LMFDB - Embedded newform 1440.2.bj.a.17.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1440,2,Mod(17,1440)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1440, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 2, 2, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1440.17");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1440 = 2^{5} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1440.bj (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:	$11.4984578911$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$24$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 360)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			17.1
Character	$\chi$	$=$	1440.17
Dual form			1440.2.bj.a.593.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.23604 - 0.0113158i) q^{5} +(-0.471963 - 0.471963i) q^{7} +O(q^{10})$	$q+(-2.23604 - 0.0113158i) q^{5} +(-0.471963 - 0.471963i) q^{7} -0.335652 q^{11} +(3.50404 + 3.50404i) q^{13} +(2.53299 - 2.53299i) q^{17} -4.07474 q^{19} +(-6.20627 - 6.20627i) q^{23} +(4.99974 + 0.0506050i) q^{25} +2.42367i q^{29} +6.41004 q^{31} +(1.04999 + 1.06067i) q^{35} +(2.24893 - 2.24893i) q^{37} -5.80736i q^{41} +(-4.87603 - 4.87603i) q^{43} +(1.68276 - 1.68276i) q^{47} -6.55450i q^{49} +(-3.05444 + 3.05444i) q^{53} +(0.750530 + 0.00379815i) q^{55} -12.2950i q^{59} -7.49787i q^{61} +(-7.79553 - 7.87483i) q^{65} +(5.55519 - 5.55519i) q^{67} -13.4793i q^{71} +(5.05035 - 5.05035i) q^{73} +(0.158415 + 0.158415i) q^{77} +8.85503i q^{79} +(-4.78510 + 4.78510i) q^{83} +(-5.69252 + 5.63520i) q^{85} -8.33405 q^{89} -3.30755i q^{91} +(9.11128 + 0.0461088i) q^{95} +(10.1367 + 10.1367i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q+O(q^{10}) $ 48 * q	$ 48 q - 32 q^{31} - 32 q^{97}+O(q^{100}) $ 48 * q - 32 * q^31 - 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times$.

$n$	$577$	$641$	$901$	$991$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$-1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−2.23604	−	0.0113158i	−0.999987	−	0.00506057i
$5$	−2.23604	−	0.0113158i	−0.999987	−	0.00506057i
$6$		0			0
$7$	−0.471963	−	0.471963i	−0.178385	−	0.178385i	0.612266	−	0.790652i	$-0.290258\pi$
$7$	−0.471963	−	0.471963i	−0.178385	−	0.178385i	−0.790652	+	0.612266i	$0.790258\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.335652			−0.101203			−0.0506014	−	0.998719i	$-0.516114\pi$
$11$	−0.335652			−0.101203			−0.0506014	+	0.998719i	$0.516114\pi$
$12$		0			0
$13$	3.50404	+	3.50404i	0.971846	+	0.971846i	0.999614	−	0.0277680i	$-0.00883995\pi$
$13$	3.50404	+	3.50404i	0.971846	+	0.971846i	−0.0277680	+	0.999614i	$0.508840\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.53299	−	2.53299i	0.614340	−	0.614340i	−0.329734	−	0.944074i	$-0.606959\pi$
$17$	2.53299	−	2.53299i	0.614340	−	0.614340i	0.944074	+	0.329734i	$0.106959\pi$
$18$		0			0
$19$	−4.07474			−0.934809			−0.467405	−	0.884044i	$-0.654811\pi$
$19$	−4.07474			−0.934809			−0.467405	+	0.884044i	$0.654811\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−6.20627	−	6.20627i	−1.29410	−	1.29410i	−0.932229	−	0.361868i	$-0.882139\pi$
$23$	−6.20627	−	6.20627i	−1.29410	−	1.29410i	−0.361868	−	0.932229i	$-0.617861\pi$
$24$		0			0
$25$	4.99974	+	0.0506050i	0.999949	+	0.0101210i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			2.42367i			0.450065i	0.974351	+	0.225032i	$0.0722488\pi$
$29$			2.42367i			0.450065i	−0.974351	+	0.225032i	$0.927751\pi$
$30$		0			0
$31$	6.41004			1.15128			0.575639	−	0.817704i	$-0.304753\pi$
$31$	6.41004			1.15128			0.575639	+	0.817704i	$0.304753\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.04999	+	1.06067i	0.177480	+	0.179286i
$36$		0			0
$37$	2.24893	−	2.24893i	0.369721	−	0.369721i	−0.497654	−	0.867375i	$-0.665805\pi$
$37$	2.24893	−	2.24893i	0.369721	−	0.369721i	0.867375	+	0.497654i	$0.165805\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	5.80736i		−	0.906957i	−0.891267	−	0.453479i	$-0.850183\pi$
$41$		−	5.80736i		−	0.906957i	0.891267	−	0.453479i	$-0.149817\pi$
$42$		0			0
$43$	−4.87603	−	4.87603i	−0.743588	−	0.743588i	0.229678	−	0.973267i	$-0.426233\pi$
$43$	−4.87603	−	4.87603i	−0.743588	−	0.743588i	−0.973267	+	0.229678i	$0.926233\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.68276	−	1.68276i	0.245455	−	0.245455i	−0.573647	−	0.819102i	$-0.694472\pi$
$47$	1.68276	−	1.68276i	0.245455	−	0.245455i	0.819102	+	0.573647i	$0.194472\pi$
$48$		0			0
$49$		−	6.55450i		−	0.936358i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−3.05444	+	3.05444i	−0.419560	+	0.419560i	−0.885052	−	0.465492i	$-0.845877\pi$
$53$	−3.05444	+	3.05444i	−0.419560	+	0.419560i	0.465492	+	0.885052i	$0.345877\pi$
$54$		0			0
$55$	0.750530	+	0.00379815i	0.101201	+	0.000512143i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	12.2950i		−	1.60067i	−0.599553	−	0.800335i	$-0.704655\pi$
$59$		−	12.2950i		−	1.60067i	0.599553	−	0.800335i	$-0.295345\pi$
$60$		0			0
$61$		−	7.49787i		−	0.960004i	−0.877267	−	0.480002i	$-0.840636\pi$
$61$		−	7.49787i		−	0.960004i	0.877267	−	0.480002i	$-0.159364\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−7.79553	−	7.87483i	−0.966916	−	0.976752i
$66$		0			0
$67$	5.55519	−	5.55519i	0.678674	−	0.678674i	−0.281026	−	0.959700i	$-0.590675\pi$
$67$	5.55519	−	5.55519i	0.678674	−	0.678674i	0.959700	+	0.281026i	$0.0906748\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	13.4793i		−	1.59970i	−0.600199	−	0.799851i	$-0.704912\pi$
$71$		−	13.4793i		−	1.59970i	0.600199	−	0.799851i	$-0.295088\pi$
$72$		0			0
$73$	5.05035	−	5.05035i	0.591099	−	0.591099i	−0.346830	−	0.937928i	$-0.612742\pi$
$73$	5.05035	−	5.05035i	0.591099	−	0.591099i	0.937928	+	0.346830i	$0.112742\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.158415	+	0.158415i	0.0180531	+	0.0180531i
$78$		0			0
$79$			8.85503i			0.996268i	0.867100	+	0.498134i	$0.165981\pi$
$79$			8.85503i			0.996268i	−0.867100	+	0.498134i	$0.834019\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−4.78510	+	4.78510i	−0.525233	+	0.525233i	−0.919147	−	0.393914i	$-0.871121\pi$
$83$	−4.78510	+	4.78510i	−0.525233	+	0.525233i	0.393914	+	0.919147i	$0.371121\pi$
$84$		0			0
$85$	−5.69252	+	5.63520i	−0.617441	+	0.611223i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−8.33405			−0.883408			−0.441704	−	0.897161i	$-0.645626\pi$
$89$	−8.33405			−0.883408			−0.441704	+	0.897161i	$0.645626\pi$
$90$		0			0
$91$		−	3.30755i		−	0.346726i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	9.11128	+	0.0461088i	0.934797	+	0.00473066i
$96$		0			0
$97$	10.1367	+	10.1367i	1.02922	+	1.02922i	0.999560	+	0.0296647i	$0.00944397\pi$
$97$	10.1367	+	10.1367i	1.02922	+	1.02922i	0.0296647	+	0.999560i	$0.490556\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	9.88830			0.983922			0.491961	−	0.870617i	$-0.336280\pi$
$101$	9.88830			0.983922			0.491961	+	0.870617i	$0.336280\pi$
$102$		0			0
$103$	10.0033	−	10.0033i	0.985653	−	0.985653i	−0.0142455	−	0.999899i	$-0.504535\pi$
$103$	10.0033	−	10.0033i	0.985653	−	0.985653i	0.999899	+	0.0142455i	$0.00453463\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	10.2013	+	10.2013i	0.986198	+	0.986198i	0.999906	−	0.0137076i	$-0.00436339\pi$
$107$	10.2013	+	10.2013i	0.986198	+	0.986198i	−0.0137076	+	0.999906i	$0.504363\pi$
$108$		0			0
$109$	−11.0144			−1.05498			−0.527492	−	0.849560i	$-0.676867\pi$
$109$	−11.0144			−1.05498			−0.527492	+	0.849560i	$0.676867\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	2.45369	+	2.45369i	0.230823	+	0.230823i	0.813036	−	0.582213i	$-0.197813\pi$
$113$	2.45369	+	2.45369i	0.230823	+	0.230823i	−0.582213	+	0.813036i	$0.697813\pi$
$114$		0			0
$115$	13.8072	+	13.9477i	1.28753	+	1.30063i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−2.39095			−0.219178
$120$		0			0
$121$	−10.8873			−0.989758
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−11.1791	−	0.169731i	−0.999885	−	0.0151812i
$126$		0			0
$127$	−6.00329	−	6.00329i	−0.532705	−	0.532705i	0.388671	−	0.921377i	$-0.372934\pi$
$127$	−6.00329	−	6.00329i	−0.532705	−	0.532705i	−0.921377	+	0.388671i	$0.872934\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−11.3345			−0.990301			−0.495151	−	0.868807i	$-0.664887\pi$
$131$	−11.3345			−0.990301			−0.495151	+	0.868807i	$0.664887\pi$
$132$		0			0
$133$	1.92312	+	1.92312i	0.166756	+	0.166756i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	10.8670	−	10.8670i	0.928434	−	0.928434i	−0.0691711	−	0.997605i	$-0.522035\pi$
$137$	10.8670	−	10.8670i	0.928434	−	0.928434i	0.997605	+	0.0691711i	$0.0220354\pi$
$138$		0			0
$139$	−13.3432			−1.13176			−0.565878	−	0.824489i	$-0.691463\pi$
$139$	−13.3432			−1.13176			−0.565878	+	0.824489i	$0.691463\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1.17614	−	1.17614i	−0.0983535	−	0.0983535i
$144$		0			0
$145$	0.0274257	−	5.41943i	0.00227758	−	0.450059i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	13.5961i		−	1.11384i	−0.830566	−	0.556919i	$-0.811983\pi$
$149$		−	13.5961i		−	1.11384i	0.830566	−	0.556919i	$-0.188017\pi$
$150$		0			0
$151$	9.64289			0.784727			0.392364	−	0.919810i	$-0.371657\pi$
$151$	9.64289			0.784727			0.392364	+	0.919810i	$0.371657\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−14.3331	−	0.0725345i	−1.15126	−	0.00582611i
$156$		0			0
$157$	7.62403	−	7.62403i	0.608464	−	0.608464i	−0.334080	−	0.942545i	$-0.608426\pi$
$157$	7.62403	−	7.62403i	0.608464	−	0.608464i	0.942545	+	0.334080i	$0.108426\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			5.85826i			0.461695i
$162$		0			0
$163$	16.2490	+	16.2490i	1.27272	+	1.27272i	0.944655	+	0.328065i	$0.106397\pi$
$163$	16.2490	+	16.2490i	1.27272	+	1.27272i	0.328065	+	0.944655i	$0.393603\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−13.7067	+	13.7067i	−1.06066	+	1.06066i	−0.0626204	+	0.998037i	$0.519946\pi$
$167$	−13.7067	+	13.7067i	−1.06066	+	1.06066i	−0.998037	+	0.0626204i	$0.980054\pi$
$168$		0			0
$169$			11.5566i			0.888971i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	3.35678	−	3.35678i	0.255211	−	0.255211i	−0.567892	−	0.823103i	$-0.692241\pi$
$173$	3.35678	−	3.35678i	0.255211	−	0.255211i	0.823103	+	0.567892i	$0.192241\pi$
$174$		0			0
$175$	−2.33581	−	2.38358i	−0.176571	−	0.180181i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			10.7537i			0.803772i	0.915690	+	0.401886i	$0.131645\pi$
$179$			10.7537i			0.803772i	−0.915690	+	0.401886i	$0.868355\pi$
$180$		0			0
$181$		−	10.1290i		−	0.752883i	−0.926440	−	0.376441i	$-0.877148\pi$
$181$		−	10.1290i		−	0.752883i	0.926440	−	0.376441i	$-0.122852\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−5.05414	+	5.00324i	−0.371587	+	0.367845i
$186$		0			0
$187$	−0.850201	+	0.850201i	−0.0621728	+	0.0621728i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			4.32194i			0.312725i	0.987700	+	0.156362i	$0.0499768\pi$
$191$			4.32194i			0.312725i	−0.987700	+	0.156362i	$0.950023\pi$
$192$		0			0
$193$	−7.97419	+	7.97419i	−0.573995	+	0.573995i	−0.933242	−	0.359248i	$-0.883033\pi$
$193$	−7.97419	+	7.97419i	−0.573995	+	0.573995i	0.359248	+	0.933242i	$0.383033\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−4.30206	−	4.30206i	−0.306509	−	0.306509i	0.537045	−	0.843554i	$-0.319541\pi$
$197$	−4.30206	−	4.30206i	−0.306509	−	0.306509i	−0.843554	+	0.537045i	$0.819541\pi$
$198$		0			0
$199$		−	3.02113i		−	0.214162i	−0.994250	−	0.107081i	$-0.965850\pi$
$199$		−	3.02113i		−	0.214162i	0.994250	−	0.107081i	$-0.0341505\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	1.14388	−	1.14388i	0.0802849	−	0.0802849i
$204$		0			0
$205$	−0.0657147	+	12.9855i	−0.00458972	+	0.906946i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1.36769			0.0946052
$210$		0			0
$211$		−	8.87754i		−	0.611155i	−0.952167	−	0.305577i	$-0.901151\pi$
$211$		−	8.87754i		−	0.611155i	0.952167	−	0.305577i	$-0.0988495\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	10.8478	+	10.9582i	0.739816	+	0.747342i
$216$		0			0
$217$	−3.02530	−	3.02530i	−0.205371	−	0.205371i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	17.7514			1.19409
$222$		0			0
$223$	−9.93808	+	9.93808i	−0.665503	+	0.665503i	−0.956672	−	0.291169i	$-0.905956\pi$
$223$	−9.93808	+	9.93808i	−0.665503	+	0.665503i	0.291169	+	0.956672i	$0.405956\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	7.21442	+	7.21442i	0.478838	+	0.478838i	0.904760	−	0.425922i	$-0.140050\pi$
$227$	7.21442	+	7.21442i	0.478838	+	0.478838i	−0.425922	+	0.904760i	$0.640050\pi$
$228$		0			0
$229$	−21.8687			−1.44512			−0.722562	−	0.691306i	$-0.757035\pi$
$229$	−21.8687			−1.44512			−0.722562	+	0.691306i	$0.757035\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	6.78300	+	6.78300i	0.444369	+	0.444369i	0.893477	−	0.449108i	$-0.148258\pi$
$233$	6.78300	+	6.78300i	0.444369	+	0.444369i	−0.449108	+	0.893477i	$0.648258\pi$
$234$		0			0
$235$	−3.78175	+	3.74367i	−0.246694	+	0.244210i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	4.63287			0.299676			0.149838	−	0.988711i	$-0.452125\pi$
$239$	4.63287			0.299676			0.149838	+	0.988711i	$0.452125\pi$
$240$		0			0
$241$	−9.72150			−0.626217			−0.313109	−	0.949717i	$-0.601370\pi$
$241$	−9.72150			−0.626217			−0.313109	+	0.949717i	$0.601370\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−0.0741692	+	14.6561i	−0.00473850	+	0.936346i
$246$		0			0
$247$	−14.2781	−	14.2781i	−0.908491	−	0.908491i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	30.6190			1.93265			0.966327	−	0.257316i	$-0.0828382\pi$
$251$	30.6190			1.93265			0.966327	+	0.257316i	$0.0828382\pi$
$252$		0			0
$253$	2.08314	+	2.08314i	0.130966	+	0.130966i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	1.63773	−	1.63773i	0.102158	−	0.102158i	−0.654180	−	0.756339i	$-0.726986\pi$
$257$	1.63773	−	1.63773i	0.102158	−	0.102158i	0.756339	+	0.654180i	$0.226986\pi$
$258$		0			0
$259$	−2.12282			−0.131905
$260$		0			0
$261$		0			0
$262$		0			0
$263$	7.77099	+	7.77099i	0.479180	+	0.479180i	0.904869	−	0.425689i	$-0.139968\pi$
$263$	7.77099	+	7.77099i	0.479180	+	0.479180i	−0.425689	+	0.904869i	$0.639968\pi$
$264$		0			0
$265$	6.86441	−	6.79529i	0.421678	−	0.417431i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	16.1341i		−	0.983714i	−0.870676	−	0.491857i	$-0.836318\pi$
$269$		−	16.1341i		−	0.983714i	0.870676	−	0.491857i	$-0.163682\pi$
$270$		0			0
$271$	−16.8474			−1.02341			−0.511704	−	0.859162i	$-0.670986\pi$
$271$	−16.8474			−1.02341			−0.511704	+	0.859162i	$0.670986\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.67817	−	0.0169856i	−0.101198	−	0.00102427i
$276$		0			0
$277$	16.3206	−	16.3206i	0.980611	−	0.980611i	−0.0192042	−	0.999816i	$-0.506113\pi$
$277$	16.3206	−	16.3206i	0.980611	−	0.980611i	0.999816	+	0.0192042i	$0.00611328\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			31.2898i			1.86659i	0.359106	+	0.933297i	$0.383082\pi$
$281$			31.2898i			1.86659i	−0.359106	+	0.933297i	$0.616918\pi$
$282$		0			0
$283$	−9.97554	−	9.97554i	−0.592984	−	0.592984i	0.345452	−	0.938436i	$-0.387726\pi$
$283$	−9.97554	−	9.97554i	−0.592984	−	0.592984i	−0.938436	+	0.345452i	$0.887726\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−2.74086	+	2.74086i	−0.161788	+	0.161788i
$288$		0			0
$289$			4.16795i			0.245174i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−5.26253	+	5.26253i	−0.307440	+	0.307440i	−0.843916	−	0.536476i	$-0.819755\pi$
$293$	−5.26253	+	5.26253i	−0.307440	+	0.307440i	0.536476	+	0.843916i	$0.319755\pi$
$294$		0			0
$295$	−0.139127	+	27.4921i	−0.00810030	+	1.60065i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	43.4941i		−	2.51533i
$300$		0			0
$301$			4.60261i			0.265290i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−0.0848442	+	16.7655i	−0.00485816	+	0.959992i
$306$		0			0
$307$	9.54586	−	9.54586i	0.544811	−	0.544811i	−0.380124	−	0.924935i	$-0.624119\pi$
$307$	9.54586	−	9.54586i	0.544811	−	0.544811i	0.924935	+	0.380124i	$0.124119\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			4.08587i			0.231688i	0.993267	+	0.115844i	$0.0369573\pi$
$311$			4.08587i			0.231688i	−0.993267	+	0.115844i	$0.963043\pi$
$312$		0			0
$313$	−14.4004	+	14.4004i	−0.813958	+	0.813958i	−0.985225	−	0.171266i	$-0.945214\pi$
$313$	−14.4004	+	14.4004i	−0.813958	+	0.813958i	0.171266	+	0.985225i	$0.445214\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	12.0841	+	12.0841i	0.678710	+	0.678710i	0.959708	−	0.280998i	$-0.0906656\pi$
$317$	12.0841	+	12.0841i	0.678710	+	0.678710i	−0.280998	+	0.959708i	$0.590666\pi$
$318$		0			0
$319$		−	0.813510i		−	0.0455478i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−10.3213	+	10.3213i	−0.574290	+	0.574290i
$324$		0			0
$325$	17.3420	+	17.6966i	0.961961	+	0.981633i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1.58840			−0.0875711
$330$		0			0
$331$		−	33.6119i		−	1.84747i	−0.383027	−	0.923737i	$-0.625118\pi$
$331$		−	33.6119i		−	1.84747i	0.383027	−	0.923737i	$-0.374882\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−12.4845	+	12.3588i	−0.682100	+	0.675231i
$336$		0			0
$337$	−20.2142	−	20.2142i	−1.10114	−	1.10114i	−0.994274	−	0.106864i	$-0.965919\pi$
$337$	−20.2142	−	20.2142i	−1.10114	−	1.10114i	−0.106864	−	0.994274i	$-0.534081\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−2.15154			−0.116512
$342$		0			0
$343$	−6.39722	+	6.39722i	−0.345417	+	0.345417i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−0.443163	−	0.443163i	−0.0237903	−	0.0237903i	0.695112	−	0.718902i	$-0.255355\pi$
$347$	−0.443163	−	0.443163i	−0.0237903	−	0.0237903i	−0.718902	+	0.695112i	$0.755355\pi$
$348$		0			0
$349$	−10.8766			−0.582212			−0.291106	−	0.956691i	$-0.594023\pi$
$349$	−10.8766			−0.582212			−0.291106	+	0.956691i	$0.594023\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	10.0437	+	10.0437i	0.534572	+	0.534572i	0.921930	−	0.387357i	$-0.126612\pi$
$353$	10.0437	+	10.0437i	0.534572	+	0.534572i	−0.387357	+	0.921930i	$0.626612\pi$
$354$		0			0
$355$	−0.152529	+	30.1403i	−0.00809540	+	1.59968i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	19.6724			1.03827			0.519136	−	0.854692i	$-0.326254\pi$
$359$	19.6724			1.03827			0.519136	+	0.854692i	$0.326254\pi$
$360$		0			0
$361$	−2.39651			−0.126132
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−11.3499	+	11.2356i	−0.594082	+	0.588100i
$366$		0			0
$367$	9.86005	+	9.86005i	0.514690	+	0.514690i	0.915960	−	0.401270i	$-0.131431\pi$
$367$	9.86005	+	9.86005i	0.514690	+	0.514690i	−0.401270	+	0.915960i	$0.631431\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	2.88316			0.149686
$372$		0			0
$373$	12.6526	+	12.6526i	0.655127	+	0.655127i	0.954223	−	0.299096i	$-0.0966852\pi$
$373$	12.6526	+	12.6526i	0.655127	+	0.655127i	−0.299096	+	0.954223i	$0.596685\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−8.49265	+	8.49265i	−0.437394	+	0.437394i
$378$		0			0
$379$	−37.9577			−1.94975			−0.974877	−	0.222743i	$-0.928499\pi$
$379$	−37.9577			−1.94975			−0.974877	+	0.222743i	$0.928499\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	9.18320	+	9.18320i	0.469240	+	0.469240i	0.901668	−	0.432428i	$-0.142343\pi$
$383$	9.18320	+	9.18320i	0.469240	+	0.469240i	−0.432428	+	0.901668i	$0.642343\pi$
$384$		0			0
$385$	−0.352429	−	0.356015i	−0.0179615	−	0.0181442i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			4.22258i			0.214093i	0.994254	+	0.107047i	$0.0341394\pi$
$389$			4.22258i			0.214093i	−0.994254	+	0.107047i	$0.965861\pi$
$390$		0			0
$391$	−31.4408			−1.59003
$392$		0			0
$393$		0			0
$394$		0			0
$395$	0.100201	−	19.8002i	0.00504168	−	0.996256i
$396$		0			0
$397$	−1.27679	+	1.27679i	−0.0640801	+	0.0640801i	−0.738421	−	0.674340i	$-0.764428\pi$
$397$	−1.27679	+	1.27679i	−0.0640801	+	0.0640801i	0.674340	+	0.738421i	$0.264428\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			0.899990i			0.0449434i	0.999747	+	0.0224717i	$0.00715356\pi$
$401$			0.899990i			0.0449434i	−0.999747	+	0.0224717i	$0.992846\pi$
$402$		0			0
$403$	22.4611	+	22.4611i	1.11886	+	1.11886i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−0.754855	+	0.754855i	−0.0374168	+	0.0374168i
$408$		0			0
$409$		−	24.1722i		−	1.19524i	−0.801781	−	0.597618i	$-0.796114\pi$
$409$		−	24.1722i		−	1.19524i	0.801781	−	0.597618i	$-0.203886\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−5.80277	+	5.80277i	−0.285536	+	0.285536i
$414$		0			0
$415$	10.7538	−	10.6455i	0.527884	−	0.522568i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			8.45513i			0.413060i	0.978440	+	0.206530i	$0.0662171\pi$
$419$			8.45513i			0.413060i	−0.978440	+	0.206530i	$0.933783\pi$
$420$		0			0
$421$			13.8561i			0.675307i	0.941270	+	0.337654i	$0.109633\pi$
$421$			13.8561i			0.675307i	−0.941270	+	0.337654i	$0.890367\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	12.7925	−	12.5361i	0.620526	−	0.608090i
$426$		0			0
$427$	−3.53872	+	3.53872i	−0.171250	+	0.171250i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			5.84351i			0.281472i	0.990047	+	0.140736i	$0.0449469\pi$
$431$			5.84351i			0.281472i	−0.990047	+	0.140736i	$0.955053\pi$
$432$		0			0
$433$	10.0632	−	10.0632i	0.483605	−	0.483605i	−0.422676	−	0.906281i	$-0.638909\pi$
$433$	10.0632	−	10.0632i	0.483605	−	0.483605i	0.906281	+	0.422676i	$0.138909\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	25.2889	+	25.2889i	1.20973	+	1.20973i
$438$		0			0
$439$			34.0222i			1.62379i	0.583802	+	0.811896i	$0.301564\pi$
$439$			34.0222i			1.62379i	−0.583802	+	0.811896i	$0.698436\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	6.45259	−	6.45259i	0.306572	−	0.306572i	−0.537006	−	0.843578i	$-0.680445\pi$
$443$	6.45259	−	6.45259i	0.306572	−	0.306572i	0.843578	+	0.537006i	$0.180445\pi$
$444$		0			0
$445$	18.6353	+	0.0943062i	0.883396	+	0.00447054i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−5.72541			−0.270199			−0.135099	−	0.990832i	$-0.543135\pi$
$449$	−5.72541			−0.270199			−0.135099	+	0.990832i	$0.543135\pi$
$450$		0			0
$451$			1.94925i			0.0917866i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−0.0374275	+	7.39582i	−0.00175463	+	0.346721i
$456$		0			0
$457$	14.9718	+	14.9718i	0.700352	+	0.700352i	0.964486	−	0.264134i	$-0.0850862\pi$
$457$	14.9718	+	14.9718i	0.700352	+	0.700352i	−0.264134	+	0.964486i	$0.585086\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−25.8187			−1.20250			−0.601249	−	0.799062i	$-0.705330\pi$
$461$	−25.8187			−1.20250			−0.601249	+	0.799062i	$0.705330\pi$
$462$		0			0
$463$	20.5140	−	20.5140i	0.953368	−	0.953368i	−0.0455925	−	0.998960i	$-0.514518\pi$
$463$	20.5140	−	20.5140i	0.953368	−	0.953368i	0.998960	+	0.0455925i	$0.0145176\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	3.07112	+	3.07112i	0.142114	+	0.142114i	0.774585	−	0.632470i	$-0.217959\pi$
$467$	3.07112	+	3.07112i	0.142114	+	0.142114i	−0.632470	+	0.774585i	$0.717959\pi$
$468$		0			0
$469$	−5.24368			−0.242131
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1.63665	+	1.63665i	0.0752532	+	0.0752532i
$474$		0			0
$475$	−20.3726	−	0.206202i	−0.934761	−	0.00946120i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−33.2621			−1.51979			−0.759893	−	0.650049i	$-0.774748\pi$
$479$	−33.2621			−1.51979			−0.759893	+	0.650049i	$0.774748\pi$
$480$		0			0
$481$	15.7607			0.718624
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−22.5513	−	22.7807i	−1.02400	−	1.03442i
$486$		0			0
$487$	−1.39322	−	1.39322i	−0.0631329	−	0.0631329i	0.674835	−	0.737968i	$-0.264215\pi$
$487$	−1.39322	−	1.39322i	−0.0631329	−	0.0631329i	−0.737968	+	0.674835i	$0.764215\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	14.4709			0.653061			0.326530	−	0.945187i	$-0.394120\pi$
$491$	14.4709			0.653061			0.326530	+	0.945187i	$0.394120\pi$
$492$		0			0
$493$	6.13913	+	6.13913i	0.276493	+	0.276493i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−6.36174	+	6.36174i	−0.285363	+	0.285363i
$498$		0			0
$499$	20.4107			0.913710			0.456855	−	0.889541i	$-0.348976\pi$
$499$	20.4107			0.913710			0.456855	+	0.889541i	$0.348976\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−2.54697	−	2.54697i	−0.113564	−	0.113564i	0.648041	−	0.761605i	$-0.275588\pi$
$503$	−2.54697	−	2.54697i	−0.113564	−	0.113564i	−0.761605	+	0.648041i	$0.775588\pi$
$504$		0			0
$505$	−22.1106	−	0.111894i	−0.983910	−	0.00497920i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	13.5045i		−	0.598575i	−0.954163	−	0.299288i	$-0.903251\pi$
$509$		−	13.5045i		−	0.598575i	0.954163	−	0.299288i	$-0.0967490\pi$
$510$		0			0
$511$	−4.76715			−0.210886
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−22.4809	+	22.2545i	−0.990628	+	0.980652i
$516$		0			0
$517$	−0.564820	+	0.564820i	−0.0248407	+	0.0248407i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	13.5151i		−	0.592108i	−0.955171	−	0.296054i	$-0.904329\pi$
$521$		−	13.5151i		−	0.592108i	0.955171	−	0.296054i	$-0.0956708\pi$
$522$		0			0
$523$	−12.6641	−	12.6641i	−0.553760	−	0.553760i	0.373764	−	0.927524i	$-0.378067\pi$
$523$	−12.6641	−	12.6641i	−0.553760	−	0.553760i	−0.927524	+	0.373764i	$0.878067\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	16.2365	−	16.2365i	0.707275	−	0.707275i
$528$		0			0
$529$			54.0356i			2.34938i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	20.3492	−	20.3492i	0.881423	−	0.881423i
$534$		0			0
$535$	−22.6951	−	22.9260i	−0.981195	−	0.991177i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			2.20003i			0.0947619i
$540$		0			0
$541$		−	17.1804i		−	0.738644i	−0.929301	−	0.369322i	$-0.879590\pi$
$541$		−	17.1804i		−	0.738644i	0.929301	−	0.369322i	$-0.120410\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	24.6285	+	0.124636i	1.05497	+	0.00533881i
$546$		0			0
$547$	17.4105	−	17.4105i	0.744417	−	0.744417i	−0.229007	−	0.973425i	$-0.573548\pi$
$547$	17.4105	−	17.4105i	0.744417	−	0.744417i	0.973425	+	0.229007i	$0.0735480\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	9.87584i		−	0.420725i
$552$		0			0
$553$	4.17924	−	4.17924i	0.177719	−	0.177719i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−9.22732	−	9.22732i	−0.390974	−	0.390974i	0.484060	−	0.875035i	$-0.339162\pi$
$557$	−9.22732	−	9.22732i	−0.390974	−	0.390974i	−0.875035	+	0.484060i	$0.839162\pi$
$558$		0			0
$559$		−	34.1717i		−	1.44531i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−19.8727	+	19.8727i	−0.837535	+	0.837535i	−0.988534	−	0.150999i	$-0.951751\pi$
$563$	−19.8727	+	19.8727i	−0.837535	+	0.837535i	0.150999	+	0.988534i	$0.451751\pi$
$564$		0			0
$565$	−5.45877	−	5.51430i	−0.229652	−	0.231988i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	4.73031			0.198305			0.0991524	−	0.995072i	$-0.468387\pi$
$569$	4.73031			0.198305			0.0991524	+	0.995072i	$0.468387\pi$
$570$		0			0
$571$		−	5.54039i		−	0.231858i	−0.993257	−	0.115929i	$-0.963015\pi$
$571$		−	5.54039i		−	0.231858i	0.993257	−	0.115929i	$-0.0369845\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−30.7157	−	31.3438i	−1.28093	−	1.30713i
$576$		0			0
$577$	−12.9105	−	12.9105i	−0.537472	−	0.537472i	0.385314	−	0.922786i	$-0.374093\pi$
$577$	−12.9105	−	12.9105i	−0.537472	−	0.537472i	−0.922786	+	0.385314i	$0.874093\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	4.51677			0.187387
$582$		0			0
$583$	1.02523	−	1.02523i	0.0424606	−	0.0424606i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−1.90309	−	1.90309i	−0.0785487	−	0.0785487i	0.666741	−	0.745290i	$-0.267689\pi$
$587$	−1.90309	−	1.90309i	−0.0785487	−	0.0785487i	−0.745290	+	0.666741i	$0.767689\pi$
$588$		0			0
$589$	−26.1192			−1.07622
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−11.7312	−	11.7312i	−0.481745	−	0.481745i	0.423944	−	0.905688i	$-0.360645\pi$
$593$	−11.7312	−	11.7312i	−0.481745	−	0.481745i	−0.905688	+	0.423944i	$0.860645\pi$
$594$		0			0
$595$	5.34626	+	0.0270554i	0.219175	+	0.00110916i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	2.61594			0.106884			0.0534422	−	0.998571i	$-0.482981\pi$
$599$	2.61594			0.106884			0.0534422	+	0.998571i	$0.482981\pi$
$600$		0			0
$601$	20.1722			0.822840			0.411420	−	0.911446i	$-0.365033\pi$
$601$	20.1722			0.822840			0.411420	+	0.911446i	$0.365033\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	24.3445	+	0.123199i	0.989745	+	0.00500873i
$606$		0			0
$607$	17.3218	+	17.3218i	0.703069	+	0.703069i	0.965068	−	0.261999i	$-0.0843819\pi$
$607$	17.3218	+	17.3218i	0.703069	+	0.703069i	−0.261999	+	0.965068i	$0.584382\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	11.7929			0.477090
$612$		0			0
$613$	−16.8299	−	16.8299i	−0.679754	−	0.679754i	0.280191	−	0.959944i	$-0.409602\pi$
$613$	−16.8299	−	16.8299i	−0.679754	−	0.679754i	−0.959944	+	0.280191i	$0.909602\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	2.56139	−	2.56139i	0.103118	−	0.103118i	−0.653666	−	0.756783i	$-0.726770\pi$
$617$	2.56139	−	2.56139i	0.103118	−	0.103118i	0.756783	+	0.653666i	$0.226770\pi$
$618$		0			0
$619$	−16.9662			−0.681931			−0.340965	−	0.940076i	$-0.610754\pi$
$619$	−16.9662			−0.681931			−0.340965	+	0.940076i	$0.610754\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	3.93336	+	3.93336i	0.157587	+	0.157587i
$624$		0			0
$625$	24.9949	+	0.506024i	0.999795	+	0.0202410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	11.3930i		−	0.454269i
$630$		0			0
$631$	39.5630			1.57498			0.787488	−	0.616330i	$-0.211381\pi$
$631$	39.5630			1.57498			0.787488	+	0.616330i	$0.211381\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	13.3557	+	13.4915i	0.530003	+	0.535394i
$636$		0			0
$637$	22.9673	−	22.9673i	0.909996	−	0.909996i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	47.3441i		−	1.86998i	−0.354676	−	0.934989i	$-0.615409\pi$
$641$		−	47.3441i		−	1.86998i	0.354676	−	0.934989i	$-0.384591\pi$
$642$		0			0
$643$	14.8250	+	14.8250i	0.584639	+	0.584639i	0.936175	−	0.351535i	$-0.114340\pi$
$643$	14.8250	+	14.8250i	0.584639	+	0.584639i	−0.351535	+	0.936175i	$0.614340\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	4.86574	−	4.86574i	0.191292	−	0.191292i	−0.604962	−	0.796254i	$-0.706812\pi$
$647$	4.86574	−	4.86574i	0.191292	−	0.191292i	0.796254	+	0.604962i	$0.206812\pi$
$648$		0			0
$649$			4.12683i			0.161992i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−11.8214	+	11.8214i	−0.462609	+	0.462609i	−0.899510	−	0.436901i	$-0.856076\pi$
$653$	−11.8214	+	11.8214i	−0.462609	+	0.462609i	0.436901	+	0.899510i	$0.356076\pi$
$654$		0			0
$655$	25.3444	+	0.128259i	0.990289	+	0.00501149i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	6.02230i		−	0.234595i	−0.993097	−	0.117298i	$-0.962577\pi$
$659$		−	6.02230i		−	0.234595i	0.993097	−	0.117298i	$-0.0374232\pi$
$660$		0			0
$661$		−	37.9790i		−	1.47721i	−0.674137	−	0.738606i	$-0.735484\pi$
$661$		−	37.9790i		−	1.47721i	0.674137	−	0.738606i	$-0.264516\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−4.27842	−	4.32194i	−0.165910	−	0.167598i
$666$		0			0
$667$	15.0420	−	15.0420i	0.582428	−	0.582428i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			2.51667i			0.0971551i
$672$		0			0
$673$	7.99686	−	7.99686i	0.308256	−	0.308256i	−0.535977	−	0.844233i	$-0.680057\pi$
$673$	7.99686	−	7.99686i	0.308256	−	0.308256i	0.844233	+	0.535977i	$0.180057\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−7.11103	−	7.11103i	−0.273299	−	0.273299i	0.557128	−	0.830427i	$-0.311903\pi$
$677$	−7.11103	−	7.11103i	−0.273299	−	0.273299i	−0.830427	+	0.557128i	$0.811903\pi$
$678$		0			0
$679$		−	9.56827i		−	0.367197i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	7.99061	−	7.99061i	0.305752	−	0.305752i	−0.537507	−	0.843259i	$-0.680634\pi$
$683$	7.99061	−	7.99061i	0.305752	−	0.305752i	0.843259	+	0.537507i	$0.180634\pi$
$684$		0			0
$685$	−24.4221	+	24.1762i	−0.933120	+	0.923723i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−21.4058			−0.815495
$690$		0			0
$691$			32.5931i			1.23990i	0.784641	+	0.619951i	$0.212848\pi$
$691$			32.5931i			1.23990i	−0.784641	+	0.619951i	$0.787152\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	29.8359	+	0.150989i	1.13174	+	0.00572732i
$696$		0			0
$697$	−14.7100	−	14.7100i	−0.557180	−	0.557180i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	42.9064			1.62055			0.810277	−	0.586047i	$-0.199317\pi$
$701$	42.9064			1.62055			0.810277	+	0.586047i	$0.199317\pi$
$702$		0			0
$703$	−9.16379	+	9.16379i	−0.345619	+	0.345619i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−4.66691	−	4.66691i	−0.175517	−	0.175517i
$708$		0			0
$709$	−10.3299			−0.387949			−0.193975	−	0.981007i	$-0.562138\pi$
$709$	−10.3299			−0.387949			−0.193975	+	0.981007i	$0.562138\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−39.7825	−	39.7825i	−1.48986	−	1.48986i
$714$		0			0
$715$	2.61658	+	2.64320i	0.0978545	+	0.0988500i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−6.49219			−0.242118			−0.121059	−	0.992645i	$-0.538629\pi$
$719$	−6.49219			−0.242118			−0.121059	+	0.992645i	$0.538629\pi$
$720$		0			0
$721$	−9.44235			−0.351652
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−0.122650	+	12.1177i	−0.00455511	+	0.450042i
$726$		0			0
$727$	11.0479	+	11.0479i	0.409744	+	0.409744i	0.881649	−	0.471905i	$-0.156434\pi$
$727$	11.0479	+	11.0479i	0.409744	+	0.409744i	−0.471905	+	0.881649i	$0.656434\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−24.7019			−0.913631
$732$		0			0
$733$	−19.5376	−	19.5376i	−0.721636	−	0.721636i	0.247302	−	0.968938i	$-0.420456\pi$
$733$	−19.5376	−	19.5376i	−0.721636	−	0.721636i	−0.968938	+	0.247302i	$0.920456\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1.86461	+	1.86461i	−0.0686837	+	0.0686837i
$738$		0			0
$739$	24.8980			0.915890			0.457945	−	0.888981i	$-0.348586\pi$
$739$	24.8980			0.915890			0.457945	+	0.888981i	$0.348586\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−15.8961	−	15.8961i	−0.583171	−	0.583171i	0.352602	−	0.935773i	$-0.385297\pi$
$743$	−15.8961	−	15.8961i	−0.583171	−	0.583171i	−0.935773	+	0.352602i	$0.885297\pi$
$744$		0			0
$745$	−0.153851	+	30.4015i	−0.00563665	+	1.11382i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	9.62928i		−	0.351846i
$750$		0			0
$751$	−11.5535			−0.421594			−0.210797	−	0.977530i	$-0.567606\pi$
$751$	−11.5535			−0.421594			−0.210797	+	0.977530i	$0.567606\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−21.5619	−	0.109117i	−0.784717	−	0.00397116i
$756$		0			0
$757$	−18.8612	+	18.8612i	−0.685524	+	0.685524i	−0.961239	−	0.275716i	$-0.911085\pi$
$757$	−18.8612	+	18.8612i	−0.685524	+	0.685524i	0.275716	+	0.961239i	$0.411085\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	33.0567i		−	1.19830i	−0.800636	−	0.599151i	$-0.795505\pi$
$761$		−	33.0567i		−	1.19830i	0.800636	−	0.599151i	$-0.204495\pi$
$762$		0			0
$763$	5.19836	+	5.19836i	0.188193	+	0.188193i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	43.0821	−	43.0821i	1.55561	−	1.55561i
$768$		0			0
$769$		−	44.9240i		−	1.62000i	−0.586429	−	0.810001i	$-0.699467\pi$
$769$		−	44.9240i		−	1.62000i	0.586429	−	0.810001i	$-0.300533\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−28.3672	+	28.3672i	−1.02030	+	1.02030i	−0.0205060	+	0.999790i	$0.506528\pi$
$773$	−28.3672	+	28.3672i	−1.02030	+	1.02030i	−0.999790	+	0.0205060i	$0.993472\pi$
$774$		0			0
$775$	32.0486	+	0.324380i	1.15122	+	0.0116521i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			23.6635i			0.847832i
$780$		0			0
$781$			4.52436i			0.161894i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−17.1339	+	16.9614i	−0.611536	+	0.605377i
$786$		0			0
$787$	−4.79302	+	4.79302i	−0.170853	+	0.170853i	−0.787354	−	0.616501i	$-0.788550\pi$
$787$	−4.79302	+	4.79302i	−0.170853	+	0.170853i	0.616501	+	0.787354i	$0.288550\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	2.31610i		−	0.0823509i
$792$		0			0
$793$	26.2729	−	26.2729i	0.932977	−	0.932977i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	7.30136	+	7.30136i	0.258627	+	0.258627i	0.824496	−	0.565868i	$-0.191459\pi$
$797$	7.30136	+	7.30136i	0.258627	+	0.258627i	−0.565868	+	0.824496i	$0.691459\pi$
$798$		0			0
$799$		−	8.52480i		−	0.301586i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1.69516	+	1.69516i	−0.0598208	+	0.0598208i
$804$		0			0
$805$	0.0662907	−	13.0993i	0.00233644	−	0.461689i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	26.9078			0.946026			0.473013	−	0.881055i	$-0.343166\pi$
$809$	26.9078			0.946026			0.473013	+	0.881055i	$0.343166\pi$
$810$		0			0
$811$		−	4.38227i		−	0.153882i	−0.997036	−	0.0769412i	$-0.975485\pi$
$811$		−	4.38227i		−	0.153882i	0.997036	−	0.0769412i	$-0.0245154\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−36.1495	−	36.5173i	−1.26626	−	1.27914i
$816$		0			0
$817$	19.8686	+	19.8686i	0.695113	+	0.695113i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−34.5890			−1.20716			−0.603582	−	0.797301i	$-0.706261\pi$
$821$	−34.5890			−1.20716			−0.603582	+	0.797301i	$0.706261\pi$
$822$		0			0
$823$	5.63496	−	5.63496i	0.196422	−	0.196422i	−0.602042	−	0.798464i	$-0.705646\pi$
$823$	5.63496	−	5.63496i	0.196422	−	0.196422i	0.798464	+	0.602042i	$0.205646\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−29.8906	−	29.8906i	−1.03940	−	1.03940i	−0.999191	−	0.0402073i	$-0.987198\pi$
$827$	−29.8906	−	29.8906i	−1.03940	−	1.03940i	−0.0402073	−	0.999191i	$-0.512802\pi$
$828$		0			0
$829$	45.0207			1.56363			0.781817	−	0.623508i	$-0.214293\pi$
$829$	45.0207			1.56363			0.781817	+	0.623508i	$0.214293\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−16.6025	−	16.6025i	−0.575241	−	0.575241i
$834$		0			0
$835$	30.8039	−	30.4937i	1.06601	−	1.05528i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−6.14853			−0.212271			−0.106135	−	0.994352i	$-0.533848\pi$
$839$	−6.14853			−0.212271			−0.106135	+	0.994352i	$0.533848\pi$
$840$		0			0
$841$	23.1258			0.797442
$842$		0			0
$843$		0			0
$844$		0			0
$845$	0.130772	−	25.8411i	0.00449870	−	0.888960i
$846$		0			0
$847$	5.13842	+	5.13842i	0.176558	+	0.176558i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−27.9149			−0.956910
$852$		0			0
$853$	11.5962	+	11.5962i	0.397046	+	0.397046i	0.877190	−	0.480144i	$-0.159415\pi$
$853$	11.5962	+	11.5962i	0.397046	+	0.397046i	−0.480144	+	0.877190i	$0.659415\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−23.8521	+	23.8521i	−0.814772	+	0.814772i	−0.985345	−	0.170573i	$-0.945438\pi$
$857$	−23.8521	+	23.8521i	−0.814772	+	0.814772i	0.170573	+	0.985345i	$0.445438\pi$
$858$		0			0
$859$	−31.3817			−1.07073			−0.535365	−	0.844621i	$-0.679826\pi$
$859$	−31.3817			−1.07073			−0.535365	+	0.844621i	$0.679826\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	12.4212	+	12.4212i	0.422823	+	0.422823i	0.886175	−	0.463351i	$-0.153353\pi$
$863$	12.4212	+	12.4212i	0.422823	+	0.422823i	−0.463351	+	0.886175i	$0.653353\pi$
$864$		0			0
$865$	−7.54388	+	7.46791i	−0.256500	+	0.253917i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	2.97220i		−	0.100825i
$870$		0			0
$871$	38.9312			1.31913
$872$		0			0
$873$		0			0
$874$		0			0
$875$	5.19599	+	5.35620i	0.175656	+	0.181073i
$876$		0			0
$877$	−38.0074	+	38.0074i	−1.28342	+	1.28342i	−0.344708	+	0.938710i	$0.612022\pi$
$877$	−38.0074	+	38.0074i	−1.28342	+	1.28342i	−0.938710	+	0.344708i	$0.887978\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	10.9118i		−	0.367626i	−0.982961	−	0.183813i	$-0.941156\pi$
$881$		−	10.9118i		−	0.367626i	0.982961	−	0.183813i	$-0.0588442\pi$
$882$		0			0
$883$	1.85770	+	1.85770i	0.0625165	+	0.0625165i	0.737674	−	0.675157i	$-0.235924\pi$
$883$	1.85770	+	1.85770i	0.0625165	+	0.0625165i	−0.675157	+	0.737674i	$0.735924\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−3.17163	+	3.17163i	−0.106493	+	0.106493i	−0.758346	−	0.651853i	$-0.773992\pi$
$887$	−3.17163	+	3.17163i	−0.106493	+	0.106493i	0.651853	+	0.758346i	$0.273992\pi$
$888$		0			0
$889$			5.66665i			0.190053i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−6.85679	+	6.85679i	−0.229454	+	0.229454i
$894$		0			0
$895$	0.121687	−	24.0458i	0.00406754	−	0.803762i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			15.5358i			0.518150i
$900$		0			0
$901$			15.4737i			0.515504i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−0.114617	+	22.6488i	−0.00381001	+	0.752873i
$906$		0			0
$907$	−5.88501	+	5.88501i	−0.195408	+	0.195408i	−0.798028	−	0.602620i	$-0.794124\pi$
$907$	−5.88501	+	5.88501i	−0.195408	+	0.195408i	0.602620	+	0.798028i	$0.294124\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			10.3886i			0.344188i	0.985080	+	0.172094i	$0.0550534\pi$
$911$			10.3886i			0.344188i	−0.985080	+	0.172094i	$0.944947\pi$
$912$		0			0
$913$	1.60613	−	1.60613i	0.0531550	−	0.0531550i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	5.34947	+	5.34947i	0.176655	+	0.176655i
$918$		0			0
$919$		−	14.5186i		−	0.478924i	−0.970906	−	0.239462i	$-0.923029\pi$
$919$		−	14.5186i		−	0.478924i	0.970906	−	0.239462i	$-0.0769710\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	47.2321	−	47.2321i	1.55466	−	1.55466i
$924$		0			0
$925$	11.3579	−	11.1302i	0.373444	−	0.365960i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	44.3689			1.45570			0.727848	−	0.685738i	$-0.240520\pi$
$929$	44.3689			1.45570			0.727848	+	0.685738i	$0.240520\pi$
$930$		0			0
$931$			26.7079i			0.875316i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	1.91070	−	1.89146i	0.0624867	−	0.0618574i
$936$		0			0
$937$	−2.66665	−	2.66665i	−0.0871157	−	0.0871157i	0.662206	−	0.749322i	$-0.269620\pi$
$937$	−2.66665	−	2.66665i	−0.0871157	−	0.0871157i	−0.749322	+	0.662206i	$0.769620\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−41.4675			−1.35180			−0.675901	−	0.736992i	$-0.736245\pi$
$941$	−41.4675			−1.35180			−0.675901	+	0.736992i	$0.736245\pi$
$942$		0			0
$943$	−36.0421	+	36.0421i	−1.17369	+	1.17369i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	34.3858	+	34.3858i	1.11739	+	1.11739i	0.992123	+	0.125264i	$0.0399779\pi$
$947$	34.3858	+	34.3858i	1.11739	+	1.11739i	0.125264	+	0.992123i	$0.460022\pi$
$948$		0			0
$949$	35.3933			1.14891
$950$		0			0
$951$		0			0
$952$		0			0
$953$	5.82947	+	5.82947i	0.188835	+	0.188835i	0.795192	−	0.606357i	$-0.207370\pi$
$953$	5.82947	+	5.82947i	0.188835	+	0.188835i	−0.606357	+	0.795192i	$0.707370\pi$
$954$		0			0
$955$	0.0489061	−	9.66403i	0.00158256	−	0.312721i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−10.2577			−0.331237
$960$		0			0
$961$	10.0886			0.325439
$962$		0			0
$963$		0			0
$964$		0			0
$965$	17.9208	−	17.7404i	0.576892	−	0.571083i
$966$		0			0
$967$	−18.1550	−	18.1550i	−0.583826	−	0.583826i	0.352126	−	0.935952i	$-0.385459\pi$
$967$	−18.1550	−	18.1550i	−0.583826	−	0.583826i	−0.935952	+	0.352126i	$0.885459\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−6.87264			−0.220554			−0.110277	−	0.993901i	$-0.535174\pi$
$971$	−6.87264			−0.220554			−0.110277	+	0.993901i	$0.535174\pi$
$972$		0			0
$973$	6.29749	+	6.29749i	0.201888	+	0.201888i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	14.8533	−	14.8533i	0.475200	−	0.475200i	−0.428393	−	0.903593i	$-0.640920\pi$
$977$	14.8533	−	14.8533i	0.475200	−	0.475200i	0.903593	+	0.428393i	$0.140920\pi$
$978$		0			0
$979$	2.79734			0.0894033
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−21.3770	−	21.3770i	−0.681822	−	0.681822i	0.278589	−	0.960410i	$-0.410133\pi$
$983$	−21.3770	−	21.3770i	−0.681822	−	0.681822i	−0.960410	+	0.278589i	$0.910133\pi$
$984$		0			0
$985$	9.57090	+	9.66827i	0.304954	+	0.308057i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			60.5240i			1.92455i
$990$		0			0
$991$	−27.1515			−0.862497			−0.431249	−	0.902233i	$-0.641927\pi$
$991$	−27.1515			−0.862497			−0.431249	+	0.902233i	$0.641927\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−0.0341864	+	6.75537i	−0.00108378	+	0.214160i
$996$		0			0
$997$	−16.8850	+	16.8850i	−0.534754	+	0.534754i	−0.921983	−	0.387230i	$-0.873432\pi$
$997$	−16.8850	+	16.8850i	−0.534754	+	0.534754i	0.387230	+	0.921983i	$0.373432\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1440.2.bj.a.17.1		48
3.2	odd	2	inner	1440.2.bj.a.17.24		48
4.3	odd	2		360.2.x.a.197.8	yes	48
5.3	odd	4	inner	1440.2.bj.a.593.2		48
8.3	odd	2		360.2.x.a.197.20	yes	48
8.5	even	2	inner	1440.2.bj.a.17.23		48
12.11	even	2		360.2.x.a.197.17	yes	48
15.8	even	4	inner	1440.2.bj.a.593.23		48
20.3	even	4		360.2.x.a.53.5	✓	48
24.5	odd	2	inner	1440.2.bj.a.17.2		48
24.11	even	2		360.2.x.a.197.5	yes	48
40.3	even	4		360.2.x.a.53.17	yes	48
40.13	odd	4	inner	1440.2.bj.a.593.24		48
60.23	odd	4		360.2.x.a.53.20	yes	48
120.53	even	4	inner	1440.2.bj.a.593.1		48
120.83	odd	4		360.2.x.a.53.8	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.x.a.53.5	✓	48	20.3	even	4
360.2.x.a.53.8	yes	48	120.83	odd	4
360.2.x.a.53.17	yes	48	40.3	even	4
360.2.x.a.53.20	yes	48	60.23	odd	4
360.2.x.a.197.5	yes	48	24.11	even	2
360.2.x.a.197.8	yes	48	4.3	odd	2
360.2.x.a.197.17	yes	48	12.11	even	2
360.2.x.a.197.20	yes	48	8.3	odd	2
1440.2.bj.a.17.1		48	1.1	even	1	trivial
1440.2.bj.a.17.2		48	24.5	odd	2	inner
1440.2.bj.a.17.23		48	8.5	even	2	inner
1440.2.bj.a.17.24		48	3.2	odd	2	inner
1440.2.bj.a.593.1		48	120.53	even	4	inner
1440.2.bj.a.593.2		48	5.3	odd	4	inner
1440.2.bj.a.593.23		48	15.8	even	4	inner
1440.2.bj.a.593.24		48	40.13	odd	4	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1440.bj (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.23604 - 0.0113158i) q^{5} +(-0.471963 - 0.471963i) q^{7} +O(q^{10})\)	\(q+(-2.23604 - 0.0113158i) q^{5} +(-0.471963 - 0.471963i) q^{7} -0.335652 q^{11} +(3.50404 + 3.50404i) q^{13} +(2.53299 - 2.53299i) q^{17} -4.07474 q^{19} +(-6.20627 - 6.20627i) q^{23} +(4.99974 + 0.0506050i) q^{25} +2.42367i q^{29} +6.41004 q^{31} +(1.04999 + 1.06067i) q^{35} +(2.24893 - 2.24893i) q^{37} -5.80736i q^{41} +(-4.87603 - 4.87603i) q^{43} +(1.68276 - 1.68276i) q^{47} -6.55450i q^{49} +(-3.05444 + 3.05444i) q^{53} +(0.750530 + 0.00379815i) q^{55} -12.2950i q^{59} -7.49787i q^{61} +(-7.79553 - 7.87483i) q^{65} +(5.55519 - 5.55519i) q^{67} -13.4793i q^{71} +(5.05035 - 5.05035i) q^{73} +(0.158415 + 0.158415i) q^{77} +8.85503i q^{79} +(-4.78510 + 4.78510i) q^{83} +(-5.69252 + 5.63520i) q^{85} -8.33405 q^{89} -3.30755i q^{91} +(9.11128 + 0.0461088i) q^{95} +(10.1367 + 10.1367i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q+O(q^{10}) \) 48 * q	\( 48 q - 32 q^{31} - 32 q^{97}+O(q^{100}) \) 48 * q - 32 * q^31 - 32 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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