LMFDB - Embedded newform 507.4.b.k.337.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [507,4,Mod(337,507)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("507.337"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(507, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 507 = 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	507.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [18,0,54,-88,0,0,0,0,162,108,0,-264,0,316,0,432,-356] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$29.9139683729$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} + 112 x^{16} + 5026 x^{14} + 114847 x^{12} + 1397921 x^{10} + 8545747 x^{8} + 21033277 x^{6} + \cdots + 729 $ x^18 + 112x^16 + 5026x^14 + 114847x^12 + 1397921x^10 + 8545747x^8 + 21033277x^6 + 6703200x^4 + 137781x^2 + 729
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 13^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.2
Root			$-5.39246i$ of defining polynomial
Character	$\chi$	$=$	507.337
Dual form			507.4.b.k.337.17

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.83750i q^{2} +3.00000 q^{3} -15.4014 q^{4} +21.1983i q^{5} -14.5125i q^{6} -16.2806i q^{7} +35.8043i q^{8} +9.00000 q^{9} +102.547 q^{10} -30.7532i q^{11} -46.2042 q^{12} -78.7575 q^{14} +63.5948i q^{15} +49.9922 q^{16} -46.2371 q^{17} -43.5375i q^{18} +144.865i q^{19} -326.483i q^{20} -48.8418i q^{21} -148.769 q^{22} -8.38045 q^{23} +107.413i q^{24} -324.366 q^{25} +27.0000 q^{27} +250.744i q^{28} -242.958 q^{29} +307.640 q^{30} +87.9353i q^{31} +44.5973i q^{32} -92.2597i q^{33} +223.672i q^{34} +345.121 q^{35} -138.613 q^{36} +49.6950i q^{37} +700.783 q^{38} -758.990 q^{40} -107.947i q^{41} -236.272 q^{42} +35.4166 q^{43} +473.643i q^{44} +190.784i q^{45} +40.5404i q^{46} +374.815i q^{47} +149.977 q^{48} +77.9418 q^{49} +1569.12i q^{50} -138.711 q^{51} -348.583 q^{53} -130.613i q^{54} +651.915 q^{55} +582.916 q^{56} +434.594i q^{57} +1175.31i q^{58} +679.430i q^{59} -979.450i q^{60} -230.403 q^{61} +425.387 q^{62} -146.525i q^{63} +615.677 q^{64} -446.306 q^{66} +295.642i q^{67} +712.117 q^{68} -25.1413 q^{69} -1669.52i q^{70} -329.215i q^{71} +322.239i q^{72} +48.9973i q^{73} +240.399 q^{74} -973.099 q^{75} -2231.12i q^{76} -500.681 q^{77} -107.942 q^{79} +1059.75i q^{80} +81.0000 q^{81} -522.192 q^{82} -515.654i q^{83} +752.233i q^{84} -980.147i q^{85} -171.328i q^{86} -728.874 q^{87} +1101.10 q^{88} +984.453i q^{89} +922.919 q^{90} +129.071 q^{92} +263.806i q^{93} +1813.17 q^{94} -3070.88 q^{95} +133.792i q^{96} -487.072i q^{97} -377.043i q^{98} -276.779i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 54 q^{3} - 88 q^{4} + 162 q^{9} + 108 q^{10} - 264 q^{12} + 316 q^{14} + 432 q^{16} - 356 q^{17} - 1260 q^{22} - 300 q^{23} + 40 q^{25} + 486 q^{27} - 194 q^{29} + 324 q^{30} - 836 q^{35} - 792 q^{36}+ \cdots - 8316 q^{95}+O(q^{100}) $ 18 * q + 54 * q^3 - 88 * q^4 + 162 * q^9 + 108 * q^10 - 264 * q^12 + 316 * q^14 + 432 * q^16 - 356 * q^17 - 1260 * q^22 - 300 * q^23 + 40 * q^25 + 486 * q^27 - 194 * q^29 + 324 * q^30 - 836 * q^35 - 792 * q^36 + 1320 * q^38 - 3012 * q^40 + 948 * q^42 - 484 * q^43 + 1296 * q^48 + 76 * q^49 - 1068 * q^51 - 302 * q^53 + 4128 * q^55 - 4552 * q^56 - 2680 * q^61 - 694 * q^62 - 1786 * q^64 - 3780 * q^66 + 5570 * q^68 - 900 * q^69 - 2382 * q^74 + 120 * q^75 + 4284 * q^77 - 3182 * q^79 + 1458 * q^81 - 3034 * q^82 - 582 * q^87 + 7432 * q^88 + 972 * q^90 + 1030 * q^92 - 1384 * q^94 - 8316 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/507\mathbb{Z}\right)^\times$.

$n$	$170$	$340$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 4.83750i	− 1.71031i	−0.518368	−	0.855157i	$-0.673460\pi$
$2$	− 4.83750i	− 1.71031i	0.518368	−	0.855157i	$-0.326540\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−15.4014	−1.92518
$5$	21.1983i	1.89603i	0.318225	+	0.948015i	$0.396913\pi$
$5$	21.1983i	1.89603i	−0.318225	+	0.948015i	$0.603087\pi$
$6$	− 14.5125i	− 0.987451i
$7$	− 16.2806i	− 0.879070i	−0.898225	−	0.439535i	$-0.855143\pi$
$7$	− 16.2806i	− 0.879070i	0.898225	−	0.439535i	$-0.144857\pi$
$8$	35.8043i	1.58234i
$9$	9.00000	0.333333
$10$	102.547	3.24281
$11$	− 30.7532i	− 0.842950i	−0.906840	−	0.421475i	$-0.861513\pi$
$11$	− 30.7532i	− 0.842950i	0.906840	−	0.421475i	$-0.138487\pi$
$12$	−46.2042	−1.11150
$13$	0	0
$13$	0	0
$14$	−78.7575	−1.50349
$15$	63.5948i	1.09467i
$16$	49.9922	0.781128
$17$	−46.2371	−0.659656	−0.329828	−	0.944041i	$-0.606991\pi$
$17$	−46.2371	−0.659656	−0.329828	+	0.944041i	$0.606991\pi$
$18$	− 43.5375i	− 0.570105i
$19$	144.865i	1.74917i	0.484873	+	0.874585i	$0.338866\pi$
$19$	144.865i	1.74917i	−0.484873	+	0.874585i	$0.661134\pi$
$20$	− 326.483i	− 3.65019i
$21$	− 48.8418i	− 0.507531i
$22$	−148.769	−1.44171
$23$	−8.38045	−0.0759758	−0.0379879	−	0.999278i	$-0.512095\pi$
$23$	−8.38045	−0.0759758	−0.0379879	+	0.999278i	$0.512095\pi$
$24$	107.413i	0.913566i
$25$	−324.366	−2.59493
$26$	0	0
$27$	27.0000	0.192450
$28$	250.744i	1.69237i
$29$	−242.958	−1.55573	−0.777865	−	0.628431i	$-0.783697\pi$
$29$	−242.958	−1.55573	−0.777865	+	0.628431i	$0.783697\pi$
$30$	307.640	1.87224
$31$	87.9353i	0.509472i	0.967011	+	0.254736i	$0.0819886\pi$
$31$	87.9353i	0.509472i	−0.967011	+	0.254736i	$0.918011\pi$
$32$	44.5973i	0.246368i
$33$	− 92.2597i	− 0.486677i
$34$	223.672i	1.12822i
$35$	345.121	1.66674
$36$	−138.613	−0.641726
$37$	49.6950i	0.220806i	0.993887	+	0.110403i	$0.0352141\pi$
$37$	49.6950i	0.220806i	−0.993887	+	0.110403i	$0.964786\pi$
$38$	700.783	2.99163
$39$	0	0
$40$	−758.990	−3.00017
$41$	− 107.947i	− 0.411181i	−0.978638	−	0.205591i	$-0.934088\pi$
$41$	− 107.947i	− 0.411181i	0.978638	−	0.205591i	$-0.0659116\pi$
$42$	−236.272	−0.868039
$43$	35.4166	0.125604	0.0628021	−	0.998026i	$-0.479996\pi$
$43$	35.4166	0.125604	0.0628021	+	0.998026i	$0.479996\pi$
$44$	473.643i	1.62283i
$45$	190.784i	0.632010i
$46$	40.5404i	0.129943i
$47$	374.815i	1.16324i	0.813460	+	0.581621i	$0.197581\pi$
$47$	374.815i	1.16324i	−0.813460	+	0.581621i	$0.802419\pi$
$48$	149.977	0.450985
$49$	77.9418	0.227236
$50$	1569.12i	4.43815i
$51$	−138.711	−0.380853
$52$	0	0
$53$	−348.583	−0.903426	−0.451713	−	0.892163i	$-0.649187\pi$
$53$	−348.583	−0.903426	−0.451713	+	0.892163i	$0.649187\pi$
$54$	− 130.613i	− 0.329150i
$55$	651.915	1.59826
$56$	582.916	1.39099
$57$	434.594i	1.00988i
$58$	1175.31i	2.66079i
$59$	679.430i	1.49923i	0.661877	+	0.749613i	$0.269760\pi$
$59$	679.430i	1.49923i	−0.661877	+	0.749613i	$0.730240\pi$
$60$	− 979.450i	− 2.10744i
$61$	−230.403	−0.483608	−0.241804	−	0.970325i	$-0.577739\pi$
$61$	−230.403	−0.483608	−0.241804	+	0.970325i	$0.577739\pi$
$62$	425.387	0.871358
$63$	− 146.525i	− 0.293023i
$64$	615.677	1.20249
$65$	0	0
$66$	−446.306	−0.832372
$67$	295.642i	0.539082i	0.962989	+	0.269541i	$0.0868719\pi$
$67$	295.642i	0.539082i	−0.962989	+	0.269541i	$0.913128\pi$
$68$	712.117	1.26995
$69$	−25.1413	−0.0438647
$70$	− 1669.52i	− 2.85066i
$71$	− 329.215i	− 0.550290i	−0.961403	−	0.275145i	$-0.911274\pi$
$71$	− 329.215i	− 0.550290i	0.961403	−	0.275145i	$-0.0887259\pi$
$72$	322.239i	0.527448i
$73$	48.9973i	0.0785575i	0.999228	+	0.0392787i	$0.0125060\pi$
$73$	48.9973i	0.0785575i	−0.999228	+	0.0392787i	$0.987494\pi$
$74$	240.399	0.377647
$75$	−973.099	−1.49818
$76$	− 2231.12i	− 3.36746i
$77$	−500.681	−0.741012
$78$	0	0
$79$	−107.942	−0.153727	−0.0768636	−	0.997042i	$-0.524491\pi$
$79$	−107.942	−0.153727	−0.0768636	+	0.997042i	$0.524491\pi$
$80$	1059.75i	1.48104i
$81$	81.0000	0.111111
$82$	−522.192	−0.703250
$83$	− 515.654i	− 0.681932i	−0.940076	−	0.340966i	$-0.889246\pi$
$83$	− 515.654i	− 0.681932i	0.940076	−	0.340966i	$-0.110754\pi$
$84$	752.233i	0.977088i
$85$	− 980.147i	− 1.25073i
$86$	− 171.328i	− 0.214823i
$87$	−728.874	−0.898201
$88$	1101.10	1.33384
$89$	984.453i	1.17249i	0.810133	+	0.586246i	$0.199395\pi$
$89$	984.453i	1.17249i	−0.810133	+	0.586246i	$0.800605\pi$
$90$	922.919	1.08094
$91$	0	0
$92$	129.071	0.146267
$93$	263.806i	0.294144i
$94$	1813.17	1.98951
$95$	−3070.88	−3.31648
$96$	133.792i	0.142241i
$97$	− 487.072i	− 0.509842i	−0.966962	−	0.254921i	$-0.917951\pi$
$97$	− 487.072i	− 0.509842i	0.966962	−	0.254921i	$-0.0820494\pi$
$98$	− 377.043i	− 0.388644i
$99$	− 276.779i	− 0.280983i
$100$	4995.70	4.99570
$101$	−766.375	−0.755021	−0.377511	−	0.926005i	$-0.623220\pi$
$101$	−766.375	−0.755021	−0.377511	+	0.926005i	$0.623220\pi$
$102$	671.016i	0.651378i
$103$	−1229.87	−1.17653	−0.588266	−	0.808668i	$-0.700189\pi$
$103$	−1229.87	−1.17653	−0.588266	+	0.808668i	$0.700189\pi$
$104$	0	0
$105$	1035.36	0.962295
$106$	1686.27i	1.54514i
$107$	76.8261	0.0694117	0.0347059	−	0.999398i	$-0.488951\pi$
$107$	76.8261	0.0694117	0.0347059	+	0.999398i	$0.488951\pi$
$108$	−415.838	−0.370500
$109$	626.001i	0.550092i	0.961431	+	0.275046i	$0.0886930\pi$
$109$	626.001i	0.550092i	−0.961431	+	0.275046i	$0.911307\pi$
$110$	− 3153.64i	− 2.73353i
$111$	149.085i	0.127482i
$112$	− 813.904i	− 0.686667i
$113$	−1343.21	−1.11822	−0.559108	−	0.829095i	$-0.688856\pi$
$113$	−1343.21	−1.11822	−0.559108	+	0.829095i	$0.688856\pi$
$114$	2102.35	1.72722
$115$	− 177.651i	− 0.144052i
$116$	3741.90	2.99506
$117$	0	0
$118$	3286.74	2.56415
$119$	752.769i	0.579884i
$120$	−2276.97	−1.73215
$121$	385.238	0.289435
$122$	1114.58i	0.827122i
$123$	− 323.840i	− 0.237396i
$124$	− 1354.33i	− 0.980824i
$125$	− 4226.22i	− 3.02404i
$126$	−708.817	−0.501162
$127$	2146.69	1.49990	0.749952	−	0.661492i	$-0.230077\pi$
$127$	2146.69	1.49990	0.749952	+	0.661492i	$0.230077\pi$
$128$	− 2621.56i	− 1.81028i
$129$	106.250	0.0725177
$130$	0	0
$131$	−798.626	−0.532644	−0.266322	−	0.963884i	$-0.585808\pi$
$131$	−798.626	−0.532644	−0.266322	+	0.963884i	$0.585808\pi$
$132$	1420.93i	0.936940i
$133$	2358.48	1.53764
$134$	1430.17	0.921999
$135$	572.353i	0.364891i
$136$	− 1655.49i	− 1.04380i
$137$	601.153i	0.374891i	0.982275	+	0.187445i	$0.0600207\pi$
$137$	601.153i	0.374891i	−0.982275	+	0.187445i	$0.939979\pi$
$138$	121.621i	0.0750224i
$139$	−2134.18	−1.30229	−0.651146	−	0.758953i	$-0.725711\pi$
$139$	−2134.18	−1.30229	−0.651146	+	0.758953i	$0.725711\pi$
$140$	−5315.35	−3.20878
$141$	1124.44i	0.671598i
$142$	−1592.58	−0.941169
$143$	0	0
$144$	449.930	0.260376
$145$	− 5150.29i	− 2.94971i
$146$	237.024	0.134358
$147$	233.825	0.131194
$148$	− 765.373i	− 0.425090i
$149$	3439.93i	1.89134i	0.325127	+	0.945670i	$0.394593\pi$
$149$	3439.93i	1.89134i	−0.325127	+	0.945670i	$0.605407\pi$
$150$	4707.37i	2.56237i
$151$	− 2224.04i	− 1.19861i	−0.800521	−	0.599305i	$-0.795444\pi$
$151$	− 2224.04i	− 1.19861i	0.800521	−	0.599305i	$-0.204556\pi$
$152$	−5186.78	−2.76779
$153$	−416.134	−0.219885
$154$	2422.05i	1.26736i
$155$	−1864.07	−0.965975
$156$	0	0
$157$	2465.32	1.25321	0.626604	−	0.779338i	$-0.284444\pi$
$157$	2465.32	1.25321	0.626604	+	0.779338i	$0.284444\pi$
$158$	522.171i	0.262922i
$159$	−1045.75	−0.521593
$160$	−945.386	−0.467121
$161$	136.439i	0.0667881i
$162$	− 391.838i	− 0.190035i
$163$	243.565i	0.117040i	0.998286	+	0.0585199i	$0.0186381\pi$
$163$	243.565i	0.117040i	−0.998286	+	0.0585199i	$0.981362\pi$
$164$	1662.53i	0.791597i
$165$	1955.75	0.922755
$166$	−2494.47	−1.16632
$167$	− 409.099i	− 0.189563i	−0.995498	−	0.0947814i	$-0.969785\pi$
$167$	− 409.099i	− 0.189563i	0.995498	−	0.0947814i	$-0.0302152\pi$
$168$	1748.75	0.803089
$169$	0	0
$170$	−4741.46	−2.13914
$171$	1303.78i	0.583056i
$172$	−545.466	−0.241810
$173$	2618.97	1.15096	0.575481	−	0.817815i	$-0.304815\pi$
$173$	2618.97	1.15096	0.575481	+	0.817815i	$0.304815\pi$
$174$	3525.93i	1.53621i
$175$	5280.88i	2.28113i
$176$	− 1537.42i	− 0.658452i
$177$	2038.29i	0.865578i
$178$	4762.29	2.00533
$179$	−163.311	−0.0681925	−0.0340963	−	0.999419i	$-0.510855\pi$
$179$	−163.311	−0.0681925	−0.0340963	+	0.999419i	$0.510855\pi$
$180$	− 2938.35i	− 1.21673i
$181$	3313.07	1.36054	0.680272	−	0.732960i	$-0.261862\pi$
$181$	3313.07	1.36054	0.680272	+	0.732960i	$0.261862\pi$
$182$	0	0
$183$	−691.209	−0.279211
$184$	− 300.056i	− 0.120220i
$185$	−1053.45	−0.418654
$186$	1276.16	0.503079
$187$	1421.94i	0.556057i
$188$	− 5772.68i	− 2.23945i
$189$	− 439.576i	− 0.169177i
$190$	14855.4i	5.67222i
$191$	−4281.90	−1.62213	−0.811066	−	0.584954i	$-0.801112\pi$
$191$	−4281.90	−1.62213	−0.811066	+	0.584954i	$0.801112\pi$
$192$	1847.03	0.694261
$193$	− 1877.33i	− 0.700171i	−0.936718	−	0.350086i	$-0.886152\pi$
$193$	− 1877.33i	− 0.700171i	0.936718	−	0.350086i	$-0.113848\pi$
$194$	−2356.21	−0.871990
$195$	0	0
$196$	−1200.41	−0.437468
$197$	− 1991.26i	− 0.720158i	−0.932922	−	0.360079i	$-0.882750\pi$
$197$	− 1991.26i	− 0.720158i	0.932922	−	0.360079i	$-0.117250\pi$
$198$	−1338.92	−0.480570
$199$	−1345.05	−0.479137	−0.239568	−	0.970879i	$-0.577006\pi$
$199$	−1345.05	−0.479137	−0.239568	+	0.970879i	$0.577006\pi$
$200$	− 11613.7i	− 4.10607i
$201$	886.927i	0.311239i
$202$	3707.34i	1.29132i
$203$	3955.51i	1.36760i
$204$	2136.35	0.733208
$205$	2288.28	0.779612
$206$	5949.50i	2.01224i
$207$	−75.4240	−0.0253253
$208$	0	0
$209$	4455.05	1.47446
$210$	− 5008.56i	− 1.64583i
$211$	−288.763	−0.0942147	−0.0471073	−	0.998890i	$-0.515000\pi$
$211$	−288.763	−0.0942147	−0.0471073	+	0.998890i	$0.515000\pi$
$212$	5368.67	1.73925
$213$	− 987.644i	− 0.317710i
$214$	− 371.646i	− 0.118716i
$215$	750.771i	0.238150i
$216$	966.717i	0.304522i
$217$	1431.64	0.447862
$218$	3028.28	0.940830
$219$	146.992i	0.0453552i
$220$	−10040.4	−3.07693
$221$	0	0
$222$	721.198	0.218035
$223$	− 1798.41i	− 0.540046i	−0.962854	−	0.270023i	$-0.912969\pi$
$223$	− 1798.41i	− 0.540046i	0.962854	−	0.270023i	$-0.0870314\pi$
$224$	726.072	0.216575
$225$	−2919.30	−0.864977
$226$	6497.77i	1.91250i
$227$	2486.10i	0.726909i	0.931612	+	0.363454i	$0.118403\pi$
$227$	2486.10i	0.726909i	−0.931612	+	0.363454i	$0.881597\pi$
$228$	− 6693.36i	− 1.94420i
$229$	− 6074.71i	− 1.75296i	−0.481438	−	0.876480i	$-0.659885\pi$
$229$	− 6074.71i	− 1.75296i	0.481438	−	0.876480i	$-0.340115\pi$
$230$	−859.386	−0.246375
$231$	−1502.04	−0.427824
$232$	− 8698.95i	− 2.46170i
$233$	−6367.16	−1.79024	−0.895121	−	0.445822i	$-0.852911\pi$
$233$	−6367.16	−1.79024	−0.895121	+	0.445822i	$0.852911\pi$
$234$	0	0
$235$	−7945.42	−2.20554
$236$	− 10464.2i	− 2.88627i
$237$	−323.827	−0.0887544
$238$	3641.52	0.991784
$239$	− 1886.43i	− 0.510556i	−0.966868	−	0.255278i	$-0.917833\pi$
$239$	− 1886.43i	− 0.510556i	0.966868	−	0.255278i	$-0.0821670\pi$
$240$	3179.24i	0.855081i
$241$	− 5847.91i	− 1.56306i	−0.623869	−	0.781529i	$-0.714440\pi$
$241$	− 5847.91i	− 1.56306i	0.623869	−	0.781529i	$-0.285560\pi$
$242$	− 1863.59i	− 0.495025i
$243$	243.000	0.0641500
$244$	3548.53	0.931031
$245$	1652.23i	0.430845i
$246$	−1566.58	−0.406021
$247$	0	0
$248$	−3148.46	−0.806160
$249$	− 1546.96i	− 0.393713i
$250$	−20444.3	−5.17205
$251$	2388.65	0.600678	0.300339	−	0.953832i	$-0.402900\pi$
$251$	2388.65	0.600678	0.300339	+	0.953832i	$0.402900\pi$
$252$	2256.70i	0.564122i
$253$	257.726i	0.0640438i
$254$	− 10384.6i	− 2.56531i
$255$	− 2940.44i	− 0.722108i
$256$	−7756.38	−1.89365
$257$	−4447.21	−1.07941	−0.539707	−	0.841853i	$-0.681465\pi$
$257$	−4447.21	−1.07941	−0.539707	+	0.841853i	$0.681465\pi$
$258$	− 513.984i	− 0.124028i
$259$	809.064	0.194104
$260$	0	0
$261$	−2186.62	−0.518577
$262$	3863.36i	0.910988i
$263$	7590.87	1.77975	0.889873	−	0.456208i	$-0.150793\pi$
$263$	7590.87	1.77975	0.889873	+	0.456208i	$0.150793\pi$
$264$	3303.30	0.770091
$265$	− 7389.36i	− 1.71292i
$266$	− 11409.2i	− 2.62985i
$267$	2953.36i	0.676939i
$268$	− 4553.31i	− 1.03783i
$269$	−557.911	−0.126455	−0.0632275	−	0.997999i	$-0.520139\pi$
$269$	−557.911	−0.126455	−0.0632275	+	0.997999i	$0.520139\pi$
$270$	2768.76	0.624079
$271$	2707.41i	0.606877i	0.952851	+	0.303439i	$0.0981347\pi$
$271$	2707.41i	0.606877i	−0.952851	+	0.303439i	$0.901865\pi$
$272$	−2311.50	−0.515276
$273$	0	0
$274$	2908.08	0.641181
$275$	9975.31i	2.18740i
$276$	387.212	0.0844472
$277$	1231.35	0.267092	0.133546	−	0.991043i	$-0.457364\pi$
$277$	1231.35	0.267092	0.133546	+	0.991043i	$0.457364\pi$
$278$	10324.1i	2.22733i
$279$	791.417i	0.169824i
$280$	12356.8i	2.63736i
$281$	4601.44i	0.976864i	0.872602	+	0.488432i	$0.162431\pi$
$281$	4601.44i	0.976864i	−0.872602	+	0.488432i	$0.837569\pi$
$282$	5439.50	1.14864
$283$	54.5677	0.0114619	0.00573094	−	0.999984i	$-0.498176\pi$
$283$	54.5677	0.0114619	0.00573094	+	0.999984i	$0.498176\pi$
$284$	5070.37i	1.05941i
$285$	−9212.63	−1.91477
$286$	0	0
$287$	−1757.44	−0.361457
$288$	401.376i	0.0821226i
$289$	−2775.13	−0.564854
$290$	−24914.5	−5.04494
$291$	− 1461.22i	− 0.294357i
$292$	− 754.627i	− 0.151237i
$293$	− 4745.24i	− 0.946144i	−0.881024	−	0.473072i	$-0.843145\pi$
$293$	− 4745.24i	− 0.946144i	0.881024	−	0.473072i	$-0.156855\pi$
$294$	− 1131.13i	− 0.224384i
$295$	−14402.7	−2.84258
$296$	−1779.30	−0.349390
$297$	− 830.337i	− 0.162226i
$298$	16640.6	3.23479
$299$	0	0
$300$	14987.1	2.88427
$301$	− 576.604i	− 0.110415i
$302$	−10758.8	−2.05000
$303$	−2299.13	−0.435912
$304$	7242.10i	1.36633i
$305$	− 4884.15i	− 0.916936i
$306$	2013.05i	0.376073i
$307$	− 1028.82i	− 0.191264i	−0.995417	−	0.0956320i	$-0.969513\pi$
$307$	− 1028.82i	− 0.191264i	0.995417	−	0.0956320i	$-0.0304872\pi$
$308$	7711.20	1.42658
$309$	−3689.61	−0.679271
$310$	9017.46i	1.65212i
$311$	2437.84	0.444492	0.222246	−	0.974991i	$-0.428661\pi$
$311$	2437.84	0.444492	0.222246	+	0.974991i	$0.428661\pi$
$312$	0	0
$313$	7934.20	1.43280	0.716402	−	0.697688i	$-0.245788\pi$
$313$	7934.20	1.43280	0.716402	+	0.697688i	$0.245788\pi$
$314$	− 11926.0i	− 2.14338i
$315$	3106.09	0.555581
$316$	1662.46	0.295952
$317$	6942.72i	1.23010i	0.788488	+	0.615050i	$0.210864\pi$
$317$	6942.72i	1.23010i	−0.788488	+	0.615050i	$0.789136\pi$
$318$	5058.81i	0.892089i
$319$	7471.75i	1.31140i
$320$	13051.3i	2.27997i
$321$	230.478	0.0400749
$322$	660.023	0.114229
$323$	− 6698.12i	− 1.15385i
$324$	−1247.51	−0.213909
$325$	0	0
$326$	1178.25	0.200175
$327$	1878.00i	0.317596i
$328$	3864.96	0.650630
$329$	6102.21	1.02257
$330$	− 9460.92i	− 1.57820i
$331$	− 5439.27i	− 0.903230i	−0.892213	−	0.451615i	$-0.850848\pi$
$331$	− 5439.27i	− 0.903230i	0.892213	−	0.451615i	$-0.149152\pi$
$332$	7941.79i	1.31284i
$333$	447.255i	0.0736018i
$334$	−1979.01	−0.324212
$335$	−6267.10	−1.02211
$336$	− 2441.71i	− 0.396447i
$337$	1663.87	0.268952	0.134476	−	0.990917i	$-0.457065\pi$
$337$	1663.87	0.268952	0.134476	+	0.990917i	$0.457065\pi$
$338$	0	0
$339$	−4029.63	−0.645602
$340$	15095.6i	2.40787i
$341$	2704.29	0.429460
$342$	6307.04	0.997210
$343$	− 6853.19i	− 1.07883i
$344$	1268.07i	0.198749i
$345$	− 532.953i	− 0.0831687i
$346$	− 12669.3i	− 1.96851i
$347$	9268.10	1.43383	0.716913	−	0.697163i	$-0.245554\pi$
$347$	9268.10	1.43383	0.716913	+	0.697163i	$0.245554\pi$
$348$	11225.7	1.72920
$349$	− 5239.45i	− 0.803614i	−0.915724	−	0.401807i	$-0.868382\pi$
$349$	− 5239.45i	− 0.803614i	0.915724	−	0.401807i	$-0.131618\pi$
$350$	25546.3	3.90144
$351$	0	0
$352$	1371.51	0.207676
$353$	1218.01i	0.183649i	0.995775	+	0.0918244i	$0.0292698\pi$
$353$	1218.01i	0.183649i	−0.995775	+	0.0918244i	$0.970730\pi$
$354$	9860.23	1.48041
$355$	6978.78	1.04337
$356$	− 15162.0i	− 2.25726i
$357$	2258.31i	0.334796i
$358$	790.018i	0.116631i
$359$	− 5316.06i	− 0.781534i	−0.920490	−	0.390767i	$-0.872210\pi$
$359$	− 5316.06i	− 0.781534i	0.920490	−	0.390767i	$-0.127790\pi$
$360$	−6830.91	−1.00006
$361$	−14126.7	−2.05959
$362$	− 16027.0i	− 2.32696i
$363$	1155.72	0.167106
$364$	0	0
$365$	−1038.66	−0.148947
$366$	3343.73i	0.477539i
$367$	−289.101	−0.0411197	−0.0205599	−	0.999789i	$-0.506545\pi$
$367$	−289.101	−0.0411197	−0.0205599	+	0.999789i	$0.506545\pi$
$368$	−418.957	−0.0593469
$369$	− 971.520i	− 0.137060i
$370$	5096.05i	0.716030i
$371$	5675.15i	0.794175i
$372$	− 4062.98i	− 0.566279i
$373$	6163.98	0.855654	0.427827	−	0.903861i	$-0.359279\pi$
$373$	6163.98	0.855654	0.427827	+	0.903861i	$0.359279\pi$
$374$	6878.64	0.951032
$375$	− 12678.7i	− 1.74593i
$376$	−13420.0	−1.84065
$377$	0	0
$378$	−2126.45	−0.289346
$379$	− 2440.03i	− 0.330702i	−0.986235	−	0.165351i	$-0.947124\pi$
$379$	− 2440.03i	− 0.330702i	0.986235	−	0.165351i	$-0.0528756\pi$
$380$	47295.9	6.38481
$381$	6440.06	0.865970
$382$	20713.7i	2.77436i
$383$	− 6577.28i	− 0.877502i	−0.898609	−	0.438751i	$-0.855421\pi$
$383$	− 6577.28i	− 0.877502i	0.898609	−	0.438751i	$-0.144579\pi$
$384$	− 7864.68i	− 1.04516i
$385$	− 10613.6i	− 1.40498i
$386$	−9081.58	−1.19751
$387$	318.750	0.0418681
$388$	7501.59i	0.981535i
$389$	−4964.94	−0.647127	−0.323563	−	0.946206i	$-0.604881\pi$
$389$	−4964.94	−0.647127	−0.323563	+	0.946206i	$0.604881\pi$
$390$	0	0
$391$	387.488	0.0501179
$392$	2790.65i	0.359565i
$393$	−2395.88	−0.307522
$394$	−9632.70	−1.23170
$395$	− 2288.19i	− 0.291471i
$396$	4262.79i	0.540943i
$397$	12073.9i	1.52637i	0.646178	+	0.763187i	$0.276366\pi$
$397$	12073.9i	1.52637i	−0.646178	+	0.763187i	$0.723634\pi$
$398$	6506.69i	0.819475i
$399$	7075.45	0.887758
$400$	−16215.8	−2.02697
$401$	4916.90i	0.612315i	0.951981	+	0.306158i	$0.0990434\pi$
$401$	4916.90i	0.612315i	−0.951981	+	0.306158i	$0.900957\pi$
$402$	4290.51	0.532316
$403$	0	0
$404$	11803.3	1.45355
$405$	1717.06i	0.210670i
$406$	19134.8	2.33902
$407$	1528.28	0.186128
$408$	− 4966.47i	− 0.602639i
$409$	15350.9i	1.85588i	0.372733	+	0.927939i	$0.378421\pi$
$409$	15350.9i	1.85588i	−0.372733	+	0.927939i	$0.621579\pi$
$410$	− 11069.6i	− 1.33338i
$411$	1803.46i	0.216443i
$412$	18941.7	2.26503
$413$	11061.5	1.31792
$414$	364.864i	0.0433142i
$415$	10931.0	1.29296
$416$	0	0
$417$	−6402.53	−0.751878
$418$	− 21551.3i	− 2.52179i
$419$	−5488.58	−0.639939	−0.319970	−	0.947428i	$-0.603673\pi$
$419$	−5488.58	−0.639939	−0.319970	+	0.947428i	$0.603673\pi$
$420$	−15946.0	−1.85259
$421$	− 927.681i	− 0.107393i	−0.998557	−	0.0536964i	$-0.982900\pi$
$421$	− 927.681i	− 0.107393i	0.998557	−	0.0536964i	$-0.0171003\pi$
$422$	1396.89i	0.161137i
$423$	3373.33i	0.387747i
$424$	− 12480.8i	− 1.42953i
$425$	14997.8	1.71176
$426$	−4777.73	−0.543384
$427$	3751.10i	0.425126i
$428$	−1183.23	−0.133630
$429$	0	0
$430$	3631.85	0.407311
$431$	11002.3i	1.22961i	0.788678	+	0.614806i	$0.210766\pi$
$431$	11002.3i	1.22961i	−0.788678	+	0.614806i	$0.789234\pi$
$432$	1349.79	0.150328
$433$	9596.76	1.06511	0.532553	−	0.846397i	$-0.321233\pi$
$433$	9596.76	1.06511	0.532553	+	0.846397i	$0.321233\pi$
$434$	− 6925.56i	− 0.765985i
$435$	− 15450.9i	− 1.70302i
$436$	− 9641.30i	− 1.05902i
$437$	− 1214.03i	− 0.132895i
$438$	711.073	0.0775716
$439$	10346.3	1.12483	0.562416	−	0.826854i	$-0.309872\pi$
$439$	10346.3	1.12483	0.562416	+	0.826854i	$0.309872\pi$
$440$	23341.4i	2.52899i
$441$	701.476	0.0757452
$442$	0	0
$443$	5650.38	0.606000	0.303000	−	0.952991i	$-0.402012\pi$
$443$	5650.38	0.606000	0.303000	+	0.952991i	$0.402012\pi$
$444$	− 2296.12i	− 0.245426i
$445$	−20868.7	−2.22308
$446$	−8699.80	−0.923649
$447$	10319.8i	1.09197i
$448$	− 10023.6i	− 1.05708i
$449$	11987.8i	1.25999i	0.776597	+	0.629997i	$0.216944\pi$
$449$	11987.8i	1.25999i	−0.776597	+	0.629997i	$0.783056\pi$
$450$	14122.1i	1.47938i
$451$	−3319.71	−0.346605
$452$	20687.3	2.15276
$453$	− 6672.13i	− 0.692018i
$454$	12026.5	1.24324
$455$	0	0
$456$	−15560.3	−1.59798
$457$	16437.6i	1.68253i	0.540623	+	0.841265i	$0.318189\pi$
$457$	16437.6i	1.68253i	−0.540623	+	0.841265i	$0.681811\pi$
$458$	−29386.4	−2.99811
$459$	−1248.40	−0.126951
$460$	2736.08i	0.277326i
$461$	− 8847.70i	− 0.893880i	−0.894564	−	0.446940i	$-0.852514\pi$
$461$	− 8847.70i	− 0.893880i	0.894564	−	0.446940i	$-0.147486\pi$
$462$	7266.14i	0.731713i
$463$	10269.6i	1.03081i	0.856945	+	0.515407i	$0.172359\pi$
$463$	10269.6i	1.03081i	−0.856945	+	0.515407i	$0.827641\pi$
$464$	−12146.0	−1.21523
$465$	−5592.22	−0.557706
$466$	30801.1i	3.06188i
$467$	−14730.4	−1.45962	−0.729811	−	0.683649i	$-0.760392\pi$
$467$	−14730.4	−1.45962	−0.729811	+	0.683649i	$0.760392\pi$
$468$	0	0
$469$	4813.24	0.473891
$470$	38436.0i	3.77217i
$471$	7395.95	0.723540
$472$	−24326.6	−2.37229
$473$	− 1089.18i	− 0.105878i
$474$	1566.51i	0.151798i
$475$	− 46989.2i	− 4.53897i
$476$	− 11593.7i	− 1.11638i
$477$	−3137.25	−0.301142
$478$	−9125.59	−0.873211
$479$	635.077i	0.0605792i	0.999541	+	0.0302896i	$0.00964295\pi$
$479$	635.077i	0.0605792i	−0.999541	+	0.0302896i	$0.990357\pi$
$480$	−2836.16	−0.269692
$481$	0	0
$482$	−28289.3	−2.67332
$483$	409.316i	0.0385601i
$484$	−5933.22	−0.557214
$485$	10325.1	0.966675
$486$	− 1175.51i	− 0.109717i
$487$	16611.1i	1.54563i	0.634632	+	0.772815i	$0.281152\pi$
$487$	16611.1i	1.54563i	−0.634632	+	0.772815i	$0.718848\pi$
$488$	− 8249.43i	− 0.765234i
$489$	730.696i	0.0675730i
$490$	7992.66	0.736881
$491$	2584.14	0.237516	0.118758	−	0.992923i	$-0.462109\pi$
$491$	2584.14	0.237516	0.118758	+	0.992923i	$0.462109\pi$
$492$	4987.59i	0.457029i
$493$	11233.7	1.02625
$494$	0	0
$495$	5867.24	0.532753
$496$	4396.08i	0.397963i
$497$	−5359.82	−0.483744
$498$	−7483.42	−0.673374
$499$	− 2183.88i	− 0.195919i	−0.995190	−	0.0979597i	$-0.968768\pi$
$499$	− 2183.88i	− 0.195919i	0.995190	−	0.0979597i	$-0.0312316\pi$
$500$	65089.8i	5.82180i
$501$	− 1227.30i	− 0.109444i
$502$	− 11555.1i	− 1.02735i
$503$	−17214.9	−1.52600	−0.762998	−	0.646401i	$-0.776273\pi$
$503$	−17214.9	−1.52600	−0.762998	+	0.646401i	$0.776273\pi$
$504$	5246.25	0.463664
$505$	− 16245.8i	− 1.43154i
$506$	1246.75	0.109535
$507$	0	0
$508$	−33062.0	−2.88758
$509$	− 20260.6i	− 1.76431i	−0.470955	−	0.882157i	$-0.656091\pi$
$509$	− 20260.6i	− 1.76431i	0.470955	−	0.882157i	$-0.343909\pi$
$510$	−14224.4	−1.23503
$511$	797.705	0.0690575
$512$	16549.0i	1.42846i
$513$	3911.34i	0.336628i
$514$	21513.4i	1.84614i
$515$	− 26071.1i	− 2.23074i
$516$	−1636.40	−0.139609
$517$	11526.8	0.980554
$518$	− 3913.85i	− 0.331978i
$519$	7856.90	0.664508
$520$	0	0
$521$	−20731.5	−1.74331	−0.871653	−	0.490124i	$-0.836952\pi$
$521$	−20731.5	−1.74331	−0.871653	+	0.490124i	$0.836952\pi$
$522$	10577.8i	0.886930i
$523$	−944.353	−0.0789554	−0.0394777	−	0.999220i	$-0.512569\pi$
$523$	−944.353	−0.0789554	−0.0394777	+	0.999220i	$0.512569\pi$
$524$	12300.0	1.02543
$525$	15842.6i	1.31701i
$526$	− 36720.9i	− 3.04393i
$527$	− 4065.87i	− 0.336076i
$528$	− 4612.27i	− 0.380158i
$529$	−12096.8	−0.994228
$530$	−35746.0	−2.92964
$531$	6114.87i	0.499742i
$532$	−36324.0	−2.96023
$533$	0	0
$534$	14286.9	1.15778
$535$	1628.58i	0.131607i
$536$	−10585.3	−0.853012
$537$	−489.934	−0.0393710
$538$	2698.89i	0.216278i
$539$	− 2396.96i	− 0.191548i
$540$	− 8815.05i	− 0.702480i
$541$	4883.06i	0.388058i	0.980996	+	0.194029i	$0.0621556\pi$
$541$	4883.06i	0.388058i	−0.980996	+	0.194029i	$0.937844\pi$
$542$	13097.1	1.03795
$543$	9939.21	0.785511
$544$	− 2062.05i	− 0.162518i
$545$	−13270.1	−1.04299
$546$	0	0
$547$	−16269.1	−1.27169	−0.635847	−	0.771815i	$-0.719349\pi$
$547$	−16269.1	−1.27169	−0.635847	+	0.771815i	$0.719349\pi$
$548$	− 9258.61i	− 0.721730i
$549$	−2073.63	−0.161203
$550$	48255.6	3.74114
$551$	− 35196.0i	− 2.72124i
$552$	− 900.169i	− 0.0694090i
$553$	1757.37i	0.135137i
$554$	− 5956.63i	− 0.456811i
$555$	−3160.34	−0.241710
$556$	32869.3	2.50714
$557$	− 15661.0i	− 1.19134i	−0.803229	−	0.595670i	$-0.796887\pi$
$557$	− 15661.0i	− 1.19134i	0.803229	−	0.595670i	$-0.203113\pi$
$558$	3828.48	0.290453
$559$	0	0
$560$	17253.3	1.30194
$561$	4265.82i	0.321040i
$562$	22259.5	1.67074
$563$	9915.18	0.742230	0.371115	−	0.928587i	$-0.378976\pi$
$563$	9915.18	0.742230	0.371115	+	0.928587i	$0.378976\pi$
$564$	− 17318.0i	− 1.29294i
$565$	− 28473.7i	− 2.12017i
$566$	− 263.971i	− 0.0196034i
$567$	− 1318.73i	− 0.0976745i
$568$	11787.3	0.870748
$569$	11299.1	0.832482	0.416241	−	0.909254i	$-0.363347\pi$
$569$	11299.1	0.832482	0.416241	+	0.909254i	$0.363347\pi$
$570$	44566.1i	3.27486i
$571$	17619.6	1.29134	0.645672	−	0.763615i	$-0.276577\pi$
$571$	17619.6	1.29134	0.645672	+	0.763615i	$0.276577\pi$
$572$	0	0
$573$	−12845.7	−0.936539
$574$	8501.61i	0.618206i
$575$	2718.33	0.197152
$576$	5541.10	0.400832
$577$	− 22153.5i	− 1.59837i	−0.601083	−	0.799186i	$-0.705264\pi$
$577$	− 22153.5i	− 1.59837i	0.601083	−	0.799186i	$-0.294736\pi$
$578$	13424.7i	0.966078i
$579$	− 5631.99i	− 0.404244i
$580$	79321.7i	5.67872i
$581$	−8395.15	−0.599466
$582$	−7068.63	−0.503444
$583$	10720.1i	0.761543i
$584$	−1754.31	−0.124305
$585$	0	0
$586$	−22955.1	−1.61820
$587$	− 9891.28i	− 0.695497i	−0.937588	−	0.347748i	$-0.886946\pi$
$587$	− 9891.28i	− 0.695497i	0.937588	−	0.347748i	$-0.113054\pi$
$588$	−3601.24	−0.252573
$589$	−12738.7	−0.891153
$590$	69673.3i	4.86170i
$591$	− 5973.77i	− 0.415783i
$592$	2484.36i	0.172477i
$593$	9746.23i	0.674924i	0.941339	+	0.337462i	$0.109568\pi$
$593$	9746.23i	0.674924i	−0.941339	+	0.337462i	$0.890432\pi$
$594$	−4016.76	−0.277457
$595$	−15957.4	−1.09948
$596$	− 52979.7i	− 3.64116i
$597$	−4035.16	−0.276630
$598$	0	0
$599$	−8460.53	−0.577109	−0.288554	−	0.957464i	$-0.593175\pi$
$599$	−8460.53	−0.577109	−0.288554	+	0.957464i	$0.593175\pi$
$600$	− 34841.2i	− 2.37064i
$601$	6792.99	0.461052	0.230526	−	0.973066i	$-0.425955\pi$
$601$	6792.99	0.461052	0.230526	+	0.973066i	$0.425955\pi$
$602$	−2789.32	−0.188844
$603$	2660.78i	0.179694i
$604$	34253.4i	2.30754i
$605$	8166.38i	0.548778i
$606$	11122.0i	0.745546i
$607$	19073.4	1.27539	0.637697	−	0.770287i	$-0.279887\pi$
$607$	19073.4	1.27539	0.637697	+	0.770287i	$0.279887\pi$
$608$	−6460.58	−0.430939
$609$	11866.5i	0.789582i
$610$	−23627.1	−1.56825
$611$	0	0
$612$	6409.05	0.423318
$613$	− 14465.2i	− 0.953091i	−0.879150	−	0.476545i	$-0.841889\pi$
$613$	− 14465.2i	− 0.953091i	0.879150	−	0.476545i	$-0.158111\pi$
$614$	−4976.93	−0.327122
$615$	6864.85	0.450109
$616$	− 17926.6i	− 1.17254i
$617$	12897.3i	0.841536i	0.907168	+	0.420768i	$0.138239\pi$
$617$	12897.3i	0.841536i	−0.907168	+	0.420768i	$0.861761\pi$
$618$	17848.5i	1.16177i
$619$	27735.8i	1.80096i	0.434895	+	0.900481i	$0.356786\pi$
$619$	27735.8i	1.80096i	−0.434895	+	0.900481i	$0.643214\pi$
$620$	28709.4	1.85967
$621$	−226.272	−0.0146216
$622$	− 11793.0i	− 0.760222i
$623$	16027.5	1.03070
$624$	0	0
$625$	49042.7	3.13873
$626$	− 38381.7i	− 2.45055i
$627$	13365.2	0.851281
$628$	−37969.3	−2.41265
$629$	− 2297.75i	− 0.145656i
$630$	− 15025.7i	− 0.950219i
$631$	− 28560.1i	− 1.80184i	−0.433990	−	0.900918i	$-0.642895\pi$
$631$	− 28560.1i	− 1.80184i	0.433990	−	0.900918i	$-0.357105\pi$
$632$	− 3864.80i	− 0.243249i
$633$	−866.290	−0.0543949
$634$	33585.4	2.10386
$635$	45506.1i	2.84386i
$636$	16106.0	1.00416
$637$	0	0
$638$	36144.6	2.24291
$639$	− 2962.93i	− 0.183430i
$640$	55572.5	3.43234
$641$	−1862.53	−0.114767	−0.0573833	−	0.998352i	$-0.518276\pi$
$641$	−1862.53	−0.114767	−0.0573833	+	0.998352i	$0.518276\pi$
$642$	− 1114.94i	− 0.0685407i
$643$	29495.2i	1.80899i	0.426487	+	0.904494i	$0.359751\pi$
$643$	29495.2i	1.80899i	−0.426487	+	0.904494i	$0.640249\pi$
$644$	− 2101.35i	− 0.128579i
$645$	2252.31i	0.137496i
$646$	−32402.2	−1.97345
$647$	−27780.7	−1.68805	−0.844027	−	0.536301i	$-0.819821\pi$
$647$	−27780.7	−1.68805	−0.844027	+	0.536301i	$0.819821\pi$
$648$	2900.15i	0.175816i
$649$	20894.7	1.26377
$650$	0	0
$651$	4294.92	0.258573
$652$	− 3751.25i	− 0.225322i
$653$	276.678	0.0165808	0.00829039	−	0.999966i	$-0.497361\pi$
$653$	276.678	0.0165808	0.00829039	+	0.999966i	$0.497361\pi$
$654$	9084.84	0.543188
$655$	− 16929.5i	− 1.00991i
$656$	− 5396.49i	− 0.321185i
$657$	440.975i	0.0261858i
$658$	− 29519.5i	− 1.74892i
$659$	18114.9	1.07080	0.535398	−	0.844600i	$-0.320162\pi$
$659$	18114.9	1.07080	0.535398	+	0.844600i	$0.320162\pi$
$660$	−30121.2	−1.77647
$661$	− 27094.4i	− 1.59433i	−0.603764	−	0.797163i	$-0.706333\pi$
$661$	− 27094.4i	− 1.59433i	0.603764	−	0.797163i	$-0.293667\pi$
$662$	−26312.5	−1.54481
$663$	0	0
$664$	18462.6	1.07905
$665$	49995.8i	2.91542i
$666$	2163.60	0.125882
$667$	2036.10	0.118198
$668$	6300.70i	0.364942i
$669$	− 5395.23i	− 0.311796i
$670$	30317.1i	1.74814i
$671$	7085.64i	0.407657i
$672$	2178.22	0.125039
$673$	−2410.31	−0.138054	−0.0690272	−	0.997615i	$-0.521990\pi$
$673$	−2410.31	−0.138054	−0.0690272	+	0.997615i	$0.521990\pi$
$674$	− 8048.99i	− 0.459993i
$675$	−8757.89	−0.499395
$676$	0	0
$677$	12508.3	0.710091	0.355046	−	0.934849i	$-0.384465\pi$
$677$	12508.3	0.710091	0.355046	+	0.934849i	$0.384465\pi$
$678$	19493.3i	1.10418i
$679$	−7929.83	−0.448187
$680$	35093.5	1.97908
$681$	7458.30i	0.419681i
$682$	− 13082.0i	− 0.734511i
$683$	16793.8i	0.940844i	0.882442	+	0.470422i	$0.155898\pi$
$683$	16793.8i	0.940844i	−0.882442	+	0.470422i	$0.844102\pi$
$684$	− 20080.1i	− 1.12249i
$685$	−12743.4	−0.710804
$686$	−33152.3	−1.84513
$687$	− 18224.1i	− 1.01207i
$688$	1770.56	0.0981131
$689$	0	0
$690$	−2578.16	−0.142245
$691$	− 26026.9i	− 1.43286i	−0.697657	−	0.716432i	$-0.745774\pi$
$691$	− 26026.9i	− 1.43286i	0.697657	−	0.716432i	$-0.254226\pi$
$692$	−40335.8	−2.21580
$693$	−4506.13	−0.247004
$694$	− 44834.4i	− 2.45229i
$695$	− 45240.8i	− 2.46918i
$696$	− 26096.9i	− 1.42126i
$697$	4991.14i	0.271238i
$698$	−25345.8	−1.37443
$699$	−19101.5	−1.03360
$700$	− 81333.0i	− 4.39157i
$701$	−5079.08	−0.273658	−0.136829	−	0.990595i	$-0.543691\pi$
$701$	−5079.08	−0.273658	−0.136829	+	0.990595i	$0.543691\pi$
$702$	0	0
$703$	−7199.04	−0.386226
$704$	− 18934.1i	− 1.01364i
$705$	−23836.3	−1.27337
$706$	5892.11	0.314097
$707$	12477.1i	0.663717i
$708$	− 31392.6i	− 1.66639i
$709$	2530.72i	0.134052i	0.997751	+	0.0670262i	$0.0213511\pi$
$709$	2530.72i	0.134052i	−0.997751	+	0.0670262i	$0.978649\pi$
$710$	− 33759.9i	− 1.78449i
$711$	−971.480	−0.0512424
$712$	−35247.7	−1.85529
$713$	− 736.937i	− 0.0387076i
$714$	10924.6	0.572607
$715$	0	0
$716$	2515.22	0.131283
$717$	− 5659.28i	− 0.294770i
$718$	−25716.4	−1.33667
$719$	−9503.67	−0.492944	−0.246472	−	0.969150i	$-0.579271\pi$
$719$	−9503.67	−0.492944	−0.246472	+	0.969150i	$0.579271\pi$
$720$	9537.73i	0.493681i
$721$	20023.0i	1.03425i
$722$	68338.2i	3.52255i
$723$	− 17543.7i	− 0.902432i
$724$	−51026.0	−2.61929
$725$	78807.4	4.03701
$726$	− 5590.77i	− 0.285803i
$727$	−33343.1	−1.70100	−0.850500	−	0.525975i	$-0.823700\pi$
$727$	−33343.1	−1.70100	−0.850500	+	0.525975i	$0.823700\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	5024.50i	0.254747i
$731$	−1637.56	−0.0828556
$732$	10645.6	0.537531
$733$	32288.1i	1.62699i	0.581569	+	0.813497i	$0.302439\pi$
$733$	32288.1i	1.62699i	−0.581569	+	0.813497i	$0.697561\pi$
$734$	1398.53i	0.0703277i
$735$	4956.69i	0.248749i
$736$	− 373.746i	− 0.0187180i
$737$	9091.96	0.454419
$738$	−4699.73	−0.234417
$739$	14168.8i	0.705288i	0.935758	+	0.352644i	$0.114717\pi$
$739$	14168.8i	0.705288i	−0.935758	+	0.352644i	$0.885283\pi$
$740$	16224.6	0.805983
$741$	0	0
$742$	27453.5	1.35829
$743$	13248.6i	0.654165i	0.944996	+	0.327083i	$0.106066\pi$
$743$	13248.6i	0.654165i	−0.944996	+	0.327083i	$0.893934\pi$
$744$	−9445.39	−0.465437
$745$	−72920.5	−3.58604
$746$	− 29818.3i	− 1.46344i
$747$	− 4640.88i	− 0.227311i
$748$	− 21899.9i	− 1.07051i
$749$	− 1250.78i	− 0.0610178i
$750$	−61333.0	−2.98609
$751$	−5113.98	−0.248484	−0.124242	−	0.992252i	$-0.539650\pi$
$751$	−5113.98	−0.248484	−0.124242	+	0.992252i	$0.539650\pi$
$752$	18737.8i	0.908641i
$753$	7165.95	0.346802
$754$	0	0
$755$	47145.9	2.27260
$756$	6770.10i	0.325696i
$757$	27380.4	1.31461	0.657303	−	0.753626i	$-0.271697\pi$
$757$	27380.4	1.31461	0.657303	+	0.753626i	$0.271697\pi$
$758$	−11803.6	−0.565604
$759$	773.178i	0.0369757i
$760$	− 109951.i	− 5.24781i
$761$	10613.8i	0.505586i	0.967520	+	0.252793i	$0.0813492\pi$
$761$	10613.8i	0.505586i	−0.967520	+	0.252793i	$0.918651\pi$
$762$	− 31153.8i	− 1.48108i
$763$	10191.7	0.483569
$764$	65947.3	3.12289
$765$	− 8821.32i	− 0.416909i
$766$	−31817.6	−1.50080
$767$	0	0
$768$	−23269.2	−1.09330
$769$	9172.22i	0.430115i	0.976601	+	0.215058i	$0.0689939\pi$
$769$	9172.22i	0.430115i	−0.976601	+	0.215058i	$0.931006\pi$
$770$	−51343.2	−2.40296
$771$	−13341.6	−0.623200
$772$	28913.5i	1.34795i
$773$	14952.1i	0.695718i	0.937547	+	0.347859i	$0.113091\pi$
$773$	14952.1i	0.695718i	−0.937547	+	0.347859i	$0.886909\pi$
$774$	− 1541.95i	− 0.0716076i
$775$	− 28523.2i	− 1.32205i
$776$	17439.3	0.806745
$777$	2427.19	0.112066
$778$	24017.9i	1.10679i
$779$	15637.7	0.719226
$780$	0	0
$781$	−10124.4	−0.463867
$782$	− 1874.47i	− 0.0857174i
$783$	−6559.87	−0.299400
$784$	3896.48	0.177500
$785$	52260.4i	2.37612i
$786$	11590.1i	0.525959i
$787$	2671.52i	0.121003i	0.998168	+	0.0605015i	$0.0192700\pi$
$787$	2671.52i	0.121003i	−0.998168	+	0.0605015i	$0.980730\pi$
$788$	30668.1i	1.38643i
$789$	22772.6	1.02754
$790$	−11069.1	−0.498508
$791$	21868.3i	0.982991i
$792$	9909.89	0.444612
$793$	0	0
$794$	58407.4	2.61058
$795$	− 22168.1i	− 0.988957i
$796$	20715.7	0.922423
$797$	12949.7	0.575536	0.287768	−	0.957700i	$-0.407087\pi$
$797$	12949.7	0.575536	0.287768	+	0.957700i	$0.407087\pi$
$798$	− 34227.5i	− 1.51835i
$799$	− 17330.4i	− 0.767339i
$800$	− 14465.9i	− 0.639307i
$801$	8860.08i	0.390831i
$802$	23785.5	1.04725
$803$	1506.82	0.0662200
$804$	− 13659.9i	− 0.599190i
$805$	−2892.27	−0.126632
$806$	0	0
$807$	−1673.73	−0.0730089
$808$	− 27439.6i	− 1.19470i
$809$	−11640.4	−0.505876	−0.252938	−	0.967482i	$-0.581397\pi$
$809$	−11640.4	−0.505876	−0.252938	+	0.967482i	$0.581397\pi$
$810$	8306.28	0.360312
$811$	18135.3i	0.785225i	0.919704	+	0.392613i	$0.128429\pi$
$811$	18135.3i	0.785225i	−0.919704	+	0.392613i	$0.871571\pi$
$812$	− 60920.4i	− 2.63286i
$813$	8122.24i	0.350381i
$814$	− 7393.06i	− 0.318337i
$815$	−5163.16	−0.221911
$816$	−6934.49	−0.297495
$817$	5130.61i	0.219703i
$818$	74260.0	3.17413
$819$	0	0
$820$	−35242.8	−1.50089
$821$	− 10789.1i	− 0.458640i	−0.973351	−	0.229320i	$-0.926350\pi$
$821$	− 10789.1i	− 0.458640i	0.973351	−	0.229320i	$-0.0736503\pi$
$822$	8724.24	0.370186
$823$	12284.7	0.520312	0.260156	−	0.965567i	$-0.416226\pi$
$823$	12284.7	0.520312	0.260156	+	0.965567i	$0.416226\pi$
$824$	− 44034.7i	− 1.86168i
$825$	29925.9i	1.26289i
$826$	− 53510.2i	− 2.25407i
$827$	36077.2i	1.51696i	0.651696	+	0.758480i	$0.274058\pi$
$827$	36077.2i	1.51696i	−0.651696	+	0.758480i	$0.725942\pi$
$828$	1161.64	0.0487556
$829$	8861.83	0.371271	0.185636	−	0.982619i	$-0.440566\pi$
$829$	8861.83	0.371271	0.185636	+	0.982619i	$0.440566\pi$
$830$	− 52878.5i	− 2.21137i
$831$	3694.04	0.154205
$832$	0	0
$833$	−3603.80	−0.149897
$834$	30972.2i	1.28595i
$835$	8672.18	0.359417
$836$	−68614.1	−2.83860
$837$	2374.25i	0.0980480i
$838$	26551.0i	1.09450i
$839$	5833.37i	0.240036i	0.992772	+	0.120018i	$0.0382952\pi$
$839$	5833.37i	0.240036i	−0.992772	+	0.120018i	$0.961705\pi$
$840$	37070.4i	1.52268i
$841$	34639.6	1.42030
$842$	−4487.66	−0.183676
$843$	13804.3i	0.563993i
$844$	4447.37	0.181380
$845$	0	0
$846$	16318.5	0.663170
$847$	− 6271.92i	− 0.254434i
$848$	−17426.4	−0.705692
$849$	163.703	0.00661752
$850$	− 72551.7i	− 2.92765i
$851$	− 416.466i	− 0.0167759i
$852$	15211.1i	0.611648i
$853$	− 28649.5i	− 1.14999i	−0.818157	−	0.574995i	$-0.805004\pi$
$853$	− 28649.5i	− 1.14999i	0.818157	−	0.574995i	$-0.194996\pi$
$854$	18146.0	0.727098
$855$	−27637.9	−1.10549
$856$	2750.71i	0.109833i
$857$	−32336.1	−1.28889	−0.644445	−	0.764651i	$-0.722911\pi$
$857$	−32336.1	−1.28889	−0.644445	+	0.764651i	$0.722911\pi$
$858$	0	0
$859$	−13878.5	−0.551254	−0.275627	−	0.961265i	$-0.588885\pi$
$859$	−13878.5	−0.551254	−0.275627	+	0.961265i	$0.588885\pi$
$860$	− 11562.9i	− 0.458480i
$861$	−5272.31	−0.208688
$862$	53223.7	2.10302
$863$	32934.7i	1.29908i	0.760326	+	0.649542i	$0.225039\pi$
$863$	32934.7i	1.29908i	−0.760326	+	0.649542i	$0.774961\pi$
$864$	1204.13i	0.0474135i
$865$	55517.5i	2.18226i
$866$	− 46424.3i	− 1.82167i
$867$	−8325.38	−0.326119
$868$	−22049.3	−0.862213
$869$	3319.57i	0.129584i
$870$	−74743.6	−2.91270
$871$	0	0
$872$	−22413.5	−0.870434
$873$	− 4383.65i	− 0.169947i
$874$	−5872.87	−0.227292
$875$	−68805.4	−2.65834
$876$	− 2263.88i	− 0.0873167i
$877$	18375.7i	0.707531i	0.935334	+	0.353765i	$0.115099\pi$
$877$	18375.7i	0.707531i	−0.935334	+	0.353765i	$0.884901\pi$
$878$	− 50050.2i	− 1.92382i
$879$	− 14235.7i	− 0.546256i
$880$	32590.7	1.24845
$881$	−46883.4	−1.79290	−0.896448	−	0.443150i	$-0.853861\pi$
$881$	−46883.4	−1.79290	−0.896448	+	0.443150i	$0.853861\pi$
$882$	− 3393.39i	− 0.129548i
$883$	−1050.05	−0.0400194	−0.0200097	−	0.999800i	$-0.506370\pi$
$883$	−1050.05	−0.0400194	−0.0200097	+	0.999800i	$0.506370\pi$
$884$	0	0
$885$	−43208.2	−1.64116
$886$	− 27333.7i	− 1.03645i
$887$	27916.8	1.05677	0.528385	−	0.849005i	$-0.322798\pi$
$887$	27916.8	1.05677	0.528385	+	0.849005i	$0.322798\pi$
$888$	−5337.89	−0.201720
$889$	− 34949.4i	− 1.31852i
$890$	100952.i	3.80217i
$891$	− 2491.01i	− 0.0936611i
$892$	27698.0i	1.03968i
$893$	−54297.4	−2.03471
$894$	49921.9	1.86761
$895$	− 3461.92i	− 0.129295i
$896$	−42680.6	−1.59136
$897$	0	0
$898$	57990.8	2.15499
$899$	− 21364.6i	− 0.792601i
$900$	44961.3	1.66523
$901$	16117.5	0.595950
$902$	16059.1i	0.592804i
$903$	− 1729.81i	− 0.0637481i
$904$	− 48092.7i	− 1.76940i
$905$	70231.3i	2.57963i
$906$	−32276.4	−1.18357
$907$	−23220.1	−0.850066	−0.425033	−	0.905178i	$-0.639737\pi$
$907$	−23220.1	−0.850066	−0.425033	+	0.905178i	$0.639737\pi$
$908$	− 38289.5i	− 1.39943i
$909$	−6897.38	−0.251674
$910$	0	0
$911$	16344.9	0.594435	0.297217	−	0.954810i	$-0.403941\pi$
$911$	16344.9	0.594435	0.297217	+	0.954810i	$0.403941\pi$
$912$	21726.3i	0.788848i
$913$	−15858.0	−0.574834
$914$	79516.7	2.87766
$915$	− 14652.4i	− 0.529393i
$916$	93559.0i	3.37476i
$917$	13002.1i	0.468231i
$918$	6039.15i	0.217126i
$919$	−32931.7	−1.18206	−0.591032	−	0.806648i	$-0.701279\pi$
$919$	−32931.7	−1.18206	−0.591032	+	0.806648i	$0.701279\pi$
$920$	6360.67	0.227940
$921$	− 3086.47i	− 0.110426i
$922$	−42800.8	−1.52882
$923$	0	0
$924$	23133.6	0.823636
$925$	− 16119.4i	− 0.572975i
$926$	49679.0	1.76302
$927$	−11068.8	−0.392177
$928$	− 10835.3i	− 0.383282i
$929$	56305.5i	1.98851i	0.107056	+	0.994253i	$0.465858\pi$
$929$	56305.5i	1.98851i	−0.107056	+	0.994253i	$0.534142\pi$
$930$	27052.4i	0.953852i
$931$	11291.0i	0.397473i
$932$	98063.3	3.44653
$933$	7313.51	0.256628
$934$	71258.5i	2.49641i
$935$	−30142.7	−1.05430
$936$	0	0
$937$	7095.61	0.247389	0.123695	−	0.992320i	$-0.460526\pi$
$937$	7095.61	0.247389	0.123695	+	0.992320i	$0.460526\pi$
$938$	− 23284.0i	− 0.810502i
$939$	23802.6	0.827230
$940$	122371.	4.24606
$941$	5185.17i	0.179630i	0.995958	+	0.0898150i	$0.0286276\pi$
$941$	5185.17i	0.179630i	−0.995958	+	0.0898150i	$0.971372\pi$
$942$	− 35777.9i	− 1.23748i
$943$	904.641i	0.0312398i
$944$	33966.2i	1.17109i
$945$	9318.26	0.320765
$946$	−5268.89	−0.181085
$947$	38451.7i	1.31944i	0.751510	+	0.659722i	$0.229326\pi$
$947$	38451.7i	1.31944i	−0.751510	+	0.659722i	$0.770674\pi$
$948$	4987.39	0.170868
$949$	0	0
$950$	−227310.	−7.76307
$951$	20828.2i	0.710199i
$952$	−26952.4	−0.917575
$953$	20930.5	0.711445	0.355722	−	0.934592i	$-0.384235\pi$
$953$	20930.5	0.711445	0.355722	+	0.934592i	$0.384235\pi$
$954$	15176.4i	0.515048i
$955$	− 90768.8i	− 3.07561i
$956$	29053.7i	0.982910i
$957$	22415.2i	0.757139i
$958$	3072.19	0.103609
$959$	9787.14	0.329555
$960$	39153.9i	1.31634i
$961$	22058.4	0.740438
$962$	0	0
$963$	691.435	0.0231372
$964$	90066.1i	3.00916i
$965$	39796.1	1.32755
$966$	1980.07	0.0659499
$967$	50788.1i	1.68897i	0.535578	+	0.844486i	$0.320094\pi$
$967$	50788.1i	1.68897i	−0.535578	+	0.844486i	$0.679906\pi$
$968$	13793.2i	0.457986i
$969$	− 20094.4i	− 0.666175i
$970$	− 49947.6i	− 1.65332i
$971$	15277.5	0.504922	0.252461	−	0.967607i	$-0.418760\pi$
$971$	15277.5	0.504922	0.252461	+	0.967607i	$0.418760\pi$
$972$	−3742.54	−0.123500
$973$	34745.7i	1.14481i
$974$	80356.3	2.64351
$975$	0	0
$976$	−11518.4	−0.377760
$977$	− 1293.19i	− 0.0423469i	−0.999776	−	0.0211734i	$-0.993260\pi$
$977$	− 1293.19i	− 0.0423469i	0.999776	−	0.0211734i	$-0.00674022\pi$
$978$	3534.74	0.115571
$979$	30275.1	0.988353
$980$	− 25446.7i	− 0.829453i
$981$	5634.01i	0.183364i
$982$	− 12500.8i	− 0.406227i
$983$	8474.63i	0.274973i	0.990504	+	0.137487i	$0.0439024\pi$
$983$	8474.63i	0.274973i	−0.990504	+	0.137487i	$0.956098\pi$
$984$	11594.9	0.375641
$985$	42211.1	1.36544
$986$	− 54343.0i	− 1.75521i
$987$	18306.6	0.590382
$988$	0	0
$989$	−296.807	−0.00954289
$990$	− 28382.8i	− 0.911175i
$991$	−7080.71	−0.226969	−0.113484	−	0.993540i	$-0.536201\pi$
$991$	−7080.71	−0.226969	−0.113484	+	0.993540i	$0.536201\pi$
$992$	−3921.68	−0.125518
$993$	− 16317.8i	− 0.521480i
$994$	25928.1i	0.827354i
$995$	− 28512.8i	− 0.908458i
$996$	23825.4i	0.757968i
$997$	19423.1	0.616985	0.308493	−	0.951227i	$-0.400175\pi$
$997$	19423.1	0.616985	0.308493	+	0.951227i	$0.400175\pi$
$998$	−10564.5	−0.335084
$999$	1341.76i	0.0424940i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.b.k.337.2		18
13.5	odd	4		507.4.a.p.1.1	yes	9
13.8	odd	4		507.4.a.o.1.9	✓	9
13.12	even	2	inner	507.4.b.k.337.17		18
39.5	even	4		1521.4.a.bf.1.9		9
39.8	even	4		1521.4.a.bi.1.1		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.4.a.o.1.9	✓	9	13.8	odd	4
507.4.a.p.1.1	yes	9	13.5	odd	4
507.4.b.k.337.2		18	1.1	even	1	trivial
507.4.b.k.337.17		18	13.12	even	2	inner
1521.4.a.bf.1.9		9	39.5	even	4
1521.4.a.bi.1.1		9	39.8	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	507.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.83750i q^{2} +3.00000 q^{3} -15.4014 q^{4} +21.1983i q^{5} -14.5125i q^{6} -16.2806i q^{7} +35.8043i q^{8} +9.00000 q^{9} +102.547 q^{10} -30.7532i q^{11} -46.2042 q^{12} -78.7575 q^{14} +63.5948i q^{15} +49.9922 q^{16} -46.2371 q^{17} -43.5375i q^{18} +144.865i q^{19} -326.483i q^{20} -48.8418i q^{21} -148.769 q^{22} -8.38045 q^{23} +107.413i q^{24} -324.366 q^{25} +27.0000 q^{27} +250.744i q^{28} -242.958 q^{29} +307.640 q^{30} +87.9353i q^{31} +44.5973i q^{32} -92.2597i q^{33} +223.672i q^{34} +345.121 q^{35} -138.613 q^{36} +49.6950i q^{37} +700.783 q^{38} -758.990 q^{40} -107.947i q^{41} -236.272 q^{42} +35.4166 q^{43} +473.643i q^{44} +190.784i q^{45} +40.5404i q^{46} +374.815i q^{47} +149.977 q^{48} +77.9418 q^{49} +1569.12i q^{50} -138.711 q^{51} -348.583 q^{53} -130.613i q^{54} +651.915 q^{55} +582.916 q^{56} +434.594i q^{57} +1175.31i q^{58} +679.430i q^{59} -979.450i q^{60} -230.403 q^{61} +425.387 q^{62} -146.525i q^{63} +615.677 q^{64} -446.306 q^{66} +295.642i q^{67} +712.117 q^{68} -25.1413 q^{69} -1669.52i q^{70} -329.215i q^{71} +322.239i q^{72} +48.9973i q^{73} +240.399 q^{74} -973.099 q^{75} -2231.12i q^{76} -500.681 q^{77} -107.942 q^{79} +1059.75i q^{80} +81.0000 q^{81} -522.192 q^{82} -515.654i q^{83} +752.233i q^{84} -980.147i q^{85} -171.328i q^{86} -728.874 q^{87} +1101.10 q^{88} +984.453i q^{89} +922.919 q^{90} +129.071 q^{92} +263.806i q^{93} +1813.17 q^{94} -3070.88 q^{95} +133.792i q^{96} -487.072i q^{97} -377.043i q^{98} -276.779i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 54 q^{3} - 88 q^{4} + 162 q^{9} + 108 q^{10} - 264 q^{12} + 316 q^{14} + 432 q^{16} - 356 q^{17} - 1260 q^{22} - 300 q^{23} + 40 q^{25} + 486 q^{27} - 194 q^{29} + 324 q^{30} - 836 q^{35} - 792 q^{36}+ \cdots - 8316 q^{95}+O(q^{100}) \) 18 * q + 54 * q^3 - 88 * q^4 + 162 * q^9 + 108 * q^10 - 264 * q^12 + 316 * q^14 + 432 * q^16 - 356 * q^17 - 1260 * q^22 - 300 * q^23 + 40 * q^25 + 486 * q^27 - 194 * q^29 + 324 * q^30 - 836 * q^35 - 792 * q^36 + 1320 * q^38 - 3012 * q^40 + 948 * q^42 - 484 * q^43 + 1296 * q^48 + 76 * q^49 - 1068 * q^51 - 302 * q^53 + 4128 * q^55 - 4552 * q^56 - 2680 * q^61 - 694 * q^62 - 1786 * q^64 - 3780 * q^66 + 5570 * q^68 - 900 * q^69 - 2382 * q^74 + 120 * q^75 + 4284 * q^77 - 3182 * q^79 + 1458 * q^81 - 3034 * q^82 - 582 * q^87 + 7432 * q^88 + 972 * q^90 + 1030 * q^92 - 1384 * q^94 - 8316 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more