LMFDB - Embedded newform 560.4.e.c.559.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [560,4,Mod(559,560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(560, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("560.559");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 560 = 2^{4} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	560.e (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$33.0410696032$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.121550625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 4x^{6} - 9x^{5} + 23x^{4} + 18x^{3} - 16x^{2} + 8x + 16 $ x^8 - x^7 - 4x^6 - 9x^5 + 23x^4 + 18x^3 - 16x^2 + 8x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{16} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			559.4
Root			$0.553538 - 0.676408i$ of defining polynomial
Character	$\chi$	$=$	560.559
Dual form			560.4.e.c.559.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.10022i q^{3} +11.1803 q^{5} -18.5203i q^{7} +0.987803 q^{9} +O(q^{10})$	$q-5.10022i q^{3} +11.1803 q^{5} -18.5203i q^{7} +0.987803 q^{9} +56.4377i q^{11} -93.4458 q^{13} -57.0221i q^{15} -133.916 q^{17} -94.4573 q^{21} +125.000 q^{25} -142.744i q^{27} -293.421 q^{29} +287.845 q^{33} -207.063i q^{35} +476.594i q^{39} +11.0440 q^{45} -464.472i q^{47} -343.000 q^{49} +683.000i q^{51} +630.993i q^{55} -18.2944i q^{63} -1044.76 q^{65} -863.748i q^{71} -523.240 q^{73} -637.527i q^{75} +1045.24 q^{77} +487.799i q^{79} -701.354 q^{81} -47.6235i q^{83} -1497.23 q^{85} +1496.51i q^{87} +1730.64i q^{91} -8.91969 q^{97} +55.7494i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 320 q^{9}+O(q^{10}) $ 8 * q - 320 * q^9	$ 8 q - 320 q^{9} + 392 q^{21} + 1000 q^{25} - 216 q^{29} - 2744 q^{49} - 1800 q^{65} + 11768 q^{81} - 4600 q^{85}+O(q^{100}) $ 8 * q - 320 * q^9 + 392 * q^21 + 1000 * q^25 - 216 * q^29 - 2744 * q^49 - 1800 * q^65 + 11768 * q^81 - 4600 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/560\mathbb{Z}\right)^\times$.

$n$	$241$	$337$	$351$	$421$
$\chi(n)$	$-1$	$-1$	$-1$	$1$

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	0
$2$	0	0
$3$	− 5.10022i	− 0.981537i	−0.871290	−	0.490768i	$-0.836716\pi$
$3$	− 5.10022i	− 0.981537i	0.871290	−	0.490768i	$-0.163284\pi$
$4$	0	0
$5$	11.1803	1.00000
$5$	11.1803	1.00000
$6$	0	0
$7$	− 18.5203i	− 1.00000i
$7$	− 18.5203i	− 1.00000i
$8$	0	0
$9$	0.987803	0.0365853
$10$	0	0
$11$	56.4377i	1.54697i	0.633817	+	0.773483i	$0.281487\pi$
$11$	56.4377i	1.54697i	−0.633817	+	0.773483i	$0.718513\pi$
$12$	0	0
$13$	−93.4458	−1.99363	−0.996816	−	0.0797400i	$-0.974591\pi$
$13$	−93.4458	−1.99363	−0.996816	+	0.0797400i	$0.974591\pi$
$14$	0	0
$15$	− 57.0221i	− 0.981537i
$16$	0	0
$17$	−133.916	−1.91055	−0.955276	−	0.295716i	$-0.904442\pi$
$17$	−133.916	−1.91055	−0.955276	+	0.295716i	$0.904442\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	−94.4573	−0.981537
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	125.000	1.00000
$26$	0	0
$27$	− 142.744i	− 1.01745i
$28$	0	0
$29$	−293.421	−1.87886	−0.939429	−	0.342745i	$-0.888643\pi$
$29$	−293.421	−1.87886	−0.939429	+	0.342745i	$0.888643\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	287.845	1.51840
$34$	0	0
$35$	− 207.063i	− 1.00000i
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	476.594i	1.95682i
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	11.0440	0.0365853
$46$	0	0
$47$	− 464.472i	− 1.44149i	−0.693198	−	0.720747i	$-0.743799\pi$
$47$	− 464.472i	− 1.44149i	0.693198	−	0.720747i	$-0.256201\pi$
$48$	0	0
$49$	−343.000	−1.00000
$50$	0	0
$51$	683.000i	1.87528i
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	630.993i	1.54697i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	− 18.2944i	− 0.0365853i
$64$	0	0
$65$	−1044.76	−1.99363
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 863.748i	− 1.44377i	−0.692011	−	0.721887i	$-0.743275\pi$
$71$	− 863.748i	− 1.44377i	0.692011	−	0.721887i	$-0.256725\pi$
$72$	0	0
$73$	−523.240	−0.838912	−0.419456	−	0.907776i	$-0.637779\pi$
$73$	−523.240	−0.838912	−0.419456	+	0.907776i	$0.637779\pi$
$74$	0	0
$75$	− 637.527i	− 0.981537i
$76$	0	0
$77$	1045.24	1.54697
$78$	0	0
$79$	487.799i	0.694705i	0.937735	+	0.347353i	$0.112919\pi$
$79$	487.799i	0.694705i	−0.937735	+	0.347353i	$0.887081\pi$
$80$	0	0
$81$	−701.354	−0.962076
$82$	0	0
$83$	− 47.6235i	− 0.0629803i	−0.999504	−	0.0314901i	$-0.989975\pi$
$83$	− 47.6235i	− 0.0629803i	0.999504	−	0.0314901i	$-0.0100253\pi$
$84$	0	0
$85$	−1497.23	−1.91055
$86$	0	0
$87$	1496.51i	1.84417i
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	1730.64i	1.99363i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−8.91969	−0.00933668	−0.00466834	−	0.999989i	$-0.501486\pi$
$97$	−8.91969	−0.00933668	−0.00466834	+	0.999989i	$0.501486\pi$
$98$	0	0
$99$	55.7494i	0.0565962i
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	2072.64i	1.98275i	0.131064	+	0.991374i	$0.458161\pi$
$103$	2072.64i	1.98275i	−0.131064	+	0.991374i	$0.541839\pi$
$104$	0	0
$105$	−1056.06	−0.981537
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	1317.45	1.15769	0.578846	−	0.815437i	$-0.303503\pi$
$109$	1317.45	1.15769	0.578846	+	0.815437i	$0.303503\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−92.3061	−0.0729376
$118$	0	0
$119$	2480.16i	1.91055i
$120$	0	0
$121$	−1854.22	−1.39310
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1397.54	1.00000
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	− 1595.92i	− 1.01745i
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	−2368.91	−1.41488
$142$	0	0
$143$	− 5273.87i	− 3.08408i
$144$	0	0
$145$	−3280.54	−1.87886
$146$	0	0
$147$	1749.37i	0.981537i
$148$	0	0
$149$	−2466.00	−1.35586	−0.677928	−	0.735128i	$-0.737122\pi$
$149$	−2466.00	−1.35586	−0.677928	+	0.735128i	$0.737122\pi$
$150$	0	0
$151$	− 3639.11i	− 1.96123i	−0.195934	−	0.980617i	$-0.562774\pi$
$151$	− 3639.11i	− 1.96123i	0.195934	−	0.980617i	$-0.437226\pi$
$152$	0	0
$153$	−132.283	−0.0698981
$154$	0	0
$155$	0	0
$156$	0	0
$157$	3904.17	1.98463	0.992315	−	0.123734i	$-0.0394868\pi$
$157$	3904.17	1.98463	0.992315	+	0.123734i	$0.0394868\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	0	0
$165$	3218.20	1.51840
$166$	0	0
$167$	− 3953.61i	− 1.83197i	−0.401211	−	0.915985i	$-0.631411\pi$
$167$	− 3953.61i	− 1.83197i	0.401211	−	0.915985i	$-0.368589\pi$
$168$	0	0
$169$	6535.12	2.97457
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−1212.69	−0.532943	−0.266472	−	0.963843i	$-0.585858\pi$
$173$	−1212.69	−0.532943	−0.266472	+	0.963843i	$0.585858\pi$
$174$	0	0
$175$	− 2315.03i	− 1.00000i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	4697.37i	1.96144i	0.195419	+	0.980720i	$0.437393\pi$
$179$	4697.37i	1.96144i	−0.195419	+	0.980720i	$0.562607\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 7557.91i	− 2.95556i
$188$	0	0
$189$	−2643.65	−1.01745
$190$	0	0
$191$	− 2056.27i	− 0.778988i	−0.921029	−	0.389494i	$-0.872650\pi$
$191$	− 2056.27i	− 0.778988i	0.921029	−	0.389494i	$-0.127350\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	5328.48i	1.95682i
$196$	0	0
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	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
$203$	5434.23i	1.87886i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	− 750.437i	− 0.244845i	−0.992478	−	0.122422i	$-0.960934\pi$
$211$	− 750.437i	− 0.244845i	0.992478	−	0.122422i	$-0.0390662\pi$
$212$	0	0
$213$	−4405.30	−1.41712
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	2668.64i	0.823423i
$220$	0	0
$221$	12513.9	3.80894
$222$	0	0
$223$	6215.30i	1.86640i	0.359356	+	0.933201i	$0.382996\pi$
$223$	6215.30i	1.86640i	−0.359356	+	0.933201i	$0.617004\pi$
$224$	0	0
$225$	123.475	0.0365853
$226$	0	0
$227$	1675.99i	0.490041i	0.969518	+	0.245020i	$0.0787947\pi$
$227$	1675.99i	0.490041i	−0.969518	+	0.245020i	$0.921205\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	− 5330.96i	− 1.51840i
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	− 5192.96i	− 1.44149i
$236$	0	0
$237$	2487.88	0.681879
$238$	0	0
$239$	− 7293.98i	− 1.97409i	−0.160432	−	0.987047i	$-0.551289\pi$
$239$	− 7293.98i	− 1.97409i	0.160432	−	0.987047i	$-0.448711\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	− 277.029i	− 0.0731334i
$244$	0	0
$245$	−3834.86	−1.00000
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−242.890	−0.0618174
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	7636.17i	1.87528i
$256$	0	0
$257$	3358.57	0.815183	0.407592	−	0.913164i	$-0.366369\pi$
$257$	3358.57	0.815183	0.407592	+	0.913164i	$0.366369\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−289.842	−0.0687386
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	8826.64	1.95682
$274$	0	0
$275$	7054.72i	1.54697i
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−80.7681	−0.0171467	−0.00857335	−	0.999963i	$-0.502729\pi$
$281$	−80.7681	−0.0171467	−0.00857335	+	0.999963i	$0.502729\pi$
$282$	0	0
$283$	− 6903.21i	− 1.45001i	−0.688742	−	0.725006i	$-0.741837\pi$
$283$	− 6903.21i	− 1.45001i	0.688742	−	0.725006i	$-0.258163\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	13020.5	2.65021
$290$	0	0
$291$	45.4924i	0.00916429i
$292$	0	0
$293$	−3064.18	−0.610960	−0.305480	−	0.952198i	$-0.598817\pi$
$293$	−3064.18	−0.610960	−0.305480	+	0.952198i	$0.598817\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	8056.14	1.57396
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	583.386i	0.108455i	0.998529	+	0.0542273i	$0.0172696\pi$
$307$	583.386i	0.108455i	−0.998529	+	0.0542273i	$0.982730\pi$
$308$	0	0
$309$	10570.9	1.94614
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	−3054.49	−0.551597	−0.275799	−	0.961215i	$-0.588942\pi$
$313$	−3054.49	−0.551597	−0.275799	+	0.961215i	$0.588942\pi$
$314$	0	0
$315$	− 204.537i	− 0.0365853i
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	− 16560.0i	− 2.90653i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−11680.7	−1.99363
$326$	0	0
$327$	− 6719.25i	− 1.13632i
$328$	0	0
$329$	−8602.15	−1.44149
$330$	0	0
$331$	9477.56i	1.57382i	0.617069	+	0.786909i	$0.288320\pi$
$331$	9477.56i	1.57382i	−0.617069	+	0.786909i	$0.711680\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	6352.45i	1.00000i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	13338.8i	2.02841i
$352$	0	0
$353$	−13256.2	−1.99874	−0.999371	−	0.0354610i	$-0.988710\pi$
$353$	−13256.2	−1.99874	−0.999371	+	0.0354610i	$0.988710\pi$
$354$	0	0
$355$	− 9656.99i	− 1.44377i
$356$	0	0
$357$	12649.3	1.87528
$358$	0	0
$359$	− 11086.7i	− 1.62990i	−0.579529	−	0.814952i	$-0.696763\pi$
$359$	− 11086.7i	− 1.62990i	0.579529	−	0.814952i	$-0.303237\pi$
$360$	0	0
$361$	−6859.00	−1.00000
$362$	0	0
$363$	9456.92i	1.36738i
$364$	0	0
$365$	−5850.00	−0.838912
$366$	0	0
$367$	− 7716.32i	− 1.09752i	−0.835981	−	0.548758i	$-0.815101\pi$
$367$	− 7716.32i	− 1.09752i	0.835981	−	0.548758i	$-0.184899\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	− 7127.77i	− 0.981537i
$376$	0	0
$377$	27418.9	3.74575
$378$	0	0
$379$	4366.07i	0.591741i	0.955228	+	0.295870i	$0.0956097\pi$
$379$	4366.07i	0.591741i	−0.955228	+	0.295870i	$0.904390\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 14239.4i	− 1.89974i	−0.312643	−	0.949871i	$-0.601214\pi$
$383$	− 14239.4i	− 1.89974i	0.312643	−	0.949871i	$-0.398786\pi$
$384$	0	0
$385$	11686.2	1.54697
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−12702.8	−1.65567	−0.827836	−	0.560971i	$-0.810428\pi$
$389$	−12702.8	−1.65567	−0.827836	+	0.560971i	$0.810428\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	5453.76i	0.694705i
$396$	0	0
$397$	−8147.27	−1.02997	−0.514987	−	0.857198i	$-0.672203\pi$
$397$	−8147.27	−1.02997	−0.514987	+	0.857198i	$0.672203\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−2003.50	−0.249501	−0.124751	−	0.992188i	$-0.539813\pi$
$401$	−2003.50	−0.249501	−0.124751	+	0.992188i	$0.539813\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−7841.37	−0.962076
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	− 532.447i	− 0.0629803i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	16532.2	1.91384	0.956922	−	0.290345i	$-0.0937700\pi$
$421$	16532.2	1.91384	0.956922	+	0.290345i	$0.0937700\pi$
$422$	0	0
$423$	− 458.807i	− 0.0527375i
$424$	0	0
$425$	−16739.5	−1.91055
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−26897.9	−3.02714
$430$	0	0
$431$	12074.0i	1.34938i	0.738101	+	0.674690i	$0.235723\pi$
$431$	12074.0i	1.34938i	−0.738101	+	0.674690i	$0.764277\pi$
$432$	0	0
$433$	−12920.0	−1.43394	−0.716970	−	0.697105i	$-0.754471\pi$
$433$	−12920.0	−1.43394	−0.716970	+	0.697105i	$0.754471\pi$
$434$	0	0
$435$	16731.5i	1.84417i
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	−338.816	−0.0365853
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	12577.1i	1.33082i
$448$	0	0
$449$	−18895.1	−1.98600	−0.993000	−	0.118118i	$-0.962314\pi$
$449$	−18895.1	−1.98600	−0.993000	+	0.118118i	$0.962314\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−18560.2	−1.92502
$454$	0	0
$455$	19349.2i	1.99363i
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	19115.7i	1.94388i
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 20174.5i	− 1.99907i	−0.0304797	−	0.999535i	$-0.509704\pi$
$467$	− 20174.5i	− 1.99907i	0.0304797	−	0.999535i	$-0.490296\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	− 19912.1i	− 1.94799i
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	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	0
$484$	0	0
$485$	−99.7252	−0.00933668
$486$	0	0
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 2129.73i	− 0.195751i	−0.995199	−	0.0978753i	$-0.968795\pi$
$491$	− 2129.73i	− 0.195751i	0.995199	−	0.0978753i	$-0.0312046\pi$
$492$	0	0
$493$	39293.7	3.58965
$494$	0	0
$495$	623.297i	0.0565962i
$496$	0	0
$497$	−15996.8	−1.44377
$498$	0	0
$499$	− 3955.91i	− 0.354892i	−0.984131	−	0.177446i	$-0.943217\pi$
$499$	− 3955.91i	− 0.354892i	0.984131	−	0.177446i	$-0.0567835\pi$
$500$	0	0
$501$	−20164.2	−1.79815
$502$	0	0
$503$	9108.44i	0.807406i	0.914890	+	0.403703i	$0.132277\pi$
$503$	9108.44i	0.807406i	−0.914890	+	0.403703i	$0.867723\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 33330.5i	− 2.91965i
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	9690.54i	0.838912i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	23172.8i	1.98275i
$516$	0	0
$517$	26213.8	2.22994
$518$	0	0
$519$	6184.98i	0.523103i
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	7296.98i	0.610086i	0.952339	+	0.305043i	$0.0986708\pi$
$523$	7296.98i	0.610086i	−0.952339	+	0.305043i	$0.901329\pi$
$524$	0	0
$525$	−11807.2	−0.981537
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−12167.0	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	23957.6	1.92523
$538$	0	0
$539$	− 19358.1i	− 1.54697i
$540$	0	0
$541$	15245.5	1.21156	0.605782	−	0.795631i	$-0.292860\pi$
$541$	15245.5	1.21156	0.605782	+	0.795631i	$0.292860\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	14729.5	1.15769
$546$	0	0
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	9034.17	0.694705
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−38547.0	−2.90099
$562$	0	0
$563$	22811.7i	1.70763i	0.520574	+	0.853816i	$0.325718\pi$
$563$	22811.7i	1.70763i	−0.520574	+	0.853816i	$0.674282\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	12989.2i	0.962076i
$568$	0	0
$569$	10026.0	0.738685	0.369343	−	0.929293i	$-0.379583\pi$
$569$	10026.0	0.738685	0.369343	+	0.929293i	$0.379583\pi$
$570$	0	0
$571$	− 16079.9i	− 1.17850i	−0.807951	−	0.589250i	$-0.799423\pi$
$571$	− 16079.9i	− 1.17850i	0.807951	−	0.589250i	$-0.200577\pi$
$572$	0	0
$573$	−10487.4	−0.764606
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−6160.21	−0.444459	−0.222229	−	0.974994i	$-0.571333\pi$
$577$	−6160.21	−0.444459	−0.222229	+	0.974994i	$0.571333\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−882.000	−0.0629803
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−1032.01	−0.0729376
$586$	0	0
$587$	24145.1i	1.69774i	0.528598	+	0.848872i	$0.322718\pi$
$587$	24145.1i	1.69774i	−0.528598	+	0.848872i	$0.677282\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−25545.7	−1.76903	−0.884515	−	0.466511i	$-0.845511\pi$
$593$	−25545.7	−1.76903	−0.884515	+	0.466511i	$0.845511\pi$
$594$	0	0
$595$	27729.0i	1.91055i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	22958.5i	1.56604i	0.621997	+	0.783019i	$0.286322\pi$
$599$	22958.5i	1.56604i	−0.621997	+	0.783019i	$0.713678\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−20730.8	−1.39310
$606$	0	0
$607$	− 29221.4i	− 1.95397i	−0.213306	−	0.976985i	$-0.568423\pi$
$607$	− 29221.4i	− 1.95397i	0.213306	−	0.976985i	$-0.431577\pi$
$608$	0	0
$609$	27715.7	1.84417
$610$	0	0
$611$	43403.0i	2.87381i
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	15625.0	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	− 26865.6i	− 1.69493i	−0.530850	−	0.847465i	$-0.678127\pi$
$631$	− 26865.6i	− 1.69493i	0.530850	−	0.847465i	$-0.321873\pi$
$632$	0	0
$633$	−3827.39	−0.240324
$634$	0	0
$635$	0	0
$636$	0	0
$637$	32051.9	1.99363
$638$	0	0
$639$	− 853.213i	− 0.0528209i
$640$	0	0
$641$	−28782.0	−1.77351	−0.886756	−	0.462239i	$-0.847046\pi$
$641$	−28782.0	−1.77351	−0.886756	+	0.462239i	$0.847046\pi$
$642$	0	0
$643$	25975.6i	1.59312i	0.604559	+	0.796560i	$0.293349\pi$
$643$	25975.6i	1.59312i	−0.604559	+	0.796560i	$0.706651\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 15572.9i	− 0.946265i	−0.880991	−	0.473133i	$-0.843123\pi$
$647$	− 15572.9i	− 0.946265i	0.880991	−	0.473133i	$-0.156877\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−516.858	−0.0306919
$658$	0	0
$659$	− 21815.9i	− 1.28957i	−0.764364	−	0.644785i	$-0.776947\pi$
$659$	− 21815.9i	− 1.28957i	0.764364	−	0.644785i	$-0.223053\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	− 63823.5i	− 3.73861i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	31699.4	1.83194
$670$	0	0
$671$	0	0
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	− 17843.0i	− 1.01745i
$676$	0	0
$677$	−2261.11	−0.128363	−0.0641814	−	0.997938i	$-0.520444\pi$
$677$	−2261.11	−0.128363	−0.0641814	+	0.997938i	$0.520444\pi$
$678$	0	0
$679$	165.195i	0.00933668i
$680$	0	0
$681$	8547.90	0.480993
$682$	0	0
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	0	0		−	1.00000i	$-0.5\pi$
$691$	0	0			1.00000i	$0.5\pi$
$692$	0	0
$693$	1032.49	0.0565962
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−2815.64	−0.151705	−0.0758526	−	0.997119i	$-0.524168\pi$
$701$	−2815.64	−0.151705	−0.0758526	+	0.997119i	$0.524168\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−26485.2	−1.41488
$706$	0	0
$707$	0	0
$708$	0	0
$709$	31583.8	1.67300	0.836498	−	0.547970i	$-0.184599\pi$
$709$	31583.8	1.67300	0.836498	+	0.547970i	$0.184599\pi$
$710$	0	0
$711$	481.850i	0.0254160i
$712$	0	0
$713$	0	0
$714$	0	0
$715$	− 58963.7i	− 3.08408i
$716$	0	0
$717$	−37200.9	−1.93765
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	38385.8	1.98275
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−36677.6	−1.87886
$726$	0	0
$727$	11392.6i	0.581194i	0.956845	+	0.290597i	$0.0938540\pi$
$727$	11392.6i	0.581194i	−0.956845	+	0.290597i	$0.906146\pi$
$728$	0	0
$729$	−20349.5	−1.03386
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−32331.1	−1.62916	−0.814581	−	0.580050i	$-0.803033\pi$
$733$	−32331.1	−1.62916	−0.814581	+	0.580050i	$0.803033\pi$
$734$	0	0
$735$	19558.6i	0.981537i
$736$	0	0
$737$	0	0
$738$	0	0
$739$	3514.46i	0.174941i	0.996167	+	0.0874707i	$0.0278784\pi$
$739$	3514.46i	0.174941i	−0.996167	+	0.0874707i	$0.972122\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	−27570.7	−1.35586
$746$	0	0
$747$	− 47.0427i	− 0.00230415i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	− 38956.6i	− 1.89287i	−0.322894	−	0.946435i	$-0.604656\pi$
$751$	− 38956.6i	− 1.89287i	0.322894	−	0.946435i	$-0.395344\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 40686.5i	− 1.96123i
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	− 24399.4i	− 1.15769i
$764$	0	0
$765$	−1478.96	−0.0698981
$766$	0	0
$767$	0	0
$768$	0	0
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	− 17129.5i	− 0.800132i
$772$	0	0
$773$	34366.8	1.59908	0.799538	−	0.600615i	$-0.205078\pi$
$773$	34366.8	1.59908	0.799538	+	0.600615i	$0.205078\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	48748.0	2.23347
$782$	0	0
$783$	41884.0i	1.91164i
$784$	0	0
$785$	43650.0	1.98463
$786$	0	0
$787$	43503.8i	1.97045i	0.171269	+	0.985224i	$0.445213\pi$
$787$	43503.8i	1.97045i	−0.171269	+	0.985224i	$0.554787\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	36518.3	1.62302	0.811508	−	0.584341i	$-0.198647\pi$
$797$	36518.3	1.62302	0.811508	+	0.584341i	$0.198647\pi$
$798$	0	0
$799$	62200.2i	2.75405i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 29530.5i	− 1.29777i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	43202.4	1.87752	0.938761	−	0.344569i	$-0.111975\pi$
$809$	43202.4	1.87752	0.938761	+	0.344569i	$0.111975\pi$
$810$	0	0
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	1709.53i	0.0729376i
$820$	0	0
$821$	9961.20	0.423445	0.211722	−	0.977330i	$-0.432093\pi$
$821$	9961.20	0.423445	0.211722	+	0.977330i	$0.432093\pi$
$822$	0	0
$823$	0	0		−	1.00000i	$-0.5\pi$
$823$	0	0			1.00000i	$0.5\pi$
$824$	0	0
$825$	35980.6	1.51840
$826$	0	0
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	45933.2	1.91055
$834$	0	0
$835$	− 44202.6i	− 1.83197i
$836$	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
$841$	61706.7	2.53010
$842$	0	0
$843$	411.935i	0.0168301i
$844$	0	0
$845$	73064.9	2.97457
$846$	0	0
$847$	34340.6i	1.39310i
$848$	0	0
$849$	−35207.9	−1.42324
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−37794.0	−1.51705	−0.758524	−	0.651645i	$-0.774079\pi$
$853$	−37794.0	−1.51705	−0.758524	+	0.651645i	$0.774079\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−7428.22	−0.296083	−0.148041	−	0.988981i	$-0.547297\pi$
$857$	−7428.22	−0.296083	−0.148041	+	0.988981i	$0.547297\pi$
$858$	0	0
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	−13558.3	−0.532943
$866$	0	0
$867$	− 66407.2i	− 2.60128i
$868$	0	0
$869$	−27530.3	−1.07469
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−8.81090	−0.000341585 0
$874$	0	0
$875$	− 25882.8i	− 1.00000i
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	15628.0i	0.599680i
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0	0
$883$	0	0		−	1.00000i	$-0.5\pi$
$883$	0	0			1.00000i	$0.5\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	18716.0i	0.708482i	0.935154	+	0.354241i	$0.115261\pi$
$887$	18716.0i	0.708482i	−0.935154	+	0.354241i	$0.884739\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	− 39582.8i	− 1.48830i
$892$	0	0
$893$	0	0
$894$	0	0
$895$	52518.2i	1.96144i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	0	0		−	1.00000i	$-0.5\pi$
$907$	0	0			1.00000i	$0.5\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	45376.3i	1.65026i	0.564944	+	0.825129i	$0.308898\pi$
$911$	45376.3i	1.65026i	−0.564944	+	0.825129i	$0.691102\pi$
$912$	0	0
$913$	2687.76	0.0974283
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	− 50196.4i	− 1.80177i	−0.434059	−	0.900885i	$-0.642919\pi$
$919$	− 50196.4i	− 1.80177i	0.434059	−	0.900885i	$-0.357081\pi$
$920$	0	0
$921$	2975.39	0.106452
$922$	0	0
$923$	80713.6i	2.87835i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	2047.36i	0.0725394i
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 84500.0i	− 2.95556i
$936$	0	0
$937$	−37730.6	−1.31548	−0.657740	−	0.753245i	$-0.728488\pi$
$937$	−37730.6	−1.31548	−0.657740	+	0.753245i	$0.728488\pi$
$938$	0	0
$939$	15578.5i	0.541413i
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−29556.9	−1.01745
$946$	0	0
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	48894.6	1.67248
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	− 22989.8i	− 0.778988i
$956$	0	0
$957$	−84459.6	−2.85286
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−29791.0	−1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	59574.2i	1.95682i
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	1301.38	0.0423545
$982$	0	0
$983$	− 56800.6i	− 1.84299i	−0.388391	−	0.921495i	$-0.626969\pi$
$983$	− 56800.6i	− 1.84299i	0.388391	−	0.921495i	$-0.373031\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	43872.8i	1.41488i
$988$	0	0
$989$	0	0
$990$	0	0
$991$	60805.5i	1.94909i	0.224189	+	0.974546i	$0.428027\pi$
$991$	60805.5i	1.94909i	−0.224189	+	0.974546i	$0.571973\pi$
$992$	0	0
$993$	48337.6	1.54476
$994$	0	0
$995$	0	0
$996$	0	0
$997$	22729.1	0.722003	0.361001	−	0.932565i	$-0.382435\pi$
$997$	22729.1	0.722003	0.361001	+	0.932565i	$0.382435\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.4.e.c.559.4	yes	8
4.3	odd	2	inner	560.4.e.c.559.6	yes	8
5.4	even	2	inner	560.4.e.c.559.5	yes	8
7.6	odd	2	inner	560.4.e.c.559.5	yes	8
20.19	odd	2	inner	560.4.e.c.559.3	✓	8
28.27	even	2	inner	560.4.e.c.559.3	✓	8
35.34	odd	2	CM	560.4.e.c.559.4	yes	8
140.139	even	2	inner	560.4.e.c.559.6	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
560.4.e.c.559.3	✓	8	20.19	odd	2	inner
560.4.e.c.559.3	✓	8	28.27	even	2	inner
560.4.e.c.559.4	yes	8	1.1	even	1	trivial
560.4.e.c.559.4	yes	8	35.34	odd	2	CM
560.4.e.c.559.5	yes	8	5.4	even	2	inner
560.4.e.c.559.5	yes	8	7.6	odd	2	inner
560.4.e.c.559.6	yes	8	4.3	odd	2	inner
560.4.e.c.559.6	yes	8	140.139	even	2	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:	\( N \)	\(=\)	\( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	560.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-5.10022i q^{3} +11.1803 q^{5} -18.5203i q^{7} +0.987803 q^{9} +O(q^{10})\)	\(q-5.10022i q^{3} +11.1803 q^{5} -18.5203i q^{7} +0.987803 q^{9} +56.4377i q^{11} -93.4458 q^{13} -57.0221i q^{15} -133.916 q^{17} -94.4573 q^{21} +125.000 q^{25} -142.744i q^{27} -293.421 q^{29} +287.845 q^{33} -207.063i q^{35} +476.594i q^{39} +11.0440 q^{45} -464.472i q^{47} -343.000 q^{49} +683.000i q^{51} +630.993i q^{55} -18.2944i q^{63} -1044.76 q^{65} -863.748i q^{71} -523.240 q^{73} -637.527i q^{75} +1045.24 q^{77} +487.799i q^{79} -701.354 q^{81} -47.6235i q^{83} -1497.23 q^{85} +1496.51i q^{87} +1730.64i q^{91} -8.91969 q^{97} +55.7494i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 320 q^{9}+O(q^{10}) \) 8 * q - 320 * q^9	\( 8 q - 320 q^{9} + 392 q^{21} + 1000 q^{25} - 216 q^{29} - 2744 q^{49} - 1800 q^{65} + 11768 q^{81} - 4600 q^{85}+O(q^{100}) \) 8 * q - 320 * q^9 + 392 * q^21 + 1000 * q^25 - 216 * q^29 - 2744 * q^49 - 1800 * q^65 + 11768 * q^81 - 4600 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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