LMFDB - Embedded newform 1120.2.w.b.657.4

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:	$ 5 $
Twist minimal:	no (minimal twist has level 280)
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			657.4
Root			$1.03179 + 1.39119i$ of defining polynomial
Character	$\chi$	$=$	1120.657
Dual form			1120.2.w.b.433.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.42298 - 2.42298i) q^{3} +(1.75060 + 1.39119i) q^{5} +(-1.87083 - 1.87083i) q^{7} -8.74166i q^{9} +O(q^{10})$	$q+(2.42298 - 2.42298i) q^{3} +(1.75060 + 1.39119i) q^{5} +(-1.87083 - 1.87083i) q^{7} -8.74166i q^{9} +(2.42298 - 2.42298i) q^{13} +(7.61249 - 0.870829i) q^{15} +4.22000i q^{19} -9.06596 q^{21} +(-0.741657 + 0.741657i) q^{23} +(1.12917 + 4.87083i) q^{25} +(-13.9119 - 13.9119i) q^{27} +(-0.672384 - 5.87775i) q^{35} -11.7417i q^{39} +(12.1613 - 15.3031i) q^{45} +7.00000i q^{49} +(10.2250 + 10.2250i) q^{57} -0.625959i q^{59} +5.47192 q^{61} +(-16.3541 + 16.3541i) q^{63} +(7.61249 - 0.870829i) q^{65} +3.59404i q^{69} +7.22497 q^{71} +(14.5379 + 9.06596i) q^{75} +15.7417i q^{79} -41.1916 q^{81} +(11.4889 - 11.4889i) q^{83} -9.06596 q^{91} +(-5.87083 + 7.38751i) q^{95} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 16 q^{15} + 24 q^{23} + 24 q^{25} - 8 q^{57} - 56 q^{63} + 16 q^{65} - 32 q^{71} - 120 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 + 16 * q^65 - 32 * q^71 - 120 * q^81 - 32 * q^95

Display 100 coefficients 10 coefficients

Character values

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

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

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
$3$	2.42298	−	2.42298i	1.39891	−	1.39891i	0.595703	−	0.803205i	$-0.296874\pi$
$3$	2.42298	−	2.42298i	1.39891	−	1.39891i	0.803205	−	0.595703i	$-0.203126\pi$
$4$		0			0
$5$	1.75060	+	1.39119i	0.782890	+	0.622160i
$5$	1.75060	+	1.39119i	0.782890	+	0.622160i
$6$		0			0
$7$	−1.87083	−	1.87083i	−0.707107	−	0.707107i
$7$	−1.87083	−	1.87083i	−0.707107	−	0.707107i
$8$		0			0
$9$		−	8.74166i		−	2.91389i
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$	2.42298	−	2.42298i	0.672014	−	0.672014i	−0.286166	−	0.958180i	$-0.592381\pi$
$13$	2.42298	−	2.42298i	0.672014	−	0.672014i	0.958180	+	0.286166i	$0.0923810\pi$
$14$		0			0
$15$	7.61249	−	0.870829i	1.96554	−	0.224847i
$16$		0			0
$17$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$17$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$18$		0			0
$19$			4.22000i			0.968134i	0.875031	+	0.484067i	$0.160841\pi$
$19$			4.22000i			0.968134i	−0.875031	+	0.484067i	$0.839159\pi$
$20$		0			0
$21$	−9.06596			−1.97835
$22$		0			0
$23$	−0.741657	+	0.741657i	−0.154646	+	0.154646i	−0.780189	−	0.625543i	$-0.784877\pi$
$23$	−0.741657	+	0.741657i	−0.154646	+	0.154646i	0.625543	+	0.780189i	$0.284877\pi$
$24$		0			0
$25$	1.12917	+	4.87083i	0.225834	+	0.974166i
$26$		0			0
$27$	−13.9119	−	13.9119i	−2.67735	−	2.67735i
$28$		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
$35$	−0.672384	−	5.87775i	−0.113654	−	0.993520i
$36$		0			0
$37$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$37$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$38$		0			0
$39$		−	11.7417i		−	1.88017i
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$43$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$44$		0			0
$45$	12.1613	−	15.3031i	1.81290	−	2.28125i
$46$		0			0
$47$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$47$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$48$		0			0
$49$			7.00000i			1.00000i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$53$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	10.2250	+	10.2250i	1.35433	+	1.35433i
$58$		0			0
$59$		−	0.625959i		−	0.0814929i	−0.999170	−	0.0407464i	$-0.987026\pi$
$59$		−	0.625959i		−	0.0814929i	0.999170	−	0.0407464i	$-0.0129736\pi$
$60$		0			0
$61$	5.47192			0.700607			0.350304	−	0.936636i	$-0.386078\pi$
$61$	5.47192			0.700607			0.350304	+	0.936636i	$0.386078\pi$
$62$		0			0
$63$	−16.3541	+	16.3541i	−2.06043	+	2.06043i
$64$		0			0
$65$	7.61249	−	0.870829i	0.944213	−	0.108013i
$66$		0			0
$67$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$67$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$68$		0			0
$69$			3.59404i			0.432672i
$70$		0			0
$71$	7.22497			0.857446			0.428723	−	0.903436i	$-0.358964\pi$
$71$	7.22497			0.857446			0.428723	+	0.903436i	$0.358964\pi$
$72$		0			0
$73$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$73$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$74$		0			0
$75$	14.5379	+	9.06596i	1.67869	+	1.04685i
$76$		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
$81$	−41.1916			−4.57684
$82$		0			0
$83$	11.4889	−	11.4889i	1.26107	−	1.26107i	0.310502	−	0.950573i	$-0.399503\pi$
$83$	11.4889	−	11.4889i	1.26107	−	1.26107i	0.950573	−	0.310502i	$-0.100497\pi$
$84$		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
$91$	−9.06596			−0.950371
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−5.87083	+	7.38751i	−0.602334	+	0.757943i
$96$		0			0
$97$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$97$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−15.1638			−1.50886			−0.754429	−	0.656382i	$-0.772086\pi$
$101$	−15.1638			−1.50886			−0.754429	+	0.656382i	$0.772086\pi$
$102$		0			0
$103$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$103$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$104$		0			0
$105$	−15.8708	−	12.6125i	−1.54883	−	1.23085i
$106$		0			0
$107$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$107$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$108$		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	0.0576178	−	0.998339i	$-0.481650\pi$
$113$	11.2250	−	11.2250i	1.05596	−	1.05596i	0.998339	−	0.0576178i	$-0.0183505\pi$
$114$		0			0
$115$	−2.33013	+	0.266555i	−0.217286	+	0.0248564i
$116$		0			0
$117$	−21.1809	−	21.1809i	−1.95817	−	1.95817i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	11.0000			1.00000
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−4.79953	+	10.0977i	−0.429283	+	0.903170i
$126$		0			0
$127$	12.2250	+	12.2250i	1.08479	+	1.08479i	0.996055	+	0.0887357i	$0.0282826\pi$
$127$	12.2250	+	12.2250i	1.08479	+	1.08479i	0.0887357	+	0.996055i	$0.471717\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−22.3519			−1.95290			−0.976448	−	0.215753i	$-0.930779\pi$
$131$	−22.3519			−1.95290			−0.976448	+	0.215753i	$0.930779\pi$
$132$		0			0
$133$	7.89490	−	7.89490i	0.684574	−	0.684574i
$134$		0			0
$135$	−5.00000	−	43.7083i	−0.430331	−	3.76181i
$136$		0			0
$137$	16.4833	+	16.4833i	1.40826	+	1.40826i	0.768922	+	0.639343i	$0.220793\pi$
$137$	16.4833	+	16.4833i	1.40826	+	1.40826i	0.639343	+	0.768922i	$0.279207\pi$
$138$		0			0
$139$			20.0098i			1.69721i	0.529028	+	0.848604i	$0.322557\pi$
$139$			20.0098i			1.69721i	−0.529028	+	0.848604i	$0.677443\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	16.9609	+	16.9609i	1.39891	+	1.39891i
$148$		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
$157$	−6.64298	−	6.64298i	−0.530167	−	0.530167i	0.390455	−	0.920622i	$-0.372318\pi$
$157$	−6.64298	−	6.64298i	−0.530167	−	0.530167i	−0.920622	+	0.390455i	$0.872318\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	2.77503			0.218703
$162$		0			0
$163$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$163$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$167$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$168$		0			0
$169$			1.25834i			0.0967956i
$170$		0			0
$171$	36.8898			2.82103
$172$		0			0
$173$	−12.7409	+	12.7409i	−0.968669	+	0.968669i	−0.999524	−	0.0308546i	$-0.990177\pi$
$173$	−12.7409	+	12.7409i	−0.968669	+	0.968669i	0.0308546	+	0.999524i	$0.490177\pi$
$174$		0			0
$175$	7.00000	−	11.2250i	0.529150	−	0.848528i
$176$		0			0
$177$	−1.51669	−	1.51669i	−0.114001	−	0.114001i
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	18.7579			1.39426			0.697131	−	0.716944i	$-0.254460\pi$
$181$	18.7579			1.39426			0.697131	+	0.716944i	$0.254460\pi$
$182$		0			0
$183$	13.2583	−	13.2583i	0.980085	−	0.980085i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$			52.0536i			3.78634i
$190$		0			0
$191$	27.2250			1.96993			0.984965	−	0.172754i	$-0.0552667\pi$
$191$	27.2250			1.96993			0.984965	+	0.172754i	$0.0552667\pi$
$192$		0			0
$193$	6.00000	−	6.00000i	0.431889	−	0.431889i	−0.457381	−	0.889271i	$-0.651213\pi$
$193$	6.00000	−	6.00000i	0.431889	−	0.431889i	0.889271	+	0.457381i	$0.151213\pi$
$194$		0			0
$195$	16.3349	−	20.5549i	1.16977	−	1.47197i
$196$		0			0
$197$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$197$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$198$		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
$207$	6.48331	+	6.48331i	0.450622	+	0.450622i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$	17.5060	−	17.5060i	1.19949	−	1.19949i
$214$		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	$-0.250000\pi$
$223$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$224$		0			0
$225$	42.5791	−	9.87083i	2.83861	−	0.658055i
$226$		0			0
$227$	−3.04894	−	3.04894i	−0.202365	−	0.202365i	0.598647	−	0.801013i	$-0.295705\pi$
$227$	−3.04894	−	3.04894i	−0.202365	−	0.202365i	−0.801013	+	0.598647i	$0.795705\pi$
$228$		0			0
$229$		−	23.6038i		−	1.55979i	−0.625913	−	0.779893i	$-0.715274\pi$
$229$		−	23.6038i		−	1.55979i	0.625913	−	0.779893i	$-0.284726\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−17.9666	+	17.9666i	−1.17703	+	1.17703i	−0.196537	+	0.980497i	$0.562969\pi$
$233$	−17.9666	+	17.9666i	−1.17703	+	1.17703i	−0.980497	+	0.196537i	$0.937031\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	38.1417	+	38.1417i	2.47757	+	2.47757i
$238$		0			0
$239$		−	7.48331i		−	0.484055i	−0.970269	−	0.242028i	$-0.922188\pi$
$239$		−	7.48331i		−	0.484055i	0.970269	−	0.242028i	$-0.0778125\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$	−58.0706	+	58.0706i	−3.72523	+	3.72523i
$244$		0			0
$245$	−9.73834	+	12.2542i	−0.622160	+	0.782890i
$246$		0			0
$247$	10.2250	+	10.2250i	0.650599	+	0.650599i
$248$		0			0
$249$		−	55.6749i		−	3.52825i
$250$		0			0
$251$	−29.7017			−1.87476			−0.937378	−	0.348315i	$-0.886754\pi$
$251$	−29.7017			−1.87476			−0.937378	+	0.348315i	$0.886754\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$257$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$258$		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	$-0.912796\pi$
$263$	−11.2250	+	11.2250i	−0.692161	+	0.692161i	0.270546	+	0.962707i	$0.412796\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	7.81404i		−	0.476430i	−0.971212	−	0.238215i	$-0.923438\pi$
$269$		−	7.81404i		−	0.476430i	0.971212	−	0.238215i	$-0.0765624\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$	−21.9666	+	21.9666i	−1.32948	+	1.32948i
$274$		0			0
$275$		0			0
$276$		0			0
$277$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$277$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$278$		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
$283$	15.0830	−	15.0830i	0.896590	−	0.896590i	−0.0985428	−	0.995133i	$-0.531418\pi$
$283$	15.0830	−	15.0830i	0.896590	−	0.896590i	0.995133	+	0.0985428i	$0.0314181\pi$
$284$		0			0
$285$	3.67490	+	32.1247i	0.217682	+	1.90290i
$286$		0			0
$287$		0			0
$288$		0			0
$289$			17.0000i			1.00000i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−16.3349	+	16.3349i	−0.954295	+	0.954295i	−0.999000	−	0.0447054i	$-0.985765\pi$
$293$	−16.3349	+	16.3349i	−0.954295	+	0.954295i	0.0447054	+	0.999000i	$0.485765\pi$
$294$		0			0
$295$	0.870829	−	1.09580i	0.0507016	−	0.0638000i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			3.59404i			0.207849i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−36.7417	+	36.7417i	−2.11075	+	2.11075i
$304$		0			0
$305$	9.57912	+	7.61249i	0.548499	+	0.435890i
$306$		0			0
$307$	24.7749	+	24.7749i	1.41398	+	1.41398i	0.719895	+	0.694083i	$0.244190\pi$
$307$	24.7749	+	24.7749i	1.41398	+	1.41398i	0.694083	+	0.719895i	$0.255810\pi$
$308$		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	$-0.250000\pi$
$313$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$314$		0			0
$315$	−51.3812	+	5.87775i	−2.89501	+	0.331173i
$316$		0			0
$317$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$317$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	14.5379	+	9.06596i	0.806416	+	0.502889i
$326$		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	$-0.493859\pi$
$337$	−18.0000	−	18.0000i	−0.980522	−	0.980522i	−0.999814	+	0.0192914i	$0.993859\pi$
$338$		0			0
$339$		−	54.3958i		−	2.95437i
$340$		0			0
$341$		0			0
$342$		0			0
$343$	13.0958	−	13.0958i	0.707107	−	0.707107i
$344$		0			0
$345$	−5.00000	+	6.29171i	−0.269191	+	0.338734i
$346$		0			0
$347$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$347$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$348$		0			0
$349$		−	35.6379i		−	1.90765i	−0.300360	−	0.953826i	$-0.597107\pi$
$349$		−	35.6379i		−	1.90765i	0.300360	−	0.953826i	$-0.402893\pi$
$350$		0			0
$351$	−67.4166			−3.59843
$352$		0			0
$353$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$353$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$354$		0			0
$355$	12.6480	+	10.0513i	0.671286	+	0.533469i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			6.00000i			0.316668i	0.987386	+	0.158334i	$0.0506123\pi$
$359$			6.00000i			0.316668i	−0.987386	+	0.158334i	$0.949388\pi$
$360$		0			0
$361$	1.19160			0.0627159
$362$		0			0
$363$	26.6528	−	26.6528i	1.39891	−	1.39891i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$367$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$373$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$374$		0			0
$375$	12.8375	+	36.0958i	0.662923	+	1.86398i
$376$		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
$381$	59.2417			3.03504
$382$		0			0
$383$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$383$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$384$		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
$393$	−54.1582	+	54.1582i	−2.73192	+	2.73192i
$394$		0			0
$395$	−21.8997	+	27.5573i	−1.10189	+	1.38656i
$396$		0			0
$397$	21.1809	+	21.1809i	1.06304	+	1.06304i	0.997875	+	0.0651619i	$0.0207564\pi$
$397$	21.1809	+	21.1809i	1.06304	+	1.06304i	0.0651619	+	0.997875i	$0.479244\pi$
$398$		0			0
$399$		−	38.2583i		−	1.91531i
$400$		0			0
$401$	−37.2250			−1.85893			−0.929463	−	0.368915i	$-0.879729\pi$
$401$	−37.2250			−1.85893			−0.929463	+	0.368915i	$0.879729\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−72.1098	−	57.3054i	−3.58317	−	2.84753i
$406$		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
$411$	79.8775			3.94006
$412$		0			0
$413$	−1.17106	+	1.17106i	−0.0576242	+	0.0576242i
$414$		0			0
$415$	36.0958	−	4.12917i	1.77187	−	0.202693i
$416$		0			0
$417$	48.4833	+	48.4833i	2.37424	+	2.37424i
$418$		0			0
$419$		−	2.96808i		−	0.145000i	−0.997368	−	0.0725002i	$-0.976902\pi$
$419$		−	2.96808i		−	0.145000i	0.997368	−	0.0725002i	$-0.0230978\pi$
$420$		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
$427$	−10.2370	−	10.2370i	−0.495404	−	0.495404i
$428$		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	$-0.250000\pi$
$433$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−3.12979	−	3.12979i	−0.149718	−	0.149718i
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	61.1916			2.91389
$442$		0			0
$443$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$443$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	29.9333i		−	1.41264i	−0.707894	−	0.706319i	$-0.750354\pi$
$449$		−	29.9333i		−	1.41264i	0.707894	−	0.706319i	$-0.249646\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−54.3958	+	54.3958i	−2.55574	+	2.55574i
$454$		0			0
$455$	−15.8708	−	12.6125i	−0.744036	−	0.591282i
$456$		0			0
$457$	3.74166	+	3.74166i	0.175027	+	0.175027i	0.789184	−	0.614157i	$-0.210504\pi$
$457$	3.74166	+	3.74166i	0.175027	+	0.175027i	−0.614157	+	0.789184i	$0.710504\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−36.8898			−1.71813			−0.859064	−	0.511867i	$-0.828954\pi$
$461$	−36.8898			−1.71813			−0.859064	+	0.511867i	$0.828954\pi$
$462$		0			0
$463$	24.0000	−	24.0000i	1.11537	−	1.11537i	0.122963	−	0.992411i	$-0.460760\pi$
$463$	24.0000	−	24.0000i	1.11537	−	1.11537i	0.992411	−	0.122963i	$-0.0392398\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−3.67490	−	3.67490i	−0.170054	−	0.170054i	0.616949	−	0.787003i	$-0.288368\pi$
$467$	−3.67490	−	3.67490i	−0.170054	−	0.170054i	−0.787003	+	0.616949i	$0.788368\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−32.1916			−1.48331
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−20.5549	+	4.76510i	−0.943123	+	0.218638i
$476$		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
$483$	6.72384	−	6.72384i	0.305945	−	0.305945i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−7.77503	−	7.77503i	−0.352320	−	0.352320i	0.508652	−	0.860972i	$-0.330144\pi$
$487$	−7.77503	−	7.77503i	−0.352320	−	0.352320i	−0.860972	+	0.508652i	$0.830144\pi$
$488$		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	$-0.250000\pi$
$503$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$504$		0			0
$505$	−26.5457	−	21.0958i	−1.18127	−	0.938751i
$506$		0			0
$507$	3.04894	+	3.04894i	0.135408	+	0.135408i
$508$		0			0
$509$		−	44.2396i		−	1.96089i	−0.196805	−	0.980443i	$-0.563057\pi$
$509$		−	44.2396i		−	1.96089i	0.196805	−	0.980443i	$-0.436943\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	58.7083	−	58.7083i	2.59203	−	2.59203i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$			61.7417i			2.71016i
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$	6.01702	−	6.01702i	0.263106	−	0.263106i	−0.563209	−	0.826315i	$-0.690433\pi$
$523$	6.01702	−	6.01702i	0.263106	−	0.263106i	0.826315	+	0.563209i	$0.190433\pi$
$524$		0			0
$525$	−10.2370	−	44.1587i	−0.446780	−	1.92724i
$526$		0			0
$527$		0			0
$528$		0			0
$529$			21.8999i			0.952169i
$530$		0			0
$531$	−5.47192			−0.237461
$532$		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
$543$	45.4499	−	45.4499i	1.95044	−	1.95044i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$547$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$548$		0			0
$549$		−	47.8336i		−	2.04149i
$550$		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	$-0.750000\pi$
$557$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−18.2128	+	18.2128i	−0.767577	+	0.767577i	−0.977679	−	0.210103i	$-0.932620\pi$
$563$	−18.2128	+	18.2128i	−0.767577	+	0.767577i	0.210103	+	0.977679i	$0.432620\pi$
$564$		0			0
$565$	35.2665	−	4.03430i	1.48367	−	0.169724i
$566$		0			0
$567$	77.0624	+	77.0624i	3.23632	+	3.23632i
$568$		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
$573$	65.9655	−	65.9655i	2.75575	−	2.75575i
$574$		0			0
$575$	−4.44994	−	2.77503i	−0.185576	−	0.115727i
$576$		0			0
$577$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$577$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$578$		0			0
$579$		−	29.0758i		−	1.20835i
$580$		0			0
$581$	−42.9877			−1.78343
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−7.61249	−	66.5457i	−0.314738	−	2.75133i
$586$		0			0
$587$	−10.8630	−	10.8630i	−0.448363	−	0.448363i	0.446447	−	0.894810i	$-0.352689\pi$
$587$	−10.8630	−	10.8630i	−0.448363	−	0.448363i	−0.894810	+	0.446447i	$0.852689\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$593$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$594$		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
$605$	19.2566	+	15.3031i	0.782890	+	0.622160i
$606$		0			0
$607$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$607$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$613$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−33.6749	−	33.6749i	−1.35570	−	1.35570i	−0.879143	−	0.476558i	$-0.841884\pi$
$617$	−33.6749	−	33.6749i	−1.35570	−	1.35570i	−0.476558	−	0.879143i	$-0.658116\pi$
$618$		0			0
$619$		−	49.0855i		−	1.97291i	−0.164018	−	0.986457i	$-0.552446\pi$
$619$		−	49.0855i		−	1.97291i	0.164018	−	0.986457i	$-0.447554\pi$
$620$		0			0
$621$	20.6358			0.828084
$622$		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	$-0.986110\pi$
$631$	−50.1916			−1.99810			−0.999048	+	0.0436231i	$0.986110\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	4.39371	+	38.4083i	0.174359	+	1.52419i
$636$		0			0
$637$	16.9609	+	16.9609i	0.672014	+	0.672014i
$638$		0			0
$639$		−	63.1582i		−	2.49850i
$640$		0			0
$641$	22.7750			0.899560			0.449780	−	0.893140i	$-0.351502\pi$
$641$	22.7750			0.899560			0.449780	+	0.893140i	$0.351502\pi$
$642$		0			0
$643$	23.0587	−	23.0587i	0.909348	−	0.909348i	−0.0868719	−	0.996219i	$-0.527687\pi$
$643$	23.0587	−	23.0587i	0.909348	−	0.909348i	0.996219	+	0.0868719i	$0.0276871\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$647$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$653$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$654$		0			0
$655$	−39.1292	−	31.0958i	−1.52890	−	1.21501i
$656$		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
$661$	−44.0779			−1.71443			−0.857215	−	0.514958i	$-0.827807\pi$
$661$	−44.0779			−1.71443			−0.857215	+	0.514958i	$0.827807\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	24.8041	−	2.83746i	0.961861	−	0.110032i
$666$		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	−0.501113	−	0.865382i	$-0.667076\pi$
$673$	9.44994	−	9.44994i	0.364269	−	0.364269i	0.865382	+	0.501113i	$0.167076\pi$
$674$		0			0
$675$	52.0536	−	83.4715i	2.00355	−	3.21282i
$676$		0			0
$677$	−30.8728	−	30.8728i	−1.18654	−	1.18654i	−0.978018	−	0.208519i	$-0.933136\pi$
$677$	−30.8728	−	30.8728i	−1.18654	−	1.18654i	−0.208519	−	0.978018i	$-0.566864\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−14.7750			−0.566180
$682$		0			0
$683$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$683$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$684$		0			0
$685$	5.92417	+	51.7871i	0.226351	+	1.97868i
$686$		0			0
$687$	−57.1916	−	57.1916i	−2.18200	−	2.18200i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−50.1758			−1.90878			−0.954388	−	0.298570i	$-0.903490\pi$
$691$	−50.1758			−1.90878			−0.954388	+	0.298570i	$0.903490\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−27.8375	+	35.0291i	−1.05593	+	1.32873i
$696$		0			0
$697$		0			0
$698$		0			0
$699$			87.0655i			3.29312i
$700$		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
$707$	28.3689	+	28.3689i	1.06692	+	1.06692i
$708$		0			0
$709$		0			0			−	1.00000i	$-0.5\pi$
$709$		0			0				1.00000i	$0.5\pi$
$710$		0			0
$711$	137.608			5.16071
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−18.1319	−	18.1319i	−0.677149	−	0.677149i
$718$		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	$-0.750000\pi$
$727$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$728$		0			0
$729$			157.833i			5.84567i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−36.9706	+	36.9706i	−1.36554	+	1.36554i	−0.498859	+	0.866683i	$0.666247\pi$
$733$	−36.9706	+	36.9706i	−1.36554	+	1.36554i	−0.866683	+	0.498859i	$0.833753\pi$
$734$		0			0
$735$	6.09580	+	53.2874i	0.224847	+	1.96554i
$736$		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
$741$	49.5498			1.82026
$742$		0			0
$743$	−30.7417	+	30.7417i	−1.12780	+	1.12780i	−0.137268	+	0.990534i	$0.543832\pi$
$743$	−30.7417	+	30.7417i	−1.12780	+	1.12780i	−0.990534	+	0.137268i	$0.956168\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−100.432	−	100.432i	−3.67463	−	3.67463i
$748$		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
$753$	−71.9666	+	71.9666i	−2.62261	+	2.62261i
$754$		0			0
$755$	−39.3008	−	31.2322i	−1.43030	−	1.13666i
$756$		0			0
$757$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$757$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$758$		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
$767$	−1.51669	−	1.51669i	−0.0547643	−	0.0547643i
$768$		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
$773$	30.2468	−	30.2468i	1.08790	−	1.08790i	0.0921578	−	0.995744i	$-0.470624\pi$
$773$	30.2468	−	30.2468i	1.08790	−	1.08790i	0.995744	−	0.0921578i	$-0.0293764\pi$
$774$		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
$785$	−2.38751	−	20.8708i	−0.0852140	−	0.744912i
$786$		0			0
$787$	20.5549	+	20.5549i	0.732703	+	0.732703i	0.971154	−	0.238451i	$-0.0766398\pi$
$787$	20.5549	+	20.5549i	0.732703	+	0.732703i	−0.238451	+	0.971154i	$0.576640\pi$
$788$		0			0
$789$			54.3958i			1.93654i
$790$		0			0
$791$	−42.0000			−1.49335
$792$		0			0
$793$	13.2583	−	13.2583i	0.470818	−	0.470818i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−38.6868	−	38.6868i	−1.37036	−	1.37036i	−0.859908	−	0.510449i	$-0.829479\pi$
$797$	−38.6868	−	38.6868i	−1.37036	−	1.37036i	−0.510449	−	0.859908i	$-0.670521\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$	4.85795	+	3.86060i	0.171220	+	0.136068i
$806$		0			0
$807$	−18.9333	−	18.9333i	−0.666482	−	0.666482i
$808$		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
$811$	33.2958			1.16917			0.584586	−	0.811332i	$-0.301257\pi$
$811$	33.2958			1.16917			0.584586	+	0.811332i	$0.301257\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$			79.2515i			2.76927i
$820$		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	−0.301376	+	0.953506i	$0.597446\pi$
$823$	−36.0000	+	36.0000i	−1.25488	+	1.25488i	−0.953506	+	0.301376i	$0.902554\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$827$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$828$		0			0
$829$			52.6796i			1.82964i	0.403864	+	0.914819i	$0.367667\pi$
$829$			52.6796i			1.82964i	−0.403864	+	0.914819i	$0.632333\pi$
$830$		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
$843$	−36.2638	+	36.2638i	−1.24899	+	1.24899i
$844$		0			0
$845$	−1.75060	+	2.20285i	−0.0602223	+	0.0757803i
$846$		0			0
$847$	−20.5791	−	20.5791i	−0.707107	−	0.707107i
$848$		0			0
$849$		−	73.0915i		−	2.50849i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	35.7187	−	35.7187i	1.22299	−	1.22299i	0.256421	−	0.966565i	$-0.417457\pi$
$853$	35.7187	−	35.7187i	1.22299	−	1.22299i	0.966565	−	0.256421i	$-0.0825433\pi$
$854$		0			0
$855$	64.5791	+	51.3208i	2.20856	+	1.75513i
$856$		0			0
$857$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$857$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$858$		0			0
$859$		−	58.6158i		−	1.99994i	−0.00751074	−	0.999972i	$-0.502391\pi$
$859$		−	58.6158i		−	1.99994i	0.00751074	−	0.999972i	$-0.497609\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−11.2250	+	11.2250i	−0.382102	+	0.382102i	−0.871859	−	0.489757i	$-0.837086\pi$
$863$	−11.2250	+	11.2250i	−0.382102	+	0.382102i	0.489757	+	0.871859i	$0.337086\pi$
$864$		0			0
$865$	−40.0291	+	4.57912i	−1.36103	+	0.155695i
$866$		0			0
$867$	41.1906	+	41.1906i	1.39891	+	1.39891i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	27.8703	−	9.91205i	0.942187	−	0.335088i
$876$		0			0
$877$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$877$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$878$		0			0
$879$			79.1582i			2.66994i
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$883$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$884$		0			0
$885$	−0.545103	−	4.76510i	−0.0183234	−	0.160177i
$886$		0			0
$887$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$887$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$888$		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
$897$	8.70829	+	8.70829i	0.290761	+	0.290761i
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	32.8375	+	26.0958i	1.09155	+	0.867454i
$906$		0			0
$907$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$907$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$908$		0			0
$909$			132.557i			4.39664i
$910$		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
$915$	41.6549	−	4.76510i	1.37707	−	0.157529i
$916$		0			0
$917$	41.8166	+	41.8166i	1.38091	+	1.38091i
$918$		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
$921$	120.058			3.95605
$922$		0			0
$923$	17.5060	−	17.5060i	0.576216	−	0.576216i
$924$		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
$931$	−29.5400			−0.968134
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$937$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	46.5817			1.51852			0.759260	−	0.650787i	$-0.225561\pi$
$941$	46.5817			1.51852			0.759260	+	0.650787i	$0.225561\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$	−72.4166	+	91.1249i	−2.35571	+	2.96429i
$946$		0			0
$947$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$947$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	12.0334	−	12.0334i	0.389799	−	0.389799i	−0.484817	−	0.874616i	$-0.661114\pi$
$953$	12.0334	−	12.0334i	0.389799	−	0.389799i	0.874616	+	0.484817i	$0.161114\pi$
$954$		0			0
$955$	47.6599	+	37.8752i	1.54224	+	1.22561i
$956$		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
$965$	18.8507	−	2.15642i	0.606826	−	0.0694178i
$966$		0			0
$967$	−17.7750	−	17.7750i	−0.571606	−	0.571606i	0.360971	−	0.932577i	$-0.382445\pi$
$967$	−17.7750	−	17.7750i	−0.571606	−	0.571606i	−0.932577	+	0.360971i	$0.882445\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−57.5255			−1.84608			−0.923041	−	0.384701i	$-0.874305\pi$
$971$	−57.5255			−1.84608			−0.923041	+	0.384701i	$0.874305\pi$
$972$		0			0
$973$	37.4349	−	37.4349i	1.20011	−	1.20011i
$974$		0			0
$975$	57.1916	−	13.2583i	1.83160	−	0.424607i
$976$		0			0
$977$	−20.9333	−	20.9333i	−0.669714	−	0.669714i	0.287936	−	0.957650i	$-0.407031\pi$
$977$	−20.9333	−	20.9333i	−0.669714	−	0.669714i	−0.957650	+	0.287936i	$0.907031\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$983$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$984$		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	$-0.450209\pi$
$991$	9.80840			0.311574			0.155787	+	0.987791i	$0.450209\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−42.2809	−	42.2809i	−1.33905	−	1.33905i	−0.896983	−	0.442065i	$-0.854246\pi$
$997$	−42.2809	−	42.2809i	−1.33905	−	1.33905i	−0.442065	−	0.896983i	$-0.645754\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1120.2.w.b.657.4		8
4.3	odd	2		280.2.s.b.237.1	yes	8
5.3	odd	4	inner	1120.2.w.b.433.4		8
7.6	odd	2	inner	1120.2.w.b.657.1		8
8.3	odd	2		280.2.s.b.237.4	yes	8
8.5	even	2	inner	1120.2.w.b.657.1		8
20.3	even	4		280.2.s.b.13.1	✓	8
28.27	even	2		280.2.s.b.237.4	yes	8
35.13	even	4	inner	1120.2.w.b.433.1		8
40.3	even	4		280.2.s.b.13.4	yes	8
40.13	odd	4	inner	1120.2.w.b.433.1		8
56.13	odd	2	CM	1120.2.w.b.657.4		8
56.27	even	2		280.2.s.b.237.1	yes	8
140.83	odd	4		280.2.s.b.13.4	yes	8
280.13	even	4	inner	1120.2.w.b.433.4		8
280.83	odd	4		280.2.s.b.13.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.s.b.13.1	✓	8	20.3	even	4
280.2.s.b.13.1	✓	8	280.83	odd	4
280.2.s.b.13.4	yes	8	40.3	even	4
280.2.s.b.13.4	yes	8	140.83	odd	4
280.2.s.b.237.1	yes	8	4.3	odd	2
280.2.s.b.237.1	yes	8	56.27	even	2
280.2.s.b.237.4	yes	8	8.3	odd	2
280.2.s.b.237.4	yes	8	28.27	even	2
1120.2.w.b.433.1		8	35.13	even	4	inner
1120.2.w.b.433.1		8	40.13	odd	4	inner
1120.2.w.b.433.4		8	5.3	odd	4	inner
1120.2.w.b.433.4		8	280.13	even	4	inner
1120.2.w.b.657.1		8	7.6	odd	2	inner
1120.2.w.b.657.1		8	8.5	even	2	inner
1120.2.w.b.657.4		8	1.1	even	1	trivial
1120.2.w.b.657.4		8	56.13	odd	2	CM

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)

\(f(q)\)	\(=\)	\(q+(2.42298 - 2.42298i) q^{3} +(1.75060 + 1.39119i) q^{5} +(-1.87083 - 1.87083i) q^{7} -8.74166i q^{9} +O(q^{10})\)	\(q+(2.42298 - 2.42298i) q^{3} +(1.75060 + 1.39119i) q^{5} +(-1.87083 - 1.87083i) q^{7} -8.74166i q^{9} +(2.42298 - 2.42298i) q^{13} +(7.61249 - 0.870829i) q^{15} +4.22000i q^{19} -9.06596 q^{21} +(-0.741657 + 0.741657i) q^{23} +(1.12917 + 4.87083i) q^{25} +(-13.9119 - 13.9119i) q^{27} +(-0.672384 - 5.87775i) q^{35} -11.7417i q^{39} +(12.1613 - 15.3031i) q^{45} +7.00000i q^{49} +(10.2250 + 10.2250i) q^{57} -0.625959i q^{59} +5.47192 q^{61} +(-16.3541 + 16.3541i) q^{63} +(7.61249 - 0.870829i) q^{65} +3.59404i q^{69} +7.22497 q^{71} +(14.5379 + 9.06596i) q^{75} +15.7417i q^{79} -41.1916 q^{81} +(11.4889 - 11.4889i) q^{83} -9.06596 q^{91} +(-5.87083 + 7.38751i) q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 16 q^{15} + 24 q^{23} + 24 q^{25} - 8 q^{57} - 56 q^{63} + 16 q^{65} - 32 q^{71} - 120 q^{81} - 32 q^{95}+O(q^{100}) \) 8 * q + 16 * q^15 + 24 * q^23 + 24 * q^25 - 8 * q^57 - 56 * q^63 + 16 * q^65 - 32 * q^71 - 120 * q^81 - 32 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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