LMFDB - Embedded newform 45.2.b.a.19.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [45,2,Mod(19,45)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(45, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("45.19");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 45 = 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	45.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:	$0.359326809096$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-5}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 5 $ x^2 + 5
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			19.1
Root			$-2.23607i$ of defining polynomial
Character	$\chi$	$=$	45.19
Dual form			45.2.b.a.19.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23607i q^{2} -3.00000 q^{4} +2.23607i q^{5} +2.23607i q^{8} +O(q^{10})$	$q-2.23607i q^{2} -3.00000 q^{4} +2.23607i q^{5} +2.23607i q^{8} +5.00000 q^{10} -1.00000 q^{16} +4.47214i q^{17} -4.00000 q^{19} -6.70820i q^{20} -8.94427i q^{23} -5.00000 q^{25} +8.00000 q^{31} +6.70820i q^{32} +10.0000 q^{34} +8.94427i q^{38} -5.00000 q^{40} -20.0000 q^{46} -8.94427i q^{47} +7.00000 q^{49} +11.1803i q^{50} +4.47214i q^{53} +2.00000 q^{61} -17.8885i q^{62} +13.0000 q^{64} -13.4164i q^{68} +12.0000 q^{76} -16.0000 q^{79} -2.23607i q^{80} +17.8885i q^{83} -10.0000 q^{85} +26.8328i q^{92} -20.0000 q^{94} -8.94427i q^{95} -15.6525i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{4}+O(q^{10}) $ 2 * q - 6 * q^4	$ 2 q - 6 q^{4} + 10 q^{10} - 2 q^{16} - 8 q^{19} - 10 q^{25} + 16 q^{31} + 20 q^{34} - 10 q^{40} - 40 q^{46} + 14 q^{49} + 4 q^{61} + 26 q^{64} + 24 q^{76} - 32 q^{79} - 20 q^{85} - 40 q^{94}+O(q^{100}) $ 2 * q - 6 * q^4 + 10 * q^10 - 2 * q^16 - 8 * q^19 - 10 * q^25 + 16 * q^31 + 20 * q^34 - 10 * q^40 - 40 * q^46 + 14 * q^49 + 4 * q^61 + 26 * q^64 + 24 * q^76 - 32 * q^79 - 20 * q^85 - 40 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/45\mathbb{Z}\right)^\times$.

$n$	$11$	$37$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 2.23607i	− 1.58114i	−0.612372	−	0.790569i	$-0.709785\pi$
$2$	− 2.23607i	− 1.58114i	0.612372	−	0.790569i	$-0.290215\pi$
$3$	0	0
$3$	0	0
$4$	−3.00000	−1.50000
$5$	2.23607i	1.00000i
$5$	2.23607i	1.00000i
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	2.23607i	0.790569i
$9$	0	0
$10$	5.00000	1.58114
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	−1.00000	−0.250000
$17$	4.47214i	1.08465i	0.840168	+	0.542326i	$0.182456\pi$
$17$	4.47214i	1.08465i	−0.840168	+	0.542326i	$0.817544\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	− 6.70820i	− 1.50000i
$21$	0	0
$22$	0	0
$23$	− 8.94427i	− 1.86501i	−0.361158	−	0.932505i	$-0.617618\pi$
$23$	− 8.94427i	− 1.86501i	0.361158	−	0.932505i	$-0.382382\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	6.70820i	1.18585i
$33$	0	0
$34$	10.0000	1.71499
$35$	0	0
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	8.94427i	1.45095i
$39$	0	0
$40$	−5.00000	−0.790569
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	0	0
$46$	−20.0000	−2.94884
$47$	− 8.94427i	− 1.30466i	−0.757937	−	0.652328i	$-0.773792\pi$
$47$	− 8.94427i	− 1.30466i	0.757937	−	0.652328i	$-0.226208\pi$
$48$	0	0
$49$	7.00000	1.00000
$50$	11.1803i	1.58114i
$51$	0	0
$52$	0	0
$53$	4.47214i	0.614295i	0.951662	+	0.307148i	$0.0993745\pi$
$53$	4.47214i	0.614295i	−0.951662	+	0.307148i	$0.900625\pi$
$54$	0	0
$55$	0	0
$56$	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
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	− 17.8885i	− 2.27185i
$63$	0	0
$64$	13.0000	1.62500
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	− 13.4164i	− 1.62698i
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	0	0
$76$	12.0000	1.37649
$77$	0	0
$78$	0	0
$79$	−16.0000	−1.80014	−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−16.0000	−1.80014	−0.900070	+	0.435745i	$0.856485\pi$
$80$	− 2.23607i	− 0.250000i
$81$	0	0
$82$	0	0
$83$	17.8885i	1.96352i	0.190117	+	0.981761i	$0.439113\pi$
$83$	17.8885i	1.96352i	−0.190117	+	0.981761i	$0.560887\pi$
$84$	0	0
$85$	−10.0000	−1.08465
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	0	0
$92$	26.8328i	2.79751i
$93$	0	0
$94$	−20.0000	−2.06284
$95$	− 8.94427i	− 0.917663i
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	− 15.6525i	− 1.58114i
$99$	0	0
$100$	15.0000	1.50000
$101$	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			$\pi$
$104$	0	0
$105$	0	0
$106$	10.0000	0.971286
$107$	17.8885i	1.72935i	0.502331	+	0.864675i	$0.332476\pi$
$107$	17.8885i	1.72935i	−0.502331	+	0.864675i	$0.667524\pi$
$108$	0	0
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	4.47214i	0.420703i	0.977626	+	0.210352i	$0.0674609\pi$
$113$	4.47214i	0.420703i	−0.977626	+	0.210352i	$0.932539\pi$
$114$	0	0
$115$	20.0000	1.86501
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	− 4.47214i	− 0.404888i
$123$	0	0
$124$	−24.0000	−2.15526
$125$	− 11.1803i	− 1.00000i
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	− 15.6525i	− 1.38350i
$129$	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
$136$	−10.0000	−0.857493
$137$	− 22.3607i	− 1.91040i	−0.295958	−	0.955201i	$-0.595639\pi$
$137$	− 22.3607i	− 1.91040i	0.295958	−	0.955201i	$-0.404361\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	− 8.94427i	− 0.725476i
$153$	0	0
$154$	0	0
$155$	17.8885i	1.43684i
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	35.7771i	2.84627i
$159$	0	0
$160$	−15.0000	−1.18585
$161$	0	0
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	0	0
$166$	40.0000	3.10460
$167$	− 8.94427i	− 0.692129i	−0.938211	−	0.346064i	$-0.887518\pi$
$167$	− 8.94427i	− 0.692129i	0.938211	−	0.346064i	$-0.112482\pi$
$168$	0	0
$169$	13.0000	1.00000
$170$	22.3607i	1.71499i
$171$	0	0
$172$	0	0
$173$	− 22.3607i	− 1.70005i	−0.526742	−	0.850026i	$-0.676586\pi$
$173$	− 22.3607i	− 1.70005i	0.526742	−	0.850026i	$-0.323414\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	0	0
$183$	0	0
$184$	20.0000	1.47442
$185$	0	0
$186$	0	0
$187$	0	0
$188$	26.8328i	1.95698i
$189$	0	0
$190$	−20.0000	−1.45095
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	−21.0000	−1.50000
$197$	4.47214i	0.318626i	0.987228	+	0.159313i	$0.0509280\pi$
$197$	4.47214i	0.318626i	−0.987228	+	0.159313i	$0.949072\pi$
$198$	0	0
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	− 11.1803i	− 0.790569i
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−28.0000	−1.92760	−0.963800	−	0.266627i	$-0.914091\pi$
$211$	−28.0000	−1.92760	−0.963800	+	0.266627i	$0.914091\pi$
$212$	− 13.4164i	− 0.921443i
$213$	0	0
$214$	40.0000	2.73434
$215$	0	0
$216$	0	0
$217$	0	0
$218$	− 31.3050i	− 2.12024i
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	0	0
$226$	10.0000	0.665190
$227$	17.8885i	1.18730i	0.804722	+	0.593652i	$0.202314\pi$
$227$	17.8885i	1.18730i	−0.804722	+	0.593652i	$0.797686\pi$
$228$	0	0
$229$	26.0000	1.71813	0.859064	−	0.511868i	$-0.171046\pi$
$229$	26.0000	1.71813	0.859064	+	0.511868i	$0.171046\pi$
$230$	− 44.7214i	− 2.94884i
$231$	0	0
$232$	0	0
$233$	− 22.3607i	− 1.46490i	−0.680823	−	0.732448i	$-0.738378\pi$
$233$	− 22.3607i	− 1.46490i	0.680823	−	0.732448i	$-0.261622\pi$
$234$	0	0
$235$	20.0000	1.30466
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	24.5967i	1.58114i
$243$	0	0
$244$	−6.00000	−0.384111
$245$	15.6525i	1.00000i
$246$	0	0
$247$	0	0
$248$	17.8885i	1.13592i
$249$	0	0
$250$	−25.0000	−1.58114
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	−9.00000	−0.562500
$257$	31.3050i	1.95275i	0.216085	+	0.976375i	$0.430671\pi$
$257$	31.3050i	1.95275i	−0.216085	+	0.976375i	$0.569329\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 8.94427i	− 0.551527i	−0.961225	−	0.275764i	$-0.911069\pi$
$263$	− 8.94427i	− 0.551527i	0.961225	−	0.275764i	$-0.0889307\pi$
$264$	0	0
$265$	−10.0000	−0.614295
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	32.0000	1.94386	0.971931	−	0.235267i	$-0.0755965\pi$
$271$	32.0000	1.94386	0.971931	+	0.235267i	$0.0755965\pi$
$272$	− 4.47214i	− 0.271163i
$273$	0	0
$274$	−50.0000	−3.02061
$275$	0	0
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	8.94427i	0.536442i
$279$	0	0
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−3.00000	−0.176471
$290$	0	0
$291$	0	0
$292$	0	0
$293$	31.3050i	1.82885i	0.404750	+	0.914427i	$0.367359\pi$
$293$	31.3050i	1.82885i	−0.404750	+	0.914427i	$0.632641\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	− 17.8885i	− 1.02937i
$303$	0	0
$304$	4.00000	0.229416
$305$	4.47214i	0.256074i
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	40.0000	2.27185
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	0	0
$316$	48.0000	2.70021
$317$	− 22.3607i	− 1.25590i	−0.778253	−	0.627950i	$-0.783894\pi$
$317$	− 22.3607i	− 1.25590i	0.778253	−	0.627950i	$-0.216106\pi$
$318$	0	0
$319$	0	0
$320$	29.0689i	1.62500i
$321$	0	0
$322$	0	0
$323$	− 17.8885i	− 0.995345i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	− 53.6656i	− 2.94528i
$333$	0	0
$334$	−20.0000	−1.09435
$335$	0	0
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	− 29.0689i	− 1.58114i
$339$	0	0
$340$	30.0000	1.62698
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	−50.0000	−2.68802
$347$	− 35.7771i	− 1.92061i	−0.278944	−	0.960307i	$-0.589984\pi$
$347$	− 35.7771i	− 1.92061i	0.278944	−	0.960307i	$-0.410016\pi$
$348$	0	0
$349$	−34.0000	−1.81998	−0.909989	−	0.414632i	$-0.863910\pi$
$349$	−34.0000	−1.81998	−0.909989	+	0.414632i	$0.863910\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	31.3050i	1.66619i	0.553127	+	0.833097i	$0.313435\pi$
$353$	31.3050i	1.66619i	−0.553127	+	0.833097i	$0.686565\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	49.1935i	2.58555i
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	8.94427i	0.466252i
$369$	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
$375$	0	0
$376$	20.0000	1.03142
$377$	0	0
$378$	0	0
$379$	−4.00000	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$380$	26.8328i	1.37649i
$381$	0	0
$382$	0	0
$383$	− 8.94427i	− 0.457031i	−0.973540	−	0.228515i	$-0.926613\pi$
$383$	− 8.94427i	− 0.457031i	0.973540	−	0.228515i	$-0.0733872\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	40.0000	2.02289
$392$	15.6525i	0.790569i
$393$	0	0
$394$	10.0000	0.503793
$395$	− 35.7771i	− 1.80014i
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	35.7771i	1.79334i
$399$	0	0
$400$	5.00000	0.250000
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	26.0000	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−40.0000	−1.96352
$416$	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
$421$	38.0000	1.85201	0.926003	−	0.377515i	$-0.123221\pi$
$421$	38.0000	1.85201	0.926003	+	0.377515i	$0.123221\pi$
$422$	62.6099i	3.04780i
$423$	0	0
$424$	−10.0000	−0.485643
$425$	− 22.3607i	− 1.08465i
$426$	0	0
$427$	0	0
$428$	− 53.6656i	− 2.59403i
$429$	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	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	−42.0000	−2.01144
$437$	35.7771i	1.71145i
$438$	0	0
$439$	−16.0000	−0.763638	−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000	−0.763638	−0.381819	+	0.924237i	$0.624702\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	17.8885i	0.849910i	0.905214	+	0.424955i	$0.139710\pi$
$443$	17.8885i	0.849910i	−0.905214	+	0.424955i	$0.860290\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	0	0
$451$	0	0
$452$	− 13.4164i	− 0.631055i
$453$	0	0
$454$	40.0000	1.87729
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	− 58.1378i	− 2.71660i
$459$	0	0
$460$	−60.0000	−2.79751
$461$	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
$466$	−50.0000	−2.31621
$467$	− 35.7771i	− 1.65557i	−0.561048	−	0.827783i	$-0.689602\pi$
$467$	− 35.7771i	− 1.65557i	0.561048	−	0.827783i	$-0.310398\pi$
$468$	0	0
$469$	0	0
$470$	− 44.7214i	− 2.06284i
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	20.0000	0.917663
$476$	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
$482$	− 4.47214i	− 0.203700i
$483$	0	0
$484$	33.0000	1.50000
$485$	0	0
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	4.47214i	0.202444i
$489$	0	0
$490$	35.0000	1.58114
$491$	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
$496$	−8.00000	−0.359211
$497$	0	0
$498$	0	0
$499$	44.0000	1.96971	0.984855	−	0.173379i	$-0.0554684\pi$
$499$	44.0000	1.96971	0.984855	+	0.173379i	$0.0554684\pi$
$500$	33.5410i	1.50000i
$501$	0	0
$502$	0	0
$503$	44.7214i	1.99403i	0.0772283	+	0.997013i	$0.475393\pi$
$503$	44.7214i	1.99403i	−0.0772283	+	0.997013i	$0.524607\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	0	0
$512$	− 11.1803i	− 0.494106i
$513$	0	0
$514$	70.0000	3.08757
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0	0
$525$	0	0
$526$	−20.0000	−0.872041
$527$	35.7771i	1.55847i
$528$	0	0
$529$	−57.0000	−2.47826
$530$	22.3607i	0.971286i
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−40.0000	−1.72935
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	− 71.5542i	− 3.07352i
$543$	0	0
$544$	−30.0000	−1.28624
$545$	31.3050i	1.34096i
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	67.0820i	2.86560i
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	12.0000	0.508913
$557$	− 22.3607i	− 0.947452i	−0.880672	−	0.473726i	$-0.842909\pi$
$557$	− 22.3607i	− 0.947452i	0.880672	−	0.473726i	$-0.157091\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 35.7771i	− 1.50782i	−0.656975	−	0.753912i	$-0.728164\pi$
$563$	− 35.7771i	− 1.50782i	0.656975	−	0.753912i	$-0.271836\pi$
$564$	0	0
$565$	−10.0000	−0.420703
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	44.7214i	1.86501i
$576$	0	0
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	6.70820i	0.279024i
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	70.0000	2.89167
$587$	17.8885i	0.738339i	0.929362	+	0.369170i	$0.120358\pi$
$587$	17.8885i	0.738339i	−0.929362	+	0.369170i	$0.879642\pi$
$588$	0	0
$589$	−32.0000	−1.31854
$590$	0	0
$591$	0	0
$592$	0	0
$593$	4.47214i	0.183649i	0.995775	+	0.0918243i	$0.0292698\pi$
$593$	4.47214i	0.183649i	−0.995775	+	0.0918243i	$0.970730\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	38.0000	1.55005	0.775026	−	0.631929i	$-0.217737\pi$
$601$	38.0000	1.55005	0.775026	+	0.631929i	$0.217737\pi$
$602$	0	0
$603$	0	0
$604$	−24.0000	−0.976546
$605$	− 24.5967i	− 1.00000i
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	− 26.8328i	− 1.08821i
$609$	0	0
$610$	10.0000	0.404888
$611$	0	0
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 49.1935i	− 1.98046i	−0.139459	−	0.990228i	$-0.544536\pi$
$617$	− 49.1935i	− 1.98046i	0.139459	−	0.990228i	$-0.455464\pi$
$618$	0	0
$619$	44.0000	1.76851	0.884255	−	0.467005i	$-0.154667\pi$
$619$	44.0000	1.76851	0.884255	+	0.467005i	$0.154667\pi$
$620$	− 53.6656i	− 2.15526i
$621$	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
$631$	32.0000	1.27390	0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000	1.27390	0.636950	+	0.770905i	$0.280196\pi$
$632$	− 35.7771i	− 1.42314i
$633$	0	0
$634$	−50.0000	−1.98575
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	35.0000	1.38350
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	0	0
$646$	−40.0000	−1.57378
$647$	44.7214i	1.75818i	0.476658	+	0.879089i	$0.341848\pi$
$647$	44.7214i	1.75818i	−0.476658	+	0.879089i	$0.658152\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 49.1935i	− 1.92509i	−0.271122	−	0.962545i	$-0.587395\pi$
$653$	− 49.1935i	− 1.92509i	0.271122	−	0.962545i	$-0.412605\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	−22.0000	−0.855701	−0.427850	−	0.903850i	$-0.640729\pi$
$661$	−22.0000	−0.855701	−0.427850	+	0.903850i	$0.640729\pi$
$662$	62.6099i	2.43340i
$663$	0	0
$664$	−40.0000	−1.55230
$665$	0	0
$666$	0	0
$667$	0	0
$668$	26.8328i	1.03819i
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	0	0
$676$	−39.0000	−1.50000
$677$	31.3050i	1.20315i	0.798817	+	0.601574i	$0.205459\pi$
$677$	31.3050i	1.20315i	−0.798817	+	0.601574i	$0.794541\pi$
$678$	0	0
$679$	0	0
$680$	− 22.3607i	− 0.857493i
$681$	0	0
$682$	0	0
$683$	− 35.7771i	− 1.36897i	−0.729026	−	0.684486i	$-0.760027\pi$
$683$	− 35.7771i	− 1.36897i	0.729026	−	0.684486i	$-0.239973\pi$
$684$	0	0
$685$	50.0000	1.91040
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−52.0000	−1.97817	−0.989087	−	0.147335i	$-0.952930\pi$
$691$	−52.0000	−1.97817	−0.989087	+	0.147335i	$0.952930\pi$
$692$	67.0820i	2.55008i
$693$	0	0
$694$	−80.0000	−3.03676
$695$	− 8.94427i	− 0.339276i
$696$	0	0
$697$	0	0
$698$	76.0263i	2.87764i
$699$	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
$706$	70.0000	2.63448
$707$	0	0
$708$	0	0
$709$	26.0000	0.976450	0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000	0.976450	0.488225	+	0.872718i	$0.337644\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 71.5542i	− 2.67972i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	0	0
$722$	6.70820i	0.249653i
$723$	0	0
$724$	66.0000	2.45287
$725$	0	0
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	0	0
$736$	60.0000	2.21163
$737$	0	0
$738$	0	0
$739$	−4.00000	−0.147142	−0.0735712	−	0.997290i	$-0.523440\pi$
$739$	−4.00000	−0.147142	−0.0735712	+	0.997290i	$0.523440\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	44.7214i	1.64067i	0.571885	+	0.820334i	$0.306212\pi$
$743$	44.7214i	1.64067i	−0.571885	+	0.820334i	$0.693788\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	8.94427i	0.326164i
$753$	0	0
$754$	0	0
$755$	17.8885i	0.651031i
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	8.94427i	0.324871i
$759$	0	0
$760$	20.0000	0.725476
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	−20.0000	−0.722629
$767$	0	0
$768$	0	0
$769$	−46.0000	−1.65880	−0.829401	−	0.558653i	$-0.811318\pi$
$769$	−46.0000	−1.65880	−0.829401	+	0.558653i	$0.811318\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	4.47214i	0.160852i	0.996761	+	0.0804258i	$0.0256280\pi$
$773$	4.47214i	0.160852i	−0.996761	+	0.0804258i	$0.974372\pi$
$774$	0	0
$775$	−40.0000	−1.43684
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	− 89.4427i	− 3.19847i
$783$	0	0
$784$	−7.00000	−0.250000
$785$	0	0
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	− 13.4164i	− 0.477940i
$789$	0	0
$790$	−80.0000	−2.84627
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	48.0000	1.70131
$797$	− 49.1935i	− 1.74252i	−0.490819	−	0.871262i	$-0.663302\pi$
$797$	− 49.1935i	− 1.74252i	0.490819	−	0.871262i	$-0.336698\pi$
$798$	0	0
$799$	40.0000	1.41510
$800$	− 33.5410i	− 1.18585i
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	0	0
$811$	−52.0000	−1.82597	−0.912983	−	0.407997i	$-0.866228\pi$
$811$	−52.0000	−1.82597	−0.912983	+	0.407997i	$0.866228\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	− 58.1378i	− 2.03274i
$819$	0	0
$820$	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
$826$	0	0
$827$	− 35.7771i	− 1.24409i	−0.782981	−	0.622046i	$-0.786302\pi$
$827$	− 35.7771i	− 1.24409i	0.782981	−	0.622046i	$-0.213698\pi$
$828$	0	0
$829$	−34.0000	−1.18087	−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000	−1.18087	−0.590434	+	0.807086i	$0.701044\pi$
$830$	89.4427i	3.10460i
$831$	0	0
$832$	0	0
$833$	31.3050i	1.08465i
$834$	0	0
$835$	20.0000	0.692129
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	− 84.9706i	− 2.92828i
$843$	0	0
$844$	84.0000	2.89140
$845$	29.0689i	1.00000i
$846$	0	0
$847$	0	0
$848$	− 4.47214i	− 0.153574i
$849$	0	0
$850$	−50.0000	−1.71499
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	−40.0000	−1.36717
$857$	58.1378i	1.98595i	0.118331	+	0.992974i	$0.462245\pi$
$857$	58.1378i	1.98595i	−0.118331	+	0.992974i	$0.537755\pi$
$858$	0	0
$859$	44.0000	1.50126	0.750630	−	0.660722i	$-0.229750\pi$
$859$	44.0000	1.50126	0.750630	+	0.660722i	$0.229750\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	44.7214i	1.52233i	0.648557	+	0.761166i	$0.275373\pi$
$863$	44.7214i	1.52233i	−0.648557	+	0.761166i	$0.724627\pi$
$864$	0	0
$865$	50.0000	1.70005
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	31.3050i	1.06012i
$873$	0	0
$874$	80.0000	2.70604
$875$	0	0
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	35.7771i	1.20742i
$879$	0	0
$880$	0	0
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	40.0000	1.34383
$887$	− 8.94427i	− 0.300319i	−0.988662	−	0.150160i	$-0.952021\pi$
$887$	− 8.94427i	− 0.300319i	0.988662	−	0.150160i	$-0.0479788\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	35.7771i	1.19723i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−20.0000	−0.666297
$902$	0	0
$903$	0	0
$904$	−10.0000	−0.332595
$905$	− 49.1935i	− 1.63525i
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	− 53.6656i	− 1.78096i
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	−78.0000	−2.57719
$917$	0	0
$918$	0	0
$919$	56.0000	1.84727	0.923635	−	0.383274i	$-0.125203\pi$
$919$	56.0000	1.84727	0.923635	+	0.383274i	$0.125203\pi$
$920$	44.7214i	1.47442i
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	0	0
$931$	−28.0000	−0.917663
$932$	67.0820i	2.19735i
$933$	0	0
$934$	−80.0000	−2.61768
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0	0
$940$	−60.0000	−1.95698
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	17.8885i	0.581300i	0.956830	+	0.290650i	$0.0938715\pi$
$947$	17.8885i	0.581300i	−0.956830	+	0.290650i	$0.906129\pi$
$948$	0	0
$949$	0	0
$950$	− 44.7214i	− 1.45095i
$951$	0	0
$952$	0	0
$953$	58.1378i	1.88327i	0.336640	+	0.941634i	$0.390710\pi$
$953$	58.1378i	1.88327i	−0.336640	+	0.941634i	$0.609290\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	0	0
$964$	−6.00000	−0.193247
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	− 24.5967i	− 0.790569i
$969$	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
$975$	0	0
$976$	−2.00000	−0.0640184
$977$	4.47214i	0.143076i	0.997438	+	0.0715382i	$0.0227908\pi$
$977$	4.47214i	0.143076i	−0.997438	+	0.0715382i	$0.977209\pi$
$978$	0	0
$979$	0	0
$980$	− 46.9574i	− 1.50000i
$981$	0	0
$982$	0	0
$983$	− 62.6099i	− 1.99695i	−0.0552438	−	0.998473i	$-0.517594\pi$
$983$	− 62.6099i	− 1.99695i	0.0552438	−	0.998473i	$-0.482406\pi$
$984$	0	0
$985$	−10.0000	−0.318626
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	53.6656i	1.70389i
$993$	0	0
$994$	0	0
$995$	− 35.7771i	− 1.13421i
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	− 98.3870i	− 3.11439i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.2.b.a.19.1	✓	2
3.2	odd	2	inner	45.2.b.a.19.2	yes	2
4.3	odd	2		720.2.f.d.289.2		2
5.2	odd	4		225.2.a.f.1.2		2
5.3	odd	4		225.2.a.f.1.1		2
5.4	even	2	inner	45.2.b.a.19.2	yes	2
7.6	odd	2		2205.2.d.a.1324.1		2
8.3	odd	2		2880.2.f.j.1729.1		2
8.5	even	2		2880.2.f.k.1729.1		2
9.2	odd	6		405.2.j.c.109.2		4
9.4	even	3		405.2.j.c.379.2		4
9.5	odd	6		405.2.j.c.379.1		4
9.7	even	3		405.2.j.c.109.1		4
12.11	even	2		720.2.f.d.289.1		2
15.2	even	4		225.2.a.f.1.1		2
15.8	even	4		225.2.a.f.1.2		2
15.14	odd	2	CM	45.2.b.a.19.1	✓	2
20.3	even	4		3600.2.a.bs.1.2		2
20.7	even	4		3600.2.a.bs.1.1		2
20.19	odd	2		720.2.f.d.289.1		2
21.20	even	2		2205.2.d.a.1324.2		2
24.5	odd	2		2880.2.f.k.1729.2		2
24.11	even	2		2880.2.f.j.1729.2		2
35.34	odd	2		2205.2.d.a.1324.2		2
40.19	odd	2		2880.2.f.j.1729.2		2
40.29	even	2		2880.2.f.k.1729.2		2
45.4	even	6		405.2.j.c.379.1		4
45.14	odd	6		405.2.j.c.379.2		4
45.29	odd	6		405.2.j.c.109.1		4
45.34	even	6		405.2.j.c.109.2		4
60.23	odd	4		3600.2.a.bs.1.1		2
60.47	odd	4		3600.2.a.bs.1.2		2
60.59	even	2		720.2.f.d.289.2		2
105.104	even	2		2205.2.d.a.1324.1		2
120.29	odd	2		2880.2.f.k.1729.1		2
120.59	even	2		2880.2.f.j.1729.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.2.b.a.19.1	✓	2	1.1	even	1	trivial
45.2.b.a.19.1	✓	2	15.14	odd	2	CM
45.2.b.a.19.2	yes	2	3.2	odd	2	inner
45.2.b.a.19.2	yes	2	5.4	even	2	inner
225.2.a.f.1.1		2	5.3	odd	4
225.2.a.f.1.1		2	15.2	even	4
225.2.a.f.1.2		2	5.2	odd	4
225.2.a.f.1.2		2	15.8	even	4
405.2.j.c.109.1		4	9.7	even	3
405.2.j.c.109.1		4	45.29	odd	6
405.2.j.c.109.2		4	9.2	odd	6
405.2.j.c.109.2		4	45.34	even	6
405.2.j.c.379.1		4	9.5	odd	6
405.2.j.c.379.1		4	45.4	even	6
405.2.j.c.379.2		4	9.4	even	3
405.2.j.c.379.2		4	45.14	odd	6
720.2.f.d.289.1		2	12.11	even	2
720.2.f.d.289.1		2	20.19	odd	2
720.2.f.d.289.2		2	4.3	odd	2
720.2.f.d.289.2		2	60.59	even	2
2205.2.d.a.1324.1		2	7.6	odd	2
2205.2.d.a.1324.1		2	105.104	even	2
2205.2.d.a.1324.2		2	21.20	even	2
2205.2.d.a.1324.2		2	35.34	odd	2
2880.2.f.j.1729.1		2	8.3	odd	2
2880.2.f.j.1729.1		2	120.59	even	2
2880.2.f.j.1729.2		2	24.11	even	2
2880.2.f.j.1729.2		2	40.19	odd	2
2880.2.f.k.1729.1		2	8.5	even	2
2880.2.f.k.1729.1		2	120.29	odd	2
2880.2.f.k.1729.2		2	24.5	odd	2
2880.2.f.k.1729.2		2	40.29	even	2
3600.2.a.bs.1.1		2	20.7	even	4
3600.2.a.bs.1.1		2	60.23	odd	4
3600.2.a.bs.1.2		2	20.3	even	4
3600.2.a.bs.1.2		2	60.47	odd	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:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	45.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.23607i q^{2} -3.00000 q^{4} +2.23607i q^{5} +2.23607i q^{8} +O(q^{10})\)	\(q-2.23607i q^{2} -3.00000 q^{4} +2.23607i q^{5} +2.23607i q^{8} +5.00000 q^{10} -1.00000 q^{16} +4.47214i q^{17} -4.00000 q^{19} -6.70820i q^{20} -8.94427i q^{23} -5.00000 q^{25} +8.00000 q^{31} +6.70820i q^{32} +10.0000 q^{34} +8.94427i q^{38} -5.00000 q^{40} -20.0000 q^{46} -8.94427i q^{47} +7.00000 q^{49} +11.1803i q^{50} +4.47214i q^{53} +2.00000 q^{61} -17.8885i q^{62} +13.0000 q^{64} -13.4164i q^{68} +12.0000 q^{76} -16.0000 q^{79} -2.23607i q^{80} +17.8885i q^{83} -10.0000 q^{85} +26.8328i q^{92} -20.0000 q^{94} -8.94427i q^{95} -15.6525i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{4}+O(q^{10}) \) 2 * q - 6 * q^4	\( 2 q - 6 q^{4} + 10 q^{10} - 2 q^{16} - 8 q^{19} - 10 q^{25} + 16 q^{31} + 20 q^{34} - 10 q^{40} - 40 q^{46} + 14 q^{49} + 4 q^{61} + 26 q^{64} + 24 q^{76} - 32 q^{79} - 20 q^{85} - 40 q^{94}+O(q^{100}) \) 2 * q - 6 * q^4 + 10 * q^10 - 2 * q^16 - 8 * q^19 - 10 * q^25 + 16 * q^31 + 20 * q^34 - 10 * q^40 - 40 * q^46 + 14 * q^49 + 4 * q^61 + 26 * q^64 + 24 * q^76 - 32 * q^79 - 20 * q^85 - 40 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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