LMFDB - Embedded newform 507.4.b.j.337.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [507,4,Mod(337,507)]

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

lf = mfeigenbasis(mf)

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")

//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);

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

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} + 97 x^{16} + 3906 x^{14} + 84743 x^{12} + 1077128 x^{10} + 8187552 x^{8} + 36483705 x^{6} + \cdots + 26460736 $ x^18 + 97x^16 + 3906x^14 + 84743x^12 + 1077128x^10 + 8187552x^8 + 36483705x^6 + 88861676x^4 + 98825392x^2 + 26460736
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 13^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.6
Root			$-1.73419i$ of defining polynomial
Character	$\chi$	$=$	507.337
Dual form			507.4.b.j.337.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.73419i q^{2} -3.00000 q^{3} +0.524213 q^{4} -21.1246i q^{5} +8.20257i q^{6} +25.8618i q^{7} -23.3068i q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-2.73419i q^{2} -3.00000 q^{3} +0.524213 q^{4} -21.1246i q^{5} +8.20257i q^{6} +25.8618i q^{7} -23.3068i q^{8} +9.00000 q^{9} -57.7586 q^{10} +6.96892i q^{11} -1.57264 q^{12} +70.7111 q^{14} +63.3738i q^{15} -59.5315 q^{16} -122.879 q^{17} -24.6077i q^{18} +43.1340i q^{19} -11.0738i q^{20} -77.5854i q^{21} +19.0543 q^{22} +75.5492 q^{23} +69.9204i q^{24} -321.248 q^{25} -27.0000 q^{27} +13.5571i q^{28} -163.764 q^{29} +173.276 q^{30} +139.421i q^{31} -23.6841i q^{32} -20.9068i q^{33} +335.975i q^{34} +546.320 q^{35} +4.71792 q^{36} -2.80616i q^{37} +117.936 q^{38} -492.347 q^{40} -300.555i q^{41} -212.133 q^{42} -363.145 q^{43} +3.65320i q^{44} -190.121i q^{45} -206.566i q^{46} +41.2660i q^{47} +178.594 q^{48} -325.834 q^{49} +878.354i q^{50} +368.638 q^{51} -125.763 q^{53} +73.8231i q^{54} +147.216 q^{55} +602.756 q^{56} -129.402i q^{57} +447.762i q^{58} +407.311i q^{59} +33.2214i q^{60} +536.710 q^{61} +381.204 q^{62} +232.756i q^{63} -541.009 q^{64} -57.1630 q^{66} +340.155i q^{67} -64.4150 q^{68} -226.648 q^{69} -1493.74i q^{70} +514.831i q^{71} -209.761i q^{72} +491.231i q^{73} -7.67257 q^{74} +963.745 q^{75} +22.6114i q^{76} -180.229 q^{77} +762.869 q^{79} +1257.58i q^{80} +81.0000 q^{81} -821.774 q^{82} +345.948i q^{83} -40.6713i q^{84} +2595.78i q^{85} +992.907i q^{86} +491.292 q^{87} +162.423 q^{88} -362.482i q^{89} -519.828 q^{90} +39.6039 q^{92} -418.264i q^{93} +112.829 q^{94} +911.188 q^{95} +71.0523i q^{96} -276.297i q^{97} +890.890i q^{98} +62.7203i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 54 q^{3} - 64 q^{4} + 162 q^{9}+O(q^{10}) $ 18 * q - 54 * q^3 - 64 * q^4 + 162 * q^9	$ 18 q - 54 q^{3} - 64 q^{4} + 162 q^{9} - 396 q^{10} + 192 q^{12} + 196 q^{14} + 64 q^{16} + 268 q^{17} + 548 q^{22} - 452 q^{23} - 1224 q^{25} - 486 q^{27} - 1094 q^{29} + 1188 q^{30} + 276 q^{35} - 576 q^{36} + 832 q^{38} + 2684 q^{40} - 588 q^{42} - 316 q^{43} - 192 q^{48} - 1284 q^{49} - 804 q^{51} + 2798 q^{53} - 2816 q^{55} + 1232 q^{56} + 4184 q^{61} + 586 q^{62} - 4962 q^{64} - 1644 q^{66} - 3158 q^{68} + 1356 q^{69} + 2074 q^{74} + 3672 q^{75} + 3372 q^{77} - 230 q^{79} + 1458 q^{81} + 10294 q^{82} + 3282 q^{87} + 968 q^{88} - 3564 q^{90} + 4174 q^{92} - 936 q^{94} + 444 q^{95}+O(q^{100}) $ 18 * q - 54 * q^3 - 64 * q^4 + 162 * q^9 - 396 * q^10 + 192 * q^12 + 196 * q^14 + 64 * q^16 + 268 * q^17 + 548 * q^22 - 452 * q^23 - 1224 * q^25 - 486 * q^27 - 1094 * q^29 + 1188 * q^30 + 276 * q^35 - 576 * q^36 + 832 * q^38 + 2684 * q^40 - 588 * q^42 - 316 * q^43 - 192 * q^48 - 1284 * q^49 - 804 * q^51 + 2798 * q^53 - 2816 * q^55 + 1232 * q^56 + 4184 * q^61 + 586 * q^62 - 4962 * q^64 - 1644 * q^66 - 3158 * q^68 + 1356 * q^69 + 2074 * q^74 + 3672 * q^75 + 3372 * q^77 - 230 * q^79 + 1458 * q^81 + 10294 * q^82 + 3282 * q^87 + 968 * q^88 - 3564 * q^90 + 4174 * q^92 - 936 * q^94 + 444 * q^95

Display 100 coefficients 10 coefficients

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 $

$2$	− 2.73419i	− 0.966682i	−0.875432	−	0.483341i	$-0.839423\pi$
$2$	− 2.73419i	− 0.966682i	0.875432	−	0.483341i	$-0.160577\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0.524213	0.0655266
$5$	− 21.1246i	− 1.88944i	−0.327877	−	0.944721i	$-0.606333\pi$
$5$	− 21.1246i	− 1.88944i	0.327877	−	0.944721i	$-0.393667\pi$
$6$	8.20257i	0.558114i
$7$	25.8618i	1.39641i	0.715899	+	0.698203i	$0.246017\pi$
$7$	25.8618i	1.39641i	−0.715899	+	0.698203i	$0.753983\pi$
$8$	− 23.3068i	− 1.03003i
$9$	9.00000	0.333333
$10$	−57.7586	−1.82649
$11$	6.96892i	0.191019i	0.995429	+	0.0955095i	$0.0304480\pi$
$11$	6.96892i	0.191019i	−0.995429	+	0.0955095i	$0.969552\pi$
$12$	−1.57264	−0.0378318
$13$	0	0
$13$	0	0
$14$	70.7111	1.34988
$15$	63.3738i	1.09087i
$16$	−59.5315	−0.930180
$17$	−122.879	−1.75310	−0.876548	−	0.481315i	$-0.840159\pi$
$17$	−122.879	−1.75310	−0.876548	+	0.481315i	$0.840159\pi$
$18$	− 24.6077i	− 0.322227i
$19$	43.1340i	0.520822i	0.965498	+	0.260411i	$0.0838580\pi$
$19$	43.1340i	0.520822i	−0.965498	+	0.260411i	$0.916142\pi$
$20$	− 11.0738i	− 0.123809i
$21$	− 77.5854i	− 0.806216i
$22$	19.0543	0.184655
$23$	75.5492	0.684918	0.342459	−	0.939533i	$-0.388740\pi$
$23$	75.5492	0.684918	0.342459	+	0.939533i	$0.388740\pi$
$24$	69.9204i	0.594685i
$25$	−321.248	−2.56999
$26$	0	0
$27$	−27.0000	−0.192450
$28$	13.5571i	0.0915018i
$29$	−163.764	−1.04863	−0.524314	−	0.851525i	$-0.675678\pi$
$29$	−163.764	−1.04863	−0.524314	+	0.851525i	$0.675678\pi$
$30$	173.276	1.05452
$31$	139.421i	0.807767i	0.914810	+	0.403884i	$0.132340\pi$
$31$	139.421i	0.807767i	−0.914810	+	0.403884i	$0.867660\pi$
$32$	− 23.6841i	− 0.130837i
$33$	− 20.9068i	− 0.110285i
$34$	335.975i	1.69469i
$35$	546.320	2.63843
$36$	4.71792	0.0218422
$37$	− 2.80616i	− 0.0124684i	−0.999981	−	0.00623419i	$-0.998016\pi$
$37$	− 2.80616i	− 0.0124684i	0.999981	−	0.00623419i	$-0.00198442\pi$
$38$	117.936	0.503469
$39$	0	0
$40$	−492.347	−1.94617
$41$	− 300.555i	− 1.14485i	−0.819957	−	0.572425i	$-0.806003\pi$
$41$	− 300.555i	− 1.14485i	0.819957	−	0.572425i	$-0.193997\pi$
$42$	−212.133	−0.779354
$43$	−363.145	−1.28789	−0.643943	−	0.765073i	$-0.722703\pi$
$43$	−363.145	−1.28789	−0.643943	+	0.765073i	$0.722703\pi$
$44$	3.65320i	0.0125168i
$45$	− 190.121i	− 0.629814i
$46$	− 206.566i	− 0.662097i
$47$	41.2660i	0.128070i	0.997948	+	0.0640348i	$0.0203969\pi$
$47$	41.2660i	0.128070i	−0.997948	+	0.0640348i	$0.979603\pi$
$48$	178.594	0.537039
$49$	−325.834	−0.949952
$50$	878.354i	2.48436i
$51$	368.638	1.01215
$52$	0	0
$53$	−125.763	−0.325941	−0.162971	−	0.986631i	$-0.552108\pi$
$53$	−125.763	−0.325941	−0.162971	+	0.986631i	$0.552108\pi$
$54$	73.8231i	0.186038i
$55$	147.216	0.360919
$56$	602.756	1.43833
$57$	− 129.402i	− 0.300697i
$58$	447.762i	1.01369i
$59$	407.311i	0.898768i	0.893339	+	0.449384i	$0.148357\pi$
$59$	407.311i	0.898768i	−0.893339	+	0.449384i	$0.851643\pi$
$60$	33.2214i	0.0714810i
$61$	536.710	1.12654	0.563268	−	0.826274i	$-0.309544\pi$
$61$	536.710	1.12654	0.563268	+	0.826274i	$0.309544\pi$
$62$	381.204	0.780854
$63$	232.756i	0.465469i
$64$	−541.009	−1.05666
$65$	0	0
$66$	−57.1630	−0.106610
$67$	340.155i	0.620247i	0.950696	+	0.310124i	$0.100370\pi$
$67$	340.155i	0.620247i	−0.950696	+	0.310124i	$0.899630\pi$
$68$	−64.4150	−0.114874
$69$	−226.648	−0.395437
$70$	− 1493.74i	− 2.55052i
$71$	514.831i	0.860552i	0.902697	+	0.430276i	$0.141584\pi$
$71$	514.831i	0.860552i	−0.902697	+	0.430276i	$0.858416\pi$
$72$	− 209.761i	− 0.343342i
$73$	491.231i	0.787592i	0.919198	+	0.393796i	$0.128838\pi$
$73$	491.231i	0.787592i	−0.919198	+	0.393796i	$0.871162\pi$
$74$	−7.67257	−0.0120529
$75$	963.745	1.48378
$76$	22.6114i	0.0341277i
$77$	−180.229	−0.266740
$78$	0	0
$79$	762.869	1.08645	0.543225	−	0.839587i	$-0.317203\pi$
$79$	762.869	1.08645	0.543225	+	0.839587i	$0.317203\pi$
$80$	1257.58i	1.75752i
$81$	81.0000	0.111111
$82$	−821.774	−1.10670
$83$	345.948i	0.457503i	0.973485	+	0.228752i	$0.0734643\pi$
$83$	345.948i	0.457503i	−0.973485	+	0.228752i	$0.926536\pi$
$84$	− 40.6713i	− 0.0528286i
$85$	2595.78i	3.31237i
$86$	992.907i	1.24498i
$87$	491.292	0.605426
$88$	162.423	0.196754
$89$	− 362.482i	− 0.431720i	−0.976424	−	0.215860i	$-0.930745\pi$
$89$	− 362.482i	− 0.431720i	0.976424	−	0.215860i	$-0.0692554\pi$
$90$	−519.828	−0.608829
$91$	0	0
$92$	39.6039	0.0448803
$93$	− 418.264i	− 0.466365i
$94$	112.829	0.123803
$95$	911.188	0.984062
$96$	71.0523i	0.0755390i
$97$	− 276.297i	− 0.289213i	−0.989489	−	0.144607i	$-0.953808\pi$
$97$	− 276.297i	− 0.289213i	0.989489	−	0.144607i	$-0.0461916\pi$
$98$	890.890i	0.918301i
$99$	62.7203i	0.0636730i
$100$	−168.403	−0.168403
$101$	−313.657	−0.309010	−0.154505	−	0.987992i	$-0.549378\pi$
$101$	−313.657	−0.309010	−0.154505	+	0.987992i	$0.549378\pi$
$102$	− 1007.93i	− 0.978427i
$103$	−507.209	−0.485212	−0.242606	−	0.970125i	$-0.578002\pi$
$103$	−507.209	−0.485212	−0.242606	+	0.970125i	$0.578002\pi$
$104$	0	0
$105$	−1638.96	−1.52330
$106$	343.860i	0.315082i
$107$	−1262.50	−1.14066	−0.570330	−	0.821416i	$-0.693185\pi$
$107$	−1262.50	−1.14066	−0.570330	+	0.821416i	$0.693185\pi$
$108$	−14.1537	−0.0126106
$109$	− 469.356i	− 0.412442i	−0.978505	−	0.206221i	$-0.933883\pi$
$109$	− 469.356i	− 0.412442i	0.978505	−	0.206221i	$-0.0661165\pi$
$110$	− 402.515i	− 0.348894i
$111$	8.41848i	0.00719862i
$112$	− 1539.59i	− 1.29891i
$113$	−1110.69	−0.924649	−0.462325	−	0.886711i	$-0.652985\pi$
$113$	−1110.69	−0.924649	−0.462325	+	0.886711i	$0.652985\pi$
$114$	−353.809	−0.290678
$115$	− 1595.95i	− 1.29411i
$116$	−85.8472	−0.0687131
$117$	0	0
$118$	1113.66	0.868823
$119$	− 3177.88i	− 2.44803i
$120$	1477.04	1.12362
$121$	1282.43	0.963512
$122$	− 1467.47i	− 1.08900i
$123$	901.665i	0.660979i
$124$	73.0864i	0.0529303i
$125$	4145.67i	2.96640i
$126$	636.400	0.449960
$127$	2547.15	1.77971	0.889853	−	0.456248i	$-0.150807\pi$
$127$	2547.15	1.77971	0.889853	+	0.456248i	$0.150807\pi$
$128$	1289.75i	0.890614i
$129$	1089.44	0.743561
$130$	0	0
$131$	−701.027	−0.467550	−0.233775	−	0.972291i	$-0.575108\pi$
$131$	−701.027	−0.467550	−0.233775	+	0.972291i	$0.575108\pi$
$132$	− 10.9596i	− 0.00722659i
$133$	−1115.52	−0.727279
$134$	930.048	0.599581
$135$	570.364i	0.363623i
$136$	2863.93i	1.80573i
$137$	− 2576.33i	− 1.60665i	−0.595541	−	0.803325i	$-0.703062\pi$
$137$	− 2576.33i	− 1.60665i	0.595541	−	0.803325i	$-0.296938\pi$
$138$	619.698i	0.382262i
$139$	233.539	0.142508	0.0712538	−	0.997458i	$-0.477300\pi$
$139$	233.539	0.142508	0.0712538	+	0.997458i	$0.477300\pi$
$140$	286.388	0.172887
$141$	− 123.798i	− 0.0739410i
$142$	1407.64	0.831880
$143$	0	0
$144$	−535.783	−0.310060
$145$	3459.45i	1.98132i
$146$	1343.12	0.761350
$147$	977.501	0.548455
$148$	− 1.47102i	0 0.000817010i
$149$	1595.97i	0.877495i	0.898610	+	0.438748i	$0.144578\pi$
$149$	1595.97i	0.877495i	−0.898610	+	0.438748i	$0.855422\pi$
$150$	− 2635.06i	− 1.43435i
$151$	− 3494.13i	− 1.88310i	−0.336869	−	0.941552i	$-0.609368\pi$
$151$	− 3494.13i	− 1.88310i	0.336869	−	0.941552i	$-0.390632\pi$
$152$	1005.32	0.536459
$153$	−1105.91	−0.584365
$154$	492.780i	0.257853i
$155$	2945.22	1.52623
$156$	0	0
$157$	−2203.71	−1.12022	−0.560112	−	0.828417i	$-0.689242\pi$
$157$	−2203.71	−1.12022	−0.560112	+	0.828417i	$0.689242\pi$
$158$	− 2085.83i	− 1.05025i
$159$	377.289	0.188182
$160$	−500.317	−0.247210
$161$	1953.84i	0.956424i
$162$	− 221.469i	− 0.107409i
$163$	− 758.697i	− 0.364575i	−0.983245	−	0.182288i	$-0.941650\pi$
$163$	− 758.697i	− 0.364575i	0.983245	−	0.182288i	$-0.0583502\pi$
$164$	− 157.555i	− 0.0750181i
$165$	−441.647	−0.208377
$166$	945.888	0.442260
$167$	855.272i	0.396305i	0.980171	+	0.198152i	$0.0634941\pi$
$167$	855.272i	0.396305i	−0.980171	+	0.198152i	$0.936506\pi$
$168$	−1808.27	−0.830422
$169$	0	0
$170$	7097.34	3.20201
$171$	388.206i	0.173607i
$172$	−190.365	−0.0843908
$173$	76.6923	0.0337041	0.0168521	−	0.999858i	$-0.494636\pi$
$173$	76.6923	0.0337041	0.0168521	+	0.999858i	$0.494636\pi$
$174$	− 1343.29i	− 0.585254i
$175$	− 8308.07i	− 3.58875i
$176$	− 414.870i	− 0.177682i
$177$	− 1221.93i	− 0.518904i
$178$	−991.095	−0.417335
$179$	−3533.09	−1.47528	−0.737642	−	0.675193i	$-0.764061\pi$
$179$	−3533.09	−1.47528	−0.737642	+	0.675193i	$0.764061\pi$
$180$	− 99.6641i	− 0.0412696i
$181$	−2352.77	−0.966189	−0.483095	−	0.875568i	$-0.660487\pi$
$181$	−2352.77	−0.966189	−0.483095	+	0.875568i	$0.660487\pi$
$182$	0	0
$183$	−1610.13	−0.650406
$184$	− 1760.81i	− 0.705482i
$185$	−59.2790	−0.0235583
$186$	−1143.61	−0.450826
$187$	− 856.337i	− 0.334875i
$188$	21.6322i	0.00839197i
$189$	− 698.269i	− 0.268739i
$190$	− 2491.36i	− 0.951275i
$191$	−1532.91	−0.580719	−0.290360	−	0.956918i	$-0.593775\pi$
$191$	−1532.91	−0.580719	−0.290360	+	0.956918i	$0.593775\pi$
$192$	1623.03	0.610062
$193$	− 4611.82i	− 1.72003i	−0.510269	−	0.860015i	$-0.670454\pi$
$193$	− 4611.82i	− 1.72003i	0.510269	−	0.860015i	$-0.329546\pi$
$194$	−755.447	−0.279577
$195$	0	0
$196$	−170.806	−0.0622471
$197$	− 1914.37i	− 0.692353i	−0.938169	−	0.346176i	$-0.887480\pi$
$197$	− 1914.37i	− 0.692353i	0.938169	−	0.346176i	$-0.112520\pi$
$198$	171.489	0.0615515
$199$	−2304.94	−0.821070	−0.410535	−	0.911845i	$-0.634658\pi$
$199$	−2304.94	−0.821070	−0.410535	+	0.911845i	$0.634658\pi$
$200$	7487.27i	2.64715i
$201$	− 1020.47i	− 0.358100i
$202$	857.596i	0.298714i
$203$	− 4235.24i	− 1.46431i
$204$	193.245	0.0663228
$205$	−6349.10	−2.16312
$206$	1386.81i	0.469045i
$207$	679.943	0.228306
$208$	0	0
$209$	−300.597	−0.0994868
$210$	4481.23i	1.47254i
$211$	−3500.56	−1.14213	−0.571064	−	0.820906i	$-0.693469\pi$
$211$	−3500.56	−1.14213	−0.571064	+	0.820906i	$0.693469\pi$
$212$	−65.9266	−0.0213578
$213$	− 1544.49i	− 0.496840i
$214$	3451.92i	1.10266i
$215$	7671.29i	2.43339i
$216$	629.284i	0.198228i
$217$	−3605.69	−1.12797
$218$	−1283.31	−0.398700
$219$	− 1473.69i	− 0.454716i
$220$	77.1723	0.0236498
$221$	0	0
$222$	23.0177	0.00695877
$223$	1255.99i	0.377163i	0.982058	+	0.188582i	$0.0603890\pi$
$223$	1255.99i	0.377163i	−0.982058	+	0.188582i	$0.939611\pi$
$224$	612.514	0.182702
$225$	−2891.24	−0.856662
$226$	3036.85i	0.893842i
$227$	− 3408.52i	− 0.996615i	−0.867000	−	0.498307i	$-0.833955\pi$
$227$	− 3408.52i	− 0.996615i	0.867000	−	0.498307i	$-0.166045\pi$
$228$	− 67.8342i	− 0.0197036i
$229$	4135.13i	1.19326i	0.802515	+	0.596631i	$0.203495\pi$
$229$	4135.13i	1.19326i	−0.802515	+	0.596631i	$0.796505\pi$
$230$	−4363.62	−1.25099
$231$	540.687	0.154003
$232$	3816.82i	1.08011i
$233$	−3788.88	−1.06531	−0.532656	−	0.846332i	$-0.678806\pi$
$233$	−3788.88	−1.06531	−0.532656	+	0.846332i	$0.678806\pi$
$234$	0	0
$235$	871.728	0.241980
$236$	213.518i	0.0588933i
$237$	−2288.61	−0.627262
$238$	−8688.93	−2.36647
$239$	1543.75i	0.417810i	0.977936	+	0.208905i	$0.0669900\pi$
$239$	1543.75i	0.417810i	−0.977936	+	0.208905i	$0.933010\pi$
$240$	− 3772.74i	− 1.01470i
$241$	− 6295.40i	− 1.68266i	−0.540518	−	0.841332i	$-0.681772\pi$
$241$	− 6295.40i	− 1.68266i	0.540518	−	0.841332i	$-0.318228\pi$
$242$	− 3506.42i	− 0.931409i
$243$	−243.000	−0.0641500
$244$	281.350	0.0738181
$245$	6883.10i	1.79488i
$246$	2465.32	0.638956
$247$	0	0
$248$	3249.46	0.832021
$249$	− 1037.84i	− 0.264140i
$250$	11335.0	2.86756
$251$	−2241.93	−0.563781	−0.281891	−	0.959447i	$-0.590962\pi$
$251$	−2241.93	−0.563781	−0.281891	+	0.959447i	$0.590962\pi$
$252$	122.014i	0.0305006i
$253$	526.497i	0.130832i
$254$	− 6964.38i	− 1.72041i
$255$	− 7787.33i	− 1.91240i
$256$	−801.658	−0.195717
$257$	−3603.80	−0.874703	−0.437351	−	0.899291i	$-0.644083\pi$
$257$	−3603.80	−0.874703	−0.437351	+	0.899291i	$0.644083\pi$
$258$	− 2978.72i	− 0.718787i
$259$	72.5724	0.0174109
$260$	0	0
$261$	−1473.88	−0.349543
$262$	1916.74i	0.451972i
$263$	−2131.06	−0.499646	−0.249823	−	0.968292i	$-0.580372\pi$
$263$	−2131.06	−0.499646	−0.249823	+	0.968292i	$0.580372\pi$
$264$	−487.270	−0.113596
$265$	2656.69i	0.615847i
$266$	3050.05i	0.703047i
$267$	1087.45i	0.249253i
$268$	178.314i	0.0406427i
$269$	−2505.79	−0.567958	−0.283979	−	0.958830i	$-0.591655\pi$
$269$	−2505.79	−0.567958	−0.283979	+	0.958830i	$0.591655\pi$
$270$	1559.48	0.351508
$271$	230.220i	0.0516046i	0.999667	+	0.0258023i	$0.00821404\pi$
$271$	230.220i	0.0516046i	−0.999667	+	0.0258023i	$0.991786\pi$
$272$	7315.19	1.63069
$273$	0	0
$274$	−7044.18	−1.55312
$275$	− 2238.75i	− 0.490916i
$276$	−118.812	−0.0259117
$277$	7750.38	1.68114	0.840569	−	0.541705i	$-0.182221\pi$
$277$	7750.38	1.68114	0.840569	+	0.541705i	$0.182221\pi$
$278$	− 638.541i	− 0.137760i
$279$	1254.79i	0.269256i
$280$	− 12733.0i	− 2.71765i
$281$	5179.21i	1.09952i	0.835322	+	0.549761i	$0.185281\pi$
$281$	5179.21i	1.09952i	−0.835322	+	0.549761i	$0.814719\pi$
$282$	−338.487	−0.0714774
$283$	−3799.93	−0.798171	−0.399085	−	0.916914i	$-0.630672\pi$
$283$	−3799.93	−0.798171	−0.399085	+	0.916914i	$0.630672\pi$
$284$	269.881i	0.0563891i
$285$	−2733.56	−0.568148
$286$	0	0
$287$	7772.90	1.59868
$288$	− 213.157i	− 0.0436125i
$289$	10186.3	2.07334
$290$	9458.78	1.91531
$291$	828.890i	0.166977i
$292$	257.509i	0.0516082i
$293$	5840.85i	1.16459i	0.812976	+	0.582297i	$0.197846\pi$
$293$	5840.85i	1.16459i	−0.812976	+	0.582297i	$0.802154\pi$
$294$	− 2672.67i	− 0.530181i
$295$	8604.27	1.69817
$296$	−65.4026	−0.0128427
$297$	− 188.161i	− 0.0367616i
$298$	4363.68	0.848258
$299$	0	0
$300$	505.208	0.0972273
$301$	− 9391.59i	− 1.79841i
$302$	−9553.62	−1.82036
$303$	940.970	0.178407
$304$	− 2567.83i	− 0.484458i
$305$	− 11337.8i	− 2.12852i
$306$	3023.78i	0.564895i
$307$	− 1686.29i	− 0.313490i	−0.987639	−	0.156745i	$-0.949900\pi$
$307$	− 1686.29i	− 0.313490i	0.987639	−	0.156745i	$-0.0501001\pi$
$308$	−94.4783	−0.0174786
$309$	1521.63	0.280137
$310$	− 8052.78i	− 1.47538i
$311$	10601.1	1.93290	0.966449	−	0.256860i	$-0.0826881\pi$
$311$	10601.1	1.93290	0.966449	+	0.256860i	$0.0826881\pi$
$312$	0	0
$313$	−5889.99	−1.06365	−0.531824	−	0.846855i	$-0.678493\pi$
$313$	−5889.99	−1.06365	−0.531824	+	0.846855i	$0.678493\pi$
$314$	6025.35i	1.08290i
$315$	4916.88	0.879476
$316$	399.906	0.0711913
$317$	− 2432.98i	− 0.431071i	−0.976496	−	0.215536i	$-0.930850\pi$
$317$	− 2432.98i	− 0.431071i	0.976496	−	0.215536i	$-0.0691498\pi$
$318$	− 1031.58i	− 0.181912i
$319$	− 1141.26i	− 0.200308i
$320$	11428.6i	1.99649i
$321$	3787.50	0.658560
$322$	5342.17	0.924557
$323$	− 5300.28i	− 0.913050i
$324$	42.4612	0.00728073
$325$	0	0
$326$	−2074.42	−0.352428
$327$	1408.07i	0.238123i
$328$	−7004.98	−1.17922
$329$	−1067.21	−0.178837
$330$	1207.55i	0.201434i
$331$	− 757.534i	− 0.125794i	−0.998020	−	0.0628971i	$-0.979966\pi$
$331$	− 757.534i	− 0.125794i	0.998020	−	0.0628971i	$-0.0200340\pi$
$332$	181.351i	0.0299786i
$333$	− 25.2554i	− 0.00415612i
$334$	2338.47	0.383101
$335$	7185.64	1.17192
$336$	4618.78i	0.749926i
$337$	889.307	0.143750	0.0718748	−	0.997414i	$-0.477102\pi$
$337$	889.307	0.143750	0.0718748	+	0.997414i	$0.477102\pi$
$338$	0	0
$339$	3332.08	0.533847
$340$	1360.74i	0.217048i
$341$	−971.615	−0.154299
$342$	1061.43	0.167823
$343$	443.956i	0.0698874i
$344$	8463.75i	1.32655i
$345$	4787.84i	0.747156i
$346$	− 209.691i	− 0.0325811i
$347$	−9644.48	−1.49205	−0.746027	−	0.665916i	$-0.768041\pi$
$347$	−9644.48	−1.49205	−0.746027	+	0.665916i	$0.768041\pi$
$348$	257.542	0.0396715
$349$	− 1807.89i	− 0.277289i	−0.990342	−	0.138645i	$-0.955725\pi$
$349$	− 1807.89i	− 0.277289i	0.990342	−	0.138645i	$-0.0442746\pi$
$350$	−22715.8	−3.46918
$351$	0	0
$352$	165.053	0.0249924
$353$	− 10681.8i	− 1.61058i	−0.592883	−	0.805289i	$-0.702010\pi$
$353$	− 10681.8i	− 1.61058i	0.592883	−	0.805289i	$-0.297990\pi$
$354$	−3340.99	−0.501615
$355$	10875.6	1.62596
$356$	− 190.018i	− 0.0282891i
$357$	9533.65i	1.41337i
$358$	9660.14i	1.42613i
$359$	− 8195.95i	− 1.20492i	−0.798149	−	0.602459i	$-0.794188\pi$
$359$	− 8195.95i	− 1.20492i	0.798149	−	0.602459i	$-0.205812\pi$
$360$	−4431.12	−0.648724
$361$	4998.46	0.728745
$362$	6432.92i	0.933997i
$363$	−3847.30	−0.556284
$364$	0	0
$365$	10377.0	1.48811
$366$	4402.40i	0.628736i
$367$	−2555.29	−0.363447	−0.181724	−	0.983350i	$-0.558168\pi$
$367$	−2555.29	−0.363447	−0.181724	+	0.983350i	$0.558168\pi$
$368$	−4497.56	−0.637096
$369$	− 2705.00i	− 0.381616i
$370$	162.080i	0.0227733i
$371$	− 3252.46i	− 0.455147i
$372$	− 219.259i	− 0.0305593i
$373$	−600.757	−0.0833941	−0.0416971	−	0.999130i	$-0.513276\pi$
$373$	−600.757	−0.0833941	−0.0416971	+	0.999130i	$0.513276\pi$
$374$	−2341.39	−0.323717
$375$	− 12437.0i	− 1.71265i
$376$	961.779	0.131915
$377$	0	0
$378$	−1909.20	−0.259785
$379$	863.612i	0.117047i	0.998286	+	0.0585234i	$0.0186392\pi$
$379$	863.612i	0.117047i	−0.998286	+	0.0585234i	$0.981361\pi$
$380$	477.656	0.0644822
$381$	−7641.44	−1.02751
$382$	4191.26i	0.561371i
$383$	9089.38i	1.21265i	0.795216	+	0.606326i	$0.207357\pi$
$383$	9089.38i	1.21265i	−0.795216	+	0.606326i	$0.792643\pi$
$384$	− 3869.24i	− 0.514196i
$385$	3807.26i	0.503990i
$386$	−12609.6	−1.66272
$387$	−3268.31	−0.429295
$388$	− 144.838i	− 0.0189511i
$389$	1513.16	0.197224	0.0986120	−	0.995126i	$-0.468560\pi$
$389$	1513.16	0.197224	0.0986120	+	0.995126i	$0.468560\pi$
$390$	0	0
$391$	−9283.44	−1.20073
$392$	7594.14i	0.978474i
$393$	2103.08	0.269940
$394$	−5234.26	−0.669285
$395$	− 16115.3i	− 2.05278i
$396$	32.8788i	0.00417228i
$397$	7572.77i	0.957346i	0.877993	+	0.478673i	$0.158882\pi$
$397$	7572.77i	0.957346i	−0.877993	+	0.478673i	$0.841118\pi$
$398$	6302.15i	0.793714i
$399$	3346.57	0.419895
$400$	19124.4	2.39055
$401$	− 11075.5i	− 1.37926i	−0.724162	−	0.689630i	$-0.757773\pi$
$401$	− 11075.5i	− 1.37926i	0.724162	−	0.689630i	$-0.242227\pi$
$402$	−2790.14	−0.346168
$403$	0	0
$404$	−164.423	−0.0202484
$405$	− 1711.09i	− 0.209938i
$406$	−11579.9	−1.41552
$407$	19.5559	0.00238170
$408$	− 8591.78i	− 1.04254i
$409$	11082.8i	1.33988i	0.742415	+	0.669940i	$0.233680\pi$
$409$	11082.8i	1.33988i	−0.742415	+	0.669940i	$0.766320\pi$
$410$	17359.6i	2.09105i
$411$	7729.00i	0.927600i
$412$	−265.886	−0.0317943
$413$	−10533.8	−1.25505
$414$	− 1859.09i	− 0.220699i
$415$	7308.02	0.864425
$416$	0	0
$417$	−700.618	−0.0822768
$418$	821.890i	0.0961721i
$419$	13035.6	1.51988	0.759940	−	0.649993i	$-0.225228\pi$
$419$	13035.6	1.51988	0.759940	+	0.649993i	$0.225228\pi$
$420$	−859.164	−0.0998165
$421$	− 14664.0i	− 1.69757i	−0.528735	−	0.848787i	$-0.677334\pi$
$421$	− 14664.0i	− 1.69757i	0.528735	−	0.848787i	$-0.322666\pi$
$422$	9571.20i	1.10407i
$423$	371.394i	0.0426899i
$424$	2931.14i	0.335728i
$425$	39474.8	4.50543
$426$	−4222.93	−0.480286
$427$	13880.3i	1.57310i
$428$	−661.820	−0.0747436
$429$	0	0
$430$	20974.8	2.35231
$431$	1454.87i	0.162596i	0.996690	+	0.0812979i	$0.0259065\pi$
$431$	1454.87i	0.162596i	−0.996690	+	0.0812979i	$0.974093\pi$
$432$	1607.35	0.179013
$433$	8982.19	0.996897	0.498449	−	0.866919i	$-0.333903\pi$
$433$	8982.19	0.996897	0.498449	+	0.866919i	$0.333903\pi$
$434$	9858.62i	1.09039i
$435$	− 10378.3i	− 1.14392i
$436$	− 246.042i	− 0.0270259i
$437$	3258.74i	0.356720i
$438$	−4029.35	−0.439566
$439$	−3348.05	−0.363995	−0.181997	−	0.983299i	$-0.558256\pi$
$439$	−3348.05	−0.363995	−0.181997	+	0.983299i	$0.558256\pi$
$440$	− 3431.13i	− 0.371756i
$441$	−2932.50	−0.316651
$442$	0	0
$443$	−15671.4	−1.68075	−0.840375	−	0.542006i	$-0.817665\pi$
$443$	−15671.4	−1.68075	−0.840375	+	0.542006i	$0.817665\pi$
$444$	4.41307i	0 0.000471701i
$445$	−7657.29	−0.815709
$446$	3434.12	0.364597
$447$	− 4787.90i	− 0.506622i
$448$	− 13991.5i	− 1.47552i
$449$	12265.8i	1.28922i	0.764511	+	0.644611i	$0.222981\pi$
$449$	12265.8i	1.28922i	−0.764511	+	0.644611i	$0.777019\pi$
$450$	7905.18i	0.828120i
$451$	2094.54	0.218688
$452$	−582.240	−0.0605891
$453$	10482.4i	1.08721i
$454$	−9319.54	−0.963409
$455$	0	0
$456$	−3015.95	−0.309725
$457$	8382.54i	0.858027i	0.903298	+	0.429014i	$0.141139\pi$
$457$	8382.54i	0.858027i	−0.903298	+	0.429014i	$0.858861\pi$
$458$	11306.2	1.15351
$459$	3317.74	0.337383
$460$	− 836.616i	− 0.0847987i
$461$	− 2308.36i	− 0.233212i	−0.993178	−	0.116606i	$-0.962798\pi$
$461$	− 2308.36i	− 0.233212i	0.993178	−	0.116606i	$-0.0372015\pi$
$462$	− 1478.34i	− 0.148871i
$463$	− 2330.53i	− 0.233928i	−0.993136	−	0.116964i	$-0.962684\pi$
$463$	− 2330.53i	− 0.233928i	0.993136	−	0.116964i	$-0.0373163\pi$
$464$	9749.12	0.975413
$465$	−8835.65	−0.881169
$466$	10359.5i	1.02982i
$467$	−3437.89	−0.340657	−0.170328	−	0.985387i	$-0.554483\pi$
$467$	−3437.89	−0.340657	−0.170328	+	0.985387i	$0.554483\pi$
$468$	0	0
$469$	−8797.03	−0.866117
$470$	− 2383.47i	− 0.233918i
$471$	6611.12	0.646761
$472$	9493.11	0.925754
$473$	− 2530.73i	− 0.246011i
$474$	6257.48i	0.606362i
$475$	− 13856.7i	− 1.33851i
$476$	− 1665.89i	− 0.160411i
$477$	−1131.87	−0.108647
$478$	4220.89	0.403889
$479$	15820.0i	1.50905i	0.656272	+	0.754525i	$0.272133\pi$
$479$	15820.0i	1.50905i	−0.656272	+	0.754525i	$0.727867\pi$
$480$	1500.95	0.142727
$481$	0	0
$482$	−17212.8	−1.62660
$483$	− 5861.52i	− 0.552191i
$484$	672.268	0.0631357
$485$	−5836.65	−0.546451
$486$	664.408i	0.0620127i
$487$	− 14504.1i	− 1.34957i	−0.738012	−	0.674787i	$-0.764235\pi$
$487$	− 14504.1i	− 1.34957i	0.738012	−	0.674787i	$-0.235765\pi$
$488$	− 12509.0i	− 1.16036i
$489$	2276.09i	0.210488i
$490$	18819.7	1.73508
$491$	17513.3	1.60970	0.804850	−	0.593479i	$-0.202246\pi$
$491$	17513.3	1.60970	0.804850	+	0.593479i	$0.202246\pi$
$492$	472.665i	0.0433117i
$493$	20123.2	1.83835
$494$	0	0
$495$	1324.94	0.120306
$496$	− 8299.95i	− 0.751369i
$497$	−13314.5	−1.20168
$498$	−2837.66	−0.255339
$499$	3856.26i	0.345952i	0.984926	+	0.172976i	$0.0553383\pi$
$499$	3856.26i	0.345952i	−0.984926	+	0.172976i	$0.944662\pi$
$500$	2173.21i	0.194378i
$501$	− 2565.82i	− 0.228807i
$502$	6129.85i	0.544997i
$503$	−1822.27	−0.161533	−0.0807666	−	0.996733i	$-0.525737\pi$
$503$	−1822.27	−0.161533	−0.0807666	+	0.996733i	$0.525737\pi$
$504$	5424.81	0.479445
$505$	6625.87i	0.583856i
$506$	1439.54	0.126473
$507$	0	0
$508$	1335.25	0.116618
$509$	− 3738.01i	− 0.325510i	−0.986667	−	0.162755i	$-0.947962\pi$
$509$	− 3738.01i	− 0.325510i	0.986667	−	0.162755i	$-0.0520380\pi$
$510$	−21292.0	−1.84868
$511$	−12704.1	−1.09980
$512$	12509.9i	1.07981i
$513$	− 1164.62i	− 0.100232i
$514$	9853.46i	0.845559i
$515$	10714.6i	0.916779i
$516$	571.096	0.0487231
$517$	−287.580	−0.0244637
$518$	− 198.427i	− 0.0168308i
$519$	−230.077	−0.0194591
$520$	0	0
$521$	17735.5	1.49138	0.745690	−	0.666294i	$-0.232120\pi$
$521$	17735.5	1.49138	0.745690	+	0.666294i	$0.232120\pi$
$522$	4029.86i	0.337897i
$523$	−6099.99	−0.510007	−0.255004	−	0.966940i	$-0.582077\pi$
$523$	−6099.99	−0.510007	−0.255004	+	0.966940i	$0.582077\pi$
$524$	−367.487	−0.0306370
$525$	24924.2i	2.07196i
$526$	5826.72i	0.482998i
$527$	− 17132.0i	− 1.41609i
$528$	1244.61i	0.102585i
$529$	−6459.31	−0.530888
$530$	7263.90	0.595328
$531$	3665.80i	0.299589i
$532$	−584.772	−0.0476561
$533$	0	0
$534$	2973.28	0.240949
$535$	26669.8i	2.15521i
$536$	7927.93	0.638870
$537$	10599.3	0.851755
$538$	6851.30i	0.549035i
$539$	− 2270.71i	− 0.181459i
$540$	298.992i	0.0238270i
$541$	− 4453.47i	− 0.353918i	−0.984218	−	0.176959i	$-0.943374\pi$
$541$	− 4453.47i	− 0.353918i	0.984218	−	0.176959i	$-0.0566260\pi$
$542$	629.464	0.0498852
$543$	7058.32	0.557830
$544$	2910.29i	0.229371i
$545$	−9914.95	−0.779284
$546$	0	0
$547$	−17915.1	−1.40035	−0.700176	−	0.713970i	$-0.746895\pi$
$547$	−17915.1	−1.40035	−0.700176	+	0.713970i	$0.746895\pi$
$548$	− 1350.55i	− 0.105278i
$549$	4830.39	0.375512
$550$	−6121.18	−0.474560
$551$	− 7063.79i	− 0.546148i
$552$	5282.43i	0.407310i
$553$	19729.2i	1.51712i
$554$	− 21191.0i	− 1.62513i
$555$	177.837	0.0136014
$556$	122.424	0.00933804
$557$	3127.12i	0.237882i	0.992901	+	0.118941i	$0.0379500\pi$
$557$	3127.12i	0.237882i	−0.992901	+	0.118941i	$0.962050\pi$
$558$	3430.83	0.260285
$559$	0	0
$560$	−32523.3	−2.45421
$561$	2569.01i	0.193340i
$562$	14160.9	1.06289
$563$	−12978.0	−0.971507	−0.485753	−	0.874096i	$-0.661455\pi$
$563$	−12978.0	−0.971507	−0.485753	+	0.874096i	$0.661455\pi$
$564$	− 64.8966i	− 0.00484510i
$565$	23463.0i	1.74707i
$566$	10389.7i	0.771577i
$567$	2094.81i	0.155156i
$568$	11999.1	0.886390
$569$	−1275.65	−0.0939857	−0.0469928	−	0.998895i	$-0.514964\pi$
$569$	−1275.65	−0.0939857	−0.0469928	+	0.998895i	$0.514964\pi$
$570$	7474.08i	0.549219i
$571$	−26406.4	−1.93533	−0.967663	−	0.252245i	$-0.918831\pi$
$571$	−26406.4	−1.93533	−0.967663	+	0.252245i	$0.918831\pi$
$572$	0	0
$573$	4598.73	0.335278
$574$	− 21252.6i	− 1.54541i
$575$	−24270.1	−1.76023
$576$	−4869.08	−0.352219
$577$	− 14705.2i	− 1.06098i	−0.847692	−	0.530489i	$-0.822008\pi$
$577$	− 14705.2i	− 1.06098i	0.847692	−	0.530489i	$-0.177992\pi$
$578$	− 27851.4i	− 2.00426i
$579$	13835.4i	0.993060i
$580$	1813.49i	0.129829i
$581$	−8946.85	−0.638860
$582$	2266.34	0.161414
$583$	− 876.433i	− 0.0622610i
$584$	11449.0	0.811239
$585$	0	0
$586$	15970.0	1.12579
$587$	18627.3i	1.30976i	0.755732	+	0.654881i	$0.227281\pi$
$587$	18627.3i	1.30976i	−0.755732	+	0.654881i	$0.772719\pi$
$588$	512.418	0.0359384
$589$	−6013.79	−0.420703
$590$	− 23525.7i	− 1.64159i
$591$	5743.12i	0.399730i
$592$	167.055i	0.0115978i
$593$	− 4495.16i	− 0.311288i	−0.987813	−	0.155644i	$-0.950255\pi$
$593$	− 4495.16i	− 0.311288i	0.987813	−	0.155644i	$-0.0497453\pi$
$594$	−514.467	−0.0355368
$595$	−67131.5	−4.62542
$596$	836.627i	0.0574993i
$597$	6914.83	0.474045
$598$	0	0
$599$	3262.10	0.222514	0.111257	−	0.993792i	$-0.464512\pi$
$599$	3262.10	0.222514	0.111257	+	0.993792i	$0.464512\pi$
$600$	− 22461.8i	− 1.52833i
$601$	−5618.35	−0.381327	−0.190663	−	0.981655i	$-0.561064\pi$
$601$	−5618.35	−0.381327	−0.190663	+	0.981655i	$0.561064\pi$
$602$	−25678.4	−1.73849
$603$	3061.40i	0.206749i
$604$	− 1831.67i	− 0.123393i
$605$	− 27090.9i	− 1.82050i
$606$	− 2572.79i	− 0.172463i
$607$	−19129.2	−1.27913	−0.639563	−	0.768738i	$-0.720885\pi$
$607$	−19129.2	−1.27913	−0.639563	+	0.768738i	$0.720885\pi$
$608$	1021.59	0.0681430
$609$	12705.7i	0.845421i
$610$	−30999.6	−2.05760
$611$	0	0
$612$	−579.735	−0.0382915
$613$	− 16054.6i	− 1.05781i	−0.848679	−	0.528907i	$-0.822602\pi$
$613$	− 16054.6i	− 1.05781i	0.848679	−	0.528907i	$-0.177398\pi$
$614$	−4610.62	−0.303045
$615$	19047.3	1.24888
$616$	4200.56i	0.274749i
$617$	− 18669.4i	− 1.21816i	−0.793110	−	0.609079i	$-0.791539\pi$
$617$	− 18669.4i	− 1.21816i	0.793110	−	0.609079i	$-0.208461\pi$
$618$	− 4160.42i	− 0.270803i
$619$	− 2488.87i	− 0.161609i	−0.996730	−	0.0808045i	$-0.974251\pi$
$619$	− 2488.87i	− 0.161609i	0.996730	−	0.0808045i	$-0.0257489\pi$
$620$	1543.92	0.100009
$621$	−2039.83	−0.131812
$622$	− 28985.3i	− 1.86850i
$623$	9374.45	0.602856
$624$	0	0
$625$	47419.5	3.03485
$626$	16104.4i	1.02821i
$627$	901.792	0.0574388
$628$	−1155.21	−0.0734044
$629$	344.819i	0.0218582i
$630$	− 13443.7i	− 0.850173i
$631$	3595.22i	0.226820i	0.993548	+	0.113410i	$0.0361774\pi$
$631$	3595.22i	0.226820i	−0.993548	+	0.113410i	$0.963823\pi$
$632$	− 17780.0i	− 1.11907i
$633$	10501.7	0.659407
$634$	−6652.22	−0.416709
$635$	− 53807.4i	− 3.36265i
$636$	197.780	0.0123309
$637$	0	0
$638$	−3120.42	−0.193634
$639$	4633.48i	0.286851i
$640$	27245.4	1.68276
$641$	4048.14	0.249441	0.124721	−	0.992192i	$-0.460197\pi$
$641$	4048.14	0.249441	0.124721	+	0.992192i	$0.460197\pi$
$642$	− 10355.8i	− 0.636618i
$643$	20768.9i	1.27379i	0.770951	+	0.636894i	$0.219781\pi$
$643$	20768.9i	1.27379i	−0.770951	+	0.636894i	$0.780219\pi$
$644$	1024.23i	0.0626712i
$645$	− 23013.9i	− 1.40492i
$646$	−14492.0	−0.882629
$647$	−14788.6	−0.898610	−0.449305	−	0.893378i	$-0.648328\pi$
$647$	−14788.6	−0.898610	−0.449305	+	0.893378i	$0.648328\pi$
$648$	− 1887.85i	− 0.114447i
$649$	−2838.52	−0.171682
$650$	0	0
$651$	10817.1	0.651235
$652$	− 397.719i	− 0.0238894i
$653$	13026.1	0.780628	0.390314	−	0.920682i	$-0.372367\pi$
$653$	13026.1	0.780628	0.390314	+	0.920682i	$0.372367\pi$
$654$	3849.92	0.230189
$655$	14808.9i	0.883408i
$656$	17892.5i	1.06492i
$657$	4421.07i	0.262531i
$658$	2917.97i	0.172879i
$659$	1633.07	0.0965330	0.0482665	−	0.998834i	$-0.484630\pi$
$659$	1633.07	0.0965330	0.0482665	+	0.998834i	$0.484630\pi$
$660$	−231.517	−0.0136542
$661$	14205.6i	0.835908i	0.908468	+	0.417954i	$0.137253\pi$
$661$	14205.6i	0.835908i	−0.908468	+	0.417954i	$0.862747\pi$
$662$	−2071.24	−0.121603
$663$	0	0
$664$	8062.95	0.471240
$665$	23565.0i	1.37415i
$666$	−69.0531	−0.00401765
$667$	−12372.2	−0.718224
$668$	448.345i	0.0259685i
$669$	− 3767.97i	− 0.217755i
$670$	− 19646.9i	− 1.13287i
$671$	3740.29i	0.215190i
$672$	−1837.54	−0.105483
$673$	27400.1	1.56939	0.784693	−	0.619885i	$-0.212821\pi$
$673$	27400.1	1.56939	0.784693	+	0.619885i	$0.212821\pi$
$674$	− 2431.53i	− 0.138960i
$675$	8673.71	0.494594
$676$	0	0
$677$	−2463.68	−0.139862	−0.0699311	−	0.997552i	$-0.522278\pi$
$677$	−2463.68	−0.139862	−0.0699311	+	0.997552i	$0.522278\pi$
$678$	− 9110.55i	− 0.516060i
$679$	7145.53	0.403859
$680$	60499.3	3.41182
$681$	10225.6i	0.575396i
$682$	2656.58i	0.149158i
$683$	− 21663.1i	− 1.21364i	−0.794839	−	0.606820i	$-0.792445\pi$
$683$	− 21663.1i	− 1.21364i	0.794839	−	0.606820i	$-0.207555\pi$
$684$	203.502i	0.0113759i
$685$	−54424.0	−3.03567
$686$	1213.86	0.0675589
$687$	− 12405.4i	− 0.688931i
$688$	21618.6	1.19797
$689$	0	0
$690$	13090.9	0.722262
$691$	3787.41i	0.208509i	0.994551	+	0.104254i	$0.0332456\pi$
$691$	3787.41i	0.208509i	−0.994551	+	0.104254i	$0.966754\pi$
$692$	40.2031	0.00220852
$693$	−1622.06	−0.0889134
$694$	26369.8i	1.44234i
$695$	− 4933.43i	− 0.269260i
$696$	− 11450.4i	− 0.623604i
$697$	36932.0i	2.00703i
$698$	−4943.10	−0.268051
$699$	11366.6	0.615058
$700$	− 4355.20i	− 0.235158i
$701$	12593.4	0.678527	0.339263	−	0.940691i	$-0.389822\pi$
$701$	12593.4	0.678527	0.339263	+	0.940691i	$0.389822\pi$
$702$	0	0
$703$	121.041	0.00649380
$704$	− 3770.25i	− 0.201842i
$705$	−2615.18	−0.139707
$706$	−29206.0	−1.55692
$707$	− 8111.73i	− 0.431504i
$708$	− 640.553i	− 0.0340020i
$709$	22849.8i	1.21036i	0.796090	+	0.605178i	$0.206898\pi$
$709$	22849.8i	1.21036i	−0.796090	+	0.605178i	$0.793102\pi$
$710$	− 29735.9i	− 1.57179i
$711$	6865.82	0.362150
$712$	−8448.30	−0.444682
$713$	10533.2i	0.553254i
$714$	26066.8	1.36628
$715$	0	0
$716$	−1852.09	−0.0966703
$717$	− 4631.24i	− 0.241223i
$718$	−22409.3	−1.16477
$719$	−7269.04	−0.377037	−0.188518	−	0.982070i	$-0.560368\pi$
$719$	−7269.04	−0.377037	−0.188518	+	0.982070i	$0.560368\pi$
$720$	11318.2i	0.585840i
$721$	− 13117.4i	− 0.677553i
$722$	− 13666.7i	− 0.704464i
$723$	18886.2i	0.971487i
$724$	−1233.35	−0.0633111
$725$	52608.9	2.69496
$726$	10519.3i	0.537749i
$727$	−21380.1	−1.09071	−0.545355	−	0.838205i	$-0.683605\pi$
$727$	−21380.1	−1.09071	−0.545355	+	0.838205i	$0.683605\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	− 28372.8i	− 1.43853i
$731$	44623.0	2.25779
$732$	−844.051	−0.0426189
$733$	− 5430.11i	− 0.273623i	−0.990597	−	0.136811i	$-0.956315\pi$
$733$	− 5430.11i	− 0.273623i	0.990597	−	0.136811i	$-0.0436854\pi$
$734$	6986.65i	0.351338i
$735$	− 20649.3i	− 1.03627i
$736$	− 1789.32i	− 0.0896128i
$737$	−2370.51	−0.118479
$738$	−7395.97	−0.368902
$739$	− 30838.6i	− 1.53507i	−0.641008	−	0.767534i	$-0.721483\pi$
$739$	− 30838.6i	− 1.53507i	0.641008	−	0.767534i	$-0.278517\pi$
$740$	−31.0748	−0.00154369
$741$	0	0
$742$	−8892.85	−0.439982
$743$	− 31665.8i	− 1.56353i	−0.623571	−	0.781767i	$-0.714319\pi$
$743$	− 31665.8i	− 1.56353i	0.623571	−	0.781767i	$-0.285681\pi$
$744$	−9748.39	−0.480367
$745$	33714.2	1.65798
$746$	1642.58i	0.0806156i
$747$	3113.53i	0.152501i
$748$	− 448.903i	− 0.0219432i
$749$	− 32650.6i	− 1.59283i
$750$	−34005.1	−1.65559
$751$	23714.1	1.15225	0.576124	−	0.817362i	$-0.304564\pi$
$751$	23714.1	1.15225	0.576124	+	0.817362i	$0.304564\pi$
$752$	− 2456.63i	− 0.119128i
$753$	6725.78	0.325499
$754$	0	0
$755$	−73812.2	−3.55801
$756$	− 366.042i	− 0.0176095i
$757$	−19213.1	−0.922474	−0.461237	−	0.887277i	$-0.652594\pi$
$757$	−19213.1	−0.922474	−0.461237	+	0.887277i	$0.652594\pi$
$758$	2361.28	0.113147
$759$	− 1579.49i	− 0.0755360i
$760$	− 21236.9i	− 1.01361i
$761$	− 11406.5i	− 0.543345i	−0.962390	−	0.271673i	$-0.912423\pi$
$761$	− 11406.5i	− 0.543345i	0.962390	−	0.271673i	$-0.0875768\pi$
$762$	20893.1i	0.993279i
$763$	12138.4	0.575936
$764$	−803.570	−0.0380526
$765$	23362.0i	1.10412i
$766$	24852.1	1.17225
$767$	0	0
$768$	2404.97	0.112997
$769$	− 6384.27i	− 0.299379i	−0.988733	−	0.149690i	$-0.952173\pi$
$769$	− 6384.27i	− 0.299379i	0.988733	−	0.149690i	$-0.0478275\pi$
$770$	10409.8	0.487198
$771$	10811.4	0.505010
$772$	− 2417.57i	− 0.112708i
$773$	− 1561.96i	− 0.0726776i	−0.999340	−	0.0363388i	$-0.988430\pi$
$773$	− 1561.96i	− 0.0726776i	0.999340	−	0.0363388i	$-0.0115695\pi$
$774$	8936.16i	0.414992i
$775$	− 44788.8i	− 2.07595i
$776$	−6439.59	−0.297897
$777$	−217.717	−0.0100522
$778$	− 4137.26i	− 0.190653i
$779$	12964.1	0.596262
$780$	0	0
$781$	−3587.82	−0.164382
$782$	25382.7i	1.16072i
$783$	4421.63	0.201809
$784$	19397.4	0.883626
$785$	46552.4i	2.11660i
$786$	− 5750.22i	− 0.260946i
$787$	− 11134.6i	− 0.504326i	−0.967685	−	0.252163i	$-0.918858\pi$
$787$	− 11134.6i	− 0.504326i	0.967685	−	0.252163i	$-0.0811419\pi$
$788$	− 1003.54i	− 0.0453675i
$789$	6393.18	0.288471
$790$	−44062.3	−1.98439
$791$	− 28724.6i	− 1.29119i
$792$	1461.81	0.0655848
$793$	0	0
$794$	20705.4	0.925448
$795$	− 7970.08i	− 0.355559i
$796$	−1208.28	−0.0538020
$797$	−13798.3	−0.613250	−0.306625	−	0.951830i	$-0.599200\pi$
$797$	−13798.3	−0.613250	−0.306625	+	0.951830i	$0.599200\pi$
$798$	− 9150.15i	− 0.405905i
$799$	− 5070.74i	− 0.224518i
$800$	7608.48i	0.336251i
$801$	− 3262.34i	− 0.143907i
$802$	−30282.5	−1.33331
$803$	−3423.35	−0.150445
$804$	− 534.941i	− 0.0234651i
$805$	41274.1	1.80711
$806$	0	0
$807$	7517.37	0.327911
$808$	7310.33i	0.318288i
$809$	8271.72	0.359479	0.179739	−	0.983714i	$-0.442475\pi$
$809$	8271.72	0.359479	0.179739	+	0.983714i	$0.442475\pi$
$810$	−4678.45	−0.202943
$811$	31496.6i	1.36374i	0.731472	+	0.681872i	$0.238834\pi$
$811$	31496.6i	1.36374i	−0.731472	+	0.681872i	$0.761166\pi$
$812$	− 2220.16i	− 0.0959514i
$813$	− 690.659i	− 0.0297939i
$814$	− 53.4695i	− 0.00230234i
$815$	−16027.2	−0.688843
$816$	−21945.6	−0.941482
$817$	− 15663.9i	− 0.670759i
$818$	30302.6	1.29524
$819$	0	0
$820$	−3328.28	−0.141742
$821$	− 12125.3i	− 0.515441i	−0.966219	−	0.257721i	$-0.917029\pi$
$821$	− 12125.3i	− 0.515441i	0.966219	−	0.257721i	$-0.0829714\pi$
$822$	21132.5	0.896693
$823$	−6237.45	−0.264184	−0.132092	−	0.991237i	$-0.542169\pi$
$823$	−6237.45	−0.264184	−0.132092	+	0.991237i	$0.542169\pi$
$824$	11821.4i	0.499780i
$825$	6716.26i	0.283431i
$826$	28801.4i	1.21323i
$827$	31790.0i	1.33669i	0.743850	+	0.668347i	$0.232998\pi$
$827$	31790.0i	1.33669i	−0.743850	+	0.668347i	$0.767002\pi$
$828$	356.435	0.0149601
$829$	23138.6	0.969404	0.484702	−	0.874679i	$-0.338928\pi$
$829$	23138.6	0.969404	0.484702	+	0.874679i	$0.338928\pi$
$830$	− 19981.5i	− 0.835624i
$831$	−23251.1	−0.970605
$832$	0	0
$833$	40038.2	1.66536
$834$	1915.62i	0.0795355i
$835$	18067.3	0.748795
$836$	−157.577	−0.00651904
$837$	− 3764.37i	− 0.155455i
$838$	− 35641.7i	− 1.46924i
$839$	3357.69i	0.138165i	0.997611	+	0.0690825i	$0.0220072\pi$
$839$	3357.69i	0.138165i	−0.997611	+	0.0690825i	$0.977993\pi$
$840$	38198.9i	1.56903i
$841$	2429.66	0.0996211
$842$	−40094.1	−1.64101
$843$	− 15537.6i	− 0.634809i
$844$	−1835.04	−0.0748397
$845$	0	0
$846$	1015.46	0.0412675
$847$	33166.1i	1.34545i
$848$	7486.87	0.303184
$849$	11399.8	0.460824
$850$	− 107932.i	− 4.35532i
$851$	− 212.003i	− 0.00853981i
$852$	− 809.643i	− 0.0325562i
$853$	− 16262.6i	− 0.652778i	−0.945236	−	0.326389i	$-0.894168\pi$
$853$	− 16262.6i	− 0.652778i	0.945236	−	0.326389i	$-0.105832\pi$
$854$	37951.4	1.52069
$855$	8200.69	0.328021
$856$	29424.9i	1.17491i
$857$	−33293.4	−1.32705	−0.663524	−	0.748155i	$-0.730940\pi$
$857$	−33293.4	−1.32705	−0.663524	+	0.748155i	$0.730940\pi$
$858$	0	0
$859$	26170.5	1.03950	0.519748	−	0.854320i	$-0.326026\pi$
$859$	26170.5	1.03950	0.519748	+	0.854320i	$0.326026\pi$
$860$	4021.39i	0.159451i
$861$	−23318.7	−0.922996
$862$	3977.90	0.157178
$863$	− 37186.2i	− 1.46678i	−0.679807	−	0.733391i	$-0.737936\pi$
$863$	− 37186.2i	− 1.46678i	0.679807	−	0.733391i	$-0.262064\pi$
$864$	639.471i	0.0251797i
$865$	− 1620.09i	− 0.0636819i
$866$	− 24559.0i	− 0.963682i
$867$	−30559.0	−1.19705
$868$	−1890.15	−0.0739122
$869$	5316.37i	0.207532i
$870$	−28376.4	−1.10580
$871$	0	0
$872$	−10939.2	−0.424825
$873$	− 2486.67i	− 0.0964043i
$874$	8910.01	0.344835
$875$	−107215.	−4.14230
$876$	− 772.528i	− 0.0297960i
$877$	19008.1i	0.731880i	0.930638	+	0.365940i	$0.119252\pi$
$877$	19008.1i	0.731880i	−0.930638	+	0.365940i	$0.880748\pi$
$878$	9154.20i	0.351867i
$879$	− 17522.5i	− 0.672379i
$880$	−8763.97	−0.335720
$881$	11259.4	0.430578	0.215289	−	0.976550i	$-0.430931\pi$
$881$	11259.4	0.430578	0.215289	+	0.976550i	$0.430931\pi$
$882$	8018.01i	0.306100i
$883$	−34604.5	−1.31884	−0.659419	−	0.751776i	$-0.729198\pi$
$883$	−34604.5	−1.31884	−0.659419	+	0.751776i	$0.729198\pi$
$884$	0	0
$885$	−25812.8	−0.980439
$886$	42848.6i	1.62475i
$887$	36352.6	1.37610	0.688050	−	0.725663i	$-0.258467\pi$
$887$	36352.6	1.37610	0.688050	+	0.725663i	$0.258467\pi$
$888$	196.208	0.00741476
$889$	65873.8i	2.48519i
$890$	20936.5i	0.788531i
$891$	564.483i	0.0212243i
$892$	658.407i	0.0247142i
$893$	−1779.97	−0.0667014
$894$	−13091.0	−0.489742
$895$	74635.1i	2.78746i
$896$	−33355.2	−1.24366
$897$	0	0
$898$	33537.1	1.24627
$899$	− 22832.2i	− 0.847048i
$900$	−1515.62	−0.0561342
$901$	15453.7	0.571406
$902$	− 5726.88i	− 0.211402i
$903$	28174.8i	1.03831i
$904$	25886.7i	0.952412i
$905$	49701.4i	1.82556i
$906$	28660.9	1.05099
$907$	36358.6	1.33106	0.665528	−	0.746373i	$-0.268206\pi$
$907$	36358.6	1.33106	0.665528	+	0.746373i	$0.268206\pi$
$908$	− 1786.79i	− 0.0653048i
$909$	−2822.91	−0.103003
$910$	0	0
$911$	16789.4	0.610603	0.305301	−	0.952256i	$-0.401243\pi$
$911$	16789.4	0.610603	0.305301	+	0.952256i	$0.401243\pi$
$912$	7703.49i	0.279702i
$913$	−2410.89	−0.0873918
$914$	22919.4	0.829439
$915$	34013.4i	1.22890i
$916$	2167.69i	0.0781905i
$917$	− 18129.8i	− 0.652890i
$918$	− 9071.34i	− 0.326142i
$919$	42322.8	1.51915	0.759575	−	0.650419i	$-0.225407\pi$
$919$	42322.8	1.51915	0.759575	+	0.650419i	$0.225407\pi$
$920$	−37196.4	−1.33297
$921$	5058.86i	0.180993i
$922$	−6311.48	−0.225442
$923$	0	0
$924$	283.435	0.0100913
$925$	901.474i	0.0320436i
$926$	−6372.11	−0.226134
$927$	−4564.88	−0.161737
$928$	3878.60i	0.137200i
$929$	− 15413.1i	− 0.544337i	−0.962250	−	0.272168i	$-0.912259\pi$
$929$	− 15413.1i	− 0.544337i	0.962250	−	0.272168i	$-0.0877408\pi$
$930$	24158.3i	0.851810i
$931$	− 14054.5i	− 0.494756i
$932$	−1986.18	−0.0698063
$933$	−31803.2	−1.11596
$934$	9399.85i	0.329307i
$935$	−18089.8	−0.632726
$936$	0	0
$937$	45360.4	1.58149	0.790747	−	0.612143i	$-0.209692\pi$
$937$	45360.4	1.58149	0.790747	+	0.612143i	$0.209692\pi$
$938$	24052.7i	0.837260i
$939$	17670.0	0.614098
$940$	456.971	0.0158561
$941$	− 37604.6i	− 1.30274i	−0.758762	−	0.651368i	$-0.774195\pi$
$941$	− 37604.6i	− 1.30274i	0.758762	−	0.651368i	$-0.225805\pi$
$942$	− 18076.1i	− 0.625212i
$943$	− 22706.7i	− 0.784127i
$944$	− 24247.8i	− 0.836016i
$945$	−14750.6	−0.507766
$946$	−6919.49	−0.237814
$947$	28270.9i	0.970095i	0.874488	+	0.485048i	$0.161198\pi$
$947$	28270.9i	0.970095i	−0.874488	+	0.485048i	$0.838802\pi$
$948$	−1199.72	−0.0411023
$949$	0	0
$950$	−37886.9	−1.29391
$951$	7298.93i	0.248879i
$952$	−74066.3	−2.52154
$953$	−21396.1	−0.727268	−0.363634	−	0.931542i	$-0.618464\pi$
$953$	−21396.1	−0.727268	−0.363634	+	0.931542i	$0.618464\pi$
$954$	3094.74i	0.105027i
$955$	32382.1i	1.09723i
$956$	809.252i	0.0273777i
$957$	3423.78i	0.115648i
$958$	43254.9	1.45877
$959$	66628.7	2.24354
$960$	− 34285.8i	− 1.15268i
$961$	10352.7	0.347512
$962$	0	0
$963$	−11362.5	−0.380220
$964$	− 3300.13i	− 0.110259i
$965$	−97422.7	−3.24990
$966$	−16026.5	−0.533793
$967$	898.300i	0.0298732i	0.999888	+	0.0149366i	$0.00475464\pi$
$967$	898.300i	0.0298732i	−0.999888	+	0.0149366i	$0.995245\pi$
$968$	− 29889.4i	− 0.992441i
$969$	15900.8i	0.527150i
$970$	15958.5i	0.528244i
$971$	3749.51	0.123921	0.0619607	−	0.998079i	$-0.480265\pi$
$971$	3749.51	0.123921	0.0619607	+	0.998079i	$0.480265\pi$
$972$	−127.384	−0.00420353
$973$	6039.76i	0.198999i
$974$	−39656.9	−1.30461
$975$	0	0
$976$	−31951.2	−1.04788
$977$	2717.19i	0.0889771i	0.999010	+	0.0444885i	$0.0141658\pi$
$977$	2717.19i	0.0889771i	−0.999010	+	0.0444885i	$0.985834\pi$
$978$	6223.26	0.203474
$979$	2526.11	0.0824666
$980$	3608.21i	0.117612i
$981$	− 4224.20i	− 0.137481i
$982$	− 47884.6i	− 1.55607i
$983$	− 4179.98i	− 0.135626i	−0.997698	−	0.0678132i	$-0.978398\pi$
$983$	− 4179.98i	− 0.135626i	0.997698	−	0.0678132i	$-0.0216022\pi$
$984$	21014.9	0.680825
$985$	−40440.4	−1.30816
$986$	− 55020.7i	− 1.77709i
$987$	3201.64	0.103252
$988$	0	0
$989$	−27435.3	−0.882096
$990$	− 3622.64i	− 0.116298i
$991$	−15297.9	−0.490367	−0.245184	−	0.969477i	$-0.578848\pi$
$991$	−15297.9	−0.490367	−0.245184	+	0.969477i	$0.578848\pi$
$992$	3302.07	0.105686
$993$	2272.60i	0.0726273i
$994$	36404.3i	1.16164i
$995$	48691.0i	1.55136i
$996$	− 544.052i	− 0.0173082i
$997$	9164.26	0.291108	0.145554	−	0.989350i	$-0.453504\pi$
$997$	9164.26	0.291108	0.145554	+	0.989350i	$0.453504\pi$
$998$	10543.7	0.334425
$999$	75.7663i	0.00239954i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.b.j.337.6		18
13.5	odd	4		507.4.a.n.1.4	✓	9
13.8	odd	4		507.4.a.q.1.6	yes	9
13.12	even	2	inner	507.4.b.j.337.13		18
39.5	even	4		1521.4.a.bj.1.6		9
39.8	even	4		1521.4.a.be.1.4		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.4.a.n.1.4	✓	9	13.5	odd	4
507.4.a.q.1.6	yes	9	13.8	odd	4
507.4.b.j.337.6		18	1.1	even	1	trivial
507.4.b.j.337.13		18	13.12	even	2	inner
1521.4.a.be.1.4		9	39.8	even	4
1521.4.a.bj.1.6		9	39.5	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-2.73419i q^{2} -3.00000 q^{3} +0.524213 q^{4} -21.1246i q^{5} +8.20257i q^{6} +25.8618i q^{7} -23.3068i q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-2.73419i q^{2} -3.00000 q^{3} +0.524213 q^{4} -21.1246i q^{5} +8.20257i q^{6} +25.8618i q^{7} -23.3068i q^{8} +9.00000 q^{9} -57.7586 q^{10} +6.96892i q^{11} -1.57264 q^{12} +70.7111 q^{14} +63.3738i q^{15} -59.5315 q^{16} -122.879 q^{17} -24.6077i q^{18} +43.1340i q^{19} -11.0738i q^{20} -77.5854i q^{21} +19.0543 q^{22} +75.5492 q^{23} +69.9204i q^{24} -321.248 q^{25} -27.0000 q^{27} +13.5571i q^{28} -163.764 q^{29} +173.276 q^{30} +139.421i q^{31} -23.6841i q^{32} -20.9068i q^{33} +335.975i q^{34} +546.320 q^{35} +4.71792 q^{36} -2.80616i q^{37} +117.936 q^{38} -492.347 q^{40} -300.555i q^{41} -212.133 q^{42} -363.145 q^{43} +3.65320i q^{44} -190.121i q^{45} -206.566i q^{46} +41.2660i q^{47} +178.594 q^{48} -325.834 q^{49} +878.354i q^{50} +368.638 q^{51} -125.763 q^{53} +73.8231i q^{54} +147.216 q^{55} +602.756 q^{56} -129.402i q^{57} +447.762i q^{58} +407.311i q^{59} +33.2214i q^{60} +536.710 q^{61} +381.204 q^{62} +232.756i q^{63} -541.009 q^{64} -57.1630 q^{66} +340.155i q^{67} -64.4150 q^{68} -226.648 q^{69} -1493.74i q^{70} +514.831i q^{71} -209.761i q^{72} +491.231i q^{73} -7.67257 q^{74} +963.745 q^{75} +22.6114i q^{76} -180.229 q^{77} +762.869 q^{79} +1257.58i q^{80} +81.0000 q^{81} -821.774 q^{82} +345.948i q^{83} -40.6713i q^{84} +2595.78i q^{85} +992.907i q^{86} +491.292 q^{87} +162.423 q^{88} -362.482i q^{89} -519.828 q^{90} +39.6039 q^{92} -418.264i q^{93} +112.829 q^{94} +911.188 q^{95} +71.0523i q^{96} -276.297i q^{97} +890.890i q^{98} +62.7203i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 54 q^{3} - 64 q^{4} + 162 q^{9}+O(q^{10}) \) 18 * q - 54 * q^3 - 64 * q^4 + 162 * q^9	\( 18 q - 54 q^{3} - 64 q^{4} + 162 q^{9} - 396 q^{10} + 192 q^{12} + 196 q^{14} + 64 q^{16} + 268 q^{17} + 548 q^{22} - 452 q^{23} - 1224 q^{25} - 486 q^{27} - 1094 q^{29} + 1188 q^{30} + 276 q^{35} - 576 q^{36} + 832 q^{38} + 2684 q^{40} - 588 q^{42} - 316 q^{43} - 192 q^{48} - 1284 q^{49} - 804 q^{51} + 2798 q^{53} - 2816 q^{55} + 1232 q^{56} + 4184 q^{61} + 586 q^{62} - 4962 q^{64} - 1644 q^{66} - 3158 q^{68} + 1356 q^{69} + 2074 q^{74} + 3672 q^{75} + 3372 q^{77} - 230 q^{79} + 1458 q^{81} + 10294 q^{82} + 3282 q^{87} + 968 q^{88} - 3564 q^{90} + 4174 q^{92} - 936 q^{94} + 444 q^{95}+O(q^{100}) \) 18 * q - 54 * q^3 - 64 * q^4 + 162 * q^9 - 396 * q^10 + 192 * q^12 + 196 * q^14 + 64 * q^16 + 268 * q^17 + 548 * q^22 - 452 * q^23 - 1224 * q^25 - 486 * q^27 - 1094 * q^29 + 1188 * q^30 + 276 * q^35 - 576 * q^36 + 832 * q^38 + 2684 * q^40 - 588 * q^42 - 316 * q^43 - 192 * q^48 - 1284 * q^49 - 804 * q^51 + 2798 * q^53 - 2816 * q^55 + 1232 * q^56 + 4184 * q^61 + 586 * q^62 - 4962 * q^64 - 1644 * q^66 - 3158 * q^68 + 1356 * q^69 + 2074 * q^74 + 3672 * q^75 + 3372 * q^77 - 230 * q^79 + 1458 * q^81 + 10294 * q^82 + 3282 * q^87 + 968 * q^88 - 3564 * q^90 + 4174 * q^92 - 936 * q^94 + 444 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more