LMFDB - Embedded newform 507.4.b.k.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} + 112 x^{16} + 5026 x^{14} + 114847 x^{12} + 1397921 x^{10} + 8545747 x^{8} + 21033277 x^{6} + \cdots + 729 $ x^18 + 112x^16 + 5026x^14 + 114847x^12 + 1397921x^10 + 8545747x^8 + 21033277x^6 + 6703200x^4 + 137781x^2 + 729
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 13^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.1
Root			$-3.27560i$ of defining polynomial
Character	$\chi$	$=$	507.337
Dual form			507.4.b.k.337.18

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.52257i q^{2} +3.00000 q^{3} -22.4988 q^{4} +6.08065i q^{5} -16.5677i q^{6} +20.2718i q^{7} +80.0709i q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-5.52257i q^{2} +3.00000 q^{3} -22.4988 q^{4} +6.08065i q^{5} -16.5677i q^{6} +20.2718i q^{7} +80.0709i q^{8} +9.00000 q^{9} +33.5808 q^{10} -48.8284i q^{11} -67.4965 q^{12} +111.953 q^{14} +18.2419i q^{15} +262.207 q^{16} +37.7513 q^{17} -49.7032i q^{18} -120.837i q^{19} -136.807i q^{20} +60.8155i q^{21} -269.658 q^{22} -74.8543 q^{23} +240.213i q^{24} +88.0257 q^{25} +27.0000 q^{27} -456.092i q^{28} -112.710 q^{29} +100.742 q^{30} -113.134i q^{31} -807.490i q^{32} -146.485i q^{33} -208.485i q^{34} -123.266 q^{35} -202.489 q^{36} -85.7704i q^{37} -667.331 q^{38} -486.883 q^{40} +133.993i q^{41} +335.858 q^{42} +319.135 q^{43} +1098.58i q^{44} +54.7258i q^{45} +413.388i q^{46} -401.982i q^{47} +786.620 q^{48} -67.9471 q^{49} -486.129i q^{50} +113.254 q^{51} -384.493 q^{53} -149.110i q^{54} +296.908 q^{55} -1623.18 q^{56} -362.511i q^{57} +622.450i q^{58} -121.629i q^{59} -410.422i q^{60} +220.043 q^{61} -624.790 q^{62} +182.446i q^{63} -2361.77 q^{64} -808.975 q^{66} -975.363i q^{67} -849.361 q^{68} -224.563 q^{69} +680.745i q^{70} -106.725i q^{71} +720.638i q^{72} +43.2566i q^{73} -473.673 q^{74} +264.077 q^{75} +2718.69i q^{76} +989.841 q^{77} +539.339 q^{79} +1594.39i q^{80} +81.0000 q^{81} +739.986 q^{82} -811.183i q^{83} -1368.28i q^{84} +229.553i q^{85} -1762.45i q^{86} -338.130 q^{87} +3909.73 q^{88} -1130.64i q^{89} +302.227 q^{90} +1684.13 q^{92} -339.401i q^{93} -2219.98 q^{94} +734.767 q^{95} -2422.47i q^{96} +229.088i q^{97} +375.243i q^{98} -439.456i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 54 q^{3} - 88 q^{4} + 162 q^{9}+O(q^{10}) $ 18 * q + 54 * q^3 - 88 * q^4 + 162 * q^9	$ 18 q + 54 q^{3} - 88 q^{4} + 162 q^{9} + 108 q^{10} - 264 q^{12} + 316 q^{14} + 432 q^{16} - 356 q^{17} - 1260 q^{22} - 300 q^{23} + 40 q^{25} + 486 q^{27} - 194 q^{29} + 324 q^{30} - 836 q^{35} - 792 q^{36} + 1320 q^{38} - 3012 q^{40} + 948 q^{42} - 484 q^{43} + 1296 q^{48} + 76 q^{49} - 1068 q^{51} - 302 q^{53} + 4128 q^{55} - 4552 q^{56} - 2680 q^{61} - 694 q^{62} - 1786 q^{64} - 3780 q^{66} + 5570 q^{68} - 900 q^{69} - 2382 q^{74} + 120 q^{75} + 4284 q^{77} - 3182 q^{79} + 1458 q^{81} - 3034 q^{82} - 582 q^{87} + 7432 q^{88} + 972 q^{90} + 1030 q^{92} - 1384 q^{94} - 8316 q^{95}+O(q^{100}) $ 18 * q + 54 * q^3 - 88 * q^4 + 162 * q^9 + 108 * q^10 - 264 * q^12 + 316 * q^14 + 432 * q^16 - 356 * q^17 - 1260 * q^22 - 300 * q^23 + 40 * q^25 + 486 * q^27 - 194 * q^29 + 324 * q^30 - 836 * q^35 - 792 * q^36 + 1320 * q^38 - 3012 * q^40 + 948 * q^42 - 484 * q^43 + 1296 * q^48 + 76 * q^49 - 1068 * q^51 - 302 * q^53 + 4128 * q^55 - 4552 * q^56 - 2680 * q^61 - 694 * q^62 - 1786 * q^64 - 3780 * q^66 + 5570 * q^68 - 900 * q^69 - 2382 * q^74 + 120 * q^75 + 4284 * q^77 - 3182 * q^79 + 1458 * q^81 - 3034 * q^82 - 582 * q^87 + 7432 * q^88 + 972 * q^90 + 1030 * q^92 - 1384 * q^94 - 8316 * q^95

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 $

$2$	− 5.52257i	− 1.95253i	−0.216591	−	0.976263i	$-0.569494\pi$
$2$	− 5.52257i	− 1.95253i	0.216591	−	0.976263i	$-0.430506\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−22.4988	−2.81235
$5$	6.08065i	0.543870i	0.962316	+	0.271935i	$0.0876635\pi$
$5$	6.08065i	0.543870i	−0.962316	+	0.271935i	$0.912336\pi$
$6$	− 16.5677i	− 1.12729i
$7$	20.2718i	1.09458i	0.836944	+	0.547288i	$0.184340\pi$
$7$	20.2718i	1.09458i	−0.836944	+	0.547288i	$0.815660\pi$
$8$	80.0709i	3.53867i
$9$	9.00000	0.333333
$10$	33.5808	1.06192
$11$	− 48.8284i	− 1.33839i	−0.743086	−	0.669196i	$-0.766639\pi$
$11$	− 48.8284i	− 1.33839i	0.743086	−	0.669196i	$-0.233361\pi$
$12$	−67.4965	−1.62371
$13$	0	0
$13$	0	0
$14$	111.953	2.13719
$15$	18.2419i	0.314003i
$16$	262.207	4.09698
$17$	37.7513	0.538591	0.269295	−	0.963058i	$-0.413209\pi$
$17$	37.7513	0.538591	0.269295	+	0.963058i	$0.413209\pi$
$18$	− 49.7032i	− 0.650842i
$19$	− 120.837i	− 1.45905i	−0.683955	−	0.729524i	$-0.739742\pi$
$19$	− 120.837i	− 1.45905i	0.683955	−	0.729524i	$-0.260258\pi$
$20$	− 136.807i	− 1.52955i
$21$	60.8155i	0.631954i
$22$	−269.658	−2.61324
$23$	−74.8543	−0.678617	−0.339309	−	0.940675i	$-0.610193\pi$
$23$	−74.8543	−0.678617	−0.339309	+	0.940675i	$0.610193\pi$
$24$	240.213i	2.04305i
$25$	88.0257	0.704206
$26$	0	0
$27$	27.0000	0.192450
$28$	− 456.092i	− 3.07834i
$29$	−112.710	−0.721715	−0.360858	−	0.932621i	$-0.617516\pi$
$29$	−112.710	−0.721715	−0.360858	+	0.932621i	$0.617516\pi$
$30$	100.742	0.613099
$31$	− 113.134i	− 0.655465i	−0.944770	−	0.327733i	$-0.893715\pi$
$31$	− 113.134i	− 0.655465i	0.944770	−	0.327733i	$-0.106285\pi$
$32$	− 807.490i	− 4.46079i
$33$	− 146.485i	− 0.772721i
$34$	− 208.485i	− 1.05161i
$35$	−123.266	−0.595307
$36$	−202.489	−0.937451
$37$	− 85.7704i	− 0.381096i	−0.981678	−	0.190548i	$-0.938973\pi$
$37$	− 85.7704i	− 0.381096i	0.981678	−	0.190548i	$-0.0610265\pi$
$38$	−667.331	−2.84883
$39$	0	0
$40$	−486.883	−1.92457
$41$	133.993i	0.510395i	0.966889	+	0.255197i	$0.0821404\pi$
$41$	133.993i	0.510395i	−0.966889	+	0.255197i	$0.917860\pi$
$42$	335.858	1.23391
$43$	319.135	1.13181	0.565903	−	0.824472i	$-0.308528\pi$
$43$	319.135	1.13181	0.565903	+	0.824472i	$0.308528\pi$
$44$	1098.58i	3.76403i
$45$	54.7258i	0.181290i
$46$	413.388i	1.32502i
$47$	− 401.982i	− 1.24755i	−0.781602	−	0.623777i	$-0.785597\pi$
$47$	− 401.982i	− 1.24755i	0.781602	−	0.623777i	$-0.214403\pi$
$48$	786.620	2.36539
$49$	−67.9471	−0.198096
$50$	− 486.129i	− 1.37498i
$51$	113.254	0.310955
$52$	0	0
$53$	−384.493	−0.996494	−0.498247	−	0.867035i	$-0.666023\pi$
$53$	−384.493	−0.996494	−0.498247	+	0.867035i	$0.666023\pi$
$54$	− 149.110i	− 0.375764i
$55$	296.908	0.727911
$56$	−1623.18	−3.87334
$57$	− 362.511i	− 0.842382i
$58$	622.450i	1.40917i
$59$	− 121.629i	− 0.268385i	−0.990955	−	0.134192i	$-0.957156\pi$
$59$	− 121.629i	− 0.268385i	0.990955	−	0.134192i	$-0.0428440\pi$
$60$	− 410.422i	− 0.883088i
$61$	220.043	0.461864	0.230932	−	0.972970i	$-0.425823\pi$
$61$	220.043	0.461864	0.230932	+	0.972970i	$0.425823\pi$
$62$	−624.790	−1.27981
$63$	182.446i	0.364859i
$64$	−2361.77	−4.61283
$65$	0	0
$66$	−808.975	−1.50876
$67$	− 975.363i	− 1.77850i	−0.457421	−	0.889250i	$-0.651227\pi$
$67$	− 975.363i	− 1.77850i	0.457421	−	0.889250i	$-0.348773\pi$
$68$	−849.361	−1.51471
$69$	−224.563	−0.391800
$70$	680.745i	1.16235i
$71$	− 106.725i	− 0.178394i	−0.996014	−	0.0891969i	$-0.971570\pi$
$71$	− 106.725i	− 0.178394i	0.996014	−	0.0891969i	$-0.0284300\pi$
$72$	720.638i	1.17956i
$73$	43.2566i	0.0693535i	0.999399	+	0.0346767i	$0.0110402\pi$
$73$	43.2566i	0.0693535i	−0.999399	+	0.0346767i	$0.988960\pi$
$74$	−473.673	−0.744100
$75$	264.077	0.406573
$76$	2718.69i	4.10336i
$77$	989.841	1.46497
$78$	0	0
$79$	539.339	0.768106	0.384053	−	0.923311i	$-0.374528\pi$
$79$	539.339	0.768106	0.384053	+	0.923311i	$0.374528\pi$
$80$	1594.39i	2.22822i
$81$	81.0000	0.111111
$82$	739.986	0.996558
$83$	− 811.183i	− 1.07276i	−0.843977	−	0.536379i	$-0.819792\pi$
$83$	− 811.183i	− 1.07276i	0.843977	−	0.536379i	$-0.180208\pi$
$84$	− 1368.28i	− 1.77728i
$85$	229.553i	0.292923i
$86$	− 1762.45i	− 2.20988i
$87$	−338.130	−0.416683
$88$	3909.73	4.73612
$89$	− 1130.64i	− 1.34661i	−0.739366	−	0.673304i	$-0.764875\pi$
$89$	− 1130.64i	− 1.34661i	0.739366	−	0.673304i	$-0.235125\pi$
$90$	302.227	0.353973
$91$	0	0
$92$	1684.13	1.90851
$93$	− 339.401i	− 0.378433i
$94$	−2219.98	−2.43588
$95$	734.767	0.793532
$96$	− 2422.47i	− 2.57544i
$97$	229.088i	0.239797i	0.992786	+	0.119899i	$0.0382569\pi$
$97$	229.088i	0.239797i	−0.992786	+	0.119899i	$0.961743\pi$
$98$	375.243i	0.386788i
$99$	− 439.456i	− 0.446131i
$100$	−1980.48	−1.98048
$101$	845.077	0.832557	0.416279	−	0.909237i	$-0.363334\pi$
$101$	845.077	0.832557	0.416279	+	0.909237i	$0.363334\pi$
$102$	− 625.454i	− 0.607148i
$103$	1095.09	1.04760	0.523800	−	0.851841i	$-0.324514\pi$
$103$	1095.09	1.04760	0.523800	+	0.851841i	$0.324514\pi$
$104$	0	0
$105$	−369.798	−0.343700
$106$	2123.39i	1.94568i
$107$	1183.33	1.06913	0.534563	−	0.845129i	$-0.320476\pi$
$107$	1183.33	1.06913	0.534563	+	0.845129i	$0.320476\pi$
$108$	−607.468	−0.541238
$109$	239.968i	0.210870i	0.994426	+	0.105435i	$0.0336234\pi$
$109$	239.968i	0.210870i	−0.994426	+	0.105435i	$0.966377\pi$
$110$	− 1639.70i	− 1.42126i
$111$	− 257.311i	− 0.220026i
$112$	5315.41i	4.48446i
$113$	−1031.22	−0.858490	−0.429245	−	0.903188i	$-0.641220\pi$
$113$	−1031.22	−0.858490	−0.429245	+	0.903188i	$0.641220\pi$
$114$	−2001.99	−1.64477
$115$	− 455.162i	− 0.369079i
$116$	2535.85	2.02972
$117$	0	0
$118$	−671.703	−0.524028
$119$	765.288i	0.589528i
$120$	−1460.65	−1.11115
$121$	−1053.21	−0.791294
$122$	− 1215.21i	− 0.901800i
$123$	401.979i	0.294676i
$124$	2545.38i	1.84340i
$125$	1295.33i	0.926866i
$126$	1007.57	0.712396
$127$	18.0201	0.0125908	0.00629539	−	0.999980i	$-0.497996\pi$
$127$	18.0201	0.0125908	0.00629539	+	0.999980i	$0.497996\pi$
$128$	6583.12i	4.54587i
$129$	957.406	0.653449
$130$	0	0
$131$	−1282.52	−0.855378	−0.427689	−	0.903926i	$-0.640672\pi$
$131$	−1282.52	−0.855378	−0.427689	+	0.903926i	$0.640672\pi$
$132$	3295.75i	2.17317i
$133$	2449.59	1.59704
$134$	−5386.51	−3.47257
$135$	164.177i	0.104668i
$136$	3022.78i	1.90589i
$137$	1547.36i	0.964965i	0.875906	+	0.482482i	$0.160265\pi$
$137$	1547.36i	0.964965i	−0.875906	+	0.482482i	$0.839735\pi$
$138$	1240.16i	0.764999i
$139$	−3096.87	−1.88973	−0.944867	−	0.327455i	$-0.893809\pi$
$139$	−3096.87	−1.88973	−0.944867	+	0.327455i	$0.893809\pi$
$140$	2773.34	1.67421
$141$	− 1205.95i	− 0.720276i
$142$	−589.398	−0.348318
$143$	0	0
$144$	2359.86	1.36566
$145$	− 685.351i	− 0.392519i
$146$	238.888	0.135414
$147$	−203.841	−0.114371
$148$	1929.73i	1.07178i
$149$	− 420.879i	− 0.231408i	−0.993284	−	0.115704i	$-0.963088\pi$
$149$	− 420.879i	− 0.231408i	0.993284	−	0.115704i	$-0.0369123\pi$
$150$	− 1458.39i	− 0.793845i
$151$	− 2801.17i	− 1.50964i	−0.655931	−	0.754821i	$-0.727724\pi$
$151$	− 2801.17i	− 1.50964i	0.655931	−	0.754821i	$-0.272276\pi$
$152$	9675.52	5.16308
$153$	339.762	0.179530
$154$	− 5466.47i	− 2.86039i
$155$	687.927	0.356488
$156$	0	0
$157$	−2344.20	−1.19164	−0.595820	−	0.803118i	$-0.703173\pi$
$157$	−2344.20	−1.19164	−0.595820	+	0.803118i	$0.703173\pi$
$158$	− 2978.54i	− 1.49975i
$159$	−1153.48	−0.575326
$160$	4910.06	2.42609
$161$	− 1517.43i	− 0.742798i
$162$	− 447.329i	− 0.216947i
$163$	3658.41i	1.75797i	0.476851	+	0.878984i	$0.341778\pi$
$163$	3658.41i	1.75797i	−0.476851	+	0.878984i	$0.658222\pi$
$164$	− 3014.68i	− 1.43541i
$165$	890.725	0.420260
$166$	−4479.82	−2.09459
$167$	1987.85i	0.921105i	0.887632	+	0.460552i	$0.152349\pi$
$167$	1987.85i	0.921105i	−0.887632	+	0.460552i	$0.847651\pi$
$168$	−4869.55	−2.23627
$169$	0	0
$170$	1267.72	0.571940
$171$	− 1087.53i	− 0.486349i
$172$	−7180.17	−3.18304
$173$	3577.11	1.57204	0.786019	−	0.618202i	$-0.212139\pi$
$173$	3577.11	1.57204	0.786019	+	0.618202i	$0.212139\pi$
$174$	1867.35i	0.813583i
$175$	1784.44i	0.770807i
$176$	− 12803.1i	− 5.48337i
$177$	− 364.886i	− 0.154952i
$178$	−6244.07	−2.62928
$179$	−770.144	−0.321582	−0.160791	−	0.986988i	$-0.551405\pi$
$179$	−770.144	−0.321582	−0.160791	+	0.986988i	$0.551405\pi$
$180$	− 1231.27i	− 0.509851i
$181$	1895.69	0.778482	0.389241	−	0.921136i	$-0.372737\pi$
$181$	1895.69	0.778482	0.389241	+	0.921136i	$0.372737\pi$
$182$	0	0
$183$	660.130	0.266657
$184$	− 5993.65i	− 2.40140i
$185$	521.539	0.207267
$186$	−1874.37	−0.738900
$187$	− 1843.34i	− 0.720846i
$188$	9044.12i	3.50857i
$189$	547.339i	0.210651i
$190$	− 4057.81i	− 1.54939i
$191$	2203.83	0.834887	0.417443	−	0.908703i	$-0.362926\pi$
$191$	2203.83	0.834887	0.417443	+	0.908703i	$0.362926\pi$
$192$	−7085.30	−2.66322
$193$	1016.38i	0.379069i	0.981874	+	0.189534i	$0.0606978\pi$
$193$	1016.38i	0.379069i	−0.981874	+	0.189534i	$0.939302\pi$
$194$	1265.15	0.468210
$195$	0	0
$196$	1528.73	0.557117
$197$	− 4318.58i	− 1.56186i	−0.624619	−	0.780930i	$-0.714746\pi$
$197$	− 4318.58i	− 1.56186i	0.624619	−	0.780930i	$-0.285254\pi$
$198$	−2426.93	−0.871082
$199$	−1363.40	−0.485672	−0.242836	−	0.970067i	$-0.578078\pi$
$199$	−1363.40	−0.485672	−0.242836	+	0.970067i	$0.578078\pi$
$200$	7048.30i	2.49195i
$201$	− 2926.09i	− 1.02682i
$202$	− 4667.00i	− 1.62559i
$203$	− 2284.84i	− 0.789972i
$204$	−2548.08	−0.874517
$205$	−814.764	−0.277588
$206$	− 6047.74i	− 2.04546i
$207$	−673.688	−0.226206
$208$	0	0
$209$	−5900.28	−1.95278
$210$	2042.23i	0.671084i
$211$	5288.34	1.72542	0.862711	−	0.505697i	$-0.168765\pi$
$211$	5288.34	1.72542	0.862711	+	0.505697i	$0.168765\pi$
$212$	8650.64	2.80249
$213$	− 320.176i	− 0.102996i
$214$	− 6535.00i	− 2.08749i
$215$	1940.55i	0.615555i
$216$	2161.91i	0.681017i
$217$	2293.43	0.717457
$218$	1325.24	0.411728
$219$	129.770i	0.0400412i
$220$	−6680.09	−2.04714
$221$	0	0
$222$	−1421.02	−0.429606
$223$	1258.01i	0.377769i	0.981999	+	0.188884i	$0.0604872\pi$
$223$	1258.01i	0.377769i	−0.981999	+	0.188884i	$0.939513\pi$
$224$	16369.3	4.88268
$225$	792.232	0.234735
$226$	5695.01i	1.67622i
$227$	− 1135.21i	− 0.331922i	−0.986132	−	0.165961i	$-0.946927\pi$
$227$	− 1135.21i	− 0.331922i	0.986132	−	0.165961i	$-0.0530726\pi$
$228$	8156.07i	2.36908i
$229$	− 218.002i	− 0.0629083i	−0.999505	−	0.0314541i	$-0.989986\pi$
$229$	− 218.002i	− 0.0629083i	0.999505	−	0.0314541i	$-0.0100138\pi$
$230$	−2513.67	−0.720636
$231$	2969.52	0.845802
$232$	− 9024.80i	− 2.55391i
$233$	2.56367	0.000720823 0	0.000360412	−	1.00000i	$-0.499885\pi$
$233$	2.56367	0.000720823 0	0.000360412		1.00000i	$0.499885\pi$
$234$	0	0
$235$	2444.31	0.678507
$236$	2736.50i	0.754793i
$237$	1618.02	0.443466
$238$	4226.36	1.15107
$239$	− 3971.13i	− 1.07477i	−0.843336	−	0.537387i	$-0.819411\pi$
$239$	− 3971.13i	− 1.07477i	0.843336	−	0.537387i	$-0.180589\pi$
$240$	4783.16i	1.28647i
$241$	− 658.032i	− 0.175882i	−0.996126	−	0.0879411i	$-0.971971\pi$
$241$	− 658.032i	− 0.175882i	0.996126	−	0.0879411i	$-0.0280287\pi$
$242$	5816.44i	1.54502i
$243$	243.000	0.0641500
$244$	−4950.72	−1.29892
$245$	− 413.162i	− 0.107739i
$246$	2219.96	0.575363
$247$	0	0
$248$	9058.72	2.31947
$249$	− 2433.55i	− 0.619357i
$250$	7153.58	1.80973
$251$	−7091.01	−1.78319	−0.891595	−	0.452833i	$-0.850413\pi$
$251$	−7091.01	−1.78319	−0.891595	+	0.452833i	$0.850413\pi$
$252$	− 4104.83i	− 1.02611i
$253$	3655.01i	0.908256i
$254$	− 99.5176i	− 0.0245838i
$255$	688.658i	0.169119i
$256$	17461.6	4.26309
$257$	−792.530	−0.192361	−0.0961803	−	0.995364i	$-0.530663\pi$
$257$	−792.530	−0.192361	−0.0961803	+	0.995364i	$0.530663\pi$
$258$	− 5287.35i	− 1.27588i
$259$	1738.72	0.417139
$260$	0	0
$261$	−1014.39	−0.240572
$262$	7082.83i	1.67015i
$263$	−337.467	−0.0791221	−0.0395610	−	0.999217i	$-0.512596\pi$
$263$	−337.467	−0.0791221	−0.0395610	+	0.999217i	$0.512596\pi$
$264$	11729.2	2.73440
$265$	− 2337.97i	− 0.541963i
$266$	− 13528.0i	− 3.11826i
$267$	− 3391.93i	− 0.777464i
$268$	21944.5i	5.00177i
$269$	−3523.93	−0.798728	−0.399364	−	0.916792i	$-0.630769\pi$
$269$	−3523.93	−0.798728	−0.399364	+	0.916792i	$0.630769\pi$
$270$	906.682	0.204366
$271$	8313.62i	1.86353i	0.363063	+	0.931765i	$0.381731\pi$
$271$	8313.62i	1.86353i	−0.363063	+	0.931765i	$0.618269\pi$
$272$	9898.65	2.20660
$273$	0	0
$274$	8545.43	1.88412
$275$	− 4298.16i	− 0.942504i
$276$	5052.40	1.10188
$277$	2500.33	0.542347	0.271174	−	0.962530i	$-0.412588\pi$
$277$	2500.33	0.542347	0.271174	+	0.962530i	$0.412588\pi$
$278$	17102.7i	3.68975i
$279$	− 1018.20i	− 0.218488i
$280$	− 9870.01i	− 2.10659i
$281$	− 3053.13i	− 0.648165i	−0.946029	−	0.324083i	$-0.894944\pi$
$281$	− 3053.13i	− 0.648165i	0.946029	−	0.324083i	$-0.105056\pi$
$282$	−6659.93	−1.40636
$283$	−1158.80	−0.243404	−0.121702	−	0.992567i	$-0.538835\pi$
$283$	−1158.80	−0.243404	−0.121702	+	0.992567i	$0.538835\pi$
$284$	2401.19i	0.501706i
$285$	2204.30	0.458146
$286$	0	0
$287$	−2716.28	−0.558666
$288$	− 7267.41i	− 1.48693i
$289$	−3487.84	−0.709920
$290$	−3784.90	−0.766403
$291$	687.263i	0.138447i
$292$	− 973.223i	− 0.195046i
$293$	− 8357.45i	− 1.66637i	−0.552991	−	0.833187i	$-0.686514\pi$
$293$	− 8357.45i	− 1.66637i	0.552991	−	0.833187i	$-0.313486\pi$
$294$	1125.73i	0.223312i
$295$	739.581	0.145966
$296$	6867.71	1.34857
$297$	− 1318.37i	− 0.257574i
$298$	−2324.33	−0.451829
$299$	0	0
$300$	−5941.43	−1.14343
$301$	6469.46i	1.23885i
$302$	−15469.7	−2.94761
$303$	2535.23	0.480677
$304$	− 31684.3i	− 5.97769i
$305$	1338.01i	0.251194i
$306$	− 1876.36i	− 0.350537i
$307$	− 8429.30i	− 1.56705i	−0.621357	−	0.783527i	$-0.713418\pi$
$307$	− 8429.30i	− 1.56705i	0.621357	−	0.783527i	$-0.286582\pi$
$308$	−22270.3	−4.12002
$309$	3285.28	0.604832
$310$	− 3799.13i	− 0.696051i
$311$	−3602.87	−0.656914	−0.328457	−	0.944519i	$-0.606529\pi$
$311$	−3602.87	−0.656914	−0.328457	+	0.944519i	$0.606529\pi$
$312$	0	0
$313$	6626.98	1.19674	0.598369	−	0.801220i	$-0.295816\pi$
$313$	6626.98	1.19674	0.598369	+	0.801220i	$0.295816\pi$
$314$	12946.0i	2.32671i
$315$	−1109.39	−0.198436
$316$	−12134.5	−2.16019
$317$	− 3983.87i	− 0.705857i	−0.935650	−	0.352928i	$-0.885186\pi$
$317$	− 3983.87i	− 0.705857i	0.935650	−	0.352928i	$-0.114814\pi$
$318$	6370.17i	1.12334i
$319$	5503.46i	0.965938i
$320$	− 14361.1i	− 2.50878i
$321$	3549.98	0.617260
$322$	−8380.14	−1.45033
$323$	− 4561.76i	− 0.785829i
$324$	−1822.41	−0.312484
$325$	0	0
$326$	20203.8	3.43248
$327$	719.905i	0.121746i
$328$	−10728.9	−1.80612
$329$	8148.91	1.36554
$330$	− 4919.09i	− 0.820567i
$331$	− 4172.82i	− 0.692927i	−0.938063	−	0.346464i	$-0.887382\pi$
$331$	− 4172.82i	− 0.692927i	0.938063	−	0.346464i	$-0.112618\pi$
$332$	18250.7i	3.01697i
$333$	− 771.933i	− 0.127032i
$334$	10978.1	1.79848
$335$	5930.84	0.967272
$336$	15946.2i	2.58910i
$337$	−4157.51	−0.672030	−0.336015	−	0.941857i	$-0.609079\pi$
$337$	−4157.51	−0.672030	−0.336015	+	0.941857i	$0.609079\pi$
$338$	0	0
$339$	−3093.67	−0.495649
$340$	− 5164.66i	− 0.823804i
$341$	−5524.14	−0.877270
$342$	−6005.98	−0.949609
$343$	5575.83i	0.877744i
$344$	25553.4i	4.00509i
$345$	− 1365.49i	− 0.213088i
$346$	− 19754.9i	− 3.06944i
$347$	4667.98	0.722162	0.361081	−	0.932535i	$-0.382408\pi$
$347$	4667.98	0.722162	0.361081	+	0.932535i	$0.382408\pi$
$348$	7607.54	1.17186
$349$	11905.6i	1.82606i	0.407896	+	0.913028i	$0.366263\pi$
$349$	11905.6i	1.82606i	−0.407896	+	0.913028i	$0.633737\pi$
$350$	9854.72	1.50502
$351$	0	0
$352$	−39428.4	−5.97029
$353$	1662.12i	0.250610i	0.992118	+	0.125305i	$0.0399910\pi$
$353$	1662.12i	0.250610i	−0.992118	+	0.125305i	$0.960009\pi$
$354$	−2015.11	−0.302548
$355$	648.959	0.0970229
$356$	25438.2i	3.78714i
$357$	2295.87i	0.340364i
$358$	4253.18i	0.627898i
$359$	12754.4i	1.87507i	0.347890	+	0.937535i	$0.386898\pi$
$359$	12754.4i	1.87507i	−0.347890	+	0.937535i	$0.613102\pi$
$360$	−4381.95	−0.641524
$361$	−7742.58	−1.12882
$362$	− 10469.1i	− 1.52001i
$363$	−3159.64	−0.456854
$364$	0	0
$365$	−263.028	−0.0377192
$366$	− 3645.62i	− 0.520655i
$367$	4463.81	0.634901	0.317451	−	0.948275i	$-0.397173\pi$
$367$	4463.81	0.634901	0.317451	+	0.948275i	$0.397173\pi$
$368$	−19627.3	−2.78028
$369$	1205.94i	0.170132i
$370$	− 2880.24i	− 0.404693i
$371$	− 7794.38i	− 1.09074i
$372$	7636.13i	1.06429i
$373$	9454.46	1.31242	0.656211	−	0.754578i	$-0.272158\pi$
$373$	9454.46	1.31242	0.656211	+	0.754578i	$0.272158\pi$
$374$	−10180.0	−1.40747
$375$	3886.00i	0.535126i
$376$	32187.0	4.41468
$377$	0	0
$378$	3022.72	0.411302
$379$	− 14085.9i	− 1.90908i	−0.298077	−	0.954542i	$-0.596345\pi$
$379$	− 14085.9i	− 1.90908i	0.298077	−	0.954542i	$-0.403655\pi$
$380$	−16531.4	−2.23169
$381$	54.0604	0.00726929
$382$	− 12170.8i	− 1.63014i
$383$	8663.84i	1.15588i	0.816080	+	0.577939i	$0.196143\pi$
$383$	8663.84i	1.15588i	−0.816080	+	0.577939i	$0.803857\pi$
$384$	19749.4i	2.62456i
$385$	6018.87i	0.796754i
$386$	5613.01	0.740141
$387$	2872.22	0.377269
$388$	− 5154.20i	− 0.674395i
$389$	−5085.58	−0.662852	−0.331426	−	0.943481i	$-0.607530\pi$
$389$	−5085.58	−0.662852	−0.331426	+	0.943481i	$0.607530\pi$
$390$	0	0
$391$	−2825.85	−0.365497
$392$	− 5440.58i	− 0.700997i
$393$	−3847.57	−0.493853
$394$	−23849.7	−3.04957
$395$	3279.53i	0.417750i
$396$	9887.24i	1.25468i
$397$	12983.7i	1.64140i	0.571360	+	0.820700i	$0.306416\pi$
$397$	12983.7i	1.64140i	−0.571360	+	0.820700i	$0.693584\pi$
$398$	7529.48i	0.948288i
$399$	7348.76	0.922051
$400$	23080.9	2.88512
$401$	6493.66i	0.808673i	0.914610	+	0.404337i	$0.132498\pi$
$401$	6493.66i	0.808673i	−0.914610	+	0.404337i	$0.867502\pi$
$402$	−16159.5	−2.00489
$403$	0	0
$404$	−19013.2	−2.34145
$405$	492.532i	0.0604300i
$406$	−12618.2	−1.54244
$407$	−4188.03	−0.510056
$408$	9068.35i	1.10037i
$409$	12546.2i	1.51680i	0.651792	+	0.758398i	$0.274017\pi$
$409$	12546.2i	1.51680i	−0.651792	+	0.758398i	$0.725983\pi$
$410$	4499.59i	0.541998i
$411$	4642.09i	0.557123i
$412$	−24638.3	−2.94622
$413$	2465.63	0.293767
$414$	3720.49i	0.441672i
$415$	4932.52	0.583440
$416$	0	0
$417$	−9290.61	−1.09104
$418$	32584.7i	3.81285i
$419$	7945.58	0.926413	0.463206	−	0.886250i	$-0.346699\pi$
$419$	7945.58	0.926413	0.463206	+	0.886250i	$0.346699\pi$
$420$	8320.01	0.966607
$421$	284.758i	0.0329650i	0.999864	+	0.0164825i	$0.00524678\pi$
$421$	284.758i	0.0329650i	−0.999864	+	0.0164825i	$0.994753\pi$
$422$	− 29205.2i	− 3.36893i
$423$	− 3617.84i	− 0.415852i
$424$	− 30786.7i	− 3.52626i
$425$	3323.09	0.379279
$426$	−1768.19	−0.201102
$427$	4460.68i	0.505545i
$428$	−26623.4	−3.00676
$429$	0	0
$430$	10716.8	1.20189
$431$	12512.1i	1.39835i	0.714952	+	0.699173i	$0.246448\pi$
$431$	12512.1i	1.39835i	−0.714952	+	0.699173i	$0.753552\pi$
$432$	7079.58	0.788464
$433$	13217.2	1.46692	0.733460	−	0.679732i	$-0.237904\pi$
$433$	13217.2	1.46692	0.733460	+	0.679732i	$0.237904\pi$
$434$	− 12665.6i	− 1.40085i
$435$	− 2056.05i	− 0.226621i
$436$	− 5399.01i	− 0.593040i
$437$	9045.16i	0.990135i
$438$	716.663	0.0781815
$439$	−4635.40	−0.503953	−0.251977	−	0.967733i	$-0.581081\pi$
$439$	−4635.40	−0.503953	−0.251977	+	0.967733i	$0.581081\pi$
$440$	23773.7i	2.57583i
$441$	−611.524	−0.0660321
$442$	0	0
$443$	−2945.92	−0.315947	−0.157974	−	0.987443i	$-0.550496\pi$
$443$	−2945.92	−0.315947	−0.157974	+	0.987443i	$0.550496\pi$
$444$	5789.20i	0.618791i
$445$	6875.05	0.732379
$446$	6947.45	0.737603
$447$	− 1262.64i	− 0.133603i
$448$	− 47877.3i	− 5.04909i
$449$	− 8668.29i	− 0.911095i	−0.890211	−	0.455548i	$-0.849443\pi$
$449$	− 8668.29i	− 0.911095i	0.890211	−	0.455548i	$-0.150557\pi$
$450$	− 4375.16i	− 0.458326i
$451$	6542.66	0.683108
$452$	23201.3	2.41438
$453$	− 8403.51i	− 0.871592i
$454$	−6269.27	−0.648087
$455$	0	0
$456$	29026.6	2.98091
$457$	5195.36i	0.531791i	0.964002	+	0.265896i	$0.0856677\pi$
$457$	5195.36i	0.531791i	−0.964002	+	0.265896i	$0.914332\pi$
$458$	−1203.93	−0.122830
$459$	1019.29	0.103652
$460$	10240.6i	1.03798i
$461$	2931.19i	0.296136i	0.988977	+	0.148068i	$0.0473055\pi$
$461$	2931.19i	0.296136i	−0.988977	+	0.148068i	$0.952694\pi$
$462$	− 16399.4i	− 1.65145i
$463$	− 396.589i	− 0.0398079i	−0.999802	−	0.0199039i	$-0.993664\pi$
$463$	− 396.589i	− 0.0398079i	0.999802	−	0.0199039i	$-0.00633604\pi$
$464$	−29553.4	−2.95685
$465$	2063.78	0.205818
$466$	− 14.1581i	− 0.00140743i
$467$	−16627.5	−1.64760	−0.823801	−	0.566880i	$-0.808150\pi$
$467$	−16627.5	−1.64760	−0.823801	+	0.566880i	$0.808150\pi$
$468$	0	0
$469$	19772.4	1.94670
$470$	− 13498.9i	− 1.32480i
$471$	−7032.60	−0.687994
$472$	9738.91	0.949724
$473$	− 15582.9i	− 1.51480i
$474$	− 8935.62i	− 0.865879i
$475$	− 10636.8i	− 1.02747i
$476$	− 17218.1i	− 1.65796i
$477$	−3460.44	−0.332165
$478$	−21930.9	−2.09852
$479$	8903.28i	0.849272i	0.905364	+	0.424636i	$0.139598\pi$
$479$	8903.28i	0.849272i	−0.905364	+	0.424636i	$0.860402\pi$
$480$	14730.2	1.40070
$481$	0	0
$482$	−3634.03	−0.343414
$483$	− 4552.30i	− 0.428855i
$484$	23696.1	2.22540
$485$	−1393.00	−0.130418
$486$	− 1341.99i	− 0.125255i
$487$	469.526i	0.0436884i	0.999761	+	0.0218442i	$0.00695377\pi$
$487$	469.526i	0.0436884i	−0.999761	+	0.0218442i	$0.993046\pi$
$488$	17619.1i	1.63438i
$489$	10975.2i	1.01496i
$490$	−2281.72	−0.210362
$491$	18823.8	1.73015	0.865077	−	0.501639i	$-0.167269\pi$
$491$	18823.8	1.73015	0.865077	+	0.501639i	$0.167269\pi$
$492$	− 9044.05i	− 0.828734i
$493$	−4254.96	−0.388709
$494$	0	0
$495$	2672.17	0.242637
$496$	− 29664.4i	− 2.68543i
$497$	2163.52	0.195265
$498$	−13439.5	−1.20931
$499$	5651.95i	0.507046i	0.967329	+	0.253523i	$0.0815894\pi$
$499$	5651.95i	0.507046i	−0.967329	+	0.253523i	$0.918411\pi$
$500$	− 29143.5i	− 2.60667i
$501$	5963.55i	0.531800i
$502$	39160.7i	3.48172i
$503$	8022.35	0.711131	0.355565	−	0.934651i	$-0.384288\pi$
$503$	8022.35	0.711131	0.355565	+	0.934651i	$0.384288\pi$
$504$	−14608.7	−1.29111
$505$	5138.62i	0.452803i
$506$	20185.1	1.77339
$507$	0	0
$508$	−405.432	−0.0354097
$509$	− 3940.56i	− 0.343148i	−0.985171	−	0.171574i	$-0.945115\pi$
$509$	− 3940.56i	− 0.343148i	0.985171	−	0.171574i	$-0.0548853\pi$
$510$	3803.16	0.330210
$511$	−876.890	−0.0759126
$512$	− 43768.2i	− 3.77793i
$513$	− 3262.60i	− 0.280794i
$514$	4376.81i	0.375589i
$515$	6658.88i	0.569758i
$516$	−21540.5	−1.83773
$517$	−19628.1	−1.66972
$518$	− 9602.22i	− 0.814474i
$519$	10731.3	0.907617
$520$	0	0
$521$	8644.84	0.726943	0.363472	−	0.931605i	$-0.381591\pi$
$521$	8644.84	0.726943	0.363472	+	0.931605i	$0.381591\pi$
$522$	5602.05i	0.469722i
$523$	1956.76	0.163600	0.0818001	−	0.996649i	$-0.473933\pi$
$523$	1956.76	0.163600	0.0818001	+	0.996649i	$0.473933\pi$
$524$	28855.3	2.40563
$525$	5353.33i	0.445025i
$526$	1863.69i	0.154488i
$527$	− 4270.95i	− 0.353028i
$528$	− 38409.4i	− 3.16582i
$529$	−6563.84	−0.539479
$530$	−12911.6	−1.05820
$531$	− 1094.66i	− 0.0894615i
$532$	−55112.8	−4.49144
$533$	0	0
$534$	−18732.2	−1.51802
$535$	7195.39i	0.581465i
$536$	78098.2	6.29352
$537$	−2310.43	−0.185666
$538$	19461.2i	1.55954i
$539$	3317.75i	0.265131i
$540$	− 3693.80i	− 0.294363i
$541$	7005.37i	0.556718i	0.960477	+	0.278359i	$0.0897904\pi$
$541$	7005.37i	0.556718i	−0.960477	+	0.278359i	$0.910210\pi$
$542$	45912.6	3.63859
$543$	5687.06	0.449457
$544$	− 30483.8i	− 2.40254i
$545$	−1459.16	−0.114686
$546$	0	0
$547$	23216.7	1.81476	0.907382	−	0.420307i	$-0.138077\pi$
$547$	23216.7	1.81476	0.907382	+	0.420307i	$0.138077\pi$
$548$	− 34813.9i	− 2.71382i
$549$	1980.39	0.153955
$550$	−23736.9	−1.84026
$551$	13619.6i	1.05302i
$552$	− 17980.9i	− 1.38645i
$553$	10933.4i	0.840751i
$554$	− 13808.2i	− 1.05895i
$555$	1564.62	0.119665
$556$	69675.9	5.31460
$557$	− 14912.9i	− 1.13443i	−0.823568	−	0.567217i	$-0.808020\pi$
$557$	− 14912.9i	− 1.13443i	0.823568	−	0.567217i	$-0.191980\pi$
$558$	−5623.11	−0.426604
$559$	0	0
$560$	−32321.1	−2.43896
$561$	− 5530.01i	− 0.416180i
$562$	−16861.1	−1.26556
$563$	−10854.3	−0.812531	−0.406266	−	0.913755i	$-0.633169\pi$
$563$	−10854.3	−0.812531	−0.406266	+	0.913755i	$0.633169\pi$
$564$	27132.4i	2.02567i
$565$	− 6270.50i	− 0.466906i
$566$	6399.55i	0.475253i
$567$	1642.02i	0.121620i
$568$	8545.58	0.631276
$569$	−17892.5	−1.31827	−0.659134	−	0.752026i	$-0.729077\pi$
$569$	−17892.5	−1.31827	−0.659134	+	0.752026i	$0.729077\pi$
$570$	− 12173.4i	− 0.894541i
$571$	−17319.8	−1.26937	−0.634686	−	0.772770i	$-0.718870\pi$
$571$	−17319.8	−1.26937	−0.634686	+	0.772770i	$0.718870\pi$
$572$	0	0
$573$	6611.48	0.482022
$574$	15000.9i	1.09081i
$575$	−6589.10	−0.477886
$576$	−21255.9	−1.53761
$577$	9738.48i	0.702632i	0.936257	+	0.351316i	$0.114266\pi$
$577$	9738.48i	0.702632i	−0.936257	+	0.351316i	$0.885734\pi$
$578$	19261.8i	1.38614i
$579$	3049.13i	0.218855i
$580$	15419.6i	1.10390i
$581$	16444.2	1.17421
$582$	3795.46	0.270321
$583$	18774.2i	1.33370i
$584$	−3463.59	−0.245419
$585$	0	0
$586$	−46154.7	−3.25364
$587$	5960.31i	0.419095i	0.977799	+	0.209547i	$0.0671990\pi$
$587$	5960.31i	0.419095i	−0.977799	+	0.209547i	$0.932801\pi$
$588$	4586.19	0.321652
$589$	−13670.7	−0.956355
$590$	− 4084.39i	− 0.285003i
$591$	− 12955.7i	− 0.901740i
$592$	− 22489.6i	− 1.56134i
$593$	− 17870.2i	− 1.23751i	−0.785586	−	0.618753i	$-0.787638\pi$
$593$	− 17870.2i	− 1.23751i	0.785586	−	0.618753i	$-0.212362\pi$
$594$	−7280.78	−0.502919
$595$	−4653.45	−0.320627
$596$	9469.28i	0.650800i
$597$	−4090.20	−0.280403
$598$	0	0
$599$	−17943.7	−1.22397	−0.611987	−	0.790868i	$-0.709629\pi$
$599$	−17943.7	−1.22397	−0.611987	+	0.790868i	$0.709629\pi$
$600$	21144.9i	1.43873i
$601$	6548.20	0.444437	0.222219	−	0.974997i	$-0.428670\pi$
$601$	6548.20	0.444437	0.222219	+	0.974997i	$0.428670\pi$
$602$	35728.1	2.41888
$603$	− 8778.27i	− 0.592834i
$604$	63023.0i	4.24565i
$605$	− 6404.21i	− 0.430361i
$606$	− 14001.0i	− 0.938534i
$607$	−14886.0	−0.995391	−0.497696	−	0.867352i	$-0.665820\pi$
$607$	−14886.0	−0.995391	−0.497696	+	0.867352i	$0.665820\pi$
$608$	−97574.6	−6.50851
$609$	− 6854.52i	− 0.456091i
$610$	7389.24	0.490462
$611$	0	0
$612$	−7644.25	−0.504903
$613$	8774.65i	0.578148i	0.957307	+	0.289074i	$0.0933473\pi$
$613$	8774.65i	0.578148i	−0.957307	+	0.289074i	$0.906653\pi$
$614$	−46551.5	−3.05971
$615$	−2444.29	−0.160266
$616$	79257.4i	5.18405i
$617$	3810.41i	0.248625i	0.992243	+	0.124312i	$0.0396725\pi$
$617$	3810.41i	0.248625i	−0.992243	+	0.124312i	$0.960327\pi$
$618$	− 18143.2i	− 1.18095i
$619$	− 16392.6i	− 1.06441i	−0.846614	−	0.532207i	$-0.821363\pi$
$619$	− 16392.6i	− 1.06441i	0.846614	−	0.532207i	$-0.178637\pi$
$620$	−15477.5	−1.00257
$621$	−2021.07	−0.130600
$622$	19897.1i	1.28264i
$623$	22920.2	1.47396
$624$	0	0
$625$	3126.74	0.200112
$626$	− 36598.0i	− 2.33666i
$627$	−17700.8	−1.12744
$628$	52741.8	3.35131
$629$	− 3237.95i	− 0.205255i
$630$	6126.70i	0.387450i
$631$	− 11077.3i	− 0.698858i	−0.936963	−	0.349429i	$-0.886376\pi$
$631$	− 11077.3i	− 0.698858i	0.936963	−	0.349429i	$-0.113624\pi$
$632$	43185.4i	2.71807i
$633$	15865.0	0.996173
$634$	−22001.2	−1.37820
$635$	109.574i	0.00684775i
$636$	25951.9	1.61802
$637$	0	0
$638$	30393.2	1.88602
$639$	− 960.527i	− 0.0594646i
$640$	−40029.6	−2.47236
$641$	2143.45	0.132077	0.0660383	−	0.997817i	$-0.478964\pi$
$641$	2143.45	0.132077	0.0660383	+	0.997817i	$0.478964\pi$
$642$	− 19605.0i	− 1.20522i
$643$	− 9774.93i	− 0.599511i	−0.954016	−	0.299755i	$-0.903095\pi$
$643$	− 9774.93i	− 0.599511i	0.954016	−	0.299755i	$-0.0969051\pi$
$644$	34140.5i	2.08901i
$645$	5821.65i	0.355391i
$646$	−25192.6	−1.53435
$647$	22447.6	1.36400	0.681999	−	0.731353i	$-0.261111\pi$
$647$	22447.6	1.36400	0.681999	+	0.731353i	$0.261111\pi$
$648$	6485.74i	0.393185i
$649$	−5938.93	−0.359204
$650$	0	0
$651$	6880.29	0.414224
$652$	− 82310.0i	− 4.94403i
$653$	10582.0	0.634157	0.317079	−	0.948399i	$-0.397298\pi$
$653$	10582.0	0.634157	0.317079	+	0.948399i	$0.397298\pi$
$654$	3975.73	0.237711
$655$	− 7798.57i	− 0.465214i
$656$	35133.9i	2.09108i
$657$	389.309i	0.0231178i
$658$	− 45003.0i	− 2.66626i
$659$	−15114.6	−0.893445	−0.446722	−	0.894673i	$-0.647409\pi$
$659$	−15114.6	−0.893445	−0.446722	+	0.894673i	$0.647409\pi$
$660$	−20040.3	−1.18192
$661$	2229.61i	0.131198i	0.997846	+	0.0655990i	$0.0208958\pi$
$661$	2229.61i	0.131198i	−0.997846	+	0.0655990i	$0.979104\pi$
$662$	−23044.7	−1.35296
$663$	0	0
$664$	64952.1	3.79613
$665$	14895.1i	0.868581i
$666$	−4263.06	−0.248033
$667$	8436.83	0.489768
$668$	− 44724.3i	− 2.59047i
$669$	3774.03i	0.218105i
$670$	− 32753.5i	− 1.88862i
$671$	− 10744.4i	− 0.618155i
$672$	49107.9	2.81901
$673$	3588.22	0.205521	0.102761	−	0.994706i	$-0.467232\pi$
$673$	3588.22	0.205521	0.102761	+	0.994706i	$0.467232\pi$
$674$	22960.2i	1.31216i
$675$	2376.69	0.135524
$676$	0	0
$677$	−26458.4	−1.50204	−0.751019	−	0.660280i	$-0.770438\pi$
$677$	−26458.4	−1.50204	−0.751019	+	0.660280i	$0.770438\pi$
$678$	17085.0i	0.967767i
$679$	−4644.03	−0.262476
$680$	−18380.5	−1.03656
$681$	− 3405.62i	− 0.191635i
$682$	30507.5i	1.71289i
$683$	− 23598.4i	− 1.32206i	−0.750358	−	0.661031i	$-0.770119\pi$
$683$	− 23598.4i	− 1.32206i	0.750358	−	0.661031i	$-0.229881\pi$
$684$	24468.2i	1.36779i
$685$	−9408.97	−0.524815
$686$	30792.9	1.71382
$687$	− 654.007i	− 0.0363201i
$688$	83679.4	4.63699
$689$	0	0
$690$	−7541.01	−0.416060
$691$	16433.6i	0.904722i	0.891835	+	0.452361i	$0.149418\pi$
$691$	16433.6i	0.904722i	−0.891835	+	0.452361i	$0.850582\pi$
$692$	−80480.8	−4.42113
$693$	8908.57	0.488324
$694$	− 25779.2i	− 1.41004i
$695$	− 18831.0i	− 1.02777i
$696$	− 27074.4i	− 1.47450i
$697$	5058.41i	0.274894i
$698$	65749.7	3.56542
$699$	7.69102	0.000416167 0
$700$	− 40147.9i	− 2.16778i
$701$	−13899.4	−0.748892	−0.374446	−	0.927249i	$-0.622167\pi$
$701$	−13899.4	−0.748892	−0.374446	+	0.927249i	$0.622167\pi$
$702$	0	0
$703$	−10364.2	−0.556038
$704$	115321.i	6.17377i
$705$	7332.93	0.391736
$706$	9179.16	0.489323
$707$	17131.3i	0.911297i
$708$	8209.50i	0.435780i
$709$	32829.6i	1.73899i	0.493944	+	0.869494i	$0.335555\pi$
$709$	32829.6i	1.73899i	−0.493944	+	0.869494i	$0.664445\pi$
$710$	− 3583.92i	− 0.189440i
$711$	4854.05	0.256035
$712$	90531.7	4.76519
$713$	8468.55i	0.444810i
$714$	12679.1	0.664570
$715$	0	0
$716$	17327.3	0.904403
$717$	− 11913.4i	− 0.620521i
$718$	70437.0	3.66112
$719$	4683.94	0.242951	0.121475	−	0.992594i	$-0.461237\pi$
$719$	4683.94	0.242951	0.121475	+	0.992594i	$0.461237\pi$
$720$	14349.5i	0.742741i
$721$	22199.5i	1.14668i
$722$	42759.0i	2.20405i
$723$	− 1974.10i	− 0.101546i
$724$	−42650.7	−2.18937
$725$	−9921.39	−0.508236
$726$	17449.3i	0.892019i
$727$	35368.3	1.80432	0.902159	−	0.431405i	$-0.141982\pi$
$727$	35368.3	1.80432	0.902159	+	0.431405i	$0.141982\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	1452.59i	0.0736478i
$731$	12047.8	0.609580
$732$	−14852.2	−0.749934
$733$	− 33988.4i	− 1.71268i	−0.516416	−	0.856338i	$-0.672734\pi$
$733$	− 33988.4i	− 1.71268i	0.516416	−	0.856338i	$-0.327266\pi$
$734$	− 24651.7i	− 1.23966i
$735$	− 1239.49i	− 0.0622029i
$736$	60444.0i	3.02717i
$737$	−47625.4	−2.38033
$738$	6659.87	0.332186
$739$	− 7777.31i	− 0.387135i	−0.981087	−	0.193568i	$-0.937994\pi$
$739$	− 7777.31i	− 0.387135i	0.981087	−	0.193568i	$-0.0620059\pi$
$740$	−11734.0	−0.582907
$741$	0	0
$742$	−43045.0	−2.12969
$743$	7022.08i	0.346723i	0.984858	+	0.173361i	$0.0554629\pi$
$743$	7022.08i	0.346723i	−0.984858	+	0.173361i	$0.944537\pi$
$744$	27176.2	1.33915
$745$	2559.22	0.125856
$746$	− 52212.9i	− 2.56254i
$747$	− 7300.64i	− 0.357586i
$748$	41472.9i	2.02727i
$749$	23988.2i	1.17024i
$750$	21460.7	1.04485
$751$	−8214.69	−0.399146	−0.199573	−	0.979883i	$-0.563955\pi$
$751$	−8214.69	−0.399146	−0.199573	+	0.979883i	$0.563955\pi$
$752$	− 105402.i	− 5.11121i
$753$	−21273.0	−1.02953
$754$	0	0
$755$	17032.9	0.821048
$756$	− 12314.5i	− 0.592426i
$757$	−20846.9	−1.00092	−0.500459	−	0.865760i	$-0.666835\pi$
$757$	−20846.9	−1.00092	−0.500459	+	0.865760i	$0.666835\pi$
$758$	−77790.3	−3.72753
$759$	10965.0i	0.524382i
$760$	58833.5i	2.80804i
$761$	− 3563.06i	− 0.169725i	−0.996393	−	0.0848627i	$-0.972955\pi$
$761$	− 3563.06i	− 0.169725i	0.996393	−	0.0848627i	$-0.0270451\pi$
$762$	− 298.553i	− 0.0141935i
$763$	−4864.60	−0.230813
$764$	−49583.6	−2.34800
$765$	2065.97i	0.0976410i
$766$	47846.7	2.25688
$767$	0	0
$768$	52384.9	2.46130
$769$	− 32557.9i	− 1.52675i	−0.645958	−	0.763373i	$-0.723542\pi$
$769$	− 32557.9i	− 1.52675i	0.645958	−	0.763373i	$-0.276458\pi$
$770$	33239.7	1.55568
$771$	−2377.59	−0.111059
$772$	− 22867.3i	− 1.06608i
$773$	− 20126.4i	− 0.936477i	−0.883602	−	0.468238i	$-0.844889\pi$
$773$	− 20126.4i	− 0.936477i	0.883602	−	0.468238i	$-0.155111\pi$
$774$	− 15862.0i	− 0.736627i
$775$	− 9958.68i	− 0.461583i
$776$	−18343.2	−0.848562
$777$	5216.17	0.240835
$778$	28085.5i	1.29423i
$779$	16191.3	0.744690
$780$	0	0
$781$	−5211.22	−0.238761
$782$	15606.0i	0.713642i
$783$	−3043.17	−0.138894
$784$	−17816.2	−0.811597
$785$	− 14254.3i	− 0.648097i
$786$	21248.5i	0.964260i
$787$	18828.4i	0.852806i	0.904533	+	0.426403i	$0.140219\pi$
$787$	18828.4i	0.852806i	−0.904533	+	0.426403i	$0.859781\pi$
$788$	97163.1i	4.39250i
$789$	−1012.40	−0.0456812
$790$	18111.5	0.815667
$791$	− 20904.8i	− 0.939682i
$792$	35187.6	1.57871
$793$	0	0
$794$	71703.7	3.20487
$795$	− 7013.90i	− 0.312902i
$796$	30674.9	1.36588
$797$	−7288.52	−0.323930	−0.161965	−	0.986796i	$-0.551783\pi$
$797$	−7288.52	−0.323930	−0.161965	+	0.986796i	$0.551783\pi$
$798$	− 40584.1i	− 1.80033i
$799$	− 15175.3i	− 0.671921i
$800$	− 71079.9i	− 3.14132i
$801$	− 10175.8i	− 0.448869i
$802$	35861.7	1.57896
$803$	2112.15	0.0928221
$804$	65833.6i	2.88778i
$805$	9226.97	0.403985
$806$	0	0
$807$	−10571.8	−0.461146
$808$	67666.1i	2.94614i
$809$	−33799.0	−1.46886	−0.734430	−	0.678684i	$-0.762551\pi$
$809$	−33799.0	−1.46886	−0.734430	+	0.678684i	$0.762551\pi$
$810$	2720.05	0.117991
$811$	29055.3i	1.25804i	0.777390	+	0.629019i	$0.216543\pi$
$811$	29055.3i	1.25804i	−0.777390	+	0.629019i	$0.783457\pi$
$812$	51406.2i	2.22168i
$813$	24940.9i	1.07591i
$814$	23128.7i	0.995898i
$815$	−22245.5	−0.956106
$816$	29696.0	1.27398
$817$	− 38563.3i	− 1.65136i
$818$	69287.3	2.96158
$819$	0	0
$820$	18331.2	0.780676
$821$	8921.98i	0.379268i	0.981855	+	0.189634i	$0.0607302\pi$
$821$	8921.98i	0.379268i	−0.981855	+	0.189634i	$0.939270\pi$
$822$	25636.3	1.08780
$823$	9148.48	0.387480	0.193740	−	0.981053i	$-0.437938\pi$
$823$	9148.48	0.387480	0.193740	+	0.981053i	$0.437938\pi$
$824$	87685.1i	3.70711i
$825$	− 12894.5i	− 0.544155i
$826$	− 13616.6i	− 0.573588i
$827$	− 20206.0i	− 0.849615i	−0.905284	−	0.424807i	$-0.860342\pi$
$827$	− 20206.0i	− 0.849615i	0.905284	−	0.424807i	$-0.139658\pi$
$828$	15157.2	0.636170
$829$	10832.4	0.453830	0.226915	−	0.973915i	$-0.427136\pi$
$829$	10832.4	0.453830	0.226915	+	0.973915i	$0.427136\pi$
$830$	− 27240.2i	− 1.13918i
$831$	7500.98	0.313124
$832$	0	0
$833$	−2565.09	−0.106693
$834$	51308.1i	2.13028i
$835$	−12087.4	−0.500961
$836$	132749.	5.49190
$837$	− 3054.61i	− 0.126144i
$838$	− 43880.1i	− 1.80884i
$839$	7047.44i	0.289994i	0.989432	+	0.144997i	$0.0463172\pi$
$839$	7047.44i	0.289994i	−0.989432	+	0.144997i	$0.953683\pi$
$840$	− 29610.0i	− 1.21624i
$841$	−11685.4	−0.479127
$842$	1572.60	0.0643650
$843$	− 9159.39i	− 0.374218i
$844$	−118981.	−4.85250
$845$	0	0
$846$	−19979.8	−0.811961
$847$	− 21350.5i	− 0.866132i
$848$	−100817.	−4.08262
$849$	−3476.40	−0.140530
$850$	− 18352.0i	− 0.740551i
$851$	6420.28i	0.258618i
$852$	7203.58i	0.289660i
$853$	− 24085.5i	− 0.966790i	−0.875402	−	0.483395i	$-0.839404\pi$
$853$	− 24085.5i	− 0.966790i	0.875402	−	0.483395i	$-0.160596\pi$
$854$	24634.5	0.987089
$855$	6612.90	0.264511
$856$	94749.9i	3.78328i
$857$	4161.31	0.165866	0.0829332	−	0.996555i	$-0.473571\pi$
$857$	4161.31	0.165866	0.0829332	+	0.996555i	$0.473571\pi$
$858$	0	0
$859$	27387.5	1.08784	0.543918	−	0.839139i	$-0.316940\pi$
$859$	27387.5	1.08784	0.543918	+	0.839139i	$0.316940\pi$
$860$	− 43660.1i	− 1.73116i
$861$	−8148.84	−0.322546
$862$	69099.1	2.73031
$863$	19420.4i	0.766022i	0.923744	+	0.383011i	$0.125113\pi$
$863$	19420.4i	0.766022i	−0.923744	+	0.383011i	$0.874887\pi$
$864$	− 21802.2i	− 0.858480i
$865$	21751.1i	0.854984i
$866$	− 72992.8i	− 2.86420i
$867$	−10463.5	−0.409873
$868$	−51599.5	−2.01774
$869$	− 26335.1i	− 1.02803i
$870$	−11354.7	−0.442483
$871$	0	0
$872$	−19214.5	−0.746198
$873$	2061.79i	0.0799324i
$874$	49952.6	1.93326
$875$	−26258.8	−1.01453
$876$	− 2919.67i	− 0.112610i
$877$	− 9908.85i	− 0.381526i	−0.981636	−	0.190763i	$-0.938904\pi$
$877$	− 9908.85i	− 0.381526i	0.981636	−	0.190763i	$-0.0610961\pi$
$878$	25599.3i	0.983981i
$879$	− 25072.4i	− 0.962081i
$880$	77851.4	2.98224
$881$	−20323.3	−0.777197	−0.388598	−	0.921407i	$-0.627041\pi$
$881$	−20323.3	−0.777197	−0.388598	+	0.921407i	$0.627041\pi$
$882$	3377.19i	0.128929i
$883$	−12443.3	−0.474238	−0.237119	−	0.971481i	$-0.576203\pi$
$883$	−12443.3	−0.474238	−0.237119	+	0.971481i	$0.576203\pi$
$884$	0	0
$885$	2218.74	0.0842737
$886$	16269.0i	0.616895i
$887$	42313.5	1.60174	0.800872	−	0.598835i	$-0.204369\pi$
$887$	42313.5	1.60174	0.800872	+	0.598835i	$0.204369\pi$
$888$	20603.1	0.778599
$889$	365.301i	0.0137816i
$890$	− 37968.0i	− 1.42999i
$891$	− 3955.10i	− 0.148710i
$892$	− 28303.7i	− 1.06242i
$893$	−48574.3	−1.82024
$894$	−6973.00	−0.260864
$895$	− 4682.97i	− 0.174899i
$896$	−133452.	−4.97580
$897$	0	0
$898$	−47871.3	−1.77894
$899$	12751.3i	0.473059i
$900$	−17824.3	−0.660159
$901$	−14515.1	−0.536702
$902$	− 36132.3i	− 1.33379i
$903$	19408.4i	0.715249i
$904$	− 82571.0i	− 3.03791i
$905$	11527.0i	0.423393i
$906$	−46409.0	−1.70181
$907$	11002.3	0.402786	0.201393	−	0.979511i	$-0.435453\pi$
$907$	11002.3	0.402786	0.201393	+	0.979511i	$0.435453\pi$
$908$	25540.8i	0.933483i
$909$	7605.69	0.277519
$910$	0	0
$911$	−40803.4	−1.48395	−0.741974	−	0.670428i	$-0.766110\pi$
$911$	−40803.4	−1.48395	−0.741974	+	0.670428i	$0.766110\pi$
$912$	− 95052.8i	− 3.45122i
$913$	−39608.8	−1.43577
$914$	28691.8	1.03834
$915$	4014.02i	0.145027i
$916$	4904.80i	0.176920i
$917$	− 25999.1i	− 0.936276i
$918$	− 5629.08i	− 0.202383i
$919$	2642.69	0.0948579	0.0474290	−	0.998875i	$-0.484897\pi$
$919$	2642.69	0.0948579	0.0474290	+	0.998875i	$0.484897\pi$
$920$	36445.3	1.30605
$921$	− 25287.9i	− 0.904739i
$922$	16187.7	0.578214
$923$	0	0
$924$	−66810.8	−2.37869
$925$	− 7550.00i	− 0.268370i
$926$	−2190.19	−0.0777258
$927$	9855.84	0.349200
$928$	91012.3i	3.21942i
$929$	12687.7i	0.448083i	0.974580	+	0.224042i	$0.0719252\pi$
$929$	12687.7i	0.448083i	−0.974580	+	0.224042i	$0.928075\pi$
$930$	− 11397.4i	− 0.401865i
$931$	8210.52i	0.289032i
$932$	−57.6796	−0.00202721
$933$	−10808.6	−0.379269
$934$	91826.7i	3.21698i
$935$	11208.7	0.392046
$936$	0	0
$937$	−23066.2	−0.804205	−0.402103	−	0.915595i	$-0.631721\pi$
$937$	−23066.2	−0.804205	−0.402103	+	0.915595i	$0.631721\pi$
$938$	− 109195.i	− 3.80099i
$939$	19880.9	0.690937
$940$	−54994.1	−1.90820
$941$	12660.7i	0.438605i	0.975657	+	0.219303i	$0.0703782\pi$
$941$	12660.7i	0.438605i	−0.975657	+	0.219303i	$0.929622\pi$
$942$	38838.1i	1.34332i
$943$	− 10029.9i	− 0.346362i
$944$	− 31891.8i	− 1.09957i
$945$	−3328.18	−0.114567
$946$	−86057.5	−2.95769
$947$	− 21059.1i	− 0.722629i	−0.932444	−	0.361315i	$-0.882328\pi$
$947$	− 21059.1i	− 0.722629i	0.932444	−	0.361315i	$-0.117672\pi$
$948$	−36403.5	−1.24718
$949$	0	0
$950$	−58742.3	−2.00616
$951$	− 11951.6i	− 0.407526i
$952$	−61277.3	−2.08614
$953$	−35646.2	−1.21164	−0.605820	−	0.795602i	$-0.707155\pi$
$953$	−35646.2	−1.21164	−0.605820	+	0.795602i	$0.707155\pi$
$954$	19110.5i	0.648560i
$955$	13400.7i	0.454070i
$956$	89345.8i	3.02265i
$957$	16510.4i	0.557685i
$958$	49169.0	1.65822
$959$	−31367.9	−1.05623
$960$	− 43083.2i	− 1.44844i
$961$	16991.7	0.570365
$962$	0	0
$963$	10649.9	0.356375
$964$	14805.0i	0.494643i
$965$	−6180.22	−0.206164
$966$	−25140.4	−0.837349
$967$	53452.1i	1.77756i	0.458329	+	0.888782i	$0.348448\pi$
$967$	53452.1i	1.77756i	−0.458329	+	0.888782i	$0.651552\pi$
$968$	− 84331.7i	− 2.80013i
$969$	− 13685.3i	− 0.453699i
$970$	7692.95i	0.254645i
$971$	6502.60	0.214911	0.107455	−	0.994210i	$-0.465730\pi$
$971$	6502.60	0.214911	0.107455	+	0.994210i	$0.465730\pi$
$972$	−5467.22	−0.180413
$973$	− 62779.2i	− 2.06846i
$974$	2592.99	0.0853027
$975$	0	0
$976$	57696.9	1.89225
$977$	− 4351.84i	− 0.142505i	−0.997458	−	0.0712527i	$-0.977300\pi$
$977$	− 4351.84i	− 0.142505i	0.997458	−	0.0712527i	$-0.0226997\pi$
$978$	60611.5	1.98174
$979$	−55207.5	−1.80229
$980$	9295.67i	0.302999i
$981$	2159.71i	0.0702899i
$982$	− 103956.i	− 3.37817i
$983$	− 48698.2i	− 1.58009i	−0.613046	−	0.790047i	$-0.710056\pi$
$983$	− 48698.2i	− 1.58009i	0.613046	−	0.790047i	$-0.289944\pi$
$984$	−32186.8	−1.04276
$985$	26259.8	0.849448
$986$	23498.3i	0.758964i
$987$	24446.7	0.788397
$988$	0	0
$989$	−23888.6	−0.768063
$990$	− 14757.3i	− 0.473755i
$991$	46630.0	1.49470	0.747351	−	0.664429i	$-0.231325\pi$
$991$	46630.0	1.49470	0.747351	+	0.664429i	$0.231325\pi$
$992$	−91354.3	−2.92389
$993$	− 12518.5i	− 0.400062i
$994$	− 11948.2i	− 0.381261i
$995$	− 8290.35i	− 0.264143i
$996$	54752.0i	1.74185i
$997$	37175.5	1.18090	0.590452	−	0.807073i	$-0.298950\pi$
$997$	37175.5	1.18090	0.590452	+	0.807073i	$0.298950\pi$
$998$	31213.3	0.990021
$999$	− 2315.80i	− 0.0733420i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.b.k.337.1		18
13.5	odd	4		507.4.a.o.1.1	✓	9
13.8	odd	4		507.4.a.p.1.9	yes	9
13.12	even	2	inner	507.4.b.k.337.18		18
39.5	even	4		1521.4.a.bi.1.9		9
39.8	even	4		1521.4.a.bf.1.1		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.4.a.o.1.1	✓	9	13.5	odd	4
507.4.a.p.1.9	yes	9	13.8	odd	4
507.4.b.k.337.1		18	1.1	even	1	trivial
507.4.b.k.337.18		18	13.12	even	2	inner
1521.4.a.bf.1.1		9	39.8	even	4
1521.4.a.bi.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)

\(f(q)\)	\(=\)	\(q-5.52257i q^{2} +3.00000 q^{3} -22.4988 q^{4} +6.08065i q^{5} -16.5677i q^{6} +20.2718i q^{7} +80.0709i q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-5.52257i q^{2} +3.00000 q^{3} -22.4988 q^{4} +6.08065i q^{5} -16.5677i q^{6} +20.2718i q^{7} +80.0709i q^{8} +9.00000 q^{9} +33.5808 q^{10} -48.8284i q^{11} -67.4965 q^{12} +111.953 q^{14} +18.2419i q^{15} +262.207 q^{16} +37.7513 q^{17} -49.7032i q^{18} -120.837i q^{19} -136.807i q^{20} +60.8155i q^{21} -269.658 q^{22} -74.8543 q^{23} +240.213i q^{24} +88.0257 q^{25} +27.0000 q^{27} -456.092i q^{28} -112.710 q^{29} +100.742 q^{30} -113.134i q^{31} -807.490i q^{32} -146.485i q^{33} -208.485i q^{34} -123.266 q^{35} -202.489 q^{36} -85.7704i q^{37} -667.331 q^{38} -486.883 q^{40} +133.993i q^{41} +335.858 q^{42} +319.135 q^{43} +1098.58i q^{44} +54.7258i q^{45} +413.388i q^{46} -401.982i q^{47} +786.620 q^{48} -67.9471 q^{49} -486.129i q^{50} +113.254 q^{51} -384.493 q^{53} -149.110i q^{54} +296.908 q^{55} -1623.18 q^{56} -362.511i q^{57} +622.450i q^{58} -121.629i q^{59} -410.422i q^{60} +220.043 q^{61} -624.790 q^{62} +182.446i q^{63} -2361.77 q^{64} -808.975 q^{66} -975.363i q^{67} -849.361 q^{68} -224.563 q^{69} +680.745i q^{70} -106.725i q^{71} +720.638i q^{72} +43.2566i q^{73} -473.673 q^{74} +264.077 q^{75} +2718.69i q^{76} +989.841 q^{77} +539.339 q^{79} +1594.39i q^{80} +81.0000 q^{81} +739.986 q^{82} -811.183i q^{83} -1368.28i q^{84} +229.553i q^{85} -1762.45i q^{86} -338.130 q^{87} +3909.73 q^{88} -1130.64i q^{89} +302.227 q^{90} +1684.13 q^{92} -339.401i q^{93} -2219.98 q^{94} +734.767 q^{95} -2422.47i q^{96} +229.088i q^{97} +375.243i q^{98} -439.456i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 54 q^{3} - 88 q^{4} + 162 q^{9}+O(q^{10}) \) 18 * q + 54 * q^3 - 88 * q^4 + 162 * q^9	\( 18 q + 54 q^{3} - 88 q^{4} + 162 q^{9} + 108 q^{10} - 264 q^{12} + 316 q^{14} + 432 q^{16} - 356 q^{17} - 1260 q^{22} - 300 q^{23} + 40 q^{25} + 486 q^{27} - 194 q^{29} + 324 q^{30} - 836 q^{35} - 792 q^{36} + 1320 q^{38} - 3012 q^{40} + 948 q^{42} - 484 q^{43} + 1296 q^{48} + 76 q^{49} - 1068 q^{51} - 302 q^{53} + 4128 q^{55} - 4552 q^{56} - 2680 q^{61} - 694 q^{62} - 1786 q^{64} - 3780 q^{66} + 5570 q^{68} - 900 q^{69} - 2382 q^{74} + 120 q^{75} + 4284 q^{77} - 3182 q^{79} + 1458 q^{81} - 3034 q^{82} - 582 q^{87} + 7432 q^{88} + 972 q^{90} + 1030 q^{92} - 1384 q^{94} - 8316 q^{95}+O(q^{100}) \) 18 * q + 54 * q^3 - 88 * q^4 + 162 * q^9 + 108 * q^10 - 264 * q^12 + 316 * q^14 + 432 * q^16 - 356 * q^17 - 1260 * q^22 - 300 * q^23 + 40 * q^25 + 486 * q^27 - 194 * q^29 + 324 * q^30 - 836 * q^35 - 792 * q^36 + 1320 * q^38 - 3012 * q^40 + 948 * q^42 - 484 * q^43 + 1296 * q^48 + 76 * q^49 - 1068 * q^51 - 302 * q^53 + 4128 * q^55 - 4552 * q^56 - 2680 * q^61 - 694 * q^62 - 1786 * q^64 - 3780 * q^66 + 5570 * q^68 - 900 * q^69 - 2382 * q^74 + 120 * q^75 + 4284 * q^77 - 3182 * q^79 + 1458 * q^81 - 3034 * q^82 - 582 * q^87 + 7432 * q^88 + 972 * q^90 + 1030 * q^92 - 1384 * q^94 - 8316 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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