LMFDB - Embedded newform 2496.2.j.c.287.1

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 = [12,0,-4,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0] 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:	$12$
Coefficient field:	12.0.18918897714528256.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 3x^{10} + 5x^{8} + 4x^{6} + 20x^{4} + 48x^{2} + 64 $ x^12 + 3x^10 + 5x^8 + 4x^6 + 20x^4 + 48*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			287.1
Root			$1.28568 - 0.589097i$ of defining polynomial
Character	$\chi$	$=$	2496.287
Dual form			2496.2.j.c.287.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.71683 - 0.229100i) q^{3} -3.12891 q^{5} +1.43366i q^{7} +(2.89503 + 0.786653i) q^{9} -1.55560i q^{11} -1.00000i q^{13} +(5.37181 + 0.716832i) q^{15} -1.57331i q^{17} +(0.328453 - 2.46136i) q^{21} -5.34141 q^{23} +4.79005 q^{25} +(-4.79005 - 2.01380i) q^{27} -7.17421 q^{29} +4.00000i q^{31} +(-0.356388 + 2.67071i) q^{33} -4.48580i q^{35} -4.65738i q^{37} +(-0.229100 + 1.71683i) q^{39} -9.64621i q^{41} -6.14644 q^{43} +(-9.05827 - 2.46136i) q^{45} -1.09740 q^{47} +4.94461 q^{49} +(-0.360444 + 2.70110i) q^{51} +10.2854 q^{53} +4.86733i q^{55} +2.47200i q^{59} -2.71278i q^{61} +(-1.12780 + 4.15050i) q^{63} +3.12891i q^{65} +6.86733 q^{67} +(9.17031 + 1.22372i) q^{69} -1.09740 q^{71} -2.71278 q^{73} +(-8.22372 - 1.09740i) q^{75} +2.23021 q^{77} +10.8673i q^{79} +(7.76236 + 4.55476i) q^{81} +12.7574i q^{83} +4.92272i q^{85} +(12.3169 + 1.64361i) q^{87} +11.8410i q^{89} +1.43366 q^{91} +(0.916400 - 6.86733i) q^{93} +8.86733 q^{97} +(1.22372 - 4.50351i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{3} + 8 q^{9} + 4 q^{25} - 4 q^{27} + 16 q^{33} - 20 q^{49} - 8 q^{51} + 16 q^{67} + 8 q^{73} - 12 q^{75} - 16 q^{91} + 40 q^{97} - 72 q^{99}+O(q^{100}) $ 12 * q - 4 * q^3 + 8 * q^9 + 4 * q^25 - 4 * q^27 + 16 * q^33 - 20 * q^49 - 8 * q^51 + 16 * q^67 + 8 * q^73 - 12 * q^75 - 16 * q^91 + 40 * q^97 - 72 * q^99

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
$3$	−1.71683	−	0.229100i	−0.991214	−	0.132271i
$3$	−1.71683	−	0.229100i	−0.991214	−	0.132271i
$4$		0			0
$5$	−3.12891			−1.39929			−0.699645	−	0.714491i	$-0.746658\pi$
$5$	−3.12891			−1.39929			−0.699645	+	0.714491i	$0.746658\pi$
$6$		0			0
$7$			1.43366i			0.541874i	0.962597	+	0.270937i	$0.0873336\pi$
$7$			1.43366i			0.541874i	−0.962597	+	0.270937i	$0.912666\pi$
$8$		0			0
$9$	2.89503	+	0.786653i	0.965009	+	0.262218i
$10$		0			0
$11$		−	1.55560i		−	0.469031i	−0.972112	−	0.234516i	$-0.924650\pi$
$11$		−	1.55560i		−	0.469031i	0.972112	−	0.234516i	$-0.0753504\pi$
$12$		0			0
$13$		−	1.00000i		−	0.277350i
$13$		−	1.00000i		−	0.277350i
$14$		0			0
$15$	5.37181	+	0.716832i	1.38699	+	0.185085i
$16$		0			0
$17$		−	1.57331i		−	0.381583i	−0.981631	−	0.190791i	$-0.938895\pi$
$17$		−	1.57331i		−	0.381583i	0.981631	−	0.190791i	$-0.0611054\pi$
$18$		0			0
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$	0.328453	−	2.46136i	0.0716742	−	0.537113i
$22$		0			0
$23$	−5.34141			−1.11376			−0.556881	−	0.830592i	$-0.688002\pi$
$23$	−5.34141			−1.11376			−0.556881	+	0.830592i	$0.688002\pi$
$24$		0			0
$25$	4.79005			0.958011
$26$		0			0
$27$	−4.79005	−	2.01380i	−0.921846	−	0.387556i
$28$		0			0
$29$	−7.17421			−1.33222			−0.666109	−	0.745855i	$-0.732041\pi$
$29$	−7.17421			−1.33222			−0.666109	+	0.745855i	$0.732041\pi$
$30$		0			0
$31$			4.00000i			0.718421i	0.933257	+	0.359211i	$0.116954\pi$
$31$			4.00000i			0.718421i	−0.933257	+	0.359211i	$0.883046\pi$
$32$		0			0
$33$	−0.356388	+	2.67071i	−0.0620392	+	0.464910i
$34$		0			0
$35$		−	4.48580i		−	0.758239i
$36$		0			0
$37$		−	4.65738i		−	0.765669i	−0.923817	−	0.382834i	$-0.874948\pi$
$37$		−	4.65738i		−	0.765669i	0.923817	−	0.382834i	$-0.125052\pi$
$38$		0			0
$39$	−0.229100	+	1.71683i	−0.0366854	+	0.274913i
$40$		0			0
$41$		−	9.64621i		−	1.50649i	−0.657743	−	0.753243i	$-0.728489\pi$
$41$		−	9.64621i		−	1.50649i	0.657743	−	0.753243i	$-0.271511\pi$
$42$		0			0
$43$	−6.14644			−0.937323			−0.468662	−	0.883378i	$-0.655264\pi$
$43$	−6.14644			−0.937323			−0.468662	+	0.883378i	$0.655264\pi$
$44$		0			0
$45$	−9.05827	−	2.46136i	−1.35033	−	0.366918i
$46$		0			0
$47$	−1.09740			−0.160072			−0.0800362	−	0.996792i	$-0.525504\pi$
$47$	−1.09740			−0.160072			−0.0800362	+	0.996792i	$0.525504\pi$
$48$		0			0
$49$	4.94461			0.706372
$50$		0			0
$51$	−0.360444	+	2.70110i	−0.0504723	+	0.378230i
$52$		0			0
$53$	10.2854			1.41281			0.706405	−	0.707808i	$-0.250316\pi$
$53$	10.2854			1.41281			0.706405	+	0.707808i	$0.250316\pi$
$54$		0			0
$55$			4.86733i			0.656311i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			2.47200i			0.321827i	0.986969	+	0.160914i	$0.0514440\pi$
$59$			2.47200i			0.321827i	−0.986969	+	0.160914i	$0.948556\pi$
$60$		0			0
$61$		−	2.71278i		−	0.347335i	−0.984804	−	0.173668i	$-0.944438\pi$
$61$		−	2.71278i		−	0.347335i	0.984804	−	0.173668i	$-0.0555619\pi$
$62$		0			0
$63$	−1.12780	+	4.15050i	−0.142089	+	0.522913i
$64$		0			0
$65$			3.12891i			0.388093i
$66$		0			0
$67$	6.86733			0.838978			0.419489	−	0.907760i	$-0.362209\pi$
$67$	6.86733			0.838978			0.419489	+	0.907760i	$0.362209\pi$
$68$		0			0
$69$	9.17031	+	1.22372i	1.10398	+	0.147318i
$70$		0			0
$71$	−1.09740			−0.130238			−0.0651188	−	0.997878i	$-0.520743\pi$
$71$	−1.09740			−0.130238			−0.0651188	+	0.997878i	$0.520743\pi$
$72$		0			0
$73$	−2.71278			−0.317506			−0.158753	−	0.987318i	$-0.550747\pi$
$73$	−2.71278			−0.317506			−0.158753	+	0.987318i	$0.550747\pi$
$74$		0			0
$75$	−8.22372	−	1.09740i	−0.949593	−	0.126717i
$76$		0			0
$77$	2.23021			0.254156
$78$		0			0
$79$			10.8673i			1.22267i	0.791372	+	0.611335i	$0.209367\pi$
$79$			10.8673i			1.22267i	−0.791372	+	0.611335i	$0.790633\pi$
$80$		0			0
$81$	7.76236	+	4.55476i	0.862484	+	0.506084i
$82$		0			0
$83$			12.7574i			1.40031i	0.713992	+	0.700154i	$0.246885\pi$
$83$			12.7574i			1.40031i	−0.713992	+	0.700154i	$0.753115\pi$
$84$		0			0
$85$			4.92272i			0.533944i
$86$		0			0
$87$	12.3169	+	1.64361i	1.32051	+	0.176214i
$88$		0			0
$89$			11.8410i			1.25515i	0.778558	+	0.627573i	$0.215951\pi$
$89$			11.8410i			1.25515i	−0.778558	+	0.627573i	$0.784049\pi$
$90$		0			0
$91$	1.43366			0.150289
$92$		0			0
$93$	0.916400	−	6.86733i	0.0950262	−	0.712109i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	8.86733			0.900341			0.450170	−	0.892943i	$-0.351363\pi$
$97$	8.86733			0.900341			0.450170	+	0.892943i	$0.351363\pi$
$98$		0			0
$99$	1.22372	−	4.50351i	0.122988	−	0.452619i
$100$		0			0
$101$	4.94400			0.491947			0.245973	−	0.969277i	$-0.420892\pi$
$101$	4.94400			0.491947			0.245973	+	0.969277i	$0.420892\pi$
$102$		0			0
$103$		−	2.71278i		−	0.267298i	−0.991029	−	0.133649i	$-0.957331\pi$
$103$		−	2.71278i		−	0.267298i	0.991029	−	0.133649i	$-0.0426694\pi$
$104$		0			0
$105$	−1.02770	+	7.70137i	−0.100293	+	0.751577i
$106$		0			0
$107$			11.5992i			1.12134i	0.828040	+	0.560670i	$0.189456\pi$
$107$			11.5992i			1.12134i	−0.828040	+	0.560670i	$0.810544\pi$
$108$		0			0
$109$			14.5028i			1.38912i	0.719435	+	0.694560i	$0.244401\pi$
$109$			14.5028i			1.38912i	−0.719435	+	0.694560i	$0.755599\pi$
$110$		0			0
$111$	−1.06701	+	7.99594i	−0.101276	+	0.758941i
$112$		0			0
$113$		−	19.2924i		−	1.81488i	−0.420183	−	0.907439i	$-0.638034\pi$
$113$		−	19.2924i		−	1.81488i	0.420183	−	0.907439i	$-0.361966\pi$
$114$		0			0
$115$	16.7128			1.55847
$116$		0			0
$117$	0.786653	−	2.89503i	0.0727261	−	0.267645i
$118$		0			0
$119$	2.25559			0.206770
$120$		0			0
$121$	8.58011			0.780010
$122$		0			0
$123$	−2.20995	+	16.5609i	−0.199264	+	1.49325i
$124$		0			0
$125$	0.656905			0.0587554
$126$		0			0
$127$			10.4474i			0.927060i	0.886081	+	0.463530i	$0.153417\pi$
$127$			10.4474i			0.927060i	−0.886081	+	0.463530i	$0.846583\pi$
$128$		0			0
$129$	10.5524	+	1.40815i	0.929088	+	0.123981i
$130$		0			0
$131$		−	8.94622i		−	0.781635i	−0.920468	−	0.390818i	$-0.872192\pi$
$131$		−	8.94622i		−	0.781635i	0.920468	−	0.390818i	$-0.127808\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	14.9876	+	6.30099i	1.28993	+	0.542303i
$136$		0			0
$137$		−	17.1824i		−	1.46799i	−0.679153	−	0.733997i	$-0.737653\pi$
$137$		−	17.1824i		−	1.46799i	0.679153	−	0.733997i	$-0.262347\pi$
$138$		0			0
$139$	6.14644			0.521334			0.260667	−	0.965429i	$-0.416057\pi$
$139$	6.14644			0.521334			0.260667	+	0.965429i	$0.416057\pi$
$140$		0			0
$141$	1.88405	+	0.251415i	0.158666	+	0.0211729i
$142$		0			0
$143$	−1.55560			−0.130086
$144$		0			0
$145$	22.4474			1.86416
$146$		0			0
$147$	−8.48906	−	1.13281i	−0.700166	−	0.0934325i
$148$		0			0
$149$	7.45141			0.610443			0.305222	−	0.952281i	$-0.401269\pi$
$149$	7.45141			0.610443			0.305222	+	0.952281i	$0.401269\pi$
$150$		0			0
$151$			20.5939i			1.67591i	0.545743	+	0.837953i	$0.316248\pi$
$151$			20.5939i			1.67591i	−0.545743	+	0.837953i	$0.683752\pi$
$152$		0			0
$153$	1.23764	−	4.55476i	0.100058	−	0.368230i
$154$		0			0
$155$		−	12.5156i		−	1.00528i
$156$		0			0
$157$			6.86733i			0.548073i	0.961719	+	0.274036i	$0.0883588\pi$
$157$			6.86733i			0.548073i	−0.961719	+	0.274036i	$0.911641\pi$
$158$		0			0
$159$	−17.6583	−	2.35639i	−1.40040	−	0.186874i
$160$		0			0
$161$		−	7.65779i		−	0.603519i
$162$		0			0
$163$	16.7128			1.30905			0.654523	−	0.756042i	$-0.272870\pi$
$163$	16.7128			1.30905			0.654523	+	0.756042i	$0.272870\pi$
$164$		0			0
$165$	1.11511	−	8.35639i	0.0868108	−	0.650544i
$166$		0			0
$167$	−20.3290			−1.57311			−0.786554	−	0.617521i	$-0.788137\pi$
$167$	−20.3290			−1.57311			−0.786554	+	0.617521i	$0.788137\pi$
$168$		0			0
$169$	−1.00000			−0.0769231
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−4.42501			−0.336427			−0.168214	−	0.985751i	$-0.553800\pi$
$173$	−4.42501			−0.336427			−0.168214	+	0.985751i	$0.553800\pi$
$174$		0			0
$175$			6.86733i			0.519121i
$176$		0			0
$177$	0.566335	−	4.24401i	0.0425684	−	0.318999i
$178$		0			0
$179$			7.59700i			0.567827i	0.958850	+	0.283913i	$0.0916328\pi$
$179$			7.59700i			0.567827i	−0.958850	+	0.283913i	$0.908367\pi$
$180$		0			0
$181$			23.5801i			1.75270i	0.481679	+	0.876348i	$0.340027\pi$
$181$			23.5801i			1.75270i	−0.481679	+	0.876348i	$0.659973\pi$
$182$		0			0
$183$	−0.621497	+	4.65738i	−0.0459424	+	0.344284i
$184$		0			0
$185$			14.5725i			1.07139i
$186$		0			0
$187$	−2.44743			−0.178974
$188$		0			0
$189$	2.88712	−	6.86733i	0.210007	−	0.499525i
$190$		0			0
$191$	18.3406			1.32708			0.663540	−	0.748141i	$-0.269053\pi$
$191$	18.3406			1.32708			0.663540	+	0.748141i	$0.269053\pi$
$192$		0			0
$193$	11.7347			0.844679			0.422340	−	0.906438i	$-0.361209\pi$
$193$	11.7347			0.844679			0.422340	+	0.906438i	$0.361209\pi$
$194$		0			0
$195$	0.716832	−	5.37181i	0.0513334	−	0.384683i
$196$		0			0
$197$	−0.536694			−0.0382378			−0.0191189	−	0.999817i	$-0.506086\pi$
$197$	−0.536694			−0.0382378			−0.0191189	+	0.999817i	$0.506086\pi$
$198$		0			0
$199$		−	12.2929i		−	0.871419i	−0.900087	−	0.435710i	$-0.856497\pi$
$199$		−	12.2929i		−	0.871419i	0.900087	−	0.435710i	$-0.143503\pi$
$200$		0			0
$201$	−11.7901	−	1.57331i	−0.831606	−	0.110972i
$202$		0			0
$203$		−	10.2854i		−	0.721895i
$204$		0			0
$205$			30.1821i			2.10801i
$206$		0			0
$207$	−15.4635	−	4.20184i	−1.07479	−	0.292048i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	17.1683			1.18192			0.590958	−	0.806702i	$-0.298750\pi$
$211$	17.1683			1.18192			0.590958	+	0.806702i	$0.298750\pi$
$212$		0			0
$213$	1.88405	+	0.251415i	0.129093	+	0.0172266i
$214$		0			0
$215$	19.2316			1.31159
$216$		0			0
$217$	−5.73466			−0.389294
$218$		0			0
$219$	4.65738	+	0.621497i	0.314717	+	0.0419969i
$220$		0			0
$221$	−1.57331			−0.105832
$222$		0			0
$223$		−	20.5939i		−	1.37907i	−0.724253	−	0.689534i	$-0.757815\pi$
$223$		−	20.5939i		−	1.37907i	0.724253	−	0.689534i	$-0.242185\pi$
$224$		0			0
$225$	13.8673	+	3.76811i	0.924489	+	0.251207i
$226$		0			0
$227$		−	21.2100i		−	1.40776i	−0.710319	−	0.703880i	$-0.751449\pi$
$227$		−	21.2100i		−	1.40776i	0.710319	−	0.703880i	$-0.248551\pi$
$228$		0			0
$229$		−	4.65738i		−	0.307768i	−0.988089	−	0.153884i	$-0.950822\pi$
$229$		−	4.65738i		−	0.307768i	0.988089	−	0.153884i	$-0.0491783\pi$
$230$		0			0
$231$	−3.82890	−	0.510941i	−0.251923	−	0.0336175i
$232$		0			0
$233$			25.2553i			1.65453i	0.561810	+	0.827266i	$0.310105\pi$
$233$			25.2553i			1.65453i	−0.561810	+	0.827266i	$0.689895\pi$
$234$		0			0
$235$	3.43366			0.223988
$236$		0			0
$237$	2.48970	−	18.6574i	0.161724	−	1.21193i
$238$		0			0
$239$	26.6830			1.72598			0.862991	−	0.505219i	$-0.168588\pi$
$239$	26.6830			1.72598			0.862991	+	0.505219i	$0.168588\pi$
$240$		0			0
$241$	0.154553			0.00995563			0.00497782	−	0.999988i	$-0.498416\pi$
$241$	0.154553			0.00995563			0.00497782	+	0.999988i	$0.498416\pi$
$242$		0			0
$243$	−12.2832	−	9.59811i	−0.787966	−	0.615719i
$244$		0			0
$245$	−15.4712			−0.988419
$246$		0			0
$247$		0			0
$248$		0			0
$249$	2.92272	−	21.9023i	0.185220	−	1.38800i
$250$		0			0
$251$			5.37682i			0.339382i	0.985497	+	0.169691i	$0.0542769\pi$
$251$			5.37682i			0.339382i	−0.985497	+	0.169691i	$0.945723\pi$
$252$		0			0
$253$			8.30911i			0.522389i
$254$		0			0
$255$	1.12780	−	8.45149i	0.0706253	−	0.529253i
$256$		0			0
$257$			25.2553i			1.57538i	0.616069	+	0.787692i	$0.288724\pi$
$257$			25.2553i			1.57538i	−0.616069	+	0.787692i	$0.711276\pi$
$258$		0			0
$259$	6.67712			0.414896
$260$		0			0
$261$	−20.7695	−	5.64361i	−1.28560	−	0.349331i
$262$		0			0
$263$	−16.1458			−0.995594			−0.497797	−	0.867294i	$-0.665857\pi$
$263$	−16.1458			−0.995594			−0.497797	+	0.867294i	$0.665857\pi$
$264$		0			0
$265$	−32.1821			−1.97693
$266$		0			0
$267$	2.71278	−	20.3290i	0.166019	−	1.24412i
$268$		0			0
$269$	−14.7104			−0.896911			−0.448455	−	0.893805i	$-0.648026\pi$
$269$	−14.7104			−0.896911			−0.448455	+	0.893805i	$0.648026\pi$
$270$		0			0
$271$			4.30099i			0.261267i	0.991431	+	0.130633i	$0.0417011\pi$
$271$			4.30099i			0.261267i	−0.991431	+	0.130633i	$0.958299\pi$
$272$		0			0
$273$	−2.46136	−	0.328453i	−0.148968	−	0.0198789i
$274$		0			0
$275$		−	7.45141i		−	0.449337i
$276$		0			0
$277$			6.86733i			0.412618i	0.978487	+	0.206309i	$0.0661452\pi$
$277$			6.86733i			0.412618i	−0.978487	+	0.206309i	$0.933855\pi$
$278$		0			0
$279$	−3.14661	+	11.5801i	−0.188383	+	0.693283i
$280$		0			0
$281$		−	5.54779i		−	0.330954i	−0.986214	−	0.165477i	$-0.947084\pi$
$281$		−	5.54779i		−	0.330954i	0.986214	−	0.165477i	$-0.0529163\pi$
$282$		0			0
$283$	−23.3148			−1.38592			−0.692959	−	0.720977i	$-0.743694\pi$
$283$	−23.3148			−1.38592			−0.692959	+	0.720977i	$0.743694\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	13.8294			0.816326
$288$		0			0
$289$	14.5247			0.854395
$290$		0			0
$291$	−15.2237	−	2.03150i	−0.892430	−	0.119089i
$292$		0			0
$293$	−14.3307			−0.837210			−0.418605	−	0.908168i	$-0.637481\pi$
$293$	−14.3307			−0.837210			−0.418605	+	0.908168i	$0.637481\pi$
$294$		0			0
$295$		−	7.73466i		−	0.450329i
$296$		0			0
$297$	−3.13267	+	7.45141i	−0.181776	+	0.432375i
$298$		0			0
$299$			5.34141i			0.308902i
$300$		0			0
$301$		−	8.81193i		−	0.507911i
$302$		0			0
$303$	−8.48802	−	1.13267i	−0.487624	−	0.0650702i
$304$		0			0
$305$			8.48802i			0.486023i
$306$		0			0
$307$	18.1546			1.03614			0.518068	−	0.855340i	$-0.326652\pi$
$307$	18.1546			1.03614			0.518068	+	0.855340i	$0.326652\pi$
$308$		0			0
$309$	−0.621497	+	4.65738i	−0.0353557	+	0.264949i
$310$		0			0
$311$	−19.2924			−1.09397			−0.546987	−	0.837141i	$-0.684225\pi$
$311$	−19.2924			−1.09397			−0.546987	+	0.837141i	$0.684225\pi$
$312$		0			0
$313$	−29.1048			−1.64510			−0.822551	−	0.568692i	$-0.807450\pi$
$313$	−29.1048			−1.64510			−0.822551	+	0.568692i	$0.807450\pi$
$314$		0			0
$315$	3.52877	−	12.9865i	0.198824	−	0.731707i
$316$		0			0
$317$	14.9522			0.839800			0.419900	−	0.907570i	$-0.362065\pi$
$317$	14.9522			0.839800			0.419900	+	0.907570i	$0.362065\pi$
$318$		0			0
$319$			11.1602i			0.624852i
$320$		0			0
$321$	2.65738	−	19.9139i	0.148321	−	1.11149i
$322$		0			0
$323$		0			0
$324$		0			0
$325$		−	4.79005i		−	0.265704i
$326$		0			0
$327$	3.32260	−	24.8989i	0.183740	−	1.37691i
$328$		0			0
$329$		−	1.57331i		−	0.0867391i
$330$		0			0
$331$	−20.6020			−1.13239			−0.566194	−	0.824272i	$-0.691585\pi$
$331$	−20.6020			−1.13239			−0.566194	+	0.824272i	$0.691585\pi$
$332$		0			0
$333$	3.66374	−	13.4832i	0.200772	−	0.738877i
$334$		0			0
$335$	−21.4872			−1.17397
$336$		0			0
$337$	13.6355			0.742773			0.371387	−	0.928478i	$-0.378882\pi$
$337$	13.6355			0.742773			0.371387	+	0.928478i	$0.378882\pi$
$338$		0			0
$339$	−4.41989	+	33.1219i	−0.240056	+	1.79893i
$340$		0			0
$341$	6.22240			0.336962
$342$		0			0
$343$			17.1246i			0.924639i
$344$		0			0
$345$	−28.6930	−	3.82890i	−1.54478	−	0.206141i
$346$		0			0
$347$			35.3775i			1.89916i	0.313523	+	0.949581i	$0.398491\pi$
$347$			35.3775i			1.89916i	−0.313523	+	0.949581i	$0.601509\pi$
$348$		0			0
$349$			14.5028i			0.776319i	0.921592	+	0.388159i	$0.126889\pi$
$349$			14.5028i			0.776319i	−0.921592	+	0.388159i	$0.873111\pi$
$350$		0			0
$351$	−2.01380	+	4.79005i	−0.107489	+	0.255674i
$352$		0			0
$353$		−	21.2808i		−	1.13267i	−0.824177	−	0.566333i	$-0.808362\pi$
$353$		−	21.2808i		−	1.13267i	0.824177	−	0.566333i	$-0.191638\pi$
$354$		0			0
$355$	3.43366			0.182240
$356$		0			0
$357$	−3.87247	−	0.516756i	−0.204953	−	0.0273496i
$358$		0			0
$359$	15.9394			0.841251			0.420626	−	0.907234i	$-0.361811\pi$
$359$	15.9394			0.841251			0.420626	+	0.907234i	$0.361811\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$		0			0
$363$	−14.7306	−	1.96570i	−0.773156	−	0.103173i
$364$		0			0
$365$	8.48802			0.444283
$366$		0			0
$367$		−	20.4474i		−	1.06735i	−0.845690	−	0.533674i	$-0.820811\pi$
$367$		−	20.4474i		−	1.06735i	0.845690	−	0.533674i	$-0.179189\pi$
$368$		0			0
$369$	7.58822	−	27.9260i	0.395027	−	1.45377i
$370$		0			0
$371$			14.7458i			0.765566i
$372$		0			0
$373$			9.58011i			0.496039i	0.968755	+	0.248020i	$0.0797797\pi$
$373$			9.58011i			0.496039i	−0.968755	+	0.248020i	$0.920220\pi$
$374$		0			0
$375$	−1.12780	−	0.150497i	−0.0582391	−	0.00777163i
$376$		0			0
$377$			7.17421i			0.369491i
$378$		0			0
$379$	30.4474			1.56398			0.781990	−	0.623291i	$-0.214205\pi$
$379$	30.4474			1.56398			0.781990	+	0.623291i	$0.214205\pi$
$380$		0			0
$381$	2.39351	−	17.9365i	0.122623	−	0.918914i
$382$		0			0
$383$	9.58542			0.489792			0.244896	−	0.969549i	$-0.421246\pi$
$383$	9.58542			0.489792			0.244896	+	0.969549i	$0.421246\pi$
$384$		0			0
$385$	−6.97812			−0.355638
$386$		0			0
$387$	−17.7941	−	4.83511i	−0.904525	−	0.245783i
$388$		0			0
$389$	10.8044			0.547805			0.273903	−	0.961757i	$-0.411685\pi$
$389$	10.8044			0.547805			0.273903	+	0.961757i	$0.411685\pi$
$390$		0			0
$391$			8.40367i			0.424992i
$392$		0			0
$393$	−2.04958	+	15.3592i	−0.103388	+	0.774767i
$394$		0			0
$395$		−	34.0029i		−	1.71087i
$396$		0			0
$397$		−	5.16021i		−	0.258984i	−0.991580	−	0.129492i	$-0.958665\pi$
$397$		−	5.16021i		−	0.258984i	0.991580	−	0.129492i	$-0.0413346\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			17.1824i			0.858049i	0.903293	+	0.429025i	$0.141143\pi$
$401$			17.1824i			0.858049i	−0.903293	+	0.429025i	$0.858857\pi$
$402$		0			0
$403$	4.00000			0.199254
$404$		0			0
$405$	−24.2877	−	14.2514i	−1.20686	−	0.708158i
$406$		0			0
$407$	−7.24503			−0.359123
$408$		0			0
$409$	−10.6020			−0.524235			−0.262117	−	0.965036i	$-0.584421\pi$
$409$	−10.6020			−0.524235			−0.262117	+	0.965036i	$0.584421\pi$
$410$		0			0
$411$	−3.93649	+	29.4993i	−0.194173	+	1.45510i
$412$		0			0
$413$	−3.54402			−0.174390
$414$		0			0
$415$		−	39.9168i		−	1.95944i
$416$		0			0
$417$	−10.5524	−	1.40815i	−0.516754	−	0.0689574i
$418$		0			0
$419$		−	24.2110i		−	1.18279i	−0.806383	−	0.591393i	$-0.798578\pi$
$419$		−	24.2110i		−	1.18279i	0.806383	−	0.591393i	$-0.201422\pi$
$420$		0			0
$421$		−	10.0829i		−	0.491412i	−0.969344	−	0.245706i	$-0.920980\pi$
$421$		−	10.0829i		−	0.491412i	0.969344	−	0.245706i	$-0.0790198\pi$
$422$		0			0
$423$	−3.17700	−	0.863273i	−0.154471	−	0.0419738i
$424$		0			0
$425$		−	7.53621i		−	0.365560i
$426$		0			0
$427$	3.88921			0.188212
$428$		0			0
$429$	2.67071	+	0.356388i	0.128943	+	0.0172066i
$430$		0			0
$431$	24.7794			1.19358			0.596792	−	0.802396i	$-0.296442\pi$
$431$	24.7794			1.19358			0.596792	+	0.802396i	$0.296442\pi$
$432$		0			0
$433$	7.37016			0.354187			0.177094	−	0.984194i	$-0.443330\pi$
$433$	7.37016			0.354187			0.177094	+	0.984194i	$0.443330\pi$
$434$		0			0
$435$	−38.5385	−	5.14271i	−1.84778	−	0.246574i
$436$		0			0
$437$		0			0
$438$		0			0
$439$			2.30911i			0.110208i	0.998481	+	0.0551038i	$0.0175490\pi$
$439$			2.30911i			0.110208i	−0.998481	+	0.0551038i	$0.982451\pi$
$440$		0			0
$441$	14.3148	+	3.88969i	0.681655	+	0.185223i
$442$		0			0
$443$			4.48580i			0.213127i	0.994306	+	0.106563i	$0.0339847\pi$
$443$			4.48580i			0.213127i	−0.994306	+	0.106563i	$0.966015\pi$
$444$		0			0
$445$		−	37.0494i		−	1.75631i
$446$		0			0
$447$	−12.7928	−	1.70712i	−0.605080	−	0.0807439i
$448$		0			0
$449$			23.4756i			1.10788i	0.832555	+	0.553942i	$0.186877\pi$
$449$			23.4756i			1.10788i	−0.832555	+	0.553942i	$0.813123\pi$
$450$		0			0
$451$	−15.0057			−0.706589
$452$		0			0
$453$	4.71806	−	35.3562i	0.221674	−	1.66118i
$454$		0			0
$455$	−4.48580			−0.210298
$456$		0			0
$457$	5.27100			0.246567			0.123283	−	0.992371i	$-0.460658\pi$
$457$	5.27100			0.246567			0.123283	+	0.992371i	$0.460658\pi$
$458$		0			0
$459$	−3.16832	+	7.53621i	−0.147885	+	0.351760i
$460$		0			0
$461$	23.7351			1.10546			0.552728	−	0.833362i	$-0.313587\pi$
$461$	23.7351			1.10546			0.552728	+	0.833362i	$0.313587\pi$
$462$		0			0
$463$		−	3.69089i		−	0.171530i	−0.996315	−	0.0857652i	$-0.972667\pi$
$463$		−	3.69089i		−	0.171530i	0.996315	−	0.0857652i	$-0.0273335\pi$
$464$		0			0
$465$	−2.86733	+	21.4872i	−0.132969	+	0.996446i
$466$		0			0
$467$			39.5013i			1.82790i	0.405827	+	0.913950i	$0.366984\pi$
$467$			39.5013i			1.82790i	−0.405827	+	0.913950i	$0.633016\pi$
$468$		0			0
$469$			9.84545i			0.454621i
$470$		0			0
$471$	1.57331	−	11.7901i	0.0724941	−	0.543257i
$472$		0			0
$473$			9.56141i			0.439634i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	29.7765	+	8.09105i	1.36337	+	0.370464i
$478$		0			0
$479$	−18.1950			−0.831352			−0.415676	−	0.909513i	$-0.636455\pi$
$479$	−18.1950			−0.831352			−0.415676	+	0.909513i	$0.636455\pi$
$480$		0			0
$481$	−4.65738			−0.212358
$482$		0			0
$483$	−1.75440	+	13.1471i	−0.0798280	+	0.598216i
$484$		0			0
$485$	−27.7450			−1.25984
$486$		0			0
$487$		−	29.7347i		−	1.34741i	−0.739002	−	0.673703i	$-0.764703\pi$
$487$		−	29.7347i		−	1.34741i	0.739002	−	0.673703i	$-0.235297\pi$
$488$		0			0
$489$	−28.6930	−	3.82890i	−1.29754	−	0.173149i
$490$		0			0
$491$			24.2110i			1.09263i	0.837580	+	0.546315i	$0.183970\pi$
$491$			24.2110i			1.09263i	−0.837580	+	0.546315i	$0.816030\pi$
$492$		0			0
$493$			11.2872i			0.508351i
$494$		0			0
$495$	−3.82890	+	14.0910i	−0.172096	+	0.633345i
$496$		0			0
$497$		−	1.57331i		−	0.0705724i
$498$		0			0
$499$	23.0494			1.03183			0.515917	−	0.856639i	$-0.327451\pi$
$499$	23.0494			1.03183			0.515917	+	0.856639i	$0.327451\pi$
$500$		0			0
$501$	34.9015	+	4.65738i	1.55929	+	0.208076i
$502$		0			0
$503$	29.9752			1.33653			0.668265	−	0.743923i	$-0.267037\pi$
$503$	29.9752			1.33653			0.668265	+	0.743923i	$0.267037\pi$
$504$		0			0
$505$	−15.4693			−0.688376
$506$		0			0
$507$	1.71683	+	0.229100i	0.0762472	+	0.0101747i
$508$		0			0
$509$	30.5790			1.35539			0.677696	−	0.735342i	$-0.262978\pi$
$509$	30.5790			1.35539			0.677696	+	0.735342i	$0.262978\pi$
$510$		0			0
$511$		−	3.88921i		−	0.172049i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			8.48802i			0.374027i
$516$		0			0
$517$			1.70712i			0.0750790i
$518$		0			0
$519$	7.59700	+	1.01377i	0.333471	+	0.0444996i
$520$		0			0
$521$			13.3295i			0.583977i	0.956422	+	0.291988i	$0.0943169\pi$
$521$			13.3295i			0.583977i	−0.956422	+	0.291988i	$0.905683\pi$
$522$		0			0
$523$	8.30911			0.363332			0.181666	−	0.983360i	$-0.441851\pi$
$523$	8.30911			0.363332			0.181666	+	0.983360i	$0.441851\pi$
$524$		0			0
$525$	1.57331	−	11.7901i	0.0686647	−	0.514560i
$526$		0			0
$527$	6.29322			0.274137
$528$		0			0
$529$	5.53068			0.240464
$530$		0			0
$531$	−1.94461	+	7.15651i	−0.0843887	+	0.310566i
$532$		0			0
$533$	−9.64621			−0.417824
$534$		0			0
$535$		−	36.2929i		−	1.56908i
$536$		0			0
$537$	1.74047	−	13.0428i	0.0751070	−	0.562838i
$538$		0			0
$539$		−	7.69183i		−	0.331311i
$540$		0			0
$541$		−	12.9665i		−	0.557473i	−0.960368	−	0.278736i	$-0.910084\pi$
$541$		−	12.9665i		−	0.557473i	0.960368	−	0.278736i	$-0.0899156\pi$
$542$		0			0
$543$	5.40220	−	40.4831i	0.231831	−	1.73730i
$544$		0			0
$545$		−	45.3780i		−	1.94378i
$546$		0			0
$547$	1.25157			0.0535133			0.0267567	−	0.999642i	$-0.491482\pi$
$547$	1.25157			0.0535133			0.0267567	+	0.999642i	$0.491482\pi$
$548$		0			0
$549$	2.13401	−	7.85356i	0.0910774	−	0.335182i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−15.5801			−0.662533
$554$		0			0
$555$	3.33856	−	25.0186i	0.141714	−	1.06198i
$556$		0			0
$557$	19.2747			0.816696			0.408348	−	0.912826i	$-0.366105\pi$
$557$	19.2747			0.816696			0.408348	+	0.912826i	$0.366105\pi$
$558$		0			0
$559$			6.14644i			0.259967i
$560$		0			0
$561$	4.20184	+	0.560707i	0.177402	+	0.0236731i
$562$		0			0
$563$			4.91862i			0.207295i	0.994614	+	0.103648i	$0.0330514\pi$
$563$			4.91862i			0.207295i	−0.994614	+	0.103648i	$0.966949\pi$
$564$		0			0
$565$			60.3642i			2.53954i
$566$		0			0
$567$	−6.53000	+	11.1286i	−0.274234	+	0.467358i
$568$		0			0
$569$		−	25.2553i		−	1.05876i	−0.848385	−	0.529379i	$-0.822425\pi$
$569$		−	25.2553i		−	1.05876i	0.848385	−	0.529379i	$-0.177575\pi$
$570$		0			0
$571$	−31.4337			−1.31546			−0.657729	−	0.753255i	$-0.728483\pi$
$571$	−31.4337			−1.31546			−0.657729	+	0.753255i	$0.728483\pi$
$572$		0			0
$573$	−31.4878	−	4.20184i	−1.31542	−	0.175534i
$574$		0			0
$575$	−25.5856			−1.06700
$576$		0			0
$577$	−45.3423			−1.88762			−0.943812	−	0.330482i	$-0.892789\pi$
$577$	−45.3423			−1.88762			−0.943812	+	0.330482i	$0.892789\pi$
$578$		0			0
$579$	−20.1464	−	2.68841i	−0.837258	−	0.111727i
$580$		0			0
$581$	−18.2899			−0.758791
$582$		0			0
$583$		−	16.0000i		−	0.662652i
$584$		0			0
$585$	−2.46136	+	9.05827i	−0.101765	+	0.374513i
$586$		0			0
$587$		−	15.9748i		−	0.659352i	−0.944094	−	0.329676i	$-0.893060\pi$
$587$		−	15.9748i		−	0.659352i	0.944094	−	0.329676i	$-0.106940\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$	0.921413	+	0.122957i	0.0379019	+	0.00505776i
$592$		0			0
$593$			47.1577i			1.93653i	0.249919	+	0.968267i	$0.419596\pi$
$593$			47.1577i			1.93653i	−0.249919	+	0.968267i	$0.580404\pi$
$594$		0			0
$595$	−7.05753			−0.289331
$596$		0			0
$597$	−2.81630	+	21.1048i	−0.115263	+	0.863763i
$598$		0			0
$599$	−37.5115			−1.53268			−0.766338	−	0.642437i	$-0.777923\pi$
$599$	−37.5115			−1.53268			−0.766338	+	0.642437i	$0.777923\pi$
$600$		0			0
$601$	−13.9446			−0.568812			−0.284406	−	0.958704i	$-0.591796\pi$
$601$	−13.9446			−0.568812			−0.284406	+	0.958704i	$0.591796\pi$
$602$		0			0
$603$	19.8811	+	5.40220i	0.809621	+	0.219995i
$604$		0			0
$605$	−26.8463			−1.09146
$606$		0			0
$607$		−	47.2040i		−	1.91595i	−0.286851	−	0.957975i	$-0.592609\pi$
$607$		−	47.2040i		−	1.91595i	0.286851	−	0.957975i	$-0.407391\pi$
$608$		0			0
$609$	−2.35639	+	17.6583i	−0.0954857	+	0.715552i
$610$		0			0
$611$			1.09740i			0.0443961i
$612$		0			0
$613$		−	32.6295i		−	1.31789i	−0.752189	−	0.658947i	$-0.771002\pi$
$613$		−	32.6295i		−	1.31789i	0.752189	−	0.658947i	$-0.228998\pi$
$614$		0			0
$615$	6.91472	−	51.8176i	0.278828	−	2.08949i
$616$		0			0
$617$		−	3.23141i		−	0.130092i	−0.997882	−	0.0650459i	$-0.979281\pi$
$617$		−	3.23141i		−	0.130092i	0.997882	−	0.0650459i	$-0.0207194\pi$
$618$		0			0
$619$	42.2096			1.69655			0.848274	−	0.529557i	$-0.177642\pi$
$619$	42.2096			1.69655			0.848274	+	0.529557i	$0.177642\pi$
$620$		0			0
$621$	25.5856	+	10.7565i	1.02672	+	0.431645i
$622$		0			0
$623$	−16.9760			−0.680131
$624$		0			0
$625$	−26.0057			−1.04023
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−7.32748			−0.292166
$630$		0			0
$631$		−	9.72655i		−	0.387208i	−0.981080	−	0.193604i	$-0.937982\pi$
$631$		−	9.72655i		−	0.387208i	0.981080	−	0.193604i	$-0.0620176\pi$
$632$		0			0
$633$	−29.4751	−	3.93326i	−1.17153	−	0.156333i
$634$		0			0
$635$		−	32.6890i		−	1.29723i
$636$		0			0
$637$		−	4.94461i		−	0.195912i
$638$		0			0
$639$	−3.17700	−	0.863273i	−0.125680	−	0.0341506i
$640$		0			0
$641$		−	16.9760i		−	0.670513i	−0.942127	−	0.335257i	$-0.891177\pi$
$641$		−	16.9760i		−	0.670513i	0.942127	−	0.335257i	$-0.108823\pi$
$642$		0			0
$643$	−34.3366			−1.35411			−0.677053	−	0.735935i	$-0.736743\pi$
$643$	−34.3366			−1.35411			−0.677053	+	0.735935i	$0.736743\pi$
$644$		0			0
$645$	−33.0175	−	4.40597i	−1.30006	−	0.173485i
$646$		0			0
$647$	18.2190			0.716264			0.358132	−	0.933671i	$-0.383414\pi$
$647$	18.2190			0.716264			0.358132	+	0.933671i	$0.383414\pi$
$648$		0			0
$649$	3.84545			0.150947
$650$		0			0
$651$	9.84545	+	1.31381i	0.385873	+	0.0514923i
$652$		0			0
$653$	24.4769			0.957853			0.478927	−	0.877855i	$-0.341026\pi$
$653$	24.4769			0.957853			0.478927	+	0.877855i	$0.341026\pi$
$654$		0			0
$655$			27.9919i			1.09373i
$656$		0			0
$657$	−7.85356	−	2.13401i	−0.306397	−	0.0832557i
$658$		0			0
$659$			8.48802i			0.330646i	0.986239	+	0.165323i	$0.0528667\pi$
$659$			8.48802i			0.330646i	−0.986239	+	0.165323i	$0.947133\pi$
$660$		0			0
$661$		−	41.4693i		−	1.61297i	−0.591255	−	0.806485i	$-0.701367\pi$
$661$		−	41.4693i		−	1.61297i	0.591255	−	0.806485i	$-0.298633\pi$
$662$		0			0
$663$	2.70110	+	0.360444i	0.104902	+	0.0139985i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	38.3204			1.48377
$668$		0			0
$669$	−4.71806	+	35.3562i	−0.182411	+	1.36695i
$670$		0			0
$671$	−4.22000			−0.162911
$672$		0			0
$673$	22.6849			0.874439			0.437220	−	0.899355i	$-0.355963\pi$
$673$	22.6849			0.874439			0.437220	+	0.899355i	$0.355963\pi$
$674$		0			0
$675$	−22.9446	−	9.64621i	−0.883138	−	0.371283i
$676$		0			0
$677$	26.2742			1.00980			0.504901	−	0.863178i	$-0.331529\pi$
$677$	26.2742			1.00980			0.504901	+	0.863178i	$0.331529\pi$
$678$		0			0
$679$			12.7128i			0.487872i
$680$		0			0
$681$	−4.85922	+	36.4141i	−0.186206	+	1.39539i
$682$		0			0
$683$			9.09182i			0.347889i	0.984755	+	0.173944i	$0.0556513\pi$
$683$			9.09182i			0.347889i	−0.984755	+	0.173944i	$0.944349\pi$
$684$		0			0
$685$			53.7622i			2.05415i
$686$		0			0
$687$	−1.06701	+	7.99594i	−0.0407088	+	0.305064i
$688$		0			0
$689$		−	10.2854i		−	0.391843i
$690$		0			0
$691$	22.0438			0.838584			0.419292	−	0.907851i	$-0.362278\pi$
$691$	22.0438			0.838584			0.419292	+	0.907851i	$0.362278\pi$
$692$		0			0
$693$	6.45652	+	1.75440i	0.245263	+	0.0666442i
$694$		0			0
$695$	−19.2316			−0.729498
$696$		0			0
$697$	−15.1764			−0.574848
$698$		0			0
$699$	5.78600	−	43.3592i	0.218846	−	1.63999i
$700$		0			0
$701$	−33.4839			−1.26467			−0.632334	−	0.774696i	$-0.717903\pi$
$701$	−33.4839			−1.26467			−0.632334	+	0.774696i	$0.717903\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	−5.89503	−	0.786653i	−0.222020	−	0.0296270i
$706$		0			0
$707$			7.08804i			0.266573i
$708$		0			0
$709$			38.0551i			1.42919i	0.699539	+	0.714594i	$0.253389\pi$
$709$			38.0551i			1.42919i	−0.699539	+	0.714594i	$0.746611\pi$
$710$		0			0
$711$	−8.54881	+	31.4612i	−0.320605	+	1.17989i
$712$		0			0
$713$		−	21.3656i		−	0.800150i
$714$		0			0
$715$	4.86733			0.182028
$716$		0			0
$717$	−45.8103	−	6.11309i	−1.71082	−	0.228297i
$718$		0			0
$719$	4.09842			0.152845			0.0764226	−	0.997076i	$-0.475650\pi$
$719$	4.09842			0.152845			0.0764226	+	0.997076i	$0.475650\pi$
$720$		0			0
$721$	3.88921			0.144842
$722$		0			0
$723$	−0.265341	−	0.0354081i	−0.00986816	−	0.00131684i
$724$		0			0
$725$	−34.3649			−1.27628
$726$		0			0
$727$		−	32.1821i		−	1.19357i	−0.802402	−	0.596784i	$-0.796445\pi$
$727$		−	32.1821i		−	1.19357i	0.802402	−	0.596784i	$-0.203555\pi$
$728$		0			0
$729$	18.8892	+	19.2924i	0.699600	+	0.714534i
$730$		0			0
$731$			9.67023i			0.357666i
$732$		0			0
$733$		−	32.6574i		−	1.20623i	−0.797655	−	0.603114i	$-0.793926\pi$
$733$		−	32.6574i		−	1.20623i	0.797655	−	0.603114i	$-0.206074\pi$
$734$		0			0
$735$	26.5615	+	3.54445i	0.979734	+	0.130739i
$736$		0			0
$737$		−	10.6828i		−	0.393507i
$738$		0			0
$739$	31.8892			1.17306			0.586532	−	0.809926i	$-0.300493\pi$
$739$	31.8892			1.17306			0.586532	+	0.809926i	$0.300493\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−5.19582			−0.190616			−0.0953081	−	0.995448i	$-0.530384\pi$
$743$	−5.19582			−0.190616			−0.0953081	+	0.995448i	$0.530384\pi$
$744$		0			0
$745$	−23.3148			−0.854187
$746$		0			0
$747$	−10.0357	+	36.9331i	−0.367185	+	1.35131i
$748$		0			0
$749$	−16.6294			−0.607625
$750$		0			0
$751$			31.7347i			1.15801i	0.815323	+	0.579007i	$0.196560\pi$
$751$			31.7347i			1.15801i	−0.815323	+	0.579007i	$0.803440\pi$
$752$		0			0
$753$	1.23183	−	9.23110i	0.0448904	−	0.336400i
$754$		0			0
$755$		−	64.4363i		−	2.34508i
$756$		0			0
$757$		−	1.17644i		−	0.0427583i	−0.999771	−	0.0213791i	$-0.993194\pi$
$757$		−	1.17644i		−	0.0427583i	0.999771	−	0.0213791i	$-0.00680571\pi$
$758$		0			0
$759$	1.90362	−	14.2653i	0.0690969	−	0.517799i
$760$		0			0
$761$		−	3.35299i		−	0.121546i	−0.998152	−	0.0607729i	$-0.980643\pi$
$761$		−	3.35299i		−	0.121546i	0.998152	−	0.0607729i	$-0.0193566\pi$
$762$		0			0
$763$	−20.7922			−0.752728
$764$		0			0
$765$	−3.87247	+	14.2514i	−0.140010	+	0.515261i
$766$		0			0
$767$	2.47200			0.0892588
$768$		0			0
$769$	21.0494			0.759062			0.379531	−	0.925179i	$-0.376085\pi$
$769$	21.0494			0.759062			0.379531	+	0.925179i	$0.376085\pi$
$770$		0			0
$771$	5.78600	−	43.3592i	0.208378	−	1.56154i
$772$		0			0
$773$	18.0317			0.648556			0.324278	−	0.945962i	$-0.394879\pi$
$773$	18.0317			0.648556			0.324278	+	0.945962i	$0.394879\pi$
$774$		0			0
$775$			19.1602i			0.688255i
$776$		0			0
$777$	−11.4635	−	1.52973i	−0.411251	−	0.0548787i
$778$		0			0
$779$		0			0
$780$		0			0
$781$			1.70712i			0.0610855i
$782$		0			0
$783$	34.3649	+	14.4474i	1.22810	+	0.516309i
$784$		0			0
$785$		−	21.4872i		−	0.766912i
$786$		0			0
$787$	−29.0057			−1.03394			−0.516970	−	0.856004i	$-0.672940\pi$
$787$	−29.0057			−1.03394			−0.516970	+	0.856004i	$0.672940\pi$
$788$		0			0
$789$	27.7197	+	3.69901i	0.986846	+	0.131688i
$790$		0			0
$791$	27.6589			0.983436
$792$		0			0
$793$	−2.71278			−0.0963335
$794$		0			0
$795$	55.2513	+	7.37292i	1.95956	+	0.261490i
$796$		0			0
$797$	−14.2776			−0.505739			−0.252869	−	0.967500i	$-0.581374\pi$
$797$	−14.2776			−0.505739			−0.252869	+	0.967500i	$0.581374\pi$
$798$		0			0
$799$			1.72655i			0.0610808i
$800$		0			0
$801$	−9.31476	+	34.2801i	−0.329121	+	1.21123i
$802$		0			0
$803$			4.22000i			0.148920i
$804$		0			0
$805$			23.9605i			0.844497i
$806$		0			0
$807$	25.2553	+	3.37016i	0.889030	+	0.118635i
$808$		0			0
$809$		−	7.03630i		−	0.247383i	−0.992321	−	0.123692i	$-0.960527\pi$
$809$		−	7.03630i		−	0.247383i	0.992321	−	0.123692i	$-0.0394733\pi$
$810$		0			0
$811$	24.5858			0.863323			0.431661	−	0.902036i	$-0.357928\pi$
$811$	24.5858			0.863323			0.431661	+	0.902036i	$0.357928\pi$
$812$		0			0
$813$	0.985358	−	7.38409i	0.0345580	−	0.258971i
$814$		0			0
$815$	−52.2927			−1.83173
$816$		0			0
$817$		0			0
$818$		0			0
$819$	4.15050	+	1.12780i	0.145030	+	0.0394084i
$820$		0			0
$821$	29.4740			1.02865			0.514324	−	0.857596i	$-0.328043\pi$
$821$	29.4740			1.02865			0.514324	+	0.857596i	$0.328043\pi$
$822$		0			0
$823$		−	1.16021i		−	0.0404424i	−0.999796	−	0.0202212i	$-0.993563\pi$
$823$		−	1.16021i		−	0.0404424i	0.999796	−	0.0202212i	$-0.00643705\pi$
$824$		0			0
$825$	−1.70712	+	12.7928i	−0.0594342	+	0.445389i
$826$		0			0
$827$		−	45.5527i		−	1.58402i	−0.610507	−	0.792011i	$-0.709034\pi$
$827$		−	45.5527i		−	1.58402i	0.610507	−	0.792011i	$-0.290966\pi$
$828$		0			0
$829$			41.4693i			1.44029i	0.693824	+	0.720144i	$0.255924\pi$
$829$			41.4693i			1.44029i	−0.693824	+	0.720144i	$0.744076\pi$
$830$		0			0
$831$	1.57331	−	11.7901i	0.0545773	−	0.408992i
$832$		0			0
$833$		−	7.77937i		−	0.269539i
$834$		0			0
$835$	63.6076			2.20123
$836$		0			0
$837$	8.05520	−	19.1602i	0.278429	−	0.662274i
$838$		0			0
$839$	−26.6223			−0.919102			−0.459551	−	0.888151i	$-0.651990\pi$
$839$	−26.6223			−0.919102			−0.459551	+	0.888151i	$0.651990\pi$
$840$		0			0
$841$	22.4693			0.774804
$842$		0			0
$843$	−1.27100	+	9.52463i	−0.0437756	+	0.328046i
$844$		0			0
$845$	3.12891			0.107638
$846$		0			0
$847$			12.3010i			0.422667i
$848$		0			0
$849$	40.0275	+	5.34141i	1.37374	+	0.183317i
$850$		0			0
$851$			24.8770i			0.852772i
$852$		0			0
$853$		−	5.18807i		−	0.177636i	−0.996048	−	0.0888180i	$-0.971691\pi$
$853$		−	5.18807i		−	0.177636i	0.996048	−	0.0888180i	$-0.0283089\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			23.6820i			0.808963i	0.914546	+	0.404481i	$0.132548\pi$
$857$			23.6820i			0.808963i	−0.914546	+	0.404481i	$0.867452\pi$
$858$		0			0
$859$	−18.4199			−0.628479			−0.314239	−	0.949344i	$-0.601749\pi$
$859$	−18.4199			−0.628479			−0.314239	+	0.949344i	$0.601749\pi$
$860$		0			0
$861$	−23.7428	−	3.16832i	−0.809153	−	0.107976i
$862$		0			0
$863$	−14.0966			−0.479854			−0.239927	−	0.970791i	$-0.577124\pi$
$863$	−14.0966			−0.479854			−0.239927	+	0.970791i	$0.577124\pi$
$864$		0			0
$865$	13.8454			0.470759
$866$		0			0
$867$	−24.9365	−	3.32761i	−0.846888	−	0.113012i
$868$		0			0
$869$	16.9052			0.573471
$870$		0			0
$871$		−	6.86733i		−	0.232691i
$872$		0			0
$873$	25.6712	+	6.97551i	0.868837	+	0.236085i
$874$		0			0
$875$			0.941782i			0.0318380i
$876$		0			0
$877$			47.3977i			1.60051i	0.599662	+	0.800253i	$0.295302\pi$
$877$			47.3977i			1.60051i	−0.599662	+	0.800253i	$0.704698\pi$
$878$		0			0
$879$	24.6034	+	3.28317i	0.829854	+	0.110738i
$880$		0			0
$881$			20.8657i			0.702984i	0.936191	+	0.351492i	$0.114326\pi$
$881$			20.8657i			0.702984i	−0.936191	+	0.351492i	$0.885674\pi$
$882$		0			0
$883$	−29.4612			−0.991448			−0.495724	−	0.868480i	$-0.665097\pi$
$883$	−29.4612			−0.991448			−0.495724	+	0.868480i	$0.665097\pi$
$884$		0			0
$885$	−1.77201	+	13.2791i	−0.0595655	+	0.446372i
$886$		0			0
$887$	24.7554			0.831206			0.415603	−	0.909546i	$-0.363571\pi$
$887$	24.7554			0.831206			0.415603	+	0.909546i	$0.363571\pi$
$888$		0			0
$889$	−14.9781			−0.502350
$890$		0			0
$891$	7.08539	−	12.0751i	0.237369	−	0.404532i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		−	23.7703i		−	0.794554i
$896$		0			0
$897$	1.22372	−	9.17031i	0.0408587	−	0.306188i
$898$		0			0
$899$		−	28.6968i		−	0.957093i
$900$		0			0
$901$		−	16.1821i		−	0.539104i
$902$		0			0
$903$	−2.01881	+	15.1286i	−0.0671819	+	0.503449i
$904$		0			0
$905$		−	73.7799i		−	2.45253i
$906$		0			0
$907$	−26.0081			−0.863585			−0.431793	−	0.901973i	$-0.642119\pi$
$907$	−26.0081			−0.863585			−0.431793	+	0.901973i	$0.642119\pi$
$908$		0			0
$909$	14.3130	+	3.88921i	0.474733	+	0.128997i
$910$		0			0
$911$	20.2442			0.670721			0.335361	−	0.942090i	$-0.391142\pi$
$911$	20.2442			0.670721			0.335361	+	0.942090i	$0.391142\pi$
$912$		0			0
$913$	19.8454			0.656788
$914$		0			0
$915$	1.94461	−	14.5725i	0.0642867	−	0.481752i
$916$		0			0
$917$	12.8259			0.423548
$918$		0			0
$919$			19.4693i			0.642234i	0.947040	+	0.321117i	$0.104058\pi$
$919$			19.4693i			0.642234i	−0.947040	+	0.321117i	$0.895942\pi$
$920$		0			0
$921$	−31.1683	−	4.15921i	−1.02703	−	0.137051i
$922$		0			0
$923$			1.09740i			0.0361214i
$924$		0			0
$925$		−	22.3091i		−	0.733519i
$926$		0			0
$927$	2.13401	−	7.85356i	0.0700902	−	0.257945i
$928$		0			0
$929$		−	13.7446i		−	0.450947i	−0.974249	−	0.225473i	$-0.927607\pi$
$929$		−	13.7446i		−	0.450947i	0.974249	−	0.225473i	$-0.0723929\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	33.1219	+	4.41989i	1.08436	+	0.144701i
$934$		0			0
$935$	7.65779			0.250437
$936$		0			0
$937$	−29.6239			−0.967770			−0.483885	−	0.875132i	$-0.660775\pi$
$937$	−29.6239			−0.967770			−0.483885	+	0.875132i	$0.660775\pi$
$938$		0			0
$939$	49.9681	+	6.66791i	1.63065	+	0.217599i
$940$		0			0
$941$	−51.1182			−1.66641			−0.833203	−	0.552968i	$-0.813495\pi$
$941$	−51.1182			−1.66641			−0.833203	+	0.552968i	$0.813495\pi$
$942$		0			0
$943$			51.5244i			1.67787i
$944$		0			0
$945$	−9.03351	+	21.4872i	−0.293860	+	0.698980i
$946$		0			0
$947$			8.57283i			0.278579i	0.990252	+	0.139290i	$0.0444819\pi$
$947$			8.57283i			0.278579i	−0.990252	+	0.139290i	$0.955518\pi$
$948$		0			0
$949$			2.71278i			0.0880604i
$950$		0			0
$951$	−25.6704	−	3.42555i	−0.832421	−	0.111081i
$952$		0			0
$953$			35.9382i			1.16415i	0.813135	+	0.582076i	$0.197759\pi$
$953$			35.9382i			1.16415i	−0.813135	+	0.582076i	$0.802241\pi$
$954$		0			0
$955$	−57.3861			−1.85697
$956$		0			0
$957$	2.55680	−	19.1602i	0.0826497	−	0.619362i
$958$		0			0
$959$	24.6338			0.795468
$960$		0			0
$961$	15.0000			0.483871
$962$		0			0
$963$	−9.12456	+	33.5801i	−0.294035	+	1.08210i
$964$		0			0
$965$	−36.7166			−1.18195
$966$		0			0
$967$			2.56634i			0.0825278i	0.999148	+	0.0412639i	$0.0131384\pi$
$967$			2.56634i			0.0825278i	−0.999148	+	0.0412639i	$0.986862\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			5.95660i			0.191156i	0.995422	+	0.0955782i	$0.0304700\pi$
$971$			5.95660i			0.191156i	−0.995422	+	0.0955782i	$0.969530\pi$
$972$		0			0
$973$			8.81193i			0.282498i
$974$		0			0
$975$	−1.09740	+	8.22372i	−0.0351450	+	0.263370i
$976$		0			0
$977$			17.3040i			0.553604i	0.960927	+	0.276802i	$0.0892746\pi$
$977$			17.3040i			0.553604i	−0.960927	+	0.276802i	$0.910725\pi$
$978$		0			0
$979$	18.4199			0.588702
$980$		0			0
$981$	−11.4087	+	41.9861i	−0.364251	+	1.34051i
$982$		0			0
$983$	0.806215			0.0257143			0.0128571	−	0.999917i	$-0.495907\pi$
$983$	0.806215			0.0257143			0.0128571	+	0.999917i	$0.495907\pi$
$984$		0			0
$985$	1.67926			0.0535058
$986$		0			0
$987$	−0.360444	+	2.70110i	−0.0114731	+	0.0859770i
$988$		0			0
$989$	32.8307			1.04395
$990$		0			0
$991$		−	50.1659i		−	1.59357i	−0.604262	−	0.796786i	$-0.706532\pi$
$991$		−	50.1659i		−	1.59357i	0.604262	−	0.796786i	$-0.293468\pi$
$992$		0			0
$993$	35.3702	+	4.71992i	1.12244	+	0.149782i
$994$		0			0
$995$			38.4633i			1.21937i
$996$		0			0
$997$		−	2.44743i		−	0.0775110i	−0.999249	−	0.0387555i	$-0.987661\pi$
$997$		−	2.44743i		−	0.0775110i	0.999249	−	0.0387555i	$-0.0123394\pi$
$998$		0			0
$999$	−9.37904	+	22.3091i	−0.296740	+	0.705829i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2496.2.j.c.287.1	✓	12
3.2	odd	2	inner	2496.2.j.c.287.4	yes	12
4.3	odd	2		2496.2.j.d.287.11	yes	12
8.3	odd	2	inner	2496.2.j.c.287.2	yes	12
8.5	even	2		2496.2.j.d.287.12	yes	12
12.11	even	2		2496.2.j.d.287.10	yes	12
24.5	odd	2		2496.2.j.d.287.9	yes	12
24.11	even	2	inner	2496.2.j.c.287.3	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2496.2.j.c.287.1	✓	12	1.1	even	1	trivial
2496.2.j.c.287.2	yes	12	8.3	odd	2	inner
2496.2.j.c.287.3	yes	12	24.11	even	2	inner
2496.2.j.c.287.4	yes	12	3.2	odd	2	inner
2496.2.j.d.287.9	yes	12	24.5	odd	2
2496.2.j.d.287.10	yes	12	12.11	even	2
2496.2.j.d.287.11	yes	12	4.3	odd	2
2496.2.j.d.287.12	yes	12	8.5	even	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)

\(f(q)\)	\(=\)	\(q+(-1.71683 - 0.229100i) q^{3} -3.12891 q^{5} +1.43366i q^{7} +(2.89503 + 0.786653i) q^{9} -1.55560i q^{11} -1.00000i q^{13} +(5.37181 + 0.716832i) q^{15} -1.57331i q^{17} +(0.328453 - 2.46136i) q^{21} -5.34141 q^{23} +4.79005 q^{25} +(-4.79005 - 2.01380i) q^{27} -7.17421 q^{29} +4.00000i q^{31} +(-0.356388 + 2.67071i) q^{33} -4.48580i q^{35} -4.65738i q^{37} +(-0.229100 + 1.71683i) q^{39} -9.64621i q^{41} -6.14644 q^{43} +(-9.05827 - 2.46136i) q^{45} -1.09740 q^{47} +4.94461 q^{49} +(-0.360444 + 2.70110i) q^{51} +10.2854 q^{53} +4.86733i q^{55} +2.47200i q^{59} -2.71278i q^{61} +(-1.12780 + 4.15050i) q^{63} +3.12891i q^{65} +6.86733 q^{67} +(9.17031 + 1.22372i) q^{69} -1.09740 q^{71} -2.71278 q^{73} +(-8.22372 - 1.09740i) q^{75} +2.23021 q^{77} +10.8673i q^{79} +(7.76236 + 4.55476i) q^{81} +12.7574i q^{83} +4.92272i q^{85} +(12.3169 + 1.64361i) q^{87} +11.8410i q^{89} +1.43366 q^{91} +(0.916400 - 6.86733i) q^{93} +8.86733 q^{97} +(1.22372 - 4.50351i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{3} + 8 q^{9} + 4 q^{25} - 4 q^{27} + 16 q^{33} - 20 q^{49} - 8 q^{51} + 16 q^{67} + 8 q^{73} - 12 q^{75} - 16 q^{91} + 40 q^{97} - 72 q^{99}+O(q^{100}) \) 12 * q - 4 * q^3 + 8 * q^9 + 4 * q^25 - 4 * q^27 + 16 * q^33 - 20 * q^49 - 8 * q^51 + 16 * q^67 + 8 * q^73 - 12 * q^75 - 16 * q^91 + 40 * q^97 - 72 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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