LMFDB - Embedded newform 507.4.b.i.337.10

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 = [10,0,30,-60,0,0,0,0,90,-80,0,-180,0,-60,0,500,-210] 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:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 70x^{8} + 1645x^{6} + 14700x^{4} + 44100x^{2} + 27648 $ x^10 + 70x^8 + 1645x^6 + 14700x^4 + 44100x^2 + 27648
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.10
Root			$5.36472i$ of defining polynomial
Character	$\chi$	$=$	507.337
Dual form			507.4.b.i.337.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+5.36472i q^{2} +3.00000 q^{3} -20.7803 q^{4} -2.69631i q^{5} +16.0942i q^{6} +15.2025i q^{7} -68.5626i q^{8} +9.00000 q^{9} +14.4650 q^{10} -66.8848i q^{11} -62.3408 q^{12} -81.5570 q^{14} -8.08894i q^{15} +201.577 q^{16} -4.16354 q^{17} +48.2825i q^{18} -26.0850i q^{19} +56.0301i q^{20} +45.6074i q^{21} +358.819 q^{22} -47.3242 q^{23} -205.688i q^{24} +117.730 q^{25} +27.0000 q^{27} -315.911i q^{28} +257.007 q^{29} +43.3949 q^{30} -206.242i q^{31} +532.906i q^{32} -200.655i q^{33} -22.3362i q^{34} +40.9906 q^{35} -187.022 q^{36} +175.686i q^{37} +139.939 q^{38} -184.866 q^{40} -156.463i q^{41} -244.671 q^{42} -51.9845 q^{43} +1389.88i q^{44} -24.2668i q^{45} -253.881i q^{46} -354.222i q^{47} +604.732 q^{48} +111.885 q^{49} +631.588i q^{50} -12.4906 q^{51} -10.4723 q^{53} +144.848i q^{54} -180.342 q^{55} +1042.32 q^{56} -78.2550i q^{57} +1378.77i q^{58} +445.114i q^{59} +168.090i q^{60} +119.696 q^{61} +1106.43 q^{62} +136.822i q^{63} -1246.28 q^{64} +1076.46 q^{66} +22.4078i q^{67} +86.5195 q^{68} -141.973 q^{69} +219.903i q^{70} -285.207i q^{71} -617.064i q^{72} +740.989i q^{73} -942.507 q^{74} +353.190 q^{75} +542.053i q^{76} +1016.81 q^{77} -547.679 q^{79} -543.516i q^{80} +81.0000 q^{81} +839.378 q^{82} -603.056i q^{83} -947.734i q^{84} +11.2262i q^{85} -278.882i q^{86} +771.021 q^{87} -4585.80 q^{88} +215.668i q^{89} +130.185 q^{90} +983.409 q^{92} -618.726i q^{93} +1900.31 q^{94} -70.3333 q^{95} +1598.72i q^{96} -1447.50i q^{97} +600.233i q^{98} -601.964i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 30 q^{3} - 60 q^{4} + 90 q^{9} - 80 q^{10} - 180 q^{12} - 60 q^{14} + 500 q^{16} - 210 q^{17} + 580 q^{22} + 120 q^{23} - 960 q^{25} + 270 q^{27} + 990 q^{29} - 240 q^{30} - 120 q^{35} - 540 q^{36}+ \cdots + 2760 q^{95}+O(q^{100}) $ 10 * q + 30 * q^3 - 60 * q^4 + 90 * q^9 - 80 * q^10 - 180 * q^12 - 60 * q^14 + 500 * q^16 - 210 * q^17 + 580 * q^22 + 120 * q^23 - 960 * q^25 + 270 * q^27 + 990 * q^29 - 240 * q^30 - 120 * q^35 - 540 * q^36 - 1380 * q^38 + 2000 * q^40 - 180 * q^42 + 740 * q^43 + 1500 * q^48 - 1550 * q^49 - 630 * q^51 + 330 * q^53 + 520 * q^55 + 5340 * q^56 + 2750 * q^61 + 1560 * q^62 - 3140 * q^64 + 1740 * q^66 + 1200 * q^68 + 360 * q^69 - 4380 * q^74 - 2880 * q^75 - 4320 * q^77 + 1100 * q^79 + 810 * q^81 + 4780 * q^82 + 2970 * q^87 - 6340 * q^88 - 720 * q^90 - 1740 * q^92 + 6460 * q^94 + 2760 * 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 $

$2$	5.36472i	1.89672i	0.317201	+	0.948358i	$0.397257\pi$
$2$	5.36472i	1.89672i	−0.317201	+	0.948358i	$0.602743\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−20.7803	−2.59753
$5$	− 2.69631i	− 0.241165i	−0.992703	−	0.120583i	$-0.961524\pi$
$5$	− 2.69631i	− 0.241165i	0.992703	−	0.120583i	$-0.0384763\pi$
$6$	16.0942i	1.09507i
$7$	15.2025i	0.820856i	0.911893	+	0.410428i	$0.134621\pi$
$7$	15.2025i	0.820856i	−0.911893	+	0.410428i	$0.865379\pi$
$8$	− 68.5626i	− 3.03007i
$9$	9.00000	0.333333
$10$	14.4650	0.457423
$11$	− 66.8848i	− 1.83332i	−0.399666	−	0.916661i	$-0.630874\pi$
$11$	− 66.8848i	− 1.83332i	0.399666	−	0.916661i	$-0.369126\pi$
$12$	−62.3408	−1.49969
$13$	0	0
$13$	0	0
$14$	−81.5570	−1.55693
$15$	− 8.08894i	− 0.139237i
$16$	201.577	3.14965
$17$	−4.16354	−0.0594004	−0.0297002	−	0.999559i	$-0.509455\pi$
$17$	−4.16354	−0.0594004	−0.0297002	+	0.999559i	$0.509455\pi$
$18$	48.2825i	0.632239i
$19$	− 26.0850i	− 0.314963i	−0.987522	−	0.157482i	$-0.949662\pi$
$19$	− 26.0850i	− 0.314963i	0.987522	−	0.157482i	$-0.0503375\pi$
$20$	56.0301i	0.626436i
$21$	45.6074i	0.473921i
$22$	358.819	3.47729
$23$	−47.3242	−0.429034	−0.214517	−	0.976720i	$-0.568818\pi$
$23$	−47.3242	−0.429034	−0.214517	+	0.976720i	$0.568818\pi$
$24$	− 205.688i	− 1.74941i
$25$	117.730	0.941839
$26$	0	0
$27$	27.0000	0.192450
$28$	− 315.911i	− 2.13220i
$29$	257.007	1.64569	0.822845	−	0.568266i	$-0.192386\pi$
$29$	257.007	1.64569	0.822845	+	0.568266i	$0.192386\pi$
$30$	43.3949	0.264093
$31$	− 206.242i	− 1.19491i	−0.801903	−	0.597455i	$-0.796179\pi$
$31$	− 206.242i	− 1.19491i	0.801903	−	0.597455i	$-0.203821\pi$
$32$	532.906i	2.94392i
$33$	− 200.655i	− 1.05847i
$34$	− 22.3362i	− 0.112666i
$35$	40.9906	0.197962
$36$	−187.022	−0.865845
$37$	175.686i	0.780611i	0.920685	+	0.390305i	$0.127631\pi$
$37$	175.686i	0.780611i	−0.920685	+	0.390305i	$0.872369\pi$
$38$	139.939	0.597396
$39$	0	0
$40$	−184.866	−0.730748
$41$	− 156.463i	− 0.595984i	−0.954568	−	0.297992i	$-0.903683\pi$
$41$	− 156.463i	− 0.595984i	0.954568	−	0.297992i	$-0.0963169\pi$
$42$	−244.671	−0.898895
$43$	−51.9845	−0.184362	−0.0921809	−	0.995742i	$-0.529384\pi$
$43$	−51.9845	−0.184362	−0.0921809	+	0.995742i	$0.529384\pi$
$44$	1389.88i	4.76211i
$45$	− 24.2668i	− 0.0803885i
$46$	− 253.881i	− 0.813755i
$47$	− 354.222i	− 1.09933i	−0.835384	−	0.549666i	$-0.814755\pi$
$47$	− 354.222i	− 1.09933i	0.835384	−	0.549666i	$-0.185245\pi$
$48$	604.732	1.81845
$49$	111.885	0.326196
$50$	631.588i	1.78640i
$51$	−12.4906	−0.0342948
$52$	0	0
$53$	−10.4723	−0.0271412	−0.0135706	−	0.999908i	$-0.504320\pi$
$53$	−10.4723	−0.0271412	−0.0135706	+	0.999908i	$0.504320\pi$
$54$	144.848i	0.365023i
$55$	−180.342	−0.442134
$56$	1042.32	2.48725
$57$	− 78.2550i	− 0.181844i
$58$	1378.77i	3.12141i
$59$	445.114i	0.982185i	0.871107	+	0.491092i	$0.163402\pi$
$59$	445.114i	0.982185i	−0.871107	+	0.491092i	$0.836598\pi$
$60$	168.090i	0.361673i
$61$	119.696	0.251238	0.125619	−	0.992079i	$-0.459908\pi$
$61$	119.696	0.251238	0.125619	+	0.992079i	$0.459908\pi$
$62$	1106.43	2.26640
$63$	136.822i	0.273619i
$64$	−1246.28	−2.43413
$65$	0	0
$66$	1076.46	2.00761
$67$	22.4078i	0.0408589i	0.999791	+	0.0204294i	$0.00650334\pi$
$67$	22.4078i	0.0408589i	−0.999791	+	0.0204294i	$0.993497\pi$
$68$	86.5195	0.154295
$69$	−141.973	−0.247703
$70$	219.903i	0.375478i
$71$	− 285.207i	− 0.476731i	−0.971176	−	0.238365i	$-0.923388\pi$
$71$	− 285.207i	− 0.476731i	0.971176	−	0.238365i	$-0.0766116\pi$
$72$	− 617.064i	− 1.01002i
$73$	740.989i	1.18803i	0.804454	+	0.594015i	$0.202458\pi$
$73$	740.989i	1.18803i	−0.804454	+	0.594015i	$0.797542\pi$
$74$	−942.507	−1.48060
$75$	353.190	0.543771
$76$	542.053i	0.818128i
$77$	1016.81	1.50489
$78$	0	0
$79$	−547.679	−0.779983	−0.389992	−	0.920818i	$-0.627522\pi$
$79$	−547.679	−0.779983	−0.389992	+	0.920818i	$0.627522\pi$
$80$	− 543.516i	− 0.759586i
$81$	81.0000	0.111111
$82$	839.378	1.13041
$83$	− 603.056i	− 0.797518i	−0.917056	−	0.398759i	$-0.869441\pi$
$83$	− 603.056i	− 0.797518i	0.917056	−	0.398759i	$-0.130559\pi$
$84$	− 947.734i	− 1.23103i
$85$	11.2262i	0.0143253i
$86$	− 278.882i	− 0.349682i
$87$	771.021	0.950139
$88$	−4585.80	−5.55509
$89$	215.668i	0.256863i	0.991718	+	0.128431i	$0.0409942\pi$
$89$	215.668i	0.256863i	−0.991718	+	0.128431i	$0.959006\pi$
$90$	130.185	0.152474
$91$	0	0
$92$	983.409	1.11443
$93$	− 618.726i	− 0.689881i
$94$	1900.31	2.08512
$95$	−70.3333	−0.0759583
$96$	1598.72i	1.69967i
$97$	− 1447.50i	− 1.51517i	−0.652735	−	0.757586i	$-0.726378\pi$
$97$	− 1447.50i	− 1.51517i	0.652735	−	0.757586i	$-0.273622\pi$
$98$	600.233i	0.618700i
$99$	− 601.964i	− 0.611107i
$100$	−2446.46	−2.44646
$101$	883.450	0.870362	0.435181	−	0.900343i	$-0.356684\pi$
$101$	883.450	0.870362	0.435181	+	0.900343i	$0.356684\pi$
$102$	− 67.0087i	− 0.0650476i
$103$	−1251.74	−1.19745	−0.598726	−	0.800954i	$-0.704326\pi$
$103$	−1251.74	−1.19745	−0.598726	+	0.800954i	$0.704326\pi$
$104$	0	0
$105$	122.972	0.114293
$106$	− 56.1812i	− 0.0514792i
$107$	−341.614	−0.308645	−0.154323	−	0.988021i	$-0.549319\pi$
$107$	−341.614	−0.308645	−0.154323	+	0.988021i	$0.549319\pi$
$108$	−561.067	−0.499896
$109$	− 775.177i	− 0.681179i	−0.940212	−	0.340589i	$-0.889373\pi$
$109$	− 775.177i	− 0.681179i	0.940212	−	0.340589i	$-0.110627\pi$
$110$	− 967.487i	− 0.838603i
$111$	527.058i	0.450686i
$112$	3064.47i	2.58541i
$113$	1279.05	1.06480	0.532402	−	0.846492i	$-0.321290\pi$
$113$	1279.05	1.06480	0.532402	+	0.846492i	$0.321290\pi$
$114$	419.816	0.344907
$115$	127.601i	0.103468i
$116$	−5340.67	−4.27473
$117$	0	0
$118$	−2387.91	−1.86293
$119$	− 63.2961i	− 0.0487592i
$120$	−554.599	−0.421898
$121$	−3142.58	−2.36107
$122$	642.137i	0.476528i
$123$	− 469.388i	− 0.344091i
$124$	4285.77i	3.10382i
$125$	− 654.476i	− 0.468305i
$126$	−734.013	−0.518977
$127$	1113.82	0.778233	0.389117	−	0.921188i	$-0.372780\pi$
$127$	1113.82	0.778233	0.389117	+	0.921188i	$0.372780\pi$
$128$	− 2422.68i	− 1.67294i
$129$	−155.953	−0.106441
$130$	0	0
$131$	2100.12	1.40068	0.700339	−	0.713811i	$-0.253032\pi$
$131$	2100.12	1.40068	0.700339	+	0.713811i	$0.253032\pi$
$132$	4169.65i	2.74941i
$133$	396.556	0.258540
$134$	−120.211	−0.0774977
$135$	− 72.8004i	− 0.0464123i
$136$	285.463i	0.179987i
$137$	− 1205.02i	− 0.751471i	−0.926727	−	0.375736i	$-0.877390\pi$
$137$	− 1205.02i	− 0.751471i	0.926727	−	0.375736i	$-0.122610\pi$
$138$	− 761.643i	− 0.469822i
$139$	322.890	0.197030	0.0985149	−	0.995136i	$-0.468591\pi$
$139$	322.890	0.197030	0.0985149	+	0.995136i	$0.468591\pi$
$140$	−851.796	−0.514213
$141$	− 1062.67i	− 0.634700i
$142$	1530.06	0.904223
$143$	0	0
$144$	1814.20	1.04988
$145$	− 692.971i	− 0.396884i
$146$	−3975.20	−2.25336
$147$	335.655	0.188329
$148$	− 3650.80i	− 2.02766i
$149$	− 1128.86i	− 0.620669i	−0.950627	−	0.310335i	$-0.899559\pi$
$149$	− 1128.86i	− 0.620669i	0.950627	−	0.310335i	$-0.100441\pi$
$150$	1894.77i	1.03138i
$151$	− 2940.44i	− 1.58470i	−0.610066	−	0.792350i	$-0.708857\pi$
$151$	− 2940.44i	− 1.58470i	0.610066	−	0.792350i	$-0.291143\pi$
$152$	−1788.46	−0.954361
$153$	−37.4719	−0.0198001
$154$	5454.93i	2.85436i
$155$	−556.093	−0.288171
$156$	0	0
$157$	629.388	0.319940	0.159970	−	0.987122i	$-0.448860\pi$
$157$	629.388	0.319940	0.159970	+	0.987122i	$0.448860\pi$
$158$	− 2938.15i	− 1.47941i
$159$	−31.4170	−0.0156700
$160$	1436.88	0.709972
$161$	− 719.444i	− 0.352175i
$162$	434.543i	0.210746i
$163$	− 394.912i	− 0.189766i	−0.995488	−	0.0948832i	$-0.969752\pi$
$163$	− 394.912i	− 0.189766i	0.995488	−	0.0948832i	$-0.0302478\pi$
$164$	3251.33i	1.54809i
$165$	−541.027	−0.255266
$166$	3235.23	1.51267
$167$	− 151.860i	− 0.0703669i	−0.999381	−	0.0351834i	$-0.988798\pi$
$167$	− 151.860i	− 0.0703669i	0.999381	−	0.0351834i	$-0.0112015\pi$
$168$	3126.96	1.43601
$169$	0	0
$170$	−60.2255	−0.0271711
$171$	− 234.765i	− 0.104988i
$172$	1080.25	0.478886
$173$	−538.813	−0.236793	−0.118397	−	0.992966i	$-0.537775\pi$
$173$	−538.813	−0.236793	−0.118397	+	0.992966i	$0.537775\pi$
$174$	4136.31i	1.80214i
$175$	1789.78i	0.773114i
$176$	− 13482.5i	− 5.77432i
$177$	1335.34i	0.567065i
$178$	−1157.00	−0.487196
$179$	2220.80	0.927319	0.463659	−	0.886014i	$-0.346536\pi$
$179$	2220.80	0.927319	0.463659	+	0.886014i	$0.346536\pi$
$180$	504.271i	0.208812i
$181$	3822.78	1.56986	0.784932	−	0.619582i	$-0.212698\pi$
$181$	3822.78	1.56986	0.784932	+	0.619582i	$0.212698\pi$
$182$	0	0
$183$	359.089	0.145052
$184$	3244.67i	1.30000i
$185$	473.704	0.188256
$186$	3319.30	1.30851
$187$	278.478i	0.108900i
$188$	7360.84i	2.85555i
$189$	410.467i	0.157974i
$190$	− 377.319i	− 0.144071i
$191$	3464.19	1.31236	0.656178	−	0.754606i	$-0.272172\pi$
$191$	3464.19	1.31236	0.656178	+	0.754606i	$0.272172\pi$
$192$	−3738.83	−1.40535
$193$	4697.40i	1.75195i	0.482357	+	0.875975i	$0.339781\pi$
$193$	4697.40i	1.75195i	−0.482357	+	0.875975i	$0.660219\pi$
$194$	7765.46	2.87385
$195$	0	0
$196$	−2325.00	−0.847304
$197$	2887.89i	1.04443i	0.852813	+	0.522217i	$0.174895\pi$
$197$	2887.89i	1.04443i	−0.852813	+	0.522217i	$0.825105\pi$
$198$	3229.37	1.15910
$199$	−63.0092	−0.0224453	−0.0112226	−	0.999937i	$-0.503572\pi$
$199$	−63.0092	−0.0224453	−0.0112226	+	0.999937i	$0.503572\pi$
$200$	− 8071.87i	− 2.85384i
$201$	67.2233i	0.0235899i
$202$	4739.47i	1.65083i
$203$	3907.14i	1.35087i
$204$	259.558	0.0890820
$205$	−421.872	−0.143731
$206$	− 6715.24i	− 2.27123i
$207$	−425.918	−0.143011
$208$	0	0
$209$	−1744.69	−0.577429
$210$	659.710i	0.216782i
$211$	−1049.70	−0.342484	−0.171242	−	0.985229i	$-0.554778\pi$
$211$	−1049.70	−0.342484	−0.171242	+	0.985229i	$0.554778\pi$
$212$	217.618	0.0705002
$213$	− 855.622i	− 0.275241i
$214$	− 1832.66i	− 0.585412i
$215$	140.166i	0.0444617i
$216$	− 1851.19i	− 0.583137i
$217$	3135.39	0.980848
$218$	4158.61	1.29200
$219$	2222.97i	0.685909i
$220$	3747.56	1.14846
$221$	0	0
$222$	−2827.52	−0.854823
$223$	− 2313.49i	− 0.694722i	−0.937732	−	0.347361i	$-0.887078\pi$
$223$	− 2313.49i	− 0.694722i	0.937732	−	0.347361i	$-0.112922\pi$
$224$	−8101.49	−2.41653
$225$	1059.57	0.313946
$226$	6861.74i	2.01963i
$227$	− 3799.46i	− 1.11092i	−0.831543	−	0.555460i	$-0.812542\pi$
$227$	− 3799.46i	− 1.11092i	0.831543	−	0.555460i	$-0.187458\pi$
$228$	1626.16i	0.472347i
$229$	− 4321.07i	− 1.24692i	−0.781856	−	0.623459i	$-0.785727\pi$
$229$	− 4321.07i	− 1.24692i	0.781856	−	0.623459i	$-0.214273\pi$
$230$	−684.543	−0.196250
$231$	3050.44	0.868850
$232$	− 17621.1i	− 4.98655i
$233$	−5279.77	−1.48450	−0.742251	−	0.670122i	$-0.766242\pi$
$233$	−5279.77	−1.48450	−0.742251	+	0.670122i	$0.766242\pi$
$234$	0	0
$235$	−955.094	−0.265121
$236$	− 9249.59i	− 2.55126i
$237$	−1643.04	−0.450323
$238$	339.566	0.0924823
$239$	− 1547.92i	− 0.418939i	−0.977815	−	0.209469i	$-0.932826\pi$
$239$	− 1547.92i	− 0.418939i	0.977815	−	0.209469i	$-0.0671737\pi$
$240$	− 1630.55i	− 0.438547i
$241$	− 4918.01i	− 1.31451i	−0.753669	−	0.657255i	$-0.771718\pi$
$241$	− 4918.01i	− 1.31451i	0.753669	−	0.657255i	$-0.228282\pi$
$242$	− 16859.1i	− 4.47828i
$243$	243.000	0.0641500
$244$	−2487.32	−0.652600
$245$	− 301.677i	− 0.0786671i
$246$	2518.14	0.652644
$247$	0	0
$248$	−14140.5	−3.62066
$249$	− 1809.17i	− 0.460447i
$250$	3511.08	0.888241
$251$	−1155.78	−0.290646	−0.145323	−	0.989384i	$-0.546422\pi$
$251$	−1155.78	−0.290646	−0.145323	+	0.989384i	$0.546422\pi$
$252$	− 2843.20i	− 0.710734i
$253$	3165.27i	0.786556i
$254$	5975.34i	1.47609i
$255$	33.6786i	0.00827073i
$256$	3026.80	0.738965
$257$	−2351.95	−0.570859	−0.285429	−	0.958400i	$-0.592136\pi$
$257$	−2351.95	−0.570859	−0.285429	+	0.958400i	$0.592136\pi$
$258$	− 836.647i	− 0.201889i
$259$	−2670.86	−0.640769
$260$	0	0
$261$	2313.06	0.548563
$262$	11266.6i	2.65669i
$263$	5521.88	1.29465	0.647326	−	0.762213i	$-0.275887\pi$
$263$	5521.88	1.29465	0.647326	+	0.762213i	$0.275887\pi$
$264$	−13757.4	−3.20723
$265$	28.2367i	0.00654553i
$266$	2127.41i	0.490376i
$267$	647.004i	0.148300i
$268$	− 465.639i	− 0.106132i
$269$	3916.95	0.887810	0.443905	−	0.896074i	$-0.353593\pi$
$269$	3916.95	0.887810	0.443905	+	0.896074i	$0.353593\pi$
$270$	390.554	0.0880310
$271$	− 2777.53i	− 0.622593i	−0.950313	−	0.311297i	$-0.899237\pi$
$271$	− 2777.53i	− 0.622593i	0.950313	−	0.311297i	$-0.100763\pi$
$272$	−839.276	−0.187090
$273$	0	0
$274$	6464.58	1.42533
$275$	− 7874.34i	− 1.72669i
$276$	2950.23	0.643416
$277$	−6583.08	−1.42794	−0.713969	−	0.700177i	$-0.753104\pi$
$277$	−6583.08	−1.42794	−0.713969	+	0.700177i	$0.753104\pi$
$278$	1732.21i	0.373710i
$279$	− 1856.18i	− 0.398303i
$280$	− 2810.42i	− 0.599839i
$281$	2871.66i	0.609640i	0.952410	+	0.304820i	$0.0985963\pi$
$281$	2871.66i	0.609640i	−0.952410	+	0.304820i	$0.901404\pi$
$282$	5700.92	1.20385
$283$	−7518.04	−1.57916	−0.789578	−	0.613651i	$-0.789700\pi$
$283$	−7518.04	−1.57916	−0.789578	+	0.613651i	$0.789700\pi$
$284$	5926.69i	1.23832i
$285$	−211.000	−0.0438546
$286$	0	0
$287$	2378.62	0.489217
$288$	4796.16i	0.981307i
$289$	−4895.66	−0.996472
$290$	3717.60	0.752776
$291$	− 4342.51i	− 0.874785i
$292$	− 15397.9i	− 3.08595i
$293$	4506.57i	0.898555i	0.893392	+	0.449278i	$0.148319\pi$
$293$	4506.57i	0.898555i	−0.893392	+	0.449278i	$0.851681\pi$
$294$	1800.70i	0.357207i
$295$	1200.17	0.236869
$296$	12045.5	2.36530
$297$	− 1805.89i	− 0.352823i
$298$	6056.02	1.17723
$299$	0	0
$300$	−7339.38	−1.41246
$301$	− 790.292i	− 0.151335i
$302$	15774.7	3.00573
$303$	2650.35	0.502504
$304$	− 5258.15i	− 0.992024i
$305$	− 322.738i	− 0.0605900i
$306$	− 201.026i	− 0.0375552i
$307$	9538.89i	1.77333i	0.462409	+	0.886667i	$0.346985\pi$
$307$	9538.89i	1.77333i	−0.462409	+	0.886667i	$0.653015\pi$
$308$	−21129.7	−3.90901
$309$	−3755.22	−0.691350
$310$	− 2983.29i	− 0.546578i
$311$	7466.28	1.36133	0.680666	−	0.732594i	$-0.261691\pi$
$311$	7466.28	1.36133	0.680666	+	0.732594i	$0.261691\pi$
$312$	0	0
$313$	−1821.65	−0.328964	−0.164482	−	0.986380i	$-0.552595\pi$
$313$	−1821.65	−0.328964	−0.164482	+	0.986380i	$0.552595\pi$
$314$	3376.49i	0.606836i
$315$	368.915	0.0659874
$316$	11380.9	2.02603
$317$	− 3125.14i	− 0.553708i	−0.960912	−	0.276854i	$-0.910708\pi$
$317$	− 3125.14i	− 0.553708i	0.960912	−	0.276854i	$-0.0892918\pi$
$318$	− 168.543i	− 0.0297215i
$319$	− 17189.9i	− 3.01708i
$320$	3360.35i	0.587029i
$321$	−1024.84	−0.178196
$322$	3859.62	0.667976
$323$	108.606i	0.0187090i
$324$	−1683.20	−0.288615
$325$	0	0
$326$	2118.60	0.359933
$327$	− 2325.53i	− 0.393279i
$328$	−10727.5	−1.80587
$329$	5385.05	0.902394
$330$	− 2902.46i	− 0.484167i
$331$	1553.67i	0.257999i	0.991645	+	0.128999i	$0.0411765\pi$
$331$	1553.67i	0.257999i	−0.991645	+	0.128999i	$0.958823\pi$
$332$	12531.7i	2.07158i
$333$	1581.17i	0.260204i
$334$	814.686	0.133466
$335$	60.4183	0.00985375
$336$	9193.42i	1.49269i
$337$	3190.43	0.515709	0.257855	−	0.966184i	$-0.416984\pi$
$337$	3190.43	0.515709	0.257855	+	0.966184i	$0.416984\pi$
$338$	0	0
$339$	3837.15	0.614764
$340$	− 233.283i	− 0.0372105i
$341$	−13794.5	−2.19065
$342$	1259.45	0.199132
$343$	6915.37i	1.08862i
$344$	3564.19i	0.558629i
$345$	382.802i	0.0597373i
$346$	− 2890.59i	− 0.449130i
$347$	−5718.68	−0.884712	−0.442356	−	0.896840i	$-0.645857\pi$
$347$	−5718.68	−0.884712	−0.442356	+	0.896840i	$0.645857\pi$
$348$	−16022.0	−2.46802
$349$	− 3328.46i	− 0.510511i	−0.966874	−	0.255256i	$-0.917840\pi$
$349$	− 3328.46i	− 0.510511i	0.966874	−	0.255256i	$-0.0821596\pi$
$350$	−9601.70	−1.46638
$351$	0	0
$352$	35643.4	5.39715
$353$	12306.5i	1.85555i	0.373142	+	0.927774i	$0.378281\pi$
$353$	12306.5i	1.85555i	−0.373142	+	0.927774i	$0.621719\pi$
$354$	−7163.74	−1.07556
$355$	−769.008	−0.114971
$356$	− 4481.64i	− 0.667210i
$357$	− 189.888i	− 0.0281511i
$358$	11914.0i	1.75886i
$359$	8539.97i	1.25549i	0.778418	+	0.627747i	$0.216023\pi$
$359$	8539.97i	1.25549i	−0.778418	+	0.627747i	$0.783977\pi$
$360$	−1663.80	−0.243583
$361$	6178.57	0.900798
$362$	20508.2i	2.97759i
$363$	−9427.74	−1.36316
$364$	0	0
$365$	1997.94	0.286512
$366$	1926.41i	0.275123i
$367$	2496.65	0.355107	0.177553	−	0.984111i	$-0.443182\pi$
$367$	2496.65	0.355107	0.177553	+	0.984111i	$0.443182\pi$
$368$	−9539.49	−1.35130
$369$	− 1408.16i	− 0.198661i
$370$	2541.29i	0.357069i
$371$	− 159.205i	− 0.0222790i
$372$	12857.3i	1.79199i
$373$	−1142.91	−0.158653	−0.0793264	−	0.996849i	$-0.525277\pi$
$373$	−1142.91	−0.158653	−0.0793264	+	0.996849i	$0.525277\pi$
$374$	−1493.96	−0.206552
$375$	− 1963.43i	− 0.270376i
$376$	−24286.4	−3.33105
$377$	0	0
$378$	−2202.04	−0.299632
$379$	12181.8i	1.65102i	0.564384	+	0.825512i	$0.309114\pi$
$379$	12181.8i	1.65102i	−0.564384	+	0.825512i	$0.690886\pi$
$380$	1461.54	0.197304
$381$	3341.46	0.449313
$382$	18584.4i	2.48917i
$383$	10180.6i	1.35824i	0.734029	+	0.679118i	$0.237638\pi$
$383$	10180.6i	1.35824i	−0.734029	+	0.679118i	$0.762362\pi$
$384$	− 7268.04i	− 0.965874i
$385$	− 2741.65i	− 0.362928i
$386$	−25200.3	−3.32295
$387$	−467.860	−0.0614539
$388$	30079.5i	3.93571i
$389$	−5845.83	−0.761941	−0.380971	−	0.924587i	$-0.624410\pi$
$389$	−5845.83	−0.761941	−0.380971	+	0.924587i	$0.624410\pi$
$390$	0	0
$391$	197.036	0.0254848
$392$	− 7671.13i	− 0.988395i
$393$	6300.37	0.808681
$394$	−15492.7	−1.98100
$395$	1476.71i	0.188105i
$396$	12509.0i	1.58737i
$397$	− 2500.92i	− 0.316166i	−0.987426	−	0.158083i	$-0.949469\pi$
$397$	− 2500.92i	− 0.316166i	0.987426	−	0.158083i	$-0.0505313\pi$
$398$	− 338.027i	− 0.0425723i
$399$	1189.67	0.149268
$400$	23731.7	2.96646
$401$	− 9189.25i	− 1.14436i	−0.820127	−	0.572181i	$-0.806098\pi$
$401$	− 9189.25i	− 1.14436i	0.820127	−	0.572181i	$-0.193902\pi$
$402$	−360.634	−0.0447433
$403$	0	0
$404$	−18358.3	−2.26080
$405$	− 218.401i	− 0.0267962i
$406$	−20960.7	−2.56223
$407$	11750.7	1.43111
$408$	856.390i	0.103916i
$409$	− 9214.38i	− 1.11399i	−0.830516	−	0.556995i	$-0.811954\pi$
$409$	− 9214.38i	− 1.11399i	0.830516	−	0.556995i	$-0.188046\pi$
$410$	− 2263.23i	− 0.272617i
$411$	− 3615.05i	− 0.433862i
$412$	26011.5	3.11042
$413$	−6766.83	−0.806232
$414$	− 2284.93i	− 0.271252i
$415$	−1626.03	−0.192334
$416$	0	0
$417$	968.669	0.113755
$418$	− 9359.78i	− 1.09522i
$419$	−6494.41	−0.757214	−0.378607	−	0.925558i	$-0.623597\pi$
$419$	−6494.41	−0.757214	−0.378607	+	0.925558i	$0.623597\pi$
$420$	−2555.39	−0.296881
$421$	− 3059.56i	− 0.354190i	−0.984194	−	0.177095i	$-0.943330\pi$
$421$	− 3059.56i	− 0.354190i	0.984194	−	0.177095i	$-0.0566699\pi$
$422$	− 5631.33i	− 0.649595i
$423$	− 3188.00i	− 0.366444i
$424$	718.010i	0.0822398i
$425$	−490.173	−0.0559456
$426$	4590.18	0.522054
$427$	1819.68i	0.206230i
$428$	7098.82	0.801716
$429$	0	0
$430$	−751.954	−0.0843313
$431$	− 7937.05i	− 0.887040i	−0.896264	−	0.443520i	$-0.853729\pi$
$431$	− 7937.05i	− 0.887040i	0.896264	−	0.443520i	$-0.146271\pi$
$432$	5442.59	0.606150
$433$	−7294.37	−0.809573	−0.404786	−	0.914411i	$-0.632654\pi$
$433$	−7294.37	−0.809573	−0.404786	+	0.914411i	$0.632654\pi$
$434$	16820.5i	1.86039i
$435$	− 2078.91i	− 0.229141i
$436$	16108.4i	1.76938i
$437$	1234.45i	0.135130i
$438$	−11925.6	−1.30098
$439$	−15214.7	−1.65412	−0.827059	−	0.562115i	$-0.809988\pi$
$439$	−15214.7	−1.65412	−0.827059	+	0.562115i	$0.809988\pi$
$440$	12364.7i	1.33970i
$441$	1006.97	0.108732
$442$	0	0
$443$	1517.05	0.162703	0.0813515	−	0.996685i	$-0.474076\pi$
$443$	1517.05	0.162703	0.0813515	+	0.996685i	$0.474076\pi$
$444$	− 10952.4i	− 1.17067i
$445$	581.509	0.0619464
$446$	12411.3	1.31769
$447$	− 3386.58i	− 0.358343i
$448$	− 18946.5i	− 1.99807i
$449$	− 705.247i	− 0.0741262i	−0.999313	−	0.0370631i	$-0.988200\pi$
$449$	− 705.247i	− 0.0741262i	0.999313	−	0.0370631i	$-0.0118002\pi$
$450$	5684.30i	0.595467i
$451$	−10465.0	−1.09263
$452$	−26579.0	−2.76586
$453$	− 8821.33i	− 0.914927i
$454$	20383.1	2.10710
$455$	0	0
$456$	−5365.37	−0.551001
$457$	7277.73i	0.744940i	0.928044	+	0.372470i	$0.121489\pi$
$457$	7277.73i	0.744940i	−0.928044	+	0.372470i	$0.878511\pi$
$458$	23181.4	2.36505
$459$	−112.416	−0.0114316
$460$	− 2651.58i	− 0.268762i
$461$	− 1961.88i	− 0.198208i	−0.995077	−	0.0991041i	$-0.968402\pi$
$461$	− 1961.88i	− 0.198208i	0.995077	−	0.0991041i	$-0.0315977\pi$
$462$	16364.8i	1.64796i
$463$	− 10374.1i	− 1.04131i	−0.853768	−	0.520653i	$-0.825689\pi$
$463$	− 10374.1i	− 1.04131i	0.853768	−	0.520653i	$-0.174311\pi$
$464$	51806.8	5.18334
$465$	−1668.28	−0.166376
$466$	− 28324.5i	− 2.81568i
$467$	−8788.92	−0.870883	−0.435442	−	0.900217i	$-0.643408\pi$
$467$	−8788.92	−0.870883	−0.435442	+	0.900217i	$0.643408\pi$
$468$	0	0
$469$	−340.653	−0.0335392
$470$	− 5123.82i	− 0.502860i
$471$	1888.16	0.184718
$472$	30518.2	2.97609
$473$	3476.97i	0.337995i
$474$	− 8814.44i	− 0.854136i
$475$	− 3070.98i	− 0.296645i
$476$	1315.31i	0.126654i
$477$	−94.2510	−0.00904707
$478$	8304.14	0.794608
$479$	− 11141.7i	− 1.06279i	−0.847125	−	0.531394i	$-0.821668\pi$
$479$	− 11141.7i	− 1.06279i	0.847125	−	0.531394i	$-0.178332\pi$
$480$	4310.65	0.409903
$481$	0	0
$482$	26383.8	2.49325
$483$	− 2158.33i	− 0.203328i
$484$	65303.7	6.13295
$485$	−3902.92	−0.365407
$486$	1303.63i	0.121674i
$487$	19640.5i	1.82750i	0.406273	+	0.913752i	$0.366828\pi$
$487$	19640.5i	1.82750i	−0.406273	+	0.913752i	$0.633172\pi$
$488$	− 8206.69i	− 0.761269i
$489$	− 1184.74i	− 0.109562i
$490$	1618.41	0.149209
$491$	3410.31	0.313453	0.156726	−	0.987642i	$-0.449906\pi$
$491$	3410.31	0.313453	0.156726	+	0.987642i	$0.449906\pi$
$492$	9754.00i	0.893789i
$493$	−1070.06	−0.0977546
$494$	0	0
$495$	−1623.08	−0.147378
$496$	− 41573.8i	− 3.76354i
$497$	4335.86	0.391327
$498$	9705.68	0.873338
$499$	5032.44i	0.451469i	0.974189	+	0.225735i	$0.0724782\pi$
$499$	5032.44i	0.451469i	−0.974189	+	0.225735i	$0.927522\pi$
$500$	13600.2i	1.21644i
$501$	− 455.580i	− 0.0406263i
$502$	− 6200.44i	− 0.551274i
$503$	17189.4	1.52373	0.761866	−	0.647735i	$-0.224284\pi$
$503$	17189.4	1.52373	0.761866	+	0.647735i	$0.224284\pi$
$504$	9380.89	0.829083
$505$	− 2382.06i	− 0.209901i
$506$	−16980.8	−1.49187
$507$	0	0
$508$	−23145.5	−2.02149
$509$	930.560i	0.0810341i	0.999179	+	0.0405170i	$0.0129005\pi$
$509$	930.560i	0.0810341i	−0.999179	+	0.0405170i	$0.987099\pi$
$510$	−180.676	−0.0156872
$511$	−11264.9	−0.975201
$512$	− 3143.50i	− 0.271337i
$513$	− 704.295i	− 0.0606148i
$514$	− 12617.6i	− 1.08276i
$515$	3375.08i	0.288784i
$516$	3240.75	0.276485
$517$	−23692.1	−2.01543
$518$	− 14328.4i	− 1.21536i
$519$	−1616.44	−0.136713
$520$	0	0
$521$	−9869.60	−0.829933	−0.414966	−	0.909837i	$-0.636207\pi$
$521$	−9869.60	−0.829933	−0.414966	+	0.909837i	$0.636207\pi$
$522$	12408.9i	1.04047i
$523$	−21420.6	−1.79093	−0.895466	−	0.445129i	$-0.853158\pi$
$523$	−21420.6	−1.79093	−0.895466	+	0.445129i	$0.853158\pi$
$524$	−43641.1	−3.63831
$525$	5369.35i	0.446358i
$526$	29623.4i	2.45559i
$527$	858.697i	0.0709781i
$528$	− 40447.4i	− 3.33380i
$529$	−9927.42	−0.815930
$530$	−151.482	−0.0124150
$531$	4006.03i	0.327395i
$532$	−8240.54	−0.671565
$533$	0	0
$534$	−3471.00	−0.281283
$535$	921.097i	0.0744345i
$536$	1536.34	0.123805
$537$	6662.39	0.535388
$538$	21013.4i	1.68392i
$539$	− 7483.41i	− 0.598021i
$540$	1512.81i	0.120558i
$541$	7771.50i	0.617602i	0.951127	+	0.308801i	$0.0999277\pi$
$541$	7771.50i	0.617602i	−0.951127	+	0.308801i	$0.900072\pi$
$542$	14900.7	1.18088
$543$	11468.3	0.906361
$544$	− 2218.78i	− 0.174870i
$545$	−2090.12	−0.164277
$546$	0	0
$547$	−15577.5	−1.21763	−0.608817	−	0.793310i	$-0.708356\pi$
$547$	−15577.5	−1.21763	−0.608817	+	0.793310i	$0.708356\pi$
$548$	25040.6i	1.95197i
$549$	1077.27	0.0837461
$550$	42243.7	3.27505
$551$	− 6704.02i	− 0.518332i
$552$	9734.01i	0.750556i
$553$	− 8326.07i	− 0.640254i
$554$	− 35316.4i	− 2.70840i
$555$	1421.11	0.108690
$556$	−6709.74	−0.511792
$557$	− 22804.5i	− 1.73475i	−0.497652	−	0.867377i	$-0.665804\pi$
$557$	− 22804.5i	− 1.73475i	0.497652	−	0.867377i	$-0.334196\pi$
$558$	9957.89	0.755468
$559$	0	0
$560$	8262.78	0.623511
$561$	835.433i	0.0628734i
$562$	−15405.7	−1.15631
$563$	3517.41	0.263306	0.131653	−	0.991296i	$-0.457972\pi$
$563$	3517.41	0.263306	0.131653	+	0.991296i	$0.457972\pi$
$564$	22082.5i	1.64865i
$565$	− 3448.71i	− 0.256794i
$566$	− 40332.2i	− 2.99521i
$567$	1231.40i	0.0912062i
$568$	−19554.6	−1.44453
$569$	6093.44	0.448946	0.224473	−	0.974480i	$-0.427934\pi$
$569$	6093.44	0.448946	0.224473	+	0.974480i	$0.427934\pi$
$570$	− 1131.96i	− 0.0831797i
$571$	−10460.2	−0.766630	−0.383315	−	0.923618i	$-0.625218\pi$
$571$	−10460.2	−0.766630	−0.383315	+	0.923618i	$0.625218\pi$
$572$	0	0
$573$	10392.6	0.757689
$574$	12760.6i	0.927906i
$575$	−5571.47	−0.404081
$576$	−11216.5	−0.811378
$577$	9648.19i	0.696117i	0.937473	+	0.348058i	$0.113159\pi$
$577$	9648.19i	0.696117i	−0.937473	+	0.348058i	$0.886841\pi$
$578$	− 26263.9i	− 1.89002i
$579$	14092.2i	1.01149i
$580$	14400.1i	1.03092i
$581$	9167.93	0.654647
$582$	23296.4	1.65922
$583$	700.440i	0.0497586i
$584$	50804.1	3.59981
$585$	0	0
$586$	−24176.5	−1.70430
$587$	− 2170.73i	− 0.152633i	−0.997084	−	0.0763166i	$-0.975684\pi$
$587$	− 2170.73i	− 0.152633i	0.997084	−	0.0763166i	$-0.0243160\pi$
$588$	−6975.01	−0.489191
$589$	−5379.82	−0.376353
$590$	6438.56i	0.449274i
$591$	8663.66i	0.603004i
$592$	35414.3i	2.45865i
$593$	22885.9i	1.58484i	0.609975	+	0.792421i	$0.291180\pi$
$593$	22885.9i	1.58484i	−0.609975	+	0.792421i	$0.708820\pi$
$594$	9688.11	0.669205
$595$	−170.666	−0.0117590
$596$	23458.0i	1.61221i
$597$	−189.028	−0.0129588
$598$	0	0
$599$	−23978.7	−1.63563	−0.817815	−	0.575482i	$-0.804815\pi$
$599$	−23978.7	−1.63563	−0.817815	+	0.575482i	$0.804815\pi$
$600$	− 24215.6i	− 1.64766i
$601$	12873.2	0.873728	0.436864	−	0.899528i	$-0.356089\pi$
$601$	12873.2	0.873728	0.436864	+	0.899528i	$0.356089\pi$
$602$	4239.70	0.287039
$603$	201.670i	0.0136196i
$604$	61103.2i	4.11631i
$605$	8473.38i	0.569408i
$606$	14218.4i	0.953107i
$607$	−7117.15	−0.475908	−0.237954	−	0.971276i	$-0.576477\pi$
$607$	−7117.15	−0.475908	−0.237954	+	0.971276i	$0.576477\pi$
$608$	13900.9	0.927227
$609$	11721.4i	0.779927i
$610$	1731.40	0.114922
$611$	0	0
$612$	778.675	0.0514315
$613$	− 173.297i	− 0.0114183i	−0.999984	−	0.00570913i	$-0.998183\pi$
$613$	− 173.297i	− 0.0114183i	0.999984	−	0.00570913i	$-0.00181728\pi$
$614$	−51173.5	−3.36351
$615$	−1265.62	−0.0829830
$616$	− 69715.5i	− 4.55993i
$617$	6102.75i	0.398197i	0.979979	+	0.199099i	$0.0638014\pi$
$617$	6102.75i	0.398197i	−0.979979	+	0.199099i	$0.936199\pi$
$618$	− 20145.7i	− 1.31129i
$619$	− 14867.8i	− 0.965409i	−0.875783	−	0.482705i	$-0.839654\pi$
$619$	− 14867.8i	− 0.965409i	0.875783	−	0.482705i	$-0.160346\pi$
$620$	11555.8	0.748534
$621$	−1277.75	−0.0825676
$622$	40054.6i	2.58206i
$623$	−3278.69	−0.210847
$624$	0	0
$625$	12951.6	0.828900
$626$	− 9772.64i	− 0.623951i
$627$	−5234.07	−0.333379
$628$	−13078.9	−0.831056
$629$	− 731.475i	− 0.0463686i
$630$	1979.13i	0.125159i
$631$	− 17210.6i	− 1.08580i	−0.839796	−	0.542902i	$-0.817325\pi$
$631$	− 17210.6i	− 1.08580i	0.839796	−	0.542902i	$-0.182675\pi$
$632$	37550.3i	2.36340i
$633$	−3149.09	−0.197733
$634$	16765.5	1.05023
$635$	− 3003.21i	− 0.187683i
$636$	652.853	0.0407033
$637$	0	0
$638$	92218.9	5.72254
$639$	− 2566.87i	− 0.158910i
$640$	−6532.30	−0.403456
$641$	12636.0	0.778616	0.389308	−	0.921108i	$-0.372714\pi$
$641$	12636.0	0.778616	0.389308	+	0.921108i	$0.372714\pi$
$642$	− 5497.99i	− 0.337988i
$643$	9586.37i	0.587947i	0.955814	+	0.293973i	$0.0949777\pi$
$643$	9586.37i	0.587947i	−0.955814	+	0.293973i	$0.905022\pi$
$644$	14950.2i	0.914786i
$645$	420.499i	0.0256700i
$646$	−582.641	−0.0354856
$647$	−5244.32	−0.318664	−0.159332	−	0.987225i	$-0.550934\pi$
$647$	−5244.32	−0.318664	−0.159332	+	0.987225i	$0.550934\pi$
$648$	− 5553.57i	− 0.336674i
$649$	29771.4	1.80066
$650$	0	0
$651$	9406.17	0.566293
$652$	8206.39i	0.492925i
$653$	18869.0	1.13078	0.565392	−	0.824822i	$-0.308725\pi$
$653$	18869.0	1.13078	0.565392	+	0.824822i	$0.308725\pi$
$654$	12475.8	0.745938
$655$	− 5662.59i	− 0.337795i
$656$	− 31539.3i	− 1.87714i
$657$	6668.90i	0.396010i
$658$	28889.3i	1.71159i
$659$	−24299.8	−1.43639	−0.718197	−	0.695839i	$-0.755033\pi$
$659$	−24299.8	−1.43639	−0.718197	+	0.695839i	$0.755033\pi$
$660$	11242.7	0.663062
$661$	29915.0i	1.76030i	0.474694	+	0.880151i	$0.342559\pi$
$661$	29915.0i	1.76030i	−0.474694	+	0.880151i	$0.657441\pi$
$662$	−8335.03	−0.489350
$663$	0	0
$664$	−41347.1	−2.41653
$665$	− 1069.24i	− 0.0623508i
$666$	−8482.56	−0.493532
$667$	−12162.6	−0.706056
$668$	3155.69i	0.182780i
$669$	− 6940.48i	− 0.401098i
$670$	324.128i	0.0186898i
$671$	− 8005.86i	− 0.460600i
$672$	−24304.5	−1.39519
$673$	−15493.7	−0.887426	−0.443713	−	0.896169i	$-0.646339\pi$
$673$	−15493.7	−0.887426	−0.443713	+	0.896169i	$0.646339\pi$
$674$	17115.8i	0.978154i
$675$	3178.71	0.181257
$676$	0	0
$677$	11729.7	0.665891	0.332945	−	0.942946i	$-0.391958\pi$
$677$	11729.7	0.665891	0.332945	+	0.942946i	$0.391958\pi$
$678$	20585.2i	1.16603i
$679$	22005.6	1.24374
$680$	769.698	0.0434067
$681$	− 11398.4i	− 0.641390i
$682$	− 74003.5i	− 4.15505i
$683$	− 1168.42i	− 0.0654587i	−0.999464	−	0.0327294i	$-0.989580\pi$
$683$	− 1168.42i	− 0.0654587i	0.999464	−	0.0327294i	$-0.0104199\pi$
$684$	4878.48i	0.272709i
$685$	−3249.10	−0.181229
$686$	−37099.1	−2.06480
$687$	− 12963.2i	− 0.719909i
$688$	−10478.9	−0.580675
$689$	0	0
$690$	−2053.63	−0.113305
$691$	− 32992.9i	− 1.81637i	−0.418574	−	0.908183i	$-0.637470\pi$
$691$	− 32992.9i	− 1.81637i	0.418574	−	0.908183i	$-0.362530\pi$
$692$	11196.7	0.615078
$693$	9151.33	0.501631
$694$	− 30679.2i	− 1.67805i
$695$	− 870.612i	− 0.0475168i
$696$	− 52863.2i	− 2.87899i
$697$	651.438i	0.0354017i
$698$	17856.3	0.968295
$699$	−15839.3	−0.857078
$700$	− 37192.2i	− 2.00819i
$701$	14785.8	0.796651	0.398326	−	0.917244i	$-0.369591\pi$
$701$	14785.8	0.796651	0.398326	+	0.917244i	$0.369591\pi$
$702$	0	0
$703$	4582.77	0.245864
$704$	83357.0i	4.46255i
$705$	−2865.28	−0.153068
$706$	−66021.0	−3.51945
$707$	13430.6i	0.714442i
$708$	− 27748.8i	− 1.47297i
$709$	12634.0i	0.669225i	0.942356	+	0.334612i	$0.108605\pi$
$709$	12634.0i	0.669225i	−0.942356	+	0.334612i	$0.891395\pi$
$710$	− 4125.52i	− 0.218067i
$711$	−4929.11	−0.259994
$712$	14786.8	0.778312
$713$	9760.24i	0.512656i
$714$	1018.70	0.0533947
$715$	0	0
$716$	−46148.7	−2.40874
$717$	− 4643.75i	− 0.241874i
$718$	−45814.6	−2.38132
$719$	27296.5	1.41584	0.707920	−	0.706292i	$-0.249634\pi$
$719$	27296.5	1.41584	0.707920	+	0.706292i	$0.249634\pi$
$720$	− 4891.64i	− 0.253195i
$721$	− 19029.5i	− 0.982936i
$722$	33146.3i	1.70856i
$723$	− 14754.0i	− 0.758932i
$724$	−79438.5	−4.07777
$725$	30257.4	1.54997
$726$	− 50577.2i	− 2.58553i
$727$	−4658.21	−0.237639	−0.118819	−	0.992916i	$-0.537911\pi$
$727$	−4658.21	−0.237639	−0.118819	+	0.992916i	$0.537911\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	10718.4i	0.543432i
$731$	216.439	0.0109512
$732$	−7461.96	−0.376779
$733$	166.474i	0.00838864i	0.999991	+	0.00419432i	$0.00133510\pi$
$733$	166.474i	0.00838864i	−0.999991	+	0.00419432i	$0.998665\pi$
$734$	13393.9i	0.673537i
$735$	− 905.031i	− 0.0454185i
$736$	− 25219.4i	− 1.26304i
$737$	1498.74	0.0749074
$738$	7554.41	0.376804
$739$	12738.1i	0.634069i	0.948414	+	0.317035i	$0.102687\pi$
$739$	12738.1i	0.634069i	−0.948414	+	0.317035i	$0.897313\pi$
$740$	−9843.70	−0.489002
$741$	0	0
$742$	854.092	0.0422570
$743$	30724.5i	1.51705i	0.651641	+	0.758527i	$0.274081\pi$
$743$	30724.5i	1.51705i	−0.651641	+	0.758527i	$0.725919\pi$
$744$	−42421.5	−2.09039
$745$	−3043.76	−0.149684
$746$	− 6131.38i	− 0.300919i
$747$	− 5427.50i	− 0.265839i
$748$	− 5786.84i	− 0.282871i
$749$	− 5193.37i	− 0.253353i
$750$	10533.2	0.512826
$751$	39538.6	1.92115	0.960575	−	0.278021i	$-0.0896785\pi$
$751$	39538.6	1.92115	0.960575	+	0.278021i	$0.0896785\pi$
$752$	− 71403.2i	− 3.46251i
$753$	−3467.34	−0.167805
$754$	0	0
$755$	−7928.35	−0.382175
$756$	− 8529.61i	− 0.410342i
$757$	23035.1	1.10598	0.552990	−	0.833188i	$-0.313487\pi$
$757$	23035.1	1.10598	0.552990	+	0.833188i	$0.313487\pi$
$758$	−65352.1	−3.13153
$759$	9495.81i	0.454119i
$760$	4822.23i	0.230159i
$761$	32454.0i	1.54594i	0.634445	+	0.772968i	$0.281229\pi$
$761$	32454.0i	1.54594i	−0.634445	+	0.772968i	$0.718771\pi$
$762$	17926.0i	0.852220i
$763$	11784.6	0.559150
$764$	−71986.8	−3.40889
$765$	101.036i	0.00477511i
$766$	−54616.1	−2.57619
$767$	0	0
$768$	9080.40	0.426641
$769$	32216.2i	1.51072i	0.655307	+	0.755362i	$0.272539\pi$
$769$	32216.2i	1.51072i	−0.655307	+	0.755362i	$0.727461\pi$
$770$	14708.2	0.688372
$771$	−7055.86	−0.329586
$772$	− 97613.2i	− 4.55075i
$773$	− 2924.60i	− 0.136081i	−0.997683	−	0.0680404i	$-0.978325\pi$
$773$	− 2924.60i	− 0.136081i	0.997683	−	0.0680404i	$-0.0216747\pi$
$774$	− 2509.94i	− 0.116561i
$775$	− 24280.9i	− 1.12541i
$776$	−99244.7	−4.59108
$777$	−8012.58	−0.369948
$778$	− 31361.2i	− 1.44519i
$779$	−4081.32	−0.187713
$780$	0	0
$781$	−19076.1	−0.874001
$782$	1057.04i	0.0483374i
$783$	6939.19	0.316713
$784$	22553.5	1.02740
$785$	− 1697.03i	− 0.0771586i
$786$	33799.8i	1.53384i
$787$	25507.1i	1.15531i	0.816280	+	0.577656i	$0.196033\pi$
$787$	25507.1i	1.15531i	−0.816280	+	0.577656i	$0.803967\pi$
$788$	− 60011.1i	− 2.71295i
$789$	16565.6	0.747468
$790$	−7922.16	−0.356782
$791$	19444.7i	0.874050i
$792$	−41272.2	−1.85170
$793$	0	0
$794$	13416.8	0.599677
$795$	84.7100i	0.00377906i
$796$	1309.35	0.0583023
$797$	−5448.98	−0.242174	−0.121087	−	0.992642i	$-0.538638\pi$
$797$	−5448.98	−0.242174	−0.121087	+	0.992642i	$0.538638\pi$
$798$	6382.24i	0.283119i
$799$	1474.82i	0.0653008i
$800$	62739.0i	2.77270i
$801$	1941.01i	0.0856209i
$802$	49297.8	2.17053
$803$	49560.9	2.17804
$804$	− 1396.92i	− 0.0612755i
$805$	−1939.85	−0.0849324
$806$	0	0
$807$	11750.9	0.512577
$808$	− 60571.7i	− 2.63726i
$809$	−2453.12	−0.106610	−0.0533048	−	0.998578i	$-0.516975\pi$
$809$	−2453.12	−0.106610	−0.0533048	+	0.998578i	$0.516975\pi$
$810$	1171.66	0.0508247
$811$	5133.85i	0.222286i	0.993804	+	0.111143i	$0.0354512\pi$
$811$	5133.85i	0.222286i	−0.993804	+	0.111143i	$0.964549\pi$
$812$	− 81191.4i	− 3.50894i
$813$	− 8332.58i	− 0.359454i
$814$	63039.4i	2.71441i
$815$	−1064.81	−0.0457651
$816$	−2517.83	−0.108017
$817$	1356.01i	0.0580673i
$818$	49432.6	2.11292
$819$	0	0
$820$	8766.61	0.373345
$821$	23854.0i	1.01402i	0.861941	+	0.507009i	$0.169249\pi$
$821$	23854.0i	1.01402i	−0.861941	+	0.507009i	$0.830751\pi$
$822$	19393.8	0.822913
$823$	−757.156	−0.0320690	−0.0160345	−	0.999871i	$-0.505104\pi$
$823$	−757.156	−0.0320690	−0.0160345	+	0.999871i	$0.505104\pi$
$824$	85822.6i	3.62836i
$825$	− 23623.0i	− 0.996907i
$826$	− 36302.2i	− 1.52919i
$827$	28621.2i	1.20345i	0.798702	+	0.601726i	$0.205520\pi$
$827$	28621.2i	1.20345i	−0.798702	+	0.601726i	$0.794480\pi$
$828$	8850.68	0.371476
$829$	−27429.2	−1.14916	−0.574582	−	0.818447i	$-0.694835\pi$
$829$	−27429.2	−1.14916	−0.574582	+	0.818447i	$0.694835\pi$
$830$	− 8723.19i	− 0.364803i
$831$	−19749.2	−0.824421
$832$	0	0
$833$	−465.838	−0.0193761
$834$	5196.64i	0.215761i
$835$	−409.462	−0.0169701
$836$	36255.1	1.49989
$837$	− 5568.54i	− 0.229960i
$838$	− 34840.7i	− 1.43622i
$839$	− 4633.62i	− 0.190668i	−0.995445	−	0.0953339i	$-0.969608\pi$
$839$	− 4633.62i	− 0.190668i	0.995445	−	0.0953339i	$-0.0303919\pi$
$840$	− 8431.27i	− 0.346317i
$841$	41663.6	1.70829
$842$	16413.7	0.671798
$843$	8614.98i	0.351976i
$844$	21813.0	0.889614
$845$	0	0
$846$	17102.7	0.695041
$847$	− 47775.0i	− 1.93810i
$848$	−2110.99	−0.0854853
$849$	−22554.1	−0.911726
$850$	− 2629.64i	− 0.106113i
$851$	− 8314.19i	− 0.334908i
$852$	17780.1i	0.714947i
$853$	− 14854.6i	− 0.596261i	−0.954525	−	0.298131i	$-0.903637\pi$
$853$	− 14854.6i	− 0.596261i	0.954525	−	0.298131i	$-0.0963631\pi$
$854$	−9762.07	−0.391161
$855$	−632.999	−0.0253194
$856$	23421.9i	0.935216i
$857$	42799.5	1.70595	0.852977	−	0.521948i	$-0.174794\pi$
$857$	42799.5	1.70595	0.852977	+	0.521948i	$0.174794\pi$
$858$	0	0
$859$	−8246.47	−0.327551	−0.163775	−	0.986498i	$-0.552367\pi$
$859$	−8246.47	−0.327551	−0.163775	+	0.986498i	$0.552367\pi$
$860$	− 2912.70i	− 0.115491i
$861$	7135.85	0.282450
$862$	42580.1	1.68246
$863$	− 17695.4i	− 0.697983i	−0.937126	−	0.348991i	$-0.886524\pi$
$863$	− 17695.4i	− 0.697983i	0.937126	−	0.348991i	$-0.113476\pi$
$864$	14388.5i	0.566558i
$865$	1452.81i	0.0571064i
$866$	− 39132.3i	− 1.53553i
$867$	−14687.0	−0.575313
$868$	−65154.2	−2.54779
$869$	36631.4i	1.42996i
$870$	11152.8	0.434615
$871$	0	0
$872$	−53148.2	−2.06402
$873$	− 13027.5i	− 0.505058i
$874$	−6622.49	−0.256303
$875$	9949.64	0.384411
$876$	− 46193.8i	− 1.78167i
$877$	14346.8i	0.552401i	0.961100	+	0.276200i	$0.0890754\pi$
$877$	14346.8i	0.552401i	−0.961100	+	0.276200i	$0.910925\pi$
$878$	− 81622.7i	− 3.13739i
$879$	13519.7i	0.518781i
$880$	−36353.0	−1.39257
$881$	2063.51	0.0789121	0.0394560	−	0.999221i	$-0.487437\pi$
$881$	2063.51	0.0789121	0.0394560	+	0.999221i	$0.487437\pi$
$882$	5402.09i	0.206233i
$883$	34137.1	1.30103	0.650513	−	0.759495i	$-0.274554\pi$
$883$	34137.1	1.30103	0.650513	+	0.759495i	$0.274554\pi$
$884$	0	0
$885$	3600.50	0.136756
$886$	8138.58i	0.308602i
$887$	−22238.5	−0.841821	−0.420910	−	0.907102i	$-0.638289\pi$
$887$	−22238.5	−0.841821	−0.420910	+	0.907102i	$0.638289\pi$
$888$	36136.5	1.36561
$889$	16932.8i	0.638817i
$890$	3119.63i	0.117495i
$891$	− 5417.67i	− 0.203702i
$892$	48075.0i	1.80456i
$893$	−9239.89	−0.346250
$894$	18168.0	0.679676
$895$	− 5987.96i	− 0.223637i
$896$	36830.7	1.37325
$897$	0	0
$898$	3783.45	0.140596
$899$	− 53005.7i	− 1.96645i
$900$	−22018.1	−0.815486
$901$	43.6019	0.00161220
$902$	− 56141.7i	− 2.07241i
$903$	− 2370.88i	− 0.0873730i
$904$	− 87694.9i	− 3.22643i
$905$	− 10307.4i	− 0.378597i
$906$	47324.0	1.73536
$907$	44160.3	1.61667	0.808335	−	0.588723i	$-0.200369\pi$
$907$	44160.3	1.61667	0.808335	+	0.588723i	$0.200369\pi$
$908$	78953.8i	2.88565i
$909$	7951.05	0.290121
$910$	0	0
$911$	11916.3	0.433376	0.216688	−	0.976241i	$-0.430475\pi$
$911$	11916.3	0.433376	0.216688	+	0.976241i	$0.430475\pi$
$912$	− 15774.4i	− 0.572745i
$913$	−40335.3	−1.46211
$914$	−39043.0	−1.41294
$915$	− 968.215i	− 0.0349816i
$916$	89793.0i	3.23891i
$917$	31927.1i	1.14975i
$918$	− 603.078i	− 0.0216825i
$919$	−20128.9	−0.722516	−0.361258	−	0.932466i	$-0.617653\pi$
$919$	−20128.9	−0.722516	−0.361258	+	0.932466i	$0.617653\pi$
$920$	8748.64	0.313515
$921$	28616.7i	1.02383i
$922$	10525.0	0.375945
$923$	0	0
$924$	−63389.0	−2.25687
$925$	20683.5i	0.735210i
$926$	55654.1	1.97506
$927$	−11265.7	−0.399151
$928$	136961.i	4.84478i
$929$	31428.4i	1.10994i	0.831871	+	0.554969i	$0.187270\pi$
$929$	31428.4i	1.10994i	−0.831871	+	0.554969i	$0.812730\pi$
$930$	− 8949.86i	− 0.315567i
$931$	− 2918.52i	− 0.102740i
$932$	109715.	3.85605
$933$	22398.9	0.785965
$934$	− 47150.1i	− 1.65182i
$935$	750.863	0.0262629
$936$	0	0
$937$	42473.2	1.48083	0.740416	−	0.672149i	$-0.234629\pi$
$937$	42473.2	1.48083	0.740416	+	0.672149i	$0.234629\pi$
$938$	− 1827.51i	− 0.0636144i
$939$	−5464.94	−0.189927
$940$	19847.1	0.688661
$941$	42644.0i	1.47732i	0.674079	+	0.738659i	$0.264541\pi$
$941$	42644.0i	1.47732i	−0.674079	+	0.738659i	$0.735459\pi$
$942$	10129.5i	0.350357i
$943$	7404.46i	0.255697i
$944$	89725.0i	3.09354i
$945$	1106.75	0.0380978
$946$	−18653.0	−0.641080
$947$	4282.95i	0.146966i	0.997296	+	0.0734831i	$0.0234115\pi$
$947$	4282.95i	0.146966i	−0.997296	+	0.0734831i	$0.976588\pi$
$948$	34142.7	1.16973
$949$	0	0
$950$	16475.0	0.562651
$951$	− 9375.42i	− 0.319683i
$952$	−4339.74	−0.147744
$953$	−40593.9	−1.37982	−0.689908	−	0.723897i	$-0.742349\pi$
$953$	−40593.9	−1.37982	−0.689908	+	0.723897i	$0.742349\pi$
$954$	− 505.630i	− 0.0171597i
$955$	− 9340.54i	− 0.316495i
$956$	32166.1i	1.08821i
$957$	− 51569.6i	− 1.74191i
$958$	59771.9	2.01581
$959$	18319.2	0.616850
$960$	10081.1i	0.338922i
$961$	−12744.8	−0.427808
$962$	0	0
$963$	−3074.52	−0.102882
$964$	102198.i	3.41448i
$965$	12665.7	0.422510
$966$	11578.9	0.385656
$967$	− 17709.4i	− 0.588931i	−0.955662	−	0.294466i	$-0.904858\pi$
$967$	− 17709.4i	− 0.588931i	0.955662	−	0.294466i	$-0.0951416\pi$
$968$	215464.i	7.15420i
$969$	325.818i	0.0108016i
$970$	− 20938.1i	− 0.693074i
$971$	4038.97	0.133488	0.0667440	−	0.997770i	$-0.478739\pi$
$971$	4038.97	0.133488	0.0667440	+	0.997770i	$0.478739\pi$
$972$	−5049.61	−0.166632
$973$	4908.72i	0.161733i
$974$	−105366.	−3.46626
$975$	0	0
$976$	24128.1	0.791312
$977$	2764.30i	0.0905197i	0.998975	+	0.0452598i	$0.0144116\pi$
$977$	2764.30i	0.0905197i	−0.998975	+	0.0452598i	$0.985588\pi$
$978$	6355.79	0.207807
$979$	14424.9	0.470912
$980$	6268.93i	0.204340i
$981$	− 6976.59i	− 0.227060i
$982$	18295.4i	0.594531i
$983$	− 17804.5i	− 0.577695i	−0.957375	−	0.288848i	$-0.906728\pi$
$983$	− 17804.5i	− 0.577695i	0.957375	−	0.288848i	$-0.0932721\pi$
$984$	−32182.4	−1.04262
$985$	7786.65	0.251881
$986$	− 5740.57i	− 0.185413i
$987$	16155.2	0.520997
$988$	0	0
$989$	2460.12	0.0790974
$990$	− 8707.39i	− 0.279534i
$991$	−8129.69	−0.260593	−0.130297	−	0.991475i	$-0.541593\pi$
$991$	−8129.69	−0.260593	−0.130297	+	0.991475i	$0.541593\pi$
$992$	109908.	3.51772
$993$	4661.02i	0.148956i
$994$	23260.7i	0.742237i
$995$	169.893i	0.00541302i
$996$	37595.0i	1.19603i
$997$	−29452.9	−0.935589	−0.467795	−	0.883837i	$-0.654951\pi$
$997$	−29452.9	−0.935589	−0.467795	+	0.883837i	$0.654951\pi$
$998$	−26997.7	−0.856309
$999$	4743.52i	0.150229i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.b.i.337.10		10
13.3	even	3		39.4.j.c.4.1	✓	10
13.4	even	6		39.4.j.c.10.1	yes	10
13.5	odd	4		507.4.a.r.1.10		10
13.8	odd	4		507.4.a.r.1.1		10
13.12	even	2	inner	507.4.b.i.337.1		10
39.5	even	4		1521.4.a.bk.1.1		10
39.8	even	4		1521.4.a.bk.1.10		10
39.17	odd	6		117.4.q.e.10.5		10
39.29	odd	6		117.4.q.e.82.5		10
52.3	odd	6		624.4.bv.h.433.3		10
52.43	odd	6		624.4.bv.h.49.3		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.j.c.4.1	✓	10	13.3	even	3
39.4.j.c.10.1	yes	10	13.4	even	6
117.4.q.e.10.5		10	39.17	odd	6
117.4.q.e.82.5		10	39.29	odd	6
507.4.a.r.1.1		10	13.8	odd	4
507.4.a.r.1.10		10	13.5	odd	4
507.4.b.i.337.1		10	13.12	even	2	inner
507.4.b.i.337.10		10	1.1	even	1	trivial
624.4.bv.h.49.3		10	52.43	odd	6
624.4.bv.h.433.3		10	52.3	odd	6
1521.4.a.bk.1.1		10	39.5	even	4
1521.4.a.bk.1.10		10	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)

\(f(q)\)	\(=\)	\(q+5.36472i q^{2} +3.00000 q^{3} -20.7803 q^{4} -2.69631i q^{5} +16.0942i q^{6} +15.2025i q^{7} -68.5626i q^{8} +9.00000 q^{9} +14.4650 q^{10} -66.8848i q^{11} -62.3408 q^{12} -81.5570 q^{14} -8.08894i q^{15} +201.577 q^{16} -4.16354 q^{17} +48.2825i q^{18} -26.0850i q^{19} +56.0301i q^{20} +45.6074i q^{21} +358.819 q^{22} -47.3242 q^{23} -205.688i q^{24} +117.730 q^{25} +27.0000 q^{27} -315.911i q^{28} +257.007 q^{29} +43.3949 q^{30} -206.242i q^{31} +532.906i q^{32} -200.655i q^{33} -22.3362i q^{34} +40.9906 q^{35} -187.022 q^{36} +175.686i q^{37} +139.939 q^{38} -184.866 q^{40} -156.463i q^{41} -244.671 q^{42} -51.9845 q^{43} +1389.88i q^{44} -24.2668i q^{45} -253.881i q^{46} -354.222i q^{47} +604.732 q^{48} +111.885 q^{49} +631.588i q^{50} -12.4906 q^{51} -10.4723 q^{53} +144.848i q^{54} -180.342 q^{55} +1042.32 q^{56} -78.2550i q^{57} +1378.77i q^{58} +445.114i q^{59} +168.090i q^{60} +119.696 q^{61} +1106.43 q^{62} +136.822i q^{63} -1246.28 q^{64} +1076.46 q^{66} +22.4078i q^{67} +86.5195 q^{68} -141.973 q^{69} +219.903i q^{70} -285.207i q^{71} -617.064i q^{72} +740.989i q^{73} -942.507 q^{74} +353.190 q^{75} +542.053i q^{76} +1016.81 q^{77} -547.679 q^{79} -543.516i q^{80} +81.0000 q^{81} +839.378 q^{82} -603.056i q^{83} -947.734i q^{84} +11.2262i q^{85} -278.882i q^{86} +771.021 q^{87} -4585.80 q^{88} +215.668i q^{89} +130.185 q^{90} +983.409 q^{92} -618.726i q^{93} +1900.31 q^{94} -70.3333 q^{95} +1598.72i q^{96} -1447.50i q^{97} +600.233i q^{98} -601.964i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 30 q^{3} - 60 q^{4} + 90 q^{9} - 80 q^{10} - 180 q^{12} - 60 q^{14} + 500 q^{16} - 210 q^{17} + 580 q^{22} + 120 q^{23} - 960 q^{25} + 270 q^{27} + 990 q^{29} - 240 q^{30} - 120 q^{35} - 540 q^{36}+ \cdots + 2760 q^{95}+O(q^{100}) \) 10 * q + 30 * q^3 - 60 * q^4 + 90 * q^9 - 80 * q^10 - 180 * q^12 - 60 * q^14 + 500 * q^16 - 210 * q^17 + 580 * q^22 + 120 * q^23 - 960 * q^25 + 270 * q^27 + 990 * q^29 - 240 * q^30 - 120 * q^35 - 540 * q^36 - 1380 * q^38 + 2000 * q^40 - 180 * q^42 + 740 * q^43 + 1500 * q^48 - 1550 * q^49 - 630 * q^51 + 330 * q^53 + 520 * q^55 + 5340 * q^56 + 2750 * q^61 + 1560 * q^62 - 3140 * q^64 + 1740 * q^66 + 1200 * q^68 + 360 * q^69 - 4380 * q^74 - 2880 * q^75 - 4320 * q^77 + 1100 * q^79 + 810 * q^81 + 4780 * q^82 + 2970 * q^87 - 6340 * q^88 - 720 * q^90 - 1740 * q^92 + 6460 * q^94 + 2760 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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