LMFDB - Embedded newform 1152.4.l.b.287.20

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			287.20
Character	$\chi$	$=$	1152.287
Dual form			1152.4.l.b.863.20

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(9.40057 - 9.40057i) q^{5} -3.57327 q^{7} +O(q^{10})$	$q+(9.40057 - 9.40057i) q^{5} -3.57327 q^{7} +(3.36191 + 3.36191i) q^{11} +(-26.9332 + 26.9332i) q^{13} +12.7764i q^{17} +(50.0373 + 50.0373i) q^{19} -208.377i q^{23} -51.7415i q^{25} +(134.170 + 134.170i) q^{29} -80.1170i q^{31} +(-33.5907 + 33.5907i) q^{35} +(308.445 + 308.445i) q^{37} +172.761 q^{41} +(87.0630 - 87.0630i) q^{43} +525.793 q^{47} -330.232 q^{49} +(127.310 - 127.310i) q^{53} +63.2078 q^{55} +(-172.640 - 172.640i) q^{59} +(332.465 - 332.465i) q^{61} +506.376i q^{65} +(-556.127 - 556.127i) q^{67} -450.409i q^{71} -797.249i q^{73} +(-12.0130 - 12.0130i) q^{77} +70.1603i q^{79} +(-636.357 + 636.357i) q^{83} +(120.105 + 120.105i) q^{85} -925.357 q^{89} +(96.2396 - 96.2396i) q^{91} +940.759 q^{95} +1263.11 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	9.40057	−	9.40057i	0.840813	−	0.840813i	−0.148152	−	0.988965i	$-0.547332\pi$
$5$	9.40057	−	9.40057i	0.840813	−	0.840813i	0.988965	+	0.148152i	$0.0473324\pi$
$6$		0			0
$7$	−3.57327			−0.192938			−0.0964691	−	0.995336i	$-0.530755\pi$
$7$	−3.57327			−0.192938			−0.0964691	+	0.995336i	$0.530755\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	3.36191	+	3.36191i	0.0921504	+	0.0921504i	0.751679	−	0.659529i	$-0.229244\pi$
$11$	3.36191	+	3.36191i	0.0921504	+	0.0921504i	−0.659529	+	0.751679i	$0.729244\pi$
$12$		0			0
$13$	−26.9332	+	26.9332i	−0.574611	+	0.574611i	−0.933413	−	0.358803i	$-0.883185\pi$
$13$	−26.9332	+	26.9332i	−0.574611	+	0.574611i	0.358803	+	0.933413i	$0.383185\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			12.7764i			0.182278i	0.995838	+	0.0911390i	$0.0290507\pi$
$17$			12.7764i			0.182278i	−0.995838	+	0.0911390i	$0.970949\pi$
$18$		0			0
$19$	50.0373	+	50.0373i	0.604176	+	0.604176i	0.941418	−	0.337242i	$-0.109494\pi$
$19$	50.0373	+	50.0373i	0.604176	+	0.604176i	−0.337242	+	0.941418i	$0.609494\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	208.377i		−	1.88912i	−0.328344	−	0.944558i	$-0.606491\pi$
$23$		−	208.377i		−	1.88912i	0.328344	−	0.944558i	$-0.393509\pi$
$24$		0			0
$25$		−	51.7415i		−	0.413932i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	134.170	+	134.170i	0.859129	+	0.859129i	0.991235	−	0.132107i	$-0.0421742\pi$
$29$	134.170	+	134.170i	0.859129	+	0.859129i	−0.132107	+	0.991235i	$0.542174\pi$
$30$		0			0
$31$		−	80.1170i		−	0.464175i	−0.972695	−	0.232088i	$-0.925444\pi$
$31$		−	80.1170i		−	0.464175i	0.972695	−	0.232088i	$-0.0745556\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−33.5907	+	33.5907i	−0.162225	+	0.162225i
$36$		0			0
$37$	308.445	+	308.445i	1.37049	+	1.37049i	0.859714	+	0.510775i	$0.170642\pi$
$37$	308.445	+	308.445i	1.37049	+	1.37049i	0.510775	+	0.859714i	$0.329358\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	172.761			0.658068			0.329034	−	0.944318i	$-0.393277\pi$
$41$	172.761			0.658068			0.329034	+	0.944318i	$0.393277\pi$
$42$		0			0
$43$	87.0630	−	87.0630i	0.308767	−	0.308767i	−0.535664	−	0.844431i	$-0.679939\pi$
$43$	87.0630	−	87.0630i	0.308767	−	0.308767i	0.844431	+	0.535664i	$0.179939\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	525.793			1.63180			0.815902	−	0.578191i	$-0.196241\pi$
$47$	525.793			1.63180			0.815902	+	0.578191i	$0.196241\pi$
$48$		0			0
$49$	−330.232			−0.962775
$50$		0			0
$51$		0			0
$52$		0			0
$53$	127.310	−	127.310i	0.329951	−	0.329951i	−0.522617	−	0.852568i	$-0.675044\pi$
$53$	127.310	−	127.310i	0.329951	−	0.329951i	0.852568	+	0.522617i	$0.175044\pi$
$54$		0			0
$55$	63.2078			0.154962
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−172.640	−	172.640i	−0.380945	−	0.380945i	0.490498	−	0.871443i	$-0.336815\pi$
$59$	−172.640	−	172.640i	−0.380945	−	0.380945i	−0.871443	+	0.490498i	$0.836815\pi$
$60$		0			0
$61$	332.465	−	332.465i	0.697832	−	0.697832i	−0.266111	−	0.963943i	$-0.585739\pi$
$61$	332.465	−	332.465i	0.697832	−	0.697832i	0.963943	+	0.266111i	$0.0857386\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			506.376i			0.966280i
$66$		0			0
$67$	−556.127	−	556.127i	−1.01406	−	1.01406i	−0.999900	−	0.0141558i	$-0.995494\pi$
$67$	−556.127	−	556.127i	−1.01406	−	1.01406i	−0.0141558	−	0.999900i	$-0.504506\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	450.409i		−	0.752868i	−0.926443	−	0.376434i	$-0.877150\pi$
$71$		−	450.409i		−	0.752868i	0.926443	−	0.376434i	$-0.122850\pi$
$72$		0			0
$73$		−	797.249i		−	1.27823i	−0.769111	−	0.639116i	$-0.779300\pi$
$73$		−	797.249i		−	1.27823i	0.769111	−	0.639116i	$-0.220700\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−12.0130	−	12.0130i	−0.0177793	−	0.0177793i
$78$		0			0
$79$			70.1603i			0.0999197i	0.998751	+	0.0499598i	$0.0159093\pi$
$79$			70.1603i			0.0999197i	−0.998751	+	0.0499598i	$0.984091\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−636.357	+	636.357i	−0.841558	+	0.841558i	−0.989062	−	0.147504i	$-0.952876\pi$
$83$	−636.357	+	636.357i	−0.841558	+	0.841558i	0.147504	+	0.989062i	$0.452876\pi$
$84$		0			0
$85$	120.105	+	120.105i	0.153262	+	0.153262i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−925.357			−1.10211			−0.551054	−	0.834470i	$-0.685774\pi$
$89$	−925.357			−1.10211			−0.551054	+	0.834470i	$0.685774\pi$
$90$		0			0
$91$	96.2396	−	96.2396i	0.110864	−	0.110864i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	940.759			1.01600
$96$		0			0
$97$	1263.11			1.32216			0.661081	−	0.750315i	$-0.270098\pi$
$97$	1263.11			1.32216			0.661081	+	0.750315i	$0.270098\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	643.591	−	643.591i	0.634057	−	0.634057i	−0.315026	−	0.949083i	$-0.602013\pi$
$101$	643.591	−	643.591i	0.634057	−	0.634057i	0.949083	+	0.315026i	$0.102013\pi$
$102$		0			0
$103$	−977.275			−0.934891			−0.467445	−	0.884022i	$-0.654826\pi$
$103$	−977.275			−0.934891			−0.467445	+	0.884022i	$0.654826\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	982.491	+	982.491i	0.887672	+	0.887672i	0.994299	−	0.106627i	$-0.0340050\pi$
$107$	982.491	+	982.491i	0.887672	+	0.887672i	−0.106627	+	0.994299i	$0.534005\pi$
$108$		0			0
$109$	849.380	−	849.380i	0.746384	−	0.746384i	−0.227414	−	0.973798i	$-0.573027\pi$
$109$	849.380	−	849.380i	0.746384	−	0.746384i	0.973798	+	0.227414i	$0.0730271\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			1760.93i			1.46597i	0.680247	+	0.732983i	$0.261873\pi$
$113$			1760.93i			1.46597i	−0.680247	+	0.732983i	$0.738127\pi$
$114$		0			0
$115$	−1958.87	−	1958.87i	−1.58839	−	1.58839i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	45.6534i		−	0.0351684i
$120$		0			0
$121$		−	1308.40i		−	0.983017i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	688.672	+	688.672i	0.492773	+	0.492773i
$126$		0			0
$127$		−	1625.59i		−	1.13581i	−0.823094	−	0.567905i	$-0.807754\pi$
$127$		−	1625.59i		−	1.13581i	0.823094	−	0.567905i	$-0.192246\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	1291.78	−	1291.78i	0.861550	−	0.861550i	−0.129969	−	0.991518i	$-0.541488\pi$
$131$	1291.78	−	1291.78i	0.861550	−	0.861550i	0.991518	+	0.129969i	$0.0414876\pi$
$132$		0			0
$133$	−178.797	−	178.797i	−0.116569	−	0.116569i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1257.75			−0.784357			−0.392179	−	0.919889i	$-0.628278\pi$
$137$	−1257.75			−0.784357			−0.392179	+	0.919889i	$0.628278\pi$
$138$		0			0
$139$	−90.6450	+	90.6450i	−0.0553123	+	0.0553123i	−0.734222	−	0.678910i	$-0.762453\pi$
$139$	−90.6450	+	90.6450i	−0.0553123	+	0.0553123i	0.678910	+	0.734222i	$0.262453\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−181.094			−0.105901
$144$		0			0
$145$	2522.55			1.44473
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−27.0087	+	27.0087i	−0.0148499	+	0.0148499i	−0.714493	−	0.699643i	$-0.753342\pi$
$149$	−27.0087	+	27.0087i	−0.0148499	+	0.0148499i	0.699643	+	0.714493i	$0.253342\pi$
$150$		0			0
$151$	1808.06			0.974422			0.487211	−	0.873284i	$-0.338014\pi$
$151$	1808.06			0.974422			0.487211	+	0.873284i	$0.338014\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−753.145	−	753.145i	−0.390284	−	0.390284i
$156$		0			0
$157$	1223.22	−	1223.22i	0.621809	−	0.621809i	−0.324185	−	0.945994i	$-0.605090\pi$
$157$	1223.22	−	1223.22i	0.621809	−	0.621809i	0.945994	+	0.324185i	$0.105090\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			744.587i			0.364483i
$162$		0			0
$163$	−2395.40	−	2395.40i	−1.15106	−	1.15106i	−0.986341	−	0.164715i	$-0.947330\pi$
$163$	−2395.40	−	2395.40i	−1.15106	−	1.15106i	−0.164715	−	0.986341i	$-0.552670\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	895.247i		−	0.414828i	−0.978253	−	0.207414i	$-0.933495\pi$
$167$		−	895.247i		−	0.414828i	0.978253	−	0.207414i	$-0.0665047\pi$
$168$		0			0
$169$			746.201i			0.339645i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	2272.33	+	2272.33i	0.998624	+	0.998624i	0.999999	−	0.00137536i	$-0.000437792\pi$
$173$	2272.33	+	2272.33i	0.998624	+	0.998624i	−0.00137536	+	0.999999i	$0.500438\pi$
$174$		0			0
$175$			184.886i			0.0798633i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	2206.38	−	2206.38i	0.921301	−	0.921301i	−0.0758204	−	0.997121i	$-0.524158\pi$
$179$	2206.38	−	2206.38i	0.921301	−	0.921301i	0.997121	+	0.0758204i	$0.0241576\pi$
$180$		0			0
$181$	986.710	+	986.710i	0.405202	+	0.405202i	0.880062	−	0.474860i	$-0.157501\pi$
$181$	986.710	+	986.710i	0.405202	+	0.405202i	−0.474860	+	0.880062i	$0.657501\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	5799.13			2.30465
$186$		0			0
$187$	−42.9530	+	42.9530i	−0.0167970	+	0.0167970i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	3104.61			1.17613			0.588067	−	0.808812i	$-0.299889\pi$
$191$	3104.61			1.17613			0.588067	+	0.808812i	$0.299889\pi$
$192$		0			0
$193$	−441.815			−0.164780			−0.0823900	−	0.996600i	$-0.526255\pi$
$193$	−441.815			−0.164780			−0.0823900	+	0.996600i	$0.526255\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1039.58	+	1039.58i	−0.375976	+	0.375976i	−0.869648	−	0.493672i	$-0.835654\pi$
$197$	−1039.58	+	1039.58i	−0.375976	+	0.375976i	0.493672	+	0.869648i	$0.335654\pi$
$198$		0			0
$199$	−4338.31			−1.54540			−0.772700	−	0.634772i	$-0.781094\pi$
$199$	−4338.31			−1.54540			−0.772700	+	0.634772i	$0.781094\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−479.425	−	479.425i	−0.165759	−	0.165759i
$204$		0			0
$205$	1624.05	−	1624.05i	0.553312	−	0.553312i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			336.442i			0.111350i
$210$		0			0
$211$	3907.28	+	3907.28i	1.27483	+	1.27483i	0.943525	+	0.331301i	$0.107488\pi$
$211$	3907.28	+	3907.28i	1.27483	+	1.27483i	0.331301	+	0.943525i	$0.392512\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	1636.88i		−	0.519231i
$216$		0			0
$217$			286.279i			0.0895571i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−344.109	−	344.109i	−0.104739	−	0.104739i
$222$		0			0
$223$		−	6191.70i		−	1.85931i	−0.368427	−	0.929657i	$-0.620104\pi$
$223$		−	6191.70i		−	1.85931i	0.368427	−	0.929657i	$-0.379896\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	2584.86	−	2584.86i	0.755785	−	0.755785i	−0.219767	−	0.975552i	$-0.570530\pi$
$227$	2584.86	−	2584.86i	0.755785	−	0.755785i	0.975552	+	0.219767i	$0.0705298\pi$
$228$		0			0
$229$	−727.799	−	727.799i	−0.210019	−	0.210019i	0.594257	−	0.804275i	$-0.297446\pi$
$229$	−727.799	−	727.799i	−0.210019	−	0.210019i	−0.804275	+	0.594257i	$0.797446\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	4374.36			1.22993			0.614965	−	0.788554i	$-0.289170\pi$
$233$	4374.36			1.22993			0.614965	+	0.788554i	$0.289170\pi$
$234$		0			0
$235$	4942.75	−	4942.75i	1.37204	−	1.37204i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−918.623			−0.248622			−0.124311	−	0.992243i	$-0.539672\pi$
$239$	−918.623			−0.248622			−0.124311	+	0.992243i	$0.539672\pi$
$240$		0			0
$241$	3401.67			0.909215			0.454608	−	0.890692i	$-0.349779\pi$
$241$	3401.67			0.909215			0.454608	+	0.890692i	$0.349779\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−3104.37	+	3104.37i	−0.809513	+	0.809513i
$246$		0			0
$247$	−2695.33			−0.694332
$248$		0			0
$249$		0			0
$250$		0			0
$251$	310.888	+	310.888i	0.0781796	+	0.0781796i	0.745115	−	0.666936i	$-0.232394\pi$
$251$	310.888	+	310.888i	0.0781796	+	0.0781796i	−0.666936	+	0.745115i	$0.732394\pi$
$252$		0			0
$253$	700.546	−	700.546i	0.174083	−	0.174083i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	558.847i		−	0.135642i	−0.997698	−	0.0678208i	$-0.978395\pi$
$257$		−	558.847i		−	0.135642i	0.997698	−	0.0678208i	$-0.0216046\pi$
$258$		0			0
$259$	−1102.16	−	1102.16i	−0.264420	−	0.264420i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	1037.09i		−	0.243155i	−0.992582	−	0.121577i	$-0.961205\pi$
$263$		−	1037.09i		−	0.243155i	0.992582	−	0.121577i	$-0.0387953\pi$
$264$		0			0
$265$		−	2393.58i		−	0.554855i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	1664.16	+	1664.16i	0.377196	+	0.377196i	0.870090	−	0.492894i	$-0.164061\pi$
$269$	1664.16	+	1664.16i	0.377196	+	0.377196i	−0.492894	+	0.870090i	$0.664061\pi$
$270$		0			0
$271$		−	3882.80i		−	0.870344i	−0.900347	−	0.435172i	$-0.856688\pi$
$271$		−	3882.80i		−	0.870344i	0.900347	−	0.435172i	$-0.143312\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	173.950	−	173.950i	0.0381440	−	0.0381440i
$276$		0			0
$277$	−519.147	−	519.147i	−0.112608	−	0.112608i	0.648557	−	0.761166i	$-0.275373\pi$
$277$	−519.147	−	519.147i	−0.112608	−	0.112608i	−0.761166	+	0.648557i	$0.775373\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−1938.11			−0.411452			−0.205726	−	0.978610i	$-0.565956\pi$
$281$	−1938.11			−0.411452			−0.205726	+	0.978610i	$0.565956\pi$
$282$		0			0
$283$	−5546.76	+	5546.76i	−1.16509	+	1.16509i	−0.181745	+	0.983346i	$0.558175\pi$
$283$	−5546.76	+	5546.76i	−1.16509	+	1.16509i	−0.983346	+	0.181745i	$0.941825\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−617.322			−0.126966
$288$		0			0
$289$	4749.76			0.966775
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−3176.54	+	3176.54i	−0.633363	+	0.633363i	−0.948910	−	0.315547i	$-0.897812\pi$
$293$	−3176.54	+	3176.54i	−0.633363	+	0.633363i	0.315547	+	0.948910i	$0.397812\pi$
$294$		0			0
$295$	−3245.82			−0.640607
$296$		0			0
$297$		0			0
$298$		0			0
$299$	5612.28	+	5612.28i	1.08551	+	1.08551i
$300$		0			0
$301$	−311.099	+	311.099i	−0.0595730	+	0.0595730i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	6250.72i		−	1.17349i
$306$		0			0
$307$	2494.33	+	2494.33i	0.463710	+	0.463710i	0.899869	−	0.436159i	$-0.143662\pi$
$307$	2494.33	+	2494.33i	0.463710	+	0.463710i	−0.436159	+	0.899869i	$0.643662\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			8539.18i			1.55695i	0.627674	+	0.778476i	$0.284007\pi$
$311$			8539.18i			1.55695i	−0.627674	+	0.778476i	$0.715993\pi$
$312$		0			0
$313$			256.378i			0.0462982i	0.999732	+	0.0231491i	$0.00736925\pi$
$313$			256.378i			0.0462982i	−0.999732	+	0.0231491i	$0.992631\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	46.2282	+	46.2282i	0.00819065	+	0.00819065i	0.711190	−	0.703000i	$-0.248156\pi$
$317$	46.2282	+	46.2282i	0.00819065	+	0.00819065i	−0.703000	+	0.711190i	$0.748156\pi$
$318$		0			0
$319$			902.134i			0.158338i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−639.295	+	639.295i	−0.110128	+	0.110128i
$324$		0			0
$325$	1393.57	+	1393.57i	0.237850	+	0.237850i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1878.80			−0.314837
$330$		0			0
$331$	−2514.21	+	2514.21i	−0.417503	+	0.417503i	−0.884342	−	0.466839i	$-0.845393\pi$
$331$	−2514.21	+	2514.21i	−0.417503	+	0.417503i	0.466839	+	0.884342i	$0.345393\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−10455.8			−1.70526
$336$		0			0
$337$	−9524.70			−1.53959			−0.769797	−	0.638288i	$-0.779643\pi$
$337$	−9524.70			−1.53959			−0.769797	+	0.638288i	$0.779643\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	269.346	−	269.346i	0.0427739	−	0.0427739i
$342$		0			0
$343$	2405.64			0.378694
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−6076.54	−	6076.54i	−0.940074	−	0.940074i	0.0582292	−	0.998303i	$-0.481455\pi$
$347$	−6076.54	−	6076.54i	−0.940074	−	0.940074i	−0.998303	+	0.0582292i	$0.981455\pi$
$348$		0			0
$349$	2783.60	−	2783.60i	0.426942	−	0.426942i	−0.460643	−	0.887585i	$-0.652381\pi$
$349$	2783.60	−	2783.60i	0.426942	−	0.426942i	0.887585	+	0.460643i	$0.152381\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	2070.44i		−	0.312177i	−0.987743	−	0.156088i	$-0.950112\pi$
$353$		−	2070.44i		−	0.312177i	0.987743	−	0.156088i	$-0.0498885\pi$
$354$		0			0
$355$	−4234.10	−	4234.10i	−0.633021	−	0.633021i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			792.325i			0.116483i	0.998303	+	0.0582414i	$0.0185493\pi$
$359$			792.325i			0.116483i	−0.998303	+	0.0582414i	$0.981451\pi$
$360$		0			0
$361$		−	1851.53i		−	0.269942i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−7494.59	−	7494.59i	−1.07475	−	1.07475i
$366$		0			0
$367$			4380.56i			0.623061i	0.950236	+	0.311530i	$0.100842\pi$
$367$			4380.56i			0.623061i	−0.950236	+	0.311530i	$0.899158\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−454.914	+	454.914i	−0.0636602	+	0.0636602i
$372$		0			0
$373$	−6602.76	−	6602.76i	−0.916563	−	0.916563i	0.0802143	−	0.996778i	$-0.474440\pi$
$373$	−6602.76	−	6602.76i	−0.916563	−	0.916563i	−0.996778	+	0.0802143i	$0.974440\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−7227.26			−0.987329
$378$		0			0
$379$	3757.17	−	3757.17i	0.509216	−	0.509216i	−0.405070	−	0.914286i	$-0.632753\pi$
$379$	3757.17	−	3757.17i	0.509216	−	0.509216i	0.914286	+	0.405070i	$0.132753\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	8220.76			1.09677			0.548383	−	0.836227i	$-0.315244\pi$
$383$	8220.76			1.09677			0.548383	+	0.836227i	$0.315244\pi$
$384$		0			0
$385$	−225.858			−0.0298982
$386$		0			0
$387$		0			0
$388$		0			0
$389$	1269.20	−	1269.20i	0.165427	−	0.165427i	−0.619539	−	0.784966i	$-0.712680\pi$
$389$	1269.20	−	1269.20i	0.165427	−	0.165427i	0.784966	+	0.619539i	$0.212680\pi$
$390$		0			0
$391$	2662.31			0.344344
$392$		0			0
$393$		0			0
$394$		0			0
$395$	659.547	+	659.547i	0.0840137	+	0.0840137i
$396$		0			0
$397$	−5888.81	+	5888.81i	−0.744460	+	0.744460i	−0.973433	−	0.228973i	$-0.926463\pi$
$397$	−5888.81	+	5888.81i	−0.744460	+	0.744460i	0.228973	+	0.973433i	$0.426463\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			7300.97i			0.909210i	0.890693	+	0.454605i	$0.150220\pi$
$401$			7300.97i			0.909210i	−0.890693	+	0.454605i	$0.849780\pi$
$402$		0			0
$403$	2157.81	+	2157.81i	0.266720	+	0.266720i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			2073.93i			0.252582i
$408$		0			0
$409$			10278.9i			1.24268i	0.783540	+	0.621341i	$0.213412\pi$
$409$			10278.9i			1.24268i	−0.783540	+	0.621341i	$0.786588\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	616.887	+	616.887i	0.0734988	+	0.0734988i
$414$		0			0
$415$			11964.2i			1.41519i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−5762.20	+	5762.20i	−0.671842	+	0.671842i	−0.958141	−	0.286298i	$-0.907575\pi$
$419$	−5762.20	+	5762.20i	−0.671842	+	0.671842i	0.286298	+	0.958141i	$0.407575\pi$
$420$		0			0
$421$	100.297	+	100.297i	0.0116109	+	0.0116109i	0.712888	−	0.701278i	$-0.247387\pi$
$421$	100.297	+	100.297i	0.0116109	+	0.0116109i	−0.701278	+	0.712888i	$0.747387\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	661.069			0.0754507
$426$		0			0
$427$	−1187.98	+	1187.98i	−0.134638	+	0.134638i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−4634.34			−0.517932			−0.258966	−	0.965886i	$-0.583382\pi$
$431$	−4634.34			−0.517932			−0.258966	+	0.965886i	$0.583382\pi$
$432$		0			0
$433$	8964.00			0.994878			0.497439	−	0.867499i	$-0.334274\pi$
$433$	8964.00			0.994878			0.497439	+	0.867499i	$0.334274\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	10426.6	−	10426.6i	1.14136	−	1.14136i
$438$		0			0
$439$	16072.0			1.74732			0.873662	−	0.486533i	$-0.161739\pi$
$439$	16072.0			1.74732			0.873662	+	0.486533i	$0.161739\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	834.705	+	834.705i	0.0895215	+	0.0895215i	0.750449	−	0.660928i	$-0.229837\pi$
$443$	834.705	+	834.705i	0.0895215	+	0.0895215i	−0.660928	+	0.750449i	$0.729837\pi$
$444$		0			0
$445$	−8698.88	+	8698.88i	−0.926666	+	0.926666i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			3827.41i			0.402286i	0.979562	+	0.201143i	$0.0644657\pi$
$449$			3827.41i			0.402286i	−0.979562	+	0.201143i	$0.935534\pi$
$450$		0			0
$451$	580.808	+	580.808i	0.0606412	+	0.0606412i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	1809.42i		−	0.186432i
$456$		0			0
$457$			11649.3i			1.19241i	0.802834	+	0.596203i	$0.203325\pi$
$457$			11649.3i			1.19241i	−0.802834	+	0.596203i	$0.796675\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−3254.45	−	3254.45i	−0.328796	−	0.328796i	0.523333	−	0.852128i	$-0.324689\pi$
$461$	−3254.45	−	3254.45i	−0.328796	−	0.328796i	−0.852128	+	0.523333i	$0.824689\pi$
$462$		0			0
$463$			13371.1i			1.34213i	0.741397	+	0.671067i	$0.234164\pi$
$463$			13371.1i			1.34213i	−0.741397	+	0.671067i	$0.765836\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−2009.07	+	2009.07i	−0.199076	+	0.199076i	−0.799604	−	0.600528i	$-0.794957\pi$
$467$	−2009.07	+	2009.07i	−0.199076	+	0.199076i	0.600528	+	0.799604i	$0.294957\pi$
$468$		0			0
$469$	1987.19	+	1987.19i	0.195650	+	0.195650i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	585.396			0.0569060
$474$		0			0
$475$	2589.01	−	2589.01i	0.250088	−	0.250088i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	3070.02			0.292845			0.146422	−	0.989222i	$-0.453224\pi$
$479$	3070.02			0.292845			0.146422	+	0.989222i	$0.453224\pi$
$480$		0			0
$481$	−16614.9			−1.57500
$482$		0			0
$483$		0			0
$484$		0			0
$485$	11874.0	−	11874.0i	1.11169	−	1.11169i
$486$		0			0
$487$	−113.794			−0.0105883			−0.00529416	−	0.999986i	$-0.501685\pi$
$487$	−113.794			−0.0105883			−0.00529416	+	0.999986i	$0.501685\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−8646.06	−	8646.06i	−0.794686	−	0.794686i	0.187566	−	0.982252i	$-0.439940\pi$
$491$	−8646.06	−	8646.06i	−0.794686	−	0.794686i	−0.982252	+	0.187566i	$0.939940\pi$
$492$		0			0
$493$	−1714.20	+	1714.20i	−0.156600	+	0.156600i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			1609.43i			0.145257i
$498$		0			0
$499$	8311.88	+	8311.88i	0.745673	+	0.745673i	0.973663	−	0.227991i	$-0.0732155\pi$
$499$	8311.88	+	8311.88i	0.745673	+	0.745673i	−0.227991	+	0.973663i	$0.573216\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	19595.5i		−	1.73702i	−0.495675	−	0.868508i	$-0.665079\pi$
$503$		−	19595.5i		−	1.73702i	0.495675	−	0.868508i	$-0.334921\pi$
$504$		0			0
$505$		−	12100.3i		−	1.06625i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−964.944	−	964.944i	−0.0840283	−	0.0840283i	0.663843	−	0.747872i	$-0.268924\pi$
$509$	−964.944	−	964.944i	−0.0840283	−	0.0840283i	−0.747872	+	0.663843i	$0.768924\pi$
$510$		0			0
$511$			2848.78i			0.246620i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−9186.94	+	9186.94i	−0.786068	+	0.786068i
$516$		0			0
$517$	1767.67	+	1767.67i	0.150371	+	0.150371i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−11558.6			−0.971962			−0.485981	−	0.873969i	$-0.661538\pi$
$521$	−11558.6			−0.971962			−0.485981	+	0.873969i	$0.661538\pi$
$522$		0			0
$523$	−174.386	+	174.386i	−0.0145801	+	0.0145801i	−0.714359	−	0.699779i	$-0.753282\pi$
$523$	−174.386	+	174.386i	−0.0145801	+	0.0145801i	0.699779	+	0.714359i	$0.253282\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	1023.60			0.0846089
$528$		0			0
$529$	−31254.1			−2.56876
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−4653.02	+	4653.02i	−0.378133	+	0.378133i
$534$		0			0
$535$	18471.9			1.49273
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−1110.21	−	1110.21i	−0.0887201	−	0.0887201i
$540$		0			0
$541$	−17450.8	+	17450.8i	−1.38682	+	1.38682i	−0.554899	+	0.831918i	$0.687243\pi$
$541$	−17450.8	+	17450.8i	−1.38682	+	1.38682i	−0.831918	+	0.554899i	$0.812757\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	15969.3i		−	1.25514i
$546$		0			0
$547$	8294.19	+	8294.19i	0.648325	+	0.648325i	0.952588	−	0.304263i	$-0.0984101\pi$
$547$	8294.19	+	8294.19i	0.648325	+	0.648325i	−0.304263	+	0.952588i	$0.598410\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			13427.0i			1.03813i
$552$		0			0
$553$		−	250.701i		−	0.0192783i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	8685.44	+	8685.44i	0.660707	+	0.660707i	0.955547	−	0.294840i	$-0.0952663\pi$
$557$	8685.44	+	8685.44i	0.660707	+	0.660707i	−0.294840	+	0.955547i	$0.595266\pi$
$558$		0			0
$559$			4689.78i			0.354842i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−6079.67	+	6079.67i	−0.455111	+	0.455111i	−0.897047	−	0.441936i	$-0.854292\pi$
$563$	−6079.67	+	6079.67i	−0.455111	+	0.455111i	0.441936	+	0.897047i	$0.354292\pi$
$564$		0			0
$565$	16553.7	+	16553.7i	1.23260	+	1.23260i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−8341.99			−0.614612			−0.307306	−	0.951611i	$-0.599428\pi$
$569$	−8341.99			−0.614612			−0.307306	+	0.951611i	$0.599428\pi$
$570$		0			0
$571$	−7249.79	+	7249.79i	−0.531338	+	0.531338i	−0.920970	−	0.389632i	$-0.872602\pi$
$571$	−7249.79	+	7249.79i	−0.531338	+	0.531338i	0.389632	+	0.920970i	$0.372602\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−10781.8			−0.781966
$576$		0			0
$577$	−7557.93			−0.545305			−0.272652	−	0.962113i	$-0.587901\pi$
$577$	−7557.93			−0.545305			−0.272652	+	0.962113i	$0.587901\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	2273.87	−	2273.87i	0.162369	−	0.162369i
$582$		0			0
$583$	856.012			0.0608103
$584$		0			0
$585$		0			0
$586$		0			0
$587$	16053.1	+	16053.1i	1.12876	+	1.12876i	0.990379	+	0.138378i	$0.0441889\pi$
$587$	16053.1	+	16053.1i	1.12876	+	1.12876i	0.138378	+	0.990379i	$0.455811\pi$
$588$		0			0
$589$	4008.84	−	4008.84i	0.280444	−	0.280444i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	17978.4i		−	1.24500i	−0.782621	−	0.622499i	$-0.786118\pi$
$593$		−	17978.4i		−	1.24500i	0.782621	−	0.622499i	$-0.213882\pi$
$594$		0			0
$595$	−429.168	−	429.168i	−0.0295700	−	0.0295700i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	12555.7i		−	0.856446i	−0.903673	−	0.428223i	$-0.859140\pi$
$599$		−	12555.7i		−	0.856446i	0.903673	−	0.428223i	$-0.140860\pi$
$600$		0			0
$601$		−	1874.61i		−	0.127233i	−0.997974	−	0.0636164i	$-0.979737\pi$
$601$		−	1874.61i		−	0.127233i	0.997974	−	0.0636164i	$-0.0202634\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−12299.7	−	12299.7i	−0.826533	−	0.826533i
$606$		0			0
$607$			10180.2i			0.680731i	0.940293	+	0.340365i	$0.110551\pi$
$607$			10180.2i			0.680731i	−0.940293	+	0.340365i	$0.889449\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−14161.3	+	14161.3i	−0.937651	+	0.937651i
$612$		0			0
$613$	−4822.54	−	4822.54i	−0.317750	−	0.317750i	0.530152	−	0.847902i	$-0.322135\pi$
$613$	−4822.54	−	4822.54i	−0.317750	−	0.317750i	−0.847902	+	0.530152i	$0.822135\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	2024.64			0.132105			0.0660526	−	0.997816i	$-0.478959\pi$
$617$	2024.64			0.132105			0.0660526	+	0.997816i	$0.478959\pi$
$618$		0			0
$619$	−17426.4	+	17426.4i	−1.13154	+	1.13154i	−0.141624	+	0.989921i	$0.545232\pi$
$619$	−17426.4	+	17426.4i	−1.13154	+	1.13154i	−0.989921	+	0.141624i	$0.954768\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	3306.55			0.212639
$624$		0			0
$625$	19415.5			1.24259
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−3940.81	+	3940.81i	−0.249810	+	0.249810i
$630$		0			0
$631$	−18780.7			−1.18486			−0.592432	−	0.805620i	$-0.701832\pi$
$631$	−18780.7			−1.18486			−0.592432	+	0.805620i	$0.701832\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−15281.5	−	15281.5i	−0.955003	−	0.955003i
$636$		0			0
$637$	8894.21	−	8894.21i	0.553221	−	0.553221i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			12312.7i			0.758690i	0.925255	+	0.379345i	$0.123851\pi$
$641$			12312.7i			0.758690i	−0.925255	+	0.379345i	$0.876149\pi$
$642$		0			0
$643$	1553.86	+	1553.86i	0.0953008	+	0.0953008i	0.753150	−	0.657849i	$-0.228534\pi$
$643$	1553.86	+	1553.86i	0.0953008	+	0.0953008i	−0.657849	+	0.753150i	$0.728534\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			23625.3i			1.43556i	0.696269	+	0.717781i	$0.254842\pi$
$647$			23625.3i			1.43556i	−0.696269	+	0.717781i	$0.745158\pi$
$648$		0			0
$649$		−	1160.80i		−	0.0702084i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−10161.7	−	10161.7i	−0.608971	−	0.608971i	0.333706	−	0.942677i	$-0.391701\pi$
$653$	−10161.7	−	10161.7i	−0.608971	−	0.608971i	−0.942677	+	0.333706i	$0.891701\pi$
$654$		0			0
$655$		−	24286.9i		−	1.44880i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−957.830	+	957.830i	−0.0566187	+	0.0566187i	−0.734849	−	0.678230i	$-0.762747\pi$
$659$	−957.830	+	957.830i	−0.0566187	+	0.0566187i	0.678230	+	0.734849i	$0.262747\pi$
$660$		0			0
$661$	−2569.18	−	2569.18i	−0.151179	−	0.151179i	0.627465	−	0.778644i	$-0.284092\pi$
$661$	−2569.18	−	2569.18i	−0.151179	−	0.151179i	−0.778644	+	0.627465i	$0.784092\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−3361.58			−0.196025
$666$		0			0
$667$	27958.0	−	27958.0i	1.62299	−	1.62299i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	2235.43			0.128611
$672$		0			0
$673$	−15018.0			−0.860183			−0.430091	−	0.902785i	$-0.641519\pi$
$673$	−15018.0			−0.860183			−0.430091	+	0.902785i	$0.641519\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−18508.2	+	18508.2i	−1.05071	+	1.05071i	−0.0520645	+	0.998644i	$0.516580\pi$
$677$	−18508.2	+	18508.2i	−1.05071	+	1.05071i	−0.998644	+	0.0520645i	$0.983420\pi$
$678$		0			0
$679$	−4513.44			−0.255096
$680$		0			0
$681$		0			0
$682$		0			0
$683$	5041.00	+	5041.00i	0.282414	+	0.282414i	0.834071	−	0.551657i	$-0.186004\pi$
$683$	5041.00	+	5041.00i	0.282414	+	0.282414i	−0.551657	+	0.834071i	$0.686004\pi$
$684$		0			0
$685$	−11823.6	+	11823.6i	−0.659498	+	0.659498i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			6857.76i			0.379187i
$690$		0			0
$691$	8172.76	+	8172.76i	0.449937	+	0.449937i	0.895333	−	0.445396i	$-0.146937\pi$
$691$	8172.76	+	8172.76i	0.449937	+	0.449937i	−0.445396	+	0.895333i	$0.646937\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			1704.23i			0.0930146i
$696$		0			0
$697$			2207.26i			0.119951i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−10848.0	−	10848.0i	−0.584486	−	0.584486i	0.351647	−	0.936133i	$-0.385622\pi$
$701$	−10848.0	−	10848.0i	−0.584486	−	0.584486i	−0.936133	+	0.351647i	$0.885622\pi$
$702$		0			0
$703$			30867.6i			1.65603i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−2299.72	+	2299.72i	−0.122334	+	0.122334i
$708$		0			0
$709$	−9117.64	−	9117.64i	−0.482962	−	0.482962i	0.423114	−	0.906076i	$-0.360937\pi$
$709$	−9117.64	−	9117.64i	−0.482962	−	0.482962i	−0.906076	+	0.423114i	$0.860937\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−16694.6			−0.876881
$714$		0			0
$715$	−1702.39	+	1702.39i	−0.0890430	+	0.0890430i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−6812.47			−0.353355			−0.176678	−	0.984269i	$-0.556535\pi$
$719$	−6812.47			−0.353355			−0.176678	+	0.984269i	$0.556535\pi$
$720$		0			0
$721$	3492.06			0.180376
$722$		0			0
$723$		0			0
$724$		0			0
$725$	6942.16	−	6942.16i	0.355621	−	0.355621i
$726$		0			0
$727$	−18664.2			−0.952153			−0.476076	−	0.879404i	$-0.657941\pi$
$727$	−18664.2			−0.952153			−0.476076	+	0.879404i	$0.657941\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	1112.35	+	1112.35i	0.0562814	+	0.0562814i
$732$		0			0
$733$	14517.0	−	14517.0i	0.731510	−	0.731510i	−0.239408	−	0.970919i	$-0.576953\pi$
$733$	14517.0	−	14517.0i	0.731510	−	0.731510i	0.970919	+	0.239408i	$0.0769535\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	3739.30i		−	0.186891i
$738$		0			0
$739$	5245.53	+	5245.53i	0.261110	+	0.261110i	0.825505	−	0.564395i	$-0.190891\pi$
$739$	5245.53	+	5245.53i	0.261110	+	0.261110i	−0.564395	+	0.825505i	$0.690891\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			10157.8i			0.501552i	0.968045	+	0.250776i	$0.0806858\pi$
$743$			10157.8i			0.501552i	−0.968045	+	0.250776i	$0.919314\pi$
$744$		0			0
$745$			507.794i			0.0249720i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−3510.70	−	3510.70i	−0.171266	−	0.171266i
$750$		0			0
$751$		−	35581.6i		−	1.72888i	−0.502734	−	0.864441i	$-0.667672\pi$
$751$		−	35581.6i		−	1.72888i	0.502734	−	0.864441i	$-0.332328\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	16996.8	−	16996.8i	0.819307	−	0.819307i
$756$		0			0
$757$	−17559.8	−	17559.8i	−0.843094	−	0.843094i	0.146166	−	0.989260i	$-0.453307\pi$
$757$	−17559.8	−	17559.8i	−0.843094	−	0.843094i	−0.989260	+	0.146166i	$0.953307\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	38591.1			1.83828			0.919138	−	0.393936i	$-0.128887\pi$
$761$	38591.1			1.83828			0.919138	+	0.393936i	$0.128887\pi$
$762$		0			0
$763$	−3035.06	+	3035.06i	−0.144006	+	0.144006i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	9299.48			0.437790
$768$		0			0
$769$	21740.3			1.01948			0.509738	−	0.860330i	$-0.329742\pi$
$769$	21740.3			1.01948			0.509738	+	0.860330i	$0.329742\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−2913.88	+	2913.88i	−0.135582	+	0.135582i	−0.771641	−	0.636058i	$-0.780564\pi$
$773$	−2913.88	+	2913.88i	−0.135582	+	0.135582i	0.636058	+	0.771641i	$0.280564\pi$
$774$		0			0
$775$	−4145.37			−0.192137
$776$		0			0
$777$		0			0
$778$		0			0
$779$	8644.51	+	8644.51i	0.397589	+	0.397589i
$780$		0			0
$781$	1514.23	−	1514.23i	0.0693771	−	0.0693771i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	22998.0i		−	1.04565i
$786$		0			0
$787$	−15745.4	−	15745.4i	−0.713168	−	0.713168i	0.254028	−	0.967197i	$-0.418244\pi$
$787$	−15745.4	−	15745.4i	−0.713168	−	0.713168i	−0.967197	+	0.254028i	$0.918244\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	6292.26i		−	0.282841i
$792$		0			0
$793$			17908.7i			0.801963i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	26542.1	+	26542.1i	1.17964	+	1.17964i	0.979837	+	0.199799i	$0.0640289\pi$
$797$	26542.1	+	26542.1i	1.17964	+	1.17964i	0.199799	+	0.979837i	$0.435971\pi$
$798$		0			0
$799$			6717.72i			0.297442i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	2680.28	−	2680.28i	0.117790	−	0.117790i
$804$		0			0
$805$	6999.55	+	6999.55i	0.306462	+	0.306462i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	36199.3			1.57318			0.786588	−	0.617479i	$-0.211846\pi$
$809$	36199.3			1.57318			0.786588	+	0.617479i	$0.211846\pi$
$810$		0			0
$811$	−20813.2	+	20813.2i	−0.901170	+	0.901170i	−0.995537	−	0.0943671i	$-0.969917\pi$
$811$	−20813.2	+	20813.2i	−0.901170	+	0.901170i	0.0943671	+	0.995537i	$0.469917\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−45036.2			−1.93565
$816$		0			0
$817$	8712.80			0.373099
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−4975.98	+	4975.98i	−0.211526	+	0.211526i	−0.804916	−	0.593389i	$-0.797789\pi$
$821$	−4975.98	+	4975.98i	−0.211526	+	0.211526i	0.593389	+	0.804916i	$0.297789\pi$
$822$		0			0
$823$	26415.2			1.11880			0.559401	−	0.828897i	$-0.311031\pi$
$823$	26415.2			1.11880			0.559401	+	0.828897i	$0.311031\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−11809.1	−	11809.1i	−0.496546	−	0.496546i	0.413815	−	0.910361i	$-0.364196\pi$
$827$	−11809.1	−	11809.1i	−0.496546	−	0.496546i	−0.910361	+	0.413815i	$0.864196\pi$
$828$		0			0
$829$	−5975.62	+	5975.62i	−0.250352	+	0.250352i	−0.821115	−	0.570763i	$-0.806648\pi$
$829$	−5975.62	+	5975.62i	−0.250352	+	0.250352i	0.570763	+	0.821115i	$0.306648\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	4219.16i		−	0.175493i
$834$		0			0
$835$	−8415.84	−	8415.84i	−0.348793	−	0.348793i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			34562.9i			1.42222i	0.703079	+	0.711111i	$0.251808\pi$
$839$			34562.9i			1.42222i	−0.703079	+	0.711111i	$0.748192\pi$
$840$		0			0
$841$			11614.1i			0.476204i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	7014.72	+	7014.72i	0.285578	+	0.285578i
$846$		0			0
$847$			4675.24i			0.189661i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	64273.0	−	64273.0i	2.58901	−	2.58901i
$852$		0			0
$853$	−16216.8	−	16216.8i	−0.650941	−	0.650941i	0.302279	−	0.953220i	$-0.402253\pi$
$853$	−16216.8	−	16216.8i	−0.650941	−	0.650941i	−0.953220	+	0.302279i	$0.902253\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−41556.4			−1.65640			−0.828202	−	0.560430i	$-0.810636\pi$
$857$	−41556.4			−1.65640			−0.828202	+	0.560430i	$0.810636\pi$
$858$		0			0
$859$	19887.3	−	19887.3i	0.789927	−	0.789927i	−0.191555	−	0.981482i	$-0.561353\pi$
$859$	19887.3	−	19887.3i	0.789927	−	0.789927i	0.981482	+	0.191555i	$0.0613529\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	2405.75			0.0948929			0.0474464	−	0.998874i	$-0.484892\pi$
$863$	2405.75			0.0948929			0.0474464	+	0.998874i	$0.484892\pi$
$864$		0			0
$865$	42722.4			1.67931
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−235.873	+	235.873i	−0.00920763	+	0.00920763i
$870$		0			0
$871$	29956.6			1.16537
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−2460.81	−	2460.81i	−0.0950748	−	0.0950748i
$876$		0			0
$877$	10981.7	−	10981.7i	0.422835	−	0.422835i	−0.463344	−	0.886179i	$-0.653351\pi$
$877$	10981.7	−	10981.7i	0.422835	−	0.422835i	0.886179	+	0.463344i	$0.153351\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			34086.6i			1.30352i	0.758423	+	0.651762i	$0.225970\pi$
$881$			34086.6i			1.30352i	−0.758423	+	0.651762i	$0.774030\pi$
$882$		0			0
$883$	1641.73	+	1641.73i	0.0625691	+	0.0625691i	0.737699	−	0.675130i	$-0.235912\pi$
$883$	1641.73	+	1641.73i	0.0625691	+	0.0625691i	−0.675130	+	0.737699i	$0.735912\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	44633.8i		−	1.68958i	−0.535098	−	0.844790i	$-0.679725\pi$
$887$		−	44633.8i		−	1.68958i	0.535098	−	0.844790i	$-0.320275\pi$
$888$		0			0
$889$			5808.67i			0.219141i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	26309.3	+	26309.3i	0.985897	+	0.985897i
$894$		0			0
$895$		−	41482.5i		−	1.54928i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	10749.3	−	10749.3i	0.398786	−	0.398786i
$900$		0			0
$901$	1626.56	+	1626.56i	0.0601428	+	0.0601428i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	18551.3			0.681398
$906$		0			0
$907$	−9652.37	+	9652.37i	−0.353364	+	0.353364i	−0.861360	−	0.507995i	$-0.830387\pi$
$907$	−9652.37	+	9652.37i	−0.353364	+	0.353364i	0.507995	+	0.861360i	$0.330387\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	26739.3			0.972463			0.486232	−	0.873830i	$-0.338371\pi$
$911$	26739.3			0.972463			0.486232	+	0.873830i	$0.338371\pi$
$912$		0			0
$913$	−4278.75			−0.155100
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−4615.86	+	4615.86i	−0.166226	+	0.166226i
$918$		0			0
$919$	−18146.3			−0.651351			−0.325676	−	0.945482i	$-0.605592\pi$
$919$	−18146.3			−0.651351			−0.325676	+	0.945482i	$0.605592\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	12131.0	+	12131.0i	0.432606	+	0.432606i
$924$		0			0
$925$	15959.4	−	15959.4i	0.567290	−	0.567290i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			34344.9i			1.21294i	0.795107	+	0.606469i	$0.207415\pi$
$929$			34344.9i			1.21294i	−0.795107	+	0.606469i	$0.792585\pi$
$930$		0			0
$931$	−16523.9	−	16523.9i	−0.581686	−	0.581686i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			807.566i			0.0282462i
$936$		0			0
$937$		−	39419.7i		−	1.37437i	−0.726482	−	0.687185i	$-0.758846\pi$
$937$		−	39419.7i		−	1.37437i	0.726482	−	0.687185i	$-0.241154\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	13204.3	+	13204.3i	0.457435	+	0.457435i	0.897813	−	0.440378i	$-0.145155\pi$
$941$	13204.3	+	13204.3i	0.457435	+	0.457435i	−0.440378	+	0.897813i	$0.645155\pi$
$942$		0			0
$943$		−	35999.5i		−	1.24317i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	23749.6	−	23749.6i	0.814952	−	0.814952i	−0.170420	−	0.985372i	$-0.554512\pi$
$947$	23749.6	−	23749.6i	0.814952	−	0.814952i	0.985372	+	0.170420i	$0.0545124\pi$
$948$		0			0
$949$	21472.5	+	21472.5i	0.734485	+	0.734485i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	51211.8			1.74073			0.870363	−	0.492410i	$-0.163884\pi$
$953$	51211.8			1.74073			0.870363	+	0.492410i	$0.163884\pi$
$954$		0			0
$955$	29185.1	−	29185.1i	0.988909	−	0.988909i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	4494.28			0.151332
$960$		0			0
$961$	23372.3			0.784541
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−4153.32	+	4153.32i	−0.138549	+	0.138549i
$966$		0			0
$967$	−46964.2			−1.56181			−0.780903	−	0.624653i	$-0.785241\pi$
$967$	−46964.2			−1.56181			−0.780903	+	0.624653i	$0.785241\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−11140.6	−	11140.6i	−0.368198	−	0.368198i	0.498622	−	0.866820i	$-0.333840\pi$
$971$	−11140.6	−	11140.6i	−0.368198	−	0.368198i	−0.866820	+	0.498622i	$0.833840\pi$
$972$		0			0
$973$	323.899	−	323.899i	0.0106719	−	0.0106719i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	32688.3i		−	1.07041i	−0.844722	−	0.535205i	$-0.820234\pi$
$977$		−	32688.3i		−	1.07041i	0.844722	−	0.535205i	$-0.179766\pi$
$978$		0			0
$979$	−3110.97	−	3110.97i	−0.101560	−	0.101560i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	10414.9i		−	0.337929i	−0.985622	−	0.168965i	$-0.945958\pi$
$983$		−	10414.9i		−	0.337929i	0.985622	−	0.168965i	$-0.0540423\pi$
$984$		0			0
$985$			19545.4i			0.632251i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−18142.0	−	18142.0i	−0.583297	−	0.583297i
$990$		0			0
$991$			27162.5i			0.870682i	0.900266	+	0.435341i	$0.143372\pi$
$991$			27162.5i			0.870682i	−0.900266	+	0.435341i	$0.856628\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−40782.6	+	40782.6i	−1.29939	+	1.29939i
$996$		0			0
$997$	−4529.97	−	4529.97i	−0.143897	−	0.143897i	0.631488	−	0.775386i	$-0.282444\pi$
$997$	−4529.97	−	4529.97i	−0.143897	−	0.143897i	−0.775386	+	0.631488i	$0.782444\pi$
$998$		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.287.20		48
3.2	odd	2	inner	1152.4.l.b.287.5		48
4.3	odd	2		1152.4.l.a.287.20		48
8.3	odd	2		576.4.l.a.143.5		48
8.5	even	2		144.4.l.a.107.2	yes	48
12.11	even	2		1152.4.l.a.287.5		48
16.3	odd	4	inner	1152.4.l.b.863.5		48
16.5	even	4		576.4.l.a.431.20		48
16.11	odd	4		144.4.l.a.35.23	yes	48
16.13	even	4		1152.4.l.a.863.5		48
24.5	odd	2		144.4.l.a.107.23	yes	48
24.11	even	2		576.4.l.a.143.20		48
48.5	odd	4		576.4.l.a.431.5		48
48.11	even	4		144.4.l.a.35.2	✓	48
48.29	odd	4		1152.4.l.a.863.20		48
48.35	even	4	inner	1152.4.l.b.863.20		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.4.l.a.35.2	✓	48	48.11	even	4
144.4.l.a.35.23	yes	48	16.11	odd	4
144.4.l.a.107.2	yes	48	8.5	even	2
144.4.l.a.107.23	yes	48	24.5	odd	2
576.4.l.a.143.5		48	8.3	odd	2
576.4.l.a.143.20		48	24.11	even	2
576.4.l.a.431.5		48	48.5	odd	4
576.4.l.a.431.20		48	16.5	even	4
1152.4.l.a.287.5		48	12.11	even	2
1152.4.l.a.287.20		48	4.3	odd	2
1152.4.l.a.863.5		48	16.13	even	4
1152.4.l.a.863.20		48	48.29	odd	4
1152.4.l.b.287.5		48	3.2	odd	2	inner
1152.4.l.b.287.20		48	1.1	even	1	trivial
1152.4.l.b.863.5		48	16.3	odd	4	inner
1152.4.l.b.863.20		48	48.35	even	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)

\(f(q)\)	\(=\)	\(q+(9.40057 - 9.40057i) q^{5} -3.57327 q^{7} +O(q^{10})\)	\(q+(9.40057 - 9.40057i) q^{5} -3.57327 q^{7} +(3.36191 + 3.36191i) q^{11} +(-26.9332 + 26.9332i) q^{13} +12.7764i q^{17} +(50.0373 + 50.0373i) q^{19} -208.377i q^{23} -51.7415i q^{25} +(134.170 + 134.170i) q^{29} -80.1170i q^{31} +(-33.5907 + 33.5907i) q^{35} +(308.445 + 308.445i) q^{37} +172.761 q^{41} +(87.0630 - 87.0630i) q^{43} +525.793 q^{47} -330.232 q^{49} +(127.310 - 127.310i) q^{53} +63.2078 q^{55} +(-172.640 - 172.640i) q^{59} +(332.465 - 332.465i) q^{61} +506.376i q^{65} +(-556.127 - 556.127i) q^{67} -450.409i q^{71} -797.249i q^{73} +(-12.0130 - 12.0130i) q^{77} +70.1603i q^{79} +(-636.357 + 636.357i) q^{83} +(120.105 + 120.105i) q^{85} -925.357 q^{89} +(96.2396 - 96.2396i) q^{91} +940.759 q^{95} +1263.11 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q+O(q^{10}) \) 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

Related objects

Downloads

Learn more