LMFDB - Embedded newform 256.4.b.i.129.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [256,4,Mod(129,256)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("256.129");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 256 = 2^{8} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	256.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$15.1044889615$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 128)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.1
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	256.129
Dual form			256.4.b.i.129.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.92820i q^{3} -11.8564i q^{5} -9.85641 q^{7} -52.7128 q^{9} +O(q^{10})$	$q-8.92820i q^{3} -11.8564i q^{5} -9.85641 q^{7} -52.7128 q^{9} +39.0718i q^{11} -91.5692i q^{13} -105.856 q^{15} -37.1384 q^{17} +46.4974i q^{19} +88.0000i q^{21} +120.708 q^{23} -15.5744 q^{25} +229.569i q^{27} +27.2820i q^{29} -81.1487 q^{31} +348.841 q^{33} +116.862i q^{35} +10.9948i q^{37} -817.549 q^{39} +205.426 q^{41} +115.359i q^{43} +624.985i q^{45} -312.841 q^{47} -245.851 q^{49} +331.580i q^{51} +90.9948i q^{53} +463.251 q^{55} +415.138 q^{57} -550.631i q^{59} -630.974i q^{61} +519.559 q^{63} -1085.68 q^{65} +661.041i q^{67} -1077.70i q^{69} -494.985 q^{71} -566.267 q^{73} +139.051i q^{75} -385.108i q^{77} +49.4153 q^{79} +626.395 q^{81} -564.067i q^{83} +440.328i q^{85} +243.580 q^{87} -1089.67 q^{89} +902.543i q^{91} +724.513i q^{93} +551.292 q^{95} +464.605 q^{97} -2059.58i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 16 q^{7} - 100 q^{9}+O(q^{10}) $ 4 * q + 16 * q^7 - 100 * q^9	$ 4 q + 16 q^{7} - 100 q^{9} - 368 q^{15} + 184 q^{17} - 16 q^{23} - 284 q^{25} - 768 q^{31} + 176 q^{33} - 1552 q^{39} + 600 q^{41} - 32 q^{47} - 540 q^{49} - 752 q^{55} + 1328 q^{57} + 1136 q^{63} - 1904 q^{65} - 816 q^{71} - 824 q^{73} - 800 q^{79} - 44 q^{81} + 1584 q^{87} - 1144 q^{89} + 2704 q^{95} + 4408 q^{97}+O(q^{100}) $ 4 * q + 16 * q^7 - 100 * q^9 - 368 * q^15 + 184 * q^17 - 16 * q^23 - 284 * q^25 - 768 * q^31 + 176 * q^33 - 1552 * q^39 + 600 * q^41 - 32 * q^47 - 540 * q^49 - 752 * q^55 + 1328 * q^57 + 1136 * q^63 - 1904 * q^65 - 816 * q^71 - 824 * q^73 - 800 * q^79 - 44 * q^81 + 1584 * q^87 - 1144 * q^89 + 2704 * q^95 + 4408 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	− 8.92820i	− 1.71823i	−0.511780	−	0.859117i	$-0.671014\pi$
$3$	− 8.92820i	− 1.71823i	0.511780	−	0.859117i	$-0.328986\pi$
$4$	0	0
$5$	− 11.8564i	− 1.06047i	−0.847851	−	0.530235i	$-0.822104\pi$
$5$	− 11.8564i	− 1.06047i	0.847851	−	0.530235i	$-0.177896\pi$
$6$	0	0
$7$	−9.85641	−0.532196	−0.266098	−	0.963946i	$-0.585734\pi$
$7$	−9.85641	−0.532196	−0.266098	+	0.963946i	$0.585734\pi$
$8$	0	0
$9$	−52.7128	−1.95233
$10$	0	0
$11$	39.0718i	1.07096i	0.844547	+	0.535481i	$0.179870\pi$
$11$	39.0718i	1.07096i	−0.844547	+	0.535481i	$0.820130\pi$
$12$	0	0
$13$	− 91.5692i	− 1.95359i	−0.214166	−	0.976797i	$-0.568703\pi$
$13$	− 91.5692i	− 1.95359i	0.214166	−	0.976797i	$-0.431297\pi$
$14$	0	0
$15$	−105.856	−1.82213
$16$	0	0
$17$	−37.1384	−0.529847	−0.264923	−	0.964269i	$-0.585347\pi$
$17$	−37.1384	−0.529847	−0.264923	+	0.964269i	$0.585347\pi$
$18$	0	0
$19$	46.4974i	0.561434i	0.959791	+	0.280717i	$0.0905722\pi$
$19$	46.4974i	0.561434i	−0.959791	+	0.280717i	$0.909428\pi$
$20$	0	0
$21$	88.0000i	0.914437i
$22$	0	0
$23$	120.708	1.09432	0.547158	−	0.837029i	$-0.315710\pi$
$23$	120.708	1.09432	0.547158	+	0.837029i	$0.315710\pi$
$24$	0	0
$25$	−15.5744	−0.124595
$26$	0	0
$27$	229.569i	1.63632i
$28$	0	0
$29$	27.2820i	0.174695i	0.996178	+	0.0873473i	$0.0278390\pi$
$29$	27.2820i	0.174695i	−0.996178	+	0.0873473i	$0.972161\pi$
$30$	0	0
$31$	−81.1487	−0.470153	−0.235077	−	0.971977i	$-0.575534\pi$
$31$	−81.1487	−0.470153	−0.235077	+	0.971977i	$0.575534\pi$
$32$	0	0
$33$	348.841	1.84016
$34$	0	0
$35$	116.862i	0.564377i
$36$	0	0
$37$	10.9948i	0.0488525i	0.999702	+	0.0244262i	$0.00777589\pi$
$37$	10.9948i	0.0488525i	−0.999702	+	0.0244262i	$0.992224\pi$
$38$	0	0
$39$	−817.549	−3.35673
$40$	0	0
$41$	205.426	0.782490	0.391245	−	0.920287i	$-0.372044\pi$
$41$	205.426	0.782490	0.391245	+	0.920287i	$0.372044\pi$
$42$	0	0
$43$	115.359i	0.409118i	0.978854	+	0.204559i	$0.0655760\pi$
$43$	115.359i	0.409118i	−0.978854	+	0.204559i	$0.934424\pi$
$44$	0	0
$45$	624.985i	2.07038i
$46$	0	0
$47$	−312.841	−0.970905	−0.485453	−	0.874263i	$-0.661345\pi$
$47$	−312.841	−0.970905	−0.485453	+	0.874263i	$0.661345\pi$
$48$	0	0
$49$	−245.851	−0.716767
$50$	0	0
$51$	331.580i	0.910400i
$52$	0	0
$53$	90.9948i	0.235832i	0.993024	+	0.117916i	$0.0376214\pi$
$53$	90.9948i	0.235832i	−0.993024	+	0.117916i	$0.962379\pi$
$54$	0	0
$55$	463.251	1.13572
$56$	0	0
$57$	415.138	0.964674
$58$	0	0
$59$	− 550.631i	− 1.21502i	−0.794313	−	0.607509i	$-0.792169\pi$
$59$	− 550.631i	− 1.21502i	0.794313	−	0.607509i	$-0.207831\pi$
$60$	0	0
$61$	− 630.974i	− 1.32439i	−0.749330	−	0.662196i	$-0.769624\pi$
$61$	− 630.974i	− 1.32439i	0.749330	−	0.662196i	$-0.230376\pi$
$62$	0	0
$63$	519.559	1.03902
$64$	0	0
$65$	−1085.68	−2.07173
$66$	0	0
$67$	661.041i	1.20536i	0.797984	+	0.602679i	$0.205900\pi$
$67$	661.041i	1.20536i	−0.797984	+	0.602679i	$0.794100\pi$
$68$	0	0
$69$	− 1077.70i	− 1.88029i
$70$	0	0
$71$	−494.985	−0.827378	−0.413689	−	0.910418i	$-0.635760\pi$
$71$	−494.985	−0.827378	−0.413689	+	0.910418i	$0.635760\pi$
$72$	0	0
$73$	−566.267	−0.907897	−0.453949	−	0.891028i	$-0.649985\pi$
$73$	−566.267	−0.907897	−0.453949	+	0.891028i	$0.649985\pi$
$74$	0	0
$75$	139.051i	0.214083i
$76$	0	0
$77$	− 385.108i	− 0.569962i
$78$	0	0
$79$	49.4153	0.0703754	0.0351877	−	0.999381i	$-0.488797\pi$
$79$	49.4153	0.0703754	0.0351877	+	0.999381i	$0.488797\pi$
$80$	0	0
$81$	626.395	0.859252
$82$	0	0
$83$	− 564.067i	− 0.745956i	−0.927840	−	0.372978i	$-0.878337\pi$
$83$	− 564.067i	− 0.745956i	0.927840	−	0.372978i	$-0.121663\pi$
$84$	0	0
$85$	440.328i	0.561886i
$86$	0	0
$87$	243.580	0.300166
$88$	0	0
$89$	−1089.67	−1.29781	−0.648904	−	0.760870i	$-0.724772\pi$
$89$	−1089.67	−1.29781	−0.648904	+	0.760870i	$0.724772\pi$
$90$	0	0
$91$	902.543i	1.03970i
$92$	0	0
$93$	724.513i	0.807833i
$94$	0	0
$95$	551.292	0.595383
$96$	0	0
$97$	464.605	0.486325	0.243162	−	0.969986i	$-0.421815\pi$
$97$	464.605	0.486325	0.243162	+	0.969986i	$0.421815\pi$
$98$	0	0
$99$	− 2059.58i	− 2.09087i
$100$	0	0
$101$	− 965.005i	− 0.950709i	−0.879794	−	0.475354i	$-0.842320\pi$
$101$	− 965.005i	− 0.950709i	0.879794	−	0.475354i	$-0.157680\pi$
$102$	0	0
$103$	1828.99	1.74967	0.874836	−	0.484419i	$-0.160969\pi$
$103$	1828.99	1.74967	0.874836	+	0.484419i	$0.160969\pi$
$104$	0	0
$105$	1043.36	0.969732
$106$	0	0
$107$	− 1726.59i	− 1.55996i	−0.625805	−	0.779980i	$-0.715229\pi$
$107$	− 1726.59i	− 1.55996i	0.625805	−	0.779980i	$-0.284771\pi$
$108$	0	0
$109$	− 423.015i	− 0.371720i	−0.982576	−	0.185860i	$-0.940493\pi$
$109$	− 423.015i	− 0.371720i	0.982576	−	0.185860i	$-0.0595072\pi$
$110$	0	0
$111$	98.1642	0.0839400
$112$	0	0
$113$	763.108	0.635284	0.317642	−	0.948211i	$-0.397109\pi$
$113$	763.108	0.635284	0.317642	+	0.948211i	$0.397109\pi$
$114$	0	0
$115$	− 1431.16i	− 1.16049i
$116$	0	0
$117$	4826.87i	3.81405i
$118$	0	0
$119$	366.052	0.281982
$120$	0	0
$121$	−195.605	−0.146961
$122$	0	0
$123$	− 1834.08i	− 1.34450i
$124$	0	0
$125$	− 1297.39i	− 0.928340i
$126$	0	0
$127$	−2159.38	−1.50877	−0.754387	−	0.656430i	$-0.772066\pi$
$127$	−2159.38	−1.50877	−0.754387	+	0.656430i	$0.772066\pi$
$128$	0	0
$129$	1029.95	0.702961
$130$	0	0
$131$	710.723i	0.474016i	0.971508	+	0.237008i	$0.0761668\pi$
$131$	710.723i	0.474016i	−0.971508	+	0.237008i	$0.923833\pi$
$132$	0	0
$133$	− 458.297i	− 0.298793i
$134$	0	0
$135$	2721.87	1.73527
$136$	0	0
$137$	987.128	0.615592	0.307796	−	0.951452i	$-0.400409\pi$
$137$	987.128	0.615592	0.307796	+	0.951452i	$0.400409\pi$
$138$	0	0
$139$	2334.15i	1.42432i	0.702019	+	0.712158i	$0.252282\pi$
$139$	2334.15i	1.42432i	−0.702019	+	0.712158i	$0.747718\pi$
$140$	0	0
$141$	2793.11i	1.66824i
$142$	0	0
$143$	3577.77	2.09223
$144$	0	0
$145$	323.467	0.185258
$146$	0	0
$147$	2195.01i	1.23157i
$148$	0	0
$149$	− 1792.29i	− 0.985435i	−0.870189	−	0.492718i	$-0.836004\pi$
$149$	− 1792.29i	− 0.985435i	0.870189	−	0.492718i	$-0.163996\pi$
$150$	0	0
$151$	−2864.32	−1.54367	−0.771837	−	0.635820i	$-0.780662\pi$
$151$	−2864.32	−1.54367	−0.771837	+	0.635820i	$0.780662\pi$
$152$	0	0
$153$	1957.67	1.03443
$154$	0	0
$155$	962.133i	0.498583i
$156$	0	0
$157$	− 2123.57i	− 1.07949i	−0.841830	−	0.539743i	$-0.818521\pi$
$157$	− 2123.57i	− 1.07949i	0.841830	−	0.539743i	$-0.181479\pi$
$158$	0	0
$159$	812.420	0.405215
$160$	0	0
$161$	−1189.74	−0.582391
$162$	0	0
$163$	− 3164.21i	− 1.52049i	−0.649636	−	0.760245i	$-0.725079\pi$
$163$	− 3164.21i	− 1.52049i	0.649636	−	0.760245i	$-0.274921\pi$
$164$	0	0
$165$	− 4136.00i	− 1.95144i
$166$	0	0
$167$	1340.34	0.621069	0.310534	−	0.950562i	$-0.399492\pi$
$167$	1340.34	0.621069	0.310534	+	0.950562i	$0.399492\pi$
$168$	0	0
$169$	−6187.92	−2.81653
$170$	0	0
$171$	− 2451.01i	− 1.09610i
$172$	0	0
$173$	2802.13i	1.23146i	0.787958	+	0.615729i	$0.211138\pi$
$173$	2802.13i	1.23146i	−0.787958	+	0.615729i	$0.788862\pi$
$174$	0	0
$175$	153.507	0.0663089
$176$	0	0
$177$	−4916.14	−2.08768
$178$	0	0
$179$	− 1074.84i	− 0.448810i	−0.974496	−	0.224405i	$-0.927956\pi$
$179$	− 1074.84i	− 0.448810i	0.974496	−	0.224405i	$-0.0720438\pi$
$180$	0	0
$181$	571.036i	0.234502i	0.993102	+	0.117251i	$0.0374081\pi$
$181$	571.036i	0.234502i	−0.993102	+	0.117251i	$0.962592\pi$
$182$	0	0
$183$	−5633.47	−2.27562
$184$	0	0
$185$	130.359	0.0518065
$186$	0	0
$187$	− 1451.07i	− 0.567446i
$188$	0	0
$189$	− 2262.73i	− 0.870842i
$190$	0	0
$191$	872.657	0.330593	0.165296	−	0.986244i	$-0.447142\pi$
$191$	872.657	0.330593	0.165296	+	0.986244i	$0.447142\pi$
$192$	0	0
$193$	672.523	0.250825	0.125413	−	0.992105i	$-0.459975\pi$
$193$	672.523	0.250825	0.125413	+	0.992105i	$0.459975\pi$
$194$	0	0
$195$	9693.19i	3.55971i
$196$	0	0
$197$	− 3129.56i	− 1.13184i	−0.824461	−	0.565918i	$-0.808522\pi$
$197$	− 3129.56i	− 1.13184i	0.824461	−	0.565918i	$-0.191478\pi$
$198$	0	0
$199$	−1502.37	−0.535176	−0.267588	−	0.963533i	$-0.586227\pi$
$199$	−1502.37	−0.535176	−0.267588	+	0.963533i	$0.586227\pi$
$200$	0	0
$201$	5901.91	2.07109
$202$	0	0
$203$	− 268.903i	− 0.0929718i
$204$	0	0
$205$	− 2435.61i	− 0.829807i
$206$	0	0
$207$	−6362.84	−2.13646
$208$	0	0
$209$	−1816.74	−0.601275
$210$	0	0
$211$	1773.70i	0.578703i	0.957223	+	0.289352i	$0.0934397\pi$
$211$	1773.70i	0.578703i	−0.957223	+	0.289352i	$0.906560\pi$
$212$	0	0
$213$	4419.32i	1.42163i
$214$	0	0
$215$	1367.74	0.433857
$216$	0	0
$217$	799.835	0.250214
$218$	0	0
$219$	5055.74i	1.55998i
$220$	0	0
$221$	3400.74i	1.03511i
$222$	0	0
$223$	−5413.66	−1.62568	−0.812838	−	0.582490i	$-0.802078\pi$
$223$	−5413.66	−1.62568	−0.812838	+	0.582490i	$0.802078\pi$
$224$	0	0
$225$	820.969	0.243250
$226$	0	0
$227$	− 3744.27i	− 1.09478i	−0.836876	−	0.547392i	$-0.815621\pi$
$227$	− 3744.27i	− 1.09478i	0.836876	−	0.547392i	$-0.184379\pi$
$228$	0	0
$229$	680.739i	0.196439i	0.995165	+	0.0982194i	$0.0313147\pi$
$229$	680.739i	0.196439i	−0.995165	+	0.0982194i	$0.968685\pi$
$230$	0	0
$231$	−3438.32	−0.979328
$232$	0	0
$233$	679.271	0.190989	0.0954947	−	0.995430i	$-0.469557\pi$
$233$	679.271	0.190989	0.0954947	+	0.995430i	$0.469557\pi$
$234$	0	0
$235$	3709.17i	1.02962i
$236$	0	0
$237$	− 441.190i	− 0.120921i
$238$	0	0
$239$	3901.27	1.05587	0.527934	−	0.849286i	$-0.322967\pi$
$239$	3901.27	1.05587	0.527934	+	0.849286i	$0.322967\pi$
$240$	0	0
$241$	5859.72	1.56622	0.783108	−	0.621886i	$-0.213633\pi$
$241$	5859.72	1.56622	0.783108	+	0.621886i	$0.213633\pi$
$242$	0	0
$243$	605.790i	0.159924i
$244$	0	0
$245$	2914.91i	0.760110i
$246$	0	0
$247$	4257.73	1.09681
$248$	0	0
$249$	−5036.10	−1.28173
$250$	0	0
$251$	− 3381.48i	− 0.850348i	−0.905112	−	0.425174i	$-0.860213\pi$
$251$	− 3381.48i	− 0.850348i	0.905112	−	0.425174i	$-0.139787\pi$
$252$	0	0
$253$	4716.27i	1.17197i
$254$	0	0
$255$	3931.34	0.965452
$256$	0	0
$257$	3652.22	0.886455	0.443227	−	0.896409i	$-0.353833\pi$
$257$	3652.22	0.886455	0.443227	+	0.896409i	$0.353833\pi$
$258$	0	0
$259$	− 108.370i	− 0.0259991i
$260$	0	0
$261$	− 1438.11i	− 0.341061i
$262$	0	0
$263$	−2462.74	−0.577410	−0.288705	−	0.957418i	$-0.593225\pi$
$263$	−2462.74	−0.577410	−0.288705	+	0.957418i	$0.593225\pi$
$264$	0	0
$265$	1078.87	0.250093
$266$	0	0
$267$	9728.81i	2.22994i
$268$	0	0
$269$	− 482.503i	− 0.109363i	−0.998504	−	0.0546816i	$-0.982586\pi$
$269$	− 482.503i	− 0.109363i	0.998504	−	0.0546816i	$-0.0174144\pi$
$270$	0	0
$271$	−3602.48	−0.807510	−0.403755	−	0.914867i	$-0.632295\pi$
$271$	−3602.48	−0.807510	−0.403755	+	0.914867i	$0.632295\pi$
$272$	0	0
$273$	8058.09	1.78644
$274$	0	0
$275$	− 608.519i	− 0.133437i
$276$	0	0
$277$	6177.72i	1.34001i	0.742356	+	0.670006i	$0.233709\pi$
$277$	6177.72i	1.34001i	−0.742356	+	0.670006i	$0.766291\pi$
$278$	0	0
$279$	4277.58	0.917892
$280$	0	0
$281$	543.477	0.115378	0.0576888	−	0.998335i	$-0.481627\pi$
$281$	543.477	0.115378	0.0576888	+	0.998335i	$0.481627\pi$
$282$	0	0
$283$	6582.11i	1.38256i	0.722585	+	0.691282i	$0.242954\pi$
$283$	6582.11i	1.38256i	−0.722585	+	0.691282i	$0.757046\pi$
$284$	0	0
$285$	− 4922.05i	− 1.02301i
$286$	0	0
$287$	−2024.76	−0.416438
$288$	0	0
$289$	−3533.74	−0.719262
$290$	0	0
$291$	− 4148.09i	− 0.835620i
$292$	0	0
$293$	6753.68i	1.34660i	0.739369	+	0.673301i	$0.235124\pi$
$293$	6753.68i	1.34660i	−0.739369	+	0.673301i	$0.764876\pi$
$294$	0	0
$295$	−6528.50	−1.28849
$296$	0	0
$297$	−8969.68	−1.75244
$298$	0	0
$299$	− 11053.1i	− 2.13785i
$300$	0	0
$301$	− 1137.03i	− 0.217731i
$302$	0	0
$303$	−8615.76	−1.63354
$304$	0	0
$305$	−7481.09	−1.40448
$306$	0	0
$307$	1821.88i	0.338698i	0.985556	+	0.169349i	$0.0541665\pi$
$307$	1821.88i	0.338698i	−0.985556	+	0.169349i	$0.945833\pi$
$308$	0	0
$309$	− 16329.6i	− 3.00635i
$310$	0	0
$311$	10598.2	1.93237	0.966184	−	0.257852i	$-0.0830148\pi$
$311$	10598.2	1.93237	0.966184	+	0.257852i	$0.0830148\pi$
$312$	0	0
$313$	7969.69	1.43921	0.719606	−	0.694382i	$-0.244322\pi$
$313$	7969.69	1.43921	0.719606	+	0.694382i	$0.244322\pi$
$314$	0	0
$315$	− 6160.10i	− 1.10185i
$316$	0	0
$317$	− 7773.66i	− 1.37733i	−0.725082	−	0.688663i	$-0.758198\pi$
$317$	− 7773.66i	− 1.37733i	0.725082	−	0.688663i	$-0.241802\pi$
$318$	0	0
$319$	−1065.96	−0.187092
$320$	0	0
$321$	−15415.3	−2.68038
$322$	0	0
$323$	− 1726.84i	− 0.297474i
$324$	0	0
$325$	1426.13i	0.243408i
$326$	0	0
$327$	−3776.77	−0.638703
$328$	0	0
$329$	3083.49	0.516712
$330$	0	0
$331$	− 8212.87i	− 1.36381i	−0.731442	−	0.681903i	$-0.761152\pi$
$331$	− 8212.87i	− 1.36381i	0.731442	−	0.681903i	$-0.238848\pi$
$332$	0	0
$333$	− 579.569i	− 0.0953760i
$334$	0	0
$335$	7837.57	1.27825
$336$	0	0
$337$	5260.17	0.850267	0.425133	−	0.905131i	$-0.360227\pi$
$337$	5260.17	0.850267	0.425133	+	0.905131i	$0.360227\pi$
$338$	0	0
$339$	− 6813.18i	− 1.09157i
$340$	0	0
$341$	− 3170.63i	− 0.503516i
$342$	0	0
$343$	5803.96	0.913657
$344$	0	0
$345$	−12777.7	−1.99399
$346$	0	0
$347$	− 2162.12i	− 0.334492i	−0.985915	−	0.167246i	$-0.946513\pi$
$347$	− 2162.12i	− 0.334492i	0.985915	−	0.167246i	$-0.0534873\pi$
$348$	0	0
$349$	− 2888.90i	− 0.443093i	−0.975150	−	0.221546i	$-0.928890\pi$
$349$	− 2888.90i	− 0.443093i	0.975150	−	0.221546i	$-0.0711104\pi$
$350$	0	0
$351$	21021.5	3.19670
$352$	0	0
$353$	11550.2	1.74152	0.870760	−	0.491709i	$-0.163628\pi$
$353$	11550.2	1.74152	0.870760	+	0.491709i	$0.163628\pi$
$354$	0	0
$355$	5868.74i	0.877409i
$356$	0	0
$357$	− 3268.18i	− 0.484511i
$358$	0	0
$359$	6523.97	0.959114	0.479557	−	0.877511i	$-0.340797\pi$
$359$	6523.97	0.959114	0.479557	+	0.877511i	$0.340797\pi$
$360$	0	0
$361$	4696.99	0.684792
$362$	0	0
$363$	1746.40i	0.252514i
$364$	0	0
$365$	6713.89i	0.962797i
$366$	0	0
$367$	6856.96	0.975287	0.487644	−	0.873043i	$-0.337857\pi$
$367$	6856.96	0.975287	0.487644	+	0.873043i	$0.337857\pi$
$368$	0	0
$369$	−10828.6	−1.52768
$370$	0	0
$371$	− 896.882i	− 0.125509i
$372$	0	0
$373$	− 5001.68i	− 0.694309i	−0.937808	−	0.347155i	$-0.887148\pi$
$373$	− 5001.68i	− 0.694309i	0.937808	−	0.347155i	$-0.112852\pi$
$374$	0	0
$375$	−11583.4	−1.59511
$376$	0	0
$377$	2498.19	0.341283
$378$	0	0
$379$	84.9775i	0.0115172i	0.999983	+	0.00575858i	$0.00183302\pi$
$379$	84.9775i	0.0115172i	−0.999983	+	0.00575858i	$0.998167\pi$
$380$	0	0
$381$	19279.4i	2.59243i
$382$	0	0
$383$	2596.76	0.346444	0.173222	−	0.984883i	$-0.444582\pi$
$383$	2596.76	0.346444	0.173222	+	0.984883i	$0.444582\pi$
$384$	0	0
$385$	−4565.99	−0.604427
$386$	0	0
$387$	− 6080.90i	− 0.798732i
$388$	0	0
$389$	9213.21i	1.20084i	0.799683	+	0.600422i	$0.205001\pi$
$389$	9213.21i	1.20084i	−0.799683	+	0.600422i	$0.794999\pi$
$390$	0	0
$391$	−4482.89	−0.579820
$392$	0	0
$393$	6345.48	0.814471
$394$	0	0
$395$	− 585.888i	− 0.0746310i
$396$	0	0
$397$	6614.19i	0.836163i	0.908409	+	0.418082i	$0.137297\pi$
$397$	6614.19i	0.836163i	−0.908409	+	0.418082i	$0.862703\pi$
$398$	0	0
$399$	−4091.77	−0.513396
$400$	0	0
$401$	3434.94	0.427763	0.213881	−	0.976860i	$-0.431389\pi$
$401$	3434.94	0.427763	0.213881	+	0.976860i	$0.431389\pi$
$402$	0	0
$403$	7430.73i	0.918489i
$404$	0	0
$405$	− 7426.79i	− 0.911210i
$406$	0	0
$407$	−429.588	−0.0523192
$408$	0	0
$409$	6435.82	0.778071	0.389035	−	0.921223i	$-0.372808\pi$
$409$	6435.82	0.778071	0.389035	+	0.921223i	$0.372808\pi$
$410$	0	0
$411$	− 8813.28i	− 1.05773i
$412$	0	0
$413$	5427.24i	0.646627i
$414$	0	0
$415$	−6687.80	−0.791064
$416$	0	0
$417$	20839.8	2.44731
$418$	0	0
$419$	− 4222.04i	− 0.492267i	−0.969236	−	0.246134i	$-0.920840\pi$
$419$	− 4222.04i	− 0.492267i	0.969236	−	0.246134i	$-0.0791602\pi$
$420$	0	0
$421$	5164.92i	0.597917i	0.954266	+	0.298958i	$0.0966391\pi$
$421$	5164.92i	0.597917i	−0.954266	+	0.298958i	$0.903361\pi$
$422$	0	0
$423$	16490.7	1.89552
$424$	0	0
$425$	578.408	0.0660163
$426$	0	0
$427$	6219.14i	0.704837i
$428$	0	0
$429$	− 31943.1i	− 3.59493i
$430$	0	0
$431$	7476.41	0.835559	0.417780	−	0.908548i	$-0.362808\pi$
$431$	7476.41	0.835559	0.417780	+	0.908548i	$0.362808\pi$
$432$	0	0
$433$	−11289.7	−1.25299	−0.626497	−	0.779424i	$-0.715512\pi$
$433$	−11289.7	−1.25299	−0.626497	+	0.779424i	$0.715512\pi$
$434$	0	0
$435$	− 2887.98i	− 0.318317i
$436$	0	0
$437$	5612.59i	0.614386i
$438$	0	0
$439$	1800.79	0.195779	0.0978895	−	0.995197i	$-0.468791\pi$
$439$	1800.79	0.195779	0.0978895	+	0.995197i	$0.468791\pi$
$440$	0	0
$441$	12959.5	1.39936
$442$	0	0
$443$	− 7020.76i	− 0.752972i	−0.926422	−	0.376486i	$-0.877132\pi$
$443$	− 7020.76i	− 0.752972i	0.926422	−	0.376486i	$-0.122868\pi$
$444$	0	0
$445$	12919.6i	1.37629i
$446$	0	0
$447$	−16001.9	−1.69321
$448$	0	0
$449$	−3318.29	−0.348774	−0.174387	−	0.984677i	$-0.555794\pi$
$449$	−3318.29	−0.348774	−0.174387	+	0.984677i	$0.555794\pi$
$450$	0	0
$451$	8026.35i	0.838018i
$452$	0	0
$453$	25573.2i	2.65239i
$454$	0	0
$455$	10700.9	1.10256
$456$	0	0
$457$	−4468.38	−0.457378	−0.228689	−	0.973500i	$-0.573444\pi$
$457$	−4468.38	−0.457378	−0.228689	+	0.973500i	$0.573444\pi$
$458$	0	0
$459$	− 8525.84i	− 0.866998i
$460$	0	0
$461$	− 1179.45i	− 0.119159i	−0.998224	−	0.0595795i	$-0.981024\pi$
$461$	− 1179.45i	− 0.119159i	0.998224	−	0.0595795i	$-0.0189760\pi$
$462$	0	0
$463$	13763.1	1.38148	0.690742	−	0.723101i	$-0.257284\pi$
$463$	13763.1	1.38148	0.690742	+	0.723101i	$0.257284\pi$
$464$	0	0
$465$	8590.11	0.856682
$466$	0	0
$467$	− 17014.1i	− 1.68591i	−0.537984	−	0.842955i	$-0.680814\pi$
$467$	− 17014.1i	− 1.68591i	0.537984	−	0.842955i	$-0.319186\pi$
$468$	0	0
$469$	− 6515.49i	− 0.641487i
$470$	0	0
$471$	−18959.7	−1.85481
$472$	0	0
$473$	−4507.28	−0.438150
$474$	0	0
$475$	− 724.168i	− 0.0699518i
$476$	0	0
$477$	− 4796.59i	− 0.460421i
$478$	0	0
$479$	−18976.7	−1.81017	−0.905083	−	0.425236i	$-0.860191\pi$
$479$	−18976.7	−1.81017	−0.905083	+	0.425236i	$0.860191\pi$
$480$	0	0
$481$	1006.79	0.0954379
$482$	0	0
$483$	10622.3i	1.00068i
$484$	0	0
$485$	− 5508.55i	− 0.515733i
$486$	0	0
$487$	−10579.6	−0.984408	−0.492204	−	0.870480i	$-0.663809\pi$
$487$	−10579.6	−0.984408	−0.492204	+	0.870480i	$0.663809\pi$
$488$	0	0
$489$	−28250.7	−2.61256
$490$	0	0
$491$	15200.9i	1.39716i	0.715533	+	0.698579i	$0.246184\pi$
$491$	15200.9i	1.39716i	−0.715533	+	0.698579i	$0.753816\pi$
$492$	0	0
$493$	− 1013.21i	− 0.0925614i
$494$	0	0
$495$	−24419.3	−2.21730
$496$	0	0
$497$	4878.77	0.440327
$498$	0	0
$499$	2553.92i	0.229117i	0.993417	+	0.114558i	$0.0365453\pi$
$499$	2553.92i	0.229117i	−0.993417	+	0.114558i	$0.963455\pi$
$500$	0	0
$501$	− 11966.8i	− 1.06714i
$502$	0	0
$503$	−4863.70	−0.431137	−0.215568	−	0.976489i	$-0.569160\pi$
$503$	−4863.70	−0.431137	−0.215568	+	0.976489i	$0.569160\pi$
$504$	0	0
$505$	−11441.5	−1.00820
$506$	0	0
$507$	55247.0i	4.83946i
$508$	0	0
$509$	− 17659.5i	− 1.53781i	−0.639366	−	0.768903i	$-0.720803\pi$
$509$	− 17659.5i	− 1.53781i	0.639366	−	0.768903i	$-0.279197\pi$
$510$	0	0
$511$	5581.35	0.483179
$512$	0	0
$513$	−10674.4	−0.918685
$514$	0	0
$515$	− 21685.3i	− 1.85547i
$516$	0	0
$517$	− 12223.3i	− 1.03980i
$518$	0	0
$519$	25018.0	2.11593
$520$	0	0
$521$	3778.19	0.317707	0.158853	−	0.987302i	$-0.449220\pi$
$521$	3778.19	0.317707	0.158853	+	0.987302i	$0.449220\pi$
$522$	0	0
$523$	2792.67i	0.233489i	0.993162	+	0.116745i	$0.0372459\pi$
$523$	2792.67i	0.233489i	−0.993162	+	0.116745i	$0.962754\pi$
$524$	0	0
$525$	− 1370.54i	− 0.113934i
$526$	0	0
$527$	3013.74	0.249109
$528$	0	0
$529$	2403.34	0.197529
$530$	0	0
$531$	29025.3i	2.37211i
$532$	0	0
$533$	− 18810.7i	− 1.52867i
$534$	0	0
$535$	−20471.1	−1.65429
$536$	0	0
$537$	−9596.35	−0.771160
$538$	0	0
$539$	− 9605.85i	− 0.767631i
$540$	0	0
$541$	2062.12i	0.163877i	0.996637	+	0.0819383i	$0.0261111\pi$
$541$	2062.12i	0.163877i	−0.996637	+	0.0819383i	$0.973889\pi$
$542$	0	0
$543$	5098.33	0.402928
$544$	0	0
$545$	−5015.44	−0.394198
$546$	0	0
$547$	16562.0i	1.29459i	0.762239	+	0.647296i	$0.224100\pi$
$547$	16562.0i	1.29459i	−0.762239	+	0.647296i	$0.775900\pi$
$548$	0	0
$549$	33260.4i	2.58565i
$550$	0	0
$551$	−1268.54	−0.0980795
$552$	0	0
$553$	−487.057	−0.0374535
$554$	0	0
$555$	− 1163.87i	− 0.0890157i
$556$	0	0
$557$	9272.94i	0.705399i	0.935737	+	0.352699i	$0.114736\pi$
$557$	9272.94i	0.705399i	−0.935737	+	0.352699i	$0.885264\pi$
$558$	0	0
$559$	10563.3	0.799251
$560$	0	0
$561$	−12955.4	−0.975005
$562$	0	0
$563$	− 62.0181i	− 0.00464254i	−0.999997	−	0.00232127i	$-0.999261\pi$
$563$	− 62.0181i	− 0.00464254i	0.999997	−	0.00232127i	$-0.000738884\pi$
$564$	0	0
$565$	− 9047.71i	− 0.673699i
$566$	0	0
$567$	−6174.00	−0.457290
$568$	0	0
$569$	−10015.9	−0.737938	−0.368969	−	0.929442i	$-0.620289\pi$
$569$	−10015.9	−0.737938	−0.368969	+	0.929442i	$0.620289\pi$
$570$	0	0
$571$	− 15543.3i	− 1.13917i	−0.821931	−	0.569587i	$-0.807103\pi$
$571$	− 15543.3i	− 1.13917i	0.821931	−	0.569587i	$-0.192897\pi$
$572$	0	0
$573$	− 7791.26i	− 0.568036i
$574$	0	0
$575$	−1879.95	−0.136346
$576$	0	0
$577$	3776.77	0.272494	0.136247	−	0.990675i	$-0.456496\pi$
$577$	3776.77	0.272494	0.136247	+	0.990675i	$0.456496\pi$
$578$	0	0
$579$	− 6004.42i	− 0.430976i
$580$	0	0
$581$	5559.67i	0.396995i
$582$	0	0
$583$	−3555.33	−0.252567
$584$	0	0
$585$	57229.3	4.04469
$586$	0	0
$587$	596.285i	0.0419273i	0.999780	+	0.0209637i	$0.00667343\pi$
$587$	596.285i	0.0419273i	−0.999780	+	0.0209637i	$0.993327\pi$
$588$	0	0
$589$	− 3773.21i	− 0.263960i
$590$	0	0
$591$	−27941.3	−1.94476
$592$	0	0
$593$	20179.6	1.39743	0.698717	−	0.715398i	$-0.253755\pi$
$593$	20179.6	1.39743	0.698717	+	0.715398i	$0.253755\pi$
$594$	0	0
$595$	− 4340.06i	− 0.299034i
$596$	0	0
$597$	13413.5i	0.919558i
$598$	0	0
$599$	13076.3	0.891960	0.445980	−	0.895043i	$-0.352855\pi$
$599$	13076.3	0.891960	0.445980	+	0.895043i	$0.352855\pi$
$600$	0	0
$601$	16866.9	1.14479	0.572393	−	0.819980i	$-0.306015\pi$
$601$	16866.9	1.14479	0.572393	+	0.819980i	$0.306015\pi$
$602$	0	0
$603$	− 34845.3i	− 2.35325i
$604$	0	0
$605$	2319.18i	0.155848i
$606$	0	0
$607$	−19953.0	−1.33421	−0.667107	−	0.744962i	$-0.732468\pi$
$607$	−19953.0	−1.33421	−0.667107	+	0.744962i	$0.732468\pi$
$608$	0	0
$609$	−2400.82	−0.159747
$610$	0	0
$611$	28646.6i	1.89676i
$612$	0	0
$613$	− 7214.32i	− 0.475340i	−0.971346	−	0.237670i	$-0.923616\pi$
$613$	− 7214.32i	− 0.475340i	0.971346	−	0.237670i	$-0.0763837\pi$
$614$	0	0
$615$	−21745.6	−1.42580
$616$	0	0
$617$	−22212.9	−1.44936	−0.724681	−	0.689084i	$-0.758013\pi$
$617$	−22212.9	−1.44936	−0.724681	+	0.689084i	$0.758013\pi$
$618$	0	0
$619$	4349.02i	0.282394i	0.989982	+	0.141197i	$0.0450951\pi$
$619$	4349.02i	0.282394i	−0.989982	+	0.141197i	$0.954905\pi$
$620$	0	0
$621$	27710.8i	1.79065i
$622$	0	0
$623$	10740.2	0.690688
$624$	0	0
$625$	−17329.2	−1.10907
$626$	0	0
$627$	16220.2i	1.03313i
$628$	0	0
$629$	− 408.331i	− 0.0258843i
$630$	0	0
$631$	−10127.9	−0.638966	−0.319483	−	0.947592i	$-0.603509\pi$
$631$	−10127.9	−0.638966	−0.319483	+	0.947592i	$0.603509\pi$
$632$	0	0
$633$	15835.9	0.994347
$634$	0	0
$635$	25602.5i	1.60001i
$636$	0	0
$637$	22512.4i	1.40027i
$638$	0	0
$639$	26092.0	1.61531
$640$	0	0
$641$	−17624.6	−1.08601	−0.543003	−	0.839731i	$-0.682713\pi$
$641$	−17624.6	−1.08601	−0.543003	+	0.839731i	$0.682713\pi$
$642$	0	0
$643$	− 27238.1i	− 1.67055i	−0.549829	−	0.835277i	$-0.685307\pi$
$643$	− 27238.1i	− 1.67055i	0.549829	−	0.835277i	$-0.314693\pi$
$644$	0	0
$645$	− 12211.5i	− 0.745468i
$646$	0	0
$647$	−28702.1	−1.74404	−0.872021	−	0.489469i	$-0.837191\pi$
$647$	−28702.1	−1.74404	−0.872021	+	0.489469i	$0.837191\pi$
$648$	0	0
$649$	21514.1	1.30124
$650$	0	0
$651$	− 7141.09i	− 0.429925i
$652$	0	0
$653$	− 7937.84i	− 0.475700i	−0.971302	−	0.237850i	$-0.923557\pi$
$653$	− 7937.84i	− 0.475700i	0.971302	−	0.237850i	$-0.0764426\pi$
$654$	0	0
$655$	8426.62	0.502680
$656$	0	0
$657$	29849.5	1.77251
$658$	0	0
$659$	− 3675.29i	− 0.217252i	−0.994083	−	0.108626i	$-0.965355\pi$
$659$	− 3675.29i	− 0.217252i	0.994083	−	0.108626i	$-0.0346451\pi$
$660$	0	0
$661$	− 4205.49i	− 0.247466i	−0.992316	−	0.123733i	$-0.960513\pi$
$661$	− 4205.49i	− 0.247466i	0.992316	−	0.123733i	$-0.0394866\pi$
$662$	0	0
$663$	30362.5	1.77855
$664$	0	0
$665$	−5433.76	−0.316860
$666$	0	0
$667$	3293.15i	0.191171i
$668$	0	0
$669$	48334.3i	2.79329i
$670$	0	0
$671$	24653.3	1.41838
$672$	0	0
$673$	−4634.55	−0.265452	−0.132726	−	0.991153i	$-0.542373\pi$
$673$	−4634.55	−0.265452	−0.132726	+	0.991153i	$0.542373\pi$
$674$	0	0
$675$	− 3575.40i	− 0.203877i
$676$	0	0
$677$	5327.42i	0.302437i	0.988500	+	0.151218i	$0.0483196\pi$
$677$	5327.42i	0.302437i	−0.988500	+	0.151218i	$0.951680\pi$
$678$	0	0
$679$	−4579.34	−0.258820
$680$	0	0
$681$	−33429.6	−1.88109
$682$	0	0
$683$	− 2816.05i	− 0.157764i	−0.996884	−	0.0788822i	$-0.974865\pi$
$683$	− 2816.05i	− 0.157764i	0.996884	−	0.0788822i	$-0.0251351\pi$
$684$	0	0
$685$	− 11703.8i	− 0.652816i
$686$	0	0
$687$	6077.77	0.337528
$688$	0	0
$689$	8332.33	0.460720
$690$	0	0
$691$	− 16910.1i	− 0.930955i	−0.885060	−	0.465477i	$-0.845883\pi$
$691$	− 16910.1i	− 0.930955i	0.885060	−	0.465477i	$-0.154117\pi$
$692$	0	0
$693$	20300.1i	1.11275i
$694$	0	0
$695$	27674.6	1.51044
$696$	0	0
$697$	−7629.19	−0.414600
$698$	0	0
$699$	− 6064.67i	− 0.328164i
$700$	0	0
$701$	− 30588.4i	− 1.64808i	−0.566528	−	0.824042i	$-0.691714\pi$
$701$	− 30588.4i	− 1.64808i	0.566528	−	0.824042i	$-0.308286\pi$
$702$	0	0
$703$	−511.232	−0.0274274
$704$	0	0
$705$	33116.2	1.76912
$706$	0	0
$707$	9511.48i	0.505963i
$708$	0	0
$709$	− 21242.9i	− 1.12524i	−0.826716	−	0.562619i	$-0.809794\pi$
$709$	− 21242.9i	− 1.12524i	0.826716	−	0.562619i	$-0.190206\pi$
$710$	0	0
$711$	−2604.82	−0.137396
$712$	0	0
$713$	−9795.28	−0.514496
$714$	0	0
$715$	− 42419.5i	− 2.21874i
$716$	0	0
$717$	− 34831.3i	− 1.81423i
$718$	0	0
$719$	12452.7	0.645907	0.322954	−	0.946415i	$-0.395324\pi$
$719$	12452.7	0.645907	0.322954	+	0.946415i	$0.395324\pi$
$720$	0	0
$721$	−18027.3	−0.931168
$722$	0	0
$723$	− 52316.8i	− 2.69112i
$724$	0	0
$725$	− 424.901i	− 0.0217661i
$726$	0	0
$727$	27178.3	1.38650	0.693251	−	0.720696i	$-0.256178\pi$
$727$	27178.3	1.38650	0.693251	+	0.720696i	$0.256178\pi$
$728$	0	0
$729$	22321.3	1.13404
$730$	0	0
$731$	− 4284.25i	− 0.216770i
$732$	0	0
$733$	− 21999.7i	− 1.10856i	−0.832329	−	0.554282i	$-0.812993\pi$
$733$	− 21999.7i	− 1.10856i	0.832329	−	0.554282i	$-0.187007\pi$
$734$	0	0
$735$	26024.9	1.30605
$736$	0	0
$737$	−25828.1	−1.29089
$738$	0	0
$739$	13644.0i	0.679166i	0.940576	+	0.339583i	$0.110286\pi$
$739$	13644.0i	0.679166i	−0.940576	+	0.339583i	$0.889714\pi$
$740$	0	0
$741$	− 38013.9i	− 1.88458i
$742$	0	0
$743$	13980.2	0.690286	0.345143	−	0.938550i	$-0.387830\pi$
$743$	13980.2	0.690286	0.345143	+	0.938550i	$0.387830\pi$
$744$	0	0
$745$	−21250.1	−1.04502
$746$	0	0
$747$	29733.5i	1.45635i
$748$	0	0
$749$	17018.0i	0.830204i
$750$	0	0
$751$	−21138.7	−1.02711	−0.513557	−	0.858055i	$-0.671673\pi$
$751$	−21138.7	−1.02711	−0.513557	+	0.858055i	$0.671673\pi$
$752$	0	0
$753$	−30190.6	−1.46110
$754$	0	0
$755$	33960.5i	1.63702i
$756$	0	0
$757$	28328.1i	1.36011i	0.733162	+	0.680054i	$0.238044\pi$
$757$	28328.1i	1.36011i	−0.733162	+	0.680054i	$0.761956\pi$
$758$	0	0
$759$	42107.8	2.01372
$760$	0	0
$761$	1093.79	0.0521023	0.0260512	−	0.999661i	$-0.491707\pi$
$761$	1093.79	0.0521023	0.0260512	+	0.999661i	$0.491707\pi$
$762$	0	0
$763$	4169.41i	0.197828i
$764$	0	0
$765$	− 23210.9i	− 1.09699i
$766$	0	0
$767$	−50420.8	−2.37365
$768$	0	0
$769$	−1659.21	−0.0778059	−0.0389029	−	0.999243i	$-0.512386\pi$
$769$	−1659.21	−0.0778059	−0.0389029	+	0.999243i	$0.512386\pi$
$770$	0	0
$771$	− 32607.7i	− 1.52314i
$772$	0	0
$773$	7537.89i	0.350736i	0.984503	+	0.175368i	$0.0561115\pi$
$773$	7537.89i	0.350736i	−0.984503	+	0.175368i	$0.943888\pi$
$774$	0	0
$775$	1263.84	0.0585787
$776$	0	0
$777$	−967.546	−0.0446725
$778$	0	0
$779$	9551.76i	0.439316i
$780$	0	0
$781$	− 19339.9i	− 0.886091i
$782$	0	0
$783$	−6263.11	−0.285856
$784$	0	0
$785$	−25177.9	−1.14476
$786$	0	0
$787$	9304.20i	0.421422i	0.977548	+	0.210711i	$0.0675778\pi$
$787$	9304.20i	0.421422i	−0.977548	+	0.210711i	$0.932422\pi$
$788$	0	0
$789$	21987.8i	0.992126i
$790$	0	0
$791$	−7521.50	−0.338096
$792$	0	0
$793$	−57777.8	−2.58733
$794$	0	0
$795$	− 9632.39i	− 0.429718i
$796$	0	0
$797$	− 38619.5i	− 1.71640i	−0.513314	−	0.858201i	$-0.671582\pi$
$797$	− 38619.5i	− 1.71640i	0.513314	−	0.858201i	$-0.328418\pi$
$798$	0	0
$799$	11618.4	0.514431
$800$	0	0
$801$	57439.7	2.53375
$802$	0	0
$803$	− 22125.1i	− 0.972324i
$804$	0	0
$805$	14106.1i	0.617608i
$806$	0	0
$807$	−4307.88	−0.187912
$808$	0	0
$809$	18550.5	0.806179	0.403090	−	0.915160i	$-0.367936\pi$
$809$	18550.5	0.806179	0.403090	+	0.915160i	$0.367936\pi$
$810$	0	0
$811$	− 3704.80i	− 0.160411i	−0.996778	−	0.0802054i	$-0.974442\pi$
$811$	− 3704.80i	− 0.160411i	0.996778	−	0.0802054i	$-0.0255576\pi$
$812$	0	0
$813$	32163.7i	1.38749i
$814$	0	0
$815$	−37516.2	−1.61243
$816$	0	0
$817$	−5363.90	−0.229693
$818$	0	0
$819$	− 47575.6i	− 2.02982i
$820$	0	0
$821$	− 5544.86i	− 0.235709i	−0.993031	−	0.117854i	$-0.962398\pi$
$821$	− 5544.86i	− 0.235709i	0.993031	−	0.117854i	$-0.0376016\pi$
$822$	0	0
$823$	−3915.39	−0.165835	−0.0829174	−	0.996556i	$-0.526424\pi$
$823$	−3915.39	−0.165835	−0.0829174	+	0.996556i	$0.526424\pi$
$824$	0	0
$825$	−5432.98	−0.229275
$826$	0	0
$827$	− 35091.2i	− 1.47550i	−0.675073	−	0.737751i	$-0.735888\pi$
$827$	− 35091.2i	− 1.47550i	0.675073	−	0.737751i	$-0.264112\pi$
$828$	0	0
$829$	3532.71i	0.148005i	0.997258	+	0.0740025i	$0.0235773\pi$
$829$	3532.71i	0.148005i	−0.997258	+	0.0740025i	$0.976423\pi$
$830$	0	0
$831$	55156.0	2.30245
$832$	0	0
$833$	9130.53	0.379777
$834$	0	0
$835$	− 15891.6i	− 0.658624i
$836$	0	0
$837$	− 18629.3i	− 0.769320i
$838$	0	0
$839$	−30081.1	−1.23780	−0.618900	−	0.785470i	$-0.712421\pi$
$839$	−30081.1	−1.23780	−0.618900	+	0.785470i	$0.712421\pi$
$840$	0	0
$841$	23644.7	0.969482
$842$	0	0
$843$	− 4852.27i	− 0.198246i
$844$	0	0
$845$	73366.5i	2.98685i
$846$	0	0
$847$	1927.97	0.0782121
$848$	0	0
$849$	58766.4	2.37557
$850$	0	0
$851$	1327.16i	0.0534601i
$852$	0	0
$853$	− 11222.1i	− 0.450453i	−0.974306	−	0.225226i	$-0.927688\pi$
$853$	− 11222.1i	− 0.450453i	0.974306	−	0.225226i	$-0.0723122\pi$
$854$	0	0
$855$	−29060.2	−1.16238
$856$	0	0
$857$	27486.0	1.09557	0.547785	−	0.836619i	$-0.315471\pi$
$857$	27486.0	1.09557	0.547785	+	0.836619i	$0.315471\pi$
$858$	0	0
$859$	14976.2i	0.594855i	0.954744	+	0.297427i	$0.0961287\pi$
$859$	14976.2i	0.594855i	−0.954744	+	0.297427i	$0.903871\pi$
$860$	0	0
$861$	18077.5i	0.715538i
$862$	0	0
$863$	9419.49	0.371545	0.185772	−	0.982593i	$-0.440521\pi$
$863$	9419.49	0.371545	0.185772	+	0.982593i	$0.440521\pi$
$864$	0	0
$865$	33223.2	1.30592
$866$	0	0
$867$	31549.9i	1.23586i
$868$	0	0
$869$	1930.75i	0.0753694i
$870$	0	0
$871$	60531.0	2.35478
$872$	0	0
$873$	−24490.7	−0.949465
$874$	0	0
$875$	12787.6i	0.494059i
$876$	0	0
$877$	− 21069.8i	− 0.811263i	−0.914037	−	0.405631i	$-0.867052\pi$
$877$	− 21069.8i	− 0.811263i	0.914037	−	0.405631i	$-0.132948\pi$
$878$	0	0
$879$	60298.2	2.31378
$880$	0	0
$881$	−14511.7	−0.554952	−0.277476	−	0.960733i	$-0.589498\pi$
$881$	−14511.7	−0.554952	−0.277476	+	0.960733i	$0.589498\pi$
$882$	0	0
$883$	26515.9i	1.01057i	0.862953	+	0.505284i	$0.168612\pi$
$883$	26515.9i	1.01057i	−0.862953	+	0.505284i	$0.831388\pi$
$884$	0	0
$885$	58287.8i	2.21392i
$886$	0	0
$887$	−29223.6	−1.10624	−0.553119	−	0.833102i	$-0.686563\pi$
$887$	−29223.6	−1.10624	−0.553119	+	0.833102i	$0.686563\pi$
$888$	0	0
$889$	21283.8	0.802964
$890$	0	0
$891$	24474.4i	0.920227i
$892$	0	0
$893$	− 14546.3i	− 0.545099i
$894$	0	0
$895$	−12743.7	−0.475949
$896$	0	0
$897$	−98684.4	−3.67333
$898$	0	0
$899$	− 2213.90i	− 0.0821332i
$900$	0	0
$901$	− 3379.41i	− 0.124955i
$902$	0	0
$903$	−10151.6	−0.374113
$904$	0	0
$905$	6770.44	0.248682
$906$	0	0
$907$	32139.3i	1.17659i	0.808646	+	0.588296i	$0.200201\pi$
$907$	32139.3i	1.17659i	−0.808646	+	0.588296i	$0.799799\pi$
$908$	0	0
$909$	50868.1i	1.85609i
$910$	0	0
$911$	−21430.7	−0.779396	−0.389698	−	0.920943i	$-0.627421\pi$
$911$	−21430.7	−0.779396	−0.389698	+	0.920943i	$0.627421\pi$
$912$	0	0
$913$	22039.1	0.798891
$914$	0	0
$915$	66792.7i	2.41322i
$916$	0	0
$917$	− 7005.17i	− 0.252270i
$918$	0	0
$919$	11676.1	0.419106	0.209553	−	0.977797i	$-0.432799\pi$
$919$	11676.1	0.419106	0.209553	+	0.977797i	$0.432799\pi$
$920$	0	0
$921$	16266.1	0.581962
$922$	0	0
$923$	45325.3i	1.61636i
$924$	0	0
$925$	− 171.238i	− 0.00608677i
$926$	0	0
$927$	−96411.5	−3.41593
$928$	0	0
$929$	41969.7	1.48222	0.741110	−	0.671383i	$-0.234300\pi$
$929$	41969.7	1.48222	0.741110	+	0.671383i	$0.234300\pi$
$930$	0	0
$931$	− 11431.4i	− 0.402417i
$932$	0	0
$933$	− 94622.6i	− 3.32026i
$934$	0	0
$935$	−17204.4	−0.601759
$936$	0	0
$937$	45948.6	1.60200	0.801001	−	0.598663i	$-0.204301\pi$
$937$	45948.6	1.60200	0.801001	+	0.598663i	$0.204301\pi$
$938$	0	0
$939$	− 71155.0i	− 2.47290i
$940$	0	0
$941$	− 15000.1i	− 0.519648i	−0.965656	−	0.259824i	$-0.916335\pi$
$941$	− 15000.1i	− 0.519648i	0.965656	−	0.259824i	$-0.0836645\pi$
$942$	0	0
$943$	24796.4	0.856292
$944$	0	0
$945$	−26827.8	−0.923502
$946$	0	0
$947$	37423.2i	1.28415i	0.766642	+	0.642075i	$0.221926\pi$
$947$	37423.2i	1.28415i	−0.766642	+	0.642075i	$0.778074\pi$
$948$	0	0
$949$	51852.6i	1.77366i
$950$	0	0
$951$	−69404.8	−2.36657
$952$	0	0
$953$	55403.0	1.88319	0.941594	−	0.336750i	$-0.109328\pi$
$953$	55403.0	1.88319	0.941594	+	0.336750i	$0.109328\pi$
$954$	0	0
$955$	− 10346.6i	− 0.350584i
$956$	0	0
$957$	9517.09i	0.321467i
$958$	0	0
$959$	−9729.54	−0.327615
$960$	0	0
$961$	−23205.9	−0.778956
$962$	0	0
$963$	91013.4i	3.04555i
$964$	0	0
$965$	− 7973.70i	− 0.265992i
$966$	0	0
$967$	31440.5	1.04556	0.522780	−	0.852467i	$-0.324895\pi$
$967$	31440.5	1.04556	0.522780	+	0.852467i	$0.324895\pi$
$968$	0	0
$969$	−15417.6	−0.511129
$970$	0	0
$971$	− 19010.3i	− 0.628290i	−0.949375	−	0.314145i	$-0.898282\pi$
$971$	− 19010.3i	− 0.628290i	0.949375	−	0.314145i	$-0.101718\pi$
$972$	0	0
$973$	− 23006.3i	− 0.758015i
$974$	0	0
$975$	12732.8	0.418232
$976$	0	0
$977$	−8504.01	−0.278472	−0.139236	−	0.990259i	$-0.544465\pi$
$977$	−8504.01	−0.278472	−0.139236	+	0.990259i	$0.544465\pi$
$978$	0	0
$979$	− 42575.4i	− 1.38990i
$980$	0	0
$981$	22298.3i	0.725720i
$982$	0	0
$983$	−52828.3	−1.71410	−0.857051	−	0.515232i	$-0.827706\pi$
$983$	−52828.3	−1.71410	−0.857051	+	0.515232i	$0.827706\pi$
$984$	0	0
$985$	−37105.3	−1.20028
$986$	0	0
$987$	− 27530.0i	− 0.887831i
$988$	0	0
$989$	13924.7i	0.447705i
$990$	0	0
$991$	13275.5	0.425541	0.212770	−	0.977102i	$-0.431751\pi$
$991$	13275.5	0.425541	0.212770	+	0.977102i	$0.431751\pi$
$992$	0	0
$993$	−73326.1	−2.34334
$994$	0	0
$995$	17812.7i	0.567538i
$996$	0	0
$997$	20858.1i	0.662570i	0.943531	+	0.331285i	$0.107482\pi$
$997$	20858.1i	0.662570i	−0.943531	+	0.331285i	$0.892518\pi$
$998$	0	0
$999$	−2524.08	−0.0799382

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.4.b.i.129.1		4
4.3	odd	2		256.4.b.h.129.4		4
8.3	odd	2		256.4.b.h.129.1		4
8.5	even	2	inner	256.4.b.i.129.4		4
16.3	odd	4		128.4.a.f.1.1	yes	2
16.5	even	4		128.4.a.e.1.1	✓	2
16.11	odd	4		128.4.a.g.1.2	yes	2
16.13	even	4		128.4.a.h.1.2	yes	2
48.5	odd	4		1152.4.a.s.1.1		2
48.11	even	4		1152.4.a.t.1.1		2
48.29	odd	4		1152.4.a.q.1.2		2
48.35	even	4		1152.4.a.r.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.4.a.e.1.1	✓	2	16.5	even	4
128.4.a.f.1.1	yes	2	16.3	odd	4
128.4.a.g.1.2	yes	2	16.11	odd	4
128.4.a.h.1.2	yes	2	16.13	even	4
256.4.b.h.129.1		4	8.3	odd	2
256.4.b.h.129.4		4	4.3	odd	2
256.4.b.i.129.1		4	1.1	even	1	trivial
256.4.b.i.129.4		4	8.5	even	2	inner
1152.4.a.q.1.2		2	48.29	odd	4
1152.4.a.r.1.2		2	48.35	even	4
1152.4.a.s.1.1		2	48.5	odd	4
1152.4.a.t.1.1		2	48.11	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 256 = 2^{8} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	256.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-8.92820i q^{3} -11.8564i q^{5} -9.85641 q^{7} -52.7128 q^{9} +O(q^{10})\)	\(q-8.92820i q^{3} -11.8564i q^{5} -9.85641 q^{7} -52.7128 q^{9} +39.0718i q^{11} -91.5692i q^{13} -105.856 q^{15} -37.1384 q^{17} +46.4974i q^{19} +88.0000i q^{21} +120.708 q^{23} -15.5744 q^{25} +229.569i q^{27} +27.2820i q^{29} -81.1487 q^{31} +348.841 q^{33} +116.862i q^{35} +10.9948i q^{37} -817.549 q^{39} +205.426 q^{41} +115.359i q^{43} +624.985i q^{45} -312.841 q^{47} -245.851 q^{49} +331.580i q^{51} +90.9948i q^{53} +463.251 q^{55} +415.138 q^{57} -550.631i q^{59} -630.974i q^{61} +519.559 q^{63} -1085.68 q^{65} +661.041i q^{67} -1077.70i q^{69} -494.985 q^{71} -566.267 q^{73} +139.051i q^{75} -385.108i q^{77} +49.4153 q^{79} +626.395 q^{81} -564.067i q^{83} +440.328i q^{85} +243.580 q^{87} -1089.67 q^{89} +902.543i q^{91} +724.513i q^{93} +551.292 q^{95} +464.605 q^{97} -2059.58i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 16 q^{7} - 100 q^{9}+O(q^{10}) \) 4 * q + 16 * q^7 - 100 * q^9	\( 4 q + 16 q^{7} - 100 q^{9} - 368 q^{15} + 184 q^{17} - 16 q^{23} - 284 q^{25} - 768 q^{31} + 176 q^{33} - 1552 q^{39} + 600 q^{41} - 32 q^{47} - 540 q^{49} - 752 q^{55} + 1328 q^{57} + 1136 q^{63} - 1904 q^{65} - 816 q^{71} - 824 q^{73} - 800 q^{79} - 44 q^{81} + 1584 q^{87} - 1144 q^{89} + 2704 q^{95} + 4408 q^{97}+O(q^{100}) \) 4 * q + 16 * q^7 - 100 * q^9 - 368 * q^15 + 184 * q^17 - 16 * q^23 - 284 * q^25 - 768 * q^31 + 176 * q^33 - 1552 * q^39 + 600 * q^41 - 32 * q^47 - 540 * q^49 - 752 * q^55 + 1328 * q^57 + 1136 * q^63 - 1904 * q^65 - 816 * q^71 - 824 * q^73 - 800 * q^79 - 44 * q^81 + 1584 * q^87 - 1144 * q^89 + 2704 * q^95 + 4408 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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