LMFDB - Embedded newform 169.10.a.f.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [169,10,Mod(1,169)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("169.1"); S:= CuspForms(chi, 10); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(169, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 10, names="a")

Level:	$N$	$=$	$169 = 13^{2}$
Weight:	$k$	$=$	$10$
Character orbit:	$[\chi]$	$=$	169.a (trivial)

Newform invariants

comment:select newform

sage:traces = [20,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$87.0410563117$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{20} - 7679 x^{18} + 24599364 x^{16} - 42662336000 x^{14} + 43527566862400 x^{12} + \cdots + 25\!\cdots\!36$ x^20 - 7679x^18 + 24599364x^16 - 42662336000x^14 + 43527566862400x^12 - 26625453181634304x^10 + 9550565219960082432x^8 - 1876223091685443895296x^6 + 172106193081772189679616x^4 - 4841822934658519419322368*x^2 + 25162285487879223389454336
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$2^{25}\cdot 3^{10}\cdot 13^{12}$
Twist minimal:	no (minimal twist has level 13)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-34.2355$ of defining polynomial
Character	$\chi$	$=$	169.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-34.2355 q^{2} +264.006 q^{3} +660.066 q^{4} -402.267 q^{5} -9038.36 q^{6} -9369.20 q^{7} -5069.12 q^{8} +50016.1 q^{9} +13771.8 q^{10} -17817.3 q^{11} +174261. q^{12} +320759. q^{14} -106201. q^{15} -164410. q^{16} +46549.2 q^{17} -1.71232e6 q^{18} +290646. q^{19} -265523. q^{20} -2.47352e6 q^{21} +609983. q^{22} -1.63303e6 q^{23} -1.33828e6 q^{24} -1.79131e6 q^{25} +8.00811e6 q^{27} -6.18429e6 q^{28} +351739. q^{29} +3.63583e6 q^{30} +6.33165e6 q^{31} +8.22405e6 q^{32} -4.70387e6 q^{33} -1.59363e6 q^{34} +3.76892e6 q^{35} +3.30139e7 q^{36} +1.22757e7 q^{37} -9.95040e6 q^{38} +2.03914e6 q^{40} -2.91222e6 q^{41} +8.46822e7 q^{42} -5.66418e6 q^{43} -1.17606e7 q^{44} -2.01198e7 q^{45} +5.59075e7 q^{46} +1.35751e7 q^{47} -4.34053e7 q^{48} +4.74283e7 q^{49} +6.13262e7 q^{50} +1.22893e7 q^{51} +5.34838e7 q^{53} -2.74161e8 q^{54} +7.16730e6 q^{55} +4.74936e7 q^{56} +7.67323e7 q^{57} -1.20420e7 q^{58} -9.40252e7 q^{59} -7.00996e7 q^{60} +1.56423e8 q^{61} -2.16767e8 q^{62} -4.68611e8 q^{63} -1.97376e8 q^{64} +1.61039e8 q^{66} -1.74234e8 q^{67} +3.07256e7 q^{68} -4.31129e8 q^{69} -1.29031e8 q^{70} +1.13072e7 q^{71} -2.53537e8 q^{72} +8.79707e7 q^{73} -4.20264e8 q^{74} -4.72915e8 q^{75} +1.91846e8 q^{76} +1.66934e8 q^{77} +3.00238e8 q^{79} +6.61369e7 q^{80} +1.12972e9 q^{81} +9.97013e7 q^{82} +7.05637e8 q^{83} -1.63269e9 q^{84} -1.87252e7 q^{85} +1.93916e8 q^{86} +9.28612e7 q^{87} +9.03179e7 q^{88} +5.36589e8 q^{89} +6.88811e8 q^{90} -1.07791e9 q^{92} +1.67159e9 q^{93} -4.64748e8 q^{94} -1.16917e8 q^{95} +2.17120e9 q^{96} -2.24820e8 q^{97} -1.62373e9 q^{98} -8.91151e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$20 q + 326 q^{3} + 5118 q^{4} + 129526 q^{9} + 88390 q^{10} + 427652 q^{12} + 473556 q^{14} + 1189618 q^{16} - 99312 q^{17} - 5073532 q^{22} + 6252378 q^{23} + 1529274 q^{25} + 18052718 q^{27} + 5424828 q^{29}+ \cdots + 9251202540 q^{95}+O(q^{100})$ 20 * q + 326 * q^3 + 5118 * q^4 + 129526 * q^9 + 88390 * q^10 + 427652 * q^12 + 473556 * q^14 + 1189618 * q^16 - 99312 * q^17 - 5073532 * q^22 + 6252378 * q^23 + 1529274 * q^25 + 18052718 * q^27 + 5424828 * q^29 + 30045516 * q^30 - 15098160 * q^35 + 54078886 * q^36 + 39021096 * q^38 + 134360674 * q^40 + 121600068 * q^42 + 53242058 * q^43 + 82771076 * q^48 + 14055298 * q^49 + 10208934 * q^51 - 36429786 * q^53 - 131454160 * q^55 + 127272672 * q^56 + 536425792 * q^61 - 890759796 * q^62 - 343337066 * q^64 - 445758060 * q^66 + 336594090 * q^68 + 1083885582 * q^69 + 1083869262 * q^74 - 1218826882 * q^75 + 1470187374 * q^77 + 1556703616 * q^79 + 2139127516 * q^81 + 3719322754 * q^82 + 1288246146 * q^87 - 4109160928 * q^88 + 6489878934 * q^90 + 10148843820 * q^92 + 4894712828 * q^94 + 9251202540 * q^95

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

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	−34.2355	−1.51301	−0.756504	−	0.653989i	$-0.773094\pi$
$2$	−34.2355	−1.51301	−0.756504	+	0.653989i	$0.773094\pi$
$3$	264.006	1.88178	0.940888	−	0.338718i	$-0.109993\pi$
$3$	264.006	1.88178	0.940888	+	0.338718i	$0.109993\pi$
$4$	660.066	1.28919
$5$	−402.267	−0.287839	−0.143919	−	0.989589i	$-0.545971\pi$
$5$	−402.267	−0.287839	−0.143919	+	0.989589i	$0.545971\pi$
$6$	−9038.36	−2.84714
$7$	−9369.20	−1.47490	−0.737448	−	0.675404i	$-0.763969\pi$
$7$	−9369.20	−1.47490	−0.737448	+	0.675404i	$0.763969\pi$
$8$	−5069.12	−0.437550
$9$	50016.1	2.54108
$10$	13771.8	0.435502
$11$	−17817.3	−0.366923	−0.183461	−	0.983027i	$-0.558730\pi$
$11$	−17817.3	−0.366923	−0.183461	+	0.983027i	$0.558730\pi$
$12$	174261.	2.42597
$13$	0	0
$13$	0	0
$14$	320759.	2.23153
$15$	−106201.	−0.541648
$16$	−164410.	−0.627176
$17$	46549.2	0.135174	0.0675868	−	0.997713i	$-0.478470\pi$
$17$	46549.2	0.135174	0.0675868	+	0.997713i	$0.478470\pi$
$18$	−1.71232e6	−3.84467
$19$	290646.	0.511650	0.255825	−	0.966723i	$-0.417653\pi$
$19$	290646.	0.511650	0.255825	+	0.966723i	$0.417653\pi$
$20$	−265523.	−0.371079
$21$	−2.47352e6	−2.77542
$22$	609983.	0.555157
$23$	−1.63303e6	−1.21680	−0.608399	−	0.793631i	$-0.708188\pi$
$23$	−1.63303e6	−1.21680	−0.608399	+	0.793631i	$0.708188\pi$
$24$	−1.33828e6	−0.823370
$25$	−1.79131e6	−0.917149
$26$	0	0
$27$	8.00811e6	2.89997
$28$	−6.18429e6	−1.90142
$29$	351739.	0.0923485	0.0461743	−	0.998933i	$-0.485297\pi$
$29$	351739.	0.0923485	0.0461743	+	0.998933i	$0.485297\pi$
$30$	3.63583e6	0.819518
$31$	6.33165e6	1.23137	0.615686	−	0.787992i	$-0.288879\pi$
$31$	6.33165e6	1.23137	0.615686	+	0.787992i	$0.288879\pi$
$32$	8.22405e6	1.38647
$33$	−4.70387e6	−0.690466
$34$	−1.59363e6	−0.204519
$35$	3.76892e6	0.424532
$36$	3.30139e7	3.27594
$37$	1.22757e7	1.07681	0.538404	−	0.842687i	$-0.319027\pi$
$37$	1.22757e7	1.07681	0.538404	+	0.842687i	$0.319027\pi$
$38$	−9.95040e6	−0.774131
$39$	0	0
$40$	2.03914e6	0.125944
$41$	−2.91222e6	−0.160952	−0.0804762	−	0.996757i	$-0.525644\pi$
$41$	−2.91222e6	−0.160952	−0.0804762	+	0.996757i	$0.525644\pi$
$42$	8.46822e7	4.19924
$43$	−5.66418e6	−0.252656	−0.126328	−	0.991989i	$-0.540319\pi$
$43$	−5.66418e6	−0.252656	−0.126328	+	0.991989i	$0.540319\pi$
$44$	−1.17606e7	−0.473034
$45$	−2.01198e7	−0.731421
$46$	5.59075e7	1.84102
$47$	1.35751e7	0.405790	0.202895	−	0.979201i	$-0.434965\pi$
$47$	1.35751e7	0.405790	0.202895	+	0.979201i	$0.434965\pi$
$48$	−4.34053e7	−1.18020
$49$	4.74283e7	1.17532
$50$	6.13262e7	1.38765
$51$	1.22893e7	0.254367
$52$	0	0
$53$	5.34838e7	0.931067	0.465534	−	0.885030i	$-0.345862\pi$
$53$	5.34838e7	0.931067	0.465534	+	0.885030i	$0.345862\pi$
$54$	−2.74161e8	−4.38767
$55$	7.16730e6	0.105615
$56$	4.74936e7	0.645340
$57$	7.67323e7	0.962811
$58$	−1.20420e7	−0.139724
$59$	−9.40252e7	−1.01021	−0.505104	−	0.863059i	$-0.668546\pi$
$59$	−9.40252e7	−1.01021	−0.505104	+	0.863059i	$0.668546\pi$
$60$	−7.00996e7	−0.698288
$61$	1.56423e8	1.44649	0.723247	−	0.690590i	$-0.242649\pi$
$61$	1.56423e8	1.44649	0.723247	+	0.690590i	$0.242649\pi$
$62$	−2.16767e8	−1.86307
$63$	−4.68611e8	−3.74783
$64$	−1.97376e8	−1.47057
$65$	0	0
$66$	1.61039e8	1.04468
$67$	−1.74234e8	−1.05632	−0.528161	−	0.849144i	$-0.677118\pi$
$67$	−1.74234e8	−1.05632	−0.528161	+	0.849144i	$0.677118\pi$
$68$	3.07256e7	0.174265
$69$	−4.31129e8	−2.28974
$70$	−1.29031e8	−0.642320
$71$	1.13072e7	0.0528072	0.0264036	−	0.999651i	$-0.491594\pi$
$71$	1.13072e7	0.0528072	0.0264036	+	0.999651i	$0.491594\pi$
$72$	−2.53537e8	−1.11185
$73$	8.79707e7	0.362564	0.181282	−	0.983431i	$-0.441975\pi$
$73$	8.79707e7	0.362564	0.181282	+	0.983431i	$0.441975\pi$
$74$	−4.20264e8	−1.62922
$75$	−4.72915e8	−1.72587
$76$	1.91846e8	0.659616
$77$	1.66934e8	0.541173
$78$	0	0
$79$	3.00238e8	0.867249	0.433625	−	0.901094i	$-0.357234\pi$
$79$	3.00238e8	0.867249	0.433625	+	0.901094i	$0.357234\pi$
$80$	6.61369e7	0.180526
$81$	1.12972e9	2.91601
$82$	9.97013e7	0.243522
$83$	7.05637e8	1.63204	0.816018	−	0.578027i	$-0.196177\pi$
$83$	7.05637e8	1.63204	0.816018	+	0.578027i	$0.196177\pi$
$84$	−1.63269e9	−3.57805
$85$	−1.87252e7	−0.0389082
$86$	1.93916e8	0.382270
$87$	9.28612e7	0.173779
$88$	9.03179e7	0.160547
$89$	5.36589e8	0.906540	0.453270	−	0.891373i	$-0.350257\pi$
$89$	5.36589e8	0.906540	0.453270	+	0.891373i	$0.350257\pi$
$90$	6.88811e8	1.10665
$91$	0	0
$92$	−1.07791e9	−1.56869
$93$	1.67159e9	2.31717
$94$	−4.64748e8	−0.613963
$95$	−1.16917e8	−0.147273
$96$	2.17120e9	2.60903
$97$	−2.24820e8	−0.257848	−0.128924	−	0.991655i	$-0.541152\pi$
$97$	−2.24820e8	−0.257848	−0.128924	+	0.991655i	$0.541152\pi$
$98$	−1.62373e9	−1.77826
$99$	−8.91151e8	−0.932380
$100$	−1.18238e9	−1.18238
$101$	−1.04690e8	−0.100106	−0.0500529	−	0.998747i	$-0.515939\pi$
$101$	−1.04690e8	−0.100106	−0.0500529	+	0.998747i	$0.515939\pi$
$102$	−4.20728e8	−0.384859
$103$	1.11352e9	0.974837	0.487419	−	0.873168i	$-0.337939\pi$
$103$	1.11352e9	0.974837	0.487419	+	0.873168i	$0.337939\pi$
$104$	0	0
$105$	9.95017e8	0.798874
$106$	−1.83104e9	−1.40871
$107$	−9.69547e8	−0.715059	−0.357530	−	0.933902i	$-0.616381\pi$
$107$	−9.69547e8	−0.715059	−0.357530	+	0.933902i	$0.616381\pi$
$108$	5.28588e9	3.73862
$109$	−4.07096e8	−0.276234	−0.138117	−	0.990416i	$-0.544105\pi$
$109$	−4.07096e8	−0.276234	−0.138117	+	0.990416i	$0.544105\pi$
$110$	−2.45376e8	−0.159796
$111$	3.24085e9	2.02631
$112$	1.54039e9	0.925019
$113$	−9.61939e8	−0.555002	−0.277501	−	0.960725i	$-0.589506\pi$
$113$	−9.61939e8	−0.555002	−0.277501	+	0.960725i	$0.589506\pi$
$114$	−2.62696e9	−1.45674
$115$	6.56913e8	0.350242
$116$	2.32171e8	0.119055
$117$	0	0
$118$	3.21900e9	1.52845
$119$	−4.36129e8	−0.199367
$120$	5.38344e8	0.236998
$121$	−2.04049e9	−0.865368
$122$	−5.35521e9	−2.18856
$123$	−7.68844e8	−0.302876
$124$	4.17931e9	1.58747
$125$	1.50626e9	0.551830
$126$	1.60431e10	5.67049
$127$	3.40790e9	1.16244	0.581220	−	0.813747i	$-0.302576\pi$
$127$	3.40790e9	1.16244	0.581220	+	0.813747i	$0.302576\pi$
$128$	2.54654e9	0.838506
$129$	−1.49538e9	−0.475442
$130$	0	0
$131$	4.49297e9	1.33295	0.666474	−	0.745529i	$-0.267803\pi$
$131$	4.49297e9	1.33295	0.666474	+	0.745529i	$0.267803\pi$
$132$	−3.10486e9	−0.890143
$133$	−2.72312e9	−0.754631
$134$	5.96498e9	1.59822
$135$	−3.22140e9	−0.834723
$136$	−2.35963e8	−0.0591452
$137$	−2.94051e9	−0.713148	−0.356574	−	0.934267i	$-0.616055\pi$
$137$	−2.94051e9	−0.713148	−0.356574	+	0.934267i	$0.616055\pi$
$138$	1.47599e10	3.46439
$139$	−2.35074e9	−0.534118	−0.267059	−	0.963680i	$-0.586052\pi$
$139$	−2.35074e9	−0.534118	−0.267059	+	0.963680i	$0.586052\pi$
$140$	2.48774e9	0.547303
$141$	3.58389e9	0.763606
$142$	−3.87108e8	−0.0798977
$143$	0	0
$144$	−8.22317e9	−1.59370
$145$	−1.41493e8	−0.0265815
$146$	−3.01172e9	−0.548563
$147$	1.25213e10	2.21168
$148$	8.10277e9	1.38821
$149$	3.82776e9	0.636219	0.318109	−	0.948054i	$-0.396952\pi$
$149$	3.82776e9	0.636219	0.318109	+	0.948054i	$0.396952\pi$
$150$	1.61905e10	2.61125
$151$	1.91204e9	0.299296	0.149648	−	0.988739i	$-0.452186\pi$
$151$	1.91204e9	0.299296	0.149648	+	0.988739i	$0.452186\pi$
$152$	−1.47332e9	−0.223872
$153$	2.32821e9	0.343487
$154$	−5.71505e9	−0.818798
$155$	−2.54701e9	−0.354436
$156$	0	0
$157$	−8.77429e9	−1.15256	−0.576280	−	0.817252i	$-0.695496\pi$
$157$	−8.77429e9	−1.15256	−0.576280	+	0.817252i	$0.695496\pi$
$158$	−1.02788e10	−1.31215
$159$	1.41200e10	1.75206
$160$	−3.30826e9	−0.399080
$161$	1.53002e10	1.79465
$162$	−3.86765e10	−4.41194
$163$	−3.38365e9	−0.375440	−0.187720	−	0.982223i	$-0.560110\pi$
$163$	−3.38365e9	−0.375440	−0.187720	+	0.982223i	$0.560110\pi$
$164$	−1.92226e9	−0.207499
$165$	1.89221e9	0.198743
$166$	−2.41578e10	−2.46928
$167$	1.33203e10	1.32523	0.662613	−	0.748962i	$-0.269447\pi$
$167$	1.33203e10	1.32523	0.662613	+	0.748962i	$0.269447\pi$
$168$	1.25386e10	1.21439
$169$	0	0
$170$	6.41066e8	0.0588684
$171$	1.45370e10	1.30014
$172$	−3.73874e9	−0.325722
$173$	2.08941e10	1.77344	0.886718	−	0.462311i	$-0.152980\pi$
$173$	2.08941e10	1.77344	0.886718	+	0.462311i	$0.152980\pi$
$174$	−3.17915e9	−0.262929
$175$	1.67831e10	1.35270
$176$	2.92935e9	0.230125
$177$	−2.48232e10	−1.90098
$178$	−1.83704e10	−1.37160
$179$	2.26281e10	1.64744	0.823719	−	0.566998i	$-0.191895\pi$
$179$	2.26281e10	1.64744	0.823719	+	0.566998i	$0.191895\pi$
$180$	−1.32804e10	−0.942943
$181$	2.31052e10	1.60013	0.800065	−	0.599913i	$-0.204798\pi$
$181$	2.31052e10	1.60013	0.800065	+	0.599913i	$0.204798\pi$
$182$	0	0
$183$	4.12966e10	2.72198
$184$	8.27801e9	0.532409
$185$	−4.93810e9	−0.309947
$186$	−5.72277e10	−3.50589
$187$	−8.29381e8	−0.0495983
$188$	8.96044e9	0.523141
$189$	−7.50296e10	−4.27715
$190$	4.00272e9	0.222825
$191$	−5.13927e9	−0.279416	−0.139708	−	0.990193i	$-0.544616\pi$
$191$	−5.13927e9	−0.279416	−0.139708	+	0.990193i	$0.544616\pi$
$192$	−5.21084e10	−2.76728
$193$	−2.11555e9	−0.109753	−0.0548763	−	0.998493i	$-0.517476\pi$
$193$	−2.11555e9	−0.109753	−0.0548763	+	0.998493i	$0.517476\pi$
$194$	7.69683e9	0.390125
$195$	0	0
$196$	3.13058e10	1.51521
$197$	5.84400e9	0.276447	0.138224	−	0.990401i	$-0.455861\pi$
$197$	5.84400e9	0.276447	0.138224	+	0.990401i	$0.455861\pi$
$198$	3.05090e10	1.41070
$199$	−3.09186e9	−0.139759	−0.0698796	−	0.997555i	$-0.522262\pi$
$199$	−3.09186e9	−0.139759	−0.0698796	+	0.997555i	$0.522262\pi$
$200$	9.08034e9	0.401298
$201$	−4.59988e10	−1.98776
$202$	3.58411e9	0.151461
$203$	−3.29552e9	−0.136204
$204$	8.11173e9	0.327927
$205$	1.17149e9	0.0463283
$206$	−3.81220e10	−1.47494
$207$	−8.16777e10	−3.09198
$208$	0	0
$209$	−5.17852e9	−0.187736
$210$	−3.40648e10	−1.20870
$211$	4.05641e10	1.40887	0.704435	−	0.709769i	$-0.251201\pi$
$211$	4.05641e10	1.40887	0.704435	+	0.709769i	$0.251201\pi$
$212$	3.53029e10	1.20032
$213$	2.98517e9	0.0993714
$214$	3.31929e10	1.08189
$215$	2.27851e9	0.0727241
$216$	−4.05941e10	−1.26888
$217$	−5.93225e10	−1.81615
$218$	1.39371e10	0.417944
$219$	2.32248e10	0.682265
$220$	4.73090e9	0.136157
$221$	0	0
$222$	−1.10952e11	−3.06582
$223$	−3.48948e10	−0.944907	−0.472453	−	0.881356i	$-0.656631\pi$
$223$	−3.48948e10	−0.944907	−0.472453	+	0.881356i	$0.656631\pi$
$224$	−7.70528e10	−2.04490
$225$	−8.95941e10	−2.33055
$226$	3.29324e10	0.839723
$227$	3.62831e10	0.906961	0.453481	−	0.891266i	$-0.350182\pi$
$227$	3.62831e10	0.906961	0.453481	+	0.891266i	$0.350182\pi$
$228$	5.06484e10	1.24125
$229$	1.36563e10	0.328151	0.164075	−	0.986448i	$-0.447536\pi$
$229$	1.36563e10	0.328151	0.164075	+	0.986448i	$0.447536\pi$
$230$	−2.24897e10	−0.529918
$231$	4.40715e10	1.01837
$232$	−1.78301e9	−0.0404071
$233$	2.78187e10	0.618352	0.309176	−	0.951005i	$-0.399947\pi$
$233$	2.78187e10	0.618352	0.309176	+	0.951005i	$0.399947\pi$
$234$	0	0
$235$	−5.46080e9	−0.116802
$236$	−6.20629e10	−1.30235
$237$	7.92646e10	1.63197
$238$	1.49311e10	0.301644
$239$	3.65446e10	0.724491	0.362245	−	0.932083i	$-0.382010\pi$
$239$	3.65446e10	0.724491	0.362245	+	0.932083i	$0.382010\pi$
$240$	1.74605e10	0.339709
$241$	9.51271e10	1.81647	0.908233	−	0.418464i	$-0.137431\pi$
$241$	9.51271e10	1.81647	0.908233	+	0.418464i	$0.137431\pi$
$242$	6.98572e10	1.30931
$243$	1.40629e11	2.58731
$244$	1.03250e11	1.86481
$245$	−1.90788e10	−0.338302
$246$	2.63217e10	0.458254
$247$	0	0
$248$	−3.20959e10	−0.538786
$249$	1.86292e11	3.07113
$250$	−5.15675e10	−0.834923
$251$	3.59087e9	0.0571042	0.0285521	−	0.999592i	$-0.490910\pi$
$251$	3.59087e9	0.0571042	0.0285521	+	0.999592i	$0.490910\pi$
$252$	−3.09314e11	−4.83167
$253$	2.90961e10	0.446471
$254$	−1.16671e11	−1.75878
$255$	−4.94356e9	−0.0732165
$256$	1.38745e10	0.201900
$257$	−1.65746e10	−0.236998	−0.118499	−	0.992954i	$-0.537808\pi$
$257$	−1.65746e10	−0.236998	−0.118499	+	0.992954i	$0.537808\pi$
$258$	5.11949e10	0.719347
$259$	−1.15013e11	−1.58818
$260$	0	0
$261$	1.75926e10	0.234665
$262$	−1.53819e11	−2.01676
$263$	2.05425e9	0.0264760	0.0132380	−	0.999912i	$-0.495786\pi$
$263$	2.05425e9	0.0264760	0.0132380	+	0.999912i	$0.495786\pi$
$264$	2.38445e10	0.302113
$265$	−2.15148e10	−0.267997
$266$	9.32273e10	1.14176
$267$	1.41663e11	1.70590
$268$	−1.15006e11	−1.36180
$269$	−1.46969e10	−0.171135	−0.0855677	−	0.996332i	$-0.527270\pi$
$269$	−1.46969e10	−0.171135	−0.0855677	+	0.996332i	$0.527270\pi$
$270$	1.10286e11	1.26294
$271$	3.59723e10	0.405141	0.202570	−	0.979268i	$-0.435071\pi$
$271$	3.59723e10	0.405141	0.202570	+	0.979268i	$0.435071\pi$
$272$	−7.65317e9	−0.0847777
$273$	0	0
$274$	1.00670e11	1.07900
$275$	3.19162e10	0.336523
$276$	−2.84574e11	−2.95192
$277$	−6.98271e10	−0.712632	−0.356316	−	0.934366i	$-0.615967\pi$
$277$	−6.98271e10	−0.712632	−0.356316	+	0.934366i	$0.615967\pi$
$278$	8.04786e10	0.808125
$279$	3.16684e11	3.12901
$280$	−1.91051e10	−0.185754
$281$	1.20371e11	1.15171	0.575857	−	0.817550i	$-0.304669\pi$
$281$	1.20371e11	1.15171	0.575857	+	0.817550i	$0.304669\pi$
$282$	−1.22696e11	−1.15534
$283$	−9.32655e10	−0.864335	−0.432167	−	0.901793i	$-0.642251\pi$
$283$	−9.32655e10	−0.864335	−0.432167	+	0.901793i	$0.642251\pi$
$284$	7.46352e9	0.0680787
$285$	−3.08669e10	−0.277134
$286$	0	0
$287$	2.72852e10	0.237388
$288$	4.11335e11	3.52314
$289$	−1.16421e11	−0.981728
$290$	4.84408e9	0.0402180
$291$	−5.93539e10	−0.485211
$292$	5.80665e10	0.467415
$293$	−2.33838e11	−1.85358	−0.926789	−	0.375583i	$-0.877442\pi$
$293$	−2.33838e11	−1.85358	−0.926789	+	0.375583i	$0.877442\pi$
$294$	−4.28674e11	−3.34630
$295$	3.78232e10	0.290777
$296$	−6.22269e10	−0.471157
$297$	−1.42683e11	−1.06406
$298$	−1.31045e11	−0.962604
$299$	0	0
$300$	−3.12155e11	−2.22498
$301$	5.30689e10	0.372641
$302$	−6.54595e10	−0.452837
$303$	−2.76388e10	−0.188377
$304$	−4.77853e10	−0.320895
$305$	−6.29238e10	−0.416357
$306$	−7.97073e10	−0.519699
$307$	1.36494e11	0.876980	0.438490	−	0.898736i	$-0.355513\pi$
$307$	1.36494e11	0.876980	0.438490	+	0.898736i	$0.355513\pi$
$308$	1.10187e11	0.697675
$309$	2.93977e11	1.83442
$310$	8.71981e10	0.536265
$311$	2.73365e11	1.65699	0.828496	−	0.559994i	$-0.189197\pi$
$311$	2.73365e11	1.65699	0.828496	+	0.559994i	$0.189197\pi$
$312$	0	0
$313$	−1.15929e11	−0.682717	−0.341359	−	0.939933i	$-0.610887\pi$
$313$	−1.15929e11	−0.682717	−0.341359	+	0.939933i	$0.610887\pi$
$314$	3.00392e11	1.74383
$315$	1.88507e11	1.07877
$316$	1.98177e11	1.11805
$317$	3.44477e11	1.91599	0.957995	−	0.286786i	$-0.0925868\pi$
$317$	3.44477e11	1.91599	0.957995	+	0.286786i	$0.0925868\pi$
$318$	−4.83406e11	−2.65088
$319$	−6.26704e9	−0.0338848
$320$	7.93979e10	0.423286
$321$	−2.55966e11	−1.34558
$322$	−5.23808e11	−2.71532
$323$	1.35293e10	0.0691617
$324$	7.45691e11	3.75930
$325$	0	0
$326$	1.15841e11	0.568044
$327$	−1.07476e11	−0.519810
$328$	1.47624e10	0.0704247
$329$	−1.27187e11	−0.598498
$330$	−6.47807e10	−0.300700
$331$	−1.84177e11	−0.843353	−0.421676	−	0.906746i	$-0.638558\pi$
$331$	−1.84177e11	−0.843353	−0.421676	+	0.906746i	$0.638558\pi$
$332$	4.65767e11	2.10401
$333$	6.13982e11	2.73626
$334$	−4.56027e11	−2.00508
$335$	7.00886e10	0.304050
$336$	4.06673e11	1.74068
$337$	−3.39813e11	−1.43518	−0.717588	−	0.696468i	$-0.754754\pi$
$337$	−3.39813e11	−1.43518	−0.717588	+	0.696468i	$0.754754\pi$
$338$	0	0
$339$	−2.53958e11	−1.04439
$340$	−1.23599e10	−0.0501602
$341$	−1.12813e11	−0.451818
$342$	−4.97680e11	−1.96713
$343$	−6.62842e10	−0.258575
$344$	2.87124e10	0.110549
$345$	1.73429e11	0.659076
$346$	−7.15318e11	−2.68322
$347$	4.61731e11	1.70965	0.854823	−	0.518920i	$-0.173666\pi$
$347$	4.61731e11	1.70965	0.854823	+	0.518920i	$0.173666\pi$
$348$	6.12946e10	0.224035
$349$	−4.21467e11	−1.52072	−0.760361	−	0.649501i	$-0.774978\pi$
$349$	−4.21467e11	−1.52072	−0.760361	+	0.649501i	$0.774978\pi$
$350$	−5.74577e11	−2.04664
$351$	0	0
$352$	−1.46530e11	−0.508728
$353$	−3.66040e11	−1.25471	−0.627354	−	0.778734i	$-0.715862\pi$
$353$	−3.66040e11	−1.25471	−0.627354	+	0.778734i	$0.715862\pi$
$354$	8.49834e11	2.87620
$355$	−4.54852e9	−0.0152000
$356$	3.54184e11	1.16870
$357$	−1.15141e11	−0.375164
$358$	−7.74683e11	−2.49259
$359$	−2.41644e11	−0.767806	−0.383903	−	0.923373i	$-0.625420\pi$
$359$	−2.41644e11	−0.767806	−0.383903	+	0.923373i	$0.625420\pi$
$360$	1.01990e11	0.320033
$361$	−2.38213e11	−0.738214
$362$	−7.91016e11	−2.42101
$363$	−5.38702e11	−1.62843
$364$	0	0
$365$	−3.53877e10	−0.104360
$366$	−1.41381e12	−4.11837
$367$	−2.14146e11	−0.616187	−0.308094	−	0.951356i	$-0.599691\pi$
$367$	−2.14146e11	−0.616187	−0.308094	+	0.951356i	$0.599691\pi$
$368$	2.68487e11	0.763146
$369$	−1.45658e11	−0.408993
$370$	1.69058e11	0.468952
$371$	−5.01101e11	−1.37323
$372$	1.10336e12	2.98727
$373$	5.41568e11	1.44865	0.724325	−	0.689459i	$-0.242152\pi$
$373$	5.41568e11	1.44865	0.724325	+	0.689459i	$0.242152\pi$
$374$	2.83942e10	0.0750426
$375$	3.97662e11	1.03842
$376$	−6.88135e10	−0.177553
$377$	0	0
$378$	2.56867e12	6.47136
$379$	−1.99906e11	−0.497678	−0.248839	−	0.968545i	$-0.580049\pi$
$379$	−1.99906e11	−0.497678	−0.248839	+	0.968545i	$0.580049\pi$
$380$	−7.71732e10	−0.189863
$381$	8.99706e11	2.18745
$382$	1.75945e11	0.422759
$383$	5.10094e11	1.21131	0.605656	−	0.795727i	$-0.292911\pi$
$383$	5.10094e11	1.21131	0.605656	+	0.795727i	$0.292911\pi$
$384$	6.72302e11	1.57788
$385$	−6.71519e10	−0.155770
$386$	7.24267e10	0.166057
$387$	−2.83300e11	−0.642019
$388$	−1.48396e11	−0.332415
$389$	2.72583e10	0.0603566	0.0301783	−	0.999545i	$-0.490392\pi$
$389$	2.72583e10	0.0603566	0.0301783	+	0.999545i	$0.490392\pi$
$390$	0	0
$391$	−7.60162e10	−0.164479
$392$	−2.40420e11	−0.514260
$393$	1.18617e12	2.50831
$394$	−2.00072e11	−0.418267
$395$	−1.20776e11	−0.249628
$396$	−5.88219e11	−1.20202
$397$	4.53705e11	0.916676	0.458338	−	0.888778i	$-0.348445\pi$
$397$	4.53705e11	0.916676	0.458338	+	0.888778i	$0.348445\pi$
$398$	1.05851e11	0.211457
$399$	−7.18920e11	−1.42005
$400$	2.94509e11	0.575214
$401$	−2.17568e11	−0.420190	−0.210095	−	0.977681i	$-0.567377\pi$
$401$	−2.17568e11	−0.420190	−0.210095	+	0.977681i	$0.567377\pi$
$402$	1.57479e12	3.00750
$403$	0	0
$404$	−6.91024e10	−0.129056
$405$	−4.54450e11	−0.839340
$406$	1.12823e11	0.206078
$407$	−2.18720e11	−0.395105
$408$	−6.22957e10	−0.111298
$409$	−8.33992e11	−1.47369	−0.736847	−	0.676060i	$-0.763686\pi$
$409$	−8.33992e11	−1.47369	−0.736847	+	0.676060i	$0.763686\pi$
$410$	−4.01065e10	−0.0700951
$411$	−7.76312e11	−1.34199
$412$	7.35000e11	1.25675
$413$	8.80941e11	1.48995
$414$	2.79627e12	4.67819
$415$	−2.83854e11	−0.469763
$416$	0	0
$417$	−6.20609e11	−1.00509
$418$	1.77289e11	0.284046
$419$	8.95743e11	1.41978	0.709889	−	0.704314i	$-0.248745\pi$
$419$	8.95743e11	1.41978	0.709889	+	0.704314i	$0.248745\pi$
$420$	6.56777e11	1.02990
$421$	−8.15504e11	−1.26519	−0.632597	−	0.774481i	$-0.718011\pi$
$421$	−8.15504e11	−1.26519	−0.632597	+	0.774481i	$0.718011\pi$
$422$	−1.38873e12	−2.13163
$423$	6.78971e11	1.03114
$424$	−2.71116e11	−0.407388
$425$	−8.33839e10	−0.123974
$426$	−1.02199e11	−0.150350
$427$	−1.46556e12	−2.13343
$428$	−6.39965e11	−0.921848
$429$	0	0
$430$	−7.80059e10	−0.110032
$431$	8.43709e11	1.17773	0.588864	−	0.808232i	$-0.299575\pi$
$431$	8.43709e11	1.17773	0.588864	+	0.808232i	$0.299575\pi$
$432$	−1.31662e12	−1.81879
$433$	2.19189e11	0.299656	0.149828	−	0.988712i	$-0.452128\pi$
$433$	2.19189e11	0.299656	0.149828	+	0.988712i	$0.452128\pi$
$434$	2.03093e12	2.74784
$435$	−3.73550e10	−0.0500204
$436$	−2.68710e11	−0.356119
$437$	−4.74633e11	−0.622575
$438$	−7.95111e11	−1.03227
$439$	−3.68387e11	−0.473385	−0.236692	−	0.971585i	$-0.576063\pi$
$439$	−3.68387e11	−0.473385	−0.236692	+	0.971585i	$0.576063\pi$
$440$	−3.63319e10	−0.0462116
$441$	2.37218e12	2.98658
$442$	0	0
$443$	−1.22853e12	−1.51554	−0.757771	−	0.652521i	$-0.773712\pi$
$443$	−1.22853e12	−1.51554	−0.757771	+	0.652521i	$0.773712\pi$
$444$	2.13918e12	2.61230
$445$	−2.15852e11	−0.260937
$446$	1.19464e12	1.42965
$447$	1.01055e12	1.19722
$448$	1.84926e12	2.16893
$449$	4.87703e11	0.566300	0.283150	−	0.959076i	$-0.408621\pi$
$449$	4.87703e11	0.566300	0.283150	+	0.959076i	$0.408621\pi$
$450$	3.06730e12	3.52614
$451$	5.18879e10	0.0590571
$452$	−6.34944e11	−0.715504
$453$	5.04789e11	0.563207
$454$	−1.24217e12	−1.37224
$455$	0	0
$456$	−3.88965e11	−0.421278
$457$	−8.36514e11	−0.897120	−0.448560	−	0.893753i	$-0.648063\pi$
$457$	−8.36514e11	−0.897120	−0.448560	+	0.893753i	$0.648063\pi$
$458$	−4.67530e11	−0.496495
$459$	3.72771e11	0.391999
$460$	4.33606e11	0.451529
$461$	−6.92865e10	−0.0714487	−0.0357243	−	0.999362i	$-0.511374\pi$
$461$	−6.92865e10	−0.0714487	−0.0357243	+	0.999362i	$0.511374\pi$
$462$	−1.50881e12	−1.54079
$463$	2.51464e11	0.254309	0.127155	−	0.991883i	$-0.459416\pi$
$463$	2.51464e11	0.254309	0.127155	+	0.991883i	$0.459416\pi$
$464$	−5.78296e10	−0.0579188
$465$	−6.72426e11	−0.666970
$466$	−9.52386e11	−0.935571
$467$	−3.44156e11	−0.334834	−0.167417	−	0.985886i	$-0.553543\pi$
$467$	−3.44156e11	−0.334834	−0.167417	+	0.985886i	$0.553543\pi$
$468$	0	0
$469$	1.63243e12	1.55796
$470$	1.86953e11	0.176722
$471$	−2.31646e12	−2.16886
$472$	4.76625e11	0.442016
$473$	1.00920e11	0.0927051
$474$	−2.71366e12	−2.46918
$475$	−5.20636e11	−0.469260
$476$	−2.87874e11	−0.257022
$477$	2.67505e12	2.36592
$478$	−1.25112e12	−1.09616
$479$	−3.21391e11	−0.278948	−0.139474	−	0.990226i	$-0.544541\pi$
$479$	−3.21391e11	−0.278948	−0.139474	+	0.990226i	$0.544541\pi$
$480$	−8.73401e11	−0.750980
$481$	0	0
$482$	−3.25672e12	−2.74833
$483$	4.03933e12	3.37713
$484$	−1.34686e12	−1.11563
$485$	9.04378e10	0.0742185
$486$	−4.81451e12	−3.91462
$487$	1.89681e12	1.52807	0.764037	−	0.645172i	$-0.223214\pi$
$487$	1.89681e12	1.52807	0.764037	+	0.645172i	$0.223214\pi$
$488$	−7.92927e11	−0.632913
$489$	−8.93303e11	−0.706494
$490$	6.53173e11	0.511853
$491$	−1.50545e12	−1.16896	−0.584479	−	0.811409i	$-0.698701\pi$
$491$	−1.50545e12	−1.16896	−0.584479	+	0.811409i	$0.698701\pi$
$492$	−5.07488e11	−0.390466
$493$	1.63732e10	0.0124831
$494$	0	0
$495$	3.58481e11	0.268375
$496$	−1.04099e12	−0.772287
$497$	−1.05940e11	−0.0778852
$498$	−6.37780e12	−4.64664
$499$	−1.38015e12	−0.996495	−0.498247	−	0.867035i	$-0.666023\pi$
$499$	−1.38015e12	−0.996495	−0.498247	+	0.867035i	$0.666023\pi$
$500$	9.94232e11	0.711414
$501$	3.51664e12	2.49378
$502$	−1.22935e11	−0.0863991
$503$	−1.83180e12	−1.27592	−0.637958	−	0.770071i	$-0.720221\pi$
$503$	−1.83180e12	−1.27592	−0.637958	+	0.770071i	$0.720221\pi$
$504$	2.37544e12	1.63986
$505$	4.21133e10	0.0288143
$506$	−9.96119e11	−0.675513
$507$	0	0
$508$	2.24944e12	1.49861
$509$	−1.85488e12	−1.22486	−0.612430	−	0.790525i	$-0.709808\pi$
$509$	−1.85488e12	−1.22486	−0.612430	+	0.790525i	$0.709808\pi$
$510$	1.69245e11	0.110777
$511$	−8.24215e11	−0.534745
$512$	−1.77883e12	−1.14398
$513$	2.32753e12	1.48377
$514$	5.67440e11	0.358580
$515$	−4.47934e11	−0.280596
$516$	−9.87048e11	−0.612935
$517$	−2.41871e11	−0.148894
$518$	3.93754e12	2.40293
$519$	5.51616e12	3.33721
$520$	0	0
$521$	1.06110e12	0.630939	0.315469	−	0.948936i	$-0.397838\pi$
$521$	1.06110e12	0.630939	0.315469	+	0.948936i	$0.397838\pi$
$522$	−6.02292e11	−0.355050
$523$	−1.42342e12	−0.831907	−0.415953	−	0.909386i	$-0.636552\pi$
$523$	−1.42342e12	−0.831907	−0.415953	+	0.909386i	$0.636552\pi$
$524$	2.96566e12	1.71842
$525$	4.43084e12	2.54548
$526$	−7.03283e10	−0.0400584
$527$	2.94733e11	0.166449
$528$	7.73365e11	0.433044
$529$	8.65628e11	0.480597
$530$	7.36568e11	0.405482
$531$	−4.70277e12	−2.56702
$532$	−1.79744e12	−0.972864
$533$	0	0
$534$	−4.84988e12	−2.58105
$535$	3.90017e11	0.205822
$536$	8.83212e11	0.462193
$537$	5.97395e12	3.10011
$538$	5.03155e11	0.258929
$539$	−8.45044e11	−0.431251
$540$	−2.12634e12	−1.07612
$541$	−1.66674e12	−0.836526	−0.418263	−	0.908326i	$-0.637361\pi$
$541$	−1.66674e12	−0.836526	−0.418263	+	0.908326i	$0.637361\pi$
$542$	−1.23153e12	−0.612981
$543$	6.09990e12	3.01109
$544$	3.82823e11	0.187414
$545$	1.63761e11	0.0795108
$546$	0	0
$547$	9.55167e11	0.456180	0.228090	−	0.973640i	$-0.426752\pi$
$547$	9.55167e11	0.456180	0.228090	+	0.973640i	$0.426752\pi$
$548$	−1.94093e12	−0.919385
$549$	7.82367e12	3.67566
$550$	−1.09267e12	−0.509161
$551$	1.02232e11	0.0472502
$552$	2.18544e12	1.00188
$553$	−2.81299e12	−1.27910
$554$	2.39056e12	1.07822
$555$	−1.30369e12	−0.583251
$556$	−1.55164e12	−0.688581
$557$	1.16406e12	0.512420	0.256210	−	0.966621i	$-0.417526\pi$
$557$	1.16406e12	0.512420	0.256210	+	0.966621i	$0.417526\pi$
$558$	−1.08418e13	−4.73422
$559$	0	0
$560$	−6.19650e11	−0.266256
$561$	−2.18961e11	−0.0933328
$562$	−4.12097e12	−1.74255
$563$	−3.44795e12	−1.44635	−0.723175	−	0.690665i	$-0.757318\pi$
$563$	−3.44795e12	−1.44635	−0.723175	+	0.690665i	$0.757318\pi$
$564$	2.36561e12	0.984434
$565$	3.86956e11	0.159751
$566$	3.19299e12	1.30775
$567$	−1.05846e13	−4.30081
$568$	−5.73176e10	−0.0231058
$569$	3.22482e12	1.28973	0.644867	−	0.764295i	$-0.276913\pi$
$569$	3.22482e12	1.28973	0.644867	+	0.764295i	$0.276913\pi$
$570$	1.05674e12	0.419306
$571$	3.76387e12	1.48174	0.740870	−	0.671649i	$-0.234414\pi$
$571$	3.76387e12	1.48174	0.740870	+	0.671649i	$0.234414\pi$
$572$	0	0
$573$	−1.35680e12	−0.525798
$574$	−9.34122e11	−0.359170
$575$	2.92525e12	1.11598
$576$	−9.87198e12	−3.73683
$577$	−6.76317e11	−0.254015	−0.127007	−	0.991902i	$-0.540537\pi$
$577$	−6.76317e11	−0.254015	−0.127007	+	0.991902i	$0.540537\pi$
$578$	3.98573e12	1.48536
$579$	−5.58517e11	−0.206530
$580$	−9.33948e10	−0.0342686
$581$	−6.61125e12	−2.40708
$582$	2.03201e12	0.734128
$583$	−9.52937e11	−0.341630
$584$	−4.45934e11	−0.158640
$585$	0	0
$586$	8.00555e12	2.80448
$587$	−4.53820e12	−1.57765	−0.788827	−	0.614615i	$-0.789312\pi$
$587$	−4.53820e12	−1.57765	−0.788827	+	0.614615i	$0.789312\pi$
$588$	8.26492e12	2.85129
$589$	1.84027e12	0.630032
$590$	−1.29490e12	−0.439947
$591$	1.54285e12	0.520212
$592$	−2.01825e12	−0.675348
$593$	2.06956e12	0.687278	0.343639	−	0.939102i	$-0.388340\pi$
$593$	2.06956e12	0.687278	0.343639	+	0.939102i	$0.388340\pi$
$594$	4.88481e12	1.60994
$595$	1.75440e11	0.0573856
$596$	2.52658e12	0.820208
$597$	−8.16268e11	−0.262996
$598$	0	0
$599$	−6.11477e11	−0.194071	−0.0970353	−	0.995281i	$-0.530936\pi$
$599$	−6.11477e11	−0.194071	−0.0970353	+	0.995281i	$0.530936\pi$
$600$	2.39726e12	0.755153
$601$	−4.53199e12	−1.41695	−0.708474	−	0.705737i	$-0.750616\pi$
$601$	−4.53199e12	−1.41695	−0.708474	+	0.705737i	$0.750616\pi$
$602$	−1.81684e12	−0.563809
$603$	−8.71450e12	−2.68420
$604$	1.26207e12	0.385849
$605$	8.20822e11	0.249086
$606$	9.46227e11	0.285015
$607$	−1.22324e12	−0.365731	−0.182866	−	0.983138i	$-0.558537\pi$
$607$	−1.22324e12	−0.365731	−0.182866	+	0.983138i	$0.558537\pi$
$608$	2.39029e12	0.709389
$609$	−8.70036e11	−0.256306
$610$	2.15423e12	0.629951
$611$	0	0
$612$	1.53677e12	0.442821
$613$	1.31919e12	0.377343	0.188671	−	0.982040i	$-0.439582\pi$
$613$	1.31919e12	0.377343	0.188671	+	0.982040i	$0.439582\pi$
$614$	−4.67292e12	−1.32688
$615$	3.09281e11	0.0871796
$616$	−8.46207e11	−0.236790
$617$	−5.58462e12	−1.55135	−0.775676	−	0.631132i	$-0.782591\pi$
$617$	−5.58462e12	−1.55135	−0.775676	+	0.631132i	$0.782591\pi$
$618$	−1.00644e13	−2.77550
$619$	−1.77467e11	−0.0485858	−0.0242929	−	0.999705i	$-0.507733\pi$
$619$	−1.77467e11	−0.0485858	−0.0242929	+	0.999705i	$0.507733\pi$
$620$	−1.68120e12	−0.456937
$621$	−1.30775e13	−3.52867
$622$	−9.35876e12	−2.50704
$623$	−5.02741e12	−1.33705
$624$	0	0
$625$	2.89273e12	0.758311
$626$	3.96887e12	1.03296
$627$	−1.36716e12	−0.353277
$628$	−5.79161e12	−1.48587
$629$	5.71424e11	0.145556
$630$	−6.45361e12	−1.63219
$631$	−3.65671e12	−0.918245	−0.459122	−	0.888373i	$-0.651836\pi$
$631$	−3.65671e12	−0.918245	−0.459122	+	0.888373i	$0.651836\pi$
$632$	−1.52194e12	−0.379465
$633$	1.07092e13	2.65118
$634$	−1.17933e13	−2.89891
$635$	−1.37089e12	−0.334595
$636$	9.32017e12	2.25874
$637$	0	0
$638$	2.14555e11	0.0512679
$639$	5.65543e11	0.134187
$640$	−1.02439e12	−0.241355
$641$	5.32688e12	1.24627	0.623134	−	0.782115i	$-0.285859\pi$
$641$	5.32688e12	1.24627	0.623134	+	0.782115i	$0.285859\pi$
$642$	8.76312e12	2.03587
$643$	7.82345e12	1.80488	0.902441	−	0.430814i	$-0.141773\pi$
$643$	7.82345e12	1.80488	0.902441	+	0.430814i	$0.141773\pi$
$644$	1.00991e13	2.31365
$645$	6.01541e11	0.136851
$646$	−4.63183e11	−0.104642
$647$	5.12091e12	1.14889	0.574444	−	0.818544i	$-0.305218\pi$
$647$	5.12091e12	1.14889	0.574444	+	0.818544i	$0.305218\pi$
$648$	−5.72669e12	−1.27590
$649$	1.67527e12	0.370668
$650$	0	0
$651$	−1.56615e13	−3.41758
$652$	−2.23343e12	−0.484015
$653$	2.15976e12	0.464833	0.232417	−	0.972616i	$-0.425337\pi$
$653$	2.15976e12	0.464833	0.232417	+	0.972616i	$0.425337\pi$
$654$	3.67948e12	0.786477
$655$	−1.80737e12	−0.383674
$656$	4.78800e11	0.100946
$657$	4.39995e12	0.921305
$658$	4.35432e12	0.905532
$659$	−5.20929e12	−1.07596	−0.537978	−	0.842959i	$-0.680812\pi$
$659$	−5.20929e12	−1.07596	−0.537978	+	0.842959i	$0.680812\pi$
$660$	1.24898e12	0.256218
$661$	2.88146e12	0.587092	0.293546	−	0.955945i	$-0.405165\pi$
$661$	2.88146e12	0.587092	0.293546	+	0.955945i	$0.405165\pi$
$662$	6.30538e12	1.27600
$663$	0	0
$664$	−3.57695e12	−0.714097
$665$	1.09542e12	0.217212
$666$	−2.10200e13	−4.13997
$667$	−5.74400e11	−0.112369
$668$	8.79229e12	1.70847
$669$	−9.21243e12	−1.77810
$670$	−2.39951e12	−0.460030
$671$	−2.78703e12	−0.530751
$672$	−2.03424e13	−3.84805
$673$	1.08571e12	0.204008	0.102004	−	0.994784i	$-0.467475\pi$
$673$	1.08571e12	0.204008	0.102004	+	0.994784i	$0.467475\pi$
$674$	1.16336e13	2.17143
$675$	−1.43450e13	−2.65970
$676$	0	0
$677$	−6.38065e12	−1.16739	−0.583695	−	0.811973i	$-0.698394\pi$
$677$	−6.38065e12	−1.16739	−0.583695	+	0.811973i	$0.698394\pi$
$678$	8.69435e12	1.58017
$679$	2.10639e12	0.380298
$680$	9.49202e10	0.0170243
$681$	9.57896e12	1.70670
$682$	3.86220e12	0.683604
$683$	7.35340e12	1.29299	0.646495	−	0.762918i	$-0.276234\pi$
$683$	7.35340e12	1.29299	0.646495	+	0.762918i	$0.276234\pi$
$684$	9.59537e12	1.67614
$685$	1.18287e12	0.205272
$686$	2.26927e12	0.391226
$687$	3.60535e12	0.617507
$688$	9.31251e11	0.158460
$689$	0	0
$690$	−5.93742e12	−0.997187
$691$	−6.16812e11	−0.102920	−0.0514602	−	0.998675i	$-0.516388\pi$
$691$	−6.16812e11	−0.102920	−0.0514602	+	0.998675i	$0.516388\pi$
$692$	1.37915e13	2.28630
$693$	8.34937e12	1.37516
$694$	−1.58076e13	−2.58671
$695$	9.45624e11	0.153740
$696$	−4.70724e11	−0.0760370
$697$	−1.35562e11	−0.0217565
$698$	1.44291e13	2.30086
$699$	7.34430e12	1.16360
$700$	1.10780e13	1.74389
$701$	9.64887e12	1.50920	0.754598	−	0.656188i	$-0.227832\pi$
$701$	9.64887e12	1.50920	0.754598	+	0.656188i	$0.227832\pi$
$702$	0	0
$703$	3.56788e12	0.550949
$704$	3.51671e12	0.539584
$705$	−1.44168e12	−0.219795
$706$	1.25316e13	1.89838
$707$	9.80862e11	0.147646
$708$	−1.63850e13	−2.45073
$709$	−1.96064e12	−0.291400	−0.145700	−	0.989329i	$-0.546543\pi$
$709$	−1.96064e12	−0.291400	−0.145700	+	0.989329i	$0.546543\pi$
$710$	1.55721e11	0.0229977
$711$	1.50167e13	2.20375
$712$	−2.72003e12	−0.396656
$713$	−1.03398e13	−1.49833
$714$	3.94189e12	0.567626
$715$	0	0
$716$	1.49360e13	2.12386
$717$	9.64799e12	1.36333
$718$	8.27280e12	1.16170
$719$	−1.37403e13	−1.91742	−0.958710	−	0.284385i	$-0.908211\pi$
$719$	−1.37403e13	−1.91742	−0.958710	+	0.284385i	$0.908211\pi$
$720$	3.30791e12	0.458730
$721$	−1.04328e13	−1.43778
$722$	8.15531e12	1.11692
$723$	2.51141e13	3.41818
$724$	1.52509e13	2.06288
$725$	−6.30073e11	−0.0846973
$726$	1.84427e13	2.46382
$727$	4.47091e12	0.593596	0.296798	−	0.954940i	$-0.404081\pi$
$727$	4.47091e12	0.593596	0.296798	+	0.954940i	$0.404081\pi$
$728$	0	0
$729$	1.48907e13	1.95272
$730$	1.21151e12	0.157898
$731$	−2.63663e11	−0.0341524
$732$	2.72585e13	3.50915
$733$	−4.24429e12	−0.543046	−0.271523	−	0.962432i	$-0.587527\pi$
$733$	−4.24429e12	−0.543046	−0.271523	+	0.962432i	$0.587527\pi$
$734$	7.33139e12	0.932296
$735$	−5.03692e12	−0.636608
$736$	−1.34301e13	−1.68706
$737$	3.10438e12	0.387588
$738$	4.98667e12	0.618810
$739$	−1.30576e13	−1.61052	−0.805258	−	0.592925i	$-0.797973\pi$
$739$	−1.30576e13	−1.61052	−0.805258	+	0.592925i	$0.797973\pi$
$740$	−3.25948e12	−0.399581
$741$	0	0
$742$	1.71554e13	2.07770
$743$	1.38712e12	0.166980	0.0834898	−	0.996509i	$-0.473393\pi$
$743$	1.38712e12	0.166980	0.0834898	+	0.996509i	$0.473393\pi$
$744$	−8.47350e12	−1.01388
$745$	−1.53978e12	−0.183128
$746$	−1.85408e13	−2.19182
$747$	3.52932e13	4.14713
$748$	−5.47446e11	−0.0639417
$749$	9.08388e12	1.05464
$750$	−1.36141e13	−1.57114
$751$	1.37227e13	1.57420	0.787101	−	0.616824i	$-0.211581\pi$
$751$	1.37227e13	1.57420	0.787101	+	0.616824i	$0.211581\pi$
$752$	−2.23188e12	−0.254502
$753$	9.48011e11	0.107457
$754$	0	0
$755$	−7.69150e11	−0.0861489
$756$	−4.95245e13	−5.51407
$757$	−1.91628e12	−0.212093	−0.106047	−	0.994361i	$-0.533819\pi$
$757$	−1.91628e12	−0.212093	−0.106047	+	0.994361i	$0.533819\pi$
$758$	6.84386e12	0.752990
$759$	7.68155e12	0.840158
$760$	5.92667e11	0.0644392
$761$	4.43436e11	0.0479291	0.0239646	−	0.999713i	$-0.492371\pi$
$761$	4.43436e11	0.0479291	0.0239646	+	0.999713i	$0.492371\pi$
$762$	−3.08018e13	−3.30963
$763$	3.81416e12	0.407416
$764$	−3.39226e12	−0.360221
$765$	−9.36561e11	−0.0988689
$766$	−1.74633e13	−1.83272
$767$	0	0
$768$	3.66294e12	0.379931
$769$	−7.05194e12	−0.727177	−0.363588	−	0.931560i	$-0.618449\pi$
$769$	−7.05194e12	−0.727177	−0.363588	+	0.931560i	$0.618449\pi$
$770$	2.29898e12	0.235682
$771$	−4.37580e12	−0.445977
$772$	−1.39640e12	−0.141492
$773$	−6.17676e12	−0.622233	−0.311117	−	0.950372i	$-0.600703\pi$
$773$	−6.17676e12	−0.622233	−0.311117	+	0.950372i	$0.600703\pi$
$774$	9.69891e12	0.971379
$775$	−1.13419e13	−1.12935
$776$	1.13964e12	0.112821
$777$	−3.03642e13	−2.98860
$778$	−9.33199e11	−0.0913200
$779$	−8.46427e11	−0.0823514
$780$	0	0
$781$	−2.01464e11	−0.0193762
$782$	2.60245e12	0.248858
$783$	2.81677e12	0.267808
$784$	−7.79771e12	−0.737131
$785$	3.52961e12	0.331751
$786$	−4.06091e13	−3.79509
$787$	1.52369e13	1.41583	0.707915	−	0.706297i	$-0.249636\pi$
$787$	1.52369e13	1.41583	0.707915	+	0.706297i	$0.249636\pi$
$788$	3.85743e12	0.356393
$789$	5.42335e11	0.0498220
$790$	4.13482e12	0.377689
$791$	9.01260e12	0.818570
$792$	4.51735e12	0.407962
$793$	0	0
$794$	−1.55328e13	−1.38694
$795$	−5.68003e12	−0.504311
$796$	−2.04083e12	−0.180177
$797$	2.09725e12	0.184115	0.0920574	−	0.995754i	$-0.470656\pi$
$797$	2.09725e12	0.184115	0.0920574	+	0.995754i	$0.470656\pi$
$798$	2.46126e13	2.14854
$799$	6.31908e11	0.0548521
$800$	−1.47318e13	−1.27160
$801$	2.68381e13	2.30359
$802$	7.44854e12	0.635750
$803$	−1.56740e12	−0.133033
$804$	−3.03622e13	−2.56261
$805$	−6.15475e12	−0.516570
$806$	0	0
$807$	−3.88006e12	−0.322039
$808$	5.30686e11	0.0438013
$809$	2.02578e13	1.66274	0.831369	−	0.555721i	$-0.187558\pi$
$809$	2.02578e13	1.66274	0.831369	+	0.555721i	$0.187558\pi$
$810$	1.55583e13	1.26993
$811$	−1.14386e13	−0.928492	−0.464246	−	0.885706i	$-0.653675\pi$
$811$	−1.14386e13	−0.928492	−0.464246	+	0.885706i	$0.653675\pi$
$812$	−2.17526e12	−0.175594
$813$	9.49689e12	0.762384
$814$	7.48796e12	0.597797
$815$	1.36113e12	0.108066
$816$	−2.02048e12	−0.159533
$817$	−1.64627e12	−0.129271
$818$	2.85521e13	2.22971
$819$	0	0
$820$	7.73262e11	0.0597261
$821$	−1.06586e13	−0.818760	−0.409380	−	0.912364i	$-0.634255\pi$
$821$	−1.06586e13	−0.818760	−0.409380	+	0.912364i	$0.634255\pi$
$822$	2.65774e13	2.03043
$823$	−1.71114e13	−1.30013	−0.650063	−	0.759881i	$-0.725257\pi$
$823$	−1.71114e13	−1.30013	−0.650063	+	0.759881i	$0.725257\pi$
$824$	−5.64458e12	−0.426540
$825$	8.42607e12	0.633260
$826$	−3.01594e13	−2.25431
$827$	1.83371e13	1.36319	0.681595	−	0.731729i	$-0.261286\pi$
$827$	1.83371e13	1.36319	0.681595	+	0.731729i	$0.261286\pi$
$828$	−5.39127e13	−3.98616
$829$	−4.12665e12	−0.303460	−0.151730	−	0.988422i	$-0.548484\pi$
$829$	−4.12665e12	−0.303460	−0.151730	+	0.988422i	$0.548484\pi$
$830$	9.71788e12	0.710755
$831$	−1.84348e13	−1.34101
$832$	0	0
$833$	2.20775e12	0.158872
$834$	2.12468e13	1.52071
$835$	−5.35832e12	−0.381452
$836$	−3.41817e12	−0.242028
$837$	5.07046e13	3.57094
$838$	−3.06662e13	−2.14813
$839$	−1.08660e13	−0.757081	−0.378540	−	0.925585i	$-0.623574\pi$
$839$	−1.08660e13	−0.757081	−0.378540	+	0.925585i	$0.623574\pi$
$840$	−5.04386e12	−0.349547
$841$	−1.43834e13	−0.991472
$842$	2.79192e13	1.91425
$843$	3.17787e13	2.16727
$844$	2.67750e13	1.81630
$845$	0	0
$846$	−2.32449e13	−1.56013
$847$	1.91178e13	1.27633
$848$	−8.79330e12	−0.583943
$849$	−2.46226e13	−1.62648
$850$	2.85469e12	0.187574
$851$	−2.00465e13	−1.31026
$852$	1.97041e12	0.128109
$853$	1.12824e13	0.729679	0.364839	−	0.931071i	$-0.381124\pi$
$853$	1.12824e13	0.729679	0.364839	+	0.931071i	$0.381124\pi$
$854$	5.01741e13	3.22789
$855$	−5.84775e12	−0.374232
$856$	4.91475e12	0.312874
$857$	−2.23215e13	−1.41355	−0.706774	−	0.707440i	$-0.749850\pi$
$857$	−2.23215e13	−1.41355	−0.706774	+	0.707440i	$0.749850\pi$
$858$	0	0
$859$	−2.62619e12	−0.164572	−0.0822862	−	0.996609i	$-0.526222\pi$
$859$	−2.62619e12	−0.164572	−0.0822862	+	0.996609i	$0.526222\pi$
$860$	1.50397e12	0.0937554
$861$	7.20346e12	0.446711
$862$	−2.88848e13	−1.78191
$863$	−2.05499e13	−1.26113	−0.630566	−	0.776136i	$-0.717177\pi$
$863$	−2.05499e13	−1.26113	−0.630566	+	0.776136i	$0.717177\pi$
$864$	6.58591e13	4.02072
$865$	−8.40499e12	−0.510464
$866$	−7.50402e12	−0.453381
$867$	−3.07358e13	−1.84739
$868$	−3.91568e13	−2.34136
$869$	−5.34943e12	−0.318213
$870$	1.27887e12	0.0756812
$871$	0	0
$872$	2.06361e12	0.120866
$873$	−1.12446e13	−0.655211
$874$	1.62493e13	0.941961
$875$	−1.41125e13	−0.813891
$876$	1.53299e13	0.879570
$877$	−2.62941e13	−1.50093	−0.750464	−	0.660911i	$-0.770170\pi$
$877$	−2.62941e13	−1.50093	−0.750464	+	0.660911i	$0.770170\pi$
$878$	1.26119e13	0.716235
$879$	−6.17346e13	−3.48802
$880$	−1.17838e12	−0.0662389
$881$	2.30187e13	1.28733	0.643665	−	0.765308i	$-0.277413\pi$
$881$	2.30187e13	1.28733	0.643665	+	0.765308i	$0.277413\pi$
$882$	−8.12126e13	−4.51871
$883$	1.03930e12	0.0575329	0.0287665	−	0.999586i	$-0.490842\pi$
$883$	1.03930e12	0.0575329	0.0287665	+	0.999586i	$0.490842\pi$
$884$	0	0
$885$	9.98556e12	0.547177
$886$	4.20592e13	2.29303
$887$	5.00948e11	0.0271729	0.0135865	−	0.999908i	$-0.495675\pi$
$887$	5.00948e11	0.0271729	0.0135865	+	0.999908i	$0.495675\pi$
$888$	−1.64283e13	−0.886612
$889$	−3.19293e13	−1.71448
$890$	7.38979e12	0.394800
$891$	−2.01286e13	−1.06995
$892$	−2.30329e13	−1.21817
$893$	3.94554e12	0.207623
$894$	−3.45967e13	−1.81140
$895$	−9.10253e12	−0.474197
$896$	−2.38591e13	−1.23671
$897$	0	0
$898$	−1.66967e13	−0.856816
$899$	2.22709e12	0.113715
$900$	−5.91381e13	−3.00453
$901$	2.48963e12	0.125856
$902$	−1.77641e12	−0.0893538
$903$	1.40105e13	0.701227
$904$	4.87618e12	0.242841
$905$	−9.29444e12	−0.460580
$906$	−1.72817e13	−0.852137
$907$	−1.56921e13	−0.769927	−0.384963	−	0.922932i	$-0.625786\pi$
$907$	−1.56921e13	−0.769927	−0.384963	+	0.922932i	$0.625786\pi$
$908$	2.39493e13	1.16925
$909$	−5.23619e12	−0.254377
$910$	0	0
$911$	2.93154e13	1.41014	0.705072	−	0.709135i	$-0.250915\pi$
$911$	2.93154e13	1.41014	0.705072	+	0.709135i	$0.250915\pi$
$912$	−1.26156e13	−0.603852
$913$	−1.25725e13	−0.598831
$914$	2.86384e13	1.35735
$915$	−1.66123e13	−0.783490
$916$	9.01407e12	0.423050
$917$	−4.20956e13	−1.96596
$918$	−1.27620e13	−0.593098
$919$	−2.25362e13	−1.04222	−0.521112	−	0.853488i	$-0.674483\pi$
$919$	−2.25362e13	−1.04222	−0.521112	+	0.853488i	$0.674483\pi$
$920$	−3.32997e12	−0.153248
$921$	3.60351e13	1.65028
$922$	2.37205e12	0.108102
$923$	0	0
$924$	2.90901e13	1.31287
$925$	−2.19895e13	−0.987593
$926$	−8.60900e12	−0.384772
$927$	5.56941e13	2.47714
$928$	2.89272e12	0.128039
$929$	1.08694e13	0.478780	0.239390	−	0.970923i	$-0.423053\pi$
$929$	1.08694e13	0.478780	0.239390	+	0.970923i	$0.423053\pi$
$930$	2.30208e13	1.00913
$931$	1.37849e13	0.601352
$932$	1.83622e13	0.797174
$933$	7.21699e13	3.11809
$934$	1.17823e13	0.506606
$935$	3.33632e11	0.0142763
$936$	0	0
$937$	−1.87914e13	−0.796398	−0.398199	−	0.917299i	$-0.630365\pi$
$937$	−1.87914e13	−0.796398	−0.398199	+	0.917299i	$0.630365\pi$
$938$	−5.58871e13	−2.35721
$939$	−3.06058e13	−1.28472
$940$	−3.60449e12	−0.150580
$941$	−9.61664e12	−0.399825	−0.199913	−	0.979814i	$-0.564066\pi$
$941$	−9.61664e12	−0.399825	−0.199913	+	0.979814i	$0.564066\pi$
$942$	7.93052e13	3.28150
$943$	4.75574e12	0.195847
$944$	1.54587e13	0.633578
$945$	3.01819e13	1.23113
$946$	−3.45505e12	−0.140264
$947$	−3.70974e13	−1.49889	−0.749444	−	0.662068i	$-0.769679\pi$
$947$	−3.70974e13	−1.49889	−0.749444	+	0.662068i	$0.769679\pi$
$948$	5.23199e13	2.10392
$949$	0	0
$950$	1.78242e13	0.709993
$951$	9.09438e13	3.60546
$952$	2.21079e12	0.0872330
$953$	4.35750e13	1.71127	0.855636	−	0.517577i	$-0.173166\pi$
$953$	4.35750e13	1.71127	0.855636	+	0.517577i	$0.173166\pi$
$954$	−9.15816e13	−3.57965
$955$	2.06736e12	0.0804268
$956$	2.41219e13	0.934008
$957$	−1.65454e12	−0.0637635
$958$	1.10030e13	0.422051
$959$	2.75502e13	1.05182
$960$	2.09615e13	0.796529
$961$	1.36502e13	0.516276
$962$	0	0
$963$	−4.84930e13	−1.81702
$964$	6.27902e13	2.34177
$965$	8.51015e11	0.0315911
$966$	−1.38288e14	−5.10962
$967$	3.87930e13	1.42670	0.713352	−	0.700806i	$-0.247176\pi$
$967$	3.87930e13	1.42670	0.713352	+	0.700806i	$0.247176\pi$
$968$	1.03435e13	0.378641
$969$	3.57183e12	0.130147
$970$	−3.09618e12	−0.112293
$971$	2.76084e13	0.996678	0.498339	−	0.866982i	$-0.333943\pi$
$971$	2.76084e13	0.996678	0.498339	+	0.866982i	$0.333943\pi$
$972$	9.28248e13	3.33554
$973$	2.20245e13	0.787769
$974$	−6.49383e13	−2.31199
$975$	0	0
$976$	−2.57176e13	−0.907206
$977$	−1.64814e13	−0.578721	−0.289360	−	0.957220i	$-0.593443\pi$
$977$	−1.64814e13	−0.578721	−0.289360	+	0.957220i	$0.593443\pi$
$978$	3.05826e13	1.06893
$979$	−9.56056e12	−0.332630
$980$	−1.25933e13	−0.436136
$981$	−2.03613e13	−0.701933
$982$	5.15397e13	1.76864
$983$	−4.86777e13	−1.66280	−0.831399	−	0.555676i	$-0.812459\pi$
$983$	−4.86777e13	−1.66280	−0.831399	+	0.555676i	$0.812459\pi$
$984$	3.89736e12	0.132523
$985$	−2.35085e12	−0.0795722
$986$	−5.60543e11	−0.0188870
$987$	−3.35782e13	−1.12624
$988$	0	0
$989$	9.24977e12	0.307431
$990$	−1.22727e13	−0.406053
$991$	1.07473e13	0.353970	0.176985	−	0.984214i	$-0.443366\pi$
$991$	1.07473e13	0.353970	0.176985	+	0.984214i	$0.443366\pi$
$992$	5.20718e13	1.70726
$993$	−4.86238e13	−1.58700
$994$	3.62689e12	0.117841
$995$	1.24375e12	0.0402281
$996$	1.22965e14	3.95927
$997$	1.10239e13	0.353353	0.176676	−	0.984269i	$-0.443465\pi$
$997$	1.10239e13	0.353353	0.176676	+	0.984269i	$0.443465\pi$
$998$	4.72502e13	1.50770
$999$	9.83051e13	3.12271

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.10.a.f.1.4		20
13.2	odd	12		13.10.e.a.4.2	✓	20
13.7	odd	12		13.10.e.a.10.2	yes	20
13.12	even	2	inner	169.10.a.f.1.17		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.10.e.a.4.2	✓	20	13.2	odd	12
13.10.e.a.10.2	yes	20	13.7	odd	12
169.10.a.f.1.4		20	1.1	even	1	trivial
169.10.a.f.1.17		20	13.12	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	169.10.a.f.1.4

Level	$169$
Weight	$10$
Character	169.1
Self dual	yes
Analytic conductor	$87.041$
Analytic rank	$0$
Dimension	$20$
Inner twists	$2$