LMFDB - Embedded newform 169.10.a.f.1.8

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			1.8
Root			$-13.9570$ 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)

$f(q)$	$=$	$q-13.9570 q^{2} +25.6151 q^{3} -317.202 q^{4} -792.583 q^{5} -357.510 q^{6} +900.794 q^{7} +11573.2 q^{8} -19026.9 q^{9} +11062.1 q^{10} -93909.0 q^{11} -8125.15 q^{12} -12572.4 q^{14} -20302.1 q^{15} +880.307 q^{16} -270555. q^{17} +265558. q^{18} +76689.5 q^{19} +251409. q^{20} +23073.9 q^{21} +1.31069e6 q^{22} -1.13631e6 q^{23} +296448. q^{24} -1.32494e6 q^{25} -991556. q^{27} -285733. q^{28} +1.50238e6 q^{29} +283356. q^{30} -6.12021e6 q^{31} -5.93775e6 q^{32} -2.40549e6 q^{33} +3.77614e6 q^{34} -713954. q^{35} +6.03536e6 q^{36} -9.31155e6 q^{37} -1.07036e6 q^{38} -9.17271e6 q^{40} -2.47362e7 q^{41} -322043. q^{42} -8.43526e6 q^{43} +2.97881e7 q^{44} +1.50804e7 q^{45} +1.58595e7 q^{46} -3.11511e6 q^{47} +22549.1 q^{48} -3.95422e7 q^{49} +1.84922e7 q^{50} -6.93028e6 q^{51} +8.73907e7 q^{53} +1.38392e7 q^{54} +7.44307e7 q^{55} +1.04250e7 q^{56} +1.96441e6 q^{57} -2.09688e7 q^{58} -3.70438e7 q^{59} +6.43986e6 q^{60} -1.01770e8 q^{61} +8.54199e7 q^{62} -1.71393e7 q^{63} +8.24226e7 q^{64} +3.35734e7 q^{66} -1.94333e8 q^{67} +8.58205e7 q^{68} -2.91068e7 q^{69} +9.96467e6 q^{70} -2.97949e8 q^{71} -2.20201e8 q^{72} -1.91567e8 q^{73} +1.29961e8 q^{74} -3.39384e7 q^{75} -2.43261e7 q^{76} -8.45927e7 q^{77} +9.03350e7 q^{79} -697717. q^{80} +3.49107e8 q^{81} +3.45244e8 q^{82} +5.79960e8 q^{83} -7.31908e6 q^{84} +2.14437e8 q^{85} +1.17731e8 q^{86} +3.84837e7 q^{87} -1.08683e9 q^{88} -1.20357e8 q^{89} -2.10477e8 q^{90} +3.60441e8 q^{92} -1.56770e8 q^{93} +4.34777e7 q^{94} -6.07828e7 q^{95} -1.52096e8 q^{96} -1.31027e9 q^{97} +5.51891e8 q^{98} +1.78679e9 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$20 q + 326 q^{3} + 5118 q^{4} + 129526 q^{9} + 88390 q^{10} + 427652 q^{12} + 473556 q^{14} + 1189618 q^{16} - 99312 q^{17} - 5073532 q^{22} + 6252378 q^{23} + 1529274 q^{25} + 18052718 q^{27} + 5424828 q^{29}+ \cdots + 9251202540 q^{95}+O(q^{100})$ 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

)

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

$2$	−13.9570	−0.616819	−0.308409	−	0.951254i	$-0.599797\pi$
$2$	−13.9570	−0.616819	−0.308409	+	0.951254i	$0.599797\pi$
$3$	25.6151	0.182579	0.0912893	−	0.995824i	$-0.470901\pi$
$3$	25.6151	0.182579	0.0912893	+	0.995824i	$0.470901\pi$
$4$	−317.202	−0.619535
$5$	−792.583	−0.567126	−0.283563	−	0.958954i	$-0.591517\pi$
$5$	−792.583	−0.567126	−0.283563	+	0.958954i	$0.591517\pi$
$6$	−357.510	−0.112618
$7$	900.794	0.141803	0.0709013	−	0.997483i	$-0.477412\pi$
$7$	900.794	0.141803	0.0709013	+	0.997483i	$0.477412\pi$
$8$	11573.2	0.998959
$9$	−19026.9	−0.966665
$10$	11062.1	0.349814
$11$	−93909.0	−1.93393	−0.966964	−	0.254914i	$-0.917953\pi$
$11$	−93909.0	−1.93393	−0.966964	+	0.254914i	$0.917953\pi$
$12$	−8125.15	−0.113114
$13$	0	0
$13$	0	0
$14$	−12572.4	−0.0874665
$15$	−20302.1	−0.103545
$16$	880.307	0.00335811
$17$	−270555.	−0.785661	−0.392831	−	0.919611i	$-0.628504\pi$
$17$	−270555.	−0.785661	−0.392831	+	0.919611i	$0.628504\pi$
$18$	265558.	0.596257
$19$	76689.5	0.135003	0.0675017	−	0.997719i	$-0.478497\pi$
$19$	76689.5	0.135003	0.0675017	+	0.997719i	$0.478497\pi$
$20$	251409.	0.351355
$21$	23073.9	0.0258901
$22$	1.31069e6	1.19288
$23$	−1.13631e6	−0.846687	−0.423344	−	0.905969i	$-0.639144\pi$
$23$	−1.13631e6	−0.846687	−0.423344	+	0.905969i	$0.639144\pi$
$24$	296448.	0.182389
$25$	−1.32494e6	−0.678368
$26$	0	0
$27$	−991556.	−0.359071
$28$	−285733.	−0.0878517
$29$	1.50238e6	0.394448	0.197224	−	0.980358i	$-0.436807\pi$
$29$	1.50238e6	0.394448	0.197224	+	0.980358i	$0.436807\pi$
$30$	283356.	0.0638686
$31$	−6.12021e6	−1.19025	−0.595126	−	0.803632i	$-0.702898\pi$
$31$	−6.12021e6	−1.19025	−0.595126	+	0.803632i	$0.702898\pi$
$32$	−5.93775e6	−1.00103
$33$	−2.40549e6	−0.353094
$34$	3.77614e6	0.484610
$35$	−713954.	−0.0804200
$36$	6.03536e6	0.598883
$37$	−9.31155e6	−0.816797	−0.408399	−	0.912804i	$-0.633913\pi$
$37$	−9.31155e6	−0.816797	−0.408399	+	0.912804i	$0.633913\pi$
$38$	−1.07036e6	−0.0832726
$39$	0	0
$40$	−9.17271e6	−0.566536
$41$	−2.47362e7	−1.36712	−0.683559	−	0.729896i	$-0.739569\pi$
$41$	−2.47362e7	−1.36712	−0.683559	+	0.729896i	$0.739569\pi$
$42$	−322043.	−0.0159695
$43$	−8.43526e6	−0.376262	−0.188131	−	0.982144i	$-0.560243\pi$
$43$	−8.43526e6	−0.376262	−0.188131	+	0.982144i	$0.560243\pi$
$44$	2.97881e7	1.19814
$45$	1.50804e7	0.548221
$46$	1.58595e7	0.522253
$47$	−3.11511e6	−0.0931180	−0.0465590	−	0.998916i	$-0.514826\pi$
$47$	−3.11511e6	−0.0931180	−0.0465590	+	0.998916i	$0.514826\pi$
$48$	22549.1	0.000613118 0
$49$	−3.95422e7	−0.979892
$50$	1.84922e7	0.418430
$51$	−6.93028e6	−0.143445
$52$	0	0
$53$	8.73907e7	1.52133	0.760666	−	0.649144i	$-0.224873\pi$
$53$	8.73907e7	1.52133	0.760666	+	0.649144i	$0.224873\pi$
$54$	1.38392e7	0.221482
$55$	7.44307e7	1.09678
$56$	1.04250e7	0.141655
$57$	1.96441e6	0.0246487
$58$	−2.09688e7	−0.243303
$59$	−3.70438e7	−0.397999	−0.198999	−	0.980000i	$-0.563769\pi$
$59$	−3.70438e7	−0.397999	−0.198999	+	0.980000i	$0.563769\pi$
$60$	6.43986e6	0.0641498
$61$	−1.01770e8	−0.941103	−0.470551	−	0.882373i	$-0.655945\pi$
$61$	−1.01770e8	−0.941103	−0.470551	+	0.882373i	$0.655945\pi$
$62$	8.54199e7	0.734170
$63$	−1.71393e7	−0.137076
$64$	8.24226e7	0.614096
$65$	0	0
$66$	3.35734e7	0.217795
$67$	−1.94333e8	−1.17817	−0.589087	−	0.808070i	$-0.700513\pi$
$67$	−1.94333e8	−1.17817	−0.589087	+	0.808070i	$0.700513\pi$
$68$	8.58205e7	0.486744
$69$	−2.91068e7	−0.154587
$70$	9.96467e6	0.0496046
$71$	−2.97949e8	−1.39149	−0.695745	−	0.718289i	$-0.744926\pi$
$71$	−2.97949e8	−1.39149	−0.695745	+	0.718289i	$0.744926\pi$
$72$	−2.20201e8	−0.965659
$73$	−1.91567e8	−0.789529	−0.394765	−	0.918782i	$-0.629174\pi$
$73$	−1.91567e8	−0.789529	−0.394765	+	0.918782i	$0.629174\pi$
$74$	1.29961e8	0.503816
$75$	−3.39384e7	−0.123855
$76$	−2.43261e7	−0.0836393
$77$	−8.45927e7	−0.274236
$78$	0	0
$79$	9.03350e7	0.260936	0.130468	−	0.991453i	$-0.458352\pi$
$79$	9.03350e7	0.260936	0.130468	+	0.991453i	$0.458352\pi$
$80$	−697717.	−0.00190447
$81$	3.49107e8	0.901106
$82$	3.45244e8	0.843263
$83$	5.79960e8	1.34136	0.670682	−	0.741745i	$-0.266002\pi$
$83$	5.79960e8	1.34136	0.670682	+	0.741745i	$0.266002\pi$
$84$	−7.31908e6	−0.0160398
$85$	2.14437e8	0.445569
$86$	1.17731e8	0.232085
$87$	3.84837e7	0.0720178
$88$	−1.08683e9	−1.93191
$89$	−1.20357e8	−0.203336	−0.101668	−	0.994818i	$-0.532418\pi$
$89$	−1.20357e8	−0.203336	−0.101668	+	0.994818i	$0.532418\pi$
$90$	−2.10477e8	−0.338153
$91$	0	0
$92$	3.60441e8	0.524552
$93$	−1.56770e8	−0.217315
$94$	4.34777e7	0.0574369
$95$	−6.07828e7	−0.0765640
$96$	−1.52096e8	−0.182767
$97$	−1.31027e9	−1.50276	−0.751380	−	0.659870i	$-0.770611\pi$
$97$	−1.31027e9	−1.50276	−0.751380	+	0.659870i	$0.770611\pi$
$98$	5.51891e8	0.604416
$99$	1.78679e9	1.86946
$100$	4.20272e8	0.420272
$101$	7.10019e8	0.678928	0.339464	−	0.940619i	$-0.389754\pi$
$101$	7.10019e8	0.678928	0.339464	+	0.940619i	$0.389754\pi$
$102$	9.67261e7	0.0884795
$103$	1.73311e9	1.51726	0.758628	−	0.651524i	$-0.225870\pi$
$103$	1.73311e9	1.51726	0.758628	+	0.651524i	$0.225870\pi$
$104$	0	0
$105$	−1.82880e7	−0.0146830
$106$	−1.21971e9	−0.938385
$107$	4.45091e7	0.0328263	0.0164131	−	0.999865i	$-0.494775\pi$
$107$	4.45091e7	0.0328263	0.0164131	+	0.999865i	$0.494775\pi$
$108$	3.14523e8	0.222457
$109$	3.30751e8	0.224431	0.112215	−	0.993684i	$-0.464205\pi$
$109$	3.30751e8	0.224431	0.112215	+	0.993684i	$0.464205\pi$
$110$	−1.03883e9	−0.676515
$111$	−2.38516e8	−0.149130
$112$	792975.	0.000476188 0
$113$	8.19291e8	0.472700	0.236350	−	0.971668i	$-0.424049\pi$
$113$	8.19291e8	0.472700	0.236350	+	0.971668i	$0.424049\pi$
$114$	−2.74173e7	−0.0152038
$115$	9.00623e8	0.480179
$116$	−4.76559e8	−0.244374
$117$	0	0
$118$	5.17021e8	0.245493
$119$	−2.43714e8	−0.111409
$120$	−2.34960e8	−0.103437
$121$	6.46095e9	2.74008
$122$	1.42041e9	0.580490
$123$	−6.33620e8	−0.249606
$124$	1.94134e9	0.737403
$125$	2.59814e9	0.951847
$126$	2.39213e8	0.0845508
$127$	−3.79954e9	−1.29603	−0.648014	−	0.761628i	$-0.724400\pi$
$127$	−3.79954e9	−1.29603	−0.648014	+	0.761628i	$0.724400\pi$
$128$	1.88976e9	0.622245
$129$	−2.16070e8	−0.0686974
$130$	0	0
$131$	3.47951e9	1.03228	0.516140	−	0.856504i	$-0.327368\pi$
$131$	3.47951e9	1.03228	0.516140	+	0.856504i	$0.327368\pi$
$132$	7.63025e8	0.218754
$133$	6.90815e7	0.0191438
$134$	2.71231e9	0.726720
$135$	7.85891e8	0.203639
$136$	−3.13118e9	−0.784844
$137$	1.42545e9	0.345709	0.172854	−	0.984947i	$-0.444701\pi$
$137$	1.42545e9	0.345709	0.172854	+	0.984947i	$0.444701\pi$
$138$	4.06243e8	0.0953522
$139$	5.35614e9	1.21698	0.608492	−	0.793560i	$-0.291775\pi$
$139$	5.35614e9	1.21698	0.608492	+	0.793560i	$0.291775\pi$
$140$	2.26468e8	0.0498230
$141$	−7.97939e7	−0.0170014
$142$	4.15848e9	0.858297
$143$	0	0
$144$	−1.67495e7	−0.00324616
$145$	−1.19076e9	−0.223702
$146$	2.67371e9	0.486996
$147$	−1.01288e9	−0.178907
$148$	2.95364e9	0.506034
$149$	−8.36102e9	−1.38970	−0.694850	−	0.719155i	$-0.744529\pi$
$149$	−8.36102e9	−1.38970	−0.694850	+	0.719155i	$0.744529\pi$
$150$	4.73678e8	0.0763963
$151$	1.96431e9	0.307478	0.153739	−	0.988111i	$-0.450868\pi$
$151$	1.96431e9	0.307478	0.153739	+	0.988111i	$0.450868\pi$
$152$	8.87542e8	0.134863
$153$	5.14781e9	0.759471
$154$	1.18066e9	0.169154
$155$	4.85078e9	0.675023
$156$	0	0
$157$	6.03427e9	0.792640	0.396320	−	0.918112i	$-0.370287\pi$
$157$	6.03427e9	0.792640	0.396320	+	0.918112i	$0.370287\pi$
$158$	−1.26081e9	−0.160950
$159$	2.23852e9	0.277763
$160$	4.70617e9	0.567711
$161$	−1.02358e9	−0.120062
$162$	−4.87249e9	−0.555819
$163$	6.50794e8	0.0722104	0.0361052	−	0.999348i	$-0.488505\pi$
$163$	6.50794e8	0.0722104	0.0361052	+	0.999348i	$0.488505\pi$
$164$	7.84637e9	0.846977
$165$	1.90655e9	0.200249
$166$	−8.09450e9	−0.827378
$167$	2.16697e9	0.215590	0.107795	−	0.994173i	$-0.465621\pi$
$167$	2.16697e9	0.215590	0.107795	+	0.994173i	$0.465621\pi$
$168$	2.67038e8	0.0258632
$169$	0	0
$170$	−2.99290e9	−0.274835
$171$	−1.45916e9	−0.130503
$172$	2.67568e9	0.233107
$173$	1.33630e10	1.13422	0.567110	−	0.823642i	$-0.308062\pi$
$173$	1.33630e10	1.13422	0.567110	+	0.823642i	$0.308062\pi$
$174$	−5.37117e8	−0.0444219
$175$	−1.19349e9	−0.0961943
$176$	−8.26688e7	−0.00649433
$177$	−9.48881e8	−0.0726661
$178$	1.67982e9	0.125422
$179$	−1.51263e10	−1.10127	−0.550635	−	0.834746i	$-0.685614\pi$
$179$	−1.51263e10	−1.10127	−0.550635	+	0.834746i	$0.685614\pi$
$180$	−4.78352e9	−0.339642
$181$	−2.06849e10	−1.43252	−0.716260	−	0.697834i	$-0.754148\pi$
$181$	−2.06849e10	−1.43252	−0.716260	+	0.697834i	$0.754148\pi$
$182$	0	0
$183$	−2.60686e9	−0.171825
$184$	−1.31508e10	−0.845806
$185$	7.38018e9	0.463227
$186$	2.18804e9	0.134044
$187$	2.54075e10	1.51941
$188$	9.88120e8	0.0576898
$189$	−8.93188e8	−0.0509172
$190$	8.48347e8	0.0472261
$191$	−1.68096e10	−0.913919	−0.456960	−	0.889487i	$-0.651062\pi$
$191$	−1.68096e10	−0.913919	−0.456960	+	0.889487i	$0.651062\pi$
$192$	2.11126e9	0.112121
$193$	5.39946e9	0.280119	0.140060	−	0.990143i	$-0.455271\pi$
$193$	5.39946e9	0.280119	0.140060	+	0.990143i	$0.455271\pi$
$194$	1.82875e10	0.926930
$195$	0	0
$196$	1.25428e10	0.607077
$197$	−1.33679e10	−0.632362	−0.316181	−	0.948699i	$-0.602401\pi$
$197$	−1.33679e10	−0.632362	−0.316181	+	0.948699i	$0.602401\pi$
$198$	−2.49383e10	−1.15312
$199$	−3.72907e10	−1.68563	−0.842813	−	0.538206i	$-0.819102\pi$
$199$	−3.72907e10	−1.68563	−0.842813	+	0.538206i	$0.819102\pi$
$200$	−1.53337e10	−0.677662
$201$	−4.97785e9	−0.215109
$202$	−9.90974e9	−0.418775
$203$	1.35334e9	0.0559338
$204$	2.19830e9	0.0888691
$205$	1.96055e10	0.775328
$206$	−2.41891e10	−0.935872
$207$	2.16205e10	0.818463
$208$	0	0
$209$	−7.20184e9	−0.261087
$210$	2.55246e8	0.00905673
$211$	4.41630e10	1.53387	0.766933	−	0.641728i	$-0.221782\pi$
$211$	4.41630e10	1.53387	0.766933	+	0.641728i	$0.221782\pi$
$212$	−2.77205e10	−0.942518
$213$	−7.63200e9	−0.254056
$214$	−6.21214e8	−0.0202479
$215$	6.68564e9	0.213388
$216$	−1.14755e10	−0.358697
$217$	−5.51305e9	−0.168781
$218$	−4.61630e9	−0.138433
$219$	−4.90701e9	−0.144151
$220$	−2.36096e10	−0.679494
$221$	0	0
$222$	3.32897e9	0.0919860
$223$	4.64388e10	1.25750	0.628751	−	0.777607i	$-0.283566\pi$
$223$	4.64388e10	1.25750	0.628751	+	0.777607i	$0.283566\pi$
$224$	−5.34869e9	−0.141949
$225$	2.52094e10	0.655754
$226$	−1.14349e10	−0.291570
$227$	4.79078e10	1.19754	0.598770	−	0.800921i	$-0.295656\pi$
$227$	4.79078e10	1.19754	0.598770	+	0.800921i	$0.295656\pi$
$228$	−6.23114e8	−0.0152708
$229$	−2.52672e10	−0.607152	−0.303576	−	0.952807i	$-0.598181\pi$
$229$	−2.52672e10	−0.607152	−0.303576	+	0.952807i	$0.598181\pi$
$230$	−1.25700e10	−0.296183
$231$	−2.16685e9	−0.0500696
$232$	1.73874e10	0.394038
$233$	−5.41842e10	−1.20440	−0.602201	−	0.798345i	$-0.705709\pi$
$233$	−5.41842e10	−1.20440	−0.602201	+	0.798345i	$0.705709\pi$
$234$	0	0
$235$	2.46899e9	0.0528097
$236$	1.17504e10	0.246574
$237$	2.31394e9	0.0476414
$238$	3.40152e9	0.0687190
$239$	−1.55035e10	−0.307355	−0.153677	−	0.988121i	$-0.549112\pi$
$239$	−1.55035e10	−0.307355	−0.153677	+	0.988121i	$0.549112\pi$
$240$	−1.78721e7	−0.000347716 0
$241$	−7.54174e10	−1.44011	−0.720054	−	0.693918i	$-0.755883\pi$
$241$	−7.54174e10	−1.44011	−0.720054	+	0.693918i	$0.755883\pi$
$242$	−9.01756e10	−1.69013
$243$	2.84592e10	0.523594
$244$	3.22817e10	0.583046
$245$	3.13405e10	0.555723
$246$	8.84344e9	0.153962
$247$	0	0
$248$	−7.08303e10	−1.18901
$249$	1.48557e10	0.244904
$250$	−3.62622e10	−0.587117
$251$	−5.95728e10	−0.947363	−0.473682	−	0.880696i	$-0.657075\pi$
$251$	−5.95728e10	−0.947363	−0.473682	+	0.880696i	$0.657075\pi$
$252$	5.43661e9	0.0849231
$253$	1.06710e11	1.63743
$254$	5.30303e10	0.799415
$255$	5.49283e9	0.0813514
$256$	−6.85757e10	−0.997908
$257$	8.69774e10	1.24368	0.621838	−	0.783146i	$-0.286386\pi$
$257$	8.69774e10	1.24368	0.621838	+	0.783146i	$0.286386\pi$
$258$	3.01569e9	0.0423738
$259$	−8.38779e9	−0.115824
$260$	0	0
$261$	−2.85857e10	−0.381299
$262$	−4.85636e10	−0.636729
$263$	−1.27752e11	−1.64652	−0.823262	−	0.567662i	$-0.807848\pi$
$263$	−1.27752e11	−1.64652	−0.823262	+	0.567662i	$0.807848\pi$
$264$	−2.78391e10	−0.352726
$265$	−6.92644e10	−0.862787
$266$	−9.64171e8	−0.0118083
$267$	−3.08295e9	−0.0371249
$268$	6.16427e10	0.729920
$269$	−7.77282e9	−0.0905093	−0.0452547	−	0.998975i	$-0.514410\pi$
$269$	−7.77282e9	−0.0905093	−0.0452547	+	0.998975i	$0.514410\pi$
$270$	−1.09687e10	−0.125608
$271$	−1.55085e11	−1.74665	−0.873327	−	0.487135i	$-0.838042\pi$
$271$	−1.55085e11	−1.74665	−0.873327	+	0.487135i	$0.838042\pi$
$272$	−2.38171e8	−0.00263833
$273$	0	0
$274$	−1.98951e10	−0.213240
$275$	1.24423e11	1.31191
$276$	9.23272e9	0.0957720
$277$	−9.67202e10	−0.987093	−0.493547	−	0.869719i	$-0.664300\pi$
$277$	−9.67202e10	−0.987093	−0.493547	+	0.869719i	$0.664300\pi$
$278$	−7.47556e10	−0.750659
$279$	1.16448e11	1.15058
$280$	−8.26272e9	−0.0803363
$281$	8.18438e9	0.0783082	0.0391541	−	0.999233i	$-0.487534\pi$
$281$	8.18438e9	0.0783082	0.0391541	+	0.999233i	$0.487534\pi$
$282$	1.11368e9	0.0104868
$283$	3.10558e10	0.287809	0.143904	−	0.989592i	$-0.454034\pi$
$283$	3.10558e10	0.287809	0.143904	+	0.989592i	$0.454034\pi$
$284$	9.45101e10	0.862076
$285$	−1.55696e9	−0.0139790
$286$	0	0
$287$	−2.22822e10	−0.193861
$288$	1.12977e11	0.967661
$289$	−4.53879e10	−0.382736
$290$	1.66195e10	0.137984
$291$	−3.35628e10	−0.274372
$292$	6.07654e10	0.489141
$293$	1.83800e11	1.45694	0.728468	−	0.685080i	$-0.240233\pi$
$293$	1.83800e11	1.45694	0.728468	+	0.685080i	$0.240233\pi$
$294$	1.41367e10	0.110353
$295$	2.93603e10	0.225716
$296$	−1.07764e11	−0.815947
$297$	9.31160e10	0.694417
$298$	1.16695e11	0.857193
$299$	0	0
$300$	1.07653e10	0.0767327
$301$	−7.59843e9	−0.0533549
$302$	−2.74160e10	−0.189658
$303$	1.81872e10	0.123958
$304$	6.75103e7	0.000453356 0
$305$	8.06615e10	0.533724
$306$	−7.18481e10	−0.468456
$307$	−1.10486e11	−0.709877	−0.354938	−	0.934890i	$-0.615498\pi$
$307$	−1.10486e11	−0.709877	−0.354938	+	0.934890i	$0.615498\pi$
$308$	2.68329e10	0.169899
$309$	4.43938e10	0.277019
$310$	−6.77024e10	−0.416367
$311$	−2.30844e11	−1.39926	−0.699629	−	0.714506i	$-0.746651\pi$
$311$	−2.30844e11	−1.39926	−0.699629	+	0.714506i	$0.746651\pi$
$312$	0	0
$313$	−2.06281e11	−1.21481	−0.607406	−	0.794392i	$-0.707790\pi$
$313$	−2.06281e11	−1.21481	−0.607406	+	0.794392i	$0.707790\pi$
$314$	−8.42203e10	−0.488915
$315$	1.35843e10	0.0777392
$316$	−2.86544e10	−0.161659
$317$	1.14975e11	0.639492	0.319746	−	0.947503i	$-0.396402\pi$
$317$	1.14975e11	0.639492	0.319746	+	0.947503i	$0.396402\pi$
$318$	−3.12430e10	−0.171329
$319$	−1.41087e11	−0.762834
$320$	−6.53268e10	−0.348270
$321$	1.14010e9	0.00599338
$322$	1.42862e10	0.0740568
$323$	−2.07487e10	−0.106067
$324$	−1.10737e11	−0.558267
$325$	0	0
$326$	−9.08314e9	−0.0445407
$327$	8.47222e9	0.0409762
$328$	−2.86277e11	−1.36569
$329$	−2.80608e9	−0.0132044
$330$	−2.66097e10	−0.123517
$331$	−2.87360e11	−1.31583	−0.657915	−	0.753092i	$-0.728561\pi$
$331$	−2.87360e11	−1.31583	−0.657915	+	0.753092i	$0.728561\pi$
$332$	−1.83964e11	−0.831021
$333$	1.77170e11	0.789570
$334$	−3.02445e10	−0.132980
$335$	1.54025e11	0.668174
$336$	2.03121e7	8.69418e−5 0
$337$	2.34936e11	0.992237	0.496118	−	0.868255i	$-0.334758\pi$
$337$	2.34936e11	0.992237	0.496118	+	0.868255i	$0.334758\pi$
$338$	0	0
$339$	2.09862e10	0.0863048
$340$	−6.80199e10	−0.276046
$341$	5.74743e11	2.30186
$342$	2.03655e10	0.0804968
$343$	−7.19696e10	−0.280754
$344$	−9.76227e10	−0.375870
$345$	2.30695e10	0.0876704
$346$	−1.86508e11	−0.699608
$347$	−2.59073e11	−0.959267	−0.479633	−	0.877469i	$-0.659230\pi$
$347$	−2.59073e11	−0.959267	−0.479633	+	0.877469i	$0.659230\pi$
$348$	−1.22071e10	−0.0446175
$349$	−2.38690e11	−0.861233	−0.430616	−	0.902535i	$-0.641704\pi$
$349$	−2.38690e11	−0.861233	−0.430616	+	0.902535i	$0.641704\pi$
$350$	1.66576e10	0.0593344
$351$	0	0
$352$	5.57609e11	1.93592
$353$	5.68374e10	0.194827	0.0974133	−	0.995244i	$-0.468943\pi$
$353$	5.68374e10	0.194827	0.0974133	+	0.995244i	$0.468943\pi$
$354$	1.32435e10	0.0448218
$355$	2.36150e11	0.789151
$356$	3.81774e10	0.125974
$357$	−6.24276e9	−0.0203409
$358$	2.11118e11	0.679284
$359$	5.45231e10	0.173243	0.0866214	−	0.996241i	$-0.472393\pi$
$359$	5.45231e10	0.173243	0.0866214	+	0.996241i	$0.472393\pi$
$360$	1.74528e11	0.547651
$361$	−3.16806e11	−0.981774
$362$	2.88700e11	0.883605
$363$	1.65498e11	0.500279
$364$	0	0
$365$	1.51833e11	0.447763
$366$	3.63839e10	0.105985
$367$	3.50164e11	1.00757	0.503785	−	0.863829i	$-0.331941\pi$
$367$	3.50164e11	1.00757	0.503785	+	0.863829i	$0.331941\pi$
$368$	−1.00031e9	−0.00284327
$369$	4.70653e11	1.32154
$370$	−1.03005e11	−0.285727
$371$	7.87210e10	0.215729
$372$	4.97276e10	0.134634
$373$	4.00096e11	1.07022	0.535112	−	0.844781i	$-0.320269\pi$
$373$	4.00096e11	1.07022	0.535112	+	0.844781i	$0.320269\pi$
$374$	−3.54613e11	−0.937201
$375$	6.65515e10	0.173787
$376$	−3.60518e10	−0.0930211
$377$	0	0
$378$	1.24662e10	0.0314067
$379$	−5.52350e11	−1.37511	−0.687556	−	0.726132i	$-0.741316\pi$
$379$	−5.52350e11	−1.37511	−0.687556	+	0.726132i	$0.741316\pi$
$380$	1.92804e10	0.0474341
$381$	−9.73256e10	−0.236627
$382$	2.34612e11	0.563722
$383$	−1.14780e11	−0.272565	−0.136282	−	0.990670i	$-0.543515\pi$
$383$	−1.14780e11	−0.272565	−0.136282	+	0.990670i	$0.543515\pi$
$384$	4.84063e10	0.113609
$385$	6.70467e10	0.155526
$386$	−7.53604e10	−0.172783
$387$	1.60497e11	0.363719
$388$	4.15621e11	0.931012
$389$	2.37272e11	0.525380	0.262690	−	0.964880i	$-0.415390\pi$
$389$	2.37272e11	0.525380	0.262690	+	0.964880i	$0.415390\pi$
$390$	0	0
$391$	3.07435e11	0.665209
$392$	−4.57629e11	−0.978872
$393$	8.91279e10	0.188472
$394$	1.86576e11	0.390053
$395$	−7.15980e10	−0.147984
$396$	−5.66774e11	−1.15820
$397$	6.75774e10	0.136535	0.0682676	−	0.997667i	$-0.478253\pi$
$397$	6.75774e10	0.136535	0.0682676	+	0.997667i	$0.478253\pi$
$398$	5.20466e11	1.03973
$399$	1.76953e9	0.00349526
$400$	−1.16635e9	−0.00227803
$401$	−5.33264e11	−1.02989	−0.514947	−	0.857222i	$-0.672188\pi$
$401$	−5.33264e11	−1.02989	−0.514947	+	0.857222i	$0.672188\pi$
$402$	6.94759e10	0.132684
$403$	0	0
$404$	−2.25219e11	−0.420619
$405$	−2.76696e11	−0.511041
$406$	−1.88886e10	−0.0345010
$407$	8.74439e11	1.57963
$408$	−8.02054e10	−0.143296
$409$	−2.09851e11	−0.370814	−0.185407	−	0.982662i	$-0.559360\pi$
$409$	−2.09851e11	−0.370814	−0.185407	+	0.982662i	$0.559360\pi$
$410$	−2.73634e11	−0.478237
$411$	3.65131e10	0.0631191
$412$	−5.49746e11	−0.939993
$413$	−3.33689e10	−0.0564373
$414$	−3.01758e11	−0.504843
$415$	−4.59666e11	−0.760723
$416$	0	0
$417$	1.37198e11	0.222195
$418$	1.00516e11	0.161043
$419$	−8.01511e10	−0.127042	−0.0635209	−	0.997981i	$-0.520233\pi$
$419$	−8.01511e10	−0.127042	−0.0635209	+	0.997981i	$0.520233\pi$
$420$	5.80098e9	0.00909661
$421$	3.45188e11	0.535533	0.267766	−	0.963484i	$-0.413714\pi$
$421$	3.45188e11	0.535533	0.267766	+	0.963484i	$0.413714\pi$
$422$	−6.16383e11	−0.946117
$423$	5.92709e10	0.0900139
$424$	1.01139e12	1.51975
$425$	3.58468e11	0.532967
$426$	1.06520e11	0.156707
$427$	−9.16741e10	−0.133451
$428$	−1.41184e10	−0.0203370
$429$	0	0
$430$	−9.33116e10	−0.131622
$431$	−3.08222e11	−0.430245	−0.215122	−	0.976587i	$-0.569015\pi$
$431$	−3.08222e11	−0.430245	−0.215122	+	0.976587i	$0.569015\pi$
$432$	−8.72874e8	−0.00120580
$433$	6.15726e11	0.841767	0.420883	−	0.907115i	$-0.361720\pi$
$433$	6.15726e11	0.841767	0.420883	+	0.907115i	$0.361720\pi$
$434$	7.69457e10	0.104107
$435$	−3.05015e10	−0.0408432
$436$	−1.04915e11	−0.139043
$437$	−8.71434e10	−0.114306
$438$	6.84872e10	0.0889151
$439$	−8.58421e11	−1.10309	−0.551544	−	0.834146i	$-0.685961\pi$
$439$	−8.58421e11	−1.10309	−0.551544	+	0.834146i	$0.685961\pi$
$440$	8.61400e11	1.09564
$441$	7.52364e11	0.947227
$442$	0	0
$443$	−5.70446e10	−0.0703717	−0.0351859	−	0.999381i	$-0.511202\pi$
$443$	−5.70446e10	−0.0703717	−0.0351859	+	0.999381i	$0.511202\pi$
$444$	7.56577e10	0.0923911
$445$	9.53927e10	0.115317
$446$	−6.48146e11	−0.775651
$447$	−2.14168e11	−0.253729
$448$	7.42458e10	0.0870804
$449$	1.29769e12	1.50683	0.753414	−	0.657546i	$-0.228406\pi$
$449$	1.29769e12	1.50683	0.753414	+	0.657546i	$0.228406\pi$
$450$	−3.51848e11	−0.404481
$451$	2.32295e12	2.64391
$452$	−2.59881e11	−0.292854
$453$	5.03160e10	0.0561390
$454$	−6.68650e11	−0.738665
$455$	0	0
$456$	2.27344e10	0.0246231
$457$	1.77521e12	1.90382	0.951912	−	0.306373i	$-0.0991153\pi$
$457$	1.77521e12	1.90382	0.951912	+	0.306373i	$0.0991153\pi$
$458$	3.52655e11	0.374503
$459$	2.68270e11	0.282108
$460$	−2.85679e11	−0.297487
$461$	−3.65491e11	−0.376897	−0.188448	−	0.982083i	$-0.560346\pi$
$461$	−3.65491e11	−0.376897	−0.188448	+	0.982083i	$0.560346\pi$
$462$	3.02427e10	0.0308839
$463$	−6.08125e11	−0.615005	−0.307502	−	0.951547i	$-0.599493\pi$
$463$	−6.08125e11	−0.615005	−0.307502	+	0.951547i	$0.599493\pi$
$464$	1.32256e9	0.00132460
$465$	1.24253e11	0.123245
$466$	7.56250e11	0.742897
$467$	1.29905e12	1.26386	0.631932	−	0.775024i	$-0.282262\pi$
$467$	1.29905e12	1.26386	0.631932	+	0.775024i	$0.282262\pi$
$468$	0	0
$469$	−1.75054e11	−0.167068
$470$	−3.44597e10	−0.0325740
$471$	1.54568e11	0.144719
$472$	−4.28715e11	−0.397585
$473$	7.92147e11	0.727663
$474$	−3.22957e10	−0.0293861
$475$	−1.01609e11	−0.0915820
$476$	7.73066e10	0.0690216
$477$	−1.66277e12	−1.47062
$478$	2.16383e11	0.189582
$479$	−1.08153e12	−0.938706	−0.469353	−	0.883011i	$-0.655513\pi$
$479$	−1.08153e12	−0.938706	−0.469353	+	0.883011i	$0.655513\pi$
$480$	1.20549e11	0.103652
$481$	0	0
$482$	1.05260e12	0.888285
$483$	−2.62192e10	−0.0219208
$484$	−2.04943e12	−1.69757
$485$	1.03850e12	0.852254
$486$	−3.97205e11	−0.322962
$487$	2.07131e12	1.66865	0.834325	−	0.551273i	$-0.185858\pi$
$487$	2.07131e12	1.66865	0.834325	+	0.551273i	$0.185858\pi$
$488$	−1.17781e12	−0.940123
$489$	1.66701e10	0.0131841
$490$	−4.37419e11	−0.342780
$491$	1.55641e12	1.20853	0.604263	−	0.796785i	$-0.293468\pi$
$491$	1.55641e12	1.20853	0.604263	+	0.796785i	$0.293468\pi$
$492$	2.00985e11	0.154640
$493$	−4.06477e11	−0.309903
$494$	0	0
$495$	−1.41618e12	−1.06022
$496$	−5.38767e9	−0.00399699
$497$	−2.68391e11	−0.197317
$498$	−2.07341e11	−0.151062
$499$	−1.31914e12	−0.952442	−0.476221	−	0.879326i	$-0.657994\pi$
$499$	−1.31914e12	−0.952442	−0.476221	+	0.879326i	$0.657994\pi$
$500$	−8.24134e11	−0.589702
$501$	5.55071e10	0.0393622
$502$	8.31459e11	0.584351
$503$	−1.27999e12	−0.891560	−0.445780	−	0.895143i	$-0.647074\pi$
$503$	−1.27999e12	−0.891560	−0.445780	+	0.895143i	$0.647074\pi$
$504$	−1.98356e11	−0.136933
$505$	−5.62749e11	−0.385038
$506$	−1.48935e12	−1.01000
$507$	0	0
$508$	1.20522e12	0.802935
$509$	1.32059e11	0.0872042	0.0436021	−	0.999049i	$-0.486117\pi$
$509$	1.32059e11	0.0872042	0.0436021	+	0.999049i	$0.486117\pi$
$510$	−7.66635e10	−0.0501791
$511$	−1.72563e11	−0.111957
$512$	−1.04433e10	−0.00671618
$513$	−7.60420e10	−0.0484758
$514$	−1.21395e12	−0.767123
$515$	−1.37364e12	−0.860476
$516$	6.85377e10	0.0425604
$517$	2.92537e11	0.180083
$518$	1.17068e11	0.0714424
$519$	3.42295e11	0.207084
$520$	0	0
$521$	−2.69675e12	−1.60351	−0.801753	−	0.597655i	$-0.796099\pi$
$521$	−2.69675e12	−1.60351	−0.801753	+	0.597655i	$0.796099\pi$
$522$	3.98970e11	0.235192
$523$	9.70396e11	0.567141	0.283571	−	0.958951i	$-0.408481\pi$
$523$	9.70396e11	0.567141	0.283571	+	0.958951i	$0.408481\pi$
$524$	−1.10371e12	−0.639533
$525$	−3.05715e10	−0.0175630
$526$	1.78304e12	1.01561
$527$	1.65585e12	0.935135
$528$	−2.11757e9	−0.00118573
$529$	−5.09943e11	−0.283121
$530$	9.66724e11	0.532183
$531$	7.04828e11	0.384732
$532$	−2.19128e10	−0.0118603
$533$	0	0
$534$	4.30287e10	0.0228993
$535$	−3.52772e10	−0.0186167
$536$	−2.24905e12	−1.17695
$537$	−3.87461e11	−0.201068
$538$	1.08485e11	0.0558278
$539$	3.71337e12	1.89504
$540$	−2.49286e11	−0.126161
$541$	−3.28705e12	−1.64975	−0.824876	−	0.565313i	$-0.808755\pi$
$541$	−3.28705e12	−1.64975	−0.824876	+	0.565313i	$0.808755\pi$
$542$	2.16452e12	1.07737
$543$	−5.29846e11	−0.261548
$544$	1.60649e12	0.786471
$545$	−2.62148e11	−0.127281
$546$	0	0
$547$	4.30933e11	0.205810	0.102905	−	0.994691i	$-0.467186\pi$
$547$	4.30933e11	0.205810	0.102905	+	0.994691i	$0.467186\pi$
$548$	−4.52157e11	−0.214179
$549$	1.93637e12	0.909731
$550$	−1.73658e12	−0.809213
$551$	1.15217e11	0.0532519
$552$	−3.36858e11	−0.154426
$553$	8.13732e10	0.0370014
$554$	1.34992e12	0.608858
$555$	1.89044e11	0.0845754
$556$	−1.69898e12	−0.753964
$557$	−1.67022e12	−0.735236	−0.367618	−	0.929977i	$-0.619827\pi$
$557$	−1.67022e12	−0.735236	−0.367618	+	0.929977i	$0.619827\pi$
$558$	−1.62527e12	−0.709696
$559$	0	0
$560$	−6.28499e8	−0.000270059 0
$561$	6.50816e11	0.277412
$562$	−1.14229e11	−0.0483020
$563$	−3.90774e12	−1.63922	−0.819611	−	0.572920i	$-0.805810\pi$
$563$	−3.90774e12	−1.63922	−0.819611	+	0.572920i	$0.805810\pi$
$564$	2.53108e10	0.0105329
$565$	−6.49356e11	−0.268080
$566$	−4.33447e11	−0.177526
$567$	3.14474e11	0.127779
$568$	−3.44822e12	−1.39004
$569$	2.65735e12	1.06278	0.531390	−	0.847128i	$-0.321670\pi$
$569$	2.65735e12	1.06278	0.531390	+	0.847128i	$0.321670\pi$
$570$	2.17305e10	0.00862248
$571$	1.61259e12	0.634837	0.317418	−	0.948286i	$-0.397184\pi$
$571$	1.61259e12	0.634837	0.317418	+	0.948286i	$0.397184\pi$
$572$	0	0
$573$	−4.30580e11	−0.166862
$574$	3.10993e11	0.119577
$575$	1.50554e12	0.574365
$576$	−1.56824e12	−0.593625
$577$	−4.67341e12	−1.75526	−0.877632	−	0.479334i	$-0.840878\pi$
$577$	−4.67341e12	−1.75526	−0.877632	+	0.479334i	$0.840878\pi$
$578$	6.33480e11	0.236079
$579$	1.38308e11	0.0511437
$580$	3.77713e11	0.138591
$581$	5.22424e11	0.190209
$582$	4.68436e11	0.169238
$583$	−8.20677e12	−2.94214
$584$	−2.21704e12	−0.788707
$585$	0	0
$586$	−2.56529e12	−0.898665
$587$	4.75074e11	0.165154	0.0825771	−	0.996585i	$-0.473685\pi$
$587$	4.75074e11	0.165154	0.0825771	+	0.996585i	$0.473685\pi$
$588$	3.21286e11	0.110839
$589$	−4.69356e11	−0.160688
$590$	−4.09782e11	−0.139226
$591$	−3.42420e11	−0.115456
$592$	−8.19703e9	−0.00274289
$593$	−9.71669e11	−0.322680	−0.161340	−	0.986899i	$-0.551582\pi$
$593$	−9.71669e11	−0.322680	−0.161340	+	0.986899i	$0.551582\pi$
$594$	−1.29962e12	−0.428330
$595$	1.93164e11	0.0631829
$596$	2.65213e12	0.860967
$597$	−9.55203e11	−0.307759
$598$	0	0
$599$	8.98292e11	0.285100	0.142550	−	0.989788i	$-0.454470\pi$
$599$	8.98292e11	0.285100	0.142550	+	0.989788i	$0.454470\pi$
$600$	−3.92775e11	−0.123727
$601$	−3.67532e12	−1.14911	−0.574553	−	0.818467i	$-0.694824\pi$
$601$	−3.67532e12	−1.14911	−0.574553	+	0.818467i	$0.694824\pi$
$602$	1.06051e11	0.0329103
$603$	3.69754e12	1.13890
$604$	−6.23084e11	−0.190494
$605$	−5.12084e12	−1.55397
$606$	−2.53839e11	−0.0764594
$607$	3.58702e11	0.107247	0.0536234	−	0.998561i	$-0.482923\pi$
$607$	3.58702e11	0.107247	0.0536234	+	0.998561i	$0.482923\pi$
$608$	−4.55364e11	−0.135143
$609$	3.46659e10	0.0102123
$610$	−1.12579e12	−0.329211
$611$	0	0
$612$	−1.63290e12	−0.470519
$613$	−5.33392e12	−1.52572	−0.762859	−	0.646565i	$-0.776205\pi$
$613$	−5.33392e12	−1.52572	−0.762859	+	0.646565i	$0.776205\pi$
$614$	1.54205e12	0.437865
$615$	5.02196e11	0.141558
$616$	−9.79006e11	−0.273951
$617$	3.91640e12	1.08794	0.543969	−	0.839105i	$-0.316921\pi$
$617$	3.91640e12	1.08794	0.543969	+	0.839105i	$0.316921\pi$
$618$	−6.19605e11	−0.170870
$619$	−7.14576e12	−1.95632	−0.978161	−	0.207848i	$-0.933354\pi$
$619$	−7.14576e12	−1.95632	−0.978161	+	0.207848i	$0.933354\pi$
$620$	−1.53868e12	−0.418200
$621$	1.12672e12	0.304021
$622$	3.22190e12	0.863088
$623$	−1.08417e11	−0.0288336
$624$	0	0
$625$	5.28527e11	0.138550
$626$	2.87906e12	0.749318
$627$	−1.84476e11	−0.0476689
$628$	−1.91408e12	−0.491068
$629$	2.51929e12	0.641726
$630$	−1.89596e11	−0.0479510
$631$	1.75398e12	0.440447	0.220223	−	0.975449i	$-0.429321\pi$
$631$	1.75398e12	0.440447	0.220223	+	0.975449i	$0.429321\pi$
$632$	1.04546e12	0.260665
$633$	1.13124e12	0.280051
$634$	−1.60470e12	−0.394451
$635$	3.01145e12	0.735012
$636$	−7.10062e11	−0.172084
$637$	0	0
$638$	1.96916e12	0.470530
$639$	5.66905e12	1.34510
$640$	−1.49779e12	−0.352891
$641$	6.62289e12	1.54948	0.774740	−	0.632279i	$-0.217881\pi$
$641$	6.62289e12	1.54948	0.774740	+	0.632279i	$0.217881\pi$
$642$	−1.59124e10	−0.00369683
$643$	6.11769e10	0.0141136	0.00705681	−	0.999975i	$-0.497754\pi$
$643$	6.11769e10	0.0141136	0.00705681	+	0.999975i	$0.497754\pi$
$644$	3.24683e11	0.0743829
$645$	1.71253e11	0.0389601
$646$	2.89590e11	0.0654241
$647$	7.69987e11	0.172749	0.0863743	−	0.996263i	$-0.472472\pi$
$647$	7.69987e11	0.172749	0.0863743	+	0.996263i	$0.472472\pi$
$648$	4.04028e12	0.900169
$649$	3.47875e12	0.769701
$650$	0	0
$651$	−1.41217e11	−0.0308158
$652$	−2.06433e11	−0.0447368
$653$	5.93443e12	1.27723	0.638616	−	0.769526i	$-0.279507\pi$
$653$	5.93443e12	1.27723	0.638616	+	0.769526i	$0.279507\pi$
$654$	−1.18247e11	−0.0252749
$655$	−2.75780e12	−0.585433
$656$	−2.17755e10	−0.00459092
$657$	3.64492e12	0.763210
$658$	3.91644e10	0.00814470
$659$	−4.90282e10	−0.0101266	−0.00506328	−	0.999987i	$-0.501612\pi$
$659$	−4.90282e10	−0.0101266	−0.00506328	+	0.999987i	$0.501612\pi$
$660$	−6.04760e11	−0.124061
$661$	−4.06587e12	−0.828414	−0.414207	−	0.910183i	$-0.635941\pi$
$661$	−4.06587e12	−0.828414	−0.414207	+	0.910183i	$0.635941\pi$
$662$	4.01068e12	0.811628
$663$	0	0
$664$	6.71198e12	1.33997
$665$	−5.47528e10	−0.0108570
$666$	−2.47276e12	−0.487021
$667$	−1.70718e12	−0.333974
$668$	−6.87367e11	−0.133566
$669$	1.18953e12	0.229593
$670$	−2.14973e12	−0.412142
$671$	9.55715e12	1.82002
$672$	−1.37007e11	−0.0259168
$673$	−3.13223e10	−0.00588553	−0.00294276	−	0.999996i	$-0.500937\pi$
$673$	−3.13223e10	−0.00588553	−0.00294276	+	0.999996i	$0.500937\pi$
$674$	−3.27901e12	−0.612030
$675$	1.31375e12	0.243582
$676$	0	0
$677$	−3.47299e12	−0.635411	−0.317706	−	0.948189i	$-0.602912\pi$
$677$	−3.47299e12	−0.635411	−0.317706	+	0.948189i	$0.602912\pi$
$678$	−2.92905e11	−0.0532344
$679$	−1.18029e12	−0.213095
$680$	2.48172e12	0.445106
$681$	1.22716e12	0.218645
$682$	−8.02170e12	−1.41983
$683$	3.59628e12	0.632354	0.316177	−	0.948700i	$-0.397601\pi$
$683$	3.59628e12	0.632354	0.316177	+	0.948700i	$0.397601\pi$
$684$	4.62849e11	0.0808512
$685$	−1.12979e12	−0.196061
$686$	1.00448e12	0.173174
$687$	−6.47221e11	−0.110853
$688$	−7.42562e9	−0.00126353
$689$	0	0
$690$	−3.21982e11	−0.0540767
$691$	3.06940e12	0.512157	0.256078	−	0.966656i	$-0.417569\pi$
$691$	3.06940e12	0.512157	0.256078	+	0.966656i	$0.417569\pi$
$692$	−4.23877e12	−0.702688
$693$	1.60953e12	0.265094
$694$	3.61588e12	0.591694
$695$	−4.24518e12	−0.690184
$696$	4.45378e11	0.0719429
$697$	6.69250e12	1.07409
$698$	3.33140e12	0.531224
$699$	−1.38793e12	−0.219898
$700$	3.78579e11	0.0595957
$701$	2.53719e12	0.396846	0.198423	−	0.980116i	$-0.436418\pi$
$701$	2.53719e12	0.396846	0.198423	+	0.980116i	$0.436418\pi$
$702$	0	0
$703$	−7.14099e11	−0.110270
$704$	−7.74022e12	−1.18762
$705$	6.32433e10	0.00964192
$706$	−7.93281e11	−0.120173
$707$	6.39580e11	0.0962737
$708$	3.00987e11	0.0450192
$709$	1.03456e12	0.153761	0.0768804	−	0.997040i	$-0.475504\pi$
$709$	1.03456e12	0.153761	0.0768804	+	0.997040i	$0.475504\pi$
$710$	−3.29595e12	−0.486763
$711$	−1.71879e12	−0.252238
$712$	−1.39291e12	−0.203125
$713$	6.95448e12	1.00777
$714$	8.71302e10	0.0125466
$715$	0	0
$716$	4.79809e12	0.682275
$717$	−3.97124e11	−0.0561164
$718$	−7.60979e11	−0.106859
$719$	1.13888e13	1.58927	0.794633	−	0.607090i	$-0.207663\pi$
$719$	1.13888e13	1.58927	0.794633	+	0.607090i	$0.207663\pi$
$720$	1.32754e10	0.00184098
$721$	1.56118e12	0.215151
$722$	4.42167e12	0.605577
$723$	−1.93182e12	−0.262933
$724$	6.56130e12	0.887496
$725$	−1.99056e12	−0.267581
$726$	−2.30985e12	−0.308582
$727$	−4.40512e12	−0.584861	−0.292430	−	0.956287i	$-0.594464\pi$
$727$	−4.40512e12	−0.584861	−0.292430	+	0.956287i	$0.594464\pi$
$728$	0	0
$729$	−6.14249e12	−0.805509
$730$	−2.11913e12	−0.276188
$731$	2.28220e12	0.295614
$732$	8.26899e11	0.106452
$733$	6.80311e12	0.870442	0.435221	−	0.900324i	$-0.356670\pi$
$733$	6.80311e12	0.870442	0.435221	+	0.900324i	$0.356670\pi$
$734$	−4.88725e12	−0.621487
$735$	8.02788e11	0.101463
$736$	6.74715e12	0.847560
$737$	1.82496e13	2.27850
$738$	−6.56890e12	−0.815153
$739$	5.03824e12	0.621411	0.310705	−	0.950506i	$-0.399435\pi$
$739$	5.03824e12	0.621411	0.310705	+	0.950506i	$0.399435\pi$
$740$	−2.34101e12	−0.286986
$741$	0	0
$742$	−1.09871e12	−0.133066
$743$	−3.43023e12	−0.412927	−0.206464	−	0.978454i	$-0.566196\pi$
$743$	−3.43023e12	−0.412927	−0.206464	+	0.978454i	$0.566196\pi$
$744$	−1.81432e12	−0.217088
$745$	6.62680e12	0.788135
$746$	−5.58415e12	−0.660134
$747$	−1.10348e13	−1.29665
$748$	−8.05932e12	−0.941328
$749$	4.00935e10	0.00465485
$750$	−9.28860e11	−0.107195
$751$	−8.21725e12	−0.942642	−0.471321	−	0.881962i	$-0.656223\pi$
$751$	−8.21725e12	−0.942642	−0.471321	+	0.881962i	$0.656223\pi$
$752$	−2.74226e9	−0.000312700 0
$753$	−1.52596e12	−0.172968
$754$	0	0
$755$	−1.55688e12	−0.174379
$756$	2.83321e11	0.0315450
$757$	1.22618e13	1.35714	0.678569	−	0.734537i	$-0.262601\pi$
$757$	1.22618e13	1.35714	0.678569	+	0.734537i	$0.262601\pi$
$758$	7.70916e12	0.848194
$759$	2.73339e12	0.298960
$760$	−7.03451e11	−0.0764843
$761$	1.25645e13	1.35804	0.679022	−	0.734118i	$-0.262404\pi$
$761$	1.25645e13	1.35804	0.679022	+	0.734118i	$0.262404\pi$
$762$	1.35837e12	0.145956
$763$	2.97939e11	0.0318249
$764$	5.33204e12	0.566205
$765$	−4.08007e12	−0.430716
$766$	1.60198e12	0.168123
$767$	0	0
$768$	−1.75657e12	−0.182197
$769$	2.74768e11	0.0283334	0.0141667	−	0.999900i	$-0.495490\pi$
$769$	2.74768e11	0.0283334	0.0141667	+	0.999900i	$0.495490\pi$
$770$	−9.35772e11	−0.0959316
$771$	2.22793e12	0.227069
$772$	−1.71272e12	−0.173543
$773$	−6.65784e12	−0.670696	−0.335348	−	0.942094i	$-0.608854\pi$
$773$	−6.65784e12	−0.670696	−0.335348	+	0.942094i	$0.608854\pi$
$774$	−2.24005e12	−0.224349
$775$	8.10890e12	0.807428
$776$	−1.51640e13	−1.50120
$777$	−2.14854e11	−0.0211470
$778$	−3.31161e12	−0.324064
$779$	−1.89701e12	−0.184566
$780$	0	0
$781$	2.79801e13	2.69104
$782$	−4.29088e12	−0.410314
$783$	−1.48970e12	−0.141635
$784$	−3.48093e10	−0.00329058
$785$	−4.78266e12	−0.449527
$786$	−1.24396e12	−0.116253
$787$	−9.72333e12	−0.903501	−0.451751	−	0.892144i	$-0.649200\pi$
$787$	−9.72333e12	−0.903501	−0.451751	+	0.892144i	$0.649200\pi$
$788$	4.24033e12	0.391770
$789$	−3.27239e12	−0.300620
$790$	9.99295e11	0.0912792
$791$	7.38012e11	0.0670300
$792$	2.06789e13	1.86751
$793$	0	0
$794$	−9.43179e11	−0.0842174
$795$	−1.77421e12	−0.157526
$796$	1.18287e13	1.04430
$797$	−2.01479e13	−1.76876	−0.884379	−	0.466770i	$-0.845418\pi$
$797$	−2.01479e13	−1.76876	−0.884379	+	0.466770i	$0.845418\pi$
$798$	−2.46973e10	−0.00215594
$799$	8.42809e11	0.0731592
$800$	7.86715e12	0.679067
$801$	2.29001e12	0.196558
$802$	7.44277e12	0.635258
$803$	1.79899e13	1.52689
$804$	1.57898e12	0.133268
$805$	8.11276e11	0.0680906
$806$	0	0
$807$	−1.99101e11	−0.0165251
$808$	8.21717e12	0.678221
$809$	−1.58485e12	−0.130083	−0.0650413	−	0.997883i	$-0.520718\pi$
$809$	−1.58485e12	−0.130083	−0.0650413	+	0.997883i	$0.520718\pi$
$810$	3.86186e12	0.315220
$811$	1.09246e13	0.886771	0.443386	−	0.896331i	$-0.353777\pi$
$811$	1.09246e13	0.886771	0.443386	+	0.896331i	$0.353777\pi$
$812$	−4.29281e11	−0.0346529
$813$	−3.97250e12	−0.318902
$814$	−1.22046e13	−0.974343
$815$	−5.15809e11	−0.0409524
$816$	−6.10078e9	−0.000481703 0
$817$	−6.46896e11	−0.0507967
$818$	2.92889e12	0.228725
$819$	0	0
$820$	−6.21890e12	−0.480343
$821$	−1.92888e13	−1.48170	−0.740851	−	0.671669i	$-0.765578\pi$
$821$	−1.92888e13	−1.48170	−0.740851	+	0.671669i	$0.765578\pi$
$822$	−5.09614e11	−0.0389330
$823$	−2.97941e12	−0.226376	−0.113188	−	0.993574i	$-0.536106\pi$
$823$	−2.97941e12	−0.226376	−0.113188	+	0.993574i	$0.536106\pi$
$824$	2.00576e13	1.51568
$825$	3.18712e12	0.239527
$826$	4.65730e11	0.0348116
$827$	−9.84829e12	−0.732126	−0.366063	−	0.930590i	$-0.619295\pi$
$827$	−9.84829e12	−0.732126	−0.366063	+	0.930590i	$0.619295\pi$
$828$	−6.85806e12	−0.507066
$829$	−2.42595e13	−1.78396	−0.891981	−	0.452072i	$-0.850685\pi$
$829$	−2.42595e13	−1.78396	−0.891981	+	0.452072i	$0.850685\pi$
$830$	6.41557e12	0.469228
$831$	−2.47749e12	−0.180222
$832$	0	0
$833$	1.06983e13	0.769863
$834$	−1.91487e12	−0.137054
$835$	−1.71751e12	−0.122267
$836$	2.28444e12	0.161752
$837$	6.06853e12	0.427385
$838$	1.11867e12	0.0783617
$839$	−5.96000e12	−0.415257	−0.207629	−	0.978208i	$-0.566575\pi$
$839$	−5.96000e12	−0.415257	−0.207629	+	0.978208i	$0.566575\pi$
$840$	−2.11650e11	−0.0146677
$841$	−1.22500e13	−0.844411
$842$	−4.81779e12	−0.330327
$843$	2.09643e11	0.0142974
$844$	−1.40086e13	−0.950283
$845$	0	0
$846$	−8.27244e11	−0.0555222
$847$	5.81999e12	0.388550
$848$	7.69307e10	0.00510879
$849$	7.95497e11	0.0525477
$850$	−5.00314e12	−0.328744
$851$	1.05808e13	0.691572
$852$	2.42088e12	0.157397
$853$	−5.87984e12	−0.380272	−0.190136	−	0.981758i	$-0.560893\pi$
$853$	−5.87984e12	−0.380272	−0.190136	+	0.981758i	$0.560893\pi$
$854$	1.27950e12	0.0823150
$855$	1.15651e12	0.0740118
$856$	5.15112e11	0.0327921
$857$	−1.71300e13	−1.08478	−0.542392	−	0.840125i	$-0.682481\pi$
$857$	−1.71300e13	−1.08478	−0.542392	+	0.840125i	$0.682481\pi$
$858$	0	0
$859$	−1.36880e13	−0.857772	−0.428886	−	0.903359i	$-0.641094\pi$
$859$	−1.36880e13	−0.857772	−0.428886	+	0.903359i	$0.641094\pi$
$860$	−2.12070e12	−0.132201
$861$	−5.70761e11	−0.0353948
$862$	4.30185e12	0.265383
$863$	−2.02454e13	−1.24245	−0.621223	−	0.783634i	$-0.713364\pi$
$863$	−2.02454e13	−1.24245	−0.621223	+	0.783634i	$0.713364\pi$
$864$	5.88762e12	0.359441
$865$	−1.05913e13	−0.643246
$866$	−8.59369e12	−0.519217
$867$	−1.16261e12	−0.0698795
$868$	1.74875e12	0.104566
$869$	−8.48327e12	−0.504632
$870$	4.25710e11	0.0251928
$871$	0	0
$872$	3.82784e12	0.224197
$873$	2.49304e13	1.45266
$874$	1.21626e12	0.0705059
$875$	2.34039e12	0.134974
$876$	1.55651e12	0.0893067
$877$	−1.02953e12	−0.0587678	−0.0293839	−	0.999568i	$-0.509355\pi$
$877$	−1.02953e12	−0.0587678	−0.0293839	+	0.999568i	$0.509355\pi$
$878$	1.19810e13	0.680405
$879$	4.70804e12	0.266005
$880$	6.55219e10	0.00368311
$881$	1.54322e13	0.863053	0.431527	−	0.902100i	$-0.357975\pi$
$881$	1.54322e13	0.863053	0.431527	+	0.902100i	$0.357975\pi$
$882$	−1.05008e13	−0.584267
$883$	1.62351e13	0.898735	0.449368	−	0.893347i	$-0.351649\pi$
$883$	1.62351e13	0.898735	0.449368	+	0.893347i	$0.351649\pi$
$884$	0	0
$885$	7.52067e11	0.0412109
$886$	7.96173e11	0.0434066
$887$	2.39023e13	1.29653	0.648265	−	0.761414i	$-0.275495\pi$
$887$	2.39023e13	1.29653	0.648265	+	0.761414i	$0.275495\pi$
$888$	−2.76039e12	−0.148975
$889$	−3.42261e12	−0.183780
$890$	−1.33140e12	−0.0711300
$891$	−3.27843e13	−1.74267
$892$	−1.47305e13	−0.779066
$893$	−2.38897e11	−0.0125712
$894$	2.98915e12	0.156505
$895$	1.19888e13	0.624559
$896$	1.70228e12	0.0882359
$897$	0	0
$898$	−1.81119e13	−0.929440
$899$	−9.19491e12	−0.469493
$900$	−7.99647e12	−0.406263
$901$	−2.36440e13	−1.19525
$902$	−3.24215e13	−1.63081
$903$	−1.94634e11	−0.00974147
$904$	9.48180e12	0.472208
$905$	1.63945e13	0.812420
$906$	−7.02262e11	−0.0346276
$907$	1.58890e13	0.779584	0.389792	−	0.920903i	$-0.372547\pi$
$907$	1.58890e13	0.779584	0.389792	+	0.920903i	$0.372547\pi$
$908$	−1.51964e13	−0.741918
$909$	−1.35094e13	−0.656296
$910$	0	0
$911$	−1.66855e13	−0.802613	−0.401307	−	0.915944i	$-0.631444\pi$
$911$	−1.66855e13	−0.802613	−0.401307	+	0.915944i	$0.631444\pi$
$912$	1.72928e9	8.27731e−5 0
$913$	−5.44634e13	−2.59410
$914$	−2.47766e13	−1.17431
$915$	2.06615e12	0.0974467
$916$	8.01480e12	0.376152
$917$	3.13432e12	0.146380
$918$	−3.74425e12	−0.174010
$919$	−5.26862e12	−0.243656	−0.121828	−	0.992551i	$-0.538876\pi$
$919$	−5.26862e12	−0.243656	−0.121828	+	0.992551i	$0.538876\pi$
$920$	1.04231e13	0.479679
$921$	−2.83010e12	−0.129608
$922$	5.10116e12	0.232477
$923$	0	0
$924$	6.87328e11	0.0310199
$925$	1.23372e13	0.554089
$926$	8.48761e12	0.379346
$927$	−3.29757e13	−1.46668
$928$	−8.92079e12	−0.394855
$929$	2.00971e13	0.885244	0.442622	−	0.896708i	$-0.354048\pi$
$929$	2.00971e13	0.885244	0.442622	+	0.896708i	$0.354048\pi$
$930$	−1.73420e12	−0.0760197
$931$	−3.03247e12	−0.132289
$932$	1.71873e13	0.746169
$933$	−5.91310e12	−0.255475
$934$	−1.81309e13	−0.779574
$935$	−2.01376e13	−0.861699
$936$	0	0
$937$	−2.24046e13	−0.949532	−0.474766	−	0.880112i	$-0.657467\pi$
$937$	−2.24046e13	−0.949532	−0.474766	+	0.880112i	$0.657467\pi$
$938$	2.44323e12	0.103051
$939$	−5.28389e12	−0.221799
$940$	−7.83167e11	−0.0327174
$941$	−3.85186e12	−0.160146	−0.0800731	−	0.996789i	$-0.525515\pi$
$941$	−3.85186e12	−0.160146	−0.0800731	+	0.996789i	$0.525515\pi$
$942$	−2.15731e12	−0.0892655
$943$	2.81081e13	1.15752
$944$	−3.26100e10	−0.00133652
$945$	7.07926e11	0.0288765
$946$	−1.10560e13	−0.448836
$947$	2.77272e13	1.12029	0.560146	−	0.828394i	$-0.310745\pi$
$947$	2.77272e13	1.12029	0.560146	+	0.828394i	$0.310745\pi$
$948$	−7.33985e11	−0.0295155
$949$	0	0
$950$	1.41815e12	0.0564895
$951$	2.94508e12	0.116758
$952$	−2.82055e12	−0.111293
$953$	1.07795e13	0.423330	0.211665	−	0.977342i	$-0.432111\pi$
$953$	1.07795e13	0.423330	0.211665	+	0.977342i	$0.432111\pi$
$954$	2.32073e13	0.907104
$955$	1.33230e13	0.518308
$956$	4.91775e12	0.190417
$957$	−3.61396e12	−0.139277
$958$	1.50950e13	0.579011
$959$	1.28404e12	0.0490224
$960$	−1.67335e12	−0.0635867
$961$	1.10174e13	0.416700
$962$	0	0
$963$	−8.46869e11	−0.0317320
$964$	2.39225e13	0.892197
$965$	−4.27952e12	−0.158863
$966$	3.65942e11	0.0135212
$967$	−1.11541e13	−0.410219	−0.205110	−	0.978739i	$-0.565755\pi$
$967$	−1.11541e13	−0.410219	−0.205110	+	0.978739i	$0.565755\pi$
$968$	7.47738e13	2.73722
$969$	−5.31480e11	−0.0193656
$970$	−1.44944e13	−0.525686
$971$	−1.63758e13	−0.591175	−0.295587	−	0.955316i	$-0.595515\pi$
$971$	−1.63758e13	−0.591175	−0.295587	+	0.955316i	$0.595515\pi$
$972$	−9.02731e12	−0.324385
$973$	4.82477e12	0.172572
$974$	−2.89093e13	−1.02925
$975$	0	0
$976$	−8.95892e10	−0.00316032
$977$	7.18430e12	0.252266	0.126133	−	0.992013i	$-0.459743\pi$
$977$	7.18430e12	0.252266	0.126133	+	0.992013i	$0.459743\pi$
$978$	−2.32665e11	−0.00813218
$979$	1.13026e13	0.393238
$980$	−9.94125e12	−0.344290
$981$	−6.29316e12	−0.216949
$982$	−2.17228e13	−0.745442
$983$	8.50949e12	0.290679	0.145339	−	0.989382i	$-0.453573\pi$
$983$	8.50949e12	0.290679	0.145339	+	0.989382i	$0.453573\pi$
$984$	−7.33300e12	−0.249347
$985$	1.05952e13	0.358629
$986$	5.67321e12	0.191154
$987$	−7.18778e10	−0.00241084
$988$	0	0
$989$	9.58510e12	0.318576
$990$	1.97657e13	0.653964
$991$	−4.38653e13	−1.44474	−0.722371	−	0.691506i	$-0.756947\pi$
$991$	−4.38653e13	−1.44474	−0.722371	+	0.691506i	$0.756947\pi$
$992$	3.63403e13	1.19148
$993$	−7.36074e12	−0.240242
$994$	3.74594e12	0.121709
$995$	2.95560e13	0.955964
$996$	−4.71226e12	−0.151727
$997$	−5.03805e13	−1.61486	−0.807428	−	0.589966i	$-0.799141\pi$
$997$	−5.03805e13	−1.61486	−0.807428	+	0.589966i	$0.799141\pi$
$998$	1.84113e13	0.587484
$999$	9.23293e12	0.293288

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.1.8		20
13.2	odd	12		13.10.e.a.4.4	✓	20
13.7	odd	12		13.10.e.a.10.4	yes	20
13.12	even	2	inner	169.10.a.f.1.13		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.10.e.a.4.4	✓	20	13.2	odd	12
13.10.e.a.10.4	yes	20	13.7	odd	12
169.10.a.f.1.8		20	1.1	even	1	trivial
169.10.a.f.1.13		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.1.8

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