LMFDB - Embedded newform 325.6.b.g.274.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [325,6,Mod(274,325)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(325, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("325.274");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 325 = 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	325.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$52.1247414392$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 326x^{10} + 38809x^{8} + 2034064x^{6} + 43897824x^{4} + 281822976x^{2} + 377913600 $ x^12 + 326x^10 + 38809x^8 + 2034064x^6 + 43897824x^4 + 281822976*x^2 + 377913600
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{14}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 65)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			274.4
Root			$-5.93318i$ of defining polynomial
Character	$\chi$	$=$	325.274
Dual form			325.6.b.g.274.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-5.93318i q^{2} +7.05430i q^{3} -3.20258 q^{4} +41.8544 q^{6} -185.746i q^{7} -170.860i q^{8} +193.237 q^{9} -353.912 q^{11} -22.5920i q^{12} -169.000i q^{13} -1102.06 q^{14} -1116.23 q^{16} +634.652i q^{17} -1146.51i q^{18} -1118.29 q^{19} +1310.31 q^{21} +2099.83i q^{22} -3509.85i q^{23} +1205.30 q^{24} -1002.71 q^{26} +3077.35i q^{27} +594.867i q^{28} +3765.67 q^{29} +2906.63 q^{31} +1155.24i q^{32} -2496.60i q^{33} +3765.50 q^{34} -618.857 q^{36} +283.305i q^{37} +6635.04i q^{38} +1192.18 q^{39} -13563.6 q^{41} -7774.29i q^{42} +5184.47i q^{43} +1133.43 q^{44} -20824.6 q^{46} -6781.50i q^{47} -7874.19i q^{48} -17694.6 q^{49} -4477.03 q^{51} +541.237i q^{52} -7664.43i q^{53} +18258.4 q^{54} -31736.6 q^{56} -7888.78i q^{57} -22342.4i q^{58} -2806.29 q^{59} -13764.7 q^{61} -17245.5i q^{62} -35893.0i q^{63} -28865.0 q^{64} -14812.8 q^{66} +67744.1i q^{67} -2032.53i q^{68} +24759.5 q^{69} -66519.0 q^{71} -33016.5i q^{72} -75902.7i q^{73} +1680.90 q^{74} +3581.43 q^{76} +65737.9i q^{77} -7073.39i q^{78} -101641. q^{79} +25248.0 q^{81} +80475.1i q^{82} +50882.7i q^{83} -4196.37 q^{84} +30760.4 q^{86} +26564.2i q^{87} +60469.5i q^{88} +52439.2 q^{89} -31391.1 q^{91} +11240.6i q^{92} +20504.2i q^{93} -40235.8 q^{94} -8149.42 q^{96} -142557. i q^{97} +104985. i q^{98} -68388.9 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 268 q^{4} + 636 q^{6} - 1036 q^{9} - 340 q^{11} + 2880 q^{14} + 7012 q^{16} - 2436 q^{19} - 792 q^{21} - 25236 q^{24} - 16728 q^{29} + 5724 q^{31} + 42968 q^{34} - 4276 q^{36} - 12844 q^{39} + 4496 q^{41}+ \cdots + 64540 q^{99}+O(q^{100}) $ 12 * q - 268 * q^4 + 636 * q^6 - 1036 * q^9 - 340 * q^11 + 2880 * q^14 + 7012 * q^16 - 2436 * q^19 - 792 * q^21 - 25236 * q^24 - 16728 * q^29 + 5724 * q^31 + 42968 * q^34 - 4276 * q^36 - 12844 * q^39 + 4496 * q^41 + 146964 * q^44 - 79772 * q^46 - 171804 * q^49 - 110056 * q^51 - 91008 * q^54 - 377600 * q^56 + 57748 * q^59 + 1944 * q^61 - 28044 * q^64 - 316080 * q^66 - 306176 * q^69 - 148348 * q^71 - 452224 * q^74 - 186844 * q^76 - 429016 * q^79 + 232012 * q^81 + 470536 * q^84 - 609332 * q^86 - 188056 * q^89 + 74360 * q^91 + 251840 * q^94 - 771716 * q^96 + 64540 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/325\mathbb{Z}\right)^\times$.

$n$	$27$	$301$
$\chi(n)$	$-1$	$1$

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 $

$2$	− 5.93318i	− 1.04885i	−0.851457	−	0.524424i	$-0.824281\pi$
$2$	− 5.93318i	− 1.04885i	0.851457	−	0.524424i	$-0.175719\pi$
$3$	7.05430i	0.452534i	0.974065	+	0.226267i	$0.0726521\pi$
$3$	7.05430i	0.452534i	−0.974065	+	0.226267i	$0.927348\pi$
$4$	−3.20258	−0.100081
$5$	0	0
$5$	0	0
$6$	41.8544	0.474639
$7$	− 185.746i	− 1.43276i	−0.697708	−	0.716382i	$-0.745797\pi$
$7$	− 185.746i	− 1.43276i	0.697708	−	0.716382i	$-0.254203\pi$
$8$	− 170.860i	− 0.943878i
$9$	193.237	0.795213
$10$	0	0
$11$	−353.912	−0.881889	−0.440945	−	0.897534i	$-0.645357\pi$
$11$	−353.912	−0.881889	−0.440945	+	0.897534i	$0.645357\pi$
$12$	− 22.5920i	− 0.0452899i
$13$	− 169.000i	− 0.277350i
$13$	− 169.000i	− 0.277350i
$14$	−1102.06	−1.50275
$15$	0	0
$16$	−1116.23	−1.09006
$17$	634.652i	0.532615i	0.963888	+	0.266308i	$0.0858037\pi$
$17$	634.652i	0.532615i	−0.963888	+	0.266308i	$0.914196\pi$
$18$	− 1146.51i	− 0.834057i
$19$	−1118.29	−0.710677	−0.355338	−	0.934738i	$-0.615634\pi$
$19$	−1118.29	−0.710677	−0.355338	+	0.934738i	$0.615634\pi$
$20$	0	0
$21$	1310.31	0.648374
$22$	2099.83i	0.924967i
$23$	− 3509.85i	− 1.38347i	−0.722153	−	0.691734i	$-0.756847\pi$
$23$	− 3509.85i	− 1.38347i	0.722153	−	0.691734i	$-0.243153\pi$
$24$	1205.30	0.427136
$25$	0	0
$26$	−1002.71	−0.290898
$27$	3077.35i	0.812394i
$28$	594.867i	0.143392i
$29$	3765.67	0.831471	0.415735	−	0.909486i	$-0.363524\pi$
$29$	3765.67	0.831471	0.415735	+	0.909486i	$0.363524\pi$
$30$	0	0
$31$	2906.63	0.543232	0.271616	−	0.962406i	$-0.412442\pi$
$31$	2906.63	0.543232	0.271616	+	0.962406i	$0.412442\pi$
$32$	1155.24i	0.199433i
$33$	− 2496.60i	− 0.399085i
$34$	3765.50	0.558632
$35$	0	0
$36$	−618.857	−0.0795855
$37$	283.305i	0.0340213i	0.999855	+	0.0170106i	$0.00541491\pi$
$37$	283.305i	0.0340213i	−0.999855	+	0.0170106i	$0.994585\pi$
$38$	6635.04i	0.745391i
$39$	1192.18	0.125510
$40$	0	0
$41$	−13563.6	−1.26013	−0.630064	−	0.776544i	$-0.716971\pi$
$41$	−13563.6	−1.26013	−0.630064	+	0.776544i	$0.716971\pi$
$42$	− 7774.29i	− 0.680045i
$43$	5184.47i	0.427596i	0.976878	+	0.213798i	$0.0685834\pi$
$43$	5184.47i	0.427596i	−0.976878	+	0.213798i	$0.931417\pi$
$44$	1133.43	0.0882601
$45$	0	0
$46$	−20824.6	−1.45105
$47$	− 6781.50i	− 0.447797i	−0.974612	−	0.223898i	$-0.928122\pi$
$47$	− 6781.50i	− 0.447797i	0.974612	−	0.223898i	$-0.0718784\pi$
$48$	− 7874.19i	− 0.493291i
$49$	−17694.6	−1.05281
$50$	0	0
$51$	−4477.03	−0.241026
$52$	541.237i	0.0277574i
$53$	− 7664.43i	− 0.374792i	−0.982284	−	0.187396i	$-0.939995\pi$
$53$	− 7664.43i	− 0.374792i	0.982284	−	0.187396i	$-0.0600048\pi$
$54$	18258.4	0.852078
$55$	0	0
$56$	−31736.6	−1.35235
$57$	− 7888.78i	− 0.321605i
$58$	− 22342.4i	− 0.872086i
$59$	−2806.29	−0.104955	−0.0524773	−	0.998622i	$-0.516712\pi$
$59$	−2806.29	−0.104955	−0.0524773	+	0.998622i	$0.516712\pi$
$60$	0	0
$61$	−13764.7	−0.473634	−0.236817	−	0.971554i	$-0.576104\pi$
$61$	−13764.7	−0.473634	−0.236817	+	0.971554i	$0.576104\pi$
$62$	− 17245.5i	− 0.569768i
$63$	− 35893.0i	− 1.13935i
$64$	−28865.0	−0.880889
$65$	0	0
$66$	−14812.8	−0.418579
$67$	67744.1i	1.84368i	0.387576	+	0.921838i	$0.373313\pi$
$67$	67744.1i	1.84368i	−0.387576	+	0.921838i	$0.626687\pi$
$68$	− 2032.53i	− 0.0533045i
$69$	24759.5	0.626065
$70$	0	0
$71$	−66519.0	−1.56603	−0.783014	−	0.622004i	$-0.786319\pi$
$71$	−66519.0	−1.56603	−0.783014	+	0.622004i	$0.786319\pi$
$72$	− 33016.5i	− 0.750584i
$73$	− 75902.7i	− 1.66706i	−0.552478	−	0.833528i	$-0.686318\pi$
$73$	− 75902.7i	− 1.66706i	0.552478	−	0.833528i	$-0.313682\pi$
$74$	1680.90	0.0356831
$75$	0	0
$76$	3581.43	0.0711250
$77$	65737.9i	1.26354i
$78$	− 7073.39i	− 0.131641i
$79$	−101641.	−1.83233	−0.916163	−	0.400806i	$-0.868730\pi$
$79$	−101641.	−1.83233	−0.916163	+	0.400806i	$0.868730\pi$
$80$	0	0
$81$	25248.0	0.427578
$82$	80475.1i	1.32168i
$83$	50882.7i	0.810727i	0.914156	+	0.405363i	$0.132855\pi$
$83$	50882.7i	0.810727i	−0.914156	+	0.405363i	$0.867145\pi$
$84$	−4196.37	−0.0648897
$85$	0	0
$86$	30760.4	0.448483
$87$	26564.2i	0.376269i
$88$	60469.5i	0.832396i
$89$	52439.2	0.701748	0.350874	−	0.936423i	$-0.385885\pi$
$89$	52439.2	0.701748	0.350874	+	0.936423i	$0.385885\pi$
$90$	0	0
$91$	−31391.1	−0.397377
$92$	11240.6i	0.138458i
$93$	20504.2i	0.245831i
$94$	−40235.8	−0.469671
$95$	0	0
$96$	−8149.42	−0.0902503
$97$	− 142557.i	− 1.53837i	−0.639028	−	0.769183i	$-0.720663\pi$
$97$	− 142557.i	− 1.53837i	0.639028	−	0.769183i	$-0.279337\pi$
$98$	104985.i	1.10424i
$99$	−68388.9	−0.701290
$100$	0	0
$101$	4751.74	0.0463499	0.0231750	−	0.999731i	$-0.492623\pi$
$101$	4751.74	0.0463499	0.0231750	+	0.999731i	$0.492623\pi$
$102$	26563.0i	0.252800i
$103$	59290.6i	0.550672i	0.961348	+	0.275336i	$0.0887890\pi$
$103$	59290.6i	0.550672i	−0.961348	+	0.275336i	$0.911211\pi$
$104$	−28875.4	−0.261785
$105$	0	0
$106$	−45474.4	−0.393100
$107$	157927.i	1.33351i	0.745276	+	0.666756i	$0.232318\pi$
$107$	157927.i	1.33351i	−0.745276	+	0.666756i	$0.767682\pi$
$108$	− 9855.46i	− 0.0813050i
$109$	−58878.4	−0.474668	−0.237334	−	0.971428i	$-0.576274\pi$
$109$	−58878.4	−0.474668	−0.237334	+	0.971428i	$0.576274\pi$
$110$	0	0
$111$	−1998.52	−0.0153958
$112$	207335.i	1.56180i
$113$	− 179734.i	− 1.32414i	−0.749443	−	0.662069i	$-0.769678\pi$
$113$	− 179734.i	− 1.32414i	0.749443	−	0.662069i	$-0.230322\pi$
$114$	−46805.5	−0.337315
$115$	0	0
$116$	−12059.9	−0.0832142
$117$	− 32657.0i	− 0.220553i
$118$	16650.2i	0.110081i
$119$	117884.	0.763112
$120$	0	0
$121$	−35797.0	−0.222271
$122$	81668.5i	0.496770i
$123$	− 95681.5i	− 0.570250i
$124$	−9308.72	−0.0543671
$125$	0	0
$126$	−212959.	−1.19501
$127$	− 123741.i	− 0.680774i	−0.940286	−	0.340387i	$-0.889442\pi$
$127$	− 123741.i	− 0.680774i	0.940286	−	0.340387i	$-0.110558\pi$
$128$	208229.i	1.12335i
$129$	−36572.8	−0.193501
$130$	0	0
$131$	−43205.0	−0.219966	−0.109983	−	0.993933i	$-0.535080\pi$
$131$	−43205.0	−0.219966	−0.109983	+	0.993933i	$0.535080\pi$
$132$	7995.58i	0.0399407i
$133$	207719.i	1.01823i
$134$	401938.	1.93373
$135$	0	0
$136$	108437.	0.502724
$137$	− 188517.i	− 0.858120i	−0.903276	−	0.429060i	$-0.858845\pi$
$137$	− 188517.i	− 0.858120i	0.903276	−	0.429060i	$-0.141155\pi$
$138$	− 146903.i	− 0.656647i
$139$	−344148.	−1.51081	−0.755403	−	0.655260i	$-0.772559\pi$
$139$	−344148.	−1.51081	−0.755403	+	0.655260i	$0.772559\pi$
$140$	0	0
$141$	47838.7	0.202643
$142$	394669.i	1.64253i
$143$	59811.2i	0.244592i
$144$	−215696.	−0.866834
$145$	0	0
$146$	−450344.	−1.74849
$147$	− 124823.i	− 0.476433i
$148$	− 907.309i	− 0.00340487i
$149$	177809.	0.656126	0.328063	−	0.944656i	$-0.393604\pi$
$149$	177809.	0.656126	0.328063	+	0.944656i	$0.393604\pi$
$150$	0	0
$151$	554784.	1.98008	0.990038	−	0.140803i	$-0.0449685\pi$
$151$	554784.	1.98008	0.990038	+	0.140803i	$0.0449685\pi$
$152$	191072.i	0.670792i
$153$	122638.i	0.423543i
$154$	390034.	1.32526
$155$	0	0
$156$	−3818.05	−0.0125612
$157$	255896.i	0.828542i	0.910154	+	0.414271i	$0.135963\pi$
$157$	255896.i	0.828542i	−0.910154	+	0.414271i	$0.864037\pi$
$158$	603056.i	1.92183i
$159$	54067.2	0.169606
$160$	0	0
$161$	−651941.	−1.98218
$162$	− 149801.i	− 0.448464i
$163$	262686.i	0.774404i	0.921995	+	0.387202i	$0.126558\pi$
$163$	262686.i	0.774404i	−0.921995	+	0.387202i	$0.873442\pi$
$164$	43438.5	0.126114
$165$	0	0
$166$	301896.	0.850329
$167$	287069.i	0.796517i	0.917273	+	0.398259i	$0.130385\pi$
$167$	287069.i	0.796517i	−0.917273	+	0.398259i	$0.869615\pi$
$168$	− 223880.i	− 0.611986i
$169$	−28561.0	−0.0769231
$170$	0	0
$171$	−216096.	−0.565140
$172$	− 16603.7i	− 0.0427941i
$173$	− 719663.i	− 1.82816i	−0.405534	−	0.914080i	$-0.632915\pi$
$173$	− 719663.i	− 1.82816i	0.405534	−	0.914080i	$-0.367085\pi$
$174$	157610.	0.394648
$175$	0	0
$176$	395046.	0.961316
$177$	− 19796.4i	− 0.0474955i
$178$	− 311131.i	− 0.736026i
$179$	−779772.	−1.81901	−0.909505	−	0.415693i	$-0.863539\pi$
$179$	−779772.	−1.81901	−0.909505	+	0.415693i	$0.863539\pi$
$180$	0	0
$181$	336065.	0.762478	0.381239	−	0.924477i	$-0.375498\pi$
$181$	336065.	0.762478	0.381239	+	0.924477i	$0.375498\pi$
$182$	186249.i	0.416788i
$183$	− 97100.5i	− 0.214335i
$184$	−599693.	−1.30582
$185$	0	0
$186$	121655.	0.257839
$187$	− 224611.i	− 0.469708i
$188$	21718.3i	0.0448158i
$189$	571605.	1.16397
$190$	0	0
$191$	409479.	0.812173	0.406086	−	0.913835i	$-0.366893\pi$
$191$	409479.	0.812173	0.406086	+	0.913835i	$0.366893\pi$
$192$	− 203622.i	− 0.398632i
$193$	− 339565.i	− 0.656189i	−0.944645	−	0.328095i	$-0.893593\pi$
$193$	− 339565.i	− 0.656189i	0.944645	−	0.328095i	$-0.106407\pi$
$194$	−845817.	−1.61351
$195$	0	0
$196$	56668.4	0.105366
$197$	− 871469.i	− 1.59988i	−0.600082	−	0.799938i	$-0.704865\pi$
$197$	− 871469.i	− 1.59988i	0.600082	−	0.799938i	$-0.295135\pi$
$198$	405764.i	0.735546i
$199$	−270952.	−0.485019	−0.242510	−	0.970149i	$-0.577971\pi$
$199$	−270952.	−0.485019	−0.242510	+	0.970149i	$0.577971\pi$
$200$	0	0
$201$	−477887.	−0.834325
$202$	− 28192.9i	− 0.0486140i
$203$	− 699458.i	− 1.19130i
$204$	14338.1	0.0241221
$205$	0	0
$206$	351781.	0.577570
$207$	− 678232.i	− 1.10015i
$208$	188642.i	0.302330i
$209$	395778.	0.626738
$210$	0	0
$211$	181455.	0.280583	0.140292	−	0.990110i	$-0.455196\pi$
$211$	181455.	0.280583	0.140292	+	0.990110i	$0.455196\pi$
$212$	24546.0i	0.0375095i
$213$	− 469245.i	− 0.708681i
$214$	937009.	1.39865
$215$	0	0
$216$	525796.	0.766801
$217$	− 539895.i	− 0.778323i
$218$	349336.i	0.497854i
$219$	535440.	0.754398
$220$	0	0
$221$	107256.	0.147721
$222$	11857.6i	0.0161478i
$223$	− 1.38761e6i	− 1.86855i	−0.356552	−	0.934276i	$-0.616048\pi$
$223$	− 1.38761e6i	− 1.86855i	0.356552	−	0.934276i	$-0.383952\pi$
$224$	214582.	0.285741
$225$	0	0
$226$	−1.06639e6	−1.38882
$227$	− 690397.i	− 0.889271i	−0.895711	−	0.444636i	$-0.853333\pi$
$227$	− 690397.i	− 0.889271i	0.895711	−	0.444636i	$-0.146667\pi$
$228$	25264.5i	0.0321865i
$229$	1.38257e6	1.74221	0.871104	−	0.491099i	$-0.163405\pi$
$229$	1.38257e6	1.74221	0.871104	+	0.491099i	$0.163405\pi$
$230$	0	0
$231$	−463735.	−0.571794
$232$	− 643403.i	− 0.784807i
$233$	71911.6i	0.0867780i	0.999058	+	0.0433890i	$0.0138155\pi$
$233$	71911.6i	0.0867780i	−0.999058	+	0.0433890i	$0.986185\pi$
$234$	−193760.	−0.231326
$235$	0	0
$236$	8987.36	0.0105039
$237$	− 717009.i	− 0.829189i
$238$	− 699427.i	− 0.800387i
$239$	825442.	0.934743	0.467371	−	0.884061i	$-0.345201\pi$
$239$	825442.	0.934743	0.467371	+	0.884061i	$0.345201\pi$
$240$	0	0
$241$	−615086.	−0.682171	−0.341086	−	0.940032i	$-0.610795\pi$
$241$	−615086.	−0.682171	−0.341086	+	0.940032i	$0.610795\pi$
$242$	212390.i	0.233128i
$243$	925902.i	1.00589i
$244$	44082.7	0.0474016
$245$	0	0
$246$	−567695.	−0.598105
$247$	188992.i	0.197106i
$248$	− 496627.i	− 0.512745i
$249$	−358942.	−0.366881
$250$	0	0
$251$	622589.	0.623759	0.311879	−	0.950122i	$-0.399041\pi$
$251$	622589.	0.623759	0.311879	+	0.950122i	$0.399041\pi$
$252$	114950.i	0.114027i
$253$	1.24218e6i	1.22007i
$254$	−734174.	−0.714028
$255$	0	0
$256$	311779.	0.297335
$257$	− 1.04334e6i	− 0.985359i	−0.870211	−	0.492680i	$-0.836017\pi$
$257$	− 1.04334e6i	− 0.985359i	0.870211	−	0.492680i	$-0.163983\pi$
$258$	216993.i	0.202953i
$259$	52622.9	0.0487444
$260$	0	0
$261$	727666.	0.661197
$262$	256343.i	0.230711i
$263$	1.31940e6i	1.17621i	0.808784	+	0.588106i	$0.200126\pi$
$263$	1.31940e6i	1.17621i	−0.808784	+	0.588106i	$0.799874\pi$
$264$	−426570.	−0.376687
$265$	0	0
$266$	1.23243e6	1.06797
$267$	369922.i	0.317564i
$268$	− 216956.i	− 0.184516i
$269$	−369633.	−0.311452	−0.155726	−	0.987800i	$-0.549772\pi$
$269$	−369633.	−0.311452	−0.155726	+	0.987800i	$0.549772\pi$
$270$	0	0
$271$	749291.	0.619765	0.309883	−	0.950775i	$-0.399710\pi$
$271$	749291.	0.619765	0.309883	+	0.950775i	$0.399710\pi$
$272$	− 708415.i	− 0.580585i
$273$	− 221442.i	− 0.179826i
$274$	−1.11850e6	−0.900037
$275$	0	0
$276$	−79294.5	−0.0626571
$277$	1.64757e6i	1.29016i	0.764114	+	0.645081i	$0.223176\pi$
$277$	1.64757e6i	1.29016i	−0.764114	+	0.645081i	$0.776824\pi$
$278$	2.04189e6i	1.58461i
$279$	561668.	0.431986
$280$	0	0
$281$	1.21917e6	0.921080	0.460540	−	0.887639i	$-0.347656\pi$
$281$	1.21917e6	0.921080	0.460540	+	0.887639i	$0.347656\pi$
$282$	− 283836.i	− 0.212542i
$283$	997517.i	0.740379i	0.928956	+	0.370190i	$0.120707\pi$
$283$	997517.i	0.740379i	−0.928956	+	0.370190i	$0.879293\pi$
$284$	213033.	0.156729
$285$	0	0
$286$	354870.	0.256540
$287$	2.51938e6i	1.80546i
$288$	223235.i	0.158592i
$289$	1.01707e6	0.716321
$290$	0	0
$291$	1.00564e6	0.696162
$292$	243085.i	0.166840i
$293$	− 1.80793e6i	− 1.23031i	−0.788408	−	0.615153i	$-0.789094\pi$
$293$	− 1.80793e6i	− 1.23031i	0.788408	−	0.615153i	$-0.210906\pi$
$294$	−740597.	−0.499705
$295$	0	0
$296$	48405.6	0.0321119
$297$	− 1.08911e6i	− 0.716442i
$298$	− 1.05497e6i	− 0.688176i
$299$	−593165.	−0.383705
$300$	0	0
$301$	962995.	0.612644
$302$	− 3.29163e6i	− 2.07680i
$303$	33520.2i	0.0209749i
$304$	1.24827e6	0.774683
$305$	0	0
$306$	727634.	0.444232
$307$	− 24494.5i	− 0.0148328i	−0.999972	−	0.00741638i	$-0.997639\pi$
$307$	− 24494.5i	− 0.0148328i	0.999972	−	0.00741638i	$-0.00236073\pi$
$308$	− 210531.i	− 0.126456i
$309$	−418254.	−0.249197
$310$	0	0
$311$	1.48212e6	0.868924	0.434462	−	0.900690i	$-0.356939\pi$
$311$	1.48212e6	0.868924	0.434462	+	0.900690i	$0.356939\pi$
$312$	− 203695.i	− 0.118466i
$313$	348766.i	0.201221i	0.994926	+	0.100611i	$0.0320796\pi$
$313$	348766.i	0.201221i	−0.994926	+	0.100611i	$0.967920\pi$
$314$	1.51828e6	0.869014
$315$	0	0
$316$	325515.	0.183380
$317$	− 406486.i	− 0.227194i	−0.993527	−	0.113597i	$-0.963763\pi$
$317$	− 406486.i	− 0.227194i	0.993527	−	0.113597i	$-0.0362373\pi$
$318$	− 320790.i	− 0.177891i
$319$	−1.33272e6	−0.733265
$320$	0	0
$321$	−1.11406e6	−0.603459
$322$	3.86808e6i	2.07901i
$323$	− 709728.i	− 0.378517i
$324$	−80858.9	−0.0427923
$325$	0	0
$326$	1.55856e6	0.812231
$327$	− 415346.i	− 0.214803i
$328$	2.31747e6i	1.18941i
$329$	−1.25964e6	−0.641587
$330$	0	0
$331$	1.58614e6	0.795743	0.397871	−	0.917441i	$-0.369749\pi$
$331$	1.58614e6	0.795743	0.397871	+	0.917441i	$0.369749\pi$
$332$	− 162956.i	− 0.0811381i
$333$	54745.0i	0.0270542i
$334$	1.70323e6	0.835425
$335$	0	0
$336$	−1.46260e6	−0.706769
$337$	2.43587e6i	1.16837i	0.811621	+	0.584185i	$0.198586\pi$
$337$	2.43587e6i	1.16837i	−0.811621	+	0.584185i	$0.801414\pi$
$338$	169457.i	0.0806806i
$339$	1.26790e6	0.599217
$340$	0	0
$341$	−1.02869e6	−0.479071
$342$	1.28213e6i	0.592745i
$343$	164869.i	0.0756664i
$344$	885820.	0.403598
$345$	0	0
$346$	−4.26989e6	−1.91746
$347$	− 1.17786e6i	− 0.525132i	−0.964914	−	0.262566i	$-0.915431\pi$
$347$	− 1.17786e6i	− 0.525132i	0.964914	−	0.262566i	$-0.0845687\pi$
$348$	− 85073.9i	− 0.0376572i
$349$	338854.	0.148919	0.0744594	−	0.997224i	$-0.476277\pi$
$349$	338854.	0.148919	0.0744594	+	0.997224i	$0.476277\pi$
$350$	0	0
$351$	520071.	0.225318
$352$	− 408854.i	− 0.175878i
$353$	− 3.25607e6i	− 1.39077i	−0.718635	−	0.695387i	$-0.755233\pi$
$353$	− 3.25607e6i	− 1.39077i	0.718635	−	0.695387i	$-0.244767\pi$
$354$	−117455.	−0.0498156
$355$	0	0
$356$	−167941.	−0.0702314
$357$	831590.i	0.345334i
$358$	4.62652e6i	1.90786i
$359$	−2.81818e6	−1.15407	−0.577036	−	0.816719i	$-0.695791\pi$
$359$	−2.81818e6	−1.15407	−0.577036	+	0.816719i	$0.695791\pi$
$360$	0	0
$361$	−1.22552e6	−0.494939
$362$	− 1.99393e6i	− 0.799723i
$363$	− 252522.i	− 0.100585i
$364$	100533.	0.0397698
$365$	0	0
$366$	−576114.	−0.224805
$367$	3.09661e6i	1.20011i	0.799958	+	0.600056i	$0.204855\pi$
$367$	3.09661e6i	1.20011i	−0.799958	+	0.600056i	$0.795145\pi$
$368$	3.91779e6i	1.50807i
$369$	−2.62098e6	−1.00207
$370$	0	0
$371$	−1.42364e6	−0.536988
$372$	− 65666.5i	− 0.0246029i
$373$	4.21455e6i	1.56848i	0.620457	+	0.784240i	$0.286947\pi$
$373$	4.21455e6i	1.56848i	−0.620457	+	0.784240i	$0.713053\pi$
$374$	−1.33266e6	−0.492652
$375$	0	0
$376$	−1.15869e6	−0.422666
$377$	− 636398.i	− 0.230609i
$378$	− 3.39143e6i	− 1.22083i
$379$	1.26649e6	0.452903	0.226452	−	0.974022i	$-0.427288\pi$
$379$	1.26649e6	0.452903	0.226452	+	0.974022i	$0.427288\pi$
$380$	0	0
$381$	872903.	0.308073
$382$	− 2.42951e6i	− 0.851845i
$383$	− 5.66939e6i	− 1.97487i	−0.158017	−	0.987436i	$-0.550510\pi$
$383$	− 5.66939e6i	− 1.97487i	0.158017	−	0.987436i	$-0.449490\pi$
$384$	−1.46891e6	−0.508354
$385$	0	0
$386$	−2.01470e6	−0.688242
$387$	1.00183e6i	0.340030i
$388$	456551.i	0.153961i
$389$	5.49114e6	1.83988	0.919938	−	0.392063i	$-0.128238\pi$
$389$	5.49114e6	1.83988	0.919938	+	0.392063i	$0.128238\pi$
$390$	0	0
$391$	2.22753e6	0.736856
$392$	3.02330e6i	0.993726i
$393$	− 304781.i	− 0.0995420i
$394$	−5.17058e6	−1.67803
$395$	0	0
$396$	219021.	0.0701856
$397$	− 1.53688e6i	− 0.489398i	−0.969599	−	0.244699i	$-0.921311\pi$
$397$	− 1.53688e6i	− 0.489398i	0.969599	−	0.244699i	$-0.0786892\pi$
$398$	1.60760e6i	0.508711i
$399$	−1.46531e6	−0.460784
$400$	0	0
$401$	30028.4	0.00932547	0.00466273	−	0.999989i	$-0.498516\pi$
$401$	30028.4	0.00932547	0.00466273	+	0.999989i	$0.498516\pi$
$402$	2.83539e6i	0.875080i
$403$	− 491220.i	− 0.150666i
$404$	−15217.8	−0.00463873
$405$	0	0
$406$	−4.15001e6	−1.24949
$407$	− 100265.i	− 0.0300030i
$408$	764946.i	0.227499i
$409$	5.54413e6	1.63880	0.819398	−	0.573225i	$-0.194308\pi$
$409$	5.54413e6	1.63880	0.819398	+	0.573225i	$0.194308\pi$
$410$	0	0
$411$	1.32985e6	0.388328
$412$	− 189883.i	− 0.0551116i
$413$	521257.i	0.150375i
$414$	−4.02407e6	−1.15389
$415$	0	0
$416$	195236.	0.0553129
$417$	− 2.42773e6i	− 0.683691i
$418$	− 2.34822e6i	− 0.657353i
$419$	1.29360e6	0.359968	0.179984	−	0.983670i	$-0.442395\pi$
$419$	1.29360e6	0.359968	0.179984	+	0.983670i	$0.442395\pi$
$420$	0	0
$421$	−3.68620e6	−1.01362	−0.506809	−	0.862058i	$-0.669175\pi$
$421$	−3.68620e6	−1.01362	−0.506809	+	0.862058i	$0.669175\pi$
$422$	− 1.07660e6i	− 0.294289i
$423$	− 1.31044e6i	− 0.356094i
$424$	−1.30955e6	−0.353758
$425$	0	0
$426$	−2.78411e6	−0.743298
$427$	2.55674e6i	0.678606i
$428$	− 505774.i	− 0.133459i
$429$	−421926.	−0.110686
$430$	0	0
$431$	1.22645e6	0.318021	0.159010	−	0.987277i	$-0.449170\pi$
$431$	1.22645e6	0.318021	0.159010	+	0.987277i	$0.449170\pi$
$432$	− 3.43501e6i	− 0.885562i
$433$	− 1.02459e6i	− 0.262621i	−0.991341	−	0.131311i	$-0.958081\pi$
$433$	− 1.02459e6i	− 0.262621i	0.991341	−	0.131311i	$-0.0419185\pi$
$434$	−3.20329e6	−0.816342
$435$	0	0
$436$	188563.	0.0475051
$437$	3.92504e6i	0.983198i
$438$	− 3.17686e6i	− 0.791249i
$439$	5.04951e6	1.25051	0.625256	−	0.780420i	$-0.284995\pi$
$439$	5.04951e6	1.25051	0.625256	+	0.780420i	$0.284995\pi$
$440$	0	0
$441$	−3.41925e6	−0.837210
$442$	− 636370.i	− 0.154937i
$443$	− 6.30848e6i	− 1.52727i	−0.645649	−	0.763634i	$-0.723413\pi$
$443$	− 6.30848e6i	− 1.52727i	0.645649	−	0.763634i	$-0.276587\pi$
$444$	6400.43	0.00154082
$445$	0	0
$446$	−8.23293e6	−1.95983
$447$	1.25432e6i	0.296919i
$448$	5.36156e6i	1.26211i
$449$	−1.16391e6	−0.272461	−0.136231	−	0.990677i	$-0.543499\pi$
$449$	−1.16391e6	−0.272461	−0.136231	+	0.990677i	$0.543499\pi$
$450$	0	0
$451$	4.80032e6	1.11129
$452$	575612.i	0.132521i
$453$	3.91361e6i	0.896050i
$454$	−4.09625e6	−0.932710
$455$	0	0
$456$	−1.34788e6	−0.303556
$457$	1.48156e6i	0.331840i	0.986139	+	0.165920i	$0.0530594\pi$
$457$	1.48156e6i	0.331840i	−0.986139	+	0.165920i	$0.946941\pi$
$458$	− 8.20306e6i	− 1.82731i
$459$	−1.95304e6	−0.432693
$460$	0	0
$461$	−5.65392e6	−1.23908	−0.619538	−	0.784967i	$-0.712680\pi$
$461$	−5.65392e6	−1.23908	−0.619538	+	0.784967i	$0.712680\pi$
$462$	2.75142e6i	0.599724i
$463$	− 2.09215e6i	− 0.453566i	−0.973945	−	0.226783i	$-0.927179\pi$
$463$	− 2.09215e6i	− 0.453566i	0.973945	−	0.226783i	$-0.0728208\pi$
$464$	−4.20334e6	−0.906357
$465$	0	0
$466$	426664.	0.0910168
$467$	− 7.48481e6i	− 1.58814i	−0.607827	−	0.794069i	$-0.707959\pi$
$467$	− 7.48481e6i	− 1.58814i	0.607827	−	0.794069i	$-0.292041\pi$
$468$	104587.i	0.0220731i
$469$	1.25832e7	2.64155
$470$	0	0
$471$	−1.80517e6	−0.374943
$472$	479482.i	0.0990644i
$473$	− 1.83485e6i	− 0.377092i
$474$	−4.25414e6	−0.869693
$475$	0	0
$476$	−377534.	−0.0763728
$477$	− 1.48105e6i	− 0.298040i
$478$	− 4.89750e6i	− 0.980402i
$479$	−2.54779e6	−0.507371	−0.253685	−	0.967287i	$-0.581643\pi$
$479$	−2.54779e6	−0.507371	−0.253685	+	0.967287i	$0.581643\pi$
$480$	0	0
$481$	47878.6	0.00943580
$482$	3.64941e6i	0.715493i
$483$	− 4.59899e6i	− 0.897004i
$484$	114643.	0.0222450
$485$	0	0
$486$	5.49354e6	1.05502
$487$	3.67779e6i	0.702692i	0.936246	+	0.351346i	$0.114276\pi$
$487$	3.67779e6i	0.702692i	−0.936246	+	0.351346i	$0.885724\pi$
$488$	2.35184e6i	0.447053i
$489$	−1.85306e6	−0.350444
$490$	0	0
$491$	−7.23294e6	−1.35398	−0.676988	−	0.735994i	$-0.736715\pi$
$491$	−7.23294e6	−1.35398	−0.676988	+	0.735994i	$0.736715\pi$
$492$	306428.i	0.0570710i
$493$	2.38989e6i	0.442854i
$494$	1.12132e6	0.206734
$495$	0	0
$496$	−3.24446e6	−0.592158
$497$	1.23556e7i	2.24375i
$498$	2.12966e6i	0.384802i
$499$	−875124.	−0.157332	−0.0786662	−	0.996901i	$-0.525066\pi$
$499$	−875124.	−0.157332	−0.0786662	+	0.996901i	$0.525066\pi$
$500$	0	0
$501$	−2.02507e6	−0.360451
$502$	− 3.69393e6i	− 0.654228i
$503$	2.95982e6i	0.521609i	0.965392	+	0.260805i	$0.0839878\pi$
$503$	2.95982e6i	0.521609i	−0.965392	+	0.260805i	$0.916012\pi$
$504$	−6.13268e6	−1.07541
$505$	0	0
$506$	7.37007e6	1.27966
$507$	− 201478.i	− 0.0348103i
$508$	396289.i	0.0681323i
$509$	1.12208e7	1.91968	0.959842	−	0.280540i	$-0.0905134\pi$
$509$	1.12208e7	1.91968	0.959842	+	0.280540i	$0.0905134\pi$
$510$	0	0
$511$	−1.40986e7	−2.38850
$512$	4.81348e6i	0.811493i
$513$	− 3.44138e6i	− 0.577350i
$514$	−6.19034e6	−1.03349
$515$	0	0
$516$	117128.	0.0193658
$517$	2.40006e6i	0.394907i
$518$	− 312221.i	− 0.0511255i
$519$	5.07672e6	0.827304
$520$	0	0
$521$	8.92586e6	1.44064	0.720321	−	0.693641i	$-0.243995\pi$
$521$	8.92586e6	1.44064	0.720321	+	0.693641i	$0.243995\pi$
$522$	− 4.31737e6i	− 0.693495i
$523$	− 6.44897e6i	− 1.03095i	−0.856906	−	0.515473i	$-0.827616\pi$
$523$	− 6.44897e6i	− 1.03095i	0.856906	−	0.515473i	$-0.172384\pi$
$524$	138368.	0.0220144
$525$	0	0
$526$	7.82821e6	1.23367
$527$	1.84470e6i	0.289334i
$528$	2.78678e6i	0.435028i
$529$	−5.88270e6	−0.913982
$530$	0	0
$531$	−542278.	−0.0834614
$532$	− 665237.i	− 0.101905i
$533$	2.29224e6i	0.349496i
$534$	2.19481e6	0.333077
$535$	0	0
$536$	1.15748e7	1.74020
$537$	− 5.50075e6i	− 0.823163i
$538$	2.19310e6i	0.326665i
$539$	6.26234e6	0.928463
$540$	0	0
$541$	6.01652e6	0.883796	0.441898	−	0.897065i	$-0.354305\pi$
$541$	6.01652e6	0.883796	0.441898	+	0.897065i	$0.354305\pi$
$542$	− 4.44567e6i	− 0.650039i
$543$	2.37071e6i	0.345047i
$544$	−733177.	−0.106221
$545$	0	0
$546$	−1.31386e6	−0.188611
$547$	928354.i	0.132662i	0.997798	+	0.0663308i	$0.0211293\pi$
$547$	928354.i	0.132662i	−0.997798	+	0.0663308i	$0.978871\pi$
$548$	603740.i	0.0858813i
$549$	−2.65985e6	−0.376640
$550$	0	0
$551$	−4.21113e6	−0.590907
$552$	− 4.23042e6i	− 0.590929i
$553$	1.88795e7i	2.62529i
$554$	9.77532e6	1.35318
$555$	0	0
$556$	1.10216e6	0.151203
$557$	652347.i	0.0890924i	0.999007	+	0.0445462i	$0.0141842\pi$
$557$	652347.i	0.0890924i	−0.999007	+	0.0445462i	$0.985816\pi$
$558$	− 3.33248e6i	− 0.453087i
$559$	876176.	0.118594
$560$	0	0
$561$	1.58448e6	0.212558
$562$	− 7.23353e6i	− 0.966072i
$563$	992674.i	0.131988i	0.997820	+	0.0659942i	$0.0210219\pi$
$563$	992674.i	0.131988i	−0.997820	+	0.0659942i	$0.978978\pi$
$564$	−153208.	−0.0202807
$565$	0	0
$566$	5.91844e6	0.776545
$567$	− 4.68972e6i	− 0.612618i
$568$	1.13654e7i	1.47814i
$569$	−8.79703e6	−1.13908	−0.569541	−	0.821963i	$-0.692879\pi$
$569$	−8.79703e6	−1.13908	−0.569541	+	0.821963i	$0.692879\pi$
$570$	0	0
$571$	−6.18261e6	−0.793563	−0.396782	−	0.917913i	$-0.629873\pi$
$571$	−6.18261e6	−0.793563	−0.396782	+	0.917913i	$0.629873\pi$
$572$	− 191550.i	− 0.0244790i
$573$	2.88859e6i	0.367535i
$574$	1.49479e7	1.89366
$575$	0	0
$576$	−5.57778e6	−0.700495
$577$	1.42671e7i	1.78401i	0.452030	+	0.892003i	$0.350700\pi$
$577$	1.42671e7i	1.78401i	−0.452030	+	0.892003i	$0.649300\pi$
$578$	− 6.03448e6i	− 0.751312i
$579$	2.39539e6	0.296948
$580$	0	0
$581$	9.45125e6	1.16158
$582$	− 5.96665e6i	− 0.730168i
$583$	2.71254e6i	0.330525i
$584$	−1.29687e7	−1.57350
$585$	0	0
$586$	−1.07268e7	−1.29040
$587$	− 7.84422e6i	− 0.939625i	−0.882766	−	0.469812i	$-0.844322\pi$
$587$	− 7.84422e6i	− 0.939625i	0.882766	−	0.469812i	$-0.155678\pi$
$588$	399756.i	0.0476817i
$589$	−3.25047e6	−0.386062
$590$	0	0
$591$	6.14761e6	0.723998
$592$	− 316233.i	− 0.0370854i
$593$	1.85555e6i	0.216689i	0.994113	+	0.108344i	$0.0345550\pi$
$593$	1.85555e6i	0.216689i	−0.994113	+	0.108344i	$0.965445\pi$
$594$	−6.46189e6	−0.751438
$595$	0	0
$596$	−569447.	−0.0656656
$597$	− 1.91137e6i	− 0.219487i
$598$	3.51935e6i	0.402448i
$599$	1.54479e7	1.75915	0.879573	−	0.475764i	$-0.157828\pi$
$599$	1.54479e7	1.75915	0.879573	+	0.475764i	$0.157828\pi$
$600$	0	0
$601$	431785.	0.0487619	0.0243810	−	0.999703i	$-0.492239\pi$
$601$	431785.	0.0487619	0.0243810	+	0.999703i	$0.492239\pi$
$602$	− 5.71362e6i	− 0.642570i
$603$	1.30907e7i	1.46612i
$604$	−1.77674e6	−0.198167
$605$	0	0
$606$	198881.	0.0219995
$607$	1.21071e7i	1.33373i	0.745180	+	0.666863i	$0.232363\pi$
$607$	1.21071e7i	1.33373i	−0.745180	+	0.666863i	$0.767637\pi$
$608$	− 1.29190e6i	− 0.141733i
$609$	4.93419e6	0.539104
$610$	0	0
$611$	−1.14607e6	−0.124197
$612$	− 392759.i	− 0.0423885i
$613$	− 9.44151e6i	− 1.01482i	−0.861704	−	0.507411i	$-0.830602\pi$
$613$	− 9.44151e6i	− 1.01482i	0.861704	−	0.507411i	$-0.169398\pi$
$614$	−145330.	−0.0155573
$615$	0	0
$616$	1.12320e7	1.19263
$617$	− 9.98389e6i	− 1.05581i	−0.849303	−	0.527906i	$-0.822977\pi$
$617$	− 9.98389e6i	− 1.05581i	0.849303	−	0.527906i	$-0.177023\pi$
$618$	2.48157e6i	0.261370i
$619$	6.44689e6	0.676275	0.338138	−	0.941097i	$-0.390203\pi$
$619$	6.44689e6	0.676275	0.338138	+	0.941097i	$0.390203\pi$
$620$	0	0
$621$	1.08010e7	1.12392
$622$	− 8.79367e6i	− 0.911369i
$623$	− 9.74037e6i	− 1.00544i
$624$	−1.33074e6	−0.136814
$625$	0	0
$626$	2.06929e6	0.211050
$627$	2.79194e6i	0.283620i
$628$	− 819528.i	− 0.0829211i
$629$	−179800.	−0.0181202
$630$	0	0
$631$	4.35897e6	0.435823	0.217912	−	0.975969i	$-0.430076\pi$
$631$	4.35897e6	0.435823	0.217912	+	0.975969i	$0.430076\pi$
$632$	1.73665e7i	1.72949i
$633$	1.28003e6i	0.126973i
$634$	−2.41176e6	−0.238292
$635$	0	0
$636$	−173155.	−0.0169743
$637$	2.99039e6i	0.291997i
$638$	7.90725e6i	0.769084i
$639$	−1.28539e7	−1.24533
$640$	0	0
$641$	1.63272e7	1.56952	0.784758	−	0.619803i	$-0.212787\pi$
$641$	1.63272e7	1.56952	0.784758	+	0.619803i	$0.212787\pi$
$642$	6.60994e6i	0.632936i
$643$	− 1.31929e7i	− 1.25838i	−0.777250	−	0.629192i	$-0.783386\pi$
$643$	− 1.31929e7i	− 1.25838i	0.777250	−	0.629192i	$-0.216614\pi$
$644$	2.08789e6	0.198378
$645$	0	0
$646$	−4.21094e6	−0.397007
$647$	− 9.42830e6i	− 0.885468i	−0.896653	−	0.442734i	$-0.854009\pi$
$647$	− 9.42830e6i	− 0.885468i	0.896653	−	0.442734i	$-0.145991\pi$
$648$	− 4.31388e6i	− 0.403581i
$649$	993180.	0.0925584
$650$	0	0
$651$	3.80858e6	0.352217
$652$	− 841273.i	− 0.0775029i
$653$	− 1.60701e7i	− 1.47481i	−0.675450	−	0.737406i	$-0.736050\pi$
$653$	− 1.60701e7i	− 1.47481i	0.675450	−	0.737406i	$-0.263950\pi$
$654$	−2.46432e6	−0.225296
$655$	0	0
$656$	1.51400e7	1.37362
$657$	− 1.46672e7i	− 1.32566i
$658$	7.47365e6i	0.672927i
$659$	−6.63639e6	−0.595276	−0.297638	−	0.954679i	$-0.596199\pi$
$659$	−6.63639e6	−0.595276	−0.297638	+	0.954679i	$0.596199\pi$
$660$	0	0
$661$	2.01198e7	1.79110	0.895552	−	0.444956i	$-0.146781\pi$
$661$	2.01198e7	1.79110	0.895552	+	0.444956i	$0.146781\pi$
$662$	− 9.41087e6i	− 0.834613i
$663$	756618.i	0.0668486i
$664$	8.69382e6	0.765227
$665$	0	0
$666$	324812.	0.0283757
$667$	− 1.32169e7i	− 1.15031i
$668$	− 919363.i	− 0.0797160i
$669$	9.78861e6	0.845582
$670$	0	0
$671$	4.87151e6	0.417693
$672$	1.51372e6i	0.129307i
$673$	9.52533e6i	0.810667i	0.914169	+	0.405334i	$0.132845\pi$
$673$	9.52533e6i	0.810667i	−0.914169	+	0.405334i	$0.867155\pi$
$674$	1.44525e7	1.22544
$675$	0	0
$676$	91469.0	0.00769852
$677$	− 3.37825e6i	− 0.283283i	−0.989918	−	0.141641i	$-0.954762\pi$
$677$	− 3.37825e6i	− 0.283283i	0.989918	−	0.141641i	$-0.0452380\pi$
$678$	− 7.52265e6i	− 0.628487i
$679$	−2.64794e7	−2.20412
$680$	0	0
$681$	4.87027e6	0.402425
$682$	6.10341e6i	0.502472i
$683$	8.86253e6i	0.726953i	0.931603	+	0.363476i	$0.118410\pi$
$683$	8.86253e6i	0.726953i	−0.931603	+	0.363476i	$0.881590\pi$
$684$	692064.	0.0565596
$685$	0	0
$686$	978195.	0.0793625
$687$	9.75310e6i	0.788407i
$688$	− 5.78704e6i	− 0.466107i
$689$	−1.29529e6	−0.103949
$690$	0	0
$691$	25025.8	0.00199385	0.000996925	−	1.00000i	$-0.499683\pi$
$691$	25025.8	0.00199385	0.000996925		1.00000i	$0.499683\pi$
$692$	2.30478e6i	0.182964i
$693$	1.27030e7i	1.00478i
$694$	−6.98842e6	−0.550783
$695$	0	0
$696$	4.53876e6	0.355152
$697$	− 8.60815e6i	− 0.671163i
$698$	− 2.01048e6i	− 0.156193i
$699$	−507286.	−0.0392699
$700$	0	0
$701$	−2.15506e7	−1.65640	−0.828198	−	0.560436i	$-0.810634\pi$
$701$	−2.15506e7	−1.65640	−0.828198	+	0.560436i	$0.810634\pi$
$702$	− 3.08568e6i	− 0.236324i
$703$	− 316819.i	− 0.0241781i
$704$	1.02157e7	0.776847
$705$	0	0
$706$	−1.93188e7	−1.45871
$707$	− 882617.i	− 0.0664085i
$708$	63399.6i	0.00475339i
$709$	−2.07938e7	−1.55352	−0.776761	−	0.629796i	$-0.783139\pi$
$709$	−2.07938e7	−1.55352	−0.776761	+	0.629796i	$0.783139\pi$
$710$	0	0
$711$	−1.96409e7	−1.45709
$712$	− 8.95977e6i	− 0.662364i
$713$	− 1.02018e7i	− 0.751544i
$714$	4.93397e6	0.362202
$715$	0	0
$716$	2.49728e6	0.182048
$717$	5.82292e6i	0.423002i
$718$	1.67208e7i	1.21045i
$719$	3.65717e6	0.263829	0.131915	−	0.991261i	$-0.457887\pi$
$719$	3.65717e6	0.263829	0.131915	+	0.991261i	$0.457887\pi$
$720$	0	0
$721$	1.10130e7	0.788982
$722$	7.27121e6i	0.519115i
$723$	− 4.33900e6i	− 0.308705i
$724$	−1.07628e6	−0.0763093
$725$	0	0
$726$	−1.49826e6	−0.105498
$727$	− 8.36880e6i	− 0.587256i	−0.955920	−	0.293628i	$-0.905137\pi$
$727$	− 8.36880e6i	− 0.587256i	0.955920	−	0.293628i	$-0.0948626\pi$
$728$	5.36349e6i	0.375075i
$729$	−396319.	−0.0276202
$730$	0	0
$731$	−3.29034e6	−0.227744
$732$	310972.i	0.0214508i
$733$	− 1.81111e7i	− 1.24504i	−0.782602	−	0.622522i	$-0.786108\pi$
$733$	− 1.81111e7i	− 1.24504i	0.782602	−	0.622522i	$-0.213892\pi$
$734$	1.83727e7	1.25873
$735$	0	0
$736$	4.05472e6	0.275910
$737$	− 2.39755e7i	− 1.62592i
$738$	1.55507e7i	1.05102i
$739$	−7.31705e6	−0.492861	−0.246431	−	0.969160i	$-0.579258\pi$
$739$	−7.31705e6	−0.492861	−0.246431	+	0.969160i	$0.579258\pi$
$740$	0	0
$741$	−1.33320e6	−0.0891972
$742$	8.44670e6i	0.563219i
$743$	− 1.04179e7i	− 0.692322i	−0.938175	−	0.346161i	$-0.887485\pi$
$743$	− 1.04179e7i	− 0.692322i	0.938175	−	0.346161i	$-0.112515\pi$
$744$	3.50336e6	0.232034
$745$	0	0
$746$	2.50057e7	1.64510
$747$	9.83240e6i	0.644701i
$748$	719336.i	0.0470087i
$749$	2.93343e7	1.91061
$750$	0	0
$751$	−1.16729e6	−0.0755229	−0.0377615	−	0.999287i	$-0.512023\pi$
$751$	−1.16729e6	−0.0755229	−0.0377615	+	0.999287i	$0.512023\pi$
$752$	7.56969e6i	0.488128i
$753$	4.39193e6i	0.282272i
$754$	−3.77586e6	−0.241873
$755$	0	0
$756$	−1.83061e6	−0.116491
$757$	4.75104e6i	0.301334i	0.988585	+	0.150667i	$0.0481422\pi$
$757$	4.75104e6i	0.301334i	−0.988585	+	0.150667i	$0.951858\pi$
$758$	− 7.51433e6i	− 0.475026i
$759$	−8.76271e6	−0.552120
$760$	0	0
$761$	−5.92209e6	−0.370692	−0.185346	−	0.982673i	$-0.559341\pi$
$761$	−5.92209e6	−0.370692	−0.185346	+	0.982673i	$0.559341\pi$
$762$	− 5.17909e6i	− 0.323121i
$763$	1.09364e7i	0.680087i
$764$	−1.31139e6	−0.0812828
$765$	0	0
$766$	−3.36375e7	−2.07134
$767$	474262.i	0.0291092i
$768$	2.19938e6i	0.134554i
$769$	5.07027e6	0.309183	0.154591	−	0.987979i	$-0.450594\pi$
$769$	5.07027e6	0.309183	0.154591	+	0.987979i	$0.450594\pi$
$770$	0	0
$771$	7.36006e6	0.445908
$772$	1.08748e6i	0.0656719i
$773$	− 2.31839e7i	− 1.39552i	−0.716330	−	0.697761i	$-0.754180\pi$
$773$	− 2.31839e7i	− 1.39552i	0.716330	−	0.697761i	$-0.245820\pi$
$774$	5.94404e6	0.356639
$775$	0	0
$776$	−2.43573e7	−1.45203
$777$	371217.i	0.0220585i
$778$	− 3.25799e7i	− 1.92975i
$779$	1.51681e7	0.895543
$780$	0	0
$781$	2.35419e7	1.38106
$782$	− 1.32163e7i	− 0.772849i
$783$	1.15883e7i	0.675482i
$784$	1.97512e7	1.14763
$785$	0	0
$786$	−1.80832e6	−0.104404
$787$	− 6.07650e6i	− 0.349717i	−0.984594	−	0.174858i	$-0.944053\pi$
$787$	− 6.07650e6i	− 0.349717i	0.984594	−	0.174858i	$-0.0559468\pi$
$788$	2.79095e6i	0.160117i
$789$	−9.30742e6	−0.532276
$790$	0	0
$791$	−3.33848e7	−1.89718
$792$	1.16849e7i	0.661932i
$793$	2.32624e6i	0.131362i
$794$	−9.11855e6	−0.513304
$795$	0	0
$796$	867745.	0.0485411
$797$	2.78805e7i	1.55473i	0.629052	+	0.777363i	$0.283443\pi$
$797$	2.78805e7i	1.55473i	−0.629052	+	0.777363i	$0.716557\pi$
$798$	8.69394e6i	0.483292i
$799$	4.30389e6	0.238503
$800$	0	0
$801$	1.01332e7	0.558039
$802$	− 178164.i	− 0.00978099i
$803$	2.68629e7i	1.47016i
$804$	1.53047e6	0.0834999
$805$	0	0
$806$	−2.91450e6	−0.158025
$807$	− 2.60750e6i	− 0.140942i
$808$	− 811883.i	− 0.0437487i
$809$	−6.10438e6	−0.327922	−0.163961	−	0.986467i	$-0.552427\pi$
$809$	−6.10438e6	−0.327922	−0.163961	+	0.986467i	$0.552427\pi$
$810$	0	0
$811$	−2.23956e7	−1.19567	−0.597835	−	0.801619i	$-0.703972\pi$
$811$	−2.23956e7	−1.19567	−0.597835	+	0.801619i	$0.703972\pi$
$812$	2.24007e6i	0.119226i
$813$	5.28572e6i	0.280465i
$814$	−594892.	−0.0314686
$815$	0	0
$816$	4.99737e6	0.262734
$817$	− 5.79777e6i	− 0.303882i
$818$	− 3.28943e7i	− 1.71885i
$819$	−6.06591e6	−0.316000
$820$	0	0
$821$	−2.42967e7	−1.25803	−0.629014	−	0.777394i	$-0.716541\pi$
$821$	−2.42967e7	−1.25803	−0.629014	+	0.777394i	$0.716541\pi$
$822$	− 7.89025e6i	− 0.407297i
$823$	− 3.64578e7i	− 1.87625i	−0.346293	−	0.938127i	$-0.612560\pi$
$823$	− 3.64578e7i	− 1.87625i	0.346293	−	0.938127i	$-0.387440\pi$
$824$	1.01304e7	0.519767
$825$	0	0
$826$	3.09271e6	0.157721
$827$	− 2.81247e7i	− 1.42996i	−0.699143	−	0.714981i	$-0.746435\pi$
$827$	− 2.81247e7i	− 1.42996i	0.699143	−	0.714981i	$-0.253565\pi$
$828$	2.17210e6i	0.110104i
$829$	−2.68734e7	−1.35812	−0.679058	−	0.734085i	$-0.737611\pi$
$829$	−2.68734e7	−1.35812	−0.679058	+	0.734085i	$0.737611\pi$
$830$	0	0
$831$	−1.16224e7	−0.583841
$832$	4.87818e6i	0.244315i
$833$	− 1.12299e7i	− 0.560743i
$834$	−1.44041e7	−0.717087
$835$	0	0
$836$	−1.26751e6	−0.0627244
$837$	8.94470e6i	0.441319i
$838$	− 7.67514e6i	− 0.377552i
$839$	−3.46774e7	−1.70076	−0.850378	−	0.526172i	$-0.823627\pi$
$839$	−3.46774e7	−1.70076	−0.850378	+	0.526172i	$0.823627\pi$
$840$	0	0
$841$	−6.33089e6	−0.308656
$842$	2.18709e7i	1.06313i
$843$	8.60037e6i	0.416819i
$844$	−581123.	−0.0280810
$845$	0	0
$846$	−7.77505e6	−0.373488
$847$	6.64914e6i	0.318462i
$848$	8.55524e6i	0.408548i
$849$	−7.03678e6	−0.335046
$850$	0	0
$851$	994359.	0.0470673
$852$	1.50280e6i	0.0709253i
$853$	− 1.54571e6i	− 0.0727368i	−0.999338	−	0.0363684i	$-0.988421\pi$
$853$	− 1.54571e6i	− 0.0727368i	0.999338	−	0.0363684i	$-0.0115790\pi$
$854$	1.51696e7	0.711754
$855$	0	0
$856$	2.69834e7	1.25867
$857$	− 1.27926e7i	− 0.594987i	−0.954724	−	0.297493i	$-0.903849\pi$
$857$	− 1.27926e7i	− 0.594987i	0.954724	−	0.297493i	$-0.0961506\pi$
$858$	2.50336e6i	0.116093i
$859$	−2.66940e6	−0.123433	−0.0617165	−	0.998094i	$-0.519657\pi$
$859$	−2.66940e6	−0.123433	−0.0617165	+	0.998094i	$0.519657\pi$
$860$	0	0
$861$	−1.77725e7	−0.817033
$862$	− 7.27673e6i	− 0.333555i
$863$	3.37798e7i	1.54394i	0.635659	+	0.771970i	$0.280728\pi$
$863$	3.37798e7i	1.54394i	−0.635659	+	0.771970i	$0.719272\pi$
$864$	−3.55508e6	−0.162019
$865$	0	0
$866$	−6.07906e6	−0.275449
$867$	7.17474e6i	0.324159i
$868$	1.72906e6i	0.0778952i
$869$	3.59721e7	1.61591
$870$	0	0
$871$	1.14488e7	0.511344
$872$	1.00600e7i	0.448029i
$873$	− 2.75473e7i	− 1.22333i
$874$	2.32880e7	1.03122
$875$	0	0
$876$	−1.71479e6	−0.0755007
$877$	3.97308e7i	1.74433i	0.489214	+	0.872164i	$0.337284\pi$
$877$	3.97308e7i	1.74433i	−0.489214	+	0.872164i	$0.662716\pi$
$878$	− 2.99596e7i	− 1.31160i
$879$	1.27537e7	0.556755
$880$	0	0
$881$	−1.67058e7	−0.725149	−0.362574	−	0.931955i	$-0.618102\pi$
$881$	−1.67058e7	−0.725149	−0.362574	+	0.931955i	$0.618102\pi$
$882$	2.02870e7i	0.878105i
$883$	1.67930e7i	0.724816i	0.932020	+	0.362408i	$0.118045\pi$
$883$	1.67930e7i	0.724816i	−0.932020	+	0.362408i	$0.881955\pi$
$884$	−343497.	−0.0147840
$885$	0	0
$886$	−3.74293e7	−1.60187
$887$	8.61531e6i	0.367673i	0.982957	+	0.183837i	$0.0588517\pi$
$887$	8.61531e6i	0.367673i	−0.982957	+	0.183837i	$0.941148\pi$
$888$	341468.i	0.0145317i
$889$	−2.29843e7	−0.975388
$890$	0	0
$891$	−8.93559e6	−0.377076
$892$	4.44393e6i	0.187006i
$893$	7.58371e6i	0.318239i
$894$	7.44208e6	0.311423
$895$	0	0
$896$	3.86777e7	1.60950
$897$	− 4.18436e6i	− 0.173639i
$898$	6.90571e6i	0.285770i
$899$	1.09454e7	0.451682
$900$	0	0
$901$	4.86425e6	0.199620
$902$	− 2.84811e7i	− 1.16558i
$903$	6.79326e6i	0.277242i
$904$	−3.07093e7	−1.24983
$905$	0	0
$906$	2.32202e7	0.939820
$907$	7.34436e6i	0.296439i	0.988954	+	0.148220i	$0.0473543\pi$
$907$	7.34436e6i	0.296439i	−0.988954	+	0.148220i	$0.952646\pi$
$908$	2.21105e6i	0.0889989i
$909$	918211.	0.0368581
$910$	0	0
$911$	3.63225e7	1.45004	0.725019	−	0.688729i	$-0.241831\pi$
$911$	3.63225e7	1.45004	0.725019	+	0.688729i	$0.241831\pi$
$912$	8.80567e6i	0.350570i
$913$	− 1.80080e7i	− 0.714971i
$914$	8.79036e6	0.348049
$915$	0	0
$916$	−4.42781e6	−0.174361
$917$	8.02515e6i	0.315159i
$918$	1.15878e7i	0.453829i
$919$	−2.25278e7	−0.879892	−0.439946	−	0.898024i	$-0.645002\pi$
$919$	−2.25278e7	−0.879892	−0.439946	+	0.898024i	$0.645002\pi$
$920$	0	0
$921$	172791.	0.00671232
$922$	3.35457e7i	1.29960i
$923$	1.12417e7i	0.434338i
$924$	1.48515e6	0.0572255
$925$	0	0
$926$	−1.24131e7	−0.475721
$927$	1.14571e7i	0.437901i
$928$	4.35026e6i	0.165823i
$929$	−3.93312e7	−1.49520	−0.747598	−	0.664152i	$-0.768793\pi$
$929$	−3.93312e7	−1.49520	−0.747598	+	0.664152i	$0.768793\pi$
$930$	0	0
$931$	1.97878e7	0.748209
$932$	− 230303.i	− 0.00868480i
$933$	1.04553e7i	0.393217i
$934$	−4.44087e7	−1.66571
$935$	0	0
$936$	−5.57978e6	−0.208175
$937$	− 1.36354e7i	− 0.507362i	−0.967288	−	0.253681i	$-0.918359\pi$
$937$	− 1.36354e7i	− 0.507362i	0.967288	−	0.253681i	$-0.0816414\pi$
$938$	− 7.46584e7i	− 2.77058i
$939$	−2.46030e6	−0.0910593
$940$	0	0
$941$	2.28438e7	0.840995	0.420498	−	0.907294i	$-0.361856\pi$
$941$	2.28438e7	0.840995	0.420498	+	0.907294i	$0.361856\pi$
$942$	1.07104e7i	0.393258i
$943$	4.76061e7i	1.74334i
$944$	3.13245e6	0.114407
$945$	0	0
$946$	−1.08865e7	−0.395512
$947$	− 6.46944e6i	− 0.234418i	−0.993107	−	0.117209i	$-0.962605\pi$
$947$	− 6.46944e6i	− 0.234418i	0.993107	−	0.117209i	$-0.0373948\pi$
$948$	2.29628e6i	0.0829858i
$949$	−1.28276e7	−0.462358
$950$	0	0
$951$	2.86748e6	0.102813
$952$	− 2.01417e7i	− 0.720284i
$953$	− 2.58068e6i	− 0.0920452i	−0.998940	−	0.0460226i	$-0.985345\pi$
$953$	− 2.58068e6i	− 0.0920452i	0.998940	−	0.0460226i	$-0.0146546\pi$
$954$	−8.78734e6	−0.312598
$955$	0	0
$956$	−2.64355e6	−0.0935497
$957$	− 9.40139e6i	− 0.331827i
$958$	1.51165e7i	0.532154i
$959$	−3.50162e7	−1.22948
$960$	0	0
$961$	−2.01807e7	−0.704899
$962$	− 284072.i	− 0.00989672i
$963$	3.05173e7i	1.06043i
$964$	1.96986e6	0.0682722
$965$	0	0
$966$	−2.72866e7	−0.940820
$967$	− 1.88591e6i	− 0.0648568i	−0.999474	−	0.0324284i	$-0.989676\pi$
$967$	− 1.88591e6i	− 0.0648568i	0.999474	−	0.0324284i	$-0.0103241\pi$
$968$	6.11627e6i	0.209797i
$969$	5.00663e6	0.171292
$970$	0	0
$971$	−2.49003e7	−0.847534	−0.423767	−	0.905771i	$-0.639292\pi$
$971$	−2.49003e7	−0.847534	−0.423767	+	0.905771i	$0.639292\pi$
$972$	− 2.96528e6i	− 0.100670i
$973$	6.39242e7i	2.16463i
$974$	2.18210e7	0.737016
$975$	0	0
$976$	1.53645e7	0.516292
$977$	1.01689e7i	0.340829i	0.985372	+	0.170415i	$0.0545107\pi$
$977$	1.01689e7i	0.340829i	−0.985372	+	0.170415i	$0.945489\pi$
$978$	1.09946e7i	0.367562i
$979$	−1.85589e7	−0.618864
$980$	0	0
$981$	−1.13775e7	−0.377462
$982$	4.29143e7i	1.42011i
$983$	− 4.29289e6i	− 0.141699i	−0.997487	−	0.0708494i	$-0.977429\pi$
$983$	− 4.29289e6i	− 0.141699i	0.997487	−	0.0708494i	$-0.0225710\pi$
$984$	−1.63482e7	−0.538246
$985$	0	0
$986$	1.41796e7	0.464486
$987$	− 8.88586e6i	− 0.290340i
$988$	− 605262.i	− 0.0197265i
$989$	1.81967e7	0.591565
$990$	0	0
$991$	−2.64765e7	−0.856398	−0.428199	−	0.903684i	$-0.640852\pi$
$991$	−2.64765e7	−0.856398	−0.428199	+	0.903684i	$0.640852\pi$
$992$	3.35786e6i	0.108339i
$993$	1.11891e7i	0.360100i
$994$	7.33082e7	2.35335
$995$	0	0
$996$	1.14954e6	0.0367177
$997$	4.21089e6i	0.134164i	0.997747	+	0.0670820i	$0.0213689\pi$
$997$	4.21089e6i	0.134164i	−0.997747	+	0.0670820i	$0.978631\pi$
$998$	5.19227e6i	0.165018i
$999$	−871829.	−0.0276387

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.6.b.g.274.4		12
5.2	odd	4		325.6.a.g.1.5		6
5.3	odd	4		65.6.a.d.1.2	✓	6
5.4	even	2	inner	325.6.b.g.274.9		12
15.8	even	4		585.6.a.m.1.5		6
20.3	even	4		1040.6.a.q.1.5		6
65.38	odd	4		845.6.a.h.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.6.a.d.1.2	✓	6	5.3	odd	4
325.6.a.g.1.5		6	5.2	odd	4
325.6.b.g.274.4		12	1.1	even	1	trivial
325.6.b.g.274.9		12	5.4	even	2	inner
585.6.a.m.1.5		6	15.8	even	4
845.6.a.h.1.5		6	65.38	odd	4
1040.6.a.q.1.5		6	20.3	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 325 = 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	325.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-5.93318i q^{2} +7.05430i q^{3} -3.20258 q^{4} +41.8544 q^{6} -185.746i q^{7} -170.860i q^{8} +193.237 q^{9} -353.912 q^{11} -22.5920i q^{12} -169.000i q^{13} -1102.06 q^{14} -1116.23 q^{16} +634.652i q^{17} -1146.51i q^{18} -1118.29 q^{19} +1310.31 q^{21} +2099.83i q^{22} -3509.85i q^{23} +1205.30 q^{24} -1002.71 q^{26} +3077.35i q^{27} +594.867i q^{28} +3765.67 q^{29} +2906.63 q^{31} +1155.24i q^{32} -2496.60i q^{33} +3765.50 q^{34} -618.857 q^{36} +283.305i q^{37} +6635.04i q^{38} +1192.18 q^{39} -13563.6 q^{41} -7774.29i q^{42} +5184.47i q^{43} +1133.43 q^{44} -20824.6 q^{46} -6781.50i q^{47} -7874.19i q^{48} -17694.6 q^{49} -4477.03 q^{51} +541.237i q^{52} -7664.43i q^{53} +18258.4 q^{54} -31736.6 q^{56} -7888.78i q^{57} -22342.4i q^{58} -2806.29 q^{59} -13764.7 q^{61} -17245.5i q^{62} -35893.0i q^{63} -28865.0 q^{64} -14812.8 q^{66} +67744.1i q^{67} -2032.53i q^{68} +24759.5 q^{69} -66519.0 q^{71} -33016.5i q^{72} -75902.7i q^{73} +1680.90 q^{74} +3581.43 q^{76} +65737.9i q^{77} -7073.39i q^{78} -101641. q^{79} +25248.0 q^{81} +80475.1i q^{82} +50882.7i q^{83} -4196.37 q^{84} +30760.4 q^{86} +26564.2i q^{87} +60469.5i q^{88} +52439.2 q^{89} -31391.1 q^{91} +11240.6i q^{92} +20504.2i q^{93} -40235.8 q^{94} -8149.42 q^{96} -142557. i q^{97} +104985. i q^{98} -68388.9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 268 q^{4} + 636 q^{6} - 1036 q^{9} - 340 q^{11} + 2880 q^{14} + 7012 q^{16} - 2436 q^{19} - 792 q^{21} - 25236 q^{24} - 16728 q^{29} + 5724 q^{31} + 42968 q^{34} - 4276 q^{36} - 12844 q^{39} + 4496 q^{41}+ \cdots + 64540 q^{99}+O(q^{100}) \) 12 * q - 268 * q^4 + 636 * q^6 - 1036 * q^9 - 340 * q^11 + 2880 * q^14 + 7012 * q^16 - 2436 * q^19 - 792 * q^21 - 25236 * q^24 - 16728 * q^29 + 5724 * q^31 + 42968 * q^34 - 4276 * q^36 - 12844 * q^39 + 4496 * q^41 + 146964 * q^44 - 79772 * q^46 - 171804 * q^49 - 110056 * q^51 - 91008 * q^54 - 377600 * q^56 + 57748 * q^59 + 1944 * q^61 - 28044 * q^64 - 316080 * q^66 - 306176 * q^69 - 148348 * q^71 - 452224 * q^74 - 186844 * q^76 - 429016 * q^79 + 232012 * q^81 + 470536 * q^84 - 609332 * q^86 - 188056 * q^89 + 74360 * q^91 + 251840 * q^94 - 771716 * q^96 + 64540 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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