LMFDB - Embedded newform 64.5.d.b.31.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [64,5,Mod(31,64)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(64, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("64.31");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 64 = 2^{6} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	64.d (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:	$6.61567763737$
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			31.4
Root			$-3.57071 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	64.31
Dual form			64.5.d.b.31.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+14.2829 q^{3} +28.5657i q^{5} +56.0000i q^{7} +123.000 q^{9} +O(q^{10})$	$q+14.2829 q^{3} +28.5657i q^{5} +56.0000i q^{7} +123.000 q^{9} -99.9800 q^{11} -257.091i q^{13} +408.000i q^{15} +378.000 q^{17} +128.546 q^{19} +799.840i q^{21} -216.000i q^{23} -191.000 q^{25} +599.880 q^{27} -599.880i q^{29} -224.000i q^{31} -1428.00 q^{33} -1599.68 q^{35} -1799.64i q^{37} -3672.00i q^{39} -1134.00 q^{41} -899.820 q^{43} +3513.58i q^{45} +3024.00i q^{47} -735.000 q^{49} +5398.92 q^{51} -999.800i q^{53} -2856.00i q^{55} +1836.00 q^{57} -214.243 q^{59} +771.274i q^{61} +6888.00i q^{63} +7344.00 q^{65} -8098.38 q^{67} -3085.10i q^{69} +4536.00i q^{71} +490.000 q^{73} -2728.03 q^{75} -5598.88i q^{77} +2800.00i q^{79} -1395.00 q^{81} +8241.21 q^{83} +10797.8i q^{85} -8568.00i q^{87} +9450.00 q^{89} +14397.1 q^{91} -3199.36i q^{93} +3672.00i q^{95} -16198.0 q^{97} -12297.5 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/64\mathbb{Z}\right)^\times$.

$n$	$5$	$63$
$\chi(n)$	$-1$	$-1$

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
$3$	14.2829	1.58698	0.793492	−	0.608581i	$-0.208261\pi$
$3$	14.2829	1.58698	0.793492	+	0.608581i	$0.208261\pi$
$4$	0	0
$5$	28.5657i	1.14263i	0.820731	+	0.571314i	$0.193566\pi$
$5$	28.5657i	1.14263i	−0.820731	+	0.571314i	$0.806434\pi$
$6$	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
$11$	−99.9800	−0.826281	−0.413140	−	0.910667i	$-0.635568\pi$
$11$	−99.9800	−0.826281	−0.413140	+	0.910667i	$0.635568\pi$
$12$	0	0
$13$	− 257.091i	− 1.52125i	−0.649191	−	0.760626i	$-0.724892\pi$
$13$	− 257.091i	− 1.52125i	0.649191	−	0.760626i	$-0.275108\pi$
$14$	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
$19$	128.546	0.356082	0.178041	−	0.984023i	$-0.443024\pi$
$19$	128.546	0.356082	0.178041	+	0.984023i	$0.443024\pi$
$20$	0	0
$21$	799.840i	1.81370i
$22$	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
$27$	599.880	0.822881
$28$	0	0
$29$	− 599.880i	− 0.713294i	−0.934239	−	0.356647i	$-0.883920\pi$
$29$	− 599.880i	− 0.713294i	0.934239	−	0.356647i	$-0.116080\pi$
$30$	0	0
$31$	− 224.000i	− 0.233091i	−0.993185	−	0.116545i	$-0.962818\pi$
$31$	− 224.000i	− 0.233091i	0.993185	−	0.116545i	$-0.0371820\pi$
$32$	0	0
$33$	−1428.00	−1.31129
$34$	0	0
$35$	−1599.68	−1.30586
$36$	0	0
$37$	− 1799.64i	− 1.31457i	−0.753644	−	0.657283i	$-0.771706\pi$
$37$	− 1799.64i	− 1.31457i	0.753644	−	0.657283i	$-0.228294\pi$
$38$	0	0
$39$	− 3672.00i	− 2.41420i
$40$	0	0
$41$	−1134.00	−0.674598	−0.337299	−	0.941397i	$-0.609513\pi$
$41$	−1134.00	−0.674598	−0.337299	+	0.941397i	$0.609513\pi$
$42$	0	0
$43$	−899.820	−0.486652	−0.243326	−	0.969945i	$-0.578239\pi$
$43$	−899.820	−0.486652	−0.243326	+	0.969945i	$0.578239\pi$
$44$	0	0
$45$	3513.58i	1.73510i
$46$	0	0
$47$	3024.00i	1.36895i	0.729039	+	0.684473i	$0.239967\pi$
$47$	3024.00i	1.36895i	−0.729039	+	0.684473i	$0.760033\pi$
$48$	0	0
$49$	−735.000	−0.306122
$50$	0	0
$51$	5398.92	2.07571
$52$	0	0
$53$	− 999.800i	− 0.355927i	−0.984037	−	0.177964i	$-0.943049\pi$
$53$	− 999.800i	− 0.355927i	0.984037	−	0.177964i	$-0.0569510\pi$
$54$	0	0
$55$	− 2856.00i	− 0.944132i
$56$	0	0
$57$	1836.00	0.565097
$58$	0	0
$59$	−214.243	−0.0615464	−0.0307732	−	0.999526i	$-0.509797\pi$
$59$	−214.243	−0.0615464	−0.0307732	+	0.999526i	$0.509797\pi$
$60$	0	0
$61$	771.274i	0.207276i	0.994615	+	0.103638i	$0.0330484\pi$
$61$	771.274i	0.207276i	−0.994615	+	0.103638i	$0.966952\pi$
$62$	0	0
$63$	6888.00i	1.73545i
$64$	0	0
$65$	7344.00	1.73822
$66$	0	0
$67$	−8098.38	−1.80405	−0.902025	−	0.431684i	$-0.857920\pi$
$67$	−8098.38	−1.80405	−0.902025	+	0.431684i	$0.857920\pi$
$68$	0	0
$69$	− 3085.10i	− 0.647994i
$70$	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	$-0.485361\pi$
$73$	490.000	0.0919497	0.0459749	+	0.998943i	$0.485361\pi$
$74$	0	0
$75$	−2728.03	−0.484982
$76$	0	0
$77$	− 5598.88i	− 0.944321i
$78$	0	0
$79$	2800.00i	0.448646i	0.974515	+	0.224323i	$0.0720171\pi$
$79$	2800.00i	0.448646i	−0.974515	+	0.224323i	$0.927983\pi$
$80$	0	0
$81$	−1395.00	−0.212620
$82$	0	0
$83$	8241.21	1.19629	0.598143	−	0.801390i	$-0.295906\pi$
$83$	8241.21	1.19629	0.598143	+	0.801390i	$0.295906\pi$
$84$	0	0
$85$	10797.8i	1.49451i
$86$	0	0
$87$	− 8568.00i	− 1.13199i
$88$	0	0
$89$	9450.00	1.19303	0.596516	−	0.802601i	$-0.296551\pi$
$89$	9450.00	1.19303	0.596516	+	0.802601i	$0.296551\pi$
$90$	0	0
$91$	14397.1	1.73857
$92$	0	0
$93$	− 3199.36i	− 0.369911i
$94$	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
$99$	−12297.5	−1.25472
$100$	0	0
$101$	16482.4i	1.61576i	0.589344	+	0.807882i	$0.299386\pi$
$101$	16482.4i	1.61576i	−0.589344	+	0.807882i	$0.700614\pi$
$102$	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
$107$	−11297.7	−0.986788	−0.493394	−	0.869806i	$-0.664244\pi$
$107$	−11297.7	−0.986788	−0.493394	+	0.869806i	$0.664244\pi$
$108$	0	0
$109$	− 12597.5i	− 1.06030i	−0.847902	−	0.530152i	$-0.822135\pi$
$109$	− 12597.5i	− 1.06030i	0.847902	−	0.530152i	$-0.177865\pi$
$110$	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
$115$	6170.19	0.466555
$116$	0	0
$117$	− 31622.2i	− 2.31005i
$118$	0	0
$119$	21168.0i	1.49481i
$120$	0	0
$121$	−4645.00	−0.317260
$122$	0	0
$123$	−16196.8	−1.07058
$124$	0	0
$125$	12397.5i	0.793441i
$126$	0	0
$127$	− 22720.0i	− 1.40864i	−0.709881	−	0.704321i	$-0.751251\pi$
$127$	− 22720.0i	− 1.40864i	0.709881	−	0.704321i	$-0.248749\pi$
$128$	0	0
$129$	−12852.0	−0.772309
$130$	0	0
$131$	20467.3	1.19267	0.596333	−	0.802737i	$-0.296624\pi$
$131$	20467.3	1.19267	0.596333	+	0.802737i	$0.296624\pi$
$132$	0	0
$133$	7198.56i	0.406951i
$134$	0	0
$135$	17136.0i	0.940247i
$136$	0	0
$137$	10098.0	0.538015	0.269007	−	0.963138i	$-0.413304\pi$
$137$	10098.0	0.538015	0.269007	+	0.963138i	$0.413304\pi$
$138$	0	0
$139$	15554.0	0.805032	0.402516	−	0.915413i	$-0.368136\pi$
$139$	15554.0	0.805032	0.402516	+	0.915413i	$0.368136\pi$
$140$	0	0
$141$	43191.4i	2.17249i
$142$	0	0
$143$	25704.0i	1.25698i
$144$	0	0
$145$	17136.0	0.815030
$146$	0	0
$147$	−10497.9	−0.485811
$148$	0	0
$149$	− 7398.52i	− 0.333252i	−0.986020	−	0.166626i	$-0.946713\pi$
$149$	− 7398.52i	− 0.333252i	0.986020	−	0.166626i	$-0.0532872\pi$
$150$	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
$155$	6398.72	0.266336
$156$	0	0
$157$	− 9512.38i	− 0.385914i	−0.981207	−	0.192957i	$-0.938192\pi$
$157$	− 9512.38i	− 0.385914i	0.981207	−	0.192957i	$-0.0618077\pi$
$158$	0	0
$159$	− 14280.0i	− 0.564851i
$160$	0	0
$161$	12096.0	0.466649
$162$	0	0
$163$	−15296.9	−0.575744	−0.287872	−	0.957669i	$-0.592948\pi$
$163$	−15296.9	−0.575744	−0.287872	+	0.957669i	$0.592948\pi$
$164$	0	0
$165$	− 40791.8i	− 1.49832i
$166$	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
$171$	15811.1	0.540718
$172$	0	0
$173$	− 38420.9i	− 1.28373i	−0.766816	−	0.641867i	$-0.778160\pi$
$173$	− 38420.9i	− 1.28373i	0.766816	−	0.641867i	$-0.221840\pi$
$174$	0	0
$175$	− 10696.0i	− 0.349257i
$176$	0	0
$177$	−3060.00	−0.0976731
$178$	0	0
$179$	−56888.6	−1.77549	−0.887747	−	0.460331i	$-0.847731\pi$
$179$	−56888.6	−1.77549	−0.887747	+	0.460331i	$0.847731\pi$
$180$	0	0
$181$	32136.4i	0.980936i	0.871459	+	0.490468i	$0.163174\pi$
$181$	32136.4i	0.980936i	−0.871459	+	0.490468i	$0.836826\pi$
$182$	0	0
$183$	11016.0i	0.328944i
$184$	0	0
$185$	51408.0	1.50206
$186$	0	0
$187$	−37792.4	−1.08074
$188$	0	0
$189$	33593.3i	0.940435i
$190$	0	0
$191$	− 48384.0i	− 1.32628i	−0.748496	−	0.663140i	$-0.769224\pi$
$191$	− 48384.0i	− 1.32628i	0.748496	−	0.663140i	$-0.230776\pi$
$192$	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.	2.75854
$196$	0	0
$197$	42991.4i	1.10777i	0.832594	+	0.553884i	$0.186855\pi$
$197$	42991.4i	1.10777i	−0.832594	+	0.553884i	$0.813145\pi$
$198$	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
$203$	33593.3	0.815193
$204$	0	0
$205$	− 32393.5i	− 0.770815i
$206$	0	0
$207$	− 26568.0i	− 0.620038i
$208$	0	0
$209$	−12852.0	−0.294224
$210$	0	0
$211$	−15296.9	−0.343589	−0.171795	−	0.985133i	$-0.554957\pi$
$211$	−15296.9	−0.343589	−0.171795	+	0.985133i	$0.554957\pi$
$212$	0	0
$213$	64787.0i	1.42800i
$214$	0	0
$215$	− 25704.0i	− 0.556063i
$216$	0	0
$217$	12544.0	0.266389
$218$	0	0
$219$	6998.60	0.145923
$220$	0	0
$221$	− 97180.6i	− 1.98973i
$222$	0	0
$223$	64736.0i	1.30178i	0.759174	+	0.650888i	$0.225603\pi$
$223$	64736.0i	1.30178i	−0.759174	+	0.650888i	$0.774397\pi$
$224$	0	0
$225$	−23493.0	−0.464059
$226$	0	0
$227$	−6384.44	−0.123900	−0.0619499	−	0.998079i	$-0.519732\pi$
$227$	−6384.44	−0.123900	−0.0619499	+	0.998079i	$0.519732\pi$
$228$	0	0
$229$	− 36764.1i	− 0.701056i	−0.936552	−	0.350528i	$-0.886002\pi$
$229$	− 36764.1i	− 0.701056i	0.936552	−	0.350528i	$-0.113998\pi$
$230$	0	0
$231$	− 79968.0i	− 1.49862i
$232$	0	0
$233$	−16470.0	−0.303376	−0.151688	−	0.988428i	$-0.548471\pi$
$233$	−16470.0	−0.303376	−0.151688	+	0.988428i	$0.548471\pi$
$234$	0	0
$235$	−86382.7	−1.56420
$236$	0	0
$237$	39992.0i	0.711994i
$238$	0	0
$239$	− 17712.0i	− 0.310079i	−0.987908	−	0.155039i	$-0.950450\pi$
$239$	− 17712.0i	− 0.310079i	0.987908	−	0.155039i	$-0.0495504\pi$
$240$	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
$243$	−68514.9	−1.16031
$244$	0	0
$245$	− 20995.8i	− 0.349784i
$246$	0	0
$247$	− 33048.0i	− 0.541691i
$248$	0	0
$249$	117708.	1.89849
$250$	0	0
$251$	−31750.8	−0.503973	−0.251986	−	0.967731i	$-0.581084\pi$
$251$	−31750.8	−0.503973	−0.251986	+	0.967731i	$0.581084\pi$
$252$	0	0
$253$	21595.7i	0.337385i
$254$	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.	1.50236
$260$	0	0
$261$	− 73785.2i	− 1.08315i
$262$	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.	1.89332
$268$	0	0
$269$	132059.i	1.82501i	0.409069	+	0.912503i	$0.365853\pi$
$269$	132059.i	1.82501i	−0.409069	+	0.912503i	$0.634147\pi$
$270$	0	0
$271$	50288.0i	0.684740i	0.939565	+	0.342370i	$0.111230\pi$
$271$	50288.0i	0.684740i	−0.939565	+	0.342370i	$0.888770\pi$
$272$	0	0
$273$	205632.	2.75909
$274$	0	0
$275$	19096.2	0.252511
$276$	0	0
$277$	− 8998.20i	− 0.117272i	−0.998279	−	0.0586362i	$-0.981325\pi$
$277$	− 8998.20i	− 0.117272i	0.998279	−	0.0586362i	$-0.0186752\pi$
$278$	0	0
$279$	− 27552.0i	− 0.353952i
$280$	0	0
$281$	−47574.0	−0.602500	−0.301250	−	0.953545i	$-0.597404\pi$
$281$	−47574.0	−0.602500	−0.301250	+	0.953545i	$0.597404\pi$
$282$	0	0
$283$	111192.	1.38836	0.694178	−	0.719803i	$-0.255768\pi$
$283$	111192.	1.38836	0.694178	+	0.719803i	$0.255768\pi$
$284$	0	0
$285$	52446.7i	0.645696i
$286$	0	0
$287$	− 63504.0i	− 0.770970i
$288$	0	0
$289$	59363.0	0.710755
$290$	0	0
$291$	−231354.	−2.73206
$292$	0	0
$293$	24937.9i	0.290485i	0.989396	+	0.145243i	$0.0463963\pi$
$293$	24937.9i	0.290485i	−0.989396	+	0.145243i	$0.953604\pi$
$294$	0	0
$295$	− 6120.00i	− 0.0703246i
$296$	0	0
$297$	−59976.0	−0.679931
$298$	0	0
$299$	−55531.7	−0.621154
$300$	0	0
$301$	− 50389.9i	− 0.556174i
$302$	0	0
$303$	235416.i	2.56419i
$304$	0	0
$305$	−22032.0	−0.236840
$306$	0	0
$307$	90624.7	0.961546	0.480773	−	0.876845i	$-0.340356\pi$
$307$	90624.7	0.961546	0.480773	+	0.876845i	$0.340356\pi$
$308$	0	0
$309$	− 149570.i	− 1.56649i
$310$	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	$-0.272672\pi$
$313$	128338.	1.30999	0.654993	+	0.755635i	$0.272672\pi$
$314$	0	0
$315$	−196761.	−1.98297
$316$	0	0
$317$	20196.0i	0.200977i	0.994938	+	0.100488i	$0.0320405\pi$
$317$	20196.0i	0.200977i	−0.994938	+	0.100488i	$0.967959\pi$
$318$	0	0
$319$	59976.0i	0.589381i
$320$	0	0
$321$	−161364.	−1.56602
$322$	0	0
$323$	48590.3	0.465741
$324$	0	0
$325$	49104.5i	0.464894i
$326$	0	0
$327$	− 179928.i	− 1.68269i
$328$	0	0
$329$	−169344.	−1.56451
$330$	0	0
$331$	−65686.9	−0.599546	−0.299773	−	0.954011i	$-0.596911\pi$
$331$	−65686.9	−0.599546	−0.299773	+	0.954011i	$0.596911\pi$
$332$	0	0
$333$	− 221356.i	− 1.99619i
$334$	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.	1.73824
$340$	0	0
$341$	22395.5i	0.192598i
$342$	0	0
$343$	93296.0i	0.793003i
$344$	0	0
$345$	88128.0	0.740416
$346$	0	0
$347$	216657.	1.79934	0.899670	−	0.436571i	$-0.143807\pi$
$347$	216657.	1.79934	0.899670	+	0.436571i	$0.143807\pi$
$348$	0	0
$349$	169423.i	1.39098i	0.718534	+	0.695492i	$0.244814\pi$
$349$	169423.i	1.39098i	−0.718534	+	0.695492i	$0.755186\pi$
$350$	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.	−1.02816
$356$	0	0
$357$	302340.i	2.37224i
$358$	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
$363$	−66343.9	−0.503486
$364$	0	0
$365$	13997.2i	0.105064i
$366$	0	0
$367$	27664.0i	0.205392i	0.994713	+	0.102696i	$0.0327468\pi$
$367$	27664.0i	0.205392i	−0.994713	+	0.102696i	$0.967253\pi$
$368$	0	0
$369$	−139482.	−1.02439
$370$	0	0
$371$	55988.8	0.406774
$372$	0	0
$373$	120576.i	0.866648i	0.901238	+	0.433324i	$0.142659\pi$
$373$	120576.i	0.866648i	−0.901238	+	0.433324i	$0.857341\pi$
$374$	0	0
$375$	177072.i	1.25918i
$376$	0	0
$377$	−154224.	−1.08510
$378$	0	0
$379$	27894.4	0.194195	0.0970977	−	0.995275i	$-0.469044\pi$
$379$	27894.4	0.194195	0.0970977	+	0.995275i	$0.469044\pi$
$380$	0	0
$381$	− 324507.i	− 2.23549i
$382$	0	0
$383$	− 217728.i	− 1.48428i	−0.670243	−	0.742142i	$-0.733810\pi$
$383$	− 217728.i	− 1.48428i	0.670243	−	0.742142i	$-0.266190\pi$
$384$	0	0
$385$	159936.	1.07901
$386$	0	0
$387$	−110678.	−0.738990
$388$	0	0
$389$	− 144171.i	− 0.952750i	−0.879242	−	0.476375i	$-0.841950\pi$
$389$	− 144171.i	− 0.952750i	0.879242	−	0.476375i	$-0.158050\pi$
$390$	0	0
$391$	− 81648.0i	− 0.534062i
$392$	0	0
$393$	292332.	1.89274
$394$	0	0
$395$	−79984.0	−0.512636
$396$	0	0
$397$	− 105150.i	− 0.667160i	−0.942722	−	0.333580i	$-0.891743\pi$
$397$	− 105150.i	− 0.667160i	0.942722	−	0.333580i	$-0.108257\pi$
$398$	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
$403$	−57588.5	−0.354589
$404$	0	0
$405$	− 39849.2i	− 0.242946i
$406$	0	0
$407$	179928.i	1.08620i
$408$	0	0
$409$	53074.0	0.317275	0.158637	−	0.987337i	$-0.449290\pi$
$409$	53074.0	0.317275	0.158637	+	0.987337i	$0.449290\pi$
$410$	0	0
$411$	144228.	0.853821
$412$	0	0
$413$	− 11997.6i	− 0.0703387i
$414$	0	0
$415$	235416.i	1.36691i
$416$	0	0
$417$	222156.	1.27757
$418$	0	0
$419$	−231597.	−1.31918	−0.659590	−	0.751626i	$-0.729270\pi$
$419$	−231597.	−1.31918	−0.659590	+	0.751626i	$0.729270\pi$
$420$	0	0
$421$	− 160168.i	− 0.903673i	−0.892101	−	0.451837i	$-0.850769\pi$
$421$	− 160168.i	− 0.903673i	0.892101	−	0.451837i	$-0.149231\pi$
$422$	0	0
$423$	371952.i	2.07877i
$424$	0	0
$425$	−72198.0	−0.399712
$426$	0	0
$427$	−43191.4	−0.236887
$428$	0	0
$429$	367127.i	1.99481i
$430$	0	0
$431$	− 209520.i	− 1.12790i	−0.825809	−	0.563950i	$-0.809281\pi$
$431$	− 209520.i	− 1.12790i	0.825809	−	0.563950i	$-0.190719\pi$
$432$	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.	1.29344
$436$	0	0
$437$	− 27765.9i	− 0.145395i
$438$	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
$443$	−22495.5	−0.114627	−0.0573137	−	0.998356i	$-0.518254\pi$
$443$	−22495.5	−0.114627	−0.0573137	+	0.998356i	$0.518254\pi$
$444$	0	0
$445$	269946.i	1.36319i
$446$	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.	0.557408
$452$	0	0
$453$	229097.i	1.11641i
$454$	0	0
$455$	411264.i	1.98654i
$456$	0	0
$457$	350098.	1.67632	0.838161	−	0.545424i	$-0.183631\pi$
$457$	350098.	1.67632	0.838161	+	0.545424i	$0.183631\pi$
$458$	0	0
$459$	226755.	1.07629
$460$	0	0
$461$	203816.i	0.959041i	0.877531	+	0.479521i	$0.159189\pi$
$461$	203816.i	0.959041i	−0.877531	+	0.479521i	$0.840811\pi$
$462$	0	0
$463$	− 89488.0i	− 0.417448i	−0.977975	−	0.208724i	$-0.933069\pi$
$463$	− 89488.0i	− 0.417448i	0.977975	−	0.208724i	$-0.0669311\pi$
$464$	0	0
$465$	91392.0	0.422671
$466$	0	0
$467$	−17467.9	−0.0800954	−0.0400477	−	0.999198i	$-0.512751\pi$
$467$	−17467.9	−0.0800954	−0.0400477	+	0.999198i	$0.512751\pi$
$468$	0	0
$469$	− 453509.i	− 2.06177i
$470$	0	0
$471$	− 135864.i	− 0.612439i
$472$	0	0
$473$	89964.0	0.402111
$474$	0	0
$475$	−24552.2	−0.108819
$476$	0	0
$477$	− 122975.i	− 0.540482i
$478$	0	0
$479$	− 175392.i	− 0.764432i	−0.924073	−	0.382216i	$-0.875161\pi$
$479$	− 175392.i	− 0.764432i	0.924073	−	0.382216i	$-0.124839\pi$
$480$	0	0
$481$	−462672.	−1.99978
$482$	0	0
$483$	172765.	0.740564
$484$	0	0
$485$	− 462707.i	− 1.96708i
$486$	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.	−1.20475	−0.602374	−	0.798214i	$-0.705778\pi$
$491$	−290442.	−1.20475	−0.602374	+	0.798214i	$0.705778\pi$
$492$	0	0
$493$	− 226755.i	− 0.932959i
$494$	0	0
$495$	− 351288.i	− 1.43368i
$496$	0	0
$497$	−254016.	−1.02837
$498$	0	0
$499$	−375225.	−1.50692	−0.753461	−	0.657493i	$-0.771617\pi$
$499$	−375225.	−1.50692	−0.753461	+	0.657493i	$0.771617\pi$
$500$	0	0
$501$	669466.i	2.66718i
$502$	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.	−2.08562
$508$	0	0
$509$	− 8369.75i	− 0.0323055i	−0.999870	−	0.0161528i	$-0.994858\pi$
$509$	− 8369.75i	− 0.0323055i	0.999870	−	0.0161528i	$-0.00514181\pi$
$510$	0	0
$511$	27440.0i	0.105085i
$512$	0	0
$513$	77112.0	0.293013
$514$	0	0
$515$	299140.	1.12787
$516$	0	0
$517$	− 302340.i	− 1.13113i
$518$	0	0
$519$	− 548760.i	− 2.03727i
$520$	0	0
$521$	476658.	1.75603	0.878014	−	0.478635i	$-0.158868\pi$
$521$	476658.	1.75603	0.878014	+	0.478635i	$0.158868\pi$
$522$	0	0
$523$	214029.	0.782471	0.391236	−	0.920291i	$-0.372048\pi$
$523$	214029.	0.782471	0.391236	+	0.920291i	$0.372048\pi$
$524$	0	0
$525$	− 152769.i	− 0.554266i
$526$	0	0
$527$	− 84672.0i	− 0.304873i
$528$	0	0
$529$	233185.	0.833277
$530$	0	0
$531$	−26351.9	−0.0934593
$532$	0	0
$533$	291542.i	1.02623i
$534$	0	0
$535$	− 322728.i	− 1.12753i
$536$	0	0
$537$	−812532.	−2.81768
$538$	0	0
$539$	73485.3	0.252943
$540$	0	0
$541$	282543.i	0.965363i	0.875796	+	0.482682i	$0.160337\pi$
$541$	282543.i	0.965363i	−0.875796	+	0.482682i	$0.839663\pi$
$542$	0	0
$543$	459000.i	1.55673i
$544$	0	0
$545$	359856.	1.21153
$546$	0	0
$547$	164667.	0.550341	0.275171	−	0.961395i	$-0.411266\pi$
$547$	164667.	0.550341	0.275171	+	0.961395i	$0.411266\pi$
$548$	0	0
$549$	94866.7i	0.314753i
$550$	0	0
$551$	− 77112.0i	− 0.253991i
$552$	0	0
$553$	−156800.	−0.512738
$554$	0	0
$555$	734253.	2.38375
$556$	0	0
$557$	− 52589.5i	− 0.169507i	−0.996402	−	0.0847537i	$-0.972990\pi$
$557$	− 52589.5i	− 0.169507i	0.996402	−	0.0847537i	$-0.0270103\pi$
$558$	0	0
$559$	231336.i	0.740320i
$560$	0	0
$561$	−539784.	−1.71512
$562$	0	0
$563$	−287357.	−0.906577	−0.453288	−	0.891364i	$-0.649749\pi$
$563$	−287357.	−0.906577	−0.453288	+	0.891364i	$0.649749\pi$
$564$	0	0
$565$	399520.i	1.25153i
$566$	0	0
$567$	− 78120.0i	− 0.242994i
$568$	0	0
$569$	391986.	1.21073	0.605363	−	0.795949i	$-0.293028\pi$
$569$	391986.	1.21073	0.605363	+	0.795949i	$0.293028\pi$
$570$	0	0
$571$	582184.	1.78561	0.892807	−	0.450439i	$-0.148732\pi$
$571$	582184.	1.78561	0.892807	+	0.450439i	$0.148732\pi$
$572$	0	0
$573$	− 691062.i	− 2.10478i
$574$	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.	−0.575590
$580$	0	0
$581$	461508.i	1.36718i
$582$	0	0
$583$	99960.0i	0.294096i
$584$	0	0
$585$	903312.	2.63953
$586$	0	0
$587$	−383566.	−1.11318	−0.556588	−	0.830788i	$-0.687890\pi$
$587$	−383566.	−1.11318	−0.556588	+	0.830788i	$0.687890\pi$
$588$	0	0
$589$	− 28794.2i	− 0.0829994i
$590$	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.	−1.70801
$596$	0	0
$597$	− 581484.i	− 1.63151i
$598$	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	$-0.524483\pi$
$601$	−55510.0	−0.153682	−0.0768409	+	0.997043i	$0.524483\pi$
$602$	0	0
$603$	−996101.	−2.73948
$604$	0	0
$605$	− 132688.i	− 0.362510i
$606$	0	0
$607$	− 388192.i	− 1.05358i	−0.849994	−	0.526792i	$-0.823395\pi$
$607$	− 388192.i	− 1.05358i	0.849994	−	0.526792i	$-0.176605\pi$
$608$	0	0
$609$	479808.	1.29370
$610$	0	0
$611$	777444.	2.08251
$612$	0	0
$613$	502100.i	1.33619i	0.744075	+	0.668096i	$0.232890\pi$
$613$	502100.i	1.33619i	−0.744075	+	0.668096i	$0.767110\pi$
$614$	0	0
$615$	− 462672.i	− 1.22327i
$616$	0	0
$617$	190890.	0.501433	0.250716	−	0.968061i	$-0.419334\pi$
$617$	190890.	0.501433	0.250716	+	0.968061i	$0.419334\pi$
$618$	0	0
$619$	−335119.	−0.874616	−0.437308	−	0.899312i	$-0.644068\pi$
$619$	−335119.	−0.874616	−0.437308	+	0.899312i	$0.644068\pi$
$620$	0	0
$621$	− 129574.i	− 0.335997i
$622$	0	0
$623$	529200.i	1.36346i
$624$	0	0
$625$	−473519.	−1.21221
$626$	0	0
$627$	−183563.	−0.466929
$628$	0	0
$629$	− 680264.i	− 1.71940i
$630$	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.	1.60956
$636$	0	0
$637$	188962.i	0.465689i
$638$	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.	1.05492	0.527460	−	0.849580i	$-0.323144\pi$
$643$	436156.	1.05492	0.527460	+	0.849580i	$0.323144\pi$
$644$	0	0
$645$	− 367127.i	− 0.882463i
$646$	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.	0.422755
$652$	0	0
$653$	459308.i	1.07715i	0.842576	+	0.538577i	$0.181038\pi$
$653$	459308.i	1.07715i	−0.842576	+	0.538577i	$0.818962\pi$
$654$	0	0
$655$	584664.i	1.36277i
$656$	0	0
$657$	60270.0	0.139627
$658$	0	0
$659$	187862.	0.432583	0.216291	−	0.976329i	$-0.430604\pi$
$659$	187862.	0.432583	0.216291	+	0.976329i	$0.430604\pi$
$660$	0	0
$661$	116462.i	0.266553i	0.991079	+	0.133276i	$0.0425498\pi$
$661$	116462.i	0.266553i	−0.991079	+	0.133276i	$0.957450\pi$
$662$	0	0
$663$	− 1.38802e6i	− 3.15767i
$664$	0	0
$665$	−205632.	−0.464994
$666$	0	0
$667$	−129574.	−0.291250
$668$	0	0
$669$	924615.i	2.06590i
$670$	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.	−0.251472
$676$	0	0
$677$	316308.i	0.690133i	0.938578	+	0.345067i	$0.112144\pi$
$677$	316308.i	0.690133i	−0.938578	+	0.345067i	$0.887856\pi$
$678$	0	0
$679$	− 907088.i	− 1.96748i
$680$	0	0
$681$	−91188.0	−0.196627
$682$	0	0
$683$	339832.	0.728489	0.364244	−	0.931303i	$-0.381327\pi$
$683$	339832.	0.728489	0.364244	+	0.931303i	$0.381327\pi$
$684$	0	0
$685$	288457.i	0.614751i
$686$	0	0
$687$	− 525096.i	− 1.11256i
$688$	0	0
$689$	−257040.	−0.541455
$690$	0	0
$691$	437184.	0.915605	0.457802	−	0.889054i	$-0.348637\pi$
$691$	437184.	0.915605	0.457802	+	0.889054i	$0.348637\pi$
$692$	0	0
$693$	− 688662.i	− 1.43397i
$694$	0	0
$695$	444312.i	0.919853i
$696$	0	0
$697$	−428652.	−0.882347
$698$	0	0
$699$	−235239.	−0.481453
$700$	0	0
$701$	136973.i	0.278739i	0.990240	+	0.139369i	$0.0445076\pi$
$701$	136973.i	0.278739i	−0.990240	+	0.139369i	$0.955492\pi$
$702$	0	0
$703$	− 231336.i	− 0.468093i
$704$	0	0
$705$	−1.23379e6	−2.48235
$706$	0	0
$707$	−923015.	−1.84659
$708$	0	0
$709$	− 390522.i	− 0.776878i	−0.921474	−	0.388439i	$-0.873014\pi$
$709$	− 390522.i	− 0.776878i	0.921474	−	0.388439i	$-0.126986\pi$
$710$	0	0
$711$	344400.i	0.681277i
$712$	0	0
$713$	−48384.0	−0.0951750
$714$	0	0
$715$	−734253.	−1.43626
$716$	0	0
$717$	− 252978.i	− 0.492090i
$718$	0	0
$719$	− 595728.i	− 1.15237i	−0.817321	−	0.576183i	$-0.804542\pi$
$719$	− 595728.i	− 1.15237i	0.817321	−	0.576183i	$-0.195458\pi$
$720$	0	0
$721$	586432.	1.12810
$722$	0	0
$723$	88582.3	0.169461
$724$	0	0
$725$	114577.i	0.217983i
$726$	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.	−0.636521
$732$	0	0
$733$	− 978233.i	− 1.82068i	−0.413858	−	0.910341i	$-0.635819\pi$
$733$	− 978233.i	− 1.82068i	0.413858	−	0.910341i	$-0.364181\pi$
$734$	0	0
$735$	− 299880.i	− 0.555102i
$736$	0	0
$737$	809676.	1.49065
$738$	0	0
$739$	373425.	0.683778	0.341889	−	0.939740i	$-0.388933\pi$
$739$	373425.	0.683778	0.341889	+	0.939740i	$0.388933\pi$
$740$	0	0
$741$	− 472020.i	− 0.859654i
$742$	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
$747$	1.01367e6	1.81658
$748$	0	0
$749$	− 632673.i	− 1.12776i
$750$	0	0
$751$	− 258992.i	− 0.459205i	−0.973285	−	0.229602i	$-0.926257\pi$
$751$	− 258992.i	− 0.459205i	0.973285	−	0.229602i	$-0.0737426\pi$
$752$	0	0
$753$	−453492.	−0.799797
$754$	0	0
$755$	−458194.	−0.803814
$756$	0	0
$757$	221356.i	0.386277i	0.981171	+	0.193139i	$0.0618667\pi$
$757$	221356.i	0.386277i	−0.981171	+	0.193139i	$0.938133\pi$
$758$	0	0
$759$	308448.i	0.535425i
$760$	0	0
$761$	83538.0	0.144250	0.0721248	−	0.997396i	$-0.477022\pi$
$761$	83538.0	0.144250	0.0721248	+	0.997396i	$0.477022\pi$
$762$	0	0
$763$	705459.	1.21178
$764$	0	0
$765$	1.32813e6i	2.26944i
$766$	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
$771$	−1.18236e6	−1.98903
$772$	0	0
$773$	803753.i	1.34513i	0.740039	+	0.672564i	$0.234807\pi$
$773$	803753.i	1.34513i	−0.740039	+	0.672564i	$0.765193\pi$
$774$	0	0
$775$	42784.0i	0.0712325i
$776$	0	0
$777$	1.43942e6	2.38422
$778$	0	0
$779$	−145771.	−0.240213
$780$	0	0
$781$	− 453509.i	− 0.743505i
$782$	0	0
$783$	− 359856.i	− 0.586956i
$784$	0	0
$785$	271728.	0.440956
$786$	0	0
$787$	939026.	1.51610	0.758051	−	0.652195i	$-0.226152\pi$
$787$	939026.	1.51610	0.758051	+	0.652195i	$0.226152\pi$
$788$	0	0
$789$	1.27415e6i	2.04675i
$790$	0	0
$791$	783216.i	1.25178i
$792$	0	0
$793$	198288.	0.315319
$794$	0	0
$795$	407918.	0.645415
$796$	0	0
$797$	− 370583.i	− 0.583403i	−0.956509	−	0.291702i	$-0.905779\pi$
$797$	− 370583.i	− 0.583403i	0.956509	−	0.291702i	$-0.0942214\pi$
$798$	0	0
$799$	1.14307e6i	1.79052i
$800$	0	0
$801$	1.16235e6	1.81164
$802$	0	0
$803$	−48990.2	−0.0759763
$804$	0	0
$805$	345531.i	0.533206i
$806$	0	0
$807$	1.88618e6i	2.89626i
$808$	0	0
$809$	−243054.	−0.371369	−0.185685	−	0.982609i	$-0.559450\pi$
$809$	−243054.	−0.371369	−0.185685	+	0.982609i	$0.559450\pi$
$810$	0	0
$811$	−812280.	−1.23499	−0.617496	−	0.786574i	$-0.711853\pi$
$811$	−812280.	−1.23499	−0.617496	+	0.786574i	$0.711853\pi$
$812$	0	0
$813$	718256.i	1.08667i
$814$	0	0
$815$	− 436968.i	− 0.657861i
$816$	0	0
$817$	−115668.	−0.173288
$818$	0	0
$819$	1.77085e6	2.64005
$820$	0	0
$821$	− 484103.i	− 0.718210i	−0.933297	−	0.359105i	$-0.883082\pi$
$821$	− 484103.i	− 0.718210i	0.933297	−	0.359105i	$-0.116918\pi$
$822$	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.	0.942453	0.471227	−	0.882012i	$-0.343811\pi$
$827$	644571.	0.942453	0.471227	+	0.882012i	$0.343811\pi$
$828$	0	0
$829$	− 1.20447e6i	− 1.75262i	−0.481746	−	0.876311i	$-0.659997\pi$
$829$	− 1.20447e6i	− 1.75262i	0.481746	−	0.876311i	$-0.340003\pi$
$830$	0	0
$831$	− 128520.i	− 0.186110i
$832$	0	0
$833$	−277830.	−0.400395
$834$	0	0
$835$	−1.33893e6	−1.92037
$836$	0	0
$837$	− 134373.i	− 0.191806i
$838$	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.	−0.956158
$844$	0	0
$845$	− 1.07221e6i	− 1.50165i
$846$	0	0
$847$	− 260120.i	− 0.362583i
$848$	0	0
$849$	1.58814e6	2.20330
$850$	0	0
$851$	−388722.	−0.536760
$852$	0	0
$853$	209015.i	0.287263i	0.989631	+	0.143632i	$0.0458780\pi$
$853$	209015.i	0.287263i	−0.989631	+	0.143632i	$0.954122\pi$
$854$	0	0
$855$	451656.i	0.617839i
$856$	0	0
$857$	−140238.	−0.190943	−0.0954716	−	0.995432i	$-0.530436\pi$
$857$	−140238.	−0.190943	−0.0954716	+	0.995432i	$0.530436\pi$
$858$	0	0
$859$	−1.07451e6	−1.45622	−0.728108	−	0.685463i	$-0.759600\pi$
$859$	−1.07451e6	−1.45622	−0.728108	+	0.685463i	$0.759600\pi$
$860$	0	0
$861$	− 907019.i	− 1.22352i
$862$	0	0
$863$	− 514080.i	− 0.690254i	−0.938556	−	0.345127i	$-0.887836\pi$
$863$	− 514080.i	− 0.690254i	0.938556	−	0.345127i	$-0.112164\pi$
$864$	0	0
$865$	1.09752e6	1.46683
$866$	0	0
$867$	847873.	1.12796
$868$	0	0
$869$	− 279944.i	− 0.370708i
$870$	0	0
$871$	2.08202e6i	2.74441i
$872$	0	0
$873$	−1.99235e6	−2.61420
$874$	0	0
$875$	−694261.	−0.906790
$876$	0	0
$877$	937612.i	1.21906i	0.792764	+	0.609529i	$0.208641\pi$
$877$	937612.i	1.21906i	−0.792764	+	0.609529i	$0.791359\pi$
$878$	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.	0.303522	0.151761	−	0.988417i	$-0.451506\pi$
$883$	236653.	0.303522	0.151761	+	0.988417i	$0.451506\pi$
$884$	0	0
$885$	− 87411.1i	− 0.111604i
$886$	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.	0.175684
$892$	0	0
$893$	388722.i	0.487457i
$894$	0	0
$895$	− 1.62506e6i	− 2.02873i
$896$	0	0
$897$	−793152.	−0.985761
$898$	0	0
$899$	−134373.	−0.166262
$900$	0	0
$901$	− 377924.i	− 0.465538i
$902$	0	0
$903$	− 719712.i	− 0.882639i
$904$	0	0
$905$	−918000.	−1.12084
$906$	0	0
$907$	128674.	0.156415	0.0782073	−	0.996937i	$-0.475080\pi$
$907$	128674.	0.156415	0.0782073	+	0.996937i	$0.475080\pi$
$908$	0	0
$909$	2.02734e6i	2.45357i
$910$	0	0
$911$	− 495504.i	− 0.597050i	−0.954402	−	0.298525i	$-0.903505\pi$
$911$	− 495504.i	− 0.597050i	0.954402	−	0.298525i	$-0.0964946\pi$
$912$	0	0
$913$	−823956.	−0.988468
$914$	0	0
$915$	−314680.	−0.375861
$916$	0	0
$917$	1.14617e6i	1.36305i
$918$	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
$923$	1.16617e6	1.36885
$924$	0	0
$925$	343731.i	0.401731i
$926$	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
$931$	−94481.1	−0.109005
$932$	0	0
$933$	− 2.13797e6i	− 2.45606i
$934$	0	0
$935$	− 1.07957e6i	− 1.23489i
$936$	0	0
$937$	−487382.	−0.555124	−0.277562	−	0.960708i	$-0.589526\pi$
$937$	−487382.	−0.555124	−0.277562	+	0.960708i	$0.589526\pi$
$938$	0	0
$939$	1.83303e6	2.07893
$940$	0	0
$941$	− 226040.i	− 0.255274i	−0.991821	−	0.127637i	$-0.959261\pi$
$941$	− 226040.i	− 0.255274i	0.991821	−	0.127637i	$-0.0407393\pi$
$942$	0	0
$943$	244944.i	0.275450i
$944$	0	0
$945$	−959616.	−1.07457
$946$	0	0
$947$	814137.	0.907816	0.453908	−	0.891049i	$-0.350030\pi$
$947$	814137.	0.907816	0.453908	+	0.891049i	$0.350030\pi$
$948$	0	0
$949$	− 125975.i	− 0.139879i
$950$	0	0
$951$	288456.i	0.318947i
$952$	0	0
$953$	−675918.	−0.744232	−0.372116	−	0.928186i	$-0.621368\pi$
$953$	−675918.	−0.744232	−0.372116	+	0.928186i	$0.621368\pi$
$954$	0	0
$955$	1.38212e6	1.51544
$956$	0	0
$957$	856629.i	0.935338i
$958$	0	0
$959$	565488.i	0.614874i
$960$	0	0
$961$	873345.	0.945669
$962$	0	0
$963$	−1.38962e6	−1.49846
$964$	0	0
$965$	− 385923.i	− 0.414425i
$966$	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
$971$	−1.32566e6	−1.40603	−0.703015	−	0.711175i	$-0.748163\pi$
$971$	−1.32566e6	−1.40603	−0.703015	+	0.711175i	$0.748163\pi$
$972$	0	0
$973$	871026.i	0.920037i
$974$	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.	−0.985779
$980$	0	0
$981$	− 1.54949e6i	− 1.61009i
$982$	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
$987$	−2.41872e6	−2.48285
$988$	0	0
$989$	194361.i	0.198709i
$990$	0	0
$991$	1.28733e6i	1.31082i	0.755275	+	0.655408i	$0.227503\pi$
$991$	1.28733e6i	1.31082i	−0.755275	+	0.655408i	$0.772497\pi$
$992$	0	0
$993$	−938196.	−0.951470
$994$	0	0
$995$	1.16297e6	1.17468
$996$	0	0
$997$	1.03891e6i	1.04517i	0.852588	+	0.522584i	$0.175032\pi$
$997$	1.03891e6i	1.04517i	−0.852588	+	0.522584i	$0.824968\pi$
$998$	0	0
$999$	− 1.07957e6i	− 1.08173i

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
64.5.d.b.31.1	✓	4	8.5	even	2	inner
64.5.d.b.31.2	yes	4	4.3	odd	2	inner
64.5.d.b.31.3	yes	4	8.3	odd	2	inner
64.5.d.b.31.4	yes	4	1.1	even	1	trivial
256.5.c.h.255.1		4	16.5	even	4
256.5.c.h.255.2		4	16.3	odd	4
256.5.c.h.255.3		4	16.11	odd	4
256.5.c.h.255.4		4	16.13	even	4
576.5.b.a.415.1		4	12.11	even	2
576.5.b.a.415.2		4	3.2	odd	2
576.5.b.a.415.3		4	24.11	even	2
576.5.b.a.415.4		4	24.5	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 64 = 2^{6} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	64.d (of order \(2\), degree \(1\), minimal)

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