LMFDB - Embedded newform 648.3.e.b.161.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [648,3,Mod(161,648)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(648, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("648.161");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$N$	$=$	$648 = 2^{3} \cdot 3^{4}$
Weight:	$k$	$=$	$3$
Character orbit:	$[\chi]$	$=$	648.e (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:	$17.6567211305$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - 2x^{2} + 4$ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2^{6}$
Twist minimal:	no (minimal twist has level 72)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.1
Root			$-1.22474 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	648.161
Dual form			648.3.e.b.161.4

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-7.38891i q^{5} -6.79796 q^{7} +6.11756i q^{11} +16.7980 q^{13} -25.1701i q^{17} -17.5959 q^{19} -14.3171i q^{23} -29.5959 q^{25} +18.7026i q^{29} -46.7980 q^{31} +50.2295i q^{35} -49.5959 q^{37} +39.8372i q^{41} -44.1918 q^{43} +33.2589i q^{47} -2.78775 q^{49} -10.1708i q^{53} +45.2020 q^{55} +16.5099i q^{59} +21.2020 q^{61} -124.119i q^{65} -86.9796 q^{67} +30.2555i q^{71} -48.7878 q^{73} -41.5869i q^{77} +111.586 q^{79} -98.2485i q^{83} -185.980 q^{85} -75.5103i q^{89} -114.192 q^{91} +130.015i q^{95} -140.596 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q + 12 q^{7} + 28 q^{13} + 8 q^{19} - 40 q^{25} - 148 q^{31} - 120 q^{37} - 20 q^{43} + 224 q^{49} + 220 q^{55} + 124 q^{61} + 44 q^{67} + 40 q^{73} + 172 q^{79} - 352 q^{85} - 300 q^{91} - 484 q^{97}+O(q^{100})$ 4 * q + 12 * q^7 + 28 * q^13 + 8 * q^19 - 40 * q^25 - 148 * q^31 - 120 * q^37 - 20 * q^43 + 224 * q^49 + 220 * q^55 + 124 * q^61 + 44 * q^67 + 40 * q^73 + 172 * q^79 - 352 * q^85 - 300 * q^91 - 484 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/648\mathbb{Z}\right)^\times$ .

$n$	$325$	$487$	$569$
$\chi(n)$	$1$	$1$	$-1$

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
$5$	− 7.38891i	− 1.47778i	−0.673826	−	0.738891i	$-0.735350\pi$
$5$	− 7.38891i	− 1.47778i	0.673826	−	0.738891i	$-0.264650\pi$
$6$	0	0
$7$	−6.79796	−0.971137	−0.485568	−	0.874199i	$-0.661387\pi$
$7$	−6.79796	−0.971137	−0.485568	+	0.874199i	$0.661387\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	6.11756i	0.556141i	0.960561	+	0.278071i	$0.0896950\pi$
$11$	6.11756i	0.556141i	−0.960561	+	0.278071i	$0.910305\pi$
$12$	0	0
$13$	16.7980	1.29215	0.646075	−	0.763274i	$-0.276409\pi$
$13$	16.7980	1.29215	0.646075	+	0.763274i	$0.276409\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 25.1701i	− 1.48059i	−0.672279	−	0.740297i	$-0.734685\pi$
$17$	− 25.1701i	− 1.48059i	0.672279	−	0.740297i	$-0.265315\pi$
$18$	0	0
$19$	−17.5959	−0.926101	−0.463050	−	0.886332i	$-0.653245\pi$
$19$	−17.5959	−0.926101	−0.463050	+	0.886332i	$0.653245\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 14.3171i	− 0.622483i	−0.950331	−	0.311241i	$-0.899255\pi$
$23$	− 14.3171i	− 0.622483i	0.950331	−	0.311241i	$-0.100745\pi$
$24$	0	0
$25$	−29.5959	−1.18384
$26$	0	0
$27$	0	0
$28$	0	0
$29$	18.7026i	0.644918i	0.946583	+	0.322459i	$0.104509\pi$
$29$	18.7026i	0.644918i	−0.946583	+	0.322459i	$0.895491\pi$
$30$	0	0
$31$	−46.7980	−1.50961	−0.754806	−	0.655948i	$-0.772269\pi$
$31$	−46.7980	−1.50961	−0.754806	+	0.655948i	$0.772269\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	50.2295i	1.43513i
$36$	0	0
$37$	−49.5959	−1.34043	−0.670215	−	0.742167i	$-0.733798\pi$
$37$	−49.5959	−1.34043	−0.670215	+	0.742167i	$0.733798\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	39.8372i	0.971638i	0.874059	+	0.485819i	$0.161479\pi$
$41$	39.8372i	0.971638i	−0.874059	+	0.485819i	$0.838521\pi$
$42$	0	0
$43$	−44.1918	−1.02772	−0.513859	−	0.857875i	$-0.671784\pi$
$43$	−44.1918	−1.02772	−0.513859	+	0.857875i	$0.671784\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	33.2589i	0.707636i	0.935314	+	0.353818i	$0.115117\pi$
$47$	33.2589i	0.707636i	−0.935314	+	0.353818i	$0.884883\pi$
$48$	0	0
$49$	−2.78775	−0.0568929
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 10.1708i	− 0.191902i	−0.995386	−	0.0959509i	$-0.969411\pi$
$53$	− 10.1708i	− 0.191902i	0.995386	−	0.0959509i	$-0.0305892\pi$
$54$	0	0
$55$	45.2020	0.821855
$56$	0	0
$57$	0	0
$58$	0	0
$59$	16.5099i	0.279828i	0.990164	+	0.139914i	$0.0446826\pi$
$59$	16.5099i	0.279828i	−0.990164	+	0.139914i	$0.955317\pi$
$60$	0	0
$61$	21.2020	0.347574	0.173787	−	0.984783i	$-0.444400\pi$
$61$	21.2020	0.347574	0.173787	+	0.984783i	$0.444400\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 124.119i	− 1.90952i
$66$	0	0
$67$	−86.9796	−1.29820	−0.649101	−	0.760702i	$-0.724855\pi$
$67$	−86.9796	−1.29820	−0.649101	+	0.760702i	$0.724855\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	30.2555i	0.426134i	0.977038	+	0.213067i	$0.0683453\pi$
$71$	30.2555i	0.426134i	−0.977038	+	0.213067i	$0.931655\pi$
$72$	0	0
$73$	−48.7878	−0.668325	−0.334163	−	0.942515i	$-0.608454\pi$
$73$	−48.7878	−0.668325	−0.334163	+	0.942515i	$0.608454\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 41.5869i	− 0.540090i
$78$	0	0
$79$	111.586	1.41248	0.706239	−	0.707974i	$-0.250391\pi$
$79$	111.586	1.41248	0.706239	+	0.707974i	$0.250391\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 98.2485i	− 1.18372i	−0.806042	−	0.591859i	$-0.798394\pi$
$83$	− 98.2485i	− 1.18372i	0.806042	−	0.591859i	$-0.201606\pi$
$84$	0	0
$85$	−185.980	−2.18800
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 75.5103i	− 0.848431i	−0.905561	−	0.424215i	$-0.860550\pi$
$89$	− 75.5103i	− 0.848431i	0.905561	−	0.424215i	$-0.139450\pi$
$90$	0	0
$91$	−114.192	−1.25486
$92$	0	0
$93$	0	0
$94$	0	0
$95$	130.015i	1.36857i
$96$	0	0
$97$	−140.596	−1.44944	−0.724721	−	0.689042i	$-0.758031\pi$
$97$	−140.596	−1.44944	−0.724721	+	0.689042i	$0.758031\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	32.5590i	0.322367i	0.986924	+	0.161183i	$0.0515310\pi$
$101$	32.5590i	0.322367i	−0.986924	+	0.161183i	$0.948469\pi$
$102$	0	0
$103$	135.586	1.31637	0.658183	−	0.752858i	$-0.271325\pi$
$103$	135.586	1.31637	0.658183	+	0.752858i	$0.271325\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 35.3409i	− 0.330289i	−0.986269	−	0.165144i	$-0.947191\pi$
$107$	− 35.3409i	− 0.330289i	0.986269	−	0.165144i	$-0.0528090\pi$
$108$	0	0
$109$	53.5959	0.491706	0.245853	−	0.969307i	$-0.420932\pi$
$109$	53.5959	0.491706	0.245853	+	0.969307i	$0.420932\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 165.209i	− 1.46203i	−0.682362	−	0.731015i	$-0.739047\pi$
$113$	− 165.209i	− 1.46203i	0.682362	−	0.731015i	$-0.260953\pi$
$114$	0	0
$115$	−105.788	−0.919894
$116$	0	0
$117$	0	0
$118$	0	0
$119$	171.105i	1.43786i
$120$	0	0
$121$	83.5755	0.690707
$122$	0	0
$123$	0	0
$124$	0	0
$125$	33.9588i	0.271670i
$126$	0	0
$127$	11.9796	0.0943275	0.0471637	−	0.998887i	$-0.484982\pi$
$127$	11.9796	0.0943275	0.0471637	+	0.998887i	$0.484982\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	15.3669i	0.117305i	0.998278	+	0.0586525i	$0.0186804\pi$
$131$	15.3669i	0.117305i	−0.998278	+	0.0586525i	$0.981320\pi$
$132$	0	0
$133$	119.616	0.899371
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 55.0934i	− 0.402141i	−0.979577	−	0.201071i	$-0.935558\pi$
$137$	− 55.0934i	− 0.402141i	0.979577	−	0.201071i	$-0.0644421\pi$
$138$	0	0
$139$	100.980	0.726472	0.363236	−	0.931697i	$-0.381672\pi$
$139$	100.980	0.726472	0.363236	+	0.931697i	$0.381672\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	102.762i	0.718619i
$144$	0	0
$145$	138.192	0.953047
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 216.378i	− 1.45220i	−0.687589	−	0.726100i	$-0.741331\pi$
$149$	− 216.378i	− 1.45220i	0.687589	−	0.726100i	$-0.258669\pi$
$150$	0	0
$151$	−153.586	−1.01712	−0.508562	−	0.861025i	$-0.669823\pi$
$151$	−153.586	−1.01712	−0.508562	+	0.861025i	$0.669823\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	345.786i	2.23088i
$156$	0	0
$157$	81.9694	0.522098	0.261049	−	0.965326i	$-0.415932\pi$
$157$	81.9694	0.522098	0.261049	+	0.965326i	$0.415932\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	97.3271i	0.604516i
$162$	0	0
$163$	55.2122	0.338725	0.169363	−	0.985554i	$-0.445829\pi$
$163$	55.2122	0.338725	0.169363	+	0.985554i	$0.445829\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	154.502i	0.925164i	0.886576	+	0.462582i	$0.153077\pi$
$167$	154.502i	0.925164i	−0.886576	+	0.462582i	$0.846923\pi$
$168$	0	0
$169$	113.171	0.669653
$170$	0	0
$171$	0	0
$172$	0	0
$173$	26.3307i	0.152201i	0.997100	+	0.0761003i	$0.0242469\pi$
$173$	26.3307i	0.152201i	−0.997100	+	0.0761003i	$0.975753\pi$
$174$	0	0
$175$	201.192	1.14967
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 266.700i	− 1.48995i	−0.667094	−	0.744973i	$-0.732462\pi$
$179$	− 266.700i	− 1.48995i	0.667094	−	0.744973i	$-0.267538\pi$
$180$	0	0
$181$	−58.4041	−0.322674	−0.161337	−	0.986899i	$-0.551581\pi$
$181$	−58.4041	−0.322674	−0.161337	+	0.986899i	$0.551581\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	366.460i	1.98086i
$186$	0	0
$187$	153.980	0.823420
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 114.962i	− 0.601896i	−0.953640	−	0.300948i	$-0.902697\pi$
$191$	− 114.962i	− 0.601896i	0.953640	−	0.300948i	$-0.0973031\pi$
$192$	0	0
$193$	216.980	1.12425	0.562123	−	0.827053i	$-0.309985\pi$
$193$	216.980	1.12425	0.562123	+	0.827053i	$0.309985\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 171.105i	− 0.868555i	−0.900779	−	0.434278i	$-0.857004\pi$
$197$	− 171.105i	− 0.868555i	0.900779	−	0.434278i	$-0.142996\pi$
$198$	0	0
$199$	−62.0000	−0.311558	−0.155779	−	0.987792i	$-0.549789\pi$
$199$	−62.0000	−0.311558	−0.155779	+	0.987792i	$0.549789\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 127.140i	− 0.626303i
$204$	0	0
$205$	294.353	1.43587
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 107.644i	− 0.515043i
$210$	0	0
$211$	−129.404	−0.613289	−0.306645	−	0.951824i	$-0.599206\pi$
$211$	−129.404	−0.613289	−0.306645	+	0.951824i	$0.599206\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	326.529i	1.51874i
$216$	0	0
$217$	318.131	1.46604
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 422.807i	− 1.91315i
$222$	0	0
$223$	98.3735	0.441137	0.220568	−	0.975372i	$-0.429209\pi$
$223$	98.3735	0.441137	0.220568	+	0.975372i	$0.429209\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 443.002i	− 1.95155i	−0.218775	−	0.975775i	$-0.570206\pi$
$227$	− 443.002i	− 1.95155i	0.218775	−	0.975775i	$-0.429794\pi$
$228$	0	0
$229$	−112.010	−0.489128	−0.244564	−	0.969633i	$-0.578645\pi$
$229$	−112.010	−0.489128	−0.244564	+	0.969633i	$0.578645\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 80.8526i	− 0.347007i	−0.984833	−	0.173503i	$-0.944491\pi$
$233$	− 80.8526i	− 0.347007i	0.984833	−	0.173503i	$-0.0555088\pi$
$234$	0	0
$235$	245.747	1.04573
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 43.1728i	− 0.180639i	−0.995913	−	0.0903197i	$-0.971211\pi$
$239$	− 43.1728i	− 0.180639i	0.995913	−	0.0903197i	$-0.0287889\pi$
$240$	0	0
$241$	−281.808	−1.16933	−0.584664	−	0.811275i	$-0.698774\pi$
$241$	−281.808	−1.16933	−0.584664	+	0.811275i	$0.698774\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	20.5984i	0.0840753i
$246$	0	0
$247$	−295.576	−1.19666
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 15.5131i	− 0.0618050i	−0.999522	−	0.0309025i	$-0.990162\pi$
$251$	− 15.5131i	− 0.0618050i	0.999522	−	0.0309025i	$-0.00983814\pi$
$252$	0	0
$253$	87.5857	0.346189
$254$	0	0
$255$	0	0
$256$	0	0
$257$	360.307i	1.40197i	0.713175	+	0.700986i	$0.247256\pi$
$257$	360.307i	1.40197i	−0.713175	+	0.700986i	$0.752744\pi$
$258$	0	0
$259$	337.151	1.30174
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 48.9935i	− 0.186287i	−0.995653	−	0.0931435i	$-0.970308\pi$
$263$	− 48.9935i	− 0.186287i	0.995653	−	0.0931435i	$-0.0296915\pi$
$264$	0	0
$265$	−75.1510	−0.283589
$266$	0	0
$267$	0	0
$268$	0	0
$269$	281.700i	1.04721i	0.851961	+	0.523606i	$0.175413\pi$
$269$	281.700i	1.04721i	−0.851961	+	0.523606i	$0.824587\pi$
$270$	0	0
$271$	−89.5959	−0.330612	−0.165306	−	0.986242i	$-0.552861\pi$
$271$	−89.5959	−0.330612	−0.165306	+	0.986242i	$0.552861\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 181.055i	− 0.658381i
$276$	0	0
$277$	84.3939	0.304671	0.152336	−	0.988329i	$-0.451321\pi$
$277$	84.3939	0.304671	0.152336	+	0.988329i	$0.451321\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 26.8668i	− 0.0956114i	−0.998857	−	0.0478057i	$-0.984777\pi$
$281$	− 26.8668i	− 0.0956114i	0.998857	−	0.0478057i	$-0.0152228\pi$
$282$	0	0
$283$	180.980	0.639504	0.319752	−	0.947501i	$-0.396400\pi$
$283$	180.980	0.639504	0.319752	+	0.947501i	$0.396400\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 270.811i	− 0.943594i
$288$	0	0
$289$	−344.535	−1.19216
$290$	0	0
$291$	0	0
$292$	0	0
$293$	471.100i	1.60785i	0.594730	+	0.803925i	$0.297259\pi$
$293$	471.100i	1.60785i	−0.594730	+	0.803925i	$0.702741\pi$
$294$	0	0
$295$	121.990	0.413525
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 240.498i	− 0.804342i
$300$	0	0
$301$	300.414	0.998054
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 156.660i	− 0.513639i
$306$	0	0
$307$	464.747	1.51383	0.756917	−	0.653511i	$-0.226705\pi$
$307$	464.747	1.51383	0.756917	+	0.653511i	$0.226705\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	252.126i	0.810696i	0.914162	+	0.405348i	$0.132850\pi$
$311$	252.126i	0.810696i	−0.914162	+	0.405348i	$0.867150\pi$
$312$	0	0
$313$	196.212	0.626876	0.313438	−	0.949609i	$-0.398519\pi$
$313$	196.212	0.626876	0.313438	+	0.949609i	$0.398519\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	114.298i	0.360560i	0.983615	+	0.180280i	$0.0577005\pi$
$317$	114.298i	0.360560i	−0.983615	+	0.180280i	$0.942300\pi$
$318$	0	0
$319$	−114.414	−0.358665
$320$	0	0
$321$	0	0
$322$	0	0
$323$	442.891i	1.37118i
$324$	0	0
$325$	−497.151	−1.52970
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 226.093i	− 0.687212i
$330$	0	0
$331$	54.5959	0.164942	0.0824712	−	0.996593i	$-0.473719\pi$
$331$	54.5959	0.164942	0.0824712	+	0.996593i	$0.473719\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	642.684i	1.91846i
$336$	0	0
$337$	−237.767	−0.705541	−0.352771	−	0.935710i	$-0.614760\pi$
$337$	−237.767	−0.705541	−0.352771	+	0.935710i	$0.614760\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 286.289i	− 0.839558i
$342$	0	0
$343$	352.051	1.02639
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 125.111i	− 0.360549i	−0.983616	−	0.180275i	$-0.942301\pi$
$347$	− 125.111i	− 0.360549i	0.983616	−	0.180275i	$-0.0576987\pi$
$348$	0	0
$349$	−539.969	−1.54719	−0.773595	−	0.633680i	$-0.781544\pi$
$349$	−539.969	−1.54719	−0.773595	+	0.633680i	$0.781544\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	293.860i	0.832463i	0.909259	+	0.416232i	$0.136649\pi$
$353$	293.860i	0.832463i	−0.909259	+	0.416232i	$0.863351\pi$
$354$	0	0
$355$	223.555	0.629733
$356$	0	0
$357$	0	0
$358$	0	0
$359$	422.550i	1.17702i	0.808490	+	0.588509i	$0.200285\pi$
$359$	422.550i	1.17702i	−0.808490	+	0.588509i	$0.799715\pi$
$360$	0	0
$361$	−51.3837	−0.142337
$362$	0	0
$363$	0	0
$364$	0	0
$365$	360.488i	0.987639i
$366$	0	0
$367$	−262.716	−0.715848	−0.357924	−	0.933751i	$-0.616515\pi$
$367$	−262.716	−0.715848	−0.357924	+	0.933751i	$0.616515\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	69.1406i	0.186363i
$372$	0	0
$373$	121.969	0.326996	0.163498	−	0.986544i	$-0.447722\pi$
$373$	121.969	0.326996	0.163498	+	0.986544i	$0.447722\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	314.166i	0.833331i
$378$	0	0
$379$	−641.151	−1.69169	−0.845846	−	0.533428i	$-0.820904\pi$
$379$	−641.151	−1.69169	−0.845846	+	0.533428i	$0.820904\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 739.794i	− 1.93158i	−0.259330	−	0.965789i	$-0.583502\pi$
$383$	− 739.794i	− 1.93158i	0.259330	−	0.965789i	$-0.416498\pi$
$384$	0	0
$385$	−307.282	−0.798134
$386$	0	0
$387$	0	0
$388$	0	0
$389$	87.4709i	0.224861i	0.993660	+	0.112430i	$0.0358636\pi$
$389$	87.4709i	0.224861i	−0.993660	+	0.112430i	$0.964136\pi$
$390$	0	0
$391$	−360.363	−0.921645
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 824.496i	− 2.08733i
$396$	0	0
$397$	483.090	1.21685	0.608425	−	0.793611i	$-0.291801\pi$
$397$	483.090	1.21685	0.608425	+	0.793611i	$0.291801\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 366.827i	− 0.914781i	−0.889266	−	0.457390i	$-0.848784\pi$
$401$	− 366.827i	− 0.914781i	0.889266	−	0.457390i	$-0.151216\pi$
$402$	0	0
$403$	−786.110	−1.95065
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 303.406i	− 0.745469i
$408$	0	0
$409$	535.282	1.30876	0.654379	−	0.756167i	$-0.272930\pi$
$409$	535.282	1.30876	0.654379	+	0.756167i	$0.272930\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 112.233i	− 0.271751i
$414$	0	0
$415$	−725.949	−1.74927
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 699.310i	− 1.66900i	−0.551009	−	0.834499i	$-0.685757\pi$
$419$	− 699.310i	− 1.66900i	0.551009	−	0.834499i	$-0.314243\pi$
$420$	0	0
$421$	−361.545	−0.858776	−0.429388	−	0.903120i	$-0.641271\pi$
$421$	−361.545	−0.858776	−0.429388	+	0.903120i	$0.641271\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	744.933i	1.75278i
$426$	0	0
$427$	−144.131	−0.337542
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 463.747i	− 1.07598i	−0.842952	−	0.537989i	$-0.819184\pi$
$431$	− 463.747i	− 1.07598i	0.842952	−	0.537989i	$-0.180816\pi$
$432$	0	0
$433$	−689.514	−1.59241	−0.796206	−	0.605026i	$-0.793163\pi$
$433$	−689.514	−1.59241	−0.796206	+	0.605026i	$0.793163\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	251.923i	0.576482i
$438$	0	0
$439$	−621.545	−1.41582	−0.707910	−	0.706303i	$-0.750362\pi$
$439$	−621.545	−1.41582	−0.707910	+	0.706303i	$0.750362\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 806.954i	− 1.82157i	−0.412884	−	0.910784i	$-0.635479\pi$
$443$	− 806.954i	− 1.82157i	0.412884	−	0.910784i	$-0.364521\pi$
$444$	0	0
$445$	−557.939	−1.25379
$446$	0	0
$447$	0	0
$448$	0	0
$449$	317.554i	0.707248i	0.935388	+	0.353624i	$0.115051\pi$
$449$	317.554i	0.707248i	−0.935388	+	0.353624i	$0.884949\pi$
$450$	0	0
$451$	−243.706	−0.540368
$452$	0	0
$453$	0	0
$454$	0	0
$455$	843.753i	1.85440i
$456$	0	0
$457$	571.686	1.25095	0.625477	−	0.780243i	$-0.284905\pi$
$457$	571.686	1.25095	0.625477	+	0.780243i	$0.284905\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 552.582i	− 1.19866i	−0.800502	−	0.599330i	$-0.795434\pi$
$461$	− 552.582i	− 1.19866i	0.800502	−	0.599330i	$-0.204566\pi$
$462$	0	0
$463$	−120.333	−0.259898	−0.129949	−	0.991521i	$-0.541481\pi$
$463$	−120.333	−0.259898	−0.129949	+	0.991521i	$0.541481\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 880.440i	− 1.88531i	−0.333767	−	0.942656i	$-0.608320\pi$
$467$	− 880.440i	− 1.88531i	0.333767	−	0.942656i	$-0.391680\pi$
$468$	0	0
$469$	591.284	1.26073
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 270.346i	− 0.571556i
$474$	0	0
$475$	520.767	1.09635
$476$	0	0
$477$	0	0
$478$	0	0
$479$	685.653i	1.43143i	0.698395	+	0.715713i	$0.253898\pi$
$479$	685.653i	1.43143i	−0.698395	+	0.715713i	$0.746102\pi$
$480$	0	0
$481$	−833.110	−1.73204
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1038.85i	2.14196i
$486$	0	0
$487$	391.131	0.803143	0.401571	−	0.915828i	$-0.368464\pi$
$487$	391.131	0.803143	0.401571	+	0.915828i	$0.368464\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 192.222i	− 0.391491i	−0.980655	−	0.195746i	$-0.937287\pi$
$491$	− 192.222i	− 0.391491i	0.980655	−	0.195746i	$-0.0627127\pi$
$492$	0	0
$493$	470.747	0.954862
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 205.676i	− 0.413834i
$498$	0	0
$499$	609.384	1.22121	0.610605	−	0.791935i	$-0.290926\pi$
$499$	609.384	1.22121	0.610605	+	0.791935i	$0.290926\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	232.130i	0.461491i	0.973014	+	0.230746i	$0.0741165\pi$
$503$	232.130i	0.461491i	−0.973014	+	0.230746i	$0.925883\pi$
$504$	0	0
$505$	240.576	0.476387
$506$	0	0
$507$	0	0
$508$	0	0
$509$	257.655i	0.506198i	0.967440	+	0.253099i	$0.0814499\pi$
$509$	257.655i	0.506198i	−0.967440	+	0.253099i	$0.918550\pi$
$510$	0	0
$511$	331.657	0.649036
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 1001.83i	− 1.94530i
$516$	0	0
$517$	−203.463	−0.393546
$518$	0	0
$519$	0	0
$520$	0	0
$521$	484.088i	0.929152i	0.885533	+	0.464576i	$0.153793\pi$
$521$	484.088i	0.929152i	−0.885533	+	0.464576i	$0.846207\pi$
$522$	0	0
$523$	−644.384	−1.23209	−0.616046	−	0.787711i	$-0.711266\pi$
$523$	−644.384	−1.23209	−0.616046	+	0.787711i	$0.711266\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1177.91i	2.23512i
$528$	0	0
$529$	324.020	0.612515
$530$	0	0
$531$	0	0
$532$	0	0
$533$	669.183i	1.25550i
$534$	0	0
$535$	−261.131	−0.488095
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 17.0542i	− 0.0316405i
$540$	0	0
$541$	−332.302	−0.614237	−0.307118	−	0.951671i	$-0.599365\pi$
$541$	−332.302	−0.614237	−0.307118	+	0.951671i	$0.599365\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 396.015i	− 0.726633i
$546$	0	0
$547$	−314.657	−0.575242	−0.287621	−	0.957744i	$-0.592864\pi$
$547$	−314.657	−0.575242	−0.287621	+	0.957744i	$0.592864\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 329.090i	− 0.597259i
$552$	0	0
$553$	−758.555	−1.37171
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 664.080i	− 1.19224i	−0.802894	−	0.596122i	$-0.796707\pi$
$557$	− 664.080i	− 1.19224i	0.802894	−	0.596122i	$-0.203293\pi$
$558$	0	0
$559$	−742.333	−1.32797
$560$	0	0
$561$	0	0
$562$	0	0
$563$	243.670i	0.432807i	0.976304	+	0.216403i	$0.0694326\pi$
$563$	243.670i	0.432807i	−0.976304	+	0.216403i	$0.930567\pi$
$564$	0	0
$565$	−1220.72	−2.16056
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 579.839i	− 1.01905i	−0.860456	−	0.509524i	$-0.829821\pi$
$569$	− 579.839i	− 1.01905i	0.860456	−	0.509524i	$-0.170179\pi$
$570$	0	0
$571$	−713.686	−1.24989	−0.624944	−	0.780670i	$-0.714878\pi$
$571$	−713.686	−1.24989	−0.624944	+	0.780670i	$0.714878\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	423.728i	0.736918i
$576$	0	0
$577$	829.433	1.43749	0.718746	−	0.695273i	$-0.244717\pi$
$577$	829.433	1.43749	0.718746	+	0.695273i	$0.244717\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	667.889i	1.14955i
$582$	0	0
$583$	62.2204	0.106725
$584$	0	0
$585$	0	0
$586$	0	0
$587$	897.756i	1.52940i	0.644388	+	0.764699i	$0.277112\pi$
$587$	897.756i	1.52940i	−0.644388	+	0.764699i	$0.722888\pi$
$588$	0	0
$589$	823.453	1.39805
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 378.065i	− 0.637547i	−0.947831	−	0.318774i	$-0.896729\pi$
$593$	− 378.065i	− 0.637547i	0.947831	−	0.318774i	$-0.103271\pi$
$594$	0	0
$595$	1264.28	2.12484
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 949.669i	− 1.58542i	−0.609596	−	0.792712i	$-0.708668\pi$
$599$	− 949.669i	− 1.58542i	0.609596	−	0.792712i	$-0.291332\pi$
$600$	0	0
$601$	504.616	0.839628	0.419814	−	0.907610i	$-0.362095\pi$
$601$	504.616	0.839628	0.419814	+	0.907610i	$0.362095\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 617.531i	− 1.02071i
$606$	0	0
$607$	859.908	1.41665	0.708326	−	0.705885i	$-0.249451\pi$
$607$	859.908	1.41665	0.708326	+	0.705885i	$0.249451\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	558.682i	0.914373i
$612$	0	0
$613$	−655.253	−1.06893	−0.534464	−	0.845191i	$-0.679487\pi$
$613$	−655.253	−1.06893	−0.534464	+	0.845191i	$0.679487\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	170.003i	0.275531i	0.990465	+	0.137765i	$0.0439920\pi$
$617$	170.003i	0.275531i	−0.990465	+	0.137765i	$0.956008\pi$
$618$	0	0
$619$	541.061	0.874089	0.437045	−	0.899440i	$-0.356025\pi$
$619$	541.061	0.874089	0.437045	+	0.899440i	$0.356025\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	513.316i	0.823943i
$624$	0	0
$625$	−488.980	−0.782367
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1248.33i	1.98463i
$630$	0	0
$631$	260.788	0.413293	0.206646	−	0.978416i	$-0.433745\pi$
$631$	260.788	0.413293	0.206646	+	0.978416i	$0.433745\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 88.5161i	− 0.139395i
$636$	0	0
$637$	−46.8286	−0.0735142
$638$	0	0
$639$	0	0
$640$	0	0
$641$	675.983i	1.05458i	0.849687	+	0.527288i	$0.176791\pi$
$641$	675.983i	1.05458i	−0.849687	+	0.527288i	$0.823209\pi$
$642$	0	0
$643$	−756.637	−1.17673	−0.588364	−	0.808596i	$-0.700228\pi$
$643$	−756.637	−1.17673	−0.588364	+	0.808596i	$0.700228\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	554.770i	0.857450i	0.903435	+	0.428725i	$0.141037\pi$
$647$	554.770i	0.857450i	−0.903435	+	0.428725i	$0.858963\pi$
$648$	0	0
$649$	−101.000	−0.155624
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 201.494i	− 0.308567i	−0.988027	−	0.154283i	$-0.950693\pi$
$653$	− 201.494i	− 0.308567i	0.988027	−	0.154283i	$-0.0493069\pi$
$654$	0	0
$655$	113.545	0.173351
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 103.591i	− 0.157194i	−0.996906	−	0.0785969i	$-0.974956\pi$
$659$	− 103.591i	− 0.157194i	0.996906	−	0.0785969i	$-0.0250440\pi$
$660$	0	0
$661$	218.414	0.330430	0.165215	−	0.986258i	$-0.447168\pi$
$661$	218.414	0.330430	0.165215	+	0.986258i	$0.447168\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 883.834i	− 1.32907i
$666$	0	0
$667$	267.767	0.401450
$668$	0	0
$669$	0	0
$670$	0	0
$671$	129.705i	0.193301i
$672$	0	0
$673$	788.857	1.17215	0.586075	−	0.810257i	$-0.300672\pi$
$673$	788.857	1.17215	0.586075	+	0.810257i	$0.300672\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 732.715i	− 1.08230i	−0.840927	−	0.541149i	$-0.817990\pi$
$677$	− 732.715i	− 1.08230i	0.840927	−	0.541149i	$-0.182010\pi$
$678$	0	0
$679$	955.765	1.40761
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 1353.33i	− 1.98145i	−0.135883	−	0.990725i	$-0.543387\pi$
$683$	− 1353.33i	− 1.98145i	0.135883	−	0.990725i	$-0.456613\pi$
$684$	0	0
$685$	−407.080	−0.594277
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 170.849i	− 0.247966i
$690$	0	0
$691$	−736.514	−1.06587	−0.532934	−	0.846157i	$-0.678910\pi$
$691$	−736.514	−1.06587	−0.532934	+	0.846157i	$0.678910\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 746.129i	− 1.07357i
$696$	0	0
$697$	1002.71	1.43860
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 1068.34i	− 1.52403i	−0.647561	−	0.762014i	$-0.724211\pi$
$701$	− 1068.34i	− 1.52403i	0.647561	−	0.762014i	$-0.275789\pi$
$702$	0	0
$703$	872.686	1.24137
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 221.335i	− 0.313062i
$708$	0	0
$709$	273.888	0.386301	0.193151	−	0.981169i	$-0.438129\pi$
$709$	273.888	0.386301	0.193151	+	0.981169i	$0.438129\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	670.011i	0.939707i
$714$	0	0
$715$	759.302	1.06196
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 654.423i	− 0.910185i	−0.890444	−	0.455092i	$-0.849606\pi$
$719$	− 654.423i	− 0.910185i	0.890444	−	0.455092i	$-0.150394\pi$
$720$	0	0
$721$	−921.706	−1.27837
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 553.521i	− 0.763477i
$726$	0	0
$727$	1166.33	1.60431	0.802155	−	0.597117i	$-0.203687\pi$
$727$	1166.33	1.60431	0.802155	+	0.597117i	$0.203687\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1112.31i	1.52163i
$732$	0	0
$733$	878.292	1.19822	0.599108	−	0.800668i	$-0.295522\pi$
$733$	878.292	1.19822	0.599108	+	0.800668i	$0.295522\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 532.103i	− 0.721984i
$738$	0	0
$739$	−593.151	−0.802640	−0.401320	−	0.915938i	$-0.631448\pi$
$739$	−593.151	−0.802640	−0.401320	+	0.915938i	$0.631448\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	54.3357i	0.0731302i	0.999331	+	0.0365651i	$0.0116416\pi$
$743$	54.3357i	0.0731302i	−0.999331	+	0.0365651i	$0.988358\pi$
$744$	0	0
$745$	−1598.80	−2.14603
$746$	0	0
$747$	0	0
$748$	0	0
$749$	240.246i	0.320756i
$750$	0	0
$751$	911.141	1.21324	0.606618	−	0.794993i	$-0.292526\pi$
$751$	911.141	1.21324	0.606618	+	0.794993i	$0.292526\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1134.83i	1.50309i
$756$	0	0
$757$	1272.22	1.68061	0.840304	−	0.542115i	$-0.182376\pi$
$757$	1272.22	1.68061	0.840304	+	0.542115i	$0.182376\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 461.279i	− 0.606149i	−0.952967	−	0.303074i	$-0.901987\pi$
$761$	− 461.279i	− 0.606149i	0.952967	−	0.303074i	$-0.0980131\pi$
$762$	0	0
$763$	−364.343	−0.477514
$764$	0	0
$765$	0	0
$766$	0	0
$767$	277.332i	0.361580i
$768$	0	0
$769$	538.878	0.700751	0.350376	−	0.936609i	$-0.386054\pi$
$769$	538.878	0.700751	0.350376	+	0.936609i	$0.386054\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 1036.29i	− 1.34061i	−0.742087	−	0.670304i	$-0.766164\pi$
$773$	− 1036.29i	− 1.34061i	0.742087	−	0.670304i	$-0.233836\pi$
$774$	0	0
$775$	1385.03	1.78713
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 700.972i	− 0.899835i
$780$	0	0
$781$	−185.090	−0.236991
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 605.664i	− 0.771546i
$786$	0	0
$787$	−1412.19	−1.79440	−0.897199	−	0.441626i	$-0.854402\pi$
$787$	−1412.19	−1.79440	−0.897199	+	0.441626i	$0.854402\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1123.09i	1.41983i
$792$	0	0
$793$	356.151	0.449119
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 69.8134i	− 0.0875953i	−0.999040	−	0.0437976i	$-0.986054\pi$
$797$	− 69.8134i	− 0.0875953i	0.999040	−	0.0437976i	$-0.0139457\pi$
$798$	0	0
$799$	837.131	1.04772
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 298.462i	− 0.371683i
$804$	0	0
$805$	719.141	0.893343
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1263.33i	1.56160i	0.624781	+	0.780800i	$0.285188\pi$
$809$	1263.33i	1.56160i	−0.624781	+	0.780800i	$0.714812\pi$
$810$	0	0
$811$	−442.241	−0.545303	−0.272652	−	0.962113i	$-0.587901\pi$
$811$	−442.241	−0.545303	−0.272652	+	0.962113i	$0.587901\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 407.958i	− 0.500562i
$816$	0	0
$817$	777.596	0.951770
$818$	0	0
$819$	0	0
$820$	0	0
$821$	575.129i	0.700523i	0.936652	+	0.350261i	$0.113907\pi$
$821$	575.129i	0.700523i	−0.936652	+	0.350261i	$0.886093\pi$
$822$	0	0
$823$	−23.2429	−0.0282416	−0.0141208	−	0.999900i	$-0.504495\pi$
$823$	−23.2429	−0.0282416	−0.0141208	+	0.999900i	$0.504495\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	790.958i	0.956418i	0.878246	+	0.478209i	$0.158714\pi$
$827$	790.958i	0.956418i	−0.878246	+	0.478209i	$0.841286\pi$
$828$	0	0
$829$	1159.78	1.39901	0.699503	−	0.714630i	$-0.253405\pi$
$829$	1159.78	1.39901	0.699503	+	0.714630i	$0.253405\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	70.1681i	0.0842354i
$834$	0	0
$835$	1141.60	1.36719
$836$	0	0
$837$	0	0
$838$	0	0
$839$	546.717i	0.651629i	0.945434	+	0.325814i	$0.105638\pi$
$839$	546.717i	0.651629i	−0.945434	+	0.325814i	$0.894362\pi$
$840$	0	0
$841$	491.212	0.584081
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 836.213i	− 0.989601i
$846$	0	0
$847$	−568.143	−0.670771
$848$	0	0
$849$	0	0
$850$	0	0
$851$	710.070i	0.834395i
$852$	0	0
$853$	216.635	0.253968	0.126984	−	0.991905i	$-0.459470\pi$
$853$	216.635	0.253968	0.126984	+	0.991905i	$0.459470\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	565.504i	0.659865i	0.944005	+	0.329932i	$0.107026\pi$
$857$	565.504i	0.659865i	−0.944005	+	0.329932i	$0.892974\pi$
$858$	0	0
$859$	375.767	0.437447	0.218724	−	0.975787i	$-0.429811\pi$
$859$	375.767	0.437447	0.218724	+	0.975787i	$0.429811\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1429.35i	1.65626i	0.560534	+	0.828131i	$0.310596\pi$
$863$	1429.35i	1.65626i	−0.560534	+	0.828131i	$0.689404\pi$
$864$	0	0
$865$	194.555	0.224919
$866$	0	0
$867$	0	0
$868$	0	0
$869$	682.632i	0.785537i
$870$	0	0
$871$	−1461.08	−1.67747
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 230.851i	− 0.263829i
$876$	0	0
$877$	−841.627	−0.959665	−0.479833	−	0.877360i	$-0.659303\pi$
$877$	−841.627	−0.959665	−0.479833	+	0.877360i	$0.659303\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 449.261i	− 0.509944i	−0.966948	−	0.254972i	$-0.917934\pi$
$881$	− 449.261i	− 0.509944i	0.966948	−	0.254972i	$-0.0820663\pi$
$882$	0	0
$883$	122.445	0.138669	0.0693346	−	0.997593i	$-0.477912\pi$
$883$	122.445	0.138669	0.0693346	+	0.997593i	$0.477912\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1571.66i	1.77188i	0.463800	+	0.885940i	$0.346485\pi$
$887$	1571.66i	1.77188i	−0.463800	+	0.885940i	$0.653515\pi$
$888$	0	0
$889$	−81.4368	−0.0916049
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 585.221i	− 0.655343i
$894$	0	0
$895$	−1970.62	−2.20182
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 875.244i	− 0.973575i
$900$	0	0
$901$	−256.000	−0.284129
$902$	0	0
$903$	0	0
$904$	0	0
$905$	431.542i	0.476842i
$906$	0	0
$907$	−698.576	−0.770205	−0.385102	−	0.922874i	$-0.625834\pi$
$907$	−698.576	−0.770205	−0.385102	+	0.922874i	$0.625834\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 81.3934i	− 0.0893451i	−0.999002	−	0.0446726i	$-0.985776\pi$
$911$	− 81.3934i	− 0.0893451i	0.999002	−	0.0446726i	$-0.0142245\pi$
$912$	0	0
$913$	601.041	0.658314
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 104.464i	− 0.113919i
$918$	0	0
$919$	−348.665	−0.379396	−0.189698	−	0.981842i	$-0.560751\pi$
$919$	−348.665	−0.379396	−0.189698	+	0.981842i	$0.560751\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	508.231i	0.550629i
$924$	0	0
$925$	1467.84	1.58685
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 1076.52i	− 1.15880i	−0.815043	−	0.579400i	$-0.803287\pi$
$929$	− 1076.52i	− 1.15880i	0.815043	−	0.579400i	$-0.196713\pi$
$930$	0	0
$931$	49.0531	0.0526886
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 1137.74i	− 1.21683i
$936$	0	0
$937$	−1437.39	−1.53404	−0.767018	−	0.641626i	$-0.778260\pi$
$937$	−1437.39	−1.53404	−0.767018	+	0.641626i	$0.778260\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	215.457i	0.228965i	0.993425	+	0.114483i	$0.0365211\pi$
$941$	215.457i	0.228965i	−0.993425	+	0.114483i	$0.963479\pi$
$942$	0	0
$943$	570.353	0.604828
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 470.715i	− 0.497059i	−0.968624	−	0.248529i	$-0.920053\pi$
$947$	− 470.715i	− 0.497059i	0.968624	−	0.248529i	$-0.0799473\pi$
$948$	0	0
$949$	−819.535	−0.863577
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1192.14i	1.25093i	0.780251	+	0.625466i	$0.215091\pi$
$953$	1192.14i	1.25093i	−0.780251	+	0.625466i	$0.784909\pi$
$954$	0	0
$955$	−849.445	−0.889471
$956$	0	0
$957$	0	0
$958$	0	0
$959$	374.522i	0.390534i
$960$	0	0
$961$	1229.05	1.27893
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 1603.24i	− 1.66139i
$966$	0	0
$967$	−192.778	−0.199356	−0.0996782	−	0.995020i	$-0.531781\pi$
$967$	−192.778	−0.199356	−0.0996782	+	0.995020i	$0.531781\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	202.388i	0.208433i	0.994555	+	0.104216i	$0.0332335\pi$
$971$	202.388i	0.208433i	−0.994555	+	0.104216i	$0.966767\pi$
$972$	0	0
$973$	−686.455	−0.705504
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 40.6431i	− 0.0415999i	−0.999784	−	0.0208000i	$-0.993379\pi$
$977$	− 40.6431i	− 0.0415999i	0.999784	−	0.0208000i	$-0.00662131\pi$
$978$	0	0
$979$	461.939	0.471848
$980$	0	0
$981$	0	0
$982$	0	0
$983$	728.737i	0.741340i	0.928765	+	0.370670i	$0.120872\pi$
$983$	728.737i	0.741340i	−0.928765	+	0.370670i	$0.879128\pi$
$984$	0	0
$985$	−1264.28	−1.28353
$986$	0	0
$987$	0	0
$988$	0	0
$989$	632.699i	0.639736i
$990$	0	0
$991$	746.527	0.753306	0.376653	−	0.926354i	$-0.377075\pi$
$991$	746.527	0.753306	0.376653	+	0.926354i	$0.377075\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	458.112i	0.460414i
$996$	0	0
$997$	238.092	0.238808	0.119404	−	0.992846i	$-0.461902\pi$
$997$	238.092	0.238808	0.119404	+	0.992846i	$0.461902\pi$
$998$	0	0
$999$	0	0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.3.e.b.161.1		4
3.2	odd	2	inner	648.3.e.b.161.4		4
4.3	odd	2		1296.3.e.c.161.1		4
9.2	odd	6		216.3.m.a.17.1		4
9.4	even	3		216.3.m.a.89.1		4
9.5	odd	6		72.3.m.a.65.2	yes	4
9.7	even	3		72.3.m.a.41.2	✓	4
12.11	even	2		1296.3.e.c.161.4		4
36.7	odd	6		144.3.q.d.113.2		4
36.11	even	6		432.3.q.c.17.1		4
36.23	even	6		144.3.q.d.65.2		4
36.31	odd	6		432.3.q.c.305.1		4
72.5	odd	6		576.3.q.h.65.1		4
72.11	even	6		1728.3.q.f.449.2		4
72.13	even	6		1728.3.q.e.1601.2		4
72.29	odd	6		1728.3.q.e.449.2		4
72.43	odd	6		576.3.q.c.257.1		4
72.59	even	6		576.3.q.c.65.1		4
72.61	even	6		576.3.q.h.257.1		4
72.67	odd	6		1728.3.q.f.1601.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.3.m.a.41.2	✓	4	9.7	even	3
72.3.m.a.65.2	yes	4	9.5	odd	6
144.3.q.d.65.2		4	36.23	even	6
144.3.q.d.113.2		4	36.7	odd	6
216.3.m.a.17.1		4	9.2	odd	6
216.3.m.a.89.1		4	9.4	even	3
432.3.q.c.17.1		4	36.11	even	6
432.3.q.c.305.1		4	36.31	odd	6
576.3.q.c.65.1		4	72.59	even	6
576.3.q.c.257.1		4	72.43	odd	6
576.3.q.h.65.1		4	72.5	odd	6
576.3.q.h.257.1		4	72.61	even	6
648.3.e.b.161.1		4	1.1	even	1	trivial
648.3.e.b.161.4		4	3.2	odd	2	inner
1296.3.e.c.161.1		4	4.3	odd	2
1296.3.e.c.161.4		4	12.11	even	2
1728.3.q.e.449.2		4	72.29	odd	6
1728.3.q.e.1601.2		4	72.13	even	6
1728.3.q.f.449.2		4	72.11	even	6
1728.3.q.f.1601.2		4	72.67	odd	6

Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Introduction

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

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Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	648.3.e.b.161.1

Level	$648$
Weight	$3$
Character	648.161
Analytic conductor	$17.657$
Analytic rank	$0$
Dimension	$4$
Inner twists	$2$