
$670$	1.52111i	0.0587655i
$671$	− 14.7463i	− 0.569275i
$672$	30.8702	1.19085
$673$	7.50232	0.289193	0.144597	−	0.989491i	$-0.453812\pi$
$673$	7.50232	0.289193	0.144597	+	0.989491i	$0.453812\pi$
$674$	− 22.3327i	− 0.860225i
$675$	10.5550	0.406261
$676$	0	0
$677$	−35.0315	−1.34637	−0.673184	−	0.739475i	$-0.735074\pi$
$677$	−35.0315	−1.34637	−0.673184	+	0.739475i	$0.735074\pi$
$678$	21.7114i	0.833821i
$679$	18.9933	0.728896
$680$	1.43535	0.0550433
$681$	− 15.6528i	− 0.599816i
$682$	− 22.5840i	− 0.864787i
$683$	− 24.0834i	− 0.921524i	−0.887524	−	0.460762i	$-0.847576\pi$
$683$	− 24.0834i	− 0.921524i	0.887524	−	0.460762i	$-0.152424\pi$
$684$	0.245061i	0.00937013i
$685$	1.53617	0.0586941
$686$	−15.9618	−0.609426
$687$	54.3183i	2.07237i
$688$	0.164210	0.00626046
$689$	0	0
$690$	−0.664874	−0.0253113
$691$	− 2.01447i	− 0.0766342i	−0.999266	−	0.0383171i	$-0.987800\pi$
$691$	− 2.01447i	− 0.0766342i	0.999266	−	0.0383171i	$-0.0121997\pi$
$692$	29.2223	1.11087
$693$	−20.5090	−0.779073
$694$	− 1.20583i	− 0.0457728i
$695$	3.63235i	0.137783i
$696$	28.0127i	1.06182i
$697$	− 25.0248i	− 0.947880i
$698$	−11.3763	−0.430598
$699$	−6.87800	−0.260150
$700$	− 15.7952i	− 0.597004i
$701$	48.8189	1.84387	0.921933	−	0.387350i	$-0.126610\pi$
$701$	48.8189	1.84387	0.921933	+	0.387350i	$0.126610\pi$
$702$	0	0
$703$	−0.501729	−0.0189231
$704$	15.1390i	0.570572i
$705$	4.08277	0.153766
$706$	5.74871	0.216355
$707$	− 31.4752i	− 1.18375i
$708$	20.6722i	0.776908i
$709$	− 20.8060i	− 0.781385i	−0.920521	−	0.390693i	$-0.872236\pi$
$709$	− 20.8060i	− 0.781385i	0.920521	−	0.390693i	$-0.127764\pi$
$710$	− 1.71618i	− 0.0644073i
$711$	20.4286	0.766134
$712$	−7.76569	−0.291032
$713$	− 9.90648i	− 0.371000i
$714$	−9.16852	−0.343123
$715$	0	0
$716$	−15.5133	−0.579761
$717$	− 56.5042i	− 2.11019i
$718$	15.9468	0.595128
$719$	−21.4306	−0.799225	−0.399613	−	0.916684i	$-0.630855\pi$
$719$	−21.4306	−0.799225	−0.399613	+	0.916684i	$0.630855\pi$
$720$	− 0.280831i	− 0.0104660i
$721$	3.21073i	0.119574i
$722$	− 15.2306i	− 0.566824i
$723$	45.5381i	1.69358i
$724$	−28.4547	−1.05751
$725$	22.8726	0.849468
$726$	− 12.6799i	− 0.470597i
$727$	−13.4862	−0.500175	−0.250088	−	0.968223i	$-0.580459\pi$
$727$	−13.4862	−0.500175	−0.250088	+	0.968223i	$0.580459\pi$
$728$	0	0
$729$	−8.02715	−0.297302
$730$	− 1.33406i	− 0.0493758i
$731$	−0.638792	−0.0236266
$732$	−10.5864	−0.391285
$733$	43.5424i	1.60828i	0.594443	+	0.804138i	$0.297373\pi$
$733$	43.5424i	1.60828i	−0.594443	+	0.804138i	$0.702627\pi$
$734$	− 0.869167i	− 0.0320816i
$735$	0.801938i	0.0295799i
$736$	8.70841i	0.320996i
$737$	−32.6165	−1.20145
$738$	19.0465	0.701112
$739$	20.0543i	0.737709i	0.929487	+	0.368855i	$0.120250\pi$
$739$	20.0543i	0.737709i	−0.929487	+	0.368855i	$0.879750\pi$
$740$	1.90754	0.0701226
$741$	0	0
$742$	19.6450	0.721191
$743$	33.1685i	1.21684i	0.793617	+	0.608418i	$0.208195\pi$
$743$	33.1685i	1.21684i	−0.793617	+	0.608418i	$0.791805\pi$
$744$	−40.1105	−1.47052
$745$	−1.07069	−0.0392270
$746$	4.91617i	0.179994i
$747$	− 3.29722i	− 0.120639i
$748$	12.4407i	0.454878i
$749$	− 7.70410i	− 0.281502i
$750$	4.42327	0.161515
$751$	−39.2814	−1.43340	−0.716700	−	0.697382i	$-0.754348\pi$
$751$	−39.2814	−1.43340	−0.716700	+	0.697382i	$0.754348\pi$
$752$	− 4.08277i	− 0.148883i
$753$	−53.3008	−1.94239
$754$	0	0
$755$	0.974181	0.0354541
$756$	− 6.83446i	− 0.248567i
$757$	−46.6426	−1.69526	−0.847628	−	0.530592i	$-0.821970\pi$
$757$	−46.6426	−1.69526	−0.847628	+	0.530592i	$0.821970\pi$
$758$	1.93171	0.0701627
$759$	− 14.2567i	− 0.517484i
$760$	− 0.0586060i	− 0.00212586i
$761$	− 21.8984i	− 0.793818i	−0.917858	−	0.396909i	$-0.870083\pi$
$761$	− 21.8984i	− 0.793818i	0.917858	−	0.396909i	$-0.129917\pi$
$762$	17.6722i	0.640195i
$763$	−37.0073	−1.33975
$764$	−19.5899	−0.708737
$765$	1.09246i	0.0394979i
$766$	24.3720	0.880595
$767$	0	0
$768$	31.8756	1.15021
$769$	− 46.7096i	− 1.68439i	−0.539172	−	0.842196i	$-0.681263\pi$
$769$	− 46.7096i	− 1.68439i	0.539172	−	0.842196i	$-0.318737\pi$
$770$	1.98254	0.0714458
$771$	31.9614	1.15106
$772$	− 18.4263i	− 0.663175i
$773$	30.2416i	1.08771i	0.839178	+	0.543857i	$0.183037\pi$
$773$	30.2416i	1.08771i	−0.839178	+	0.543857i	$0.816963\pi$
$774$	− 0.486189i	− 0.0174757i
$775$	32.7506i	1.17644i
$776$	21.6939	0.778767
$777$	−30.1444	−1.08142
$778$	− 12.7836i	− 0.458315i
$779$	−1.02177	−0.0366087
$780$	0	0
$781$	36.7995	1.31679
$782$	− 2.58642i	− 0.0924901i
$783$	9.89679	0.353682
$784$	0.801938	0.0286406
$785$	− 1.10023i	− 0.0392688i
$786$	− 11.8509i	− 0.422706i
$787$	28.7023i	1.02313i	0.859246	+	0.511563i	$0.170933\pi$
$787$	28.7023i	1.02313i	−0.859246	+	0.511563i	$0.829067\pi$
$788$	0.760316i	0.0270851i
$789$	38.4131	1.36754
$790$	−1.97477	−0.0702592
$791$	− 28.3980i	− 1.00972i
$792$	−23.4252	−0.832378
$793$	0	0
$794$	13.5646	0.481391
$795$	− 5.76809i	− 0.204573i
$796$	−15.5929	−0.552674
$797$	18.5418	0.656785	0.328392	−	0.944541i	$-0.393493\pi$
$797$	18.5418	0.656785	0.328392	+	0.944541i	$0.393493\pi$
$798$	0.374354i	0.0132520i
$799$	15.8823i	0.561876i
$800$	− 28.7899i	− 1.01788i
$801$	− 5.91053i	− 0.208838i
$802$	21.3817	0.755012
$803$	28.6058	1.00948
$804$	23.4155i	0.825801i
$805$	0.869641	0.0306508
$806$	0	0
$807$	14.5308	0.511508
$808$	− 35.9506i	− 1.26474i
$809$	−10.0677	−0.353962	−0.176981	−	0.984214i	$-0.556633\pi$
$809$	−10.0677	−0.353962	−0.176981	+	0.984214i	$0.556633\pi$
$810$	2.16852	0.0761941
$811$	10.0285i	0.352147i	0.984377	+	0.176074i	$0.0563397\pi$
$811$	10.0285i	0.352147i	−0.984377	+	0.176074i	$0.943660\pi$
$812$	− 14.8103i	− 0.519740i
$813$	− 14.4886i	− 0.508137i
$814$	− 19.3860i	− 0.679478i
$815$	3.99090	0.139795
$816$	2.69202	0.0942396
$817$	0.0260821i	0 0.000912498i
$818$	−22.8683	−0.799572
$819$	0	0
$820$	3.88471	0.135660
$821$	26.1704i	0.913355i	0.889632	+	0.456677i	$0.150961\pi$
$821$	26.1704i	0.913355i	−0.889632	+	0.456677i	$0.849039\pi$
$822$	−11.2078	−0.390915
$823$	−1.82238	−0.0635242	−0.0317621	−	0.999495i	$-0.510112\pi$
$823$	−1.82238	−0.0635242	−0.0317621	+	0.999495i	$0.510112\pi$
$824$	3.66727i	0.127755i
$825$	47.1323i	1.64094i
$826$	12.8151i	0.445894i
$827$	− 32.2941i	− 1.12298i	−0.827485	−	0.561488i	$-0.810229\pi$
$827$	− 32.2941i	− 1.12298i	0.827485	−	0.561488i	$-0.189771\pi$
$828$	−4.15346	−0.144343
$829$	−15.1002	−0.524453	−0.262226	−	0.965006i	$-0.584457\pi$
$829$	−15.1002	−0.524453	−0.262226	+	0.965006i	$0.584457\pi$
$830$	0.318732i	0.0110634i
$831$	30.2446	1.04917
$832$	0	0
$833$	−3.11960	−0.108088
$834$	− 26.5013i	− 0.917663i
$835$	3.98062	0.137755
$836$	0.507960	0.0175682
$837$	14.1709i	0.489818i
$838$	23.7448i	0.820250i
$839$	− 32.9965i	− 1.13917i	−0.821933	−	0.569584i	$-0.807104\pi$
$839$	− 32.9965i	− 1.13917i	0.821933	−	0.569584i	$-0.192896\pi$
$840$	− 3.52111i	− 0.121490i
$841$	−7.55363	−0.260470
$842$	9.35105	0.322258
$843$	− 11.3177i	− 0.389801i
$844$	11.9196	0.410290
$845$	0	0
$846$	−12.0881	−0.415599
$847$	16.5851i	0.569870i
$848$	−5.76809	−0.198077
$849$	49.7211	1.70642
$850$	8.55065i	0.293285i
$851$	− 8.50365i	− 0.291501i
$852$	− 26.4185i	− 0.905082i
$853$	− 37.7802i	− 1.29357i	−0.762673	−	0.646784i	$-0.776113\pi$
$853$	− 37.7802i	− 1.29357i	0.762673	−	0.646784i	$-0.223887\pi$
$854$	−6.56273	−0.224572
$855$	0.0446055	0.00152547
$856$	− 8.79954i	− 0.300762i
$857$	−27.3623	−0.934677	−0.467339	−	0.884078i	$-0.654787\pi$
$857$	−27.3623	−0.934677	−0.467339	+	0.884078i	$0.654787\pi$
$858$	0	0
$859$	−20.0629	−0.684538	−0.342269	−	0.939602i	$-0.611195\pi$
$859$	−20.0629	−0.684538	−0.342269	+	0.939602i	$0.611195\pi$
$860$	− 0.0991626i	− 0.00338142i
$861$	−61.3889	−2.09213
$862$	−3.48773	−0.118792
$863$	6.14483i	0.209173i	0.994516	+	0.104586i	$0.0333518\pi$
$863$	6.14483i	0.209173i	−0.994516	+	0.104586i	$0.966648\pi$
$864$	− 12.4571i	− 0.423800i
$865$	− 5.31900i	− 0.180851i
$866$	− 11.5386i	− 0.392096i
$867$	27.7265	0.941640
$868$	21.2064	0.719793
$869$	− 42.3443i	− 1.43643i
$870$	2.06100	0.0698744
$871$	0	0
$872$	−42.2693	−1.43142
$873$	16.5114i	0.558827i
$874$	−0.105604	−0.00357212
$875$	−5.78554	−0.195587
$876$	− 20.5362i	− 0.693853i
$877$	13.5077i	0.456123i	0.973647	+	0.228061i	$0.0732386\pi$
$877$	13.5077i	0.456123i	−0.973647	+	0.228061i	$0.926761\pi$
$878$	− 16.2252i	− 0.547574i
$879$	− 33.5840i	− 1.13276i
$880$	−0.582105	−0.0196228
$881$	5.23431	0.176348	0.0881741	−	0.996105i	$-0.471897\pi$
$881$	5.23431	0.176348	0.0881741	+	0.996105i	$0.471897\pi$
$882$	− 2.37435i	− 0.0799487i
$883$	4.57301	0.153894	0.0769470	−	0.997035i	$-0.475483\pi$
$883$	4.57301	0.153894	0.0769470	+	0.997035i	$0.475483\pi$
$884$	0	0
$885$	3.76271	0.126482
$886$	− 6.51334i	− 0.218820i
$887$	−1.64071	−0.0550897	−0.0275448	−	0.999621i	$-0.508769\pi$
$887$	−1.64071	−0.0550897	−0.0275448	+	0.999621i	$0.508769\pi$
$888$	−34.4306	−1.15541
$889$	− 23.1148i	− 0.775246i
$890$	0.571352i	0.0191517i
$891$	46.4989i	1.55777i
$892$	3.06531i	0.102634i
$893$	0.648481	0.0217006
$894$	7.81163	0.261260
$895$	2.82371i	0.0943861i
$896$	−20.7396	−0.692862
$897$	0	0
$898$	10.0175	0.334287
$899$	30.7084i	1.02418i
$900$	13.7313	0.457708
$901$	22.4383	0.747529
$902$	− 39.4795i	− 1.31452i
$903$	1.56704i	0.0521478i
$904$	− 32.4359i	− 1.07880i
$905$	5.17928i	0.172165i
$906$	−7.10752	−0.236132
$907$	−8.10215	−0.269027	−0.134514	−	0.990912i	$-0.542947\pi$
$907$	−8.10215	−0.269027	−0.134514	+	0.990912i	$0.542947\pi$
$908$	9.45234i	0.313687i
$909$	27.3623	0.907549
$910$	0	0
$911$	−9.18119	−0.304187	−0.152093	−	0.988366i	$-0.548601\pi$
$911$	−9.18119	−0.304187	−0.152093	+	0.988366i	$0.548601\pi$
$912$	− 0.109916i	− 0.00363969i
$913$	−6.83446	−0.226188
$914$	−4.79656	−0.158656
$915$	1.92692i	0.0637020i
$916$	− 32.8015i	− 1.08379i
$917$	15.5007i	0.511877i
$918$	3.69979i	0.122111i
$919$	27.5036	0.907262	0.453631	−	0.891190i	$-0.350128\pi$
$919$	27.5036	0.907262	0.453631	+	0.891190i	$0.350128\pi$
$920$	0.993295	0.0327480
$921$	− 42.9831i	− 1.41634i
$922$	1.64933	0.0543180
$923$	0	0
$924$	30.5187	1.00399
$925$	28.1129i	0.924346i
$926$	−6.77586	−0.222668
$927$	−2.79118	−0.0916745
$928$	− 26.9946i	− 0.886142i
$929$	− 24.2131i	− 0.794407i	−0.917731	−	0.397203i	$-0.869981\pi$
$929$	− 24.2131i	− 0.794407i	0.917731	−	0.397203i	$-0.130019\pi$
$930$	2.95108i	0.0967698i
$931$	0.127375i	0.00417454i
$932$	4.15346	0.136051
$933$	−0.606268	−0.0198483
$934$	26.8761i	0.879412i
$935$	2.26444	0.0740550
$936$	0	0
$937$	11.1830	0.365333	0.182666	−	0.983175i	$-0.441527\pi$
$937$	11.1830	0.365333	0.182666	+	0.983175i	$0.441527\pi$
$938$	14.5157i	0.473955i
$939$	52.5652	1.71540
$940$	−2.46548	−0.0804152
$941$	15.9638i	0.520404i	0.965554	+	0.260202i	$0.0837891\pi$
$941$	15.9638i	0.520404i	−0.965554	+	0.260202i	$0.916211\pi$
$942$	8.02715i	0.261539i
$943$	− 17.3177i	− 0.563941i
$944$	− 3.76271i	− 0.122466i
$945$	−1.24400	−0.0404672
$946$	−1.00777	−0.0327654
$947$	6.51466i	0.211698i	0.994382	+	0.105849i	$0.0337560\pi$
$947$	6.51466i	0.211698i	−0.994382	+	0.105849i	$0.966244\pi$
$948$	−30.3991	−0.987317
$949$	0	0
$950$	0.349126	0.0113271
$951$	− 31.4470i	− 1.01974i
$952$	13.6974	0.443935
$953$	−47.6469	−1.54344	−0.771718	−	0.635965i	$-0.780602\pi$
$953$	−47.6469	−1.54344	−0.771718	+	0.635965i	$0.780602\pi$
$954$	17.0780i	0.552920i
$955$	3.56571i	0.115384i
$956$	34.1215i	1.10357i
$957$	44.1933i	1.42857i
$958$	−19.8329	−0.640773
$959$	14.6595	0.473380
$960$	− 1.97823i	− 0.0638471i
$961$	−12.9705	−0.418402
$962$	0	0
$963$	6.69740	0.215821
$964$	− 27.4993i	− 0.885694i
$965$	−3.35391	−0.107966
$966$	−6.34481	−0.204141
$967$	43.8122i	1.40891i	0.709751	+	0.704453i	$0.248808\pi$
$967$	43.8122i	1.40891i	−0.709751	+	0.704453i	$0.751192\pi$
$968$	18.9433i	0.608861i
$969$	0.427583i	0.0137360i
$970$	− 1.59611i	− 0.0512479i
$971$	4.29483	0.137828	0.0689139	−	0.997623i	$-0.478047\pi$
$971$	4.29483	0.137828	0.0689139	+	0.997623i	$0.478047\pi$
$972$	24.6823	0.791686
$973$	34.6631i	1.11125i
$974$	30.2776	0.970156
$975$	0	0
$976$	1.92692	0.0616792
$977$	− 26.8019i	− 0.857470i	−0.903430	−	0.428735i	$-0.858959\pi$
$977$	− 26.8019i	− 0.857470i	0.903430	−	0.428735i	$-0.141041\pi$
$978$	−29.1172	−0.931066
$979$	−12.2513	−0.391553
$980$	− 0.484271i	− 0.0154695i
$981$	− 32.1715i	− 1.02716i
$982$	25.1094i	0.801275i
$983$	27.2495i	0.869124i	0.900642	+	0.434562i	$0.143097\pi$
$983$	27.2495i	0.869124i	−0.900642	+	0.434562i	$0.856903\pi$
$984$	−70.1178	−2.23527
$985$	0.138391	0.00440951
$986$	8.01746i	0.255328i
$987$	38.9614	1.24015
$988$	0	0
$989$	−0.442058	−0.0140566
$990$	1.72348i	0.0547758i
$991$	24.3889	0.774740	0.387370	−	0.921924i	$-0.373384\pi$
$991$	24.3889	0.774740	0.387370	+	0.921924i	$0.373384\pi$
$992$	38.6528	1.22723
$993$	40.0441i	1.27076i
$994$	− 16.3773i	− 0.519458i
$995$	2.83818i	0.0899764i
$996$	4.90648i	0.155468i
$997$	31.3207	0.991935	0.495967	−	0.868341i	$-0.334814\pi$
$997$	31.3207	0.991935	0.495967	+	0.868341i	$0.334814\pi$
$998$	−17.2314	−0.545452
$999$	12.1642i	0.384859i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.2.b.b.168.2		6
3.2	odd	2		1521.2.b.l.1351.5		6
4.3	odd	2		2704.2.f.o.337.5		6
13.2	odd	12		169.2.c.b.22.3		6
13.3	even	3		169.2.e.b.147.5		12
13.4	even	6		169.2.e.b.23.5		12
13.5	odd	4		169.2.a.c.1.1	yes	3
13.6	odd	12		169.2.c.b.146.3		6
13.7	odd	12		169.2.c.c.146.1		6
13.8	odd	4		169.2.a.b.1.3	✓	3
13.9	even	3		169.2.e.b.23.2		12
13.10	even	6		169.2.e.b.147.2		12
13.11	odd	12		169.2.c.c.22.1		6
13.12	even	2	inner	169.2.b.b.168.5		6
39.5	even	4		1521.2.a.o.1.3		3
39.8	even	4		1521.2.a.r.1.1		3
39.38	odd	2		1521.2.b.l.1351.2		6
52.31	even	4		2704.2.a.ba.1.3		3
52.47	even	4		2704.2.a.z.1.3		3
52.51	odd	2		2704.2.f.o.337.6		6
65.34	odd	4		4225.2.a.bg.1.1		3
65.44	odd	4		4225.2.a.bb.1.3		3
91.34	even	4		8281.2.a.bf.1.3		3
91.83	even	4		8281.2.a.bj.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.2.a.b.1.3	✓	3	13.8	odd	4
169.2.a.c.1.1	yes	3	13.5	odd	4
169.2.b.b.168.2		6	1.1	even	1	trivial
169.2.b.b.168.5		6	13.12	even	2	inner
169.2.c.b.22.3		6	13.2	odd	12
169.2.c.b.146.3		6	13.6	odd	12
169.2.c.c.22.1		6	13.11	odd	12
169.2.c.c.146.1		6	13.7	odd	12
169.2.e.b.23.2		12	13.9	even	3
169.2.e.b.23.5		12	13.4	even	6
169.2.e.b.147.2		12	13.10	even	6
169.2.e.b.147.5		12	13.3	even	3
1521.2.a.o.1.3		3	39.5	even	4
1521.2.a.r.1.1		3	39.8	even	4
1521.2.b.l.1351.2		6	39.38	odd	2
1521.2.b.l.1351.5		6	3.2	odd	2
2704.2.a.z.1.3		3	52.47	even	4
2704.2.a.ba.1.3		3	52.31	even	4
2704.2.f.o.337.5		6	4.3	odd	2
2704.2.f.o.337.6		6	52.51	odd	2
4225.2.a.bb.1.3		3	65.44	odd	4
4225.2.a.bg.1.1		3	65.34	odd	4
8281.2.a.bf.1.3		3	91.34	even	4
8281.2.a.bj.1.1		3	91.83	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	169.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.801938i q^{2} -2.24698 q^{3} +1.35690 q^{4} -0.246980i q^{5} +1.80194i q^{6} -2.35690i q^{7} -2.69202i q^{8} +2.04892 q^{9} +O(q^{10})\)	\(q-0.801938i q^{2} -2.24698 q^{3} +1.35690 q^{4} -0.246980i q^{5} +1.80194i q^{6} -2.35690i q^{7} -2.69202i q^{8} +2.04892 q^{9} -0.198062 q^{10} -4.24698i q^{11} -3.04892 q^{12} -1.89008 q^{14} +0.554958i q^{15} +0.554958 q^{16} -2.15883 q^{17} -1.64310i q^{18} +0.0881460i q^{19} -0.335126i q^{20} +5.29590i q^{21} -3.40581 q^{22} -1.49396 q^{23} +6.04892i q^{24} +4.93900 q^{25} +2.13706 q^{27} -3.19806i q^{28} +4.63102 q^{29} +0.445042 q^{30} +6.63102i q^{31} -5.82908i q^{32} +9.54288i q^{33} +1.73125i q^{34} -0.582105 q^{35} +2.78017 q^{36} +5.69202i q^{37} +0.0706876 q^{38} -0.664874 q^{40} +11.5918i q^{41} +4.24698 q^{42} +0.295897 q^{43} -5.76271i q^{44} -0.506041i q^{45} +1.19806i q^{46} -7.35690i q^{47} -1.24698 q^{48} +1.44504 q^{49} -3.96077i q^{50} +4.85086 q^{51} -10.3937 q^{53} -1.71379i q^{54} -1.04892 q^{55} -6.34481 q^{56} -0.198062i q^{57} -3.71379i q^{58} -6.78017i q^{59} +0.753020i q^{60} +3.47219 q^{61} +5.31767 q^{62} -4.82908i q^{63} -3.56465 q^{64} +7.65279 q^{66} -7.67994i q^{67} -2.92931 q^{68} +3.35690 q^{69} +0.466812i q^{70} +8.66487i q^{71} -5.51573i q^{72} +6.73556i q^{73} +4.56465 q^{74} -11.0978 q^{75} +0.119605i q^{76} -10.0097 q^{77} +9.97046 q^{79} -0.137063i q^{80} -10.9487 q^{81} +9.29590 q^{82} -1.60925i q^{83} +7.18598i q^{84} +0.533188i q^{85} -0.237291i q^{86} -10.4058 q^{87} -11.4330 q^{88} -2.88471i q^{89} -0.405813 q^{90} -2.02715 q^{92} -14.8998i q^{93} -5.89977 q^{94} +0.0217703 q^{95} +13.0978i q^{96} +8.05861i q^{97} -1.15883i q^{98} -8.70171i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{3} - 6 q^{9}+O(q^{10}) \) 6 * q - 4 * q^3 - 6 * q^9	\( 6 q - 4 q^{3} - 6 q^{9} - 10 q^{10} - 10 q^{14} + 4 q^{16} + 4 q^{17} + 6 q^{22} + 10 q^{23} + 10 q^{25} + 2 q^{27} - 2 q^{29} + 2 q^{30} + 8 q^{35} + 14 q^{36} - 24 q^{38} - 6 q^{40} + 16 q^{42} - 26 q^{43} + 2 q^{48} + 8 q^{49} + 2 q^{51} + 2 q^{53} + 12 q^{55} + 8 q^{56} + 8 q^{61} - 2 q^{62} + 22 q^{64} + 10 q^{66} - 42 q^{68} + 12 q^{69} - 16 q^{74} - 30 q^{75} - 16 q^{77} - 10 q^{79} - 2 q^{81} + 28 q^{82} - 36 q^{87} - 30 q^{88} + 24 q^{90} + 10 q^{94} - 6 q^{95}+O(q^{100}) \) 6 * q - 4 * q^3 - 6 * q^9 - 10 * q^10 - 10 * q^14 + 4 * q^16 + 4 * q^17 + 6 * q^22 + 10 * q^23 + 10 * q^25 + 2 * q^27 - 2 * q^29 + 2 * q^30 + 8 * q^35 + 14 * q^36 - 24 * q^38 - 6 * q^40 + 16 * q^42 - 26 * q^43 + 2 * q^48 + 8 * q^49 + 2 * q^51 + 2 * q^53 + 12 * q^55 + 8 * q^56 + 8 * q^61 - 2 * q^62 + 22 * q^64 + 10 * q^66 - 42 * q^68 + 12 * q^69 - 16 * q^74 - 30 * q^75 - 16 * q^77 - 10 * q^79 - 2 * q^81 + 28 * q^82 - 36 * q^87 - 30 * q^88 + 24 * q^90 + 10 * q^94 - 6 * q^95

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