LMFDB - Embedded newform 112.10.f.b.111.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [112,10,Mod(111,112)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 10, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("112.111"); S:= CuspForms(chi, 10); N := Newforms(S);

Level:	$ N $	$=$	$ 112 = 2^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	112.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$57.6840136504$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			111.5
Character	$\chi$	$=$	112.111
Dual form			112.10.f.b.111.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-168.927 q^{3} +1693.73i q^{5} +(-6136.85 - 1640.94i) q^{7} +8853.27 q^{9} -45208.8i q^{11} +49206.6i q^{13} -286117. i q^{15} -295750. i q^{17} -460692. q^{19} +(1.03668e6 + 277199. i) q^{21} -1.79721e6i q^{23} -915608. q^{25} +1.82943e6 q^{27} -5.43628e6 q^{29} -4.36627e6 q^{31} +7.63697e6i q^{33} +(2.77932e6 - 1.03942e7i) q^{35} -1.38090e7 q^{37} -8.31232e6i q^{39} -6.11229e6i q^{41} +1.02988e7i q^{43} +1.49951e7i q^{45} -1.82144e6 q^{47} +(3.49682e7 + 2.01404e7i) q^{49} +4.99602e7i q^{51} +8.73449e7 q^{53} +7.65716e7 q^{55} +7.78233e7 q^{57} +1.13887e8 q^{59} +3.81369e7i q^{61} +(-5.43312e7 - 1.45277e7i) q^{63} -8.33429e7 q^{65} +2.09793e8i q^{67} +3.03598e8i q^{69} -1.24067e8i q^{71} +2.09070e8i q^{73} +1.54671e8 q^{75} +(-7.41851e7 + 2.77439e8i) q^{77} -4.05189e8i q^{79} -4.83299e8 q^{81} +1.21182e8 q^{83} +5.00922e8 q^{85} +9.18334e8 q^{87} +1.05652e9i q^{89} +(8.07453e7 - 3.01974e8i) q^{91} +7.37580e8 q^{93} -7.80290e8i q^{95} +7.91112e8i q^{97} -4.00246e8i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 155768 q^{9} + 220672 q^{21} - 9187656 q^{25} + 14881104 q^{29} + 2829456 q^{37} - 214802472 q^{49} + 327087120 q^{53} + 238245440 q^{57} - 495797952 q^{65} + 347010000 q^{77} + 1816013720 q^{81}+ \cdots + 288442240 q^{93}+O(q^{100}) $ 24 * q + 155768 * q^9 + 220672 * q^21 - 9187656 * q^25 + 14881104 * q^29 + 2829456 * q^37 - 214802472 * q^49 + 327087120 * q^53 + 238245440 * q^57 - 495797952 * q^65 + 347010000 * q^77 + 1816013720 * q^81 - 1291803264 * q^85 + 288442240 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/112\mathbb{Z}\right)^\times$.

$n$	$15$	$17$	$85$
$\chi(n)$	$-1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−168.927			−1.20407			−0.602037	−	0.798468i	$-0.705644\pi$
$3$	−168.927			−1.20407			−0.602037	+	0.798468i	$0.705644\pi$
$4$		0			0
$5$			1693.73i			1.21194i	0.795489	+	0.605969i	$0.207214\pi$
$5$			1693.73i			1.21194i	−0.795489	+	0.605969i	$0.792786\pi$
$6$		0			0
$7$	−6136.85	−	1640.94i	−0.966060	−	0.258317i
$7$	−6136.85	−	1640.94i	−0.966060	−	0.258317i
$8$		0			0
$9$	8853.27			0.449793
$10$		0			0
$11$		−	45208.8i		−	0.931013i	−0.885045	−	0.465506i	$-0.845872\pi$
$11$		−	45208.8i		−	0.931013i	0.885045	−	0.465506i	$-0.154128\pi$
$12$		0			0
$13$			49206.6i			0.477836i	0.971040	+	0.238918i	$0.0767926\pi$
$13$			49206.6i			0.477836i	−0.971040	+	0.238918i	$0.923207\pi$
$14$		0			0
$15$		−	286117.i		−	1.45926i
$16$		0			0
$17$		−	295750.i		−	0.858826i	−0.903108	−	0.429413i	$-0.858720\pi$
$17$		−	295750.i		−	0.858826i	0.903108	−	0.429413i	$-0.141280\pi$
$18$		0			0
$19$	−460692.			−0.810998			−0.405499	−	0.914096i	$-0.632902\pi$
$19$	−460692.			−0.810998			−0.405499	+	0.914096i	$0.632902\pi$
$20$		0			0
$21$	1.03668e6	+	277199.i	1.16321	+	0.311032i
$22$		0			0
$23$		−	1.79721e6i		−	1.33914i	−0.742751	−	0.669568i	$-0.766479\pi$
$23$		−	1.79721e6i		−	1.33914i	0.742751	−	0.669568i	$-0.233521\pi$
$24$		0			0
$25$	−915608.			−0.468791
$26$		0			0
$27$	1.82943e6			0.662490
$28$		0			0
$29$	−5.43628e6			−1.42729			−0.713643	−	0.700510i	$-0.752956\pi$
$29$	−5.43628e6			−1.42729			−0.713643	+	0.700510i	$0.752956\pi$
$30$		0			0
$31$	−4.36627e6			−0.849147			−0.424574	−	0.905393i	$-0.639576\pi$
$31$	−4.36627e6			−0.849147			−0.424574	+	0.905393i	$0.639576\pi$
$32$		0			0
$33$			7.63697e6i			1.12101i
$34$		0			0
$35$	2.77932e6	−	1.03942e7i	0.313064	−	1.17080i
$36$		0			0
$37$	−1.38090e7			−1.21131			−0.605655	−	0.795728i	$-0.707089\pi$
$37$	−1.38090e7			−1.21131			−0.605655	+	0.795728i	$0.707089\pi$
$38$		0			0
$39$		−	8.31232e6i		−	0.575349i
$40$		0			0
$41$		−	6.11229e6i		−	0.337813i	−0.985632	−	0.168907i	$-0.945976\pi$
$41$		−	6.11229e6i		−	0.337813i	0.985632	−	0.168907i	$-0.0540237\pi$
$42$		0			0
$43$			1.02988e7i			0.459389i	0.973263	+	0.229694i	$0.0737727\pi$
$43$			1.02988e7i			0.459389i	−0.973263	+	0.229694i	$0.926227\pi$
$44$		0			0
$45$			1.49951e7i			0.545121i
$46$		0			0
$47$	−1.82144e6			−0.0544471			−0.0272236	−	0.999629i	$-0.508667\pi$
$47$	−1.82144e6			−0.0544471			−0.0272236	+	0.999629i	$0.508667\pi$
$48$		0			0
$49$	3.49682e7	+	2.01404e7i	0.866545	+	0.499099i
$50$		0			0
$51$			4.99602e7i			1.03409i
$52$		0			0
$53$	8.73449e7			1.52053			0.760267	−	0.649611i	$-0.225068\pi$
$53$	8.73449e7			1.52053			0.760267	+	0.649611i	$0.225068\pi$
$54$		0			0
$55$	7.65716e7			1.12833
$56$		0			0
$57$	7.78233e7			0.976501
$58$		0			0
$59$	1.13887e8			1.22360			0.611801	−	0.791012i	$-0.290446\pi$
$59$	1.13887e8			1.22360			0.611801	+	0.791012i	$0.290446\pi$
$60$		0			0
$61$			3.81369e7i			0.352664i	0.984331	+	0.176332i	$0.0564233\pi$
$61$			3.81369e7i			0.352664i	−0.984331	+	0.176332i	$0.943577\pi$
$62$		0			0
$63$	−5.43312e7	−	1.45277e7i	−0.434527	−	0.116189i
$64$		0			0
$65$	−8.33429e7			−0.579107
$66$		0			0
$67$			2.09793e8i			1.27190i	0.771729	+	0.635952i	$0.219392\pi$
$67$			2.09793e8i			1.27190i	−0.771729	+	0.635952i	$0.780608\pi$
$68$		0			0
$69$			3.03598e8i			1.61242i
$70$		0			0
$71$		−	1.24067e8i		−	0.579422i	−0.957114	−	0.289711i	$-0.906441\pi$
$71$		−	1.24067e8i		−	0.579422i	0.957114	−	0.289711i	$-0.0935592\pi$
$72$		0			0
$73$			2.09070e8i			0.861665i	0.902432	+	0.430833i	$0.141780\pi$
$73$			2.09070e8i			0.861665i	−0.902432	+	0.430833i	$0.858220\pi$
$74$		0			0
$75$	1.54671e8			0.564459
$76$		0			0
$77$	−7.41851e7	+	2.77439e8i	−0.240496	+	0.899414i
$78$		0			0
$79$		−	4.05189e8i		−	1.17040i	−0.810887	−	0.585202i	$-0.801015\pi$
$79$		−	4.05189e8i		−	1.17040i	0.810887	−	0.585202i	$-0.198985\pi$
$80$		0			0
$81$	−4.83299e8			−1.24748
$82$		0			0
$83$	1.21182e8			0.280276			0.140138	−	0.990132i	$-0.455245\pi$
$83$	1.21182e8			0.280276			0.140138	+	0.990132i	$0.455245\pi$
$84$		0			0
$85$	5.00922e8			1.04084
$86$		0			0
$87$	9.18334e8			1.71856
$88$		0			0
$89$			1.05652e9i			1.78494i	0.451110	+	0.892469i	$0.351028\pi$
$89$			1.05652e9i			1.78494i	−0.451110	+	0.892469i	$0.648972\pi$
$90$		0			0
$91$	8.07453e7	−	3.01974e8i	0.123433	−	0.461618i
$92$		0			0
$93$	7.37580e8			1.02244
$94$		0			0
$95$		−	7.80290e8i		−	0.982878i
$96$		0			0
$97$			7.91112e8i			0.907330i	0.891172	+	0.453665i	$0.149884\pi$
$97$			7.91112e8i			0.907330i	−0.891172	+	0.453665i	$0.850116\pi$
$98$		0			0
$99$		−	4.00246e8i		−	0.418763i
$100$		0			0
$101$		−	4.89619e8i		−	0.468180i	−0.972215	−	0.234090i	$-0.924789\pi$
$101$		−	4.89619e8i		−	0.468180i	0.972215	−	0.234090i	$-0.0752110\pi$
$102$		0			0
$103$	−1.74244e9			−1.52542			−0.762710	−	0.646741i	$-0.776132\pi$
$103$	−1.74244e9			−1.52542			−0.762710	+	0.646741i	$0.776132\pi$
$104$		0			0
$105$	−4.69502e8	+	1.75586e9i	−0.376952	+	1.40973i
$106$		0			0
$107$		−	9.55807e7i		−	0.0704925i	−0.999379	−	0.0352463i	$-0.988778\pi$
$107$		−	9.55807e7i		−	0.0704925i	0.999379	−	0.0352463i	$-0.0112216\pi$
$108$		0			0
$109$	2.59562e9			1.76126			0.880628	−	0.473808i	$-0.157121\pi$
$109$	2.59562e9			1.76126			0.880628	+	0.473808i	$0.157121\pi$
$110$		0			0
$111$	2.33271e9			1.45851
$112$		0			0
$113$	−2.15286e9			−1.24212			−0.621059	−	0.783764i	$-0.713297\pi$
$113$	−2.15286e9			−1.24212			−0.621059	+	0.783764i	$0.713297\pi$
$114$		0			0
$115$	3.04400e9			1.62295
$116$		0			0
$117$			4.35640e8i			0.214927i
$118$		0			0
$119$	−4.85310e8	+	1.81498e9i	−0.221849	+	0.829678i
$120$		0			0
$121$	3.14115e8			0.133215
$122$		0			0
$123$			1.03253e9i			0.406752i
$124$		0			0
$125$			1.75728e9i			0.643791i
$126$		0			0
$127$			4.00213e9i			1.36513i	0.730825	+	0.682565i	$0.239136\pi$
$127$			4.00213e9i			1.36513i	−0.730825	+	0.682565i	$0.760864\pi$
$128$		0			0
$129$		−	1.73975e9i		−	0.553138i
$130$		0			0
$131$	−2.97589e9			−0.882869			−0.441434	−	0.897294i	$-0.645530\pi$
$131$	−2.97589e9			−0.882869			−0.441434	+	0.897294i	$0.645530\pi$
$132$		0			0
$133$	2.82720e9	+	7.55970e8i	0.783473	+	0.209494i
$134$		0			0
$135$			3.09857e9i			0.802896i
$136$		0			0
$137$	−2.51690e9			−0.610413			−0.305207	−	0.952286i	$-0.598726\pi$
$137$	−2.51690e9			−0.610413			−0.305207	+	0.952286i	$0.598726\pi$
$138$		0			0
$139$	2.36033e9			0.536298			0.268149	−	0.963377i	$-0.413588\pi$
$139$	2.36033e9			0.536298			0.268149	+	0.963377i	$0.413588\pi$
$140$		0			0
$141$	3.07691e8			0.0655584
$142$		0			0
$143$	2.22457e9			0.444871
$144$		0			0
$145$		−	9.20761e9i		−	1.72978i
$146$		0			0
$147$	−5.90707e9	−	3.40226e9i	−1.04338	−	0.600952i
$148$		0			0
$149$	1.02809e9			0.170881			0.0854404	−	0.996343i	$-0.472770\pi$
$149$	1.02809e9			0.170881			0.0854404	+	0.996343i	$0.472770\pi$
$150$		0			0
$151$		−	7.43840e9i		−	1.16435i	−0.813064	−	0.582175i	$-0.802202\pi$
$151$		−	7.43840e9i		−	1.16435i	0.813064	−	0.582175i	$-0.197798\pi$
$152$		0			0
$153$		−	2.61836e9i		−	0.386294i
$154$		0			0
$155$		−	7.39530e9i		−	1.02911i
$156$		0			0
$157$		−	1.13686e10i		−	1.49334i	−0.665196	−	0.746669i	$-0.731652\pi$
$157$		−	1.13686e10i		−	1.49334i	0.665196	−	0.746669i	$-0.268348\pi$
$158$		0			0
$159$	−1.47549e10			−1.83084
$160$		0			0
$161$	−2.94913e9	+	1.10292e10i	−0.345921	+	1.29369i
$162$		0			0
$163$		−	7.02745e9i		−	0.779747i	−0.920868	−	0.389873i	$-0.872519\pi$
$163$		−	7.02745e9i		−	0.779747i	0.920868	−	0.389873i	$-0.127481\pi$
$164$		0			0
$165$	−1.29350e10			−1.35859
$166$		0			0
$167$	−4.19723e9			−0.417579			−0.208790	−	0.977961i	$-0.566952\pi$
$167$	−4.19723e9			−0.417579			−0.208790	+	0.977961i	$0.566952\pi$
$168$		0			0
$169$	8.18321e9			0.771673
$170$		0			0
$171$	−4.07863e9			−0.364781
$172$		0			0
$173$		−	2.31245e10i		−	1.96275i	−0.192104	−	0.981375i	$-0.561531\pi$
$173$		−	2.31245e10i		−	1.96275i	0.192104	−	0.981375i	$-0.438469\pi$
$174$		0			0
$175$	5.61895e9	+	1.50246e9i	0.452881	+	0.121097i
$176$		0			0
$177$	−1.92386e10			−1.47331
$178$		0			0
$179$			2.18114e10i			1.58798i	0.607929	+	0.793992i	$0.292001\pi$
$179$			2.18114e10i			1.58798i	−0.607929	+	0.793992i	$0.707999\pi$
$180$		0			0
$181$			8.40374e9i			0.581995i	0.956724	+	0.290998i	$0.0939871\pi$
$181$			8.40374e9i			0.581995i	−0.956724	+	0.290998i	$0.906013\pi$
$182$		0			0
$183$		−	6.44235e9i		−	0.424634i
$184$		0			0
$185$		−	2.33888e10i		−	1.46803i
$186$		0			0
$187$	−1.33705e10			−0.799578
$188$		0			0
$189$	−1.12269e10	−	3.00199e9i	−0.640005	−	0.171132i
$190$		0			0
$191$			2.50676e10i			1.36289i	0.731867	+	0.681447i	$0.238649\pi$
$191$			2.50676e10i			1.36289i	−0.731867	+	0.681447i	$0.761351\pi$
$192$		0			0
$193$	1.56589e10			0.812368			0.406184	−	0.913791i	$-0.366859\pi$
$193$	1.56589e10			0.812368			0.406184	+	0.913791i	$0.366859\pi$
$194$		0			0
$195$	1.40789e10			0.697287
$196$		0			0
$197$	−1.28676e9			−0.0608694			−0.0304347	−	0.999537i	$-0.509689\pi$
$197$	−1.28676e9			−0.0608694			−0.0304347	+	0.999537i	$0.509689\pi$
$198$		0			0
$199$	−3.28256e10			−1.48379			−0.741897	−	0.670514i	$-0.766074\pi$
$199$	−3.28256e10			−1.48379			−0.741897	+	0.670514i	$0.766074\pi$
$200$		0			0
$201$		−	3.54396e10i		−	1.53146i
$202$		0			0
$203$	3.33616e10	+	8.92063e9i	1.37884	+	0.368692i
$204$		0			0
$205$	1.03526e10			0.409409
$206$		0			0
$207$		−	1.59112e10i		−	0.602334i
$208$		0			0
$209$			2.08273e10i			0.755049i
$210$		0			0
$211$			4.02405e10i			1.39763i	0.715302	+	0.698815i	$0.246289\pi$
$211$			4.02405e10i			1.39763i	−0.715302	+	0.698815i	$0.753711\pi$
$212$		0			0
$213$			2.09583e10i			0.697666i
$214$		0			0
$215$	−1.74435e10			−0.556751
$216$		0			0
$217$	2.67951e10	+	7.16481e9i	0.820328	+	0.219349i
$218$		0			0
$219$		−	3.53175e10i		−	1.03751i
$220$		0			0
$221$	1.45529e10			0.410378
$222$		0			0
$223$	−1.30506e9			−0.0353394			−0.0176697	−	0.999844i	$-0.505625\pi$
$223$	−1.30506e9			−0.0353394			−0.0176697	+	0.999844i	$0.505625\pi$
$224$		0			0
$225$	−8.10613e9			−0.210859
$226$		0			0
$227$	3.49416e9			0.0873426			0.0436713	−	0.999046i	$-0.486095\pi$
$227$	3.49416e9			0.0873426			0.0436713	+	0.999046i	$0.486095\pi$
$228$		0			0
$229$		−	6.06601e10i		−	1.45762i	−0.684717	−	0.728809i	$-0.740074\pi$
$229$		−	6.06601e10i		−	1.45762i	0.684717	−	0.728809i	$-0.259926\pi$
$230$		0			0
$231$	1.25318e10	−	4.68670e10i	0.289575	−	1.08296i
$232$		0			0
$233$	5.36507e10			1.19254			0.596271	−	0.802783i	$-0.296648\pi$
$233$	5.36507e10			1.19254			0.596271	+	0.802783i	$0.296648\pi$
$234$		0			0
$235$		−	3.08504e9i		−	0.0659865i
$236$		0			0
$237$			6.84473e10i			1.40925i
$238$		0			0
$239$			4.89762e10i			0.970944i	0.874252	+	0.485472i	$0.161352\pi$
$239$			4.89762e10i			0.970944i	−0.874252	+	0.485472i	$0.838648\pi$
$240$		0			0
$241$			4.99585e10i			0.953966i	0.878913	+	0.476983i	$0.158270\pi$
$241$			4.99585e10i			0.953966i	−0.878913	+	0.476983i	$0.841730\pi$
$242$		0			0
$243$	4.56335e10			0.839567
$244$		0			0
$245$	−3.41125e10	+	5.92268e10i	−0.604877	+	1.05020i
$246$		0			0
$247$		−	2.26691e10i		−	0.387524i
$248$		0			0
$249$	−2.04709e10			−0.337473
$250$		0			0
$251$	6.39457e10			1.01690			0.508451	−	0.861091i	$-0.330218\pi$
$251$	6.39457e10			1.01690			0.508451	+	0.861091i	$0.330218\pi$
$252$		0			0
$253$	−8.12498e10			−1.24675
$254$		0			0
$255$	−8.46192e10			−1.25325
$256$		0			0
$257$		−	5.38824e10i		−	0.770455i	−0.922822	−	0.385228i	$-0.874123\pi$
$257$		−	5.38824e10i		−	0.770455i	0.922822	−	0.385228i	$-0.125877\pi$
$258$		0			0
$259$	8.47438e10	+	2.26598e10i	1.17020	+	0.312901i
$260$		0			0
$261$	−4.81289e10			−0.641983
$262$		0			0
$263$		−	4.22799e10i		−	0.544920i	−0.962167	−	0.272460i	$-0.912163\pi$
$263$		−	4.22799e10i		−	0.544920i	0.962167	−	0.272460i	$-0.0878373\pi$
$264$		0			0
$265$			1.47939e11i			1.84279i
$266$		0			0
$267$		−	1.78475e11i		−	2.14920i
$268$		0			0
$269$			1.05246e11i			1.22552i	0.790268	+	0.612761i	$0.209941\pi$
$269$			1.05246e11i			1.22552i	−0.790268	+	0.612761i	$0.790059\pi$
$270$		0			0
$271$	6.03804e10			0.680039			0.340020	−	0.940418i	$-0.389566\pi$
$271$	6.03804e10			0.680039			0.340020	+	0.940418i	$0.389566\pi$
$272$		0			0
$273$	−1.36400e10	+	5.10114e10i	−0.148622	+	0.555822i
$274$		0			0
$275$			4.13935e10i			0.436451i
$276$		0			0
$277$	−9.12817e8			−0.00931590			−0.00465795	−	0.999989i	$-0.501483\pi$
$277$	−9.12817e8			−0.00931590			−0.00465795	+	0.999989i	$0.501483\pi$
$278$		0			0
$279$	−3.86558e10			−0.381940
$280$		0			0
$281$	1.98039e11			1.89484			0.947418	−	0.319999i	$-0.103683\pi$
$281$	1.98039e11			1.89484			0.947418	+	0.319999i	$0.103683\pi$
$282$		0			0
$283$	−3.67215e10			−0.340315			−0.170158	−	0.985417i	$-0.554428\pi$
$283$	−3.67215e10			−0.340315			−0.170158	+	0.985417i	$0.554428\pi$
$284$		0			0
$285$			1.31812e11i			1.18346i
$286$		0			0
$287$	−1.00299e10	+	3.75102e10i	−0.0872628	+	0.326348i
$288$		0			0
$289$	3.11196e10			0.262418
$290$		0			0
$291$		−	1.33640e11i		−	1.09249i
$292$		0			0
$293$		−	8.62876e10i		−	0.683981i	−0.939703	−	0.341990i	$-0.888899\pi$
$293$		−	8.62876e10i		−	0.683981i	0.939703	−	0.341990i	$-0.111101\pi$
$294$		0			0
$295$			1.92894e11i			1.48293i
$296$		0			0
$297$		−	8.27063e10i		−	0.616786i
$298$		0			0
$299$	8.84348e10			0.639887
$300$		0			0
$301$	1.68998e10	−	6.32025e10i	0.118668	−	0.443797i
$302$		0			0
$303$			8.27099e10i			0.563723i
$304$		0			0
$305$	−6.45938e10			−0.427407
$306$		0			0
$307$	2.00085e11			1.28556			0.642781	−	0.766050i	$-0.277781\pi$
$307$	2.00085e11			1.28556			0.642781	+	0.766050i	$0.277781\pi$
$308$		0			0
$309$	2.94344e11			1.83672
$310$		0			0
$311$	1.11203e11			0.674057			0.337028	−	0.941494i	$-0.390578\pi$
$311$	1.11203e11			0.674057			0.337028	+	0.941494i	$0.390578\pi$
$312$		0			0
$313$		−	3.94588e10i		−	0.232378i	−0.993227	−	0.116189i	$-0.962932\pi$
$313$		−	3.94588e10i		−	0.232378i	0.993227	−	0.116189i	$-0.0370678\pi$
$314$		0			0
$315$	2.46061e10	−	9.20226e10i	0.140814	−	0.526619i
$316$		0			0
$317$	1.34898e11			0.750307			0.375154	−	0.926963i	$-0.377590\pi$
$317$	1.34898e11			0.750307			0.375154	+	0.926963i	$0.377590\pi$
$318$		0			0
$319$			2.45768e11i			1.32882i
$320$		0			0
$321$			1.61461e10i			0.0848782i
$322$		0			0
$323$			1.36250e11i			0.696506i
$324$		0			0
$325$		−	4.50540e10i		−	0.224005i
$326$		0			0
$327$	−4.38471e11			−2.12068
$328$		0			0
$329$	1.11779e10	+	2.98888e9i	0.0525992	+	0.0140646i
$330$		0			0
$331$			1.60155e11i			0.733356i	0.930348	+	0.366678i	$0.119505\pi$
$331$			1.60155e11i			0.733356i	−0.930348	+	0.366678i	$0.880495\pi$
$332$		0			0
$333$	−1.22255e11			−0.544838
$334$		0			0
$335$	−3.55333e11			−1.54147
$336$		0			0
$337$	−3.33346e11			−1.40787			−0.703933	−	0.710267i	$-0.748574\pi$
$337$	−3.33346e11			−1.40787			−0.703933	+	0.710267i	$0.748574\pi$
$338$		0			0
$339$	3.63676e11			1.49560
$340$		0			0
$341$			1.97394e11i			0.790567i
$342$		0			0
$343$	−1.81545e11	−	1.80980e11i	−0.708209	−	0.706003i
$344$		0			0
$345$	−5.14213e11			−1.95415
$346$		0			0
$347$		−	3.13544e11i		−	1.16096i	−0.814276	−	0.580478i	$-0.802866\pi$
$347$		−	3.13544e11i		−	1.16096i	0.814276	−	0.580478i	$-0.197134\pi$
$348$		0			0
$349$		−	4.90813e11i		−	1.77093i	−0.464705	−	0.885466i	$-0.653840\pi$
$349$		−	4.90813e11i		−	1.77093i	0.464705	−	0.885466i	$-0.346160\pi$
$350$		0			0
$351$			9.00201e10i			0.316561i
$352$		0			0
$353$		−	5.10486e10i		−	0.174984i	−0.996165	−	0.0874918i	$-0.972115\pi$
$353$		−	5.10486e10i		−	0.174984i	0.996165	−	0.0874918i	$-0.0278852\pi$
$354$		0			0
$355$	2.10137e11			0.702223
$356$		0			0
$357$	8.19818e10	−	3.06598e11i	0.267123	−	0.998993i
$358$		0			0
$359$		−	2.94013e11i		−	0.934204i	−0.884203	−	0.467102i	$-0.845298\pi$
$359$		−	2.94013e11i		−	0.934204i	0.884203	−	0.467102i	$-0.154702\pi$
$360$		0			0
$361$	−1.10450e11			−0.342283
$362$		0			0
$363$	−5.30624e10			−0.160401
$364$		0			0
$365$	−3.54109e11			−1.04428
$366$		0			0
$367$	−2.97585e11			−0.856276			−0.428138	−	0.903713i	$-0.640830\pi$
$367$	−2.97585e11			−0.856276			−0.428138	+	0.903713i	$0.640830\pi$
$368$		0			0
$369$		−	5.41138e10i		−	0.151946i
$370$		0			0
$371$	−5.36023e11	−	1.43328e11i	−1.46893	−	0.392779i
$372$		0			0
$373$	4.58974e11			1.22772			0.613858	−	0.789416i	$-0.289617\pi$
$373$	4.58974e11			1.22772			0.613858	+	0.789416i	$0.289617\pi$
$374$		0			0
$375$		−	2.96851e11i		−	0.775172i
$376$		0			0
$377$		−	2.67501e11i		−	0.682008i
$378$		0			0
$379$			3.16136e11i			0.787041i	0.919316	+	0.393521i	$0.128743\pi$
$379$			3.16136e11i			0.787041i	−0.919316	+	0.393521i	$0.871257\pi$
$380$		0			0
$381$		−	6.76066e11i		−	1.64372i
$382$		0			0
$383$	−1.12180e11			−0.266393			−0.133196	−	0.991090i	$-0.542524\pi$
$383$	−1.12180e11			−0.266393			−0.133196	+	0.991090i	$0.542524\pi$
$384$		0			0
$385$	−4.69908e11	−	1.25650e11i	−1.09003	−	0.291466i
$386$		0			0
$387$			9.11785e10i			0.206630i
$388$		0			0
$389$	4.61511e11			1.02190			0.510950	−	0.859610i	$-0.329294\pi$
$389$	4.61511e11			1.02190			0.510950	+	0.859610i	$0.329294\pi$
$390$		0			0
$391$	−5.31527e11			−1.15008
$392$		0			0
$393$	5.02708e11			1.06304
$394$		0			0
$395$	6.86283e11			1.41846
$396$		0			0
$397$			9.26586e11i			1.87210i	0.351870	+	0.936049i	$0.385546\pi$
$397$			9.26586e11i			1.87210i	−0.351870	+	0.936049i	$0.614454\pi$
$398$		0			0
$399$	−4.77590e11	−	1.27704e11i	−0.943359	−	0.252247i
$400$		0			0
$401$	1.51836e11			0.293241			0.146620	−	0.989193i	$-0.453160\pi$
$401$	1.51836e11			0.293241			0.146620	+	0.989193i	$0.453160\pi$
$402$		0			0
$403$		−	2.14849e11i		−	0.405753i
$404$		0			0
$405$		−	8.18580e11i		−	1.51187i
$406$		0			0
$407$			6.24289e11i			1.12774i
$408$		0			0
$409$		−	2.02579e11i		−	0.357965i	−0.983852	−	0.178982i	$-0.942719\pi$
$409$		−	2.02579e11i		−	0.357965i	0.983852	−	0.178982i	$-0.0572805\pi$
$410$		0			0
$411$	4.25173e11			0.734983
$412$		0			0
$413$	−6.98907e11	−	1.86882e11i	−1.18207	−	0.316077i
$414$		0			0
$415$			2.05250e11i			0.339677i
$416$		0			0
$417$	−3.98723e11			−0.645742
$418$		0			0
$419$	4.65377e11			0.737635			0.368818	−	0.929502i	$-0.379763\pi$
$419$	4.65377e11			0.737635			0.368818	+	0.929502i	$0.379763\pi$
$420$		0			0
$421$	9.20844e11			1.42862			0.714310	−	0.699829i	$-0.246741\pi$
$421$	9.20844e11			1.42862			0.714310	+	0.699829i	$0.246741\pi$
$422$		0			0
$423$	−1.61257e10			−0.0244899
$424$		0			0
$425$			2.70791e11i			0.402610i
$426$		0			0
$427$	6.25805e10	−	2.34041e11i	0.0910991	−	0.340695i
$428$		0			0
$429$	−3.75790e11			−0.535657
$430$		0			0
$431$			2.52962e11i			0.353108i	0.984291	+	0.176554i	$0.0564950\pi$
$431$			2.52962e11i			0.353108i	−0.984291	+	0.176554i	$0.943505\pi$
$432$		0			0
$433$		−	4.62645e11i		−	0.632488i	−0.948678	−	0.316244i	$-0.897578\pi$
$433$		−	4.62645e11i		−	0.632488i	0.948678	−	0.316244i	$-0.102422\pi$
$434$		0			0
$435$			1.55541e12i			2.08278i
$436$		0			0
$437$			8.27962e11i			1.08604i
$438$		0			0
$439$	−2.34018e11			−0.300718			−0.150359	−	0.988631i	$-0.548043\pi$
$439$	−2.34018e11			−0.300718			−0.150359	+	0.988631i	$0.548043\pi$
$440$		0			0
$441$	3.09583e11	+	1.78309e11i	0.389766	+	0.224491i
$442$		0			0
$443$		−	1.50254e11i		−	0.185357i	−0.995696	−	0.0926787i	$-0.970457\pi$
$443$		−	1.50254e11i		−	0.185357i	0.995696	−	0.0926787i	$-0.0295430\pi$
$444$		0			0
$445$	−1.78946e12			−2.16323
$446$		0			0
$447$	−1.73672e11			−0.205753
$448$		0			0
$449$	4.37081e11			0.507520			0.253760	−	0.967267i	$-0.418333\pi$
$449$	4.37081e11			0.507520			0.253760	+	0.967267i	$0.418333\pi$
$450$		0			0
$451$	−2.76329e11			−0.314509
$452$		0			0
$453$			1.25655e12i			1.40196i
$454$		0			0
$455$	5.11463e11	+	1.36761e11i	0.559452	+	0.149593i
$456$		0			0
$457$	1.80825e12			1.93925			0.969626	−	0.244591i	$-0.0786537\pi$
$457$	1.80825e12			1.93925			0.969626	+	0.244591i	$0.0786537\pi$
$458$		0			0
$459$		−	5.41055e11i		−	0.568963i
$460$		0			0
$461$			7.40235e11i			0.763335i	0.924300	+	0.381668i	$0.124650\pi$
$461$			7.40235e11i			0.763335i	−0.924300	+	0.381668i	$0.875350\pi$
$462$		0			0
$463$		−	9.62518e11i		−	0.973407i	−0.873567	−	0.486703i	$-0.838199\pi$
$463$		−	9.62518e11i		−	0.973407i	0.873567	−	0.486703i	$-0.161801\pi$
$464$		0			0
$465$			1.24926e12i			1.23913i
$466$		0			0
$467$	1.51772e12			1.47661			0.738306	−	0.674466i	$-0.235626\pi$
$467$	1.51772e12			1.47661			0.738306	+	0.674466i	$0.235626\pi$
$468$		0			0
$469$	3.44258e11	−	1.28747e12i	0.328554	−	1.22874i
$470$		0			0
$471$			1.92046e12i			1.79809i
$472$		0			0
$473$	4.65598e11			0.427697
$474$		0			0
$475$	4.21813e11			0.380189
$476$		0			0
$477$	7.73288e11			0.683926
$478$		0			0
$479$	3.87370e11			0.336214			0.168107	−	0.985769i	$-0.446235\pi$
$479$	3.87370e11			0.336214			0.168107	+	0.985769i	$0.446235\pi$
$480$		0			0
$481$		−	6.79495e11i		−	0.578807i
$482$		0			0
$483$	4.98187e11	−	1.86313e12i	0.416514	−	1.55769i
$484$		0			0
$485$	−1.33993e12			−1.09963
$486$		0			0
$487$		−	2.31594e12i		−	1.86572i	−0.360240	−	0.932860i	$-0.617305\pi$
$487$		−	2.31594e12i		−	1.86572i	0.360240	−	0.932860i	$-0.382695\pi$
$488$		0			0
$489$			1.18713e12i			0.938872i
$490$		0			0
$491$			4.04008e11i			0.313706i	0.987622	+	0.156853i	$0.0501349\pi$
$491$			4.04008e11i			0.313706i	−0.987622	+	0.156853i	$0.949865\pi$
$492$		0			0
$493$			1.60778e12i			1.22579i
$494$		0			0
$495$	6.77909e11			0.507514
$496$		0			0
$497$	−2.03587e11	+	7.61382e11i	−0.149674	+	0.559756i
$498$		0			0
$499$		−	1.85192e12i		−	1.33712i	−0.743660	−	0.668558i	$-0.766912\pi$
$499$		−	1.85192e12i		−	1.33712i	0.743660	−	0.668558i	$-0.233088\pi$
$500$		0			0
$501$	7.09025e11			0.502796
$502$		0			0
$503$	−1.38084e12			−0.961805			−0.480903	−	0.876774i	$-0.659691\pi$
$503$	−1.38084e12			−0.961805			−0.480903	+	0.876774i	$0.659691\pi$
$504$		0			0
$505$	8.29285e11			0.567404
$506$		0			0
$507$	−1.38236e12			−0.929151
$508$		0			0
$509$			2.41684e12i			1.59594i	0.602694	+	0.797972i	$0.294094\pi$
$509$			2.41684e12i			1.59594i	−0.602694	+	0.797972i	$0.705906\pi$
$510$		0			0
$511$	3.43072e11	−	1.28303e12i	0.222582	−	0.832420i
$512$		0			0
$513$	−8.42805e11			−0.537278
$514$		0			0
$515$		−	2.95122e12i		−	1.84871i
$516$		0			0
$517$			8.23452e10i			0.0506910i
$518$		0			0
$519$			3.90635e12i			2.36329i
$520$		0			0
$521$		−	4.01142e11i		−	0.238522i	−0.992863	−	0.119261i	$-0.961948\pi$
$521$		−	4.01142e11i		−	0.238522i	0.992863	−	0.119261i	$-0.0380525\pi$
$522$		0			0
$523$	1.62838e11			0.0951696			0.0475848	−	0.998867i	$-0.484848\pi$
$523$	1.62838e11			0.0951696			0.0475848	+	0.998867i	$0.484848\pi$
$524$		0			0
$525$	−9.49191e11	−	2.53806e11i	−0.545302	−	0.145809i
$526$		0			0
$527$			1.29133e12i			0.729270i
$528$		0			0
$529$	−1.42883e12			−0.793284
$530$		0			0
$531$	1.00827e12			0.550367
$532$		0			0
$533$	3.00765e11			0.161419
$534$		0			0
$535$	1.61888e11			0.0854325
$536$		0			0
$537$		−	3.68454e12i		−	1.91205i
$538$		0			0
$539$	9.10525e11	−	1.58087e12i	0.464668	−	0.806764i
$540$		0			0
$541$	−1.25807e11			−0.0631416			−0.0315708	−	0.999502i	$-0.510051\pi$
$541$	−1.25807e11			−0.0631416			−0.0315708	+	0.999502i	$0.510051\pi$
$542$		0			0
$543$		−	1.41962e12i		−	0.700765i
$544$		0			0
$545$			4.39630e12i			2.13453i
$546$		0			0
$547$			4.18042e12i			1.99654i	0.0588179	+	0.998269i	$0.481267\pi$
$547$			4.18042e12i			1.99654i	−0.0588179	+	0.998269i	$0.518733\pi$
$548$		0			0
$549$			3.37637e11i			0.158626i
$550$		0			0
$551$	2.50445e12			1.15753
$552$		0			0
$553$	−6.64893e11	+	2.48659e12i	−0.302335	+	1.13068i
$554$		0			0
$555$			3.95099e12i			1.76762i
$556$		0			0
$557$	−9.36295e11			−0.412159			−0.206079	−	0.978535i	$-0.566071\pi$
$557$	−9.36295e11			−0.412159			−0.206079	+	0.978535i	$0.566071\pi$
$558$		0			0
$559$	−5.06772e11			−0.219512
$560$		0			0
$561$	2.25864e12			0.962751
$562$		0			0
$563$	−3.68677e12			−1.54653			−0.773265	−	0.634084i	$-0.781378\pi$
$563$	−3.68677e12			−1.54653			−0.773265	+	0.634084i	$0.781378\pi$
$564$		0			0
$565$		−	3.64637e12i		−	1.50537i
$566$		0			0
$567$	2.96593e12	+	7.93066e11i	1.20514	+	0.322245i
$568$		0			0
$569$	3.09650e12			1.23841			0.619207	−	0.785228i	$-0.287454\pi$
$569$	3.09650e12			1.23841			0.619207	+	0.785228i	$0.287454\pi$
$570$		0			0
$571$			3.20987e12i			1.26365i	0.775113	+	0.631823i	$0.217693\pi$
$571$			3.20987e12i			1.26365i	−0.775113	+	0.631823i	$0.782307\pi$
$572$		0			0
$573$		−	4.23459e12i		−	1.64103i
$574$		0			0
$575$			1.64554e12i			0.627775i
$576$		0			0
$577$		−	2.37360e12i		−	0.891491i	−0.895160	−	0.445745i	$-0.852939\pi$
$577$		−	2.37360e12i		−	0.891491i	0.895160	−	0.445745i	$-0.147061\pi$
$578$		0			0
$579$	−2.64520e12			−0.978150
$580$		0			0
$581$	−7.43674e11	−	1.98853e11i	−0.270764	−	0.0724000i
$582$		0			0
$583$		−	3.94876e12i		−	1.41564i
$584$		0			0
$585$	−7.37858e11			−0.260478
$586$		0			0
$587$	2.76975e12			0.962874			0.481437	−	0.876481i	$-0.340115\pi$
$587$	2.76975e12			0.962874			0.481437	+	0.876481i	$0.340115\pi$
$588$		0			0
$589$	2.01151e12			0.688657
$590$		0			0
$591$	2.17368e11			0.0732913
$592$		0			0
$593$		−	2.42580e12i		−	0.805579i	−0.915293	−	0.402790i	$-0.868041\pi$
$593$		−	2.42580e12i		−	0.805579i	0.915293	−	0.402790i	$-0.131959\pi$
$594$		0			0
$595$	−3.07408e12	−	8.21985e11i	−1.00552	−	0.268867i
$596$		0			0
$597$	5.54512e12			1.78660
$598$		0			0
$599$		−	2.69284e12i		−	0.854652i	−0.904098	−	0.427326i	$-0.859456\pi$
$599$		−	2.69284e12i		−	0.854652i	0.904098	−	0.427326i	$-0.140544\pi$
$600$		0			0
$601$		−	1.31161e12i		−	0.410081i	−0.978754	−	0.205040i	$-0.934267\pi$
$601$		−	1.31161e12i		−	0.410081i	0.978754	−	0.205040i	$-0.0657326\pi$
$602$		0			0
$603$			1.85735e12i			0.572093i
$604$		0			0
$605$			5.32026e11i			0.161449i
$606$		0			0
$607$	−5.89476e12			−1.76245			−0.881225	−	0.472697i	$-0.843281\pi$
$607$	−5.89476e12			−1.76245			−0.881225	+	0.472697i	$0.843281\pi$
$608$		0			0
$609$	−5.63568e12	−	1.50693e12i	−1.66023	−	0.443932i
$610$		0			0
$611$		−	8.96270e10i		−	0.0260168i
$612$		0			0
$613$	−4.43845e12			−1.26958			−0.634788	−	0.772686i	$-0.718913\pi$
$613$	−4.43845e12			−1.26958			−0.634788	+	0.772686i	$0.718913\pi$
$614$		0			0
$615$	−1.74883e12			−0.492958
$616$		0			0
$617$	2.76164e12			0.767157			0.383578	−	0.923508i	$-0.374692\pi$
$617$	2.76164e12			0.767157			0.383578	+	0.923508i	$0.374692\pi$
$618$		0			0
$619$	−2.78619e12			−0.762785			−0.381392	−	0.924413i	$-0.624555\pi$
$619$	−2.78619e12			−0.762785			−0.381392	+	0.924413i	$0.624555\pi$
$620$		0			0
$621$		−	3.28788e12i		−	0.887164i
$622$		0			0
$623$	1.73369e12	−	6.48371e12i	0.461079	−	1.72436i
$624$		0			0
$625$	−4.76466e12			−1.24903
$626$		0			0
$627$		−	3.51829e12i		−	0.909135i
$628$		0			0
$629$			4.08402e12i			1.04030i
$630$		0			0
$631$			6.34324e12i			1.59287i	0.604727	+	0.796433i	$0.293282\pi$
$631$			6.34324e12i			1.59287i	−0.604727	+	0.796433i	$0.706718\pi$
$632$		0			0
$633$		−	6.79770e12i		−	1.68285i
$634$		0			0
$635$	−6.77853e12			−1.65445
$636$		0			0
$637$	−9.91043e11	+	1.72067e12i	−0.238487	+	0.414066i
$638$		0			0
$639$		−	1.09840e12i		−	0.260620i
$640$		0			0
$641$	−3.00469e12			−0.702972			−0.351486	−	0.936193i	$-0.614323\pi$
$641$	−3.00469e12			−0.702972			−0.351486	+	0.936193i	$0.614323\pi$
$642$		0			0
$643$	4.27099e12			0.985324			0.492662	−	0.870221i	$-0.336024\pi$
$643$	4.27099e12			0.985324			0.492662	+	0.870221i	$0.336024\pi$
$644$		0			0
$645$	2.94668e12			0.670369
$646$		0			0
$647$	8.01078e12			1.79724			0.898619	−	0.438729i	$-0.144571\pi$
$647$	8.01078e12			1.79724			0.898619	+	0.438729i	$0.144571\pi$
$648$		0			0
$649$		−	5.14869e12i		−	1.13919i
$650$		0			0
$651$	−4.52642e12	−	1.21033e12i	−0.987735	−	0.264112i
$652$		0			0
$653$	−4.20387e12			−0.904774			−0.452387	−	0.891822i	$-0.649427\pi$
$653$	−4.20387e12			−0.904774			−0.452387	+	0.891822i	$0.649427\pi$
$654$		0			0
$655$		−	5.04037e12i		−	1.06998i
$656$		0			0
$657$			1.85095e12i			0.387571i
$658$		0			0
$659$			3.07348e12i			0.634813i	0.948290	+	0.317406i	$0.102812\pi$
$659$			3.07348e12i			0.634813i	−0.948290	+	0.317406i	$0.897188\pi$
$660$		0			0
$661$		−	4.83796e12i		−	0.985725i	−0.870107	−	0.492863i	$-0.835951\pi$
$661$		−	4.83796e12i		−	0.985725i	0.870107	−	0.492863i	$-0.164049\pi$
$662$		0			0
$663$	−2.45837e12			−0.494125
$664$		0			0
$665$	−1.28041e12	+	4.78852e12i	−0.253894	+	0.949520i
$666$		0			0
$667$			9.77016e12i			1.91133i
$668$		0			0
$669$	2.20460e11			0.0425512
$670$		0			0
$671$	1.72412e12			0.328335
$672$		0			0
$673$	−4.43989e12			−0.834265			−0.417133	−	0.908846i	$-0.636965\pi$
$673$	−4.43989e12			−0.834265			−0.417133	+	0.908846i	$0.636965\pi$
$674$		0			0
$675$	−1.67504e12			−0.310569
$676$		0			0
$677$		−	4.81080e11i		−	0.0880174i	−0.999031	−	0.0440087i	$-0.985987\pi$
$677$		−	4.81080e11i		−	0.0880174i	0.999031	−	0.0440087i	$-0.0140129\pi$
$678$		0			0
$679$	1.29817e12	−	4.85494e12i	0.234379	−	0.876536i
$680$		0			0
$681$	−5.90257e11			−0.105167
$682$		0			0
$683$		−	1.00344e13i		−	1.76441i	−0.470867	−	0.882204i	$-0.656059\pi$
$683$		−	1.00344e13i		−	1.76441i	0.470867	−	0.882204i	$-0.343941\pi$
$684$		0			0
$685$		−	4.26296e12i		−	0.739783i
$686$		0			0
$687$			1.02471e13i			1.75508i
$688$		0			0
$689$			4.29795e12i			0.726565i
$690$		0			0
$691$	−1.35628e12			−0.226306			−0.113153	−	0.993578i	$-0.536095\pi$
$691$	−1.35628e12			−0.226306			−0.113153	+	0.993578i	$0.536095\pi$
$692$		0			0
$693$	−6.56781e11	+	2.45625e12i	−0.108173	+	0.404550i
$694$		0			0
$695$			3.99777e12i			0.649960i
$696$		0			0
$697$	−1.80771e12			−0.290123
$698$		0			0
$699$	−9.06305e12			−1.43591
$700$		0			0
$701$	1.18722e13			1.85695			0.928477	−	0.371391i	$-0.121119\pi$
$701$	1.18722e13			1.85695			0.928477	+	0.371391i	$0.121119\pi$
$702$		0			0
$703$	6.36171e12			0.982369
$704$		0			0
$705$			5.21146e11i			0.0794526i
$706$		0			0
$707$	−8.03438e11	+	3.00472e12i	−0.120939	+	0.452290i
$708$		0			0
$709$	8.19927e11			0.121862			0.0609308	−	0.998142i	$-0.480593\pi$
$709$	8.19927e11			0.121862			0.0609308	+	0.998142i	$0.480593\pi$
$710$		0			0
$711$		−	3.58725e12i		−	0.526440i
$712$		0			0
$713$			7.84712e12i			1.13712i
$714$		0			0
$715$			3.76783e12i			0.539156i
$716$		0			0
$717$		−	8.27339e12i		−	1.16909i
$718$		0			0
$719$	4.47269e12			0.624149			0.312075	−	0.950058i	$-0.398976\pi$
$719$	4.47269e12			0.624149			0.312075	+	0.950058i	$0.398976\pi$
$720$		0			0
$721$	1.06931e13	+	2.85924e12i	1.47365	+	0.394041i
$722$		0			0
$723$		−	8.43933e12i		−	1.14864i
$724$		0			0
$725$	4.97750e12			0.669099
$726$		0			0
$727$	5.53064e12			0.734295			0.367147	−	0.930163i	$-0.380334\pi$
$727$	5.53064e12			0.734295			0.367147	+	0.930163i	$0.380334\pi$
$728$		0			0
$729$	1.80406e12			0.236579
$730$		0			0
$731$	3.04589e12			0.394535
$732$		0			0
$733$			6.69160e12i			0.856174i	0.903738	+	0.428087i	$0.140812\pi$
$733$			6.69160e12i			0.856174i	−0.903738	+	0.428087i	$0.859188\pi$
$734$		0			0
$735$	5.76252e12	−	1.00050e13i	0.728316	−	1.26452i
$736$		0			0
$737$	9.48448e12			1.18416
$738$		0			0
$739$			4.40470e12i			0.543271i	0.962400	+	0.271635i	$0.0875645\pi$
$739$			4.40470e12i			0.543271i	−0.962400	+	0.271635i	$0.912436\pi$
$740$		0			0
$741$			3.82942e12i			0.466607i
$742$		0			0
$743$			2.47583e12i			0.298038i	0.988834	+	0.149019i	$0.0476116\pi$
$743$			2.47583e12i			0.298038i	−0.988834	+	0.149019i	$0.952388\pi$
$744$		0			0
$745$			1.74131e12i			0.207097i
$746$		0			0
$747$	1.07286e12			0.126066
$748$		0			0
$749$	−1.56842e11	+	5.86564e11i	−0.0182094	+	0.0681000i
$750$		0			0
$751$			1.37467e13i			1.57696i	0.615062	+	0.788479i	$0.289131\pi$
$751$			1.37467e13i			1.57696i	−0.615062	+	0.788479i	$0.710869\pi$
$752$		0			0
$753$	−1.08021e13			−1.22443
$754$		0			0
$755$	1.25987e13			1.41112
$756$		0			0
$757$	4.67655e11			0.0517600			0.0258800	−	0.999665i	$-0.491761\pi$
$757$	4.67655e11			0.0517600			0.0258800	+	0.999665i	$0.491761\pi$
$758$		0			0
$759$	1.37253e13			1.50118
$760$		0			0
$761$			1.71021e13i			1.84850i	0.381793	+	0.924248i	$0.375307\pi$
$761$			1.71021e13i			1.84850i	−0.381793	+	0.924248i	$0.624693\pi$
$762$		0			0
$763$	−1.59290e13	−	4.25927e12i	−1.70148	−	0.454962i
$764$		0			0
$765$	4.43480e12			0.468164
$766$		0			0
$767$			5.60399e12i			0.584680i
$768$		0			0
$769$		−	1.03252e13i		−	1.06470i	−0.846523	−	0.532352i	$-0.821308\pi$
$769$		−	1.03252e13i		−	1.06470i	0.846523	−	0.532352i	$-0.178692\pi$
$770$		0			0
$771$			9.10218e12i			0.927685i
$772$		0			0
$773$			1.78597e13i			1.79915i	0.436771	+	0.899573i	$0.356122\pi$
$773$			1.78597e13i			1.79915i	−0.436771	+	0.899573i	$0.643878\pi$
$774$		0			0
$775$	3.99779e12			0.398073
$776$		0			0
$777$	−1.43155e13	−	3.82785e12i	−1.40900	−	0.376756i
$778$		0			0
$779$			2.81589e12i			0.273966i
$780$		0			0
$781$	−5.60893e12			−0.539449
$782$		0			0
$783$	−9.94530e12			−0.945562
$784$		0			0
$785$	1.92554e13			1.80983
$786$		0			0
$787$	−1.78811e13			−1.66153			−0.830764	−	0.556624i	$-0.812096\pi$
$787$	−1.78811e13			−1.66153			−0.830764	+	0.556624i	$0.812096\pi$
$788$		0			0
$789$			7.14221e12i			0.656124i
$790$		0			0
$791$	1.32118e13	+	3.53272e12i	1.19996	+	0.320860i
$792$		0			0
$793$	−1.87659e12			−0.168515
$794$		0			0
$795$		−	2.49909e13i		−	2.21886i
$796$		0			0
$797$			1.20603e13i			1.05876i	0.848386	+	0.529378i	$0.177575\pi$
$797$			1.20603e13i			1.05876i	−0.848386	+	0.529378i	$0.822425\pi$
$798$		0			0
$799$			5.38692e11i			0.0467606i
$800$		0			0
$801$			9.35367e12i			0.802852i
$802$		0			0
$803$	9.45179e12			0.802221
$804$		0			0
$805$	−1.86806e13	−	4.99503e12i	−1.56787	−	0.419235i
$806$		0			0
$807$		−	1.77789e13i		−	1.47562i
$808$		0			0
$809$	−3.38799e12			−0.278082			−0.139041	−	0.990287i	$-0.544402\pi$
$809$	−3.38799e12			−0.278082			−0.139041	+	0.990287i	$0.544402\pi$
$810$		0			0
$811$	1.43126e13			1.16178			0.580891	−	0.813982i	$-0.302704\pi$
$811$	1.43126e13			1.16178			0.580891	+	0.813982i	$0.302704\pi$
$812$		0			0
$813$	−1.01999e13			−0.818817
$814$		0			0
$815$	1.19026e13			0.945004
$816$		0			0
$817$		−	4.74460e12i		−	0.372563i
$818$		0			0
$819$	7.14860e11	−	2.67345e12i	0.0555192	−	0.207632i
$820$		0			0
$821$	−1.80121e12			−0.138363			−0.0691816	−	0.997604i	$-0.522039\pi$
$821$	−1.80121e12			−0.138363			−0.0691816	+	0.997604i	$0.522039\pi$
$822$		0			0
$823$		−	2.14788e13i		−	1.63196i	−0.578078	−	0.815982i	$-0.696197\pi$
$823$		−	2.14788e13i		−	1.63196i	0.578078	−	0.815982i	$-0.303803\pi$
$824$		0			0
$825$		−	6.99248e12i		−	0.525519i
$826$		0			0
$827$			3.85779e12i			0.286790i	0.989666	+	0.143395i	$0.0458019\pi$
$827$			3.85779e12i			0.286790i	−0.989666	+	0.143395i	$0.954198\pi$
$828$		0			0
$829$			8.01507e12i			0.589402i	0.955589	+	0.294701i	$0.0952201\pi$
$829$			8.01507e12i			0.589402i	−0.955589	+	0.294701i	$0.904780\pi$
$830$		0			0
$831$	1.54199e11			0.0112170
$832$		0			0
$833$	5.95654e12	−	1.03419e13i	0.428639	−	0.744211i
$834$		0			0
$835$		−	7.10899e12i		−	0.506080i
$836$		0			0
$837$	−7.98779e12			−0.562552
$838$		0			0
$839$	−2.29199e13			−1.59692			−0.798460	−	0.602047i	$-0.794352\pi$
$839$	−2.29199e13			−1.59692			−0.798460	+	0.602047i	$0.794352\pi$
$840$		0			0
$841$	1.50460e13			1.03714
$842$		0			0
$843$	−3.34540e13			−2.28152
$844$		0			0
$845$			1.38602e13i			0.935219i
$846$		0			0
$847$	−1.92767e12	−	5.15444e11i	−0.128694	−	0.0344117i
$848$		0			0
$849$	6.20324e12			0.409764
$850$		0			0
$851$			2.48178e13i			1.62211i
$852$		0			0
$853$		−	1.41724e13i		−	0.916584i	−0.888802	−	0.458292i	$-0.848461\pi$
$853$		−	1.41724e13i		−	0.916584i	0.888802	−	0.458292i	$-0.151539\pi$
$854$		0			0
$855$		−	6.90812e12i		−	0.442092i
$856$		0			0
$857$			1.84260e13i			1.16686i	0.812164	+	0.583430i	$0.198290\pi$
$857$			1.84260e13i			1.16686i	−0.812164	+	0.583430i	$0.801710\pi$
$858$		0			0
$859$	2.29934e13			1.44090			0.720449	−	0.693508i	$-0.243936\pi$
$859$	2.29934e13			1.44090			0.720449	+	0.693508i	$0.243936\pi$
$860$		0			0
$861$	1.69432e12	−	6.33648e12i	0.105071	−	0.392947i
$862$		0			0
$863$		−	4.67630e12i		−	0.286981i	−0.989652	−	0.143491i	$-0.954167\pi$
$863$		−	4.67630e12i		−	0.286981i	0.989652	−	0.143491i	$-0.0458327\pi$
$864$		0			0
$865$	3.91667e13			2.37873
$866$		0			0
$867$	−5.25693e12			−0.315971
$868$		0			0
$869$	−1.83181e13			−1.08966
$870$		0			0
$871$	−1.03232e13			−0.607761
$872$		0			0
$873$			7.00393e12i			0.408111i
$874$		0			0
$875$	2.88359e12	−	1.07841e13i	0.166302	−	0.621941i
$876$		0			0
$877$	6.01431e12			0.343311			0.171655	−	0.985157i	$-0.445088\pi$
$877$	6.01431e12			0.343311			0.171655	+	0.985157i	$0.445088\pi$
$878$		0			0
$879$			1.45763e13i			0.823563i
$880$		0			0
$881$		−	2.75835e13i		−	1.54261i	−0.636464	−	0.771307i	$-0.719603\pi$
$881$		−	2.75835e13i		−	1.54261i	0.636464	−	0.771307i	$-0.280397\pi$
$882$		0			0
$883$			1.63458e13i			0.904862i	0.891799	+	0.452431i	$0.149443\pi$
$883$			1.63458e13i			0.904862i	−0.891799	+	0.452431i	$0.850557\pi$
$884$		0			0
$885$		−	3.25850e13i		−	1.78555i
$886$		0			0
$887$	2.26877e13			1.23065			0.615323	−	0.788275i	$-0.289025\pi$
$887$	2.26877e13			1.23065			0.615323	+	0.788275i	$0.289025\pi$
$888$		0			0
$889$	6.56726e12	−	2.45604e13i	0.352636	−	1.31880i
$890$		0			0
$891$			2.18494e13i			1.16142i
$892$		0			0
$893$	8.39124e11			0.0441565
$894$		0			0
$895$	−3.69428e13			−1.92454
$896$		0			0
$897$	−1.49390e13			−0.770470
$898$		0			0
$899$	2.37363e13			1.21198
$900$		0			0
$901$		−	2.58323e13i		−	1.30587i
$902$		0			0
$903$	−2.85483e12	+	1.06766e13i	−0.142885	+	0.534365i
$904$		0			0
$905$	−1.42337e13			−0.705341
$906$		0			0
$907$			1.63940e13i			0.804365i	0.915559	+	0.402183i	$0.131748\pi$
$907$			1.63940e13i			0.804365i	−0.915559	+	0.402183i	$0.868252\pi$
$908$		0			0
$909$		−	4.33473e12i		−	0.210584i
$910$		0			0
$911$			1.82806e13i			0.879344i	0.898159	+	0.439672i	$0.144905\pi$
$911$			1.82806e13i			0.879344i	−0.898159	+	0.439672i	$0.855095\pi$
$912$		0			0
$913$		−	5.47848e12i		−	0.260941i
$914$		0			0
$915$	1.09116e13			0.514629
$916$		0			0
$917$	1.82626e13	+	4.88327e12i	0.852904	+	0.228060i
$918$		0			0
$919$			2.05279e13i			0.949345i	0.880163	+	0.474672i	$0.157433\pi$
$919$			2.05279e13i			0.949345i	−0.880163	+	0.474672i	$0.842567\pi$
$920$		0			0
$921$	−3.37998e13			−1.54791
$922$		0			0
$923$	6.10493e12			0.276868
$924$		0			0
$925$	1.26436e13			0.567851
$926$		0			0
$927$	−1.54263e13			−0.686123
$928$		0			0
$929$			4.06890e13i			1.79228i	0.443771	+	0.896140i	$0.353640\pi$
$929$			4.06890e13i			1.79228i	−0.443771	+	0.896140i	$0.646360\pi$
$930$		0			0
$931$	−1.61096e13	−	9.27855e12i	−0.702766	−	0.404768i
$932$		0			0
$933$	−1.87852e13			−0.811614
$934$		0			0
$935$		−	2.26461e13i		−	0.969038i
$936$		0			0
$937$		−	1.47151e13i		−	0.623641i	−0.950141	−	0.311821i	$-0.899061\pi$
$937$		−	1.47151e13i		−	0.623641i	0.950141	−	0.311821i	$-0.100939\pi$
$938$		0			0
$939$			6.66565e12i			0.279800i
$940$		0			0
$941$			3.74200e12i			0.155579i	0.996970	+	0.0777895i	$0.0247862\pi$
$941$			3.74200e12i			0.155579i	−0.996970	+	0.0777895i	$0.975214\pi$
$942$		0			0
$943$	−1.09851e13			−0.452378
$944$		0			0
$945$	5.08458e12	−	1.90154e13i	0.207401	−	0.775646i
$946$		0			0
$947$		−	6.99898e12i		−	0.282787i	−0.989953	−	0.141394i	$-0.954842\pi$
$947$		−	6.99898e12i		−	0.282787i	0.989953	−	0.141394i	$-0.0451583\pi$
$948$		0			0
$949$	−1.02876e13			−0.411734
$950$		0			0
$951$	−2.27879e13			−0.903425
$952$		0			0
$953$	−2.18454e13			−0.857911			−0.428955	−	0.903326i	$-0.641118\pi$
$953$	−2.18454e13			−0.857911			−0.428955	+	0.903326i	$0.641118\pi$
$954$		0			0
$955$	−4.24578e13			−1.65174
$956$		0			0
$957$		−	4.15167e13i		−	1.60000i
$958$		0			0
$959$	1.54459e13	+	4.13010e12i	0.589696	+	0.157680i
$960$		0			0
$961$	−7.37530e12			−0.278949
$962$		0			0
$963$		−	8.46202e11i		−	0.0317070i
$964$		0			0
$965$			2.65220e13i			0.984539i
$966$		0			0
$967$		−	3.85871e13i		−	1.41913i	−0.704638	−	0.709567i	$-0.748890\pi$
$967$		−	3.85871e13i		−	1.41913i	0.704638	−	0.709567i	$-0.251110\pi$
$968$		0			0
$969$		−	2.30163e13i		−	0.838644i
$970$		0			0
$971$	−2.99182e13			−1.08006			−0.540030	−	0.841645i	$-0.681587\pi$
$971$	−2.99182e13			−1.08006			−0.540030	+	0.841645i	$0.681587\pi$
$972$		0			0
$973$	−1.44850e13	−	3.87317e12i	−0.518096	−	0.138535i
$974$		0			0
$975$			7.61083e12i			0.269719i
$976$		0			0
$977$	−7.34454e12			−0.257893			−0.128946	−	0.991652i	$-0.541160\pi$
$977$	−7.34454e12			−0.257893			−0.128946	+	0.991652i	$0.541160\pi$
$978$		0			0
$979$	4.77640e13			1.66180
$980$		0			0
$981$	2.29798e13			0.792201
$982$		0			0
$983$	1.79793e13			0.614159			0.307080	−	0.951684i	$-0.400648\pi$
$983$	1.79793e13			0.614159			0.307080	+	0.951684i	$0.400648\pi$
$984$		0			0
$985$		−	2.17943e12i		−	0.0737699i
$986$		0			0
$987$	−1.88825e12	−	5.04903e11i	−0.0633333	−	0.0169348i
$988$		0			0
$989$	1.85092e13			0.615184
$990$		0			0
$991$			7.25494e12i			0.238947i	0.992837	+	0.119474i	$0.0381207\pi$
$991$			7.25494e12i			0.238947i	−0.992837	+	0.119474i	$0.961879\pi$
$992$		0			0
$993$		−	2.70545e13i		−	0.883014i
$994$		0			0
$995$		−	5.55978e13i		−	1.79827i
$996$		0			0
$997$			2.82088e13i			0.904183i	0.891972	+	0.452091i	$0.149322\pi$
$997$			2.82088e13i			0.904183i	−0.891972	+	0.452091i	$0.850678\pi$
$998$		0			0
$999$	−2.52626e13			−0.802480

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.10.f.b.111.5	✓	24
4.3	odd	2	inner	112.10.f.b.111.19	yes	24
7.6	odd	2	inner	112.10.f.b.111.20	yes	24
28.27	even	2	inner	112.10.f.b.111.6	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.10.f.b.111.5	✓	24	1.1	even	1	trivial
112.10.f.b.111.6	yes	24	28.27	even	2	inner
112.10.f.b.111.19	yes	24	4.3	odd	2	inner
112.10.f.b.111.20	yes	24	7.6	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	112.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-168.927 q^{3} +1693.73i q^{5} +(-6136.85 - 1640.94i) q^{7} +8853.27 q^{9} -45208.8i q^{11} +49206.6i q^{13} -286117. i q^{15} -295750. i q^{17} -460692. q^{19} +(1.03668e6 + 277199. i) q^{21} -1.79721e6i q^{23} -915608. q^{25} +1.82943e6 q^{27} -5.43628e6 q^{29} -4.36627e6 q^{31} +7.63697e6i q^{33} +(2.77932e6 - 1.03942e7i) q^{35} -1.38090e7 q^{37} -8.31232e6i q^{39} -6.11229e6i q^{41} +1.02988e7i q^{43} +1.49951e7i q^{45} -1.82144e6 q^{47} +(3.49682e7 + 2.01404e7i) q^{49} +4.99602e7i q^{51} +8.73449e7 q^{53} +7.65716e7 q^{55} +7.78233e7 q^{57} +1.13887e8 q^{59} +3.81369e7i q^{61} +(-5.43312e7 - 1.45277e7i) q^{63} -8.33429e7 q^{65} +2.09793e8i q^{67} +3.03598e8i q^{69} -1.24067e8i q^{71} +2.09070e8i q^{73} +1.54671e8 q^{75} +(-7.41851e7 + 2.77439e8i) q^{77} -4.05189e8i q^{79} -4.83299e8 q^{81} +1.21182e8 q^{83} +5.00922e8 q^{85} +9.18334e8 q^{87} +1.05652e9i q^{89} +(8.07453e7 - 3.01974e8i) q^{91} +7.37580e8 q^{93} -7.80290e8i q^{95} +7.91112e8i q^{97} -4.00246e8i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 155768 q^{9} + 220672 q^{21} - 9187656 q^{25} + 14881104 q^{29} + 2829456 q^{37} - 214802472 q^{49} + 327087120 q^{53} + 238245440 q^{57} - 495797952 q^{65} + 347010000 q^{77} + 1816013720 q^{81}+ \cdots + 288442240 q^{93}+O(q^{100}) \) 24 * q + 155768 * q^9 + 220672 * q^21 - 9187656 * q^25 + 14881104 * q^29 + 2829456 * q^37 - 214802472 * q^49 + 327087120 * q^53 + 238245440 * q^57 - 495797952 * q^65 + 347010000 * q^77 + 1816013720 * q^81 - 1291803264 * q^85 + 288442240 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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