LMFDB - Embedded newform 285.2.b.b.284.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [285,2,Mod(284,285)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(285, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("285.284");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 285 = 3 \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	285.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:	$2.27573645761$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 13x^{8} + 256 $ x^16 + 13*x^8 + 256
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2}\cdot 3^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			284.16
Root			$-0.348022 - 1.37072i$ of defining polynomial
Character	$\chi$	$=$	285.284
Dual form			285.2.b.b.284.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.74145i q^{2} +(1.14551 + 1.29916i) q^{3} -5.51552 q^{4} +2.23607i q^{5} +(-3.56158 + 3.14034i) q^{6} -9.63761i q^{8} +(-0.375634 + 2.97639i) q^{9} +O(q^{10})$	\(q+2.74145i q^{2} +(1.14551 + 1.29916i) q^{3} -5.51552 q^{4} +2.23607i q^{5} +(-3.56158 + 3.14034i) q^{6} -9.63761i q^{8} +(-0.375634 + 2.97639i) q^{9} -6.13006 q^{10} -2.13645i q^{11} +(-6.31806 - 7.16555i) q^{12} +7.20211 q^{13} +(-2.90501 + 2.56143i) q^{15} +15.3899 q^{16} +(-8.15961 - 1.02978i) q^{18} -4.35890 q^{19} -12.3331i q^{20} +5.85695 q^{22} +(12.5208 - 11.0399i) q^{24} -5.00000 q^{25} +19.7442i q^{26} +(-4.29710 + 2.92146i) q^{27} +(-7.02202 - 7.96393i) q^{30} +22.9155i q^{32} +(2.77559 - 2.44731i) q^{33} +(2.07182 - 16.4163i) q^{36} +5.38708 q^{37} -11.9497i q^{38} +(8.25006 + 9.35669i) q^{39} +21.5504 q^{40} +11.7836i q^{44} +(-6.65541 - 0.839944i) q^{45} +(17.6293 + 19.9940i) q^{48} +7.00000 q^{49} -13.7072i q^{50} -39.7234 q^{52} +0.514484i q^{53} +(-8.00903 - 11.7803i) q^{54} +4.77724 q^{55} +(-4.99314 - 5.66291i) q^{57} +(16.0227 - 14.1276i) q^{60} -11.8083 q^{61} -32.0416 q^{64} +16.1044i q^{65} +(6.70917 + 7.60912i) q^{66} +3.02584 q^{67} +(28.6853 + 3.62022i) q^{72} +14.7684i q^{74} +(-5.72753 - 6.49580i) q^{75} +24.0416 q^{76} +(-25.6509 + 22.6171i) q^{78} +34.4130i q^{80} +(-8.71780 - 2.23607i) q^{81} -20.5902 q^{88} +(2.30266 - 18.2454i) q^{90} -9.74679i q^{95} +(-29.7709 + 26.2498i) q^{96} +19.1331 q^{97} +19.1901i q^{98} +(6.35890 + 0.802522i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 32 q^{4}+O(q^{10}) $ 16 * q - 32 * q^4	$ 16 q - 32 q^{4} + 64 q^{16} + 8 q^{24} - 80 q^{25} - 40 q^{30} + 56 q^{36} + 112 q^{49} - 88 q^{54} - 128 q^{64} + 104 q^{66} - 152 q^{96} + 32 q^{99}+O(q^{100}) $ 16 * q - 32 * q^4 + 64 * q^16 + 8 * q^24 - 80 * q^25 - 40 * q^30 + 56 * q^36 + 112 * q^49 - 88 * q^54 - 128 * q^64 + 104 * q^66 - 152 * q^96 + 32 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/285\mathbb{Z}\right)^\times$.

$n$	$172$	$191$	$211$
$\chi(n)$	$-1$	$-1$	$-1$

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 $

$2$			2.74145i			1.93849i	0.246089	+	0.969247i	$0.420855\pi$
$2$			2.74145i			1.93849i	−0.246089	+	0.969247i	$0.579145\pi$
$3$	1.14551	+	1.29916i	0.661358	+	0.750070i
$3$	1.14551	+	1.29916i	0.661358	+	0.750070i
$4$	−5.51552			−2.75776
$5$			2.23607i			1.00000i
$5$			2.23607i			1.00000i
$6$	−3.56158	+	3.14034i	−1.45401	+	1.28204i
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		−	9.63761i		−	3.40741i
$9$	−0.375634	+	2.97639i	−0.125211	+	0.992130i
$10$	−6.13006			−1.93849
$11$		−	2.13645i		−	0.644163i	−0.946712	−	0.322081i	$-0.895618\pi$
$11$		−	2.13645i		−	0.644163i	0.946712	−	0.322081i	$-0.104382\pi$
$12$	−6.31806	−	7.16555i	−1.82387	−	2.06852i
$13$	7.20211			1.99751			0.998753	−	0.0499278i	$-0.0158991\pi$
$13$	7.20211			1.99751			0.998753	+	0.0499278i	$0.0158991\pi$
$14$		0			0
$15$	−2.90501	+	2.56143i	−0.750070	+	0.661358i
$16$	15.3899			3.84749
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$	−8.15961	−	1.02978i	−1.92324	−	0.242722i
$19$	−4.35890			−1.00000
$19$	−4.35890			−1.00000
$20$		−	12.3331i		−	2.75776i
$21$		0			0
$22$	5.85695			1.24871
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$	12.5208	−	11.0399i	2.55580	−	2.25352i
$25$	−5.00000			−1.00000
$26$			19.7442i			3.87215i
$27$	−4.29710	+	2.92146i	−0.826977	+	0.562236i
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$	−7.02202	−	7.96393i	−1.28204	−	1.45401i
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$			22.9155i			4.05092i
$33$	2.77559	−	2.44731i	0.483168	−	0.426022i
$34$		0			0
$35$		0			0
$36$	2.07182	−	16.4163i	0.345303	−	2.73606i
$37$	5.38708			0.885631			0.442816	−	0.896613i	$-0.353980\pi$
$37$	5.38708			0.885631			0.442816	+	0.896613i	$0.353980\pi$
$38$		−	11.9497i		−	1.93849i
$39$	8.25006	+	9.35669i	1.32107	+	1.49827i
$40$	21.5504			3.40741
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$			11.7836i			1.77645i
$45$	−6.65541	−	0.839944i	−0.992130	−	0.125211i
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$	17.6293	+	19.9940i	2.54457	+	2.88589i
$49$	7.00000			1.00000
$50$		−	13.7072i		−	1.93849i
$51$		0			0
$52$	−39.7234			−5.50864
$53$			0.514484i			0.0706698i	0.999376	+	0.0353349i	$0.0112498\pi$
$53$			0.514484i			0.0706698i	−0.999376	+	0.0353349i	$0.988750\pi$
$54$	−8.00903	−	11.7803i	−1.08989	−	1.60309i
$55$	4.77724			0.644163
$56$		0			0
$57$	−4.99314	−	5.66291i	−0.661358	−	0.750070i
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$	16.0227	−	14.1276i	2.06852	−	1.82387i
$61$	−11.8083			−1.51190			−0.755948	−	0.654632i	$-0.772824\pi$
$61$	−11.8083			−1.51190			−0.755948	+	0.654632i	$0.772824\pi$
$62$		0			0
$63$		0			0
$64$	−32.0416			−4.00520
$65$			16.1044i			1.99751i
$66$	6.70917	+	7.60912i	0.825842	+	0.936618i
$67$	3.02584			0.369666			0.184833	−	0.982770i	$-0.440826\pi$
$67$	3.02584			0.369666			0.184833	+	0.982770i	$0.440826\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$	28.6853	+	3.62022i	3.38059	+	0.426647i
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$			14.7684i			1.71679i
$75$	−5.72753	−	6.49580i	−0.661358	−	0.750070i
$76$	24.0416			2.75776
$77$		0			0
$78$	−25.6509	+	22.6171i	−2.90439	+	2.56088i
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$			34.4130i			3.84749i
$81$	−8.71780	−	2.23607i	−0.968644	−	0.248452i
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$	−20.5902			−2.19493
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$	2.30266	−	18.2454i	0.242722	−	1.92324i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	9.74679i		−	1.00000i
$96$	−29.7709	+	26.2498i	−3.03848	+	2.67911i
$97$	19.1331			1.94268			0.971338	−	0.237701i	$-0.0763937\pi$
$97$	19.1331			1.94268			0.971338	+	0.237701i	$0.0763937\pi$
$98$			19.1901i			1.93849i
$99$	6.35890	+	0.802522i	0.639093	+	0.0806565i
$100$	27.5776			2.75776
$101$		−	13.5854i		−	1.35180i	−0.736992	−	0.675901i	$-0.763755\pi$
$101$		−	13.5854i		−	1.35180i	0.736992	−	0.675901i	$-0.236245\pi$
$102$		0			0
$103$	11.3784			1.12114			0.560572	−	0.828106i	$-0.310581\pi$
$103$	11.3784			1.12114			0.560572	+	0.828106i	$0.310581\pi$
$104$		−	69.4111i		−	6.80632i
$105$		0			0
$106$	−1.41043			−0.136993
$107$		−	4.40279i		−	0.425634i	−0.977092	−	0.212817i	$-0.931736\pi$
$107$		−	4.40279i		−	0.425634i	0.977092	−	0.212817i	$-0.0682637\pi$
$108$	23.7007	−	16.1134i	2.28061	−	1.55051i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$			13.0965i			1.24871i
$111$	6.17093	+	6.99868i	0.585719	+	0.664286i
$112$		0			0
$113$			9.88568i			0.929966i	0.885319	+	0.464983i	$0.153940\pi$
$113$			9.88568i			0.929966i	−0.885319	+	0.464983i	$0.846060\pi$
$114$	15.5246	−	13.6884i	1.45401	−	1.28204i
$115$		0			0
$116$		0			0
$117$	−2.70536	+	21.4363i	−0.250110	+	1.98179i
$118$		0			0
$119$		0			0
$120$	24.6861	+	27.9974i	2.25352	+	2.55580i
$121$	6.43560			0.585054
$122$		−	32.3718i		−	2.93080i
$123$		0			0
$124$		0			0
$125$		−	11.1803i		−	1.00000i
$126$		0			0
$127$	−17.6472			−1.56594			−0.782968	−	0.622062i	$-0.786295\pi$
$127$	−17.6472			−1.56594			−0.782968	+	0.622062i	$0.786295\pi$
$128$		−	42.0094i		−	3.71314i
$129$		0			0
$130$	−44.1493			−3.87215
$131$		−	19.4936i		−	1.70316i	−0.524222	−	0.851581i	$-0.675644\pi$
$131$		−	19.4936i		−	1.70316i	0.524222	−	0.851581i	$-0.324356\pi$
$132$	−15.3088	+	13.4982i	−1.33246	+	1.17487i
$133$		0			0
$134$			8.29519i			0.716595i
$135$	−6.53259	−	9.60860i	−0.562236	−	0.826977i
$136$		0			0
$137$		0			0			−	1.00000i	$-0.5\pi$
$137$		0			0				1.00000i	$0.5\pi$
$138$		0			0
$139$	9.28485			0.787531			0.393765	−	0.919211i	$-0.371172\pi$
$139$	9.28485			0.787531			0.393765	+	0.919211i	$0.371172\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	15.3869i		−	1.28672i
$144$	−5.78099	+	45.8065i	−0.481749	+	3.81721i
$145$		0			0
$146$		0			0
$147$	8.01854	+	9.09412i	0.661358	+	0.750070i
$148$	−29.7126			−2.44236
$149$			22.1312i			1.81306i	0.422140	+	0.906531i	$0.361279\pi$
$149$			22.1312i			1.81306i	−0.422140	+	0.906531i	$0.638721\pi$
$150$	17.8079	−	15.7017i	1.45401	−	1.28204i
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$			42.0094i			3.40741i
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−45.5034	−	51.6071i	−3.64319	−	4.13187i
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		0			0
$159$	−0.668397	+	0.589344i	−0.0530073	+	0.0467380i
$160$	−51.2405			−4.05092
$161$		0			0
$162$	6.13006	−	23.8994i	0.481623	−	1.87771i
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		0			0
$165$	5.47235	+	6.20640i	0.426022	+	0.483168i
$166$		0			0
$167$		−	24.4139i		−	1.88920i	−0.328223	−	0.944600i	$-0.606450\pi$
$167$		−	24.4139i		−	1.88920i	0.328223	−	0.944600i	$-0.393550\pi$
$168$		0			0
$169$	38.8704			2.99003
$170$		0			0
$171$	1.63735	−	12.9738i	0.125211	−	0.992130i
$172$		0			0
$173$		−	20.8515i		−	1.58531i	−0.609672	−	0.792654i	$-0.708699\pi$
$173$		−	20.8515i		−	1.58531i	0.609672	−	0.792654i	$-0.291301\pi$
$174$		0			0
$175$		0			0
$176$		−	32.8798i		−	2.47841i
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	36.7081	+	4.63273i	2.73606	+	0.345303i
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$	−13.5265	−	15.3409i	−0.999904	−	1.13403i
$184$		0			0
$185$			12.0459i			0.885631i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$	26.7203			1.93849
$191$		−	8.94427i		−	0.647185i	−0.946197	−	0.323592i	$-0.895109\pi$
$191$		−	8.94427i		−	0.647185i	0.946197	−	0.323592i	$-0.104891\pi$
$192$	−36.7038	−	41.6272i	−2.64887	−	3.00418i
$193$	−8.35898			−0.601693			−0.300846	−	0.953673i	$-0.597269\pi$
$193$	−8.35898			−0.601693			−0.300846	+	0.953673i	$0.597269\pi$
$194$			52.4525i			3.76587i
$195$	−20.9222	+	18.4477i	−1.49827	+	1.32107i
$196$	−38.6087			−2.75776
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$	−2.20007	+	17.4326i	−0.156352	+	1.23888i
$199$	−8.71780			−0.617988			−0.308994	−	0.951064i	$-0.599992\pi$
$199$	−8.71780			−0.617988			−0.308994	+	0.951064i	$0.599992\pi$
$200$			48.1881i			3.40741i
$201$	3.46612	+	3.93106i	0.244481	+	0.277275i
$202$	37.2438			2.62046
$203$		0			0
$204$		0			0
$205$		0			0
$206$			31.1932i			2.17333i
$207$		0			0
$208$	110.840			7.68538
$209$			9.31255i			0.644163i
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		−	2.83765i		−	0.194890i
$213$		0			0
$214$	12.0700			0.825088
$215$		0			0
$216$	28.1559	+	41.4138i	1.91577	+	2.81785i
$217$		0			0
$218$		0			0
$219$		0			0
$220$	−26.3490			−1.77645
$221$		0			0
$222$	−19.1865	+	16.9173i	−1.28771	+	1.13541i
$223$	−3.90113			−0.261239			−0.130620	−	0.991433i	$-0.541697\pi$
$223$	−3.90113			−0.261239			−0.130620	+	0.991433i	$0.541697\pi$
$224$		0			0
$225$	1.87817	−	14.8820i	0.125211	−	0.992130i
$226$	−27.1010			−1.80273
$227$			26.3344i			1.74787i	0.486041	+	0.873936i	$0.338441\pi$
$227$			26.3344i			1.74787i	−0.486041	+	0.873936i	$0.661559\pi$
$228$	27.5398	+	31.2339i	1.82387	+	2.06852i
$229$	−16.3159			−1.07818			−0.539092	−	0.842247i	$-0.681233\pi$
$229$	−16.3159			−1.07818			−0.539092	+	0.842247i	$0.681233\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$	−58.7664	−	7.41659i	−3.84168	−	0.484838i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	19.4936i		−	1.26094i	−0.776215	−	0.630468i	$-0.782863\pi$
$239$		−	19.4936i		−	1.26094i	0.776215	−	0.630468i	$-0.217137\pi$
$240$	−44.7079	+	39.4202i	−2.88589	+	2.54457i
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$			17.6428i			1.13412i
$243$	−7.08128	−	13.8872i	−0.454264	−	0.890867i
$244$	65.1289			4.16945
$245$			15.6525i			1.00000i
$246$		0			0
$247$	−31.3933			−1.99751
$248$		0			0
$249$		0			0
$250$	30.6503			1.93849
$251$			17.8885i			1.12911i	0.825394	+	0.564557i	$0.190953\pi$
$251$			17.8885i			1.12911i	−0.825394	+	0.564557i	$0.809047\pi$
$252$		0			0
$253$		0			0
$254$		−	48.3788i		−	3.03556i
$255$		0			0
$256$	51.0832			3.19270
$257$		−	15.0754i		−	0.940380i	−0.882565	−	0.470190i	$-0.844185\pi$
$257$		−	15.0754i		−	0.940380i	0.882565	−	0.470190i	$-0.155815\pi$
$258$		0			0
$259$		0			0
$260$		−	88.8242i		−	5.50864i
$261$		0			0
$262$	53.4406			3.30157
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$	−23.5862	−	26.7500i	−1.45163	−	1.64635i
$265$	−1.15042			−0.0706698
$266$		0			0
$267$		0			0
$268$	−16.6891			−1.01945
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$	26.3415	−	17.9087i	1.60309	−	1.08989i
$271$	−32.9014			−1.99862			−0.999309	−	0.0371559i	$-0.988170\pi$
$271$	−32.9014			−1.99862			−0.999309	+	0.0371559i	$0.988170\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			10.6822i			0.644163i
$276$		0			0
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$			25.4539i			1.52662i
$279$		0			0
$280$		0			0
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$		0			0
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$		0			0
$285$	12.6626	−	11.1650i	0.750070	−	0.661358i
$286$	42.1824			2.49430
$287$		0			0
$288$	−68.2054	−	8.60783i	−4.01904	−	0.507221i
$289$	−17.0000			−1.00000
$290$		0			0
$291$	21.9171	+	24.8570i	1.28480	+	1.45714i
$292$		0			0
$293$			31.6943i			1.85160i	0.378011	+	0.925801i	$0.376608\pi$
$293$			31.6943i			1.85160i	−0.378011	+	0.925801i	$0.623392\pi$
$294$	−24.9310	+	21.9824i	−1.45401	+	1.28204i
$295$		0			0
$296$		−	51.9186i		−	3.01771i
$297$	6.24155	+	9.18052i	0.362171	+	0.532708i
$298$	−60.6716			−3.51461
$299$		0			0
$300$	31.5903	+	35.8277i	1.82387	+	2.06852i
$301$		0			0
$302$		0			0
$303$	17.6497	−	15.5622i	1.01395	−	0.894025i
$304$	−67.0832			−3.84749
$305$		−	26.4041i		−	1.51190i
$306$		0			0
$307$	−31.8343			−1.81688			−0.908439	−	0.418017i	$-0.862725\pi$
$307$	−31.8343			−1.81688			−0.908439	+	0.418017i	$0.862725\pi$
$308$		0			0
$309$	13.0340	+	14.7823i	0.741478	+	0.840937i
$310$		0			0
$311$			33.5802i			1.90416i	0.305848	+	0.952080i	$0.401060\pi$
$311$			33.5802i			1.90416i	−0.305848	+	0.952080i	$0.598940\pi$
$312$	90.1762	−	79.5108i	5.10522	−	4.50142i
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	23.0117i		−	1.29246i	−0.763141	−	0.646232i	$-0.776344\pi$
$317$		−	23.0117i		−	1.29246i	0.763141	−	0.646232i	$-0.223656\pi$
$318$	−1.61565	−	1.83237i	−0.0906014	−	0.102754i
$319$		0			0
$320$		−	71.6472i		−	4.00520i
$321$	5.71993	−	5.04342i	0.319255	−	0.281496i
$322$		0			0
$323$		0			0
$324$	48.0832	+	12.3331i	2.67129	+	0.685171i
$325$	−36.0105			−1.99751
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$	−17.0145	+	15.0022i	−0.936618	+	0.825842i
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		0			0
$333$	−2.02357	+	16.0341i	−0.110891	+	0.878661i
$334$	66.9292			3.66220
$335$			6.76599i			0.369666i
$336$		0			0
$337$	15.5546			0.847315			0.423658	−	0.905822i	$-0.360746\pi$
$337$	15.5546			0.847315			0.423658	+	0.905822i	$0.360746\pi$
$338$			106.561i			5.79615i
$339$	−12.8431	+	11.3241i	−0.697540	+	0.615041i
$340$		0			0
$341$		0			0
$342$	35.5669	+	4.48871i	1.92324	+	0.242722i
$343$		0			0
$344$		0			0
$345$		0			0
$346$	57.1631			3.07311
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$		0			0
$349$	−34.8712			−1.86661			−0.933306	−	0.359082i	$-0.883090\pi$
$349$	−34.8712			−1.86661			−0.933306	+	0.359082i	$0.883090\pi$
$350$		0			0
$351$	−30.9482	+	21.0407i	−1.65189	+	1.12307i
$352$	48.9577			2.60945
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	37.8531i		−	1.99781i	−0.0467657	−	0.998906i	$-0.514891\pi$
$359$		−	37.8531i		−	1.99781i	0.0467657	−	0.998906i	$-0.485109\pi$
$360$	−8.09505	+	64.1423i	−0.426647	+	3.38059i
$361$	19.0000			1.00000
$362$		0			0
$363$	7.37201	+	8.36087i	0.386930	+	0.438832i
$364$		0			0
$365$		0			0
$366$	42.0561	−	37.0820i	2.19831	−	1.93831i
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$		0			0
$369$		0			0
$370$	−33.0231			−1.71679
$371$		0			0
$372$		0			0
$373$	32.8792			1.70242			0.851211	−	0.524824i	$-0.175869\pi$
$373$	32.8792			1.70242			0.851211	+	0.524824i	$0.175869\pi$
$374$		0			0
$375$	14.5251	−	12.8071i	0.750070	−	0.661358i
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$			53.7587i			2.75776i
$381$	−20.2150	−	22.9265i	−1.03564	−	1.17456i
$382$	24.5202			1.25456
$383$		−	37.3001i		−	1.90595i	−0.303053	−	0.952974i	$-0.598006\pi$
$383$		−	37.3001i		−	1.90595i	0.303053	−	0.952974i	$-0.401994\pi$
$384$	54.5769	−	48.1220i	2.78512	−	2.45571i
$385$		0			0
$386$		−	22.9157i		−	1.16638i
$387$		0			0
$388$	−105.529			−5.35744
$389$			38.9872i			1.97673i	0.152106	+	0.988364i	$0.451394\pi$
$389$			38.9872i			1.97673i	−0.152106	+	0.988364i	$0.548606\pi$
$390$	−50.5733	−	57.3571i	−2.56088	−	2.90439i
$391$		0			0
$392$		−	67.4633i		−	3.40741i
$393$	25.3253	−	22.3300i	1.27749	−	1.12640i
$394$		0			0
$395$		0			0
$396$	−35.0727	−	4.42633i	−1.76247	−	0.222431i
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		−	23.8994i		−	1.19797i
$399$		0			0
$400$	−76.9497			−3.84749
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$	−10.7768	+	9.50218i	−0.537497	+	0.473926i
$403$		0			0
$404$			74.9308i			3.72795i
$405$	5.00000	−	19.4936i	0.248452	−	0.968644i
$406$		0			0
$407$		−	11.5092i		−	0.570491i
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$		0			0
$412$	−62.7577			−3.09185
$413$		0			0
$414$		0			0
$415$		0			0
$416$			165.040i			8.09174i
$417$	10.6358	+	12.0625i	0.520840	+	0.590704i
$418$	−25.5299			−1.24871
$419$		−	19.4936i		−	0.952324i	−0.879358	−	0.476162i	$-0.842028\pi$
$419$		−	19.4936i		−	0.952324i	0.879358	−	0.476162i	$-0.157972\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$	4.95839			0.240801
$425$		0			0
$426$		0			0
$427$		0			0
$428$			24.2837i			1.17380i
$429$	19.9901	−	17.6258i	0.965130	−	0.850982i
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$	−66.1321	+	44.9611i	−3.18178	+	2.16319i
$433$	−27.6580			−1.32916			−0.664580	−	0.747217i	$-0.731389\pi$
$433$	−27.6580			−1.32916			−0.664580	+	0.747217i	$0.731389\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		−	46.0412i		−	2.19493i
$441$	−2.62944	+	20.8347i	−0.125211	+	0.992130i
$442$		0			0
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$	−34.0359	−	38.6014i	−1.61527	−	1.83194i
$445$		0			0
$446$		−	10.6947i		−	0.506411i
$447$	−28.7520	+	25.3515i	−1.35992	+	1.19908i
$448$		0			0
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$	40.7981	+	5.14890i	1.92324	+	0.242722i
$451$		0			0
$452$		−	54.5247i		−	2.56463i
$453$		0			0
$454$	−72.1942			−3.38824
$455$		0			0
$456$	−54.5769	+	48.1220i	−2.55580	+	2.25352i
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$		−	44.7291i		−	2.09006i
$459$		0			0
$460$		0			0
$461$			38.9872i			1.81581i	0.419172	+	0.907907i	$0.362320\pi$
$461$			38.9872i			1.81581i	−0.419172	+	0.907907i	$0.637680\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$	14.9215	−	118.232i	0.689745	−	5.46529i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	21.7945			1.00000
$476$		0			0
$477$	−1.53130	−	0.193258i	−0.0701136	−	0.00884866i
$478$	53.4406			2.44432
$479$		−	25.0344i		−	1.14385i	−0.820305	−	0.571927i	$-0.806196\pi$
$479$		−	25.0344i		−	1.14385i	0.820305	−	0.571927i	$-0.193804\pi$
$480$	−58.6963	−	66.5697i	−2.67911	−	3.03848i
$481$	38.7984			1.76905
$482$		0			0
$483$		0			0
$484$	−35.4957			−1.61344
$485$			42.7830i			1.94268i
$486$	38.0711	−	19.4129i	1.72694	−	0.880588i
$487$	−40.1868			−1.82104			−0.910519	−	0.413467i	$-0.864318\pi$
$487$	−40.1868			−1.82104			−0.910519	+	0.413467i	$0.864318\pi$
$488$			113.804i			5.15165i
$489$		0			0
$490$	−42.9104			−1.93849
$491$		−	35.7771i		−	1.61460i	−0.590143	−	0.807299i	$-0.700929\pi$
$491$		−	35.7771i		−	1.61460i	0.590143	−	0.807299i	$-0.299071\pi$
$492$		0			0
$493$		0			0
$494$		−	86.0629i		−	3.87215i
$495$	−1.79449	+	14.2189i	−0.0806565	+	0.639093i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−28.3938			−1.27108			−0.635541	−	0.772067i	$-0.719223\pi$
$499$	−28.3938			−1.27108			−0.635541	+	0.772067i	$0.719223\pi$
$500$			61.6654i			2.75776i
$501$	31.7175	−	27.9662i	1.41703	−	1.24944i
$502$	−49.0405			−2.18878
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$	30.3780			1.35180
$506$		0			0
$507$	44.5262	+	50.4988i	1.97748	+	2.24273i
$508$	97.3335			4.31848
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$		0			0
$512$			56.0231i			2.47589i
$513$	18.7306	−	12.7344i	0.826977	−	0.562236i
$514$	41.3285			1.82292
$515$			25.4428i			1.12114i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	27.0894	−	23.8855i	1.18909	−	1.04846i
$520$	155.208			6.80632
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$		0			0
$523$	28.4214			1.24278			0.621390	−	0.783502i	$-0.286568\pi$
$523$	28.4214			1.24278			0.621390	+	0.783502i	$0.286568\pi$
$524$			107.517i			4.69692i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	42.7161	−	37.6640i	1.85898	−	1.63911i
$529$	−23.0000			−1.00000
$530$		−	3.15381i		−	0.136993i
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	9.84493			0.425634
$536$		−	29.1619i		−	1.25960i
$537$		0			0
$538$		0			0
$539$		−	14.9551i		−	0.644163i
$540$	36.0306	+	52.9965i	1.55051	+	2.28061i
$541$	−7.30067			−0.313881			−0.156940	−	0.987608i	$-0.550163\pi$
$541$	−7.30067			−0.313881			−0.156940	+	0.987608i	$0.550163\pi$
$542$		−	90.1974i		−	3.87431i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	42.1674			1.80295			0.901475	−	0.432832i	$-0.142486\pi$
$547$	42.1674			1.80295			0.901475	+	0.432832i	$0.142486\pi$
$548$		0			0
$549$	4.43560	−	35.1461i	0.189307	−	1.50000i
$550$	−29.2848			−1.24871
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−15.6495	+	13.7986i	−0.664286	+	0.585719i
$556$	−51.2108			−2.17182
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$			41.0327i			1.72932i	0.502354	+	0.864662i	$0.332467\pi$
$563$			41.0327i			1.72932i	−0.502354	+	0.864662i	$0.667533\pi$
$564$		0			0
$565$	−22.1050			−0.929966
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$	30.6083	+	34.7140i	1.28204	+	1.45401i
$571$	46.9635			1.96536			0.982681	−	0.185306i	$-0.0593276\pi$
$571$	46.9635			1.96536			0.982681	+	0.185306i	$0.0593276\pi$
$572$			84.8669i			3.54846i
$573$	11.6200	−	10.2457i	0.485434	−	0.428021i
$574$		0			0
$575$		0			0
$576$	12.0359	−	95.3683i	0.501497	−	3.97368i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$		−	46.6046i		−	1.93849i
$579$	−9.57526	−	10.8597i	−0.397934	−	0.451312i
$580$		0			0
$581$		0			0
$582$	−68.1442	+	60.0846i	−2.82467	+	2.49059i
$583$	1.09917			0.0455228
$584$		0			0
$585$	−47.9330	−	6.04937i	−1.98179	−	0.250110i
$586$	−86.8883			−3.58932
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$	−44.2264	−	50.1588i	−1.82387	−	2.06852i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	82.9069			3.40745
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$	−25.1679	+	17.1109i	−1.03265	+	0.702067i
$595$		0			0
$596$		−	122.065i		−	4.99999i
$597$	−9.98629	−	11.3258i	−0.408711	−	0.463535i
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$	−62.6040	+	55.1997i	−2.55580	+	2.25352i
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$	−1.13661	+	9.00609i	−0.0462864	+	0.366756i
$604$		0			0
$605$			14.3904i			0.585054i
$606$	42.6629	+	48.3856i	1.73306	+	1.96553i
$607$	−5.32669			−0.216203			−0.108102	−	0.994140i	$-0.534477\pi$
$607$	−5.32669			−0.216203			−0.108102	+	0.994140i	$0.534477\pi$
$608$		−	99.8862i		−	4.05092i
$609$		0			0
$610$	72.3855			2.93080
$611$		0			0
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		−	87.2719i		−	3.52201i
$615$		0			0
$616$		0			0
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$	−40.5249	+	35.7320i	−1.63015	+	1.43735i
$619$	0.269629			0.0108373			0.00541866	−	0.999985i	$-0.498275\pi$
$619$	0.269629			0.0108373			0.00541866	+	0.999985i	$0.498275\pi$
$620$		0			0
$621$		0			0
$622$	−92.0584			−3.69120
$623$		0			0
$624$	126.968	+	143.999i	5.08278	+	5.76457i
$625$	25.0000			1.00000
$626$		0			0
$627$	−12.0985	+	10.6676i	−0.483168	+	0.426022i
$628$		0			0
$629$		0			0
$630$		0			0
$631$	42.4559			1.69014			0.845071	−	0.534654i	$-0.179558\pi$
$631$	42.4559			1.69014			0.845071	+	0.534654i	$0.179558\pi$
$632$		0			0
$633$		0			0
$634$	63.0852			2.50543
$635$		−	39.4603i		−	1.56594i
$636$	3.68656	−	3.25054i	0.146181	−	0.128892i
$637$	50.4148			1.99751
$638$		0			0
$639$		0			0
$640$	93.9358			3.71314
$641$		0			0			−	1.00000i	$-0.5\pi$
$641$		0			0				1.00000i	$0.5\pi$
$642$	13.8263	+	15.6809i	0.545679	+	0.618874i
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$	−21.5504	+	84.0188i	−0.846578	+	3.30057i
$649$		0			0
$650$		−	98.7209i		−	3.87215i
$651$		0			0
$652$		0			0
$653$		0			0			−	1.00000i	$-0.5\pi$
$653$		0			0				1.00000i	$0.5\pi$
$654$		0			0
$655$	43.5890			1.70316
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$	−30.1829	−	34.2315i	−1.17487	−	1.33246i
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−43.9565	−	5.54751i	−1.70328	−	0.214962i
$667$		0			0
$668$			134.655i			5.20996i
$669$	−4.46877	−	5.06820i	−0.172773	−	0.195948i
$670$	−18.5486			−0.716595
$671$			25.2278i			0.973907i
$672$		0			0
$673$	42.0622			1.62138			0.810690	−	0.585476i	$-0.199092\pi$
$673$	42.0622			1.62138			0.810690	+	0.585476i	$0.199092\pi$
$674$			42.6422i			1.64252i
$675$	21.4855	−	14.6073i	0.826977	−	0.562236i
$676$	−214.390			−8.24579
$677$		−	30.6654i		−	1.17857i	−0.807927	−	0.589283i	$-0.799410\pi$
$677$		−	30.6654i		−	1.17857i	0.807927	−	0.589283i	$-0.200590\pi$
$678$	−31.0444	−	35.2086i	−1.19225	−	1.35218i
$679$		0			0
$680$		0			0
$681$	−34.2125	+	30.1661i	−1.31103	+	1.15597i
$682$		0			0
$683$			9.85290i			0.377011i	0.982072	+	0.188505i	$0.0603643\pi$
$683$			9.85290i			0.377011i	−0.982072	+	0.188505i	$0.939636\pi$
$684$	−9.03085	+	71.5572i	−0.345303	+	2.73606i
$685$		0			0
$686$		0			0
$687$	−18.6899	−	21.1970i	−0.713066	−	0.808715i
$688$		0			0
$689$			3.70537i			0.141163i
$690$		0			0
$691$	−37.4090			−1.42311			−0.711553	−	0.702632i	$-0.752008\pi$
$691$	−37.4090			−1.42311			−0.711553	+	0.702632i	$0.752008\pi$
$692$			115.007i			4.37190i
$693$		0			0
$694$		0			0
$695$			20.7616i			0.787531i
$696$		0			0
$697$		0			0
$698$		−	95.5975i		−	3.61842i
$699$		0			0
$700$		0			0
$701$		−	49.3021i		−	1.86212i	−0.364871	−	0.931058i	$-0.618887\pi$
$701$		−	49.3021i		−	1.86212i	0.364871	−	0.931058i	$-0.381113\pi$
$702$	−57.6819	−	84.8427i	−2.17706	−	3.20218i
$703$	−23.4818			−0.885631
$704$			68.4552i			2.58000i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−34.8712			−1.30962			−0.654808	−	0.755796i	$-0.727250\pi$
$709$	−34.8712			−1.30962			−0.654808	+	0.755796i	$0.727250\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$	34.4062			1.28672
$716$		0			0
$717$	25.3253	−	22.3300i	0.945790	−	0.833930i
$718$	103.772			3.87275
$719$			46.3989i			1.73039i	0.501438	+	0.865194i	$0.332805\pi$
$719$			46.3989i			1.73039i	−0.501438	+	0.865194i	$0.667195\pi$
$720$	−102.426	−	12.9267i	−3.81721	−	0.481749i
$721$		0			0
$722$			52.0875i			1.93849i
$723$		0			0
$724$		0			0
$725$		0			0
$726$	−22.9209	+	20.2100i	−0.850673	+	0.750062i
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	9.93011	−	25.1076i	0.367782	−	0.929912i
$730$		0			0
$731$		0			0
$732$	74.6055	+	84.6128i	2.75750	+	3.12738i
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		0			0
$735$	−20.3351	+	17.9300i	−0.750070	+	0.661358i
$736$		0			0
$737$		−	6.46455i		−	0.238125i
$738$		0			0
$739$	−8.71780			−0.320689			−0.160345	−	0.987061i	$-0.551261\pi$
$739$	−8.71780			−0.320689			−0.160345	+	0.987061i	$0.551261\pi$
$740$		−	66.4393i		−	2.44236i
$741$	−35.9612	−	40.7849i	−1.32107	−	1.49827i
$742$		0			0
$743$			22.3559i			0.820159i	0.912050	+	0.410079i	$0.134499\pi$
$743$			22.3559i			0.820159i	−0.912050	+	0.410079i	$0.865501\pi$
$744$		0			0
$745$	−49.4869			−1.81306
$746$			90.1366i			3.30013i
$747$		0			0
$748$		0			0
$749$		0			0
$750$	35.1101	+	39.8196i	1.28204	+	1.45401i
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		0			0
$753$	−23.2401	+	20.4914i	−0.846916	+	0.746749i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$	−93.9358			−3.40741
$761$			45.0292i			1.63231i	0.577834	+	0.816154i	$0.303898\pi$
$761$			45.0292i			1.63231i	−0.577834	+	0.816154i	$0.696102\pi$
$762$	62.8519	−	55.4182i	2.27688	−	2.00759i
$763$		0			0
$764$			49.3323i			1.78478i
$765$		0			0
$766$	102.256			3.69467
$767$		0			0
$768$	58.5161	+	66.3653i	2.11152	+	2.39475i
$769$	−53.9946			−1.94709			−0.973547	−	0.228488i	$-0.926622\pi$
$769$	−53.9946			−1.94709			−0.973547	+	0.228488i	$0.926622\pi$
$770$		0			0
$771$	19.5854	−	17.2690i	0.705351	−	0.621928i
$772$	46.1042			1.65933
$773$			44.9432i			1.61650i	0.588843	+	0.808248i	$0.299584\pi$
$773$			44.9432i			1.61650i	−0.588843	+	0.808248i	$0.700416\pi$
$774$		0			0
$775$		0			0
$776$		−	184.398i		−	6.61950i
$777$		0			0
$778$	−106.881			−3.83188
$779$		0			0
$780$	115.397	−	101.749i	4.13187	−	3.64319i
$781$		0			0
$782$		0			0
$783$		0			0
$784$	107.730			3.84749
$785$		0			0
$786$	61.2165	+	69.4279i	2.18352	+	2.47641i
$787$	55.9135			1.99310			0.996550	−	0.0829888i	$-0.0264466\pi$
$787$	55.9135			1.99310			0.996550	+	0.0829888i	$0.0264466\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	7.73440	−	61.2846i	0.274830	−	2.17765i
$793$	−85.0446			−3.02002
$794$		0			0
$795$	−1.31781	−	1.49458i	−0.0467380	−	0.0530073i
$796$	48.0832			1.70426
$797$		−	33.7523i		−	1.19557i	−0.801658	−	0.597783i	$-0.796048\pi$
$797$		−	33.7523i		−	1.19557i	0.801658	−	0.597783i	$-0.203952\pi$
$798$		0			0
$799$		0			0
$800$		−	114.577i		−	4.05092i
$801$		0			0
$802$		0			0
$803$		0			0
$804$	−19.1175	−	21.6818i	−0.674221	−	0.764659i
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−130.931			−4.60615
$809$		−	22.3607i		−	0.786160i	−0.919504	−	0.393080i	$-0.871410\pi$
$809$		−	22.3607i		−	0.786160i	0.919504	−	0.393080i	$-0.128590\pi$
$810$	53.4406	+	13.7072i	1.87771	+	0.481623i
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$	−37.6888	−	42.7442i	−1.32180	−	1.49911i
$814$	31.5519			1.10589
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$			38.9872i			1.36066i	0.732905	+	0.680331i	$0.238164\pi$
$821$			38.9872i			1.36066i	−0.732905	+	0.680331i	$0.761836\pi$
$822$		0			0
$823$		0			0		1.00000			$0$
$823$		0			0		−1.00000			$\pi$
$824$		−	109.660i		−	3.82020i
$825$	−13.8779	+	12.2366i	−0.483168	+	0.426022i
$826$		0			0
$827$			56.6227i			1.96896i	0.175489	+	0.984481i	$0.443849\pi$
$827$			56.6227i			1.96896i	−0.175489	+	0.984481i	$0.556151\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$	−230.767			−8.00041
$833$		0			0
$834$	−33.0687	+	29.1576i	−1.14508	+	1.00964i
$835$	54.5910			1.88920
$836$		−	51.3636i		−	1.77645i
$837$		0			0
$838$	53.4406			1.84608
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$			86.9168i			2.99003i
$846$		0			0
$847$		0			0
$848$			7.91787i			0.271901i
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$		0			0
$855$	29.0103	+	3.66123i	0.992130	+	0.125211i
$856$	−42.4324			−1.45031
$857$		−	49.3422i		−	1.68550i	−0.538308	−	0.842748i	$-0.680936\pi$
$857$		−	49.3422i		−	1.68550i	0.538308	−	0.842748i	$-0.319064\pi$
$858$	48.3202	+	54.8017i	1.64962	+	1.87090i
$859$	−4.00000			−0.136478			−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000			−0.136478			−0.0682391	+	0.997669i	$0.521738\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	55.5937i		−	1.89243i	−0.323537	−	0.946216i	$-0.604872\pi$
$863$		−	55.5937i		−	1.89243i	0.323537	−	0.946216i	$-0.395128\pi$
$864$	−66.9467	−	98.4700i	−2.27757	−	3.35002i
$865$	46.6253			1.58531
$866$		−	75.8229i		−	2.57657i
$867$	−19.4736	−	22.0857i	−0.661358	−	0.750070i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	21.7925			0.738409
$872$		0			0
$873$	−7.18707	+	56.9477i	−0.243245	+	1.92739i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−54.4275			−1.83789			−0.918944	−	0.394388i	$-0.870956\pi$
$877$	−54.4275			−1.83789			−0.918944	+	0.394388i	$0.870956\pi$
$878$		0			0
$879$	−41.1760	+	36.3060i	−1.38883	+	1.22457i
$880$	73.5215			2.47841
$881$			34.9499i			1.17749i	0.808318	+	0.588746i	$0.200378\pi$
$881$			34.9499i			1.17749i	−0.808318	+	0.588746i	$0.799622\pi$
$882$	−57.1173	−	7.20846i	−1.92324	−	0.242722i
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			59.2317i			1.98881i	0.105655	+	0.994403i	$0.466306\pi$
$887$			59.2317i			1.98881i	−0.105655	+	0.994403i	$0.533694\pi$
$888$	67.4506	−	59.4731i	2.26349	−	1.99579i
$889$		0			0
$890$		0			0
$891$	−4.77724	+	18.6251i	−0.160044	+	0.623965i
$892$	21.5168			0.720435
$893$		0			0
$894$	−69.4996	−	78.8221i	−2.32441	−	2.63620i
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−10.3591	+	82.0817i	−0.345303	+	2.73606i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	95.2743			3.16878
$905$		0			0
$906$		0			0
$907$	37.8860			1.25798			0.628991	−	0.777412i	$-0.283468\pi$
$907$	37.8860			1.25798			0.628991	+	0.777412i	$0.283468\pi$
$908$		−	145.248i		−	4.82021i
$909$	40.4356	+	5.10316i	1.34116	+	0.169261i
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$	−76.8442	−	87.1518i	−2.54457	−	2.88589i
$913$		0			0
$914$		0			0
$915$	34.3032	−	30.2461i	1.13403	−	0.999904i
$916$	89.9907			2.97338
$917$		0			0
$918$		0			0
$919$	−16.0000			−0.527791			−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000			−0.527791			−0.263896	+	0.964551i	$0.585007\pi$
$920$		0			0
$921$	−36.4663	−	41.3578i	−1.20161	−	1.36279i
$922$	−106.881			−3.51995
$923$		0			0
$924$		0			0
$925$	−26.9354			−0.885631
$926$		0			0
$927$	−4.27411	+	33.8665i	−0.140380	+	1.11232i
$928$		0			0
$929$			31.3050i			1.02708i	0.858065	+	0.513541i	$0.171667\pi$
$929$			31.3050i			1.02708i	−0.858065	+	0.513541i	$0.828333\pi$
$930$		0			0
$931$	−30.5123			−1.00000
$932$		0			0
$933$	−43.6261	+	38.4663i	−1.42825	+	1.25933i
$934$		0			0
$935$		0			0
$936$	206.595	+	26.0732i	6.75276	+	0.852229i
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0			−	1.00000i	$-0.5\pi$
$947$		0			0				1.00000i	$0.5\pi$
$948$		0			0
$949$		0			0
$950$			59.7484i			1.93849i
$951$	29.8958	−	26.3600i	0.969439	−	0.854781i
$952$		0			0
$953$		−	18.1623i		−	0.588336i	−0.955754	−	0.294168i	$-0.904957\pi$
$953$		−	18.1623i		−	0.588336i	0.955754	−	0.294168i	$-0.0950425\pi$
$954$	0.529805	−	4.19799i	0.0171531	−	0.135915i
$955$	20.0000			0.647185
$956$			107.517i			3.47736i
$957$		0			0
$958$	68.6306			2.21735
$959$		0			0
$960$	93.0812	−	82.0723i	3.00418	−	2.64887i
$961$	31.0000			1.00000
$962$			106.364i			3.42930i
$963$	13.1044	+	1.65384i	0.422284	+	0.0532942i
$964$		0			0
$965$		−	18.6913i		−	0.601693i
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$		−	62.0238i		−	1.99352i
$969$		0			0
$970$	−117.287			−3.76587
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$	39.0569	+	76.5954i	1.25275	+	2.45680i
$973$		0			0
$974$		−	110.170i		−	3.53007i
$975$	−41.2503	−	46.7835i	−1.32107	−	1.49827i
$976$	−181.729			−5.81700
$977$			7.72547i			0.247160i	0.992335	+	0.123580i	$0.0394375\pi$
$977$			7.72547i			0.247160i	−0.992335	+	0.123580i	$0.960562\pi$
$978$		0			0
$979$		0			0
$980$		−	86.3316i		−	2.75776i
$981$		0			0
$982$	98.0809			3.12989
$983$			24.1741i			0.771035i	0.922700	+	0.385518i	$0.125977\pi$
$983$			24.1741i			0.771035i	−0.922700	+	0.385518i	$0.874023\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	173.150			5.50864
$989$		0			0
$990$	−38.9804	−	4.91951i	−1.23888	−	0.156352i
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	19.4936i		−	0.617988i
$996$		0			0
$997$		0			0		1.00000			$0$
$997$		0			0		−1.00000			$\pi$
$998$		−	77.8401i		−	2.46398i
$999$	−23.1488	+	15.7382i	−0.732397	+	0.497933i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	285.2.b.b.284.16	yes	16
3.2	odd	2	inner	285.2.b.b.284.2	yes	16
5.4	even	2	inner	285.2.b.b.284.1	✓	16
15.14	odd	2	inner	285.2.b.b.284.15	yes	16
19.18	odd	2	inner	285.2.b.b.284.1	✓	16
57.56	even	2	inner	285.2.b.b.284.15	yes	16
95.94	odd	2	CM	285.2.b.b.284.16	yes	16
285.284	even	2	inner	285.2.b.b.284.2	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
285.2.b.b.284.1	✓	16	5.4	even	2	inner
285.2.b.b.284.1	✓	16	19.18	odd	2	inner
285.2.b.b.284.2	yes	16	3.2	odd	2	inner
285.2.b.b.284.2	yes	16	285.284	even	2	inner
285.2.b.b.284.15	yes	16	15.14	odd	2	inner
285.2.b.b.284.15	yes	16	57.56	even	2	inner
285.2.b.b.284.16	yes	16	1.1	even	1	trivial
285.2.b.b.284.16	yes	16	95.94	odd	2	CM

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:	\( N \)	\(=\)	\( 285 = 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	285.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.74145i q^{2} +(1.14551 + 1.29916i) q^{3} -5.51552 q^{4} +2.23607i q^{5} +(-3.56158 + 3.14034i) q^{6} -9.63761i q^{8} +(-0.375634 + 2.97639i) q^{9} +O(q^{10})\)	\(q+2.74145i q^{2} +(1.14551 + 1.29916i) q^{3} -5.51552 q^{4} +2.23607i q^{5} +(-3.56158 + 3.14034i) q^{6} -9.63761i q^{8} +(-0.375634 + 2.97639i) q^{9} -6.13006 q^{10} -2.13645i q^{11} +(-6.31806 - 7.16555i) q^{12} +7.20211 q^{13} +(-2.90501 + 2.56143i) q^{15} +15.3899 q^{16} +(-8.15961 - 1.02978i) q^{18} -4.35890 q^{19} -12.3331i q^{20} +5.85695 q^{22} +(12.5208 - 11.0399i) q^{24} -5.00000 q^{25} +19.7442i q^{26} +(-4.29710 + 2.92146i) q^{27} +(-7.02202 - 7.96393i) q^{30} +22.9155i q^{32} +(2.77559 - 2.44731i) q^{33} +(2.07182 - 16.4163i) q^{36} +5.38708 q^{37} -11.9497i q^{38} +(8.25006 + 9.35669i) q^{39} +21.5504 q^{40} +11.7836i q^{44} +(-6.65541 - 0.839944i) q^{45} +(17.6293 + 19.9940i) q^{48} +7.00000 q^{49} -13.7072i q^{50} -39.7234 q^{52} +0.514484i q^{53} +(-8.00903 - 11.7803i) q^{54} +4.77724 q^{55} +(-4.99314 - 5.66291i) q^{57} +(16.0227 - 14.1276i) q^{60} -11.8083 q^{61} -32.0416 q^{64} +16.1044i q^{65} +(6.70917 + 7.60912i) q^{66} +3.02584 q^{67} +(28.6853 + 3.62022i) q^{72} +14.7684i q^{74} +(-5.72753 - 6.49580i) q^{75} +24.0416 q^{76} +(-25.6509 + 22.6171i) q^{78} +34.4130i q^{80} +(-8.71780 - 2.23607i) q^{81} -20.5902 q^{88} +(2.30266 - 18.2454i) q^{90} -9.74679i q^{95} +(-29.7709 + 26.2498i) q^{96} +19.1331 q^{97} +19.1901i q^{98} +(6.35890 + 0.802522i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 32 q^{4}+O(q^{10}) \) 16 * q - 32 * q^4	\( 16 q - 32 q^{4} + 64 q^{16} + 8 q^{24} - 80 q^{25} - 40 q^{30} + 56 q^{36} + 112 q^{49} - 88 q^{54} - 128 q^{64} + 104 q^{66} - 152 q^{96} + 32 q^{99}+O(q^{100}) \) 16 * q - 32 * q^4 + 64 * q^16 + 8 * q^24 - 80 * q^25 - 40 * q^30 + 56 * q^36 + 112 * q^49 - 88 * q^54 - 128 * q^64 + 104 * q^66 - 152 * q^96 + 32 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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