LMFDB - Embedded newform 98.5.b.a.97.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [98,5,Mod(97,98)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(98, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("98.97");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 98 = 2 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	98.b (of order $2$, degree $1$, 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:	$10.1302563822$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.2048.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 2 $ x^4 + 4*x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			97.4
Root			$-1.84776i$ of defining polynomial
Character	$\chi$	$=$	98.97
Dual form			98.5.b.a.97.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82843 q^{2} +2.03347i q^{3} +8.00000 q^{4} +0.710974i q^{5} +5.75152i q^{6} +22.6274 q^{8} +76.8650 q^{9} +O(q^{10})$	\(q+2.82843 q^{2} +2.03347i q^{3} +8.00000 q^{4} +0.710974i q^{5} +5.75152i q^{6} +22.6274 q^{8} +76.8650 q^{9} +2.01094i q^{10} +151.664 q^{11} +16.2677i q^{12} +260.864i q^{13} -1.44574 q^{15} +64.0000 q^{16} -385.598i q^{17} +217.407 q^{18} +390.140i q^{19} +5.68779i q^{20} +428.971 q^{22} -177.647 q^{23} +46.0121i q^{24} +624.495 q^{25} +737.836i q^{26} +321.013i q^{27} -320.887 q^{29} -4.08918 q^{30} -1346.37i q^{31} +181.019 q^{32} +308.404i q^{33} -1090.64i q^{34} +614.920 q^{36} -797.088 q^{37} +1103.48i q^{38} -530.459 q^{39} +16.0875i q^{40} +815.856i q^{41} -2167.70 q^{43} +1213.31 q^{44} +54.6490i q^{45} -502.461 q^{46} -4285.61i q^{47} +130.142i q^{48} +1766.34 q^{50} +784.102 q^{51} +2086.91i q^{52} -3171.57 q^{53} +907.963i q^{54} +107.829i q^{55} -793.337 q^{57} -907.606 q^{58} +4706.39i q^{59} -11.5659 q^{60} -2534.78i q^{61} -3808.12i q^{62} +512.000 q^{64} -185.468 q^{65} +872.298i q^{66} -4092.43 q^{67} -3084.78i q^{68} -361.239i q^{69} -2255.28 q^{71} +1739.26 q^{72} -4653.19i q^{73} -2254.51 q^{74} +1269.89i q^{75} +3121.12i q^{76} -1500.37 q^{78} -4193.74 q^{79} +45.5023i q^{80} +5573.29 q^{81} +2307.59i q^{82} -7799.12i q^{83} +274.150 q^{85} -6131.18 q^{86} -652.514i q^{87} +3431.76 q^{88} +9469.11i q^{89} +154.571i q^{90} -1421.17 q^{92} +2737.81 q^{93} -12121.5i q^{94} -277.379 q^{95} +368.097i q^{96} +9945.39i q^{97} +11657.7 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 32 q^{4} - 196 q^{9}+O(q^{10}) $ 4 * q + 32 * q^4 - 196 * q^9	$ 4 q + 32 q^{4} - 196 q^{9} + 24 q^{11} + 888 q^{15} + 256 q^{16} + 1424 q^{18} + 1648 q^{22} + 104 q^{23} + 948 q^{25} - 1408 q^{29} - 2528 q^{30} - 1568 q^{36} - 3392 q^{37} + 2200 q^{39} - 2024 q^{43} + 192 q^{44} - 2304 q^{46} + 4384 q^{50} + 18936 q^{51} - 16680 q^{53} - 6064 q^{57} + 352 q^{58} + 7104 q^{60} + 2048 q^{64} - 6048 q^{65} - 20816 q^{67} - 1984 q^{71} + 11392 q^{72} + 576 q^{74} - 12224 q^{78} - 29616 q^{79} + 30852 q^{81} - 29688 q^{85} - 18800 q^{86} + 13184 q^{88} + 832 q^{92} + 18192 q^{93} + 7240 q^{95} + 72160 q^{99}+O(q^{100}) $ 4 * q + 32 * q^4 - 196 * q^9 + 24 * q^11 + 888 * q^15 + 256 * q^16 + 1424 * q^18 + 1648 * q^22 + 104 * q^23 + 948 * q^25 - 1408 * q^29 - 2528 * q^30 - 1568 * q^36 - 3392 * q^37 + 2200 * q^39 - 2024 * q^43 + 192 * q^44 - 2304 * q^46 + 4384 * q^50 + 18936 * q^51 - 16680 * q^53 - 6064 * q^57 + 352 * q^58 + 7104 * q^60 + 2048 * q^64 - 6048 * q^65 - 20816 * q^67 - 1984 * q^71 + 11392 * q^72 + 576 * q^74 - 12224 * q^78 - 29616 * q^79 + 30852 * q^81 - 29688 * q^85 - 18800 * q^86 + 13184 * q^88 + 832 * q^92 + 18192 * q^93 + 7240 * q^95 + 72160 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/98\mathbb{Z}\right)^\times$.

$n$	$3$
$\chi(n)$	$-1$

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 $

$2$	2.82843	0.707107
$2$	2.82843	0.707107
$3$	2.03347i	0.225941i	0.993598	+	0.112970i	$0.0360365\pi$
$3$	2.03347i	0.225941i	−0.993598	+	0.112970i	$0.963963\pi$
$4$	8.00000	0.500000
$5$	0.710974i	0.0284390i	0.999899	+	0.0142195i	$0.00452635\pi$
$5$	0.710974i	0.0284390i	−0.999899	+	0.0142195i	$0.995474\pi$
$6$	5.75152i	0.159764i
$7$	0	0
$7$	0	0
$8$	22.6274	0.353553
$9$	76.8650	0.948951
$10$	2.01094i	0.0201094i
$11$	151.664	1.25342	0.626711	−	0.779252i	$-0.284401\pi$
$11$	151.664	1.25342	0.626711	+	0.779252i	$0.284401\pi$
$12$	16.2677i	0.112970i
$13$	260.864i	1.54358i	0.635880	+	0.771788i	$0.280637\pi$
$13$	260.864i	1.54358i	−0.635880	+	0.771788i	$0.719363\pi$
$14$	0	0
$15$	−1.44574	−0.00642552
$16$	64.0000	0.250000
$17$	− 385.598i	− 1.33425i	−0.744946	−	0.667125i	$-0.767525\pi$
$17$	− 385.598i	− 1.33425i	0.744946	−	0.667125i	$-0.232475\pi$
$18$	217.407	0.671009
$19$	390.140i	1.08072i	0.841434	+	0.540360i	$0.181712\pi$
$19$	390.140i	1.08072i	−0.841434	+	0.540360i	$0.818288\pi$
$20$	5.68779i	0.0142195i
$21$	0	0
$22$	428.971	0.886303
$23$	−177.647	−0.335816	−0.167908	−	0.985803i	$-0.553701\pi$
$23$	−177.647	−0.335816	−0.167908	+	0.985803i	$0.553701\pi$
$24$	46.0121i	0.0798822i
$25$	624.495	0.999191
$26$	737.836i	1.09147i
$27$	321.013i	0.440348i
$28$	0	0
$29$	−320.887	−0.381554	−0.190777	−	0.981633i	$-0.561101\pi$
$29$	−320.887	−0.381554	−0.190777	+	0.981633i	$0.561101\pi$
$30$	−4.08918	−0.00454353
$31$	− 1346.37i	− 1.40101i	−0.713646	−	0.700506i	$-0.752958\pi$
$31$	− 1346.37i	− 1.40101i	0.713646	−	0.700506i	$-0.247042\pi$
$32$	181.019	0.176777
$33$	308.404i	0.283199i
$34$	− 1090.64i	− 0.943457i
$35$	0	0
$36$	614.920	0.474475
$37$	−797.088	−0.582241	−0.291121	−	0.956686i	$-0.594028\pi$
$37$	−797.088	−0.582241	−0.291121	+	0.956686i	$0.594028\pi$
$38$	1103.48i	0.764184i
$39$	−530.459	−0.348757
$40$	16.0875i	0.0100547i
$41$	815.856i	0.485340i	0.970109	+	0.242670i	$0.0780232\pi$
$41$	815.856i	0.485340i	−0.970109	+	0.242670i	$0.921977\pi$
$42$	0	0
$43$	−2167.70	−1.17236	−0.586182	−	0.810179i	$-0.699370\pi$
$43$	−2167.70	−1.17236	−0.586182	+	0.810179i	$0.699370\pi$
$44$	1213.31	0.626711
$45$	54.6490i	0.0269872i
$46$	−502.461	−0.237458
$47$	− 4285.61i	− 1.94007i	−0.242970	−	0.970034i	$-0.578122\pi$
$47$	− 4285.61i	− 1.94007i	0.242970	−	0.970034i	$-0.421878\pi$
$48$	130.142i	0.0564852i
$49$	0	0
$50$	1766.34	0.706535
$51$	784.102	0.301462
$52$	2086.91i	0.771788i
$53$	−3171.57	−1.12907	−0.564536	−	0.825408i	$-0.690945\pi$
$53$	−3171.57	−1.12907	−0.564536	+	0.825408i	$0.690945\pi$
$54$	907.963i	0.311373i
$55$	107.829i	0.0356460i
$56$	0	0
$57$	−793.337	−0.244179
$58$	−907.606	−0.269800
$59$	4706.39i	1.35202i	0.736891	+	0.676012i	$0.236293\pi$
$59$	4706.39i	1.35202i	−0.736891	+	0.676012i	$0.763707\pi$
$60$	−11.5659	−0.00321276
$61$	− 2534.78i	− 0.681209i	−0.940207	−	0.340604i	$-0.889368\pi$
$61$	− 2534.78i	− 0.681209i	0.940207	−	0.340604i	$-0.110632\pi$
$62$	− 3808.12i	− 0.990665i
$63$	0	0
$64$	512.000	0.125000
$65$	−185.468	−0.0438977
$66$	872.298i	0.200252i
$67$	−4092.43	−0.911657	−0.455828	−	0.890068i	$-0.650657\pi$
$67$	−4092.43	−0.911657	−0.455828	+	0.890068i	$0.650657\pi$
$68$	− 3084.78i	− 0.667125i
$69$	− 361.239i	− 0.0758746i
$70$	0	0
$71$	−2255.28	−0.447388	−0.223694	−	0.974659i	$-0.571812\pi$
$71$	−2255.28	−0.447388	−0.223694	+	0.974659i	$0.571812\pi$
$72$	1739.26	0.335505
$73$	− 4653.19i	− 0.873183i	−0.899660	−	0.436591i	$-0.856186\pi$
$73$	− 4653.19i	− 0.873183i	0.899660	−	0.436591i	$-0.143814\pi$
$74$	−2254.51	−0.411707
$75$	1269.89i	0.225758i
$76$	3121.12i	0.540360i
$77$	0	0
$78$	−1500.37	−0.246608
$79$	−4193.74	−0.671965	−0.335983	−	0.941868i	$-0.609068\pi$
$79$	−4193.74	−0.671965	−0.335983	+	0.941868i	$0.609068\pi$
$80$	45.5023i	0.00710974i
$81$	5573.29	0.849458
$82$	2307.59i	0.343187i
$83$	− 7799.12i	− 1.13211i	−0.824367	−	0.566056i	$-0.808468\pi$
$83$	− 7799.12i	− 1.13211i	0.824367	−	0.566056i	$-0.191532\pi$
$84$	0	0
$85$	274.150	0.0379447
$86$	−6131.18	−0.828986
$87$	− 652.514i	− 0.0862088i
$88$	3431.76	0.443151
$89$	9469.11i	1.19544i	0.801704	+	0.597722i	$0.203927\pi$
$89$	9469.11i	1.19544i	−0.801704	+	0.597722i	$0.796073\pi$
$90$	154.571i	0.0190828i
$91$	0	0
$92$	−1421.17	−0.167908
$93$	2737.81	0.316546
$94$	− 12121.5i	− 1.37183i
$95$	−277.379	−0.0307345
$96$	368.097i	0.0399411i
$97$	9945.39i	1.05701i	0.848931	+	0.528504i	$0.177247\pi$
$97$	9945.39i	1.05701i	−0.848931	+	0.528504i	$0.822753\pi$
$98$	0	0
$99$	11657.7	1.18944
$100$	4995.96	0.499596
$101$	2068.32i	0.202756i	0.994848	+	0.101378i	$0.0323252\pi$
$101$	2068.32i	0.202756i	−0.994848	+	0.101378i	$0.967675\pi$
$102$	2217.77	0.213166
$103$	2504.76i	0.236098i	0.993008	+	0.118049i	$0.0376639\pi$
$103$	2504.76i	0.236098i	−0.993008	+	0.118049i	$0.962336\pi$
$104$	5902.68i	0.545736i
$105$	0	0
$106$	−8970.54	−0.798375
$107$	6443.39	0.562791	0.281395	−	0.959592i	$-0.409203\pi$
$107$	6443.39	0.562791	0.281395	+	0.959592i	$0.409203\pi$
$108$	2568.11i	0.220174i
$109$	23546.0	1.98182	0.990912	−	0.134515i	$-0.0429477\pi$
$109$	23546.0	1.98182	0.990912	+	0.134515i	$0.0429477\pi$
$110$	304.987i	0.0252055i
$111$	− 1620.85i	− 0.131552i
$112$	0	0
$113$	19692.2	1.54219	0.771096	−	0.636719i	$-0.219709\pi$
$113$	19692.2	1.54219	0.771096	+	0.636719i	$0.219709\pi$
$114$	−2243.90	−0.172660
$115$	− 126.302i	− 0.00955026i
$116$	−2567.10	−0.190777
$117$	20051.3i	1.46478i
$118$	13311.7i	0.956025i
$119$	0	0
$120$	−32.7134	−0.00227177
$121$	8360.97	0.571065
$122$	− 7169.44i	− 0.481687i
$123$	−1659.02	−0.109658
$124$	− 10771.0i	− 0.700506i
$125$	888.358i	0.0568549i
$126$	0	0
$127$	−12550.5	−0.778135	−0.389067	−	0.921209i	$-0.627203\pi$
$127$	−12550.5	−0.778135	−0.389067	+	0.921209i	$0.627203\pi$
$128$	1448.15	0.0883883
$129$	− 4407.95i	− 0.264885i
$130$	−524.582	−0.0310403
$131$	− 19323.3i	− 1.12600i	−0.826457	−	0.562999i	$-0.809647\pi$
$131$	− 19323.3i	− 1.12600i	0.826457	−	0.562999i	$-0.190353\pi$
$132$	2467.23i	0.141600i
$133$	0	0
$134$	−11575.1	−0.644639
$135$	−228.232	−0.0125230
$136$	− 8725.09i	− 0.471728i
$137$	3708.59	0.197591	0.0987955	−	0.995108i	$-0.468501\pi$
$137$	3708.59	0.197591	0.0987955	+	0.995108i	$0.468501\pi$
$138$	− 1021.74i	− 0.0536515i
$139$	1096.09i	0.0567307i	0.999598	+	0.0283653i	$0.00903018\pi$
$139$	1096.09i	0.0567307i	−0.999598	+	0.0283653i	$0.990970\pi$
$140$	0	0
$141$	8714.65	0.438341
$142$	−6378.90	−0.316351
$143$	39563.7i	1.93475i
$144$	4919.36	0.237238
$145$	− 228.143i	− 0.0108510i
$146$	− 13161.2i	− 0.617433i
$147$	0	0
$148$	−6376.71	−0.291121
$149$	−1233.82	−0.0555748	−0.0277874	−	0.999614i	$-0.508846\pi$
$149$	−1233.82	−0.0555748	−0.0277874	+	0.999614i	$0.508846\pi$
$150$	3591.79i	0.159635i
$151$	−42672.8	−1.87153	−0.935766	−	0.352622i	$-0.885290\pi$
$151$	−42672.8	−1.87153	−0.935766	+	0.352622i	$0.885290\pi$
$152$	8827.86i	0.382092i
$153$	− 29639.0i	− 1.26614i
$154$	0	0
$155$	957.236	0.0398433
$156$	−4243.67	−0.174378
$157$	− 29884.9i	− 1.21242i	−0.795305	−	0.606209i	$-0.792689\pi$
$157$	− 29884.9i	− 1.21242i	0.795305	−	0.606209i	$-0.207311\pi$
$158$	−11861.7	−0.475151
$159$	− 6449.28i	− 0.255104i
$160$	128.700i	0.00502734i
$161$	0	0
$162$	15763.7	0.600658
$163$	−26205.2	−0.986308	−0.493154	−	0.869942i	$-0.664156\pi$
$163$	−26205.2	−0.986308	−0.493154	+	0.869942i	$0.664156\pi$
$164$	6526.85i	0.242670i
$165$	−219.267	−0.00805389
$166$	− 22059.3i	− 0.800525i
$167$	− 35887.5i	− 1.28680i	−0.765532	−	0.643398i	$-0.777524\pi$
$167$	− 35887.5i	− 1.28680i	0.765532	−	0.643398i	$-0.222476\pi$
$168$	0	0
$169$	−39489.2	−1.38263
$170$	775.414	0.0268309
$171$	29988.1i	1.02555i
$172$	−17341.6	−0.586182
$173$	10972.7i	0.366625i	0.983055	+	0.183312i	$0.0586820\pi$
$173$	10972.7i	0.366625i	−0.983055	+	0.183312i	$0.941318\pi$
$174$	− 1845.59i	− 0.0609588i
$175$	0	0
$176$	9706.50	0.313355
$177$	−9570.30	−0.305477
$178$	26782.7i	0.845306i
$179$	−33844.1	−1.05628	−0.528138	−	0.849159i	$-0.677109\pi$
$179$	−33844.1	−1.05628	−0.528138	+	0.849159i	$0.677109\pi$
$180$	437.192i	0.0134936i
$181$	50094.5i	1.52909i	0.644571	+	0.764545i	$0.277036\pi$
$181$	50094.5i	1.52909i	−0.644571	+	0.764545i	$0.722964\pi$
$182$	0	0
$183$	5154.39	0.153913
$184$	−4019.69	−0.118729
$185$	− 566.709i	− 0.0165583i
$186$	7743.69	0.223832
$187$	− 58481.4i	− 1.67238i
$188$	− 34284.9i	− 0.970034i
$189$	0	0
$190$	−784.547	−0.0217326
$191$	17612.9	0.482798	0.241399	−	0.970426i	$-0.422394\pi$
$191$	17612.9	0.482798	0.241399	+	0.970426i	$0.422394\pi$
$192$	1041.14i	0.0282426i
$193$	13264.2	0.356097	0.178048	−	0.984022i	$-0.443022\pi$
$193$	13264.2	0.356097	0.178048	+	0.984022i	$0.443022\pi$
$194$	28129.8i	0.747417i
$195$	− 377.143i	− 0.00991828i
$196$	0	0
$197$	5362.20	0.138169	0.0690844	−	0.997611i	$-0.477992\pi$
$197$	5362.20	0.138169	0.0690844	+	0.997611i	$0.477992\pi$
$198$	32972.8	0.841058
$199$	− 42230.6i	− 1.06640i	−0.845988	−	0.533202i	$-0.820989\pi$
$199$	− 42230.6i	− 1.06640i	0.845988	−	0.533202i	$-0.179011\pi$
$200$	14130.7	0.353267
$201$	− 8321.82i	− 0.205981i
$202$	5850.08i	0.143370i
$203$	0	0
$204$	6272.81	0.150731
$205$	−580.053	−0.0138026
$206$	7084.53i	0.166946i
$207$	−13654.8	−0.318673
$208$	16695.3i	0.385894i
$209$	59170.2i	1.35460i
$210$	0	0
$211$	−77049.0	−1.73062	−0.865311	−	0.501236i	$-0.832879\pi$
$211$	−77049.0	−1.73062	−0.865311	+	0.501236i	$0.832879\pi$
$212$	−25372.5	−0.564536
$213$	− 4586.04i	− 0.101083i
$214$	18224.7	0.397953
$215$	− 1541.18i	− 0.0333408i
$216$	7263.71i	0.155686i
$217$	0	0
$218$	66598.3	1.40136
$219$	9462.11	0.197288
$220$	862.633i	0.0178230i
$221$	100589.	2.05951
$222$	− 4584.47i	− 0.0930214i
$223$	3702.60i	0.0744555i	0.999307	+	0.0372278i	$0.0118527\pi$
$223$	3702.60i	0.0744555i	−0.999307	+	0.0372278i	$0.988147\pi$
$224$	0	0
$225$	48001.8	0.948183
$226$	55698.1	1.09049
$227$	− 12018.8i	− 0.233243i	−0.993176	−	0.116621i	$-0.962794\pi$
$227$	− 12018.8i	− 0.233243i	0.993176	−	0.116621i	$-0.0372064\pi$
$228$	−6346.69	−0.122089
$229$	33360.5i	0.636153i	0.948065	+	0.318076i	$0.103037\pi$
$229$	33360.5i	0.636153i	−0.948065	+	0.318076i	$0.896963\pi$
$230$	− 357.237i	− 0.00675305i
$231$	0	0
$232$	−7260.85	−0.134900
$233$	54138.9	0.997235	0.498618	−	0.866822i	$-0.333841\pi$
$233$	54138.9	0.997235	0.498618	+	0.866822i	$0.333841\pi$
$234$	56713.7i	1.03575i
$235$	3046.96	0.0551735
$236$	37651.2i	0.676012i
$237$	− 8527.83i	− 0.151824i
$238$	0	0
$239$	29165.4	0.510590	0.255295	−	0.966863i	$-0.417827\pi$
$239$	29165.4	0.510590	0.255295	+	0.966863i	$0.417827\pi$
$240$	−92.5275	−0.00160638
$241$	− 100130.i	− 1.72398i	−0.506929	−	0.861988i	$-0.669219\pi$
$241$	− 100130.i	− 1.72398i	0.506929	−	0.861988i	$-0.330781\pi$
$242$	23648.4	0.403804
$243$	37335.2i	0.632275i
$244$	− 20278.2i	− 0.340604i
$245$	0	0
$246$	−4692.41	−0.0775400
$247$	−101774.	−1.66817
$248$	− 30464.9i	− 0.495333i
$249$	15859.3	0.255791
$250$	2512.66i	0.0402025i
$251$	81988.2i	1.30138i	0.759344	+	0.650690i	$0.225520\pi$
$251$	81988.2i	1.30138i	−0.759344	+	0.650690i	$0.774480\pi$
$252$	0	0
$253$	−26942.6	−0.420919
$254$	−35498.3	−0.550224
$255$	557.476i	0.00857325i
$256$	4096.00	0.0625000
$257$	73774.8i	1.11697i	0.829515	+	0.558485i	$0.188617\pi$
$257$	73774.8i	1.11697i	−0.829515	+	0.558485i	$0.811383\pi$
$258$	− 12467.6i	− 0.187302i
$259$	0	0
$260$	−1483.74	−0.0219488
$261$	−24665.0	−0.362076
$262$	− 54654.4i	− 0.796201i
$263$	39243.6	0.567358	0.283679	−	0.958919i	$-0.408445\pi$
$263$	39243.6	0.567358	0.283679	+	0.958919i	$0.408445\pi$
$264$	6978.38i	0.100126i
$265$	− 2254.90i	− 0.0321096i
$266$	0	0
$267$	−19255.1	−0.270100
$268$	−32739.4	−0.455828
$269$	− 10311.6i	− 0.142502i	−0.997458	−	0.0712509i	$-0.977301\pi$
$269$	− 10311.6i	− 0.142502i	0.997458	−	0.0712509i	$-0.0226991\pi$
$270$	−645.538	−0.00885512
$271$	− 36969.3i	− 0.503388i	−0.967807	−	0.251694i	$-0.919012\pi$
$271$	− 36969.3i	− 0.503388i	0.967807	−	0.251694i	$-0.0809876\pi$
$272$	− 24678.3i	− 0.333562i
$273$	0	0
$274$	10489.5	0.139718
$275$	94713.3	1.25241
$276$	− 2889.91i	− 0.0379373i
$277$	−92405.8	−1.20431	−0.602157	−	0.798378i	$-0.705692\pi$
$277$	−92405.8	−1.20431	−0.602157	+	0.798378i	$0.705692\pi$
$278$	3100.22i	0.0401146i
$279$	− 103489.i	− 1.32949i
$280$	0	0
$281$	21171.6	0.268128	0.134064	−	0.990973i	$-0.457197\pi$
$281$	21171.6	0.268128	0.134064	+	0.990973i	$0.457197\pi$
$282$	24648.8	0.309954
$283$	48065.2i	0.600147i	0.953916	+	0.300074i	$0.0970112\pi$
$283$	48065.2i	0.600147i	−0.953916	+	0.300074i	$0.902989\pi$
$284$	−18042.3	−0.223694
$285$	− 564.042i	− 0.00694419i
$286$	111903.i	1.36808i
$287$	0	0
$288$	13914.1	0.167752
$289$	−65164.9	−0.780222
$290$	− 645.284i	− 0.00767282i
$291$	−20223.6	−0.238821
$292$	− 37225.5i	− 0.436591i
$293$	− 26781.1i	− 0.311956i	−0.987761	−	0.155978i	$-0.950147\pi$
$293$	− 26781.1i	− 0.311956i	0.987761	−	0.155978i	$-0.0498528\pi$
$294$	0	0
$295$	−3346.12	−0.0384501
$296$	−18036.0	−0.205853
$297$	48686.2i	0.551941i
$298$	−3489.76	−0.0392973
$299$	− 46341.7i	− 0.518358i
$300$	10159.1i	0.112879i
$301$	0	0
$302$	−120697.	−1.32337
$303$	−4205.85	−0.0458109
$304$	24968.9i	0.270180i
$305$	1802.16	0.0193729
$306$	− 83831.8i	− 0.895294i
$307$	− 69996.0i	− 0.742671i	−0.928499	−	0.371336i	$-0.878900\pi$
$307$	− 69996.0i	− 0.742671i	0.928499	−	0.371336i	$-0.121100\pi$
$308$	0	0
$309$	−5093.35	−0.0533441
$310$	2707.47	0.0281735
$311$	86961.6i	0.899097i	0.893256	+	0.449549i	$0.148415\pi$
$311$	86961.6i	0.899097i	−0.893256	+	0.449549i	$0.851585\pi$
$312$	−12002.9	−0.123304
$313$	− 33163.2i	− 0.338507i	−0.985573	−	0.169253i	$-0.945864\pi$
$313$	− 33163.2i	− 0.338507i	0.985573	−	0.169253i	$-0.0541357\pi$
$314$	− 84527.3i	− 0.857310i
$315$	0	0
$316$	−33549.9	−0.335983
$317$	29206.0	0.290639	0.145319	−	0.989385i	$-0.453579\pi$
$317$	29206.0	0.290639	0.145319	+	0.989385i	$0.453579\pi$
$318$	− 18241.3i	− 0.180386i
$319$	−48667.1	−0.478249
$320$	364.019i	0.00355487i
$321$	13102.4i	0.127157i
$322$	0	0
$323$	150437.	1.44195
$324$	44586.4	0.424729
$325$	162908.i	1.54233i
$326$	−74119.5	−0.697425
$327$	47880.1i	0.447775i
$328$	18460.7i	0.171594i
$329$	0	0
$330$	−620.181	−0.00569496
$331$	166993.	1.52420	0.762102	−	0.647457i	$-0.224168\pi$
$331$	166993.	1.52420	0.762102	+	0.647457i	$0.224168\pi$
$332$	− 62393.0i	− 0.566056i
$333$	−61268.2	−0.552518
$334$	− 101505.i	− 0.909902i
$335$	− 2909.61i	− 0.0259266i
$336$	0	0
$337$	16126.0	0.141993	0.0709964	−	0.997477i	$-0.477382\pi$
$337$	16126.0	0.141993	0.0709964	+	0.997477i	$0.477382\pi$
$338$	−111692.	−0.977664
$339$	40043.5i	0.348444i
$340$	2193.20	0.0189723
$341$	− 204196.i	− 1.75606i
$342$	84819.2i	0.725173i
$343$	0	0
$344$	−49049.5	−0.414493
$345$	256.832	0.00215779
$346$	31035.5i	0.259243i
$347$	10071.3	0.0836428	0.0418214	−	0.999125i	$-0.486684\pi$
$347$	10071.3	0.0836428	0.0418214	+	0.999125i	$0.486684\pi$
$348$	− 5220.11i	− 0.0431044i
$349$	− 97674.1i	− 0.801915i	−0.916097	−	0.400958i	$-0.868677\pi$
$349$	− 97674.1i	− 0.801915i	0.916097	−	0.400958i	$-0.131323\pi$
$350$	0	0
$351$	−83740.9	−0.679710
$352$	27454.1	0.221576
$353$	85255.3i	0.684182i	0.939667	+	0.342091i	$0.111135\pi$
$353$	85255.3i	0.684182i	−0.939667	+	0.342091i	$0.888865\pi$
$354$	−27068.9	−0.216005
$355$	− 1603.45i	− 0.0127232i
$356$	75752.9i	0.597722i
$357$	0	0
$358$	−95725.6	−0.746899
$359$	−142240.	−1.10365	−0.551826	−	0.833959i	$-0.686069\pi$
$359$	−142240.	−1.10365	−0.551826	+	0.833959i	$0.686069\pi$
$360$	1236.57i	0.00954140i
$361$	−21888.1	−0.167955
$362$	141689.i	1.08123i
$363$	17001.8i	0.129027i
$364$	0	0
$365$	3308.30	0.0248324
$366$	14578.8	0.108833
$367$	− 177841.i	− 1.32038i	−0.751098	−	0.660191i	$-0.770475\pi$
$367$	− 177841.i	− 1.32038i	0.751098	−	0.660191i	$-0.229525\pi$
$368$	−11369.4	−0.0839540
$369$	62710.8i	0.460564i
$370$	− 1602.90i	− 0.0117085i
$371$	0	0
$372$	21902.5	0.158273
$373$	−21670.8	−0.155761	−0.0778803	−	0.996963i	$-0.524815\pi$
$373$	−21670.8	−0.155761	−0.0778803	+	0.996963i	$0.524815\pi$
$374$	− 165410.i	− 1.18255i
$375$	−1806.45	−0.0128459
$376$	− 96972.3i	− 0.685917i
$377$	− 83708.0i	− 0.588958i
$378$	0	0
$379$	125199.	0.871608	0.435804	−	0.900042i	$-0.356464\pi$
$379$	125199.	0.871608	0.435804	+	0.900042i	$0.356464\pi$
$380$	−2219.03	−0.0153673
$381$	− 25521.1i	− 0.175812i
$382$	49816.9	0.341390
$383$	211410.i	1.44121i	0.693345	+	0.720605i	$0.256136\pi$
$383$	211410.i	1.44121i	−0.693345	+	0.720605i	$0.743864\pi$
$384$	2944.78i	0.0199705i
$385$	0	0
$386$	37516.9	0.251798
$387$	−166620.	−1.11252
$388$	79563.1i	0.528504i
$389$	−39160.8	−0.258793	−0.129396	−	0.991593i	$-0.541304\pi$
$389$	−39160.8	−0.258793	−0.129396	+	0.991593i	$0.541304\pi$
$390$	− 1066.72i	− 0.00701328i
$391$	68500.3i	0.448063i
$392$	0	0
$393$	39293.2	0.254409
$394$	15166.6	0.0977001
$395$	− 2981.64i	− 0.0191100i
$396$	93261.2	0.594718
$397$	196525.i	1.24691i	0.781858	+	0.623456i	$0.214272\pi$
$397$	196525.i	1.24691i	−0.781858	+	0.623456i	$0.785728\pi$
$398$	− 119446.i	− 0.754061i
$399$	0	0
$400$	39967.6	0.249798
$401$	213650.	1.32866	0.664330	−	0.747439i	$-0.268717\pi$
$401$	213650.	1.32866	0.664330	+	0.747439i	$0.268717\pi$
$402$	− 23537.7i	− 0.145650i
$403$	351221.	2.16257
$404$	16546.5i	0.101378i
$405$	3962.47i	0.0241577i
$406$	0	0
$407$	−120890.	−0.729794
$408$	17742.2	0.106583
$409$	− 35604.1i	− 0.212840i	−0.994321	−	0.106420i	$-0.966061\pi$
$409$	− 35604.1i	− 0.212840i	0.994321	−	0.106420i	$-0.0339388\pi$
$410$	−1640.64	−0.00975988
$411$	7541.29i	0.0446439i
$412$	20038.1i	0.118049i
$413$	0	0
$414$	−38621.7	−0.225336
$415$	5544.97	0.0321961
$416$	47221.5i	0.272868i
$417$	−2228.87	−0.0128178
$418$	167358.i	0.957845i
$419$	254947.i	1.45219i	0.687597	+	0.726093i	$0.258666\pi$
$419$	254947.i	1.45219i	−0.687597	+	0.726093i	$0.741334\pi$
$420$	0	0
$421$	107186.	0.604748	0.302374	−	0.953189i	$-0.402221\pi$
$421$	107186.	0.604748	0.302374	+	0.953189i	$0.402221\pi$
$422$	−217927.	−1.22373
$423$	− 329413.i	− 1.84103i
$424$	−71764.3	−0.399187
$425$	− 240804.i	− 1.33317i
$426$	− 12971.3i	− 0.0714766i
$427$	0	0
$428$	51547.1	0.281395
$429$	−80451.6	−0.437139
$430$	− 4359.11i	− 0.0235755i
$431$	119958.	0.645768	0.322884	−	0.946439i	$-0.395348\pi$
$431$	119958.	0.645768	0.322884	+	0.946439i	$0.395348\pi$
$432$	20544.9i	0.110087i
$433$	− 110159.i	− 0.587547i	−0.955875	−	0.293773i	$-0.905089\pi$
$433$	− 110159.i	− 0.587547i	0.955875	−	0.293773i	$-0.0949111\pi$
$434$	0	0
$435$	463.921	0.00245169
$436$	188368.	0.990912
$437$	− 69307.1i	− 0.362923i
$438$	26762.9	0.139503
$439$	352295.i	1.82801i	0.405708	+	0.914003i	$0.367025\pi$
$439$	352295.i	1.82801i	−0.405708	+	0.914003i	$0.632975\pi$
$440$	2439.90i	0.0126028i
$441$	0	0
$442$	284508.	1.45630
$443$	285732.	1.45597	0.727984	−	0.685594i	$-0.240457\pi$
$443$	285732.	1.45597	0.727984	+	0.685594i	$0.240457\pi$
$444$	− 12966.8i	− 0.0657761i
$445$	−6732.29	−0.0339972
$446$	10472.5i	0.0526480i
$447$	− 2508.92i	− 0.0125566i
$448$	0	0
$449$	−342184.	−1.69733	−0.848667	−	0.528927i	$-0.822594\pi$
$449$	−342184.	−1.69733	−0.848667	+	0.528927i	$0.822594\pi$
$450$	135770.	0.670467
$451$	123736.i	0.608335i
$452$	157538.	0.771096
$453$	− 86773.8i	− 0.422856i
$454$	− 33994.2i	− 0.164927i
$455$	0	0
$456$	−17951.2	−0.0863302
$457$	−277624.	−1.32931	−0.664653	−	0.747152i	$-0.731421\pi$
$457$	−277624.	−1.32931	−0.664653	+	0.747152i	$0.731421\pi$
$458$	94357.7i	0.449828i
$459$	123782.	0.587534
$460$	− 1010.42i	− 0.00477513i
$461$	361934.i	1.70305i	0.524315	+	0.851524i	$0.324322\pi$
$461$	361934.i	1.70305i	−0.524315	+	0.851524i	$0.675678\pi$
$462$	0	0
$463$	146876.	0.685153	0.342577	−	0.939490i	$-0.388700\pi$
$463$	146876.	0.685153	0.342577	+	0.939490i	$0.388700\pi$
$464$	−20536.8	−0.0953886
$465$	1946.51i	0.00900224i
$466$	153128.	0.705152
$467$	− 268410.i	− 1.23073i	−0.788241	−	0.615367i	$-0.789008\pi$
$467$	− 268410.i	− 1.23073i	0.788241	−	0.615367i	$-0.210992\pi$
$468$	160411.i	0.732389i
$469$	0	0
$470$	8618.09	0.0390136
$471$	60770.0	0.273935
$472$	106494.i	0.478013i
$473$	−328762.	−1.46947
$474$	− 24120.3i	− 0.107356i
$475$	243640.i	1.07985i
$476$	0	0
$477$	−243782.	−1.07143
$478$	82492.2	0.361042
$479$	− 155285.i	− 0.676797i	−0.941003	−	0.338399i	$-0.890115\pi$
$479$	− 155285.i	− 0.676797i	0.941003	−	0.338399i	$-0.109885\pi$
$480$	−261.707	−0.00113588
$481$	− 207932.i	− 0.898733i
$482$	− 283211.i	− 1.21903i
$483$	0	0
$484$	66887.7	0.285533
$485$	−7070.91	−0.0300602
$486$	105600.i	0.447086i
$487$	−155628.	−0.656190	−0.328095	−	0.944645i	$-0.606407\pi$
$487$	−155628.	−0.656190	−0.328095	+	0.944645i	$0.606407\pi$
$488$	− 57355.5i	− 0.240844i
$489$	− 53287.5i	− 0.222847i
$490$	0	0
$491$	37550.7	0.155760	0.0778798	−	0.996963i	$-0.475185\pi$
$491$	37550.7	0.155760	0.0778798	+	0.996963i	$0.475185\pi$
$492$	−13272.1	−0.0548291
$493$	123734.i	0.509089i
$494$	−287859.	−1.17958
$495$	8288.29i	0.0338263i
$496$	− 86167.9i	− 0.350253i
$497$	0	0
$498$	44856.8	0.180871
$499$	167820.	0.673975	0.336987	−	0.941509i	$-0.390592\pi$
$499$	167820.	0.673975	0.336987	+	0.941509i	$0.390592\pi$
$500$	7106.86i	0.0284275i
$501$	72976.0	0.290740
$502$	231898.i	0.920214i
$503$	− 133035.i	− 0.525810i	−0.964822	−	0.262905i	$-0.915319\pi$
$503$	− 133035.i	− 0.525810i	0.964822	−	0.262905i	$-0.0846806\pi$
$504$	0	0
$505$	−1470.52	−0.00576617
$506$	−76205.2	−0.297635
$507$	− 80299.9i	− 0.312392i
$508$	−100404.	−0.389067
$509$	− 227994.i	− 0.880011i	−0.897995	−	0.440005i	$-0.854976\pi$
$509$	− 227994.i	− 0.880011i	0.897995	−	0.440005i	$-0.145024\pi$
$510$	1576.78i	0.00606220i
$511$	0	0
$512$	11585.2	0.0441942
$513$	−125240.	−0.475892
$514$	208667.i	0.789817i
$515$	−1780.82	−0.00671437
$516$	− 35263.6i	− 0.132442i
$517$	− 649973.i	− 2.43172i
$518$	0	0
$519$	−22312.7	−0.0828355
$520$	−4196.65	−0.0155202
$521$	341411.i	1.25777i	0.777498	+	0.628885i	$0.216489\pi$
$521$	341411.i	1.25777i	−0.777498	+	0.628885i	$0.783511\pi$
$522$	−69763.2	−0.256027
$523$	62912.0i	0.230001i	0.993365	+	0.115001i	$0.0366870\pi$
$523$	62912.0i	0.230001i	−0.993365	+	0.115001i	$0.963313\pi$
$524$	− 154586.i	− 0.562999i
$525$	0	0
$526$	110998.	0.401182
$527$	−519159.	−1.86930
$528$	19737.9i	0.0707998i
$529$	−248283.	−0.887228
$530$	− 6377.82i	− 0.0227050i
$531$	361757.i	1.28300i
$532$	0	0
$533$	−212828.	−0.749159
$534$	−54461.7	−0.190989
$535$	4581.08i	0.0160052i
$536$	−92601.1	−0.322319
$537$	− 68820.9i	− 0.238656i
$538$	− 29165.5i	− 0.100764i
$539$	0	0
$540$	−1825.86	−0.00626151
$541$	251203.	0.858282	0.429141	−	0.903237i	$-0.358816\pi$
$541$	251203.	0.858282	0.429141	+	0.903237i	$0.358816\pi$
$542$	− 104565.i	− 0.355949i
$543$	−101866.	−0.345484
$544$	− 69800.7i	− 0.235864i
$545$	16740.6i	0.0563610i
$546$	0	0
$547$	−153431.	−0.512789	−0.256395	−	0.966572i	$-0.582535\pi$
$547$	−153431.	−0.512789	−0.256395	+	0.966572i	$0.582535\pi$
$548$	29668.7	0.0987955
$549$	− 194836.i	− 0.646434i
$550$	267890.	0.885586
$551$	− 125191.i	− 0.412353i
$552$	− 8173.91i	− 0.0268257i
$553$	0	0
$554$	−261363.	−0.851579
$555$	1152.38	0.00374121
$556$	8768.75i	0.0283653i
$557$	−475151.	−1.53152	−0.765758	−	0.643129i	$-0.777636\pi$
$557$	−475151.	−1.53152	−0.765758	+	0.643129i	$0.777636\pi$
$558$	− 292711.i	− 0.940093i
$559$	− 565476.i	− 1.80963i
$560$	0	0
$561$	118920.	0.377858
$562$	59882.5	0.189595
$563$	− 39487.0i	− 0.124577i	−0.998058	−	0.0622884i	$-0.980160\pi$
$563$	− 39487.0i	− 0.124577i	0.998058	−	0.0622884i	$-0.0198398\pi$
$564$	69717.2	0.219170
$565$	14000.7i	0.0438583i
$566$	135949.i	0.424368i
$567$	0	0
$568$	−51031.2	−0.158175
$569$	−85307.7	−0.263490	−0.131745	−	0.991284i	$-0.542058\pi$
$569$	−85307.7	−0.263490	−0.131745	+	0.991284i	$0.542058\pi$
$570$	− 1595.35i	− 0.00491028i
$571$	89099.0	0.273276	0.136638	−	0.990621i	$-0.456370\pi$
$571$	89099.0	0.273276	0.136638	+	0.990621i	$0.456370\pi$
$572$	316510.i	0.967375i
$573$	35815.4i	0.109084i
$574$	0	0
$575$	−110939.	−0.335545
$576$	39354.9	0.118619
$577$	311076.i	0.934362i	0.884162	+	0.467181i	$0.154730\pi$
$577$	311076.i	0.934362i	−0.884162	+	0.467181i	$0.845270\pi$
$578$	−184314.	−0.551700
$579$	26972.4i	0.0804568i
$580$	− 1825.14i	− 0.00542551i
$581$	0	0
$582$	−57201.1	−0.168872
$583$	−481012.	−1.41520
$584$	− 105290.i	− 0.308717i
$585$	−14256.0	−0.0416567
$586$	− 75748.3i	− 0.220586i
$587$	62466.3i	0.181288i	0.995883	+	0.0906440i	$0.0288926\pi$
$587$	62466.3i	0.181288i	−0.995883	+	0.0906440i	$0.971107\pi$
$588$	0	0
$589$	525274.	1.51410
$590$	−9464.27	−0.0271884
$591$	10903.9i	0.0312180i
$592$	−51013.7	−0.145560
$593$	− 47567.3i	− 0.135269i	−0.997710	−	0.0676346i	$-0.978455\pi$
$593$	− 47567.3i	− 0.135269i	0.997710	−	0.0676346i	$-0.0215452\pi$
$594$	137705.i	0.390281i
$595$	0	0
$596$	−9870.52	−0.0277874
$597$	85874.6	0.240944
$598$	− 131074.i	− 0.366534i
$599$	564035.	1.57200	0.785999	−	0.618227i	$-0.212149\pi$
$599$	564035.	1.57200	0.785999	+	0.618227i	$0.212149\pi$
$600$	28734.3i	0.0798176i
$601$	255637.i	0.707740i	0.935295	+	0.353870i	$0.115135\pi$
$601$	255637.i	0.707740i	−0.935295	+	0.353870i	$0.884865\pi$
$602$	0	0
$603$	−314565.	−0.865118
$604$	−341382.	−0.935766
$605$	5944.43i	0.0162405i
$606$	−11896.0	−0.0323932
$607$	572699.i	1.55435i	0.629284	+	0.777176i	$0.283348\pi$
$607$	572699.i	1.55435i	−0.629284	+	0.777176i	$0.716652\pi$
$608$	70622.8i	0.191046i
$609$	0	0
$610$	5097.28	0.0136987
$611$	1.11796e6	2.99464
$612$	− 237112.i	− 0.633069i
$613$	283002.	0.753127	0.376563	−	0.926391i	$-0.377106\pi$
$613$	283002.	0.753127	0.376563	+	0.926391i	$0.377106\pi$
$614$	− 197979.i	− 0.525148i
$615$	− 1179.52i	− 0.00311856i
$616$	0	0
$617$	248170.	0.651896	0.325948	−	0.945388i	$-0.394317\pi$
$617$	248170.	0.651896	0.325948	+	0.945388i	$0.394317\pi$
$618$	−14406.2	−0.0377200
$619$	56588.1i	0.147688i	0.997270	+	0.0738438i	$0.0235266\pi$
$619$	56588.1i	0.147688i	−0.997270	+	0.0738438i	$0.976473\pi$
$620$	7657.89	0.0199217
$621$	− 57027.0i	− 0.147876i
$622$	245964.i	0.635758i
$623$	0	0
$624$	−33949.4	−0.0871892
$625$	389677.	0.997574
$626$	− 93799.7i	− 0.239361i
$627$	−120321.	−0.306059
$628$	− 239079.i	− 0.606209i
$629$	307356.i	0.776855i
$630$	0	0
$631$	681448.	1.71149	0.855744	−	0.517399i	$-0.173100\pi$
$631$	681448.	1.71149	0.855744	+	0.517399i	$0.173100\pi$
$632$	−94893.4	−0.237576
$633$	− 156677.i	− 0.391018i
$634$	82607.0	0.205513
$635$	− 8923.10i	− 0.0221293i
$636$	− 51594.2i	− 0.127552i
$637$	0	0
$638$	−137651.	−0.338173
$639$	−173352.	−0.424549
$640$	1029.60i	0.00251367i
$641$	678114.	1.65039	0.825195	−	0.564848i	$-0.191065\pi$
$641$	678114.	1.65039	0.825195	+	0.564848i	$0.191065\pi$
$642$	37059.3i	0.0899139i
$643$	345520.i	0.835701i	0.908516	+	0.417851i	$0.137216\pi$
$643$	345520.i	0.835701i	−0.908516	+	0.417851i	$0.862784\pi$
$644$	0	0
$645$	3133.94	0.00753305
$646$	425501.	1.01961
$647$	− 7757.50i	− 0.0185316i	−0.999957	−	0.00926581i	$-0.997051\pi$
$647$	− 7757.50i	− 0.0185316i	0.999957	−	0.00926581i	$-0.00294944\pi$
$648$	126109.	0.300329
$649$	713791.i	1.69466i
$650$	460774.i	1.09059i
$651$	0	0
$652$	−209642.	−0.493154
$653$	−448473.	−1.05174	−0.525872	−	0.850564i	$-0.676261\pi$
$653$	−448473.	−1.05174	−0.525872	+	0.850564i	$0.676261\pi$
$654$	135425.i	0.316625i
$655$	13738.3	0.0320222
$656$	52214.8i	0.121335i
$657$	− 357667.i	− 0.828607i
$658$	0	0
$659$	583660.	1.34397	0.671984	−	0.740565i	$-0.265442\pi$
$659$	583660.	1.34397	0.671984	+	0.740565i	$0.265442\pi$
$660$	−1754.14	−0.00402694
$661$	382886.i	0.876327i	0.898895	+	0.438164i	$0.144371\pi$
$661$	382886.i	0.876327i	−0.898895	+	0.438164i	$0.855629\pi$
$662$	472328.	1.07777
$663$	204544.i	0.465329i
$664$	− 176474.i	− 0.400262i
$665$	0	0
$666$	−173293.	−0.390689
$667$	57004.6	0.128132
$668$	− 287100.i	− 0.643398i
$669$	−7529.12	−0.0168226
$670$	− 8229.62i	− 0.0183329i
$671$	− 384435.i	− 0.853842i
$672$	0	0
$673$	−323802.	−0.714907	−0.357453	−	0.933931i	$-0.616355\pi$
$673$	−323802.	−0.714907	−0.357453	+	0.933931i	$0.616355\pi$
$674$	45611.2	0.100404
$675$	200471.i	0.439992i
$676$	−315913.	−0.691313
$677$	374803.i	0.817760i	0.912588	+	0.408880i	$0.134081\pi$
$677$	374803.i	0.817760i	−0.912588	+	0.408880i	$0.865919\pi$
$678$	113260.i	0.246387i
$679$	0	0
$680$	6203.31	0.0134155
$681$	24439.8	0.0526990
$682$	− 577554.i	− 1.24172i
$683$	97364.5	0.208718	0.104359	−	0.994540i	$-0.466721\pi$
$683$	97364.5	0.208718	0.104359	+	0.994540i	$0.466721\pi$
$684$	239905.i	0.512775i
$685$	2636.71i	0.00561928i
$686$	0	0
$687$	−67837.5	−0.143733
$688$	−138733.	−0.293091
$689$	− 827348.i	− 1.74281i
$690$	726.429	0.00152579
$691$	− 787116.i	− 1.64848i	−0.566244	−	0.824238i	$-0.691604\pi$
$691$	− 787116.i	− 1.64848i	0.566244	−	0.824238i	$-0.308396\pi$
$692$	87781.7i	0.183312i
$693$	0	0
$694$	28486.1	0.0591444
$695$	−779.294	−0.00161336
$696$	− 14764.7i	− 0.0304794i
$697$	314593.	0.647564
$698$	− 276264.i	− 0.567040i
$699$	110090.i	0.225316i
$700$	0	0
$701$	−232522.	−0.473182	−0.236591	−	0.971609i	$-0.576030\pi$
$701$	−232522.	−0.473182	−0.236591	+	0.971609i	$0.576030\pi$
$702$	−236855.	−0.480627
$703$	− 310976.i	− 0.629240i
$704$	77652.0	0.156678
$705$	6195.89i	0.0124659i
$706$	241138.i	0.483790i
$707$	0	0
$708$	−76562.4	−0.152739
$709$	151049.	0.300487	0.150243	−	0.988649i	$-0.451994\pi$
$709$	151049.	0.300487	0.150243	+	0.988649i	$0.451994\pi$
$710$	− 4535.23i	− 0.00899669i
$711$	−322351.	−0.637662
$712$	214261.i	0.422653i
$713$	239179.i	0.470483i
$714$	0	0
$715$	−28128.8	−0.0550223
$716$	−270753.	−0.528138
$717$	59306.9i	0.115363i
$718$	−402315.	−0.780400
$719$	712110.i	1.37749i	0.725002	+	0.688746i	$0.241839\pi$
$719$	712110.i	1.37749i	−0.725002	+	0.688746i	$0.758161\pi$
$720$	3497.54i	0.00674679i
$721$	0	0
$722$	−61908.8	−0.118762
$723$	203612.	0.389517
$724$	400756.i	0.764545i
$725$	−200392.	−0.381246
$726$	48088.2i	0.0912359i
$727$	− 693160.i	− 1.31149i	−0.754983	−	0.655744i	$-0.772355\pi$
$727$	− 693160.i	− 1.31149i	0.754983	−	0.655744i	$-0.227645\pi$
$728$	0	0
$729$	375517.	0.706601
$730$	9357.28	0.0175592
$731$	835861.i	1.56423i
$732$	41235.1	0.0769565
$733$	− 269107.i	− 0.500861i	−0.968135	−	0.250431i	$-0.919428\pi$
$733$	− 269107.i	− 0.500861i	0.968135	−	0.250431i	$-0.0805722\pi$
$734$	− 503010.i	− 0.933651i
$735$	0	0
$736$	−32157.5	−0.0593645
$737$	−620674.	−1.14269
$738$	177373.i	0.325668i
$739$	86185.1	0.157813	0.0789066	−	0.996882i	$-0.474857\pi$
$739$	86185.1	0.157813	0.0789066	+	0.996882i	$0.474857\pi$
$740$	− 4533.67i	− 0.00827917i
$741$	− 206953.i	− 0.376908i
$742$	0	0
$743$	−457439.	−0.828620	−0.414310	−	0.910136i	$-0.635977\pi$
$743$	−457439.	−0.828620	−0.414310	+	0.910136i	$0.635977\pi$
$744$	61949.5	0.111916
$745$	− 877.210i	− 0.00158049i
$746$	−61294.3	−0.110139
$747$	− 599480.i	− 1.07432i
$748$	− 467851.i	− 0.836189i
$749$	0	0
$750$	−5109.41	−0.00908339
$751$	−581442.	−1.03092	−0.515462	−	0.856913i	$-0.672380\pi$
$751$	−581442.	−1.03092	−0.515462	+	0.856913i	$0.672380\pi$
$752$	− 274279.i	− 0.485017i
$753$	−166720.	−0.294035
$754$	− 236762.i	− 0.416456i
$755$	− 30339.2i	− 0.0532244i
$756$	0	0
$757$	398071.	0.694655	0.347327	−	0.937744i	$-0.387089\pi$
$757$	398071.	0.694655	0.347327	+	0.937744i	$0.387089\pi$
$758$	354115.	0.616320
$759$	− 54787.0i	− 0.0951029i
$760$	−6276.38	−0.0108663
$761$	323965.i	0.559408i	0.960086	+	0.279704i	$0.0902363\pi$
$761$	323965.i	0.559408i	−0.960086	+	0.279704i	$0.909764\pi$
$762$	− 72184.6i	− 0.124318i
$763$	0	0
$764$	140904.	0.241399
$765$	21072.6	0.0360076
$766$	597957.i	1.01909i
$767$	−1.22773e6	−2.08695
$768$	8329.09i	0.0141213i
$769$	318602.i	0.538760i	0.963034	+	0.269380i	$0.0868187\pi$
$769$	318602.i	0.538760i	−0.963034	+	0.269380i	$0.913181\pi$
$770$	0	0
$771$	−150019.	−0.252369
$772$	106114.	0.178048
$773$	− 512015.i	− 0.856887i	−0.903569	−	0.428443i	$-0.859062\pi$
$773$	− 512015.i	− 0.856887i	0.903569	−	0.428443i	$-0.140938\pi$
$774$	−471274.	−0.786667
$775$	− 840803.i	− 1.39988i
$776$	225038.i	0.373709i
$777$	0	0
$778$	−110763.	−0.182994
$779$	−318298.	−0.524516
$780$	− 3017.14i	− 0.00495914i
$781$	−342045.	−0.560765
$782$	193748.i	0.316828i
$783$	− 103009.i	− 0.168017i
$784$	0	0
$785$	21247.4	0.0344799
$786$	111138.	0.179894
$787$	644919.i	1.04125i	0.853785	+	0.520626i	$0.174301\pi$
$787$	644919.i	1.04125i	−0.853785	+	0.520626i	$0.825699\pi$
$788$	42897.6	0.0690844
$789$	79800.5i	0.128189i
$790$	− 8433.34i	− 0.0135128i
$791$	0	0
$792$	263783.	0.420529
$793$	661233.	1.05150
$794$	555856.i	0.881700i
$795$	4585.27	0.00725488
$796$	− 337845.i	− 0.533202i
$797$	− 278844.i	− 0.438980i	−0.975615	−	0.219490i	$-0.929561\pi$
$797$	− 278844.i	− 0.438980i	0.975615	−	0.219490i	$-0.0704393\pi$
$798$	0	0
$799$	−1.65252e6	−2.58853
$800$	113046.	0.176634
$801$	727843.i	1.13442i
$802$	604293.	0.939505
$803$	− 705721.i	− 1.09447i
$804$	− 66574.6i	− 0.102990i
$805$	0	0
$806$	993402.	1.52917
$807$	20968.2	0.0321970
$808$	46800.6i	0.0716851i
$809$	662520.	1.01228	0.506142	−	0.862450i	$-0.331071\pi$
$809$	662520.	1.01228	0.506142	+	0.862450i	$0.331071\pi$
$810$	11207.5i	0.0170821i
$811$	586078.i	0.891073i	0.895264	+	0.445537i	$0.146987\pi$
$811$	586078.i	0.891073i	−0.895264	+	0.445537i	$0.853013\pi$
$812$	0	0
$813$	75175.9	0.113736
$814$	−341927.	−0.516042
$815$	− 18631.2i	− 0.0280496i
$816$	50182.5	0.0753654
$817$	− 845706.i	− 1.26700i
$818$	− 100704.i	− 0.150501i
$819$	0	0
$820$	−4640.42	−0.00690128
$821$	790534.	1.17283	0.586414	−	0.810012i	$-0.300539\pi$
$821$	790534.	1.17283	0.586414	+	0.810012i	$0.300539\pi$
$822$	21330.0i	0.0315680i
$823$	−1.15645e6	−1.70737	−0.853684	−	0.520791i	$-0.825637\pi$
$823$	−1.15645e6	−1.70737	−0.853684	+	0.520791i	$0.825637\pi$
$824$	56676.3i	0.0834731i
$825$	192597.i	0.282970i
$826$	0	0
$827$	390408.	0.570832	0.285416	−	0.958404i	$-0.407868\pi$
$827$	390408.	0.570832	0.285416	+	0.958404i	$0.407868\pi$
$828$	−109239.	−0.159336
$829$	− 531226.i	− 0.772984i	−0.922293	−	0.386492i	$-0.873687\pi$
$829$	− 531226.i	− 0.772984i	0.922293	−	0.386492i	$-0.126313\pi$
$830$	15683.6	0.0227661
$831$	− 187904.i	− 0.272104i
$832$	133562.i	0.192947i
$833$	0	0
$834$	−6304.20	−0.00906354
$835$	25515.1	0.0365951
$836$	473361.i	0.677299i
$837$	432204.	0.616933
$838$	721099.i	1.02685i
$839$	− 263842.i	− 0.374817i	−0.982282	−	0.187409i	$-0.939991\pi$
$839$	− 263842.i	− 0.374817i	0.982282	−	0.187409i	$-0.0600089\pi$
$840$	0	0
$841$	−604312.	−0.854416
$842$	303168.	0.427622
$843$	43051.9i	0.0605811i
$844$	−616392.	−0.865311
$845$	− 28075.8i	− 0.0393204i
$846$	− 931722.i	− 1.30180i
$847$	0	0
$848$	−202980.	−0.282268
$849$	−97739.0	−0.135598
$850$	− 681096.i	− 0.942694i
$851$	141600.	0.195526
$852$	− 36688.3i	− 0.0505416i
$853$	1.17089e6i	1.60923i	0.593798	+	0.804614i	$0.297628\pi$
$853$	1.17089e6i	1.60923i	−0.593798	+	0.804614i	$0.702372\pi$
$854$	0	0
$855$	−21320.8	−0.0291656
$856$	145797.	0.198977
$857$	− 424840.i	− 0.578447i	−0.957262	−	0.289223i	$-0.906603\pi$
$857$	− 424840.i	− 0.578447i	0.957262	−	0.289223i	$-0.0933970\pi$
$858$	−227551.	−0.309104
$859$	− 408175.i	− 0.553172i	−0.960989	−	0.276586i	$-0.910797\pi$
$859$	− 408175.i	− 0.553172i	0.960989	−	0.276586i	$-0.0892030\pi$
$860$	− 12329.4i	− 0.0166704i
$861$	0	0
$862$	339294.	0.456627
$863$	−882741.	−1.18525	−0.592627	−	0.805477i	$-0.701909\pi$
$863$	−882741.	−1.18525	−0.592627	+	0.805477i	$0.701909\pi$
$864$	58109.6i	0.0778432i
$865$	−7801.31	−0.0104264
$866$	− 311575.i	− 0.415458i
$867$	− 132511.i	− 0.176284i
$868$	0	0
$869$	−636039.	−0.842256
$870$	1312.17	0.00173360
$871$	− 1.06757e6i	− 1.40721i
$872$	532786.	0.700680
$873$	764452.i	1.00305i
$874$	− 196030.i	− 0.256625i
$875$	0	0
$876$	75696.9	0.0986438
$877$	−595392.	−0.774112	−0.387056	−	0.922056i	$-0.626508\pi$
$877$	−595392.	−0.774112	−0.387056	+	0.922056i	$0.626508\pi$
$878$	996441.i	1.29260i
$879$	54458.5	0.0704835
$880$	6901.07i	0.00891150i
$881$	238444.i	0.307209i	0.988132	+	0.153605i	$0.0490882\pi$
$881$	238444.i	0.307209i	−0.988132	+	0.153605i	$0.950912\pi$
$882$	0	0
$883$	189523.	0.243076	0.121538	−	0.992587i	$-0.461217\pi$
$883$	189523.	0.243076	0.121538	+	0.992587i	$0.461217\pi$
$884$	804710.	1.02976
$885$	− 6804.24i	− 0.00868746i
$886$	808173.	1.02953
$887$	819553.i	1.04167i	0.853658	+	0.520834i	$0.174379\pi$
$887$	819553.i	1.04167i	−0.853658	+	0.520834i	$0.825621\pi$
$888$	− 36675.7i	− 0.0465107i
$889$	0	0
$890$	−19041.8	−0.0240396
$891$	845268.	1.06473
$892$	29620.8i	0.0372278i
$893$	1.67199e6	2.09667
$894$	− 7096.31i	− 0.00887886i
$895$	− 24062.3i	− 0.0300394i
$896$	0	0
$897$	94234.3	0.117118
$898$	−967843.	−1.20020
$899$	432034.i	0.534563i
$900$	384014.	0.474092
$901$	1.22295e6i	1.50646i
$902$	349978.i	0.430158i
$903$	0	0
$904$	445585.	0.545247
$905$	−35615.9	−0.0434857
$906$	− 245433.i	− 0.299004i
$907$	−657619.	−0.799392	−0.399696	−	0.916648i	$-0.630884\pi$
$907$	−657619.	−0.799392	−0.399696	+	0.916648i	$0.630884\pi$
$908$	− 96150.0i	− 0.116621i
$909$	158981.i	0.192406i
$910$	0	0
$911$	1.32632e6	1.59812	0.799061	−	0.601250i	$-0.205330\pi$
$911$	1.32632e6	1.59812	0.799061	+	0.601250i	$0.205330\pi$
$912$	−50773.6	−0.0610447
$913$	− 1.18285e6i	− 1.41901i
$914$	−785240.	−0.939961
$915$	3664.64i	0.00437712i
$916$	266884.i	0.318076i
$917$	0	0
$918$	350109.	0.415449
$919$	−236310.	−0.279803	−0.139901	−	0.990165i	$-0.544679\pi$
$919$	−236310.	−0.279803	−0.139901	+	0.990165i	$0.544679\pi$
$920$	− 2857.89i	− 0.00337653i
$921$	142335.	0.167800
$922$	1.02370e6i	1.20424i
$923$	− 588322.i	− 0.690577i
$924$	0	0
$925$	−497777.	−0.581770
$926$	415427.	0.484476
$927$	192528.i	0.224045i
$928$	−58086.8	−0.0674499
$929$	352762.i	0.408743i	0.978893	+	0.204372i	$0.0655151\pi$
$929$	352762.i	0.408743i	−0.978893	+	0.204372i	$0.934485\pi$
$930$	5505.56i	0.00636554i
$931$	0	0
$932$	433111.	0.498618
$933$	−176834.	−0.203143
$934$	− 759177.i	− 0.870261i
$935$	41578.7	0.0475607
$936$	453710.i	0.517877i
$937$	− 113445.i	− 0.129213i	−0.997911	−	0.0646066i	$-0.979421\pi$
$937$	− 113445.i	− 0.129213i	0.997911	−	0.0646066i	$-0.0205793\pi$
$938$	0	0
$939$	67436.3	0.0764826
$940$	24375.6	0.0275867
$941$	− 1.17351e6i	− 1.32528i	−0.748937	−	0.662642i	$-0.769435\pi$
$941$	− 1.17351e6i	− 1.32528i	0.748937	−	0.662642i	$-0.230565\pi$
$942$	171884.	0.193701
$943$	− 144934.i	− 0.162985i
$944$	301209.i	0.338006i
$945$	0	0
$946$	−929880.	−1.03907
$947$	271958.	0.303250	0.151625	−	0.988438i	$-0.451549\pi$
$947$	271958.	0.303250	0.151625	+	0.988438i	$0.451549\pi$
$948$	− 68222.6i	− 0.0759122i
$949$	1.21385e6	1.34782
$950$	689118.i	0.763566i
$951$	59389.5i	0.0656672i
$952$	0	0
$953$	465476.	0.512520	0.256260	−	0.966608i	$-0.417510\pi$
$953$	465476.	0.512520	0.256260	+	0.966608i	$0.417510\pi$
$954$	−689521.	−0.757618
$955$	12522.3i	0.0137303i
$956$	233323.	0.255295
$957$	− 98962.9i	− 0.108056i
$958$	− 439212.i	− 0.478568i
$959$	0	0
$960$	−740.220	−0.000803190 0
$961$	−889199.	−0.962836
$962$	− 588120.i	− 0.635500i
$963$	495271.	0.534061
$964$	− 801042.i	− 0.861988i
$965$	9430.53i	0.0101270i
$966$	0	0
$967$	−1.49573e6	−1.59956	−0.799782	−	0.600291i	$-0.795051\pi$
$967$	−1.49573e6	−1.59956	−0.799782	+	0.600291i	$0.795051\pi$
$968$	189187.	0.201902
$969$	305909.i	0.325795i
$970$	−19999.6	−0.0212558
$971$	404524.i	0.429048i	0.976719	+	0.214524i	$0.0688201\pi$
$971$	404524.i	0.429048i	−0.976719	+	0.214524i	$0.931180\pi$
$972$	298682.i	0.316138i
$973$	0	0
$974$	−440182.	−0.463996
$975$	−331269.	−0.348475
$976$	− 162226.i	− 0.170302i
$977$	−1.65194e6	−1.73063	−0.865315	−	0.501229i	$-0.832882\pi$
$977$	−1.65194e6	−1.73063	−0.865315	+	0.501229i	$0.832882\pi$
$978$	− 150720.i	− 0.157577i
$979$	1.43612e6i	1.49839i
$980$	0	0
$981$	1.80987e6	1.88065
$982$	106209.	0.110139
$983$	978562.i	1.01270i	0.862328	+	0.506350i	$0.169006\pi$
$983$	978562.i	1.01270i	−0.862328	+	0.506350i	$0.830994\pi$
$984$	−37539.3	−0.0387700
$985$	3812.38i	0.00392938i
$986$	349971.i	0.359980i
$987$	0	0
$988$	−814188.	−0.834086
$989$	385085.	0.393699
$990$	23442.8i	0.0239188i
$991$	1.26854e6	1.29168	0.645841	−	0.763472i	$-0.276507\pi$
$991$	1.26854e6	1.29168	0.645841	+	0.763472i	$0.276507\pi$
$992$	− 243720.i	− 0.247666i
$993$	339575.i	0.344380i
$994$	0	0
$995$	30024.9	0.0303274
$996$	126874.	0.127895
$997$	2071.33i	0.00208382i	0.999999	+	0.00104191i	$0.000331650\pi$
$997$	2071.33i	0.00208382i	−0.999999	+	0.00104191i	$0.999668\pi$
$998$	474668.	0.476572
$999$	− 255876.i	− 0.256389i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	98.5.b.a.97.4	yes	4
3.2	odd	2		882.5.c.c.685.1		4
4.3	odd	2		784.5.c.a.97.2		4
7.2	even	3		98.5.d.c.31.1		8
7.3	odd	6		98.5.d.c.19.1		8
7.4	even	3		98.5.d.c.19.2		8
7.5	odd	6		98.5.d.c.31.2		8
7.6	odd	2	inner	98.5.b.a.97.3	✓	4
21.20	even	2		882.5.c.c.685.2		4
28.27	even	2		784.5.c.a.97.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
98.5.b.a.97.3	✓	4	7.6	odd	2	inner
98.5.b.a.97.4	yes	4	1.1	even	1	trivial
98.5.d.c.19.1		8	7.3	odd	6
98.5.d.c.19.2		8	7.4	even	3
98.5.d.c.31.1		8	7.2	even	3
98.5.d.c.31.2		8	7.5	odd	6
784.5.c.a.97.2		4	4.3	odd	2
784.5.c.a.97.3		4	28.27	even	2
882.5.c.c.685.1		4	3.2	odd	2
882.5.c.c.685.2		4	21.20	even	2

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:	\( N \)	\(=\)	\( 98 = 2 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	98.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.82843 q^{2} +2.03347i q^{3} +8.00000 q^{4} +0.710974i q^{5} +5.75152i q^{6} +22.6274 q^{8} +76.8650 q^{9} +O(q^{10})\)	\(q+2.82843 q^{2} +2.03347i q^{3} +8.00000 q^{4} +0.710974i q^{5} +5.75152i q^{6} +22.6274 q^{8} +76.8650 q^{9} +2.01094i q^{10} +151.664 q^{11} +16.2677i q^{12} +260.864i q^{13} -1.44574 q^{15} +64.0000 q^{16} -385.598i q^{17} +217.407 q^{18} +390.140i q^{19} +5.68779i q^{20} +428.971 q^{22} -177.647 q^{23} +46.0121i q^{24} +624.495 q^{25} +737.836i q^{26} +321.013i q^{27} -320.887 q^{29} -4.08918 q^{30} -1346.37i q^{31} +181.019 q^{32} +308.404i q^{33} -1090.64i q^{34} +614.920 q^{36} -797.088 q^{37} +1103.48i q^{38} -530.459 q^{39} +16.0875i q^{40} +815.856i q^{41} -2167.70 q^{43} +1213.31 q^{44} +54.6490i q^{45} -502.461 q^{46} -4285.61i q^{47} +130.142i q^{48} +1766.34 q^{50} +784.102 q^{51} +2086.91i q^{52} -3171.57 q^{53} +907.963i q^{54} +107.829i q^{55} -793.337 q^{57} -907.606 q^{58} +4706.39i q^{59} -11.5659 q^{60} -2534.78i q^{61} -3808.12i q^{62} +512.000 q^{64} -185.468 q^{65} +872.298i q^{66} -4092.43 q^{67} -3084.78i q^{68} -361.239i q^{69} -2255.28 q^{71} +1739.26 q^{72} -4653.19i q^{73} -2254.51 q^{74} +1269.89i q^{75} +3121.12i q^{76} -1500.37 q^{78} -4193.74 q^{79} +45.5023i q^{80} +5573.29 q^{81} +2307.59i q^{82} -7799.12i q^{83} +274.150 q^{85} -6131.18 q^{86} -652.514i q^{87} +3431.76 q^{88} +9469.11i q^{89} +154.571i q^{90} -1421.17 q^{92} +2737.81 q^{93} -12121.5i q^{94} -277.379 q^{95} +368.097i q^{96} +9945.39i q^{97} +11657.7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 32 q^{4} - 196 q^{9}+O(q^{10}) \) 4 * q + 32 * q^4 - 196 * q^9	\( 4 q + 32 q^{4} - 196 q^{9} + 24 q^{11} + 888 q^{15} + 256 q^{16} + 1424 q^{18} + 1648 q^{22} + 104 q^{23} + 948 q^{25} - 1408 q^{29} - 2528 q^{30} - 1568 q^{36} - 3392 q^{37} + 2200 q^{39} - 2024 q^{43} + 192 q^{44} - 2304 q^{46} + 4384 q^{50} + 18936 q^{51} - 16680 q^{53} - 6064 q^{57} + 352 q^{58} + 7104 q^{60} + 2048 q^{64} - 6048 q^{65} - 20816 q^{67} - 1984 q^{71} + 11392 q^{72} + 576 q^{74} - 12224 q^{78} - 29616 q^{79} + 30852 q^{81} - 29688 q^{85} - 18800 q^{86} + 13184 q^{88} + 832 q^{92} + 18192 q^{93} + 7240 q^{95} + 72160 q^{99}+O(q^{100}) \) 4 * q + 32 * q^4 - 196 * q^9 + 24 * q^11 + 888 * q^15 + 256 * q^16 + 1424 * q^18 + 1648 * q^22 + 104 * q^23 + 948 * q^25 - 1408 * q^29 - 2528 * q^30 - 1568 * q^36 - 3392 * q^37 + 2200 * q^39 - 2024 * q^43 + 192 * q^44 - 2304 * q^46 + 4384 * q^50 + 18936 * q^51 - 16680 * q^53 - 6064 * q^57 + 352 * q^58 + 7104 * q^60 + 2048 * q^64 - 6048 * q^65 - 20816 * q^67 - 1984 * q^71 + 11392 * q^72 + 576 * q^74 - 12224 * q^78 - 29616 * q^79 + 30852 * q^81 - 29688 * q^85 - 18800 * q^86 + 13184 * q^88 + 832 * q^92 + 18192 * q^93 + 7240 * q^95 + 72160 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more