LMFDB - Embedded newform 1120.2.w.a.433.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1120,2,Mod(433,1120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1120, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 2, 3, 2]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1120.433");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1120 = 2^{5} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1120.w (of order $4$, degree $2$, 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:	$8.94324502638$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.40282095616.8
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{6} + 8x^{4} - 36x^{2} + 81 $ x^8 - 4x^6 + 8x^4 - 36*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 280)
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			433.2
Root			$-1.03179 - 1.39119i$ of defining polynomial
Character	$\chi$	$=$	1120.433
Dual form			1120.2.w.a.657.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.359404 - 0.359404i) q^{3} +(-1.39119 - 1.75060i) q^{5} +(1.87083 - 1.87083i) q^{7} -2.74166i q^{9} +O(q^{10})$	$q+(-0.359404 - 0.359404i) q^{3} +(-1.39119 - 1.75060i) q^{5} +(1.87083 - 1.87083i) q^{7} -2.74166i q^{9} +(-4.48655 - 4.48655i) q^{13} +(-0.129171 + 1.12917i) q^{15} +7.62834i q^{19} -1.34477 q^{21} +(-0.741657 - 0.741657i) q^{23} +(-1.12917 + 4.87083i) q^{25} +(-2.06358 + 2.06358i) q^{27} +(-5.87775 - 0.672384i) q^{35} +3.22497i q^{39} +(-4.79953 + 3.81417i) q^{45} -7.00000i q^{49} +(2.74166 - 2.74166i) q^{57} -15.3495i q^{59} -14.6307 q^{61} +(-5.12917 - 5.12917i) q^{63} +(-1.61249 + 14.0958i) q^{65} +0.533109i q^{69} -7.22497 q^{71} +(2.15642 - 1.34477i) q^{75} +15.7417i q^{79} -6.74166 q^{81} +(-5.83132 - 5.83132i) q^{83} -16.7871 q^{91} +(13.3541 - 10.6125i) q^{95} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 16 q^{15} + 24 q^{23} - 24 q^{25} - 8 q^{57} - 56 q^{63} + 32 q^{65} + 32 q^{71} - 24 q^{81} + 32 q^{95}+O(q^{100}) $ 8 * q - 16 * q^15 + 24 * q^23 - 24 * q^25 - 8 * q^57 - 56 * q^63 + 32 * q^65 + 32 * q^71 - 24 * q^81 + 32 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1120\mathbb{Z}\right)^\times$.

$n$	$351$	$421$	$801$	$897$
$\chi(n)$	$1$	$-1$	$-1$	$e\left(\frac{3}{4}\right)$

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
$3$	−0.359404	−	0.359404i	−0.207502	−	0.207502i	0.595703	−	0.803205i	$-0.296874\pi$
$3$	−0.359404	−	0.359404i	−0.207502	−	0.207502i	−0.803205	+	0.595703i	$0.796874\pi$
$4$		0			0
$5$	−1.39119	−	1.75060i	−0.622160	−	0.782890i
$5$	−1.39119	−	1.75060i	−0.622160	−	0.782890i
$6$		0			0
$7$	1.87083	−	1.87083i	0.707107	−	0.707107i
$7$	1.87083	−	1.87083i	0.707107	−	0.707107i
$8$		0			0
$9$		−	2.74166i		−	0.913886i
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$	−4.48655	−	4.48655i	−1.24435	−	1.24435i	−0.958180	−	0.286166i	$-0.907619\pi$
$13$	−4.48655	−	4.48655i	−1.24435	−	1.24435i	−0.286166	−	0.958180i	$-0.592381\pi$
$14$		0			0
$15$	−0.129171	+	1.12917i	−0.0333519	+	0.291551i
$16$		0			0
$17$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$17$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$18$		0			0
$19$			7.62834i			1.75006i	0.484067	+	0.875031i	$0.339159\pi$
$19$			7.62834i			1.75006i	−0.484067	+	0.875031i	$0.660841\pi$
$20$		0			0
$21$	−1.34477			−0.293452
$22$		0			0
$23$	−0.741657	−	0.741657i	−0.154646	−	0.154646i	0.625543	−	0.780189i	$-0.284877\pi$
$23$	−0.741657	−	0.741657i	−0.154646	−	0.154646i	−0.780189	+	0.625543i	$0.784877\pi$
$24$		0			0
$25$	−1.12917	+	4.87083i	−0.225834	+	0.974166i
$26$		0			0
$27$	−2.06358	+	2.06358i	−0.397135	+	0.397135i
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−5.87775	−	0.672384i	−0.993520	−	0.113654i
$36$		0			0
$37$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$37$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$38$		0			0
$39$			3.22497i			0.516409i
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$43$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$44$		0			0
$45$	−4.79953	+	3.81417i	−0.715472	+	0.568583i
$46$		0			0
$47$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$47$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$48$		0			0
$49$		−	7.00000i		−	1.00000i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$53$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	2.74166	−	2.74166i	0.363141	−	0.363141i
$58$		0			0
$59$		−	15.3495i		−	1.99834i	−0.0407464	−	0.999170i	$-0.512974\pi$
$59$		−	15.3495i		−	1.99834i	0.0407464	−	0.999170i	$-0.487026\pi$
$60$		0			0
$61$	−14.6307			−1.87327			−0.936636	−	0.350304i	$-0.886078\pi$
$61$	−14.6307			−1.87327			−0.936636	+	0.350304i	$0.886078\pi$
$62$		0			0
$63$	−5.12917	−	5.12917i	−0.646215	−	0.646215i
$64$		0			0
$65$	−1.61249	+	14.0958i	−0.200004	+	1.74837i
$66$		0			0
$67$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$67$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$68$		0			0
$69$			0.533109i			0.0641788i
$70$		0			0
$71$	−7.22497			−0.857446			−0.428723	−	0.903436i	$-0.641036\pi$
$71$	−7.22497			−0.857446			−0.428723	+	0.903436i	$0.641036\pi$
$72$		0			0
$73$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$73$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$74$		0			0
$75$	2.15642	−	1.34477i	0.249002	−	0.155280i
$76$		0			0
$77$		0			0
$78$		0			0
$79$			15.7417i			1.77107i	0.464568	+	0.885537i	$0.346210\pi$
$79$			15.7417i			1.77107i	−0.464568	+	0.885537i	$0.653790\pi$
$80$		0			0
$81$	−6.74166			−0.749073
$82$		0			0
$83$	−5.83132	−	5.83132i	−0.640071	−	0.640071i	0.310502	−	0.950573i	$-0.399503\pi$
$83$	−5.83132	−	5.83132i	−0.640071	−	0.640071i	−0.950573	+	0.310502i	$0.899503\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$	−16.7871			−1.75977
$92$		0			0
$93$		0			0
$94$		0			0
$95$	13.3541	−	10.6125i	1.37011	−	1.08882i
$96$		0			0
$97$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$97$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	13.1931			1.31276			0.656382	−	0.754429i	$-0.272086\pi$
$101$	13.1931			1.31276			0.656382	+	0.754429i	$0.272086\pi$
$102$		0			0
$103$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$103$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$104$		0			0
$105$	1.87083	+	2.35414i	0.182574	+	0.229741i
$106$		0			0
$107$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$107$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$108$		0			0
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−11.2250	−	11.2250i	−1.05596	−	1.05596i	−0.998339	−	0.0576178i	$-0.981650\pi$
$113$	−11.2250	−	11.2250i	−1.05596	−	1.05596i	−0.0576178	−	0.998339i	$-0.518350\pi$
$114$		0			0
$115$	−0.266555	+	2.33013i	−0.0248564	+	0.217286i
$116$		0			0
$117$	−12.3006	+	12.3006i	−1.13719	+	1.13719i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	11.0000			1.00000
$122$		0			0
$123$		0			0
$124$		0			0
$125$	10.0977	−	4.79953i	0.903170	−	0.429283i
$126$		0			0
$127$	12.2250	−	12.2250i	1.08479	−	1.08479i	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	12.2250	−	12.2250i	1.08479	−	1.08479i	0.996055	−	0.0887357i	$-0.0282826\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	4.93881			0.431506			0.215753	−	0.976448i	$-0.430779\pi$
$131$	4.93881			0.431506			0.215753	+	0.976448i	$0.430779\pi$
$132$		0			0
$133$	14.2713	+	14.2713i	1.23748	+	1.23748i
$134$		0			0
$135$	6.48331	+	0.741657i	0.557995	+	0.0638317i
$136$		0			0
$137$	16.4833	−	16.4833i	1.40826	−	1.40826i	0.639343	−	0.768922i	$-0.279207\pi$
$137$	16.4833	−	16.4833i	1.40826	−	1.40826i	0.768922	−	0.639343i	$-0.220793\pi$
$138$		0			0
$139$			12.4743i			1.05806i	0.848604	+	0.529028i	$0.177443\pi$
$139$			12.4743i			1.05806i	−0.848604	+	0.529028i	$0.822557\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−2.51583	+	2.51583i	−0.207502	+	0.207502i
$148$		0			0
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		0			0
$151$	−22.4499			−1.82695			−0.913475	−	0.406894i	$-0.866612\pi$
$151$	−22.4499			−1.82695			−0.913475	+	0.406894i	$0.866612\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	16.4277	−	16.4277i	1.31108	−	1.31108i	0.390455	−	0.920622i	$-0.372318\pi$
$157$	16.4277	−	16.4277i	1.31108	−	1.31108i	0.920622	−	0.390455i	$-0.127682\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−2.77503			−0.218703
$162$		0			0
$163$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$163$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$167$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$168$		0			0
$169$			27.2583i			2.09680i
$170$		0			0
$171$	20.9143			1.59936
$172$		0			0
$173$	−13.5525	−	13.5525i	−1.03038	−	1.03038i	−0.999524	−	0.0308546i	$-0.990177\pi$
$173$	−13.5525	−	13.5525i	−1.03038	−	1.03038i	−0.0308546	−	0.999524i	$-0.509823\pi$
$174$		0			0
$175$	7.00000	+	11.2250i	0.529150	+	0.848528i
$176$		0			0
$177$	−5.51669	+	5.51669i	−0.414659	+	0.414659i
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	19.2910			1.43389			0.716944	−	0.697131i	$-0.245540\pi$
$181$	19.2910			1.43389			0.716944	+	0.697131i	$0.245540\pi$
$182$		0			0
$183$	5.25834	+	5.25834i	0.388708	+	0.388708i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$			7.72119i			0.561634i
$190$		0			0
$191$	−27.2250			−1.96993			−0.984965	−	0.172754i	$-0.944733\pi$
$191$	−27.2250			−1.96993			−0.984965	+	0.172754i	$0.944733\pi$
$192$		0			0
$193$	−6.00000	−	6.00000i	−0.431889	−	0.431889i	0.457381	−	0.889271i	$-0.348787\pi$
$193$	−6.00000	−	6.00000i	−0.431889	−	0.431889i	−0.889271	+	0.457381i	$0.848787\pi$
$194$		0			0
$195$	5.64562	−	4.48655i	0.404291	−	0.321289i
$196$		0			0
$197$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$197$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−2.03337	+	2.03337i	−0.141329	+	0.141329i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$	2.59668	+	2.59668i	0.177922	+	0.177922i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$223$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$224$		0			0
$225$	13.3541	+	3.09580i	0.890276	+	0.206387i
$226$		0			0
$227$	21.0880	−	21.0880i	1.39966	−	1.39966i	0.598647	−	0.801013i	$-0.295705\pi$
$227$	21.0880	−	21.0880i	1.39966	−	1.39966i	0.801013	−	0.598647i	$-0.204295\pi$
$228$		0			0
$229$			18.9436i			1.25183i	0.779893	+	0.625913i	$0.215274\pi$
$229$			18.9436i			1.25183i	−0.779893	+	0.625913i	$0.784726\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−17.9666	−	17.9666i	−1.17703	−	1.17703i	−0.980497	−	0.196537i	$-0.937031\pi$
$233$	−17.9666	−	17.9666i	−1.17703	−	1.17703i	−0.196537	−	0.980497i	$-0.562969\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	5.65762	−	5.65762i	0.367502	−	0.367502i
$238$		0			0
$239$			7.48331i			0.484055i	0.970269	+	0.242028i	$0.0778125\pi$
$239$			7.48331i			0.484055i	−0.970269	+	0.242028i	$0.922188\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$	8.61370	+	8.61370i	0.552569	+	0.552569i
$244$		0			0
$245$	−12.2542	+	9.73834i	−0.782890	+	0.622160i
$246$		0			0
$247$	34.2250	−	34.2250i	2.17768	−	2.17768i
$248$		0			0
$249$			4.19160i			0.265632i
$250$		0			0
$251$	11.0367			0.696629			0.348315	−	0.937378i	$-0.386754\pi$
$251$	11.0367			0.696629			0.348315	+	0.937378i	$0.386754\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$257$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	11.2250	+	11.2250i	0.692161	+	0.692161i	0.962707	−	0.270546i	$-0.0872041\pi$
$263$	11.2250	+	11.2250i	0.692161	+	0.692161i	−0.270546	+	0.962707i	$0.587204\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	31.8581i		−	1.94242i	−0.238215	−	0.971212i	$-0.576562\pi$
$269$		−	31.8581i		−	1.94242i	0.238215	−	0.971212i	$-0.423438\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$	6.03337	+	6.03337i	0.365156	+	0.365156i
$274$		0			0
$275$		0			0
$276$		0			0
$277$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$277$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−14.9666			−0.892834			−0.446417	−	0.894825i	$-0.647300\pi$
$281$	−14.9666			−0.892834			−0.446417	+	0.894825i	$0.647300\pi$
$282$		0			0
$283$	18.3985	+	18.3985i	1.09368	+	1.09368i	0.995133	+	0.0985428i	$0.0314181\pi$
$283$	18.3985	+	18.3985i	1.09368	+	1.09368i	0.0985428	+	0.995133i	$0.468582\pi$
$284$		0			0
$285$	−8.61370	−	0.985363i	−0.510232	−	0.0583679i
$286$		0			0
$287$		0			0
$288$		0			0
$289$		−	17.0000i		−	1.00000i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	17.8654	+	17.8654i	1.04371	+	1.04371i	0.999000	+	0.0447054i	$0.0142349\pi$
$293$	17.8654	+	17.8654i	1.04371	+	1.04371i	0.0447054	+	0.999000i	$0.485765\pi$
$294$		0			0
$295$	−26.8708	+	21.3541i	−1.56448	+	1.24329i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			6.65497i			0.384867i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−4.74166	−	4.74166i	−0.272401	−	0.272401i
$304$		0			0
$305$	20.3541	+	25.6125i	1.16547	+	1.46657i
$306$		0			0
$307$	0.452253	−	0.452253i	0.0258115	−	0.0258115i	−0.694083	−	0.719895i	$-0.744190\pi$
$307$	0.452253	−	0.452253i	0.0258115	−	0.0258115i	0.719895	+	0.694083i	$0.244190\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$313$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$314$		0			0
$315$	−1.84345	+	16.1148i	−0.103866	+	0.907964i
$316$		0			0
$317$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$317$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	26.9193	−	16.7871i	1.49322	−	0.931184i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	18.0000	−	18.0000i	0.980522	−	0.980522i	−0.0192914	−	0.999814i	$-0.506141\pi$
$337$	18.0000	−	18.0000i	0.980522	−	0.980522i	0.999814	+	0.0192914i	$0.00614103\pi$
$338$		0			0
$339$			8.06860i			0.438226i
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−13.0958	−	13.0958i	−0.707107	−	0.707107i
$344$		0			0
$345$	0.933259	−	0.741657i	0.0502450	−	0.0399295i
$346$		0			0
$347$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$347$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$348$		0			0
$349$		−	11.2224i		−	0.600720i	−0.953826	−	0.300360i	$-0.902893\pi$
$349$		−	11.2224i		−	0.600720i	0.953826	−	0.300360i	$-0.0971069\pi$
$350$		0			0
$351$	18.5167			0.988348
$352$		0			0
$353$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$353$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$354$		0			0
$355$	10.0513	+	12.6480i	0.533469	+	0.671286i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	6.00000i		−	0.316668i	−0.987386	−	0.158334i	$-0.949388\pi$
$359$		−	6.00000i		−	0.316668i	0.987386	−	0.158334i	$-0.0506123\pi$
$360$		0			0
$361$	−39.1916			−2.06272
$362$		0			0
$363$	−3.95345	−	3.95345i	−0.207502	−	0.207502i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$367$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$373$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$374$		0			0
$375$	−5.35414	−	1.90420i	−0.276487	−	0.0983324i
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$	−8.78741			−0.450193
$382$		0			0
$383$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$383$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	−1.77503	−	1.77503i	−0.0895383	−	0.0895383i
$394$		0			0
$395$	27.5573	−	21.8997i	1.38656	−	1.10189i
$396$		0			0
$397$	−18.5842	+	18.5842i	−0.932713	+	0.932713i	−0.997875	−	0.0651619i	$-0.979244\pi$
$397$	−18.5842	+	18.5842i	−0.932713	+	0.932713i	0.0651619	+	0.997875i	$0.479244\pi$
$398$		0			0
$399$		−	10.2583i		−	0.513559i
$400$		0			0
$401$	37.2250			1.85893			0.929463	−	0.368915i	$-0.120271\pi$
$401$	37.2250			1.85893			0.929463	+	0.368915i	$0.120271\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	9.37894	+	11.8019i	0.466043	+	0.586442i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0			−	1.00000i	$-0.5\pi$
$409$		0			0				1.00000i	$0.5\pi$
$410$		0			0
$411$	−11.8483			−0.584436
$412$		0			0
$413$	−28.7163	−	28.7163i	−1.41304	−	1.41304i
$414$		0			0
$415$	−2.09580	+	18.3208i	−0.102879	+	0.899331i
$416$		0			0
$417$	4.48331	−	4.48331i	0.219549	−	0.219549i
$418$		0			0
$419$		−	40.8312i		−	1.99474i	−0.0725002	−	0.997368i	$-0.523098\pi$
$419$		−	40.8312i		−	1.99474i	0.0725002	−	0.997368i	$-0.476902\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−27.3716	+	27.3716i	−1.32460	+	1.32460i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	18.0000			0.867029			0.433515	−	0.901146i	$-0.357273\pi$
$431$	18.0000			0.867029			0.433515	+	0.901146i	$0.357273\pi$
$432$		0			0
$433$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$433$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	5.65762	−	5.65762i	0.270640	−	0.270640i
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	−19.1916			−0.913886
$442$		0			0
$443$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$443$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$			29.9333i			1.41264i	0.707894	+	0.706319i	$0.249646\pi$
$449$			29.9333i			1.41264i	−0.707894	+	0.706319i	$0.750354\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	8.06860	+	8.06860i	0.379096	+	0.379096i
$454$		0			0
$455$	23.3541	+	29.3875i	1.09486	+	1.37771i
$456$		0			0
$457$	−3.74166	+	3.74166i	−0.175027	+	0.175027i	−0.789184	−	0.614157i	$-0.789496\pi$
$457$	−3.74166	+	3.74166i	−0.175027	+	0.175027i	0.614157	+	0.789184i	$0.289496\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−21.9805			−1.02373			−0.511867	−	0.859064i	$-0.671046\pi$
$461$	−21.9805			−1.02373			−0.511867	+	0.859064i	$0.671046\pi$
$462$		0			0
$463$	−24.0000	−	24.0000i	−1.11537	−	1.11537i	−0.992411	−	0.122963i	$-0.960760\pi$
$463$	−24.0000	−	24.0000i	−1.11537	−	1.11537i	−0.122963	−	0.992411i	$-0.539240\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−30.3397	+	30.3397i	−1.40395	+	1.40395i	−0.616949	+	0.787003i	$0.711632\pi$
$467$	−30.3397	+	30.3397i	−1.40395	+	1.40395i	−0.787003	+	0.616949i	$0.788368\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−11.8084			−0.544102
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−37.1563	−	8.61370i	−1.70485	−	0.395224i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$	0.997356	+	0.997356i	0.0453813	+	0.0453813i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−7.77503	+	7.77503i	−0.352320	+	0.352320i	−0.860972	−	0.508652i	$-0.830144\pi$
$487$	−7.77503	+	7.77503i	−0.352320	+	0.352320i	0.508652	+	0.860972i	$0.330144\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−13.5167	+	13.5167i	−0.606306	+	0.606306i
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$503$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$504$		0			0
$505$	−18.3541	−	23.0958i	−0.816749	−	1.02775i
$506$		0			0
$507$	9.79676	−	9.79676i	0.435089	−	0.435089i
$508$		0			0
$509$		−	8.88026i		−	0.393611i	−0.980443	−	0.196805i	$-0.936943\pi$
$509$		−	8.88026i		−	0.393611i	0.980443	−	0.196805i	$-0.0630567\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−15.7417	−	15.7417i	−0.695011	−	0.695011i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$			9.74166i			0.427611i
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$	−31.7773	−	31.7773i	−1.38952	−	1.38952i	−0.826315	−	0.563209i	$-0.809567\pi$
$523$	−31.7773	−	31.7773i	−1.38952	−	1.38952i	−0.563209	−	0.826315i	$-0.690433\pi$
$524$		0			0
$525$	1.51847	−	6.55013i	0.0662716	−	0.285871i
$526$		0			0
$527$		0			0
$528$		0			0
$529$		−	21.8999i		−	0.952169i
$530$		0			0
$531$	−42.0832			−1.82625
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$	−6.93326	−	6.93326i	−0.297535	−	0.297535i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$547$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$548$		0			0
$549$			40.1124i			1.71196i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	29.4499	+	29.4499i	1.25234	+	1.25234i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$557$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−28.1832	−	28.1832i	−1.18778	−	1.18778i	−0.977679	−	0.210103i	$-0.932620\pi$
$563$	−28.1832	−	28.1832i	−1.18778	−	1.18778i	−0.210103	−	0.977679i	$-0.567380\pi$
$564$		0			0
$565$	−4.03430	+	35.2665i	−0.169724	+	1.48367i
$566$		0			0
$567$	−12.6125	+	12.6125i	−0.529675	+	0.529675i
$568$		0			0
$569$			31.6749i			1.32788i	0.747785	+	0.663941i	$0.231117\pi$
$569$			31.6749i			1.32788i	−0.747785	+	0.663941i	$0.768883\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$	9.78477	+	9.78477i	0.408764	+	0.408764i
$574$		0			0
$575$	4.44994	−	2.77503i	0.185576	−	0.115727i
$576$		0			0
$577$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$577$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$578$		0			0
$579$			4.31285i			0.179236i
$580$		0			0
$581$	−21.8188			−0.905197
$582$		0			0
$583$		0			0
$584$		0			0
$585$	38.6459	+	4.42088i	1.59781	+	0.182781i
$586$		0			0
$587$	32.4961	−	32.4961i	1.34126	−	1.34126i	0.446447	−	0.894810i	$-0.352689\pi$
$587$	32.4961	−	32.4961i	1.34126	−	1.34126i	0.894810	−	0.446447i	$-0.147311\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$593$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	41.6749i		−	1.70279i	−0.524524	−	0.851395i	$-0.675757\pi$
$599$		−	41.6749i		−	1.70279i	0.524524	−	0.851395i	$-0.324243\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−15.3031	−	19.2566i	−0.622160	−	0.782890i
$606$		0			0
$607$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$607$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$613$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	33.6749	−	33.6749i	1.35570	−	1.35570i	0.476558	−	0.879143i	$-0.341884\pi$
$617$	33.6749	−	33.6749i	1.35570	−	1.35570i	0.879143	−	0.476558i	$-0.158116\pi$
$618$		0			0
$619$		−	8.16145i		−	0.328036i	−0.986457	−	0.164018i	$-0.947554\pi$
$619$		−	8.16145i		−	0.328036i	0.986457	−	0.164018i	$-0.0524456\pi$
$620$		0			0
$621$	3.06093			0.122831
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−22.4499	−	11.0000i	−0.897998	−	0.440000i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	50.1916			1.99810			0.999048	−	0.0436231i	$-0.0138901\pi$
$631$	50.1916			1.99810			0.999048	+	0.0436231i	$0.0138901\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−38.4083	−	4.39371i	−1.52419	−	0.174359i
$636$		0			0
$637$	−31.4059	+	31.4059i	−1.24435	+	1.24435i
$638$		0			0
$639$			19.8084i			0.783608i
$640$		0			0
$641$	−22.7750			−0.899560			−0.449780	−	0.893140i	$-0.648498\pi$
$641$	−22.7750			−0.899560			−0.449780	+	0.893140i	$0.648498\pi$
$642$		0			0
$643$	27.4644	+	27.4644i	1.08309	+	1.08309i	0.996219	+	0.0868719i	$0.0276871\pi$
$643$	27.4644	+	27.4644i	1.08309	+	1.08309i	0.0868719	+	0.996219i	$0.472313\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$647$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$653$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$654$		0			0
$655$	−6.87083	−	8.64586i	−0.268465	−	0.337822i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	26.4791			1.02992			0.514958	−	0.857215i	$-0.327807\pi$
$661$	26.4791			1.02992			0.514958	+	0.857215i	$0.327807\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	5.12917	−	44.8375i	0.198901	−	1.73872i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	9.44994	+	9.44994i	0.364269	+	0.364269i	0.865382	−	0.501113i	$-0.167076\pi$
$673$	9.44994	+	9.44994i	0.364269	+	0.364269i	−0.501113	+	0.865382i	$0.667076\pi$
$674$		0			0
$675$	−7.72119	−	12.3815i	−0.297189	−	0.476562i
$676$		0			0
$677$	20.0218	−	20.0218i	0.769500	−	0.769500i	−0.208519	−	0.978018i	$-0.566864\pi$
$677$	20.0218	−	20.0218i	0.769500	−	0.769500i	0.978018	+	0.208519i	$0.0668642\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−15.1582			−0.580865
$682$		0			0
$683$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$683$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$684$		0			0
$685$	−51.7871	−	5.92417i	−1.97868	−	0.226351i
$686$		0			0
$687$	6.80840	−	6.80840i	0.259757	−	0.259757i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−15.6969			−0.597140			−0.298570	−	0.954388i	$-0.596510\pi$
$691$	−15.6969			−0.597140			−0.298570	+	0.954388i	$0.596510\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	21.8375	−	17.3541i	0.828342	−	0.658280i
$696$		0			0
$697$		0			0
$698$		0			0
$699$			12.9146i			0.488474i
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	24.6820	−	24.6820i	0.928264	−	0.928264i
$708$		0			0
$709$		0			0			−	1.00000i	$-0.5\pi$
$709$		0			0				1.00000i	$0.5\pi$
$710$		0			0
$711$	43.1582			1.61856
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	2.68953	−	2.68953i	0.100442	−	0.100442i
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$727$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$728$		0			0
$729$			14.0334i			0.519754i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−9.95847	−	9.95847i	−0.367825	−	0.367825i	0.498859	−	0.866683i	$-0.333753\pi$
$733$	−9.95847	−	9.95847i	−0.367825	−	0.367825i	−0.866683	+	0.498859i	$0.833753\pi$
$734$		0			0
$735$	7.90420	+	0.904199i	0.291551	+	0.0333519i
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$		0			0
$741$	−24.6012			−0.903747
$742$		0			0
$743$	−30.7417	−	30.7417i	−1.12780	−	1.12780i	−0.990534	−	0.137268i	$-0.956168\pi$
$743$	−30.7417	−	30.7417i	−1.12780	−	1.12780i	−0.137268	−	0.990534i	$-0.543832\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−15.9875	+	15.9875i	−0.584952	+	0.584952i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−22.4499			−0.819210			−0.409605	−	0.912263i	$-0.634333\pi$
$751$	−22.4499			−0.819210			−0.409605	+	0.912263i	$0.634333\pi$
$752$		0			0
$753$	−3.96663	−	3.96663i	−0.144552	−	0.144552i
$754$		0			0
$755$	31.2322	+	39.3008i	1.13666	+	1.43030i
$756$		0			0
$757$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$757$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−68.8665	+	68.8665i	−2.48663	+	2.48663i
$768$		0			0
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−25.1223	−	25.1223i	−0.903587	−	0.903587i	0.0921578	−	0.995744i	$-0.470624\pi$
$773$	−25.1223	−	25.1223i	−0.903587	−	0.903587i	−0.995744	+	0.0921578i	$0.970624\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−51.6125	−	5.90420i	−1.84213	−	0.210730i
$786$		0			0
$787$	−33.9337	+	33.9337i	−1.20961	+	1.20961i	−0.238451	+	0.971154i	$0.576640\pi$
$787$	−33.9337	+	33.9337i	−1.20961	+	1.20961i	−0.971154	+	0.238451i	$0.923360\pi$
$788$		0			0
$789$		−	8.06860i		−	0.287250i
$790$		0			0
$791$	−42.0000			−1.49335
$792$		0			0
$793$	65.6415	+	65.6415i	2.33100	+	2.33100i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	9.86562	−	9.86562i	0.349458	−	0.349458i	−0.510449	−	0.859908i	$-0.670521\pi$
$797$	9.86562	−	9.86562i	0.349458	−	0.349458i	0.859908	+	0.510449i	$0.170521\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$	3.86060	+	4.85795i	0.136068	+	0.171220i
$806$		0			0
$807$	−11.4499	+	11.4499i	−0.403057	+	0.403057i
$808$		0			0
$809$			11.6749i			0.410468i	0.978713	+	0.205234i	$0.0657956\pi$
$809$			11.6749i			0.410468i	−0.978713	+	0.205234i	$0.934204\pi$
$810$		0			0
$811$	46.2103			1.62266			0.811332	−	0.584586i	$-0.198743\pi$
$811$	46.2103			1.62266			0.811332	+	0.584586i	$0.198743\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$			46.0246i			1.60823i
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$		0			0
$823$	36.0000	+	36.0000i	1.25488	+	1.25488i	0.953506	+	0.301376i	$0.0974458\pi$
$823$	36.0000	+	36.0000i	1.25488	+	1.25488i	0.301376	+	0.953506i	$0.402554\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$827$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$828$		0			0
$829$		−	23.2564i		−	0.807729i	−0.914819	−	0.403864i	$-0.867667\pi$
$829$		−	23.2564i		−	0.807729i	0.914819	−	0.403864i	$-0.132333\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$		0			0
$843$	5.37907	+	5.37907i	0.185265	+	0.185265i
$844$		0			0
$845$	47.7183	−	37.9216i	1.64156	−	1.30454i
$846$		0			0
$847$	20.5791	−	20.5791i	0.707107	−	0.707107i
$848$		0			0
$849$		−	13.2250i		−	0.453880i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−20.7406	−	20.7406i	−0.710144	−	0.710144i	0.256421	−	0.966565i	$-0.417457\pi$
$853$	−20.7406	−	20.7406i	−0.710144	−	0.710144i	−0.966565	+	0.256421i	$0.917457\pi$
$854$		0			0
$855$	−29.0958	−	36.6125i	−0.995055	−	1.25212i
$856$		0			0
$857$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$857$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$858$		0			0
$859$			0.440260i			0.0150215i	0.999972	+	0.00751074i	$0.00239076\pi$
$859$			0.440260i			0.0150215i	−0.999972	+	0.00751074i	$0.997609\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	11.2250	+	11.2250i	0.382102	+	0.382102i	0.871859	−	0.489757i	$-0.162914\pi$
$863$	11.2250	+	11.2250i	0.382102	+	0.382102i	−0.489757	+	0.871859i	$0.662914\pi$
$864$		0			0
$865$	−4.87083	+	42.5791i	−0.165613	+	1.44773i
$866$		0			0
$867$	−6.10987	+	6.10987i	−0.207502	+	0.207502i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	9.91205	−	27.8703i	0.335088	−	0.942187i
$876$		0			0
$877$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$877$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$878$		0			0
$879$		−	12.8418i		−	0.433142i
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$883$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$884$		0			0
$885$	17.3323	+	1.98272i	0.582617	+	0.0666484i
$886$		0			0
$887$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$887$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$888$		0			0
$889$		−	45.7417i		−	1.53413i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$	2.39182	−	2.39182i	0.0798607	−	0.0798607i
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−26.8375	−	33.7707i	−0.892107	−	1.12258i
$906$		0			0
$907$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$907$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$908$		0			0
$909$		−	36.1710i		−	1.19972i
$910$		0			0
$911$	52.3832			1.73553			0.867766	−	0.496972i	$-0.165555\pi$
$911$	52.3832			1.73553			0.867766	+	0.496972i	$0.165555\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$	1.88987	−	16.5206i	0.0624772	−	0.546154i
$916$		0			0
$917$	9.23966	−	9.23966i	0.305121	−	0.305121i
$918$		0			0
$919$			53.1582i			1.75353i	0.480921	+	0.876764i	$0.340303\pi$
$919$			53.1582i			1.75353i	−0.480921	+	0.876764i	$0.659697\pi$
$920$		0			0
$921$	−0.325084			−0.0107119
$922$		0			0
$923$	32.4152	+	32.4152i	1.06696	+	1.06696i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$	53.3984			1.75006
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$937$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	39.9267			1.30157			0.650787	−	0.759260i	$-0.274439\pi$
$941$	39.9267			1.30157			0.650787	+	0.759260i	$0.274439\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$	13.5167	−	10.7417i	0.439698	−	0.349426i
$946$		0			0
$947$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$947$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	12.0334	+	12.0334i	0.389799	+	0.389799i	0.874616	−	0.484817i	$-0.161114\pi$
$953$	12.0334	+	12.0334i	0.389799	+	0.389799i	−0.484817	+	0.874616i	$0.661114\pi$
$954$		0			0
$955$	37.8752	+	47.6599i	1.22561	+	1.54224i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	61.6749i		−	1.99159i
$960$		0			0
$961$	31.0000			1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−2.15642	+	18.8507i	−0.0694178	+	0.606826i
$966$		0			0
$967$	−17.7750	+	17.7750i	−0.571606	+	0.571606i	−0.932577	−	0.360971i	$-0.882445\pi$
$967$	−17.7750	+	17.7750i	−0.571606	+	0.571606i	0.360971	+	0.932577i	$0.382445\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−23.9752			−0.769402			−0.384701	−	0.923041i	$-0.625695\pi$
$971$	−23.9752			−0.769402			−0.384701	+	0.923041i	$0.625695\pi$
$972$		0			0
$973$	23.3373	+	23.3373i	0.748159	+	0.748159i
$974$		0			0
$975$	−15.7083	−	3.64155i	−0.503068	−	0.116623i
$976$		0			0
$977$	−20.9333	+	20.9333i	−0.669714	+	0.669714i	−0.957650	−	0.287936i	$-0.907031\pi$
$977$	−20.9333	+	20.9333i	−0.669714	+	0.669714i	0.287936	+	0.957650i	$0.407031\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$983$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−9.80840			−0.311574			−0.155787	−	0.987791i	$-0.549791\pi$
$991$	−9.80840			−0.311574			−0.155787	+	0.987791i	$0.549791\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−14.3642	+	14.3642i	−0.454918	+	0.454918i	−0.896983	−	0.442065i	$-0.854246\pi$
$997$	−14.3642	+	14.3642i	−0.454918	+	0.454918i	0.442065	+	0.896983i	$0.354246\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1120.2.w.a.433.2		8
4.3	odd	2		280.2.s.a.13.3	yes	8
5.2	odd	4	inner	1120.2.w.a.657.2		8
7.6	odd	2	inner	1120.2.w.a.433.3		8
8.3	odd	2		280.2.s.a.13.2	✓	8
8.5	even	2	inner	1120.2.w.a.433.3		8
20.7	even	4		280.2.s.a.237.3	yes	8
28.27	even	2		280.2.s.a.13.2	✓	8
35.27	even	4	inner	1120.2.w.a.657.3		8
40.27	even	4		280.2.s.a.237.2	yes	8
40.37	odd	4	inner	1120.2.w.a.657.3		8
56.13	odd	2	CM	1120.2.w.a.433.2		8
56.27	even	2		280.2.s.a.13.3	yes	8
140.27	odd	4		280.2.s.a.237.2	yes	8
280.27	odd	4		280.2.s.a.237.3	yes	8
280.237	even	4	inner	1120.2.w.a.657.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.s.a.13.2	✓	8	8.3	odd	2
280.2.s.a.13.2	✓	8	28.27	even	2
280.2.s.a.13.3	yes	8	4.3	odd	2
280.2.s.a.13.3	yes	8	56.27	even	2
280.2.s.a.237.2	yes	8	40.27	even	4
280.2.s.a.237.2	yes	8	140.27	odd	4
280.2.s.a.237.3	yes	8	20.7	even	4
280.2.s.a.237.3	yes	8	280.27	odd	4
1120.2.w.a.433.2		8	1.1	even	1	trivial
1120.2.w.a.433.2		8	56.13	odd	2	CM
1120.2.w.a.433.3		8	7.6	odd	2	inner
1120.2.w.a.433.3		8	8.5	even	2	inner
1120.2.w.a.657.2		8	5.2	odd	4	inner
1120.2.w.a.657.2		8	280.237	even	4	inner
1120.2.w.a.657.3		8	35.27	even	4	inner
1120.2.w.a.657.3		8	40.37	odd	4	inner

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:	\( N \)	\(=\)	\( 1120 = 2^{5} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1120.w (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.359404 - 0.359404i) q^{3} +(-1.39119 - 1.75060i) q^{5} +(1.87083 - 1.87083i) q^{7} -2.74166i q^{9} +O(q^{10})\)	\(q+(-0.359404 - 0.359404i) q^{3} +(-1.39119 - 1.75060i) q^{5} +(1.87083 - 1.87083i) q^{7} -2.74166i q^{9} +(-4.48655 - 4.48655i) q^{13} +(-0.129171 + 1.12917i) q^{15} +7.62834i q^{19} -1.34477 q^{21} +(-0.741657 - 0.741657i) q^{23} +(-1.12917 + 4.87083i) q^{25} +(-2.06358 + 2.06358i) q^{27} +(-5.87775 - 0.672384i) q^{35} +3.22497i q^{39} +(-4.79953 + 3.81417i) q^{45} -7.00000i q^{49} +(2.74166 - 2.74166i) q^{57} -15.3495i q^{59} -14.6307 q^{61} +(-5.12917 - 5.12917i) q^{63} +(-1.61249 + 14.0958i) q^{65} +0.533109i q^{69} -7.22497 q^{71} +(2.15642 - 1.34477i) q^{75} +15.7417i q^{79} -6.74166 q^{81} +(-5.83132 - 5.83132i) q^{83} -16.7871 q^{91} +(13.3541 - 10.6125i) q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 16 q^{15} + 24 q^{23} - 24 q^{25} - 8 q^{57} - 56 q^{63} + 32 q^{65} + 32 q^{71} - 24 q^{81} + 32 q^{95}+O(q^{100}) \) 8 * q - 16 * q^15 + 24 * q^23 - 24 * q^25 - 8 * q^57 - 56 * q^63 + 32 * q^65 + 32 * q^71 - 24 * q^81 + 32 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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