LMFDB - Embedded newform 2240.2.e.d.2239.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,2,Mod(2239,2240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2240, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2240.2239");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2240.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$17.8864900528$
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^{8} $
Twist minimal:	no (minimal twist has level 560)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			2239.5
Root			$2.25820 + 0.369600i$ of defining polynomial
Character	$\chi$	$=$	2240.2239
Dual form			2240.2.e.d.2239.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.613616i q^{3} -2.23607 q^{5} +2.64575i q^{7} +2.62348 q^{9} +O(q^{10})$	$q+0.613616i q^{3} -2.23607 q^{5} +2.64575i q^{7} +2.62348 q^{9} -5.55612i q^{11} +1.06281 q^{13} -1.37209i q^{15} -5.75583 q^{17} -1.62348 q^{21} +5.00000 q^{25} +3.45065i q^{27} -9.62348 q^{29} +3.40932 q^{33} -5.91608i q^{35} +0.652160i q^{39} -5.86627 q^{45} -13.6511i q^{47} -7.00000 q^{49} -3.53187i q^{51} +12.4239i q^{55} +6.94106i q^{63} -2.37652 q^{65} -11.8322i q^{71} +13.4164 q^{73} +3.06808i q^{75} +14.7001 q^{77} -9.74015i q^{79} +5.75305 q^{81} -15.8745i q^{83} +12.8704 q^{85} -5.90512i q^{87} +2.81194i q^{91} -19.3931 q^{97} -14.5763i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 20 q^{9}+O(q^{10}) $ 8 * q - 20 * q^9	$ 8 q - 20 q^{9} + 28 q^{21} + 40 q^{25} - 36 q^{29} - 56 q^{49} - 60 q^{65} + 128 q^{81} - 20 q^{85}+O(q^{100}) $ 8 * q - 20 * q^9 + 28 * q^21 + 40 * q^25 - 36 * q^29 - 56 * q^49 - 60 * q^65 + 128 * q^81 - 20 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times$.

$n$	$897$	$1471$	$1541$	$1921$
$\chi(n)$	$-1$	$-1$	$1$	$-1$

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
$3$	0.613616i	0.354271i	0.984186	+	0.177136i	$0.0566831\pi$
$3$	0.613616i	0.354271i	−0.984186	+	0.177136i	$0.943317\pi$
$4$	0	0
$5$	−2.23607	−1.00000
$5$	−2.23607	−1.00000
$6$	0	0
$7$	2.64575i	1.00000i
$7$	2.64575i	1.00000i
$8$	0	0
$9$	2.62348	0.874492
$10$	0	0
$11$	− 5.55612i	− 1.67523i	−0.546259	−	0.837616i	$-0.683949\pi$
$11$	− 5.55612i	− 1.67523i	0.546259	−	0.837616i	$-0.316051\pi$
$12$	0	0
$13$	1.06281	0.294772	0.147386	−	0.989079i	$-0.452914\pi$
$13$	1.06281	0.294772	0.147386	+	0.989079i	$0.452914\pi$
$14$	0	0
$15$	− 1.37209i	− 0.354271i
$16$	0	0
$17$	−5.75583	−1.39599	−0.697997	−	0.716101i	$-0.745925\pi$
$17$	−5.75583	−1.39599	−0.697997	+	0.716101i	$0.745925\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	−1.62348	−0.354271
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	5.00000	1.00000
$26$	0	0
$27$	3.45065i	0.664079i
$28$	0	0
$29$	−9.62348	−1.78703	−0.893517	−	0.449029i	$-0.851770\pi$
$29$	−9.62348	−1.78703	−0.893517	+	0.449029i	$0.851770\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	3.40932	0.593487
$34$	0	0
$35$	− 5.91608i	− 1.00000i
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0.652160i	0.104429i
$40$	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
$45$	−5.86627	−0.874492
$46$	0	0
$47$	− 13.6511i	− 1.99122i	−0.0936230	−	0.995608i	$-0.529845\pi$
$47$	− 13.6511i	− 1.99122i	0.0936230	−	0.995608i	$-0.470155\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	− 3.53187i	− 0.494561i
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	12.4239i	1.67523i
$56$	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
$63$	6.94106i	0.874492i
$64$	0	0
$65$	−2.37652	−0.294772
$66$	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
$71$	− 11.8322i	− 1.40422i	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	− 11.8322i	− 1.40422i	0.712069	−	0.702109i	$-0.247758\pi$
$72$	0	0
$73$	13.4164	1.57027	0.785136	−	0.619324i	$-0.212593\pi$
$73$	13.4164	1.57027	0.785136	+	0.619324i	$0.212593\pi$
$74$	0	0
$75$	3.06808i	0.354271i
$76$	0	0
$77$	14.7001	1.67523
$78$	0	0
$79$	− 9.74015i	− 1.09585i	−0.836527	−	0.547926i	$-0.815418\pi$
$79$	− 9.74015i	− 1.09585i	0.836527	−	0.547926i	$-0.184582\pi$
$80$	0	0
$81$	5.75305	0.639228
$82$	0	0
$83$	− 15.8745i	− 1.74245i	−0.490881	−	0.871227i	$-0.663325\pi$
$83$	− 15.8745i	− 1.74245i	0.490881	−	0.871227i	$-0.336675\pi$
$84$	0	0
$85$	12.8704	1.39599
$86$	0	0
$87$	− 5.90512i	− 0.633095i
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	2.81194i	0.294772i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−19.3931	−1.96907	−0.984536	−	0.175180i	$-0.943949\pi$
$97$	−19.3931	−1.96907	−0.984536	+	0.175180i	$0.943949\pi$
$98$	0	0
$99$	− 14.5763i	− 1.46498i
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	− 16.1055i	− 1.58693i	−0.608618	−	0.793463i	$-0.708276\pi$
$103$	− 16.1055i	− 1.58693i	0.608618	−	0.793463i	$-0.291724\pi$
$104$	0	0
$105$	3.63020	0.354271
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	20.8704	1.99902	0.999512	−	0.0312328i	$-0.00994332\pi$
$109$	20.8704	1.99902	0.999512	+	0.0312328i	$0.00994332\pi$
$110$	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
$117$	2.78827	0.257775
$118$	0	0
$119$	− 15.2285i	− 1.39599i
$120$	0	0
$121$	−19.8704	−1.80640
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−11.1803	−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
$135$	− 7.71590i	− 0.664079i
$136$	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
$141$	8.37652	0.705431
$142$	0	0
$143$	− 5.90512i	− 0.493811i
$144$	0	0
$145$	21.5187	1.78703
$146$	0	0
$147$	− 4.29531i	− 0.354271i
$148$	0	0
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	− 23.5966i	− 1.92026i	−0.279554	−	0.960130i	$-0.590186\pi$
$151$	− 23.5966i	− 1.92026i	0.279554	−	0.960130i	$-0.409814\pi$
$152$	0	0
$153$	−15.1003	−1.22079
$154$	0	0
$155$	0	0
$156$	0	0
$157$	13.4164	1.07075	0.535373	−	0.844616i	$-0.320171\pi$
$157$	13.4164	1.07075	0.535373	+	0.844616i	$0.320171\pi$
$158$	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
$165$	−7.62348	−0.593487
$166$	0	0
$167$	17.3328i	1.34125i	0.741796	+	0.670625i	$0.233974\pi$
$167$	17.3328i	1.34125i	−0.741796	+	0.670625i	$0.766026\pi$
$168$	0	0
$169$	−11.8704	−0.913110
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−26.2118	−1.99284	−0.996422	−	0.0845218i	$-0.973064\pi$
$173$	−26.2118	−1.99284	−0.996422	+	0.0845218i	$0.973064\pi$
$174$	0	0
$175$	13.2288i	1.00000i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 11.8322i	− 0.884377i	−0.896922	−	0.442189i	$-0.854202\pi$
$179$	− 11.8322i	− 0.884377i	0.896922	−	0.442189i	$-0.145798\pi$
$180$	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
$187$	31.9801i	2.33861i
$188$	0	0
$189$	−9.12957	−0.664079
$190$	0	0
$191$	26.3407i	1.90595i	0.303052	+	0.952974i	$0.401994\pi$
$191$	26.3407i	1.90595i	−0.303052	+	0.952974i	$0.598006\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	− 1.45827i	− 0.104429i
$196$	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
$203$	− 25.4613i	− 1.78703i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	− 11.0445i	− 0.760332i	−0.924918	−	0.380166i	$-0.875867\pi$
$211$	− 11.0445i	− 0.760332i	0.924918	−	0.380166i	$-0.124133\pi$
$212$	0	0
$213$	7.26040	0.497475
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	8.23252i	0.556302i
$220$	0	0
$221$	−6.11738	−0.411499
$222$	0	0
$223$	− 8.74216i	− 0.585418i	−0.956202	−	0.292709i	$-0.905443\pi$
$223$	− 8.74216i	− 0.585418i	0.956202	−	0.292709i	$-0.0945567\pi$
$224$	0	0
$225$	13.1174	0.874492
$226$	0	0
$227$	29.1430i	1.93429i	0.254225	+	0.967145i	$0.418180\pi$
$227$	29.1430i	1.93429i	−0.254225	+	0.967145i	$0.581820\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	9.02022i	0.593487i
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	30.5248i	1.99122i
$236$	0	0
$237$	5.97671	0.388229
$238$	0	0
$239$	6.99597i	0.452532i	0.974066	+	0.226266i	$0.0726518\pi$
$239$	6.99597i	0.452532i	−0.974066	+	0.226266i	$0.927348\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	13.8821i	0.890539i
$244$	0	0
$245$	15.6525	1.00000
$246$	0	0
$247$	0	0
$248$	0	0
$249$	9.74085	0.617301
$250$	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
$255$	7.89750i	0.494561i
$256$	0	0
$257$	−4.47214	−0.278964	−0.139482	−	0.990225i	$-0.544544\pi$
$257$	−4.47214	−0.278964	−0.139482	+	0.990225i	$0.544544\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−25.2470	−1.56275
$262$	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
$273$	−1.72545	−0.104429
$274$	0	0
$275$	− 27.7806i	− 1.67523i
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	21.6235	1.28995	0.644974	−	0.764204i	$-0.276868\pi$
$281$	21.6235	1.28995	0.644974	+	0.764204i	$0.276868\pi$
$282$	0	0
$283$	− 30.3703i	− 1.80532i	−0.430350	−	0.902662i	$-0.641610\pi$
$283$	− 30.3703i	− 1.80532i	0.430350	−	0.902662i	$-0.358390\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	16.1296	0.948798
$290$	0	0
$291$	− 11.8999i	− 0.697586i
$292$	0	0
$293$	−8.32322	−0.486248	−0.243124	−	0.969995i	$-0.578172\pi$
$293$	−8.32322	−0.486248	−0.243124	+	0.969995i	$0.578172\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	19.1722	1.11249
$298$	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
$307$	− 23.0069i	− 1.31307i	−0.754295	−	0.656535i	$-0.772021\pi$
$307$	− 23.0069i	− 1.31307i	0.754295	−	0.656535i	$-0.227979\pi$
$308$	0	0
$309$	9.88262	0.562203
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	−15.1419	−0.855869	−0.427934	−	0.903810i	$-0.640759\pi$
$313$	−15.1419	−0.855869	−0.427934	+	0.903810i	$0.640759\pi$
$314$	0	0
$315$	− 15.5207i	− 0.874492i
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	53.4691i	2.99370i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	5.31407	0.294772
$326$	0	0
$327$	12.8064i	0.708197i
$328$	0	0
$329$	36.1174	1.99122
$330$	0	0
$331$	35.4965i	1.95106i	0.219860	+	0.975531i	$0.429440\pi$
$331$	35.4965i	1.95106i	−0.219860	+	0.975531i	$0.570560\pi$
$332$	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
$343$	− 18.5203i	− 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
$351$	3.66740i	0.195752i
$352$	0	0
$353$	35.1560	1.87117	0.935583	−	0.353106i	$-0.114874\pi$
$353$	35.1560	1.87117	0.935583	+	0.353106i	$0.114874\pi$
$354$	0	0
$355$	26.4575i	1.40422i
$356$	0	0
$357$	9.34445	0.494561
$358$	0	0
$359$	− 11.8322i	− 0.624477i	−0.950004	−	0.312239i	$-0.898921\pi$
$359$	− 11.8322i	− 0.624477i	0.950004	−	0.312239i	$-0.101079\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	− 12.1928i	− 0.639957i
$364$	0	0
$365$	−30.0000	−1.57027
$366$	0	0
$367$	19.7872i	1.03289i	0.856322	+	0.516443i	$0.172744\pi$
$367$	19.7872i	1.03289i	−0.856322	+	0.516443i	$0.827256\pi$
$368$	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
$375$	− 6.86044i	− 0.354271i
$376$	0	0
$377$	−10.2280	−0.526767
$378$	0	0
$379$	35.4965i	1.82333i	0.410932	+	0.911666i	$0.365203\pi$
$379$	35.4965i	1.82333i	−0.410932	+	0.911666i	$0.634797\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 15.8745i	− 0.811149i	−0.914062	−	0.405575i	$-0.867071\pi$
$383$	− 15.8745i	− 0.811149i	0.914062	−	0.405575i	$-0.132929\pi$
$384$	0	0
$385$	−32.8704	−1.67523
$386$	0	0
$387$	0	0
$388$	0	0
$389$	14.3765	0.728919	0.364459	−	0.931219i	$-0.381254\pi$
$389$	14.3765	0.728919	0.364459	+	0.931219i	$0.381254\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	21.7796i	1.09585i
$396$	0	0
$397$	−39.8490	−1.99997	−0.999983	−	0.00579782i	$-0.998154\pi$
$397$	−39.8490	−1.99997	−0.999983	+	0.00579782i	$0.998154\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	12.1174	0.605113	0.302556	−	0.953131i	$-0.402160\pi$
$401$	12.1174	0.605113	0.302556	+	0.953131i	$0.402160\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−12.8642	−0.639228
$406$	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
$415$	35.4965i	1.74245i
$416$	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
$421$	−3.12957	−0.152526	−0.0762630	−	0.997088i	$-0.524299\pi$
$421$	−3.12957	−0.152526	−0.0762630	+	0.997088i	$0.524299\pi$
$422$	0	0
$423$	− 35.8133i	− 1.74130i
$424$	0	0
$425$	−28.7791	−1.39599
$426$	0	0
$427$	0	0
$428$	0	0
$429$	3.62348	0.174943
$430$	0	0
$431$	− 18.1082i	− 0.872241i	−0.899888	−	0.436121i	$-0.856352\pi$
$431$	− 18.1082i	− 0.872241i	0.899888	−	0.436121i	$-0.143648\pi$
$432$	0	0
$433$	−40.2492	−1.93425	−0.967127	−	0.254293i	$-0.918157\pi$
$433$	−40.2492	−1.93425	−0.967127	+	0.254293i	$0.918157\pi$
$434$	0	0
$435$	13.2042i	0.633095i
$436$	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
$441$	−18.3643	−0.874492
$442$	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
$447$	− 3.68170i	− 0.174138i
$448$	0	0
$449$	−31.3643	−1.48017	−0.740087	−	0.672511i	$-0.765216\pi$
$449$	−31.3643	−1.48017	−0.740087	+	0.672511i	$0.765216\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	14.4792	0.680293
$454$	0	0
$455$	− 6.28769i	− 0.294772i
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	− 19.8614i	− 0.927050i
$460$	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
$467$	− 32.8247i	− 1.51895i	−0.650538	−	0.759473i	$-0.725457\pi$
$467$	− 32.8247i	− 1.51895i	0.650538	−	0.759473i	$-0.274543\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	8.23252i	0.379335i
$472$	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
$485$	43.3643	1.96907
$486$	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
$491$	13.7886i	0.622273i	0.950365	+	0.311136i	$0.100710\pi$
$491$	13.7886i	0.622273i	−0.950365	+	0.311136i	$0.899290\pi$
$492$	0	0
$493$	55.3911	2.49469
$494$	0	0
$495$	32.5937i	1.46498i
$496$	0	0
$497$	31.3050	1.40422
$498$	0	0
$499$	44.3812i	1.98677i	0.114816	+	0.993387i	$0.463372\pi$
$499$	44.3812i	1.98677i	−0.114816	+	0.993387i	$0.536628\pi$
$500$	0	0
$501$	−10.6357	−0.475167
$502$	0	0
$503$	− 21.0145i	− 0.936989i	−0.883466	−	0.468495i	$-0.844797\pi$
$503$	− 21.0145i	− 0.936989i	0.883466	−	0.468495i	$-0.155203\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 7.28388i	− 0.323489i
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	35.4965i	1.57027i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	36.0131i	1.58693i
$516$	0	0
$517$	−75.8470	−3.33575
$518$	0	0
$519$	− 16.0840i	− 0.706007i
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	− 37.0405i	− 1.61967i	−0.586659	−	0.809834i	$-0.699557\pi$
$523$	− 37.0405i	− 1.61967i	0.586659	−	0.809834i	$-0.300443\pi$
$524$	0	0
$525$	−8.11738	−0.354271
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	7.26040	0.313309
$538$	0	0
$539$	38.8928i	1.67523i
$540$	0	0
$541$	36.8704	1.58518	0.792592	−	0.609753i	$-0.208731\pi$
$541$	36.8704	1.58518	0.792592	+	0.609753i	$0.208731\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−46.6677	−1.99902
$546$	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
$553$	25.7700	1.09585
$554$	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
$561$	−19.6235	−0.828504
$562$	0	0
$563$	− 15.8745i	− 0.669031i	−0.942390	−	0.334515i	$-0.891427\pi$
$563$	− 15.8745i	− 0.669031i	0.942390	−	0.334515i	$-0.108573\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	15.2211i	0.639228i
$568$	0	0
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	35.4965i	1.48548i	0.669579	+	0.742741i	$0.266474\pi$
$571$	35.4965i	1.48548i	−0.669579	+	0.742741i	$0.733526\pi$
$572$	0	0
$573$	−16.1631	−0.675223
$574$	0	0
$575$	0	0
$576$	0	0
$577$	13.0162	0.541873	0.270936	−	0.962597i	$-0.412667\pi$
$577$	13.0162	0.541873	0.270936	+	0.962597i	$0.412667\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	42.0000	1.74245
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−6.23475	−0.257775
$586$	0	0
$587$	47.6235i	1.96563i	0.184585	+	0.982817i	$0.440906\pi$
$587$	47.6235i	1.96563i	−0.184585	+	0.982817i	$0.559094\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−0.621055	−0.0255037	−0.0127518	−	0.999919i	$-0.504059\pi$
$593$	−0.621055	−0.0255037	−0.0127518	+	0.999919i	$0.504059\pi$
$594$	0	0
$595$	34.0519i	1.39599i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	48.5652i	1.98432i	0.124975	+	0.992160i	$0.460115\pi$
$599$	48.5652i	1.98432i	−0.124975	+	0.992160i	$0.539885\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	44.4316	1.80640
$606$	0	0
$607$	5.06046i	0.205398i	0.994712	+	0.102699i	$0.0327478\pi$
$607$	5.06046i	0.205398i	−0.994712	+	0.102699i	$0.967252\pi$
$608$	0	0
$609$	15.6235	0.633095
$610$	0	0
$611$	− 14.5086i	− 0.586954i
$612$	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
$625$	25.0000	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	31.8291i	1.26710i	0.773704	+	0.633548i	$0.218402\pi$
$631$	31.8291i	1.26710i	−0.773704	+	0.633548i	$0.781598\pi$
$632$	0	0
$633$	6.77706	0.269364
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−7.43970	−0.294772
$638$	0	0
$639$	− 31.0414i	− 1.22798i
$640$	0	0
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	0	0
$643$	19.3252i	0.762110i	0.924552	+	0.381055i	$0.124439\pi$
$643$	19.3252i	0.762110i	−0.924552	+	0.381055i	$0.875561\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	47.6235i	1.87227i	0.351636	+	0.936137i	$0.385626\pi$
$647$	47.6235i	1.87227i	−0.351636	+	0.936137i	$0.614374\pi$
$648$	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
$657$	35.1976	1.37319
$658$	0	0
$659$	− 47.1253i	− 1.83574i	−0.396878	−	0.917871i	$-0.629907\pi$
$659$	− 47.1253i	− 1.83574i	0.396878	−	0.917871i	$-0.370093\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	− 3.75372i	− 0.145782i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	5.36433	0.207397
$670$	0	0
$671$	0	0
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	17.2533i	0.664079i
$676$	0	0
$677$	32.5886	1.25248	0.626242	−	0.779629i	$-0.284592\pi$
$677$	32.5886	1.25248	0.626242	+	0.779629i	$0.284592\pi$
$678$	0	0
$679$	− 51.3094i	− 1.96907i
$680$	0	0
$681$	−17.8826	−0.685264
$682$	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
$693$	38.5654	1.46498
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	19.3643	0.731381	0.365690	−	0.930737i	$-0.380833\pi$
$701$	19.3643	0.731381	0.365690	+	0.930737i	$0.380833\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−18.7305	−0.705431
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−46.6113	−1.75052	−0.875262	−	0.483650i	$-0.839311\pi$
$709$	−46.6113	−1.75052	−0.875262	+	0.483650i	$0.839311\pi$
$710$	0	0
$711$	− 25.5530i	− 0.958314i
$712$	0	0
$713$	0	0
$714$	0	0
$715$	13.2042i	0.493811i
$716$	0	0
$717$	−4.29284	−0.160319
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	42.6113	1.58693
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−48.1174	−1.78703
$726$	0	0
$727$	5.29150i	0.196251i	0.995174	+	0.0981255i	$0.0312847\pi$
$727$	5.29150i	0.196251i	−0.995174	+	0.0981255i	$0.968715\pi$
$728$	0	0
$729$	8.74085	0.323735
$730$	0	0
$731$	0	0
$732$	0	0
$733$	33.4722	1.23632	0.618161	−	0.786051i	$-0.287878\pi$
$733$	33.4722	1.23632	0.618161	+	0.786051i	$0.287878\pi$
$734$	0	0
$735$	9.60461i	0.354271i
$736$	0	0
$737$	0	0
$738$	0	0
$739$	− 36.1486i	− 1.32975i	−0.746955	−	0.664875i	$-0.768485\pi$
$739$	− 36.1486i	− 1.32975i	0.746955	−	0.664875i	$-0.231515\pi$
$740$	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
$745$	13.4164	0.491539
$746$	0	0
$747$	− 41.6464i	− 1.52376i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	15.3640i	0.560641i	0.959906	+	0.280321i	$0.0904408\pi$
$751$	15.3640i	0.560641i	−0.959906	+	0.280321i	$0.909559\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	52.7635i	1.92026i
$756$	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
$763$	55.2180i	1.99902i
$764$	0	0
$765$	33.7652	1.22079
$766$	0	0
$767$	0	0
$768$	0	0
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	− 2.74417i	− 0.0988290i
$772$	0	0
$773$	−49.2351	−1.77086	−0.885431	−	0.464770i	$-0.846137\pi$
$773$	−49.2351	−1.77086	−0.885431	+	0.464770i	$0.846137\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−65.7409	−2.35239
$782$	0	0
$783$	− 33.2073i	− 1.18673i
$784$	0	0
$785$	−30.0000	−1.07075
$786$	0	0
$787$	− 51.5363i	− 1.83707i	−0.395340	−	0.918535i	$-0.629373\pi$
$787$	− 51.5363i	− 1.83707i	0.395340	−	0.918535i	$-0.370627\pi$
$788$	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
$797$	−21.0770	−0.746585	−0.373293	−	0.927714i	$-0.621771\pi$
$797$	−21.0770	−0.746585	−0.373293	+	0.927714i	$0.621771\pi$
$798$	0	0
$799$	78.5733i	2.77972i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 74.5431i	− 2.63057i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−55.3643	−1.94651	−0.973253	−	0.229736i	$-0.926214\pi$
$809$	−55.3643	−1.94651	−0.973253	+	0.229736i	$0.926214\pi$
$810$	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
$819$	7.37706i	0.257775i
$820$	0	0
$821$	−33.6235	−1.17347	−0.586734	−	0.809780i	$-0.699586\pi$
$821$	−33.6235	−1.17347	−0.586734	+	0.809780i	$0.699586\pi$
$822$	0	0
$823$	0	0		−	1.00000i	$-0.5\pi$
$823$	0	0			1.00000i	$0.5\pi$
$824$	0	0
$825$	17.0466	0.593487
$826$	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
$833$	40.2908	1.39599
$834$	0	0
$835$	− 38.7573i	− 1.34125i
$836$	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
$841$	63.6113	2.19349
$842$	0	0
$843$	13.2685i	0.456992i
$844$	0	0
$845$	26.5431	0.913110
$846$	0	0
$847$	− 52.5722i	− 1.80640i
$848$	0	0
$849$	18.6357	0.639575
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−40.2492	−1.37811	−0.689054	−	0.724710i	$-0.741974\pi$
$853$	−40.2492	−1.37811	−0.689054	+	0.724710i	$0.741974\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	49.1935	1.68042	0.840209	−	0.542263i	$-0.182432\pi$
$857$	49.1935	1.68042	0.840209	+	0.542263i	$0.182432\pi$
$858$	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
$865$	58.6113	1.99284
$866$	0	0
$867$	9.89736i	0.336132i
$868$	0	0
$869$	−54.1174	−1.83581
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−50.8774	−1.72194
$874$	0	0
$875$	− 29.5804i	− 1.00000i
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	− 5.10726i	− 0.172264i
$880$	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
$887$	47.6235i	1.59904i	0.600639	+	0.799521i	$0.294913\pi$
$887$	47.6235i	1.59904i	−0.600639	+	0.799521i	$0.705087\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	− 31.9646i	− 1.07085i
$892$	0	0
$893$	0	0
$894$	0	0
$895$	26.4575i	0.884377i
$896$	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
$911$	− 59.1608i	− 1.96008i	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	− 59.1608i	− 1.96008i	0.198789	−	0.980042i	$-0.436299\pi$
$912$	0	0
$913$	−88.2006	−2.91901
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	1.50762i	0.0497319i	0.999691	+	0.0248659i	$0.00791589\pi$
$919$	1.50762i	0.0497319i	−0.999691	+	0.0248659i	$0.992084\pi$
$920$	0	0
$921$	14.1174	0.465183
$922$	0	0
$923$	− 12.5754i	− 0.413924i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 42.2525i	− 1.38775i
$928$	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
$935$	− 71.5096i	− 2.33861i
$936$	0	0
$937$	−56.0537	−1.83120	−0.915598	−	0.402096i	$-0.868282\pi$
$937$	−56.0537	−1.83120	−0.915598	+	0.402096i	$0.868282\pi$
$938$	0	0
$939$	− 9.29129i	− 0.303210i
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	20.4143	0.664079
$946$	0	0
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	14.2591	0.462872
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	− 58.8997i	− 1.90595i
$956$	0	0
$957$	−32.8095	−1.06058
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−31.0000	−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
$975$	3.26080i	0.104429i
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	54.7530	1.74813
$982$	0	0
$983$	2.60600i	0.0831184i	0.999136	+	0.0415592i	$0.0132325\pi$
$983$	2.60600i	0.0831184i	−0.999136	+	0.0415592i	$0.986767\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	22.1622i	0.705431i
$988$	0	0
$989$	0	0
$990$	0	0
$991$	35.4965i	1.12758i	0.825917	+	0.563791i	$0.190658\pi$
$991$	35.4965i	1.12758i	−0.825917	+	0.563791i	$0.809342\pi$
$992$	0	0
$993$	−21.7812	−0.691206
$994$	0	0
$995$	0	0
$996$	0	0
$997$	46.2259	1.46399	0.731995	−	0.681310i	$-0.238589\pi$
$997$	46.2259	1.46399	0.731995	+	0.681310i	$0.238589\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.e.d.2239.5		8
4.3	odd	2	inner	2240.2.e.d.2239.3		8
5.4	even	2	inner	2240.2.e.d.2239.4		8
7.6	odd	2	inner	2240.2.e.d.2239.4		8
8.3	odd	2		560.2.e.c.559.6	yes	8
8.5	even	2		560.2.e.c.559.4	yes	8
20.19	odd	2	inner	2240.2.e.d.2239.6		8
28.27	even	2	inner	2240.2.e.d.2239.6		8
35.34	odd	2	CM	2240.2.e.d.2239.5		8
40.3	even	4		2800.2.k.p.2351.3		8
40.13	odd	4		2800.2.k.p.2351.6		8
40.19	odd	2		560.2.e.c.559.3	✓	8
40.27	even	4		2800.2.k.p.2351.5		8
40.29	even	2		560.2.e.c.559.5	yes	8
40.37	odd	4		2800.2.k.p.2351.4		8
56.13	odd	2		560.2.e.c.559.5	yes	8
56.27	even	2		560.2.e.c.559.3	✓	8
140.139	even	2	inner	2240.2.e.d.2239.3		8
280.13	even	4		2800.2.k.p.2351.4		8
280.27	odd	4		2800.2.k.p.2351.3		8
280.69	odd	2		560.2.e.c.559.4	yes	8
280.83	odd	4		2800.2.k.p.2351.5		8
280.139	even	2		560.2.e.c.559.6	yes	8
280.237	even	4		2800.2.k.p.2351.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
560.2.e.c.559.3	✓	8	40.19	odd	2
560.2.e.c.559.3	✓	8	56.27	even	2
560.2.e.c.559.4	yes	8	8.5	even	2
560.2.e.c.559.4	yes	8	280.69	odd	2
560.2.e.c.559.5	yes	8	40.29	even	2
560.2.e.c.559.5	yes	8	56.13	odd	2
560.2.e.c.559.6	yes	8	8.3	odd	2
560.2.e.c.559.6	yes	8	280.139	even	2
2240.2.e.d.2239.3		8	4.3	odd	2	inner
2240.2.e.d.2239.3		8	140.139	even	2	inner
2240.2.e.d.2239.4		8	5.4	even	2	inner
2240.2.e.d.2239.4		8	7.6	odd	2	inner
2240.2.e.d.2239.5		8	1.1	even	1	trivial
2240.2.e.d.2239.5		8	35.34	odd	2	CM
2240.2.e.d.2239.6		8	20.19	odd	2	inner
2240.2.e.d.2239.6		8	28.27	even	2	inner
2800.2.k.p.2351.3		8	40.3	even	4
2800.2.k.p.2351.3		8	280.27	odd	4
2800.2.k.p.2351.4		8	40.37	odd	4
2800.2.k.p.2351.4		8	280.13	even	4
2800.2.k.p.2351.5		8	40.27	even	4
2800.2.k.p.2351.5		8	280.83	odd	4
2800.2.k.p.2351.6		8	40.13	odd	4
2800.2.k.p.2351.6		8	280.237	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2240.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.613616i q^{3} -2.23607 q^{5} +2.64575i q^{7} +2.62348 q^{9} +O(q^{10})\)	\(q+0.613616i q^{3} -2.23607 q^{5} +2.64575i q^{7} +2.62348 q^{9} -5.55612i q^{11} +1.06281 q^{13} -1.37209i q^{15} -5.75583 q^{17} -1.62348 q^{21} +5.00000 q^{25} +3.45065i q^{27} -9.62348 q^{29} +3.40932 q^{33} -5.91608i q^{35} +0.652160i q^{39} -5.86627 q^{45} -13.6511i q^{47} -7.00000 q^{49} -3.53187i q^{51} +12.4239i q^{55} +6.94106i q^{63} -2.37652 q^{65} -11.8322i q^{71} +13.4164 q^{73} +3.06808i q^{75} +14.7001 q^{77} -9.74015i q^{79} +5.75305 q^{81} -15.8745i q^{83} +12.8704 q^{85} -5.90512i q^{87} +2.81194i q^{91} -19.3931 q^{97} -14.5763i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 20 q^{9}+O(q^{10}) \) 8 * q - 20 * q^9	\( 8 q - 20 q^{9} + 28 q^{21} + 40 q^{25} - 36 q^{29} - 56 q^{49} - 60 q^{65} + 128 q^{81} - 20 q^{85}+O(q^{100}) \) 8 * q - 20 * q^9 + 28 * q^21 + 40 * q^25 - 36 * q^29 - 56 * q^49 - 60 * q^65 + 128 * q^81 - 20 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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