LMFDB - Embedded newform 169.4.b.g.168.5

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			168.5
Root			$-2.16135i$ of defining polynomial
Character	$\chi$	$=$	169.168
Dual form			169.4.b.g.168.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.16135i q^{2} +7.08883 q^{3} -1.99415 q^{4} -13.6039i q^{5} -22.4103i q^{6} -14.3315i q^{7} -18.9866i q^{8} +23.2516 q^{9} +O(q^{10})$	\(q-3.16135i q^{2} +7.08883 q^{3} -1.99415 q^{4} -13.6039i q^{5} -22.4103i q^{6} -14.3315i q^{7} -18.9866i q^{8} +23.2516 q^{9} -43.0068 q^{10} +67.7223i q^{11} -14.1362 q^{12} -45.3068 q^{14} -96.4360i q^{15} -75.9766 q^{16} -0.337795 q^{17} -73.5064i q^{18} +40.5229i q^{19} +27.1282i q^{20} -101.593i q^{21} +214.094 q^{22} +155.632 q^{23} -134.593i q^{24} -60.0670 q^{25} -26.5720 q^{27} +28.5791i q^{28} -33.7925 q^{29} -304.868 q^{30} -157.397i q^{31} +88.2957i q^{32} +480.072i q^{33} +1.06789i q^{34} -194.964 q^{35} -46.3670 q^{36} +58.6204i q^{37} +128.107 q^{38} -258.293 q^{40} +59.3488i q^{41} -321.173 q^{42} +208.311 q^{43} -135.048i q^{44} -316.313i q^{45} -492.006i q^{46} +221.212i q^{47} -538.585 q^{48} +137.609 q^{49} +189.893i q^{50} -2.39457 q^{51} -409.639 q^{53} +84.0035i q^{54} +921.289 q^{55} -272.106 q^{56} +287.260i q^{57} +106.830i q^{58} -173.587i q^{59} +192.308i q^{60} +560.796 q^{61} -497.586 q^{62} -333.229i q^{63} -328.679 q^{64} +1517.68 q^{66} -269.074i q^{67} +0.673613 q^{68} +1103.25 q^{69} +616.351i q^{70} +60.9754i q^{71} -441.469i q^{72} -282.066i q^{73} +185.320 q^{74} -425.805 q^{75} -80.8086i q^{76} +970.560 q^{77} +984.026 q^{79} +1033.58i q^{80} -816.157 q^{81} +187.622 q^{82} +1201.86i q^{83} +202.592i q^{84} +4.59534i q^{85} -658.543i q^{86} -239.550 q^{87} +1285.82 q^{88} +539.899i q^{89} -999.976 q^{90} -310.352 q^{92} -1115.76i q^{93} +699.328 q^{94} +551.271 q^{95} +625.913i q^{96} +1587.25i q^{97} -435.030i q^{98} +1574.65i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	− 3.16135i	− 1.11771i	−0.829267	−	0.558853i	$-0.811241\pi$
$2$	− 3.16135i	− 1.11771i	0.829267	−	0.558853i	$-0.188759\pi$
$3$	7.08883	1.36425	0.682123	−	0.731237i	$-0.261057\pi$
$3$	7.08883	1.36425	0.682123	+	0.731237i	$0.261057\pi$
$4$	−1.99415	−0.249268
$5$	− 13.6039i	− 1.21677i	−0.793641	−	0.608386i	$-0.791817\pi$
$5$	− 13.6039i	− 1.21677i	0.793641	−	0.608386i	$-0.208183\pi$
$6$	− 22.4103i	− 1.52483i
$7$	− 14.3315i	− 0.773827i	−0.922116	−	0.386913i	$-0.873541\pi$
$7$	− 14.3315i	− 0.773827i	0.922116	−	0.386913i	$-0.126459\pi$
$8$	− 18.9866i	− 0.839098i
$9$	23.2516	0.861169
$10$	−43.0068	−1.36000
$11$	67.7223i	1.85628i	0.372237	+	0.928138i	$0.378591\pi$
$11$	67.7223i	1.85628i	−0.372237	+	0.928138i	$0.621409\pi$
$12$	−14.1362	−0.340063
$13$	0	0
$13$	0	0
$14$	−45.3068	−0.864911
$15$	− 96.4360i	− 1.65998i
$16$	−75.9766	−1.18713
$17$	−0.337795	−0.00481925	−0.00240963	−	0.999997i	$-0.500767\pi$
$17$	−0.337795	−0.00481925	−0.00240963	+	0.999997i	$0.500767\pi$
$18$	− 73.5064i	− 0.962535i
$19$	40.5229i	0.489294i	0.969612	+	0.244647i	$0.0786721\pi$
$19$	40.5229i	0.489294i	−0.969612	+	0.244647i	$0.921328\pi$
$20$	27.1282i	0.303303i
$21$	− 101.593i	− 1.05569i
$22$	214.094	2.07477
$23$	155.632	1.41093	0.705466	−	0.708744i	$-0.250738\pi$
$23$	155.632	1.41093	0.705466	+	0.708744i	$0.250738\pi$
$24$	− 134.593i	− 1.14474i
$25$	−60.0670	−0.480536
$26$	0	0
$27$	−26.5720	−0.189399
$28$	28.5791i	0.192890i
$29$	−33.7925	−0.216383	−0.108192	−	0.994130i	$-0.534506\pi$
$29$	−33.7925	−0.216383	−0.108192	+	0.994130i	$0.534506\pi$
$30$	−304.868	−1.85537
$31$	− 157.397i	− 0.911911i	−0.890003	−	0.455956i	$-0.849298\pi$
$31$	− 157.397i	− 0.911911i	0.890003	−	0.455956i	$-0.150702\pi$
$32$	88.2957i	0.487769i
$33$	480.072i	2.53242i
$34$	1.06789i	0.00538651i
$35$	−194.964	−0.941571
$36$	−46.3670	−0.214662
$37$	58.6204i	0.260463i	0.991484	+	0.130231i	$0.0415720\pi$
$37$	58.6204i	0.260463i	−0.991484	+	0.130231i	$0.958428\pi$
$38$	128.107	0.546888
$39$	0	0
$40$	−258.293	−1.02099
$41$	59.3488i	0.226066i	0.993591	+	0.113033i	$0.0360566\pi$
$41$	59.3488i	0.226066i	−0.993591	+	0.113033i	$0.963943\pi$
$42$	−321.173	−1.17995
$43$	208.311	0.738769	0.369385	−	0.929277i	$-0.379569\pi$
$43$	208.311	0.738769	0.369385	+	0.929277i	$0.379569\pi$
$44$	− 135.048i	− 0.462711i
$45$	− 316.313i	− 1.04785i
$46$	− 492.006i	− 1.57701i
$47$	221.212i	0.686533i	0.939238	+	0.343266i	$0.111533\pi$
$47$	221.212i	0.686533i	−0.939238	+	0.343266i	$0.888467\pi$
$48$	−538.585	−1.61954
$49$	137.609	0.401192
$50$	189.893i	0.537098i
$51$	−2.39457	−0.00657465
$52$	0	0
$53$	−409.639	−1.06167	−0.530833	−	0.847477i	$-0.678121\pi$
$53$	−409.639	−1.06167	−0.530833	+	0.847477i	$0.678121\pi$
$54$	84.0035i	0.211693i
$55$	921.289	2.25867
$56$	−272.106	−0.649316
$57$	287.260i	0.667518i
$58$	106.830i	0.241853i
$59$	− 173.587i	− 0.383036i	−0.981489	−	0.191518i	$-0.938659\pi$
$59$	− 173.587i	− 0.383036i	0.981489	−	0.191518i	$-0.0613411\pi$
$60$	192.308i	0.413780i
$61$	560.796	1.17709	0.588546	−	0.808464i	$-0.299701\pi$
$61$	560.796	1.17709	0.588546	+	0.808464i	$0.299701\pi$
$62$	−497.586	−1.01925
$63$	− 333.229i	− 0.666396i
$64$	−328.679	−0.641951
$65$	0	0
$66$	1517.68	2.83050
$67$	− 269.074i	− 0.490635i	−0.969443	−	0.245318i	$-0.921108\pi$
$67$	− 269.074i	− 0.490635i	0.969443	−	0.245318i	$-0.0788923\pi$
$68$	0.673613	0.00120129
$69$	1103.25	1.92486
$70$	616.351i	1.05240i
$71$	60.9754i	0.101922i	0.998701	+	0.0509609i	$0.0162284\pi$
$71$	60.9754i	0.101922i	−0.998701	+	0.0509609i	$0.983772\pi$
$72$	− 441.469i	− 0.722605i
$73$	− 282.066i	− 0.452238i	−0.974100	−	0.226119i	$-0.927396\pi$
$73$	− 282.066i	− 0.452238i	0.974100	−	0.226119i	$-0.0726038\pi$
$74$	185.320	0.291121
$75$	−425.805	−0.655570
$76$	− 80.8086i	− 0.121966i
$77$	970.560	1.43644
$78$	0	0
$79$	984.026	1.40141	0.700706	−	0.713450i	$-0.252868\pi$
$79$	984.026	1.40141	0.700706	+	0.713450i	$0.252868\pi$
$80$	1033.58i	1.44447i
$81$	−816.157	−1.11956
$82$	187.622	0.252676
$83$	1201.86i	1.58942i	0.606991	+	0.794709i	$0.292376\pi$
$83$	1201.86i	1.58942i	−0.606991	+	0.794709i	$0.707624\pi$
$84$	202.592i	0.263150i
$85$	4.59534i	0.00586394i
$86$	− 658.543i	− 0.825727i
$87$	−239.550	−0.295200
$88$	1285.82	1.55760
$89$	539.899i	0.643024i	0.946906	+	0.321512i	$0.104191\pi$
$89$	539.899i	0.643024i	−0.946906	+	0.321512i	$0.895809\pi$
$90$	−999.976	−1.17119
$91$	0	0
$92$	−310.352	−0.351701
$93$	− 1115.76i	− 1.24407i
$94$	699.328	0.767342
$95$	551.271	0.595360
$96$	625.913i	0.665438i
$97$	1587.25i	1.66146i	0.556678	+	0.830728i	$0.312076\pi$
$97$	1587.25i	1.66146i	−0.556678	+	0.830728i	$0.687924\pi$
$98$	− 435.030i	− 0.448415i
$99$	1574.65i	1.59857i
$100$	119.782	0.119782
$101$	−1160.73	−1.14353	−0.571765	−	0.820417i	$-0.693741\pi$
$101$	−1160.73	−1.14353	−0.571765	+	0.820417i	$0.693741\pi$
$102$	7.57009i	0.00734853i
$103$	82.2962	0.0787270	0.0393635	−	0.999225i	$-0.487467\pi$
$103$	82.2962	0.0787270	0.0393635	+	0.999225i	$0.487467\pi$
$104$	0	0
$105$	−1382.07	−1.28454
$106$	1295.01i	1.18663i
$107$	1322.04	1.19446	0.597229	−	0.802071i	$-0.296269\pi$
$107$	1322.04	1.19446	0.597229	+	0.802071i	$0.296269\pi$
$108$	52.9885	0.0472113
$109$	− 2073.41i	− 1.82199i	−0.412420	−	0.910994i	$-0.635316\pi$
$109$	− 2073.41i	− 1.82199i	0.412420	−	0.910994i	$-0.364684\pi$
$110$	− 2912.52i	− 2.52453i
$111$	415.550i	0.355336i
$112$	1088.86i	0.918636i
$113$	−1602.82	−1.33434	−0.667172	−	0.744904i	$-0.732495\pi$
$113$	−1602.82	−1.33434	−0.667172	+	0.744904i	$0.732495\pi$
$114$	908.131	0.746090
$115$	− 2117.20i	− 1.71678i
$116$	67.3872	0.0539375
$117$	0	0
$118$	−548.771	−0.428122
$119$	4.84110i	0.00372927i
$120$	−1830.99	−1.39288
$121$	−3255.30	−2.44576
$122$	− 1772.87i	− 1.31564i
$123$	420.714i	0.308410i
$124$	313.872i	0.227311i
$125$	− 883.344i	− 0.632070i
$126$	−1053.45	−0.744835
$127$	−614.495	−0.429352	−0.214676	−	0.976685i	$-0.568869\pi$
$127$	−614.495	−0.429352	−0.214676	+	0.976685i	$0.568869\pi$
$128$	1745.43i	1.20528i
$129$	1476.68	1.00786
$130$	0	0
$131$	−330.171	−0.220208	−0.110104	−	0.993920i	$-0.535118\pi$
$131$	−330.171	−0.220208	−0.110104	+	0.993920i	$0.535118\pi$
$132$	− 957.333i	− 0.631251i
$133$	580.753	0.378629
$134$	−850.636	−0.548386
$135$	361.484i	0.230456i
$136$	6.41358i	0.00404383i
$137$	2302.54i	1.43591i	0.696091	+	0.717953i	$0.254921\pi$
$137$	2302.54i	1.43591i	−0.696091	+	0.717953i	$0.745079\pi$
$138$	− 3487.75i	− 2.15143i
$139$	1207.22	0.736656	0.368328	−	0.929696i	$-0.379930\pi$
$139$	1207.22	0.736656	0.368328	+	0.929696i	$0.379930\pi$
$140$	388.788	0.234704
$141$	1568.13i	0.936600i
$142$	192.765	0.113919
$143$	0	0
$144$	−1766.57	−1.02232
$145$	459.711i	0.263289i
$146$	−891.711	−0.505470
$147$	975.486	0.547325
$148$	− 116.898i	− 0.0649251i
$149$	− 338.938i	− 0.186355i	−0.995650	−	0.0931774i	$-0.970298\pi$
$149$	− 338.938i	− 0.186355i	0.995650	−	0.0931774i	$-0.0297024\pi$
$150$	1346.12i	0.732734i
$151$	1694.81i	0.913386i	0.889624	+	0.456693i	$0.150966\pi$
$151$	1694.81i	0.913386i	−0.889624	+	0.456693i	$0.849034\pi$
$152$	769.393	0.410566
$153$	−7.85426	−0.00415019
$154$	− 3068.28i	− 1.60551i
$155$	−2141.21	−1.10959
$156$	0	0
$157$	9.59250	0.00487621	0.00243811	−	0.999997i	$-0.499224\pi$
$157$	9.59250	0.00487621	0.00243811	+	0.999997i	$0.499224\pi$
$158$	− 3110.85i	− 1.56637i
$159$	−2903.86	−1.44837
$160$	1201.17	0.593504
$161$	− 2230.43i	− 1.09182i
$162$	2580.16i	1.25134i
$163$	122.100i	0.0586725i	0.999570	+	0.0293363i	$0.00933936\pi$
$163$	122.100i	0.0586725i	−0.999570	+	0.0293363i	$0.990661\pi$
$164$	− 118.350i	− 0.0563512i
$165$	6530.87	3.08138
$166$	3799.51	1.77650
$167$	2211.25i	1.02462i	0.858801	+	0.512309i	$0.171210\pi$
$167$	2211.25i	1.02462i	−0.858801	+	0.512309i	$0.828790\pi$
$168$	−1928.92	−0.885828
$169$	0	0
$170$	14.5275	0.00655416
$171$	942.221i	0.421365i
$172$	−415.402	−0.184152
$173$	213.874	0.0939914	0.0469957	−	0.998895i	$-0.485035\pi$
$173$	213.874	0.0939914	0.0469957	+	0.998895i	$0.485035\pi$
$174$	757.301i	0.329947i
$175$	860.848i	0.371852i
$176$	− 5145.30i	− 2.20365i
$177$	− 1230.53i	− 0.522556i
$178$	1706.81	0.718712
$179$	−2967.41	−1.23907	−0.619537	−	0.784967i	$-0.712680\pi$
$179$	−2967.41	−1.23907	−0.619537	+	0.784967i	$0.712680\pi$
$180$	630.774i	0.261195i
$181$	−2329.44	−0.956609	−0.478304	−	0.878194i	$-0.658748\pi$
$181$	−2329.44	−0.956609	−0.478304	+	0.878194i	$0.658748\pi$
$182$	0	0
$183$	3975.39	1.60584
$184$	− 2954.92i	− 1.18391i
$185$	797.467	0.316924
$186$	−3527.30	−1.39051
$187$	− 22.8762i	− 0.00894586i
$188$	− 441.128i	− 0.171131i
$189$	380.816i	0.146562i
$190$	− 1742.76i	− 0.665438i
$191$	−4479.76	−1.69709	−0.848544	−	0.529125i	$-0.822520\pi$
$191$	−4479.76	−1.69709	−0.848544	+	0.529125i	$0.822520\pi$
$192$	−2329.95	−0.875779
$193$	1302.63i	0.485830i	0.970048	+	0.242915i	$0.0781035\pi$
$193$	1302.63i	0.485830i	−0.970048	+	0.242915i	$0.921896\pi$
$194$	5017.87	1.85702
$195$	0	0
$196$	−274.412	−0.100004
$197$	740.954i	0.267974i	0.990983	+	0.133987i	$0.0427780\pi$
$197$	740.954i	0.267974i	−0.990983	+	0.133987i	$0.957222\pi$
$198$	4978.02	1.78673
$199$	4688.74	1.67023	0.835115	−	0.550075i	$-0.185401\pi$
$199$	4688.74	1.67023	0.835115	+	0.550075i	$0.185401\pi$
$200$	1140.47i	0.403217i
$201$	− 1907.42i	− 0.669348i
$202$	3669.46i	1.27813i
$203$	484.297i	0.167443i
$204$	4.77513	0.00163885
$205$	807.377	0.275071
$206$	− 260.167i	− 0.0879937i
$207$	3618.68	1.21505
$208$	0	0
$209$	−2744.30	−0.908265
$210$	4369.21i	1.43573i
$211$	−3205.11	−1.04573	−0.522865	−	0.852416i	$-0.675137\pi$
$211$	−3205.11	−1.04573	−0.522865	+	0.852416i	$0.675137\pi$
$212$	816.880	0.264640
$213$	432.244i	0.139046i
$214$	− 4179.45i	− 1.33505i
$215$	− 2833.84i	− 0.898914i
$216$	504.513i	0.158925i
$217$	−2255.72	−0.705661
$218$	−6554.78	−2.03645
$219$	− 1999.52i	− 0.616964i
$220$	−1837.18	−0.563014
$221$	0	0
$222$	1313.70	0.397161
$223$	− 1057.39i	− 0.317525i	−0.987317	−	0.158762i	$-0.949250\pi$
$223$	− 1057.39i	− 0.317525i	0.987317	−	0.158762i	$-0.0507504\pi$
$224$	1265.41	0.377449
$225$	−1396.65	−0.413823
$226$	5067.09i	1.49141i
$227$	− 5159.68i	− 1.50863i	−0.656510	−	0.754317i	$-0.727968\pi$
$227$	− 5159.68i	− 1.50863i	0.656510	−	0.754317i	$-0.272032\pi$
$228$	− 572.839i	− 0.166391i
$229$	1698.25i	0.490059i	0.969516	+	0.245030i	$0.0787977\pi$
$229$	1698.25i	0.490059i	−0.969516	+	0.245030i	$0.921202\pi$
$230$	−6693.22	−1.91886
$231$	6880.14	1.95965
$232$	641.606i	0.181567i
$233$	−3162.23	−0.889119	−0.444559	−	0.895749i	$-0.646640\pi$
$233$	−3162.23	−0.889119	−0.444559	+	0.895749i	$0.646640\pi$
$234$	0	0
$235$	3009.35	0.835354
$236$	346.159i	0.0954788i
$237$	6975.60	1.91187
$238$	15.3044	0.00416823
$239$	− 2350.04i	− 0.636030i	−0.948086	−	0.318015i	$-0.896984\pi$
$239$	− 2350.04i	− 0.636030i	0.948086	−	0.318015i	$-0.103016\pi$
$240$	7326.88i	1.97062i
$241$	− 5167.18i	− 1.38111i	−0.723280	−	0.690555i	$-0.757367\pi$
$241$	− 5167.18i	− 1.38111i	0.723280	−	0.690555i	$-0.242633\pi$
$242$	10291.2i	2.73364i
$243$	−5068.16	−1.33795
$244$	−1118.31	−0.293412
$245$	− 1872.02i	− 0.488159i
$246$	1330.02	0.344712
$247$	0	0
$248$	−2988.43	−0.765183
$249$	8519.81i	2.16836i
$250$	−2792.56	−0.706469
$251$	−2894.01	−0.727761	−0.363880	−	0.931446i	$-0.618548\pi$
$251$	−2894.01	−0.727761	−0.363880	+	0.931446i	$0.618548\pi$
$252$	664.508i	0.166111i
$253$	10539.7i	2.61908i
$254$	1942.64i	0.479889i
$255$	32.5756i	0.00799986i
$256$	2888.50	0.705201
$257$	6162.63	1.49577	0.747887	−	0.663826i	$-0.231068\pi$
$257$	6162.63	1.49577	0.747887	+	0.663826i	$0.231068\pi$
$258$	− 4668.30i	− 1.12650i
$259$	840.116	0.201553
$260$	0	0
$261$	−785.729	−0.186343
$262$	1043.79i	0.246127i
$263$	−199.427	−0.0467574	−0.0233787	−	0.999727i	$-0.507442\pi$
$263$	−199.427	−0.0467574	−0.0233787	+	0.999727i	$0.507442\pi$
$264$	9114.94	2.12495
$265$	5572.70i	1.29181i
$266$	− 1835.96i	− 0.423196i
$267$	3827.25i	0.877243i
$268$	536.572i	0.122300i
$269$	1108.00	0.251138	0.125569	−	0.992085i	$-0.459924\pi$
$269$	1108.00	0.251138	0.125569	+	0.992085i	$0.459924\pi$
$270$	1142.78	0.257582
$271$	406.054i	0.0910186i	0.998964	+	0.0455093i	$0.0144911\pi$
$271$	406.054i	0.0910186i	−0.998964	+	0.0455093i	$0.985509\pi$
$272$	25.6645	0.00572110
$273$	0	0
$274$	7279.13	1.60492
$275$	− 4067.87i	− 0.892007i
$276$	−2200.04	−0.479806
$277$	−288.810	−0.0626460	−0.0313230	−	0.999509i	$-0.509972\pi$
$277$	−288.810	−0.0626460	−0.0313230	+	0.999509i	$0.509972\pi$
$278$	− 3816.45i	− 0.823366i
$279$	− 3659.72i	− 0.785310i
$280$	3701.71i	0.790071i
$281$	− 5134.93i	− 1.09012i	−0.838396	−	0.545061i	$-0.816506\pi$
$281$	− 5134.93i	− 1.09012i	0.838396	−	0.545061i	$-0.183494\pi$
$282$	4957.42	1.04684
$283$	−5274.40	−1.10788	−0.553940	−	0.832556i	$-0.686876\pi$
$283$	−5274.40	−1.10788	−0.553940	+	0.832556i	$0.686876\pi$
$284$	− 121.594i	− 0.0254059i
$285$	3907.87	0.812218
$286$	0	0
$287$	850.556	0.174936
$288$	2053.01i	0.420052i
$289$	−4912.89	−0.999977
$290$	1453.31	0.294280
$291$	11251.8i	2.26664i
$292$	562.482i	0.112729i
$293$	− 1445.69i	− 0.288253i	−0.989559	−	0.144127i	$-0.953963\pi$
$293$	− 1445.69i	− 0.288253i	0.989559	−	0.144127i	$-0.0460372\pi$
$294$	− 3083.86i	− 0.611749i
$295$	−2361.47	−0.466068
$296$	1113.00	0.218554
$297$	− 1799.52i	− 0.351578i
$298$	−1071.50	−0.208290
$299$	0	0
$300$	849.117	0.163413
$301$	− 2985.40i	− 0.571679i
$302$	5357.88	1.02090
$303$	−8228.19	−1.56006
$304$	− 3078.79i	− 0.580858i
$305$	− 7629.04i	− 1.43225i
$306$	24.8301i	0.00463870i
$307$	− 9421.42i	− 1.75149i	−0.482770	−	0.875747i	$-0.660369\pi$
$307$	− 9421.42i	− 1.75149i	0.482770	−	0.875747i	$-0.339631\pi$
$308$	−1935.44	−0.358058
$309$	583.384	0.107403
$310$	6769.12i	1.24019i
$311$	−7885.87	−1.43784	−0.718918	−	0.695095i	$-0.755362\pi$
$311$	−7885.87	−1.43784	−0.718918	+	0.695095i	$0.755362\pi$
$312$	0	0
$313$	−550.423	−0.0993986	−0.0496993	−	0.998764i	$-0.515826\pi$
$313$	−550.423	−0.0993986	−0.0496993	+	0.998764i	$0.515826\pi$
$314$	− 30.3253i	− 0.00545017i
$315$	−4533.23	−0.810852
$316$	−1962.29	−0.349328
$317$	− 150.974i	− 0.0267494i	−0.999911	−	0.0133747i	$-0.995743\pi$
$317$	− 150.974i	− 0.0267494i	0.999911	−	0.0133747i	$-0.00425742\pi$
$318$	9180.14i	1.61886i
$319$	− 2288.51i	− 0.401667i
$320$	4471.32i	0.781108i
$321$	9371.76	1.62953
$322$	−7051.18	−1.22033
$323$	− 13.6884i	− 0.00235803i
$324$	1627.54	0.279070
$325$	0	0
$326$	386.002	0.0655787
$327$	− 14698.1i	− 2.48564i
$328$	1126.83	0.189692
$329$	3170.29	0.531257
$330$	− 20646.4i	− 3.44408i
$331$	− 7079.29i	− 1.17557i	−0.809018	−	0.587784i	$-0.800001\pi$
$331$	− 7079.29i	− 1.17557i	0.809018	−	0.587784i	$-0.199999\pi$
$332$	− 2396.69i	− 0.396191i
$333$	1363.02i	0.224303i
$334$	6990.52	1.14522
$335$	−3660.46	−0.596992
$336$	7718.72i	1.25325i
$337$	6633.34	1.07223	0.536114	−	0.844145i	$-0.319892\pi$
$337$	6633.34	1.07223	0.536114	+	0.844145i	$0.319892\pi$
$338$	0	0
$339$	−11362.1	−1.82037
$340$	− 9.16378i	− 0.00146169i
$341$	10659.2	1.69276
$342$	2978.69	0.470963
$343$	− 6887.83i	− 1.08428i
$344$	− 3955.11i	− 0.619900i
$345$	− 15008.5i	− 2.34212i
$346$	− 676.130i	− 0.105055i
$347$	6404.17	0.990760	0.495380	−	0.868676i	$-0.335029\pi$
$347$	6404.17	0.990760	0.495380	+	0.868676i	$0.335029\pi$
$348$	477.697	0.0735840
$349$	168.098i	0.0257825i	0.999917	+	0.0128913i	$0.00410353\pi$
$349$	168.098i	0.0257825i	−0.999917	+	0.0128913i	$0.995896\pi$
$350$	2721.45	0.415621
$351$	0	0
$352$	−5979.58	−0.905434
$353$	8451.92i	1.27436i	0.770713	+	0.637182i	$0.219900\pi$
$353$	8451.92i	1.27436i	−0.770713	+	0.637182i	$0.780100\pi$
$354$	−3890.14	−0.584065
$355$	829.505	0.124016
$356$	− 1076.64i	− 0.160285i
$357$	34.3178i	0.00508764i
$358$	9381.01i	1.38492i
$359$	2631.92i	0.386929i	0.981107	+	0.193464i	$0.0619724\pi$
$359$	2631.92i	0.386929i	−0.981107	+	0.193464i	$0.938028\pi$
$360$	−6005.71	−0.879246
$361$	5216.89	0.760591
$362$	7364.19i	1.06921i
$363$	−23076.3	−3.33662
$364$	0	0
$365$	−3837.21	−0.550271
$366$	− 12567.6i	− 1.79486i
$367$	−13269.9	−1.88742	−0.943708	−	0.330780i	$-0.892688\pi$
$367$	−13269.9	−1.88742	−0.943708	+	0.330780i	$0.892688\pi$
$368$	−11824.4	−1.67496
$369$	1379.95i	0.194681i
$370$	− 2521.07i	− 0.354228i
$371$	5870.73i	0.821545i
$372$	2224.98i	0.310108i
$373$	−11680.1	−1.62137	−0.810684	−	0.585483i	$-0.800905\pi$
$373$	−11680.1	−1.62137	−0.810684	+	0.585483i	$0.800905\pi$
$374$	−72.3198	−0.00999885
$375$	− 6261.88i	− 0.862299i
$376$	4200.06	0.576068
$377$	0	0
$378$	1203.89	0.163814
$379$	3378.23i	0.457857i	0.973443	+	0.228928i	$0.0735222\pi$
$379$	3378.23i	0.457857i	−0.973443	+	0.228928i	$0.926478\pi$
$380$	−1099.31	−0.148404
$381$	−4356.06	−0.585742
$382$	14162.1i	1.89685i
$383$	4380.92i	0.584476i	0.956346	+	0.292238i	$0.0944000\pi$
$383$	4380.92i	0.584476i	−0.956346	+	0.292238i	$0.905600\pi$
$384$	12373.1i	1.64430i
$385$	− 13203.4i	− 1.74782i
$386$	4118.06	0.543015
$387$	4843.55	0.636205
$388$	− 3165.22i	− 0.414148i
$389$	12484.0	1.62716	0.813578	−	0.581456i	$-0.197517\pi$
$389$	12484.0	1.62716	0.813578	+	0.581456i	$0.197517\pi$
$390$	0	0
$391$	−52.5716	−0.00679964
$392$	− 2612.73i	− 0.336639i
$393$	−2340.53	−0.300417
$394$	2342.42	0.299516
$395$	− 13386.6i	− 1.70520i
$396$	− 3140.08i	− 0.398472i
$397$	9450.48i	1.19473i	0.801971	+	0.597363i	$0.203785\pi$
$397$	9450.48i	1.19473i	−0.801971	+	0.597363i	$0.796215\pi$
$398$	− 14822.8i	− 1.86683i
$399$	4116.86	0.516544
$400$	4563.68	0.570460
$401$	2906.03i	0.361895i	0.983493	+	0.180948i	$0.0579165\pi$
$401$	2906.03i	0.361895i	−0.983493	+	0.180948i	$0.942084\pi$
$402$	−6030.02	−0.748134
$403$	0	0
$404$	2314.66	0.285046
$405$	11102.9i	1.36225i
$406$	1531.03	0.187152
$407$	−3969.90	−0.483491
$408$	45.4648i	0.00551678i
$409$	5795.93i	0.700710i	0.936617	+	0.350355i	$0.113939\pi$
$409$	5795.93i	0.700710i	−0.936617	+	0.350355i	$0.886061\pi$
$410$	− 2552.40i	− 0.307449i
$411$	16322.3i	1.95893i
$412$	−164.111	−0.0196242
$413$	−2487.76	−0.296404
$414$	− 11439.9i	− 1.35807i
$415$	16350.1	1.93396
$416$	0	0
$417$	8557.79	1.00498
$418$	8675.71i	1.01517i
$419$	−4142.47	−0.482990	−0.241495	−	0.970402i	$-0.577638\pi$
$419$	−4142.47	−0.482990	−0.241495	+	0.970402i	$0.577638\pi$
$420$	2756.05	0.320194
$421$	1426.92i	0.165187i	0.996583	+	0.0825935i	$0.0263203\pi$
$421$	1426.92i	0.165187i	−0.996583	+	0.0825935i	$0.973680\pi$
$422$	10132.5i	1.16882i
$423$	5143.52i	0.591221i
$424$	7777.66i	0.890841i
$425$	20.2903	0.00231582
$426$	1366.48	0.155413
$427$	− 8037.04i	− 0.910865i
$428$	−2636.35	−0.297740
$429$	0	0
$430$	−8958.78	−1.00472
$431$	− 2543.08i	− 0.284213i	−0.989851	−	0.142106i	$-0.954612\pi$
$431$	− 2543.08i	− 0.284213i	0.989851	−	0.142106i	$-0.0453875\pi$
$432$	2018.85	0.224842
$433$	3448.44	0.382728	0.191364	−	0.981519i	$-0.438709\pi$
$433$	3448.44	0.382728	0.191364	+	0.981519i	$0.438709\pi$
$434$	7131.14i	0.788723i
$435$	3258.82i	0.359191i
$436$	4134.68i	0.454164i
$437$	6306.65i	0.690361i
$438$	−6321.19	−0.689585
$439$	12642.6	1.37448	0.687240	−	0.726431i	$-0.258822\pi$
$439$	12642.6	1.37448	0.687240	+	0.726431i	$0.258822\pi$
$440$	− 17492.2i	− 1.89524i
$441$	3199.62	0.345494
$442$	0	0
$443$	8486.59	0.910181	0.455091	−	0.890445i	$-0.349607\pi$
$443$	8486.59	0.910181	0.455091	+	0.890445i	$0.349607\pi$
$444$	− 828.667i	− 0.0885739i
$445$	7344.74	0.782414
$446$	−3342.78	−0.354900
$447$	− 2402.67i	− 0.254234i
$448$	4710.45i	0.496759i
$449$	− 11369.5i	− 1.19502i	−0.801863	−	0.597508i	$-0.796158\pi$
$449$	− 11369.5i	− 1.19502i	0.801863	−	0.597508i	$-0.203842\pi$
$450$	4415.31i	0.462532i
$451$	−4019.23	−0.419642
$452$	3196.26	0.332610
$453$	12014.2i	1.24608i
$454$	−16311.6	−1.68621
$455$	0	0
$456$	5454.10	0.560113
$457$	11289.0i	1.15553i	0.816202	+	0.577766i	$0.196075\pi$
$457$	11289.0i	1.15553i	−0.816202	+	0.577766i	$0.803925\pi$
$458$	5368.77	0.547742
$459$	8.97589	0.000912764 0
$460$	4222.01i	0.427940i
$461$	3834.12i	0.387359i	0.981065	+	0.193680i	$0.0620422\pi$
$461$	3834.12i	0.387359i	−0.981065	+	0.193680i	$0.937958\pi$
$462$	− 21750.5i	− 2.19032i
$463$	− 1294.44i	− 0.129930i	−0.997888	−	0.0649651i	$-0.979306\pi$
$463$	− 1294.44i	− 0.129930i	0.997888	−	0.0649651i	$-0.0206936\pi$
$464$	2567.44	0.256876
$465$	−15178.7	−1.51375
$466$	9996.93i	0.993774i
$467$	−13861.5	−1.37352	−0.686760	−	0.726884i	$-0.740968\pi$
$467$	−13861.5	−1.37352	−0.686760	+	0.726884i	$0.740968\pi$
$468$	0	0
$469$	−3856.22	−0.379667
$470$	− 9513.61i	− 0.933681i
$471$	67.9997	0.00665235
$472$	−3295.84	−0.321405
$473$	14107.3i	1.37136i
$474$	− 22052.3i	− 2.13691i
$475$	− 2434.09i	− 0.235124i
$476$	− 9.65386i	0 0.000929588i
$477$	−9524.75	−0.914274
$478$	−7429.29	−0.710895
$479$	9627.66i	0.918369i	0.888341	+	0.459185i	$0.151858\pi$
$479$	9627.66i	0.918369i	−0.888341	+	0.459185i	$0.848142\pi$
$480$	8514.88	0.809686
$481$	0	0
$482$	−16335.3	−1.54368
$483$	− 15811.2i	− 1.48951i
$484$	6491.55	0.609650
$485$	21592.9	2.02162
$486$	16022.2i	1.49544i
$487$	− 11952.8i	− 1.11219i	−0.831120	−	0.556094i	$-0.812300\pi$
$487$	− 11952.8i	− 1.11219i	0.831120	−	0.556094i	$-0.187700\pi$
$488$	− 10647.6i	− 0.987695i
$489$	865.548i	0.0800438i
$490$	−5918.12	−0.545619
$491$	−5410.12	−0.497261	−0.248631	−	0.968598i	$-0.579980\pi$
$491$	−5410.12	−0.497261	−0.248631	+	0.968598i	$0.579980\pi$
$492$	− 838.965i	− 0.0768769i
$493$	11.4149	0.00104281
$494$	0	0
$495$	21421.4	1.94509
$496$	11958.4i	1.08256i
$497$	873.867	0.0788698
$498$	26934.1	2.42359
$499$	− 14472.9i	− 1.29838i	−0.760624	−	0.649192i	$-0.775107\pi$
$499$	− 14472.9i	− 1.29838i	0.760624	−	0.649192i	$-0.224893\pi$
$500$	1761.52i	0.157555i
$501$	15675.2i	1.39783i
$502$	9148.97i	0.813423i
$503$	−601.940	−0.0533582	−0.0266791	−	0.999644i	$-0.508493\pi$
$503$	−601.940	−0.0533582	−0.0266791	+	0.999644i	$0.508493\pi$
$504$	−6326.90	−0.559171
$505$	15790.4i	1.39142i
$506$	33319.8	2.92736
$507$	0	0
$508$	1225.39	0.107024
$509$	17420.8i	1.51702i	0.651659	+	0.758512i	$0.274073\pi$
$509$	17420.8i	1.51702i	−0.651659	+	0.758512i	$0.725927\pi$
$510$	102.983	0.00894149
$511$	−4042.43	−0.349954
$512$	4831.90i	0.417074i
$513$	− 1076.78i	− 0.0926721i
$514$	− 19482.2i	− 1.67184i
$515$	− 1119.55i	− 0.0957929i
$516$	−2944.71	−0.251228
$517$	−14981.0	−1.27439
$518$	− 2655.90i	− 0.225277i
$519$	1516.11	0.128227
$520$	0	0
$521$	12881.5	1.08320	0.541601	−	0.840636i	$-0.317818\pi$
$521$	12881.5	1.08320	0.541601	+	0.840636i	$0.317818\pi$
$522$	2483.97i	0.208276i
$523$	16534.0	1.38238	0.691188	−	0.722675i	$-0.257088\pi$
$523$	16534.0	1.38238	0.691188	+	0.722675i	$0.257088\pi$
$524$	658.410	0.0548907
$525$	6102.41i	0.507297i
$526$	630.459i	0.0522611i
$527$	53.1678i	0.00439473i
$528$	− 36474.2i	− 3.00632i
$529$	12054.2	0.990730
$530$	17617.3	1.44386
$531$	− 4036.18i	− 0.329859i
$532$	−1158.11	−0.0943802
$533$	0	0
$534$	12099.3	0.980501
$535$	− 17985.0i	− 1.45338i
$536$	−5108.80	−0.411691
$537$	−21035.4	−1.69040
$538$	− 3502.79i	− 0.280699i
$539$	9319.18i	0.744723i
$540$	− 720.852i	− 0.0574454i
$541$	19026.9i	1.51207i	0.654531	+	0.756035i	$0.272866\pi$
$541$	19026.9i	1.51207i	−0.654531	+	0.756035i	$0.727134\pi$
$542$	1283.68	0.101732
$543$	−16513.0	−1.30505
$544$	− 29.8258i	− 0.00235068i
$545$	−28206.5	−2.21694
$546$	0	0
$547$	8153.61	0.637336	0.318668	−	0.947866i	$-0.396764\pi$
$547$	8153.61	0.637336	0.318668	+	0.947866i	$0.396764\pi$
$548$	− 4591.60i	− 0.357926i
$549$	13039.4	1.01368
$550$	−12860.0	−0.997002
$551$	− 1369.37i	− 0.105875i
$552$	− 20946.9i	− 1.61515i
$553$	− 14102.5i	− 1.08445i
$554$	913.031i	0.0700198i
$555$	5653.11	0.432363
$556$	−2407.38	−0.183625
$557$	− 1308.06i	− 0.0995051i	−0.998762	−	0.0497525i	$-0.984157\pi$
$557$	− 1308.06i	− 0.0995051i	0.998762	−	0.0497525i	$-0.0158433\pi$
$558$	−11569.7	−0.877746
$559$	0	0
$560$	14812.7	1.11777
$561$	− 162.166i	− 0.0122044i
$562$	−16233.3	−1.21844
$563$	20736.1	1.55226	0.776129	−	0.630575i	$-0.217181\pi$
$563$	20736.1	1.55226	0.776129	+	0.630575i	$0.217181\pi$
$564$	− 3127.09i	− 0.233465i
$565$	21804.7i	1.62359i
$566$	16674.2i	1.23829i
$567$	11696.7i	0.866343i
$568$	1157.72	0.0855224
$569$	14982.0	1.10383	0.551913	−	0.833902i	$-0.313898\pi$
$569$	14982.0	1.10383	0.551913	+	0.833902i	$0.313898\pi$
$570$	− 12354.1i	− 0.907822i
$571$	5668.79	0.415467	0.207734	−	0.978185i	$-0.433391\pi$
$571$	5668.79	0.415467	0.207734	+	0.978185i	$0.433391\pi$
$572$	0	0
$573$	−31756.2	−2.31525
$574$	− 2688.91i	− 0.195527i
$575$	−9348.32	−0.678004
$576$	−7642.30	−0.552828
$577$	6872.94i	0.495882i	0.968775	+	0.247941i	$0.0797540\pi$
$577$	6872.94i	0.495882i	−0.968775	+	0.247941i	$0.920246\pi$
$578$	15531.4i	1.11768i
$579$	9234.10i	0.662791i
$580$	− 916.731i	− 0.0656297i
$581$	17224.5	1.22993
$582$	35570.9	2.53344
$583$	− 27741.7i	− 1.97074i
$584$	−5355.49	−0.379472
$585$	0	0
$586$	−4570.34	−0.322182
$587$	18593.4i	1.30738i	0.756762	+	0.653691i	$0.226780\pi$
$587$	18593.4i	1.30738i	−0.756762	+	0.653691i	$0.773220\pi$
$588$	−1945.26	−0.136431
$589$	6378.17	0.446193
$590$	7465.44i	0.520928i
$591$	5252.50i	0.365582i
$592$	− 4453.77i	− 0.309204i
$593$	− 15612.6i	− 1.08117i	−0.841291	−	0.540583i	$-0.818204\pi$
$593$	− 15612.6i	− 1.08117i	0.841291	−	0.540583i	$-0.181796\pi$
$594$	−5688.91	−0.392961
$595$	65.8580	0.00453767
$596$	675.892i	0.0464523i
$597$	33237.7	2.27861
$598$	0	0
$599$	−22979.8	−1.56749	−0.783747	−	0.621081i	$-0.786694\pi$
$599$	−22979.8	−1.56749	−0.783747	+	0.621081i	$0.786694\pi$
$600$	8084.60i	0.550087i
$601$	2781.77	0.188803	0.0944015	−	0.995534i	$-0.469906\pi$
$601$	2781.77	0.188803	0.0944015	+	0.995534i	$0.469906\pi$
$602$	−9437.89	−0.638970
$603$	− 6256.38i	− 0.422520i
$604$	− 3379.69i	− 0.227678i
$605$	44284.9i	2.97593i
$606$	26012.2i	1.74369i
$607$	−4078.57	−0.272725	−0.136363	−	0.990659i	$-0.543541\pi$
$607$	−4078.57	−0.272725	−0.136363	+	0.990659i	$0.543541\pi$
$608$	−3578.00	−0.238663
$609$	3433.10i	0.228434i
$610$	−24118.1	−1.60084
$611$	0	0
$612$	15.6625	0.00103451
$613$	− 26290.1i	− 1.73221i	−0.499858	−	0.866107i	$-0.666615\pi$
$613$	− 26290.1i	− 1.73221i	0.499858	−	0.866107i	$-0.333385\pi$
$614$	−29784.4	−1.95766
$615$	5723.36	0.375265
$616$	− 18427.6i	− 1.20531i
$617$	4851.69i	0.316567i	0.987394	+	0.158283i	$0.0505960\pi$
$617$	4851.69i	0.316567i	−0.987394	+	0.158283i	$0.949404\pi$
$618$	− 1844.28i	− 0.120045i
$619$	− 4957.05i	− 0.321875i	−0.986965	−	0.160937i	$-0.948548\pi$
$619$	− 4957.05i	− 0.321875i	0.986965	−	0.160937i	$-0.0514517\pi$
$620$	4269.89	0.276585
$621$	−4135.45	−0.267230
$622$	24930.0i	1.60708i
$623$	7737.54	0.497589
$624$	0	0
$625$	−19525.3	−1.24962
$626$	1740.08i	0.111098i
$627$	−19453.9	−1.23910
$628$	−19.1289	−0.00121548
$629$	− 19.8017i	− 0.00125524i
$630$	14331.1i	0.906295i
$631$	− 2725.11i	− 0.171925i	−0.996298	−	0.0859627i	$-0.972603\pi$
$631$	− 2725.11i	− 0.171925i	0.996298	−	0.0859627i	$-0.0273966\pi$
$632$	− 18683.3i	− 1.17592i
$633$	−22720.5	−1.42663
$634$	−477.283	−0.0298980
$635$	8359.55i	0.522423i
$636$	5790.73	0.361034
$637$	0	0
$638$	−7234.77	−0.448946
$639$	1417.77i	0.0877719i
$640$	23744.8	1.46655
$641$	−21396.3	−1.31841	−0.659207	−	0.751961i	$-0.729108\pi$
$641$	−21396.3	−1.31841	−0.659207	+	0.751961i	$0.729108\pi$
$642$	− 29627.4i	− 1.82134i
$643$	− 21833.8i	− 1.33910i	−0.742768	−	0.669549i	$-0.766487\pi$
$643$	− 21833.8i	− 1.33910i	0.742768	−	0.669549i	$-0.233513\pi$
$644$	4447.80i	0.272155i
$645$	− 20088.6i	− 1.22634i
$646$	−43.2740	−0.00263559
$647$	4869.06	0.295861	0.147931	−	0.988998i	$-0.452739\pi$
$647$	4869.06	0.295861	0.147931	+	0.988998i	$0.452739\pi$
$648$	15496.1i	0.939418i
$649$	11755.7	0.711021
$650$	0	0
$651$	−15990.5	−0.962696
$652$	− 243.486i	− 0.0146252i
$653$	−16847.0	−1.00961	−0.504803	−	0.863235i	$-0.668435\pi$
$653$	−16847.0	−1.00961	−0.504803	+	0.863235i	$0.668435\pi$
$654$	−46465.7	−2.77822
$655$	4491.63i	0.267943i
$656$	− 4509.12i	− 0.268371i
$657$	− 6558.49i	− 0.389454i
$658$	− 10022.4i	− 0.593790i
$659$	−24741.0	−1.46248	−0.731240	−	0.682121i	$-0.761058\pi$
$659$	−24741.0	−1.46248	−0.731240	+	0.682121i	$0.761058\pi$
$660$	−13023.5	−0.768089
$661$	− 16522.0i	− 0.972210i	−0.873900	−	0.486105i	$-0.838417\pi$
$661$	− 16522.0i	− 0.972210i	0.873900	−	0.486105i	$-0.161583\pi$
$662$	−22380.1	−1.31394
$663$	0	0
$664$	22819.3	1.33368
$665$	− 7900.53i	− 0.460706i
$666$	4308.97	0.250705
$667$	−5259.19	−0.305302
$668$	− 4409.55i	− 0.255405i
$669$	− 7495.66i	− 0.433182i
$670$	11572.0i	0.667262i
$671$	37978.4i	2.18501i
$672$	8970.26	0.514934
$673$	20721.5	1.18686	0.593429	−	0.804886i	$-0.297774\pi$
$673$	20721.5	1.18686	0.593429	+	0.804886i	$0.297774\pi$
$674$	− 20970.3i	− 1.19844i
$675$	1596.10	0.0910133
$676$	0	0
$677$	32407.3	1.83975	0.919877	−	0.392207i	$-0.128288\pi$
$677$	32407.3	1.83975	0.919877	+	0.392207i	$0.128288\pi$
$678$	35919.7i	2.03465i
$679$	22747.7	1.28568
$680$	87.2500	0.00492042
$681$	− 36576.1i	− 2.05815i
$682$	− 33697.6i	− 1.89201i
$683$	10123.8i	0.567168i	0.958947	+	0.283584i	$0.0915234\pi$
$683$	10123.8i	0.567168i	−0.958947	+	0.283584i	$0.908477\pi$
$684$	− 1878.93i	− 0.105033i
$685$	31323.6	1.74717
$686$	−21774.9	−1.21191
$687$	12038.6i	0.668562i
$688$	−15826.7	−0.877018
$689$	0	0
$690$	−47447.1	−2.61780
$691$	− 29704.5i	− 1.63533i	−0.575695	−	0.817665i	$-0.695268\pi$
$691$	− 29704.5i	− 1.63533i	0.575695	−	0.817665i	$-0.304732\pi$
$692$	−426.495	−0.0234291
$693$	22567.0	1.23701
$694$	− 20245.8i	− 1.10738i
$695$	− 16423.0i	− 0.896343i
$696$	4548.24i	0.247702i
$697$	− 20.0477i	− 0.00108947i
$698$	531.418	0.0288173
$699$	−22416.5	−1.21298
$700$	− 1716.66i	− 0.0926908i
$701$	20585.2	1.10912	0.554558	−	0.832145i	$-0.312887\pi$
$701$	20585.2	1.10912	0.554558	+	0.832145i	$0.312887\pi$
$702$	0	0
$703$	−2375.47	−0.127443
$704$	− 22258.9i	− 1.19164i
$705$	21332.8	1.13963
$706$	26719.5	1.42437
$707$	16634.9i	0.884894i
$708$	2453.86i	0.130257i
$709$	− 29660.2i	− 1.57110i	−0.618796	−	0.785552i	$-0.712379\pi$
$709$	− 29660.2i	− 1.57110i	0.618796	−	0.785552i	$-0.287621\pi$
$710$	− 2622.36i	− 0.138613i
$711$	22880.1	1.20685
$712$	10250.8	0.539560
$713$	− 24495.9i	− 1.28664i
$714$	108.491	0.00568649
$715$	0	0
$716$	5917.44	0.308862
$717$	− 16659.0i	− 0.867702i
$718$	8320.43	0.432473
$719$	−1279.68	−0.0663758	−0.0331879	−	0.999449i	$-0.510566\pi$
$719$	−1279.68	−0.0663758	−0.0331879	+	0.999449i	$0.510566\pi$
$720$	24032.4i	1.24393i
$721$	− 1179.43i	− 0.0609211i
$722$	− 16492.4i	− 0.850118i
$723$	− 36629.3i	− 1.88417i
$724$	4645.25	0.238452
$725$	2029.82	0.103980
$726$	72952.3i	3.72936i
$727$	−6202.77	−0.316435	−0.158217	−	0.987404i	$-0.550575\pi$
$727$	−6202.77	−0.316435	−0.158217	+	0.987404i	$0.550575\pi$
$728$	0	0
$729$	−13891.1	−0.705740
$730$	12130.8i	0.615042i
$731$	−70.3663	−0.00356032
$732$	−7927.51	−0.400286
$733$	− 35501.2i	− 1.78891i	−0.447162	−	0.894453i	$-0.647565\pi$
$733$	− 35501.2i	− 1.78891i	0.447162	−	0.894453i	$-0.352435\pi$
$734$	41950.7i	2.10958i
$735$	− 13270.5i	− 0.665970i
$736$	13741.6i	0.688209i
$737$	18222.3	0.910754
$738$	4362.52	0.217597
$739$	3893.52i	0.193810i	0.995294	+	0.0969050i	$0.0308943\pi$
$739$	3893.52i	0.193810i	−0.995294	+	0.0969050i	$0.969106\pi$
$740$	−1590.27	−0.0789991
$741$	0	0
$742$	18559.5	0.918247
$743$	− 2214.99i	− 0.109368i	−0.998504	−	0.0546839i	$-0.982585\pi$
$743$	− 2214.99i	− 0.109368i	0.998504	−	0.0546839i	$-0.0174151\pi$
$744$	−21184.5	−1.04390
$745$	−4610.89	−0.226751
$746$	36924.8i	1.81221i
$747$	27945.2i	1.36876i
$748$	45.6186i	0.00222992i
$749$	− 18946.9i	− 0.924303i
$750$	−19796.0	−0.963797
$751$	−2604.34	−0.126543	−0.0632714	−	0.997996i	$-0.520153\pi$
$751$	−2604.34	−0.126543	−0.0632714	+	0.997996i	$0.520153\pi$
$752$	− 16806.9i	− 0.815006i
$753$	−20515.1	−0.992845
$754$	0	0
$755$	23056.0	1.11138
$756$	− 759.403i	− 0.0365334i
$757$	−9685.72	−0.465037	−0.232519	−	0.972592i	$-0.574697\pi$
$757$	−9685.72	−0.465037	−0.232519	+	0.972592i	$0.574697\pi$
$758$	10679.8	0.511750
$759$	74714.4i	3.57307i
$760$	− 10466.8i	− 0.499565i
$761$	10105.6i	0.481378i	0.970602	+	0.240689i	$0.0773734\pi$
$761$	10105.6i	0.481378i	−0.970602	+	0.240689i	$0.922627\pi$
$762$	13771.0i	0.654687i
$763$	−29715.0	−1.40990
$764$	8933.29	0.423030
$765$	106.849i	0.00504984i
$766$	13849.6	0.653273
$767$	0	0
$768$	20476.1	0.962068
$769$	− 4573.20i	− 0.214452i	−0.994235	−	0.107226i	$-0.965803\pi$
$769$	− 4573.20i	− 0.214452i	0.994235	−	0.107226i	$-0.0341969\pi$
$770$	−41740.7	−1.95355
$771$	43685.8	2.04061
$772$	− 2597.63i	− 0.121102i
$773$	7446.13i	0.346467i	0.984881	+	0.173233i	$0.0554215\pi$
$773$	7446.13i	0.346467i	−0.984881	+	0.173233i	$0.944579\pi$
$774$	− 15312.2i	− 0.711091i
$775$	9454.34i	0.438206i
$776$	30136.6	1.39412
$777$	5955.44	0.274968
$778$	− 39466.3i	− 1.81868i
$779$	−2404.99	−0.110613
$780$	0	0
$781$	−4129.39	−0.189195
$782$	166.197i	0.00760000i
$783$	897.935	0.0409829
$784$	−10455.0	−0.476269
$785$	− 130.496i	− 0.00593324i
$786$	7399.23i	0.335779i
$787$	− 3267.11i	− 0.147980i	−0.997259	−	0.0739898i	$-0.976427\pi$
$787$	− 3267.11i	− 0.147980i	0.997259	−	0.0739898i	$-0.0235732\pi$
$788$	− 1477.57i	− 0.0667973i
$789$	−1413.71	−0.0637887
$790$	−42319.8	−1.90591
$791$	22970.8i	1.03255i
$792$	29897.3	1.34135
$793$	0	0
$794$	29876.3	1.33535
$795$	39504.0i	1.76234i
$796$	−9350.03	−0.416336
$797$	3165.75	0.140698	0.0703492	−	0.997522i	$-0.477589\pi$
$797$	3165.75	0.140698	0.0703492	+	0.997522i	$0.477589\pi$
$798$	− 13014.9i	− 0.577344i
$799$	− 74.7242i	− 0.00330857i
$800$	− 5303.66i	− 0.234391i
$801$	12553.5i	0.553752i
$802$	9186.97	0.404493
$803$	19102.2	0.839478
$804$	3803.67i	0.166847i
$805$	−30342.6	−1.32849
$806$	0	0
$807$	7854.45	0.342614
$808$	22038.3i	0.959534i
$809$	13589.7	0.590592	0.295296	−	0.955406i	$-0.404582\pi$
$809$	13589.7	0.590592	0.295296	+	0.955406i	$0.404582\pi$
$810$	35100.3	1.52259
$811$	24977.7i	1.08149i	0.841188	+	0.540744i	$0.181857\pi$
$811$	24977.7i	1.08149i	−0.841188	+	0.540744i	$0.818143\pi$
$812$	− 965.758i	− 0.0417383i
$813$	2878.45i	0.124172i
$814$	12550.3i	0.540401i
$815$	1661.04	0.0713911
$816$	181.931	0.00780499
$817$	8441.35i	0.361476i
$818$	18323.0	0.783188
$819$	0	0
$820$	−1610.03	−0.0685666
$821$	− 8317.38i	− 0.353567i	−0.984250	−	0.176784i	$-0.943431\pi$
$821$	− 8317.38i	− 0.353567i	0.984250	−	0.176784i	$-0.0565693\pi$
$822$	51600.6	2.18951
$823$	−14462.9	−0.612571	−0.306286	−	0.951940i	$-0.599086\pi$
$823$	−14462.9	−0.612571	−0.306286	+	0.951940i	$0.599086\pi$
$824$	− 1562.53i	− 0.0660597i
$825$	− 28836.5i	− 1.21692i
$826$	7864.69i	0.331293i
$827$	− 17881.0i	− 0.751854i	−0.926649	−	0.375927i	$-0.877324\pi$
$827$	− 17881.0i	− 0.751854i	0.926649	−	0.375927i	$-0.122676\pi$
$828$	−7216.18	−0.302874
$829$	−351.169	−0.0147124	−0.00735620	−	0.999973i	$-0.502342\pi$
$829$	−351.169	−0.0147124	−0.00735620	+	0.999973i	$0.502342\pi$
$830$	− 51688.3i	− 2.16160i
$831$	−2047.33	−0.0854646
$832$	0	0
$833$	−46.4836	−0.00193345
$834$	− 27054.2i	− 1.12327i
$835$	30081.6	1.24673
$836$	5472.54	0.226402
$837$	4182.34i	0.172716i
$838$	13095.8i	0.539841i
$839$	33425.9i	1.37543i	0.725978	+	0.687717i	$0.241387\pi$
$839$	33425.9i	1.37543i	−0.725978	+	0.687717i	$0.758613\pi$
$840$	26240.8i	1.07785i
$841$	−23247.1	−0.953178
$842$	4510.99	0.184631
$843$	− 36400.7i	− 1.48720i
$844$	6391.46	0.260667
$845$	0	0
$846$	16260.5	0.660811
$847$	46653.3i	1.89259i
$848$	31123.0	1.26034
$849$	−37389.3	−1.51142
$850$	− 64.1449i	− 0.00258841i
$851$	9123.18i	0.367495i
$852$	− 861.959i	− 0.0346599i
$853$	35097.5i	1.40881i	0.709797	+	0.704406i	$0.248787\pi$
$853$	35097.5i	1.40881i	−0.709797	+	0.704406i	$0.751213\pi$
$854$	−25407.9	−1.01808
$855$	12817.9	0.512706
$856$	− 25101.2i	− 1.00227i
$857$	−15015.8	−0.598519	−0.299259	−	0.954172i	$-0.596740\pi$
$857$	−15015.8	−0.598519	−0.299259	+	0.954172i	$0.596740\pi$
$858$	0	0
$859$	19647.1	0.780383	0.390192	−	0.920734i	$-0.372409\pi$
$859$	19647.1	0.780383	0.390192	+	0.920734i	$0.372409\pi$
$860$	5651.10i	0.224071i
$861$	6029.45	0.238656
$862$	−8039.57	−0.317667
$863$	− 9035.14i	− 0.356385i	−0.983996	−	0.178192i	$-0.942975\pi$
$863$	− 9035.14i	− 0.356385i	0.983996	−	0.178192i	$-0.0570249\pi$
$864$	− 2346.19i	− 0.0923833i
$865$	− 2909.52i	− 0.114366i
$866$	− 10901.7i	− 0.427778i
$867$	−34826.6	−1.36422
$868$	4498.24	0.175899
$869$	66640.5i	2.60141i
$870$	10302.3	0.401471
$871$	0	0
$872$	−39367.0	−1.52883
$873$	36906.2i	1.43080i
$874$	19937.5	0.771621
$875$	−12659.6	−0.489113
$876$	3987.34i	0.153790i
$877$	− 8254.16i	− 0.317814i	−0.987294	−	0.158907i	$-0.949203\pi$
$877$	− 8254.16i	− 0.317814i	0.987294	−	0.158907i	$-0.0507971\pi$
$878$	− 39967.6i	− 1.53627i
$879$	− 10248.3i	− 0.393248i
$880$	−69996.4	−2.68134
$881$	42107.4	1.61026	0.805128	−	0.593101i	$-0.202097\pi$
$881$	42107.4	1.61026	0.805128	+	0.593101i	$0.202097\pi$
$882$	− 10115.1i	− 0.386161i
$883$	17584.7	0.670183	0.335091	−	0.942186i	$-0.391233\pi$
$883$	17584.7	0.670183	0.335091	+	0.942186i	$0.391233\pi$
$884$	0	0
$885$	−16740.1	−0.635832
$886$	− 26829.1i	− 1.01732i
$887$	−23845.8	−0.902666	−0.451333	−	0.892356i	$-0.649051\pi$
$887$	−23845.8	−0.902666	−0.451333	+	0.892356i	$0.649051\pi$
$888$	7889.89	0.298161
$889$	8806.63i	0.332244i
$890$	− 23219.3i	− 0.874509i
$891$	− 55272.0i	− 2.07821i
$892$	2108.59i	0.0791488i
$893$	−8964.14	−0.335917
$894$	−7595.70	−0.284159
$895$	40368.4i	1.50767i
$896$	25014.6	0.932679
$897$	0	0
$898$	−35943.1	−1.33568
$899$	5318.83i	0.197322i
$900$	2785.13	0.103153
$901$	138.374	0.00511644
$902$	12706.2i	0.469036i
$903$	− 21163.0i	− 0.779912i
$904$	30432.2i	1.11965i
$905$	31689.6i	1.16398i
$906$	37981.1	1.39276
$907$	−30565.7	−1.11898	−0.559492	−	0.828836i	$-0.689004\pi$
$907$	−30565.7	−1.11898	−0.559492	+	0.828836i	$0.689004\pi$
$908$	10289.2i	0.376055i
$909$	−26988.7	−0.984773
$910$	0	0
$911$	18556.9	0.674882	0.337441	−	0.941347i	$-0.390439\pi$
$911$	18556.9	0.674882	0.337441	+	0.941347i	$0.390439\pi$
$912$	− 21825.0i	− 0.792433i
$913$	−81392.9	−2.95040
$914$	35688.6	1.29155
$915$	− 54081.0i	− 1.95395i
$916$	− 3386.56i	− 0.122156i
$917$	4731.84i	0.170402i
$918$	− 28.3760i	− 0.00102020i
$919$	−9938.82	−0.356748	−0.178374	−	0.983963i	$-0.557084\pi$
$919$	−9938.82	−0.356748	−0.178374	+	0.983963i	$0.557084\pi$
$920$	−40198.5	−1.44055
$921$	− 66786.9i	− 2.38947i
$922$	12121.0	0.432954
$923$	0	0
$924$	−13720.0	−0.488479
$925$	− 3521.15i	− 0.125162i
$926$	−4092.18	−0.145224
$927$	1913.52	0.0677973
$928$	− 2983.73i	− 0.105545i
$929$	− 5953.61i	− 0.210260i	−0.994458	−	0.105130i	$-0.966474\pi$
$929$	− 5953.61i	− 0.210260i	0.994458	−	0.105130i	$-0.0335259\pi$
$930$	47985.2i	1.69193i
$931$	5576.31i	0.196301i
$932$	6305.95	0.221629
$933$	−55901.6	−1.96156
$934$	43821.1i	1.53519i
$935$	−311.207	−0.0108851
$936$	0	0
$937$	−24568.0	−0.856564	−0.428282	−	0.903645i	$-0.640881\pi$
$937$	−24568.0	−0.856564	−0.428282	+	0.903645i	$0.640881\pi$
$938$	12190.9i	0.424356i
$939$	−3901.86	−0.135604
$940$	−6001.08	−0.208227
$941$	− 11024.8i	− 0.381931i	−0.981597	−	0.190965i	$-0.938838\pi$
$941$	− 11024.8i	− 0.381931i	0.981597	−	0.190965i	$-0.0611618\pi$
$942$	− 214.971i	− 0.00743538i
$943$	9236.55i	0.318964i
$944$	13188.6i	0.454715i
$945$	5180.60	0.178333
$946$	44598.0	1.53278
$947$	− 6171.46i	− 0.211769i	−0.994378	−	0.105885i	$-0.966233\pi$
$947$	− 6171.46i	− 0.211769i	0.994378	−	0.105885i	$-0.0337674\pi$
$948$	−13910.4	−0.476569
$949$	0	0
$950$	−7695.01	−0.262799
$951$	− 1070.23i	− 0.0364928i
$952$	91.9161	0.00312922
$953$	18838.4	0.640333	0.320166	−	0.947361i	$-0.396261\pi$
$953$	18838.4	0.640333	0.320166	+	0.947361i	$0.396261\pi$
$954$	30111.1i	1.02189i
$955$	60942.3i	2.06497i
$956$	4686.32i	0.158542i
$957$	− 16222.8i	− 0.547973i
$958$	30436.4	1.02647
$959$	32998.8	1.11114
$960$	31696.5i	1.06562i
$961$	5017.33	0.168418
$962$	0	0
$963$	30739.6	1.02863
$964$	10304.1i	0.344267i
$965$	17720.8	0.591144
$966$	−49984.6	−1.66483
$967$	24934.5i	0.829203i	0.910003	+	0.414602i	$0.136079\pi$
$967$	24934.5i	0.829203i	−0.910003	+	0.414602i	$0.863921\pi$
$968$	61807.2i	2.05223i
$969$	− 97.0351i	− 0.00321694i
$970$	− 68262.8i	− 2.25957i
$971$	32161.2	1.06293	0.531463	−	0.847082i	$-0.321643\pi$
$971$	32161.2	1.06293	0.531463	+	0.847082i	$0.321643\pi$
$972$	10106.6	0.333509
$973$	− 17301.3i	− 0.570044i
$974$	−37787.2	−1.24310
$975$	0	0
$976$	−42607.4	−1.39737
$977$	− 18864.0i	− 0.617720i	−0.951107	−	0.308860i	$-0.900052\pi$
$977$	− 18864.0i	− 0.617720i	0.951107	−	0.308860i	$-0.0999475\pi$
$978$	2736.30	0.0894655
$979$	−36563.1	−1.19363
$980$	3733.08i	0.121683i
$981$	− 48210.0i	− 1.56904i
$982$	17103.3i	0.555792i
$983$	7883.83i	0.255804i	0.991787	+	0.127902i	$0.0408243\pi$
$983$	7883.83i	0.255804i	−0.991787	+	0.127902i	$0.959176\pi$
$984$	7987.93	0.258786
$985$	10079.9	0.326063
$986$	− 36.0867i	− 0.00116555i
$987$	22473.6	0.724766
$988$	0	0
$989$	32419.7	1.04235
$990$	− 67720.6i	− 2.17404i
$991$	14172.3	0.454286	0.227143	−	0.973861i	$-0.427062\pi$
$991$	14172.3	0.454286	0.227143	+	0.973861i	$0.427062\pi$
$992$	13897.4	0.444802
$993$	− 50183.9i	− 1.60376i
$994$	− 2762.60i	− 0.0881533i
$995$	− 63785.3i	− 2.03229i
$996$	− 16989.7i	− 0.540503i
$997$	52462.7	1.66651	0.833256	−	0.552888i	$-0.186474\pi$
$997$	52462.7	1.66651	0.833256	+	0.552888i	$0.186474\pi$
$998$	−45753.8	−1.45121
$999$	− 1557.66i	− 0.0493315i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.4.b.g.168.5		18
13.2	odd	12		169.4.c.l.22.7		18
13.3	even	3		169.4.e.h.147.14		36
13.4	even	6		169.4.e.h.23.14		36
13.5	odd	4		169.4.a.k.1.3	✓	9
13.6	odd	12		169.4.c.l.146.7		18
13.7	odd	12		169.4.c.k.146.3		18
13.8	odd	4		169.4.a.l.1.7	yes	9
13.9	even	3		169.4.e.h.23.5		36
13.10	even	6		169.4.e.h.147.5		36
13.11	odd	12		169.4.c.k.22.3		18
13.12	even	2	inner	169.4.b.g.168.14		18
39.5	even	4		1521.4.a.bh.1.7		9
39.8	even	4		1521.4.a.bg.1.3		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.4.a.k.1.3	✓	9	13.5	odd	4
169.4.a.l.1.7	yes	9	13.8	odd	4
169.4.b.g.168.5		18	1.1	even	1	trivial
169.4.b.g.168.14		18	13.12	even	2	inner
169.4.c.k.22.3		18	13.11	odd	12
169.4.c.k.146.3		18	13.7	odd	12
169.4.c.l.22.7		18	13.2	odd	12
169.4.c.l.146.7		18	13.6	odd	12
169.4.e.h.23.5		36	13.9	even	3
169.4.e.h.23.14		36	13.4	even	6
169.4.e.h.147.5		36	13.10	even	6
169.4.e.h.147.14		36	13.3	even	3
1521.4.a.bg.1.3		9	39.8	even	4
1521.4.a.bh.1.7		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)

\(f(q)\)	\(=\)	\(q-3.16135i q^{2} +7.08883 q^{3} -1.99415 q^{4} -13.6039i q^{5} -22.4103i q^{6} -14.3315i q^{7} -18.9866i q^{8} +23.2516 q^{9} +O(q^{10})\)	\(q-3.16135i q^{2} +7.08883 q^{3} -1.99415 q^{4} -13.6039i q^{5} -22.4103i q^{6} -14.3315i q^{7} -18.9866i q^{8} +23.2516 q^{9} -43.0068 q^{10} +67.7223i q^{11} -14.1362 q^{12} -45.3068 q^{14} -96.4360i q^{15} -75.9766 q^{16} -0.337795 q^{17} -73.5064i q^{18} +40.5229i q^{19} +27.1282i q^{20} -101.593i q^{21} +214.094 q^{22} +155.632 q^{23} -134.593i q^{24} -60.0670 q^{25} -26.5720 q^{27} +28.5791i q^{28} -33.7925 q^{29} -304.868 q^{30} -157.397i q^{31} +88.2957i q^{32} +480.072i q^{33} +1.06789i q^{34} -194.964 q^{35} -46.3670 q^{36} +58.6204i q^{37} +128.107 q^{38} -258.293 q^{40} +59.3488i q^{41} -321.173 q^{42} +208.311 q^{43} -135.048i q^{44} -316.313i q^{45} -492.006i q^{46} +221.212i q^{47} -538.585 q^{48} +137.609 q^{49} +189.893i q^{50} -2.39457 q^{51} -409.639 q^{53} +84.0035i q^{54} +921.289 q^{55} -272.106 q^{56} +287.260i q^{57} +106.830i q^{58} -173.587i q^{59} +192.308i q^{60} +560.796 q^{61} -497.586 q^{62} -333.229i q^{63} -328.679 q^{64} +1517.68 q^{66} -269.074i q^{67} +0.673613 q^{68} +1103.25 q^{69} +616.351i q^{70} +60.9754i q^{71} -441.469i q^{72} -282.066i q^{73} +185.320 q^{74} -425.805 q^{75} -80.8086i q^{76} +970.560 q^{77} +984.026 q^{79} +1033.58i q^{80} -816.157 q^{81} +187.622 q^{82} +1201.86i q^{83} +202.592i q^{84} +4.59534i q^{85} -658.543i q^{86} -239.550 q^{87} +1285.82 q^{88} +539.899i q^{89} -999.976 q^{90} -310.352 q^{92} -1115.76i q^{93} +699.328 q^{94} +551.271 q^{95} +625.913i q^{96} +1587.25i q^{97} -435.030i q^{98} +1574.65i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 2 q^{3} - 74 q^{4} + 132 q^{9}+O(q^{10}) \) 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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