LMFDB - Embedded newform 1152.5.b.f.703.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,5,Mod(703,1152)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1152, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("1152.703");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$N$	$=$	$1152 = 2^{7} \cdot 3^{2}$
Weight:	$k$	$=$	$5$
Character orbit:	$[\chi]$	$=$	1152.b (of order $2$ , degree $1$ , minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$119.082197473$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - 16x^{2} + 81$ x^4 - 16*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2^{7}\cdot 3^{4}$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			703.3
Root			$-2.91548 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1152.703
Dual form			1152.5.b.f.703.2

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+24.7386i q^{5} -50.9117i q^{7} -139.943 q^{11} +98.9545i q^{13} -16.0000 q^{17} +559.771 q^{19} -814.587i q^{23} +13.0000 q^{25} +420.557i q^{29} +661.852i q^{31} +1259.49 q^{35} +2473.86i q^{37} -464.000 q^{41} -559.771 q^{43} -2443.76i q^{47} -191.000 q^{49} -3191.28i q^{53} -3461.99i q^{55} -3638.51 q^{59} +4255.05i q^{61} -2448.00 q^{65} -5597.71 q^{67} +3258.35i q^{71} +898.000 q^{73} +7124.73i q^{77} +8604.08i q^{79} +419.829 q^{83} -395.818i q^{85} -7072.00 q^{89} +5037.94 q^{91} +13848.0i q^{95} -2366.00 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 64 q^{17} + 52 q^{25} - 1856 q^{41} - 764 q^{49} - 9792 q^{65} + 3592 q^{73} - 28288 q^{89} - 9464 q^{97}+O(q^{100})$ 4 * q - 64 * q^17 + 52 * q^25 - 1856 * q^41 - 764 * q^49 - 9792 * q^65 + 3592 * q^73 - 28288 * q^89 - 9464 * q^97

Character values

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

$n$	$127$	$641$	$901$
$\chi(n)$	$-1$	$1$	$-1$

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
$5$	24.7386i	0.989545i	0.869022	+	0.494773i	$0.164749\pi$
$5$	24.7386i	0.989545i	−0.869022	+	0.494773i	$0.835251\pi$
$6$	0	0
$7$	− 50.9117i	− 1.03901i	−0.854466	−	0.519507i	$-0.826116\pi$
$7$	− 50.9117i	− 1.03901i	0.854466	−	0.519507i	$-0.173884\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−139.943	−1.15655	−0.578276	−	0.815841i	$-0.696274\pi$
$11$	−139.943	−1.15655	−0.578276	+	0.815841i	$0.696274\pi$
$12$	0	0
$13$	98.9545i	0.585530i	0.956184	+	0.292765i	$0.0945753\pi$
$13$	98.9545i	0.585530i	−0.956184	+	0.292765i	$0.905425\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−16.0000	−0.0553633	−0.0276817	−	0.999617i	$-0.508812\pi$
$17$	−16.0000	−0.0553633	−0.0276817	+	0.999617i	$0.508812\pi$
$18$	0	0
$19$	559.771	1.55061	0.775307	−	0.631585i	$-0.217595\pi$
$19$	559.771	1.55061	0.775307	+	0.631585i	$0.217595\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 814.587i	− 1.53986i	−0.638127	−	0.769931i	$-0.720291\pi$
$23$	− 814.587i	− 1.53986i	0.638127	−	0.769931i	$-0.279709\pi$
$24$	0	0
$25$	13.0000	0.0208000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	420.557i	0.500068i	0.968237	+	0.250034i	$0.0804417\pi$
$29$	420.557i	0.500068i	−0.968237	+	0.250034i	$0.919558\pi$
$30$	0	0
$31$	661.852i	0.688712i	0.938839	+	0.344356i	$0.111903\pi$
$31$	661.852i	0.688712i	−0.938839	+	0.344356i	$0.888097\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1259.49	1.02815
$36$	0	0
$37$	2473.86i	1.80706i	0.428526	+	0.903529i	$0.359033\pi$
$37$	2473.86i	1.80706i	−0.428526	+	0.903529i	$0.640967\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−464.000	−0.276026	−0.138013	−	0.990430i	$-0.544072\pi$
$41$	−464.000	−0.276026	−0.138013	+	0.990430i	$0.544072\pi$
$42$	0	0
$43$	−559.771	−0.302743	−0.151371	−	0.988477i	$-0.548369\pi$
$43$	−559.771	−0.302743	−0.151371	+	0.988477i	$0.548369\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 2443.76i	− 1.10627i	−0.833090	−	0.553137i	$-0.813430\pi$
$47$	− 2443.76i	− 1.10627i	0.833090	−	0.553137i	$-0.186570\pi$
$48$	0	0
$49$	−191.000	−0.0795502
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 3191.28i	− 1.13609i	−0.822997	−	0.568046i	$-0.807699\pi$
$53$	− 3191.28i	− 1.13609i	0.822997	−	0.568046i	$-0.192301\pi$
$54$	0	0
$55$	− 3461.99i	− 1.14446i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−3638.51	−1.04525	−0.522625	−	0.852563i	$-0.675047\pi$
$59$	−3638.51	−1.04525	−0.522625	+	0.852563i	$0.675047\pi$
$60$	0	0
$61$	4255.05i	1.14352i	0.820420	+	0.571761i	$0.193740\pi$
$61$	4255.05i	1.14352i	−0.820420	+	0.571761i	$0.806260\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2448.00	−0.579408
$66$	0	0
$67$	−5597.71	−1.24698	−0.623492	−	0.781829i	$-0.714287\pi$
$67$	−5597.71	−1.24698	−0.623492	+	0.781829i	$0.714287\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	3258.35i	0.646369i	0.946336	+	0.323185i	$0.104754\pi$
$71$	3258.35i	0.646369i	−0.946336	+	0.323185i	$0.895246\pi$
$72$	0	0
$73$	898.000	0.168512	0.0842560	−	0.996444i	$-0.473149\pi$
$73$	898.000	0.168512	0.0842560	+	0.996444i	$0.473149\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	7124.73i	1.20167i
$78$	0	0
$79$	8604.08i	1.37864i	0.724458	+	0.689319i	$0.242090\pi$
$79$	8604.08i	1.37864i	−0.724458	+	0.689319i	$0.757910\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	419.829	0.0609419	0.0304709	−	0.999536i	$-0.490299\pi$
$83$	419.829	0.0609419	0.0304709	+	0.999536i	$0.490299\pi$
$84$	0	0
$85$	− 395.818i	− 0.0547845i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−7072.00	−0.892817	−0.446408	−	0.894829i	$-0.647297\pi$
$89$	−7072.00	−0.892817	−0.446408	+	0.894829i	$0.647297\pi$
$90$	0	0
$91$	5037.94	0.608374
$92$	0	0
$93$	0	0
$94$	0	0
$95$	13848.0i	1.53440i
$96$	0	0
$97$	−2366.00	−0.251461	−0.125731	−	0.992064i	$-0.540128\pi$
$97$	−2366.00	−0.251461	−0.125731	+	0.992064i	$0.540128\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 7941.10i	− 0.778463i	−0.921140	−	0.389232i	$-0.872741\pi$
$101$	− 7941.10i	− 0.778463i	0.921140	−	0.389232i	$-0.127259\pi$
$102$	0	0
$103$	− 7076.72i	− 0.667049i	−0.942741	−	0.333525i	$-0.891762\pi$
$103$	− 7076.72i	− 0.667049i	0.942741	−	0.333525i	$-0.108238\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	16513.3	1.44233	0.721166	−	0.692762i	$-0.243607\pi$
$107$	16513.3	1.44233	0.721166	+	0.692762i	$0.243607\pi$
$108$	0	0
$109$	− 9004.86i	− 0.757921i	−0.925413	−	0.378961i	$-0.876282\pi$
$109$	− 9004.86i	− 0.757921i	0.925413	−	0.378961i	$-0.123718\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−15584.0	−1.22046	−0.610228	−	0.792226i	$-0.708922\pi$
$113$	−15584.0	−1.22046	−0.610228	+	0.792226i	$0.708922\pi$
$114$	0	0
$115$	20151.8	1.52376
$116$	0	0
$117$	0	0
$118$	0	0
$119$	814.587i	0.0575233i
$120$	0	0
$121$	4943.00	0.337614
$122$	0	0
$123$	0	0
$124$	0	0
$125$	15783.2i	1.01013i
$126$	0	0
$127$	− 11251.5i	− 0.697593i	−0.937198	−	0.348797i	$-0.886590\pi$
$127$	− 11251.5i	− 0.697593i	0.937198	−	0.348797i	$-0.113410\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−21831.1	−1.27213	−0.636067	−	0.771634i	$-0.719440\pi$
$131$	−21831.1	−1.27213	−0.636067	+	0.771634i	$0.719440\pi$
$132$	0	0
$133$	− 28498.9i	− 1.61111i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−27440.0	−1.46199	−0.730993	−	0.682385i	$-0.760943\pi$
$137$	−27440.0	−1.46199	−0.730993	+	0.682385i	$0.760943\pi$
$138$	0	0
$139$	−14554.1	−0.753277	−0.376638	−	0.926360i	$-0.622920\pi$
$139$	−14554.1	−0.753277	−0.376638	+	0.926360i	$0.622920\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 13848.0i	− 0.677196i
$144$	0	0
$145$	−10404.0	−0.494839
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 24516.0i	− 1.10427i	−0.833753	−	0.552137i	$-0.813813\pi$
$149$	− 24516.0i	− 1.10427i	0.833753	−	0.552137i	$-0.186187\pi$
$150$	0	0
$151$	23368.5i	1.02489i	0.858721	+	0.512444i	$0.171260\pi$
$151$	23368.5i	1.02489i	−0.858721	+	0.512444i	$0.828740\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−16373.3	−0.681511
$156$	0	0
$157$	− 13161.0i	− 0.533935i	−0.963706	−	0.266967i	$-0.913978\pi$
$157$	− 13161.0i	− 0.533935i	0.963706	−	0.266967i	$-0.0860216\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−41472.0	−1.59994
$162$	0	0
$163$	−21831.1	−0.821675	−0.410838	−	0.911709i	$-0.634764\pi$
$163$	−21831.1	−0.821675	−0.410838	+	0.911709i	$0.634764\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 10589.6i	− 0.379706i	−0.981812	−	0.189853i	$-0.939199\pi$
$167$	− 10589.6i	− 0.379706i	0.981812	−	0.189853i	$-0.0608012\pi$
$168$	0	0
$169$	18769.0	0.657155
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 17044.9i	− 0.569512i	−0.958600	−	0.284756i	$-0.908087\pi$
$173$	− 17044.9i	− 0.569512i	0.958600	−	0.284756i	$-0.0919125\pi$
$174$	0	0
$175$	− 661.852i	− 0.0216115i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	31067.3	0.969611	0.484806	−	0.874622i	$-0.338890\pi$
$179$	31067.3	0.969611	0.484806	+	0.874622i	$0.338890\pi$
$180$	0	0
$181$	22858.5i	0.697735i	0.937172	+	0.348868i	$0.113434\pi$
$181$	22858.5i	0.697735i	−0.937172	+	0.348868i	$0.886566\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−61200.0	−1.78817
$186$	0	0
$187$	2239.09	0.0640306
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 52948.2i	− 1.45139i	−0.688017	−	0.725695i	$-0.741518\pi$
$191$	− 52948.2i	− 1.45139i	0.688017	−	0.725695i	$-0.258482\pi$
$192$	0	0
$193$	−25250.0	−0.677871	−0.338935	−	0.940810i	$-0.610067\pi$
$193$	−25250.0	−0.677871	−0.338935	+	0.940810i	$0.610067\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	57764.7i	1.48844i	0.667937	+	0.744218i	$0.267178\pi$
$197$	57764.7i	1.48844i	−0.667937	+	0.744218i	$0.732822\pi$
$198$	0	0
$199$	7993.14i	0.201842i	0.994894	+	0.100921i	$0.0321789\pi$
$199$	7993.14i	0.201842i	−0.994894	+	0.100921i	$0.967821\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	21411.3	0.519577
$204$	0	0
$205$	− 11478.7i	− 0.273140i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−78336.0	−1.79337
$210$	0	0
$211$	−43662.2	−0.980710	−0.490355	−	0.871523i	$-0.663133\pi$
$211$	−43662.2	−0.980710	−0.490355	+	0.871523i	$0.663133\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 13848.0i	− 0.299578i
$216$	0	0
$217$	33696.0	0.715581
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 1583.27i	− 0.0324169i
$222$	0	0
$223$	− 88026.3i	− 1.77012i	−0.465477	−	0.885060i	$-0.654117\pi$
$223$	− 88026.3i	− 1.77012i	0.465477	−	0.885060i	$-0.345883\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	16933.1	0.328613	0.164306	−	0.986409i	$-0.447461\pi$
$227$	16933.1	0.328613	0.164306	+	0.986409i	$0.447461\pi$
$228$	0	0
$229$	63627.8i	1.21332i	0.794961	+	0.606660i	$0.207491\pi$
$229$	63627.8i	1.21332i	−0.794961	+	0.606660i	$0.792509\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−89440.0	−1.64748	−0.823740	−	0.566968i	$-0.808116\pi$
$233$	−89440.0	−1.64748	−0.823740	+	0.566968i	$0.808116\pi$
$234$	0	0
$235$	60455.3	1.09471
$236$	0	0
$237$	0	0
$238$	0	0
$239$	108340.i	1.89668i	0.317261	+	0.948338i	$0.397237\pi$
$239$	108340.i	1.89668i	−0.317261	+	0.948338i	$0.602763\pi$
$240$	0	0
$241$	−56222.0	−0.967993	−0.483996	−	0.875070i	$-0.660815\pi$
$241$	−56222.0	−0.967993	−0.483996	+	0.875070i	$0.660815\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 4725.08i	− 0.0787185i
$246$	0	0
$247$	55391.9i	0.907930i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−110975.	−1.76147	−0.880737	−	0.473605i	$-0.842952\pi$
$251$	−110975.	−1.76147	−0.880737	+	0.473605i	$0.842952\pi$
$252$	0	0
$253$	113996.i	1.78093i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−112064.	−1.69668	−0.848340	−	0.529452i	$-0.822398\pi$
$257$	−112064.	−1.69668	−0.848340	+	0.529452i	$0.822398\pi$
$258$	0	0
$259$	125949.	1.87756
$260$	0	0
$261$	0	0
$262$	0	0
$263$	9775.04i	0.141321i	0.997500	+	0.0706606i	$0.0225107\pi$
$263$	9775.04i	0.141321i	−0.997500	+	0.0706606i	$0.977489\pi$
$264$	0	0
$265$	78948.0	1.12422
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 96159.1i	− 1.32888i	−0.747342	−	0.664440i	$-0.768670\pi$
$269$	− 96159.1i	− 1.32888i	0.747342	−	0.664440i	$-0.231330\pi$
$270$	0	0
$271$	123766.i	1.68525i	0.538502	+	0.842624i	$0.318990\pi$
$271$	123766.i	1.68525i	−0.538502	+	0.842624i	$0.681010\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1819.26	−0.0240563
$276$	0	0
$277$	− 37701.7i	− 0.491362i	−0.969351	−	0.245681i	$-0.920989\pi$
$277$	− 37701.7i	− 0.491362i	0.969351	−	0.245681i	$-0.0790115\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−116416.	−1.47435	−0.737174	−	0.675703i	$-0.763840\pi$
$281$	−116416.	−1.47435	−0.737174	+	0.675703i	$0.763840\pi$
$282$	0	0
$283$	−67172.6	−0.838724	−0.419362	−	0.907819i	$-0.637746\pi$
$283$	−67172.6	−0.838724	−0.419362	+	0.907819i	$0.637746\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	23623.0i	0.286795i
$288$	0	0
$289$	−83265.0	−0.996935
$290$	0	0
$291$	0	0
$292$	0	0
$293$	116346.i	1.35524i	0.735413	+	0.677619i	$0.236988\pi$
$293$	116346.i	1.35524i	−0.735413	+	0.677619i	$0.763012\pi$
$294$	0	0
$295$	− 90011.9i	− 1.03432i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	80607.1	0.901635
$300$	0	0
$301$	28498.9i	0.314554i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−105264.	−1.13157
$306$	0	0
$307$	−45901.3	−0.487021	−0.243511	−	0.969898i	$-0.578299\pi$
$307$	−45901.3	−0.487021	−0.243511	+	0.969898i	$0.578299\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	135221.i	1.39806i	0.715094	+	0.699028i	$0.246384\pi$
$311$	135221.i	1.39806i	−0.715094	+	0.699028i	$0.753616\pi$
$312$	0	0
$313$	98306.0	1.00344	0.501720	−	0.865030i	$-0.332701\pi$
$313$	98306.0	1.00344	0.501720	+	0.865030i	$0.332701\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 46286.0i	− 0.460607i	−0.973119	−	0.230304i	$-0.926028\pi$
$317$	− 46286.0i	− 0.460607i	0.973119	−	0.230304i	$-0.0739720\pi$
$318$	0	0
$319$	− 58853.9i	− 0.578354i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−8956.34	−0.0858471
$324$	0	0
$325$	1286.41i	0.0121790i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−124416.	−1.14944
$330$	0	0
$331$	−87324.3	−0.797039	−0.398519	−	0.917160i	$-0.630476\pi$
$331$	−87324.3	−0.797039	−0.398519	+	0.917160i	$0.630476\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 138480.i	− 1.23395i
$336$	0	0
$337$	−144418.	−1.27163	−0.635816	−	0.771841i	$-0.719336\pi$
$337$	−144418.	−1.27163	−0.635816	+	0.771841i	$0.719336\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 92621.4i	− 0.796531i
$342$	0	0
$343$	− 112515.i	− 0.956360i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	198859.	1.65153	0.825764	−	0.564016i	$-0.190744\pi$
$347$	198859.	1.65153	0.825764	+	0.564016i	$0.190744\pi$
$348$	0	0
$349$	− 212257.i	− 1.74266i	−0.490699	−	0.871329i	$-0.663259\pi$
$349$	− 212257.i	− 1.74266i	0.490699	−	0.871329i	$-0.336741\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−94688.0	−0.759881	−0.379940	−	0.925011i	$-0.624056\pi$
$353$	−94688.0	−0.759881	−0.379940	+	0.925011i	$0.624056\pi$
$354$	0	0
$355$	−80607.1	−0.639612
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 55391.9i	− 0.429791i	−0.976637	−	0.214896i	$-0.931059\pi$
$359$	− 55391.9i	− 0.429791i	0.976637	−	0.214896i	$-0.0689411\pi$
$360$	0	0
$361$	183023.	1.40440
$362$	0	0
$363$	0	0
$364$	0	0
$365$	22215.3i	0.166750i
$366$	0	0
$367$	197385.i	1.46548i	0.680506	+	0.732742i	$0.261760\pi$
$367$	197385.i	1.46548i	−0.680506	+	0.732742i	$0.738240\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−162474.	−1.18042
$372$	0	0
$373$	− 76293.9i	− 0.548368i	−0.961677	−	0.274184i	$-0.911592\pi$
$373$	− 76293.9i	− 0.548368i	0.961677	−	0.274184i	$-0.0884078\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−41616.0	−0.292804
$378$	0	0
$379$	80047.3	0.557273	0.278637	−	0.960397i	$-0.410117\pi$
$379$	80047.3	0.557273	0.278637	+	0.960397i	$0.410117\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	31768.9i	0.216573i	0.994120	+	0.108287i	$0.0345364\pi$
$383$	31768.9i	0.216573i	−0.994120	+	0.108287i	$0.965464\pi$
$384$	0	0
$385$	−176256.	−1.18911
$386$	0	0
$387$	0	0
$388$	0	0
$389$	219308.i	1.44929i	0.689122	+	0.724645i	$0.257996\pi$
$389$	219308.i	1.44929i	−0.689122	+	0.724645i	$0.742004\pi$
$390$	0	0
$391$	13033.4i	0.0852519i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−212853.	−1.36422
$396$	0	0
$397$	39878.7i	0.253023i	0.991965	+	0.126511i	$0.0403780\pi$
$397$	39878.7i	0.253023i	−0.991965	+	0.126511i	$0.959622\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−58288.0	−0.362485	−0.181243	−	0.983438i	$-0.558012\pi$
$401$	−58288.0	−0.362485	−0.181243	+	0.983438i	$0.558012\pi$
$402$	0	0
$403$	−65493.3	−0.403261
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 346199.i	− 2.08996i
$408$	0	0
$409$	163166.	0.975401	0.487700	−	0.873011i	$-0.337836\pi$
$409$	163166.	0.975401	0.487700	+	0.873011i	$0.337836\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	185243.i	1.08603i
$414$	0	0
$415$	10386.0i	0.0603047i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−27288.9	−0.155438	−0.0777190	−	0.996975i	$-0.524764\pi$
$419$	−27288.9	−0.155438	−0.0777190	+	0.996975i	$0.524764\pi$
$420$	0	0
$421$	− 236402.i	− 1.33379i	−0.745151	−	0.666895i	$-0.767623\pi$
$421$	− 236402.i	− 1.33379i	0.745151	−	0.666895i	$-0.232377\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−208.000	−0.00115156
$426$	0	0
$427$	216632.	1.18814
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 83902.5i	− 0.451669i	−0.974166	−	0.225834i	$-0.927489\pi$
$431$	− 83902.5i	− 0.451669i	0.974166	−	0.225834i	$-0.0725108\pi$
$432$	0	0
$433$	−245378.	−1.30876	−0.654380	−	0.756166i	$-0.727070\pi$
$433$	−245378.	−1.30876	−0.654380	+	0.756166i	$0.727070\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 455982.i	− 2.38773i
$438$	0	0
$439$	70309.0i	0.364823i	0.983222	+	0.182411i	$0.0583903\pi$
$439$	70309.0i	0.364823i	−0.983222	+	0.182411i	$0.941610\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	89143.6	0.454237	0.227119	−	0.973867i	$-0.427069\pi$
$443$	89143.6	0.454237	0.227119	+	0.973867i	$0.427069\pi$
$444$	0	0
$445$	− 174952.i	− 0.883482i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	99632.0	0.494204	0.247102	−	0.968989i	$-0.420522\pi$
$449$	99632.0	0.494204	0.247102	+	0.968989i	$0.420522\pi$
$450$	0	0
$451$	64933.5	0.319239
$452$	0	0
$453$	0	0
$454$	0	0
$455$	124632.i	0.602013i
$456$	0	0
$457$	336158.	1.60957	0.804787	−	0.593563i	$-0.202279\pi$
$457$	336158.	1.60957	0.804787	+	0.593563i	$0.202279\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 42377.3i	− 0.199403i	−0.995017	−	0.0997014i	$-0.968211\pi$
$461$	− 42377.3i	− 0.199403i	0.995017	−	0.0997014i	$-0.0317888\pi$
$462$	0	0
$463$	− 188831.i	− 0.880871i	−0.897784	−	0.440436i	$-0.854824\pi$
$463$	− 188831.i	− 0.880871i	0.897784	−	0.440436i	$-0.145176\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	334323.	1.53297	0.766484	−	0.642263i	$-0.222004\pi$
$467$	334323.	1.53297	0.766484	+	0.642263i	$0.222004\pi$
$468$	0	0
$469$	284989.i	1.29563i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	78336.0	0.350138
$474$	0	0
$475$	7277.03	0.0322528
$476$	0	0
$477$	0	0
$478$	0	0
$479$	169434.i	0.738465i	0.929337	+	0.369232i	$0.120379\pi$
$479$	169434.i	0.738465i	−0.929337	+	0.369232i	$0.879621\pi$
$480$	0	0
$481$	−244800.	−1.05809
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 58531.6i	− 0.248832i
$486$	0	0
$487$	− 174780.i	− 0.736942i	−0.929639	−	0.368471i	$-0.879881\pi$
$487$	− 174780.i	− 0.736942i	0.929639	−	0.368471i	$-0.120119\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−158415.	−0.657104	−0.328552	−	0.944486i	$-0.606561\pi$
$491$	−158415.	−0.657104	−0.328552	+	0.944486i	$0.606561\pi$
$492$	0	0
$493$	− 6728.91i	− 0.0276854i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	165888.	0.671587
$498$	0	0
$499$	465730.	1.87039	0.935197	−	0.354129i	$-0.115223\pi$
$499$	465730.	1.87039	0.935197	+	0.354129i	$0.115223\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	236230.i	0.933683i	0.884341	+	0.466842i	$0.154608\pi$
$503$	236230.i	0.933683i	−0.884341	+	0.466842i	$0.845392\pi$
$504$	0	0
$505$	196452.	0.770324
$506$	0	0
$507$	0	0
$508$	0	0
$509$	301242.i	1.16273i	0.813641	+	0.581367i	$0.197482\pi$
$509$	301242.i	1.16273i	−0.813641	+	0.581367i	$0.802518\pi$
$510$	0	0
$511$	− 45718.7i	− 0.175086i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	175068.	0.660075
$516$	0	0
$517$	341987.i	1.27946i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	329200.	1.21279	0.606393	−	0.795165i	$-0.292616\pi$
$521$	329200.	1.21279	0.606393	+	0.795165i	$0.292616\pi$
$522$	0	0
$523$	−522267.	−1.90937	−0.954683	−	0.297626i	$-0.903805\pi$
$523$	−522267.	−1.90937	−0.954683	+	0.297626i	$0.903805\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 10589.6i	− 0.0381294i
$528$	0	0
$529$	−383711.	−1.37118
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 45914.9i	− 0.161622i
$534$	0	0
$535$	408515.i	1.42725i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	26729.1	0.0920040
$540$	0	0
$541$	476268.i	1.62726i	0.581383	+	0.813630i	$0.302512\pi$
$541$	476268.i	1.62726i	−0.581383	+	0.813630i	$0.697488\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	222768.	0.749997
$546$	0	0
$547$	−533462.	−1.78291	−0.891454	−	0.453111i	$-0.850314\pi$
$547$	−533462.	−1.78291	−0.891454	+	0.453111i	$0.850314\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	235416.i	0.775411i
$552$	0	0
$553$	438048.	1.43242
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 33669.3i	− 0.108523i	−0.998527	−	0.0542617i	$-0.982719\pi$
$557$	− 33669.3i	− 0.108523i	0.998527	−	0.0542617i	$-0.0172805\pi$
$558$	0	0
$559$	− 55391.9i	− 0.177265i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−456634.	−1.44063	−0.720313	−	0.693650i	$-0.756002\pi$
$563$	−456634.	−1.44063	−0.720313	+	0.693650i	$0.756002\pi$
$564$	0	0
$565$	− 385527.i	− 1.20770i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	317296.	0.980032	0.490016	−	0.871714i	$-0.336991\pi$
$569$	317296.	0.980032	0.490016	+	0.871714i	$0.336991\pi$
$570$	0	0
$571$	−12315.0	−0.0377712	−0.0188856	−	0.999822i	$-0.506012\pi$
$571$	−12315.0	−0.0377712	−0.0188856	+	0.999822i	$0.506012\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 10589.6i	− 0.0320291i
$576$	0	0
$577$	−249118.	−0.748262	−0.374131	−	0.927376i	$-0.622059\pi$
$577$	−249118.	−0.748262	−0.374131	+	0.927376i	$0.622059\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 21374.2i	− 0.0633195i
$582$	0	0
$583$	446597.i	1.31395i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−363851.	−1.05596	−0.527980	−	0.849257i	$-0.677051\pi$
$587$	−363851.	−1.05596	−0.527980	+	0.849257i	$0.677051\pi$
$588$	0	0
$589$	370486.i	1.06793i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	562912.	1.60078	0.800389	−	0.599481i	$-0.204626\pi$
$593$	562912.	1.60078	0.800389	+	0.599481i	$0.204626\pi$
$594$	0	0
$595$	−20151.8	−0.0569219
$596$	0	0
$597$	0	0
$598$	0	0
$599$	180024.i	0.501737i	0.968021	+	0.250868i	$0.0807162\pi$
$599$	180024.i	0.501737i	−0.968021	+	0.250868i	$0.919284\pi$
$600$	0	0
$601$	185662.	0.514013	0.257006	−	0.966410i	$-0.417264\pi$
$601$	185662.	0.514013	0.257006	+	0.966410i	$0.417264\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	122283.i	0.334084i
$606$	0	0
$607$	− 291877.i	− 0.792177i	−0.918212	−	0.396088i	$-0.870367\pi$
$607$	− 291877.i	− 0.792177i	0.918212	−	0.396088i	$-0.129633\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	241821.	0.647757
$612$	0	0
$613$	244319.i	0.650183i	0.945683	+	0.325092i	$0.105395\pi$
$613$	244319.i	0.650183i	−0.945683	+	0.325092i	$0.894605\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	341312.	0.896564	0.448282	−	0.893892i	$-0.352036\pi$
$617$	341312.	0.896564	0.448282	+	0.893892i	$0.352036\pi$
$618$	0	0
$619$	320189.	0.835652	0.417826	−	0.908527i	$-0.362792\pi$
$619$	320189.	0.835652	0.417826	+	0.908527i	$0.362792\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	360047.i	0.927649i
$624$	0	0
$625$	−382331.	−0.978767
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 39581.8i	− 0.100045i
$630$	0	0
$631$	− 474548.i	− 1.19185i	−0.803040	−	0.595925i	$-0.796786\pi$
$631$	− 474548.i	− 1.19185i	0.803040	−	0.595925i	$-0.203214\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	278346.	0.690300
$636$	0	0
$637$	− 18900.3i	− 0.0465790i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	484208.	1.17846	0.589231	−	0.807964i	$-0.299431\pi$
$641$	484208.	1.17846	0.589231	+	0.807964i	$0.299431\pi$
$642$	0	0
$643$	50939.2	0.123206	0.0616028	−	0.998101i	$-0.480379\pi$
$643$	50939.2	0.123206	0.0616028	+	0.998101i	$0.480379\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 667147.i	− 1.59372i	−0.604162	−	0.796861i	$-0.706492\pi$
$647$	− 667147.i	− 1.59372i	0.604162	−	0.796861i	$-0.293508\pi$
$648$	0	0
$649$	509184.	1.20889
$650$	0	0
$651$	0	0
$652$	0	0
$653$	209363.i	0.490991i	0.969398	+	0.245496i	$0.0789507\pi$
$653$	209363.i	0.490991i	−0.969398	+	0.245496i	$0.921049\pi$
$654$	0	0
$655$	− 540071.i	− 1.25883i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	43662.2	0.100539	0.0502695	−	0.998736i	$-0.483992\pi$
$659$	43662.2	0.100539	0.0502695	+	0.998736i	$0.483992\pi$
$660$	0	0
$661$	95095.3i	0.217649i	0.994061	+	0.108824i	$0.0347086\pi$
$661$	95095.3i	0.217649i	−0.994061	+	0.108824i	$0.965291\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	705024.	1.59427
$666$	0	0
$667$	342580.	0.770035
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 595463.i	− 1.32254i
$672$	0	0
$673$	−232514.	−0.513356	−0.256678	−	0.966497i	$-0.582628\pi$
$673$	−232514.	−0.513356	−0.256678	+	0.966497i	$0.582628\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	269181.i	0.587310i	0.955912	+	0.293655i	$0.0948716\pi$
$677$	269181.i	0.587310i	−0.955912	+	0.293655i	$0.905128\pi$
$678$	0	0
$679$	120457.i	0.261272i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	776823.	1.66525	0.832627	−	0.553834i	$-0.186836\pi$
$683$	776823.	1.66525	0.832627	+	0.553834i	$0.186836\pi$
$684$	0	0
$685$	− 678828.i	− 1.44670i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	315792.	0.665216
$690$	0	0
$691$	−263652.	−0.552173	−0.276087	−	0.961133i	$-0.589038\pi$
$691$	−263652.	−0.552173	−0.276087	+	0.961133i	$0.589038\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 360047.i	− 0.745401i
$696$	0	0
$697$	7424.00	0.0152817
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 806257.i	− 1.64073i	−0.571840	−	0.820365i	$-0.693770\pi$
$701$	− 806257.i	− 1.64073i	0.571840	−	0.820365i	$-0.306230\pi$
$702$	0	0
$703$	1.38480e6i	2.80205i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−404295.	−0.808834
$708$	0	0
$709$	− 310420.i	− 0.617530i	−0.951138	−	0.308765i	$-0.900084\pi$
$709$	− 310420.i	− 0.617530i	0.951138	−	0.308765i	$-0.0999156\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	539136.	1.06052
$714$	0	0
$715$	342580.	0.670116
$716$	0	0
$717$	0	0
$718$	0	0
$719$	190613.i	0.368719i	0.982859	+	0.184360i	$0.0590211\pi$
$719$	190613.i	0.368719i	−0.982859	+	0.184360i	$0.940979\pi$
$720$	0	0
$721$	−360288.	−0.693073
$722$	0	0
$723$	0	0
$724$	0	0
$725$	5467.24i	0.0104014i
$726$	0	0
$727$	− 405919.i	− 0.768016i	−0.923330	−	0.384008i	$-0.874543\pi$
$727$	− 405919.i	− 0.768016i	0.923330	−	0.384008i	$-0.125457\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	8956.34	0.0167608
$732$	0	0
$733$	− 355346.i	− 0.661368i	−0.943742	−	0.330684i	$-0.892721\pi$
$733$	− 355346.i	− 0.661368i	0.943742	−	0.330684i	$-0.107279\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	783360.	1.44220
$738$	0	0
$739$	858689.	1.57234	0.786171	−	0.618009i	$-0.212060\pi$
$739$	858689.	1.57234	0.786171	+	0.618009i	$0.212060\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 349458.i	− 0.633020i	−0.948589	−	0.316510i	$-0.897489\pi$
$743$	− 349458.i	− 0.633020i	0.948589	−	0.316510i	$-0.102511\pi$
$744$	0	0
$745$	606492.	1.09273
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 840718.i	− 1.49860i
$750$	0	0
$751$	− 92812.0i	− 0.164560i	−0.996609	−	0.0822800i	$-0.973780\pi$
$751$	− 92812.0i	− 0.164560i	0.996609	−	0.0822800i	$-0.0262202\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−578104.	−1.01417
$756$	0	0
$757$	− 88762.2i	− 0.154895i	−0.996996	−	0.0774473i	$-0.975323\pi$
$757$	− 88762.2i	− 0.154895i	0.996996	−	0.0774473i	$-0.0246770\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	242320.	0.418427	0.209214	−	0.977870i	$-0.432910\pi$
$761$	242320.	0.418427	0.209214	+	0.977870i	$0.432910\pi$
$762$	0	0
$763$	−458453.	−0.787491
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 360047.i	− 0.612025i
$768$	0	0
$769$	−2210.00	−0.00373714	−0.00186857	−	0.999998i	$-0.500595\pi$
$769$	−2210.00	−0.00373714	−0.00186857	+	0.999998i	$0.500595\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	677616.i	1.13403i	0.823707	+	0.567015i	$0.191902\pi$
$773$	677616.i	1.13403i	−0.823707	+	0.567015i	$0.808098\pi$
$774$	0	0
$775$	8604.08i	0.0143252i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−259734.	−0.428010
$780$	0	0
$781$	− 455982.i	− 0.747560i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	325584.	0.528352
$786$	0	0
$787$	−414791.	−0.669699	−0.334849	−	0.942272i	$-0.608685\pi$
$787$	−414791.	−0.669699	−0.334849	+	0.942272i	$0.608685\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	793408.i	1.26807i
$792$	0	0
$793$	−421056.	−0.669566
$794$	0	0
$795$	0	0
$796$	0	0
$797$	424688.i	0.668580i	0.942470	+	0.334290i	$0.108497\pi$
$797$	424688.i	0.668580i	−0.942470	+	0.334290i	$0.891503\pi$
$798$	0	0
$799$	39100.2i	0.0612470i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−125669.	−0.194893
$804$	0	0
$805$	− 1.02596e6i	− 1.58321i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	657232.	1.00420	0.502102	−	0.864809i	$-0.332560\pi$
$809$	657232.	1.00420	0.502102	+	0.864809i	$0.332560\pi$
$810$	0	0
$811$	−778642.	−1.18385	−0.591924	−	0.805994i	$-0.701632\pi$
$811$	−778642.	−1.18385	−0.591924	+	0.805994i	$0.701632\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 540071.i	− 0.813085i
$816$	0	0
$817$	−313344.	−0.469437
$818$	0	0
$819$	0	0
$820$	0	0
$821$	914315.i	1.35647i	0.734846	+	0.678234i	$0.237254\pi$
$821$	914315.i	1.35647i	−0.734846	+	0.678234i	$0.762746\pi$
$822$	0	0
$823$	− 458663.i	− 0.677165i	−0.940937	−	0.338582i	$-0.890053\pi$
$823$	− 458663.i	− 0.677165i	0.940937	−	0.338582i	$-0.109947\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−1.14613e6	−1.67581	−0.837903	−	0.545820i	$-0.816218\pi$
$827$	−1.14613e6	−1.67581	−0.837903	+	0.545820i	$0.816218\pi$
$828$	0	0
$829$	275193.i	0.400431i	0.979752	+	0.200215i	$0.0641642\pi$
$829$	275193.i	0.400431i	−0.979752	+	0.200215i	$0.935836\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	3056.00	0.00440416
$834$	0	0
$835$	261973.	0.375737
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1.39783e6i	1.98578i	0.119041	+	0.992889i	$0.462018\pi$
$839$	1.39783e6i	1.98578i	−0.119041	+	0.992889i	$0.537982\pi$
$840$	0	0
$841$	530413.	0.749932
$842$	0	0
$843$	0	0
$844$	0	0
$845$	464319.i	0.650285i
$846$	0	0
$847$	− 251656.i	− 0.350785i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	2.01518e6	2.78262
$852$	0	0
$853$	925918.i	1.27255i	0.771463	+	0.636274i	$0.219525\pi$
$853$	925918.i	1.27255i	−0.771463	+	0.636274i	$0.780475\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	110512.	0.150469	0.0752346	−	0.997166i	$-0.476029\pi$
$857$	110512.	0.150469	0.0752346	+	0.997166i	$0.476029\pi$
$858$	0	0
$859$	−897314.	−1.21607	−0.608034	−	0.793911i	$-0.708042\pi$
$859$	−897314.	−1.21607	−0.608034	+	0.793911i	$0.708042\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	844727.i	1.13421i	0.823644	+	0.567107i	$0.191937\pi$
$863$	844727.i	1.13421i	−0.823644	+	0.567107i	$0.808063\pi$
$864$	0	0
$865$	421668.	0.563558
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 1.20408e6i	− 1.59447i
$870$	0	0
$871$	− 553919.i	− 0.730147i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	803552.	1.04954
$876$	0	0
$877$	282119.i	0.366804i	0.983038	+	0.183402i	$0.0587110\pi$
$877$	282119.i	0.366804i	−0.983038	+	0.183402i	$0.941289\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	632896.	0.815418	0.407709	−	0.913112i	$-0.366328\pi$
$881$	632896.	0.815418	0.407709	+	0.913112i	$0.366328\pi$
$882$	0	0
$883$	−749534.	−0.961324	−0.480662	−	0.876906i	$-0.659604\pi$
$883$	−749534.	−0.961324	−0.480662	+	0.876906i	$0.659604\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1.24958e6i	− 1.58824i	−0.607762	−	0.794119i	$-0.707933\pi$
$887$	− 1.24958e6i	− 1.58824i	0.607762	−	0.794119i	$-0.292067\pi$
$888$	0	0
$889$	−572832.	−0.724809
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 1.36795e6i	− 1.71540i
$894$	0	0
$895$	768563.i	0.959474i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−278346.	−0.344402
$900$	0	0
$901$	51060.5i	0.0628979i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−565488.	−0.690440
$906$	0	0
$907$	1.14249e6	1.38880	0.694399	−	0.719590i	$-0.255670\pi$
$907$	1.14249e6	1.38880	0.694399	+	0.719590i	$0.255670\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	857760.i	1.03354i	0.856123	+	0.516772i	$0.172867\pi$
$911$	857760.i	1.03354i	−0.856123	+	0.516772i	$0.827133\pi$
$912$	0	0
$913$	−58752.0	−0.0704825
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.11146e6i	1.32176i
$918$	0	0
$919$	123766.i	0.146545i	0.997312	+	0.0732726i	$0.0233443\pi$
$919$	123766.i	0.146545i	−0.997312	+	0.0732726i	$0.976656\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−322428.	−0.378469
$924$	0	0
$925$	32160.2i	0.0375868i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−443440.	−0.513811	−0.256906	−	0.966437i	$-0.582703\pi$
$929$	−443440.	−0.513811	−0.256906	+	0.966437i	$0.582703\pi$
$930$	0	0
$931$	−106916.	−0.123352
$932$	0	0
$933$	0	0
$934$	0	0
$935$	55391.9i	0.0633612i
$936$	0	0
$937$	256606.	0.292272	0.146136	−	0.989264i	$-0.453316\pi$
$937$	256606.	0.292272	0.146136	+	0.989264i	$0.453316\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	723927.i	0.817552i	0.912635	+	0.408776i	$0.134044\pi$
$941$	723927.i	0.817552i	−0.912635	+	0.408776i	$0.865956\pi$
$942$	0	0
$943$	377968.i	0.425042i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−83685.8	−0.0933151	−0.0466576	−	0.998911i	$-0.514857\pi$
$947$	−83685.8	−0.0933151	−0.0466576	+	0.998911i	$0.514857\pi$
$948$	0	0
$949$	88861.2i	0.0986687i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	401200.	0.441749	0.220874	−	0.975302i	$-0.429109\pi$
$953$	401200.	0.441749	0.220874	+	0.975302i	$0.429109\pi$
$954$	0	0
$955$	1.30987e6	1.43622
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1.39702e6i	1.51902i
$960$	0	0
$961$	485473.	0.525676
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 624651.i	− 0.670784i
$966$	0	0
$967$	1.11278e6i	1.19002i	0.803717	+	0.595011i	$0.202852\pi$
$967$	1.11278e6i	1.19002i	−0.803717	+	0.595011i	$0.797148\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−16373.3	−0.0173659	−0.00868297	−	0.999962i	$-0.502764\pi$
$971$	−16373.3	−0.0173659	−0.00868297	+	0.999962i	$0.502764\pi$
$972$	0	0
$973$	740972.i	0.782665i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−994288.	−1.04165	−0.520827	−	0.853663i	$-0.674376\pi$
$977$	−994288.	−1.04165	−0.520827	+	0.853663i	$0.674376\pi$
$978$	0	0
$979$	989676.	1.03259
$980$	0	0
$981$	0	0
$982$	0	0
$983$	529482.i	0.547954i	0.961736	+	0.273977i	$0.0883392\pi$
$983$	529482.i	0.547954i	−0.961736	+	0.273977i	$0.911661\pi$
$984$	0	0
$985$	−1.42902e6	−1.47287
$986$	0	0
$987$	0	0
$988$	0	0
$989$	455982.i	0.466182i
$990$	0	0
$991$	− 625959.i	− 0.637380i	−0.947859	−	0.318690i	$-0.896757\pi$
$991$	− 625959.i	− 0.637380i	0.947859	−	0.318690i	$-0.103243\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−197739.	−0.199732
$996$	0	0
$997$	1.82324e6i	1.83423i	0.398627	+	0.917113i	$0.369487\pi$
$997$	1.82324e6i	1.83423i	−0.398627	+	0.917113i	$0.630513\pi$
$998$	0	0
$999$	0	0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.5.b.f.703.3	yes	4
3.2	odd	2		1152.5.b.h.703.1	yes	4
4.3	odd	2	inner	1152.5.b.f.703.4	yes	4
8.3	odd	2	inner	1152.5.b.f.703.2	yes	4
8.5	even	2	inner	1152.5.b.f.703.1	✓	4
12.11	even	2		1152.5.b.h.703.2	yes	4
24.5	odd	2		1152.5.b.h.703.3	yes	4
24.11	even	2		1152.5.b.h.703.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1152.5.b.f.703.1	✓	4	8.5	even	2	inner
1152.5.b.f.703.2	yes	4	8.3	odd	2	inner
1152.5.b.f.703.3	yes	4	1.1	even	1	trivial
1152.5.b.f.703.4	yes	4	4.3	odd	2	inner
1152.5.b.h.703.1	yes	4	3.2	odd	2
1152.5.b.h.703.2	yes	4	12.11	even	2
1152.5.b.h.703.3	yes	4	24.5	odd	2
1152.5.b.h.703.4	yes	4	24.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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Newspace parameters

Newform invariants

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$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1152.5.b.f.703.3

Level	$1152$
Weight	$5$
Character	1152.703
Analytic conductor	$119.082$
Analytic rank	$0$
Dimension	$4$
Inner twists	$4$