LMFDB - Embedded newform 112.10.f.b.111.15

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			111.15
Character	$\chi$	$=$	112.111
Dual form			112.10.f.b.111.16

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+64.1982 q^{3} +287.809i q^{5} +(-6348.43 + 225.841i) q^{7} -15561.6 q^{9} +34079.3i q^{11} +69110.3i q^{13} +18476.8i q^{15} -247175. i q^{17} +266515. q^{19} +(-407558. + 14498.6i) q^{21} -2.00717e6i q^{23} +1.87029e6 q^{25} -2.26264e6 q^{27} +1.51576e6 q^{29} +4.35389e6 q^{31} +2.18783e6i q^{33} +(-64999.0 - 1.82713e6i) q^{35} -4.73155e6 q^{37} +4.43676e6i q^{39} -2.17361e7i q^{41} +5.28283e6i q^{43} -4.47876e6i q^{45} +1.61100e7 q^{47} +(4.02516e7 - 2.86747e6i) q^{49} -1.58682e7i q^{51} -7.33344e7 q^{53} -9.80832e6 q^{55} +1.71098e7 q^{57} +5.42398e7 q^{59} -1.54440e8i q^{61} +(9.87917e7 - 3.51444e6i) q^{63} -1.98905e7 q^{65} +1.59079e8i q^{67} -1.28857e8i q^{69} +1.43343e8i q^{71} -3.33960e8i q^{73} +1.20069e8 q^{75} +(-7.69651e6 - 2.16350e8i) q^{77} -4.15669e8i q^{79} +1.61041e8 q^{81} +6.30458e8 q^{83} +7.11390e7 q^{85} +9.73092e7 q^{87} -5.95650e8i q^{89} +(-1.56079e7 - 4.38742e8i) q^{91} +2.79512e8 q^{93} +7.67054e7i q^{95} -1.06914e9i q^{97} -5.30328e8i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	64.1982			0.457591			0.228796	−	0.973475i	$-0.426521\pi$
$3$	64.1982			0.457591			0.228796	+	0.973475i	$0.426521\pi$
$4$		0			0
$5$			287.809i			0.205939i	0.994685	+	0.102970i	$0.0328344\pi$
$5$			287.809i			0.205939i	−0.994685	+	0.102970i	$0.967166\pi$
$6$		0			0
$7$	−6348.43	+	225.841i	−0.999368	+	0.0355518i
$7$	−6348.43	+	225.841i	−0.999368	+	0.0355518i
$8$		0			0
$9$	−15561.6			−0.790610
$10$		0			0
$11$			34079.3i			0.701817i	0.936410	+	0.350909i	$0.114127\pi$
$11$			34079.3i			0.701817i	−0.936410	+	0.350909i	$0.885873\pi$
$12$		0			0
$13$			69110.3i			0.671116i	0.942019	+	0.335558i	$0.108925\pi$
$13$			69110.3i			0.671116i	−0.942019	+	0.335558i	$0.891075\pi$
$14$		0			0
$15$			18476.8i			0.0942358i
$16$		0			0
$17$		−	247175.i		−	0.717768i	−0.933382	−	0.358884i	$-0.883157\pi$
$17$		−	247175.i		−	0.717768i	0.933382	−	0.358884i	$-0.116843\pi$
$18$		0			0
$19$	266515.			0.469171			0.234586	−	0.972095i	$-0.424627\pi$
$19$	266515.			0.469171			0.234586	+	0.972095i	$0.424627\pi$
$20$		0			0
$21$	−407558.	+	14498.6i	−0.457302	+	0.0162682i
$22$		0			0
$23$		−	2.00717e6i		−	1.49558i	−0.663937	−	0.747789i	$-0.731116\pi$
$23$		−	2.00717e6i		−	1.49558i	0.663937	−	0.747789i	$-0.268884\pi$
$24$		0			0
$25$	1.87029e6			0.957589
$26$		0			0
$27$	−2.26264e6			−0.819367
$28$		0			0
$29$	1.51576e6			0.397960			0.198980	−	0.980004i	$-0.436237\pi$
$29$	1.51576e6			0.397960			0.198980	+	0.980004i	$0.436237\pi$
$30$		0			0
$31$	4.35389e6			0.846739			0.423370	−	0.905957i	$-0.360847\pi$
$31$	4.35389e6			0.846739			0.423370	+	0.905957i	$0.360847\pi$
$32$		0			0
$33$			2.18783e6i			0.321145i
$34$		0			0
$35$	−64999.0	−	1.82713e6i	−0.00732151	−	0.205809i
$36$		0			0
$37$	−4.73155e6			−0.415046			−0.207523	−	0.978230i	$-0.566540\pi$
$37$	−4.73155e6			−0.415046			−0.207523	+	0.978230i	$0.566540\pi$
$38$		0			0
$39$			4.43676e6i			0.307097i
$40$		0			0
$41$		−	2.17361e7i		−	1.20131i	−0.799509	−	0.600654i	$-0.794907\pi$
$41$		−	2.17361e7i		−	1.20131i	0.799509	−	0.600654i	$-0.205093\pi$
$42$		0			0
$43$			5.28283e6i			0.235645i	0.993035	+	0.117823i	$0.0375914\pi$
$43$			5.28283e6i			0.235645i	−0.993035	+	0.117823i	$0.962409\pi$
$44$		0			0
$45$		−	4.47876e6i		−	0.162818i
$46$		0			0
$47$	1.61100e7			0.481566			0.240783	−	0.970579i	$-0.422596\pi$
$47$	1.61100e7			0.481566			0.240783	+	0.970579i	$0.422596\pi$
$48$		0			0
$49$	4.02516e7	−	2.86747e6i	0.997472	−	0.0710587i
$50$		0			0
$51$		−	1.58682e7i		−	0.328444i
$52$		0			0
$53$	−7.33344e7			−1.27663			−0.638317	−	0.769774i	$-0.720369\pi$
$53$	−7.33344e7			−1.27663			−0.638317	+	0.769774i	$0.720369\pi$
$54$		0			0
$55$	−9.80832e6			−0.144532
$56$		0			0
$57$	1.71098e7			0.214688
$58$		0			0
$59$	5.42398e7			0.582752			0.291376	−	0.956609i	$-0.405887\pi$
$59$	5.42398e7			0.582752			0.291376	+	0.956609i	$0.405887\pi$
$60$		0			0
$61$		−	1.54440e8i		−	1.42816i	−0.700065	−	0.714079i	$-0.746846\pi$
$61$		−	1.54440e8i		−	1.42816i	0.700065	−	0.714079i	$-0.253154\pi$
$62$		0			0
$63$	9.87917e7	−	3.51444e6i	0.790111	−	0.0281076i
$64$		0			0
$65$	−1.98905e7			−0.138209
$66$		0			0
$67$			1.59079e8i			0.964444i	0.876049	+	0.482222i	$0.160170\pi$
$67$			1.59079e8i			0.964444i	−0.876049	+	0.482222i	$0.839830\pi$
$68$		0			0
$69$		−	1.28857e8i		−	0.684363i
$70$		0			0
$71$			1.43343e8i			0.669445i	0.942317	+	0.334722i	$0.108643\pi$
$71$			1.43343e8i			0.669445i	−0.942317	+	0.334722i	$0.891357\pi$
$72$		0			0
$73$		−	3.33960e8i		−	1.37639i	−0.725526	−	0.688194i	$-0.758404\pi$
$73$		−	3.33960e8i		−	1.37639i	0.725526	−	0.688194i	$-0.241596\pi$
$74$		0			0
$75$	1.20069e8			0.438184
$76$		0			0
$77$	−7.69651e6	−	2.16350e8i	−0.0249509	−	0.701373i
$78$		0			0
$79$		−	4.15669e8i		−	1.20068i	−0.799746	−	0.600338i	$-0.795033\pi$
$79$		−	4.15669e8i		−	1.20068i	0.799746	−	0.600338i	$-0.204967\pi$
$80$		0			0
$81$	1.61041e8			0.415675
$82$		0			0
$83$	6.30458e8			1.45816			0.729079	−	0.684430i	$-0.239949\pi$
$83$	6.30458e8			1.45816			0.729079	+	0.684430i	$0.239949\pi$
$84$		0			0
$85$	7.11390e7			0.147816
$86$		0			0
$87$	9.73092e7			0.182103
$88$		0			0
$89$		−	5.95650e8i		−	1.00632i	−0.864193	−	0.503160i	$-0.832171\pi$
$89$		−	5.95650e8i		−	1.00632i	0.864193	−	0.503160i	$-0.167829\pi$
$90$		0			0
$91$	−1.56079e7	−	4.38742e8i	−0.0238594	−	0.670692i
$92$		0			0
$93$	2.79512e8			0.387460
$94$		0			0
$95$			7.67054e7i			0.0966206i
$96$		0			0
$97$		−	1.06914e9i		−	1.22621i	−0.790003	−	0.613103i	$-0.789921\pi$
$97$		−	1.06914e9i		−	1.22621i	0.790003	−	0.613103i	$-0.210079\pi$
$98$		0			0
$99$		−	5.30328e8i		−	0.554864i
$100$		0			0
$101$		−	1.48238e9i		−	1.41747i	−0.705476	−	0.708734i	$-0.749267\pi$
$101$		−	1.48238e9i		−	1.41747i	0.705476	−	0.708734i	$-0.250733\pi$
$102$		0			0
$103$	1.58348e9			1.38626			0.693132	−	0.720811i	$-0.256230\pi$
$103$	1.58348e9			1.38626			0.693132	+	0.720811i	$0.256230\pi$
$104$		0			0
$105$	−4.17282e6	−	1.17299e8i	−0.00335025	−	0.0941763i
$106$		0			0
$107$			1.79438e9i			1.32339i	0.749773	+	0.661695i	$0.230162\pi$
$107$			1.79438e9i			1.32339i	−0.749773	+	0.661695i	$0.769838\pi$
$108$		0			0
$109$	−6.50786e8			−0.441590			−0.220795	−	0.975320i	$-0.570865\pi$
$109$	−6.50786e8			−0.441590			−0.220795	+	0.975320i	$0.570865\pi$
$110$		0			0
$111$	−3.03757e8			−0.189921
$112$		0			0
$113$	−8.96788e8			−0.517413			−0.258706	−	0.965956i	$-0.583296\pi$
$113$	−8.96788e8			−0.517413			−0.258706	+	0.965956i	$0.583296\pi$
$114$		0			0
$115$	5.77681e8			0.307998
$116$		0			0
$117$		−	1.07547e9i		−	0.530592i
$118$		0			0
$119$	5.58222e7	+	1.56917e9i	0.0255179	+	0.717314i
$120$		0			0
$121$	1.19655e9			0.507453
$122$		0			0
$123$		−	1.39542e9i		−	0.549707i
$124$		0			0
$125$			1.10041e9i			0.403144i
$126$		0			0
$127$		−	1.04283e7i		−	0.00355710i	−0.999998	−	0.00177855i	$-0.999434\pi$
$127$		−	1.04283e7i		−	0.00355710i	0.999998	−	0.00177855i	$-0.000566130\pi$
$128$		0			0
$129$			3.39148e8i			0.107829i
$130$		0			0
$131$	−6.46180e6			−0.00191705			−0.000958523	−	1.00000i	$-0.500305\pi$
$131$	−6.46180e6			−0.00191705			−0.000958523		1.00000i	$0.500305\pi$
$132$		0			0
$133$	−1.69196e9	+	6.01901e7i	−0.468874	+	0.0166799i
$134$		0			0
$135$		−	6.51207e8i		−	0.168740i
$136$		0			0
$137$	3.63532e9			0.881659			0.440829	−	0.897591i	$-0.354684\pi$
$137$	3.63532e9			0.881659			0.440829	+	0.897591i	$0.354684\pi$
$138$		0			0
$139$	−2.48715e9			−0.565113			−0.282556	−	0.959251i	$-0.591182\pi$
$139$	−2.48715e9			−0.565113			−0.282556	+	0.959251i	$0.591182\pi$
$140$		0			0
$141$	1.03424e9			0.220360
$142$		0			0
$143$	−2.35523e9			−0.471001
$144$		0			0
$145$			4.36249e8i			0.0819556i
$146$		0			0
$147$	2.58408e9	−	1.84087e8i	0.456434	−	0.0325158i
$148$		0			0
$149$	−6.75454e9			−1.12268			−0.561342	−	0.827584i	$-0.689715\pi$
$149$	−6.75454e9			−1.12268			−0.561342	+	0.827584i	$0.689715\pi$
$150$		0			0
$151$		−	8.66313e8i		−	0.135606i	−0.997699	−	0.0678030i	$-0.978401\pi$
$151$		−	8.66313e8i		−	0.135606i	0.997699	−	0.0678030i	$-0.0215989\pi$
$152$		0			0
$153$			3.84643e9i			0.567475i
$154$		0			0
$155$			1.25309e9i			0.174377i
$156$		0			0
$157$			7.59365e9i			0.997475i	0.866753	+	0.498738i	$0.166203\pi$
$157$			7.59365e9i			0.997475i	−0.866753	+	0.498738i	$0.833797\pi$
$158$		0			0
$159$	−4.70794e9			−0.584176
$160$		0			0
$161$	4.53301e8	+	1.27424e10i	0.0531705	+	1.49463i
$162$		0			0
$163$		−	2.06374e9i		−	0.228987i	−0.993424	−	0.114494i	$-0.963475\pi$
$163$		−	2.06374e9i		−	0.228987i	0.993424	−	0.114494i	$-0.0365245\pi$
$164$		0			0
$165$	−6.29677e8			−0.0661363
$166$		0			0
$167$	−7.54280e9			−0.750427			−0.375214	−	0.926938i	$-0.622431\pi$
$167$	−7.54280e9			−0.750427			−0.375214	+	0.926938i	$0.622431\pi$
$168$		0			0
$169$	5.82826e9			0.549603
$170$		0			0
$171$	−4.14740e9			−0.370932
$172$		0			0
$173$			2.70454e9i			0.229554i	0.993391	+	0.114777i	$0.0366154\pi$
$173$			2.70454e9i			0.229554i	−0.993391	+	0.114777i	$0.963385\pi$
$174$		0			0
$175$	−1.18734e10	+	4.22389e8i	−0.956984	+	0.0340440i
$176$		0			0
$177$	3.48210e9			0.266662
$178$		0			0
$179$		−	2.66970e9i		−	0.194367i	−0.995266	−	0.0971837i	$-0.969017\pi$
$179$		−	2.66970e9i		−	0.194367i	0.995266	−	0.0971837i	$-0.0309834\pi$
$180$		0			0
$181$			2.42989e10i			1.68281i	0.540409	+	0.841403i	$0.318270\pi$
$181$			2.42989e10i			1.68281i	−0.540409	+	0.841403i	$0.681730\pi$
$182$		0			0
$183$		−	9.91480e9i		−	0.653513i
$184$		0			0
$185$		−	1.36178e9i		−	0.0854741i
$186$		0			0
$187$	8.42355e9			0.503742
$188$		0			0
$189$	1.43642e10	−	5.10997e8i	0.818849	−	0.0291300i
$190$		0			0
$191$			1.18264e8i			0.00642986i	0.999995	+	0.00321493i	$0.00102335\pi$
$191$			1.18264e8i			0.00642986i	−0.999995	+	0.00321493i	$0.998977\pi$
$192$		0			0
$193$	−2.37032e10			−1.22970			−0.614851	−	0.788643i	$-0.710784\pi$
$193$	−2.37032e10			−1.22970			−0.614851	+	0.788643i	$0.710784\pi$
$194$		0			0
$195$	−1.27694e9			−0.0632432
$196$		0			0
$197$	−2.95896e9			−0.139972			−0.0699861	−	0.997548i	$-0.522295\pi$
$197$	−2.95896e9			−0.139972			−0.0699861	+	0.997548i	$0.522295\pi$
$198$		0			0
$199$	−6.90176e9			−0.311976			−0.155988	−	0.987759i	$-0.549856\pi$
$199$	−6.90176e9			−0.311976			−0.155988	+	0.987759i	$0.549856\pi$
$200$		0			0
$201$			1.02126e10i			0.441321i
$202$		0			0
$203$	−9.62271e9	+	3.42321e8i	−0.397709	+	0.0141482i
$204$		0			0
$205$	6.25583e9			0.247396
$206$		0			0
$207$			3.12347e10i			1.18242i
$208$		0			0
$209$			9.08267e9i			0.329272i
$210$		0			0
$211$			3.12457e10i			1.08522i	0.839984	+	0.542611i	$0.182564\pi$
$211$			3.12457e10i			1.08522i	−0.839984	+	0.542611i	$0.817436\pi$
$212$		0			0
$213$			9.20239e9i			0.306332i
$214$		0			0
$215$	−1.52044e9			−0.0485286
$216$		0			0
$217$	−2.76404e10	+	9.83287e8i	−0.846204	+	0.0301031i
$218$		0			0
$219$		−	2.14396e10i		−	0.629823i
$220$		0			0
$221$	1.70823e10			0.481706
$222$		0			0
$223$	−1.52691e10			−0.413467			−0.206733	−	0.978397i	$-0.566283\pi$
$223$	−1.52691e10			−0.413467			−0.206733	+	0.978397i	$0.566283\pi$
$224$		0			0
$225$	−2.91047e10			−0.757080
$226$		0			0
$227$	2.79932e10			0.699738			0.349869	−	0.936799i	$-0.386226\pi$
$227$	2.79932e10			0.699738			0.349869	+	0.936799i	$0.386226\pi$
$228$		0			0
$229$		−	6.19384e10i		−	1.48833i	−0.667993	−	0.744167i	$-0.732847\pi$
$229$		−	6.19384e10i		−	1.48833i	0.667993	−	0.744167i	$-0.267153\pi$
$230$		0			0
$231$	−4.94103e8	−	1.38893e10i	−0.0114173	−	0.320942i
$232$		0			0
$233$	2.11411e10			0.469923			0.234961	−	0.972005i	$-0.424504\pi$
$233$	2.11411e10			0.469923			0.234961	+	0.972005i	$0.424504\pi$
$234$		0			0
$235$			4.63660e9i			0.0991733i
$236$		0			0
$237$		−	2.66852e10i		−	0.549419i
$238$		0			0
$239$		−	6.13506e10i		−	1.21627i	−0.793835	−	0.608133i	$-0.791919\pi$
$239$		−	6.13506e10i		−	1.21627i	0.793835	−	0.608133i	$-0.208081\pi$
$240$		0			0
$241$		−	3.31588e10i		−	0.633173i	−0.948564	−	0.316587i	$-0.897463\pi$
$241$		−	3.31588e10i		−	0.633173i	0.948564	−	0.316587i	$-0.102537\pi$
$242$		0			0
$243$	5.48741e10			1.00958
$244$		0			0
$245$	8.25283e8	+	1.15848e10i	0.0146338	+	0.205418i
$246$		0			0
$247$			1.84190e10i			0.314868i
$248$		0			0
$249$	4.04743e10			0.667240
$250$		0			0
$251$	6.31734e10			1.00462			0.502311	−	0.864687i	$-0.332483\pi$
$251$	6.31734e10			1.00462			0.502311	+	0.864687i	$0.332483\pi$
$252$		0			0
$253$	6.84030e10			1.04962
$254$		0			0
$255$	4.56700e9			0.0676395
$256$		0			0
$257$		−	1.06241e11i		−	1.51912i	−0.650436	−	0.759561i	$-0.725414\pi$
$257$		−	1.06241e11i		−	1.51912i	0.650436	−	0.759561i	$-0.274586\pi$
$258$		0			0
$259$	3.00379e10	−	1.06858e9i	0.414783	−	0.0147556i
$260$		0			0
$261$	−2.35877e10			−0.314632
$262$		0			0
$263$		−	9.82440e9i		−	0.126621i	−0.997994	−	0.0633104i	$-0.979834\pi$
$263$		−	9.82440e9i		−	0.126621i	0.997994	−	0.0633104i	$-0.0201658\pi$
$264$		0			0
$265$		−	2.11063e10i		−	0.262909i
$266$		0			0
$267$		−	3.82397e10i		−	0.460483i
$268$		0			0
$269$		−	6.10530e10i		−	0.710922i	−0.934691	−	0.355461i	$-0.884324\pi$
$269$		−	6.10530e10i		−	0.710922i	0.934691	−	0.355461i	$-0.115676\pi$
$270$		0			0
$271$	−6.84815e10			−0.771279			−0.385639	−	0.922650i	$-0.626019\pi$
$271$	−6.84815e10			−0.771279			−0.385639	+	0.922650i	$0.626019\pi$
$272$		0			0
$273$	−1.00200e9	−	2.81665e10i	−0.0109178	−	0.306903i
$274$		0			0
$275$			6.37383e10i			0.672052i
$276$		0			0
$277$	−1.08197e11			−1.10422			−0.552112	−	0.833770i	$-0.686178\pi$
$277$	−1.08197e11			−1.10422			−0.552112	+	0.833770i	$0.686178\pi$
$278$		0			0
$279$	−6.77534e10			−0.669441
$280$		0			0
$281$	−2.84930e10			−0.272621			−0.136311	−	0.990666i	$-0.543524\pi$
$281$	−2.84930e10			−0.272621			−0.136311	+	0.990666i	$0.543524\pi$
$282$		0			0
$283$	1.97471e11			1.83005			0.915027	−	0.403393i	$-0.132169\pi$
$283$	1.97471e11			1.83005			0.915027	+	0.403393i	$0.132169\pi$
$284$		0			0
$285$			4.92435e9i			0.0442127i
$286$		0			0
$287$	4.90890e9	+	1.37990e11i	0.0427086	+	1.20055i
$288$		0			0
$289$	5.74925e10			0.484809
$290$		0			0
$291$		−	6.86372e10i		−	0.561101i
$292$		0			0
$293$		−	1.11667e11i		−	0.885159i	−0.896729	−	0.442580i	$-0.854063\pi$
$293$		−	1.11667e11i		−	0.885159i	0.896729	−	0.442580i	$-0.145937\pi$
$294$		0			0
$295$			1.56107e10i			0.120011i
$296$		0			0
$297$		−	7.71093e10i		−	0.575046i
$298$		0			0
$299$	1.38716e11			1.00371
$300$		0			0
$301$	−1.19308e9	−	3.35377e10i	−0.00837761	−	0.235496i
$302$		0			0
$303$		−	9.51662e10i		−	0.648621i
$304$		0			0
$305$	4.44492e10			0.294114
$306$		0			0
$307$	8.44746e10			0.542754			0.271377	−	0.962473i	$-0.412521\pi$
$307$	8.44746e10			0.542754			0.271377	+	0.962473i	$0.412521\pi$
$308$		0			0
$309$	1.01657e11			0.634342
$310$		0			0
$311$	−2.39332e11			−1.45070			−0.725352	−	0.688379i	$-0.758323\pi$
$311$	−2.39332e11			−1.45070			−0.725352	+	0.688379i	$0.758323\pi$
$312$		0			0
$313$			5.65996e10i			0.333322i	0.986014	+	0.166661i	$0.0532986\pi$
$313$			5.65996e10i			0.333322i	−0.986014	+	0.166661i	$0.946701\pi$
$314$		0			0
$315$	1.01149e9	+	2.84331e10i	0.00578846	+	0.162715i
$316$		0			0
$317$	−2.10774e11			−1.17233			−0.586166	−	0.810191i	$-0.699363\pi$
$317$	−2.10774e11			−1.17233			−0.586166	+	0.810191i	$0.699363\pi$
$318$		0			0
$319$			5.16561e10i			0.279295i
$320$		0			0
$321$			1.15196e11i			0.605571i
$322$		0			0
$323$		−	6.58759e10i		−	0.336756i
$324$		0			0
$325$			1.29256e11i			0.642654i
$326$		0			0
$327$	−4.17793e10			−0.202068
$328$		0			0
$329$	−1.02273e11	+	3.63831e9i	−0.481262	+	0.0171206i
$330$		0			0
$331$		−	2.54637e9i		−	0.0116599i	−0.999983	−	0.00582997i	$-0.998144\pi$
$331$		−	2.54637e9i		−	0.0116599i	0.999983	−	0.00582997i	$-0.00185575\pi$
$332$		0			0
$333$	7.36305e10			0.328139
$334$		0			0
$335$	−4.57844e10			−0.198617
$336$		0			0
$337$	1.13465e11			0.479212			0.239606	−	0.970870i	$-0.422982\pi$
$337$	1.13465e11			0.479212			0.239606	+	0.970870i	$0.422982\pi$
$338$		0			0
$339$	−5.75722e10			−0.236763
$340$		0			0
$341$			1.48378e11i			0.594256i
$342$		0			0
$343$	−2.54887e11	+	2.72944e10i	−0.994315	+	0.106476i
$344$		0			0
$345$	3.70861e10			0.140937
$346$		0			0
$347$			3.94996e11i			1.46255i	0.682083	+	0.731275i	$0.261074\pi$
$347$			3.94996e11i			1.46255i	−0.682083	+	0.731275i	$0.738926\pi$
$348$		0			0
$349$			1.44178e11i			0.520217i	0.965579	+	0.260109i	$0.0837584\pi$
$349$			1.44178e11i			0.520217i	−0.965579	+	0.260109i	$0.916242\pi$
$350$		0			0
$351$		−	1.56372e11i		−	0.549891i
$352$		0			0
$353$			2.69572e11i			0.924035i	0.886871	+	0.462018i	$0.152874\pi$
$353$			2.69572e11i			0.924035i	−0.886871	+	0.462018i	$0.847126\pi$
$354$		0			0
$355$	−4.12554e10			−0.137865
$356$		0			0
$357$	3.58369e9	+	1.00738e11i	0.0116768	+	0.328237i
$358$		0			0
$359$		−	1.80651e11i		−	0.574004i	−0.957930	−	0.287002i	$-0.907341\pi$
$359$		−	1.80651e11i		−	0.574004i	0.957930	−	0.287002i	$-0.0926586\pi$
$360$		0			0
$361$	−2.51657e11			−0.779879
$362$		0			0
$363$	7.68162e10			0.232206
$364$		0			0
$365$	9.61164e10			0.283452
$366$		0			0
$367$	−2.91586e11			−0.839013			−0.419507	−	0.907752i	$-0.637797\pi$
$367$	−2.91586e11			−0.839013			−0.419507	+	0.907752i	$0.637797\pi$
$368$		0			0
$369$			3.38248e11i			0.949766i
$370$		0			0
$371$	4.65559e11	−	1.65619e10i	1.27583	−	0.0453866i
$372$		0			0
$373$	3.13721e11			0.839176			0.419588	−	0.907715i	$-0.362174\pi$
$373$	3.13721e11			0.839176			0.419588	+	0.907715i	$0.362174\pi$
$374$		0			0
$375$			7.06445e10i			0.184475i
$376$		0			0
$377$			1.04755e11i			0.267078i
$378$		0			0
$379$		−	6.63289e11i		−	1.65130i	−0.564182	−	0.825650i	$-0.690808\pi$
$379$		−	6.63289e11i		−	1.65130i	0.564182	−	0.825650i	$-0.309192\pi$
$380$		0			0
$381$		−	6.69477e8i		−	0.00162769i
$382$		0			0
$383$	−7.45087e11			−1.76935			−0.884673	−	0.466212i	$-0.845618\pi$
$383$	−7.45087e11			−1.76935			−0.884673	+	0.466212i	$0.845618\pi$
$384$		0			0
$385$	6.22675e10	−	2.21512e9i	0.144440	−	0.00513836i
$386$		0			0
$387$		−	8.22092e10i		−	0.186304i
$388$		0			0
$389$	−4.14783e11			−0.918434			−0.459217	−	0.888324i	$-0.651870\pi$
$389$	−4.14783e11			−0.918434			−0.459217	+	0.888324i	$0.651870\pi$
$390$		0			0
$391$	−4.96122e11			−1.07348
$392$		0			0
$393$	−4.14836e8			−0.000877223
$394$		0			0
$395$	1.19633e11			0.247266
$396$		0			0
$397$			7.63768e11i			1.54314i	0.636147	+	0.771568i	$0.280527\pi$
$397$			7.63768e11i			1.54314i	−0.636147	+	0.771568i	$0.719473\pi$
$398$		0			0
$399$	−1.08621e11	+	3.86410e9i	−0.214553	+	0.00763256i
$400$		0			0
$401$	5.31196e11			1.02590			0.512951	−	0.858418i	$-0.328552\pi$
$401$	5.31196e11			1.02590			0.512951	+	0.858418i	$0.328552\pi$
$402$		0			0
$403$			3.00899e11i			0.568261i
$404$		0			0
$405$			4.63490e10i			0.0856038i
$406$		0			0
$407$		−	1.61248e11i		−	0.291286i
$408$		0			0
$409$			3.93122e11i			0.694660i	0.937743	+	0.347330i	$0.112912\pi$
$409$			3.93122e11i			0.694660i	−0.937743	+	0.347330i	$0.887088\pi$
$410$		0			0
$411$	2.33381e11			0.403439
$412$		0			0
$413$	−3.44338e11	+	1.22496e10i	−0.582384	+	0.0207179i
$414$		0			0
$415$			1.81451e11i			0.300292i
$416$		0			0
$417$	−1.59671e11			−0.258590
$418$		0			0
$419$	5.50843e11			0.873101			0.436551	−	0.899680i	$-0.356200\pi$
$419$	5.50843e11			0.873101			0.436551	+	0.899680i	$0.356200\pi$
$420$		0			0
$421$	−1.66826e11			−0.258817			−0.129409	−	0.991591i	$-0.541308\pi$
$421$	−1.66826e11			−0.258817			−0.129409	+	0.991591i	$0.541308\pi$
$422$		0			0
$423$	−2.50698e11			−0.380731
$424$		0			0
$425$		−	4.62289e11i		−	0.687327i
$426$		0			0
$427$	3.48790e10	+	9.80454e11i	0.0507736	+	1.42726i
$428$		0			0
$429$	−1.51202e11			−0.215526
$430$		0			0
$431$			7.48705e11i			1.04511i	0.852605	+	0.522556i	$0.175021\pi$
$431$			7.48705e11i			1.04511i	−0.852605	+	0.522556i	$0.824979\pi$
$432$		0			0
$433$		−	6.54583e11i		−	0.894889i	−0.894312	−	0.447444i	$-0.852334\pi$
$433$		−	6.54583e11i		−	0.894889i	0.894312	−	0.447444i	$-0.147666\pi$
$434$		0			0
$435$			2.80064e10i			0.0375021i
$436$		0			0
$437$		−	5.34942e11i		−	0.701681i
$438$		0			0
$439$	1.27719e11			0.164121			0.0820605	−	0.996627i	$-0.473850\pi$
$439$	1.27719e11			0.164121			0.0820605	+	0.996627i	$0.473850\pi$
$440$		0			0
$441$	−6.26379e11	+	4.46224e10i	−0.788612	+	0.0561797i
$442$		0			0
$443$		−	7.65538e11i		−	0.944387i	−0.881495	−	0.472193i	$-0.843462\pi$
$443$		−	7.65538e11i		−	0.944387i	0.881495	−	0.472193i	$-0.156538\pi$
$444$		0			0
$445$	1.71433e11			0.207240
$446$		0			0
$447$	−4.33629e11			−0.513730
$448$		0			0
$449$	1.39923e12			1.62473			0.812367	−	0.583147i	$-0.198179\pi$
$449$	1.39923e12			1.62473			0.812367	+	0.583147i	$0.198179\pi$
$450$		0			0
$451$	7.40751e11			0.843098
$452$		0			0
$453$		−	5.56158e10i		−	0.0620520i
$454$		0			0
$455$	1.26274e11	−	4.49210e9i	0.138122	−	0.00491358i
$456$		0			0
$457$	−8.65465e11			−0.928168			−0.464084	−	0.885791i	$-0.653616\pi$
$457$	−8.65465e11			−0.928168			−0.464084	+	0.885791i	$0.653616\pi$
$458$		0			0
$459$			5.59268e11i			0.588116i
$460$		0			0
$461$		−	9.53906e11i		−	0.983675i	−0.870687	−	0.491837i	$-0.836325\pi$
$461$		−	9.53906e11i		−	0.983675i	0.870687	−	0.491837i	$-0.163675\pi$
$462$		0			0
$463$		−	1.91649e12i		−	1.93817i	−0.246731	−	0.969084i	$-0.579356\pi$
$463$		−	1.91649e12i		−	1.93817i	0.246731	−	0.969084i	$-0.420644\pi$
$464$		0			0
$465$			8.04459e10i			0.0797932i
$466$		0			0
$467$	5.51891e11			0.536942			0.268471	−	0.963288i	$-0.413482\pi$
$467$	5.51891e11			0.536942			0.268471	+	0.963288i	$0.413482\pi$
$468$		0			0
$469$	−3.59266e10	−	1.00990e12i	−0.0342877	−	0.963834i
$470$		0			0
$471$			4.87499e11i			0.456436i
$472$		0			0
$473$	−1.80035e11			−0.165380
$474$		0			0
$475$	4.98461e11			0.449273
$476$		0			0
$477$	1.14120e12			1.00932
$478$		0			0
$479$	4.76583e10			0.0413646			0.0206823	−	0.999786i	$-0.493416\pi$
$479$	4.76583e10			0.0413646			0.0206823	+	0.999786i	$0.493416\pi$
$480$		0			0
$481$		−	3.26999e11i		−	0.278544i
$482$		0			0
$483$	2.91011e10	+	8.18038e11i	0.0243303	+	0.683930i
$484$		0			0
$485$	3.07709e11			0.252524
$486$		0			0
$487$		−	1.14633e11i		−	0.0923482i	−0.998933	−	0.0461741i	$-0.985297\pi$
$487$		−	1.14633e11i		−	0.0923482i	0.998933	−	0.0461741i	$-0.0147029\pi$
$488$		0			0
$489$		−	1.32489e11i		−	0.104782i
$490$		0			0
$491$		−	2.26182e12i		−	1.75627i	−0.478410	−	0.878137i	$-0.658787\pi$
$491$		−	2.26182e12i		−	1.75627i	0.478410	−	0.878137i	$-0.341213\pi$
$492$		0			0
$493$		−	3.74658e11i		−	0.285643i
$494$		0			0
$495$	1.52633e11			0.114268
$496$		0			0
$497$	−3.23728e10	−	9.10005e11i	−0.0238000	−	0.669022i
$498$		0			0
$499$		−	2.67880e11i		−	0.193414i	−0.995313	−	0.0967071i	$-0.969169\pi$
$499$		−	2.67880e11i		−	0.193414i	0.995313	−	0.0967071i	$-0.0308310\pi$
$500$		0			0
$501$	−4.84235e11			−0.343389
$502$		0			0
$503$	2.28139e12			1.58907			0.794537	−	0.607216i	$-0.207714\pi$
$503$	2.28139e12			1.58907			0.794537	+	0.607216i	$0.207714\pi$
$504$		0			0
$505$	4.26642e11			0.291912
$506$		0			0
$507$	3.74164e11			0.251493
$508$		0			0
$509$		−	1.54692e12i		−	1.02150i	−0.859729	−	0.510750i	$-0.829368\pi$
$509$		−	1.54692e12i		−	1.02150i	0.859729	−	0.510750i	$-0.170632\pi$
$510$		0			0
$511$	7.54218e10	+	2.12012e12i	0.0489331	+	1.37552i
$512$		0			0
$513$	−6.03029e11			−0.384423
$514$		0			0
$515$			4.55740e11i			0.285486i
$516$		0			0
$517$			5.49019e11i			0.337971i
$518$		0			0
$519$			1.73627e11i			0.105042i
$520$		0			0
$521$		−	3.13568e12i		−	1.86450i	−0.361817	−	0.932249i	$-0.617844\pi$
$521$		−	3.13568e12i		−	1.86450i	0.361817	−	0.932249i	$-0.382156\pi$
$522$		0			0
$523$	−3.78585e11			−0.221261			−0.110631	−	0.993862i	$-0.535287\pi$
$523$	−3.78585e11			−0.221261			−0.110631	+	0.993862i	$0.535287\pi$
$524$		0			0
$525$	−7.62253e11	+	2.71166e10i	−0.437907	+	0.0155782i
$526$		0			0
$527$		−	1.07617e12i		−	0.607762i
$528$		0			0
$529$	−2.22758e12			−1.23675
$530$		0			0
$531$	−8.44057e11			−0.460730
$532$		0			0
$533$	1.50219e12			0.806217
$534$		0			0
$535$	−5.16438e11			−0.272537
$536$		0			0
$537$		−	1.71390e11i		−	0.0889408i
$538$		0			0
$539$	9.77216e10	+	1.37175e12i	0.0498702	+	0.700043i
$540$		0			0
$541$	3.00642e12			1.50890			0.754452	−	0.656355i	$-0.227903\pi$
$541$	3.00642e12			1.50890			0.754452	+	0.656355i	$0.227903\pi$
$542$		0			0
$543$			1.55995e12i			0.770037i
$544$		0			0
$545$		−	1.87302e11i		−	0.0909406i
$546$		0			0
$547$		−	2.99127e12i		−	1.42861i	−0.699837	−	0.714303i	$-0.746744\pi$
$547$		−	2.99127e12i		−	1.42861i	0.699837	−	0.714303i	$-0.253256\pi$
$548$		0			0
$549$			2.40334e12i			1.12912i
$550$		0			0
$551$	4.03974e11			0.186712
$552$		0			0
$553$	9.38752e10	+	2.63885e12i	0.0426862	+	1.19992i
$554$		0			0
$555$		−	8.74240e10i		−	0.0391122i
$556$		0			0
$557$	−4.99847e11			−0.220033			−0.110017	−	0.993930i	$-0.535090\pi$
$557$	−4.99847e11			−0.220033			−0.110017	+	0.993930i	$0.535090\pi$
$558$		0			0
$559$	−3.65098e11			−0.158145
$560$		0			0
$561$	5.40777e11			0.230508
$562$		0			0
$563$	2.04877e12			0.859421			0.429710	−	0.902967i	$-0.358616\pi$
$563$	2.04877e12			0.859421			0.429710	+	0.902967i	$0.358616\pi$
$564$		0			0
$565$		−	2.58103e11i		−	0.106555i
$566$		0			0
$567$	−1.02236e12	+	3.63697e10i	−0.415413	+	0.0147780i
$568$		0			0
$569$	−5.34800e11			−0.213888			−0.106944	−	0.994265i	$-0.534107\pi$
$569$	−5.34800e11			−0.213888			−0.106944	+	0.994265i	$0.534107\pi$
$570$		0			0
$571$			4.82904e12i			1.90107i	0.310614	+	0.950536i	$0.399465\pi$
$571$			4.82904e12i			1.90107i	−0.310614	+	0.950536i	$0.600535\pi$
$572$		0			0
$573$			7.59233e9i			0.00294225i
$574$		0			0
$575$		−	3.75399e12i		−	1.43215i
$576$		0			0
$577$			2.09003e12i			0.784987i	0.919755	+	0.392493i	$0.128387\pi$
$577$			2.09003e12i			0.784987i	−0.919755	+	0.392493i	$0.871613\pi$
$578$		0			0
$579$	−1.52171e12			−0.562700
$580$		0			0
$581$	−4.00242e12	+	1.42383e11i	−1.45724	+	0.0518401i
$582$		0			0
$583$		−	2.49919e12i		−	0.895964i
$584$		0			0
$585$	3.09528e11			0.109270
$586$		0			0
$587$	−3.60556e12			−1.25343			−0.626717	−	0.779247i	$-0.715602\pi$
$587$	−3.60556e12			−1.25343			−0.626717	+	0.779247i	$0.715602\pi$
$588$		0			0
$589$	1.16038e12			0.397265
$590$		0			0
$591$	−1.89960e11			−0.0640500
$592$		0			0
$593$		−	4.18309e11i		−	0.138916i	−0.997585	−	0.0694579i	$-0.977873\pi$
$593$		−	4.18309e11i		−	0.138916i	0.997585	−	0.0694579i	$-0.0221269\pi$
$594$		0			0
$595$	−4.51621e11	+	1.60661e10i	−0.147723	+	0.00525514i
$596$		0			0
$597$	−4.43081e11			−0.142757
$598$		0			0
$599$			5.90002e12i			1.87255i	0.351271	+	0.936274i	$0.385750\pi$
$599$			5.90002e12i			1.87255i	−0.351271	+	0.936274i	$0.614250\pi$
$600$		0			0
$601$			3.83011e12i			1.19750i	0.800935	+	0.598752i	$0.204336\pi$
$601$			3.83011e12i			1.19750i	−0.800935	+	0.598752i	$0.795664\pi$
$602$		0			0
$603$		−	2.47552e12i		−	0.762499i
$604$		0			0
$605$			3.44376e11i			0.104504i
$606$		0			0
$607$	1.69105e12			0.505600			0.252800	−	0.967519i	$-0.418649\pi$
$607$	1.69105e12			0.505600			0.252800	+	0.967519i	$0.418649\pi$
$608$		0			0
$609$	−6.17761e11	+	2.19764e10i	−0.181988	+	0.00647410i
$610$		0			0
$611$			1.11337e12i			0.323187i
$612$		0			0
$613$	−3.90342e12			−1.11654			−0.558269	−	0.829660i	$-0.688534\pi$
$613$	−3.90342e12			−1.11654			−0.558269	+	0.829660i	$0.688534\pi$
$614$		0			0
$615$	4.01613e11			0.113206
$616$		0			0
$617$	3.98086e12			1.10584			0.552921	−	0.833234i	$-0.313513\pi$
$617$	3.98086e12			1.10584			0.552921	+	0.833234i	$0.313513\pi$
$618$		0			0
$619$	−5.63198e12			−1.54189			−0.770945	−	0.636902i	$-0.780215\pi$
$619$	−5.63198e12			−1.54189			−0.770945	+	0.636902i	$0.780215\pi$
$620$		0			0
$621$			4.54150e12i			1.22543i
$622$		0			0
$623$	1.34522e11	+	3.78144e12i	0.0357765	+	1.00568i
$624$		0			0
$625$	3.33620e12			0.874566
$626$		0			0
$627$			5.83091e11i			0.150672i
$628$		0			0
$629$			1.16952e12i			0.297906i
$630$		0			0
$631$		−	6.26554e12i		−	1.57335i	−0.617365	−	0.786677i	$-0.711800\pi$
$631$		−	6.26554e12i		−	1.57335i	0.617365	−	0.786677i	$-0.288200\pi$
$632$		0			0
$633$			2.00592e12i			0.496588i
$634$		0			0
$635$	3.00134e9			0.000732545
$636$		0			0
$637$	1.98172e11	+	2.78180e12i	0.0476886	+	0.669420i
$638$		0			0
$639$		−	2.23065e12i		−	0.529270i
$640$		0			0
$641$	−2.49293e12			−0.583243			−0.291622	−	0.956534i	$-0.594195\pi$
$641$	−2.49293e12			−0.583243			−0.291622	+	0.956534i	$0.594195\pi$
$642$		0			0
$643$	−6.89253e12			−1.59012			−0.795058	−	0.606533i	$-0.792560\pi$
$643$	−6.89253e12			−1.59012			−0.795058	+	0.606533i	$0.792560\pi$
$644$		0			0
$645$	−9.76098e10			−0.0222062
$646$		0			0
$647$	4.30867e12			0.966661			0.483330	−	0.875438i	$-0.339427\pi$
$647$	4.30867e12			0.966661			0.483330	+	0.875438i	$0.339427\pi$
$648$		0			0
$649$			1.84845e12i			0.408985i
$650$		0			0
$651$	−1.77446e12	+	6.31253e10i	−0.387215	+	0.0137749i
$652$		0			0
$653$	5.34169e11			0.114966			0.0574830	−	0.998346i	$-0.481692\pi$
$653$	5.34169e11			0.114966			0.0574830	+	0.998346i	$0.481692\pi$
$654$		0			0
$655$		−	1.85976e9i		−	0.000394795i
$656$		0			0
$657$			5.19694e12i			1.08819i
$658$		0			0
$659$		−	5.44371e12i		−	1.12437i	−0.827010	−	0.562187i	$-0.809960\pi$
$659$		−	5.44371e12i		−	1.12437i	0.827010	−	0.562187i	$-0.190040\pi$
$660$		0			0
$661$			2.00297e12i			0.408102i	0.978960	+	0.204051i	$0.0654108\pi$
$661$			2.00297e12i			0.408102i	−0.978960	+	0.204051i	$0.934589\pi$
$662$		0			0
$663$	1.09666e12			0.220424
$664$		0			0
$665$	−1.73232e10	−	4.86959e11i	−0.00343504	−	0.0965595i
$666$		0			0
$667$		−	3.04239e12i		−	0.595181i
$668$		0			0
$669$	−9.80247e11			−0.189199
$670$		0			0
$671$	5.26322e12			1.00231
$672$		0			0
$673$	7.56935e12			1.42230			0.711149	−	0.703041i	$-0.248175\pi$
$673$	7.56935e12			1.42230			0.711149	+	0.703041i	$0.248175\pi$
$674$		0			0
$675$	−4.23180e12			−0.784617
$676$		0			0
$677$		−	9.06404e12i		−	1.65834i	−0.558999	−	0.829169i	$-0.688814\pi$
$677$		−	9.06404e12i		−	1.65834i	0.558999	−	0.829169i	$-0.311186\pi$
$678$		0			0
$679$	2.41457e11	+	6.78739e12i	0.0435939	+	1.22543i
$680$		0			0
$681$	1.79711e12			0.320194
$682$		0			0
$683$			1.56867e12i			0.275829i	0.990444	+	0.137914i	$0.0440399\pi$
$683$			1.56867e12i			0.275829i	−0.990444	+	0.137914i	$0.955960\pi$
$684$		0			0
$685$			1.04628e12i			0.181568i
$686$		0			0
$687$		−	3.97634e12i		−	0.681048i
$688$		0			0
$689$		−	5.06817e12i		−	0.856770i
$690$		0			0
$691$	−3.57705e12			−0.596862			−0.298431	−	0.954431i	$-0.596463\pi$
$691$	−3.57705e12			−0.596862			−0.298431	+	0.954431i	$0.596463\pi$
$692$		0			0
$693$	1.19770e11	+	3.36675e12i	0.0197264	+	0.554513i
$694$		0			0
$695$		−	7.15822e11i		−	0.116379i
$696$		0			0
$697$	−5.37261e12			−0.862260
$698$		0			0
$699$	1.35722e12			0.215032
$700$		0			0
$701$	1.48104e12			0.231652			0.115826	−	0.993270i	$-0.463048\pi$
$701$	1.48104e12			0.231652			0.115826	+	0.993270i	$0.463048\pi$
$702$		0			0
$703$	−1.26103e12			−0.194727
$704$		0			0
$705$			2.97662e11i			0.0453808i
$706$		0			0
$707$	3.34782e11	+	9.41079e12i	0.0503935	+	1.41657i
$708$		0			0
$709$	−5.10442e12			−0.758644			−0.379322	−	0.925265i	$-0.623843\pi$
$709$	−5.10442e12			−0.758644			−0.379322	+	0.925265i	$0.623843\pi$
$710$		0			0
$711$			6.46848e12i			0.949268i
$712$		0			0
$713$		−	8.73899e12i		−	1.26636i
$714$		0			0
$715$		−	6.77856e11i		−	0.0969975i
$716$		0			0
$717$		−	3.93860e12i		−	0.556552i
$718$		0			0
$719$	9.65335e12			1.34709			0.673547	−	0.739144i	$-0.264770\pi$
$719$	9.65335e12			1.34709			0.673547	+	0.739144i	$0.264770\pi$
$720$		0			0
$721$	−1.00526e13	+	3.57616e11i	−1.38539	+	0.0492842i
$722$		0			0
$723$		−	2.12874e12i		−	0.289734i
$724$		0			0
$725$	2.83492e12			0.381083
$726$		0			0
$727$	8.02535e12			1.06551			0.532757	−	0.846268i	$-0.321156\pi$
$727$	8.02535e12			1.06551			0.532757	+	0.846268i	$0.321156\pi$
$728$		0			0
$729$	3.53048e11			0.0462978
$730$		0			0
$731$	1.30578e12			0.169139
$732$		0			0
$733$			5.83361e12i			0.746396i	0.927752	+	0.373198i	$0.121739\pi$
$733$			5.83361e12i			0.746396i	−0.927752	+	0.373198i	$0.878261\pi$
$734$		0			0
$735$	5.29817e10	+	7.43721e11i	0.00669627	+	0.0939976i
$736$		0			0
$737$	−5.42131e12			−0.676863
$738$		0			0
$739$			4.02467e12i			0.496399i	0.968709	+	0.248199i	$0.0798388\pi$
$739$			4.02467e12i			0.496399i	−0.968709	+	0.248199i	$0.920161\pi$
$740$		0			0
$741$			1.18247e12i			0.144081i
$742$		0			0
$743$		−	1.35541e13i		−	1.63162i	−0.578319	−	0.815811i	$-0.696291\pi$
$743$		−	1.35541e13i		−	1.63162i	0.578319	−	0.815811i	$-0.303709\pi$
$744$		0			0
$745$		−	1.94401e12i		−	0.231204i
$746$		0			0
$747$	−9.81092e12			−1.15283
$748$		0			0
$749$	−4.05245e11	−	1.13915e13i	−0.0470489	−	1.32255i
$750$		0			0
$751$			8.57077e12i			0.983196i	0.870822	+	0.491598i	$0.163587\pi$
$751$			8.57077e12i			0.983196i	−0.870822	+	0.491598i	$0.836413\pi$
$752$		0			0
$753$	4.05562e12			0.459706
$754$		0			0
$755$	2.49332e11			0.0279265
$756$		0			0
$757$	1.65330e13			1.82987			0.914933	−	0.403605i	$-0.132243\pi$
$757$	1.65330e13			1.82987			0.914933	+	0.403605i	$0.132243\pi$
$758$		0			0
$759$	4.39135e12			0.480297
$760$		0			0
$761$		−	1.17954e13i		−	1.27492i	−0.770483	−	0.637461i	$-0.779985\pi$
$761$		−	1.17954e13i		−	1.27492i	0.770483	−	0.637461i	$-0.220015\pi$
$762$		0			0
$763$	4.13147e12	−	1.46974e11i	0.441311	−	0.0156993i
$764$		0			0
$765$	−1.10704e12			−0.116865
$766$		0			0
$767$			3.74853e12i			0.391094i
$768$		0			0
$769$		−	6.20355e12i		−	0.639694i	−0.947469	−	0.319847i	$-0.896369\pi$
$769$		−	6.20355e12i		−	0.639694i	0.947469	−	0.319847i	$-0.103631\pi$
$770$		0			0
$771$		−	6.82048e12i		−	0.695137i
$772$		0			0
$773$		−	1.14032e13i		−	1.14873i	−0.818600	−	0.574364i	$-0.805249\pi$
$773$		−	1.14032e13i		−	1.14873i	0.818600	−	0.574364i	$-0.194751\pi$
$774$		0			0
$775$	8.14304e12			0.810828
$776$		0			0
$777$	1.92838e12	−	6.86009e10i	0.189801	−	0.00675204i
$778$		0			0
$779$		−	5.79300e12i		−	0.563619i
$780$		0			0
$781$	−4.88504e12			−0.469828
$782$		0			0
$783$	−3.42962e12			−0.326076
$784$		0			0
$785$	−2.18552e12			−0.205419
$786$		0			0
$787$	9.61146e12			0.893106			0.446553	−	0.894757i	$-0.352651\pi$
$787$	9.61146e12			0.893106			0.446553	+	0.894757i	$0.352651\pi$
$788$		0			0
$789$		−	6.30709e11i		−	0.0579406i
$790$		0			0
$791$	5.69320e12	−	2.02532e11i	0.517085	−	0.0183950i
$792$		0			0
$793$	1.06734e13			0.958461
$794$		0			0
$795$		−	1.35499e12i		−	0.120305i
$796$		0			0
$797$		−	1.75710e13i		−	1.54253i	−0.636512	−	0.771267i	$-0.719623\pi$
$797$		−	1.75710e13i		−	1.54253i	0.636512	−	0.771267i	$-0.280377\pi$
$798$		0			0
$799$		−	3.98199e12i		−	0.345653i
$800$		0			0
$801$			9.26925e12i			0.795607i
$802$		0			0
$803$	1.13811e13			0.965973
$804$		0			0
$805$	−3.66737e12	+	1.30464e11i	−0.307803	+	0.0109499i
$806$		0			0
$807$		−	3.91950e12i		−	0.325311i
$808$		0			0
$809$	−7.36795e12			−0.604753			−0.302377	−	0.953189i	$-0.597780\pi$
$809$	−7.36795e12			−0.604753			−0.302377	+	0.953189i	$0.597780\pi$
$810$		0			0
$811$	−1.66690e13			−1.35306			−0.676529	−	0.736416i	$-0.736517\pi$
$811$	−1.66690e13			−1.35306			−0.676529	+	0.736416i	$0.736517\pi$
$812$		0			0
$813$	−4.39639e12			−0.352930
$814$		0			0
$815$	5.93962e11			0.0471574
$816$		0			0
$817$			1.40796e12i			0.110558i
$818$		0			0
$819$	2.42884e11	+	6.82753e12i	0.0188635	+	0.530256i
$820$		0			0
$821$	1.60784e13			1.23509			0.617545	−	0.786535i	$-0.288127\pi$
$821$	1.60784e13			1.23509			0.617545	+	0.786535i	$0.288127\pi$
$822$		0			0
$823$			8.77883e12i			0.667017i	0.942747	+	0.333509i	$0.108233\pi$
$823$			8.77883e12i			0.667017i	−0.942747	+	0.333509i	$0.891767\pi$
$824$		0			0
$825$			4.09189e12i			0.307525i
$826$		0			0
$827$		−	1.57071e12i		−	0.116767i	−0.998294	−	0.0583836i	$-0.981405\pi$
$827$		−	1.57071e12i		−	0.116767i	0.998294	−	0.0583836i	$-0.0185947\pi$
$828$		0			0
$829$			5.52683e12i			0.406425i	0.979135	+	0.203213i	$0.0651383\pi$
$829$			5.52683e12i			0.406425i	−0.979135	+	0.203213i	$0.934862\pi$
$830$		0			0
$831$	−6.94607e12			−0.505283
$832$		0			0
$833$	−7.08767e11	−	9.94918e12i	−0.0510036	−	0.715953i
$834$		0			0
$835$		−	2.17088e12i		−	0.154542i
$836$		0			0
$837$	−9.85128e12			−0.693790
$838$		0			0
$839$	2.39028e13			1.66540			0.832702	−	0.553721i	$-0.186793\pi$
$839$	2.39028e13			1.66540			0.832702	+	0.553721i	$0.186793\pi$
$840$		0			0
$841$	−1.22096e13			−0.841627
$842$		0			0
$843$	−1.82920e12			−0.124749
$844$		0			0
$845$			1.67742e12i			0.113185i
$846$		0			0
$847$	−7.59620e12	+	2.70229e11i	−0.507132	+	0.0180409i
$848$		0			0
$849$	1.26773e13			0.837416
$850$		0			0
$851$			9.49703e12i			0.620733i
$852$		0			0
$853$			1.07468e13i			0.695038i	0.937673	+	0.347519i	$0.112976\pi$
$853$			1.07468e13i			0.695038i	−0.937673	+	0.347519i	$0.887024\pi$
$854$		0			0
$855$		−	1.19366e12i		−	0.0763893i
$856$		0			0
$857$			1.91474e12i			0.121254i	0.998160	+	0.0606270i	$0.0193100\pi$
$857$			1.91474e12i			0.121254i	−0.998160	+	0.0606270i	$0.980690\pi$
$858$		0			0
$859$	−2.48486e13			−1.55716			−0.778578	−	0.627548i	$-0.784059\pi$
$859$	−2.48486e13			−1.55716			−0.778578	+	0.627548i	$0.784059\pi$
$860$		0			0
$861$	3.15143e11	+	8.85872e12i	0.0195431	+	0.549360i
$862$		0			0
$863$		−	6.46899e12i		−	0.396997i	−0.980101	−	0.198499i	$-0.936393\pi$
$863$		−	6.46899e12i		−	0.396997i	0.980101	−	0.198499i	$-0.0636066\pi$
$864$		0			0
$865$	−7.78389e11			−0.0472742
$866$		0			0
$867$	3.69092e12			0.221844
$868$		0			0
$869$	1.41657e13			0.842656
$870$		0			0
$871$	−1.09940e13			−0.647254
$872$		0			0
$873$			1.66376e13i			0.969452i
$874$		0			0
$875$	−2.48518e11	−	6.98589e12i	−0.0143325	−	0.402889i
$876$		0			0
$877$	−2.20170e13			−1.25678			−0.628392	−	0.777897i	$-0.716286\pi$
$877$	−2.20170e13			−1.25678			−0.628392	+	0.777897i	$0.716286\pi$
$878$		0			0
$879$		−	7.16884e12i		−	0.405041i
$880$		0			0
$881$		−	1.93564e13i		−	1.08251i	−0.840858	−	0.541256i	$-0.817949\pi$
$881$		−	1.93564e13i		−	1.08251i	0.840858	−	0.541256i	$-0.182051\pi$
$882$		0			0
$883$		−	1.42982e13i		−	0.791514i	−0.918355	−	0.395757i	$-0.870482\pi$
$883$		−	1.42982e13i		−	0.791514i	0.918355	−	0.395757i	$-0.129518\pi$
$884$		0			0
$885$			1.00218e12i			0.0549161i
$886$		0			0
$887$	−2.49671e12			−0.135429			−0.0677144	−	0.997705i	$-0.521571\pi$
$887$	−2.49671e12			−0.135429			−0.0677144	+	0.997705i	$0.521571\pi$
$888$		0			0
$889$	2.35513e9	+	6.62032e10i	0.000126461	+	0.00355485i
$890$		0			0
$891$			5.48817e12i			0.291728i
$892$		0			0
$893$	4.29357e12			0.225937
$894$		0			0
$895$	7.68362e11			0.0400278
$896$		0			0
$897$	8.90533e12			0.459287
$898$		0			0
$899$	6.59946e12			0.336969
$900$		0			0
$901$			1.81264e13i			0.916327i
$902$		0			0
$903$	−7.65936e10	−	2.15306e12i	−0.00383352	−	0.107761i
$904$		0			0
$905$	−6.99344e12			−0.346555
$906$		0			0
$907$			2.22910e13i			1.09369i	0.837232	+	0.546847i	$0.184172\pi$
$907$			2.22910e13i			1.09369i	−0.837232	+	0.546847i	$0.815828\pi$
$908$		0			0
$909$			2.30682e13i			1.12066i
$910$		0			0
$911$			2.49419e13i			1.19977i	0.800087	+	0.599884i	$0.204787\pi$
$911$			2.49419e13i			1.19977i	−0.800087	+	0.599884i	$0.795213\pi$
$912$		0			0
$913$			2.14856e13i			1.02336i
$914$		0			0
$915$	2.85356e12			0.134584
$916$		0			0
$917$	4.10223e10	−	1.45934e9i	0.00191583	−	6.81545e-5i
$918$		0			0
$919$		−	1.42887e13i		−	0.660803i	−0.943841	−	0.330401i	$-0.892816\pi$
$919$		−	1.42887e13i		−	0.660803i	0.943841	−	0.330401i	$-0.107184\pi$
$920$		0			0
$921$	5.42312e12			0.248360
$922$		0			0
$923$	−9.90650e12			−0.449275
$924$		0			0
$925$	−8.84938e12			−0.397443
$926$		0			0
$927$	−2.46415e13			−1.09599
$928$		0			0
$929$			1.57456e13i			0.693568i	0.937945	+	0.346784i	$0.112726\pi$
$929$			1.57456e13i			0.693568i	−0.937945	+	0.346784i	$0.887274\pi$
$930$		0			0
$931$	1.07277e13	−	7.64226e11i	0.467985	−	0.0333387i
$932$		0			0
$933$	−1.53647e13			−0.663829
$934$		0			0
$935$			2.42437e12i			0.103740i
$936$		0			0
$937$			3.54742e13i			1.50344i	0.659485	+	0.751718i	$0.270774\pi$
$937$			3.54742e13i			1.50344i	−0.659485	+	0.751718i	$0.729226\pi$
$938$		0			0
$939$			3.63360e12i			0.152525i
$940$		0			0
$941$		−	6.19787e11i		−	0.0257685i	−0.999917	−	0.0128843i	$-0.995899\pi$
$941$		−	6.19787e11i		−	0.0257685i	0.999917	−	0.0128843i	$-0.00410130\pi$
$942$		0			0
$943$	−4.36280e13			−1.79665
$944$		0			0
$945$	1.47069e11	+	4.13415e12i	0.00599900	+	0.168633i
$946$		0			0
$947$			2.04685e13i			0.827010i	0.910502	+	0.413505i	$0.135696\pi$
$947$			2.04685e13i			0.827010i	−0.910502	+	0.413505i	$0.864304\pi$
$948$		0			0
$949$	2.30801e13			0.923717
$950$		0			0
$951$	−1.35313e13			−0.536449
$952$		0			0
$953$	−9.58358e12			−0.376366			−0.188183	−	0.982134i	$-0.560260\pi$
$953$	−9.58358e12			−0.376366			−0.188183	+	0.982134i	$0.560260\pi$
$954$		0			0
$955$	−3.40373e10			−0.00132416
$956$		0			0
$957$			3.31623e12i			0.127803i
$958$		0			0
$959$	−2.30786e13	+	8.21005e11i	−0.881101	+	0.0313446i
$960$		0			0
$961$	−7.48328e12			−0.283033
$962$		0			0
$963$		−	2.79234e13i		−	1.04629i
$964$		0			0
$965$		−	6.82199e12i		−	0.253244i
$966$		0			0
$967$			1.42925e13i			0.525641i	0.964845	+	0.262820i	$0.0846526\pi$
$967$			1.42925e13i			0.525641i	−0.964845	+	0.262820i	$0.915347\pi$
$968$		0			0
$969$		−	4.22912e12i		−	0.154096i
$970$		0			0
$971$	3.25124e13			1.17371			0.586856	−	0.809691i	$-0.300365\pi$
$971$	3.25124e13			1.17371			0.586856	+	0.809691i	$0.300365\pi$
$972$		0			0
$973$	1.57895e13	−	5.61700e11i	0.564755	−	0.0200908i
$974$		0			0
$975$			8.29804e12i			0.294073i
$976$		0			0
$977$	−2.86797e13			−1.00704			−0.503522	−	0.863982i	$-0.667963\pi$
$977$	−2.86797e13			−1.00704			−0.503522	+	0.863982i	$0.667963\pi$
$978$		0			0
$979$	2.02993e13			0.706252
$980$		0			0
$981$	1.01273e13			0.349125
$982$		0			0
$983$	4.65199e12			0.158909			0.0794545	−	0.996838i	$-0.474682\pi$
$983$	4.65199e12			0.158909			0.0794545	+	0.996838i	$0.474682\pi$
$984$		0			0
$985$		−	8.51615e11i		−	0.0288257i
$986$		0			0
$987$	−6.56578e12	+	2.33573e11i	−0.220221	+	0.00783421i
$988$		0			0
$989$	1.06035e13			0.352426
$990$		0			0
$991$		−	5.56778e13i		−	1.83380i	−0.399122	−	0.916898i	$-0.630685\pi$
$991$		−	5.56778e13i		−	1.83380i	0.399122	−	0.916898i	$-0.369315\pi$
$992$		0			0
$993$		−	1.63473e11i		−	0.00533548i
$994$		0			0
$995$		−	1.98639e12i		−	0.0642480i
$996$		0			0
$997$			4.00732e13i			1.28448i	0.766505	+	0.642238i	$0.221994\pi$
$997$			4.00732e13i			1.28448i	−0.766505	+	0.642238i	$0.778006\pi$
$998$		0			0
$999$	1.07058e13			0.340075

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.10.f.b.111.15	yes	24
4.3	odd	2	inner	112.10.f.b.111.9	✓	24
7.6	odd	2	inner	112.10.f.b.111.10	yes	24
28.27	even	2	inner	112.10.f.b.111.16	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.10.f.b.111.9	✓	24	4.3	odd	2	inner
112.10.f.b.111.10	yes	24	7.6	odd	2	inner
112.10.f.b.111.15	yes	24	1.1	even	1	trivial
112.10.f.b.111.16	yes	24	28.27	even	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)

\(f(q)\)	\(=\)	\(q+64.1982 q^{3} +287.809i q^{5} +(-6348.43 + 225.841i) q^{7} -15561.6 q^{9} +34079.3i q^{11} +69110.3i q^{13} +18476.8i q^{15} -247175. i q^{17} +266515. q^{19} +(-407558. + 14498.6i) q^{21} -2.00717e6i q^{23} +1.87029e6 q^{25} -2.26264e6 q^{27} +1.51576e6 q^{29} +4.35389e6 q^{31} +2.18783e6i q^{33} +(-64999.0 - 1.82713e6i) q^{35} -4.73155e6 q^{37} +4.43676e6i q^{39} -2.17361e7i q^{41} +5.28283e6i q^{43} -4.47876e6i q^{45} +1.61100e7 q^{47} +(4.02516e7 - 2.86747e6i) q^{49} -1.58682e7i q^{51} -7.33344e7 q^{53} -9.80832e6 q^{55} +1.71098e7 q^{57} +5.42398e7 q^{59} -1.54440e8i q^{61} +(9.87917e7 - 3.51444e6i) q^{63} -1.98905e7 q^{65} +1.59079e8i q^{67} -1.28857e8i q^{69} +1.43343e8i q^{71} -3.33960e8i q^{73} +1.20069e8 q^{75} +(-7.69651e6 - 2.16350e8i) q^{77} -4.15669e8i q^{79} +1.61041e8 q^{81} +6.30458e8 q^{83} +7.11390e7 q^{85} +9.73092e7 q^{87} -5.95650e8i q^{89} +(-1.56079e7 - 4.38742e8i) q^{91} +2.79512e8 q^{93} +7.67054e7i q^{95} -1.06914e9i q^{97} -5.30328e8i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 155768 q^{9} + 220672 q^{21} - 9187656 q^{25} + 14881104 q^{29} + 2829456 q^{37} - 214802472 q^{49} + 327087120 q^{53} + 238245440 q^{57} - 495797952 q^{65} + 347010000 q^{77} + 1816013720 q^{81}+ \cdots + 288442240 q^{93}+O(q^{100}) \) 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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