LMFDB - Embedded newform 49.5.b.b.48.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [49,5,Mod(48,49)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(49, 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("49.48");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 49 = 7^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	49.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:	$5.06512819111$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 66x^{6} + 212x^{5} + 2021x^{4} - 4400x^{3} - 25028x^{2} + 27264x + 127778 $ x^8 - 4x^7 - 66x^6 + 212x^5 + 2021x^4 - 4400x^3 - 25028x^2 + 27264*x + 127778
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 7^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			48.5
Root			$-2.71722 - 0.765367i$ of defining polynomial
Character	$\chi$	$=$	49.48
Dual form			49.5.b.b.48.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.71722 q^{2} -16.1337i q^{3} -13.0512 q^{4} +7.83726i q^{5} -27.7052i q^{6} -49.8872 q^{8} -179.298 q^{9} +O(q^{10})$	\(q+1.71722 q^{2} -16.1337i q^{3} -13.0512 q^{4} +7.83726i q^{5} -27.7052i q^{6} -49.8872 q^{8} -179.298 q^{9} +13.4583i q^{10} +30.5126 q^{11} +210.564i q^{12} -162.301i q^{13} +126.444 q^{15} +123.151 q^{16} -345.619i q^{17} -307.894 q^{18} -263.892i q^{19} -102.285i q^{20} +52.3969 q^{22} +213.179 q^{23} +804.868i q^{24} +563.577 q^{25} -278.707i q^{26} +1585.91i q^{27} -1011.68 q^{29} +217.133 q^{30} +273.802i q^{31} +1009.67 q^{32} -492.283i q^{33} -593.504i q^{34} +2340.04 q^{36} +1546.00 q^{37} -453.160i q^{38} -2618.52 q^{39} -390.979i q^{40} -531.628i q^{41} -1470.93 q^{43} -398.225 q^{44} -1405.20i q^{45} +366.075 q^{46} +2383.59i q^{47} -1986.89i q^{48} +967.786 q^{50} -5576.13 q^{51} +2118.22i q^{52} +3206.62 q^{53} +2723.36i q^{54} +239.135i q^{55} -4257.56 q^{57} -1737.27 q^{58} -3118.51i q^{59} -1650.25 q^{60} -3944.63i q^{61} +470.179i q^{62} -236.587 q^{64} +1272.00 q^{65} -845.358i q^{66} -6919.27 q^{67} +4510.73i q^{68} -3439.38i q^{69} +8580.05 q^{71} +8944.67 q^{72} -359.285i q^{73} +2654.82 q^{74} -9092.62i q^{75} +3444.09i q^{76} -4496.58 q^{78} +2087.16 q^{79} +965.167i q^{80} +11063.6 q^{81} -912.922i q^{82} +10594.2i q^{83} +2708.71 q^{85} -2525.91 q^{86} +16322.2i q^{87} -1522.19 q^{88} +10431.1i q^{89} -2413.04i q^{90} -2782.23 q^{92} +4417.46 q^{93} +4093.14i q^{94} +2068.19 q^{95} -16289.8i q^{96} -1403.85i q^{97} -5470.85 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 12 q^{2} + 52 q^{4} - 372 q^{8} - 352 q^{9}+O(q^{10}) $ 8 * q - 12 * q^2 + 52 * q^4 - 372 * q^8 - 352 * q^9	$ 8 q - 12 q^{2} + 52 q^{4} - 372 q^{8} - 352 q^{9} + 120 q^{11} - 632 q^{15} - 300 q^{16} + 340 q^{18} + 1952 q^{22} + 1752 q^{23} - 2192 q^{25} + 1248 q^{29} + 456 q^{30} + 3156 q^{32} + 4940 q^{36} + 2368 q^{37} - 7672 q^{39} - 8552 q^{43} + 264 q^{44} + 7208 q^{46} - 5556 q^{50} - 11976 q^{51} + 5496 q^{53} - 9200 q^{57} - 17496 q^{58} + 2856 q^{60} + 3980 q^{64} + 30240 q^{65} - 7440 q^{67} + 9984 q^{71} - 1508 q^{72} - 1080 q^{74} + 22456 q^{78} - 14096 q^{79} + 3432 q^{81} + 11912 q^{85} + 44496 q^{86} - 44464 q^{88} - 32232 q^{92} + 9584 q^{93} + 22488 q^{95} + 30144 q^{99}+O(q^{100}) $ 8 * q - 12 * q^2 + 52 * q^4 - 372 * q^8 - 352 * q^9 + 120 * q^11 - 632 * q^15 - 300 * q^16 + 340 * q^18 + 1952 * q^22 + 1752 * q^23 - 2192 * q^25 + 1248 * q^29 + 456 * q^30 + 3156 * q^32 + 4940 * q^36 + 2368 * q^37 - 7672 * q^39 - 8552 * q^43 + 264 * q^44 + 7208 * q^46 - 5556 * q^50 - 11976 * q^51 + 5496 * q^53 - 9200 * q^57 - 17496 * q^58 + 2856 * q^60 + 3980 * q^64 + 30240 * q^65 - 7440 * q^67 + 9984 * q^71 - 1508 * q^72 - 1080 * q^74 + 22456 * q^78 - 14096 * q^79 + 3432 * q^81 + 11912 * q^85 + 44496 * q^86 - 44464 * q^88 - 32232 * q^92 + 9584 * q^93 + 22488 * q^95 + 30144 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/49\mathbb{Z}\right)^\times$.

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

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 $

$2$	1.71722	0.429305	0.214653	−	0.976690i	$-0.431138\pi$
$2$	1.71722	0.429305	0.214653	+	0.976690i	$0.431138\pi$
$3$	− 16.1337i	− 1.79264i	−0.443409	−	0.896319i	$-0.646231\pi$
$3$	− 16.1337i	− 1.79264i	0.443409	−	0.896319i	$-0.353769\pi$
$4$	−13.0512	−0.815697
$5$	7.83726i	0.313490i	0.987639	+	0.156745i	$0.0501001\pi$
$5$	7.83726i	0.313490i	−0.987639	+	0.156745i	$0.949900\pi$
$6$	− 27.7052i	− 0.769589i
$7$	0	0
$7$	0	0
$8$	−49.8872	−0.779488
$9$	−179.298	−2.21355
$10$	13.4583i	0.134583i
$11$	30.5126	0.252170	0.126085	−	0.992019i	$-0.459759\pi$
$11$	30.5126	0.252170	0.126085	+	0.992019i	$0.459759\pi$
$12$	210.564i	1.46225i
$13$	− 162.301i	− 0.960361i	−0.877170	−	0.480181i	$-0.840571\pi$
$13$	− 162.301i	− 0.960361i	0.877170	−	0.480181i	$-0.159429\pi$
$14$	0	0
$15$	126.444	0.561975
$16$	123.151	0.481059
$17$	− 345.619i	− 1.19591i	−0.801528	−	0.597957i	$-0.795979\pi$
$17$	− 345.619i	− 1.19591i	0.801528	−	0.597957i	$-0.204021\pi$
$18$	−307.894	−0.950290
$19$	− 263.892i	− 0.731002i	−0.930811	−	0.365501i	$-0.880898\pi$
$19$	− 263.892i	− 0.731002i	0.930811	−	0.365501i	$-0.119102\pi$
$20$	− 102.285i	− 0.255713i
$21$	0	0
$22$	52.3969	0.108258
$23$	213.179	0.402985	0.201492	−	0.979490i	$-0.435421\pi$
$23$	213.179	0.402985	0.201492	+	0.979490i	$0.435421\pi$
$24$	804.868i	1.39734i
$25$	563.577	0.901724
$26$	− 278.707i	− 0.412288i
$27$	1585.91i	2.17546i
$28$	0	0
$29$	−1011.68	−1.20295	−0.601474	−	0.798893i	$-0.705419\pi$
$29$	−1011.68	−1.20295	−0.601474	+	0.798893i	$0.705419\pi$
$30$	217.133	0.241259
$31$	273.802i	0.284914i	0.989801	+	0.142457i	$0.0455002\pi$
$31$	273.802i	0.284914i	−0.989801	+	0.142457i	$0.954500\pi$
$32$	1009.67	0.986009
$33$	− 492.283i	− 0.452050i
$34$	− 593.504i	− 0.513412i
$35$	0	0
$36$	2340.04	1.80559
$37$	1546.00	1.12929	0.564646	−	0.825333i	$-0.309013\pi$
$37$	1546.00	1.12929	0.564646	+	0.825333i	$0.309013\pi$
$38$	− 453.160i	− 0.313823i
$39$	−2618.52	−1.72158
$40$	− 390.979i	− 0.244362i
$41$	− 531.628i	− 0.316257i	−0.987419	−	0.158128i	$-0.949454\pi$
$41$	− 531.628i	− 0.316257i	0.987419	−	0.158128i	$-0.0505460\pi$
$42$	0	0
$43$	−1470.93	−0.795526	−0.397763	−	0.917488i	$-0.630213\pi$
$43$	−1470.93	−0.795526	−0.397763	+	0.917488i	$0.630213\pi$
$44$	−398.225	−0.205695
$45$	− 1405.20i	− 0.693928i
$46$	366.075	0.173003
$47$	2383.59i	1.07903i	0.841975	+	0.539517i	$0.181393\pi$
$47$	2383.59i	1.07903i	−0.841975	+	0.539517i	$0.818607\pi$
$48$	− 1986.89i	− 0.862365i
$49$	0	0
$50$	967.786	0.387115
$51$	−5576.13	−2.14384
$52$	2118.22i	0.783364i
$53$	3206.62	1.14155	0.570777	−	0.821105i	$-0.306642\pi$
$53$	3206.62	1.14155	0.570777	+	0.821105i	$0.306642\pi$
$54$	2723.36i	0.933938i
$55$	239.135i	0.0790530i
$56$	0	0
$57$	−4257.56	−1.31042
$58$	−1737.27	−0.516431
$59$	− 3118.51i	− 0.895865i	−0.894067	−	0.447932i	$-0.852160\pi$
$59$	− 3118.51i	− 0.895865i	0.894067	−	0.447932i	$-0.147840\pi$
$60$	−1650.25	−0.458401
$61$	− 3944.63i	− 1.06010i	−0.847966	−	0.530050i	$-0.822173\pi$
$61$	− 3944.63i	− 1.06010i	0.847966	−	0.530050i	$-0.177827\pi$
$62$	470.179i	0.122315i
$63$	0	0
$64$	−236.587	−0.0577605
$65$	1272.00	0.301064
$66$	− 845.358i	− 0.194067i
$67$	−6919.27	−1.54138	−0.770692	−	0.637208i	$-0.780089\pi$
$67$	−6919.27	−1.54138	−0.770692	+	0.637208i	$0.780089\pi$
$68$	4510.73i	0.975504i
$69$	− 3439.38i	− 0.722406i
$70$	0	0
$71$	8580.05	1.70205	0.851027	−	0.525122i	$-0.175980\pi$
$71$	8580.05	1.70205	0.851027	+	0.525122i	$0.175980\pi$
$72$	8944.67	1.72544
$73$	− 359.285i	− 0.0674207i	−0.999432	−	0.0337104i	$-0.989268\pi$
$73$	− 359.285i	− 0.0674207i	0.999432	−	0.0337104i	$-0.0107324\pi$
$74$	2654.82	0.484811
$75$	− 9092.62i	− 1.61647i
$76$	3444.09i	0.596276i
$77$	0	0
$78$	−4496.58	−0.739083
$79$	2087.16	0.334427	0.167214	−	0.985921i	$-0.446523\pi$
$79$	2087.16	0.334427	0.167214	+	0.985921i	$0.446523\pi$
$80$	965.167i	0.150807i
$81$	11063.6	1.68627
$82$	− 912.922i	− 0.135771i
$83$	10594.2i	1.53784i	0.639344	+	0.768921i	$0.279206\pi$
$83$	10594.2i	1.53784i	−0.639344	+	0.768921i	$0.720794\pi$
$84$	0	0
$85$	2708.71	0.374908
$86$	−2525.91	−0.341523
$87$	16322.2i	2.15645i
$88$	−1522.19	−0.196564
$89$	10431.1i	1.31689i	0.752628	+	0.658446i	$0.228786\pi$
$89$	10431.1i	1.31689i	−0.752628	+	0.658446i	$0.771214\pi$
$90$	− 2413.04i	− 0.297907i
$91$	0	0
$92$	−2782.23	−0.328714
$93$	4417.46	0.510748
$94$	4093.14i	0.463235i
$95$	2068.19	0.229162
$96$	− 16289.8i	− 1.76756i
$97$	− 1403.85i	− 0.149203i	−0.997213	−	0.0746016i	$-0.976231\pi$
$97$	− 1403.85i	− 0.149203i	0.997213	−	0.0746016i	$-0.0237685\pi$
$98$	0	0
$99$	−5470.85	−0.558193
$100$	−7355.34	−0.735534
$101$	− 14051.5i	− 1.37747i	−0.725014	−	0.688734i	$-0.758167\pi$
$101$	− 14051.5i	− 1.37747i	0.725014	−	0.688734i	$-0.241833\pi$
$102$	−9575.45	−0.920362
$103$	1748.62i	0.164824i	0.996598	+	0.0824121i	$0.0262624\pi$
$103$	1748.62i	0.164824i	−0.996598	+	0.0824121i	$0.973738\pi$
$104$	8096.75i	0.748590i
$105$	0	0
$106$	5506.48	0.490075
$107$	8463.37	0.739224	0.369612	−	0.929186i	$-0.379491\pi$
$107$	8463.37	0.739224	0.369612	+	0.929186i	$0.379491\pi$
$108$	− 20698.0i	− 1.77452i
$109$	−10952.5	−0.921852	−0.460926	−	0.887439i	$-0.652483\pi$
$109$	−10952.5	−0.921852	−0.460926	+	0.887439i	$0.652483\pi$
$110$	410.648i	0.0339378i
$111$	− 24942.8i	− 2.02441i
$112$	0	0
$113$	5884.58	0.460849	0.230424	−	0.973090i	$-0.425989\pi$
$113$	5884.58	0.460849	0.230424	+	0.973090i	$0.425989\pi$
$114$	−7311.17	−0.562571
$115$	1670.74i	0.126332i
$116$	13203.6	0.981241
$117$	29100.2i	2.12581i
$118$	− 5355.16i	− 0.384599i
$119$	0	0
$120$	−6307.96	−0.438053
$121$	−13710.0	−0.936410
$122$	− 6773.81i	− 0.455107i
$123$	−8577.15	−0.566934
$124$	− 3573.44i	− 0.232403i
$125$	9315.19i	0.596172i
$126$	0	0
$127$	−8451.12	−0.523971	−0.261985	−	0.965072i	$-0.584377\pi$
$127$	−8451.12	−0.523971	−0.261985	+	0.965072i	$0.584377\pi$
$128$	−16561.0	−1.01081
$129$	23731.6i	1.42609i
$130$	2184.30	0.129248
$131$	801.065i	0.0466794i	0.999728	+	0.0233397i	$0.00742993\pi$
$131$	801.065i	0.0466794i	−0.999728	+	0.0233397i	$0.992570\pi$
$132$	6424.86i	0.368736i
$133$	0	0
$134$	−11881.9	−0.661723
$135$	−12429.2	−0.681987
$136$	17242.0i	0.932201i
$137$	22808.9	1.21524	0.607621	−	0.794227i	$-0.292124\pi$
$137$	22808.9	1.21524	0.607621	+	0.794227i	$0.292124\pi$
$138$	− 5906.16i	− 0.310133i
$139$	− 32895.2i	− 1.70256i	−0.524709	−	0.851281i	$-0.675826\pi$
$139$	− 32895.2i	− 1.70256i	0.524709	−	0.851281i	$-0.324174\pi$
$140$	0	0
$141$	38456.2	1.93432
$142$	14733.8	0.730700
$143$	− 4952.23i	− 0.242175i
$144$	−22080.7	−1.06485
$145$	− 7928.79i	− 0.377112i
$146$	− 616.971i	− 0.0289440i
$147$	0	0
$148$	−20177.1	−0.921161
$149$	29961.9	1.34957	0.674786	−	0.738013i	$-0.264236\pi$
$149$	29961.9	1.34957	0.674786	+	0.738013i	$0.264236\pi$
$150$	− 15614.0i	− 0.693957i
$151$	−4517.62	−0.198132	−0.0990662	−	0.995081i	$-0.531586\pi$
$151$	−4517.62	−0.198132	−0.0990662	+	0.995081i	$0.531586\pi$
$152$	13164.8i	0.569807i
$153$	61968.8i	2.64722i
$154$	0	0
$155$	−2145.86	−0.0893178
$156$	34174.8	1.40429
$157$	− 31715.2i	− 1.28667i	−0.765583	−	0.643337i	$-0.777550\pi$
$157$	− 31715.2i	− 1.28667i	0.765583	−	0.643337i	$-0.222450\pi$
$158$	3584.12	0.143571
$159$	− 51734.9i	− 2.04639i
$160$	7913.07i	0.309104i
$161$	0	0
$162$	18998.6	0.723923
$163$	13505.2	0.508306	0.254153	−	0.967164i	$-0.418203\pi$
$163$	13505.2	0.508306	0.254153	+	0.967164i	$0.418203\pi$
$164$	6938.36i	0.257970i
$165$	3858.15	0.141713
$166$	18192.6i	0.660203i
$167$	− 24965.3i	− 0.895168i	−0.894242	−	0.447584i	$-0.852285\pi$
$167$	− 24965.3i	− 0.895168i	0.894242	−	0.447584i	$-0.147715\pi$
$168$	0	0
$169$	2219.36	0.0777060
$170$	4651.45	0.160950
$171$	47315.2i	1.61811i
$172$	19197.3	0.648908
$173$	− 25428.2i	− 0.849618i	−0.905283	−	0.424809i	$-0.860341\pi$
$173$	− 25428.2i	− 0.849618i	0.905283	−	0.424809i	$-0.139659\pi$
$174$	28028.8i	0.925775i
$175$	0	0
$176$	3757.66	0.121309
$177$	−50313.2	−1.60596
$178$	17912.5i	0.565348i
$179$	−3612.57	−0.112748	−0.0563742	−	0.998410i	$-0.517954\pi$
$179$	−3612.57	−0.112748	−0.0563742	+	0.998410i	$0.517954\pi$
$180$	18339.5i	0.566035i
$181$	15458.2i	0.471847i	0.971772	+	0.235923i	$0.0758114\pi$
$181$	15458.2i	0.471847i	−0.971772	+	0.235923i	$0.924189\pi$
$182$	0	0
$183$	−63641.8	−1.90038
$184$	−10634.9	−0.314122
$185$	12116.4i	0.354022i
$186$	7585.75	0.219267
$187$	− 10545.7i	− 0.301574i
$188$	− 31108.5i	− 0.880165i
$189$	0	0
$190$	3551.53	0.0983804
$191$	13375.6	0.366646	0.183323	−	0.983053i	$-0.441315\pi$
$191$	13375.6	0.366646	0.183323	+	0.983053i	$0.441315\pi$
$192$	3817.04i	0.103544i
$193$	13763.6	0.369501	0.184751	−	0.982785i	$-0.440852\pi$
$193$	13763.6	0.369501	0.184751	+	0.982785i	$0.440852\pi$
$194$	− 2410.73i	− 0.0640537i
$195$	− 20522.1i	− 0.539699i
$196$	0	0
$197$	25150.4	0.648057	0.324028	−	0.946047i	$-0.394963\pi$
$197$	25150.4	0.648057	0.324028	+	0.946047i	$0.394963\pi$
$198$	−9394.65	−0.239635
$199$	− 42790.6i	− 1.08054i	−0.841491	−	0.540271i	$-0.818322\pi$
$199$	− 42790.6i	− 1.08054i	0.841491	−	0.540271i	$-0.181678\pi$
$200$	−28115.3	−0.702883
$201$	111634.i	2.76314i
$202$	− 24129.6i	− 0.591354i
$203$	0	0
$204$	72775.0	1.74873
$205$	4166.50	0.0991435
$206$	3002.77i	0.0707599i
$207$	−38222.5	−0.892028
$208$	− 19987.6i	− 0.461991i
$209$	− 8052.02i	− 0.184337i
$210$	0	0
$211$	−3516.62	−0.0789879	−0.0394940	−	0.999220i	$-0.512575\pi$
$211$	−3516.62	−0.0789879	−0.0394940	+	0.999220i	$0.512575\pi$
$212$	−41850.1	−0.931162
$213$	− 138428.i	− 3.05117i
$214$	14533.5	0.317352
$215$	− 11528.0i	− 0.249390i
$216$	− 79116.8i	− 1.69575i
$217$	0	0
$218$	−18807.9	−0.395756
$219$	−5796.61	−0.120861
$220$	− 3120.99i	− 0.0644833i
$221$	−56094.4	−1.14851
$222$	− 42832.3i	− 0.869091i
$223$	67281.8i	1.35297i	0.736457	+	0.676485i	$0.236497\pi$
$223$	67281.8i	1.35297i	−0.736457	+	0.676485i	$0.763503\pi$
$224$	0	0
$225$	−101048.	−1.99601
$226$	10105.1	0.197845
$227$	74331.3i	1.44251i	0.692667	+	0.721257i	$0.256435\pi$
$227$	74331.3i	1.44251i	−0.692667	+	0.721257i	$0.743565\pi$
$228$	55566.1	1.06891
$229$	27965.0i	0.533266i	0.963798	+	0.266633i	$0.0859112\pi$
$229$	27965.0i	0.533266i	−0.963798	+	0.266633i	$0.914089\pi$
$230$	2869.03i	0.0542349i
$231$	0	0
$232$	50469.8	0.937683
$233$	−39880.8	−0.734602	−0.367301	−	0.930102i	$-0.619718\pi$
$233$	−39880.8	−0.734602	−0.367301	+	0.930102i	$0.619718\pi$
$234$	49971.5i	0.912622i
$235$	−18680.8	−0.338267
$236$	40700.1i	0.730755i
$237$	− 33673.7i	− 0.599508i
$238$	0	0
$239$	−29726.5	−0.520413	−0.260206	−	0.965553i	$-0.583791\pi$
$239$	−29726.5	−0.520413	−0.260206	+	0.965553i	$0.583791\pi$
$240$	15571.8	0.270343
$241$	62828.5i	1.08174i	0.841106	+	0.540870i	$0.181905\pi$
$241$	62828.5i	1.08174i	−0.841106	+	0.540870i	$0.818095\pi$
$242$	−23543.1	−0.402006
$243$	− 50038.4i	− 0.847405i
$244$	51482.0i	0.864721i
$245$	0	0
$246$	−14728.9	−0.243388
$247$	−42829.9	−0.702026
$248$	− 13659.2i	− 0.222087i
$249$	170924.	2.75679
$250$	15996.2i	0.255940i
$251$	62181.9i	0.986999i	0.869746	+	0.493499i	$0.164282\pi$
$251$	62181.9i	0.986999i	−0.869746	+	0.493499i	$0.835718\pi$
$252$	0	0
$253$	6504.64	0.101621
$254$	−14512.4	−0.224943
$255$	− 43701.6i	− 0.672074i
$256$	−24653.6	−0.376184
$257$	− 114684.i	− 1.73634i	−0.496264	−	0.868172i	$-0.665295\pi$
$257$	− 114684.i	− 1.73634i	0.496264	−	0.868172i	$-0.334705\pi$
$258$	40752.3i	0.612228i
$259$	0	0
$260$	−16601.0	−0.245577
$261$	181392.	2.66279
$262$	1375.61i	0.0200397i
$263$	−83591.6	−1.20851	−0.604256	−	0.796790i	$-0.706530\pi$
$263$	−83591.6	−1.20851	−0.604256	+	0.796790i	$0.706530\pi$
$264$	24558.6i	0.352368i
$265$	25131.1i	0.357866i
$266$	0	0
$267$	168293.	2.36071
$268$	90304.4	1.25730
$269$	6858.72i	0.0947848i	0.998876	+	0.0473924i	$0.0150911\pi$
$269$	6858.72i	0.0947848i	−0.998876	+	0.0473924i	$0.984909\pi$
$270$	−21343.7	−0.292780
$271$	50356.0i	0.685666i	0.939396	+	0.342833i	$0.111387\pi$
$271$	50356.0i	0.685666i	−0.939396	+	0.342833i	$0.888613\pi$
$272$	− 42563.4i	− 0.575305i
$273$	0	0
$274$	39167.9	0.521710
$275$	17196.2	0.227388
$276$	44887.8i	0.589265i
$277$	84305.6	1.09875	0.549373	−	0.835577i	$-0.314867\pi$
$277$	84305.6	1.09875	0.549373	+	0.835577i	$0.314867\pi$
$278$	− 56488.3i	− 0.730919i
$279$	− 49092.2i	− 0.630672i
$280$	0	0
$281$	9852.07	0.124771	0.0623857	−	0.998052i	$-0.480129\pi$
$281$	9852.07	0.124771	0.0623857	+	0.998052i	$0.480129\pi$
$282$	66037.7	0.830412
$283$	− 31733.2i	− 0.396225i	−0.980179	−	0.198112i	$-0.936519\pi$
$283$	− 31733.2i	− 0.396225i	0.980179	−	0.198112i	$-0.0634811\pi$
$284$	−111980.	−1.38836
$285$	− 33367.6i	− 0.410805i
$286$	− 8504.07i	− 0.103967i
$287$	0	0
$288$	−181032.	−2.18258
$289$	−35931.6	−0.430211
$290$	− 13615.5i	− 0.161896i
$291$	−22649.4	−0.267468
$292$	4689.08i	0.0549949i
$293$	− 17750.9i	− 0.206769i	−0.994641	−	0.103384i	$-0.967033\pi$
$293$	− 17750.9i	− 0.206769i	0.994641	−	0.103384i	$-0.0329672\pi$
$294$	0	0
$295$	24440.5	0.280845
$296$	−77125.7	−0.880270
$297$	48390.3i	0.548587i
$298$	51451.1	0.579378
$299$	− 34599.2i	− 0.387011i
$300$	118669.i	1.31855i
$301$	0	0
$302$	−7757.74	−0.0850592
$303$	−226704.	−2.46930
$304$	− 32498.6i	− 0.351655i
$305$	30915.1	0.332331
$306$	106414.i	1.13647i
$307$	5360.84i	0.0568795i	0.999596	+	0.0284398i	$0.00905388\pi$
$307$	5360.84i	0.0568795i	−0.999596	+	0.0284398i	$0.990946\pi$
$308$	0	0
$309$	28211.8	0.295470
$310$	−3684.91	−0.0383446
$311$	138382.i	1.43073i	0.698751	+	0.715365i	$0.253740\pi$
$311$	138382.i	1.43073i	−0.698751	+	0.715365i	$0.746260\pi$
$312$	130631.	1.34195
$313$	45760.7i	0.467094i	0.972346	+	0.233547i	$0.0750333\pi$
$313$	45760.7i	0.467094i	−0.972346	+	0.233547i	$0.924967\pi$
$314$	− 54462.0i	− 0.552375i
$315$	0	0
$316$	−27239.9	−0.272792
$317$	39438.7	0.392468	0.196234	−	0.980557i	$-0.437129\pi$
$317$	39438.7	0.392468	0.196234	+	0.980557i	$0.437129\pi$
$318$	− 88840.1i	− 0.878527i
$319$	−30868.9	−0.303348
$320$	− 1854.19i	− 0.0181074i
$321$	− 136546.i	− 1.32516i
$322$	0	0
$323$	−91206.0	−0.874215
$324$	−144393.	−1.37548
$325$	− 91469.2i	− 0.865981i
$326$	23191.4	0.218218
$327$	176705.i	1.65255i
$328$	26521.4i	0.246518i
$329$	0	0
$330$	6625.29	0.0608383
$331$	39807.2	0.363333	0.181667	−	0.983360i	$-0.441851\pi$
$331$	39807.2	0.363333	0.181667	+	0.983360i	$0.441851\pi$
$332$	− 138266.i	− 1.25441i
$333$	−277195.	−2.49975
$334$	− 42871.0i	− 0.384300i
$335$	− 54228.1i	− 0.483209i
$336$	0	0
$337$	−71094.3	−0.626001	−0.313001	−	0.949753i	$-0.601334\pi$
$337$	−71094.3	−0.626001	−0.313001	+	0.949753i	$0.601334\pi$
$338$	3811.13	0.0333596
$339$	− 94940.3i	− 0.826136i
$340$	−35351.8	−0.305811
$341$	8354.42i	0.0718468i
$342$	81250.6i	0.694664i
$343$	0	0
$344$	73380.5	0.620103
$345$	26955.3	0.226467
$346$	− 43665.8i	− 0.364745i
$347$	66258.0	0.550275	0.275137	−	0.961405i	$-0.411277\pi$
$347$	66258.0	0.550275	0.275137	+	0.961405i	$0.411277\pi$
$348$	− 213023.i	− 1.75901i
$349$	− 225684.i	− 1.85289i	−0.376429	−	0.926445i	$-0.622848\pi$
$349$	− 225684.i	− 1.85289i	0.376429	−	0.926445i	$-0.377152\pi$
$350$	0	0
$351$	257395.	2.08923
$352$	30807.8	0.248642
$353$	132999.i	1.06733i	0.845697	+	0.533664i	$0.179185\pi$
$353$	132999.i	1.06733i	−0.845697	+	0.533664i	$0.820815\pi$
$354$	−86398.8	−0.689448
$355$	67244.1i	0.533578i
$356$	− 136138.i	− 1.07419i
$357$	0	0
$358$	−6203.58	−0.0484034
$359$	232730.	1.80578	0.902888	−	0.429875i	$-0.141443\pi$
$359$	232730.	1.80578	0.902888	+	0.429875i	$0.141443\pi$
$360$	70101.7i	0.540908i
$361$	60682.2	0.465636
$362$	26545.1i	0.202566i
$363$	221193.i	1.67865i
$364$	0	0
$365$	2815.81	0.0211357
$366$	−109287.	−0.815842
$367$	108571.i	0.806084i	0.915181	+	0.403042i	$0.132047\pi$
$367$	108571.i	0.806084i	−0.915181	+	0.403042i	$0.867953\pi$
$368$	26253.2	0.193859
$369$	95319.7i	0.700052i
$370$	20806.5i	0.151984i
$371$	0	0
$372$	−57652.9	−0.416615
$373$	−169883.	−1.22105	−0.610523	−	0.791999i	$-0.709041\pi$
$373$	−169883.	−1.22105	−0.610523	+	0.791999i	$0.709041\pi$
$374$	− 18109.4i	− 0.129467i
$375$	150289.	1.06872
$376$	− 118910.i	− 0.841094i
$377$	164197.i	1.15526i
$378$	0	0
$379$	−224839.	−1.56528	−0.782641	−	0.622474i	$-0.786128\pi$
$379$	−224839.	−1.56528	−0.782641	+	0.622474i	$0.786128\pi$
$380$	−26992.2	−0.186927
$381$	136348.i	0.939290i
$382$	22968.9	0.157403
$383$	24577.7i	0.167550i	0.996485	+	0.0837750i	$0.0266977\pi$
$383$	24577.7i	0.167550i	−0.996485	+	0.0837750i	$0.973302\pi$
$384$	267192.i	1.81201i
$385$	0	0
$386$	23635.1	0.158629
$387$	263734.	1.76094
$388$	18321.9i	0.121705i
$389$	144136.	0.952520	0.476260	−	0.879305i	$-0.341992\pi$
$389$	144136.	0.952520	0.476260	+	0.879305i	$0.341992\pi$
$390$	− 35240.9i	− 0.231696i
$391$	− 73678.7i	− 0.481935i
$392$	0	0
$393$	12924.2	0.0836793
$394$	43188.8	0.278214
$395$	16357.6i	0.104840i
$396$	71400.8	0.455316
$397$	− 153247.i	− 0.972327i	−0.873868	−	0.486164i	$-0.838396\pi$
$397$	− 153247.i	− 0.972327i	0.873868	−	0.486164i	$-0.161604\pi$
$398$	− 73480.8i	− 0.463882i
$399$	0	0
$400$	69405.2	0.433782
$401$	−45426.3	−0.282500	−0.141250	−	0.989974i	$-0.545112\pi$
$401$	−45426.3	−0.282500	−0.141250	+	0.989974i	$0.545112\pi$
$402$	191700.i	1.18623i
$403$	44438.4	0.273620
$404$	183389.i	1.12360i
$405$	86708.3i	0.528629i
$406$	0	0
$407$	47172.5	0.284774
$408$	278178.	1.67110
$409$	− 10421.7i	− 0.0623003i	−0.999515	−	0.0311502i	$-0.990083\pi$
$409$	− 10421.7i	− 0.0623003i	0.999515	−	0.0311502i	$-0.00991701\pi$
$410$	7154.80	0.0425628
$411$	− 367993.i	− 2.17849i
$412$	− 22821.5i	− 0.134447i
$413$	0	0
$414$	−65636.5	−0.382952
$415$	−83029.4	−0.482099
$416$	− 163871.i	− 0.946925i
$417$	−530723.	−3.05208
$418$	− 13827.1i	− 0.0791368i
$419$	148730.i	0.847167i	0.905857	+	0.423584i	$0.139228\pi$
$419$	148730.i	0.847167i	−0.905857	+	0.423584i	$0.860772\pi$
$420$	0	0
$421$	238275.	1.34436	0.672178	−	0.740389i	$-0.265359\pi$
$421$	238275.	1.34436	0.672178	+	0.740389i	$0.265359\pi$
$422$	−6038.81	−0.0339099
$423$	− 427372.i	− 2.38850i
$424$	−159970.	−0.889827
$425$	− 194783.i	− 1.07838i
$426$	− 237712.i	− 1.30988i
$427$	0	0
$428$	−110457.	−0.602983
$429$	−79898.0	−0.434132
$430$	− 19796.2i	− 0.107064i
$431$	280105.	1.50788	0.753938	−	0.656945i	$-0.228152\pi$
$431$	280105.	1.50788	0.753938	+	0.656945i	$0.228152\pi$
$432$	195307.i	1.04653i
$433$	78101.0i	0.416563i	0.978069	+	0.208281i	$0.0667870\pi$
$433$	78101.0i	0.416563i	−0.978069	+	0.208281i	$0.933213\pi$
$434$	0	0
$435$	−127921.	−0.676026
$436$	142943.	0.751952
$437$	− 56256.1i	− 0.294583i
$438$	−9954.06	−0.0518862
$439$	310618.i	1.61175i	0.592087	+	0.805874i	$0.298304\pi$
$439$	310618.i	1.61175i	−0.592087	+	0.805874i	$0.701696\pi$
$440$	− 11929.8i	− 0.0616208i
$441$	0	0
$442$	−96326.4	−0.493061
$443$	−5382.84	−0.0274286	−0.0137143	−	0.999906i	$-0.504366\pi$
$443$	−5382.84	−0.0274286	−0.0137143	+	0.999906i	$0.504366\pi$
$444$	325532.i	1.65131i
$445$	−81751.3	−0.412833
$446$	115538.i	0.580836i
$447$	− 483397.i	− 2.41930i
$448$	0	0
$449$	167992.	0.833292	0.416646	−	0.909069i	$-0.363206\pi$
$449$	167992.	0.833292	0.416646	+	0.909069i	$0.363206\pi$
$450$	−173522.	−0.856899
$451$	− 16221.3i	− 0.0797506i
$452$	−76800.6	−0.375913
$453$	72886.1i	0.355180i
$454$	127643.i	0.619279i
$455$	0	0
$456$	212398.	1.02146
$457$	−195647.	−0.936787	−0.468394	−	0.883520i	$-0.655167\pi$
$457$	−195647.	−0.936787	−0.468394	+	0.883520i	$0.655167\pi$
$458$	48022.1i	0.228934i
$459$	548122.	2.60167
$460$	− 21805.1i	− 0.103049i
$461$	355715.i	1.67379i	0.547365	+	0.836894i	$0.315631\pi$
$461$	355715.i	1.67379i	−0.547365	+	0.836894i	$0.684369\pi$
$462$	0	0
$463$	−103798.	−0.484205	−0.242102	−	0.970251i	$-0.577837\pi$
$463$	−103798.	−0.484205	−0.242102	+	0.970251i	$0.577837\pi$
$464$	−124589.	−0.578689
$465$	34620.8i	0.160114i
$466$	−68484.1	−0.315368
$467$	294112.i	1.34859i	0.738463	+	0.674295i	$0.235552\pi$
$467$	294112.i	1.34859i	−0.738463	+	0.674295i	$0.764448\pi$
$468$	− 379792.i	− 1.73402i
$469$	0	0
$470$	−32079.0	−0.145220
$471$	−511685.	−2.30654
$472$	155574.i	0.698316i
$473$	−44881.8	−0.200608
$474$	− 57825.2i	− 0.257372i
$475$	− 148723.i	− 0.659162i
$476$	0	0
$477$	−574941.	−2.52689
$478$	−51046.9	−0.223416
$479$	− 367747.i	− 1.60279i	−0.598133	−	0.801397i	$-0.704091\pi$
$479$	− 367747.i	− 1.60279i	0.598133	−	0.801397i	$-0.295909\pi$
$480$	127668.	0.554112
$481$	− 250918.i	− 1.08453i
$482$	107890.i	0.464396i
$483$	0	0
$484$	178931.	0.763827
$485$	11002.4	0.0467738
$486$	− 85926.9i	− 0.363795i
$487$	−342314.	−1.44333	−0.721667	−	0.692240i	$-0.756624\pi$
$487$	−342314.	−1.44333	−0.721667	+	0.692240i	$0.756624\pi$
$488$	196787.i	0.826336i
$489$	− 217889.i	− 0.911209i
$490$	0	0
$491$	41579.7	0.172472	0.0862360	−	0.996275i	$-0.472516\pi$
$491$	41579.7	0.172472	0.0862360	+	0.996275i	$0.472516\pi$
$492$	111942.	0.462447
$493$	349655.i	1.43862i
$494$	−73548.4	−0.301383
$495$	− 42876.4i	− 0.174988i
$496$	33719.1i	0.137060i
$497$	0	0
$498$	293514.	1.18351
$499$	355975.	1.42961	0.714807	−	0.699322i	$-0.246515\pi$
$499$	355975.	1.42961	0.714807	+	0.699322i	$0.246515\pi$
$500$	− 121574.i	− 0.486296i
$501$	−402785.	−1.60471
$502$	106780.i	0.423723i
$503$	196448.i	0.776446i	0.921566	+	0.388223i	$0.126911\pi$
$503$	196448.i	0.776446i	−0.921566	+	0.388223i	$0.873089\pi$
$504$	0	0
$505$	110126.	0.431823
$506$	11169.9	0.0436263
$507$	− 35806.6i	− 0.139299i
$508$	110297.	0.427401
$509$	− 133569.i	− 0.515551i	−0.966205	−	0.257776i	$-0.917010\pi$
$509$	− 133569.i	− 0.515551i	0.966205	−	0.257776i	$-0.0829895\pi$
$510$	− 75045.3i	− 0.288525i
$511$	0	0
$512$	222641.	0.849308
$513$	418509.	1.59027
$514$	− 196937.i	− 0.745421i
$515$	−13704.4	−0.0516708
$516$	− 309724.i	− 1.16326i
$517$	72729.4i	0.272100i
$518$	0	0
$519$	−410253.	−1.52306
$520$	−63456.3	−0.234676
$521$	− 359413.i	− 1.32409i	−0.749463	−	0.662046i	$-0.769688\pi$
$521$	− 359413.i	− 1.32409i	0.749463	−	0.662046i	$-0.230312\pi$
$522$	311490.	1.14315
$523$	− 149926.i	− 0.548119i	−0.961713	−	0.274060i	$-0.911633\pi$
$523$	− 149926.i	− 0.548119i	0.961713	−	0.274060i	$-0.0883665\pi$
$524$	− 10454.8i	− 0.0380763i
$525$	0	0
$526$	−143545.	−0.518821
$527$	94631.3	0.340733
$528$	− 60625.2i	− 0.217463i
$529$	−234396.	−0.837603
$530$	43155.7i	0.153634i
$531$	559142.i	1.98305i
$532$	0	0
$533$	−86283.7	−0.303721
$534$	288996.	1.01347
$535$	66329.6i	0.231739i
$536$	345183.	1.20149
$537$	58284.3i	0.202117i
$538$	11777.9i	0.0406916i
$539$	0	0
$540$	162216.	0.556295
$541$	−168242.	−0.574830	−0.287415	−	0.957806i	$-0.592796\pi$
$541$	−168242.	−0.574830	−0.287415	+	0.957806i	$0.592796\pi$
$542$	86472.4i	0.294360i
$543$	249398.	0.845851
$544$	− 348962.i	− 1.17918i
$545$	− 85837.7i	− 0.288992i
$546$	0	0
$547$	297422.	0.994027	0.497013	−	0.867743i	$-0.334430\pi$
$547$	297422.	0.994027	0.497013	+	0.867743i	$0.334430\pi$
$548$	−297682.	−0.991270
$549$	707265.i	2.34659i
$550$	29529.7	0.0976188
$551$	266974.i	0.879357i
$552$	171581.i	0.563107i
$553$	0	0
$554$	144771.	0.471697
$555$	195483.	0.634634
$556$	429321.i	1.38878i
$557$	451666.	1.45582	0.727909	−	0.685674i	$-0.240492\pi$
$557$	451666.	1.45582	0.727909	+	0.685674i	$0.240492\pi$
$558$	− 84302.0i	− 0.270751i
$559$	238733.i	0.763992i
$560$	0	0
$561$	−170142.	−0.540613
$562$	16918.2	0.0535650
$563$	550994.i	1.73832i	0.494531	+	0.869160i	$0.335340\pi$
$563$	550994.i	1.73832i	−0.494531	+	0.869160i	$0.664660\pi$
$564$	−501897.	−1.57782
$565$	46119.0i	0.144472i
$566$	− 54493.0i	− 0.170101i
$567$	0	0
$568$	−428035.	−1.32673
$569$	143742.	0.443975	0.221988	−	0.975049i	$-0.428746\pi$
$569$	143742.	0.443975	0.221988	+	0.975049i	$0.428746\pi$
$570$	− 57299.5i	− 0.176361i
$571$	118245.	0.362668	0.181334	−	0.983422i	$-0.441958\pi$
$571$	118245.	0.362668	0.181334	+	0.983422i	$0.441958\pi$
$572$	64632.3i	0.197541i
$573$	− 215799.i	− 0.657264i
$574$	0	0
$575$	120143.	0.363381
$576$	42419.5	0.127856
$577$	− 451941.i	− 1.35747i	−0.734383	−	0.678735i	$-0.762529\pi$
$577$	− 451941.i	− 1.35747i	0.734383	−	0.678735i	$-0.237471\pi$
$578$	−61702.5	−0.184692
$579$	− 222058.i	− 0.662383i
$580$	103480.i	0.307609i
$581$	0	0
$582$	−38894.0	−0.114825
$583$	97842.4	0.287866
$584$	17923.7i	0.0525536i
$585$	−228066.	−0.666422
$586$	− 30482.2i	− 0.0887670i
$587$	− 59990.7i	− 0.174104i	−0.996204	−	0.0870518i	$-0.972255\pi$
$587$	− 59990.7i	− 0.174104i	0.996204	−	0.0870518i	$-0.0277446\pi$
$588$	0	0
$589$	72254.1	0.208273
$590$	41969.8	0.120568
$591$	− 405771.i	− 1.16173i
$592$	190392.	0.543256
$593$	− 77447.2i	− 0.220240i	−0.993918	−	0.110120i	$-0.964876\pi$
$593$	− 77447.2i	− 0.220240i	0.993918	−	0.110120i	$-0.0351235\pi$
$594$	83096.9i	0.235511i
$595$	0	0
$596$	−391037.	−1.10084
$597$	−690372.	−1.93702
$598$	− 59414.4i	− 0.166146i
$599$	−407051.	−1.13448	−0.567238	−	0.823554i	$-0.691988\pi$
$599$	−407051.	−1.13448	−0.567238	+	0.823554i	$0.691988\pi$
$600$	453605.i	1.26002i
$601$	− 681741.i	− 1.88743i	−0.330761	−	0.943715i	$-0.607305\pi$
$601$	− 681741.i	− 1.88743i	0.330761	−	0.943715i	$-0.392695\pi$
$602$	0	0
$603$	1.24061e6	3.41193
$604$	58960.1	0.161616
$605$	− 107449.i	− 0.293556i
$606$	−389301.	−1.06008
$607$	− 96785.6i	− 0.262684i	−0.991337	−	0.131342i	$-0.958071\pi$
$607$	− 96785.6i	− 0.262684i	0.991337	−	0.131342i	$-0.0419286\pi$
$608$	− 266444.i	− 0.720774i
$609$	0	0
$610$	53088.1	0.142672
$611$	386858.	1.03626
$612$	− 808764.i	− 2.15933i
$613$	478095.	1.27231	0.636156	−	0.771561i	$-0.280524\pi$
$613$	478095.	1.27231	0.636156	+	0.771561i	$0.280524\pi$
$614$	9205.74i	0.0244187i
$615$	− 67221.3i	− 0.177728i
$616$	0	0
$617$	−55323.0	−0.145323	−0.0726617	−	0.997357i	$-0.523149\pi$
$617$	−55323.0	−0.145323	−0.0726617	+	0.997357i	$0.523149\pi$
$618$	48445.9	0.126847
$619$	− 142719.i	− 0.372479i	−0.982504	−	0.186239i	$-0.940370\pi$
$619$	− 142719.i	− 0.372479i	0.982504	−	0.186239i	$-0.0596300\pi$
$620$	28005.9	0.0728562
$621$	338083.i	0.876679i
$622$	237632.i	0.614220i
$623$	0	0
$624$	−322474.	−0.828182
$625$	279230.	0.714830
$626$	78581.2i	0.200526i
$627$	−129909.	−0.330450
$628$	413920.i	1.04954i
$629$	− 534328.i	− 1.35054i
$630$	0	0
$631$	−667428.	−1.67628	−0.838138	−	0.545459i	$-0.816356\pi$
$631$	−667428.	−1.67628	−0.838138	+	0.545459i	$0.816356\pi$
$632$	−104123.	−0.260682
$633$	56736.3i	0.141597i
$634$	67724.9	0.168488
$635$	− 66233.6i	− 0.164260i
$636$	675200.i	1.66924i
$637$	0	0
$638$	−53008.8	−0.130229
$639$	−1.53839e6	−3.76759
$640$	− 129793.i	− 0.316878i
$641$	11099.0	0.0270127	0.0135063	−	0.999909i	$-0.495701\pi$
$641$	11099.0	0.0270127	0.0135063	+	0.999909i	$0.495701\pi$
$642$	− 234479.i	− 0.568898i
$643$	136149.i	0.329300i	0.986352	+	0.164650i	$0.0526494\pi$
$643$	136149.i	0.329300i	−0.986352	+	0.164650i	$0.947351\pi$
$644$	0	0
$645$	−185991.	−0.447066
$646$	−156621.	−0.375305
$647$	110825.i	0.264746i	0.991200	+	0.132373i	$0.0422597\pi$
$647$	110825.i	0.264746i	−0.991200	+	0.132373i	$0.957740\pi$
$648$	−551932.	−1.31443
$649$	− 95153.7i	− 0.225911i
$650$	− 157073.i	− 0.371770i
$651$	0	0
$652$	−176258.	−0.414624
$653$	−88318.2	−0.207121	−0.103560	−	0.994623i	$-0.533024\pi$
$653$	−88318.2	−0.207121	−0.103560	+	0.994623i	$0.533024\pi$
$654$	303442.i	0.709447i
$655$	−6278.16	−0.0146335
$656$	− 65470.6i	− 0.152138i
$657$	64419.0i	0.149239i
$658$	0	0
$659$	136014.	0.313194	0.156597	−	0.987663i	$-0.449948\pi$
$659$	136014.	0.313194	0.156597	+	0.987663i	$0.449948\pi$
$660$	−50353.3	−0.115595
$661$	138133.i	0.316150i	0.987427	+	0.158075i	$0.0505288\pi$
$661$	138133.i	0.316150i	−0.987427	+	0.158075i	$0.949471\pi$
$662$	68357.7	0.155981
$663$	905012.i	2.05886i
$664$	− 528515.i	− 1.19873i
$665$	0	0
$666$	−476004.	−1.07316
$667$	−215669.	−0.484769
$668$	325827.i	0.730186i
$669$	1.08551e6	2.42539
$670$	− 93121.6i	− 0.207444i
$671$	− 120361.i	− 0.267326i
$672$	0	0
$673$	752500.	1.66141	0.830704	−	0.556714i	$-0.187938\pi$
$673$	752500.	1.66141	0.830704	+	0.556714i	$0.187938\pi$
$674$	−122085.	−0.268745
$675$	893785.i	1.96167i
$676$	−28965.2	−0.0633846
$677$	− 774498.i	− 1.68983i	−0.534901	−	0.844914i	$-0.679651\pi$
$677$	− 774498.i	− 1.68983i	0.534901	−	0.844914i	$-0.320349\pi$
$678$	− 163033.i	− 0.354664i
$679$	0	0
$680$	−135130.	−0.292236
$681$	1.19924e6	2.58591
$682$	14346.4i	0.0308442i
$683$	−330088.	−0.707600	−0.353800	−	0.935321i	$-0.615111\pi$
$683$	−330088.	−0.707600	−0.353800	+	0.935321i	$0.615111\pi$
$684$	− 617518.i	− 1.31989i
$685$	178759.i	0.380967i
$686$	0	0
$687$	451181.	0.955954
$688$	−181146.	−0.382695
$689$	− 520438.i	− 1.09630i
$690$	46288.1	0.0972236
$691$	705962.i	1.47851i	0.673424	+	0.739257i	$0.264823\pi$
$691$	705962.i	1.47851i	−0.673424	+	0.739257i	$0.735177\pi$
$692$	331868.i	0.693031i
$693$	0	0
$694$	113780.	0.236236
$695$	257808.	0.533737
$696$	− 814268.i	− 1.68093i
$697$	−183741.	−0.378216
$698$	− 387549.i	− 0.795455i
$699$	643427.i	1.31688i
$700$	0	0
$701$	−387191.	−0.787933	−0.393966	−	0.919125i	$-0.628897\pi$
$701$	−387191.	−0.787933	−0.393966	+	0.919125i	$0.628897\pi$
$702$	442005.	0.896918
$703$	− 407977.i	− 0.825515i
$704$	−7218.89	−0.0145655
$705$	301391.i	0.606390i
$706$	228388.i	0.458209i
$707$	0	0
$708$	656645.	1.30998
$709$	650735.	1.29453	0.647264	−	0.762266i	$-0.275913\pi$
$709$	650735.	1.29453	0.647264	+	0.762266i	$0.275913\pi$
$710$	115473.i	0.229068i
$711$	−374224.	−0.740273
$712$	− 520379.i	− 1.02650i
$713$	58368.9i	0.114816i
$714$	0	0
$715$	38811.9	0.0759194
$716$	47148.2	0.0919685
$717$	479600.i	0.932912i
$718$	399649.	0.775229
$719$	− 84700.1i	− 0.163842i	−0.996639	−	0.0819212i	$-0.973894\pi$
$719$	− 84700.1i	− 0.163842i	0.996639	−	0.0819212i	$-0.0261056\pi$
$720$	− 173052.i	− 0.333820i
$721$	0	0
$722$	104205.	0.199900
$723$	1.01366e6	1.93917
$724$	− 201747.i	− 0.384884i
$725$	−570159.	−1.08473
$726$	379838.i	0.720651i
$727$	− 602305.i	− 1.13959i	−0.821788	−	0.569794i	$-0.807023\pi$
$727$	− 602305.i	− 1.13959i	0.821788	−	0.569794i	$-0.192977\pi$
$728$	0	0
$729$	88844.8	0.167177
$730$	4835.36	0.00907368
$731$	508381.i	0.951381i
$732$	830598.	1.55013
$733$	471815.i	0.878140i	0.898453	+	0.439070i	$0.144692\pi$
$733$	471815.i	0.878140i	−0.898453	+	0.439070i	$0.855308\pi$
$734$	186440.i	0.346056i
$735$	0	0
$736$	215241.	0.397347
$737$	−211125.	−0.388691
$738$	163685.i	0.300536i
$739$	−328102.	−0.600786	−0.300393	−	0.953815i	$-0.597118\pi$
$739$	−328102.	−0.600786	−0.300393	+	0.953815i	$0.597118\pi$
$740$	− 158133.i	− 0.288775i
$741$	691007.i	1.25848i
$742$	0	0
$743$	−237585.	−0.430370	−0.215185	−	0.976573i	$-0.569035\pi$
$743$	−237585.	−0.430370	−0.215185	+	0.976573i	$0.569035\pi$
$744$	−220375.	−0.398122
$745$	234819.i	0.423078i
$746$	−291726.	−0.524201
$747$	− 1.89952e6i	− 3.40410i
$748$	137634.i	0.245993i
$749$	0	0
$750$	258079.	0.458807
$751$	−364285.	−0.645894	−0.322947	−	0.946417i	$-0.604674\pi$
$751$	−364285.	−0.645894	−0.322947	+	0.946417i	$0.604674\pi$
$752$	293541.i	0.519079i
$753$	1.00323e6	1.76933
$754$	281962.i	0.495961i
$755$	− 35405.7i	− 0.0621126i
$756$	0	0
$757$	−590661.	−1.03073	−0.515367	−	0.856969i	$-0.672345\pi$
$757$	−590661.	−1.03073	−0.515367	+	0.856969i	$0.672345\pi$
$758$	−386097.	−0.671983
$759$	− 104944.i	− 0.182169i
$760$	−103176.	−0.178629
$761$	− 272715.i	− 0.470912i	−0.971885	−	0.235456i	$-0.924342\pi$
$761$	− 272715.i	− 0.470912i	0.971885	−	0.235456i	$-0.0756584\pi$
$762$	234140.i	0.403242i
$763$	0	0
$764$	−174567.	−0.299072
$765$	−485665.	−0.829878
$766$	42205.4i	0.0719301i
$767$	−506137.	−0.860354
$768$	397754.i	0.674361i
$769$	725929.i	1.22756i	0.789478	+	0.613778i	$0.210351\pi$
$769$	725929.i	1.22756i	−0.789478	+	0.613778i	$0.789649\pi$
$770$	0	0
$771$	−1.85028e6	−3.11264
$772$	−179630.	−0.301401
$773$	392620.i	0.657073i	0.944491	+	0.328536i	$0.106555\pi$
$773$	392620.i	0.657073i	−0.944491	+	0.328536i	$0.893445\pi$
$774$	452890.	0.755980
$775$	154309.i	0.256914i
$776$	70034.4i	0.116302i
$777$	0	0
$778$	247514.	0.408922
$779$	−140292.	−0.231184
$780$	267837.i	0.440231i
$781$	261800.	0.429207
$782$	− 126523.i	− 0.206897i
$783$	− 1.60443e6i	− 2.61697i
$784$	0	0
$785$	248560.	0.403360
$786$	22193.7	0.0359239
$787$	− 324279.i	− 0.523563i	−0.965127	−	0.261782i	$-0.915690\pi$
$787$	− 324279.i	− 0.523563i	0.965127	−	0.261782i	$-0.0843100\pi$
$788$	−328242.	−0.528618
$789$	1.34865e6i	2.16643i
$790$	28089.6i	0.0450082i
$791$	0	0
$792$	272925.	0.435104
$793$	−640218.	−1.01808
$794$	− 263160.i	− 0.417425i
$795$	405459.	0.641524
$796$	558466.i	0.881396i
$797$	179577.i	0.282706i	0.989959	+	0.141353i	$0.0451453\pi$
$797$	179577.i	0.282706i	−0.989959	+	0.141353i	$0.954855\pi$
$798$	0	0
$799$	823813.	1.29043
$800$	569029.	0.889108
$801$	− 1.87027e6i	− 2.91501i
$802$	−78006.9	−0.121279
$803$	− 10962.7i	− 0.0170015i
$804$	− 1.45695e6i	− 2.25389i
$805$	0	0
$806$	76310.5	0.117467
$807$	110657.	0.169915
$808$	700993.i	1.07372i
$809$	−982618.	−1.50137	−0.750685	−	0.660660i	$-0.770276\pi$
$809$	−982618.	−1.50137	−0.750685	+	0.660660i	$0.770276\pi$
$810$	148897.i	0.226943i
$811$	994497.i	1.51203i	0.654552	+	0.756017i	$0.272857\pi$
$811$	994497.i	1.51203i	−0.654552	+	0.756017i	$0.727143\pi$
$812$	0	0
$813$	812431.	1.22915
$814$	81005.6	0.122255
$815$	105844.i	0.159349i
$816$	−686707.	−1.03131
$817$	388166.i	0.581531i
$818$	− 17896.3i	− 0.0267459i
$819$	0	0
$820$	−54377.7	−0.0808710
$821$	−1.02275e6	−1.51734	−0.758668	−	0.651477i	$-0.774150\pi$
$821$	−1.02275e6	−1.51734	−0.758668	+	0.651477i	$0.774150\pi$
$822$	− 631925.i	− 0.935237i
$823$	317711.	0.469065	0.234532	−	0.972108i	$-0.424644\pi$
$823$	317711.	0.469065	0.234532	+	0.972108i	$0.424644\pi$
$824$	− 87233.8i	− 0.128478i
$825$	− 277439.i	− 0.407624i
$826$	0	0
$827$	−50540.8	−0.0738977	−0.0369489	−	0.999317i	$-0.511764\pi$
$827$	−50540.8	−0.0738977	−0.0369489	+	0.999317i	$0.511764\pi$
$828$	498848.	0.727625
$829$	− 1.07529e6i	− 1.56465i	−0.622868	−	0.782327i	$-0.714033\pi$
$829$	− 1.07529e6i	− 1.56465i	0.622868	−	0.782327i	$-0.285967\pi$
$830$	−142580.	−0.206967
$831$	− 1.36017e6i	− 1.96965i
$832$	38398.3i	0.0554710i
$833$	0	0
$834$	−911369.	−1.31027
$835$	195660.	0.280627
$836$	105088.i	0.150363i
$837$	−434227.	−0.619820
$838$	255401.i	0.363693i
$839$	− 966211.i	− 1.37261i	−0.727312	−	0.686307i	$-0.759231\pi$
$839$	− 966211.i	− 1.37261i	0.727312	−	0.686307i	$-0.240769\pi$
$840$	0	0
$841$	316212.	0.447082
$842$	409171.	0.577139
$843$	− 158951.i	− 0.223670i
$844$	45896.0	0.0644302
$845$	17393.7i	0.0243601i
$846$	− 733892.i	− 1.02539i
$847$	0	0
$848$	394899.	0.549155
$849$	−511976.	−0.710288
$850$	− 334486.i	− 0.462956i
$851$	329575.	0.455088
$852$	1.80665e6i	2.48883i
$853$	− 395709.i	− 0.543848i	−0.962319	−	0.271924i	$-0.912340\pi$
$853$	− 395709.i	− 0.543848i	0.962319	−	0.271924i	$-0.0876600\pi$
$854$	0	0
$855$	−370822.	−0.507263
$856$	−422214.	−0.576216
$857$	787067.i	1.07164i	0.844331	+	0.535822i	$0.179998\pi$
$857$	787067.i	1.07164i	−0.844331	+	0.535822i	$0.820002\pi$
$858$	−137202.	−0.186375
$859$	− 1.32116e6i	− 1.79047i	−0.445592	−	0.895236i	$-0.647007\pi$
$859$	− 1.32116e6i	− 1.79047i	0.445592	−	0.895236i	$-0.352993\pi$
$860$	150454.i	0.203426i
$861$	0	0
$862$	481001.	0.647339
$863$	−648112.	−0.870219	−0.435110	−	0.900377i	$-0.643290\pi$
$863$	−648112.	−0.870219	−0.435110	+	0.900377i	$0.643290\pi$
$864$	1.60125e6i	2.14503i
$865$	199288.	0.266347
$866$	134117.i	0.178833i
$867$	579712.i	0.771212i
$868$	0	0
$869$	63684.7	0.0843327
$870$	−219669.	−0.290221
$871$	1.12300e6i	1.48028i
$872$	546391.	0.718572
$873$	251708.i	0.330270i
$874$	− 96604.2i	− 0.126466i
$875$	0	0
$876$	75652.5	0.0985860
$877$	35839.1	0.0465969	0.0232985	−	0.999729i	$-0.492583\pi$
$877$	35839.1	0.0465969	0.0232985	+	0.999729i	$0.492583\pi$
$878$	533399.i	0.691931i
$879$	−286389.	−0.370662
$880$	29449.8i	0.0380291i
$881$	830960.i	1.07060i	0.844661	+	0.535302i	$0.179802\pi$
$881$	830960.i	1.07060i	−0.844661	+	0.535302i	$0.820198\pi$
$882$	0	0
$883$	−1.54585e6	−1.98265	−0.991327	−	0.131415i	$-0.958048\pi$
$883$	−1.54585e6	−1.98265	−0.991327	+	0.131415i	$0.958048\pi$
$884$	732096.	0.936836
$885$	− 394318.i	− 0.503454i
$886$	−9243.52	−0.0117752
$887$	853705.i	1.08508i	0.840031	+	0.542539i	$0.182537\pi$
$887$	853705.i	1.08508i	−0.840031	+	0.542539i	$0.817463\pi$
$888$	1.24433e6i	1.57801i
$889$	0	0
$890$	−140385.	−0.177231
$891$	337579.	0.425227
$892$	− 878105.i	− 1.10361i
$893$	629008.	0.788776
$894$	− 830099.i	− 1.03862i
$895$	− 28312.7i	− 0.0353455i
$896$	0	0
$897$	−558214.	−0.693771
$898$	288480.	0.357736
$899$	− 277000.i	− 0.342736i
$900$	1.31880e6	1.62814
$901$	− 1.10827e6i	− 1.36520i
$902$	− 27855.6i	− 0.0342373i
$903$	0	0
$904$	−293565.	−0.359226
$905$	−121150.	−0.147919
$906$	125161.i	0.152480i
$907$	194393.	0.236302	0.118151	−	0.992996i	$-0.462303\pi$
$907$	194393.	0.236302	0.118151	+	0.992996i	$0.462303\pi$
$908$	− 970110.i	− 1.17665i
$909$	2.51941e6i	3.04910i
$910$	0	0
$911$	433279.	0.522072	0.261036	−	0.965329i	$-0.415936\pi$
$911$	433279.	0.522072	0.261036	+	0.965329i	$0.415936\pi$
$912$	−524324.	−0.630391
$913$	323256.i	0.387798i
$914$	−335969.	−0.402168
$915$	− 498777.i	− 0.595750i
$916$	− 364976.i	− 0.434984i
$917$	0	0
$918$	941246.	1.11691
$919$	935259.	1.10739	0.553695	−	0.832719i	$-0.313217\pi$
$919$	935259.	1.10739	0.553695	+	0.832719i	$0.313217\pi$
$920$	− 83348.5i	− 0.0984741i
$921$	86490.4	0.101964
$922$	610841.i	0.718566i
$923$	− 1.39255e6i	− 1.63459i
$924$	0	0
$925$	871291.	1.01831
$926$	−178245.	−0.207872
$927$	− 313524.i	− 0.364847i
$928$	−1.02146e6	−1.18612
$929$	1.16877e6i	1.35425i	0.735867	+	0.677126i	$0.236775\pi$
$929$	1.16877e6i	1.35425i	−0.735867	+	0.677126i	$0.763225\pi$
$930$	59451.5i	0.0687380i
$931$	0	0
$932$	520491.	0.599213
$933$	2.23262e6	2.56478
$934$	505056.i	0.578956i
$935$	82649.7	0.0945405
$936$	− 1.45173e6i	− 1.65704i
$937$	695595.i	0.792277i	0.918191	+	0.396139i	$0.129650\pi$
$937$	695595.i	0.792277i	−0.918191	+	0.396139i	$0.870350\pi$
$938$	0	0
$939$	738292.	0.837331
$940$	243806.	0.275923
$941$	− 1.55308e6i	− 1.75394i	−0.480543	−	0.876971i	$-0.659560\pi$
$941$	− 1.55308e6i	− 1.75394i	0.480543	−	0.876971i	$-0.340440\pi$
$942$	−878676.	−0.990210
$943$	− 113332.i	− 0.127447i
$944$	− 384048.i	− 0.430964i
$945$	0	0
$946$	−77072.0	−0.0861220
$947$	1.04501e6	1.16525	0.582627	−	0.812740i	$-0.302025\pi$
$947$	1.04501e6	1.16525	0.582627	+	0.812740i	$0.302025\pi$
$948$	439481.i	0.489017i
$949$	−58312.3	−0.0647482
$950$	− 255391.i	− 0.282981i
$951$	− 636294.i	− 0.703553i
$952$	0	0
$953$	−855805.	−0.942300	−0.471150	−	0.882053i	$-0.656161\pi$
$953$	−855805.	−0.942300	−0.471150	+	0.882053i	$0.656161\pi$
$954$	−987300.	−1.08481
$955$	104828.i	0.114940i
$956$	387965.	0.424499
$957$	498032.i	0.543793i
$958$	− 631502.i	− 0.688087i
$959$	0	0
$960$	−29915.1	−0.0324600
$961$	848553.	0.918824
$962$	− 430881.i	− 0.465594i
$963$	−1.51746e6	−1.63631
$964$	− 819985.i	− 0.882372i
$965$	107869.i	0.115835i
$966$	0	0
$967$	−1.73940e6	−1.86015	−0.930074	−	0.367371i	$-0.880258\pi$
$967$	−1.73940e6	−1.86015	−0.930074	+	0.367371i	$0.880258\pi$
$968$	683953.	0.729920
$969$	1.47150e6i	1.56715i
$970$	18893.5	0.0200802
$971$	93336.5i	0.0989950i	0.998774	+	0.0494975i	$0.0157620\pi$
$971$	93336.5i	0.0989950i	−0.998774	+	0.0494975i	$0.984238\pi$
$972$	653059.i	0.691226i
$973$	0	0
$974$	−587829.	−0.619630
$975$	−1.47574e6	−1.55239
$976$	− 485786.i	− 0.509971i
$977$	1.05190e6	1.10201	0.551004	−	0.834503i	$-0.314245\pi$
$977$	1.05190e6	1.10201	0.551004	+	0.834503i	$0.314245\pi$
$978$	− 374164.i	− 0.391187i
$979$	318280.i	0.332081i
$980$	0	0
$981$	1.96376e6	2.04057
$982$	71401.5	0.0740431
$983$	− 32971.7i	− 0.0341219i	−0.999854	−	0.0170610i	$-0.994569\pi$
$983$	− 32971.7i	− 0.0341219i	0.999854	−	0.0170610i	$-0.00543094\pi$
$984$	427890.	0.441918
$985$	197110.i	0.203159i
$986$	600435.i	0.617607i
$987$	0	0
$988$	558980.	0.572641
$989$	−313571.	−0.320585
$990$	− 73628.3i	− 0.0751232i
$991$	638020.	0.649661	0.324831	−	0.945772i	$-0.394693\pi$
$991$	638020.	0.649661	0.324831	+	0.945772i	$0.394693\pi$
$992$	276451.i	0.280928i
$993$	− 642239.i	− 0.651325i
$994$	0	0
$995$	335361.	0.338740
$996$	−2.23076e6	−2.24871
$997$	1.72164e6i	1.73202i	0.500026	+	0.866010i	$0.333324\pi$
$997$	1.72164e6i	1.73202i	−0.500026	+	0.866010i	$0.666676\pi$
$998$	611288.	0.613740
$999$	2.45182e6i	2.45674i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.5.b.b.48.5	✓	8
3.2	odd	2		441.5.d.g.244.3		8
4.3	odd	2		784.5.c.g.97.8		8
7.2	even	3		49.5.d.c.31.4		16
7.3	odd	6		49.5.d.c.19.4		16
7.4	even	3		49.5.d.c.19.3		16
7.5	odd	6		49.5.d.c.31.3		16
7.6	odd	2	inner	49.5.b.b.48.6	yes	8
21.20	even	2		441.5.d.g.244.4		8
28.27	even	2		784.5.c.g.97.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.5.b.b.48.5	✓	8	1.1	even	1	trivial
49.5.b.b.48.6	yes	8	7.6	odd	2	inner
49.5.d.c.19.3		16	7.4	even	3
49.5.d.c.19.4		16	7.3	odd	6
49.5.d.c.31.3		16	7.5	odd	6
49.5.d.c.31.4		16	7.2	even	3
441.5.d.g.244.3		8	3.2	odd	2
441.5.d.g.244.4		8	21.20	even	2
784.5.c.g.97.1		8	28.27	even	2
784.5.c.g.97.8		8	4.3	odd	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:	\( N \)	\(=\)	\( 49 = 7^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	49.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.71722 q^{2} -16.1337i q^{3} -13.0512 q^{4} +7.83726i q^{5} -27.7052i q^{6} -49.8872 q^{8} -179.298 q^{9} +O(q^{10})\)	\(q+1.71722 q^{2} -16.1337i q^{3} -13.0512 q^{4} +7.83726i q^{5} -27.7052i q^{6} -49.8872 q^{8} -179.298 q^{9} +13.4583i q^{10} +30.5126 q^{11} +210.564i q^{12} -162.301i q^{13} +126.444 q^{15} +123.151 q^{16} -345.619i q^{17} -307.894 q^{18} -263.892i q^{19} -102.285i q^{20} +52.3969 q^{22} +213.179 q^{23} +804.868i q^{24} +563.577 q^{25} -278.707i q^{26} +1585.91i q^{27} -1011.68 q^{29} +217.133 q^{30} +273.802i q^{31} +1009.67 q^{32} -492.283i q^{33} -593.504i q^{34} +2340.04 q^{36} +1546.00 q^{37} -453.160i q^{38} -2618.52 q^{39} -390.979i q^{40} -531.628i q^{41} -1470.93 q^{43} -398.225 q^{44} -1405.20i q^{45} +366.075 q^{46} +2383.59i q^{47} -1986.89i q^{48} +967.786 q^{50} -5576.13 q^{51} +2118.22i q^{52} +3206.62 q^{53} +2723.36i q^{54} +239.135i q^{55} -4257.56 q^{57} -1737.27 q^{58} -3118.51i q^{59} -1650.25 q^{60} -3944.63i q^{61} +470.179i q^{62} -236.587 q^{64} +1272.00 q^{65} -845.358i q^{66} -6919.27 q^{67} +4510.73i q^{68} -3439.38i q^{69} +8580.05 q^{71} +8944.67 q^{72} -359.285i q^{73} +2654.82 q^{74} -9092.62i q^{75} +3444.09i q^{76} -4496.58 q^{78} +2087.16 q^{79} +965.167i q^{80} +11063.6 q^{81} -912.922i q^{82} +10594.2i q^{83} +2708.71 q^{85} -2525.91 q^{86} +16322.2i q^{87} -1522.19 q^{88} +10431.1i q^{89} -2413.04i q^{90} -2782.23 q^{92} +4417.46 q^{93} +4093.14i q^{94} +2068.19 q^{95} -16289.8i q^{96} -1403.85i q^{97} -5470.85 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 12 q^{2} + 52 q^{4} - 372 q^{8} - 352 q^{9}+O(q^{10}) \) 8 * q - 12 * q^2 + 52 * q^4 - 372 * q^8 - 352 * q^9	\( 8 q - 12 q^{2} + 52 q^{4} - 372 q^{8} - 352 q^{9} + 120 q^{11} - 632 q^{15} - 300 q^{16} + 340 q^{18} + 1952 q^{22} + 1752 q^{23} - 2192 q^{25} + 1248 q^{29} + 456 q^{30} + 3156 q^{32} + 4940 q^{36} + 2368 q^{37} - 7672 q^{39} - 8552 q^{43} + 264 q^{44} + 7208 q^{46} - 5556 q^{50} - 11976 q^{51} + 5496 q^{53} - 9200 q^{57} - 17496 q^{58} + 2856 q^{60} + 3980 q^{64} + 30240 q^{65} - 7440 q^{67} + 9984 q^{71} - 1508 q^{72} - 1080 q^{74} + 22456 q^{78} - 14096 q^{79} + 3432 q^{81} + 11912 q^{85} + 44496 q^{86} - 44464 q^{88} - 32232 q^{92} + 9584 q^{93} + 22488 q^{95} + 30144 q^{99}+O(q^{100}) \) 8 * q - 12 * q^2 + 52 * q^4 - 372 * q^8 - 352 * q^9 + 120 * q^11 - 632 * q^15 - 300 * q^16 + 340 * q^18 + 1952 * q^22 + 1752 * q^23 - 2192 * q^25 + 1248 * q^29 + 456 * q^30 + 3156 * q^32 + 4940 * q^36 + 2368 * q^37 - 7672 * q^39 - 8552 * q^43 + 264 * q^44 + 7208 * q^46 - 5556 * q^50 - 11976 * q^51 + 5496 * q^53 - 9200 * q^57 - 17496 * q^58 + 2856 * q^60 + 3980 * q^64 + 30240 * q^65 - 7440 * q^67 + 9984 * q^71 - 1508 * q^72 - 1080 * q^74 + 22456 * q^78 - 14096 * q^79 + 3432 * q^81 + 11912 * q^85 + 44496 * q^86 - 44464 * q^88 - 32232 * q^92 + 9584 * q^93 + 22488 * q^95 + 30144 * q^99

Introduction

L-functions

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Database

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