LMFDB - Embedded newform 169.10.a.f.1.20

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			1.20
Root			$41.3599$ 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)

$f(q)$	$=$	$q+41.3599 q^{2} +214.052 q^{3} +1198.64 q^{4} +1662.90 q^{5} +8853.18 q^{6} +124.522 q^{7} +28399.6 q^{8} +26135.3 q^{9} +68777.5 q^{10} -72425.9 q^{11} +256572. q^{12} +5150.23 q^{14} +355947. q^{15} +560898. q^{16} -281440. q^{17} +1.08096e6 q^{18} +157910. q^{19} +1.99323e6 q^{20} +26654.3 q^{21} -2.99553e6 q^{22} +2.60475e6 q^{23} +6.07899e6 q^{24} +812113. q^{25} +1.38113e6 q^{27} +149258. q^{28} -998522. q^{29} +1.47220e7 q^{30} +2.62786e6 q^{31} +8.65813e6 q^{32} -1.55029e7 q^{33} -1.16403e7 q^{34} +207068. q^{35} +3.13270e7 q^{36} +552605. q^{37} +6.53116e6 q^{38} +4.72256e7 q^{40} -5.25427e6 q^{41} +1.10242e6 q^{42} +1.33968e7 q^{43} -8.68129e7 q^{44} +4.34605e7 q^{45} +1.07732e8 q^{46} -2.46747e7 q^{47} +1.20061e8 q^{48} -4.03381e7 q^{49} +3.35889e7 q^{50} -6.02427e7 q^{51} -3.60475e7 q^{53} +5.71236e7 q^{54} -1.20437e8 q^{55} +3.53638e6 q^{56} +3.38010e7 q^{57} -4.12988e7 q^{58} -5.37869e7 q^{59} +4.26654e8 q^{60} +1.04085e8 q^{61} +1.08688e8 q^{62} +3.25443e6 q^{63} +7.09198e7 q^{64} -6.41200e8 q^{66} -5.45531e7 q^{67} -3.37346e8 q^{68} +5.57553e8 q^{69} +8.56432e6 q^{70} -1.40628e8 q^{71} +7.42232e8 q^{72} +4.29589e7 q^{73} +2.28557e7 q^{74} +1.73835e8 q^{75} +1.89278e8 q^{76} -9.01864e6 q^{77} -2.82937e8 q^{79} +9.32717e8 q^{80} -2.18787e8 q^{81} -2.17316e8 q^{82} -4.41851e8 q^{83} +3.19490e7 q^{84} -4.68006e8 q^{85} +5.54092e8 q^{86} -2.13736e8 q^{87} -2.05686e9 q^{88} +7.17027e8 q^{89} +1.79752e9 q^{90} +3.12217e9 q^{92} +5.62499e8 q^{93} -1.02054e9 q^{94} +2.62589e8 q^{95} +1.85329e9 q^{96} +1.99252e8 q^{97} -1.66838e9 q^{98} -1.89288e9 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$20 q + 326 q^{3} + 5118 q^{4} + 129526 q^{9} + 88390 q^{10} + 427652 q^{12} + 473556 q^{14} + 1189618 q^{16} - 99312 q^{17} - 5073532 q^{22} + 6252378 q^{23} + 1529274 q^{25} + 18052718 q^{27} + 5424828 q^{29}+ \cdots + 9251202540 q^{95}+O(q^{100})$ 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

)

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

$2$	41.3599	1.82787	0.913934	−	0.405863i	$-0.133029\pi$
$2$	41.3599	1.82787	0.913934	+	0.405863i	$0.133029\pi$
$3$	214.052	1.52572	0.762858	−	0.646565i	$-0.223795\pi$
$3$	214.052	1.52572	0.762858	+	0.646565i	$0.223795\pi$
$4$	1198.64	2.34110
$5$	1662.90	1.18987	0.594937	−	0.803772i	$-0.297177\pi$
$5$	1662.90	1.18987	0.594937	+	0.803772i	$0.297177\pi$
$6$	8853.18	2.78881
$7$	124.522	0.0196022	0.00980112	−	0.999952i	$-0.496880\pi$
$7$	124.522	0.0196022	0.00980112	+	0.999952i	$0.496880\pi$
$8$	28399.6	2.45136
$9$	26135.3	1.32781
$10$	68777.5	2.17493
$11$	−72425.9	−1.49151	−0.745757	−	0.666219i	$-0.767912\pi$
$11$	−72425.9	−1.49151	−0.745757	+	0.666219i	$0.767912\pi$
$12$	256572.	3.57186
$13$	0	0
$13$	0	0
$14$	5150.23	0.0358303
$15$	355947.	1.81541
$16$	560898.	2.13966
$17$	−281440.	−0.817269	−0.408634	−	0.912698i	$-0.633995\pi$
$17$	−281440.	−0.817269	−0.408634	+	0.912698i	$0.633995\pi$
$18$	1.08096e6	2.42707
$19$	157910.	0.277984	0.138992	−	0.990294i	$-0.455614\pi$
$19$	157910.	0.277984	0.138992	+	0.990294i	$0.455614\pi$
$20$	1.99323e6	2.78562
$21$	26654.3	0.0299075
$22$	−2.99553e6	−2.72629
$23$	2.60475e6	1.94085	0.970423	−	0.241411i	$-0.0776103\pi$
$23$	2.60475e6	1.94085	0.970423	+	0.241411i	$0.0776103\pi$
$24$	6.07899e6	3.74008
$25$	812113.	0.415802
$26$	0	0
$27$	1.38113e6	0.500149
$28$	149258.	0.0458908
$29$	−998522.	−0.262160	−0.131080	−	0.991372i	$-0.541845\pi$
$29$	−998522.	−0.262160	−0.131080	+	0.991372i	$0.541845\pi$
$30$	1.47220e7	3.31833
$31$	2.62786e6	0.511063	0.255531	−	0.966801i	$-0.417750\pi$
$31$	2.62786e6	0.511063	0.255531	+	0.966801i	$0.417750\pi$
$32$	8.65813e6	1.45965
$33$	−1.55029e7	−2.27563
$34$	−1.16403e7	−1.49386
$35$	207068.	0.0233242
$36$	3.13270e7	3.10854
$37$	552605.	0.0484738	0.0242369	−	0.999706i	$-0.492284\pi$
$37$	552605.	0.0484738	0.0242369	+	0.999706i	$0.492284\pi$
$38$	6.53116e6	0.508118
$39$	0	0
$40$	4.72256e7	2.91681
$41$	−5.25427e6	−0.290392	−0.145196	−	0.989403i	$-0.546381\pi$
$41$	−5.25427e6	−0.290392	−0.145196	+	0.989403i	$0.546381\pi$
$42$	1.10242e6	0.0546669
$43$	1.33968e7	0.597577	0.298789	−	0.954319i	$-0.403417\pi$
$43$	1.33968e7	0.597577	0.298789	+	0.954319i	$0.403417\pi$
$44$	−8.68129e7	−3.49178
$45$	4.34605e7	1.57993
$46$	1.07732e8	3.54761
$47$	−2.46747e7	−0.737583	−0.368791	−	0.929512i	$-0.620228\pi$
$47$	−2.46747e7	−0.737583	−0.368791	+	0.929512i	$0.620228\pi$
$48$	1.20061e8	3.26451
$49$	−4.03381e7	−0.999616
$50$	3.35889e7	0.760031
$51$	−6.02427e7	−1.24692
$52$	0	0
$53$	−3.60475e7	−0.627529	−0.313764	−	0.949501i	$-0.601590\pi$
$53$	−3.60475e7	−0.627529	−0.313764	+	0.949501i	$0.601590\pi$
$54$	5.71236e7	0.914206
$55$	−1.20437e8	−1.77471
$56$	3.53638e6	0.0480521
$57$	3.38010e7	0.424124
$58$	−4.12988e7	−0.479194
$59$	−5.37869e7	−0.577887	−0.288943	−	0.957346i	$-0.593304\pi$
$59$	−5.37869e7	−0.577887	−0.288943	+	0.957346i	$0.593304\pi$
$60$	4.26654e8	4.25006
$61$	1.04085e8	0.962506	0.481253	−	0.876582i	$-0.340182\pi$
$61$	1.04085e8	0.962506	0.481253	+	0.876582i	$0.340182\pi$
$62$	1.08688e8	0.934155
$63$	3.25443e6	0.0260281
$64$	7.09198e7	0.528394
$65$	0	0
$66$	−6.41200e8	−4.15955
$67$	−5.45531e7	−0.330737	−0.165368	−	0.986232i	$-0.552881\pi$
$67$	−5.45531e7	−0.330737	−0.165368	+	0.986232i	$0.552881\pi$
$68$	−3.37346e8	−1.91331
$69$	5.57553e8	2.96118
$70$	8.56432e6	0.0426336
$71$	−1.40628e8	−0.656762	−0.328381	−	0.944545i	$-0.606503\pi$
$71$	−1.40628e8	−0.656762	−0.328381	+	0.944545i	$0.606503\pi$
$72$	7.42232e8	3.25494
$73$	4.29589e7	0.177052	0.0885258	−	0.996074i	$-0.471784\pi$
$73$	4.29589e7	0.177052	0.0885258	+	0.996074i	$0.471784\pi$
$74$	2.28557e7	0.0886038
$75$	1.73835e8	0.634396
$76$	1.89278e8	0.650788
$77$	−9.01864e6	−0.0292370
$78$	0	0
$79$	−2.82937e8	−0.817273	−0.408637	−	0.912697i	$-0.633996\pi$
$79$	−2.82937e8	−0.817273	−0.408637	+	0.912697i	$0.633996\pi$
$80$	9.32717e8	2.54592
$81$	−2.18787e8	−0.564727
$82$	−2.17316e8	−0.530798
$83$	−4.41851e8	−1.02194	−0.510969	−	0.859599i	$-0.670713\pi$
$83$	−4.41851e8	−1.02194	−0.510969	+	0.859599i	$0.670713\pi$
$84$	3.19490e7	0.0700164
$85$	−4.68006e8	−0.972447
$86$	5.54092e8	1.09229
$87$	−2.13736e8	−0.399982
$88$	−2.05686e9	−3.65623
$89$	7.17027e8	1.21138	0.605690	−	0.795701i	$-0.292897\pi$
$89$	7.17027e8	1.21138	0.605690	+	0.795701i	$0.292897\pi$
$90$	1.79752e9	2.88790
$91$	0	0
$92$	3.12217e9	4.54372
$93$	5.62499e8	0.779737
$94$	−1.02054e9	−1.34820
$95$	2.62589e8	0.330766
$96$	1.85329e9	2.22701
$97$	1.99252e8	0.228523	0.114261	−	0.993451i	$-0.463550\pi$
$97$	1.99252e8	0.228523	0.114261	+	0.993451i	$0.463550\pi$
$98$	−1.66838e9	−1.82717
$99$	−1.89288e9	−1.98045
$100$	9.73434e8	0.973434
$101$	1.14193e9	1.09192	0.545961	−	0.837810i	$-0.316165\pi$
$101$	1.14193e9	1.09192	0.545961	+	0.837810i	$0.316165\pi$
$102$	−2.49164e9	−2.27921
$103$	−1.06178e9	−0.929538	−0.464769	−	0.885432i	$-0.653863\pi$
$103$	−1.06178e9	−0.929538	−0.464769	+	0.885432i	$0.653863\pi$
$104$	0	0
$105$	4.43234e7	0.0355861
$106$	−1.49092e9	−1.14704
$107$	2.01890e9	1.48897	0.744487	−	0.667637i	$-0.232694\pi$
$107$	2.01890e9	1.48897	0.744487	+	0.667637i	$0.232694\pi$
$108$	1.65549e9	1.17090
$109$	−8.62088e8	−0.584969	−0.292484	−	0.956270i	$-0.594482\pi$
$109$	−8.62088e8	−0.584969	−0.292484	+	0.956270i	$0.594482\pi$
$110$	−4.98127e9	−3.24394
$111$	1.18286e8	0.0739573
$112$	6.98442e7	0.0419420
$113$	−1.82041e9	−1.05031	−0.525154	−	0.851007i	$-0.675992\pi$
$113$	−1.82041e9	−1.05031	−0.525154	+	0.851007i	$0.675992\pi$
$114$	1.39801e9	0.775244
$115$	4.33144e9	2.30936
$116$	−1.19687e9	−0.613744
$117$	0	0
$118$	−2.22462e9	−1.05630
$119$	−3.50455e7	−0.0160203
$120$	1.01087e10	4.45022
$121$	2.88757e9	1.22461
$122$	4.30494e9	1.75933
$123$	−1.12469e9	−0.443056
$124$	3.14987e9	1.19645
$125$	−1.89739e9	−0.695123
$126$	1.34603e8	0.0475759
$127$	−3.39438e9	−1.15783	−0.578914	−	0.815389i	$-0.696523\pi$
$127$	−3.39438e9	−1.15783	−0.578914	+	0.815389i	$0.696523\pi$
$128$	−1.49972e9	−0.493817
$129$	2.86762e9	0.911734
$130$	0	0
$131$	1.31505e9	0.390139	0.195070	−	0.980789i	$-0.437507\pi$
$131$	1.31505e9	0.390139	0.195070	+	0.980789i	$0.437507\pi$
$132$	−1.85825e10	−5.32747
$133$	1.96633e7	0.00544910
$134$	−2.25631e9	−0.604543
$135$	2.29669e9	0.595114
$136$	−7.99276e9	−2.00342
$137$	−2.22029e9	−0.538478	−0.269239	−	0.963073i	$-0.586772\pi$
$137$	−2.22029e9	−0.538478	−0.269239	+	0.963073i	$0.586772\pi$
$138$	2.30603e10	5.41265
$139$	6.80605e9	1.54642	0.773212	−	0.634147i	$-0.218649\pi$
$139$	6.80605e9	1.54642	0.773212	+	0.634147i	$0.218649\pi$
$140$	2.48201e8	0.0546043
$141$	−5.28166e9	−1.12534
$142$	−5.81635e9	−1.20047
$143$	0	0
$144$	1.46592e10	2.84106
$145$	−1.66044e9	−0.311938
$146$	1.77678e9	0.323627
$147$	−8.63446e9	−1.52513
$148$	6.62377e8	0.113482
$149$	7.13055e9	1.18518	0.592591	−	0.805504i	$-0.298105\pi$
$149$	7.13055e9	1.18518	0.592591	+	0.805504i	$0.298105\pi$
$150$	7.18978e9	1.15959
$151$	−9.69848e9	−1.51812	−0.759062	−	0.651018i	$-0.774342\pi$
$151$	−9.69848e9	−1.51812	−0.759062	+	0.651018i	$0.774342\pi$
$152$	4.48458e9	0.681437
$153$	−7.35551e9	−1.08518
$154$	−3.73010e8	−0.0534414
$155$	4.36987e9	0.608101
$156$	0	0
$157$	6.71182e9	0.881642	0.440821	−	0.897595i	$-0.354687\pi$
$157$	6.71182e9	0.881642	0.440821	+	0.897595i	$0.354687\pi$
$158$	−1.17022e10	−1.49387
$159$	−7.71604e9	−0.957431
$160$	1.43976e10	1.73680
$161$	3.24349e8	0.0380449
$162$	−9.04901e9	−1.03225
$163$	8.32930e9	0.924196	0.462098	−	0.886829i	$-0.347097\pi$
$163$	8.32930e9	0.924196	0.462098	+	0.886829i	$0.347097\pi$
$164$	−6.29799e9	−0.679837
$165$	−2.57798e10	−2.70771
$166$	−1.82749e10	−1.86797
$167$	−1.73805e10	−1.72917	−0.864587	−	0.502483i	$-0.832420\pi$
$167$	−1.73805e10	−1.72917	−0.864587	+	0.502483i	$0.832420\pi$
$168$	7.56969e8	0.0733139
$169$	0	0
$170$	−1.93567e10	−1.77751
$171$	4.12704e9	0.369110
$172$	1.60580e10	1.39899
$173$	1.50016e10	1.27330	0.636648	−	0.771154i	$-0.280320\pi$
$173$	1.50016e10	1.27330	0.636648	+	0.771154i	$0.280320\pi$
$174$	−8.84010e9	−0.731115
$175$	1.01126e8	0.00815065
$176$	−4.06236e10	−3.19132
$177$	−1.15132e10	−0.881692
$178$	2.96562e10	2.21424
$179$	2.63296e10	1.91693	0.958465	−	0.285210i	$-0.0920634\pi$
$179$	2.63296e10	1.91693	0.958465	+	0.285210i	$0.0920634\pi$
$180$	5.20936e10	3.69878
$181$	1.33696e10	0.925903	0.462951	−	0.886384i	$-0.346790\pi$
$181$	1.33696e10	0.925903	0.462951	+	0.886384i	$0.346790\pi$
$182$	0	0
$183$	2.22796e10	1.46851
$184$	7.39738e10	4.75771
$185$	9.18928e8	0.0576778
$186$	2.32649e10	1.42526
$187$	2.03835e10	1.21897
$188$	−2.95761e10	−1.72676
$189$	1.71982e8	0.00980404
$190$	1.08607e10	0.604596
$191$	2.17230e10	1.18105	0.590527	−	0.807018i	$-0.298920\pi$
$191$	2.17230e10	1.18105	0.590527	+	0.807018i	$0.298920\pi$
$192$	1.51805e10	0.806179
$193$	1.89241e10	0.981762	0.490881	−	0.871227i	$-0.336675\pi$
$193$	1.89241e10	0.981762	0.490881	+	0.871227i	$0.336675\pi$
$194$	8.24104e9	0.417709
$195$	0	0
$196$	−4.83510e10	−2.34020
$197$	3.68204e10	1.74177	0.870885	−	0.491488i	$-0.163547\pi$
$197$	3.68204e10	1.74177	0.870885	+	0.491488i	$0.163547\pi$
$198$	−7.82892e10	−3.62000
$199$	−1.06136e10	−0.479762	−0.239881	−	0.970802i	$-0.577108\pi$
$199$	−1.06136e10	−0.479762	−0.239881	+	0.970802i	$0.577108\pi$
$200$	2.30636e10	1.01928
$201$	−1.16772e10	−0.504611
$202$	4.72300e10	1.99589
$203$	−1.24338e8	−0.00513893
$204$	−7.22096e10	−2.91917
$205$	−8.73732e9	−0.345530
$206$	−4.39152e10	−1.69907
$207$	6.80760e10	2.57708
$208$	0	0
$209$	−1.14368e10	−0.414616
$210$	1.83321e9	0.0650468
$211$	1.38838e10	0.482211	0.241106	−	0.970499i	$-0.422490\pi$
$211$	1.38838e10	0.482211	0.241106	+	0.970499i	$0.422490\pi$
$212$	−4.32081e10	−1.46911
$213$	−3.01017e10	−1.00203
$214$	8.35014e10	2.72165
$215$	2.22776e10	0.711042
$216$	3.92236e10	1.22604
$217$	3.27227e8	0.0100180
$218$	−3.56559e10	−1.06925
$219$	9.19544e9	0.270131
$220$	−1.44361e11	−4.15479
$221$	0	0
$222$	4.89232e9	0.135184
$223$	−4.30970e10	−1.16701	−0.583506	−	0.812109i	$-0.698319\pi$
$223$	−4.30970e10	−1.16701	−0.583506	+	0.812109i	$0.698319\pi$
$224$	1.07813e9	0.0286124
$225$	2.12248e10	0.552107
$226$	−7.52921e10	−1.91982
$227$	1.76035e10	0.440032	0.220016	−	0.975496i	$-0.429389\pi$
$227$	1.76035e10	0.440032	0.220016	+	0.975496i	$0.429389\pi$
$228$	4.05154e10	0.992918
$229$	−5.00585e10	−1.20287	−0.601434	−	0.798923i	$-0.705404\pi$
$229$	−5.00585e10	−1.20287	−0.601434	+	0.798923i	$0.705404\pi$
$230$	1.79148e11	4.22121
$231$	−1.93046e9	−0.0446074
$232$	−2.83576e10	−0.642648
$233$	4.03325e10	0.896506	0.448253	−	0.893907i	$-0.352046\pi$
$233$	4.03325e10	0.896506	0.448253	+	0.893907i	$0.352046\pi$
$234$	0	0
$235$	−4.10315e10	−0.877631
$236$	−6.44714e10	−1.35289
$237$	−6.05632e10	−1.24693
$238$	−1.44948e9	−0.0292830
$239$	−7.01980e10	−1.39166	−0.695831	−	0.718205i	$-0.744964\pi$
$239$	−7.01980e10	−1.39166	−0.695831	+	0.718205i	$0.744964\pi$
$240$	1.99650e11	3.88436
$241$	−4.54620e10	−0.868105	−0.434052	−	0.900888i	$-0.642917\pi$
$241$	−4.54620e10	−0.868105	−0.434052	+	0.900888i	$0.642917\pi$
$242$	1.19430e11	2.23843
$243$	−7.40167e10	−1.36176
$244$	1.24761e11	2.25332
$245$	−6.70782e10	−1.18942
$246$	−4.65170e10	−0.809848
$247$	0	0
$248$	7.46300e10	1.25280
$249$	−9.45791e10	−1.55919
$250$	−7.84759e10	−1.27059
$251$	8.88333e10	1.41268	0.706341	−	0.707872i	$-0.250345\pi$
$251$	8.88333e10	1.41268	0.706341	+	0.707872i	$0.250345\pi$
$252$	3.90090e9	0.0609344
$253$	−1.88652e11	−2.89480
$254$	−1.40391e11	−2.11636
$255$	−1.00178e11	−1.48368
$256$	−9.83393e10	−1.43103
$257$	−3.57366e10	−0.510993	−0.255496	−	0.966810i	$-0.582239\pi$
$257$	−3.57366e10	−0.510993	−0.255496	+	0.966810i	$0.582239\pi$
$258$	1.18605e11	1.66653
$259$	6.88116e7	0.000950196 0
$260$	0	0
$261$	−2.60967e10	−0.348100
$262$	5.43902e10	0.713123
$263$	7.02163e8	0.00904976	0.00452488	−	0.999990i	$-0.498560\pi$
$263$	7.02163e8	0.00904976	0.00452488	+	0.999990i	$0.498560\pi$
$264$	−4.40276e11	−5.57837
$265$	−5.99434e10	−0.746681
$266$	8.13275e8	0.00996024
$267$	1.53481e11	1.84822
$268$	−6.53897e10	−0.774289
$269$	−4.37432e9	−0.0509360	−0.0254680	−	0.999676i	$-0.508108\pi$
$269$	−4.37432e9	−0.0509360	−0.0254680	+	0.999676i	$0.508108\pi$
$270$	9.49909e10	1.08779
$271$	8.18198e9	0.0921503	0.0460752	−	0.998938i	$-0.485329\pi$
$271$	8.18198e9	0.0921503	0.0460752	+	0.998938i	$0.485329\pi$
$272$	−1.57859e11	−1.74867
$273$	0	0
$274$	−9.18312e10	−0.984266
$275$	−5.88180e10	−0.620174
$276$	6.68307e11	6.93243
$277$	6.44866e10	0.658128	0.329064	−	0.944308i	$-0.393267\pi$
$277$	6.44866e10	0.658128	0.329064	+	0.944308i	$0.393267\pi$
$278$	2.81498e11	2.82666
$279$	6.86799e10	0.678595
$280$	5.88064e9	0.0571760
$281$	−1.56666e11	−1.49898	−0.749489	−	0.662017i	$-0.769701\pi$
$281$	−1.56666e11	−1.49898	−0.749489	+	0.662017i	$0.769701\pi$
$282$	−2.18449e11	−2.05698
$283$	1.55686e10	0.144281	0.0721406	−	0.997394i	$-0.477017\pi$
$283$	1.55686e10	0.144281	0.0721406	+	0.997394i	$0.477017\pi$
$284$	−1.68562e11	−1.53755
$285$	5.62078e10	0.504655
$286$	0	0
$287$	−6.54273e8	−0.00569233
$288$	2.26283e11	1.93814
$289$	−3.93797e10	−0.332072
$290$	−6.86758e10	−0.570181
$291$	4.26503e10	0.348661
$292$	5.14924e10	0.414496
$293$	−2.38890e11	−1.89363	−0.946813	−	0.321784i	$-0.895718\pi$
$293$	−2.38890e11	−1.89363	−0.946813	+	0.321784i	$0.895718\pi$
$294$	−3.57121e11	−2.78774
$295$	−8.94423e10	−0.687613
$296$	1.56937e10	0.118827
$297$	−1.00030e11	−0.745978
$298$	2.94919e11	2.16636
$299$	0	0
$300$	2.08366e11	1.48519
$301$	1.66820e9	0.0117139
$302$	−4.01128e11	−2.77493
$303$	2.44432e11	1.66596
$304$	8.85716e10	0.594789
$305$	1.73083e11	1.14526
$306$	−3.04224e11	−1.98357
$307$	4.63794e10	0.297991	0.148995	−	0.988838i	$-0.452396\pi$
$307$	4.63794e10	0.297991	0.148995	+	0.988838i	$0.452396\pi$
$308$	−1.08101e10	−0.0684468
$309$	−2.27276e11	−1.41821
$310$	1.80737e11	1.11153
$311$	−6.35548e10	−0.385236	−0.192618	−	0.981274i	$-0.561698\pi$
$311$	−6.35548e10	−0.385236	−0.192618	+	0.981274i	$0.561698\pi$
$312$	0	0
$313$	−4.62147e10	−0.272164	−0.136082	−	0.990698i	$-0.543451\pi$
$313$	−4.62147e10	−0.272164	−0.136082	+	0.990698i	$0.543451\pi$
$314$	2.77601e11	1.61152
$315$	5.41179e9	0.0309702
$316$	−3.39140e11	−1.91332
$317$	1.57383e11	0.875367	0.437684	−	0.899129i	$-0.355799\pi$
$317$	1.57383e11	0.875367	0.437684	+	0.899129i	$0.355799\pi$
$318$	−3.19135e11	−1.75006
$319$	7.23189e10	0.391015
$320$	1.17933e11	0.628722
$321$	4.32149e11	2.27175
$322$	1.34151e10	0.0695411
$323$	−4.44422e10	−0.227187
$324$	−2.62247e11	−1.32208
$325$	0	0
$326$	3.44499e11	1.68931
$327$	−1.84532e11	−0.892496
$328$	−1.49219e11	−0.711854
$329$	−3.07254e9	−0.0144583
$330$	−1.06625e12	−4.94934
$331$	−1.89830e11	−0.869238	−0.434619	−	0.900614i	$-0.643117\pi$
$331$	−1.89830e11	−0.869238	−0.434619	+	0.900614i	$0.643117\pi$
$332$	−5.29622e11	−2.39246
$333$	1.44425e10	0.0643641
$334$	−7.18857e11	−3.16070
$335$	−9.07163e10	−0.393535
$336$	1.49503e10	0.0639917
$337$	6.99731e10	0.295526	0.147763	−	0.989023i	$-0.452793\pi$
$337$	6.99731e10	0.295526	0.147763	+	0.989023i	$0.452793\pi$
$338$	0	0
$339$	−3.89663e11	−1.60247
$340$	−5.60972e11	−2.27660
$341$	−1.90325e11	−0.762257
$342$	1.70694e11	0.674685
$343$	−1.00479e10	−0.0391969
$344$	3.80464e11	1.46488
$345$	9.27155e11	3.52343
$346$	6.20464e11	2.32742
$347$	−1.03906e10	−0.0384730	−0.0192365	−	0.999815i	$-0.506124\pi$
$347$	−1.03906e10	−0.0384730	−0.0192365	+	0.999815i	$0.506124\pi$
$348$	−2.56193e11	−0.936399
$349$	−2.36535e11	−0.853457	−0.426729	−	0.904380i	$-0.640334\pi$
$349$	−2.36535e11	−0.853457	−0.426729	+	0.904380i	$0.640334\pi$
$350$	4.18257e9	0.0148983
$351$	0	0
$352$	−6.27073e11	−2.17709
$353$	−3.53919e10	−0.121316	−0.0606580	−	0.998159i	$-0.519320\pi$
$353$	−3.53919e10	−0.121316	−0.0606580	+	0.998159i	$0.519320\pi$
$354$	−4.76186e11	−1.61162
$355$	−2.33850e11	−0.781465
$356$	8.59460e11	2.83596
$357$	−7.50156e9	−0.0244424
$358$	1.08899e12	3.50390
$359$	3.89544e11	1.23775	0.618873	−	0.785491i	$-0.287590\pi$
$359$	3.89544e11	1.23775	0.618873	+	0.785491i	$0.287590\pi$
$360$	1.23426e12	3.87297
$361$	−2.97752e11	−0.922725
$362$	5.52966e11	1.69243
$363$	6.18090e11	1.86841
$364$	0	0
$365$	7.14363e10	0.210669
$366$	9.21482e11	2.68424
$367$	1.68314e11	0.484309	0.242154	−	0.970238i	$-0.422146\pi$
$367$	1.68314e11	0.484309	0.242154	+	0.970238i	$0.422146\pi$
$368$	1.46100e12	4.15274
$369$	−1.37322e11	−0.385586
$370$	3.80068e10	0.105427
$371$	−4.48871e9	−0.0123010
$372$	6.74236e11	1.82544
$373$	2.15603e11	0.576721	0.288361	−	0.957522i	$-0.406890\pi$
$373$	2.15603e11	0.576721	0.288361	+	0.957522i	$0.406890\pi$
$374$	8.43061e11	2.22811
$375$	−4.06140e11	−1.06056
$376$	−7.00749e11	−1.80808
$377$	0	0
$378$	7.11316e9	0.0179205
$379$	5.68776e11	1.41600	0.708002	−	0.706210i	$-0.249597\pi$
$379$	5.68776e11	1.41600	0.708002	+	0.706210i	$0.249597\pi$
$380$	3.14751e11	0.774356
$381$	−7.26575e11	−1.76652
$382$	8.98462e11	2.15881
$383$	3.38261e11	0.803262	0.401631	−	0.915802i	$-0.368443\pi$
$383$	3.38261e11	0.803262	0.401631	+	0.915802i	$0.368443\pi$
$384$	−3.21019e11	−0.753425
$385$	−1.49971e10	−0.0347884
$386$	7.82698e11	1.79453
$387$	3.50131e11	0.793471
$388$	2.38832e11	0.534995
$389$	6.44312e11	1.42667	0.713334	−	0.700825i	$-0.247184\pi$
$389$	6.44312e11	1.42667	0.713334	+	0.700825i	$0.247184\pi$
$390$	0	0
$391$	−7.33080e11	−1.58619
$392$	−1.14558e12	−2.45041
$393$	2.81488e11	0.595242
$394$	1.52289e12	3.18372
$395$	−4.70495e11	−0.972453
$396$	−2.26888e12	−4.63643
$397$	4.08070e11	0.824474	0.412237	−	0.911077i	$-0.364748\pi$
$397$	4.08070e11	0.824474	0.412237	+	0.911077i	$0.364748\pi$
$398$	−4.38980e11	−0.876941
$399$	4.20898e9	0.00831379
$400$	4.55512e11	0.889673
$401$	−5.71961e10	−0.110463	−0.0552315	−	0.998474i	$-0.517590\pi$
$401$	−5.71961e10	−0.110463	−0.0552315	+	0.998474i	$0.517590\pi$
$402$	−4.82968e11	−0.922362
$403$	0	0
$404$	1.36876e12	2.55630
$405$	−3.63821e11	−0.671954
$406$	−5.14262e9	−0.00939328
$407$	−4.00230e10	−0.0722993
$408$	−1.71087e12	−3.05665
$409$	5.61035e11	0.991368	0.495684	−	0.868503i	$-0.334917\pi$
$409$	5.61035e11	0.991368	0.495684	+	0.868503i	$0.334917\pi$
$410$	−3.61375e11	−0.631583
$411$	−4.75259e11	−0.821565
$412$	−1.27270e12	−2.17614
$413$	−6.69767e9	−0.0113279
$414$	2.81562e12	4.71056
$415$	−7.34754e11	−1.21598
$416$	0	0
$417$	1.45685e12	2.35941
$418$	−4.73025e11	−0.757864
$419$	−3.10894e11	−0.492776	−0.246388	−	0.969171i	$-0.579244\pi$
$419$	−3.10894e11	−0.492776	−0.246388	+	0.969171i	$0.579244\pi$
$420$	5.31279e10	0.0833108
$421$	2.32067e10	0.0360034	0.0180017	−	0.999838i	$-0.494270\pi$
$421$	2.32067e10	0.0360034	0.0180017	+	0.999838i	$0.494270\pi$
$422$	5.74233e11	0.881418
$423$	−6.44880e11	−0.979372
$424$	−1.02373e12	−1.53830
$425$	−2.28561e11	−0.339822
$426$	−1.24500e12	−1.83158
$427$	1.29609e10	0.0188673
$428$	2.41994e12	3.48584
$429$	0	0
$430$	9.21400e11	1.29969
$431$	8.87725e11	1.23917	0.619585	−	0.784930i	$-0.287301\pi$
$431$	8.87725e11	1.23917	0.619585	+	0.784930i	$0.287301\pi$
$432$	7.74676e11	1.07015
$433$	1.18770e12	1.62372	0.811862	−	0.583850i	$-0.198454\pi$
$433$	1.18770e12	1.62372	0.811862	+	0.583850i	$0.198454\pi$
$434$	1.35341e10	0.0183115
$435$	−3.55421e11	−0.475929
$436$	−1.03334e12	−1.36947
$437$	4.11317e11	0.539524
$438$	3.80323e11	0.493763
$439$	3.27305e11	0.420593	0.210297	−	0.977638i	$-0.432557\pi$
$439$	3.27305e11	0.420593	0.210297	+	0.977638i	$0.432557\pi$
$440$	−3.42036e12	−4.35046
$441$	−1.05425e12	−1.32730
$442$	0	0
$443$	−5.47305e11	−0.675169	−0.337584	−	0.941295i	$-0.609610\pi$
$443$	−5.47305e11	−0.675169	−0.337584	+	0.941295i	$0.609610\pi$
$444$	1.41783e11	0.173142
$445$	1.19234e12	1.44139
$446$	−1.78249e12	−2.13314
$447$	1.52631e12	1.80825
$448$	8.83109e9	0.0103577
$449$	1.93461e11	0.224639	0.112320	−	0.993672i	$-0.464172\pi$
$449$	1.93461e11	0.224639	0.112320	+	0.993672i	$0.464172\pi$
$450$	8.77858e11	1.00918
$451$	3.80545e11	0.433123
$452$	−2.18203e12	−2.45888
$453$	−2.07598e12	−2.31623
$454$	7.28082e11	0.804320
$455$	0	0
$456$	9.59935e11	1.03968
$457$	−8.87358e11	−0.951647	−0.475824	−	0.879541i	$-0.657850\pi$
$457$	−8.87358e11	−0.951647	−0.475824	+	0.879541i	$0.657850\pi$
$458$	−2.07041e12	−2.19868
$459$	−3.88706e11	−0.408756
$460$	5.19186e12	5.40645
$461$	6.75642e11	0.696727	0.348363	−	0.937360i	$-0.386738\pi$
$461$	6.75642e11	0.696727	0.348363	+	0.937360i	$0.386738\pi$
$462$	−7.98437e10	−0.0815364
$463$	−8.76098e11	−0.886009	−0.443005	−	0.896519i	$-0.646088\pi$
$463$	−8.76098e11	−0.886009	−0.443005	+	0.896519i	$0.646088\pi$
$464$	−5.60069e11	−0.560932
$465$	9.35379e11	0.927789
$466$	1.66815e12	1.63870
$467$	−1.66467e12	−1.61958	−0.809791	−	0.586718i	$-0.800420\pi$
$467$	−1.66467e12	−1.61958	−0.809791	+	0.586718i	$0.800420\pi$
$468$	0	0
$469$	−6.79307e9	−0.00648318
$470$	−1.69706e12	−1.60419
$471$	1.43668e12	1.34514
$472$	−1.52753e12	−1.41661
$473$	−9.70278e11	−0.891295
$474$	−2.50489e12	−2.27922
$475$	1.28241e11	0.115586
$476$	−4.20071e10	−0.0375052
$477$	−9.42113e11	−0.833241
$478$	−2.90338e12	−2.54378
$479$	6.46122e11	0.560795	0.280398	−	0.959884i	$-0.409534\pi$
$479$	6.46122e11	0.560795	0.280398	+	0.959884i	$0.409534\pi$
$480$	3.08184e12	2.64987
$481$	0	0
$482$	−1.88031e12	−1.58678
$483$	6.94277e10	0.0580458
$484$	3.46117e12	2.86694
$485$	3.31336e11	0.271913
$486$	−3.06132e12	−2.48912
$487$	1.40768e10	0.0113403	0.00567014	−	0.999984i	$-0.498195\pi$
$487$	1.40768e10	0.0113403	0.00567014	+	0.999984i	$0.498195\pi$
$488$	2.95596e12	2.35944
$489$	1.78290e12	1.41006
$490$	−2.77435e12	−2.17410
$491$	2.01349e12	1.56345	0.781724	−	0.623624i	$-0.214341\pi$
$491$	2.01349e12	1.56345	0.781724	+	0.623624i	$0.214341\pi$
$492$	−1.34810e12	−1.03724
$493$	2.81024e11	0.214255
$494$	0	0
$495$	−3.14766e12	−2.35649
$496$	1.47396e12	1.09350
$497$	−1.75113e10	−0.0128740
$498$	−3.91179e12	−2.84999
$499$	1.30123e12	0.939512	0.469756	−	0.882796i	$-0.344342\pi$
$499$	1.30123e12	0.939512	0.469756	+	0.882796i	$0.344342\pi$
$500$	−2.27429e12	−1.62735
$501$	−3.72034e12	−2.63823
$502$	3.67414e12	2.58220
$503$	−1.87081e12	−1.30309	−0.651543	−	0.758612i	$-0.725878\pi$
$503$	−1.87081e12	−1.30309	−0.651543	+	0.758612i	$0.725878\pi$
$504$	9.24243e10	0.0638042
$505$	1.89891e12	1.29925
$506$	−7.80262e12	−5.29131
$507$	0	0
$508$	−4.06865e12	−2.71059
$509$	−9.71726e11	−0.641673	−0.320836	−	0.947135i	$-0.603964\pi$
$509$	−9.71726e11	−0.641673	−0.320836	+	0.947135i	$0.603964\pi$
$510$	−4.14334e12	−2.71197
$511$	5.34933e9	0.00347061
$512$	−3.29945e12	−2.12191
$513$	2.18095e11	0.139033
$514$	−1.47807e12	−0.934027
$515$	−1.76564e12	−1.10603
$516$	3.43726e12	2.13446
$517$	1.78709e12	1.10011
$518$	2.84605e9	0.00173683
$519$	3.21112e12	1.94269
$520$	0	0
$521$	1.09705e12	0.652316	0.326158	−	0.945315i	$-0.394246\pi$
$521$	1.09705e12	0.652316	0.326158	+	0.945315i	$0.394246\pi$
$522$	−1.07936e12	−0.636280
$523$	1.24282e12	0.726359	0.363180	−	0.931719i	$-0.381691\pi$
$523$	1.24282e12	0.726359	0.363180	+	0.931719i	$0.381691\pi$
$524$	1.57627e12	0.913356
$525$	2.16463e10	0.0124356
$526$	2.90414e10	0.0165418
$527$	−7.39583e11	−0.417676
$528$	−8.69556e12	−4.86906
$529$	4.98358e12	2.76688
$530$	−2.47926e12	−1.36483
$531$	−1.40574e12	−0.767325
$532$	2.35694e10	0.0127569
$533$	0	0
$534$	6.34797e12	3.37831
$535$	3.35722e12	1.77169
$536$	−1.54928e12	−0.810754
$537$	5.63592e12	2.92469
$538$	−1.80921e11	−0.0931043
$539$	2.92152e12	1.49094
$540$	2.75291e12	1.39322
$541$	−1.82157e11	−0.0914238	−0.0457119	−	0.998955i	$-0.514556\pi$
$541$	−1.82157e11	−0.0914238	−0.0457119	+	0.998955i	$0.514556\pi$
$542$	3.38406e11	0.168439
$543$	2.86180e12	1.41267
$544$	−2.43674e12	−1.19293
$545$	−1.43357e12	−0.696039
$546$	0	0
$547$	2.12370e12	1.01426	0.507130	−	0.861869i	$-0.330706\pi$
$547$	2.12370e12	1.01426	0.507130	+	0.861869i	$0.330706\pi$
$548$	−2.66134e12	−1.26063
$549$	2.72029e12	1.27803
$550$	−2.43271e12	−1.13360
$551$	−1.57677e11	−0.0728763
$552$	1.58342e13	7.25891
$553$	−3.52319e10	−0.0160204
$554$	2.66716e12	1.20297
$555$	1.96698e11	0.0880000
$556$	8.15803e12	3.62034
$557$	2.76104e11	0.121541	0.0607706	−	0.998152i	$-0.480644\pi$
$557$	2.76104e11	0.121541	0.0607706	+	0.998152i	$0.480644\pi$
$558$	2.84060e12	1.24038
$559$	0	0
$560$	1.16144e11	0.0499058
$561$	4.36314e12	1.85980
$562$	−6.47968e12	−2.73993
$563$	−3.90784e12	−1.63927	−0.819633	−	0.572889i	$-0.805823\pi$
$563$	−3.90784e12	−1.63927	−0.819633	+	0.572889i	$0.805823\pi$
$564$	−6.33084e12	−2.63454
$565$	−3.02716e12	−1.24973
$566$	6.43915e11	0.263727
$567$	−2.72438e10	−0.0110699
$568$	−3.99376e12	−1.60996
$569$	−4.41437e12	−1.76548	−0.882742	−	0.469858i	$-0.844305\pi$
$569$	−4.41437e12	−1.76548	−0.882742	+	0.469858i	$0.844305\pi$
$570$	2.32475e12	0.922443
$571$	−1.60238e12	−0.630818	−0.315409	−	0.948956i	$-0.602142\pi$
$571$	−1.60238e12	−0.630818	−0.315409	+	0.948956i	$0.602142\pi$
$572$	0	0
$573$	4.64985e12	1.80195
$574$	−2.70607e10	−0.0104048
$575$	2.11535e12	0.807007
$576$	1.85351e12	0.701608
$577$	3.55043e12	1.33349	0.666745	−	0.745286i	$-0.267687\pi$
$577$	3.55043e12	1.33349	0.666745	+	0.745286i	$0.267687\pi$
$578$	−1.62874e12	−0.606983
$579$	4.05073e12	1.49789
$580$	−1.99028e12	−0.730278
$581$	−5.50202e10	−0.0200323
$582$	1.76401e12	0.637306
$583$	2.61077e12	0.935968
$584$	1.22001e12	0.434017
$585$	0	0
$586$	−9.88049e12	−3.46130
$587$	4.57130e12	1.58916	0.794581	−	0.607158i	$-0.207691\pi$
$587$	4.57130e12	1.58916	0.794581	+	0.607158i	$0.207691\pi$
$588$	−1.03496e13	−3.57049
$589$	4.14966e11	0.142067
$590$	−3.69933e12	−1.25687
$591$	7.88149e12	2.65745
$592$	3.09955e11	0.103717
$593$	−4.62392e12	−1.53555	−0.767775	−	0.640720i	$-0.778636\pi$
$593$	−4.62392e12	−1.53555	−0.767775	+	0.640720i	$0.778636\pi$
$594$	−4.13723e12	−1.36355
$595$	−5.82771e10	−0.0190622
$596$	8.54700e12	2.77463
$597$	−2.27187e12	−0.731981
$598$	0	0
$599$	7.99528e11	0.253754	0.126877	−	0.991918i	$-0.459505\pi$
$599$	7.99528e11	0.253754	0.126877	+	0.991918i	$0.459505\pi$
$600$	4.93682e12	1.55513
$601$	3.40928e12	1.06593	0.532963	−	0.846138i	$-0.321078\pi$
$601$	3.40928e12	1.06593	0.532963	+	0.846138i	$0.321078\pi$
$602$	6.89968e10	0.0214114
$603$	−1.42576e12	−0.439157
$604$	−1.16250e13	−3.55408
$605$	4.80174e12	1.45713
$606$	1.01097e13	3.04516
$607$	−1.67978e12	−0.502232	−0.251116	−	0.967957i	$-0.580798\pi$
$607$	−1.67978e12	−0.502232	−0.251116	+	0.967957i	$0.580798\pi$
$608$	1.36721e12	0.405759
$609$	−2.66149e10	−0.00784055
$610$	7.15869e12	2.09339
$611$	0	0
$612$	−8.81664e12	−2.54052
$613$	−1.99546e12	−0.570782	−0.285391	−	0.958411i	$-0.592124\pi$
$613$	−1.99546e12	−0.570782	−0.285391	+	0.958411i	$0.592124\pi$
$614$	1.91825e12	0.544688
$615$	−1.87024e12	−0.527181
$616$	−2.56125e11	−0.0716703
$617$	1.43732e11	0.0399274	0.0199637	−	0.999801i	$-0.493645\pi$
$617$	1.43732e11	0.0399274	0.0199637	+	0.999801i	$0.493645\pi$
$618$	−9.40014e12	−2.59231
$619$	6.27659e12	1.71837	0.859184	−	0.511667i	$-0.170972\pi$
$619$	6.27659e12	1.71837	0.859184	+	0.511667i	$0.170972\pi$
$620$	5.23791e12	1.42363
$621$	3.59751e12	0.970712
$622$	−2.62862e12	−0.704160
$623$	8.92858e10	0.0237458
$624$	0	0
$625$	−4.74133e12	−1.24291
$626$	−1.91144e12	−0.497480
$627$	−2.44807e12	−0.632587
$628$	8.04509e12	2.06401
$629$	−1.55525e11	−0.0396161
$630$	2.23831e11	0.0566094
$631$	−4.60505e11	−0.115639	−0.0578193	−	0.998327i	$-0.518415\pi$
$631$	−4.60505e11	−0.115639	−0.0578193	+	0.998327i	$0.518415\pi$
$632$	−8.03527e12	−2.00343
$633$	2.97186e12	0.735718
$634$	6.50934e12	1.60006
$635$	−5.64452e12	−1.37767
$636$	−9.24879e12	−2.24144
$637$	0	0
$638$	2.99111e12	0.714725
$639$	−3.67535e12	−0.872057
$640$	−2.49389e12	−0.587580
$641$	−7.95795e11	−0.186183	−0.0930915	−	0.995658i	$-0.529675\pi$
$641$	−7.95795e11	−0.186183	−0.0930915	+	0.995658i	$0.529675\pi$
$642$	1.78737e13	4.15246
$643$	2.78399e12	0.642271	0.321136	−	0.947033i	$-0.395935\pi$
$643$	2.78399e12	0.642271	0.321136	+	0.947033i	$0.395935\pi$
$644$	3.88780e11	0.0890670
$645$	4.76857e12	1.08485
$646$	−1.83813e12	−0.415269
$647$	−4.43614e12	−0.995259	−0.497630	−	0.867390i	$-0.665796\pi$
$647$	−4.43614e12	−0.995259	−0.497630	+	0.867390i	$0.665796\pi$
$648$	−6.21345e12	−1.38435
$649$	3.89557e12	0.861926
$650$	0	0
$651$	7.00436e10	0.0152846
$652$	9.98386e12	2.16364
$653$	4.13689e12	0.890357	0.445178	−	0.895442i	$-0.353140\pi$
$653$	4.13689e12	0.890357	0.445178	+	0.895442i	$0.353140\pi$
$654$	−7.63223e12	−1.63137
$655$	2.18679e12	0.464217
$656$	−2.94711e12	−0.621339
$657$	1.12274e12	0.235091
$658$	−1.27080e11	−0.0264278
$659$	4.90170e12	1.01242	0.506212	−	0.862409i	$-0.331045\pi$
$659$	4.90170e12	1.01242	0.506212	+	0.862409i	$0.331045\pi$
$660$	−3.09008e13	−6.33903
$661$	−6.28240e12	−1.28003	−0.640013	−	0.768364i	$-0.721071\pi$
$661$	−6.28240e12	−1.28003	−0.640013	+	0.768364i	$0.721071\pi$
$662$	−7.85135e12	−1.58885
$663$	0	0
$664$	−1.25484e13	−2.50513
$665$	3.26982e10	0.00648375
$666$	5.97342e11	0.117649
$667$	−2.60090e12	−0.508813
$668$	−2.08331e13	−4.04817
$669$	−9.22501e12	−1.78053
$670$	−3.75202e12	−0.719331
$671$	−7.53844e12	−1.43559
$672$	2.30776e11	0.0436545
$673$	−6.99253e12	−1.31391	−0.656956	−	0.753929i	$-0.728156\pi$
$673$	−6.99253e12	−1.31391	−0.656956	+	0.753929i	$0.728156\pi$
$674$	2.89408e12	0.540183
$675$	1.12164e12	0.207963
$676$	0	0
$677$	−2.24164e12	−0.410126	−0.205063	−	0.978749i	$-0.565740\pi$
$677$	−2.24164e12	−0.410126	−0.205063	+	0.978749i	$0.565740\pi$
$678$	−1.61164e13	−2.92911
$679$	2.48113e10	0.00447956
$680$	−1.32912e13	−2.38382
$681$	3.76808e12	0.671364
$682$	−7.87183e12	−1.39330
$683$	4.85320e12	0.853365	0.426683	−	0.904401i	$-0.359682\pi$
$683$	4.85320e12	0.853365	0.426683	+	0.904401i	$0.359682\pi$
$684$	4.94685e12	0.864124
$685$	−3.69213e12	−0.640721
$686$	−4.15581e11	−0.0716469
$687$	−1.07151e13	−1.83524
$688$	7.51426e12	1.27861
$689$	0	0
$690$	3.83471e13	6.44037
$691$	−1.09872e12	−0.183330	−0.0916652	−	0.995790i	$-0.529219\pi$
$691$	−1.09872e12	−0.183330	−0.0916652	+	0.995790i	$0.529219\pi$
$692$	1.79816e13	2.98092
$693$	−2.35705e11	−0.0388212
$694$	−4.29753e11	−0.0703236
$695$	1.13178e13	1.84005
$696$	−6.07000e12	−0.980499
$697$	1.47876e12	0.237328
$698$	−9.78309e12	−1.56001
$699$	8.63326e12	1.36781
$700$	1.21214e11	0.0190815
$701$	5.77807e12	0.903758	0.451879	−	0.892079i	$-0.350754\pi$
$701$	5.77807e12	0.903758	0.451879	+	0.892079i	$0.350754\pi$
$702$	0	0
$703$	8.72621e10	0.0134749
$704$	−5.13643e12	−0.788106
$705$	−8.78288e12	−1.33902
$706$	−1.46381e12	−0.221749
$707$	1.42195e11	0.0214041
$708$	−1.38002e13	−2.06413
$709$	1.79240e11	0.0266395	0.0133198	−	0.999911i	$-0.495760\pi$
$709$	1.79240e11	0.0266395	0.0133198	+	0.999911i	$0.495760\pi$
$710$	−9.67201e12	−1.42841
$711$	−7.39464e12	−1.08519
$712$	2.03632e13	2.96952
$713$	6.84492e12	0.991894
$714$	−3.10264e11	−0.0446776
$715$	0	0
$716$	3.15599e13	4.48773
$717$	−1.50260e13	−2.12328
$718$	1.61115e13	2.26243
$719$	5.40775e12	0.754634	0.377317	−	0.926084i	$-0.376847\pi$
$719$	5.40775e12	0.754634	0.377317	+	0.926084i	$0.376847\pi$
$720$	2.43769e13	3.38051
$721$	−1.32215e11	−0.0182210
$722$	−1.23150e13	−1.68662
$723$	−9.73124e12	−1.32448
$724$	1.60254e13	2.16763
$725$	−8.10913e11	−0.109007
$726$	2.55642e13	3.41521
$727$	−1.10551e13	−1.46777	−0.733883	−	0.679276i	$-0.762294\pi$
$727$	−1.10551e13	−1.46777	−0.733883	+	0.679276i	$0.762294\pi$
$728$	0	0
$729$	−1.15370e13	−1.51294
$730$	2.95460e12	0.385076
$731$	−3.77040e12	−0.488381
$732$	2.67053e13	3.43793
$733$	8.63452e12	1.10477	0.552383	−	0.833590i	$-0.313719\pi$
$733$	8.63452e12	1.10477	0.552383	+	0.833590i	$0.313719\pi$
$734$	6.96145e12	0.885253
$735$	−1.43582e13	−1.81471
$736$	2.25523e13	2.83296
$737$	3.95106e12	0.493298
$738$	−5.67963e12	−0.704800
$739$	7.56910e12	0.933565	0.466782	−	0.884372i	$-0.345413\pi$
$739$	7.56910e12	0.933565	0.466782	+	0.884372i	$0.345413\pi$
$740$	1.10147e12	0.135030
$741$	0	0
$742$	−1.85653e11	−0.0224846
$743$	1.07973e12	0.129976	0.0649882	−	0.997886i	$-0.479299\pi$
$743$	1.07973e12	0.129976	0.0649882	+	0.997886i	$0.479299\pi$
$744$	1.59747e13	1.91141
$745$	1.18574e13	1.41022
$746$	8.91735e12	1.05417
$747$	−1.15479e13	−1.35694
$748$	2.44326e13	2.85373
$749$	2.51397e11	0.0291872
$750$	−1.67979e13	−1.93856
$751$	3.09147e12	0.354639	0.177319	−	0.984153i	$-0.443257\pi$
$751$	3.09147e12	0.354639	0.177319	+	0.984153i	$0.443257\pi$
$752$	−1.38400e13	−1.57817
$753$	1.90150e13	2.15535
$754$	0	0
$755$	−1.61276e13	−1.80638
$756$	2.06145e11	0.0229522
$757$	1.12889e13	1.24945	0.624725	−	0.780845i	$-0.285211\pi$
$757$	1.12889e13	1.24945	0.624725	+	0.780845i	$0.285211\pi$
$758$	2.35245e13	2.58827
$759$	−4.03813e13	−4.41664
$760$	7.45741e12	0.810825
$761$	5.47810e12	0.592105	0.296053	−	0.955172i	$-0.404330\pi$
$761$	5.47810e12	0.592105	0.296053	+	0.955172i	$0.404330\pi$
$762$	−3.00511e13	−3.22896
$763$	−1.07349e11	−0.0114667
$764$	2.60381e13	2.76497
$765$	−1.22315e13	−1.29123
$766$	1.39905e13	1.46826
$767$	0	0
$768$	−2.10497e13	−2.18334
$769$	1.38364e13	1.42678	0.713388	−	0.700769i	$-0.247160\pi$
$769$	1.38364e13	1.42678	0.713388	+	0.700769i	$0.247160\pi$
$770$	−6.20279e11	−0.0635885
$771$	−7.64951e12	−0.779630
$772$	2.26832e13	2.29840
$773$	−1.83558e13	−1.84912	−0.924562	−	0.381031i	$-0.875569\pi$
$773$	−1.83558e13	−1.84912	−0.924562	+	0.381031i	$0.875569\pi$
$774$	1.44814e13	1.45036
$775$	2.13412e12	0.212501
$776$	5.65866e12	0.560190
$777$	1.47293e10	0.00144973
$778$	2.66487e13	2.60776
$779$	−8.29703e11	−0.0807242
$780$	0	0
$781$	1.01851e13	0.979569
$782$	−3.03201e13	−2.89935
$783$	−1.37909e12	−0.131119
$784$	−2.26256e13	−2.13883
$785$	1.11611e13	1.04904
$786$	1.16423e13	1.08802
$787$	4.95419e12	0.460348	0.230174	−	0.973149i	$-0.426070\pi$
$787$	4.95419e12	0.460348	0.230174	+	0.973149i	$0.426070\pi$
$788$	4.41346e13	4.07766
$789$	1.50300e11	0.0138074
$790$	−1.94597e13	−1.77751
$791$	−2.26682e11	−0.0205884
$792$	−5.37568e13	−4.85479
$793$	0	0
$794$	1.68777e13	1.50703
$795$	−1.28310e13	−1.13922
$796$	−1.27220e13	−1.12317
$797$	7.77957e12	0.682957	0.341478	−	0.939890i	$-0.389072\pi$
$797$	7.77957e12	0.682957	0.341478	+	0.939890i	$0.389072\pi$
$798$	1.74083e11	0.0151965
$799$	6.94442e12	0.602804
$800$	7.03138e12	0.606925
$801$	1.87397e13	1.60848
$802$	−2.36563e12	−0.201912
$803$	−3.11134e12	−0.264075
$804$	−1.39968e13	−1.18135
$805$	5.39361e11	0.0452687
$806$	0	0
$807$	−9.36332e11	−0.0777139
$808$	3.24302e13	2.67669
$809$	−2.25302e13	−1.84925	−0.924627	−	0.380873i	$-0.875624\pi$
$809$	−2.25302e13	−1.84925	−0.924627	+	0.380873i	$0.875624\pi$
$810$	−1.50476e13	−1.22824
$811$	−2.07312e13	−1.68279	−0.841394	−	0.540422i	$-0.818265\pi$
$811$	−2.07312e13	−1.68279	−0.841394	+	0.540422i	$0.818265\pi$
$812$	−1.49037e11	−0.0120308
$813$	1.75137e12	0.140595
$814$	−1.65535e12	−0.132154
$815$	1.38508e13	1.09968
$816$	−3.37900e13	−2.66798
$817$	2.11550e12	0.166117
$818$	2.32044e13	1.81209
$819$	0	0
$820$	−1.04729e13	−0.808921
$821$	−2.10879e13	−1.61990	−0.809952	−	0.586496i	$-0.800507\pi$
$821$	−2.10879e13	−1.61990	−0.809952	+	0.586496i	$0.800507\pi$
$822$	−1.96567e13	−1.50171
$823$	−1.46114e11	−0.0111017	−0.00555087	−	0.999985i	$-0.501767\pi$
$823$	−1.46114e11	−0.0111017	−0.00555087	+	0.999985i	$0.501767\pi$
$824$	−3.01541e13	−2.27863
$825$	−1.25901e13	−0.946210
$826$	−2.77015e11	−0.0207059
$827$	−5.25930e12	−0.390979	−0.195490	−	0.980706i	$-0.562630\pi$
$827$	−5.25930e12	−0.390979	−0.195490	+	0.980706i	$0.562630\pi$
$828$	8.15989e13	6.03320
$829$	−5.68705e12	−0.418207	−0.209104	−	0.977894i	$-0.567055\pi$
$829$	−5.68705e12	−0.418207	−0.209104	+	0.977894i	$0.567055\pi$
$830$	−3.03894e13	−2.22265
$831$	1.38035e13	1.00412
$832$	0	0
$833$	1.13527e13	0.816955
$834$	6.02552e13	4.31268
$835$	−2.89021e13	−2.05750
$836$	−1.37087e13	−0.970659
$837$	3.62943e12	0.255607
$838$	−1.28586e13	−0.900730
$839$	−9.71153e12	−0.676641	−0.338321	−	0.941031i	$-0.609859\pi$
$839$	−9.71153e12	−0.676641	−0.338321	+	0.941031i	$0.609859\pi$
$840$	1.25876e12	0.0872343
$841$	−1.35101e13	−0.931272
$842$	9.59826e11	0.0658094
$843$	−3.35346e13	−2.28701
$844$	1.66417e13	1.12891
$845$	0	0
$846$	−2.66722e13	−1.79016
$847$	3.59567e11	0.0240051
$848$	−2.02190e13	−1.34270
$849$	3.33249e12	0.220132
$850$	−9.45325e12	−0.621150
$851$	1.43940e12	0.0940802
$852$	−3.60812e13	−2.34586
$853$	−1.13294e13	−0.732715	−0.366357	−	0.930474i	$-0.619395\pi$
$853$	−1.13294e13	−0.732715	−0.366357	+	0.930474i	$0.619395\pi$
$854$	5.36061e11	0.0344869
$855$	6.86285e12	0.439195
$856$	5.73358e13	3.65001
$857$	−5.84955e12	−0.370432	−0.185216	−	0.982698i	$-0.559299\pi$
$857$	−5.84955e12	−0.370432	−0.185216	+	0.982698i	$0.559299\pi$
$858$	0	0
$859$	−1.91819e13	−1.20205	−0.601026	−	0.799229i	$-0.705241\pi$
$859$	−1.91819e13	−1.20205	−0.601026	+	0.799229i	$0.705241\pi$
$860$	2.67029e13	1.66462
$861$	−1.40049e11	−0.00868489
$862$	3.67163e13	2.26504
$863$	−5.38209e12	−0.330295	−0.165148	−	0.986269i	$-0.552810\pi$
$863$	−5.38209e12	−0.330295	−0.165148	+	0.986269i	$0.552810\pi$
$864$	1.19580e13	0.730042
$865$	2.49461e13	1.51506
$866$	4.91233e13	2.96795
$867$	−8.42930e12	−0.506647
$868$	3.92228e11	0.0234531
$869$	2.04919e13	1.21897
$870$	−1.47002e13	−0.869935
$871$	0	0
$872$	−2.44829e13	−1.43397
$873$	5.20751e12	0.303435
$874$	1.70121e13	0.986178
$875$	−2.36267e11	−0.0136260
$876$	1.10221e13	0.632403
$877$	2.71005e13	1.54696	0.773481	−	0.633819i	$-0.218514\pi$
$877$	2.71005e13	1.54696	0.773481	+	0.633819i	$0.218514\pi$
$878$	1.35373e13	0.768789
$879$	−5.11350e13	−2.88914
$880$	−6.75529e13	−3.79728
$881$	−2.75007e13	−1.53799	−0.768993	−	0.639257i	$-0.779242\pi$
$881$	−2.75007e13	−1.53799	−0.768993	+	0.639257i	$0.779242\pi$
$882$	−4.36037e13	−2.42613
$883$	−2.51515e13	−1.39233	−0.696163	−	0.717884i	$-0.745111\pi$
$883$	−2.51515e13	−1.39233	−0.696163	+	0.717884i	$0.745111\pi$
$884$	0	0
$885$	−1.91453e13	−1.04910
$886$	−2.26365e13	−1.23412
$887$	2.07778e12	0.112705	0.0563524	−	0.998411i	$-0.482053\pi$
$887$	2.07778e12	0.112705	0.0563524	+	0.998411i	$0.482053\pi$
$888$	3.35928e12	0.181296
$889$	−4.22676e11	−0.0226960
$890$	4.93153e13	2.63467
$891$	1.58458e13	0.842298
$892$	−5.16580e13	−2.73209
$893$	−3.89638e12	−0.205036
$894$	6.31281e13	3.30525
$895$	4.37836e13	2.28091
$896$	−1.86749e11	−0.00967992
$897$	0	0
$898$	8.00154e12	0.410610
$899$	−2.62397e12	−0.133980
$900$	2.54410e13	1.29254
$901$	1.01452e13	0.512860
$902$	1.57393e13	0.791693
$903$	3.57083e11	0.0178720
$904$	−5.16989e13	−2.57468
$905$	2.22323e13	1.10171
$906$	−8.58624e13	−4.23376
$907$	−3.02045e13	−1.48197	−0.740985	−	0.671522i	$-0.765641\pi$
$907$	−3.02045e13	−1.48197	−0.740985	+	0.671522i	$0.765641\pi$
$908$	2.11004e13	1.03016
$909$	2.98446e13	1.44987
$910$	0	0
$911$	3.35277e13	1.61276	0.806382	−	0.591395i	$-0.201423\pi$
$911$	3.35277e13	1.61276	0.806382	+	0.591395i	$0.201423\pi$
$912$	1.89589e13	0.907480
$913$	3.20015e13	1.52423
$914$	−3.67011e13	−1.73949
$915$	3.70487e13	1.74734
$916$	−6.00023e13	−2.81603
$917$	1.63752e11	0.00764761
$918$	−1.60769e13	−0.747152
$919$	−3.78731e12	−0.175150	−0.0875751	−	0.996158i	$-0.527912\pi$
$919$	−3.78731e12	−0.175150	−0.0875751	+	0.996158i	$0.527912\pi$
$920$	1.23011e14	5.66107
$921$	9.92762e12	0.454650
$922$	2.79445e13	1.27352
$923$	0	0
$924$	−2.31393e12	−0.104430
$925$	4.48778e11	0.0201555
$926$	−3.62354e13	−1.61951
$927$	−2.77500e13	−1.23425
$928$	−8.64533e12	−0.382662
$929$	−3.73602e13	−1.64565	−0.822826	−	0.568293i	$-0.807604\pi$
$929$	−3.73602e13	−1.64565	−0.822826	+	0.568293i	$0.807604\pi$
$930$	3.86872e13	1.69588
$931$	−6.36980e12	−0.277877
$932$	4.83443e13	2.09881
$933$	−1.36040e13	−0.587761
$934$	−6.88508e13	−2.96038
$935$	3.38958e13	1.45042
$936$	0	0
$937$	1.56665e13	0.663964	0.331982	−	0.943286i	$-0.392283\pi$
$937$	1.56665e13	0.663964	0.331982	+	0.943286i	$0.392283\pi$
$938$	−2.80961e11	−0.0118504
$939$	−9.89237e12	−0.415246
$940$	−4.91822e13	−2.05462
$941$	3.66219e13	1.52261	0.761303	−	0.648396i	$-0.224560\pi$
$941$	3.66219e13	1.52261	0.761303	+	0.648396i	$0.224560\pi$
$942$	5.94210e13	2.45873
$943$	−1.36861e13	−0.563606
$944$	−3.01690e13	−1.23648
$945$	2.85989e11	0.0116656
$946$	−4.01307e13	−1.62917
$947$	4.08341e12	0.164986	0.0824931	−	0.996592i	$-0.473712\pi$
$947$	4.08341e12	0.164986	0.0824931	+	0.996592i	$0.473712\pi$
$948$	−7.25937e13	−2.91918
$949$	0	0
$950$	5.30404e12	0.211276
$951$	3.36881e13	1.33556
$952$	−9.95276e11	−0.0392715
$953$	1.60101e13	0.628749	0.314374	−	0.949299i	$-0.398205\pi$
$953$	1.60101e13	0.628749	0.314374	+	0.949299i	$0.398205\pi$
$954$	−3.89657e13	−1.52305
$955$	3.61232e13	1.40531
$956$	−8.41424e13	−3.25802
$957$	1.54800e13	0.596579
$958$	2.67235e13	1.02506
$959$	−2.76476e11	−0.0105554
$960$	2.52437e13	0.959252
$961$	−1.95340e13	−0.738815
$962$	0	0
$963$	5.27645e13	1.97708
$964$	−5.44928e13	−2.03232
$965$	3.14688e13	1.16817
$966$	2.87153e12	0.106100
$967$	3.50361e13	1.28854	0.644268	−	0.764800i	$-0.277162\pi$
$967$	3.50361e13	1.28854	0.644268	+	0.764800i	$0.277162\pi$
$968$	8.20057e13	3.00196
$969$	−9.51295e12	−0.346624
$970$	1.37040e13	0.497022
$971$	1.50255e13	0.542428	0.271214	−	0.962519i	$-0.412575\pi$
$971$	1.50255e13	0.542428	0.271214	+	0.962519i	$0.412575\pi$
$972$	−8.87196e13	−3.18802
$973$	8.47505e11	0.0303134
$974$	5.82215e11	0.0207285
$975$	0	0
$976$	5.83810e13	2.05943
$977$	5.63886e12	0.198000	0.0990001	−	0.995087i	$-0.468436\pi$
$977$	5.63886e12	0.198000	0.0990001	+	0.995087i	$0.468436\pi$
$978$	7.37408e13	2.57741
$979$	−5.19313e13	−1.80679
$980$	−8.04029e13	−2.78455
$981$	−2.25310e13	−0.776728
$982$	8.32780e13	2.85778
$983$	7.12741e12	0.243468	0.121734	−	0.992563i	$-0.461155\pi$
$983$	7.12741e12	0.243468	0.121734	+	0.992563i	$0.461155\pi$
$984$	−3.19406e13	−1.08609
$985$	6.12287e13	2.07249
$986$	1.16231e13	0.391631
$987$	−6.57685e11	−0.0220592
$988$	0	0
$989$	3.48954e13	1.15981
$990$	−1.30187e14	−4.30735
$991$	−1.35139e13	−0.445092	−0.222546	−	0.974922i	$-0.571437\pi$
$991$	−1.35139e13	−0.445092	−0.222546	+	0.974922i	$0.571437\pi$
$992$	2.27523e13	0.745973
$993$	−4.06335e13	−1.32621
$994$	−7.24265e11	−0.0235320
$995$	−1.76494e13	−0.570856
$996$	−1.13367e14	−3.65021
$997$	−3.09398e11	−0.00991721	−0.00495861	−	0.999988i	$-0.501578\pi$
$997$	−3.09398e11	−0.00991721	−0.00495861	+	0.999988i	$0.501578\pi$
$998$	5.38189e13	1.71730
$999$	7.63222e11	0.0242441

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.1.20		20
13.2	odd	12		13.10.e.a.4.10	✓	20
13.7	odd	12		13.10.e.a.10.10	yes	20
13.12	even	2	inner	169.10.a.f.1.1		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.10.e.a.4.10	✓	20	13.2	odd	12
13.10.e.a.10.10	yes	20	13.7	odd	12
169.10.a.f.1.1		20	13.12	even	2	inner
169.10.a.f.1.20		20	1.1	even	1	trivial

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

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Varieties

Fields

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Groups

Database

Properties

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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.1.20

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