LMFDB - Embedded newform 2496.2.j.e.287.18

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2496,2,Mod(287,2496)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2496.287"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2496, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2496 = 2^{6} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2496.j (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [32,0,0,0,0,0,0,0,-4,0,0,0,0,0,0,0,0,0,-32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$19.9306603445$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			287.18
Character	$\chi$	$=$	2496.287
Dual form			2496.2.j.e.287.20

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.0969806 - 1.72933i) q^{3} +3.03737 q^{5} -2.31810i q^{7} +(-2.98119 - 0.335424i) q^{9} -1.37788i q^{11} -1.00000i q^{13} +(0.294566 - 5.25262i) q^{15} -2.25342i q^{17} -7.84800 q^{19} +(-4.00876 - 0.224810i) q^{21} -5.17983 q^{23} +4.22559 q^{25} +(-0.869177 + 5.12294i) q^{27} -0.308414 q^{29} -5.62113i q^{31} +(-2.38281 - 0.133628i) q^{33} -7.04090i q^{35} +5.13447i q^{37} +(-1.72933 - 0.0969806i) q^{39} +6.69261i q^{41} +0.695678 q^{43} +(-9.05496 - 1.01880i) q^{45} -5.99870 q^{47} +1.62643 q^{49} +(-3.89691 - 0.218538i) q^{51} -11.8413 q^{53} -4.18512i q^{55} +(-0.761104 + 13.5718i) q^{57} -13.7784i q^{59} -10.6018i q^{61} +(-0.777544 + 6.91068i) q^{63} -3.03737i q^{65} +10.9844 q^{67} +(-0.502343 + 8.95765i) q^{69} +11.4819 q^{71} +12.3889 q^{73} +(0.409800 - 7.30745i) q^{75} -3.19406 q^{77} +7.61698i q^{79} +(8.77498 + 1.99992i) q^{81} -4.21772i q^{83} -6.84445i q^{85} +(-0.0299102 + 0.533351i) q^{87} +9.89782i q^{89} -2.31810 q^{91} +(-9.72080 - 0.545140i) q^{93} -23.8372 q^{95} -1.59572 q^{97} +(-0.462173 + 4.10772i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 4 q^{9} - 32 q^{19} + 56 q^{25} + 24 q^{27} + 32 q^{33} - 16 q^{43} + 8 q^{49} + 40 q^{51} - 8 q^{57} + 16 q^{67} - 16 q^{73} + 120 q^{75} - 44 q^{81} - 80 q^{97}+O(q^{100}) $ 32 * q - 4 * q^9 - 32 * q^19 + 56 * q^25 + 24 * q^27 + 32 * q^33 - 16 * q^43 + 8 * q^49 + 40 * q^51 - 8 * q^57 + 16 * q^67 - 16 * q^73 + 120 * q^75 - 44 * q^81 - 80 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2496\mathbb{Z}\right)^\times$.

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

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
$3$	0.0969806	−	1.72933i	0.0559918	−	0.998431i
$3$	0.0969806	−	1.72933i	0.0559918	−	0.998431i
$4$		0			0
$5$	3.03737			1.35835			0.679176	−	0.733976i	$-0.262337\pi$
$5$	3.03737			1.35835			0.679176	+	0.733976i	$0.262337\pi$
$6$		0			0
$7$		−	2.31810i		−	0.876158i	−0.898937	−	0.438079i	$-0.855659\pi$
$7$		−	2.31810i		−	0.876158i	0.898937	−	0.438079i	$-0.144341\pi$
$8$		0			0
$9$	−2.98119	−	0.335424i	−0.993730	−	0.111808i
$10$		0			0
$11$		−	1.37788i		−	0.415446i	−0.978188	−	0.207723i	$-0.933395\pi$
$11$		−	1.37788i		−	0.415446i	0.978188	−	0.207723i	$-0.0666053\pi$
$12$		0			0
$13$		−	1.00000i		−	0.277350i
$13$		−	1.00000i		−	0.277350i
$14$		0			0
$15$	0.294566	−	5.25262i	0.0760565	−	1.35622i
$16$		0			0
$17$		−	2.25342i		−	0.546534i	−0.961938	−	0.273267i	$-0.911896\pi$
$17$		−	2.25342i		−	0.546534i	0.961938	−	0.273267i	$-0.0881042\pi$
$18$		0			0
$19$	−7.84800			−1.80045			−0.900227	−	0.435420i	$-0.856600\pi$
$19$	−7.84800			−1.80045			−0.900227	+	0.435420i	$0.856600\pi$
$20$		0			0
$21$	−4.00876	−	0.224810i	−0.874783	−	0.0490576i
$22$		0			0
$23$	−5.17983			−1.08007			−0.540035	−	0.841643i	$-0.681589\pi$
$23$	−5.17983			−1.08007			−0.540035	+	0.841643i	$0.681589\pi$
$24$		0			0
$25$	4.22559			0.845118
$26$		0			0
$27$	−0.869177	+	5.12294i	−0.167273	+	0.985911i
$28$		0			0
$29$	−0.308414			−0.0572711			−0.0286355	−	0.999590i	$-0.509116\pi$
$29$	−0.308414			−0.0572711			−0.0286355	+	0.999590i	$0.509116\pi$
$30$		0			0
$31$		−	5.62113i		−	1.00958i	−0.863241	−	0.504792i	$-0.831569\pi$
$31$		−	5.62113i		−	1.00958i	0.863241	−	0.504792i	$-0.168431\pi$
$32$		0			0
$33$	−2.38281	−	0.133628i	−0.414795	−	0.0232616i
$34$		0			0
$35$		−	7.04090i		−	1.19013i
$36$		0			0
$37$			5.13447i			0.844101i	0.906572	+	0.422051i	$0.138690\pi$
$37$			5.13447i			0.844101i	−0.906572	+	0.422051i	$0.861310\pi$
$38$		0			0
$39$	−1.72933	−	0.0969806i	−0.276915	−	0.0155293i
$40$		0			0
$41$			6.69261i			1.04521i	0.852575	+	0.522605i	$0.175040\pi$
$41$			6.69261i			1.04521i	−0.852575	+	0.522605i	$0.824960\pi$
$42$		0			0
$43$	0.695678			0.106090			0.0530449	−	0.998592i	$-0.483107\pi$
$43$	0.695678			0.106090			0.0530449	+	0.998592i	$0.483107\pi$
$44$		0			0
$45$	−9.05496	−	1.01880i	−1.34983	−	0.151874i
$46$		0			0
$47$	−5.99870			−0.875001			−0.437500	−	0.899218i	$-0.644136\pi$
$47$	−5.99870			−0.875001			−0.437500	+	0.899218i	$0.644136\pi$
$48$		0			0
$49$	1.62643			0.232348
$50$		0			0
$51$	−3.89691	−	0.218538i	−0.545676	−	0.0306014i
$52$		0			0
$53$	−11.8413			−1.62652			−0.813261	−	0.581899i	$-0.802310\pi$
$53$	−11.8413			−1.62652			−0.813261	+	0.581899i	$0.802310\pi$
$54$		0			0
$55$		−	4.18512i		−	0.564322i
$56$		0			0
$57$	−0.761104	+	13.5718i	−0.100811	+	1.79763i
$58$		0			0
$59$		−	13.7784i		−	1.79379i	−0.442243	−	0.896895i	$-0.645817\pi$
$59$		−	13.7784i		−	1.79379i	0.442243	−	0.896895i	$-0.354183\pi$
$60$		0			0
$61$		−	10.6018i		−	1.35742i	−0.734408	−	0.678708i	$-0.762540\pi$
$61$		−	10.6018i		−	1.35742i	0.734408	−	0.678708i	$-0.237460\pi$
$62$		0			0
$63$	−0.777544	+	6.91068i	−0.0979613	+	0.870664i
$64$		0			0
$65$		−	3.03737i		−	0.376739i
$66$		0			0
$67$	10.9844			1.34196			0.670978	−	0.741477i	$-0.265875\pi$
$67$	10.9844			1.34196			0.670978	+	0.741477i	$0.265875\pi$
$68$		0			0
$69$	−0.502343	+	8.95765i	−0.0604750	+	1.07837i
$70$		0			0
$71$	11.4819			1.36265			0.681323	−	0.731983i	$-0.261405\pi$
$71$	11.4819			1.36265			0.681323	+	0.731983i	$0.261405\pi$
$72$		0			0
$73$	12.3889			1.45001			0.725005	−	0.688743i	$-0.241837\pi$
$73$	12.3889			1.45001			0.725005	+	0.688743i	$0.241837\pi$
$74$		0			0
$75$	0.409800	−	7.30745i	0.0473197	−	0.843792i
$76$		0			0
$77$	−3.19406			−0.363996
$78$		0			0
$79$			7.61698i			0.856977i	0.903547	+	0.428489i	$0.140954\pi$
$79$			7.61698i			0.856977i	−0.903547	+	0.428489i	$0.859046\pi$
$80$		0			0
$81$	8.77498	+	1.99992i	0.974998	+	0.222214i
$82$		0			0
$83$		−	4.21772i		−	0.462955i	−0.972840	−	0.231477i	$-0.925644\pi$
$83$		−	4.21772i		−	0.462955i	0.972840	−	0.231477i	$-0.0743560\pi$
$84$		0			0
$85$		−	6.84445i		−	0.742385i
$86$		0			0
$87$	−0.0299102	+	0.533351i	−0.00320671	+	0.0571812i
$88$		0			0
$89$			9.89782i			1.04917i	0.851359	+	0.524584i	$0.175779\pi$
$89$			9.89782i			1.04917i	−0.851359	+	0.524584i	$0.824221\pi$
$90$		0			0
$91$	−2.31810			−0.243002
$92$		0			0
$93$	−9.72080	−	0.545140i	−1.00800	−	0.0565284i
$94$		0			0
$95$	−23.8372			−2.44565
$96$		0			0
$97$	−1.59572			−0.162021			−0.0810105	−	0.996713i	$-0.525815\pi$
$97$	−1.59572			−0.162021			−0.0810105	+	0.996713i	$0.525815\pi$
$98$		0			0
$99$	−0.462173	+	4.10772i	−0.0464502	+	0.412841i
$100$		0			0
$101$	−0.589367			−0.0586442			−0.0293221	−	0.999570i	$-0.509335\pi$
$101$	−0.589367			−0.0586442			−0.0293221	+	0.999570i	$0.509335\pi$
$102$		0			0
$103$		−	0.375101i		−	0.0369598i	−0.999829	−	0.0184799i	$-0.994117\pi$
$103$		−	0.375101i		−	0.0369598i	0.999829	−	0.0184799i	$-0.00588267\pi$
$104$		0			0
$105$	−12.1761	−	0.682831i	−1.18826	−	0.0666375i
$106$		0			0
$107$		−	3.23268i		−	0.312515i	−0.987716	−	0.156257i	$-0.950057\pi$
$107$		−	3.23268i		−	0.312515i	0.987716	−	0.156257i	$-0.0499429\pi$
$108$		0			0
$109$		−	16.9994i		−	1.62825i	−0.580689	−	0.814125i	$-0.697217\pi$
$109$		−	16.9994i		−	1.62825i	0.580689	−	0.814125i	$-0.302783\pi$
$110$		0			0
$111$	8.87921	+	0.497944i	0.842777	+	0.0472627i
$112$		0			0
$113$		−	15.8565i		−	1.49165i	−0.666139	−	0.745827i	$-0.732054\pi$
$113$		−	15.8565i		−	1.49165i	0.666139	−	0.745827i	$-0.267946\pi$
$114$		0			0
$115$	−15.7330			−1.46711
$116$		0			0
$117$	−0.335424	+	2.98119i	−0.0310099	+	0.275611i
$118$		0			0
$119$	−5.22363			−0.478850
$120$		0			0
$121$	9.10145			0.827404
$122$		0			0
$123$	11.5737	+	0.649053i	1.04357	+	0.0585231i
$124$		0			0
$125$	−2.35217			−0.210384
$126$		0			0
$127$			3.72234i			0.330304i	0.986268	+	0.165152i	$0.0528115\pi$
$127$			3.72234i			0.330304i	−0.986268	+	0.165152i	$0.947189\pi$
$128$		0			0
$129$	0.0674673	−	1.20306i	0.00594016	−	0.105923i
$130$		0			0
$131$		−	0.729041i		−	0.0636966i	−0.999493	−	0.0318483i	$-0.989861\pi$
$131$		−	0.729041i		−	0.0636966i	0.999493	−	0.0318483i	$-0.0101393\pi$
$132$		0			0
$133$			18.1924i			1.57748i
$134$		0			0
$135$	−2.64001	+	15.5602i	−0.227216	+	1.33921i
$136$		0			0
$137$			2.55115i			0.217959i	0.994044	+	0.108980i	$0.0347584\pi$
$137$			2.55115i			0.217959i	−0.994044	+	0.108980i	$0.965242\pi$
$138$		0			0
$139$	20.4368			1.73342			0.866712	−	0.498808i	$-0.166229\pi$
$139$	20.4368			1.73342			0.866712	+	0.498808i	$0.166229\pi$
$140$		0			0
$141$	−0.581758	+	10.3738i	−0.0489929	+	0.873628i
$142$		0			0
$143$	−1.37788			−0.115224
$144$		0			0
$145$	−0.936767			−0.0777942
$146$		0			0
$147$	0.157733	−	2.81265i	0.0130096	−	0.231983i
$148$		0			0
$149$	−10.0240			−0.821201			−0.410600	−	0.911815i	$-0.634681\pi$
$149$	−10.0240			−0.821201			−0.410600	+	0.911815i	$0.634681\pi$
$150$		0			0
$151$			4.30865i			0.350633i	0.984512	+	0.175316i	$0.0560948\pi$
$151$			4.30865i			0.350633i	−0.984512	+	0.175316i	$0.943905\pi$
$152$		0			0
$153$	−0.755849	+	6.71786i	−0.0611068	+	0.543107i
$154$		0			0
$155$		−	17.0734i		−	1.37137i
$156$		0			0
$157$		−	11.8001i		−	0.941749i	−0.882200	−	0.470875i	$-0.843938\pi$
$157$		−	11.8001i		−	0.941749i	0.882200	−	0.470875i	$-0.156062\pi$
$158$		0			0
$159$	−1.14837	+	20.4775i	−0.0910719	+	1.62397i
$160$		0			0
$161$			12.0073i			0.946311i
$162$		0			0
$163$	−6.95849			−0.545031			−0.272515	−	0.962151i	$-0.587856\pi$
$163$	−6.95849			−0.545031			−0.272515	+	0.962151i	$0.587856\pi$
$164$		0			0
$165$	−7.23748	−	0.405876i	−0.563437	−	0.0315974i
$166$		0			0
$167$	15.5317			1.20188			0.600938	−	0.799296i	$-0.294794\pi$
$167$	15.5317			1.20188			0.600938	+	0.799296i	$0.294794\pi$
$168$		0			0
$169$	−1.00000			−0.0769231
$170$		0			0
$171$	23.3964	+	2.63241i	1.78917	+	0.201305i
$172$		0			0
$173$	−3.04950			−0.231849			−0.115925	−	0.993258i	$-0.536983\pi$
$173$	−3.04950			−0.231849			−0.115925	+	0.993258i	$0.536983\pi$
$174$		0			0
$175$		−	9.79532i		−	0.740457i
$176$		0			0
$177$	−23.8274	−	1.33623i	−1.79098	−	0.100438i
$178$		0			0
$179$			5.46947i			0.408808i	0.978887	+	0.204404i	$0.0655256\pi$
$179$			5.46947i			0.408808i	−0.978887	+	0.204404i	$0.934474\pi$
$180$		0			0
$181$			13.2616i			0.985730i	0.870106	+	0.492865i	$0.164050\pi$
$181$			13.2616i			0.985730i	−0.870106	+	0.492865i	$0.835950\pi$
$182$		0			0
$183$	−18.3340	−	1.02817i	−1.35529	−	0.0760042i
$184$		0			0
$185$			15.5953i			1.14659i
$186$		0			0
$187$	−3.10494			−0.227055
$188$		0			0
$189$	11.8755	+	2.01484i	0.863813	+	0.146558i
$190$		0			0
$191$	−5.05470			−0.365745			−0.182873	−	0.983137i	$-0.558540\pi$
$191$	−5.05470			−0.365745			−0.182873	+	0.983137i	$0.558540\pi$
$192$		0			0
$193$	10.3602			0.745742			0.372871	−	0.927883i	$-0.378373\pi$
$193$	10.3602			0.745742			0.372871	+	0.927883i	$0.378373\pi$
$194$		0			0
$195$	−5.25262	−	0.294566i	−0.376148	−	0.0210943i
$196$		0			0
$197$	11.6235			0.828138			0.414069	−	0.910245i	$-0.364107\pi$
$197$	11.6235			0.828138			0.414069	+	0.910245i	$0.364107\pi$
$198$		0			0
$199$		−	3.88415i		−	0.275340i	−0.990478	−	0.137670i	$-0.956039\pi$
$199$		−	3.88415i		−	0.275340i	0.990478	−	0.137670i	$-0.0439613\pi$
$200$		0			0
$201$	1.06527	−	18.9957i	0.0751385	−	1.33985i
$202$		0			0
$203$			0.714934i			0.0501785i
$204$		0			0
$205$			20.3279i			1.41976i
$206$		0			0
$207$	15.4421	+	1.73744i	1.07330	+	0.120760i
$208$		0			0
$209$			10.8136i			0.747992i
$210$		0			0
$211$	−16.4286			−1.13099			−0.565495	−	0.824751i	$-0.691315\pi$
$211$	−16.4286			−1.13099			−0.565495	+	0.824751i	$0.691315\pi$
$212$		0			0
$213$	1.11352	−	19.8560i	0.0762970	−	1.36051i
$214$		0			0
$215$	2.11303			0.144107
$216$		0			0
$217$	−13.0303			−0.884555
$218$		0			0
$219$	1.20148	−	21.4245i	0.0811887	−	1.44774i
$220$		0			0
$221$	−2.25342			−0.151581
$222$		0			0
$223$		−	22.6772i		−	1.51858i	−0.650752	−	0.759290i	$-0.725546\pi$
$223$		−	22.6772i		−	1.51858i	0.650752	−	0.759290i	$-0.274454\pi$
$224$		0			0
$225$	−12.5973	−	1.41736i	−0.839819	−	0.0944909i
$226$		0			0
$227$		−	21.7066i		−	1.44072i	−0.693602	−	0.720359i	$-0.743977\pi$
$227$		−	21.7066i		−	1.44072i	0.693602	−	0.720359i	$-0.256023\pi$
$228$		0			0
$229$		−	6.05572i		−	0.400173i	−0.979778	−	0.200087i	$-0.935878\pi$
$229$		−	6.05572i		−	0.400173i	0.979778	−	0.200087i	$-0.0641223\pi$
$230$		0			0
$231$	−0.309762	+	5.52359i	−0.0203808	+	0.363425i
$232$		0			0
$233$			7.25738i			0.475447i	0.971333	+	0.237723i	$0.0764012\pi$
$233$			7.25738i			0.475447i	−0.971333	+	0.237723i	$0.923599\pi$
$234$		0			0
$235$	−18.2203			−1.18856
$236$		0			0
$237$	13.1723	+	0.738699i	0.855633	+	0.0479837i
$238$		0			0
$239$	−7.11715			−0.460370			−0.230185	−	0.973147i	$-0.573933\pi$
$239$	−7.11715			−0.460370			−0.230185	+	0.973147i	$0.573933\pi$
$240$		0			0
$241$	−12.3453			−0.795232			−0.397616	−	0.917552i	$-0.630162\pi$
$241$	−12.3453			−0.795232			−0.397616	+	0.917552i	$0.630162\pi$
$242$		0			0
$243$	4.30954	−	14.9809i	0.276457	−	0.961026i
$244$		0			0
$245$	4.94008			0.315610
$246$		0			0
$247$			7.84800i			0.499356i
$248$		0			0
$249$	−7.29385	−	0.409037i	−0.462229	−	0.0259217i
$250$		0			0
$251$			21.6490i			1.36647i	0.730198	+	0.683235i	$0.239428\pi$
$251$			21.6490i			1.36647i	−0.730198	+	0.683235i	$0.760572\pi$
$252$		0			0
$253$			7.13718i			0.448711i
$254$		0			0
$255$	−11.8363	−	0.663779i	−0.741220	−	0.0415674i
$256$		0			0
$257$		−	23.1241i		−	1.44244i	−0.692704	−	0.721222i	$-0.743581\pi$
$257$		−	23.1241i		−	1.44244i	0.692704	−	0.721222i	$-0.256419\pi$
$258$		0			0
$259$	11.9022			0.739566
$260$		0			0
$261$	0.919441	+	0.103449i	0.0569120	+	0.00640336i
$262$		0			0
$263$	−26.4383			−1.63025			−0.815127	−	0.579282i	$-0.803333\pi$
$263$	−26.4383			−1.63025			−0.815127	+	0.579282i	$0.803333\pi$
$264$		0			0
$265$	−35.9663			−2.20939
$266$		0			0
$267$	17.1166	+	0.959897i	1.04752	+	0.0587447i
$268$		0			0
$269$	−27.2047			−1.65870			−0.829351	−	0.558728i	$-0.811289\pi$
$269$	−27.2047			−1.65870			−0.829351	+	0.558728i	$0.811289\pi$
$270$		0			0
$271$		−	2.81994i		−	0.171299i	−0.996325	−	0.0856495i	$-0.972703\pi$
$271$		−	2.81994i		−	0.171299i	0.996325	−	0.0856495i	$-0.0272965\pi$
$272$		0			0
$273$	−0.224810	+	4.00876i	−0.0136061	+	0.242621i
$274$		0			0
$275$		−	5.82235i		−	0.351101i
$276$		0			0
$277$		−	0.749703i		−	0.0450453i	−0.999746	−	0.0225226i	$-0.992830\pi$
$277$		−	0.749703i		−	0.0450453i	0.999746	−	0.0225226i	$-0.00716978\pi$
$278$		0			0
$279$	−1.88546	+	16.7576i	−0.112879	+	1.00325i
$280$		0			0
$281$			12.3283i			0.735444i	0.929936	+	0.367722i	$0.119862\pi$
$281$			12.3283i			0.735444i	−0.929936	+	0.367722i	$0.880138\pi$
$282$		0			0
$283$	−21.3365			−1.26833			−0.634163	−	0.773199i	$-0.718655\pi$
$283$	−21.3365			−1.26833			−0.634163	+	0.773199i	$0.718655\pi$
$284$		0			0
$285$	−2.31175	+	41.2226i	−0.136936	+	2.44181i
$286$		0			0
$287$	15.5141			0.915768
$288$		0			0
$289$	11.9221			0.701301
$290$		0			0
$291$	−0.154754	+	2.75954i	−0.00907185	+	0.161767i
$292$		0			0
$293$	−23.7964			−1.39020			−0.695100	−	0.718913i	$-0.744640\pi$
$293$	−23.7964			−1.39020			−0.695100	+	0.718913i	$0.744640\pi$
$294$		0			0
$295$		−	41.8499i		−	2.43660i
$296$		0			0
$297$	7.05880	+	1.19762i	0.409593	+	0.0694930i
$298$		0			0
$299$			5.17983i			0.299557i
$300$		0			0
$301$		−	1.61265i		−	0.0929514i
$302$		0			0
$303$	−0.0571572	+	1.01921i	−0.00328359	+	0.0585522i
$304$		0			0
$305$		−	32.2014i		−	1.84385i
$306$		0			0
$307$	−8.35764			−0.476996			−0.238498	−	0.971143i	$-0.576655\pi$
$307$	−8.35764			−0.476996			−0.238498	+	0.971143i	$0.576655\pi$
$308$		0			0
$309$	−0.648674	−	0.0363775i	−0.0369018	−	0.00206944i
$310$		0			0
$311$	24.5264			1.39077			0.695384	−	0.718639i	$-0.255234\pi$
$311$	24.5264			1.39077			0.695384	+	0.718639i	$0.255234\pi$
$312$		0			0
$313$	15.0553			0.850973			0.425487	−	0.904965i	$-0.360103\pi$
$313$	15.0553			0.850973			0.425487	+	0.904965i	$0.360103\pi$
$314$		0			0
$315$	−2.36169	+	20.9903i	−0.133066	+	1.18267i
$316$		0			0
$317$	2.30671			0.129558			0.0647790	−	0.997900i	$-0.479366\pi$
$317$	2.30671			0.129558			0.0647790	+	0.997900i	$0.479366\pi$
$318$		0			0
$319$			0.424958i			0.0237931i
$320$		0			0
$321$	−5.59038	−	0.313507i	−0.312024	−	0.0174983i
$322$		0			0
$323$			17.6848i			0.984009i
$324$		0			0
$325$		−	4.22559i		−	0.234394i
$326$		0			0
$327$	−29.3977	−	1.64862i	−1.62570	−	0.0911687i
$328$		0			0
$329$			13.9056i			0.766639i
$330$		0			0
$331$	−12.4922			−0.686635			−0.343318	−	0.939219i	$-0.611551\pi$
$331$	−12.4922			−0.686635			−0.343318	+	0.939219i	$0.611551\pi$
$332$		0			0
$333$	1.72222	−	15.3068i	0.0943772	−	0.838809i
$334$		0			0
$335$	33.3636			1.82285
$336$		0			0
$337$	32.6356			1.77777			0.888887	−	0.458127i	$-0.151480\pi$
$337$	32.6356			1.77777			0.888887	+	0.458127i	$0.151480\pi$
$338$		0			0
$339$	−27.4212	−	1.53777i	−1.48931	−	0.0835204i
$340$		0			0
$341$	−7.74523			−0.419428
$342$		0			0
$343$		−	19.9969i		−	1.07973i
$344$		0			0
$345$	−1.52580	+	27.2077i	−0.0821463	+	1.46481i
$346$		0			0
$347$			33.6017i			1.80383i	0.431910	+	0.901917i	$0.357840\pi$
$347$			33.6017i			1.80383i	−0.431910	+	0.901917i	$0.642160\pi$
$348$		0			0
$349$		−	3.36050i		−	0.179883i	−0.995947	−	0.0899416i	$-0.971332\pi$
$349$		−	3.36050i		−	0.179883i	0.995947	−	0.0899416i	$-0.0286680\pi$
$350$		0			0
$351$	5.12294	+	0.869177i	0.273442	+	0.0463932i
$352$		0			0
$353$			7.86078i			0.418387i	0.977874	+	0.209193i	$0.0670838\pi$
$353$			7.86078i			0.418387i	−0.977874	+	0.209193i	$0.932916\pi$
$354$		0			0
$355$	34.8746			1.85095
$356$		0			0
$357$	−0.506591	+	9.03340i	−0.0268116	+	0.478098i
$358$		0			0
$359$	5.39566			0.284772			0.142386	−	0.989811i	$-0.454523\pi$
$359$	5.39566			0.284772			0.142386	+	0.989811i	$0.454523\pi$
$360$		0			0
$361$	42.5911			2.24164
$362$		0			0
$363$	0.882664	−	15.7394i	0.0463278	−	0.826106i
$364$		0			0
$365$	37.6296			1.96962
$366$		0			0
$367$		−	11.7595i		−	0.613839i	−0.951736	−	0.306919i	$-0.900702\pi$
$367$		−	11.7595i		−	0.613839i	0.951736	−	0.306919i	$-0.0992982\pi$
$368$		0			0
$369$	2.24486	−	19.9519i	0.116863	−	1.03866i
$370$		0			0
$371$			27.4492i			1.42509i
$372$		0			0
$373$		−	37.4755i		−	1.94041i	−0.242289	−	0.970204i	$-0.577898\pi$
$373$		−	37.4755i		−	1.94041i	0.242289	−	0.970204i	$-0.422102\pi$
$374$		0			0
$375$	−0.228115	+	4.06768i	−0.0117798	+	0.210054i
$376$		0			0
$377$			0.308414i			0.0158841i
$378$		0			0
$379$	21.5977			1.10940			0.554700	−	0.832050i	$-0.312833\pi$
$379$	21.5977			1.10940			0.554700	+	0.832050i	$0.312833\pi$
$380$		0			0
$381$	6.43716	+	0.360995i	0.329786	+	0.0184943i
$382$		0			0
$383$	9.37754			0.479170			0.239585	−	0.970875i	$-0.422989\pi$
$383$	9.37754			0.479170			0.239585	+	0.970875i	$0.422989\pi$
$384$		0			0
$385$	−9.70152			−0.494435
$386$		0			0
$387$	−2.07395	−	0.233347i	−0.105425	−	0.0118617i
$388$		0			0
$389$	21.3532			1.08265			0.541326	−	0.840813i	$-0.317922\pi$
$389$	21.3532			1.08265			0.541326	+	0.840813i	$0.317922\pi$
$390$		0			0
$391$			11.6723i			0.590294i
$392$		0			0
$393$	−1.26076	−	0.0707029i	−0.0635967	−	0.00356649i
$394$		0			0
$395$			23.1355i			1.16408i
$396$		0			0
$397$		−	34.8270i		−	1.74792i	−0.486002	−	0.873958i	$-0.661545\pi$
$397$		−	34.8270i		−	1.74792i	0.486002	−	0.873958i	$-0.338455\pi$
$398$		0			0
$399$	31.4608	+	1.76431i	1.57501	+	0.0883261i
$400$		0			0
$401$			3.00045i			0.149835i	0.997190	+	0.0749176i	$0.0238694\pi$
$401$			3.00045i			0.149835i	−0.997190	+	0.0749176i	$0.976131\pi$
$402$		0			0
$403$	−5.62113			−0.280008
$404$		0			0
$405$	26.6528	+	6.07450i	1.32439	+	0.301844i
$406$		0			0
$407$	7.07468			0.350679
$408$		0			0
$409$	−12.8978			−0.637753			−0.318877	−	0.947796i	$-0.603306\pi$
$409$	−12.8978			−0.637753			−0.318877	+	0.947796i	$0.603306\pi$
$410$		0			0
$411$	4.41179	+	0.247412i	0.217617	+	0.0122039i
$412$		0			0
$413$	−31.9396			−1.57164
$414$		0			0
$415$		−	12.8108i		−	0.628855i
$416$		0			0
$417$	1.98197	−	35.3420i	0.0970576	−	1.73071i
$418$		0			0
$419$			29.5342i			1.44284i	0.692497	+	0.721421i	$0.256511\pi$
$419$			29.5342i			1.44284i	−0.692497	+	0.721421i	$0.743489\pi$
$420$		0			0
$421$		−	19.8386i		−	0.966873i	−0.875379	−	0.483437i	$-0.839388\pi$
$421$		−	19.8386i		−	0.966873i	0.875379	−	0.483437i	$-0.160612\pi$
$422$		0			0
$423$	17.8833	+	2.01211i	0.869514	+	0.0978320i
$424$		0			0
$425$		−	9.52201i		−	0.461885i
$426$		0			0
$427$	−24.5759			−1.18931
$428$		0			0
$429$	−0.133628	+	2.38281i	−0.00645160	+	0.115043i
$430$		0			0
$431$	11.6413			0.560743			0.280372	−	0.959892i	$-0.409542\pi$
$431$	11.6413			0.560743			0.280372	+	0.959892i	$0.409542\pi$
$432$		0			0
$433$	35.1179			1.68766			0.843829	−	0.536612i	$-0.180296\pi$
$433$	35.1179			1.68766			0.843829	+	0.536612i	$0.180296\pi$
$434$		0			0
$435$	−0.0908482	+	1.61998i	−0.00435584	+	0.0776722i
$436$		0			0
$437$	40.6513			1.94462
$438$		0			0
$439$		−	6.33417i		−	0.302313i	−0.988510	−	0.151157i	$-0.951700\pi$
$439$		−	6.33417i		−	0.302313i	0.988510	−	0.151157i	$-0.0482998\pi$
$440$		0			0
$441$	−4.84871	−	0.545545i	−0.230891	−	0.0259783i
$442$		0			0
$443$		−	26.1460i		−	1.24223i	−0.783718	−	0.621116i	$-0.786679\pi$
$443$		−	26.1460i		−	1.24223i	0.783718	−	0.621116i	$-0.213321\pi$
$444$		0			0
$445$			30.0633i			1.42514i
$446$		0			0
$447$	−0.972137	+	17.3349i	−0.0459805	+	0.819913i
$448$		0			0
$449$		−	8.80186i		−	0.415386i	−0.978194	−	0.207693i	$-0.933405\pi$
$449$		−	8.80186i		−	0.415386i	0.978194	−	0.207693i	$-0.0665954\pi$
$450$		0			0
$451$	9.22161			0.434228
$452$		0			0
$453$	7.45109	+	0.417855i	0.350083	+	0.0196325i
$454$		0			0
$455$	−7.04090			−0.330083
$456$		0			0
$457$	38.1526			1.78470			0.892351	−	0.451342i	$-0.149054\pi$
$457$	38.1526			1.78470			0.892351	+	0.451342i	$0.149054\pi$
$458$		0			0
$459$	11.5441	+	1.95862i	0.538833	+	0.0914204i
$460$		0			0
$461$	5.81449			0.270808			0.135404	−	0.990790i	$-0.456767\pi$
$461$	5.81449			0.270808			0.135404	+	0.990790i	$0.456767\pi$
$462$		0			0
$463$			21.2965i			0.989734i	0.868969	+	0.494867i	$0.164783\pi$
$463$			21.2965i			0.989734i	−0.868969	+	0.494867i	$0.835217\pi$
$464$		0			0
$465$	−29.5256	−	1.65579i	−1.36922	−	0.0767854i
$466$		0			0
$467$			15.2886i			0.707474i	0.935345	+	0.353737i	$0.115089\pi$
$467$			15.2886i			0.707474i	−0.935345	+	0.353737i	$0.884911\pi$
$468$		0			0
$469$		−	25.4628i		−	1.17576i
$470$		0			0
$471$	−20.4063	−	1.14438i	−0.940272	−	0.0527302i
$472$		0			0
$473$		−	0.958560i		−	0.0440746i
$474$		0			0
$475$	−33.1624			−1.52160
$476$		0			0
$477$	35.3011	+	3.97184i	1.61632	+	0.181858i
$478$		0			0
$479$	−40.9222			−1.86978			−0.934892	−	0.354932i	$-0.884504\pi$
$479$	−40.9222			−1.86978			−0.934892	+	0.354932i	$0.884504\pi$
$480$		0			0
$481$	5.13447			0.234112
$482$		0			0
$483$	20.7647	+	1.16448i	0.944826	+	0.0529856i
$484$		0			0
$485$	−4.84679			−0.220081
$486$		0			0
$487$			9.42575i			0.427121i	0.976930	+	0.213561i	$0.0685061\pi$
$487$			9.42575i			0.427121i	−0.976930	+	0.213561i	$0.931494\pi$
$488$		0			0
$489$	−0.674838	+	12.0335i	−0.0305172	+	0.544176i
$490$		0			0
$491$		−	11.3969i		−	0.514336i	−0.966367	−	0.257168i	$-0.917211\pi$
$491$		−	11.3969i		−	0.514336i	0.966367	−	0.257168i	$-0.0827893\pi$
$492$		0			0
$493$			0.694986i			0.0313006i
$494$		0			0
$495$	−1.40379	+	12.4766i	−0.0630957	+	0.560784i
$496$		0			0
$497$		−	26.6160i		−	1.19389i
$498$		0			0
$499$	21.1418			0.946439			0.473219	−	0.880945i	$-0.343092\pi$
$499$	21.1418			0.946439			0.473219	+	0.880945i	$0.343092\pi$
$500$		0			0
$501$	1.50627	−	26.8594i	0.0672951	−	1.19999i
$502$		0			0
$503$	5.34204			0.238190			0.119095	−	0.992883i	$-0.462001\pi$
$503$	5.34204			0.238190			0.119095	+	0.992883i	$0.462001\pi$
$504$		0			0
$505$	−1.79012			−0.0796594
$506$		0			0
$507$	−0.0969806	+	1.72933i	−0.00430706	+	0.0768024i
$508$		0			0
$509$	21.7402			0.963618			0.481809	−	0.876276i	$-0.339980\pi$
$509$	21.7402			0.963618			0.481809	+	0.876276i	$0.339980\pi$
$510$		0			0
$511$		−	28.7186i		−	1.27044i
$512$		0			0
$513$	6.82130	−	40.2049i	0.301168	−	1.77509i
$514$		0			0
$515$		−	1.13932i		−	0.0502044i
$516$		0			0
$517$			8.26549i			0.363516i
$518$		0			0
$519$	−0.295743	+	5.27361i	−0.0129817	+	0.231486i
$520$		0			0
$521$			28.4493i			1.24638i	0.782069	+	0.623192i	$0.214165\pi$
$521$			28.4493i			1.24638i	−0.782069	+	0.623192i	$0.785835\pi$
$522$		0			0
$523$	−9.58879			−0.419289			−0.209644	−	0.977778i	$-0.567231\pi$
$523$	−9.58879			−0.419289			−0.209644	+	0.977778i	$0.567231\pi$
$524$		0			0
$525$	−16.9394	−	0.949956i	−0.739295	−	0.0414595i
$526$		0			0
$527$	−12.6667			−0.551772
$528$		0			0
$529$	3.83064			0.166550
$530$		0			0
$531$	−4.62159	+	41.0759i	−0.200560	+	1.78254i
$532$		0			0
$533$	6.69261			0.289889
$534$		0			0
$535$		−	9.81882i		−	0.424505i
$536$		0			0
$537$	9.45854	+	0.530433i	0.408166	+	0.0228899i
$538$		0			0
$539$		−	2.24103i		−	0.0965280i
$540$		0			0
$541$		−	10.4390i		−	0.448808i	−0.974496	−	0.224404i	$-0.927956\pi$
$541$		−	10.4390i		−	0.448808i	0.974496	−	0.224404i	$-0.0720436\pi$
$542$		0			0
$543$	22.9338	+	1.28612i	0.984184	+	0.0551928i
$544$		0			0
$545$		−	51.6335i		−	2.21174i
$546$		0			0
$547$	11.5237			0.492716			0.246358	−	0.969179i	$-0.420766\pi$
$547$	11.5237			0.492716			0.246358	+	0.969179i	$0.420766\pi$
$548$		0			0
$549$	−3.55608	+	31.6059i	−0.151770	+	1.34890i
$550$		0			0
$551$	2.42044			0.103114
$552$		0			0
$553$	17.6569			0.750847
$554$		0			0
$555$	26.9694	+	1.51244i	1.14479	+	0.0641994i
$556$		0			0
$557$	23.8856			1.01206			0.506032	−	0.862514i	$-0.331112\pi$
$557$	23.8856			1.01206			0.506032	+	0.862514i	$0.331112\pi$
$558$		0			0
$559$		−	0.695678i		−	0.0294240i
$560$		0			0
$561$	−0.301119	+	5.36947i	−0.0127132	+	0.226699i
$562$		0			0
$563$		−	24.5358i		−	1.03406i	−0.855967	−	0.517031i	$-0.827037\pi$
$563$		−	24.5358i		−	1.03406i	0.855967	−	0.517031i	$-0.172963\pi$
$564$		0			0
$565$		−	48.1620i		−	2.02619i
$566$		0			0
$567$	4.63601	−	20.3412i	0.194694	−	0.854252i
$568$		0			0
$569$		−	39.2414i		−	1.64509i	−0.568702	−	0.822544i	$-0.692554\pi$
$569$		−	39.2414i		−	1.64509i	0.568702	−	0.822544i	$-0.307446\pi$
$570$		0			0
$571$	12.2726			0.513591			0.256795	−	0.966466i	$-0.417333\pi$
$571$	12.2726			0.513591			0.256795	+	0.966466i	$0.417333\pi$
$572$		0			0
$573$	−0.490208	+	8.74126i	−0.0204787	+	0.365171i
$574$		0			0
$575$	−21.8878			−0.912786
$576$		0			0
$577$	27.0428			1.12581			0.562904	−	0.826522i	$-0.309684\pi$
$577$	27.0428			1.12581			0.562904	+	0.826522i	$0.309684\pi$
$578$		0			0
$579$	1.00474	−	17.9162i	0.0417554	−	0.744572i
$580$		0			0
$581$	−9.77708			−0.405621
$582$		0			0
$583$			16.3158i			0.675733i
$584$		0			0
$585$	−1.01880	+	9.05496i	−0.0421224	+	0.374377i
$586$		0			0
$587$			15.0685i			0.621945i	0.950419	+	0.310973i	$0.100655\pi$
$587$			15.0685i			0.621945i	−0.950419	+	0.310973i	$0.899345\pi$
$588$		0			0
$589$			44.1146i			1.81771i
$590$		0			0
$591$	1.12725	−	20.1009i	0.0463689	−	0.826839i
$592$		0			0
$593$			34.9320i			1.43449i	0.696823	+	0.717243i	$0.254596\pi$
$593$			34.9320i			1.43449i	−0.696823	+	0.717243i	$0.745404\pi$
$594$		0			0
$595$	−15.8661			−0.650446
$596$		0			0
$597$	−6.71698	−	0.376687i	−0.274908	−	0.0154168i
$598$		0			0
$599$	39.0572			1.59583			0.797916	−	0.602769i	$-0.205936\pi$
$599$	39.0572			1.59583			0.797916	+	0.602769i	$0.205936\pi$
$600$		0			0
$601$	13.9053			0.567211			0.283605	−	0.958941i	$-0.408469\pi$
$601$	13.9053			0.567211			0.283605	+	0.958941i	$0.408469\pi$
$602$		0			0
$603$	−32.7465	−	3.68442i	−1.33354	−	0.150041i
$604$		0			0
$605$	27.6444			1.12391
$606$		0			0
$607$			34.5423i			1.40203i	0.713148	+	0.701014i	$0.247269\pi$
$607$			34.5423i			1.40203i	−0.713148	+	0.701014i	$0.752731\pi$
$608$		0			0
$609$	1.23636	+	0.0693347i	0.0500998	+	0.00280958i
$610$		0			0
$611$			5.99870i			0.242682i
$612$		0			0
$613$			16.2895i			0.657925i	0.944343	+	0.328963i	$0.106699\pi$
$613$			16.2895i			0.657925i	−0.944343	+	0.328963i	$0.893301\pi$
$614$		0			0
$615$	35.1537	+	1.97141i	1.41753	+	0.0794950i
$616$		0			0
$617$			0.850612i			0.0342443i	0.999853	+	0.0171222i	$0.00545042\pi$
$617$			0.850612i			0.0342443i	−0.999853	+	0.0171222i	$0.994550\pi$
$618$		0			0
$619$	−39.2985			−1.57954			−0.789770	−	0.613404i	$-0.789800\pi$
$619$	−39.2985			−1.57954			−0.789770	+	0.613404i	$0.789800\pi$
$620$		0			0
$621$	4.50219	−	26.5360i	0.180667	−	1.06485i
$622$		0			0
$623$	22.9441			0.919236
$624$		0			0
$625$	−28.2723			−1.13089
$626$		0			0
$627$	18.7003	+	1.04871i	0.746819	+	0.0418814i
$628$		0			0
$629$	11.5701			0.461330
$630$		0			0
$631$			37.0533i			1.47507i	0.675310	+	0.737534i	$0.264010\pi$
$631$			37.0533i			1.47507i	−0.675310	+	0.737534i	$0.735990\pi$
$632$		0			0
$633$	−1.59325	+	28.4105i	−0.0633262	+	1.12922i
$634$		0			0
$635$			11.3061i			0.448669i
$636$		0			0
$637$		−	1.62643i		−	0.0644417i
$638$		0			0
$639$	−34.2296	−	3.85129i	−1.35410	−	0.152355i
$640$		0			0
$641$		−	41.0568i		−	1.62165i	−0.585291	−	0.810824i	$-0.699020\pi$
$641$		−	41.0568i		−	1.62165i	0.585291	−	0.810824i	$-0.300980\pi$
$642$		0			0
$643$	−6.44016			−0.253975			−0.126988	−	0.991904i	$-0.540531\pi$
$643$	−6.44016			−0.253975			−0.126988	+	0.991904i	$0.540531\pi$
$644$		0			0
$645$	0.204923	−	3.65413i	0.00806882	−	0.143881i
$646$		0			0
$647$	5.48794			0.215753			0.107877	−	0.994164i	$-0.465595\pi$
$647$	5.48794			0.215753			0.107877	+	0.994164i	$0.465595\pi$
$648$		0			0
$649$	−18.9849			−0.745224
$650$		0			0
$651$	−1.26369	+	22.5337i	−0.0495278	+	0.883167i
$652$		0			0
$653$	40.8712			1.59942			0.799708	−	0.600390i	$-0.204988\pi$
$653$	40.8712			1.59942			0.799708	+	0.600390i	$0.204988\pi$
$654$		0			0
$655$		−	2.21436i		−	0.0865224i
$656$		0			0
$657$	−36.9336	−	4.15553i	−1.44092	−	0.162123i
$658$		0			0
$659$		−	19.8369i		−	0.772736i	−0.922345	−	0.386368i	$-0.873730\pi$
$659$		−	19.8369i		−	0.772736i	0.922345	−	0.386368i	$-0.126270\pi$
$660$		0			0
$661$			20.2239i			0.786620i	0.919406	+	0.393310i	$0.128670\pi$
$661$			20.2239i			0.786620i	−0.919406	+	0.393310i	$0.871330\pi$
$662$		0			0
$663$	−0.218538	+	3.89691i	−0.00848730	+	0.151343i
$664$		0			0
$665$			55.2570i			2.14278i
$666$		0			0
$667$	1.59753			0.0618567
$668$		0			0
$669$	−39.2165	−	2.19925i	−1.51620	−	0.0850281i
$670$		0			0
$671$	−14.6079			−0.563934
$672$		0			0
$673$	−45.4100			−1.75043			−0.875213	−	0.483738i	$-0.839279\pi$
$673$	−45.4100			−1.75043			−0.875213	+	0.483738i	$0.839279\pi$
$674$		0			0
$675$	−3.67279	+	21.6474i	−0.141366	+	0.833211i
$676$		0			0
$677$	−14.0654			−0.540578			−0.270289	−	0.962779i	$-0.587119\pi$
$677$	−14.0654			−0.540578			−0.270289	+	0.962779i	$0.587119\pi$
$678$		0			0
$679$			3.69904i			0.141956i
$680$		0			0
$681$	−37.5379	−	2.10512i	−1.43846	−	0.0806683i
$682$		0			0
$683$			28.4191i			1.08743i	0.839271	+	0.543713i	$0.182982\pi$
$683$			28.4191i			1.08743i	−0.839271	+	0.543713i	$0.817018\pi$
$684$		0			0
$685$			7.74877i			0.296065i
$686$		0			0
$687$	−10.4724	−	0.587287i	−0.399545	−	0.0224064i
$688$		0			0
$689$			11.8413i			0.451116i
$690$		0			0
$691$	38.8094			1.47638			0.738190	−	0.674593i	$-0.235681\pi$
$691$	38.8094			1.47638			0.738190	+	0.674593i	$0.235681\pi$
$692$		0			0
$693$	9.52209	+	1.07136i	0.361714	+	0.0406977i
$694$		0			0
$695$	62.0740			2.35460
$696$		0			0
$697$	15.0812			0.571242
$698$		0			0
$699$	12.5504	+	0.703825i	0.474701	+	0.0266211i
$700$		0			0
$701$	−32.6457			−1.23301			−0.616505	−	0.787351i	$-0.711452\pi$
$701$	−32.6457			−1.23301			−0.616505	+	0.787351i	$0.711452\pi$
$702$		0			0
$703$		−	40.2953i		−	1.51977i
$704$		0			0
$705$	−1.76701	+	31.5089i	−0.0665495	+	1.18669i
$706$		0			0
$707$			1.36621i			0.0513816i
$708$		0			0
$709$			35.4529i			1.33146i	0.746191	+	0.665732i	$0.231880\pi$
$709$			35.4529i			1.33146i	−0.746191	+	0.665732i	$0.768120\pi$
$710$		0			0
$711$	2.55492	−	22.7077i	0.0958168	−	0.851604i
$712$		0			0
$713$			29.1165i			1.09042i
$714$		0			0
$715$	−4.18512			−0.156515
$716$		0			0
$717$	−0.690225	+	12.3079i	−0.0257769	+	0.459648i
$718$		0			0
$719$	19.8333			0.739656			0.369828	−	0.929100i	$-0.379417\pi$
$719$	19.8333			0.739656			0.369828	+	0.929100i	$0.379417\pi$
$720$		0			0
$721$	−0.869519			−0.0323826
$722$		0			0
$723$	−1.19726	+	21.3492i	−0.0445265	+	0.793985i
$724$		0			0
$725$	−1.30323			−0.0484008
$726$		0			0
$727$		−	45.1449i		−	1.67433i	−0.546948	−	0.837167i	$-0.684210\pi$
$727$		−	45.1449i		−	1.67433i	0.546948	−	0.837167i	$-0.315790\pi$
$728$		0			0
$729$	−25.4891	−	8.90549i	−0.944039	−	0.329833i
$730$		0			0
$731$		−	1.56765i		−	0.0579817i
$732$		0			0
$733$		−	12.1472i		−	0.448666i	−0.974513	−	0.224333i	$-0.927980\pi$
$733$		−	12.1472i		−	0.448666i	0.974513	−	0.224333i	$-0.0720203\pi$
$734$		0			0
$735$	0.479092	−	8.54304i	0.0176716	−	0.315115i
$736$		0			0
$737$		−	15.1352i		−	0.557511i
$738$		0			0
$739$	33.6165			1.23660			0.618301	−	0.785941i	$-0.287821\pi$
$739$	33.6165			1.23660			0.618301	+	0.785941i	$0.287821\pi$
$740$		0			0
$741$	13.5718	+	0.761104i	0.498573	+	0.0279599i
$742$		0			0
$743$	−17.7701			−0.651922			−0.325961	−	0.945383i	$-0.605688\pi$
$743$	−17.7701			−0.651922			−0.325961	+	0.945383i	$0.605688\pi$
$744$		0			0
$745$	−30.4467			−1.11548
$746$		0			0
$747$	−1.41472	+	12.5738i	−0.0517620	+	0.460052i
$748$		0			0
$749$	−7.49365			−0.273812
$750$		0			0
$751$			25.3471i			0.924929i	0.886638	+	0.462465i	$0.153035\pi$
$751$			25.3471i			0.924929i	−0.886638	+	0.462465i	$0.846965\pi$
$752$		0			0
$753$	37.4383	+	2.09953i	1.36433	+	0.0765111i
$754$		0			0
$755$			13.0869i			0.476282i
$756$		0			0
$757$		−	39.7251i		−	1.44383i	−0.691979	−	0.721917i	$-0.743261\pi$
$757$		−	39.7251i		−	1.44383i	0.691979	−	0.721917i	$-0.256739\pi$
$758$		0			0
$759$	12.3426	+	0.692168i	0.448007	+	0.0251241i
$760$		0			0
$761$		−	38.9389i		−	1.41153i	−0.708444	−	0.705767i	$-0.750602\pi$
$761$		−	38.9389i		−	1.41153i	0.708444	−	0.705767i	$-0.249398\pi$
$762$		0			0
$763$	−39.4063			−1.42660
$764$		0			0
$765$	−2.29579	+	20.4046i	−0.0830045	+	0.737730i
$766$		0			0
$767$	−13.7784			−0.497508
$768$		0			0
$769$	−26.8577			−0.968514			−0.484257	−	0.874926i	$-0.660910\pi$
$769$	−26.8577			−0.968514			−0.484257	+	0.874926i	$0.660910\pi$
$770$		0			0
$771$	−39.9893	−	2.24259i	−1.44018	−	0.0807650i
$772$		0			0
$773$	9.30563			0.334700			0.167350	−	0.985898i	$-0.446479\pi$
$773$	9.30563			0.334700			0.167350	+	0.985898i	$0.446479\pi$
$774$		0			0
$775$		−	23.7526i		−	0.853218i
$776$		0			0
$777$	1.15428	−	20.5828i	0.0414096	−	0.738406i
$778$		0			0
$779$		−	52.5236i		−	1.88185i
$780$		0			0
$781$		−	15.8206i		−	0.566106i
$782$		0			0
$783$	0.268067	−	1.57999i	0.00957992	−	0.0564642i
$784$		0			0
$785$		−	35.8412i		−	1.27923i
$786$		0			0
$787$	−4.38908			−0.156454			−0.0782269	−	0.996936i	$-0.524926\pi$
$787$	−4.38908			−0.156454			−0.0782269	+	0.996936i	$0.524926\pi$
$788$		0			0
$789$	−2.56400	+	45.7206i	−0.0912809	+	1.62770i
$790$		0			0
$791$	−36.7569			−1.30692
$792$		0			0
$793$	−10.6018			−0.376480
$794$		0			0
$795$	−3.48803	+	62.1976i	−0.123708	+	2.20592i
$796$		0			0
$797$	27.0980			0.959862			0.479931	−	0.877306i	$-0.340662\pi$
$797$	27.0980			0.959862			0.479931	+	0.877306i	$0.340662\pi$
$798$		0			0
$799$			13.5176i			0.478217i
$800$		0			0
$801$	3.31996	−	29.5073i	0.117305	−	1.04259i
$802$		0			0
$803$		−	17.0704i		−	0.602402i
$804$		0			0
$805$			36.4707i			1.28542i
$806$		0			0
$807$	−2.63833	+	47.0461i	−0.0928737	+	1.65610i
$808$		0			0
$809$			28.9080i			1.01635i	0.861254	+	0.508175i	$0.169680\pi$
$809$			28.9080i			1.01635i	−0.861254	+	0.508175i	$0.830320\pi$
$810$		0			0
$811$	8.03330			0.282087			0.141044	−	0.990003i	$-0.454954\pi$
$811$	8.03330			0.282087			0.141044	+	0.990003i	$0.454954\pi$
$812$		0			0
$813$	−4.87661	−	0.273479i	−0.171030	−	0.00959134i
$814$		0			0
$815$	−21.1355			−0.740343
$816$		0			0
$817$	−5.45968			−0.191010
$818$		0			0
$819$	6.91068	+	0.777544i	0.241479	+	0.0271696i
$820$		0			0
$821$	17.5416			0.612206			0.306103	−	0.951998i	$-0.400975\pi$
$821$	17.5416			0.612206			0.306103	+	0.951998i	$0.400975\pi$
$822$		0			0
$823$			22.2423i			0.775317i	0.921803	+	0.387659i	$0.126716\pi$
$823$			22.2423i			0.775317i	−0.921803	+	0.387659i	$0.873284\pi$
$824$		0			0
$825$	−10.0688	−	0.564656i	−0.350550	−	0.0196588i
$826$		0			0
$827$			46.5476i			1.61862i	0.587382	+	0.809310i	$0.300159\pi$
$827$			46.5476i			1.61862i	−0.587382	+	0.809310i	$0.699841\pi$
$828$		0			0
$829$			10.1596i			0.352857i	0.984313	+	0.176428i	$0.0564544\pi$
$829$			10.1596i			0.352857i	−0.984313	+	0.176428i	$0.943546\pi$
$830$		0			0
$831$	−1.29649	−	0.0727066i	−0.0449746	−	0.00252217i
$832$		0			0
$833$		−	3.66503i		−	0.126986i
$834$		0			0
$835$	47.1753			1.63257
$836$		0			0
$837$	28.7967	+	4.88575i	0.995360	+	0.168876i
$838$		0			0
$839$	−47.1544			−1.62795			−0.813976	−	0.580899i	$-0.802701\pi$
$839$	−47.1544			−1.62795			−0.813976	+	0.580899i	$0.802701\pi$
$840$		0			0
$841$	−28.9049			−0.996720
$842$		0			0
$843$	21.3197	+	1.19560i	0.734290	+	0.0411788i
$844$		0			0
$845$	−3.03737			−0.104489
$846$		0			0
$847$		−	21.0980i		−	0.724937i
$848$		0			0
$849$	−2.06923	+	36.8980i	−0.0710158	+	1.26634i
$850$		0			0
$851$		−	26.5957i		−	0.911688i
$852$		0			0
$853$		−	6.49512i		−	0.222389i	−0.993799	−	0.111194i	$-0.964532\pi$
$853$		−	6.49512i		−	0.222389i	0.993799	−	0.111194i	$-0.0354676\pi$
$854$		0			0
$855$	71.0634	+	7.99558i	2.43032	+	0.273443i
$856$		0			0
$857$		−	33.6702i		−	1.15015i	−0.818100	−	0.575077i	$-0.804972\pi$
$857$		−	33.6702i		−	1.15015i	0.818100	−	0.575077i	$-0.195028\pi$
$858$		0			0
$859$	−1.93323			−0.0659610			−0.0329805	−	0.999456i	$-0.510500\pi$
$859$	−1.93323			−0.0659610			−0.0329805	+	0.999456i	$0.510500\pi$
$860$		0			0
$861$	1.50457	−	26.8291i	0.0512755	−	0.914332i
$862$		0			0
$863$	2.21988			0.0755657			0.0377829	−	0.999286i	$-0.487970\pi$
$863$	2.21988			0.0755657			0.0377829	+	0.999286i	$0.487970\pi$
$864$		0			0
$865$	−9.26246			−0.314933
$866$		0			0
$867$	1.15621	−	20.6173i	0.0392671	−	0.700201i
$868$		0			0
$869$	10.4953			0.356028
$870$		0			0
$871$		−	10.9844i		−	0.372191i
$872$		0			0
$873$	4.75715	+	0.535243i	0.161005	+	0.0181152i
$874$		0			0
$875$			5.45255i			0.184330i
$876$		0			0
$877$			19.5292i			0.659454i	0.944076	+	0.329727i	$0.106957\pi$
$877$			19.5292i			0.659454i	−0.944076	+	0.329727i	$0.893043\pi$
$878$		0			0
$879$	−2.30779	+	41.1519i	−0.0778398	+	1.38802i
$880$		0			0
$881$		−	54.3755i		−	1.83196i	−0.401229	−	0.915978i	$-0.631417\pi$
$881$		−	54.3755i		−	1.83196i	0.401229	−	0.915978i	$-0.368583\pi$
$882$		0			0
$883$	39.8004			1.33939			0.669695	−	0.742636i	$-0.266425\pi$
$883$	39.8004			1.33939			0.669695	+	0.742636i	$0.266425\pi$
$884$		0			0
$885$	−72.3725	−	4.05863i	−2.43277	−	0.136429i
$886$		0			0
$887$	13.0627			0.438602			0.219301	−	0.975657i	$-0.429622\pi$
$887$	13.0627			0.438602			0.219301	+	0.975657i	$0.429622\pi$
$888$		0			0
$889$	8.62873			0.289398
$890$		0			0
$891$	2.75565	−	12.0909i	0.0923179	−	0.405059i
$892$		0			0
$893$	47.0778			1.57540
$894$		0			0
$895$			16.6128i			0.555304i
$896$		0			0
$897$	8.95765	+	0.502343i	0.299087	+	0.0167727i
$898$		0			0
$899$			1.73364i			0.0578200i
$900$		0			0
$901$			26.6833i			0.888950i
$902$		0			0
$903$	−2.78881	−	0.156396i	−0.0928056	−	0.00520452i
$904$		0			0
$905$			40.2804i			1.33897i
$906$		0			0
$907$	−3.12450			−0.103747			−0.0518737	−	0.998654i	$-0.516519\pi$
$907$	−3.12450			−0.103747			−0.0518737	+	0.998654i	$0.516519\pi$
$908$		0			0
$909$	1.75701	+	0.197688i	0.0582765	+	0.00655688i
$910$		0			0
$911$	36.7677			1.21817			0.609084	−	0.793105i	$-0.291537\pi$
$911$	36.7677			1.21817			0.609084	+	0.793105i	$0.291537\pi$
$912$		0			0
$913$	−5.81151			−0.192333
$914$		0			0
$915$	−55.6870	−	3.12291i	−1.84096	−	0.103240i
$916$		0			0
$917$	−1.68999			−0.0558083
$918$		0			0
$919$			24.5079i			0.808440i	0.914662	+	0.404220i	$0.132457\pi$
$919$			24.5079i			0.808440i	−0.914662	+	0.404220i	$0.867543\pi$
$920$		0			0
$921$	−0.810529	+	14.4531i	−0.0267078	+	0.476247i
$922$		0			0
$923$		−	11.4819i		−	0.377930i
$924$		0			0
$925$			21.6962i			0.713365i
$926$		0			0
$927$	−0.125818	+	1.11825i	−0.00413239	+	0.0367280i
$928$		0			0
$929$			53.3725i			1.75109i	0.483133	+	0.875547i	$0.339499\pi$
$929$			53.3725i			1.75109i	−0.483133	+	0.875547i	$0.660501\pi$
$930$		0			0
$931$	−12.7643			−0.418332
$932$		0			0
$933$	2.37859	−	42.4144i	0.0778715	−	1.38859i
$934$		0			0
$935$	−9.43083			−0.308421
$936$		0			0
$937$	−39.4667			−1.28932			−0.644660	−	0.764469i	$-0.723001\pi$
$937$	−39.4667			−1.28932			−0.644660	+	0.764469i	$0.723001\pi$
$938$		0			0
$939$	1.46007	−	26.0356i	0.0476475	−	0.849638i
$940$		0			0
$941$	−59.2831			−1.93258			−0.966288	−	0.257465i	$-0.917113\pi$
$941$	−59.2831			−1.93258			−0.966288	+	0.257465i	$0.917113\pi$
$942$		0			0
$943$		−	34.6666i		−	1.12890i
$944$		0			0
$945$	36.0701	+	6.11979i	1.17336	+	0.199077i
$946$		0			0
$947$		−	0.391710i		−	0.0127289i	−0.999980	−	0.00636443i	$-0.997974\pi$
$947$		−	0.391710i		−	0.0127289i	0.999980	−	0.00636443i	$-0.00202587\pi$
$948$		0			0
$949$		−	12.3889i		−	0.402161i
$950$		0			0
$951$	0.223706	−	3.98908i	0.00725418	−	0.129355i
$952$		0			0
$953$		−	17.4300i		−	0.564615i	−0.959324	−	0.282307i	$-0.908900\pi$
$953$		−	17.4300i		−	0.564615i	0.959324	−	0.282307i	$-0.0910998\pi$
$954$		0			0
$955$	−15.3530			−0.496810
$956$		0			0
$957$	0.734894	+	0.0412127i	0.0237557	+	0.00133222i
$958$		0			0
$959$	5.91381			0.190967
$960$		0			0
$961$	−0.597054			−0.0192598
$962$		0			0
$963$	−1.08432	+	9.63723i	−0.0349416	+	0.310555i
$964$		0			0
$965$	31.4677			1.01298
$966$		0			0
$967$			22.1983i			0.713849i	0.934133	+	0.356924i	$0.116175\pi$
$967$			22.1983i			0.713849i	−0.934133	+	0.356924i	$0.883825\pi$
$968$		0			0
$969$	30.5829	+	1.71508i	0.982466	+	0.0550964i
$970$		0			0
$971$		−	14.1933i		−	0.455485i	−0.973721	−	0.227742i	$-0.926866\pi$
$971$		−	14.1933i		−	0.455485i	0.973721	−	0.227742i	$-0.0731344\pi$
$972$		0			0
$973$		−	47.3744i		−	1.51875i
$974$		0			0
$975$	−7.30745	−	0.409800i	−0.234026	−	0.0131241i
$976$		0			0
$977$			23.5519i			0.753493i	0.926316	+	0.376746i	$0.122957\pi$
$977$			23.5519i			0.753493i	−0.926316	+	0.376746i	$0.877043\pi$
$978$		0			0
$979$	13.6380			0.435873
$980$		0			0
$981$	−5.70201	+	50.6785i	−0.182051	+	1.61804i
$982$		0			0
$983$	−50.7863			−1.61983			−0.809916	−	0.586546i	$-0.800487\pi$
$983$	−50.7863			−1.61983			−0.809916	+	0.586546i	$0.800487\pi$
$984$		0			0
$985$	35.3047			1.12490
$986$		0			0
$987$	24.0474	+	1.34857i	0.765436	+	0.0429255i
$988$		0			0
$989$	−3.60349			−0.114584
$990$		0			0
$991$			30.0634i			0.954995i	0.878633	+	0.477498i	$0.158456\pi$
$991$			30.0634i			0.954995i	−0.878633	+	0.477498i	$0.841544\pi$
$992$		0			0
$993$	−1.21150	+	21.6032i	−0.0384459	+	0.685558i
$994$		0			0
$995$		−	11.7976i		−	0.374008i
$996$		0			0
$997$			4.86064i			0.153938i	0.997033	+	0.0769690i	$0.0245242\pi$
$997$			4.86064i			0.153938i	−0.997033	+	0.0769690i	$0.975476\pi$
$998$		0			0
$999$	−26.3036	−	4.46276i	−0.832208	−	0.141196i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2496.2.j.e.287.18	yes	32
3.2	odd	2	inner	2496.2.j.e.287.19	yes	32
4.3	odd	2		2496.2.j.f.287.16	yes	32
8.3	odd	2	inner	2496.2.j.e.287.17	✓	32
8.5	even	2		2496.2.j.f.287.15	yes	32
12.11	even	2		2496.2.j.f.287.13	yes	32
24.5	odd	2		2496.2.j.f.287.14	yes	32
24.11	even	2	inner	2496.2.j.e.287.20	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2496.2.j.e.287.17	✓	32	8.3	odd	2	inner
2496.2.j.e.287.18	yes	32	1.1	even	1	trivial
2496.2.j.e.287.19	yes	32	3.2	odd	2	inner
2496.2.j.e.287.20	yes	32	24.11	even	2	inner
2496.2.j.f.287.13	yes	32	12.11	even	2
2496.2.j.f.287.14	yes	32	24.5	odd	2
2496.2.j.f.287.15	yes	32	8.5	even	2
2496.2.j.f.287.16	yes	32	4.3	odd	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2496 = 2^{6} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2496.j (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(0.0969806 - 1.72933i) q^{3} +3.03737 q^{5} -2.31810i q^{7} +(-2.98119 - 0.335424i) q^{9} -1.37788i q^{11} -1.00000i q^{13} +(0.294566 - 5.25262i) q^{15} -2.25342i q^{17} -7.84800 q^{19} +(-4.00876 - 0.224810i) q^{21} -5.17983 q^{23} +4.22559 q^{25} +(-0.869177 + 5.12294i) q^{27} -0.308414 q^{29} -5.62113i q^{31} +(-2.38281 - 0.133628i) q^{33} -7.04090i q^{35} +5.13447i q^{37} +(-1.72933 - 0.0969806i) q^{39} +6.69261i q^{41} +0.695678 q^{43} +(-9.05496 - 1.01880i) q^{45} -5.99870 q^{47} +1.62643 q^{49} +(-3.89691 - 0.218538i) q^{51} -11.8413 q^{53} -4.18512i q^{55} +(-0.761104 + 13.5718i) q^{57} -13.7784i q^{59} -10.6018i q^{61} +(-0.777544 + 6.91068i) q^{63} -3.03737i q^{65} +10.9844 q^{67} +(-0.502343 + 8.95765i) q^{69} +11.4819 q^{71} +12.3889 q^{73} +(0.409800 - 7.30745i) q^{75} -3.19406 q^{77} +7.61698i q^{79} +(8.77498 + 1.99992i) q^{81} -4.21772i q^{83} -6.84445i q^{85} +(-0.0299102 + 0.533351i) q^{87} +9.89782i q^{89} -2.31810 q^{91} +(-9.72080 - 0.545140i) q^{93} -23.8372 q^{95} -1.59572 q^{97} +(-0.462173 + 4.10772i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 4 q^{9} - 32 q^{19} + 56 q^{25} + 24 q^{27} + 32 q^{33} - 16 q^{43} + 8 q^{49} + 40 q^{51} - 8 q^{57} + 16 q^{67} - 16 q^{73} + 120 q^{75} - 44 q^{81} - 80 q^{97}+O(q^{100}) \) 32 * q - 4 * q^9 - 32 * q^19 + 56 * q^25 + 24 * q^27 + 32 * q^33 - 16 * q^43 + 8 * q^49 + 40 * q^51 - 8 * q^57 + 16 * q^67 - 16 * q^73 + 120 * q^75 - 44 * q^81 - 80 * q^97

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