LMFDB - Embedded newform 2304.3.g.z.1279.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2304,3,Mod(1279,2304)]

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([1, 0, 0]))

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

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

chi := DirichletCharacter("2304.1279");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2304 = 2^{8} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2304.g (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:	$8$
Coefficient field:	8.0.22581504.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 $ x^8 - 4x^7 + 5x^6 + 2x^5 - 11x^4 + 4x^3 + 20x^2 - 32*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{24} $
Twist minimal:	no (minimal twist has level 24)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1279.3
Root			$1.40994 + 0.109843i$ of defining polynomial
Character	$\chi$	$=$	2304.1279
Dual form			2304.3.g.z.1279.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.87875 q^{5} -10.7436i q^{7} +O(q^{10})$	$q-2.87875 q^{5} -10.7436i q^{7} +8.00000i q^{11} -15.7298 q^{13} +15.8564 q^{17} -1.07180i q^{19} -21.4873i q^{23} -16.7128 q^{25} -40.0958 q^{29} +9.20092i q^{31} +30.9282i q^{35} +9.97227 q^{37} +51.5692 q^{41} -12.7846i q^{43} -1.54272i q^{47} -66.4256 q^{49} -28.5808 q^{53} -23.0300i q^{55} +11.2154i q^{59} -1.54272 q^{61} +45.2820 q^{65} +43.2154i q^{67} +84.4063i q^{71} -105.426 q^{73} +85.9491 q^{77} -73.6627i q^{79} +12.2872i q^{83} -45.6466 q^{85} +33.1384 q^{89} +168.995i q^{91} +3.08543i q^{95} -69.1384 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 16 q^{17} + 88 q^{25} + 80 q^{41} - 88 q^{49} - 192 q^{65} - 400 q^{73} - 400 q^{89} + 112 q^{97}+O(q^{100}) $ 8 * q + 16 * q^17 + 88 * q^25 + 80 * q^41 - 88 * q^49 - 192 * q^65 - 400 * q^73 - 400 * q^89 + 112 * 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
$5$	−2.87875	−0.575749	−0.287875	−	0.957668i	$-0.592949\pi$
$5$	−2.87875	−0.575749	−0.287875	+	0.957668i	$0.592949\pi$
$6$	0	0
$7$	− 10.7436i	− 1.53480i	−0.641166	−	0.767402i	$-0.721549\pi$
$7$	− 10.7436i	− 1.53480i	0.641166	−	0.767402i	$-0.278451\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	8.00000i	0.727273i	0.931541	+	0.363636i	$0.118465\pi$
$11$	8.00000i	0.727273i	−0.931541	+	0.363636i	$0.881535\pi$
$12$	0	0
$13$	−15.7298	−1.20998	−0.604991	−	0.796232i	$-0.706823\pi$
$13$	−15.7298	−1.20998	−0.604991	+	0.796232i	$0.706823\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	15.8564	0.932730	0.466365	−	0.884592i	$-0.345563\pi$
$17$	15.8564	0.932730	0.466365	+	0.884592i	$0.345563\pi$
$18$	0	0
$19$	− 1.07180i	− 0.0564104i	−0.999602	−	0.0282052i	$-0.991021\pi$
$19$	− 1.07180i	− 0.0564104i	0.999602	−	0.0282052i	$-0.00897918\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 21.4873i	− 0.934229i	−0.884197	−	0.467114i	$-0.845294\pi$
$23$	− 21.4873i	− 0.934229i	0.884197	−	0.467114i	$-0.154706\pi$
$24$	0	0
$25$	−16.7128	−0.668513
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−40.0958	−1.38261	−0.691307	−	0.722562i	$-0.742965\pi$
$29$	−40.0958	−1.38261	−0.691307	+	0.722562i	$0.742965\pi$
$30$	0	0
$31$	9.20092i	0.296804i	0.988927	+	0.148402i	$0.0474129\pi$
$31$	9.20092i	0.296804i	−0.988927	+	0.148402i	$0.952587\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	30.9282i	0.883663i
$36$	0	0
$37$	9.97227	0.269521	0.134760	−	0.990878i	$-0.456974\pi$
$37$	9.97227	0.269521	0.134760	+	0.990878i	$0.456974\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	51.5692	1.25779	0.628893	−	0.777492i	$-0.283508\pi$
$41$	51.5692	1.25779	0.628893	+	0.777492i	$0.283508\pi$
$42$	0	0
$43$	− 12.7846i	− 0.297317i	−0.988889	−	0.148658i	$-0.952505\pi$
$43$	− 12.7846i	− 0.297317i	0.988889	−	0.148658i	$-0.0474954\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 1.54272i	− 0.0328237i	−0.999865	−	0.0164119i	$-0.994776\pi$
$47$	− 1.54272i	− 0.0328237i	0.999865	−	0.0164119i	$-0.00522430\pi$
$48$	0	0
$49$	−66.4256	−1.35563
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−28.5808	−0.539260	−0.269630	−	0.962964i	$-0.586901\pi$
$53$	−28.5808	−0.539260	−0.269630	+	0.962964i	$0.586901\pi$
$54$	0	0
$55$	− 23.0300i	− 0.418727i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	11.2154i	0.190091i	0.995473	+	0.0950457i	$0.0302997\pi$
$59$	11.2154i	0.190091i	−0.995473	+	0.0950457i	$0.969700\pi$
$60$	0	0
$61$	−1.54272	−0.0252904	−0.0126452	−	0.999920i	$-0.504025\pi$
$61$	−1.54272	−0.0252904	−0.0126452	+	0.999920i	$0.504025\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	45.2820	0.696647
$66$	0	0
$67$	43.2154i	0.645006i	0.946568	+	0.322503i	$0.104524\pi$
$67$	43.2154i	0.645006i	−0.946568	+	0.322503i	$0.895476\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	84.4063i	1.18882i	0.804162	+	0.594411i	$0.202615\pi$
$71$	84.4063i	1.18882i	−0.804162	+	0.594411i	$0.797385\pi$
$72$	0	0
$73$	−105.426	−1.44419	−0.722093	−	0.691796i	$-0.756820\pi$
$73$	−105.426	−1.44419	−0.722093	+	0.691796i	$0.756820\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	85.9491	1.11622
$78$	0	0
$79$	− 73.6627i	− 0.932439i	−0.884669	−	0.466220i	$-0.845616\pi$
$79$	− 73.6627i	− 0.932439i	0.884669	−	0.466220i	$-0.154384\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	12.2872i	0.148038i	0.997257	+	0.0740192i	$0.0235826\pi$
$83$	12.2872i	0.148038i	−0.997257	+	0.0740192i	$0.976417\pi$
$84$	0	0
$85$	−45.6466	−0.537019
$86$	0	0
$87$	0	0
$88$	0	0
$89$	33.1384	0.372342	0.186171	−	0.982517i	$-0.440392\pi$
$89$	33.1384	0.372342	0.186171	+	0.982517i	$0.440392\pi$
$90$	0	0
$91$	168.995i	1.85709i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	3.08543i	0.0324782i
$96$	0	0
$97$	−69.1384	−0.712767	−0.356384	−	0.934340i	$-0.615990\pi$
$97$	−69.1384	−0.712767	−0.356384	+	0.934340i	$0.615990\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	97.2574	0.962944	0.481472	−	0.876461i	$-0.340102\pi$
$101$	97.2574	0.962944	0.481472	+	0.876461i	$0.340102\pi$
$102$	0	0
$103$	139.667i	1.35599i	0.735065	+	0.677996i	$0.237151\pi$
$103$	139.667i	1.35599i	−0.735065	+	0.677996i	$0.762849\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	197.779i	1.84841i	0.381901	+	0.924203i	$0.375269\pi$
$107$	197.779i	1.84841i	−0.381901	+	0.924203i	$0.624731\pi$
$108$	0	0
$109$	190.713	1.74966	0.874832	−	0.484427i	$-0.160972\pi$
$109$	190.713	1.74966	0.874832	+	0.484427i	$0.160972\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	113.713	1.00631	0.503154	−	0.864197i	$-0.332173\pi$
$113$	113.713	1.00631	0.503154	+	0.864197i	$0.332173\pi$
$114$	0	0
$115$	61.8564i	0.537882i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 170.355i	− 1.43156i
$120$	0	0
$121$	57.0000	0.471074
$122$	0	0
$123$	0	0
$124$	0	0
$125$	120.081	0.960645
$126$	0	0
$127$	36.8590i	0.290229i	0.989415	+	0.145114i	$0.0463550\pi$
$127$	36.8590i	0.290229i	−0.989415	+	0.145114i	$0.953645\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	194.641i	1.48581i	0.669397	+	0.742905i	$0.266552\pi$
$131$	194.641i	1.48581i	−0.669397	+	0.742905i	$0.733448\pi$
$132$	0	0
$133$	−11.5150	−0.0865789
$134$	0	0
$135$	0	0
$136$	0	0
$137$	16.4308	0.119933	0.0599664	−	0.998200i	$-0.480901\pi$
$137$	16.4308	0.119933	0.0599664	+	0.998200i	$0.480901\pi$
$138$	0	0
$139$	− 113.492i	− 0.816491i	−0.912872	−	0.408246i	$-0.866141\pi$
$139$	− 113.492i	− 0.816491i	0.912872	−	0.408246i	$-0.133859\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 125.838i	− 0.879987i
$144$	0	0
$145$	115.426	0.796039
$146$	0	0
$147$	0	0
$148$	0	0
$149$	31.6662	0.212525	0.106262	−	0.994338i	$-0.466112\pi$
$149$	31.6662	0.212525	0.106262	+	0.994338i	$0.466112\pi$
$150$	0	0
$151$	− 90.5218i	− 0.599482i	−0.954021	−	0.299741i	$-0.903100\pi$
$151$	− 90.5218i	− 0.599482i	0.954021	−	0.299741i	$-0.0969003\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 26.4871i	− 0.170885i
$156$	0	0
$157$	−231.016	−1.47144	−0.735719	−	0.677287i	$-0.763156\pi$
$157$	−231.016	−1.47144	−0.735719	+	0.677287i	$0.763156\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−230.851	−1.43386
$162$	0	0
$163$	− 22.3538i	− 0.137140i	−0.997646	−	0.0685700i	$-0.978156\pi$
$163$	− 22.3538i	− 0.137140i	0.997646	−	0.0685700i	$-0.0218437\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	221.044i	1.32361i	0.749674	+	0.661807i	$0.230210\pi$
$167$	221.044i	1.32361i	−0.749674	+	0.661807i	$0.769790\pi$
$168$	0	0
$169$	78.4256	0.464057
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−231.525	−1.33830	−0.669148	−	0.743130i	$-0.733341\pi$
$173$	−231.525	−1.33830	−0.669148	+	0.743130i	$0.733341\pi$
$174$	0	0
$175$	179.556i	1.02604i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	193.646i	1.08182i	0.841080	+	0.540911i	$0.181920\pi$
$179$	193.646i	1.08182i	−0.841080	+	0.540911i	$0.818080\pi$
$180$	0	0
$181$	270.492	1.49443	0.747214	−	0.664583i	$-0.231391\pi$
$181$	270.492	1.49443	0.747214	+	0.664583i	$0.231391\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−28.7077	−0.155177
$186$	0	0
$187$	126.851i	0.678349i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	311.510i	1.63094i	0.578798	+	0.815471i	$0.303522\pi$
$191$	311.510i	1.63094i	−0.578798	+	0.815471i	$0.696478\pi$
$192$	0	0
$193$	48.2769	0.250139	0.125070	−	0.992148i	$-0.460085\pi$
$193$	48.2769	0.250139	0.125070	+	0.992148i	$0.460085\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−251.883	−1.27859	−0.639297	−	0.768960i	$-0.720775\pi$
$197$	−251.883	−1.27859	−0.639297	+	0.768960i	$0.720775\pi$
$198$	0	0
$199$	− 72.1200i	− 0.362412i	−0.983445	−	0.181206i	$-0.942000\pi$
$199$	− 72.1200i	− 0.362412i	0.983445	−	0.181206i	$-0.0580001\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	430.774i	2.12204i
$204$	0	0
$205$	−148.455	−0.724170
$206$	0	0
$207$	0	0
$208$	0	0
$209$	8.57437	0.0410257
$210$	0	0
$211$	264.918i	1.25554i	0.778401	+	0.627768i	$0.216031\pi$
$211$	264.918i	1.25554i	−0.778401	+	0.627768i	$0.783969\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	36.8037i	0.171180i
$216$	0	0
$217$	98.8513	0.455536
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−249.418	−1.12859
$222$	0	0
$223$	30.6882i	0.137615i	0.997630	+	0.0688076i	$0.0219195\pi$
$223$	30.6882i	0.137615i	−0.997630	+	0.0688076i	$0.978081\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	295.846i	1.30329i	0.758525	+	0.651643i	$0.225920\pi$
$227$	295.846i	1.30329i	−0.758525	+	0.651643i	$0.774080\pi$
$228$	0	0
$229$	−256.718	−1.12104	−0.560519	−	0.828141i	$-0.689398\pi$
$229$	−256.718	−1.12104	−0.560519	+	0.828141i	$0.689398\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−404.564	−1.73633	−0.868163	−	0.496279i	$-0.834699\pi$
$233$	−404.564	−1.73633	−0.868163	+	0.496279i	$0.834699\pi$
$234$	0	0
$235$	4.44109i	0.0188983i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	115.150i	0.481799i	0.970550	+	0.240899i	$0.0774424\pi$
$239$	115.150i	0.481799i	−0.970550	+	0.240899i	$0.922558\pi$
$240$	0	0
$241$	251.415	1.04322	0.521609	−	0.853185i	$-0.325332\pi$
$241$	251.415	1.04322	0.521609	+	0.853185i	$0.325332\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	191.223	0.780500
$246$	0	0
$247$	16.8591i	0.0682555i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 6.85125i	− 0.0272958i	−0.999907	−	0.0136479i	$-0.995656\pi$
$251$	− 6.85125i	− 0.0272958i	0.999907	−	0.0136479i	$-0.00434440\pi$
$252$	0	0
$253$	171.898	0.679439
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−248.277	−0.966058	−0.483029	−	0.875604i	$-0.660463\pi$
$257$	−248.277	−0.966058	−0.483029	+	0.875604i	$0.660463\pi$
$258$	0	0
$259$	− 107.138i	− 0.413662i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 498.835i	− 1.89671i	−0.317208	−	0.948356i	$-0.602745\pi$
$263$	− 498.835i	− 1.89671i	0.317208	−	0.948356i	$-0.397255\pi$
$264$	0	0
$265$	82.2769	0.310479
$266$	0	0
$267$	0	0
$268$	0	0
$269$	234.611	0.872158	0.436079	−	0.899908i	$-0.356367\pi$
$269$	234.611	0.872158	0.436079	+	0.899908i	$0.356367\pi$
$270$	0	0
$271$	− 101.321i	− 0.373878i	−0.982372	−	0.186939i	$-0.940143\pi$
$271$	− 101.321i	− 0.373878i	0.982372	−	0.186939i	$-0.0598566\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 133.703i	− 0.486191i
$276$	0	0
$277$	−169.942	−0.613509	−0.306755	−	0.951789i	$-0.599243\pi$
$277$	−169.942	−0.613509	−0.306755	+	0.951789i	$0.599243\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−111.128	−0.395474	−0.197737	−	0.980255i	$-0.563359\pi$
$281$	−111.128	−0.395474	−0.197737	+	0.980255i	$0.563359\pi$
$282$	0	0
$283$	550.620i	1.94566i	0.231531	+	0.972828i	$0.425627\pi$
$283$	550.620i	1.94566i	−0.231531	+	0.972828i	$0.574373\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 554.041i	− 1.93046i
$288$	0	0
$289$	−37.5744	−0.130015
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−223.509	−0.762829	−0.381414	−	0.924404i	$-0.624563\pi$
$293$	−223.509	−0.762829	−0.381414	+	0.924404i	$0.624563\pi$
$294$	0	0
$295$	− 32.2863i	− 0.109445i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	337.990i	1.13040i
$300$	0	0
$301$	−137.353	−0.456323
$302$	0	0
$303$	0	0
$304$	0	0
$305$	4.44109	0.0145610
$306$	0	0
$307$	371.790i	1.21104i	0.795829	+	0.605521i	$0.207035\pi$
$307$	371.790i	1.21104i	−0.795829	+	0.605521i	$0.792965\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	330.023i	1.06117i	0.847633	+	0.530583i	$0.178027\pi$
$311$	330.023i	1.06117i	−0.847633	+	0.530583i	$0.821973\pi$
$312$	0	0
$313$	80.2769	0.256476	0.128238	−	0.991743i	$-0.459068\pi$
$313$	80.2769	0.256476	0.128238	+	0.991743i	$0.459068\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	68.8833	0.217297	0.108649	−	0.994080i	$-0.465348\pi$
$317$	68.8833	0.217297	0.108649	+	0.994080i	$0.465348\pi$
$318$	0	0
$319$	− 320.766i	− 1.00554i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 16.9948i	− 0.0526156i
$324$	0	0
$325$	262.889	0.808888
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−16.5744	−0.0503780
$330$	0	0
$331$	− 396.056i	− 1.19654i	−0.801293	−	0.598272i	$-0.795854\pi$
$331$	− 396.056i	− 1.19654i	0.801293	−	0.598272i	$-0.204146\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 124.406i	− 0.371362i
$336$	0	0
$337$	231.723	0.687606	0.343803	−	0.939042i	$-0.388285\pi$
$337$	231.723	0.687606	0.343803	+	0.939042i	$0.388285\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−73.6073	−0.215857
$342$	0	0
$343$	187.215i	0.545815i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 462.123i	− 1.33177i	−0.746056	−	0.665883i	$-0.768055\pi$
$347$	− 462.123i	− 1.33177i	0.746056	−	0.665883i	$-0.231945\pi$
$348$	0	0
$349$	266.993	0.765022	0.382511	−	0.923951i	$-0.375059\pi$
$349$	266.993	0.765022	0.382511	+	0.923951i	$0.375059\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−262.862	−0.744650	−0.372325	−	0.928102i	$-0.621439\pi$
$353$	−262.862	−0.744650	−0.372325	+	0.928102i	$0.621439\pi$
$354$	0	0
$355$	− 242.985i	− 0.684463i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	164.185i	0.457339i	0.973504	+	0.228669i	$0.0734374\pi$
$359$	164.185i	0.457339i	−0.973504	+	0.228669i	$0.926563\pi$
$360$	0	0
$361$	359.851	0.996818
$362$	0	0
$363$	0	0
$364$	0	0
$365$	303.494	0.831490
$366$	0	0
$367$	− 242.475i	− 0.660696i	−0.943859	−	0.330348i	$-0.892834\pi$
$367$	− 242.475i	− 0.660696i	0.943859	−	0.330348i	$-0.107166\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	307.061i	0.827659i
$372$	0	0
$373$	−328.480	−0.880643	−0.440321	−	0.897840i	$-0.645136\pi$
$373$	−328.480	−0.880643	−0.440321	+	0.897840i	$0.645136\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	630.697	1.67294
$378$	0	0
$379$	36.2102i	0.0955415i	0.998858	+	0.0477708i	$0.0152117\pi$
$379$	36.2102i	0.0955415i	−0.998858	+	0.0477708i	$0.984788\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	164.295i	0.428969i	0.976727	+	0.214485i	$0.0688072\pi$
$383$	164.295i	0.428969i	−0.976727	+	0.214485i	$0.931193\pi$
$384$	0	0
$385$	−247.426	−0.642664
$386$	0	0
$387$	0	0
$388$	0	0
$389$	604.936	1.55510	0.777552	−	0.628819i	$-0.216461\pi$
$389$	604.936	1.55510	0.777552	+	0.628819i	$0.216461\pi$
$390$	0	0
$391$	− 340.711i	− 0.871383i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	212.056i	0.536851i
$396$	0	0
$397$	541.699	1.36448	0.682241	−	0.731128i	$-0.261006\pi$
$397$	541.699	1.36448	0.682241	+	0.731128i	$0.261006\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	379.569	0.946557	0.473278	−	0.880913i	$-0.343070\pi$
$401$	379.569	0.946557	0.473278	+	0.880913i	$0.343070\pi$
$402$	0	0
$403$	− 144.728i	− 0.359127i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	79.7782i	0.196015i
$408$	0	0
$409$	−251.415	−0.614707	−0.307354	−	0.951595i	$-0.599443\pi$
$409$	−251.415	−0.614707	−0.307354	+	0.951595i	$0.599443\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	120.494	0.291753
$414$	0	0
$415$	− 35.3717i	− 0.0852330i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	268.133i	0.639936i	0.947428	+	0.319968i	$0.103672\pi$
$419$	268.133i	0.639936i	−0.947428	+	0.319968i	$0.896328\pi$
$420$	0	0
$421$	218.261	0.518434	0.259217	−	0.965819i	$-0.416536\pi$
$421$	218.261	0.518434	0.259217	+	0.965819i	$0.416536\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−265.005	−0.623542
$426$	0	0
$427$	16.5744i	0.0388159i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	550.955i	1.27832i	0.769074	+	0.639159i	$0.220718\pi$
$431$	550.955i	1.27832i	−0.769074	+	0.639159i	$0.779282\pi$
$432$	0	0
$433$	−263.128	−0.607686	−0.303843	−	0.952722i	$-0.598270\pi$
$433$	−263.128	−0.607686	−0.303843	+	0.952722i	$0.598270\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−23.0300	−0.0527002
$438$	0	0
$439$	440.489i	1.00339i	0.865044	+	0.501696i	$0.167290\pi$
$439$	440.489i	1.00339i	−0.865044	+	0.501696i	$0.832710\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	228.708i	0.516270i	0.966109	+	0.258135i	$0.0831080\pi$
$443$	228.708i	0.516270i	−0.966109	+	0.258135i	$0.916892\pi$
$444$	0	0
$445$	−95.3972	−0.214376
$446$	0	0
$447$	0	0
$448$	0	0
$449$	108.410	0.241448	0.120724	−	0.992686i	$-0.461478\pi$
$449$	108.410	0.241448	0.120724	+	0.992686i	$0.461478\pi$
$450$	0	0
$451$	412.554i	0.914753i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 486.493i	− 1.06922i
$456$	0	0
$457$	−561.692	−1.22909	−0.614543	−	0.788883i	$-0.710660\pi$
$457$	−561.692	−1.22909	−0.614543	+	0.788883i	$0.710660\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−335.160	−0.727028	−0.363514	−	0.931589i	$-0.618423\pi$
$461$	−335.160	−0.727028	−0.363514	+	0.931589i	$0.618423\pi$
$462$	0	0
$463$	389.912i	0.842142i	0.907028	+	0.421071i	$0.138346\pi$
$463$	389.912i	0.842142i	−0.907028	+	0.421071i	$0.861654\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 546.410i	− 1.17004i	−0.811018	−	0.585022i	$-0.801086\pi$
$467$	− 546.410i	− 1.17004i	0.811018	−	0.585022i	$-0.198914\pi$
$468$	0	0
$469$	464.290	0.989958
$470$	0	0
$471$	0	0
$472$	0	0
$473$	102.277	0.216230
$474$	0	0
$475$	17.9127i	0.0377110i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 368.369i	− 0.769037i	−0.923117	−	0.384519i	$-0.874367\pi$
$479$	− 368.369i	− 0.769037i	0.923117	−	0.384519i	$-0.125633\pi$
$480$	0	0
$481$	−156.862	−0.326116
$482$	0	0
$483$	0	0
$484$	0	0
$485$	199.032	0.410375
$486$	0	0
$487$	90.6326i	0.186104i	0.995661	+	0.0930519i	$0.0296623\pi$
$487$	90.6326i	0.186104i	−0.995661	+	0.0930519i	$0.970338\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	17.6462i	0.0359392i	0.999839	+	0.0179696i	$0.00572022\pi$
$491$	17.6462i	0.0359392i	−0.999839	+	0.0179696i	$0.994280\pi$
$492$	0	0
$493$	−635.775	−1.28960
$494$	0	0
$495$	0	0
$496$	0	0
$497$	906.831	1.82461
$498$	0	0
$499$	− 548.631i	− 1.09946i	−0.835342	−	0.549730i	$-0.814731\pi$
$499$	− 548.631i	− 1.09946i	0.835342	−	0.549730i	$-0.185269\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	262.475i	0.521820i	0.965363	+	0.260910i	$0.0840225\pi$
$503$	262.475i	0.521820i	−0.965363	+	0.260910i	$0.915977\pi$
$504$	0	0
$505$	−279.979	−0.554415
$506$	0	0
$507$	0	0
$508$	0	0
$509$	230.093	0.452049	0.226025	−	0.974122i	$-0.427427\pi$
$509$	230.093	0.452049	0.226025	+	0.974122i	$0.427427\pi$
$510$	0	0
$511$	1132.65i	2.21654i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 402.067i	− 0.780712i
$516$	0	0
$517$	12.3417	0.0238718
$518$	0	0
$519$	0	0
$520$	0	0
$521$	164.144	0.315055	0.157527	−	0.987515i	$-0.449648\pi$
$521$	164.144	0.315055	0.157527	+	0.987515i	$0.449648\pi$
$522$	0	0
$523$	− 185.492i	− 0.354670i	−0.984151	−	0.177335i	$-0.943252\pi$
$523$	− 185.492i	− 0.354670i	0.984151	−	0.177335i	$-0.0567476\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	145.893i	0.276838i
$528$	0	0
$529$	67.2975	0.127216
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−811.172	−1.52190
$534$	0	0
$535$	− 569.357i	− 1.06422i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 531.405i	− 0.985909i
$540$	0	0
$541$	−891.253	−1.64742	−0.823709	−	0.567013i	$-0.808099\pi$
$541$	−891.253	−1.64742	−0.823709	+	0.567013i	$0.808099\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−549.015	−1.00737
$546$	0	0
$547$	− 524.631i	− 0.959105i	−0.877513	−	0.479553i	$-0.840799\pi$
$547$	− 524.631i	− 0.959105i	0.877513	−	0.479553i	$-0.159201\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	42.9745i	0.0779937i
$552$	0	0
$553$	−791.405	−1.43111
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−570.804	−1.02478	−0.512391	−	0.858752i	$-0.671240\pi$
$557$	−570.804	−1.02478	−0.512391	+	0.858752i	$0.671240\pi$
$558$	0	0
$559$	201.099i	0.359748i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 161.877i	− 0.287526i	−0.989612	−	0.143763i	$-0.954080\pi$
$563$	− 161.877i	− 0.287526i	0.989612	−	0.143763i	$-0.0459203\pi$
$564$	0	0
$565$	−327.350	−0.579381
$566$	0	0
$567$	0	0
$568$	0	0
$569$	624.123	1.09688	0.548438	−	0.836191i	$-0.315222\pi$
$569$	624.123	1.09688	0.548438	+	0.836191i	$0.315222\pi$
$570$	0	0
$571$	− 593.031i	− 1.03858i	−0.854597	−	0.519291i	$-0.826196\pi$
$571$	− 593.031i	− 1.03858i	0.854597	−	0.519291i	$-0.173804\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	359.113i	0.624544i
$576$	0	0
$577$	−1003.68	−1.73948	−0.869742	−	0.493507i	$-0.835715\pi$
$577$	−1003.68	−1.73948	−0.869742	+	0.493507i	$0.835715\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	132.009	0.227210
$582$	0	0
$583$	− 228.646i	− 0.392189i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 859.215i	− 1.46374i	−0.681444	−	0.731870i	$-0.738648\pi$
$587$	− 859.215i	− 1.46374i	0.681444	−	0.731870i	$-0.261352\pi$
$588$	0	0
$589$	9.86151	0.0167428
$590$	0	0
$591$	0	0
$592$	0	0
$593$	1007.42	1.69885	0.849423	−	0.527713i	$-0.176950\pi$
$593$	1007.42	1.69885	0.849423	+	0.527713i	$0.176950\pi$
$594$	0	0
$595$	490.410i	0.824219i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	86.0598i	0.143672i	0.997416	+	0.0718362i	$0.0228859\pi$
$599$	86.0598i	0.143672i	−0.997416	+	0.0718362i	$0.977114\pi$
$600$	0	0
$601$	406.000	0.675541	0.337770	−	0.941229i	$-0.390327\pi$
$601$	406.000	0.675541	0.337770	+	0.941229i	$0.390327\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−164.089	−0.271221
$606$	0	0
$607$	− 1187.80i	− 1.95684i	−0.206617	−	0.978422i	$-0.566245\pi$
$607$	− 1187.80i	− 1.95684i	0.206617	−	0.978422i	$-0.433755\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	24.2666i	0.0397162i
$612$	0	0
$613$	−500.378	−0.816277	−0.408139	−	0.912920i	$-0.633822\pi$
$613$	−500.378	−0.816277	−0.408139	+	0.912920i	$0.633822\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	88.8306	0.143972	0.0719859	−	0.997406i	$-0.477066\pi$
$617$	88.8306	0.143972	0.0719859	+	0.997406i	$0.477066\pi$
$618$	0	0
$619$	− 424.231i	− 0.685349i	−0.939454	−	0.342674i	$-0.888667\pi$
$619$	− 424.231i	− 0.685349i	0.939454	−	0.342674i	$-0.111333\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 356.027i	− 0.571472i
$624$	0	0
$625$	72.1384	0.115422
$626$	0	0
$627$	0	0
$628$	0	0
$629$	158.124	0.251390
$630$	0	0
$631$	90.6326i	0.143633i	0.997418	+	0.0718166i	$0.0228796\pi$
$631$	90.6326i	0.143633i	−0.997418	+	0.0718166i	$0.977120\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 106.108i	− 0.167099i
$636$	0	0
$637$	1044.86	1.64028
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−1090.40	−1.70109	−0.850546	−	0.525901i	$-0.823728\pi$
$641$	−1090.40	−1.70109	−0.850546	+	0.525901i	$0.823728\pi$
$642$	0	0
$643$	454.200i	0.706376i	0.935552	+	0.353188i	$0.114902\pi$
$643$	454.200i	0.706376i	−0.935552	+	0.353188i	$0.885098\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 610.789i	− 0.944032i	−0.881590	−	0.472016i	$-0.843526\pi$
$647$	− 610.789i	− 0.944032i	0.881590	−	0.472016i	$-0.156474\pi$
$648$	0	0
$649$	−89.7231	−0.138248
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−673.612	−1.03157	−0.515783	−	0.856720i	$-0.672499\pi$
$653$	−673.612	−1.03157	−0.515783	+	0.856720i	$0.672499\pi$
$654$	0	0
$655$	− 560.322i	− 0.855454i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 819.328i	− 1.24329i	−0.783299	−	0.621645i	$-0.786465\pi$
$659$	− 819.328i	− 1.24329i	0.783299	−	0.621645i	$-0.213535\pi$
$660$	0	0
$661$	370.628	0.560707	0.280354	−	0.959897i	$-0.409548\pi$
$661$	370.628	0.560707	0.280354	+	0.959897i	$0.409548\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	33.1487	0.0498477
$666$	0	0
$667$	861.549i	1.29168i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 12.3417i	− 0.0183930i
$672$	0	0
$673$	−255.703	−0.379944	−0.189972	−	0.981789i	$-0.560840\pi$
$673$	−255.703	−0.379944	−0.189972	+	0.981789i	$0.560840\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−934.323	−1.38009	−0.690047	−	0.723765i	$-0.742410\pi$
$677$	−934.323	−1.38009	−0.690047	+	0.723765i	$0.742410\pi$
$678$	0	0
$679$	742.798i	1.09396i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 1142.54i	− 1.67283i	−0.548096	−	0.836415i	$-0.684647\pi$
$683$	− 1142.54i	− 1.67283i	0.548096	−	0.836415i	$-0.315353\pi$
$684$	0	0
$685$	−47.3001	−0.0690512
$686$	0	0
$687$	0	0
$688$	0	0
$689$	449.569	0.652495
$690$	0	0
$691$	− 1316.90i	− 1.90578i	−0.303309	−	0.952892i	$-0.598091\pi$
$691$	− 1316.90i	− 1.90578i	0.303309	−	0.952892i	$-0.401909\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	326.716i	0.470094i
$696$	0	0
$697$	817.703	1.17317
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−318.493	−0.454341	−0.227170	−	0.973855i	$-0.572947\pi$
$701$	−318.493	−0.454341	−0.227170	+	0.973855i	$0.572947\pi$
$702$	0	0
$703$	− 10.6883i	− 0.0152038i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 1044.90i	− 1.47793i
$708$	0	0
$709$	289.831	0.408788	0.204394	−	0.978889i	$-0.434478\pi$
$709$	289.831	0.408788	0.204394	+	0.978889i	$0.434478\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	197.703	0.277283
$714$	0	0
$715$	362.256i	0.506652i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 491.122i	− 0.683062i	−0.939870	−	0.341531i	$-0.889055\pi$
$719$	− 491.122i	− 0.683062i	0.939870	−	0.341531i	$-0.110945\pi$
$720$	0	0
$721$	1500.53	2.08118
$722$	0	0
$723$	0	0
$724$	0	0
$725$	670.113	0.924294
$726$	0	0
$727$	774.918i	1.06591i	0.846143	+	0.532956i	$0.178919\pi$
$727$	774.918i	1.06591i	−0.846143	+	0.532956i	$0.821081\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 202.718i	− 0.277316i
$732$	0	0
$733$	858.966	1.17185	0.585925	−	0.810365i	$-0.300731\pi$
$733$	858.966	1.17185	0.585925	+	0.810365i	$0.300731\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−345.723	−0.469095
$738$	0	0
$739$	− 63.1948i	− 0.0855139i	−0.999086	−	0.0427569i	$-0.986386\pi$
$739$	− 63.1948i	− 0.0855139i	0.999086	−	0.0427569i	$-0.0136141\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1105.00i	1.48721i	0.668620	+	0.743604i	$0.266885\pi$
$743$	1105.00i	1.48721i	−0.668620	+	0.743604i	$0.733115\pi$
$744$	0	0
$745$	−91.1591	−0.122361
$746$	0	0
$747$	0	0
$748$	0	0
$749$	2124.87	2.83694
$750$	0	0
$751$	804.119i	1.07073i	0.844621	+	0.535365i	$0.179826\pi$
$751$	804.119i	1.07073i	−0.844621	+	0.535365i	$0.820174\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	260.589i	0.345152i
$756$	0	0
$757$	−1351.03	−1.78471	−0.892355	−	0.451334i	$-0.850948\pi$
$757$	−1351.03	−1.78471	−0.892355	+	0.451334i	$0.850948\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−709.805	−0.932726	−0.466363	−	0.884593i	$-0.654436\pi$
$761$	−709.805	−0.932726	−0.466363	+	0.884593i	$0.654436\pi$
$762$	0	0
$763$	− 2048.95i	− 2.68539i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 176.415i	− 0.230007i
$768$	0	0
$769$	−195.703	−0.254490	−0.127245	−	0.991871i	$-0.540613\pi$
$769$	−195.703	−0.254490	−0.127245	+	0.991871i	$0.540613\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−12.5484	−0.0162334	−0.00811670	−	0.999967i	$-0.502584\pi$
$773$	−12.5484	−0.0162334	−0.00811670	+	0.999967i	$0.502584\pi$
$774$	0	0
$775$	− 153.773i	− 0.198417i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 55.2717i	− 0.0709521i
$780$	0	0
$781$	−675.251	−0.864598
$782$	0	0
$783$	0	0
$784$	0	0
$785$	665.036	0.847180
$786$	0	0
$787$	432.918i	0.550086i	0.961432	+	0.275043i	$0.0886921\pi$
$787$	432.918i	0.550086i	−0.961432	+	0.275043i	$0.911308\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 1221.69i	− 1.54449i
$792$	0	0
$793$	24.2666	0.0306010
$794$	0	0
$795$	0	0
$796$	0	0
$797$	700.746	0.879230	0.439615	−	0.898186i	$-0.355115\pi$
$797$	700.746	0.879230	0.439615	+	0.898186i	$0.355115\pi$
$798$	0	0
$799$	− 24.4619i	− 0.0306157i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 843.405i	− 1.05032i
$804$	0	0
$805$	664.562	0.825543
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−848.102	−1.04833	−0.524167	−	0.851615i	$-0.675623\pi$
$809$	−848.102	−1.04833	−0.524167	+	0.851615i	$0.675623\pi$
$810$	0	0
$811$	1242.18i	1.53166i	0.643041	+	0.765832i	$0.277673\pi$
$811$	1242.18i	1.53166i	−0.643041	+	0.765832i	$0.722327\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	64.3510i	0.0789583i
$816$	0	0
$817$	−13.7025	−0.0167717
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−208.303	−0.253719	−0.126859	−	0.991921i	$-0.540490\pi$
$821$	−208.303	−0.253719	−0.126859	+	0.991921i	$0.540490\pi$
$822$	0	0
$823$	− 778.114i	− 0.945461i	−0.881207	−	0.472730i	$-0.843268\pi$
$823$	− 778.114i	− 0.945461i	0.881207	−	0.472730i	$-0.156732\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1280.57i	1.54845i	0.632910	+	0.774225i	$0.281860\pi$
$827$	1280.57i	1.54845i	−0.632910	+	0.774225i	$0.718140\pi$
$828$	0	0
$829$	9.78043	0.0117979	0.00589893	−	0.999983i	$-0.498122\pi$
$829$	9.78043	0.0117979	0.00589893	+	0.999983i	$0.498122\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−1053.27	−1.26443
$834$	0	0
$835$	− 636.328i	− 0.762070i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1232.38i	1.46886i	0.678682	+	0.734432i	$0.262551\pi$
$839$	1232.38i	1.46886i	−0.678682	+	0.734432i	$0.737449\pi$
$840$	0	0
$841$	766.672	0.911619
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−225.768	−0.267181
$846$	0	0
$847$	− 612.387i	− 0.723007i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 214.277i	− 0.251794i
$852$	0	0
$853$	412.170	0.483201	0.241600	−	0.970376i	$-0.422328\pi$
$853$	412.170	0.483201	0.241600	+	0.970376i	$0.422328\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−136.451	−0.159220	−0.0796099	−	0.996826i	$-0.525367\pi$
$857$	−136.451	−0.159220	−0.0796099	+	0.996826i	$0.525367\pi$
$858$	0	0
$859$	− 649.646i	− 0.756282i	−0.925748	−	0.378141i	$-0.876563\pi$
$859$	− 649.646i	− 0.756282i	0.925748	−	0.378141i	$-0.123437\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1192.38i	1.38167i	0.723015	+	0.690833i	$0.242756\pi$
$863$	1192.38i	1.38167i	−0.723015	+	0.690833i	$0.757244\pi$
$864$	0	0
$865$	666.502	0.770523
$866$	0	0
$867$	0	0
$868$	0	0
$869$	589.302	0.678138
$870$	0	0
$871$	− 679.768i	− 0.780446i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 1290.10i	− 1.47440i
$876$	0	0
$877$	744.451	0.848861	0.424431	−	0.905460i	$-0.360474\pi$
$877$	744.451	0.848861	0.424431	+	0.905460i	$0.360474\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−651.108	−0.739055	−0.369528	−	0.929220i	$-0.620480\pi$
$881$	−651.108	−0.739055	−0.369528	+	0.929220i	$0.620480\pi$
$882$	0	0
$883$	− 1171.44i	− 1.32666i	−0.748327	−	0.663330i	$-0.769143\pi$
$883$	− 1171.44i	− 1.32666i	0.748327	−	0.663330i	$-0.230857\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1026.87i	− 1.15769i	−0.815437	−	0.578845i	$-0.803504\pi$
$887$	− 1026.87i	− 1.15769i	0.815437	−	0.578845i	$-0.196496\pi$
$888$	0	0
$889$	396.000	0.445444
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−1.65348	−0.00185160
$894$	0	0
$895$	− 557.458i	− 0.622859i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 368.918i	− 0.410365i
$900$	0	0
$901$	−453.189	−0.502984
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−778.677	−0.860416
$906$	0	0
$907$	− 470.508i	− 0.518752i	−0.965776	−	0.259376i	$-0.916483\pi$
$907$	− 470.508i	− 0.518752i	0.965776	−	0.259376i	$-0.0835168\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 553.930i	− 0.608046i	−0.952665	−	0.304023i	$-0.901670\pi$
$911$	− 553.930i	− 0.608046i	0.952665	−	0.304023i	$-0.0983300\pi$
$912$	0	0
$913$	−98.2975	−0.107664
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2091.15	2.28043
$918$	0	0
$919$	− 910.123i	− 0.990341i	−0.868796	−	0.495170i	$-0.835106\pi$
$919$	− 910.123i	− 0.990341i	0.868796	−	0.495170i	$-0.164894\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 1327.69i	− 1.43845i
$924$	0	0
$925$	−166.665	−0.180178
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−1278.69	−1.37641	−0.688206	−	0.725515i	$-0.741602\pi$
$929$	−1278.69	−1.37641	−0.688206	+	0.725515i	$0.741602\pi$
$930$	0	0
$931$	71.1948i	0.0764713i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 365.173i	− 0.390559i
$936$	0	0
$937$	−634.554	−0.677219	−0.338609	−	0.940927i	$-0.609956\pi$
$937$	−634.554	−0.677219	−0.338609	+	0.940927i	$0.609956\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−1136.25	−1.20749	−0.603745	−	0.797177i	$-0.706326\pi$
$941$	−1136.25	−1.20749	−0.603745	+	0.797177i	$0.706326\pi$
$942$	0	0
$943$	− 1108.08i	− 1.17506i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	775.615i	0.819023i	0.912305	+	0.409512i	$0.134301\pi$
$947$	775.615i	0.819023i	−0.912305	+	0.409512i	$0.865699\pi$
$948$	0	0
$949$	1658.32	1.74744
$950$	0	0
$951$	0	0
$952$	0	0
$953$	599.590	0.629160	0.314580	−	0.949231i	$-0.398136\pi$
$953$	599.590	0.629160	0.314580	+	0.949231i	$0.398136\pi$
$954$	0	0
$955$	− 896.758i	− 0.939014i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 176.526i	− 0.184073i
$960$	0	0
$961$	876.343	0.911908
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−138.977	−0.144018
$966$	0	0
$967$	1407.53i	1.45556i	0.685811	+	0.727780i	$0.259448\pi$
$967$	1407.53i	1.45556i	−0.685811	+	0.727780i	$0.740552\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 653.661i	− 0.673184i	−0.941651	−	0.336592i	$-0.890726\pi$
$971$	− 653.661i	− 0.673184i	0.941651	−	0.336592i	$-0.109274\pi$
$972$	0	0
$973$	−1219.32	−1.25315
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1003.57	1.02719	0.513597	−	0.858031i	$-0.328313\pi$
$977$	1003.57	1.02719	0.513597	+	0.858031i	$0.328313\pi$
$978$	0	0
$979$	265.108i	0.270794i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 105.672i	− 0.107500i	−0.998554	−	0.0537498i	$-0.982883\pi$
$983$	− 105.672i	− 0.107500i	0.998554	−	0.0537498i	$-0.0171173\pi$
$984$	0	0
$985$	725.108	0.736150
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−274.706	−0.277762
$990$	0	0
$991$	1728.18i	1.74388i	0.489615	+	0.871938i	$0.337137\pi$
$991$	1728.18i	1.74388i	−0.489615	+	0.871938i	$0.662863\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	207.615i	0.208659i
$996$	0	0
$997$	−135.205	−0.135612	−0.0678060	−	0.997699i	$-0.521600\pi$
$997$	−135.205	−0.135612	−0.0678060	+	0.997699i	$0.521600\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2304.3.g.z.1279.3		8
3.2	odd	2		768.3.g.h.511.3		8
4.3	odd	2	inner	2304.3.g.z.1279.4		8
8.3	odd	2	inner	2304.3.g.z.1279.6		8
8.5	even	2	inner	2304.3.g.z.1279.5		8
12.11	even	2		768.3.g.h.511.7		8
16.3	odd	4		288.3.b.b.271.3		4
16.5	even	4		288.3.b.b.271.2		4
16.11	odd	4		72.3.b.b.19.3		4
16.13	even	4		72.3.b.b.19.4		4
24.5	odd	2		768.3.g.h.511.6		8
24.11	even	2		768.3.g.h.511.2		8
48.5	odd	4		96.3.b.a.79.2		4
48.11	even	4		24.3.b.a.19.2	yes	4
48.29	odd	4		24.3.b.a.19.1	✓	4
48.35	even	4		96.3.b.a.79.1		4
240.29	odd	4		600.3.g.a.451.4		4
240.53	even	4		2400.3.p.a.1999.4		8
240.59	even	4		600.3.g.a.451.3		4
240.77	even	4		600.3.p.a.499.7		8
240.83	odd	4		2400.3.p.a.1999.1		8
240.107	odd	4		600.3.p.a.499.1		8
240.149	odd	4		2400.3.g.a.751.3		4
240.173	even	4		600.3.p.a.499.2		8
240.179	even	4		2400.3.g.a.751.4		4
240.197	even	4		2400.3.p.a.1999.5		8
240.203	odd	4		600.3.p.a.499.8		8
240.227	odd	4		2400.3.p.a.1999.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.3.b.a.19.1	✓	4	48.29	odd	4
24.3.b.a.19.2	yes	4	48.11	even	4
72.3.b.b.19.3		4	16.11	odd	4
72.3.b.b.19.4		4	16.13	even	4
96.3.b.a.79.1		4	48.35	even	4
96.3.b.a.79.2		4	48.5	odd	4
288.3.b.b.271.2		4	16.5	even	4
288.3.b.b.271.3		4	16.3	odd	4
600.3.g.a.451.3		4	240.59	even	4
600.3.g.a.451.4		4	240.29	odd	4
600.3.p.a.499.1		8	240.107	odd	4
600.3.p.a.499.2		8	240.173	even	4
600.3.p.a.499.7		8	240.77	even	4
600.3.p.a.499.8		8	240.203	odd	4
768.3.g.h.511.2		8	24.11	even	2
768.3.g.h.511.3		8	3.2	odd	2
768.3.g.h.511.6		8	24.5	odd	2
768.3.g.h.511.7		8	12.11	even	2
2304.3.g.z.1279.3		8	1.1	even	1	trivial
2304.3.g.z.1279.4		8	4.3	odd	2	inner
2304.3.g.z.1279.5		8	8.5	even	2	inner
2304.3.g.z.1279.6		8	8.3	odd	2	inner
2400.3.g.a.751.3		4	240.149	odd	4
2400.3.g.a.751.4		4	240.179	even	4
2400.3.p.a.1999.1		8	240.83	odd	4
2400.3.p.a.1999.4		8	240.53	even	4
2400.3.p.a.1999.5		8	240.197	even	4
2400.3.p.a.1999.8		8	240.227	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:	\( N \)	\(=\)	\( 2304 = 2^{8} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2304.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.87875 q^{5} -10.7436i q^{7} +O(q^{10})\)	\(q-2.87875 q^{5} -10.7436i q^{7} +8.00000i q^{11} -15.7298 q^{13} +15.8564 q^{17} -1.07180i q^{19} -21.4873i q^{23} -16.7128 q^{25} -40.0958 q^{29} +9.20092i q^{31} +30.9282i q^{35} +9.97227 q^{37} +51.5692 q^{41} -12.7846i q^{43} -1.54272i q^{47} -66.4256 q^{49} -28.5808 q^{53} -23.0300i q^{55} +11.2154i q^{59} -1.54272 q^{61} +45.2820 q^{65} +43.2154i q^{67} +84.4063i q^{71} -105.426 q^{73} +85.9491 q^{77} -73.6627i q^{79} +12.2872i q^{83} -45.6466 q^{85} +33.1384 q^{89} +168.995i q^{91} +3.08543i q^{95} -69.1384 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 16 q^{17} + 88 q^{25} + 80 q^{41} - 88 q^{49} - 192 q^{65} - 400 q^{73} - 400 q^{89} + 112 q^{97}+O(q^{100}) \) 8 * q + 16 * q^17 + 88 * q^25 + 80 * q^41 - 88 * q^49 - 192 * q^65 - 400 * q^73 - 400 * q^89 + 112 * q^97

Introduction

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