LMFDB - Embedded newform 50.4.b.a.49.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [50,4,Mod(49,50)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(50, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("50.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 50 = 2 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	50.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:	$2.95009550029$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 10)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	50.49
Dual form			50.4.b.a.49.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.00000i q^{2} +8.00000i q^{3} -4.00000 q^{4} -16.0000 q^{6} -4.00000i q^{7} -8.00000i q^{8} -37.0000 q^{9} +12.0000 q^{11} -32.0000i q^{12} +58.0000i q^{13} +8.00000 q^{14} +16.0000 q^{16} +66.0000i q^{17} -74.0000i q^{18} +100.000 q^{19} +32.0000 q^{21} +24.0000i q^{22} -132.000i q^{23} +64.0000 q^{24} -116.000 q^{26} -80.0000i q^{27} +16.0000i q^{28} +90.0000 q^{29} +152.000 q^{31} +32.0000i q^{32} +96.0000i q^{33} -132.000 q^{34} +148.000 q^{36} -34.0000i q^{37} +200.000i q^{38} -464.000 q^{39} -438.000 q^{41} +64.0000i q^{42} -32.0000i q^{43} -48.0000 q^{44} +264.000 q^{46} -204.000i q^{47} +128.000i q^{48} +327.000 q^{49} -528.000 q^{51} -232.000i q^{52} -222.000i q^{53} +160.000 q^{54} -32.0000 q^{56} +800.000i q^{57} +180.000i q^{58} -420.000 q^{59} +902.000 q^{61} +304.000i q^{62} +148.000i q^{63} -64.0000 q^{64} -192.000 q^{66} -1024.00i q^{67} -264.000i q^{68} +1056.00 q^{69} +432.000 q^{71} +296.000i q^{72} -362.000i q^{73} +68.0000 q^{74} -400.000 q^{76} -48.0000i q^{77} -928.000i q^{78} +160.000 q^{79} -359.000 q^{81} -876.000i q^{82} -72.0000i q^{83} -128.000 q^{84} +64.0000 q^{86} +720.000i q^{87} -96.0000i q^{88} -810.000 q^{89} +232.000 q^{91} +528.000i q^{92} +1216.00i q^{93} +408.000 q^{94} -256.000 q^{96} +1106.00i q^{97} +654.000i q^{98} -444.000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{4} - 32 q^{6} - 74 q^{9} + 24 q^{11} + 16 q^{14} + 32 q^{16} + 200 q^{19} + 64 q^{21} + 128 q^{24} - 232 q^{26} + 180 q^{29} + 304 q^{31} - 264 q^{34} + 296 q^{36} - 928 q^{39} - 876 q^{41} - 96 q^{44}+ \cdots - 888 q^{99}+O(q^{100}) $ 2 * q - 8 * q^4 - 32 * q^6 - 74 * q^9 + 24 * q^11 + 16 * q^14 + 32 * q^16 + 200 * q^19 + 64 * q^21 + 128 * q^24 - 232 * q^26 + 180 * q^29 + 304 * q^31 - 264 * q^34 + 296 * q^36 - 928 * q^39 - 876 * q^41 - 96 * q^44 + 528 * q^46 + 654 * q^49 - 1056 * q^51 + 320 * q^54 - 64 * q^56 - 840 * q^59 + 1804 * q^61 - 128 * q^64 - 384 * q^66 + 2112 * q^69 + 864 * q^71 + 136 * q^74 - 800 * q^76 + 320 * q^79 - 718 * q^81 - 256 * q^84 + 128 * q^86 - 1620 * q^89 + 464 * q^91 + 816 * q^94 - 512 * q^96 - 888 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/50\mathbb{Z}\right)^\times$.

$n$	$27$
$\chi(n)$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.00000i	0.707107i
$2$	2.00000i	0.707107i
$3$	8.00000i	1.53960i	0.638285	+	0.769800i	$0.279644\pi$
$3$	8.00000i	1.53960i	−0.638285	+	0.769800i	$0.720356\pi$
$4$	−4.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−16.0000	−1.08866
$7$	− 4.00000i	− 0.215980i	−0.994152	−	0.107990i	$-0.965559\pi$
$7$	− 4.00000i	− 0.215980i	0.994152	−	0.107990i	$-0.0344414\pi$
$8$	− 8.00000i	− 0.353553i
$9$	−37.0000	−1.37037
$10$	0	0
$11$	12.0000	0.328921	0.164461	−	0.986384i	$-0.447412\pi$
$11$	12.0000	0.328921	0.164461	+	0.986384i	$0.447412\pi$
$12$	− 32.0000i	− 0.769800i
$13$	58.0000i	1.23741i	0.785624	+	0.618704i	$0.212342\pi$
$13$	58.0000i	1.23741i	−0.785624	+	0.618704i	$0.787658\pi$
$14$	8.00000	0.152721
$15$	0	0
$16$	16.0000	0.250000
$17$	66.0000i	0.941609i	0.882238	+	0.470804i	$0.156036\pi$
$17$	66.0000i	0.941609i	−0.882238	+	0.470804i	$0.843964\pi$
$18$	− 74.0000i	− 0.968998i
$19$	100.000	1.20745	0.603726	−	0.797192i	$-0.293682\pi$
$19$	100.000	1.20745	0.603726	+	0.797192i	$0.293682\pi$
$20$	0	0
$21$	32.0000	0.332522
$22$	24.0000i	0.232583i
$23$	− 132.000i	− 1.19669i	−0.801238	−	0.598346i	$-0.795825\pi$
$23$	− 132.000i	− 1.19669i	0.801238	−	0.598346i	$-0.204175\pi$
$24$	64.0000	0.544331
$25$	0	0
$26$	−116.000	−0.874980
$27$	− 80.0000i	− 0.570222i
$28$	16.0000i	0.107990i
$29$	90.0000	0.576296	0.288148	−	0.957586i	$-0.406961\pi$
$29$	90.0000	0.576296	0.288148	+	0.957586i	$0.406961\pi$
$30$	0	0
$31$	152.000	0.880645	0.440323	−	0.897840i	$-0.354864\pi$
$31$	152.000	0.880645	0.440323	+	0.897840i	$0.354864\pi$
$32$	32.0000i	0.176777i
$33$	96.0000i	0.506408i
$34$	−132.000	−0.665818
$35$	0	0
$36$	148.000	0.685185
$37$	− 34.0000i	− 0.151069i	−0.997143	−	0.0755347i	$-0.975934\pi$
$37$	− 34.0000i	− 0.151069i	0.997143	−	0.0755347i	$-0.0240664\pi$
$38$	200.000i	0.853797i
$39$	−464.000	−1.90511
$40$	0	0
$41$	−438.000	−1.66839	−0.834196	−	0.551467i	$-0.814068\pi$
$41$	−438.000	−1.66839	−0.834196	+	0.551467i	$0.814068\pi$
$42$	64.0000i	0.235129i
$43$	− 32.0000i	− 0.113487i	−0.998389	−	0.0567437i	$-0.981928\pi$
$43$	− 32.0000i	− 0.113487i	0.998389	−	0.0567437i	$-0.0180718\pi$
$44$	−48.0000	−0.164461
$45$	0	0
$46$	264.000	0.846189
$47$	− 204.000i	− 0.633116i	−0.948573	−	0.316558i	$-0.897473\pi$
$47$	− 204.000i	− 0.633116i	0.948573	−	0.316558i	$-0.102527\pi$
$48$	128.000i	0.384900i
$49$	327.000	0.953353
$50$	0	0
$51$	−528.000	−1.44970
$52$	− 232.000i	− 0.618704i
$53$	− 222.000i	− 0.575359i	−0.957727	−	0.287680i	$-0.907116\pi$
$53$	− 222.000i	− 0.575359i	0.957727	−	0.287680i	$-0.0928838\pi$
$54$	160.000	0.403208
$55$	0	0
$56$	−32.0000	−0.0763604
$57$	800.000i	1.85899i
$58$	180.000i	0.407503i
$59$	−420.000	−0.926769	−0.463384	−	0.886157i	$-0.653365\pi$
$59$	−420.000	−0.926769	−0.463384	+	0.886157i	$0.653365\pi$
$60$	0	0
$61$	902.000	1.89327	0.946633	−	0.322312i	$-0.104460\pi$
$61$	902.000	1.89327	0.946633	+	0.322312i	$0.104460\pi$
$62$	304.000i	0.622710i
$63$	148.000i	0.295972i
$64$	−64.0000	−0.125000
$65$	0	0
$66$	−192.000	−0.358084
$67$	− 1024.00i	− 1.86719i	−0.358334	−	0.933593i	$-0.616655\pi$
$67$	− 1024.00i	− 1.86719i	0.358334	−	0.933593i	$-0.383345\pi$
$68$	− 264.000i	− 0.470804i
$69$	1056.00	1.84243
$70$	0	0
$71$	432.000	0.722098	0.361049	−	0.932547i	$-0.382419\pi$
$71$	432.000	0.722098	0.361049	+	0.932547i	$0.382419\pi$
$72$	296.000i	0.484499i
$73$	− 362.000i	− 0.580396i	−0.956967	−	0.290198i	$-0.906279\pi$
$73$	− 362.000i	− 0.580396i	0.956967	−	0.290198i	$-0.0937211\pi$
$74$	68.0000	0.106822
$75$	0	0
$76$	−400.000	−0.603726
$77$	− 48.0000i	− 0.0710404i
$78$	− 928.000i	− 1.34712i
$79$	160.000	0.227866	0.113933	−	0.993488i	$-0.463655\pi$
$79$	160.000	0.227866	0.113933	+	0.993488i	$0.463655\pi$
$80$	0	0
$81$	−359.000	−0.492455
$82$	− 876.000i	− 1.17973i
$83$	− 72.0000i	− 0.0952172i	−0.998866	−	0.0476086i	$-0.984840\pi$
$83$	− 72.0000i	− 0.0952172i	0.998866	−	0.0476086i	$-0.0151600\pi$
$84$	−128.000	−0.166261
$85$	0	0
$86$	64.0000	0.0802476
$87$	720.000i	0.887266i
$88$	− 96.0000i	− 0.116291i
$89$	−810.000	−0.964717	−0.482359	−	0.875974i	$-0.660220\pi$
$89$	−810.000	−0.964717	−0.482359	+	0.875974i	$0.660220\pi$
$90$	0	0
$91$	232.000	0.267255
$92$	528.000i	0.598346i
$93$	1216.00i	1.35584i
$94$	408.000	0.447681
$95$	0	0
$96$	−256.000	−0.272166
$97$	1106.00i	1.15770i	0.815433	+	0.578852i	$0.196499\pi$
$97$	1106.00i	1.15770i	−0.815433	+	0.578852i	$0.803501\pi$
$98$	654.000i	0.674122i
$99$	−444.000	−0.450744
$100$	0	0
$101$	−258.000	−0.254178	−0.127089	−	0.991891i	$-0.540563\pi$
$101$	−258.000	−0.254178	−0.127089	+	0.991891i	$0.540563\pi$
$102$	− 1056.00i	− 1.02509i
$103$	988.000i	0.945151i	0.881290	+	0.472575i	$0.156676\pi$
$103$	988.000i	0.945151i	−0.881290	+	0.472575i	$0.843324\pi$
$104$	464.000	0.437490
$105$	0	0
$106$	444.000	0.406840
$107$	− 24.0000i	− 0.0216838i	−0.999941	−	0.0108419i	$-0.996549\pi$
$107$	− 24.0000i	− 0.0216838i	0.999941	−	0.0108419i	$-0.00345115\pi$
$108$	320.000i	0.285111i
$109$	−950.000	−0.834803	−0.417401	−	0.908722i	$-0.637059\pi$
$109$	−950.000	−0.834803	−0.417401	+	0.908722i	$0.637059\pi$
$110$	0	0
$111$	272.000	0.232586
$112$	− 64.0000i	− 0.0539949i
$113$	1038.00i	0.864131i	0.901842	+	0.432066i	$0.142215\pi$
$113$	1038.00i	0.864131i	−0.901842	+	0.432066i	$0.857785\pi$
$114$	−1600.00	−1.31451
$115$	0	0
$116$	−360.000	−0.288148
$117$	− 2146.00i	− 1.69571i
$118$	− 840.000i	− 0.655324i
$119$	264.000	0.203368
$120$	0	0
$121$	−1187.00	−0.891811
$122$	1804.00i	1.33874i
$123$	− 3504.00i	− 2.56866i
$124$	−608.000	−0.440323
$125$	0	0
$126$	−296.000	−0.209284
$127$	− 124.000i	− 0.0866395i	−0.999061	−	0.0433198i	$-0.986207\pi$
$127$	− 124.000i	− 0.0866395i	0.999061	−	0.0433198i	$-0.0137934\pi$
$128$	− 128.000i	− 0.0883883i
$129$	256.000	0.174725
$130$	0	0
$131$	132.000	0.0880374	0.0440187	−	0.999031i	$-0.485984\pi$
$131$	132.000	0.0880374	0.0440187	+	0.999031i	$0.485984\pi$
$132$	− 384.000i	− 0.253204i
$133$	− 400.000i	− 0.260785i
$134$	2048.00	1.32030
$135$	0	0
$136$	528.000	0.332909
$137$	− 1254.00i	− 0.782018i	−0.920387	−	0.391009i	$-0.872126\pi$
$137$	− 1254.00i	− 0.782018i	0.920387	−	0.391009i	$-0.127874\pi$
$138$	2112.00i	1.30279i
$139$	2860.00	1.74519	0.872597	−	0.488440i	$-0.162434\pi$
$139$	2860.00	1.74519	0.872597	+	0.488440i	$0.162434\pi$
$140$	0	0
$141$	1632.00	0.974746
$142$	864.000i	0.510600i
$143$	696.000i	0.407010i
$144$	−592.000	−0.342593
$145$	0	0
$146$	724.000	0.410402
$147$	2616.00i	1.46778i
$148$	136.000i	0.0755347i
$149$	−750.000	−0.412365	−0.206183	−	0.978514i	$-0.566104\pi$
$149$	−750.000	−0.412365	−0.206183	+	0.978514i	$0.566104\pi$
$150$	0	0
$151$	−448.000	−0.241442	−0.120721	−	0.992686i	$-0.538521\pi$
$151$	−448.000	−0.241442	−0.120721	+	0.992686i	$0.538521\pi$
$152$	− 800.000i	− 0.426898i
$153$	− 2442.00i	− 1.29035i
$154$	96.0000	0.0502331
$155$	0	0
$156$	1856.00	0.952557
$157$	2246.00i	1.14172i	0.821047	+	0.570861i	$0.193390\pi$
$157$	2246.00i	1.14172i	−0.821047	+	0.570861i	$0.806610\pi$
$158$	320.000i	0.161126i
$159$	1776.00	0.885824
$160$	0	0
$161$	−528.000	−0.258461
$162$	− 718.000i	− 0.348219i
$163$	568.000i	0.272940i	0.990644	+	0.136470i	$0.0435757\pi$
$163$	568.000i	0.272940i	−0.990644	+	0.136470i	$0.956424\pi$
$164$	1752.00	0.834196
$165$	0	0
$166$	144.000	0.0673287
$167$	− 1524.00i	− 0.706172i	−0.935591	−	0.353086i	$-0.885132\pi$
$167$	− 1524.00i	− 0.706172i	0.935591	−	0.353086i	$-0.114868\pi$
$168$	− 256.000i	− 0.117564i
$169$	−1167.00	−0.531179
$170$	0	0
$171$	−3700.00	−1.65466
$172$	128.000i	0.0567437i
$173$	− 3702.00i	− 1.62692i	−0.581618	−	0.813462i	$-0.697580\pi$
$173$	− 3702.00i	− 1.62692i	0.581618	−	0.813462i	$-0.302420\pi$
$174$	−1440.00	−0.627391
$175$	0	0
$176$	192.000	0.0822304
$177$	− 3360.00i	− 1.42685i
$178$	− 1620.00i	− 0.682158i
$179$	−3180.00	−1.32785	−0.663923	−	0.747801i	$-0.731110\pi$
$179$	−3180.00	−1.32785	−0.663923	+	0.747801i	$0.731110\pi$
$180$	0	0
$181$	−2098.00	−0.861564	−0.430782	−	0.902456i	$-0.641762\pi$
$181$	−2098.00	−0.861564	−0.430782	+	0.902456i	$0.641762\pi$
$182$	464.000i	0.188978i
$183$	7216.00i	2.91487i
$184$	−1056.00	−0.423094
$185$	0	0
$186$	−2432.00	−0.958725
$187$	792.000i	0.309715i
$188$	816.000i	0.316558i
$189$	−320.000	−0.123156
$190$	0	0
$191$	4392.00	1.66384	0.831921	−	0.554894i	$-0.187241\pi$
$191$	4392.00	1.66384	0.831921	+	0.554894i	$0.187241\pi$
$192$	− 512.000i	− 0.192450i
$193$	2158.00i	0.804851i	0.915453	+	0.402425i	$0.131833\pi$
$193$	2158.00i	0.804851i	−0.915453	+	0.402425i	$0.868167\pi$
$194$	−2212.00	−0.818620
$195$	0	0
$196$	−1308.00	−0.476676
$197$	− 1074.00i	− 0.388423i	−0.980960	−	0.194212i	$-0.937785\pi$
$197$	− 1074.00i	− 0.388423i	0.980960	−	0.194212i	$-0.0622148\pi$
$198$	− 888.000i	− 0.318724i
$199$	−2840.00	−1.01167	−0.505835	−	0.862630i	$-0.668815\pi$
$199$	−2840.00	−1.01167	−0.505835	+	0.862630i	$0.668815\pi$
$200$	0	0
$201$	8192.00	2.87472
$202$	− 516.000i	− 0.179731i
$203$	− 360.000i	− 0.124468i
$204$	2112.00	0.724851
$205$	0	0
$206$	−1976.00	−0.668323
$207$	4884.00i	1.63991i
$208$	928.000i	0.309352i
$209$	1200.00	0.397157
$210$	0	0
$211$	−2668.00	−0.870487	−0.435243	−	0.900313i	$-0.643338\pi$
$211$	−2668.00	−0.870487	−0.435243	+	0.900313i	$0.643338\pi$
$212$	888.000i	0.287680i
$213$	3456.00i	1.11174i
$214$	48.0000	0.0153328
$215$	0	0
$216$	−640.000	−0.201604
$217$	− 608.000i	− 0.190202i
$218$	− 1900.00i	− 0.590295i
$219$	2896.00	0.893578
$220$	0	0
$221$	−3828.00	−1.16515
$222$	544.000i	0.164463i
$223$	− 1772.00i	− 0.532116i	−0.963957	−	0.266058i	$-0.914279\pi$
$223$	− 1772.00i	− 0.532116i	0.963957	−	0.266058i	$-0.0857213\pi$
$224$	128.000	0.0381802
$225$	0	0
$226$	−2076.00	−0.611033
$227$	− 2784.00i	− 0.814011i	−0.913426	−	0.407006i	$-0.866573\pi$
$227$	− 2784.00i	− 0.814011i	0.913426	−	0.407006i	$-0.133427\pi$
$228$	− 3200.00i	− 0.929496i
$229$	−350.000	−0.100998	−0.0504992	−	0.998724i	$-0.516081\pi$
$229$	−350.000	−0.100998	−0.0504992	+	0.998724i	$0.516081\pi$
$230$	0	0
$231$	384.000	0.109374
$232$	− 720.000i	− 0.203751i
$233$	− 1962.00i	− 0.551652i	−0.961208	−	0.275826i	$-0.911049\pi$
$233$	− 1962.00i	− 0.551652i	0.961208	−	0.275826i	$-0.0889513\pi$
$234$	4292.00	1.19905
$235$	0	0
$236$	1680.00	0.463384
$237$	1280.00i	0.350823i
$238$	528.000i	0.143803i
$239$	4320.00	1.16919	0.584597	−	0.811324i	$-0.301252\pi$
$239$	4320.00	1.16919	0.584597	+	0.811324i	$0.301252\pi$
$240$	0	0
$241$	−478.000	−0.127762	−0.0638811	−	0.997958i	$-0.520348\pi$
$241$	−478.000	−0.127762	−0.0638811	+	0.997958i	$0.520348\pi$
$242$	− 2374.00i	− 0.630605i
$243$	− 5032.00i	− 1.32841i
$244$	−3608.00	−0.946633
$245$	0	0
$246$	7008.00	1.81632
$247$	5800.00i	1.49411i
$248$	− 1216.00i	− 0.311355i
$249$	576.000	0.146596
$250$	0	0
$251$	2652.00	0.666903	0.333452	−	0.942767i	$-0.391787\pi$
$251$	2652.00	0.666903	0.333452	+	0.942767i	$0.391787\pi$
$252$	− 592.000i	− 0.147986i
$253$	− 1584.00i	− 0.393617i
$254$	248.000	0.0612634
$255$	0	0
$256$	256.000	0.0625000
$257$	− 2334.00i	− 0.566502i	−0.959046	−	0.283251i	$-0.908587\pi$
$257$	− 2334.00i	− 0.566502i	0.959046	−	0.283251i	$-0.0914129\pi$
$258$	512.000i	0.123549i
$259$	−136.000	−0.0326279
$260$	0	0
$261$	−3330.00	−0.789739
$262$	264.000i	0.0622518i
$263$	3948.00i	0.925643i	0.886451	+	0.462822i	$0.153163\pi$
$263$	3948.00i	0.925643i	−0.886451	+	0.462822i	$0.846837\pi$
$264$	768.000	0.179042
$265$	0	0
$266$	800.000	0.184403
$267$	− 6480.00i	− 1.48528i
$268$	4096.00i	0.933593i
$269$	−1590.00	−0.360387	−0.180193	−	0.983631i	$-0.557672\pi$
$269$	−1590.00	−0.360387	−0.180193	+	0.983631i	$0.557672\pi$
$270$	0	0
$271$	4952.00	1.11001	0.555005	−	0.831847i	$-0.312716\pi$
$271$	4952.00	1.11001	0.555005	+	0.831847i	$0.312716\pi$
$272$	1056.00i	0.235402i
$273$	1856.00i	0.411466i
$274$	2508.00	0.552970
$275$	0	0
$276$	−4224.00	−0.921213
$277$	1646.00i	0.357034i	0.983937	+	0.178517i	$0.0571300\pi$
$277$	1646.00i	0.357034i	−0.983937	+	0.178517i	$0.942870\pi$
$278$	5720.00i	1.23404i
$279$	−5624.00	−1.20681
$280$	0	0
$281$	−1158.00	−0.245838	−0.122919	−	0.992417i	$-0.539226\pi$
$281$	−1158.00	−0.245838	−0.122919	+	0.992417i	$0.539226\pi$
$282$	3264.00i	0.689250i
$283$	− 6992.00i	− 1.46866i	−0.678792	−	0.734331i	$-0.737496\pi$
$283$	− 6992.00i	− 1.46866i	0.678792	−	0.734331i	$-0.262504\pi$
$284$	−1728.00	−0.361049
$285$	0	0
$286$	−1392.00	−0.287800
$287$	1752.00i	0.360339i
$288$	− 1184.00i	− 0.242250i
$289$	557.000	0.113373
$290$	0	0
$291$	−8848.00	−1.78240
$292$	1448.00i	0.290198i
$293$	258.000i	0.0514421i	0.999669	+	0.0257210i	$0.00818816\pi$
$293$	258.000i	0.0514421i	−0.999669	+	0.0257210i	$0.991812\pi$
$294$	−5232.00	−1.03788
$295$	0	0
$296$	−272.000	−0.0534111
$297$	− 960.000i	− 0.187558i
$298$	− 1500.00i	− 0.291586i
$299$	7656.00	1.48080
$300$	0	0
$301$	−128.000	−0.0245110
$302$	− 896.000i	− 0.170725i
$303$	− 2064.00i	− 0.391332i
$304$	1600.00	0.301863
$305$	0	0
$306$	4884.00	0.912417
$307$	− 8944.00i	− 1.66274i	−0.555720	−	0.831370i	$-0.687557\pi$
$307$	− 8944.00i	− 1.66274i	0.555720	−	0.831370i	$-0.312443\pi$
$308$	192.000i	0.0355202i
$309$	−7904.00	−1.45515
$310$	0	0
$311$	1392.00	0.253804	0.126902	−	0.991915i	$-0.459497\pi$
$311$	1392.00	0.253804	0.126902	+	0.991915i	$0.459497\pi$
$312$	3712.00i	0.673560i
$313$	5878.00i	1.06148i	0.847534	+	0.530742i	$0.178087\pi$
$313$	5878.00i	1.06148i	−0.847534	+	0.530742i	$0.821913\pi$
$314$	−4492.00	−0.807319
$315$	0	0
$316$	−640.000	−0.113933
$317$	10326.0i	1.82955i	0.403969	+	0.914773i	$0.367630\pi$
$317$	10326.0i	1.82955i	−0.403969	+	0.914773i	$0.632370\pi$
$318$	3552.00i	0.626372i
$319$	1080.00	0.189556
$320$	0	0
$321$	192.000	0.0333844
$322$	− 1056.00i	− 0.182760i
$323$	6600.00i	1.13695i
$324$	1436.00	0.246228
$325$	0	0
$326$	−1136.00	−0.192998
$327$	− 7600.00i	− 1.28526i
$328$	3504.00i	0.589866i
$329$	−816.000	−0.136740
$330$	0	0
$331$	−4228.00	−0.702090	−0.351045	−	0.936359i	$-0.614174\pi$
$331$	−4228.00	−0.702090	−0.351045	+	0.936359i	$0.614174\pi$
$332$	288.000i	0.0476086i
$333$	1258.00i	0.207021i
$334$	3048.00	0.499339
$335$	0	0
$336$	512.000	0.0831306
$337$	1106.00i	0.178776i	0.995997	+	0.0893882i	$0.0284912\pi$
$337$	1106.00i	0.178776i	−0.995997	+	0.0893882i	$0.971509\pi$
$338$	− 2334.00i	− 0.375600i
$339$	−8304.00	−1.33042
$340$	0	0
$341$	1824.00	0.289663
$342$	− 7400.00i	− 1.17002i
$343$	− 2680.00i	− 0.421885i
$344$	−256.000	−0.0401238
$345$	0	0
$346$	7404.00	1.15041
$347$	9336.00i	1.44433i	0.691720	+	0.722165i	$0.256853\pi$
$347$	9336.00i	1.44433i	−0.691720	+	0.722165i	$0.743147\pi$
$348$	− 2880.00i	− 0.443633i
$349$	11770.0	1.80525	0.902627	−	0.430424i	$-0.141636\pi$
$349$	11770.0	1.80525	0.902627	+	0.430424i	$0.141636\pi$
$350$	0	0
$351$	4640.00	0.705598
$352$	384.000i	0.0581456i
$353$	− 8322.00i	− 1.25477i	−0.778707	−	0.627387i	$-0.784124\pi$
$353$	− 8322.00i	− 1.25477i	0.778707	−	0.627387i	$-0.215876\pi$
$354$	6720.00	1.00894
$355$	0	0
$356$	3240.00	0.482359
$357$	2112.00i	0.313106i
$358$	− 6360.00i	− 0.938929i
$359$	−10680.0	−1.57011	−0.785054	−	0.619427i	$-0.787365\pi$
$359$	−10680.0	−1.57011	−0.785054	+	0.619427i	$0.787365\pi$
$360$	0	0
$361$	3141.00	0.457938
$362$	− 4196.00i	− 0.609218i
$363$	− 9496.00i	− 1.37303i
$364$	−928.000	−0.133628
$365$	0	0
$366$	−14432.0	−2.06113
$367$	− 5884.00i	− 0.836900i	−0.908240	−	0.418450i	$-0.862574\pi$
$367$	− 5884.00i	− 0.836900i	0.908240	−	0.418450i	$-0.137426\pi$
$368$	− 2112.00i	− 0.299173i
$369$	16206.0	2.28632
$370$	0	0
$371$	−888.000	−0.124266
$372$	− 4864.00i	− 0.677921i
$373$	2098.00i	0.291234i	0.989341	+	0.145617i	$0.0465167\pi$
$373$	2098.00i	0.291234i	−0.989341	+	0.145617i	$0.953483\pi$
$374$	−1584.00	−0.219002
$375$	0	0
$376$	−1632.00	−0.223840
$377$	5220.00i	0.713113i
$378$	− 640.000i	− 0.0870848i
$379$	−3860.00	−0.523153	−0.261576	−	0.965183i	$-0.584242\pi$
$379$	−3860.00	−0.523153	−0.261576	+	0.965183i	$0.584242\pi$
$380$	0	0
$381$	992.000	0.133390
$382$	8784.00i	1.17651i
$383$	9588.00i	1.27917i	0.768718	+	0.639587i	$0.220895\pi$
$383$	9588.00i	1.27917i	−0.768718	+	0.639587i	$0.779105\pi$
$384$	1024.00	0.136083
$385$	0	0
$386$	−4316.00	−0.569116
$387$	1184.00i	0.155520i
$388$	− 4424.00i	− 0.578852i
$389$	13410.0	1.74785	0.873925	−	0.486060i	$-0.161566\pi$
$389$	13410.0	1.74785	0.873925	+	0.486060i	$0.161566\pi$
$390$	0	0
$391$	8712.00	1.12682
$392$	− 2616.00i	− 0.337061i
$393$	1056.00i	0.135542i
$394$	2148.00	0.274657
$395$	0	0
$396$	1776.00	0.225372
$397$	− 13114.0i	− 1.65787i	−0.559348	−	0.828933i	$-0.688948\pi$
$397$	− 13114.0i	− 1.65787i	0.559348	−	0.828933i	$-0.311052\pi$
$398$	− 5680.00i	− 0.715358i
$399$	3200.00	0.401505
$400$	0	0
$401$	−5838.00	−0.727022	−0.363511	−	0.931590i	$-0.618422\pi$
$401$	−5838.00	−0.727022	−0.363511	+	0.931590i	$0.618422\pi$
$402$	16384.0i	2.03274i
$403$	8816.00i	1.08972i
$404$	1032.00	0.127089
$405$	0	0
$406$	720.000	0.0880123
$407$	− 408.000i	− 0.0496899i
$408$	4224.00i	0.512547i
$409$	−9530.00	−1.15215	−0.576074	−	0.817398i	$-0.695416\pi$
$409$	−9530.00	−1.15215	−0.576074	+	0.817398i	$0.695416\pi$
$410$	0	0
$411$	10032.0	1.20400
$412$	− 3952.00i	− 0.472575i
$413$	1680.00i	0.200163i
$414$	−9768.00	−1.15959
$415$	0	0
$416$	−1856.00	−0.218745
$417$	22880.0i	2.68690i
$418$	2400.00i	0.280832i
$419$	−7260.00	−0.846478	−0.423239	−	0.906018i	$-0.639107\pi$
$419$	−7260.00	−0.846478	−0.423239	+	0.906018i	$0.639107\pi$
$420$	0	0
$421$	12062.0	1.39636	0.698178	−	0.715924i	$-0.253994\pi$
$421$	12062.0	1.39636	0.698178	+	0.715924i	$0.253994\pi$
$422$	− 5336.00i	− 0.615527i
$423$	7548.00i	0.867604i
$424$	−1776.00	−0.203420
$425$	0	0
$426$	−6912.00	−0.786121
$427$	− 3608.00i	− 0.408907i
$428$	96.0000i	0.0108419i
$429$	−5568.00	−0.626633
$430$	0	0
$431$	−13608.0	−1.52082	−0.760411	−	0.649442i	$-0.775002\pi$
$431$	−13608.0	−1.52082	−0.760411	+	0.649442i	$0.775002\pi$
$432$	− 1280.00i	− 0.142556i
$433$	3838.00i	0.425964i	0.977056	+	0.212982i	$0.0683176\pi$
$433$	3838.00i	0.425964i	−0.977056	+	0.212982i	$0.931682\pi$
$434$	1216.00	0.134493
$435$	0	0
$436$	3800.00	0.417401
$437$	− 13200.0i	− 1.44495i
$438$	5792.00i	0.631855i
$439$	−7400.00	−0.804516	−0.402258	−	0.915526i	$-0.631775\pi$
$439$	−7400.00	−0.804516	−0.402258	+	0.915526i	$0.631775\pi$
$440$	0	0
$441$	−12099.0	−1.30645
$442$	− 7656.00i	− 0.823889i
$443$	− 8352.00i	− 0.895746i	−0.894097	−	0.447873i	$-0.852182\pi$
$443$	− 8352.00i	− 0.895746i	0.894097	−	0.447873i	$-0.147818\pi$
$444$	−1088.00	−0.116293
$445$	0	0
$446$	3544.00	0.376263
$447$	− 6000.00i	− 0.634878i
$448$	256.000i	0.0269975i
$449$	−10770.0	−1.13200	−0.566000	−	0.824405i	$-0.691510\pi$
$449$	−10770.0	−1.13200	−0.566000	+	0.824405i	$0.691510\pi$
$450$	0	0
$451$	−5256.00	−0.548770
$452$	− 4152.00i	− 0.432066i
$453$	− 3584.00i	− 0.371724i
$454$	5568.00	0.575593
$455$	0	0
$456$	6400.00	0.657253
$457$	− 6694.00i	− 0.685191i	−0.939483	−	0.342595i	$-0.888694\pi$
$457$	− 6694.00i	− 0.685191i	0.939483	−	0.342595i	$-0.111306\pi$
$458$	− 700.000i	− 0.0714167i
$459$	5280.00	0.536927
$460$	0	0
$461$	−3018.00	−0.304907	−0.152454	−	0.988311i	$-0.548717\pi$
$461$	−3018.00	−0.304907	−0.152454	+	0.988311i	$0.548717\pi$
$462$	768.000i	0.0773389i
$463$	− 14492.0i	− 1.45464i	−0.686296	−	0.727322i	$-0.740765\pi$
$463$	− 14492.0i	− 1.45464i	0.686296	−	0.727322i	$-0.259235\pi$
$464$	1440.00	0.144074
$465$	0	0
$466$	3924.00	0.390077
$467$	7776.00i	0.770515i	0.922809	+	0.385257i	$0.125887\pi$
$467$	7776.00i	0.770515i	−0.922809	+	0.385257i	$0.874113\pi$
$468$	8584.00i	0.847854i
$469$	−4096.00	−0.403274
$470$	0	0
$471$	−17968.0	−1.75780
$472$	3360.00i	0.327662i
$473$	− 384.000i	− 0.0373284i
$474$	−2560.00	−0.248069
$475$	0	0
$476$	−1056.00	−0.101684
$477$	8214.00i	0.788455i
$478$	8640.00i	0.826746i
$479$	13680.0	1.30492	0.652458	−	0.757825i	$-0.273738\pi$
$479$	13680.0	1.30492	0.652458	+	0.757825i	$0.273738\pi$
$480$	0	0
$481$	1972.00	0.186934
$482$	− 956.000i	− 0.0903415i
$483$	− 4224.00i	− 0.397927i
$484$	4748.00	0.445905
$485$	0	0
$486$	10064.0	0.939326
$487$	7916.00i	0.736567i	0.929714	+	0.368284i	$0.120054\pi$
$487$	7916.00i	0.736567i	−0.929714	+	0.368284i	$0.879946\pi$
$488$	− 7216.00i	− 0.669371i
$489$	−4544.00	−0.420218
$490$	0	0
$491$	13932.0	1.28053	0.640267	−	0.768152i	$-0.278824\pi$
$491$	13932.0	1.28053	0.640267	+	0.768152i	$0.278824\pi$
$492$	14016.0i	1.28433i
$493$	5940.00i	0.542645i
$494$	−11600.0	−1.05650
$495$	0	0
$496$	2432.00	0.220161
$497$	− 1728.00i	− 0.155959i
$498$	1152.00i	0.103659i
$499$	8260.00	0.741019	0.370509	−	0.928829i	$-0.379183\pi$
$499$	8260.00	0.741019	0.370509	+	0.928829i	$0.379183\pi$
$500$	0	0
$501$	12192.0	1.08722
$502$	5304.00i	0.471572i
$503$	11148.0i	0.988200i	0.869405	+	0.494100i	$0.164502\pi$
$503$	11148.0i	0.988200i	−0.869405	+	0.494100i	$0.835498\pi$
$504$	1184.00	0.104642
$505$	0	0
$506$	3168.00	0.278330
$507$	− 9336.00i	− 0.817803i
$508$	496.000i	0.0433198i
$509$	9690.00	0.843815	0.421907	−	0.906639i	$-0.361361\pi$
$509$	9690.00	0.843815	0.421907	+	0.906639i	$0.361361\pi$
$510$	0	0
$511$	−1448.00	−0.125354
$512$	512.000i	0.0441942i
$513$	− 8000.00i	− 0.688516i
$514$	4668.00	0.400577
$515$	0	0
$516$	−1024.00	−0.0873626
$517$	− 2448.00i	− 0.208245i
$518$	− 272.000i	− 0.0230714i
$519$	29616.0	2.50481
$520$	0	0
$521$	−16038.0	−1.34863	−0.674316	−	0.738443i	$-0.735562\pi$
$521$	−16038.0	−1.34863	−0.674316	+	0.738443i	$0.735562\pi$
$522$	− 6660.00i	− 0.558430i
$523$	− 992.000i	− 0.0829391i	−0.999140	−	0.0414695i	$-0.986796\pi$
$523$	− 992.000i	− 0.0829391i	0.999140	−	0.0414695i	$-0.0132039\pi$
$524$	−528.000	−0.0440187
$525$	0	0
$526$	−7896.00	−0.654528
$527$	10032.0i	0.829223i
$528$	1536.00i	0.126602i
$529$	−5257.00	−0.432070
$530$	0	0
$531$	15540.0	1.27002
$532$	1600.00i	0.130392i
$533$	− 25404.0i	− 2.06448i
$534$	12960.0	1.05025
$535$	0	0
$536$	−8192.00	−0.660150
$537$	− 25440.0i	− 2.04435i
$538$	− 3180.00i	− 0.254832i
$539$	3924.00	0.313578
$540$	0	0
$541$	7142.00	0.567576	0.283788	−	0.958887i	$-0.408409\pi$
$541$	7142.00	0.567576	0.283788	+	0.958887i	$0.408409\pi$
$542$	9904.00i	0.784895i
$543$	− 16784.0i	− 1.32646i
$544$	−2112.00	−0.166455
$545$	0	0
$546$	−3712.00	−0.290950
$547$	7616.00i	0.595314i	0.954673	+	0.297657i	$0.0962051\pi$
$547$	7616.00i	0.595314i	−0.954673	+	0.297657i	$0.903795\pi$
$548$	5016.00i	0.391009i
$549$	−33374.0	−2.59448
$550$	0	0
$551$	9000.00	0.695849
$552$	− 8448.00i	− 0.651396i
$553$	− 640.000i	− 0.0492144i
$554$	−3292.00	−0.252462
$555$	0	0
$556$	−11440.0	−0.872597
$557$	− 10314.0i	− 0.784593i	−0.919839	−	0.392296i	$-0.871681\pi$
$557$	− 10314.0i	− 0.784593i	0.919839	−	0.392296i	$-0.128319\pi$
$558$	− 11248.0i	− 0.853344i
$559$	1856.00	0.140430
$560$	0	0
$561$	−6336.00	−0.476838
$562$	− 2316.00i	− 0.173834i
$563$	7128.00i	0.533587i	0.963754	+	0.266793i	$0.0859641\pi$
$563$	7128.00i	0.533587i	−0.963754	+	0.266793i	$0.914036\pi$
$564$	−6528.00	−0.487373
$565$	0	0
$566$	13984.0	1.03850
$567$	1436.00i	0.106360i
$568$	− 3456.00i	− 0.255300i
$569$	−2010.00	−0.148091	−0.0740453	−	0.997255i	$-0.523591\pi$
$569$	−2010.00	−0.148091	−0.0740453	+	0.997255i	$0.523591\pi$
$570$	0	0
$571$	−23188.0	−1.69945	−0.849726	−	0.527224i	$-0.823233\pi$
$571$	−23188.0	−1.69945	−0.849726	+	0.527224i	$0.823233\pi$
$572$	− 2784.00i	− 0.203505i
$573$	35136.0i	2.56165i
$574$	−3504.00	−0.254798
$575$	0	0
$576$	2368.00	0.171296
$577$	22466.0i	1.62092i	0.585793	+	0.810461i	$0.300783\pi$
$577$	22466.0i	1.62092i	−0.585793	+	0.810461i	$0.699217\pi$
$578$	1114.00i	0.0801666i
$579$	−17264.0	−1.23915
$580$	0	0
$581$	−288.000	−0.0205650
$582$	− 17696.0i	− 1.26035i
$583$	− 2664.00i	− 0.189248i
$584$	−2896.00	−0.205201
$585$	0	0
$586$	−516.000	−0.0363750
$587$	22776.0i	1.60148i	0.599015	+	0.800738i	$0.295559\pi$
$587$	22776.0i	1.60148i	−0.599015	+	0.800738i	$0.704441\pi$
$588$	− 10464.0i	− 0.733891i
$589$	15200.0	1.06334
$590$	0	0
$591$	8592.00	0.598016
$592$	− 544.000i	− 0.0377673i
$593$	21198.0i	1.46796i	0.679174	+	0.733978i	$0.262338\pi$
$593$	21198.0i	1.46796i	−0.679174	+	0.733978i	$0.737662\pi$
$594$	1920.00	0.132624
$595$	0	0
$596$	3000.00	0.206183
$597$	− 22720.0i	− 1.55757i
$598$	15312.0i	1.04708i
$599$	−15960.0	−1.08866	−0.544330	−	0.838871i	$-0.683216\pi$
$599$	−15960.0	−1.08866	−0.544330	+	0.838871i	$0.683216\pi$
$600$	0	0
$601$	5882.00	0.399221	0.199610	−	0.979875i	$-0.436032\pi$
$601$	5882.00	0.399221	0.199610	+	0.979875i	$0.436032\pi$
$602$	− 256.000i	− 0.0173319i
$603$	37888.0i	2.55874i
$604$	1792.00	0.120721
$605$	0	0
$606$	4128.00	0.276714
$607$	8516.00i	0.569446i	0.958610	+	0.284723i	$0.0919016\pi$
$607$	8516.00i	0.569446i	−0.958610	+	0.284723i	$0.908098\pi$
$608$	3200.00i	0.213449i
$609$	2880.00	0.191631
$610$	0	0
$611$	11832.0	0.783423
$612$	9768.00i	0.645176i
$613$	− 8462.00i	− 0.557548i	−0.960357	−	0.278774i	$-0.910072\pi$
$613$	− 8462.00i	− 0.557548i	0.960357	−	0.278774i	$-0.0899280\pi$
$614$	17888.0	1.17573
$615$	0	0
$616$	−384.000	−0.0251166
$617$	− 11094.0i	− 0.723870i	−0.932203	−	0.361935i	$-0.882116\pi$
$617$	− 11094.0i	− 0.723870i	0.932203	−	0.361935i	$-0.117884\pi$
$618$	− 15808.0i	− 1.02895i
$619$	−2180.00	−0.141553	−0.0707767	−	0.997492i	$-0.522548\pi$
$619$	−2180.00	−0.141553	−0.0707767	+	0.997492i	$0.522548\pi$
$620$	0	0
$621$	−10560.0	−0.682380
$622$	2784.00i	0.179467i
$623$	3240.00i	0.208359i
$624$	−7424.00	−0.476279
$625$	0	0
$626$	−11756.0	−0.750582
$627$	9600.00i	0.611463i
$628$	− 8984.00i	− 0.570861i
$629$	2244.00	0.142248
$630$	0	0
$631$	−26848.0	−1.69382	−0.846911	−	0.531734i	$-0.821541\pi$
$631$	−26848.0	−1.69382	−0.846911	+	0.531734i	$0.821541\pi$
$632$	− 1280.00i	− 0.0805628i
$633$	− 21344.0i	− 1.34020i
$634$	−20652.0	−1.29368
$635$	0	0
$636$	−7104.00	−0.442912
$637$	18966.0i	1.17969i
$638$	2160.00i	0.134036i
$639$	−15984.0	−0.989542
$640$	0	0
$641$	26322.0	1.62193	0.810965	−	0.585095i	$-0.198943\pi$
$641$	26322.0	1.62193	0.810965	+	0.585095i	$0.198943\pi$
$642$	384.000i	0.0236063i
$643$	10168.0i	0.623619i	0.950145	+	0.311809i	$0.100935\pi$
$643$	10168.0i	0.623619i	−0.950145	+	0.311809i	$0.899065\pi$
$644$	2112.00	0.129231
$645$	0	0
$646$	−13200.0	−0.803943
$647$	− 23604.0i	− 1.43426i	−0.696937	−	0.717132i	$-0.745454\pi$
$647$	− 23604.0i	− 1.43426i	0.696937	−	0.717132i	$-0.254546\pi$
$648$	2872.00i	0.174109i
$649$	−5040.00	−0.304834
$650$	0	0
$651$	4864.00	0.292834
$652$	− 2272.00i	− 0.136470i
$653$	− 16422.0i	− 0.984139i	−0.870556	−	0.492069i	$-0.836241\pi$
$653$	− 16422.0i	− 0.984139i	0.870556	−	0.492069i	$-0.163759\pi$
$654$	15200.0	0.908818
$655$	0	0
$656$	−7008.00	−0.417098
$657$	13394.0i	0.795357i
$658$	− 1632.00i	− 0.0966899i
$659$	26100.0	1.54281	0.771405	−	0.636345i	$-0.219554\pi$
$659$	26100.0	1.54281	0.771405	+	0.636345i	$0.219554\pi$
$660$	0	0
$661$	−3058.00	−0.179943	−0.0899716	−	0.995944i	$-0.528678\pi$
$661$	−3058.00	−0.179943	−0.0899716	+	0.995944i	$0.528678\pi$
$662$	− 8456.00i	− 0.496453i
$663$	− 30624.0i	− 1.79387i
$664$	−576.000	−0.0336644
$665$	0	0
$666$	−2516.00	−0.146386
$667$	− 11880.0i	− 0.689648i
$668$	6096.00i	0.353086i
$669$	14176.0	0.819246
$670$	0	0
$671$	10824.0	0.622736
$672$	1024.00i	0.0587822i
$673$	− 10802.0i	− 0.618702i	−0.950948	−	0.309351i	$-0.899888\pi$
$673$	− 10802.0i	− 0.618702i	0.950948	−	0.309351i	$-0.100112\pi$
$674$	−2212.00	−0.126414
$675$	0	0
$676$	4668.00	0.265589
$677$	− 10674.0i	− 0.605960i	−0.952997	−	0.302980i	$-0.902018\pi$
$677$	− 10674.0i	− 0.605960i	0.952997	−	0.302980i	$-0.0979816\pi$
$678$	− 16608.0i	− 0.940747i
$679$	4424.00	0.250041
$680$	0	0
$681$	22272.0	1.25325
$682$	3648.00i	0.204823i
$683$	28608.0i	1.60272i	0.598185	+	0.801358i	$0.295889\pi$
$683$	28608.0i	1.60272i	−0.598185	+	0.801358i	$0.704111\pi$
$684$	14800.0	0.827328
$685$	0	0
$686$	5360.00	0.298317
$687$	− 2800.00i	− 0.155497i
$688$	− 512.000i	− 0.0283718i
$689$	12876.0	0.711954
$690$	0	0
$691$	−2428.00	−0.133669	−0.0668346	−	0.997764i	$-0.521290\pi$
$691$	−2428.00	−0.133669	−0.0668346	+	0.997764i	$0.521290\pi$
$692$	14808.0i	0.813462i
$693$	1776.00i	0.0973516i
$694$	−18672.0	−1.02130
$695$	0	0
$696$	5760.00	0.313696
$697$	− 28908.0i	− 1.57097i
$698$	23540.0i	1.27651i
$699$	15696.0	0.849324
$700$	0	0
$701$	−6618.00	−0.356574	−0.178287	−	0.983979i	$-0.557056\pi$
$701$	−6618.00	−0.356574	−0.178287	+	0.983979i	$0.557056\pi$
$702$	9280.00i	0.498933i
$703$	− 3400.00i	− 0.182409i
$704$	−768.000	−0.0411152
$705$	0	0
$706$	16644.0	0.887259
$707$	1032.00i	0.0548972i
$708$	13440.0i	0.713427i
$709$	−20510.0	−1.08642	−0.543208	−	0.839598i	$-0.682791\pi$
$709$	−20510.0	−1.08642	−0.543208	+	0.839598i	$0.682791\pi$
$710$	0	0
$711$	−5920.00	−0.312261
$712$	6480.00i	0.341079i
$713$	− 20064.0i	− 1.05386i
$714$	−4224.00	−0.221399
$715$	0	0
$716$	12720.0	0.663923
$717$	34560.0i	1.80009i
$718$	− 21360.0i	− 1.11023i
$719$	−31680.0	−1.64321	−0.821603	−	0.570061i	$-0.806920\pi$
$719$	−31680.0	−1.64321	−0.821603	+	0.570061i	$0.806920\pi$
$720$	0	0
$721$	3952.00	0.204133
$722$	6282.00i	0.323811i
$723$	− 3824.00i	− 0.196703i
$724$	8392.00	0.430782
$725$	0	0
$726$	18992.0	0.970880
$727$	13196.0i	0.673195i	0.941649	+	0.336597i	$0.109276\pi$
$727$	13196.0i	0.673195i	−0.941649	+	0.336597i	$0.890724\pi$
$728$	− 1856.00i	− 0.0944889i
$729$	30563.0	1.55276
$730$	0	0
$731$	2112.00	0.106861
$732$	− 28864.0i	− 1.45744i
$733$	− 8102.00i	− 0.408259i	−0.978944	−	0.204130i	$-0.934564\pi$
$733$	− 8102.00i	− 0.408259i	0.978944	−	0.204130i	$-0.0654364\pi$
$734$	11768.0	0.591778
$735$	0	0
$736$	4224.00	0.211547
$737$	− 12288.0i	− 0.614158i
$738$	32412.0i	1.61667i
$739$	12580.0	0.626201	0.313101	−	0.949720i	$-0.398632\pi$
$739$	12580.0	0.626201	0.313101	+	0.949720i	$0.398632\pi$
$740$	0	0
$741$	−46400.0	−2.30033
$742$	− 1776.00i	− 0.0878693i
$743$	− 29892.0i	− 1.47595i	−0.674828	−	0.737975i	$-0.735782\pi$
$743$	− 29892.0i	− 1.47595i	0.674828	−	0.737975i	$-0.264218\pi$
$744$	9728.00	0.479363
$745$	0	0
$746$	−4196.00	−0.205934
$747$	2664.00i	0.130483i
$748$	− 3168.00i	− 0.154858i
$749$	−96.0000	−0.00468326
$750$	0	0
$751$	−40408.0	−1.96339	−0.981697	−	0.190450i	$-0.939005\pi$
$751$	−40408.0	−1.96339	−0.981697	+	0.190450i	$0.939005\pi$
$752$	− 3264.00i	− 0.158279i
$753$	21216.0i	1.02676i
$754$	−10440.0	−0.504247
$755$	0	0
$756$	1280.00	0.0615782
$757$	32366.0i	1.55398i	0.629513	+	0.776990i	$0.283254\pi$
$757$	32366.0i	1.55398i	−0.629513	+	0.776990i	$0.716746\pi$
$758$	− 7720.00i	− 0.369925i
$759$	12672.0	0.606014
$760$	0	0
$761$	−17238.0	−0.821126	−0.410563	−	0.911832i	$-0.634668\pi$
$761$	−17238.0	−0.821126	−0.410563	+	0.911832i	$0.634668\pi$
$762$	1984.00i	0.0943212i
$763$	3800.00i	0.180300i
$764$	−17568.0	−0.831921
$765$	0	0
$766$	−19176.0	−0.904513
$767$	− 24360.0i	− 1.14679i
$768$	2048.00i	0.0962250i
$769$	−10850.0	−0.508792	−0.254396	−	0.967100i	$-0.581877\pi$
$769$	−10850.0	−0.508792	−0.254396	+	0.967100i	$0.581877\pi$
$770$	0	0
$771$	18672.0	0.872186
$772$	− 8632.00i	− 0.402425i
$773$	− 9102.00i	− 0.423514i	−0.977322	−	0.211757i	$-0.932081\pi$
$773$	− 9102.00i	− 0.423514i	0.977322	−	0.211757i	$-0.0679185\pi$
$774$	−2368.00	−0.109969
$775$	0	0
$776$	8848.00	0.409310
$777$	− 1088.00i	− 0.0502340i
$778$	26820.0i	1.23592i
$779$	−43800.0	−2.01450
$780$	0	0
$781$	5184.00	0.237514
$782$	17424.0i	0.796779i
$783$	− 7200.00i	− 0.328617i
$784$	5232.00	0.238338
$785$	0	0
$786$	−2112.00	−0.0958429
$787$	− 25504.0i	− 1.15517i	−0.816330	−	0.577585i	$-0.803995\pi$
$787$	− 25504.0i	− 1.15517i	0.816330	−	0.577585i	$-0.196005\pi$
$788$	4296.00i	0.194212i
$789$	−31584.0	−1.42512
$790$	0	0
$791$	4152.00	0.186635
$792$	3552.00i	0.159362i
$793$	52316.0i	2.34274i
$794$	26228.0	1.17229
$795$	0	0
$796$	11360.0	0.505835
$797$	14166.0i	0.629593i	0.949159	+	0.314796i	$0.101936\pi$
$797$	14166.0i	0.629593i	−0.949159	+	0.314796i	$0.898064\pi$
$798$	6400.00i	0.283907i
$799$	13464.0	0.596148
$800$	0	0
$801$	29970.0	1.32202
$802$	− 11676.0i	− 0.514082i
$803$	− 4344.00i	− 0.190905i
$804$	−32768.0	−1.43736
$805$	0	0
$806$	−17632.0	−0.770547
$807$	− 12720.0i	− 0.554852i
$808$	2064.00i	0.0898654i
$809$	−33210.0	−1.44327	−0.721633	−	0.692276i	$-0.756608\pi$
$809$	−33210.0	−1.44327	−0.721633	+	0.692276i	$0.756608\pi$
$810$	0	0
$811$	39212.0	1.69780	0.848902	−	0.528550i	$-0.177264\pi$
$811$	39212.0	1.69780	0.848902	+	0.528550i	$0.177264\pi$
$812$	1440.00i	0.0622341i
$813$	39616.0i	1.70897i
$814$	816.000	0.0351361
$815$	0	0
$816$	−8448.00	−0.362425
$817$	− 3200.00i	− 0.137030i
$818$	− 19060.0i	− 0.814691i
$819$	−8584.00	−0.366238
$820$	0	0
$821$	6222.00	0.264494	0.132247	−	0.991217i	$-0.457781\pi$
$821$	6222.00	0.264494	0.132247	+	0.991217i	$0.457781\pi$
$822$	20064.0i	0.851353i
$823$	− 31172.0i	− 1.32028i	−0.751144	−	0.660138i	$-0.770498\pi$
$823$	− 31172.0i	− 1.32028i	0.751144	−	0.660138i	$-0.229502\pi$
$824$	7904.00	0.334161
$825$	0	0
$826$	−3360.00	−0.141537
$827$	− 264.000i	− 0.0111006i	−0.999985	−	0.00555029i	$-0.998233\pi$
$827$	− 264.000i	− 0.0111006i	0.999985	−	0.00555029i	$-0.00176672\pi$
$828$	− 19536.0i	− 0.819955i
$829$	29050.0	1.21707	0.608533	−	0.793528i	$-0.291758\pi$
$829$	29050.0	1.21707	0.608533	+	0.793528i	$0.291758\pi$
$830$	0	0
$831$	−13168.0	−0.549691
$832$	− 3712.00i	− 0.154676i
$833$	21582.0i	0.897685i
$834$	−45760.0	−1.89993
$835$	0	0
$836$	−4800.00	−0.198578
$837$	− 12160.0i	− 0.502164i
$838$	− 14520.0i	− 0.598550i
$839$	21720.0	0.893752	0.446876	−	0.894596i	$-0.352537\pi$
$839$	21720.0	0.893752	0.446876	+	0.894596i	$0.352537\pi$
$840$	0	0
$841$	−16289.0	−0.667883
$842$	24124.0i	0.987373i
$843$	− 9264.00i	− 0.378492i
$844$	10672.0	0.435243
$845$	0	0
$846$	−15096.0	−0.613488
$847$	4748.00i	0.192613i
$848$	− 3552.00i	− 0.143840i
$849$	55936.0	2.26115
$850$	0	0
$851$	−4488.00	−0.180783
$852$	− 13824.0i	− 0.555871i
$853$	6658.00i	0.267252i	0.991032	+	0.133626i	$0.0426620\pi$
$853$	6658.00i	0.267252i	−0.991032	+	0.133626i	$0.957338\pi$
$854$	7216.00	0.289141
$855$	0	0
$856$	−192.000	−0.00766638
$857$	− 13974.0i	− 0.556993i	−0.960437	−	0.278496i	$-0.910164\pi$
$857$	− 13974.0i	− 0.556993i	0.960437	−	0.278496i	$-0.0898360\pi$
$858$	− 11136.0i	− 0.443096i
$859$	−23780.0	−0.944544	−0.472272	−	0.881453i	$-0.656566\pi$
$859$	−23780.0	−0.944544	−0.472272	+	0.881453i	$0.656566\pi$
$860$	0	0
$861$	−14016.0	−0.554778
$862$	− 27216.0i	− 1.07538i
$863$	12228.0i	0.482324i	0.970485	+	0.241162i	$0.0775286\pi$
$863$	12228.0i	0.482324i	−0.970485	+	0.241162i	$0.922471\pi$
$864$	2560.00	0.100802
$865$	0	0
$866$	−7676.00	−0.301202
$867$	4456.00i	0.174549i
$868$	2432.00i	0.0951008i
$869$	1920.00	0.0749500
$870$	0	0
$871$	59392.0	2.31047
$872$	7600.00i	0.295147i
$873$	− 40922.0i	− 1.58648i
$874$	26400.0	1.02173
$875$	0	0
$876$	−11584.0	−0.446789
$877$	11606.0i	0.446872i	0.974719	+	0.223436i	$0.0717274\pi$
$877$	11606.0i	0.446872i	−0.974719	+	0.223436i	$0.928273\pi$
$878$	− 14800.0i	− 0.568879i
$879$	−2064.00	−0.0792002
$880$	0	0
$881$	−32958.0	−1.26037	−0.630183	−	0.776446i	$-0.717020\pi$
$881$	−32958.0	−1.26037	−0.630183	+	0.776446i	$0.717020\pi$
$882$	− 24198.0i	− 0.923797i
$883$	− 8072.00i	− 0.307638i	−0.988099	−	0.153819i	$-0.950843\pi$
$883$	− 8072.00i	− 0.307638i	0.988099	−	0.153819i	$-0.0491573\pi$
$884$	15312.0	0.582577
$885$	0	0
$886$	16704.0	0.633388
$887$	15756.0i	0.596431i	0.954498	+	0.298216i	$0.0963915\pi$
$887$	15756.0i	0.596431i	−0.954498	+	0.298216i	$0.903609\pi$
$888$	− 2176.00i	− 0.0822317i
$889$	−496.000	−0.0187124
$890$	0	0
$891$	−4308.00	−0.161979
$892$	7088.00i	0.266058i
$893$	− 20400.0i	− 0.764457i
$894$	12000.0	0.448926
$895$	0	0
$896$	−512.000	−0.0190901
$897$	61248.0i	2.27983i
$898$	− 21540.0i	− 0.800444i
$899$	13680.0	0.507512
$900$	0	0
$901$	14652.0	0.541763
$902$	− 10512.0i	− 0.388039i
$903$	− 1024.00i	− 0.0377371i
$904$	8304.00	0.305517
$905$	0	0
$906$	7168.00	0.262849
$907$	18776.0i	0.687372i	0.939085	+	0.343686i	$0.111676\pi$
$907$	18776.0i	0.687372i	−0.939085	+	0.343686i	$0.888324\pi$
$908$	11136.0i	0.407006i
$909$	9546.00	0.348318
$910$	0	0
$911$	−20568.0	−0.748022	−0.374011	−	0.927424i	$-0.622018\pi$
$911$	−20568.0	−0.748022	−0.374011	+	0.927424i	$0.622018\pi$
$912$	12800.0i	0.464748i
$913$	− 864.000i	− 0.0313190i
$914$	13388.0	0.484503
$915$	0	0
$916$	1400.00	0.0504992
$917$	− 528.000i	− 0.0190143i
$918$	10560.0i	0.379664i
$919$	6280.00	0.225417	0.112708	−	0.993628i	$-0.464047\pi$
$919$	6280.00	0.225417	0.112708	+	0.993628i	$0.464047\pi$
$920$	0	0
$921$	71552.0	2.55996
$922$	− 6036.00i	− 0.215602i
$923$	25056.0i	0.893530i
$924$	−1536.00	−0.0546869
$925$	0	0
$926$	28984.0	1.02859
$927$	− 36556.0i	− 1.29521i
$928$	2880.00i	0.101876i
$929$	20430.0	0.721514	0.360757	−	0.932660i	$-0.382518\pi$
$929$	20430.0	0.721514	0.360757	+	0.932660i	$0.382518\pi$
$930$	0	0
$931$	32700.0	1.15113
$932$	7848.00i	0.275826i
$933$	11136.0i	0.390757i
$934$	−15552.0	−0.544836
$935$	0	0
$936$	−17168.0	−0.599523
$937$	8906.00i	0.310508i	0.987875	+	0.155254i	$0.0496197\pi$
$937$	8906.00i	0.310508i	−0.987875	+	0.155254i	$0.950380\pi$
$938$	− 8192.00i	− 0.285158i
$939$	−47024.0	−1.63426
$940$	0	0
$941$	−17418.0	−0.603412	−0.301706	−	0.953401i	$-0.597556\pi$
$941$	−17418.0	−0.603412	−0.301706	+	0.953401i	$0.597556\pi$
$942$	− 35936.0i	− 1.24295i
$943$	57816.0i	1.99655i
$944$	−6720.00	−0.231692
$945$	0	0
$946$	768.000	0.0263952
$947$	− 2544.00i	− 0.0872956i	−0.999047	−	0.0436478i	$-0.986102\pi$
$947$	− 2544.00i	− 0.0872956i	0.999047	−	0.0436478i	$-0.0138979\pi$
$948$	− 5120.00i	− 0.175411i
$949$	20996.0	0.718187
$950$	0	0
$951$	−82608.0	−2.81677
$952$	− 2112.00i	− 0.0719016i
$953$	− 15402.0i	− 0.523525i	−0.965132	−	0.261763i	$-0.915696\pi$
$953$	− 15402.0i	− 0.523525i	0.965132	−	0.261763i	$-0.0843038\pi$
$954$	−16428.0	−0.557522
$955$	0	0
$956$	−17280.0	−0.584597
$957$	8640.00i	0.291841i
$958$	27360.0i	0.922716i
$959$	−5016.00	−0.168900
$960$	0	0
$961$	−6687.00	−0.224464
$962$	3944.00i	0.132183i
$963$	888.000i	0.0297148i
$964$	1912.00	0.0638811
$965$	0	0
$966$	8448.00	0.281377
$967$	− 49444.0i	− 1.64427i	−0.569291	−	0.822136i	$-0.692782\pi$
$967$	− 49444.0i	− 1.64427i	0.569291	−	0.822136i	$-0.307218\pi$
$968$	9496.00i	0.315303i
$969$	−52800.0	−1.75044
$970$	0	0
$971$	−25188.0	−0.832463	−0.416231	−	0.909259i	$-0.636649\pi$
$971$	−25188.0	−0.832463	−0.416231	+	0.909259i	$0.636649\pi$
$972$	20128.0i	0.664204i
$973$	− 11440.0i	− 0.376927i
$974$	−15832.0	−0.520832
$975$	0	0
$976$	14432.0	0.473317
$977$	2946.00i	0.0964697i	0.998836	+	0.0482348i	$0.0153596\pi$
$977$	2946.00i	0.0964697i	−0.998836	+	0.0482348i	$0.984640\pi$
$978$	− 9088.00i	− 0.297139i
$979$	−9720.00	−0.317316
$980$	0	0
$981$	35150.0	1.14399
$982$	27864.0i	0.905475i
$983$	− 15012.0i	− 0.487089i	−0.969890	−	0.243544i	$-0.921690\pi$
$983$	− 15012.0i	− 0.487089i	0.969890	−	0.243544i	$-0.0783102\pi$
$984$	−28032.0	−0.908158
$985$	0	0
$986$	−11880.0	−0.383708
$987$	− 6528.00i	− 0.210525i
$988$	− 23200.0i	− 0.747055i
$989$	−4224.00	−0.135809
$990$	0	0
$991$	−5128.00	−0.164376	−0.0821878	−	0.996617i	$-0.526191\pi$
$991$	−5128.00	−0.164376	−0.0821878	+	0.996617i	$0.526191\pi$
$992$	4864.00i	0.155678i
$993$	− 33824.0i	− 1.08094i
$994$	3456.00	0.110279
$995$	0	0
$996$	−2304.00	−0.0732982
$997$	− 49714.0i	− 1.57920i	−0.613625	−	0.789598i	$-0.710289\pi$
$997$	− 49714.0i	− 1.57920i	0.613625	−	0.789598i	$-0.289711\pi$
$998$	16520.0i	0.523979i
$999$	−2720.00	−0.0861431

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	50.4.b.a.49.2		2
3.2	odd	2		450.4.c.d.199.1		2
4.3	odd	2		400.4.c.c.49.1		2
5.2	odd	4		50.4.a.c.1.1		1
5.3	odd	4		10.4.a.a.1.1	✓	1
5.4	even	2	inner	50.4.b.a.49.1		2
15.2	even	4		450.4.a.q.1.1		1
15.8	even	4		90.4.a.a.1.1		1
15.14	odd	2		450.4.c.d.199.2		2
20.3	even	4		80.4.a.f.1.1		1
20.7	even	4		400.4.a.b.1.1		1
20.19	odd	2		400.4.c.c.49.2		2
35.3	even	12		490.4.e.a.471.1		2
35.13	even	4		490.4.a.o.1.1		1
35.18	odd	12		490.4.e.i.471.1		2
35.23	odd	12		490.4.e.i.361.1		2
35.27	even	4		2450.4.a.b.1.1		1
35.33	even	12		490.4.e.a.361.1		2
40.3	even	4		320.4.a.b.1.1		1
40.13	odd	4		320.4.a.m.1.1		1
40.27	even	4		1600.4.a.bx.1.1		1
40.37	odd	4		1600.4.a.d.1.1		1
45.13	odd	12		810.4.e.c.541.1		2
45.23	even	12		810.4.e.w.541.1		2
45.38	even	12		810.4.e.w.271.1		2
45.43	odd	12		810.4.e.c.271.1		2
55.43	even	4		1210.4.a.b.1.1		1
60.23	odd	4		720.4.a.j.1.1		1
65.38	odd	4		1690.4.a.a.1.1		1
80.3	even	4		1280.4.d.g.641.2		2
80.13	odd	4		1280.4.d.j.641.1		2
80.43	even	4		1280.4.d.g.641.1		2
80.53	odd	4		1280.4.d.j.641.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.4.a.a.1.1	✓	1	5.3	odd	4
50.4.a.c.1.1		1	5.2	odd	4
50.4.b.a.49.1		2	5.4	even	2	inner
50.4.b.a.49.2		2	1.1	even	1	trivial
80.4.a.f.1.1		1	20.3	even	4
90.4.a.a.1.1		1	15.8	even	4
320.4.a.b.1.1		1	40.3	even	4
320.4.a.m.1.1		1	40.13	odd	4
400.4.a.b.1.1		1	20.7	even	4
400.4.c.c.49.1		2	4.3	odd	2
400.4.c.c.49.2		2	20.19	odd	2
450.4.a.q.1.1		1	15.2	even	4
450.4.c.d.199.1		2	3.2	odd	2
450.4.c.d.199.2		2	15.14	odd	2
490.4.a.o.1.1		1	35.13	even	4
490.4.e.a.361.1		2	35.33	even	12
490.4.e.a.471.1		2	35.3	even	12
490.4.e.i.361.1		2	35.23	odd	12
490.4.e.i.471.1		2	35.18	odd	12
720.4.a.j.1.1		1	60.23	odd	4
810.4.e.c.271.1		2	45.43	odd	12
810.4.e.c.541.1		2	45.13	odd	12
810.4.e.w.271.1		2	45.38	even	12
810.4.e.w.541.1		2	45.23	even	12
1210.4.a.b.1.1		1	55.43	even	4
1280.4.d.g.641.1		2	80.43	even	4
1280.4.d.g.641.2		2	80.3	even	4
1280.4.d.j.641.1		2	80.13	odd	4
1280.4.d.j.641.2		2	80.53	odd	4
1600.4.a.d.1.1		1	40.37	odd	4
1600.4.a.bx.1.1		1	40.27	even	4
1690.4.a.a.1.1		1	65.38	odd	4
2450.4.a.b.1.1		1	35.27	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:	\( N \)	\(=\)	\( 50 = 2 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	50.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{2} +8.00000i q^{3} -4.00000 q^{4} -16.0000 q^{6} -4.00000i q^{7} -8.00000i q^{8} -37.0000 q^{9} +12.0000 q^{11} -32.0000i q^{12} +58.0000i q^{13} +8.00000 q^{14} +16.0000 q^{16} +66.0000i q^{17} -74.0000i q^{18} +100.000 q^{19} +32.0000 q^{21} +24.0000i q^{22} -132.000i q^{23} +64.0000 q^{24} -116.000 q^{26} -80.0000i q^{27} +16.0000i q^{28} +90.0000 q^{29} +152.000 q^{31} +32.0000i q^{32} +96.0000i q^{33} -132.000 q^{34} +148.000 q^{36} -34.0000i q^{37} +200.000i q^{38} -464.000 q^{39} -438.000 q^{41} +64.0000i q^{42} -32.0000i q^{43} -48.0000 q^{44} +264.000 q^{46} -204.000i q^{47} +128.000i q^{48} +327.000 q^{49} -528.000 q^{51} -232.000i q^{52} -222.000i q^{53} +160.000 q^{54} -32.0000 q^{56} +800.000i q^{57} +180.000i q^{58} -420.000 q^{59} +902.000 q^{61} +304.000i q^{62} +148.000i q^{63} -64.0000 q^{64} -192.000 q^{66} -1024.00i q^{67} -264.000i q^{68} +1056.00 q^{69} +432.000 q^{71} +296.000i q^{72} -362.000i q^{73} +68.0000 q^{74} -400.000 q^{76} -48.0000i q^{77} -928.000i q^{78} +160.000 q^{79} -359.000 q^{81} -876.000i q^{82} -72.0000i q^{83} -128.000 q^{84} +64.0000 q^{86} +720.000i q^{87} -96.0000i q^{88} -810.000 q^{89} +232.000 q^{91} +528.000i q^{92} +1216.00i q^{93} +408.000 q^{94} -256.000 q^{96} +1106.00i q^{97} +654.000i q^{98} -444.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{4} - 32 q^{6} - 74 q^{9} + 24 q^{11} + 16 q^{14} + 32 q^{16} + 200 q^{19} + 64 q^{21} + 128 q^{24} - 232 q^{26} + 180 q^{29} + 304 q^{31} - 264 q^{34} + 296 q^{36} - 928 q^{39} - 876 q^{41} - 96 q^{44}+ \cdots - 888 q^{99}+O(q^{100}) \) 2 * q - 8 * q^4 - 32 * q^6 - 74 * q^9 + 24 * q^11 + 16 * q^14 + 32 * q^16 + 200 * q^19 + 64 * q^21 + 128 * q^24 - 232 * q^26 + 180 * q^29 + 304 * q^31 - 264 * q^34 + 296 * q^36 - 928 * q^39 - 876 * q^41 - 96 * q^44 + 528 * q^46 + 654 * q^49 - 1056 * q^51 + 320 * q^54 - 64 * q^56 - 840 * q^59 + 1804 * q^61 - 128 * q^64 - 384 * q^66 + 2112 * q^69 + 864 * q^71 + 136 * q^74 - 800 * q^76 + 320 * q^79 - 718 * q^81 - 256 * q^84 + 128 * q^86 - 1620 * q^89 + 464 * q^91 + 816 * q^94 - 512 * q^96 - 888 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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