LMFDB - Embedded newform 3025.2.a.bb.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3025,2,Mod(1,3025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3025, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3025 = 5^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3025.a (trivial)

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:	yes
Analytic conductor:	$24.1547466114$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.4400.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 7x^{2} + 11 $ x^4 - 7*x^2 + 11
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 605)
Fricke sign:	$-1$
Sato-Tate group:	$N(\mathrm{U}(1))$

Embedding invariants

Embedding label			1.4
Root			$1.54336$ of defining polynomial
Character	$\chi$	$=$	3025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.49721 q^{2} +4.23607 q^{4} +3.08672 q^{7} +5.58394 q^{8} -3.00000 q^{9} +O(q^{10})$	$q+2.49721 q^{2} +4.23607 q^{4} +3.08672 q^{7} +5.58394 q^{8} -3.00000 q^{9} +1.90770 q^{13} +7.70820 q^{14} +5.47214 q^{16} +8.08115 q^{17} -7.49164 q^{18} +4.76393 q^{26} +13.0756 q^{28} -8.94427 q^{31} +2.49721 q^{32} +20.1803 q^{34} -12.7082 q^{36} +13.0756 q^{43} +2.52786 q^{49} +8.08115 q^{52} +17.2361 q^{56} +4.00000 q^{59} -22.3357 q^{62} -9.26017 q^{63} -4.70820 q^{64} +34.2323 q^{68} +8.00000 q^{71} -16.7518 q^{72} -11.8965 q^{73} +9.00000 q^{81} +0.728677 q^{83} +32.6525 q^{86} -13.4164 q^{89} +5.88854 q^{91} +6.31261 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{4} - 12 q^{9}+O(q^{10}) $ 4 * q + 8 * q^4 - 12 * q^9	$ 4 q + 8 q^{4} - 12 q^{9} + 4 q^{14} + 4 q^{16} + 28 q^{26} + 36 q^{34} - 24 q^{36} + 28 q^{49} + 60 q^{56} + 16 q^{59} + 8 q^{64} + 32 q^{71} + 36 q^{81} + 68 q^{86} - 48 q^{91}+O(q^{100}) $ 4 * q + 8 * q^4 - 12 * q^9 + 4 * q^14 + 4 * q^16 + 28 * q^26 + 36 * q^34 - 24 * q^36 + 28 * q^49 + 60 * q^56 + 16 * q^59 + 8 * q^64 + 32 * q^71 + 36 * q^81 + 68 * q^86 - 48 * q^91

Display 100 coefficients 10 coefficients

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 $

$2$	2.49721	1.76580	0.882898	−	0.469565i	$-0.155589\pi$
$2$	2.49721	1.76580	0.882898	+	0.469565i	$0.155589\pi$
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	4.23607	2.11803
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.08672	1.16667	0.583336	−	0.812231i	$-0.301747\pi$
$7$	3.08672	1.16667	0.583336	+	0.812231i	$0.301747\pi$
$8$	5.58394	1.97422
$9$	−3.00000	−1.00000
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	1.90770	0.529101	0.264550	−	0.964372i	$-0.414776\pi$
$13$	1.90770	0.529101	0.264550	+	0.964372i	$0.414776\pi$
$14$	7.70820	2.06010
$15$	0	0
$16$	5.47214	1.36803
$17$	8.08115	1.95997	0.979983	−	0.199081i	$-0.0637955\pi$
$17$	8.08115	1.95997	0.979983	+	0.199081i	$0.0637955\pi$
$18$	−7.49164	−1.76580
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	0	0
$26$	4.76393	0.934284
$27$	0	0
$28$	13.0756	2.47105
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−8.94427	−1.60644	−0.803219	−	0.595683i	$-0.796881\pi$
$31$	−8.94427	−1.60644	−0.803219	+	0.595683i	$0.796881\pi$
$32$	2.49721	0.441449
$33$	0	0
$34$	20.1803	3.46090
$35$	0	0
$36$	−12.7082	−2.11803
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	13.0756	1.99401	0.997003	−	0.0773627i	$-0.0246499\pi$
$43$	13.0756	1.99401	0.997003	+	0.0773627i	$0.0246499\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	2.52786	0.361123
$50$	0	0
$51$	0	0
$52$	8.08115	1.12065
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	0	0
$56$	17.2361	2.30327
$57$	0	0
$58$	0	0
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	−22.3357	−2.83664
$63$	−9.26017	−1.16667
$64$	−4.70820	−0.588525
$65$	0	0
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	34.2323	4.15128
$69$	0	0
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	−16.7518	−1.97422
$73$	−11.8965	−1.39239	−0.696193	−	0.717855i	$-0.745124\pi$
$73$	−11.8965	−1.39239	−0.696193	+	0.717855i	$0.745124\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	0.728677	0.0799827	0.0399913	−	0.999200i	$-0.487267\pi$
$83$	0.728677	0.0799827	0.0399913	+	0.999200i	$0.487267\pi$
$84$	0	0
$85$	0	0
$86$	32.6525	3.52101
$87$	0	0
$88$	0	0
$89$	−13.4164	−1.42214	−0.711068	−	0.703123i	$-0.751788\pi$
$89$	−13.4164	−1.42214	−0.711068	+	0.703123i	$0.751788\pi$
$90$	0	0
$91$	5.88854	0.617287
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	6.31261	0.637670
$99$	0	0
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	10.6525	1.04456
$105$	0	0
$106$	0	0
$107$	−19.2490	−1.86087	−0.930436	−	0.366453i	$-0.880572\pi$
$107$	−19.2490	−1.86087	−0.930436	+	0.366453i	$0.880572\pi$
$108$	0	0
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	0	0
$111$	0	0
$112$	16.8910	1.59605
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−5.72310	−0.529101
$118$	9.98885	0.919548
$119$	24.9443	2.28664
$120$	0	0
$121$	0	0
$122$	0	0
$123$	0	0
$124$	−37.8885	−3.40249
$125$	0	0
$126$	−23.1246	−2.06010
$127$	16.8910	1.49883	0.749416	−	0.662100i	$-0.230334\pi$
$127$	16.8910	1.49883	0.749416	+	0.662100i	$0.230334\pi$
$128$	−16.7518	−1.48066
$129$	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
$135$	0	0
$136$	45.1246	3.86940
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	0	0
$142$	19.9777	1.67649
$143$	0	0
$144$	−16.4164	−1.36803
$145$	0	0
$146$	−29.7082	−2.45867
$147$	0	0
$148$	0	0
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	−24.2434	−1.95997
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	22.4749	1.76580
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	0	0
$165$	0	0
$166$	1.81966	0.141233
$167$	−10.7175	−0.829347	−0.414673	−	0.909970i	$-0.636104\pi$
$167$	−10.7175	−0.829347	−0.414673	+	0.909970i	$0.636104\pi$
$168$	0	0
$169$	−9.36068	−0.720052
$170$	0	0
$171$	0	0
$172$	55.3890	4.22337
$173$	10.4392	0.793677	0.396839	−	0.917888i	$-0.370107\pi$
$173$	10.4392	0.793677	0.396839	+	0.917888i	$0.370107\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	−33.5036	−2.51120
$179$	−17.8885	−1.33705	−0.668526	−	0.743689i	$-0.733075\pi$
$179$	−17.8885	−1.33705	−0.668526	+	0.743689i	$0.733075\pi$
$180$	0	0
$181$	4.47214	0.332411	0.166206	−	0.986091i	$-0.446848\pi$
$181$	4.47214	0.332411	0.166206	+	0.986091i	$0.446848\pi$
$182$	14.7049	1.09000
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−26.8328	−1.94155	−0.970777	−	0.239983i	$-0.922858\pi$
$191$	−26.8328	−1.94155	−0.970777	+	0.239983i	$0.922858\pi$
$192$	0	0
$193$	20.4280	1.47044	0.735221	−	0.677827i	$-0.237078\pi$
$193$	20.4280	1.47044	0.735221	+	0.677827i	$0.237078\pi$
$194$	0	0
$195$	0	0
$196$	10.7082	0.764872
$197$	−21.8854	−1.55927	−0.779635	−	0.626234i	$-0.784595\pi$
$197$	−21.8854	−1.55927	−0.779635	+	0.626234i	$0.784595\pi$
$198$	0	0
$199$	−24.0000	−1.70131	−0.850657	−	0.525720i	$-0.823796\pi$
$199$	−24.0000	−1.70131	−0.850657	+	0.525720i	$0.823796\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	10.4392	0.723828
$209$	0	0
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	0	0
$213$	0	0
$214$	−48.0689	−3.28592
$215$	0	0
$216$	0	0
$217$	−27.6085	−1.87419
$218$	0	0
$219$	0	0
$220$	0	0
$221$	15.4164	1.03702
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	7.70820	0.515026
$225$	0	0
$226$	0	0
$227$	−25.4225	−1.68735	−0.843674	−	0.536855i	$-0.819612\pi$
$227$	−25.4225	−1.68735	−0.843674	+	0.536855i	$0.819612\pi$
$228$	0	0
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−15.7119	−1.02932	−0.514662	−	0.857393i	$-0.672083\pi$
$233$	−15.7119	−1.02932	−0.514662	+	0.857393i	$0.672083\pi$
$234$	−14.2918	−0.934284
$235$	0	0
$236$	16.9443	1.10298
$237$	0	0
$238$	62.2911	4.03773
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	−49.9442	−3.17146
$249$	0	0
$250$	0	0
$251$	28.0000	1.76734	0.883672	−	0.468106i	$-0.155064\pi$
$251$	28.0000	1.76734	0.883672	+	0.468106i	$0.155064\pi$
$252$	−39.2267	−2.47105
$253$	0	0
$254$	42.1803	2.64663
$255$	0	0
$256$	−32.4164	−2.02603
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−4.54408	−0.280200	−0.140100	−	0.990137i	$-0.544742\pi$
$263$	−4.54408	−0.280200	−0.140100	+	0.990137i	$0.544742\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	14.0000	0.853595	0.426798	−	0.904347i	$-0.359642\pi$
$269$	14.0000	0.853595	0.426798	+	0.904347i	$0.359642\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	44.2211	2.68130
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−30.4169	−1.82757	−0.913787	−	0.406194i	$-0.866856\pi$
$277$	−30.4169	−1.82757	−0.913787	+	0.406194i	$0.866856\pi$
$278$	0	0
$279$	26.8328	1.60644
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	−26.8798	−1.59784	−0.798920	−	0.601438i	$-0.794595\pi$
$283$	−26.8798	−1.59784	−0.798920	+	0.601438i	$0.794595\pi$
$284$	33.8885	2.01092
$285$	0	0
$286$	0	0
$287$	0	0
$288$	−7.49164	−0.441449
$289$	48.3050	2.84147
$290$	0	0
$291$	0	0
$292$	−50.3946	−2.94912
$293$	34.2323	1.99987	0.999936	−	0.0113203i	$-0.00360343\pi$
$293$	34.2323	1.99987	0.999936	+	0.0113203i	$0.00360343\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	40.3607	2.32635
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	−60.5410	−3.46090
$307$	33.0533	1.88645	0.943225	−	0.332155i	$-0.107776\pi$
$307$	33.0533	1.88645	0.943225	+	0.332155i	$0.107776\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	32.0000	1.81455	0.907277	−	0.420534i	$-0.138157\pi$
$311$	32.0000	1.81455	0.907277	+	0.420534i	$0.138157\pi$
$312$	0	0
$313$	0	0		−	1.00000i	$-0.5\pi$
$313$	0	0			1.00000i	$0.5\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	38.1246	2.11803
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−35.7771	−1.96649	−0.983243	−	0.182298i	$-0.941646\pi$
$331$	−35.7771	−1.96649	−0.983243	+	0.182298i	$0.941646\pi$
$332$	3.08672	0.169406
$333$	0	0
$334$	−26.7639	−1.46446
$335$	0	0
$336$	0	0
$337$	−16.6126	−0.904948	−0.452474	−	0.891778i	$-0.649459\pi$
$337$	−16.6126	−0.904948	−0.452474	+	0.891778i	$0.649459\pi$
$338$	−23.3756	−1.27147
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−13.8042	−0.745359
$344$	73.0132	3.93661
$345$	0	0
$346$	26.0689	1.40147
$347$	−11.6182	−0.623699	−0.311849	−	0.950132i	$-0.600948\pi$
$347$	−11.6182	−0.623699	−0.311849	+	0.950132i	$0.600948\pi$
$348$	0	0
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	0	0
$356$	−56.8328	−3.01213
$357$	0	0
$358$	−44.6715	−2.36096
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	11.1679	0.586970
$363$	0	0
$364$	24.9443	1.30744
$365$	0	0
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−6.62379	−0.342967	−0.171484	−	0.985187i	$-0.554856\pi$
$373$	−6.62379	−0.342967	−0.171484	+	0.985187i	$0.554856\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	36.0000	1.84920	0.924598	−	0.380945i	$-0.124401\pi$
$379$	36.0000	1.84920	0.924598	+	0.380945i	$0.124401\pi$
$380$	0	0
$381$	0	0
$382$	−67.0072	−3.42839
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	51.0132	2.59650
$387$	−39.2267	−1.99401
$388$	0	0
$389$	26.0000	1.31825	0.659126	−	0.752032i	$-0.270926\pi$
$389$	26.0000	1.31825	0.659126	+	0.752032i	$0.270926\pi$
$390$	0	0
$391$	0	0
$392$	14.1154	0.712937
$393$	0	0
$394$	−54.6525	−2.75335
$395$	0	0
$396$	0	0
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	−59.9331	−3.00417
$399$	0	0
$400$	0	0
$401$	4.47214	0.223328	0.111664	−	0.993746i	$-0.464382\pi$
$401$	4.47214	0.223328	0.111664	+	0.993746i	$0.464382\pi$
$402$	0	0
$403$	−17.0630	−0.849968
$404$	0	0
$405$	0	0
$406$	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
$411$	0	0
$412$	0	0
$413$	12.3469	0.607551
$414$	0	0
$415$	0	0
$416$	4.76393	0.233571
$417$	0	0
$418$	0	0
$419$	35.7771	1.74783	0.873913	−	0.486083i	$-0.161575\pi$
$419$	35.7771	1.74783	0.873913	+	0.486083i	$0.161575\pi$
$420$	0	0
$421$	−31.3050	−1.52571	−0.762855	−	0.646570i	$-0.776203\pi$
$421$	−31.3050	−1.52571	−0.762855	+	0.646570i	$0.776203\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	−81.5402	−3.94139
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	0	0		−	1.00000i	$-0.5\pi$
$433$	0	0			1.00000i	$0.5\pi$
$434$	−68.9443	−3.30943
$435$	0	0
$436$	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
$441$	−7.58359	−0.361123
$442$	38.4980	1.83116
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	−14.5329	−0.686616
$449$	−40.2492	−1.89948	−0.949739	−	0.313042i	$-0.898652\pi$
$449$	−40.2492	−1.89948	−0.949739	+	0.313042i	$0.898652\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	−63.4853	−2.97951
$455$	0	0
$456$	0	0
$457$	36.5903	1.71162	0.855812	−	0.517287i	$-0.173058\pi$
$457$	36.5903	1.71162	0.855812	+	0.517287i	$0.173058\pi$
$458$	−14.9833	−0.700122
$459$	0	0
$460$	0	0
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	0	0
$465$	0	0
$466$	−39.2361	−1.81758
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	−24.2434	−1.12065
$469$	0	0
$470$	0	0
$471$	0	0
$472$	22.3357	1.02809
$473$	0	0
$474$	0	0
$475$	0	0
$476$	105.666	4.84318
$477$	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
$485$	0	0
$486$	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
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	−48.9443	−2.19766
$497$	24.6938	1.10767
$498$	0	0
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	0	0
$501$	0	0
$502$	69.9219	3.12077
$503$	36.8687	1.64389	0.821946	−	0.569565i	$-0.192888\pi$
$503$	36.8687	1.64389	0.821946	+	0.569565i	$0.192888\pi$
$504$	−51.7082	−2.30327
$505$	0	0
$506$	0	0
$507$	0	0
$508$	71.5513	3.17458
$509$	34.0000	1.50702	0.753512	−	0.657434i	$-0.228358\pi$
$509$	34.0000	1.50702	0.753512	+	0.657434i	$0.228358\pi$
$510$	0	0
$511$	−36.7214	−1.62446
$512$	−47.4470	−2.09688
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	22.3607	0.979639	0.489820	−	0.871824i	$-0.337063\pi$
$521$	22.3607	0.979639	0.489820	+	0.871824i	$0.337063\pi$
$522$	0	0
$523$	−45.4002	−1.98521	−0.992605	−	0.121387i	$-0.961266\pi$
$523$	−45.4002	−1.98521	−0.992605	+	0.121387i	$0.961266\pi$
$524$	0	0
$525$	0	0
$526$	−11.3475	−0.494776
$527$	−72.2800	−3.14857
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	−12.0000	−0.520756
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	34.9610	1.50727
$539$	0	0
$540$	0	0
$541$	0	0		−	1.00000i	$-0.5\pi$
$541$	0	0			1.00000i	$0.5\pi$
$542$	0	0
$543$	0	0
$544$	20.1803	0.865225
$545$	0	0
$546$	0	0
$547$	−8.35948	−0.357425	−0.178713	−	0.983901i	$-0.557193\pi$
$547$	−8.35948	−0.357425	−0.178713	+	0.983901i	$0.557193\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	−75.9574	−3.22712
$555$	0	0
$556$	0	0
$557$	22.7861	0.965478	0.482739	−	0.875764i	$-0.339642\pi$
$557$	22.7861	0.965478	0.482739	+	0.875764i	$0.339642\pi$
$558$	67.0072	2.83664
$559$	24.9443	1.05503
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−17.7917	−0.749829	−0.374915	−	0.927059i	$-0.622328\pi$
$563$	−17.7917	−0.749829	−0.374915	+	0.927059i	$0.622328\pi$
$564$	0	0
$565$	0	0
$566$	−67.1246	−2.82146
$567$	27.7805	1.16667
$568$	44.6715	1.87437
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	0	0		−	1.00000i	$-0.5\pi$
$571$	0	0			1.00000i	$0.5\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	14.1246	0.588525
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	120.628	5.01745
$579$	0	0
$580$	0	0
$581$	2.24922	0.0933135
$582$	0	0
$583$	0	0
$584$	−66.4296	−2.74887
$585$	0	0
$586$	85.4853	3.53136
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	32.7749	1.34591	0.672953	−	0.739686i	$-0.265026\pi$
$593$	32.7749	1.34591	0.672953	+	0.739686i	$0.265026\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	44.7214	1.82727	0.913633	−	0.406541i	$-0.133265\pi$
$599$	44.7214	1.82727	0.913633	+	0.406541i	$0.133265\pi$
$600$	0	0
$601$	0	0		−	1.00000i	$-0.5\pi$
$601$	0	0			1.00000i	$0.5\pi$
$602$	100.789	4.10786
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	24.5218	0.995308	0.497654	−	0.867376i	$-0.334195\pi$
$607$	24.5218	0.995308	0.497654	+	0.867376i	$0.334195\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	−102.697	−4.15128
$613$	−46.5792	−1.88132	−0.940658	−	0.339357i	$-0.889791\pi$
$613$	−46.5792	−1.88132	−0.940658	+	0.339357i	$0.889791\pi$
$614$	82.5410	3.33108
$615$	0	0
$616$	0	0
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	0	0
$619$	−17.8885	−0.719001	−0.359501	−	0.933145i	$-0.617053\pi$
$619$	−17.8885	−0.719001	−0.359501	+	0.933145i	$0.617053\pi$
$620$	0	0
$621$	0	0
$622$	79.9108	3.20413
$623$	−41.4127	−1.65917
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−48.0000	−1.91085	−0.955425	−	0.295234i	$-0.904602\pi$
$631$	−48.0000	−1.91085	−0.955425	+	0.295234i	$0.904602\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	4.82241	0.191071
$638$	0	0
$639$	−24.0000	−0.949425
$640$	0	0
$641$	−31.3050	−1.23647	−0.618236	−	0.785993i	$-0.712152\pi$
$641$	−31.3050	−1.23647	−0.618236	+	0.785993i	$0.712152\pi$
$642$	0	0
$643$	0	0		−	1.00000i	$-0.5\pi$
$643$	0	0			1.00000i	$0.5\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	50.2554	1.97422
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	35.6896	1.39239
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	−42.0000	−1.63361	−0.816805	−	0.576913i	$-0.804257\pi$
$661$	−42.0000	−1.63361	−0.816805	+	0.576913i	$0.804257\pi$
$662$	−89.3430	−3.47241
$663$	0	0
$664$	4.06888	0.157903
$665$	0	0
$666$	0	0
$667$	0	0
$668$	−45.4002	−1.75659
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	48.0365	1.85167	0.925836	−	0.377925i	$-0.123362\pi$
$673$	48.0365	1.85167	0.925836	+	0.377925i	$0.123362\pi$
$674$	−41.4853	−1.59795
$675$	0	0
$676$	−39.6525	−1.52510
$677$	−2.80839	−0.107935	−0.0539677	−	0.998543i	$-0.517187\pi$
$677$	−2.80839	−0.107935	−0.0539677	+	0.998543i	$0.517187\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	0	0
$685$	0	0
$686$	−34.4721	−1.31615
$687$	0	0
$688$	71.5513	2.72787
$689$	0	0
$690$	0	0
$691$	28.0000	1.06517	0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000	1.06517	0.532585	+	0.846376i	$0.321221\pi$
$692$	44.2211	1.68104
$693$	0	0
$694$	−29.0132	−1.10132
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−4.47214	−0.167955	−0.0839773	−	0.996468i	$-0.526762\pi$
$709$	−4.47214	−0.167955	−0.0839773	+	0.996468i	$0.526762\pi$
$710$	0	0
$711$	0	0
$712$	−74.9164	−2.80761
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−75.7771	−2.83192
$717$	0	0
$718$	0	0
$719$	26.8328	1.00070	0.500348	−	0.865825i	$-0.333206\pi$
$719$	26.8328	1.00070	0.500348	+	0.865825i	$0.333206\pi$
$720$	0	0
$721$	0	0
$722$	−47.4470	−1.76580
$723$	0	0
$724$	18.9443	0.704058
$725$	0	0
$726$	0	0
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	32.8813	1.21866
$729$	−27.0000	−1.00000
$730$	0	0
$731$	105.666	3.90818
$732$	0	0
$733$	−29.5162	−1.09021	−0.545103	−	0.838369i	$-0.683509\pi$
$733$	−29.5162	−1.09021	−0.545103	+	0.838369i	$0.683509\pi$
$734$	0	0
$735$	0	0
$736$	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
$741$	0	0
$742$	0	0
$743$	18.3483	0.673135	0.336567	−	0.941659i	$-0.390734\pi$
$743$	18.3483	0.673135	0.336567	+	0.941659i	$0.390734\pi$
$744$	0	0
$745$	0	0
$746$	−16.5410	−0.605610
$747$	−2.18603	−0.0799827
$748$	0	0
$749$	−59.4164	−2.17103
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0		−	1.00000i	$-0.5\pi$
$757$	0	0			1.00000i	$0.5\pi$
$758$	89.8996	3.26530
$759$	0	0
$760$	0	0
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	0	0
$764$	−113.666	−4.11228
$765$	0	0
$766$	0	0
$767$	7.63080	0.275532
$768$	0	0
$769$	0	0		−	1.00000i	$-0.5\pi$
$769$	0	0			1.00000i	$0.5\pi$
$770$	0	0
$771$	0	0
$772$	86.5346	3.11445
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	−97.9574	−3.52101
$775$	0	0
$776$	0	0
$777$	0	0
$778$	64.9275	2.32776
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	13.8328	0.494029
$785$	0	0
$786$	0	0
$787$	−46.8575	−1.67029	−0.835145	−	0.550030i	$-0.814616\pi$
$787$	−46.8575	−1.67029	−0.835145	+	0.550030i	$0.814616\pi$
$788$	−92.7080	−3.30259
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	−101.666	−3.60344
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	40.2492	1.42214
$802$	11.1679	0.394351
$803$	0	0
$804$	0	0
$805$	0	0
$806$	−42.6099	−1.50087
$807$	0	0
$808$	0	0
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	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
$819$	−17.6656	−0.617287
$820$	0	0
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	0	0
$823$	0	0		−	1.00000i	$-0.5\pi$
$823$	0	0			1.00000i	$0.5\pi$
$824$	0	0
$825$	0	0
$826$	30.8328	1.07281
$827$	53.0310	1.84407	0.922034	−	0.387110i	$-0.126527\pi$
$827$	53.0310	1.84407	0.922034	+	0.387110i	$0.126527\pi$
$828$	0	0
$829$	−22.3607	−0.776619	−0.388309	−	0.921529i	$-0.626941\pi$
$829$	−22.3607	−0.776619	−0.388309	+	0.921529i	$0.626941\pi$
$830$	0	0
$831$	0	0
$832$	−8.98184	−0.311389
$833$	20.4280	0.707790
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	89.3430	3.08630
$839$	56.0000	1.93333	0.966667	−	0.256036i	$-0.0824164\pi$
$839$	56.0000	1.93333	0.966667	+	0.256036i	$0.0824164\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	−78.1751	−2.69409
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−18.9707	−0.649544	−0.324772	−	0.945792i	$-0.605288\pi$
$853$	−18.9707	−0.649544	−0.324772	+	0.945792i	$0.605288\pi$
$854$	0	0
$855$	0	0
$856$	−107.485	−3.67377
$857$	52.7526	1.80200	0.900998	−	0.433824i	$-0.142836\pi$
$857$	52.7526	1.80200	0.900998	+	0.433824i	$0.142836\pi$
$858$	0	0
$859$	35.7771	1.22070	0.610349	−	0.792132i	$-0.291029\pi$
$859$	35.7771	1.22070	0.610349	+	0.792132i	$0.291029\pi$
$860$	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
$865$	0	0
$866$	0	0
$867$	0	0
$868$	−116.951	−3.96959
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−38.9484	−1.31519	−0.657597	−	0.753370i	$-0.728427\pi$
$877$	−38.9484	−1.31519	−0.657597	+	0.753370i	$0.728427\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	2.00000	0.0673817	0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000	0.0673817	0.0336909	+	0.999432i	$0.489274\pi$
$882$	−18.9378	−0.637670
$883$	0	0		−	1.00000i	$-0.5\pi$
$883$	0	0			1.00000i	$0.5\pi$
$884$	65.3050	2.19644
$885$	0	0
$886$	0	0
$887$	13.9763	0.469277	0.234639	−	0.972083i	$-0.424609\pi$
$887$	13.9763	0.469277	0.234639	+	0.972083i	$0.424609\pi$
$888$	0	0
$889$	52.1378	1.74864
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	−51.7082	−1.72745
$897$	0	0
$898$	−100.511	−3.35409
$899$	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	$0.5\pi$
$908$	−107.691	−3.57386
$909$	0	0
$910$	0	0
$911$	−8.94427	−0.296337	−0.148168	−	0.988962i	$-0.547338\pi$
$911$	−8.94427	−0.296337	−0.148168	+	0.988962i	$0.547338\pi$
$912$	0	0
$913$	0	0
$914$	91.3738	3.02238
$915$	0	0
$916$	−25.4164	−0.839782
$917$	0	0
$918$	0	0
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	15.2616	0.502342
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	14.0000	0.459325	0.229663	−	0.973270i	$-0.426238\pi$
$929$	14.0000	0.459325	0.229663	+	0.973270i	$0.426238\pi$
$930$	0	0
$931$	0	0
$932$	−66.5569	−2.18014
$933$	0	0
$934$	0	0
$935$	0	0
$936$	−31.9574	−1.04456
$937$	8.98184	0.293424	0.146712	−	0.989179i	$-0.453131\pi$
$937$	8.98184	0.293424	0.146712	+	0.989179i	$0.453131\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	0	0
$944$	21.8885	0.712411
$945$	0	0
$946$	0	0
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	−22.6950	−0.736712
$950$	0	0
$951$	0	0
$952$	139.287	4.51432
$953$	51.8519	1.67965	0.839825	−	0.542858i	$-0.182658\pi$
$953$	51.8519	1.67965	0.839825	+	0.542858i	$0.182658\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	49.0000	1.58065
$962$	0	0
$963$	57.7471	1.86087
$964$	0	0
$965$	0	0
$966$	0	0
$967$	61.5625	1.97972	0.989858	−	0.142063i	$-0.0453736\pi$
$967$	61.5625	1.97972	0.989858	+	0.142063i	$0.0453736\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	17.8885	0.574071	0.287035	−	0.957920i	$-0.407330\pi$
$971$	17.8885	0.574071	0.287035	+	0.957920i	$0.407330\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	62.6099	1.98887	0.994435	−	0.105356i	$-0.0335982\pi$
$991$	62.6099	1.98887	0.994435	+	0.105356i	$0.0335982\pi$
$992$	−22.3357	−0.709161
$993$	0	0
$994$	61.6656	1.95592
$995$	0	0
$996$	0	0
$997$	58.0254	1.83768	0.918841	−	0.394627i	$-0.129126\pi$
$997$	58.0254	1.83768	0.918841	+	0.394627i	$0.129126\pi$
$998$	9.98885	0.316191
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3025.2.a.bb.1.4		4
5.2	odd	4		605.2.b.b.364.4	yes	4
5.3	odd	4		605.2.b.b.364.1	✓	4
5.4	even	2	inner	3025.2.a.bb.1.1		4
11.10	odd	2	inner	3025.2.a.bb.1.1		4
55.2	even	20		605.2.j.a.444.1		8
55.3	odd	20		605.2.j.c.9.1		8
55.7	even	20		605.2.j.c.269.2		8
55.8	even	20		605.2.j.c.9.2		8
55.13	even	20		605.2.j.a.444.2		8
55.17	even	20		605.2.j.a.124.2		8
55.18	even	20		605.2.j.c.269.1		8
55.27	odd	20		605.2.j.a.124.1		8
55.28	even	20		605.2.j.a.124.1		8
55.32	even	4		605.2.b.b.364.1	✓	4
55.37	odd	20		605.2.j.c.269.1		8
55.38	odd	20		605.2.j.a.124.2		8
55.42	odd	20		605.2.j.a.444.2		8
55.43	even	4		605.2.b.b.364.4	yes	4
55.47	odd	20		605.2.j.c.9.2		8
55.48	odd	20		605.2.j.c.269.2		8
55.52	even	20		605.2.j.c.9.1		8
55.53	odd	20		605.2.j.a.444.1		8
55.54	odd	2	CM	3025.2.a.bb.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.b.b.364.1	✓	4	5.3	odd	4
605.2.b.b.364.1	✓	4	55.32	even	4
605.2.b.b.364.4	yes	4	5.2	odd	4
605.2.b.b.364.4	yes	4	55.43	even	4
605.2.j.a.124.1		8	55.27	odd	20
605.2.j.a.124.1		8	55.28	even	20
605.2.j.a.124.2		8	55.17	even	20
605.2.j.a.124.2		8	55.38	odd	20
605.2.j.a.444.1		8	55.2	even	20
605.2.j.a.444.1		8	55.53	odd	20
605.2.j.a.444.2		8	55.13	even	20
605.2.j.a.444.2		8	55.42	odd	20
605.2.j.c.9.1		8	55.3	odd	20
605.2.j.c.9.1		8	55.52	even	20
605.2.j.c.9.2		8	55.8	even	20
605.2.j.c.9.2		8	55.47	odd	20
605.2.j.c.269.1		8	55.18	even	20
605.2.j.c.269.1		8	55.37	odd	20
605.2.j.c.269.2		8	55.7	even	20
605.2.j.c.269.2		8	55.48	odd	20
3025.2.a.bb.1.1		4	5.4	even	2	inner
3025.2.a.bb.1.1		4	11.10	odd	2	inner
3025.2.a.bb.1.4		4	1.1	even	1	trivial
3025.2.a.bb.1.4		4	55.54	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:	\( N \)	\(=\)	\( 3025 = 5^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3025.a (trivial)

\(f(q)\)	\(=\)	\(q+2.49721 q^{2} +4.23607 q^{4} +3.08672 q^{7} +5.58394 q^{8} -3.00000 q^{9} +O(q^{10})\)	\(q+2.49721 q^{2} +4.23607 q^{4} +3.08672 q^{7} +5.58394 q^{8} -3.00000 q^{9} +1.90770 q^{13} +7.70820 q^{14} +5.47214 q^{16} +8.08115 q^{17} -7.49164 q^{18} +4.76393 q^{26} +13.0756 q^{28} -8.94427 q^{31} +2.49721 q^{32} +20.1803 q^{34} -12.7082 q^{36} +13.0756 q^{43} +2.52786 q^{49} +8.08115 q^{52} +17.2361 q^{56} +4.00000 q^{59} -22.3357 q^{62} -9.26017 q^{63} -4.70820 q^{64} +34.2323 q^{68} +8.00000 q^{71} -16.7518 q^{72} -11.8965 q^{73} +9.00000 q^{81} +0.728677 q^{83} +32.6525 q^{86} -13.4164 q^{89} +5.88854 q^{91} +6.31261 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{4} - 12 q^{9}+O(q^{10}) \) 4 * q + 8 * q^4 - 12 * q^9	\( 4 q + 8 q^{4} - 12 q^{9} + 4 q^{14} + 4 q^{16} + 28 q^{26} + 36 q^{34} - 24 q^{36} + 28 q^{49} + 60 q^{56} + 16 q^{59} + 8 q^{64} + 32 q^{71} + 36 q^{81} + 68 q^{86} - 48 q^{91}+O(q^{100}) \) 4 * q + 8 * q^4 - 12 * q^9 + 4 * q^14 + 4 * q^16 + 28 * q^26 + 36 * q^34 - 24 * q^36 + 28 * q^49 + 60 * q^56 + 16 * q^59 + 8 * q^64 + 32 * q^71 + 36 * q^81 + 68 * q^86 - 48 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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