LMFDB - Embedded newform 169.4.b.g.168.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [169,4,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, 4, names="a")

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

chi := DirichletCharacter("169.168");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 169 = 13^{2} $
Weight:	$ k $	$=$	$ 4 $
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:	$9.97132279097$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} + 108 x^{16} + 4636 x^{14} + 101999 x^{12} + 1237806 x^{10} + 8358937 x^{8} + 30682857 x^{6} + \cdots + 16777216 $ x^18 + 108x^16 + 4636x^14 + 101999x^12 + 1237806x^10 + 8358937x^8 + 30682857x^6 + 58141952x^4 + 52035584x^2 + 16777216
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 13^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			168.11
Root			$1.39012i$ of defining polynomial
Character	$\chi$	$=$	169.168
Dual form			169.4.b.g.168.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.390115i q^{2} +3.60967 q^{3} +7.84781 q^{4} +7.52136i q^{5} +1.40819i q^{6} +19.5446i q^{7} +6.18247i q^{8} -13.9703 q^{9} +O(q^{10})$	\(q+0.390115i q^{2} +3.60967 q^{3} +7.84781 q^{4} +7.52136i q^{5} +1.40819i q^{6} +19.5446i q^{7} +6.18247i q^{8} -13.9703 q^{9} -2.93420 q^{10} +45.8243i q^{11} +28.3280 q^{12} -7.62463 q^{14} +27.1496i q^{15} +60.3706 q^{16} -86.5200 q^{17} -5.45003i q^{18} -148.737i q^{19} +59.0262i q^{20} +70.5494i q^{21} -17.8767 q^{22} +91.5351 q^{23} +22.3167i q^{24} +68.4292 q^{25} -147.889 q^{27} +153.382i q^{28} +258.901 q^{29} -10.5915 q^{30} -31.2317i q^{31} +73.0113i q^{32} +165.410i q^{33} -33.7528i q^{34} -147.002 q^{35} -109.636 q^{36} +148.436i q^{37} +58.0245 q^{38} -46.5006 q^{40} -95.9135i q^{41} -27.5224 q^{42} +80.9930 q^{43} +359.620i q^{44} -105.076i q^{45} +35.7093i q^{46} +94.3777i q^{47} +217.918 q^{48} -38.9898 q^{49} +26.6953i q^{50} -312.308 q^{51} +493.555 q^{53} -57.6938i q^{54} -344.661 q^{55} -120.834 q^{56} -536.891i q^{57} +101.001i q^{58} -575.686i q^{59} +213.065i q^{60} -40.2865 q^{61} +12.1840 q^{62} -273.043i q^{63} +454.482 q^{64} -64.5291 q^{66} -601.141i q^{67} -678.992 q^{68} +330.411 q^{69} -57.3476i q^{70} +518.800i q^{71} -86.3710i q^{72} -1055.21i q^{73} -57.9072 q^{74} +247.007 q^{75} -1167.26i q^{76} -895.615 q^{77} -320.840 q^{79} +454.069i q^{80} -156.633 q^{81} +37.4173 q^{82} +32.4841i q^{83} +553.658i q^{84} -650.748i q^{85} +31.5966i q^{86} +934.547 q^{87} -283.307 q^{88} +450.795i q^{89} +40.9916 q^{90} +718.350 q^{92} -112.736i q^{93} -36.8182 q^{94} +1118.70 q^{95} +263.546i q^{96} -231.743i q^{97} -15.2105i q^{98} -640.179i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 2 q^{3} - 74 q^{4} + 132 q^{9}+O(q^{10}) $ 18 * q + 2 * q^3 - 74 * q^4 + 132 * q^9	$ 18 q + 2 q^{3} - 74 q^{4} + 132 q^{9} + 294 q^{10} - 78 q^{12} - 294 q^{14} + 538 q^{16} + 110 q^{17} + 680 q^{22} + 408 q^{23} - 614 q^{25} - 1336 q^{27} + 560 q^{29} - 1042 q^{30} + 40 q^{35} + 1818 q^{36} + 1478 q^{38} + 26 q^{40} - 8 q^{42} + 1066 q^{43} - 264 q^{48} - 806 q^{49} - 940 q^{51} - 556 q^{53} - 500 q^{55} - 500 q^{56} - 272 q^{61} - 4070 q^{62} - 568 q^{64} + 6558 q^{66} - 3072 q^{68} + 4100 q^{69} - 3980 q^{74} - 4786 q^{75} + 1436 q^{77} + 824 q^{79} - 1670 q^{81} - 5514 q^{82} - 1572 q^{87} + 1272 q^{88} + 2560 q^{90} + 8020 q^{92} - 5062 q^{94} + 3228 q^{95}+O(q^{100}) $ 18 * q + 2 * q^3 - 74 * q^4 + 132 * q^9 + 294 * q^10 - 78 * q^12 - 294 * q^14 + 538 * q^16 + 110 * q^17 + 680 * q^22 + 408 * q^23 - 614 * q^25 - 1336 * q^27 + 560 * q^29 - 1042 * q^30 + 40 * q^35 + 1818 * q^36 + 1478 * q^38 + 26 * q^40 - 8 * q^42 + 1066 * q^43 - 264 * q^48 - 806 * q^49 - 940 * q^51 - 556 * q^53 - 500 * q^55 - 500 * q^56 - 272 * q^61 - 4070 * q^62 - 568 * q^64 + 6558 * q^66 - 3072 * q^68 + 4100 * q^69 - 3980 * q^74 - 4786 * q^75 + 1436 * q^77 + 824 * q^79 - 1670 * q^81 - 5514 * q^82 - 1572 * q^87 + 1272 * q^88 + 2560 * q^90 + 8020 * q^92 - 5062 * q^94 + 3228 * 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 $

$2$	0.390115i	0.137927i	0.997619	+	0.0689633i	$0.0219691\pi$
$2$	0.390115i	0.137927i	−0.997619	+	0.0689633i	$0.978031\pi$
$3$	3.60967	0.694681	0.347340	−	0.937739i	$-0.387085\pi$
$3$	3.60967	0.694681	0.347340	+	0.937739i	$0.387085\pi$
$4$	7.84781	0.980976
$5$	7.52136i	0.672731i	0.941732	+	0.336365i	$0.109198\pi$
$5$	7.52136i	0.672731i	−0.941732	+	0.336365i	$0.890802\pi$
$6$	1.40819i	0.0958150i
$7$	19.5446i	1.05531i	0.849460	+	0.527654i	$0.176928\pi$
$7$	19.5446i	1.05531i	−0.849460	+	0.527654i	$0.823072\pi$
$8$	6.18247i	0.273229i
$9$	−13.9703	−0.517418
$10$	−2.93420	−0.0927875
$11$	45.8243i	1.25605i	0.778194	+	0.628024i	$0.216136\pi$
$11$	45.8243i	1.25605i	−0.778194	+	0.628024i	$0.783864\pi$
$12$	28.3280	0.681465
$13$	0	0
$13$	0	0
$14$	−7.62463	−0.145555
$15$	27.1496i	0.467333i
$16$	60.3706	0.943291
$17$	−86.5200	−1.23436	−0.617182	−	0.786821i	$-0.711726\pi$
$17$	−86.5200	−1.23436	−0.617182	+	0.786821i	$0.711726\pi$
$18$	− 5.45003i	− 0.0713658i
$19$	− 148.737i	− 1.79593i	−0.440072	−	0.897963i	$-0.645047\pi$
$19$	− 148.737i	− 1.79593i	0.440072	−	0.897963i	$-0.354953\pi$
$20$	59.0262i	0.659933i
$21$	70.5494i	0.733102i
$22$	−17.8767	−0.173242
$23$	91.5351	0.829843	0.414922	−	0.909857i	$-0.363809\pi$
$23$	91.5351	0.829843	0.414922	+	0.909857i	$0.363809\pi$
$24$	22.3167i	0.189807i
$25$	68.4292	0.547433
$26$	0	0
$27$	−147.889	−1.05412
$28$	153.382i	1.03523i
$29$	258.901	1.65782	0.828909	−	0.559383i	$-0.188962\pi$
$29$	258.901	1.65782	0.828909	+	0.559383i	$0.188962\pi$
$30$	−10.5915	−0.0644577
$31$	− 31.2317i	− 0.180948i	−0.995899	−	0.0904739i	$-0.971162\pi$
$31$	− 31.2317i	− 0.180948i	0.995899	−	0.0904739i	$-0.0288382\pi$
$32$	73.0113i	0.403334i
$33$	165.410i	0.872553i
$34$	− 33.7528i	− 0.170252i
$35$	−147.002	−0.709938
$36$	−109.636	−0.507575
$37$	148.436i	0.659533i	0.944062	+	0.329767i	$0.106970\pi$
$37$	148.436i	0.659533i	−0.944062	+	0.329767i	$0.893030\pi$
$38$	58.0245	0.247706
$39$	0	0
$40$	−46.5006	−0.183810
$41$	− 95.9135i	− 0.365346i	−0.983174	−	0.182673i	$-0.941525\pi$
$41$	− 95.9135i	− 0.365346i	0.983174	−	0.182673i	$-0.0584749\pi$
$42$	−27.5224	−0.101114
$43$	80.9930	0.287240	0.143620	−	0.989633i	$-0.454126\pi$
$43$	80.9930	0.287240	0.143620	+	0.989633i	$0.454126\pi$
$44$	359.620i	1.23215i
$45$	− 105.076i	− 0.348083i
$46$	35.7093i	0.114457i
$47$	94.3777i	0.292902i	0.989218	+	0.146451i	$0.0467851\pi$
$47$	94.3777i	0.292902i	−0.989218	+	0.146451i	$0.953215\pi$
$48$	217.918	0.655286
$49$	−38.9898	−0.113673
$50$	26.6953i	0.0755056i
$51$	−312.308	−0.857489
$52$	0	0
$53$	493.555	1.27915	0.639575	−	0.768729i	$-0.279110\pi$
$53$	493.555	1.27915	0.639575	+	0.768729i	$0.279110\pi$
$54$	− 57.6938i	− 0.145391i
$55$	−344.661	−0.844983
$56$	−120.834	−0.288341
$57$	− 536.891i	− 1.24759i
$58$	101.001i	0.228657i
$59$	− 575.686i	− 1.27030i	−0.772387	−	0.635152i	$-0.780938\pi$
$59$	− 575.686i	− 1.27030i	0.772387	−	0.635152i	$-0.219062\pi$
$60$	213.065i	0.458443i
$61$	−40.2865	−0.0845599	−0.0422800	−	0.999106i	$-0.513462\pi$
$61$	−40.2865	−0.0845599	−0.0422800	+	0.999106i	$0.513462\pi$
$62$	12.1840	0.0249575
$63$	− 273.043i	− 0.546035i
$64$	454.482	0.887660
$65$	0	0
$66$	−64.5291	−0.120348
$67$	− 601.141i	− 1.09613i	−0.836434	−	0.548067i	$-0.815364\pi$
$67$	− 601.141i	− 1.09613i	0.836434	−	0.548067i	$-0.184636\pi$
$68$	−678.992	−1.21088
$69$	330.411	0.576476
$70$	− 57.3476i	− 0.0979193i
$71$	518.800i	0.867187i	0.901109	+	0.433593i	$0.142755\pi$
$71$	518.800i	0.867187i	−0.901109	+	0.433593i	$0.857245\pi$
$72$	− 86.3710i	− 0.141374i
$73$	− 1055.21i	− 1.69182i	−0.533328	−	0.845908i	$-0.679059\pi$
$73$	− 1055.21i	− 1.69182i	0.533328	−	0.845908i	$-0.320941\pi$
$74$	−57.9072	−0.0909672
$75$	247.007	0.380291
$76$	− 1167.26i	− 1.76176i
$77$	−895.615	−1.32552
$78$	0	0
$79$	−320.840	−0.456928	−0.228464	−	0.973552i	$-0.573370\pi$
$79$	−320.840	−0.456928	−0.228464	+	0.973552i	$0.573370\pi$
$80$	454.069i	0.634581i
$81$	−156.633	−0.214860
$82$	37.4173	0.0503909
$83$	32.4841i	0.0429590i	0.999769	+	0.0214795i	$0.00683766\pi$
$83$	32.4841i	0.0429590i	−0.999769	+	0.0214795i	$0.993162\pi$
$84$	553.658i	0.719155i
$85$	− 650.748i	− 0.830394i
$86$	31.5966i	0.0396180i
$87$	934.547	1.15165
$88$	−283.307	−0.343189
$89$	450.795i	0.536901i	0.963293	+	0.268450i	$0.0865116\pi$
$89$	450.795i	0.536901i	−0.963293	+	0.268450i	$0.913488\pi$
$90$	40.9916	0.0480099
$91$	0	0
$92$	718.350	0.814057
$93$	− 112.736i	− 0.125701i
$94$	−36.8182	−0.0403990
$95$	1118.70	1.20817
$96$	263.546i	0.280189i
$97$	− 231.743i	− 0.242576i	−0.992617	−	0.121288i	$-0.961298\pi$
$97$	− 231.743i	− 0.242576i	0.992617	−	0.121288i	$-0.0387025\pi$
$98$	− 15.2105i	− 0.0156785i
$99$	− 640.179i	− 0.649903i
$100$	537.019	0.537019
$101$	−570.125	−0.561679	−0.280840	−	0.959755i	$-0.590613\pi$
$101$	−570.125	−0.561679	−0.280840	+	0.959755i	$0.590613\pi$
$102$	− 121.836i	− 0.118270i
$103$	−969.551	−0.927502	−0.463751	−	0.885965i	$-0.653497\pi$
$103$	−969.551	−0.927502	−0.463751	+	0.885965i	$0.653497\pi$
$104$	0	0
$105$	−530.627	−0.493180
$106$	192.543i	0.176429i
$107$	−343.156	−0.310039	−0.155019	−	0.987911i	$-0.549544\pi$
$107$	−343.156	−0.310039	−0.155019	+	0.987911i	$0.549544\pi$
$108$	−1160.61	−1.03407
$109$	83.1640i	0.0730795i	0.999332	+	0.0365398i	$0.0116336\pi$
$109$	83.1640i	0.0730795i	−0.999332	+	0.0365398i	$0.988366\pi$
$110$	− 134.457i	− 0.116546i
$111$	535.805i	0.458165i
$112$	1179.92i	0.995461i
$113$	−2116.18	−1.76171	−0.880856	−	0.473384i	$-0.843032\pi$
$113$	−2116.18	−1.76171	−0.880856	+	0.473384i	$0.843032\pi$
$114$	209.449	0.172077
$115$	688.469i	0.558261i
$116$	2031.81	1.62628
$117$	0	0
$118$	224.584	0.175209
$119$	− 1691.00i	− 1.30263i
$120$	−167.852	−0.127689
$121$	−768.863	−0.577658
$122$	− 15.7164i	− 0.0116631i
$123$	− 346.216i	− 0.253799i
$124$	− 245.101i	− 0.177506i
$125$	1454.85i	1.04101i
$126$	106.518	0.0753128
$127$	1176.69	0.822161	0.411081	−	0.911599i	$-0.365152\pi$
$127$	1176.69	0.822161	0.411081	+	0.911599i	$0.365152\pi$
$128$	761.391i	0.525766i
$129$	292.358	0.199540
$130$	0	0
$131$	775.336	0.517110	0.258555	−	0.965996i	$-0.416754\pi$
$131$	775.336	0.517110	0.258555	+	0.965996i	$0.416754\pi$
$132$	1298.11i	0.855954i
$133$	2907.00	1.89525
$134$	234.514	0.151186
$135$	− 1112.33i	− 0.709140i
$136$	− 534.907i	− 0.337264i
$137$	2542.62i	1.58563i	0.609464	+	0.792814i	$0.291385\pi$
$137$	2542.62i	1.58563i	−0.609464	+	0.792814i	$0.708615\pi$
$138$	128.899i	0.0795114i
$139$	−286.317	−0.174713	−0.0873564	−	0.996177i	$-0.527842\pi$
$139$	−286.317	−0.174713	−0.0873564	+	0.996177i	$0.527842\pi$
$140$	−1153.64	−0.696432
$141$	340.672i	0.203473i
$142$	−202.392	−0.119608
$143$	0	0
$144$	−843.395	−0.488076
$145$	1947.29i	1.11527i
$146$	411.652	0.233346
$147$	−140.740	−0.0789664
$148$	1164.90i	0.646987i
$149$	− 2354.13i	− 1.29435i	−0.762343	−	0.647173i	$-0.775951\pi$
$149$	− 2354.13i	− 1.29435i	0.762343	−	0.647173i	$-0.224049\pi$
$150$	96.3610i	0.0524523i
$151$	− 165.158i	− 0.0890089i	−0.999009	−	0.0445045i	$-0.985829\pi$
$151$	− 165.158i	− 0.0890089i	0.999009	−	0.0445045i	$-0.0141709\pi$
$152$	919.562	0.490699
$153$	1208.71	0.638682
$154$	− 349.393i	− 0.182824i
$155$	234.905	0.121729
$156$	0	0
$157$	−3095.72	−1.57367	−0.786833	−	0.617166i	$-0.788281\pi$
$157$	−3095.72	−1.57367	−0.786833	+	0.617166i	$0.788281\pi$
$158$	− 125.165i	− 0.0630226i
$159$	1781.57	0.888601
$160$	−549.144	−0.271335
$161$	1789.01i	0.875740i
$162$	− 61.1048i	− 0.0296349i
$163$	− 299.310i	− 0.143827i	−0.997411	−	0.0719134i	$-0.977089\pi$
$163$	− 299.310i	− 0.143827i	0.997411	−	0.0719134i	$-0.0229105\pi$
$164$	− 752.711i	− 0.358395i
$165$	−1244.11	−0.586993
$166$	−12.6726	−0.00592519
$167$	3005.17i	1.39250i	0.717801	+	0.696249i	$0.245149\pi$
$167$	3005.17i	1.39250i	−0.717801	+	0.696249i	$0.754851\pi$
$168$	−436.170	−0.200305
$169$	0	0
$170$	253.867	0.114533
$171$	2077.90i	0.929245i
$172$	635.618	0.281776
$173$	−455.853	−0.200334	−0.100167	−	0.994971i	$-0.531938\pi$
$173$	−455.853	−0.200334	−0.100167	+	0.994971i	$0.531938\pi$
$174$	364.581i	0.158844i
$175$	1337.42i	0.577710i
$176$	2766.44i	1.18482i
$177$	− 2078.03i	− 0.882455i
$178$	−175.862	−0.0740529
$179$	−1364.19	−0.569635	−0.284818	−	0.958582i	$-0.591933\pi$
$179$	−1364.19	−0.569635	−0.284818	+	0.958582i	$0.591933\pi$
$180$	− 824.614i	− 0.341461i
$181$	2026.11	0.832041	0.416021	−	0.909355i	$-0.363424\pi$
$181$	2026.11	0.832041	0.416021	+	0.909355i	$0.363424\pi$
$182$	0	0
$183$	−145.421	−0.0587422
$184$	565.914i	0.226738i
$185$	−1116.44	−0.443688
$186$	43.9801	0.0173375
$187$	− 3964.71i	− 1.55042i
$188$	740.658i	0.287330i
$189$	− 2890.43i	− 1.11242i
$190$	436.423i	0.166639i
$191$	2161.06	0.818686	0.409343	−	0.912380i	$-0.365758\pi$
$191$	2161.06	0.818686	0.409343	+	0.912380i	$0.365758\pi$
$192$	1640.53	0.616641
$193$	− 1207.88i	− 0.450491i	−0.974302	−	0.225246i	$-0.927682\pi$
$193$	− 1207.88i	− 0.450491i	0.974302	−	0.225246i	$-0.0723185\pi$
$194$	90.4063	0.0334577
$195$	0	0
$196$	−305.985	−0.111510
$197$	− 4926.76i	− 1.78181i	−0.454187	−	0.890906i	$-0.650070\pi$
$197$	− 4926.76i	− 1.78181i	0.454187	−	0.890906i	$-0.349930\pi$
$198$	249.743	0.0896389
$199$	−1009.45	−0.359589	−0.179795	−	0.983704i	$-0.557543\pi$
$199$	−1009.45	−0.359589	−0.179795	+	0.983704i	$0.557543\pi$
$200$	423.061i	0.149575i
$201$	− 2169.92i	− 0.761464i
$202$	− 222.415i	− 0.0774705i
$203$	5060.11i	1.74951i
$204$	−2450.94	−0.841176
$205$	721.400	0.245779
$206$	− 378.237i	− 0.127927i
$207$	−1278.77	−0.429376
$208$	0	0
$209$	6815.76	2.25577
$210$	− 207.006i	− 0.0680226i
$211$	−4911.32	−1.60241	−0.801206	−	0.598388i	$-0.795808\pi$
$211$	−4911.32	−1.60241	−0.801206	+	0.598388i	$0.795808\pi$
$212$	3873.32	1.25482
$213$	1872.70i	0.602418i
$214$	− 133.871i	− 0.0427626i
$215$	609.178i	0.193235i
$216$	− 914.321i	− 0.288017i
$217$	610.410	0.190956
$218$	−32.4435	−0.0100796
$219$	− 3808.95i	− 1.17527i
$220$	−2704.83	−0.828908
$221$	0	0
$222$	−209.026	−0.0631932
$223$	1364.58i	0.409771i	0.978786	+	0.204885i	$0.0656822\pi$
$223$	1364.58i	0.409771i	−0.978786	+	0.204885i	$0.934318\pi$
$224$	−1426.97	−0.425641
$225$	−955.976	−0.283252
$226$	− 825.554i	− 0.242987i
$227$	4169.88i	1.21923i	0.792699	+	0.609614i	$0.208675\pi$
$227$	4169.88i	1.21923i	−0.792699	+	0.609614i	$0.791325\pi$
$228$	− 4213.42i	− 1.22386i
$229$	− 3506.89i	− 1.01197i	−0.862541	−	0.505987i	$-0.831128\pi$
$229$	− 3506.89i	− 1.01197i	0.862541	−	0.505987i	$-0.168872\pi$
$230$	−268.582	−0.0769991
$231$	−3232.87	−0.920811
$232$	1600.65i	0.452965i
$233$	570.253	0.160337	0.0801684	−	0.996781i	$-0.474454\pi$
$233$	570.253	0.160337	0.0801684	+	0.996781i	$0.474454\pi$
$234$	0	0
$235$	−709.848	−0.197044
$236$	− 4517.87i	− 1.24614i
$237$	−1158.13	−0.317419
$238$	659.683	0.179668
$239$	231.056i	0.0625347i	0.999511	+	0.0312674i	$0.00995433\pi$
$239$	231.056i	0.0625347i	−0.999511	+	0.0312674i	$0.990046\pi$
$240$	1639.04i	0.440831i
$241$	− 3088.50i	− 0.825508i	−0.910842	−	0.412754i	$-0.864567\pi$
$241$	− 3088.50i	− 0.825508i	0.910842	−	0.412754i	$-0.135433\pi$
$242$	− 299.945i	− 0.0796744i
$243$	3427.62	0.904863
$244$	−316.161	−0.0829513
$245$	− 293.256i	− 0.0764713i
$246$	135.064	0.0350056
$247$	0	0
$248$	193.089	0.0494403
$249$	117.257i	0.0298428i
$250$	−567.559	−0.143582
$251$	2445.64	0.615009	0.307504	−	0.951547i	$-0.400506\pi$
$251$	2445.64	0.615009	0.307504	+	0.951547i	$0.400506\pi$
$252$	− 2142.79i	− 0.535648i
$253$	4194.53i	1.04232i
$254$	459.045i	0.113398i
$255$	− 2348.98i	− 0.576859i
$256$	3338.83	0.815143
$257$	−3273.99	−0.794654	−0.397327	−	0.917677i	$-0.630062\pi$
$257$	−3273.99	−0.794654	−0.397327	+	0.917677i	$0.630062\pi$
$258$	114.053i	0.0275219i
$259$	−2901.12	−0.696010
$260$	0	0
$261$	−3616.93	−0.857786
$262$	302.471i	0.0713233i
$263$	−4631.27	−1.08584	−0.542921	−	0.839784i	$-0.682682\pi$
$263$	−4631.27	−1.08584	−0.542921	+	0.839784i	$0.682682\pi$
$264$	−1022.65	−0.238407
$265$	3712.20i	0.860524i
$266$	1134.06i	0.261406i
$267$	1627.22i	0.372975i
$268$	− 4717.64i	− 1.07528i
$269$	−2838.51	−0.643373	−0.321686	−	0.946846i	$-0.604250\pi$
$269$	−2838.51	−0.643373	−0.321686	+	0.946846i	$0.604250\pi$
$270$	433.936	0.0978093
$271$	− 7052.31i	− 1.58080i	−0.612589	−	0.790401i	$-0.709872\pi$
$271$	− 7052.31i	− 1.58080i	0.612589	−	0.790401i	$-0.290128\pi$
$272$	−5223.26	−1.16436
$273$	0	0
$274$	−991.917	−0.218700
$275$	3135.72i	0.687603i
$276$	2593.01	0.565510
$277$	1938.31	0.420439	0.210220	−	0.977654i	$-0.432582\pi$
$277$	1938.31	0.420439	0.210220	+	0.977654i	$0.432582\pi$
$278$	− 111.697i	− 0.0240975i
$279$	436.317i	0.0936258i
$280$	− 908.834i	− 0.193976i
$281$	3290.74i	0.698609i	0.937009	+	0.349305i	$0.113582\pi$
$281$	3290.74i	0.698609i	−0.937009	+	0.349305i	$0.886418\pi$
$282$	−132.901	−0.0280644
$283$	7918.08	1.66318	0.831592	−	0.555388i	$-0.187430\pi$
$283$	7918.08	1.66318	0.831592	+	0.555388i	$0.187430\pi$
$284$	4071.45i	0.850690i
$285$	4038.15	0.839296
$286$	0	0
$287$	1874.59	0.385552
$288$	− 1019.99i	− 0.208693i
$289$	2572.71	0.523653
$290$	−759.667	−0.153825
$291$	− 836.514i	− 0.168513i
$292$	− 8281.06i	− 1.65963i
$293$	− 5675.43i	− 1.13161i	−0.824539	−	0.565806i	$-0.808565\pi$
$293$	− 5675.43i	− 1.13161i	0.824539	−	0.565806i	$-0.191435\pi$
$294$	− 54.9049i	− 0.0108916i
$295$	4329.94	0.854572
$296$	−917.702	−0.180204
$297$	− 6776.91i	− 1.32403i
$298$	918.381	0.178525
$299$	0	0
$300$	1938.46	0.373057
$301$	1582.97i	0.303126i
$302$	64.4306	0.0122767
$303$	−2057.96	−0.390188
$304$	− 8979.33i	− 1.69408i
$305$	− 303.009i	− 0.0568861i
$306$	471.536i	0.0880913i
$307$	4338.86i	0.806618i	0.915064	+	0.403309i	$0.132140\pi$
$307$	4338.86i	0.806618i	−0.915064	+	0.403309i	$0.867860\pi$
$308$	−7028.62	−1.30030
$309$	−3499.76	−0.644318
$310$	91.6401i	0.0167897i
$311$	5234.75	0.954454	0.477227	−	0.878780i	$-0.341642\pi$
$311$	5234.75	0.954454	0.477227	+	0.878780i	$0.341642\pi$
$312$	0	0
$313$	2167.86	0.391484	0.195742	−	0.980655i	$-0.437288\pi$
$313$	2167.86	0.391484	0.195742	+	0.980655i	$0.437288\pi$
$314$	− 1207.69i	− 0.217050i
$315$	2053.66	0.367335
$316$	−2517.89	−0.448236
$317$	4863.71i	0.861744i	0.902413	+	0.430872i	$0.141794\pi$
$317$	4863.71i	0.861744i	−0.902413	+	0.430872i	$0.858206\pi$
$318$	695.017i	0.122562i
$319$	11864.0i	2.08230i
$320$	3418.32i	0.597156i
$321$	−1238.68	−0.215378
$322$	−697.922	−0.120788
$323$	12868.7i	2.21682i
$324$	−1229.22	−0.210772
$325$	0	0
$326$	116.765	0.0198375
$327$	300.194i	0.0507669i
$328$	592.983	0.0998231
$329$	−1844.57	−0.309102
$330$	− 485.347i	− 0.0809620i
$331$	− 2685.26i	− 0.445908i	−0.974829	−	0.222954i	$-0.928430\pi$
$331$	− 2685.26i	− 0.445908i	0.974829	−	0.222954i	$-0.0715699\pi$
$332$	254.929i	0.0421417i
$333$	− 2073.70i	− 0.341255i
$334$	−1172.36	−0.192062
$335$	4521.39	0.737404
$336$	4259.11i	0.691528i
$337$	6518.36	1.05364	0.526821	−	0.849976i	$-0.323384\pi$
$337$	6518.36	1.05364	0.526821	+	0.849976i	$0.323384\pi$
$338$	0	0
$339$	−7638.71	−1.22383
$340$	− 5106.95i	− 0.814597i
$341$	1431.17	0.227279
$342$	−810.620	−0.128168
$343$	5941.75i	0.935347i
$344$	500.737i	0.0784824i
$345$	2485.14i	0.387813i
$346$	− 177.835i	− 0.0276314i
$347$	75.7963	0.0117261	0.00586305	−	0.999983i	$-0.498134\pi$
$347$	75.7963	0.0117261	0.00586305	+	0.999983i	$0.498134\pi$
$348$	7334.15	1.12975
$349$	3682.09i	0.564750i	0.959304	+	0.282375i	$0.0911222\pi$
$349$	3682.09i	0.564750i	−0.959304	+	0.282375i	$0.908878\pi$
$350$	−521.747	−0.0796816
$351$	0	0
$352$	−3345.69	−0.506607
$353$	10031.5i	1.51254i	0.654261	+	0.756268i	$0.272980\pi$
$353$	10031.5i	1.51254i	−0.654261	+	0.756268i	$0.727020\pi$
$354$	810.673	0.121714
$355$	−3902.08	−0.583383
$356$	3537.75i	0.526687i
$357$	− 6103.93i	− 0.904914i
$358$	− 532.193i	− 0.0785678i
$359$	6869.76i	1.00995i	0.863134	+	0.504975i	$0.168498\pi$
$359$	6869.76i	1.00995i	−0.863134	+	0.504975i	$0.831502\pi$
$360$	649.627	0.0951066
$361$	−15263.7	−2.22535
$362$	790.416i	0.114761i
$363$	−2775.34	−0.401288
$364$	0	0
$365$	7936.59	1.13814
$366$	− 56.7309i	− 0.00810211i
$367$	8883.29	1.26350	0.631749	−	0.775173i	$-0.282337\pi$
$367$	8883.29	1.26350	0.631749	+	0.775173i	$0.282337\pi$
$368$	5526.03	0.782784
$369$	1339.94i	0.189037i
$370$	− 435.541i	− 0.0611964i
$371$	9646.31i	1.34990i
$372$	− 884.732i	− 0.123310i
$373$	−5454.50	−0.757166	−0.378583	−	0.925567i	$-0.623589\pi$
$373$	−5454.50	−0.757166	−0.378583	+	0.925567i	$0.623589\pi$
$374$	1546.70	0.213844
$375$	5251.53i	0.723167i
$376$	−583.487	−0.0800294
$377$	0	0
$378$	1127.60	0.153433
$379$	6642.15i	0.900222i	0.892973	+	0.450111i	$0.148616\pi$
$379$	6642.15i	0.900222i	−0.892973	+	0.450111i	$0.851384\pi$
$380$	8779.37	1.18519
$381$	4247.46	0.571140
$382$	843.064i	0.112919i
$383$	1262.33i	0.168413i	0.996448	+	0.0842066i	$0.0268356\pi$
$383$	1262.33i	0.168413i	−0.996448	+	0.0842066i	$0.973164\pi$
$384$	2748.37i	0.365240i
$385$	− 6736.24i	− 0.891716i
$386$	471.211	0.0621347
$387$	−1131.50	−0.148623
$388$	− 1818.67i	− 0.237962i
$389$	2793.42	0.364093	0.182046	−	0.983290i	$-0.441728\pi$
$389$	2793.42	0.364093	0.182046	+	0.983290i	$0.441728\pi$
$390$	0	0
$391$	−7919.62	−1.02433
$392$	− 241.053i	− 0.0310588i
$393$	2798.71	0.359227
$394$	1922.00	0.245759
$395$	− 2413.15i	− 0.307390i
$396$	− 5024.00i	− 0.637539i
$397$	− 6848.91i	− 0.865836i	−0.901433	−	0.432918i	$-0.857484\pi$
$397$	− 6848.91i	− 0.865836i	0.901433	−	0.432918i	$-0.142516\pi$
$398$	− 393.803i	− 0.0495969i
$399$	10493.3	1.31660
$400$	4131.11	0.516389
$401$	− 11024.5i	− 1.37291i	−0.727171	−	0.686456i	$-0.759165\pi$
$401$	− 11024.5i	− 1.37291i	0.727171	−	0.686456i	$-0.240835\pi$
$402$	846.518	0.105026
$403$	0	0
$404$	−4474.24	−0.550994
$405$	− 1178.09i	− 0.144543i
$406$	−1974.03	−0.241304
$407$	−6801.97	−0.828406
$408$	− 1930.84i	− 0.234291i
$409$	3530.56i	0.426833i	0.976961	+	0.213417i	$0.0684592\pi$
$409$	3530.56i	0.426833i	−0.976961	+	0.213417i	$0.931541\pi$
$410$	281.429i	0.0338995i
$411$	9178.03i	1.10151i
$412$	−7608.86	−0.909858
$413$	11251.5	1.34056
$414$	− 498.869i	− 0.0592224i
$415$	−244.325	−0.0288998
$416$	0	0
$417$	−1033.51	−0.121370
$418$	2658.93i	0.311131i
$419$	−4179.22	−0.487275	−0.243637	−	0.969866i	$-0.578341\pi$
$419$	−4179.22	−0.487275	−0.243637	+	0.969866i	$0.578341\pi$
$420$	−4164.26	−0.483798
$421$	− 6209.31i	− 0.718820i	−0.933180	−	0.359410i	$-0.882978\pi$
$421$	− 6209.31i	− 0.718820i	0.933180	−	0.359410i	$-0.117022\pi$
$422$	− 1915.98i	− 0.221015i
$423$	− 1318.48i	− 0.151553i
$424$	3051.39i	0.349501i
$425$	−5920.49	−0.675732
$426$	−730.568	−0.0830895
$427$	− 787.382i	− 0.0892367i
$428$	−2693.03	−0.304141
$429$	0	0
$430$	−237.650	−0.0266523
$431$	− 11880.2i	− 1.32773i	−0.747854	−	0.663864i	$-0.768916\pi$
$431$	− 11880.2i	− 1.32773i	0.747854	−	0.663864i	$-0.231084\pi$
$432$	−8928.16	−0.994343
$433$	−8725.71	−0.968432	−0.484216	−	0.874949i	$-0.660895\pi$
$433$	−8725.71	−0.968432	−0.484216	+	0.874949i	$0.660895\pi$
$434$	238.130i	0.0263378i
$435$	7029.06i	0.774754i
$436$	652.655i	0.0716893i
$437$	− 13614.7i	− 1.49034i
$438$	1485.93	0.162101
$439$	1200.37	0.130503	0.0652514	−	0.997869i	$-0.479215\pi$
$439$	1200.37	0.130503	0.0652514	+	0.997869i	$0.479215\pi$
$440$	− 2130.86i	− 0.230874i
$441$	544.699	0.0588165
$442$	0	0
$443$	2258.86	0.242261	0.121130	−	0.992637i	$-0.461348\pi$
$443$	2258.86	0.242261	0.121130	+	0.992637i	$0.461348\pi$
$444$	4204.90i	0.449449i
$445$	−3390.59	−0.361190
$446$	−532.343	−0.0565183
$447$	− 8497.62i	− 0.899158i
$448$	8882.65i	0.936754i
$449$	− 14662.4i	− 1.54112i	−0.637367	−	0.770561i	$-0.719976\pi$
$449$	− 14662.4i	− 1.54112i	0.637367	−	0.770561i	$-0.280024\pi$
$450$	− 372.941i	− 0.0390680i
$451$	4395.16	0.458892
$452$	−16607.4	−1.72820
$453$	− 596.165i	− 0.0618328i
$454$	−1626.73	−0.168164
$455$	0	0
$456$	3319.31	0.340879
$457$	− 9334.94i	− 0.955514i	−0.878492	−	0.477757i	$-0.841450\pi$
$457$	− 9334.94i	− 0.955514i	0.878492	−	0.477757i	$-0.158550\pi$
$458$	1368.09	0.139578
$459$	12795.4	1.30117
$460$	5402.97i	0.547641i
$461$	− 10736.9i	− 1.08474i	−0.840140	−	0.542370i	$-0.817527\pi$
$461$	− 10736.9i	− 1.08474i	0.840140	−	0.542370i	$-0.182473\pi$
$462$	− 1261.19i	− 0.127004i
$463$	10650.0i	1.06900i	0.845168	+	0.534501i	$0.179501\pi$
$463$	10650.0i	1.06900i	−0.845168	+	0.534501i	$0.820499\pi$
$464$	15630.0	1.56380
$465$	847.929	0.0845630
$466$	222.464i	0.0221147i
$467$	2638.11	0.261407	0.130703	−	0.991422i	$-0.458276\pi$
$467$	2638.11	0.261407	0.130703	+	0.991422i	$0.458276\pi$
$468$	0	0
$469$	11749.0	1.15676
$470$	− 276.923i	− 0.0271776i
$471$	−11174.5	−1.09320
$472$	3559.16	0.347084
$473$	3711.45i	0.360787i
$474$	− 451.803i	− 0.0437806i
$475$	− 10177.9i	− 0.983149i
$476$	− 13270.6i	− 1.27785i
$477$	−6895.11	−0.661856
$478$	−90.1386	−0.00862520
$479$	− 3225.31i	− 0.307658i	−0.988097	−	0.153829i	$-0.950840\pi$
$479$	− 3225.31i	− 0.307658i	0.988097	−	0.153829i	$-0.0491605\pi$
$480$	−1982.23	−0.188491
$481$	0	0
$482$	1204.87	0.113860
$483$	6457.75i	0.608360i
$484$	−6033.89	−0.566669
$485$	1743.02	0.163189
$486$	1337.17i	0.124805i
$487$	1496.39i	0.139236i	0.997574	+	0.0696181i	$0.0221781\pi$
$487$	1496.39i	0.139236i	−0.997574	+	0.0696181i	$0.977822\pi$
$488$	− 249.070i	− 0.0231043i
$489$	− 1080.41i	− 0.0999137i
$490$	114.404	0.0105474
$491$	518.408	0.0476485	0.0238243	−	0.999716i	$-0.492416\pi$
$491$	518.408	0.0476485	0.0238243	+	0.999716i	$0.492416\pi$
$492$	− 2717.04i	− 0.248970i
$493$	−22400.1	−2.04635
$494$	0	0
$495$	4815.01	0.437210
$496$	− 1885.48i	− 0.170686i
$497$	−10139.7	−0.915148
$498$	−45.7437	−0.00411611
$499$	2405.01i	0.215757i	0.994164	+	0.107879i	$0.0344058\pi$
$499$	2405.01i	0.215757i	−0.994164	+	0.107879i	$0.965594\pi$
$500$	11417.4i	1.02120i
$501$	10847.7i	0.967341i
$502$	954.080i	0.0848260i
$503$	2413.76	0.213965	0.106983	−	0.994261i	$-0.465881\pi$
$503$	2413.76	0.213965	0.106983	+	0.994261i	$0.465881\pi$
$504$	1688.08	0.149193
$505$	− 4288.12i	− 0.377859i
$506$	−1636.35	−0.143764
$507$	0	0
$508$	9234.45	0.806520
$509$	− 15678.3i	− 1.36528i	−0.730754	−	0.682641i	$-0.760831\pi$
$509$	− 15678.3i	− 1.36528i	0.730754	−	0.682641i	$-0.239169\pi$
$510$	916.374	0.0795642
$511$	20623.6	1.78539
$512$	7393.65i	0.638196i
$513$	21996.6i	1.89312i
$514$	− 1277.24i	− 0.109604i
$515$	− 7292.34i	− 0.623959i
$516$	2294.37	0.195744
$517$	−4324.79	−0.367899
$518$	− 1131.77i	− 0.0959983i
$519$	−1645.48	−0.139168
$520$	0	0
$521$	−11691.5	−0.983135	−0.491568	−	0.870839i	$-0.663576\pi$
$521$	−11691.5	−0.983135	−0.491568	+	0.870839i	$0.663576\pi$
$522$	− 1411.02i	− 0.118311i
$523$	−3878.95	−0.324311	−0.162156	−	0.986765i	$-0.551845\pi$
$523$	−3878.95	−0.324311	−0.162156	+	0.986765i	$0.551845\pi$
$524$	6084.69	0.507273
$525$	4827.63i	0.401324i
$526$	− 1806.73i	− 0.149767i
$527$	2702.17i	0.223355i
$528$	9985.92i	0.823071i
$529$	−3788.32	−0.311360
$530$	−1448.19	−0.118689
$531$	8042.50i	0.657278i
$532$	22813.6	1.85920
$533$	0	0
$534$	−634.804	−0.0514431
$535$	− 2581.00i	− 0.208573i
$536$	3716.54	0.299496
$537$	−4924.29	−0.395715
$538$	− 1107.35i	− 0.0887382i
$539$	− 1786.68i	− 0.142779i
$540$	− 8729.34i	− 0.695650i
$541$	− 16353.0i	− 1.29958i	−0.760115	−	0.649788i	$-0.774858\pi$
$541$	− 16353.0i	− 1.29958i	0.760115	−	0.649788i	$-0.225142\pi$
$542$	2751.22	0.218035
$543$	7313.58	0.578003
$544$	− 6316.93i	− 0.497861i
$545$	−625.506	−0.0491628
$546$	0	0
$547$	2748.67	0.214853	0.107426	−	0.994213i	$-0.465739\pi$
$547$	2748.67	0.214853	0.107426	+	0.994213i	$0.465739\pi$
$548$	19954.0i	1.55546i
$549$	562.814	0.0437529
$550$	−1223.29	−0.0948387
$551$	− 38508.1i	− 2.97732i
$552$	2042.76i	0.157510i
$553$	− 6270.68i	− 0.482200i
$554$	756.163i	0.0579897i
$555$	−4029.98	−0.308222
$556$	−2246.96	−0.171389
$557$	− 16765.6i	− 1.27537i	−0.770297	−	0.637686i	$-0.779892\pi$
$557$	− 16765.6i	− 1.27537i	0.770297	−	0.637686i	$-0.220108\pi$
$558$	−170.214	−0.0129135
$559$	0	0
$560$	−8874.58	−0.669677
$561$	− 14311.3i	− 1.07705i
$562$	−1283.77	−0.0963568
$563$	−18492.4	−1.38430	−0.692151	−	0.721752i	$-0.743337\pi$
$563$	−18492.4	−1.38430	−0.692151	+	0.721752i	$0.743337\pi$
$564$	2673.53i	0.199603i
$565$	− 15916.5i	− 1.18516i
$566$	3088.96i	0.229397i
$567$	− 3061.32i	− 0.226743i
$568$	−3207.47	−0.236941
$569$	−1563.28	−0.115178	−0.0575888	−	0.998340i	$-0.518341\pi$
$569$	−1563.28	−0.115178	−0.0575888	+	0.998340i	$0.518341\pi$
$570$	1575.34i	0.115761i
$571$	−9165.98	−0.671776	−0.335888	−	0.941902i	$-0.609036\pi$
$571$	−9165.98	−0.671776	−0.335888	+	0.941902i	$0.609036\pi$
$572$	0	0
$573$	7800.72	0.568726
$574$	731.305i	0.0531778i
$575$	6263.67	0.454284
$576$	−6349.25	−0.459292
$577$	18762.5i	1.35372i	0.736114	+	0.676858i	$0.236659\pi$
$577$	18762.5i	1.35372i	−0.736114	+	0.676858i	$0.763341\pi$
$578$	1003.65i	0.0722257i
$579$	− 4360.03i	− 0.312948i
$580$	15281.9i	1.09405i
$581$	−634.888	−0.0453349
$582$	326.337	0.0232424
$583$	22616.8i	1.60667i
$584$	6523.79	0.462254
$585$	0	0
$586$	2214.07	0.156079
$587$	12646.3i	0.889212i	0.895726	+	0.444606i	$0.146656\pi$
$587$	12646.3i	0.889212i	−0.895726	+	0.444606i	$0.853344\pi$
$588$	−1104.50	−0.0774642
$589$	−4645.31	−0.324969
$590$	1689.18i	0.117868i
$591$	− 17784.0i	− 1.23779i
$592$	8961.17i	0.622132i
$593$	9662.74i	0.669142i	0.942371	+	0.334571i	$0.108591\pi$
$593$	9662.74i	0.669142i	−0.942371	+	0.334571i	$0.891409\pi$
$594$	2643.78	0.182619
$595$	12718.6	0.876321
$596$	− 18474.7i	− 1.26972i
$597$	−3643.79	−0.249800
$598$	0	0
$599$	−26968.7	−1.83959	−0.919794	−	0.392402i	$-0.871644\pi$
$599$	−26968.7	−1.83959	−0.919794	+	0.392402i	$0.871644\pi$
$600$	1527.11i	0.103907i
$601$	11280.1	0.765600	0.382800	−	0.923831i	$-0.374960\pi$
$601$	11280.1	0.765600	0.382800	+	0.923831i	$0.374960\pi$
$602$	−617.542	−0.0418092
$603$	8398.11i	0.567160i
$604$	− 1296.13i	− 0.0873156i
$605$	− 5782.89i	− 0.388608i
$606$	− 802.843i	− 0.0538173i
$607$	12052.6	0.805931	0.402965	−	0.915215i	$-0.367979\pi$
$607$	12052.6	0.805931	0.402965	+	0.915215i	$0.367979\pi$
$608$	10859.5	0.724358
$609$	18265.3i	1.21535i
$610$	118.208	0.00784610
$611$	0	0
$612$	9485.73	0.626532
$613$	11259.7i	0.741883i	0.928656	+	0.370941i	$0.120965\pi$
$613$	11259.7i	0.741883i	−0.928656	+	0.370941i	$0.879035\pi$
$614$	−1692.65	−0.111254
$615$	2604.01	0.170738
$616$	− 5537.12i	− 0.362170i
$617$	− 24058.2i	− 1.56976i	−0.619645	−	0.784882i	$-0.712723\pi$
$617$	− 24058.2i	− 1.56976i	0.619645	−	0.784882i	$-0.287277\pi$
$618$	− 1365.31i	− 0.0888686i
$619$	2793.41i	0.181384i	0.995879	+	0.0906919i	$0.0289079\pi$
$619$	2793.41i	0.181384i	−0.995879	+	0.0906919i	$0.971092\pi$
$620$	1843.49	0.119413
$621$	−13537.1	−0.874756
$622$	2042.15i	0.131645i
$623$	−8810.59	−0.566595
$624$	0	0
$625$	−2388.81	−0.152884
$626$	845.714i	0.0539960i
$627$	24602.6	1.56704
$628$	−24294.6	−1.54373
$629$	− 12842.7i	− 0.814104i
$630$	801.163i	0.0506652i
$631$	24850.0i	1.56777i	0.620908	+	0.783883i	$0.286764\pi$
$631$	24850.0i	1.56777i	−0.620908	+	0.783883i	$0.713236\pi$
$632$	− 1983.59i	− 0.124846i
$633$	−17728.2	−1.11317
$634$	−1897.41	−0.118857
$635$	8850.32i	0.553093i
$636$	13981.4	0.871696
$637$	0	0
$638$	−4628.31	−0.287205
$639$	− 7247.79i	− 0.448698i
$640$	−5726.69	−0.353699
$641$	7794.71	0.480300	0.240150	−	0.970736i	$-0.422803\pi$
$641$	7794.71	0.480300	0.240150	+	0.970736i	$0.422803\pi$
$642$	− 483.228i	− 0.0297064i
$643$	27709.9i	1.69949i	0.527195	+	0.849744i	$0.323244\pi$
$643$	27709.9i	1.69949i	−0.527195	+	0.849744i	$0.676756\pi$
$644$	14039.8i	0.859080i
$645$	2198.93i	0.134237i
$646$	−5020.28	−0.305759
$647$	−11150.9	−0.677571	−0.338786	−	0.940864i	$-0.610016\pi$
$647$	−11150.9	−0.677571	−0.338786	+	0.940864i	$0.610016\pi$
$648$	− 968.378i	− 0.0587060i
$649$	26380.4	1.59556
$650$	0	0
$651$	2203.38	0.132653
$652$	− 2348.93i	− 0.141091i
$653$	29137.5	1.74615	0.873077	−	0.487583i	$-0.162121\pi$
$653$	29137.5	1.74615	0.873077	+	0.487583i	$0.162121\pi$
$654$	−117.110	−0.00700211
$655$	5831.58i	0.347876i
$656$	− 5790.36i	− 0.344627i
$657$	14741.6i	0.875377i
$658$	− 719.595i	− 0.0426333i
$659$	−5300.12	−0.313298	−0.156649	−	0.987654i	$-0.550069\pi$
$659$	−5300.12	−0.313298	−0.156649	+	0.987654i	$0.550069\pi$
$660$	−9763.54	−0.575826
$661$	27412.5i	1.61305i	0.591203	+	0.806523i	$0.298653\pi$
$661$	27412.5i	1.61305i	−0.591203	+	0.806523i	$0.701347\pi$
$662$	1047.56	0.0615025
$663$	0	0
$664$	−200.832	−0.0117377
$665$	21864.6i	1.27499i
$666$	808.981	0.0470681
$667$	23698.6	1.37573
$668$	23584.0i	1.36601i
$669$	4925.67i	0.284660i
$670$	1763.87i	0.101708i
$671$	− 1846.10i	− 0.106211i
$672$	−5150.90	−0.295685
$673$	21282.1	1.21896	0.609482	−	0.792800i	$-0.291377\pi$
$673$	21282.1	1.21896	0.609482	+	0.792800i	$0.291377\pi$
$674$	2542.91i	0.145325i
$675$	−10119.9	−0.577061
$676$	0	0
$677$	−13544.2	−0.768904	−0.384452	−	0.923145i	$-0.625610\pi$
$677$	−13544.2	−0.768904	−0.384452	+	0.923145i	$0.625610\pi$
$678$	− 2979.98i	− 0.168798i
$679$	4529.31	0.255992
$680$	4023.23	0.226888
$681$	15051.9i	0.846974i
$682$	558.322i	0.0313479i
$683$	− 10350.1i	− 0.579849i	−0.957050	−	0.289924i	$-0.906370\pi$
$683$	− 10350.1i	− 0.579849i	0.957050	−	0.289924i	$-0.0936302\pi$
$684$	16307.0i	0.911567i
$685$	−19124.0	−1.06670
$686$	−2317.97	−0.129009
$687$	− 12658.7i	− 0.702999i
$688$	4889.60	0.270951
$689$	0	0
$690$	−969.492	−0.0534898
$691$	− 25714.8i	− 1.41569i	−0.706370	−	0.707843i	$-0.749669\pi$
$691$	− 25714.8i	− 1.41569i	0.706370	−	0.707843i	$-0.250331\pi$
$692$	−3577.45	−0.196523
$693$	12512.0	0.685847
$694$	29.5693i	0.00161734i
$695$	− 2153.49i	− 0.117535i
$696$	5777.81i	0.314666i
$697$	8298.43i	0.450969i
$698$	−1436.44	−0.0778940
$699$	2058.42	0.111383
$700$	10495.8i	0.566720i
$701$	−7431.30	−0.400394	−0.200197	−	0.979756i	$-0.564158\pi$
$701$	−7431.30	−0.400394	−0.200197	+	0.979756i	$0.564158\pi$
$702$	0	0
$703$	22077.9	1.18447
$704$	20826.3i	1.11494i
$705$	−2562.32	−0.136883
$706$	−3913.46	−0.208619
$707$	− 11142.8i	− 0.592744i
$708$	− 16308.0i	− 0.865668i
$709$	19986.8i	1.05870i	0.848403	+	0.529351i	$0.177565\pi$
$709$	19986.8i	1.05870i	−0.848403	+	0.529351i	$0.822435\pi$
$710$	− 1522.26i	− 0.0804641i
$711$	4482.23	0.236423
$712$	−2787.03	−0.146697
$713$	− 2858.80i	− 0.150158i
$714$	2381.24	0.124812
$715$	0	0
$716$	−10705.9	−0.558798
$717$	834.037i	0.0434417i
$718$	−2680.00	−0.139299
$719$	−36101.9	−1.87256	−0.936281	−	0.351251i	$-0.885756\pi$
$719$	−36101.9	−1.87256	−0.936281	+	0.351251i	$0.885756\pi$
$720$	− 6343.48i	− 0.328344i
$721$	− 18949.5i	− 0.978800i
$722$	− 5954.59i	− 0.306935i
$723$	− 11148.4i	− 0.573465i
$724$	15900.5	0.816213
$725$	17716.4	0.907545
$726$	− 1082.70i	− 0.0553483i
$727$	1751.90	0.0893735	0.0446868	−	0.999001i	$-0.485771\pi$
$727$	1751.90	0.0893735	0.0446868	+	0.999001i	$0.485771\pi$
$728$	0	0
$729$	16601.6	0.843451
$730$	3096.18i	0.156979i
$731$	−7007.52	−0.354559
$732$	−1141.24	−0.0576247
$733$	− 20031.3i	− 1.00938i	−0.863302	−	0.504688i	$-0.831607\pi$
$733$	− 20031.3i	− 1.00938i	0.863302	−	0.504688i	$-0.168393\pi$
$734$	3465.51i	0.174270i
$735$	− 1058.56i	− 0.0531231i
$736$	6683.10i	0.334704i
$737$	27546.8	1.37680
$738$	−522.731	−0.0260732
$739$	− 18632.9i	− 0.927502i	−0.885966	−	0.463751i	$-0.846503\pi$
$739$	− 18632.9i	− 0.927502i	0.885966	−	0.463751i	$-0.153497\pi$
$740$	−8761.62	−0.435248
$741$	0	0
$742$	−3763.17	−0.186187
$743$	4907.69i	0.242323i	0.992633	+	0.121161i	$0.0386619\pi$
$743$	4907.69i	0.242323i	−0.992633	+	0.121161i	$0.961338\pi$
$744$	696.988	0.0343452
$745$	17706.2	0.870747
$746$	− 2127.88i	− 0.104433i
$747$	− 453.813i	− 0.0222278i
$748$	− 31114.3i	− 1.52093i
$749$	− 6706.84i	− 0.327186i
$750$	−2048.70	−0.0997440
$751$	−31156.9	−1.51389	−0.756945	−	0.653479i	$-0.773309\pi$
$751$	−31156.9	−1.51389	−0.756945	+	0.653479i	$0.773309\pi$
$752$	5697.64i	0.276292i
$753$	8827.93	0.427235
$754$	0	0
$755$	1242.21	0.0598790
$756$	− 22683.5i	− 1.09126i
$757$	−11047.0	−0.530396	−0.265198	−	0.964194i	$-0.585437\pi$
$757$	−11047.0	−0.530396	−0.265198	+	0.964194i	$0.585437\pi$
$758$	−2591.20	−0.124165
$759$	15140.9i	0.724082i
$760$	6916.35i	0.330109i
$761$	− 27606.1i	− 1.31501i	−0.753452	−	0.657503i	$-0.771613\pi$
$761$	− 27606.1i	− 1.31501i	0.753452	−	0.657503i	$-0.228387\pi$
$762$	1657.00i	0.0787753i
$763$	−1625.40	−0.0771213
$764$	16959.6	0.803112
$765$	9091.14i	0.429661i
$766$	−492.456	−0.0232287
$767$	0	0
$768$	12052.1	0.566264
$769$	− 3694.72i	− 0.173257i	−0.996241	−	0.0866287i	$-0.972391\pi$
$769$	− 3694.72i	− 0.173257i	0.996241	−	0.0866287i	$-0.0276094\pi$
$770$	2627.91	0.122991
$771$	−11818.0	−0.552031
$772$	− 9479.18i	− 0.441921i
$773$	19808.0i	0.921662i	0.887488	+	0.460831i	$0.152449\pi$
$773$	19808.0i	0.921662i	−0.887488	+	0.460831i	$0.847551\pi$
$774$	− 441.414i	− 0.0204991i
$775$	− 2137.16i	− 0.0990569i
$776$	1432.74	0.0662789
$777$	−10472.1	−0.483505
$778$	1089.76i	0.0502181i
$779$	−14265.9	−0.656133
$780$	0	0
$781$	−23773.6	−1.08923
$782$	− 3089.56i	− 0.141282i
$783$	−38288.7	−1.74754
$784$	−2353.84	−0.107227
$785$	− 23284.0i	− 1.05865i
$786$	1091.82i	0.0495469i
$787$	4116.79i	0.186464i	0.995644	+	0.0932322i	$0.0297199\pi$
$787$	4116.79i	0.186464i	−0.995644	+	0.0932322i	$0.970280\pi$
$788$	− 38664.3i	− 1.74792i
$789$	−16717.4	−0.754314
$790$	941.408	0.0423972
$791$	− 41359.8i	− 1.85915i
$792$	3957.89	0.177572
$793$	0	0
$794$	2671.86	0.119422
$795$	13399.8i	0.597789i
$796$	−7922.00	−0.352748
$797$	−25359.3	−1.12707	−0.563533	−	0.826094i	$-0.690558\pi$
$797$	−25359.3	−1.12707	−0.563533	+	0.826094i	$0.690558\pi$
$798$	4093.59i	0.181594i
$799$	− 8165.55i	− 0.361548i
$800$	4996.10i	0.220799i
$801$	− 6297.74i	− 0.277802i
$802$	4300.83	0.189361
$803$	48354.1	2.12500
$804$	− 17029.1i	− 0.746978i
$805$	−13455.8	−0.589137
$806$	0	0
$807$	−10246.1	−0.446939
$808$	− 3524.78i	− 0.153467i
$809$	−5558.73	−0.241576	−0.120788	−	0.992678i	$-0.538542\pi$
$809$	−5558.73	−0.241576	−0.120788	+	0.992678i	$0.538542\pi$
$810$	459.591	0.0199363
$811$	15021.4i	0.650399i	0.945646	+	0.325199i	$0.105431\pi$
$811$	15021.4i	0.650399i	−0.945646	+	0.325199i	$0.894569\pi$
$812$	39710.8i	1.71623i
$813$	− 25456.5i	− 1.09815i
$814$	− 2653.55i	− 0.114259i
$815$	2251.22	0.0967567
$816$	−18854.2	−0.808861
$817$	− 12046.6i	− 0.515862i
$818$	−1377.32	−0.0588717
$819$	0	0
$820$	5661.41	0.241104
$821$	− 15901.6i	− 0.675966i	−0.941152	−	0.337983i	$-0.890255\pi$
$821$	− 15901.6i	− 0.675966i	0.941152	−	0.337983i	$-0.109745\pi$
$822$	−3580.49	−0.151927
$823$	40279.3	1.70601	0.853006	−	0.521901i	$-0.174777\pi$
$823$	40279.3	1.70601	0.853006	+	0.521901i	$0.174777\pi$
$824$	− 5994.23i	− 0.253421i
$825$	11318.9i	0.477664i
$826$	4389.39i	0.184899i
$827$	5251.09i	0.220796i	0.993887	+	0.110398i	$0.0352125\pi$
$827$	5251.09i	0.220796i	−0.993887	+	0.110398i	$0.964787\pi$
$828$	−10035.6	−0.421208
$829$	−33964.4	−1.42296	−0.711479	−	0.702708i	$-0.751974\pi$
$829$	−33964.4	−1.42296	−0.711479	+	0.702708i	$0.751974\pi$
$830$	− 95.3148i	− 0.00398606i
$831$	6996.65	0.292071
$832$	0	0
$833$	3373.40	0.140314
$834$	− 403.188i	− 0.0167401i
$835$	−22603.0	−0.936776
$836$	53488.8	2.21286
$837$	4618.83i	0.190741i
$838$	− 1630.38i	− 0.0672081i
$839$	− 15485.0i	− 0.637188i	−0.947891	−	0.318594i	$-0.896789\pi$
$839$	− 15485.0i	− 0.637188i	0.947891	−	0.318594i	$-0.103211\pi$
$840$	− 3280.59i	− 0.134751i
$841$	42640.8	1.74836
$842$	2422.35	0.0991444
$843$	11878.5i	0.485311i
$844$	−38543.1	−1.57193
$845$	0	0
$846$	514.361	0.0209032
$847$	− 15027.1i	− 0.609606i
$848$	29796.2	1.20661
$849$	28581.6	1.15538
$850$	− 2309.67i	− 0.0932014i
$851$	13587.1i	0.547309i
$852$	14696.6i	0.590958i
$853$	20057.8i	0.805118i	0.915394	+	0.402559i	$0.131879\pi$
$853$	20057.8i	0.805118i	−0.915394	+	0.402559i	$0.868121\pi$
$854$	307.170	0.0123081
$855$	−15628.6	−0.625132
$856$	− 2121.55i	− 0.0847117i
$857$	8066.23	0.321514	0.160757	−	0.986994i	$-0.448606\pi$
$857$	8066.23	0.321514	0.160757	+	0.986994i	$0.448606\pi$
$858$	0	0
$859$	39719.0	1.57764	0.788821	−	0.614623i	$-0.210692\pi$
$859$	39719.0	1.57764	0.788821	+	0.614623i	$0.210692\pi$
$860$	4780.71i	0.189559i
$861$	6766.64	0.267835
$862$	4634.66	0.183129
$863$	24473.8i	0.965351i	0.875799	+	0.482676i	$0.160335\pi$
$863$	24473.8i	0.965351i	−0.875799	+	0.482676i	$0.839665\pi$
$864$	− 10797.6i	− 0.425163i
$865$	− 3428.63i	− 0.134771i
$866$	− 3404.03i	− 0.133572i
$867$	9286.62	0.363772
$868$	4790.38	0.187323
$869$	− 14702.3i	− 0.573924i
$870$	−2742.15	−0.106859
$871$	0	0
$872$	−514.159	−0.0199675
$873$	3237.51i	0.125513i
$874$	5311.28	0.205557
$875$	−28434.4	−1.09858
$876$	− 29891.9i	− 1.15291i
$877$	25326.6i	0.975162i	0.873078	+	0.487581i	$0.162121\pi$
$877$	25326.6i	0.975162i	−0.873078	+	0.487581i	$0.837879\pi$
$878$	468.284i	0.0179998i
$879$	− 20486.4i	− 0.786109i
$880$	−20807.4	−0.797064
$881$	−1327.44	−0.0507634	−0.0253817	−	0.999678i	$-0.508080\pi$
$881$	−1327.44	−0.0507634	−0.0253817	+	0.999678i	$0.508080\pi$
$882$	212.496i	0.00811235i
$883$	2112.05	0.0804941	0.0402470	−	0.999190i	$-0.487186\pi$
$883$	2112.05	0.0804941	0.0402470	+	0.999190i	$0.487186\pi$
$884$	0	0
$885$	15629.6	0.593655
$886$	881.214i	0.0334142i
$887$	−40935.4	−1.54958	−0.774790	−	0.632219i	$-0.782144\pi$
$887$	−40935.4	−1.54958	−0.774790	+	0.632219i	$0.782144\pi$
$888$	−3312.60	−0.125184
$889$	22997.9i	0.867632i
$890$	− 1322.72i	− 0.0498177i
$891$	− 7177.58i	− 0.269874i
$892$	10708.9i	0.401975i
$893$	14037.4	0.526030
$894$	3315.05	0.124018
$895$	− 10260.6i	− 0.383211i
$896$	−14881.0	−0.554845
$897$	0	0
$898$	5720.04	0.212562
$899$	− 8085.93i	− 0.299979i
$900$	−7502.32	−0.277864
$901$	−42702.3	−1.57894
$902$	1714.62i	0.0632934i
$903$	5714.01i	0.210576i
$904$	− 13083.2i	− 0.481351i
$905$	15239.1i	0.559740i
$906$	232.573	0.00852838
$907$	−153.968	−0.00563663	−0.00281831	−	0.999996i	$-0.500897\pi$
$907$	−153.968	−0.00563663	−0.00281831	+	0.999996i	$0.500897\pi$
$908$	32724.4i	1.19603i
$909$	7964.82	0.290623
$910$	0	0
$911$	−733.607	−0.0266800	−0.0133400	−	0.999911i	$-0.504246\pi$
$911$	−733.607	−0.0266800	−0.0133400	+	0.999911i	$0.504246\pi$
$912$	− 32412.4i	− 1.17684i
$913$	−1488.56	−0.0539586
$914$	3641.70	0.131791
$915$	− 1093.76i	− 0.0395177i
$916$	− 27521.4i	− 0.992722i
$917$	15153.6i	0.545710i
$918$	4991.67i	0.179466i
$919$	−51106.4	−1.83443	−0.917216	−	0.398390i	$-0.869569\pi$
$919$	−51106.4	−1.83443	−0.917216	+	0.398390i	$0.869569\pi$
$920$	−4256.44	−0.152533
$921$	15661.8i	0.560342i
$922$	4188.61	0.149615
$923$	0	0
$924$	−25371.0	−0.903294
$925$	10157.4i	0.361051i
$926$	−4154.73	−0.147444
$927$	13544.9	0.479907
$928$	18902.7i	0.668655i
$929$	− 30645.2i	− 1.08228i	−0.840933	−	0.541139i	$-0.817993\pi$
$929$	− 30645.2i	− 1.08228i	0.840933	−	0.541139i	$-0.182007\pi$
$930$	330.790i	0.0116635i
$931$	5799.22i	0.204148i
$932$	4475.23	0.157287
$933$	18895.7	0.663041
$934$	1029.17i	0.0360550i
$935$	29820.0	1.04302
$936$	0	0
$937$	−24422.2	−0.851482	−0.425741	−	0.904845i	$-0.639987\pi$
$937$	−24422.2	−0.851482	−0.425741	+	0.904845i	$0.639987\pi$
$938$	4583.48i	0.159548i
$939$	7825.24	0.271956
$940$	−5570.75	−0.193296
$941$	− 16475.9i	− 0.570774i	−0.958412	−	0.285387i	$-0.907878\pi$
$941$	− 16475.9i	− 0.570774i	0.958412	−	0.285387i	$-0.0921221\pi$
$942$	− 4359.35i	− 0.150781i
$943$	− 8779.46i	− 0.303180i
$944$	− 34754.5i	− 1.19827i
$945$	21740.0	0.748361
$946$	−1447.89	−0.0497622
$947$	10122.1i	0.347332i	0.984805	+	0.173666i	$0.0555613\pi$
$947$	10122.1i	0.347332i	−0.984805	+	0.173666i	$0.944439\pi$
$948$	−9088.76	−0.311381
$949$	0	0
$950$	3970.57	0.135602
$951$	17556.4i	0.598637i
$952$	10454.5	0.355917
$953$	−38097.2	−1.29495	−0.647475	−	0.762086i	$-0.724175\pi$
$953$	−38097.2	−1.29495	−0.647475	+	0.762086i	$0.724175\pi$
$954$	− 2689.89i	− 0.0912875i
$955$	16254.1i	0.550756i
$956$	1813.29i	0.0613451i
$957$	42824.9i	1.44653i
$958$	1258.24	0.0424342
$959$	−49694.5	−1.67332
$960$	12339.0i	0.414833i
$961$	28815.6	0.967258
$962$	0	0
$963$	4794.00	0.160420
$964$	− 24237.9i	− 0.809804i
$965$	9084.87	0.303059
$966$	−2519.27	−0.0839090
$967$	− 44515.5i	− 1.48037i	−0.672401	−	0.740187i	$-0.734737\pi$
$967$	− 44515.5i	− 1.48037i	0.672401	−	0.740187i	$-0.265263\pi$
$968$	− 4753.47i	− 0.157833i
$969$	46451.8i	1.53999i
$970$	679.978i	0.0225080i
$971$	−25444.3	−0.840934	−0.420467	−	0.907308i	$-0.638134\pi$
$971$	−25444.3	−0.840934	−0.420467	+	0.907308i	$0.638134\pi$
$972$	26899.3	0.887649
$973$	− 5595.94i	− 0.184376i
$974$	−583.765	−0.0192044
$975$	0	0
$976$	−2432.12	−0.0797646
$977$	39467.9i	1.29242i	0.763161	+	0.646208i	$0.223646\pi$
$977$	39467.9i	1.29242i	−0.763161	+	0.646208i	$0.776354\pi$
$978$	421.484	0.0137808
$979$	−20657.4	−0.674374
$980$	− 2301.42i	− 0.0750165i
$981$	− 1161.83i	− 0.0378127i
$982$	202.239i	0.00657200i
$983$	14970.4i	0.485740i	0.970059	+	0.242870i	$0.0780888\pi$
$983$	14970.4i	0.485740i	−0.970059	+	0.242870i	$0.921911\pi$
$984$	2140.47	0.0693452
$985$	37055.9	1.19868
$986$	− 8738.63i	− 0.282246i
$987$	−6658.28	−0.214727
$988$	0	0
$989$	7413.71	0.238364
$990$	1878.41i	0.0603028i
$991$	58422.4	1.87270	0.936352	−	0.351063i	$-0.114180\pi$
$991$	58422.4	1.87270	0.936352	+	0.351063i	$0.114180\pi$
$992$	2280.27	0.0729825
$993$	− 9692.91i	− 0.309763i
$994$	− 3955.66i	− 0.126223i
$995$	− 7592.46i	− 0.241907i
$996$	920.210i	0.0292751i
$997$	20131.6	0.639493	0.319746	−	0.947503i	$-0.396402\pi$
$997$	20131.6	0.639493	0.319746	+	0.947503i	$0.396402\pi$
$998$	−938.230	−0.0297587
$999$	− 21952.1i	− 0.695228i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.4.b.g.168.11		18
13.2	odd	12		169.4.c.l.22.4		18
13.3	even	3		169.4.e.h.147.8		36
13.4	even	6		169.4.e.h.23.8		36
13.5	odd	4		169.4.a.k.1.6	✓	9
13.6	odd	12		169.4.c.l.146.4		18
13.7	odd	12		169.4.c.k.146.6		18
13.8	odd	4		169.4.a.l.1.4	yes	9
13.9	even	3		169.4.e.h.23.11		36
13.10	even	6		169.4.e.h.147.11		36
13.11	odd	12		169.4.c.k.22.6		18
13.12	even	2	inner	169.4.b.g.168.8		18
39.5	even	4		1521.4.a.bh.1.4		9
39.8	even	4		1521.4.a.bg.1.6		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.4.a.k.1.6	✓	9	13.5	odd	4
169.4.a.l.1.4	yes	9	13.8	odd	4
169.4.b.g.168.8		18	13.12	even	2	inner
169.4.b.g.168.11		18	1.1	even	1	trivial
169.4.c.k.22.6		18	13.11	odd	12
169.4.c.k.146.6		18	13.7	odd	12
169.4.c.l.22.4		18	13.2	odd	12
169.4.c.l.146.4		18	13.6	odd	12
169.4.e.h.23.8		36	13.4	even	6
169.4.e.h.23.11		36	13.9	even	3
169.4.e.h.147.8		36	13.3	even	3
169.4.e.h.147.11		36	13.10	even	6
1521.4.a.bg.1.6		9	39.8	even	4
1521.4.a.bh.1.4		9	39.5	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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	169.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.390115i q^{2} +3.60967 q^{3} +7.84781 q^{4} +7.52136i q^{5} +1.40819i q^{6} +19.5446i q^{7} +6.18247i q^{8} -13.9703 q^{9} +O(q^{10})\)	\(q+0.390115i q^{2} +3.60967 q^{3} +7.84781 q^{4} +7.52136i q^{5} +1.40819i q^{6} +19.5446i q^{7} +6.18247i q^{8} -13.9703 q^{9} -2.93420 q^{10} +45.8243i q^{11} +28.3280 q^{12} -7.62463 q^{14} +27.1496i q^{15} +60.3706 q^{16} -86.5200 q^{17} -5.45003i q^{18} -148.737i q^{19} +59.0262i q^{20} +70.5494i q^{21} -17.8767 q^{22} +91.5351 q^{23} +22.3167i q^{24} +68.4292 q^{25} -147.889 q^{27} +153.382i q^{28} +258.901 q^{29} -10.5915 q^{30} -31.2317i q^{31} +73.0113i q^{32} +165.410i q^{33} -33.7528i q^{34} -147.002 q^{35} -109.636 q^{36} +148.436i q^{37} +58.0245 q^{38} -46.5006 q^{40} -95.9135i q^{41} -27.5224 q^{42} +80.9930 q^{43} +359.620i q^{44} -105.076i q^{45} +35.7093i q^{46} +94.3777i q^{47} +217.918 q^{48} -38.9898 q^{49} +26.6953i q^{50} -312.308 q^{51} +493.555 q^{53} -57.6938i q^{54} -344.661 q^{55} -120.834 q^{56} -536.891i q^{57} +101.001i q^{58} -575.686i q^{59} +213.065i q^{60} -40.2865 q^{61} +12.1840 q^{62} -273.043i q^{63} +454.482 q^{64} -64.5291 q^{66} -601.141i q^{67} -678.992 q^{68} +330.411 q^{69} -57.3476i q^{70} +518.800i q^{71} -86.3710i q^{72} -1055.21i q^{73} -57.9072 q^{74} +247.007 q^{75} -1167.26i q^{76} -895.615 q^{77} -320.840 q^{79} +454.069i q^{80} -156.633 q^{81} +37.4173 q^{82} +32.4841i q^{83} +553.658i q^{84} -650.748i q^{85} +31.5966i q^{86} +934.547 q^{87} -283.307 q^{88} +450.795i q^{89} +40.9916 q^{90} +718.350 q^{92} -112.736i q^{93} -36.8182 q^{94} +1118.70 q^{95} +263.546i q^{96} -231.743i q^{97} -15.2105i q^{98} -640.179i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 2 q^{3} - 74 q^{4} + 132 q^{9}+O(q^{10}) \) 18 * q + 2 * q^3 - 74 * q^4 + 132 * q^9	\( 18 q + 2 q^{3} - 74 q^{4} + 132 q^{9} + 294 q^{10} - 78 q^{12} - 294 q^{14} + 538 q^{16} + 110 q^{17} + 680 q^{22} + 408 q^{23} - 614 q^{25} - 1336 q^{27} + 560 q^{29} - 1042 q^{30} + 40 q^{35} + 1818 q^{36} + 1478 q^{38} + 26 q^{40} - 8 q^{42} + 1066 q^{43} - 264 q^{48} - 806 q^{49} - 940 q^{51} - 556 q^{53} - 500 q^{55} - 500 q^{56} - 272 q^{61} - 4070 q^{62} - 568 q^{64} + 6558 q^{66} - 3072 q^{68} + 4100 q^{69} - 3980 q^{74} - 4786 q^{75} + 1436 q^{77} + 824 q^{79} - 1670 q^{81} - 5514 q^{82} - 1572 q^{87} + 1272 q^{88} + 2560 q^{90} + 8020 q^{92} - 5062 q^{94} + 3228 q^{95}+O(q^{100}) \) 18 * q + 2 * q^3 - 74 * q^4 + 132 * q^9 + 294 * q^10 - 78 * q^12 - 294 * q^14 + 538 * q^16 + 110 * q^17 + 680 * q^22 + 408 * q^23 - 614 * q^25 - 1336 * q^27 + 560 * q^29 - 1042 * q^30 + 40 * q^35 + 1818 * q^36 + 1478 * q^38 + 26 * q^40 - 8 * q^42 + 1066 * q^43 - 264 * q^48 - 806 * q^49 - 940 * q^51 - 556 * q^53 - 500 * q^55 - 500 * q^56 - 272 * q^61 - 4070 * q^62 - 568 * q^64 + 6558 * q^66 - 3072 * q^68 + 4100 * q^69 - 3980 * q^74 - 4786 * q^75 + 1436 * q^77 + 824 * q^79 - 1670 * q^81 - 5514 * q^82 - 1572 * q^87 + 1272 * q^88 + 2560 * q^90 + 8020 * q^92 - 5062 * q^94 + 3228 * q^95

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