LMFDB - Embedded newform 256.8.b.c.129.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [256,8,Mod(129,256)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("256.129"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(256, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 8, names="a")

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

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,-912] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$79.9705665239$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 8)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	256.129
Dual form			256.8.b.c.129.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+84.0000i q^{3} +82.0000i q^{5} -456.000 q^{7} -4869.00 q^{9} -2524.00i q^{11} -10778.0i q^{13} -6888.00 q^{15} -11150.0 q^{17} -4124.00i q^{19} -38304.0i q^{21} +81704.0 q^{23} +71401.0 q^{25} -225288. i q^{27} +99798.0i q^{29} +40480.0 q^{31} +212016. q^{33} -37392.0i q^{35} +419442. i q^{37} +905352. q^{39} -141402. q^{41} -690428. i q^{43} -399258. i q^{45} +682032. q^{47} -615607. q^{49} -936600. i q^{51} -1.81312e6i q^{53} +206968. q^{55} +346416. q^{57} -966028. i q^{59} +1.88767e6i q^{61} +2.22026e6 q^{63} +883796. q^{65} -2.96587e6i q^{67} +6.86314e6i q^{69} -2.54823e6 q^{71} +1.68033e6 q^{73} +5.99768e6i q^{75} +1.15094e6i q^{77} -4.03806e6 q^{79} +8.27569e6 q^{81} +5.38576e6i q^{83} -914300. i q^{85} -8.38303e6 q^{87} +6.47305e6 q^{89} +4.91477e6i q^{91} +3.40032e6i q^{93} +338168. q^{95} -6.06576e6 q^{97} +1.22894e7i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 912 q^{7} - 9738 q^{9} - 13776 q^{15} - 22300 q^{17} + 163408 q^{23} + 142802 q^{25} + 80960 q^{31} + 424032 q^{33} + 1810704 q^{39} - 282804 q^{41} + 1364064 q^{47} - 1231214 q^{49} + 413936 q^{55}+ \cdots - 12131516 q^{97}+O(q^{100}) $ 2 * q - 912 * q^7 - 9738 * q^9 - 13776 * q^15 - 22300 * q^17 + 163408 * q^23 + 142802 * q^25 + 80960 * q^31 + 424032 * q^33 + 1810704 * q^39 - 282804 * q^41 + 1364064 * q^47 - 1231214 * q^49 + 413936 * q^55 + 692832 * q^57 + 4440528 * q^63 + 1767592 * q^65 - 5096464 * q^71 + 3360652 * q^73 - 8076128 * q^79 + 16551378 * q^81 - 16766064 * q^87 + 12946092 * q^89 + 676336 * q^95 - 12131516 * q^97

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
$3$	84.0000i	1.79620i	0.439790	+	0.898100i	$0.355053\pi$
$3$	84.0000i	1.79620i	−0.439790	+	0.898100i	$0.644947\pi$
$4$	0	0
$5$	82.0000i	0.293372i	0.989183	+	0.146686i	$0.0468607\pi$
$5$	82.0000i	0.293372i	−0.989183	+	0.146686i	$0.953139\pi$
$6$	0	0
$7$	−456.000	−0.502483	−0.251242	−	0.967924i	$-0.580839\pi$
$7$	−456.000	−0.502483	−0.251242	+	0.967924i	$0.580839\pi$
$8$	0	0
$9$	−4869.00	−2.22634
$10$	0	0
$11$	− 2524.00i	− 0.571762i	−0.958265	−	0.285881i	$-0.907714\pi$
$11$	− 2524.00i	− 0.571762i	0.958265	−	0.285881i	$-0.0922861\pi$
$12$	0	0
$13$	− 10778.0i	− 1.36062i	−0.732925	−	0.680309i	$-0.761845\pi$
$13$	− 10778.0i	− 1.36062i	0.732925	−	0.680309i	$-0.238155\pi$
$14$	0	0
$15$	−6888.00	−0.526955
$16$	0	0
$17$	−11150.0	−0.550432	−0.275216	−	0.961382i	$-0.588749\pi$
$17$	−11150.0	−0.550432	−0.275216	+	0.961382i	$0.588749\pi$
$18$	0	0
$19$	− 4124.00i	− 0.137937i	−0.997619	−	0.0689685i	$-0.978029\pi$
$19$	− 4124.00i	− 0.137937i	0.997619	−	0.0689685i	$-0.0219708\pi$
$20$	0	0
$21$	− 38304.0i	− 0.902561i
$22$	0	0
$23$	81704.0	1.40022	0.700109	−	0.714036i	$-0.253135\pi$
$23$	81704.0	1.40022	0.700109	+	0.714036i	$0.253135\pi$
$24$	0	0
$25$	71401.0	0.913933
$26$	0	0
$27$	− 225288.i	− 2.20275i
$28$	0	0
$29$	99798.0i	0.759852i	0.925017	+	0.379926i	$0.124051\pi$
$29$	99798.0i	0.759852i	−0.925017	+	0.379926i	$0.875949\pi$
$30$	0	0
$31$	40480.0	0.244048	0.122024	−	0.992527i	$-0.461062\pi$
$31$	40480.0	0.244048	0.122024	+	0.992527i	$0.461062\pi$
$32$	0	0
$33$	212016.	1.02700
$34$	0	0
$35$	− 37392.0i	− 0.147415i
$36$	0	0
$37$	419442.i	1.36134i	0.732591	+	0.680669i	$0.238311\pi$
$37$	419442.i	1.36134i	−0.732591	+	0.680669i	$0.761689\pi$
$38$	0	0
$39$	905352.	2.44394
$40$	0	0
$41$	−141402.	−0.320414	−0.160207	−	0.987083i	$-0.551216\pi$
$41$	−141402.	−0.320414	−0.160207	+	0.987083i	$0.551216\pi$
$42$	0	0
$43$	− 690428.i	− 1.32428i	−0.749382	−	0.662138i	$-0.769649\pi$
$43$	− 690428.i	− 1.32428i	0.749382	−	0.662138i	$-0.230351\pi$
$44$	0	0
$45$	− 399258.i	− 0.653145i
$46$	0	0
$47$	682032.	0.958213	0.479107	−	0.877757i	$-0.340961\pi$
$47$	682032.	0.958213	0.479107	+	0.877757i	$0.340961\pi$
$48$	0	0
$49$	−615607.	−0.747510
$50$	0	0
$51$	− 936600.i	− 0.988686i
$52$	0	0
$53$	− 1.81312e6i	− 1.67286i	−0.548071	−	0.836432i	$-0.684638\pi$
$53$	− 1.81312e6i	− 1.67286i	0.548071	−	0.836432i	$-0.315362\pi$
$54$	0	0
$55$	206968.	0.167739
$56$	0	0
$57$	346416.	0.247763
$58$	0	0
$59$	− 966028.i	− 0.612361i	−0.951973	−	0.306181i	$-0.900949\pi$
$59$	− 966028.i	− 0.612361i	0.951973	−	0.306181i	$-0.0990511\pi$
$60$	0	0
$61$	1.88767e6i	1.06481i	0.846490	+	0.532404i	$0.178711\pi$
$61$	1.88767e6i	1.06481i	−0.846490	+	0.532404i	$0.821289\pi$
$62$	0	0
$63$	2.22026e6	1.11870
$64$	0	0
$65$	883796.	0.399168
$66$	0	0
$67$	− 2.96587e6i	− 1.20473i	−0.798220	−	0.602365i	$-0.794225\pi$
$67$	− 2.96587e6i	− 1.20473i	0.798220	−	0.602365i	$-0.205775\pi$
$68$	0	0
$69$	6.86314e6i	2.51507i
$70$	0	0
$71$	−2.54823e6	−0.844957	−0.422479	−	0.906373i	$-0.638840\pi$
$71$	−2.54823e6	−0.844957	−0.422479	+	0.906373i	$0.638840\pi$
$72$	0	0
$73$	1.68033e6	0.505549	0.252775	−	0.967525i	$-0.418657\pi$
$73$	1.68033e6	0.505549	0.252775	+	0.967525i	$0.418657\pi$
$74$	0	0
$75$	5.99768e6i	1.64161i
$76$	0	0
$77$	1.15094e6i	0.287301i
$78$	0	0
$79$	−4.03806e6	−0.921464	−0.460732	−	0.887539i	$-0.652413\pi$
$79$	−4.03806e6	−0.921464	−0.460732	+	0.887539i	$0.652413\pi$
$80$	0	0
$81$	8.27569e6	1.73024
$82$	0	0
$83$	5.38576e6i	1.03389i	0.856019	+	0.516945i	$0.172931\pi$
$83$	5.38576e6i	1.03389i	−0.856019	+	0.516945i	$0.827069\pi$
$84$	0	0
$85$	− 914300.i	− 0.161481i
$86$	0	0
$87$	−8.38303e6	−1.36485
$88$	0	0
$89$	6.47305e6	0.973293	0.486647	−	0.873599i	$-0.338220\pi$
$89$	6.47305e6	0.973293	0.486647	+	0.873599i	$0.338220\pi$
$90$	0	0
$91$	4.91477e6i	0.683688i
$92$	0	0
$93$	3.40032e6i	0.438359i
$94$	0	0
$95$	338168.	0.0404669
$96$	0	0
$97$	−6.06576e6	−0.674814	−0.337407	−	0.941359i	$-0.609550\pi$
$97$	−6.06576e6	−0.674814	−0.337407	+	0.941359i	$0.609550\pi$
$98$	0	0
$99$	1.22894e7i	1.27293i
$100$	0	0
$101$	− 9.70069e6i	− 0.936866i	−0.883499	−	0.468433i	$-0.844819\pi$
$101$	− 9.70069e6i	− 0.936866i	0.883499	−	0.468433i	$-0.155181\pi$
$102$	0	0
$103$	4.10159e6	0.369847	0.184924	−	0.982753i	$-0.440796\pi$
$103$	4.10159e6	0.369847	0.184924	+	0.982753i	$0.440796\pi$
$104$	0	0
$105$	3.14093e6	0.264786
$106$	0	0
$107$	72900.0i	0.00575287i	0.999996	+	0.00287643i	$0.000915598\pi$
$107$	72900.0i	0.00575287i	−0.999996	+	0.00287643i	$0.999084\pi$
$108$	0	0
$109$	9.55841e6i	0.706957i	0.935443	+	0.353478i	$0.115001\pi$
$109$	9.55841e6i	0.706957i	−0.935443	+	0.353478i	$0.884999\pi$
$110$	0	0
$111$	−3.52331e7	−2.44524
$112$	0	0
$113$	9.33890e6	0.608865	0.304433	−	0.952534i	$-0.401533\pi$
$113$	9.33890e6	0.608865	0.304433	+	0.952534i	$0.401533\pi$
$114$	0	0
$115$	6.69973e6i	0.410785i
$116$	0	0
$117$	5.24781e7i	3.02920i
$118$	0	0
$119$	5.08440e6	0.276583
$120$	0	0
$121$	1.31166e7	0.673089
$122$	0	0
$123$	− 1.18778e7i	− 0.575529i
$124$	0	0
$125$	1.22611e7i	0.561495i
$126$	0	0
$127$	3.59794e7	1.55862	0.779311	−	0.626637i	$-0.215569\pi$
$127$	3.59794e7	1.55862	0.779311	+	0.626637i	$0.215569\pi$
$128$	0	0
$129$	5.79960e7	2.37867
$130$	0	0
$131$	676052.i	0.0262743i	0.999914	+	0.0131371i	$0.00418180\pi$
$131$	676052.i	0.0262743i	−0.999914	+	0.0131371i	$0.995818\pi$
$132$	0	0
$133$	1.88054e6i	0.0693111i
$134$	0	0
$135$	1.84736e7	0.646225
$136$	0	0
$137$	2.95841e7	0.982962	0.491481	−	0.870888i	$-0.336456\pi$
$137$	2.95841e7	0.982962	0.491481	+	0.870888i	$0.336456\pi$
$138$	0	0
$139$	− 3.19084e7i	− 1.00775i	−0.863776	−	0.503876i	$-0.831907\pi$
$139$	− 3.19084e7i	− 1.00775i	0.863776	−	0.503876i	$-0.168093\pi$
$140$	0	0
$141$	5.72907e7i	1.72114i
$142$	0	0
$143$	−2.72037e7	−0.777949
$144$	0	0
$145$	−8.18344e6	−0.222919
$146$	0	0
$147$	− 5.17110e7i	− 1.34268i
$148$	0	0
$149$	1.16603e7i	0.288773i	0.989521	+	0.144386i	$0.0461208\pi$
$149$	1.16603e7i	0.288773i	−0.989521	+	0.144386i	$0.953879\pi$
$150$	0	0
$151$	−1.76295e7	−0.416698	−0.208349	−	0.978055i	$-0.566809\pi$
$151$	−1.76295e7	−0.416698	−0.208349	+	0.978055i	$0.566809\pi$
$152$	0	0
$153$	5.42894e7	1.22545
$154$	0	0
$155$	3.31936e6i	0.0715968i
$156$	0	0
$157$	6.34658e6i	0.130885i	0.997856	+	0.0654427i	$0.0208460\pi$
$157$	6.34658e6i	0.130885i	−0.997856	+	0.0654427i	$0.979154\pi$
$158$	0	0
$159$	1.52302e8	3.00480
$160$	0	0
$161$	−3.72570e7	−0.703587
$162$	0	0
$163$	− 8.04234e7i	− 1.45454i	−0.686351	−	0.727271i	$-0.740789\pi$
$163$	− 8.04234e7i	− 1.45454i	0.686351	−	0.727271i	$-0.259211\pi$
$164$	0	0
$165$	1.73853e7i	0.301293i
$166$	0	0
$167$	1.14767e8	1.90682	0.953411	−	0.301673i	$-0.0975451\pi$
$167$	1.14767e8	1.90682	0.953411	+	0.301673i	$0.0975451\pi$
$168$	0	0
$169$	−5.34168e7	−0.851283
$170$	0	0
$171$	2.00798e7i	0.307095i
$172$	0	0
$173$	− 6.33755e7i	− 0.930594i	−0.885155	−	0.465297i	$-0.845947\pi$
$173$	− 6.33755e7i	− 0.930594i	0.885155	−	0.465297i	$-0.154053\pi$
$174$	0	0
$175$	−3.25589e7	−0.459236
$176$	0	0
$177$	8.11464e7	1.09992
$178$	0	0
$179$	1.13228e7i	0.147559i	0.997275	+	0.0737796i	$0.0235061\pi$
$179$	1.13228e7i	0.147559i	−0.997275	+	0.0737796i	$0.976494\pi$
$180$	0	0
$181$	5.22650e6i	0.0655143i	0.999463	+	0.0327571i	$0.0104288\pi$
$181$	5.22650e6i	0.0655143i	−0.999463	+	0.0327571i	$0.989571\pi$
$182$	0	0
$183$	−1.58564e8	−1.91261
$184$	0	0
$185$	−3.43942e7	−0.399379
$186$	0	0
$187$	2.81426e7i	0.314716i
$188$	0	0
$189$	1.02731e8i	1.10684i
$190$	0	0
$191$	8.50301e7	0.882990	0.441495	−	0.897264i	$-0.354448\pi$
$191$	8.50301e7	0.882990	0.441495	+	0.897264i	$0.354448\pi$
$192$	0	0
$193$	1.15092e8	1.15237	0.576186	−	0.817319i	$-0.304540\pi$
$193$	1.15092e8	1.15237	0.576186	+	0.817319i	$0.304540\pi$
$194$	0	0
$195$	7.42389e7i	0.716985i
$196$	0	0
$197$	1.38522e8i	1.29088i	0.763810	+	0.645441i	$0.223326\pi$
$197$	1.38522e8i	1.29088i	−0.763810	+	0.645441i	$0.776674\pi$
$198$	0	0
$199$	−2.19614e7	−0.197548	−0.0987742	−	0.995110i	$-0.531492\pi$
$199$	−2.19614e7	−0.197548	−0.0987742	+	0.995110i	$0.531492\pi$
$200$	0	0
$201$	2.49133e8	2.16394
$202$	0	0
$203$	− 4.55079e7i	− 0.381813i
$204$	0	0
$205$	− 1.15950e7i	− 0.0940007i
$206$	0	0
$207$	−3.97817e8	−3.11736
$208$	0	0
$209$	−1.04090e7	−0.0788671
$210$	0	0
$211$	6.10208e7i	0.447187i	0.974682	+	0.223594i	$0.0717789\pi$
$211$	6.10208e7i	0.447187i	−0.974682	+	0.223594i	$0.928221\pi$
$212$	0	0
$213$	− 2.14051e8i	− 1.51771i
$214$	0	0
$215$	5.66151e7	0.388506
$216$	0	0
$217$	−1.84589e7	−0.122630
$218$	0	0
$219$	1.41147e8i	0.908068i
$220$	0	0
$221$	1.20175e8i	0.748928i
$222$	0	0
$223$	4.22448e7	0.255098	0.127549	−	0.991832i	$-0.459289\pi$
$223$	4.22448e7	0.255098	0.127549	+	0.991832i	$0.459289\pi$
$224$	0	0
$225$	−3.47651e8	−2.03472
$226$	0	0
$227$	2.39102e8i	1.35673i	0.734726	+	0.678364i	$0.237311\pi$
$227$	2.39102e8i	1.35673i	−0.734726	+	0.678364i	$0.762689\pi$
$228$	0	0
$229$	4.67889e7i	0.257465i	0.991679	+	0.128733i	$0.0410909\pi$
$229$	4.67889e7i	0.257465i	−0.991679	+	0.128733i	$0.958909\pi$
$230$	0	0
$231$	−9.66793e7	−0.516050
$232$	0	0
$233$	−3.45225e8	−1.78795	−0.893977	−	0.448113i	$-0.852096\pi$
$233$	−3.45225e8	−1.78795	−0.893977	+	0.448113i	$0.852096\pi$
$234$	0	0
$235$	5.59266e7i	0.281113i
$236$	0	0
$237$	− 3.39197e8i	− 1.65513i
$238$	0	0
$239$	−2.34413e8	−1.11068	−0.555340	−	0.831624i	$-0.687412\pi$
$239$	−2.34413e8	−1.11068	−0.555340	+	0.831624i	$0.687412\pi$
$240$	0	0
$241$	−1.09557e8	−0.504175	−0.252087	−	0.967705i	$-0.581117\pi$
$241$	−1.09557e8	−0.504175	−0.252087	+	0.967705i	$0.581117\pi$
$242$	0	0
$243$	2.02453e8i	0.905112i
$244$	0	0
$245$	− 5.04798e7i	− 0.219299i
$246$	0	0
$247$	−4.44485e7	−0.187680
$248$	0	0
$249$	−4.52404e8	−1.85707
$250$	0	0
$251$	− 3.94031e8i	− 1.57280i	−0.617720	−	0.786398i	$-0.711943\pi$
$251$	− 3.94031e8i	− 1.57280i	0.617720	−	0.786398i	$-0.288057\pi$
$252$	0	0
$253$	− 2.06221e8i	− 0.800591i
$254$	0	0
$255$	7.68012e7	0.290053
$256$	0	0
$257$	3.19064e8	1.17250	0.586248	−	0.810131i	$-0.300604\pi$
$257$	3.19064e8	1.17250	0.586248	+	0.810131i	$0.300604\pi$
$258$	0	0
$259$	− 1.91266e8i	− 0.684050i
$260$	0	0
$261$	− 4.85916e8i	− 1.69169i
$262$	0	0
$263$	2.19359e8	0.743549	0.371774	−	0.928323i	$-0.378750\pi$
$263$	2.19359e8	0.743549	0.371774	+	0.928323i	$0.378750\pi$
$264$	0	0
$265$	1.48676e8	0.490772
$266$	0	0
$267$	5.43736e8i	1.74823i
$268$	0	0
$269$	− 1.48033e8i	− 0.463687i	−0.972753	−	0.231844i	$-0.925524\pi$
$269$	− 1.48033e8i	− 0.463687i	0.972753	−	0.231844i	$-0.0744757\pi$
$270$	0	0
$271$	3.69934e8	1.12910	0.564549	−	0.825399i	$-0.309050\pi$
$271$	3.69934e8	1.12910	0.564549	+	0.825399i	$0.309050\pi$
$272$	0	0
$273$	−4.12841e8	−1.22804
$274$	0	0
$275$	− 1.80216e8i	− 0.522552i
$276$	0	0
$277$	3.95860e8i	1.11908i	0.828803	+	0.559541i	$0.189023\pi$
$277$	3.95860e8i	1.11908i	−0.828803	+	0.559541i	$0.810977\pi$
$278$	0	0
$279$	−1.97097e8	−0.543332
$280$	0	0
$281$	5.97760e8	1.60714	0.803572	−	0.595208i	$-0.202930\pi$
$281$	5.97760e8	1.60714	0.803572	+	0.595208i	$0.202930\pi$
$282$	0	0
$283$	8.05797e7i	0.211336i	0.994401	+	0.105668i	$0.0336981\pi$
$283$	8.05797e7i	0.211336i	−0.994401	+	0.105668i	$0.966302\pi$
$284$	0	0
$285$	2.84061e7i	0.0726867i
$286$	0	0
$287$	6.44793e7	0.161003
$288$	0	0
$289$	−2.86016e8	−0.697025
$290$	0	0
$291$	− 5.09524e8i	− 1.21210i
$292$	0	0
$293$	− 7.54530e8i	− 1.75243i	−0.481924	−	0.876213i	$-0.660062\pi$
$293$	− 7.54530e8i	− 1.75243i	0.481924	−	0.876213i	$-0.339938\pi$
$294$	0	0
$295$	7.92143e7	0.179650
$296$	0	0
$297$	−5.68627e8	−1.25945
$298$	0	0
$299$	− 8.80606e8i	− 1.90516i
$300$	0	0
$301$	3.14835e8i	0.665427i
$302$	0	0
$303$	8.14858e8	1.68280
$304$	0	0
$305$	−1.54789e8	−0.312385
$306$	0	0
$307$	− 8.20472e8i	− 1.61838i	−0.587549	−	0.809188i	$-0.699907\pi$
$307$	− 8.20472e8i	− 1.61838i	0.587549	−	0.809188i	$-0.300093\pi$
$308$	0	0
$309$	3.44534e8i	0.664320i
$310$	0	0
$311$	6.53503e8	1.23193	0.615965	−	0.787773i	$-0.288766\pi$
$311$	6.53503e8	1.23193	0.615965	+	0.787773i	$0.288766\pi$
$312$	0	0
$313$	−6.63587e8	−1.22319	−0.611594	−	0.791172i	$-0.709471\pi$
$313$	−6.63587e8	−1.22319	−0.611594	+	0.791172i	$0.709471\pi$
$314$	0	0
$315$	1.82062e8i	0.328195i
$316$	0	0
$317$	− 3.54718e8i	− 0.625426i	−0.949848	−	0.312713i	$-0.898762\pi$
$317$	− 3.54718e8i	− 0.625426i	0.949848	−	0.312713i	$-0.101238\pi$
$318$	0	0
$319$	2.51890e8	0.434454
$320$	0	0
$321$	−6.12360e6	−0.0103333
$322$	0	0
$323$	4.59826e7i	0.0759250i
$324$	0	0
$325$	− 7.69560e8i	− 1.24351i
$326$	0	0
$327$	−8.02906e8	−1.26984
$328$	0	0
$329$	−3.11007e8	−0.481486
$330$	0	0
$331$	− 3.05543e8i	− 0.463100i	−0.972823	−	0.231550i	$-0.925620\pi$
$331$	− 3.05543e8i	− 0.463100i	0.972823	−	0.231550i	$-0.0743797\pi$
$332$	0	0
$333$	− 2.04226e9i	− 3.03080i
$334$	0	0
$335$	2.43201e8	0.353434
$336$	0	0
$337$	3.54965e7	0.0505220	0.0252610	−	0.999681i	$-0.491958\pi$
$337$	3.54965e7	0.0505220	0.0252610	+	0.999681i	$0.491958\pi$
$338$	0	0
$339$	7.84467e8i	1.09364i
$340$	0	0
$341$	− 1.02172e8i	− 0.139537i
$342$	0	0
$343$	6.56252e8	0.878095
$344$	0	0
$345$	−5.62777e8	−0.737853
$346$	0	0
$347$	− 1.90594e8i	− 0.244882i	−0.992476	−	0.122441i	$-0.960928\pi$
$347$	− 1.90594e8i	− 0.244882i	0.992476	−	0.122441i	$-0.0390723\pi$
$348$	0	0
$349$	8.60864e8i	1.08404i	0.840366	+	0.542020i	$0.182340\pi$
$349$	8.60864e8i	1.08404i	−0.840366	+	0.542020i	$0.817660\pi$
$350$	0	0
$351$	−2.42815e9	−2.99710
$352$	0	0
$353$	−1.04544e9	−1.26500	−0.632498	−	0.774562i	$-0.717970\pi$
$353$	−1.04544e9	−1.26500	−0.632498	+	0.774562i	$0.717970\pi$
$354$	0	0
$355$	− 2.08955e8i	− 0.247887i
$356$	0	0
$357$	4.27090e8i	0.496798i
$358$	0	0
$359$	7.63303e8	0.870696	0.435348	−	0.900262i	$-0.356625\pi$
$359$	7.63303e8	0.870696	0.435348	+	0.900262i	$0.356625\pi$
$360$	0	0
$361$	8.76864e8	0.980973
$362$	0	0
$363$	1.10179e9i	1.20900i
$364$	0	0
$365$	1.37787e8i	0.148314i
$366$	0	0
$367$	1.38692e9	1.46460	0.732302	−	0.680980i	$-0.238446\pi$
$367$	1.38692e9	1.46460	0.732302	+	0.680980i	$0.238446\pi$
$368$	0	0
$369$	6.88486e8	0.713351
$370$	0	0
$371$	8.26782e8i	0.840586i
$372$	0	0
$373$	− 4.77105e8i	− 0.476029i	−0.971262	−	0.238015i	$-0.923503\pi$
$373$	− 4.77105e8i	− 0.476029i	0.971262	−	0.238015i	$-0.0764966\pi$
$374$	0	0
$375$	−1.02994e9	−1.00856
$376$	0	0
$377$	1.07562e9	1.03387
$378$	0	0
$379$	− 3.92468e8i	− 0.370311i	−0.982709	−	0.185156i	$-0.940721\pi$
$379$	− 3.92468e8i	− 0.370311i	0.982709	−	0.185156i	$-0.0592789\pi$
$380$	0	0
$381$	3.02227e9i	2.79960i
$382$	0	0
$383$	−2.10409e9	−1.91368	−0.956839	−	0.290617i	$-0.906139\pi$
$383$	−2.10409e9	−1.91368	−0.956839	+	0.290617i	$0.906139\pi$
$384$	0	0
$385$	−9.43774e7	−0.0842860
$386$	0	0
$387$	3.36169e9i	2.94829i
$388$	0	0
$389$	1.26019e9i	1.08546i	0.839907	+	0.542730i	$0.182609\pi$
$389$	1.26019e9i	1.08546i	−0.839907	+	0.542730i	$0.817391\pi$
$390$	0	0
$391$	−9.11000e8	−0.770725
$392$	0	0
$393$	−5.67884e7	−0.0471939
$394$	0	0
$395$	− 3.31121e8i	− 0.270332i
$396$	0	0
$397$	− 9.81298e8i	− 0.787107i	−0.919302	−	0.393554i	$-0.871246\pi$
$397$	− 9.81298e8i	− 0.787107i	0.919302	−	0.393554i	$-0.128754\pi$
$398$	0	0
$399$	−1.57966e8	−0.124497
$400$	0	0
$401$	9.09981e8	0.704737	0.352369	−	0.935861i	$-0.385376\pi$
$401$	9.09981e8	0.704737	0.352369	+	0.935861i	$0.385376\pi$
$402$	0	0
$403$	− 4.36293e8i	− 0.332056i
$404$	0	0
$405$	6.78606e8i	0.507604i
$406$	0	0
$407$	1.05867e9	0.778361
$408$	0	0
$409$	3.55609e7	0.0257004	0.0128502	−	0.999917i	$-0.495910\pi$
$409$	3.55609e7	0.0257004	0.0128502	+	0.999917i	$0.495910\pi$
$410$	0	0
$411$	2.48507e9i	1.76560i
$412$	0	0
$413$	4.40509e8i	0.307701i
$414$	0	0
$415$	−4.41633e8	−0.303314
$416$	0	0
$417$	2.68031e9	1.81013
$418$	0	0
$419$	− 2.65360e9i	− 1.76233i	−0.472813	−	0.881163i	$-0.656761\pi$
$419$	− 2.65360e9i	− 1.76233i	0.472813	−	0.881163i	$-0.343239\pi$
$420$	0	0
$421$	1.12113e9i	0.732264i	0.930563	+	0.366132i	$0.119318\pi$
$421$	1.12113e9i	0.732264i	−0.930563	+	0.366132i	$0.880682\pi$
$422$	0	0
$423$	−3.32081e9	−2.13331
$424$	0	0
$425$	−7.96121e8	−0.503058
$426$	0	0
$427$	− 8.60778e8i	− 0.535049i
$428$	0	0
$429$	− 2.28511e9i	− 1.39735i
$430$	0	0
$431$	1.06344e9	0.639799	0.319900	−	0.947451i	$-0.396351\pi$
$431$	1.06344e9	0.639799	0.319900	+	0.947451i	$0.396351\pi$
$432$	0	0
$433$	−7.05962e8	−0.417901	−0.208951	−	0.977926i	$-0.567005\pi$
$433$	−7.05962e8	−0.417901	−0.208951	+	0.977926i	$0.567005\pi$
$434$	0	0
$435$	− 6.87409e8i	− 0.400408i
$436$	0	0
$437$	− 3.36947e8i	− 0.193142i
$438$	0	0
$439$	1.48506e9	0.837760	0.418880	−	0.908042i	$-0.362423\pi$
$439$	1.48506e9	0.837760	0.418880	+	0.908042i	$0.362423\pi$
$440$	0	0
$441$	2.99739e9	1.66421
$442$	0	0
$443$	− 7.22153e8i	− 0.394654i	−0.980338	−	0.197327i	$-0.936774\pi$
$443$	− 7.22153e8i	− 0.394654i	0.980338	−	0.197327i	$-0.0632260\pi$
$444$	0	0
$445$	5.30790e8i	0.285537i
$446$	0	0
$447$	−9.79462e8	−0.518694
$448$	0	0
$449$	−1.22968e9	−0.641109	−0.320554	−	0.947230i	$-0.603869\pi$
$449$	−1.22968e9	−0.641109	−0.320554	+	0.947230i	$0.603869\pi$
$450$	0	0
$451$	3.56899e8i	0.183201i
$452$	0	0
$453$	− 1.48088e9i	− 0.748473i
$454$	0	0
$455$	−4.03011e8	−0.200575
$456$	0	0
$457$	8.85551e7	0.0434017	0.0217009	−	0.999765i	$-0.493092\pi$
$457$	8.85551e7	0.0434017	0.0217009	+	0.999765i	$0.493092\pi$
$458$	0	0
$459$	2.51196e9i	1.21246i
$460$	0	0
$461$	2.10937e8i	0.100277i	0.998742	+	0.0501384i	$0.0159662\pi$
$461$	2.10937e8i	0.100277i	−0.998742	+	0.0501384i	$0.984034\pi$
$462$	0	0
$463$	3.29775e9	1.54413	0.772066	−	0.635543i	$-0.219224\pi$
$463$	3.29775e9	1.54413	0.772066	+	0.635543i	$0.219224\pi$
$464$	0	0
$465$	−2.78826e8	−0.128602
$466$	0	0
$467$	− 8.82873e7i	− 0.0401134i	−0.999799	−	0.0200567i	$-0.993615\pi$
$467$	− 8.82873e7i	− 0.0401134i	0.999799	−	0.0200567i	$-0.00638467\pi$
$468$	0	0
$469$	1.35244e9i	0.605357i
$470$	0	0
$471$	−5.33113e8	−0.235096
$472$	0	0
$473$	−1.74264e9	−0.757171
$474$	0	0
$475$	− 2.94458e8i	− 0.126065i
$476$	0	0
$477$	8.82807e9i	3.72436i
$478$	0	0
$479$	4.51507e9	1.87711	0.938557	−	0.345125i	$-0.112164\pi$
$479$	4.51507e9	1.87711	0.938557	+	0.345125i	$0.112164\pi$
$480$	0	0
$481$	4.52075e9	1.85226
$482$	0	0
$483$	− 3.12959e9i	− 1.26378i
$484$	0	0
$485$	− 4.97392e8i	− 0.197972i
$486$	0	0
$487$	3.31338e9	1.29993	0.649964	−	0.759965i	$-0.274784\pi$
$487$	3.31338e9	1.29993	0.649964	+	0.759965i	$0.274784\pi$
$488$	0	0
$489$	6.75557e9	2.61265
$490$	0	0
$491$	− 4.01694e9i	− 1.53147i	−0.643154	−	0.765737i	$-0.722374\pi$
$491$	− 4.01694e9i	− 1.53147i	0.643154	−	0.765737i	$-0.277626\pi$
$492$	0	0
$493$	− 1.11275e9i	− 0.418247i
$494$	0	0
$495$	−1.00773e9	−0.373443
$496$	0	0
$497$	1.16199e9	0.424577
$498$	0	0
$499$	− 2.70976e9i	− 0.976290i	−0.872763	−	0.488145i	$-0.837674\pi$
$499$	− 2.70976e9i	− 0.976290i	0.872763	−	0.488145i	$-0.162326\pi$
$500$	0	0
$501$	9.64045e9i	3.42504i
$502$	0	0
$503$	3.04579e8	0.106712	0.0533558	−	0.998576i	$-0.483008\pi$
$503$	3.04579e8	0.106712	0.0533558	+	0.998576i	$0.483008\pi$
$504$	0	0
$505$	7.95456e8	0.274850
$506$	0	0
$507$	− 4.48701e9i	− 1.52908i
$508$	0	0
$509$	− 1.88202e8i	− 0.0632575i	−0.999500	−	0.0316287i	$-0.989931\pi$
$509$	− 1.88202e8i	− 0.0632575i	0.999500	−	0.0316287i	$-0.0100694\pi$
$510$	0	0
$511$	−7.66229e8	−0.254030
$512$	0	0
$513$	−9.29088e8	−0.303841
$514$	0	0
$515$	3.36331e8i	0.108503i
$516$	0	0
$517$	− 1.72145e9i	− 0.547870i
$518$	0	0
$519$	5.32355e9	1.67153
$520$	0	0
$521$	−4.14963e9	−1.28552	−0.642758	−	0.766069i	$-0.722210\pi$
$521$	−4.14963e9	−1.28552	−0.642758	+	0.766069i	$0.722210\pi$
$522$	0	0
$523$	2.51360e9i	0.768318i	0.923267	+	0.384159i	$0.125509\pi$
$523$	2.51360e9i	0.768318i	−0.923267	+	0.384159i	$0.874491\pi$
$524$	0	0
$525$	− 2.73494e9i	− 0.824880i
$526$	0	0
$527$	−4.51352e8	−0.134332
$528$	0	0
$529$	3.27072e9	0.960613
$530$	0	0
$531$	4.70359e9i	1.36332i
$532$	0	0
$533$	1.52403e9i	0.435962i
$534$	0	0
$535$	−5.97780e6	−0.00168773
$536$	0	0
$537$	−9.51112e8	−0.265046
$538$	0	0
$539$	1.55379e9i	0.427398i
$540$	0	0
$541$	− 1.32416e9i	− 0.359543i	−0.983708	−	0.179772i	$-0.942464\pi$
$541$	− 1.32416e9i	− 0.359543i	0.983708	−	0.179772i	$-0.0575358\pi$
$542$	0	0
$543$	−4.39026e8	−0.117677
$544$	0	0
$545$	−7.83789e8	−0.207401
$546$	0	0
$547$	− 5.58047e8i	− 0.145786i	−0.997340	−	0.0728929i	$-0.976777\pi$
$547$	− 5.58047e8i	− 0.145786i	0.997340	−	0.0728929i	$-0.0232231\pi$
$548$	0	0
$549$	− 9.19107e9i	− 2.37062i
$550$	0	0
$551$	4.11567e8	0.104812
$552$	0	0
$553$	1.84136e9	0.463020
$554$	0	0
$555$	− 2.88912e9i	− 0.717364i
$556$	0	0
$557$	− 3.30331e9i	− 0.809946i	−0.914329	−	0.404973i	$-0.867281\pi$
$557$	− 3.30331e9i	− 0.809946i	0.914329	−	0.404973i	$-0.132719\pi$
$558$	0	0
$559$	−7.44143e9	−1.80184
$560$	0	0
$561$	−2.36398e9	−0.565293
$562$	0	0
$563$	1.22011e8i	0.0288152i	0.999896	+	0.0144076i	$0.00458623\pi$
$563$	1.22011e8i	0.0288152i	−0.999896	+	0.0144076i	$0.995414\pi$
$564$	0	0
$565$	7.65790e8i	0.178624i
$566$	0	0
$567$	−3.77371e9	−0.869417
$568$	0	0
$569$	−5.00925e8	−0.113993	−0.0569967	−	0.998374i	$-0.518152\pi$
$569$	−5.00925e8	−0.113993	−0.0569967	+	0.998374i	$0.518152\pi$
$570$	0	0
$571$	− 6.98702e9i	− 1.57060i	−0.619116	−	0.785300i	$-0.712509\pi$
$571$	− 6.98702e9i	− 1.57060i	0.619116	−	0.785300i	$-0.287491\pi$
$572$	0	0
$573$	7.14253e9i	1.58603i
$574$	0	0
$575$	5.83375e9	1.27971
$576$	0	0
$577$	−8.16573e9	−1.76962	−0.884809	−	0.465954i	$-0.845711\pi$
$577$	−8.16573e9	−1.76962	−0.884809	+	0.465954i	$0.845711\pi$
$578$	0	0
$579$	9.66769e9i	2.06989i
$580$	0	0
$581$	− 2.45591e9i	− 0.519512i
$582$	0	0
$583$	−4.57631e9	−0.956479
$584$	0	0
$585$	−4.30320e9	−0.888682
$586$	0	0
$587$	8.53182e9i	1.74104i	0.492135	+	0.870519i	$0.336217\pi$
$587$	8.53182e9i	1.74104i	−0.492135	+	0.870519i	$0.663783\pi$
$588$	0	0
$589$	− 1.66940e8i	− 0.0336632i
$590$	0	0
$591$	−1.16358e10	−2.31868
$592$	0	0
$593$	−1.71175e9	−0.337092	−0.168546	−	0.985694i	$-0.553907\pi$
$593$	−1.71175e9	−0.337092	−0.168546	+	0.985694i	$0.553907\pi$
$594$	0	0
$595$	4.16921e8i	0.0811417i
$596$	0	0
$597$	− 1.84475e9i	− 0.354836i
$598$	0	0
$599$	−4.77362e9	−0.907516	−0.453758	−	0.891125i	$-0.649917\pi$
$599$	−4.77362e9	−0.907516	−0.453758	+	0.891125i	$0.649917\pi$
$600$	0	0
$601$	−7.89998e8	−0.148445	−0.0742224	−	0.997242i	$-0.523647\pi$
$601$	−7.89998e8	−0.148445	−0.0742224	+	0.997242i	$0.523647\pi$
$602$	0	0
$603$	1.44408e10i	2.68214i
$604$	0	0
$605$	1.07556e9i	0.197465i
$606$	0	0
$607$	1.82652e9	0.331485	0.165743	−	0.986169i	$-0.446998\pi$
$607$	1.82652e9	0.331485	0.165743	+	0.986169i	$0.446998\pi$
$608$	0	0
$609$	3.82266e9	0.685813
$610$	0	0
$611$	− 7.35094e9i	− 1.30376i
$612$	0	0
$613$	6.90339e9i	1.21046i	0.796050	+	0.605231i	$0.206919\pi$
$613$	6.90339e9i	1.21046i	−0.796050	+	0.605231i	$0.793081\pi$
$614$	0	0
$615$	9.73977e8	0.168844
$616$	0	0
$617$	5.69235e9	0.975649	0.487825	−	0.872942i	$-0.337791\pi$
$617$	5.69235e9	0.975649	0.487825	+	0.872942i	$0.337791\pi$
$618$	0	0
$619$	− 4.28594e9i	− 0.726321i	−0.931727	−	0.363161i	$-0.881698\pi$
$619$	− 4.28594e9i	− 0.726321i	0.931727	−	0.363161i	$-0.118302\pi$
$620$	0	0
$621$	− 1.84069e10i	− 3.08433i
$622$	0	0
$623$	−2.95171e9	−0.489064
$624$	0	0
$625$	4.57279e9	0.749206
$626$	0	0
$627$	− 8.74354e8i	− 0.141661i
$628$	0	0
$629$	− 4.67678e9i	− 0.749324i
$630$	0	0
$631$	−5.61602e8	−0.0889869	−0.0444935	−	0.999010i	$-0.514167\pi$
$631$	−5.61602e8	−0.0889869	−0.0444935	+	0.999010i	$0.514167\pi$
$632$	0	0
$633$	−5.12575e9	−0.803238
$634$	0	0
$635$	2.95031e9i	0.457256i
$636$	0	0
$637$	6.63501e9i	1.01708i
$638$	0	0
$639$	1.24073e10	1.88116
$640$	0	0
$641$	5.17445e9	0.775998	0.387999	−	0.921660i	$-0.373166\pi$
$641$	5.17445e9	0.775998	0.387999	+	0.921660i	$0.373166\pi$
$642$	0	0
$643$	1.04374e10i	1.54830i	0.633004	+	0.774148i	$0.281822\pi$
$643$	1.04374e10i	1.54830i	−0.633004	+	0.774148i	$0.718178\pi$
$644$	0	0
$645$	4.75567e9i	0.697835i
$646$	0	0
$647$	−9.71623e8	−0.141037	−0.0705185	−	0.997510i	$-0.522465\pi$
$647$	−9.71623e8	−0.141037	−0.0705185	+	0.997510i	$0.522465\pi$
$648$	0	0
$649$	−2.43825e9	−0.350125
$650$	0	0
$651$	− 1.55055e9i	− 0.220268i
$652$	0	0
$653$	− 7.25223e9i	− 1.01924i	−0.860400	−	0.509619i	$-0.829786\pi$
$653$	− 7.25223e9i	− 1.01924i	0.860400	−	0.509619i	$-0.170214\pi$
$654$	0	0
$655$	−5.54363e7	−0.00770814
$656$	0	0
$657$	−8.18151e9	−1.12552
$658$	0	0
$659$	− 3.81924e9i	− 0.519851i	−0.965629	−	0.259925i	$-0.916302\pi$
$659$	− 3.81924e9i	− 0.519851i	0.965629	−	0.259925i	$-0.0836979\pi$
$660$	0	0
$661$	− 1.07881e10i	− 1.45292i	−0.687210	−	0.726459i	$-0.741165\pi$
$661$	− 1.07881e10i	− 1.45292i	0.687210	−	0.726459i	$-0.258835\pi$
$662$	0	0
$663$	−1.00947e10	−1.34523
$664$	0	0
$665$	−1.54205e8	−0.0203339
$666$	0	0
$667$	8.15390e9i	1.06396i
$668$	0	0
$669$	3.54857e9i	0.458207i
$670$	0	0
$671$	4.76448e9	0.608817
$672$	0	0
$673$	−6.34833e9	−0.802798	−0.401399	−	0.915903i	$-0.631476\pi$
$673$	−6.34833e9	−0.802798	−0.401399	+	0.915903i	$0.631476\pi$
$674$	0	0
$675$	− 1.60858e10i	− 2.01316i
$676$	0	0
$677$	− 8.82566e9i	− 1.09317i	−0.837404	−	0.546584i	$-0.815928\pi$
$677$	− 8.82566e9i	− 1.09317i	0.837404	−	0.546584i	$-0.184072\pi$
$678$	0	0
$679$	2.76599e9	0.339083
$680$	0	0
$681$	−2.00846e10	−2.43696
$682$	0	0
$683$	4.92331e9i	0.591268i	0.955301	+	0.295634i	$0.0955309\pi$
$683$	4.92331e9i	0.591268i	−0.955301	+	0.295634i	$0.904469\pi$
$684$	0	0
$685$	2.42590e9i	0.288374i
$686$	0	0
$687$	−3.93027e9	−0.462459
$688$	0	0
$689$	−1.95418e10	−2.27613
$690$	0	0
$691$	5.68449e9i	0.655418i	0.944779	+	0.327709i	$0.106277\pi$
$691$	5.68449e9i	0.655418i	−0.944779	+	0.327709i	$0.893723\pi$
$692$	0	0
$693$	− 5.60395e9i	− 0.639628i
$694$	0	0
$695$	2.61649e9	0.295646
$696$	0	0
$697$	1.57663e9	0.176366
$698$	0	0
$699$	− 2.89989e10i	− 3.21152i
$700$	0	0
$701$	− 1.70567e9i	− 0.187017i	−0.995618	−	0.0935085i	$-0.970192\pi$
$701$	− 1.70567e9i	− 0.187017i	0.995618	−	0.0935085i	$-0.0298082\pi$
$702$	0	0
$703$	1.72978e9	0.187779
$704$	0	0
$705$	−4.69784e9	−0.504936
$706$	0	0
$707$	4.42351e9i	0.470760i
$708$	0	0
$709$	− 4.52189e9i	− 0.476495i	−0.971204	−	0.238248i	$-0.923427\pi$
$709$	− 4.52189e9i	− 0.476495i	0.971204	−	0.238248i	$-0.0765730\pi$
$710$	0	0
$711$	1.96613e10	2.05149
$712$	0	0
$713$	3.30738e9	0.341720
$714$	0	0
$715$	− 2.23070e9i	− 0.228229i
$716$	0	0
$717$	− 1.96907e10i	− 1.99500i
$718$	0	0
$719$	3.09206e9	0.310239	0.155120	−	0.987896i	$-0.450424\pi$
$719$	3.09206e9	0.310239	0.155120	+	0.987896i	$0.450424\pi$
$720$	0	0
$721$	−1.87033e9	−0.185842
$722$	0	0
$723$	− 9.20280e9i	− 0.905599i
$724$	0	0
$725$	7.12568e9i	0.694453i
$726$	0	0
$727$	1.44622e10	1.39593	0.697965	−	0.716132i	$-0.254089\pi$
$727$	1.44622e10	1.39593	0.697965	+	0.716132i	$0.254089\pi$
$728$	0	0
$729$	1.09288e9	0.104478
$730$	0	0
$731$	7.69827e9i	0.728924i
$732$	0	0
$733$	3.15415e9i	0.295814i	0.989001	+	0.147907i	$0.0472536\pi$
$733$	3.15415e9i	0.295814i	−0.989001	+	0.147907i	$0.952746\pi$
$734$	0	0
$735$	4.24030e9	0.393905
$736$	0	0
$737$	−7.48585e9	−0.688819
$738$	0	0
$739$	− 1.54236e10i	− 1.40582i	−0.711277	−	0.702912i	$-0.751883\pi$
$739$	− 1.54236e10i	− 1.40582i	0.711277	−	0.702912i	$-0.248117\pi$
$740$	0	0
$741$	− 3.73367e9i	− 0.337111i
$742$	0	0
$743$	−1.59520e10	−1.42677	−0.713385	−	0.700772i	$-0.752839\pi$
$743$	−1.59520e10	−1.42677	−0.713385	+	0.700772i	$0.752839\pi$
$744$	0	0
$745$	−9.56141e8	−0.0847179
$746$	0	0
$747$	− 2.62233e10i	− 2.30179i
$748$	0	0
$749$	− 3.32424e7i	− 0.00289072i
$750$	0	0
$751$	−6.13964e9	−0.528936	−0.264468	−	0.964395i	$-0.585196\pi$
$751$	−6.13964e9	−0.528936	−0.264468	+	0.964395i	$0.585196\pi$
$752$	0	0
$753$	3.30986e10	2.82506
$754$	0	0
$755$	− 1.44562e9i	− 0.122248i
$756$	0	0
$757$	1.42818e10i	1.19660i	0.801273	+	0.598299i	$0.204157\pi$
$757$	1.42818e10i	1.19660i	−0.801273	+	0.598299i	$0.795843\pi$
$758$	0	0
$759$	1.73226e10	1.43802
$760$	0	0
$761$	−1.47536e10	−1.21353	−0.606767	−	0.794880i	$-0.707534\pi$
$761$	−1.47536e10	−1.21353	−0.606767	+	0.794880i	$0.707534\pi$
$762$	0	0
$763$	− 4.35863e9i	− 0.355234i
$764$	0	0
$765$	4.45173e9i	0.359512i
$766$	0	0
$767$	−1.04118e10	−0.833190
$768$	0	0
$769$	1.97592e10	1.56685	0.783424	−	0.621487i	$-0.213471\pi$
$769$	1.97592e10	1.56685	0.783424	+	0.621487i	$0.213471\pi$
$770$	0	0
$771$	2.68014e10i	2.10604i
$772$	0	0
$773$	− 1.01370e10i	− 0.789374i	−0.918816	−	0.394687i	$-0.870853\pi$
$773$	− 1.01370e10i	− 0.789374i	0.918816	−	0.394687i	$-0.129147\pi$
$774$	0	0
$775$	2.89031e9	0.223043
$776$	0	0
$777$	1.60663e10	1.22869
$778$	0	0
$779$	5.83142e8i	0.0441970i
$780$	0	0
$781$	6.43174e9i	0.483114i
$782$	0	0
$783$	2.24833e10	1.67376
$784$	0	0
$785$	−5.20420e8	−0.0383981
$786$	0	0
$787$	1.27882e10i	0.935188i	0.883943	+	0.467594i	$0.154879\pi$
$787$	1.27882e10i	0.935188i	−0.883943	+	0.467594i	$0.845121\pi$
$788$	0	0
$789$	1.84261e10i	1.33556i
$790$	0	0
$791$	−4.25854e9	−0.305945
$792$	0	0
$793$	2.03453e10	1.44880
$794$	0	0
$795$	1.24888e10i	0.881524i
$796$	0	0
$797$	− 7.38617e9i	− 0.516791i	−0.966039	−	0.258396i	$-0.916806\pi$
$797$	− 7.38617e9i	− 0.516791i	0.966039	−	0.258396i	$-0.0831938\pi$
$798$	0	0
$799$	−7.60466e9	−0.527431
$800$	0	0
$801$	−3.15173e10	−2.16688
$802$	0	0
$803$	− 4.24114e9i	− 0.289054i
$804$	0	0
$805$	− 3.05508e9i	− 0.206413i
$806$	0	0
$807$	1.24348e10	0.832875
$808$	0	0
$809$	−1.53742e10	−1.02087	−0.510437	−	0.859915i	$-0.670516\pi$
$809$	−1.53742e10	−1.02087	−0.510437	+	0.859915i	$0.670516\pi$
$810$	0	0
$811$	9.77882e9i	0.643744i	0.946783	+	0.321872i	$0.104312\pi$
$811$	9.77882e9i	0.643744i	−0.946783	+	0.321872i	$0.895688\pi$
$812$	0	0
$813$	3.10745e10i	2.02809i
$814$	0	0
$815$	6.59472e9	0.426722
$816$	0	0
$817$	−2.84733e9	−0.182667
$818$	0	0
$819$	− 2.39300e10i	− 1.52212i
$820$	0	0
$821$	− 1.83470e10i	− 1.15708i	−0.815654	−	0.578540i	$-0.803623\pi$
$821$	− 1.83470e10i	− 1.15708i	0.815654	−	0.578540i	$-0.196377\pi$
$822$	0	0
$823$	−3.16960e10	−1.98201	−0.991004	−	0.133829i	$-0.957273\pi$
$823$	−3.16960e10	−1.98201	−0.991004	+	0.133829i	$0.957273\pi$
$824$	0	0
$825$	1.51382e10	0.938608
$826$	0	0
$827$	− 6.12845e9i	− 0.376774i	−0.982095	−	0.188387i	$-0.939674\pi$
$827$	− 6.12845e9i	− 0.376774i	0.982095	−	0.188387i	$-0.0603260\pi$
$828$	0	0
$829$	− 1.24652e10i	− 0.759904i	−0.925006	−	0.379952i	$-0.875940\pi$
$829$	− 1.24652e10i	− 0.759904i	0.925006	−	0.379952i	$-0.124060\pi$
$830$	0	0
$831$	−3.32522e10	−2.01010
$832$	0	0
$833$	6.86402e9	0.411454
$834$	0	0
$835$	9.41091e9i	0.559409i
$836$	0	0
$837$	− 9.11966e9i	− 0.537575i
$838$	0	0
$839$	−1.82237e10	−1.06530	−0.532648	−	0.846337i	$-0.678803\pi$
$839$	−1.82237e10	−1.06530	−0.532648	+	0.846337i	$0.678803\pi$
$840$	0	0
$841$	7.29024e9	0.422625
$842$	0	0
$843$	5.02118e10i	2.88675i
$844$	0	0
$845$	− 4.38017e9i	− 0.249743i
$846$	0	0
$847$	−5.98117e9	−0.338216
$848$	0	0
$849$	−6.76870e9	−0.379602
$850$	0	0
$851$	3.42701e10i	1.90617i
$852$	0	0
$853$	2.48619e10i	1.37155i	0.727812	+	0.685777i	$0.240537\pi$
$853$	2.48619e10i	1.37155i	−0.727812	+	0.685777i	$0.759463\pi$
$854$	0	0
$855$	−1.64654e9	−0.0900930
$856$	0	0
$857$	−2.96761e10	−1.61055	−0.805275	−	0.592902i	$-0.797982\pi$
$857$	−2.96761e10	−1.61055	−0.805275	+	0.592902i	$0.797982\pi$
$858$	0	0
$859$	1.14772e10i	0.617819i	0.951091	+	0.308910i	$0.0999640\pi$
$859$	1.14772e10i	0.617819i	−0.951091	+	0.308910i	$0.900036\pi$
$860$	0	0
$861$	5.41626e9i	0.289194i
$862$	0	0
$863$	−2.13485e10	−1.13066	−0.565328	−	0.824866i	$-0.691250\pi$
$863$	−2.13485e10	−1.13066	−0.565328	+	0.824866i	$0.691250\pi$
$864$	0	0
$865$	5.19679e9	0.273010
$866$	0	0
$867$	− 2.40254e10i	− 1.25200i
$868$	0	0
$869$	1.01921e10i	0.526858i
$870$	0	0
$871$	−3.19661e10	−1.63918
$872$	0	0
$873$	2.95342e10	1.50236
$874$	0	0
$875$	− 5.59108e9i	− 0.282142i
$876$	0	0
$877$	− 7.92753e9i	− 0.396862i	−0.980115	−	0.198431i	$-0.936415\pi$
$877$	− 7.92753e9i	− 0.396862i	0.980115	−	0.198431i	$-0.0635846\pi$
$878$	0	0
$879$	6.33805e10	3.14771
$880$	0	0
$881$	7.32045e9	0.360680	0.180340	−	0.983604i	$-0.442280\pi$
$881$	7.32045e9	0.360680	0.180340	+	0.983604i	$0.442280\pi$
$882$	0	0
$883$	3.54988e9i	0.173521i	0.996229	+	0.0867604i	$0.0276514\pi$
$883$	3.54988e9i	0.173521i	−0.996229	+	0.0867604i	$0.972349\pi$
$884$	0	0
$885$	6.65400e9i	0.322687i
$886$	0	0
$887$	5.80634e9	0.279364	0.139682	−	0.990196i	$-0.455392\pi$
$887$	5.80634e9	0.279364	0.139682	+	0.990196i	$0.455392\pi$
$888$	0	0
$889$	−1.64066e10	−0.783182
$890$	0	0
$891$	− 2.08878e10i	− 0.989285i
$892$	0	0
$893$	− 2.81270e9i	− 0.132173i
$894$	0	0
$895$	−9.28466e8	−0.0432898
$896$	0	0
$897$	7.39709e10	3.42206
$898$	0	0
$899$	4.03982e9i	0.185440i
$900$	0	0
$901$	2.02163e10i	0.920798i
$902$	0	0
$903$	−2.64462e10	−1.19524
$904$	0	0
$905$	−4.28573e8	−0.0192201
$906$	0	0
$907$	1.78240e10i	0.793196i	0.917992	+	0.396598i	$0.129809\pi$
$907$	1.78240e10i	0.793196i	−0.917992	+	0.396598i	$0.870191\pi$
$908$	0	0
$909$	4.72326e10i	2.08578i
$910$	0	0
$911$	−1.87703e10	−0.822538	−0.411269	−	0.911514i	$-0.634914\pi$
$911$	−1.87703e10	−0.822538	−0.411269	+	0.911514i	$0.634914\pi$
$912$	0	0
$913$	1.35937e10	0.591138
$914$	0	0
$915$	− 1.30023e10i	− 0.561107i
$916$	0	0
$917$	− 3.08280e8i	− 0.0132024i
$918$	0	0
$919$	3.75844e10	1.59736	0.798681	−	0.601754i	$-0.205531\pi$
$919$	3.75844e10	1.59736	0.798681	+	0.601754i	$0.205531\pi$
$920$	0	0
$921$	6.89197e10	2.90693
$922$	0	0
$923$	2.74648e10i	1.14966i
$924$	0	0
$925$	2.99486e10i	1.24417i
$926$	0	0
$927$	−1.99707e10	−0.823404
$928$	0	0
$929$	−1.92372e10	−0.787205	−0.393602	−	0.919281i	$-0.628771\pi$
$929$	−1.92372e10	−0.787205	−0.393602	+	0.919281i	$0.628771\pi$
$930$	0	0
$931$	2.53876e9i	0.103109i
$932$	0	0
$933$	5.48943e10i	2.21279i
$934$	0	0
$935$	−2.30769e9	−0.0923289
$936$	0	0
$937$	−1.04732e9	−0.0415900	−0.0207950	−	0.999784i	$-0.506620\pi$
$937$	−1.04732e9	−0.0415900	−0.0207950	+	0.999784i	$0.506620\pi$
$938$	0	0
$939$	− 5.57413e10i	− 2.19709i
$940$	0	0
$941$	− 7.97861e9i	− 0.312150i	−0.987745	−	0.156075i	$-0.950116\pi$
$941$	− 7.97861e9i	− 0.312150i	0.987745	−	0.156075i	$-0.0498842\pi$
$942$	0	0
$943$	−1.15531e10	−0.448650
$944$	0	0
$945$	−8.42397e9	−0.324717
$946$	0	0
$947$	4.26943e9i	0.163360i	0.996659	+	0.0816799i	$0.0260285\pi$
$947$	4.26943e9i	0.163360i	−0.996659	+	0.0816799i	$0.973971\pi$
$948$	0	0
$949$	− 1.81106e10i	− 0.687860i
$950$	0	0
$951$	2.97963e10	1.12339
$952$	0	0
$953$	1.06048e10	0.396897	0.198449	−	0.980111i	$-0.436410\pi$
$953$	1.06048e10	0.396897	0.198449	+	0.980111i	$0.436410\pi$
$954$	0	0
$955$	6.97247e9i	0.259045i
$956$	0	0
$957$	2.11588e10i	0.780367i
$958$	0	0
$959$	−1.34904e10	−0.493922
$960$	0	0
$961$	−2.58740e10	−0.940441
$962$	0	0
$963$	− 3.54950e8i	− 0.0128078i
$964$	0	0
$965$	9.43750e9i	0.338074i
$966$	0	0
$967$	−1.65090e10	−0.587123	−0.293562	−	0.955940i	$-0.594841\pi$
$967$	−1.65090e10	−0.587123	−0.293562	+	0.955940i	$0.594841\pi$
$968$	0	0
$969$	−3.86254e9	−0.136377
$970$	0	0
$971$	− 2.46094e10i	− 0.862649i	−0.902197	−	0.431324i	$-0.858046\pi$
$971$	− 2.46094e10i	− 0.862649i	0.902197	−	0.431324i	$-0.141954\pi$
$972$	0	0
$973$	1.45503e10i	0.506379i
$974$	0	0
$975$	6.46430e10	2.23360
$976$	0	0
$977$	−2.53886e10	−0.870979	−0.435489	−	0.900194i	$-0.643425\pi$
$977$	−2.53886e10	−0.870979	−0.435489	+	0.900194i	$0.643425\pi$
$978$	0	0
$979$	− 1.63380e10i	− 0.556492i
$980$	0	0
$981$	− 4.65399e10i	− 1.57392i
$982$	0	0
$983$	1.87585e10	0.629884	0.314942	−	0.949111i	$-0.398015\pi$
$983$	1.87585e10	0.629884	0.314942	+	0.949111i	$0.398015\pi$
$984$	0	0
$985$	−1.13588e10	−0.378709
$986$	0	0
$987$	− 2.61246e10i	− 0.864846i
$988$	0	0
$989$	− 5.64107e10i	− 1.85428i
$990$	0	0
$991$	3.59792e9	0.117434	0.0587170	−	0.998275i	$-0.481299\pi$
$991$	3.59792e9	0.117434	0.0587170	+	0.998275i	$0.481299\pi$
$992$	0	0
$993$	2.56656e10	0.831821
$994$	0	0
$995$	− 1.80083e9i	− 0.0579552i
$996$	0	0
$997$	1.34287e10i	0.429143i	0.976708	+	0.214571i	$0.0688354\pi$
$997$	1.34287e10i	0.429143i	−0.976708	+	0.214571i	$0.931165\pi$
$998$	0	0
$999$	9.44952e10	2.99868

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.8.b.c.129.2		2
4.3	odd	2		256.8.b.e.129.1		2
8.3	odd	2		256.8.b.e.129.2		2
8.5	even	2	inner	256.8.b.c.129.1		2
16.3	odd	4		64.8.a.g.1.1		1
16.5	even	4		16.8.a.c.1.1		1
16.11	odd	4		8.8.a.a.1.1	✓	1
16.13	even	4		64.8.a.a.1.1		1
48.5	odd	4		144.8.a.g.1.1		1
48.11	even	4		72.8.a.d.1.1		1
48.29	odd	4		576.8.a.k.1.1		1
48.35	even	4		576.8.a.j.1.1		1
80.27	even	4		200.8.c.a.49.2		2
80.37	odd	4		400.8.c.b.49.1		2
80.43	even	4		200.8.c.a.49.1		2
80.53	odd	4		400.8.c.b.49.2		2
80.59	odd	4		200.8.a.i.1.1		1
80.69	even	4		400.8.a.b.1.1		1
112.27	even	4		392.8.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.8.a.a.1.1	✓	1	16.11	odd	4
16.8.a.c.1.1		1	16.5	even	4
64.8.a.a.1.1		1	16.13	even	4
64.8.a.g.1.1		1	16.3	odd	4
72.8.a.d.1.1		1	48.11	even	4
144.8.a.g.1.1		1	48.5	odd	4
200.8.a.i.1.1		1	80.59	odd	4
200.8.c.a.49.1		2	80.43	even	4
200.8.c.a.49.2		2	80.27	even	4
256.8.b.c.129.1		2	8.5	even	2	inner
256.8.b.c.129.2		2	1.1	even	1	trivial
256.8.b.e.129.1		2	4.3	odd	2
256.8.b.e.129.2		2	8.3	odd	2
392.8.a.d.1.1		1	112.27	even	4
400.8.a.b.1.1		1	80.69	even	4
400.8.c.b.49.1		2	80.37	odd	4
400.8.c.b.49.2		2	80.53	odd	4
576.8.a.j.1.1		1	48.35	even	4
576.8.a.k.1.1		1	48.29	odd	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+84.0000i q^{3} +82.0000i q^{5} -456.000 q^{7} -4869.00 q^{9} -2524.00i q^{11} -10778.0i q^{13} -6888.00 q^{15} -11150.0 q^{17} -4124.00i q^{19} -38304.0i q^{21} +81704.0 q^{23} +71401.0 q^{25} -225288. i q^{27} +99798.0i q^{29} +40480.0 q^{31} +212016. q^{33} -37392.0i q^{35} +419442. i q^{37} +905352. q^{39} -141402. q^{41} -690428. i q^{43} -399258. i q^{45} +682032. q^{47} -615607. q^{49} -936600. i q^{51} -1.81312e6i q^{53} +206968. q^{55} +346416. q^{57} -966028. i q^{59} +1.88767e6i q^{61} +2.22026e6 q^{63} +883796. q^{65} -2.96587e6i q^{67} +6.86314e6i q^{69} -2.54823e6 q^{71} +1.68033e6 q^{73} +5.99768e6i q^{75} +1.15094e6i q^{77} -4.03806e6 q^{79} +8.27569e6 q^{81} +5.38576e6i q^{83} -914300. i q^{85} -8.38303e6 q^{87} +6.47305e6 q^{89} +4.91477e6i q^{91} +3.40032e6i q^{93} +338168. q^{95} -6.06576e6 q^{97} +1.22894e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 912 q^{7} - 9738 q^{9} - 13776 q^{15} - 22300 q^{17} + 163408 q^{23} + 142802 q^{25} + 80960 q^{31} + 424032 q^{33} + 1810704 q^{39} - 282804 q^{41} + 1364064 q^{47} - 1231214 q^{49} + 413936 q^{55}+ \cdots - 12131516 q^{97}+O(q^{100}) \) 2 * q - 912 * q^7 - 9738 * q^9 - 13776 * q^15 - 22300 * q^17 + 163408 * q^23 + 142802 * q^25 + 80960 * q^31 + 424032 * q^33 + 1810704 * q^39 - 282804 * q^41 + 1364064 * q^47 - 1231214 * q^49 + 413936 * q^55 + 692832 * q^57 + 4440528 * q^63 + 1767592 * q^65 - 5096464 * q^71 + 3360652 * q^73 - 8076128 * q^79 + 16551378 * q^81 - 16766064 * q^87 + 12946092 * q^89 + 676336 * q^95 - 12131516 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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