LMFDB - Embedded newform 128.5.c.a.127.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [128,5,Mod(127,128)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(128, 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("128.127");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 128 = 2^{7} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	128.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$13.2313552747$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.205520896.4
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 12x^{6} - 12x^{5} - 8x^{4} + 12x^{3} + 12x^{2} + 4x + 1 $ x^8 - 4x^7 + 12x^6 - 12x^5 - 8x^4 + 12x^3 + 12x^2 + 4*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{39} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.6
Root			$-0.550501 + 0.228025i$ of defining polynomial
Character	$\chi$	$=$	128.127
Dual form			128.5.c.a.127.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+7.05664i q^{3} +9.50277 q^{5} -6.49140i q^{7} +31.2038 q^{9} +O(q^{10})$	$q+7.05664i q^{3} +9.50277 q^{5} -6.49140i q^{7} +31.2038 q^{9} +128.436i q^{11} +220.919 q^{13} +67.0576i q^{15} -382.628 q^{17} +77.3763i q^{19} +45.8075 q^{21} +951.968i q^{23} -534.697 q^{25} +791.782i q^{27} +561.558 q^{29} +321.171i q^{31} -906.325 q^{33} -61.6862i q^{35} +1090.63 q^{37} +1558.94i q^{39} -437.095 q^{41} +2558.67i q^{43} +296.523 q^{45} -3832.88i q^{47} +2358.86 q^{49} -2700.07i q^{51} +1524.58 q^{53} +1220.50i q^{55} -546.017 q^{57} +1757.10i q^{59} -4766.53 q^{61} -202.556i q^{63} +2099.34 q^{65} -6919.35i q^{67} -6717.70 q^{69} -337.001i q^{71} +4239.85 q^{73} -3773.17i q^{75} +833.728 q^{77} -11095.6i q^{79} -3059.81 q^{81} -3972.05i q^{83} -3636.02 q^{85} +3962.71i q^{87} +5526.64 q^{89} -1434.07i q^{91} -2266.39 q^{93} +735.289i q^{95} +4730.94 q^{97} +4007.69i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 48 q^{5} - 216 q^{9}+O(q^{10}) $ 8 * q - 48 * q^5 - 216 * q^9	$ 8 q - 48 q^{5} - 216 q^{9} - 240 q^{13} + 240 q^{17} + 1216 q^{21} + 664 q^{25} - 432 q^{29} + 992 q^{33} - 2800 q^{37} - 2928 q^{41} + 4880 q^{45} - 5752 q^{49} - 1776 q^{53} + 8608 q^{57} - 12656 q^{61} + 672 q^{65} + 4416 q^{69} + 560 q^{73} + 31296 q^{77} + 14696 q^{81} + 14432 q^{85} - 22992 q^{89} - 56320 q^{93} - 3728 q^{97}+O(q^{100}) $ 8 * q - 48 * q^5 - 216 * q^9 - 240 * q^13 + 240 * q^17 + 1216 * q^21 + 664 * q^25 - 432 * q^29 + 992 * q^33 - 2800 * q^37 - 2928 * q^41 + 4880 * q^45 - 5752 * q^49 - 1776 * q^53 + 8608 * q^57 - 12656 * q^61 + 672 * q^65 + 4416 * q^69 + 560 * q^73 + 31296 * q^77 + 14696 * q^81 + 14432 * q^85 - 22992 * q^89 - 56320 * q^93 - 3728 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/128\mathbb{Z}\right)^\times$.

$n$	$5$	$127$
$\chi(n)$	$1$	$-1$

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
$3$	7.05664i	0.784071i	0.919950	+	0.392036i	$0.128229\pi$
$3$	7.05664i	0.784071i	−0.919950	+	0.392036i	$0.871771\pi$
$4$	0	0
$5$	9.50277	0.380111	0.190055	−	0.981773i	$-0.439133\pi$
$5$	9.50277	0.380111	0.190055	+	0.981773i	$0.439133\pi$
$6$	0	0
$7$	− 6.49140i	− 0.132478i	−0.997804	−	0.0662388i	$-0.978900\pi$
$7$	− 6.49140i	− 0.132478i	0.997804	−	0.0662388i	$-0.0210999\pi$
$8$	0	0
$9$	31.2038	0.385232
$10$	0	0
$11$	128.436i	1.06145i	0.847543	+	0.530727i	$0.178081\pi$
$11$	128.436i	1.06145i	−0.847543	+	0.530727i	$0.821919\pi$
$12$	0	0
$13$	220.919	1.30721	0.653606	−	0.756835i	$-0.273256\pi$
$13$	220.919	1.30721	0.653606	+	0.756835i	$0.273256\pi$
$14$	0	0
$15$	67.0576i	0.298034i
$16$	0	0
$17$	−382.628	−1.32397	−0.661986	−	0.749516i	$-0.730286\pi$
$17$	−382.628	−1.32397	−0.661986	+	0.749516i	$0.730286\pi$
$18$	0	0
$19$	77.3763i	0.214339i	0.994241	+	0.107169i	$0.0341787\pi$
$19$	77.3763i	0.214339i	−0.994241	+	0.107169i	$0.965821\pi$
$20$	0	0
$21$	45.8075	0.103872
$22$	0	0
$23$	951.968i	1.79956i	0.436342	+	0.899781i	$0.356274\pi$
$23$	951.968i	1.79956i	−0.436342	+	0.899781i	$0.643726\pi$
$24$	0	0
$25$	−534.697	−0.855516
$26$	0	0
$27$	791.782i	1.08612i
$28$	0	0
$29$	561.558	0.667726	0.333863	−	0.942622i	$-0.391648\pi$
$29$	561.558	0.667726	0.333863	+	0.942622i	$0.391648\pi$
$30$	0	0
$31$	321.171i	0.334205i	0.985940	+	0.167102i	$0.0534410\pi$
$31$	321.171i	0.334205i	−0.985940	+	0.167102i	$0.946559\pi$
$32$	0	0
$33$	−906.325	−0.832255
$34$	0	0
$35$	− 61.6862i	− 0.0503561i
$36$	0	0
$37$	1090.63	0.796659	0.398330	−	0.917242i	$-0.369590\pi$
$37$	1090.63	0.796659	0.398330	+	0.917242i	$0.369590\pi$
$38$	0	0
$39$	1558.94i	1.02495i
$40$	0	0
$41$	−437.095	−0.260021	−0.130011	−	0.991513i	$-0.541501\pi$
$41$	−437.095	−0.260021	−0.130011	+	0.991513i	$0.541501\pi$
$42$	0	0
$43$	2558.67i	1.38381i	0.721988	+	0.691906i	$0.243229\pi$
$43$	2558.67i	1.38381i	−0.721988	+	0.691906i	$0.756771\pi$
$44$	0	0
$45$	296.523	0.146431
$46$	0	0
$47$	− 3832.88i	− 1.73512i	−0.497332	−	0.867560i	$-0.665687\pi$
$47$	− 3832.88i	− 1.73512i	0.497332	−	0.867560i	$-0.334313\pi$
$48$	0	0
$49$	2358.86	0.982450
$50$	0	0
$51$	− 2700.07i	− 1.03809i
$52$	0	0
$53$	1524.58	0.542747	0.271373	−	0.962474i	$-0.412522\pi$
$53$	1524.58	0.542747	0.271373	+	0.962474i	$0.412522\pi$
$54$	0	0
$55$	1220.50i	0.403470i
$56$	0	0
$57$	−546.017	−0.168057
$58$	0	0
$59$	1757.10i	0.504770i	0.967627	+	0.252385i	$0.0812149\pi$
$59$	1757.10i	0.504770i	−0.967627	+	0.252385i	$0.918785\pi$
$60$	0	0
$61$	−4766.53	−1.28098	−0.640490	−	0.767966i	$-0.721269\pi$
$61$	−4766.53	−1.28098	−0.640490	+	0.767966i	$0.721269\pi$
$62$	0	0
$63$	− 202.556i	− 0.0510346i
$64$	0	0
$65$	2099.34	0.496885
$66$	0	0
$67$	− 6919.35i	− 1.54140i	−0.637198	−	0.770700i	$-0.719907\pi$
$67$	− 6919.35i	− 1.54140i	0.637198	−	0.770700i	$-0.280093\pi$
$68$	0	0
$69$	−6717.70	−1.41098
$70$	0	0
$71$	− 337.001i	− 0.0668521i	−0.999441	−	0.0334260i	$-0.989358\pi$
$71$	− 337.001i	− 0.0668521i	0.999441	−	0.0334260i	$-0.0106418\pi$
$72$	0	0
$73$	4239.85	0.795619	0.397809	−	0.917468i	$-0.369771\pi$
$73$	4239.85	0.795619	0.397809	+	0.917468i	$0.369771\pi$
$74$	0	0
$75$	− 3773.17i	− 0.670785i
$76$	0	0
$77$	833.728	0.140619
$78$	0	0
$79$	− 11095.6i	− 1.77785i	−0.458050	−	0.888926i	$-0.651452\pi$
$79$	− 11095.6i	− 1.77785i	0.458050	−	0.888926i	$-0.348548\pi$
$80$	0	0
$81$	−3059.81	−0.466364
$82$	0	0
$83$	− 3972.05i	− 0.576579i	−0.957543	−	0.288289i	$-0.906913\pi$
$83$	− 3972.05i	− 0.576579i	0.957543	−	0.288289i	$-0.0930865\pi$
$84$	0	0
$85$	−3636.02	−0.503256
$86$	0	0
$87$	3962.71i	0.523545i
$88$	0	0
$89$	5526.64	0.697720	0.348860	−	0.937175i	$-0.386569\pi$
$89$	5526.64	0.697720	0.348860	+	0.937175i	$0.386569\pi$
$90$	0	0
$91$	− 1434.07i	− 0.173176i
$92$	0	0
$93$	−2266.39	−0.262040
$94$	0	0
$95$	735.289i	0.0814724i
$96$	0	0
$97$	4730.94	0.502810	0.251405	−	0.967882i	$-0.419107\pi$
$97$	4730.94	0.502810	0.251405	+	0.967882i	$0.419107\pi$
$98$	0	0
$99$	4007.69i	0.408906i
$100$	0	0
$101$	1509.47	0.147973	0.0739865	−	0.997259i	$-0.476428\pi$
$101$	1509.47	0.147973	0.0739865	+	0.997259i	$0.476428\pi$
$102$	0	0
$103$	− 11777.5i	− 1.11014i	−0.831803	−	0.555071i	$-0.812691\pi$
$103$	− 11777.5i	− 1.11014i	0.831803	−	0.555071i	$-0.187309\pi$
$104$	0	0
$105$	435.298	0.0394828
$106$	0	0
$107$	6628.96i	0.578999i	0.957178	+	0.289500i	$0.0934889\pi$
$107$	6628.96i	0.578999i	−0.957178	+	0.289500i	$0.906511\pi$
$108$	0	0
$109$	13286.3	1.11828	0.559142	−	0.829072i	$-0.311131\pi$
$109$	13286.3	1.11828	0.559142	+	0.829072i	$0.311131\pi$
$110$	0	0
$111$	7696.16i	0.624638i
$112$	0	0
$113$	−1399.22	−0.109579	−0.0547896	−	0.998498i	$-0.517449\pi$
$113$	−1399.22	−0.109579	−0.0547896	+	0.998498i	$0.517449\pi$
$114$	0	0
$115$	9046.33i	0.684033i
$116$	0	0
$117$	6893.51	0.503580
$118$	0	0
$119$	2483.79i	0.175397i
$120$	0	0
$121$	−1854.76	−0.126683
$122$	0	0
$123$	− 3084.43i	− 0.203875i
$124$	0	0
$125$	−11020.3	−0.705301
$126$	0	0
$127$	18940.8i	1.17433i	0.809467	+	0.587165i	$0.199756\pi$
$127$	18940.8i	1.17433i	−0.809467	+	0.587165i	$0.800244\pi$
$128$	0	0
$129$	−18055.6	−1.08501
$130$	0	0
$131$	16937.9i	0.987001i	0.869746	+	0.493500i	$0.164283\pi$
$131$	16937.9i	0.987001i	−0.869746	+	0.493500i	$0.835717\pi$
$132$	0	0
$133$	502.280	0.0283951
$134$	0	0
$135$	7524.12i	0.412846i
$136$	0	0
$137$	−32620.9	−1.73802	−0.869010	−	0.494795i	$-0.835243\pi$
$137$	−32620.9	−1.73802	−0.869010	+	0.494795i	$0.835243\pi$
$138$	0	0
$139$	− 16900.2i	− 0.874705i	−0.899290	−	0.437353i	$-0.855916\pi$
$139$	− 16900.2i	− 0.874705i	0.899290	−	0.437353i	$-0.144084\pi$
$140$	0	0
$141$	27047.3	1.36046
$142$	0	0
$143$	28373.9i	1.38754i
$144$	0	0
$145$	5336.35	0.253810
$146$	0	0
$147$	16645.6i	0.770310i
$148$	0	0
$149$	33124.9	1.49205	0.746023	−	0.665920i	$-0.231961\pi$
$149$	33124.9	1.49205	0.746023	+	0.665920i	$0.231961\pi$
$150$	0	0
$151$	− 22546.4i	− 0.988833i	−0.869225	−	0.494416i	$-0.835382\pi$
$151$	− 22546.4i	− 0.988833i	0.869225	−	0.494416i	$-0.164618\pi$
$152$	0	0
$153$	−11939.5	−0.510037
$154$	0	0
$155$	3052.01i	0.127035i
$156$	0	0
$157$	−49129.2	−1.99315	−0.996576	−	0.0826841i	$-0.973651\pi$
$157$	−49129.2	−1.99315	−0.996576	+	0.0826841i	$0.973651\pi$
$158$	0	0
$159$	10758.4i	0.425552i
$160$	0	0
$161$	6179.60	0.238401
$162$	0	0
$163$	8068.82i	0.303693i	0.988404	+	0.151847i	$0.0485220\pi$
$163$	8068.82i	0.303693i	−0.988404	+	0.151847i	$0.951478\pi$
$164$	0	0
$165$	−8612.60	−0.316349
$166$	0	0
$167$	15371.0i	0.551149i	0.961280	+	0.275575i	$0.0888681\pi$
$167$	15371.0i	0.551149i	−0.961280	+	0.275575i	$0.911132\pi$
$168$	0	0
$169$	20244.1	0.708801
$170$	0	0
$171$	2414.44i	0.0825702i
$172$	0	0
$173$	43716.8	1.46068	0.730341	−	0.683083i	$-0.239361\pi$
$173$	43716.8	1.46068	0.730341	+	0.683083i	$0.239361\pi$
$174$	0	0
$175$	3470.93i	0.113337i
$176$	0	0
$177$	−12399.2	−0.395775
$178$	0	0
$179$	28396.2i	0.886246i	0.896461	+	0.443123i	$0.146129\pi$
$179$	28396.2i	0.886246i	−0.896461	+	0.443123i	$0.853871\pi$
$180$	0	0
$181$	−6646.43	−0.202876	−0.101438	−	0.994842i	$-0.532344\pi$
$181$	−6646.43	−0.202876	−0.101438	+	0.994842i	$0.532344\pi$
$182$	0	0
$183$	− 33635.7i	− 1.00438i
$184$	0	0
$185$	10364.0	0.302819
$186$	0	0
$187$	− 49143.1i	− 1.40533i
$188$	0	0
$189$	5139.77	0.143887
$190$	0	0
$191$	− 39571.8i	− 1.08472i	−0.840145	−	0.542362i	$-0.817530\pi$
$191$	− 39571.8i	− 1.08472i	0.840145	−	0.542362i	$-0.182470\pi$
$192$	0	0
$193$	31206.5	0.837781	0.418890	−	0.908037i	$-0.362419\pi$
$193$	31206.5	0.837781	0.418890	+	0.908037i	$0.362419\pi$
$194$	0	0
$195$	14814.3i	0.389593i
$196$	0	0
$197$	−39072.5	−1.00679	−0.503394	−	0.864057i	$-0.667916\pi$
$197$	−39072.5	−1.00679	−0.503394	+	0.864057i	$0.667916\pi$
$198$	0	0
$199$	− 45057.3i	− 1.13778i	−0.822413	−	0.568891i	$-0.807373\pi$
$199$	− 45057.3i	− 1.13778i	0.822413	−	0.568891i	$-0.192627\pi$
$200$	0	0
$201$	48827.3	1.20857
$202$	0	0
$203$	− 3645.30i	− 0.0884587i
$204$	0	0
$205$	−4153.62	−0.0988368
$206$	0	0
$207$	29705.0i	0.693249i
$208$	0	0
$209$	−9937.89	−0.227511
$210$	0	0
$211$	− 88509.2i	− 1.98803i	−0.109228	−	0.994017i	$-0.534838\pi$
$211$	− 88509.2i	− 1.98803i	0.109228	−	0.994017i	$-0.465162\pi$
$212$	0	0
$213$	2378.10	0.0524168
$214$	0	0
$215$	24314.4i	0.526001i
$216$	0	0
$217$	2084.85	0.0442746
$218$	0	0
$219$	29919.1i	0.623822i
$220$	0	0
$221$	−84529.7	−1.73071
$222$	0	0
$223$	− 63144.7i	− 1.26978i	−0.772604	−	0.634888i	$-0.781046\pi$
$223$	− 63144.7i	− 1.26978i	0.772604	−	0.634888i	$-0.218954\pi$
$224$	0	0
$225$	−16684.6	−0.329572
$226$	0	0
$227$	− 79318.8i	− 1.53930i	−0.638464	−	0.769652i	$-0.720430\pi$
$227$	− 79318.8i	− 1.53930i	0.638464	−	0.769652i	$-0.279570\pi$
$228$	0	0
$229$	−73199.3	−1.39584	−0.697921	−	0.716175i	$-0.745891\pi$
$229$	−73199.3	−1.39584	−0.697921	+	0.716175i	$0.745891\pi$
$230$	0	0
$231$	5883.32i	0.110255i
$232$	0	0
$233$	−15770.0	−0.290482	−0.145241	−	0.989396i	$-0.546396\pi$
$233$	−15770.0	−0.290482	−0.145241	+	0.989396i	$0.546396\pi$
$234$	0	0
$235$	− 36423.0i	− 0.659538i
$236$	0	0
$237$	78297.5	1.39396
$238$	0	0
$239$	− 3598.13i	− 0.0629913i	−0.999504	−	0.0314957i	$-0.989973\pi$
$239$	− 3598.13i	− 0.0629913i	0.999504	−	0.0314957i	$-0.0100270\pi$
$240$	0	0
$241$	66204.6	1.13987	0.569933	−	0.821691i	$-0.306969\pi$
$241$	66204.6	1.13987	0.569933	+	0.821691i	$0.306969\pi$
$242$	0	0
$243$	42542.4i	0.720459i
$244$	0	0
$245$	22415.7	0.373440
$246$	0	0
$247$	17093.9i	0.280186i
$248$	0	0
$249$	28029.3	0.452079
$250$	0	0
$251$	42650.1i	0.676975i	0.940971	+	0.338488i	$0.109915\pi$
$251$	42650.1i	0.676975i	−0.940971	+	0.338488i	$0.890085\pi$
$252$	0	0
$253$	−122267.	−1.91015
$254$	0	0
$255$	− 25658.1i	− 0.394589i
$256$	0	0
$257$	−49938.6	−0.756084	−0.378042	−	0.925789i	$-0.623402\pi$
$257$	−49938.6	−0.756084	−0.378042	+	0.925789i	$0.623402\pi$
$258$	0	0
$259$	− 7079.69i	− 0.105539i
$260$	0	0
$261$	17522.8	0.257230
$262$	0	0
$263$	87275.8i	1.26178i	0.775874	+	0.630888i	$0.217309\pi$
$263$	87275.8i	1.26178i	−0.775874	+	0.630888i	$0.782691\pi$
$264$	0	0
$265$	14487.7	0.206304
$266$	0	0
$267$	38999.5i	0.547062i
$268$	0	0
$269$	110723.	1.53015	0.765073	−	0.643943i	$-0.222703\pi$
$269$	110723.	1.53015	0.765073	+	0.643943i	$0.222703\pi$
$270$	0	0
$271$	− 82671.9i	− 1.12569i	−0.826562	−	0.562846i	$-0.809706\pi$
$271$	− 82671.9i	− 1.12569i	0.826562	−	0.562846i	$-0.190294\pi$
$272$	0	0
$273$	10119.7	0.135782
$274$	0	0
$275$	− 68674.3i	− 0.908090i
$276$	0	0
$277$	46204.8	0.602181	0.301091	−	0.953596i	$-0.402649\pi$
$277$	46204.8	0.602181	0.301091	+	0.953596i	$0.402649\pi$
$278$	0	0
$279$	10021.7i	0.128746i
$280$	0	0
$281$	−84998.0	−1.07646	−0.538228	−	0.842800i	$-0.680906\pi$
$281$	−84998.0	−1.07646	−0.538228	+	0.842800i	$0.680906\pi$
$282$	0	0
$283$	31233.9i	0.389990i	0.980804	+	0.194995i	$0.0624690\pi$
$283$	31233.9i	0.389990i	−0.980804	+	0.194995i	$0.937531\pi$
$284$	0	0
$285$	−5188.67	−0.0638802
$286$	0	0
$287$	2837.36i	0.0344469i
$288$	0	0
$289$	62883.2	0.752903
$290$	0	0
$291$	33384.5i	0.394239i
$292$	0	0
$293$	108954.	1.26914	0.634570	−	0.772866i	$-0.281177\pi$
$293$	108954.	1.26914	0.634570	+	0.772866i	$0.281177\pi$
$294$	0	0
$295$	16697.3i	0.191868i
$296$	0	0
$297$	−101693.	−1.15287
$298$	0	0
$299$	210308.i	2.35241i
$300$	0	0
$301$	16609.3	0.183324
$302$	0	0
$303$	10651.8i	0.116021i
$304$	0	0
$305$	−45295.2	−0.486915
$306$	0	0
$307$	49054.1i	0.520474i	0.965545	+	0.260237i	$0.0838006\pi$
$307$	49054.1i	0.520474i	−0.965545	+	0.260237i	$0.916199\pi$
$308$	0	0
$309$	83109.5	0.870430
$310$	0	0
$311$	116875.i	1.20837i	0.796845	+	0.604184i	$0.206501\pi$
$311$	116875.i	1.20837i	−0.796845	+	0.604184i	$0.793499\pi$
$312$	0	0
$313$	44685.2	0.456116	0.228058	−	0.973648i	$-0.426762\pi$
$313$	44685.2	0.456116	0.228058	+	0.973648i	$0.426762\pi$
$314$	0	0
$315$	− 1924.85i	− 0.0193988i
$316$	0	0
$317$	2512.62	0.0250039	0.0125019	−	0.999922i	$-0.496020\pi$
$317$	2512.62	0.0250039	0.0125019	+	0.999922i	$0.496020\pi$
$318$	0	0
$319$	72124.2i	0.708760i
$320$	0	0
$321$	−46778.2	−0.453977
$322$	0	0
$323$	− 29606.3i	− 0.283779i
$324$	0	0
$325$	−118125.	−1.11834
$326$	0	0
$327$	93756.9i	0.876814i
$328$	0	0
$329$	−24880.8	−0.229864
$330$	0	0
$331$	− 137925.i	− 1.25889i	−0.777046	−	0.629444i	$-0.783283\pi$
$331$	− 137925.i	− 1.25889i	0.777046	−	0.629444i	$-0.216717\pi$
$332$	0	0
$333$	34031.7	0.306899
$334$	0	0
$335$	− 65752.9i	− 0.585903i
$336$	0	0
$337$	−150041.	−1.32115	−0.660574	−	0.750761i	$-0.729687\pi$
$337$	−150041.	−1.32115	−0.660574	+	0.750761i	$0.729687\pi$
$338$	0	0
$339$	− 9873.78i	− 0.0859180i
$340$	0	0
$341$	−41249.8	−0.354742
$342$	0	0
$343$	− 30898.2i	− 0.262630i
$344$	0	0
$345$	−63836.7	−0.536330
$346$	0	0
$347$	142965.i	1.18733i	0.804714	+	0.593663i	$0.202319\pi$
$347$	142965.i	1.18733i	−0.804714	+	0.593663i	$0.797681\pi$
$348$	0	0
$349$	64364.5	0.528440	0.264220	−	0.964462i	$-0.414885\pi$
$349$	64364.5	0.528440	0.264220	+	0.964462i	$0.414885\pi$
$350$	0	0
$351$	174919.i	1.41979i
$352$	0	0
$353$	17129.7	0.137467	0.0687337	−	0.997635i	$-0.478104\pi$
$353$	17129.7	0.137467	0.0687337	+	0.997635i	$0.478104\pi$
$354$	0	0
$355$	− 3202.44i	− 0.0254112i
$356$	0	0
$357$	−17527.2	−0.137523
$358$	0	0
$359$	− 29116.6i	− 0.225919i	−0.993600	−	0.112959i	$-0.963967\pi$
$359$	− 29116.6i	− 0.225919i	0.993600	−	0.112959i	$-0.0360330\pi$
$360$	0	0
$361$	124334.	0.954059
$362$	0	0
$363$	− 13088.4i	− 0.0993282i
$364$	0	0
$365$	40290.3	0.302423
$366$	0	0
$367$	− 4255.09i	− 0.0315919i	−0.999875	−	0.0157960i	$-0.994972\pi$
$367$	− 4255.09i	− 0.0315919i	0.999875	−	0.0157960i	$-0.00502822\pi$
$368$	0	0
$369$	−13639.1	−0.100169
$370$	0	0
$371$	− 9896.63i	− 0.0719017i
$372$	0	0
$373$	−696.069	−0.00500305	−0.00250152	−	0.999997i	$-0.500796\pi$
$373$	−696.069	−0.00500305	−0.00250152	+	0.999997i	$0.500796\pi$
$374$	0	0
$375$	− 77766.5i	− 0.553006i
$376$	0	0
$377$	124059.	0.872860
$378$	0	0
$379$	− 169548.i	− 1.18036i	−0.807272	−	0.590179i	$-0.799057\pi$
$379$	− 169548.i	− 1.18036i	0.807272	−	0.590179i	$-0.200943\pi$
$380$	0	0
$381$	−133658.	−0.920758
$382$	0	0
$383$	− 57233.5i	− 0.390169i	−0.980786	−	0.195084i	$-0.937502\pi$
$383$	− 57233.5i	− 0.390169i	0.980786	−	0.195084i	$-0.0624981\pi$
$384$	0	0
$385$	7922.72	0.0534506
$386$	0	0
$387$	79840.2i	0.533089i
$388$	0	0
$389$	−23301.7	−0.153988	−0.0769941	−	0.997032i	$-0.524532\pi$
$389$	−23301.7	−0.153988	−0.0769941	+	0.997032i	$0.524532\pi$
$390$	0	0
$391$	− 364250.i	− 2.38257i
$392$	0	0
$393$	−119525.	−0.773879
$394$	0	0
$395$	− 105439.i	− 0.675781i
$396$	0	0
$397$	42551.3	0.269980	0.134990	−	0.990847i	$-0.456900\pi$
$397$	42551.3	0.269980	0.134990	+	0.990847i	$0.456900\pi$
$398$	0	0
$399$	3544.41i	0.0222637i
$400$	0	0
$401$	171821.	1.06853	0.534266	−	0.845317i	$-0.320588\pi$
$401$	171821.	1.06853	0.534266	+	0.845317i	$0.320588\pi$
$402$	0	0
$403$	70952.6i	0.436876i
$404$	0	0
$405$	−29076.7	−0.177270
$406$	0	0
$407$	140076.i	0.845616i
$408$	0	0
$409$	91529.4	0.547160	0.273580	−	0.961849i	$-0.411792\pi$
$409$	91529.4	0.547160	0.273580	+	0.961849i	$0.411792\pi$
$410$	0	0
$411$	− 230194.i	− 1.36273i
$412$	0	0
$413$	11406.1	0.0668706
$414$	0	0
$415$	− 37745.5i	− 0.219164i
$416$	0	0
$417$	119259.	0.685831
$418$	0	0
$419$	65318.4i	0.372056i	0.982544	+	0.186028i	$0.0595614\pi$
$419$	65318.4i	0.372056i	−0.982544	+	0.186028i	$0.940439\pi$
$420$	0	0
$421$	−108601.	−0.612729	−0.306364	−	0.951914i	$-0.599113\pi$
$421$	−108601.	−0.612729	−0.306364	+	0.951914i	$0.599113\pi$
$422$	0	0
$423$	− 119601.i	− 0.668425i
$424$	0	0
$425$	204590.	1.13268
$426$	0	0
$427$	30941.4i	0.169701i
$428$	0	0
$429$	−200224.	−1.08793
$430$	0	0
$431$	− 193826.i	− 1.04342i	−0.853124	−	0.521708i	$-0.825295\pi$
$431$	− 193826.i	− 1.04342i	0.853124	−	0.521708i	$-0.174705\pi$
$432$	0	0
$433$	197671.	1.05431	0.527154	−	0.849770i	$-0.323259\pi$
$433$	197671.	1.05431	0.527154	+	0.849770i	$0.323259\pi$
$434$	0	0
$435$	37656.7i	0.199005i
$436$	0	0
$437$	−73659.8	−0.385716
$438$	0	0
$439$	− 168788.i	− 0.875815i	−0.899020	−	0.437908i	$-0.855720\pi$
$439$	− 168788.i	− 0.875815i	0.899020	−	0.437908i	$-0.144280\pi$
$440$	0	0
$441$	73605.5	0.378471
$442$	0	0
$443$	− 33462.6i	− 0.170511i	−0.996359	−	0.0852555i	$-0.972829\pi$
$443$	− 33462.6i	− 0.170511i	0.996359	−	0.0852555i	$-0.0271707\pi$
$444$	0	0
$445$	52518.4	0.265211
$446$	0	0
$447$	233751.i	1.16987i
$448$	0	0
$449$	−245538.	−1.21794	−0.608969	−	0.793194i	$-0.708417\pi$
$449$	−245538.	−1.21794	−0.608969	+	0.793194i	$0.708417\pi$
$450$	0	0
$451$	− 56138.7i	− 0.276000i
$452$	0	0
$453$	159102.	0.775315
$454$	0	0
$455$	− 13627.6i	− 0.0658261i
$456$	0	0
$457$	−133704.	−0.640196	−0.320098	−	0.947385i	$-0.603716\pi$
$457$	−133704.	−0.640196	−0.320098	+	0.947385i	$0.603716\pi$
$458$	0	0
$459$	− 302958.i	− 1.43799i
$460$	0	0
$461$	−275191.	−1.29489	−0.647445	−	0.762112i	$-0.724162\pi$
$461$	−275191.	−1.29489	−0.647445	+	0.762112i	$0.724162\pi$
$462$	0	0
$463$	345853.i	1.61335i	0.590994	+	0.806676i	$0.298736\pi$
$463$	345853.i	1.61335i	−0.590994	+	0.806676i	$0.701264\pi$
$464$	0	0
$465$	−21536.9	−0.0996042
$466$	0	0
$467$	− 95388.2i	− 0.437382i	−0.975794	−	0.218691i	$-0.929821\pi$
$467$	− 95388.2i	− 0.437382i	0.975794	−	0.218691i	$-0.0701786\pi$
$468$	0	0
$469$	−44916.2	−0.204201
$470$	0	0
$471$	− 346687.i	− 1.56277i
$472$	0	0
$473$	−328624.	−1.46885
$474$	0	0
$475$	− 41372.9i	− 0.183370i
$476$	0	0
$477$	47572.6	0.209084
$478$	0	0
$479$	− 13002.6i	− 0.0566707i	−0.999598	−	0.0283354i	$-0.990979\pi$
$479$	− 13002.6i	− 0.0566707i	0.999598	−	0.0283354i	$-0.00902063\pi$
$480$	0	0
$481$	240940.	1.04140
$482$	0	0
$483$	43607.2i	0.186924i
$484$	0	0
$485$	44957.0	0.191123
$486$	0	0
$487$	140830.i	0.593794i	0.954909	+	0.296897i	$0.0959519\pi$
$487$	140830.i	0.593794i	−0.954909	+	0.296897i	$0.904048\pi$
$488$	0	0
$489$	−56938.8	−0.238117
$490$	0	0
$491$	− 19020.5i	− 0.0788969i	−0.999222	−	0.0394484i	$-0.987440\pi$
$491$	− 19020.5i	− 0.0788969i	0.999222	−	0.0394484i	$-0.0125601\pi$
$492$	0	0
$493$	−214868.	−0.884051
$494$	0	0
$495$	38084.1i	0.155430i
$496$	0	0
$497$	−2187.61	−0.00885640
$498$	0	0
$499$	288185.i	1.15736i	0.815553	+	0.578682i	$0.196433\pi$
$499$	288185.i	1.15736i	−0.815553	+	0.578682i	$0.803567\pi$
$500$	0	0
$501$	−108468.	−0.432140
$502$	0	0
$503$	128399.i	0.507486i	0.967272	+	0.253743i	$0.0816618\pi$
$503$	128399.i	0.507486i	−0.967272	+	0.253743i	$0.918338\pi$
$504$	0	0
$505$	14344.2	0.0562461
$506$	0	0
$507$	142855.i	0.555750i
$508$	0	0
$509$	452490.	1.74652	0.873259	−	0.487256i	$-0.162002\pi$
$509$	452490.	1.74652	0.873259	+	0.487256i	$0.162002\pi$
$510$	0	0
$511$	− 27522.6i	− 0.105402i
$512$	0	0
$513$	−61265.2	−0.232798
$514$	0	0
$515$	− 111919.i	− 0.421976i
$516$	0	0
$517$	492279.	1.84175
$518$	0	0
$519$	308493.i	1.14528i
$520$	0	0
$521$	233443.	0.860015	0.430008	−	0.902825i	$-0.358511\pi$
$521$	233443.	0.860015	0.430008	+	0.902825i	$0.358511\pi$
$522$	0	0
$523$	235862.i	0.862293i	0.902282	+	0.431146i	$0.141891\pi$
$523$	235862.i	0.862293i	−0.902282	+	0.431146i	$0.858109\pi$
$524$	0	0
$525$	−24493.1	−0.0888640
$526$	0	0
$527$	− 122889.i	− 0.442478i
$528$	0	0
$529$	−626402.	−2.23842
$530$	0	0
$531$	54828.4i	0.194454i
$532$	0	0
$533$	−96562.6	−0.339902
$534$	0	0
$535$	62993.5i	0.220084i
$536$	0	0
$537$	−200382.	−0.694880
$538$	0	0
$539$	302962.i	1.04282i
$540$	0	0
$541$	335400.	1.14596	0.572978	−	0.819571i	$-0.305788\pi$
$541$	335400.	1.14596	0.572978	+	0.819571i	$0.305788\pi$
$542$	0	0
$543$	− 46901.5i	− 0.159070i
$544$	0	0
$545$	126257.	0.425072
$546$	0	0
$547$	333389.i	1.11423i	0.830434	+	0.557117i	$0.188092\pi$
$547$	333389.i	1.11423i	−0.830434	+	0.557117i	$0.811908\pi$
$548$	0	0
$549$	−148734.	−0.493475
$550$	0	0
$551$	43451.3i	0.143120i
$552$	0	0
$553$	−72025.8	−0.235525
$554$	0	0
$555$	73134.8i	0.237431i
$556$	0	0
$557$	144667.	0.466294	0.233147	−	0.972441i	$-0.425098\pi$
$557$	144667.	0.466294	0.233147	+	0.972441i	$0.425098\pi$
$558$	0	0
$559$	565257.i	1.80893i
$560$	0	0
$561$	346785.	1.10188
$562$	0	0
$563$	− 333299.i	− 1.05152i	−0.850634	−	0.525759i	$-0.823781\pi$
$563$	− 333299.i	− 1.05152i	0.850634	−	0.525759i	$-0.176219\pi$
$564$	0	0
$565$	−13296.4	−0.0416523
$566$	0	0
$567$	19862.5i	0.0617827i
$568$	0	0
$569$	303109.	0.936212	0.468106	−	0.883672i	$-0.344937\pi$
$569$	303109.	0.936212	0.468106	+	0.883672i	$0.344937\pi$
$570$	0	0
$571$	24945.5i	0.0765102i	0.999268	+	0.0382551i	$0.0121799\pi$
$571$	24945.5i	0.0765102i	−0.999268	+	0.0382551i	$0.987820\pi$
$572$	0	0
$573$	279244.	0.850501
$574$	0	0
$575$	− 509015.i	− 1.53955i
$576$	0	0
$577$	−532837.	−1.60045	−0.800226	−	0.599698i	$-0.795287\pi$
$577$	−532837.	−1.60045	−0.800226	+	0.599698i	$0.795287\pi$
$578$	0	0
$579$	220213.i	0.656880i
$580$	0	0
$581$	−25784.2	−0.0763837
$582$	0	0
$583$	195810.i	0.576100i
$584$	0	0
$585$	65507.4	0.191416
$586$	0	0
$587$	624244.i	1.81167i	0.423634	+	0.905833i	$0.360754\pi$
$587$	624244.i	1.81167i	−0.423634	+	0.905833i	$0.639246\pi$
$588$	0	0
$589$	−24851.0	−0.0716330
$590$	0	0
$591$	− 275720.i	− 0.789394i
$592$	0	0
$593$	−158609.	−0.451045	−0.225522	−	0.974238i	$-0.572409\pi$
$593$	−158609.	−0.451045	−0.225522	+	0.974238i	$0.572409\pi$
$594$	0	0
$595$	23602.9i	0.0666701i
$596$	0	0
$597$	317953.	0.892102
$598$	0	0
$599$	514165.i	1.43301i	0.697583	+	0.716504i	$0.254259\pi$
$599$	514165.i	1.43301i	−0.697583	+	0.716504i	$0.745741\pi$
$600$	0	0
$601$	−407561.	−1.12835	−0.564175	−	0.825655i	$-0.690806\pi$
$601$	−407561.	−1.12835	−0.564175	+	0.825655i	$0.690806\pi$
$602$	0	0
$603$	− 215910.i	− 0.593797i
$604$	0	0
$605$	−17625.4	−0.0481534
$606$	0	0
$607$	− 177651.i	− 0.482158i	−0.970505	−	0.241079i	$-0.922499\pi$
$607$	− 177651.i	− 0.482158i	0.970505	−	0.241079i	$-0.0775013\pi$
$608$	0	0
$609$	25723.5	0.0693579
$610$	0	0
$611$	− 846755.i	− 2.26817i
$612$	0	0
$613$	132291.	0.352055	0.176028	−	0.984385i	$-0.443675\pi$
$613$	132291.	0.352055	0.176028	+	0.984385i	$0.443675\pi$
$614$	0	0
$615$	− 29310.6i	− 0.0774951i
$616$	0	0
$617$	−351562.	−0.923488	−0.461744	−	0.887013i	$-0.652776\pi$
$617$	−351562.	−0.923488	−0.461744	+	0.887013i	$0.652776\pi$
$618$	0	0
$619$	− 318145.i	− 0.830318i	−0.909749	−	0.415159i	$-0.863726\pi$
$619$	− 318145.i	− 0.830318i	0.909749	−	0.415159i	$-0.136274\pi$
$620$	0	0
$621$	−753751.	−1.95454
$622$	0	0
$623$	− 35875.6i	− 0.0924322i
$624$	0	0
$625$	229462.	0.587423
$626$	0	0
$627$	− 70128.1i	− 0.178384i
$628$	0	0
$629$	−417304.	−1.05475
$630$	0	0
$631$	260248.i	0.653626i	0.945089	+	0.326813i	$0.105975\pi$
$631$	260248.i	0.653626i	−0.945089	+	0.326813i	$0.894025\pi$
$632$	0	0
$633$	624578.	1.55876
$634$	0	0
$635$	179990.i	0.446375i
$636$	0	0
$637$	521117.	1.28427
$638$	0	0
$639$	− 10515.7i	− 0.0257536i
$640$	0	0
$641$	26771.0	0.0651550	0.0325775	−	0.999469i	$-0.489628\pi$
$641$	26771.0	0.0651550	0.0325775	+	0.999469i	$0.489628\pi$
$642$	0	0
$643$	5813.96i	0.0140621i	0.999975	+	0.00703105i	$0.00223807\pi$
$643$	5813.96i	0.0140621i	−0.999975	+	0.00703105i	$0.997762\pi$
$644$	0	0
$645$	−171578.	−0.412423
$646$	0	0
$647$	150358.i	0.359185i	0.983741	+	0.179592i	$0.0574779\pi$
$647$	150358.i	0.359185i	−0.983741	+	0.179592i	$0.942522\pi$
$648$	0	0
$649$	−225675.	−0.535789
$650$	0	0
$651$	14712.0i	0.0347144i
$652$	0	0
$653$	−807136.	−1.89287	−0.946434	−	0.322898i	$-0.895343\pi$
$653$	−807136.	−1.89287	−0.946434	+	0.322898i	$0.895343\pi$
$654$	0	0
$655$	160957.i	0.375169i
$656$	0	0
$657$	132300.	0.306498
$658$	0	0
$659$	− 327592.i	− 0.754331i	−0.926146	−	0.377166i	$-0.876899\pi$
$659$	− 327592.i	− 0.754331i	0.926146	−	0.377166i	$-0.123101\pi$
$660$	0	0
$661$	−482222.	−1.10368	−0.551842	−	0.833949i	$-0.686075\pi$
$661$	−482222.	−1.10368	−0.551842	+	0.833949i	$0.686075\pi$
$662$	0	0
$663$	− 596496.i	− 1.35700i
$664$	0	0
$665$	4773.05	0.0107933
$666$	0	0
$667$	534585.i	1.20161i
$668$	0	0
$669$	445589.	0.995595
$670$	0	0
$671$	− 612193.i	− 1.35970i
$672$	0	0
$673$	350805.	0.774526	0.387263	−	0.921969i	$-0.373421\pi$
$673$	350805.	0.774526	0.387263	+	0.921969i	$0.373421\pi$
$674$	0	0
$675$	− 423364.i	− 0.929194i
$676$	0	0
$677$	468433.	1.02204	0.511022	−	0.859567i	$-0.329267\pi$
$677$	468433.	1.02204	0.511022	+	0.859567i	$0.329267\pi$
$678$	0	0
$679$	− 30710.4i	− 0.0666110i
$680$	0	0
$681$	559724.	1.20692
$682$	0	0
$683$	− 254774.i	− 0.546153i	−0.961992	−	0.273076i	$-0.911959\pi$
$683$	− 254774.i	− 0.546153i	0.961992	−	0.273076i	$-0.0880412\pi$
$684$	0	0
$685$	−309989.	−0.660640
$686$	0	0
$687$	− 516541.i	− 1.09444i
$688$	0	0
$689$	336807.	0.709485
$690$	0	0
$691$	− 416717.i	− 0.872741i	−0.899767	−	0.436371i	$-0.856264\pi$
$691$	− 416717.i	− 0.872741i	0.899767	−	0.436371i	$-0.143736\pi$
$692$	0	0
$693$	26015.5	0.0541709
$694$	0	0
$695$	− 160598.i	− 0.332485i
$696$	0	0
$697$	167245.	0.344261
$698$	0	0
$699$	− 111283.i	− 0.227759i
$700$	0	0
$701$	−554976.	−1.12938	−0.564688	−	0.825305i	$-0.691003\pi$
$701$	−554976.	−1.12938	−0.564688	+	0.825305i	$0.691003\pi$
$702$	0	0
$703$	84388.6i	0.170755i
$704$	0	0
$705$	257024.	0.517125
$706$	0	0
$707$	− 9798.58i	− 0.0196031i
$708$	0	0
$709$	208869.	0.415511	0.207755	−	0.978181i	$-0.433384\pi$
$709$	208869.	0.415511	0.207755	+	0.978181i	$0.433384\pi$
$710$	0	0
$711$	− 346224.i	− 0.684886i
$712$	0	0
$713$	−305744.	−0.601422
$714$	0	0
$715$	269630.i	0.527420i
$716$	0	0
$717$	25390.7	0.0493897
$718$	0	0
$719$	− 245345.i	− 0.474590i	−0.971438	−	0.237295i	$-0.923739\pi$
$719$	− 245345.i	− 0.474590i	0.971438	−	0.237295i	$-0.0762608\pi$
$720$	0	0
$721$	−76452.3	−0.147069
$722$	0	0
$723$	467182.i	0.893737i
$724$	0	0
$725$	−300264.	−0.571251
$726$	0	0
$727$	462973.i	0.875965i	0.898983	+	0.437983i	$0.144307\pi$
$727$	462973.i	0.875965i	−0.898983	+	0.437983i	$0.855693\pi$
$728$	0	0
$729$	−548051.	−1.03125
$730$	0	0
$731$	− 979018.i	− 1.83213i
$732$	0	0
$733$	275737.	0.513201	0.256600	−	0.966518i	$-0.417398\pi$
$733$	275737.	0.513201	0.256600	+	0.966518i	$0.417398\pi$
$734$	0	0
$735$	158180.i	0.292803i
$736$	0	0
$737$	888692.	1.63612
$738$	0	0
$739$	751459.i	1.37599i	0.725714	+	0.687997i	$0.241510\pi$
$739$	751459.i	1.37599i	−0.725714	+	0.687997i	$0.758490\pi$
$740$	0	0
$741$	−120625.	−0.219686
$742$	0	0
$743$	33291.6i	0.0603055i	0.999545	+	0.0301527i	$0.00959937\pi$
$743$	33291.6i	0.0603055i	−0.999545	+	0.0301527i	$0.990401\pi$
$744$	0	0
$745$	314778.	0.567143
$746$	0	0
$747$	− 123943.i	− 0.222117i
$748$	0	0
$749$	43031.2	0.0767044
$750$	0	0
$751$	− 356855.i	− 0.632721i	−0.948639	−	0.316361i	$-0.897539\pi$
$751$	− 356855.i	− 0.632721i	0.948639	−	0.316361i	$-0.102461\pi$
$752$	0	0
$753$	−300967.	−0.530797
$754$	0	0
$755$	− 214253.i	− 0.375866i
$756$	0	0
$757$	622463.	1.08623	0.543115	−	0.839658i	$-0.317245\pi$
$757$	622463.	1.08623	0.543115	+	0.839658i	$0.317245\pi$
$758$	0	0
$759$	− 862793.i	− 1.49769i
$760$	0	0
$761$	967703.	1.67099	0.835493	−	0.549502i	$-0.185182\pi$
$761$	967703.	1.67099	0.835493	+	0.549502i	$0.185182\pi$
$762$	0	0
$763$	− 86246.9i	− 0.148147i
$764$	0	0
$765$	−113458.	−0.193871
$766$	0	0
$767$	388177.i	0.659841i
$768$	0	0
$769$	−1.06634e6	−1.80320	−0.901600	−	0.432572i	$-0.857606\pi$
$769$	−1.06634e6	−1.80320	−0.901600	+	0.432572i	$0.857606\pi$
$770$	0	0
$771$	− 352399.i	− 0.592823i
$772$	0	0
$773$	−314448.	−0.526247	−0.263123	−	0.964762i	$-0.584753\pi$
$773$	−314448.	−0.526247	−0.263123	+	0.964762i	$0.584753\pi$
$774$	0	0
$775$	− 171729.i	− 0.285917i
$776$	0	0
$777$	49958.8	0.0827504
$778$	0	0
$779$	− 33820.8i	− 0.0557326i
$780$	0	0
$781$	43283.0	0.0709603
$782$	0	0
$783$	444632.i	0.725232i
$784$	0	0
$785$	−466863.	−0.757618
$786$	0	0
$787$	505795.i	0.816629i	0.912841	+	0.408315i	$0.133883\pi$
$787$	505795.i	0.816629i	−0.912841	+	0.408315i	$0.866117\pi$
$788$	0	0
$789$	−615874.	−0.989323
$790$	0	0
$791$	9082.88i	0.0145168i
$792$	0	0
$793$	−1.05302e6	−1.67451
$794$	0	0
$795$	102234.i	0.161757i
$796$	0	0
$797$	324326.	0.510581	0.255290	−	0.966864i	$-0.417829\pi$
$797$	324326.	0.510581	0.255290	+	0.966864i	$0.417829\pi$
$798$	0	0
$799$	1.46657e6i	2.29725i
$800$	0	0
$801$	172452.	0.268784
$802$	0	0
$803$	544549.i	0.844512i
$804$	0	0
$805$	58723.3	0.0906189
$806$	0	0
$807$	781332.i	1.19974i
$808$	0	0
$809$	−35692.7	−0.0545359	−0.0272679	−	0.999628i	$-0.508681\pi$
$809$	−35692.7	−0.0545359	−0.0272679	+	0.999628i	$0.508681\pi$
$810$	0	0
$811$	− 448798.i	− 0.682353i	−0.939999	−	0.341177i	$-0.889175\pi$
$811$	− 448798.i	− 0.682353i	0.939999	−	0.341177i	$-0.110825\pi$
$812$	0	0
$813$	583386.	0.882622
$814$	0	0
$815$	76676.2i	0.115437i
$816$	0	0
$817$	−197980.	−0.296604
$818$	0	0
$819$	− 44748.5i	− 0.0667130i
$820$	0	0
$821$	−389526.	−0.577897	−0.288949	−	0.957345i	$-0.593306\pi$
$821$	−389526.	−0.577897	−0.288949	+	0.957345i	$0.593306\pi$
$822$	0	0
$823$	538138.i	0.794500i	0.917711	+	0.397250i	$0.130035\pi$
$823$	538138.i	0.794500i	−0.917711	+	0.397250i	$0.869965\pi$
$824$	0	0
$825$	484610.	0.712007
$826$	0	0
$827$	1.07219e6i	1.56769i	0.620956	+	0.783845i	$0.286744\pi$
$827$	1.07219e6i	1.56769i	−0.620956	+	0.783845i	$0.713256\pi$
$828$	0	0
$829$	193379.	0.281385	0.140693	−	0.990053i	$-0.455067\pi$
$829$	193379.	0.281385	0.140693	+	0.990053i	$0.455067\pi$
$830$	0	0
$831$	326050.i	0.472153i
$832$	0	0
$833$	−902567.	−1.30074
$834$	0	0
$835$	146067.i	0.209498i
$836$	0	0
$837$	−254297.	−0.362986
$838$	0	0
$839$	− 744976.i	− 1.05832i	−0.848521	−	0.529161i	$-0.822507\pi$
$839$	− 744976.i	− 1.05832i	0.848521	−	0.529161i	$-0.177493\pi$
$840$	0	0
$841$	−391934.	−0.554141
$842$	0	0
$843$	− 599800.i	− 0.844018i
$844$	0	0
$845$	192375.	0.269423
$846$	0	0
$847$	12040.0i	0.0167826i
$848$	0	0
$849$	−220406.	−0.305780
$850$	0	0
$851$	1.03824e6i	1.43364i
$852$	0	0
$853$	1.23891e6	1.70271	0.851355	−	0.524589i	$-0.175781\pi$
$853$	1.23891e6	1.70271	0.851355	+	0.524589i	$0.175781\pi$
$854$	0	0
$855$	22943.8i	0.0313858i
$856$	0	0
$857$	−726182.	−0.988744	−0.494372	−	0.869251i	$-0.664602\pi$
$857$	−726182.	−0.988744	−0.494372	+	0.869251i	$0.664602\pi$
$858$	0	0
$859$	− 656830.i	− 0.890156i	−0.895492	−	0.445078i	$-0.853176\pi$
$859$	− 656830.i	− 0.890156i	0.895492	−	0.445078i	$-0.146824\pi$
$860$	0	0
$861$	−20022.2	−0.0270089
$862$	0	0
$863$	273884.i	0.367743i	0.982950	+	0.183872i	$0.0588631\pi$
$863$	273884.i	0.367743i	−0.982950	+	0.183872i	$0.941137\pi$
$864$	0	0
$865$	415430.	0.555221
$866$	0	0
$867$	443744.i	0.590329i
$868$	0	0
$869$	1.42507e6	1.88711
$870$	0	0
$871$	− 1.52861e6i	− 2.01494i
$872$	0	0
$873$	147623.	0.193699
$874$	0	0
$875$	71537.4i	0.0934366i
$876$	0	0
$877$	213538.	0.277636	0.138818	−	0.990318i	$-0.455670\pi$
$877$	213538.	0.277636	0.138818	+	0.990318i	$0.455670\pi$
$878$	0	0
$879$	768852.i	0.995096i
$880$	0	0
$881$	1.04959e6	1.35228	0.676140	−	0.736773i	$-0.263651\pi$
$881$	1.04959e6	1.35228	0.676140	+	0.736773i	$0.263651\pi$
$882$	0	0
$883$	783260.i	1.00458i	0.864699	+	0.502290i	$0.167509\pi$
$883$	783260.i	1.00458i	−0.864699	+	0.502290i	$0.832491\pi$
$884$	0	0
$885$	−117827.	−0.150438
$886$	0	0
$887$	653199.i	0.830230i	0.909769	+	0.415115i	$0.136259\pi$
$887$	653199.i	0.830230i	−0.909769	+	0.415115i	$0.863741\pi$
$888$	0	0
$889$	122952.	0.155572
$890$	0	0
$891$	− 392989.i	− 0.495023i
$892$	0	0
$893$	296574.	0.371904
$894$	0	0
$895$	269842.i	0.336871i
$896$	0	0
$897$	−1.48406e6	−1.84445
$898$	0	0
$899$	180356.i	0.223157i
$900$	0	0
$901$	−583345.	−0.718582
$902$	0	0
$903$	117206.i	0.143739i
$904$	0	0
$905$	−63159.5	−0.0771155
$906$	0	0
$907$	1.12141e6i	1.36317i	0.731741	+	0.681583i	$0.238708\pi$
$907$	1.12141e6i	1.36317i	−0.731741	+	0.681583i	$0.761292\pi$
$908$	0	0
$909$	47101.3	0.0570040
$910$	0	0
$911$	87118.2i	0.104972i	0.998622	+	0.0524859i	$0.0167144\pi$
$911$	87118.2i	0.104972i	−0.998622	+	0.0524859i	$0.983286\pi$
$912$	0	0
$913$	510154.	0.612011
$914$	0	0
$915$	− 319632.i	− 0.381776i
$916$	0	0
$917$	109951.	0.130755
$918$	0	0
$919$	− 262218.i	− 0.310478i	−0.987877	−	0.155239i	$-0.950385\pi$
$919$	− 262218.i	− 0.310478i	0.987877	−	0.155239i	$-0.0496148\pi$
$920$	0	0
$921$	−346157.	−0.408088
$922$	0	0
$923$	− 74449.9i	− 0.0873898i
$924$	0	0
$925$	−583155.	−0.681555
$926$	0	0
$927$	− 367503.i	− 0.427662i
$928$	0	0
$929$	780426.	0.904274	0.452137	−	0.891948i	$-0.350662\pi$
$929$	780426.	0.904274	0.452137	+	0.891948i	$0.350662\pi$
$930$	0	0
$931$	182520.i	0.210577i
$932$	0	0
$933$	−824742.	−0.947447
$934$	0	0
$935$	− 466996.i	− 0.534183i
$936$	0	0
$937$	−122296.	−0.139294	−0.0696471	−	0.997572i	$-0.522187\pi$
$937$	−122296.	−0.139294	−0.0696471	+	0.997572i	$0.522187\pi$
$938$	0	0
$939$	315328.i	0.357627i
$940$	0	0
$941$	368852.	0.416556	0.208278	−	0.978070i	$-0.433214\pi$
$941$	368852.	0.416556	0.208278	+	0.978070i	$0.433214\pi$
$942$	0	0
$943$	− 416101.i	− 0.467924i
$944$	0	0
$945$	48842.1	0.0546928
$946$	0	0
$947$	− 1.28300e6i	− 1.43063i	−0.698801	−	0.715317i	$-0.746283\pi$
$947$	− 1.28300e6i	− 1.43063i	0.698801	−	0.715317i	$-0.253717\pi$
$948$	0	0
$949$	936663.	1.04004
$950$	0	0
$951$	17730.6i	0.0196048i
$952$	0	0
$953$	640178.	0.704880	0.352440	−	0.935834i	$-0.385352\pi$
$953$	640178.	0.704880	0.352440	+	0.935834i	$0.385352\pi$
$954$	0	0
$955$	− 376042.i	− 0.412315i
$956$	0	0
$957$	−508954.	−0.555719
$958$	0	0
$959$	211755.i	0.230249i
$960$	0	0
$961$	820370.	0.888307
$962$	0	0
$963$	206849.i	0.223049i
$964$	0	0
$965$	296548.	0.318449
$966$	0	0
$967$	1.34359e6i	1.43686i	0.695599	+	0.718430i	$0.255139\pi$
$967$	1.34359e6i	1.43686i	−0.695599	+	0.718430i	$0.744861\pi$
$968$	0	0
$969$	208921.	0.222503
$970$	0	0
$971$	− 1.07804e6i	− 1.14340i	−0.820463	−	0.571699i	$-0.806284\pi$
$971$	− 1.07804e6i	− 1.14340i	0.820463	−	0.571699i	$-0.193716\pi$
$972$	0	0
$973$	−109706.	−0.115879
$974$	0	0
$975$	− 833563.i	− 0.876858i
$976$	0	0
$977$	−1.48550e6	−1.55626	−0.778132	−	0.628101i	$-0.783833\pi$
$977$	−1.48550e6	−1.55626	−0.778132	+	0.628101i	$0.783833\pi$
$978$	0	0
$979$	709818.i	0.740597i
$980$	0	0
$981$	414584.	0.430799
$982$	0	0
$983$	− 1.05481e6i	− 1.09161i	−0.837912	−	0.545805i	$-0.816224\pi$
$983$	− 1.05481e6i	− 1.09161i	0.837912	−	0.545805i	$-0.183776\pi$
$984$	0	0
$985$	−371297.	−0.382691
$986$	0	0
$987$	− 175575.i	− 0.180230i
$988$	0	0
$989$	−2.43577e6	−2.49025
$990$	0	0
$991$	− 818767.i	− 0.833706i	−0.908974	−	0.416853i	$-0.863133\pi$
$991$	− 818767.i	− 0.833706i	0.908974	−	0.416853i	$-0.136867\pi$
$992$	0	0
$993$	973288.	0.987058
$994$	0	0
$995$	− 428169.i	− 0.432483i
$996$	0	0
$997$	−1.77533e6	−1.78603	−0.893014	−	0.450029i	$-0.851414\pi$
$997$	−1.77533e6	−1.78603	−0.893014	+	0.450029i	$0.851414\pi$
$998$	0	0
$999$	863539.i	0.865268i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.5.c.a.127.6	yes	8
3.2	odd	2		1152.5.g.b.127.3		8
4.3	odd	2	inner	128.5.c.a.127.3	✓	8
8.3	odd	2		128.5.c.b.127.6	yes	8
8.5	even	2		128.5.c.b.127.3	yes	8
12.11	even	2		1152.5.g.b.127.4		8
16.3	odd	4		256.5.d.g.127.5		8
16.5	even	4		256.5.d.g.127.6		8
16.11	odd	4		256.5.d.h.127.4		8
16.13	even	4		256.5.d.h.127.3		8
24.5	odd	2		1152.5.g.a.127.5		8
24.11	even	2		1152.5.g.a.127.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.5.c.a.127.3	✓	8	4.3	odd	2	inner
128.5.c.a.127.6	yes	8	1.1	even	1	trivial
128.5.c.b.127.3	yes	8	8.5	even	2
128.5.c.b.127.6	yes	8	8.3	odd	2
256.5.d.g.127.5		8	16.3	odd	4
256.5.d.g.127.6		8	16.5	even	4
256.5.d.h.127.3		8	16.13	even	4
256.5.d.h.127.4		8	16.11	odd	4
1152.5.g.a.127.5		8	24.5	odd	2
1152.5.g.a.127.6		8	24.11	even	2
1152.5.g.b.127.3		8	3.2	odd	2
1152.5.g.b.127.4		8	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 128 = 2^{7} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	128.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+7.05664i q^{3} +9.50277 q^{5} -6.49140i q^{7} +31.2038 q^{9} +O(q^{10})\)	\(q+7.05664i q^{3} +9.50277 q^{5} -6.49140i q^{7} +31.2038 q^{9} +128.436i q^{11} +220.919 q^{13} +67.0576i q^{15} -382.628 q^{17} +77.3763i q^{19} +45.8075 q^{21} +951.968i q^{23} -534.697 q^{25} +791.782i q^{27} +561.558 q^{29} +321.171i q^{31} -906.325 q^{33} -61.6862i q^{35} +1090.63 q^{37} +1558.94i q^{39} -437.095 q^{41} +2558.67i q^{43} +296.523 q^{45} -3832.88i q^{47} +2358.86 q^{49} -2700.07i q^{51} +1524.58 q^{53} +1220.50i q^{55} -546.017 q^{57} +1757.10i q^{59} -4766.53 q^{61} -202.556i q^{63} +2099.34 q^{65} -6919.35i q^{67} -6717.70 q^{69} -337.001i q^{71} +4239.85 q^{73} -3773.17i q^{75} +833.728 q^{77} -11095.6i q^{79} -3059.81 q^{81} -3972.05i q^{83} -3636.02 q^{85} +3962.71i q^{87} +5526.64 q^{89} -1434.07i q^{91} -2266.39 q^{93} +735.289i q^{95} +4730.94 q^{97} +4007.69i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 48 q^{5} - 216 q^{9}+O(q^{10}) \) 8 * q - 48 * q^5 - 216 * q^9	\( 8 q - 48 q^{5} - 216 q^{9} - 240 q^{13} + 240 q^{17} + 1216 q^{21} + 664 q^{25} - 432 q^{29} + 992 q^{33} - 2800 q^{37} - 2928 q^{41} + 4880 q^{45} - 5752 q^{49} - 1776 q^{53} + 8608 q^{57} - 12656 q^{61} + 672 q^{65} + 4416 q^{69} + 560 q^{73} + 31296 q^{77} + 14696 q^{81} + 14432 q^{85} - 22992 q^{89} - 56320 q^{93} - 3728 q^{97}+O(q^{100}) \) 8 * q - 48 * q^5 - 216 * q^9 - 240 * q^13 + 240 * q^17 + 1216 * q^21 + 664 * q^25 - 432 * q^29 + 992 * q^33 - 2800 * q^37 - 2928 * q^41 + 4880 * q^45 - 5752 * q^49 - 1776 * q^53 + 8608 * q^57 - 12656 * q^61 + 672 * q^65 + 4416 * q^69 + 560 * q^73 + 31296 * q^77 + 14696 * q^81 + 14432 * q^85 - 22992 * q^89 - 56320 * q^93 - 3728 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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