LMFDB - Embedded newform 96.6.c.a.95.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [96,6,Mod(95,96)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(96, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("96.95");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 96 = 2^{5} \cdot 3 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	96.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$15.3968467020$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 416x^{16} + 57056x^{12} + 3187216x^{8} + 63121536x^{4} + 49787136 $ x^20 + 416x^16 + 57056x^12 + 3187216x^8 + 63121536x^4 + 49787136
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{134}\cdot 3^{14} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			95.9
Root			$1.85069 + 1.85069i$ of defining polynomial
Character	$\chi$	$=$	96.95
Dual form			96.6.c.a.95.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-6.21400 - 14.2964i) q^{3} +36.1786i q^{5} -57.8847i q^{7} +(-165.772 + 177.675i) q^{9} +O(q^{10})$	\(q+(-6.21400 - 14.2964i) q^{3} +36.1786i q^{5} -57.8847i q^{7} +(-165.772 + 177.675i) q^{9} -114.418 q^{11} -295.579 q^{13} +(517.223 - 224.814i) q^{15} +1155.61i q^{17} +1955.80i q^{19} +(-827.541 + 359.696i) q^{21} +4540.43 q^{23} +1816.11 q^{25} +(3570.22 + 1265.87i) q^{27} +4911.55i q^{29} +1701.31i q^{31} +(710.991 + 1635.76i) q^{33} +2094.19 q^{35} -14990.4 q^{37} +(1836.73 + 4225.70i) q^{39} +19071.6i q^{41} -5939.24i q^{43} +(-6428.05 - 5997.42i) q^{45} -13287.7 q^{47} +13456.4 q^{49} +(16521.0 - 7180.96i) q^{51} -13202.0i q^{53} -4139.47i q^{55} +(27960.8 - 12153.3i) q^{57} -25888.9 q^{59} -5568.84 q^{61} +(10284.7 + 9595.68i) q^{63} -10693.6i q^{65} +20870.4i q^{67} +(-28214.2 - 64911.6i) q^{69} -51334.6 q^{71} +73309.4 q^{73} +(-11285.3 - 25963.7i) q^{75} +6623.03i q^{77} -46469.0i q^{79} +(-4088.08 - 58907.3i) q^{81} -95537.3 q^{83} -41808.4 q^{85} +(70217.3 - 30520.4i) q^{87} +87567.6i q^{89} +17109.5i q^{91} +(24322.5 - 10571.9i) q^{93} -70758.2 q^{95} -50886.1 q^{97} +(18967.3 - 20329.2i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 44 q^{9}+O(q^{10}) $ 20 * q - 44 * q^9	$ 20 q - 44 q^{9} - 232 q^{13} - 1640 q^{21} - 15228 q^{25} - 13264 q^{33} + 17208 q^{37} + 24224 q^{45} - 51780 q^{49} - 37512 q^{57} + 123416 q^{61} + 100192 q^{69} - 48120 q^{73} - 42508 q^{81} + 168960 q^{85} + 34840 q^{93} + 39368 q^{97}+O(q^{100}) $ 20 * q - 44 * q^9 - 232 * q^13 - 1640 * q^21 - 15228 * q^25 - 13264 * q^33 + 17208 * q^37 + 24224 * q^45 - 51780 * q^49 - 37512 * q^57 + 123416 * q^61 + 100192 * q^69 - 48120 * q^73 - 42508 * q^81 + 168960 * q^85 + 34840 * q^93 + 39368 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/96\mathbb{Z}\right)^\times$.

$n$	$31$	$37$	$65$
$\chi(n)$	$-1$	$1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$	−6.21400	−	14.2964i	−0.398629	−	0.917112i
$3$	−6.21400	−	14.2964i	−0.398629	−	0.917112i
$4$		0			0
$5$			36.1786i			0.647183i	0.946197	+	0.323592i	$0.104890\pi$
$5$			36.1786i			0.647183i	−0.946197	+	0.323592i	$0.895110\pi$
$6$		0			0
$7$		−	57.8847i		−	0.446497i	−0.974762	−	0.223249i	$-0.928334\pi$
$7$		−	57.8847i		−	0.446497i	0.974762	−	0.223249i	$-0.0716662\pi$
$8$		0			0
$9$	−165.772	+	177.675i	−0.682191	+	0.731174i
$10$		0			0
$11$	−114.418			−0.285109			−0.142554	−	0.989787i	$-0.545532\pi$
$11$	−114.418			−0.285109			−0.142554	+	0.989787i	$0.545532\pi$
$12$		0			0
$13$	−295.579			−0.485082			−0.242541	−	0.970141i	$-0.577981\pi$
$13$	−295.579			−0.485082			−0.242541	+	0.970141i	$0.577981\pi$
$14$		0			0
$15$	517.223	−	224.814i	0.593540	−	0.257986i
$16$		0			0
$17$			1155.61i			0.969815i	0.874566	+	0.484907i	$0.161147\pi$
$17$			1155.61i			0.969815i	−0.874566	+	0.484907i	$0.838853\pi$
$18$		0			0
$19$			1955.80i			1.24291i	0.783449	+	0.621456i	$0.213459\pi$
$19$			1955.80i			1.24291i	−0.783449	+	0.621456i	$0.786541\pi$
$20$		0			0
$21$	−827.541	+	359.696i	−0.409488	+	0.177987i
$22$		0			0
$23$	4540.43			1.78969			0.894843	−	0.446380i	$-0.147287\pi$
$23$	4540.43			1.78969			0.894843	+	0.446380i	$0.147287\pi$
$24$		0			0
$25$	1816.11			0.581154
$26$		0			0
$27$	3570.22	+	1265.87i	0.942510	+	0.334179i
$28$		0			0
$29$			4911.55i			1.08448i	0.840222	+	0.542242i	$0.182425\pi$
$29$			4911.55i			1.08448i	−0.840222	+	0.542242i	$0.817575\pi$
$30$		0			0
$31$			1701.31i			0.317965i	0.987281	+	0.158982i	$0.0508213\pi$
$31$			1701.31i			0.317965i	−0.987281	+	0.158982i	$0.949179\pi$
$32$		0			0
$33$	710.991	+	1635.76i	0.113653	+	0.261477i
$34$		0			0
$35$	2094.19			0.288965
$36$		0			0
$37$	−14990.4			−1.80015			−0.900076	−	0.435733i	$-0.856489\pi$
$37$	−14990.4			−1.80015			−0.900076	+	0.435733i	$0.856489\pi$
$38$		0			0
$39$	1836.73	+	4225.70i	0.193367	+	0.444875i
$40$		0			0
$41$			19071.6i			1.77185i	0.463824	+	0.885927i	$0.346477\pi$
$41$			19071.6i			1.77185i	−0.463824	+	0.885927i	$0.653523\pi$
$42$		0			0
$43$		−	5939.24i		−	0.489846i	−0.969543	−	0.244923i	$-0.921237\pi$
$43$		−	5939.24i		−	0.489846i	0.969543	−	0.244923i	$-0.0787627\pi$
$44$		0			0
$45$	−6428.05	−	5997.42i	−0.473204	−	0.441502i
$46$		0			0
$47$	−13287.7			−0.877416			−0.438708	−	0.898630i	$-0.644564\pi$
$47$	−13287.7			−0.877416			−0.438708	+	0.898630i	$0.644564\pi$
$48$		0			0
$49$	13456.4			0.800640
$50$		0			0
$51$	16521.0	−	7180.96i	0.889429	−	0.386596i
$52$		0			0
$53$		−	13202.0i		−	0.645582i	−0.946470	−	0.322791i	$-0.895379\pi$
$53$		−	13202.0i		−	0.645582i	0.946470	−	0.322791i	$-0.104621\pi$
$54$		0			0
$55$		−	4139.47i		−	0.184518i
$56$		0			0
$57$	27960.8	−	12153.3i	1.13989	−	0.495460i
$58$		0			0
$59$	−25888.9			−0.968242			−0.484121	−	0.875001i	$-0.660861\pi$
$59$	−25888.9			−0.968242			−0.484121	+	0.875001i	$0.660861\pi$
$60$		0			0
$61$	−5568.84			−0.191620			−0.0958099	−	0.995400i	$-0.530544\pi$
$61$	−5568.84			−0.191620			−0.0958099	+	0.995400i	$0.530544\pi$
$62$		0			0
$63$	10284.7	+	9595.68i	0.326467	+	0.304596i
$64$		0			0
$65$		−	10693.6i		−	0.313937i
$66$		0			0
$67$			20870.4i			0.567994i	0.958825	+	0.283997i	$0.0916606\pi$
$67$			20870.4i			0.567994i	−0.958825	+	0.283997i	$0.908339\pi$
$68$		0			0
$69$	−28214.2	−	64911.6i	−0.713420	−	1.64134i
$70$		0			0
$71$	−51334.6			−1.20855			−0.604275	−	0.796776i	$-0.706537\pi$
$71$	−51334.6			−1.20855			−0.604275	+	0.796776i	$0.706537\pi$
$72$		0			0
$73$	73309.4			1.61010			0.805049	−	0.593208i	$-0.202139\pi$
$73$	73309.4			1.61010			0.805049	+	0.593208i	$0.202139\pi$
$74$		0			0
$75$	−11285.3	−	25963.7i	−0.231665	−	0.532984i
$76$		0			0
$77$			6623.03i			0.127300i
$78$		0			0
$79$		−	46469.0i		−	0.837715i	−0.908052	−	0.418857i	$-0.862431\pi$
$79$		−	46469.0i		−	0.837715i	0.908052	−	0.418857i	$-0.137569\pi$
$80$		0			0
$81$	−4088.08	−	58907.3i	−0.0692320	−	0.997601i
$82$		0			0
$83$	−95537.3			−1.52222			−0.761111	−	0.648622i	$-0.775346\pi$
$83$	−95537.3			−1.52222			−0.761111	+	0.648622i	$0.775346\pi$
$84$		0			0
$85$	−41808.4			−0.627648
$86$		0			0
$87$	70217.3	−	30520.4i	0.994595	−	0.432307i
$88$		0			0
$89$			87567.6i			1.17184i	0.810369	+	0.585920i	$0.199267\pi$
$89$			87567.6i			1.17184i	−0.810369	+	0.585920i	$0.800733\pi$
$90$		0			0
$91$			17109.5i			0.216588i
$92$		0			0
$93$	24322.5	−	10571.9i	0.291610	−	0.126750i
$94$		0			0
$95$	−70758.2			−0.804392
$96$		0			0
$97$	−50886.1			−0.549124			−0.274562	−	0.961569i	$-0.588533\pi$
$97$	−50886.1			−0.549124			−0.274562	+	0.961569i	$0.588533\pi$
$98$		0			0
$99$	18967.3	−	20329.2i	0.194499	−	0.208464i
$100$		0			0
$101$			69801.1i			0.680862i	0.940270	+	0.340431i	$0.110573\pi$
$101$			69801.1i			0.680862i	−0.940270	+	0.340431i	$0.889427\pi$
$102$		0			0
$103$		−	82496.6i		−	0.766201i	−0.923707	−	0.383101i	$-0.874856\pi$
$103$		−	82496.6i		−	0.766201i	0.923707	−	0.383101i	$-0.125144\pi$
$104$		0			0
$105$	−13013.3	−	29939.3i	−0.115190	−	0.265014i
$106$		0			0
$107$	75209.7			0.635059			0.317530	−	0.948248i	$-0.397147\pi$
$107$	75209.7			0.635059			0.317530	+	0.948248i	$0.397147\pi$
$108$		0			0
$109$	48849.4			0.393816			0.196908	−	0.980422i	$-0.436910\pi$
$109$	48849.4			0.393816			0.196908	+	0.980422i	$0.436910\pi$
$110$		0			0
$111$	93150.5	+	214308.i	0.717592	+	1.65094i
$112$		0			0
$113$			45527.2i			0.335409i	0.985837	+	0.167705i	$0.0536355\pi$
$113$			45527.2i			0.335409i	−0.985837	+	0.167705i	$0.946364\pi$
$114$		0			0
$115$			164266.i			1.15826i
$116$		0			0
$117$	48998.8	−	52517.1i	0.330918	−	0.354679i
$118$		0			0
$119$	66892.1			0.433020
$120$		0			0
$121$	−147960.			−0.918713
$122$		0			0
$123$	272655.	−	118511.i	1.62499	−	0.706312i
$124$		0			0
$125$			178762.i			1.02330i
$126$		0			0
$127$			113221.i			0.622899i	0.950263	+	0.311450i	$0.100815\pi$
$127$			113221.i			0.622899i	−0.950263	+	0.311450i	$0.899185\pi$
$128$		0			0
$129$	−84909.5	+	36906.4i	−0.449244	+	0.195267i
$130$		0			0
$131$	73591.9			0.374672			0.187336	−	0.982296i	$-0.440015\pi$
$131$	73591.9			0.374672			0.187336	+	0.982296i	$0.440015\pi$
$132$		0			0
$133$	113211.			0.554957
$134$		0			0
$135$	−45797.3	+	129166.i	−0.216275	+	0.609976i
$136$		0			0
$137$			56044.1i			0.255110i	0.991831	+	0.127555i	$0.0407130\pi$
$137$			56044.1i			0.255110i	−0.991831	+	0.127555i	$0.959287\pi$
$138$		0			0
$139$		−	61631.4i		−	0.270561i	−0.990807	−	0.135280i	$-0.956806\pi$
$139$		−	61631.4i		−	0.270561i	0.990807	−	0.135280i	$-0.0431935\pi$
$140$		0			0
$141$	82569.8	+	189966.i	0.349763	+	0.804689i
$142$		0			0
$143$	33819.4			0.138301
$144$		0			0
$145$	−177693.			−0.701860
$146$		0			0
$147$	−83617.9	−	192377.i	−0.319158	−	0.734277i
$148$		0			0
$149$			318680.i			1.17595i	0.808878	+	0.587976i	$0.200075\pi$
$149$			318680.i			1.17595i	−0.808878	+	0.587976i	$0.799925\pi$
$150$		0			0
$151$		−	433550.i		−	1.54738i	−0.633566	−	0.773689i	$-0.718409\pi$
$151$		−	433550.i		−	1.54738i	0.633566	−	0.773689i	$-0.281591\pi$
$152$		0			0
$153$	−205323.	−	191568.i	−0.709104	−	0.661598i
$154$		0			0
$155$	−61551.1			−0.205781
$156$		0			0
$157$	137096.			0.443890			0.221945	−	0.975059i	$-0.428759\pi$
$157$	137096.			0.443890			0.221945	+	0.975059i	$0.428759\pi$
$158$		0			0
$159$	−188741.	+	82037.5i	−0.592071	+	0.257347i
$160$		0			0
$161$		−	262821.i		−	0.799090i
$162$		0			0
$163$			659320.i			1.94369i	0.235618	+	0.971846i	$0.424288\pi$
$163$			659320.i			1.94369i	−0.235618	+	0.971846i	$0.575712\pi$
$164$		0			0
$165$	−59179.4	+	25722.7i	−0.169223	+	0.0735540i
$166$		0			0
$167$	163704.			0.454222			0.227111	−	0.973869i	$-0.427072\pi$
$167$	163704.			0.454222			0.227111	+	0.973869i	$0.427072\pi$
$168$		0			0
$169$	−283926.			−0.764696
$170$		0			0
$171$	−347498.	−	324218.i	−0.908786	−	0.847903i
$172$		0			0
$173$		−	640742.i		−	1.62768i	−0.581091	−	0.813838i	$-0.697374\pi$
$173$		−	640742.i		−	1.62768i	0.581091	−	0.813838i	$-0.302626\pi$
$174$		0			0
$175$		−	105125.i		−	0.259484i
$176$		0			0
$177$	160874.	+	370118.i	0.385969	+	0.887987i
$178$		0			0
$179$	−338416.			−0.789439			−0.394719	−	0.918802i	$-0.629158\pi$
$179$	−338416.			−0.789439			−0.394719	+	0.918802i	$0.629158\pi$
$180$		0			0
$181$	692113.			1.57029			0.785146	−	0.619311i	$-0.212588\pi$
$181$	692113.			1.57029			0.785146	+	0.619311i	$0.212588\pi$
$182$		0			0
$183$	34604.8	+	79614.2i	0.0763851	+	0.175737i
$184$		0			0
$185$		−	542333.i		−	1.16503i
$186$		0			0
$187$		−	132222.i		−	0.276503i
$188$		0			0
$189$	73274.3	−	206661.i	0.149210	−	0.420828i
$190$		0			0
$191$	−24688.5			−0.0489678			−0.0244839	−	0.999700i	$-0.507794\pi$
$191$	−24688.5			−0.0489678			−0.0244839	+	0.999700i	$0.507794\pi$
$192$		0			0
$193$	564427.			1.09072			0.545361	−	0.838201i	$-0.316392\pi$
$193$	564427.			1.09072			0.545361	+	0.838201i	$0.316392\pi$
$194$		0			0
$195$	−152880.	+	66450.3i	−0.287915	+	0.125144i
$196$		0			0
$197$		−	630840.i		−	1.15812i	−0.815285	−	0.579061i	$-0.803419\pi$
$197$		−	630840.i		−	1.15812i	0.815285	−	0.579061i	$-0.196581\pi$
$198$		0			0
$199$		−	947253.i		−	1.69564i	−0.530286	−	0.847819i	$-0.677915\pi$
$199$		−	947253.i		−	1.69564i	0.530286	−	0.847819i	$-0.322085\pi$
$200$		0			0
$201$	298371.	−	129689.i	0.520914	−	0.226419i
$202$		0			0
$203$	284304.			0.484219
$204$		0			0
$205$	−689985.			−1.14671
$206$		0			0
$207$	−752677.	+	806722.i	−1.22091	+	1.30857i
$208$		0			0
$209$		−	223778.i		−	0.354365i
$210$		0			0
$211$			522194.i			0.807469i	0.914876	+	0.403734i	$0.132288\pi$
$211$			522194.i			0.807469i	−0.914876	+	0.403734i	$0.867712\pi$
$212$		0			0
$213$	318993.	+	733898.i	0.481762	+	1.10838i
$214$		0			0
$215$	214873.			0.317020
$216$		0			0
$217$	98479.8			0.141970
$218$		0			0
$219$	−455545.	−	1.04806e6i	−0.641831	−	1.47664i
$220$		0			0
$221$		−	341574.i		−	0.470439i
$222$		0			0
$223$			335095.i			0.451238i	0.974216	+	0.225619i	$0.0724404\pi$
$223$			335095.i			0.451238i	−0.974216	+	0.225619i	$0.927560\pi$
$224$		0			0
$225$	−301060.	+	322677.i	−0.396458	+	0.424925i
$226$		0			0
$227$	542830.			0.699196			0.349598	−	0.936900i	$-0.386318\pi$
$227$	542830.			0.699196			0.349598	+	0.936900i	$0.386318\pi$
$228$		0			0
$229$	−1.13784e6			−1.43381			−0.716904	−	0.697172i	$-0.754441\pi$
$229$	−1.13784e6			−1.43381			−0.716904	+	0.697172i	$0.754441\pi$
$230$		0			0
$231$	94685.2	−	41155.5i	0.116749	−	0.0507456i
$232$		0			0
$233$		−	36611.5i		−	0.0441803i	−0.999756	−	0.0220901i	$-0.992968\pi$
$233$		−	36611.5i		−	0.0441803i	0.999756	−	0.0220901i	$-0.00703208\pi$
$234$		0			0
$235$		−	480731.i		−	0.567848i
$236$		0			0
$237$	−664339.	+	288759.i	−0.768278	+	0.333937i
$238$		0			0
$239$	−822358.			−0.931250			−0.465625	−	0.884982i	$-0.654170\pi$
$239$	−822358.			−0.931250			−0.465625	+	0.884982i	$0.654170\pi$
$240$		0			0
$241$	707488.			0.784651			0.392325	−	0.919826i	$-0.371671\pi$
$241$	707488.			0.784651			0.392325	+	0.919826i	$0.371671\pi$
$242$		0			0
$243$	−816757.	+	424495.i	−0.887314	+	0.461166i
$244$		0			0
$245$			486833.i			0.518161i
$246$		0			0
$247$		−	578093.i		−	0.602914i
$248$		0			0
$249$	593669.	+	1.36584e6i	0.606801	+	1.39605i
$250$		0			0
$251$	1.89976e6			1.90333			0.951667	−	0.307132i	$-0.0993692\pi$
$251$	1.89976e6			1.90333			0.951667	+	0.307132i	$0.0993692\pi$
$252$		0			0
$253$	−519504.			−0.510256
$254$		0			0
$255$	259797.	+	597708.i	0.250198	+	0.575623i
$256$		0			0
$257$			1.12906e6i			1.06631i	0.846017	+	0.533156i	$0.178994\pi$
$257$			1.12906e6i			1.06631i	−0.846017	+	0.533156i	$0.821006\pi$
$258$		0			0
$259$			867716.i			0.803763i
$260$		0			0
$261$	−872661.	−	814199.i	−0.792947	−	0.739825i
$262$		0			0
$263$	1.90594e6			1.69910			0.849550	−	0.527508i	$-0.176873\pi$
$263$	1.90594e6			1.69910			0.849550	+	0.527508i	$0.176873\pi$
$264$		0			0
$265$	477632.			0.417810
$266$		0			0
$267$	1.25190e6	−	544145.i	1.07471	−	0.467129i
$268$		0			0
$269$		−	220717.i		−	0.185975i	−0.995667	−	0.0929877i	$-0.970358\pi$
$269$		−	220717.i		−	0.185975i	0.995667	−	0.0929877i	$-0.0296417\pi$
$270$		0			0
$271$			1.67935e6i			1.38905i	0.719468	+	0.694525i	$0.244386\pi$
$271$			1.67935e6i			1.38905i	−0.719468	+	0.694525i	$0.755614\pi$
$272$		0			0
$273$	244604.	−	106318.i	0.198635	−	0.0863380i
$274$		0			0
$275$	−207794.			−0.165692
$276$		0			0
$277$	371115.			0.290609			0.145304	−	0.989387i	$-0.453584\pi$
$277$	371115.			0.290609			0.145304	+	0.989387i	$0.453584\pi$
$278$		0			0
$279$	−302281.	−	282030.i	−0.232488	−	0.216913i
$280$		0			0
$281$			437573.i			0.330586i	0.986244	+	0.165293i	$0.0528570\pi$
$281$			437573.i			0.330586i	−0.986244	+	0.165293i	$0.947143\pi$
$282$		0			0
$283$			421090.i			0.312543i	0.987714	+	0.156271i	$0.0499474\pi$
$283$			421090.i			0.312543i	−0.987714	+	0.156271i	$0.950053\pi$
$284$		0			0
$285$	439692.	+	1.01159e6i	0.320654	+	0.737718i
$286$		0			0
$287$	1.10396e6			0.791128
$288$		0			0
$289$	84424.0			0.0594595
$290$		0			0
$291$	316207.	+	727487.i	0.218896	+	0.503608i
$292$		0			0
$293$		−	913641.i		−	0.621737i	−0.950453	−	0.310868i	$-0.899380\pi$
$293$		−	913641.i		−	0.621737i	0.950453	−	0.310868i	$-0.100620\pi$
$294$		0			0
$295$		−	936626.i		−	0.626630i
$296$		0			0
$297$	−408496.	−	144837.i	−0.268718	−	0.0952773i
$298$		0			0
$299$	−1.34205e6			−0.868145
$300$		0			0
$301$	−343791.			−0.218715
$302$		0			0
$303$	997902.	−	433744.i	0.624427	−	0.271411i
$304$		0			0
$305$		−	201473.i		−	0.124013i
$306$		0			0
$307$			1.04561e6i			0.633172i	0.948564	+	0.316586i	$0.102537\pi$
$307$			1.04561e6i			0.633172i	−0.948564	+	0.316586i	$0.897463\pi$
$308$		0			0
$309$	−1.17940e6	+	512634.i	−0.702693	+	0.305430i
$310$		0			0
$311$	−1.15394e6			−0.676522			−0.338261	−	0.941052i	$-0.609839\pi$
$311$	−1.15394e6			−0.676522			−0.338261	+	0.941052i	$0.609839\pi$
$312$		0			0
$313$	−423979.			−0.244615			−0.122308	−	0.992492i	$-0.539029\pi$
$313$	−423979.			−0.244615			−0.122308	+	0.992492i	$0.539029\pi$
$314$		0			0
$315$	−347159.	+	372086.i	−0.197130	+	0.211284i
$316$		0			0
$317$			1.89609e6i			1.05977i	0.848070	+	0.529885i	$0.177765\pi$
$317$			1.89609e6i			1.05977i	−0.848070	+	0.529885i	$0.822235\pi$
$318$		0			0
$319$		−	561967.i		−	0.309196i
$320$		0			0
$321$	−467353.	−	1.07523e6i	−0.253153	−	0.582421i
$322$		0			0
$323$	−2.26014e6			−1.20539
$324$		0			0
$325$	−536803.			−0.281907
$326$		0			0
$327$	−303550.	−	698369.i	−0.156986	−	0.361173i
$328$		0			0
$329$			769155.i			0.391764i
$330$		0			0
$331$		−	2.53608e6i		−	1.27231i	−0.771561	−	0.636156i	$-0.780524\pi$
$331$		−	2.53608e6i		−	1.27231i	0.771561	−	0.636156i	$-0.219476\pi$
$332$		0			0
$333$	2.48500e6	−	2.66343e6i	1.22805	−	1.31623i
$334$		0			0
$335$	−755063.			−0.367596
$336$		0			0
$337$	−1.10104e6			−0.528113			−0.264056	−	0.964507i	$-0.585061\pi$
$337$	−1.10104e6			−0.528113			−0.264056	+	0.964507i	$0.585061\pi$
$338$		0			0
$339$	650874.	−	282906.i	0.307608	−	0.133704i
$340$		0			0
$341$		−	194660.i		−	0.0906546i
$342$		0			0
$343$		−	1.75179e6i		−	0.803981i
$344$		0			0
$345$	2.34841e6	−	1.02075e6i	1.06225	−	0.461714i
$346$		0			0
$347$	−1.22857e6			−0.547742			−0.273871	−	0.961766i	$-0.588304\pi$
$347$	−1.22857e6			−0.547742			−0.273871	+	0.961766i	$0.588304\pi$
$348$		0			0
$349$	2.38685e6			1.04897			0.524483	−	0.851421i	$-0.324259\pi$
$349$	2.38685e6			1.04897			0.524483	+	0.851421i	$0.324259\pi$
$350$		0			0
$351$	−1.05528e6	−	374163.i	−0.457194	−	0.162104i
$352$		0			0
$353$		−	2.30477e6i		−	0.984444i	−0.870470	−	0.492222i	$-0.836185\pi$
$353$		−	2.30477e6i		−	0.984444i	0.870470	−	0.492222i	$-0.163815\pi$
$354$		0			0
$355$		−	1.85722e6i		−	0.782153i
$356$		0			0
$357$	−415668.	−	956314.i	−0.172614	−	0.397128i
$358$		0			0
$359$	−3.08815e6			−1.26463			−0.632314	−	0.774712i	$-0.717895\pi$
$359$	−3.08815e6			−1.26463			−0.632314	+	0.774712i	$0.717895\pi$
$360$		0			0
$361$	−1.34906e6			−0.544831
$362$		0			0
$363$	919422.	+	2.11529e6i	0.366225	+	0.842563i
$364$		0			0
$365$			2.65223e6i			1.04203i
$366$		0			0
$367$			4.24714e6i			1.64601i	0.568036	+	0.823004i	$0.307703\pi$
$367$			4.24714e6i			1.64601i	−0.568036	+	0.823004i	$0.692297\pi$
$368$		0			0
$369$	−3.38856e6	−	3.16155e6i	−1.29553	−	1.20874i
$370$		0			0
$371$	−764196.			−0.288251
$372$		0			0
$373$	4.46771e6			1.66270			0.831349	−	0.555751i	$-0.187569\pi$
$373$	4.46771e6			1.66270			0.831349	+	0.555751i	$0.187569\pi$
$374$		0			0
$375$	2.55565e6	−	1.11083e6i	0.938478	−	0.407915i
$376$		0			0
$377$		−	1.45175e6i		−	0.526064i
$378$		0			0
$379$		−	2.52600e6i		−	0.903307i	−0.892193	−	0.451654i	$-0.850834\pi$
$379$		−	2.52600e6i		−	0.903307i	0.892193	−	0.451654i	$-0.149166\pi$
$380$		0			0
$381$	1.61865e6	−	703556.i	0.571269	−	0.248305i
$382$		0			0
$383$	−3.52845e6			−1.22910			−0.614551	−	0.788877i	$-0.710663\pi$
$383$	−3.52845e6			−1.22910			−0.614551	+	0.788877i	$0.710663\pi$
$384$		0			0
$385$	−239612.			−0.0823866
$386$		0			0
$387$	1.05526e6	+	984561.i	0.358163	+	0.334168i
$388$		0			0
$389$		−	2.79143e6i		−	0.935303i	−0.883913	−	0.467652i	$-0.845100\pi$
$389$		−	2.79143e6i		−	0.935303i	0.883913	−	0.467652i	$-0.154900\pi$
$390$		0			0
$391$			5.24696e6i			1.73566i
$392$		0			0
$393$	−457300.	−	1.05210e6i	−0.149355	−	0.343617i
$394$		0			0
$395$	1.68119e6			0.542155
$396$		0			0
$397$	−1.12421e6			−0.357989			−0.178995	−	0.983850i	$-0.557284\pi$
$397$	−1.12421e6			−0.357989			−0.178995	+	0.983850i	$0.557284\pi$
$398$		0			0
$399$	−703493.	−	1.61851e6i	−0.221222	−	0.508958i
$400$		0			0
$401$			3.16095e6i			0.981651i	0.871258	+	0.490825i	$0.163305\pi$
$401$			3.16095e6i			0.981651i	−0.871258	+	0.490825i	$0.836695\pi$
$402$		0			0
$403$		−	502871.i		−	0.154239i
$404$		0			0
$405$	2.13119e6	−	147901.i	0.645630	−	0.0448058i
$406$		0			0
$407$	1.71517e6			0.513240
$408$		0			0
$409$	−1.26449e6			−0.373772			−0.186886	−	0.982382i	$-0.559839\pi$
$409$	−1.26449e6			−0.373772			−0.186886	+	0.982382i	$0.559839\pi$
$410$		0			0
$411$	801227.	−	348258.i	0.233965	−	0.101694i
$412$		0			0
$413$			1.49857e6i			0.432317i
$414$		0			0
$415$		−	3.45641e6i		−	0.985156i
$416$		0			0
$417$	−881105.	+	382978.i	−0.248135	+	0.107853i
$418$		0			0
$419$	−984075.			−0.273838			−0.136919	−	0.990582i	$-0.543720\pi$
$419$	−984075.			−0.273838			−0.136919	+	0.990582i	$0.543720\pi$
$420$		0			0
$421$	994173.			0.273374			0.136687	−	0.990614i	$-0.456355\pi$
$421$	994173.			0.273374			0.136687	+	0.990614i	$0.456355\pi$
$422$		0			0
$423$	2.20273e6	−	2.36090e6i	0.598565	−	0.641544i
$424$		0			0
$425$			2.09871e6i			0.563612i
$426$		0			0
$427$			322351.i			0.0855577i
$428$		0			0
$429$	−210154.	−	483495.i	−0.0551308	−	0.126838i
$430$		0			0
$431$	243593.			0.0631643			0.0315822	−	0.999501i	$-0.489945\pi$
$431$	243593.			0.0631643			0.0315822	+	0.999501i	$0.489945\pi$
$432$		0			0
$433$	585515.			0.150078			0.0750392	−	0.997181i	$-0.476092\pi$
$433$	585515.			0.150078			0.0750392	+	0.997181i	$0.476092\pi$
$434$		0			0
$435$	1.10419e6	+	2.54037e6i	0.279781	+	0.643685i
$436$		0			0
$437$			8.88017e6i			2.22442i
$438$		0			0
$439$			2.89925e6i			0.717999i	0.933338	+	0.358999i	$0.116882\pi$
$439$			2.89925e6i			0.717999i	−0.933338	+	0.358999i	$0.883118\pi$
$440$		0			0
$441$	−2.23069e6	+	2.39086e6i	−0.546189	+	0.585408i
$442$		0			0
$443$	−3.02564e6			−0.732501			−0.366251	−	0.930516i	$-0.619359\pi$
$443$	−3.02564e6			−0.732501			−0.366251	+	0.930516i	$0.619359\pi$
$444$		0			0
$445$	−3.16808e6			−0.758395
$446$		0			0
$447$	4.55597e6	−	1.98028e6i	1.07848	−	0.468768i
$448$		0			0
$449$			339862.i			0.0795585i	0.999208	+	0.0397793i	$0.0126655\pi$
$449$			339862.i			0.0795585i	−0.999208	+	0.0397793i	$0.987335\pi$
$450$		0			0
$451$		−	2.18213e6i		−	0.505172i
$452$		0			0
$453$	−6.19819e6	+	2.69408e6i	−1.41912	+	0.616829i
$454$		0			0
$455$	−618998.			−0.140172
$456$		0			0
$457$	−4.12317e6			−0.923508			−0.461754	−	0.887008i	$-0.652780\pi$
$457$	−4.12317e6			−0.923508			−0.461754	+	0.887008i	$0.652780\pi$
$458$		0			0
$459$	−1.46285e6	+	4.12578e6i	−0.324091	+	0.914060i
$460$		0			0
$461$		−	1.40750e6i		−	0.308457i	−0.988035	−	0.154229i	$-0.950711\pi$
$461$		−	1.40750e6i		−	0.308457i	0.988035	−	0.154229i	$-0.0492892\pi$
$462$		0			0
$463$			373619.i			0.0809985i	0.999180	+	0.0404992i	$0.0128948\pi$
$463$			373619.i			0.0809985i	−0.999180	+	0.0404992i	$0.987105\pi$
$464$		0			0
$465$	382478.	+	879957.i	0.0820304	+	0.188725i
$466$		0			0
$467$	8.73587e6			1.85359			0.926795	−	0.375567i	$-0.122552\pi$
$467$	8.73587e6			1.85359			0.926795	+	0.375567i	$0.122552\pi$
$468$		0			0
$469$	1.20808e6			0.253608
$470$		0			0
$471$	−851914.	−	1.95997e6i	−0.176947	−	0.407097i
$472$		0			0
$473$			679553.i			0.139659i
$474$		0			0
$475$			3.55194e6i			0.722324i
$476$		0			0
$477$	2.34568e6	+	2.18853e6i	0.472033	+	0.440410i
$478$		0			0
$479$	6.56616e6			1.30759			0.653797	−	0.756670i	$-0.273175\pi$
$479$	6.56616e6			1.30759			0.653797	+	0.756670i	$0.273175\pi$
$480$		0			0
$481$	4.43085e6			0.873221
$482$		0			0
$483$	−3.75739e6	+	1.63317e6i	−0.732856	+	0.318540i
$484$		0			0
$485$		−	1.84099e6i		−	0.355383i
$486$		0			0
$487$			6.03619e6i			1.15329i	0.816993	+	0.576647i	$0.195639\pi$
$487$			6.03619e6i			1.15329i	−0.816993	+	0.576647i	$0.804361\pi$
$488$		0			0
$489$	9.42588e6	−	4.09702e6i	1.78258	−	0.774811i
$490$		0			0
$491$	−1.73841e6			−0.325424			−0.162712	−	0.986674i	$-0.552024\pi$
$491$	−1.73841e6			−0.325424			−0.162712	+	0.986674i	$0.552024\pi$
$492$		0			0
$493$	−5.67583e6			−1.05175
$494$		0			0
$495$	735482.	+	686210.i	0.134915	+	0.125876i
$496$		0			0
$497$			2.97149e6i			0.539614i
$498$		0			0
$499$		−	5.57823e6i		−	1.00287i	−0.865195	−	0.501436i	$-0.832805\pi$
$499$		−	5.57823e6i		−	1.00287i	0.865195	−	0.501436i	$-0.167195\pi$
$500$		0			0
$501$	−1.01726e6	−	2.34037e6i	−0.181066	−	0.416573i
$502$		0			0
$503$	5.61205e6			0.989012			0.494506	−	0.869174i	$-0.335349\pi$
$503$	5.61205e6			0.989012			0.494506	+	0.869174i	$0.335349\pi$
$504$		0			0
$505$	−2.52531e6			−0.440642
$506$		0			0
$507$	1.76432e6	+	4.05911e6i	0.304829	+	0.701312i
$508$		0			0
$509$		−	6.81201e6i		−	1.16542i	−0.812682	−	0.582708i	$-0.801993\pi$
$509$		−	6.81201e6i		−	1.16542i	0.812682	−	0.582708i	$-0.198007\pi$
$510$		0			0
$511$		−	4.24349e6i		−	0.718904i
$512$		0			0
$513$	−2.47578e6	+	6.98264e6i	−0.415355	+	1.17146i
$514$		0			0
$515$	2.98461e6			0.495873
$516$		0			0
$517$	1.52035e6			0.250159
$518$		0			0
$519$	−9.16029e6	+	3.98157e6i	−1.49276	+	0.648838i
$520$		0			0
$521$		−	3.72615e6i		−	0.601404i	−0.953718	−	0.300702i	$-0.902779\pi$
$521$		−	3.72615e6i		−	0.601404i	0.953718	−	0.300702i	$-0.0972209\pi$
$522$		0			0
$523$			1.32730e6i			0.212185i	0.994356	+	0.106092i	$0.0338339\pi$
$523$			1.32730e6i			0.212185i	−0.994356	+	0.106092i	$0.966166\pi$
$524$		0			0
$525$	−1.50290e6	+	653246.i	−0.237976	+	0.103438i
$526$		0			0
$527$	−1.96605e6			−0.308367
$528$		0			0
$529$	1.41791e7			2.20298
$530$		0			0
$531$	4.29167e6	−	4.59983e6i	0.660526	−	0.707954i
$532$		0			0
$533$		−	5.63717e6i		−	0.859494i
$534$		0			0
$535$			2.72098e6i			0.411000i
$536$		0			0
$537$	2.10292e6	+	4.83812e6i	0.314693	+	0.724004i
$538$		0			0
$539$	−1.53964e6			−0.228270
$540$		0			0
$541$	−7.17248e6			−1.05360			−0.526801	−	0.849989i	$-0.676609\pi$
$541$	−7.17248e6			−1.05360			−0.526801	+	0.849989i	$0.676609\pi$
$542$		0			0
$543$	−4.30079e6	−	9.89470e6i	−0.625963	−	1.44013i
$544$		0			0
$545$			1.76730e6i			0.254871i
$546$		0			0
$547$		−	9.86107e6i		−	1.40914i	−0.709632	−	0.704572i	$-0.751139\pi$
$547$		−	9.86107e6i		−	1.40914i	0.709632	−	0.704572i	$-0.248861\pi$
$548$		0			0
$549$	923160.	−	989446.i	0.130721	−	0.140107i
$550$		0			0
$551$	−9.60601e6			−1.34792
$552$		0			0
$553$	−2.68985e6			−0.374037
$554$		0			0
$555$	−7.75339e6	+	3.37006e6i	−1.06846	+	0.464413i
$556$		0			0
$557$		−	9.22581e6i		−	1.25999i	−0.776600	−	0.629994i	$-0.783057\pi$
$557$		−	9.22581e6i		−	1.25999i	0.776600	−	0.629994i	$-0.216943\pi$
$558$		0			0
$559$			1.75551e6i			0.237615i
$560$		0			0
$561$	−1.89029e6	+	821628.i	−0.253584	+	0.110222i
$562$		0			0
$563$	−1.00543e7			−1.33685			−0.668425	−	0.743780i	$-0.733031\pi$
$563$	−1.00543e7			−1.33685			−0.668425	+	0.743780i	$0.733031\pi$
$564$		0			0
$565$	−1.64711e6			−0.217071
$566$		0			0
$567$	−3.40983e6	+	236637.i	−0.445426	+	0.0309119i
$568$		0			0
$569$		−	1.29750e7i		−	1.68007i	−0.542530	−	0.840036i	$-0.682534\pi$
$569$		−	1.29750e7i		−	1.68007i	0.542530	−	0.840036i	$-0.317466\pi$
$570$		0			0
$571$			102836.i			0.0131994i	0.999978	+	0.00659972i	$0.00210077\pi$
$571$			102836.i			0.0131994i	−0.999978	+	0.00659972i	$0.997899\pi$
$572$		0			0
$573$	153414.	+	352956.i	0.0195200	+	0.0449090i
$574$		0			0
$575$	8.24590e6			1.04008
$576$		0			0
$577$	6.88650e6			0.861111			0.430556	−	0.902564i	$-0.358318\pi$
$577$	6.88650e6			0.861111			0.430556	+	0.902564i	$0.358318\pi$
$578$		0			0
$579$	−3.50735e6	−	8.06925e6i	−0.434793	−	1.00032i
$580$		0			0
$581$			5.53015e6i			0.679668i
$582$		0			0
$583$			1.51054e6i			0.184061i
$584$		0			0
$585$	1.90000e6	+	1.77271e6i	0.229543	+	0.214165i
$586$		0			0
$587$	1.07376e7			1.28621			0.643103	−	0.765780i	$-0.277647\pi$
$587$	1.07376e7			1.28621			0.643103	+	0.765780i	$0.277647\pi$
$588$		0			0
$589$	−3.32742e6			−0.395202
$590$		0			0
$591$	−9.01873e6	+	3.92004e6i	−1.06213	+	0.461660i
$592$		0			0
$593$		−	1.12656e7i		−	1.31558i	−0.753199	−	0.657792i	$-0.771490\pi$
$593$		−	1.12656e7i		−	1.31558i	0.753199	−	0.657792i	$-0.228510\pi$
$594$		0			0
$595$			2.42007e6i			0.280243i
$596$		0			0
$597$	−1.35423e7	+	5.88623e6i	−1.55509	+	0.675930i
$598$		0			0
$599$	−9.18305e6			−1.04573			−0.522865	−	0.852415i	$-0.675137\pi$
$599$	−9.18305e6			−1.04573			−0.522865	+	0.852415i	$0.675137\pi$
$600$		0			0
$601$	−27562.1			−0.00311261			−0.00155631	−	0.999999i	$-0.500495\pi$
$601$	−27562.1			−0.00311261			−0.00155631	+	0.999999i	$0.500495\pi$
$602$		0			0
$603$	−3.70816e6	−	3.45973e6i	−0.415303	−	0.387480i
$604$		0			0
$605$		−	5.35298e6i		−	0.594575i
$606$		0			0
$607$		−	9.94174e6i		−	1.09519i	−0.836743	−	0.547596i	$-0.815543\pi$
$607$		−	9.94174e6i		−	1.09519i	0.836743	−	0.547596i	$-0.184457\pi$
$608$		0			0
$609$	−1.76666e6	−	4.06451e6i	−0.193024	−	0.444084i
$610$		0			0
$611$	3.92756e6			0.425618
$612$		0			0
$613$	1.39606e7			1.50056			0.750281	−	0.661119i	$-0.229918\pi$
$613$	1.39606e7			1.50056			0.750281	+	0.661119i	$0.229918\pi$
$614$		0			0
$615$	4.28757e6	+	9.86428e6i	0.457113	+	1.05167i
$616$		0			0
$617$			1.58730e7i			1.67860i	0.543671	+	0.839299i	$0.317034\pi$
$617$			1.58730e7i			1.67860i	−0.543671	+	0.839299i	$0.682966\pi$
$618$		0			0
$619$			7.64068e6i			0.801504i	0.916187	+	0.400752i	$0.131251\pi$
$619$			7.64068e6i			0.801504i	−0.916187	+	0.400752i	$0.868749\pi$
$620$		0			0
$621$	1.62103e7	+	5.74758e6i	1.68680	+	0.598075i
$622$		0			0
$623$	5.06882e6			0.523223
$624$		0			0
$625$	−792050.			−0.0811059
$626$		0			0
$627$	−3.19921e6	+	1.39056e6i	−0.324993	+	0.141260i
$628$		0			0
$629$		−	1.73231e7i		−	1.74581i
$630$		0			0
$631$		−	4.52822e6i		−	0.452746i	−0.974041	−	0.226373i	$-0.927313\pi$
$631$		−	4.52822e6i		−	0.452746i	0.974041	−	0.226373i	$-0.0726868\pi$
$632$		0			0
$633$	7.46548e6	−	3.24492e6i	0.740540	−	0.321880i
$634$		0			0
$635$	−4.09618e6			−0.403130
$636$		0			0
$637$	−3.97742e6			−0.388376
$638$		0			0
$639$	8.50986e6	−	9.12089e6i	0.824461	−	0.883660i
$640$		0			0
$641$		−	9.34885e6i		−	0.898696i	−0.893357	−	0.449348i	$-0.851656\pi$
$641$		−	9.34885e6i		−	0.898696i	0.893357	−	0.449348i	$-0.148344\pi$
$642$		0			0
$643$			2.98305e6i			0.284533i	0.989828	+	0.142267i	$0.0454390\pi$
$643$			2.98305e6i			0.284533i	−0.989828	+	0.142267i	$0.954561\pi$
$644$		0			0
$645$	−1.33522e6	−	3.07191e6i	−0.126373	−	0.290743i
$646$		0			0
$647$	1.92362e7			1.80659			0.903295	−	0.429020i	$-0.141141\pi$
$647$	1.92362e7			1.80659			0.903295	+	0.429020i	$0.141141\pi$
$648$		0			0
$649$	2.96215e6			0.276055
$650$		0			0
$651$	−611954.	−	1.40790e6i	−0.0565935	−	0.130203i
$652$		0			0
$653$			1.55600e7i			1.42799i	0.700149	+	0.713997i	$0.253117\pi$
$653$			1.55600e7i			1.42799i	−0.700149	+	0.713997i	$0.746883\pi$
$654$		0			0
$655$			2.66245e6i			0.242482i
$656$		0			0
$657$	−1.21527e7	+	1.30253e7i	−1.09839	+	1.17726i
$658$		0			0
$659$	8.76971e6			0.786632			0.393316	−	0.919403i	$-0.371328\pi$
$659$	8.76971e6			0.786632			0.393316	+	0.919403i	$0.371328\pi$
$660$		0			0
$661$	1.72564e6			0.153619			0.0768096	−	0.997046i	$-0.475527\pi$
$661$	1.72564e6			0.153619			0.0768096	+	0.997046i	$0.475527\pi$
$662$		0			0
$663$	−4.88326e6	+	2.12254e6i	−0.431446	+	0.187531i
$664$		0			0
$665$			4.09582e6i			0.359159i
$666$		0			0
$667$			2.23005e7i			1.94089i
$668$		0			0
$669$	4.79064e6	−	2.08228e6i	0.413836	−	0.179876i
$670$		0			0
$671$	637173.			0.0546325
$672$		0			0
$673$	−1.72691e7			−1.46971			−0.734855	−	0.678225i	$-0.762750\pi$
$673$	−1.72691e7			−1.46971			−0.734855	+	0.678225i	$0.762750\pi$
$674$		0			0
$675$	6.48390e6	+	2.29895e6i	0.547743	+	0.194209i
$676$		0			0
$677$			1.63692e7i			1.37264i	0.727300	+	0.686320i	$0.240775\pi$
$677$			1.63692e7i			1.37264i	−0.727300	+	0.686320i	$0.759225\pi$
$678$		0			0
$679$			2.94553e6i			0.245182i
$680$		0			0
$681$	−3.37315e6	−	7.76049e6i	−0.278720	−	0.641241i
$682$		0			0
$683$	−6.24061e6			−0.511889			−0.255944	−	0.966692i	$-0.582386\pi$
$683$	−6.24061e6			−0.511889			−0.255944	+	0.966692i	$0.582386\pi$
$684$		0			0
$685$	−2.02760e6			−0.165103
$686$		0			0
$687$	7.07052e6	+	1.62669e7i	0.571557	+	1.31496i
$688$		0			0
$689$			3.90224e6i			0.313160i
$690$		0			0
$691$		−	1.18638e7i		−	0.945211i	−0.881274	−	0.472605i	$-0.843314\pi$
$691$		−	1.18638e7i		−	0.945211i	0.881274	−	0.472605i	$-0.156686\pi$
$692$		0			0
$693$	−1.17675e6	−	1.09791e6i	−0.0930788	−	0.0868431i
$694$		0			0
$695$	2.22974e6			0.175102
$696$		0			0
$697$	−2.20393e7			−1.71837
$698$		0			0
$699$	−523412.	+	227504.i	−0.0405183	+	0.0176115i
$700$		0			0
$701$			1.12864e7i			0.867482i	0.901038	+	0.433741i	$0.142807\pi$
$701$			1.12864e7i			0.867482i	−0.901038	+	0.433741i	$0.857193\pi$
$702$		0			0
$703$		−	2.93182e7i		−	2.23743i
$704$		0			0
$705$	−6.87271e6	+	2.98726e6i	−0.520781	+	0.226361i
$706$		0			0
$707$	4.04042e6			0.304003
$708$		0			0
$709$	1.04073e7			0.777538			0.388769	−	0.921335i	$-0.372900\pi$
$709$	1.04073e7			0.777538			0.388769	+	0.921335i	$0.372900\pi$
$710$		0			0
$711$	8.25641e6	+	7.70328e6i	0.612515	+	0.571481i
$712$		0			0
$713$			7.72467e6i			0.569058i
$714$		0			0
$715$			1.22354e6i			0.0895062i
$716$		0			0
$717$	5.11014e6	+	1.17567e7i	0.371223	+	0.854061i
$718$		0			0
$719$	−1.42216e7			−1.02595			−0.512977	−	0.858403i	$-0.671457\pi$
$719$	−1.42216e7			−1.02595			−0.512977	+	0.858403i	$0.671457\pi$
$720$		0			0
$721$	−4.77529e6			−0.342107
$722$		0			0
$723$	−4.39633e6	−	1.01145e7i	−0.312784	−	0.719613i
$724$		0			0
$725$			8.91990e6i			0.630253i
$726$		0			0
$727$			2.21710e6i			0.155578i	0.996970	+	0.0777892i	$0.0247861\pi$
$727$			2.21710e6i			0.155578i	−0.996970	+	0.0777892i	$0.975214\pi$
$728$		0			0
$729$	1.11441e7	+	9.03885e6i	0.776649	+	0.629933i
$730$		0			0
$731$	6.86344e6			0.475060
$732$		0			0
$733$	1.06428e7			0.731635			0.365818	−	0.930687i	$-0.380789\pi$
$733$	1.06428e7			0.731635			0.365818	+	0.930687i	$0.380789\pi$
$734$		0			0
$735$	6.95994e6	−	3.02518e6i	0.475212	−	0.206554i
$736$		0			0
$737$		−	2.38794e6i		−	0.161940i
$738$		0			0
$739$			1.00821e7i			0.679111i	0.940586	+	0.339555i	$0.110277\pi$
$739$			1.00821e7i			0.679111i	−0.940586	+	0.339555i	$0.889723\pi$
$740$		0			0
$741$	−8.26463e6	+	3.59227e6i	−0.552940	+	0.240339i
$742$		0			0
$743$	−2.11350e7			−1.40453			−0.702265	−	0.711916i	$-0.747828\pi$
$743$	−2.11350e7			−1.40453			−0.702265	+	0.711916i	$0.747828\pi$
$744$		0			0
$745$	−1.15294e7			−0.761056
$746$		0			0
$747$	1.58374e7	−	1.69746e7i	1.03845	−	1.11301i
$748$		0			0
$749$		−	4.35349e6i		−	0.283552i
$750$		0			0
$751$			1.67578e7i			1.08422i	0.840308	+	0.542110i	$0.182374\pi$
$751$			1.67578e7i			1.08422i	−0.840308	+	0.542110i	$0.817626\pi$
$752$		0			0
$753$	−1.18051e7	−	2.71597e7i	−0.758723	−	1.74557i
$754$		0			0
$755$	1.56852e7			1.00144
$756$		0			0
$757$	1.55144e6			0.0983998			0.0491999	−	0.998789i	$-0.484333\pi$
$757$	1.55144e6			0.0983998			0.0491999	+	0.998789i	$0.484333\pi$
$758$		0			0
$759$	3.22820e6	+	7.42703e6i	0.203403	+	0.467962i
$760$		0			0
$761$			4.06490e6i			0.254441i	0.991874	+	0.127221i	$0.0406057\pi$
$761$			4.06490e6i			0.254441i	−0.991874	+	0.127221i	$0.959394\pi$
$762$		0			0
$763$		−	2.82763e6i		−	0.175838i
$764$		0			0
$765$	6.93067e6	−	7.42832e6i	0.428175	−	0.458920i
$766$		0			0
$767$	7.65222e6			0.469677
$768$		0			0
$769$	2.26089e7			1.37868			0.689339	−	0.724439i	$-0.257901\pi$
$769$	2.26089e7			1.37868			0.689339	+	0.724439i	$0.257901\pi$
$770$		0			0
$771$	1.61415e7	−	7.01599e6i	0.977929	−	0.425063i
$772$		0			0
$773$		−	1.50729e6i		−	0.0907293i	−0.998970	−	0.0453647i	$-0.985555\pi$
$773$		−	1.50729e6i		−	0.0907293i	0.998970	−	0.0453647i	$-0.0144450\pi$
$774$		0			0
$775$			3.08976e6i			0.184787i
$776$		0			0
$777$	1.24052e7	−	5.39199e6i	0.737141	−	0.320403i
$778$		0			0
$779$	−3.73003e7			−2.20226
$780$		0			0
$781$	5.87358e6			0.344568
$782$		0			0
$783$	−6.21737e6	+	1.75353e7i	−0.362412	+	1.02214i
$784$		0			0
$785$			4.95994e6i			0.287278i
$786$		0			0
$787$			1.44318e7i			0.830585i	0.909688	+	0.415293i	$0.136321\pi$
$787$			1.44318e7i			0.830585i	−0.909688	+	0.415293i	$0.863679\pi$
$788$		0			0
$789$	−1.18435e7	−	2.72480e7i	−0.677310	−	1.55827i
$790$		0			0
$791$	2.63533e6			0.149759
$792$		0			0
$793$	1.64603e6			0.0929512
$794$		0			0
$795$	−2.96801e6	−	6.82840e6i	−0.166551	−	0.383178i
$796$		0			0
$797$			1.87022e7i			1.04291i	0.853278	+	0.521456i	$0.174611\pi$
$797$			1.87022e7i			1.04291i	−0.853278	+	0.521456i	$0.825389\pi$
$798$		0			0
$799$		−	1.53554e7i		−	0.850930i
$800$		0			0
$801$	−1.55586e7	−	1.45163e7i	−0.856820	−	0.799418i
$802$		0			0
$803$	−8.38788e6			−0.459053
$804$		0			0
$805$	9.50852e6			0.517158
$806$		0			0
$807$	−3.15545e6	+	1.37154e6i	−0.170560	+	0.0741351i
$808$		0			0
$809$		−	2.72136e7i		−	1.46189i	−0.682435	−	0.730946i	$-0.739079\pi$
$809$		−	2.72136e7i		−	1.46189i	0.682435	−	0.730946i	$-0.260921\pi$
$810$		0			0
$811$			1.16663e7i			0.622847i	0.950271	+	0.311424i	$0.100806\pi$
$811$			1.16663e7i			0.622847i	−0.950271	+	0.311424i	$0.899194\pi$
$812$		0			0
$813$	2.40086e7	−	1.04355e7i	1.27392	−	0.553715i
$814$		0			0
$815$	−2.38533e7			−1.25792
$816$		0			0
$817$	1.16160e7			0.608836
$818$		0			0
$819$	−3.03994e6	−	2.83628e6i	−0.158363	−	0.147754i
$820$		0			0
$821$		−	1.60653e7i		−	0.831822i	−0.909405	−	0.415911i	$-0.863463\pi$
$821$		−	1.60653e7i		−	0.831822i	0.909405	−	0.415911i	$-0.136537\pi$
$822$		0			0
$823$		−	3.14851e7i		−	1.62034i	−0.586197	−	0.810169i	$-0.699375\pi$
$823$		−	3.14851e7i		−	1.62034i	0.586197	−	0.810169i	$-0.300625\pi$
$824$		0			0
$825$	1.29124e6	+	2.97071e6i	0.0660496	+	0.151958i
$826$		0			0
$827$	1.13442e7			0.576779			0.288389	−	0.957513i	$-0.406880\pi$
$827$	1.13442e7			0.576779			0.288389	+	0.957513i	$0.406880\pi$
$828$		0			0
$829$	−2.00733e7			−1.01446			−0.507228	−	0.861812i	$-0.669330\pi$
$829$	−2.00733e7			−1.01446			−0.507228	+	0.861812i	$0.669330\pi$
$830$		0			0
$831$	−2.30611e6	−	5.30559e6i	−0.115845	−	0.266521i
$832$		0			0
$833$			1.55503e7i			0.776473i
$834$		0			0
$835$			5.92259e6i			0.293965i
$836$		0			0
$837$	−2.15363e6	+	6.07405e6i	−0.106257	+	0.299685i
$838$		0			0
$839$	−1.22032e7			−0.598508			−0.299254	−	0.954174i	$-0.596738\pi$
$839$	−1.22032e7			−0.598508			−0.299254	+	0.954174i	$0.596738\pi$
$840$		0			0
$841$	−3.61216e6			−0.176107
$842$		0			0
$843$	6.25571e6	−	2.71908e6i	0.303185	−	0.131781i
$844$		0			0
$845$		−	1.02721e7i		−	0.494898i
$846$		0			0
$847$			8.56460e6i			0.410203i
$848$		0			0
$849$	6.02006e6	−	2.61666e6i	0.286637	−	0.124588i
$850$		0			0
$851$	−6.80629e7			−3.22171
$852$		0			0
$853$	−708002.			−0.0333167			−0.0166583	−	0.999861i	$-0.505303\pi$
$853$	−708002.			−0.0333167			−0.0166583	+	0.999861i	$0.505303\pi$
$854$		0			0
$855$	1.17297e7	−	1.25720e7i	0.548749	−	0.588151i
$856$		0			0
$857$			1.01954e7i			0.474192i	0.971486	+	0.237096i	$0.0761956\pi$
$857$			1.01954e7i			0.474192i	−0.971486	+	0.237096i	$0.923804\pi$
$858$		0			0
$859$		−	3.14505e7i		−	1.45427i	−0.686496	−	0.727134i	$-0.740852\pi$
$859$		−	3.14505e7i		−	1.45427i	0.686496	−	0.727134i	$-0.259148\pi$
$860$		0			0
$861$	−6.85998e6	−	1.57826e7i	−0.315366	−	0.725553i
$862$		0			0
$863$	1.33323e7			0.609364			0.304682	−	0.952454i	$-0.401450\pi$
$863$	1.33323e7			0.609364			0.304682	+	0.952454i	$0.401450\pi$
$864$		0			0
$865$	2.31812e7			1.05340
$866$		0			0
$867$	−524611.	−	1.20696e6i	−0.0237023	−	0.0545311i
$868$		0			0
$869$			5.31687e6i			0.238840i
$870$		0			0
$871$		−	6.16885e6i		−	0.275524i
$872$		0			0
$873$	8.43551e6	−	9.04121e6i	0.374607	−	0.401505i
$874$		0			0
$875$	1.03476e7			0.456899
$876$		0			0
$877$	−3.28216e7			−1.44099			−0.720495	−	0.693460i	$-0.756085\pi$
$877$	−3.28216e7			−1.44099			−0.720495	+	0.693460i	$0.756085\pi$
$878$		0			0
$879$	−1.30617e7	+	5.67737e6i	−0.570202	+	0.247842i
$880$		0			0
$881$			1.61293e7i			0.700125i	0.936726	+	0.350062i	$0.113840\pi$
$881$			1.61293e7i			0.700125i	−0.936726	+	0.350062i	$0.886160\pi$
$882$		0			0
$883$			1.75882e7i			0.759138i	0.925163	+	0.379569i	$0.123928\pi$
$883$			1.75882e7i			0.759138i	−0.925163	+	0.379569i	$0.876072\pi$
$884$		0			0
$885$	−1.33904e7	+	5.82020e6i	−0.574690	+	0.249793i
$886$		0			0
$887$	1.13047e7			0.482447			0.241224	−	0.970470i	$-0.422451\pi$
$887$	1.13047e7			0.482447			0.241224	+	0.970470i	$0.422451\pi$
$888$		0			0
$889$	6.55377e6			0.278123
$890$		0			0
$891$	467748.	+	6.74003e6i	0.0197387	+	0.284425i
$892$		0			0
$893$		−	2.59881e7i		−	1.09055i
$894$		0			0
$895$		−	1.22434e7i		−	0.510911i
$896$		0			0
$897$	8.33953e6	+	1.91865e7i	0.346067	+	0.796186i
$898$		0			0
$899$	−8.35606e6			−0.344828
$900$		0			0
$901$	1.52564e7			0.626095
$902$		0			0
$903$	2.13632e6	+	4.91496e6i	0.0871860	+	0.200586i
$904$		0			0
$905$			2.50397e7i			1.01627i
$906$		0			0
$907$		−	8.83914e6i		−	0.356773i	−0.983961	−	0.178386i	$-0.942912\pi$
$907$		−	8.83914e6i		−	0.356773i	0.983961	−	0.178386i	$-0.0570877\pi$
$908$		0			0
$909$	−1.24019e7	−	1.15711e7i	−0.497829	−	0.464477i
$910$		0			0
$911$	−1.27645e7			−0.509575			−0.254788	−	0.966997i	$-0.582006\pi$
$911$	−1.27645e7			−0.509575			−0.254788	+	0.966997i	$0.582006\pi$
$912$		0			0
$913$	1.09311e7			0.433999
$914$		0			0
$915$	−2.88033e6	+	1.25195e6i	−0.113734	+	0.0494351i
$916$		0			0
$917$		−	4.25984e6i		−	0.167290i
$918$		0			0
$919$		−	4.57548e7i		−	1.78710i	−0.448967	−	0.893548i	$-0.648208\pi$
$919$		−	4.57548e7i		−	1.78710i	0.448967	−	0.893548i	$-0.351792\pi$
$920$		0			0
$921$	1.49484e7	−	6.49740e6i	0.580690	−	0.252401i
$922$		0			0
$923$	1.51734e7			0.586245
$924$		0			0
$925$	−2.72242e7			−1.04617
$926$		0			0
$927$	1.46576e7	+	1.36757e7i	0.560227	+	0.522695i
$928$		0			0
$929$		−	1.32383e7i		−	0.503259i	−0.967824	−	0.251630i	$-0.919034\pi$
$929$		−	1.32383e7i		−	0.503259i	0.967824	−	0.251630i	$-0.0809665\pi$
$930$		0			0
$931$			2.63180e7i			0.995126i
$932$		0			0
$933$	7.17058e6	+	1.64971e7i	0.269681	+	0.620446i
$934$		0			0
$935$	4.78361e6			0.178948
$936$		0			0
$937$	−2.71022e7			−1.00845			−0.504226	−	0.863572i	$-0.668222\pi$
$937$	−2.71022e7			−1.00845			−0.504226	+	0.863572i	$0.668222\pi$
$938$		0			0
$939$	2.63461e6	+	6.06136e6i	0.0975107	+	0.224340i
$940$		0			0
$941$			3.12202e7i			1.14938i	0.818373	+	0.574688i	$0.194876\pi$
$941$			3.12202e7i			1.14938i	−0.818373	+	0.574688i	$0.805124\pi$
$942$		0			0
$943$			8.65933e7i			3.17106i
$944$		0			0
$945$	7.47672e6	+	2.65097e6i	0.272353	+	0.0965661i
$946$		0			0
$947$	3.42016e7			1.23929			0.619643	−	0.784883i	$-0.287277\pi$
$947$	3.42016e7			1.23929			0.619643	+	0.784883i	$0.287277\pi$
$948$		0			0
$949$	−2.16687e7			−0.781029
$950$		0			0
$951$	2.71072e7	−	1.17823e7i	0.971928	−	0.422454i
$952$		0			0
$953$		−	4.23251e7i		−	1.50961i	−0.655947	−	0.754807i	$-0.727730\pi$
$953$		−	4.23251e7i		−	1.50961i	0.655947	−	0.754807i	$-0.272270\pi$
$954$		0			0
$955$		−	893196.i		−	0.0316912i
$956$		0			0
$957$	−8.03409e6	+	3.49207e6i	−0.283568	+	0.123254i
$958$		0			0
$959$	3.24410e6			0.113906
$960$		0			0
$961$	2.57347e7			0.898898
$962$		0			0
$963$	−1.24677e7	+	1.33629e7i	−0.433231	+	0.464339i
$964$		0			0
$965$			2.04202e7i			0.705897i
$966$		0			0
$967$			3.63922e7i			1.25153i	0.780010	+	0.625767i	$0.215214\pi$
$967$			3.63922e7i			1.25153i	−0.780010	+	0.625767i	$0.784786\pi$
$968$		0			0
$969$	1.40445e7	+	3.23118e7i	0.480505	+	1.10548i
$970$		0			0
$971$	4.11273e7			1.39985			0.699925	−	0.714216i	$-0.253217\pi$
$971$	4.11273e7			1.39985			0.699925	+	0.714216i	$0.253217\pi$
$972$		0			0
$973$	−3.56751e6			−0.120805
$974$		0			0
$975$	3.33569e6	+	7.67433e6i	0.112376	+	0.258541i
$976$		0			0
$977$			2.71917e7i			0.911380i	0.890138	+	0.455690i	$0.150608\pi$
$977$			2.71917e7i			0.911380i	−0.890138	+	0.455690i	$0.849392\pi$
$978$		0			0
$979$		−	1.00193e7i		−	0.334102i
$980$		0			0
$981$	−8.09788e6	+	8.67934e6i	−0.268657	+	0.287948i
$982$		0			0
$983$	2.74580e7			0.906326			0.453163	−	0.891428i	$-0.350295\pi$
$983$	2.74580e7			0.906326			0.453163	+	0.891428i	$0.350295\pi$
$984$		0			0
$985$	2.28229e7			0.749516
$986$		0			0
$987$	1.09961e7	−	4.77953e6i	0.359291	−	0.156168i
$988$		0			0
$989$		−	2.69667e7i		−	0.876671i
$990$		0			0
$991$		−	4.59414e7i		−	1.48601i	−0.669288	−	0.743003i	$-0.733401\pi$
$991$		−	4.59414e7i		−	1.48601i	0.669288	−	0.743003i	$-0.266599\pi$
$992$		0			0
$993$	−3.62568e7	+	1.57592e7i	−1.16685	+	0.507180i
$994$		0			0
$995$	3.42703e7			1.09739
$996$		0			0
$997$	2.10427e7			0.670445			0.335223	−	0.942139i	$-0.391188\pi$
$997$	2.10427e7			0.670445			0.335223	+	0.942139i	$0.391188\pi$
$998$		0			0
$999$	−5.35191e7	−	1.89759e7i	−1.69666	−	0.601572i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	96.6.c.a.95.9	✓	20
3.2	odd	2	inner	96.6.c.a.95.11	yes	20
4.3	odd	2	inner	96.6.c.a.95.12	yes	20
8.3	odd	2		192.6.c.f.191.9		20
8.5	even	2		192.6.c.f.191.12		20
12.11	even	2	inner	96.6.c.a.95.10	yes	20
24.5	odd	2		192.6.c.f.191.10		20
24.11	even	2		192.6.c.f.191.11		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
96.6.c.a.95.9	✓	20	1.1	even	1	trivial
96.6.c.a.95.10	yes	20	12.11	even	2	inner
96.6.c.a.95.11	yes	20	3.2	odd	2	inner
96.6.c.a.95.12	yes	20	4.3	odd	2	inner
192.6.c.f.191.9		20	8.3	odd	2
192.6.c.f.191.10		20	24.5	odd	2
192.6.c.f.191.11		20	24.11	even	2
192.6.c.f.191.12		20	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 96 = 2^{5} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	96.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-6.21400 - 14.2964i) q^{3} +36.1786i q^{5} -57.8847i q^{7} +(-165.772 + 177.675i) q^{9} +O(q^{10})\)	\(q+(-6.21400 - 14.2964i) q^{3} +36.1786i q^{5} -57.8847i q^{7} +(-165.772 + 177.675i) q^{9} -114.418 q^{11} -295.579 q^{13} +(517.223 - 224.814i) q^{15} +1155.61i q^{17} +1955.80i q^{19} +(-827.541 + 359.696i) q^{21} +4540.43 q^{23} +1816.11 q^{25} +(3570.22 + 1265.87i) q^{27} +4911.55i q^{29} +1701.31i q^{31} +(710.991 + 1635.76i) q^{33} +2094.19 q^{35} -14990.4 q^{37} +(1836.73 + 4225.70i) q^{39} +19071.6i q^{41} -5939.24i q^{43} +(-6428.05 - 5997.42i) q^{45} -13287.7 q^{47} +13456.4 q^{49} +(16521.0 - 7180.96i) q^{51} -13202.0i q^{53} -4139.47i q^{55} +(27960.8 - 12153.3i) q^{57} -25888.9 q^{59} -5568.84 q^{61} +(10284.7 + 9595.68i) q^{63} -10693.6i q^{65} +20870.4i q^{67} +(-28214.2 - 64911.6i) q^{69} -51334.6 q^{71} +73309.4 q^{73} +(-11285.3 - 25963.7i) q^{75} +6623.03i q^{77} -46469.0i q^{79} +(-4088.08 - 58907.3i) q^{81} -95537.3 q^{83} -41808.4 q^{85} +(70217.3 - 30520.4i) q^{87} +87567.6i q^{89} +17109.5i q^{91} +(24322.5 - 10571.9i) q^{93} -70758.2 q^{95} -50886.1 q^{97} +(18967.3 - 20329.2i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 44 q^{9}+O(q^{10}) \) 20 * q - 44 * q^9	\( 20 q - 44 q^{9} - 232 q^{13} - 1640 q^{21} - 15228 q^{25} - 13264 q^{33} + 17208 q^{37} + 24224 q^{45} - 51780 q^{49} - 37512 q^{57} + 123416 q^{61} + 100192 q^{69} - 48120 q^{73} - 42508 q^{81} + 168960 q^{85} + 34840 q^{93} + 39368 q^{97}+O(q^{100}) \) 20 * q - 44 * q^9 - 232 * q^13 - 1640 * q^21 - 15228 * q^25 - 13264 * q^33 + 17208 * q^37 + 24224 * q^45 - 51780 * q^49 - 37512 * q^57 + 123416 * q^61 + 100192 * q^69 - 48120 * q^73 - 42508 * q^81 + 168960 * q^85 + 34840 * q^93 + 39368 * q^97

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