LMFDB - Embedded newform 1014.4.b.q.337.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1014,4,Mod(337,1014)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1014, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1014.337");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1014 = 2 \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1014.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$59.8279367458$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 845x^{10} + 287958x^{8} + 50362537x^{6} + 4731667920x^{4} + 224458698240x^{2} + 4178851762176 $ x^12 + 845x^10 + 287958x^8 + 50362537x^6 + 4731667920x^4 + 224458698240*x^2 + 4178851762176
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 13^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.3
Root			$-14.7697i$ of defining polynomial
Character	$\chi$	$=$	1014.337
Dual form			1014.4.b.q.337.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{2} +3.00000 q^{3} -4.00000 q^{4} -9.80890i q^{5} -6.00000i q^{6} +11.4763i q^{7} +8.00000i q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-2.00000i q^{2} +3.00000 q^{3} -4.00000 q^{4} -9.80890i q^{5} -6.00000i q^{6} +11.4763i q^{7} +8.00000i q^{8} +9.00000 q^{9} -19.6178 q^{10} +19.6826i q^{11} -12.0000 q^{12} +22.9525 q^{14} -29.4267i q^{15} +16.0000 q^{16} -25.5010 q^{17} -18.0000i q^{18} +4.32083i q^{19} +39.2356i q^{20} +34.4288i q^{21} +39.3653 q^{22} +33.5416 q^{23} +24.0000i q^{24} +28.7854 q^{25} +27.0000 q^{27} -45.9051i q^{28} +116.838 q^{29} -58.8534 q^{30} -288.350i q^{31} -32.0000i q^{32} +59.0479i q^{33} +51.0020i q^{34} +112.570 q^{35} -36.0000 q^{36} +124.747i q^{37} +8.64165 q^{38} +78.4712 q^{40} +142.268i q^{41} +68.8576 q^{42} +228.871 q^{43} -78.7305i q^{44} -88.2801i q^{45} -67.0831i q^{46} -598.729i q^{47} +48.0000 q^{48} +211.295 q^{49} -57.5708i q^{50} -76.5030 q^{51} +119.901 q^{53} -54.0000i q^{54} +193.065 q^{55} -91.8101 q^{56} +12.9625i q^{57} -233.676i q^{58} +499.881i q^{59} +117.707i q^{60} +696.612 q^{61} -576.701 q^{62} +103.286i q^{63} -64.0000 q^{64} +118.096 q^{66} +67.8564i q^{67} +102.004 q^{68} +100.625 q^{69} -225.139i q^{70} -538.072i q^{71} +72.0000i q^{72} -1214.82i q^{73} +249.495 q^{74} +86.3562 q^{75} -17.2833i q^{76} -225.883 q^{77} +345.554 q^{79} -156.942i q^{80} +81.0000 q^{81} +284.536 q^{82} +1160.79i q^{83} -137.715i q^{84} +250.137i q^{85} -457.741i q^{86} +350.514 q^{87} -157.461 q^{88} -726.029i q^{89} -176.560 q^{90} -134.166 q^{92} -865.051i q^{93} -1197.46 q^{94} +42.3826 q^{95} -96.0000i q^{96} -598.944i q^{97} -422.591i q^{98} +177.144i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 36 q^{3} - 48 q^{4} + 108 q^{9}+O(q^{10}) $ 12 * q + 36 * q^3 - 48 * q^4 + 108 * q^9	$ 12 q + 36 q^{3} - 48 q^{4} + 108 q^{9} + 12 q^{10} - 144 q^{12} - 4 q^{14} + 192 q^{16} - 102 q^{17} + 256 q^{22} - 444 q^{23} - 370 q^{25} + 324 q^{27} + 658 q^{29} + 36 q^{30} + 1688 q^{35} - 432 q^{36} - 852 q^{38} - 48 q^{40} - 12 q^{42} - 982 q^{43} + 576 q^{48} - 2266 q^{49} - 306 q^{51} + 4604 q^{53} - 658 q^{55} + 16 q^{56} + 690 q^{61} - 1156 q^{62} - 768 q^{64} + 768 q^{66} + 408 q^{68} - 1332 q^{69} + 1880 q^{74} - 1110 q^{75} - 6582 q^{77} + 6200 q^{79} + 972 q^{81} + 1284 q^{82} + 1974 q^{87} - 1024 q^{88} + 108 q^{90} + 1776 q^{92} - 2564 q^{94} + 1330 q^{95}+O(q^{100}) $ 12 * q + 36 * q^3 - 48 * q^4 + 108 * q^9 + 12 * q^10 - 144 * q^12 - 4 * q^14 + 192 * q^16 - 102 * q^17 + 256 * q^22 - 444 * q^23 - 370 * q^25 + 324 * q^27 + 658 * q^29 + 36 * q^30 + 1688 * q^35 - 432 * q^36 - 852 * q^38 - 48 * q^40 - 12 * q^42 - 982 * q^43 + 576 * q^48 - 2266 * q^49 - 306 * q^51 + 4604 * q^53 - 658 * q^55 + 16 * q^56 + 690 * q^61 - 1156 * q^62 - 768 * q^64 + 768 * q^66 + 408 * q^68 - 1332 * q^69 + 1880 * q^74 - 1110 * q^75 - 6582 * q^77 + 6200 * q^79 + 972 * q^81 + 1284 * q^82 + 1974 * q^87 - 1024 * q^88 + 108 * q^90 + 1776 * q^92 - 2564 * q^94 + 1330 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1014\mathbb{Z}\right)^\times$.

$n$	$677$	$847$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 2.00000i	− 0.707107i
$2$	− 2.00000i	− 0.707107i
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−4.00000	−0.500000
$5$	− 9.80890i	− 0.877335i	−0.898649	−	0.438667i	$-0.855451\pi$
$5$	− 9.80890i	− 0.877335i	0.898649	−	0.438667i	$-0.144549\pi$
$6$	− 6.00000i	− 0.408248i
$7$	11.4763i	0.619660i	0.950792	+	0.309830i	$0.100272\pi$
$7$	11.4763i	0.619660i	−0.950792	+	0.309830i	$0.899728\pi$
$8$	8.00000i	0.353553i
$9$	9.00000	0.333333
$10$	−19.6178	−0.620370
$11$	19.6826i	0.539503i	0.962930	+	0.269752i	$0.0869416\pi$
$11$	19.6826i	0.539503i	−0.962930	+	0.269752i	$0.913058\pi$
$12$	−12.0000	−0.288675
$13$	0	0
$13$	0	0
$14$	22.9525	0.438166
$15$	− 29.4267i	− 0.506530i
$16$	16.0000	0.250000
$17$	−25.5010	−0.363818	−0.181909	−	0.983315i	$-0.558228\pi$
$17$	−25.5010	−0.363818	−0.181909	+	0.983315i	$0.558228\pi$
$18$	− 18.0000i	− 0.235702i
$19$	4.32083i	0.0521719i	0.999660	+	0.0260859i	$0.00830435\pi$
$19$	4.32083i	0.0521719i	−0.999660	+	0.0260859i	$0.991696\pi$
$20$	39.2356i	0.438667i
$21$	34.4288i	0.357761i
$22$	39.3653	0.381487
$23$	33.5416	0.304083	0.152041	−	0.988374i	$-0.451415\pi$
$23$	33.5416	0.304083	0.152041	+	0.988374i	$0.451415\pi$
$24$	24.0000i	0.204124i
$25$	28.7854	0.230283
$26$	0	0
$27$	27.0000	0.192450
$28$	− 45.9051i	− 0.309830i
$29$	116.838	0.748147	0.374074	−	0.927399i	$-0.377961\pi$
$29$	116.838	0.748147	0.374074	+	0.927399i	$0.377961\pi$
$30$	−58.8534	−0.358171
$31$	− 288.350i	− 1.67062i	−0.549778	−	0.835311i	$-0.685288\pi$
$31$	− 288.350i	− 1.67062i	0.549778	−	0.835311i	$-0.314712\pi$
$32$	− 32.0000i	− 0.176777i
$33$	59.0479i	0.311482i
$34$	51.0020i	0.257258i
$35$	112.570	0.543649
$36$	−36.0000	−0.166667
$37$	124.747i	0.554279i	0.960830	+	0.277140i	$0.0893865\pi$
$37$	124.747i	0.554279i	−0.960830	+	0.277140i	$0.910614\pi$
$38$	8.64165	0.0368911
$39$	0	0
$40$	78.4712	0.310185
$41$	142.268i	0.541916i	0.962591	+	0.270958i	$0.0873405\pi$
$41$	142.268i	0.541916i	−0.962591	+	0.270958i	$0.912660\pi$
$42$	68.8576	0.252975
$43$	228.871	0.811685	0.405842	−	0.913943i	$-0.366978\pi$
$43$	228.871	0.811685	0.405842	+	0.913943i	$0.366978\pi$
$44$	− 78.7305i	− 0.269752i
$45$	− 88.2801i	− 0.292445i
$46$	− 67.0831i	− 0.215019i
$47$	− 598.729i	− 1.85816i	−0.369877	−	0.929081i	$-0.620600\pi$
$47$	− 598.729i	− 1.85816i	0.369877	−	0.929081i	$-0.379400\pi$
$48$	48.0000	0.144338
$49$	211.295	0.616021
$50$	− 57.5708i	− 0.162835i
$51$	−76.5030	−0.210050
$52$	0	0
$53$	119.901	0.310750	0.155375	−	0.987856i	$-0.450341\pi$
$53$	119.901	0.310750	0.155375	+	0.987856i	$0.450341\pi$
$54$	− 54.0000i	− 0.136083i
$55$	193.065	0.473325
$56$	−91.8101	−0.219083
$57$	12.9625i	0.0301214i
$58$	− 233.676i	− 0.529020i
$59$	499.881i	1.10303i	0.834164	+	0.551517i	$0.185951\pi$
$59$	499.881i	1.10303i	−0.834164	+	0.551517i	$0.814049\pi$
$60$	117.707i	0.253265i
$61$	696.612	1.46216	0.731082	−	0.682290i	$-0.239016\pi$
$61$	696.612	1.46216	0.731082	+	0.682290i	$0.239016\pi$
$62$	−576.701	−1.18131
$63$	103.286i	0.206553i
$64$	−64.0000	−0.125000
$65$	0	0
$66$	118.096	0.220251
$67$	67.8564i	0.123731i	0.998084	+	0.0618655i	$0.0197050\pi$
$67$	67.8564i	0.123731i	−0.998084	+	0.0618655i	$0.980295\pi$
$68$	102.004	0.181909
$69$	100.625	0.175562
$70$	− 225.139i	− 0.384418i
$71$	− 538.072i	− 0.899399i	−0.893180	−	0.449700i	$-0.851531\pi$
$71$	− 538.072i	− 0.899399i	0.893180	−	0.449700i	$-0.148469\pi$
$72$	72.0000i	0.117851i
$73$	− 1214.82i	− 1.94773i	−0.227124	−	0.973866i	$-0.572932\pi$
$73$	− 1214.82i	− 1.94773i	0.227124	−	0.973866i	$-0.427068\pi$
$74$	249.495	0.391935
$75$	86.3562	0.132954
$76$	− 17.2833i	− 0.0260859i
$77$	−225.883	−0.334309
$78$	0	0
$79$	345.554	0.492124	0.246062	−	0.969254i	$-0.420863\pi$
$79$	345.554	0.492124	0.246062	+	0.969254i	$0.420863\pi$
$80$	− 156.942i	− 0.219334i
$81$	81.0000	0.111111
$82$	284.536	0.383192
$83$	1160.79i	1.53510i	0.640989	+	0.767550i	$0.278524\pi$
$83$	1160.79i	1.53510i	−0.640989	+	0.767550i	$0.721476\pi$
$84$	− 137.715i	− 0.178880i
$85$	250.137i	0.319190i
$86$	− 457.741i	− 0.573948i
$87$	350.514	0.431943
$88$	−157.461	−0.190743
$89$	− 726.029i	− 0.864707i	−0.901704	−	0.432353i	$-0.857683\pi$
$89$	− 726.029i	− 0.864707i	0.901704	−	0.432353i	$-0.142317\pi$
$90$	−176.560	−0.206790
$91$	0	0
$92$	−134.166	−0.152041
$93$	− 865.051i	− 0.964534i
$94$	−1197.46	−1.31392
$95$	42.3826	0.0457722
$96$	− 96.0000i	− 0.102062i
$97$	− 598.944i	− 0.626944i	−0.949598	−	0.313472i	$-0.898508\pi$
$97$	− 598.944i	− 0.626944i	0.949598	−	0.313472i	$-0.101492\pi$
$98$	− 422.591i	− 0.435593i
$99$	177.144i	0.179834i
$100$	−115.142	−0.115142
$101$	−1769.08	−1.74287	−0.871437	−	0.490508i	$-0.836811\pi$
$101$	−1769.08	−1.74287	−0.871437	+	0.490508i	$0.836811\pi$
$102$	153.006i	0.148528i
$103$	−973.249	−0.931040	−0.465520	−	0.885037i	$-0.654133\pi$
$103$	−973.249	−0.931040	−0.465520	+	0.885037i	$0.654133\pi$
$104$	0	0
$105$	337.709	0.313876
$106$	− 239.803i	− 0.219733i
$107$	−1318.13	−1.19092	−0.595460	−	0.803385i	$-0.703030\pi$
$107$	−1318.13	−1.19092	−0.595460	+	0.803385i	$0.703030\pi$
$108$	−108.000	−0.0962250
$109$	− 325.349i	− 0.285897i	−0.989730	−	0.142949i	$-0.954342\pi$
$109$	− 325.349i	− 0.285897i	0.989730	−	0.142949i	$-0.0456584\pi$
$110$	− 386.130i	− 0.334691i
$111$	374.242i	0.320013i
$112$	183.620i	0.154915i
$113$	732.485	0.609791	0.304895	−	0.952386i	$-0.401378\pi$
$113$	732.485	0.609791	0.304895	+	0.952386i	$0.401378\pi$
$114$	25.9250	0.0212991
$115$	− 329.006i	− 0.266782i
$116$	−467.352	−0.374074
$117$	0	0
$118$	999.763	0.779963
$119$	− 292.656i	− 0.225443i
$120$	235.414	0.179085
$121$	943.594	0.708936
$122$	− 1393.22i	− 1.03391i
$123$	426.805i	0.312875i
$124$	1153.40i	0.835311i
$125$	− 1508.47i	− 1.07937i
$126$	206.573	0.146055
$127$	2259.84	1.57896	0.789480	−	0.613776i	$-0.210350\pi$
$127$	2259.84	1.57896	0.789480	+	0.613776i	$0.210350\pi$
$128$	128.000i	0.0883883i
$129$	686.612	0.468627
$130$	0	0
$131$	1615.48	1.07745	0.538723	−	0.842483i	$-0.318907\pi$
$131$	1615.48	1.07745	0.538723	+	0.842483i	$0.318907\pi$
$132$	− 236.192i	− 0.155741i
$133$	−49.5869	−0.0323288
$134$	135.713	0.0874911
$135$	− 264.840i	− 0.168843i
$136$	− 204.008i	− 0.128629i
$137$	389.403i	0.242839i	0.992601	+	0.121420i	$0.0387447\pi$
$137$	389.403i	0.242839i	−0.992601	+	0.121420i	$0.961255\pi$
$138$	− 201.249i	− 0.124141i
$139$	1727.37	1.05405	0.527027	−	0.849848i	$-0.323307\pi$
$139$	1727.37	1.05405	0.527027	+	0.849848i	$0.323307\pi$
$140$	−450.278	−0.271825
$141$	− 1796.19i	− 1.07281i
$142$	−1076.14	−0.635971
$143$	0	0
$144$	144.000	0.0833333
$145$	− 1146.05i	− 0.656376i
$146$	−2429.65	−1.37725
$147$	633.886	0.355660
$148$	− 498.989i	− 0.277140i
$149$	− 278.594i	− 0.153177i	−0.997063	−	0.0765883i	$-0.975597\pi$
$149$	− 278.594i	− 0.153177i	0.997063	−	0.0765883i	$-0.0244027\pi$
$150$	− 172.712i	− 0.0940128i
$151$	− 776.493i	− 0.418477i	−0.977865	−	0.209239i	$-0.932901\pi$
$151$	− 776.493i	− 0.418477i	0.977865	−	0.209239i	$-0.0670986\pi$
$152$	−34.5666	−0.0184455
$153$	−229.509	−0.121273
$154$	451.766i	0.236392i
$155$	−2828.40	−1.46569
$156$	0	0
$157$	398.605	0.202625	0.101313	−	0.994855i	$-0.467696\pi$
$157$	398.605	0.202625	0.101313	+	0.994855i	$0.467696\pi$
$158$	− 691.107i	− 0.347984i
$159$	359.704	0.179411
$160$	−313.885	−0.155092
$161$	384.932i	0.188428i
$162$	− 162.000i	− 0.0785674i
$163$	2777.81i	1.33482i	0.744693	+	0.667408i	$0.232596\pi$
$163$	2777.81i	1.33482i	−0.744693	+	0.667408i	$0.767404\pi$
$164$	− 569.073i	− 0.270958i
$165$	579.195	0.273274
$166$	2321.58	1.08548
$167$	− 3774.00i	− 1.74875i	−0.485253	−	0.874374i	$-0.661272\pi$
$167$	− 3774.00i	− 1.74875i	0.485253	−	0.874374i	$-0.338728\pi$
$168$	−275.430	−0.126488
$169$	0	0
$170$	500.274	0.225701
$171$	38.8874i	0.0173906i
$172$	−915.483	−0.405842
$173$	1692.90	0.743982	0.371991	−	0.928236i	$-0.378675\pi$
$173$	1692.90	0.743982	0.371991	+	0.928236i	$0.378675\pi$
$174$	− 701.028i	− 0.305430i
$175$	330.349i	0.142697i
$176$	314.922i	0.134876i
$177$	1499.64i	0.636837i
$178$	−1452.06	−0.611440
$179$	367.052	0.153267	0.0766333	−	0.997059i	$-0.475583\pi$
$179$	367.052	0.153267	0.0766333	+	0.997059i	$0.475583\pi$
$180$	353.121i	0.146222i
$181$	−999.648	−0.410515	−0.205258	−	0.978708i	$-0.565803\pi$
$181$	−999.648	−0.410515	−0.205258	+	0.978708i	$0.565803\pi$
$182$	0	0
$183$	2089.84	0.844181
$184$	268.333i	0.107509i
$185$	1223.63	0.486289
$186$	−1730.10	−0.682028
$187$	− 501.927i	− 0.196281i
$188$	2394.92i	0.929081i
$189$	309.859i	0.119254i
$190$	− 84.7651i	− 0.0323658i
$191$	1562.34	0.591870	0.295935	−	0.955208i	$-0.404369\pi$
$191$	1562.34	0.591870	0.295935	+	0.955208i	$0.404369\pi$
$192$	−192.000	−0.0721688
$193$	− 4656.25i	− 1.73660i	−0.496038	−	0.868301i	$-0.665213\pi$
$193$	− 4656.25i	− 1.73660i	0.496038	−	0.868301i	$-0.334787\pi$
$194$	−1197.89	−0.443316
$195$	0	0
$196$	−845.181	−0.308011
$197$	− 1181.41i	− 0.427269i	−0.976914	−	0.213634i	$-0.931470\pi$
$197$	− 1181.41i	− 0.427269i	0.976914	−	0.213634i	$-0.0685301\pi$
$198$	354.287	0.127162
$199$	−3341.44	−1.19029	−0.595147	−	0.803617i	$-0.702906\pi$
$199$	−3341.44	−1.19029	−0.595147	+	0.803617i	$0.702906\pi$
$200$	230.283i	0.0814174i
$201$	203.569i	0.0714362i
$202$	3538.16i	1.23240i
$203$	1340.86i	0.463597i
$204$	306.012	0.105025
$205$	1395.49	0.475442
$206$	1946.50i	0.658344i
$207$	301.874	0.101361
$208$	0	0
$209$	−85.0452	−0.0281469
$210$	− 675.417i	− 0.221944i
$211$	1395.08	0.455172	0.227586	−	0.973758i	$-0.426917\pi$
$211$	1395.08	0.455172	0.227586	+	0.973758i	$0.426917\pi$
$212$	−479.606	−0.155375
$213$	− 1614.22i	− 0.519269i
$214$	2636.26i	0.842107i
$215$	− 2244.97i	− 0.712120i
$216$	216.000i	0.0680414i
$217$	3309.19	1.03522
$218$	−650.698	−0.202160
$219$	− 3644.47i	− 1.12452i
$220$	−772.260	−0.236663
$221$	0	0
$222$	748.484	0.226284
$223$	3511.30i	1.05441i	0.849737	+	0.527207i	$0.176761\pi$
$223$	3511.30i	1.05441i	−0.849737	+	0.527207i	$0.823239\pi$
$224$	367.240	0.109541
$225$	259.069	0.0767611
$226$	− 1464.97i	− 0.431187i
$227$	590.297i	0.172596i	0.996269	+	0.0862982i	$0.0275038\pi$
$227$	590.297i	0.172596i	−0.996269	+	0.0862982i	$0.972496\pi$
$228$	− 51.8499i	− 0.0150607i
$229$	− 3432.98i	− 0.990645i	−0.868709	−	0.495322i	$-0.835050\pi$
$229$	− 3432.98i	− 0.990645i	0.868709	−	0.495322i	$-0.164950\pi$
$230$	−658.012	−0.188644
$231$	−677.649	−0.193013
$232$	934.704i	0.264510i
$233$	−5376.00	−1.51156	−0.755780	−	0.654826i	$-0.772742\pi$
$233$	−5376.00	−1.51156	−0.755780	+	0.654826i	$0.772742\pi$
$234$	0	0
$235$	−5872.87	−1.63023
$236$	− 1999.53i	− 0.551517i
$237$	1036.66	0.284128
$238$	−585.312	−0.159412
$239$	4342.72i	1.17534i	0.809099	+	0.587672i	$0.199955\pi$
$239$	4342.72i	1.17534i	−0.809099	+	0.587672i	$0.800045\pi$
$240$	− 470.827i	− 0.126632i
$241$	493.464i	0.131895i	0.997823	+	0.0659477i	$0.0210071\pi$
$241$	493.464i	0.131895i	−0.997823	+	0.0659477i	$0.978993\pi$
$242$	− 1887.19i	− 0.501293i
$243$	243.000	0.0641500
$244$	−2786.45	−0.731082
$245$	− 2072.58i	− 0.540457i
$246$	853.609	0.221236
$247$	0	0
$248$	2306.80	0.590654
$249$	3482.37i	0.886291i
$250$	−3016.93	−0.763230
$251$	1116.07	0.280661	0.140331	−	0.990105i	$-0.455183\pi$
$251$	1116.07	0.280661	0.140331	+	0.990105i	$0.455183\pi$
$252$	− 413.145i	− 0.103277i
$253$	660.186i	0.164054i
$254$	− 4519.67i	− 1.11649i
$255$	750.410i	0.184284i
$256$	256.000	0.0625000
$257$	−2998.66	−0.727825	−0.363913	−	0.931433i	$-0.618559\pi$
$257$	−2998.66	−0.727825	−0.363913	+	0.931433i	$0.618559\pi$
$258$	− 1373.22i	− 0.331369i
$259$	−1431.63	−0.343465
$260$	0	0
$261$	1051.54	0.249382
$262$	− 3230.97i	− 0.761870i
$263$	2234.09	0.523802	0.261901	−	0.965095i	$-0.415651\pi$
$263$	2234.09	0.523802	0.261901	+	0.965095i	$0.415651\pi$
$264$	−472.383	−0.110126
$265$	− 1176.10i	− 0.272632i
$266$	99.1739i	0.0228599i
$267$	− 2178.09i	− 0.499239i
$268$	− 271.426i	− 0.0618655i
$269$	−1476.38	−0.334635	−0.167317	−	0.985903i	$-0.553510\pi$
$269$	−1476.38	−0.334635	−0.167317	+	0.985903i	$0.553510\pi$
$270$	−529.681	−0.119390
$271$	2073.87i	0.464866i	0.972612	+	0.232433i	$0.0746686\pi$
$271$	2073.87i	0.464866i	−0.972612	+	0.232433i	$0.925331\pi$
$272$	−408.016	−0.0909544
$273$	0	0
$274$	778.807	0.171713
$275$	566.573i	0.124239i
$276$	−402.499	−0.0877811
$277$	385.634	0.0836481	0.0418240	−	0.999125i	$-0.486683\pi$
$277$	385.634	0.0836481	0.0418240	+	0.999125i	$0.486683\pi$
$278$	− 3454.74i	− 0.745329i
$279$	− 2595.15i	− 0.556874i
$280$	900.556i	0.192209i
$281$	− 50.9106i	− 0.0108081i	−0.999985	−	0.00540405i	$-0.998280\pi$
$281$	− 50.9106i	− 0.0108081i	0.999985	−	0.00540405i	$-0.00172017\pi$
$282$	−3592.37	−0.758591
$283$	7079.05	1.48695	0.743474	−	0.668765i	$-0.233177\pi$
$283$	7079.05	1.48695	0.743474	+	0.668765i	$0.233177\pi$
$284$	2152.29i	0.449700i
$285$	127.148	0.0264266
$286$	0	0
$287$	−1632.71	−0.335804
$288$	− 288.000i	− 0.0589256i
$289$	−4262.70	−0.867637
$290$	−2292.11	−0.464128
$291$	− 1796.83i	− 0.361966i
$292$	4859.30i	0.973866i
$293$	− 7424.87i	− 1.48043i	−0.672371	−	0.740214i	$-0.734724\pi$
$293$	− 7424.87i	− 1.48043i	0.672371	−	0.740214i	$-0.265276\pi$
$294$	− 1267.77i	− 0.251490i
$295$	4903.29	0.967730
$296$	−997.979	−0.195967
$297$	531.431i	0.103827i
$298$	−557.188	−0.108312
$299$	0	0
$300$	−345.425	−0.0664771
$301$	2626.58i	0.502969i
$302$	−1552.99	−0.295908
$303$	−5307.24	−1.00625
$304$	69.1332i	0.0130430i
$305$	− 6833.00i	− 1.28281i
$306$	459.018i	0.0857526i
$307$	− 1513.08i	− 0.281290i	−0.990060	−	0.140645i	$-0.955082\pi$
$307$	− 1513.08i	− 0.281290i	0.990060	−	0.140645i	$-0.0449176\pi$
$308$	903.532	0.167154
$309$	−2919.75	−0.537536
$310$	5656.80i	1.03640i
$311$	−4122.14	−0.751592	−0.375796	−	0.926702i	$-0.622631\pi$
$311$	−4122.14	−0.751592	−0.375796	+	0.926702i	$0.622631\pi$
$312$	0	0
$313$	−9089.64	−1.64146	−0.820730	−	0.571316i	$-0.806433\pi$
$313$	−9089.64	−1.64146	−0.820730	+	0.571316i	$0.806433\pi$
$314$	− 797.211i	− 0.143278i
$315$	1013.13	0.181216
$316$	−1382.21	−0.246062
$317$	5598.11i	0.991864i	0.868361	+	0.495932i	$0.165174\pi$
$317$	5598.11i	0.991864i	−0.868361	+	0.495932i	$0.834826\pi$
$318$	− 719.409i	− 0.126863i
$319$	2299.68i	0.403628i
$320$	627.770i	0.109667i
$321$	−3954.39	−0.687578
$322$	769.864	0.133239
$323$	− 110.185i	− 0.0189810i
$324$	−324.000	−0.0555556
$325$	0	0
$326$	5555.62	0.943857
$327$	− 976.047i	− 0.165063i
$328$	−1138.15	−0.191596
$329$	6871.17	1.15143
$330$	− 1158.39i	− 0.193234i
$331$	4375.60i	0.726601i	0.931672	+	0.363301i	$0.118350\pi$
$331$	4375.60i	0.726601i	−0.931672	+	0.363301i	$0.881650\pi$
$332$	− 4643.16i	− 0.767550i
$333$	1122.73i	0.184760i
$334$	−7548.00	−1.23655
$335$	665.597	0.108554
$336$	550.861i	0.0894402i
$337$	9930.42	1.60518	0.802588	−	0.596534i	$-0.203456\pi$
$337$	9930.42	1.60518	0.802588	+	0.596534i	$0.203456\pi$
$338$	0	0
$339$	2197.45	0.352063
$340$	− 1000.55i	− 0.159595i
$341$	5675.50	0.901306
$342$	77.7749	0.0122970
$343$	6361.24i	1.00138i
$344$	1830.97i	0.286974i
$345$	− 987.018i	− 0.154027i
$346$	− 3385.80i	− 0.526074i
$347$	−2527.04	−0.390947	−0.195474	−	0.980709i	$-0.562624\pi$
$347$	−2527.04	−0.390947	−0.195474	+	0.980709i	$0.562624\pi$
$348$	−1402.06	−0.215972
$349$	9477.40i	1.45362i	0.686838	+	0.726810i	$0.258998\pi$
$349$	9477.40i	1.45362i	−0.686838	+	0.726810i	$0.741002\pi$
$350$	660.698	0.100902
$351$	0	0
$352$	629.844	0.0953716
$353$	− 4890.71i	− 0.737411i	−0.929546	−	0.368706i	$-0.879801\pi$
$353$	− 4890.71i	− 0.737411i	0.929546	−	0.368706i	$-0.120199\pi$
$354$	2999.29	0.450312
$355$	−5277.89	−0.789075
$356$	2904.11i	0.432353i
$357$	− 877.968i	− 0.130160i
$358$	− 734.103i	− 0.108376i
$359$	6750.73i	0.992451i	0.868194	+	0.496226i	$0.165281\pi$
$359$	6750.73i	0.992451i	−0.868194	+	0.496226i	$0.834719\pi$
$360$	706.241	0.103395
$361$	6840.33	0.997278
$362$	1999.30i	0.290278i
$363$	2830.78	0.409304
$364$	0	0
$365$	−11916.1	−1.70881
$366$	− 4179.67i	− 0.596926i
$367$	−1226.22	−0.174409	−0.0872046	−	0.996190i	$-0.527793\pi$
$367$	−1226.22	−0.174409	−0.0872046	+	0.996190i	$0.527793\pi$
$368$	536.665	0.0760207
$369$	1280.41i	0.180639i
$370$	− 2447.27i	− 0.343858i
$371$	1376.02i	0.192559i
$372$	3460.21i	0.482267i
$373$	10148.0	1.40870	0.704349	−	0.709854i	$-0.251239\pi$
$373$	10148.0	1.40870	0.704349	+	0.709854i	$0.251239\pi$
$374$	−1003.85	−0.138792
$375$	− 4525.40i	− 0.623175i
$376$	4789.83	0.656959
$377$	0	0
$378$	619.718	0.0843250
$379$	9857.27i	1.33597i	0.744173	+	0.667987i	$0.232844\pi$
$379$	9857.27i	1.33597i	−0.744173	+	0.667987i	$0.767156\pi$
$380$	−169.530	−0.0228861
$381$	6779.51	0.911613
$382$	− 3124.69i	− 0.418515i
$383$	− 6761.43i	− 0.902070i	−0.892506	−	0.451035i	$-0.851055\pi$
$383$	− 6761.43i	− 0.902070i	0.892506	−	0.451035i	$-0.148945\pi$
$384$	384.000i	0.0510310i
$385$	2215.67i	0.293301i
$386$	−9312.49	−1.22796
$387$	2059.84	0.270562
$388$	2395.77i	0.313472i
$389$	13560.8	1.76750	0.883750	−	0.467960i	$-0.155011\pi$
$389$	13560.8	1.76750	0.883750	+	0.467960i	$0.155011\pi$
$390$	0	0
$391$	−855.343	−0.110631
$392$	1690.36i	0.217796i
$393$	4846.45	0.622064
$394$	−2362.82	−0.302125
$395$	− 3389.50i	− 0.431758i
$396$	− 708.575i	− 0.0899172i
$397$	8154.41i	1.03088i	0.856926	+	0.515439i	$0.172371\pi$
$397$	8154.41i	1.03088i	−0.856926	+	0.515439i	$0.827629\pi$
$398$	6682.89i	0.841665i
$399$	−148.761	−0.0186651
$400$	460.567	0.0575708
$401$	820.960i	0.102236i	0.998693	+	0.0511182i	$0.0162785\pi$
$401$	820.960i	0.102236i	−0.998693	+	0.0511182i	$0.983721\pi$
$402$	407.139	0.0505130
$403$	0	0
$404$	7076.33	0.871437
$405$	− 794.521i	− 0.0974817i
$406$	2681.73	0.327813
$407$	−2455.36	−0.299036
$408$	− 612.024i	− 0.0742640i
$409$	4598.95i	0.555999i	0.960581	+	0.278000i	$0.0896714\pi$
$409$	4598.95i	0.555999i	−0.960581	+	0.278000i	$0.910329\pi$
$410$	− 2790.99i	− 0.336188i
$411$	1168.21i	0.140203i
$412$	3893.00	0.465520
$413$	−5736.77	−0.683506
$414$	− 603.748i	− 0.0716730i
$415$	11386.1	1.34680
$416$	0	0
$417$	5182.11	0.608559
$418$	170.090i	0.0199029i
$419$	10274.3	1.19793	0.598965	−	0.800775i	$-0.295579\pi$
$419$	10274.3	1.19793	0.598965	+	0.800775i	$0.295579\pi$
$420$	−1350.83	−0.156938
$421$	12730.1i	1.47370i	0.676054	+	0.736852i	$0.263689\pi$
$421$	12730.1i	1.47370i	−0.676054	+	0.736852i	$0.736311\pi$
$422$	− 2790.16i	− 0.321855i
$423$	− 5388.56i	− 0.619387i
$424$	959.212i	0.109867i
$425$	−734.057	−0.0837811
$426$	−3228.43	−0.367178
$427$	7994.50i	0.906045i
$428$	5272.52	0.595460
$429$	0	0
$430$	−4489.94	−0.503545
$431$	960.330i	0.107326i	0.998559	+	0.0536630i	$0.0170897\pi$
$431$	960.330i	0.107326i	−0.998559	+	0.0536630i	$0.982910\pi$
$432$	432.000	0.0481125
$433$	13800.9	1.53171	0.765853	−	0.643016i	$-0.222317\pi$
$433$	13800.9	1.53171	0.765853	+	0.643016i	$0.222317\pi$
$434$	− 6618.37i	− 0.732009i
$435$	− 3438.16i	− 0.378959i
$436$	1301.40i	0.142949i
$437$	144.927i	0.0158646i
$438$	−7288.95	−0.795158
$439$	5932.37	0.644958	0.322479	−	0.946577i	$-0.395484\pi$
$439$	5932.37	0.644958	0.322479	+	0.946577i	$0.395484\pi$
$440$	1544.52i	0.167346i
$441$	1901.66	0.205340
$442$	0	0
$443$	−3164.11	−0.339349	−0.169674	−	0.985500i	$-0.554272\pi$
$443$	−3164.11	−0.339349	−0.169674	+	0.985500i	$0.554272\pi$
$444$	− 1496.97i	− 0.160007i
$445$	−7121.54	−0.758637
$446$	7022.60	0.745583
$447$	− 835.782i	− 0.0884365i
$448$	− 734.481i	− 0.0774575i
$449$	4040.22i	0.424654i	0.977199	+	0.212327i	$0.0681042\pi$
$449$	4040.22i	0.424654i	−0.977199	+	0.212327i	$0.931896\pi$
$450$	− 518.137i	− 0.0542783i
$451$	−2800.21	−0.292366
$452$	−2929.94	−0.304895
$453$	− 2329.48i	− 0.241608i
$454$	1180.59	0.122044
$455$	0	0
$456$	−103.700	−0.0106495
$457$	10702.3i	1.09548i	0.836649	+	0.547740i	$0.184512\pi$
$457$	10702.3i	1.09548i	−0.836649	+	0.547740i	$0.815488\pi$
$458$	−6865.96	−0.700492
$459$	−688.527	−0.0700167
$460$	1316.02i	0.133391i
$461$	− 6301.39i	− 0.636627i	−0.947985	−	0.318314i	$-0.896884\pi$
$461$	− 6301.39i	− 0.636627i	0.947985	−	0.318314i	$-0.103116\pi$
$462$	1355.30i	0.136481i
$463$	12433.2i	1.24799i	0.781429	+	0.623995i	$0.214491\pi$
$463$	12433.2i	1.24799i	−0.781429	+	0.623995i	$0.785509\pi$
$464$	1869.41	0.187037
$465$	−8485.21	−0.846219
$466$	10752.0i	1.06883i
$467$	16155.0	1.60078	0.800390	−	0.599479i	$-0.204626\pi$
$467$	16155.0	1.60078	0.800390	+	0.599479i	$0.204626\pi$
$468$	0	0
$469$	−778.738	−0.0766712
$470$	11745.7i	1.15275i
$471$	1195.82	0.116986
$472$	−3999.05	−0.389981
$473$	4504.78i	0.437907i
$474$	− 2073.32i	− 0.200909i
$475$	124.377i	0.0120143i
$476$	1170.62i	0.112722i
$477$	1079.11	0.103583
$478$	8685.44	0.831093
$479$	− 12044.8i	− 1.14894i	−0.818527	−	0.574468i	$-0.805209\pi$
$479$	− 12044.8i	− 1.14894i	0.818527	−	0.574468i	$-0.194791\pi$
$480$	−941.655	−0.0895426
$481$	0	0
$482$	986.928	0.0932642
$483$	1154.80i	0.108789i
$484$	−3774.38	−0.354468
$485$	−5874.98	−0.550039
$486$	− 486.000i	− 0.0453609i
$487$	9695.94i	0.902187i	0.892477	+	0.451093i	$0.148966\pi$
$487$	9695.94i	0.902187i	−0.892477	+	0.451093i	$0.851034\pi$
$488$	5572.89i	0.516953i
$489$	8333.43i	0.770656i
$490$	−4145.15	−0.382161
$491$	−6776.62	−0.622861	−0.311431	−	0.950269i	$-0.600808\pi$
$491$	−6776.62	−0.622861	−0.311431	+	0.950269i	$0.600808\pi$
$492$	− 1707.22i	− 0.156438i
$493$	−2979.48	−0.272189
$494$	0	0
$495$	1737.59	0.157775
$496$	− 4613.61i	− 0.417655i
$497$	6175.05	0.557322
$498$	6964.74	0.626702
$499$	− 7886.70i	− 0.707530i	−0.935334	−	0.353765i	$-0.884901\pi$
$499$	− 7886.70i	− 0.707530i	0.935334	−	0.353765i	$-0.115099\pi$
$500$	6033.87i	0.539685i
$501$	− 11322.0i	− 1.00964i
$502$	− 2232.15i	− 0.198458i
$503$	−22143.4	−1.96288	−0.981439	−	0.191775i	$-0.938576\pi$
$503$	−22143.4	−1.96288	−0.981439	+	0.191775i	$0.938576\pi$
$504$	−826.291	−0.0730276
$505$	17352.7i	1.52908i
$506$	1320.37	0.116003
$507$	0	0
$508$	−9039.34	−0.789480
$509$	11807.8i	1.02823i	0.857720	+	0.514116i	$0.171880\pi$
$509$	11807.8i	1.02823i	−0.857720	+	0.514116i	$0.828120\pi$
$510$	1500.82	0.130309
$511$	13941.6	1.20693
$512$	− 512.000i	− 0.0441942i
$513$	116.662i	0.0100405i
$514$	5997.31i	0.514650i
$515$	9546.51i	0.816834i
$516$	−2746.45	−0.234313
$517$	11784.6	1.00248
$518$	2863.27i	0.242866i
$519$	5078.70	0.429538
$520$	0	0
$521$	5614.28	0.472104	0.236052	−	0.971740i	$-0.424147\pi$
$521$	5614.28	0.472104	0.236052	+	0.971740i	$0.424147\pi$
$522$	− 2103.08i	− 0.176340i
$523$	8738.77	0.730630	0.365315	−	0.930884i	$-0.380961\pi$
$523$	8738.77	0.730630	0.365315	+	0.930884i	$0.380961\pi$
$524$	−6461.94	−0.538723
$525$	991.047i	0.0823864i
$526$	− 4468.18i	− 0.370384i
$527$	7353.22i	0.607802i
$528$	944.766i	0.0778706i
$529$	−11042.0	−0.907534
$530$	−2352.20	−0.192780
$531$	4498.93i	0.367678i
$532$	198.348	0.0161644
$533$	0	0
$534$	−4356.17	−0.353015
$535$	12929.4i	1.04484i
$536$	−542.851	−0.0437455
$537$	1101.15	0.0884885
$538$	2952.77i	0.236623i
$539$	4158.85i	0.332346i
$540$	1059.36i	0.0844216i
$541$	− 10863.1i	− 0.863291i	−0.902043	−	0.431645i	$-0.857933\pi$
$541$	− 10863.1i	− 0.863291i	0.902043	−	0.431645i	$-0.142067\pi$
$542$	4147.74	0.328710
$543$	−2998.94	−0.237011
$544$	816.032i	0.0643145i
$545$	−3191.32	−0.250828
$546$	0	0
$547$	−12943.4	−1.01174	−0.505868	−	0.862611i	$-0.668828\pi$
$547$	−12943.4	−1.01174	−0.505868	+	0.862611i	$0.668828\pi$
$548$	− 1557.61i	− 0.121420i
$549$	6269.51	0.487388
$550$	1133.15	0.0878500
$551$	504.837i	0.0390322i
$552$	804.998i	0.0620706i
$553$	3965.66i	0.304950i
$554$	− 771.268i	− 0.0591481i
$555$	3670.90	0.280759
$556$	−6909.48	−0.527027
$557$	17075.0i	1.29891i	0.760402	+	0.649453i	$0.225002\pi$
$557$	17075.0i	1.29891i	−0.760402	+	0.649453i	$0.774998\pi$
$558$	−5190.31	−0.393769
$559$	0	0
$560$	1801.11	0.135912
$561$	− 1505.78i	− 0.113323i
$562$	−101.821	−0.00764247
$563$	−4614.20	−0.345409	−0.172704	−	0.984974i	$-0.555251\pi$
$563$	−4614.20	−0.345409	−0.172704	+	0.984974i	$0.555251\pi$
$564$	7184.75i	0.536405i
$565$	− 7184.87i	− 0.534991i
$566$	− 14158.1i	− 1.05143i
$567$	929.577i	0.0688511i
$568$	4304.57	0.317986
$569$	−18131.4	−1.33586	−0.667932	−	0.744222i	$-0.732820\pi$
$569$	−18131.4	−1.33586	−0.667932	+	0.744222i	$0.732820\pi$
$570$	− 254.295i	− 0.0186864i
$571$	−1526.19	−0.111855	−0.0559274	−	0.998435i	$-0.517812\pi$
$571$	−1526.19	−0.111855	−0.0559274	+	0.998435i	$0.517812\pi$
$572$	0	0
$573$	4687.03	0.341716
$574$	3265.41i	0.237449i
$575$	965.508	0.0700252
$576$	−576.000	−0.0416667
$577$	23094.3i	1.66625i	0.553082	+	0.833127i	$0.313452\pi$
$577$	23094.3i	1.66625i	−0.553082	+	0.833127i	$0.686548\pi$
$578$	8525.40i	0.613512i
$579$	− 13968.7i	− 1.00263i
$580$	4584.21i	0.328188i
$581$	−13321.5	−0.951240
$582$	−3593.66	−0.255949
$583$	2359.98i	0.167651i
$584$	9718.59	0.688627
$585$	0	0
$586$	−14849.7	−1.04682
$587$	− 22084.0i	− 1.55282i	−0.630228	−	0.776410i	$-0.717039\pi$
$587$	− 22084.0i	− 1.55282i	0.630228	−	0.776410i	$-0.282961\pi$
$588$	−2535.54	−0.177830
$589$	1245.91	0.0871594
$590$	− 9806.58i	− 0.684289i
$591$	− 3544.23i	− 0.246684i
$592$	1995.96i	0.138570i
$593$	− 4484.19i	− 0.310529i	−0.987873	−	0.155265i	$-0.950377\pi$
$593$	− 4484.19i	− 0.310529i	0.987873	−	0.155265i	$-0.0496230\pi$
$594$	1062.86	0.0734171
$595$	−2870.64	−0.197789
$596$	1114.38i	0.0765883i
$597$	−10024.3	−0.687217
$598$	0	0
$599$	12402.2	0.845978	0.422989	−	0.906135i	$-0.360981\pi$
$599$	12402.2	0.845978	0.422989	+	0.906135i	$0.360981\pi$
$600$	690.850i	0.0470064i
$601$	−25204.8	−1.71069	−0.855344	−	0.518060i	$-0.826655\pi$
$601$	−25204.8	−1.71069	−0.855344	+	0.518060i	$0.826655\pi$
$602$	5253.16	0.355653
$603$	610.708i	0.0412437i
$604$	3105.97i	0.209239i
$605$	− 9255.62i	− 0.621974i
$606$	10614.5i	0.711525i
$607$	−1715.02	−0.114680	−0.0573399	−	0.998355i	$-0.518262\pi$
$607$	−1715.02	−0.114680	−0.0573399	+	0.998355i	$0.518262\pi$
$608$	138.266	0.00922277
$609$	4022.59i	0.267658i
$610$	−13666.0	−0.907082
$611$	0	0
$612$	918.036	0.0606363
$613$	− 11803.0i	− 0.777679i	−0.921305	−	0.388840i	$-0.872876\pi$
$613$	− 11803.0i	− 0.777679i	0.921305	−	0.388840i	$-0.127124\pi$
$614$	−3026.16	−0.198902
$615$	4186.48	0.274497
$616$	− 1807.06i	− 0.118196i
$617$	4891.04i	0.319134i	0.987187	+	0.159567i	$0.0510098\pi$
$617$	4891.04i	0.319134i	−0.987187	+	0.159567i	$0.948990\pi$
$618$	5839.49i	0.380095i
$619$	− 11692.6i	− 0.759235i	−0.925144	−	0.379618i	$-0.876056\pi$
$619$	− 11692.6i	− 0.759235i	0.925144	−	0.379618i	$-0.123944\pi$
$620$	11313.6	0.732847
$621$	905.622	0.0585207
$622$	8244.28i	0.531456i
$623$	8332.10	0.535824
$624$	0	0
$625$	−11198.2	−0.716686
$626$	18179.3i	1.16069i
$627$	−255.136	−0.0162506
$628$	−1594.42	−0.101313
$629$	− 3181.18i	− 0.201657i
$630$	− 2026.25i	− 0.128139i
$631$	1559.27i	0.0983731i	0.998790	+	0.0491866i	$0.0156629\pi$
$631$	1559.27i	0.0983731i	−0.998790	+	0.0491866i	$0.984337\pi$
$632$	2764.43i	0.173992i
$633$	4185.24	0.262794
$634$	11196.2	0.701354
$635$	− 22166.5i	− 1.38528i
$636$	−1438.82	−0.0897057
$637$	0	0
$638$	4599.36	0.285408
$639$	− 4842.65i	− 0.299800i
$640$	1255.54	0.0775462
$641$	−15946.8	−0.982621	−0.491310	−	0.870985i	$-0.663482\pi$
$641$	−15946.8	−0.982621	−0.491310	+	0.870985i	$0.663482\pi$
$642$	7908.78i	0.486191i
$643$	24409.1i	1.49705i	0.663108	+	0.748524i	$0.269237\pi$
$643$	24409.1i	1.49705i	−0.663108	+	0.748524i	$0.730763\pi$
$644$	− 1539.73i	− 0.0942139i
$645$	− 6734.91i	− 0.411142i
$646$	−220.371	−0.0134216
$647$	17248.1	1.04805	0.524027	−	0.851701i	$-0.324429\pi$
$647$	17248.1	1.04805	0.524027	+	0.851701i	$0.324429\pi$
$648$	648.000i	0.0392837i
$649$	−9838.98	−0.595091
$650$	0	0
$651$	9927.56	0.597683
$652$	− 11111.2i	− 0.667408i
$653$	31685.0	1.89882	0.949410	−	0.314039i	$-0.101682\pi$
$653$	31685.0	1.89882	0.949410	+	0.314039i	$0.101682\pi$
$654$	−1952.09	−0.116717
$655$	− 15846.1i	− 0.945282i
$656$	2276.29i	0.135479i
$657$	− 10933.4i	− 0.649244i
$658$	− 13742.3i	− 0.814183i
$659$	−43.1572	−0.00255109	−0.00127554	−	0.999999i	$-0.500406\pi$
$659$	−43.1572	−0.00255109	−0.00127554	+	0.999999i	$0.500406\pi$
$660$	−2316.78	−0.136637
$661$	− 1356.07i	− 0.0797960i	−0.999204	−	0.0398980i	$-0.987297\pi$
$661$	− 1356.07i	− 0.0797960i	0.999204	−	0.0398980i	$-0.0127033\pi$
$662$	8751.21	0.513785
$663$	0	0
$664$	−9286.33	−0.542740
$665$	486.393i	0.0283632i
$666$	2245.45	0.130645
$667$	3918.93	0.227499
$668$	15096.0i	0.874374i
$669$	10533.9i	0.608766i
$670$	− 1331.19i	− 0.0767590i
$671$	13711.2i	0.788843i
$672$	1101.72	0.0632438
$673$	7091.49	0.406177	0.203088	−	0.979160i	$-0.434902\pi$
$673$	7091.49	0.406177	0.203088	+	0.979160i	$0.434902\pi$
$674$	− 19860.8i	− 1.13503i
$675$	777.206	0.0443180
$676$	0	0
$677$	2245.58	0.127481	0.0637405	−	0.997967i	$-0.479697\pi$
$677$	2245.58	0.127481	0.0637405	+	0.997967i	$0.479697\pi$
$678$	− 4394.91i	− 0.248946i
$679$	6873.64	0.388492
$680$	−2001.09	−0.112851
$681$	1770.89i	0.0996486i
$682$	− 11351.0i	− 0.637320i
$683$	8548.77i	0.478931i	0.970905	+	0.239465i	$0.0769721\pi$
$683$	8548.77i	0.478931i	−0.970905	+	0.239465i	$0.923028\pi$
$684$	− 155.550i	− 0.00869531i
$685$	3819.62	0.213051
$686$	12722.5	0.708085
$687$	− 10298.9i	− 0.571949i
$688$	3661.93	0.202921
$689$	0	0
$690$	−1974.04	−0.108913
$691$	− 6392.96i	− 0.351953i	−0.984394	−	0.175977i	$-0.943692\pi$
$691$	− 6392.96i	− 0.351953i	0.984394	−	0.175977i	$-0.0563083\pi$
$692$	−6771.60	−0.371991
$693$	−2032.95	−0.111436
$694$	5054.08i	0.276441i
$695$	− 16943.6i	− 0.924759i
$696$	2804.11i	0.152715i
$697$	− 3627.98i	− 0.197159i
$698$	18954.8	1.02786
$699$	−16128.0	−0.872699
$700$	− 1321.40i	− 0.0713487i
$701$	−22777.7	−1.22725	−0.613625	−	0.789597i	$-0.710289\pi$
$701$	−22777.7	−1.22725	−0.613625	+	0.789597i	$0.710289\pi$
$702$	0	0
$703$	−539.011	−0.0289178
$704$	− 1259.69i	− 0.0674379i
$705$	−17618.6	−0.941214
$706$	−9781.42	−0.521429
$707$	− 20302.4i	− 1.07999i
$708$	− 5998.58i	− 0.318419i
$709$	− 35220.3i	− 1.86562i	−0.360365	−	0.932812i	$-0.617348\pi$
$709$	− 35220.3i	− 1.86562i	0.360365	−	0.932812i	$-0.382652\pi$
$710$	10555.8i	0.557960i
$711$	3109.98	0.164041
$712$	5808.23	0.305720
$713$	− 9671.73i	− 0.508007i
$714$	−1755.94	−0.0920368
$715$	0	0
$716$	−1468.21	−0.0766333
$717$	13028.2i	0.678585i
$718$	13501.5	0.701769
$719$	−25790.8	−1.33774	−0.668869	−	0.743381i	$-0.733221\pi$
$719$	−25790.8	−1.33774	−0.668869	+	0.743381i	$0.733221\pi$
$720$	− 1412.48i	− 0.0731112i
$721$	− 11169.3i	− 0.576928i
$722$	− 13680.7i	− 0.705182i
$723$	1480.39i	0.0761499i
$724$	3998.59	0.205258
$725$	3363.23	0.172286
$726$	− 5661.56i	− 0.289422i
$727$	−12236.4	−0.624239	−0.312119	−	0.950043i	$-0.601039\pi$
$727$	−12236.4	−0.624239	−0.312119	+	0.950043i	$0.601039\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	23832.2i	1.20831i
$731$	−5836.43	−0.295305
$732$	−8359.34	−0.422090
$733$	− 6703.90i	− 0.337809i	−0.985632	−	0.168905i	$-0.945977\pi$
$733$	− 6703.90i	− 0.337809i	0.985632	−	0.168905i	$-0.0540230\pi$
$734$	2452.44i	0.123326i
$735$	− 6217.73i	− 0.312033i
$736$	− 1073.33i	− 0.0537547i
$737$	−1335.59	−0.0667533
$738$	2560.83	0.127731
$739$	− 6779.55i	− 0.337469i	−0.985662	−	0.168735i	$-0.946032\pi$
$739$	− 6779.55i	− 0.337469i	0.985662	−	0.168735i	$-0.0539681\pi$
$740$	−4894.54	−0.243144
$741$	0	0
$742$	2752.04	0.136160
$743$	− 25938.1i	− 1.28072i	−0.768074	−	0.640361i	$-0.778785\pi$
$743$	− 25938.1i	− 1.28072i	0.768074	−	0.640361i	$-0.221215\pi$
$744$	6920.41	0.341014
$745$	−2732.70	−0.134387
$746$	− 20296.0i	− 0.996100i
$747$	10447.1i	0.511700i
$748$	2007.71i	0.0981404i
$749$	− 15127.2i	− 0.737965i
$750$	−9050.80	−0.440651
$751$	35204.5	1.71056	0.855279	−	0.518168i	$-0.173386\pi$
$751$	35204.5	1.71056	0.855279	+	0.518168i	$0.173386\pi$
$752$	− 9579.66i	− 0.464540i
$753$	3348.22	0.162040
$754$	0	0
$755$	−7616.54	−0.367145
$756$	− 1239.44i	− 0.0596268i
$757$	−23447.7	−1.12579	−0.562895	−	0.826529i	$-0.690312\pi$
$757$	−23447.7	−1.12579	−0.562895	+	0.826529i	$0.690312\pi$
$758$	19714.5	0.944676
$759$	1980.56i	0.0947164i
$760$	339.060i	0.0161829i
$761$	25158.2i	1.19840i	0.800599	+	0.599200i	$0.204515\pi$
$761$	25158.2i	1.19840i	−0.800599	+	0.599200i	$0.795485\pi$
$762$	− 13559.0i	− 0.644608i
$763$	3733.79	0.177159
$764$	−6249.37	−0.295935
$765$	2251.23i	0.106397i
$766$	−13522.9	−0.637860
$767$	0	0
$768$	768.000	0.0360844
$769$	− 3708.37i	− 0.173898i	−0.996213	−	0.0869488i	$-0.972288\pi$
$769$	− 3708.37i	− 0.173898i	0.996213	−	0.0869488i	$-0.0277117\pi$
$770$	4431.33	0.207395
$771$	−8995.97	−0.420210
$772$	18625.0i	0.868301i
$773$	− 3567.51i	− 0.165995i	−0.996550	−	0.0829976i	$-0.973551\pi$
$773$	− 3567.51i	− 0.165995i	0.996550	−	0.0829976i	$-0.0264494\pi$
$774$	− 4119.67i	− 0.191316i
$775$	− 8300.29i	− 0.384716i
$776$	4791.55	0.221658
$777$	−4294.90	−0.198299
$778$	− 27121.5i	− 1.24981i
$779$	−614.716	−0.0282728
$780$	0	0
$781$	10590.7	0.485229
$782$	1710.69i	0.0782277i
$783$	3154.63	0.143981
$784$	3380.73	0.154005
$785$	− 3909.88i	− 0.177770i
$786$	− 9692.91i	− 0.439866i
$787$	34978.8i	1.58432i	0.610313	+	0.792160i	$0.291044\pi$
$787$	34978.8i	1.58432i	−0.610313	+	0.792160i	$0.708956\pi$
$788$	4725.64i	0.213634i
$789$	6702.27	0.302417
$790$	−6779.00	−0.305299
$791$	8406.19i	0.377863i
$792$	−1417.15	−0.0635811
$793$	0	0
$794$	16308.8	0.728940
$795$	− 3528.31i	− 0.157404i
$796$	13365.8	0.595147
$797$	−27780.2	−1.23466	−0.617330	−	0.786704i	$-0.711786\pi$
$797$	−27780.2	−1.23466	−0.617330	+	0.786704i	$0.711786\pi$
$798$	297.522i	0.0131982i
$799$	15268.2i	0.676032i
$800$	− 921.133i	− 0.0407087i
$801$	− 6534.26i	− 0.288236i
$802$	1641.92	0.0722920
$803$	23910.9	1.05081
$804$	− 814.277i	− 0.0357181i
$805$	3775.76	0.165314
$806$	0	0
$807$	−4429.15	−0.193202
$808$	− 14152.7i	− 0.616199i
$809$	−19900.8	−0.864865	−0.432433	−	0.901666i	$-0.642345\pi$
$809$	−19900.8	−0.864865	−0.432433	+	0.901666i	$0.642345\pi$
$810$	−1589.04	−0.0689299
$811$	− 43420.9i	− 1.88004i	−0.341114	−	0.940022i	$-0.610804\pi$
$811$	− 43420.9i	− 1.88004i	0.341114	−	0.940022i	$-0.389196\pi$
$812$	− 5363.45i	− 0.231798i
$813$	6221.61i	0.268390i
$814$	4910.71i	0.211450i
$815$	27247.3	1.17108
$816$	−1224.05	−0.0525125
$817$	988.910i	0.0423471i
$818$	9197.91	0.393151
$819$	0	0
$820$	−5581.98	−0.237721
$821$	− 20320.2i	− 0.863799i	−0.901922	−	0.431899i	$-0.857844\pi$
$821$	− 20320.2i	− 0.863799i	0.901922	−	0.431899i	$-0.142156\pi$
$822$	2336.42	0.0991387
$823$	−31446.2	−1.33189	−0.665944	−	0.746002i	$-0.731971\pi$
$823$	−31446.2	−1.33189	−0.665944	+	0.746002i	$0.731971\pi$
$824$	− 7785.99i	− 0.329172i
$825$	1699.72i	0.0717292i
$826$	11473.5i	0.483312i
$827$	18625.1i	0.783143i	0.920148	+	0.391572i	$0.128068\pi$
$827$	18625.1i	0.783143i	−0.920148	+	0.391572i	$0.871932\pi$
$828$	−1207.50	−0.0506804
$829$	−27595.2	−1.15612	−0.578058	−	0.815996i	$-0.696189\pi$
$829$	−27595.2	−1.15612	−0.578058	+	0.815996i	$0.696189\pi$
$830$	− 22772.2i	− 0.952330i
$831$	1156.90	0.0482942
$832$	0	0
$833$	−5388.24	−0.224119
$834$	− 10364.2i	− 0.430316i
$835$	−37018.8	−1.53424
$836$	340.181	0.0140734
$837$	− 7785.46i	− 0.321511i
$838$	− 20548.6i	− 0.847064i
$839$	34829.9i	1.43321i	0.697479	+	0.716605i	$0.254305\pi$
$839$	34829.9i	1.43321i	−0.697479	+	0.716605i	$0.745695\pi$
$840$	2701.67i	0.110972i
$841$	−10737.9	−0.440276
$842$	25460.3	1.04207
$843$	− 152.732i	− 0.00624005i
$844$	−5580.32	−0.227586
$845$	0	0
$846$	−10777.1	−0.437973
$847$	10828.9i	0.439299i
$848$	1918.42	0.0776874
$849$	21237.2	0.858489
$850$	1468.11i	0.0592422i
$851$	4184.22i	0.168547i
$852$	6456.86i	0.259634i
$853$	− 35675.7i	− 1.43202i	−0.698090	−	0.716010i	$-0.745966\pi$
$853$	− 35675.7i	− 1.43202i	0.698090	−	0.716010i	$-0.254034\pi$
$854$	15989.0	0.640670
$855$	381.443	0.0152574
$856$	− 10545.0i	− 0.421054i
$857$	−48343.4	−1.92693	−0.963465	−	0.267833i	$-0.913692\pi$
$857$	−48343.4	−1.92693	−0.963465	+	0.267833i	$0.913692\pi$
$858$	0	0
$859$	34409.4	1.36674	0.683372	−	0.730070i	$-0.260513\pi$
$859$	34409.4	1.36674	0.683372	+	0.730070i	$0.260513\pi$
$860$	8979.88i	0.356060i
$861$	−4898.12	−0.193876
$862$	1920.66	0.0758909
$863$	− 7065.25i	− 0.278684i	−0.990244	−	0.139342i	$-0.955501\pi$
$863$	− 7065.25i	− 0.278684i	0.990244	−	0.139342i	$-0.0444987\pi$
$864$	− 864.000i	− 0.0340207i
$865$	− 16605.5i	− 0.652721i
$866$	− 27601.8i	− 1.08308i
$867$	−12788.1	−0.500930
$868$	−13236.7	−0.517609
$869$	6801.40i	0.265503i
$870$	−6876.32	−0.267964
$871$	0	0
$872$	2602.79	0.101080
$873$	− 5390.49i	− 0.208981i
$874$	289.855	0.0112179
$875$	17311.6	0.668843
$876$	14577.9i	0.562262i
$877$	18998.8i	0.731521i	0.930709	+	0.365760i	$0.119191\pi$
$877$	18998.8i	0.731521i	−0.930709	+	0.365760i	$0.880809\pi$
$878$	− 11864.7i	− 0.456054i
$879$	− 22274.6i	− 0.854726i
$880$	3089.04	0.118331
$881$	−12897.2	−0.493211	−0.246605	−	0.969116i	$-0.579315\pi$
$881$	−12897.2	−0.493211	−0.246605	+	0.969116i	$0.579315\pi$
$882$	− 3803.32i	− 0.145198i
$883$	2504.93	0.0954672	0.0477336	−	0.998860i	$-0.484800\pi$
$883$	2504.93	0.0954672	0.0477336	+	0.998860i	$0.484800\pi$
$884$	0	0
$885$	14709.9	0.558719
$886$	6328.22i	0.239956i
$887$	11715.5	0.443481	0.221740	−	0.975106i	$-0.428826\pi$
$887$	11715.5	0.443481	0.221740	+	0.975106i	$0.428826\pi$
$888$	−2993.94	−0.113142
$889$	25934.5i	0.978418i
$890$	14243.1i	0.536438i
$891$	1594.29i	0.0599448i
$892$	− 14045.2i	− 0.527207i
$893$	2587.00	0.0969438
$894$	−1671.56	−0.0625340
$895$	− 3600.37i	− 0.134466i
$896$	−1468.96	−0.0547707
$897$	0	0
$898$	8080.44	0.300276
$899$	− 33690.3i	− 1.24987i
$900$	−1036.27	−0.0383806
$901$	−3057.61	−0.113056
$902$	5600.43i	0.206734i
$903$	7879.74i	0.290389i
$904$	5859.88i	0.215594i
$905$	9805.45i	0.360159i
$906$	−4658.96	−0.170843
$907$	−32397.6	−1.18605	−0.593023	−	0.805185i	$-0.702066\pi$
$907$	−32397.6	−1.18605	−0.593023	+	0.805185i	$0.702066\pi$
$908$	− 2361.19i	− 0.0862982i
$909$	−15921.7	−0.580958
$910$	0	0
$911$	6809.26	0.247641	0.123820	−	0.992305i	$-0.460485\pi$
$911$	6809.26	0.247641	0.123820	+	0.992305i	$0.460485\pi$
$912$	207.400i	0.00753036i
$913$	−22847.4	−0.828192
$914$	21404.7	0.774621
$915$	− 20499.0i	− 0.740629i
$916$	13731.9i	0.495322i
$917$	18539.7i	0.667651i
$918$	1377.05i	0.0495093i
$919$	50361.8	1.80771	0.903854	−	0.427841i	$-0.140726\pi$
$919$	50361.8	1.80771	0.903854	+	0.427841i	$0.140726\pi$
$920$	2632.05	0.0943218
$921$	− 4539.24i	− 0.162403i
$922$	−12602.8	−0.450163
$923$	0	0
$924$	2710.60	0.0965066
$925$	3590.90i	0.127641i
$926$	24866.4	0.882462
$927$	−8759.24	−0.310347
$928$	− 3738.82i	− 0.132255i
$929$	− 3689.36i	− 0.130295i	−0.997876	−	0.0651475i	$-0.979248\pi$
$929$	− 3689.36i	− 0.130295i	0.997876	−	0.0651475i	$-0.0207518\pi$
$930$	16970.4i	0.598367i
$931$	912.970i	0.0321390i
$932$	21504.0	0.755780
$933$	−12366.4	−0.433932
$934$	− 32310.0i	− 1.13192i
$935$	−4923.35	−0.172204
$936$	0	0
$937$	43698.7	1.52356	0.761779	−	0.647837i	$-0.224326\pi$
$937$	43698.7	1.52356	0.761779	+	0.647837i	$0.224326\pi$
$938$	1557.48i	0.0542147i
$939$	−27268.9	−0.947697
$940$	23491.5	0.815115
$941$	7298.16i	0.252830i	0.991977	+	0.126415i	$0.0403471\pi$
$941$	7298.16i	0.252830i	−0.991977	+	0.126415i	$0.959653\pi$
$942$	− 2391.63i	− 0.0827214i
$943$	4771.90i	0.164787i
$944$	7998.10i	0.275759i
$945$	3039.38	0.104625
$946$	9009.56	0.309647
$947$	12875.6i	0.441818i	0.975294	+	0.220909i	$0.0709024\pi$
$947$	12875.6i	0.441818i	−0.975294	+	0.220909i	$0.929098\pi$
$948$	−4146.64	−0.142064
$949$	0	0
$950$	248.754	0.00849540
$951$	16794.3i	0.572653i
$952$	2341.25	0.0797062
$953$	34020.0	1.15636	0.578182	−	0.815908i	$-0.303762\pi$
$953$	34020.0	1.15636	0.578182	+	0.815908i	$0.303762\pi$
$954$	− 2158.23i	− 0.0732444i
$955$	− 15324.9i	− 0.519268i
$956$	− 17370.9i	− 0.587672i
$957$	6899.04i	0.233035i
$958$	−24089.6	−0.812421
$959$	−4468.90	−0.150478
$960$	1883.31i	0.0633162i
$961$	−53355.0	−1.79098
$962$	0	0
$963$	−11863.2	−0.396973
$964$	− 1973.86i	− 0.0659477i
$965$	−45672.7	−1.52358
$966$	2309.59	0.0769253
$967$	− 26847.5i	− 0.892820i	−0.894829	−	0.446410i	$-0.852702\pi$
$967$	− 26847.5i	− 0.892820i	0.894829	−	0.446410i	$-0.147298\pi$
$968$	7548.75i	0.250647i
$969$	− 330.556i	− 0.0109587i
$970$	11750.0i	0.388937i
$971$	−30144.5	−0.996275	−0.498138	−	0.867098i	$-0.665983\pi$
$971$	−30144.5	−0.996275	−0.498138	+	0.867098i	$0.665983\pi$
$972$	−972.000	−0.0320750
$973$	19823.7i	0.653155i
$974$	19391.9	0.637942
$975$	0	0
$976$	11145.8	0.365541
$977$	− 2834.23i	− 0.0928097i	−0.998923	−	0.0464049i	$-0.985224\pi$
$977$	− 2834.23i	− 0.0928097i	0.998923	−	0.0464049i	$-0.0147764\pi$
$978$	16666.9	0.544936
$979$	14290.2	0.466512
$980$	8290.30i	0.270229i
$981$	− 2928.14i	− 0.0952991i
$982$	13553.2i	0.440429i
$983$	− 15556.3i	− 0.504749i	−0.967630	−	0.252374i	$-0.918789\pi$
$983$	− 15556.3i	− 0.504749i	0.967630	−	0.252374i	$-0.0812115\pi$
$984$	−3414.44	−0.110618
$985$	−11588.3	−0.374858
$986$	5958.97i	0.192467i
$987$	20613.5	0.664778
$988$	0	0
$989$	7676.68	0.246819
$990$	− 3475.17i	− 0.111564i
$991$	19781.4	0.634084	0.317042	−	0.948412i	$-0.397310\pi$
$991$	19781.4	0.634084	0.317042	+	0.948412i	$0.397310\pi$
$992$	−9227.22	−0.295327
$993$	13126.8i	0.419503i
$994$	− 12350.1i	− 0.394086i
$995$	32775.9i	1.04429i
$996$	− 13929.5i	− 0.443145i
$997$	20397.3	0.647932	0.323966	−	0.946069i	$-0.394984\pi$
$997$	20397.3	0.647932	0.323966	+	0.946069i	$0.394984\pi$
$998$	−15773.4	−0.500299
$999$	3368.18i	0.106671i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1014.4.b.q.337.3		12
13.5	odd	4		1014.4.a.bc.1.3	✓	6
13.8	odd	4		1014.4.a.be.1.4	yes	6
13.12	even	2	inner	1014.4.b.q.337.10		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1014.4.a.bc.1.3	✓	6	13.5	odd	4
1014.4.a.be.1.4	yes	6	13.8	odd	4
1014.4.b.q.337.3		12	1.1	even	1	trivial
1014.4.b.q.337.10		12	13.12	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1014 = 2 \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1014.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{2} +3.00000 q^{3} -4.00000 q^{4} -9.80890i q^{5} -6.00000i q^{6} +11.4763i q^{7} +8.00000i q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-2.00000i q^{2} +3.00000 q^{3} -4.00000 q^{4} -9.80890i q^{5} -6.00000i q^{6} +11.4763i q^{7} +8.00000i q^{8} +9.00000 q^{9} -19.6178 q^{10} +19.6826i q^{11} -12.0000 q^{12} +22.9525 q^{14} -29.4267i q^{15} +16.0000 q^{16} -25.5010 q^{17} -18.0000i q^{18} +4.32083i q^{19} +39.2356i q^{20} +34.4288i q^{21} +39.3653 q^{22} +33.5416 q^{23} +24.0000i q^{24} +28.7854 q^{25} +27.0000 q^{27} -45.9051i q^{28} +116.838 q^{29} -58.8534 q^{30} -288.350i q^{31} -32.0000i q^{32} +59.0479i q^{33} +51.0020i q^{34} +112.570 q^{35} -36.0000 q^{36} +124.747i q^{37} +8.64165 q^{38} +78.4712 q^{40} +142.268i q^{41} +68.8576 q^{42} +228.871 q^{43} -78.7305i q^{44} -88.2801i q^{45} -67.0831i q^{46} -598.729i q^{47} +48.0000 q^{48} +211.295 q^{49} -57.5708i q^{50} -76.5030 q^{51} +119.901 q^{53} -54.0000i q^{54} +193.065 q^{55} -91.8101 q^{56} +12.9625i q^{57} -233.676i q^{58} +499.881i q^{59} +117.707i q^{60} +696.612 q^{61} -576.701 q^{62} +103.286i q^{63} -64.0000 q^{64} +118.096 q^{66} +67.8564i q^{67} +102.004 q^{68} +100.625 q^{69} -225.139i q^{70} -538.072i q^{71} +72.0000i q^{72} -1214.82i q^{73} +249.495 q^{74} +86.3562 q^{75} -17.2833i q^{76} -225.883 q^{77} +345.554 q^{79} -156.942i q^{80} +81.0000 q^{81} +284.536 q^{82} +1160.79i q^{83} -137.715i q^{84} +250.137i q^{85} -457.741i q^{86} +350.514 q^{87} -157.461 q^{88} -726.029i q^{89} -176.560 q^{90} -134.166 q^{92} -865.051i q^{93} -1197.46 q^{94} +42.3826 q^{95} -96.0000i q^{96} -598.944i q^{97} -422.591i q^{98} +177.144i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 36 q^{3} - 48 q^{4} + 108 q^{9}+O(q^{10}) \) 12 * q + 36 * q^3 - 48 * q^4 + 108 * q^9	\( 12 q + 36 q^{3} - 48 q^{4} + 108 q^{9} + 12 q^{10} - 144 q^{12} - 4 q^{14} + 192 q^{16} - 102 q^{17} + 256 q^{22} - 444 q^{23} - 370 q^{25} + 324 q^{27} + 658 q^{29} + 36 q^{30} + 1688 q^{35} - 432 q^{36} - 852 q^{38} - 48 q^{40} - 12 q^{42} - 982 q^{43} + 576 q^{48} - 2266 q^{49} - 306 q^{51} + 4604 q^{53} - 658 q^{55} + 16 q^{56} + 690 q^{61} - 1156 q^{62} - 768 q^{64} + 768 q^{66} + 408 q^{68} - 1332 q^{69} + 1880 q^{74} - 1110 q^{75} - 6582 q^{77} + 6200 q^{79} + 972 q^{81} + 1284 q^{82} + 1974 q^{87} - 1024 q^{88} + 108 q^{90} + 1776 q^{92} - 2564 q^{94} + 1330 q^{95}+O(q^{100}) \) 12 * q + 36 * q^3 - 48 * q^4 + 108 * q^9 + 12 * q^10 - 144 * q^12 - 4 * q^14 + 192 * q^16 - 102 * q^17 + 256 * q^22 - 444 * q^23 - 370 * q^25 + 324 * q^27 + 658 * q^29 + 36 * q^30 + 1688 * q^35 - 432 * q^36 - 852 * q^38 - 48 * q^40 - 12 * q^42 - 982 * q^43 + 576 * q^48 - 2266 * q^49 - 306 * q^51 + 4604 * q^53 - 658 * q^55 + 16 * q^56 + 690 * q^61 - 1156 * q^62 - 768 * q^64 + 768 * q^66 + 408 * q^68 - 1332 * q^69 + 1880 * q^74 - 1110 * q^75 - 6582 * q^77 + 6200 * q^79 + 972 * q^81 + 1284 * q^82 + 1974 * q^87 - 1024 * q^88 + 108 * q^90 + 1776 * q^92 - 2564 * q^94 + 1330 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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