LMFDB - Embedded newform 169.10.a.f.1.11

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			1.11
Root			$2.58166$ 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)

$f(q)$	$=$	\(q+2.58166 q^{2} +184.202 q^{3} -505.335 q^{4} +581.690 q^{5} +475.547 q^{6} +10922.8 q^{7} -2626.41 q^{8} +14247.4 q^{9} +1501.73 q^{10} +78280.9 q^{11} -93083.8 q^{12} +28199.0 q^{14} +107149. q^{15} +251951. q^{16} -169969. q^{17} +36782.0 q^{18} +398015. q^{19} -293949. q^{20} +2.01201e6 q^{21} +202094. q^{22} +1.48328e6 q^{23} -483791. q^{24} -1.61476e6 q^{25} -1.00124e6 q^{27} -5.51968e6 q^{28} -2.10711e6 q^{29} +276621. q^{30} -1.23336e6 q^{31} +1.99517e6 q^{32} +1.44195e7 q^{33} -438802. q^{34} +6.35370e6 q^{35} -7.19973e6 q^{36} -148663. q^{37} +1.02754e6 q^{38} -1.52776e6 q^{40} -3.23215e7 q^{41} +5.19431e6 q^{42} +2.06939e7 q^{43} -3.95581e7 q^{44} +8.28760e6 q^{45} +3.82931e6 q^{46} +3.31297e7 q^{47} +4.64099e7 q^{48} +7.89542e7 q^{49} -4.16876e6 q^{50} -3.13086e7 q^{51} +4.81647e7 q^{53} -2.58486e6 q^{54} +4.55352e7 q^{55} -2.86878e7 q^{56} +7.33153e7 q^{57} -5.43983e6 q^{58} +7.37915e6 q^{59} -5.41460e7 q^{60} -3.78696e7 q^{61} -3.18412e6 q^{62} +1.55622e8 q^{63} -1.23848e8 q^{64} +3.72262e7 q^{66} +8.03450e7 q^{67} +8.58912e7 q^{68} +2.73223e8 q^{69} +1.64031e7 q^{70} +4.97348e7 q^{71} -3.74196e7 q^{72} -7.08914e6 q^{73} -383796. q^{74} -2.97443e8 q^{75} -2.01131e8 q^{76} +8.55047e8 q^{77} +1.85274e8 q^{79} +1.46558e8 q^{80} -4.64863e8 q^{81} -8.34430e7 q^{82} +2.38643e8 q^{83} -1.01674e9 q^{84} -9.88693e7 q^{85} +5.34245e7 q^{86} -3.88134e8 q^{87} -2.05598e8 q^{88} +3.02574e6 q^{89} +2.13957e7 q^{90} -7.49551e8 q^{92} -2.27188e8 q^{93} +8.55294e7 q^{94} +2.31522e8 q^{95} +3.67515e8 q^{96} -5.53994e8 q^{97} +2.03833e8 q^{98} +1.11530e9 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.58166	0.114094	0.0570471	−	0.998371i	$-0.481831\pi$
$2$	2.58166	0.114094	0.0570471	+	0.998371i	$0.481831\pi$
$3$	184.202	1.31295	0.656476	−	0.754347i	$-0.272046\pi$
$3$	184.202	1.31295	0.656476	+	0.754347i	$0.272046\pi$
$4$	−505.335	−0.986983
$5$	581.690	0.416224	0.208112	−	0.978105i	$-0.433268\pi$
$5$	581.690	0.416224	0.208112	+	0.978105i	$0.433268\pi$
$6$	475.547	0.149800
$7$	10922.8	1.71946	0.859732	−	0.510745i	$-0.170630\pi$
$7$	10922.8	1.71946	0.859732	+	0.510745i	$0.170630\pi$
$8$	−2626.41	−0.226703
$9$	14247.4	0.723845
$10$	1501.73	0.0474887
$11$	78280.9	1.61209	0.806044	−	0.591856i	$-0.201605\pi$
$11$	78280.9	1.61209	0.806044	+	0.591856i	$0.201605\pi$
$12$	−93083.8	−1.29586
$13$	0	0
$13$	0	0
$14$	28199.0	0.196181
$15$	107149.	0.546482
$16$	251951.	0.961117
$17$	−169969.	−0.493571	−0.246785	−	0.969070i	$-0.579374\pi$
$17$	−169969.	−0.493571	−0.246785	+	0.969070i	$0.579374\pi$
$18$	36782.0	0.0825865
$19$	398015.	0.700662	0.350331	−	0.936626i	$-0.386069\pi$
$19$	398015.	0.700662	0.350331	+	0.936626i	$0.386069\pi$
$20$	−293949.	−0.410806
$21$	2.01201e6	2.25758
$22$	202094.	0.183930
$23$	1.48328e6	1.10521	0.552607	−	0.833442i	$-0.313633\pi$
$23$	1.48328e6	1.10521	0.552607	+	0.833442i	$0.313633\pi$
$24$	−483791.	−0.297651
$25$	−1.61476e6	−0.826758
$26$	0	0
$27$	−1.00124e6	−0.362578
$28$	−5.51968e6	−1.69708
$29$	−2.10711e6	−0.553217	−0.276609	−	0.960983i	$-0.589211\pi$
$29$	−2.10711e6	−0.553217	−0.276609	+	0.960983i	$0.589211\pi$
$30$	276621.	0.0623505
$31$	−1.23336e6	−0.239863	−0.119931	−	0.992782i	$-0.538267\pi$
$31$	−1.23336e6	−0.239863	−0.119931	+	0.992782i	$0.538267\pi$
$32$	1.99517e6	0.336361
$33$	1.44195e7	2.11659
$34$	−438802.	−0.0563136
$35$	6.35370e6	0.715682
$36$	−7.19973e6	−0.714422
$37$	−148663.	−0.0130405	−0.00652025	−	0.999979i	$-0.502075\pi$
$37$	−148663.	−0.0130405	−0.00652025	+	0.999979i	$0.502075\pi$
$38$	1.02754e6	0.0799415
$39$	0	0
$40$	−1.52776e6	−0.0943593
$41$	−3.23215e7	−1.78634	−0.893170	−	0.449719i	$-0.851524\pi$
$41$	−3.23215e7	−1.78634	−0.893170	+	0.449719i	$0.851524\pi$
$42$	5.19431e6	0.257576
$43$	2.06939e7	0.923069	0.461534	−	0.887122i	$-0.347299\pi$
$43$	2.06939e7	0.923069	0.461534	+	0.887122i	$0.347299\pi$
$44$	−3.95581e7	−1.59110
$45$	8.28760e6	0.301282
$46$	3.82931e6	0.126099
$47$	3.31297e7	0.990322	0.495161	−	0.868801i	$-0.335109\pi$
$47$	3.31297e7	0.990322	0.495161	+	0.868801i	$0.335109\pi$
$48$	4.64099e7	1.26190
$49$	7.89542e7	1.95656
$50$	−4.16876e6	−0.0943283
$51$	−3.13086e7	−0.648035
$52$	0	0
$53$	4.81647e7	0.838469	0.419235	−	0.907878i	$-0.362298\pi$
$53$	4.81647e7	0.838469	0.419235	+	0.907878i	$0.362298\pi$
$54$	−2.58486e6	−0.0413681
$55$	4.55352e7	0.670989
$56$	−2.86878e7	−0.389808
$57$	7.33153e7	0.919936
$58$	−5.43983e6	−0.0631189
$59$	7.37915e6	0.0792816	0.0396408	−	0.999214i	$-0.487379\pi$
$59$	7.37915e6	0.0792816	0.0396408	+	0.999214i	$0.487379\pi$
$60$	−5.41460e7	−0.539368
$61$	−3.78696e7	−0.350192	−0.175096	−	0.984551i	$-0.556024\pi$
$61$	−3.78696e7	−0.350192	−0.175096	+	0.984551i	$0.556024\pi$
$62$	−3.18412e6	−0.0273670
$63$	1.55622e8	1.24463
$64$	−1.23848e8	−0.922740
$65$	0	0
$66$	3.72262e7	0.241491
$67$	8.03450e7	0.487105	0.243552	−	0.969888i	$-0.421687\pi$
$67$	8.03450e7	0.487105	0.243552	+	0.969888i	$0.421687\pi$
$68$	8.58912e7	0.487146
$69$	2.73223e8	1.45109
$70$	1.64031e7	0.0816552
$71$	4.97348e7	0.232272	0.116136	−	0.993233i	$-0.462949\pi$
$71$	4.97348e7	0.232272	0.116136	+	0.993233i	$0.462949\pi$
$72$	−3.74196e7	−0.164098
$73$	−7.08914e6	−0.0292173	−0.0146087	−	0.999893i	$-0.504650\pi$
$73$	−7.08914e6	−0.0292173	−0.0146087	+	0.999893i	$0.504650\pi$
$74$	−383796.	−0.00148784
$75$	−2.97443e8	−1.08549
$76$	−2.01131e8	−0.691541
$77$	8.55047e8	2.77193
$78$	0	0
$79$	1.85274e8	0.535172	0.267586	−	0.963534i	$-0.413774\pi$
$79$	1.85274e8	0.535172	0.267586	+	0.963534i	$0.413774\pi$
$80$	1.46558e8	0.400040
$81$	−4.64863e8	−1.19989
$82$	−8.34430e7	−0.203811
$83$	2.38643e8	0.551948	0.275974	−	0.961165i	$-0.411000\pi$
$83$	2.38643e8	0.551948	0.275974	+	0.961165i	$0.411000\pi$
$84$	−1.01674e9	−2.22819
$85$	−9.88693e7	−0.205436
$86$	5.34245e7	0.105317
$87$	−3.88134e8	−0.726348
$88$	−2.05598e8	−0.365465
$89$	3.02574e6	0.00511183	0.00255591	−	0.999997i	$-0.499186\pi$
$89$	3.02574e6	0.00511183	0.00255591	+	0.999997i	$0.499186\pi$
$90$	2.13957e7	0.0343745
$91$	0	0
$92$	−7.49551e8	−1.09083
$93$	−2.27188e8	−0.314929
$94$	8.55294e7	0.112990
$95$	2.31522e8	0.291632
$96$	3.67515e8	0.441626
$97$	−5.53994e8	−0.635379	−0.317689	−	0.948195i	$-0.602907\pi$
$97$	−5.53994e8	−0.635379	−0.317689	+	0.948195i	$0.602907\pi$
$98$	2.03833e8	0.223232
$99$	1.11530e9	1.16690
$100$	8.15995e8	0.815995
$101$	−1.83903e9	−1.75850	−0.879250	−	0.476361i	$-0.841956\pi$
$101$	−1.83903e9	−1.75850	−0.879250	+	0.476361i	$0.841956\pi$
$102$	−8.08282e7	−0.0739370
$103$	−1.49590e9	−1.30959	−0.654796	−	0.755806i	$-0.727245\pi$
$103$	−1.49590e9	−1.30959	−0.654796	+	0.755806i	$0.727245\pi$
$104$	0	0
$105$	1.17036e9	0.939657
$106$	1.24345e8	0.0956645
$107$	1.73735e9	1.28133	0.640663	−	0.767822i	$-0.278660\pi$
$107$	1.73735e9	1.28133	0.640663	+	0.767822i	$0.278660\pi$
$108$	5.05963e8	0.357859
$109$	1.40396e9	0.952657	0.476329	−	0.879267i	$-0.341967\pi$
$109$	1.40396e9	0.952657	0.476329	+	0.879267i	$0.341967\pi$
$110$	1.17556e8	0.0765560
$111$	−2.73840e7	−0.0171216
$112$	2.75201e9	1.65261
$113$	1.83385e9	1.05806	0.529031	−	0.848603i	$-0.322556\pi$
$113$	1.83385e9	1.05806	0.529031	+	0.848603i	$0.322556\pi$
$114$	1.89275e8	0.104959
$115$	8.62807e8	0.460016
$116$	1.06480e9	0.546016
$117$	0	0
$118$	1.90504e7	0.00904557
$119$	−1.85654e9	−0.848677
$120$	−2.81416e8	−0.123889
$121$	3.76995e9	1.59883
$122$	−9.77664e7	−0.0399549
$123$	−5.95369e9	−2.34538
$124$	6.23261e8	0.236740
$125$	−2.07541e9	−0.760340
$126$	4.01763e8	0.142005
$127$	3.59174e9	1.22515	0.612574	−	0.790414i	$-0.290134\pi$
$127$	3.59174e9	1.22515	0.612574	+	0.790414i	$0.290134\pi$
$128$	−1.34126e9	−0.441640
$129$	3.81186e9	1.21195
$130$	0	0
$131$	5.90939e9	1.75316	0.876581	−	0.481255i	$-0.159819\pi$
$131$	5.90939e9	1.75316	0.876581	+	0.481255i	$0.159819\pi$
$132$	−7.28668e9	−2.08904
$133$	4.34745e9	1.20476
$134$	2.07423e8	0.0555758
$135$	−5.82413e8	−0.150914
$136$	4.46408e8	0.111894
$137$	6.38102e9	1.54756	0.773780	−	0.633455i	$-0.218364\pi$
$137$	6.38102e9	1.54756	0.773780	+	0.633455i	$0.218364\pi$
$138$	7.05367e8	0.165561
$139$	1.22325e9	0.277938	0.138969	−	0.990297i	$-0.455621\pi$
$139$	1.22325e9	0.277938	0.138969	+	0.990297i	$0.455621\pi$
$140$	−3.21075e9	−0.706366
$141$	6.10255e9	1.30025
$142$	1.28398e8	0.0265009
$143$	0	0
$144$	3.58966e9	0.695700
$145$	−1.22568e9	−0.230262
$146$	−1.83017e7	−0.00333353
$147$	1.45435e10	2.56887
$148$	7.51244e7	0.0128707
$149$	−2.05209e9	−0.341081	−0.170541	−	0.985351i	$-0.554551\pi$
$149$	−2.05209e9	−0.341081	−0.170541	+	0.985351i	$0.554551\pi$
$150$	−7.67895e8	−0.123849
$151$	−6.34687e7	−0.00993490	−0.00496745	−	0.999988i	$-0.501581\pi$
$151$	−6.34687e7	−0.00993490	−0.00496745	+	0.999988i	$0.501581\pi$
$152$	−1.04535e9	−0.158842
$153$	−2.42162e9	−0.357269
$154$	2.20744e9	0.316261
$155$	−7.17435e8	−0.0998366
$156$	0	0
$157$	3.65041e9	0.479506	0.239753	−	0.970834i	$-0.422934\pi$
$157$	3.65041e9	0.479506	0.239753	+	0.970834i	$0.422934\pi$
$158$	4.78315e8	0.0610600
$159$	8.87203e9	1.10087
$160$	1.16057e9	0.140002
$161$	1.62015e10	1.90038
$162$	−1.20012e9	−0.136901
$163$	−1.45792e10	−1.61766	−0.808832	−	0.588040i	$-0.799900\pi$
$163$	−1.45792e10	−1.61766	−0.808832	+	0.588040i	$0.799900\pi$
$164$	1.63332e10	1.76309
$165$	8.38769e9	0.880977
$166$	6.16096e8	0.0629741
$167$	−6.22517e9	−0.619337	−0.309668	−	0.950845i	$-0.600218\pi$
$167$	−6.22517e9	−0.619337	−0.309668	+	0.950845i	$0.600218\pi$
$168$	−5.28435e9	−0.511800
$169$	0	0
$170$	−2.55247e8	−0.0234390
$171$	5.67070e9	0.507171
$172$	−1.04573e10	−0.911053
$173$	−1.86116e10	−1.57971	−0.789853	−	0.613296i	$-0.789843\pi$
$173$	−1.86116e10	−1.57971	−0.789853	+	0.613296i	$0.789843\pi$
$174$	−1.00203e9	−0.0828721
$175$	−1.76377e10	−1.42158
$176$	1.97230e10	1.54940
$177$	1.35926e9	0.104093
$178$	7.81142e6	0.000583230 0
$179$	−7.33587e9	−0.534088	−0.267044	−	0.963684i	$-0.586047\pi$
$179$	−7.33587e9	−0.534088	−0.267044	+	0.963684i	$0.586047\pi$
$180$	−4.18802e9	−0.297360
$181$	1.48037e10	1.02522	0.512611	−	0.858621i	$-0.328678\pi$
$181$	1.48037e10	1.02522	0.512611	+	0.858621i	$0.328678\pi$
$182$	0	0
$183$	−6.97567e9	−0.459786
$184$	−3.89569e9	−0.250556
$185$	−8.64756e7	−0.00542776
$186$	−5.86521e8	−0.0359315
$187$	−1.33053e10	−0.795679
$188$	−1.67416e10	−0.977431
$189$	−1.09364e10	−0.623441
$190$	5.97710e8	0.0332735
$191$	−1.67160e10	−0.908827	−0.454413	−	0.890791i	$-0.650151\pi$
$191$	−1.67160e10	−0.908827	−0.454413	+	0.890791i	$0.650151\pi$
$192$	−2.28131e10	−1.21151
$193$	−9.86376e7	−0.00511722	−0.00255861	−	0.999997i	$-0.500814\pi$
$193$	−9.86376e7	−0.00511722	−0.00255861	+	0.999997i	$0.500814\pi$
$194$	−1.43022e9	−0.0724930
$195$	0	0
$196$	−3.98983e10	−1.93109
$197$	2.14840e10	1.01629	0.508145	−	0.861272i	$-0.330332\pi$
$197$	2.14840e10	1.01629	0.508145	+	0.861272i	$0.330332\pi$
$198$	2.87933e9	0.133137
$199$	3.06053e10	1.38343	0.691716	−	0.722170i	$-0.256855\pi$
$199$	3.06053e10	1.38343	0.691716	+	0.722170i	$0.256855\pi$
$200$	4.24103e9	0.187429
$201$	1.47997e10	0.639545
$202$	−4.74774e9	−0.200635
$203$	−2.30155e10	−0.951238
$204$	1.58214e10	0.639599
$205$	−1.88011e10	−0.743517
$206$	−3.86191e9	−0.149417
$207$	2.11329e10	0.800004
$208$	0	0
$209$	3.11570e10	1.12953
$210$	3.02148e9	0.107209
$211$	−4.48475e10	−1.55764	−0.778820	−	0.627247i	$-0.784182\pi$
$211$	−4.48475e10	−1.55764	−0.778820	+	0.627247i	$0.784182\pi$
$212$	−2.43393e10	−0.827554
$213$	9.16126e9	0.304963
$214$	4.48523e9	0.146192
$215$	1.20374e10	0.384203
$216$	2.62967e9	0.0821977
$217$	−1.34718e10	−0.412436
$218$	3.62455e9	0.108693
$219$	−1.30583e9	−0.0383610
$220$	−2.30106e10	−0.662255
$221$	0	0
$222$	−7.06960e7	−0.00195347
$223$	−1.54810e10	−0.419205	−0.209602	−	0.977787i	$-0.567217\pi$
$223$	−1.54810e10	−0.419205	−0.209602	+	0.977787i	$0.567217\pi$
$224$	2.17929e10	0.578361
$225$	−2.30062e10	−0.598444
$226$	4.73437e9	0.120719
$227$	6.66609e10	1.66631	0.833153	−	0.553042i	$-0.186533\pi$
$227$	6.66609e10	1.66631	0.833153	+	0.553042i	$0.186533\pi$
$228$	−3.70488e10	−0.907961
$229$	−3.06730e10	−0.737050	−0.368525	−	0.929618i	$-0.620137\pi$
$229$	−3.06730e10	−0.737050	−0.368525	+	0.929618i	$0.620137\pi$
$230$	2.22747e9	0.0524852
$231$	1.57502e11	3.63941
$232$	5.53413e9	0.125416
$233$	−4.69078e10	−1.04266	−0.521331	−	0.853355i	$-0.674564\pi$
$233$	−4.69078e10	−1.04266	−0.521331	+	0.853355i	$0.674564\pi$
$234$	0	0
$235$	1.92712e10	0.412196
$236$	−3.72894e9	−0.0782495
$237$	3.41279e10	0.702655
$238$	−4.79295e9	−0.0968292
$239$	−5.84061e10	−1.15789	−0.578945	−	0.815366i	$-0.696535\pi$
$239$	−5.84061e10	−1.15789	−0.578945	+	0.815366i	$0.696535\pi$
$240$	2.69962e10	0.525233
$241$	2.87141e10	0.548301	0.274150	−	0.961687i	$-0.411603\pi$
$241$	2.87141e10	0.548301	0.274150	+	0.961687i	$0.411603\pi$
$242$	9.73272e9	0.182417
$243$	−6.59214e10	−1.21282
$244$	1.91368e10	0.345634
$245$	4.59269e10	0.814367
$246$	−1.53704e10	−0.267594
$247$	0	0
$248$	3.23931e9	0.0543777
$249$	4.39586e10	0.724682
$250$	−5.35799e9	−0.0867504
$251$	−4.94425e10	−0.786265	−0.393132	−	0.919482i	$-0.628609\pi$
$251$	−4.94425e10	−0.786265	−0.393132	+	0.919482i	$0.628609\pi$
$252$	−7.86413e10	−1.22842
$253$	1.16112e11	1.78170
$254$	9.27264e9	0.139782
$255$	−1.82119e10	−0.269728
$256$	5.99475e10	0.872352
$257$	−5.44970e10	−0.779244	−0.389622	−	0.920975i	$-0.627394\pi$
$257$	−5.44970e10	−0.779244	−0.389622	+	0.920975i	$0.627394\pi$
$258$	9.84091e9	0.138276
$259$	−1.62381e9	−0.0224227
$260$	0	0
$261$	−3.00209e10	−0.400444
$262$	1.52560e10	0.200026
$263$	2.94548e10	0.379626	0.189813	−	0.981820i	$-0.439212\pi$
$263$	2.94548e10	0.379626	0.189813	+	0.981820i	$0.439212\pi$
$264$	−3.78716e10	−0.479839
$265$	2.80169e10	0.348991
$266$	1.12236e10	0.137457
$267$	5.57347e8	0.00671159
$268$	−4.06012e10	−0.480764
$269$	−2.80590e9	−0.0326729	−0.0163364	−	0.999867i	$-0.505200\pi$
$269$	−2.80590e9	−0.0326729	−0.0163364	+	0.999867i	$0.505200\pi$
$270$	−1.50359e9	−0.0172184
$271$	−1.01493e11	−1.14308	−0.571538	−	0.820576i	$-0.693653\pi$
$271$	−1.01493e11	−1.14308	−0.571538	+	0.820576i	$0.693653\pi$
$272$	−4.28238e10	−0.474379
$273$	0	0
$274$	1.64736e10	0.176568
$275$	−1.26405e11	−1.33281
$276$	−1.38069e11	−1.43220
$277$	1.71349e10	0.174873	0.0874366	−	0.996170i	$-0.472132\pi$
$277$	1.71349e10	0.174873	0.0874366	+	0.996170i	$0.472132\pi$
$278$	3.15800e9	0.0317111
$279$	−1.75723e10	−0.173623
$280$	−1.66874e10	−0.162247
$281$	−1.52845e11	−1.46242	−0.731210	−	0.682152i	$-0.761044\pi$
$281$	−1.52845e11	−1.46242	−0.731210	+	0.682152i	$0.761044\pi$
$282$	1.57547e10	0.148351
$283$	6.76540e10	0.626981	0.313490	−	0.949591i	$-0.398502\pi$
$283$	6.76540e10	0.626981	0.313490	+	0.949591i	$0.398502\pi$
$284$	−2.51327e10	−0.229249
$285$	4.26468e10	0.382899
$286$	0	0
$287$	−3.53042e11	−3.07155
$288$	2.84261e10	0.243473
$289$	−8.96984e10	−0.756388
$290$	−3.16430e9	−0.0262716
$291$	−1.02047e11	−0.834222
$292$	3.58239e9	0.0288370
$293$	−1.77482e10	−0.140686	−0.0703429	−	0.997523i	$-0.522409\pi$
$293$	−1.77482e10	−0.140686	−0.0703429	+	0.997523i	$0.522409\pi$
$294$	3.75464e10	0.293093
$295$	4.29238e9	0.0329989
$296$	3.90449e8	0.00295632
$297$	−7.83781e10	−0.584508
$298$	−5.29779e9	−0.0389154
$299$	0	0
$300$	1.50308e11	1.07136
$301$	2.26035e11	1.58718
$302$	−1.63855e8	−0.00113351
$303$	−3.38753e11	−2.30883
$304$	1.00280e11	0.673418
$305$	−2.20284e10	−0.145758
$306$	−6.25180e9	−0.0407623
$307$	2.09930e11	1.34881	0.674405	−	0.738362i	$-0.264400\pi$
$307$	2.09930e11	1.34881	0.674405	+	0.738362i	$0.264400\pi$
$308$	−4.32085e11	−2.73584
$309$	−2.75549e11	−1.71943
$310$	−1.85217e9	−0.0113908
$311$	−1.33748e11	−0.810708	−0.405354	−	0.914160i	$-0.632852\pi$
$311$	−1.33748e11	−0.810708	−0.405354	+	0.914160i	$0.632852\pi$
$312$	0	0
$313$	−5.76088e10	−0.339265	−0.169633	−	0.985507i	$-0.554258\pi$
$313$	−5.76088e10	−0.339265	−0.169633	+	0.985507i	$0.554258\pi$
$314$	9.42412e9	0.0547088
$315$	9.05239e10	0.518043
$316$	−9.36256e10	−0.528205
$317$	−4.68801e10	−0.260749	−0.130374	−	0.991465i	$-0.541618\pi$
$317$	−4.68801e10	−0.260749	−0.130374	+	0.991465i	$0.541618\pi$
$318$	2.29046e10	0.125603
$319$	−1.64946e11	−0.891835
$320$	−7.20413e10	−0.384066
$321$	3.20023e11	1.68232
$322$	4.18268e10	0.216822
$323$	−6.76502e10	−0.345826
$324$	2.34912e11	1.18427
$325$	0	0
$326$	−3.76384e10	−0.184566
$327$	2.58613e11	1.25079
$328$	8.48895e10	0.404969
$329$	3.61869e11	1.70282
$330$	2.16541e10	0.100514
$331$	1.03157e11	0.472360	0.236180	−	0.971709i	$-0.424105\pi$
$331$	1.03157e11	0.472360	0.236180	+	0.971709i	$0.424105\pi$
$332$	−1.20595e11	−0.544763
$333$	−2.11806e9	−0.00943930
$334$	−1.60712e10	−0.0706627
$335$	4.67359e10	0.202745
$336$	5.06927e11	2.16979
$337$	−1.41335e11	−0.596917	−0.298458	−	0.954423i	$-0.596472\pi$
$337$	−1.41335e11	−0.596917	−0.298458	+	0.954423i	$0.596472\pi$
$338$	0	0
$339$	3.37799e11	1.38918
$340$	4.99621e10	0.202762
$341$	−9.65487e10	−0.386680
$342$	1.46398e10	0.0578652
$343$	4.21627e11	1.64477
$344$	−5.43506e10	−0.209263
$345$	1.58931e11	0.603980
$346$	−4.80488e10	−0.180235
$347$	1.44749e10	0.0535961	0.0267981	−	0.999641i	$-0.491469\pi$
$347$	1.44749e10	0.0535961	0.0267981	+	0.999641i	$0.491469\pi$
$348$	1.96138e11	0.716893
$349$	−2.05497e11	−0.741467	−0.370734	−	0.928739i	$-0.620894\pi$
$349$	−2.05497e11	−0.741467	−0.370734	+	0.928739i	$0.620894\pi$
$350$	−4.55346e10	−0.162194
$351$	0	0
$352$	1.56184e11	0.542244
$353$	−3.94613e11	−1.35265	−0.676325	−	0.736604i	$-0.736428\pi$
$353$	−3.94613e11	−1.35265	−0.676325	+	0.736604i	$0.736428\pi$
$354$	3.50913e9	0.0118764
$355$	2.89303e10	0.0966773
$356$	−1.52901e9	−0.00504528
$357$	−3.41978e11	−1.11427
$358$	−1.89387e10	−0.0609364
$359$	−2.88696e11	−0.917310	−0.458655	−	0.888614i	$-0.651669\pi$
$359$	−2.88696e11	−0.917310	−0.458655	+	0.888614i	$0.651669\pi$
$360$	−2.17666e10	−0.0683015
$361$	−1.64272e11	−0.509073
$362$	3.82182e10	0.116972
$363$	6.94433e11	2.09918
$364$	0	0
$365$	−4.12368e9	−0.0121609
$366$	−1.80088e10	−0.0524589
$367$	1.69825e10	0.0488657	0.0244329	−	0.999701i	$-0.492222\pi$
$367$	1.69825e10	0.0488657	0.0244329	+	0.999701i	$0.492222\pi$
$368$	3.73713e11	1.06224
$369$	−4.60499e11	−1.29303
$370$	−2.23250e8	−0.000619276 0
$371$	5.26094e11	1.44172
$372$	1.14806e11	0.310829
$373$	3.56271e10	0.0952995	0.0476497	−	0.998864i	$-0.484827\pi$
$373$	3.56271e10	0.0952995	0.0476497	+	0.998864i	$0.484827\pi$
$374$	−3.43498e10	−0.0907824
$375$	−3.82294e11	−0.998291
$376$	−8.70121e10	−0.224509
$377$	0	0
$378$	−2.82340e10	−0.0711310
$379$	−6.80085e11	−1.69312	−0.846558	−	0.532296i	$-0.821329\pi$
$379$	−6.80085e11	−1.69312	−0.846558	+	0.532296i	$0.821329\pi$
$380$	−1.16996e11	−0.287836
$381$	6.61606e11	1.60856
$382$	−4.31549e10	−0.103692
$383$	−2.63055e11	−0.624671	−0.312336	−	0.949972i	$-0.601111\pi$
$383$	−2.63055e11	−0.624671	−0.312336	+	0.949972i	$0.601111\pi$
$384$	−2.47063e11	−0.579853
$385$	4.97373e11	1.15374
$386$	−2.54648e8	−0.000583846 0
$387$	2.94835e11	0.668159
$388$	2.79953e11	0.627108
$389$	−5.59712e11	−1.23934	−0.619671	−	0.784862i	$-0.712734\pi$
$389$	−5.59712e11	−1.23934	−0.619671	+	0.784862i	$0.712734\pi$
$390$	0	0
$391$	−2.52111e11	−0.545501
$392$	−2.07366e11	−0.443558
$393$	1.08852e12	2.30182
$394$	5.54643e10	0.115953
$395$	1.07772e11	0.222751
$396$	−5.63601e11	−1.15171
$397$	5.10139e11	1.03070	0.515349	−	0.856981i	$-0.327662\pi$
$397$	5.10139e11	1.03070	0.515349	+	0.856981i	$0.327662\pi$
$398$	7.90124e10	0.157842
$399$	8.00809e11	1.58180
$400$	−4.06841e11	−0.794611
$401$	5.30662e11	1.02487	0.512434	−	0.858727i	$-0.328744\pi$
$401$	5.30662e11	1.02487	0.512434	+	0.858727i	$0.328744\pi$
$402$	3.82078e10	0.0729684
$403$	0	0
$404$	9.29326e11	1.73561
$405$	−2.70407e11	−0.499424
$406$	−5.94183e10	−0.108531
$407$	−1.16374e10	−0.0210224
$408$	8.22294e10	0.146912
$409$	3.12122e11	0.551530	0.275765	−	0.961225i	$-0.411069\pi$
$409$	3.12122e11	0.551530	0.275765	+	0.961225i	$0.411069\pi$
$410$	−4.85380e10	−0.0848310
$411$	1.17540e12	2.03187
$412$	7.55932e11	1.29254
$413$	8.06011e10	0.136322
$414$	5.45578e10	0.0912758
$415$	1.38817e11	0.229734
$416$	0	0
$417$	2.25325e11	0.364919
$418$	8.04367e10	0.128873
$419$	−1.43253e11	−0.227061	−0.113530	−	0.993535i	$-0.536216\pi$
$419$	−1.43253e11	−0.227061	−0.113530	+	0.993535i	$0.536216\pi$
$420$	−5.91426e11	−0.927425
$421$	1.01293e12	1.57148	0.785742	−	0.618555i	$-0.212282\pi$
$421$	1.01293e12	1.57148	0.785742	+	0.618555i	$0.212282\pi$
$422$	−1.15781e11	−0.177718
$423$	4.72013e11	0.716840
$424$	−1.26500e11	−0.190084
$425$	2.74459e11	0.408063
$426$	2.36512e10	0.0347945
$427$	−4.13643e11	−0.602143
$428$	−8.77942e11	−1.26465
$429$	0	0
$430$	3.10765e10	0.0438354
$431$	3.60765e11	0.503590	0.251795	−	0.967781i	$-0.418979\pi$
$431$	3.60765e11	0.503590	0.251795	+	0.967781i	$0.418979\pi$
$432$	−2.52264e11	−0.348480
$433$	8.47280e11	1.15833	0.579164	−	0.815211i	$-0.303379\pi$
$433$	8.47280e11	1.15833	0.579164	+	0.815211i	$0.303379\pi$
$434$	−3.47795e10	−0.0470565
$435$	−2.25774e11	−0.302323
$436$	−7.09472e11	−0.940256
$437$	5.90366e11	0.774382
$438$	−3.37122e9	−0.00437676
$439$	−4.94585e11	−0.635551	−0.317776	−	0.948166i	$-0.602936\pi$
$439$	−4.94585e11	−0.635551	−0.317776	+	0.948166i	$0.602936\pi$
$440$	−1.19594e11	−0.152115
$441$	1.12490e12	1.41625
$442$	0	0
$443$	−1.03664e12	−1.27882	−0.639411	−	0.768865i	$-0.720822\pi$
$443$	−1.03664e12	−1.27882	−0.639411	+	0.768865i	$0.720822\pi$
$444$	1.38381e10	0.0168987
$445$	1.76004e9	0.00212766
$446$	−3.99666e10	−0.0478288
$447$	−3.77999e11	−0.447823
$448$	−1.35277e12	−1.58662
$449$	−3.60756e11	−0.418895	−0.209448	−	0.977820i	$-0.567167\pi$
$449$	−3.60756e11	−0.418895	−0.209448	+	0.977820i	$0.567167\pi$
$450$	−5.93942e10	−0.0682791
$451$	−2.53016e12	−2.87974
$452$	−9.26709e11	−1.04429
$453$	−1.16911e10	−0.0130441
$454$	1.72096e11	0.190116
$455$	0	0
$456$	−1.92556e11	−0.208552
$457$	−8.46499e11	−0.907828	−0.453914	−	0.891046i	$-0.649973\pi$
$457$	−8.46499e11	−0.907828	−0.453914	+	0.891046i	$0.649973\pi$
$458$	−7.91872e10	−0.0840931
$459$	1.70180e11	0.178958
$460$	−4.36007e11	−0.454028
$461$	6.81855e10	0.0703133	0.0351567	−	0.999382i	$-0.488807\pi$
$461$	6.81855e10	0.0703133	0.0351567	+	0.999382i	$0.488807\pi$
$462$	4.06615e11	0.415236
$463$	−1.76445e12	−1.78442	−0.892208	−	0.451626i	$-0.850844\pi$
$463$	−1.76445e12	−1.78442	−0.892208	+	0.451626i	$0.850844\pi$
$464$	−5.30888e11	−0.531707
$465$	−1.32153e11	−0.131081
$466$	−1.21100e11	−0.118962
$467$	1.15511e12	1.12382	0.561910	−	0.827198i	$-0.310067\pi$
$467$	1.15511e12	1.12382	0.561910	+	0.827198i	$0.310067\pi$
$468$	0	0
$469$	8.77594e11	0.837559
$470$	4.97517e10	0.0470292
$471$	6.72414e11	0.629568
$472$	−1.93807e10	−0.0179734
$473$	1.61994e12	1.48807
$474$	8.81066e10	0.0801689
$475$	−6.42700e11	−0.579278
$476$	9.38174e11	0.837630
$477$	6.86223e11	0.606922
$478$	−1.50784e11	−0.132109
$479$	−1.39164e12	−1.20786	−0.603929	−	0.797038i	$-0.706399\pi$
$479$	−1.39164e12	−1.20786	−0.603929	+	0.797038i	$0.706399\pi$
$480$	2.13780e11	0.183815
$481$	0	0
$482$	7.41300e10	0.0625580
$483$	2.98436e12	2.49511
$484$	−1.90509e12	−1.57801
$485$	−3.22253e11	−0.264460
$486$	−1.70186e11	−0.138376
$487$	2.30306e11	0.185534	0.0927672	−	0.995688i	$-0.470429\pi$
$487$	2.30306e11	0.185534	0.0927672	+	0.995688i	$0.470429\pi$
$488$	9.94612e10	0.0793897
$489$	−2.68551e12	−2.12392
$490$	1.18568e11	0.0929145
$491$	2.54903e11	0.197929	0.0989644	−	0.995091i	$-0.468447\pi$
$491$	2.54903e11	0.197929	0.0989644	+	0.995091i	$0.468447\pi$
$492$	3.00861e12	2.31485
$493$	3.58143e11	0.273052
$494$	0	0
$495$	6.48761e11	0.485692
$496$	−3.10747e11	−0.230536
$497$	5.43244e11	0.399384
$498$	1.13486e11	0.0826820
$499$	−3.16708e11	−0.228669	−0.114334	−	0.993442i	$-0.536474\pi$
$499$	−3.16708e11	−0.228669	−0.114334	+	0.993442i	$0.536474\pi$
$500$	1.04878e12	0.750442
$501$	−1.14669e12	−0.813160
$502$	−1.27644e11	−0.0897082
$503$	1.62741e12	1.13355	0.566776	−	0.823872i	$-0.308190\pi$
$503$	1.62741e12	1.13355	0.566776	+	0.823872i	$0.308190\pi$
$504$	−4.08728e11	−0.282161
$505$	−1.06975e12	−0.731930
$506$	2.99762e11	0.203282
$507$	0	0
$508$	−1.81503e12	−1.20920
$509$	5.77669e11	0.381460	0.190730	−	0.981643i	$-0.438914\pi$
$509$	5.77669e11	0.381460	0.190730	+	0.981643i	$0.438914\pi$
$510$	−4.70170e10	−0.0307744
$511$	−7.74333e10	−0.0502382
$512$	8.41490e11	0.541171
$513$	−3.98509e11	−0.254045
$514$	−1.40693e11	−0.0889072
$515$	−8.70153e11	−0.545083
$516$	−1.92627e12	−1.19617
$517$	2.59342e12	1.59649
$518$	−4.19213e9	−0.00255830
$519$	−3.42830e12	−2.07408
$520$	0	0
$521$	7.78716e11	0.463030	0.231515	−	0.972831i	$-0.425632\pi$
$521$	7.78716e11	0.463030	0.231515	+	0.972831i	$0.425632\pi$
$522$	−7.75037e10	−0.0456883
$523$	1.82550e12	1.06690	0.533451	−	0.845831i	$-0.320895\pi$
$523$	1.82550e12	1.06690	0.533451	+	0.845831i	$0.320895\pi$
$524$	−2.98622e12	−1.73034
$525$	−3.24891e12	−1.86647
$526$	7.60423e10	0.0433131
$527$	2.09633e11	0.118389
$528$	3.63301e12	2.03430
$529$	3.98952e11	0.221498
$530$	7.23301e10	0.0398178
$531$	1.05134e11	0.0573876
$532$	−2.19692e12	−1.18908
$533$	0	0
$534$	1.43888e9	0.000765753 0
$535$	1.01060e12	0.533318
$536$	−2.11019e11	−0.110428
$537$	−1.35128e12	−0.701233
$538$	−7.24388e9	−0.00372778
$539$	6.18061e12	3.15415
$540$	2.94314e11	0.148949
$541$	3.71466e11	0.186437	0.0932184	−	0.995646i	$-0.470285\pi$
$541$	3.71466e11	0.186437	0.0932184	+	0.995646i	$0.470285\pi$
$542$	−2.62021e11	−0.130418
$543$	2.72688e12	1.34607
$544$	−3.39117e11	−0.166018
$545$	8.16672e11	0.396519
$546$	0	0
$547$	−2.69915e12	−1.28909	−0.644547	−	0.764564i	$-0.722954\pi$
$547$	−2.69915e12	−1.28909	−0.644547	+	0.764564i	$0.722954\pi$
$548$	−3.22455e12	−1.52741
$549$	−5.39545e11	−0.253485
$550$	−3.26334e11	−0.152065
$551$	−8.38661e11	−0.387618
$552$	−7.17594e11	−0.328968
$553$	2.02372e12	0.920209
$554$	4.42365e10	0.0199520
$555$	−1.59290e10	−0.00712640
$556$	−6.18149e11	−0.274320
$557$	−3.29600e12	−1.45091	−0.725453	−	0.688272i	$-0.758370\pi$
$557$	−3.29600e12	−1.45091	−0.725453	+	0.688272i	$0.758370\pi$
$558$	−4.53655e10	−0.0198094
$559$	0	0
$560$	1.60082e12	0.687854
$561$	−2.45087e12	−1.04469
$562$	−3.94593e11	−0.166854
$563$	2.45081e12	1.02807	0.514034	−	0.857770i	$-0.328150\pi$
$563$	2.45081e12	1.02807	0.514034	+	0.857770i	$0.328150\pi$
$564$	−3.08383e12	−1.28332
$565$	1.06673e12	0.440390
$566$	1.74659e11	0.0715349
$567$	−5.07761e12	−2.06317
$568$	−1.30624e11	−0.0526569
$569$	1.15195e12	0.460710	0.230355	−	0.973107i	$-0.426011\pi$
$569$	1.15195e12	0.460710	0.230355	+	0.973107i	$0.426011\pi$
$570$	1.10099e11	0.0436866
$571$	−2.60662e12	−1.02616	−0.513081	−	0.858340i	$-0.671496\pi$
$571$	−2.60662e12	−1.02616	−0.513081	+	0.858340i	$0.671496\pi$
$572$	0	0
$573$	−3.07912e12	−1.19325
$574$	−9.11433e11	−0.350446
$575$	−2.39514e12	−0.913744
$576$	−1.76452e12	−0.667921
$577$	6.62796e11	0.248936	0.124468	−	0.992224i	$-0.460278\pi$
$577$	6.62796e11	0.248936	0.124468	+	0.992224i	$0.460278\pi$
$578$	−2.31571e11	−0.0862995
$579$	−1.81693e10	−0.00671867
$580$	6.19381e11	0.227265
$581$	2.60666e12	0.949055
$582$	−2.63450e11	−0.0951799
$583$	3.77037e12	1.35169
$584$	1.86190e10	0.00662366
$585$	0	0
$586$	−4.58198e10	−0.0160514
$587$	2.95869e12	1.02856	0.514278	−	0.857623i	$-0.328060\pi$
$587$	2.95869e12	1.02856	0.514278	+	0.857623i	$0.328060\pi$
$588$	−7.34936e12	−2.53543
$589$	−4.90897e11	−0.168063
$590$	1.10815e10	0.00376498
$591$	3.95740e12	1.33434
$592$	−3.74557e10	−0.0125334
$593$	1.38060e12	0.458482	0.229241	−	0.973370i	$-0.426376\pi$
$593$	1.38060e12	0.458482	0.229241	+	0.973370i	$0.426376\pi$
$594$	−2.02345e11	−0.0666890
$595$	−1.07993e12	−0.353240
$596$	1.03699e12	0.336641
$597$	5.63756e12	1.81638
$598$	0	0
$599$	−1.25780e12	−0.399202	−0.199601	−	0.979877i	$-0.563965\pi$
$599$	−1.25780e12	−0.399202	−0.199601	+	0.979877i	$0.563965\pi$
$600$	7.81206e11	0.246085
$601$	4.79769e12	1.50002	0.750011	−	0.661426i	$-0.230048\pi$
$601$	4.79769e12	1.50002	0.750011	+	0.661426i	$0.230048\pi$
$602$	5.83546e11	0.181089
$603$	1.14471e12	0.352588
$604$	3.20730e10	0.00980557
$605$	2.19294e12	0.665470
$606$	−8.74545e11	−0.263424
$607$	−2.66431e12	−0.796592	−0.398296	−	0.917257i	$-0.630398\pi$
$607$	−2.66431e12	−0.796592	−0.398296	+	0.917257i	$0.630398\pi$
$608$	7.94110e11	0.235675
$609$	−4.23951e12	−1.24893
$610$	−5.68698e10	−0.0166302
$611$	0	0
$612$	1.22373e12	0.352618
$613$	3.53410e12	1.01090	0.505449	−	0.862857i	$-0.331327\pi$
$613$	3.53410e12	1.01090	0.505449	+	0.862857i	$0.331327\pi$
$614$	5.41966e11	0.153891
$615$	−3.46320e12	−0.976203
$616$	−2.24571e12	−0.628405
$617$	−2.74865e12	−0.763548	−0.381774	−	0.924256i	$-0.624687\pi$
$617$	−2.74865e12	−0.763548	−0.381774	+	0.924256i	$0.624687\pi$
$618$	−7.11372e11	−0.196177
$619$	−2.63721e12	−0.721999	−0.360999	−	0.932566i	$-0.617564\pi$
$619$	−2.63721e12	−0.721999	−0.360999	+	0.932566i	$0.617564\pi$
$620$	3.62545e11	0.0985370
$621$	−1.48512e12	−0.400727
$622$	−3.45290e11	−0.0924970
$623$	3.30496e10	0.00878961
$624$	0	0
$625$	1.94659e12	0.510286
$626$	−1.48726e11	−0.0387082
$627$	5.73919e12	1.48302
$628$	−1.84468e12	−0.473264
$629$	2.52680e10	0.00643640
$630$	2.33702e11	0.0591057
$631$	−3.28043e12	−0.823756	−0.411878	−	0.911239i	$-0.635127\pi$
$631$	−3.28043e12	−0.823756	−0.411878	+	0.911239i	$0.635127\pi$
$632$	−4.86606e11	−0.121325
$633$	−8.26101e12	−2.04511
$634$	−1.21028e11	−0.0297499
$635$	2.08928e12	0.509935
$636$	−4.48335e12	−1.08654
$637$	0	0
$638$	−4.25835e11	−0.101753
$639$	7.08594e11	0.168129
$640$	−7.80199e11	−0.183821
$641$	2.35043e12	0.549902	0.274951	−	0.961458i	$-0.411338\pi$
$641$	2.35043e12	0.549902	0.274951	+	0.961458i	$0.411338\pi$
$642$	8.26190e11	0.191943
$643$	3.47432e12	0.801530	0.400765	−	0.916181i	$-0.368744\pi$
$643$	3.47432e12	0.801530	0.400765	+	0.916181i	$0.368744\pi$
$644$	−8.18720e12	−1.87564
$645$	2.21732e12	0.504441
$646$	−1.74650e11	−0.0394568
$647$	5.29323e12	1.18755	0.593775	−	0.804631i	$-0.297637\pi$
$647$	5.29323e12	1.18755	0.593775	+	0.804631i	$0.297637\pi$
$648$	1.22092e12	0.272020
$649$	5.77646e11	0.127809
$650$	0	0
$651$	−2.48153e12	−0.541509
$652$	7.36736e12	1.59661
$653$	−3.38625e12	−0.728803	−0.364401	−	0.931242i	$-0.618726\pi$
$653$	−3.38625e12	−0.728803	−0.364401	+	0.931242i	$0.618726\pi$
$654$	6.67651e11	0.142708
$655$	3.43744e12	0.729708
$656$	−8.14343e12	−1.71688
$657$	−1.01002e11	−0.0211488
$658$	9.34222e11	0.194282
$659$	2.94724e12	0.608738	0.304369	−	0.952554i	$-0.401554\pi$
$659$	2.94724e12	0.608738	0.304369	+	0.952554i	$0.401554\pi$
$660$	−4.23859e12	−0.869509
$661$	−2.08486e11	−0.0424786	−0.0212393	−	0.999774i	$-0.506761\pi$
$661$	−2.08486e11	−0.0424786	−0.0212393	+	0.999774i	$0.506761\pi$
$662$	2.66316e11	0.0538935
$663$	0	0
$664$	−6.26776e11	−0.125128
$665$	2.52887e12	0.501451
$666$	−5.46811e9	−0.00107697
$667$	−3.12542e12	−0.611424
$668$	3.14579e12	0.611275
$669$	−2.85163e12	−0.550396
$670$	1.20656e11	0.0231320
$671$	−2.96447e12	−0.564541
$672$	4.01430e12	0.759361
$673$	−5.65219e12	−1.06206	−0.531030	−	0.847353i	$-0.678195\pi$
$673$	−5.65219e12	−1.06206	−0.531030	+	0.847353i	$0.678195\pi$
$674$	−3.64877e11	−0.0681047
$675$	1.61677e12	0.299765
$676$	0	0
$677$	6.03672e12	1.10447	0.552233	−	0.833690i	$-0.313776\pi$
$677$	6.03672e12	1.10447	0.552233	+	0.833690i	$0.313776\pi$
$678$	8.72082e11	0.158498
$679$	−6.05118e12	−1.09251
$680$	2.59671e11	0.0465730
$681$	1.22791e13	2.18778
$682$	−2.49256e11	−0.0441179
$683$	2.45213e12	0.431173	0.215586	−	0.976485i	$-0.430834\pi$
$683$	2.45213e12	0.431173	0.215586	+	0.976485i	$0.430834\pi$
$684$	−2.86560e12	−0.500569
$685$	3.71178e12	0.644131
$686$	1.08850e12	0.187659
$687$	−5.65004e12	−0.967712
$688$	5.21385e12	0.887177
$689$	0	0
$690$	4.10305e11	0.0689106
$691$	−4.98454e12	−0.831714	−0.415857	−	0.909430i	$-0.636518\pi$
$691$	−4.98454e12	−0.831714	−0.415857	+	0.909430i	$0.636518\pi$
$692$	9.40509e12	1.55914
$693$	1.21822e13	2.00645
$694$	3.73693e10	0.00611501
$695$	7.11551e11	0.115684
$696$	1.01940e12	0.164665
$697$	5.49365e12	0.881685
$698$	−5.30524e11	−0.0845971
$699$	−8.64052e12	−1.36897
$700$	8.91296e12	1.40308
$701$	−1.40532e12	−0.219808	−0.109904	−	0.993942i	$-0.535054\pi$
$701$	−1.40532e12	−0.219808	−0.109904	+	0.993942i	$0.535054\pi$
$702$	0	0
$703$	−5.91700e10	−0.00913698
$704$	−9.69494e12	−1.48754
$705$	3.54980e12	0.541194
$706$	−1.01876e12	−0.154329
$707$	−2.00874e13	−3.02368
$708$	−6.86879e11	−0.102738
$709$	7.02930e12	1.04473	0.522365	−	0.852722i	$-0.325050\pi$
$709$	7.02930e12	1.04473	0.522365	+	0.852722i	$0.325050\pi$
$710$	7.46880e10	0.0110303
$711$	2.63968e12	0.387381
$712$	−7.94683e9	−0.00115887
$713$	−1.82941e12	−0.265100
$714$	−8.82871e11	−0.127132
$715$	0	0
$716$	3.70707e12	0.527136
$717$	−1.07585e13	−1.52026
$718$	−7.45315e11	−0.104660
$719$	6.77735e12	0.945758	0.472879	−	0.881127i	$-0.343215\pi$
$719$	6.77735e12	0.945758	0.472879	+	0.881127i	$0.343215\pi$
$720$	2.08807e12	0.289567
$721$	−1.63395e13	−2.25180
$722$	−4.24093e11	−0.0580823
$723$	5.28920e12	0.719893
$724$	−7.48084e12	−1.01188
$725$	3.40248e12	0.457377
$726$	1.79279e12	0.239505
$727$	1.09901e13	1.45914	0.729570	−	0.683907i	$-0.239720\pi$
$727$	1.09901e13	1.45914	0.729570	+	0.683907i	$0.239720\pi$
$728$	0	0
$729$	−2.99296e12	−0.392488
$730$	−1.06459e10	−0.00138749
$731$	−3.51732e12	−0.455600
$732$	3.52505e12	0.453801
$733$	−3.08386e12	−0.394573	−0.197286	−	0.980346i	$-0.563213\pi$
$733$	−3.08386e12	−0.394573	−0.197286	+	0.980346i	$0.563213\pi$
$734$	4.38430e10	0.00557530
$735$	8.45984e12	1.06922
$736$	2.95939e12	0.371751
$737$	6.28948e12	0.785255
$738$	−1.18885e12	−0.147528
$739$	−1.06215e13	−1.31005	−0.655025	−	0.755608i	$-0.727342\pi$
$739$	−1.06215e13	−1.31005	−0.655025	+	0.755608i	$0.727342\pi$
$740$	4.36992e10	0.00535711
$741$	0	0
$742$	1.35819e12	0.164492
$743$	−1.46900e13	−1.76837	−0.884186	−	0.467136i	$-0.845286\pi$
$743$	−1.46900e13	−1.76837	−0.884186	+	0.467136i	$0.845286\pi$
$744$	5.96689e11	0.0713953
$745$	−1.19368e12	−0.141966
$746$	9.19769e10	0.0108731
$747$	3.40006e12	0.399525
$748$	6.72364e12	0.785321
$749$	1.89767e13	2.20319
$750$	−9.86953e11	−0.113899
$751$	3.27035e12	0.375159	0.187579	−	0.982249i	$-0.439936\pi$
$751$	3.27035e12	0.375159	0.187579	+	0.982249i	$0.439936\pi$
$752$	8.34705e12	0.951816
$753$	−9.10742e12	−1.03233
$754$	0	0
$755$	−3.69192e10	−0.00413514
$756$	5.52653e12	0.615325
$757$	1.22014e13	1.35045	0.675223	−	0.737613i	$-0.264047\pi$
$757$	1.22014e13	1.35045	0.675223	+	0.737613i	$0.264047\pi$
$758$	−1.75575e12	−0.193175
$759$	2.13881e13	2.33929
$760$	−6.08071e11	−0.0661140
$761$	−1.85616e12	−0.200625	−0.100312	−	0.994956i	$-0.531984\pi$
$761$	−1.85616e12	−0.200625	−0.100312	+	0.994956i	$0.531984\pi$
$762$	1.70804e12	0.183527
$763$	1.53352e13	1.63806
$764$	8.44716e12	0.896996
$765$	−1.40863e12	−0.148704
$766$	−6.79117e11	−0.0712714
$767$	0	0
$768$	1.10425e13	1.14536
$769$	−4.43851e12	−0.457687	−0.228843	−	0.973463i	$-0.573494\pi$
$769$	−4.43851e12	−0.457687	−0.228843	+	0.973463i	$0.573494\pi$
$770$	1.28405e12	0.131635
$771$	−1.00385e13	−1.02311
$772$	4.98450e10	0.00505061
$773$	3.37609e11	0.0340100	0.0170050	−	0.999855i	$-0.494587\pi$
$773$	3.37609e11	0.0340100	0.0170050	+	0.999855i	$0.494587\pi$
$774$	7.61163e11	0.0762330
$775$	1.99158e12	0.198308
$776$	1.45502e12	0.144042
$777$	−2.99110e11	−0.0294399
$778$	−1.44498e12	−0.141402
$779$	−1.28644e13	−1.25162
$780$	0	0
$781$	3.89328e12	0.374444
$782$	−6.50863e11	−0.0622385
$783$	2.10972e12	0.200585
$784$	1.98926e13	1.88048
$785$	2.12341e12	0.199582
$786$	2.81019e12	0.262624
$787$	−3.42768e12	−0.318503	−0.159252	−	0.987238i	$-0.550908\pi$
$787$	−3.42768e12	−0.318503	−0.159252	+	0.987238i	$0.550908\pi$
$788$	−1.08566e13	−1.00306
$789$	5.42564e12	0.498431
$790$	2.78231e11	0.0254146
$791$	2.00308e13	1.81930
$792$	−2.92924e12	−0.264540
$793$	0	0
$794$	1.31700e12	0.117597
$795$	5.16078e12	0.458208
$796$	−1.54659e13	−1.36542
$797$	2.09842e13	1.84218	0.921088	−	0.389355i	$-0.127302\pi$
$797$	2.09842e13	1.84218	0.921088	+	0.389355i	$0.127302\pi$
$798$	2.06741e12	0.180474
$799$	−5.63101e12	−0.488794
$800$	−3.22173e12	−0.278089
$801$	4.31090e10	0.00370017
$802$	1.36999e12	0.116932
$803$	−5.54944e11	−0.0471009
$804$	−7.47882e12	−0.631220
$805$	9.42428e12	0.790982
$806$	0	0
$807$	−5.16853e11	−0.0428979
$808$	4.83004e12	0.398658
$809$	2.08611e13	1.71226	0.856130	−	0.516761i	$-0.172862\pi$
$809$	2.08611e13	1.71226	0.856130	+	0.516761i	$0.172862\pi$
$810$	−6.98097e11	−0.0569814
$811$	−1.47113e13	−1.19415	−0.597073	−	0.802187i	$-0.703670\pi$
$811$	−1.47113e13	−1.19415	−0.597073	+	0.802187i	$0.703670\pi$
$812$	1.16306e13	0.938855
$813$	−1.86953e13	−1.50080
$814$	−3.00439e10	−0.00239854
$815$	−8.48056e12	−0.673310
$816$	−7.88825e12	−0.622837
$817$	8.23648e12	0.646759
$818$	8.05792e11	0.0629264
$819$	0	0
$820$	9.50086e12	0.733838
$821$	3.30223e11	0.0253667	0.0126833	−	0.999920i	$-0.495963\pi$
$821$	3.30223e11	0.0253667	0.0126833	+	0.999920i	$0.495963\pi$
$822$	3.03447e12	0.231825
$823$	−1.88707e13	−1.43380	−0.716898	−	0.697178i	$-0.754439\pi$
$823$	−1.88707e13	−1.43380	−0.716898	+	0.697178i	$0.754439\pi$
$824$	3.92886e12	0.296889
$825$	−2.32841e13	−1.74991
$826$	2.08084e11	0.0155535
$827$	−7.40416e12	−0.550428	−0.275214	−	0.961383i	$-0.588749\pi$
$827$	−7.40416e12	−0.550428	−0.275214	+	0.961383i	$0.588749\pi$
$828$	−1.06792e13	−0.789590
$829$	−1.44577e13	−1.06318	−0.531588	−	0.847003i	$-0.678405\pi$
$829$	−1.44577e13	−1.06318	−0.531588	+	0.847003i	$0.678405\pi$
$830$	3.58377e11	0.0262113
$831$	3.15629e12	0.229600
$832$	0	0
$833$	−1.34198e13	−0.965700
$834$	5.81711e11	0.0416351
$835$	−3.62112e12	−0.257783
$836$	−1.57447e13	−1.11482
$837$	1.23489e12	0.0869691
$838$	−3.69831e11	−0.0259063
$839$	2.38597e13	1.66241	0.831203	−	0.555969i	$-0.187653\pi$
$839$	2.38597e13	1.66241	0.831203	+	0.555969i	$0.187653\pi$
$840$	−3.07386e12	−0.213023
$841$	−1.00672e13	−0.693951
$842$	2.61504e12	0.179297
$843$	−2.81543e13	−1.92009
$844$	2.26630e13	1.53736
$845$	0	0
$846$	1.21858e12	0.0817873
$847$	4.11785e13	2.74913
$848$	1.21351e13	0.805867
$849$	1.24620e13	0.823196
$850$	7.08560e11	0.0465577
$851$	−2.20508e11	−0.0144125
$852$	−4.62950e12	−0.300993
$853$	−2.68153e13	−1.73425	−0.867126	−	0.498089i	$-0.834035\pi$
$853$	−2.68153e13	−1.73425	−0.867126	+	0.498089i	$0.834035\pi$
$854$	−1.06788e12	−0.0687011
$855$	3.29859e12	0.211097
$856$	−4.56299e12	−0.290481
$857$	2.11963e13	1.34229	0.671146	−	0.741326i	$-0.265803\pi$
$857$	2.11963e13	1.34229	0.671146	+	0.741326i	$0.265803\pi$
$858$	0	0
$859$	−2.07483e13	−1.30021	−0.650105	−	0.759845i	$-0.725275\pi$
$859$	−2.07483e13	−1.30021	−0.650105	+	0.759845i	$0.725275\pi$
$860$	−6.08294e12	−0.379202
$861$	−6.50310e13	−4.03280
$862$	9.31372e11	0.0574567
$863$	−2.10953e13	−1.29461	−0.647304	−	0.762232i	$-0.724103\pi$
$863$	−2.10953e13	−1.29461	−0.647304	+	0.762232i	$0.724103\pi$
$864$	−1.99765e12	−0.121957
$865$	−1.08262e13	−0.657511
$866$	2.18739e12	0.132159
$867$	−1.65226e13	−0.993102
$868$	6.80776e12	0.407067
$869$	1.45034e13	0.862744
$870$	−5.82871e11	−0.0344934
$871$	0	0
$872$	−3.68739e12	−0.215970
$873$	−7.89300e12	−0.459916
$874$	1.52412e12	0.0883525
$875$	−2.26693e13	−1.30738
$876$	6.59884e11	0.0378616
$877$	4.15688e12	0.237284	0.118642	−	0.992937i	$-0.462146\pi$
$877$	4.15688e12	0.237284	0.118642	+	0.992937i	$0.462146\pi$
$878$	−1.27685e12	−0.0725127
$879$	−3.26926e12	−0.184714
$880$	1.14727e13	0.644899
$881$	−9.05261e12	−0.506270	−0.253135	−	0.967431i	$-0.581462\pi$
$881$	−9.05261e12	−0.506270	−0.253135	+	0.967431i	$0.581462\pi$
$882$	2.90410e12	0.161585
$883$	−1.99914e13	−1.10667	−0.553337	−	0.832957i	$-0.686646\pi$
$883$	−1.99914e13	−1.10667	−0.553337	+	0.832957i	$0.686646\pi$
$884$	0	0
$885$	7.90666e11	0.0433260
$886$	−2.67624e12	−0.145906
$887$	2.32772e10	0.00126262	0.000631311	−	1.00000i	$-0.499799\pi$
$887$	2.32772e10	0.00126262	0.000631311		1.00000i	$0.499799\pi$
$888$	7.19216e10	0.00388151
$889$	3.92319e13	2.10660
$890$	4.54383e9	0.000242754 0
$891$	−3.63899e13	−1.93433
$892$	7.82308e12	0.413748
$893$	1.31861e13	0.693881
$894$	−9.75864e11	−0.0510940
$895$	−4.26721e12	−0.222300
$896$	−1.46504e13	−0.759385
$897$	0	0
$898$	−9.31349e11	−0.0477935
$899$	2.59883e12	0.132696
$900$	1.16258e13	0.590654
$901$	−8.18649e12	−0.413844
$902$	−6.53199e12	−0.328561
$903$	4.16362e13	2.08390
$904$	−4.81644e12	−0.239866
$905$	8.61119e12	0.426721
$906$	−3.01824e10	−0.00148825
$907$	1.40339e13	0.688568	0.344284	−	0.938866i	$-0.388122\pi$
$907$	1.40339e13	0.688568	0.344284	+	0.938866i	$0.388122\pi$
$908$	−3.36861e13	−1.64462
$909$	−2.62015e13	−1.27288
$910$	0	0
$911$	3.49044e13	1.67899	0.839494	−	0.543368i	$-0.182851\pi$
$911$	3.49044e13	1.67899	0.839494	+	0.543368i	$0.182851\pi$
$912$	1.84719e13	0.884166
$913$	1.86812e13	0.889789
$914$	−2.18537e12	−0.103578
$915$	−4.05768e12	−0.191374
$916$	1.55002e13	0.727455
$917$	6.45472e13	3.01450
$918$	4.39346e11	0.0204181
$919$	−3.60628e13	−1.66779	−0.833893	−	0.551926i	$-0.813893\pi$
$919$	−3.60628e13	−1.66779	−0.833893	+	0.551926i	$0.813893\pi$
$920$	−2.26609e12	−0.104287
$921$	3.86695e13	1.77092
$922$	1.76032e11	0.00802234
$923$	0	0
$924$	−7.95911e13	−3.59203
$925$	2.40055e11	0.0107813
$926$	−4.55522e12	−0.203591
$927$	−2.13128e13	−0.947941
$928$	−4.20405e12	−0.186081
$929$	3.08860e13	1.36047	0.680237	−	0.732992i	$-0.261877\pi$
$929$	3.08860e13	1.36047	0.680237	+	0.732992i	$0.261877\pi$
$930$	−3.41174e11	−0.0149556
$931$	3.14250e13	1.37089
$932$	2.37042e13	1.02909
$933$	−2.46366e13	−1.06442
$934$	2.98209e12	0.128221
$935$	−7.73958e12	−0.331181
$936$	0	0
$937$	1.01258e12	0.0429143	0.0214572	−	0.999770i	$-0.493169\pi$
$937$	1.01258e12	0.0429143	0.0214572	+	0.999770i	$0.493169\pi$
$938$	2.26565e12	0.0955607
$939$	−1.06117e13	−0.445439
$940$	−9.73842e12	−0.406830
$941$	−3.84597e13	−1.59901	−0.799507	−	0.600656i	$-0.794906\pi$
$941$	−3.84597e13	−1.59901	−0.799507	+	0.600656i	$0.794906\pi$
$942$	1.73594e12	0.0718301
$943$	−4.79417e13	−1.97429
$944$	1.85918e12	0.0761989
$945$	−6.36159e12	−0.259491
$946$	4.18212e12	0.169780
$947$	−1.37376e13	−0.555055	−0.277528	−	0.960718i	$-0.589515\pi$
$947$	−1.37376e13	−0.555055	−0.277528	+	0.960718i	$0.589515\pi$
$948$	−1.72460e13	−0.693508
$949$	0	0
$950$	−1.65923e12	−0.0660922
$951$	−8.63542e12	−0.342351
$952$	4.87603e12	0.192398
$953$	−2.01820e13	−0.792587	−0.396293	−	0.918124i	$-0.629704\pi$
$953$	−2.01820e13	−0.792587	−0.396293	+	0.918124i	$0.629704\pi$
$954$	1.77159e12	0.0692463
$955$	−9.72351e12	−0.378275
$956$	2.95146e13	1.14282
$957$	−3.03835e13	−1.17094
$958$	−3.59273e12	−0.137810
$959$	6.96987e13	2.66097
$960$	−1.32702e13	−0.504261
$961$	−2.49184e13	−0.942466
$962$	0	0
$963$	2.47528e13	0.927481
$964$	−1.45103e13	−0.541163
$965$	−5.73765e10	−0.00212991
$966$	7.70459e12	0.284677
$967$	−4.05710e13	−1.49210	−0.746048	−	0.665893i	$-0.768051\pi$
$967$	−4.05710e13	−1.49210	−0.746048	+	0.665893i	$0.768051\pi$
$968$	−9.90144e12	−0.362459
$969$	−1.24613e13	−0.454053
$970$	−8.31948e11	−0.0301733
$971$	−1.06583e13	−0.384770	−0.192385	−	0.981319i	$-0.561622\pi$
$971$	−1.06583e13	−0.384770	−0.192385	+	0.981319i	$0.561622\pi$
$972$	3.33124e13	1.19704
$973$	1.33613e13	0.477904
$974$	5.94571e11	0.0211684
$975$	0	0
$976$	−9.54129e12	−0.336576
$977$	−3.94428e13	−1.38498	−0.692488	−	0.721429i	$-0.743486\pi$
$977$	−3.94428e13	−1.38498	−0.692488	+	0.721429i	$0.743486\pi$
$978$	−6.93307e12	−0.242327
$979$	2.36857e11	0.00824071
$980$	−2.32085e13	−0.803766
$981$	2.00029e13	0.689576
$982$	6.58073e11	0.0225825
$983$	−4.21803e13	−1.44085	−0.720426	−	0.693532i	$-0.756054\pi$
$983$	−4.21803e13	−1.44085	−0.720426	+	0.693532i	$0.756054\pi$
$984$	1.56368e13	0.531705
$985$	1.24970e13	0.423004
$986$	9.24602e11	0.0311536
$987$	6.66571e13	2.23573
$988$	0	0
$989$	3.06947e13	1.02019
$990$	1.67488e12	0.0554147
$991$	−3.05842e13	−1.00731	−0.503657	−	0.863904i	$-0.668013\pi$
$991$	−3.05842e13	−1.00731	−0.503657	+	0.863904i	$0.668013\pi$
$992$	−2.46077e12	−0.0806805
$993$	1.90018e13	0.620186
$994$	1.40247e12	0.0455674
$995$	1.78028e13	0.575817
$996$	−2.22138e13	−0.715248
$997$	3.48998e13	1.11865	0.559325	−	0.828948i	$-0.311060\pi$
$997$	3.48998e13	1.11865	0.559325	+	0.828948i	$0.311060\pi$
$998$	−8.17632e11	−0.0260898
$999$	1.48847e11	0.00472820

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.10.a.f.1.11		20
13.2	odd	12		13.10.e.a.4.6	✓	20
13.7	odd	12		13.10.e.a.10.6	yes	20
13.12	even	2	inner	169.10.a.f.1.10		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.10.e.a.4.6	✓	20	13.2	odd	12
13.10.e.a.10.6	yes	20	13.7	odd	12
169.10.a.f.1.10		20	13.12	even	2	inner
169.10.a.f.1.11		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)

\(f(q)\)	\(=\)	\(q+2.58166 q^{2} +184.202 q^{3} -505.335 q^{4} +581.690 q^{5} +475.547 q^{6} +10922.8 q^{7} -2626.41 q^{8} +14247.4 q^{9} +1501.73 q^{10} +78280.9 q^{11} -93083.8 q^{12} +28199.0 q^{14} +107149. q^{15} +251951. q^{16} -169969. q^{17} +36782.0 q^{18} +398015. q^{19} -293949. q^{20} +2.01201e6 q^{21} +202094. q^{22} +1.48328e6 q^{23} -483791. q^{24} -1.61476e6 q^{25} -1.00124e6 q^{27} -5.51968e6 q^{28} -2.10711e6 q^{29} +276621. q^{30} -1.23336e6 q^{31} +1.99517e6 q^{32} +1.44195e7 q^{33} -438802. q^{34} +6.35370e6 q^{35} -7.19973e6 q^{36} -148663. q^{37} +1.02754e6 q^{38} -1.52776e6 q^{40} -3.23215e7 q^{41} +5.19431e6 q^{42} +2.06939e7 q^{43} -3.95581e7 q^{44} +8.28760e6 q^{45} +3.82931e6 q^{46} +3.31297e7 q^{47} +4.64099e7 q^{48} +7.89542e7 q^{49} -4.16876e6 q^{50} -3.13086e7 q^{51} +4.81647e7 q^{53} -2.58486e6 q^{54} +4.55352e7 q^{55} -2.86878e7 q^{56} +7.33153e7 q^{57} -5.43983e6 q^{58} +7.37915e6 q^{59} -5.41460e7 q^{60} -3.78696e7 q^{61} -3.18412e6 q^{62} +1.55622e8 q^{63} -1.23848e8 q^{64} +3.72262e7 q^{66} +8.03450e7 q^{67} +8.58912e7 q^{68} +2.73223e8 q^{69} +1.64031e7 q^{70} +4.97348e7 q^{71} -3.74196e7 q^{72} -7.08914e6 q^{73} -383796. q^{74} -2.97443e8 q^{75} -2.01131e8 q^{76} +8.55047e8 q^{77} +1.85274e8 q^{79} +1.46558e8 q^{80} -4.64863e8 q^{81} -8.34430e7 q^{82} +2.38643e8 q^{83} -1.01674e9 q^{84} -9.88693e7 q^{85} +5.34245e7 q^{86} -3.88134e8 q^{87} -2.05598e8 q^{88} +3.02574e6 q^{89} +2.13957e7 q^{90} -7.49551e8 q^{92} -2.27188e8 q^{93} +8.55294e7 q^{94} +2.31522e8 q^{95} +3.67515e8 q^{96} -5.53994e8 q^{97} +2.03833e8 q^{98} +1.11530e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 326 q^{3} + 5118 q^{4} + 129526 q^{9} + 88390 q^{10} + 427652 q^{12} + 473556 q^{14} + 1189618 q^{16} - 99312 q^{17} - 5073532 q^{22} + 6252378 q^{23} + 1529274 q^{25} + 18052718 q^{27} + 5424828 q^{29}+ \cdots + 9251202540 q^{95}+O(q^{100}) \) 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

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