LMFDB - Embedded newform 750.6.c.a.499.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [750,6,Mod(499,750)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(750, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("750.499");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 750 = 2 \cdot 3 \cdot 5^{3} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	750.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$120.287864860$
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} + 1810x^{4} + 801025 $ x^8 + 1810*x^4 + 801025
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6}\cdot 5^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			499.8
Root			$-3.72585 - 3.72585i$ of defining polynomial
Character	$\chi$	$=$	750.499
Dual form			750.6.c.a.499.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000i q^{2} +9.00000i q^{3} -16.0000 q^{4} -36.0000 q^{6} +94.1683i q^{7} -64.0000i q^{8} -81.0000 q^{9} +O(q^{10})$	\(q+4.00000i q^{2} +9.00000i q^{3} -16.0000 q^{4} -36.0000 q^{6} +94.1683i q^{7} -64.0000i q^{8} -81.0000 q^{9} +132.150 q^{11} -144.000i q^{12} +534.158i q^{13} -376.673 q^{14} +256.000 q^{16} +533.666i q^{17} -324.000i q^{18} +1889.84 q^{19} -847.515 q^{21} +528.601i q^{22} +1735.07i q^{23} +576.000 q^{24} -2136.63 q^{26} -729.000i q^{27} -1506.69i q^{28} +5257.77 q^{29} +2617.22 q^{31} +1024.00i q^{32} +1189.35i q^{33} -2134.66 q^{34} +1296.00 q^{36} -4222.34i q^{37} +7559.35i q^{38} -4807.43 q^{39} +15485.4 q^{41} -3390.06i q^{42} +6323.17i q^{43} -2114.40 q^{44} -6940.26 q^{46} +14526.5i q^{47} +2304.00i q^{48} +7939.33 q^{49} -4802.99 q^{51} -8546.53i q^{52} -492.367i q^{53} +2916.00 q^{54} +6026.77 q^{56} +17008.5i q^{57} +21031.1i q^{58} +14574.3 q^{59} -20092.5 q^{61} +10468.9i q^{62} -7627.63i q^{63} -4096.00 q^{64} -4757.41 q^{66} +13069.6i q^{67} -8538.65i q^{68} -15615.6 q^{69} +6152.88 q^{71} +5184.00i q^{72} +23673.4i q^{73} +16889.4 q^{74} -30237.4 q^{76} +12444.4i q^{77} -19229.7i q^{78} +37254.6 q^{79} +6561.00 q^{81} +61941.4i q^{82} -4806.14i q^{83} +13560.2 q^{84} -25292.7 q^{86} +47319.9i q^{87} -8457.62i q^{88} +46719.6 q^{89} -50300.8 q^{91} -27761.1i q^{92} +23554.9i q^{93} -58106.0 q^{94} -9216.00 q^{96} +116564. i q^{97} +31757.3i q^{98} -10704.2 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 128 q^{4} - 288 q^{6} - 648 q^{9}+O(q^{10}) $ 8 * q - 128 * q^4 - 288 * q^6 - 648 * q^9	$ 8 q - 128 q^{4} - 288 q^{6} - 648 q^{9} - 520 q^{11} + 688 q^{14} + 2048 q^{16} + 2672 q^{19} + 1548 q^{21} + 4608 q^{24} - 1216 q^{26} - 7072 q^{29} - 26636 q^{31} - 8592 q^{34} + 10368 q^{36} - 2736 q^{39} + 21172 q^{41} + 8320 q^{44} - 7680 q^{46} + 75868 q^{49} - 19332 q^{51} + 23328 q^{54} - 11008 q^{56} + 31180 q^{59} - 76620 q^{61} - 32768 q^{64} + 18720 q^{66} - 17280 q^{69} - 164820 q^{71} - 114944 q^{74} - 42752 q^{76} + 342456 q^{79} + 52488 q^{81} - 24768 q^{84} + 52432 q^{86} + 356856 q^{89} - 340324 q^{91} + 118720 q^{94} - 73728 q^{96} + 42120 q^{99}+O(q^{100}) $ 8 * q - 128 * q^4 - 288 * q^6 - 648 * q^9 - 520 * q^11 + 688 * q^14 + 2048 * q^16 + 2672 * q^19 + 1548 * q^21 + 4608 * q^24 - 1216 * q^26 - 7072 * q^29 - 26636 * q^31 - 8592 * q^34 + 10368 * q^36 - 2736 * q^39 + 21172 * q^41 + 8320 * q^44 - 7680 * q^46 + 75868 * q^49 - 19332 * q^51 + 23328 * q^54 - 11008 * q^56 + 31180 * q^59 - 76620 * q^61 - 32768 * q^64 + 18720 * q^66 - 17280 * q^69 - 164820 * q^71 - 114944 * q^74 - 42752 * q^76 + 342456 * q^79 + 52488 * q^81 - 24768 * q^84 + 52432 * q^86 + 356856 * q^89 - 340324 * q^91 + 118720 * q^94 - 73728 * q^96 + 42120 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/750\mathbb{Z}\right)^\times$.

$n$	$127$	$251$
$\chi(n)$	$-1$	$1$

Coefficient data

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

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

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

$2$	4.00000i	0.707107i
$2$	4.00000i	0.707107i
$3$	9.00000i	0.577350i
$3$	9.00000i	0.577350i
$4$	−16.0000	−0.500000
$5$	0	0
$5$	0	0
$6$	−36.0000	−0.408248
$7$	94.1683i	0.726373i	0.931717	+	0.363186i	$0.118311\pi$
$7$	94.1683i	0.726373i	−0.931717	+	0.363186i	$0.881689\pi$
$8$	− 64.0000i	− 0.353553i
$9$	−81.0000	−0.333333
$10$	0	0
$11$	132.150	0.329296	0.164648	−	0.986352i	$-0.447351\pi$
$11$	132.150	0.329296	0.164648	+	0.986352i	$0.447351\pi$
$12$	− 144.000i	− 0.288675i
$13$	534.158i	0.876621i	0.898824	+	0.438310i	$0.144423\pi$
$13$	534.158i	0.876621i	−0.898824	+	0.438310i	$0.855577\pi$
$14$	−376.673	−0.513623
$15$	0	0
$16$	256.000	0.250000
$17$	533.666i	0.447865i	0.974605	+	0.223932i	$0.0718895\pi$
$17$	533.666i	0.447865i	−0.974605	+	0.223932i	$0.928111\pi$
$18$	− 324.000i	− 0.235702i
$19$	1889.84	1.20099	0.600496	−	0.799628i	$-0.294970\pi$
$19$	1889.84	1.20099	0.600496	+	0.799628i	$0.294970\pi$
$20$	0	0
$21$	−847.515	−0.419372
$22$	528.601i	0.232847i
$23$	1735.07i	0.683906i	0.939717	+	0.341953i	$0.111088\pi$
$23$	1735.07i	0.683906i	−0.939717	+	0.341953i	$0.888912\pi$
$24$	576.000	0.204124
$25$	0	0
$26$	−2136.63	−0.619864
$27$	− 729.000i	− 0.192450i
$28$	− 1506.69i	− 0.363186i
$29$	5257.77	1.16093	0.580466	−	0.814285i	$-0.302871\pi$
$29$	5257.77	1.16093	0.580466	+	0.814285i	$0.302871\pi$
$30$	0	0
$31$	2617.22	0.489142	0.244571	−	0.969631i	$-0.421353\pi$
$31$	2617.22	0.489142	0.244571	+	0.969631i	$0.421353\pi$
$32$	1024.00i	0.176777i
$33$	1189.35i	0.190119i
$34$	−2134.66	−0.316688
$35$	0	0
$36$	1296.00	0.166667
$37$	− 4222.34i	− 0.507048i	−0.967329	−	0.253524i	$-0.918410\pi$
$37$	− 4222.34i	− 0.507048i	0.967329	−	0.253524i	$-0.0815896\pi$
$38$	7559.35i	0.849230i
$39$	−4807.43	−0.506117
$40$	0	0
$41$	15485.4	1.43867	0.719336	−	0.694663i	$-0.244446\pi$
$41$	15485.4	1.43867	0.719336	+	0.694663i	$0.244446\pi$
$42$	− 3390.06i	− 0.296540i
$43$	6323.17i	0.521511i	0.965405	+	0.260756i	$0.0839717\pi$
$43$	6323.17i	0.521511i	−0.965405	+	0.260756i	$0.916028\pi$
$44$	−2114.40	−0.164648
$45$	0	0
$46$	−6940.26	−0.483595
$47$	14526.5i	0.959215i	0.877483	+	0.479608i	$0.159221\pi$
$47$	14526.5i	0.959215i	−0.877483	+	0.479608i	$0.840779\pi$
$48$	2304.00i	0.144338i
$49$	7939.33	0.472383
$50$	0	0
$51$	−4802.99	−0.258575
$52$	− 8546.53i	− 0.438310i
$53$	− 492.367i	− 0.0240768i	−0.999928	−	0.0120384i	$-0.996168\pi$
$53$	− 492.367i	− 0.0240768i	0.999928	−	0.0120384i	$-0.00383204\pi$
$54$	2916.00	0.136083
$55$	0	0
$56$	6026.77	0.256812
$57$	17008.5i	0.693393i
$58$	21031.1i	0.820902i
$59$	14574.3	0.545076	0.272538	−	0.962145i	$-0.412137\pi$
$59$	14574.3	0.545076	0.272538	+	0.962145i	$0.412137\pi$
$60$	0	0
$61$	−20092.5	−0.691370	−0.345685	−	0.938351i	$-0.612353\pi$
$61$	−20092.5	−0.691370	−0.345685	+	0.938351i	$0.612353\pi$
$62$	10468.9i	0.345876i
$63$	− 7627.63i	− 0.242124i
$64$	−4096.00	−0.125000
$65$	0	0
$66$	−4757.41	−0.134434
$67$	13069.6i	0.355693i	0.984058	+	0.177846i	$0.0569130\pi$
$67$	13069.6i	0.355693i	−0.984058	+	0.177846i	$0.943087\pi$
$68$	− 8538.65i	− 0.223932i
$69$	−15615.6	−0.394853
$70$	0	0
$71$	6152.88	0.144855	0.0724273	−	0.997374i	$-0.476925\pi$
$71$	6152.88	0.144855	0.0724273	+	0.997374i	$0.476925\pi$
$72$	5184.00i	0.117851i
$73$	23673.4i	0.519941i	0.965617	+	0.259970i	$0.0837128\pi$
$73$	23673.4i	0.519941i	−0.965617	+	0.259970i	$0.916287\pi$
$74$	16889.4	0.358537
$75$	0	0
$76$	−30237.4	−0.600496
$77$	12444.4i	0.239192i
$78$	− 19229.7i	− 0.357879i
$79$	37254.6	0.671603	0.335801	−	0.941933i	$-0.390993\pi$
$79$	37254.6	0.671603	0.335801	+	0.941933i	$0.390993\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	61941.4i	1.01729i
$83$	− 4806.14i	− 0.0765775i	−0.999267	−	0.0382887i	$-0.987809\pi$
$83$	− 4806.14i	− 0.0765775i	0.999267	−	0.0382887i	$-0.0121907\pi$
$84$	13560.2	0.209686
$85$	0	0
$86$	−25292.7	−0.368764
$87$	47319.9i	0.670264i
$88$	− 8457.62i	− 0.116424i
$89$	46719.6	0.625208	0.312604	−	0.949884i	$-0.398799\pi$
$89$	46719.6	0.625208	0.312604	+	0.949884i	$0.398799\pi$
$90$	0	0
$91$	−50300.8	−0.636753
$92$	− 27761.1i	− 0.341953i
$93$	23554.9i	0.282406i
$94$	−58106.0	−0.678268
$95$	0	0
$96$	−9216.00	−0.102062
$97$	116564.i	1.25787i	0.777459	+	0.628934i	$0.216508\pi$
$97$	116564.i	1.25787i	−0.777459	+	0.628934i	$0.783492\pi$
$98$	31757.3i	0.334025i
$99$	−10704.2	−0.109765
$100$	0	0
$101$	−88277.5	−0.861086	−0.430543	−	0.902570i	$-0.641678\pi$
$101$	−88277.5	−0.861086	−0.430543	+	0.902570i	$0.641678\pi$
$102$	− 19212.0i	− 0.182840i
$103$	− 11574.3i	− 0.107499i	−0.998554	−	0.0537494i	$-0.982883\pi$
$103$	− 11574.3i	− 0.107499i	0.998554	−	0.0537494i	$-0.0171172\pi$
$104$	34186.1	0.309932
$105$	0	0
$106$	1969.47	0.0170249
$107$	− 167770.i	− 1.41662i	−0.705899	−	0.708312i	$-0.749457\pi$
$107$	− 167770.i	− 1.41662i	0.705899	−	0.708312i	$-0.250543\pi$
$108$	11664.0i	0.0962250i
$109$	−77794.6	−0.627167	−0.313583	−	0.949561i	$-0.601530\pi$
$109$	−77794.6	−0.627167	−0.313583	+	0.949561i	$0.601530\pi$
$110$	0	0
$111$	38001.1	0.292744
$112$	24107.1i	0.181593i
$113$	224429.i	1.65342i	0.562631	+	0.826708i	$0.309789\pi$
$113$	224429.i	1.65342i	−0.562631	+	0.826708i	$0.690211\pi$
$114$	−68034.1	−0.490303
$115$	0	0
$116$	−84124.3	−0.580466
$117$	− 43266.8i	− 0.292207i
$118$	58297.1i	0.385427i
$119$	−50254.4	−0.325317
$120$	0	0
$121$	−143587.	−0.891564
$122$	− 80370.2i	− 0.488872i
$123$	139368.i	0.830617i
$124$	−41875.4	−0.244571
$125$	0	0
$126$	30510.5	0.171208
$127$	− 188778.i	− 1.03858i	−0.854597	−	0.519292i	$-0.826196\pi$
$127$	− 188778.i	− 1.03858i	0.854597	−	0.519292i	$-0.173804\pi$
$128$	− 16384.0i	− 0.0883883i
$129$	−56908.5	−0.301095
$130$	0	0
$131$	−160700.	−0.818158	−0.409079	−	0.912499i	$-0.634150\pi$
$131$	−160700.	−0.818158	−0.409079	+	0.912499i	$0.634150\pi$
$132$	− 19029.6i	− 0.0950595i
$133$	177963.i	0.872368i
$134$	−52278.4	−0.251513
$135$	0	0
$136$	34154.6	0.158344
$137$	254974.i	1.16063i	0.814392	+	0.580316i	$0.197071\pi$
$137$	254974.i	1.16063i	−0.814392	+	0.580316i	$0.802929\pi$
$138$	− 62462.4i	− 0.279203i
$139$	189439.	0.831634	0.415817	−	0.909448i	$-0.363496\pi$
$139$	189439.	0.831634	0.415817	+	0.909448i	$0.363496\pi$
$140$	0	0
$141$	−130738.	−0.553803
$142$	24611.5i	0.102428i
$143$	70589.2i	0.288668i
$144$	−20736.0	−0.0833333
$145$	0	0
$146$	−94693.7	−0.367654
$147$	71454.0i	0.272730i
$148$	67557.4i	0.253524i
$149$	265733.	0.980574	0.490287	−	0.871561i	$-0.336892\pi$
$149$	265733.	0.980574	0.490287	+	0.871561i	$0.336892\pi$
$150$	0	0
$151$	19381.8	0.0691755	0.0345878	−	0.999402i	$-0.488988\pi$
$151$	19381.8	0.0691755	0.0345878	+	0.999402i	$0.488988\pi$
$152$	− 120950.i	− 0.424615i
$153$	− 43226.9i	− 0.149288i
$154$	−49777.4	−0.169134
$155$	0	0
$156$	76918.8	0.253059
$157$	44209.1i	0.143140i	0.997436	+	0.0715702i	$0.0228010\pi$
$157$	44209.1i	0.143140i	−0.997436	+	0.0715702i	$0.977199\pi$
$158$	149018.i	0.474895i
$159$	4431.30	0.0139008
$160$	0	0
$161$	−163388.	−0.496771
$162$	26244.0i	0.0785674i
$163$	100514.i	0.296317i	0.988964	+	0.148158i	$0.0473345\pi$
$163$	100514.i	0.296317i	−0.988964	+	0.148158i	$0.952665\pi$
$164$	−247766.	−0.719336
$165$	0	0
$166$	19224.6	0.0541485
$167$	123580.i	0.342890i	0.985194	+	0.171445i	$0.0548437\pi$
$167$	123580.i	0.342890i	−0.985194	+	0.171445i	$0.945156\pi$
$168$	54240.9i	0.148270i
$169$	85967.8	0.231536
$170$	0	0
$171$	−153077.	−0.400331
$172$	− 101171.i	− 0.260756i
$173$	79665.4i	0.202374i	0.994867	+	0.101187i	$0.0322640\pi$
$173$	79665.4i	0.202374i	−0.994867	+	0.101187i	$0.967736\pi$
$174$	−189280.	−0.473948
$175$	0	0
$176$	33830.5	0.0823240
$177$	131169.i	0.314700i
$178$	186878.i	0.442088i
$179$	−458713.	−1.07006	−0.535030	−	0.844833i	$-0.679700\pi$
$179$	−458713.	−1.07006	−0.535030	+	0.844833i	$0.679700\pi$
$180$	0	0
$181$	−776670.	−1.76214	−0.881070	−	0.472986i	$-0.843176\pi$
$181$	−776670.	−1.76214	−0.881070	+	0.472986i	$0.843176\pi$
$182$	− 201203.i	− 0.450253i
$183$	− 180833.i	− 0.399162i
$184$	111044.	0.241797
$185$	0	0
$186$	−94219.8	−0.199692
$187$	70524.0i	0.147480i
$188$	− 232424.i	− 0.479608i
$189$	68648.7	0.139791
$190$	0	0
$191$	−713979.	−1.41613	−0.708063	−	0.706149i	$-0.750431\pi$
$191$	−713979.	−1.41613	−0.708063	+	0.706149i	$0.750431\pi$
$192$	− 36864.0i	− 0.0721688i
$193$	− 1.02726e6i	− 1.98512i	−0.121754	−	0.992560i	$-0.538852\pi$
$193$	− 1.02726e6i	− 1.98512i	0.121754	−	0.992560i	$-0.461148\pi$
$194$	−466256.	−0.889447
$195$	0	0
$196$	−127029.	−0.236191
$197$	− 856571.i	− 1.57253i	−0.617892	−	0.786263i	$-0.712013\pi$
$197$	− 856571.i	− 1.57253i	0.617892	−	0.786263i	$-0.287987\pi$
$198$	− 42816.7i	− 0.0776158i
$199$	−125414.	−0.224499	−0.112250	−	0.993680i	$-0.535806\pi$
$199$	−125414.	−0.224499	−0.112250	+	0.993680i	$0.535806\pi$
$200$	0	0
$201$	−117626.	−0.205359
$202$	− 353110.i	− 0.608880i
$203$	495115.i	0.843269i
$204$	76847.9	0.129287
$205$	0	0
$206$	46297.4	0.0760131
$207$	− 140540.i	− 0.227969i
$208$	136745.i	0.219155i
$209$	249742.	0.395482
$210$	0	0
$211$	−161585.	−0.249859	−0.124930	−	0.992166i	$-0.539871\pi$
$211$	−161585.	−0.249859	−0.124930	+	0.992166i	$0.539871\pi$
$212$	7877.87i	0.0120384i
$213$	55375.9i	0.0836319i
$214$	671080.	1.00170
$215$	0	0
$216$	−46656.0	−0.0680414
$217$	246459.i	0.355300i
$218$	− 311178.i	− 0.443474i
$219$	−213061.	−0.300188
$220$	0	0
$221$	−285062.	−0.392608
$222$	152004.i	0.207001i
$223$	− 484595.i	− 0.652554i	−0.945274	−	0.326277i	$-0.894206\pi$
$223$	− 484595.i	− 0.652554i	0.945274	−	0.326277i	$-0.105794\pi$
$224$	−96428.3	−0.128406
$225$	0	0
$226$	−897715.	−1.16914
$227$	− 123903.i	− 0.159595i	−0.996811	−	0.0797974i	$-0.974573\pi$
$227$	− 123903.i	− 0.159595i	0.996811	−	0.0797974i	$-0.0254273\pi$
$228$	− 272137.i	− 0.346697i
$229$	1.37970e6	1.73859	0.869293	−	0.494297i	$-0.164575\pi$
$229$	1.37970e6	1.73859	0.869293	+	0.494297i	$0.164575\pi$
$230$	0	0
$231$	−111999.	−0.138097
$232$	− 336497.i	− 0.410451i
$233$	289015.i	0.348763i	0.984678	+	0.174381i	$0.0557926\pi$
$233$	289015.i	0.348763i	−0.984678	+	0.174381i	$0.944207\pi$
$234$	173067.	0.206621
$235$	0	0
$236$	−233189.	−0.272538
$237$	335292.i	0.387750i
$238$	− 201018.i	− 0.230034i
$239$	38150.9	0.0432026	0.0216013	−	0.999767i	$-0.493124\pi$
$239$	38150.9	0.0432026	0.0216013	+	0.999767i	$0.493124\pi$
$240$	0	0
$241$	−1.12982e6	−1.25304	−0.626520	−	0.779405i	$-0.715521\pi$
$241$	−1.12982e6	−1.25304	−0.626520	+	0.779405i	$0.715521\pi$
$242$	− 574349.i	− 0.630431i
$243$	59049.0i	0.0641500i
$244$	321481.	0.345685
$245$	0	0
$246$	−557473.	−0.587335
$247$	1.00947e6i	1.05281i
$248$	− 167502.i	− 0.172938i
$249$	43255.2	0.0442120
$250$	0	0
$251$	−1.17040e6	−1.17260	−0.586301	−	0.810093i	$-0.699416\pi$
$251$	−1.17040e6	−1.17260	−0.586301	+	0.810093i	$0.699416\pi$
$252$	122042.i	0.121062i
$253$	229289.i	0.225207i
$254$	755111.	0.734390
$255$	0	0
$256$	65536.0	0.0625000
$257$	− 252515.i	− 0.238481i	−0.992865	−	0.119241i	$-0.961954\pi$
$257$	− 252515.i	− 0.238481i	0.992865	−	0.119241i	$-0.0380460\pi$
$258$	− 227634.i	− 0.212906i
$259$	397611.	0.368306
$260$	0	0
$261$	−425879.	−0.386977
$262$	− 642799.i	− 0.578525i
$263$	− 1.86430e6i	− 1.66198i	−0.556285	−	0.830992i	$-0.687774\pi$
$263$	− 1.86430e6i	− 1.66198i	0.556285	−	0.830992i	$-0.312226\pi$
$264$	76118.5	0.0672172
$265$	0	0
$266$	−711851.	−0.616858
$267$	420477.i	0.360964i
$268$	− 209113.i	− 0.177846i
$269$	−1.43904e6	−1.21253	−0.606263	−	0.795264i	$-0.707332\pi$
$269$	−1.43904e6	−1.21253	−0.606263	+	0.795264i	$0.707332\pi$
$270$	0	0
$271$	582891.	0.482130	0.241065	−	0.970509i	$-0.422503\pi$
$271$	582891.	0.482130	0.241065	+	0.970509i	$0.422503\pi$
$272$	136618.i	0.111966i
$273$	− 452707.i	− 0.367630i
$274$	−1.01990e6	−0.820690
$275$	0	0
$276$	249850.	0.197427
$277$	586076.i	0.458939i	0.973316	+	0.229469i	$0.0736991\pi$
$277$	586076.i	0.458939i	−0.973316	+	0.229469i	$0.926301\pi$
$278$	757755.i	0.588054i
$279$	−211994.	−0.163047
$280$	0	0
$281$	767061.	0.579514	0.289757	−	0.957100i	$-0.406425\pi$
$281$	767061.	0.579514	0.289757	+	0.957100i	$0.406425\pi$
$282$	− 522954.i	− 0.391598i
$283$	− 398361.i	− 0.295672i	−0.989012	−	0.147836i	$-0.952769\pi$
$283$	− 398361.i	− 0.295672i	0.989012	−	0.147836i	$-0.0472308\pi$
$284$	−98446.1	−0.0724273
$285$	0	0
$286$	−282357.	−0.204119
$287$	1.45823e6i	1.04501i
$288$	− 82944.0i	− 0.0589256i
$289$	1.13506e6	0.799417
$290$	0	0
$291$	−1.04908e6	−0.726230
$292$	− 378775.i	− 0.259970i
$293$	920055.i	0.626102i	0.949736	+	0.313051i	$0.101351\pi$
$293$	920055.i	0.626102i	−0.949736	+	0.313051i	$0.898649\pi$
$294$	−285816.	−0.192849
$295$	0	0
$296$	−270230.	−0.179268
$297$	− 96337.5i	− 0.0633730i
$298$	1.06293e6i	0.693371i
$299$	−926800.	−0.599526
$300$	0	0
$301$	−595442.	−0.378812
$302$	77527.3i	0.0489145i
$303$	− 794498.i	− 0.497148i
$304$	483798.	0.300248
$305$	0	0
$306$	172908.	0.105563
$307$	162209.i	0.0982268i	0.998793	+	0.0491134i	$0.0156396\pi$
$307$	162209.i	0.0982268i	−0.998793	+	0.0491134i	$0.984360\pi$
$308$	− 199110.i	− 0.119596i
$309$	104169.	0.0620644
$310$	0	0
$311$	−2.73905e6	−1.60583	−0.802914	−	0.596095i	$-0.796718\pi$
$311$	−2.73905e6	−1.60583	−0.802914	+	0.596095i	$0.796718\pi$
$312$	307675.i	0.178939i
$313$	− 2.46085e6i	− 1.41979i	−0.704308	−	0.709895i	$-0.748742\pi$
$313$	− 2.46085e6i	− 1.41979i	0.704308	−	0.709895i	$-0.251258\pi$
$314$	−176836.	−0.101216
$315$	0	0
$316$	−596074.	−0.335801
$317$	913335.i	0.510484i	0.966877	+	0.255242i	$0.0821551\pi$
$317$	913335.i	0.510484i	−0.966877	+	0.255242i	$0.917845\pi$
$318$	17725.2i	0.00982932i
$319$	694816.	0.382290
$320$	0	0
$321$	1.50993e6	0.817888
$322$	− 653553.i	− 0.351270i
$323$	1.00854e6i	0.537882i
$324$	−104976.	−0.0555556
$325$	0	0
$326$	−402055.	−0.209527
$327$	− 700151.i	− 0.362095i
$328$	− 991063.i	− 0.508647i
$329$	−1.36794e6	−0.696748
$330$	0	0
$331$	−1.90238e6	−0.954393	−0.477197	−	0.878797i	$-0.658347\pi$
$331$	−1.90238e6	−0.954393	−0.477197	+	0.878797i	$0.658347\pi$
$332$	76898.2i	0.0382887i
$333$	342010.i	0.169016i
$334$	−494318.	−0.242460
$335$	0	0
$336$	−216964.	−0.104843
$337$	2.33979e6i	1.12228i	0.827720	+	0.561141i	$0.189637\pi$
$337$	2.33979e6i	1.12228i	−0.827720	+	0.561141i	$0.810363\pi$
$338$	343871.i	0.163721i
$339$	−2.01986e6	−0.954601
$340$	0	0
$341$	345866.	0.161073
$342$	− 612307.i	− 0.283077i
$343$	2.33032e6i	1.06950i
$344$	404683.	0.184382
$345$	0	0
$346$	−318662.	−0.143100
$347$	3.61592e6i	1.61211i	0.591839	+	0.806056i	$0.298402\pi$
$347$	3.61592e6i	1.61211i	−0.591839	+	0.806056i	$0.701598\pi$
$348$	− 757119.i	− 0.335132i
$349$	2.73344e6	1.20128	0.600642	−	0.799518i	$-0.294912\pi$
$349$	2.73344e6	1.20128	0.600642	+	0.799518i	$0.294912\pi$
$350$	0	0
$351$	389401.	0.168706
$352$	135322.i	0.0582118i
$353$	1.85909e6i	0.794078i	0.917802	+	0.397039i	$0.129962\pi$
$353$	1.85909e6i	0.794078i	−0.917802	+	0.397039i	$0.870038\pi$
$354$	−524674.	−0.222526
$355$	0	0
$356$	−747514.	−0.312604
$357$	− 452289.i	− 0.187822i
$358$	− 1.83485e6i	− 0.756647i
$359$	−379501.	−0.155409	−0.0777046	−	0.996976i	$-0.524759\pi$
$359$	−379501.	−0.155409	−0.0777046	+	0.996976i	$0.524759\pi$
$360$	0	0
$361$	1.09539e6	0.442383
$362$	− 3.10668e6i	− 1.24602i
$363$	− 1.29229e6i	− 0.514745i
$364$	804813.	0.318377
$365$	0	0
$366$	723332.	0.282250
$367$	98592.9i	0.0382103i	0.999817	+	0.0191051i	$0.00608173\pi$
$367$	98592.9i	0.0382103i	−0.999817	+	0.0191051i	$0.993918\pi$
$368$	444177.i	0.170977i
$369$	−1.25431e6	−0.479557
$370$	0	0
$371$	46365.3	0.0174887
$372$	− 376879.i	− 0.141203i
$373$	− 2.85580e6i	− 1.06281i	−0.847118	−	0.531405i	$-0.821664\pi$
$373$	− 2.85580e6i	− 1.06281i	0.847118	−	0.531405i	$-0.178336\pi$
$374$	−282096.	−0.104284
$375$	0	0
$376$	929696.	0.339134
$377$	2.80848e6i	1.01770i
$378$	274595.i	0.0988468i
$379$	−1.73057e6	−0.618858	−0.309429	−	0.950923i	$-0.600138\pi$
$379$	−1.73057e6	−0.618858	−0.309429	+	0.950923i	$0.600138\pi$
$380$	0	0
$381$	1.69900e6	0.599627
$382$	− 2.85591e6i	− 1.00135i
$383$	700460.i	0.243998i	0.992530	+	0.121999i	$0.0389305\pi$
$383$	700460.i	0.243998i	−0.992530	+	0.121999i	$0.961070\pi$
$384$	147456.	0.0510310
$385$	0	0
$386$	4.10904e6	1.40369
$387$	− 512177.i	− 0.173837i
$388$	− 1.86502e6i	− 0.628934i
$389$	491786.	0.164779	0.0823896	−	0.996600i	$-0.473745\pi$
$389$	491786.	0.164779	0.0823896	+	0.996600i	$0.473745\pi$
$390$	0	0
$391$	−925945.	−0.306297
$392$	− 508117.i	− 0.167012i
$393$	− 1.44630e6i	− 0.472364i
$394$	3.42629e6	1.11194
$395$	0	0
$396$	171267.	0.0548826
$397$	3.63285e6i	1.15683i	0.815741	+	0.578417i	$0.196329\pi$
$397$	3.63285e6i	1.15683i	−0.815741	+	0.578417i	$0.803671\pi$
$398$	− 501657.i	− 0.158745i
$399$	−1.60166e6	−0.503662
$400$	0	0
$401$	−209443.	−0.0650436	−0.0325218	−	0.999471i	$-0.510354\pi$
$401$	−209443.	−0.0650436	−0.0325218	+	0.999471i	$0.510354\pi$
$402$	− 470505.i	− 0.145211i
$403$	1.39801e6i	0.428792i
$404$	1.41244e6	0.430543
$405$	0	0
$406$	−1.98046e6	−0.596281
$407$	− 557983.i	− 0.166969i
$408$	307391.i	0.0914200i
$409$	−2.50986e6	−0.741893	−0.370947	−	0.928654i	$-0.620967\pi$
$409$	−2.50986e6	−0.741893	−0.370947	+	0.928654i	$0.620967\pi$
$410$	0	0
$411$	−2.29477e6	−0.670091
$412$	185190.i	0.0537494i
$413$	1.37244e6i	0.395929i
$414$	562161.	0.161198
$415$	0	0
$416$	−546978.	−0.154966
$417$	1.70495e6i	0.480144i
$418$	998970.i	0.279648i
$419$	−1.11169e6	−0.309350	−0.154675	−	0.987965i	$-0.549433\pi$
$419$	−1.11169e6	−0.309350	−0.154675	+	0.987965i	$0.549433\pi$
$420$	0	0
$421$	4.13266e6	1.13638	0.568191	−	0.822897i	$-0.307643\pi$
$421$	4.13266e6	1.13638	0.568191	+	0.822897i	$0.307643\pi$
$422$	− 646341.i	− 0.176677i
$423$	− 1.17665e6i	− 0.319738i
$424$	−31511.5	−0.00851244
$425$	0	0
$426$	−221504.	−0.0591367
$427$	− 1.89208e6i	− 0.502192i
$428$	2.68432e6i	0.708312i
$429$	−635302.	−0.166662
$430$	0	0
$431$	−174538.	−0.0452583	−0.0226291	−	0.999744i	$-0.507204\pi$
$431$	−174538.	−0.0452583	−0.0226291	+	0.999744i	$0.507204\pi$
$432$	− 186624.i	− 0.0481125i
$433$	− 1.25259e6i	− 0.321062i	−0.987031	−	0.160531i	$-0.948679\pi$
$433$	− 1.25259e6i	− 0.321062i	0.987031	−	0.160531i	$-0.0513207\pi$
$434$	−985835.	−0.251235
$435$	0	0
$436$	1.24471e6	0.313583
$437$	3.27899e6i	0.821366i
$438$	− 852243.i	− 0.212265i
$439$	1.35406e6	0.335334	0.167667	−	0.985844i	$-0.446377\pi$
$439$	1.35406e6	0.335334	0.167667	+	0.985844i	$0.446377\pi$
$440$	0	0
$441$	−643086.	−0.157461
$442$	− 1.14025e6i	− 0.277615i
$443$	5.99479e6i	1.45132i	0.688051	+	0.725662i	$0.258467\pi$
$443$	5.99479e6i	1.45132i	−0.688051	+	0.725662i	$0.741533\pi$
$444$	−608017.	−0.146372
$445$	0	0
$446$	1.93838e6	0.461426
$447$	2.39160e6i	0.566135i
$448$	− 385713.i	− 0.0907966i
$449$	−1.46774e6	−0.343585	−0.171792	−	0.985133i	$-0.554956\pi$
$449$	−1.46774e6	−0.343585	−0.171792	+	0.985133i	$0.554956\pi$
$450$	0	0
$451$	2.04639e6	0.473748
$452$	− 3.59086e6i	− 0.826708i
$453$	174436.i	0.0399385i
$454$	495614.	0.112851
$455$	0	0
$456$	1.08855e6	0.245152
$457$	− 4.62531e6i	− 1.03598i	−0.855387	−	0.517989i	$-0.826681\pi$
$457$	− 4.62531e6i	− 1.03598i	0.855387	−	0.517989i	$-0.173319\pi$
$458$	5.51880e6i	1.22937i
$459$	389042.	0.0861916
$460$	0	0
$461$	−392691.	−0.0860595	−0.0430297	−	0.999074i	$-0.513701\pi$
$461$	−392691.	−0.0860595	−0.0430297	+	0.999074i	$0.513701\pi$
$462$	− 447997.i	− 0.0976495i
$463$	− 4.11602e6i	− 0.892328i	−0.894951	−	0.446164i	$-0.852790\pi$
$463$	− 4.11602e6i	− 0.892328i	0.894951	−	0.446164i	$-0.147210\pi$
$464$	1.34599e6	0.290233
$465$	0	0
$466$	−1.15606e6	−0.246613
$467$	1.50755e6i	0.319875i	0.987127	+	0.159938i	$0.0511293\pi$
$467$	1.50755e6i	0.319875i	−0.987127	+	0.159938i	$0.948871\pi$
$468$	692269.i	0.146103i
$469$	−1.23074e6	−0.258365
$470$	0	0
$471$	−397881.	−0.0826421
$472$	− 932754.i	− 0.192714i
$473$	835608.i	0.171732i
$474$	−1.34117e6	−0.274181
$475$	0	0
$476$	804070.	0.162658
$477$	39881.7i	0.00802561i
$478$	152603.i	0.0305488i
$479$	5.70951e6	1.13700	0.568500	−	0.822683i	$-0.307524\pi$
$479$	5.70951e6	1.13700	0.568500	+	0.822683i	$0.307524\pi$
$480$	0	0
$481$	2.25540e6	0.444489
$482$	− 4.51926e6i	− 0.886033i
$483$	− 1.47049e6i	− 0.286811i
$484$	2.29740e6	0.445782
$485$	0	0
$486$	−236196.	−0.0453609
$487$	− 2.94829e6i	− 0.563310i	−0.959516	−	0.281655i	$-0.909117\pi$
$487$	− 2.94829e6i	− 0.563310i	0.959516	−	0.281655i	$-0.0908834\pi$
$488$	1.28592e6i	0.244436i
$489$	−904623.	−0.171078
$490$	0	0
$491$	7.50888e6	1.40563	0.702816	−	0.711372i	$-0.251926\pi$
$491$	7.50888e6	1.40563	0.702816	+	0.711372i	$0.251926\pi$
$492$	− 2.22989e6i	− 0.415309i
$493$	2.80589e6i	0.519940i
$494$	−4.03789e6	−0.744453
$495$	0	0
$496$	670007.	0.122286
$497$	579406.i	0.105219i
$498$	173021.i	0.0312626i
$499$	1.72114e6	0.309433	0.154716	−	0.987959i	$-0.450554\pi$
$499$	1.72114e6	0.309433	0.154716	+	0.987959i	$0.450554\pi$
$500$	0	0
$501$	−1.11222e6	−0.197968
$502$	− 4.68161e6i	− 0.829155i
$503$	8.14115e6i	1.43471i	0.696705	+	0.717357i	$0.254648\pi$
$503$	8.14115e6i	1.43471i	−0.696705	+	0.717357i	$0.745352\pi$
$504$	−488168.	−0.0856039
$505$	0	0
$506$	−917158.	−0.159246
$507$	773710.i	0.133678i
$508$	3.02044e6i	0.519292i
$509$	2.85708e6	0.488796	0.244398	−	0.969675i	$-0.421410\pi$
$509$	2.85708e6	0.488796	0.244398	+	0.969675i	$0.421410\pi$
$510$	0	0
$511$	−2.22929e6	−0.377671
$512$	262144.i	0.0441942i
$513$	− 1.37769e6i	− 0.231131i
$514$	1.01006e6	0.168632
$515$	0	0
$516$	910537.	0.150547
$517$	1.91968e6i	0.315866i
$518$	1.59044e6i	0.260431i
$519$	−716989.	−0.116841
$520$	0	0
$521$	6.73914e6	1.08770	0.543852	−	0.839181i	$-0.316965\pi$
$521$	6.73914e6	1.08770	0.543852	+	0.839181i	$0.316965\pi$
$522$	− 1.70352e6i	− 0.273634i
$523$	7.13702e6i	1.14094i	0.821319	+	0.570470i	$0.193239\pi$
$523$	7.13702e6i	1.14094i	−0.821319	+	0.570470i	$0.806761\pi$
$524$	2.57120e6	0.409079
$525$	0	0
$526$	7.45720e6	1.17520
$527$	1.39672e6i	0.219070i
$528$	304474.i	0.0475298i
$529$	3.42589e6	0.532273
$530$	0	0
$531$	−1.18052e6	−0.181692
$532$	− 2.84740e6i	− 0.436184i
$533$	8.27163e6i	1.26117i
$534$	−1.68191e6	−0.255240
$535$	0	0
$536$	836454.	0.125756
$537$	− 4.12842e6i	− 0.617800i
$538$	− 5.75615e6i	− 0.857386i
$539$	1.04918e6	0.155554
$540$	0	0
$541$	4.97039e6	0.730125	0.365063	−	0.930983i	$-0.381048\pi$
$541$	4.97039e6	0.730125	0.365063	+	0.930983i	$0.381048\pi$
$542$	2.33157e6i	0.340918i
$543$	− 6.99003e6i	− 1.01737i
$544$	−546474.	−0.0791721
$545$	0	0
$546$	1.81083e6	0.259953
$547$	7.52689e6i	1.07559i	0.843075	+	0.537796i	$0.180743\pi$
$547$	7.52689e6i	1.07559i	−0.843075	+	0.537796i	$0.819257\pi$
$548$	− 4.07958e6i	− 0.580316i
$549$	1.62750e6	0.230457
$550$	0	0
$551$	9.93633e6	1.39427
$552$	999398.i	0.139602i
$553$	3.50820e6i	0.487834i
$554$	−2.34431e6	−0.324519
$555$	0	0
$556$	−3.03102e6	−0.415817
$557$	− 4.00163e6i	− 0.546511i	−0.961942	−	0.273255i	$-0.911900\pi$
$557$	− 4.00163e6i	− 0.546511i	0.961942	−	0.273255i	$-0.0881004\pi$
$558$	− 847978.i	− 0.115292i
$559$	−3.37757e6	−0.457168
$560$	0	0
$561$	−634716.	−0.0851476
$562$	3.06824e6i	0.409779i
$563$	4.39306e6i	0.584112i	0.956401	+	0.292056i	$0.0943393\pi$
$563$	4.39306e6i	0.584112i	−0.956401	+	0.292056i	$0.905661\pi$
$564$	2.09182e6	0.276902
$565$	0	0
$566$	1.59344e6	0.209072
$567$	617838.i	0.0807081i
$568$	− 393784.i	− 0.0512139i
$569$	−9.39680e6	−1.21674	−0.608372	−	0.793652i	$-0.708177\pi$
$569$	−9.39680e6	−1.21674	−0.608372	+	0.793652i	$0.708177\pi$
$570$	0	0
$571$	1.00548e7	1.29058	0.645288	−	0.763940i	$-0.276738\pi$
$571$	1.00548e7	1.29058	0.645288	+	0.763940i	$0.276738\pi$
$572$	− 1.12943e6i	− 0.144334i
$573$	− 6.42581e6i	− 0.817601i
$574$	−5.83292e6	−0.738935
$575$	0	0
$576$	331776.	0.0416667
$577$	4.96902e6i	0.621342i	0.950517	+	0.310671i	$0.100554\pi$
$577$	4.96902e6i	0.621342i	−0.950517	+	0.310671i	$0.899446\pi$
$578$	4.54023e6i	0.565273i
$579$	9.24534e6	1.14611
$580$	0	0
$581$	452586.	0.0556238
$582$	− 4.19630e6i	− 0.513522i
$583$	− 65066.4i	− 0.00792839i
$584$	1.51510e6	0.183827
$585$	0	0
$586$	−3.68022e6	−0.442721
$587$	1.09341e7i	1.30975i	0.755738	+	0.654874i	$0.227278\pi$
$587$	1.09341e7i	1.30975i	−0.755738	+	0.654874i	$0.772722\pi$
$588$	− 1.14326e6i	− 0.136365i
$589$	4.94611e6	0.587456
$590$	0	0
$591$	7.70914e6	0.907899
$592$	− 1.08092e6i	− 0.126762i
$593$	1.20154e7i	1.40314i	0.712599	+	0.701572i	$0.247518\pi$
$593$	1.20154e7i	1.40314i	−0.712599	+	0.701572i	$0.752482\pi$
$594$	385350.	0.0448115
$595$	0	0
$596$	−4.25173e6	−0.490287
$597$	− 1.12873e6i	− 0.129615i
$598$	− 3.70720e6i	− 0.423929i
$599$	3.94517e6	0.449260	0.224630	−	0.974444i	$-0.427883\pi$
$599$	3.94517e6	0.449260	0.224630	+	0.974444i	$0.427883\pi$
$600$	0	0
$601$	1.43093e7	1.61596	0.807982	−	0.589207i	$-0.200560\pi$
$601$	1.43093e7	1.61596	0.807982	+	0.589207i	$0.200560\pi$
$602$	− 2.38177e6i	− 0.267860i
$603$	− 1.05864e6i	− 0.118564i
$604$	−310109.	−0.0345878
$605$	0	0
$606$	3.17799e6	0.351537
$607$	4.72163e6i	0.520140i	0.965590	+	0.260070i	$0.0837457\pi$
$607$	4.72163e6i	0.520140i	−0.965590	+	0.260070i	$0.916254\pi$
$608$	1.93519e6i	0.212308i
$609$	−4.45604e6	−0.486862
$610$	0	0
$611$	−7.75945e6	−0.840868
$612$	691631.i	0.0746441i
$613$	− 1.48426e7i	− 1.59536i	−0.603083	−	0.797679i	$-0.706061\pi$
$613$	− 1.48426e7i	− 1.59536i	0.603083	−	0.797679i	$-0.293939\pi$
$614$	−648838.	−0.0694569
$615$	0	0
$616$	796439.	0.0845670
$617$	4.46412e6i	0.472088i	0.971742	+	0.236044i	$0.0758509\pi$
$617$	4.46412e6i	0.472088i	−0.971742	+	0.236044i	$0.924149\pi$
$618$	416677.i	0.0438862i
$619$	1.66157e7	1.74298	0.871490	−	0.490414i	$-0.163154\pi$
$619$	1.66157e7	1.74298	0.871490	+	0.490414i	$0.163154\pi$
$620$	0	0
$621$	1.26486e6	0.131618
$622$	− 1.09562e7i	− 1.13549i
$623$	4.39951e6i	0.454134i
$624$	−1.23070e6	−0.126529
$625$	0	0
$626$	9.84339e6	1.00394
$627$	2.24768e6i	0.228332i
$628$	− 707345.i	− 0.0715702i
$629$	2.25332e6	0.227089
$630$	0	0
$631$	1.63515e7	1.63488	0.817439	−	0.576015i	$-0.195393\pi$
$631$	1.63515e7	1.63488	0.817439	+	0.576015i	$0.195393\pi$
$632$	− 2.38430e6i	− 0.237447i
$633$	− 1.45427e6i	− 0.144256i
$634$	−3.65334e6	−0.360967
$635$	0	0
$636$	−70900.8	−0.00695038
$637$	4.24086e6i	0.414100i
$638$	2.77926e6i	0.270320i
$639$	−498383.	−0.0482849
$640$	0	0
$641$	6.00155e6	0.576924	0.288462	−	0.957491i	$-0.406856\pi$
$641$	6.00155e6	0.576924	0.288462	+	0.957491i	$0.406856\pi$
$642$	6.03972e6i	0.578334i
$643$	− 8.77891e6i	− 0.837362i	−0.908133	−	0.418681i	$-0.862493\pi$
$643$	− 8.77891e6i	− 0.837362i	0.908133	−	0.418681i	$-0.137507\pi$
$644$	2.61421e6	0.248385
$645$	0	0
$646$	−4.03416e6	−0.380340
$647$	2.17512e6i	0.204278i	0.994770	+	0.102139i	$0.0325686\pi$
$647$	2.17512e6i	0.204278i	−0.994770	+	0.102139i	$0.967431\pi$
$648$	− 419904.i	− 0.0392837i
$649$	1.92600e6	0.179491
$650$	0	0
$651$	−2.21813e6	−0.205132
$652$	− 1.60822e6i	− 0.148158i
$653$	− 1.09217e7i	− 1.00232i	−0.865355	−	0.501160i	$-0.832907\pi$
$653$	− 1.09217e7i	− 1.00232i	0.865355	−	0.501160i	$-0.167093\pi$
$654$	2.80060e6	0.256040
$655$	0	0
$656$	3.96425e6	0.359668
$657$	− 1.91755e6i	− 0.173314i
$658$	− 5.47174e6i	− 0.492675i
$659$	−2.04495e7	−1.83429	−0.917147	−	0.398550i	$-0.869513\pi$
$659$	−2.04495e7	−1.83429	−0.917147	+	0.398550i	$0.869513\pi$
$660$	0	0
$661$	1.26852e7	1.12926	0.564628	−	0.825346i	$-0.309020\pi$
$661$	1.26852e7	1.12926	0.564628	+	0.825346i	$0.309020\pi$
$662$	− 7.60952e6i	− 0.674858i
$663$	− 2.56556e6i	− 0.226672i
$664$	−307593.	−0.0270742
$665$	0	0
$666$	−1.36804e6	−0.119512
$667$	9.12258e6i	0.793968i
$668$	− 1.97727e6i	− 0.171445i
$669$	4.36135e6	0.376752
$670$	0	0
$671$	−2.65523e6	−0.227665
$672$	− 867855.i	− 0.0741351i
$673$	− 1.71634e7i	− 1.46072i	−0.683064	−	0.730359i	$-0.739353\pi$
$673$	− 1.71634e7i	− 1.46072i	0.683064	−	0.730359i	$-0.260647\pi$
$674$	−9.35915e6	−0.793573
$675$	0	0
$676$	−1.37548e6	−0.115768
$677$	− 1.48089e7i	− 1.24180i	−0.783889	−	0.620901i	$-0.786767\pi$
$677$	− 1.48089e7i	− 1.24180i	0.783889	−	0.620901i	$-0.213233\pi$
$678$	− 8.07943e6i	− 0.675005i
$679$	−1.09766e7	−0.913681
$680$	0	0
$681$	1.11513e6	0.0921421
$682$	1.38346e6i	0.113895i
$683$	9.07252e6i	0.744177i	0.928197	+	0.372089i	$0.121358\pi$
$683$	9.07252e6i	0.744177i	−0.928197	+	0.372089i	$0.878642\pi$
$684$	2.44923e6	0.200165
$685$	0	0
$686$	−9.32128e6	−0.756250
$687$	1.24173e7i	1.00377i
$688$	1.61873e6i	0.130378i
$689$	263002.	0.0211062
$690$	0	0
$691$	−2.18302e7	−1.73925	−0.869626	−	0.493711i	$-0.835640\pi$
$691$	−2.18302e7	−1.73925	−0.869626	+	0.493711i	$0.835640\pi$
$692$	− 1.27465e6i	− 0.101187i
$693$	− 1.00799e6i	− 0.0797305i
$694$	−1.44637e7	−1.13994
$695$	0	0
$696$	3.02848e6	0.236974
$697$	8.26400e6i	0.644330i
$698$	1.09338e7i	0.849437i
$699$	−2.60113e6	−0.201358
$700$	0	0
$701$	1.13810e6	0.0874755	0.0437378	−	0.999043i	$-0.486073\pi$
$701$	1.13810e6	0.0874755	0.0437378	+	0.999043i	$0.486073\pi$
$702$	1.55761e6i	0.119293i
$703$	− 7.97953e6i	− 0.608961i
$704$	−541287.	−0.0411620
$705$	0	0
$706$	−7.43635e6	−0.561498
$707$	− 8.31294e6i	− 0.625470i
$708$	− 2.09870e6i	− 0.157350i
$709$	−2.32785e7	−1.73916	−0.869579	−	0.493795i	$-0.835609\pi$
$709$	−2.32785e7	−1.73916	−0.869579	+	0.493795i	$0.835609\pi$
$710$	0	0
$711$	−3.01762e6	−0.223868
$712$	− 2.99006e6i	− 0.221044i
$713$	4.54104e6i	0.334527i
$714$	1.80916e6	0.132810
$715$	0	0
$716$	7.33941e6	0.535030
$717$	343358.i	0.0249430i
$718$	− 1.51800e6i	− 0.109891i
$719$	5.01088e6	0.361486	0.180743	−	0.983530i	$-0.442150\pi$
$719$	5.01088e6	0.361486	0.180743	+	0.983530i	$0.442150\pi$
$720$	0	0
$721$	1.08994e6	0.0780842
$722$	4.38154e6i	0.312812i
$723$	− 1.01683e7i	− 0.723443i
$724$	1.24267e7	0.881070
$725$	0	0
$726$	5.16914e6	0.363980
$727$	− 1.91868e7i	− 1.34638i	−0.739471	−	0.673189i	$-0.764924\pi$
$727$	− 1.91868e7i	− 1.34638i	0.739471	−	0.673189i	$-0.235076\pi$
$728$	3.21925e6i	0.225126i
$729$	−531441.	−0.0370370
$730$	0	0
$731$	−3.37446e6	−0.233567
$732$	2.89333e6i	0.199581i
$733$	2.82844e6i	0.194440i	0.995263	+	0.0972202i	$0.0309951\pi$
$733$	2.82844e6i	0.194440i	−0.995263	+	0.0972202i	$0.969005\pi$
$734$	−394372.	−0.0270188
$735$	0	0
$736$	−1.77671e6	−0.120899
$737$	1.72715e6i	0.117128i
$738$	− 5.01726e6i	− 0.339098i
$739$	6.55599e6	0.441598	0.220799	−	0.975319i	$-0.429134\pi$
$739$	6.55599e6	0.441598	0.220799	+	0.975319i	$0.429134\pi$
$740$	0	0
$741$	−9.08525e6	−0.607843
$742$	185461.i	0.0123664i
$743$	2.36773e7i	1.57347i	0.617289	+	0.786737i	$0.288231\pi$
$743$	2.36773e7i	1.57347i	−0.617289	+	0.786737i	$0.711769\pi$
$744$	1.50752e6	0.0998458
$745$	0	0
$746$	1.14232e7	0.751520
$747$	389297.i	0.0255258i
$748$	− 1.12838e6i	− 0.0737400i
$749$	1.57986e7	1.02900
$750$	0	0
$751$	1.44205e7	0.932999	0.466500	−	0.884521i	$-0.345515\pi$
$751$	1.44205e7	0.932999	0.466500	+	0.884521i	$0.345515\pi$
$752$	3.71878e6i	0.239804i
$753$	− 1.05336e7i	− 0.677002i
$754$	−1.12339e7	−0.719620
$755$	0	0
$756$	−1.09838e6	−0.0698953
$757$	2.25600e7i	1.43087i	0.698679	+	0.715435i	$0.253771\pi$
$757$	2.25600e7i	1.43087i	−0.698679	+	0.715435i	$0.746229\pi$
$758$	− 6.92227e6i	− 0.437598i
$759$	−2.06360e6	−0.130024
$760$	0	0
$761$	−7.43324e6	−0.465283	−0.232641	−	0.972563i	$-0.574737\pi$
$761$	−7.43324e6	−0.465283	−0.232641	+	0.972563i	$0.574737\pi$
$762$	6.79600e6i	0.424000i
$763$	− 7.32578e6i	− 0.455557i
$764$	1.14237e7	0.708063
$765$	0	0
$766$	−2.80184e6	−0.172533
$767$	7.78498e6i	0.477825i
$768$	589824.i	0.0360844i
$769$	−1.22108e7	−0.744611	−0.372306	−	0.928110i	$-0.621433\pi$
$769$	−1.22108e7	−0.744611	−0.372306	+	0.928110i	$0.621433\pi$
$770$	0	0
$771$	2.27263e6	0.137687
$772$	1.64362e7i	0.992560i
$773$	7.65633e6i	0.460863i	0.973089	+	0.230431i	$0.0740138\pi$
$773$	7.65633e6i	0.460863i	−0.973089	+	0.230431i	$0.925986\pi$
$774$	2.04871e6	0.122921
$775$	0	0
$776$	7.46009e6	0.444723
$777$	3.57849e6i	0.212641i
$778$	1.96715e6i	0.116517i
$779$	2.92648e7	1.72783
$780$	0	0
$781$	813105.	0.0477000
$782$	− 3.70378e6i	− 0.216585i
$783$	− 3.83291e6i	− 0.223421i
$784$	2.03247e6	0.118096
$785$	0	0
$786$	5.78519e6	0.334012
$787$	− 8.60423e6i	− 0.495194i	−0.968863	−	0.247597i	$-0.920359\pi$
$787$	− 8.60423e6i	− 0.495194i	0.968863	−	0.247597i	$-0.0796409\pi$
$788$	1.37051e7i	0.786263i
$789$	1.67787e7	0.959547
$790$	0	0
$791$	−2.11341e7	−1.20100
$792$	685067.i	0.0388079i
$793$	− 1.07326e7i	− 0.606069i
$794$	−1.45314e7	−0.818005
$795$	0	0
$796$	2.00663e6	0.112250
$797$	7.83737e6i	0.437044i	0.975832	+	0.218522i	$0.0701235\pi$
$797$	7.83737e6i	0.437044i	−0.975832	+	0.218522i	$0.929877\pi$
$798$	− 6.40666e6i	− 0.356143i
$799$	−7.75229e6	−0.429599
$800$	0	0
$801$	−3.78429e6	−0.208403
$802$	− 837771.i	− 0.0459928i
$803$	3.12845e6i	0.171214i
$804$	1.88202e6	0.102680
$805$	0	0
$806$	−5.59203e6	−0.303202
$807$	− 1.29513e7i	− 0.700053i
$808$	5.64976e6i	0.304440i
$809$	−2.13498e7	−1.14689	−0.573447	−	0.819243i	$-0.694394\pi$
$809$	−2.13498e7	−1.14689	−0.573447	+	0.819243i	$0.694394\pi$
$810$	0	0
$811$	−3.66547e7	−1.95694	−0.978470	−	0.206391i	$-0.933828\pi$
$811$	−3.66547e7	−1.95694	−0.978470	+	0.206391i	$0.933828\pi$
$812$	− 7.92184e6i	− 0.421634i
$813$	5.24602e6i	0.278358i
$814$	2.23193e6	0.118065
$815$	0	0
$816$	−1.22957e6	−0.0646437
$817$	1.19498e7i	0.626331i
$818$	− 1.00394e7i	− 0.524598i
$819$	4.07436e6	0.212251
$820$	0	0
$821$	1.26184e7	0.653352	0.326676	−	0.945136i	$-0.394071\pi$
$821$	1.26184e7	0.653352	0.326676	+	0.945136i	$0.394071\pi$
$822$	− 9.17906e6i	− 0.473826i
$823$	− 3.91395e6i	− 0.201426i	−0.994916	−	0.100713i	$-0.967888\pi$
$823$	− 3.91395e6i	− 0.201426i	0.994916	−	0.100713i	$-0.0321124\pi$
$824$	−740758.	−0.0380066
$825$	0	0
$826$	−5.48974e6	−0.279964
$827$	− 7.38912e6i	− 0.375689i	−0.982199	−	0.187845i	$-0.939850\pi$
$827$	− 7.38912e6i	− 0.375689i	0.982199	−	0.187845i	$-0.0601501\pi$
$828$	2.24865e6i	0.113984i
$829$	−885494.	−0.0447506	−0.0223753	−	0.999750i	$-0.507123\pi$
$829$	−885494.	−0.0447506	−0.0223753	+	0.999750i	$0.507123\pi$
$830$	0	0
$831$	−5.27469e6	−0.264968
$832$	− 2.18791e6i	− 0.109578i
$833$	4.23695e6i	0.211564i
$834$	−6.81980e6	−0.339513
$835$	0	0
$836$	−3.99588e6	−0.197741
$837$	− 1.90795e6i	− 0.0941355i
$838$	− 4.44678e6i	− 0.218744i
$839$	1.76812e7	0.867175	0.433588	−	0.901111i	$-0.357247\pi$
$839$	1.76812e7	0.867175	0.433588	+	0.901111i	$0.357247\pi$
$840$	0	0
$841$	7.13299e6	0.347762
$842$	1.65306e7i	0.803544i
$843$	6.90355e6i	0.334583i
$844$	2.58536e6	0.124930
$845$	0	0
$846$	4.70658e6	0.226089
$847$	− 1.35214e7i	− 0.647608i
$848$	− 126046.i	− 0.00601920i
$849$	3.58525e6	0.170706
$850$	0	0
$851$	7.32604e6	0.346773
$852$	− 886015.i	− 0.0418159i
$853$	− 2.56521e7i	− 1.20712i	−0.797319	−	0.603559i	$-0.793749\pi$
$853$	− 2.56521e7i	− 1.20712i	0.797319	−	0.603559i	$-0.206251\pi$
$854$	7.56832e6	0.355103
$855$	0	0
$856$	−1.07373e7	−0.500852
$857$	− 1.45961e6i	− 0.0678866i	−0.999424	−	0.0339433i	$-0.989193\pi$
$857$	− 1.45961e6i	− 0.0678866i	0.999424	−	0.0339433i	$-0.0108066\pi$
$858$	− 2.54121e6i	− 0.117848i
$859$	−3.05323e7	−1.41181	−0.705905	−	0.708307i	$-0.749459\pi$
$859$	−3.05323e7	−1.41181	−0.705905	+	0.708307i	$0.749459\pi$
$860$	0	0
$861$	−1.31241e7	−0.603338
$862$	− 698154.i	− 0.0320024i
$863$	7.36238e6i	0.336505i	0.985744	+	0.168252i	$0.0538123\pi$
$863$	7.36238e6i	0.336505i	−0.985744	+	0.168252i	$0.946188\pi$
$864$	746496.	0.0340207
$865$	0	0
$866$	5.01036e6	0.227025
$867$	1.02155e7i	0.461544i
$868$	− 3.94334e6i	− 0.177650i
$869$	4.92321e6	0.221156
$870$	0	0
$871$	−6.98123e6	−0.311808
$872$	4.97885e6i	0.221737i
$873$	− 9.44168e6i	− 0.419289i
$874$	−1.31160e7	−0.580794
$875$	0	0
$876$	3.40897e6	0.150094
$877$	− 2.22205e7i	− 0.975561i	−0.872966	−	0.487781i	$-0.837807\pi$
$877$	− 2.22205e7i	− 0.975561i	0.872966	−	0.487781i	$-0.162193\pi$
$878$	5.41625e6i	0.237117i
$879$	−8.28050e6	−0.361480
$880$	0	0
$881$	−1.26864e7	−0.550679	−0.275340	−	0.961347i	$-0.588790\pi$
$881$	−1.26864e7	−0.550679	−0.275340	+	0.961347i	$0.588790\pi$
$882$	− 2.57234e6i	− 0.111342i
$883$	− 2.90790e7i	− 1.25510i	−0.778578	−	0.627548i	$-0.784059\pi$
$883$	− 2.90790e7i	− 1.25510i	0.778578	−	0.627548i	$-0.215941\pi$
$884$	4.56099e6	0.196304
$885$	0	0
$886$	−2.39792e7	−1.02624
$887$	− 1.20570e7i	− 0.514555i	−0.966338	−	0.257278i	$-0.917175\pi$
$887$	− 1.20570e7i	− 0.514555i	0.966338	−	0.257278i	$-0.0828255\pi$
$888$	− 2.43207e6i	− 0.103501i
$889$	1.77769e7	0.754399
$890$	0	0
$891$	867038.	0.0365884
$892$	7.75352e6i	0.326277i
$893$	2.74527e7i	1.15201i
$894$	−9.56640e6	−0.400318
$895$	0	0
$896$	1.54285e6	0.0642029
$897$	− 8.34120e6i	− 0.346137i
$898$	− 5.87097e6i	− 0.242951i
$899$	1.37607e7	0.567861
$900$	0	0
$901$	262759.	0.0107832
$902$	8.18557e6i	0.334991i
$903$	− 5.35898e6i	− 0.218707i
$904$	1.43634e7	0.584571
$905$	0	0
$906$	−697746.	−0.0282408
$907$	− 9.81240e6i	− 0.396056i	−0.980196	−	0.198028i	$-0.936546\pi$
$907$	− 9.81240e6i	− 0.396056i	0.980196	−	0.198028i	$-0.0634538\pi$
$908$	1.98246e6i	0.0797974i
$909$	7.15048e6	0.287029
$910$	0	0
$911$	−2.61374e7	−1.04344	−0.521720	−	0.853117i	$-0.674709\pi$
$911$	−2.61374e7	−1.04344	−0.521720	+	0.853117i	$0.674709\pi$
$912$	4.35418e6i	0.173348i
$913$	− 635132.i	− 0.0252166i
$914$	1.85013e7	0.732547
$915$	0	0
$916$	−2.20752e7	−0.869293
$917$	− 1.51328e7i	− 0.594288i
$918$	1.55617e6i	0.0609467i
$919$	−2.32169e7	−0.906808	−0.453404	−	0.891305i	$-0.649791\pi$
$919$	−2.32169e7	−0.906808	−0.453404	+	0.891305i	$0.649791\pi$
$920$	0	0
$921$	−1.45988e6	−0.0567113
$922$	− 1.57076e6i	− 0.0608532i
$923$	3.28661e6i	0.126983i
$924$	1.79199e6	0.0690486
$925$	0	0
$926$	1.64641e7	0.630971
$927$	937522.i	0.0358329i
$928$	5.38396e6i	0.205226i
$929$	3.44336e7	1.30901	0.654506	−	0.756057i	$-0.272877\pi$
$929$	3.44336e7	1.30901	0.654506	+	0.756057i	$0.272877\pi$
$930$	0	0
$931$	1.50040e7	0.567328
$932$	− 4.62424e6i	− 0.174381i
$933$	− 2.46514e7i	− 0.927125i
$934$	−6.03022e6	−0.226186
$935$	0	0
$936$	−2.76908e6	−0.103311
$937$	− 4.03500e7i	− 1.50139i	−0.660647	−	0.750697i	$-0.729718\pi$
$937$	− 4.03500e7i	− 1.50139i	0.660647	−	0.750697i	$-0.270282\pi$
$938$	− 4.92296e6i	− 0.182692i
$939$	2.21476e7	0.819716
$940$	0	0
$941$	−2.34697e7	−0.864040	−0.432020	−	0.901864i	$-0.642199\pi$
$941$	−2.34697e7	−0.864040	−0.432020	+	0.901864i	$0.642199\pi$
$942$	− 1.59153e6i	− 0.0584368i
$943$	2.68681e7i	0.983916i
$944$	3.73102e6	0.136269
$945$	0	0
$946$	−3.34243e6	−0.121433
$947$	− 3.95011e7i	− 1.43131i	−0.698454	−	0.715655i	$-0.746128\pi$
$947$	− 3.95011e7i	− 1.43131i	0.698454	−	0.715655i	$-0.253872\pi$
$948$	− 5.36466e6i	− 0.193875i
$949$	−1.26454e7	−0.455791
$950$	0	0
$951$	−8.22002e6	−0.294728
$952$	3.21628e6i	0.115017i
$953$	− 1.16553e7i	− 0.415710i	−0.978160	−	0.207855i	$-0.933352\pi$
$953$	− 1.16553e7i	− 0.415710i	0.978160	−	0.207855i	$-0.0666482\pi$
$954$	−159527.	−0.00567496
$955$	0	0
$956$	−610414.	−0.0216013
$957$	6.25334e6i	0.220715i
$958$	2.28381e7i	0.803980i
$959$	−2.40105e7	−0.843051
$960$	0	0
$961$	−2.17793e7	−0.760740
$962$	9.02159e6i	0.314301i
$963$	1.35894e7i	0.472208i
$964$	1.80770e7	0.626520
$965$	0	0
$966$	5.88198e6	0.202806
$967$	− 2.23640e7i	− 0.769101i	−0.923104	−	0.384550i	$-0.874356\pi$
$967$	− 2.23640e7i	− 0.769101i	0.923104	−	0.384550i	$-0.125644\pi$
$968$	9.18959e6i	0.315216i
$969$	−9.07687e6	−0.310547
$970$	0	0
$971$	−4.24936e7	−1.44636	−0.723178	−	0.690662i	$-0.757319\pi$
$971$	−4.24936e7	−1.44636	−0.723178	+	0.690662i	$0.757319\pi$
$972$	− 944784.i	− 0.0320750i
$973$	1.78391e7i	0.604076i
$974$	1.17932e7	0.398321
$975$	0	0
$976$	−5.14369e6	−0.172842
$977$	9.08345e6i	0.304449i	0.988346	+	0.152225i	$0.0486437\pi$
$977$	9.08345e6i	0.304449i	−0.988346	+	0.152225i	$0.951356\pi$
$978$	− 3.61849e6i	− 0.120971i
$979$	6.17401e6	0.205878
$980$	0	0
$981$	6.30136e6	0.209056
$982$	3.00355e7i	0.993931i
$983$	4.64721e7i	1.53394i	0.641683	+	0.766970i	$0.278237\pi$
$983$	4.64721e7i	1.53394i	−0.641683	+	0.766970i	$0.721763\pi$
$984$	8.91956e6	0.293668
$985$	0	0
$986$	−1.12236e7	−0.367653
$987$	− 1.23114e7i	− 0.402268i
$988$	− 1.61516e7i	− 0.526407i
$989$	−1.09711e7	−0.356665
$990$	0	0
$991$	1.58030e7	0.511159	0.255580	−	0.966788i	$-0.417734\pi$
$991$	1.58030e7	0.511159	0.255580	+	0.966788i	$0.417734\pi$
$992$	2.68003e6i	0.0864690i
$993$	− 1.71214e7i	− 0.551019i
$994$	−2.31762e6	−0.0744007
$995$	0	0
$996$	−692084.	−0.0221060
$997$	2.45497e7i	0.782181i	0.920352	+	0.391091i	$0.127902\pi$
$997$	2.45497e7i	0.782181i	−0.920352	+	0.391091i	$0.872098\pi$
$998$	6.88458e6i	0.218802i
$999$	−3.07809e6	−0.0975814

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	750.6.c.a.499.8		8
5.2	odd	4		750.6.a.d.1.1	✓	4
5.3	odd	4		750.6.a.e.1.4	yes	4
5.4	even	2	inner	750.6.c.a.499.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
750.6.a.d.1.1	✓	4	5.2	odd	4
750.6.a.e.1.4	yes	4	5.3	odd	4
750.6.c.a.499.1		8	5.4	even	2	inner
750.6.c.a.499.8		8	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 750 = 2 \cdot 3 \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	750.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.00000i q^{2} +9.00000i q^{3} -16.0000 q^{4} -36.0000 q^{6} +94.1683i q^{7} -64.0000i q^{8} -81.0000 q^{9} +O(q^{10})\)	\(q+4.00000i q^{2} +9.00000i q^{3} -16.0000 q^{4} -36.0000 q^{6} +94.1683i q^{7} -64.0000i q^{8} -81.0000 q^{9} +132.150 q^{11} -144.000i q^{12} +534.158i q^{13} -376.673 q^{14} +256.000 q^{16} +533.666i q^{17} -324.000i q^{18} +1889.84 q^{19} -847.515 q^{21} +528.601i q^{22} +1735.07i q^{23} +576.000 q^{24} -2136.63 q^{26} -729.000i q^{27} -1506.69i q^{28} +5257.77 q^{29} +2617.22 q^{31} +1024.00i q^{32} +1189.35i q^{33} -2134.66 q^{34} +1296.00 q^{36} -4222.34i q^{37} +7559.35i q^{38} -4807.43 q^{39} +15485.4 q^{41} -3390.06i q^{42} +6323.17i q^{43} -2114.40 q^{44} -6940.26 q^{46} +14526.5i q^{47} +2304.00i q^{48} +7939.33 q^{49} -4802.99 q^{51} -8546.53i q^{52} -492.367i q^{53} +2916.00 q^{54} +6026.77 q^{56} +17008.5i q^{57} +21031.1i q^{58} +14574.3 q^{59} -20092.5 q^{61} +10468.9i q^{62} -7627.63i q^{63} -4096.00 q^{64} -4757.41 q^{66} +13069.6i q^{67} -8538.65i q^{68} -15615.6 q^{69} +6152.88 q^{71} +5184.00i q^{72} +23673.4i q^{73} +16889.4 q^{74} -30237.4 q^{76} +12444.4i q^{77} -19229.7i q^{78} +37254.6 q^{79} +6561.00 q^{81} +61941.4i q^{82} -4806.14i q^{83} +13560.2 q^{84} -25292.7 q^{86} +47319.9i q^{87} -8457.62i q^{88} +46719.6 q^{89} -50300.8 q^{91} -27761.1i q^{92} +23554.9i q^{93} -58106.0 q^{94} -9216.00 q^{96} +116564. i q^{97} +31757.3i q^{98} -10704.2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 128 q^{4} - 288 q^{6} - 648 q^{9}+O(q^{10}) \) 8 * q - 128 * q^4 - 288 * q^6 - 648 * q^9	\( 8 q - 128 q^{4} - 288 q^{6} - 648 q^{9} - 520 q^{11} + 688 q^{14} + 2048 q^{16} + 2672 q^{19} + 1548 q^{21} + 4608 q^{24} - 1216 q^{26} - 7072 q^{29} - 26636 q^{31} - 8592 q^{34} + 10368 q^{36} - 2736 q^{39} + 21172 q^{41} + 8320 q^{44} - 7680 q^{46} + 75868 q^{49} - 19332 q^{51} + 23328 q^{54} - 11008 q^{56} + 31180 q^{59} - 76620 q^{61} - 32768 q^{64} + 18720 q^{66} - 17280 q^{69} - 164820 q^{71} - 114944 q^{74} - 42752 q^{76} + 342456 q^{79} + 52488 q^{81} - 24768 q^{84} + 52432 q^{86} + 356856 q^{89} - 340324 q^{91} + 118720 q^{94} - 73728 q^{96} + 42120 q^{99}+O(q^{100}) \) 8 * q - 128 * q^4 - 288 * q^6 - 648 * q^9 - 520 * q^11 + 688 * q^14 + 2048 * q^16 + 2672 * q^19 + 1548 * q^21 + 4608 * q^24 - 1216 * q^26 - 7072 * q^29 - 26636 * q^31 - 8592 * q^34 + 10368 * q^36 - 2736 * q^39 + 21172 * q^41 + 8320 * q^44 - 7680 * q^46 + 75868 * q^49 - 19332 * q^51 + 23328 * q^54 - 11008 * q^56 + 31180 * q^59 - 76620 * q^61 - 32768 * q^64 + 18720 * q^66 - 17280 * q^69 - 164820 * q^71 - 114944 * q^74 - 42752 * q^76 + 342456 * q^79 + 52488 * q^81 - 24768 * q^84 + 52432 * q^86 + 356856 * q^89 - 340324 * q^91 + 118720 * q^94 - 73728 * q^96 + 42120 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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