LMFDB - Embedded newform 605.2.b.b.364.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [605,2,Mod(364,605)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(605, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("605.364");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 605 = 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	605.b (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:	$4.83094932229$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.4400.1
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:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			364.2
Root			$-2.14896i$ of defining polynomial
Character	$\chi$	$=$	605.364
Dual form			605.2.b.b.364.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.32813i q^{2} +0.236068 q^{4} -2.23607 q^{5} +4.29792i q^{7} -2.96979i q^{8} +3.00000 q^{9} +O(q^{10})$	$q-1.32813i q^{2} +0.236068 q^{4} -2.23607 q^{5} +4.29792i q^{7} -2.96979i q^{8} +3.00000 q^{9} +2.96979i q^{10} +6.95418i q^{13} +5.70820 q^{14} -3.47214 q^{16} +1.64166i q^{17} -3.98439i q^{18} -0.527864 q^{20} +5.00000 q^{25} +9.23607 q^{26} +1.01460i q^{28} +8.94427 q^{31} -1.32813i q^{32} +2.18034 q^{34} -9.61045i q^{35} +0.708204 q^{36} +6.64066i q^{40} +1.01460i q^{43} -6.70820 q^{45} -11.4721 q^{49} -6.64066i q^{50} +1.64166i q^{52} +12.7639 q^{56} -4.00000 q^{59} -11.8792i q^{62} +12.8938i q^{63} -8.70820 q^{64} -15.5500i q^{65} +0.387543i q^{68} -12.7639 q^{70} +8.00000 q^{71} -8.90937i q^{72} -12.2667i q^{73} +7.76393 q^{80} +9.00000 q^{81} +18.2063i q^{83} -3.67086i q^{85} +1.34752 q^{86} -13.4164 q^{89} +8.90937i q^{90} -29.8885 q^{91} +15.2365i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 20 q^{20} + 20 q^{25} + 28 q^{26} - 36 q^{34} - 24 q^{36} - 28 q^{49} + 60 q^{56} - 16 q^{59} - 8 q^{64} - 60 q^{70} + 32 q^{71} + 40 q^{80} + 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 - 20 * q^20 + 20 * q^25 + 28 * q^26 - 36 * q^34 - 24 * q^36 - 28 * q^49 + 60 * q^56 - 16 * q^59 - 8 * q^64 - 60 * q^70 + 32 * q^71 + 40 * q^80 + 36 * q^81 + 68 * q^86 - 48 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/605\mathbb{Z}\right)^\times$.

$n$	$122$	$486$
$\chi(n)$	$-1$	$1$

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 $

$2$	− 1.32813i	− 0.939130i	−0.882898	−	0.469565i	$-0.844411\pi$
$2$	− 1.32813i	− 0.939130i	0.882898	−	0.469565i	$-0.155589\pi$
$3$	0	0	1.00000			$0$
$3$	0	0	−1.00000			$\pi$
$4$	0.236068	0.118034
$5$	−2.23607	−1.00000
$5$	−2.23607	−1.00000
$6$	0	0
$7$	4.29792i	1.62446i	0.583336	+	0.812231i	$0.301747\pi$
$7$	4.29792i	1.62446i	−0.583336	+	0.812231i	$0.698253\pi$
$8$	− 2.96979i	− 1.04998i
$9$	3.00000	1.00000
$10$	2.96979i	0.939130i
$11$	0	0
$11$	0	0
$12$	0	0
$13$	6.95418i	1.92874i	0.264550	+	0.964372i	$0.414776\pi$
$13$	6.95418i	1.92874i	−0.264550	+	0.964372i	$0.585224\pi$
$14$	5.70820	1.52558
$15$	0	0
$16$	−3.47214	−0.868034
$17$	1.64166i	0.398161i	0.979983	+	0.199081i	$0.0637955\pi$
$17$	1.64166i	0.398161i	−0.979983	+	0.199081i	$0.936204\pi$
$18$	− 3.98439i	− 0.939130i
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	−0.527864	−0.118034
$21$	0	0
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	5.00000	1.00000
$26$	9.23607	1.81134
$27$	0	0
$28$	1.01460i	0.191742i
$29$	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	$-0.203119\pi$
$31$	8.94427	1.60644	0.803219	+	0.595683i	$0.203119\pi$
$32$	− 1.32813i	− 0.234783i
$33$	0	0
$34$	2.18034	0.373925
$35$	− 9.61045i	− 1.62446i
$36$	0.708204	0.118034
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	6.64066i	1.04998i
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	1.01460i	0.154725i	0.997003	+	0.0773627i	$0.0246499\pi$
$43$	1.01460i	0.154725i	−0.997003	+	0.0773627i	$0.975350\pi$
$44$	0	0
$45$	−6.70820	−1.00000
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	−11.4721	−1.63888
$50$	− 6.64066i	− 0.939130i
$51$	0	0
$52$	1.64166i	0.227657i
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	0	0
$56$	12.7639	1.70565
$57$	0	0
$58$	0	0
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	− 11.8792i	− 1.50866i
$63$	12.8938i	1.62446i
$64$	−8.70820	−1.08853
$65$	− 15.5500i	− 1.92874i
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0.387543i	0.0469965i
$69$	0	0
$70$	−12.7639	−1.52558
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	− 8.90937i	− 1.04998i
$73$	− 12.2667i	− 1.43571i	−0.696193	−	0.717855i	$-0.745124\pi$
$73$	− 12.2667i	− 1.43571i	0.696193	−	0.717855i	$-0.254876\pi$
$74$	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	$0.5\pi$
$80$	7.76393	0.868034
$81$	9.00000	1.00000
$82$	0	0
$83$	18.2063i	1.99840i	0.0399913	+	0.999200i	$0.487267\pi$
$83$	18.2063i	1.99840i	−0.0399913	+	0.999200i	$0.512733\pi$
$84$	0	0
$85$	− 3.67086i	− 0.398161i
$86$	1.34752	0.145307
$87$	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	$0.751788\pi$
$90$	8.90937i	0.939130i
$91$	−29.8885	−3.13317
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	15.2365i	1.53912i
$99$	0	0
$100$	1.18034	0.118034
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	20.6525	2.02514
$105$	0	0
$106$	0	0
$107$	− 7.58124i	− 0.732906i	−0.930436	−	0.366453i	$-0.880572\pi$
$107$	− 7.58124i	− 0.732906i	0.930436	−	0.366453i	$-0.119428\pi$
$108$	0	0
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	0	0
$111$	0	0
$112$	− 14.9230i	− 1.41009i
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	20.8626i	1.92874i
$118$	5.31252i	0.489057i
$119$	−7.05573	−0.646798
$120$	0	0
$121$	0	0
$122$	0	0
$123$	0	0
$124$	2.11146	0.189614
$125$	−11.1803	−1.00000
$126$	17.1246	1.52558
$127$	− 14.9230i	− 1.32420i	−0.749416	−	0.662100i	$-0.769666\pi$
$127$	− 14.9230i	− 1.32420i	0.749416	−	0.662100i	$-0.230334\pi$
$128$	8.90937i	0.787485i
$129$	0	0
$130$	−20.6525	−1.81134
$131$	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
$136$	4.87539	0.418061
$137$	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	$0.5\pi$
$140$	− 2.26872i	− 0.191742i
$141$	0	0
$142$	− 10.6250i	− 0.891634i
$143$	0	0
$144$	−10.4164	−0.868034
$145$	0	0
$146$	−16.2918	−1.34832
$147$	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
$153$	4.92498i	0.398161i
$154$	0	0
$155$	−20.0000	−1.60644
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	2.96979i	0.234783i
$161$	0	0
$162$	− 11.9532i	− 0.939130i
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	0	0
$166$	24.1803	1.87676
$167$	23.5188i	1.81994i	0.414673	+	0.909970i	$0.363896\pi$
$167$	23.5188i	1.81994i	−0.414673	+	0.909970i	$0.636104\pi$
$168$	0	0
$169$	−35.3607	−2.72005
$170$	−4.87539	−0.373925
$171$	0	0
$172$	0.239515i	0.0182628i
$173$	− 24.1459i	− 1.83578i	−0.396839	−	0.917888i	$-0.629893\pi$
$173$	− 24.1459i	− 1.83578i	0.396839	−	0.917888i	$-0.370107\pi$
$174$	0	0
$175$	21.4896i	1.62446i
$176$	0	0
$177$	0	0
$178$	17.8187i	1.33557i
$179$	−17.8885	−1.33705	−0.668526	−	0.743689i	$-0.733075\pi$
$179$	−17.8885	−1.33705	−0.668526	+	0.743689i	$0.733075\pi$
$180$	−1.58359	−0.118034
$181$	−4.47214	−0.332411	−0.166206	−	0.986091i	$-0.553152\pi$
$181$	−4.47214	−0.332411	−0.166206	+	0.986091i	$0.553152\pi$
$182$	39.6959i	2.94246i
$183$	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	$-0.0771417\pi$
$191$	26.8328	1.94155	0.970777	+	0.239983i	$0.0771417\pi$
$192$	0	0
$193$	− 18.8333i	− 1.35565i	−0.735221	−	0.677827i	$-0.762922\pi$
$193$	− 18.8333i	− 1.35565i	0.735221	−	0.677827i	$-0.237078\pi$
$194$	0	0
$195$	0	0
$196$	−2.70820	−0.193443
$197$	17.5792i	1.25247i	0.779635	+	0.626234i	$0.215405\pi$
$197$	17.5792i	1.25247i	−0.779635	+	0.626234i	$0.784595\pi$
$198$	0	0
$199$	24.0000	1.70131	0.850657	−	0.525720i	$-0.176204\pi$
$199$	24.0000	1.70131	0.850657	+	0.525720i	$0.176204\pi$
$200$	− 14.8490i	− 1.04998i
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	− 24.1459i	− 1.67422i
$209$	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
$214$	−10.0689	−0.688295
$215$	− 2.26872i	− 0.154725i
$216$	0	0
$217$	38.4418i	2.60960i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−11.4164	−0.767951
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	5.70820	0.381395
$225$	15.0000	1.00000
$226$	0	0
$227$	− 16.1771i	− 1.07371i	−0.843674	−	0.536855i	$-0.819612\pi$
$227$	− 16.1771i	− 1.07371i	0.843674	−	0.536855i	$-0.180388\pi$
$228$	0	0
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 26.1751i	− 1.71479i	−0.514662	−	0.857393i	$-0.672083\pi$
$233$	− 26.1751i	− 1.71479i	0.514662	−	0.857393i	$-0.327917\pi$
$234$	27.7082	1.81134
$235$	0	0
$236$	−0.944272	−0.0614669
$237$	0	0
$238$	9.37093i	0.607427i
$239$	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
$245$	25.6525	1.63888
$246$	0	0
$247$	0	0
$248$	− 26.5626i	− 1.68673i
$249$	0	0
$250$	14.8490i	0.939130i
$251$	28.0000	1.76734	0.883672	−	0.468106i	$-0.155064\pi$
$251$	28.0000	1.76734	0.883672	+	0.468106i	$0.155064\pi$
$252$	3.04381i	0.191742i
$253$	0	0
$254$	−19.8197	−1.24360
$255$	0	0
$256$	−5.58359	−0.348975
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	0	0
$260$	− 3.67086i	− 0.227657i
$261$	0	0
$262$	0	0
$263$	− 32.1147i	− 1.98027i	−0.140100	−	0.990137i	$-0.544742\pi$
$263$	− 32.1147i	− 1.98027i	0.140100	−	0.990137i	$-0.455258\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−14.0000	−0.853595	−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000	−0.853595	−0.426798	+	0.904347i	$0.640358\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	− 5.70007i	− 0.345617i
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 13.5208i	− 0.812388i	−0.913787	−	0.406194i	$-0.866856\pi$
$277$	− 13.5208i	− 0.812388i	0.913787	−	0.406194i	$-0.133144\pi$
$278$	0	0
$279$	26.8328	1.60644
$280$	−28.5410	−1.70565
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	− 20.2355i	− 1.20288i	−0.798920	−	0.601438i	$-0.794595\pi$
$283$	− 20.2355i	− 1.20288i	0.798920	−	0.601438i	$-0.205405\pi$
$284$	1.88854	0.112064
$285$	0	0
$286$	0	0
$287$	0	0
$288$	− 3.98439i	− 0.234783i
$289$	14.3050	0.841468
$290$	0	0
$291$	0	0
$292$	− 2.89578i	− 0.169463i
$293$	0.387543i	0.0226405i	0.999936	+	0.0113203i	$0.00360343\pi$
$293$	0.387543i	0.0226405i	−0.999936	+	0.0113203i	$0.996397\pi$
$294$	0	0
$295$	8.94427	0.520756
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−4.36068	−0.251345
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	6.54102	0.373925
$307$	− 11.6397i	− 0.664310i	−0.943225	−	0.332155i	$-0.892224\pi$
$307$	− 11.6397i	− 0.664310i	0.943225	−	0.332155i	$-0.107776\pi$
$308$	0	0
$309$	0	0
$310$	26.5626i	1.50866i
$311$	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	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	− 28.8313i	− 1.62446i
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	19.4721	1.08853
$321$	0	0
$322$	0	0
$323$	0	0
$324$	2.12461	0.118034
$325$	34.7709i	1.92874i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	35.7771	1.96649	0.983243	−	0.182298i	$-0.0583536\pi$
$331$	35.7771	1.96649	0.983243	+	0.182298i	$0.0583536\pi$
$332$	4.29792i	0.235879i
$333$	0	0
$334$	31.2361	1.70916
$335$	0	0
$336$	0	0
$337$	− 32.7417i	− 1.78356i	−0.452474	−	0.891778i	$-0.649459\pi$
$337$	− 32.7417i	− 1.78356i	0.452474	−	0.891778i	$-0.350541\pi$
$338$	46.9636i	2.55448i
$339$	0	0
$340$	− 0.866573i	− 0.0469965i
$341$	0	0
$342$	0	0
$343$	− 19.2209i	− 1.03783i
$344$	3.01316	0.162458
$345$	0	0
$346$	−32.0689	−1.72403
$347$	− 35.3980i	− 1.90026i	−0.311849	−	0.950132i	$-0.600948\pi$
$347$	− 35.3980i	− 1.90026i	0.311849	−	0.950132i	$-0.399052\pi$
$348$	0	0
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	28.5410	1.52558
$351$	0	0
$352$	0	0
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	−17.8885	−0.949425
$356$	−3.16718	−0.167860
$357$	0	0
$358$	23.7583i	1.25567i
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	19.9220i	1.04998i
$361$	−19.0000	−1.00000
$362$	5.93958i	0.312178i
$363$	0	0
$364$	−7.05573	−0.369821
$365$	27.4292i	1.43571i
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	38.0542i	1.97037i	0.171484	+	0.985187i	$0.445144\pi$
$373$	38.0542i	1.97037i	−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	$-0.875599\pi$
$379$	−36.0000	−1.84920	−0.924598	+	0.380945i	$0.875599\pi$
$380$	0	0
$381$	0	0
$382$	− 35.6375i	− 1.82337i
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	0	0
$386$	−25.0132	−1.27314
$387$	3.04381i	0.154725i
$388$	0	0
$389$	−26.0000	−1.31825	−0.659126	−	0.752032i	$-0.729074\pi$
$389$	−26.0000	−1.31825	−0.659126	+	0.752032i	$0.729074\pi$
$390$	0	0
$391$	0	0
$392$	34.0698i	1.72079i
$393$	0	0
$394$	23.3475	1.17623
$395$	0	0
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	− 31.8751i	− 1.59776i
$399$	0	0
$400$	−17.3607	−0.868034
$401$	−4.47214	−0.223328	−0.111664	−	0.993746i	$-0.535618\pi$
$401$	−4.47214	−0.223328	−0.111664	+	0.993746i	$0.535618\pi$
$402$	0	0
$403$	62.2001i	3.09841i
$404$	0	0
$405$	−20.1246	−1.00000
$406$	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
$413$	− 17.1917i	− 0.845948i
$414$	0	0
$415$	− 40.7105i	− 1.99840i
$416$	9.23607	0.452835
$417$	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	$-0.223797\pi$
$421$	31.3050	1.52571	0.762855	+	0.646570i	$0.223797\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	8.20830i	0.398161i
$426$	0	0
$427$	0	0
$428$	− 1.78969i	− 0.0865079i
$429$	0	0
$430$	−3.01316	−0.145307
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	51.0557	2.45075
$435$	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
$441$	−34.4164	−1.63888
$442$	15.1625i	0.721206i
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	30.0000	1.42214
$446$	0	0
$447$	0	0
$448$	− 37.4272i	− 1.76827i
$449$	−40.2492	−1.89948	−0.949739	−	0.313042i	$-0.898652\pi$
$449$	−40.2492	−1.89948	−0.949739	+	0.313042i	$0.898652\pi$
$450$	− 19.9220i	− 0.939130i
$451$	0	0
$452$	0	0
$453$	0	0
$454$	−21.4853	−1.00835
$455$	66.8328	3.13317
$456$	0	0
$457$	22.1167i	1.03457i	0.855812	+	0.517287i	$0.173058\pi$
$457$	22.1167i	1.03457i	−0.855812	+	0.517287i	$0.826942\pi$
$458$	− 7.96879i	− 0.372357i
$459$	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	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	−34.7639	−1.61041
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	4.92498i	0.227657i
$469$	0	0
$470$	0	0
$471$	0	0
$472$	11.8792i	0.546783i
$473$	0	0
$474$	0	0
$475$	0	0
$476$	−1.66563	−0.0763441
$477$	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	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	− 34.0698i	− 1.53912i
$491$	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
$496$	−31.0557	−1.39444
$497$	34.3834i	1.54231i
$498$	0	0
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	−2.63932	−0.118034
$501$	0	0
$502$	− 37.1877i	− 1.65977i
$503$	25.5480i	1.13913i	0.821946	+	0.569565i	$0.192888\pi$
$503$	25.5480i	1.13913i	−0.821946	+	0.569565i	$0.807112\pi$
$504$	38.2918	1.70565
$505$	0	0
$506$	0	0
$507$	0	0
$508$	− 3.52284i	− 0.156301i
$509$	−34.0000	−1.50702	−0.753512	−	0.657434i	$-0.771642\pi$
$509$	−34.0000	−1.50702	−0.753512	+	0.657434i	$0.771642\pi$
$510$	0	0
$511$	52.7214	2.33226
$512$	25.2345i	1.11522i
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	−46.1803	−2.02514
$521$	−22.3607	−0.979639	−0.489820	−	0.871824i	$-0.662937\pi$
$521$	−22.3607	−0.979639	−0.489820	+	0.871824i	$0.662937\pi$
$522$	0	0
$523$	5.55204i	0.242774i	0.992605	+	0.121387i	$0.0387342\pi$
$523$	5.55204i	0.242774i	−0.992605	+	0.121387i	$0.961266\pi$
$524$	0	0
$525$	0	0
$526$	−42.6525	−1.85974
$527$	14.6835i	0.639621i
$528$	0	0
$529$	23.0000	1.00000
$530$	0	0
$531$	−12.0000	−0.520756
$532$	0	0
$533$	0	0
$534$	0	0
$535$	16.9522i	0.732906i
$536$	0	0
$537$	0	0
$538$	18.5938i	0.801637i
$539$	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
$544$	2.18034	0.0934813
$545$	0	0
$546$	0	0
$547$	46.0230i	1.96780i	0.178713	+	0.983901i	$0.442807\pi$
$547$	46.0230i	1.96780i	−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
$554$	−17.9574	−0.762938
$555$	0	0
$556$	0	0
$557$	41.3376i	1.75153i	0.482739	+	0.875764i	$0.339642\pi$
$557$	41.3376i	1.75153i	−0.482739	+	0.875764i	$0.660358\pi$
$558$	− 35.6375i	− 1.50866i
$559$	−7.05573	−0.298426
$560$	33.3688i	1.41009i
$561$	0	0
$562$	0	0
$563$	43.9938i	1.85412i	0.374915	+	0.927059i	$0.377672\pi$
$563$	43.9938i	1.85412i	−0.374915	+	0.927059i	$0.622328\pi$
$564$	0	0
$565$	0	0
$566$	−26.8754	−1.12966
$567$	38.6813i	1.62446i
$568$	− 23.7583i	− 0.996877i
$569$	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
$576$	−26.1246	−1.08853
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	− 18.9989i	− 0.790248i
$579$	0	0
$580$	0	0
$581$	−78.2492	−3.24632
$582$	0	0
$583$	0	0
$584$	−36.4296	−1.50747
$585$	− 46.6501i	− 1.92874i
$586$	0.514708	0.0212624
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0	0
$589$	0	0
$590$	− 11.8792i	− 0.489057i
$591$	0	0
$592$	0	0
$593$	− 36.0250i	− 1.47937i	−0.672953	−	0.739686i	$-0.734974\pi$
$593$	− 36.0250i	− 1.47937i	0.672953	−	0.739686i	$-0.265026\pi$
$594$	0	0
$595$	15.7771	0.646798
$596$	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	$0.5\pi$
$602$	5.79155i	0.236046i
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 42.7397i	− 1.73475i	−0.497654	−	0.867376i	$-0.665805\pi$
$607$	− 42.7397i	− 1.73475i	0.497654	−	0.867376i	$-0.334195\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	1.16263i	0.0469965i
$613$	16.8041i	0.678713i	0.940658	+	0.339357i	$0.110209\pi$
$613$	16.8041i	0.678713i	−0.940658	+	0.339357i	$0.889791\pi$
$614$	−15.4590	−0.623874
$615$	0	0
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	−17.8885	−0.719001	−0.359501	−	0.933145i	$-0.617053\pi$
$619$	−17.8885	−0.719001	−0.359501	+	0.933145i	$0.617053\pi$
$620$	−4.72136	−0.189614
$621$	0	0
$622$	− 42.5002i	− 1.70410i
$623$	− 57.6627i	− 2.31021i
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	−38.2918	−1.52558
$631$	−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
$635$	33.3688i	1.32420i
$636$	0	0
$637$	− 79.7793i	− 3.16097i
$638$	0	0
$639$	24.0000	0.949425
$640$	− 19.9220i	− 0.787485i
$641$	31.3050	1.23647	0.618236	−	0.785993i	$-0.287848\pi$
$641$	31.3050	1.23647	0.618236	+	0.785993i	$0.287848\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	− 26.7281i	− 1.04998i
$649$	0	0
$650$	46.1803	1.81134
$651$	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
$657$	− 36.8001i	− 1.43571i
$658$	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	$0.804257\pi$
$662$	− 47.5167i	− 1.84679i
$663$	0	0
$664$	54.0689	2.09828
$665$	0	0
$666$	0	0
$667$	0	0
$668$	5.55204i	0.214815i
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	19.6084i	0.755850i	0.925836	+	0.377925i	$0.123362\pi$
$673$	19.6084i	0.755850i	−0.925836	+	0.377925i	$0.876638\pi$
$674$	−43.4853	−1.67499
$675$	0	0
$676$	−8.34752	−0.321059
$677$	− 51.9626i	− 1.99709i	−0.0539677	−	0.998543i	$-0.517187\pi$
$677$	− 51.9626i	− 1.99709i	0.0539677	−	0.998543i	$-0.482813\pi$
$678$	0	0
$679$	0	0
$680$	−10.9017	−0.418061
$681$	0	0
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0	0
$685$	0	0
$686$	−25.5279	−0.974658
$687$	0	0
$688$	− 3.52284i	− 0.134307i
$689$	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	$0.321221\pi$
$692$	− 5.70007i	− 0.216684i
$693$	0	0
$694$	−47.0132	−1.78459
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	5.07301i	0.191742i
$701$	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	$0.526762\pi$
$710$	23.7583i	0.891634i
$711$	0	0
$712$	39.8439i	1.49321i
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−4.22291	−0.157818
$717$	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	$0.333206\pi$
$720$	23.2918	0.868034
$721$	0	0
$722$	25.2345i	0.939130i
$723$	0	0
$724$	−1.05573	−0.0392358
$725$	0	0
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	88.7627i	3.28977i
$729$	27.0000	1.00000
$730$	36.4296	1.34832
$731$	−1.66563	−0.0616056
$732$	0	0
$733$	− 45.3960i	− 1.67674i	−0.545103	−	0.838369i	$-0.683509\pi$
$733$	− 45.3960i	− 1.67674i	0.545103	−	0.838369i	$-0.316491\pi$
$734$	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
$743$	51.3356i	1.88332i	0.336567	+	0.941659i	$0.390734\pi$
$743$	51.3356i	1.88332i	−0.336567	+	0.941659i	$0.609266\pi$
$744$	0	0
$745$	0	0
$746$	50.5410	1.85044
$747$	54.6189i	1.99840i
$748$	0	0
$749$	32.5836	1.19058
$750$	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	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	47.8127i	1.73664i
$759$	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
$764$	6.33437	0.229169
$765$	− 11.0126i	− 0.398161i
$766$	0	0
$767$	− 27.8167i	− 1.00440i
$768$	0	0
$769$	0	0		−	1.00000i	$-0.5\pi$
$769$	0	0			1.00000i	$0.5\pi$
$770$	0	0
$771$	0	0
$772$	− 4.44595i	− 0.160013i
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	4.04257	0.145307
$775$	44.7214	1.60644
$776$	0	0
$777$	0	0
$778$	34.5314i	1.23801i
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	39.8328	1.42260
$785$	0	0
$786$	0	0
$787$	30.8605i	1.10006i	0.835145	+	0.550030i	$0.185384\pi$
$787$	30.8605i	1.10006i	−0.835145	+	0.550030i	$0.814616\pi$
$788$	4.14989i	0.147834i
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	5.66563	0.200813
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	− 6.64066i	− 0.234783i
$801$	−40.2492	−1.42214
$802$	5.93958i	0.209734i
$803$	0	0
$804$	0	0
$805$	0	0
$806$	82.6099	2.90981
$807$	0	0
$808$	0	0
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	26.7281i	0.939130i
$811$	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
$819$	−89.6656	−3.13317
$820$	0	0
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	0	0
$826$	−22.8328	−0.794455
$827$	− 22.2647i	− 0.774219i	−0.922034	−	0.387110i	$-0.873473\pi$
$827$	− 22.2647i	− 0.774219i	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	$0.626941\pi$
$830$	−54.0689	−1.87676
$831$	0	0
$832$	− 60.5585i	− 2.09949i
$833$	− 18.8333i	− 0.652537i
$834$	0	0
$835$	− 52.5897i	− 1.81994i
$836$	0	0
$837$	0	0
$838$	− 47.5167i	− 1.64144i
$839$	−56.0000	−1.93333	−0.966667	−	0.256036i	$-0.917584\pi$
$839$	−56.0000	−1.93333	−0.966667	+	0.256036i	$0.917584\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	− 41.5771i	− 1.43284i
$843$	0	0
$844$	0	0
$845$	79.0689	2.72005
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	10.9017	0.373925
$851$	0	0
$852$	0	0
$853$	55.2459i	1.89158i	0.324772	+	0.945792i	$0.394712\pi$
$853$	55.2459i	1.89158i	−0.324772	+	0.945792i	$0.605288\pi$
$854$	0	0
$855$	0	0
$856$	−22.5147	−0.769537
$857$	25.4000i	0.867647i	0.900998	+	0.433824i	$0.142836\pi$
$857$	25.4000i	0.867647i	−0.900998	+	0.433824i	$0.857164\pi$
$858$	0	0
$859$	35.7771	1.22070	0.610349	−	0.792132i	$-0.291029\pi$
$859$	35.7771	1.22070	0.610349	+	0.792132i	$0.291029\pi$
$860$	− 0.535572i	− 0.0182628i
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	53.9918i	1.83578i
$866$	0	0
$867$	0	0
$868$	9.07487i	0.308021i
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 48.0522i	− 1.62446i
$876$	0	0
$877$	− 44.6209i	− 1.50674i	−0.657597	−	0.753370i	$-0.728427\pi$
$877$	− 44.6209i	− 1.50674i	0.657597	−	0.753370i	$-0.271573\pi$
$878$	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	$0.489274\pi$
$882$	45.7095i	1.53912i
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	−2.69505	−0.0906443
$885$	0	0
$886$	0	0
$887$	57.9022i	1.94417i	0.234639	+	0.972083i	$0.424609\pi$
$887$	57.9022i	1.94417i	−0.234639	+	0.972083i	$0.575391\pi$
$888$	0	0
$889$	64.1378	2.15111
$890$	− 39.8439i	− 1.33557i
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	40.0000	1.33705
$896$	−38.2918	−1.27924
$897$	0	0
$898$	53.4562i	1.78386i
$899$	0	0
$900$	3.54102	0.118034
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	10.0000	0.332411
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	− 3.81889i	− 0.126734i
$909$	0	0
$910$	− 88.7627i	− 2.94246i
$911$	8.94427	0.296337	0.148168	−	0.988962i	$-0.452662\pi$
$911$	8.94427	0.296337	0.148168	+	0.988962i	$0.452662\pi$
$912$	0	0
$913$	0	0
$914$	29.3738	0.971600
$915$	0	0
$916$	1.41641	0.0467994
$917$	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
$923$	55.6335i	1.83120i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−14.0000	−0.459325	−0.229663	−	0.973270i	$-0.573762\pi$
$929$	−14.0000	−0.459325	−0.229663	+	0.973270i	$0.573762\pi$
$930$	0	0
$931$	0	0
$932$	− 6.17910i	− 0.202403i
$933$	0	0
$934$	0	0
$935$	0	0
$936$	61.9574	2.02514
$937$	60.5585i	1.97836i	0.146712	+	0.989179i	$0.453131\pi$
$937$	60.5585i	1.97836i	−0.146712	+	0.989179i	$0.546869\pi$
$938$	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
$944$	13.8885	0.452034
$945$	0	0
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	85.3050	2.76912
$950$	0	0
$951$	0	0
$952$	20.9540i	0.679124i
$953$	33.5168i	1.08572i	0.839825	+	0.542858i	$0.182658\pi$
$953$	33.5168i	1.08572i	−0.839825	+	0.542858i	$0.817342\pi$
$954$	0	0
$955$	−60.0000	−1.94155
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	49.0000	1.58065
$962$	0	0
$963$	− 22.7437i	− 0.732906i
$964$	0	0
$965$	42.1126i	1.35565i
$966$	0	0
$967$	8.83536i	0.284126i	0.989858	+	0.142063i	$0.0453736\pi$
$967$	8.83536i	0.284126i	−0.989858	+	0.142063i	$0.954626\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−17.8885	−0.574071	−0.287035	−	0.957920i	$-0.592670\pi$
$971$	−17.8885	−0.574071	−0.287035	+	0.957920i	$0.592670\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	6.05573	0.193443
$981$	0	0
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	− 39.3084i	− 1.25247i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−62.6099	−1.98887	−0.994435	−	0.105356i	$-0.966402\pi$
$991$	−62.6099	−1.98887	−0.994435	+	0.105356i	$0.966402\pi$
$992$	− 11.8792i	− 0.377164i
$993$	0	0
$994$	45.6656	1.44843
$995$	−53.6656	−1.70131
$996$	0	0
$997$	− 24.9210i	− 0.789255i	−0.918841	−	0.394627i	$-0.870874\pi$
$997$	− 24.9210i	− 0.789255i	0.918841	−	0.394627i	$-0.129126\pi$
$998$	5.31252i	0.168165i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.b.b.364.2	✓	4
5.2	odd	4		3025.2.a.bb.1.3		4
5.3	odd	4		3025.2.a.bb.1.2		4
5.4	even	2	inner	605.2.b.b.364.3	yes	4
11.2	odd	10		605.2.j.c.444.2		8
11.3	even	5		605.2.j.a.9.1		8
11.4	even	5		605.2.j.a.269.2		8
11.5	even	5		605.2.j.c.124.2		8
11.6	odd	10		605.2.j.c.124.1		8
11.7	odd	10		605.2.j.a.269.1		8
11.8	odd	10		605.2.j.a.9.2		8
11.9	even	5		605.2.j.c.444.1		8
11.10	odd	2	inner	605.2.b.b.364.3	yes	4
55.4	even	10		605.2.j.a.269.1		8
55.9	even	10		605.2.j.c.444.2		8
55.14	even	10		605.2.j.a.9.2		8
55.19	odd	10		605.2.j.a.9.1		8
55.24	odd	10		605.2.j.c.444.1		8
55.29	odd	10		605.2.j.a.269.2		8
55.32	even	4		3025.2.a.bb.1.2		4
55.39	odd	10		605.2.j.c.124.2		8
55.43	even	4		3025.2.a.bb.1.3		4
55.49	even	10		605.2.j.c.124.1		8
55.54	odd	2	CM	605.2.b.b.364.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.b.b.364.2	✓	4	1.1	even	1	trivial
605.2.b.b.364.2	✓	4	55.54	odd	2	CM
605.2.b.b.364.3	yes	4	5.4	even	2	inner
605.2.b.b.364.3	yes	4	11.10	odd	2	inner
605.2.j.a.9.1		8	11.3	even	5
605.2.j.a.9.1		8	55.19	odd	10
605.2.j.a.9.2		8	11.8	odd	10
605.2.j.a.9.2		8	55.14	even	10
605.2.j.a.269.1		8	11.7	odd	10
605.2.j.a.269.1		8	55.4	even	10
605.2.j.a.269.2		8	11.4	even	5
605.2.j.a.269.2		8	55.29	odd	10
605.2.j.c.124.1		8	11.6	odd	10
605.2.j.c.124.1		8	55.49	even	10
605.2.j.c.124.2		8	11.5	even	5
605.2.j.c.124.2		8	55.39	odd	10
605.2.j.c.444.1		8	11.9	even	5
605.2.j.c.444.1		8	55.24	odd	10
605.2.j.c.444.2		8	11.2	odd	10
605.2.j.c.444.2		8	55.9	even	10
3025.2.a.bb.1.2		4	5.3	odd	4
3025.2.a.bb.1.2		4	55.32	even	4
3025.2.a.bb.1.3		4	5.2	odd	4
3025.2.a.bb.1.3		4	55.43	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:	\( N \)	\(=\)	\( 605 = 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	605.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.32813i q^{2} +0.236068 q^{4} -2.23607 q^{5} +4.29792i q^{7} -2.96979i q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q-1.32813i q^{2} +0.236068 q^{4} -2.23607 q^{5} +4.29792i q^{7} -2.96979i q^{8} +3.00000 q^{9} +2.96979i q^{10} +6.95418i q^{13} +5.70820 q^{14} -3.47214 q^{16} +1.64166i q^{17} -3.98439i q^{18} -0.527864 q^{20} +5.00000 q^{25} +9.23607 q^{26} +1.01460i q^{28} +8.94427 q^{31} -1.32813i q^{32} +2.18034 q^{34} -9.61045i q^{35} +0.708204 q^{36} +6.64066i q^{40} +1.01460i q^{43} -6.70820 q^{45} -11.4721 q^{49} -6.64066i q^{50} +1.64166i q^{52} +12.7639 q^{56} -4.00000 q^{59} -11.8792i q^{62} +12.8938i q^{63} -8.70820 q^{64} -15.5500i q^{65} +0.387543i q^{68} -12.7639 q^{70} +8.00000 q^{71} -8.90937i q^{72} -12.2667i q^{73} +7.76393 q^{80} +9.00000 q^{81} +18.2063i q^{83} -3.67086i q^{85} +1.34752 q^{86} -13.4164 q^{89} +8.90937i q^{90} -29.8885 q^{91} +15.2365i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4} + 12 q^{9}+O(q^{10}) \) 4 * q - 8 * q^4 + 12 * q^9	\( 4 q - 8 q^{4} + 12 q^{9} - 4 q^{14} + 4 q^{16} - 20 q^{20} + 20 q^{25} + 28 q^{26} - 36 q^{34} - 24 q^{36} - 28 q^{49} + 60 q^{56} - 16 q^{59} - 8 q^{64} - 60 q^{70} + 32 q^{71} + 40 q^{80} + 36 q^{81} + 68 q^{86} - 48 q^{91}+O(q^{100}) \) 4 * q - 8 * q^4 + 12 * q^9 - 4 * q^14 + 4 * q^16 - 20 * q^20 + 20 * q^25 + 28 * q^26 - 36 * q^34 - 24 * q^36 - 28 * q^49 + 60 * q^56 - 16 * q^59 - 8 * q^64 - 60 * q^70 + 32 * q^71 + 40 * q^80 + 36 * q^81 + 68 * q^86 - 48 * q^91

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