LMFDB - Embedded newform 1152.4.l.b.287.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,4,Mod(287,1152)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1152, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1, 2]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1152.287");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1152.l (of order $4$, degree $2$, 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:	$67.9702003266$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$24$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 144)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			287.8
Character	$\chi$	$=$	1152.287
Dual form			1152.4.l.b.863.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-6.30986 + 6.30986i) q^{5} +27.2034 q^{7} +O(q^{10})$	$q+(-6.30986 + 6.30986i) q^{5} +27.2034 q^{7} +(4.03980 + 4.03980i) q^{11} +(-37.9212 + 37.9212i) q^{13} +79.8655i q^{17} +(-75.2269 - 75.2269i) q^{19} +25.2124i q^{23} +45.3714i q^{25} +(107.714 + 107.714i) q^{29} -237.126i q^{31} +(-171.649 + 171.649i) q^{35} +(-210.098 - 210.098i) q^{37} -378.638 q^{41} +(-191.327 + 191.327i) q^{43} +417.483 q^{47} +397.023 q^{49} +(-139.419 + 139.419i) q^{53} -50.9811 q^{55} +(282.641 + 282.641i) q^{59} +(255.821 - 255.821i) q^{61} -478.555i q^{65} +(348.674 + 348.674i) q^{67} -321.318i q^{71} -135.177i q^{73} +(109.896 + 109.896i) q^{77} +522.058i q^{79} +(-444.221 + 444.221i) q^{83} +(-503.940 - 503.940i) q^{85} -1102.27 q^{89} +(-1031.58 + 1031.58i) q^{91} +949.342 q^{95} -1069.15 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q+O(q^{10}) $ 48 * q	$ 48 q + 48 q^{19} - 864 q^{43} + 2352 q^{49} - 576 q^{55} - 1824 q^{61} - 816 q^{67} + 480 q^{85} + 3600 q^{91}+O(q^{100}) $ 48 * q + 48 * q^19 - 864 * q^43 + 2352 * q^49 - 576 * q^55 - 1824 * q^61 - 816 * q^67 + 480 * q^85 + 3600 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$.

$n$	$127$	$641$	$901$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{1}{4}\right)$

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
$5$	−6.30986	+	6.30986i	−0.564371	+	0.564371i	−0.930546	−	0.366175i	$-0.880667\pi$
$5$	−6.30986	+	6.30986i	−0.564371	+	0.564371i	0.366175	+	0.930546i	$0.380667\pi$
$6$		0			0
$7$	27.2034			1.46884			0.734422	−	0.678693i	$-0.237453\pi$
$7$	27.2034			1.46884			0.734422	+	0.678693i	$0.237453\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	4.03980	+	4.03980i	0.110731	+	0.110731i	0.760302	−	0.649570i	$-0.225051\pi$
$11$	4.03980	+	4.03980i	0.110731	+	0.110731i	−0.649570	+	0.760302i	$0.725051\pi$
$12$		0			0
$13$	−37.9212	+	37.9212i	−0.809034	+	0.809034i	−0.984488	−	0.175453i	$-0.943861\pi$
$13$	−37.9212	+	37.9212i	−0.809034	+	0.809034i	0.175453	+	0.984488i	$0.443861\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			79.8655i			1.13943i	0.821844	+	0.569713i	$0.192945\pi$
$17$			79.8655i			1.13943i	−0.821844	+	0.569713i	$0.807055\pi$
$18$		0			0
$19$	−75.2269	−	75.2269i	−0.908328	−	0.908328i	0.0878089	−	0.996137i	$-0.472014\pi$
$19$	−75.2269	−	75.2269i	−0.908328	−	0.908328i	−0.996137	+	0.0878089i	$0.972014\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			25.2124i			0.228572i	0.993448	+	0.114286i	$0.0364580\pi$
$23$			25.2124i			0.228572i	−0.993448	+	0.114286i	$0.963542\pi$
$24$		0			0
$25$			45.3714i			0.362971i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	107.714	+	107.714i	0.689726	+	0.689726i	0.962171	−	0.272445i	$-0.0878324\pi$
$29$	107.714	+	107.714i	0.689726	+	0.689726i	−0.272445	+	0.962171i	$0.587832\pi$
$30$		0			0
$31$		−	237.126i		−	1.37384i	−0.726732	−	0.686921i	$-0.758962\pi$
$31$		−	237.126i		−	1.37384i	0.726732	−	0.686921i	$-0.241038\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−171.649	+	171.649i	−0.828973	+	0.828973i
$36$		0			0
$37$	−210.098	−	210.098i	−0.933512	−	0.933512i	0.0644112	−	0.997923i	$-0.479483\pi$
$37$	−210.098	−	210.098i	−0.933512	−	0.933512i	−0.997923	+	0.0644112i	$0.979483\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−378.638			−1.44228			−0.721139	−	0.692790i	$-0.756381\pi$
$41$	−378.638			−1.44228			−0.721139	+	0.692790i	$0.756381\pi$
$42$		0			0
$43$	−191.327	+	191.327i	−0.678538	+	0.678538i	−0.959669	−	0.281131i	$-0.909290\pi$
$43$	−191.327	+	191.327i	−0.678538	+	0.678538i	0.281131	+	0.959669i	$0.409290\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	417.483			1.29566			0.647831	−	0.761784i	$-0.275676\pi$
$47$	417.483			1.29566			0.647831	+	0.761784i	$0.275676\pi$
$48$		0			0
$49$	397.023			1.15750
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−139.419	+	139.419i	−0.361335	+	0.361335i	−0.864304	−	0.502970i	$-0.832241\pi$
$53$	−139.419	+	139.419i	−0.361335	+	0.361335i	0.502970	+	0.864304i	$0.332241\pi$
$54$		0			0
$55$	−50.9811			−0.124987
$56$		0			0
$57$		0			0
$58$		0			0
$59$	282.641	+	282.641i	0.623674	+	0.623674i	0.946469	−	0.322795i	$-0.104623\pi$
$59$	282.641	+	282.641i	0.623674	+	0.623674i	−0.322795	+	0.946469i	$0.604623\pi$
$60$		0			0
$61$	255.821	−	255.821i	0.536960	−	0.536960i	−0.385675	−	0.922635i	$-0.626031\pi$
$61$	255.821	−	255.821i	0.536960	−	0.536960i	0.922635	+	0.385675i	$0.126031\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	478.555i		−	0.913191i
$66$		0			0
$67$	348.674	+	348.674i	0.635781	+	0.635781i	0.949512	−	0.313731i	$-0.101579\pi$
$67$	348.674	+	348.674i	0.635781	+	0.635781i	−0.313731	+	0.949512i	$0.601579\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	321.318i		−	0.537090i	−0.963267	−	0.268545i	$-0.913457\pi$
$71$		−	321.318i		−	0.537090i	0.963267	−	0.268545i	$-0.0865428\pi$
$72$		0			0
$73$		−	135.177i		−	0.216729i	−0.994111	−	0.108365i	$-0.965439\pi$
$73$		−	135.177i		−	0.216729i	0.994111	−	0.108365i	$-0.0345614\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	109.896	+	109.896i	0.162647	+	0.162647i
$78$		0			0
$79$			522.058i			0.743495i	0.928334	+	0.371747i	$0.121241\pi$
$79$			522.058i			0.743495i	−0.928334	+	0.371747i	$0.878759\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−444.221	+	444.221i	−0.587465	+	0.587465i	−0.936944	−	0.349479i	$-0.886359\pi$
$83$	−444.221	+	444.221i	−0.587465	+	0.587465i	0.349479	+	0.936944i	$0.386359\pi$
$84$		0			0
$85$	−503.940	−	503.940i	−0.643058	−	0.643058i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1102.27			−1.31281			−0.656405	−	0.754409i	$-0.727924\pi$
$89$	−1102.27			−1.31281			−0.656405	+	0.754409i	$0.727924\pi$
$90$		0			0
$91$	−1031.58	+	1031.58i	−1.18835	+	1.18835i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	949.342			1.02527
$96$		0			0
$97$	−1069.15			−1.11913			−0.559565	−	0.828786i	$-0.689032\pi$
$97$	−1069.15			−1.11913			−0.559565	+	0.828786i	$0.689032\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1184.31	−	1184.31i	1.16677	−	1.16677i	0.183805	−	0.982963i	$-0.441159\pi$
$101$	1184.31	−	1184.31i	1.16677	−	1.16677i	0.982963	−	0.183805i	$-0.0588415\pi$
$102$		0			0
$103$	−1973.70			−1.88810			−0.944049	−	0.329806i	$-0.893017\pi$
$103$	−1973.70			−1.88810			−0.944049	+	0.329806i	$0.893017\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1025.91	−	1025.91i	−0.926901	−	0.926901i	0.0706038	−	0.997504i	$-0.477507\pi$
$107$	−1025.91	−	1025.91i	−0.926901	−	0.926901i	−0.997504	+	0.0706038i	$0.977507\pi$
$108$		0			0
$109$	1201.58	−	1201.58i	1.05587	−	1.05587i	0.0575281	−	0.998344i	$-0.481678\pi$
$109$	1201.58	−	1201.58i	1.05587	−	1.05587i	0.998344	−	0.0575281i	$-0.0183219\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			1755.12i			1.46113i	0.682842	+	0.730566i	$0.260744\pi$
$113$			1755.12i			1.46113i	−0.682842	+	0.730566i	$0.739256\pi$
$114$		0			0
$115$	−159.087	−	159.087i	−0.128999	−	0.128999i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			2172.61i			1.67364i
$120$		0			0
$121$		−	1298.36i		−	0.975477i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1075.02	−	1075.02i	−0.769221	−	0.769221i
$126$		0			0
$127$		−	1185.87i		−	0.828575i	−0.910146	−	0.414288i	$-0.864031\pi$
$127$		−	1185.87i		−	0.828575i	0.910146	−	0.414288i	$-0.135969\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1081.59	+	1081.59i	−0.721367	+	0.721367i	−0.968884	−	0.247516i	$-0.920386\pi$
$131$	−1081.59	+	1081.59i	−0.721367	+	0.721367i	0.247516	+	0.968884i	$0.420386\pi$
$132$		0			0
$133$	−2046.43	−	2046.43i	−1.33419	−	1.33419i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−2165.11			−1.35020			−0.675101	−	0.737725i	$-0.735900\pi$
$137$	−2165.11			−1.35020			−0.675101	+	0.737725i	$0.735900\pi$
$138$		0			0
$139$	−342.051	+	342.051i	−0.208722	+	0.208722i	−0.803724	−	0.595002i	$-0.797151\pi$
$139$	−342.051	+	342.051i	−0.208722	+	0.208722i	0.595002	+	0.803724i	$0.297151\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−306.388			−0.179171
$144$		0			0
$145$	−1359.32			−0.778522
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1937.50	+	1937.50i	−1.06528	+	1.06528i	−0.0675623	+	0.997715i	$0.521522\pi$
$149$	−1937.50	+	1937.50i	−1.06528	+	1.06528i	−0.997715	+	0.0675623i	$0.978478\pi$
$150$		0			0
$151$	−2341.03			−1.26166			−0.630828	−	0.775923i	$-0.717285\pi$
$151$	−2341.03			−1.26166			−0.630828	+	0.775923i	$0.717285\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1496.23	+	1496.23i	0.775356	+	0.775356i
$156$		0			0
$157$	−856.893	+	856.893i	−0.435589	+	0.435589i	−0.890525	−	0.454935i	$-0.849662\pi$
$157$	−856.893	+	856.893i	−0.435589	+	0.435589i	0.454935	+	0.890525i	$0.349662\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			685.862i			0.335736i
$162$		0			0
$163$	−1737.64	−	1737.64i	−0.834982	−	0.834982i	0.153211	−	0.988193i	$-0.451039\pi$
$163$	−1737.64	−	1737.64i	−0.834982	−	0.834982i	−0.988193	+	0.153211i	$0.951039\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			2472.59i			1.14572i	0.819655	+	0.572858i	$0.194165\pi$
$167$			2472.59i			1.14572i	−0.819655	+	0.572858i	$0.805835\pi$
$168$		0			0
$169$		−	679.034i		−	0.309073i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1114.69	+	1114.69i	0.489875	+	0.489875i	0.908267	−	0.418392i	$-0.137406\pi$
$173$	1114.69	+	1114.69i	0.489875	+	0.489875i	−0.418392	+	0.908267i	$0.637406\pi$
$174$		0			0
$175$			1234.25i			0.533148i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	1364.12	−	1364.12i	0.569602	−	0.569602i	−0.362415	−	0.932017i	$-0.618048\pi$
$179$	1364.12	−	1364.12i	0.569602	−	0.569602i	0.932017	+	0.362415i	$0.118048\pi$
$180$		0			0
$181$	443.476	+	443.476i	0.182118	+	0.182118i	0.792278	−	0.610160i	$-0.208895\pi$
$181$	443.476	+	443.476i	0.182118	+	0.182118i	−0.610160	+	0.792278i	$0.708895\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	2651.38			1.05369
$186$		0			0
$187$	−322.641	+	322.641i	−0.126170	+	0.126170i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−642.969			−0.243579			−0.121790	−	0.992556i	$-0.538863\pi$
$191$	−642.969			−0.243579			−0.121790	+	0.992556i	$0.538863\pi$
$192$		0			0
$193$	−1038.09			−0.387168			−0.193584	−	0.981084i	$-0.562011\pi$
$193$	−1038.09			−0.387168			−0.193584	+	0.981084i	$0.562011\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−2641.47	+	2641.47i	−0.955315	+	0.955315i	−0.999043	−	0.0437288i	$-0.986076\pi$
$197$	−2641.47	+	2641.47i	−0.955315	+	0.955315i	0.0437288	+	0.999043i	$0.486076\pi$
$198$		0			0
$199$	−2547.50			−0.907476			−0.453738	−	0.891135i	$-0.649910\pi$
$199$	−2547.50			−0.907476			−0.453738	+	0.891135i	$0.649910\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	2930.19	+	2930.19i	1.01310	+	1.01310i
$204$		0			0
$205$	2389.15	−	2389.15i	0.813980	−	0.813980i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	607.804i		−	0.201161i
$210$		0			0
$211$	1159.93	+	1159.93i	0.378448	+	0.378448i	0.870542	−	0.492094i	$-0.163768\pi$
$211$	1159.93	+	1159.93i	0.378448	+	0.378448i	−0.492094	+	0.870542i	$0.663768\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	2414.50i		−	0.765894i
$216$		0			0
$217$		−	6450.63i		−	2.01796i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−3028.59	−	3028.59i	−0.921834	−	0.921834i
$222$		0			0
$223$			3613.80i			1.08519i	0.839994	+	0.542596i	$0.182559\pi$
$223$			3613.80i			1.08519i	−0.839994	+	0.542596i	$0.817441\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−444.232	+	444.232i	−0.129889	+	0.129889i	−0.769062	−	0.639174i	$-0.779276\pi$
$227$	−444.232	+	444.232i	−0.129889	+	0.129889i	0.639174	+	0.769062i	$0.279276\pi$
$228$		0			0
$229$	188.918	+	188.918i	0.0545154	+	0.0545154i	0.733839	−	0.679324i	$-0.237727\pi$
$229$	188.918	+	188.918i	0.0545154	+	0.0545154i	−0.679324	+	0.733839i	$0.737727\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−6015.55			−1.69138			−0.845691	−	0.533673i	$-0.820811\pi$
$233$	−6015.55			−1.69138			−0.845691	+	0.533673i	$0.820811\pi$
$234$		0			0
$235$	−2634.26	+	2634.26i	−0.731234	+	0.731234i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	3866.73			1.04652			0.523260	−	0.852173i	$-0.324716\pi$
$239$	3866.73			1.04652			0.523260	+	0.852173i	$0.324716\pi$
$240$		0			0
$241$	6049.74			1.61700			0.808501	−	0.588494i	$-0.200279\pi$
$241$	6049.74			1.61700			0.808501	+	0.588494i	$0.200279\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−2505.16	+	2505.16i	−0.653260	+	0.653260i
$246$		0			0
$247$	5705.39			1.46974
$248$		0			0
$249$		0			0
$250$		0			0
$251$	3748.61	+	3748.61i	0.942670	+	0.942670i	0.998443	−	0.0557730i	$-0.0177623\pi$
$251$	3748.61	+	3748.61i	0.942670	+	0.942670i	−0.0557730	+	0.998443i	$0.517762\pi$
$252$		0			0
$253$	−101.853	+	101.853i	−0.0253101	+	0.0253101i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	4593.31i		−	1.11488i	−0.830219	−	0.557438i	$-0.811785\pi$
$257$		−	4593.31i		−	1.11488i	0.830219	−	0.557438i	$-0.188215\pi$
$258$		0			0
$259$	−5715.38	−	5715.38i	−1.37118	−	1.37118i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			3477.14i			0.815246i	0.913150	+	0.407623i	$0.133642\pi$
$263$			3477.14i			0.815246i	−0.913150	+	0.407623i	$0.866358\pi$
$264$		0			0
$265$		−	1759.43i		−	0.407853i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	115.440	+	115.440i	0.0261654	+	0.0261654i	0.720068	−	0.693903i	$-0.244110\pi$
$269$	115.440	+	115.440i	0.0261654	+	0.0261654i	−0.693903	+	0.720068i	$0.744110\pi$
$270$		0			0
$271$			2252.96i			0.505009i	0.967596	+	0.252504i	$0.0812542\pi$
$271$			2252.96i			0.505009i	−0.967596	+	0.252504i	$0.918746\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−183.291	+	183.291i	−0.0401923	+	0.0401923i
$276$		0			0
$277$	−1423.92	−	1423.92i	−0.308864	−	0.308864i	0.535605	−	0.844469i	$-0.320084\pi$
$277$	−1423.92	−	1423.92i	−0.308864	−	0.308864i	−0.844469	+	0.535605i	$0.820084\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	5780.83			1.22724			0.613621	−	0.789600i	$-0.289712\pi$
$281$	5780.83			1.22724			0.613621	+	0.789600i	$0.289712\pi$
$282$		0			0
$283$	−2544.53	+	2544.53i	−0.534476	+	0.534476i	−0.921901	−	0.387425i	$-0.873365\pi$
$283$	−2544.53	+	2544.53i	−0.534476	+	0.534476i	0.387425	+	0.921901i	$0.373365\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−10300.2			−2.11848
$288$		0			0
$289$	−1465.50			−0.298289
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−423.704	+	423.704i	−0.0844815	+	0.0844815i	−0.748085	−	0.663603i	$-0.769026\pi$
$293$	−423.704	+	423.704i	−0.0844815	+	0.0844815i	0.663603	+	0.748085i	$0.269026\pi$
$294$		0			0
$295$	−3566.85			−0.703966
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−956.084	−	956.084i	−0.184922	−	0.184922i
$300$		0			0
$301$	−5204.75	+	5204.75i	−0.996667	+	0.996667i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			3228.39i			0.606089i
$306$		0			0
$307$	−687.135	−	687.135i	−0.127742	−	0.127742i	0.640345	−	0.768087i	$-0.278791\pi$
$307$	−687.135	−	687.135i	−0.127742	−	0.127742i	−0.768087	+	0.640345i	$0.778791\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	441.970i		−	0.0805846i	−0.999188	−	0.0402923i	$-0.987171\pi$
$311$		−	441.970i		−	0.0805846i	0.999188	−	0.0402923i	$-0.0128289\pi$
$312$		0			0
$313$		−	1949.31i		−	0.352018i	−0.984389	−	0.176009i	$-0.943681\pi$
$313$		−	1949.31i		−	0.352018i	0.984389	−	0.176009i	$-0.0563188\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1627.66	−	1627.66i	−0.288386	−	0.288386i	0.548056	−	0.836442i	$-0.315368\pi$
$317$	−1627.66	−	1627.66i	−0.288386	−	0.288386i	−0.836442	+	0.548056i	$0.815368\pi$
$318$		0			0
$319$			870.289i			0.152749i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	6008.03	−	6008.03i	1.03497	−	1.03497i
$324$		0			0
$325$	−1720.54	−	1720.54i	−0.293656	−	0.293656i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	11356.9			1.90313
$330$		0			0
$331$	5461.11	−	5461.11i	0.906857	−	0.906857i	−0.0891603	−	0.996017i	$-0.528418\pi$
$331$	5461.11	−	5461.11i	0.906857	−	0.906857i	0.996017	+	0.0891603i	$0.0284183\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−4400.17			−0.717632
$336$		0			0
$337$	−7450.74			−1.20436			−0.602178	−	0.798362i	$-0.705700\pi$
$337$	−7450.74			−1.20436			−0.602178	+	0.798362i	$0.705700\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	957.942	−	957.942i	0.152128	−	0.152128i
$342$		0			0
$343$	1469.61			0.231346
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−7071.55	−	7071.55i	−1.09401	−	1.09401i	−0.995096	−	0.0989122i	$-0.968464\pi$
$347$	−7071.55	−	7071.55i	−1.09401	−	1.09401i	−0.0989122	−	0.995096i	$-0.531536\pi$
$348$		0			0
$349$	−3300.15	+	3300.15i	−0.506170	+	0.506170i	−0.913348	−	0.407179i	$-0.866513\pi$
$349$	−3300.15	+	3300.15i	−0.506170	+	0.506170i	0.407179	+	0.913348i	$0.366513\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			18.3891i			0.00277268i	0.999999	+	0.00138634i	$0.000441285\pi$
$353$			18.3891i			0.00277268i	−0.999999	+	0.00138634i	$0.999559\pi$
$354$		0			0
$355$	2027.47	+	2027.47i	0.303118	+	0.303118i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	670.681i		−	0.0985995i	−0.998784	−	0.0492997i	$-0.984301\pi$
$359$		−	670.681i		−	0.0985995i	0.998784	−	0.0492997i	$-0.0156990\pi$
$360$		0			0
$361$			4459.18i			0.650121i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	852.946	+	852.946i	0.122316	+	0.122316i
$366$		0			0
$367$		−	5038.40i		−	0.716627i	−0.933601	−	0.358314i	$-0.883352\pi$
$367$		−	5038.40i		−	0.716627i	0.933601	−	0.358314i	$-0.116648\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−3792.68	+	3792.68i	−0.530744	+	0.530744i
$372$		0			0
$373$	3151.66	+	3151.66i	0.437498	+	0.437498i	0.891169	−	0.453671i	$-0.149886\pi$
$373$	3151.66	+	3151.66i	0.437498	+	0.437498i	−0.453671	+	0.891169i	$0.649886\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−8169.31			−1.11602
$378$		0			0
$379$	1494.63	−	1494.63i	0.202570	−	0.202570i	−0.598530	−	0.801100i	$-0.704248\pi$
$379$	1494.63	−	1494.63i	0.202570	−	0.202570i	0.801100	+	0.598530i	$0.204248\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	2530.45			0.337598			0.168799	−	0.985650i	$-0.446011\pi$
$383$	2530.45			0.337598			0.168799	+	0.985650i	$0.446011\pi$
$384$		0			0
$385$	−1386.86			−0.183587
$386$		0			0
$387$		0			0
$388$		0			0
$389$	6128.44	−	6128.44i	0.798776	−	0.798776i	−0.184126	−	0.982903i	$-0.558945\pi$
$389$	6128.44	−	6128.44i	0.798776	−	0.798776i	0.982903	+	0.184126i	$0.0589455\pi$
$390$		0			0
$391$	−2013.60			−0.260440
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−3294.11	−	3294.11i	−0.419607	−	0.419607i
$396$		0			0
$397$	6952.91	−	6952.91i	0.878983	−	0.878983i	−0.114446	−	0.993429i	$-0.536509\pi$
$397$	6952.91	−	6952.91i	0.878983	−	0.878983i	0.993429	+	0.114446i	$0.0365093\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			845.400i			0.105280i	0.998614	+	0.0526399i	$0.0167636\pi$
$401$			845.400i			0.105280i	−0.998614	+	0.0526399i	$0.983236\pi$
$402$		0			0
$403$	8992.10	+	8992.10i	1.11149	+	1.11149i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	1697.51i		−	0.206738i
$408$		0			0
$409$			14578.4i			1.76249i	0.472662	+	0.881244i	$0.343293\pi$
$409$			14578.4i			1.76249i	−0.472662	+	0.881244i	$0.656707\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	7688.79	+	7688.79i	0.916079	+	0.916079i
$414$		0			0
$415$		−	5605.95i		−	0.663097i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−4924.02	+	4924.02i	−0.574115	+	0.574115i	−0.933276	−	0.359161i	$-0.883063\pi$
$419$	−4924.02	+	4924.02i	−0.574115	+	0.574115i	0.359161	+	0.933276i	$0.383063\pi$
$420$		0			0
$421$	6264.27	+	6264.27i	0.725182	+	0.725182i	0.969656	−	0.244474i	$-0.0786152\pi$
$421$	6264.27	+	6264.27i	0.725182	+	0.725182i	−0.244474	+	0.969656i	$0.578615\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−3623.61			−0.413578
$426$		0			0
$427$	6959.20	−	6959.20i	0.788711	−	0.788711i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	14200.7			1.58707			0.793533	−	0.608527i	$-0.208239\pi$
$431$	14200.7			1.58707			0.793533	+	0.608527i	$0.208239\pi$
$432$		0			0
$433$	−1574.27			−0.174722			−0.0873611	−	0.996177i	$-0.527843\pi$
$433$	−1574.27			−0.174722			−0.0873611	+	0.996177i	$0.527843\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1896.65	−	1896.65i	0.207618	−	0.207618i
$438$		0			0
$439$	−1716.76			−0.186644			−0.0933220	−	0.995636i	$-0.529749\pi$
$439$	−1716.76			−0.186644			−0.0933220	+	0.995636i	$0.529749\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	6531.76	+	6531.76i	0.700527	+	0.700527i	0.964524	−	0.263997i	$-0.0850408\pi$
$443$	6531.76	+	6531.76i	0.700527	+	0.700527i	−0.263997	+	0.964524i	$0.585041\pi$
$444$		0			0
$445$	6955.15	−	6955.15i	0.740912	−	0.740912i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			2179.52i			0.229082i	0.993419	+	0.114541i	$0.0365397\pi$
$449$			2179.52i			0.229082i	−0.993419	+	0.114541i	$0.963460\pi$
$450$		0			0
$451$	−1529.62	−	1529.62i	−0.159706	−	0.159706i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	13018.3i		−	1.34133i
$456$		0			0
$457$		−	833.132i		−	0.0852785i	−0.999091	−	0.0426392i	$-0.986423\pi$
$457$		−	833.132i		−	0.0852785i	0.999091	−	0.0426392i	$-0.0135766\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−6435.20	−	6435.20i	−0.650146	−	0.650146i	0.302882	−	0.953028i	$-0.402051\pi$
$461$	−6435.20	−	6435.20i	−0.650146	−	0.650146i	−0.953028	+	0.302882i	$0.902051\pi$
$462$		0			0
$463$			2144.06i			0.215211i	0.994194	+	0.107606i	$0.0343184\pi$
$463$			2144.06i			0.215211i	−0.994194	+	0.107606i	$0.965682\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	1590.87	−	1590.87i	0.157638	−	0.157638i	−0.623881	−	0.781519i	$-0.714445\pi$
$467$	1590.87	−	1590.87i	0.157638	−	0.157638i	0.781519	+	0.623881i	$0.214445\pi$
$468$		0			0
$469$	9485.11	+	9485.11i	0.933863	+	0.933863i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−1545.85			−0.150271
$474$		0			0
$475$	3413.15	−	3413.15i	0.329697	−	0.329697i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−3087.11			−0.294475			−0.147238	−	0.989101i	$-0.547038\pi$
$479$	−3087.11			−0.294475			−0.147238	+	0.989101i	$0.547038\pi$
$480$		0			0
$481$	15934.4			1.51049
$482$		0			0
$483$		0			0
$484$		0			0
$485$	6746.18	−	6746.18i	0.631604	−	0.631604i
$486$		0			0
$487$	20640.5			1.92055			0.960276	−	0.279051i	$-0.0900198\pi$
$487$	20640.5			1.92055			0.960276	+	0.279051i	$0.0900198\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−13880.9	−	13880.9i	−1.27584	−	1.27584i	−0.942976	−	0.332861i	$-0.891986\pi$
$491$	−13880.9	−	13880.9i	−1.27584	−	1.27584i	−0.332861	−	0.942976i	$-0.608014\pi$
$492$		0			0
$493$	−8602.66	+	8602.66i	−0.785891	+	0.785891i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	8740.93i		−	0.788902i
$498$		0			0
$499$	8527.15	+	8527.15i	0.764985	+	0.764985i	0.977219	−	0.212234i	$-0.0680737\pi$
$499$	8527.15	+	8527.15i	0.764985	+	0.764985i	−0.212234	+	0.977219i	$0.568074\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			6352.16i			0.563079i	0.959550	+	0.281540i	$0.0908451\pi$
$503$			6352.16i			0.563079i	−0.959550	+	0.281540i	$0.909155\pi$
$504$		0			0
$505$			14945.7i			1.31698i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	6383.68	+	6383.68i	0.555897	+	0.555897i	0.928137	−	0.372239i	$-0.121410\pi$
$509$	6383.68	+	6383.68i	0.555897	+	0.555897i	−0.372239	+	0.928137i	$0.621410\pi$
$510$		0			0
$511$		−	3677.26i		−	0.318342i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	12453.7	−	12453.7i	1.06559	−	1.06559i
$516$		0			0
$517$	1686.55	+	1686.55i	0.143471	+	0.143471i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	14655.6			1.23238			0.616192	−	0.787596i	$-0.288674\pi$
$521$	14655.6			1.23238			0.616192	+	0.787596i	$0.288674\pi$
$522$		0			0
$523$	−5288.73	+	5288.73i	−0.442180	+	0.442180i	−0.892744	−	0.450564i	$-0.851223\pi$
$523$	−5288.73	+	5288.73i	−0.442180	+	0.442180i	0.450564	+	0.892744i	$0.351223\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	18938.2			1.56539
$528$		0			0
$529$	11531.3			0.947755
$530$		0			0
$531$		0			0
$532$		0			0
$533$	14358.4	−	14358.4i	1.16685	−	1.16685i
$534$		0			0
$535$	12946.7			1.04623
$536$		0			0
$537$		0			0
$538$		0			0
$539$	1603.89	+	1603.89i	0.128172	+	0.128172i
$540$		0			0
$541$	5155.04	−	5155.04i	0.409672	−	0.409672i	−0.471952	−	0.881624i	$-0.656450\pi$
$541$	5155.04	−	5155.04i	0.409672	−	0.409672i	0.881624	+	0.471952i	$0.156450\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			15163.5i			1.19181i
$546$		0			0
$547$	15716.4	+	15716.4i	1.22849	+	1.22849i	0.964535	+	0.263956i	$0.0850273\pi$
$547$	15716.4	+	15716.4i	1.22849	+	1.22849i	0.263956	+	0.964535i	$0.414973\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	16206.0i		−	1.25300i
$552$		0			0
$553$			14201.7i			1.09208i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	6521.03	+	6521.03i	0.496059	+	0.496059i	0.910209	−	0.414150i	$-0.135921\pi$
$557$	6521.03	+	6521.03i	0.496059	+	0.496059i	−0.414150	+	0.910209i	$0.635921\pi$
$558$		0			0
$559$		−	14510.7i		−	1.09792i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−3081.99	+	3081.99i	−0.230711	+	0.230711i	−0.812989	−	0.582278i	$-0.802161\pi$
$563$	−3081.99	+	3081.99i	−0.230711	+	0.230711i	0.582278	+	0.812989i	$0.302161\pi$
$564$		0			0
$565$	−11074.6	−	11074.6i	−0.824620	−	0.824620i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−15731.8			−1.15907			−0.579535	−	0.814948i	$-0.696766\pi$
$569$	−15731.8			−1.15907			−0.579535	+	0.814948i	$0.696766\pi$
$570$		0			0
$571$	15369.5	−	15369.5i	1.12643	−	1.12643i	0.135680	−	0.990753i	$-0.456678\pi$
$571$	15369.5	−	15369.5i	1.12643	−	1.12643i	0.990753	−	0.135680i	$-0.0433220\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1143.92			−0.0829649
$576$		0			0
$577$	−3414.08			−0.246326			−0.123163	−	0.992386i	$-0.539304\pi$
$577$	−3414.08			−0.246326			−0.123163	+	0.992386i	$0.539304\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−12084.3	+	12084.3i	−0.862895	+	0.862895i
$582$		0			0
$583$	−1126.45			−0.0800222
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−14858.9	−	14858.9i	−1.04479	−	1.04479i	−0.998949	−	0.0458432i	$-0.985403\pi$
$587$	−14858.9	−	14858.9i	−1.04479	−	1.04479i	−0.0458432	−	0.998949i	$-0.514597\pi$
$588$		0			0
$589$	−17838.3	+	17838.3i	−1.24790	+	1.24790i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	10177.1i		−	0.704765i	−0.935856	−	0.352382i	$-0.885372\pi$
$593$		−	10177.1i		−	0.704765i	0.935856	−	0.352382i	$-0.114628\pi$
$594$		0			0
$595$	−13708.9	−	13708.9i	−0.944552	−	0.944552i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			3301.30i			0.225188i	0.993641	+	0.112594i	$0.0359159\pi$
$599$			3301.30i			0.225188i	−0.993641	+	0.112594i	$0.964084\pi$
$600$		0			0
$601$			15622.6i			1.06033i	0.847894	+	0.530166i	$0.177870\pi$
$601$			15622.6i			1.06033i	−0.847894	+	0.530166i	$0.822130\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	8192.47	+	8192.47i	0.550531	+	0.550531i
$606$		0			0
$607$		−	8352.28i		−	0.558499i	−0.960219	−	0.279249i	$-0.909914\pi$
$607$		−	8352.28i		−	0.558499i	0.960219	−	0.279249i	$-0.0900856\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−15831.5	+	15831.5i	−1.04824	+	1.04824i
$612$		0			0
$613$	4415.12	+	4415.12i	0.290906	+	0.290906i	0.837438	−	0.546532i	$-0.184052\pi$
$613$	4415.12	+	4415.12i	0.290906	+	0.290906i	−0.546532	+	0.837438i	$0.684052\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−908.905			−0.0593049			−0.0296525	−	0.999560i	$-0.509440\pi$
$617$	−908.905			−0.0593049			−0.0296525	+	0.999560i	$0.509440\pi$
$618$		0			0
$619$	−12163.4	+	12163.4i	−0.789804	+	0.789804i	−0.981462	−	0.191658i	$-0.938614\pi$
$619$	−12163.4	+	12163.4i	−0.789804	+	0.789804i	0.191658	+	0.981462i	$0.438614\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−29985.4			−1.92831
$624$		0			0
$625$	7895.01			0.505281
$626$		0			0
$627$		0			0
$628$		0			0
$629$	16779.6	−	16779.6i	1.06367	−	1.06367i
$630$		0			0
$631$	26228.8			1.65476			0.827380	−	0.561643i	$-0.189830\pi$
$631$	26228.8			1.65476			0.827380	+	0.561643i	$0.189830\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	7482.68	+	7482.68i	0.467624	+	0.467624i
$636$		0			0
$637$	−15055.6	+	15055.6i	−0.936459	+	0.936459i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	9142.35i		−	0.563340i	−0.959511	−	0.281670i	$-0.909112\pi$
$641$		−	9142.35i		−	0.563340i	0.959511	−	0.281670i	$-0.0908883\pi$
$642$		0			0
$643$	−14128.8	−	14128.8i	−0.866541	−	0.866541i	0.125547	−	0.992088i	$-0.459931\pi$
$643$	−14128.8	−	14128.8i	−0.866541	−	0.866541i	−0.992088	+	0.125547i	$0.959931\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	16153.8i		−	0.981566i	−0.871282	−	0.490783i	$-0.836711\pi$
$647$		−	16153.8i		−	0.981566i	0.871282	−	0.490783i	$-0.163289\pi$
$648$		0			0
$649$			2283.63i			0.138121i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	4351.77	+	4351.77i	0.260793	+	0.260793i	0.825376	−	0.564583i	$-0.190963\pi$
$653$	4351.77	+	4351.77i	0.260793	+	0.260793i	−0.564583	+	0.825376i	$0.690963\pi$
$654$		0			0
$655$		−	13649.4i		−	0.814237i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−21049.5	+	21049.5i	−1.24427	+	1.24427i	−0.286056	+	0.958213i	$0.592344\pi$
$659$	−21049.5	+	21049.5i	−1.24427	+	1.24427i	−0.958213	+	0.286056i	$0.907656\pi$
$660$		0			0
$661$	−14180.5	−	14180.5i	−0.834428	−	0.834428i	0.153691	−	0.988119i	$-0.450884\pi$
$661$	−14180.5	−	14180.5i	−0.834428	−	0.834428i	−0.988119	+	0.153691i	$0.950884\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	25825.3			1.50596
$666$		0			0
$667$	−2715.74	+	2715.74i	−0.157652	+	0.157652i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	2066.94			0.118917
$672$		0			0
$673$	2224.60			0.127418			0.0637089	−	0.997969i	$-0.479707\pi$
$673$	2224.60			0.127418			0.0637089	+	0.997969i	$0.479707\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−11483.6	+	11483.6i	−0.651919	+	0.651919i	−0.953455	−	0.301536i	$-0.902501\pi$
$677$	−11483.6	+	11483.6i	−0.651919	+	0.651919i	0.301536	+	0.953455i	$0.402501\pi$
$678$		0			0
$679$	−29084.5			−1.64383
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1941.72	+	1941.72i	0.108781	+	0.108781i	0.759403	−	0.650621i	$-0.225491\pi$
$683$	1941.72	+	1941.72i	0.108781	+	0.108781i	−0.650621	+	0.759403i	$0.725491\pi$
$684$		0			0
$685$	13661.5	−	13661.5i	0.762014	−	0.762014i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	10573.9i		−	0.584664i
$690$		0			0
$691$	203.904	+	203.904i	0.0112256	+	0.0112256i	0.712697	−	0.701472i	$-0.247473\pi$
$691$	203.904	+	203.904i	0.0112256	+	0.0112256i	−0.701472	+	0.712697i	$0.747473\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	4316.58i		−	0.235593i
$696$		0			0
$697$		−	30240.1i		−	1.64337i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	62.5193	+	62.5193i	0.00336850	+	0.00336850i	0.708789	−	0.705421i	$-0.249242\pi$
$701$	62.5193	+	62.5193i	0.00336850	+	0.00336850i	−0.705421	+	0.708789i	$0.749242\pi$
$702$		0			0
$703$			31610.1i			1.69587i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	32217.3	−	32217.3i	1.71380	−	1.71380i
$708$		0			0
$709$	19068.4	+	19068.4i	1.01005	+	1.01005i	0.999949	+	0.0101050i	$0.00321657\pi$
$709$	19068.4	+	19068.4i	1.01005	+	1.01005i	0.0101050	+	0.999949i	$0.496783\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	5978.52			0.314021
$714$		0			0
$715$	1933.27	−	1933.27i	0.101119	−	0.101119i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−4219.83			−0.218878			−0.109439	−	0.993994i	$-0.534905\pi$
$719$	−4219.83			−0.218878			−0.109439	+	0.993994i	$0.534905\pi$
$720$		0			0
$721$	−53691.2			−2.77332
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−4887.15	+	4887.15i	−0.250351	+	0.250351i
$726$		0			0
$727$	3771.74			0.192415			0.0962077	−	0.995361i	$-0.469329\pi$
$727$	3771.74			0.192415			0.0962077	+	0.995361i	$0.469329\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−15280.5	−	15280.5i	−0.773143	−	0.773143i
$732$		0			0
$733$	5607.56	−	5607.56i	0.282565	−	0.282565i	−0.551566	−	0.834131i	$-0.685970\pi$
$733$	5607.56	−	5607.56i	0.282565	−	0.282565i	0.834131	+	0.551566i	$0.185970\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			2817.15i			0.140802i
$738$		0			0
$739$	7137.31	+	7137.31i	0.355277	+	0.355277i	0.862069	−	0.506791i	$-0.169169\pi$
$739$	7137.31	+	7137.31i	0.355277	+	0.355277i	−0.506791	+	0.862069i	$0.669169\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	17134.1i		−	0.846016i	−0.906126	−	0.423008i	$-0.860974\pi$
$743$		−	17134.1i		−	0.846016i	0.906126	−	0.423008i	$-0.139026\pi$
$744$		0			0
$745$		−	24450.7i		−	1.20242i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−27908.2	−	27908.2i	−1.36147	−	1.36147i
$750$		0			0
$751$			17533.1i			0.851922i	0.904742	+	0.425961i	$0.140064\pi$
$751$			17533.1i			0.851922i	−0.904742	+	0.425961i	$0.859936\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	14771.5	−	14771.5i	0.712042	−	0.712042i
$756$		0			0
$757$	19568.4	+	19568.4i	0.939532	+	0.939532i	0.998273	−	0.0587408i	$-0.0187086\pi$
$757$	19568.4	+	19568.4i	0.939532	+	0.939532i	−0.0587408	+	0.998273i	$0.518709\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	4059.70			0.193383			0.0966913	−	0.995314i	$-0.469174\pi$
$761$	4059.70			0.193383			0.0966913	+	0.995314i	$0.469174\pi$
$762$		0			0
$763$	32686.9	−	32686.9i	1.55091	−	1.55091i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−21436.2			−1.00915
$768$		0			0
$769$	21167.3			0.992602			0.496301	−	0.868150i	$-0.334691\pi$
$769$	21167.3			0.992602			0.496301	+	0.868150i	$0.334691\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	1992.49	−	1992.49i	0.0927102	−	0.0927102i	−0.659231	−	0.751941i	$-0.729118\pi$
$773$	1992.49	−	1992.49i	0.0927102	−	0.0927102i	0.751941	+	0.659231i	$0.229118\pi$
$774$		0			0
$775$	10758.7			0.498665
$776$		0			0
$777$		0			0
$778$		0			0
$779$	28483.8	+	28483.8i	1.31006	+	1.31006i
$780$		0			0
$781$	1298.06	−	1298.06i	0.0594728	−	0.0594728i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	10813.7i		−	0.491668i
$786$		0			0
$787$	17873.4	+	17873.4i	0.809551	+	0.809551i	0.984566	−	0.175015i	$-0.0559973\pi$
$787$	17873.4	+	17873.4i	0.809551	+	0.809551i	−0.175015	+	0.984566i	$0.555997\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			47745.2i			2.14617i
$792$		0			0
$793$			19402.1i			0.868839i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−4327.93	−	4327.93i	−0.192350	−	0.192350i	0.604361	−	0.796711i	$-0.293429\pi$
$797$	−4327.93	−	4327.93i	−0.192350	−	0.192350i	−0.796711	+	0.604361i	$0.793429\pi$
$798$		0			0
$799$			33342.5i			1.47631i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	546.087	−	546.087i	0.0239988	−	0.0239988i
$804$		0			0
$805$	−4327.69	−	4327.69i	−0.189480	−	0.189480i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	7284.89			0.316592			0.158296	−	0.987392i	$-0.449400\pi$
$809$	7284.89			0.316592			0.158296	+	0.987392i	$0.449400\pi$
$810$		0			0
$811$	−28534.9	+	28534.9i	−1.23551	+	1.23551i	−0.273688	+	0.961819i	$0.588244\pi$
$811$	−28534.9	+	28534.9i	−1.23551	+	1.23551i	−0.961819	+	0.273688i	$0.911756\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	21928.5			0.942479
$816$		0			0
$817$	28785.9			1.23267
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−17928.4	+	17928.4i	−0.762128	+	0.762128i	−0.976707	−	0.214579i	$-0.931162\pi$
$821$	−17928.4	+	17928.4i	−0.762128	+	0.762128i	0.214579	+	0.976707i	$0.431162\pi$
$822$		0			0
$823$	−25241.2			−1.06908			−0.534540	−	0.845143i	$-0.679515\pi$
$823$	−25241.2			−1.06908			−0.534540	+	0.845143i	$0.679515\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	27181.2	+	27181.2i	1.14291	+	1.14291i	0.987916	+	0.154990i	$0.0495344\pi$
$827$	27181.2	+	27181.2i	1.14291	+	1.14291i	0.154990	+	0.987916i	$0.450466\pi$
$828$		0			0
$829$	2755.03	−	2755.03i	0.115424	−	0.115424i	−0.647036	−	0.762460i	$-0.723992\pi$
$829$	2755.03	−	2755.03i	0.115424	−	0.115424i	0.762460	+	0.647036i	$0.223992\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			31708.4i			1.31889i
$834$		0			0
$835$	−15601.7	−	15601.7i	−0.646608	−	0.646608i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			20544.0i			0.845361i	0.906279	+	0.422681i	$0.138911\pi$
$839$			20544.0i			0.845361i	−0.906279	+	0.422681i	$0.861089\pi$
$840$		0			0
$841$		−	1184.24i		−	0.0485561i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	4284.61	+	4284.61i	0.174432	+	0.174432i
$846$		0			0
$847$		−	35319.8i		−	1.43282i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	5297.08	−	5297.08i	0.213374	−	0.213374i
$852$		0			0
$853$	−9553.45	−	9553.45i	−0.383475	−	0.383475i	0.488878	−	0.872352i	$-0.337406\pi$
$853$	−9553.45	−	9553.45i	−0.383475	−	0.383475i	−0.872352	+	0.488878i	$0.837406\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−21465.0			−0.855578			−0.427789	−	0.903879i	$-0.640707\pi$
$857$	−21465.0			−0.855578			−0.427789	+	0.903879i	$0.640707\pi$
$858$		0			0
$859$	−4252.94	+	4252.94i	−0.168927	+	0.168927i	−0.786508	−	0.617580i	$-0.788113\pi$
$859$	−4252.94	+	4252.94i	−0.168927	+	0.168927i	0.617580	+	0.786508i	$0.288113\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	7487.36			0.295333			0.147667	−	0.989037i	$-0.452824\pi$
$863$	7487.36			0.295333			0.147667	+	0.989037i	$0.452824\pi$
$864$		0			0
$865$	−14067.1			−0.552942
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−2109.01	+	2109.01i	−0.0823282	+	0.0823282i
$870$		0			0
$871$	−26444.3			−1.02874
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−29244.1	−	29244.1i	−1.12987	−	1.12987i
$876$		0			0
$877$	14348.4	−	14348.4i	0.552465	−	0.552465i	−0.374686	−	0.927152i	$-0.622250\pi$
$877$	14348.4	−	14348.4i	0.552465	−	0.552465i	0.927152	+	0.374686i	$0.122250\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			17497.5i			0.669131i	0.942372	+	0.334566i	$0.108590\pi$
$881$			17497.5i			0.669131i	−0.942372	+	0.334566i	$0.891410\pi$
$882$		0			0
$883$	−5399.20	−	5399.20i	−0.205773	−	0.205773i	0.596695	−	0.802468i	$-0.296480\pi$
$883$	−5399.20	−	5399.20i	−0.205773	−	0.205773i	−0.802468	+	0.596695i	$0.796480\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			32588.2i			1.23360i	0.787119	+	0.616801i	$0.211572\pi$
$887$			32588.2i			1.23360i	−0.787119	+	0.616801i	$0.788428\pi$
$888$		0			0
$889$		−	32259.7i		−	1.21705i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−31406.0	−	31406.0i	−1.17689	−	1.17689i
$894$		0			0
$895$			17214.8i			0.642934i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	25541.9	−	25541.9i	0.947575	−	0.947575i
$900$		0			0
$901$	−11134.8	−	11134.8i	−0.411714	−	0.411714i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−5596.54			−0.205564
$906$		0			0
$907$	−175.952	+	175.952i	−0.00644145	+	0.00644145i	−0.710320	−	0.703879i	$-0.751450\pi$
$907$	−175.952	+	175.952i	−0.00644145	+	0.00644145i	0.703879	+	0.710320i	$0.251450\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	13051.9			0.474676			0.237338	−	0.971427i	$-0.423725\pi$
$911$	13051.9			0.474676			0.237338	+	0.971427i	$0.423725\pi$
$912$		0			0
$913$	−3589.13			−0.130102
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−29422.9	+	29422.9i	−1.05958	+	1.05958i
$918$		0			0
$919$	−8004.68			−0.287323			−0.143662	−	0.989627i	$-0.545888\pi$
$919$	−8004.68			−0.287323			−0.143662	+	0.989627i	$0.545888\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	12184.8	+	12184.8i	0.434524	+	0.434524i
$924$		0			0
$925$	9532.46	−	9532.46i	0.338838	−	0.338838i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	19953.1i		−	0.704672i	−0.935874	−	0.352336i	$-0.885387\pi$
$929$		−	19953.1i		−	0.704672i	0.935874	−	0.352336i	$-0.114613\pi$
$930$		0			0
$931$	−29866.8	−	29866.8i	−1.05139	−	1.05139i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	4071.63i		−	0.142414i
$936$		0			0
$937$		−	47079.0i		−	1.64141i	−0.571351	−	0.820706i	$-0.693580\pi$
$937$		−	47079.0i		−	1.64141i	0.571351	−	0.820706i	$-0.306420\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	4612.54	+	4612.54i	0.159792	+	0.159792i	0.782475	−	0.622682i	$-0.213957\pi$
$941$	4612.54	+	4612.54i	0.159792	+	0.159792i	−0.622682	+	0.782475i	$0.713957\pi$
$942$		0			0
$943$		−	9546.38i		−	0.329664i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−35884.4	+	35884.4i	−1.23135	+	1.23135i	−0.267899	+	0.963447i	$0.586329\pi$
$947$	−35884.4	+	35884.4i	−1.23135	+	1.23135i	−0.963447	+	0.267899i	$0.913671\pi$
$948$		0			0
$949$	5126.07	+	5126.07i	0.175342	+	0.175342i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	2274.82			0.0773228			0.0386614	−	0.999252i	$-0.487691\pi$
$953$	2274.82			0.0773228			0.0386614	+	0.999252i	$0.487691\pi$
$954$		0			0
$955$	4057.04	−	4057.04i	0.137469	−	0.137469i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−58898.2			−1.98324
$960$		0			0
$961$	−26437.8			−0.887442
$962$		0			0
$963$		0			0
$964$		0			0
$965$	6550.21	−	6550.21i	0.218506	−	0.218506i
$966$		0			0
$967$	−11693.6			−0.388874			−0.194437	−	0.980915i	$-0.562288\pi$
$967$	−11693.6			−0.388874			−0.194437	+	0.980915i	$0.562288\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	13147.9	+	13147.9i	0.434537	+	0.434537i	0.890168	−	0.455632i	$-0.150587\pi$
$971$	13147.9	+	13147.9i	0.434537	+	0.434537i	−0.455632	+	0.890168i	$0.650587\pi$
$972$		0			0
$973$	−9304.93	+	9304.93i	−0.306580	+	0.306580i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			57554.9i			1.88469i	0.334642	+	0.942345i	$0.391385\pi$
$977$			57554.9i			1.88469i	−0.334642	+	0.942345i	$0.608615\pi$
$978$		0			0
$979$	−4452.94	−	4452.94i	−0.145369	−	0.145369i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			29857.4i			0.968773i	0.874854	+	0.484386i	$0.160957\pi$
$983$			29857.4i			0.968773i	−0.874854	+	0.484386i	$0.839043\pi$
$984$		0			0
$985$		−	33334.6i		−	1.07830i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−4823.82	−	4823.82i	−0.155095	−	0.155095i
$990$		0			0
$991$			37310.0i			1.19596i	0.801513	+	0.597978i	$0.204029\pi$
$991$			37310.0i			1.19596i	−0.801513	+	0.597978i	$0.795971\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	16074.4	−	16074.4i	0.512153	−	0.512153i
$996$		0			0
$997$	−31495.0	−	31495.0i	−1.00046	−	1.00046i	−1.00000		0.000458156i	$-0.999854\pi$
$997$	−31495.0	−	31495.0i	−1.00046	−	1.00046i	−0.000458156	−	1.00000i	$-0.500146\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.4.l.b.287.8		48
3.2	odd	2	inner	1152.4.l.b.287.17		48
4.3	odd	2		1152.4.l.a.287.8		48
8.3	odd	2		576.4.l.a.143.17		48
8.5	even	2		144.4.l.a.107.14	yes	48
12.11	even	2		1152.4.l.a.287.17		48
16.3	odd	4	inner	1152.4.l.b.863.17		48
16.5	even	4		576.4.l.a.431.8		48
16.11	odd	4		144.4.l.a.35.11	✓	48
16.13	even	4		1152.4.l.a.863.17		48
24.5	odd	2		144.4.l.a.107.11	yes	48
24.11	even	2		576.4.l.a.143.8		48
48.5	odd	4		576.4.l.a.431.17		48
48.11	even	4		144.4.l.a.35.14	yes	48
48.29	odd	4		1152.4.l.a.863.8		48
48.35	even	4	inner	1152.4.l.b.863.8		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.4.l.a.35.11	✓	48	16.11	odd	4
144.4.l.a.35.14	yes	48	48.11	even	4
144.4.l.a.107.11	yes	48	24.5	odd	2
144.4.l.a.107.14	yes	48	8.5	even	2
576.4.l.a.143.8		48	24.11	even	2
576.4.l.a.143.17		48	8.3	odd	2
576.4.l.a.431.8		48	16.5	even	4
576.4.l.a.431.17		48	48.5	odd	4
1152.4.l.a.287.8		48	4.3	odd	2
1152.4.l.a.287.17		48	12.11	even	2
1152.4.l.a.863.8		48	48.29	odd	4
1152.4.l.a.863.17		48	16.13	even	4
1152.4.l.b.287.8		48	1.1	even	1	trivial
1152.4.l.b.287.17		48	3.2	odd	2	inner
1152.4.l.b.863.8		48	48.35	even	4	inner
1152.4.l.b.863.17		48	16.3	odd	4	inner

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:	\( N \)	\(=\)	\( 1152 = 2^{7} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1152.l (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-6.30986 + 6.30986i) q^{5} +27.2034 q^{7} +O(q^{10})\)	\(q+(-6.30986 + 6.30986i) q^{5} +27.2034 q^{7} +(4.03980 + 4.03980i) q^{11} +(-37.9212 + 37.9212i) q^{13} +79.8655i q^{17} +(-75.2269 - 75.2269i) q^{19} +25.2124i q^{23} +45.3714i q^{25} +(107.714 + 107.714i) q^{29} -237.126i q^{31} +(-171.649 + 171.649i) q^{35} +(-210.098 - 210.098i) q^{37} -378.638 q^{41} +(-191.327 + 191.327i) q^{43} +417.483 q^{47} +397.023 q^{49} +(-139.419 + 139.419i) q^{53} -50.9811 q^{55} +(282.641 + 282.641i) q^{59} +(255.821 - 255.821i) q^{61} -478.555i q^{65} +(348.674 + 348.674i) q^{67} -321.318i q^{71} -135.177i q^{73} +(109.896 + 109.896i) q^{77} +522.058i q^{79} +(-444.221 + 444.221i) q^{83} +(-503.940 - 503.940i) q^{85} -1102.27 q^{89} +(-1031.58 + 1031.58i) q^{91} +949.342 q^{95} -1069.15 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q+O(q^{10}) \) 48 * q	\( 48 q + 48 q^{19} - 864 q^{43} + 2352 q^{49} - 576 q^{55} - 1824 q^{61} - 816 q^{67} + 480 q^{85} + 3600 q^{91}+O(q^{100}) \) 48 * q + 48 * q^19 - 864 * q^43 + 2352 * q^49 - 576 * q^55 - 1824 * q^61 - 816 * q^67 + 480 * q^85 + 3600 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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