LMFDB - Embedded newform 2304.3.e.g.1025.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2304,3,Mod(1025,2304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2304, 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("2304.1025");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2304 = 2^{8} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2304.e (of order $2$, degree $1$, not 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:	$62.7794529086$
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} + 4x^{2} + 1 $ x^4 + 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{3} $
Twist minimal:	no (minimal twist has level 576)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1025.2
Root			$-1.93185i$ of defining polynomial
Character	$\chi$	$=$	2304.1025
Dual form			2304.3.e.g.1025.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-7.34847i q^{5} +10.3923 q^{7} +O(q^{10})$	$q-7.34847i q^{5} +10.3923 q^{7} +8.48528i q^{11} -10.3923 q^{13} +21.2132i q^{17} -20.0000 q^{19} -14.6969i q^{23} -29.0000 q^{25} -36.7423i q^{29} +51.9615 q^{31} -76.3675i q^{35} -41.5692 q^{37} -72.1249i q^{41} +40.0000 q^{43} -73.4847i q^{47} +59.0000 q^{49} -36.7423i q^{53} +62.3538 q^{55} +33.9411i q^{59} +76.3675i q^{65} -100.000 q^{67} -73.4847i q^{71} -20.0000 q^{73} +88.1816i q^{77} -51.9615 q^{79} -127.279i q^{83} +155.885 q^{85} -12.7279i q^{89} -108.000 q^{91} +146.969i q^{95} +40.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 80 q^{19} - 116 q^{25} + 160 q^{43} + 236 q^{49} - 400 q^{67} - 80 q^{73} - 432 q^{91} + 160 q^{97}+O(q^{100}) $ 4 * q - 80 * q^19 - 116 * q^25 + 160 * q^43 + 236 * q^49 - 400 * q^67 - 80 * q^73 - 432 * q^91 + 160 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times$.

$n$	$1279$	$1793$	$2053$
$\chi(n)$	$1$	$-1$	$1$

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
$5$	− 7.34847i	− 1.46969i	−0.678233	−	0.734847i	$-0.737254\pi$
$5$	− 7.34847i	− 1.46969i	0.678233	−	0.734847i	$-0.262746\pi$
$6$	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
$11$	8.48528i	0.771389i	0.922627	+	0.385695i	$0.126038\pi$
$11$	8.48528i	0.771389i	−0.922627	+	0.385695i	$0.873962\pi$
$12$	0	0
$13$	−10.3923	−0.799408	−0.399704	−	0.916644i	$-0.630887\pi$
$13$	−10.3923	−0.799408	−0.399704	+	0.916644i	$0.630887\pi$
$14$	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
$19$	−20.0000	−1.05263	−0.526316	−	0.850289i	$-0.676427\pi$
$19$	−20.0000	−1.05263	−0.526316	+	0.850289i	$0.676427\pi$
$20$	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
$29$	− 36.7423i	− 1.26698i	−0.773752	−	0.633489i	$-0.781622\pi$
$29$	− 36.7423i	− 1.26698i	0.773752	−	0.633489i	$-0.218378\pi$
$30$	0	0
$31$	51.9615	1.67618	0.838089	−	0.545533i	$-0.183673\pi$
$31$	51.9615	1.67618	0.838089	+	0.545533i	$0.183673\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 76.3675i	− 2.18193i
$36$	0	0
$37$	−41.5692	−1.12349	−0.561746	−	0.827310i	$-0.689870\pi$
$37$	−41.5692	−1.12349	−0.561746	+	0.827310i	$0.689870\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 72.1249i	− 1.75914i	−0.475766	−	0.879572i	$-0.657829\pi$
$41$	− 72.1249i	− 1.75914i	0.475766	−	0.879572i	$-0.342171\pi$
$42$	0	0
$43$	40.0000	0.930233	0.465116	−	0.885250i	$-0.346013\pi$
$43$	40.0000	0.930233	0.465116	+	0.885250i	$0.346013\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 73.4847i	− 1.56350i	−0.623589	−	0.781752i	$-0.714326\pi$
$47$	− 73.4847i	− 1.56350i	0.623589	−	0.781752i	$-0.285674\pi$
$48$	0	0
$49$	59.0000	1.20408
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 36.7423i	− 0.693252i	−0.938003	−	0.346626i	$-0.887327\pi$
$53$	− 36.7423i	− 0.693252i	0.938003	−	0.346626i	$-0.112673\pi$
$54$	0	0
$55$	62.3538	1.13371
$56$	0	0
$57$	0	0
$58$	0	0
$59$	33.9411i	0.575273i	0.957740	+	0.287637i	$0.0928695\pi$
$59$	33.9411i	0.575273i	−0.957740	+	0.287637i	$0.907130\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	76.3675i	1.17489i
$66$	0	0
$67$	−100.000	−1.49254	−0.746269	−	0.665645i	$-0.768157\pi$
$67$	−100.000	−1.49254	−0.746269	+	0.665645i	$0.768157\pi$
$68$	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	$-0.543742\pi$
$73$	−20.0000	−0.273973	−0.136986	+	0.990573i	$0.543742\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	88.1816i	1.14522i
$78$	0	0
$79$	−51.9615	−0.657741	−0.328870	−	0.944375i	$-0.606668\pi$
$79$	−51.9615	−0.657741	−0.328870	+	0.944375i	$0.606668\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 127.279i	− 1.53348i	−0.641955	−	0.766742i	$-0.721876\pi$
$83$	− 127.279i	− 1.53348i	0.641955	−	0.766742i	$-0.278124\pi$
$84$	0	0
$85$	155.885	1.83394
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 12.7279i	− 0.143010i	−0.997440	−	0.0715052i	$-0.977220\pi$
$89$	− 12.7279i	− 0.143010i	0.997440	−	0.0715052i	$-0.0227802\pi$
$90$	0	0
$91$	−108.000	−1.18681
$92$	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
$101$	183.712i	1.81893i	0.415783	+	0.909464i	$0.363508\pi$
$101$	183.712i	1.81893i	−0.415783	+	0.909464i	$0.636492\pi$
$102$	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
$107$	− 169.706i	− 1.58603i	−0.609200	−	0.793017i	$-0.708509\pi$
$107$	− 169.706i	− 1.58603i	0.609200	−	0.793017i	$-0.291491\pi$
$108$	0	0
$109$	51.9615	0.476711	0.238356	−	0.971178i	$-0.423392\pi$
$109$	51.9615	0.476711	0.238356	+	0.971178i	$0.423392\pi$
$110$	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
$115$	−108.000	−0.939130
$116$	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
$125$	29.3939i	0.235151i
$126$	0	0
$127$	10.3923	0.0818292	0.0409146	−	0.999163i	$-0.486973\pi$
$127$	10.3923	0.0818292	0.0409146	+	0.999163i	$0.486973\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 50.9117i	− 0.388639i	−0.980938	−	0.194319i	$-0.937750\pi$
$131$	− 50.9117i	− 0.388639i	0.980938	−	0.194319i	$-0.0622498\pi$
$132$	0	0
$133$	−207.846	−1.56275
$134$	0	0
$135$	0	0
$136$	0	0
$137$	106.066i	0.774205i	0.922037	+	0.387102i	$0.126524\pi$
$137$	106.066i	0.774205i	−0.922037	+	0.387102i	$0.873476\pi$
$138$	0	0
$139$	−172.000	−1.23741	−0.618705	−	0.785623i	$-0.712342\pi$
$139$	−172.000	−1.23741	−0.618705	+	0.785623i	$0.712342\pi$
$140$	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
$149$	− 110.227i	− 0.739779i	−0.929076	−	0.369889i	$-0.879396\pi$
$149$	− 110.227i	− 0.739779i	0.929076	−	0.369889i	$-0.120604\pi$
$150$	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
$155$	− 381.838i	− 2.46347i
$156$	0	0
$157$	166.277	1.05909	0.529544	−	0.848282i	$-0.322363\pi$
$157$	166.277	1.05909	0.529544	+	0.848282i	$0.322363\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 152.735i	− 0.948665i
$162$	0	0
$163$	160.000	0.981595	0.490798	−	0.871274i	$-0.336705\pi$
$163$	160.000	0.981595	0.490798	+	0.871274i	$0.336705\pi$
$164$	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
$173$	− 36.7423i	− 0.212384i	−0.994346	−	0.106192i	$-0.966134\pi$
$173$	− 36.7423i	− 0.212384i	0.994346	−	0.106192i	$-0.0338657\pi$
$174$	0	0
$175$	−301.377	−1.72215
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 50.9117i	− 0.284423i	−0.989836	−	0.142211i	$-0.954579\pi$
$179$	− 50.9117i	− 0.284423i	0.989836	−	0.142211i	$-0.0454213\pi$
$180$	0	0
$181$	259.808	1.43540	0.717701	−	0.696352i	$-0.245195\pi$
$181$	259.808	1.43540	0.717701	+	0.696352i	$0.245195\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	305.470i	1.65119i
$186$	0	0
$187$	−180.000	−0.962567
$188$	0	0
$189$	0	0
$190$	0	0
$191$	293.939i	1.53895i	0.638679	+	0.769473i	$0.279481\pi$
$191$	293.939i	1.53895i	−0.638679	+	0.769473i	$0.720519\pi$
$192$	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
$197$	− 22.0454i	− 0.111906i	−0.998433	−	0.0559528i	$-0.982180\pi$
$197$	− 22.0454i	− 0.111906i	0.998433	−	0.0559528i	$-0.0178196\pi$
$198$	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
$203$	− 381.838i	− 1.88097i
$204$	0	0
$205$	−530.008	−2.58540
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 169.706i	− 0.811989i
$210$	0	0
$211$	−172.000	−0.815166	−0.407583	−	0.913168i	$-0.633628\pi$
$211$	−172.000	−0.815166	−0.407583	+	0.913168i	$0.633628\pi$
$212$	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
$221$	− 220.454i	− 0.997530i
$222$	0	0
$223$	10.3923	0.0466023	0.0233011	−	0.999728i	$-0.492582\pi$
$223$	10.3923	0.0466023	0.0233011	+	0.999728i	$0.492582\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 42.4264i	− 0.186900i	−0.995624	−	0.0934502i	$-0.970210\pi$
$227$	− 42.4264i	− 0.186900i	0.995624	−	0.0934502i	$-0.0297896\pi$
$228$	0	0
$229$	−259.808	−1.13453	−0.567266	−	0.823535i	$-0.691999\pi$
$229$	−259.808	−1.13453	−0.567266	+	0.823535i	$0.691999\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 106.066i	− 0.455219i	−0.973752	−	0.227609i	$-0.926909\pi$
$233$	− 106.066i	− 0.455219i	0.973752	−	0.227609i	$-0.0730910\pi$
$234$	0	0
$235$	−540.000	−2.29787
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 146.969i	− 0.614935i	−0.951559	−	0.307467i	$-0.900519\pi$
$239$	− 146.969i	− 0.614935i	0.951559	−	0.307467i	$-0.0994815\pi$
$240$	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
$245$	− 433.560i	− 1.76963i
$246$	0	0
$247$	207.846	0.841482
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 246.073i	− 0.980371i	−0.871618	−	0.490186i	$-0.836929\pi$
$251$	− 246.073i	− 0.980371i	0.871618	−	0.490186i	$-0.163071\pi$
$252$	0	0
$253$	124.708	0.492916
$254$	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
$259$	−432.000	−1.66795
$260$	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
$269$	− 110.227i	− 0.409766i	−0.978786	−	0.204883i	$-0.934319\pi$
$269$	− 110.227i	− 0.409766i	0.978786	−	0.204883i	$-0.0656814\pi$
$270$	0	0
$271$	467.654	1.72566	0.862830	−	0.505495i	$-0.168690\pi$
$271$	467.654	1.72566	0.862830	+	0.505495i	$0.168690\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 246.073i	− 0.894811i
$276$	0	0
$277$	−322.161	−1.16304	−0.581519	−	0.813533i	$-0.697541\pi$
$277$	−322.161	−1.16304	−0.581519	+	0.813533i	$0.697541\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	140.007i	0.498246i	0.968472	+	0.249123i	$0.0801424\pi$
$281$	140.007i	0.498246i	−0.968472	+	0.249123i	$0.919858\pi$
$282$	0	0
$283$	80.0000	0.282686	0.141343	−	0.989961i	$-0.454858\pi$
$283$	80.0000	0.282686	0.141343	+	0.989961i	$0.454858\pi$
$284$	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
$293$	315.984i	1.07844i	0.842164	+	0.539222i	$0.181282\pi$
$293$	315.984i	1.07844i	−0.842164	+	0.539222i	$0.818718\pi$
$294$	0	0
$295$	249.415	0.845476
$296$	0	0
$297$	0	0
$298$	0	0
$299$	152.735i	0.510820i
$300$	0	0
$301$	415.692	1.38104
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	380.000	1.23779	0.618893	−	0.785476i	$-0.287582\pi$
$307$	380.000	1.23779	0.618893	+	0.785476i	$0.287582\pi$
$308$	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	$-0.664908\pi$
$313$	−310.000	−0.990415	−0.495208	+	0.868775i	$0.664908\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	242.499i	0.764983i	0.923959	+	0.382491i	$0.124934\pi$
$317$	242.499i	0.764983i	−0.923959	+	0.382491i	$0.875066\pi$
$318$	0	0
$319$	311.769	0.977333
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 424.264i	− 1.31351i
$324$	0	0
$325$	301.377	0.927313
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 763.675i	− 2.32120i
$330$	0	0
$331$	500.000	1.51057	0.755287	−	0.655394i	$-0.227497\pi$
$331$	500.000	1.51057	0.755287	+	0.655394i	$0.227497\pi$
$332$	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
$341$	440.908i	1.29299i
$342$	0	0
$343$	103.923	0.302983
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 42.4264i	− 0.122266i	−0.998130	−	0.0611332i	$-0.980529\pi$
$347$	− 42.4264i	− 0.122266i	0.998130	−	0.0611332i	$-0.0194714\pi$
$348$	0	0
$349$	−207.846	−0.595548	−0.297774	−	0.954636i	$-0.596244\pi$
$349$	−207.846	−0.595548	−0.297774	+	0.954636i	$0.596244\pi$
$350$	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
$355$	−540.000	−1.52113
$356$	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
$365$	146.969i	0.402656i
$366$	0	0
$367$	218.238	0.594655	0.297328	−	0.954776i	$-0.403905\pi$
$367$	218.238	0.594655	0.297328	+	0.954776i	$0.403905\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 381.838i	− 1.02921i
$372$	0	0
$373$	665.108	1.78313	0.891565	−	0.452893i	$-0.149608\pi$
$373$	665.108	1.78313	0.891565	+	0.452893i	$0.149608\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	381.838i	1.01283i
$378$	0	0
$379$	−92.0000	−0.242744	−0.121372	−	0.992607i	$-0.538729\pi$
$379$	−92.0000	−0.242744	−0.121372	+	0.992607i	$0.538729\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 146.969i	− 0.383732i	−0.981421	−	0.191866i	$-0.938546\pi$
$383$	− 146.969i	− 0.383732i	0.981421	−	0.191866i	$-0.0614539\pi$
$384$	0	0
$385$	648.000	1.68312
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 257.196i	− 0.661173i	−0.943776	−	0.330587i	$-0.892753\pi$
$389$	− 257.196i	− 0.661173i	0.943776	−	0.330587i	$-0.107247\pi$
$390$	0	0
$391$	311.769	0.797364
$392$	0	0
$393$	0	0
$394$	0	0
$395$	381.838i	0.966678i
$396$	0	0
$397$	41.5692	0.104708	0.0523542	−	0.998629i	$-0.483328\pi$
$397$	41.5692	0.104708	0.0523542	+	0.998629i	$0.483328\pi$
$398$	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
$403$	−540.000	−1.33995
$404$	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	$-0.648532\pi$
$409$	−368.000	−0.899756	−0.449878	+	0.893090i	$0.648532\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	352.727i	0.854059i
$414$	0	0
$415$	−935.307	−2.25375
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 432.749i	− 1.03281i	−0.856343	−	0.516407i	$-0.827269\pi$
$419$	− 432.749i	− 1.03281i	0.856343	−	0.516407i	$-0.172731\pi$
$420$	0	0
$421$	51.9615	0.123424	0.0617120	−	0.998094i	$-0.480344\pi$
$421$	51.9615	0.123424	0.0617120	+	0.998094i	$0.480344\pi$
$422$	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	$-0.859836\pi$
$431$	− 367.423i	− 0.852491i	0.904608	−	0.426245i	$-0.140164\pi$
$432$	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
$437$	293.939i	0.672629i
$438$	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
$443$	551.543i	1.24502i	0.782612	+	0.622509i	$0.213887\pi$
$443$	551.543i	1.24502i	−0.782612	+	0.622509i	$0.786113\pi$
$444$	0	0
$445$	−93.5307	−0.210181
$446$	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
$451$	612.000	1.35698
$452$	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	$-0.767065\pi$
$457$	−680.000	−1.48796	−0.743982	+	0.668199i	$0.767065\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	330.681i	0.717313i	0.933470	+	0.358656i	$0.116765\pi$
$461$	330.681i	0.717313i	−0.933470	+	0.358656i	$0.883235\pi$
$462$	0	0
$463$	322.161	0.695813	0.347907	−	0.937529i	$-0.386893\pi$
$463$	322.161	0.695813	0.347907	+	0.937529i	$0.386893\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	466.690i	0.999337i	0.866217	+	0.499669i	$0.166545\pi$
$467$	466.690i	0.999337i	−0.866217	+	0.499669i	$0.833455\pi$
$468$	0	0
$469$	−1039.23	−2.21584
$470$	0	0
$471$	0	0
$472$	0	0
$473$	339.411i	0.717571i
$474$	0	0
$475$	580.000	1.22105
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 220.454i	− 0.460238i	−0.973162	−	0.230119i	$-0.926088\pi$
$479$	− 220.454i	− 0.460238i	0.973162	−	0.230119i	$-0.0739116\pi$
$480$	0	0
$481$	432.000	0.898129
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 293.939i	− 0.606059i
$486$	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
$491$	220.617i	0.449322i	0.974437	+	0.224661i	$0.0721275\pi$
$491$	220.617i	0.449322i	−0.974437	+	0.224661i	$0.927872\pi$
$492$	0	0
$493$	779.423	1.58098
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 763.675i	− 1.53657i
$498$	0	0
$499$	−640.000	−1.28257	−0.641283	−	0.767305i	$-0.721597\pi$
$499$	−640.000	−1.28257	−0.641283	+	0.767305i	$0.721597\pi$
$500$	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
$509$	− 624.620i	− 1.22715i	−0.789636	−	0.613576i	$-0.789731\pi$
$509$	− 624.620i	− 1.22715i	0.789636	−	0.613576i	$-0.210269\pi$
$510$	0	0
$511$	−207.846	−0.406744
$512$	0	0
$513$	0	0
$514$	0	0
$515$	687.308i	1.33458i
$516$	0	0
$517$	623.538	1.20607
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 861.256i	− 1.65308i	−0.562876	−	0.826541i	$-0.690305\pi$
$521$	− 861.256i	− 1.65308i	0.562876	−	0.826541i	$-0.309695\pi$
$522$	0	0
$523$	−20.0000	−0.0382409	−0.0191205	−	0.999817i	$-0.506087\pi$
$523$	−20.0000	−0.0382409	−0.0191205	+	0.999817i	$0.506087\pi$
$524$	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
$533$	749.544i	1.40627i
$534$	0	0
$535$	−1247.08	−2.33098
$536$	0	0
$537$	0	0
$538$	0	0
$539$	500.632i	0.928816i
$540$	0	0
$541$	−363.731	−0.672330	−0.336165	−	0.941803i	$-0.609130\pi$
$541$	−363.731	−0.672330	−0.336165	+	0.941803i	$0.609130\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 381.838i	− 0.700620i
$546$	0	0
$547$	760.000	1.38940	0.694698	−	0.719301i	$-0.255538\pi$
$547$	760.000	1.38940	0.694698	+	0.719301i	$0.255538\pi$
$548$	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
$557$	− 992.043i	− 1.78105i	−0.454937	−	0.890524i	$-0.650338\pi$
$557$	− 992.043i	− 1.78105i	0.454937	−	0.890524i	$-0.349662\pi$
$558$	0	0
$559$	−415.692	−0.743635
$560$	0	0
$561$	0	0
$562$	0	0
$563$	212.132i	0.376789i	0.982093	+	0.188394i	$0.0603283\pi$
$563$	212.132i	0.376789i	−0.982093	+	0.188394i	$0.939672\pi$
$564$	0	0
$565$	−467.654	−0.827706
$566$	0	0
$567$	0	0
$568$	0	0
$569$	479.418i	0.842563i	0.906930	+	0.421282i	$0.138420\pi$
$569$	479.418i	0.842563i	−0.906930	+	0.421282i	$0.861580\pi$
$570$	0	0
$571$	−280.000	−0.490368	−0.245184	−	0.969477i	$-0.578848\pi$
$571$	−280.000	−0.490368	−0.245184	+	0.969477i	$0.578848\pi$
$572$	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
$581$	− 1322.72i	− 2.27663i
$582$	0	0
$583$	311.769	0.534767
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 678.823i	− 1.15643i	−0.815886	−	0.578213i	$-0.803750\pi$
$587$	− 678.823i	− 1.15643i	0.815886	−	0.578213i	$-0.196250\pi$
$588$	0	0
$589$	−1039.23	−1.76440
$590$	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
$595$	1620.00	2.72269
$596$	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	$-0.366347\pi$
$601$	490.000	0.815308	0.407654	+	0.913137i	$0.366347\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 360.075i	− 0.595165i
$606$	0	0
$607$	−322.161	−0.530744	−0.265372	−	0.964146i	$-0.585495\pi$
$607$	−322.161	−0.530744	−0.265372	+	0.964146i	$0.585495\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	763.675i	1.24988i
$612$	0	0
$613$	374.123	0.610315	0.305157	−	0.952302i	$-0.401291\pi$
$613$	374.123	0.610315	0.305157	+	0.952302i	$0.401291\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	784.889i	1.27210i	0.771646	+	0.636052i	$0.219434\pi$
$617$	784.889i	1.27210i	−0.771646	+	0.636052i	$0.780566\pi$
$618$	0	0
$619$	1112.00	1.79645	0.898223	−	0.439540i	$-0.144859\pi$
$619$	1112.00	1.79645	0.898223	+	0.439540i	$0.144859\pi$
$620$	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
$629$	− 881.816i	− 1.40193i
$630$	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
$635$	− 76.3675i	− 0.120264i
$636$	0	0
$637$	−613.146	−0.962553
$638$	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
$643$	−80.0000	−0.124417	−0.0622084	−	0.998063i	$-0.519814\pi$
$643$	−80.0000	−0.124417	−0.0622084	+	0.998063i	$0.519814\pi$
$644$	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
$653$	1080.22i	1.65425i	0.562018	+	0.827125i	$0.310025\pi$
$653$	1080.22i	1.65425i	−0.562018	+	0.827125i	$0.689975\pi$
$654$	0	0
$655$	−374.123	−0.571180
$656$	0	0
$657$	0	0
$658$	0	0
$659$	712.764i	1.08158i	0.841156	+	0.540792i	$0.181875\pi$
$659$	712.764i	1.08158i	−0.841156	+	0.540792i	$0.818125\pi$
$660$	0	0
$661$	−1247.08	−1.88665	−0.943326	−	0.331868i	$-0.892321\pi$
$661$	−1247.08	−1.88665	−0.943326	+	0.331868i	$0.892321\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1527.35i	2.29677i
$666$	0	0
$667$	−540.000	−0.809595
$668$	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
$677$	169.015i	0.249653i	0.992179	+	0.124826i	$0.0398374\pi$
$677$	169.015i	0.249653i	−0.992179	+	0.124826i	$0.960163\pi$
$678$	0	0
$679$	415.692	0.612212
$680$	0	0
$681$	0	0
$682$	0	0
$683$	42.4264i	0.0621177i	0.999518	+	0.0310589i	$0.00988793\pi$
$683$	42.4264i	0.0621177i	−0.999518	+	0.0310589i	$0.990112\pi$
$684$	0	0
$685$	779.423	1.13784
$686$	0	0
$687$	0	0
$688$	0	0
$689$	381.838i	0.554191i
$690$	0	0
$691$	400.000	0.578871	0.289436	−	0.957197i	$-0.406532\pi$
$691$	400.000	0.578871	0.289436	+	0.957197i	$0.406532\pi$
$692$	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
$701$	992.043i	1.41518i	0.706622	+	0.707592i	$0.250218\pi$
$701$	992.043i	1.41518i	−0.706622	+	0.707592i	$0.749782\pi$
$702$	0	0
$703$	831.384	1.18262
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1909.19i	2.70041i
$708$	0	0
$709$	1195.12	1.68563	0.842817	−	0.538200i	$-0.180895\pi$
$709$	1195.12	1.68563	0.842817	+	0.538200i	$0.180895\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 763.675i	− 1.07107i
$714$	0	0
$715$	−648.000	−0.906294
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 661.362i	− 0.919836i	−0.887961	−	0.459918i	$-0.847879\pi$
$719$	− 661.362i	− 0.919836i	0.887961	−	0.459918i	$-0.152121\pi$
$720$	0	0
$721$	−972.000	−1.34813
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1065.53i	1.46969i
$726$	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
$731$	848.528i	1.16078i
$732$	0	0
$733$	613.146	0.836488	0.418244	−	0.908335i	$-0.362646\pi$
$733$	613.146	0.836488	0.418244	+	0.908335i	$0.362646\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 848.528i	− 1.15133i
$738$	0	0
$739$	920.000	1.24493	0.622463	−	0.782649i	$-0.286132\pi$
$739$	920.000	1.24493	0.622463	+	0.782649i	$0.286132\pi$
$740$	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
$749$	− 1763.63i	− 2.35465i
$750$	0	0
$751$	51.9615	0.0691898	0.0345949	−	0.999401i	$-0.488986\pi$
$751$	51.9615	0.0691898	0.0345949	+	0.999401i	$0.488986\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1145.51i	1.51724i
$756$	0	0
$757$	405.300	0.535403	0.267701	−	0.963502i	$-0.413736\pi$
$757$	405.300	0.535403	0.267701	+	0.963502i	$0.413736\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1030.96i	1.35475i	0.735640	+	0.677373i	$0.236882\pi$
$761$	1030.96i	1.35475i	−0.735640	+	0.677373i	$0.763118\pi$
$762$	0	0
$763$	540.000	0.707733
$764$	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
$773$	933.256i	1.20732i	0.797243	+	0.603658i	$0.206291\pi$
$773$	933.256i	1.20732i	−0.797243	+	0.603658i	$0.793709\pi$
$774$	0	0
$775$	−1506.88	−1.94437
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1442.50i	1.85173i
$780$	0	0
$781$	623.538	0.798384
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 1221.88i	− 1.55654i
$786$	0	0
$787$	100.000	0.127065	0.0635324	−	0.997980i	$-0.479763\pi$
$787$	100.000	0.127065	0.0635324	+	0.997980i	$0.479763\pi$
$788$	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
$797$	404.166i	0.507109i	0.967321	+	0.253554i	$0.0815997\pi$
$797$	404.166i	0.507109i	−0.967321	+	0.253554i	$0.918400\pi$
$798$	0	0
$799$	1558.85	1.95100
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 169.706i	− 0.211340i
$804$	0	0
$805$	−1122.37	−1.39425
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 1090.36i	− 1.34779i	−0.738829	−	0.673893i	$-0.764621\pi$
$809$	− 1090.36i	− 1.34779i	0.738829	−	0.673893i	$-0.235379\pi$
$810$	0	0
$811$	772.000	0.951911	0.475956	−	0.879469i	$-0.342102\pi$
$811$	772.000	0.951911	0.475956	+	0.879469i	$0.342102\pi$
$812$	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
$821$	477.650i	0.581791i	0.956755	+	0.290896i	$0.0939532\pi$
$821$	477.650i	0.581791i	−0.956755	+	0.290896i	$0.906047\pi$
$822$	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
$827$	− 763.675i	− 0.923428i	−0.887029	−	0.461714i	$-0.847235\pi$
$827$	− 763.675i	− 0.923428i	0.887029	−	0.461714i	$-0.152765\pi$
$828$	0	0
$829$	−363.731	−0.438758	−0.219379	−	0.975640i	$-0.570403\pi$
$829$	−363.731	−0.438758	−0.219379	+	0.975640i	$0.570403\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1251.58i	1.50250i
$834$	0	0
$835$	432.000	0.517365
$836$	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
$845$	448.257i	0.530481i
$846$	0	0
$847$	509.223	0.601208
$848$	0	0
$849$	0	0
$850$	0	0
$851$	610.940i	0.717909i
$852$	0	0
$853$	997.661	1.16959	0.584796	−	0.811181i	$-0.301175\pi$
$853$	997.661	1.16959	0.584796	+	0.811181i	$0.301175\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	572.756i	0.668327i	0.942515	+	0.334164i	$0.108454\pi$
$857$	572.756i	0.668327i	−0.942515	+	0.334164i	$0.891546\pi$
$858$	0	0
$859$	568.000	0.661234	0.330617	−	0.943765i	$-0.392743\pi$
$859$	568.000	0.661234	0.330617	+	0.943765i	$0.392743\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	823.029i	0.953683i	0.878989	+	0.476842i	$0.158218\pi$
$863$	823.029i	0.953683i	−0.878989	+	0.476842i	$0.841782\pi$
$864$	0	0
$865$	−270.000	−0.312139
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 440.908i	− 0.507374i
$870$	0	0
$871$	1039.23	1.19315
$872$	0	0
$873$	0	0
$874$	0	0
$875$	305.470i	0.349109i
$876$	0	0
$877$	41.5692	0.0473993	0.0236997	−	0.999719i	$-0.492455\pi$
$877$	41.5692	0.0473993	0.0236997	+	0.999719i	$0.492455\pi$
$878$	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
$883$	520.000	0.588901	0.294451	−	0.955667i	$-0.404863\pi$
$883$	520.000	0.588901	0.294451	+	0.955667i	$0.404863\pi$
$884$	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
$893$	1469.69i	1.64579i
$894$	0	0
$895$	−374.123	−0.418014
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 1909.19i	− 2.12368i
$900$	0	0
$901$	779.423	0.865064
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 1909.19i	− 2.10960i
$906$	0	0
$907$	40.0000	0.0441014	0.0220507	−	0.999757i	$-0.492980\pi$
$907$	40.0000	0.0441014	0.0220507	+	0.999757i	$0.492980\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 1175.76i	− 1.29062i	−0.763921	−	0.645310i	$-0.776728\pi$
$911$	− 1175.76i	− 1.29062i	0.763921	−	0.645310i	$-0.223272\pi$
$912$	0	0
$913$	1080.00	1.18291
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 529.090i	− 0.576979i
$918$	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
$923$	763.675i	0.827384i
$924$	0	0
$925$	1205.51	1.30325
$926$	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
$931$	−1180.00	−1.26745
$932$	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	$-0.419306\pi$
$937$	470.000	0.501601	0.250800	+	0.968039i	$0.419306\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	624.620i	0.663783i	0.943318	+	0.331892i	$0.107687\pi$
$941$	624.620i	0.663783i	−0.943318	+	0.331892i	$0.892313\pi$
$942$	0	0
$943$	−1060.02	−1.12409
$944$	0	0
$945$	0	0
$946$	0	0
$947$	339.411i	0.358407i	0.983812	+	0.179203i	$0.0573520\pi$
$947$	339.411i	0.358407i	−0.983812	+	0.179203i	$0.942648\pi$
$948$	0	0
$949$	207.846	0.219016
$950$	0	0
$951$	0	0
$952$	0	0
$953$	530.330i	0.556485i	0.960511	+	0.278242i	$0.0897519\pi$
$953$	530.330i	0.556485i	−0.960511	+	0.278242i	$0.910248\pi$
$954$	0	0
$955$	2160.00	2.26178
$956$	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
$965$	73.4847i	0.0761499i
$966$	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
$971$	1603.72i	1.65162i	0.563952	+	0.825808i	$0.309280\pi$
$971$	1603.72i	1.65162i	−0.563952	+	0.825808i	$0.690720\pi$
$972$	0	0
$973$	−1787.48	−1.83708
$974$	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
$979$	108.000	0.110317
$980$	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
$989$	− 587.878i	− 0.594416i
$990$	0	0
$991$	467.654	0.471901	0.235950	−	0.971765i	$-0.424180\pi$
$991$	467.654	0.471901	0.235950	+	0.971765i	$0.424180\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	381.838i	0.383756i
$996$	0	0
$997$	−249.415	−0.250166	−0.125083	−	0.992146i	$-0.539920\pi$
$997$	−249.415	−0.250166	−0.125083	+	0.992146i	$0.539920\pi$
$998$	0	0
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.3.h.b.161.1	✓	8	16.13	even	4
576.3.h.b.161.2	yes	8	48.5	odd	4
576.3.h.b.161.3	yes	8	16.3	odd	4
576.3.h.b.161.4	yes	8	48.11	even	4
576.3.h.b.161.5	yes	8	48.29	odd	4
576.3.h.b.161.6	yes	8	16.5	even	4
576.3.h.b.161.7	yes	8	48.35	even	4
576.3.h.b.161.8	yes	8	16.11	odd	4
2304.3.e.g.1025.1		4	24.11	even	2	inner
2304.3.e.g.1025.2		4	1.1	even	1	trivial
2304.3.e.g.1025.3		4	8.3	odd	2	inner
2304.3.e.g.1025.4		4	3.2	odd	2	inner
2304.3.e.j.1025.1		4	4.3	odd	2
2304.3.e.j.1025.2		4	24.5	odd	2
2304.3.e.j.1025.3		4	12.11	even	2
2304.3.e.j.1025.4		4	8.5	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:	\( N \)	\(=\)	\( 2304 = 2^{8} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2304.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-7.34847i q^{5} +10.3923 q^{7} +O(q^{10})\)	\(q-7.34847i q^{5} +10.3923 q^{7} +8.48528i q^{11} -10.3923 q^{13} +21.2132i q^{17} -20.0000 q^{19} -14.6969i q^{23} -29.0000 q^{25} -36.7423i q^{29} +51.9615 q^{31} -76.3675i q^{35} -41.5692 q^{37} -72.1249i q^{41} +40.0000 q^{43} -73.4847i q^{47} +59.0000 q^{49} -36.7423i q^{53} +62.3538 q^{55} +33.9411i q^{59} +76.3675i q^{65} -100.000 q^{67} -73.4847i q^{71} -20.0000 q^{73} +88.1816i q^{77} -51.9615 q^{79} -127.279i q^{83} +155.885 q^{85} -12.7279i q^{89} -108.000 q^{91} +146.969i q^{95} +40.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 80 q^{19} - 116 q^{25} + 160 q^{43} + 236 q^{49} - 400 q^{67} - 80 q^{73} - 432 q^{91} + 160 q^{97}+O(q^{100}) \) 4 * q - 80 * q^19 - 116 * q^25 + 160 * q^43 + 236 * q^49 - 400 * q^67 - 80 * q^73 - 432 * q^91 + 160 * q^97

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