LMFDB - Embedded newform 507.4.b.k.337.16

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} + 6703200 x^{4} + 137781 x^{2} + 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			337.16
Root			$4.14324i$ of defining polynomial
Character	$\chi$	$=$	507.337
Dual form			507.4.b.k.337.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.69820i q^{2} +3.00000 q^{3} -14.0731 q^{4} +4.47249i q^{5} +14.0946i q^{6} -27.2096i q^{7} -28.5326i q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+4.69820i q^{2} +3.00000 q^{3} -14.0731 q^{4} +4.47249i q^{5} +14.0946i q^{6} -27.2096i q^{7} -28.5326i q^{8} +9.00000 q^{9} -21.0127 q^{10} +5.99207i q^{11} -42.2193 q^{12} +127.836 q^{14} +13.4175i q^{15} +21.4672 q^{16} -105.037 q^{17} +42.2838i q^{18} -156.462i q^{19} -62.9418i q^{20} -81.6287i q^{21} -28.1519 q^{22} +175.423 q^{23} -85.5978i q^{24} +104.997 q^{25} +27.0000 q^{27} +382.923i q^{28} +204.886 q^{29} -63.0380 q^{30} +31.9570i q^{31} -127.404i q^{32} +17.9762i q^{33} -493.483i q^{34} +121.695 q^{35} -126.658 q^{36} -344.140i q^{37} +735.088 q^{38} +127.612 q^{40} +46.5921i q^{41} +383.508 q^{42} +173.286 q^{43} -84.3269i q^{44} +40.2524i q^{45} +824.175i q^{46} +265.613i q^{47} +64.4016 q^{48} -397.361 q^{49} +493.296i q^{50} -315.110 q^{51} +172.912 q^{53} +126.851i q^{54} -26.7995 q^{55} -776.360 q^{56} -469.385i q^{57} +962.596i q^{58} +137.566i q^{59} -188.825i q^{60} -58.9384 q^{61} -150.140 q^{62} -244.886i q^{63} +770.306 q^{64} -84.4558 q^{66} +211.668i q^{67} +1478.19 q^{68} +526.270 q^{69} +571.746i q^{70} -436.317i q^{71} -256.793i q^{72} -1159.11i q^{73} +1616.84 q^{74} +314.990 q^{75} +2201.90i q^{76} +163.042 q^{77} -1017.51 q^{79} +96.0118i q^{80} +81.0000 q^{81} -218.899 q^{82} -150.251i q^{83} +1148.77i q^{84} -469.775i q^{85} +814.131i q^{86} +614.659 q^{87} +170.969 q^{88} -565.984i q^{89} -189.114 q^{90} -2468.75 q^{92} +95.8709i q^{93} -1247.90 q^{94} +699.773 q^{95} -382.211i q^{96} -286.741i q^{97} -1866.88i q^{98} +53.9286i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	4.69820i	1.66106i	0.556970	+	0.830532i	$0.311964\pi$
$2$	4.69820i	1.66106i	−0.556970	+	0.830532i	$0.688036\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−14.0731	−1.75914
$5$	4.47249i	0.400032i	0.979793	+	0.200016i	$0.0640994\pi$
$5$	4.47249i	0.400032i	−0.979793	+	0.200016i	$0.935901\pi$
$6$	14.0946i	0.959016i
$7$	− 27.2096i	− 1.46918i	−0.678512	−	0.734589i	$-0.737375\pi$
$7$	− 27.2096i	− 1.46918i	0.678512	−	0.734589i	$-0.262625\pi$
$8$	− 28.5326i	− 1.26098i
$9$	9.00000	0.333333
$10$	−21.0127	−0.664479
$11$	5.99207i	0.164243i	0.996622	+	0.0821216i	$0.0261696\pi$
$11$	5.99207i	0.164243i	−0.996622	+	0.0821216i	$0.973830\pi$
$12$	−42.2193	−1.01564
$13$	0	0
$13$	0	0
$14$	127.836	2.44040
$15$	13.4175i	0.230958i
$16$	21.4672	0.335425
$17$	−105.037	−1.49854	−0.749268	−	0.662267i	$-0.769594\pi$
$17$	−105.037	−1.49854	−0.749268	+	0.662267i	$0.769594\pi$
$18$	42.2838i	0.553688i
$19$	− 156.462i	− 1.88920i	−0.328227	−	0.944599i	$-0.606451\pi$
$19$	− 156.462i	− 1.88920i	0.328227	−	0.944599i	$-0.393549\pi$
$20$	− 62.9418i	− 0.703711i
$21$	− 81.6287i	− 0.848231i
$22$	−28.1519	−0.272819
$23$	175.423	1.59036	0.795181	−	0.606372i	$-0.207376\pi$
$23$	175.423	1.59036	0.795181	+	0.606372i	$0.207376\pi$
$24$	− 85.5978i	− 0.728024i
$25$	104.997	0.839975
$26$	0	0
$27$	27.0000	0.192450
$28$	382.923i	2.58449i
$29$	204.886	1.31195	0.655973	−	0.754785i	$-0.272259\pi$
$29$	204.886	1.31195	0.655973	+	0.754785i	$0.272259\pi$
$30$	−63.0380	−0.383637
$31$	31.9570i	0.185150i	0.995706	+	0.0925748i	$0.0295097\pi$
$31$	31.9570i	0.185150i	−0.995706	+	0.0925748i	$0.970490\pi$
$32$	− 127.404i	− 0.703813i
$33$	17.9762i	0.0948259i
$34$	− 493.483i	− 2.48916i
$35$	121.695	0.587718
$36$	−126.658	−0.586379
$37$	− 344.140i	− 1.52909i	−0.644571	−	0.764545i	$-0.722964\pi$
$37$	− 344.140i	− 1.52909i	0.644571	−	0.764545i	$-0.277036\pi$
$38$	735.088	3.13808
$39$	0	0
$40$	127.612	0.504430
$41$	46.5921i	0.177475i	0.996055	+	0.0887373i	$0.0282832\pi$
$41$	46.5921i	0.177475i	−0.996055	+	0.0887373i	$0.971717\pi$
$42$	383.508	1.40897
$43$	173.286	0.614554	0.307277	−	0.951620i	$-0.400582\pi$
$43$	173.286	0.614554	0.307277	+	0.951620i	$0.400582\pi$
$44$	− 84.3269i	− 0.288926i
$45$	40.2524i	0.133344i
$46$	824.175i	2.64169i
$47$	265.613i	0.824334i	0.911108	+	0.412167i	$0.135228\pi$
$47$	265.613i	0.824334i	−0.911108	+	0.412167i	$0.864772\pi$
$48$	64.4016	0.193658
$49$	−397.361	−1.15849
$50$	493.296i	1.39525i
$51$	−315.110	−0.865180
$52$	0	0
$53$	172.912	0.448137	0.224068	−	0.974573i	$-0.428066\pi$
$53$	172.912	0.448137	0.224068	+	0.974573i	$0.428066\pi$
$54$	126.851i	0.319672i
$55$	−26.7995	−0.0657025
$56$	−776.360	−1.85260
$57$	− 469.385i	− 1.09073i
$58$	962.596i	2.17923i
$59$	137.566i	0.303551i	0.988415	+	0.151775i	$0.0484991\pi$
$59$	137.566i	0.303551i	−0.988415	+	0.151775i	$0.951501\pi$
$60$	− 188.825i	− 0.406288i
$61$	−58.9384	−0.123710	−0.0618548	−	0.998085i	$-0.519702\pi$
$61$	−58.9384	−0.123710	−0.0618548	+	0.998085i	$0.519702\pi$
$62$	−150.140	−0.307546
$63$	− 244.886i	− 0.489726i
$64$	770.306	1.50450
$65$	0	0
$66$	−84.4558	−0.157512
$67$	211.668i	0.385961i	0.981203	+	0.192980i	$0.0618154\pi$
$67$	211.668i	0.385961i	−0.981203	+	0.192980i	$0.938185\pi$
$68$	1478.19	2.63613
$69$	526.270	0.918196
$70$	571.746i	0.976238i
$71$	− 436.317i	− 0.729314i	−0.931142	−	0.364657i	$-0.881186\pi$
$71$	− 436.317i	− 0.729314i	0.931142	−	0.364657i	$-0.118814\pi$
$72$	− 256.793i	− 0.420325i
$73$	− 1159.11i	− 1.85840i	−0.369579	−	0.929199i	$-0.620498\pi$
$73$	− 1159.11i	− 1.85840i	0.369579	−	0.929199i	$-0.379502\pi$
$74$	1616.84	2.53992
$75$	314.990	0.484960
$76$	2201.90i	3.32336i
$77$	163.042	0.241303
$78$	0	0
$79$	−1017.51	−1.44910	−0.724548	−	0.689224i	$-0.757952\pi$
$79$	−1017.51	−1.44910	−0.724548	+	0.689224i	$0.757952\pi$
$80$	96.0118i	0.134181i
$81$	81.0000	0.111111
$82$	−218.899	−0.294797
$83$	− 150.251i	− 0.198701i	−0.995053	−	0.0993504i	$-0.968324\pi$
$83$	− 150.251i	− 0.198701i	0.995053	−	0.0993504i	$-0.0316765\pi$
$84$	1148.77i	1.49215i
$85$	− 469.775i	− 0.599462i
$86$	814.131i	1.02081i
$87$	614.659	0.757452
$88$	170.969	0.207107
$89$	− 565.984i	− 0.674092i	−0.941488	−	0.337046i	$-0.890572\pi$
$89$	− 565.984i	− 0.674092i	0.941488	−	0.337046i	$-0.109428\pi$
$90$	−189.114	−0.221493
$91$	0	0
$92$	−2468.75	−2.79766
$93$	95.8709i	0.106896i
$94$	−1247.90	−1.36927
$95$	699.773	0.755739
$96$	− 382.211i	− 0.406346i
$97$	− 286.741i	− 0.300145i	−0.988675	−	0.150073i	$-0.952049\pi$
$97$	− 286.741i	− 0.300145i	0.988675	−	0.150073i	$-0.0479508\pi$
$98$	− 1866.88i	− 1.92432i
$99$	53.9286i	0.0547478i
$100$	−1477.63	−1.47763
$101$	1218.57	1.20052	0.600258	−	0.799807i	$-0.295065\pi$
$101$	1218.57	1.20052	0.600258	+	0.799807i	$0.295065\pi$
$102$	− 1480.45i	− 1.43712i
$103$	−74.2485	−0.0710283	−0.0355142	−	0.999369i	$-0.511307\pi$
$103$	−74.2485	−0.0710283	−0.0355142	+	0.999369i	$0.511307\pi$
$104$	0	0
$105$	365.084	0.339319
$106$	812.374i	0.744384i
$107$	−1253.46	−1.13249	−0.566246	−	0.824237i	$-0.691605\pi$
$107$	−1253.46	−1.13249	−0.566246	+	0.824237i	$0.691605\pi$
$108$	−379.973	−0.338546
$109$	722.643i	0.635015i	0.948256	+	0.317507i	$0.102846\pi$
$109$	722.643i	0.635015i	−0.948256	+	0.317507i	$0.897154\pi$
$110$	− 125.909i	− 0.109136i
$111$	− 1032.42i	− 0.882820i
$112$	− 584.113i	− 0.492799i
$113$	−855.913	−0.712544	−0.356272	−	0.934382i	$-0.615952\pi$
$113$	−855.913	−0.712544	−0.356272	+	0.934382i	$0.615952\pi$
$114$	2205.26	1.81177
$115$	784.580i	0.636195i
$116$	−2883.38	−2.30789
$117$	0	0
$118$	−646.310	−0.504218
$119$	2858.00i	2.20162i
$120$	382.836	0.291233
$121$	1295.10	0.973024
$122$	− 276.905i	− 0.205490i
$123$	139.776i	0.102465i
$124$	− 449.733i	− 0.325703i
$125$	1028.66i	0.736048i
$126$	1150.52	0.813467
$127$	−726.104	−0.507333	−0.253667	−	0.967292i	$-0.581637\pi$
$127$	−726.104	−0.507333	−0.253667	+	0.967292i	$0.581637\pi$
$128$	2599.82i	1.79526i
$129$	519.857	0.354813
$130$	0	0
$131$	−1456.91	−0.971685	−0.485843	−	0.874046i	$-0.661487\pi$
$131$	−1456.91	−0.971685	−0.485843	+	0.874046i	$0.661487\pi$
$132$	− 252.981i	− 0.166812i
$133$	−4257.25	−2.77557
$134$	−994.460	−0.641106
$135$	120.757i	0.0769862i
$136$	2996.97i	1.88962i
$137$	− 1806.80i	− 1.12675i	−0.826200	−	0.563377i	$-0.809502\pi$
$137$	− 1806.80i	− 1.12675i	0.826200	−	0.563377i	$-0.190498\pi$
$138$	2472.52i	1.52518i
$139$	1229.28	0.750117	0.375059	−	0.927001i	$-0.377623\pi$
$139$	1229.28	0.750117	0.375059	+	0.927001i	$0.377623\pi$
$140$	−1712.62	−1.03388
$141$	796.840i	0.475929i
$142$	2049.90	1.21144
$143$	0	0
$144$	193.205	0.111808
$145$	916.352i	0.524820i
$146$	5445.71	3.08692
$147$	−1192.08	−0.668853
$148$	4843.12i	2.68988i
$149$	446.127i	0.245290i	0.992451	+	0.122645i	$0.0391376\pi$
$149$	446.127i	0.245290i	−0.992451	+	0.122645i	$0.960862\pi$
$150$	1479.89i	0.805549i
$151$	− 207.208i	− 0.111671i	−0.998440	−	0.0558355i	$-0.982218\pi$
$151$	− 207.208i	− 0.111671i	0.998440	−	0.0558355i	$-0.0177822\pi$
$152$	−4464.26	−2.38223
$153$	−945.329	−0.499512
$154$	766.002i	0.400819i
$155$	−142.927	−0.0740658
$156$	0	0
$157$	1096.60	0.557442	0.278721	−	0.960372i	$-0.410089\pi$
$157$	1096.60	0.557442	0.278721	+	0.960372i	$0.410089\pi$
$158$	− 4780.46i	− 2.40704i
$159$	518.735	0.258732
$160$	569.812	0.281547
$161$	− 4773.20i	− 2.33653i
$162$	380.554i	0.184563i
$163$	3154.25i	1.51571i	0.652425	+	0.757853i	$0.273752\pi$
$163$	3154.25i	1.51571i	−0.652425	+	0.757853i	$0.726248\pi$
$164$	− 655.694i	− 0.312202i
$165$	−80.3984	−0.0379334
$166$	705.908	0.330055
$167$	− 3679.65i	− 1.70503i	−0.522705	−	0.852514i	$-0.675077\pi$
$167$	− 3679.65i	− 1.70503i	0.522705	−	0.852514i	$-0.324923\pi$
$168$	−2329.08	−1.06960
$169$	0	0
$170$	2207.10	0.995745
$171$	− 1408.15i	− 0.629733i
$172$	−2438.66	−1.08108
$173$	3666.99	1.61154	0.805768	−	0.592231i	$-0.201753\pi$
$173$	3666.99	1.61154	0.805768	+	0.592231i	$0.201753\pi$
$174$	2887.79i	1.25818i
$175$	− 2856.92i	− 1.23407i
$176$	128.633i	0.0550913i
$177$	412.697i	0.175255i
$178$	2659.11	1.11971
$179$	−4173.50	−1.74269	−0.871347	−	0.490666i	$-0.836753\pi$
$179$	−4173.50	−1.74269	−0.871347	+	0.490666i	$0.836753\pi$
$180$	− 566.476i	− 0.234570i
$181$	499.197	0.205000	0.102500	−	0.994733i	$-0.467316\pi$
$181$	499.197	0.205000	0.102500	+	0.994733i	$0.467316\pi$
$182$	0	0
$183$	−176.815	−0.0714238
$184$	− 5005.29i	− 2.00541i
$185$	1539.16	0.611685
$186$	−450.421	−0.177562
$187$	− 629.386i	− 0.246124i
$188$	− 3738.00i	− 1.45012i
$189$	− 734.659i	− 0.282744i
$190$	3287.68i	1.25533i
$191$	−3086.68	−1.16934	−0.584671	−	0.811271i	$-0.698776\pi$
$191$	−3086.68	−1.16934	−0.584671	+	0.811271i	$0.698776\pi$
$192$	2310.92	0.868625
$193$	− 1644.14i	− 0.613201i	−0.951838	−	0.306601i	$-0.900808\pi$
$193$	− 1644.14i	− 0.613201i	0.951838	−	0.306601i	$-0.0991916\pi$
$194$	1347.17	0.498561
$195$	0	0
$196$	5592.10	2.03794
$197$	− 1371.21i	− 0.495911i	−0.968771	−	0.247956i	$-0.920241\pi$
$197$	− 1371.21i	− 0.495911i	0.968771	−	0.247956i	$-0.0797587\pi$
$198$	−253.367	−0.0909396
$199$	4627.33	1.64836	0.824178	−	0.566332i	$-0.191638\pi$
$199$	4627.33	1.64836	0.824178	+	0.566332i	$0.191638\pi$
$200$	− 2995.83i	− 1.05919i
$201$	635.004i	0.222835i
$202$	5725.08i	1.99413i
$203$	− 5574.87i	− 1.92748i
$204$	4434.57	1.52197
$205$	−208.383	−0.0709955
$206$	− 348.834i	− 0.117983i
$207$	1578.81	0.530121
$208$	0	0
$209$	937.528	0.310288
$210$	1715.24i	0.563631i
$211$	5088.31	1.66016	0.830080	−	0.557644i	$-0.188294\pi$
$211$	5088.31	1.66016	0.830080	+	0.557644i	$0.188294\pi$
$212$	−2433.40	−0.788334
$213$	− 1308.95i	− 0.421069i
$214$	− 5889.01i	− 1.88114i
$215$	775.019i	0.245841i
$216$	− 770.380i	− 0.242675i
$217$	869.535	0.272018
$218$	−3395.12	−1.05480
$219$	− 3477.32i	− 1.07295i
$220$	377.151	0.115580
$221$	0	0
$222$	4850.52	1.46642
$223$	4744.82i	1.42483i	0.701759	+	0.712415i	$0.252398\pi$
$223$	4744.82i	1.42483i	−0.701759	+	0.712415i	$0.747602\pi$
$224$	−3466.60	−1.03403
$225$	944.971	0.279992
$226$	− 4021.25i	− 1.18358i
$227$	1145.52i	0.334937i	0.985877	+	0.167469i	$0.0535593\pi$
$227$	1145.52i	0.334937i	−0.985877	+	0.167469i	$0.946441\pi$
$228$	6605.70i	1.91874i
$229$	− 1348.47i	− 0.389123i	−0.980890	−	0.194561i	$-0.937672\pi$
$229$	− 1348.47i	− 0.389123i	0.980890	−	0.194561i	$-0.0623283\pi$
$230$	−3686.12	−1.05676
$231$	489.125	0.139316
$232$	− 5845.94i	− 1.65433i
$233$	952.002	0.267673	0.133836	−	0.991003i	$-0.457270\pi$
$233$	952.002	0.267673	0.133836	+	0.991003i	$0.457270\pi$
$234$	0	0
$235$	−1187.95	−0.329760
$236$	− 1935.97i	− 0.533987i
$237$	−3052.52	−0.836636
$238$	−13427.5	−3.65703
$239$	− 3069.12i	− 0.830647i	−0.909674	−	0.415323i	$-0.863668\pi$
$239$	− 3069.12i	− 0.830647i	0.909674	−	0.415323i	$-0.136332\pi$
$240$	288.036i	0.0774692i
$241$	− 2508.17i	− 0.670396i	−0.942148	−	0.335198i	$-0.891197\pi$
$241$	− 2508.17i	− 0.670396i	0.942148	−	0.335198i	$-0.108803\pi$
$242$	6084.62i	1.61626i
$243$	243.000	0.0641500
$244$	829.446	0.217622
$245$	− 1777.19i	− 0.463432i
$246$	−656.697	−0.170201
$247$	0	0
$248$	911.815	0.233469
$249$	− 450.752i	− 0.114720i
$250$	−4832.85	−1.22262
$251$	−3405.91	−0.856490	−0.428245	−	0.903663i	$-0.640868\pi$
$251$	−3405.91	−0.856490	−0.428245	+	0.903663i	$0.640868\pi$
$252$	3446.31i	0.861495i
$253$	1051.15i	0.261206i
$254$	− 3411.38i	− 0.842714i
$255$	− 1409.33i	− 0.346100i
$256$	−6052.04	−1.47755
$257$	4733.95	1.14901	0.574506	−	0.818501i	$-0.305194\pi$
$257$	4733.95	1.14901	0.574506	+	0.818501i	$0.305194\pi$
$258$	2442.39i	0.589367i
$259$	−9363.91	−2.24651
$260$	0	0
$261$	1843.98	0.437315
$262$	− 6844.85i	− 1.61403i
$263$	−1866.11	−0.437526	−0.218763	−	0.975778i	$-0.570202\pi$
$263$	−1866.11	−0.437526	−0.218763	+	0.975778i	$0.570202\pi$
$264$	512.908	0.119573
$265$	773.346i	0.179269i
$266$	− 20001.4i	− 4.61040i
$267$	− 1697.95i	− 0.389187i
$268$	− 2978.83i	− 0.678958i
$269$	1580.19	0.358164	0.179082	−	0.983834i	$-0.442687\pi$
$269$	1580.19	0.358164	0.179082	+	0.983834i	$0.442687\pi$
$270$	−567.342	−0.127879
$271$	− 4922.09i	− 1.10330i	−0.834074	−	0.551652i	$-0.813998\pi$
$271$	− 4922.09i	− 1.10330i	0.834074	−	0.551652i	$-0.186002\pi$
$272$	−2254.84	−0.502646
$273$	0	0
$274$	8488.71	1.87161
$275$	629.148i	0.137960i
$276$	−7406.25	−1.61523
$277$	−2687.08	−0.582856	−0.291428	−	0.956593i	$-0.594130\pi$
$277$	−2687.08	−0.582856	−0.291428	+	0.956593i	$0.594130\pi$
$278$	5775.41i	1.24599i
$279$	287.613i	0.0617165i
$280$	− 3472.26i	− 0.741098i
$281$	− 883.753i	− 0.187617i	−0.995590	−	0.0938083i	$-0.970096\pi$
$281$	− 883.753i	− 0.187617i	0.995590	−	0.0938083i	$-0.0299041\pi$
$282$	−3743.71	−0.790550
$283$	469.776	0.0986759	0.0493379	−	0.998782i	$-0.484289\pi$
$283$	469.776	0.0986759	0.0493379	+	0.998782i	$0.484289\pi$
$284$	6140.33i	1.28296i
$285$	2099.32	0.436326
$286$	0	0
$287$	1267.75	0.260742
$288$	− 1146.63i	− 0.234604i
$289$	6119.68	1.24561
$290$	−4305.20	−0.871760
$291$	− 860.222i	− 0.173289i
$292$	16312.2i	3.26918i
$293$	3403.76i	0.678668i	0.940666	+	0.339334i	$0.110202\pi$
$293$	3403.76i	0.678668i	−0.940666	+	0.339334i	$0.889798\pi$
$294$	− 5600.65i	− 1.11101i
$295$	−615.261	−0.121430
$296$	−9819.22	−1.92814
$297$	161.786i	0.0316086i
$298$	−2095.99	−0.407442
$299$	0	0
$300$	−4432.89	−0.853110
$301$	− 4715.03i	− 0.902889i
$302$	973.504	0.185493
$303$	3655.70	0.693118
$304$	− 3358.79i	− 0.633684i
$305$	− 263.602i	− 0.0494878i
$306$	− 4441.35i	− 0.829722i
$307$	888.862i	0.165244i	0.996581	+	0.0826222i	$0.0263295\pi$
$307$	888.862i	0.165244i	−0.996581	+	0.0826222i	$0.973671\pi$
$308$	−2294.50	−0.424484
$309$	−222.745	−0.0410082
$310$	− 671.501i	− 0.123028i
$311$	1218.36	0.222145	0.111072	−	0.993812i	$-0.464571\pi$
$311$	1218.36	0.222145	0.111072	+	0.993812i	$0.464571\pi$
$312$	0	0
$313$	−4870.41	−0.879526	−0.439763	−	0.898114i	$-0.644938\pi$
$313$	−4870.41	−0.879526	−0.439763	+	0.898114i	$0.644938\pi$
$314$	5152.06i	0.925948i
$315$	1095.25	0.195906
$316$	14319.5	2.54916
$317$	4340.93i	0.769120i	0.923100	+	0.384560i	$0.125647\pi$
$317$	4340.93i	0.769120i	−0.923100	+	0.384560i	$0.874353\pi$
$318$	2437.12i	0.429770i
$319$	1227.69i	0.215478i
$320$	3445.19i	0.601849i
$321$	−3760.38	−0.653844
$322$	22425.4	3.88112
$323$	16434.2i	2.83103i
$324$	−1139.92	−0.195460
$325$	0	0
$326$	−14819.3	−2.51769
$327$	2167.93i	0.366626i
$328$	1329.39	0.223791
$329$	7227.23	1.21109
$330$	− 377.728i	− 0.0630098i
$331$	907.289i	0.150662i	0.997159	+	0.0753310i	$0.0240013\pi$
$331$	907.289i	0.150662i	−0.997159	+	0.0753310i	$0.975999\pi$
$332$	2114.49i	0.349542i
$333$	− 3097.26i	− 0.509697i
$334$	17287.7	2.83216
$335$	−946.684	−0.154397
$336$	− 1752.34i	− 0.284518i
$337$	−8660.88	−1.39997	−0.699983	−	0.714160i	$-0.746809\pi$
$337$	−8660.88	−1.39997	−0.699983	+	0.714160i	$0.746809\pi$
$338$	0	0
$339$	−2567.74	−0.411388
$340$	6611.19i	1.05454i
$341$	−191.488	−0.0304096
$342$	6615.79	1.04603
$343$	1479.14i	0.232846i
$344$	− 4944.29i	− 0.774937i
$345$	2353.74i	0.367308i
$346$	17228.2i	2.67687i
$347$	−347.605	−0.0537763	−0.0268882	−	0.999638i	$-0.508560\pi$
$347$	−347.605	−0.0537763	−0.0268882	+	0.999638i	$0.508560\pi$
$348$	−8650.15	−1.33246
$349$	− 10970.2i	− 1.68258i	−0.540581	−	0.841292i	$-0.681795\pi$
$349$	− 10970.2i	− 1.68258i	0.540581	−	0.841292i	$-0.318205\pi$
$350$	13422.4	2.04988
$351$	0	0
$352$	763.411	0.115596
$353$	− 10384.8i	− 1.56580i	−0.622146	−	0.782901i	$-0.713739\pi$
$353$	− 10384.8i	− 1.56580i	0.622146	−	0.782901i	$-0.286261\pi$
$354$	−1938.93	−0.291110
$355$	1951.42	0.291749
$356$	7965.15i	1.18582i
$357$	8574.00i	1.27110i
$358$	− 19608.0i	− 2.89473i
$359$	8665.80i	1.27399i	0.770867	+	0.636996i	$0.219823\pi$
$359$	8665.80i	1.27399i	−0.770867	+	0.636996i	$0.780177\pi$
$360$	1148.51	0.168143
$361$	−17621.2	−2.56907
$362$	2345.33i	0.340518i
$363$	3885.29	0.561776
$364$	0	0
$365$	5184.09	0.743419
$366$	− 830.714i	− 0.118640i
$367$	1234.22	0.175547	0.0877734	−	0.996140i	$-0.472025\pi$
$367$	1234.22	0.175547	0.0877734	+	0.996140i	$0.472025\pi$
$368$	3765.85	0.533447
$369$	419.329i	0.0591582i
$370$	7231.31i	1.01605i
$371$	− 4704.85i	− 0.658393i
$372$	− 1349.20i	− 0.188045i
$373$	−427.483	−0.0593410	−0.0296705	−	0.999560i	$-0.509446\pi$
$373$	−427.483	−0.0593410	−0.0296705	+	0.999560i	$0.509446\pi$
$374$	2956.98	0.408829
$375$	3085.98i	0.424958i
$376$	7578.64	1.03946
$377$	0	0
$378$	3451.57	0.469656
$379$	124.241i	0.0168386i	0.999965	+	0.00841929i	$0.00267998\pi$
$379$	124.241i	0.0168386i	−0.999965	+	0.00841929i	$0.997320\pi$
$380$	−9847.98	−1.32945
$381$	−2178.31	−0.292909
$382$	− 14501.8i	− 1.94235i
$383$	9341.21i	1.24625i	0.782122	+	0.623125i	$0.214137\pi$
$383$	9341.21i	1.24625i	−0.782122	+	0.623125i	$0.785863\pi$
$384$	7799.46i	1.03650i
$385$	729.202i	0.0965288i
$386$	7724.51	1.01857
$387$	1559.57	0.204851
$388$	4035.33i	0.527997i
$389$	−11368.9	−1.48182	−0.740908	−	0.671607i	$-0.765604\pi$
$389$	−11368.9	−1.48182	−0.740908	+	0.671607i	$0.765604\pi$
$390$	0	0
$391$	−18425.9	−2.38321
$392$	11337.7i	1.46082i
$393$	−4370.73	−0.561003
$394$	6442.21	0.823741
$395$	− 4550.80i	− 0.579685i
$396$	− 758.942i	− 0.0963088i
$397$	12077.8i	1.52687i	0.645883	+	0.763436i	$0.276489\pi$
$397$	12077.8i	1.52687i	−0.645883	+	0.763436i	$0.723511\pi$
$398$	21740.1i	2.73802i
$399$	−12771.8	−1.60248
$400$	2253.99	0.281748
$401$	4856.74i	0.604823i	0.953178	+	0.302411i	$0.0977916\pi$
$401$	4856.74i	0.604823i	−0.953178	+	0.302411i	$0.902208\pi$
$402$	−2983.38	−0.370143
$403$	0	0
$404$	−17149.0	−2.11187
$405$	362.272i	0.0444480i
$406$	26191.8	3.20167
$407$	2062.11	0.251143
$408$	8990.90i	1.09097i
$409$	2981.80i	0.360490i	0.983622	+	0.180245i	$0.0576890\pi$
$409$	2981.80i	0.360490i	−0.983622	+	0.180245i	$0.942311\pi$
$410$	− 979.023i	− 0.117928i
$411$	− 5420.40i	− 0.650532i
$412$	1044.91	0.124949
$413$	3743.10	0.445971
$414$	7417.57i	0.880565i
$415$	671.995	0.0794866
$416$	0	0
$417$	3687.84	0.433080
$418$	4404.70i	0.515409i
$419$	7774.01	0.906408	0.453204	−	0.891407i	$-0.350281\pi$
$419$	7774.01	0.906408	0.453204	+	0.891407i	$0.350281\pi$
$420$	−5137.86	−0.596909
$421$	3959.71i	0.458396i	0.973380	+	0.229198i	$0.0736103\pi$
$421$	3959.71i	0.458396i	−0.973380	+	0.229198i	$0.926390\pi$
$422$	23905.9i	2.75763i
$423$	2390.52i	0.274778i
$424$	− 4933.62i	− 0.565089i
$425$	−11028.5	−1.25873
$426$	6149.71	0.699424
$427$	1603.69i	0.181752i
$428$	17640.1	1.99221
$429$	0	0
$430$	−3641.19	−0.408358
$431$	4375.12i	0.488961i	0.969654	+	0.244480i	$0.0786174\pi$
$431$	4375.12i	0.488961i	−0.969654	+	0.244480i	$0.921383\pi$
$432$	579.614	0.0645525
$433$	−8992.74	−0.998068	−0.499034	−	0.866582i	$-0.666312\pi$
$433$	−8992.74	−0.998068	−0.499034	+	0.866582i	$0.666312\pi$
$434$	4085.25i	0.451839i
$435$	2749.06i	0.303005i
$436$	− 10169.8i	− 1.11708i
$437$	− 27447.0i	− 3.00451i
$438$	16337.1	1.78223
$439$	195.465	0.0212506	0.0106253	−	0.999944i	$-0.496618\pi$
$439$	195.465	0.0212506	0.0106253	+	0.999944i	$0.496618\pi$
$440$	764.659i	0.0828493i
$441$	−3576.25	−0.386162
$442$	0	0
$443$	7369.97	0.790424	0.395212	−	0.918590i	$-0.370671\pi$
$443$	7369.97	0.790424	0.395212	+	0.918590i	$0.370671\pi$
$444$	14529.4i	1.55300i
$445$	2531.36	0.269658
$446$	−22292.1	−2.36673
$447$	1338.38i	0.141618i
$448$	− 20959.7i	− 2.21038i
$449$	164.281i	0.0172670i	0.999963	+	0.00863351i	$0.00274817\pi$
$449$	164.281i	0.0172670i	−0.999963	+	0.00863351i	$0.997252\pi$
$450$	4439.67i	0.465084i
$451$	−279.183	−0.0291490
$452$	12045.3	1.25346
$453$	− 621.623i	− 0.0644733i
$454$	−5381.88	−0.556352
$455$	0	0
$456$	−13392.8	−1.37538
$457$	11687.0i	1.19627i	0.801397	+	0.598133i	$0.204091\pi$
$457$	11687.0i	1.19627i	−0.801397	+	0.598133i	$0.795909\pi$
$458$	6335.36	0.646358
$459$	−2835.99	−0.288393
$460$	− 11041.5i	− 1.11915i
$461$	1057.53i	0.106842i	0.998572	+	0.0534211i	$0.0170126\pi$
$461$	1057.53i	0.106842i	−0.998572	+	0.0534211i	$0.982987\pi$
$462$	2298.01i	0.231413i
$463$	− 8554.74i	− 0.858688i	−0.903141	−	0.429344i	$-0.858745\pi$
$463$	− 8554.74i	− 0.858688i	0.903141	−	0.429344i	$-0.141255\pi$
$464$	4398.33	0.440059
$465$	−428.782	−0.0427619
$466$	4472.70i	0.444622i
$467$	7705.26	0.763506	0.381753	−	0.924264i	$-0.375321\pi$
$467$	7705.26	0.763506	0.381753	+	0.924264i	$0.375321\pi$
$468$	0	0
$469$	5759.40	0.567046
$470$	− 5581.24i	− 0.547752i
$471$	3289.81	0.321839
$472$	3925.10	0.382770
$473$	1038.34i	0.100936i
$474$	− 14341.4i	− 1.38971i
$475$	− 16428.0i	− 1.58688i
$476$	− 40220.9i	− 3.87295i
$477$	1556.21	0.149379
$478$	14419.3	1.37976
$479$	4508.59i	0.430069i	0.976606	+	0.215034i	$0.0689863\pi$
$479$	4508.59i	0.430069i	−0.976606	+	0.215034i	$0.931014\pi$
$480$	1709.44	0.162551
$481$	0	0
$482$	11783.9	1.11357
$483$	− 14319.6i	− 1.34899i
$484$	−18226.0	−1.71168
$485$	1282.45	0.120068
$486$	1141.66i	0.106557i
$487$	6725.73i	0.625815i	0.949784	+	0.312908i	$0.101303\pi$
$487$	6725.73i	0.625815i	−0.949784	+	0.312908i	$0.898697\pi$
$488$	1681.67i	0.155995i
$489$	9462.76i	0.875094i
$490$	8349.61	0.769790
$491$	−11517.5	−1.05861	−0.529303	−	0.848433i	$-0.677546\pi$
$491$	−11517.5	−1.05861	−0.529303	+	0.848433i	$0.677546\pi$
$492$	− 1967.08i	− 0.180250i
$493$	−21520.5	−1.96600
$494$	0	0
$495$	−241.195	−0.0219008
$496$	686.026i	0.0621038i
$497$	−11872.0	−1.07149
$498$	2117.72	0.190557
$499$	19907.9i	1.78598i	0.450080	+	0.892988i	$0.351395\pi$
$499$	19907.9i	1.78598i	−0.450080	+	0.892988i	$0.648605\pi$
$500$	− 14476.4i	− 1.29481i
$501$	− 11038.9i	− 0.984398i
$502$	− 16001.6i	− 1.42269i
$503$	−5735.48	−0.508415	−0.254207	−	0.967150i	$-0.581815\pi$
$503$	−5735.48	−0.508415	−0.254207	+	0.967150i	$0.581815\pi$
$504$	−6987.24	−0.617533
$505$	5450.04i	0.480244i
$506$	−4938.51	−0.433881
$507$	0	0
$508$	10218.5	0.892469
$509$	9253.84i	0.805834i	0.915237	+	0.402917i	$0.132004\pi$
$509$	9253.84i	0.805834i	−0.915237	+	0.402917i	$0.867996\pi$
$510$	6621.29	0.574894
$511$	−31538.8	−2.73032
$512$	− 7635.12i	− 0.659039i
$513$	− 4224.46i	− 0.363576i
$514$	22241.1i	1.90858i
$515$	− 332.076i	− 0.0284136i
$516$	−7315.99	−0.624164
$517$	−1591.57	−0.135391
$518$	− 43993.5i	− 3.73159i
$519$	11001.0	0.930421
$520$	0	0
$521$	3887.42	0.326892	0.163446	−	0.986552i	$-0.447739\pi$
$521$	3887.42	0.326892	0.163446	+	0.986552i	$0.447739\pi$
$522$	8663.37i	0.726409i
$523$	4782.27	0.399836	0.199918	−	0.979813i	$-0.435932\pi$
$523$	4782.27	0.399836	0.199918	+	0.979813i	$0.435932\pi$
$524$	20503.2	1.70933
$525$	− 8570.76i	− 0.712492i
$526$	− 8767.37i	− 0.726759i
$527$	− 3356.65i	− 0.277453i
$528$	385.898i	0.0318070i
$529$	18606.4	1.52925
$530$	−3633.34	−0.297777
$531$	1238.09i	0.101184i
$532$	59912.7	4.88261
$533$	0	0
$534$	7977.32	0.646465
$535$	− 5606.09i	− 0.453033i
$536$	6039.44	0.486687
$537$	−12520.5	−1.00615
$538$	7424.06i	0.594933i
$539$	− 2381.01i	− 0.190274i
$540$	− 1699.43i	− 0.135429i
$541$	14872.6i	1.18192i	0.806699	+	0.590962i	$0.201252\pi$
$541$	14872.6i	1.18192i	−0.806699	+	0.590962i	$0.798748\pi$
$542$	23125.0	1.83266
$543$	1497.59	0.118357
$544$	13382.0i	1.05469i
$545$	−3232.01	−0.254026
$546$	0	0
$547$	16965.3	1.32611	0.663055	−	0.748570i	$-0.269259\pi$
$547$	16965.3	1.32611	0.663055	+	0.748570i	$0.269259\pi$
$548$	25427.3i	1.98212i
$549$	−530.446	−0.0412366
$550$	−2955.86	−0.229161
$551$	− 32056.8i	− 2.47852i
$552$	− 15015.9i	− 1.15782i
$553$	27686.0i	2.12898i
$554$	− 12624.5i	− 0.968162i
$555$	4617.49	0.353156
$556$	−17299.8	−1.31956
$557$	− 1934.05i	− 0.147125i	−0.997291	−	0.0735623i	$-0.976563\pi$
$557$	− 1934.05i	− 0.147125i	0.997291	−	0.0735623i	$-0.0234368\pi$
$558$	−1351.26	−0.102515
$559$	0	0
$560$	2612.44	0.197135
$561$	− 1888.16i	− 0.142100i
$562$	4152.05	0.311643
$563$	−6592.13	−0.493473	−0.246736	−	0.969083i	$-0.579358\pi$
$563$	−6592.13	−0.493473	−0.246736	+	0.969083i	$0.579358\pi$
$564$	− 11214.0i	− 0.837225i
$565$	− 3828.06i	− 0.285040i
$566$	2207.10i	0.163907i
$567$	− 2203.98i	− 0.163242i
$568$	−12449.3	−0.919646
$569$	14477.7	1.06667	0.533336	−	0.845903i	$-0.320938\pi$
$569$	14477.7	1.06667	0.533336	+	0.845903i	$0.320938\pi$
$570$	9863.03i	0.724766i
$571$	4998.21	0.366320	0.183160	−	0.983083i	$-0.441367\pi$
$571$	4998.21	0.366320	0.183160	+	0.983083i	$0.441367\pi$
$572$	0	0
$573$	−9260.04	−0.675120
$574$	5956.15i	0.433109i
$575$	18418.9	1.33586
$576$	6932.75	0.501501
$577$	9291.73i	0.670398i	0.942147	+	0.335199i	$0.108804\pi$
$577$	9291.73i	0.670398i	−0.942147	+	0.335199i	$0.891196\pi$
$578$	28751.5i	2.06904i
$579$	− 4932.42i	− 0.354032i
$580$	− 12895.9i	− 0.923230i
$581$	−4088.26	−0.291927
$582$	4041.50	0.287844
$583$	1036.10i	0.0736034i
$584$	−33072.3	−2.34339
$585$	0	0
$586$	−15991.5	−1.12731
$587$	− 5602.64i	− 0.393945i	−0.980409	−	0.196972i	$-0.936889\pi$
$587$	− 5602.64i	− 0.393945i	0.980409	−	0.196972i	$-0.0631109\pi$
$588$	16776.3	1.17660
$589$	5000.04	0.349784
$590$	− 2890.62i	− 0.201703i
$591$	− 4113.62i	− 0.286314i
$592$	− 7387.73i	− 0.512895i
$593$	10885.8i	0.753839i	0.926246	+	0.376919i	$0.123017\pi$
$593$	10885.8i	0.753839i	−0.926246	+	0.376919i	$0.876983\pi$
$594$	−760.102	−0.0525040
$595$	−12782.4	−0.880717
$596$	− 6278.38i	− 0.431498i
$597$	13882.0	0.951678
$598$	0	0
$599$	20403.2	1.39174	0.695872	−	0.718166i	$-0.255018\pi$
$599$	20403.2	1.39174	0.695872	+	0.718166i	$0.255018\pi$
$600$	− 8987.50i	− 0.611522i
$601$	−6312.19	−0.428419	−0.214209	−	0.976788i	$-0.568717\pi$
$601$	−6312.19	−0.428419	−0.214209	+	0.976788i	$0.568717\pi$
$602$	22152.2	1.49976
$603$	1905.01i	0.128654i
$604$	2916.05i	0.196445i
$605$	5792.30i	0.389241i
$606$	17175.2i	1.15131i
$607$	−21848.6	−1.46097	−0.730484	−	0.682930i	$-0.760705\pi$
$607$	−21848.6	−1.46097	−0.730484	+	0.682930i	$0.760705\pi$
$608$	−19933.8	−1.32964
$609$	− 16724.6i	− 1.11283i
$610$	1238.45	0.0822025
$611$	0	0
$612$	13303.7	0.878710
$613$	− 1335.14i	− 0.0879704i	−0.999032	−	0.0439852i	$-0.985995\pi$
$613$	− 1335.14i	− 0.0879704i	0.999032	−	0.0439852i	$-0.0140054\pi$
$614$	−4176.05	−0.274482
$615$	−625.148	−0.0409893
$616$	− 4652.00i	− 0.304277i
$617$	18908.3i	1.23374i	0.787064	+	0.616871i	$0.211600\pi$
$617$	18908.3i	1.23374i	−0.787064	+	0.616871i	$0.788400\pi$
$618$	− 1046.50i	− 0.0681173i
$619$	− 6722.49i	− 0.436510i	−0.975892	−	0.218255i	$-0.929964\pi$
$619$	− 6722.49i	− 0.436510i	0.975892	−	0.218255i	$-0.0700365\pi$
$620$	2011.43	0.130292
$621$	4736.43	0.306065
$622$	5724.11i	0.368997i
$623$	−15400.2	−0.990362
$624$	0	0
$625$	8523.93	0.545532
$626$	− 22882.2i	− 1.46095i
$627$	2812.59	0.179145
$628$	−15432.6	−0.980617
$629$	36147.3i	2.29140i
$630$	5145.71i	0.325413i
$631$	8098.44i	0.510925i	0.966819	+	0.255463i	$0.0822278\pi$
$631$	8098.44i	0.510925i	−0.966819	+	0.255463i	$0.917772\pi$
$632$	29032.2i	1.82727i
$633$	15264.9	0.958494
$634$	−20394.6	−1.27756
$635$	− 3247.50i	− 0.202950i
$636$	−7300.21	−0.455145
$637$	0	0
$638$	−5767.94	−0.357923
$639$	− 3926.85i	− 0.243105i
$640$	−11627.7	−0.718163
$641$	10955.9	0.675090	0.337545	−	0.941309i	$-0.390403\pi$
$641$	10955.9	0.675090	0.337545	+	0.941309i	$0.390403\pi$
$642$	− 17667.0i	− 1.08608i
$643$	− 28125.0i	− 1.72495i	−0.506104	−	0.862473i	$-0.668915\pi$
$643$	− 28125.0i	− 1.72495i	0.506104	−	0.862473i	$-0.331085\pi$
$644$	67173.7i	4.11027i
$645$	2325.06i	0.141936i
$646$	−77211.1	−4.70253
$647$	29001.4	1.76223	0.881115	−	0.472901i	$-0.156793\pi$
$647$	29001.4	1.76223	0.881115	+	0.472901i	$0.156793\pi$
$648$	− 2311.14i	− 0.140108i
$649$	−824.302	−0.0498562
$650$	0	0
$651$	2608.61	0.157050
$652$	− 44390.1i	− 2.66634i
$653$	19506.3	1.16898	0.584488	−	0.811402i	$-0.301295\pi$
$653$	19506.3	1.16898	0.584488	+	0.811402i	$0.301295\pi$
$654$	−10185.4	−0.608989
$655$	− 6516.01i	− 0.388705i
$656$	1000.20i	0.0595294i
$657$	− 10432.0i	− 0.619466i
$658$	33955.0i	2.01171i
$659$	5985.86	0.353833	0.176917	−	0.984226i	$-0.443388\pi$
$659$	5985.86	0.353833	0.176917	+	0.984226i	$0.443388\pi$
$660$	1131.45	0.0667300
$661$	280.836i	0.0165254i	0.999966	+	0.00826268i	$0.00263012\pi$
$661$	280.836i	0.0165254i	−0.999966	+	0.00826268i	$0.997370\pi$
$662$	−4262.63	−0.250259
$663$	0	0
$664$	−4287.04	−0.250557
$665$	− 19040.5i	− 1.11032i
$666$	14551.6	0.846639
$667$	35941.8	2.08647
$668$	51784.0i	2.99938i
$669$	14234.5i	0.822626i
$670$	− 4447.71i	− 0.256463i
$671$	− 353.163i	− 0.0203185i
$672$	−10399.8	−0.596996
$673$	14868.9	0.851640	0.425820	−	0.904808i	$-0.359986\pi$
$673$	14868.9	0.851640	0.425820	+	0.904808i	$0.359986\pi$
$674$	− 40690.6i	− 2.32543i
$675$	2834.91	0.161653
$676$	0	0
$677$	−20769.5	−1.17908	−0.589540	−	0.807739i	$-0.700691\pi$
$677$	−20769.5	−1.17908	−0.589540	+	0.807739i	$0.700691\pi$
$678$	− 12063.7i	− 0.683341i
$679$	−7802.09	−0.440967
$680$	−13403.9	−0.755907
$681$	3436.56i	0.193376i
$682$	− 899.650i	− 0.0505123i
$683$	− 14980.8i	− 0.839275i	−0.907692	−	0.419638i	$-0.862157\pi$
$683$	− 14980.8i	− 0.839275i	0.907692	−	0.419638i	$-0.137843\pi$
$684$	19817.1i	1.10779i
$685$	8080.90	0.450738
$686$	−6949.30	−0.386772
$687$	− 4045.40i	− 0.224660i
$688$	3719.96	0.206137
$689$	0	0
$690$	−11058.3	−0.610122
$691$	9472.45i	0.521489i	0.965408	+	0.260745i	$0.0839680\pi$
$691$	9472.45i	0.521489i	−0.965408	+	0.260745i	$0.916032\pi$
$692$	−51605.8	−2.83491
$693$	1467.37	0.0804342
$694$	− 1633.12i	− 0.0893260i
$695$	5497.95i	0.300071i
$696$	− 17537.8i	− 0.955128i
$697$	− 4893.87i	− 0.265952i
$698$	51540.3	2.79488
$699$	2856.01	0.154541
$700$	40205.7i	2.17090i
$701$	−1035.34	−0.0557834	−0.0278917	−	0.999611i	$-0.508879\pi$
$701$	−1035.34	−0.0557834	−0.0278917	+	0.999611i	$0.508879\pi$
$702$	0	0
$703$	−53844.8	−2.88875
$704$	4615.72i	0.247105i
$705$	−3563.86	−0.190387
$706$	48790.0	2.60090
$707$	− 33156.7i	− 1.76377i
$708$	− 5807.92i	− 0.308298i
$709$	− 16800.4i	− 0.889917i	−0.895551	−	0.444958i	$-0.853219\pi$
$709$	− 16800.4i	− 0.889917i	0.895551	−	0.444958i	$-0.146781\pi$
$710$	9168.18i	0.484614i
$711$	−9157.57	−0.483032
$712$	−16149.0	−0.850013
$713$	5606.00i	0.294455i
$714$	−40282.4	−2.11139
$715$	0	0
$716$	58734.1	3.06564
$717$	− 9207.35i	− 0.479574i
$718$	−40713.7	−2.11618
$719$	−14873.1	−0.771449	−0.385724	−	0.922614i	$-0.626048\pi$
$719$	−14873.1	−0.771449	−0.385724	+	0.922614i	$0.626048\pi$
$720$	864.107i	0.0447269i
$721$	2020.27i	0.104353i
$722$	− 82788.1i	− 4.26739i
$723$	− 7524.51i	− 0.387053i
$724$	−7025.24	−0.360623
$725$	21512.4	1.10200
$726$	18253.9i	0.933146i
$727$	2318.92	0.118300	0.0591500	−	0.998249i	$-0.481161\pi$
$727$	2318.92	0.118300	0.0591500	+	0.998249i	$0.481161\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	24355.9i	1.23487i
$731$	−18201.3	−0.920931
$732$	2488.34	0.125644
$733$	− 3350.95i	− 0.168854i	−0.996430	−	0.0844270i	$-0.973094\pi$
$733$	− 3350.95i	− 0.168854i	0.996430	−	0.0844270i	$-0.0269060\pi$
$734$	5798.61i	0.291595i
$735$	− 5331.58i	− 0.267562i
$736$	− 22349.6i	− 1.11932i
$737$	−1268.33	−0.0633915
$738$	−1970.09	−0.0982656
$739$	29348.1i	1.46088i	0.682979	+	0.730438i	$0.260684\pi$
$739$	29348.1i	1.46088i	−0.682979	+	0.730438i	$0.739316\pi$
$740$	−21660.8	−1.07604
$741$	0	0
$742$	22104.3	1.09363
$743$	23622.2i	1.16637i	0.812338	+	0.583187i	$0.198195\pi$
$743$	23622.2i	1.16637i	−0.812338	+	0.583187i	$0.801805\pi$
$744$	2735.45	0.134793
$745$	−1995.30	−0.0981236
$746$	− 2008.40i	− 0.0985693i
$747$	− 1352.26i	− 0.0662336i
$748$	8857.41i	0.432966i
$749$	34106.1i	1.66383i
$750$	−14498.5	−0.705882
$751$	−29751.4	−1.44560	−0.722798	−	0.691059i	$-0.757144\pi$
$751$	−29751.4	−1.44560	−0.722798	+	0.691059i	$0.757144\pi$
$752$	5701.97i	0.276502i
$753$	−10217.7	−0.494495
$754$	0	0
$755$	926.735	0.0446720
$756$	10338.9i	0.497385i
$757$	−36907.2	−1.77201	−0.886006	−	0.463673i	$-0.846531\pi$
$757$	−36907.2	−1.77201	−0.886006	+	0.463673i	$0.846531\pi$
$758$	−583.709	−0.0279700
$759$	3153.45i	0.150807i
$760$	− 19966.4i	− 0.952968i
$761$	20208.2i	0.962613i	0.876552	+	0.481306i	$0.159838\pi$
$761$	20208.2i	0.962613i	−0.876552	+	0.481306i	$0.840162\pi$
$762$	− 10234.2i	− 0.486541i
$763$	19662.8	0.932950
$764$	43439.1	2.05703
$765$	− 4227.98i	− 0.199821i
$766$	−43886.9	−2.07010
$767$	0	0
$768$	−18156.1	−0.853063
$769$	24751.0i	1.16066i	0.814382	+	0.580329i	$0.197076\pi$
$769$	24751.0i	1.16066i	−0.814382	+	0.580329i	$0.802924\pi$
$770$	−3425.94	−0.160341
$771$	14201.9	0.663382
$772$	23138.2i	1.07871i
$773$	− 30673.4i	− 1.42722i	−0.700541	−	0.713612i	$-0.747058\pi$
$773$	− 30673.4i	− 1.42722i	0.700541	−	0.713612i	$-0.252942\pi$
$774$	7327.18i	0.340271i
$775$	3355.38i	0.155521i
$776$	−8181.46	−0.378476
$777$	−28091.7	−1.29702
$778$	− 53413.4i	− 2.46139i
$779$	7289.87	0.335285
$780$	0	0
$781$	2614.44	0.119785
$782$	− 86568.5i	− 3.95867i
$783$	5531.93	0.252484
$784$	−8530.23	−0.388585
$785$	4904.55i	0.222995i
$786$	− 20534.6i	− 0.931862i
$787$	− 29009.9i	− 1.31397i	−0.753905	−	0.656983i	$-0.771832\pi$
$787$	− 29009.9i	− 1.31397i	0.753905	−	0.656983i	$-0.228168\pi$
$788$	19297.1i	0.872375i
$789$	−5598.33	−0.252606
$790$	21380.6	0.962894
$791$	23289.0i	1.04685i
$792$	1538.72	0.0690355
$793$	0	0
$794$	−56744.0	−2.53623
$795$	2320.04i	0.103501i
$796$	−65120.8	−2.89968
$797$	−6778.24	−0.301252	−0.150626	−	0.988591i	$-0.548129\pi$
$797$	−6778.24	−0.301252	−0.150626	+	0.988591i	$0.548129\pi$
$798$	− 60004.3i	− 2.66182i
$799$	− 27899.1i	− 1.23529i
$800$	− 13377.0i	− 0.591185i
$801$	− 5093.86i	− 0.224697i
$802$	−22817.9	−1.00465
$803$	6945.44	0.305229
$804$	− 8936.48i	− 0.391997i
$805$	21348.1	0.934685
$806$	0	0
$807$	4740.58	0.206786
$808$	− 34768.9i	− 1.51382i
$809$	−34862.9	−1.51510	−0.757549	−	0.652778i	$-0.773603\pi$
$809$	−34862.9	−1.51510	−0.757549	+	0.652778i	$0.773603\pi$
$810$	−1702.03	−0.0738310
$811$	22665.4i	0.981370i	0.871337	+	0.490685i	$0.163253\pi$
$811$	22665.4i	0.981370i	−0.871337	+	0.490685i	$0.836747\pi$
$812$	78455.6i	3.39070i
$813$	− 14766.3i	− 0.636993i
$814$	9688.21i	0.417164i
$815$	−14107.4	−0.606331
$816$	−6764.52	−0.290203
$817$	− 27112.6i	− 1.16101i
$818$	−14009.1	−0.598797
$819$	0	0
$820$	2932.59	0.124891
$821$	20748.7i	0.882016i	0.897503	+	0.441008i	$0.145379\pi$
$821$	20748.7i	0.882016i	−0.897503	+	0.441008i	$0.854621\pi$
$822$	25466.1	1.08058
$823$	−6141.41	−0.260117	−0.130058	−	0.991506i	$-0.541516\pi$
$823$	−6141.41	−0.260117	−0.130058	+	0.991506i	$0.541516\pi$
$824$	2118.50i	0.0895650i
$825$	1887.44i	0.0796513i
$826$	17585.8i	0.740786i
$827$	28383.5i	1.19346i	0.802442	+	0.596730i	$0.203533\pi$
$827$	28383.5i	1.19346i	−0.802442	+	0.596730i	$0.796467\pi$
$828$	−22218.8	−0.932555
$829$	908.734	0.0380720	0.0190360	−	0.999819i	$-0.493940\pi$
$829$	908.734	0.0380720	0.0190360	+	0.999819i	$0.493940\pi$
$830$	3157.17i	0.132032i
$831$	−8061.25	−0.336512
$832$	0	0
$833$	41737.4	1.73603
$834$	17326.2i	0.719375i
$835$	16457.2	0.682065
$836$	−13193.9	−0.545839
$837$	862.838i	0.0356321i
$838$	36523.8i	1.50560i
$839$	27820.3i	1.14477i	0.819985	+	0.572385i	$0.193982\pi$
$839$	27820.3i	1.14477i	−0.819985	+	0.572385i	$0.806018\pi$
$840$	− 10416.8i	− 0.427873i
$841$	17589.3	0.721200
$842$	−18603.5	−0.761425
$843$	− 2651.26i	− 0.108321i
$844$	−71608.3	−2.92045
$845$	0	0
$846$	−11231.1	−0.456424
$847$	− 35239.0i	− 1.42955i
$848$	3711.93	0.150316
$849$	1409.33	0.0569706
$850$	− 51814.1i	− 2.09084i
$851$	− 60370.3i	− 2.43181i
$852$	18421.0i	0.740719i
$853$	− 5802.11i	− 0.232896i	−0.993197	−	0.116448i	$-0.962849\pi$
$853$	− 5802.11i	− 0.232896i	0.993197	−	0.116448i	$-0.0371509\pi$
$854$	−7534.45	−0.301901
$855$	6297.96	0.251913
$856$	35764.5i	1.42804i
$857$	43311.1	1.72635	0.863173	−	0.504909i	$-0.168474\pi$
$857$	43311.1	1.72635	0.863173	+	0.504909i	$0.168474\pi$
$858$	0	0
$859$	−16698.2	−0.663254	−0.331627	−	0.943411i	$-0.607598\pi$
$859$	−16698.2	−0.663254	−0.331627	+	0.943411i	$0.607598\pi$
$860$	− 10906.9i	− 0.432468i
$861$	3803.25	0.150539
$862$	−20555.2	−0.812196
$863$	19429.2i	0.766369i	0.923672	+	0.383185i	$0.125173\pi$
$863$	19429.2i	0.766369i	−0.923672	+	0.383185i	$0.874827\pi$
$864$	− 3439.90i	− 0.135449i
$865$	16400.6i	0.644666i
$866$	− 42249.7i	− 1.65786i
$867$	18359.0	0.719153
$868$	−12237.0	−0.478517
$869$	− 6096.98i	− 0.238004i
$870$	−12915.6	−0.503311
$871$	0	0
$872$	20618.9	0.800738
$873$	− 2580.67i	− 0.100048i
$874$	128952.	4.99068
$875$	27989.4	1.08139
$876$	48936.6i	1.88746i
$877$	11619.7i	0.447400i	0.974658	+	0.223700i	$0.0718137\pi$
$877$	11619.7i	0.447400i	−0.974658	+	0.223700i	$0.928186\pi$
$878$	918.332i	0.0352986i
$879$	10211.3i	0.391829i
$880$	−575.309	−0.0220383
$881$	51102.0	1.95422	0.977112	−	0.212726i	$-0.0682341\pi$
$881$	51102.0	1.95422	0.977112	+	0.212726i	$0.0682341\pi$
$882$	− 16801.9i	− 0.641441i
$883$	37838.5	1.44209	0.721046	−	0.692888i	$-0.243662\pi$
$883$	37838.5	1.44209	0.721046	+	0.692888i	$0.243662\pi$
$884$	0	0
$885$	−1845.78	−0.0701077
$886$	34625.6i	1.31295i
$887$	−9626.92	−0.364420	−0.182210	−	0.983260i	$-0.558325\pi$
$887$	−9626.92	−0.364420	−0.182210	+	0.983260i	$0.558325\pi$
$888$	−29457.7	−1.11321
$889$	19757.0i	0.745364i
$890$	11892.8i	0.447920i
$891$	485.357i	0.0182493i
$892$	− 66774.3i	− 2.50647i
$893$	41558.3	1.55733
$894$	−6287.98	−0.235237
$895$	− 18666.0i	− 0.697133i
$896$	70740.0	2.63757
$897$	0	0
$898$	−771.825	−0.0286816
$899$	6547.54i	0.242906i
$900$	−13298.7	−0.492543
$901$	−18162.0	−0.671549
$902$	− 1311.66i	− 0.0484184i
$903$	− 14145.1i	− 0.521283i
$904$	24421.4i	0.898500i
$905$	2232.65i	0.0820065i
$906$	2920.51	0.107094
$907$	−17066.0	−0.624772	−0.312386	−	0.949955i	$-0.601128\pi$
$907$	−17066.0	−0.624772	−0.312386	+	0.949955i	$0.601128\pi$
$908$	− 16121.0i	− 0.589200i
$909$	10967.1	0.400172
$910$	0	0
$911$	−37423.9	−1.36104	−0.680521	−	0.732729i	$-0.738246\pi$
$911$	−37423.9	−1.36104	−0.680521	+	0.732729i	$0.738246\pi$
$912$	− 10076.4i	− 0.365858i
$913$	900.312	0.0326353
$914$	−54907.7	−1.98708
$915$	− 790.805i	− 0.0285718i
$916$	18977.1i	0.684520i
$917$	39641.9i	1.42758i
$918$	− 13324.0i	− 0.479040i
$919$	−2783.88	−0.0999256	−0.0499628	−	0.998751i	$-0.515910\pi$
$919$	−2783.88	−0.0999256	−0.0499628	+	0.998751i	$0.515910\pi$
$920$	22386.1	0.802227
$921$	2666.59i	0.0954039i
$922$	−4968.50	−0.177472
$923$	0	0
$924$	−6883.50	−0.245076
$925$	− 36133.6i	− 1.28440i
$926$	40191.9	1.42634
$927$	−668.236	−0.0236761
$928$	− 26103.3i	− 0.923363i
$929$	26585.2i	0.938894i	0.882961	+	0.469447i	$0.155547\pi$
$929$	26585.2i	0.938894i	−0.882961	+	0.469447i	$0.844453\pi$
$930$	− 2014.50i	− 0.0710303i
$931$	62171.8i	2.18861i
$932$	−13397.6	−0.470873
$933$	3655.09	0.128255
$934$	36200.9i	1.26823i
$935$	2814.92	0.0984576
$936$	0	0
$937$	34474.0	1.20194	0.600970	−	0.799272i	$-0.294781\pi$
$937$	34474.0	1.20194	0.600970	+	0.799272i	$0.294781\pi$
$938$	27058.8i	0.941900i
$939$	−14611.2	−0.507795
$940$	16718.2	0.580092
$941$	− 41994.3i	− 1.45481i	−0.686209	−	0.727404i	$-0.740726\pi$
$941$	− 41994.3i	− 1.45481i	0.686209	−	0.727404i	$-0.259274\pi$
$942$	15456.2i	0.534596i
$943$	8173.34i	0.282249i
$944$	2953.15i	0.101819i
$945$	3285.75	0.113106
$946$	−4878.32	−0.167662
$947$	49352.0i	1.69348i	0.532008	+	0.846739i	$0.321438\pi$
$947$	49352.0i	1.69348i	−0.532008	+	0.846739i	$0.678562\pi$
$948$	42958.5	1.47176
$949$	0	0
$950$	77181.9	2.63591
$951$	13022.8i	0.444052i
$952$	81546.2	2.77618
$953$	51144.5	1.73844	0.869220	−	0.494425i	$-0.164621\pi$
$953$	51144.5	1.73844	0.869220	+	0.494425i	$0.164621\pi$
$954$	7311.36i	0.248128i
$955$	− 13805.2i	− 0.467774i
$956$	43192.0i	1.46122i
$957$	3683.07i	0.124406i
$958$	−21182.3	−0.714372
$959$	−49162.3	−1.65540
$960$	10335.6i	0.347478i
$961$	28769.8	0.965720
$962$	0	0
$963$	−11281.1	−0.377497
$964$	35297.7i	1.17932i
$965$	7353.41	0.245300
$966$	67276.3	2.24077
$967$	− 24895.1i	− 0.827892i	−0.910301	−	0.413946i	$-0.864150\pi$
$967$	− 24895.1i	− 0.827892i	0.910301	−	0.413946i	$-0.135850\pi$
$968$	− 36952.4i	− 1.22696i
$969$	49302.6i	1.63450i
$970$	6025.19i	0.199440i
$971$	42942.9	1.41926	0.709630	−	0.704574i	$-0.248862\pi$
$971$	42942.9	1.41926	0.709630	+	0.704574i	$0.248862\pi$
$972$	−3419.76	−0.112849
$973$	− 33448.2i	− 1.10206i
$974$	−31598.8	−1.03952
$975$	0	0
$976$	−1265.24	−0.0414953
$977$	42555.7i	1.39353i	0.717301	+	0.696764i	$0.245377\pi$
$977$	42555.7i	1.39353i	−0.717301	+	0.696764i	$0.754623\pi$
$978$	−44458.0	−1.45359
$979$	3391.41	0.110715
$980$	25010.6i	0.815240i
$981$	6503.78i	0.211672i
$982$	− 54111.3i	− 1.75841i
$983$	4345.38i	0.140993i	0.997512	+	0.0704965i	$0.0224584\pi$
$983$	4345.38i	0.140993i	−0.997512	+	0.0704965i	$0.977542\pi$
$984$	3988.18	0.129206
$985$	6132.71	0.198380
$986$	− 101108.i	− 3.26565i
$987$	21681.7	0.699225
$988$	0	0
$989$	30398.4	0.977363
$990$	− 1133.18i	− 0.0363787i
$991$	7934.89	0.254349	0.127175	−	0.991880i	$-0.459409\pi$
$991$	7934.89	0.254349	0.127175	+	0.991880i	$0.459409\pi$
$992$	4071.43	0.130311
$993$	2721.87i	0.0869848i
$994$	− 55777.0i	− 1.77982i
$995$	20695.7i	0.659395i
$996$	6343.48i	0.201808i
$997$	2725.62	0.0865810	0.0432905	−	0.999063i	$-0.486216\pi$
$997$	2725.62	0.0865810	0.0432905	+	0.999063i	$0.486216\pi$
$998$	−93531.5	−2.96662
$999$	− 9291.79i	− 0.294273i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.b.k.337.16		18
13.5	odd	4		507.4.a.p.1.8	yes	9
13.8	odd	4		507.4.a.o.1.2	✓	9
13.12	even	2	inner	507.4.b.k.337.3		18
39.5	even	4		1521.4.a.bf.1.2		9
39.8	even	4		1521.4.a.bi.1.8		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.4.a.o.1.2	✓	9	13.8	odd	4
507.4.a.p.1.8	yes	9	13.5	odd	4
507.4.b.k.337.3		18	13.12	even	2	inner
507.4.b.k.337.16		18	1.1	even	1	trivial
1521.4.a.bf.1.2		9	39.5	even	4
1521.4.a.bi.1.8		9	39.8	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+4.69820i q^{2} +3.00000 q^{3} -14.0731 q^{4} +4.47249i q^{5} +14.0946i q^{6} -27.2096i q^{7} -28.5326i q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+4.69820i q^{2} +3.00000 q^{3} -14.0731 q^{4} +4.47249i q^{5} +14.0946i q^{6} -27.2096i q^{7} -28.5326i q^{8} +9.00000 q^{9} -21.0127 q^{10} +5.99207i q^{11} -42.2193 q^{12} +127.836 q^{14} +13.4175i q^{15} +21.4672 q^{16} -105.037 q^{17} +42.2838i q^{18} -156.462i q^{19} -62.9418i q^{20} -81.6287i q^{21} -28.1519 q^{22} +175.423 q^{23} -85.5978i q^{24} +104.997 q^{25} +27.0000 q^{27} +382.923i q^{28} +204.886 q^{29} -63.0380 q^{30} +31.9570i q^{31} -127.404i q^{32} +17.9762i q^{33} -493.483i q^{34} +121.695 q^{35} -126.658 q^{36} -344.140i q^{37} +735.088 q^{38} +127.612 q^{40} +46.5921i q^{41} +383.508 q^{42} +173.286 q^{43} -84.3269i q^{44} +40.2524i q^{45} +824.175i q^{46} +265.613i q^{47} +64.4016 q^{48} -397.361 q^{49} +493.296i q^{50} -315.110 q^{51} +172.912 q^{53} +126.851i q^{54} -26.7995 q^{55} -776.360 q^{56} -469.385i q^{57} +962.596i q^{58} +137.566i q^{59} -188.825i q^{60} -58.9384 q^{61} -150.140 q^{62} -244.886i q^{63} +770.306 q^{64} -84.4558 q^{66} +211.668i q^{67} +1478.19 q^{68} +526.270 q^{69} +571.746i q^{70} -436.317i q^{71} -256.793i q^{72} -1159.11i q^{73} +1616.84 q^{74} +314.990 q^{75} +2201.90i q^{76} +163.042 q^{77} -1017.51 q^{79} +96.0118i q^{80} +81.0000 q^{81} -218.899 q^{82} -150.251i q^{83} +1148.77i q^{84} -469.775i q^{85} +814.131i q^{86} +614.659 q^{87} +170.969 q^{88} -565.984i q^{89} -189.114 q^{90} -2468.75 q^{92} +95.8709i q^{93} -1247.90 q^{94} +699.773 q^{95} -382.211i q^{96} -286.741i q^{97} -1866.88i q^{98} +53.9286i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 54 q^{3} - 88 q^{4} + 162 q^{9}+O(q^{10}) \) 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

Groups

Database

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