LMFDB - Embedded newform 256.5.c.h.255.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [256,5,Mod(255,256)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(256, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("256.255");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 256 = 2^{8} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	256.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$26.4627105495$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{51})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 25x^{2} + 169 $ x^4 - 25*x^2 + 169
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 64)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			255.3
Root			$-3.57071 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	256.255
Dual form			256.5.c.h.255.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+14.2829i q^{3} -28.5657 q^{5} +56.0000i q^{7} -123.000 q^{9} +O(q^{10})$	$q+14.2829i q^{3} -28.5657 q^{5} +56.0000i q^{7} -123.000 q^{9} +99.9800i q^{11} -257.091 q^{13} -408.000i q^{15} +378.000 q^{17} +128.546i q^{19} -799.840 q^{21} -216.000i q^{23} +191.000 q^{25} -599.880i q^{27} -599.880 q^{29} +224.000i q^{31} -1428.00 q^{33} -1599.68i q^{35} +1799.64 q^{37} -3672.00i q^{39} +1134.00 q^{41} +899.820i q^{43} +3513.58 q^{45} -3024.00i q^{47} -735.000 q^{49} +5398.92i q^{51} +999.800 q^{53} -2856.00i q^{55} -1836.00 q^{57} +214.243i q^{59} +771.274 q^{61} -6888.00i q^{63} +7344.00 q^{65} -8098.38i q^{67} +3085.10 q^{69} +4536.00i q^{71} -490.000 q^{73} +2728.03i q^{75} -5598.88 q^{77} -2800.00i q^{79} -1395.00 q^{81} +8241.21i q^{83} -10797.8 q^{85} -8568.00i q^{87} -9450.00 q^{89} -14397.1i q^{91} -3199.36 q^{93} -3672.00i q^{95} -16198.0 q^{97} -12297.5i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 492 q^{9}+O(q^{10}) $ 4 * q - 492 * q^9	$ 4 q - 492 q^{9} + 1512 q^{17} + 764 q^{25} - 5712 q^{33} + 4536 q^{41} - 2940 q^{49} - 7344 q^{57} + 29376 q^{65} - 1960 q^{73} - 5580 q^{81} - 37800 q^{89} - 64792 q^{97}+O(q^{100}) $ 4 * q - 492 * q^9 + 1512 * q^17 + 764 * q^25 - 5712 * q^33 + 4536 * q^41 - 2940 * q^49 - 7344 * q^57 + 29376 * q^65 - 1960 * q^73 - 5580 * q^81 - 37800 * q^89 - 64792 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/256\mathbb{Z}\right)^\times$.

$n$	$5$	$255$
$\chi(n)$	$1$	$-1$

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 $

$2$	0	0
$2$	0	0
$3$	14.2829i	1.58698i	0.608581	+	0.793492i	$0.291739\pi$
$3$	14.2829i	1.58698i	−0.608581	+	0.793492i	$0.708261\pi$
$4$	0	0
$5$	−28.5657	−1.14263	−0.571314	−	0.820731i	$-0.693566\pi$
$5$	−28.5657	−1.14263	−0.571314	+	0.820731i	$0.693566\pi$
$6$	0	0
$7$	56.0000i	1.14286i	0.820652	+	0.571429i	$0.193611\pi$
$7$	56.0000i	1.14286i	−0.820652	+	0.571429i	$0.806389\pi$
$8$	0	0
$9$	−123.000	−1.51852
$10$	0	0
$11$	99.9800i	0.826281i	0.910667	+	0.413140i	$0.135568\pi$
$11$	99.9800i	0.826281i	−0.910667	+	0.413140i	$0.864432\pi$
$12$	0	0
$13$	−257.091	−1.52125	−0.760626	−	0.649191i	$-0.775108\pi$
$13$	−257.091	−1.52125	−0.760626	+	0.649191i	$0.775108\pi$
$14$	0	0
$15$	− 408.000i	− 1.81333i
$16$	0	0
$17$	378.000	1.30796	0.653979	−	0.756512i	$-0.273098\pi$
$17$	378.000	1.30796	0.653979	+	0.756512i	$0.273098\pi$
$18$	0	0
$19$	128.546i	0.356082i	0.984023	+	0.178041i	$0.0569760\pi$
$19$	128.546i	0.356082i	−0.984023	+	0.178041i	$0.943024\pi$
$20$	0	0
$21$	−799.840	−1.81370
$22$	0	0
$23$	− 216.000i	− 0.408318i	−0.978938	−	0.204159i	$-0.934554\pi$
$23$	− 216.000i	− 0.408318i	0.978938	−	0.204159i	$-0.0654459\pi$
$24$	0	0
$25$	191.000	0.305600
$26$	0	0
$27$	− 599.880i	− 0.822881i
$28$	0	0
$29$	−599.880	−0.713294	−0.356647	−	0.934239i	$-0.616080\pi$
$29$	−599.880	−0.713294	−0.356647	+	0.934239i	$0.616080\pi$
$30$	0	0
$31$	224.000i	0.233091i	0.993185	+	0.116545i	$0.0371820\pi$
$31$	224.000i	0.233091i	−0.993185	+	0.116545i	$0.962818\pi$
$32$	0	0
$33$	−1428.00	−1.31129
$34$	0	0
$35$	− 1599.68i	− 1.30586i
$36$	0	0
$37$	1799.64	1.31457	0.657283	−	0.753644i	$-0.271706\pi$
$37$	1799.64	1.31457	0.657283	+	0.753644i	$0.271706\pi$
$38$	0	0
$39$	− 3672.00i	− 2.41420i
$40$	0	0
$41$	1134.00	0.674598	0.337299	−	0.941397i	$-0.390487\pi$
$41$	1134.00	0.674598	0.337299	+	0.941397i	$0.390487\pi$
$42$	0	0
$43$	899.820i	0.486652i	0.969945	+	0.243326i	$0.0782385\pi$
$43$	899.820i	0.486652i	−0.969945	+	0.243326i	$0.921761\pi$
$44$	0	0
$45$	3513.58	1.73510
$46$	0	0
$47$	− 3024.00i	− 1.36895i	−0.729039	−	0.684473i	$-0.760033\pi$
$47$	− 3024.00i	− 1.36895i	0.729039	−	0.684473i	$-0.239967\pi$
$48$	0	0
$49$	−735.000	−0.306122
$50$	0	0
$51$	5398.92i	2.07571i
$52$	0	0
$53$	999.800	0.355927	0.177964	−	0.984037i	$-0.443049\pi$
$53$	999.800	0.355927	0.177964	+	0.984037i	$0.443049\pi$
$54$	0	0
$55$	− 2856.00i	− 0.944132i
$56$	0	0
$57$	−1836.00	−0.565097
$58$	0	0
$59$	214.243i	0.0615464i	0.999526	+	0.0307732i	$0.00979695\pi$
$59$	214.243i	0.0615464i	−0.999526	+	0.0307732i	$0.990203\pi$
$60$	0	0
$61$	771.274	0.207276	0.103638	−	0.994615i	$-0.466952\pi$
$61$	771.274	0.207276	0.103638	+	0.994615i	$0.466952\pi$
$62$	0	0
$63$	− 6888.00i	− 1.73545i
$64$	0	0
$65$	7344.00	1.73822
$66$	0	0
$67$	− 8098.38i	− 1.80405i	−0.431684	−	0.902025i	$-0.642080\pi$
$67$	− 8098.38i	− 1.80405i	0.431684	−	0.902025i	$-0.357920\pi$
$68$	0	0
$69$	3085.10	0.647994
$70$	0	0
$71$	4536.00i	0.899821i	0.893074	+	0.449911i	$0.148544\pi$
$71$	4536.00i	0.899821i	−0.893074	+	0.449911i	$0.851456\pi$
$72$	0	0
$73$	−490.000	−0.0919497	−0.0459749	−	0.998943i	$-0.514639\pi$
$73$	−490.000	−0.0919497	−0.0459749	+	0.998943i	$0.514639\pi$
$74$	0	0
$75$	2728.03i	0.484982i
$76$	0	0
$77$	−5598.88	−0.944321
$78$	0	0
$79$	− 2800.00i	− 0.448646i	−0.974515	−	0.224323i	$-0.927983\pi$
$79$	− 2800.00i	− 0.448646i	0.974515	−	0.224323i	$-0.0720171\pi$
$80$	0	0
$81$	−1395.00	−0.212620
$82$	0	0
$83$	8241.21i	1.19629i	0.801390	+	0.598143i	$0.204094\pi$
$83$	8241.21i	1.19629i	−0.801390	+	0.598143i	$0.795906\pi$
$84$	0	0
$85$	−10797.8	−1.49451
$86$	0	0
$87$	− 8568.00i	− 1.13199i
$88$	0	0
$89$	−9450.00	−1.19303	−0.596516	−	0.802601i	$-0.703449\pi$
$89$	−9450.00	−1.19303	−0.596516	+	0.802601i	$0.703449\pi$
$90$	0	0
$91$	− 14397.1i	− 1.73857i
$92$	0	0
$93$	−3199.36	−0.369911
$94$	0	0
$95$	− 3672.00i	− 0.406870i
$96$	0	0
$97$	−16198.0	−1.72154	−0.860772	−	0.508991i	$-0.830018\pi$
$97$	−16198.0	−1.72154	−0.860772	+	0.508991i	$0.830018\pi$
$98$	0	0
$99$	− 12297.5i	− 1.25472i
$100$	0	0
$101$	−16482.4	−1.61576	−0.807882	−	0.589344i	$-0.799386\pi$
$101$	−16482.4	−1.61576	−0.807882	+	0.589344i	$0.799386\pi$
$102$	0	0
$103$	− 10472.0i	− 0.987086i	−0.869721	−	0.493543i	$-0.835701\pi$
$103$	− 10472.0i	− 0.987086i	0.869721	−	0.493543i	$-0.164299\pi$
$104$	0	0
$105$	22848.0	2.07238
$106$	0	0
$107$	11297.7i	0.986788i	0.869806	+	0.493394i	$0.164244\pi$
$107$	11297.7i	0.986788i	−0.869806	+	0.493394i	$0.835756\pi$
$108$	0	0
$109$	−12597.5	−1.06030	−0.530152	−	0.847902i	$-0.677865\pi$
$109$	−12597.5	−1.06030	−0.530152	+	0.847902i	$0.677865\pi$
$110$	0	0
$111$	25704.0i	2.08619i
$112$	0	0
$113$	13986.0	1.09531	0.547654	−	0.836705i	$-0.315521\pi$
$113$	13986.0	1.09531	0.547654	+	0.836705i	$0.315521\pi$
$114$	0	0
$115$	6170.19i	0.466555i
$116$	0	0
$117$	31622.2	2.31005
$118$	0	0
$119$	21168.0i	1.49481i
$120$	0	0
$121$	4645.00	0.317260
$122$	0	0
$123$	16196.8i	1.07058i
$124$	0	0
$125$	12397.5	0.793441
$126$	0	0
$127$	22720.0i	1.40864i	0.709881	+	0.704321i	$0.248749\pi$
$127$	22720.0i	1.40864i	−0.709881	+	0.704321i	$0.751251\pi$
$128$	0	0
$129$	−12852.0	−0.772309
$130$	0	0
$131$	20467.3i	1.19267i	0.802737	+	0.596333i	$0.203376\pi$
$131$	20467.3i	1.19267i	−0.802737	+	0.596333i	$0.796624\pi$
$132$	0	0
$133$	−7198.56	−0.406951
$134$	0	0
$135$	17136.0i	0.940247i
$136$	0	0
$137$	−10098.0	−0.538015	−0.269007	−	0.963138i	$-0.586696\pi$
$137$	−10098.0	−0.538015	−0.269007	+	0.963138i	$0.586696\pi$
$138$	0	0
$139$	− 15554.0i	− 0.805032i	−0.915413	−	0.402516i	$-0.868136\pi$
$139$	− 15554.0i	− 0.805032i	0.915413	−	0.402516i	$-0.131864\pi$
$140$	0	0
$141$	43191.4	2.17249
$142$	0	0
$143$	− 25704.0i	− 1.25698i
$144$	0	0
$145$	17136.0	0.815030
$146$	0	0
$147$	− 10497.9i	− 0.485811i
$148$	0	0
$149$	7398.52	0.333252	0.166626	−	0.986020i	$-0.446713\pi$
$149$	7398.52	0.333252	0.166626	+	0.986020i	$0.446713\pi$
$150$	0	0
$151$	16040.0i	0.703478i	0.936098	+	0.351739i	$0.114410\pi$
$151$	16040.0i	0.703478i	−0.936098	+	0.351739i	$0.885590\pi$
$152$	0	0
$153$	−46494.0	−1.98616
$154$	0	0
$155$	− 6398.72i	− 0.266336i
$156$	0	0
$157$	−9512.38	−0.385914	−0.192957	−	0.981207i	$-0.561808\pi$
$157$	−9512.38	−0.385914	−0.192957	+	0.981207i	$0.561808\pi$
$158$	0	0
$159$	14280.0i	0.564851i
$160$	0	0
$161$	12096.0	0.466649
$162$	0	0
$163$	− 15296.9i	− 0.575744i	−0.957669	−	0.287872i	$-0.907052\pi$
$163$	− 15296.9i	− 0.575744i	0.957669	−	0.287872i	$-0.0929478\pi$
$164$	0	0
$165$	40791.8	1.49832
$166$	0	0
$167$	46872.0i	1.68066i	0.542073	+	0.840331i	$0.317640\pi$
$167$	46872.0i	1.68066i	−0.542073	+	0.840331i	$0.682360\pi$
$168$	0	0
$169$	37535.0	1.31420
$170$	0	0
$171$	− 15811.1i	− 0.540718i
$172$	0	0
$173$	−38420.9	−1.28373	−0.641867	−	0.766816i	$-0.721840\pi$
$173$	−38420.9	−1.28373	−0.641867	+	0.766816i	$0.721840\pi$
$174$	0	0
$175$	10696.0i	0.349257i
$176$	0	0
$177$	−3060.00	−0.0976731
$178$	0	0
$179$	− 56888.6i	− 1.77549i	−0.460331	−	0.887747i	$-0.652269\pi$
$179$	− 56888.6i	− 1.77549i	0.460331	−	0.887747i	$-0.347731\pi$
$180$	0	0
$181$	−32136.4	−0.980936	−0.490468	−	0.871459i	$-0.663174\pi$
$181$	−32136.4	−0.980936	−0.490468	+	0.871459i	$0.663174\pi$
$182$	0	0
$183$	11016.0i	0.328944i
$184$	0	0
$185$	−51408.0	−1.50206
$186$	0	0
$187$	37792.4i	1.08074i
$188$	0	0
$189$	33593.3	0.940435
$190$	0	0
$191$	48384.0i	1.32628i	0.748496	+	0.663140i	$0.230776\pi$
$191$	48384.0i	1.32628i	−0.748496	+	0.663140i	$0.769224\pi$
$192$	0	0
$193$	−13510.0	−0.362694	−0.181347	−	0.983419i	$-0.558046\pi$
$193$	−13510.0	−0.362694	−0.181347	+	0.983419i	$0.558046\pi$
$194$	0	0
$195$	104893.i	2.75854i
$196$	0	0
$197$	−42991.4	−1.10777	−0.553884	−	0.832594i	$-0.686855\pi$
$197$	−42991.4	−1.10777	−0.553884	+	0.832594i	$0.686855\pi$
$198$	0	0
$199$	− 40712.0i	− 1.02805i	−0.857774	−	0.514027i	$-0.828153\pi$
$199$	− 40712.0i	− 1.02805i	0.857774	−	0.514027i	$-0.171847\pi$
$200$	0	0
$201$	115668.	2.86300
$202$	0	0
$203$	− 33593.3i	− 0.815193i
$204$	0	0
$205$	−32393.5	−0.770815
$206$	0	0
$207$	26568.0i	0.620038i
$208$	0	0
$209$	−12852.0	−0.294224
$210$	0	0
$211$	− 15296.9i	− 0.343589i	−0.985133	−	0.171795i	$-0.945043\pi$
$211$	− 15296.9i	− 0.343589i	0.985133	−	0.171795i	$-0.0549566\pi$
$212$	0	0
$213$	−64787.0	−1.42800
$214$	0	0
$215$	− 25704.0i	− 0.556063i
$216$	0	0
$217$	−12544.0	−0.266389
$218$	0	0
$219$	− 6998.60i	− 0.145923i
$220$	0	0
$221$	−97180.6	−1.98973
$222$	0	0
$223$	− 64736.0i	− 1.30178i	−0.759174	−	0.650888i	$-0.774397\pi$
$223$	− 64736.0i	− 1.30178i	0.759174	−	0.650888i	$-0.225603\pi$
$224$	0	0
$225$	−23493.0	−0.464059
$226$	0	0
$227$	− 6384.44i	− 0.123900i	−0.998079	−	0.0619499i	$-0.980268\pi$
$227$	− 6384.44i	− 0.123900i	0.998079	−	0.0619499i	$-0.0197319\pi$
$228$	0	0
$229$	36764.1	0.701056	0.350528	−	0.936552i	$-0.386002\pi$
$229$	36764.1	0.701056	0.350528	+	0.936552i	$0.386002\pi$
$230$	0	0
$231$	− 79968.0i	− 1.49862i
$232$	0	0
$233$	16470.0	0.303376	0.151688	−	0.988428i	$-0.451529\pi$
$233$	16470.0	0.303376	0.151688	+	0.988428i	$0.451529\pi$
$234$	0	0
$235$	86382.7i	1.56420i
$236$	0	0
$237$	39992.0	0.711994
$238$	0	0
$239$	17712.0i	0.310079i	0.987908	+	0.155039i	$0.0495504\pi$
$239$	17712.0i	0.310079i	−0.987908	+	0.155039i	$0.950450\pi$
$240$	0	0
$241$	6202.00	0.106782	0.0533910	−	0.998574i	$-0.482997\pi$
$241$	6202.00	0.106782	0.0533910	+	0.998574i	$0.482997\pi$
$242$	0	0
$243$	− 68514.9i	− 1.16031i
$244$	0	0
$245$	20995.8	0.349784
$246$	0	0
$247$	− 33048.0i	− 0.541691i
$248$	0	0
$249$	−117708.	−1.89849
$250$	0	0
$251$	31750.8i	0.503973i	0.967731	+	0.251986i	$0.0810838\pi$
$251$	31750.8i	0.503973i	−0.967731	+	0.251986i	$0.918916\pi$
$252$	0	0
$253$	21595.7	0.337385
$254$	0	0
$255$	− 154224.i	− 2.37176i
$256$	0	0
$257$	−82782.0	−1.25334	−0.626671	−	0.779284i	$-0.715583\pi$
$257$	−82782.0	−1.25334	−0.626671	+	0.779284i	$0.715583\pi$
$258$	0	0
$259$	100780.i	1.50236i
$260$	0	0
$261$	73785.2	1.08315
$262$	0	0
$263$	89208.0i	1.28971i	0.764305	+	0.644855i	$0.223082\pi$
$263$	89208.0i	1.28971i	−0.764305	+	0.644855i	$0.776918\pi$
$264$	0	0
$265$	−28560.0	−0.406693
$266$	0	0
$267$	− 134973.i	− 1.89332i
$268$	0	0
$269$	132059.	1.82501	0.912503	−	0.409069i	$-0.134147\pi$
$269$	132059.	1.82501	0.912503	+	0.409069i	$0.134147\pi$
$270$	0	0
$271$	− 50288.0i	− 0.684740i	−0.939565	−	0.342370i	$-0.888770\pi$
$271$	− 50288.0i	− 0.684740i	0.939565	−	0.342370i	$-0.111230\pi$
$272$	0	0
$273$	205632.	2.75909
$274$	0	0
$275$	19096.2i	0.252511i
$276$	0	0
$277$	8998.20	0.117272	0.0586362	−	0.998279i	$-0.481325\pi$
$277$	8998.20	0.117272	0.0586362	+	0.998279i	$0.481325\pi$
$278$	0	0
$279$	− 27552.0i	− 0.353952i
$280$	0	0
$281$	47574.0	0.602500	0.301250	−	0.953545i	$-0.402596\pi$
$281$	47574.0	0.602500	0.301250	+	0.953545i	$0.402596\pi$
$282$	0	0
$283$	− 111192.i	− 1.38836i	−0.719803	−	0.694178i	$-0.755768\pi$
$283$	− 111192.i	− 1.38836i	0.719803	−	0.694178i	$-0.244232\pi$
$284$	0	0
$285$	52446.7	0.645696
$286$	0	0
$287$	63504.0i	0.770970i
$288$	0	0
$289$	59363.0	0.710755
$290$	0	0
$291$	− 231354.i	− 2.73206i
$292$	0	0
$293$	−24937.9	−0.290485	−0.145243	−	0.989396i	$-0.546396\pi$
$293$	−24937.9	−0.290485	−0.145243	+	0.989396i	$0.546396\pi$
$294$	0	0
$295$	− 6120.00i	− 0.0703246i
$296$	0	0
$297$	59976.0	0.679931
$298$	0	0
$299$	55531.7i	0.621154i
$300$	0	0
$301$	−50389.9	−0.556174
$302$	0	0
$303$	− 235416.i	− 2.56419i
$304$	0	0
$305$	−22032.0	−0.236840
$306$	0	0
$307$	90624.7i	0.961546i	0.876845	+	0.480773i	$0.159644\pi$
$307$	90624.7i	0.961546i	−0.876845	+	0.480773i	$0.840356\pi$
$308$	0	0
$309$	149570.	1.56649
$310$	0	0
$311$	− 149688.i	− 1.54763i	−0.633414	−	0.773813i	$-0.718347\pi$
$311$	− 149688.i	− 1.54763i	0.633414	−	0.773813i	$-0.281653\pi$
$312$	0	0
$313$	−128338.	−1.30999	−0.654993	−	0.755635i	$-0.727328\pi$
$313$	−128338.	−1.30999	−0.654993	+	0.755635i	$0.727328\pi$
$314$	0	0
$315$	196761.i	1.98297i
$316$	0	0
$317$	20196.0	0.200977	0.100488	−	0.994938i	$-0.467959\pi$
$317$	20196.0	0.200977	0.100488	+	0.994938i	$0.467959\pi$
$318$	0	0
$319$	− 59976.0i	− 0.589381i
$320$	0	0
$321$	−161364.	−1.56602
$322$	0	0
$323$	48590.3i	0.465741i
$324$	0	0
$325$	−49104.5	−0.464894
$326$	0	0
$327$	− 179928.i	− 1.68269i
$328$	0	0
$329$	169344.	1.56451
$330$	0	0
$331$	65686.9i	0.599546i	0.954011	+	0.299773i	$0.0969109\pi$
$331$	65686.9i	0.599546i	−0.954011	+	0.299773i	$0.903089\pi$
$332$	0	0
$333$	−221356.	−1.99619
$334$	0	0
$335$	231336.i	2.06136i
$336$	0	0
$337$	−89950.0	−0.792030	−0.396015	−	0.918244i	$-0.629607\pi$
$337$	−89950.0	−0.792030	−0.396015	+	0.918244i	$0.629607\pi$
$338$	0	0
$339$	199760.i	1.73824i
$340$	0	0
$341$	−22395.5	−0.192598
$342$	0	0
$343$	93296.0i	0.793003i
$344$	0	0
$345$	−88128.0	−0.740416
$346$	0	0
$347$	− 216657.i	− 1.79934i	−0.436571	−	0.899670i	$-0.643807\pi$
$347$	− 216657.i	− 1.79934i	0.436571	−	0.899670i	$-0.356193\pi$
$348$	0	0
$349$	169423.	1.39098	0.695492	−	0.718534i	$-0.255186\pi$
$349$	169423.	1.39098	0.695492	+	0.718534i	$0.255186\pi$
$350$	0	0
$351$	154224.i	1.25181i
$352$	0	0
$353$	32130.0	0.257847	0.128923	−	0.991655i	$-0.458848\pi$
$353$	32130.0	0.257847	0.128923	+	0.991655i	$0.458848\pi$
$354$	0	0
$355$	− 129574.i	− 1.02816i
$356$	0	0
$357$	−302340.	−2.37224
$358$	0	0
$359$	33048.0i	0.256423i	0.991747	+	0.128211i	$0.0409236\pi$
$359$	33048.0i	0.256423i	−0.991747	+	0.128211i	$0.959076\pi$
$360$	0	0
$361$	113797.	0.873205
$362$	0	0
$363$	66343.9i	0.503486i
$364$	0	0
$365$	13997.2	0.105064
$366$	0	0
$367$	− 27664.0i	− 0.205392i	−0.994713	−	0.102696i	$-0.967253\pi$
$367$	− 27664.0i	− 0.205392i	0.994713	−	0.102696i	$-0.0327468\pi$
$368$	0	0
$369$	−139482.	−1.02439
$370$	0	0
$371$	55988.8i	0.406774i
$372$	0	0
$373$	−120576.	−0.866648	−0.433324	−	0.901238i	$-0.642659\pi$
$373$	−120576.	−0.866648	−0.433324	+	0.901238i	$0.642659\pi$
$374$	0	0
$375$	177072.i	1.25918i
$376$	0	0
$377$	154224.	1.08510
$378$	0	0
$379$	− 27894.4i	− 0.194195i	−0.995275	−	0.0970977i	$-0.969044\pi$
$379$	− 27894.4i	− 0.194195i	0.995275	−	0.0970977i	$-0.0309559\pi$
$380$	0	0
$381$	−324507.	−2.23549
$382$	0	0
$383$	217728.i	1.48428i	0.670243	+	0.742142i	$0.266190\pi$
$383$	217728.i	1.48428i	−0.670243	+	0.742142i	$0.733810\pi$
$384$	0	0
$385$	159936.	1.07901
$386$	0	0
$387$	− 110678.i	− 0.738990i
$388$	0	0
$389$	144171.	0.952750	0.476375	−	0.879242i	$-0.341950\pi$
$389$	144171.	0.952750	0.476375	+	0.879242i	$0.341950\pi$
$390$	0	0
$391$	− 81648.0i	− 0.534062i
$392$	0	0
$393$	−292332.	−1.89274
$394$	0	0
$395$	79984.0i	0.512636i
$396$	0	0
$397$	−105150.	−0.667160	−0.333580	−	0.942722i	$-0.608257\pi$
$397$	−105150.	−0.667160	−0.333580	+	0.942722i	$0.608257\pi$
$398$	0	0
$399$	− 102816.i	− 0.645825i
$400$	0	0
$401$	24570.0	0.152798	0.0763988	−	0.997077i	$-0.475658\pi$
$401$	24570.0	0.152798	0.0763988	+	0.997077i	$0.475658\pi$
$402$	0	0
$403$	− 57588.5i	− 0.354589i
$404$	0	0
$405$	39849.2	0.242946
$406$	0	0
$407$	179928.i	1.08620i
$408$	0	0
$409$	−53074.0	−0.317275	−0.158637	−	0.987337i	$-0.550710\pi$
$409$	−53074.0	−0.317275	−0.158637	+	0.987337i	$0.550710\pi$
$410$	0	0
$411$	− 144228.i	− 0.853821i
$412$	0	0
$413$	−11997.6	−0.0703387
$414$	0	0
$415$	− 235416.i	− 1.36691i
$416$	0	0
$417$	222156.	1.27757
$418$	0	0
$419$	− 231597.i	− 1.31918i	−0.751626	−	0.659590i	$-0.770730\pi$
$419$	− 231597.i	− 1.31918i	0.751626	−	0.659590i	$-0.229270\pi$
$420$	0	0
$421$	160168.	0.903673	0.451837	−	0.892101i	$-0.350769\pi$
$421$	160168.	0.903673	0.451837	+	0.892101i	$0.350769\pi$
$422$	0	0
$423$	371952.i	2.07877i
$424$	0	0
$425$	72198.0	0.399712
$426$	0	0
$427$	43191.4i	0.236887i
$428$	0	0
$429$	367127.	1.99481
$430$	0	0
$431$	209520.i	1.12790i	0.825809	+	0.563950i	$0.190719\pi$
$431$	209520.i	1.12790i	−0.825809	+	0.563950i	$0.809281\pi$
$432$	0	0
$433$	124474.	0.663900	0.331950	−	0.943297i	$-0.392293\pi$
$433$	124474.	0.663900	0.331950	+	0.943297i	$0.392293\pi$
$434$	0	0
$435$	244751.i	1.29344i
$436$	0	0
$437$	27765.9	0.145395
$438$	0	0
$439$	− 80696.0i	− 0.418719i	−0.977839	−	0.209360i	$-0.932862\pi$
$439$	− 80696.0i	− 0.418719i	0.977839	−	0.209360i	$-0.0671379\pi$
$440$	0	0
$441$	90405.0	0.464853
$442$	0	0
$443$	22495.5i	0.114627i	0.998356	+	0.0573137i	$0.0182535\pi$
$443$	22495.5i	0.114627i	−0.998356	+	0.0573137i	$0.981746\pi$
$444$	0	0
$445$	269946.	1.36319
$446$	0	0
$447$	105672.i	0.528865i
$448$	0	0
$449$	−92934.0	−0.460980	−0.230490	−	0.973075i	$-0.574033\pi$
$449$	−92934.0	−0.460980	−0.230490	+	0.973075i	$0.574033\pi$
$450$	0	0
$451$	113377.i	0.557408i
$452$	0	0
$453$	−229097.	−1.11641
$454$	0	0
$455$	411264.i	1.98654i
$456$	0	0
$457$	−350098.	−1.67632	−0.838161	−	0.545424i	$-0.816369\pi$
$457$	−350098.	−1.67632	−0.838161	+	0.545424i	$0.816369\pi$
$458$	0	0
$459$	− 226755.i	− 1.07629i
$460$	0	0
$461$	203816.	0.959041	0.479521	−	0.877531i	$-0.340811\pi$
$461$	203816.	0.959041	0.479521	+	0.877531i	$0.340811\pi$
$462$	0	0
$463$	89488.0i	0.417448i	0.977975	+	0.208724i	$0.0669311\pi$
$463$	89488.0i	0.417448i	−0.977975	+	0.208724i	$0.933069\pi$
$464$	0	0
$465$	91392.0	0.422671
$466$	0	0
$467$	− 17467.9i	− 0.0800954i	−0.999198	−	0.0400477i	$-0.987249\pi$
$467$	− 17467.9i	− 0.0800954i	0.999198	−	0.0400477i	$-0.0127510\pi$
$468$	0	0
$469$	453509.	2.06177
$470$	0	0
$471$	− 135864.i	− 0.612439i
$472$	0	0
$473$	−89964.0	−0.402111
$474$	0	0
$475$	24552.2i	0.108819i
$476$	0	0
$477$	−122975.	−0.540482
$478$	0	0
$479$	175392.i	0.764432i	0.924073	+	0.382216i	$0.124839\pi$
$479$	175392.i	0.764432i	−0.924073	+	0.382216i	$0.875161\pi$
$480$	0	0
$481$	−462672.	−1.99978
$482$	0	0
$483$	172765.i	0.740564i
$484$	0	0
$485$	462707.	1.96708
$486$	0	0
$487$	− 73384.0i	− 0.309416i	−0.987960	−	0.154708i	$-0.950556\pi$
$487$	− 73384.0i	− 0.309416i	0.987960	−	0.154708i	$-0.0494438\pi$
$488$	0	0
$489$	218484.	0.913696
$490$	0	0
$491$	290442.i	1.20475i	0.798214	+	0.602374i	$0.205778\pi$
$491$	290442.i	1.20475i	−0.798214	+	0.602374i	$0.794222\pi$
$492$	0	0
$493$	−226755.	−0.932959
$494$	0	0
$495$	351288.i	1.43368i
$496$	0	0
$497$	−254016.	−1.02837
$498$	0	0
$499$	− 375225.i	− 1.50692i	−0.657493	−	0.753461i	$-0.728383\pi$
$499$	− 375225.i	− 1.50692i	0.657493	−	0.753461i	$-0.271617\pi$
$500$	0	0
$501$	−669466.	−2.66718
$502$	0	0
$503$	213192.i	0.842626i	0.906915	+	0.421313i	$0.138431\pi$
$503$	213192.i	0.842626i	−0.906915	+	0.421313i	$0.861569\pi$
$504$	0	0
$505$	470832.	1.84622
$506$	0	0
$507$	536107.i	2.08562i
$508$	0	0
$509$	−8369.75	−0.0323055	−0.0161528	−	0.999870i	$-0.505142\pi$
$509$	−8369.75	−0.0323055	−0.0161528	+	0.999870i	$0.505142\pi$
$510$	0	0
$511$	− 27440.0i	− 0.105085i
$512$	0	0
$513$	77112.0	0.293013
$514$	0	0
$515$	299140.i	1.12787i
$516$	0	0
$517$	302340.	1.13113
$518$	0	0
$519$	− 548760.i	− 2.03727i
$520$	0	0
$521$	−476658.	−1.75603	−0.878014	−	0.478635i	$-0.841132\pi$
$521$	−476658.	−1.75603	−0.878014	+	0.478635i	$0.841132\pi$
$522$	0	0
$523$	− 214029.i	− 0.782471i	−0.920291	−	0.391236i	$-0.872048\pi$
$523$	− 214029.i	− 0.782471i	0.920291	−	0.391236i	$-0.127952\pi$
$524$	0	0
$525$	−152769.	−0.554266
$526$	0	0
$527$	84672.0i	0.304873i
$528$	0	0
$529$	233185.	0.833277
$530$	0	0
$531$	− 26351.9i	− 0.0934593i
$532$	0	0
$533$	−291542.	−1.02623
$534$	0	0
$535$	− 322728.i	− 1.12753i
$536$	0	0
$537$	812532.	2.81768
$538$	0	0
$539$	− 73485.3i	− 0.252943i
$540$	0	0
$541$	282543.	0.965363	0.482682	−	0.875796i	$-0.339663\pi$
$541$	282543.	0.965363	0.482682	+	0.875796i	$0.339663\pi$
$542$	0	0
$543$	− 459000.i	− 1.55673i
$544$	0	0
$545$	359856.	1.21153
$546$	0	0
$547$	164667.i	0.550341i	0.961395	+	0.275171i	$0.0887343\pi$
$547$	164667.i	0.550341i	−0.961395	+	0.275171i	$0.911266\pi$
$548$	0	0
$549$	−94866.7	−0.314753
$550$	0	0
$551$	− 77112.0i	− 0.253991i
$552$	0	0
$553$	156800.	0.512738
$554$	0	0
$555$	− 734253.i	− 2.38375i
$556$	0	0
$557$	−52589.5	−0.169507	−0.0847537	−	0.996402i	$-0.527010\pi$
$557$	−52589.5	−0.169507	−0.0847537	+	0.996402i	$0.527010\pi$
$558$	0	0
$559$	− 231336.i	− 0.740320i
$560$	0	0
$561$	−539784.	−1.71512
$562$	0	0
$563$	− 287357.i	− 0.906577i	−0.891364	−	0.453288i	$-0.850251\pi$
$563$	− 287357.i	− 0.906577i	0.891364	−	0.453288i	$-0.149749\pi$
$564$	0	0
$565$	−399520.	−1.25153
$566$	0	0
$567$	− 78120.0i	− 0.242994i
$568$	0	0
$569$	−391986.	−1.21073	−0.605363	−	0.795949i	$-0.706972\pi$
$569$	−391986.	−1.21073	−0.605363	+	0.795949i	$0.706972\pi$
$570$	0	0
$571$	− 582184.i	− 1.78561i	−0.450439	−	0.892807i	$-0.648732\pi$
$571$	− 582184.i	− 1.78561i	0.450439	−	0.892807i	$-0.351268\pi$
$572$	0	0
$573$	−691062.	−2.10478
$574$	0	0
$575$	− 41256.0i	− 0.124782i
$576$	0	0
$577$	195874.	0.588336	0.294168	−	0.955754i	$-0.404958\pi$
$577$	195874.	0.588336	0.294168	+	0.955754i	$0.404958\pi$
$578$	0	0
$579$	− 192961.i	− 0.575590i
$580$	0	0
$581$	−461508.	−1.36718
$582$	0	0
$583$	99960.0i	0.294096i
$584$	0	0
$585$	−903312.	−2.63953
$586$	0	0
$587$	383566.i	1.11318i	0.830788	+	0.556588i	$0.187890\pi$
$587$	383566.i	1.11318i	−0.830788	+	0.556588i	$0.812110\pi$
$588$	0	0
$589$	−28794.2	−0.0829994
$590$	0	0
$591$	− 614040.i	− 1.75801i
$592$	0	0
$593$	32130.0	0.0913695	0.0456848	−	0.998956i	$-0.485453\pi$
$593$	32130.0	0.0913695	0.0456848	+	0.998956i	$0.485453\pi$
$594$	0	0
$595$	− 604679.i	− 1.70801i
$596$	0	0
$597$	581484.	1.63151
$598$	0	0
$599$	642600.i	1.79096i	0.445097	+	0.895482i	$0.353169\pi$
$599$	642600.i	1.79096i	−0.445097	+	0.895482i	$0.646831\pi$
$600$	0	0
$601$	55510.0	0.153682	0.0768409	−	0.997043i	$-0.475517\pi$
$601$	55510.0	0.153682	0.0768409	+	0.997043i	$0.475517\pi$
$602$	0	0
$603$	996101.i	2.73948i
$604$	0	0
$605$	−132688.	−0.362510
$606$	0	0
$607$	388192.i	1.05358i	0.849994	+	0.526792i	$0.176605\pi$
$607$	388192.i	1.05358i	−0.849994	+	0.526792i	$0.823395\pi$
$608$	0	0
$609$	479808.	1.29370
$610$	0	0
$611$	777444.i	2.08251i
$612$	0	0
$613$	−502100.	−1.33619	−0.668096	−	0.744075i	$-0.732890\pi$
$613$	−502100.	−1.33619	−0.668096	+	0.744075i	$0.732890\pi$
$614$	0	0
$615$	− 462672.i	− 1.22327i
$616$	0	0
$617$	−190890.	−0.501433	−0.250716	−	0.968061i	$-0.580666\pi$
$617$	−190890.	−0.501433	−0.250716	+	0.968061i	$0.580666\pi$
$618$	0	0
$619$	335119.i	0.874616i	0.899312	+	0.437308i	$0.144068\pi$
$619$	335119.i	0.874616i	−0.899312	+	0.437308i	$0.855932\pi$
$620$	0	0
$621$	−129574.	−0.335997
$622$	0	0
$623$	− 529200.i	− 1.36346i
$624$	0	0
$625$	−473519.	−1.21221
$626$	0	0
$627$	− 183563.i	− 0.466929i
$628$	0	0
$629$	680264.	1.71940
$630$	0	0
$631$	− 366520.i	− 0.920532i	−0.887781	−	0.460266i	$-0.847754\pi$
$631$	− 366520.i	− 0.920532i	0.887781	−	0.460266i	$-0.152246\pi$
$632$	0	0
$633$	218484.	0.545271
$634$	0	0
$635$	− 649013.i	− 1.60956i
$636$	0	0
$637$	188962.	0.465689
$638$	0	0
$639$	− 557928.i	− 1.36640i
$640$	0	0
$641$	447930.	1.09017	0.545085	−	0.838381i	$-0.316497\pi$
$641$	447930.	1.09017	0.545085	+	0.838381i	$0.316497\pi$
$642$	0	0
$643$	436156.i	1.05492i	0.849580	+	0.527460i	$0.176856\pi$
$643$	436156.i	1.05492i	−0.849580	+	0.527460i	$0.823144\pi$
$644$	0	0
$645$	367127.	0.882463
$646$	0	0
$647$	645624.i	1.54231i	0.636649	+	0.771154i	$0.280320\pi$
$647$	645624.i	1.54231i	−0.636649	+	0.771154i	$0.719680\pi$
$648$	0	0
$649$	−21420.0	−0.0508546
$650$	0	0
$651$	− 179164.i	− 0.422755i
$652$	0	0
$653$	459308.	1.07715	0.538577	−	0.842576i	$-0.318962\pi$
$653$	459308.	1.07715	0.538577	+	0.842576i	$0.318962\pi$
$654$	0	0
$655$	− 584664.i	− 1.36277i
$656$	0	0
$657$	60270.0	0.139627
$658$	0	0
$659$	187862.i	0.432583i	0.976329	+	0.216291i	$0.0693961\pi$
$659$	187862.i	0.432583i	−0.976329	+	0.216291i	$0.930604\pi$
$660$	0	0
$661$	−116462.	−0.266553	−0.133276	−	0.991079i	$-0.542550\pi$
$661$	−116462.	−0.266553	−0.133276	+	0.991079i	$0.542550\pi$
$662$	0	0
$663$	− 1.38802e6i	− 3.15767i
$664$	0	0
$665$	205632.	0.464994
$666$	0	0
$667$	129574.i	0.291250i
$668$	0	0
$669$	924615.	2.06590
$670$	0	0
$671$	77112.0i	0.171268i
$672$	0	0
$673$	250.000	0.000551963 0	0.000275981	−	1.00000i	$-0.499912\pi$
$673$	250.000	0.000551963 0	0.000275981		1.00000i	$0.499912\pi$
$674$	0	0
$675$	− 114577.i	− 0.251472i
$676$	0	0
$677$	−316308.	−0.690133	−0.345067	−	0.938578i	$-0.612144\pi$
$677$	−316308.	−0.690133	−0.345067	+	0.938578i	$0.612144\pi$
$678$	0	0
$679$	− 907088.i	− 1.96748i
$680$	0	0
$681$	91188.0	0.196627
$682$	0	0
$683$	− 339832.i	− 0.728489i	−0.931303	−	0.364244i	$-0.881327\pi$
$683$	− 339832.i	− 0.728489i	0.931303	−	0.364244i	$-0.118673\pi$
$684$	0	0
$685$	288457.	0.614751
$686$	0	0
$687$	525096.i	1.11256i
$688$	0	0
$689$	−257040.	−0.541455
$690$	0	0
$691$	437184.i	0.915605i	0.889054	+	0.457802i	$0.151363\pi$
$691$	437184.i	0.915605i	−0.889054	+	0.457802i	$0.848637\pi$
$692$	0	0
$693$	688662.	1.43397
$694$	0	0
$695$	444312.i	0.919853i
$696$	0	0
$697$	428652.	0.882347
$698$	0	0
$699$	235239.i	0.481453i
$700$	0	0
$701$	136973.	0.278739	0.139369	−	0.990240i	$-0.455492\pi$
$701$	136973.	0.278739	0.139369	+	0.990240i	$0.455492\pi$
$702$	0	0
$703$	231336.i	0.468093i
$704$	0	0
$705$	−1.23379e6	−2.48235
$706$	0	0
$707$	− 923015.i	− 1.84659i
$708$	0	0
$709$	390522.	0.776878	0.388439	−	0.921474i	$-0.373014\pi$
$709$	390522.	0.776878	0.388439	+	0.921474i	$0.373014\pi$
$710$	0	0
$711$	344400.i	0.681277i
$712$	0	0
$713$	48384.0	0.0951750
$714$	0	0
$715$	734253.i	1.43626i
$716$	0	0
$717$	−252978.	−0.492090
$718$	0	0
$719$	595728.i	1.15237i	0.817321	+	0.576183i	$0.195458\pi$
$719$	595728.i	1.15237i	−0.817321	+	0.576183i	$0.804542\pi$
$720$	0	0
$721$	586432.	1.12810
$722$	0	0
$723$	88582.3i	0.169461i
$724$	0	0
$725$	−114577.	−0.217983
$726$	0	0
$727$	396200.i	0.749628i	0.927100	+	0.374814i	$0.122293\pi$
$727$	396200.i	0.749628i	−0.927100	+	0.374814i	$0.877707\pi$
$728$	0	0
$729$	865593.	1.62877
$730$	0	0
$731$	340132.i	0.636521i
$732$	0	0
$733$	−978233.	−1.82068	−0.910341	−	0.413858i	$-0.864181\pi$
$733$	−978233.	−1.82068	−0.910341	+	0.413858i	$0.864181\pi$
$734$	0	0
$735$	299880.i	0.555102i
$736$	0	0
$737$	809676.	1.49065
$738$	0	0
$739$	373425.i	0.683778i	0.939740	+	0.341889i	$0.111067\pi$
$739$	373425.i	0.683778i	−0.939740	+	0.341889i	$0.888933\pi$
$740$	0	0
$741$	472020.	0.859654
$742$	0	0
$743$	− 597672.i	− 1.08264i	−0.840816	−	0.541322i	$-0.817924\pi$
$743$	− 597672.i	− 1.08264i	0.840816	−	0.541322i	$-0.182076\pi$
$744$	0	0
$745$	−211344.	−0.380783
$746$	0	0
$747$	− 1.01367e6i	− 1.81658i
$748$	0	0
$749$	−632673.	−1.12776
$750$	0	0
$751$	258992.i	0.459205i	0.973285	+	0.229602i	$0.0737426\pi$
$751$	258992.i	0.459205i	−0.973285	+	0.229602i	$0.926257\pi$
$752$	0	0
$753$	−453492.	−0.799797
$754$	0	0
$755$	− 458194.i	− 0.803814i
$756$	0	0
$757$	−221356.	−0.386277	−0.193139	−	0.981171i	$-0.561867\pi$
$757$	−221356.	−0.386277	−0.193139	+	0.981171i	$0.561867\pi$
$758$	0	0
$759$	308448.i	0.535425i
$760$	0	0
$761$	−83538.0	−0.144250	−0.0721248	−	0.997396i	$-0.522978\pi$
$761$	−83538.0	−0.144250	−0.0721248	+	0.997396i	$0.522978\pi$
$762$	0	0
$763$	− 705459.i	− 1.21178i
$764$	0	0
$765$	1.32813e6	2.26944
$766$	0	0
$767$	− 55080.0i	− 0.0936275i
$768$	0	0
$769$	−1.16263e6	−1.96602	−0.983012	−	0.183541i	$-0.941244\pi$
$769$	−1.16263e6	−1.96602	−0.983012	+	0.183541i	$0.941244\pi$
$770$	0	0
$771$	− 1.18236e6i	− 1.98903i
$772$	0	0
$773$	−803753.	−1.34513	−0.672564	−	0.740039i	$-0.734807\pi$
$773$	−803753.	−1.34513	−0.672564	+	0.740039i	$0.734807\pi$
$774$	0	0
$775$	42784.0i	0.0712325i
$776$	0	0
$777$	−1.43942e6	−2.38422
$778$	0	0
$779$	145771.i	0.240213i
$780$	0	0
$781$	−453509.	−0.743505
$782$	0	0
$783$	359856.i	0.586956i
$784$	0	0
$785$	271728.	0.440956
$786$	0	0
$787$	939026.i	1.51610i	0.652195	+	0.758051i	$0.273848\pi$
$787$	939026.i	1.51610i	−0.652195	+	0.758051i	$0.726152\pi$
$788$	0	0
$789$	−1.27415e6	−2.04675
$790$	0	0
$791$	783216.i	1.25178i
$792$	0	0
$793$	−198288.	−0.315319
$794$	0	0
$795$	− 407918.i	− 0.645415i
$796$	0	0
$797$	−370583.	−0.583403	−0.291702	−	0.956509i	$-0.594221\pi$
$797$	−370583.	−0.583403	−0.291702	+	0.956509i	$0.594221\pi$
$798$	0	0
$799$	− 1.14307e6i	− 1.79052i
$800$	0	0
$801$	1.16235e6	1.81164
$802$	0	0
$803$	− 48990.2i	− 0.0759763i
$804$	0	0
$805$	−345531.	−0.533206
$806$	0	0
$807$	1.88618e6i	2.89626i
$808$	0	0
$809$	243054.	0.371369	0.185685	−	0.982609i	$-0.440550\pi$
$809$	243054.	0.371369	0.185685	+	0.982609i	$0.440550\pi$
$810$	0	0
$811$	812280.i	1.23499i	0.786574	+	0.617496i	$0.211853\pi$
$811$	812280.i	1.23499i	−0.786574	+	0.617496i	$0.788147\pi$
$812$	0	0
$813$	718256.	1.08667
$814$	0	0
$815$	436968.i	0.657861i
$816$	0	0
$817$	−115668.	−0.173288
$818$	0	0
$819$	1.77085e6i	2.64005i
$820$	0	0
$821$	484103.	0.718210	0.359105	−	0.933297i	$-0.383082\pi$
$821$	484103.	0.718210	0.359105	+	0.933297i	$0.383082\pi$
$822$	0	0
$823$	− 474040.i	− 0.699867i	−0.936775	−	0.349933i	$-0.886204\pi$
$823$	− 474040.i	− 0.699867i	0.936775	−	0.349933i	$-0.113796\pi$
$824$	0	0
$825$	−272748.	−0.400732
$826$	0	0
$827$	− 644571.i	− 0.942453i	−0.882012	−	0.471227i	$-0.843811\pi$
$827$	− 644571.i	− 0.942453i	0.882012	−	0.471227i	$-0.156189\pi$
$828$	0	0
$829$	−1.20447e6	−1.75262	−0.876311	−	0.481746i	$-0.840003\pi$
$829$	−1.20447e6	−1.75262	−0.876311	+	0.481746i	$0.840003\pi$
$830$	0	0
$831$	128520.i	0.186110i
$832$	0	0
$833$	−277830.	−0.400395
$834$	0	0
$835$	− 1.33893e6i	− 1.92037i
$836$	0	0
$837$	134373.	0.191806
$838$	0	0
$839$	− 418824.i	− 0.594987i	−0.954724	−	0.297494i	$-0.903849\pi$
$839$	− 418824.i	− 0.594987i	0.954724	−	0.297494i	$-0.0961507\pi$
$840$	0	0
$841$	−347425.	−0.491212
$842$	0	0
$843$	679493.i	0.956158i
$844$	0	0
$845$	−1.07221e6	−1.50165
$846$	0	0
$847$	260120.i	0.362583i
$848$	0	0
$849$	1.58814e6	2.20330
$850$	0	0
$851$	− 388722.i	− 0.536760i
$852$	0	0
$853$	−209015.	−0.287263	−0.143632	−	0.989631i	$-0.545878\pi$
$853$	−209015.	−0.287263	−0.143632	+	0.989631i	$0.545878\pi$
$854$	0	0
$855$	451656.i	0.617839i
$856$	0	0
$857$	140238.	0.190943	0.0954716	−	0.995432i	$-0.469564\pi$
$857$	140238.	0.190943	0.0954716	+	0.995432i	$0.469564\pi$
$858$	0	0
$859$	1.07451e6i	1.45622i	0.685463	+	0.728108i	$0.259600\pi$
$859$	1.07451e6i	1.45622i	−0.685463	+	0.728108i	$0.740400\pi$
$860$	0	0
$861$	−907019.	−1.22352
$862$	0	0
$863$	514080.i	0.690254i	0.938556	+	0.345127i	$0.112164\pi$
$863$	514080.i	0.690254i	−0.938556	+	0.345127i	$0.887836\pi$
$864$	0	0
$865$	1.09752e6	1.46683
$866$	0	0
$867$	847873.i	1.12796i
$868$	0	0
$869$	279944.	0.370708
$870$	0	0
$871$	2.08202e6i	2.74441i
$872$	0	0
$873$	1.99235e6	2.61420
$874$	0	0
$875$	694261.i	0.906790i
$876$	0	0
$877$	937612.	1.21906	0.609529	−	0.792764i	$-0.291359\pi$
$877$	937612.	1.21906	0.609529	+	0.792764i	$0.291359\pi$
$878$	0	0
$879$	− 356184.i	− 0.460995i
$880$	0	0
$881$	−1.02627e6	−1.32224	−0.661119	−	0.750281i	$-0.729918\pi$
$881$	−1.02627e6	−1.32224	−0.661119	+	0.750281i	$0.729918\pi$
$882$	0	0
$883$	236653.i	0.303522i	0.988417	+	0.151761i	$0.0484944\pi$
$883$	236653.i	0.303522i	−0.988417	+	0.151761i	$0.951506\pi$
$884$	0	0
$885$	87411.1	0.111604
$886$	0	0
$887$	55944.0i	0.0711060i	0.999368	+	0.0355530i	$0.0113193\pi$
$887$	55944.0i	0.0711060i	−0.999368	+	0.0355530i	$0.988681\pi$
$888$	0	0
$889$	−1.27232e6	−1.60988
$890$	0	0
$891$	− 139472.i	− 0.175684i
$892$	0	0
$893$	388722.	0.487457
$894$	0	0
$895$	1.62506e6i	2.02873i
$896$	0	0
$897$	−793152.	−0.985761
$898$	0	0
$899$	− 134373.i	− 0.166262i
$900$	0	0
$901$	377924.	0.465538
$902$	0	0
$903$	− 719712.i	− 0.882639i
$904$	0	0
$905$	918000.	1.12084
$906$	0	0
$907$	− 128674.i	− 0.156415i	−0.996937	−	0.0782073i	$-0.975080\pi$
$907$	− 128674.i	− 0.156415i	0.996937	−	0.0782073i	$-0.0249196\pi$
$908$	0	0
$909$	2.02734e6	2.45357
$910$	0	0
$911$	495504.i	0.597050i	0.954402	+	0.298525i	$0.0964946\pi$
$911$	495504.i	0.597050i	−0.954402	+	0.298525i	$0.903505\pi$
$912$	0	0
$913$	−823956.	−0.988468
$914$	0	0
$915$	− 314680.i	− 0.375861i
$916$	0	0
$917$	−1.14617e6	−1.36305
$918$	0	0
$919$	− 106456.i	− 0.126049i	−0.998012	−	0.0630245i	$-0.979925\pi$
$919$	− 106456.i	− 0.126049i	0.998012	−	0.0630245i	$-0.0200746\pi$
$920$	0	0
$921$	−1.29438e6	−1.52596
$922$	0	0
$923$	− 1.16617e6i	− 1.36885i
$924$	0	0
$925$	343731.	0.401731
$926$	0	0
$927$	1.28806e6i	1.49891i
$928$	0	0
$929$	−834246.	−0.966635	−0.483318	−	0.875445i	$-0.660568\pi$
$929$	−834246.	−0.966635	−0.483318	+	0.875445i	$0.660568\pi$
$930$	0	0
$931$	− 94481.1i	− 0.109005i
$932$	0	0
$933$	2.13797e6	2.45606
$934$	0	0
$935$	− 1.07957e6i	− 1.23489i
$936$	0	0
$937$	487382.	0.555124	0.277562	−	0.960708i	$-0.410474\pi$
$937$	487382.	0.555124	0.277562	+	0.960708i	$0.410474\pi$
$938$	0	0
$939$	− 1.83303e6i	− 2.07893i
$940$	0	0
$941$	−226040.	−0.255274	−0.127637	−	0.991821i	$-0.540739\pi$
$941$	−226040.	−0.255274	−0.127637	+	0.991821i	$0.540739\pi$
$942$	0	0
$943$	− 244944.i	− 0.275450i
$944$	0	0
$945$	−959616.	−1.07457
$946$	0	0
$947$	814137.i	0.907816i	0.891049	+	0.453908i	$0.149970\pi$
$947$	814137.i	0.907816i	−0.891049	+	0.453908i	$0.850030\pi$
$948$	0	0
$949$	125975.	0.139879
$950$	0	0
$951$	288456.i	0.318947i
$952$	0	0
$953$	675918.	0.744232	0.372116	−	0.928186i	$-0.378632\pi$
$953$	675918.	0.744232	0.372116	+	0.928186i	$0.378632\pi$
$954$	0	0
$955$	− 1.38212e6i	− 1.51544i
$956$	0	0
$957$	856629.	0.935338
$958$	0	0
$959$	− 565488.i	− 0.614874i
$960$	0	0
$961$	873345.	0.945669
$962$	0	0
$963$	− 1.38962e6i	− 1.49846i
$964$	0	0
$965$	385923.	0.414425
$966$	0	0
$967$	− 726920.i	− 0.777381i	−0.921368	−	0.388690i	$-0.872928\pi$
$967$	− 726920.i	− 0.777381i	0.921368	−	0.388690i	$-0.127072\pi$
$968$	0	0
$969$	−694008.	−0.739123
$970$	0	0
$971$	1.32566e6i	1.40603i	0.711175	+	0.703015i	$0.248163\pi$
$971$	1.32566e6i	1.40603i	−0.711175	+	0.703015i	$0.751837\pi$
$972$	0	0
$973$	871026.	0.920037
$974$	0	0
$975$	− 701352.i	− 0.737780i
$976$	0	0
$977$	−67014.0	−0.0702064	−0.0351032	−	0.999384i	$-0.511176\pi$
$977$	−67014.0	−0.0702064	−0.0351032	+	0.999384i	$0.511176\pi$
$978$	0	0
$979$	− 944811.i	− 0.985779i
$980$	0	0
$981$	1.54949e6	1.61009
$982$	0	0
$983$	− 1.34114e6i	− 1.38793i	−0.720007	−	0.693966i	$-0.755862\pi$
$983$	− 1.34114e6i	− 1.38793i	0.720007	−	0.693966i	$-0.244138\pi$
$984$	0	0
$985$	1.22808e6	1.26577
$986$	0	0
$987$	2.41872e6i	2.48285i
$988$	0	0
$989$	194361.	0.198709
$990$	0	0
$991$	− 1.28733e6i	− 1.31082i	−0.755275	−	0.655408i	$-0.772497\pi$
$991$	− 1.28733e6i	− 1.31082i	0.755275	−	0.655408i	$-0.227503\pi$
$992$	0	0
$993$	−938196.	−0.951470
$994$	0	0
$995$	1.16297e6i	1.17468i
$996$	0	0
$997$	−1.03891e6	−1.04517	−0.522584	−	0.852588i	$-0.675032\pi$
$997$	−1.03891e6	−1.04517	−0.522584	+	0.852588i	$0.675032\pi$
$998$	0	0
$999$	− 1.07957e6i	− 1.08173i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.5.c.h.255.3		4
4.3	odd	2	inner	256.5.c.h.255.1		4
8.3	odd	2	inner	256.5.c.h.255.4		4
8.5	even	2	inner	256.5.c.h.255.2		4
16.3	odd	4		64.5.d.b.31.4	yes	4
16.5	even	4		64.5.d.b.31.3	yes	4
16.11	odd	4		64.5.d.b.31.1	✓	4
16.13	even	4		64.5.d.b.31.2	yes	4
48.5	odd	4		576.5.b.a.415.3		4
48.11	even	4		576.5.b.a.415.4		4
48.29	odd	4		576.5.b.a.415.1		4
48.35	even	4		576.5.b.a.415.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
64.5.d.b.31.1	✓	4	16.11	odd	4
64.5.d.b.31.2	yes	4	16.13	even	4
64.5.d.b.31.3	yes	4	16.5	even	4
64.5.d.b.31.4	yes	4	16.3	odd	4
256.5.c.h.255.1		4	4.3	odd	2	inner
256.5.c.h.255.2		4	8.5	even	2	inner
256.5.c.h.255.3		4	1.1	even	1	trivial
256.5.c.h.255.4		4	8.3	odd	2	inner
576.5.b.a.415.1		4	48.29	odd	4
576.5.b.a.415.2		4	48.35	even	4
576.5.b.a.415.3		4	48.5	odd	4
576.5.b.a.415.4		4	48.11	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:	\( N \)	\(=\)	\( 256 = 2^{8} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	256.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+14.2829i q^{3} -28.5657 q^{5} +56.0000i q^{7} -123.000 q^{9} +O(q^{10})\)	\(q+14.2829i q^{3} -28.5657 q^{5} +56.0000i q^{7} -123.000 q^{9} +99.9800i q^{11} -257.091 q^{13} -408.000i q^{15} +378.000 q^{17} +128.546i q^{19} -799.840 q^{21} -216.000i q^{23} +191.000 q^{25} -599.880i q^{27} -599.880 q^{29} +224.000i q^{31} -1428.00 q^{33} -1599.68i q^{35} +1799.64 q^{37} -3672.00i q^{39} +1134.00 q^{41} +899.820i q^{43} +3513.58 q^{45} -3024.00i q^{47} -735.000 q^{49} +5398.92i q^{51} +999.800 q^{53} -2856.00i q^{55} -1836.00 q^{57} +214.243i q^{59} +771.274 q^{61} -6888.00i q^{63} +7344.00 q^{65} -8098.38i q^{67} +3085.10 q^{69} +4536.00i q^{71} -490.000 q^{73} +2728.03i q^{75} -5598.88 q^{77} -2800.00i q^{79} -1395.00 q^{81} +8241.21i q^{83} -10797.8 q^{85} -8568.00i q^{87} -9450.00 q^{89} -14397.1i q^{91} -3199.36 q^{93} -3672.00i q^{95} -16198.0 q^{97} -12297.5i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 492 q^{9}+O(q^{10}) \) 4 * q - 492 * q^9	\( 4 q - 492 q^{9} + 1512 q^{17} + 764 q^{25} - 5712 q^{33} + 4536 q^{41} - 2940 q^{49} - 7344 q^{57} + 29376 q^{65} - 1960 q^{73} - 5580 q^{81} - 37800 q^{89} - 64792 q^{97}+O(q^{100}) \) 4 * q - 492 * q^9 + 1512 * q^17 + 764 * q^25 - 5712 * q^33 + 4536 * q^41 - 2940 * q^49 - 7344 * q^57 + 29376 * q^65 - 1960 * q^73 - 5580 * q^81 - 37800 * q^89 - 64792 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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