LMFDB - Embedded newform 360.2.m.b.179.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [360,2,Mod(179,360)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(360, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("360.179");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 360 = 2^{3} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	360.m (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:	$2.87461447277$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 9 $ x^4 - 4*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			179.4
Root			$-1.58114 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	360.179
Dual form			360.2.m.b.179.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{2} -2.00000 q^{4} +2.23607i q^{5} +5.16228 q^{7} -2.82843i q^{8} +O(q^{10})$	$q+1.41421i q^{2} -2.00000 q^{4} +2.23607i q^{5} +5.16228 q^{7} -2.82843i q^{8} -3.16228 q^{10} +3.05792i q^{11} -0.837722 q^{13} +7.30056i q^{14} +4.00000 q^{16} -6.32456 q^{19} -4.47214i q^{20} -4.32456 q^{22} +4.47214i q^{23} -5.00000 q^{25} -1.18472i q^{26} -10.3246 q^{28} +5.65685i q^{32} +11.5432i q^{35} +11.1623 q^{37} -8.94427i q^{38} +6.32456 q^{40} -10.3585i q^{41} -6.11584i q^{44} -6.32456 q^{46} +2.82843i q^{47} +19.6491 q^{49} -7.07107i q^{50} +1.67544 q^{52} -5.65685i q^{53} -6.83772 q^{55} -14.6011i q^{56} -5.42736i q^{59} -8.00000 q^{64} -1.87320i q^{65} -16.3246 q^{70} +15.7858i q^{74} +12.6491 q^{76} +15.7858i q^{77} +8.94427i q^{80} +14.6491 q^{82} +8.64911 q^{88} -18.8438i q^{89} -4.32456 q^{91} -8.94427i q^{92} -4.00000 q^{94} -14.1421i q^{95} +27.7880i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4} + 8 q^{7}+O(q^{10}) $ 4 * q - 8 * q^4 + 8 * q^7	$ 4 q - 8 q^{4} + 8 q^{7} - 16 q^{13} + 16 q^{16} + 8 q^{22} - 20 q^{25} - 16 q^{28} + 32 q^{37} + 28 q^{49} + 32 q^{52} - 40 q^{55} - 32 q^{64} - 40 q^{70} + 8 q^{82} - 16 q^{88} + 8 q^{91} - 16 q^{94}+O(q^{100}) $ 4 * q - 8 * q^4 + 8 * q^7 - 16 * q^13 + 16 * q^16 + 8 * q^22 - 20 * q^25 - 16 * q^28 + 32 * q^37 + 28 * q^49 + 32 * q^52 - 40 * q^55 - 32 * q^64 - 40 * q^70 + 8 * q^82 - 16 * q^88 + 8 * q^91 - 16 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/360\mathbb{Z}\right)^\times$.

$n$	$181$	$217$	$271$	$281$
$\chi(n)$	$-1$	$-1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.41421i	1.00000i
$2$	1.41421i	1.00000i
$3$	0	0
$3$	0	0
$4$	−2.00000	−1.00000
$5$	2.23607i	1.00000i
$5$	2.23607i	1.00000i
$6$	0	0
$7$	5.16228	1.95116	0.975579	−	0.219650i	$-0.0704915\pi$
$7$	5.16228	1.95116	0.975579	+	0.219650i	$0.0704915\pi$
$8$	− 2.82843i	− 1.00000i
$9$	0	0
$10$	−3.16228	−1.00000
$11$	3.05792i	0.921998i	0.887401	+	0.460999i	$0.152509\pi$
$11$	3.05792i	0.921998i	−0.887401	+	0.460999i	$0.847491\pi$
$12$	0	0
$13$	−0.837722	−0.232342	−0.116171	−	0.993229i	$-0.537062\pi$
$13$	−0.837722	−0.232342	−0.116171	+	0.993229i	$0.537062\pi$
$14$	7.30056i	1.95116i
$15$	0	0
$16$	4.00000	1.00000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	−6.32456	−1.45095	−0.725476	−	0.688247i	$-0.758380\pi$
$19$	−6.32456	−1.45095	−0.725476	+	0.688247i	$0.758380\pi$
$20$	− 4.47214i	− 1.00000i
$21$	0	0
$22$	−4.32456	−0.921998
$23$	4.47214i	0.932505i	0.884652	+	0.466252i	$0.154396\pi$
$23$	4.47214i	0.932505i	−0.884652	+	0.466252i	$0.845604\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	− 1.18472i	− 0.232342i
$27$	0	0
$28$	−10.3246	−1.95116
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	5.65685i	1.00000i
$33$	0	0
$34$	0	0
$35$	11.5432i	1.95116i
$36$	0	0
$37$	11.1623	1.83507	0.917534	−	0.397658i	$-0.130177\pi$
$37$	11.1623	1.83507	0.917534	+	0.397658i	$0.130177\pi$
$38$	− 8.94427i	− 1.45095i
$39$	0	0
$40$	6.32456	1.00000
$41$	− 10.3585i	− 1.61772i	−0.587999	−	0.808862i	$-0.700084\pi$
$41$	− 10.3585i	− 1.61772i	0.587999	−	0.808862i	$-0.299916\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	− 6.11584i	− 0.921998i
$45$	0	0
$46$	−6.32456	−0.932505
$47$	2.82843i	0.412568i	0.978492	+	0.206284i	$0.0661372\pi$
$47$	2.82843i	0.412568i	−0.978492	+	0.206284i	$0.933863\pi$
$48$	0	0
$49$	19.6491	2.80702
$50$	− 7.07107i	− 1.00000i
$51$	0	0
$52$	1.67544	0.232342
$53$	− 5.65685i	− 0.777029i	−0.921443	−	0.388514i	$-0.872988\pi$
$53$	− 5.65685i	− 0.777029i	0.921443	−	0.388514i	$-0.127012\pi$
$54$	0	0
$55$	−6.83772	−0.921998
$56$	− 14.6011i	− 1.95116i
$57$	0	0
$58$	0	0
$59$	− 5.42736i	− 0.706582i	−0.935513	−	0.353291i	$-0.885063\pi$
$59$	− 5.42736i	− 0.706582i	0.935513	−	0.353291i	$-0.114937\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	−8.00000	−1.00000
$65$	− 1.87320i	− 0.232342i
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0	0
$70$	−16.3246	−1.95116
$71$	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			$\pi$
$74$	15.7858i	1.83507i
$75$	0	0
$76$	12.6491	1.45095
$77$	15.7858i	1.79896i
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	8.94427i	1.00000i
$81$	0	0
$82$	14.6491	1.61772
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	8.64911	0.921998
$89$	− 18.8438i	− 1.99744i	−0.0506267	−	0.998718i	$-0.516122\pi$
$89$	− 18.8438i	− 1.99744i	0.0506267	−	0.998718i	$-0.483878\pi$
$90$	0	0
$91$	−4.32456	−0.453337
$92$	− 8.94427i	− 0.932505i
$93$	0	0
$94$	−4.00000	−0.412568
$95$	− 14.1421i	− 1.45095i
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	27.7880i	2.80702i
$99$	0	0
$100$	10.0000	1.00000
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	17.1623	1.69105	0.845525	−	0.533936i	$-0.179288\pi$
$103$	17.1623	1.69105	0.845525	+	0.533936i	$0.179288\pi$
$104$	2.36944i	0.232342i
$105$	0	0
$106$	8.00000	0.777029
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	− 9.67000i	− 0.921998i
$111$	0	0
$112$	20.6491	1.95116
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	−10.0000	−0.932505
$116$	0	0
$117$	0	0
$118$	7.67544	0.706582
$119$	0	0
$120$	0	0
$121$	1.64911	0.149919
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 11.1803i	− 1.00000i
$126$	0	0
$127$	−13.8114	−1.22556	−0.612781	−	0.790253i	$-0.709949\pi$
$127$	−13.8114	−1.22556	−0.612781	+	0.790253i	$0.709949\pi$
$128$	− 11.3137i	− 1.00000i
$129$	0	0
$130$	2.64911	0.232342
$131$	20.0285i	1.74990i	0.484216	+	0.874948i	$0.339105\pi$
$131$	20.0285i	1.74990i	−0.484216	+	0.874948i	$0.660895\pi$
$132$	0	0
$133$	−32.6491	−2.83104
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	14.0000	1.18746	0.593732	−	0.804663i	$-0.297654\pi$
$139$	14.0000	1.18746	0.593732	+	0.804663i	$0.297654\pi$
$140$	− 23.0864i	− 1.95116i
$141$	0	0
$142$	0	0
$143$	− 2.56169i	− 0.214219i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	−22.3246	−1.83507
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	17.8885i	1.45095i
$153$	0	0
$154$	−22.3246	−1.79896
$155$	0	0
$156$	0	0
$157$	−7.81139	−0.623417	−0.311708	−	0.950178i	$-0.600901\pi$
$157$	−7.81139	−0.623417	−0.311708	+	0.950178i	$0.600901\pi$
$158$	0	0
$159$	0	0
$160$	−12.6491	−1.00000
$161$	23.0864i	1.81946i
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	20.7170i	1.61772i
$165$	0	0
$166$	0	0
$167$	4.47214i	0.346064i	0.984916	+	0.173032i	$0.0553564\pi$
$167$	4.47214i	0.346064i	−0.984916	+	0.173032i	$0.944644\pi$
$168$	0	0
$169$	−12.2982	−0.946017
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 22.6274i	− 1.72033i	−0.510015	−	0.860165i	$-0.670360\pi$
$173$	− 22.6274i	− 1.72033i	0.510015	−	0.860165i	$-0.329640\pi$
$174$	0	0
$175$	−25.8114	−1.95116
$176$	12.2317i	0.921998i
$177$	0	0
$178$	26.6491	1.99744
$179$	− 13.9126i	− 1.03988i	−0.854203	−	0.519940i	$-0.825954\pi$
$179$	− 13.9126i	− 1.03988i	0.854203	−	0.519940i	$-0.174046\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	− 6.11584i	− 0.453337i
$183$	0	0
$184$	12.6491	0.932505
$185$	24.9596i	1.83507i
$186$	0	0
$187$	0	0
$188$	− 5.65685i	− 0.412568i
$189$	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
$196$	−39.2982	−2.80702
$197$	− 22.3607i	− 1.59313i	−0.604551	−	0.796566i	$-0.706648\pi$
$197$	− 22.3607i	− 1.59313i	0.604551	−	0.796566i	$-0.293352\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	14.1421i	1.00000i
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	23.1623	1.61772
$206$	24.2711i	1.69105i
$207$	0	0
$208$	−3.35089	−0.232342
$209$	− 19.3400i	− 1.33778i
$210$	0	0
$211$	−22.0000	−1.51454	−0.757271	−	0.653101i	$-0.773468\pi$
$211$	−22.0000	−1.51454	−0.757271	+	0.653101i	$0.773468\pi$
$212$	11.3137i	0.777029i
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	13.6754	0.921998
$221$	0	0
$222$	0	0
$223$	−1.81139	−0.121300	−0.0606498	−	0.998159i	$-0.519317\pi$
$223$	−1.81139	−0.121300	−0.0606498	+	0.998159i	$0.519317\pi$
$224$	29.2023i	1.95116i
$225$	0	0
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	− 14.1421i	− 0.932505i
$231$	0	0
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	−6.32456	−0.412568
$236$	10.8547i	0.706582i
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−25.2982	−1.62960	−0.814801	−	0.579741i	$-0.803154\pi$
$241$	−25.2982	−1.62960	−0.814801	+	0.579741i	$0.803154\pi$
$242$	2.33219i	0.149919i
$243$	0	0
$244$	0	0
$245$	43.9367i	2.80702i
$246$	0	0
$247$	5.29822	0.337118
$248$	0	0
$249$	0	0
$250$	15.8114	1.00000
$251$	− 23.7749i	− 1.50066i	−0.661065	−	0.750329i	$-0.729895\pi$
$251$	− 23.7749i	− 1.50066i	0.661065	−	0.750329i	$-0.270105\pi$
$252$	0	0
$253$	−13.6754	−0.859768
$254$	− 19.5323i	− 1.22556i
$255$	0	0
$256$	16.0000	1.00000
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	57.6228	3.58051
$260$	3.74641i	0.232342i
$261$	0	0
$262$	−28.3246	−1.74990
$263$	31.3050i	1.93035i	0.261612	+	0.965173i	$0.415746\pi$
$263$	31.3050i	1.93035i	−0.261612	+	0.965173i	$0.584254\pi$
$264$	0	0
$265$	12.6491	0.777029
$266$	− 46.1728i	− 2.83104i
$267$	0	0
$268$	0	0
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 15.2896i	− 0.921998i
$276$	0	0
$277$	30.1359	1.81069	0.905347	−	0.424673i	$-0.139611\pi$
$277$	30.1359	1.81069	0.905347	+	0.424673i	$0.139611\pi$
$278$	19.7990i	1.18746i
$279$	0	0
$280$	32.6491	1.95116
$281$	33.4449i	1.99516i	0.0695635	+	0.997578i	$0.477839\pi$
$281$	33.4449i	1.99516i	−0.0695635	+	0.997578i	$0.522161\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	0	0
$286$	3.62278	0.214219
$287$	− 53.4734i	− 3.15643i
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	4.47214i	0.261265i	0.991431	+	0.130632i	$0.0417008\pi$
$293$	4.47214i	0.261265i	−0.991431	+	0.130632i	$0.958299\pi$
$294$	0	0
$295$	12.1359	0.706582
$296$	− 31.5717i	− 1.83507i
$297$	0	0
$298$	0	0
$299$	− 3.74641i	− 0.216660i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	−25.2982	−1.45095
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	− 31.5717i	− 1.79896i
$309$	0	0
$310$	0	0
$311$	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			$\pi$
$314$	− 11.0470i	− 0.623417i
$315$	0	0
$316$	0	0
$317$	31.3050i	1.75826i	0.476581	+	0.879131i	$0.341876\pi$
$317$	31.3050i	1.75826i	−0.476581	+	0.879131i	$0.658124\pi$
$318$	0	0
$319$	0	0
$320$	− 17.8885i	− 1.00000i
$321$	0	0
$322$	−32.6491	−1.81946
$323$	0	0
$324$	0	0
$325$	4.18861	0.232342
$326$	0	0
$327$	0	0
$328$	−29.2982	−1.61772
$329$	14.6011i	0.804986i
$330$	0	0
$331$	31.6228	1.73814	0.869072	−	0.494685i	$-0.164716\pi$
$331$	31.6228	1.73814	0.869072	+	0.494685i	$0.164716\pi$
$332$	0	0
$333$	0	0
$334$	−6.32456	−0.346064
$335$	0	0
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	− 17.3923i	− 0.946017i
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	65.2982	3.52577
$344$	0	0
$345$	0	0
$346$	32.0000	1.72033
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	− 36.5028i	− 1.95116i
$351$	0	0
$352$	−17.2982	−0.921998
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	0	0
$356$	37.6875i	1.99744i
$357$	0	0
$358$	19.6754	1.03988
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	21.0000	1.10526
$362$	0	0
$363$	0	0
$364$	8.64911	0.453337
$365$	0	0
$366$	0	0
$367$	−37.8114	−1.97374	−0.986869	−	0.161521i	$-0.948360\pi$
$367$	−37.8114	−1.97374	−0.986869	+	0.161521i	$0.948360\pi$
$368$	17.8885i	0.932505i
$369$	0	0
$370$	−35.2982	−1.83507
$371$	− 29.2023i	− 1.51611i
$372$	0	0
$373$	6.13594	0.317707	0.158854	−	0.987302i	$-0.449220\pi$
$373$	6.13594	0.317707	0.158854	+	0.987302i	$0.449220\pi$
$374$	0	0
$375$	0	0
$376$	8.00000	0.412568
$377$	0	0
$378$	0	0
$379$	−34.0000	−1.74646	−0.873231	−	0.487306i	$-0.837980\pi$
$379$	−34.0000	−1.74646	−0.873231	+	0.487306i	$0.837980\pi$
$380$	28.2843i	1.45095i
$381$	0	0
$382$	0	0
$383$	36.7696i	1.87884i	0.342773	+	0.939418i	$0.388634\pi$
$383$	36.7696i	1.87884i	−0.342773	+	0.939418i	$0.611366\pi$
$384$	0	0
$385$	−35.2982	−1.79896
$386$	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
$391$	0	0
$392$	− 55.5761i	− 2.80702i
$393$	0	0
$394$	31.6228	1.59313
$395$	0	0
$396$	0	0
$397$	−24.8377	−1.24657	−0.623285	−	0.781995i	$-0.714202\pi$
$397$	−24.8377	−1.24657	−0.623285	+	0.781995i	$0.714202\pi$
$398$	0	0
$399$	0	0
$400$	−20.0000	−1.00000
$401$	− 35.8143i	− 1.78848i	−0.447586	−	0.894241i	$-0.647716\pi$
$401$	− 35.8143i	− 1.78848i	0.447586	−	0.894241i	$-0.352284\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	34.1334i	1.69193i
$408$	0	0
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	32.7564i	1.61772i
$411$	0	0
$412$	−34.3246	−1.69105
$413$	− 28.0175i	− 1.37865i
$414$	0	0
$415$	0	0
$416$	− 4.73887i	− 0.232342i
$417$	0	0
$418$	27.3509	1.33778
$419$	− 40.7455i	− 1.99055i	−0.0971178	−	0.995273i	$-0.530962\pi$
$419$	− 40.7455i	− 1.99055i	0.0971178	−	0.995273i	$-0.469038\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	− 31.1127i	− 1.51454i
$423$	0	0
$424$	−16.0000	−0.777029
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	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
$436$	0	0
$437$	− 28.2843i	− 1.35302i
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	19.3400i	0.921998i
$441$	0	0
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	42.1359	1.99744
$446$	− 2.56169i	− 0.121300i
$447$	0	0
$448$	−41.2982	−1.95116
$449$	41.9302i	1.97881i	0.145191	+	0.989404i	$0.453620\pi$
$449$	41.9302i	1.97881i	−0.145191	+	0.989404i	$0.546380\pi$
$450$	0	0
$451$	31.6754	1.49154
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 9.67000i	− 0.453337i
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	0	0
$460$	20.0000	0.932505
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	10.1886	0.473505	0.236752	−	0.971570i	$-0.423917\pi$
$463$	10.1886	0.473505	0.236752	+	0.971570i	$0.423917\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	0	0
$470$	− 8.94427i	− 0.412568i
$471$	0	0
$472$	−15.3509	−0.706582
$473$	0	0
$474$	0	0
$475$	31.6228	1.45095
$476$	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
$481$	−9.35089	−0.426364
$482$	− 35.7771i	− 1.62960i
$483$	0	0
$484$	−3.29822	−0.149919
$485$	0	0
$486$	0	0
$487$	0.135944	0.00616019	0.00308010	−	0.999995i	$-0.499020\pi$
$487$	0.135944	0.00616019	0.00308010	+	0.999995i	$0.499020\pi$
$488$	0	0
$489$	0	0
$490$	−62.1359	−2.80702
$491$	29.8907i	1.34895i	0.738298	+	0.674475i	$0.235630\pi$
$491$	29.8907i	1.34895i	−0.738298	+	0.674475i	$0.764370\pi$
$492$	0	0
$493$	0	0
$494$	7.49282i	0.337118i
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−44.2719	−1.98188	−0.990941	−	0.134298i	$-0.957122\pi$
$499$	−44.2719	−1.98188	−0.990941	+	0.134298i	$0.957122\pi$
$500$	22.3607i	1.00000i
$501$	0	0
$502$	33.6228	1.50066
$503$	19.7990i	0.882793i	0.897312	+	0.441397i	$0.145517\pi$
$503$	19.7990i	0.882793i	−0.897312	+	0.441397i	$0.854483\pi$
$504$	0	0
$505$	0	0
$506$	− 19.3400i	− 0.859768i
$507$	0	0
$508$	27.6228	1.22556
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	0	0
$512$	22.6274i	1.00000i
$513$	0	0
$514$	0	0
$515$	38.3760i	1.69105i
$516$	0	0
$517$	−8.64911	−0.380387
$518$	81.4909i	3.58051i
$519$	0	0
$520$	−5.29822	−0.232342
$521$	− 8.98151i	− 0.393487i	−0.980455	−	0.196744i	$-0.936963\pi$
$521$	− 8.98151i	− 0.393487i	0.980455	−	0.196744i	$-0.0630367\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	− 40.0570i	− 1.74990i
$525$	0	0
$526$	−44.2719	−1.93035
$527$	0	0
$528$	0	0
$529$	3.00000	0.130435
$530$	17.8885i	0.777029i
$531$	0	0
$532$	65.2982	2.83104
$533$	8.67753i	0.375866i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	60.0855i	2.58806i
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	0	0
$550$	21.6228	0.921998
$551$	0	0
$552$	0	0
$553$	0	0
$554$	42.6187i	1.81069i
$555$	0	0
$556$	−28.0000	−1.18746
$557$	45.2548i	1.91751i	0.284236	+	0.958754i	$0.408260\pi$
$557$	45.2548i	1.91751i	−0.284236	+	0.958754i	$0.591740\pi$
$558$	0	0
$559$	0	0
$560$	46.1728i	1.95116i
$561$	0	0
$562$	−47.2982	−1.99516
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	23.5826i	0.988636i	0.869281	+	0.494318i	$0.164582\pi$
$569$	23.5826i	0.988636i	−0.869281	+	0.494318i	$0.835418\pi$
$570$	0	0
$571$	−44.2719	−1.85272	−0.926360	−	0.376638i	$-0.877080\pi$
$571$	−44.2719	−1.85272	−0.926360	+	0.376638i	$0.877080\pi$
$572$	5.12338i	0.214219i
$573$	0	0
$574$	75.6228	3.15643
$575$	− 22.3607i	− 0.932505i
$576$	0	0
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	− 24.0416i	− 1.00000i
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	17.2982	0.716419
$584$	0	0
$585$	0	0
$586$	−6.32456	−0.261265
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	0	0
$590$	17.1628i	0.706582i
$591$	0	0
$592$	44.6491	1.83507
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	5.29822	0.216660
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	−25.2982	−1.03194	−0.515968	−	0.856608i	$-0.672568\pi$
$601$	−25.2982	−1.03194	−0.515968	+	0.856608i	$0.672568\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	3.68752i	0.149919i
$606$	0	0
$607$	−32.7851	−1.33070	−0.665352	−	0.746530i	$-0.731719\pi$
$607$	−32.7851	−1.33070	−0.665352	+	0.746530i	$0.731719\pi$
$608$	− 35.7771i	− 1.45095i
$609$	0	0
$610$	0	0
$611$	− 2.36944i	− 0.0958571i
$612$	0	0
$613$	−38.7851	−1.56651	−0.783257	−	0.621698i	$-0.786443\pi$
$613$	−38.7851	−1.56651	−0.783257	+	0.621698i	$0.786443\pi$
$614$	0	0
$615$	0	0
$616$	44.6491	1.79896
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	0	0
$619$	−46.0000	−1.84890	−0.924448	−	0.381308i	$-0.875474\pi$
$619$	−46.0000	−1.84890	−0.924448	+	0.381308i	$0.875474\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 97.2768i	− 3.89731i
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	0	0
$628$	15.6228	0.623417
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	−44.2719	−1.75826
$635$	− 30.8832i	− 1.22556i
$636$	0	0
$637$	−16.4605	−0.652189
$638$	0	0
$639$	0	0
$640$	25.2982	1.00000
$641$	− 37.1913i	− 1.46897i	−0.678626	−	0.734484i	$-0.737424\pi$
$641$	− 37.1913i	− 1.46897i	0.678626	−	0.734484i	$-0.262576\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	− 46.1728i	− 1.81946i
$645$	0	0
$646$	0	0
$647$	− 31.1127i	− 1.22317i	−0.791180	−	0.611583i	$-0.790533\pi$
$647$	− 31.1127i	− 1.22317i	0.791180	−	0.611583i	$-0.209467\pi$
$648$	0	0
$649$	16.5964	0.651467
$650$	5.92359i	0.232342i
$651$	0	0
$652$	0	0
$653$	4.47214i	0.175008i	0.996164	+	0.0875041i	$0.0278891\pi$
$653$	4.47214i	0.175008i	−0.996164	+	0.0875041i	$0.972111\pi$
$654$	0	0
$655$	−44.7851	−1.74990
$656$	− 41.4339i	− 1.61772i
$657$	0	0
$658$	−20.6491	−0.804986
$659$	12.9202i	0.503299i	0.967818	+	0.251649i	$0.0809729\pi$
$659$	12.9202i	0.503299i	−0.967818	+	0.251649i	$0.919027\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	44.7214i	1.73814i
$663$	0	0
$664$	0	0
$665$	− 73.0056i	− 2.83104i
$666$	0	0
$667$	0	0
$668$	− 8.94427i	− 0.346064i
$669$	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
$676$	24.5964	0.946017
$677$	− 49.1935i	− 1.89066i	−0.326116	−	0.945330i	$-0.605740\pi$
$677$	− 49.1935i	− 1.89066i	0.326116	−	0.945330i	$-0.394260\pi$
$678$	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
$686$	92.3456i	3.52577i
$687$	0	0
$688$	0	0
$689$	4.73887i	0.180537i
$690$	0	0
$691$	31.6228	1.20299	0.601494	−	0.798878i	$-0.294573\pi$
$691$	31.6228	1.20299	0.601494	+	0.798878i	$0.294573\pi$
$692$	45.2548i	1.72033i
$693$	0	0
$694$	0	0
$695$	31.3050i	1.18746i
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	51.6228	1.95116
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	−70.5964	−2.66260
$704$	− 24.4634i	− 0.921998i
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	−53.2982	−1.99744
$713$	0	0
$714$	0	0
$715$	5.72811	0.214219
$716$	27.8253i	1.03988i
$717$	0	0
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	88.5964	3.29950
$722$	29.6985i	1.10526i
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	41.1623	1.52662	0.763312	−	0.646030i	$-0.223572\pi$
$727$	41.1623	1.52662	0.763312	+	0.646030i	$0.223572\pi$
$728$	12.2317i	0.453337i
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−50.7851	−1.87579	−0.937894	−	0.346921i	$-0.887227\pi$
$733$	−50.7851	−1.87579	−0.937894	+	0.346921i	$0.887227\pi$
$734$	− 53.4734i	− 1.97374i
$735$	0	0
$736$	−25.2982	−0.932505
$737$	0	0
$738$	0	0
$739$	−6.32456	−0.232653	−0.116326	−	0.993211i	$-0.537112\pi$
$739$	−6.32456	−0.232653	−0.116326	+	0.993211i	$0.537112\pi$
$740$	− 49.9192i	− 1.83507i
$741$	0	0
$742$	41.2982	1.51611
$743$	36.7696i	1.34894i	0.738300	+	0.674472i	$0.235629\pi$
$743$	36.7696i	1.34894i	−0.738300	+	0.674472i	$0.764371\pi$
$744$	0	0
$745$	0	0
$746$	8.67753i	0.317707i
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	11.3137i	0.412568i
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	54.1359	1.96760	0.983802	−	0.179258i	$-0.0573696\pi$
$757$	54.1359	1.96760	0.983802	+	0.179258i	$0.0573696\pi$
$758$	− 48.0833i	− 1.74646i
$759$	0	0
$760$	−40.0000	−1.45095
$761$	− 20.2207i	− 0.733001i	−0.930418	−	0.366501i	$-0.880556\pi$
$761$	− 20.2207i	− 0.733001i	0.930418	−	0.366501i	$-0.119444\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	−52.0000	−1.87884
$767$	4.54662i	0.164169i
$768$	0	0
$769$	12.6491	0.456139	0.228069	−	0.973645i	$-0.426759\pi$
$769$	12.6491	0.456139	0.228069	+	0.973645i	$0.426759\pi$
$770$	− 49.9192i	− 1.79896i
$771$	0	0
$772$	0	0
$773$	− 22.3607i	− 0.804258i	−0.915583	−	0.402129i	$-0.868270\pi$
$773$	− 22.3607i	− 0.804258i	0.915583	−	0.402129i	$-0.131730\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	65.5128i	2.34724i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	78.5964	2.80702
$785$	− 17.4668i	− 0.623417i
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	44.7214i	1.59313i
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	− 35.1258i	− 1.24657i
$795$	0	0
$796$	0	0
$797$	− 39.5980i	− 1.40263i	−0.712850	−	0.701316i	$-0.752596\pi$
$797$	− 39.5980i	− 1.40263i	0.712850	−	0.701316i	$-0.247404\pi$
$798$	0	0
$799$	0	0
$800$	− 28.2843i	− 1.00000i
$801$	0	0
$802$	50.6491	1.78848
$803$	0	0
$804$	0	0
$805$	−51.6228	−1.81946
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 54.1619i	− 1.90423i	−0.305741	−	0.952115i	$-0.598904\pi$
$809$	− 54.1619i	− 1.90423i	0.305741	−	0.952115i	$-0.401096\pi$
$810$	0	0
$811$	2.00000	0.0702295	0.0351147	−	0.999383i	$-0.488820\pi$
$811$	2.00000	0.0702295	0.0351147	+	0.999383i	$0.488820\pi$
$812$	0	0
$813$	0	0
$814$	−48.2719	−1.69193
$815$	0	0
$816$	0	0
$817$	0	0
$818$	19.7990i	0.692255i
$819$	0	0
$820$	−46.3246	−1.61772
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	0	0
$823$	−11.8641	−0.413555	−0.206778	−	0.978388i	$-0.566298\pi$
$823$	−11.8641	−0.413555	−0.206778	+	0.978388i	$0.566298\pi$
$824$	− 48.5423i	− 1.69105i
$825$	0	0
$826$	39.6228	1.37865
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	6.70178	0.232342
$833$	0	0
$834$	0	0
$835$	−10.0000	−0.346064
$836$	38.6800i	1.33778i
$837$	0	0
$838$	57.6228	1.99055
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	0	0
$844$	44.0000	1.51454
$845$	− 27.4997i	− 0.946017i
$846$	0	0
$847$	8.51317	0.292516
$848$	− 22.6274i	− 0.777029i
$849$	0	0
$850$	0	0
$851$	49.9192i	1.71121i
$852$	0	0
$853$	37.1096	1.27061	0.635304	−	0.772262i	$-0.280875\pi$
$853$	37.1096	1.27061	0.635304	+	0.772262i	$0.280875\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	14.0000	0.477674	0.238837	−	0.971060i	$-0.423234\pi$
$859$	14.0000	0.477674	0.238837	+	0.971060i	$0.423234\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	58.1378i	1.97903i	0.144421	+	0.989516i	$0.453868\pi$
$863$	58.1378i	1.97903i	−0.144421	+	0.989516i	$0.546132\pi$
$864$	0	0
$865$	50.5964	1.72033
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	40.0000	1.35302
$875$	− 57.7160i	− 1.95116i
$876$	0	0
$877$	49.1096	1.65831	0.829157	−	0.559016i	$-0.188821\pi$
$877$	49.1096	1.65831	0.829157	+	0.559016i	$0.188821\pi$
$878$	0	0
$879$	0	0
$880$	−27.3509	−0.921998
$881$	32.0679i	1.08040i	0.841538	+	0.540198i	$0.181651\pi$
$881$	32.0679i	1.08040i	−0.841538	+	0.540198i	$0.818349\pi$
$882$	0	0
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	4.47214i	0.150160i	0.997178	+	0.0750798i	$0.0239212\pi$
$887$	4.47214i	0.150160i	−0.997178	+	0.0750798i	$0.976079\pi$
$888$	0	0
$889$	−71.2982	−2.39127
$890$	59.5892i	1.99744i
$891$	0	0
$892$	3.62278	0.121300
$893$	− 17.8885i	− 0.598617i
$894$	0	0
$895$	31.1096	1.03988
$896$	− 58.4045i	− 1.95116i
$897$	0	0
$898$	−59.2982	−1.97881
$899$	0	0
$900$	0	0
$901$	0	0
$902$	44.7958i	1.49154i
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	0	0
$910$	13.6754	0.453337
$911$	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
$916$	0	0
$917$	103.393i	3.41432i
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	28.2843i	0.932505i
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−55.8114	−1.83507
$926$	14.4089i	0.473505i
$927$	0	0
$928$	0	0
$929$	− 11.7355i	− 0.385028i	−0.981294	−	0.192514i	$-0.938336\pi$
$929$	− 11.7355i	− 0.385028i	0.981294	−	0.192514i	$-0.0616641\pi$
$930$	0	0
$931$	−124.272	−4.07285
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0	0
$940$	12.6491	0.412568
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	46.3246	1.50854
$944$	− 21.7094i	− 0.706582i
$945$	0	0
$946$	0	0
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	0	0
$950$	44.7214i	1.45095i
$951$	0	0
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	31.0000	1.00000
$962$	− 13.2242i	− 0.426364i
$963$	0	0
$964$	50.5964	1.62960
$965$	0	0
$966$	0	0
$967$	60.1359	1.93384	0.966921	−	0.255077i	$-0.0821008\pi$
$967$	60.1359	1.93384	0.966921	+	0.255077i	$0.0821008\pi$
$968$	− 4.66439i	− 0.149919i
$969$	0	0
$970$	0	0
$971$	− 39.3685i	− 1.26339i	−0.775215	−	0.631697i	$-0.782359\pi$
$971$	− 39.3685i	− 1.26339i	0.775215	−	0.631697i	$-0.217641\pi$
$972$	0	0
$973$	72.2719	2.31693
$974$	0.192253i	0.00616019i
$975$	0	0
$976$	0	0
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	0	0
$979$	57.6228	1.84163
$980$	− 87.8735i	− 2.80702i
$981$	0	0
$982$	−42.2719	−1.34895
$983$	− 48.0833i	− 1.53362i	−0.641875	−	0.766809i	$-0.721843\pi$
$983$	− 48.0833i	− 1.53362i	0.641875	−	0.766809i	$-0.278157\pi$
$984$	0	0
$985$	50.0000	1.59313
$986$	0	0
$987$	0	0
$988$	−10.5964	−0.337118
$989$	0	0
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−62.7851	−1.98842	−0.994211	−	0.107442i	$-0.965734\pi$
$997$	−62.7851	−1.98842	−0.994211	+	0.107442i	$0.965734\pi$
$998$	− 62.6099i	− 1.98188i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	360.2.m.b.179.4	yes	4
3.2	odd	2	inner	360.2.m.b.179.1	yes	4
4.3	odd	2		1440.2.m.a.719.3		4
5.2	odd	4		1800.2.b.f.251.4		8
5.3	odd	4		1800.2.b.f.251.5		8
5.4	even	2		360.2.m.a.179.1	✓	4
8.3	odd	2		360.2.m.a.179.1	✓	4
8.5	even	2		1440.2.m.b.719.2		4
12.11	even	2		1440.2.m.a.719.1		4
15.2	even	4		1800.2.b.f.251.8		8
15.8	even	4		1800.2.b.f.251.1		8
15.14	odd	2		360.2.m.a.179.4	yes	4
20.3	even	4		7200.2.b.f.4751.7		8
20.7	even	4		7200.2.b.f.4751.1		8
20.19	odd	2		1440.2.m.b.719.2		4
24.5	odd	2		1440.2.m.b.719.4		4
24.11	even	2		360.2.m.a.179.4	yes	4
40.3	even	4		1800.2.b.f.251.4		8
40.13	odd	4		7200.2.b.f.4751.1		8
40.19	odd	2	CM	360.2.m.b.179.4	yes	4
40.27	even	4		1800.2.b.f.251.5		8
40.29	even	2		1440.2.m.a.719.3		4
40.37	odd	4		7200.2.b.f.4751.7		8
60.23	odd	4		7200.2.b.f.4751.8		8
60.47	odd	4		7200.2.b.f.4751.2		8
60.59	even	2		1440.2.m.b.719.4		4
120.29	odd	2		1440.2.m.a.719.1		4
120.53	even	4		7200.2.b.f.4751.2		8
120.59	even	2	inner	360.2.m.b.179.1	yes	4
120.77	even	4		7200.2.b.f.4751.8		8
120.83	odd	4		1800.2.b.f.251.8		8
120.107	odd	4		1800.2.b.f.251.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.m.a.179.1	✓	4	5.4	even	2
360.2.m.a.179.1	✓	4	8.3	odd	2
360.2.m.a.179.4	yes	4	15.14	odd	2
360.2.m.a.179.4	yes	4	24.11	even	2
360.2.m.b.179.1	yes	4	3.2	odd	2	inner
360.2.m.b.179.1	yes	4	120.59	even	2	inner
360.2.m.b.179.4	yes	4	1.1	even	1	trivial
360.2.m.b.179.4	yes	4	40.19	odd	2	CM
1440.2.m.a.719.1		4	12.11	even	2
1440.2.m.a.719.1		4	120.29	odd	2
1440.2.m.a.719.3		4	4.3	odd	2
1440.2.m.a.719.3		4	40.29	even	2
1440.2.m.b.719.2		4	8.5	even	2
1440.2.m.b.719.2		4	20.19	odd	2
1440.2.m.b.719.4		4	24.5	odd	2
1440.2.m.b.719.4		4	60.59	even	2
1800.2.b.f.251.1		8	15.8	even	4
1800.2.b.f.251.1		8	120.107	odd	4
1800.2.b.f.251.4		8	5.2	odd	4
1800.2.b.f.251.4		8	40.3	even	4
1800.2.b.f.251.5		8	5.3	odd	4
1800.2.b.f.251.5		8	40.27	even	4
1800.2.b.f.251.8		8	15.2	even	4
1800.2.b.f.251.8		8	120.83	odd	4
7200.2.b.f.4751.1		8	20.7	even	4
7200.2.b.f.4751.1		8	40.13	odd	4
7200.2.b.f.4751.2		8	60.47	odd	4
7200.2.b.f.4751.2		8	120.53	even	4
7200.2.b.f.4751.7		8	20.3	even	4
7200.2.b.f.4751.7		8	40.37	odd	4
7200.2.b.f.4751.8		8	60.23	odd	4
7200.2.b.f.4751.8		8	120.77	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:	\( N \)	\(=\)	\( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	360.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} -2.00000 q^{4} +2.23607i q^{5} +5.16228 q^{7} -2.82843i q^{8} +O(q^{10})\)	\(q+1.41421i q^{2} -2.00000 q^{4} +2.23607i q^{5} +5.16228 q^{7} -2.82843i q^{8} -3.16228 q^{10} +3.05792i q^{11} -0.837722 q^{13} +7.30056i q^{14} +4.00000 q^{16} -6.32456 q^{19} -4.47214i q^{20} -4.32456 q^{22} +4.47214i q^{23} -5.00000 q^{25} -1.18472i q^{26} -10.3246 q^{28} +5.65685i q^{32} +11.5432i q^{35} +11.1623 q^{37} -8.94427i q^{38} +6.32456 q^{40} -10.3585i q^{41} -6.11584i q^{44} -6.32456 q^{46} +2.82843i q^{47} +19.6491 q^{49} -7.07107i q^{50} +1.67544 q^{52} -5.65685i q^{53} -6.83772 q^{55} -14.6011i q^{56} -5.42736i q^{59} -8.00000 q^{64} -1.87320i q^{65} -16.3246 q^{70} +15.7858i q^{74} +12.6491 q^{76} +15.7858i q^{77} +8.94427i q^{80} +14.6491 q^{82} +8.64911 q^{88} -18.8438i q^{89} -4.32456 q^{91} -8.94427i q^{92} -4.00000 q^{94} -14.1421i q^{95} +27.7880i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4} + 8 q^{7}+O(q^{10}) \) 4 * q - 8 * q^4 + 8 * q^7	\( 4 q - 8 q^{4} + 8 q^{7} - 16 q^{13} + 16 q^{16} + 8 q^{22} - 20 q^{25} - 16 q^{28} + 32 q^{37} + 28 q^{49} + 32 q^{52} - 40 q^{55} - 32 q^{64} - 40 q^{70} + 8 q^{82} - 16 q^{88} + 8 q^{91} - 16 q^{94}+O(q^{100}) \) 4 * q - 8 * q^4 + 8 * q^7 - 16 * q^13 + 16 * q^16 + 8 * q^22 - 20 * q^25 - 16 * q^28 + 32 * q^37 + 28 * q^49 + 32 * q^52 - 40 * q^55 - 32 * q^64 - 40 * q^70 + 8 * q^82 - 16 * q^88 + 8 * q^91 - 16 * q^94

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