LMFDB - Embedded newform 180.3.g.a.161.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [180,3,Mod(161,180)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(180, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("180.161");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 180 = 2^{2} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	180.g (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:	$4.90464475849$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.4
Root			$-1.58114 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	180.161
Dual form			180.3.g.a.161.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.23607i q^{5} +5.48683 q^{7} +O(q^{10})$	$q+2.23607i q^{5} +5.48683 q^{7} +9.17377i q^{11} +11.4868 q^{13} +16.9706i q^{17} +26.9737 q^{19} +4.93113i q^{23} -5.00000 q^{25} -20.5247i q^{29} +20.9737 q^{31} +12.2689i q^{35} -62.4605 q^{37} -40.9377i q^{41} +1.02633 q^{43} +86.2298i q^{47} -18.8947 q^{49} -96.0920i q^{53} -20.5132 q^{55} -112.374i q^{59} -66.9210 q^{61} +25.6853i q^{65} -76.0000 q^{67} -24.0789i q^{71} -18.9210 q^{73} +50.3349i q^{77} +106.921 q^{79} +45.1804i q^{83} -37.9473 q^{85} -115.928i q^{89} +63.0263 q^{91} +60.3150i q^{95} -87.0263 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 16 q^{7}+O(q^{10}) $ 4 * q - 16 * q^7	$ 4 q - 16 q^{7} + 8 q^{13} + 32 q^{19} - 20 q^{25} + 8 q^{31} - 136 q^{37} + 80 q^{43} + 228 q^{49} - 120 q^{55} - 40 q^{61} - 304 q^{67} + 152 q^{73} + 200 q^{79} + 328 q^{91} - 424 q^{97}+O(q^{100}) $ 4 * q - 16 * q^7 + 8 * q^13 + 32 * q^19 - 20 * q^25 + 8 * q^31 - 136 * q^37 + 80 * q^43 + 228 * q^49 - 120 * q^55 - 40 * q^61 - 304 * q^67 + 152 * q^73 + 200 * q^79 + 328 * q^91 - 424 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/180\mathbb{Z}\right)^\times$.

$n$	$37$	$91$	$101$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	2.23607i	0.447214i
$5$	2.23607i	0.447214i
$6$	0	0
$7$	5.48683	0.783833	0.391917	−	0.920001i	$-0.371812\pi$
$7$	5.48683	0.783833	0.391917	+	0.920001i	$0.371812\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	9.17377i	0.833979i	0.908911	+	0.416989i	$0.136915\pi$
$11$	9.17377i	0.833979i	−0.908911	+	0.416989i	$0.863085\pi$
$12$	0	0
$13$	11.4868	0.883603	0.441801	−	0.897113i	$-0.354340\pi$
$13$	11.4868	0.883603	0.441801	+	0.897113i	$0.354340\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	16.9706i	0.998268i	0.866525	+	0.499134i	$0.166349\pi$
$17$	16.9706i	0.998268i	−0.866525	+	0.499134i	$0.833651\pi$
$18$	0	0
$19$	26.9737	1.41967	0.709833	−	0.704370i	$-0.248770\pi$
$19$	26.9737	1.41967	0.709833	+	0.704370i	$0.248770\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.93113i	0.214397i	0.994238	+	0.107198i	$0.0341880\pi$
$23$	4.93113i	0.214397i	−0.994238	+	0.107198i	$0.965812\pi$
$24$	0	0
$25$	−5.00000	−0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 20.5247i	− 0.707749i	−0.935293	−	0.353874i	$-0.884864\pi$
$29$	− 20.5247i	− 0.707749i	0.935293	−	0.353874i	$-0.115136\pi$
$30$	0	0
$31$	20.9737	0.676570	0.338285	−	0.941044i	$-0.390153\pi$
$31$	20.9737	0.676570	0.338285	+	0.941044i	$0.390153\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	12.2689i	0.350541i
$36$	0	0
$37$	−62.4605	−1.68812	−0.844061	−	0.536247i	$-0.819841\pi$
$37$	−62.4605	−1.68812	−0.844061	+	0.536247i	$0.819841\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 40.9377i	− 0.998481i	−0.866464	−	0.499240i	$-0.833612\pi$
$41$	− 40.9377i	− 0.998481i	0.866464	−	0.499240i	$-0.166388\pi$
$42$	0	0
$43$	1.02633	0.0238682	0.0119341	−	0.999929i	$-0.496201\pi$
$43$	1.02633	0.0238682	0.0119341	+	0.999929i	$0.496201\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	86.2298i	1.83468i	0.398109	+	0.917338i	$0.369667\pi$
$47$	86.2298i	1.83468i	−0.398109	+	0.917338i	$0.630333\pi$
$48$	0	0
$49$	−18.8947	−0.385605
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 96.0920i	− 1.81306i	−0.422144	−	0.906529i	$-0.638722\pi$
$53$	− 96.0920i	− 1.81306i	0.422144	−	0.906529i	$-0.361278\pi$
$54$	0	0
$55$	−20.5132	−0.372967
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 112.374i	− 1.90465i	−0.305092	−	0.952323i	$-0.598687\pi$
$59$	− 112.374i	− 1.90465i	0.305092	−	0.952323i	$-0.401313\pi$
$60$	0	0
$61$	−66.9210	−1.09707	−0.548533	−	0.836129i	$-0.684813\pi$
$61$	−66.9210	−1.09707	−0.548533	+	0.836129i	$0.684813\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	25.6853i	0.395159i
$66$	0	0
$67$	−76.0000	−1.13433	−0.567164	−	0.823605i	$-0.691960\pi$
$67$	−76.0000	−1.13433	−0.567164	+	0.823605i	$0.691960\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 24.0789i	− 0.339139i	−0.985518	−	0.169570i	$-0.945762\pi$
$71$	− 24.0789i	− 0.339139i	0.985518	−	0.169570i	$-0.0542377\pi$
$72$	0	0
$73$	−18.9210	−0.259192	−0.129596	−	0.991567i	$-0.541368\pi$
$73$	−18.9210	−0.259192	−0.129596	+	0.991567i	$0.541368\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	50.3349i	0.653700i
$78$	0	0
$79$	106.921	1.35343	0.676715	−	0.736245i	$-0.263403\pi$
$79$	106.921	1.35343	0.676715	+	0.736245i	$0.263403\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	45.1804i	0.544342i	0.962249	+	0.272171i	$0.0877416\pi$
$83$	45.1804i	0.544342i	−0.962249	+	0.272171i	$0.912258\pi$
$84$	0	0
$85$	−37.9473	−0.446439
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 115.928i	− 1.30256i	−0.758835	−	0.651282i	$-0.774231\pi$
$89$	− 115.928i	− 1.30256i	0.758835	−	0.651282i	$-0.225769\pi$
$90$	0	0
$91$	63.0263	0.692597
$92$	0	0
$93$	0	0
$94$	0	0
$95$	60.3150i	0.634894i
$96$	0	0
$97$	−87.0263	−0.897179	−0.448589	−	0.893738i	$-0.648073\pi$
$97$	−87.0263	−0.897179	−0.448589	+	0.893738i	$0.648073\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	129.233i	1.27953i	0.768569	+	0.639767i	$0.220969\pi$
$101$	129.233i	1.27953i	−0.768569	+	0.639767i	$0.779031\pi$
$102$	0	0
$103$	114.302	1.10973	0.554866	−	0.831939i	$-0.312769\pi$
$103$	114.302	1.10973	0.554866	+	0.831939i	$0.312769\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	93.3381i	0.872319i	0.899869	+	0.436159i	$0.143662\pi$
$107$	93.3381i	0.872319i	−0.899869	+	0.436159i	$0.856338\pi$
$108$	0	0
$109$	120.868	1.10888	0.554442	−	0.832222i	$-0.312932\pi$
$109$	120.868	1.10888	0.554442	+	0.832222i	$0.312932\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 110.309i	− 0.976183i	−0.872793	−	0.488091i	$-0.837693\pi$
$113$	− 110.309i	− 0.976183i	0.872793	−	0.488091i	$-0.162307\pi$
$114$	0	0
$115$	−11.0263	−0.0958812
$116$	0	0
$117$	0	0
$118$	0	0
$119$	93.1146i	0.782476i
$120$	0	0
$121$	36.8420	0.304479
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 11.1803i	− 0.0894427i
$126$	0	0
$127$	105.381	0.829776	0.414888	−	0.909873i	$-0.363821\pi$
$127$	105.381	0.829776	0.414888	+	0.909873i	$0.363821\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 140.584i	− 1.07316i	−0.843850	−	0.536580i	$-0.819716\pi$
$131$	− 140.584i	− 1.07316i	0.843850	−	0.536580i	$-0.180284\pi$
$132$	0	0
$133$	148.000	1.11278
$134$	0	0
$135$	0	0
$136$	0	0
$137$	266.951i	1.94855i	0.225365	+	0.974274i	$0.427643\pi$
$137$	266.951i	1.94855i	−0.225365	+	0.974274i	$0.572357\pi$
$138$	0	0
$139$	−15.8420	−0.113971	−0.0569856	−	0.998375i	$-0.518149\pi$
$139$	−15.8420	−0.113971	−0.0569856	+	0.998375i	$0.518149\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	105.378i	0.736906i
$144$	0	0
$145$	45.8947	0.316515
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 41.6262i	− 0.279370i	−0.990196	−	0.139685i	$-0.955391\pi$
$149$	− 41.6262i	− 0.279370i	0.990196	−	0.139685i	$-0.0446091\pi$
$150$	0	0
$151$	103.842	0.687695	0.343848	−	0.939025i	$-0.388270\pi$
$151$	103.842	0.687695	0.343848	+	0.939025i	$0.388270\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	46.8985i	0.302571i
$156$	0	0
$157$	−12.5132	−0.0797017	−0.0398509	−	0.999206i	$-0.512688\pi$
$157$	−12.5132	−0.0797017	−0.0398509	+	0.999206i	$0.512688\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	27.0563i	0.168051i
$162$	0	0
$163$	−290.868	−1.78447	−0.892234	−	0.451573i	$-0.850863\pi$
$163$	−290.868	−1.78447	−0.892234	+	0.451573i	$0.850863\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 174.637i	− 1.04573i	−0.852416	−	0.522865i	$-0.824863\pi$
$167$	− 174.637i	− 1.04573i	0.852416	−	0.522865i	$-0.175137\pi$
$168$	0	0
$169$	−37.0527	−0.219247
$170$	0	0
$171$	0	0
$172$	0	0
$173$	43.5799i	0.251907i	0.992036	+	0.125954i	$0.0401990\pi$
$173$	43.5799i	0.251907i	−0.992036	+	0.125954i	$0.959801\pi$
$174$	0	0
$175$	−27.4342	−0.156767
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 88.2952i	− 0.493270i	−0.969109	−	0.246635i	$-0.920675\pi$
$179$	− 88.2952i	− 0.493270i	0.969109	−	0.246635i	$-0.0793248\pi$
$180$	0	0
$181$	−56.8683	−0.314190	−0.157095	−	0.987584i	$-0.550213\pi$
$181$	−56.8683	−0.314190	−0.157095	+	0.987584i	$0.550213\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 139.666i	− 0.754951i
$186$	0	0
$187$	−155.684	−0.832535
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 56.6430i	− 0.296560i	−0.988945	−	0.148280i	$-0.952626\pi$
$191$	− 56.6430i	− 0.296560i	0.988945	−	0.148280i	$-0.0473737\pi$
$192$	0	0
$193$	110.000	0.569948	0.284974	−	0.958535i	$-0.408015\pi$
$193$	110.000	0.569948	0.284974	+	0.958535i	$0.408015\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 245.049i	− 1.24391i	−0.783055	−	0.621953i	$-0.786339\pi$
$197$	− 245.049i	− 1.24391i	0.783055	−	0.621953i	$-0.213661\pi$
$198$	0	0
$199$	−169.895	−0.853742	−0.426871	−	0.904313i	$-0.640384\pi$
$199$	−169.895	−0.853742	−0.426871	+	0.904313i	$0.640384\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 112.616i	− 0.554757i
$204$	0	0
$205$	91.5395	0.446534
$206$	0	0
$207$	0	0
$208$	0	0
$209$	247.450i	1.18397i
$210$	0	0
$211$	−265.579	−1.25867	−0.629333	−	0.777136i	$-0.716672\pi$
$211$	−265.579	−1.25867	−0.629333	+	0.777136i	$0.716672\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	2.29495i	0.0106742i
$216$	0	0
$217$	115.079	0.530318
$218$	0	0
$219$	0	0
$220$	0	0
$221$	194.938i	0.882072i
$222$	0	0
$223$	−187.329	−0.840040	−0.420020	−	0.907515i	$-0.637977\pi$
$223$	−187.329	−0.840040	−0.420020	+	0.907515i	$0.637977\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 421.063i	− 1.85490i	−0.373942	−	0.927452i	$-0.621994\pi$
$227$	− 421.063i	− 1.85490i	0.373942	−	0.927452i	$-0.378006\pi$
$228$	0	0
$229$	−102.105	−0.445875	−0.222937	−	0.974833i	$-0.571565\pi$
$229$	−102.105	−0.445875	−0.222937	+	0.974833i	$0.571565\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 99.0694i	− 0.425191i	−0.977140	−	0.212595i	$-0.931808\pi$
$233$	− 99.0694i	− 0.425191i	0.977140	−	0.212595i	$-0.0681916\pi$
$234$	0	0
$235$	−192.816	−0.820492
$236$	0	0
$237$	0	0
$238$	0	0
$239$	77.9680i	0.326226i	0.986607	+	0.163113i	$0.0521535\pi$
$239$	77.9680i	0.326226i	−0.986607	+	0.163113i	$0.947847\pi$
$240$	0	0
$241$	−257.947	−1.07032	−0.535160	−	0.844750i	$-0.679749\pi$
$241$	−257.947	−1.07032	−0.535160	+	0.844750i	$0.679749\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 42.2498i	− 0.172448i
$246$	0	0
$247$	309.842	1.25442
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 274.971i	− 1.09550i	−0.836641	−	0.547752i	$-0.815484\pi$
$251$	− 274.971i	− 1.09550i	0.836641	−	0.547752i	$-0.184516\pi$
$252$	0	0
$253$	−45.2370	−0.178802
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 131.634i	− 0.512193i	−0.966651	−	0.256096i	$-0.917564\pi$
$257$	− 131.634i	− 0.512193i	0.966651	−	0.256096i	$-0.0824365\pi$
$258$	0	0
$259$	−342.710	−1.32321
$260$	0	0
$261$	0	0
$262$	0	0
$263$	458.782i	1.74442i	0.489133	+	0.872209i	$0.337313\pi$
$263$	458.782i	1.74442i	−0.489133	+	0.872209i	$0.662687\pi$
$264$	0	0
$265$	214.868	0.810824
$266$	0	0
$267$	0	0
$268$	0	0
$269$	301.916i	1.12236i	0.827692	+	0.561182i	$0.189653\pi$
$269$	301.916i	1.12236i	−0.827692	+	0.561182i	$0.810347\pi$
$270$	0	0
$271$	475.842	1.75587	0.877937	−	0.478776i	$-0.158919\pi$
$271$	475.842	1.75587	0.877937	+	0.478776i	$0.158919\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 45.8688i	− 0.166796i
$276$	0	0
$277$	322.039	1.16260	0.581298	−	0.813691i	$-0.302545\pi$
$277$	322.039	1.16260	0.581298	+	0.813691i	$0.302545\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	352.139i	1.25316i	0.779355	+	0.626582i	$0.215547\pi$
$281$	352.139i	1.25316i	−0.779355	+	0.626582i	$0.784453\pi$
$282$	0	0
$283$	−281.631	−0.995164	−0.497582	−	0.867417i	$-0.665779\pi$
$283$	−281.631	−0.995164	−0.497582	+	0.867417i	$0.665779\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 224.618i	− 0.782642i
$288$	0	0
$289$	1.00000	0.00346021
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 64.3281i	− 0.219550i	−0.993956	−	0.109775i	$-0.964987\pi$
$293$	− 64.3281i	− 0.219550i	0.993956	−	0.109775i	$-0.0350130\pi$
$294$	0	0
$295$	251.276	0.851784
$296$	0	0
$297$	0	0
$298$	0	0
$299$	56.6430i	0.189442i
$300$	0	0
$301$	5.63132	0.0187087
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 149.640i	− 0.490623i
$306$	0	0
$307$	230.158	0.749700	0.374850	−	0.927085i	$-0.377694\pi$
$307$	230.158	0.749700	0.374850	+	0.927085i	$0.377694\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 8.48528i	− 0.0272839i	−0.999907	−	0.0136419i	$-0.995658\pi$
$311$	− 8.48528i	− 0.0272839i	0.999907	−	0.0136419i	$-0.00434250\pi$
$312$	0	0
$313$	605.579	1.93476	0.967378	−	0.253337i	$-0.0815282\pi$
$313$	605.579	1.93476	0.967378	+	0.253337i	$0.0815282\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	87.9601i	0.277477i	0.990329	+	0.138738i	$0.0443047\pi$
$317$	87.9601i	0.277477i	−0.990329	+	0.138738i	$0.955695\pi$
$318$	0	0
$319$	188.289	0.590248
$320$	0	0
$321$	0	0
$322$	0	0
$323$	457.758i	1.41721i
$324$	0	0
$325$	−57.4342	−0.176721
$326$	0	0
$327$	0	0
$328$	0	0
$329$	473.128i	1.43808i
$330$	0	0
$331$	−49.2370	−0.148752	−0.0743761	−	0.997230i	$-0.523697\pi$
$331$	−49.2370	−0.148752	−0.0743761	+	0.997230i	$0.523697\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 169.941i	− 0.507287i
$336$	0	0
$337$	−2.71033	−0.00804251	−0.00402125	−	0.999992i	$-0.501280\pi$
$337$	−2.71033	−0.00804251	−0.00402125	+	0.999992i	$0.501280\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	192.408i	0.564245i
$342$	0	0
$343$	−372.527	−1.08608
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 74.9906i	− 0.216111i	−0.994145	−	0.108056i	$-0.965538\pi$
$347$	− 74.9906i	− 0.216111i	0.994145	−	0.108056i	$-0.0344624\pi$
$348$	0	0
$349$	−150.921	−0.432438	−0.216219	−	0.976345i	$-0.569373\pi$
$349$	−150.921	−0.432438	−0.216219	+	0.976345i	$0.569373\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 292.407i	− 0.828349i	−0.910198	−	0.414174i	$-0.864070\pi$
$353$	− 292.407i	− 0.828349i	0.910198	−	0.414174i	$-0.135930\pi$
$354$	0	0
$355$	53.8420	0.151668
$356$	0	0
$357$	0	0
$358$	0	0
$359$	576.776i	1.60662i	0.595563	+	0.803309i	$0.296929\pi$
$359$	576.776i	1.60662i	−0.595563	+	0.803309i	$0.703071\pi$
$360$	0	0
$361$	366.579	1.01545
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 42.3086i	− 0.115914i
$366$	0	0
$367$	243.540	0.663595	0.331798	−	0.943351i	$-0.392345\pi$
$367$	243.540	0.663595	0.331798	+	0.943351i	$0.392345\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 527.241i	− 1.42113i
$372$	0	0
$373$	115.908	0.310746	0.155373	−	0.987856i	$-0.450342\pi$
$373$	115.908	0.310746	0.155373	+	0.987856i	$0.450342\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 235.764i	− 0.625369i
$378$	0	0
$379$	30.2107	0.0797115	0.0398558	−	0.999205i	$-0.487310\pi$
$379$	30.2107	0.0797115	0.0398558	+	0.999205i	$0.487310\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	651.319i	1.70057i	0.526320	+	0.850286i	$0.323571\pi$
$383$	651.319i	1.70057i	−0.526320	+	0.850286i	$0.676429\pi$
$384$	0	0
$385$	−112.552	−0.292344
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 205.600i	− 0.528536i	−0.964449	−	0.264268i	$-0.914870\pi$
$389$	− 205.600i	− 0.528536i	0.964449	−	0.264268i	$-0.0851303\pi$
$390$	0	0
$391$	−83.6840	−0.214026
$392$	0	0
$393$	0	0
$394$	0	0
$395$	239.083i	0.605272i
$396$	0	0
$397$	118.355	0.298124	0.149062	−	0.988828i	$-0.452375\pi$
$397$	118.355	0.298124	0.149062	+	0.988828i	$0.452375\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	170.971i	0.426361i	0.977013	+	0.213181i	$0.0683823\pi$
$401$	170.971i	0.426361i	−0.977013	+	0.213181i	$0.931618\pi$
$402$	0	0
$403$	240.921	0.597819
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 572.998i	− 1.40786i
$408$	0	0
$409$	−335.947	−0.821387	−0.410694	−	0.911773i	$-0.634713\pi$
$409$	−335.947	−0.821387	−0.410694	+	0.911773i	$0.634713\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 616.578i	− 1.49292i
$414$	0	0
$415$	−101.026	−0.243437
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 407.535i	− 0.972637i	−0.873781	−	0.486319i	$-0.838339\pi$
$419$	− 407.535i	− 0.972637i	0.873781	−	0.486319i	$-0.161661\pi$
$420$	0	0
$421$	−771.210	−1.83185	−0.915926	−	0.401346i	$-0.868542\pi$
$421$	−771.210	−1.83185	−0.915926	+	0.401346i	$0.868542\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 84.8528i	− 0.199654i
$426$	0	0
$427$	−367.184	−0.859916
$428$	0	0
$429$	0	0
$430$	0	0
$431$	436.210i	1.01209i	0.862508	+	0.506044i	$0.168893\pi$
$431$	436.210i	1.01209i	−0.862508	+	0.506044i	$0.831107\pi$
$432$	0	0
$433$	−838.500	−1.93649	−0.968244	−	0.250006i	$-0.919568\pi$
$433$	−838.500	−1.93649	−0.968244	+	0.250006i	$0.919568\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	133.011i	0.304372i
$438$	0	0
$439$	50.0000	0.113895	0.0569476	−	0.998377i	$-0.481863\pi$
$439$	50.0000	0.113895	0.0569476	+	0.998377i	$0.481863\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 492.853i	− 1.11253i	−0.831003	−	0.556267i	$-0.812233\pi$
$443$	− 492.853i	− 1.11253i	0.831003	−	0.556267i	$-0.187767\pi$
$444$	0	0
$445$	259.223	0.582525
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 483.102i	− 1.07595i	−0.842960	−	0.537976i	$-0.819189\pi$
$449$	− 483.102i	− 1.07595i	0.842960	−	0.537976i	$-0.180811\pi$
$450$	0	0
$451$	375.553	0.832712
$452$	0	0
$453$	0	0
$454$	0	0
$455$	140.931i	0.309739i
$456$	0	0
$457$	−546.921	−1.19676	−0.598382	−	0.801211i	$-0.704189\pi$
$457$	−546.921	−1.19676	−0.598382	+	0.801211i	$0.704189\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 130.386i	− 0.282834i	−0.989950	−	0.141417i	$-0.954834\pi$
$461$	− 130.386i	− 0.282834i	0.989950	−	0.141417i	$-0.0451658\pi$
$462$	0	0
$463$	24.7765	0.0535130	0.0267565	−	0.999642i	$-0.491482\pi$
$463$	24.7765	0.0535130	0.0267565	+	0.999642i	$0.491482\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	671.491i	1.43788i	0.695071	+	0.718941i	$0.255373\pi$
$467$	671.491i	1.43788i	−0.695071	+	0.718941i	$0.744627\pi$
$468$	0	0
$469$	−416.999	−0.889124
$470$	0	0
$471$	0	0
$472$	0	0
$473$	9.41535i	0.0199056i
$474$	0	0
$475$	−134.868	−0.283933
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 538.257i	− 1.12371i	−0.827236	−	0.561855i	$-0.810088\pi$
$479$	− 538.257i	− 1.12371i	0.827236	−	0.561855i	$-0.189912\pi$
$480$	0	0
$481$	−717.473	−1.49163
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 194.597i	− 0.401231i
$486$	0	0
$487$	608.250	1.24897	0.624486	−	0.781036i	$-0.285308\pi$
$487$	608.250	1.24897	0.624486	+	0.781036i	$0.285308\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 143.561i	− 0.292386i	−0.989256	−	0.146193i	$-0.953298\pi$
$491$	− 143.561i	− 0.292386i	0.989256	−	0.146193i	$-0.0467020\pi$
$492$	0	0
$493$	348.316	0.706523
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 132.117i	− 0.265828i
$498$	0	0
$499$	616.605	1.23568	0.617841	−	0.786303i	$-0.288008\pi$
$499$	616.605	1.23568	0.617841	+	0.786303i	$0.288008\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	374.729i	0.744989i	0.928034	+	0.372494i	$0.121497\pi$
$503$	374.729i	0.744989i	−0.928034	+	0.372494i	$0.878503\pi$
$504$	0	0
$505$	−288.974	−0.572225
$506$	0	0
$507$	0	0
$508$	0	0
$509$	173.036i	0.339953i	0.985448	+	0.169977i	$0.0543693\pi$
$509$	173.036i	0.339953i	−0.985448	+	0.169977i	$0.945631\pi$
$510$	0	0
$511$	−103.816	−0.203163
$512$	0	0
$513$	0	0
$514$	0	0
$515$	255.588i	0.496288i
$516$	0	0
$517$	−791.052	−1.53008
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 455.116i	− 0.873543i	−0.899572	−	0.436772i	$-0.856122\pi$
$521$	− 455.116i	− 0.873543i	0.899572	−	0.436772i	$-0.143878\pi$
$522$	0	0
$523$	−295.395	−0.564809	−0.282404	−	0.959295i	$-0.591132\pi$
$523$	−295.395	−0.564809	−0.282404	+	0.959295i	$0.591132\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	355.935i	0.675398i
$528$	0	0
$529$	504.684	0.954034
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 470.245i	− 0.882260i
$534$	0	0
$535$	−208.710	−0.390113
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 173.335i	− 0.321587i
$540$	0	0
$541$	906.394	1.67541	0.837703	−	0.546127i	$-0.183898\pi$
$541$	906.394	1.67541	0.837703	+	0.546127i	$0.183898\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	270.270i	0.495908i
$546$	0	0
$547$	−672.921	−1.23020	−0.615101	−	0.788448i	$-0.710885\pi$
$547$	−672.921	−1.23020	−0.615101	+	0.788448i	$0.710885\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 553.627i	− 1.00477i
$552$	0	0
$553$	586.658	1.06086
$554$	0	0
$555$	0	0
$556$	0	0
$557$	870.336i	1.56254i	0.624192	+	0.781271i	$0.285428\pi$
$557$	870.336i	1.56254i	−0.624192	+	0.781271i	$0.714572\pi$
$558$	0	0
$559$	11.7893	0.0210900
$560$	0	0
$561$	0	0
$562$	0	0
$563$	23.6320i	0.0419751i	0.999780	+	0.0209875i	$0.00668103\pi$
$563$	23.6320i	0.0419751i	−0.999780	+	0.0209875i	$0.993319\pi$
$564$	0	0
$565$	246.658	0.436562
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 295.273i	− 0.518933i	−0.965752	−	0.259466i	$-0.916453\pi$
$569$	− 295.273i	− 0.518933i	0.965752	−	0.259466i	$-0.0835467\pi$
$570$	0	0
$571$	893.920	1.56553	0.782767	−	0.622314i	$-0.213807\pi$
$571$	893.920	1.56553	0.782767	+	0.622314i	$0.213807\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 24.6556i	− 0.0428794i
$576$	0	0
$577$	−264.763	−0.458861	−0.229431	−	0.973325i	$-0.573686\pi$
$577$	−264.763	−0.458861	−0.229431	+	0.973325i	$0.573686\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	247.897i	0.426673i
$582$	0	0
$583$	881.526	1.51205
$584$	0	0
$585$	0	0
$586$	0	0
$587$	791.885i	1.34904i	0.738258	+	0.674519i	$0.235649\pi$
$587$	791.885i	1.34904i	−0.738258	+	0.674519i	$0.764351\pi$
$588$	0	0
$589$	565.737	0.960504
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 1.82388i	− 0.00307567i	−0.999999	−	0.00153784i	$-0.999510\pi$
$593$	− 1.82388i	− 0.00307567i	0.999999	−	0.00153784i	$-0.000489509\pi$
$594$	0	0
$595$	−208.211	−0.349934
$596$	0	0
$597$	0	0
$598$	0	0
$599$	758.204i	1.26578i	0.774241	+	0.632891i	$0.218132\pi$
$599$	758.204i	1.26578i	−0.774241	+	0.632891i	$0.781868\pi$
$600$	0	0
$601$	−283.579	−0.471845	−0.235922	−	0.971772i	$-0.575811\pi$
$601$	−283.579	−0.471845	−0.235922	+	0.971772i	$0.575811\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	82.3812i	0.136167i
$606$	0	0
$607$	167.828	0.276488	0.138244	−	0.990398i	$-0.455854\pi$
$607$	167.828	0.276488	0.138244	+	0.990398i	$0.455854\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	990.507i	1.62112i
$612$	0	0
$613$	−395.698	−0.645510	−0.322755	−	0.946483i	$-0.604609\pi$
$613$	−395.698	−0.645510	−0.322755	+	0.946483i	$0.604609\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	855.190i	1.38604i	0.720916	+	0.693022i	$0.243721\pi$
$617$	855.190i	1.38604i	−0.720916	+	0.693022i	$0.756279\pi$
$618$	0	0
$619$	308.158	0.497832	0.248916	−	0.968525i	$-0.419926\pi$
$619$	308.158	0.497832	0.248916	+	0.968525i	$0.419926\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 636.079i	− 1.02099i
$624$	0	0
$625$	25.0000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 1059.99i	− 1.68520i
$630$	0	0
$631$	344.974	0.546709	0.273355	−	0.961913i	$-0.411867\pi$
$631$	344.974	0.546709	0.273355	+	0.961913i	$0.411867\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	235.640i	0.371087i
$636$	0	0
$637$	−217.040	−0.340722
$638$	0	0
$639$	0	0
$640$	0	0
$641$	202.641i	0.316133i	0.987428	+	0.158066i	$0.0505260\pi$
$641$	202.641i	0.316133i	−0.987428	+	0.158066i	$0.949474\pi$
$642$	0	0
$643$	724.605	1.12691	0.563456	−	0.826146i	$-0.309471\pi$
$643$	724.605	1.12691	0.563456	+	0.826146i	$0.309471\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	125.679i	0.194249i	0.995272	+	0.0971243i	$0.0309644\pi$
$647$	125.679i	0.194249i	−0.995272	+	0.0971243i	$0.969036\pi$
$648$	0	0
$649$	1030.89	1.58843
$650$	0	0
$651$	0	0
$652$	0	0
$653$	551.226i	0.844144i	0.906562	+	0.422072i	$0.138697\pi$
$653$	551.226i	0.844144i	−0.906562	+	0.422072i	$0.861303\pi$
$654$	0	0
$655$	314.355	0.479932
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 801.765i	− 1.21664i	−0.793692	−	0.608320i	$-0.791844\pi$
$659$	− 801.765i	− 1.21664i	0.793692	−	0.608320i	$-0.208156\pi$
$660$	0	0
$661$	−529.079	−0.800422	−0.400211	−	0.916423i	$-0.631063\pi$
$661$	−529.079	−0.800422	−0.400211	+	0.916423i	$0.631063\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	330.938i	0.497651i
$666$	0	0
$667$	101.210	0.151739
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 613.918i	− 0.914929i
$672$	0	0
$673$	413.395	0.614257	0.307129	−	0.951668i	$-0.400632\pi$
$673$	413.395	0.614257	0.307129	+	0.951668i	$0.400632\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	359.936i	0.531663i	0.964019	+	0.265832i	$0.0856465\pi$
$677$	359.936i	0.531663i	−0.964019	+	0.265832i	$0.914353\pi$
$678$	0	0
$679$	−477.499	−0.703239
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 21.5484i	− 0.0315496i	−0.999876	−	0.0157748i	$-0.994979\pi$
$683$	− 21.5484i	− 0.0315496i	0.999876	−	0.0157748i	$-0.00502148\pi$
$684$	0	0
$685$	−596.921	−0.871418
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 1103.79i	− 1.60202i
$690$	0	0
$691$	−384.921	−0.557049	−0.278525	−	0.960429i	$-0.589845\pi$
$691$	−384.921	−0.557049	−0.278525	+	0.960429i	$0.589845\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 35.4238i	− 0.0509695i
$696$	0	0
$697$	694.736	0.996752
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 439.057i	− 0.626330i	−0.949699	−	0.313165i	$-0.898611\pi$
$701$	− 439.057i	− 0.626330i	0.949699	−	0.313165i	$-0.101389\pi$
$702$	0	0
$703$	−1684.79	−2.39657
$704$	0	0
$705$	0	0
$706$	0	0
$707$	709.080i	1.00294i
$708$	0	0
$709$	−126.421	−0.178309	−0.0891547	−	0.996018i	$-0.528417\pi$
$709$	−126.421	−0.178309	−0.0891547	+	0.996018i	$0.528417\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	103.424i	0.145054i
$714$	0	0
$715$	−235.631	−0.329554
$716$	0	0
$717$	0	0
$718$	0	0
$719$	571.715i	0.795153i	0.917569	+	0.397576i	$0.130149\pi$
$719$	571.715i	0.795153i	−0.917569	+	0.397576i	$0.869851\pi$
$720$	0	0
$721$	627.159	0.869846
$722$	0	0
$723$	0	0
$724$	0	0
$725$	102.624i	0.141550i
$726$	0	0
$727$	−841.038	−1.15686	−0.578431	−	0.815731i	$-0.696335\pi$
$727$	−841.038	−1.15686	−0.578431	+	0.815731i	$0.696335\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	17.4175i	0.0238269i
$732$	0	0
$733$	494.749	0.674965	0.337483	−	0.941332i	$-0.390425\pi$
$733$	494.749	0.674965	0.337483	+	0.941332i	$0.390425\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 697.206i	− 0.946006i
$738$	0	0
$739$	1263.97	1.71038	0.855191	−	0.518312i	$-0.173439\pi$
$739$	1263.97	1.71038	0.855191	+	0.518312i	$0.173439\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 956.343i	− 1.28714i	−0.765389	−	0.643568i	$-0.777453\pi$
$743$	− 956.343i	− 1.28714i	0.765389	−	0.643568i	$-0.222547\pi$
$744$	0	0
$745$	93.0790	0.124938
$746$	0	0
$747$	0	0
$748$	0	0
$749$	512.131i	0.683752i
$750$	0	0
$751$	−1121.66	−1.49355	−0.746776	−	0.665076i	$-0.768399\pi$
$751$	−1121.66	−1.49355	−0.746776	+	0.665076i	$0.768399\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	232.198i	0.307547i
$756$	0	0
$757$	−466.433	−0.616160	−0.308080	−	0.951360i	$-0.599687\pi$
$757$	−466.433	−0.616160	−0.308080	+	0.951360i	$0.599687\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1223.41i	1.60763i	0.594880	+	0.803814i	$0.297199\pi$
$761$	1223.41i	1.60763i	−0.594880	+	0.803814i	$0.702801\pi$
$762$	0	0
$763$	663.184	0.869180
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 1290.82i	− 1.68295i
$768$	0	0
$769$	−1290.63	−1.67832	−0.839162	−	0.543882i	$-0.816954\pi$
$769$	−1290.63	−1.67832	−0.839162	+	0.543882i	$0.816954\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 329.455i	− 0.426204i	−0.977030	−	0.213102i	$-0.931643\pi$
$773$	− 329.455i	− 0.426204i	0.977030	−	0.213102i	$-0.0683566\pi$
$774$	0	0
$775$	−104.868	−0.135314
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 1104.24i	− 1.41751i
$780$	0	0
$781$	220.894	0.282835
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 27.9803i	− 0.0356437i
$786$	0	0
$787$	−1506.13	−1.91376	−0.956881	−	0.290480i	$-0.906185\pi$
$787$	−1506.13	−1.91376	−0.956881	+	0.290480i	$0.906185\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 605.245i	− 0.765165i
$792$	0	0
$793$	−768.710	−0.969370
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 250.874i	− 0.314773i	−0.987537	−	0.157387i	$-0.949693\pi$
$797$	− 250.874i	− 0.314773i	0.987537	−	0.157387i	$-0.0503069\pi$
$798$	0	0
$799$	−1463.37	−1.83150
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 173.577i	− 0.216160i
$804$	0	0
$805$	−60.4997	−0.0751548
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1097.02i	1.35602i	0.735053	+	0.678010i	$0.237157\pi$
$809$	1097.02i	1.35602i	−0.735053	+	0.678010i	$0.762843\pi$
$810$	0	0
$811$	−221.473	−0.273087	−0.136543	−	0.990634i	$-0.543599\pi$
$811$	−221.473	−0.273087	−0.136543	+	0.990634i	$0.543599\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 650.401i	− 0.798038i
$816$	0	0
$817$	27.6840	0.0338849
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 1003.92i	− 1.22281i	−0.791319	−	0.611403i	$-0.790605\pi$
$821$	− 1003.92i	− 1.22281i	0.791319	−	0.611403i	$-0.209395\pi$
$822$	0	0
$823$	335.013	0.407063	0.203531	−	0.979068i	$-0.434758\pi$
$823$	335.013	0.407063	0.203531	+	0.979068i	$0.434758\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1004.72i	1.21490i	0.794357	+	0.607451i	$0.207808\pi$
$827$	1004.72i	1.21490i	−0.794357	+	0.607451i	$0.792192\pi$
$828$	0	0
$829$	676.763	0.816361	0.408180	−	0.912901i	$-0.366163\pi$
$829$	676.763	0.816361	0.408180	+	0.912901i	$0.366163\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 320.653i	− 0.384938i
$834$	0	0
$835$	390.500	0.467664
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 621.286i	− 0.740507i	−0.928931	−	0.370254i	$-0.879271\pi$
$839$	− 621.286i	− 0.740507i	0.928931	−	0.370254i	$-0.120729\pi$
$840$	0	0
$841$	419.736	0.499092
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 82.8523i	− 0.0980500i
$846$	0	0
$847$	202.146	0.238661
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 308.001i	− 0.361928i
$852$	0	0
$853$	−544.591	−0.638443	−0.319221	−	0.947680i	$-0.603421\pi$
$853$	−544.591	−0.638443	−0.319221	+	0.947680i	$0.603421\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	362.149i	0.422578i	0.977424	+	0.211289i	$0.0677661\pi$
$857$	362.149i	0.422578i	−0.977424	+	0.211289i	$0.932234\pi$
$858$	0	0
$859$	−21.6840	−0.0252433	−0.0126216	−	0.999920i	$-0.504018\pi$
$859$	−21.6840	−0.0252433	−0.0126216	+	0.999920i	$0.504018\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	220.300i	0.255273i	0.991821	+	0.127636i	$0.0407390\pi$
$863$	220.300i	0.255273i	−0.991821	+	0.127636i	$0.959261\pi$
$864$	0	0
$865$	−97.4477	−0.112656
$866$	0	0
$867$	0	0
$868$	0	0
$869$	980.868i	1.12873i
$870$	0	0
$871$	−872.999	−1.00230
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 61.3447i	− 0.0701082i
$876$	0	0
$877$	1136.41	1.29579	0.647895	−	0.761730i	$-0.275650\pi$
$877$	1136.41	1.29579	0.647895	+	0.761730i	$0.275650\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 711.758i	− 0.807898i	−0.914782	−	0.403949i	$-0.867637\pi$
$881$	− 711.758i	− 0.807898i	0.914782	−	0.403949i	$-0.132363\pi$
$882$	0	0
$883$	−536.394	−0.607468	−0.303734	−	0.952757i	$-0.598233\pi$
$883$	−536.394	−0.607468	−0.303734	+	0.952757i	$0.598233\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1463.95i	− 1.65045i	−0.564801	−	0.825227i	$-0.691047\pi$
$887$	− 1463.95i	− 1.65045i	0.564801	−	0.825227i	$-0.308953\pi$
$888$	0	0
$889$	578.211	0.650406
$890$	0	0
$891$	0	0
$892$	0	0
$893$	2325.93i	2.60463i
$894$	0	0
$895$	197.434	0.220597
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 430.479i	− 0.478842i
$900$	0	0
$901$	1630.74	1.80992
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 127.161i	− 0.140510i
$906$	0	0
$907$	855.657	0.943392	0.471696	−	0.881761i	$-0.343642\pi$
$907$	855.657	0.943392	0.471696	+	0.881761i	$0.343642\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1581.02i	1.73547i	0.497024	+	0.867737i	$0.334426\pi$
$911$	1581.02i	1.73547i	−0.497024	+	0.867737i	$0.665574\pi$
$912$	0	0
$913$	−414.474	−0.453969
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 771.360i	− 0.841178i
$918$	0	0
$919$	−1489.08	−1.62033	−0.810163	−	0.586205i	$-0.800621\pi$
$919$	−1489.08	−1.62033	−0.810163	+	0.586205i	$0.800621\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 276.590i	− 0.299664i
$924$	0	0
$925$	312.302	0.337624
$926$	0	0
$927$	0	0
$928$	0	0
$929$	736.990i	0.793316i	0.917966	+	0.396658i	$0.129830\pi$
$929$	736.990i	0.793316i	−0.917966	+	0.396658i	$0.870170\pi$
$930$	0	0
$931$	−509.658	−0.547431
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 348.120i	− 0.372321i
$936$	0	0
$937$	1550.05	1.65427	0.827135	−	0.562003i	$-0.189969\pi$
$937$	1550.05	1.65427	0.827135	+	0.562003i	$0.189969\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 628.711i	− 0.668131i	−0.942550	−	0.334065i	$-0.891579\pi$
$941$	− 628.711i	− 0.668131i	0.942550	−	0.334065i	$-0.108421\pi$
$942$	0	0
$943$	201.869	0.214071
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1498.73i	1.58261i	0.611423	+	0.791304i	$0.290598\pi$
$947$	1498.73i	1.58261i	−0.611423	+	0.791304i	$0.709402\pi$
$948$	0	0
$949$	−217.342	−0.229022
$950$	0	0
$951$	0	0
$952$	0	0
$953$	224.748i	0.235832i	0.993024	+	0.117916i	$0.0376214\pi$
$953$	224.748i	0.235832i	−0.993024	+	0.117916i	$0.962379\pi$
$954$	0	0
$955$	126.658	0.132626
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1464.72i	1.52734i
$960$	0	0
$961$	−521.105	−0.542253
$962$	0	0
$963$	0	0
$964$	0	0
$965$	245.967i	0.254889i
$966$	0	0
$967$	233.986	0.241972	0.120986	−	0.992654i	$-0.461394\pi$
$967$	233.986	0.241972	0.120986	+	0.992654i	$0.461394\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	691.940i	0.712606i	0.934371	+	0.356303i	$0.115963\pi$
$971$	691.940i	0.712606i	−0.934371	+	0.356303i	$0.884037\pi$
$972$	0	0
$973$	−86.9224	−0.0893344
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 395.571i	− 0.404883i	−0.979294	−	0.202442i	$-0.935112\pi$
$977$	− 395.571i	− 0.404883i	0.979294	−	0.202442i	$-0.0648877\pi$
$978$	0	0
$979$	1063.50	1.08631
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1.82388i	0.00185542i	1.00000		0.000927709i	$0.000295299\pi$
$983$	1.82388i	0.00185542i	−1.00000		0.000927709i	$0.999705\pi$
$984$	0	0
$985$	547.947	0.556292
$986$	0	0
$987$	0	0
$988$	0	0
$989$	5.06098i	0.00511727i
$990$	0	0
$991$	506.316	0.510914	0.255457	−	0.966820i	$-0.417774\pi$
$991$	506.316	0.510914	0.255457	+	0.966820i	$0.417774\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 379.896i	− 0.381805i
$996$	0	0
$997$	−295.670	−0.296560	−0.148280	−	0.988945i	$-0.547374\pi$
$997$	−295.670	−0.296560	−0.148280	+	0.988945i	$0.547374\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	180.3.g.a.161.4	yes	4
3.2	odd	2	inner	180.3.g.a.161.2	✓	4
4.3	odd	2		720.3.l.c.161.3		4
5.2	odd	4		900.3.b.b.449.6		8
5.3	odd	4		900.3.b.b.449.4		8
5.4	even	2		900.3.g.d.701.2		4
8.3	odd	2		2880.3.l.f.1601.1		4
8.5	even	2		2880.3.l.b.1601.2		4
9.2	odd	6		1620.3.o.f.701.3		8
9.4	even	3		1620.3.o.f.1241.3		8
9.5	odd	6		1620.3.o.f.1241.1		8
9.7	even	3		1620.3.o.f.701.1		8
12.11	even	2		720.3.l.c.161.1		4
15.2	even	4		900.3.b.b.449.5		8
15.8	even	4		900.3.b.b.449.3		8
15.14	odd	2		900.3.g.d.701.1		4
20.3	even	4		3600.3.c.k.449.5		8
20.7	even	4		3600.3.c.k.449.3		8
20.19	odd	2		3600.3.l.n.1601.3		4
24.5	odd	2		2880.3.l.b.1601.4		4
24.11	even	2		2880.3.l.f.1601.3		4
60.23	odd	4		3600.3.c.k.449.6		8
60.47	odd	4		3600.3.c.k.449.4		8
60.59	even	2		3600.3.l.n.1601.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.3.g.a.161.2	✓	4	3.2	odd	2	inner
180.3.g.a.161.4	yes	4	1.1	even	1	trivial
720.3.l.c.161.1		4	12.11	even	2
720.3.l.c.161.3		4	4.3	odd	2
900.3.b.b.449.3		8	15.8	even	4
900.3.b.b.449.4		8	5.3	odd	4
900.3.b.b.449.5		8	15.2	even	4
900.3.b.b.449.6		8	5.2	odd	4
900.3.g.d.701.1		4	15.14	odd	2
900.3.g.d.701.2		4	5.4	even	2
1620.3.o.f.701.1		8	9.7	even	3
1620.3.o.f.701.3		8	9.2	odd	6
1620.3.o.f.1241.1		8	9.5	odd	6
1620.3.o.f.1241.3		8	9.4	even	3
2880.3.l.b.1601.2		4	8.5	even	2
2880.3.l.b.1601.4		4	24.5	odd	2
2880.3.l.f.1601.1		4	8.3	odd	2
2880.3.l.f.1601.3		4	24.11	even	2
3600.3.c.k.449.3		8	20.7	even	4
3600.3.c.k.449.4		8	60.47	odd	4
3600.3.c.k.449.5		8	20.3	even	4
3600.3.c.k.449.6		8	60.23	odd	4
3600.3.l.n.1601.3		4	20.19	odd	2
3600.3.l.n.1601.4		4	60.59	even	2

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:	\( N \)	\(=\)	\( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	180.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.23607i q^{5} +5.48683 q^{7} +O(q^{10})\)	\(q+2.23607i q^{5} +5.48683 q^{7} +9.17377i q^{11} +11.4868 q^{13} +16.9706i q^{17} +26.9737 q^{19} +4.93113i q^{23} -5.00000 q^{25} -20.5247i q^{29} +20.9737 q^{31} +12.2689i q^{35} -62.4605 q^{37} -40.9377i q^{41} +1.02633 q^{43} +86.2298i q^{47} -18.8947 q^{49} -96.0920i q^{53} -20.5132 q^{55} -112.374i q^{59} -66.9210 q^{61} +25.6853i q^{65} -76.0000 q^{67} -24.0789i q^{71} -18.9210 q^{73} +50.3349i q^{77} +106.921 q^{79} +45.1804i q^{83} -37.9473 q^{85} -115.928i q^{89} +63.0263 q^{91} +60.3150i q^{95} -87.0263 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 16 q^{7}+O(q^{10}) \) 4 * q - 16 * q^7	\( 4 q - 16 q^{7} + 8 q^{13} + 32 q^{19} - 20 q^{25} + 8 q^{31} - 136 q^{37} + 80 q^{43} + 228 q^{49} - 120 q^{55} - 40 q^{61} - 304 q^{67} + 152 q^{73} + 200 q^{79} + 328 q^{91} - 424 q^{97}+O(q^{100}) \) 4 * q - 16 * q^7 + 8 * q^13 + 32 * q^19 - 20 * q^25 + 8 * q^31 - 136 * q^37 + 80 * q^43 + 228 * q^49 - 120 * q^55 - 40 * q^61 - 304 * q^67 + 152 * q^73 + 200 * q^79 + 328 * q^91 - 424 * q^97

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