LMFDB - Embedded newform 2496.2.j.d.287.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2496,2,Mod(287,2496)]

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

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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			287.4
Root			$0.643948 - 1.25910i$ of defining polynomial
Character	$\chi$	$=$	2496.287
Dual form			2496.2.j.d.287.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.42976 + 0.977641i) q^{3} +2.48533 q^{5} +4.85952i q^{7} +(1.08844 - 2.79559i) q^{9} +O(q^{10})$	$q+(-1.42976 + 0.977641i) q^{3} +2.48533 q^{5} +4.85952i q^{7} +(1.08844 - 2.79559i) q^{9} +3.10584i q^{11} -1.00000i q^{13} +(-3.55343 + 2.42976i) q^{15} +5.59117i q^{17} +(-4.75087 - 6.94796i) q^{21} -8.88123 q^{23} +1.17687 q^{25} +(1.17687 + 5.06112i) q^{27} +1.06010 q^{29} -4.00000i q^{31} +(-3.03640 - 4.44061i) q^{33} +12.0775i q^{35} +11.5422i q^{37} +(0.977641 + 1.42976i) q^{39} -5.95630i q^{41} +5.21327 q^{43} +(2.70513 - 6.94796i) q^{45} +1.15056 q^{47} -16.6150 q^{49} +(-5.46616 - 7.99405i) q^{51} +5.15158 q^{53} +7.71905i q^{55} -7.01640i q^{59} -8.07279i q^{61} +(13.5852 + 5.28929i) q^{63} -2.48533i q^{65} +5.71905 q^{67} +(12.6980 - 8.68265i) q^{69} +1.15056 q^{71} -8.07279 q^{73} +(-1.68265 + 1.15056i) q^{75} -15.0929 q^{77} +1.71905i q^{79} +(-6.63061 - 6.08564i) q^{81} -12.1680i q^{83} +13.8959i q^{85} +(-1.51569 + 1.03640i) q^{87} +8.25742i q^{89} +4.85952 q^{91} +(3.91056 + 5.71905i) q^{93} -3.71905 q^{97} +(8.68265 + 3.38051i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 4 q^{3} + 8 q^{9}+O(q^{10}) $ 12 * q + 4 * q^3 + 8 * q^9	$ 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

Display 100 coefficients 10 coefficients

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
$3$	−1.42976	+	0.977641i	−0.825473	+	0.564441i
$3$	−1.42976	+	0.977641i	−0.825473	+	0.564441i
$4$		0			0
$5$	2.48533			1.11147			0.555737	−	0.831358i	$-0.312436\pi$
$5$	2.48533			1.11147			0.555737	+	0.831358i	$0.312436\pi$
$6$		0			0
$7$			4.85952i			1.83673i	0.395738	+	0.918364i	$0.370489\pi$
$7$			4.85952i			1.83673i	−0.395738	+	0.918364i	$0.629511\pi$
$8$		0			0
$9$	1.08844	−	2.79559i	0.362812	−	0.931862i
$10$		0			0
$11$			3.10584i			0.936446i	0.883610	+	0.468223i	$0.155106\pi$
$11$			3.10584i			0.936446i	−0.883610	+	0.468223i	$0.844894\pi$
$12$		0			0
$13$		−	1.00000i		−	0.277350i
$13$		−	1.00000i		−	0.277350i
$14$		0			0
$15$	−3.55343	+	2.42976i	−0.917492	+	0.627362i
$16$		0			0
$17$			5.59117i			1.35606i	0.735035	+	0.678029i	$0.237166\pi$
$17$			5.59117i			1.35606i	−0.735035	+	0.678029i	$0.762834\pi$
$18$		0			0
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$	−4.75087	−	6.94796i	−1.03672	−	1.51617i
$22$		0			0
$23$	−8.88123			−1.85186			−0.925932	−	0.377691i	$-0.876718\pi$
$23$	−8.88123			−1.85186			−0.925932	+	0.377691i	$0.876718\pi$
$24$		0			0
$25$	1.17687			0.235375
$26$		0			0
$27$	1.17687	+	5.06112i	0.226490	+	0.974014i
$28$		0			0
$29$	1.06010			0.196856			0.0984279	−	0.995144i	$-0.468619\pi$
$29$	1.06010			0.196856			0.0984279	+	0.995144i	$0.468619\pi$
$30$		0			0
$31$		−	4.00000i		−	0.718421i	−0.933257	−	0.359211i	$-0.883046\pi$
$31$		−	4.00000i		−	0.718421i	0.933257	−	0.359211i	$-0.116954\pi$
$32$		0			0
$33$	−3.03640	−	4.44061i	−0.528569	−	0.773012i
$34$		0			0
$35$			12.0775i			2.04147i
$36$		0			0
$37$			11.5422i			1.89752i	0.315995	+	0.948761i	$0.397662\pi$
$37$			11.5422i			1.89752i	−0.315995	+	0.948761i	$0.602338\pi$
$38$		0			0
$39$	0.977641	+	1.42976i	0.156548	+	0.228945i
$40$		0			0
$41$		−	5.95630i		−	0.930218i	−0.885253	−	0.465109i	$-0.846015\pi$
$41$		−	5.95630i		−	0.930218i	0.885253	−	0.465109i	$-0.153985\pi$
$42$		0			0
$43$	5.21327			0.795016			0.397508	−	0.917599i	$-0.369875\pi$
$43$	5.21327			0.795016			0.397508	+	0.917599i	$0.369875\pi$
$44$		0			0
$45$	2.70513	−	6.94796i	0.403257	−	1.03574i
$46$		0			0
$47$	1.15056			0.167826			0.0839132	−	0.996473i	$-0.473258\pi$
$47$	1.15056			0.167826			0.0839132	+	0.996473i	$0.473258\pi$
$48$		0			0
$49$	−16.6150			−2.37357
$50$		0			0
$51$	−5.46616	−	7.99405i	−0.765415	−	1.11939i
$52$		0			0
$53$	5.15158			0.707624			0.353812	−	0.935317i	$-0.384885\pi$
$53$	5.15158			0.707624			0.353812	+	0.935317i	$0.384885\pi$
$54$		0			0
$55$			7.71905i			1.04084i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	7.01640i		−	0.913458i	−0.889606	−	0.456729i	$-0.849021\pi$
$59$		−	7.01640i		−	0.913458i	0.889606	−	0.456729i	$-0.150979\pi$
$60$		0			0
$61$		−	8.07279i		−	1.03362i	−0.856101	−	0.516808i	$-0.827120\pi$
$61$		−	8.07279i		−	1.03362i	0.856101	−	0.516808i	$-0.172880\pi$
$62$		0			0
$63$	13.5852	+	5.28929i	1.71158	+	0.666387i
$64$		0			0
$65$		−	2.48533i		−	0.308267i
$66$		0			0
$67$	5.71905			0.698693			0.349346	−	0.936994i	$-0.386404\pi$
$67$	5.71905			0.698693			0.349346	+	0.936994i	$0.386404\pi$
$68$		0			0
$69$	12.6980	−	8.68265i	1.52866	−	1.04527i
$70$		0			0
$71$	1.15056			0.136546			0.0682732	−	0.997667i	$-0.478251\pi$
$71$	1.15056			0.136546			0.0682732	+	0.997667i	$0.478251\pi$
$72$		0			0
$73$	−8.07279			−0.944849			−0.472425	−	0.881371i	$-0.656621\pi$
$73$	−8.07279			−0.944849			−0.472425	+	0.881371i	$0.656621\pi$
$74$		0			0
$75$	−1.68265	+	1.15056i	−0.194296	+	0.132855i
$76$		0			0
$77$	−15.0929			−1.72000
$78$		0			0
$79$			1.71905i			0.193408i	0.995313	+	0.0967039i	$0.0308300\pi$
$79$			1.71905i			0.193408i	−0.995313	+	0.0967039i	$0.969170\pi$
$80$		0			0
$81$	−6.63061	−	6.08564i	−0.736734	−	0.676182i
$82$		0			0
$83$		−	12.1680i		−	1.33561i	−0.744336	−	0.667805i	$-0.767234\pi$
$83$		−	12.1680i		−	1.33561i	0.744336	−	0.667805i	$-0.232766\pi$
$84$		0			0
$85$			13.8959i			1.50722i
$86$		0			0
$87$	−1.51569	+	1.03640i	−0.162499	+	0.111113i
$88$		0			0
$89$			8.25742i			0.875285i	0.899149	+	0.437643i	$0.144186\pi$
$89$			8.25742i			0.875285i	−0.899149	+	0.437643i	$0.855814\pi$
$90$		0			0
$91$	4.85952			0.509416
$92$		0			0
$93$	3.91056	+	5.71905i	0.405507	+	0.593038i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−3.71905			−0.377612			−0.188806	−	0.982014i	$-0.560462\pi$
$97$	−3.71905			−0.377612			−0.188806	+	0.982014i	$0.560462\pi$
$98$		0			0
$99$	8.68265	+	3.38051i	0.872639	+	0.339754i
$100$		0			0
$101$	14.0328			1.39632			0.698158	−	0.715943i	$-0.254003\pi$
$101$	14.0328			1.39632			0.698158	+	0.715943i	$0.254003\pi$
$102$		0			0
$103$			8.07279i			0.795436i	0.917508	+	0.397718i	$0.130198\pi$
$103$			8.07279i			0.795436i	−0.917508	+	0.397718i	$0.869802\pi$
$104$		0			0
$105$	−11.8075	−	17.2680i	−1.15229	−	1.68518i
$106$		0			0
$107$			13.8519i			1.33911i	0.742761	+	0.669556i	$0.233516\pi$
$107$			13.8519i			1.33911i	−0.742761	+	0.669556i	$0.766484\pi$
$108$		0			0
$109$			16.2497i			1.55644i	0.627994	+	0.778218i	$0.283876\pi$
$109$			16.2497i			1.55644i	−0.627994	+	0.778218i	$0.716124\pi$
$110$		0			0
$111$	−11.2841	−	16.5026i	−1.07104	−	1.56635i
$112$		0			0
$113$		−	11.9126i		−	1.12064i	−0.828275	−	0.560322i	$-0.810677\pi$
$113$		−	11.9126i		−	1.12064i	0.828275	−	0.560322i	$-0.189323\pi$
$114$		0			0
$115$	−22.0728			−2.05830
$116$		0			0
$117$	−2.79559	−	1.08844i	−0.258452	−	0.100626i
$118$		0			0
$119$	−27.1704			−2.49071
$120$		0			0
$121$	1.35375			0.123068
$122$		0			0
$123$	5.82313	+	8.51610i	0.525054	+	0.767870i
$124$		0			0
$125$	−9.50174			−0.849861
$126$		0			0
$127$			9.36530i			0.831036i	0.909585	+	0.415518i	$0.136400\pi$
$127$			9.36530i			0.831036i	−0.909585	+	0.415518i	$0.863600\pi$
$128$		0			0
$129$	−7.45374	+	5.09671i	−0.656265	+	0.448740i
$130$		0			0
$131$		−	18.1083i		−	1.58213i	−0.611733	−	0.791064i	$-0.709527\pi$
$131$		−	18.1083i		−	1.58213i	0.611733	−	0.791064i	$-0.290473\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	2.92492	+	12.5786i	0.251737	+	1.08259i
$136$		0			0
$137$			0.623802i			0.0532950i	0.999645	+	0.0266475i	$0.00848317\pi$
$137$			0.623802i			0.0532950i	−0.999645	+	0.0266475i	$0.991517\pi$
$138$		0			0
$139$	−5.21327			−0.442184			−0.221092	−	0.975253i	$-0.570962\pi$
$139$	−5.21327			−0.442184			−0.221092	+	0.975253i	$0.570962\pi$
$140$		0			0
$141$	−1.64503	+	1.12483i	−0.138536	+	0.0947281i
$142$		0			0
$143$	3.10584			0.259724
$144$		0			0
$145$	2.63470			0.218800
$146$		0			0
$147$	23.7554	−	16.2435i	1.95932	−	1.33974i
$148$		0			0
$149$	3.65518			0.299444			0.149722	−	0.988728i	$-0.452162\pi$
$149$	3.65518			0.299444			0.149722	+	0.988728i	$0.452162\pi$
$150$		0			0
$151$			0.152027i			0.0123718i	0.999981	+	0.00618591i	$0.00196905\pi$
$151$			0.152027i			0.0123718i	−0.999981	+	0.00618591i	$0.998031\pi$
$152$		0			0
$153$	15.6306	+	6.08564i	1.26366	+	0.491995i
$154$		0			0
$155$		−	9.94133i		−	0.798507i
$156$		0			0
$157$		−	5.71905i		−	0.456430i	−0.973611	−	0.228215i	$-0.926711\pi$
$157$		−	5.71905i		−	0.456430i	0.973611	−	0.228215i	$-0.0732888\pi$
$158$		0			0
$159$	−7.36554	+	5.03640i	−0.584125	+	0.399412i
$160$		0			0
$161$		−	43.1585i		−	3.40137i
$162$		0			0
$163$	−22.0728			−1.72887			−0.864437	−	0.502741i	$-0.832325\pi$
$163$	−22.0728			−1.72887			−0.864437	+	0.502741i	$0.832325\pi$
$164$		0			0
$165$	−7.54645	−	11.0364i	−0.587491	−	0.859182i
$166$		0			0
$167$	−11.8061			−0.913587			−0.456794	−	0.889573i	$-0.651002\pi$
$167$	−11.8061			−0.913587			−0.456794	+	0.889573i	$0.651002\pi$
$168$		0			0
$169$	−1.00000			−0.0769231
$170$		0			0
$171$		0			0
$172$		0			0
$173$	12.7918			0.972542			0.486271	−	0.873808i	$-0.338357\pi$
$173$	12.7918			0.972542			0.486271	+	0.873808i	$0.338357\pi$
$174$		0			0
$175$			5.71905i			0.432319i
$176$		0			0
$177$	6.85952	+	10.0318i	0.515593	+	0.754035i
$178$		0			0
$179$		−	18.2892i		−	1.36700i	−0.729951	−	0.683500i	$-0.760457\pi$
$179$		−	18.2892i		−	1.36700i	0.729951	−	0.683500i	$-0.239543\pi$
$180$		0			0
$181$			16.3537i			1.21556i	0.794104	+	0.607782i	$0.207941\pi$
$181$			16.3537i			1.21556i	−0.794104	+	0.607782i	$0.792059\pi$
$182$		0			0
$183$	7.89229	+	11.5422i	0.583415	+	0.853222i
$184$		0			0
$185$			28.6861i			2.10905i
$186$		0			0
$187$	−17.3653			−1.26988
$188$		0			0
$189$	−24.5946	+	5.71905i	−1.78900	+	0.415999i
$190$		0			0
$191$	−25.3961			−1.83760			−0.918798	−	0.394729i	$-0.870839\pi$
$191$	−25.3961			−1.83760			−0.918798	+	0.394729i	$0.870839\pi$
$192$		0			0
$193$	−13.4381			−0.967295			−0.483648	−	0.875263i	$-0.660688\pi$
$193$	−13.4381			−0.967295			−0.483648	+	0.875263i	$0.660688\pi$
$194$		0			0
$195$	2.42976	+	3.55343i	0.173999	+	0.254467i
$196$		0			0
$197$	−18.1276			−1.29154			−0.645768	−	0.763533i	$-0.723463\pi$
$197$	−18.1276			−1.29154			−0.645768	+	0.763533i	$0.723463\pi$
$198$		0			0
$199$			10.4265i			0.739118i	0.929207	+	0.369559i	$0.120491\pi$
$199$			10.4265i			0.739118i	−0.929207	+	0.369559i	$0.879509\pi$
$200$		0			0
$201$	−8.17687	+	5.59117i	−0.576752	+	0.394371i
$202$		0			0
$203$			5.15158i			0.361570i
$204$		0			0
$205$		−	14.8034i		−	1.03391i
$206$		0			0
$207$	−9.66666	+	24.8282i	−0.671879	+	1.72568i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	14.2976			0.984289			0.492144	−	0.870514i	$-0.336213\pi$
$211$	14.2976			0.984289			0.492144	+	0.870514i	$0.336213\pi$
$212$		0			0
$213$	−1.64503	+	1.12483i	−0.112715	+	0.0770724i
$214$		0			0
$215$	12.9567			0.883640
$216$		0			0
$217$	19.4381			1.31954
$218$		0			0
$219$	11.5422	−	7.89229i	0.779948	−	0.533312i
$220$		0			0
$221$	5.59117			0.376103
$222$		0			0
$223$		−	0.152027i		−	0.0101805i	−0.999987	−	0.00509025i	$-0.998380\pi$
$223$		−	0.152027i		−	0.0101805i	0.999987	−	0.00509025i	$-0.00162028\pi$
$224$		0			0
$225$	1.28095	−	3.29005i	0.0853969	−	0.219337i
$226$		0			0
$227$			9.49844i			0.630434i	0.949020	+	0.315217i	$0.102077\pi$
$227$			9.49844i			0.630434i	−0.949020	+	0.315217i	$0.897923\pi$
$228$		0			0
$229$			11.5422i			0.762728i	0.924425	+	0.381364i	$0.124546\pi$
$229$			11.5422i			0.762728i	−0.924425	+	0.381364i	$0.875454\pi$
$230$		0			0
$231$	21.5793	−	14.7554i	1.41981	−	0.970837i
$232$		0			0
$233$			10.9237i			0.715634i	0.933792	+	0.357817i	$0.116479\pi$
$233$			10.9237i			0.715634i	−0.933792	+	0.357817i	$0.883521\pi$
$234$		0			0
$235$	2.85952			0.186535
$236$		0			0
$237$	−1.68061	−	2.45783i	−0.109167	−	0.159653i
$238$		0			0
$239$	9.30152			0.601666			0.300833	−	0.953677i	$-0.402735\pi$
$239$	9.30152			0.601666			0.300833	+	0.953677i	$0.402735\pi$
$240$		0			0
$241$	−17.7918			−1.14607			−0.573037	−	0.819530i	$-0.694235\pi$
$241$	−17.7918			−1.14607			−0.573037	+	0.819530i	$0.694235\pi$
$242$		0			0
$243$	15.4298	+	2.21866i	0.989820	+	0.142327i
$244$		0			0
$245$	−41.2937			−2.63816
$246$		0			0
$247$		0			0
$248$		0			0
$249$	11.8959	+	17.3973i	0.753873	+	1.10251i
$250$		0			0
$251$			26.2753i			1.65848i	0.558893	+	0.829240i	$0.311226\pi$
$251$			26.2753i			1.65848i	−0.558893	+	0.829240i	$0.688774\pi$
$252$		0			0
$253$		−	27.5837i		−	1.73417i
$254$		0			0
$255$	−13.5852	−	19.8679i	−0.850739	−	1.24417i
$256$		0			0
$257$			10.9237i			0.681400i	0.940172	+	0.340700i	$0.110664\pi$
$257$			10.9237i			0.681400i	−0.940172	+	0.340700i	$0.889336\pi$
$258$		0			0
$259$	−56.0895			−3.48523
$260$		0			0
$261$	1.15385	−	2.96360i	0.0714217	−	0.183442i
$262$		0			0
$263$	23.0950			1.42410			0.712048	−	0.702131i	$-0.247768\pi$
$263$	23.0950			1.42410			0.712048	+	0.702131i	$0.247768\pi$
$264$		0			0
$265$	12.8034			0.786506
$266$		0			0
$267$	−8.07279	−	11.8061i	−0.494047	−	0.722525i
$268$		0			0
$269$	7.64021			0.465832			0.232916	−	0.972497i	$-0.425173\pi$
$269$	7.64021			0.465832			0.232916	+	0.972497i	$0.425173\pi$
$270$		0			0
$271$			14.5786i			0.885585i	0.896624	+	0.442793i	$0.146012\pi$
$271$			14.5786i			0.885585i	−0.896624	+	0.442793i	$0.853988\pi$
$272$		0			0
$273$	−6.94796	+	4.75087i	−0.420510	+	0.287536i
$274$		0			0
$275$			3.65518i			0.220416i
$276$		0			0
$277$		−	5.71905i		−	0.343624i	−0.985130	−	0.171812i	$-0.945038\pi$
$277$		−	5.71905i		−	0.343624i	0.985130	−	0.171812i	$-0.0549622\pi$
$278$		0			0
$279$	−11.1823	−	4.35375i	−0.669470	−	0.260652i
$280$		0			0
$281$		−	30.6221i		−	1.82676i	−0.407105	−	0.913381i	$-0.633462\pi$
$281$		−	30.6221i		−	1.82676i	0.407105	−	0.913381i	$-0.366538\pi$
$282$		0			0
$283$	−9.08435			−0.540008			−0.270004	−	0.962859i	$-0.587025\pi$
$283$	−9.08435			−0.540008			−0.270004	+	0.962859i	$0.587025\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	28.9448			1.70856
$288$		0			0
$289$	−14.2612			−0.838895
$290$		0			0
$291$	5.31735	−	3.63589i	0.311709	−	0.213140i
$292$		0			0
$293$	−6.57681			−0.384222			−0.192111	−	0.981373i	$-0.561533\pi$
$293$	−6.57681			−0.384222			−0.192111	+	0.981373i	$0.561533\pi$
$294$		0			0
$295$		−	17.4381i		−	1.01529i
$296$		0			0
$297$	−15.7190	+	3.65518i	−0.912112	+	0.212095i
$298$		0			0
$299$			8.88123i			0.513615i
$300$		0			0
$301$			25.3340i			1.46023i
$302$		0			0
$303$	−20.0636	+	13.7190i	−1.15262	+	0.788139i
$304$		0			0
$305$		−	20.0636i		−	1.14884i
$306$		0			0
$307$	−0.208158			−0.0118802			−0.00594011	−	0.999982i	$-0.501891\pi$
$307$	−0.208158			−0.0118802			−0.00594011	+	0.999982i	$0.501891\pi$
$308$		0			0
$309$	−7.89229	−	11.5422i	−0.448977	−	0.656611i
$310$		0			0
$311$	11.9126			0.675502			0.337751	−	0.941235i	$-0.390334\pi$
$311$	11.9126			0.675502			0.337751	+	0.941235i	$0.390334\pi$
$312$		0			0
$313$	6.90747			0.390433			0.195217	−	0.980760i	$-0.437459\pi$
$313$	6.90747			0.390433			0.195217	+	0.980760i	$0.437459\pi$
$314$		0			0
$315$	33.7638	+	13.1456i	1.90237	+	0.740672i
$316$		0			0
$317$	14.4691			0.812666			0.406333	−	0.913725i	$-0.366807\pi$
$317$	14.4691			0.812666			0.406333	+	0.913725i	$0.366807\pi$
$318$		0			0
$319$			3.29250i			0.184345i
$320$		0			0
$321$	−13.5422	−	19.8049i	−0.755850	−	1.10540i
$322$		0			0
$323$		0			0
$324$		0			0
$325$		−	1.17687i		−	0.0652812i
$326$		0			0
$327$	−15.8863	−	23.2332i	−0.878517	−	1.28480i
$328$		0			0
$329$			5.59117i			0.308251i
$330$		0			0
$331$	−17.1571			−0.943042			−0.471521	−	0.881855i	$-0.656295\pi$
$331$	−17.1571			−0.943042			−0.471521	+	0.881855i	$0.656295\pi$
$332$		0			0
$333$	32.2671	+	12.5629i	1.76823	+	0.688444i
$334$		0			0
$335$	14.2137			0.776579
$336$		0			0
$337$	27.9687			1.52355			0.761777	−	0.647840i	$-0.224327\pi$
$337$	27.9687			1.52355			0.761777	+	0.647840i	$0.224327\pi$
$338$		0			0
$339$	11.6463	+	17.0322i	0.632538	+	0.925062i
$340$		0			0
$341$	12.4234			0.672763
$342$		0			0
$343$		−	46.7242i		−	2.52287i
$344$		0			0
$345$	31.5588	−	21.5793i	1.69907	−	1.16179i
$346$		0			0
$347$		−	10.1382i		−	0.544250i	−0.962262	−	0.272125i	$-0.912274\pi$
$347$		−	10.1382i		−	0.544250i	0.962262	−	0.272125i	$-0.0877264\pi$
$348$		0			0
$349$			16.2497i			0.869825i	0.900473	+	0.434912i	$0.143221\pi$
$349$			16.2497i			0.869825i	−0.900473	+	0.434912i	$0.856779\pi$
$350$		0			0
$351$	5.06112	−	1.17687i	0.270143	−	0.0628169i
$352$		0			0
$353$			25.2896i			1.34603i	0.739629	+	0.673015i	$0.235001\pi$
$353$			25.2896i			1.34603i	−0.739629	+	0.673015i	$0.764999\pi$
$354$		0			0
$355$	2.85952			0.151768
$356$		0			0
$357$	38.8473	−	26.5629i	2.05601	−	1.40586i
$358$		0			0
$359$	16.4084			0.866002			0.433001	−	0.901394i	$-0.357455\pi$
$359$	16.4084			0.866002			0.433001	+	0.901394i	$0.357455\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$		0			0
$363$	−1.93554	+	1.32348i	−0.101589	+	0.0694646i
$364$		0			0
$365$	−20.0636			−1.05018
$366$		0			0
$367$			0.634701i			0.0331311i	0.999863	+	0.0165656i	$0.00527323\pi$
$367$			0.634701i			0.0331311i	−0.999863	+	0.0165656i	$0.994727\pi$
$368$		0			0
$369$	−16.6514	−	6.48306i	−0.866835	−	0.337495i
$370$		0			0
$371$			25.0342i			1.29971i
$372$		0			0
$373$			2.35375i			0.121872i	0.998142	+	0.0609362i	$0.0194086\pi$
$373$			2.35375i			0.121872i	−0.998142	+	0.0609362i	$0.980591\pi$
$374$		0			0
$375$	13.5852	−	9.28929i	0.701538	−	0.479697i
$376$		0			0
$377$		−	1.06010i		−	0.0545979i
$378$		0			0
$379$	−10.6347			−0.546268			−0.273134	−	0.961976i	$-0.588060\pi$
$379$	−10.6347			−0.546268			−0.273134	+	0.961976i	$0.588060\pi$
$380$		0			0
$381$	−9.15590	−	13.3901i	−0.469071	−	0.685998i
$382$		0			0
$383$	18.9130			0.966410			0.483205	−	0.875507i	$-0.339473\pi$
$383$	18.9130			0.966410			0.483205	+	0.875507i	$0.339473\pi$
$384$		0			0
$385$	−37.5109			−1.91173
$386$		0			0
$387$	5.67432	−	14.5742i	0.288442	−	0.740846i
$388$		0			0
$389$	31.9762			1.62126			0.810628	−	0.585561i	$-0.199126\pi$
$389$	31.9762			1.62126			0.810628	+	0.585561i	$0.199126\pi$
$390$		0			0
$391$		−	49.6565i		−	2.51124i
$392$		0			0
$393$	17.7034	+	25.8905i	0.893019	+	1.30601i
$394$		0			0
$395$			4.27240i			0.214968i
$396$		0			0
$397$			9.29250i			0.466377i	0.972432	+	0.233189i	$0.0749160\pi$
$397$			9.29250i			0.466377i	−0.972432	+	0.233189i	$0.925084\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	0.623802i		−	0.0311512i	−0.999879	−	0.0155756i	$-0.995042\pi$
$401$		−	0.623802i		−	0.0311512i	0.999879	−	0.0155756i	$-0.00495807\pi$
$402$		0			0
$403$	−4.00000			−0.199254
$404$		0			0
$405$	−16.4793	−	15.1248i	−0.818861	−	0.751559i
$406$		0			0
$407$	−35.8482			−1.77693
$408$		0			0
$409$	27.1571			1.34283			0.671417	−	0.741080i	$-0.265686\pi$
$409$	27.1571			1.34283			0.671417	+	0.741080i	$0.265686\pi$
$410$		0			0
$411$	−0.609855	−	0.891889i	−0.0300819	−	0.0439936i
$412$		0			0
$413$	34.0964			1.67777
$414$		0			0
$415$		−	30.2415i		−	1.48450i
$416$		0			0
$417$	7.45374	−	5.09671i	0.365011	−	0.249587i
$418$		0			0
$419$		−	16.3179i		−	0.797183i	−0.917129	−	0.398592i	$-0.869499\pi$
$419$		−	16.3179i		−	0.797183i	0.917129	−	0.398592i	$-0.130501\pi$
$420$		0			0
$421$		−	4.60342i		−	0.224357i	−0.993688	−	0.112178i	$-0.964217\pi$
$421$		−	4.60342i		−	0.224357i	0.993688	−	0.112178i	$-0.0357828\pi$
$422$		0			0
$423$	1.25231	−	3.21649i	0.0608895	−	0.156391i
$424$		0			0
$425$			6.58011i			0.319182i
$426$		0			0
$427$	39.2299			1.89847
$428$		0			0
$429$	−4.44061	+	3.03640i	−0.214395	+	0.146599i
$430$		0			0
$431$	−17.6654			−0.850913			−0.425456	−	0.904979i	$-0.639886\pi$
$431$	−17.6654			−0.850913			−0.425456	+	0.904979i	$0.639886\pi$
$432$		0			0
$433$	−3.46938			−0.166728			−0.0833638	−	0.996519i	$-0.526566\pi$
$433$	−3.46938			−0.166728			−0.0833638	+	0.996519i	$0.526566\pi$
$434$		0			0
$435$	−3.76699	+	2.57579i	−0.180614	+	0.123500i
$436$		0			0
$437$		0			0
$438$		0			0
$439$			33.5837i			1.60286i	0.598087	+	0.801431i	$0.295928\pi$
$439$			33.5837i			1.60286i	−0.598087	+	0.801431i	$0.704072\pi$
$440$		0			0
$441$	−18.0843	+	46.4486i	−0.861159	+	2.21184i
$442$		0			0
$443$		−	12.0775i		−	0.573821i	−0.957957	−	0.286910i	$-0.907372\pi$
$443$		−	12.0775i		−	0.573821i	0.957957	−	0.286910i	$-0.0926282\pi$
$444$		0			0
$445$			20.5224i			0.972857i
$446$		0			0
$447$	−5.22604	+	3.57346i	−0.247183	+	0.169019i
$448$		0			0
$449$		−	22.9885i		−	1.08489i	−0.840090	−	0.542447i	$-0.817498\pi$
$449$		−	22.9885i		−	1.08489i	0.840090	−	0.542447i	$-0.182502\pi$
$450$		0			0
$451$	18.4993			0.871100
$452$		0			0
$453$	−0.148628	−	0.217363i	−0.00698316	−	0.0102126i
$454$		0			0
$455$	12.0775			0.566203
$456$		0			0
$457$	33.9374			1.58753			0.793763	−	0.608227i	$-0.208119\pi$
$457$	33.9374			1.58753			0.793763	+	0.608227i	$0.208119\pi$
$458$		0			0
$459$	−28.2976	+	6.58011i	−1.32082	+	0.307133i
$460$		0			0
$461$	−9.57620			−0.446008			−0.223004	−	0.974818i	$-0.571586\pi$
$461$	−9.57620			−0.446008			−0.223004	+	0.974818i	$0.571586\pi$
$462$		0			0
$463$			39.5837i			1.83961i	0.392376	+	0.919805i	$0.371653\pi$
$463$			39.5837i			1.83961i	−0.392376	+	0.919805i	$0.628347\pi$
$464$		0			0
$465$	9.71905	+	14.2137i	0.450710	+	0.659146i
$466$		0			0
$467$		−	27.7358i		−	1.28346i	−0.766931	−	0.641729i	$-0.778217\pi$
$467$		−	27.7358i		−	1.28346i	0.766931	−	0.641729i	$-0.221783\pi$
$468$		0			0
$469$			27.7918i			1.28331i
$470$		0			0
$471$	5.59117	+	8.17687i	0.257628	+	0.376770i
$472$		0			0
$473$			16.1916i			0.744490i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	5.60717	−	14.4017i	0.256735	−	0.659408i
$478$		0			0
$479$	10.7620			0.491731			0.245865	−	0.969304i	$-0.420928\pi$
$479$	10.7620			0.491731			0.245865	+	0.969304i	$0.420928\pi$
$480$		0			0
$481$	11.5422			0.526278
$482$		0			0
$483$	42.1935	+	61.7064i	1.91987	+	2.80774i
$484$		0			0
$485$	−9.24307			−0.419706
$486$		0			0
$487$			4.56191i			0.206720i	0.994644	+	0.103360i	$0.0329593\pi$
$487$			4.56191i			0.206720i	−0.994644	+	0.103360i	$0.967041\pi$
$488$		0			0
$489$	31.5588	−	21.5793i	1.42714	−	0.975848i
$490$		0			0
$491$			16.3179i			0.736418i	0.929743	+	0.368209i	$0.120029\pi$
$491$			16.3179i			0.736418i	−0.929743	+	0.368209i	$0.879971\pi$
$492$		0			0
$493$			5.92721i			0.266948i
$494$		0			0
$495$	21.5793	+	8.40170i	0.969916	+	0.377628i
$496$		0			0
$497$			5.59117i			0.250798i
$498$		0			0
$499$	34.5224			1.54544			0.772718	−	0.634749i	$-0.218897\pi$
$499$	34.5224			1.54544			0.772718	+	0.634749i	$0.218897\pi$
$500$		0			0
$501$	16.8800	−	11.5422i	0.754142	−	0.515666i
$502$		0			0
$503$	5.84984			0.260832			0.130416	−	0.991459i	$-0.458369\pi$
$503$	5.84984			0.260832			0.130416	+	0.991459i	$0.458369\pi$
$504$		0			0
$505$	34.8762			1.55197
$506$		0			0
$507$	1.42976	−	0.977641i	0.0634979	−	0.0434186i
$508$		0			0
$509$	10.7395			0.476018			0.238009	−	0.971263i	$-0.423505\pi$
$509$	10.7395			0.476018			0.238009	+	0.971263i	$0.423505\pi$
$510$		0			0
$511$		−	39.2299i		−	1.73543i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			20.0636i			0.884107i
$516$		0			0
$517$			3.57346i			0.157160i
$518$		0			0
$519$	−18.2892	+	12.5058i	−0.802807	+	0.548943i
$520$		0			0
$521$			12.9015i			0.565227i	0.959234	+	0.282613i	$0.0912014\pi$
$521$			12.9015i			0.565227i	−0.959234	+	0.282613i	$0.908799\pi$
$522$		0			0
$523$	27.5837			1.20615			0.603075	−	0.797684i	$-0.293942\pi$
$523$	27.5837			1.20615			0.603075	+	0.797684i	$0.293942\pi$
$524$		0			0
$525$	−5.59117	−	8.17687i	−0.244019	−	0.356868i
$526$		0			0
$527$	22.3647			0.974221
$528$		0			0
$529$	55.8762			2.42940
$530$		0			0
$531$	−19.6150	−	7.63691i	−0.851217	−	0.331414i
$532$		0			0
$533$	−5.95630			−0.257996
$534$		0			0
$535$			34.4265i			1.48839i
$536$		0			0
$537$	17.8803	+	26.1492i	0.771591	+	1.12842i
$538$		0			0
$539$		−	51.6035i		−	2.22272i
$540$		0			0
$541$			39.1259i			1.68215i	0.540917	+	0.841076i	$0.318077\pi$
$541$			39.1259i			1.68215i	−0.540917	+	0.841076i	$0.681923\pi$
$542$		0			0
$543$	−15.9881	−	23.3820i	−0.686115	−	1.00342i
$544$		0			0
$545$			40.3858i			1.72994i
$546$		0			0
$547$	−39.9439			−1.70788			−0.853938	−	0.520374i	$-0.825792\pi$
$547$	−39.9439			−1.70788			−0.853938	+	0.520374i	$0.825792\pi$
$548$		0			0
$549$	−22.5682	−	8.78673i	−0.963187	−	0.375008i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−8.35375			−0.355238
$554$		0			0
$555$	−28.0447	−	41.0143i	−1.19043	−	1.74096i
$556$		0			0
$557$	20.6096			0.873258			0.436629	−	0.899642i	$-0.356172\pi$
$557$	20.6096			0.873258			0.436629	+	0.899642i	$0.356172\pi$
$558$		0			0
$559$		−	5.21327i		−	0.220498i
$560$		0			0
$561$	24.8282	−	16.9770i	1.04825	−	0.716771i
$562$		0			0
$563$			28.2305i			1.18978i	0.803809	+	0.594888i	$0.202803\pi$
$563$			28.2305i			1.18978i	−0.803809	+	0.594888i	$0.797197\pi$
$564$		0			0
$565$		−	29.6068i		−	1.24557i
$566$		0			0
$567$	29.5733	−	32.2216i	1.24196	−	1.35318i
$568$		0			0
$569$		−	10.9237i		−	0.457944i	−0.973433	−	0.228972i	$-0.926464\pi$
$569$		−	10.9237i		−	0.457944i	0.973433	−	0.228972i	$-0.0735365\pi$
$570$		0			0
$571$	25.1405			1.05210			0.526048	−	0.850455i	$-0.323673\pi$
$571$	25.1405			1.05210			0.526048	+	0.850455i	$0.323673\pi$
$572$		0			0
$573$	36.3103	−	24.8282i	1.51689	−	1.03721i
$574$		0			0
$575$	−10.4521			−0.435882
$576$		0			0
$577$	14.0959			0.586820			0.293410	−	0.955987i	$-0.405210\pi$
$577$	14.0959			0.586820			0.293410	+	0.955987i	$0.405210\pi$
$578$		0			0
$579$	19.2133	−	13.1376i	0.798476	−	0.545981i
$580$		0			0
$581$	59.1306			2.45315
$582$		0			0
$583$			16.0000i			0.662652i
$584$		0			0
$585$	−6.94796	−	2.70513i	−0.287263	−	0.111843i
$586$		0			0
$587$		−	33.8024i		−	1.39518i	−0.716500	−	0.697588i	$-0.754257\pi$
$587$		−	33.8024i		−	1.39518i	0.716500	−	0.697588i	$-0.245743\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$	25.9181	−	17.7223i	1.06613	−	0.728997i
$592$		0			0
$593$		−	6.47365i		−	0.265841i	−0.991127	−	0.132920i	$-0.957565\pi$
$593$		−	6.47365i		−	0.265841i	0.991127	−	0.132920i	$-0.0424355\pi$
$594$		0			0
$595$	−67.5276			−2.76836
$596$		0			0
$597$	−10.1934	−	14.9075i	−0.417189	−	0.610122i
$598$		0			0
$599$	−12.4300			−0.507874			−0.253937	−	0.967221i	$-0.581726\pi$
$599$	−12.4300			−0.507874			−0.253937	+	0.967221i	$0.581726\pi$
$600$		0			0
$601$	7.61497			0.310621			0.155311	−	0.987866i	$-0.450362\pi$
$601$	7.61497			0.310621			0.155311	+	0.987866i	$0.450362\pi$
$602$		0			0
$603$	6.22482	−	15.9881i	0.253494	−	0.651085i
$604$		0			0
$605$	3.36451			0.136787
$606$		0			0
$607$		−	28.3143i		−	1.14924i	−0.818420	−	0.574621i	$-0.805150\pi$
$607$		−	28.3143i		−	1.14924i	0.818420	−	0.574621i	$-0.194850\pi$
$608$		0			0
$609$	−5.03640	−	7.36554i	−0.204085	−	0.298467i
$610$		0			0
$611$		−	1.15056i		−	0.0465467i
$612$		0			0
$613$			32.1687i			1.29928i	0.760241	+	0.649641i	$0.225081\pi$
$613$			32.1687i			1.29928i	−0.760241	+	0.649641i	$0.774919\pi$
$614$		0			0
$615$	14.4724	+	21.1653i	0.583583	+	0.853468i
$616$		0			0
$617$			21.4176i			0.862242i	0.902294	+	0.431121i	$0.141882\pi$
$617$			21.4176i			0.862242i	−0.902294	+	0.431121i	$0.858118\pi$
$618$		0			0
$619$	29.8149			1.19836			0.599182	−	0.800613i	$-0.295493\pi$
$619$	29.8149			1.19836			0.599182	+	0.800613i	$0.295493\pi$
$620$		0			0
$621$	−10.4521	−	44.9490i	−0.419428	−	1.80374i
$622$		0			0
$623$	−40.1271			−1.60766
$624$		0			0
$625$	−29.4993			−1.17997
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−64.5343			−2.57315
$630$		0			0
$631$			1.56702i			0.0623821i	0.999513	+	0.0311910i	$0.00993002\pi$
$631$			1.56702i			0.0623821i	−0.999513	+	0.0311910i	$0.990070\pi$
$632$		0			0
$633$	−20.4422	+	13.9779i	−0.812504	+	0.555573i
$634$		0			0
$635$			23.2759i			0.923675i
$636$		0			0
$637$			16.6150i			0.658309i
$638$		0			0
$639$	1.25231	−	3.21649i	0.0495407	−	0.127242i
$640$		0			0
$641$			40.1271i			1.58493i	0.609919	+	0.792464i	$0.291202\pi$
$641$			40.1271i			1.58493i	−0.609919	+	0.792464i	$0.708798\pi$
$642$		0			0
$643$	−28.5952			−1.12769			−0.563843	−	0.825882i	$-0.690678\pi$
$643$	−28.5952			−1.12769			−0.563843	+	0.825882i	$0.690678\pi$
$644$		0			0
$645$	−18.5250	+	12.6670i	−0.729421	+	0.498763i
$646$		0			0
$647$	24.3426			0.957005			0.478502	−	0.878086i	$-0.341180\pi$
$647$	24.3426			0.957005			0.478502	+	0.878086i	$0.341180\pi$
$648$		0			0
$649$	21.7918			0.855405
$650$		0			0
$651$	−27.7918	+	19.0035i	−1.08925	+	0.744805i
$652$		0			0
$653$	−29.3132			−1.14712			−0.573558	−	0.819165i	$-0.694437\pi$
$653$	−29.3132			−1.14712			−0.573558	+	0.819165i	$0.694437\pi$
$654$		0			0
$655$		−	45.0051i		−	1.75850i
$656$		0			0
$657$	−8.78673	+	22.5682i	−0.342803	+	0.880469i
$658$		0			0
$659$			20.0636i			0.781566i	0.920483	+	0.390783i	$0.127796\pi$
$659$			20.0636i			0.781566i	−0.920483	+	0.390783i	$0.872204\pi$
$660$		0			0
$661$			8.87619i			0.345244i	0.984988	+	0.172622i	$0.0552239\pi$
$661$			8.87619i			0.345244i	−0.984988	+	0.172622i	$0.944776\pi$
$662$		0			0
$663$	−7.99405	+	5.46616i	−0.310463	+	0.212288i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−9.41499			−0.364550
$668$		0			0
$669$	0.148628	+	0.217363i	0.00574630	+	0.00840373i
$670$		0			0
$671$	25.0728			0.967926
$672$		0			0
$673$	−20.5537			−0.792288			−0.396144	−	0.918188i	$-0.629652\pi$
$673$	−20.5537			−0.792288			−0.396144	+	0.918188i	$0.629652\pi$
$674$		0			0
$675$	1.38503	+	5.95630i	0.0533099	+	0.229258i
$676$		0			0
$677$	−4.09807			−0.157502			−0.0787508	−	0.996894i	$-0.525093\pi$
$677$	−4.09807			−0.157502			−0.0787508	+	0.996894i	$0.525093\pi$
$678$		0			0
$679$		−	18.0728i		−	0.693570i
$680$		0			0
$681$	−9.28607	−	13.5805i	−0.355843	−	0.520406i
$682$		0			0
$683$			3.47427i			0.132939i	0.997788	+	0.0664695i	$0.0211735\pi$
$683$			3.47427i			0.132939i	−0.997788	+	0.0664695i	$0.978826\pi$
$684$		0			0
$685$			1.55036i			0.0592360i
$686$		0			0
$687$	−11.2841	−	16.5026i	−0.430515	−	0.629612i
$688$		0			0
$689$		−	5.15158i		−	0.196260i
$690$		0			0
$691$	39.0218			1.48446			0.742229	−	0.670146i	$-0.233769\pi$
$691$	39.0218			1.48446			0.742229	+	0.670146i	$0.233769\pi$
$692$		0			0
$693$	−16.4277	+	42.1935i	−0.624036	+	1.60280i
$694$		0			0
$695$	−12.9567			−0.491476
$696$		0			0
$697$	33.3027			1.26143
$698$		0			0
$699$	−10.6794	−	15.6183i	−0.403933	−	0.590737i
$700$		0			0
$701$	22.5522			0.851785			0.425892	−	0.904774i	$-0.359960\pi$
$701$	22.5522			0.851785			0.425892	+	0.904774i	$0.359960\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	−4.08844	+	2.79559i	−0.153979	+	0.105288i
$706$		0			0
$707$			68.1928i			2.56465i
$708$		0			0
$709$		−	16.0231i		−	0.601760i	−0.953662	−	0.300880i	$-0.902720\pi$
$709$		−	16.0231i		−	0.601760i	0.953662	−	0.300880i	$-0.0972804\pi$
$710$		0			0
$711$	4.80574	+	1.87107i	0.180229	+	0.0701708i
$712$		0			0
$713$			35.5249i			1.33042i
$714$		0			0
$715$	7.71905			0.288676
$716$		0			0
$717$	−13.2990	+	9.09355i	−0.496659	+	0.339605i
$718$		0			0
$719$	24.6658			0.919880			0.459940	−	0.887950i	$-0.347871\pi$
$719$	24.6658			0.919880			0.459940	+	0.887950i	$0.347871\pi$
$720$		0			0
$721$	−39.2299			−1.46100
$722$		0			0
$723$	25.4381	−	17.3940i	0.946053	−	0.646891i
$724$		0			0
$725$	1.24760			0.0463349
$726$		0			0
$727$		−	12.8034i		−	0.474852i	−0.971406	−	0.237426i	$-0.923696\pi$
$727$		−	12.8034i		−	0.474852i	0.971406	−	0.237426i	$-0.0763036\pi$
$728$		0			0
$729$	−24.2299	+	11.9126i	−0.897405	+	0.441208i
$730$		0			0
$731$			29.1483i			1.07809i
$732$		0			0
$733$		−	16.4578i		−	0.607884i	−0.952691	−	0.303942i	$-0.901697\pi$
$733$		−	16.4578i		−	0.607884i	0.952691	−	0.303942i	$-0.0983029\pi$
$734$		0			0
$735$	59.0402	−	40.3704i	2.17773	−	1.48909i
$736$		0			0
$737$			17.7625i			0.654288i
$738$		0			0
$739$	11.2299			0.413100			0.206550	−	0.978436i	$-0.433776\pi$
$739$	11.2299			0.413100			0.206550	+	0.978436i	$0.433776\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−23.5153			−0.862691			−0.431345	−	0.902187i	$-0.641961\pi$
$743$	−23.5153			−0.862691			−0.431345	+	0.902187i	$0.641961\pi$
$744$		0			0
$745$	9.08435			0.332825
$746$		0			0
$747$	−34.0167	−	13.2441i	−1.24460	−	0.484576i
$748$		0			0
$749$	−67.3136			−2.45958
$750$		0			0
$751$		−	6.56191i		−	0.239447i	−0.992807	−	0.119724i	$-0.961799\pi$
$751$		−	6.56191i		−	0.239447i	0.992807	−	0.119724i	$-0.0382009\pi$
$752$		0			0
$753$	−25.6878	−	37.5674i	−0.936114	−	1.36903i
$754$		0			0
$755$			0.377838i			0.0137509i
$756$		0			0
$757$			47.3027i			1.71925i	0.510928	+	0.859623i	$0.329302\pi$
$757$			47.3027i			1.71925i	−0.510928	+	0.859623i	$0.670698\pi$
$758$		0			0
$759$	26.9669	+	39.4381i	0.978838	+	1.43151i
$760$		0			0
$761$		−	28.3210i		−	1.02664i	−0.858199	−	0.513318i	$-0.828416\pi$
$761$		−	28.3210i		−	1.02664i	0.858199	−	0.513318i	$-0.171584\pi$
$762$		0			0
$763$	−78.9656			−2.85875
$764$		0			0
$765$	38.8473	+	15.1248i	1.40453	+	0.546840i
$766$		0			0
$767$	−7.01640			−0.253348
$768$		0			0
$769$	−36.5224			−1.31703			−0.658516	−	0.752566i	$-0.728816\pi$
$769$	−36.5224			−1.31703			−0.658516	+	0.752566i	$0.728816\pi$
$770$		0			0
$771$	−10.6794	−	15.6183i	−0.384610	−	0.562478i
$772$		0			0
$773$	4.82504			0.173545			0.0867723	−	0.996228i	$-0.472345\pi$
$773$	4.82504			0.173545			0.0867723	+	0.996228i	$0.472345\pi$
$774$		0			0
$775$		−	4.70750i		−	0.169098i
$776$		0			0
$777$	80.1946	−	54.8353i	2.87696	−	1.96721i
$778$		0			0
$779$		0			0
$780$		0			0
$781$			3.57346i			0.127868i
$782$		0			0
$783$	1.24760	+	5.36530i	0.0445858	+	0.191740i
$784$		0			0
$785$		−	14.2137i		−	0.507310i
$786$		0			0
$787$	32.4993			1.15848			0.579238	−	0.815158i	$-0.303350\pi$
$787$	32.4993			1.15848			0.579238	+	0.815158i	$0.303350\pi$
$788$		0			0
$789$	−33.0203	+	22.5786i	−1.17555	+	0.803818i
$790$		0			0
$791$	57.8896			2.05832
$792$		0			0
$793$	−8.07279			−0.286673
$794$		0			0
$795$	−18.3058	+	12.5171i	−0.649240	+	0.443936i
$796$		0			0
$797$	−32.6679			−1.15715			−0.578577	−	0.815627i	$-0.696392\pi$
$797$	−32.6679			−1.15715			−0.578577	+	0.815627i	$0.696392\pi$
$798$		0			0
$799$			6.43298i			0.227582i
$800$		0			0
$801$	23.0843	+	8.98769i	0.815645	+	0.317564i
$802$		0			0
$803$		−	25.0728i		−	0.884801i
$804$		0			0
$805$		−	107.263i		−	3.78053i
$806$		0			0
$807$	−10.9237	+	7.46938i	−0.384532	+	0.262935i
$808$		0			0
$809$		−	35.2662i		−	1.23989i	−0.784643	−	0.619947i	$-0.787154\pi$
$809$		−	35.2662i		−	1.23989i	0.784643	−	0.619947i	$-0.212846\pi$
$810$		0			0
$811$	−20.8531			−0.732251			−0.366125	−	0.930566i	$-0.619316\pi$
$811$	−20.8531			−0.732251			−0.366125	+	0.930566i	$0.619316\pi$
$812$		0			0
$813$	−14.2526	−	20.8439i	−0.499861	−	0.731027i
$814$		0			0
$815$	−54.8582			−1.92160
$816$		0			0
$817$		0			0
$818$		0			0
$819$	5.28929	−	13.5852i	0.184823	−	0.474706i
$820$		0			0
$821$	−41.3715			−1.44387			−0.721937	−	0.691959i	$-0.756748\pi$
$821$	−41.3715			−1.44387			−0.721937	+	0.691959i	$0.756748\pi$
$822$		0			0
$823$		−	13.2925i		−	0.463348i	−0.972794	−	0.231674i	$-0.925580\pi$
$823$		−	13.2925i		−	0.463348i	0.972794	−	0.231674i	$-0.0744202\pi$
$824$		0			0
$825$	−3.57346	−	5.22604i	−0.124412	−	0.181947i
$826$		0			0
$827$		−	16.7382i		−	0.582045i	−0.956716	−	0.291023i	$-0.906005\pi$
$827$		−	16.7382i		−	0.582045i	0.956716	−	0.291023i	$-0.0939955\pi$
$828$		0			0
$829$		−	8.87619i		−	0.308283i	−0.988049	−	0.154141i	$-0.950739\pi$
$829$		−	8.87619i		−	0.308283i	0.988049	−	0.154141i	$-0.0492611\pi$
$830$		0			0
$831$	5.59117	+	8.17687i	0.193956	+	0.283653i
$832$		0			0
$833$		−	92.8972i		−	3.21870i
$834$		0			0
$835$	−29.3422			−1.01543
$836$		0			0
$837$	20.2445	−	4.70750i	0.699752	−	0.162715i
$838$		0			0
$839$	−34.1708			−1.17971			−0.589854	−	0.807510i	$-0.700815\pi$
$839$	−34.1708			−1.17971			−0.589854	+	0.807510i	$0.700815\pi$
$840$		0			0
$841$	−27.8762			−0.961248
$842$		0			0
$843$	29.9374	+	43.7823i	1.03110	+	1.50794i
$844$		0			0
$845$	−2.48533			−0.0854980
$846$		0			0
$847$			6.57857i			0.226042i
$848$		0			0
$849$	12.9884	−	8.88123i	0.445762	−	0.304803i
$850$		0			0
$851$		−	102.509i		−	3.51395i
$852$		0			0
$853$		−	39.3340i		−	1.34677i	−0.739292	−	0.673386i	$-0.764839\pi$
$853$		−	39.3340i		−	1.34677i	0.739292	−	0.673386i	$-0.235161\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			16.5148i			0.564136i	0.959394	+	0.282068i	$0.0910205\pi$
$857$			16.5148i			0.564136i	−0.959394	+	0.282068i	$0.908980\pi$
$858$		0			0
$859$	25.6463			0.875039			0.437519	−	0.899209i	$-0.355857\pi$
$859$	25.6463			0.875039			0.437519	+	0.899209i	$0.355857\pi$
$860$		0			0
$861$	−41.3842	+	28.2976i	−1.41037	+	0.964380i
$862$		0			0
$863$	35.4279			1.20598			0.602989	−	0.797749i	$-0.293976\pi$
$863$	35.4279			1.20598			0.602989	+	0.797749i	$0.293976\pi$
$864$		0			0
$865$	31.7918			1.08096
$866$		0			0
$867$	20.3901	−	13.9423i	0.692486	−	0.473507i
$868$		0			0
$869$	−5.33909			−0.181116
$870$		0			0
$871$		−	5.71905i		−	0.193783i
$872$		0			0
$873$	−4.04795	+	10.3969i	−0.137002	+	0.351882i
$874$		0			0
$875$		−	46.1739i		−	1.56096i
$876$		0			0
$877$			9.51907i			0.321436i	0.987000	+	0.160718i	$0.0513810\pi$
$877$			9.51907i			0.321436i	−0.987000	+	0.160718i	$0.948619\pi$
$878$		0			0
$879$	9.40328	−	6.42976i	0.317165	−	0.216870i
$880$		0			0
$881$			6.32144i			0.212975i	0.994314	+	0.106487i	$0.0339603\pi$
$881$			6.32144i			0.212975i	−0.994314	+	0.106487i	$0.966040\pi$
$882$		0			0
$883$	−3.87107			−0.130272			−0.0651360	−	0.997876i	$-0.520748\pi$
$883$	−3.87107			−0.130272			−0.0651360	+	0.997876i	$0.520748\pi$
$884$		0			0
$885$	17.0482	+	24.9323i	0.573069	+	0.838091i
$886$		0			0
$887$	−52.7700			−1.77184			−0.885922	−	0.463834i	$-0.846473\pi$
$887$	−52.7700			−1.77184			−0.885922	+	0.463834i	$0.846473\pi$
$888$		0			0
$889$	−45.5109			−1.52639
$890$		0			0
$891$	18.9010	−	20.5936i	0.633208	−	0.689912i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		−	45.4548i		−	1.51938i
$896$		0			0
$897$	−8.68265	−	12.6980i	−0.289905	−	0.423975i
$898$		0			0
$899$		−	4.24040i		−	0.141425i
$900$		0			0
$901$			28.8034i			0.959580i
$902$		0			0
$903$	−24.7676	−	36.2216i	−0.824213	−	1.20538i
$904$		0			0
$905$			40.6445i			1.35107i
$906$		0			0
$907$	8.99489			0.298670			0.149335	−	0.988787i	$-0.452287\pi$
$907$	8.99489			0.298670			0.149335	+	0.988787i	$0.452287\pi$
$908$		0			0
$909$	15.2738	−	39.2299i	0.506601	−	1.30117i
$910$		0			0
$911$	1.57086			0.0520449			0.0260224	−	0.999661i	$-0.491716\pi$
$911$	1.57086			0.0520449			0.0260224	+	0.999661i	$0.491716\pi$
$912$		0			0
$913$	37.7918			1.25073
$914$		0			0
$915$	19.6150	+	28.6861i	0.648451	+	0.948334i
$916$		0			0
$917$	87.9977			2.90594
$918$		0			0
$919$			30.8762i			1.01851i	0.860615	+	0.509256i	$0.170079\pi$
$919$			30.8762i			1.01851i	−0.860615	+	0.509256i	$0.829921\pi$
$920$		0			0
$921$	0.297617	−	0.203504i	0.00980681	−	0.00670569i
$922$		0			0
$923$		−	1.15056i		−	0.0378711i
$924$		0			0
$925$			13.5837i			0.446629i
$926$		0			0
$927$	22.5682	+	8.78673i	0.741237	+	0.288594i
$928$		0			0
$929$			18.7095i			0.613839i	0.951735	+	0.306920i	$0.0992983\pi$
$929$			18.7095i			0.613839i	−0.951735	+	0.306920i	$0.900702\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−17.0322	+	11.6463i	−0.557609	+	0.381281i
$934$		0			0
$935$	−43.1585			−1.41143
$936$		0			0
$937$	38.6680			1.26323			0.631615	−	0.775282i	$-0.282392\pi$
$937$	38.6680			1.26323			0.631615	+	0.775282i	$0.282392\pi$
$938$		0			0
$939$	−9.87604	+	6.75303i	−0.322292	+	0.220377i
$940$		0			0
$941$	−5.18688			−0.169087			−0.0845436	−	0.996420i	$-0.526943\pi$
$941$	−5.18688			−0.169087			−0.0845436	+	0.996420i	$0.526943\pi$
$942$		0			0
$943$			52.8993i			1.72264i
$944$		0			0
$945$	−61.1259	+	14.2137i	−1.98842	+	0.462373i
$946$		0			0
$947$			30.2989i			0.984581i	0.870431	+	0.492290i	$0.163840\pi$
$947$			30.2989i			0.984581i	−0.870431	+	0.492290i	$0.836160\pi$
$948$		0			0
$949$			8.07279i			0.262054i
$950$		0			0
$951$	−20.6874	+	14.1456i	−0.670834	+	0.458702i
$952$		0			0
$953$		−	6.83878i		−	0.221530i	−0.993847	−	0.110765i	$-0.964670\pi$
$953$		−	6.83878i		−	0.221530i	0.993847	−	0.110765i	$-0.0353301\pi$
$954$		0			0
$955$	−63.1177			−2.04244
$956$		0			0
$957$	−3.21889	−	4.70750i	−0.104052	−	0.152172i
$958$		0			0
$959$	−3.03138			−0.0978884
$960$		0			0
$961$	15.0000			0.483871
$962$		0			0
$963$	38.7242	+	15.0769i	1.24787	+	0.485847i
$964$		0			0
$965$	−33.3981			−1.07512
$966$		0			0
$967$		−	8.85952i		−	0.284903i	−0.989802	−	0.142452i	$-0.954501\pi$
$967$		−	8.85952i		−	0.284903i	0.989802	−	0.142452i	$-0.0454985\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	25.4187i		−	0.815724i	−0.913044	−	0.407862i	$-0.866275\pi$
$971$		−	25.4187i		−	0.815724i	0.913044	−	0.407862i	$-0.133725\pi$
$972$		0			0
$973$		−	25.3340i		−	0.812171i
$974$		0			0
$975$	1.15056	+	1.68265i	0.0368474	+	0.0538879i
$976$		0			0
$977$			49.1148i			1.57132i	0.618657	+	0.785661i	$0.287677\pi$
$977$			49.1148i			1.57132i	−0.618657	+	0.785661i	$0.712323\pi$
$978$		0			0
$979$	−25.6463			−0.819658
$980$		0			0
$981$	45.4274	+	17.6867i	1.45038	+	0.564694i
$982$		0			0
$983$	28.1175			0.896809			0.448404	−	0.893831i	$-0.351992\pi$
$983$	28.1175			0.896809			0.448404	+	0.893831i	$0.351992\pi$
$984$		0			0
$985$	−45.0531			−1.43551
$986$		0			0
$987$	−5.46616	−	7.99405i	−0.173990	−	0.254453i
$988$		0			0
$989$	−46.3002			−1.47226
$990$		0			0
$991$			39.2068i			1.24545i	0.782442	+	0.622723i	$0.213974\pi$
$991$			39.2068i			1.24545i	−0.782442	+	0.622723i	$0.786026\pi$
$992$		0			0
$993$	24.5306	−	16.7735i	0.778456	−	0.532292i
$994$		0			0
$995$			25.9134i			0.821511i
$996$		0			0
$997$			17.3653i			0.549965i	0.961449	+	0.274982i	$0.0886720\pi$
$997$			17.3653i			0.549965i	−0.961449	+	0.274982i	$0.911328\pi$
$998$		0			0
$999$	−58.4164	+	13.5837i	−1.84821	+	0.429769i

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+(-1.42976 + 0.977641i) q^{3} +2.48533 q^{5} +4.85952i q^{7} +(1.08844 - 2.79559i) q^{9} +O(q^{10})\)	\(q+(-1.42976 + 0.977641i) q^{3} +2.48533 q^{5} +4.85952i q^{7} +(1.08844 - 2.79559i) q^{9} +3.10584i q^{11} -1.00000i q^{13} +(-3.55343 + 2.42976i) q^{15} +5.59117i q^{17} +(-4.75087 - 6.94796i) q^{21} -8.88123 q^{23} +1.17687 q^{25} +(1.17687 + 5.06112i) q^{27} +1.06010 q^{29} -4.00000i q^{31} +(-3.03640 - 4.44061i) q^{33} +12.0775i q^{35} +11.5422i q^{37} +(0.977641 + 1.42976i) q^{39} -5.95630i q^{41} +5.21327 q^{43} +(2.70513 - 6.94796i) q^{45} +1.15056 q^{47} -16.6150 q^{49} +(-5.46616 - 7.99405i) q^{51} +5.15158 q^{53} +7.71905i q^{55} -7.01640i q^{59} -8.07279i q^{61} +(13.5852 + 5.28929i) q^{63} -2.48533i q^{65} +5.71905 q^{67} +(12.6980 - 8.68265i) q^{69} +1.15056 q^{71} -8.07279 q^{73} +(-1.68265 + 1.15056i) q^{75} -15.0929 q^{77} +1.71905i q^{79} +(-6.63061 - 6.08564i) q^{81} -12.1680i q^{83} +13.8959i q^{85} +(-1.51569 + 1.03640i) q^{87} +8.25742i q^{89} +4.85952 q^{91} +(3.91056 + 5.71905i) q^{93} -3.71905 q^{97} +(8.68265 + 3.38051i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{3} + 8 q^{9}+O(q^{10}) \) 12 * q + 4 * q^3 + 8 * q^9	\( 12 q + 4 q^{3} + 8 q^{9} + 4 q^{25} + 4 q^{27} + 16 q^{33} - 20 q^{49} + 8 q^{51} - 16 q^{67} + 8 q^{73} + 12 q^{75} + 16 q^{91} + 40 q^{97} + 72 q^{99}+O(q^{100}) \) 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

Related objects

Downloads

Learn more