LMFDB - Embedded newform 1248.2.h.c.623.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1248,2,Mod(623,1248)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1248, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("1248.623");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1248 = 2^{5} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1248.h (of order $2$, degree $1$, 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:	$9.96533017226$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 312)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			623.19
Character	$\chi$	$=$	1248.623
Dual form			1248.2.h.c.623.23

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.662918 - 1.60017i) q^{3} +2.38561i q^{5} +2.63829 q^{7} +(-2.12108 - 2.12156i) q^{9} +O(q^{10})$	\(q+(0.662918 - 1.60017i) q^{3} +2.38561i q^{5} +2.63829 q^{7} +(-2.12108 - 2.12156i) q^{9} +0.764084 q^{11} +(1.05175 + 3.44874i) q^{13} +(3.81737 + 1.58146i) q^{15} -0.584523i q^{17} +4.28969i q^{19} +(1.74897 - 4.22170i) q^{21} +4.39993 q^{23} -0.691126 q^{25} +(-4.80096 + 1.98766i) q^{27} -8.65308 q^{29} +7.09996 q^{31} +(0.506525 - 1.22266i) q^{33} +6.29391i q^{35} +7.63475 q^{37} +(6.21579 + 0.603265i) q^{39} -0.139556 q^{41} +6.17544 q^{43} +(5.06122 - 5.06006i) q^{45} -0.889975i q^{47} -0.0394518 q^{49} +(-0.935336 - 0.387491i) q^{51} +0.803622 q^{53} +1.82280i q^{55} +(6.86423 + 2.84371i) q^{57} -9.13099 q^{59} +9.47787i q^{61} +(-5.59601 - 5.59729i) q^{63} +(-8.22735 + 2.50905i) q^{65} -11.5441i q^{67} +(2.91679 - 7.04063i) q^{69} +2.56670i q^{71} +13.3885i q^{73} +(-0.458160 + 1.10592i) q^{75} +2.01587 q^{77} -10.9069i q^{79} +(-0.00205378 + 9.00000i) q^{81} +14.6776 q^{83} +1.39444 q^{85} +(-5.73629 + 13.8464i) q^{87} +5.68613 q^{89} +(2.77481 + 9.09877i) q^{91} +(4.70669 - 11.3611i) q^{93} -10.2335 q^{95} -10.4965i q^{97} +(-1.62068 - 1.62105i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 4 q^{3} - 28 q^{9}+O(q^{10}) $ 32 * q + 4 * q^3 - 28 * q^9	$ 32 q + 4 q^{3} - 28 q^{9} + 56 q^{25} + 16 q^{27} - 40 q^{43} + 8 q^{49} + 52 q^{51} + 8 q^{75} - 76 q^{81} + 56 q^{91}+O(q^{100}) $ 32 * q + 4 * q^3 - 28 * q^9 + 56 * q^25 + 16 * q^27 - 40 * q^43 + 8 * q^49 + 52 * q^51 + 8 * q^75 - 76 * q^81 + 56 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1248\mathbb{Z}\right)^\times$.

$n$	$703$	$769$	$833$	$1093$
$\chi(n)$	$-1$	$-1$	$-1$	$-1$

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
$3$	0.662918	−	1.60017i	0.382736	−	0.923858i
$3$	0.662918	−	1.60017i	0.382736	−	0.923858i
$4$		0			0
$5$			2.38561i			1.06688i	0.845839	+	0.533438i	$0.179100\pi$
$5$			2.38561i			1.06688i	−0.845839	+	0.533438i	$0.820900\pi$
$6$		0			0
$7$	2.63829			0.997178			0.498589	−	0.866839i	$-0.333852\pi$
$7$	2.63829			0.997178			0.498589	+	0.866839i	$0.333852\pi$
$8$		0			0
$9$	−2.12108	−	2.12156i	−0.707026	−	0.707187i
$10$		0			0
$11$	0.764084			0.230380			0.115190	−	0.993343i	$-0.463252\pi$
$11$	0.764084			0.230380			0.115190	+	0.993343i	$0.463252\pi$
$12$		0			0
$13$	1.05175	+	3.44874i	0.291702	+	0.956509i
$13$	1.05175	+	3.44874i	0.291702	+	0.956509i
$14$		0			0
$15$	3.81737	+	1.58146i	0.985642	+	0.408332i
$16$		0			0
$17$		−	0.584523i		−	0.141768i	−0.997485	−	0.0708839i	$-0.977418\pi$
$17$		−	0.584523i		−	0.141768i	0.997485	−	0.0708839i	$-0.0225820\pi$
$18$		0			0
$19$			4.28969i			0.984122i	0.870561	+	0.492061i	$0.163756\pi$
$19$			4.28969i			0.984122i	−0.870561	+	0.492061i	$0.836244\pi$
$20$		0			0
$21$	1.74897	−	4.22170i	0.381656	−	0.921251i
$22$		0			0
$23$	4.39993			0.917449			0.458724	−	0.888579i	$-0.348307\pi$
$23$	4.39993			0.917449			0.458724	+	0.888579i	$0.348307\pi$
$24$		0			0
$25$	−0.691126			−0.138225
$26$		0			0
$27$	−4.80096	+	1.98766i	−0.923945	+	0.382525i
$28$		0			0
$29$	−8.65308			−1.60684			−0.803419	−	0.595415i	$-0.796988\pi$
$29$	−8.65308			−1.60684			−0.803419	+	0.595415i	$0.796988\pi$
$30$		0			0
$31$	7.09996			1.27519			0.637595	−	0.770372i	$-0.279929\pi$
$31$	7.09996			1.27519			0.637595	+	0.770372i	$0.279929\pi$
$32$		0			0
$33$	0.506525	−	1.22266i	0.0881747	−	0.212838i
$34$		0			0
$35$			6.29391i			1.06387i
$36$		0			0
$37$	7.63475			1.25515			0.627573	−	0.778558i	$-0.284049\pi$
$37$	7.63475			1.25515			0.627573	+	0.778558i	$0.284049\pi$
$38$		0			0
$39$	6.21579	+	0.603265i	0.995323	+	0.0965997i
$40$		0			0
$41$	−0.139556			−0.0217949			−0.0108975	−	0.999941i	$-0.503469\pi$
$41$	−0.139556			−0.0217949			−0.0108975	+	0.999941i	$0.503469\pi$
$42$		0			0
$43$	6.17544			0.941746			0.470873	−	0.882201i	$-0.343939\pi$
$43$	6.17544			0.941746			0.470873	+	0.882201i	$0.343939\pi$
$44$		0			0
$45$	5.06122	−	5.06006i	0.754482	−	0.754309i
$46$		0			0
$47$		−	0.889975i		−	0.129816i	−0.997891	−	0.0649081i	$-0.979325\pi$
$47$		−	0.889975i		−	0.129816i	0.997891	−	0.0649081i	$-0.0206754\pi$
$48$		0			0
$49$	−0.0394518			−0.00563597
$50$		0			0
$51$	−0.935336	−	0.387491i	−0.130973	−	0.0542596i
$52$		0			0
$53$	0.803622			0.110386			0.0551930	−	0.998476i	$-0.482423\pi$
$53$	0.803622			0.110386			0.0551930	+	0.998476i	$0.482423\pi$
$54$		0			0
$55$			1.82280i			0.245787i
$56$		0			0
$57$	6.86423	+	2.84371i	0.909189	+	0.376659i
$58$		0			0
$59$	−9.13099			−1.18875			−0.594377	−	0.804187i	$-0.702601\pi$
$59$	−9.13099			−1.18875			−0.594377	+	0.804187i	$0.702601\pi$
$60$		0			0
$61$			9.47787i			1.21352i	0.794886	+	0.606758i	$0.207530\pi$
$61$			9.47787i			1.21352i	−0.794886	+	0.606758i	$0.792470\pi$
$62$		0			0
$63$	−5.59601	−	5.59729i	−0.705031	−	0.705192i
$64$		0			0
$65$	−8.22735	+	2.50905i	−1.02048	+	0.311210i
$66$		0			0
$67$		−	11.5441i		−	1.41034i	−0.709040	−	0.705168i	$-0.750872\pi$
$67$		−	11.5441i		−	1.41034i	0.709040	−	0.705168i	$-0.249128\pi$
$68$		0			0
$69$	2.91679	−	7.04063i	0.351141	−	0.847592i
$70$		0			0
$71$			2.56670i			0.304611i	0.988333	+	0.152305i	$0.0486697\pi$
$71$			2.56670i			0.304611i	−0.988333	+	0.152305i	$0.951330\pi$
$72$		0			0
$73$			13.3885i			1.56700i	0.621391	+	0.783500i	$0.286568\pi$
$73$			13.3885i			1.56700i	−0.621391	+	0.783500i	$0.713432\pi$
$74$		0			0
$75$	−0.458160	+	1.10592i	−0.0529037	+	0.127700i
$76$		0			0
$77$	2.01587			0.229730
$78$		0			0
$79$		−	10.9069i		−	1.22713i	−0.789646	−	0.613563i	$-0.789736\pi$
$79$		−	10.9069i		−	1.22713i	0.789646	−	0.613563i	$-0.210264\pi$
$80$		0			0
$81$	−0.00205378	+	9.00000i	−0.000228198	+	1.00000i
$82$		0			0
$83$	14.6776			1.61107			0.805536	−	0.592547i	$-0.201877\pi$
$83$	14.6776			1.61107			0.805536	+	0.592547i	$0.201877\pi$
$84$		0			0
$85$	1.39444			0.151249
$86$		0			0
$87$	−5.73629	+	13.8464i	−0.614995	+	1.48449i
$88$		0			0
$89$	5.68613			0.602729			0.301364	−	0.953509i	$-0.402558\pi$
$89$	5.68613			0.602729			0.301364	+	0.953509i	$0.402558\pi$
$90$		0			0
$91$	2.77481	+	9.09877i	0.290879	+	0.953810i
$92$		0			0
$93$	4.70669	−	11.3611i	0.488061	−	1.17809i
$94$		0			0
$95$	−10.2335			−1.04994
$96$		0			0
$97$		−	10.4965i		−	1.06576i	−0.846191	−	0.532879i	$-0.821110\pi$
$97$		−	10.4965i		−	1.06576i	0.846191	−	0.532879i	$-0.178890\pi$
$98$		0			0
$99$	−1.62068	−	1.62105i	−0.162885	−	0.162922i
$100$		0			0
$101$	15.4371			1.53605			0.768023	−	0.640422i	$-0.221241\pi$
$101$	15.4371			1.53605			0.768023	+	0.640422i	$0.221241\pi$
$102$		0			0
$103$		−	4.99712i		−	0.492381i	−0.969221	−	0.246191i	$-0.920821\pi$
$103$		−	4.99712i		−	0.492381i	0.969221	−	0.246191i	$-0.0791789\pi$
$104$		0			0
$105$	10.0713	+	4.17235i	0.982860	+	0.407180i
$106$		0			0
$107$		−	19.1495i		−	1.85126i	−0.378434	−	0.925628i	$-0.623537\pi$
$107$		−	19.1495i		−	1.85126i	0.378434	−	0.925628i	$-0.376463\pi$
$108$		0			0
$109$	−7.05520			−0.675766			−0.337883	−	0.941188i	$-0.609711\pi$
$109$	−7.05520			−0.675766			−0.337883	+	0.941188i	$0.609711\pi$
$110$		0			0
$111$	5.06122	−	12.2169i	0.480389	−	1.15958i
$112$		0			0
$113$		−	11.2283i		−	1.05627i	−0.849160	−	0.528136i	$-0.822891\pi$
$113$		−	11.2283i		−	1.05627i	0.849160	−	0.528136i	$-0.177109\pi$
$114$		0			0
$115$			10.4965i			0.978804i
$116$		0			0
$117$	5.08589	−	9.54640i	0.470191	−	0.882565i
$118$		0			0
$119$		−	1.54214i		−	0.141368i
$120$		0			0
$121$	−10.4162			−0.946925
$122$		0			0
$123$	−0.0925140	+	0.223312i	−0.00834170	+	0.0201354i
$124$		0			0
$125$			10.2793i			0.919407i
$126$		0			0
$127$			3.17432i			0.281675i	0.990033	+	0.140838i	$0.0449795\pi$
$127$			3.17432i			0.281675i	−0.990033	+	0.140838i	$0.955020\pi$
$128$		0			0
$129$	4.09381	−	9.88174i	0.360440	−	0.870039i
$130$		0			0
$131$		−	6.42017i		−	0.560933i	−0.959864	−	0.280467i	$-0.909511\pi$
$131$		−	6.42017i		−	0.560933i	0.959864	−	0.280467i	$-0.0904892\pi$
$132$		0			0
$133$			11.3174i			0.981345i
$134$		0			0
$135$	−4.74178	−	11.4532i	−0.408107	−	0.985735i
$136$		0			0
$137$	−6.69918			−0.572350			−0.286175	−	0.958177i	$-0.592384\pi$
$137$	−6.69918			−0.572350			−0.286175	+	0.958177i	$0.592384\pi$
$138$		0			0
$139$	−10.4217			−0.883957			−0.441979	−	0.897026i	$-0.645723\pi$
$139$	−10.4217			−0.883957			−0.441979	+	0.897026i	$0.645723\pi$
$140$		0			0
$141$	−1.42411	−	0.589981i	−0.119932	−	0.0496854i
$142$		0			0
$143$	0.803622	+	2.63513i	0.0672023	+	0.220361i
$144$		0			0
$145$		−	20.6429i		−	1.71430i
$146$		0			0
$147$	−0.0261533	+	0.0631295i	−0.00215709	+	0.00520683i
$148$		0			0
$149$			4.48086i			0.367086i	0.983012	+	0.183543i	$0.0587567\pi$
$149$			4.48086i			0.367086i	−0.983012	+	0.183543i	$0.941243\pi$
$150$		0			0
$151$	5.12597			0.417145			0.208573	−	0.978007i	$-0.433118\pi$
$151$	5.12597			0.417145			0.208573	+	0.978007i	$0.433118\pi$
$152$		0			0
$153$	−1.24010	+	1.23982i	−0.100256	+	0.100233i
$154$		0			0
$155$			16.9377i			1.36047i
$156$		0			0
$157$		−	7.23041i		−	0.577050i	−0.957472	−	0.288525i	$-0.906835\pi$
$157$		−	7.23041i		−	0.577050i	0.957472	−	0.288525i	$-0.0931648\pi$
$158$		0			0
$159$	0.532736	−	1.28593i	0.0422487	−	0.101981i
$160$		0			0
$161$	11.6083			0.914860
$162$		0			0
$163$		−	14.7862i		−	1.15814i	−0.815276	−	0.579072i	$-0.803415\pi$
$163$		−	14.7862i		−	1.15814i	0.815276	−	0.579072i	$-0.196585\pi$
$164$		0			0
$165$	2.91679	+	1.20837i	0.227072	+	0.0940715i
$166$		0			0
$167$			18.3295i			1.41838i	0.705019	+	0.709189i	$0.250938\pi$
$167$			18.3295i			1.41838i	−0.705019	+	0.709189i	$0.749062\pi$
$168$		0			0
$169$	−10.7877	+	7.25440i	−0.829820	+	0.558031i
$170$		0			0
$171$	9.10084	−	9.09877i	0.695959	−	0.695800i
$172$		0			0
$173$	−15.4371			−1.17366			−0.586829	−	0.809711i	$-0.699624\pi$
$173$	−15.4371			−1.17366			−0.586829	+	0.809711i	$0.699624\pi$
$174$		0			0
$175$	−1.82339			−0.137835
$176$		0			0
$177$	−6.05310	+	14.6111i	−0.454979	+	1.09824i
$178$		0			0
$179$		−	2.31859i		−	0.173300i	−0.996239	−	0.0866499i	$-0.972384\pi$
$179$		−	2.31859i		−	0.173300i	0.996239	−	0.0866499i	$-0.0276162\pi$
$180$		0			0
$181$		−	20.4624i		−	1.52096i	−0.649364	−	0.760478i	$-0.724965\pi$
$181$		−	20.4624i		−	1.52096i	0.649364	−	0.760478i	$-0.275035\pi$
$182$		0			0
$183$	15.1662	+	6.28305i	1.12112	+	0.464457i
$184$		0			0
$185$			18.2135i			1.33908i
$186$		0			0
$187$		−	0.446625i		−	0.0326604i
$188$		0			0
$189$	−12.6663	+	5.24401i	−0.921338	+	0.381446i
$190$		0			0
$191$	−21.7061			−1.57060			−0.785299	−	0.619117i	$-0.787491\pi$
$191$	−21.7061			−1.57060			−0.785299	+	0.619117i	$0.787491\pi$
$192$		0			0
$193$			4.36245i			0.314016i	0.987597	+	0.157008i	$0.0501849\pi$
$193$			4.36245i			0.314016i	−0.987597	+	0.157008i	$0.949815\pi$
$194$		0			0
$195$	−1.43915	+	14.8284i	−0.103060	+	1.06189i
$196$		0			0
$197$			23.5782i			1.67988i	0.542680	+	0.839939i	$0.317410\pi$
$197$			23.5782i			1.67988i	−0.542680	+	0.839939i	$0.682590\pi$
$198$		0			0
$199$			10.9069i			0.773172i	0.922253	+	0.386586i	$0.126346\pi$
$199$			10.9069i			0.773172i	−0.922253	+	0.386586i	$0.873654\pi$
$200$		0			0
$201$	−18.4725	−	7.65279i	−1.30295	−	0.539786i
$202$		0			0
$203$	−22.8293			−1.60230
$204$		0			0
$205$		−	0.332925i		−	0.0232525i
$206$		0			0
$207$	−9.33260	−	9.33473i	−0.648660	−	0.648808i
$208$		0			0
$209$			3.27768i			0.226722i
$210$		0			0
$211$	−19.5978			−1.34917			−0.674584	−	0.738198i	$-0.735677\pi$
$211$	−19.5978			−1.34917			−0.674584	+	0.738198i	$0.735677\pi$
$212$		0			0
$213$	4.10715	+	1.70151i	0.281417	+	0.116586i
$214$		0			0
$215$			14.7322i			1.00473i
$216$		0			0
$217$	18.7317			1.27159
$218$		0			0
$219$	21.4238	+	8.87546i	1.44769	+	0.599748i
$220$		0			0
$221$	2.01587	−	0.614770i	0.135602	−	0.0413539i
$222$		0			0
$223$	−6.46108			−0.432666			−0.216333	−	0.976320i	$-0.569410\pi$
$223$	−6.46108			−0.432666			−0.216333	+	0.976320i	$0.569410\pi$
$224$		0			0
$225$	1.46593	+	1.46627i	0.0977288	+	0.0977511i
$226$		0			0
$227$	−10.9383			−0.725998			−0.362999	−	0.931789i	$-0.618247\pi$
$227$	−10.9383			−0.725998			−0.362999	+	0.931789i	$0.618247\pi$
$228$		0			0
$229$	−22.2189			−1.46826			−0.734132	−	0.679007i	$-0.762410\pi$
$229$	−22.2189			−1.46826			−0.734132	+	0.679007i	$0.762410\pi$
$230$		0			0
$231$	1.33636	−	3.22573i	0.0879259	−	0.212238i
$232$		0			0
$233$		−	16.4500i		−	1.07768i	−0.842409	−	0.538839i	$-0.818863\pi$
$233$		−	16.4500i		−	1.07768i	0.842409	−	0.538839i	$-0.181137\pi$
$234$		0			0
$235$	2.12313			0.138498
$236$		0			0
$237$	−17.4529	−	7.23041i	−1.13369	−	0.469666i
$238$		0			0
$239$			28.2054i			1.82446i	0.409681	+	0.912229i	$0.365640\pi$
$239$			28.2054i			1.82446i	−0.409681	+	0.912229i	$0.634360\pi$
$240$		0			0
$241$			8.57938i			0.552646i	0.961065	+	0.276323i	$0.0891160\pi$
$241$			8.57938i			0.552646i	−0.961065	+	0.276323i	$0.910884\pi$
$242$		0			0
$243$	14.4002	+	5.96955i	0.923770	+	0.382947i
$244$		0			0
$245$		−	0.0941165i		−	0.00601288i
$246$		0			0
$247$	−14.7940	+	4.51166i	−0.941322	+	0.287070i
$248$		0			0
$249$	9.73003	−	23.4866i	0.616615	−	1.48840i
$250$		0			0
$251$		−	3.97532i		−	0.250920i	−0.992099	−	0.125460i	$-0.959959\pi$
$251$		−	3.97532i		−	0.250920i	0.992099	−	0.125460i	$-0.0400407\pi$
$252$		0			0
$253$	3.36192			0.211362
$254$		0			0
$255$	0.924402	−	2.23134i	0.0578883	−	0.139732i
$256$		0			0
$257$			13.5763i			0.846868i	0.905927	+	0.423434i	$0.139175\pi$
$257$			13.5763i			0.846868i	−0.905927	+	0.423434i	$0.860825\pi$
$258$		0			0
$259$	20.1426			1.25160
$260$		0			0
$261$	18.3539	+	18.3581i	1.13608	+	1.13633i
$262$		0			0
$263$	9.86533			0.608322			0.304161	−	0.952621i	$-0.401624\pi$
$263$	9.86533			0.608322			0.304161	+	0.952621i	$0.401624\pi$
$264$		0			0
$265$			1.91713i			0.117768i
$266$		0			0
$267$	3.76944	−	9.09877i	0.230686	−	0.556836i
$268$		0			0
$269$	−13.4212			−0.818305			−0.409152	−	0.912466i	$-0.634176\pi$
$269$	−13.4212			−0.818305			−0.409152	+	0.912466i	$0.634176\pi$
$270$		0			0
$271$	−10.7867			−0.655247			−0.327624	−	0.944808i	$-0.606248\pi$
$271$	−10.7867			−0.655247			−0.327624	+	0.944808i	$0.606248\pi$
$272$		0			0
$273$	16.3990	+	1.59158i	0.992515	+	0.0963271i
$274$		0			0
$275$	−0.528078			−0.0318443
$276$		0			0
$277$			4.93632i			0.296594i	0.988943	+	0.148297i	$0.0473792\pi$
$277$			4.93632i			0.296594i	−0.988943	+	0.148297i	$0.952621\pi$
$278$		0			0
$279$	−15.0596	−	15.0630i	−0.901592	−	0.901798i
$280$		0			0
$281$	1.15261			0.0687587			0.0343794	−	0.999409i	$-0.489055\pi$
$281$	1.15261			0.0687587			0.0343794	+	0.999409i	$0.489055\pi$
$282$		0			0
$283$	5.16258			0.306884			0.153442	−	0.988158i	$-0.450964\pi$
$283$	5.16258			0.306884			0.153442	+	0.988158i	$0.450964\pi$
$284$		0			0
$285$	−6.78399	+	16.3754i	−0.401849	+	0.969992i
$286$		0			0
$287$	−0.368187			−0.0217334
$288$		0			0
$289$	16.6583			0.979902
$290$		0			0
$291$	−16.7962	−	6.95833i	−0.984610	−	0.407904i
$292$		0			0
$293$		−	5.96029i		−	0.348204i	−0.984728	−	0.174102i	$-0.944298\pi$
$293$		−	5.96029i		−	0.348204i	0.984728	−	0.174102i	$-0.0557022\pi$
$294$		0			0
$295$		−	21.7830i		−	1.26825i
$296$		0			0
$297$	−3.66834	+	1.51874i	−0.212858	+	0.0881262i
$298$		0			0
$299$	4.62761	+	15.1742i	0.267622	+	0.877548i
$300$		0			0
$301$	16.2926			0.939088
$302$		0			0
$303$	10.2335	−	24.7019i	0.587900	−	1.41909i
$304$		0			0
$305$	−22.6105			−1.29467
$306$		0			0
$307$			8.30199i			0.473820i	0.971532	+	0.236910i	$0.0761346\pi$
$307$			8.30199i			0.473820i	−0.971532	+	0.236910i	$0.923865\pi$
$308$		0			0
$309$	−7.99624	−	3.31268i	−0.454890	−	0.188452i
$310$		0			0
$311$	−5.46540			−0.309914			−0.154957	−	0.987921i	$-0.549524\pi$
$311$	−5.46540			−0.309914			−0.154957	+	0.987921i	$0.549524\pi$
$312$		0			0
$313$	5.46660			0.308991			0.154495	−	0.987994i	$-0.450625\pi$
$313$	5.46660			0.308991			0.154495	+	0.987994i	$0.450625\pi$
$314$		0			0
$315$	13.3529	−	13.3499i	0.752352	−	0.752181i
$316$		0			0
$317$		−	1.11720i		−	0.0627481i	−0.999508	−	0.0313741i	$-0.990012\pi$
$317$		−	1.11720i		−	0.0627481i	0.999508	−	0.0313741i	$-0.00998831\pi$
$318$		0			0
$319$	−6.61168			−0.370183
$320$		0			0
$321$	−30.6425	−	12.6946i	−1.71030	−	0.708543i
$322$		0			0
$323$	2.50742			0.139517
$324$		0			0
$325$	−0.726889	−	2.38351i	−0.0403205	−	0.132214i
$326$		0			0
$327$	−4.67703	+	11.2895i	−0.258640	+	0.624312i
$328$		0			0
$329$		−	2.34801i		−	0.129450i
$330$		0			0
$331$		−	21.5940i		−	1.18691i	−0.804866	−	0.593456i	$-0.797763\pi$
$331$		−	21.5940i		−	1.18691i	0.804866	−	0.593456i	$-0.202237\pi$
$332$		0			0
$333$	−16.1939	−	16.1976i	−0.887420	−	0.887623i
$334$		0			0
$335$	27.5397			1.50465
$336$		0			0
$337$	−23.6501			−1.28830			−0.644152	−	0.764898i	$-0.722790\pi$
$337$	−23.6501			−1.28830			−0.644152	+	0.764898i	$0.722790\pi$
$338$		0			0
$339$	−17.9672	−	7.44346i	−0.975845	−	0.404273i
$340$		0			0
$341$	5.42496			0.293778
$342$		0			0
$343$	−18.5721			−1.00280
$344$		0			0
$345$	16.7962	+	6.95833i	0.904276	+	0.374624i
$346$		0			0
$347$			6.66848i			0.357983i	0.983851	+	0.178991i	$0.0572834\pi$
$347$			6.66848i			0.357983i	−0.983851	+	0.178991i	$0.942717\pi$
$348$		0			0
$349$	−34.9112			−1.86875			−0.934376	−	0.356289i	$-0.884042\pi$
$349$	−34.9112			−1.86875			−0.934376	+	0.356289i	$0.884042\pi$
$350$		0			0
$351$	−11.9043	−	14.4668i	−0.635405	−	0.772179i
$352$		0			0
$353$	4.70688			0.250522			0.125261	−	0.992124i	$-0.460023\pi$
$353$	4.70688			0.250522			0.125261	+	0.992124i	$0.460023\pi$
$354$		0			0
$355$	−6.12313			−0.324982
$356$		0			0
$357$	−2.46768	−	1.02231i	−0.130604	−	0.0541065i
$358$		0			0
$359$		−	17.1329i		−	0.904242i	−0.891957	−	0.452121i	$-0.850668\pi$
$359$		−	17.1329i		−	0.904242i	0.891957	−	0.452121i	$-0.149332\pi$
$360$		0			0
$361$	0.598561			0.0315032
$362$		0			0
$363$	−6.90508	+	16.6676i	−0.362422	+	0.874824i
$364$		0			0
$365$	−31.9396			−1.67180
$366$		0			0
$367$			3.48197i			0.181757i	0.995862	+	0.0908787i	$0.0289676\pi$
$367$			3.48197i			0.181757i	−0.995862	+	0.0908787i	$0.971032\pi$
$368$		0			0
$369$	0.296008	+	0.296076i	0.0154096	+	0.0154131i
$370$		0			0
$371$	2.12018			0.110074
$372$		0			0
$373$		−	22.5462i		−	1.16740i	−0.811970	−	0.583699i	$-0.801605\pi$
$373$		−	22.5462i		−	1.16740i	0.811970	−	0.583699i	$-0.198395\pi$
$374$		0			0
$375$	16.4486	+	6.81433i	0.849401	+	0.351890i
$376$		0			0
$377$	−9.10084	−	29.8423i	−0.468717	−	1.53695i
$378$		0			0
$379$		−	24.4044i		−	1.25357i	−0.779193	−	0.626784i	$-0.784371\pi$
$379$		−	24.4044i		−	1.25357i	0.779193	−	0.626784i	$-0.215629\pi$
$380$		0			0
$381$	5.07944	+	2.10431i	0.260228	+	0.107807i
$382$		0			0
$383$			10.6604i			0.544722i	0.962195	+	0.272361i	$0.0878045\pi$
$383$			10.6604i			0.544722i	−0.962195	+	0.272361i	$0.912196\pi$
$384$		0			0
$385$			4.80908i			0.245093i
$386$		0			0
$387$	−13.0986	−	13.1016i	−0.665839	−	0.665991i
$388$		0			0
$389$	29.1201			1.47645			0.738224	−	0.674555i	$-0.235665\pi$
$389$	29.1201			1.47645			0.738224	+	0.674555i	$0.235665\pi$
$390$		0			0
$391$		−	2.57186i		−	0.130065i
$392$		0			0
$393$	−10.2734	−	4.25605i	−0.518222	−	0.214689i
$394$		0			0
$395$	26.0197			1.30919
$396$		0			0
$397$	−6.54151			−0.328309			−0.164154	−	0.986435i	$-0.552490\pi$
$397$	−6.54151			−0.328309			−0.164154	+	0.986435i	$0.552490\pi$
$398$		0			0
$399$	18.1098	+	7.50253i	0.906623	+	0.375596i
$400$		0			0
$401$	28.2966			1.41307			0.706533	−	0.707681i	$-0.250258\pi$
$401$	28.2966			1.41307			0.706533	+	0.707681i	$0.250258\pi$
$402$		0			0
$403$	7.46735	+	24.4859i	0.371975	+	1.21973i
$404$		0			0
$405$	−21.4705	−	0.00489951i	−1.06688	−	0.000243459i
$406$		0			0
$407$	5.83359			0.289160
$408$		0			0
$409$		−	30.6927i		−	1.51766i	−0.651290	−	0.758829i	$-0.725772\pi$
$409$		−	30.6927i		−	1.51766i	0.651290	−	0.758829i	$-0.274228\pi$
$410$		0			0
$411$	−4.44101	+	10.7198i	−0.219059	+	0.528770i
$412$		0			0
$413$	−24.0902			−1.18540
$414$		0			0
$415$			35.0149i			1.71881i
$416$		0			0
$417$	−6.90874	+	16.6765i	−0.338322	+	0.816651i
$418$		0			0
$419$			4.78509i			0.233767i	0.993146	+	0.116883i	$0.0372904\pi$
$419$			4.78509i			0.233767i	−0.993146	+	0.116883i	$0.962710\pi$
$420$		0			0
$421$	4.18333			0.203883			0.101942	−	0.994790i	$-0.467495\pi$
$421$	4.18333			0.203883			0.101942	+	0.994790i	$0.467495\pi$
$422$		0			0
$423$	−1.88814	+	1.88771i	−0.0918044	+	0.0917835i
$424$		0			0
$425$			0.403979i			0.0195959i
$426$		0			0
$427$			25.0053i			1.21009i
$428$		0			0
$429$	4.74939	+	0.460945i	0.229303	+	0.0222546i
$430$		0			0
$431$		−	19.8528i		−	0.956274i	−0.878285	−	0.478137i	$-0.841312\pi$
$431$		−	19.8528i		−	0.956274i	0.878285	−	0.478137i	$-0.158688\pi$
$432$		0			0
$433$	24.3561			1.17048			0.585240	−	0.810860i	$-0.301000\pi$
$433$	24.3561			1.17048			0.585240	+	0.810860i	$0.301000\pi$
$434$		0			0
$435$	−33.0321	−	13.6845i	−1.58377	−	0.656123i
$436$		0			0
$437$			18.8743i			0.902882i
$438$		0			0
$439$		−	16.7139i		−	0.797712i	−0.917014	−	0.398856i	$-0.869407\pi$
$439$		−	16.7139i		−	0.797712i	0.917014	−	0.398856i	$-0.130593\pi$
$440$		0			0
$441$	0.0836803	+	0.0836994i	0.00398478	+	0.00398569i
$442$		0			0
$443$		−	27.4553i		−	1.30444i	−0.758030	−	0.652219i	$-0.773838\pi$
$443$		−	27.4553i		−	1.30444i	0.758030	−	0.652219i	$-0.226162\pi$
$444$		0			0
$445$			13.5649i			0.643037i
$446$		0			0
$447$	7.17013	+	2.97044i	0.339135	+	0.140497i
$448$		0			0
$449$	−19.3205			−0.911791			−0.455895	−	0.890033i	$-0.650681\pi$
$449$	−19.3205			−0.911791			−0.455895	+	0.890033i	$0.650681\pi$
$450$		0			0
$451$	−0.106632			−0.00502111
$452$		0			0
$453$	3.39810	−	8.20241i	0.159657	−	0.385383i
$454$		0			0
$455$	−21.7061	+	6.61960i	−1.01760	+	0.310332i
$456$		0			0
$457$		−	22.5600i		−	1.05531i	−0.849458	−	0.527656i	$-0.823071\pi$
$457$		−	22.5600i		−	1.05531i	0.849458	−	0.527656i	$-0.176929\pi$
$458$		0			0
$459$	1.16183	+	2.80627i	0.0542297	+	0.130986i
$460$		0			0
$461$		−	17.8735i		−	0.832451i	−0.909261	−	0.416226i	$-0.863353\pi$
$461$		−	17.8735i		−	0.832451i	0.909261	−	0.416226i	$-0.136647\pi$
$462$		0			0
$463$	−4.69526			−0.218207			−0.109104	−	0.994030i	$-0.534798\pi$
$463$	−4.69526			−0.218207			−0.109104	+	0.994030i	$0.534798\pi$
$464$		0			0
$465$	27.1032	+	11.2283i	1.25688	+	0.520701i
$466$		0			0
$467$			11.7976i			0.545926i	0.962025	+	0.272963i	$0.0880036\pi$
$467$			11.7976i			0.545926i	−0.962025	+	0.272963i	$0.911996\pi$
$468$		0			0
$469$		−	30.4566i		−	1.40636i
$470$		0			0
$471$	−11.5699	−	4.79317i	−0.533112	−	0.220858i
$472$		0			0
$473$	4.71855			0.216959
$474$		0			0
$475$		−	2.96471i		−	0.136030i
$476$		0			0
$477$	−1.70455	−	1.70493i	−0.0780458	−	0.0780636i
$478$		0			0
$479$		−	33.8258i		−	1.54554i	−0.634687	−	0.772770i	$-0.718871\pi$
$479$		−	33.8258i		−	1.54554i	0.634687	−	0.772770i	$-0.281129\pi$
$480$		0			0
$481$	8.02982	+	26.3303i	0.366128	+	1.20056i
$482$		0			0
$483$	7.69534	−	18.5752i	0.350150	−	0.845200i
$484$		0			0
$485$	25.0406			1.13703
$486$		0			0
$487$	12.4952			0.566211			0.283106	−	0.959089i	$-0.408635\pi$
$487$	12.4952			0.566211			0.283106	+	0.959089i	$0.408635\pi$
$488$		0			0
$489$	−23.6604	−	9.80204i	−1.06996	−	0.443264i
$490$		0			0
$491$		−	11.0921i		−	0.500582i	−0.968171	−	0.250291i	$-0.919474\pi$
$491$		−	11.0921i		−	0.500582i	0.968171	−	0.250291i	$-0.0805262\pi$
$492$		0			0
$493$			5.05793i			0.227798i
$494$		0			0
$495$	3.86719	−	3.86631i	0.173817	−	0.173778i
$496$		0			0
$497$			6.77168i			0.303751i
$498$		0			0
$499$			8.57057i			0.383671i	0.981427	+	0.191836i	$0.0614441\pi$
$499$			8.57057i			0.383671i	−0.981427	+	0.191836i	$0.938556\pi$
$500$		0			0
$501$	29.3302	+	12.1509i	1.31038	+	0.542864i
$502$		0			0
$503$	18.9134			0.843307			0.421654	−	0.906757i	$-0.361450\pi$
$503$	18.9134			0.843307			0.421654	+	0.906757i	$0.361450\pi$
$504$		0			0
$505$			36.8268i			1.63877i
$506$		0			0
$507$	4.45693	+	22.0712i	0.197939	+	0.980214i
$508$		0			0
$509$		−	21.7567i		−	0.964349i	−0.876075	−	0.482174i	$-0.839847\pi$
$509$		−	21.7567i		−	0.964349i	0.876075	−	0.482174i	$-0.160153\pi$
$510$		0			0
$511$			35.3226i			1.56258i
$512$		0			0
$513$	−8.52644	−	20.5946i	−0.376452	−	0.909275i
$514$		0			0
$515$	11.9212			0.525310
$516$		0			0
$517$		−	0.680016i		−	0.0299071i
$518$		0			0
$519$	−10.2335	+	24.7019i	−0.449202	+	1.08429i
$520$		0			0
$521$			35.7964i			1.56827i	0.620591	+	0.784134i	$0.286893\pi$
$521$			35.7964i			1.56827i	−0.620591	+	0.784134i	$0.713107\pi$
$522$		0			0
$523$	−10.4148			−0.455406			−0.227703	−	0.973731i	$-0.573121\pi$
$523$	−10.4148			−0.455406			−0.227703	+	0.973731i	$0.573121\pi$
$524$		0			0
$525$	−1.20876	+	2.91773i	−0.0527545	+	0.127340i
$526$		0			0
$527$		−	4.15009i		−	0.180781i
$528$		0			0
$529$	−3.64062			−0.158288
$530$		0			0
$531$	19.3675	+	19.3720i	0.840480	+	0.840672i
$532$		0			0
$533$	−0.146777	−	0.481291i	−0.00635762	−	0.0208470i
$534$		0			0
$535$	45.6833			1.97506
$536$		0			0
$537$	−3.71014	−	1.53704i	−0.160104	−	0.0663281i
$538$		0			0
$539$	−0.0301445			−0.00129841
$540$		0			0
$541$	42.8351			1.84162			0.920812	−	0.390006i	$-0.127527\pi$
$541$	42.8351			1.84162			0.920812	+	0.390006i	$0.127527\pi$
$542$		0			0
$543$	−32.7432	−	13.5649i	−1.40515	−	0.582125i
$544$		0			0
$545$		−	16.8310i		−	0.720959i
$546$		0			0
$547$	6.86925			0.293708			0.146854	−	0.989158i	$-0.453085\pi$
$547$	6.86925			0.293708			0.146854	+	0.989158i	$0.453085\pi$
$548$		0			0
$549$	20.1079	−	20.1033i	0.858184	−	0.857988i
$550$		0			0
$551$		−	37.1190i		−	1.58132i
$552$		0			0
$553$		−	28.7756i		−	1.22366i
$554$		0			0
$555$	29.1447	+	12.0741i	1.23712	+	0.512516i
$556$		0			0
$557$			31.9362i			1.35318i	0.736360	+	0.676590i	$0.236543\pi$
$557$			31.9362i			1.35318i	−0.736360	+	0.676590i	$0.763457\pi$
$558$		0			0
$559$	6.49499	+	21.2975i	0.274709	+	0.900788i
$560$		0			0
$561$	−0.714675	−	0.296076i	−0.0301736	−	0.0125003i
$562$		0			0
$563$			6.16766i			0.259936i	0.991518	+	0.129968i	$0.0414874\pi$
$563$			6.16766i			0.259936i	−0.991518	+	0.129968i	$0.958513\pi$
$564$		0			0
$565$	26.7864			1.12691
$566$		0			0
$567$	−0.00541845	+	23.7446i	−0.000227554	+	0.997178i
$568$		0			0
$569$			0.180544i			0.00756881i	0.999993	+	0.00378441i	$0.00120462\pi$
$569$			0.180544i			0.00756881i	−0.999993	+	0.00378441i	$0.998795\pi$
$570$		0			0
$571$	−0.551781			−0.0230913			−0.0115457	−	0.999933i	$-0.503675\pi$
$571$	−0.551781			−0.0230913			−0.0115457	+	0.999933i	$0.503675\pi$
$572$		0			0
$573$	−14.3894	+	34.7334i	−0.601125	+	1.45101i
$574$		0			0
$575$	−3.04090			−0.126814
$576$		0			0
$577$			32.0177i			1.33292i	0.745543	+	0.666458i	$0.232190\pi$
$577$			32.0177i			1.33292i	−0.745543	+	0.666458i	$0.767810\pi$
$578$		0			0
$579$	6.98066	+	2.89195i	0.290106	+	0.120185i
$580$		0			0
$581$	38.7236			1.60653
$582$		0			0
$583$	0.614035			0.0254307
$584$		0			0
$585$	22.7740	+	12.1329i	0.941588	+	0.501635i
$586$		0			0
$587$	−22.7316			−0.938232			−0.469116	−	0.883137i	$-0.655427\pi$
$587$	−22.7316			−0.938232			−0.469116	+	0.883137i	$0.655427\pi$
$588$		0			0
$589$			30.4566i			1.25494i
$590$		0			0
$591$	37.7291	+	15.6304i	1.55197	+	0.642950i
$592$		0			0
$593$	13.2588			0.544474			0.272237	−	0.962230i	$-0.412237\pi$
$593$	13.2588			0.544474			0.272237	+	0.962230i	$0.412237\pi$
$594$		0			0
$595$	3.67894			0.150822
$596$		0			0
$597$	17.4529	+	7.23041i	0.714301	+	0.295921i
$598$		0			0
$599$	−14.6334			−0.597906			−0.298953	−	0.954268i	$-0.596637\pi$
$599$	−14.6334			−0.597906			−0.298953	+	0.954268i	$0.596637\pi$
$600$		0			0
$601$	23.6583			0.965044			0.482522	−	0.875884i	$-0.339721\pi$
$601$	23.6583			0.965044			0.482522	+	0.875884i	$0.339721\pi$
$602$		0			0
$603$	−24.4915	+	24.4859i	−0.997372	+	0.997144i
$604$		0			0
$605$		−	24.8489i		−	1.01025i
$606$		0			0
$607$		−	15.9152i		−	0.645977i	−0.946403	−	0.322988i	$-0.895313\pi$
$607$		−	15.9152i		−	0.645977i	0.946403	−	0.322988i	$-0.104687\pi$
$608$		0			0
$609$	−15.1340	+	36.5307i	−0.613259	+	1.48030i
$610$		0			0
$611$	3.06930	−	0.936028i	0.124170	−	0.0378676i
$612$		0			0
$613$	−8.36313			−0.337784			−0.168892	−	0.985635i	$-0.554019\pi$
$613$	−8.36313			−0.337784			−0.168892	+	0.985635i	$0.554019\pi$
$614$		0			0
$615$	−0.532736	−	0.220702i	−0.0214820	−	0.00889957i
$616$		0			0
$617$	4.84880			0.195205			0.0976026	−	0.995225i	$-0.468883\pi$
$617$	4.84880			0.195205			0.0976026	+	0.995225i	$0.468883\pi$
$618$		0			0
$619$			9.89555i			0.397736i	0.980026	+	0.198868i	$0.0637264\pi$
$619$			9.89555i			0.397736i	−0.980026	+	0.198868i	$0.936274\pi$
$620$		0			0
$621$	−21.1239	+	8.74556i	−0.847672	+	0.350947i
$622$		0			0
$623$	15.0016			0.601028
$624$		0			0
$625$	−27.9780			−1.11912
$626$		0			0
$627$	5.24484	+	2.17284i	0.209459	+	0.0867747i
$628$		0			0
$629$		−	4.46269i		−	0.177939i
$630$		0			0
$631$	13.4569			0.535713			0.267856	−	0.963459i	$-0.413685\pi$
$631$	13.4569			0.535713			0.267856	+	0.963459i	$0.413685\pi$
$632$		0			0
$633$	−12.9918	+	31.3598i	−0.516376	+	1.24644i
$634$		0			0
$635$	−7.57268			−0.300513
$636$		0			0
$637$	−0.0414932	−	0.136059i	−0.00164402	−	0.00539086i
$638$		0			0
$639$	5.44541	−	5.44416i	0.215417	−	0.215368i
$640$		0			0
$641$		−	0.939589i		−	0.0371115i	−0.999828	−	0.0185558i	$-0.994093\pi$
$641$		−	0.939589i		−	0.0371115i	0.999828	−	0.0185558i	$-0.00590683\pi$
$642$		0			0
$643$		−	42.1553i		−	1.66244i	−0.555943	−	0.831221i	$-0.687643\pi$
$643$		−	42.1553i		−	1.66244i	0.555943	−	0.831221i	$-0.312357\pi$
$644$		0			0
$645$	23.5740	+	9.76623i	0.928224	+	0.384545i
$646$		0			0
$647$	−33.5469			−1.31886			−0.659432	−	0.751765i	$-0.729203\pi$
$647$	−33.5469			−1.31886			−0.659432	+	0.751765i	$0.729203\pi$
$648$		0			0
$649$	−6.97684			−0.273865
$650$		0			0
$651$	12.4176	−	29.9739i	0.486684	−	1.17477i
$652$		0			0
$653$	11.6194			0.454700			0.227350	−	0.973813i	$-0.426994\pi$
$653$	11.6194			0.454700			0.227350	+	0.973813i	$0.426994\pi$
$654$		0			0
$655$	15.3160			0.598446
$656$		0			0
$657$	28.4044	−	28.3980i	1.10816	−	1.10791i
$658$		0			0
$659$		−	17.4797i		−	0.680912i	−0.940261	−	0.340456i	$-0.889419\pi$
$659$		−	17.4797i		−	0.680912i	0.940261	−	0.340456i	$-0.110581\pi$
$660$		0			0
$661$	41.5530			1.61623			0.808113	−	0.589028i	$-0.200489\pi$
$661$	41.5530			1.61623			0.808113	+	0.589028i	$0.200489\pi$
$662$		0			0
$663$	0.352622	−	3.63328i	0.0136947	−	0.141105i
$664$		0			0
$665$	−26.9989			−1.04697
$666$		0			0
$667$	−38.0730			−1.47419
$668$		0			0
$669$	−4.28317	+	10.3388i	−0.165597	+	0.399722i
$670$		0			0
$671$			7.24189i			0.279570i
$672$		0			0
$673$	−23.7494			−0.915473			−0.457736	−	0.889088i	$-0.651340\pi$
$673$	−23.7494			−0.915473			−0.457736	+	0.889088i	$0.651340\pi$
$674$		0			0
$675$	3.31807	−	1.37372i	0.127712	−	0.0528746i
$676$		0			0
$677$	0.998225			0.0383649			0.0191824	−	0.999816i	$-0.493894\pi$
$677$	0.998225			0.0383649			0.0191824	+	0.999816i	$0.493894\pi$
$678$		0			0
$679$		−	27.6928i		−	1.06275i
$680$		0			0
$681$	−7.25118	+	17.5031i	−0.277866	+	0.670719i
$682$		0			0
$683$	−20.6883			−0.791615			−0.395808	−	0.918333i	$-0.629535\pi$
$683$	−20.6883			−0.791615			−0.395808	+	0.918333i	$0.629535\pi$
$684$		0			0
$685$		−	15.9816i		−	0.610626i
$686$		0			0
$687$	−14.7293	+	35.5539i	−0.561957	+	1.35647i
$688$		0			0
$689$	0.845206	+	2.77149i	0.0321998	+	0.105585i
$690$		0			0
$691$		−	28.8484i		−	1.09744i	−0.836005	−	0.548722i	$-0.815115\pi$
$691$		−	28.8484i		−	1.09744i	0.836005	−	0.548722i	$-0.184885\pi$
$692$		0			0
$693$	−4.27582	−	4.27680i	−0.162425	−	0.162462i
$694$		0			0
$695$		−	24.8621i		−	0.943073i
$696$		0			0
$697$			0.0815735i			0.00308982i
$698$		0			0
$699$	−26.3228	−	10.9050i	−0.995621	−	0.412466i
$700$		0			0
$701$	−1.46047			−0.0551611			−0.0275805	−	0.999620i	$-0.508780\pi$
$701$	−1.46047			−0.0551611			−0.0275805	+	0.999620i	$0.508780\pi$
$702$		0			0
$703$			32.7507i			1.23522i
$704$		0			0
$705$	1.40746	−	3.39737i	0.0530081	−	0.127952i
$706$		0			0
$707$	40.7274			1.53171
$708$		0			0
$709$	21.9297			0.823586			0.411793	−	0.911277i	$-0.364903\pi$
$709$	21.9297			0.823586			0.411793	+	0.911277i	$0.364903\pi$
$710$		0			0
$711$	−23.1398	+	23.1345i	−0.867809	+	0.867611i
$712$		0			0
$713$	31.2393			1.16992
$714$		0			0
$715$	−6.28639	+	1.91713i	−0.235097	+	0.0716965i
$716$		0			0
$717$	45.1334	+	18.6979i	1.68554	+	0.698286i
$718$		0			0
$719$	15.0016			0.559467			0.279733	−	0.960078i	$-0.409754\pi$
$719$	15.0016			0.559467			0.279733	+	0.960078i	$0.409754\pi$
$720$		0			0
$721$		−	13.1838i		−	0.490992i
$722$		0			0
$723$	13.7285	+	5.68743i	0.510567	+	0.211518i
$724$		0			0
$725$	5.98037			0.222105
$726$		0			0
$727$		−	24.4156i		−	0.905526i	−0.891631	−	0.452763i	$-0.850438\pi$
$727$		−	24.4156i		−	0.905526i	0.891631	−	0.452763i	$-0.149562\pi$
$728$		0			0
$729$	19.0984	−	19.0853i	0.707349	−	0.706865i
$730$		0			0
$731$		−	3.60969i		−	0.133509i
$732$		0			0
$733$	12.6996			0.469071			0.234536	−	0.972108i	$-0.424643\pi$
$733$	12.6996			0.469071			0.234536	+	0.972108i	$0.424643\pi$
$734$		0			0
$735$	−0.150602	−	0.0623915i	−0.00555505	−	0.00230135i
$736$		0			0
$737$		−	8.82066i		−	0.324913i
$738$		0			0
$739$			20.7156i			0.762037i	0.924567	+	0.381018i	$0.124427\pi$
$739$			20.7156i			0.762037i	−0.924567	+	0.381018i	$0.875573\pi$
$740$		0			0
$741$	−2.58782	+	26.6638i	−0.0950659	+	0.979520i
$742$		0			0
$743$			31.3626i			1.15058i	0.817948	+	0.575292i	$0.195111\pi$
$743$			31.3626i			1.15058i	−0.817948	+	0.575292i	$0.804889\pi$
$744$		0			0
$745$	−10.6896			−0.391636
$746$		0			0
$747$	−31.1323	−	31.1394i	−1.13907	−	1.13933i
$748$		0			0
$749$		−	50.5220i		−	1.84603i
$750$		0			0
$751$			37.1565i			1.35586i	0.735127	+	0.677930i	$0.237123\pi$
$751$			37.1565i			1.35586i	−0.735127	+	0.677930i	$0.762877\pi$
$752$		0			0
$753$	−6.36118	−	2.63531i	−0.231814	−	0.0960361i
$754$		0			0
$755$			12.2286i			0.445043i
$756$		0			0
$757$		−	10.7590i		−	0.391043i	−0.980699	−	0.195522i	$-0.937360\pi$
$757$		−	10.7590i		−	0.391043i	0.980699	−	0.195522i	$-0.0626400\pi$
$758$		0			0
$759$	2.22868	−	5.37963i	0.0808958	−	0.195268i
$760$		0			0
$761$	−36.7545			−1.33235			−0.666174	−	0.745796i	$-0.732069\pi$
$761$	−36.7545			−1.33235			−0.666174	+	0.745796i	$0.732069\pi$
$762$		0			0
$763$	−18.6136			−0.673859
$764$		0			0
$765$	−2.95772	−	2.95840i	−0.106937	−	0.106961i
$766$		0			0
$767$	−9.60348	−	31.4904i	−0.346762	−	1.13705i
$768$		0			0
$769$		−	15.7373i		−	0.567502i	−0.958898	−	0.283751i	$-0.908421\pi$
$769$		−	15.7373i		−	0.567502i	0.958898	−	0.283751i	$-0.0915789\pi$
$770$		0			0
$771$	21.7244	+	9.00000i	0.782386	+	0.324127i
$772$		0			0
$773$		−	11.9057i		−	0.428220i	−0.976810	−	0.214110i	$-0.931315\pi$
$773$		−	11.9057i		−	0.428220i	0.976810	−	0.214110i	$-0.0686851\pi$
$774$		0			0
$775$	−4.90696			−0.176263
$776$		0			0
$777$	13.3529	−	32.2316i	0.479034	−	1.15630i
$778$		0			0
$779$		−	0.598650i		−	0.0214489i
$780$		0			0
$781$			1.96117i			0.0701762i
$782$		0			0
$783$	41.5431	−	17.1994i	1.48463	−	0.614656i
$784$		0			0
$785$	17.2489			0.615641
$786$		0			0
$787$			16.1112i			0.574301i	0.957885	+	0.287151i	$0.0927080\pi$
$787$			16.1112i			0.574301i	−0.957885	+	0.287151i	$0.907292\pi$
$788$		0			0
$789$	6.53991	−	15.7862i	0.232827	−	0.562003i
$790$		0			0
$791$		−	29.6235i		−	1.05329i
$792$		0			0
$793$	−32.6867	+	9.96831i	−1.16074	+	0.353985i
$794$		0			0
$795$	3.06773	+	1.27090i	0.108801	+	0.0450741i
$796$		0			0
$797$	−14.7802			−0.523542			−0.261771	−	0.965130i	$-0.584307\pi$
$797$	−14.7802			−0.523542			−0.261771	+	0.965130i	$0.584307\pi$
$798$		0			0
$799$	−0.520211			−0.0184038
$800$		0			0
$801$	−12.0607	−	12.0635i	−0.426145	−	0.426242i
$802$		0			0
$803$			10.2299i			0.361005i
$804$		0			0
$805$			27.6928i			0.976042i
$806$		0			0
$807$	−8.89716	+	21.4762i	−0.313195	+	0.755997i
$808$		0			0
$809$			18.0231i			0.633657i	0.948483	+	0.316829i	$0.102618\pi$
$809$			18.0231i			0.633657i	−0.948483	+	0.316829i	$0.897382\pi$
$810$		0			0
$811$			45.5754i			1.60037i	0.599754	+	0.800185i	$0.295265\pi$
$811$			45.5754i			1.60037i	−0.599754	+	0.800185i	$0.704735\pi$
$812$		0			0
$813$	−7.15072	+	17.2606i	−0.250787	+	0.605355i
$814$		0			0
$815$	35.2741			1.23560
$816$		0			0
$817$			26.4907i			0.926793i
$818$		0			0
$819$	13.4180	−	25.1861i	0.468864	−	0.880074i
$820$		0			0
$821$		−	43.8260i		−	1.52954i	−0.644305	−	0.764769i	$-0.722853\pi$
$821$		−	43.8260i		−	1.52954i	0.644305	−	0.764769i	$-0.277147\pi$
$822$		0			0
$823$		−	41.8348i		−	1.45827i	−0.684369	−	0.729135i	$-0.739922\pi$
$823$		−	41.8348i		−	1.45827i	0.684369	−	0.729135i	$-0.260078\pi$
$824$		0			0
$825$	−0.350073	+	0.845014i	−0.0121880	+	0.0294196i
$826$		0			0
$827$	32.1625			1.11840			0.559200	−	0.829033i	$-0.311108\pi$
$827$	32.1625			1.11840			0.559200	+	0.829033i	$0.311108\pi$
$828$		0			0
$829$			11.0453i			0.383619i	0.981432	+	0.191810i	$0.0614356\pi$
$829$			11.0453i			0.383619i	−0.981432	+	0.191810i	$0.938564\pi$
$830$		0			0
$831$	7.89894	+	3.27237i	0.274011	+	0.113517i
$832$		0			0
$833$			0.0230605i			0.000798998i
$834$		0			0
$835$	−43.7269			−1.51323
$836$		0			0
$837$	−34.0866	+	14.1123i	−1.17821	+	0.487792i
$838$		0			0
$839$		−	32.2336i		−	1.11283i	−0.830905	−	0.556415i	$-0.812177\pi$
$839$		−	32.2336i		−	1.11283i	0.830905	−	0.556415i	$-0.187823\pi$
$840$		0			0
$841$	45.8758			1.58193
$842$		0			0
$843$	0.764084	−	1.84436i	0.0263164	−	0.0635233i
$844$		0			0
$845$	−17.3062	−	25.7351i	−0.595350	−	0.885315i
$846$		0			0
$847$	−27.4808			−0.944253
$848$		0			0
$849$	3.42237	−	8.26100i	0.117456	−	0.283517i
$850$		0			0
$851$	33.5924			1.15153
$852$		0			0
$853$	−22.1165			−0.757256			−0.378628	−	0.925549i	$-0.623604\pi$
$853$	−22.1165			−0.757256			−0.378628	+	0.925549i	$0.623604\pi$
$854$		0			0
$855$	21.7061	+	21.7110i	0.742333	+	0.742502i
$856$		0			0
$857$			2.33809i			0.0798678i	0.999202	+	0.0399339i	$0.0127147\pi$
$857$			2.33809i			0.0798678i	−0.999202	+	0.0399339i	$0.987285\pi$
$858$		0			0
$859$	43.4057			1.48098			0.740492	−	0.672065i	$-0.234592\pi$
$859$	43.4057			1.48098			0.740492	+	0.672065i	$0.234592\pi$
$860$		0			0
$861$	−0.244078	+	0.589162i	−0.00831816	+	0.0200786i
$862$		0			0
$863$			42.1353i			1.43430i	0.696917	+	0.717152i	$0.254555\pi$
$863$			42.1353i			1.43430i	−0.696917	+	0.717152i	$0.745445\pi$
$864$		0			0
$865$		−	36.8268i		−	1.25215i
$866$		0			0
$867$	11.0431	−	26.6561i	0.375044	−	0.905290i
$868$		0			0
$869$		−	8.33382i		−	0.282705i
$870$		0			0
$871$	39.8126	−	12.1415i	1.34900	−	0.411397i
$872$		0			0
$873$	−22.2690	+	22.2639i	−0.753691	+	0.753519i
$874$		0			0
$875$			27.1197i			0.916813i
$876$		0			0
$877$	−38.5223			−1.30080			−0.650402	−	0.759590i	$-0.725400\pi$
$877$	−38.5223			−1.30080			−0.650402	+	0.759590i	$0.725400\pi$
$878$		0			0
$879$	−9.53747	−	3.95119i	−0.321691	−	0.133270i
$880$		0			0
$881$			3.54012i			0.119270i	0.998220	+	0.0596348i	$0.0189936\pi$
$881$			3.54012i			0.119270i	−0.998220	+	0.0596348i	$0.981006\pi$
$882$		0			0
$883$	24.1271			0.811942			0.405971	−	0.913886i	$-0.366933\pi$
$883$	24.1271			0.811942			0.405971	+	0.913886i	$0.366933\pi$
$884$		0			0
$885$	−34.8564	−	14.4403i	−1.17169	−	0.485406i
$886$		0			0
$887$	34.9805			1.17453			0.587265	−	0.809394i	$-0.300204\pi$
$887$	34.9805			1.17453			0.587265	+	0.809394i	$0.300204\pi$
$888$		0			0
$889$			8.37475i			0.280880i
$890$		0			0
$891$	−0.00156926	+	6.87675i	−5.25721e−5	+	0.230380i
$892$		0			0
$893$	3.81772			0.127755
$894$		0			0
$895$	5.53126			0.184890
$896$		0			0
$897$	27.3490	+	2.65432i	0.913158	+	0.0886252i
$898$		0			0
$899$	−61.4365			−2.04902
$900$		0			0
$901$		−	0.469736i		−	0.0156492i
$902$		0			0
$903$	10.8006	−	26.0709i	0.359423	−	0.867584i
$904$		0			0
$905$	48.8152			1.62267
$906$		0			0
$907$	49.1062			1.63054			0.815272	−	0.579078i	$-0.196587\pi$
$907$	49.1062			1.63054			0.815272	+	0.579078i	$0.196587\pi$
$908$		0			0
$909$	−32.7432	−	32.7507i	−1.08602	−	1.08627i
$910$		0			0
$911$	29.3415			0.972128			0.486064	−	0.873923i	$-0.338432\pi$
$911$	29.3415			0.972128			0.486064	+	0.873923i	$0.338432\pi$
$912$		0			0
$913$	11.2149			0.371159
$914$		0			0
$915$	−14.9889	+	36.1806i	−0.495518	+	1.19609i
$916$		0			0
$917$		−	16.9382i		−	0.559350i
$918$		0			0
$919$		−	30.8671i		−	1.01821i	−0.860704	−	0.509106i	$-0.829976\pi$
$919$		−	30.8671i		−	1.01821i	0.860704	−	0.509106i	$-0.170024\pi$
$920$		0			0
$921$	13.2846	+	5.50354i	0.437742	+	0.181348i
$922$		0			0
$923$	−8.85188	+	2.69951i	−0.291363	+	0.0888556i
$924$		0			0
$925$	−5.27657			−0.173493
$926$		0			0
$927$	−10.6017	+	10.5993i	−0.348206	+	0.348126i
$928$		0			0
$929$	56.3252			1.84797			0.923985	−	0.382429i	$-0.124912\pi$
$929$	56.3252			1.84797			0.923985	+	0.382429i	$0.124912\pi$
$930$		0			0
$931$		−	0.169236i		−	0.00554648i
$932$		0			0
$933$	−3.62312	+	8.74556i	−0.118615	+	0.286317i
$934$		0			0
$935$	1.06547			0.0348447
$936$		0			0
$937$	−1.08988			−0.0356047			−0.0178024	−	0.999842i	$-0.505667\pi$
$937$	−1.08988			−0.0356047			−0.0178024	+	0.999842i	$0.505667\pi$
$938$		0			0
$939$	3.62391	−	8.74748i	0.118262	−	0.285463i
$940$		0			0
$941$		−	11.8789i		−	0.387240i	−0.981077	−	0.193620i	$-0.937977\pi$
$941$		−	11.8789i		−	0.387240i	0.981077	−	0.193620i	$-0.0620229\pi$
$942$		0			0
$943$	−0.614035			−0.0199957
$944$		0			0
$945$	−12.5102	−	30.2168i	−0.406956	−	0.982953i
$946$		0			0
$947$	−26.2255			−0.852216			−0.426108	−	0.904672i	$-0.640116\pi$
$947$	−26.2255			−0.852216			−0.426108	+	0.904672i	$0.640116\pi$
$948$		0			0
$949$	−46.1734	+	14.0813i	−1.49885	+	0.457097i
$950$		0			0
$951$	−1.78771	−	0.740612i	−0.0579703	−	0.0240160i
$952$		0			0
$953$		−	16.4039i		−	0.531375i	−0.964059	−	0.265687i	$-0.914401\pi$
$953$		−	16.4039i		−	0.531375i	0.964059	−	0.265687i	$-0.0855989\pi$
$954$		0			0
$955$		−	51.7822i		−	1.67563i
$956$		0			0
$957$	−4.38300	+	10.5798i	−0.141682	+	0.341996i
$958$		0			0
$959$	−17.6744			−0.570735
$960$		0			0
$961$	19.4094			0.626109
$962$		0			0
$963$	−40.6270	+	40.6177i	−1.30919	+	1.30889i
$964$		0			0
$965$	−10.4071			−0.335017
$966$		0			0
$967$	−23.4856			−0.755245			−0.377622	−	0.925960i	$-0.623258\pi$
$967$	−23.4856			−0.755245			−0.377622	+	0.925960i	$0.623258\pi$
$968$		0			0
$969$	1.66222	−	4.01230i	0.0533981	−	0.128894i
$970$		0			0
$971$			38.7343i			1.24304i	0.783397	+	0.621521i	$0.213485\pi$
$971$			38.7343i			1.24304i	−0.783397	+	0.621521i	$0.786515\pi$
$972$		0			0
$973$	−27.4954			−0.881463
$974$		0			0
$975$	−4.29589	−	0.416932i	−0.137579	−	0.0133525i
$976$		0			0
$977$	−61.1629			−1.95677			−0.978387	−	0.206781i	$-0.933701\pi$
$977$	−61.1629			−1.95677			−0.978387	+	0.206781i	$0.933701\pi$
$978$		0			0
$979$	4.34468			0.138857
$980$		0			0
$981$	14.9646	+	14.9681i	0.477784	+	0.477893i
$982$		0			0
$983$		−	3.87323i		−	0.123537i	−0.998091	−	0.0617684i	$-0.980326\pi$
$983$		−	3.87323i		−	0.123537i	0.998091	−	0.0617684i	$-0.0196740\pi$
$984$		0			0
$985$	−56.2484			−1.79222
$986$		0			0
$987$	−3.75721	−	1.55654i	−0.119593	−	0.0495452i
$988$		0			0
$989$	27.1715			0.864003
$990$		0			0
$991$			20.9226i			0.664627i	0.943169	+	0.332313i	$0.107829\pi$
$991$			20.9226i			0.664627i	−0.943169	+	0.332313i	$0.892171\pi$
$992$		0			0
$993$	−34.5540	−	14.3150i	−1.09654	−	0.454274i
$994$		0			0
$995$	−26.0197			−0.824879
$996$		0			0
$997$			46.7104i			1.47933i	0.672974	+	0.739666i	$0.265016\pi$
$997$			46.7104i			1.47933i	−0.672974	+	0.739666i	$0.734984\pi$
$998$		0			0
$999$	−36.6541	+	15.1753i	−1.15969	+	0.480125i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1248.2.h.c.623.19		32
3.2	odd	2	inner	1248.2.h.c.623.21		32
4.3	odd	2		312.2.h.c.155.4	yes	32
8.3	odd	2	inner	1248.2.h.c.623.20		32
8.5	even	2		312.2.h.c.155.2	yes	32
12.11	even	2		312.2.h.c.155.29	yes	32
13.12	even	2	inner	1248.2.h.c.623.18		32
24.5	odd	2		312.2.h.c.155.31	yes	32
24.11	even	2	inner	1248.2.h.c.623.22		32
39.38	odd	2	inner	1248.2.h.c.623.24		32
52.51	odd	2		312.2.h.c.155.30	yes	32
104.51	odd	2	inner	1248.2.h.c.623.17		32
104.77	even	2		312.2.h.c.155.32	yes	32
156.155	even	2		312.2.h.c.155.3	yes	32
312.77	odd	2		312.2.h.c.155.1	✓	32
312.155	even	2	inner	1248.2.h.c.623.23		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.2.h.c.155.1	✓	32	312.77	odd	2
312.2.h.c.155.2	yes	32	8.5	even	2
312.2.h.c.155.3	yes	32	156.155	even	2
312.2.h.c.155.4	yes	32	4.3	odd	2
312.2.h.c.155.29	yes	32	12.11	even	2
312.2.h.c.155.30	yes	32	52.51	odd	2
312.2.h.c.155.31	yes	32	24.5	odd	2
312.2.h.c.155.32	yes	32	104.77	even	2
1248.2.h.c.623.17		32	104.51	odd	2	inner
1248.2.h.c.623.18		32	13.12	even	2	inner
1248.2.h.c.623.19		32	1.1	even	1	trivial
1248.2.h.c.623.20		32	8.3	odd	2	inner
1248.2.h.c.623.21		32	3.2	odd	2	inner
1248.2.h.c.623.22		32	24.11	even	2	inner
1248.2.h.c.623.23		32	312.155	even	2	inner
1248.2.h.c.623.24		32	39.38	odd	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:	\( N \)	\(=\)	\( 1248 = 2^{5} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1248.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.662918 - 1.60017i) q^{3} +2.38561i q^{5} +2.63829 q^{7} +(-2.12108 - 2.12156i) q^{9} +O(q^{10})\)	\(q+(0.662918 - 1.60017i) q^{3} +2.38561i q^{5} +2.63829 q^{7} +(-2.12108 - 2.12156i) q^{9} +0.764084 q^{11} +(1.05175 + 3.44874i) q^{13} +(3.81737 + 1.58146i) q^{15} -0.584523i q^{17} +4.28969i q^{19} +(1.74897 - 4.22170i) q^{21} +4.39993 q^{23} -0.691126 q^{25} +(-4.80096 + 1.98766i) q^{27} -8.65308 q^{29} +7.09996 q^{31} +(0.506525 - 1.22266i) q^{33} +6.29391i q^{35} +7.63475 q^{37} +(6.21579 + 0.603265i) q^{39} -0.139556 q^{41} +6.17544 q^{43} +(5.06122 - 5.06006i) q^{45} -0.889975i q^{47} -0.0394518 q^{49} +(-0.935336 - 0.387491i) q^{51} +0.803622 q^{53} +1.82280i q^{55} +(6.86423 + 2.84371i) q^{57} -9.13099 q^{59} +9.47787i q^{61} +(-5.59601 - 5.59729i) q^{63} +(-8.22735 + 2.50905i) q^{65} -11.5441i q^{67} +(2.91679 - 7.04063i) q^{69} +2.56670i q^{71} +13.3885i q^{73} +(-0.458160 + 1.10592i) q^{75} +2.01587 q^{77} -10.9069i q^{79} +(-0.00205378 + 9.00000i) q^{81} +14.6776 q^{83} +1.39444 q^{85} +(-5.73629 + 13.8464i) q^{87} +5.68613 q^{89} +(2.77481 + 9.09877i) q^{91} +(4.70669 - 11.3611i) q^{93} -10.2335 q^{95} -10.4965i q^{97} +(-1.62068 - 1.62105i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 4 q^{3} - 28 q^{9}+O(q^{10}) \) 32 * q + 4 * q^3 - 28 * q^9	\( 32 q + 4 q^{3} - 28 q^{9} + 56 q^{25} + 16 q^{27} - 40 q^{43} + 8 q^{49} + 52 q^{51} + 8 q^{75} - 76 q^{81} + 56 q^{91}+O(q^{100}) \) 32 * q + 4 * q^3 - 28 * q^9 + 56 * q^25 + 16 * q^27 - 40 * q^43 + 8 * q^49 + 52 * q^51 + 8 * q^75 - 76 * q^81 + 56 * q^91

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