LMFDB - Embedded newform 576.3.h.b.161.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [576,3,Mod(161,576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(576, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("576.161");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 576 = 2^{6} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	576.h (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:	$15.6948632272$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.3
Root			$0.965926 + 0.258819i$ of defining polynomial
Character	$\chi$	$=$	576.161
Dual form			576.3.h.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)

$f(q)$	$=$	$q-7.34847 q^{5} +10.3923 q^{7} +O(q^{10})$	$q-7.34847 q^{5} +10.3923 q^{7} -8.48528 q^{11} -10.3923i q^{13} +21.2132i q^{17} +20.0000i q^{19} -14.6969i q^{23} +29.0000 q^{25} +36.7423 q^{29} -51.9615 q^{31} -76.3675 q^{35} +41.5692i q^{37} +72.1249i q^{41} +40.0000i q^{43} +73.4847i q^{47} +59.0000 q^{49} -36.7423 q^{53} +62.3538 q^{55} -33.9411 q^{59} +76.3675i q^{65} +100.000i q^{67} -73.4847i q^{71} +20.0000 q^{73} -88.1816 q^{77} +51.9615 q^{79} -127.279 q^{83} -155.885i q^{85} +12.7279i q^{89} -108.000i q^{91} -146.969i q^{95} +40.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 232 q^{25} + 472 q^{49} + 160 q^{73} + 320 q^{97}+O(q^{100}) $ 8 * q + 232 * q^25 + 472 * q^49 + 160 * q^73 + 320 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/576\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$325$
$\chi(n)$	$-1$	$1$	$-1$

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
$5$	−7.34847	−1.46969	−0.734847	−	0.678233i	$-0.762746\pi$
$5$	−7.34847	−1.46969	−0.734847	+	0.678233i	$0.762746\pi$
$6$	0	0
$7$	10.3923	1.48461	0.742307	−	0.670059i	$-0.233731\pi$
$7$	10.3923	1.48461	0.742307	+	0.670059i	$0.233731\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−8.48528	−0.771389	−0.385695	−	0.922627i	$-0.626038\pi$
$11$	−8.48528	−0.771389	−0.385695	+	0.922627i	$0.626038\pi$
$12$	0	0
$13$	− 10.3923i	− 0.799408i	−0.916644	−	0.399704i	$-0.869113\pi$
$13$	− 10.3923i	− 0.799408i	0.916644	−	0.399704i	$-0.130887\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	21.2132i	1.24784i	0.781490	+	0.623918i	$0.214460\pi$
$17$	21.2132i	1.24784i	−0.781490	+	0.623918i	$0.785540\pi$
$18$	0	0
$19$	20.0000i	1.05263i	0.850289	+	0.526316i	$0.176427\pi$
$19$	20.0000i	1.05263i	−0.850289	+	0.526316i	$0.823573\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 14.6969i	− 0.638997i	−0.947587	−	0.319499i	$-0.896486\pi$
$23$	− 14.6969i	− 0.638997i	0.947587	−	0.319499i	$-0.103514\pi$
$24$	0	0
$25$	29.0000	1.16000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	36.7423	1.26698	0.633489	−	0.773752i	$-0.281622\pi$
$29$	36.7423	1.26698	0.633489	+	0.773752i	$0.281622\pi$
$30$	0	0
$31$	−51.9615	−1.67618	−0.838089	−	0.545533i	$-0.816327\pi$
$31$	−51.9615	−1.67618	−0.838089	+	0.545533i	$0.816327\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−76.3675	−2.18193
$36$	0	0
$37$	41.5692i	1.12349i	0.827310	+	0.561746i	$0.189870\pi$
$37$	41.5692i	1.12349i	−0.827310	+	0.561746i	$0.810130\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	72.1249i	1.75914i	0.475766	+	0.879572i	$0.342171\pi$
$41$	72.1249i	1.75914i	−0.475766	+	0.879572i	$0.657829\pi$
$42$	0	0
$43$	40.0000i	0.930233i	0.885250	+	0.465116i	$0.153987\pi$
$43$	40.0000i	0.930233i	−0.885250	+	0.465116i	$0.846013\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	73.4847i	1.56350i	0.623589	+	0.781752i	$0.285674\pi$
$47$	73.4847i	1.56350i	−0.623589	+	0.781752i	$0.714326\pi$
$48$	0	0
$49$	59.0000	1.20408
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−36.7423	−0.693252	−0.346626	−	0.938003i	$-0.612673\pi$
$53$	−36.7423	−0.693252	−0.346626	+	0.938003i	$0.612673\pi$
$54$	0	0
$55$	62.3538	1.13371
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−33.9411	−0.575273	−0.287637	−	0.957740i	$-0.592870\pi$
$59$	−33.9411	−0.575273	−0.287637	+	0.957740i	$0.592870\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	76.3675i	1.17489i
$66$	0	0
$67$	100.000i	1.49254i	0.665645	+	0.746269i	$0.268157\pi$
$67$	100.000i	1.49254i	−0.665645	+	0.746269i	$0.731843\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 73.4847i	− 1.03500i	−0.855685	−	0.517498i	$-0.826864\pi$
$71$	− 73.4847i	− 1.03500i	0.855685	−	0.517498i	$-0.173136\pi$
$72$	0	0
$73$	20.0000	0.273973	0.136986	−	0.990573i	$-0.456258\pi$
$73$	20.0000	0.273973	0.136986	+	0.990573i	$0.456258\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−88.1816	−1.14522
$78$	0	0
$79$	51.9615	0.657741	0.328870	−	0.944375i	$-0.393332\pi$
$79$	51.9615	0.657741	0.328870	+	0.944375i	$0.393332\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−127.279	−1.53348	−0.766742	−	0.641955i	$-0.778124\pi$
$83$	−127.279	−1.53348	−0.766742	+	0.641955i	$0.778124\pi$
$84$	0	0
$85$	− 155.885i	− 1.83394i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	12.7279i	0.143010i	0.997440	+	0.0715052i	$0.0227802\pi$
$89$	12.7279i	0.143010i	−0.997440	+	0.0715052i	$0.977220\pi$
$90$	0	0
$91$	− 108.000i	− 1.18681i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 146.969i	− 1.54705i
$96$	0	0
$97$	40.0000	0.412371	0.206186	−	0.978513i	$-0.433895\pi$
$97$	40.0000	0.412371	0.206186	+	0.978513i	$0.433895\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	183.712	1.81893	0.909464	−	0.415783i	$-0.136492\pi$
$101$	183.712	1.81893	0.909464	+	0.415783i	$0.136492\pi$
$102$	0	0
$103$	−93.5307	−0.908065	−0.454033	−	0.890985i	$-0.650015\pi$
$103$	−93.5307	−0.908065	−0.454033	+	0.890985i	$0.650015\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	169.706	1.58603	0.793017	−	0.609200i	$-0.208509\pi$
$107$	169.706	1.58603	0.793017	+	0.609200i	$0.208509\pi$
$108$	0	0
$109$	51.9615i	0.476711i	0.971178	+	0.238356i	$0.0766084\pi$
$109$	51.9615i	0.476711i	−0.971178	+	0.238356i	$0.923392\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 63.6396i	− 0.563182i	−0.959534	−	0.281591i	$-0.909138\pi$
$113$	− 63.6396i	− 0.563182i	0.959534	−	0.281591i	$-0.0908622\pi$
$114$	0	0
$115$	108.000i	0.939130i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	220.454i	1.85256i
$120$	0	0
$121$	−49.0000	−0.404959
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−29.3939	−0.235151
$126$	0	0
$127$	−10.3923	−0.0818292	−0.0409146	−	0.999163i	$-0.513027\pi$
$127$	−10.3923	−0.0818292	−0.0409146	+	0.999163i	$0.513027\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−50.9117	−0.388639	−0.194319	−	0.980938i	$-0.562250\pi$
$131$	−50.9117	−0.388639	−0.194319	+	0.980938i	$0.562250\pi$
$132$	0	0
$133$	207.846i	1.56275i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 106.066i	− 0.774205i	−0.922037	−	0.387102i	$-0.873476\pi$
$137$	− 106.066i	− 0.774205i	0.922037	−	0.387102i	$-0.126524\pi$
$138$	0	0
$139$	− 172.000i	− 1.23741i	−0.785623	−	0.618705i	$-0.787658\pi$
$139$	− 172.000i	− 1.23741i	0.785623	−	0.618705i	$-0.212342\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	88.1816i	0.616655i
$144$	0	0
$145$	−270.000	−1.86207
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−110.227	−0.739779	−0.369889	−	0.929076i	$-0.620604\pi$
$149$	−110.227	−0.739779	−0.369889	+	0.929076i	$0.620604\pi$
$150$	0	0
$151$	−155.885	−1.03235	−0.516174	−	0.856484i	$-0.672644\pi$
$151$	−155.885	−1.03235	−0.516174	+	0.856484i	$0.672644\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	381.838	2.46347
$156$	0	0
$157$	166.277i	1.05909i	0.848282	+	0.529544i	$0.177637\pi$
$157$	166.277i	1.05909i	−0.848282	+	0.529544i	$0.822363\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 152.735i	− 0.948665i
$162$	0	0
$163$	− 160.000i	− 0.981595i	−0.871274	−	0.490798i	$-0.836705\pi$
$163$	− 160.000i	− 0.981595i	0.871274	−	0.490798i	$-0.163295\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	58.7878i	0.352022i	0.984388	+	0.176011i	$0.0563195\pi$
$167$	58.7878i	0.352022i	−0.984388	+	0.176011i	$0.943680\pi$
$168$	0	0
$169$	61.0000	0.360947
$170$	0	0
$171$	0	0
$172$	0	0
$173$	36.7423	0.212384	0.106192	−	0.994346i	$-0.466134\pi$
$173$	36.7423	0.212384	0.106192	+	0.994346i	$0.466134\pi$
$174$	0	0
$175$	301.377	1.72215
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−50.9117	−0.284423	−0.142211	−	0.989836i	$-0.545421\pi$
$179$	−50.9117	−0.284423	−0.142211	+	0.989836i	$0.545421\pi$
$180$	0	0
$181$	− 259.808i	− 1.43540i	−0.696352	−	0.717701i	$-0.745195\pi$
$181$	− 259.808i	− 1.43540i	0.696352	−	0.717701i	$-0.254805\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 305.470i	− 1.65119i
$186$	0	0
$187$	− 180.000i	− 0.962567i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 293.939i	− 1.53895i	−0.638679	−	0.769473i	$-0.720519\pi$
$191$	− 293.939i	− 1.53895i	0.638679	−	0.769473i	$-0.279481\pi$
$192$	0	0
$193$	−10.0000	−0.0518135	−0.0259067	−	0.999664i	$-0.508247\pi$
$193$	−10.0000	−0.0518135	−0.0259067	+	0.999664i	$0.508247\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−22.0454	−0.111906	−0.0559528	−	0.998433i	$-0.517820\pi$
$197$	−22.0454	−0.111906	−0.0559528	+	0.998433i	$0.517820\pi$
$198$	0	0
$199$	−51.9615	−0.261113	−0.130557	−	0.991441i	$-0.541676\pi$
$199$	−51.9615	−0.261113	−0.130557	+	0.991441i	$0.541676\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	381.838	1.88097
$204$	0	0
$205$	− 530.008i	− 2.58540i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 169.706i	− 0.811989i
$210$	0	0
$211$	172.000i	0.815166i	0.913168	+	0.407583i	$0.133628\pi$
$211$	172.000i	0.815166i	−0.913168	+	0.407583i	$0.866372\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 293.939i	− 1.36716i
$216$	0	0
$217$	−540.000	−2.48848
$218$	0	0
$219$	0	0
$220$	0	0
$221$	220.454	0.997530
$222$	0	0
$223$	−10.3923	−0.0466023	−0.0233011	−	0.999728i	$-0.507418\pi$
$223$	−10.3923	−0.0466023	−0.0233011	+	0.999728i	$0.507418\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−42.4264	−0.186900	−0.0934502	−	0.995624i	$-0.529790\pi$
$227$	−42.4264	−0.186900	−0.0934502	+	0.995624i	$0.529790\pi$
$228$	0	0
$229$	259.808i	1.13453i	0.823535	+	0.567266i	$0.191999\pi$
$229$	259.808i	1.13453i	−0.823535	+	0.567266i	$0.808001\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	106.066i	0.455219i	0.973752	+	0.227609i	$0.0730910\pi$
$233$	106.066i	0.455219i	−0.973752	+	0.227609i	$0.926909\pi$
$234$	0	0
$235$	− 540.000i	− 2.29787i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	146.969i	0.614935i	0.951559	+	0.307467i	$0.0994815\pi$
$239$	146.969i	0.614935i	−0.951559	+	0.307467i	$0.900519\pi$
$240$	0	0
$241$	140.000	0.580913	0.290456	−	0.956888i	$-0.406193\pi$
$241$	140.000	0.580913	0.290456	+	0.956888i	$0.406193\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−433.560	−1.76963
$246$	0	0
$247$	207.846	0.841482
$248$	0	0
$249$	0	0
$250$	0	0
$251$	246.073	0.980371	0.490186	−	0.871618i	$-0.336929\pi$
$251$	246.073	0.980371	0.490186	+	0.871618i	$0.336929\pi$
$252$	0	0
$253$	124.708i	0.492916i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 21.2132i	− 0.0825416i	−0.999148	−	0.0412708i	$-0.986859\pi$
$257$	− 21.2132i	− 0.0825416i	0.999148	−	0.0412708i	$-0.0131406\pi$
$258$	0	0
$259$	432.000i	1.66795i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	88.1816i	0.335291i	0.985847	+	0.167646i	$0.0536165\pi$
$263$	88.1816i	0.335291i	−0.985847	+	0.167646i	$0.946384\pi$
$264$	0	0
$265$	270.000	1.01887
$266$	0	0
$267$	0	0
$268$	0	0
$269$	110.227	0.409766	0.204883	−	0.978786i	$-0.434319\pi$
$269$	110.227	0.409766	0.204883	+	0.978786i	$0.434319\pi$
$270$	0	0
$271$	−467.654	−1.72566	−0.862830	−	0.505495i	$-0.831310\pi$
$271$	−467.654	−1.72566	−0.862830	+	0.505495i	$0.831310\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−246.073	−0.894811
$276$	0	0
$277$	322.161i	1.16304i	0.813533	+	0.581519i	$0.197541\pi$
$277$	322.161i	1.16304i	−0.813533	+	0.581519i	$0.802459\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 140.007i	− 0.498246i	−0.968472	−	0.249123i	$-0.919858\pi$
$281$	− 140.007i	− 0.498246i	0.968472	−	0.249123i	$-0.0801424\pi$
$282$	0	0
$283$	80.0000i	0.282686i	0.989961	+	0.141343i	$0.0451420\pi$
$283$	80.0000i	0.282686i	−0.989961	+	0.141343i	$0.954858\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	749.544i	2.61165i
$288$	0	0
$289$	−161.000	−0.557093
$290$	0	0
$291$	0	0
$292$	0	0
$293$	315.984	1.07844	0.539222	−	0.842164i	$-0.318718\pi$
$293$	315.984	1.07844	0.539222	+	0.842164i	$0.318718\pi$
$294$	0	0
$295$	249.415	0.845476
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−152.735	−0.510820
$300$	0	0
$301$	415.692i	1.38104i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 380.000i	− 1.23779i	−0.785476	−	0.618893i	$-0.787582\pi$
$307$	− 380.000i	− 1.23779i	0.785476	−	0.618893i	$-0.212418\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	146.969i	0.472570i	0.971684	+	0.236285i	$0.0759300\pi$
$311$	146.969i	0.472570i	−0.971684	+	0.236285i	$0.924070\pi$
$312$	0	0
$313$	310.000	0.990415	0.495208	−	0.868775i	$-0.335092\pi$
$313$	310.000	0.990415	0.495208	+	0.868775i	$0.335092\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−242.499	−0.764983	−0.382491	−	0.923959i	$-0.624934\pi$
$317$	−242.499	−0.764983	−0.382491	+	0.923959i	$0.624934\pi$
$318$	0	0
$319$	−311.769	−0.977333
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−424.264	−1.31351
$324$	0	0
$325$	− 301.377i	− 0.927313i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	763.675i	2.32120i
$330$	0	0
$331$	500.000i	1.51057i	0.655394	+	0.755287i	$0.272503\pi$
$331$	500.000i	1.51057i	−0.655394	+	0.755287i	$0.727497\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 734.847i	− 2.19357i
$336$	0	0
$337$	100.000	0.296736	0.148368	−	0.988932i	$-0.452598\pi$
$337$	100.000	0.296736	0.148368	+	0.988932i	$0.452598\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	440.908	1.29299
$342$	0	0
$343$	103.923	0.302983
$344$	0	0
$345$	0	0
$346$	0	0
$347$	42.4264	0.122266	0.0611332	−	0.998130i	$-0.480529\pi$
$347$	42.4264	0.122266	0.0611332	+	0.998130i	$0.480529\pi$
$348$	0	0
$349$	− 207.846i	− 0.595548i	−0.954636	−	0.297774i	$-0.903756\pi$
$349$	− 207.846i	− 0.595548i	0.954636	−	0.297774i	$-0.0962441\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	615.183i	1.74273i	0.490637	+	0.871364i	$0.336764\pi$
$353$	615.183i	1.74273i	−0.490637	+	0.871364i	$0.663236\pi$
$354$	0	0
$355$	540.000i	1.52113i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 514.393i	− 1.43285i	−0.697665	−	0.716425i	$-0.745777\pi$
$359$	− 514.393i	− 1.43285i	0.697665	−	0.716425i	$-0.254223\pi$
$360$	0	0
$361$	−39.0000	−0.108033
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−146.969	−0.402656
$366$	0	0
$367$	−218.238	−0.594655	−0.297328	−	0.954776i	$-0.596095\pi$
$367$	−218.238	−0.594655	−0.297328	+	0.954776i	$0.596095\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−381.838	−1.02921
$372$	0	0
$373$	− 665.108i	− 1.78313i	−0.452893	−	0.891565i	$-0.649608\pi$
$373$	− 665.108i	− 1.78313i	0.452893	−	0.891565i	$-0.350392\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 381.838i	− 1.01283i
$378$	0	0
$379$	− 92.0000i	− 0.242744i	−0.992607	−	0.121372i	$-0.961271\pi$
$379$	− 92.0000i	− 0.242744i	0.992607	−	0.121372i	$-0.0387294\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	146.969i	0.383732i	0.981421	+	0.191866i	$0.0614539\pi$
$383$	146.969i	0.383732i	−0.981421	+	0.191866i	$0.938546\pi$
$384$	0	0
$385$	648.000	1.68312
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−257.196	−0.661173	−0.330587	−	0.943776i	$-0.607247\pi$
$389$	−257.196	−0.661173	−0.330587	+	0.943776i	$0.607247\pi$
$390$	0	0
$391$	311.769	0.797364
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−381.838	−0.966678
$396$	0	0
$397$	41.5692i	0.104708i	0.998629	+	0.0523542i	$0.0166725\pi$
$397$	41.5692i	0.104708i	−0.998629	+	0.0523542i	$0.983328\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	224.860i	0.560748i	0.959891	+	0.280374i	$0.0904585\pi$
$401$	224.860i	0.560748i	−0.959891	+	0.280374i	$0.909542\pi$
$402$	0	0
$403$	540.000i	1.33995i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 352.727i	− 0.866650i
$408$	0	0
$409$	368.000	0.899756	0.449878	−	0.893090i	$-0.351468\pi$
$409$	368.000	0.899756	0.449878	+	0.893090i	$0.351468\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−352.727	−0.854059
$414$	0	0
$415$	935.307	2.25375
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−432.749	−1.03281	−0.516407	−	0.856343i	$-0.672731\pi$
$419$	−432.749	−1.03281	−0.516407	+	0.856343i	$0.672731\pi$
$420$	0	0
$421$	− 51.9615i	− 0.123424i	−0.998094	−	0.0617120i	$-0.980344\pi$
$421$	− 51.9615i	− 0.123424i	0.998094	−	0.0617120i	$-0.0196560\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	615.183i	1.44749i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	367.423i	0.852491i	0.904608	+	0.426245i	$0.140164\pi$
$431$	367.423i	0.852491i	−0.904608	+	0.426245i	$0.859836\pi$
$432$	0	0
$433$	−470.000	−1.08545	−0.542725	−	0.839910i	$-0.682607\pi$
$433$	−470.000	−1.08545	−0.542725	+	0.839910i	$0.682607\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	293.939	0.672629
$438$	0	0
$439$	−155.885	−0.355090	−0.177545	−	0.984113i	$-0.556816\pi$
$439$	−155.885	−0.355090	−0.177545	+	0.984113i	$0.556816\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−551.543	−1.24502	−0.622509	−	0.782612i	$-0.713887\pi$
$443$	−551.543	−1.24502	−0.622509	+	0.782612i	$0.713887\pi$
$444$	0	0
$445$	− 93.5307i	− 0.210181i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	691.550i	1.54020i	0.637922	+	0.770101i	$0.279794\pi$
$449$	691.550i	1.54020i	−0.637922	+	0.770101i	$0.720206\pi$
$450$	0	0
$451$	− 612.000i	− 1.35698i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	793.635i	1.74425i
$456$	0	0
$457$	680.000	1.48796	0.743982	−	0.668199i	$-0.232935\pi$
$457$	680.000	1.48796	0.743982	+	0.668199i	$0.232935\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−330.681	−0.717313	−0.358656	−	0.933470i	$-0.616765\pi$
$461$	−330.681	−0.717313	−0.358656	+	0.933470i	$0.616765\pi$
$462$	0	0
$463$	−322.161	−0.695813	−0.347907	−	0.937529i	$-0.613107\pi$
$463$	−322.161	−0.695813	−0.347907	+	0.937529i	$0.613107\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	466.690	0.999337	0.499669	−	0.866217i	$-0.333455\pi$
$467$	466.690	0.999337	0.499669	+	0.866217i	$0.333455\pi$
$468$	0	0
$469$	1039.23i	2.21584i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 339.411i	− 0.717571i
$474$	0	0
$475$	580.000i	1.22105i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	220.454i	0.460238i	0.973162	+	0.230119i	$0.0739116\pi$
$479$	220.454i	0.460238i	−0.973162	+	0.230119i	$0.926088\pi$
$480$	0	0
$481$	432.000	0.898129
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−293.939	−0.606059
$486$	0	0
$487$	−633.931	−1.30171	−0.650853	−	0.759204i	$-0.725588\pi$
$487$	−633.931	−1.30171	−0.650853	+	0.759204i	$0.725588\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−220.617	−0.449322	−0.224661	−	0.974437i	$-0.572128\pi$
$491$	−220.617	−0.449322	−0.224661	+	0.974437i	$0.572128\pi$
$492$	0	0
$493$	779.423i	1.58098i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 763.675i	− 1.53657i
$498$	0	0
$499$	640.000i	1.28257i	0.767305	+	0.641283i	$0.221597\pi$
$499$	640.000i	1.28257i	−0.767305	+	0.641283i	$0.778403\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	73.4847i	0.146093i	0.997329	+	0.0730464i	$0.0232721\pi$
$503$	73.4847i	0.146093i	−0.997329	+	0.0730464i	$0.976728\pi$
$504$	0	0
$505$	−1350.00	−2.67327
$506$	0	0
$507$	0	0
$508$	0	0
$509$	624.620	1.22715	0.613576	−	0.789636i	$-0.289731\pi$
$509$	624.620	1.22715	0.613576	+	0.789636i	$0.289731\pi$
$510$	0	0
$511$	207.846	0.406744
$512$	0	0
$513$	0	0
$514$	0	0
$515$	687.308	1.33458
$516$	0	0
$517$	− 623.538i	− 1.20607i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	861.256i	1.65308i	0.562876	+	0.826541i	$0.309695\pi$
$521$	861.256i	1.65308i	−0.562876	+	0.826541i	$0.690305\pi$
$522$	0	0
$523$	− 20.0000i	− 0.0382409i	−0.999817	−	0.0191205i	$-0.993913\pi$
$523$	− 20.0000i	− 0.0382409i	0.999817	−	0.0191205i	$-0.00608660\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 1102.27i	− 2.09159i
$528$	0	0
$529$	313.000	0.591682
$530$	0	0
$531$	0	0
$532$	0	0
$533$	749.544	1.40627
$534$	0	0
$535$	−1247.08	−2.33098
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−500.632	−0.928816
$540$	0	0
$541$	− 363.731i	− 0.672330i	−0.941803	−	0.336165i	$-0.890870\pi$
$541$	− 363.731i	− 0.672330i	0.941803	−	0.336165i	$-0.109130\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 381.838i	− 0.700620i
$546$	0	0
$547$	− 760.000i	− 1.38940i	−0.719301	−	0.694698i	$-0.755538\pi$
$547$	− 760.000i	− 1.38940i	0.719301	−	0.694698i	$-0.244462\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	734.847i	1.33366i
$552$	0	0
$553$	540.000	0.976492
$554$	0	0
$555$	0	0
$556$	0	0
$557$	992.043	1.78105	0.890524	−	0.454937i	$-0.150338\pi$
$557$	992.043	1.78105	0.890524	+	0.454937i	$0.150338\pi$
$558$	0	0
$559$	415.692	0.743635
$560$	0	0
$561$	0	0
$562$	0	0
$563$	212.132	0.376789	0.188394	−	0.982093i	$-0.439672\pi$
$563$	212.132	0.376789	0.188394	+	0.982093i	$0.439672\pi$
$564$	0	0
$565$	467.654i	0.827706i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 479.418i	− 0.842563i	−0.906930	−	0.421282i	$-0.861580\pi$
$569$	− 479.418i	− 0.842563i	0.906930	−	0.421282i	$-0.138420\pi$
$570$	0	0
$571$	− 280.000i	− 0.490368i	−0.969477	−	0.245184i	$-0.921152\pi$
$571$	− 280.000i	− 0.490368i	0.969477	−	0.245184i	$-0.0788483\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 426.211i	− 0.741237i
$576$	0	0
$577$	−650.000	−1.12652	−0.563258	−	0.826281i	$-0.690452\pi$
$577$	−650.000	−1.12652	−0.563258	+	0.826281i	$0.690452\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−1322.72	−2.27663
$582$	0	0
$583$	311.769	0.534767
$584$	0	0
$585$	0	0
$586$	0	0
$587$	678.823	1.15643	0.578213	−	0.815886i	$-0.303750\pi$
$587$	678.823	1.15643	0.578213	+	0.815886i	$0.303750\pi$
$588$	0	0
$589$	− 1039.23i	− 1.76440i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 445.477i	− 0.751226i	−0.926777	−	0.375613i	$-0.877432\pi$
$593$	− 445.477i	− 0.751226i	0.926777	−	0.375613i	$-0.122568\pi$
$594$	0	0
$595$	− 1620.00i	− 2.72269i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 367.423i	− 0.613395i	−0.951807	−	0.306697i	$-0.900776\pi$
$599$	− 367.423i	− 0.613395i	0.951807	−	0.306697i	$-0.0992240\pi$
$600$	0	0
$601$	−490.000	−0.815308	−0.407654	−	0.913137i	$-0.633653\pi$
$601$	−490.000	−0.815308	−0.407654	+	0.913137i	$0.633653\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	360.075	0.595165
$606$	0	0
$607$	322.161	0.530744	0.265372	−	0.964146i	$-0.414505\pi$
$607$	322.161	0.530744	0.265372	+	0.964146i	$0.414505\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	763.675	1.24988
$612$	0	0
$613$	− 374.123i	− 0.610315i	−0.952302	−	0.305157i	$-0.901291\pi$
$613$	− 374.123i	− 0.610315i	0.952302	−	0.305157i	$-0.0987091\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 784.889i	− 1.27210i	−0.771646	−	0.636052i	$-0.780566\pi$
$617$	− 784.889i	− 1.27210i	0.771646	−	0.636052i	$-0.219434\pi$
$618$	0	0
$619$	1112.00i	1.79645i	0.439540	+	0.898223i	$0.355141\pi$
$619$	1112.00i	1.79645i	−0.439540	+	0.898223i	$0.644859\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	132.272i	0.212315i
$624$	0	0
$625$	−509.000	−0.814400
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−881.816	−1.40193
$630$	0	0
$631$	987.269	1.56461	0.782305	−	0.622896i	$-0.214044\pi$
$631$	987.269	1.56461	0.782305	+	0.622896i	$0.214044\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	76.3675	0.120264
$636$	0	0
$637$	− 613.146i	− 0.962553i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 97.5807i	− 0.152232i	−0.997099	−	0.0761160i	$-0.975748\pi$
$641$	− 97.5807i	− 0.152232i	0.997099	−	0.0761160i	$-0.0242519\pi$
$642$	0	0
$643$	80.0000i	0.124417i	0.998063	+	0.0622084i	$0.0198143\pi$
$643$	80.0000i	0.124417i	−0.998063	+	0.0622084i	$0.980186\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 514.393i	− 0.795043i	−0.917593	−	0.397522i	$-0.869870\pi$
$647$	− 514.393i	− 0.795043i	0.917593	−	0.397522i	$-0.130130\pi$
$648$	0	0
$649$	288.000	0.443760
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−1080.22	−1.65425	−0.827125	−	0.562018i	$-0.810025\pi$
$653$	−1080.22	−1.65425	−0.827125	+	0.562018i	$0.810025\pi$
$654$	0	0
$655$	374.123	0.571180
$656$	0	0
$657$	0	0
$658$	0	0
$659$	712.764	1.08158	0.540792	−	0.841156i	$-0.318125\pi$
$659$	712.764	1.08158	0.540792	+	0.841156i	$0.318125\pi$
$660$	0	0
$661$	1247.08i	1.88665i	0.331868	+	0.943326i	$0.392321\pi$
$661$	1247.08i	1.88665i	−0.331868	+	0.943326i	$0.607679\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 1527.35i	− 2.29677i
$666$	0	0
$667$	− 540.000i	− 0.809595i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−50.0000	−0.0742942	−0.0371471	−	0.999310i	$-0.511827\pi$
$673$	−50.0000	−0.0742942	−0.0371471	+	0.999310i	$0.511827\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	169.015	0.249653	0.124826	−	0.992179i	$-0.460163\pi$
$677$	169.015	0.249653	0.124826	+	0.992179i	$0.460163\pi$
$678$	0	0
$679$	415.692	0.612212
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−42.4264	−0.0621177	−0.0310589	−	0.999518i	$-0.509888\pi$
$683$	−42.4264	−0.0621177	−0.0310589	+	0.999518i	$0.509888\pi$
$684$	0	0
$685$	779.423i	1.13784i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	381.838i	0.554191i
$690$	0	0
$691$	− 400.000i	− 0.578871i	−0.957197	−	0.289436i	$-0.906532\pi$
$691$	− 400.000i	− 0.578871i	0.957197	−	0.289436i	$-0.0934676\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1263.94i	1.81861i
$696$	0	0
$697$	−1530.00	−2.19512
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−992.043	−1.41518	−0.707592	−	0.706622i	$-0.750218\pi$
$701$	−992.043	−1.41518	−0.707592	+	0.706622i	$0.750218\pi$
$702$	0	0
$703$	−831.384	−1.18262
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1909.19	2.70041
$708$	0	0
$709$	− 1195.12i	− 1.68563i	−0.538200	−	0.842817i	$-0.680895\pi$
$709$	− 1195.12i	− 1.68563i	0.538200	−	0.842817i	$-0.319105\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	763.675i	1.07107i
$714$	0	0
$715$	− 648.000i	− 0.906294i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	661.362i	0.919836i	0.887961	+	0.459918i	$0.152121\pi$
$719$	661.362i	0.919836i	−0.887961	+	0.459918i	$0.847879\pi$
$720$	0	0
$721$	−972.000	−1.34813
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1065.53	1.46969
$726$	0	0
$727$	−93.5307	−0.128653	−0.0643265	−	0.997929i	$-0.520490\pi$
$727$	−93.5307	−0.128653	−0.0643265	+	0.997929i	$0.520490\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−848.528	−1.16078
$732$	0	0
$733$	613.146i	0.836488i	0.908335	+	0.418244i	$0.137354\pi$
$733$	613.146i	0.836488i	−0.908335	+	0.418244i	$0.862646\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 848.528i	− 1.15133i
$738$	0	0
$739$	− 920.000i	− 1.24493i	−0.782649	−	0.622463i	$-0.786132\pi$
$739$	− 920.000i	− 1.24493i	0.782649	−	0.622463i	$-0.213868\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 1175.76i	− 1.58244i	−0.611530	−	0.791221i	$-0.709446\pi$
$743$	− 1175.76i	− 1.58244i	0.611530	−	0.791221i	$-0.290554\pi$
$744$	0	0
$745$	810.000	1.08725
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1763.63	2.35465
$750$	0	0
$751$	−51.9615	−0.0691898	−0.0345949	−	0.999401i	$-0.511014\pi$
$751$	−51.9615	−0.0691898	−0.0345949	+	0.999401i	$0.511014\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1145.51	1.51724
$756$	0	0
$757$	− 405.300i	− 0.535403i	−0.963502	−	0.267701i	$-0.913736\pi$
$757$	− 405.300i	− 0.535403i	0.963502	−	0.267701i	$-0.0862641\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 1030.96i	− 1.35475i	−0.735640	−	0.677373i	$-0.763118\pi$
$761$	− 1030.96i	− 1.35475i	0.735640	−	0.677373i	$-0.236882\pi$
$762$	0	0
$763$	540.000i	0.707733i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	352.727i	0.459878i
$768$	0	0
$769$	890.000	1.15735	0.578674	−	0.815559i	$-0.303571\pi$
$769$	890.000	1.15735	0.578674	+	0.815559i	$0.303571\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	933.256	1.20732	0.603658	−	0.797243i	$-0.293709\pi$
$773$	933.256	1.20732	0.603658	+	0.797243i	$0.293709\pi$
$774$	0	0
$775$	−1506.88	−1.94437
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−1442.50	−1.85173
$780$	0	0
$781$	623.538i	0.798384i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 1221.88i	− 1.55654i
$786$	0	0
$787$	− 100.000i	− 0.127065i	−0.997980	−	0.0635324i	$-0.979763\pi$
$787$	− 100.000i	− 0.127065i	0.997980	−	0.0635324i	$-0.0202366\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 661.362i	− 0.836109i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−404.166	−0.507109	−0.253554	−	0.967321i	$-0.581600\pi$
$797$	−404.166	−0.507109	−0.253554	+	0.967321i	$0.581600\pi$
$798$	0	0
$799$	−1558.85	−1.95100
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−169.706	−0.211340
$804$	0	0
$805$	1122.37i	1.39425i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1090.36i	1.34779i	0.738829	+	0.673893i	$0.235379\pi$
$809$	1090.36i	1.34779i	−0.738829	+	0.673893i	$0.764621\pi$
$810$	0	0
$811$	772.000i	0.951911i	0.879469	+	0.475956i	$0.157898\pi$
$811$	772.000i	0.951911i	−0.879469	+	0.475956i	$0.842102\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1175.76i	1.44264i
$816$	0	0
$817$	−800.000	−0.979192
$818$	0	0
$819$	0	0
$820$	0	0
$821$	477.650	0.581791	0.290896	−	0.956755i	$-0.406047\pi$
$821$	477.650	0.581791	0.290896	+	0.956755i	$0.406047\pi$
$822$	0	0
$823$	1132.76	1.37638	0.688190	−	0.725530i	$-0.258405\pi$
$823$	1132.76	1.37638	0.688190	+	0.725530i	$0.258405\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	763.675	0.923428	0.461714	−	0.887029i	$-0.347235\pi$
$827$	763.675	0.923428	0.461714	+	0.887029i	$0.347235\pi$
$828$	0	0
$829$	− 363.731i	− 0.438758i	−0.975640	−	0.219379i	$-0.929597\pi$
$829$	− 363.731i	− 0.438758i	0.975640	−	0.219379i	$-0.0704032\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1251.58i	1.50250i
$834$	0	0
$835$	− 432.000i	− 0.517365i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 808.332i	− 0.963447i	−0.876323	−	0.481723i	$-0.840011\pi$
$839$	− 808.332i	− 0.963447i	0.876323	−	0.481723i	$-0.159989\pi$
$840$	0	0
$841$	509.000	0.605232
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−448.257	−0.530481
$846$	0	0
$847$	−509.223	−0.601208
$848$	0	0
$849$	0	0
$850$	0	0
$851$	610.940	0.717909
$852$	0	0
$853$	− 997.661i	− 1.16959i	−0.811181	−	0.584796i	$-0.801175\pi$
$853$	− 997.661i	− 1.16959i	0.811181	−	0.584796i	$-0.198825\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 572.756i	− 0.668327i	−0.942515	−	0.334164i	$-0.891546\pi$
$857$	− 572.756i	− 0.668327i	0.942515	−	0.334164i	$-0.108454\pi$
$858$	0	0
$859$	568.000i	0.661234i	0.943765	+	0.330617i	$0.107257\pi$
$859$	568.000i	0.661234i	−0.943765	+	0.330617i	$0.892743\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 823.029i	− 0.953683i	−0.878989	−	0.476842i	$-0.841782\pi$
$863$	− 823.029i	− 0.953683i	0.878989	−	0.476842i	$-0.158218\pi$
$864$	0	0
$865$	−270.000	−0.312139
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−440.908	−0.507374
$870$	0	0
$871$	1039.23	1.19315
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−305.470	−0.349109
$876$	0	0
$877$	41.5692i	0.0473993i	0.999719	+	0.0236997i	$0.00754455\pi$
$877$	41.5692i	0.0473993i	−0.999719	+	0.0236997i	$0.992455\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	12.7279i	0.0144471i	0.999974	+	0.00722357i	$0.00229935\pi$
$881$	12.7279i	0.0144471i	−0.999974	+	0.00722357i	$0.997701\pi$
$882$	0	0
$883$	− 520.000i	− 0.588901i	−0.955667	−	0.294451i	$-0.904863\pi$
$883$	− 520.000i	− 0.588901i	0.955667	−	0.294451i	$-0.0951367\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 88.1816i	− 0.0994156i	−0.998764	−	0.0497078i	$-0.984171\pi$
$887$	− 88.1816i	− 0.0994156i	0.998764	−	0.0497078i	$-0.0158290\pi$
$888$	0	0
$889$	−108.000	−0.121485
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−1469.69	−1.64579
$894$	0	0
$895$	374.123	0.418014
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−1909.19	−2.12368
$900$	0	0
$901$	− 779.423i	− 0.865064i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1909.19i	2.10960i
$906$	0	0
$907$	40.0000i	0.0441014i	0.999757	+	0.0220507i	$0.00701953\pi$
$907$	40.0000i	0.0441014i	−0.999757	+	0.0220507i	$0.992980\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1175.76i	1.29062i	0.763921	+	0.645310i	$0.223272\pi$
$911$	1175.76i	1.29062i	−0.763921	+	0.645310i	$0.776728\pi$
$912$	0	0
$913$	1080.00	1.18291
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−529.090	−0.576979
$918$	0	0
$919$	1714.73	1.86587	0.932933	−	0.360051i	$-0.117241\pi$
$919$	1714.73	1.86587	0.932933	+	0.360051i	$0.117241\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−763.675	−0.827384
$924$	0	0
$925$	1205.51i	1.30325i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1412.80i	1.52077i	0.649470	+	0.760387i	$0.274991\pi$
$929$	1412.80i	1.52077i	−0.649470	+	0.760387i	$0.725009\pi$
$930$	0	0
$931$	1180.00i	1.26745i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1322.72i	1.41468i
$936$	0	0
$937$	−470.000	−0.501601	−0.250800	−	0.968039i	$-0.580694\pi$
$937$	−470.000	−0.501601	−0.250800	+	0.968039i	$0.580694\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−624.620	−0.663783	−0.331892	−	0.943318i	$-0.607687\pi$
$941$	−624.620	−0.663783	−0.331892	+	0.943318i	$0.607687\pi$
$942$	0	0
$943$	1060.02	1.12409
$944$	0	0
$945$	0	0
$946$	0	0
$947$	339.411	0.358407	0.179203	−	0.983812i	$-0.442648\pi$
$947$	339.411	0.358407	0.179203	+	0.983812i	$0.442648\pi$
$948$	0	0
$949$	− 207.846i	− 0.219016i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 530.330i	− 0.556485i	−0.960511	−	0.278242i	$-0.910248\pi$
$953$	− 530.330i	− 0.556485i	0.960511	−	0.278242i	$-0.0897519\pi$
$954$	0	0
$955$	2160.00i	2.26178i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 1102.27i	− 1.14940i
$960$	0	0
$961$	1739.00	1.80957
$962$	0	0
$963$	0	0
$964$	0	0
$965$	73.4847	0.0761499
$966$	0	0
$967$	1028.84	1.06395	0.531974	−	0.846761i	$-0.321450\pi$
$967$	1028.84	1.06395	0.531974	+	0.846761i	$0.321450\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−1603.72	−1.65162	−0.825808	−	0.563952i	$-0.809280\pi$
$971$	−1603.72	−1.65162	−0.825808	+	0.563952i	$0.809280\pi$
$972$	0	0
$973$	− 1787.48i	− 1.83708i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 1378.86i	− 1.41132i	−0.708551	−	0.705659i	$-0.750651\pi$
$977$	− 1378.86i	− 1.41132i	0.708551	−	0.705659i	$-0.249349\pi$
$978$	0	0
$979$	− 108.000i	− 0.110317i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 440.908i	− 0.448533i	−0.974528	−	0.224267i	$-0.928001\pi$
$983$	− 440.908i	− 0.448533i	0.974528	−	0.224267i	$-0.0719986\pi$
$984$	0	0
$985$	162.000	0.164467
$986$	0	0
$987$	0	0
$988$	0	0
$989$	587.878	0.594416
$990$	0	0
$991$	−467.654	−0.471901	−0.235950	−	0.971765i	$-0.575820\pi$
$991$	−467.654	−0.471901	−0.235950	+	0.971765i	$0.575820\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	381.838	0.383756
$996$	0	0
$997$	249.415i	0.250166i	0.992146	+	0.125083i	$0.0399197\pi$
$997$	249.415i	0.250166i	−0.992146	+	0.125083i	$0.960080\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.3.h.b.161.3	yes	8
3.2	odd	2	inner	576.3.h.b.161.7	yes	8
4.3	odd	2	inner	576.3.h.b.161.1	✓	8
8.3	odd	2	inner	576.3.h.b.161.6	yes	8
8.5	even	2	inner	576.3.h.b.161.8	yes	8
12.11	even	2	inner	576.3.h.b.161.5	yes	8
16.3	odd	4		2304.3.e.j.1025.4		4
16.5	even	4		2304.3.e.j.1025.1		4
16.11	odd	4		2304.3.e.g.1025.2		4
16.13	even	4		2304.3.e.g.1025.3		4
24.5	odd	2	inner	576.3.h.b.161.4	yes	8
24.11	even	2	inner	576.3.h.b.161.2	yes	8
48.5	odd	4		2304.3.e.j.1025.3		4
48.11	even	4		2304.3.e.g.1025.4		4
48.29	odd	4		2304.3.e.g.1025.1		4
48.35	even	4		2304.3.e.j.1025.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.3.h.b.161.1	✓	8	4.3	odd	2	inner
576.3.h.b.161.2	yes	8	24.11	even	2	inner
576.3.h.b.161.3	yes	8	1.1	even	1	trivial
576.3.h.b.161.4	yes	8	24.5	odd	2	inner
576.3.h.b.161.5	yes	8	12.11	even	2	inner
576.3.h.b.161.6	yes	8	8.3	odd	2	inner
576.3.h.b.161.7	yes	8	3.2	odd	2	inner
576.3.h.b.161.8	yes	8	8.5	even	2	inner
2304.3.e.g.1025.1		4	48.29	odd	4
2304.3.e.g.1025.2		4	16.11	odd	4
2304.3.e.g.1025.3		4	16.13	even	4
2304.3.e.g.1025.4		4	48.11	even	4
2304.3.e.j.1025.1		4	16.5	even	4
2304.3.e.j.1025.2		4	48.35	even	4
2304.3.e.j.1025.3		4	48.5	odd	4
2304.3.e.j.1025.4		4	16.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 576 = 2^{6} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	576.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-7.34847 q^{5} +10.3923 q^{7} +O(q^{10})\)	\(q-7.34847 q^{5} +10.3923 q^{7} -8.48528 q^{11} -10.3923i q^{13} +21.2132i q^{17} +20.0000i q^{19} -14.6969i q^{23} +29.0000 q^{25} +36.7423 q^{29} -51.9615 q^{31} -76.3675 q^{35} +41.5692i q^{37} +72.1249i q^{41} +40.0000i q^{43} +73.4847i q^{47} +59.0000 q^{49} -36.7423 q^{53} +62.3538 q^{55} -33.9411 q^{59} +76.3675i q^{65} +100.000i q^{67} -73.4847i q^{71} +20.0000 q^{73} -88.1816 q^{77} +51.9615 q^{79} -127.279 q^{83} -155.885i q^{85} +12.7279i q^{89} -108.000i q^{91} -146.969i q^{95} +40.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 232 q^{25} + 472 q^{49} + 160 q^{73} + 320 q^{97}+O(q^{100}) \) 8 * q + 232 * q^25 + 472 * q^49 + 160 * q^73 + 320 * q^97

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