LMFDB - Embedded newform 1152.4.l.b.287.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1152,4,Mod(287,1152)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [48,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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			287.1
Character	$\chi$	$=$	1152.287
Dual form			1152.4.l.b.863.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-15.2065 + 15.2065i) q^{5} +24.4971 q^{7} +(20.1941 + 20.1941i) q^{11} +(26.7706 - 26.7706i) q^{13} -85.9084i q^{17} +(-53.2541 - 53.2541i) q^{19} +119.986i q^{23} -337.472i q^{25} +(-78.5071 - 78.5071i) q^{29} -200.984i q^{31} +(-372.515 + 372.515i) q^{35} +(76.9819 + 76.9819i) q^{37} -279.001 q^{41} +(-15.7637 + 15.7637i) q^{43} +470.913 q^{47} +257.110 q^{49} +(112.686 - 112.686i) q^{53} -614.162 q^{55} +(-241.182 - 241.182i) q^{59} +(-6.00098 + 6.00098i) q^{61} +814.171i q^{65} +(-273.801 - 273.801i) q^{67} -448.812i q^{71} -54.7653i q^{73} +(494.698 + 494.698i) q^{77} +29.1159i q^{79} +(893.890 - 893.890i) q^{83} +(1306.36 + 1306.36i) q^{85} -281.264 q^{89} +(655.802 - 655.802i) q^{91} +1619.61 q^{95} +188.720 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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

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
$5$	−15.2065	+	15.2065i	−1.36011	+	1.36011i	−0.486333	+	0.873774i	$0.661666\pi$
$5$	−15.2065	+	15.2065i	−1.36011	+	1.36011i	−0.873774	+	0.486333i	$0.838334\pi$
$6$		0			0
$7$	24.4971			1.32272			0.661361	−	0.750068i	$-0.269979\pi$
$7$	24.4971			1.32272			0.661361	+	0.750068i	$0.269979\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	20.1941	+	20.1941i	0.553524	+	0.553524i	0.927456	−	0.373932i	$-0.121991\pi$
$11$	20.1941	+	20.1941i	0.553524	+	0.553524i	−0.373932	+	0.927456i	$0.621991\pi$
$12$		0			0
$13$	26.7706	−	26.7706i	0.571140	−	0.571140i	−0.361307	−	0.932447i	$-0.617669\pi$
$13$	26.7706	−	26.7706i	0.571140	−	0.571140i	0.932447	+	0.361307i	$0.117669\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	85.9084i		−	1.22564i	−0.790223	−	0.612819i	$-0.790035\pi$
$17$		−	85.9084i		−	1.22564i	0.790223	−	0.612819i	$-0.209965\pi$
$18$		0			0
$19$	−53.2541	−	53.2541i	−0.643017	−	0.643017i	0.308279	−	0.951296i	$-0.400247\pi$
$19$	−53.2541	−	53.2541i	−0.643017	−	0.643017i	−0.951296	+	0.308279i	$0.900247\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			119.986i			1.08778i	0.839157	+	0.543889i	$0.183049\pi$
$23$			119.986i			1.08778i	−0.839157	+	0.543889i	$0.816951\pi$
$24$		0			0
$25$		−	337.472i		−	2.69978i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−78.5071	−	78.5071i	−0.502703	−	0.502703i	0.409574	−	0.912277i	$-0.365677\pi$
$29$	−78.5071	−	78.5071i	−0.502703	−	0.502703i	−0.912277	+	0.409574i	$0.865677\pi$
$30$		0			0
$31$		−	200.984i		−	1.16445i	−0.813029	−	0.582223i	$-0.802183\pi$
$31$		−	200.984i		−	1.16445i	0.813029	−	0.582223i	$-0.197817\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−372.515	+	372.515i	−1.79904	+	1.79904i
$36$		0			0
$37$	76.9819	+	76.9819i	0.342047	+	0.342047i	0.857137	−	0.515089i	$-0.172241\pi$
$37$	76.9819	+	76.9819i	0.342047	+	0.342047i	−0.515089	+	0.857137i	$0.672241\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−279.001			−1.06275			−0.531374	−	0.847137i	$-0.678324\pi$
$41$	−279.001			−1.06275			−0.531374	+	0.847137i	$0.678324\pi$
$42$		0			0
$43$	−15.7637	+	15.7637i	−0.0559057	+	0.0559057i	−0.734507	−	0.678601i	$-0.762586\pi$
$43$	−15.7637	+	15.7637i	−0.0559057	+	0.0559057i	0.678601	+	0.734507i	$0.262586\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	470.913			1.46148			0.730742	−	0.682654i	$-0.239174\pi$
$47$	470.913			1.46148			0.730742	+	0.682654i	$0.239174\pi$
$48$		0			0
$49$	257.110			0.749592
$50$		0			0
$51$		0			0
$52$		0			0
$53$	112.686	−	112.686i	0.292050	−	0.292050i	−0.545840	−	0.837889i	$-0.683789\pi$
$53$	112.686	−	112.686i	0.292050	−	0.292050i	0.837889	+	0.545840i	$0.183789\pi$
$54$		0			0
$55$	−614.162			−1.50570
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−241.182	−	241.182i	−0.532190	−	0.532190i	0.389033	−	0.921224i	$-0.372809\pi$
$59$	−241.182	−	241.182i	−0.532190	−	0.532190i	−0.921224	+	0.389033i	$0.872809\pi$
$60$		0			0
$61$	−6.00098	+	6.00098i	−0.0125958	+	0.0125958i	−0.713377	−	0.700781i	$-0.752835\pi$
$61$	−6.00098	+	6.00098i	−0.0125958	+	0.0125958i	0.700781	+	0.713377i	$0.252835\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			814.171i			1.55362i
$66$		0			0
$67$	−273.801	−	273.801i	−0.499255	−	0.499255i	0.411951	−	0.911206i	$-0.364847\pi$
$67$	−273.801	−	273.801i	−0.499255	−	0.499255i	−0.911206	+	0.411951i	$0.864847\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	448.812i		−	0.750201i	−0.926984	−	0.375100i	$-0.877608\pi$
$71$		−	448.812i		−	0.750201i	0.926984	−	0.375100i	$-0.122392\pi$
$72$		0			0
$73$		−	54.7653i		−	0.0878053i	−0.999036	−	0.0439027i	$-0.986021\pi$
$73$		−	54.7653i		−	0.0878053i	0.999036	−	0.0439027i	$-0.0139791\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	494.698	+	494.698i	0.732157	+	0.732157i
$78$		0			0
$79$			29.1159i			0.0414658i	0.999785	+	0.0207329i	$0.00659996\pi$
$79$			29.1159i			0.0414658i	−0.999785	+	0.0207329i	$0.993400\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	893.890	−	893.890i	1.18213	−	1.18213i	0.202945	−	0.979190i	$-0.434949\pi$
$83$	893.890	−	893.890i	1.18213	−	1.18213i	0.979190	−	0.202945i	$-0.0650512\pi$
$84$		0			0
$85$	1306.36	+	1306.36i	1.66700	+	1.66700i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−281.264			−0.334988			−0.167494	−	0.985873i	$-0.553567\pi$
$89$	−281.264			−0.334988			−0.167494	+	0.985873i	$0.553567\pi$
$90$		0			0
$91$	655.802	−	655.802i	0.755459	−	0.755459i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1619.61			1.74914
$96$		0			0
$97$	188.720			0.197542			0.0987709	−	0.995110i	$-0.468509\pi$
$97$	188.720			0.197542			0.0987709	+	0.995110i	$0.468509\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	218.904	−	218.904i	0.215661	−	0.215661i	−0.591006	−	0.806667i	$-0.701269\pi$
$101$	218.904	−	218.904i	0.215661	−	0.215661i	0.806667	+	0.591006i	$0.201269\pi$
$102$		0			0
$103$	1897.36			1.81507			0.907536	−	0.419973i	$-0.137961\pi$
$103$	1897.36			1.81507			0.907536	+	0.419973i	$0.137961\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−400.476	−	400.476i	−0.361827	−	0.361827i	0.502658	−	0.864485i	$-0.332355\pi$
$107$	−400.476	−	400.476i	−0.361827	−	0.361827i	−0.864485	+	0.502658i	$0.832355\pi$
$108$		0			0
$109$	206.240	−	206.240i	0.181231	−	0.181231i	−0.610661	−	0.791892i	$-0.709096\pi$
$109$	206.240	−	206.240i	0.181231	−	0.181231i	0.791892	+	0.610661i	$0.209096\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			385.068i			0.320568i	0.987071	+	0.160284i	$0.0512410\pi$
$113$			385.068i			0.320568i	−0.987071	+	0.160284i	$0.948759\pi$
$114$		0			0
$115$	−1824.57	−	1824.57i	−1.47949	−	1.47949i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	2104.51i		−	1.62118i
$120$		0			0
$121$		−	515.394i		−	0.387223i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	3230.95	+	3230.95i	2.31188	+	2.31188i
$126$		0			0
$127$			1420.04i			0.992189i	0.868268	+	0.496095i	$0.165233\pi$
$127$			1420.04i			0.992189i	−0.868268	+	0.496095i	$0.834767\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	202.131	−	202.131i	0.134811	−	0.134811i	−0.636481	−	0.771292i	$-0.719611\pi$
$131$	202.131	−	202.131i	0.134811	−	0.134811i	0.771292	+	0.636481i	$0.219611\pi$
$132$		0			0
$133$	−1304.57	−	1304.57i	−0.850532	−	0.850532i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	985.515			0.614586			0.307293	−	0.951615i	$-0.400577\pi$
$137$	985.515			0.614586			0.307293	+	0.951615i	$0.400577\pi$
$138$		0			0
$139$	1199.58	−	1199.58i	0.731992	−	0.731992i	−0.239022	−	0.971014i	$-0.576827\pi$
$139$	1199.58	−	1199.58i	0.731992	−	0.731992i	0.971014	+	0.239022i	$0.0768268\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	1081.22			0.632279
$144$		0			0
$145$	2387.63			1.36746
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1257.05	−	1257.05i	0.691150	−	0.691150i	−0.271335	−	0.962485i	$-0.587465\pi$
$149$	1257.05	−	1257.05i	0.691150	−	0.691150i	0.962485	+	0.271335i	$0.0874651\pi$
$150$		0			0
$151$	−1832.83			−0.987770			−0.493885	−	0.869527i	$-0.664424\pi$
$151$	−1832.83			−0.987770			−0.493885	+	0.869527i	$0.664424\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	3056.25	+	3056.25i	1.58377	+	1.58377i
$156$		0			0
$157$	2101.87	−	2101.87i	1.06845	−	1.06845i	0.0709767	−	0.997478i	$-0.477388\pi$
$157$	2101.87	−	2101.87i	1.06845	−	1.06845i	0.997478	−	0.0709767i	$-0.0226116\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			2939.33i			1.43883i
$162$		0			0
$163$	154.434	+	154.434i	0.0742099	+	0.0742099i	0.743238	−	0.669028i	$-0.233289\pi$
$163$	154.434	+	154.434i	0.0742099	+	0.0742099i	−0.669028	+	0.743238i	$0.733289\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			3482.63i			1.61374i	0.590730	+	0.806869i	$0.298840\pi$
$167$			3482.63i			1.61374i	−0.590730	+	0.806869i	$0.701160\pi$
$168$		0			0
$169$			763.674i			0.347598i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−2563.59	−	2563.59i	−1.12663	−	1.12663i	−0.990722	−	0.135903i	$-0.956606\pi$
$173$	−2563.59	−	2563.59i	−1.12663	−	1.12663i	−0.135903	−	0.990722i	$-0.543394\pi$
$174$		0			0
$175$		−	8267.11i		−	3.57106i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	580.106	−	580.106i	0.242230	−	0.242230i	−0.575542	−	0.817772i	$-0.695209\pi$
$179$	580.106	−	580.106i	0.242230	−	0.242230i	0.817772	+	0.575542i	$0.195209\pi$
$180$		0			0
$181$	−18.6302	−	18.6302i	−0.00765067	−	0.00765067i	0.703271	−	0.710922i	$-0.251722\pi$
$181$	−18.6302	−	18.6302i	−0.00765067	−	0.00765067i	−0.710922	+	0.703271i	$0.751722\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−2341.24			−0.930441
$186$		0			0
$187$	1734.85	−	1734.85i	0.678420	−	0.678420i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1727.83			−0.654561			−0.327281	−	0.944927i	$-0.606132\pi$
$191$	−1727.83			−0.654561			−0.327281	+	0.944927i	$0.606132\pi$
$192$		0			0
$193$	4097.40			1.52817			0.764087	−	0.645113i	$-0.223190\pi$
$193$	4097.40			1.52817			0.764087	+	0.645113i	$0.223190\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−3889.19	+	3889.19i	−1.40656	+	1.40656i	−0.629835	+	0.776729i	$0.716877\pi$
$197$	−3889.19	+	3889.19i	−1.40656	+	1.40656i	−0.776729	+	0.629835i	$0.783123\pi$
$198$		0			0
$199$	−3332.92			−1.18726			−0.593629	−	0.804739i	$-0.702305\pi$
$199$	−3332.92			−1.18726			−0.593629	+	0.804739i	$0.702305\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1923.20	−	1923.20i	−0.664936	−	0.664936i
$204$		0			0
$205$	4242.62	−	4242.62i	1.44545	−	1.44545i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	2150.84i		−	0.711850i
$210$		0			0
$211$	−3673.14	−	3673.14i	−1.19843	−	1.19843i	−0.974636	−	0.223796i	$-0.928155\pi$
$211$	−3673.14	−	3673.14i	−1.19843	−	1.19843i	−0.223796	−	0.974636i	$-0.571845\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	479.421i		−	0.152075i
$216$		0			0
$217$		−	4923.54i		−	1.54024i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−2299.82	−	2299.82i	−0.700011	−	0.700011i
$222$		0			0
$223$			867.243i			0.260426i	0.991486	+	0.130213i	$0.0415660\pi$
$223$			867.243i			0.260426i	−0.991486	+	0.130213i	$0.958434\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	331.276	−	331.276i	0.0968614	−	0.0968614i	−0.657016	−	0.753877i	$-0.728181\pi$
$227$	331.276	−	331.276i	0.0968614	−	0.0968614i	0.753877	+	0.657016i	$0.228181\pi$
$228$		0			0
$229$	4331.27	+	4331.27i	1.24986	+	1.24986i	0.955781	+	0.294080i	$0.0950132\pi$
$229$	4331.27	+	4331.27i	1.24986	+	1.24986i	0.294080	+	0.955781i	$0.404987\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	3952.98			1.11145			0.555726	−	0.831365i	$-0.312440\pi$
$233$	3952.98			1.11145			0.555726	+	0.831365i	$0.312440\pi$
$234$		0			0
$235$	−7160.91	+	7160.91i	−1.98777	+	1.98777i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	2530.55			0.684887			0.342443	−	0.939538i	$-0.388745\pi$
$239$	2530.55			0.684887			0.342443	+	0.939538i	$0.388745\pi$
$240$		0			0
$241$	−315.543			−0.0843398			−0.0421699	−	0.999110i	$-0.513427\pi$
$241$	−315.543			−0.0843398			−0.0421699	+	0.999110i	$0.513427\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−3909.73	+	3909.73i	−1.01953	+	1.01953i
$246$		0			0
$247$	−2851.28			−0.734505
$248$		0			0
$249$		0			0
$250$		0			0
$251$	871.154	+	871.154i	0.219071	+	0.219071i	0.808107	−	0.589036i	$-0.200492\pi$
$251$	871.154	+	871.154i	0.219071	+	0.219071i	−0.589036	+	0.808107i	$0.700492\pi$
$252$		0			0
$253$	−2423.02	+	2423.02i	−0.602111	+	0.602111i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			1822.51i			0.442353i	0.975234	+	0.221177i	$0.0709898\pi$
$257$			1822.51i			0.442353i	−0.975234	+	0.221177i	$0.929010\pi$
$258$		0			0
$259$	1885.84	+	1885.84i	0.452433	+	0.452433i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			1120.24i			0.262651i	0.991339	+	0.131325i	$0.0419233\pi$
$263$			1120.24i			0.262651i	−0.991339	+	0.131325i	$0.958077\pi$
$264$		0			0
$265$			3427.11i			0.794437i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	1477.67	+	1477.67i	0.334926	+	0.334926i	0.854454	−	0.519527i	$-0.173892\pi$
$269$	1477.67	+	1477.67i	0.334926	+	0.334926i	−0.519527	+	0.854454i	$0.673892\pi$
$270$		0			0
$271$		−	5865.36i		−	1.31474i	−0.753566	−	0.657372i	$-0.771668\pi$
$271$		−	5865.36i		−	1.31474i	0.753566	−	0.657372i	$-0.228332\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	6814.96	−	6814.96i	1.49439	−	1.49439i
$276$		0			0
$277$	−197.049	−	197.049i	−0.0427420	−	0.0427420i	0.685413	−	0.728155i	$-0.259622\pi$
$277$	−197.049	−	197.049i	−0.0427420	−	0.0427420i	−0.728155	+	0.685413i	$0.759622\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	4436.82			0.941917			0.470958	−	0.882155i	$-0.343908\pi$
$281$	4436.82			0.941917			0.470958	+	0.882155i	$0.343908\pi$
$282$		0			0
$283$	6047.06	−	6047.06i	1.27018	−	1.27018i	0.324186	−	0.945994i	$-0.394910\pi$
$283$	6047.06	−	6047.06i	1.27018	−	1.27018i	0.945994	−	0.324186i	$-0.105090\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−6834.73			−1.40572
$288$		0			0
$289$	−2467.25			−0.502189
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−365.261	+	365.261i	−0.0728287	+	0.0728287i	−0.742583	−	0.669754i	$-0.766399\pi$
$293$	−365.261	+	365.261i	−0.0728287	+	0.0728287i	0.669754	+	0.742583i	$0.266399\pi$
$294$		0			0
$295$	7335.05			1.44767
$296$		0			0
$297$		0			0
$298$		0			0
$299$	3212.11	+	3212.11i	0.621274	+	0.621274i
$300$		0			0
$301$	−386.166	+	386.166i	−0.0739477	+	0.0739477i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	182.507i		−	0.0342634i
$306$		0			0
$307$	4339.31	+	4339.31i	0.806701	+	0.806701i	0.984133	−	0.177432i	$-0.0567789\pi$
$307$	4339.31	+	4339.31i	0.806701	+	0.806701i	−0.177432	+	0.984133i	$0.556779\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	8077.43i		−	1.47276i	−0.676567	−	0.736382i	$-0.736533\pi$
$311$		−	8077.43i		−	1.47276i	0.676567	−	0.736382i	$-0.263467\pi$
$312$		0			0
$313$			6259.69i			1.13041i	0.824950	+	0.565206i	$0.191203\pi$
$313$			6259.69i			1.13041i	−0.824950	+	0.565206i	$0.808797\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	4411.87	+	4411.87i	0.781689	+	0.781689i	0.980116	−	0.198427i	$-0.0635832\pi$
$317$	4411.87	+	4411.87i	0.781689	+	0.781689i	−0.198427	+	0.980116i	$0.563583\pi$
$318$		0			0
$319$		−	3170.76i		−	0.556516i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−4574.97	+	4574.97i	−0.788106	+	0.788106i
$324$		0			0
$325$	−9034.32	−	9034.32i	−1.54195	−	1.54195i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	11536.0			1.93314
$330$		0			0
$331$	−5055.03	+	5055.03i	−0.839424	+	0.839424i	−0.988783	−	0.149359i	$-0.952279\pi$
$331$	−5055.03	+	5055.03i	−0.839424	+	0.839424i	0.149359	+	0.988783i	$0.452279\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	8327.07			1.35808
$336$		0			0
$337$	9698.71			1.56772			0.783861	−	0.620936i	$-0.213247\pi$
$337$	9698.71			1.56772			0.783861	+	0.620936i	$0.213247\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	4058.70	−	4058.70i	0.644548	−	0.644548i
$342$		0			0
$343$	−2104.06			−0.331220
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−1095.25	−	1095.25i	−0.169441	−	0.169441i	0.617293	−	0.786734i	$-0.288229\pi$
$347$	−1095.25	−	1095.25i	−0.169441	−	0.169441i	−0.786734	+	0.617293i	$0.788229\pi$
$348$		0			0
$349$	−2265.51	+	2265.51i	−0.347478	+	0.347478i	−0.859169	−	0.511691i	$-0.829019\pi$
$349$	−2265.51	+	2265.51i	−0.347478	+	0.347478i	0.511691	+	0.859169i	$0.329019\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	1670.69i		−	0.251904i	−0.992036	−	0.125952i	$-0.959801\pi$
$353$		−	1670.69i		−	0.251904i	0.992036	−	0.125952i	$-0.0401985\pi$
$354$		0			0
$355$	6824.85	+	6824.85i	1.02035	+	1.02035i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			753.959i			0.110842i	0.998463	+	0.0554212i	$0.0176502\pi$
$359$			753.959i			0.110842i	−0.998463	+	0.0554212i	$0.982350\pi$
$360$		0			0
$361$		−	1187.01i		−	0.173058i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	832.785	+	832.785i	0.119425	+	0.119425i
$366$		0			0
$367$			5281.79i			0.751246i	0.926773	+	0.375623i	$0.122571\pi$
$367$			5281.79i			0.751246i	−0.926773	+	0.375623i	$0.877429\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	2760.49	−	2760.49i	0.386300	−	0.386300i
$372$		0			0
$373$	1795.08	+	1795.08i	0.249184	+	0.249184i	0.820636	−	0.571452i	$-0.193620\pi$
$373$	1795.08	+	1795.08i	0.249184	+	0.249184i	−0.571452	+	0.820636i	$0.693620\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−4203.36			−0.574228
$378$		0			0
$379$	9113.85	−	9113.85i	1.23522	−	1.23522i	0.273283	−	0.961934i	$-0.411890\pi$
$379$	9113.85	−	9113.85i	1.23522	−	1.23522i	0.961934	−	0.273283i	$-0.0881096\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−2975.11			−0.396922			−0.198461	−	0.980109i	$-0.563594\pi$
$383$	−2975.11			−0.396922			−0.198461	+	0.980109i	$0.563594\pi$
$384$		0			0
$385$	−15045.2			−1.99162
$386$		0			0
$387$		0			0
$388$		0			0
$389$	4056.95	−	4056.95i	0.528780	−	0.528780i	−0.391429	−	0.920209i	$-0.628019\pi$
$389$	4056.95	−	4056.95i	0.528780	−	0.528780i	0.920209	+	0.391429i	$0.128019\pi$
$390$		0			0
$391$	10307.8			1.33322
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−442.750	−	442.750i	−0.0563979	−	0.0563979i
$396$		0			0
$397$	−2409.95	+	2409.95i	−0.304665	+	0.304665i	−0.842836	−	0.538171i	$-0.819115\pi$
$397$	−2409.95	+	2409.95i	−0.304665	+	0.304665i	0.538171	+	0.842836i	$0.319115\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			7830.00i			0.975091i	0.873098	+	0.487545i	$0.162108\pi$
$401$			7830.00i			0.975091i	−0.873098	+	0.487545i	$0.837892\pi$
$402$		0			0
$403$	−5380.46	−	5380.46i	−0.665061	−	0.665061i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			3109.17i			0.378662i
$408$		0			0
$409$			4518.97i			0.546329i	0.961967	+	0.273164i	$0.0880703\pi$
$409$			4518.97i			0.546329i	−0.961967	+	0.273164i	$0.911930\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−5908.27	−	5908.27i	−0.703940	−	0.703940i
$414$		0			0
$415$			27185.8i			3.21566i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	427.092	−	427.092i	0.0497967	−	0.0497967i	−0.681770	−	0.731567i	$-0.738789\pi$
$419$	427.092	−	427.092i	0.0497967	−	0.0497967i	0.731567	+	0.681770i	$0.238789\pi$
$420$		0			0
$421$	3815.11	+	3815.11i	0.441656	+	0.441656i	0.892568	−	0.450913i	$-0.148901\pi$
$421$	3815.11	+	3815.11i	0.441656	+	0.441656i	−0.450913	+	0.892568i	$0.648901\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−28991.7			−3.30895
$426$		0			0
$427$	−147.007	+	147.007i	−0.0166608	+	0.0166608i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−12742.0			−1.42404			−0.712022	−	0.702157i	$-0.752220\pi$
$431$	−12742.0			−1.42404			−0.712022	+	0.702157i	$0.752220\pi$
$432$		0			0
$433$	−14363.7			−1.59417			−0.797085	−	0.603867i	$-0.793626\pi$
$433$	−14363.7			−1.59417			−0.797085	+	0.603867i	$0.793626\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	6389.77	−	6389.77i	0.699460	−	0.699460i
$438$		0			0
$439$	−3156.53			−0.343173			−0.171586	−	0.985169i	$-0.554889\pi$
$439$	−3156.53			−0.343173			−0.171586	+	0.985169i	$0.554889\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−8113.23	−	8113.23i	−0.870139	−	0.870139i	0.122349	−	0.992487i	$-0.460957\pi$
$443$	−8113.23	−	8113.23i	−0.870139	−	0.870139i	−0.992487	+	0.122349i	$0.960957\pi$
$444$		0			0
$445$	4277.03	−	4277.03i	0.455619	−	0.455619i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	18120.9i		−	1.90463i	−0.305112	−	0.952316i	$-0.598694\pi$
$449$		−	18120.9i		−	1.90463i	0.305112	−	0.952316i	$-0.401306\pi$
$450$		0			0
$451$	−5634.19	−	5634.19i	−0.588256	−	0.588256i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			19944.9i			2.05501i
$456$		0			0
$457$		−	11253.4i		−	1.15189i	−0.817490	−	0.575943i	$-0.804635\pi$
$457$		−	11253.4i		−	1.15189i	0.817490	−	0.575943i	$-0.195365\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−1886.34	−	1886.34i	−0.190576	−	0.190576i	0.605369	−	0.795945i	$-0.293025\pi$
$461$	−1886.34	−	1886.34i	−0.190576	−	0.190576i	−0.795945	+	0.605369i	$0.793025\pi$
$462$		0			0
$463$			4690.87i			0.470849i	0.971893	+	0.235424i	$0.0756480\pi$
$463$			4690.87i			0.470849i	−0.971893	+	0.235424i	$0.924352\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−7780.38	+	7780.38i	−0.770948	+	0.770948i	−0.978272	−	0.207324i	$-0.933525\pi$
$467$	−7780.38	+	7780.38i	−0.770948	+	0.770948i	0.207324	+	0.978272i	$0.433525\pi$
$468$		0			0
$469$	−6707.34	−	6707.34i	−0.660375	−	0.660375i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−636.669			−0.0618902
$474$		0			0
$475$	−17971.8	+	17971.8i	−1.73600	+	1.73600i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	1769.08			0.168750			0.0843749	−	0.996434i	$-0.473111\pi$
$479$	1769.08			0.168750			0.0843749	+	0.996434i	$0.473111\pi$
$480$		0			0
$481$	4121.70			0.390714
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−2869.75	+	2869.75i	−0.268678	+	0.268678i
$486$		0			0
$487$	−18479.1			−1.71944			−0.859721	−	0.510765i	$-0.829362\pi$
$487$	−18479.1			−1.71944			−0.859721	+	0.510765i	$0.829362\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−3394.93	−	3394.93i	−0.312039	−	0.312039i	0.533660	−	0.845699i	$-0.320816\pi$
$491$	−3394.93	−	3394.93i	−0.312039	−	0.312039i	−0.845699	+	0.533660i	$0.820816\pi$
$492$		0			0
$493$	−6744.42	+	6744.42i	−0.616132	+	0.616132i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	10994.6i		−	0.992307i
$498$		0			0
$499$	−7006.50	−	7006.50i	−0.628565	−	0.628565i	0.319142	−	0.947707i	$-0.396605\pi$
$499$	−7006.50	−	7006.50i	−0.628565	−	0.628565i	−0.947707	+	0.319142i	$0.896605\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	15005.5i		−	1.33014i	−0.746781	−	0.665070i	$-0.768402\pi$
$503$		−	15005.5i		−	1.33014i	0.746781	−	0.665070i	$-0.231598\pi$
$504$		0			0
$505$			6657.49i			0.586643i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−15473.1	−	15473.1i	−1.34742	−	1.34742i	−0.888458	−	0.458958i	$-0.848223\pi$
$509$	−15473.1	−	15473.1i	−1.34742	−	1.34742i	−0.458958	−	0.888458i	$-0.651777\pi$
$510$		0			0
$511$		−	1341.59i		−	0.116142i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−28852.1	+	28852.1i	−2.46869	+	2.46869i
$516$		0			0
$517$	9509.68	+	9509.68i	0.808965	+	0.808965i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−8687.96			−0.730569			−0.365284	−	0.930896i	$-0.619028\pi$
$521$	−8687.96			−0.730569			−0.365284	+	0.930896i	$0.619028\pi$
$522$		0			0
$523$	6047.61	−	6047.61i	0.505628	−	0.505628i	−0.407553	−	0.913182i	$-0.633618\pi$
$523$	6047.61	−	6047.61i	0.505628	−	0.505628i	0.913182	+	0.407553i	$0.133618\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−17266.2			−1.42719
$528$		0			0
$529$	−2229.75			−0.183262
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−7469.02	+	7469.02i	−0.606978	+	0.606978i
$534$		0			0
$535$	12179.6			0.984246
$536$		0			0
$537$		0			0
$538$		0			0
$539$	5192.12	+	5192.12i	0.414917	+	0.414917i
$540$		0			0
$541$	−16062.4	+	16062.4i	−1.27648	+	1.27648i	−0.333854	+	0.942625i	$0.608349\pi$
$541$	−16062.4	+	16062.4i	−1.27648	+	1.27648i	−0.942625	+	0.333854i	$0.891651\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			6272.34i			0.492987i
$546$		0			0
$547$	10435.4	+	10435.4i	0.815697	+	0.815697i	0.985481	−	0.169784i	$-0.0543070\pi$
$547$	10435.4	+	10435.4i	0.815697	+	0.815697i	−0.169784	+	0.985481i	$0.554307\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			8361.64i			0.646493i
$552$		0			0
$553$			713.257i			0.0548477i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	6335.69	+	6335.69i	0.481960	+	0.481960i	0.905757	−	0.423797i	$-0.139303\pi$
$557$	6335.69	+	6335.69i	0.481960	+	0.481960i	−0.423797	+	0.905757i	$0.639303\pi$
$558$		0			0
$559$			844.008i			0.0638600i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−9187.42	+	9187.42i	−0.687750	+	0.687750i	−0.961734	−	0.273984i	$-0.911658\pi$
$563$	−9187.42	+	9187.42i	−0.687750	+	0.687750i	0.273984	+	0.961734i	$0.411658\pi$
$564$		0			0
$565$	−5855.52	−	5855.52i	−0.436006	−	0.436006i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−11816.3			−0.870588			−0.435294	−	0.900288i	$-0.643356\pi$
$569$	−11816.3			−0.870588			−0.435294	+	0.900288i	$0.643356\pi$
$570$		0			0
$571$	−2344.11	+	2344.11i	−0.171800	+	0.171800i	−0.787770	−	0.615970i	$-0.788764\pi$
$571$	−2344.11	+	2344.11i	−0.171800	+	0.171800i	0.615970	+	0.787770i	$0.288764\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	40492.1			2.93676
$576$		0			0
$577$	16615.2			1.19878			0.599392	−	0.800456i	$-0.295409\pi$
$577$	16615.2			1.19878			0.599392	+	0.800456i	$0.295409\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	21897.8	−	21897.8i	1.56364	−	1.56364i
$582$		0			0
$583$	4551.20			0.323313
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−1186.35	−	1186.35i	−0.0834173	−	0.0834173i	0.664167	−	0.747584i	$-0.268786\pi$
$587$	−1186.35	−	1186.35i	−0.0834173	−	0.0834173i	−0.747584	+	0.664167i	$0.768786\pi$
$588$		0			0
$589$	−10703.2	+	10703.2i	−0.748758	+	0.748758i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	1478.23i		−	0.102367i	−0.998689	−	0.0511833i	$-0.983701\pi$
$593$		−	1478.23i		−	0.102367i	0.998689	−	0.0511833i	$-0.0162993\pi$
$594$		0			0
$595$	32002.1	+	32002.1i	2.20497	+	2.20497i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	10548.3i		−	0.719522i	−0.933044	−	0.359761i	$-0.882858\pi$
$599$		−	10548.3i		−	0.719522i	0.933044	−	0.359761i	$-0.117142\pi$
$600$		0			0
$601$		−	21596.7i		−	1.46580i	−0.680335	−	0.732901i	$-0.738166\pi$
$601$		−	21596.7i		−	1.46580i	0.680335	−	0.732901i	$-0.261834\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	7837.32	+	7837.32i	0.526665	+	0.526665i
$606$		0			0
$607$			12331.9i			0.824604i	0.911047	+	0.412302i	$0.135275\pi$
$607$			12331.9i			0.824604i	−0.911047	+	0.412302i	$0.864725\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	12606.6	−	12606.6i	0.834711	−	0.834711i
$612$		0			0
$613$	−6110.69	−	6110.69i	−0.402624	−	0.402624i	0.476533	−	0.879157i	$-0.341893\pi$
$613$	−6110.69	−	6110.69i	−0.402624	−	0.402624i	−0.879157	+	0.476533i	$0.841893\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	9428.72			0.615212			0.307606	−	0.951514i	$-0.400472\pi$
$617$	9428.72			0.615212			0.307606	+	0.951514i	$0.400472\pi$
$618$		0			0
$619$	209.732	−	209.732i	0.0136185	−	0.0136185i	−0.700265	−	0.713883i	$-0.746935\pi$
$619$	209.732	−	209.732i	0.0136185	−	0.0136185i	0.713883	+	0.700265i	$0.246935\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−6890.16			−0.443096
$624$		0			0
$625$	−56078.5			−3.58903
$626$		0			0
$627$		0			0
$628$		0			0
$629$	6613.39	−	6613.39i	0.419226	−	0.419226i
$630$		0			0
$631$	17914.3			1.13020			0.565101	−	0.825022i	$-0.308837\pi$
$631$	17914.3			1.13020			0.565101	+	0.825022i	$0.308837\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−21593.7	−	21593.7i	−1.34948	−	1.34948i
$636$		0			0
$637$	6882.98	−	6882.98i	0.428122	−	0.428122i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	3322.93i		−	0.204755i	−0.994746	−	0.102377i	$-0.967355\pi$
$641$		−	3322.93i		−	0.204755i	0.994746	−	0.102377i	$-0.0326450\pi$
$642$		0			0
$643$	−11199.4	−	11199.4i	−0.686878	−	0.686878i	0.274663	−	0.961541i	$-0.411434\pi$
$643$	−11199.4	−	11199.4i	−0.686878	−	0.686878i	−0.961541	+	0.274663i	$0.911434\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	21868.3i		−	1.32879i	−0.747380	−	0.664397i	$-0.768688\pi$
$647$		−	21868.3i		−	1.32879i	0.747380	−	0.664397i	$-0.231312\pi$
$648$		0			0
$649$		−	9740.93i		−	0.589160i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	2170.27	+	2170.27i	0.130060	+	0.130060i	0.769140	−	0.639080i	$-0.220685\pi$
$653$	2170.27	+	2170.27i	0.130060	+	0.130060i	−0.639080	+	0.769140i	$0.720685\pi$
$654$		0			0
$655$			6147.38i			0.366715i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	9827.52	−	9827.52i	0.580919	−	0.580919i	−0.354237	−	0.935156i	$-0.615259\pi$
$659$	9827.52	−	9827.52i	0.580919	−	0.580919i	0.935156	+	0.354237i	$0.115259\pi$
$660$		0			0
$661$	−6022.78	−	6022.78i	−0.354401	−	0.354401i	0.507343	−	0.861744i	$-0.330628\pi$
$661$	−6022.78	−	6022.78i	−0.354401	−	0.354401i	−0.861744	+	0.507343i	$0.830628\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	39675.8			2.31363
$666$		0			0
$667$	9419.79	−	9419.79i	0.546830	−	0.546830i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−242.369			−0.0139442
$672$		0			0
$673$	−15216.4			−0.871543			−0.435771	−	0.900057i	$-0.643524\pi$
$673$	−15216.4			−0.871543			−0.435771	+	0.900057i	$0.643524\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−5754.92	+	5754.92i	−0.326705	+	0.326705i	−0.851332	−	0.524627i	$-0.824205\pi$
$677$	−5754.92	+	5754.92i	−0.326705	+	0.326705i	0.524627	+	0.851332i	$0.324205\pi$
$678$		0			0
$679$	4623.09			0.261293
$680$		0			0
$681$		0			0
$682$		0			0
$683$	10651.0	+	10651.0i	0.596705	+	0.596705i	0.939434	−	0.342729i	$-0.111351\pi$
$683$	10651.0	+	10651.0i	0.596705	+	0.596705i	−0.342729	+	0.939434i	$0.611351\pi$
$684$		0			0
$685$	−14986.2	+	14986.2i	−0.835902	+	0.835902i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	6033.34i		−	0.333602i
$690$		0			0
$691$	16585.3	+	16585.3i	0.913073	+	0.913073i	0.996513	−	0.0834401i	$-0.0265907\pi$
$691$	16585.3	+	16585.3i	0.913073	+	0.913073i	−0.0834401	+	0.996513i	$0.526591\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			36482.7i			1.99117i
$696$		0			0
$697$			23968.6i			1.30254i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−6023.16	−	6023.16i	−0.324524	−	0.324524i	0.525975	−	0.850500i	$-0.323700\pi$
$701$	−6023.16	−	6023.16i	−0.324524	−	0.324524i	−0.850500	+	0.525975i	$0.823700\pi$
$702$		0			0
$703$		−	8199.20i		−	0.439884i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	5362.51	−	5362.51i	0.285259	−	0.285259i
$708$		0			0
$709$	21886.4	+	21886.4i	1.15932	+	1.15932i	0.984621	+	0.174701i	$0.0558960\pi$
$709$	21886.4	+	21886.4i	1.15932	+	1.15932i	0.174701	+	0.984621i	$0.444104\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	24115.4			1.26666
$714$		0			0
$715$	−16441.5	+	16441.5i	−0.859966	+	0.859966i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−24967.9			−1.29506			−0.647528	−	0.762041i	$-0.724197\pi$
$719$	−24967.9			−1.29506			−0.647528	+	0.762041i	$0.724197\pi$
$720$		0			0
$721$	46479.9			2.40084
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−26494.0	+	26494.0i	−1.35719	+	1.35719i
$726$		0			0
$727$	31570.1			1.61055			0.805276	−	0.592900i	$-0.202017\pi$
$727$	31570.1			1.61055			0.805276	+	0.592900i	$0.202017\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	1354.24	+	1354.24i	0.0685202	+	0.0685202i
$732$		0			0
$733$	14686.6	−	14686.6i	0.740055	−	0.740055i	−0.232533	−	0.972588i	$-0.574701\pi$
$733$	14686.6	−	14686.6i	0.740055	−	0.740055i	0.972588	+	0.232533i	$0.0747015\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	11058.3i		−	0.552699i
$738$		0			0
$739$	−16900.6	−	16900.6i	−0.841271	−	0.841271i	0.147753	−	0.989024i	$-0.452796\pi$
$739$	−16900.6	−	16900.6i	−0.841271	−	0.841271i	−0.989024	+	0.147753i	$0.952796\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	13888.3i		−	0.685750i	−0.939381	−	0.342875i	$-0.888599\pi$
$743$		−	13888.3i		−	0.685750i	0.939381	−	0.342875i	$-0.111401\pi$
$744$		0			0
$745$			38230.5i			1.88008i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−9810.52	−	9810.52i	−0.478596	−	0.478596i
$750$		0			0
$751$			8334.75i			0.404979i	0.979284	+	0.202490i	$0.0649032\pi$
$751$			8334.75i			0.404979i	−0.979284	+	0.202490i	$0.935097\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	27870.8	−	27870.8i	1.34347	−	1.34347i
$756$		0			0
$757$	14800.7	+	14800.7i	0.710620	+	0.710620i	0.966665	−	0.256045i	$-0.0824195\pi$
$757$	14800.7	+	14800.7i	0.710620	+	0.710620i	−0.256045	+	0.966665i	$0.582420\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−5724.34			−0.272677			−0.136339	−	0.990662i	$-0.543533\pi$
$761$	−5724.34			−0.272677			−0.136339	+	0.990662i	$0.543533\pi$
$762$		0			0
$763$	5052.28	−	5052.28i	0.239718	−	0.239718i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−12913.2			−0.607910
$768$		0			0
$769$	−16257.1			−0.762348			−0.381174	−	0.924503i	$-0.624480\pi$
$769$	−16257.1			−0.762348			−0.381174	+	0.924503i	$0.624480\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	9339.94	−	9339.94i	0.434585	−	0.434585i	−0.455600	−	0.890185i	$-0.650575\pi$
$773$	9339.94	−	9339.94i	0.434585	−	0.434585i	0.890185	+	0.455600i	$0.150575\pi$
$774$		0			0
$775$	−67826.6			−3.14374
$776$		0			0
$777$		0			0
$778$		0			0
$779$	14858.0	+	14858.0i	0.683365	+	0.683365i
$780$		0			0
$781$	9063.38	−	9063.38i	0.415254	−	0.415254i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			63923.9i			2.90642i
$786$		0			0
$787$	21944.5	+	21944.5i	0.993950	+	0.993950i	0.999982	−	0.00603172i	$-0.00191997\pi$
$787$	21944.5	+	21944.5i	0.993950	+	0.993950i	−0.00603172	+	0.999982i	$0.501920\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			9433.07i			0.424022i
$792$		0			0
$793$			321.299i			0.0143880i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	21112.7	+	21112.7i	0.938331	+	0.938331i	0.998206	−	0.0598747i	$-0.0190701\pi$
$797$	21112.7	+	21112.7i	0.938331	+	0.938331i	−0.0598747	+	0.998206i	$0.519070\pi$
$798$		0			0
$799$		−	40455.4i		−	1.79125i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1105.94	−	1105.94i	0.0486023	−	0.0486023i
$804$		0			0
$805$	−44696.7	−	44696.7i	−1.95696	−	1.95696i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−39210.9			−1.70406			−0.852028	−	0.523496i	$-0.824628\pi$
$809$	−39210.9			−1.70406			−0.852028	+	0.523496i	$0.824628\pi$
$810$		0			0
$811$	7512.78	−	7512.78i	0.325289	−	0.325289i	−0.525503	−	0.850792i	$-0.676123\pi$
$811$	7512.78	−	7512.78i	0.325289	−	0.325289i	0.850792	+	0.525503i	$0.176123\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−4696.79			−0.201867
$816$		0			0
$817$	1678.97			0.0718966
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−12626.1	+	12626.1i	−0.536727	+	0.536727i	−0.922566	−	0.385839i	$-0.873912\pi$
$821$	−12626.1	+	12626.1i	−0.536727	+	0.536727i	0.385839	+	0.922566i	$0.373912\pi$
$822$		0			0
$823$	−15641.0			−0.662468			−0.331234	−	0.943549i	$-0.607465\pi$
$823$	−15641.0			−0.662468			−0.331234	+	0.943549i	$0.607465\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−26716.0	−	26716.0i	−1.12334	−	1.12334i	−0.991235	−	0.132109i	$-0.957825\pi$
$827$	−26716.0	−	26716.0i	−1.12334	−	1.12334i	−0.132109	−	0.991235i	$-0.542175\pi$
$828$		0			0
$829$	3654.34	−	3654.34i	0.153101	−	0.153101i	−0.626401	−	0.779501i	$-0.715473\pi$
$829$	3654.34	−	3654.34i	0.153101	−	0.153101i	0.779501	+	0.626401i	$0.215473\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	22087.9i		−	0.918729i
$834$		0			0
$835$	−52958.5	−	52958.5i	−2.19486	−	2.19486i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	7316.14i		−	0.301050i	−0.988606	−	0.150525i	$-0.951904\pi$
$839$		−	7316.14i		−	0.301050i	0.988606	−	0.150525i	$-0.0480965\pi$
$840$		0			0
$841$		−	12062.3i		−	0.494579i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−11612.8	−	11612.8i	−0.472771	−	0.472771i
$846$		0			0
$847$		−	12625.7i		−	0.512189i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−9236.79	+	9236.79i	−0.372072	+	0.372072i
$852$		0			0
$853$	−4362.84	−	4362.84i	−0.175124	−	0.175124i	0.614102	−	0.789226i	$-0.289518\pi$
$853$	−4362.84	−	4362.84i	−0.175124	−	0.175124i	−0.789226	+	0.614102i	$0.789518\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−10598.4			−0.422442			−0.211221	−	0.977438i	$-0.567744\pi$
$857$	−10598.4			−0.422442			−0.211221	+	0.977438i	$0.567744\pi$
$858$		0			0
$859$	−25041.9	+	25041.9i	−0.994668	+	0.994668i	−0.999986	−	0.00531813i	$-0.998307\pi$
$859$	−25041.9	+	25041.9i	−0.994668	+	0.994668i	0.00531813	+	0.999986i	$0.498307\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	11567.3			0.456265			0.228133	−	0.973630i	$-0.426738\pi$
$863$	11567.3			0.456265			0.228133	+	0.973630i	$0.426738\pi$
$864$		0			0
$865$	77966.2			3.06466
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−587.971	+	587.971i	−0.0229523	+	0.0229523i
$870$		0			0
$871$	−14659.6			−0.570289
$872$		0			0
$873$		0			0
$874$		0			0
$875$	79149.1	+	79149.1i	3.05797	+	3.05797i
$876$		0			0
$877$	29456.6	−	29456.6i	1.13418	−	1.13418i	0.144710	−	0.989474i	$-0.453775\pi$
$877$	29456.6	−	29456.6i	1.13418	−	1.13418i	0.989474	−	0.144710i	$-0.0462250\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	42700.5i		−	1.63294i	−0.577390	−	0.816468i	$-0.695929\pi$
$881$		−	42700.5i		−	1.63294i	0.577390	−	0.816468i	$-0.304071\pi$
$882$		0			0
$883$	−135.261	−	135.261i	−0.00515504	−	0.00515504i	0.704525	−	0.709680i	$-0.251160\pi$
$883$	−135.261	−	135.261i	−0.00515504	−	0.00515504i	−0.709680	+	0.704525i	$0.751160\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	27302.4i		−	1.03351i	−0.856132	−	0.516757i	$-0.827139\pi$
$887$		−	27302.4i		−	1.03351i	0.856132	−	0.516757i	$-0.172861\pi$
$888$		0			0
$889$			34786.9i			1.31239i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−25078.0	−	25078.0i	−0.939758	−	0.939758i
$894$		0			0
$895$			17642.7i			0.658917i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−15778.7	+	15778.7i	−0.585370	+	0.585370i
$900$		0			0
$901$	−9680.68	−	9680.68i	−0.357947	−	0.357947i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	566.598			0.0208114
$906$		0			0
$907$	22150.8	−	22150.8i	0.810921	−	0.810921i	−0.173851	−	0.984772i	$-0.555621\pi$
$907$	22150.8	−	22150.8i	0.810921	−	0.810921i	0.984772	+	0.173851i	$0.0556212\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−40596.8			−1.47643			−0.738217	−	0.674564i	$-0.764332\pi$
$911$	−40596.8			−1.47643			−0.738217	+	0.674564i	$0.764332\pi$
$912$		0			0
$913$	36102.7			1.30868
$914$		0			0
$915$		0			0
$916$		0			0
$917$	4951.63	−	4951.63i	0.178317	−	0.178317i
$918$		0			0
$919$	33460.1			1.20103			0.600515	−	0.799613i	$-0.294962\pi$
$919$	33460.1			1.20103			0.600515	+	0.799613i	$0.294962\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−12015.0	−	12015.0i	−0.428470	−	0.428470i
$924$		0			0
$925$	25979.3	−	25979.3i	0.923452	−	0.923452i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			22691.6i			0.801387i	0.916212	+	0.400693i	$0.131231\pi$
$929$			22691.6i			0.801387i	−0.916212	+	0.400693i	$0.868769\pi$
$930$		0			0
$931$	−13692.2	−	13692.2i	−0.482001	−	0.482001i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			52761.7i			1.84545i
$936$		0			0
$937$			5447.73i			0.189936i	0.995480	+	0.0949678i	$0.0302748\pi$
$937$			5447.73i			0.189936i	−0.995480	+	0.0949678i	$0.969725\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	20191.7	+	20191.7i	0.699501	+	0.699501i	0.964303	−	0.264802i	$-0.0853067\pi$
$941$	20191.7	+	20191.7i	0.699501	+	0.699501i	−0.264802	+	0.964303i	$0.585307\pi$
$942$		0			0
$943$		−	33476.4i		−	1.15603i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−2444.37	+	2444.37i	−0.0838768	+	0.0838768i	−0.747800	−	0.663924i	$-0.768890\pi$
$947$	−2444.37	+	2444.37i	−0.0838768	+	0.0838768i	0.663924	+	0.747800i	$0.268890\pi$
$948$		0			0
$949$	−1466.10	−	1466.10i	−0.0501491	−	0.0501491i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	9299.20			0.316087			0.158043	−	0.987432i	$-0.449481\pi$
$953$	9299.20			0.316087			0.158043	+	0.987432i	$0.449481\pi$
$954$		0			0
$955$	26274.1	−	26274.1i	0.890273	−	0.890273i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	24142.3			0.812926
$960$		0			0
$961$	−10603.6			−0.355933
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−62307.0	+	62307.0i	−2.07848	+	2.07848i
$966$		0			0
$967$	−39928.8			−1.32784			−0.663922	−	0.747802i	$-0.731109\pi$
$967$	−39928.8			−1.32784			−0.663922	+	0.747802i	$0.731109\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−18987.0	−	18987.0i	−0.627521	−	0.627521i	0.319923	−	0.947444i	$-0.396343\pi$
$971$	−18987.0	−	18987.0i	−0.627521	−	0.627521i	−0.947444	+	0.319923i	$0.896343\pi$
$972$		0			0
$973$	29386.2	−	29386.2i	0.968222	−	0.968222i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	26451.4i		−	0.866176i	−0.901352	−	0.433088i	$-0.857424\pi$
$977$		−	26451.4i		−	0.866176i	0.901352	−	0.433088i	$-0.142576\pi$
$978$		0			0
$979$	−5679.88	−	5679.88i	−0.185424	−	0.185424i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	3516.98i		−	0.114114i	−0.998371	−	0.0570571i	$-0.981828\pi$
$983$		−	3516.98i		−	0.114114i	0.998371	−	0.0570571i	$-0.0181717\pi$
$984$		0			0
$985$		−	118281.i		−	3.82615i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−1891.43	−	1891.43i	−0.0608130	−	0.0608130i
$990$		0			0
$991$		−	55027.6i		−	1.76388i	−0.471359	−	0.881942i	$-0.656236\pi$
$991$		−	55027.6i		−	1.76388i	0.471359	−	0.881942i	$-0.343764\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	50681.9	−	50681.9i	1.61480	−	1.61480i
$996$		0			0
$997$	28913.9	+	28913.9i	0.918466	+	0.918466i	0.996918	−	0.0784515i	$-0.0249976\pi$
$997$	28913.9	+	28913.9i	0.918466	+	0.918466i	−0.0784515	+	0.996918i	$0.524998\pi$
$998$		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.287.1		48
3.2	odd	2	inner	1152.4.l.b.287.24		48
4.3	odd	2		1152.4.l.a.287.1		48
8.3	odd	2		576.4.l.a.143.24		48
8.5	even	2		144.4.l.a.107.1	yes	48
12.11	even	2		1152.4.l.a.287.24		48
16.3	odd	4	inner	1152.4.l.b.863.24		48
16.5	even	4		576.4.l.a.431.1		48
16.11	odd	4		144.4.l.a.35.24	yes	48
16.13	even	4		1152.4.l.a.863.24		48
24.5	odd	2		144.4.l.a.107.24	yes	48
24.11	even	2		576.4.l.a.143.1		48
48.5	odd	4		576.4.l.a.431.24		48
48.11	even	4		144.4.l.a.35.1	✓	48
48.29	odd	4		1152.4.l.a.863.1		48
48.35	even	4	inner	1152.4.l.b.863.1		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.4.l.a.35.1	✓	48	48.11	even	4
144.4.l.a.35.24	yes	48	16.11	odd	4
144.4.l.a.107.1	yes	48	8.5	even	2
144.4.l.a.107.24	yes	48	24.5	odd	2
576.4.l.a.143.1		48	24.11	even	2
576.4.l.a.143.24		48	8.3	odd	2
576.4.l.a.431.1		48	16.5	even	4
576.4.l.a.431.24		48	48.5	odd	4
1152.4.l.a.287.1		48	4.3	odd	2
1152.4.l.a.287.24		48	12.11	even	2
1152.4.l.a.863.1		48	48.29	odd	4
1152.4.l.a.863.24		48	16.13	even	4
1152.4.l.b.287.1		48	1.1	even	1	trivial
1152.4.l.b.287.24		48	3.2	odd	2	inner
1152.4.l.b.863.1		48	48.35	even	4	inner
1152.4.l.b.863.24		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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+(-15.2065 + 15.2065i) q^{5} +24.4971 q^{7} +(20.1941 + 20.1941i) q^{11} +(26.7706 - 26.7706i) q^{13} -85.9084i q^{17} +(-53.2541 - 53.2541i) q^{19} +119.986i q^{23} -337.472i q^{25} +(-78.5071 - 78.5071i) q^{29} -200.984i q^{31} +(-372.515 + 372.515i) q^{35} +(76.9819 + 76.9819i) q^{37} -279.001 q^{41} +(-15.7637 + 15.7637i) q^{43} +470.913 q^{47} +257.110 q^{49} +(112.686 - 112.686i) q^{53} -614.162 q^{55} +(-241.182 - 241.182i) q^{59} +(-6.00098 + 6.00098i) q^{61} +814.171i q^{65} +(-273.801 - 273.801i) q^{67} -448.812i q^{71} -54.7653i q^{73} +(494.698 + 494.698i) q^{77} +29.1159i q^{79} +(893.890 - 893.890i) q^{83} +(1306.36 + 1306.36i) q^{85} -281.264 q^{89} +(655.802 - 655.802i) q^{91} +1619.61 q^{95} +188.720 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 48 q^{19} - 864 q^{43} + 2352 q^{49} - 576 q^{55} - 1824 q^{61} - 816 q^{67} + 480 q^{85} + 3600 q^{91}+O(q^{100}) \) 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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