LMFDB - Embedded newform 765.2.b.c.154.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [765,2,Mod(154,765)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(765, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("765.154"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$765 = 3^{2} \cdot 5 \cdot 17$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	765.b (of order $2$ , degree $1$ , minimal)

Newform invariants

comment:select newform

sage:traces = [8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$6.10855575463$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.619810816.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} - 2x^{5} + 14x^{4} - 8x^{3} + 2x^{2} + 2x + 1$ x^8 - 2x^5 + 14x^4 - 8x^3 + 2x^2 + 2*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$2$
Twist minimal:	no (minimal twist has level 85)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			154.4
Root			$-1.49094 - 1.49094i$ of defining polynomial
Character	$\chi$	$=$	765.154
Dual form			765.2.b.c.154.5

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.134632i q^{2} +1.98187 q^{4} +(1.29021 + 1.82630i) q^{5} +2.86537i q^{7} -0.536087i q^{8} +(0.245878 - 0.173703i) q^{10} +3.84724 q^{11} -6.22085i q^{13} +0.385770 q^{14} +3.89157 q^{16} +1.00000i q^{17} -6.62231 q^{19} +(2.55703 + 3.61949i) q^{20} -0.517961i q^{22} +4.51796i q^{23} +(-1.67072 + 4.71261i) q^{25} -0.837525 q^{26} +5.67880i q^{28} -0.658562 q^{29} +3.49984 q^{31} -1.59610i q^{32} +0.134632 q^{34} +(-5.23301 + 3.69693i) q^{35} +3.34144i q^{37} +0.891574i q^{38} +(0.979054 - 0.691665i) q^{40} -2.04189 q^{41} +1.29303i q^{43} +7.62475 q^{44} +0.608262 q^{46} -6.22085i q^{47} -1.21033 q^{49} +(0.634468 + 0.224932i) q^{50} -12.3290i q^{52} -9.92750i q^{53} +(4.96375 + 7.02621i) q^{55} +1.53609 q^{56} +0.0886634i q^{58} -2.00000 q^{59} -7.61634 q^{61} -0.471189i q^{62} +7.56826 q^{64} +(11.3611 - 8.02621i) q^{65} +0.257106i q^{67} +1.98187i q^{68} +(0.497724 + 0.704530i) q^{70} -1.18868 q^{71} -3.26330i q^{73} +0.449864 q^{74} -13.1246 q^{76} +11.0238i q^{77} +4.99756 q^{79} +(5.02095 + 7.10717i) q^{80} +0.274904i q^{82} +7.91534i q^{83} +(-1.82630 + 1.29021i) q^{85} +0.174083 q^{86} -2.06246i q^{88} +12.8013 q^{89} +17.8250 q^{91} +8.95403i q^{92} -0.837525 q^{94} +(-8.54417 - 12.0943i) q^{95} -4.08866i q^{97} +0.162950i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$8 q - 8 q^{4} + 2 q^{5} + 6 q^{10} + 4 q^{11} - 12 q^{14} - 8 q^{16} - 8 q^{19} + 2 q^{20} - 12 q^{25} - 8 q^{29} - 24 q^{31} + 4 q^{34} + 22 q^{40} + 12 q^{41} + 32 q^{44} - 8 q^{46} - 16 q^{49} - 44 q^{50}+ \cdots - 4 q^{95}+O(q^{100})$ 8 * q - 8 * q^4 + 2 * q^5 + 6 * q^10 + 4 * q^11 - 12 * q^14 - 8 * q^16 - 8 * q^19 + 2 * q^20 - 12 * q^25 - 8 * q^29 - 24 * q^31 + 4 * q^34 + 22 * q^40 + 12 * q^41 + 32 * q^44 - 8 * q^46 - 16 * q^49 - 44 * q^50 - 8 * q^55 + 8 * q^56 - 16 * q^59 + 12 * q^61 + 48 * q^64 - 20 * q^65 + 40 * q^70 + 20 * q^71 + 40 * q^74 - 24 * q^76 + 24 * q^79 + 26 * q^80 - 2 * q^85 - 40 * q^86 + 48 * q^89 + 36 * q^91 - 4 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/765\mathbb{Z}\right)^\times$ .

$n$	$307$	$496$	$596$
$\chi(n)$	$-1$	$1$	$1$

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

)

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

$2$		−	0.134632i		−	0.0951991i	−0.998866	−	0.0475996i	$-0.984843\pi$
$2$		−	0.134632i		−	0.0951991i	0.998866	−	0.0475996i	$-0.0151571\pi$
$3$		0			0
$3$		0			0
$4$	1.98187			0.990937
$5$	1.29021	+	1.82630i	0.576999	+	0.816745i
$5$	1.29021	+	1.82630i	0.576999	+	0.816745i
$6$		0			0
$7$			2.86537i			1.08301i	0.840698	+	0.541504i	$0.182145\pi$
$7$			2.86537i			1.08301i	−0.840698	+	0.541504i	$0.817855\pi$
$8$		−	0.536087i		−	0.189535i
$9$		0			0
$10$	0.245878	−	0.173703i	0.0777534	−	0.0549298i
$11$	3.84724			1.15999			0.579994	−	0.814621i	$-0.303055\pi$
$11$	3.84724			1.15999			0.579994	+	0.814621i	$0.303055\pi$
$12$		0			0
$13$		−	6.22085i		−	1.72535i	−0.505755	−	0.862677i	$-0.668786\pi$
$13$		−	6.22085i		−	1.72535i	0.505755	−	0.862677i	$-0.331214\pi$
$14$	0.385770			0.103101
$15$		0			0
$16$	3.89157			0.972894
$17$			1.00000i			0.242536i
$17$			1.00000i			0.242536i
$18$		0			0
$19$	−6.62231			−1.51926			−0.759631	−	0.650354i	$-0.774620\pi$
$19$	−6.62231			−1.51926			−0.759631	+	0.650354i	$0.774620\pi$
$20$	2.55703	+	3.61949i	0.571770	+	0.809343i
$21$		0			0
$22$		−	0.517961i		−	0.110430i
$23$			4.51796i			0.942060i	0.882117	+	0.471030i	$0.156118\pi$
$23$			4.51796i			0.942060i	−0.882117	+	0.471030i	$0.843882\pi$
$24$		0			0
$25$	−1.67072	+	4.71261i	−0.334144	+	0.942522i
$26$	−0.837525			−0.164252
$27$		0			0
$28$			5.67880i			1.07319i
$29$	−0.658562			−0.122292			−0.0611459	−	0.998129i	$-0.519476\pi$
$29$	−0.658562			−0.122292			−0.0611459	+	0.998129i	$0.519476\pi$
$30$		0			0
$31$	3.49984			0.628589			0.314295	−	0.949326i	$-0.398232\pi$
$31$	3.49984			0.628589			0.314295	+	0.949326i	$0.398232\pi$
$32$		−	1.59610i		−	0.282154i
$33$		0			0
$34$	0.134632			0.0230892
$35$	−5.23301	+	3.69693i	−0.884541	+	0.624894i
$36$		0			0
$37$			3.34144i			0.549329i	0.961540	+	0.274665i	$0.0885668\pi$
$37$			3.34144i			0.549329i	−0.961540	+	0.274665i	$0.911433\pi$
$38$			0.891574i			0.144632i
$39$		0			0
$40$	0.979054	−	0.691665i	0.154802	−	0.109362i
$41$	−2.04189			−0.318890			−0.159445	−	0.987207i	$-0.550970\pi$
$41$	−2.04189			−0.318890			−0.159445	+	0.987207i	$0.550970\pi$
$42$		0			0
$43$			1.29303i			0.197185i	0.995128	+	0.0985926i	$0.0314341\pi$
$43$			1.29303i			0.197185i	−0.995128	+	0.0985926i	$0.968566\pi$
$44$	7.62475			1.14947
$45$		0			0
$46$	0.608262			0.0896833
$47$		−	6.22085i		−	0.907405i	−0.891153	−	0.453702i	$-0.850103\pi$
$47$		−	6.22085i		−	0.907405i	0.891153	−	0.453702i	$-0.149897\pi$
$48$		0			0
$49$	−1.21033			−0.172905
$50$	0.634468	+	0.224932i	0.0897273	+	0.0318102i
$51$		0			0
$52$		−	12.3290i		−	1.70972i
$53$		−	9.92750i		−	1.36365i	−0.731517	−	0.681823i	$-0.761187\pi$
$53$		−	9.92750i		−	1.36365i	0.731517	−	0.681823i	$-0.238813\pi$
$54$		0			0
$55$	4.96375	+	7.02621i	0.669312	+	0.947413i
$56$	1.53609			0.205268
$57$		0			0
$58$			0.0886634i			0.0116421i
$59$	−2.00000			−0.260378			−0.130189	−	0.991489i	$-0.541558\pi$
$59$	−2.00000			−0.260378			−0.130189	+	0.991489i	$0.541558\pi$
$60$		0			0
$61$	−7.61634			−0.975173			−0.487586	−	0.873075i	$-0.662123\pi$
$61$	−7.61634			−0.975173			−0.487586	+	0.873075i	$0.662123\pi$
$62$		−	0.471189i		−	0.0598411i
$63$		0			0
$64$	7.56826			0.946033
$65$	11.3611	−	8.02621i	1.40917	−	0.995528i
$66$		0			0
$67$			0.257106i			0.0314106i	0.999877	+	0.0157053i	$0.00499935\pi$
$67$			0.257106i			0.0314106i	−0.999877	+	0.0157053i	$0.995001\pi$
$68$			1.98187i			0.240338i
$69$		0			0
$70$	0.497724	+	0.704530i	0.0594894	+	0.0842075i
$71$	−1.18868			−0.141070			−0.0705352	−	0.997509i	$-0.522471\pi$
$71$	−1.18868			−0.141070			−0.0705352	+	0.997509i	$0.522471\pi$
$72$		0			0
$73$		−	3.26330i		−	0.381940i	−0.981596	−	0.190970i	$-0.938837\pi$
$73$		−	3.26330i		−	0.381940i	0.981596	−	0.190970i	$-0.0611633\pi$
$74$	0.449864			0.0522956
$75$		0			0
$76$	−13.1246			−1.50549
$77$			11.0238i			1.25627i
$78$		0			0
$79$	4.99756			0.562269			0.281135	−	0.959668i	$-0.409289\pi$
$79$	4.99756			0.562269			0.281135	+	0.959668i	$0.409289\pi$
$80$	5.02095	+	7.10717i	0.561359	+	0.794606i
$81$		0			0
$82$			0.274904i			0.0303580i
$83$			7.91534i			0.868821i	0.900715	+	0.434411i	$0.143043\pi$
$83$			7.91534i			0.868821i	−0.900715	+	0.434411i	$0.856957\pi$
$84$		0			0
$85$	−1.82630	+	1.29021i	−0.198090	+	0.139943i
$86$	0.174083			0.0187719
$87$		0			0
$88$		−	2.06246i		−	0.219859i
$89$	12.8013			1.35693			0.678466	−	0.734632i	$-0.262645\pi$
$89$	12.8013			1.35693			0.678466	+	0.734632i	$0.262645\pi$
$90$		0			0
$91$	17.8250			1.86857
$92$			8.95403i			0.933522i
$93$		0			0
$94$	−0.837525			−0.0863841
$95$	−8.54417	−	12.0943i	−0.876613	−	1.24085i
$96$		0			0
$97$		−	4.08866i		−	0.415141i	−0.978220	−	0.207570i	$-0.933444\pi$
$97$		−	4.08866i		−	0.415141i	0.978220	−	0.207570i	$-0.0665556\pi$
$98$			0.162950i			0.0164604i
$99$		0			0
$100$	−3.31116	+	9.33980i	−0.331116	+	0.933980i
$101$	−12.2871			−1.22261			−0.611304	−	0.791396i	$-0.709355\pi$
$101$	−12.2871			−1.22261			−0.611304	+	0.791396i	$0.709355\pi$
$102$		0			0
$103$		−	13.5623i		−	1.33633i	−0.744012	−	0.668166i	$-0.767079\pi$
$103$		−	13.5623i		−	1.33633i	0.744012	−	0.668166i	$-0.232921\pi$
$104$	−3.33492			−0.327016
$105$		0			0
$106$	−1.33656			−0.129818
$107$			6.63800i			0.641719i	0.947127	+	0.320860i	$0.103972\pi$
$107$			6.63800i			0.641719i	−0.947127	+	0.320860i	$0.896028\pi$
$108$		0			0
$109$	2.58042			0.247159			0.123580	−	0.992335i	$-0.460563\pi$
$109$	2.58042			0.247159			0.123580	+	0.992335i	$0.460563\pi$
$110$	0.945951	−	0.668279i	0.0901929	−	0.0637179i
$111$		0			0
$112$			11.1508i			1.05365i
$113$		−	8.63283i		−	0.812108i	−0.913849	−	0.406054i	$-0.866904\pi$
$113$		−	8.63283i		−	0.812108i	0.913849	−	0.406054i	$-0.133096\pi$
$114$		0			0
$115$	−8.25114	+	5.82912i	−0.769423	+	0.543568i
$116$	−1.30519			−0.121184
$117$		0			0
$118$			0.269264i			0.0247877i
$119$	−2.86537			−0.262668
$120$		0			0
$121$	3.80127			0.345570
$122$			1.02540i			0.0928356i
$123$		0			0
$124$	6.93623			0.622892
$125$	−10.7622	+	3.02903i	−0.962601	+	0.270924i
$126$		0			0
$127$			3.24550i			0.287991i	0.989578	+	0.143996i	$0.0459951\pi$
$127$			3.24550i			0.287991i	−0.989578	+	0.143996i	$0.954005\pi$
$128$		−	4.21114i		−	0.372216i
$129$		0			0
$130$	−1.08058	−	1.52957i	−0.0947734	−	0.134152i
$131$	3.44742			0.301203			0.150601	−	0.988595i	$-0.451879\pi$
$131$	3.44742			0.301203			0.150601	+	0.988595i	$0.451879\pi$
$132$		0			0
$133$		−	18.9754i		−	1.64537i
$134$	0.0346147			0.00299026
$135$		0			0
$136$	0.536087			0.0459691
$137$		−	14.6869i		−	1.25478i	−0.778703	−	0.627392i	$-0.784122\pi$
$137$		−	14.6869i		−	1.25478i	0.778703	−	0.627392i	$-0.215878\pi$
$138$		0			0
$139$	−13.9883			−1.18647			−0.593237	−	0.805028i	$-0.702150\pi$
$139$	−13.9883			−1.18647			−0.593237	+	0.805028i	$0.702150\pi$
$140$	−10.3712	+	7.32684i	−0.876524	+	0.619231i
$141$		0			0
$142$			0.160034i			0.0134298i
$143$		−	23.9331i		−	2.00139i
$144$		0			0
$145$	−0.849683	−	1.20273i	−0.0705623	−	0.0998812i
$146$	−0.439344			−0.0363603
$147$		0			0
$148$			6.62231i			0.544351i
$149$	−17.8388			−1.46141			−0.730707	−	0.682691i	$-0.760809\pi$
$149$	−17.8388			−1.46141			−0.730707	+	0.682691i	$0.760809\pi$
$150$		0			0
$151$	−2.80291			−0.228098			−0.114049	−	0.993475i	$-0.536382\pi$
$151$	−2.80291			−0.228098			−0.114049	+	0.993475i	$0.536382\pi$
$152$			3.55014i			0.287954i
$153$		0			0
$154$	1.48415			0.119596
$155$	4.51552	+	6.39174i	0.362695	+	0.513397i
$156$		0			0
$157$			3.60582i			0.287776i	0.989594	+	0.143888i	$0.0459605\pi$
$157$			3.60582i			0.287776i	−0.989594	+	0.143888i	$0.954040\pi$
$158$		−	0.672831i		−	0.0535275i
$159$		0			0
$160$	2.91496	−	2.05931i	0.230448	−	0.162803i
$161$	−12.9456			−1.02026
$162$		0			0
$163$		−	13.7256i		−	1.07507i	−0.843242	−	0.537535i	$-0.819356\pi$
$163$		−	13.7256i		−	1.07507i	0.843242	−	0.537535i	$-0.180644\pi$
$164$	−4.04677			−0.316000
$165$		0			0
$166$	1.06566			0.0827110
$167$		−	6.32684i		−	0.489586i	−0.969575	−	0.244793i	$-0.921280\pi$
$167$		−	6.32684i		−	0.489586i	0.969575	−	0.244793i	$-0.0787199\pi$
$168$		0			0
$169$	−25.6990			−1.97685
$170$	0.173703	+	0.245878i	0.0133224	+	0.0188580i
$171$		0			0
$172$			2.56262i			0.195398i
$173$		−	1.76699i		−	0.134342i	−0.997741	−	0.0671708i	$-0.978603\pi$
$173$		−	1.76699i		−	0.134342i	0.997741	−	0.0671708i	$-0.0213972\pi$
$174$		0			0
$175$	−13.5034	−	4.78723i	−1.02076	−	0.361880i
$176$	14.9718			1.12854
$177$		0			0
$178$		−	1.72346i		−	0.129179i
$179$	1.10843			0.0828476			0.0414238	−	0.999142i	$-0.486811\pi$
$179$	1.10843			0.0828476			0.0414238	+	0.999142i	$0.486811\pi$
$180$		0			0
$181$	−10.8310			−0.805062			−0.402531	−	0.915406i	$-0.631870\pi$
$181$	−10.8310			−0.805062			−0.402531	+	0.915406i	$0.631870\pi$
$182$		−	2.39982i		−	0.177886i
$183$		0			0
$184$	2.42202			0.178554
$185$	−6.10246	+	4.31116i	−0.448662	+	0.316962i
$186$		0			0
$187$			3.84724i			0.281338i
$188$		−	12.3290i		−	0.899181i
$189$		0			0
$190$	−1.62828	+	1.15032i	−0.118128	+	0.0834528i
$191$	−3.19221			−0.230980			−0.115490	−	0.993309i	$-0.536844\pi$
$191$	−3.19221			−0.230980			−0.115490	+	0.993309i	$0.536844\pi$
$192$		0			0
$193$		−	7.11407i		−	0.512082i	−0.966666	−	0.256041i	$-0.917582\pi$
$193$		−	7.11407i		−	0.512082i	0.966666	−	0.256041i	$-0.0824182\pi$
$194$	−0.550464			−0.0395210
$195$		0			0
$196$	−2.39873			−0.171338
$197$			23.1721i			1.65095i	0.564442	+	0.825473i	$0.309091\pi$
$197$			23.1721i			1.65095i	−0.564442	+	0.825473i	$0.690909\pi$
$198$		0			0
$199$	24.7131			1.75186			0.875932	−	0.482434i	$-0.160247\pi$
$199$	24.7131			1.75186			0.875932	+	0.482434i	$0.160247\pi$
$200$	2.52637	+	0.895651i	0.178641	+	0.0633321i
$201$		0			0
$202$			1.65423i			0.116391i
$203$		−	1.88702i		−	0.132443i
$204$		0			0
$205$	−2.63447	−	3.72910i	−0.183999	−	0.260452i
$206$	−1.82592			−0.127218
$207$		0			0
$208$		−	24.2089i		−	1.67859i
$209$	−25.4776			−1.76232
$210$		0			0
$211$	−4.27138			−0.294054			−0.147027	−	0.989133i	$-0.546970\pi$
$211$	−4.27138			−0.294054			−0.147027	+	0.989133i	$0.546970\pi$
$212$		−	19.6751i		−	1.35129i
$213$		0			0
$214$	0.893686			0.0610911
$215$	−2.36146	+	1.66828i	−0.161050	+	0.113776i
$216$		0			0
$217$			10.0283i			0.680767i
$218$		−	0.347407i		−	0.0235293i
$219$		0			0
$220$	9.83753	+	13.9251i	0.663246	+	0.938827i
$221$	6.22085			0.418460
$222$		0			0
$223$		−	3.65423i		−	0.244705i	−0.992487	−	0.122353i	$-0.960956\pi$
$223$		−	3.65423i		−	0.244705i	0.992487	−	0.122353i	$-0.0390439\pi$
$224$	4.57343			0.305575
$225$		0			0
$226$	−1.16225			−0.0773120
$227$		−	19.0016i		−	1.26118i	−0.776117	−	0.630589i	$-0.782813\pi$
$227$		−	19.0016i		−	1.26118i	0.776117	−	0.630589i	$-0.217187\pi$
$228$		0			0
$229$	20.8851			1.38012			0.690062	−	0.723751i	$-0.257583\pi$
$229$	20.8851			1.38012			0.690062	+	0.723751i	$0.257583\pi$
$230$	0.784785	+	1.11087i	0.0517472	+	0.0732483i
$231$		0			0
$232$			0.353047i			0.0231786i
$233$		−	0.682876i		−	0.0447367i	−0.999750	−	0.0223684i	$-0.992879\pi$
$233$		−	0.682876i		−	0.0447367i	0.999750	−	0.0223684i	$-0.00712066\pi$
$234$		0			0
$235$	11.3611	−	8.02621i	0.741118	−	0.523572i
$236$	−3.96375			−0.258018
$237$		0			0
$238$			0.385770i			0.0250057i
$239$	28.9915			1.87531			0.937653	−	0.347574i	$-0.112994\pi$
$239$	28.9915			1.87531			0.937653	+	0.347574i	$0.112994\pi$
$240$		0			0
$241$	−18.0943			−1.16556			−0.582778	−	0.812631i	$-0.698034\pi$
$241$	−18.0943			−1.16556			−0.582778	+	0.812631i	$0.698034\pi$
$242$		−	0.511773i		−	0.0328980i
$243$		0			0
$244$	−15.0946			−0.966335
$245$	−1.56158	−	2.21043i	−0.0997660	−	0.141219i
$246$		0			0
$247$			41.1964i			2.62127i
$248$		−	1.87622i		−	0.119140i
$249$		0			0
$250$	0.407803	+	1.44894i	0.0257918	+	0.0916387i
$251$	7.21325			0.455296			0.227648	−	0.973743i	$-0.426896\pi$
$251$	7.21325			0.455296			0.227648	+	0.973743i	$0.426896\pi$
$252$		0			0
$253$			17.3817i			1.09278i
$254$	0.436948			0.0274165
$255$		0			0
$256$	14.5696			0.910598
$257$			4.87942i			0.304370i	0.988352	+	0.152185i	$0.0486309\pi$
$257$			4.87942i			0.304370i	−0.988352	+	0.152185i	$0.951369\pi$
$258$		0			0
$259$	−9.57445			−0.594927
$260$	22.5163	−	15.9069i	1.39640	−	0.986506i
$261$		0			0
$262$		−	0.464133i		−	0.0286742i
$263$			30.0234i			1.85132i	0.378351	+	0.925662i	$0.376491\pi$
$263$			30.0234i			1.85132i	−0.378351	+	0.925662i	$0.623509\pi$
$264$		0			0
$265$	18.1306	−	12.8086i	1.11375	−	0.786823i
$266$	−2.55469			−0.156638
$267$		0			0
$268$			0.509553i			0.0311259i
$269$	−29.6568			−1.80821			−0.904104	−	0.427312i	$-0.859460\pi$
$269$	−29.6568			−1.80821			−0.904104	+	0.427312i	$0.859460\pi$
$270$		0			0
$271$	3.91622			0.237893			0.118947	−	0.992901i	$-0.462048\pi$
$271$	3.91622			0.237893			0.118947	+	0.992901i	$0.462048\pi$
$272$			3.89157i			0.235961i
$273$		0			0
$274$	−1.97732			−0.119454
$275$	−6.42766	+	18.1306i	−0.387603	+	1.09331i
$276$		0			0
$277$		−	0.455504i		−	0.0273686i	−0.999906	−	0.0136843i	$-0.995644\pi$
$277$		−	0.455504i		−	0.0273686i	0.999906	−	0.0136843i	$-0.00435598\pi$
$278$			1.88327i			0.112951i
$279$		0			0
$280$	1.98187	+	2.80535i	0.118440	+	0.167652i
$281$	−16.4941			−0.983957			−0.491978	−	0.870607i	$-0.663726\pi$
$281$	−16.4941			−0.983957			−0.491978	+	0.870607i	$0.663726\pi$
$282$		0			0
$283$		−	8.79775i		−	0.522972i	−0.965207	−	0.261486i	$-0.915788\pi$
$283$		−	8.79775i		−	0.522972i	0.965207	−	0.261486i	$-0.0842125\pi$
$284$	−2.35582			−0.139792
$285$		0			0
$286$	−3.22216			−0.190531
$287$		−	5.85077i		−	0.345360i
$288$		0			0
$289$	−1.00000			−0.0588235
$290$	−0.161926	+	0.114394i	−0.00950860	+	0.00671747i
$291$		0			0
$292$		−	6.46744i		−	0.378478i
$293$		−	22.5023i		−	1.31460i	−0.753630	−	0.657299i	$-0.771699\pi$
$293$		−	22.5023i		−	1.31460i	0.753630	−	0.657299i	$-0.228301\pi$
$294$		0			0
$295$	−2.58042	−	3.65259i	−0.150238	−	0.212662i
$296$	1.79130			0.104117
$297$		0			0
$298$			2.40168i			0.139125i
$299$	28.1056			1.62539
$300$		0			0
$301$	−3.70501			−0.213553
$302$			0.377361i			0.0217147i
$303$		0			0
$304$	−25.7712			−1.47808
$305$	−9.82668	−	13.9097i	−0.562674	−	0.796467i
$306$		0			0
$307$			11.7034i			0.667947i	0.942583	+	0.333973i	$0.108390\pi$
$307$			11.7034i			0.667947i	−0.942583	+	0.333973i	$0.891610\pi$
$308$			21.8477i			1.24489i
$309$		0			0
$310$	0.860532	−	0.607933i	0.0488749	−	0.0345283i
$311$	21.0605			1.19423			0.597115	−	0.802155i	$-0.296313\pi$
$311$	21.0605			1.19423			0.597115	+	0.802155i	$0.296313\pi$
$312$		0			0
$313$		−	22.2911i		−	1.25997i	−0.776609	−	0.629983i	$-0.783062\pi$
$313$		−	22.2911i		−	1.25997i	0.776609	−	0.629983i	$-0.216938\pi$
$314$	0.485459			0.0273960
$315$		0			0
$316$	9.90453			0.557174
$317$			27.8613i			1.56485i	0.622747	+	0.782423i	$0.286016\pi$
$317$			27.8613i			1.56485i	−0.622747	+	0.782423i	$0.713984\pi$
$318$		0			0
$319$	−2.53365			−0.141857
$320$	9.76464	+	13.8219i	0.545860	+	0.772667i
$321$		0			0
$322$			1.74289i			0.0971277i
$323$		−	6.62231i		−	0.368475i
$324$		0			0
$325$	29.3165	+	10.3933i	1.62618	+	0.576517i
$326$	−1.84790			−0.102346
$327$		0			0
$328$			1.09463i			0.0604409i
$329$	17.8250			0.982726
$330$		0			0
$331$	−13.0067			−0.714914			−0.357457	−	0.933930i	$-0.616356\pi$
$331$	−13.0067			−0.714914			−0.357457	+	0.933930i	$0.616356\pi$
$332$			15.6872i			0.860947i
$333$		0			0
$334$	−0.851794			−0.0466081
$335$	−0.469553	+	0.331721i	−0.0256544	+	0.0181239i
$336$		0			0
$337$		−	15.1126i		−	0.823238i	−0.911356	−	0.411619i	$-0.864963\pi$
$337$		−	15.1126i		−	0.823238i	0.911356	−	0.411619i	$-0.135037\pi$
$338$			3.45991i			0.188194i
$339$		0			0
$340$	−3.61949	+	2.55703i	−0.196294	+	0.138675i
$341$	13.4647			0.729155
$342$		0			0
$343$			16.5895i			0.895750i
$344$	0.693177			0.0373736
$345$		0			0
$346$	−0.237893			−0.0127892
$347$			3.58828i			0.192629i	0.995351	+	0.0963144i	$0.0307054\pi$
$347$			3.58828i			0.192629i	−0.995351	+	0.0963144i	$0.969295\pi$
$348$		0			0
$349$	−31.2851			−1.67465			−0.837326	−	0.546703i	$-0.815883\pi$
$349$	−31.2851			−1.67465			−0.837326	+	0.546703i	$0.815883\pi$
$350$	−0.644513	+	1.81798i	−0.0344507	+	0.0971753i
$351$		0			0
$352$		−	6.14060i		−	0.327295i
$353$		−	24.4417i		−	1.30090i	−0.759549	−	0.650450i	$-0.774580\pi$
$353$		−	24.4417i		−	1.30090i	0.759549	−	0.650450i	$-0.225420\pi$
$354$		0			0
$355$	−1.53365	−	2.17088i	−0.0813975	−	0.115219i
$356$	25.3705			1.34463
$357$		0			0
$358$		−	0.149229i		−	0.00788702i
$359$	−0.136195			−0.00718808			−0.00359404	−	0.999994i	$-0.501144\pi$
$359$	−0.136195			−0.00718808			−0.00359404	+	0.999994i	$0.501144\pi$
$360$		0			0
$361$	24.8550			1.30816
$362$			1.45820i			0.0766412i
$363$		0			0
$364$	35.3270			1.85164
$365$	5.95975	−	4.21033i	0.311947	−	0.220379i
$366$		0			0
$367$		−	16.9896i		−	0.886851i	−0.896311	−	0.443426i	$-0.853763\pi$
$367$		−	16.9896i		−	0.886851i	0.896311	−	0.443426i	$-0.146237\pi$
$368$			17.5820i			0.916524i
$369$		0			0
$370$	0.580419	+	0.821585i	0.0301745	+	0.0427122i
$371$	28.4459			1.47684
$372$		0			0
$373$			18.3533i			0.950296i	0.879906	+	0.475148i	$0.157606\pi$
$373$			18.3533i			0.950296i	−0.879906	+	0.475148i	$0.842394\pi$
$374$	0.517961			0.0267832
$375$		0			0
$376$	−3.33492			−0.171985
$377$			4.09682i			0.210997i
$378$		0			0
$379$	35.1943			1.80781			0.903905	−	0.427732i	$-0.140687\pi$
$379$	35.1943			1.80781			0.903905	+	0.427732i	$0.140687\pi$
$380$	−16.9335	−	23.9694i	−0.868668	−	1.22960i
$381$		0			0
$382$			0.429773i			0.0219891i
$383$			10.8237i			0.553067i	0.961004	+	0.276533i	$0.0891856\pi$
$383$			10.8237i			0.553067i	−0.961004	+	0.276533i	$0.910814\pi$
$384$		0			0
$385$	−20.1327	+	14.2230i	−1.02606	+	0.724870i
$386$	−0.957780			−0.0487497
$387$		0			0
$388$		−	8.10322i		−	0.411379i
$389$	0.174083			0.00882636			0.00441318	−	0.999990i	$-0.498595\pi$
$389$	0.174083			0.00882636			0.00441318	+	0.999990i	$0.498595\pi$
$390$		0			0
$391$	−4.51796			−0.228483
$392$			0.648845i			0.0327716i
$393$		0			0
$394$	3.11971			0.157169
$395$	6.44790	+	9.12703i	0.324429	+	0.459231i
$396$		0			0
$397$			2.75287i			0.138162i	0.997611	+	0.0690812i	$0.0220067\pi$
$397$			2.75287i			0.138162i	−0.997611	+	0.0690812i	$0.977993\pi$
$398$		−	3.32717i		−	0.166776i
$399$		0			0
$400$	−6.50173	+	18.3395i	−0.325086	+	0.916974i
$401$	−33.9912			−1.69744			−0.848719	−	0.528843i	$-0.822626\pi$
$401$	−33.9912			−1.69744			−0.848719	+	0.528843i	$0.822626\pi$
$402$		0			0
$403$		−	21.7720i		−	1.08454i
$404$	−24.3514			−1.21153
$405$		0			0
$406$	−0.254053			−0.0126085
$407$			12.8553i			0.637215i
$408$		0			0
$409$	4.68288			0.231553			0.115777	−	0.993275i	$-0.463064\pi$
$409$	4.68288			0.231553			0.115777	+	0.993275i	$0.463064\pi$
$410$	−0.502056	+	0.354683i	−0.0247948	+	0.0175166i
$411$		0			0
$412$		−	26.8788i		−	1.32422i
$413$		−	5.73074i		−	0.281991i
$414$		0			0
$415$	−14.4558	+	10.2124i	−0.709605	+	0.501309i
$416$	−9.92913			−0.486816
$417$		0			0
$418$			3.43010i			0.167772i
$419$	−29.0991			−1.42158			−0.710792	−	0.703402i	$-0.751663\pi$
$419$	−29.0991			−1.42158			−0.710792	+	0.703402i	$0.751663\pi$
$420$		0			0
$421$	7.19057			0.350447			0.175224	−	0.984529i	$-0.443935\pi$
$421$	7.19057			0.350447			0.175224	+	0.984529i	$0.443935\pi$
$422$			0.575063i			0.0279936i
$423$		0			0
$424$	−5.32200			−0.258459
$425$	−4.71261	−	1.67072i	−0.228595	−	0.0810418i
$426$		0			0
$427$		−	21.8236i		−	1.05612i
$428$			13.1557i			0.635903i
$429$		0			0
$430$	0.224604	+	0.317927i	0.0108313	+	0.0153318i
$431$	−7.63181			−0.367611			−0.183806	−	0.982963i	$-0.558842\pi$
$431$	−7.63181			−0.367611			−0.183806	+	0.982963i	$0.558842\pi$
$432$		0			0
$433$			32.2894i			1.55173i	0.630898	+	0.775865i	$0.282686\pi$
$433$			32.2894i			1.55173i	−0.630898	+	0.775865i	$0.717314\pi$
$434$	1.35013			0.0648084
$435$		0			0
$436$	5.11407			0.244919
$437$		−	29.9193i		−	1.43124i
$438$		0			0
$439$	13.3974			0.639422			0.319711	−	0.947515i	$-0.396414\pi$
$439$	13.3974			0.639422			0.319711	+	0.947515i	$0.396414\pi$
$440$	3.76666	−	2.66100i	0.179568	−	0.126858i
$441$		0			0
$442$		−	0.837525i		−	0.0398370i
$443$			6.89070i			0.327387i	0.986511	+	0.163693i	$0.0523408\pi$
$443$			6.89070i			0.327387i	−0.986511	+	0.163693i	$0.947659\pi$
$444$		0			0
$445$	16.5163	+	23.3789i	0.782949	+	1.10827i
$446$	−0.491976			−0.0232957
$447$		0			0
$448$			21.6859i			1.02456i
$449$	21.1778			0.999440			0.499720	−	0.866187i	$-0.333436\pi$
$449$	21.1778			0.999440			0.499720	+	0.866187i	$0.333436\pi$
$450$		0			0
$451$	−7.85565			−0.369908
$452$		−	17.1092i		−	0.804748i
$453$		0			0
$454$	−2.55822			−0.120063
$455$	22.9980	+	32.5538i	1.07816	+	1.52615i
$456$		0			0
$457$		−	30.2321i		−	1.41420i	−0.707114	−	0.707100i	$-0.750003\pi$
$457$		−	30.2321i		−	1.41420i	0.707114	−	0.707100i	$-0.249997\pi$
$458$		−	2.81179i		−	0.131387i
$459$		0			0
$460$	−16.3527	+	11.5526i	−0.762449	+	0.538642i
$461$	−3.74267			−0.174314			−0.0871568	−	0.996195i	$-0.527778\pi$
$461$	−3.74267			−0.174314			−0.0871568	+	0.996195i	$0.527778\pi$
$462$		0			0
$463$		−	2.53452i		−	0.117789i	−0.998264	−	0.0588947i	$-0.981242\pi$
$463$		−	2.53452i		−	0.117789i	0.998264	−	0.0588947i	$-0.0187576\pi$
$464$	−2.56284			−0.118977
$465$		0			0
$466$	−0.0919369			−0.00425890
$467$			30.1170i			1.39365i	0.717242	+	0.696824i	$0.245404\pi$
$467$			30.1170i			1.39365i	−0.717242	+	0.696824i	$0.754596\pi$
$468$		0			0
$469$	−0.736705			−0.0340179
$470$	−1.08058	−	1.52957i	−0.0498436	−	0.0705538i
$471$		0			0
$472$			1.07217i			0.0493508i
$473$			4.97460i			0.228732i
$474$		0			0
$475$	11.0640	−	31.2084i	0.507652	−	1.43194i
$476$	−5.67880			−0.260287
$477$		0			0
$478$		−	3.90318i		−	0.178527i
$479$	14.4995			0.662499			0.331250	−	0.943543i	$-0.392530\pi$
$479$	14.4995			0.662499			0.331250	+	0.943543i	$0.392530\pi$
$480$		0			0
$481$	20.7866			0.947787
$482$			2.43607i			0.110960i
$483$		0			0
$484$	7.53365			0.342438
$485$	7.46711	−	5.27523i	0.339064	−	0.239536i
$486$		0			0
$487$			23.9424i			1.08493i	0.840077	+	0.542467i	$0.182510\pi$
$487$			23.9424i			1.08493i	−0.840077	+	0.542467i	$0.817490\pi$
$488$			4.08302i			0.184830i
$489$		0			0
$490$	−0.297594	+	0.210239i	−0.0134439	+	0.00949764i
$491$	21.1295			0.953559			0.476780	−	0.879023i	$-0.341804\pi$
$491$	21.1295			0.953559			0.476780	+	0.879023i	$0.341804\pi$
$492$		0			0
$493$		−	0.658562i		−	0.0296601i
$494$	5.54635			0.249542
$495$		0			0
$496$	13.6199			0.611550
$497$		−	3.40601i		−	0.152780i
$498$		0			0
$499$	−4.01027			−0.179524			−0.0897621	−	0.995963i	$-0.528611\pi$
$499$	−4.01027			−0.179524			−0.0897621	+	0.995963i	$0.528611\pi$
$500$	−21.3293	+	6.00315i	−0.953877	+	0.268469i
$501$		0			0
$502$		−	0.971133i		−	0.0433438i
$503$			1.44015i			0.0642130i	0.999484	+	0.0321065i	$0.0102216\pi$
$503$			1.44015i			0.0642130i	−0.999484	+	0.0321065i	$0.989778\pi$
$504$		0			0
$505$	−15.8529	−	22.4398i	−0.705444	−	0.998559i
$506$	2.34013			0.104031
$507$		0			0
$508$			6.43217i			0.285381i
$509$	31.1478			1.38060			0.690301	−	0.723522i	$-0.257478\pi$
$509$	31.1478			1.38060			0.690301	+	0.723522i	$0.257478\pi$
$510$		0			0
$511$	9.35054			0.413644
$512$		−	10.3838i		−	0.458904i
$513$		0			0
$514$	0.656925			0.0289757
$515$	24.7688	−	17.4982i	1.09144	−	0.771063i
$516$		0			0
$517$		−	23.9331i		−	1.05258i
$518$			1.28903i			0.0566366i
$519$		0			0
$520$	−4.30275	−	6.09055i	−0.188688	−	0.267088i
$521$	39.1197			1.71387			0.856933	−	0.515428i	$-0.172367\pi$
$521$	39.1197			1.71387			0.856933	+	0.515428i	$0.172367\pi$
$522$		0			0
$523$			40.4813i			1.77012i	0.465473	+	0.885062i	$0.345884\pi$
$523$			40.4813i			1.77012i	−0.465473	+	0.885062i	$0.654116\pi$
$524$	6.83236			0.298473
$525$		0			0
$526$	4.04211			0.176244
$527$			3.49984i			0.152455i
$528$		0			0
$529$	2.58802			0.112523
$530$	−1.72444	−	2.44095i	−0.0749049	−	0.106028i
$531$		0			0
$532$		−	37.6068i		−	1.63046i
$533$			12.7023i			0.550198i
$534$		0			0
$535$	−12.1229	+	8.56440i	−0.524121	+	0.370272i
$536$	0.137831			0.00595341
$537$		0			0
$538$			3.99275i			0.172140i
$539$	−4.65645			−0.200568
$540$		0			0
$541$	−18.5288			−0.796614			−0.398307	−	0.917252i	$-0.630402\pi$
$541$	−18.5288			−0.796614			−0.398307	+	0.917252i	$0.630402\pi$
$542$		−	0.527248i		−	0.0226472i
$543$		0			0
$544$	1.59610			0.0684324
$545$	3.32928	+	4.71261i	0.142611	+	0.201866i
$546$		0			0
$547$			43.2959i			1.85120i	0.378504	+	0.925600i	$0.376439\pi$
$547$			43.2959i			1.85120i	−0.378504	+	0.925600i	$0.623561\pi$
$548$		−	29.1075i		−	1.24341i
$549$		0			0
$550$	2.44095	+	0.865368i	0.104082	+	0.0368994i
$551$	4.36120			0.185793
$552$		0			0
$553$			14.3198i			0.608942i
$554$	−0.0613254			−0.00260546
$555$		0			0
$556$	−27.7231			−1.17572
$557$			25.3698i			1.07495i	0.843279	+	0.537476i	$0.180622\pi$
$557$			25.3698i			1.07495i	−0.843279	+	0.537476i	$0.819378\pi$
$558$		0			0
$559$	8.04375			0.340214
$560$	−20.3647	+	14.3869i	−0.860564	+	0.607956i
$561$		0			0
$562$			2.22063i			0.0936718i
$563$		−	35.6622i		−	1.50298i	−0.659742	−	0.751492i	$-0.729335\pi$
$563$		−	35.6622i		−	1.50298i	0.659742	−	0.751492i	$-0.270665\pi$
$564$		0			0
$565$	15.7661	−	11.1382i	0.663285	−	0.468586i
$566$	−1.18446			−0.0497864
$567$		0			0
$568$			0.637236i			0.0267378i
$569$	27.7944			1.16520			0.582602	−	0.812758i	$-0.302035\pi$
$569$	27.7944			1.16520			0.582602	+	0.812758i	$0.302035\pi$
$570$		0			0
$571$	−15.8578			−0.663627			−0.331813	−	0.943345i	$-0.607660\pi$
$571$	−15.8578			−0.663627			−0.331813	+	0.943345i	$0.607660\pi$
$572$		−	47.4325i		−	1.98325i
$573$		0			0
$574$	−0.787700			−0.0328780
$575$	−21.2914	−	7.54824i	−0.887912	−	0.314784i
$576$		0			0
$577$			25.2762i			1.05226i	0.850403	+	0.526131i	$0.176358\pi$
$577$			25.2762i			1.05226i	−0.850403	+	0.526131i	$0.823642\pi$
$578$			0.134632i			0.00559995i
$579$		0			0
$580$	−1.68396	−	2.38366i	−0.0699228	−	0.0989760i
$581$	−22.6804			−0.940940
$582$		0			0
$583$		−	38.1935i		−	1.58181i
$584$	−1.74941			−0.0723911
$585$		0			0
$586$	−3.02952			−0.125148
$587$		−	15.7829i		−	0.651431i	−0.945468	−	0.325716i	$-0.894395\pi$
$587$		−	15.7829i		−	0.651431i	0.945468	−	0.325716i	$-0.105605\pi$
$588$		0			0
$589$	−23.1770			−0.954992
$590$	−0.491756	+	0.347407i	−0.0202453	+	0.0143025i
$591$		0			0
$592$			13.0035i			0.534439i
$593$			19.3560i			0.794855i	0.917634	+	0.397428i	$0.130097\pi$
$593$			19.3560i			0.794855i	−0.917634	+	0.397428i	$0.869903\pi$
$594$		0			0
$595$	−3.69693	−	5.23301i	−0.151559	−	0.214533i
$596$	−35.3543			−1.44817
$597$		0			0
$598$		−	3.78391i		−	0.154735i
$599$	−15.8356			−0.647023			−0.323512	−	0.946224i	$-0.604864\pi$
$599$	−15.8356			−0.647023			−0.323512	+	0.946224i	$0.604864\pi$
$600$		0			0
$601$	9.30443			0.379536			0.189768	−	0.981829i	$-0.439226\pi$
$601$	9.30443			0.379536			0.189768	+	0.981829i	$0.439226\pi$
$602$			0.498812i			0.0203301i
$603$		0			0
$604$	−5.55502			−0.226030
$605$	4.90444	+	6.94225i	0.199394	+	0.282243i
$606$		0			0
$607$			26.6661i			1.08234i	0.840912	+	0.541172i	$0.182019\pi$
$607$			26.6661i			1.08234i	−0.840912	+	0.541172i	$0.817981\pi$
$608$			10.5699i			0.428666i
$609$		0			0
$610$	−1.87269	+	1.32298i	−0.0758230	+	0.0535661i
$611$	−38.6990			−1.56560
$612$		0			0
$613$		−	26.0929i		−	1.05388i	−0.849902	−	0.526941i	$-0.823339\pi$
$613$		−	26.0929i		−	1.05388i	0.849902	−	0.526941i	$-0.176661\pi$
$614$	1.57565			0.0635879
$615$		0			0
$616$	5.90970			0.238109
$617$		−	23.4622i		−	0.944554i	−0.881450	−	0.472277i	$-0.843432\pi$
$617$		−	23.4622i		−	0.944554i	0.881450	−	0.472277i	$-0.156568\pi$
$618$		0			0
$619$	−3.50613			−0.140923			−0.0704617	−	0.997514i	$-0.522447\pi$
$619$	−3.50613			−0.140923			−0.0704617	+	0.997514i	$0.522447\pi$
$620$	8.94920	+	12.6676i	0.359408	+	0.508744i
$621$		0			0
$622$		−	2.83541i		−	0.113690i
$623$			36.6804i			1.46957i
$624$		0			0
$625$	−19.4174	−	15.7469i	−0.776696	−	0.629876i
$626$	−3.00109			−0.119948
$627$		0			0
$628$			7.14628i			0.285168i
$629$	−3.34144			−0.133232
$630$		0			0
$631$	−7.57823			−0.301685			−0.150842	−	0.988558i	$-0.548199\pi$
$631$	−7.57823			−0.301685			−0.150842	+	0.988558i	$0.548199\pi$
$632$		−	2.67913i		−	0.106570i
$633$		0			0
$634$	3.75102			0.148972
$635$	−5.92724	+	4.18737i	−0.235215	+	0.166171i
$636$		0			0
$637$			7.52932i			0.298322i
$638$			0.341110i			0.0135047i
$639$		0			0
$640$	7.69079	−	5.43325i	0.304005	−	0.214768i
$641$	−4.21055			−0.166307			−0.0831535	−	0.996537i	$-0.526499\pi$
$641$	−4.21055			−0.166307			−0.0831535	+	0.996537i	$0.526499\pi$
$642$		0			0
$643$			39.0726i			1.54087i	0.637516	+	0.770437i	$0.279962\pi$
$643$			39.0726i			1.54087i	−0.637516	+	0.770437i	$0.720038\pi$
$644$	−25.6566			−1.01101
$645$		0			0
$646$	−0.891574			−0.0350785
$647$			27.0481i			1.06337i	0.846942	+	0.531685i	$0.178441\pi$
$647$			27.0481i			1.06337i	−0.846942	+	0.531685i	$0.821559\pi$
$648$		0			0
$649$	−7.69448			−0.302035
$650$	1.39927	−	3.94693i	0.0548839	−	0.154811i
$651$		0			0
$652$		−	27.2024i		−	1.06533i
$653$			35.6550i			1.39529i	0.716445	+	0.697643i	$0.245768\pi$
$653$			35.6550i			1.39529i	−0.716445	+	0.697643i	$0.754232\pi$
$654$		0			0
$655$	4.44790	+	6.29602i	0.173794	+	0.246006i
$656$	−7.94617			−0.310246
$657$		0			0
$658$		−	2.39982i		−	0.0935547i
$659$	0.586716			0.0228552			0.0114276	−	0.999935i	$-0.496362\pi$
$659$	0.586716			0.0228552			0.0114276	+	0.999935i	$0.496362\pi$
$660$		0			0
$661$	18.8785			0.734290			0.367145	−	0.930164i	$-0.380335\pi$
$661$	18.8785			0.734290			0.367145	+	0.930164i	$0.380335\pi$
$662$			1.75112i			0.0680592i
$663$		0			0
$664$	4.24331			0.164672
$665$	34.6546	−	24.4822i	1.34385	−	0.949378i
$666$		0			0
$667$		−	2.97536i		−	0.115206i
$668$		−	12.5390i		−	0.485149i
$669$		0			0
$670$	0.0446602	+	0.0632168i	0.00172538	+	0.00244228i
$671$	−29.3019			−1.13119
$672$		0			0
$673$			44.9680i			1.73339i	0.498841	+	0.866694i	$0.333759\pi$
$673$			44.9680i			1.73339i	−0.498841	+	0.866694i	$0.666241\pi$
$674$	−2.03464			−0.0783716
$675$		0			0
$676$	−50.9323			−1.95893
$677$		−	33.3876i		−	1.28319i	−0.767044	−	0.641594i	$-0.778273\pi$
$677$		−	33.3876i		−	1.28319i	0.767044	−	0.641594i	$-0.221727\pi$
$678$		0			0
$679$	11.7155			0.449601
$680$	0.691665	+	0.979054i	0.0265241	+	0.0375450i
$681$		0			0
$682$		−	1.81278i		−	0.0694149i
$683$		−	22.0237i		−	0.842713i	−0.906895	−	0.421357i	$-0.861554\pi$
$683$		−	22.0237i		−	0.842713i	0.906895	−	0.421357i	$-0.138446\pi$
$684$		0			0
$685$	26.8226	−	18.9492i	1.02484	−	0.724010i
$686$	2.23348			0.0852746
$687$		0			0
$688$			5.03192i			0.191840i
$689$	−61.7575			−2.35277
$690$		0			0
$691$	4.51396			0.171719			0.0858595	−	0.996307i	$-0.472636\pi$
$691$	4.51396			0.171719			0.0858595	+	0.996307i	$0.472636\pi$
$692$		−	3.50195i		−	0.133124i
$693$		0			0
$694$	0.483097			0.0183381
$695$	−18.0479	−	25.5468i	−0.684594	−	0.969046i
$696$		0			0
$697$		−	2.04189i		−	0.0773422i
$698$			4.21197i			0.159425i
$699$		0			0
$700$	−26.7620	−	9.48768i	−1.01151	−	0.358601i
$701$	29.3948			1.11023			0.555114	−	0.831774i	$-0.312675\pi$
$701$	29.3948			1.11023			0.555114	+	0.831774i	$0.312675\pi$
$702$		0			0
$703$		−	22.1280i		−	0.834575i
$704$	29.1169			1.09739
$705$		0			0
$706$	−3.29063			−0.123845
$707$		−	35.2070i		−	1.32409i
$708$		0			0
$709$	1.86205			0.0699308			0.0349654	−	0.999389i	$-0.488868\pi$
$709$	1.86205			0.0699308			0.0349654	+	0.999389i	$0.488868\pi$
$710$	−0.292270	+	0.206478i	−0.0109687	+	0.00774897i
$711$		0			0
$712$		−	6.86260i		−	0.257187i
$713$			15.8121i			0.592169i
$714$		0			0
$715$	43.7090	−	30.8788i	1.63462	−	1.15480i
$716$	2.19676			0.0820968
$717$		0			0
$718$			0.0183362i			0.000684299i
$719$	−15.8028			−0.589346			−0.294673	−	0.955598i	$-0.595211\pi$
$719$	−15.8028			−0.589346			−0.294673	+	0.955598i	$0.595211\pi$
$720$		0			0
$721$	38.8610			1.44726
$722$		−	3.34627i		−	0.124535i
$723$		0			0
$724$	−21.4657			−0.797766
$725$	1.10027	−	3.10355i	0.0408631	−	0.115263i
$726$		0			0
$727$			21.0930i			0.782296i	0.920328	+	0.391148i	$0.127922\pi$
$727$			21.0930i			0.782296i	−0.920328	+	0.391148i	$0.872078\pi$
$728$		−	9.55578i		−	0.354161i
$729$		0			0
$730$	−0.566845	−	0.802372i	−0.0209799	−	0.0296971i
$731$	−1.29303			−0.0478244
$732$		0			0
$733$		−	40.8012i		−	1.50703i	−0.657434	−	0.753513i	$-0.728358\pi$
$733$		−	40.8012i		−	1.50703i	0.657434	−	0.753513i	$-0.271642\pi$
$734$	−2.28735			−0.0844275
$735$		0			0
$736$	7.21114			0.265806
$737$			0.989151i			0.0364358i
$738$		0			0
$739$	−13.8161			−0.508234			−0.254117	−	0.967173i	$-0.581785\pi$
$739$	−13.8161			−0.508234			−0.254117	+	0.967173i	$0.581785\pi$
$740$	−12.0943	+	8.54417i	−0.444595	+	0.314090i
$741$		0			0
$742$		−	3.82973i		−	0.140594i
$743$			3.01696i			0.110682i	0.998468	+	0.0553408i	$0.0176245\pi$
$743$			3.01696i			0.110682i	−0.998468	+	0.0553408i	$0.982375\pi$
$744$		0			0
$745$	−23.0158	−	32.5790i	−0.843235	−	1.19360i
$746$	2.47093			0.0904674
$747$		0			0
$748$			7.62475i			0.278788i
$749$	−19.0203			−0.694987
$750$		0			0
$751$	−44.6641			−1.62982			−0.814909	−	0.579589i	$-0.803213\pi$
$751$	−44.6641			−1.62982			−0.814909	+	0.579589i	$0.803213\pi$
$752$		−	24.2089i		−	0.882808i
$753$		0			0
$754$	0.551562			0.0200867
$755$	−3.61634	−	5.11895i	−0.131612	−	0.186298i
$756$		0			0
$757$		−	12.3652i		−	0.449421i	−0.974426	−	0.224710i	$-0.927856\pi$
$757$		−	12.3652i		−	0.449421i	0.974426	−	0.224710i	$-0.0721436\pi$
$758$		−	4.73828i		−	0.172102i
$759$		0			0
$760$	−6.48360	+	4.58042i	−0.235185	+	0.166149i
$761$	−16.7175			−0.606009			−0.303004	−	0.952989i	$-0.597990\pi$
$761$	−16.7175			−0.606009			−0.303004	+	0.952989i	$0.597990\pi$
$762$		0			0
$763$			7.39385i			0.267675i
$764$	−6.32656			−0.228887
$765$		0			0
$766$	1.45722			0.0526515
$767$			12.4417i			0.449244i
$768$		0			0
$769$	−13.4025			−0.483308			−0.241654	−	0.970362i	$-0.577690\pi$
$769$	−13.4025			−0.483308			−0.241654	+	0.970362i	$0.577690\pi$
$770$	1.91486	+	2.71050i	0.0690069	+	0.0976796i
$771$		0			0
$772$		−	14.0992i		−	0.507441i
$773$		−	25.8358i		−	0.929248i	−0.885508	−	0.464624i	$-0.846189\pi$
$773$		−	25.8358i		−	0.929248i	0.885508	−	0.464624i	$-0.153811\pi$
$774$		0			0
$775$	−5.84724	+	16.4934i	−0.210039	+	0.592459i
$776$	−2.19188			−0.0786839
$777$		0			0
$778$		−	0.0234371i		−	0.000840261i
$779$	13.5220			0.484477
$780$		0			0
$781$	−4.57314			−0.163640
$782$			0.608262i			0.0217514i
$783$		0			0
$784$	−4.71011			−0.168218
$785$	−6.58530	+	4.65226i	−0.235039	+	0.166046i
$786$		0			0
$787$		−	45.2989i		−	1.61473i	−0.590051	−	0.807366i	$-0.700892\pi$
$787$		−	45.2989i		−	1.61473i	0.590051	−	0.807366i	$-0.299108\pi$
$788$			45.9242i			1.63598i
$789$		0			0
$790$	1.22879	−	0.868093i	0.0437183	−	0.0308854i
$791$	24.7362			0.879519
$792$		0			0
$793$			47.3802i			1.68252i
$794$	0.370623			0.0131529
$795$		0			0
$796$	48.9782			1.73599
$797$			29.3607i			1.04001i	0.854163	+	0.520005i	$0.174070\pi$
$797$			29.3607i			1.04001i	−0.854163	+	0.520005i	$0.825930\pi$
$798$		0			0
$799$	6.22085			0.220078
$800$	7.52182	+	2.66664i	0.265936	+	0.0942800i
$801$		0			0
$802$			4.57630i			0.161595i
$803$		−	12.5547i		−	0.443045i
$804$		0			0
$805$	−16.7026	−	23.6425i	−0.588688	−	0.833290i
$806$	−2.93120			−0.103247
$807$		0			0
$808$			6.58694i			0.231728i
$809$	5.04830			0.177489			0.0887444	−	0.996054i	$-0.471715\pi$
$809$	5.04830			0.177489			0.0887444	+	0.996054i	$0.471715\pi$
$810$		0			0
$811$	54.3374			1.90805			0.954023	−	0.299734i	$-0.0968981\pi$
$811$	54.3374			1.90805			0.954023	+	0.299734i	$0.0968981\pi$
$812$		−	3.73984i		−	0.131243i
$813$		0			0
$814$	1.73074			0.0606623
$815$	25.0670	−	17.7089i	0.878057	−	0.620314i
$816$		0			0
$817$		−	8.56284i		−	0.299576i
$818$		−	0.630464i		−	0.0220437i
$819$		0			0
$820$	−5.22118	−	7.39061i	−0.182332	−	0.258091i
$821$	−3.05948			−0.106777			−0.0533883	−	0.998574i	$-0.517002\pi$
$821$	−3.05948			−0.106777			−0.0533883	+	0.998574i	$0.517002\pi$
$822$		0			0
$823$			6.87523i			0.239656i	0.992795	+	0.119828i	$0.0382342\pi$
$823$			6.87523i			0.239656i	−0.992795	+	0.119828i	$0.961766\pi$
$824$	−7.27057			−0.253282
$825$		0			0
$826$	−0.771540			−0.0268453
$827$			24.2438i			0.843040i	0.906819	+	0.421520i	$0.138503\pi$
$827$			24.2438i			0.843040i	−0.906819	+	0.421520i	$0.861497\pi$
$828$		0			0
$829$	−15.7734			−0.547832			−0.273916	−	0.961754i	$-0.588319\pi$
$829$	−15.7734			−0.547832			−0.273916	+	0.961754i	$0.588319\pi$
$830$	1.37492	+	1.94621i	0.0477242	+	0.0675538i
$831$		0			0
$832$		−	47.0811i		−	1.63224i
$833$		−	1.21033i		−	0.0419356i
$834$		0			0
$835$	11.5547	−	8.16295i	0.399866	−	0.282491i
$836$	−50.4935			−1.74635
$837$		0			0
$838$			3.91767i			0.135334i
$839$	9.98832			0.344835			0.172418	−	0.985024i	$-0.444842\pi$
$839$	9.98832			0.344835			0.172418	+	0.985024i	$0.444842\pi$
$840$		0			0
$841$	−28.5663			−0.985045
$842$		−	0.968080i		−	0.0333622i
$843$		0			0
$844$	−8.46533			−0.291389
$845$	−33.1571	−	46.9341i	−1.14064	−	1.61458i
$846$		0			0
$847$			10.8920i			0.374255i
$848$		−	38.6336i		−	1.32668i
$849$		0			0
$850$	−0.224932	+	0.634468i	−0.00771511	+	0.0217621i
$851$	−15.0965			−0.517501
$852$		0			0
$853$			31.1852i			1.06776i	0.845560	+	0.533880i	$0.179266\pi$
$853$			31.1852i			1.06776i	−0.845560	+	0.533880i	$0.820734\pi$
$854$	−2.93816			−0.100542
$855$		0			0
$856$	3.55854			0.121629
$857$		−	26.4712i		−	0.904240i	−0.891957	−	0.452120i	$-0.850668\pi$
$857$		−	26.4712i		−	0.904240i	0.891957	−	0.452120i	$-0.149332\pi$
$858$		0			0
$859$	−22.7223			−0.775273			−0.387637	−	0.921812i	$-0.626708\pi$
$859$	−22.7223			−0.775273			−0.387637	+	0.921812i	$0.626708\pi$
$860$	−4.68011	+	3.30632i	−0.159590	+	0.112745i
$861$		0			0
$862$			1.02748i			0.0349963i
$863$			35.3962i			1.20490i	0.798156	+	0.602451i	$0.205809\pi$
$863$			35.3962i			1.20490i	−0.798156	+	0.602451i	$0.794191\pi$
$864$		0			0
$865$	3.22704	−	2.27978i	0.109723	−	0.0775150i
$866$	4.34719			0.147723
$867$		0			0
$868$			19.8749i			0.674597i
$869$	19.2268			0.652225
$870$		0			0
$871$	1.59942			0.0541944
$872$		−	1.38333i		−	0.0468455i
$873$		0			0
$874$	−4.02810			−0.136252
$875$	−8.67927	−	30.8377i	−0.293413	−	1.04250i
$876$		0			0
$877$			1.09791i			0.0370736i	0.999828	+	0.0185368i	$0.00590079\pi$
$877$			1.09791i			0.0370736i	−0.999828	+	0.0185368i	$0.994099\pi$
$878$		−	1.80371i		−	0.0608724i
$879$		0			0
$880$	19.3168	+	27.3430i	0.651169	+	0.921732i
$881$	16.5399			0.557245			0.278622	−	0.960401i	$-0.410122\pi$
$881$	16.5399			0.557245			0.278622	+	0.960401i	$0.410122\pi$
$882$		0			0
$883$		−	22.1505i		−	0.745425i	−0.927947	−	0.372712i	$-0.878428\pi$
$883$		−	22.1505i		−	0.745425i	0.927947	−	0.372712i	$-0.121572\pi$
$884$	12.3290			0.414668
$885$		0			0
$886$	0.927707			0.0311669
$887$			19.5836i			0.657553i	0.944408	+	0.328776i	$0.106636\pi$
$887$			19.5836i			0.657553i	−0.944408	+	0.328776i	$0.893364\pi$
$888$		0			0
$889$	−9.29955			−0.311897
$890$	3.14755	−	2.22362i	0.105506	−	0.0745360i
$891$		0			0
$892$		−	7.24222i		−	0.242488i
$893$			41.1964i			1.37859i
$894$		0			0
$895$	1.43010	+	2.02431i	0.0478030	+	0.0676654i
$896$	12.0665			0.403112
$897$		0			0
$898$		−	2.85120i		−	0.0951458i
$899$	−2.30486			−0.0768713
$900$		0			0
$901$	9.92750			0.330733
$902$			1.05762i			0.0352149i
$903$		0			0
$904$	−4.62795			−0.153923
$905$	−13.9743	−	19.7806i	−0.464520	−	0.657531i
$906$		0			0
$907$			38.3180i			1.27233i	0.771553	+	0.636165i	$0.219480\pi$
$907$			38.3180i			1.27233i	−0.771553	+	0.636165i	$0.780520\pi$
$908$		−	37.6587i		−	1.24975i
$909$		0			0
$910$	4.38278	−	3.09627i	0.145288	−	0.102640i
$911$	37.8395			1.25368			0.626840	−	0.779148i	$-0.284348\pi$
$911$	37.8395			1.25368			0.626840	+	0.779148i	$0.284348\pi$
$912$		0			0
$913$			30.4522i			1.00782i
$914$	−4.07021			−0.134631
$915$		0			0
$916$	41.3916			1.36762
$917$			9.87814i			0.326205i
$918$		0			0
$919$	28.8996			0.953309			0.476655	−	0.879091i	$-0.341849\pi$
$919$	28.8996			0.953309			0.476655	+	0.879091i	$0.341849\pi$
$920$	3.12491	+	4.42333i	0.103025	+	0.145833i
$921$		0			0
$922$			0.503883i			0.0165945i
$923$			7.39461i			0.243397i
$924$		0			0
$925$	−15.7469	−	5.58260i	−0.517755	−	0.183555i
$926$	−0.341228			−0.0112134
$927$		0			0
$928$			1.05113i			0.0345051i
$929$	−47.2978			−1.55179			−0.775895	−	0.630862i	$-0.782701\pi$
$929$	−47.2978			−1.55179			−0.775895	+	0.630862i	$0.782701\pi$
$930$		0			0
$931$	8.01521			0.262688
$932$		−	1.35338i		−	0.0443313i
$933$		0			0
$934$	4.05471			0.132674
$935$	−7.02621	+	4.96375i	−0.229782	+	0.162332i
$936$		0			0
$937$		−	1.24572i		−	0.0406958i	−0.999793	−	0.0203479i	$-0.993523\pi$
$937$		−	1.24572i		−	0.0406958i	0.999793	−	0.0203479i	$-0.00647739\pi$
$938$			0.0991839i			0.00323847i
$939$		0			0
$940$	22.5163	−	15.9069i	0.734402	−	0.518827i
$941$	−37.9458			−1.23700			−0.618499	−	0.785785i	$-0.712259\pi$
$941$	−37.9458			−1.23700			−0.618499	+	0.785785i	$0.712259\pi$
$942$		0			0
$943$		−	9.22519i		−	0.300413i
$944$	−7.78315			−0.253320
$945$		0			0
$946$	0.669739			0.0217751
$947$			52.2986i			1.69948i	0.527205	+	0.849738i	$0.323240\pi$
$947$			52.2986i			1.69948i	−0.527205	+	0.849738i	$0.676760\pi$
$948$		0			0
$949$	−20.3005			−0.658982
$950$	−4.20164	−	1.48957i	−0.136319	−	0.0483280i
$951$		0			0
$952$			1.53609i			0.0497849i
$953$			0.806914i			0.0261385i	0.999915	+	0.0130693i	$0.00416019\pi$
$953$			0.806914i			0.0261385i	−0.999915	+	0.0130693i	$0.995840\pi$
$954$		0			0
$955$	−4.11862	−	5.82992i	−0.133275	−	0.188652i
$956$	57.4575			1.85831
$957$		0			0
$958$		−	1.95210i		−	0.0630694i
$959$	42.0833			1.35894
$960$		0			0
$961$	−18.7511			−0.604876
$962$		−	2.79854i		−	0.0902285i
$963$		0			0
$964$	−35.8606			−1.15499
$965$	12.9924	−	9.17864i	0.418240	−	0.295471i
$966$		0			0
$967$			27.3892i			0.880777i	0.897807	+	0.440388i	$0.145159\pi$
$967$			27.3892i			0.880777i	−0.897807	+	0.440388i	$0.854841\pi$
$968$		−	2.03781i		−	0.0654978i
$969$		0			0
$970$	−0.710214	−	1.00531i	−0.0228036	−	0.0322786i
$971$	−26.7596			−0.858757			−0.429378	−	0.903125i	$-0.641267\pi$
$971$	−26.7596			−0.858757			−0.429378	+	0.903125i	$0.641267\pi$
$972$		0			0
$973$		−	40.0817i		−	1.28496i
$974$	3.22341			0.103285
$975$		0			0
$976$	−29.6396			−0.948739
$977$		−	2.64662i		−	0.0846730i	−0.999103	−	0.0423365i	$-0.986520\pi$
$977$		−	2.64662i		−	0.0846730i	0.999103	−	0.0423365i	$-0.0134802\pi$
$978$		0			0
$979$	49.2496			1.57402
$980$	−3.09487	−	4.38079i	−0.0988618	−	0.139939i
$981$		0			0
$982$		−	2.84470i		−	0.0907780i
$983$			33.0275i			1.05341i	0.850047	+	0.526707i	$0.176573\pi$
$983$			33.0275i			1.05341i	−0.850047	+	0.526707i	$0.823427\pi$
$984$		0			0
$985$	−42.3192	+	29.8969i	−1.34840	+	0.952594i
$986$	−0.0886634			−0.00282362
$987$		0			0
$988$			81.6461i			2.59751i
$989$	−5.84186			−0.185760
$990$		0			0
$991$	−32.2132			−1.02329			−0.511643	−	0.859198i	$-0.670963\pi$
$991$	−32.2132			−1.02329			−0.511643	+	0.859198i	$0.670963\pi$
$992$		−	5.58610i		−	0.177359i
$993$		0			0
$994$	−0.458557			−0.0145446
$995$	31.8851	+	45.1334i	1.01082	+	1.43083i
$996$		0			0
$997$			52.7919i			1.67194i	0.548777	+	0.835969i	$0.315093\pi$
$997$			52.7919i			1.67194i	−0.548777	+	0.835969i	$0.684907\pi$
$998$			0.539910i			0.0170905i
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	765.2.b.c.154.4		8
3.2	odd	2		85.2.b.a.69.5	yes	8
5.2	odd	4		3825.2.a.bj.1.2		4
5.3	odd	4		3825.2.a.bh.1.3		4
5.4	even	2	inner	765.2.b.c.154.5		8
12.11	even	2		1360.2.e.d.1089.3		8
15.2	even	4		425.2.a.g.1.3		4
15.8	even	4		425.2.a.h.1.2		4
15.14	odd	2		85.2.b.a.69.4	✓	8
51.50	odd	2		1445.2.b.e.579.5		8
60.23	odd	4		6800.2.a.bt.1.4		4
60.47	odd	4		6800.2.a.bw.1.1		4
60.59	even	2		1360.2.e.d.1089.6		8
255.152	even	4		7225.2.a.v.1.3		4
255.203	even	4		7225.2.a.w.1.2		4
255.254	odd	2		1445.2.b.e.579.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.2.b.a.69.4	✓	8	15.14	odd	2
85.2.b.a.69.5	yes	8	3.2	odd	2
425.2.a.g.1.3		4	15.2	even	4
425.2.a.h.1.2		4	15.8	even	4
765.2.b.c.154.4		8	1.1	even	1	trivial
765.2.b.c.154.5		8	5.4	even	2	inner
1360.2.e.d.1089.3		8	12.11	even	2
1360.2.e.d.1089.6		8	60.59	even	2
1445.2.b.e.579.4		8	255.254	odd	2
1445.2.b.e.579.5		8	51.50	odd	2
3825.2.a.bh.1.3		4	5.3	odd	4
3825.2.a.bj.1.2		4	5.2	odd	4
6800.2.a.bt.1.4		4	60.23	odd	4
6800.2.a.bw.1.1		4	60.47	odd	4
7225.2.a.v.1.3		4	255.152	even	4
7225.2.a.w.1.2		4	255.203	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 $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	765.2.b.c.154.4

Level	$765$
Weight	$2$
Character	765.154
Analytic conductor	$6.109$
Analytic rank	$0$
Dimension	$8$
Inner twists	$2$