LMFDB - Embedded newform 169.10.a.f.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [169,10,Mod(1,169)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(169, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 10, names="a")

//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);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	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			1.14
Root			$19.7704$ 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)

$f(q)$	$=$	\(q+19.7704 q^{2} +99.5109 q^{3} -121.130 q^{4} -1521.53 q^{5} +1967.37 q^{6} -9243.80 q^{7} -12517.3 q^{8} -9780.58 q^{9} -30081.3 q^{10} +38807.1 q^{11} -12053.7 q^{12} -182754. q^{14} -151409. q^{15} -185453. q^{16} +453281. q^{17} -193366. q^{18} -965410. q^{19} +184302. q^{20} -919859. q^{21} +767233. q^{22} +543135. q^{23} -1.24560e6 q^{24} +361927. q^{25} -2.93195e6 q^{27} +1.11970e6 q^{28} +4.05377e6 q^{29} -2.99342e6 q^{30} -2.44251e6 q^{31} +2.74234e6 q^{32} +3.86173e6 q^{33} +8.96157e6 q^{34} +1.40647e7 q^{35} +1.18472e6 q^{36} +7.60255e6 q^{37} -1.90866e7 q^{38} +1.90454e7 q^{40} -4.47020e6 q^{41} -1.81860e7 q^{42} +1.62810e7 q^{43} -4.70069e6 q^{44} +1.48814e7 q^{45} +1.07380e7 q^{46} +1.05026e7 q^{47} -1.84546e7 q^{48} +4.50942e7 q^{49} +7.15545e6 q^{50} +4.51064e7 q^{51} -4.76659e7 q^{53} -5.79659e7 q^{54} -5.90461e7 q^{55} +1.15707e8 q^{56} -9.60688e7 q^{57} +8.01449e7 q^{58} +1.11655e8 q^{59} +1.83401e7 q^{60} +6.91907e7 q^{61} -4.82895e7 q^{62} +9.04097e7 q^{63} +1.49169e8 q^{64} +7.63480e7 q^{66} +2.81909e8 q^{67} -5.49058e7 q^{68} +5.40479e7 q^{69} +2.78066e8 q^{70} -3.46433e8 q^{71} +1.22426e8 q^{72} +1.76119e8 q^{73} +1.50306e8 q^{74} +3.60157e7 q^{75} +1.16940e8 q^{76} -3.58725e8 q^{77} -2.36277e8 q^{79} +2.82172e8 q^{80} -9.92495e7 q^{81} -8.83778e7 q^{82} -2.90237e8 q^{83} +1.11422e8 q^{84} -6.89681e8 q^{85} +3.21883e8 q^{86} +4.03395e8 q^{87} -4.85758e8 q^{88} -4.52851e8 q^{89} +2.94213e8 q^{90} -6.57898e7 q^{92} -2.43056e8 q^{93} +2.07642e8 q^{94} +1.46890e9 q^{95} +2.72893e8 q^{96} -1.48537e9 q^{97} +8.91532e8 q^{98} -3.79556e8 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	19.7704	0.873738	0.436869	−	0.899525i	$-0.356087\pi$
$2$	19.7704	0.873738	0.436869	+	0.899525i	$0.356087\pi$
$3$	99.5109	0.709292	0.354646	−	0.935001i	$-0.384601\pi$
$3$	99.5109	0.709292	0.354646	+	0.935001i	$0.384601\pi$
$4$	−121.130	−0.236581
$5$	−1521.53	−1.08872	−0.544359	−	0.838852i	$-0.683227\pi$
$5$	−1521.53	−1.08872	−0.544359	+	0.838852i	$0.683227\pi$
$6$	1967.37	0.619735
$7$	−9243.80	−1.45516	−0.727578	−	0.686025i	$-0.759354\pi$
$7$	−9243.80	−1.45516	−0.727578	+	0.686025i	$0.759354\pi$
$8$	−12517.3	−1.08045
$9$	−9780.58	−0.496905
$10$	−30081.3	−0.951254
$11$	38807.1	0.799178	0.399589	−	0.916694i	$-0.369153\pi$
$11$	38807.1	0.799178	0.399589	+	0.916694i	$0.369153\pi$
$12$	−12053.7	−0.167805
$13$	0	0
$13$	0	0
$14$	−182754.	−1.27142
$15$	−151409.	−0.772219
$16$	−185453.	−0.707448
$17$	453281.	1.31628	0.658139	−	0.752896i	$-0.271344\pi$
$17$	453281.	1.31628	0.658139	+	0.752896i	$0.271344\pi$
$18$	−193366.	−0.434165
$19$	−965410.	−1.69950	−0.849749	−	0.527188i	$-0.823247\pi$
$19$	−965410.	−1.69950	−0.849749	+	0.527188i	$0.823247\pi$
$20$	184302.	0.257570
$21$	−919859.	−1.03213
$22$	767233.	0.698273
$23$	543135.	0.404699	0.202350	−	0.979313i	$-0.435142\pi$
$23$	543135.	0.404699	0.202350	+	0.979313i	$0.435142\pi$
$24$	−1.24560e6	−0.766353
$25$	361927.	0.185306
$26$	0	0
$27$	−2.93195e6	−1.06174
$28$	1.11970e6	0.344263
$29$	4.05377e6	1.06431	0.532156	−	0.846647i	$-0.321382\pi$
$29$	4.05377e6	1.06431	0.532156	+	0.846647i	$0.321382\pi$
$30$	−2.99342e6	−0.674717
$31$	−2.44251e6	−0.475016	−0.237508	−	0.971386i	$-0.576331\pi$
$31$	−2.44251e6	−0.475016	−0.237508	+	0.971386i	$0.576331\pi$
$32$	2.74234e6	0.462324
$33$	3.86173e6	0.566851
$34$	8.96157e6	1.15008
$35$	1.40647e7	1.58425
$36$	1.18472e6	0.117558
$37$	7.60255e6	0.666886	0.333443	−	0.942770i	$-0.391790\pi$
$37$	7.60255e6	0.666886	0.333443	+	0.942770i	$0.391790\pi$
$38$	−1.90866e7	−1.48492
$39$	0	0
$40$	1.90454e7	1.17630
$41$	−4.47020e6	−0.247058	−0.123529	−	0.992341i	$-0.539421\pi$
$41$	−4.47020e6	−0.247058	−0.123529	+	0.992341i	$0.539421\pi$
$42$	−1.81860e7	−0.901811
$43$	1.62810e7	0.726229	0.363115	−	0.931744i	$-0.381713\pi$
$43$	1.62810e7	0.726229	0.363115	+	0.931744i	$0.381713\pi$
$44$	−4.70069e6	−0.189071
$45$	1.48814e7	0.540989
$46$	1.07380e7	0.353601
$47$	1.05026e7	0.313948	0.156974	−	0.987603i	$-0.449826\pi$
$47$	1.05026e7	0.313948	0.156974	+	0.987603i	$0.449826\pi$
$48$	−1.84546e7	−0.501787
$49$	4.50942e7	1.11748
$50$	7.15545e6	0.161909
$51$	4.51064e7	0.933626
$52$	0	0
$53$	−4.76659e7	−0.829787	−0.414893	−	0.909870i	$-0.636181\pi$
$53$	−4.76659e7	−0.829787	−0.414893	+	0.909870i	$0.636181\pi$
$54$	−5.79659e7	−0.927685
$55$	−5.90461e7	−0.870080
$56$	1.15707e8	1.57222
$57$	−9.60688e7	−1.20544
$58$	8.01449e7	0.929929
$59$	1.11655e8	1.19963	0.599813	−	0.800140i	$-0.295242\pi$
$59$	1.11655e8	1.19963	0.599813	+	0.800140i	$0.295242\pi$
$60$	1.83401e7	0.182693
$61$	6.91907e7	0.639828	0.319914	−	0.947447i	$-0.396346\pi$
$61$	6.91907e7	0.639828	0.319914	+	0.947447i	$0.396346\pi$
$62$	−4.82895e7	−0.415040
$63$	9.04097e7	0.723074
$64$	1.49169e8	1.11140
$65$	0	0
$66$	7.63480e7	0.495279
$67$	2.81909e8	1.70912	0.854560	−	0.519352i	$-0.173827\pi$
$67$	2.81909e8	1.70912	0.854560	+	0.519352i	$0.173827\pi$
$68$	−5.49058e7	−0.311407
$69$	5.40479e7	0.287050
$70$	2.78066e8	1.38422
$71$	−3.46433e8	−1.61792	−0.808960	−	0.587864i	$-0.799969\pi$
$71$	−3.46433e8	−1.61792	−0.808960	+	0.587864i	$0.799969\pi$
$72$	1.22426e8	0.536880
$73$	1.76119e8	0.725862	0.362931	−	0.931816i	$-0.381776\pi$
$73$	1.76119e8	0.725862	0.362931	+	0.931816i	$0.381776\pi$
$74$	1.50306e8	0.582683
$75$	3.60157e7	0.131436
$76$	1.16940e8	0.402070
$77$	−3.58725e8	−1.16293
$78$	0	0
$79$	−2.36277e8	−0.682496	−0.341248	−	0.939973i	$-0.610850\pi$
$79$	−2.36277e8	−0.682496	−0.341248	+	0.939973i	$0.610850\pi$
$80$	2.82172e8	0.770211
$81$	−9.92495e7	−0.256180
$82$	−8.83778e7	−0.215864
$83$	−2.90237e8	−0.671276	−0.335638	−	0.941991i	$-0.608952\pi$
$83$	−2.90237e8	−0.671276	−0.335638	+	0.941991i	$0.608952\pi$
$84$	1.11422e8	0.244183
$85$	−6.89681e8	−1.43306
$86$	3.21883e8	0.634534
$87$	4.03395e8	0.754907
$88$	−4.85758e8	−0.863471
$89$	−4.52851e8	−0.765069	−0.382534	−	0.923941i	$-0.624949\pi$
$89$	−4.52851e8	−0.765069	−0.382534	+	0.923941i	$0.624949\pi$
$90$	2.94213e8	0.472683
$91$	0	0
$92$	−6.57898e7	−0.0957444
$93$	−2.43056e8	−0.336925
$94$	2.07642e8	0.274309
$95$	1.46890e9	1.85027
$96$	2.72893e8	0.327923
$97$	−1.48537e9	−1.70358	−0.851790	−	0.523884i	$-0.824483\pi$
$97$	−1.48537e9	−1.70358	−0.851790	+	0.523884i	$0.824483\pi$
$98$	8.91532e8	0.976382
$99$	−3.79556e8	−0.397116
$100$	−4.38401e7	−0.0438401
$101$	−5.89236e8	−0.563435	−0.281717	−	0.959497i	$-0.590904\pi$
$101$	−5.89236e8	−0.563435	−0.281717	+	0.959497i	$0.590904\pi$
$102$	8.91774e8	0.815744
$103$	1.70244e9	1.49041	0.745203	−	0.666838i	$-0.232353\pi$
$103$	1.70244e9	1.49041	0.745203	+	0.666838i	$0.232353\pi$
$104$	0	0
$105$	1.39959e9	1.12370
$106$	−9.42376e8	−0.725016
$107$	−1.70949e8	−0.126078	−0.0630389	−	0.998011i	$-0.520079\pi$
$107$	−1.70949e8	−0.126078	−0.0630389	+	0.998011i	$0.520079\pi$
$108$	3.55146e8	0.251189
$109$	1.55941e9	1.05813	0.529066	−	0.848581i	$-0.322542\pi$
$109$	1.55941e9	1.05813	0.529066	+	0.848581i	$0.322542\pi$
$110$	−1.16737e9	−0.760222
$111$	7.56536e8	0.473016
$112$	1.71429e9	1.02945
$113$	−1.21786e9	−0.702659	−0.351330	−	0.936252i	$-0.614270\pi$
$113$	−1.21786e9	−0.702659	−0.351330	+	0.936252i	$0.614270\pi$
$114$	−1.89932e9	−1.05324
$115$	−8.26396e8	−0.440603
$116$	−4.91032e8	−0.251796
$117$	0	0
$118$	2.20748e9	1.04816
$119$	−4.19004e9	−1.91539
$120$	1.89522e9	0.834343
$121$	−8.51959e8	−0.361314
$122$	1.36793e9	0.559042
$123$	−4.44834e8	−0.175237
$124$	2.95860e8	0.112380
$125$	2.42105e9	0.886971
$126$	1.78744e9	0.631777
$127$	2.82775e9	0.964548	0.482274	−	0.876020i	$-0.339811\pi$
$127$	2.82775e9	0.964548	0.482274	+	0.876020i	$0.339811\pi$
$128$	1.54506e9	0.508747
$129$	1.62014e9	0.515109
$130$	0	0
$131$	1.69529e9	0.502947	0.251473	−	0.967864i	$-0.419085\pi$
$131$	1.69529e9	0.502947	0.251473	+	0.967864i	$0.419085\pi$
$132$	−4.67770e8	−0.134106
$133$	8.92406e9	2.47303
$134$	5.57347e9	1.49332
$135$	4.46104e9	1.15594
$136$	−5.67384e9	−1.42217
$137$	−2.91520e9	−0.707009	−0.353505	−	0.935433i	$-0.615010\pi$
$137$	−2.91520e9	−0.707009	−0.353505	+	0.935433i	$0.615010\pi$
$138$	1.06855e9	0.250807
$139$	−9.42169e7	−0.0214073	−0.0107037	−	0.999943i	$-0.503407\pi$
$139$	−9.42169e7	−0.0214073	−0.0107037	+	0.999943i	$0.503407\pi$
$140$	−1.70365e9	−0.374805
$141$	1.04513e9	0.222681
$142$	−6.84914e9	−1.41364
$143$	0	0
$144$	1.81384e9	0.351534
$145$	−6.16794e9	−1.15873
$146$	3.48196e9	0.634213
$147$	4.48737e9	0.792617
$148$	−9.20894e8	−0.157773
$149$	−9.49232e9	−1.57774	−0.788868	−	0.614563i	$-0.789332\pi$
$149$	−9.49232e9	−1.57774	−0.788868	+	0.614563i	$0.789332\pi$
$150$	7.12045e8	0.114841
$151$	−3.58560e9	−0.561262	−0.280631	−	0.959816i	$-0.590544\pi$
$151$	−3.58560e9	−0.561262	−0.280631	+	0.959816i	$0.590544\pi$
$152$	1.20843e10	1.83622
$153$	−4.43336e9	−0.654065
$154$	−7.09215e9	−1.01610
$155$	3.71635e9	0.517159
$156$	0	0
$157$	1.18061e10	1.55081	0.775403	−	0.631467i	$-0.217547\pi$
$157$	1.18061e10	1.55081	0.775403	+	0.631467i	$0.217547\pi$
$158$	−4.67130e9	−0.596323
$159$	−4.74328e9	−0.588561
$160$	−4.17255e9	−0.503341
$161$	−5.02063e9	−0.588900
$162$	−1.96221e9	−0.223835
$163$	5.11424e9	0.567462	0.283731	−	0.958904i	$-0.408428\pi$
$163$	5.11424e9	0.567462	0.283731	+	0.958904i	$0.408428\pi$
$164$	5.41474e8	0.0584494
$165$	−5.87573e9	−0.617141
$166$	−5.73811e9	−0.586520
$167$	1.02165e10	1.01643	0.508215	−	0.861230i	$-0.330306\pi$
$167$	1.02165e10	1.01643	0.508215	+	0.861230i	$0.330306\pi$
$168$	1.15141e10	1.11516
$169$	0	0
$170$	−1.36353e10	−1.25212
$171$	9.44227e9	0.844489
$172$	−1.97212e9	−0.171812
$173$	1.44080e10	1.22292	0.611458	−	0.791277i	$-0.290583\pi$
$173$	1.44080e10	1.22292	0.611458	+	0.791277i	$0.290583\pi$
$174$	7.97529e9	0.659591
$175$	−3.34558e9	−0.269650
$176$	−7.19690e9	−0.565377
$177$	1.11109e10	0.850885
$178$	−8.95307e9	−0.668470
$179$	−1.15726e10	−0.842542	−0.421271	−	0.906935i	$-0.638416\pi$
$179$	−1.15726e10	−0.842542	−0.421271	+	0.906935i	$0.638416\pi$
$180$	−1.80258e9	−0.127988
$181$	1.98233e10	1.37284	0.686422	−	0.727203i	$-0.259180\pi$
$181$	1.98233e10	1.37284	0.686422	+	0.727203i	$0.259180\pi$
$182$	0	0
$183$	6.88523e9	0.453825
$184$	−6.79856e9	−0.437257
$185$	−1.15675e10	−0.726050
$186$	−4.80533e9	−0.294384
$187$	1.75905e10	1.05194
$188$	−1.27218e9	−0.0742743
$189$	2.71023e10	1.54500
$190$	2.90408e10	1.61665
$191$	6.27259e9	0.341033	0.170517	−	0.985355i	$-0.445456\pi$
$191$	6.27259e9	0.341033	0.170517	+	0.985355i	$0.445456\pi$
$192$	1.48440e10	0.788306
$193$	−1.05273e10	−0.546147	−0.273073	−	0.961993i	$-0.588040\pi$
$193$	−1.05273e10	−0.546147	−0.273073	+	0.961993i	$0.588040\pi$
$194$	−2.93665e10	−1.48848
$195$	0	0
$196$	−5.46225e9	−0.264374
$197$	3.01663e10	1.42700	0.713499	−	0.700656i	$-0.247109\pi$
$197$	3.01663e10	1.42700	0.713499	+	0.700656i	$0.247109\pi$
$198$	−7.50398e9	−0.346975
$199$	5.92900e9	0.268005	0.134002	−	0.990981i	$-0.457217\pi$
$199$	5.92900e9	0.268005	0.134002	+	0.990981i	$0.457217\pi$
$200$	−4.53033e9	−0.200214
$201$	2.80530e10	1.21227
$202$	−1.16495e10	−0.492294
$203$	−3.74723e10	−1.54874
$204$	−5.46373e9	−0.220878
$205$	6.80154e9	0.268977
$206$	3.36580e10	1.30222
$207$	−5.31218e9	−0.201097
$208$	0	0
$209$	−3.74647e10	−1.35820
$210$	2.76706e10	0.981818
$211$	−3.32122e9	−0.115352	−0.0576762	−	0.998335i	$-0.518369\pi$
$211$	−3.32122e9	−0.115352	−0.0576762	+	0.998335i	$0.518369\pi$
$212$	5.77376e9	0.196312
$213$	−3.44739e10	−1.14758
$214$	−3.37973e9	−0.110159
$215$	−2.47721e10	−0.790659
$216$	3.66999e10	1.14716
$217$	2.25781e10	0.691222
$218$	3.08301e10	0.924531
$219$	1.75258e10	0.514848
$220$	7.15224e9	0.205845
$221$	0	0
$222$	1.49571e10	0.413293
$223$	−6.95646e9	−0.188372	−0.0941860	−	0.995555i	$-0.530025\pi$
$223$	−6.95646e9	−0.188372	−0.0941860	+	0.995555i	$0.530025\pi$
$224$	−2.53497e10	−0.672754
$225$	−3.53985e9	−0.0920797
$226$	−2.40777e10	−0.613940
$227$	1.35171e10	0.337884	0.168942	−	0.985626i	$-0.445965\pi$
$227$	1.35171e10	0.337884	0.168942	+	0.985626i	$0.445965\pi$
$228$	1.16368e10	0.285185
$229$	−6.99795e10	−1.68155	−0.840777	−	0.541381i	$-0.817902\pi$
$229$	−6.99795e10	−1.68155	−0.840777	+	0.541381i	$0.817902\pi$
$230$	−1.63382e10	−0.384972
$231$	−3.56970e10	−0.824856
$232$	−5.07421e10	−1.14993
$233$	1.50692e9	0.0334956	0.0167478	−	0.999860i	$-0.494669\pi$
$233$	1.50692e9	0.0334956	0.0167478	+	0.999860i	$0.494669\pi$
$234$	0	0
$235$	−1.59801e10	−0.341801
$236$	−1.35248e10	−0.283809
$237$	−2.35122e10	−0.484089
$238$	−8.28390e10	−1.67355
$239$	−4.09671e10	−0.812165	−0.406083	−	0.913836i	$-0.633105\pi$
$239$	−4.09671e10	−0.812165	−0.406083	+	0.913836i	$0.633105\pi$
$240$	2.80792e10	0.546304
$241$	−2.37244e8	−0.00453021	−0.00226510	−	0.999997i	$-0.500721\pi$
$241$	−2.37244e8	−0.00453021	−0.00226510	+	0.999997i	$0.500721\pi$
$242$	−1.68436e10	−0.315694
$243$	4.78331e10	0.880036
$244$	−8.38105e9	−0.151371
$245$	−6.86122e10	−1.21662
$246$	−8.79456e9	−0.153111
$247$	0	0
$248$	3.05735e10	0.513230
$249$	−2.88817e10	−0.476131
$250$	4.78653e10	0.774981
$251$	3.12908e9	0.0497605	0.0248802	−	0.999690i	$-0.492080\pi$
$251$	3.12908e9	0.0497605	0.0248802	+	0.999690i	$0.492080\pi$
$252$	−1.09513e10	−0.171066
$253$	2.10775e10	0.323427
$254$	5.59058e10	0.842763
$255$	−6.86308e10	−1.01645
$256$	−4.58281e10	−0.666887
$257$	1.16093e11	1.65999	0.829996	−	0.557769i	$-0.188343\pi$
$257$	1.16093e11	1.65999	0.829996	+	0.557769i	$0.188343\pi$
$258$	3.20309e10	0.450070
$259$	−7.02764e10	−0.970422
$260$	0	0
$261$	−3.96483e10	−0.528861
$262$	3.35165e10	0.439444
$263$	1.45825e10	0.187946	0.0939728	−	0.995575i	$-0.470043\pi$
$263$	1.45825e10	0.187946	0.0939728	+	0.995575i	$0.470043\pi$
$264$	−4.83382e10	−0.612453
$265$	7.25251e10	0.903403
$266$	1.76433e11	2.16078
$267$	−4.50636e10	−0.542657
$268$	−3.41476e10	−0.404346
$269$	−1.38425e11	−1.61187	−0.805935	−	0.592004i	$-0.798337\pi$
$269$	−1.38425e11	−1.61187	−0.805935	+	0.592004i	$0.798337\pi$
$270$	8.81968e10	1.00999
$271$	4.58189e10	0.516039	0.258019	−	0.966140i	$-0.416930\pi$
$271$	4.58189e10	0.516039	0.258019	+	0.966140i	$0.416930\pi$
$272$	−8.40625e10	−0.931198
$273$	0	0
$274$	−5.76347e10	−0.617741
$275$	1.40453e10	0.148093
$276$	−6.54680e9	−0.0679107
$277$	−6.30656e10	−0.643626	−0.321813	−	0.946803i	$-0.604292\pi$
$277$	−6.30656e10	−0.643626	−0.321813	+	0.946803i	$0.604292\pi$
$278$	−1.86271e9	−0.0187044
$279$	2.38891e10	0.236038
$280$	−1.76052e11	−1.71170
$281$	9.53019e10	0.911849	0.455924	−	0.890018i	$-0.349309\pi$
$281$	9.53019e10	0.911849	0.455924	+	0.890018i	$0.349309\pi$
$282$	2.06626e10	0.194565
$283$	−8.82038e10	−0.817426	−0.408713	−	0.912663i	$-0.634022\pi$
$283$	−8.82038e10	−0.817426	−0.408713	+	0.912663i	$0.634022\pi$
$284$	4.19633e10	0.382770
$285$	1.46172e11	1.31238
$286$	0	0
$287$	4.13216e10	0.359508
$288$	−2.68217e10	−0.229731
$289$	8.68761e10	0.732589
$290$	−1.21943e11	−1.01243
$291$	−1.47811e11	−1.20834
$292$	−2.13333e10	−0.171725
$293$	4.53169e10	0.359216	0.179608	−	0.983738i	$-0.442517\pi$
$293$	4.53169e10	0.359216	0.179608	+	0.983738i	$0.442517\pi$
$294$	8.87172e10	0.692540
$295$	−1.69887e11	−1.30605
$296$	−9.51630e10	−0.720535
$297$	−1.13780e11	−0.848522
$298$	−1.87667e11	−1.37853
$299$	0	0
$300$	−4.36256e9	−0.0310954
$301$	−1.50499e11	−1.05678
$302$	−7.08889e10	−0.490396
$303$	−5.86355e10	−0.399640
$304$	1.79038e11	1.20231
$305$	−1.05276e11	−0.696592
$306$	−8.76494e10	−0.571482
$307$	−2.27379e11	−1.46093	−0.730463	−	0.682953i	$-0.760695\pi$
$307$	−2.27379e11	−1.46093	−0.730463	+	0.682953i	$0.760695\pi$
$308$	4.34522e10	0.275127
$309$	1.69411e11	1.05713
$310$	7.34738e10	0.451861
$311$	1.56295e11	0.947377	0.473688	−	0.880693i	$-0.342922\pi$
$311$	1.56295e11	0.947377	0.473688	+	0.880693i	$0.342922\pi$
$312$	0	0
$313$	−9.62686e10	−0.566937	−0.283469	−	0.958982i	$-0.591485\pi$
$313$	−9.62686e10	−0.566937	−0.283469	+	0.958982i	$0.591485\pi$
$314$	2.33411e11	1.35500
$315$	−1.37561e11	−0.787223
$316$	2.86202e10	0.161466
$317$	−8.54722e9	−0.0475399	−0.0237699	−	0.999717i	$-0.507567\pi$
$317$	−8.54722e9	−0.0475399	−0.0237699	+	0.999717i	$0.507567\pi$
$318$	−9.37767e10	−0.514248
$319$	1.57315e11	0.850574
$320$	−2.26966e11	−1.21000
$321$	−1.70113e10	−0.0894260
$322$	−9.92601e10	−0.514545
$323$	−4.37602e11	−2.23701
$324$	1.20221e10	0.0606075
$325$	0	0
$326$	1.01111e11	0.495813
$327$	1.55178e11	0.750525
$328$	5.59546e10	0.266934
$329$	−9.70843e10	−0.456844
$330$	−1.16166e11	−0.539219
$331$	3.23832e11	1.48284	0.741419	−	0.671042i	$-0.234153\pi$
$331$	3.23832e11	1.48284	0.741419	+	0.671042i	$0.234153\pi$
$332$	3.51563e10	0.158811
$333$	−7.43573e10	−0.331379
$334$	2.01984e11	0.888094
$335$	−4.28933e11	−1.86075
$336$	1.70591e11	0.730178
$337$	1.16577e11	0.492356	0.246178	−	0.969225i	$-0.420825\pi$
$337$	1.16577e11	0.492356	0.246178	+	0.969225i	$0.420825\pi$
$338$	0	0
$339$	−1.21190e11	−0.498391
$340$	8.35408e10	0.339034
$341$	−9.47866e10	−0.379623
$342$	1.86678e11	0.737862
$343$	−4.38212e10	−0.170947
$344$	−2.03794e11	−0.784654
$345$	−8.22354e10	−0.312516
$346$	2.84853e11	1.06851
$347$	3.91173e11	1.44839	0.724197	−	0.689593i	$-0.242211\pi$
$347$	3.91173e11	1.44839	0.724197	+	0.689593i	$0.242211\pi$
$348$	−4.88631e10	−0.178597
$349$	1.52168e11	0.549046	0.274523	−	0.961581i	$-0.411480\pi$
$349$	1.52168e11	0.549046	0.274523	+	0.961581i	$0.411480\pi$
$350$	−6.61435e10	−0.235603
$351$	0	0
$352$	1.06422e11	0.369480
$353$	−3.05234e11	−1.04628	−0.523138	−	0.852248i	$-0.675239\pi$
$353$	−3.05234e11	−1.04628	−0.523138	+	0.852248i	$0.675239\pi$
$354$	2.19668e11	0.743451
$355$	5.27108e11	1.76146
$356$	5.48537e10	0.181001
$357$	−4.16955e11	−1.35857
$358$	−2.28795e11	−0.736161
$359$	4.59883e11	1.46124	0.730622	−	0.682782i	$-0.239230\pi$
$359$	4.59883e11	1.46124	0.730622	+	0.682782i	$0.239230\pi$
$360$	−1.86275e11	−0.584511
$361$	6.09329e11	1.88829
$362$	3.91914e11	1.19951
$363$	−8.47792e10	−0.256277
$364$	0	0
$365$	−2.67971e11	−0.790259
$366$	1.36124e11	0.396524
$367$	−4.76454e10	−0.137096	−0.0685478	−	0.997648i	$-0.521837\pi$
$367$	−4.76454e10	−0.137096	−0.0685478	+	0.997648i	$0.521837\pi$
$368$	−1.00726e11	−0.286304
$369$	4.37212e10	0.122765
$370$	−2.28695e11	−0.634378
$371$	4.40614e11	1.20747
$372$	2.94413e10	0.0797102
$373$	−4.45233e10	−0.119096	−0.0595481	−	0.998225i	$-0.518966\pi$
$373$	−4.45233e10	−0.119096	−0.0595481	+	0.998225i	$0.518966\pi$
$374$	3.47772e11	0.919121
$375$	2.40921e11	0.629122
$376$	−1.31464e11	−0.339205
$377$	0	0
$378$	5.35825e11	1.34993
$379$	1.15114e11	0.286584	0.143292	−	0.989680i	$-0.454231\pi$
$379$	1.15114e11	0.286584	0.143292	+	0.989680i	$0.454231\pi$
$380$	−1.77927e11	−0.437740
$381$	2.81392e11	0.684146
$382$	1.24012e11	0.297974
$383$	5.21296e11	1.23791	0.618957	−	0.785425i	$-0.287556\pi$
$383$	5.21296e11	1.23791	0.618957	+	0.785425i	$0.287556\pi$
$384$	1.53751e11	0.360850
$385$	5.45810e11	1.26610
$386$	−2.08129e11	−0.477189
$387$	−1.59238e11	−0.360867
$388$	1.79923e11	0.403035
$389$	1.68464e11	0.373021	0.186510	−	0.982453i	$-0.440282\pi$
$389$	1.68464e11	0.373021	0.186510	+	0.982453i	$0.440282\pi$
$390$	0	0
$391$	2.46193e11	0.532697
$392$	−5.64456e11	−1.20738
$393$	1.68699e11	0.356736
$394$	5.96400e11	1.24682
$395$	3.59503e11	0.743045
$396$	4.59755e10	0.0939502
$397$	6.78075e11	1.37000	0.684999	−	0.728544i	$-0.259802\pi$
$397$	6.78075e11	1.37000	0.684999	+	0.728544i	$0.259802\pi$
$398$	1.17219e11	0.234166
$399$	8.88041e11	1.75410
$400$	−6.71205e10	−0.131095
$401$	−3.34719e11	−0.646443	−0.323221	−	0.946323i	$-0.604766\pi$
$401$	−3.34719e11	−0.646443	−0.323221	+	0.946323i	$0.604766\pi$
$402$	5.54621e11	1.05920
$403$	0	0
$404$	7.13740e10	0.133298
$405$	1.51011e11	0.278908
$406$	−7.40843e11	−1.35319
$407$	2.95033e11	0.532961
$408$	−5.64609e11	−1.00873
$409$	−6.86952e11	−1.21387	−0.606934	−	0.794752i	$-0.707601\pi$
$409$	−6.86952e11	−1.21387	−0.606934	+	0.794752i	$0.707601\pi$
$410$	1.34469e11	0.235015
$411$	−2.90094e11	−0.501476
$412$	−2.06216e11	−0.352602
$413$	−1.03212e12	−1.74564
$414$	−1.05024e11	−0.175706
$415$	4.41604e11	0.730830
$416$	0	0
$417$	−9.37561e9	−0.0151840
$418$	−7.40694e11	−1.18671
$419$	9.09694e11	1.44189	0.720945	−	0.692992i	$-0.243708\pi$
$419$	9.09694e11	1.44189	0.720945	+	0.692992i	$0.243708\pi$
$420$	−1.69532e11	−0.265846
$421$	7.71291e11	1.19660	0.598300	−	0.801272i	$-0.295843\pi$
$421$	7.71291e11	1.19660	0.598300	+	0.801272i	$0.295843\pi$
$422$	−6.56620e10	−0.100788
$423$	−1.02722e11	−0.156003
$424$	5.96646e11	0.896542
$425$	1.64055e11	0.243915
$426$	−6.81564e11	−1.00268
$427$	−6.39585e11	−0.931049
$428$	2.07070e10	0.0298277
$429$	0	0
$430$	−4.89755e11	−0.690829
$431$	−5.86965e11	−0.819341	−0.409671	−	0.912233i	$-0.634356\pi$
$431$	−5.86965e11	−0.819341	−0.409671	+	0.912233i	$0.634356\pi$
$432$	5.43739e11	0.751127
$433$	6.24963e11	0.854395	0.427197	−	0.904158i	$-0.359501\pi$
$433$	6.24963e11	0.854395	0.427197	+	0.904158i	$0.359501\pi$
$434$	4.46378e11	0.603947
$435$	−6.13777e11	−0.821881
$436$	−1.88890e11	−0.250334
$437$	−5.24348e11	−0.687786
$438$	3.46493e11	0.449842
$439$	4.25543e11	0.546831	0.273416	−	0.961896i	$-0.411847\pi$
$439$	4.25543e11	0.546831	0.273416	+	0.961896i	$0.411847\pi$
$440$	7.39095e11	0.940076
$441$	−4.41048e11	−0.555280
$442$	0	0
$443$	1.27293e12	1.57032	0.785161	−	0.619292i	$-0.212580\pi$
$443$	1.27293e12	1.57032	0.785161	+	0.619292i	$0.212580\pi$
$444$	−9.16390e10	−0.111907
$445$	6.89026e11	0.832944
$446$	−1.37532e11	−0.164588
$447$	−9.44590e11	−1.11908
$448$	−1.37889e12	−1.61726
$449$	−9.55517e11	−1.10951	−0.554753	−	0.832015i	$-0.687187\pi$
$449$	−9.55517e11	−1.10951	−0.554753	+	0.832015i	$0.687187\pi$
$450$	−6.99845e10	−0.0804536
$451$	−1.73475e11	−0.197444
$452$	1.47519e11	0.166236
$453$	−3.56807e11	−0.398099
$454$	2.67240e11	0.295223
$455$	0	0
$456$	1.20252e12	1.30242
$457$	1.64747e12	1.76683	0.883416	−	0.468589i	$-0.155237\pi$
$457$	1.64747e12	1.76683	0.883416	+	0.468589i	$0.155237\pi$
$458$	−1.38353e12	−1.46924
$459$	−1.32900e12	−1.39755
$460$	1.00101e11	0.104239
$461$	−2.01578e11	−0.207869	−0.103934	−	0.994584i	$-0.533143\pi$
$461$	−2.01578e11	−0.207869	−0.103934	+	0.994584i	$0.533143\pi$
$462$	−7.05746e11	−0.720708
$463$	−7.27788e11	−0.736021	−0.368011	−	0.929822i	$-0.619961\pi$
$463$	−7.27788e11	−0.736021	−0.368011	+	0.929822i	$0.619961\pi$
$464$	−7.51785e11	−0.752945
$465$	3.69817e11	0.366816
$466$	2.97924e10	0.0292664
$467$	−1.74702e12	−1.69970	−0.849851	−	0.527024i	$-0.823308\pi$
$467$	−1.74702e12	−1.69970	−0.849851	+	0.527024i	$0.823308\pi$
$468$	0	0
$469$	−2.60591e12	−2.48704
$470$	−3.15933e11	−0.298645
$471$	1.17483e12	1.09997
$472$	−1.39762e12	−1.29613
$473$	6.31819e11	0.580387
$474$	−4.64846e11	−0.422967
$475$	−3.49408e11	−0.314928
$476$	5.07538e11	0.453145
$477$	4.66200e11	0.412325
$478$	−8.09937e11	−0.709620
$479$	−3.24829e11	−0.281933	−0.140966	−	0.990014i	$-0.545021\pi$
$479$	−3.24829e11	−0.281933	−0.140966	+	0.990014i	$0.545021\pi$
$480$	−4.15215e11	−0.357015
$481$	0	0
$482$	−4.69041e9	−0.00395822
$483$	−4.99608e11	−0.417702
$484$	1.03198e11	0.0854801
$485$	2.26004e12	1.85472
$486$	9.45682e11	0.768921
$487$	1.78478e12	1.43782	0.718911	−	0.695102i	$-0.244641\pi$
$487$	1.78478e12	1.43782	0.718911	+	0.695102i	$0.244641\pi$
$488$	−8.66077e11	−0.691302
$489$	5.08923e11	0.402496
$490$	−1.35649e12	−1.06300
$491$	6.75975e11	0.524884	0.262442	−	0.964948i	$-0.415472\pi$
$491$	6.75975e11	0.524884	0.262442	+	0.964948i	$0.415472\pi$
$492$	5.38826e10	0.0414577
$493$	1.83750e12	1.40093
$494$	0	0
$495$	5.77505e11	0.432347
$496$	4.52971e11	0.336049
$497$	3.20236e12	2.35432
$498$	−5.71005e11	−0.416014
$499$	1.75261e12	1.26542	0.632708	−	0.774390i	$-0.281943\pi$
$499$	1.75261e12	1.26542	0.632708	+	0.774390i	$0.281943\pi$
$500$	−2.93262e11	−0.209841
$501$	1.01665e12	0.720945
$502$	6.18632e10	0.0434776
$503$	−3.96462e11	−0.276151	−0.138075	−	0.990422i	$-0.544092\pi$
$503$	−3.96462e11	−0.276151	−0.138075	+	0.990422i	$0.544092\pi$
$504$	−1.13168e12	−0.781244
$505$	8.96541e11	0.613421
$506$	4.16711e11	0.282591
$507$	0	0
$508$	−3.42524e11	−0.228194
$509$	−2.66748e11	−0.176145	−0.0880727	−	0.996114i	$-0.528071\pi$
$509$	−2.66748e11	−0.176145	−0.0880727	+	0.996114i	$0.528071\pi$
$510$	−1.35686e12	−0.888115
$511$	−1.62801e12	−1.05624
$512$	−1.69711e12	−1.09143
$513$	2.83053e12	1.80443
$514$	2.29520e12	1.45040
$515$	−2.59031e12	−1.62263
$516$	−1.96247e11	−0.121865
$517$	4.07577e11	0.250901
$518$	−1.38940e12	−0.847895
$519$	1.43375e12	0.867405
$520$	0	0
$521$	2.43114e12	1.44557	0.722787	−	0.691071i	$-0.242861\pi$
$521$	2.43114e12	1.44557	0.722787	+	0.691071i	$0.242861\pi$
$522$	−7.83864e11	−0.462087
$523$	−3.14301e12	−1.83691	−0.918456	−	0.395523i	$-0.870563\pi$
$523$	−3.14301e12	−1.83691	−0.918456	+	0.395523i	$0.870563\pi$
$524$	−2.05349e11	−0.118988
$525$	−3.32921e11	−0.191260
$526$	2.88303e11	0.164215
$527$	−1.10714e12	−0.625253
$528$	−7.16170e11	−0.401017
$529$	−1.50616e12	−0.836218
$530$	1.43385e12	0.789338
$531$	−1.09206e12	−0.596100
$532$	−1.08097e12	−0.585074
$533$	0	0
$534$	−8.90928e11	−0.474140
$535$	2.60104e11	0.137263
$536$	−3.52873e12	−1.84662
$537$	−1.15160e12	−0.597608
$538$	−2.73673e12	−1.40835
$539$	1.74997e12	0.893063
$540$	−5.40365e11	−0.273473
$541$	1.31906e12	0.662028	0.331014	−	0.943626i	$-0.392609\pi$
$541$	1.31906e12	0.662028	0.331014	+	0.943626i	$0.392609\pi$
$542$	9.05859e11	0.450883
$543$	1.97263e12	0.973748
$544$	1.24305e12	0.608547
$545$	−2.37268e12	−1.15201
$546$	0	0
$547$	−2.73121e12	−1.30440	−0.652202	−	0.758045i	$-0.726155\pi$
$547$	−2.73121e12	−1.30440	−0.652202	+	0.758045i	$0.726155\pi$
$548$	3.53117e11	0.167265
$549$	−6.76725e11	−0.317934
$550$	2.77682e11	0.129394
$551$	−3.91355e12	−1.80879
$552$	−6.76531e11	−0.310143
$553$	2.18410e12	0.993137
$554$	−1.24684e12	−0.562361
$555$	−1.15109e12	−0.514981
$556$	1.14125e10	0.00506458
$557$	−2.75823e11	−0.121418	−0.0607088	−	0.998156i	$-0.519336\pi$
$557$	−2.75823e11	−0.121418	−0.0607088	+	0.998156i	$0.519336\pi$
$558$	4.72299e11	0.206235
$559$	0	0
$560$	−2.60835e12	−1.12078
$561$	1.75045e12	0.746133
$562$	1.88416e12	0.796717
$563$	3.43268e8	0.000143994 0	7.19972e−5	−	1.00000i	$-0.499977\pi$
$563$	3.43268e8	0.000143994 0	7.19972e−5		1.00000i	$0.499977\pi$
$564$	−1.26596e11	−0.0526822
$565$	1.85301e12	0.764998
$566$	−1.74383e12	−0.714216
$567$	9.17443e11	0.372782
$568$	4.33639e12	1.74808
$569$	2.17713e12	0.870721	0.435361	−	0.900256i	$-0.356621\pi$
$569$	2.17713e12	0.870721	0.435361	+	0.900256i	$0.356621\pi$
$570$	2.88988e12	1.14668
$571$	−2.35383e12	−0.926641	−0.463321	−	0.886191i	$-0.653342\pi$
$571$	−2.35383e12	−0.926641	−0.463321	+	0.886191i	$0.653342\pi$
$572$	0	0
$573$	6.24191e11	0.241892
$574$	8.16947e11	0.314116
$575$	1.96575e11	0.0749934
$576$	−1.45896e12	−0.552259
$577$	2.87096e12	1.07829	0.539145	−	0.842213i	$-0.318747\pi$
$577$	2.87096e12	1.07829	0.539145	+	0.842213i	$0.318747\pi$
$578$	1.71758e12	0.640091
$579$	−1.04758e12	−0.387377
$580$	7.47120e11	0.274135
$581$	2.68289e12	0.976811
$582$	−2.92228e12	−1.05577
$583$	−1.84977e12	−0.663148
$584$	−2.20453e12	−0.784256
$585$	0	0
$586$	8.95935e11	0.313861
$587$	2.30595e12	0.801639	0.400820	−	0.916157i	$-0.368725\pi$
$587$	2.30595e12	0.801639	0.400820	+	0.916157i	$0.368725\pi$
$588$	−5.43553e11	−0.187518
$589$	2.35802e12	0.807289
$590$	−3.35874e12	−1.14115
$591$	3.00187e12	1.01216
$592$	−1.40992e12	−0.471787
$593$	−1.61589e12	−0.536620	−0.268310	−	0.963333i	$-0.586465\pi$
$593$	−1.61589e12	−0.536620	−0.268310	+	0.963333i	$0.586465\pi$
$594$	−2.24949e12	−0.741386
$595$	6.37527e12	2.08532
$596$	1.14980e12	0.373263
$597$	5.90000e11	0.190094
$598$	0	0
$599$	−4.17127e12	−1.32388	−0.661938	−	0.749558i	$-0.730266\pi$
$599$	−4.17127e12	−1.32388	−0.661938	+	0.749558i	$0.730266\pi$
$600$	−4.50817e11	−0.142010
$601$	2.82549e12	0.883404	0.441702	−	0.897162i	$-0.354375\pi$
$601$	2.82549e12	0.883404	0.441702	+	0.897162i	$0.354375\pi$
$602$	−2.97542e12	−0.923346
$603$	−2.75724e12	−0.849270
$604$	4.34323e11	0.132784
$605$	1.29628e12	0.393369
$606$	−1.15925e12	−0.349180
$607$	−1.71393e11	−0.0512441	−0.0256220	−	0.999672i	$-0.508157\pi$
$607$	−1.71393e11	−0.0512441	−0.0256220	+	0.999672i	$0.508157\pi$
$608$	−2.64748e12	−0.785719
$609$	−3.72890e12	−1.09851
$610$	−2.08135e12	−0.608640
$611$	0	0
$612$	5.37011e11	0.154740
$613$	5.97398e12	1.70880	0.854401	−	0.519615i	$-0.173925\pi$
$613$	5.97398e12	1.70880	0.854401	+	0.519615i	$0.173925\pi$
$614$	−4.49539e12	−1.27647
$615$	6.76827e11	0.190783
$616$	4.49025e12	1.25648
$617$	5.82013e11	0.161678	0.0808388	−	0.996727i	$-0.474240\pi$
$617$	5.82013e11	0.161678	0.0808388	+	0.996727i	$0.474240\pi$
$618$	3.34934e12	0.923657
$619$	−6.13432e11	−0.167942	−0.0839709	−	0.996468i	$-0.526760\pi$
$619$	−6.13432e11	−0.167942	−0.0839709	+	0.996468i	$0.526760\pi$
$620$	−4.50160e11	−0.122350
$621$	−1.59244e12	−0.429687
$622$	3.09002e12	0.827759
$623$	4.18607e12	1.11329
$624$	0	0
$625$	−4.39059e12	−1.15097
$626$	−1.90327e12	−0.495355
$627$	−3.72815e12	−0.963362
$628$	−1.43007e12	−0.366892
$629$	3.44609e12	0.877807
$630$	−2.71964e12	−0.687827
$631$	6.56659e11	0.164895	0.0824475	−	0.996595i	$-0.473726\pi$
$631$	6.56659e11	0.164895	0.0824475	+	0.996595i	$0.473726\pi$
$632$	2.95754e12	0.737401
$633$	−3.30498e11	−0.0818186
$634$	−1.68982e11	−0.0415374
$635$	−4.30250e12	−1.05012
$636$	5.74552e11	0.139243
$637$	0	0
$638$	3.11019e12	0.743179
$639$	3.38832e12	0.803952
$640$	−2.35086e12	−0.553882
$641$	−4.68212e12	−1.09542	−0.547711	−	0.836668i	$-0.684501\pi$
$641$	−4.68212e12	−1.09542	−0.547711	+	0.836668i	$0.684501\pi$
$642$	−3.36320e11	−0.0781349
$643$	−3.85913e12	−0.890307	−0.445153	−	0.895454i	$-0.646851\pi$
$643$	−3.85913e12	−0.890307	−0.445153	+	0.895454i	$0.646851\pi$
$644$	6.08147e11	0.139323
$645$	−2.46509e12	−0.560808
$646$	−8.65159e12	−1.95456
$647$	−2.95801e11	−0.0663636	−0.0331818	−	0.999449i	$-0.510564\pi$
$647$	−2.95801e11	−0.0663636	−0.0331818	+	0.999449i	$0.510564\pi$
$648$	1.24233e12	0.276790
$649$	4.33302e12	0.958716
$650$	0	0
$651$	2.24676e12	0.490278
$652$	−6.19486e11	−0.134251
$653$	2.88888e12	0.621755	0.310878	−	0.950450i	$-0.399377\pi$
$653$	2.88888e12	0.621755	0.310878	+	0.950450i	$0.399377\pi$
$654$	3.06794e12	0.655762
$655$	−2.57943e12	−0.547567
$656$	8.29013e11	0.174781
$657$	−1.72255e12	−0.360684
$658$	−1.91940e12	−0.399162
$659$	8.70413e11	0.179780	0.0898899	−	0.995952i	$-0.471348\pi$
$659$	8.70413e11	0.179780	0.0898899	+	0.995952i	$0.471348\pi$
$660$	7.11725e11	0.146004
$661$	5.17707e12	1.05482	0.527409	−	0.849612i	$-0.323164\pi$
$661$	5.17707e12	1.05482	0.527409	+	0.849612i	$0.323164\pi$
$662$	6.40230e12	1.29561
$663$	0	0
$664$	3.63297e12	0.725279
$665$	−1.35782e13	−2.69244
$666$	−1.47008e12	−0.289538
$667$	2.20175e12	0.430726
$668$	−1.23752e12	−0.240468
$669$	−6.92244e11	−0.133611
$670$	−8.48020e12	−1.62581
$671$	2.68509e12	0.511337
$672$	−2.52257e12	−0.477179
$673$	−9.85716e12	−1.85218	−0.926092	−	0.377297i	$-0.876854\pi$
$673$	−9.85716e12	−1.85218	−0.926092	+	0.377297i	$0.876854\pi$
$674$	2.30479e12	0.430191
$675$	−1.06115e12	−0.196748
$676$	0	0
$677$	8.69916e12	1.59158	0.795790	−	0.605573i	$-0.207056\pi$
$677$	8.69916e12	1.59158	0.795790	+	0.605573i	$0.207056\pi$
$678$	−2.39599e12	−0.435463
$679$	1.37305e13	2.47897
$680$	8.63291e12	1.54834
$681$	1.34510e12	0.239659
$682$	−1.87397e12	−0.331691
$683$	1.53046e12	0.269109	0.134554	−	0.990906i	$-0.457040\pi$
$683$	1.53046e12	0.269109	0.134554	+	0.990906i	$0.457040\pi$
$684$	−1.14374e12	−0.199790
$685$	4.43556e12	0.769734
$686$	−8.66364e11	−0.149363
$687$	−6.96372e12	−1.19271
$688$	−3.01937e12	−0.513769
$689$	0	0
$690$	−1.62583e12	−0.273058
$691$	1.00313e12	0.167380	0.0836902	−	0.996492i	$-0.473329\pi$
$691$	1.00313e12	0.167380	0.0836902	+	0.996492i	$0.473329\pi$
$692$	−1.74524e12	−0.289319
$693$	3.50854e12	0.577865
$694$	7.73367e12	1.26552
$695$	1.43354e11	0.0233065
$696$	−5.04939e12	−0.815638
$697$	−2.02626e12	−0.325198
$698$	3.00842e12	0.479722
$699$	1.49955e11	0.0237581
$700$	4.05249e11	0.0637941
$701$	−6.48282e12	−1.01399	−0.506994	−	0.861949i	$-0.669243\pi$
$701$	−6.48282e12	−1.01399	−0.506994	+	0.861949i	$0.669243\pi$
$702$	0	0
$703$	−7.33957e12	−1.13337
$704$	5.78883e12	0.888206
$705$	−1.59019e12	−0.242437
$706$	−6.03460e12	−0.914171
$707$	5.44678e12	0.819885
$708$	−1.34586e12	−0.201304
$709$	4.20475e12	0.624931	0.312465	−	0.949929i	$-0.398845\pi$
$709$	4.20475e12	0.624931	0.312465	+	0.949929i	$0.398845\pi$
$710$	1.04212e13	1.53905
$711$	2.31093e12	0.339135
$712$	5.66845e12	0.826617
$713$	−1.32661e12	−0.192239
$714$	−8.24338e12	−1.18703
$715$	0	0
$716$	1.40178e12	0.199330
$717$	−4.07667e12	−0.576062
$718$	9.09210e12	1.27674
$719$	−9.00975e12	−1.25728	−0.628641	−	0.777695i	$-0.716389\pi$
$719$	−9.00975e12	−1.25728	−0.628641	+	0.777695i	$0.716389\pi$
$720$	−2.75981e12	−0.382722
$721$	−1.57370e13	−2.16877
$722$	1.20467e13	1.64987
$723$	−2.36083e10	−0.00321324
$724$	−2.40118e12	−0.324790
$725$	1.46717e12	0.197224
$726$	−1.67612e12	−0.223919
$727$	4.23757e12	0.562615	0.281308	−	0.959618i	$-0.409232\pi$
$727$	4.23757e12	0.562615	0.281308	+	0.959618i	$0.409232\pi$
$728$	0	0
$729$	6.71344e12	0.880383
$730$	−5.29790e12	−0.690479
$731$	7.37989e12	0.955920
$732$	−8.34006e11	−0.107367
$733$	4.35002e12	0.556575	0.278288	−	0.960498i	$-0.410233\pi$
$733$	4.35002e12	0.556575	0.278288	+	0.960498i	$0.410233\pi$
$734$	−9.41970e11	−0.119786
$735$	−6.82766e12	−0.862936
$736$	1.48946e12	0.187102
$737$	1.09401e13	1.36589
$738$	8.64387e11	0.107264
$739$	3.68548e11	0.0454563	0.0227281	−	0.999742i	$-0.492765\pi$
$739$	3.68548e11	0.0454563	0.0227281	+	0.999742i	$0.492765\pi$
$740$	1.40117e12	0.171770
$741$	0	0
$742$	8.71113e12	1.05501
$743$	−6.34876e12	−0.764257	−0.382128	−	0.924109i	$-0.624809\pi$
$743$	−6.34876e12	−0.764257	−0.382128	+	0.924109i	$0.624809\pi$
$744$	3.04240e12	0.364030
$745$	1.44428e13	1.71771
$746$	−8.80246e11	−0.104059
$747$	2.83869e12	0.333561
$748$	−2.13073e12	−0.248870
$749$	1.58022e12	0.183463
$750$	4.76312e12	0.549688
$751$	−1.03951e13	−1.19247	−0.596236	−	0.802809i	$-0.703338\pi$
$751$	−1.03951e13	−1.19247	−0.596236	+	0.802809i	$0.703338\pi$
$752$	−1.94775e12	−0.222102
$753$	3.11377e11	0.0352947
$754$	0	0
$755$	5.45560e12	0.611056
$756$	−3.28290e12	−0.365518
$757$	−4.19405e12	−0.464197	−0.232098	−	0.972692i	$-0.574559\pi$
$757$	−4.19405e12	−0.464197	−0.232098	+	0.972692i	$0.574559\pi$
$758$	2.27586e12	0.250399
$759$	2.09744e12	0.229404
$760$	−1.83866e13	−1.99913
$761$	4.72170e12	0.510349	0.255175	−	0.966895i	$-0.417867\pi$
$761$	4.72170e12	0.510349	0.255175	+	0.966895i	$0.417867\pi$
$762$	5.56324e12	0.597765
$763$	−1.44148e13	−1.53975
$764$	−7.59797e11	−0.0806822
$765$	6.74548e12	0.712093
$766$	1.03063e13	1.08161
$767$	0	0
$768$	−4.56040e12	−0.473017
$769$	−1.30977e13	−1.35060	−0.675302	−	0.737541i	$-0.735987\pi$
$769$	−1.30977e13	−1.35060	−0.675302	+	0.737541i	$0.735987\pi$
$770$	1.07909e13	1.10624
$771$	1.15525e13	1.17742
$772$	1.27517e12	0.129208
$773$	−1.08876e11	−0.0109679	−0.00548397	−	0.999985i	$-0.501746\pi$
$773$	−1.08876e11	−0.0109679	−0.00548397	+	0.999985i	$0.501746\pi$
$774$	−3.14820e12	−0.315303
$775$	−8.84009e11	−0.0880236
$776$	1.85928e13	1.84063
$777$	−6.99327e12	−0.688312
$778$	3.33060e12	0.325923
$779$	4.31558e12	0.419875
$780$	0	0
$781$	−1.34441e13	−1.29301
$782$	4.86734e12	0.465438
$783$	−1.18855e13	−1.13002
$784$	−8.36287e12	−0.790556
$785$	−1.79633e13	−1.68839
$786$	3.33526e12	0.311694
$787$	1.73724e13	1.61426	0.807131	−	0.590372i	$-0.201019\pi$
$787$	1.73724e13	1.61426	0.807131	+	0.590372i	$0.201019\pi$
$788$	−3.65403e12	−0.337601
$789$	1.45112e12	0.133308
$790$	7.10753e12	0.649227
$791$	1.12577e13	1.02248
$792$	4.75099e12	0.429063
$793$	0	0
$794$	1.34058e13	1.19702
$795$	7.21704e12	0.640777
$796$	−7.18178e11	−0.0634050
$797$	1.88209e13	1.65226	0.826128	−	0.563483i	$-0.190539\pi$
$797$	1.88209e13	1.65226	0.826128	+	0.563483i	$0.190539\pi$
$798$	1.75570e13	1.53263
$799$	4.76065e12	0.413243
$800$	9.92527e11	0.0856717
$801$	4.42915e12	0.380166
$802$	−6.61753e12	−0.564822
$803$	6.83467e12	0.580093
$804$	−3.39806e12	−0.286799
$805$	7.63904e12	0.641146
$806$	0	0
$807$	−1.37748e13	−1.14329
$808$	7.37562e12	0.608762
$809$	−1.41885e13	−1.16458	−0.582290	−	0.812981i	$-0.697843\pi$
$809$	−1.41885e13	−1.16458	−0.582290	+	0.812981i	$0.697843\pi$
$810$	2.98556e12	0.243693
$811$	2.49677e12	0.202667	0.101334	−	0.994852i	$-0.467689\pi$
$811$	2.49677e12	0.202667	0.101334	+	0.994852i	$0.467689\pi$
$812$	4.53900e12	0.366403
$813$	4.55948e12	0.366022
$814$	5.83292e12	0.465668
$815$	−7.78147e12	−0.617806
$816$	−8.36513e12	−0.660491
$817$	−1.57179e13	−1.23423
$818$	−1.35813e13	−1.06060
$819$	0	0
$820$	−8.23869e11	−0.0636349
$821$	5.97064e12	0.458645	0.229323	−	0.973350i	$-0.426349\pi$
$821$	5.97064e12	0.458645	0.229323	+	0.973350i	$0.426349\pi$
$822$	−5.73528e12	−0.438159
$823$	1.04151e12	0.0791343	0.0395672	−	0.999217i	$-0.487402\pi$
$823$	1.04151e12	0.0791343	0.0395672	+	0.999217i	$0.487402\pi$
$824$	−2.13099e13	−1.61031
$825$	1.39766e12	0.105041
$826$	−2.04055e13	−1.52523
$827$	−4.22577e12	−0.314146	−0.157073	−	0.987587i	$-0.550206\pi$
$827$	−4.22577e12	−0.314146	−0.157073	+	0.987587i	$0.550206\pi$
$828$	6.43462e11	0.0475759
$829$	7.38832e12	0.543313	0.271656	−	0.962394i	$-0.412429\pi$
$829$	7.38832e12	0.543313	0.271656	+	0.962394i	$0.412429\pi$
$830$	8.73071e12	0.638555
$831$	−6.27572e12	−0.456519
$832$	0	0
$833$	2.04404e13	1.47091
$834$	−1.85360e11	−0.0132669
$835$	−1.55447e13	−1.10661
$836$	4.53809e12	0.321325
$837$	7.16130e12	0.504345
$838$	1.79851e13	1.25983
$839$	7.56034e12	0.526760	0.263380	−	0.964692i	$-0.415163\pi$
$839$	7.56034e12	0.526760	0.263380	+	0.964692i	$0.415163\pi$
$840$	−1.75190e13	−1.21410
$841$	1.92594e12	0.132758
$842$	1.52488e13	1.04552
$843$	9.48357e12	0.646767
$844$	4.02299e11	0.0272902
$845$	0	0
$846$	−2.03086e12	−0.136305
$847$	7.87534e12	0.525768
$848$	8.83979e12	0.587031
$849$	−8.77724e12	−0.579794
$850$	3.24343e12	0.213118
$851$	4.12921e12	0.269888
$852$	4.17581e12	0.271495
$853$	1.99879e13	1.29269	0.646347	−	0.763043i	$-0.276296\pi$
$853$	1.99879e13	1.29269	0.646347	+	0.763043i	$0.276296\pi$
$854$	−1.26449e13	−0.813494
$855$	−1.43667e13	−0.919410
$856$	2.13981e12	0.136221
$857$	8.41249e12	0.532735	0.266367	−	0.963872i	$-0.414177\pi$
$857$	8.41249e12	0.532735	0.266367	+	0.963872i	$0.414177\pi$
$858$	0	0
$859$	3.81635e12	0.239155	0.119577	−	0.992825i	$-0.461846\pi$
$859$	3.81635e12	0.239155	0.119577	+	0.992825i	$0.461846\pi$
$860$	3.00063e12	0.187055
$861$	4.11195e12	0.254996
$862$	−1.16046e13	−0.715890
$863$	1.53354e13	0.941121	0.470560	−	0.882368i	$-0.344052\pi$
$863$	1.53354e13	0.941121	0.470560	+	0.882368i	$0.344052\pi$
$864$	−8.04040e12	−0.490869
$865$	−2.19222e13	−1.33141
$866$	1.23558e13	0.746517
$867$	8.64512e12	0.519619
$868$	−2.73487e12	−0.163530
$869$	−9.16923e12	−0.545436
$870$	−1.21346e13	−0.718109
$871$	0	0
$872$	−1.95195e13	−1.14326
$873$	1.45278e13	0.846517
$874$	−1.03666e13	−0.600945
$875$	−2.23797e13	−1.29068
$876$	−2.12289e12	−0.121803
$877$	−1.52103e12	−0.0868239	−0.0434119	−	0.999057i	$-0.513823\pi$
$877$	−1.52103e12	−0.0868239	−0.0434119	+	0.999057i	$0.513823\pi$
$878$	8.41317e12	0.477787
$879$	4.50952e12	0.254789
$880$	1.09503e13	0.615536
$881$	−8.92983e11	−0.0499403	−0.0249702	−	0.999688i	$-0.507949\pi$
$881$	−8.92983e11	−0.0499403	−0.0249702	+	0.999688i	$0.507949\pi$
$882$	−8.71970e12	−0.485169
$883$	1.69912e13	0.940592	0.470296	−	0.882509i	$-0.344147\pi$
$883$	1.69912e13	0.940592	0.470296	+	0.882509i	$0.344147\pi$
$884$	0	0
$885$	−1.69056e13	−0.926374
$886$	2.51664e13	1.37205
$887$	6.57514e12	0.356655	0.178328	−	0.983971i	$-0.442931\pi$
$887$	6.57514e12	0.356655	0.178328	+	0.983971i	$0.442931\pi$
$888$	−9.46975e12	−0.511070
$889$	−2.61391e13	−1.40357
$890$	1.36224e13	0.727775
$891$	−3.85158e12	−0.204734
$892$	8.42634e11	0.0445653
$893$	−1.01394e13	−0.533555
$894$	−1.86750e13	−0.977779
$895$	1.76080e13	0.917290
$896$	−1.42823e13	−0.740306
$897$	0	0
$898$	−1.88910e13	−0.969418
$899$	−9.90138e12	−0.505565
$900$	4.28781e11	0.0217843
$901$	−2.16061e13	−1.09223
$902$	−3.42968e12	−0.172514
$903$	−1.49762e13	−0.749563
$904$	1.52443e13	0.759187
$905$	−3.01617e13	−1.49464
$906$	−7.05422e12	−0.347834
$907$	1.24839e13	0.612517	0.306258	−	0.951948i	$-0.400923\pi$
$907$	1.24839e13	0.612517	0.306258	+	0.951948i	$0.400923\pi$
$908$	−1.63733e12	−0.0799372
$909$	5.76308e12	0.279973
$910$	0	0
$911$	−1.51870e13	−0.730534	−0.365267	−	0.930903i	$-0.619022\pi$
$911$	−1.51870e13	−0.730534	−0.365267	+	0.930903i	$0.619022\pi$
$912$	1.78163e13	0.852786
$913$	−1.12632e13	−0.536470
$914$	3.25713e13	1.54375
$915$	−1.04761e13	−0.494087
$916$	8.47659e12	0.397825
$917$	−1.56709e13	−0.731866
$918$	−2.62749e13	−1.22109
$919$	2.15334e13	0.995849	0.497925	−	0.867220i	$-0.334096\pi$
$919$	2.15334e13	0.995849	0.497925	+	0.867220i	$0.334096\pi$
$920$	1.03442e13	0.476049
$921$	−2.26267e13	−1.03622
$922$	−3.98529e12	−0.181623
$923$	0	0
$924$	4.32397e12	0.195146
$925$	2.75156e12	0.123578
$926$	−1.43887e13	−0.643090
$927$	−1.66509e13	−0.740590
$928$	1.11168e13	0.492057
$929$	1.39285e13	0.613526	0.306763	−	0.951786i	$-0.400754\pi$
$929$	1.39285e13	0.613526	0.306763	+	0.951786i	$0.400754\pi$
$930$	7.31145e12	0.320501
$931$	−4.35344e13	−1.89915
$932$	−1.82532e11	−0.00792443
$933$	1.55530e13	0.671967
$934$	−3.45394e13	−1.48509
$935$	−2.67645e13	−1.14527
$936$	0	0
$937$	1.94606e13	0.824761	0.412380	−	0.911012i	$-0.364697\pi$
$937$	1.94606e13	0.824761	0.412380	+	0.911012i	$0.364697\pi$
$938$	−5.15200e13	−2.17302
$939$	−9.57977e12	−0.402124
$940$	1.93566e12	0.0808638
$941$	−2.38676e13	−0.992328	−0.496164	−	0.868229i	$-0.665258\pi$
$941$	−2.38676e13	−0.992328	−0.496164	+	0.868229i	$0.665258\pi$
$942$	2.32270e13	0.961089
$943$	−2.42792e12	−0.0999844
$944$	−2.07069e13	−0.848673
$945$	−4.12370e13	−1.68207
$946$	1.24913e13	0.507106
$947$	3.33959e13	1.34933	0.674664	−	0.738125i	$-0.264288\pi$
$947$	3.33959e13	1.34933	0.674664	+	0.738125i	$0.264288\pi$
$948$	2.84802e12	0.114526
$949$	0	0
$950$	−6.90794e12	−0.275165
$951$	−8.50541e11	−0.0337196
$952$	5.24478e13	2.06948
$953$	−1.23406e13	−0.484639	−0.242320	−	0.970196i	$-0.577908\pi$
$953$	−1.23406e13	−0.484639	−0.242320	+	0.970196i	$0.577908\pi$
$954$	9.21698e12	0.360264
$955$	−9.54394e12	−0.371289
$956$	4.96233e12	0.192143
$957$	1.56546e13	0.603306
$958$	−6.42202e12	−0.246335
$959$	2.69475e13	1.02881
$960$	−2.25855e13	−0.858243
$961$	−2.04738e13	−0.774360
$962$	0	0
$963$	1.67198e12	0.0626487
$964$	2.87373e10	0.00107176
$965$	1.60176e13	0.594600
$966$	−9.87746e12	−0.364963
$967$	−1.96551e13	−0.722863	−0.361432	−	0.932399i	$-0.617712\pi$
$967$	−1.96551e13	−0.722863	−0.361432	+	0.932399i	$0.617712\pi$
$968$	1.06642e13	0.390381
$969$	−4.35462e13	−1.58669
$970$	4.46819e13	1.62054
$971$	−3.59611e12	−0.129822	−0.0649108	−	0.997891i	$-0.520676\pi$
$971$	−3.59611e12	−0.129822	−0.0649108	+	0.997891i	$0.520676\pi$
$972$	−5.79401e12	−0.208200
$973$	8.70922e11	0.0311510
$974$	3.52859e13	1.25628
$975$	0	0
$976$	−1.28316e13	−0.452645
$977$	−5.51875e13	−1.93783	−0.968914	−	0.247398i	$-0.920424\pi$
$977$	−5.51875e13	−1.93783	−0.968914	+	0.247398i	$0.920424\pi$
$978$	1.00616e13	0.351676
$979$	−1.75738e13	−0.611426
$980$	8.31097e12	0.287829
$981$	−1.52519e13	−0.525791
$982$	1.33643e13	0.458612
$983$	2.30506e13	0.787394	0.393697	−	0.919240i	$-0.371196\pi$
$983$	2.30506e13	0.787394	0.393697	+	0.919240i	$0.371196\pi$
$984$	5.56810e12	0.189334
$985$	−4.58989e13	−1.55360
$986$	3.63282e13	1.22405
$987$	−9.66095e12	−0.324035
$988$	0	0
$989$	8.84280e12	0.293905
$990$	1.14175e13	0.377758
$991$	−3.82758e13	−1.26065	−0.630323	−	0.776333i	$-0.717077\pi$
$991$	−3.82758e13	−1.26065	−0.630323	+	0.776333i	$0.717077\pi$
$992$	−6.69819e12	−0.219611
$993$	3.22248e13	1.05177
$994$	6.33120e13	2.05706
$995$	−9.02115e12	−0.291782
$996$	3.49844e12	0.112644
$997$	−3.74284e13	−1.19970	−0.599851	−	0.800112i	$-0.704774\pi$
$997$	−3.74284e13	−1.19970	−0.599851	+	0.800112i	$0.704774\pi$
$998$	3.46499e13	1.10564
$999$	−2.22903e13	−0.708061

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.10.a.f.1.14		20
13.2	odd	12		13.10.e.a.4.7	✓	20
13.7	odd	12		13.10.e.a.10.7	yes	20
13.12	even	2	inner	169.10.a.f.1.7		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.10.e.a.4.7	✓	20	13.2	odd	12
13.10.e.a.10.7	yes	20	13.7	odd	12
169.10.a.f.1.7		20	13.12	even	2	inner
169.10.a.f.1.14		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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+19.7704 q^{2} +99.5109 q^{3} -121.130 q^{4} -1521.53 q^{5} +1967.37 q^{6} -9243.80 q^{7} -12517.3 q^{8} -9780.58 q^{9} -30081.3 q^{10} +38807.1 q^{11} -12053.7 q^{12} -182754. q^{14} -151409. q^{15} -185453. q^{16} +453281. q^{17} -193366. q^{18} -965410. q^{19} +184302. q^{20} -919859. q^{21} +767233. q^{22} +543135. q^{23} -1.24560e6 q^{24} +361927. q^{25} -2.93195e6 q^{27} +1.11970e6 q^{28} +4.05377e6 q^{29} -2.99342e6 q^{30} -2.44251e6 q^{31} +2.74234e6 q^{32} +3.86173e6 q^{33} +8.96157e6 q^{34} +1.40647e7 q^{35} +1.18472e6 q^{36} +7.60255e6 q^{37} -1.90866e7 q^{38} +1.90454e7 q^{40} -4.47020e6 q^{41} -1.81860e7 q^{42} +1.62810e7 q^{43} -4.70069e6 q^{44} +1.48814e7 q^{45} +1.07380e7 q^{46} +1.05026e7 q^{47} -1.84546e7 q^{48} +4.50942e7 q^{49} +7.15545e6 q^{50} +4.51064e7 q^{51} -4.76659e7 q^{53} -5.79659e7 q^{54} -5.90461e7 q^{55} +1.15707e8 q^{56} -9.60688e7 q^{57} +8.01449e7 q^{58} +1.11655e8 q^{59} +1.83401e7 q^{60} +6.91907e7 q^{61} -4.82895e7 q^{62} +9.04097e7 q^{63} +1.49169e8 q^{64} +7.63480e7 q^{66} +2.81909e8 q^{67} -5.49058e7 q^{68} +5.40479e7 q^{69} +2.78066e8 q^{70} -3.46433e8 q^{71} +1.22426e8 q^{72} +1.76119e8 q^{73} +1.50306e8 q^{74} +3.60157e7 q^{75} +1.16940e8 q^{76} -3.58725e8 q^{77} -2.36277e8 q^{79} +2.82172e8 q^{80} -9.92495e7 q^{81} -8.83778e7 q^{82} -2.90237e8 q^{83} +1.11422e8 q^{84} -6.89681e8 q^{85} +3.21883e8 q^{86} +4.03395e8 q^{87} -4.85758e8 q^{88} -4.52851e8 q^{89} +2.94213e8 q^{90} -6.57898e7 q^{92} -2.43056e8 q^{93} +2.07642e8 q^{94} +1.46890e9 q^{95} +2.72893e8 q^{96} -1.48537e9 q^{97} +8.91532e8 q^{98} -3.79556e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 326 q^{3} + 5118 q^{4} + 129526 q^{9} + 88390 q^{10} + 427652 q^{12} + 473556 q^{14} + 1189618 q^{16} - 99312 q^{17} - 5073532 q^{22} + 6252378 q^{23} + 1529274 q^{25} + 18052718 q^{27} + 5424828 q^{29}+ \cdots + 9251202540 q^{95}+O(q^{100}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more