LMFDB - Embedded newform 169.4.b.e.168.1

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:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 13)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			168.1
Root			$-1.56155i$ of defining polynomial
Character	$\chi$	$=$	169.168
Dual form			169.4.b.e.168.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.56155i q^{2} +8.68466 q^{3} -12.8078 q^{4} +2.80776i q^{5} -39.6155i q^{6} -9.56155i q^{7} +21.9309i q^{8} +48.4233 q^{9} +O(q^{10})$	\(q-4.56155i q^{2} +8.68466 q^{3} -12.8078 q^{4} +2.80776i q^{5} -39.6155i q^{6} -9.56155i q^{7} +21.9309i q^{8} +48.4233 q^{9} +12.8078 q^{10} -39.4233i q^{11} -111.231 q^{12} -43.6155 q^{14} +24.3845i q^{15} -2.42329 q^{16} -2.01515 q^{17} -220.885i q^{18} -60.1922i q^{19} -35.9612i q^{20} -83.0388i q^{21} -179.831 q^{22} -4.46876 q^{23} +190.462i q^{24} +117.116 q^{25} +186.054 q^{27} +122.462i q^{28} +140.693 q^{29} +111.231 q^{30} +136.155i q^{31} +186.501i q^{32} -342.378i q^{33} +9.19224i q^{34} +26.8466 q^{35} -620.194 q^{36} +185.708i q^{37} -274.570 q^{38} -61.5767 q^{40} +310.231i q^{41} -378.786 q^{42} -427.471 q^{43} +504.924i q^{44} +135.961i q^{45} +20.3845i q^{46} +258.617i q^{47} -21.0455 q^{48} +251.577 q^{49} -534.233i q^{50} -17.5009 q^{51} +612.656 q^{53} -848.695i q^{54} +110.691 q^{55} +209.693 q^{56} -522.749i q^{57} -641.779i q^{58} +517.885i q^{59} -312.311i q^{60} -161.311 q^{61} +621.080 q^{62} -463.002i q^{63} +831.348 q^{64} -1561.77 q^{66} -49.8987i q^{67} +25.8096 q^{68} -38.8096 q^{69} -122.462i q^{70} +279.963i q^{71} +1061.96i q^{72} -467.732i q^{73} +847.118 q^{74} +1017.12 q^{75} +770.928i q^{76} -376.948 q^{77} +37.5379 q^{79} -6.80403i q^{80} +308.386 q^{81} +1415.14 q^{82} -76.1553i q^{83} +1063.54i q^{84} -5.65808i q^{85} +1949.93i q^{86} +1221.87 q^{87} +864.587 q^{88} -202.806i q^{89} +620.194 q^{90} +57.2348 q^{92} +1182.46i q^{93} +1179.70 q^{94} +169.006 q^{95} +1619.70i q^{96} -1174.37i q^{97} -1147.58i q^{98} -1909.01i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 10 q^{3} - 10 q^{4} + 70 q^{9}+O(q^{10}) $ 4 * q + 10 * q^3 - 10 * q^4 + 70 * q^9	$ 4 q + 10 q^{3} - 10 q^{4} + 70 q^{9} + 10 q^{10} - 280 q^{12} - 92 q^{14} + 114 q^{16} - 140 q^{17} - 340 q^{22} - 290 q^{23} - 150 q^{25} + 670 q^{27} + 68 q^{29} + 280 q^{30} - 140 q^{35} - 1450 q^{36} - 620 q^{38} - 370 q^{40} - 740 q^{42} - 910 q^{43} - 480 q^{48} + 1130 q^{49} + 466 q^{51} + 1090 q^{53} + 1020 q^{55} + 344 q^{56} + 1004 q^{61} + 1000 q^{62} + 2542 q^{64} - 3196 q^{66} - 1010 q^{68} + 958 q^{69} + 1698 q^{74} + 3450 q^{75} - 510 q^{77} + 480 q^{79} + 244 q^{81} + 3030 q^{82} + 3230 q^{87} + 2040 q^{88} + 1450 q^{90} - 2080 q^{92} + 2080 q^{94} - 2540 q^{95}+O(q^{100}) $ 4 * q + 10 * q^3 - 10 * q^4 + 70 * q^9 + 10 * q^10 - 280 * q^12 - 92 * q^14 + 114 * q^16 - 140 * q^17 - 340 * q^22 - 290 * q^23 - 150 * q^25 + 670 * q^27 + 68 * q^29 + 280 * q^30 - 140 * q^35 - 1450 * q^36 - 620 * q^38 - 370 * q^40 - 740 * q^42 - 910 * q^43 - 480 * q^48 + 1130 * q^49 + 466 * q^51 + 1090 * q^53 + 1020 * q^55 + 344 * q^56 + 1004 * q^61 + 1000 * q^62 + 2542 * q^64 - 3196 * q^66 - 1010 * q^68 + 958 * q^69 + 1698 * q^74 + 3450 * q^75 - 510 * q^77 + 480 * q^79 + 244 * q^81 + 3030 * q^82 + 3230 * q^87 + 2040 * q^88 + 1450 * q^90 - 2080 * q^92 + 2080 * q^94 - 2540 * 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 $

$2$	− 4.56155i	− 1.61275i	−0.591403	−	0.806376i	$-0.701426\pi$
$2$	− 4.56155i	− 1.61275i	0.591403	−	0.806376i	$-0.298574\pi$
$3$	8.68466	1.67136	0.835682	−	0.549214i	$-0.185073\pi$
$3$	8.68466	1.67136	0.835682	+	0.549214i	$0.185073\pi$
$4$	−12.8078	−1.60097
$5$	2.80776i	0.251134i	0.992085	+	0.125567i	$0.0400750\pi$
$5$	2.80776i	0.251134i	−0.992085	+	0.125567i	$0.959925\pi$
$6$	− 39.6155i	− 2.69550i
$7$	− 9.56155i	− 0.516275i	−0.966108	−	0.258138i	$-0.916891\pi$
$7$	− 9.56155i	− 0.516275i	0.966108	−	0.258138i	$-0.0831088\pi$
$8$	21.9309i	0.969217i
$9$	48.4233	1.79346
$10$	12.8078	0.405017
$11$	− 39.4233i	− 1.08060i	−0.841473	−	0.540299i	$-0.818311\pi$
$11$	− 39.4233i	− 1.08060i	0.841473	−	0.540299i	$-0.181689\pi$
$12$	−111.231	−2.67580
$13$	0	0
$13$	0	0
$14$	−43.6155	−0.832624
$15$	24.3845i	0.419736i
$16$	−2.42329	−0.0378639
$17$	−2.01515	−0.0287498	−0.0143749	−	0.999897i	$-0.504576\pi$
$17$	−2.01515	−0.0287498	−0.0143749	+	0.999897i	$0.504576\pi$
$18$	− 220.885i	− 2.89240i
$19$	− 60.1922i	− 0.726792i	−0.931635	−	0.363396i	$-0.881617\pi$
$19$	− 60.1922i	− 0.726792i	0.931635	−	0.363396i	$-0.118383\pi$
$20$	− 35.9612i	− 0.402058i
$21$	− 83.0388i	− 0.862884i
$22$	−179.831	−1.74274
$23$	−4.46876	−0.0405131	−0.0202565	−	0.999795i	$-0.506448\pi$
$23$	−4.46876	−0.0405131	−0.0202565	+	0.999795i	$0.506448\pi$
$24$	190.462i	1.61991i
$25$	117.116	0.936932
$26$	0	0
$27$	186.054	1.32615
$28$	122.462i	0.826542i
$29$	140.693	0.900899	0.450449	−	0.892802i	$-0.351264\pi$
$29$	140.693	0.900899	0.450449	+	0.892802i	$0.351264\pi$
$30$	111.231	0.676931
$31$	136.155i	0.788845i	0.918929	+	0.394423i	$0.129055\pi$
$31$	136.155i	0.788845i	−0.918929	+	0.394423i	$0.870945\pi$
$32$	186.501i	1.03028i
$33$	− 342.378i	− 1.80607i
$34$	9.19224i	0.0463663i
$35$	26.8466	0.129654
$36$	−620.194	−2.87127
$37$	185.708i	0.825142i	0.910925	+	0.412571i	$0.135369\pi$
$37$	185.708i	0.825142i	−0.910925	+	0.412571i	$0.864631\pi$
$38$	−274.570	−1.17214
$39$	0	0
$40$	−61.5767	−0.243403
$41$	310.231i	1.18171i	0.806779	+	0.590853i	$0.201209\pi$
$41$	310.231i	1.18171i	−0.806779	+	0.590853i	$0.798791\pi$
$42$	−378.786	−1.39162
$43$	−427.471	−1.51602	−0.758008	−	0.652246i	$-0.773827\pi$
$43$	−427.471	−1.51602	−0.758008	+	0.652246i	$0.773827\pi$
$44$	504.924i	1.73000i
$45$	135.961i	0.450398i
$46$	20.3845i	0.0653375i
$47$	258.617i	0.802622i	0.915942	+	0.401311i	$0.131445\pi$
$47$	258.617i	0.802622i	−0.915942	+	0.401311i	$0.868555\pi$
$48$	−21.0455	−0.0632844
$49$	251.577	0.733460
$50$	− 534.233i	− 1.51104i
$51$	−17.5009	−0.0480514
$52$	0	0
$53$	612.656	1.58783	0.793913	−	0.608031i	$-0.208040\pi$
$53$	612.656	1.58783	0.793913	+	0.608031i	$0.208040\pi$
$54$	− 848.695i	− 2.13875i
$55$	110.691	0.271375
$56$	209.693	0.500383
$57$	− 522.749i	− 1.21473i
$58$	− 641.779i	− 1.45293i
$59$	517.885i	1.14276i	0.820685	+	0.571381i	$0.193592\pi$
$59$	517.885i	1.14276i	−0.820685	+	0.571381i	$0.806408\pi$
$60$	− 312.311i	− 0.671985i
$61$	−161.311	−0.338585	−0.169293	−	0.985566i	$-0.554148\pi$
$61$	−161.311	−0.338585	−0.169293	+	0.985566i	$0.554148\pi$
$62$	621.080	1.27221
$63$	− 463.002i	− 0.925917i
$64$	831.348	1.62373
$65$	0	0
$66$	−1561.77	−2.91274
$67$	− 49.8987i	− 0.0909865i	−0.998965	−	0.0454933i	$-0.985514\pi$
$67$	− 49.8987i	− 0.0909865i	0.998965	−	0.0454933i	$-0.0144860\pi$
$68$	25.8096	0.0460276
$69$	−38.8096	−0.0677120
$70$	− 122.462i	− 0.209100i
$71$	279.963i	0.467965i	0.972241	+	0.233982i	$0.0751758\pi$
$71$	279.963i	0.467965i	−0.972241	+	0.233982i	$0.924824\pi$
$72$	1061.96i	1.73825i
$73$	− 467.732i	− 0.749916i	−0.927042	−	0.374958i	$-0.877657\pi$
$73$	− 467.732i	− 0.749916i	0.927042	−	0.374958i	$-0.122343\pi$
$74$	847.118	1.33075
$75$	1017.12	1.56595
$76$	770.928i	1.16357i
$77$	−376.948	−0.557886
$78$	0	0
$79$	37.5379	0.0534600	0.0267300	−	0.999643i	$-0.491491\pi$
$79$	37.5379	0.0534600	0.0267300	+	0.999643i	$0.491491\pi$
$80$	− 6.80403i	− 0.00950892i
$81$	308.386	0.423027
$82$	1415.14	1.90580
$83$	− 76.1553i	− 0.100712i	−0.998731	−	0.0503562i	$-0.983964\pi$
$83$	− 76.1553i	− 0.100712i	0.998731	−	0.0503562i	$-0.0160357\pi$
$84$	1063.54i	1.38145i
$85$	− 5.65808i	− 0.00722006i
$86$	1949.93i	2.44496i
$87$	1221.87	1.50573
$88$	864.587	1.04733
$89$	− 202.806i	− 0.241544i	−0.992680	−	0.120772i	$-0.961463\pi$
$89$	− 202.806i	− 0.241544i	0.992680	−	0.120772i	$-0.0385369\pi$
$90$	620.194	0.726380
$91$	0	0
$92$	57.2348	0.0648602
$93$	1182.46i	1.31845i
$94$	1179.70	1.29443
$95$	169.006	0.182522
$96$	1619.70i	1.72198i
$97$	− 1174.37i	− 1.22927i	−0.788812	−	0.614634i	$-0.789304\pi$
$97$	− 1174.37i	− 1.22927i	0.788812	−	0.614634i	$-0.210696\pi$
$98$	− 1147.58i	− 1.18289i
$99$	− 1909.01i	− 1.93800i
$100$	−1500.00	−1.50000
$101$	−970.697	−0.956316	−0.478158	−	0.878274i	$-0.658695\pi$
$101$	−970.697	−0.956316	−0.478158	+	0.878274i	$0.658695\pi$
$102$	79.8314i	0.0774950i
$103$	1899.70	1.81731	0.908654	−	0.417550i	$-0.137111\pi$
$103$	1899.70	1.81731	0.908654	+	0.417550i	$0.137111\pi$
$104$	0	0
$105$	233.153	0.216699
$106$	− 2794.66i	− 2.56077i
$107$	−1906.49	−1.72250	−0.861251	−	0.508180i	$-0.830318\pi$
$107$	−1906.49	−1.72250	−0.861251	+	0.508180i	$0.830318\pi$
$108$	−2382.94	−2.12313
$109$	− 896.004i	− 0.787354i	−0.919249	−	0.393677i	$-0.871203\pi$
$109$	− 896.004i	− 0.787354i	0.919249	−	0.393677i	$-0.128797\pi$
$110$	− 504.924i	− 0.437660i
$111$	1612.81i	1.37911i
$112$	23.1704i	0.0195482i
$113$	−334.882	−0.278788	−0.139394	−	0.990237i	$-0.544516\pi$
$113$	−334.882	−0.278788	−0.139394	+	0.990237i	$0.544516\pi$
$114$	−2384.55	−1.95906
$115$	− 12.5472i	− 0.0101742i
$116$	−1801.96	−1.44231
$117$	0	0
$118$	2362.36	1.84299
$119$	19.2680i	0.0148428i
$120$	−534.773	−0.406815
$121$	−223.196	−0.167690
$122$	735.827i	0.546054i
$123$	2694.25i	1.97506i
$124$	− 1743.84i	− 1.26292i
$125$	679.806i	0.486430i
$126$	−2112.01	−1.49327
$127$	−620.893	−0.433822	−0.216911	−	0.976191i	$-0.569598\pi$
$127$	−620.893	−0.433822	−0.216911	+	0.976191i	$0.569598\pi$
$128$	− 2300.23i	− 1.58839i
$129$	−3712.44	−2.53381
$130$	0	0
$131$	−1331.70	−0.888180	−0.444090	−	0.895982i	$-0.646473\pi$
$131$	−1331.70	−0.888180	−0.444090	+	0.895982i	$0.646473\pi$
$132$	4385.09i	2.89147i
$133$	−575.531	−0.375225
$134$	−227.616	−0.146739
$135$	522.396i	0.333042i
$136$	− 44.1941i	− 0.0278648i
$137$	− 622.015i	− 0.387900i	−0.981011	−	0.193950i	$-0.937870\pi$
$137$	− 622.015i	− 0.387900i	0.981011	−	0.193950i	$-0.0621300\pi$
$138$	177.032i	0.109203i
$139$	330.580	0.201723	0.100861	−	0.994900i	$-0.467840\pi$
$139$	330.580	0.201723	0.100861	+	0.994900i	$0.467840\pi$
$140$	−343.845	−0.207573
$141$	2246.00i	1.34147i
$142$	1277.07	0.754711
$143$	0	0
$144$	−117.344	−0.0679073
$145$	395.033i	0.226246i
$146$	−2133.58	−1.20943
$147$	2184.86	1.22588
$148$	− 2378.51i	− 1.32103i
$149$	− 1810.54i	− 0.995470i	−0.867329	−	0.497735i	$-0.834165\pi$
$149$	− 1810.54i	− 0.995470i	0.867329	−	0.497735i	$-0.165835\pi$
$150$	− 4639.63i	− 2.52549i
$151$	423.239i	0.228097i	0.993475	+	0.114049i	$0.0363819\pi$
$151$	423.239i	0.228097i	−0.993475	+	0.114049i	$0.963618\pi$
$152$	1320.07	0.704419
$153$	−97.5804	−0.0515615
$154$	1719.47i	0.899732i
$155$	−382.292	−0.198106
$156$	0	0
$157$	1322.17	0.672105	0.336052	−	0.941843i	$-0.390908\pi$
$157$	1322.17	0.672105	0.336052	+	0.941843i	$0.390908\pi$
$158$	− 171.231i	− 0.0862178i
$159$	5320.71	2.65383
$160$	−523.651	−0.258739
$161$	42.7283i	0.0209159i
$162$	− 1406.72i	− 0.682237i
$163$	3606.39i	1.73297i	0.499201	+	0.866486i	$0.333627\pi$
$163$	3606.39i	1.73297i	−0.499201	+	0.866486i	$0.666373\pi$
$164$	− 3973.37i	− 1.89188i
$165$	961.316	0.453566
$166$	−347.386	−0.162424
$167$	3415.43i	1.58260i	0.611429	+	0.791300i	$0.290595\pi$
$167$	3415.43i	1.58260i	−0.611429	+	0.791300i	$0.709405\pi$
$168$	1821.11	0.836321
$169$	0	0
$170$	−25.8096	−0.0116442
$171$	− 2914.71i	− 1.30347i
$172$	5474.94	2.42710
$173$	−2342.23	−1.02934	−0.514671	−	0.857388i	$-0.672086\pi$
$173$	−2342.23	−1.02934	−0.514671	+	0.857388i	$0.672086\pi$
$174$	− 5573.63i	− 2.42837i
$175$	− 1119.82i	− 0.483715i
$176$	95.5342i	0.0409157i
$177$	4497.66i	1.90997i
$178$	−925.110	−0.389550
$179$	666.891	0.278468	0.139234	−	0.990260i	$-0.455536\pi$
$179$	666.891	0.278468	0.139234	+	0.990260i	$0.455536\pi$
$180$	− 1741.36i	− 0.721073i
$181$	701.037	0.287888	0.143944	−	0.989586i	$-0.454022\pi$
$181$	701.037	0.287888	0.143944	+	0.989586i	$0.454022\pi$
$182$	0	0
$183$	−1400.93	−0.565899
$184$	− 98.0037i	− 0.0392659i
$185$	−521.425	−0.207221
$186$	5393.86	2.12633
$187$	79.4440i	0.0310670i
$188$	− 3312.31i	− 1.28497i
$189$	− 1778.96i	− 0.684660i
$190$	− 770.928i	− 0.294363i
$191$	1300.88	0.492819	0.246409	−	0.969166i	$-0.420749\pi$
$191$	1300.88	0.492819	0.246409	+	0.969166i	$0.420749\pi$
$192$	7219.97	2.71384
$193$	519.333i	0.193691i	0.995299	+	0.0968457i	$0.0308753\pi$
$193$	519.333i	0.193691i	−0.995299	+	0.0968457i	$0.969125\pi$
$194$	−5356.94	−1.98251
$195$	0	0
$196$	−3222.14	−1.17425
$197$	3121.05i	1.12876i	0.825516	+	0.564379i	$0.190884\pi$
$197$	3121.05i	1.12876i	−0.825516	+	0.564379i	$0.809116\pi$
$198$	−8708.03	−3.12552
$199$	−1237.06	−0.440667	−0.220333	−	0.975425i	$-0.570715\pi$
$199$	−1237.06	−0.440667	−0.220333	+	0.975425i	$0.570715\pi$
$200$	2568.47i	0.908090i
$201$	− 433.353i	− 0.152072i
$202$	4427.89i	1.54230i
$203$	− 1345.25i	− 0.465112i
$204$	224.148	0.0769289
$205$	−871.056	−0.296767
$206$	− 8665.57i	− 2.93087i
$207$	−216.392	−0.0726584
$208$	0	0
$209$	−2372.98	−0.785369
$210$	− 1063.54i	− 0.349483i
$211$	−2531.67	−0.826006	−0.413003	−	0.910730i	$-0.635520\pi$
$211$	−2531.67	−0.826006	−0.413003	+	0.910730i	$0.635520\pi$
$212$	−7846.76	−2.54206
$213$	2431.38i	0.782139i
$214$	8696.57i	2.77797i
$215$	− 1200.24i	− 0.380723i
$216$	4080.33i	1.28533i
$217$	1301.86	0.407261
$218$	−4087.17	−1.26981
$219$	− 4062.09i	− 1.25338i
$220$	−1417.71	−0.434463
$221$	0	0
$222$	7356.93	2.22417
$223$	− 1195.53i	− 0.359008i	−0.983757	−	0.179504i	$-0.942551\pi$
$223$	− 1195.53i	− 0.359008i	0.983757	−	0.179504i	$-0.0574492\pi$
$224$	1783.24	0.531909
$225$	5671.16	1.68035
$226$	1527.58i	0.449617i
$227$	− 869.192i	− 0.254142i	−0.991894	−	0.127071i	$-0.959442\pi$
$227$	− 869.192i	− 0.254142i	0.991894	−	0.127071i	$-0.0405577\pi$
$228$	6695.25i	1.94475i
$229$	− 4684.64i	− 1.35183i	−0.736978	−	0.675916i	$-0.763748\pi$
$229$	− 4684.64i	− 1.35183i	0.736978	−	0.675916i	$-0.236252\pi$
$230$	−57.2348	−0.0164085
$231$	−3273.66	−0.932430
$232$	3085.52i	0.873166i
$233$	4868.99	1.36900	0.684502	−	0.729011i	$-0.260020\pi$
$233$	4868.99	1.36900	0.684502	+	0.729011i	$0.260020\pi$
$234$	0	0
$235$	−726.137	−0.201566
$236$	− 6632.95i	− 1.82953i
$237$	326.004	0.0893511
$238$	87.8920	0.0239378
$239$	4807.53i	1.30114i	0.759444	+	0.650572i	$0.225471\pi$
$239$	4807.53i	1.30114i	−0.759444	+	0.650572i	$0.774529\pi$
$240$	− 59.0907i	− 0.0158929i
$241$	− 5875.96i	− 1.57056i	−0.619143	−	0.785278i	$-0.712520\pi$
$241$	− 5875.96i	− 1.57056i	0.619143	−	0.785278i	$-0.287480\pi$
$242$	1018.12i	0.270443i
$243$	−2345.23	−0.619121
$244$	2066.03	0.542065
$245$	706.368i	0.184197i
$246$	12290.0	3.18528
$247$	0	0
$248$	−2986.00	−0.764562
$249$	− 661.383i	− 0.168327i
$250$	3100.97	0.784490
$251$	−5806.27	−1.46011	−0.730057	−	0.683386i	$-0.760506\pi$
$251$	−5806.27	−1.46011	−0.730057	+	0.683386i	$0.760506\pi$
$252$	5930.02i	1.48237i
$253$	176.173i	0.0437783i
$254$	2832.24i	0.699647i
$255$	− 49.1385i	− 0.0120673i
$256$	−3841.83	−0.937947
$257$	1195.86	0.290256	0.145128	−	0.989413i	$-0.453641\pi$
$257$	1195.86	0.290256	0.145128	+	0.989413i	$0.453641\pi$
$258$	16934.5i	4.08641i
$259$	1775.66	0.426001
$260$	0	0
$261$	6812.83	1.61572
$262$	6074.64i	1.43241i
$263$	234.184	0.0549066	0.0274533	−	0.999623i	$-0.491260\pi$
$263$	234.184	0.0549066	0.0274533	+	0.999623i	$0.491260\pi$
$264$	7508.64	1.75047
$265$	1720.19i	0.398757i
$266$	2625.32i	0.605145i
$267$	− 1761.30i	− 0.403707i
$268$	639.091i	0.145667i
$269$	−2668.27	−0.604785	−0.302393	−	0.953183i	$-0.597785\pi$
$269$	−2668.27	−0.604785	−0.302393	+	0.953183i	$0.597785\pi$
$270$	2382.94	0.537114
$271$	− 5701.28i	− 1.27796i	−0.769222	−	0.638982i	$-0.779356\pi$
$271$	− 5701.28i	− 1.27796i	0.769222	−	0.638982i	$-0.220644\pi$
$272$	4.88331	0.00108858
$273$	0	0
$274$	−2837.35	−0.625587
$275$	− 4617.12i	− 1.01245i
$276$	497.065	0.108405
$277$	7152.49	1.55145	0.775725	−	0.631072i	$-0.217385\pi$
$277$	7152.49	1.55145	0.775725	+	0.631072i	$0.217385\pi$
$278$	− 1507.96i	− 0.325329i
$279$	6593.09i	1.41476i
$280$	588.769i	0.125663i
$281$	6132.87i	1.30198i	0.759086	+	0.650990i	$0.225646\pi$
$281$	6132.87i	1.30198i	−0.759086	+	0.650990i	$0.774354\pi$
$282$	10245.3	2.16346
$283$	−3377.15	−0.709367	−0.354683	−	0.934986i	$-0.615411\pi$
$283$	−3377.15	−0.709367	−0.354683	+	0.934986i	$0.615411\pi$
$284$	− 3585.70i	− 0.749198i
$285$	1467.76	0.305061
$286$	0	0
$287$	2966.29	0.610086
$288$	9030.99i	1.84776i
$289$	−4908.94	−0.999173
$290$	1801.96	0.364879
$291$	− 10199.0i	− 2.05455i
$292$	5990.60i	1.20059i
$293$	4704.77i	0.938073i	0.883179	+	0.469037i	$0.155399\pi$
$293$	4704.77i	0.938073i	−0.883179	+	0.469037i	$0.844601\pi$
$294$	− 9966.34i	− 1.97704i
$295$	−1454.10	−0.286986
$296$	−4072.75	−0.799742
$297$	− 7334.86i	− 1.43304i
$298$	−8258.86	−1.60545
$299$	0	0
$300$	−13027.0	−2.50704
$301$	4087.28i	0.782681i
$302$	1930.62	0.367864
$303$	−8430.17	−1.59835
$304$	145.863i	0.0275192i
$305$	− 452.922i	− 0.0850303i
$306$	445.118i	0.0831560i
$307$	− 5130.49i	− 0.953787i	−0.878961	−	0.476894i	$-0.841763\pi$
$307$	− 5130.49i	− 0.953787i	0.878961	−	0.476894i	$-0.158237\pi$
$308$	4827.86	0.893159
$309$	16498.2	3.03738
$310$	1743.84i	0.319496i
$311$	−7948.94	−1.44933	−0.724667	−	0.689099i	$-0.758006\pi$
$311$	−7948.94	−1.44933	−0.724667	+	0.689099i	$0.758006\pi$
$312$	0	0
$313$	−8521.87	−1.53893	−0.769465	−	0.638689i	$-0.779477\pi$
$313$	−8521.87	−1.53893	−0.769465	+	0.638689i	$0.779477\pi$
$314$	− 6031.14i	− 1.08394i
$315$	1300.00	0.232529
$316$	−480.776	−0.0855879
$317$	− 6662.46i	− 1.18044i	−0.807241	−	0.590222i	$-0.799040\pi$
$317$	− 6662.46i	− 1.18044i	0.807241	−	0.590222i	$-0.200960\pi$
$318$	− 24270.7i	− 4.27998i
$319$	− 5546.59i	− 0.973509i
$320$	2334.23i	0.407773i
$321$	−16557.2	−2.87893
$322$	194.907	0.0337322
$323$	121.297i	0.0208951i
$324$	−3949.74	−0.677253
$325$	0	0
$326$	16450.8	2.79486
$327$	− 7781.49i	− 1.31595i
$328$	−6803.64	−1.14533
$329$	2472.78	0.414374
$330$	− 4385.09i	− 0.731489i
$331$	− 3911.77i	− 0.649578i	−0.945786	−	0.324789i	$-0.894707\pi$
$331$	− 3911.77i	− 0.649578i	0.945786	−	0.324789i	$-0.105293\pi$
$332$	975.379i	0.161238i
$333$	8992.61i	1.47986i
$334$	15579.7	2.55234
$335$	140.104	0.0228498
$336$	201.227i	0.0326722i
$337$	627.211	0.101384	0.0506919	−	0.998714i	$-0.483857\pi$
$337$	627.211	0.101384	0.0506919	+	0.998714i	$0.483857\pi$
$338$	0	0
$339$	−2908.34	−0.465957
$340$	72.4674i	0.0115591i
$341$	5367.69	0.852424
$342$	−13295.6	−2.10217
$343$	− 5685.08i	− 0.894943i
$344$	− 9374.80i	− 1.46935i
$345$	− 108.968i	− 0.0170048i
$346$	10684.2i	1.66007i
$347$	−3823.02	−0.591442	−0.295721	−	0.955274i	$-0.595560\pi$
$347$	−3823.02	−0.591442	−0.295721	+	0.955274i	$0.595560\pi$
$348$	−15649.4	−2.41063
$349$	− 3410.67i	− 0.523120i	−0.965187	−	0.261560i	$-0.915763\pi$
$349$	− 3410.67i	− 0.523120i	0.965187	−	0.261560i	$-0.0842369\pi$
$350$	−5108.10	−0.780112
$351$	0	0
$352$	7352.48	1.11332
$353$	− 5587.64i	− 0.842492i	−0.906946	−	0.421246i	$-0.861593\pi$
$353$	− 5587.64i	− 0.842492i	0.906946	−	0.421246i	$-0.138407\pi$
$354$	20516.3	3.08031
$355$	−786.070	−0.117522
$356$	2597.49i	0.386704i
$357$	167.336i	0.0248077i
$358$	− 3042.06i	− 0.449100i
$359$	− 2230.14i	− 0.327861i	−0.986472	−	0.163931i	$-0.947583\pi$
$359$	− 2230.14i	− 0.327861i	0.986472	−	0.163931i	$-0.0524173\pi$
$360$	−2981.75	−0.436533
$361$	3235.89	0.471774
$362$	− 3197.82i	− 0.464292i
$363$	−1938.38	−0.280272
$364$	0	0
$365$	1313.28	0.188330
$366$	6390.40i	0.912655i
$367$	−8699.14	−1.23731	−0.618653	−	0.785664i	$-0.712321\pi$
$367$	−8699.14	−1.23731	−0.618653	+	0.785664i	$0.712321\pi$
$368$	10.8291	0.00153398
$369$	15022.4i	2.11934i
$370$	2378.51i	0.334197i
$371$	− 5857.94i	− 0.819756i
$372$	− 15144.7i	− 2.11080i
$373$	10964.2	1.52199	0.760997	−	0.648755i	$-0.224710\pi$
$373$	10964.2	1.52199	0.760997	+	0.648755i	$0.224710\pi$
$374$	362.388	0.0501033
$375$	5903.88i	0.813000i
$376$	−5671.70	−0.777914
$377$	0	0
$378$	−8114.84	−1.10419
$379$	13910.1i	1.88526i	0.333835	+	0.942631i	$0.391657\pi$
$379$	13910.1i	1.88526i	−0.333835	+	0.942631i	$0.608343\pi$
$380$	−2164.58	−0.292213
$381$	−5392.24	−0.725074
$382$	− 5934.03i	− 0.794794i
$383$	− 494.753i	− 0.0660071i	−0.999455	−	0.0330035i	$-0.989493\pi$
$383$	− 494.753i	− 0.0660071i	0.999455	−	0.0330035i	$-0.0105073\pi$
$384$	− 19976.7i	− 2.65477i
$385$	− 1058.38i	− 0.140104i
$386$	2368.97	0.312376
$387$	−20699.5	−2.71891
$388$	15041.0i	1.96802i
$389$	4140.47	0.539666	0.269833	−	0.962907i	$-0.413032\pi$
$389$	4140.47	0.539666	0.269833	+	0.962907i	$0.413032\pi$
$390$	0	0
$391$	9.00524	0.00116474
$392$	5517.30i	0.710881i
$393$	−11565.4	−1.48447
$394$	14236.8	1.82041
$395$	105.398i	0.0134256i
$396$	24450.1i	3.10269i
$397$	1881.79i	0.237895i	0.992901	+	0.118948i	$0.0379521\pi$
$397$	1881.79i	0.237895i	−0.992901	+	0.118948i	$0.962048\pi$
$398$	5642.90i	0.710686i
$399$	−4998.29	−0.627137
$400$	−283.807	−0.0354759
$401$	− 421.765i	− 0.0525236i	−0.999655	−	0.0262618i	$-0.991640\pi$
$401$	− 421.765i	− 0.0525236i	0.999655	−	0.0262618i	$-0.00836035\pi$
$402$	−1976.76	−0.245254
$403$	0	0
$404$	12432.5	1.53103
$405$	865.876i	0.106236i
$406$	−6136.41	−0.750110
$407$	7321.23	0.891646
$408$	− 383.811i	− 0.0465722i
$409$	2550.22i	0.308313i	0.988046	+	0.154157i	$0.0492660\pi$
$409$	2550.22i	0.308313i	−0.988046	+	0.154157i	$0.950734\pi$
$410$	3973.37i	0.478611i
$411$	− 5401.99i	− 0.648322i
$412$	−24330.9	−2.90946
$413$	4951.79	0.589980
$414$	987.083i	0.117180i
$415$	213.826	0.0252923
$416$	0	0
$417$	2870.98	0.337152
$418$	10824.5i	1.26661i
$419$	12384.8	1.44400	0.722002	−	0.691891i	$-0.243222\pi$
$419$	12384.8	1.44400	0.722002	+	0.691891i	$0.243222\pi$
$420$	−2986.17	−0.346929
$421$	10463.0i	1.21124i	0.795752	+	0.605622i	$0.207076\pi$
$421$	10463.0i	1.21124i	−0.795752	+	0.605622i	$0.792924\pi$
$422$	11548.3i	1.33214i
$423$	12523.1i	1.43947i
$424$	13436.1i	1.53895i
$425$	−236.008	−0.0269366
$426$	11090.9	1.26140
$427$	1542.38i	0.174803i
$428$	24417.9	2.75767
$429$	0	0
$430$	−5474.94	−0.614012
$431$	− 3962.39i	− 0.442834i	−0.975179	−	0.221417i	$-0.928932\pi$
$431$	− 3962.39i	− 0.442834i	0.975179	−	0.221417i	$-0.0710682\pi$
$432$	−450.863	−0.0502133
$433$	8394.14	0.931632	0.465816	−	0.884882i	$-0.345761\pi$
$433$	8394.14	0.931632	0.465816	+	0.884882i	$0.345761\pi$
$434$	− 5938.48i	− 0.656812i
$435$	3430.73i	0.378140i
$436$	11475.8i	1.26053i
$437$	268.984i	0.0294446i
$438$	−18529.4	−2.02140
$439$	−10174.5	−1.10616	−0.553079	−	0.833129i	$-0.686547\pi$
$439$	−10174.5	−1.10616	−0.553079	+	0.833129i	$0.686547\pi$
$440$	2427.56i	0.263021i
$441$	12182.2	1.31543
$442$	0	0
$443$	−5880.74	−0.630705	−0.315353	−	0.948975i	$-0.602123\pi$
$443$	−5880.74	−0.630705	−0.315353	+	0.948975i	$0.602123\pi$
$444$	− 20656.5i	− 2.20792i
$445$	569.431	0.0606598
$446$	−5453.48	−0.578990
$447$	− 15723.9i	− 1.66379i
$448$	− 7948.97i	− 0.838289i
$449$	− 10664.9i	− 1.12095i	−0.828172	−	0.560475i	$-0.810619\pi$
$449$	− 10664.9i	− 1.12095i	0.828172	−	0.560475i	$-0.189381\pi$
$450$	− 25869.3i	− 2.70998i
$451$	12230.3	1.27695
$452$	4289.10	0.446332
$453$	3675.68i	0.381233i
$454$	−3964.87	−0.409869
$455$	0	0
$456$	11464.3	1.17734
$457$	− 14828.9i	− 1.51787i	−0.651169	−	0.758933i	$-0.725721\pi$
$457$	− 14828.9i	− 1.51787i	0.651169	−	0.758933i	$-0.274279\pi$
$458$	−21369.2	−2.18017
$459$	−374.928	−0.0381266
$460$	160.702i	0.0162886i
$461$	− 9711.70i	− 0.981169i	−0.871394	−	0.490585i	$-0.836783\pi$
$461$	− 9711.70i	− 0.981169i	0.871394	−	0.490585i	$-0.163217\pi$
$462$	14933.0i	1.50378i
$463$	− 11353.5i	− 1.13962i	−0.821777	−	0.569809i	$-0.807017\pi$
$463$	− 11353.5i	− 1.13962i	0.821777	−	0.569809i	$-0.192983\pi$
$464$	−340.941	−0.0341116
$465$	−3320.07	−0.331107
$466$	− 22210.1i	− 2.20787i
$467$	−6451.31	−0.639252	−0.319626	−	0.947544i	$-0.603557\pi$
$467$	−6451.31	−0.639252	−0.319626	+	0.947544i	$0.603557\pi$
$468$	0	0
$469$	−477.109	−0.0469741
$470$	3312.31i	0.325076i
$471$	11482.6	1.12333
$472$	−11357.7	−1.10758
$473$	16852.3i	1.63820i
$474$	− 1487.08i	− 0.144101i
$475$	− 7049.50i	− 0.680954i
$476$	− 246.780i	− 0.0237629i
$477$	29666.8	2.84770
$478$	21929.8	2.09842
$479$	9566.46i	0.912531i	0.889844	+	0.456266i	$0.150813\pi$
$479$	9566.46i	0.912531i	−0.889844	+	0.456266i	$0.849187\pi$
$480$	−4547.73	−0.432447
$481$	0	0
$482$	−26803.5	−2.53292
$483$	371.080i	0.0349581i
$484$	2858.64	0.268467
$485$	3297.35	0.308711
$486$	10697.9i	0.998489i
$487$	4917.11i	0.457527i	0.973482	+	0.228764i	$0.0734683\pi$
$487$	4917.11i	0.457527i	−0.973482	+	0.228764i	$0.926532\pi$
$488$	− 3537.68i	− 0.328162i
$489$	31320.3i	2.89643i
$490$	3222.14	0.297064
$491$	−2950.82	−0.271220	−0.135610	−	0.990762i	$-0.543299\pi$
$491$	−2950.82	−0.271220	−0.135610	+	0.990762i	$0.543299\pi$
$492$	− 34507.3i	− 3.16201i
$493$	−283.519	−0.0259007
$494$	0	0
$495$	5360.04	0.486699
$496$	− 329.944i	− 0.0298688i
$497$	2676.88	0.241599
$498$	−3016.93	−0.271470
$499$	13430.1i	1.20484i	0.798180	+	0.602418i	$0.205796\pi$
$499$	13430.1i	1.20484i	−0.798180	+	0.602418i	$0.794204\pi$
$500$	− 8706.79i	− 0.778759i
$501$	29661.9i	2.64510i
$502$	26485.6i	2.35480i
$503$	1320.29	0.117035	0.0585175	−	0.998286i	$-0.481363\pi$
$503$	1320.29	0.117035	0.0585175	+	0.998286i	$0.481363\pi$
$504$	10154.0	0.897414
$505$	− 2725.49i	− 0.240164i
$506$	803.623	0.0706036
$507$	0	0
$508$	7952.25	0.694536
$509$	20916.4i	1.82143i	0.413041	+	0.910713i	$0.364467\pi$
$509$	20916.4i	1.82143i	−0.413041	+	0.910713i	$0.635533\pi$
$510$	−224.148	−0.0194616
$511$	−4472.24	−0.387163
$512$	− 877.105i	− 0.0757089i
$513$	− 11199.0i	− 0.963837i
$514$	− 5454.98i	− 0.468110i
$515$	5333.90i	0.456388i
$516$	47548.0	4.05656
$517$	10195.5	0.867311
$518$	− 8099.77i	− 0.687033i
$519$	−20341.4	−1.72040
$520$	0	0
$521$	−10104.2	−0.849661	−0.424831	−	0.905273i	$-0.639666\pi$
$521$	−10104.2	−0.849661	−0.424831	+	0.905273i	$0.639666\pi$
$522$	− 31077.1i	− 2.60576i
$523$	7131.22	0.596227	0.298113	−	0.954530i	$-0.403643\pi$
$523$	7131.22	0.596227	0.298113	+	0.954530i	$0.403643\pi$
$524$	17056.2	1.42195
$525$	− 9725.21i	− 0.808463i
$526$	− 1068.24i	− 0.0885507i
$527$	− 274.374i	− 0.0226792i
$528$	829.682i	0.0683849i
$529$	−12147.0	−0.998359
$530$	7846.76	0.643097
$531$	25077.7i	2.04949i
$532$	7371.27	0.600724
$533$	0	0
$534$	−8034.26	−0.651080
$535$	− 5352.98i	− 0.432579i
$536$	1094.32	0.0881856
$537$	5791.72	0.465421
$538$	12171.5i	0.975369i
$539$	− 9917.98i	− 0.792575i
$540$	− 6690.72i	− 0.533190i
$541$	− 16831.7i	− 1.33762i	−0.743435	−	0.668809i	$-0.766805\pi$
$541$	− 16831.7i	− 1.33762i	0.743435	−	0.668809i	$-0.233195\pi$
$542$	−26006.7	−2.06104
$543$	6088.27	0.481165
$544$	− 375.828i	− 0.0296204i
$545$	2515.77	0.197731
$546$	0	0
$547$	−9560.55	−0.747312	−0.373656	−	0.927567i	$-0.621896\pi$
$547$	−9560.55	−0.747312	−0.373656	+	0.927567i	$0.621896\pi$
$548$	7966.62i	0.621017i
$549$	−7811.19	−0.607238
$550$	−21061.2	−1.63282
$551$	− 8468.64i	− 0.654766i
$552$	− 851.129i	− 0.0656276i
$553$	− 358.920i	− 0.0276001i
$554$	− 32626.5i	− 2.50210i
$555$	−4528.40	−0.346342
$556$	−4234.00	−0.322952
$557$	22827.9i	1.73653i	0.496097	+	0.868267i	$0.334766\pi$
$557$	22827.9i	1.73653i	−0.496097	+	0.868267i	$0.665234\pi$
$558$	30074.7	2.28166
$559$	0	0
$560$	−65.0571	−0.00490922
$561$	689.944i	0.0519242i
$562$	27975.4	2.09977
$563$	−21629.7	−1.61916	−0.809578	−	0.587013i	$-0.800304\pi$
$563$	−21629.7	−1.61916	−0.809578	+	0.587013i	$0.800304\pi$
$564$	− 28766.3i	− 2.14766i
$565$	− 940.271i	− 0.0700133i
$566$	15405.1i	1.14403i
$567$	− 2948.65i	− 0.218398i
$568$	−6139.83	−0.453559
$569$	10589.9	0.780229	0.390114	−	0.920766i	$-0.372436\pi$
$569$	10589.9	0.780229	0.390114	+	0.920766i	$0.372436\pi$
$570$	− 6695.25i	− 0.491988i
$571$	1757.27	0.128791	0.0643954	−	0.997924i	$-0.479488\pi$
$571$	1757.27	0.128791	0.0643954	+	0.997924i	$0.479488\pi$
$572$	0	0
$573$	11297.7	0.823679
$574$	− 13530.9i	− 0.983917i
$575$	−523.365	−0.0379580
$576$	40256.6	2.91208
$577$	− 13580.6i	− 0.979840i	−0.871767	−	0.489920i	$-0.837026\pi$
$577$	− 13580.6i	− 0.979840i	0.871767	−	0.489920i	$-0.162974\pi$
$578$	22392.4i	1.61142i
$579$	4510.23i	0.323729i
$580$	− 5059.49i	− 0.362214i
$581$	−728.163	−0.0519953
$582$	−46523.2	−3.31349
$583$	− 24152.9i	− 1.71580i
$584$	10257.8	0.726831
$585$	0	0
$586$	21461.0	1.51288
$587$	− 957.326i	− 0.0673136i	−0.999433	−	0.0336568i	$-0.989285\pi$
$587$	− 957.326i	− 0.0673136i	0.999433	−	0.0336568i	$-0.0107153\pi$
$588$	−27983.1	−1.96259
$589$	8195.49	0.573327
$590$	6632.95i	0.462838i
$591$	27105.2i	1.88657i
$592$	− 450.026i	− 0.0312431i
$593$	− 6729.49i	− 0.466015i	−0.972475	−	0.233007i	$-0.925143\pi$
$593$	− 6729.49i	− 0.466015i	0.972475	−	0.233007i	$-0.0748567\pi$
$594$	−33458.4	−2.31113
$595$	−54.1000	−0.00372754
$596$	23188.9i	1.59372i
$597$	−10743.4	−0.736514
$598$	0	0
$599$	2281.52	0.155626	0.0778132	−	0.996968i	$-0.475206\pi$
$599$	2281.52	0.155626	0.0778132	+	0.996968i	$0.475206\pi$
$600$	22306.2i	1.51775i
$601$	6401.42	0.434475	0.217237	−	0.976119i	$-0.430295\pi$
$601$	6401.42	0.434475	0.217237	+	0.976119i	$0.430295\pi$
$602$	18644.4	1.26227
$603$	− 2416.26i	− 0.163180i
$604$	− 5420.74i	− 0.365177i
$605$	− 626.682i	− 0.0421128i
$606$	38454.7i	2.57775i
$607$	2779.24	0.185841	0.0929207	−	0.995674i	$-0.470380\pi$
$607$	2779.24	0.185841	0.0929207	+	0.995674i	$0.470380\pi$
$608$	11225.9	0.748800
$609$	− 11683.0i	− 0.777371i
$610$	−2066.03	−0.137133
$611$	0	0
$612$	1249.79	0.0825485
$613$	22620.8i	1.49045i	0.666812	+	0.745226i	$0.267658\pi$
$613$	22620.8i	1.49045i	−0.666812	+	0.745226i	$0.732342\pi$
$614$	−23403.0	−1.53822
$615$	−7564.82	−0.496005
$616$	− 8266.80i	− 0.540712i
$617$	21974.0i	1.43378i	0.697187	+	0.716889i	$0.254435\pi$
$617$	21974.0i	1.43378i	−0.697187	+	0.716889i	$0.745565\pi$
$618$	− 75257.5i	− 4.89854i
$619$	− 7145.19i	− 0.463957i	−0.972721	−	0.231979i	$-0.925480\pi$
$619$	− 7145.19i	− 0.463957i	0.972721	−	0.231979i	$-0.0745199\pi$
$620$	4896.30	0.317162
$621$	−831.430	−0.0537265
$622$	36259.5i	2.33742i
$623$	−1939.14	−0.124703
$624$	0	0
$625$	12730.8	0.814773
$626$	38873.0i	2.48191i
$627$	−20608.5	−1.31264
$628$	−16934.0	−1.07602
$629$	− 374.231i	− 0.0237227i
$630$	− 5930.02i	− 0.375012i
$631$	18883.2i	1.19133i	0.803232	+	0.595666i	$0.203112\pi$
$631$	18883.2i	1.19133i	−0.803232	+	0.595666i	$0.796888\pi$
$632$	823.239i	0.0518144i
$633$	−21986.7	−1.38056
$634$	−30391.1	−1.90376
$635$	− 1743.32i	− 0.108947i
$636$	−68146.4	−4.24871
$637$	0	0
$638$	−25301.1	−1.57003
$639$	13556.7i	0.839274i
$640$	6458.50	0.398898
$641$	3631.08	0.223743	0.111871	−	0.993723i	$-0.464316\pi$
$641$	3631.08	0.223743	0.111871	+	0.993723i	$0.464316\pi$
$642$	75526.7i	4.64299i
$643$	− 10772.0i	− 0.660660i	−0.943866	−	0.330330i	$-0.892840\pi$
$643$	− 10772.0i	− 0.660660i	0.943866	−	0.330330i	$-0.107160\pi$
$644$	− 547.253i	− 0.0334857i
$645$	− 10423.6i	− 0.636327i
$646$	553.301	0.0336987
$647$	−15148.3	−0.920464	−0.460232	−	0.887799i	$-0.652234\pi$
$647$	−15148.3	−0.920464	−0.460232	+	0.887799i	$0.652234\pi$
$648$	6763.18i	0.410004i
$649$	20416.7	1.23487
$650$	0	0
$651$	11306.2	0.680682
$652$	− 46189.8i	− 2.77444i
$653$	7358.89	0.441004	0.220502	−	0.975387i	$-0.429230\pi$
$653$	7358.89	0.441004	0.220502	+	0.975387i	$0.429230\pi$
$654$	−35495.7	−2.12231
$655$	− 3739.11i	− 0.223052i
$656$	− 751.780i	− 0.0447441i
$657$	− 22649.1i	− 1.34494i
$658$	− 11279.7i	− 0.668282i
$659$	28333.3	1.67482	0.837411	−	0.546574i	$-0.184068\pi$
$659$	28333.3	1.67482	0.837411	+	0.546574i	$0.184068\pi$
$660$	−12312.3	−0.726146
$661$	− 1109.68i	− 0.0652975i	−0.999467	−	0.0326488i	$-0.989606\pi$
$661$	− 1109.68i	− 0.0652975i	0.999467	−	0.0326488i	$-0.0103943\pi$
$662$	−17843.8	−1.04761
$663$	0	0
$664$	1670.15	0.0976121
$665$	− 1615.96i	− 0.0942317i
$666$	41020.3	2.38664
$667$	−628.724	−0.0364982
$668$	− 43744.1i	− 2.53369i
$669$	− 10382.8i	− 0.600032i
$670$	− 639.091i	− 0.0368511i
$671$	6359.39i	0.365874i
$672$	15486.8	0.889013
$673$	−20979.1	−1.20161	−0.600806	−	0.799395i	$-0.705154\pi$
$673$	−20979.1	−1.20161	−0.600806	+	0.799395i	$0.705154\pi$
$674$	− 2861.06i	− 0.163507i
$675$	21790.0	1.24251
$676$	0	0
$677$	30941.9	1.75656	0.878282	−	0.478142i	$-0.158690\pi$
$677$	30941.9	1.75656	0.878282	+	0.478142i	$0.158690\pi$
$678$	13266.5i	0.751473i
$679$	−11228.8	−0.634641
$680$	124.087	0.00699780
$681$	− 7548.64i	− 0.424764i
$682$	− 24485.0i	− 1.37475i
$683$	− 5426.21i	− 0.303995i	−0.988381	−	0.151997i	$-0.951429\pi$
$683$	− 5426.21i	− 0.303995i	0.988381	−	0.151997i	$-0.0485705\pi$
$684$	37330.9i	2.08682i
$685$	1746.47	0.0974150
$686$	−25932.8	−1.44332
$687$	− 40684.5i	− 2.25940i
$688$	1035.89	0.0574023
$689$	0	0
$690$	−497.065	−0.0274245
$691$	− 33792.7i	− 1.86040i	−0.367056	−	0.930199i	$-0.619634\pi$
$691$	− 33792.7i	− 1.86040i	0.367056	−	0.930199i	$-0.380366\pi$
$692$	29998.7	1.64795
$693$	−18253.1	−1.00054
$694$	17438.9i	0.953849i
$695$	928.192i	0.0506595i
$696$	26796.7i	1.45938i
$697$	− 625.164i	− 0.0339738i
$698$	−15557.9	−0.843662
$699$	42285.5	2.28810
$700$	14342.3i	0.774413i
$701$	−6905.96	−0.372089	−0.186045	−	0.982541i	$-0.559567\pi$
$701$	−6905.96	−0.372089	−0.186045	+	0.982541i	$0.559567\pi$
$702$	0	0
$703$	11178.2	0.599707
$704$	− 32774.5i	− 1.75459i
$705$	−6306.25	−0.336889
$706$	−25488.3	−1.35873
$707$	9281.37i	0.493723i
$708$	− 57604.9i	− 3.05781i
$709$	2007.13i	0.106318i	0.998586	+	0.0531589i	$0.0169290\pi$
$709$	2007.13i	0.106318i	−0.998586	+	0.0531589i	$0.983071\pi$
$710$	3585.70i	0.189534i
$711$	1817.71	0.0958782
$712$	4447.71	0.234108
$713$	− 608.445i	− 0.0319585i
$714$	763.312	0.0400088
$715$	0	0
$716$	−8541.38	−0.445819
$717$	41751.8i	2.17469i
$718$	−10172.9	−0.528759
$719$	12787.4	0.663270	0.331635	−	0.943408i	$-0.392400\pi$
$719$	12787.4	0.663270	0.331635	+	0.943408i	$0.392400\pi$
$720$	− 329.474i	− 0.0170538i
$721$	− 18164.1i	− 0.938231i
$722$	− 14760.7i	− 0.760854i
$723$	− 51030.7i	− 2.62497i
$724$	−8978.72	−0.460900
$725$	16477.5	0.844081
$726$	8842.03i	0.452009i
$727$	6090.70	0.310717	0.155359	−	0.987858i	$-0.450347\pi$
$727$	6090.70	0.310717	0.155359	+	0.987858i	$0.450347\pi$
$728$	0	0
$729$	−28693.9	−1.45780
$730$	− 5990.60i	− 0.303729i
$731$	861.420	0.0435852
$732$	17942.7	0.905988
$733$	− 38846.5i	− 1.95747i	−0.205117	−	0.978737i	$-0.565757\pi$
$733$	− 38846.5i	− 1.95747i	0.205117	−	0.978737i	$-0.434243\pi$
$734$	39681.6i	1.99547i
$735$	6134.57i	0.307860i
$736$	− 833.427i	− 0.0417399i
$737$	−1967.17	−0.0983198
$738$	68525.5	3.41797
$739$	− 14457.5i	− 0.719661i	−0.933018	−	0.359830i	$-0.882835\pi$
$739$	− 14457.5i	− 0.719661i	0.933018	−	0.359830i	$-0.117165\pi$
$740$	6678.29	0.331755
$741$	0	0
$742$	−26721.3	−1.32206
$743$	− 1277.80i	− 0.0630929i	−0.999502	−	0.0315464i	$-0.989957\pi$
$743$	− 1277.80i	− 0.0630929i	0.999502	−	0.0315464i	$-0.0100432\pi$
$744$	−25932.4	−1.27786
$745$	5083.56	0.249996
$746$	− 50013.7i	− 2.45460i
$747$	− 3687.69i	− 0.180623i
$748$	− 1017.50i	− 0.0497373i
$749$	18229.0i	0.889285i
$750$	26930.9	1.31117
$751$	13007.9	0.632042	0.316021	−	0.948752i	$-0.397653\pi$
$751$	13007.9	0.632042	0.316021	+	0.948752i	$0.397653\pi$
$752$	− 626.706i	− 0.0303904i
$753$	−50425.5	−2.44038
$754$	0	0
$755$	−1188.35	−0.0572829
$756$	22784.6i	1.09612i
$757$	−10723.2	−0.514850	−0.257425	−	0.966298i	$-0.582874\pi$
$757$	−10723.2	−0.514850	−0.257425	+	0.966298i	$0.582874\pi$
$758$	63451.7	3.04046
$759$	1530.00i	0.0731694i
$760$	3706.44i	0.176904i
$761$	− 13621.8i	− 0.648870i	−0.945908	−	0.324435i	$-0.894826\pi$
$761$	− 13621.8i	− 0.648870i	0.945908	−	0.324435i	$-0.105174\pi$
$762$	24597.0i	1.16936i
$763$	−8567.19	−0.406491
$764$	−16661.4	−0.788988
$765$	− 273.983i	− 0.0129489i
$766$	−2256.84	−0.106453
$767$	0	0
$768$	−33365.0	−1.56765
$769$	− 8495.15i	− 0.398365i	−0.979962	−	0.199183i	$-0.936171\pi$
$769$	− 8495.15i	− 0.398365i	0.979962	−	0.199183i	$-0.0638287\pi$
$770$	−4827.86	−0.225953
$771$	10385.6	0.485123
$772$	− 6651.50i	− 0.310094i
$773$	34262.5i	1.59423i	0.603830	+	0.797113i	$0.293641\pi$
$773$	34262.5i	1.59423i	−0.603830	+	0.797113i	$0.706359\pi$
$774$	94422.0i	4.38492i
$775$	15946.0i	0.739094i
$776$	25754.9	1.19143
$777$	15421.0	0.712002
$778$	− 18887.0i	− 0.870348i
$779$	18673.5	0.858854
$780$	0	0
$781$	11037.1	0.505681
$782$	− 41.0779i	− 0.00187844i
$783$	26176.5	1.19473
$784$	−609.644	−0.0277717
$785$	3712.33i	0.168788i
$786$	52756.2i	2.39408i
$787$	12642.6i	0.572629i	0.958136	+	0.286315i	$0.0924303\pi$
$787$	12642.6i	0.572629i	−0.958136	+	0.286315i	$0.907570\pi$
$788$	− 39973.6i	− 1.80711i
$789$	2033.81	0.0917688
$790$	480.776	0.0216522
$791$	3202.00i	0.143932i
$792$	41866.2	1.87834
$793$	0	0
$794$	8583.89	0.383666
$795$	14939.3i	0.666468i
$796$	15843.9	0.705494
$797$	−19084.4	−0.848184	−0.424092	−	0.905619i	$-0.639407\pi$
$797$	−19084.4	−0.848184	−0.424092	+	0.905619i	$0.639407\pi$
$798$	22800.0i	1.01142i
$799$	− 521.154i	− 0.0230752i
$800$	21842.3i	0.965304i
$801$	− 9820.53i	− 0.433198i
$802$	−1923.90	−0.0847075
$803$	−18439.5	−0.810357
$804$	5550.28i	0.243462i
$805$	−119.971	−0.00525269
$806$	0	0
$807$	−23173.0	−1.01082
$808$	− 21288.2i	− 0.926878i
$809$	11610.0	0.504558	0.252279	−	0.967655i	$-0.418820\pi$
$809$	11610.0	0.504558	0.252279	+	0.967655i	$0.418820\pi$
$810$	3949.74	0.171333
$811$	9613.36i	0.416240i	0.978103	+	0.208120i	$0.0667345\pi$
$811$	9613.36i	0.416240i	−0.978103	+	0.208120i	$0.933266\pi$
$812$	17229.6i	0.744630i
$813$	− 49513.7i	− 2.13594i
$814$	− 33396.2i	− 1.43800i
$815$	−10125.9	−0.435208
$816$	42.4099	0.00181941
$817$	25730.4i	1.10183i
$818$	11632.9	0.497233
$819$	0	0
$820$	11156.3	0.475115
$821$	− 26481.5i	− 1.12571i	−0.826555	−	0.562856i	$-0.809703\pi$
$821$	− 26481.5i	− 1.12571i	0.826555	−	0.562856i	$-0.190297\pi$
$822$	−24641.5	−1.04558
$823$	−13814.5	−0.585107	−0.292553	−	0.956249i	$-0.594505\pi$
$823$	−13814.5	−0.585107	−0.292553	+	0.956249i	$0.594505\pi$
$824$	41662.0i	1.76136i
$825$	− 40098.1i	− 1.69216i
$826$	− 22587.8i	− 0.951491i
$827$	44401.0i	1.86696i	0.358633	+	0.933479i	$0.383243\pi$
$827$	44401.0i	1.86696i	−0.358633	+	0.933479i	$0.616757\pi$
$828$	2771.50	0.116324
$829$	−24337.4	−1.01963	−0.509815	−	0.860284i	$-0.670286\pi$
$829$	−24337.4	−1.01963	−0.509815	+	0.860284i	$0.670286\pi$
$830$	− 975.379i	− 0.0407902i
$831$	62116.9	2.59303
$832$	0	0
$833$	−506.966	−0.0210868
$834$	− 13096.1i	− 0.543743i
$835$	−9589.73	−0.397445
$836$	30392.5	1.25735
$837$	25332.2i	1.04613i
$838$	− 56494.0i	− 2.32882i
$839$	24680.1i	1.01556i	0.861488	+	0.507778i	$0.169533\pi$
$839$	24680.1i	1.01556i	−0.861488	+	0.507778i	$0.830467\pi$
$840$	5113.26i	0.210029i
$841$	−4594.43	−0.188381
$842$	47727.4	1.95344
$843$	53261.9i	2.17608i
$844$	32425.0	1.32241
$845$	0	0
$846$	57124.8	2.32150
$847$	2134.10i	0.0865744i
$848$	−1484.65	−0.0601214
$849$	−29329.4	−1.18561
$850$	1076.56i	0.0434421i
$851$	− 829.885i	− 0.0334290i
$852$	− 31140.6i	− 1.25218i
$853$	10151.7i	0.407490i	0.979024	+	0.203745i	$0.0653113\pi$
$853$	10151.7i	0.407490i	−0.979024	+	0.203745i	$0.934689\pi$
$854$	7035.65	0.281914
$855$	8183.81	0.327345
$856$	− 41811.1i	− 1.66948i
$857$	−2028.92	−0.0808713	−0.0404357	−	0.999182i	$-0.512875\pi$
$857$	−2028.92	−0.0808713	−0.0404357	+	0.999182i	$0.512875\pi$
$858$	0	0
$859$	6655.76	0.264367	0.132184	−	0.991225i	$-0.457801\pi$
$859$	6655.76	0.264367	0.132184	+	0.991225i	$0.457801\pi$
$860$	15372.3i	0.609526i
$861$	25761.2	1.01967
$862$	−18074.6	−0.714182
$863$	45690.8i	1.80224i	0.433568	+	0.901121i	$0.357254\pi$
$863$	45690.8i	1.80224i	−0.433568	+	0.901121i	$0.642746\pi$
$864$	34699.2i	1.36631i
$865$	− 6576.42i	− 0.258503i
$866$	− 38290.3i	− 1.50249i
$867$	−42632.5	−1.66998
$868$	−16673.9	−0.652014
$869$	− 1479.87i	− 0.0577688i
$870$	15649.4	0.609846
$871$	0	0
$872$	19650.1	0.763117
$873$	− 56866.8i	− 2.20464i
$874$	1226.99	0.0474868
$875$	6500.00	0.251132
$876$	52026.3i	2.00663i
$877$	30447.5i	1.17234i	0.810189	+	0.586168i	$0.199364\pi$
$877$	30447.5i	1.17234i	−0.810189	+	0.586168i	$0.800636\pi$
$878$	46411.6i	1.78396i
$879$	40859.3i	1.56786i
$880$	−268.237	−0.0102753
$881$	−32542.0	−1.24446	−0.622230	−	0.782835i	$-0.713773\pi$
$881$	−32542.0	−1.24446	−0.622230	+	0.782835i	$0.713773\pi$
$882$	− 55569.6i	− 2.12146i
$883$	−27641.9	−1.05348	−0.526741	−	0.850026i	$-0.676586\pi$
$883$	−27641.9	−1.05348	−0.526741	+	0.850026i	$0.676586\pi$
$884$	0	0
$885$	−12628.4	−0.479658
$886$	26825.3i	1.01717i
$887$	40099.9	1.51795	0.758976	−	0.651119i	$-0.225700\pi$
$887$	40099.9	1.51795	0.758976	+	0.651119i	$0.225700\pi$
$888$	−35370.4	−1.33666
$889$	5936.70i	0.223971i
$890$	− 2597.49i	− 0.0978293i
$891$	− 12157.6i	− 0.457121i
$892$	15312.1i	0.574761i
$893$	15566.8	0.583339
$894$	−71725.4	−2.68328
$895$	1872.47i	0.0699328i
$896$	−21993.8	−0.820044
$897$	0	0
$898$	−48648.4	−1.80781
$899$	19156.1i	0.710670i
$900$	−72634.9	−2.69018
$901$	−1234.60	−0.0456497
$902$	− 55789.3i	− 2.05940i
$903$	35496.7i	1.30814i
$904$	− 7344.26i	− 0.270206i
$905$	1968.35i	0.0722984i
$906$	16766.8	0.614835
$907$	36824.9	1.34812	0.674062	−	0.738674i	$-0.264548\pi$
$907$	36824.9	1.34812	0.674062	+	0.738674i	$0.264548\pi$
$908$	11132.4i	0.406874i
$909$	−47004.3	−1.71511
$910$	0	0
$911$	34520.5	1.25545	0.627725	−	0.778435i	$-0.283986\pi$
$911$	34520.5	1.25545	0.627725	+	0.778435i	$0.283986\pi$
$912$	1266.77i	0.0459946i
$913$	−3002.29	−0.108830
$914$	−67642.6	−2.44794
$915$	− 3933.47i	− 0.142117i
$916$	59999.8i	2.16424i
$917$	12733.2i	0.458545i
$918$	1710.25i	0.0614888i
$919$	23522.8	0.844336	0.422168	−	0.906518i	$-0.361269\pi$
$919$	23522.8	0.844336	0.422168	+	0.906518i	$0.361269\pi$
$920$	275.171	0.00986101
$921$	− 44556.6i	− 1.59412i
$922$	−44300.4	−1.58238
$923$	0	0
$924$	41928.3	1.49279
$925$	21749.5i	0.773102i
$926$	−51789.7	−1.83792
$927$	91989.6	3.25926
$928$	26239.4i	0.928180i
$929$	− 24563.2i	− 0.867482i	−0.901038	−	0.433741i	$-0.857193\pi$
$929$	− 24563.2i	− 0.867482i	0.901038	−	0.433741i	$-0.142807\pi$
$930$	15144.7i	0.533994i
$931$	− 15143.0i	− 0.533073i
$932$	−62360.9	−2.19174
$933$	−69033.8	−2.42236
$934$	29428.0i	1.03096i
$935$	−223.060	−0.00780197
$936$	0	0
$937$	−12115.6	−0.422411	−0.211206	−	0.977442i	$-0.567739\pi$
$937$	−12115.6	−0.422411	−0.211206	+	0.977442i	$0.567739\pi$
$938$	2176.36i	0.0757576i
$939$	−74009.6	−2.57211
$940$	9300.19	0.322701
$941$	14898.3i	0.516123i	0.966128	+	0.258062i	$0.0830837\pi$
$941$	14898.3i	0.516123i	−0.966128	+	0.258062i	$0.916916\pi$
$942$	− 52378.4i	− 1.81166i
$943$	− 1386.35i	− 0.0478745i
$944$	− 1254.99i	− 0.0432695i
$945$	4994.91	0.171941
$946$	76872.7	2.64201
$947$	− 7434.32i	− 0.255103i	−0.991832	−	0.127552i	$-0.959288\pi$
$947$	− 7434.32i	− 0.255103i	0.991832	−	0.127552i	$-0.0407119\pi$
$948$	−4175.38	−0.143049
$949$	0	0
$950$	−32156.7	−1.09821
$951$	− 57861.2i	− 1.97295i
$952$	−422.564	−0.0143859
$953$	23528.6	0.799754	0.399877	−	0.916569i	$-0.369053\pi$
$953$	23528.6	0.799754	0.399877	+	0.916569i	$0.369053\pi$
$954$	− 135327.i	− 4.59263i
$955$	3652.56i	0.123764i
$956$	− 61573.8i	− 2.08309i
$957$	− 48170.2i	− 1.62709i
$958$	43637.9	1.47169
$959$	−5947.43	−0.200263
$960$	20272.0i	0.681536i
$961$	11252.7	0.377723
$962$	0	0
$963$	−92318.7	−3.08923
$964$	75257.9i	2.51441i
$965$	−1458.17	−0.0486425
$966$	1692.70	0.0563787
$967$	23558.0i	0.783427i	0.920087	+	0.391713i	$0.128118\pi$
$967$	23558.0i	0.783427i	−0.920087	+	0.391713i	$0.871882\pi$
$968$	− 4894.88i	− 0.162528i
$969$	1053.42i	0.0349234i
$970$	− 15041.0i	− 0.497875i
$971$	−262.338	−0.00867026	−0.00433513	−	0.999991i	$-0.501380\pi$
$971$	−262.338	−0.00867026	−0.00433513	+	0.999991i	$0.501380\pi$
$972$	30037.1	0.991194
$973$	− 3160.86i	− 0.104144i
$974$	22429.7	0.737878
$975$	0	0
$976$	390.903	0.0128202
$977$	33144.4i	1.08535i	0.839944	+	0.542673i	$0.182588\pi$
$977$	33144.4i	1.08535i	−0.839944	+	0.542673i	$0.817412\pi$
$978$	142869.	4.67122
$979$	−7995.28	−0.261011
$980$	− 9047.00i	− 0.294894i
$981$	− 43387.5i	− 1.41208i
$982$	13460.3i	0.437410i
$983$	4866.80i	0.157911i	0.996878	+	0.0789557i	$0.0251586\pi$
$983$	4866.80i	0.157911i	−0.996878	+	0.0789557i	$0.974841\pi$
$984$	−59087.3	−1.91426
$985$	−8763.17	−0.283470
$986$	1293.28i	0.0417714i
$987$	21475.3	0.692569
$988$	0	0
$989$	1910.26	0.0614184
$990$	− 24450.1i	− 0.784924i
$991$	12533.9	0.401770	0.200885	−	0.979615i	$-0.435618\pi$
$991$	12533.9	0.401770	0.200885	+	0.979615i	$0.435618\pi$
$992$	−25393.1	−0.812733
$993$	− 33972.4i	− 1.08568i
$994$	− 12210.7i	− 0.389639i
$995$	− 3473.37i	− 0.110666i
$996$	8470.83i	0.269487i
$997$	−3560.92	−0.113115	−0.0565574	−	0.998399i	$-0.518012\pi$
$997$	−3560.92	−0.113115	−0.0565574	+	0.998399i	$0.518012\pi$
$998$	61262.1	1.94310
$999$	34551.8i	1.09426i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.4.b.e.168.1		4
13.2	odd	12		13.4.c.b.9.2	yes	4
13.3	even	3		169.4.e.g.147.4		8
13.4	even	6		169.4.e.g.23.4		8
13.5	odd	4		169.4.a.f.1.1		2
13.6	odd	12		13.4.c.b.3.2	✓	4
13.7	odd	12		169.4.c.f.146.1		4
13.8	odd	4		169.4.a.j.1.2		2
13.9	even	3		169.4.e.g.23.1		8
13.10	even	6		169.4.e.g.147.1		8
13.11	odd	12		169.4.c.f.22.1		4
13.12	even	2	inner	169.4.b.e.168.4		4
39.2	even	12		117.4.g.d.100.1		4
39.5	even	4		1521.4.a.t.1.2		2
39.8	even	4		1521.4.a.l.1.1		2
39.32	even	12		117.4.g.d.55.1		4
52.15	even	12		208.4.i.e.113.2		4
52.19	even	12		208.4.i.e.81.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.4.c.b.3.2	✓	4	13.6	odd	12
13.4.c.b.9.2	yes	4	13.2	odd	12
117.4.g.d.55.1		4	39.32	even	12
117.4.g.d.100.1		4	39.2	even	12
169.4.a.f.1.1		2	13.5	odd	4
169.4.a.j.1.2		2	13.8	odd	4
169.4.b.e.168.1		4	1.1	even	1	trivial
169.4.b.e.168.4		4	13.12	even	2	inner
169.4.c.f.22.1		4	13.11	odd	12
169.4.c.f.146.1		4	13.7	odd	12
169.4.e.g.23.1		8	13.9	even	3
169.4.e.g.23.4		8	13.4	even	6
169.4.e.g.147.1		8	13.10	even	6
169.4.e.g.147.4		8	13.3	even	3
208.4.i.e.81.2		4	52.19	even	12
208.4.i.e.113.2		4	52.15	even	12
1521.4.a.l.1.1		2	39.8	even	4
1521.4.a.t.1.2		2	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)

\(f(q)\)	\(=\)	\(q-4.56155i q^{2} +8.68466 q^{3} -12.8078 q^{4} +2.80776i q^{5} -39.6155i q^{6} -9.56155i q^{7} +21.9309i q^{8} +48.4233 q^{9} +O(q^{10})\)	\(q-4.56155i q^{2} +8.68466 q^{3} -12.8078 q^{4} +2.80776i q^{5} -39.6155i q^{6} -9.56155i q^{7} +21.9309i q^{8} +48.4233 q^{9} +12.8078 q^{10} -39.4233i q^{11} -111.231 q^{12} -43.6155 q^{14} +24.3845i q^{15} -2.42329 q^{16} -2.01515 q^{17} -220.885i q^{18} -60.1922i q^{19} -35.9612i q^{20} -83.0388i q^{21} -179.831 q^{22} -4.46876 q^{23} +190.462i q^{24} +117.116 q^{25} +186.054 q^{27} +122.462i q^{28} +140.693 q^{29} +111.231 q^{30} +136.155i q^{31} +186.501i q^{32} -342.378i q^{33} +9.19224i q^{34} +26.8466 q^{35} -620.194 q^{36} +185.708i q^{37} -274.570 q^{38} -61.5767 q^{40} +310.231i q^{41} -378.786 q^{42} -427.471 q^{43} +504.924i q^{44} +135.961i q^{45} +20.3845i q^{46} +258.617i q^{47} -21.0455 q^{48} +251.577 q^{49} -534.233i q^{50} -17.5009 q^{51} +612.656 q^{53} -848.695i q^{54} +110.691 q^{55} +209.693 q^{56} -522.749i q^{57} -641.779i q^{58} +517.885i q^{59} -312.311i q^{60} -161.311 q^{61} +621.080 q^{62} -463.002i q^{63} +831.348 q^{64} -1561.77 q^{66} -49.8987i q^{67} +25.8096 q^{68} -38.8096 q^{69} -122.462i q^{70} +279.963i q^{71} +1061.96i q^{72} -467.732i q^{73} +847.118 q^{74} +1017.12 q^{75} +770.928i q^{76} -376.948 q^{77} +37.5379 q^{79} -6.80403i q^{80} +308.386 q^{81} +1415.14 q^{82} -76.1553i q^{83} +1063.54i q^{84} -5.65808i q^{85} +1949.93i q^{86} +1221.87 q^{87} +864.587 q^{88} -202.806i q^{89} +620.194 q^{90} +57.2348 q^{92} +1182.46i q^{93} +1179.70 q^{94} +169.006 q^{95} +1619.70i q^{96} -1174.37i q^{97} -1147.58i q^{98} -1909.01i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 10 q^{3} - 10 q^{4} + 70 q^{9}+O(q^{10}) \) 4 * q + 10 * q^3 - 10 * q^4 + 70 * q^9	\( 4 q + 10 q^{3} - 10 q^{4} + 70 q^{9} + 10 q^{10} - 280 q^{12} - 92 q^{14} + 114 q^{16} - 140 q^{17} - 340 q^{22} - 290 q^{23} - 150 q^{25} + 670 q^{27} + 68 q^{29} + 280 q^{30} - 140 q^{35} - 1450 q^{36} - 620 q^{38} - 370 q^{40} - 740 q^{42} - 910 q^{43} - 480 q^{48} + 1130 q^{49} + 466 q^{51} + 1090 q^{53} + 1020 q^{55} + 344 q^{56} + 1004 q^{61} + 1000 q^{62} + 2542 q^{64} - 3196 q^{66} - 1010 q^{68} + 958 q^{69} + 1698 q^{74} + 3450 q^{75} - 510 q^{77} + 480 q^{79} + 244 q^{81} + 3030 q^{82} + 3230 q^{87} + 2040 q^{88} + 1450 q^{90} - 2080 q^{92} + 2080 q^{94} - 2540 q^{95}+O(q^{100}) \) 4 * q + 10 * q^3 - 10 * q^4 + 70 * q^9 + 10 * q^10 - 280 * q^12 - 92 * q^14 + 114 * q^16 - 140 * q^17 - 340 * q^22 - 290 * q^23 - 150 * q^25 + 670 * q^27 + 68 * q^29 + 280 * q^30 - 140 * q^35 - 1450 * q^36 - 620 * q^38 - 370 * q^40 - 740 * q^42 - 910 * q^43 - 480 * q^48 + 1130 * q^49 + 466 * q^51 + 1090 * q^53 + 1020 * q^55 + 344 * q^56 + 1004 * q^61 + 1000 * q^62 + 2542 * q^64 - 3196 * q^66 - 1010 * q^68 + 958 * q^69 + 1698 * q^74 + 3450 * q^75 - 510 * q^77 + 480 * q^79 + 244 * q^81 + 3030 * q^82 + 3230 * q^87 + 2040 * q^88 + 1450 * q^90 - 2080 * q^92 + 2080 * q^94 - 2540 * q^95

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