LMFDB - Embedded newform 507.4.b.j.337.1

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			337.1
Root			$-4.48584i$ of defining polynomial
Character	$\chi$	$=$	507.337
Dual form			507.4.b.j.337.18

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.48584i q^{2} -3.00000 q^{3} -22.0945 q^{4} -13.3185i q^{5} +16.4575i q^{6} +21.4234i q^{7} +77.3200i q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-5.48584i q^{2} -3.00000 q^{3} -22.0945 q^{4} -13.3185i q^{5} +16.4575i q^{6} +21.4234i q^{7} +77.3200i q^{8} +9.00000 q^{9} -73.0635 q^{10} -19.0520i q^{11} +66.2834 q^{12} +117.525 q^{14} +39.9556i q^{15} +247.410 q^{16} +71.7906 q^{17} -49.3726i q^{18} +102.134i q^{19} +294.266i q^{20} -64.2701i q^{21} -104.516 q^{22} +37.8302 q^{23} -231.960i q^{24} -52.3838 q^{25} -27.0000 q^{27} -473.338i q^{28} +40.8605 q^{29} +219.190 q^{30} +6.05542i q^{31} -738.690i q^{32} +57.1559i q^{33} -393.832i q^{34} +285.328 q^{35} -198.850 q^{36} -285.682i q^{37} +560.293 q^{38} +1029.79 q^{40} +342.705i q^{41} -352.576 q^{42} -306.458 q^{43} +420.943i q^{44} -119.867i q^{45} -207.530i q^{46} +346.863i q^{47} -742.229 q^{48} -115.961 q^{49} +287.369i q^{50} -215.372 q^{51} +398.219 q^{53} +148.118i q^{54} -253.745 q^{55} -1656.46 q^{56} -306.403i q^{57} -224.155i q^{58} -208.497i q^{59} -882.799i q^{60} +546.936 q^{61} +33.2191 q^{62} +192.810i q^{63} -2073.06 q^{64} +313.548 q^{66} +678.268i q^{67} -1586.18 q^{68} -113.490 q^{69} -1565.27i q^{70} -957.777i q^{71} +695.880i q^{72} +270.360i q^{73} -1567.21 q^{74} +157.151 q^{75} -2256.60i q^{76} +408.157 q^{77} -1032.86 q^{79} -3295.14i q^{80} +81.0000 q^{81} +1880.03 q^{82} +1065.90i q^{83} +1420.01i q^{84} -956.147i q^{85} +1681.18i q^{86} -122.582 q^{87} +1473.10 q^{88} +427.185i q^{89} -657.571 q^{90} -835.837 q^{92} -18.1663i q^{93} +1902.83 q^{94} +1360.28 q^{95} +2216.07i q^{96} -698.084i q^{97} +636.144i q^{98} -171.468i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	− 5.48584i	− 1.93954i	−0.244025	−	0.969769i	$-0.578468\pi$
$2$	− 5.48584i	− 1.93954i	0.244025	−	0.969769i	$-0.421532\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−22.0945	−2.76181
$5$	− 13.3185i	− 1.19125i	−0.803264	−	0.595624i	$-0.796905\pi$
$5$	− 13.3185i	− 1.19125i	0.803264	−	0.595624i	$-0.203095\pi$
$6$	16.4575i	1.11979i
$7$	21.4234i	1.15675i	0.815770	+	0.578377i	$0.196314\pi$
$7$	21.4234i	1.15675i	−0.815770	+	0.578377i	$0.803686\pi$
$8$	77.3200i	3.41709i
$9$	9.00000	0.333333
$10$	−73.0635	−2.31047
$11$	− 19.0520i	− 0.522217i	−0.965310	−	0.261108i	$-0.915912\pi$
$11$	− 19.0520i	− 0.522217i	0.965310	−	0.261108i	$-0.0840880\pi$
$12$	66.2834	1.59453
$13$	0	0
$13$	0	0
$14$	117.525	2.24357
$15$	39.9556i	0.687767i
$16$	247.410	3.86578
$17$	71.7906	1.02422	0.512111	−	0.858919i	$-0.328864\pi$
$17$	71.7906	1.02422	0.512111	+	0.858919i	$0.328864\pi$
$18$	− 49.3726i	− 0.646513i
$19$	102.134i	1.23322i	0.787268	+	0.616611i	$0.211495\pi$
$19$	102.134i	1.23322i	−0.787268	+	0.616611i	$0.788505\pi$
$20$	294.266i	3.29000i
$21$	− 64.2701i	− 0.667852i
$22$	−104.516	−1.01286
$23$	37.8302	0.342962	0.171481	−	0.985187i	$-0.445145\pi$
$23$	37.8302	0.342962	0.171481	+	0.985187i	$0.445145\pi$
$24$	− 231.960i	− 1.97286i
$25$	−52.3838	−0.419070
$26$	0	0
$27$	−27.0000	−0.192450
$28$	− 473.338i	− 3.19473i
$29$	40.8605	0.261642	0.130821	−	0.991406i	$-0.458239\pi$
$29$	40.8605	0.261642	0.130821	+	0.991406i	$0.458239\pi$
$30$	219.190	1.33395
$31$	6.05542i	0.0350834i	0.999846	+	0.0175417i	$0.00558399\pi$
$31$	6.05542i	0.0350834i	−0.999846	+	0.0175417i	$0.994416\pi$
$32$	− 738.690i	− 4.08073i
$33$	57.1559i	0.301502i
$34$	− 393.832i	− 1.98652i
$35$	285.328	1.37798
$36$	−198.850	−0.920603
$37$	− 285.682i	− 1.26935i	−0.772781	−	0.634673i	$-0.781135\pi$
$37$	− 285.682i	− 1.26935i	0.772781	−	0.634673i	$-0.218865\pi$
$38$	560.293	2.39188
$39$	0	0
$40$	1029.79	4.07060
$41$	342.705i	1.30540i	0.757615	+	0.652702i	$0.226364\pi$
$41$	342.705i	1.30540i	−0.757615	+	0.652702i	$0.773636\pi$
$42$	−352.576	−1.29532
$43$	−306.458	−1.08685	−0.543424	−	0.839458i	$-0.682872\pi$
$43$	−306.458	−1.08685	−0.543424	+	0.839458i	$0.682872\pi$
$44$	420.943i	1.44226i
$45$	− 119.867i	− 0.397082i
$46$	− 207.530i	− 0.665188i
$47$	346.863i	1.07649i	0.842788	+	0.538246i	$0.180913\pi$
$47$	346.863i	1.07649i	−0.842788	+	0.538246i	$0.819087\pi$
$48$	−742.229	−2.23191
$49$	−115.961	−0.338079
$50$	287.369i	0.812802i
$51$	−215.372	−0.591335
$52$	0	0
$53$	398.219	1.03207	0.516034	−	0.856568i	$-0.327408\pi$
$53$	398.219	1.03207	0.516034	+	0.856568i	$0.327408\pi$
$54$	148.118i	0.373264i
$55$	−253.745	−0.622089
$56$	−1656.46	−3.95274
$57$	− 306.403i	− 0.712001i
$58$	− 224.155i	− 0.507464i
$59$	− 208.497i	− 0.460067i	−0.973183	−	0.230034i	$-0.926116\pi$
$59$	− 208.497i	− 0.460067i	0.973183	−	0.230034i	$-0.0738836\pi$
$60$	− 882.799i	− 1.89948i
$61$	546.936	1.14800	0.574000	−	0.818855i	$-0.305391\pi$
$61$	546.936	1.14800	0.574000	+	0.818855i	$0.305391\pi$
$62$	33.2191	0.0680456
$63$	192.810i	0.385585i
$64$	−2073.06	−4.04895
$65$	0	0
$66$	313.548	0.584774
$67$	678.268i	1.23677i	0.785875	+	0.618386i	$0.212213\pi$
$67$	678.268i	1.23677i	−0.785875	+	0.618386i	$0.787787\pi$
$68$	−1586.18	−2.82871
$69$	−113.490	−0.198009
$70$	− 1565.27i	− 2.67264i
$71$	− 957.777i	− 1.60095i	−0.599368	−	0.800474i	$-0.704581\pi$
$71$	− 957.777i	− 1.60095i	0.599368	−	0.800474i	$-0.295419\pi$
$72$	695.880i	1.13903i
$73$	270.360i	0.433469i	0.976231	+	0.216734i	$0.0695405\pi$
$73$	270.360i	0.433469i	−0.976231	+	0.216734i	$0.930459\pi$
$74$	−1567.21	−2.46195
$75$	157.151	0.241950
$76$	− 2256.60i	− 3.40592i
$77$	408.157	0.604076
$78$	0	0
$79$	−1032.86	−1.47096	−0.735482	−	0.677544i	$-0.763044\pi$
$79$	−1032.86	−1.47096	−0.735482	+	0.677544i	$0.763044\pi$
$80$	− 3295.14i	− 4.60510i
$81$	81.0000	0.111111
$82$	1880.03	2.53188
$83$	1065.90i	1.40961i	0.709400	+	0.704806i	$0.248966\pi$
$83$	1065.90i	1.40961i	−0.709400	+	0.704806i	$0.751034\pi$
$84$	1420.01i	1.84448i
$85$	− 956.147i	− 1.22010i
$86$	1681.18i	2.10798i
$87$	−122.582	−0.151059
$88$	1473.10	1.78446
$89$	427.185i	0.508782i	0.967101	+	0.254391i	$0.0818750\pi$
$89$	427.185i	0.508782i	−0.967101	+	0.254391i	$0.918125\pi$
$90$	−657.571	−0.770156
$91$	0	0
$92$	−835.837	−0.947196
$93$	− 18.1663i	− 0.0202554i
$94$	1902.83	2.08790
$95$	1360.28	1.46907
$96$	2216.07i	2.35601i
$97$	− 698.084i	− 0.730718i	−0.930867	−	0.365359i	$-0.880946\pi$
$97$	− 698.084i	− 0.730718i	0.930867	−	0.365359i	$-0.119054\pi$
$98$	636.144i	0.655717i
$99$	− 171.468i	− 0.174072i
$100$	1157.39	1.15739
$101$	−88.7364	−0.0874218	−0.0437109	−	0.999044i	$-0.513918\pi$
$101$	−88.7364	−0.0874218	−0.0437109	+	0.999044i	$0.513918\pi$
$102$	1181.50i	1.14692i
$103$	1427.61	1.36570	0.682849	−	0.730560i	$-0.260741\pi$
$103$	1427.61	1.36570	0.682849	+	0.730560i	$0.260741\pi$
$104$	0	0
$105$	−855.985	−0.795577
$106$	− 2184.57i	− 2.00173i
$107$	−15.0233	−0.0135735	−0.00678673	−	0.999977i	$-0.502160\pi$
$107$	−15.0233	−0.0135735	−0.00678673	+	0.999977i	$0.502160\pi$
$108$	596.551	0.531510
$109$	− 2053.56i	− 1.80455i	−0.431163	−	0.902274i	$-0.641896\pi$
$109$	− 2053.56i	− 1.80455i	0.431163	−	0.902274i	$-0.358104\pi$
$110$	1392.00i	1.20657i
$111$	857.046i	0.732857i
$112$	5300.35i	4.47175i
$113$	717.456	0.597280	0.298640	−	0.954366i	$-0.403467\pi$
$113$	717.456	0.597280	0.298640	+	0.954366i	$0.403467\pi$
$114$	−1680.88	−1.38095
$115$	− 503.843i	− 0.408553i
$116$	−902.792	−0.722605
$117$	0	0
$118$	−1143.78	−0.892318
$119$	1538.00i	1.18477i
$120$	−3089.37	−2.35016
$121$	968.023	0.727290
$122$	− 3000.41i	− 2.22659i
$123$	− 1028.12i	− 0.753675i
$124$	− 133.791i	− 0.0968937i
$125$	− 967.143i	− 0.692031i
$126$	1057.73	0.747856
$127$	117.640	0.0821955	0.0410978	−	0.999155i	$-0.486914\pi$
$127$	117.640	0.0821955	0.0410978	+	0.999155i	$0.486914\pi$
$128$	5462.96i	3.77236i
$129$	919.375	0.627492
$130$	0	0
$131$	−262.376	−0.174991	−0.0874957	−	0.996165i	$-0.527886\pi$
$131$	−262.376	−0.174991	−0.0874957	+	0.996165i	$0.527886\pi$
$132$	− 1262.83i	− 0.832690i
$133$	−2188.06	−1.42653
$134$	3720.87	2.39876
$135$	359.601i	0.229256i
$136$	5550.85i	3.49986i
$137$	− 1317.41i	− 0.821563i	−0.911734	−	0.410782i	$-0.865256\pi$
$137$	− 1317.41i	− 0.821563i	0.911734	−	0.410782i	$-0.134744\pi$
$138$	622.591i	0.384047i
$139$	−6.43478	−0.00392656	−0.00196328	−	0.999998i	$-0.500625\pi$
$139$	−6.43478	−0.00392656	−0.00196328	+	0.999998i	$0.500625\pi$
$140$	−6304.18	−3.80572
$141$	− 1040.59i	− 0.621513i
$142$	−5254.22	−3.10510
$143$	0	0
$144$	2226.69	1.28859
$145$	− 544.203i	− 0.311680i
$146$	1483.15	0.840729
$147$	347.883	0.195190
$148$	6311.99i	3.50569i
$149$	− 548.312i	− 0.301473i	−0.988574	−	0.150736i	$-0.951836\pi$
$149$	− 548.312i	− 0.301473i	0.988574	−	0.150736i	$-0.0481645\pi$
$150$	− 862.107i	− 0.469272i
$151$	1485.90i	0.800799i	0.916341	+	0.400399i	$0.131129\pi$
$151$	1485.90i	0.800799i	−0.916341	+	0.400399i	$0.868871\pi$
$152$	−7897.03	−4.21404
$153$	646.116	0.341408
$154$	− 2239.09i	− 1.17163i
$155$	80.6494	0.0417930
$156$	0	0
$157$	2132.36	1.08396	0.541978	−	0.840393i	$-0.317676\pi$
$157$	2132.36	1.08396	0.541978	+	0.840393i	$0.317676\pi$
$158$	5666.13i	2.85299i
$159$	−1194.66	−0.595865
$160$	−9838.28	−4.86115
$161$	810.450i	0.396723i
$162$	− 444.353i	− 0.215504i
$163$	480.633i	0.230957i	0.993310	+	0.115479i	$0.0368402\pi$
$163$	480.633i	0.230957i	−0.993310	+	0.115479i	$0.963160\pi$
$164$	− 7571.88i	− 3.60527i
$165$	761.234	0.359163
$166$	5847.36	2.73400
$167$	− 2919.71i	− 1.35290i	−0.736490	−	0.676448i	$-0.763518\pi$
$167$	− 2919.71i	− 1.35290i	0.736490	−	0.676448i	$-0.236482\pi$
$168$	4969.37	2.28211
$169$	0	0
$170$	−5245.27	−2.36643
$171$	919.209i	0.411074i
$172$	6771.04	3.00167
$173$	2339.62	1.02820	0.514098	−	0.857731i	$-0.328127\pi$
$173$	2339.62	1.02820	0.514098	+	0.857731i	$0.328127\pi$
$174$	672.464i	0.292985i
$175$	− 1122.24i	− 0.484761i
$176$	− 4713.64i	− 2.01877i
$177$	625.490i	0.265620i
$178$	2343.47	0.986802
$179$	4558.71	1.90354	0.951772	−	0.306808i	$-0.0992608\pi$
$179$	4558.71	1.90354	0.951772	+	0.306808i	$0.0992608\pi$
$180$	2648.40i	1.09667i
$181$	3524.56	1.44739	0.723696	−	0.690118i	$-0.242442\pi$
$181$	3524.56	1.44739	0.723696	+	0.690118i	$0.242442\pi$
$182$	0	0
$183$	−1640.81	−0.662798
$184$	2925.03i	1.17193i
$185$	−3804.87	−1.51211
$186$	−99.6573	−0.0392862
$187$	− 1367.75i	− 0.534866i
$188$	− 7663.75i	− 2.97306i
$189$	− 578.431i	− 0.222617i
$190$	− 7462.29i	− 2.84932i
$191$	3343.27	1.26655	0.633273	−	0.773929i	$-0.281711\pi$
$191$	3343.27	1.26655	0.633273	+	0.773929i	$0.281711\pi$
$192$	6219.18	2.33766
$193$	3341.33i	1.24619i	0.782147	+	0.623094i	$0.214124\pi$
$193$	3341.33i	1.24619i	−0.782147	+	0.623094i	$0.785876\pi$
$194$	−3829.58	−1.41726
$195$	0	0
$196$	2562.10	0.933709
$197$	− 3653.95i	− 1.32149i	−0.750611	−	0.660745i	$-0.770241\pi$
$197$	− 3653.95i	− 1.32149i	0.750611	−	0.660745i	$-0.229759\pi$
$198$	−940.645	−0.337620
$199$	−1126.55	−0.401301	−0.200651	−	0.979663i	$-0.564306\pi$
$199$	−1126.55	−0.401301	−0.200651	+	0.979663i	$0.564306\pi$
$200$	− 4050.31i	− 1.43200i
$201$	− 2034.81i	− 0.714050i
$202$	486.794i	0.169558i
$203$	875.371i	0.302655i
$204$	4758.53	1.63315
$205$	4564.33	1.55506
$206$	− 7831.66i	− 2.64882i
$207$	340.471	0.114321
$208$	0	0
$209$	1945.86	0.644009
$210$	4695.80i	1.54305i
$211$	−74.5243	−0.0243150	−0.0121575	−	0.999926i	$-0.503870\pi$
$211$	−74.5243	−0.0243150	−0.0121575	+	0.999926i	$0.503870\pi$
$212$	−8798.44	−2.85037
$213$	2873.33i	0.924307i
$214$	82.4156i	0.0263262i
$215$	4081.58i	1.29471i
$216$	− 2087.64i	− 0.657620i
$217$	−129.728	−0.0405829
$218$	−11265.5	−3.49999
$219$	− 811.079i	− 0.250263i
$220$	5606.35	1.71809
$221$	0	0
$222$	4701.62	1.42140
$223$	3178.62i	0.954512i	0.878764	+	0.477256i	$0.158369\pi$
$223$	3178.62i	0.954512i	−0.878764	+	0.477256i	$0.841631\pi$
$224$	15825.2	4.72039
$225$	−471.454	−0.139690
$226$	− 3935.85i	− 1.15845i
$227$	− 1349.76i	− 0.394655i	−0.980338	−	0.197327i	$-0.936774\pi$
$227$	− 1349.76i	− 0.394655i	0.980338	−	0.197327i	$-0.0632262\pi$
$228$	6769.81i	1.96641i
$229$	4821.46i	1.39131i	0.718374	+	0.695657i	$0.244887\pi$
$229$	4821.46i	1.39131i	−0.718374	+	0.695657i	$0.755113\pi$
$230$	−2764.00	−0.792404
$231$	−1224.47	−0.348763
$232$	3159.34i	0.894055i
$233$	2400.88	0.675050	0.337525	−	0.941317i	$-0.390410\pi$
$233$	2400.88	0.675050	0.337525	+	0.941317i	$0.390410\pi$
$234$	0	0
$235$	4619.71	1.28237
$236$	4606.63i	1.27062i
$237$	3098.59	0.849262
$238$	8437.21	2.29791
$239$	− 1880.85i	− 0.509047i	−0.967067	−	0.254523i	$-0.918081\pi$
$239$	− 1880.85i	− 0.509047i	0.967067	−	0.254523i	$-0.0819186\pi$
$240$	9885.41i	2.65875i
$241$	− 5435.01i	− 1.45270i	−0.687327	−	0.726349i	$-0.741216\pi$
$241$	− 5435.01i	− 1.45270i	0.687327	−	0.726349i	$-0.258784\pi$
$242$	− 5310.42i	− 1.41061i
$243$	−243.000	−0.0641500
$244$	−12084.3	−3.17056
$245$	1544.43i	0.402736i
$246$	−5640.08	−1.46178
$247$	0	0
$248$	−468.205	−0.119883
$249$	− 3197.70i	− 0.813840i
$250$	−5305.59	−1.34222
$251$	−1256.70	−0.316024	−0.158012	−	0.987437i	$-0.550508\pi$
$251$	−1256.70	−0.316024	−0.158012	+	0.987437i	$0.550508\pi$
$252$	− 4260.04i	− 1.06491i
$253$	− 720.739i	− 0.179101i
$254$	− 645.353i	− 0.159421i
$255$	2868.44i	0.704426i
$256$	13384.5	3.26769
$257$	−5504.87	−1.33612	−0.668062	−	0.744105i	$-0.732876\pi$
$257$	−5504.87	−1.33612	−0.668062	+	0.744105i	$0.732876\pi$
$258$	− 5043.55i	− 1.21705i
$259$	6120.27	1.46832
$260$	0	0
$261$	367.745	0.0872139
$262$	1439.35i	0.339402i
$263$	2032.44	0.476522	0.238261	−	0.971201i	$-0.423423\pi$
$263$	2032.44	0.476522	0.238261	+	0.971201i	$0.423423\pi$
$264$	−4419.29	−1.03026
$265$	− 5303.70i	− 1.22945i
$266$	12003.4i	2.76682i
$267$	− 1281.56i	− 0.293745i
$268$	− 14986.0i	− 3.41572i
$269$	5523.82	1.25202	0.626009	−	0.779816i	$-0.284687\pi$
$269$	5523.82	1.25202	0.626009	+	0.779816i	$0.284687\pi$
$270$	1972.71	0.444650
$271$	− 3918.69i	− 0.878389i	−0.898392	−	0.439194i	$-0.855264\pi$
$271$	− 3918.69i	− 0.878389i	0.898392	−	0.439194i	$-0.144736\pi$
$272$	17761.7	3.95941
$273$	0	0
$274$	−7227.12	−1.59345
$275$	998.013i	0.218845i
$276$	2507.51	0.546864
$277$	2944.56	0.638705	0.319353	−	0.947636i	$-0.396535\pi$
$277$	2944.56	0.638705	0.319353	+	0.947636i	$0.396535\pi$
$278$	35.3002i	0.00761570i
$279$	54.4988i	0.0116945i
$280$	22061.6i	4.70869i
$281$	5048.76i	1.07183i	0.844273	+	0.535914i	$0.180033\pi$
$281$	5048.76i	1.07183i	−0.844273	+	0.535914i	$0.819967\pi$
$282$	−5708.50	−1.20545
$283$	−4491.66	−0.943469	−0.471734	−	0.881741i	$-0.656372\pi$
$283$	−4491.66	−0.943469	−0.471734	+	0.881741i	$0.656372\pi$
$284$	21161.6i	4.42151i
$285$	−4080.84	−0.848169
$286$	0	0
$287$	−7341.90	−1.51003
$288$	− 6648.21i	− 1.36024i
$289$	240.893	0.0490318
$290$	−2985.41	−0.604515
$291$	2094.25i	0.421880i
$292$	− 5973.45i	− 1.19716i
$293$	4754.86i	0.948061i	0.880508	+	0.474031i	$0.157201\pi$
$293$	4754.86i	0.948061i	−0.880508	+	0.474031i	$0.842799\pi$
$294$	− 1908.43i	− 0.378578i
$295$	−2776.88	−0.548054
$296$	22088.9	4.33747
$297$	514.403i	0.100501i
$298$	−3007.95	−0.584718
$299$	0	0
$300$	−3472.17	−0.668220
$301$	− 6565.38i	− 1.25722i
$302$	8151.40	1.55318
$303$	266.209	0.0504730
$304$	25269.0i	4.76736i
$305$	− 7284.40i	− 1.36755i
$306$	− 3544.49i	− 0.662173i
$307$	1623.31i	0.301783i	0.988550	+	0.150891i	$0.0482143\pi$
$307$	1623.31i	0.301783i	−0.988550	+	0.150891i	$0.951786\pi$
$308$	−9018.02	−1.66834
$309$	−4282.84	−0.788486
$310$	− 442.430i	− 0.0810592i
$311$	−6683.49	−1.21860	−0.609302	−	0.792938i	$-0.708550\pi$
$311$	−6683.49	−1.21860	−0.609302	+	0.792938i	$0.708550\pi$
$312$	0	0
$313$	1584.55	0.286147	0.143073	−	0.989712i	$-0.454301\pi$
$313$	1584.55	0.286147	0.143073	+	0.989712i	$0.454301\pi$
$314$	− 11697.8i	− 2.10237i
$315$	2567.95	0.459327
$316$	22820.6	4.06252
$317$	− 787.932i	− 0.139605i	−0.997561	−	0.0698023i	$-0.977763\pi$
$317$	− 787.932i	− 0.139605i	0.997561	−	0.0698023i	$-0.0222368\pi$
$318$	6553.70i	1.15570i
$319$	− 778.474i	− 0.136634i
$320$	27610.2i	4.82330i
$321$	45.0700	0.00783664
$322$	4446.00	0.769459
$323$	7332.28i	1.26309i
$324$	−1789.65	−0.306868
$325$	0	0
$326$	2636.67	0.447951
$327$	6160.69i	1.04186i
$328$	−26498.0	−4.46069
$329$	−7430.97	−1.24524
$330$	− 4176.01i	− 0.696611i
$331$	9493.56i	1.57647i	0.615371	+	0.788237i	$0.289006\pi$
$331$	9493.56i	1.57647i	−0.615371	+	0.788237i	$0.710994\pi$
$332$	− 23550.5i	− 3.89308i
$333$	− 2571.14i	− 0.423115i
$334$	−16017.1	−2.62399
$335$	9033.55	1.47330
$336$	− 15901.1i	− 2.58177i
$337$	4123.06	0.666461	0.333230	−	0.942845i	$-0.391861\pi$
$337$	4123.06	0.666461	0.333230	+	0.942845i	$0.391861\pi$
$338$	0	0
$339$	−2152.37	−0.344840
$340$	21125.6i	3.36969i
$341$	115.368	0.0183211
$342$	5042.63	0.797294
$343$	4863.94i	0.765680i
$344$	− 23695.4i	− 3.71386i
$345$	1511.53i	0.235878i
$346$	− 12834.8i	− 1.99423i
$347$	2320.76	0.359034	0.179517	−	0.983755i	$-0.442546\pi$
$347$	2320.76	0.359034	0.179517	+	0.983755i	$0.442546\pi$
$348$	2708.38	0.417196
$349$	3818.85i	0.585726i	0.956154	+	0.292863i	$0.0946079\pi$
$349$	3818.85i	0.585726i	−0.956154	+	0.292863i	$0.905392\pi$
$350$	−6156.41	−0.940212
$351$	0	0
$352$	−14073.5	−2.13102
$353$	6065.53i	0.914549i	0.889326	+	0.457274i	$0.151174\pi$
$353$	6065.53i	0.914549i	−0.889326	+	0.457274i	$0.848826\pi$
$354$	3431.34	0.515180
$355$	−12756.2	−1.90712
$356$	− 9438.43i	− 1.40516i
$357$	− 4613.99i	− 0.684029i
$358$	− 25008.4i	− 3.69199i
$359$	7406.02i	1.08879i	0.838830	+	0.544394i	$0.183240\pi$
$359$	7406.02i	1.08879i	−0.838830	+	0.544394i	$0.816760\pi$
$360$	9268.11	1.35687
$361$	−3572.42	−0.520836
$362$	− 19335.2i	− 2.80727i
$363$	−2904.07	−0.419901
$364$	0	0
$365$	3600.80	0.516368
$366$	9001.22i	1.28552i
$367$	−4754.23	−0.676209	−0.338105	−	0.941109i	$-0.609786\pi$
$367$	−4754.23	−0.676209	−0.338105	+	0.941109i	$0.609786\pi$
$368$	9359.55	1.32582
$369$	3084.35i	0.435134i
$370$	20872.9i	2.93279i
$371$	8531.20i	1.19385i
$372$	401.374i	0.0559416i
$373$	−4684.13	−0.650229	−0.325114	−	0.945675i	$-0.605403\pi$
$373$	−4684.13	−0.650229	−0.325114	+	0.945675i	$0.605403\pi$
$374$	−7503.27	−1.03739
$375$	2901.43i	0.399544i
$376$	−26819.4	−3.67847
$377$	0	0
$378$	−3173.18	−0.431775
$379$	− 12103.9i	− 1.64047i	−0.572030	−	0.820233i	$-0.693844\pi$
$379$	− 12103.9i	− 1.64047i	0.572030	−	0.820233i	$-0.306156\pi$
$380$	−30054.7	−4.05730
$381$	−352.919	−0.0474556
$382$	− 18340.6i	− 2.45651i
$383$	5601.56i	0.747327i	0.927564	+	0.373663i	$0.121898\pi$
$383$	5601.56i	0.747327i	−0.927564	+	0.373663i	$0.878102\pi$
$384$	− 16388.9i	− 2.17797i
$385$	− 5436.06i	− 0.719604i
$386$	18330.0	2.41703
$387$	−2758.13	−0.362283
$388$	15423.8i	2.01810i
$389$	−9450.46	−1.23177	−0.615883	−	0.787837i	$-0.711201\pi$
$389$	−9450.46	−1.23177	−0.615883	+	0.787837i	$0.711201\pi$
$390$	0	0
$391$	2715.85	0.351270
$392$	− 8966.11i	− 1.15525i
$393$	787.127	0.101031
$394$	−20045.0	−2.56308
$395$	13756.2i	1.75228i
$396$	3788.49i	0.480754i
$397$	− 2723.98i	− 0.344364i	−0.985065	−	0.172182i	$-0.944918\pi$
$397$	− 2723.98i	− 0.344364i	0.985065	−	0.172182i	$-0.0550817\pi$
$398$	6180.07i	0.778339i
$399$	6564.18	0.823610
$400$	−12960.2	−1.62003
$401$	4680.46i	0.582870i	0.956591	+	0.291435i	$0.0941328\pi$
$401$	4680.46i	0.582870i	−0.956591	+	0.291435i	$0.905867\pi$
$402$	−11162.6	−1.38493
$403$	0	0
$404$	1960.58	0.241442
$405$	− 1078.80i	− 0.132361i
$406$	4802.15	0.587011
$407$	−5442.80	−0.662874
$408$	− 16652.6i	− 2.02065i
$409$	3272.70i	0.395659i	0.980236	+	0.197830i	$0.0633893\pi$
$409$	3272.70i	0.395659i	−0.980236	+	0.197830i	$0.936611\pi$
$410$	− 25039.2i	− 3.01609i
$411$	3952.24i	0.474330i
$412$	−31542.3	−3.77179
$413$	4466.71	0.532185
$414$	− 1867.77i	− 0.221729i
$415$	14196.2	1.67920
$416$	0	0
$417$	19.3043	0.00226700
$418$	− 10674.7i	− 1.24908i
$419$	1501.48	0.175065	0.0875323	−	0.996162i	$-0.472102\pi$
$419$	1501.48	0.175065	0.0875323	+	0.996162i	$0.472102\pi$
$420$	18912.5	2.19723
$421$	16578.1i	1.91916i	0.281436	+	0.959580i	$0.409189\pi$
$421$	16578.1i	1.91916i	−0.281436	+	0.959580i	$0.590811\pi$
$422$	408.828i	0.0471598i
$423$	3121.76i	0.358831i
$424$	30790.3i	3.52667i
$425$	−3760.66	−0.429221
$426$	15762.6	1.79273
$427$	11717.2i	1.32795i
$428$	331.933	0.0374873
$429$	0	0
$430$	22390.9	2.51113
$431$	3776.55i	0.422065i	0.977479	+	0.211032i	$0.0676826\pi$
$431$	3776.55i	0.422065i	−0.977479	+	0.211032i	$0.932317\pi$
$432$	−6680.06	−0.743969
$433$	709.953	0.0787948	0.0393974	−	0.999224i	$-0.487456\pi$
$433$	709.953	0.0787948	0.0393974	+	0.999224i	$0.487456\pi$
$434$	711.665i	0.0787120i
$435$	1632.61i	0.179949i
$436$	45372.4i	4.98382i
$437$	3863.76i	0.422949i
$438$	−4449.45	−0.485395
$439$	16262.2	1.76800	0.884002	−	0.467484i	$-0.154839\pi$
$439$	16262.2	1.76800	0.884002	+	0.467484i	$0.154839\pi$
$440$	− 19619.5i	− 2.12574i
$441$	−1043.65	−0.112693
$442$	0	0
$443$	2899.45	0.310964	0.155482	−	0.987839i	$-0.450307\pi$
$443$	2899.45	0.310964	0.155482	+	0.987839i	$0.450307\pi$
$444$	− 18936.0i	− 2.02401i
$445$	5689.49	0.606085
$446$	17437.4	1.85131
$447$	1644.94i	0.174055i
$448$	− 44412.0i	− 4.68363i
$449$	7817.78i	0.821701i	0.911703	+	0.410851i	$0.134768\pi$
$449$	7817.78i	0.821701i	−0.911703	+	0.410851i	$0.865232\pi$
$450$	2586.32i	0.270934i
$451$	6529.20	0.681703
$452$	−15851.8	−1.64957
$453$	− 4457.69i	− 0.462341i
$454$	−7404.57	−0.765448
$455$	0	0
$456$	23691.1	2.43297
$457$	− 14452.9i	− 1.47939i	−0.672943	−	0.739694i	$-0.734970\pi$
$457$	− 14452.9i	− 1.47939i	0.672943	−	0.739694i	$-0.265030\pi$
$458$	26449.8	2.69851
$459$	−1938.35	−0.197112
$460$	11132.1i	1.12834i
$461$	2891.46i	0.292123i	0.989276	+	0.146061i	$0.0466597\pi$
$461$	2891.46i	0.292123i	−0.989276	+	0.146061i	$0.953340\pi$
$462$	6717.26i	0.676440i
$463$	− 9223.83i	− 0.925848i	−0.886398	−	0.462924i	$-0.846800\pi$
$463$	− 9223.83i	− 0.925848i	0.886398	−	0.462924i	$-0.153200\pi$
$464$	10109.3	1.01145
$465$	−241.948	−0.0241292
$466$	− 13170.8i	− 1.30929i
$467$	7988.25	0.791546	0.395773	−	0.918348i	$-0.370477\pi$
$467$	7988.25	0.791546	0.395773	+	0.918348i	$0.370477\pi$
$468$	0	0
$469$	−14530.8	−1.43064
$470$	− 25343.0i	− 2.48720i
$471$	−6397.09	−0.625822
$472$	16121.0	1.57209
$473$	5838.64i	0.567570i
$474$	− 16998.4i	− 1.64718i
$475$	− 5350.18i	− 0.516806i
$476$	− 33981.2i	− 3.27212i
$477$	3583.97	0.344023
$478$	−10318.1	−0.987315
$479$	6908.64i	0.659006i	0.944155	+	0.329503i	$0.106881\pi$
$479$	6908.64i	0.659006i	−0.944155	+	0.329503i	$0.893119\pi$
$480$	29514.8	2.80659
$481$	0	0
$482$	−29815.6	−2.81756
$483$	− 2431.35i	− 0.229048i
$484$	−21387.9	−2.00863
$485$	−9297.47	−0.870466
$486$	1333.06i	0.124421i
$487$	13455.0i	1.25196i	0.779841	+	0.625978i	$0.215300\pi$
$487$	13455.0i	1.25196i	−0.779841	+	0.625978i	$0.784700\pi$
$488$	42289.1i	3.92282i
$489$	− 1441.90i	− 0.133343i
$490$	8472.52	0.781121
$491$	1044.02	0.0959589	0.0479795	−	0.998848i	$-0.484722\pi$
$491$	1044.02	0.0959589	0.0479795	+	0.998848i	$0.484722\pi$
$492$	22715.7i	2.08151i
$493$	2933.40	0.267979
$494$	0	0
$495$	−2283.70	−0.207363
$496$	1498.17i	0.135625i
$497$	20518.8	1.85190
$498$	−17542.1	−1.57847
$499$	− 16190.7i	− 1.45249i	−0.687435	−	0.726246i	$-0.741263\pi$
$499$	− 16190.7i	− 1.45249i	0.687435	−	0.726246i	$-0.258737\pi$
$500$	21368.5i	1.91126i
$501$	8759.12i	0.781095i
$502$	6894.04i	0.612940i
$503$	−11854.8	−1.05086	−0.525429	−	0.850838i	$-0.676095\pi$
$503$	−11854.8	−1.05086	−0.525429	+	0.850838i	$0.676095\pi$
$504$	−14908.1	−1.31758
$505$	1181.84i	0.104141i
$506$	−3953.86	−0.347372
$507$	0	0
$508$	−2599.19	−0.227008
$509$	6647.07i	0.578834i	0.957203	+	0.289417i	$0.0934614\pi$
$509$	6647.07i	0.578834i	−0.957203	+	0.289417i	$0.906539\pi$
$510$	15735.8	1.36626
$511$	−5792.02	−0.501416
$512$	− 29721.4i	− 2.56545i
$513$	− 2757.63i	− 0.237334i
$514$	30198.8i	2.59146i
$515$	− 19013.7i	− 1.62688i
$516$	−20313.1	−1.73301
$517$	6608.42	0.562162
$518$	− 33574.8i	− 2.84786i
$519$	−7018.86	−0.593630
$520$	0	0
$521$	16839.6	1.41604	0.708019	−	0.706193i	$-0.249589\pi$
$521$	16839.6	1.41604	0.708019	+	0.706193i	$0.249589\pi$
$522$	− 2017.39i	− 0.169155i
$523$	10414.1	0.870701	0.435351	−	0.900261i	$-0.356624\pi$
$523$	10414.1	0.870701	0.435351	+	0.900261i	$0.356624\pi$
$524$	5797.05	0.483293
$525$	3366.71i	0.279877i
$526$	− 11149.6i	− 0.924233i
$527$	434.723i	0.0359332i
$528$	14140.9i	1.16554i
$529$	−10735.9	−0.882377
$530$	−29095.3	−2.38456
$531$	− 1876.47i	− 0.153356i
$532$	48344.1	3.93981
$533$	0	0
$534$	−7030.42	−0.569730
$535$	200.089i	0.0161694i
$536$	−52443.7	−4.22616
$537$	−13676.1	−1.09901
$538$	− 30302.8i	− 2.42834i
$539$	2209.29i	0.176550i
$540$	− 7945.19i	− 0.633160i
$541$	5464.03i	0.434227i	0.976146	+	0.217114i	$0.0696642\pi$
$541$	5464.03i	0.434227i	−0.976146	+	0.217114i	$0.930336\pi$
$542$	−21497.3	−1.70367
$543$	−10573.7	−0.835653
$544$	− 53031.0i	− 4.17957i
$545$	−27350.5	−2.14966
$546$	0	0
$547$	24902.3	1.94652	0.973261	−	0.229702i	$-0.0737753\pi$
$547$	24902.3	1.94652	0.973261	+	0.229702i	$0.0737753\pi$
$548$	29107.5i	2.26900i
$549$	4922.43	0.382667
$550$	5474.94	0.424459
$551$	4173.26i	0.322662i
$552$	− 8775.08i	− 0.676617i
$553$	− 22127.4i	− 1.70154i
$554$	− 16153.4i	− 1.23879i
$555$	11414.6	0.873014
$556$	142.173	0.0108444
$557$	9817.71i	0.746840i	0.927662	+	0.373420i	$0.121815\pi$
$557$	9817.71i	0.746840i	−0.927662	+	0.373420i	$0.878185\pi$
$558$	298.972	0.0226819
$559$	0	0
$560$	70593.0	5.32696
$561$	4103.26i	0.308805i
$562$	27696.7	2.07885
$563$	23333.7	1.74671	0.873356	−	0.487083i	$-0.161939\pi$
$563$	23333.7	1.74671	0.873356	+	0.487083i	$0.161939\pi$
$564$	22991.2i	1.71650i
$565$	− 9555.48i	− 0.711508i
$566$	24640.6i	1.82989i
$567$	1735.29i	0.128528i
$568$	74055.4	5.47059
$569$	−24543.9	−1.80832	−0.904160	−	0.427194i	$-0.859502\pi$
$569$	−24543.9	−1.80832	−0.904160	+	0.427194i	$0.859502\pi$
$570$	22386.9i	1.64506i
$571$	10562.5	0.774126	0.387063	−	0.922053i	$-0.373490\pi$
$571$	10562.5	0.774126	0.387063	+	0.922053i	$0.373490\pi$
$572$	0	0
$573$	−10029.8	−0.731241
$574$	40276.5i	2.92876i
$575$	−1981.69	−0.143725
$576$	−18657.5	−1.34965
$577$	− 18922.5i	− 1.36526i	−0.730765	−	0.682629i	$-0.760837\pi$
$577$	− 18922.5i	− 1.36526i	0.730765	−	0.682629i	$-0.239163\pi$
$578$	− 1321.50i	− 0.0950991i
$579$	− 10024.0i	− 0.719486i
$580$	12023.9i	0.860801i
$581$	−22835.2	−1.63057
$582$	11488.7	0.818253
$583$	− 7586.85i	− 0.538963i
$584$	−20904.2	−1.48120
$585$	0	0
$586$	26084.4	1.83880
$587$	− 7989.65i	− 0.561786i	−0.959739	−	0.280893i	$-0.909369\pi$
$587$	− 7989.65i	− 0.561786i	0.959739	−	0.280893i	$-0.0906306\pi$
$588$	−7686.29	−0.539077
$589$	−618.466	−0.0432656
$590$	15233.5i	1.06297i
$591$	10961.9i	0.762962i
$592$	− 70680.4i	− 4.90701i
$593$	25239.2i	1.74781i	0.486097	+	0.873905i	$0.338420\pi$
$593$	25239.2i	1.74781i	−0.486097	+	0.873905i	$0.661580\pi$
$594$	2821.93	0.194925
$595$	20483.9	1.41136
$596$	12114.7i	0.832610i
$597$	3379.65	0.231691
$598$	0	0
$599$	−7412.19	−0.505599	−0.252800	−	0.967519i	$-0.581351\pi$
$599$	−7412.19	−0.505599	−0.252800	+	0.967519i	$0.581351\pi$
$600$	12150.9i	0.826767i
$601$	−21459.2	−1.45647	−0.728236	−	0.685327i	$-0.759659\pi$
$601$	−21459.2	−1.45647	−0.728236	+	0.685327i	$0.759659\pi$
$602$	−36016.6	−2.43842
$603$	6104.42i	0.412257i
$604$	− 32830.1i	− 2.21165i
$605$	− 12892.7i	− 0.866382i
$606$	− 1460.38i	− 0.0978944i
$607$	−18246.7	−1.22011	−0.610057	−	0.792358i	$-0.708853\pi$
$607$	−18246.7	−1.22011	−0.610057	+	0.792358i	$0.708853\pi$
$608$	75445.6	5.03244
$609$	− 2626.11i	− 0.174738i
$610$	−39961.1	−2.65242
$611$	0	0
$612$	−14275.6	−0.942902
$613$	− 24087.5i	− 1.58709i	−0.608512	−	0.793545i	$-0.708233\pi$
$613$	− 24087.5i	− 1.58709i	0.608512	−	0.793545i	$-0.291767\pi$
$614$	8905.23	0.585319
$615$	−13693.0	−0.897813
$616$	31558.7i	2.06418i
$617$	− 14400.6i	− 0.939621i	−0.882767	−	0.469811i	$-0.844322\pi$
$617$	− 14400.6i	− 0.939621i	0.882767	−	0.469811i	$-0.155678\pi$
$618$	23495.0i	1.52930i
$619$	− 25635.8i	− 1.66460i	−0.554322	−	0.832302i	$-0.687022\pi$
$619$	− 25635.8i	− 1.66460i	0.554322	−	0.832302i	$-0.312978\pi$
$620$	−1781.91	−0.115424
$621$	−1021.41	−0.0660031
$622$	36664.6i	2.36353i
$623$	−9151.76	−0.588535
$624$	0	0
$625$	−19428.9	−1.24345
$626$	− 8692.58i	− 0.554993i
$627$	−5837.58	−0.371819
$628$	−47113.4	−2.99368
$629$	− 20509.3i	− 1.30009i
$630$	− 14087.4i	− 0.890881i
$631$	2757.17i	0.173948i	0.996211	+	0.0869741i	$0.0277197\pi$
$631$	2757.17i	0.173948i	−0.996211	+	0.0869741i	$0.972280\pi$
$632$	− 79861.0i	− 5.02643i
$633$	223.573	0.0140383
$634$	−4322.47	−0.270768
$635$	− 1566.79i	− 0.0979152i
$636$	26395.3	1.64566
$637$	0	0
$638$	−4270.58	−0.265006
$639$	− 8620.00i	− 0.533649i
$640$	72758.7	4.49382
$641$	−14317.0	−0.882196	−0.441098	−	0.897459i	$-0.645411\pi$
$641$	−14317.0	−0.882196	−0.441098	+	0.897459i	$0.645411\pi$
$642$	− 247.247i	− 0.0151995i
$643$	− 51.9171i	− 0.00318415i	−0.999999	−	0.00159208i	$-0.999493\pi$
$643$	− 51.9171i	− 0.00318415i	0.999999	−	0.00159208i	$-0.000506774\pi$
$644$	− 17906.5i	− 1.09567i
$645$	− 12244.7i	− 0.747498i
$646$	40223.8	2.44982
$647$	9735.07	0.591538	0.295769	−	0.955260i	$-0.404424\pi$
$647$	9735.07	0.591538	0.295769	+	0.955260i	$0.404424\pi$
$648$	6262.92i	0.379677i
$649$	−3972.27	−0.240255
$650$	0	0
$651$	389.183	0.0234305
$652$	− 10619.3i	− 0.637860i
$653$	10842.6	0.649778	0.324889	−	0.945752i	$-0.394673\pi$
$653$	10842.6	0.649778	0.324889	+	0.945752i	$0.394673\pi$
$654$	33796.6	2.02072
$655$	3494.46i	0.208458i
$656$	84788.5i	5.04640i
$657$	2433.24i	0.144490i
$658$	40765.1i	2.41518i
$659$	28940.9	1.71074	0.855369	−	0.518020i	$-0.173331\pi$
$659$	28940.9	1.71074	0.855369	+	0.518020i	$0.173331\pi$
$660$	−16819.0	−0.991940
$661$	3108.58i	0.182919i	0.995809	+	0.0914597i	$0.0291533\pi$
$661$	3108.58i	0.182919i	−0.995809	+	0.0914597i	$0.970847\pi$
$662$	52080.2	3.05763
$663$	0	0
$664$	−82415.4	−4.81677
$665$	29141.8i	1.69935i
$666$	−14104.8	−0.820648
$667$	1545.76	0.0897333
$668$	64509.4i	3.73644i
$669$	− 9535.86i	− 0.551088i
$670$	− 49556.6i	− 2.85752i
$671$	− 10420.2i	− 0.599505i
$672$	−47475.7	−2.72532
$673$	9950.09	0.569908	0.284954	−	0.958541i	$-0.408022\pi$
$673$	9950.09	0.569908	0.284954	+	0.958541i	$0.408022\pi$
$674$	− 22618.4i	− 1.29263i
$675$	1414.36	0.0806501
$676$	0	0
$677$	−3483.60	−0.197763	−0.0988816	−	0.995099i	$-0.531527\pi$
$677$	−3483.60	−0.197763	−0.0988816	+	0.995099i	$0.531527\pi$
$678$	11807.6i	0.668830i
$679$	14955.3	0.845261
$680$	73929.3	4.16920
$681$	4049.28i	0.227854i
$682$	− 632.889i	− 0.0355346i
$683$	− 16811.4i	− 0.941829i	−0.882179	−	0.470915i	$-0.843924\pi$
$683$	− 16811.4i	− 0.941829i	0.882179	−	0.470915i	$-0.156076\pi$
$684$	− 20309.4i	− 1.13531i
$685$	−17546.0	−0.978685
$686$	26682.8	1.48506
$687$	− 14464.4i	− 0.803276i
$688$	−75820.8	−4.20151
$689$	0	0
$690$	8292.01	0.457495
$691$	3078.78i	0.169497i	0.996402	+	0.0847484i	$0.0270086\pi$
$691$	3078.78i	0.169497i	−0.996402	+	0.0847484i	$0.972991\pi$
$692$	−51692.6	−2.83968
$693$	3673.42	0.201359
$694$	− 12731.3i	− 0.696361i
$695$	85.7020i	0.00467750i
$696$	− 9478.01i	− 0.516183i
$697$	24603.0i	1.33702i
$698$	20949.6	1.13604
$699$	−7202.63	−0.389740
$700$	24795.2i	1.33882i
$701$	18608.6	1.00262	0.501311	−	0.865267i	$-0.332851\pi$
$701$	18608.6	1.00262	0.501311	+	0.865267i	$0.332851\pi$
$702$	0	0
$703$	29177.9	1.56539
$704$	39495.9i	2.11443i
$705$	−13859.1	−0.740376
$706$	33274.5	1.77380
$707$	− 1901.03i	− 0.101126i
$708$	− 13819.9i	− 0.733591i
$709$	19046.2i	1.00888i	0.863448	+	0.504439i	$0.168301\pi$
$709$	19046.2i	1.00888i	−0.863448	+	0.504439i	$0.831699\pi$
$710$	69978.5i	3.69894i
$711$	−9295.77	−0.490322
$712$	−33030.0	−1.73855
$713$	229.078i	0.0120323i
$714$	−25311.6	−1.32670
$715$	0	0
$716$	−100722.	−5.25722
$717$	5642.55i	0.293898i
$718$	40628.2	2.11174
$719$	14013.4	0.726859	0.363430	−	0.931622i	$-0.381606\pi$
$719$	14013.4	0.726859	0.363430	+	0.931622i	$0.381606\pi$
$720$	− 29656.2i	− 1.53503i
$721$	30584.3i	1.57978i
$722$	19597.7i	1.01018i
$723$	16305.0i	0.838715i
$724$	−77873.2	−3.99742
$725$	−2140.43	−0.109646
$726$	15931.3i	0.814414i
$727$	−2578.98	−0.131567	−0.0657834	−	0.997834i	$-0.520955\pi$
$727$	−2578.98	−0.131567	−0.0657834	+	0.997834i	$0.520955\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	− 19753.4i	− 1.00152i
$731$	−22000.8	−1.11317
$732$	36252.8	1.83052
$733$	− 5029.12i	− 0.253417i	−0.991940	−	0.126709i	$-0.959559\pi$
$733$	− 5029.12i	− 0.253417i	0.991940	−	0.126709i	$-0.0404413\pi$
$734$	26081.0i	1.31153i
$735$	− 4633.30i	− 0.232520i
$736$	− 27944.8i	− 1.39954i
$737$	12922.3	0.645863
$738$	16920.2	0.843960
$739$	− 23856.4i	− 1.18751i	−0.804644	−	0.593757i	$-0.797644\pi$
$739$	− 23856.4i	− 1.18751i	0.804644	−	0.593757i	$-0.202356\pi$
$740$	84066.5	4.17614
$741$	0	0
$742$	46800.8	2.31551
$743$	− 4887.62i	− 0.241332i	−0.992693	−	0.120666i	$-0.961497\pi$
$743$	− 4887.62i	− 0.241332i	0.992693	−	0.120666i	$-0.0385030\pi$
$744$	1404.62	0.0692147
$745$	−7302.72	−0.359129
$746$	25696.4i	1.26114i
$747$	9593.10i	0.469870i
$748$	30219.8i	1.47720i
$749$	− 321.851i	− 0.0157012i
$750$	15916.8	0.774932
$751$	−24814.3	−1.20571	−0.602854	−	0.797851i	$-0.705970\pi$
$751$	−24814.3	−1.20571	−0.602854	+	0.797851i	$0.705970\pi$
$752$	85817.2i	4.16148i
$753$	3770.09	0.182456
$754$	0	0
$755$	19790.0	0.953949
$756$	12780.1i	0.614826i
$757$	−6552.91	−0.314623	−0.157311	−	0.987549i	$-0.550283\pi$
$757$	−6552.91	−0.314623	−0.157311	+	0.987549i	$0.550283\pi$
$758$	−66400.2	−3.18175
$759$	2162.22i	0.103404i
$760$	105177.i	5.01996i
$761$	20351.4i	0.969431i	0.874672	+	0.484715i	$0.161077\pi$
$761$	20351.4i	0.969431i	−0.874672	+	0.484715i	$0.838923\pi$
$762$	1936.06i	0.0920420i
$763$	43994.3	2.08742
$764$	−73867.7	−3.49796
$765$	− 8605.32i	− 0.406701i
$766$	30729.3	1.44947
$767$	0	0
$768$	−40153.4	−1.88660
$769$	22209.7i	1.04148i	0.853714	+	0.520742i	$0.174345\pi$
$769$	22209.7i	1.04148i	−0.853714	+	0.520742i	$0.825655\pi$
$770$	−29821.4	−1.39570
$771$	16514.6	0.771412
$772$	− 73824.9i	− 3.44173i
$773$	− 28496.6i	− 1.32594i	−0.748645	−	0.662971i	$-0.769295\pi$
$773$	− 28496.6i	− 1.32594i	0.748645	−	0.662971i	$-0.230705\pi$
$774$	15130.6i	0.702661i
$775$	− 317.206i	− 0.0147024i
$776$	53975.9	2.49693
$777$	−18360.8	−0.847735
$778$	51843.7i	2.38906i
$779$	−35001.9	−1.60985
$780$	0	0
$781$	−18247.5	−0.836041
$782$	− 14898.7i	− 0.681301i
$783$	−1103.23	−0.0503530
$784$	−28689.9	−1.30694
$785$	− 28400.0i	− 1.29126i
$786$	− 4318.05i	− 0.195954i
$787$	− 2394.07i	− 0.108436i	−0.998529	−	0.0542181i	$-0.982733\pi$
$787$	− 2394.07i	− 0.108436i	0.998529	−	0.0542181i	$-0.0172666\pi$
$788$	80732.2i	3.64970i
$789$	−6097.31	−0.275120
$790$	75464.6	3.39862
$791$	15370.3i	0.690906i
$792$	13257.9	0.594821
$793$	0	0
$794$	−14943.3	−0.667907
$795$	15911.1i	0.709822i
$796$	24890.5	1.10832
$797$	37996.7	1.68872	0.844362	−	0.535773i	$-0.179980\pi$
$797$	37996.7	1.68872	0.844362	+	0.535773i	$0.179980\pi$
$798$	− 36010.1i	− 1.59742i
$799$	24901.5i	1.10257i
$800$	38695.4i	1.71011i
$801$	3844.67i	0.169594i
$802$	25676.3	1.13050
$803$	5150.88	0.226364
$804$	44957.9i	1.97207i
$805$	10794.0	0.472595
$806$	0	0
$807$	−16571.4	−0.722853
$808$	− 6861.10i	− 0.298729i
$809$	−25993.4	−1.12964	−0.564821	−	0.825213i	$-0.691055\pi$
$809$	−25993.4	−1.12964	−0.564821	+	0.825213i	$0.691055\pi$
$810$	−5918.14	−0.256719
$811$	15524.4i	0.672176i	0.941831	+	0.336088i	$0.109104\pi$
$811$	15524.4i	0.672176i	−0.941831	+	0.336088i	$0.890896\pi$
$812$	− 19340.9i	− 0.835875i
$813$	11756.1i	0.507138i
$814$	29858.3i	1.28567i
$815$	6401.33	0.275127
$816$	−53285.1	−2.28597
$817$	− 31299.9i	− 1.34033i
$818$	17953.5	0.767396
$819$	0	0
$820$	−100847.	−4.29477
$821$	2029.40i	0.0862684i	0.999069	+	0.0431342i	$0.0137343\pi$
$821$	2029.40i	0.0862684i	−0.999069	+	0.0431342i	$0.986266\pi$
$822$	21681.3	0.919980
$823$	42010.4	1.77933	0.889667	−	0.456610i	$-0.150936\pi$
$823$	42010.4	1.77933	0.889667	+	0.456610i	$0.150936\pi$
$824$	110383.i	4.66672i
$825$	− 2994.04i	− 0.126350i
$826$	− 24503.6i	− 1.03219i
$827$	3941.24i	0.165720i	0.996561	+	0.0828599i	$0.0264054\pi$
$827$	3941.24i	0.165720i	−0.996561	+	0.0828599i	$0.973595\pi$
$828$	−7522.53	−0.315732
$829$	−11264.7	−0.471941	−0.235970	−	0.971760i	$-0.575827\pi$
$829$	−11264.7	−0.471941	−0.235970	+	0.971760i	$0.575827\pi$
$830$	− 77878.4i	− 3.25686i
$831$	−8833.68	−0.368757
$832$	0	0
$833$	−8324.92	−0.346268
$834$	− 105.901i	− 0.00439693i
$835$	−38886.3	−1.61163
$836$	−42992.7	−1.77863
$837$	− 163.496i	− 0.00675181i
$838$	− 8236.88i	− 0.339544i
$839$	35221.1i	1.44931i	0.689114	+	0.724653i	$0.258000\pi$
$839$	35221.1i	1.44931i	−0.689114	+	0.724653i	$0.742000\pi$
$840$	− 66184.8i	− 2.71856i
$841$	−22719.4	−0.931544
$842$	90944.8	3.72228
$843$	− 15146.3i	− 0.618820i
$844$	1646.57	0.0671533
$845$	0	0
$846$	17125.5	0.695966
$847$	20738.3i	0.841295i
$848$	98523.2	3.98974
$849$	13475.0	0.544712
$850$	20630.4i	0.832490i
$851$	− 10807.4i	− 0.435338i
$852$	− 63484.7i	− 2.55276i
$853$	− 37662.7i	− 1.51178i	−0.654701	−	0.755888i	$-0.727205\pi$
$853$	− 37662.7i	− 1.51178i	0.654701	−	0.755888i	$-0.272795\pi$
$854$	64278.8	2.57562
$855$	12242.5	0.489691
$856$	− 1161.60i	− 0.0463818i
$857$	−800.368	−0.0319020	−0.0159510	−	0.999873i	$-0.505078\pi$
$857$	−800.368	−0.0319020	−0.0159510	+	0.999873i	$0.505078\pi$
$858$	0	0
$859$	−8800.69	−0.349564	−0.174782	−	0.984607i	$-0.555922\pi$
$859$	−8800.69	−0.349564	−0.174782	+	0.984607i	$0.555922\pi$
$860$	− 90180.4i	− 3.57573i
$861$	22025.7	0.871816
$862$	20717.6	0.818611
$863$	− 17991.0i	− 0.709641i	−0.934934	−	0.354821i	$-0.884542\pi$
$863$	− 17991.0i	− 0.709641i	0.934934	−	0.354821i	$-0.115458\pi$
$864$	19944.6i	0.785336i
$865$	− 31160.3i	− 1.22484i
$866$	− 3894.69i	− 0.152826i
$867$	−722.680	−0.0283085
$868$	2866.26	0.112082
$869$	19678.1i	0.768162i
$870$	8956.24	0.349017
$871$	0	0
$872$	158782.	6.16631
$873$	− 6282.76i	− 0.243573i
$874$	21196.0	0.820325
$875$	20719.5	0.800510
$876$	17920.4i	0.691179i
$877$	− 44304.5i	− 1.70588i	−0.522008	−	0.852940i	$-0.674817\pi$
$877$	− 44304.5i	− 1.70588i	0.522008	−	0.852940i	$-0.325183\pi$
$878$	− 89212.0i	− 3.42911i
$879$	− 14264.6i	− 0.547363i
$880$	−62778.8	−2.40486
$881$	−10814.6	−0.413567	−0.206783	−	0.978387i	$-0.566300\pi$
$881$	−10814.6	−0.413567	−0.206783	+	0.978387i	$0.566300\pi$
$882$	5725.30i	0.218572i
$883$	−14530.8	−0.553793	−0.276897	−	0.960900i	$-0.589306\pi$
$883$	−14530.8	−0.553793	−0.276897	+	0.960900i	$0.589306\pi$
$884$	0	0
$885$	8330.63	0.316419
$886$	− 15905.9i	− 0.603126i
$887$	−3385.82	−0.128168	−0.0640838	−	0.997945i	$-0.520412\pi$
$887$	−3385.82	−0.128168	−0.0640838	+	0.997945i	$0.520412\pi$
$888$	−66266.8	−2.50424
$889$	2520.24i	0.0950800i
$890$	− 31211.7i	− 1.17552i
$891$	− 1543.21i	− 0.0580241i
$892$	− 70229.9i	− 2.63618i
$893$	−35426.6	−1.32755
$894$	9023.86	0.337587
$895$	− 60715.4i	− 2.26759i
$896$	−117035.	−4.36369
$897$	0	0
$898$	42887.1	1.59372
$899$	247.428i	0.00917929i
$900$	10416.5	0.385797
$901$	28588.4	1.05707
$902$	− 35818.2i	− 1.32219i
$903$	19696.1i	0.725854i
$904$	55473.7i	2.04096i
$905$	− 46942.0i	− 1.72420i
$906$	−24454.2	−0.896729
$907$	−38174.2	−1.39752	−0.698762	−	0.715354i	$-0.746265\pi$
$907$	−38174.2	−1.39752	−0.698762	+	0.715354i	$0.746265\pi$
$908$	29822.2i	1.08996i
$909$	−798.628	−0.0291406
$910$	0	0
$911$	−11699.5	−0.425489	−0.212745	−	0.977108i	$-0.568240\pi$
$911$	−11699.5	−0.425489	−0.212745	+	0.977108i	$0.568240\pi$
$912$	− 75807.0i	− 2.75244i
$913$	20307.5	0.736123
$914$	−79286.6	−2.86933
$915$	21853.2i	0.789557i
$916$	− 106528.i	− 3.84254i
$917$	− 5620.97i	− 0.202422i
$918$	10633.5i	0.382306i
$919$	21615.6	0.775879	0.387939	−	0.921685i	$-0.373187\pi$
$919$	21615.6	0.775879	0.387939	+	0.921685i	$0.373187\pi$
$920$	38957.1	1.39606
$921$	− 4869.93i	− 0.174234i
$922$	15862.1	0.566583
$923$	0	0
$924$	27054.1	0.963218
$925$	14965.1i	0.531945i
$926$	−50600.5	−1.79572
$927$	12848.5	0.455232
$928$	− 30183.3i	− 1.06769i
$929$	− 22325.9i	− 0.788471i	−0.919009	−	0.394236i	$-0.871009\pi$
$929$	− 22325.9i	− 0.788471i	0.919009	−	0.394236i	$-0.128991\pi$
$930$	1327.29i	0.0467995i
$931$	− 11843.6i	− 0.416926i
$932$	−53046.1	−1.86436
$933$	20050.5	0.703561
$934$	− 43822.3i	− 1.53523i
$935$	−18216.5	−0.637158
$936$	0	0
$937$	−29401.5	−1.02509	−0.512543	−	0.858661i	$-0.671297\pi$
$937$	−29401.5	−1.02509	−0.512543	+	0.858661i	$0.671297\pi$
$938$	79713.7i	2.77478i
$939$	−4753.65	−0.165207
$940$	−102070.	−3.54166
$941$	− 30280.7i	− 1.04902i	−0.851406	−	0.524508i	$-0.824249\pi$
$941$	− 30280.7i	− 1.04902i	0.851406	−	0.524508i	$-0.175751\pi$
$942$	35093.4i	1.21381i
$943$	12964.6i	0.447704i
$944$	− 51584.1i	− 1.77852i
$945$	−7703.86	−0.265192
$946$	32029.8	1.10082
$947$	25226.1i	0.865617i	0.901486	+	0.432809i	$0.142477\pi$
$947$	25226.1i	0.865617i	−0.901486	+	0.432809i	$0.857523\pi$
$948$	−68461.7	−2.34550
$949$	0	0
$950$	−29350.2	−1.00237
$951$	2363.80i	0.0806008i
$952$	−118918.	−4.04848
$953$	11893.7	0.404275	0.202137	−	0.979357i	$-0.435211\pi$
$953$	11893.7	0.404275	0.202137	+	0.979357i	$0.435211\pi$
$954$	− 19661.1i	− 0.667245i
$955$	− 44527.5i	− 1.50877i
$956$	41556.4i	1.40589i
$957$	2335.42i	0.0788855i
$958$	37899.7	1.27817
$959$	28223.4	0.950346
$960$	− 82830.5i	− 2.78473i
$961$	29754.3	0.998769
$962$	0	0
$963$	−135.210	−0.00452449
$964$	120084.i	4.01207i
$965$	44501.7	1.48452
$966$	−13338.0	−0.444247
$967$	− 8534.72i	− 0.283824i	−0.989879	−	0.141912i	$-0.954675\pi$
$967$	− 8534.72i	− 0.283824i	0.989879	−	0.141912i	$-0.0453250\pi$
$968$	74847.5i	2.48522i
$969$	− 21996.9i	− 0.729247i
$970$	51004.4i	1.68830i
$971$	25615.3	0.846585	0.423292	−	0.905993i	$-0.360874\pi$
$971$	25615.3	0.846585	0.423292	+	0.905993i	$0.360874\pi$
$972$	5368.95	0.177170
$973$	− 137.855i	− 0.00454206i
$974$	73811.8	2.42822
$975$	0	0
$976$	135317.	4.43791
$977$	22995.5i	0.753011i	0.926414	+	0.376506i	$0.122874\pi$
$977$	22995.5i	0.753011i	−0.926414	+	0.376506i	$0.877126\pi$
$978$	−7910.02	−0.258624
$979$	8138.72	0.265694
$980$	− 34123.4i	− 1.11228i
$981$	− 18482.1i	− 0.601516i
$982$	− 5727.31i	− 0.186116i
$983$	− 20592.1i	− 0.668144i	−0.942548	−	0.334072i	$-0.891577\pi$
$983$	− 20592.1i	− 0.668144i	0.942548	−	0.334072i	$-0.108423\pi$
$984$	79493.9	2.57538
$985$	−48665.4	−1.57422
$986$	− 16092.2i	− 0.519756i
$987$	22292.9	0.718937
$988$	0	0
$989$	−11593.4	−0.372748
$990$	12528.0i	0.402189i
$991$	−11557.9	−0.370485	−0.185242	−	0.982693i	$-0.559307\pi$
$991$	−11557.9	−0.370485	−0.185242	+	0.982693i	$0.559307\pi$
$992$	4473.08	0.143166
$993$	− 28480.7i	− 0.910178i
$994$	− 112563.i	− 3.59183i
$995$	15004.0i	0.478049i
$996$	70651.5i	2.24767i
$997$	−15481.5	−0.491778	−0.245889	−	0.969298i	$-0.579080\pi$
$997$	−15481.5	−0.491778	−0.245889	+	0.969298i	$0.579080\pi$
$998$	−88819.4	−2.81716
$999$	7713.41i	0.244286i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.b.j.337.1		18
13.5	odd	4		507.4.a.n.1.1	✓	9
13.8	odd	4		507.4.a.q.1.9	yes	9
13.12	even	2	inner	507.4.b.j.337.18		18
39.5	even	4		1521.4.a.bj.1.9		9
39.8	even	4		1521.4.a.be.1.1		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.4.a.n.1.1	✓	9	13.5	odd	4
507.4.a.q.1.9	yes	9	13.8	odd	4
507.4.b.j.337.1		18	1.1	even	1	trivial
507.4.b.j.337.18		18	13.12	even	2	inner
1521.4.a.be.1.1		9	39.8	even	4
1521.4.a.bj.1.9		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)

\(f(q)\)	\(=\)	\(q-5.48584i q^{2} -3.00000 q^{3} -22.0945 q^{4} -13.3185i q^{5} +16.4575i q^{6} +21.4234i q^{7} +77.3200i q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-5.48584i q^{2} -3.00000 q^{3} -22.0945 q^{4} -13.3185i q^{5} +16.4575i q^{6} +21.4234i q^{7} +77.3200i q^{8} +9.00000 q^{9} -73.0635 q^{10} -19.0520i q^{11} +66.2834 q^{12} +117.525 q^{14} +39.9556i q^{15} +247.410 q^{16} +71.7906 q^{17} -49.3726i q^{18} +102.134i q^{19} +294.266i q^{20} -64.2701i q^{21} -104.516 q^{22} +37.8302 q^{23} -231.960i q^{24} -52.3838 q^{25} -27.0000 q^{27} -473.338i q^{28} +40.8605 q^{29} +219.190 q^{30} +6.05542i q^{31} -738.690i q^{32} +57.1559i q^{33} -393.832i q^{34} +285.328 q^{35} -198.850 q^{36} -285.682i q^{37} +560.293 q^{38} +1029.79 q^{40} +342.705i q^{41} -352.576 q^{42} -306.458 q^{43} +420.943i q^{44} -119.867i q^{45} -207.530i q^{46} +346.863i q^{47} -742.229 q^{48} -115.961 q^{49} +287.369i q^{50} -215.372 q^{51} +398.219 q^{53} +148.118i q^{54} -253.745 q^{55} -1656.46 q^{56} -306.403i q^{57} -224.155i q^{58} -208.497i q^{59} -882.799i q^{60} +546.936 q^{61} +33.2191 q^{62} +192.810i q^{63} -2073.06 q^{64} +313.548 q^{66} +678.268i q^{67} -1586.18 q^{68} -113.490 q^{69} -1565.27i q^{70} -957.777i q^{71} +695.880i q^{72} +270.360i q^{73} -1567.21 q^{74} +157.151 q^{75} -2256.60i q^{76} +408.157 q^{77} -1032.86 q^{79} -3295.14i q^{80} +81.0000 q^{81} +1880.03 q^{82} +1065.90i q^{83} +1420.01i q^{84} -956.147i q^{85} +1681.18i q^{86} -122.582 q^{87} +1473.10 q^{88} +427.185i q^{89} -657.571 q^{90} -835.837 q^{92} -18.1663i q^{93} +1902.83 q^{94} +1360.28 q^{95} +2216.07i q^{96} -698.084i q^{97} +636.144i q^{98} -171.468i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 54 q^{3} - 64 q^{4} + 162 q^{9}+O(q^{10}) \) 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

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