LMFDB - Embedded newform 285.2.b.b.284.9

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			284.9
Root			$1.37072 - 0.348022i$ of defining polynomial
Character	$\chi$	$=$	285.284
Dual form			285.2.b.b.284.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.696044i q^{2} +(-1.29916 + 1.14551i) q^{3} +1.51552 q^{4} +2.23607i q^{5} +(-0.797322 - 0.904273i) q^{6} +2.44696i q^{8} +(0.375634 - 2.97639i) q^{9} +O(q^{10})$	\(q+0.696044i q^{2} +(-1.29916 + 1.14551i) q^{3} +1.51552 q^{4} +2.23607i q^{5} +(-0.797322 - 0.904273i) q^{6} +2.44696i q^{8} +(0.375634 - 2.97639i) q^{9} -1.55640 q^{10} +2.13645i q^{11} +(-1.96891 + 1.73604i) q^{12} -0.360034 q^{13} +(-2.56143 - 2.90501i) q^{15} +1.32785 q^{16} +(2.07170 + 0.261458i) q^{18} -4.35890 q^{19} +3.38881i q^{20} -1.48706 q^{22} +(-2.80300 - 3.17899i) q^{24} -5.00000 q^{25} -0.250600i q^{26} +(2.92146 + 4.29710i) q^{27} +(2.02202 - 1.78287i) q^{30} +5.81816i q^{32} +(-2.44731 - 2.77559i) q^{33} +(0.569282 - 4.51079i) q^{36} +10.9078 q^{37} -3.03399i q^{38} +(0.467742 - 0.412421i) q^{39} -5.47157 q^{40} +3.23783i q^{44} +(6.65541 + 0.839944i) q^{45} +(-1.72509 + 1.52106i) q^{48} +7.00000 q^{49} -3.48022i q^{50} -0.545640 q^{52} -14.5511i q^{53} +(-2.99097 + 2.03347i) q^{54} -4.77724 q^{55} +(5.66291 - 4.99314i) q^{57} +(-3.88190 - 4.40261i) q^{60} +11.8083 q^{61} -1.39399 q^{64} -0.805061i q^{65} +(1.93193 - 1.70344i) q^{66} +16.0886 q^{67} +(7.28310 + 0.919161i) q^{72} +7.59229i q^{74} +(6.49580 - 5.72753i) q^{75} -6.60601 q^{76} +(0.287063 + 0.325569i) q^{78} +2.96917i q^{80} +(-8.71780 - 2.23607i) q^{81} -5.22780 q^{88} +(-0.584638 + 4.63246i) q^{90} -9.74679i q^{95} +(-6.66474 - 7.55872i) q^{96} -4.68216 q^{97} +4.87231i q^{98} +(6.35890 + 0.802522i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$			0.696044i			0.492178i	0.969247	+	0.246089i	$0.0791455\pi$
$2$			0.696044i			0.492178i	−0.969247	+	0.246089i	$0.920855\pi$
$3$	−1.29916	+	1.14551i	−0.750070	+	0.661358i
$3$	−1.29916	+	1.14551i	−0.750070	+	0.661358i
$4$	1.51552			0.757761
$5$			2.23607i			1.00000i
$5$			2.23607i			1.00000i
$6$	−0.797322	−	0.904273i	−0.325506	−	0.369168i
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$			2.44696i			0.865131i
$9$	0.375634	−	2.97639i	0.125211	−	0.992130i
$10$	−1.55640			−0.492178
$11$			2.13645i			0.644163i	0.946712	+	0.322081i	$0.104382\pi$
$11$			2.13645i			0.644163i	−0.946712	+	0.322081i	$0.895618\pi$
$12$	−1.96891	+	1.73604i	−0.568374	+	0.501151i
$13$	−0.360034			−0.0998555			−0.0499278	−	0.998753i	$-0.515899\pi$
$13$	−0.360034			−0.0998555			−0.0499278	+	0.998753i	$0.515899\pi$
$14$		0			0
$15$	−2.56143	−	2.90501i	−0.661358	−	0.750070i
$16$	1.32785			0.331963
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$	2.07170	+	0.261458i	0.488304	+	0.0616262i
$19$	−4.35890			−1.00000
$19$	−4.35890			−1.00000
$20$			3.38881i			0.757761i
$21$		0			0
$22$	−1.48706			−0.317042
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$	−2.80300	−	3.17899i	−0.572161	−	0.648909i
$25$	−5.00000			−1.00000
$26$		−	0.250600i		−	0.0491467i
$27$	2.92146	+	4.29710i	0.562236	+	0.826977i
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$	2.02202	−	1.78287i	0.369168	−	0.325506i
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$			5.81816i			1.02852i
$33$	−2.44731	−	2.77559i	−0.426022	−	0.483168i
$34$		0			0
$35$		0			0
$36$	0.569282	−	4.51079i	0.0948804	−	0.751798i
$37$	10.9078			1.79323			0.896613	−	0.442816i	$-0.146020\pi$
$37$	10.9078			1.79323			0.896613	+	0.442816i	$0.146020\pi$
$38$		−	3.03399i		−	0.492178i
$39$	0.467742	−	0.412421i	0.0748987	−	0.0660403i
$40$	−5.47157			−0.865131
$41$		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			$\pi$
$44$			3.23783i			0.488122i
$45$	6.65541	+	0.839944i	0.992130	+	0.125211i
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$	−1.72509	+	1.52106i	−0.248996	+	0.219547i
$49$	7.00000			1.00000
$50$		−	3.48022i		−	0.492178i
$51$		0			0
$52$	−0.545640			−0.0756667
$53$		−	14.5511i		−	1.99875i	−0.0353349	−	0.999376i	$-0.511250\pi$
$53$		−	14.5511i		−	1.99875i	0.0353349	−	0.999376i	$-0.488750\pi$
$54$	−2.99097	+	2.03347i	−0.407020	+	0.276720i
$55$	−4.77724			−0.644163
$56$		0			0
$57$	5.66291	−	4.99314i	0.750070	−	0.661358i
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$	−3.88190	−	4.40261i	−0.501151	−	0.568374i
$61$	11.8083			1.51190			0.755948	−	0.654632i	$-0.227176\pi$
$61$	11.8083			1.51190			0.755948	+	0.654632i	$0.227176\pi$
$62$		0			0
$63$		0			0
$64$	−1.39399			−0.174249
$65$		−	0.805061i		−	0.0998555i
$66$	1.93193	−	1.70344i	0.237804	−	0.209679i
$67$	16.0886			1.96554			0.982770	−	0.184833i	$-0.0591744\pi$
$67$	16.0886			1.96554			0.982770	+	0.184833i	$0.0591744\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$	7.28310	+	0.919161i	0.858322	+	0.108324i
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$			7.59229i			0.882585i
$75$	6.49580	−	5.72753i	0.750070	−	0.661358i
$76$	−6.60601			−0.757761
$77$		0			0
$78$	0.287063	+	0.325569i	0.0325035	+	0.0368635i
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$			2.96917i			0.331963i
$81$	−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
$88$	−5.22780			−0.557285
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$	−0.584638	+	4.63246i	−0.0616262	+	0.488304i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	9.74679i		−	1.00000i
$96$	−6.66474	−	7.55872i	−0.680217	−	0.771459i
$97$	−4.68216			−0.475401			−0.237701	−	0.971338i	$-0.576394\pi$
$97$	−4.68216			−0.475401			−0.237701	+	0.971338i	$0.576394\pi$
$98$			4.87231i			0.492178i
$99$	6.35890	+	0.802522i	0.639093	+	0.0806565i
$100$	−7.57761			−0.757761
$101$			13.5854i			1.35180i	0.736992	+	0.675901i	$0.236245\pi$
$101$			13.5854i			1.35180i	−0.736992	+	0.675901i	$0.763755\pi$
$102$		0			0
$103$	−16.8087			−1.65621			−0.828106	−	0.560572i	$-0.810581\pi$
$103$	−16.8087			−1.65621			−0.828106	+	0.560572i	$0.810581\pi$
$104$		−	0.880989i		−	0.0863881i
$105$		0			0
$106$	10.1282			0.983740
$107$			20.2142i			1.95418i	0.212817	+	0.977092i	$0.431736\pi$
$107$			20.2142i			1.95418i	−0.212817	+	0.977092i	$0.568264\pi$
$108$	4.42754	+	6.51235i	0.426040	+	0.626651i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		−	3.32517i		−	0.317042i
$111$	−14.1709	+	12.4949i	−1.34505	+	1.18596i
$112$		0			0
$113$		−	18.8222i		−	1.77064i	−0.464983	−	0.885319i	$-0.653940\pi$
$113$		−	18.8222i		−	1.77064i	0.464983	−	0.885319i	$-0.346060\pi$
$114$	3.47545	+	3.94163i	0.325506	+	0.369168i
$115$		0			0
$116$		0			0
$117$	−0.135241	+	1.07160i	−0.0125031	+	0.0990697i
$118$		0			0
$119$		0			0
$120$	7.10844	−	6.26771i	0.648909	−	0.572161i
$121$	6.43560			0.585054
$122$			8.21909i			0.744121i
$123$		0			0
$124$		0			0
$125$		−	11.1803i		−	1.00000i
$126$		0			0
$127$	−14.0206			−1.24412			−0.622062	−	0.782968i	$-0.713705\pi$
$127$	−14.0206			−1.24412			−0.622062	+	0.782968i	$0.713705\pi$
$128$			10.6660i			0.942754i
$129$		0			0
$130$	0.560358			0.0491467
$131$		−	19.4936i		−	1.70316i	−0.524222	−	0.851581i	$-0.675644\pi$
$131$		−	19.4936i		−	1.70316i	0.524222	−	0.851581i	$-0.324356\pi$
$132$	−3.70896	−	4.20646i	−0.322823	−	0.366126i
$133$		0			0
$134$			11.1984i			0.967395i
$135$	−9.60860	+	6.53259i	−0.826977	+	0.562236i
$136$		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	$-0.628828\pi$
$139$	−9.28485			−0.787531			−0.393765	+	0.919211i	$0.628828\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	0.769194i		−	0.0643232i
$144$	0.498787	−	3.95221i	0.0415656	−	0.329351i
$145$		0			0
$146$		0			0
$147$	−9.09412	+	8.01854i	−0.750070	+	0.661358i
$148$	16.5310			1.35884
$149$		−	22.1312i		−	1.81306i	−0.422140	−	0.906531i	$-0.638721\pi$
$149$		−	22.1312i		−	1.81306i	0.422140	−	0.906531i	$-0.361279\pi$
$150$	3.98661	+	4.52136i	0.325506	+	0.369168i
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		−	10.6660i		−	0.865131i
$153$		0			0
$154$		0			0
$155$		0			0
$156$	0.708874	−	0.625034i	0.0567553	−	0.0500427i
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		0			0
$159$	16.6684	+	18.9042i	1.32189	+	1.49920i
$160$	−13.0098			−1.02852
$161$		0			0
$162$	1.55640	−	6.06797i	0.122282	−	0.476745i
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		0			0
$165$	6.20640	−	5.47235i	0.483168	−	0.426022i
$166$		0			0
$167$			8.48316i			0.656446i	0.944600	+	0.328223i	$0.106450\pi$
$167$			8.48316i			0.656446i	−0.944600	+	0.328223i	$0.893550\pi$
$168$		0			0
$169$	−12.8704			−0.990029
$170$		0			0
$171$	−1.63735	+	12.9738i	−0.125211	+	0.992130i
$172$		0			0
$173$			16.0380i			1.21934i	0.792654	+	0.609672i	$0.208699\pi$
$173$			16.0380i			1.21934i	−0.792654	+	0.609672i	$0.791301\pi$
$174$		0			0
$175$		0			0
$176$			2.83689i			0.213839i
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	10.0864	+	1.27295i	0.751798	+	0.0948804i
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$	−15.3409	+	13.5265i	−1.13403	+	0.999904i
$184$		0			0
$185$			24.3905i			1.79323i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$	6.78420			0.492178
$191$		−	8.94427i		−	0.647185i	−0.946197	−	0.323592i	$-0.895109\pi$
$191$		−	8.94427i		−	0.647185i	0.946197	−	0.323592i	$-0.104891\pi$
$192$	1.81102	−	1.59682i	0.130699	−	0.115241i
$193$	26.4977			1.90735			0.953673	−	0.300846i	$-0.0972691\pi$
$193$	26.4977			1.90735			0.953673	+	0.300846i	$0.0972691\pi$
$194$		−	3.25899i		−	0.233982i
$195$	0.922202	+	1.04590i	0.0660403	+	0.0748987i
$196$	10.6087			0.757761
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$	−0.558591	+	4.42607i	−0.0396973	+	0.314547i
$199$	−8.71780			−0.617988			−0.308994	−	0.951064i	$-0.599992\pi$
$199$	−8.71780			−0.617988			−0.308994	+	0.951064i	$0.599992\pi$
$200$		−	12.2348i		−	0.865131i
$201$	−20.9017	+	18.4296i	−1.47429	+	1.29993i
$202$	−9.45607			−0.665327
$203$		0			0
$204$		0			0
$205$		0			0
$206$		−	11.6996i		−	0.815150i
$207$		0			0
$208$	−0.478073			−0.0331484
$209$		−	9.31255i		−	0.644163i
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		−	22.0526i		−	1.51458i
$213$		0			0
$214$	−14.0700			−0.961806
$215$		0			0
$216$	−10.5148	+	7.14870i	−0.715443	+	0.486407i
$217$		0			0
$218$		0			0
$219$		0			0
$220$	−7.24001			−0.488122
$221$		0			0
$222$	−8.69701	−	9.86360i	−0.583705	−	0.662001i
$223$	−29.6105			−1.98287			−0.991433	−	0.130620i	$-0.958303\pi$
$223$	−29.6105			−1.98287			−0.991433	+	0.130620i	$0.958303\pi$
$224$		0			0
$225$	−1.87817	+	14.8820i	−0.125211	+	0.992130i
$226$	13.1010			0.871469
$227$		−	14.6459i		−	0.972082i	−0.873936	−	0.486041i	$-0.838441\pi$
$227$		−	14.6459i		−	0.972082i	0.873936	−	0.486041i	$-0.161559\pi$
$228$	8.58226	−	7.56722i	0.568374	−	0.501151i
$229$	16.3159			1.07818			0.539092	−	0.842247i	$-0.318767\pi$
$229$	16.3159			1.07818			0.539092	+	0.842247i	$0.318767\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$	−0.745883	−	0.0941338i	−0.0487599	−	0.00615372i
$235$		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	$-0.217137\pi$
$240$	−3.40120	−	3.85743i	−0.219547	−	0.248996i
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$			4.47946i			0.287951i
$243$	13.8872	−	7.08128i	0.890867	−	0.454264i
$244$	17.8957			1.14566
$245$			15.6525i			1.00000i
$246$		0			0
$247$	1.56935			0.0998555
$248$		0			0
$249$		0			0
$250$	7.78201			0.492178
$251$			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
$254$		−	9.75893i		−	0.612330i
$255$		0			0
$256$	−10.2120			−0.638251
$257$		−	28.2972i		−	1.76513i	−0.470190	−	0.882565i	$-0.655815\pi$
$257$		−	28.2972i		−	1.76513i	0.470190	−	0.882565i	$-0.344185\pi$
$258$		0			0
$259$		0			0
$260$		−	1.22009i		−	0.0756667i
$261$		0			0
$262$	13.5684			0.838258
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$	6.79175	−	5.98847i	0.418003	−	0.368565i
$265$	32.5373			1.99875
$266$		0			0
$267$		0			0
$268$	24.3827			1.48941
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$	−4.54697	−	6.68801i	−0.276720	−	0.407020i
$271$	32.9014			1.99862			0.999309	−	0.0371559i	$-0.0118298\pi$
$271$	32.9014			1.99862			0.999309	+	0.0371559i	$0.0118298\pi$
$272$		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			$\pi$
$278$		−	6.46267i		−	0.387605i
$279$		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
$285$	11.1650	+	12.6626i	0.661358	+	0.750070i
$286$	0.535393			0.0316584
$287$		0			0
$288$	17.3171	+	2.18550i	1.02042	+	0.128782i
$289$	−17.0000			−1.00000
$290$		0			0
$291$	6.08287	−	5.36344i	0.356584	−	0.314410i
$292$		0			0
$293$			12.9410i			0.756022i	0.925801	+	0.378011i	$0.123392\pi$
$293$			12.9410i			0.756022i	−0.925801	+	0.378011i	$0.876608\pi$
$294$	−5.58126	−	6.32991i	−0.325506	−	0.369168i
$295$		0			0
$296$			26.6909i			1.55137i
$297$	−9.18052	+	6.24155i	−0.532708	+	0.362171i
$298$	15.4043			0.892348
$299$		0			0
$300$	9.84453	−	8.68020i	0.568374	−	0.501151i
$301$		0			0
$302$		0			0
$303$	−15.5622	−	17.6497i	−0.894025	−	1.01395i
$304$	−5.78798			−0.331963
$305$			26.4041i			1.51190i
$306$		0			0
$307$	−14.6485			−0.836034			−0.418017	−	0.908439i	$-0.637275\pi$
$307$	−14.6485			−0.836034			−0.418017	+	0.908439i	$0.637275\pi$
$308$		0			0
$309$	21.8372	−	19.2545i	1.24227	−	1.09535i
$310$		0			0
$311$		−	33.5802i		−	1.90416i	−0.305848	−	0.952080i	$-0.598940\pi$
$311$		−	33.5802i		−	1.90416i	0.305848	−	0.952080i	$-0.401060\pi$
$312$	1.00918	+	1.14455i	0.0571334	+	0.0647971i
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	27.1747i		−	1.52628i	−0.646232	−	0.763141i	$-0.723656\pi$
$317$		−	27.1747i		−	1.52628i	0.646232	−	0.763141i	$-0.276344\pi$
$318$	−13.1582	+	11.6019i	−0.737875	+	0.650604i
$319$		0			0
$320$		−	3.11706i		−	0.174249i
$321$	−23.1555	−	26.2615i	−1.29242	−	1.46578i
$322$		0			0
$323$		0			0
$324$	−13.2120	−	3.38881i	−0.734001	−	0.188267i
$325$	1.80017			0.0998555
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$	3.80900	+	4.31993i	0.209679	+	0.237804i
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		0			0
$333$	4.09733	−	32.4658i	0.224532	−	1.77911i
$334$	−5.90465			−0.323088
$335$			35.9753i			1.96554i
$336$		0			0
$337$	−33.2574			−1.81164			−0.905822	−	0.423658i	$-0.860746\pi$
$337$	−33.2574			−1.81164			−0.905822	+	0.423658i	$0.860746\pi$
$338$		−	8.95835i		−	0.487270i
$339$	21.5609	+	24.4530i	1.17103	+	1.32810i
$340$		0			0
$341$		0			0
$342$	−9.03033	−	1.13967i	−0.488304	−	0.0616262i
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−11.1631			−0.600134
$347$		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
$351$	−1.05183	−	1.54710i	−0.0561423	−	0.0825782i
$352$	−12.4302			−0.662532
$353$		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	$0.485109\pi$
$359$			37.8531i			1.99781i	−0.0467657	+	0.998906i	$0.514891\pi$
$360$	−2.05531	+	16.2855i	−0.108324	+	0.858322i
$361$	19.0000			1.00000
$362$		0			0
$363$	−8.36087	+	7.37201i	−0.438832	+	0.386930i
$364$		0			0
$365$		0			0
$366$	−9.41501	−	10.6779i	−0.492130	−	0.558143i
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$		0			0
$369$		0			0
$370$	−16.9769			−0.882585
$371$		0			0
$372$		0			0
$373$	−20.2721			−1.04965			−0.524824	−	0.851211i	$-0.675869\pi$
$373$	−20.2721			−1.04965			−0.524824	+	0.851211i	$0.675869\pi$
$374$		0			0
$375$	12.8071	+	14.5251i	0.661358	+	0.750070i
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		−	14.7715i		−	0.757761i
$381$	18.2150	−	16.0606i	0.933181	−	0.822811i
$382$	6.22561			0.318530
$383$			11.8617i			0.606105i	0.952974	+	0.303053i	$0.0980058\pi$
$383$			11.8617i			0.606105i	−0.952974	+	0.303053i	$0.901994\pi$
$384$	−12.2180	−	13.8569i	−0.623498	−	0.707132i
$385$		0			0
$386$			18.4436i			0.938752i
$387$		0			0
$388$	−7.09591			−0.360240
$389$			38.9872i			1.97673i	0.152106	+	0.988364i	$0.451394\pi$
$389$			38.9872i			1.97673i	−0.152106	+	0.988364i	$0.548606\pi$
$390$	−0.727995	+	0.641893i	−0.0368635	+	0.0325035i
$391$		0			0
$392$			17.1287i			0.865131i
$393$	22.3300	+	25.3253i	1.12640	+	1.27749i
$394$		0			0
$395$		0			0
$396$	9.63705	+	1.21624i	0.484280	+	0.0611184i
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		−	6.06797i		−	0.304160i
$399$		0			0
$400$	−6.63927			−0.331963
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$	−12.8278	−	14.5485i	−0.639794	−	0.725614i
$403$		0			0
$404$			20.5891i			1.02434i
$405$	5.00000	−	19.4936i	0.248452	−	0.968644i
$406$		0			0
$407$			23.3039i			1.15513i
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$		0			0
$412$	−25.4740			−1.25501
$413$		0			0
$414$		0			0
$415$		0			0
$416$		−	2.09474i		−	0.102703i
$417$	12.0625	−	10.6358i	0.590704	−	0.520840i
$418$	6.48195			0.317042
$419$		−	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
$424$	35.6060			1.72918
$425$		0			0
$426$		0			0
$427$		0			0
$428$			30.6351i			1.48081i
$429$	0.881116	+	0.999306i	0.0425407	+	0.0482470i
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$	3.87928	+	5.70592i	0.186642	+	0.274526i
$433$	−31.0972			−1.49443			−0.747217	−	0.664580i	$-0.768611\pi$
$433$	−31.0972			−1.49443			−0.747217	+	0.664580i	$0.768611\pi$
$434$		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			$\pi$
$440$		−	11.6897i		−	0.557285i
$441$	2.62944	−	20.8347i	0.125211	−	0.992130i
$442$		0			0
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$	−21.4764	+	18.9363i	−1.01922	+	0.898678i
$445$		0			0
$446$		−	20.6102i		−	0.975922i
$447$	25.3515	+	28.7520i	1.19908	+	1.35992i
$448$		0			0
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$	−10.3585	−	1.30729i	−0.488304	−	0.0616262i
$451$		0			0
$452$		−	28.5254i		−	1.34172i
$453$		0			0
$454$	10.1942			0.478437
$455$		0			0
$456$	12.2180	+	13.8569i	0.572161	+	0.648909i
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$			11.3566i			0.530658i
$459$		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	$0.5\pi$
$468$	−0.204961	+	1.62404i	−0.00947433	+	0.0750712i
$469$		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
$477$	−43.3098	−	5.46590i	−1.98302	−	0.250266i
$478$	13.5684			0.620604
$479$			25.0344i			1.14385i	0.820305	+	0.571927i	$0.193804\pi$
$479$			25.0344i			1.14385i	−0.820305	+	0.571927i	$0.806196\pi$
$480$	16.9018	−	14.9028i	0.771459	−	0.680217i
$481$	−3.92717			−0.179063
$482$		0			0
$483$		0			0
$484$	9.75329			0.443331
$485$		−	10.4696i		−	0.475401i
$486$	4.92888	+	9.66613i	0.223579	+	0.438465i
$487$	18.2488			0.826934			0.413467	−	0.910519i	$-0.364318\pi$
$487$	18.2488			0.826934			0.413467	+	0.910519i	$0.364318\pi$
$488$			28.8944i			1.30799i
$489$		0			0
$490$	−10.8948			−0.492178
$491$		−	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
$494$			1.09234i			0.0491467i
$495$	−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	$-0.280777\pi$
$499$	28.3938			1.27108			0.635541	+	0.772067i	$0.280777\pi$
$500$		−	16.9441i		−	0.757761i
$501$	−9.71750	−	11.0210i	−0.434146	−	0.492381i
$502$	−12.4512			−0.555725
$503$		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
$507$	16.7207	−	14.7431i	0.742591	−	0.654763i
$508$	−21.2485			−0.942749
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$		0			0
$512$			14.2241i			0.628621i
$513$	−12.7344	−	18.7306i	−0.562236	−	0.826977i
$514$	19.6961			0.868758
$515$		−	37.5854i		−	1.65621i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−18.3716	−	20.8359i	−0.806423	−	0.914594i
$520$	1.96995			0.0863881
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$		0			0
$523$	35.8361			1.56700			0.783502	−	0.621390i	$-0.213432\pi$
$523$	35.8361			1.56700			0.783502	+	0.621390i	$0.213432\pi$
$524$		−	29.5430i		−	1.29059i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	−3.24967	−	3.68557i	−0.141424	−	0.160394i
$529$	−23.0000			−1.00000
$530$			22.6474i			0.983740i
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−45.2004			−1.95418
$536$			39.3682i			1.70045i
$537$		0			0
$538$		0			0
$539$			14.9551i			0.644163i
$540$	−14.5621	+	9.90029i	−0.626651	+	0.426040i
$541$	7.30067			0.313881			0.156940	−	0.987608i	$-0.449837\pi$
$541$	7.30067			0.313881			0.156940	+	0.987608i	$0.449837\pi$
$542$			22.9008i			0.983675i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	20.2462			0.865664			0.432832	−	0.901475i	$-0.357514\pi$
$547$	20.2462			0.865664			0.432832	+	0.901475i	$0.357514\pi$
$548$		0			0
$549$	4.43560	−	35.1461i	0.189307	−	1.50000i
$550$	7.43531			0.317042
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−27.9395	−	31.6872i	−1.18596	−	1.34505i
$556$	−14.0714			−0.596760
$557$		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
$563$		−	23.8393i		−	1.00471i	−0.864662	−	0.502354i	$-0.832467\pi$
$563$		−	23.8393i		−	1.00471i	0.864662	−	0.502354i	$-0.167533\pi$
$564$		0			0
$565$	42.0876			1.77064
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$	−8.81376	+	7.77134i	−0.369168	+	0.325506i
$571$	−46.9635			−1.96536			−0.982681	−	0.185306i	$-0.940672\pi$
$571$	−46.9635			−1.96536			−0.982681	+	0.185306i	$0.940672\pi$
$572$		−	1.16573i		−	0.0487417i
$573$	10.2457	+	11.6200i	0.428021	+	0.485434i
$574$		0			0
$575$		0			0
$576$	−0.523630	+	4.14906i	−0.0218179	+	0.172877i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$		−	11.8328i		−	0.492178i
$579$	−34.4247	+	30.3532i	−1.43064	+	1.26144i
$580$		0			0
$581$		0			0
$582$	3.73319	+	4.23395i	0.154746	+	0.175503i
$583$	31.0877			1.28752
$584$		0			0
$585$	−2.39618	−	0.302408i	−0.0990697	−	0.0125031i
$586$	−9.00751			−0.372097
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$	−13.7823	+	12.1523i	−0.568374	+	0.501151i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	14.4839			0.595285
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$	−4.34439	−	6.39005i	−0.178253	−	0.262187i
$595$		0			0
$596$		−	33.5404i		−	1.37387i
$597$	11.3258	−	9.98629i	0.463535	−	0.408711i
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$	14.0150	+	15.8950i	0.572161	+	0.648909i
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$	6.04344	−	47.8861i	0.246108	−	1.95007i
$604$		0			0
$605$			14.3904i			0.585054i
$606$	12.2850	−	10.8320i	0.499042	−	0.440019i
$607$	48.9860			1.98828			0.994140	−	0.108102i	$-0.0344772\pi$
$607$	48.9860			1.98828			0.994140	+	0.108102i	$0.0344772\pi$
$608$		−	25.3608i		−	1.02852i
$609$		0			0
$610$	−18.3784			−0.744121
$611$		0			0
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		−	10.1960i		−	0.411477i
$615$		0			0
$616$		0			0
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$	13.4020	+	15.1997i	0.539106	+	0.611420i
$619$	−0.269629			−0.0108373			−0.00541866	−	0.999985i	$-0.501725\pi$
$619$	−0.269629			−0.0108373			−0.00541866	+	0.999985i	$0.501725\pi$
$620$		0			0
$621$		0			0
$622$	23.3733			0.937185
$623$		0			0
$624$	0.621093	−	0.547635i	0.0248636	−	0.0219229i
$625$	25.0000			1.00000
$626$		0			0
$627$	10.6676	+	12.0985i	0.426022	+	0.483168i
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−42.4559			−1.69014			−0.845071	−	0.534654i	$-0.820442\pi$
$631$	−42.4559			−1.69014			−0.845071	+	0.534654i	$0.820442\pi$
$632$		0			0
$633$		0			0
$634$	18.9148			0.751202
$635$		−	31.3509i		−	1.24412i
$636$	25.2613	+	28.6498i	1.00168	+	1.13604i
$637$	−2.52024			−0.0998555
$638$		0			0
$639$		0			0
$640$	−23.8500			−0.942754
$641$		0			0			−	1.00000i	$-0.5\pi$
$641$		0			0				1.00000i	$0.5\pi$
$642$	18.2792	−	16.1173i	0.721422	−	0.636098i
$643$		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	$0.5\pi$
$648$	5.47157	−	21.3321i	0.214943	−	0.838004i
$649$		0			0
$650$			1.25300i			0.0491467i
$651$		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	$0.5\pi$
$660$	9.40594	−	8.29348i	0.366126	−	0.322823i
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	22.5976	+	2.85192i	0.875639	+	0.110510i
$667$		0			0
$668$			12.8564i			0.497430i
$669$	38.4688	−	33.9190i	1.48729	−	1.31138i
$670$	−25.0404			−0.967395
$671$			25.2278i			0.973907i
$672$		0			0
$673$	30.3771			1.17095			0.585476	−	0.810690i	$-0.300908\pi$
$673$	30.3771			1.17095			0.585476	+	0.810690i	$0.300908\pi$
$674$		−	23.1486i		−	0.891651i
$675$	−14.6073	−	21.4855i	−0.562236	−	0.826977i
$676$	−19.5053			−0.750206
$677$		−	42.0433i		−	1.61585i	−0.589283	−	0.807927i	$-0.700590\pi$
$677$		−	42.0433i		−	1.61585i	0.589283	−	0.807927i	$-0.299410\pi$
$678$	−17.0204	+	15.0073i	−0.653663	+	0.576353i
$679$		0			0
$680$		0			0
$681$	16.7769	+	19.0274i	0.642894	+	0.729130i
$682$		0			0
$683$		−	51.3315i		−	1.96414i	−0.188505	−	0.982072i	$-0.560364\pi$
$683$		−	51.3315i		−	1.96414i	0.188505	−	0.982072i	$-0.439636\pi$
$684$	−2.48144	+	19.6621i	−0.0948804	+	0.751798i
$685$		0			0
$686$		0			0
$687$	−21.1970	+	18.6899i	−0.808715	+	0.713066i
$688$		0			0
$689$			5.23890i			0.199586i
$690$		0			0
$691$	37.4090			1.42311			0.711553	−	0.702632i	$-0.247992\pi$
$691$	37.4090			1.42311			0.711553	+	0.702632i	$0.247992\pi$
$692$			24.3059i			0.923972i
$693$		0			0
$694$		0			0
$695$		−	20.7616i		−	0.787531i
$696$		0			0
$697$		0			0
$698$		−	24.2719i		−	0.918705i
$699$		0			0
$700$		0			0
$701$			49.3021i			1.86212i	0.364871	+	0.931058i	$0.381113\pi$
$701$			49.3021i			1.86212i	−0.364871	+	0.931058i	$0.618887\pi$
$702$	1.07685	−	0.732118i	0.0406432	−	0.0276320i
$703$	−47.5458			−1.79323
$704$		−	2.97819i		−	0.112245i
$705$		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
$715$	1.71997			0.0643232
$716$		0			0
$717$	22.3300	+	25.3253i	0.833930	+	0.945790i
$718$	−26.3474			−0.983278
$719$		−	46.3989i		−	1.73039i	−0.501438	−	0.865194i	$-0.667195\pi$
$719$		−	46.3989i		−	1.73039i	0.501438	−	0.865194i	$-0.332805\pi$
$720$	8.83741	+	1.11532i	0.329351	+	0.0415656i
$721$		0			0
$722$			13.2248i			0.492178i
$723$		0			0
$724$		0			0
$725$		0			0
$726$	−5.13124	−	5.81953i	−0.190438	−	0.215983i
$727$		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
$732$	−23.2494	+	20.4996i	−0.859323	+	0.757689i
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		0			0
$735$	−17.9300	−	20.3351i	−0.661358	−	0.750070i
$736$		0			0
$737$			34.3725i			1.26613i
$738$		0			0
$739$	−8.71780			−0.320689			−0.160345	−	0.987061i	$-0.551261\pi$
$739$	−8.71780			−0.320689			−0.160345	+	0.987061i	$0.551261\pi$
$740$			36.9644i			1.35884i
$741$	−2.03884	+	1.79770i	−0.0748987	+	0.0660403i
$742$		0			0
$743$			49.7214i			1.82410i	0.410079	+	0.912050i	$0.365501\pi$
$743$			49.7214i			1.82410i	−0.410079	+	0.912050i	$0.634499\pi$
$744$		0			0
$745$	49.4869			1.81306
$746$		−	14.1103i		−	0.516613i
$747$		0			0
$748$		0			0
$749$		0			0
$750$	−10.1101	+	8.91434i	−0.369168	+	0.325506i
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		0			0
$753$	−20.4914	−	23.2401i	−0.746749	−	0.846916i
$754$		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
$760$	23.8500			0.865131
$761$		−	45.0292i		−	1.63231i	−0.577834	−	0.816154i	$-0.696102\pi$
$761$		−	45.0292i		−	1.63231i	0.577834	−	0.816154i	$-0.303898\pi$
$762$	11.1789	+	12.6784i	0.404969	+	0.459291i
$763$		0			0
$764$		−	13.5552i		−	0.490412i
$765$		0			0
$766$	−8.25627			−0.298311
$767$		0			0
$768$	13.2670	−	11.6979i	0.478733	−	0.422112i
$769$	53.9946			1.94709			0.973547	−	0.228488i	$-0.0733782\pi$
$769$	53.9946			1.94709			0.973547	+	0.228488i	$0.0733782\pi$
$770$		0			0
$771$	32.4146	+	36.7626i	1.16738	+	1.32397i
$772$	40.1578			1.44531
$773$			32.7430i			1.17769i	0.808248	+	0.588843i	$0.200416\pi$
$773$			32.7430i			1.17769i	−0.808248	+	0.588843i	$0.799584\pi$
$774$		0			0
$775$		0			0
$776$		−	11.4570i		−	0.411284i
$777$		0			0
$778$	−27.1368			−0.972901
$779$		0			0
$780$	1.39762	+	1.58509i	0.0500427	+	0.0567553i
$781$		0			0
$782$		0			0
$783$		0			0
$784$	9.29498			0.331963
$785$		0			0
$786$	−17.6275	+	15.5427i	−0.628753	+	0.554389i
$787$	4.65625			0.165978			0.0829888	−	0.996550i	$-0.473553\pi$
$787$	4.65625			0.165978			0.0829888	+	0.996550i	$0.473553\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−1.96374	+	15.5600i	−0.0697784	+	0.552899i
$793$	−4.25139			−0.150971
$794$		0			0
$795$	−42.2712	+	37.2717i	−1.49920	+	1.32189i
$796$	−13.2120			−0.468288
$797$			45.2635i			1.60332i	0.597783	+	0.801658i	$0.296048\pi$
$797$			45.2635i			1.60332i	−0.597783	+	0.801658i	$0.703952\pi$
$798$		0			0
$799$		0			0
$800$		−	29.0908i		−	1.02852i
$801$		0			0
$802$		0			0
$803$		0			0
$804$	−31.6770	+	27.9305i	−1.11716	+	0.985033i
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−33.2430			−1.16949
$809$		−	22.3607i		−	0.786160i	−0.919504	−	0.393080i	$-0.871410\pi$
$809$		−	22.3607i		−	0.786160i	0.919504	−	0.393080i	$-0.128590\pi$
$810$	13.5684	+	3.48022i	0.476745	+	0.122282i
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$	−42.7442	+	37.6888i	−1.49911	+	1.32180i
$814$	−16.2205			−0.568529
$815$		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			$\pi$
$824$		−	41.1302i		−	1.43284i
$825$	12.2366	+	13.8779i	0.426022	+	0.483168i
$826$		0			0
$827$		−	10.0933i		−	0.350978i	−0.984481	−	0.175489i	$-0.943849\pi$
$827$		−	10.0933i		−	0.350978i	0.984481	−	0.175489i	$-0.0561506\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$	0.501884			0.0173997
$833$		0			0
$834$	7.40302	+	8.39604i	0.256346	+	0.290731i
$835$	−18.9689			−0.656446
$836$		−	14.1134i		−	0.488122i
$837$		0			0
$838$	13.5684			0.468713
$839$		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
$845$		−	28.7790i		−	0.990029i
$846$		0			0
$847$		0			0
$848$		−	19.3218i		−	0.663512i
$849$		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
$856$	−49.4634			−1.69062
$857$			31.5174i			1.07662i	0.842748	+	0.538308i	$0.180936\pi$
$857$			31.5174i			1.07662i	−0.842748	+	0.538308i	$0.819064\pi$
$858$	−0.695561	+	0.613296i	−0.0237461	+	0.0209376i
$859$	−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
$863$		−	19.0090i		−	0.647073i	−0.946216	−	0.323537i	$-0.895128\pi$
$863$		−	19.0090i		−	0.647073i	0.946216	−	0.323537i	$-0.104872\pi$
$864$	−25.0012	+	16.9975i	−0.850559	+	0.578268i
$865$	−35.8620			−1.21934
$866$		−	21.6450i		−	0.735527i
$867$	22.0857	−	19.4736i	0.750070	−	0.661358i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−5.79246			−0.196270
$872$		0			0
$873$	−1.75878	+	13.9359i	−0.0595256	+	0.471660i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−23.3590			−0.788777			−0.394388	−	0.918944i	$-0.629044\pi$
$877$	−23.3590			−0.788777			−0.394388	+	0.918944i	$0.629044\pi$
$878$		0			0
$879$	−14.8240	−	16.8124i	−0.500001	−	0.567069i
$880$	−6.34348			−0.213839
$881$		−	34.9499i		−	1.17749i	−0.808318	−	0.588746i	$-0.799622\pi$
$881$		−	34.9499i		−	1.17749i	0.808318	−	0.588746i	$-0.200378\pi$
$882$	14.5019	+	1.83021i	0.488304	+	0.0616262i
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	6.29336i		−	0.211310i	−0.994403	−	0.105655i	$-0.966306\pi$
$887$		−	6.29336i		−	0.211310i	0.994403	−	0.105655i	$-0.0336940\pi$
$888$	−30.5745	−	34.6757i	−1.02601	−	1.16364i
$889$		0			0
$890$		0			0
$891$	4.77724	−	18.6251i	0.160044	−	0.623965i
$892$	−44.8754			−1.50254
$893$		0			0
$894$	−20.0127	+	17.6457i	−0.669324	+	0.590161i
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−2.84641	+	22.5539i	−0.0948804	+	0.751798i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	46.0570			1.53183
$905$		0			0
$906$		0			0
$907$	46.8258			1.55482			0.777412	−	0.628991i	$-0.216532\pi$
$907$	46.8258			1.55482			0.777412	+	0.628991i	$0.216532\pi$
$908$		−	22.1962i		−	0.736606i
$909$	40.4356	+	5.10316i	1.34116	+	0.169261i
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$	7.51951	−	6.63016i	0.248996	−	0.219547i
$913$		0			0
$914$		0			0
$915$	−30.2461	−	34.3032i	−0.999904	−	1.13403i
$916$	24.7271			0.817007
$917$		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
$921$	19.0307	−	16.7799i	0.627084	−	0.552918i
$922$	−27.1368			−0.893703
$923$		0			0
$924$		0			0
$925$	−54.5388			−1.79323
$926$		0			0
$927$	−6.31393	+	50.0293i	−0.207377	+	1.64318i
$928$		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
$933$	38.4663	+	43.6261i	1.25933	+	1.42825i
$934$		0			0
$935$		0			0
$936$	−2.62217	−	0.330930i	−0.0857082	−	0.0108168i
$937$		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
$950$			15.1699i			0.492178i
$951$	31.1287	+	35.3043i	1.00942	+	1.14482i
$952$		0			0
$953$			59.0096i			1.91151i	0.294168	+	0.955754i	$0.404957\pi$
$953$			59.0096i			1.91151i	−0.294168	+	0.955754i	$0.595043\pi$
$954$	3.80451	−	30.1456i	0.123176	−	0.975998i
$955$	20.0000			0.647185
$956$		−	29.5430i		−	0.955488i
$957$		0			0
$958$	−17.4251			−0.562979
$959$		0			0
$960$	3.57061	+	4.04956i	0.115241	+	0.130699i
$961$	31.0000			1.00000
$962$		−	2.73348i		−	0.0881310i
$963$	60.1655	+	7.59316i	1.93880	+	0.244686i
$964$		0			0
$965$			59.2506i			1.90735i
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$			15.7476i			0.506148i
$969$		0			0
$970$	7.28732			0.233982
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$	21.0464	−	10.7318i	0.675065	−	0.344224i
$973$		0			0
$974$			12.7020i			0.406998i
$975$	−2.33871	+	2.06211i	−0.0748987	+	0.0660403i
$976$	15.6797			0.501894
$977$		−	62.0348i		−	1.98467i	−0.123580	−	0.992335i	$-0.539438\pi$
$977$		−	62.0348i		−	1.98467i	0.123580	−	0.992335i	$-0.460562\pi$
$978$		0			0
$979$		0			0
$980$			23.7217i			0.757761i
$981$		0			0
$982$	24.9024			0.794669
$983$		−	57.8585i		−	1.84540i	−0.385518	−	0.922700i	$-0.625977\pi$
$983$		−	57.8585i		−	1.84540i	0.385518	−	0.922700i	$-0.374023\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	2.37839			0.0756667
$989$		0			0
$990$	−9.89700	−	1.24905i	−0.314547	−	0.0396973i
$991$		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			$\pi$
$998$			19.7633i			0.625598i
$999$	31.8666	+	46.8717i	1.00822	+	1.48296i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	285.2.b.b.284.9	yes	16
3.2	odd	2	inner	285.2.b.b.284.7	✓	16
5.4	even	2	inner	285.2.b.b.284.8	yes	16
15.14	odd	2	inner	285.2.b.b.284.10	yes	16
19.18	odd	2	inner	285.2.b.b.284.8	yes	16
57.56	even	2	inner	285.2.b.b.284.10	yes	16
95.94	odd	2	CM	285.2.b.b.284.9	yes	16
285.284	even	2	inner	285.2.b.b.284.7	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
285.2.b.b.284.7	✓	16	3.2	odd	2	inner
285.2.b.b.284.7	✓	16	285.284	even	2	inner
285.2.b.b.284.8	yes	16	5.4	even	2	inner
285.2.b.b.284.8	yes	16	19.18	odd	2	inner
285.2.b.b.284.9	yes	16	1.1	even	1	trivial
285.2.b.b.284.9	yes	16	95.94	odd	2	CM
285.2.b.b.284.10	yes	16	15.14	odd	2	inner
285.2.b.b.284.10	yes	16	57.56	even	2	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:	\( N \)	\(=\)	\( 285 = 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	285.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.696044i q^{2} +(-1.29916 + 1.14551i) q^{3} +1.51552 q^{4} +2.23607i q^{5} +(-0.797322 - 0.904273i) q^{6} +2.44696i q^{8} +(0.375634 - 2.97639i) q^{9} +O(q^{10})\)	\(q+0.696044i q^{2} +(-1.29916 + 1.14551i) q^{3} +1.51552 q^{4} +2.23607i q^{5} +(-0.797322 - 0.904273i) q^{6} +2.44696i q^{8} +(0.375634 - 2.97639i) q^{9} -1.55640 q^{10} +2.13645i q^{11} +(-1.96891 + 1.73604i) q^{12} -0.360034 q^{13} +(-2.56143 - 2.90501i) q^{15} +1.32785 q^{16} +(2.07170 + 0.261458i) q^{18} -4.35890 q^{19} +3.38881i q^{20} -1.48706 q^{22} +(-2.80300 - 3.17899i) q^{24} -5.00000 q^{25} -0.250600i q^{26} +(2.92146 + 4.29710i) q^{27} +(2.02202 - 1.78287i) q^{30} +5.81816i q^{32} +(-2.44731 - 2.77559i) q^{33} +(0.569282 - 4.51079i) q^{36} +10.9078 q^{37} -3.03399i q^{38} +(0.467742 - 0.412421i) q^{39} -5.47157 q^{40} +3.23783i q^{44} +(6.65541 + 0.839944i) q^{45} +(-1.72509 + 1.52106i) q^{48} +7.00000 q^{49} -3.48022i q^{50} -0.545640 q^{52} -14.5511i q^{53} +(-2.99097 + 2.03347i) q^{54} -4.77724 q^{55} +(5.66291 - 4.99314i) q^{57} +(-3.88190 - 4.40261i) q^{60} +11.8083 q^{61} -1.39399 q^{64} -0.805061i q^{65} +(1.93193 - 1.70344i) q^{66} +16.0886 q^{67} +(7.28310 + 0.919161i) q^{72} +7.59229i q^{74} +(6.49580 - 5.72753i) q^{75} -6.60601 q^{76} +(0.287063 + 0.325569i) q^{78} +2.96917i q^{80} +(-8.71780 - 2.23607i) q^{81} -5.22780 q^{88} +(-0.584638 + 4.63246i) q^{90} -9.74679i q^{95} +(-6.66474 - 7.55872i) q^{96} -4.68216 q^{97} +4.87231i q^{98} +(6.35890 + 0.802522i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 32 q^{4}+O(q^{10}) \) 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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