LMFDB - Embedded newform 112.10.f.b.111.4

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			111.4
Character	$\chi$	$=$	112.111
Dual form			112.10.f.b.111.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-170.771 q^{3} +701.626i q^{5} +(911.052 + 6286.78i) q^{7} +9479.76 q^{9} +68199.0i q^{11} +639.722i q^{13} -119817. i q^{15} +293169. i q^{17} +957083. q^{19} +(-155581. - 1.07360e6i) q^{21} +344952. i q^{23} +1.46085e6 q^{25} +1.74242e6 q^{27} -4.41859e6 q^{29} +8.60318e6 q^{31} -1.16464e7i q^{33} +(-4.41097e6 + 639217. i) q^{35} +6.37031e6 q^{37} -109246. i q^{39} +3.94558e6i q^{41} +3.24456e7i q^{43} +6.65124e6i q^{45} -2.56748e6 q^{47} +(-3.86936e7 + 1.14552e7i) q^{49} -5.00647e7i q^{51} -796252. q^{53} -4.78502e7 q^{55} -1.63442e8 q^{57} -1.06324e8 q^{59} +1.27001e8i q^{61} +(8.63655e6 + 5.95972e7i) q^{63} -448846. q^{65} +1.97960e8i q^{67} -5.89079e7i q^{69} +1.30446e8i q^{71} -2.96623e8i q^{73} -2.49470e8 q^{75} +(-4.28752e8 + 6.21328e7i) q^{77} +1.96821e8i q^{79} -4.84145e8 q^{81} +4.73757e8 q^{83} -2.05695e8 q^{85} +7.54568e8 q^{87} -8.79838e8i q^{89} +(-4.02179e6 + 582820. i) q^{91} -1.46917e9 q^{93} +6.71514e8i q^{95} +6.60272e7i q^{97} +6.46510e8i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−170.771			−1.21722			−0.608609	−	0.793470i	$-0.708272\pi$
$3$	−170.771			−1.21722			−0.608609	+	0.793470i	$0.708272\pi$
$4$		0			0
$5$			701.626i			0.502043i	0.967982	+	0.251021i	$0.0807664\pi$
$5$			701.626i			0.502043i	−0.967982	+	0.251021i	$0.919234\pi$
$6$		0			0
$7$	911.052	+	6286.78i	0.143417	+	0.989662i
$7$	911.052	+	6286.78i	0.143417	+	0.989662i
$8$		0			0
$9$	9479.76			0.481622
$10$		0			0
$11$			68199.0i			1.40447i	0.711947	+	0.702233i	$0.247813\pi$
$11$			68199.0i			1.40447i	−0.711947	+	0.702233i	$0.752187\pi$
$12$		0			0
$13$			639.722i			0.00621222i	0.999995	+	0.00310611i	$0.000988706\pi$
$13$			639.722i			0.00621222i	−0.999995	+	0.00310611i	$0.999011\pi$
$14$		0			0
$15$		−	119817.i		−	0.611096i
$16$		0			0
$17$			293169.i			0.851329i	0.904881	+	0.425665i	$0.139960\pi$
$17$			293169.i			0.851329i	−0.904881	+	0.425665i	$0.860040\pi$
$18$		0			0
$19$	957083.			1.68484			0.842420	−	0.538822i	$-0.181130\pi$
$19$	957083.			1.68484			0.842420	+	0.538822i	$0.181130\pi$
$20$		0			0
$21$	−155581.	−	1.07360e6i	−0.174570	−	1.20464i
$22$		0			0
$23$			344952.i			0.257030i	0.991708	+	0.128515i	$0.0410211\pi$
$23$			344952.i			0.257030i	−0.991708	+	0.128515i	$0.958979\pi$
$24$		0			0
$25$	1.46085e6			0.747953
$26$		0			0
$27$	1.74242e6			0.630980
$28$		0			0
$29$	−4.41859e6			−1.16009			−0.580047	−	0.814583i	$-0.696966\pi$
$29$	−4.41859e6			−1.16009			−0.580047	+	0.814583i	$0.696966\pi$
$30$		0			0
$31$	8.60318e6			1.67314			0.836568	−	0.547863i	$-0.184559\pi$
$31$	8.60318e6			1.67314			0.836568	+	0.547863i	$0.184559\pi$
$32$		0			0
$33$		−	1.16464e7i		−	1.70954i
$34$		0			0
$35$	−4.41097e6	+	639217.i	−0.496853	+	0.0720016i
$36$		0			0
$37$	6.37031e6			0.558796			0.279398	−	0.960175i	$-0.409865\pi$
$37$	6.37031e6			0.558796			0.279398	+	0.960175i	$0.409865\pi$
$38$		0			0
$39$		−	109246.i		−	0.00756163i
$40$		0			0
$41$			3.94558e6i			0.218064i	0.994038	+	0.109032i	$0.0347750\pi$
$41$			3.94558e6i			0.218064i	−0.994038	+	0.109032i	$0.965225\pi$
$42$		0			0
$43$			3.24456e7i			1.44726i	0.690187	+	0.723631i	$0.257528\pi$
$43$			3.24456e7i			1.44726i	−0.690187	+	0.723631i	$0.742472\pi$
$44$		0			0
$45$			6.65124e6i			0.241795i
$46$		0			0
$47$	−2.56748e6			−0.0767479			−0.0383739	−	0.999263i	$-0.512218\pi$
$47$	−2.56748e6			−0.0767479			−0.0383739	+	0.999263i	$0.512218\pi$
$48$		0			0
$49$	−3.86936e7	+	1.14552e7i	−0.958863	+	0.283870i
$50$		0			0
$51$		−	5.00647e7i		−	1.03625i
$52$		0			0
$53$	−796252.			−0.0138615			−0.00693073	−	0.999976i	$-0.502206\pi$
$53$	−796252.			−0.0138615			−0.00693073	+	0.999976i	$0.502206\pi$
$54$		0			0
$55$	−4.78502e7			−0.705102
$56$		0			0
$57$	−1.63442e8			−2.05082
$58$		0			0
$59$	−1.06324e8			−1.14235			−0.571173	−	0.820830i	$-0.693511\pi$
$59$	−1.06324e8			−1.14235			−0.571173	+	0.820830i	$0.693511\pi$
$60$		0			0
$61$			1.27001e8i			1.17442i	0.809434	+	0.587211i	$0.199774\pi$
$61$			1.27001e8i			1.17442i	−0.809434	+	0.587211i	$0.800226\pi$
$62$		0			0
$63$	8.63655e6	+	5.95972e7i	0.0690729	+	0.476643i
$64$		0			0
$65$	−448846.			−0.00311880
$66$		0			0
$67$			1.97960e8i			1.20016i	0.799938	+	0.600082i	$0.204866\pi$
$67$			1.97960e8i			1.20016i	−0.799938	+	0.600082i	$0.795134\pi$
$68$		0			0
$69$		−	5.89079e7i		−	0.312862i
$70$		0			0
$71$			1.30446e8i			0.609210i	0.952479	+	0.304605i	$0.0985244\pi$
$71$			1.30446e8i			0.609210i	−0.952479	+	0.304605i	$0.901476\pi$
$72$		0			0
$73$		−	2.96623e8i		−	1.22251i	−0.791435	−	0.611253i	$-0.790666\pi$
$73$		−	2.96623e8i		−	1.22251i	0.791435	−	0.611253i	$-0.209334\pi$
$74$		0			0
$75$	−2.49470e8			−0.910423
$76$		0			0
$77$	−4.28752e8	+	6.21328e7i	−1.38995	+	0.201425i
$78$		0			0
$79$			1.96821e8i			0.568525i	0.958746	+	0.284263i	$0.0917487\pi$
$79$			1.96821e8i			0.568525i	−0.958746	+	0.284263i	$0.908251\pi$
$80$		0			0
$81$	−4.84145e8			−1.24966
$82$		0			0
$83$	4.73757e8			1.09573			0.547866	−	0.836566i	$-0.315440\pi$
$83$	4.73757e8			1.09573			0.547866	+	0.836566i	$0.315440\pi$
$84$		0			0
$85$	−2.05695e8			−0.427403
$86$		0			0
$87$	7.54568e8			1.41209
$88$		0			0
$89$		−	8.79838e8i		−	1.48644i	−0.669046	−	0.743221i	$-0.733297\pi$
$89$		−	8.79838e8i		−	1.48644i	0.669046	−	0.743221i	$-0.266703\pi$
$90$		0			0
$91$	−4.02179e6	+	582820.i	−0.00614800	+	0.000890940i
$92$		0			0
$93$	−1.46917e9			−2.03657
$94$		0			0
$95$			6.71514e8i			0.845861i
$96$		0			0
$97$			6.60272e7i			0.0757268i	0.999283	+	0.0378634i	$0.0120552\pi$
$97$			6.60272e7i			0.0757268i	−0.999283	+	0.0378634i	$0.987945\pi$
$98$		0			0
$99$			6.46510e8i			0.676421i
$100$		0			0
$101$			6.81994e8i			0.652130i	0.945347	+	0.326065i	$0.105723\pi$
$101$			6.81994e8i			0.652130i	−0.945347	+	0.326065i	$0.894277\pi$
$102$		0			0
$103$	−1.65757e7			−0.0145112			−0.00725561	−	0.999974i	$-0.502310\pi$
$103$	−1.65757e7			−0.0145112			−0.00725561	+	0.999974i	$0.502310\pi$
$104$		0			0
$105$	7.53266e8	−	1.09160e8i	0.604778	−	0.0876418i
$106$		0			0
$107$			1.33033e9i			0.981145i	0.871400	+	0.490573i	$0.163212\pi$
$107$			1.33033e9i			0.981145i	−0.871400	+	0.490573i	$0.836788\pi$
$108$		0			0
$109$	−1.56148e9			−1.05954			−0.529769	−	0.848142i	$-0.677721\pi$
$109$	−1.56148e9			−1.05954			−0.529769	+	0.848142i	$0.677721\pi$
$110$		0			0
$111$	−1.08787e9			−0.680177
$112$		0			0
$113$	−5.52232e8			−0.318617			−0.159308	−	0.987229i	$-0.550926\pi$
$113$	−5.52232e8			−0.318617			−0.159308	+	0.987229i	$0.550926\pi$
$114$		0			0
$115$	−2.42028e8			−0.129040
$116$		0			0
$117$			6.06441e6i			0.00299194i
$118$		0			0
$119$	−1.84309e9	+	2.67092e8i	−0.842528	+	0.122095i
$120$		0			0
$121$	−2.29316e9			−0.972524
$122$		0			0
$123$		−	6.73791e8i		−	0.265431i
$124$		0			0
$125$			2.39533e9i			0.877547i
$126$		0			0
$127$		−	2.51743e9i		−	0.858699i	−0.903138	−	0.429350i	$-0.858743\pi$
$127$		−	2.51743e9i		−	0.858699i	0.903138	−	0.429350i	$-0.141257\pi$
$128$		0			0
$129$		−	5.54076e9i		−	1.76163i
$130$		0			0
$131$	−6.03017e9			−1.78899			−0.894496	−	0.447075i	$-0.852466\pi$
$131$	−6.03017e9			−1.78899			−0.894496	+	0.447075i	$0.852466\pi$
$132$		0			0
$133$	8.71952e8	+	6.01697e9i	0.241635	+	1.66742i
$134$		0			0
$135$			1.22253e9i			0.316779i
$136$		0			0
$137$	1.86703e9			0.452801			0.226401	−	0.974034i	$-0.427304\pi$
$137$	1.86703e9			0.452801			0.226401	+	0.974034i	$0.427304\pi$
$138$		0			0
$139$	4.58448e9			1.04165			0.520827	−	0.853662i	$-0.325624\pi$
$139$	4.58448e9			1.04165			0.520827	+	0.853662i	$0.325624\pi$
$140$		0			0
$141$	4.38451e8			0.0934190
$142$		0			0
$143$	−4.36285e7			−0.00872484
$144$		0			0
$145$		−	3.10020e9i		−	0.582416i
$146$		0			0
$147$	6.60774e9	−	1.95621e9i	1.16715	−	0.345531i
$148$		0			0
$149$	1.05819e10			1.75884			0.879420	−	0.476047i	$-0.157931\pi$
$149$	1.05819e10			1.75884			0.879420	+	0.476047i	$0.157931\pi$
$150$		0			0
$151$		−	6.06779e9i		−	0.949804i	−0.880039	−	0.474902i	$-0.842484\pi$
$151$		−	6.06779e9i		−	0.949804i	0.880039	−	0.474902i	$-0.157516\pi$
$152$		0			0
$153$			2.77917e9i			0.410019i
$154$		0			0
$155$			6.03621e9i			0.839986i
$156$		0			0
$157$			1.30695e9i			0.171676i	0.996309	+	0.0858382i	$0.0273568\pi$
$157$			1.30695e9i			0.171676i	−0.996309	+	0.0858382i	$0.972643\pi$
$158$		0			0
$159$	1.35977e8			0.0168724
$160$		0			0
$161$	−2.16864e9	+	3.14270e8i	−0.254373	+	0.0368626i
$162$		0			0
$163$		−	9.52503e9i		−	1.05687i	−0.848973	−	0.528436i	$-0.822779\pi$
$163$		−	9.52503e9i		−	1.05687i	0.848973	−	0.528436i	$-0.177221\pi$
$164$		0			0
$165$	8.17143e9			0.858263
$166$		0			0
$167$	1.03941e10			1.03410			0.517051	−	0.855955i	$-0.327030\pi$
$167$	1.03941e10			1.03410			0.517051	+	0.855955i	$0.327030\pi$
$168$		0			0
$169$	1.06041e10			0.999961
$170$		0			0
$171$	9.07292e9			0.811455
$172$		0			0
$173$		−	2.11492e10i		−	1.79509i	−0.440926	−	0.897544i	$-0.645350\pi$
$173$		−	2.11492e10i		−	1.79509i	0.440926	−	0.897544i	$-0.354650\pi$
$174$		0			0
$175$	1.33091e9	+	9.18402e9i	0.107269	+	0.740221i
$176$		0			0
$177$	1.81571e10			1.39048
$178$		0			0
$179$		−	7.49003e8i		−	0.0545312i	−0.999628	−	0.0272656i	$-0.991320\pi$
$179$		−	7.49003e8i		−	0.0545312i	0.999628	−	0.0272656i	$-0.00867999\pi$
$180$		0			0
$181$			1.25305e10i			0.867788i	0.900964	+	0.433894i	$0.142861\pi$
$181$			1.25305e10i			0.867788i	−0.900964	+	0.433894i	$0.857139\pi$
$182$		0			0
$183$		−	2.16881e10i		−	1.42953i
$184$		0			0
$185$			4.46958e9i			0.280539i
$186$		0			0
$187$	−1.99938e10			−1.19566
$188$		0			0
$189$	1.58743e9	+	1.09542e10i	0.0904935	+	0.624457i
$190$		0			0
$191$		−	2.26420e10i		−	1.23102i	−0.788129	−	0.615511i	$-0.788950\pi$
$191$		−	2.26420e10i		−	1.23102i	0.788129	−	0.615511i	$-0.211050\pi$
$192$		0			0
$193$	1.12574e10			0.584022			0.292011	−	0.956415i	$-0.405676\pi$
$193$	1.12574e10			0.584022			0.292011	+	0.956415i	$0.405676\pi$
$194$		0			0
$195$	7.66499e7			0.00379626
$196$		0			0
$197$	9.12718e8			0.0431756			0.0215878	−	0.999767i	$-0.493128\pi$
$197$	9.12718e8			0.0431756			0.0215878	+	0.999767i	$0.493128\pi$
$198$		0			0
$199$	−1.38094e10			−0.624215			−0.312108	−	0.950047i	$-0.601035\pi$
$199$	−1.38094e10			−0.624215			−0.312108	+	0.950047i	$0.601035\pi$
$200$		0			0
$201$		−	3.38058e10i		−	1.46086i
$202$		0			0
$203$	−4.02557e9	−	2.77787e10i	−0.166378	−	1.14810i
$204$		0			0
$205$	−2.76832e9			−0.109477
$206$		0			0
$207$			3.27007e9i			0.123791i
$208$		0			0
$209$			6.52722e10i			2.36630i
$210$		0			0
$211$			3.87757e10i			1.34676i	0.739299	+	0.673378i	$0.235157\pi$
$211$			3.87757e10i			1.34676i	−0.739299	+	0.673378i	$0.764843\pi$
$212$		0			0
$213$		−	2.22763e10i		−	0.741542i
$214$		0			0
$215$	−2.27646e10			−0.726587
$216$		0			0
$217$	7.83794e9	+	5.40863e10i	0.239957	+	1.65584i
$218$		0			0
$219$			5.06546e10i			1.48806i
$220$		0			0
$221$	−1.87547e8			−0.00528864
$222$		0			0
$223$	2.88746e9			0.0781888			0.0390944	−	0.999236i	$-0.487553\pi$
$223$	2.88746e9			0.0781888			0.0390944	+	0.999236i	$0.487553\pi$
$224$		0			0
$225$	1.38485e10			0.360230
$226$		0			0
$227$	−6.56365e10			−1.64070			−0.820349	−	0.571863i	$-0.806221\pi$
$227$	−6.56365e10			−1.64070			−0.820349	+	0.571863i	$0.806221\pi$
$228$		0			0
$229$		−	3.62721e10i		−	0.871591i	−0.900046	−	0.435795i	$-0.856467\pi$
$229$		−	3.62721e10i		−	0.871591i	0.900046	−	0.435795i	$-0.143533\pi$
$230$		0			0
$231$	7.32185e10	−	1.06105e10i	1.69187	−	0.245178i
$232$		0			0
$233$	−8.11267e10			−1.80328			−0.901638	−	0.432492i	$-0.857634\pi$
$233$	−8.11267e10			−1.80328			−0.901638	+	0.432492i	$0.857634\pi$
$234$		0			0
$235$		−	1.80141e9i		−	0.0385307i
$236$		0			0
$237$		−	3.36113e10i		−	0.692019i
$238$		0			0
$239$		−	4.06555e10i		−	0.805989i	−0.915202	−	0.402995i	$-0.867969\pi$
$239$		−	4.06555e10i		−	0.805989i	0.915202	−	0.402995i	$-0.132031\pi$
$240$		0			0
$241$		−	3.59022e9i		−	0.0685558i	−0.999412	−	0.0342779i	$-0.989087\pi$
$241$		−	3.59022e9i		−	0.0685558i	0.999412	−	0.0342779i	$-0.0109131\pi$
$242$		0			0
$243$	4.83819e10			0.890133
$244$		0			0
$245$	−8.03724e9	−	2.71484e10i	−0.142515	−	0.481390i
$246$		0			0
$247$			6.12268e8i			0.0104666i
$248$		0			0
$249$	−8.09040e10			−1.33375
$250$		0			0
$251$	4.32066e10			0.687098			0.343549	−	0.939135i	$-0.388371\pi$
$251$	4.32066e10			0.687098			0.343549	+	0.939135i	$0.388371\pi$
$252$		0			0
$253$	−2.35254e10			−0.360990
$254$		0			0
$255$	3.51267e10			0.520244
$256$		0			0
$257$		−	6.66559e10i		−	0.953102i	−0.879147	−	0.476551i	$-0.841887\pi$
$257$		−	6.66559e10i		−	0.953102i	0.879147	−	0.476551i	$-0.158113\pi$
$258$		0			0
$259$	5.80369e9	+	4.00488e10i	0.0801410	+	0.553019i
$260$		0			0
$261$	−4.18872e10			−0.558726
$262$		0			0
$263$			3.42721e10i			0.441712i	0.975306	+	0.220856i	$0.0708851\pi$
$263$			3.42721e10i			0.441712i	−0.975306	+	0.220856i	$0.929115\pi$
$264$		0			0
$265$		−	5.58671e8i		−	0.00695904i
$266$		0			0
$267$			1.50251e11i			1.80932i
$268$		0			0
$269$		−	2.48906e10i		−	0.289835i	−0.989444	−	0.144917i	$-0.953708\pi$
$269$		−	2.48906e10i		−	0.289835i	0.989444	−	0.144917i	$-0.0462916\pi$
$270$		0			0
$271$	−1.35903e11			−1.53062			−0.765310	−	0.643662i	$-0.777414\pi$
$271$	−1.35903e11			−1.53062			−0.765310	+	0.643662i	$0.777414\pi$
$272$		0			0
$273$	6.86806e8	−	9.95288e7i	0.00748346	−	0.00108447i
$274$		0			0
$275$			9.96283e10i			1.05047i
$276$		0			0
$277$	−5.95160e10			−0.607400			−0.303700	−	0.952768i	$-0.598222\pi$
$277$	−5.95160e10			−0.607400			−0.303700	+	0.952768i	$0.598222\pi$
$278$		0			0
$279$	8.15560e10			0.805819
$280$		0			0
$281$	−7.40279e10			−0.708300			−0.354150	−	0.935189i	$-0.615230\pi$
$281$	−7.40279e10			−0.708300			−0.354150	+	0.935189i	$0.615230\pi$
$282$		0			0
$283$	2.24712e10			0.208251			0.104126	−	0.994564i	$-0.466796\pi$
$283$	2.24712e10			0.208251			0.104126	+	0.994564i	$0.466796\pi$
$284$		0			0
$285$		−	1.14675e11i		−	1.02960i
$286$		0			0
$287$	−2.48050e10	+	3.59463e9i	−0.215809	+	0.0312741i
$288$		0			0
$289$	3.26400e10			0.275239
$290$		0			0
$291$		−	1.12755e10i		−	0.0921761i
$292$		0			0
$293$			2.19592e11i			1.74065i	0.492474	+	0.870327i	$0.336093\pi$
$293$			2.19592e11i			1.74065i	−0.492474	+	0.870327i	$0.663907\pi$
$294$		0			0
$295$		−	7.45997e10i		−	0.573506i
$296$		0			0
$297$			1.18831e11i			0.886190i
$298$		0			0
$299$	−2.20674e8			−0.00159673
$300$		0			0
$301$	−2.03978e11	+	2.95596e10i	−1.43230	+	0.207563i
$302$		0			0
$303$		−	1.16465e11i		−	0.793785i
$304$		0			0
$305$	−8.91074e10			−0.589610
$306$		0			0
$307$	−2.09786e11			−1.34789			−0.673943	−	0.738783i	$-0.735401\pi$
$307$	−2.09786e11			−1.34789			−0.673943	+	0.738783i	$0.735401\pi$
$308$		0			0
$309$	2.83065e9			0.0176633
$310$		0			0
$311$	−1.56756e11			−0.950174			−0.475087	−	0.879939i	$-0.657583\pi$
$311$	−1.56756e11			−0.950174			−0.475087	+	0.879939i	$0.657583\pi$
$312$		0			0
$313$			5.89917e10i			0.347409i	0.984798	+	0.173705i	$0.0555738\pi$
$313$			5.89917e10i			0.347409i	−0.984798	+	0.173705i	$0.944426\pi$
$314$		0			0
$315$	−4.18149e10	+	6.05963e9i	−0.239295	+	0.0346776i
$316$		0			0
$317$	2.70184e11			1.50277			0.751387	−	0.659862i	$-0.229385\pi$
$317$	2.70184e11			1.50277			0.751387	+	0.659862i	$0.229385\pi$
$318$		0			0
$319$		−	3.01344e11i		−	1.62931i
$320$		0			0
$321$		−	2.27182e11i		−	1.19427i
$322$		0			0
$323$			2.80587e11i			1.43435i
$324$		0			0
$325$			9.34536e8i			0.00464645i
$326$		0			0
$327$	2.66655e11			1.28969
$328$		0			0
$329$	−2.33911e9	−	1.61412e10i	−0.0110070	−	0.0759545i
$330$		0			0
$331$			1.26896e11i			0.581060i	0.956866	+	0.290530i	$0.0938316\pi$
$331$			1.26896e11i			0.581060i	−0.956866	+	0.290530i	$0.906168\pi$
$332$		0			0
$333$	6.03890e10			0.269128
$334$		0			0
$335$	−1.38894e11			−0.602534
$336$		0			0
$337$	1.66309e11			0.702396			0.351198	−	0.936301i	$-0.385774\pi$
$337$	1.66309e11			0.702396			0.351198	+	0.936301i	$0.385774\pi$
$338$		0			0
$339$	9.43053e10			0.387826
$340$		0			0
$341$			5.86728e11i			2.34986i
$342$		0			0
$343$	−1.07268e11	−	2.32822e11i	−0.418453	−	0.908239i
$344$		0			0
$345$	4.13313e10			0.157070
$346$		0			0
$347$			1.03524e11i			0.383315i	0.981462	+	0.191658i	$0.0613863\pi$
$347$			1.03524e11i			0.383315i	−0.981462	+	0.191658i	$0.938614\pi$
$348$		0			0
$349$		−	3.39536e11i		−	1.22510i	−0.790432	−	0.612550i	$-0.790144\pi$
$349$		−	3.39536e11i		−	1.22510i	0.790432	−	0.612550i	$-0.209856\pi$
$350$		0			0
$351$			1.11466e9i			0.00391978i
$352$		0			0
$353$		−	4.61304e11i		−	1.58125i	−0.612299	−	0.790626i	$-0.709755\pi$
$353$		−	4.61304e11i		−	1.58125i	0.612299	−	0.790626i	$-0.290245\pi$
$354$		0			0
$355$	−9.15240e10			−0.305849
$356$		0			0
$357$	3.14746e11	−	4.56116e10i	1.02554	−	0.148617i
$358$		0			0
$359$		−	4.07793e11i		−	1.29573i	−0.761755	−	0.647865i	$-0.775662\pi$
$359$		−	4.07793e11i		−	1.29573i	0.761755	−	0.647865i	$-0.224338\pi$
$360$		0			0
$361$	5.93321e11			1.83868
$362$		0			0
$363$	3.91606e11			1.18377
$364$		0			0
$365$	2.08118e11			0.613751
$366$		0			0
$367$	6.42802e11			1.84961			0.924804	−	0.380444i	$-0.124229\pi$
$367$	6.42802e11			1.84961			0.924804	+	0.380444i	$0.124229\pi$
$368$		0			0
$369$			3.74031e10i			0.105024i
$370$		0			0
$371$	−7.25426e8	−	5.00586e9i	−0.00198797	−	0.0137182i
$372$		0			0
$373$	−1.08459e11			−0.290120			−0.145060	−	0.989423i	$-0.546337\pi$
$373$	−1.08459e11			−0.290120			−0.145060	+	0.989423i	$0.546337\pi$
$374$		0			0
$375$		−	4.09053e11i		−	1.06817i
$376$		0			0
$377$		−	2.82667e9i		−	0.00720675i
$378$		0			0
$379$			2.41986e10i			0.0602441i	0.999546	+	0.0301220i	$0.00958960\pi$
$379$			2.41986e10i			0.0602441i	−0.999546	+	0.0301220i	$0.990410\pi$
$380$		0			0
$381$			4.29905e11i			1.04522i
$382$		0			0
$383$	3.27819e11			0.778467			0.389233	−	0.921139i	$-0.372740\pi$
$383$	3.27819e11			0.778467			0.389233	+	0.921139i	$0.372740\pi$
$384$		0			0
$385$	−4.35940e10	−	3.00824e11i	−0.101124	−	0.697813i
$386$		0			0
$387$			3.07576e11i			0.697033i
$388$		0			0
$389$	−3.53885e11			−0.783590			−0.391795	−	0.920053i	$-0.628146\pi$
$389$	−3.53885e11			−0.783590			−0.391795	+	0.920053i	$0.628146\pi$
$390$		0			0
$391$	−1.01129e11			−0.218817
$392$		0			0
$393$	1.02978e12			2.17760
$394$		0			0
$395$	−1.38095e11			−0.285424
$396$		0			0
$397$			5.84709e11i			1.18136i	0.806906	+	0.590680i	$0.201140\pi$
$397$			5.84709e11i			1.18136i	−0.806906	+	0.590680i	$0.798860\pi$
$398$		0			0
$399$	−1.48904e11	−	1.02752e12i	−0.294123	−	2.02962i
$400$		0			0
$401$	9.52365e10			0.183930			0.0919652	−	0.995762i	$-0.470685\pi$
$401$	9.52365e10			0.183930			0.0919652	+	0.995762i	$0.470685\pi$
$402$		0			0
$403$			5.50365e9i			0.0103939i
$404$		0			0
$405$		−	3.39689e11i		−	0.627384i
$406$		0			0
$407$			4.34449e11i			0.784810i
$408$		0			0
$409$			5.81633e11i			1.02777i	0.857860	+	0.513883i	$0.171793\pi$
$409$			5.81633e11i			1.02777i	−0.857860	+	0.513883i	$0.828207\pi$
$410$		0			0
$411$	−3.18834e11			−0.551158
$412$		0			0
$413$	−9.68667e10	−	6.68436e11i	−0.163832	−	1.13054i
$414$		0			0
$415$			3.32400e11i			0.550104i
$416$		0			0
$417$	−7.82896e11			−1.26792
$418$		0			0
$419$	−3.53730e11			−0.560672			−0.280336	−	0.959902i	$-0.590446\pi$
$419$	−3.53730e11			−0.560672			−0.280336	+	0.959902i	$0.590446\pi$
$420$		0			0
$421$	5.79193e11			0.898575			0.449287	−	0.893387i	$-0.351678\pi$
$421$	5.79193e11			0.898575			0.449287	+	0.893387i	$0.351678\pi$
$422$		0			0
$423$	−2.43391e10			−0.0369634
$424$		0			0
$425$			4.28274e11i			0.636754i
$426$		0			0
$427$	−7.98429e11	+	1.15705e11i	−1.16228	+	0.168432i
$428$		0			0
$429$	7.45048e9			0.0106200
$430$		0			0
$431$		−	9.64808e11i		−	1.34677i	−0.739293	−	0.673384i	$-0.764840\pi$
$431$		−	9.64808e11i		−	1.34677i	0.739293	−	0.673384i	$-0.235160\pi$
$432$		0			0
$433$		−	9.35707e10i		−	0.127922i	−0.997952	−	0.0639608i	$-0.979627\pi$
$433$		−	9.35707e10i		−	0.127922i	0.997952	−	0.0639608i	$-0.0203733\pi$
$434$		0			0
$435$			5.29424e11i			0.708928i
$436$		0			0
$437$			3.30148e11i			0.433054i
$438$		0			0
$439$	5.66295e11			0.727700			0.363850	−	0.931458i	$-0.381462\pi$
$439$	5.66295e11			0.727700			0.363850	+	0.931458i	$0.381462\pi$
$440$		0			0
$441$	−3.66806e11	+	1.08592e11i	−0.461809	+	0.136718i
$442$		0			0
$443$			3.70442e11i			0.456987i	0.973545	+	0.228493i	$0.0733800\pi$
$443$			3.70442e11i			0.456987i	−0.973545	+	0.228493i	$0.926620\pi$
$444$		0			0
$445$	6.17317e11			0.746257
$446$		0			0
$447$	−1.80709e12			−2.14089
$448$		0			0
$449$	−8.37437e11			−0.972397			−0.486199	−	0.873848i	$-0.661617\pi$
$449$	−8.37437e11			−0.972397			−0.486199	+	0.873848i	$0.661617\pi$
$450$		0			0
$451$	−2.69085e11			−0.306263
$452$		0			0
$453$			1.03620e12i			1.15612i
$454$		0			0
$455$	−4.08922e8	−	2.82179e9i	−0.000447290	−	0.00308656i
$456$		0			0
$457$	−1.44000e11			−0.154433			−0.0772166	−	0.997014i	$-0.524603\pi$
$457$	−1.44000e11			−0.154433			−0.0772166	+	0.997014i	$0.524603\pi$
$458$		0			0
$459$			5.10823e11i			0.537172i
$460$		0			0
$461$		−	1.64706e12i		−	1.69846i	−0.528022	−	0.849231i	$-0.677066\pi$
$461$		−	1.64706e12i		−	1.69846i	0.528022	−	0.849231i	$-0.322934\pi$
$462$		0			0
$463$			1.32888e12i			1.34392i	0.740589	+	0.671958i	$0.234546\pi$
$463$			1.32888e12i			1.34392i	−0.740589	+	0.671958i	$0.765454\pi$
$464$		0			0
$465$		−	1.03081e12i		−	1.02245i
$466$		0			0
$467$	1.24795e12			1.21414			0.607072	−	0.794647i	$-0.292344\pi$
$467$	1.24795e12			1.21414			0.607072	+	0.794647i	$0.292344\pi$
$468$		0			0
$469$	−1.24453e12	+	1.80352e11i	−1.18776	+	0.172124i
$470$		0			0
$471$		−	2.23189e11i		−	0.208968i
$472$		0			0
$473$	−2.21276e12			−2.03263
$474$		0			0
$475$	1.39815e12			1.26018
$476$		0			0
$477$	−7.54827e9			−0.00667598
$478$		0			0
$479$	7.39354e11			0.641715			0.320858	−	0.947127i	$-0.396029\pi$
$479$	7.39354e11			0.641715			0.320858	+	0.947127i	$0.396029\pi$
$480$		0			0
$481$			4.07523e9i			0.00347136i
$482$		0			0
$483$	3.70341e11	−	5.36681e10i	0.309628	−	0.0448698i
$484$		0			0
$485$	−4.63264e10			−0.0380181
$486$		0			0
$487$		−	2.22309e12i		−	1.79092i	−0.445139	−	0.895462i	$-0.646846\pi$
$487$		−	2.22309e12i		−	1.79092i	0.445139	−	0.895462i	$-0.353154\pi$
$488$		0			0
$489$			1.62660e12i			1.28644i
$490$		0			0
$491$			2.34437e12i			1.82037i	0.414200	+	0.910186i	$0.364062\pi$
$491$			2.34437e12i			1.82037i	−0.414200	+	0.910186i	$0.635938\pi$
$492$		0			0
$493$		−	1.29539e12i		−	0.987621i
$494$		0			0
$495$	−4.53608e11			−0.339592
$496$		0			0
$497$	−8.20083e11	+	1.18843e11i	−0.602912	+	0.0873713i
$498$		0			0
$499$			6.06349e11i			0.437794i	0.975748	+	0.218897i	$0.0702459\pi$
$499$			6.06349e11i			0.437794i	−0.975748	+	0.218897i	$0.929754\pi$
$500$		0			0
$501$	−1.77501e12			−1.25873
$502$		0			0
$503$	1.04303e12			0.726512			0.363256	−	0.931689i	$-0.381665\pi$
$503$	1.04303e12			0.726512			0.363256	+	0.931689i	$0.381665\pi$
$504$		0			0
$505$	−4.78504e11			−0.327397
$506$		0			0
$507$	−1.81087e12			−1.21717
$508$		0			0
$509$		−	2.44040e12i		−	1.61150i	−0.592253	−	0.805752i	$-0.701762\pi$
$509$		−	2.44040e12i		−	1.61150i	0.592253	−	0.805752i	$-0.298238\pi$
$510$		0			0
$511$	1.86480e12	−	2.70238e11i	1.20987	−	0.175329i
$512$		0			0
$513$	1.66764e12			1.06310
$514$		0			0
$515$		−	1.16299e10i		−	0.00728525i
$516$		0			0
$517$		−	1.75100e11i		−	0.107790i
$518$		0			0
$519$			3.61166e12i			2.18501i
$520$		0			0
$521$			2.45154e12i			1.45770i	0.684671	+	0.728852i	$0.259946\pi$
$521$			2.45154e12i			1.45770i	−0.684671	+	0.728852i	$0.740054\pi$
$522$		0			0
$523$	1.43291e12			0.837454			0.418727	−	0.908112i	$-0.362476\pi$
$523$	1.43291e12			0.837454			0.418727	+	0.908112i	$0.362476\pi$
$524$		0			0
$525$	−2.27280e11	−	1.56836e12i	−0.130570	−	0.901011i
$526$		0			0
$527$			2.52218e12i			1.42439i
$528$		0			0
$529$	1.68216e12			0.933936
$530$		0			0
$531$	−1.00793e12			−0.550178
$532$		0			0
$533$	−2.52407e9			−0.00135466
$534$		0			0
$535$	−9.33396e11			−0.492577
$536$		0			0
$537$			1.27908e11i			0.0663764i
$538$		0			0
$539$	−7.81231e11	−	2.63886e12i	−0.398685	−	1.34669i
$540$		0			0
$541$	−1.31275e12			−0.658861			−0.329431	−	0.944180i	$-0.606857\pi$
$541$	−1.31275e12			−0.658861			−0.329431	+	0.944180i	$0.606857\pi$
$542$		0			0
$543$		−	2.13984e12i		−	1.05629i
$544$		0			0
$545$		−	1.09557e12i		−	0.531933i
$546$		0			0
$547$			9.83820e11i			0.469864i	0.972012	+	0.234932i	$0.0754868\pi$
$547$			9.83820e11i			0.469864i	−0.972012	+	0.234932i	$0.924513\pi$
$548$		0			0
$549$			1.20394e12i			0.565627i
$550$		0			0
$551$	−4.22896e12			−1.95457
$552$		0			0
$553$	−1.23737e12	+	1.79314e11i	−0.562648	+	0.0815364i
$554$		0			0
$555$		−	7.63275e11i		−	0.341478i
$556$		0			0
$557$	−3.38935e12			−1.49200			−0.745999	−	0.665947i	$-0.768028\pi$
$557$	−3.38935e12			−1.49200			−0.745999	+	0.665947i	$0.768028\pi$
$558$		0			0
$559$	−2.07561e10			−0.00899070
$560$		0			0
$561$	3.41437e12			1.45538
$562$		0			0
$563$	1.64016e12			0.688018			0.344009	−	0.938966i	$-0.388215\pi$
$563$	1.64016e12			0.688018			0.344009	+	0.938966i	$0.388215\pi$
$564$		0			0
$565$		−	3.87460e11i		−	0.159959i
$566$		0			0
$567$	−4.41081e11	−	3.04371e12i	−0.179223	−	1.23674i
$568$		0			0
$569$	−1.45892e12			−0.583482			−0.291741	−	0.956497i	$-0.594235\pi$
$569$	−1.45892e12			−0.583482			−0.291741	+	0.956497i	$0.594235\pi$
$570$		0			0
$571$		−	3.80118e12i		−	1.49643i	−0.663457	−	0.748215i	$-0.730911\pi$
$571$		−	3.80118e12i		−	1.49643i	0.663457	−	0.748215i	$-0.269089\pi$
$572$		0			0
$573$			3.86661e12i			1.49842i
$574$		0			0
$575$			5.03922e11i			0.192246i
$576$		0			0
$577$		−	3.15623e11i		−	0.118543i	−0.998242	−	0.0592717i	$-0.981122\pi$
$577$		−	3.15623e11i		−	0.118543i	0.998242	−	0.0592717i	$-0.0188778\pi$
$578$		0			0
$579$	−1.92243e12			−0.710883
$580$		0			0
$581$	4.31617e11	+	2.97841e12i	0.157147	+	1.08440i
$582$		0			0
$583$		−	5.43036e10i		−	0.0194679i
$584$		0			0
$585$	−4.25495e9			−0.00150208
$586$		0			0
$587$	8.15983e11			0.283667			0.141834	−	0.989890i	$-0.454700\pi$
$587$	8.15983e11			0.283667			0.141834	+	0.989890i	$0.454700\pi$
$588$		0			0
$589$	8.23396e12			2.81897
$590$		0			0
$591$	−1.55866e11			−0.0525542
$592$		0			0
$593$		−	5.57762e12i		−	1.85226i	−0.377202	−	0.926131i	$-0.623114\pi$
$593$		−	5.57762e12i		−	1.85226i	0.377202	−	0.926131i	$-0.376886\pi$
$594$		0			0
$595$	−1.87399e11	−	1.29316e12i	−0.0612971	−	0.422985i
$596$		0			0
$597$	2.35824e12			0.759807
$598$		0			0
$599$			4.79799e12i			1.52278i	0.648292	+	0.761392i	$0.275484\pi$
$599$			4.79799e12i			1.52278i	−0.648292	+	0.761392i	$0.724516\pi$
$600$		0			0
$601$			3.84252e12i			1.20138i	0.799481	+	0.600692i	$0.205108\pi$
$601$			3.84252e12i			1.20138i	−0.799481	+	0.600692i	$0.794892\pi$
$602$		0			0
$603$			1.87661e12i			0.578025i
$604$		0			0
$605$		−	1.60894e12i		−	0.488248i
$606$		0			0
$607$	1.11143e12			0.332303			0.166151	−	0.986100i	$-0.446866\pi$
$607$	1.11143e12			0.332303			0.166151	+	0.986100i	$0.446866\pi$
$608$		0			0
$609$	6.87450e11	+	4.74380e12i	0.202518	+	1.39749i
$610$		0			0
$611$		−	1.64247e9i		−	0.000476774i
$612$		0			0
$613$	3.23418e12			0.925108			0.462554	−	0.886591i	$-0.346933\pi$
$613$	3.23418e12			0.925108			0.462554	+	0.886591i	$0.346933\pi$
$614$		0			0
$615$	4.72749e11			0.133258
$616$		0			0
$617$	2.90100e12			0.805870			0.402935	−	0.915229i	$-0.367990\pi$
$617$	2.90100e12			0.805870			0.402935	+	0.915229i	$0.367990\pi$
$618$		0			0
$619$	−4.75827e12			−1.30269			−0.651346	−	0.758781i	$-0.725795\pi$
$619$	−4.75827e12			−1.30269			−0.651346	+	0.758781i	$0.725795\pi$
$620$		0			0
$621$			6.01052e11i			0.162181i
$622$		0			0
$623$	5.53135e12	−	8.01578e11i	1.47108	−	0.213182i
$624$		0			0
$625$	1.17259e12			0.307387
$626$		0			0
$627$		−	1.11466e13i		−	2.88030i
$628$		0			0
$629$			1.86758e12i			0.475719i
$630$		0			0
$631$			3.88629e12i			0.975894i	0.872873	+	0.487947i	$0.162254\pi$
$631$			3.88629e12i			0.975894i	−0.872873	+	0.487947i	$0.837746\pi$
$632$		0			0
$633$		−	6.62177e12i		−	1.63930i
$634$		0			0
$635$	1.76630e12			0.431104
$636$		0			0
$637$	−7.32812e9	−	2.47531e10i	−0.00176346	−	0.00595666i
$638$		0			0
$639$			1.23659e12i			0.293409i
$640$		0			0
$641$	−3.14655e12			−0.736161			−0.368081	−	0.929794i	$-0.619985\pi$
$641$	−3.14655e12			−0.736161			−0.368081	+	0.929794i	$0.619985\pi$
$642$		0			0
$643$	8.30067e12			1.91498			0.957489	−	0.288470i	$-0.0931464\pi$
$643$	8.30067e12			1.91498			0.957489	+	0.288470i	$0.0931464\pi$
$644$		0			0
$645$	3.88754e12			0.884416
$646$		0			0
$647$	4.77823e12			1.07201			0.536003	−	0.844216i	$-0.319933\pi$
$647$	4.77823e12			1.07201			0.536003	+	0.844216i	$0.319933\pi$
$648$		0			0
$649$		−	7.25120e12i		−	1.60438i
$650$		0			0
$651$	−1.33849e12	−	9.23637e12i	−0.292080	−	2.01552i
$652$		0			0
$653$	9.59424e11			0.206491			0.103245	−	0.994656i	$-0.467077\pi$
$653$	9.59424e11			0.206491			0.103245	+	0.994656i	$0.467077\pi$
$654$		0			0
$655$		−	4.23092e12i		−	0.898151i
$656$		0			0
$657$		−	2.81191e12i		−	0.588786i
$658$		0			0
$659$			9.04123e12i			1.86743i	0.358023	+	0.933713i	$0.383451\pi$
$659$			9.04123e12i			1.86743i	−0.358023	+	0.933713i	$0.616549\pi$
$660$		0			0
$661$			2.90021e11i			0.0590912i	0.999563	+	0.0295456i	$0.00940602\pi$
$661$			2.90021e11i			0.0590912i	−0.999563	+	0.0295456i	$0.990594\pi$
$662$		0			0
$663$	3.20275e10			0.00643743
$664$		0			0
$665$	−4.22166e12	+	6.11784e11i	−0.837117	+	0.121311i
$666$		0			0
$667$		−	1.52420e12i		−	0.298179i
$668$		0			0
$669$	−4.93095e11			−0.0951729
$670$		0			0
$671$	−8.66137e12			−1.64943
$672$		0			0
$673$	4.50836e12			0.847132			0.423566	−	0.905865i	$-0.360778\pi$
$673$	4.50836e12			0.847132			0.423566	+	0.905865i	$0.360778\pi$
$674$		0			0
$675$	2.54541e12			0.471943
$676$		0			0
$677$			4.48072e12i			0.819782i	0.912134	+	0.409891i	$0.134433\pi$
$677$			4.48072e12i			0.819782i	−0.912134	+	0.409891i	$0.865567\pi$
$678$		0			0
$679$	−4.15098e11	+	6.01542e10i	−0.0749440	+	0.0108605i
$680$		0			0
$681$	1.12088e13			1.99709
$682$		0			0
$683$		−	3.88544e12i		−	0.683199i	−0.939846	−	0.341599i	$-0.889031\pi$
$683$		−	3.88544e12i		−	0.683199i	0.939846	−	0.341599i	$-0.110969\pi$
$684$		0			0
$685$			1.30995e12i			0.227326i
$686$		0			0
$687$			6.19422e12i			1.06092i
$688$		0			0
$689$		−	5.09380e8i		−	8.61104e-5i
$690$		0			0
$691$	−1.02128e13			−1.70409			−0.852045	−	0.523468i	$-0.824638\pi$
$691$	−1.02128e13			−1.70409			−0.852045	+	0.523468i	$0.824638\pi$
$692$		0			0
$693$	−4.06447e12	+	5.89004e11i	−0.669428	+	0.0970106i
$694$		0			0
$695$			3.21659e12i			0.522955i
$696$		0			0
$697$	−1.15672e12			−0.185644
$698$		0			0
$699$	1.38541e13			2.19498
$700$		0			0
$701$	−1.09832e13			−1.71790			−0.858948	−	0.512063i	$-0.828881\pi$
$701$	−1.09832e13			−1.71790			−0.858948	+	0.512063i	$0.828881\pi$
$702$		0			0
$703$	6.09692e12			0.941481
$704$		0			0
$705$			3.07629e11i			0.0469003i
$706$		0			0
$707$	−4.28754e12	+	6.21331e11i	−0.645388	+	0.0935268i
$708$		0			0
$709$	2.44868e12			0.363936			0.181968	−	0.983304i	$-0.441753\pi$
$709$	2.44868e12			0.363936			0.181968	+	0.983304i	$0.441753\pi$
$710$		0			0
$711$			1.86582e12i			0.273814i
$712$		0			0
$713$			2.96769e12i			0.430046i
$714$		0			0
$715$		−	3.06109e10i		−	0.00438024i
$716$		0			0
$717$			6.94279e12i			0.981065i
$718$		0			0
$719$	−1.14434e13			−1.59689			−0.798445	−	0.602067i	$-0.794344\pi$
$719$	−1.14434e13			−1.59689			−0.798445	+	0.602067i	$0.794344\pi$
$720$		0			0
$721$	−1.51013e10	−	1.04208e11i	−0.00208116	−	0.0143612i
$722$		0			0
$723$			6.13105e11i			0.0834474i
$724$		0			0
$725$	−6.45488e12			−0.867695
$726$		0			0
$727$	−1.31151e13			−1.74127			−0.870635	−	0.491929i	$-0.836292\pi$
$727$	−1.31151e13			−1.74127			−0.870635	+	0.491929i	$0.836292\pi$
$728$		0			0
$729$	1.26719e12			0.166176
$730$		0			0
$731$	−9.51202e12			−1.23210
$732$		0			0
$733$		−	5.70825e12i		−	0.730357i	−0.930937	−	0.365179i	$-0.881008\pi$
$733$		−	5.70825e12i		−	0.730357i	0.930937	−	0.365179i	$-0.118992\pi$
$734$		0			0
$735$	1.37253e12	+	4.63616e12i	0.173471	+	0.585957i
$736$		0			0
$737$	−1.35007e13			−1.68559
$738$		0			0
$739$		−	1.35668e13i		−	1.67331i	−0.547731	−	0.836655i	$-0.684508\pi$
$739$		−	1.35668e13i		−	1.67331i	0.547731	−	0.836655i	$-0.315492\pi$
$740$		0			0
$741$		−	1.04558e11i		−	0.0127401i
$742$		0			0
$743$			1.62987e13i			1.96202i	0.193967	+	0.981008i	$0.437865\pi$
$743$			1.62987e13i			1.96202i	−0.193967	+	0.981008i	$0.562135\pi$
$744$		0			0
$745$			7.42455e12i			0.883012i
$746$		0			0
$747$	4.49110e12			0.527728
$748$		0			0
$749$	−8.36351e12	+	1.21200e12i	−0.971002	+	0.140713i
$750$		0			0
$751$			6.88388e12i			0.789684i	0.918749	+	0.394842i	$0.129201\pi$
$751$			6.88388e12i			0.789684i	−0.918749	+	0.394842i	$0.870799\pi$
$752$		0			0
$753$	−7.37844e12			−0.836349
$754$		0			0
$755$	4.25732e12			0.476842
$756$		0			0
$757$	2.32574e12			0.257412			0.128706	−	0.991683i	$-0.458918\pi$
$757$	2.32574e12			0.257412			0.128706	+	0.991683i	$0.458918\pi$
$758$		0			0
$759$	4.01746e12			0.439404
$760$		0			0
$761$			7.21032e12i			0.779334i	0.920956	+	0.389667i	$0.127410\pi$
$761$			7.21032e12i			0.779334i	−0.920956	+	0.389667i	$0.872590\pi$
$762$		0			0
$763$	−1.42259e12	−	9.81666e12i	−0.151956	−	1.04858i
$764$		0			0
$765$	−1.94994e12			−0.205847
$766$		0			0
$767$		−	6.80179e10i		−	0.00709649i
$768$		0			0
$769$			5.15415e11i			0.0531482i	0.999647	+	0.0265741i	$0.00845979\pi$
$769$			5.15415e11i			0.0531482i	−0.999647	+	0.0265741i	$0.991540\pi$
$770$		0			0
$771$			1.13829e13i			1.16013i
$772$		0			0
$773$		−	1.69506e13i		−	1.70757i	−0.520626	−	0.853785i	$-0.674301\pi$
$773$		−	1.69506e13i		−	1.70757i	0.520626	−	0.853785i	$-0.325699\pi$
$774$		0			0
$775$	1.25679e13			1.25143
$776$		0			0
$777$	−9.91102e11	−	6.83917e12i	−0.0975492	−	0.673145i
$778$		0			0
$779$			3.77625e12i			0.367402i
$780$		0			0
$781$	−8.89627e12			−0.855614
$782$		0			0
$783$	−7.69903e12			−0.731995
$784$		0			0
$785$	−9.16990e11			−0.0861888
$786$		0			0
$787$	−9.80859e12			−0.911423			−0.455712	−	0.890127i	$-0.650615\pi$
$787$	−9.80859e12			−0.911423			−0.455712	+	0.890127i	$0.650615\pi$
$788$		0			0
$789$		−	5.85268e12i		−	0.537661i
$790$		0			0
$791$	−5.03112e11	−	3.47176e12i	−0.0456952	−	0.315323i
$792$		0			0
$793$	−8.12456e10			−0.00729576
$794$		0			0
$795$			9.54048e10i			0.00847068i
$796$		0			0
$797$		−	1.16917e13i		−	1.02640i	−0.858269	−	0.513199i	$-0.828460\pi$
$797$		−	1.16917e13i		−	1.02640i	0.858269	−	0.513199i	$-0.171540\pi$
$798$		0			0
$799$		−	7.52704e11i		−	0.0653377i
$800$		0			0
$801$		−	8.34066e12i		−	0.715902i
$802$		0			0
$803$	2.02294e13			1.71697
$804$		0			0
$805$	−2.20500e11	−	1.52157e12i	−0.0185066	−	0.127706i
$806$		0			0
$807$			4.25059e12i			0.352792i
$808$		0			0
$809$	−2.06211e13			−1.69256			−0.846278	−	0.532742i	$-0.821162\pi$
$809$	−2.06211e13			−1.69256			−0.846278	+	0.532742i	$0.821162\pi$
$810$		0			0
$811$	8.98562e10			0.00729380			0.00364690	−	0.999993i	$-0.498839\pi$
$811$	8.98562e10			0.00729380			0.00364690	+	0.999993i	$0.498839\pi$
$812$		0			0
$813$	2.32083e13			1.86310
$814$		0			0
$815$	6.68301e12			0.530594
$816$		0			0
$817$			3.10531e13i			2.43840i
$818$		0			0
$819$	−3.81256e10	+	5.52500e9i	−0.00296101	+	0.000429096i
$820$		0			0
$821$	−1.76196e13			−1.35348			−0.676740	−	0.736222i	$-0.736608\pi$
$821$	−1.76196e13			−1.35348			−0.676740	+	0.736222i	$0.736608\pi$
$822$		0			0
$823$			1.64339e13i			1.24865i	0.781163	+	0.624327i	$0.214627\pi$
$823$			1.64339e13i			1.24865i	−0.781163	+	0.624327i	$0.785373\pi$
$824$		0			0
$825$		−	1.70136e13i		−	1.27866i
$826$		0			0
$827$		−	2.21536e12i		−	0.164691i	−0.996604	−	0.0823453i	$-0.973759\pi$
$827$		−	2.21536e12i		−	0.164691i	0.996604	−	0.0823453i	$-0.0262410\pi$
$828$		0			0
$829$			9.81244e12i			0.721575i	0.932648	+	0.360788i	$0.117492\pi$
$829$			9.81244e12i			0.721575i	−0.932648	+	0.360788i	$0.882508\pi$
$830$		0			0
$831$	1.01636e13			0.739339
$832$		0			0
$833$	−3.35829e12	−	1.13437e13i	−0.241666	−	0.816308i
$834$		0			0
$835$			7.29278e12i			0.519163i
$836$		0			0
$837$	1.49903e13			1.05572
$838$		0			0
$839$	−6.56726e12			−0.457568			−0.228784	−	0.973477i	$-0.573475\pi$
$839$	−6.56726e12			−0.457568			−0.228784	+	0.973477i	$0.573475\pi$
$840$		0			0
$841$	5.01680e12			0.345816
$842$		0			0
$843$	1.26418e13			0.862155
$844$		0			0
$845$			7.44010e12i			0.502023i
$846$		0			0
$847$	−2.08919e12	−	1.44166e13i	−0.139477	−	0.962470i
$848$		0			0
$849$	−3.83743e12			−0.253487
$850$		0			0
$851$			2.19746e12i			0.143627i
$852$		0			0
$853$		−	1.77620e13i		−	1.14874i	−0.818596	−	0.574370i	$-0.805247\pi$
$853$		−	1.77620e13i		−	1.14874i	0.818596	−	0.574370i	$-0.194753\pi$
$854$		0			0
$855$			6.36580e12i			0.407385i
$856$		0			0
$857$			1.28931e13i			0.816480i	0.912875	+	0.408240i	$0.133857\pi$
$857$			1.28931e13i			0.816480i	−0.912875	+	0.408240i	$0.866143\pi$
$858$		0			0
$859$	9.86398e11			0.0618134			0.0309067	−	0.999522i	$-0.490161\pi$
$859$	9.86398e11			0.0618134			0.0309067	+	0.999522i	$0.490161\pi$
$860$		0			0
$861$	4.23597e12	−	6.13858e11i	0.262687	−	0.0380674i
$862$		0			0
$863$			1.11654e13i			0.685210i	0.939479	+	0.342605i	$0.111309\pi$
$863$			1.11654e13i			0.685210i	−0.939479	+	0.342605i	$0.888691\pi$
$864$		0			0
$865$	1.48388e13			0.901210
$866$		0			0
$867$	−5.57396e12			−0.335026
$868$		0			0
$869$	−1.34230e13			−0.798474
$870$		0			0
$871$	−1.26639e11			−0.00745568
$872$		0			0
$873$			6.25922e11i			0.0364717i
$874$		0			0
$875$	−1.50589e13	+	2.18227e12i	−0.868475	+	0.125856i
$876$		0			0
$877$	2.73296e13			1.56004			0.780019	−	0.625755i	$-0.215209\pi$
$877$	2.73296e13			1.56004			0.780019	+	0.625755i	$0.215209\pi$
$878$		0			0
$879$		−	3.75000e13i		−	2.11876i
$880$		0			0
$881$			2.62844e13i			1.46996i	0.678087	+	0.734982i	$0.262809\pi$
$881$			2.62844e13i			1.46996i	−0.678087	+	0.734982i	$0.737191\pi$
$882$		0			0
$883$		−	1.13800e13i		−	0.629968i	−0.949097	−	0.314984i	$-0.898001\pi$
$883$		−	1.13800e13i		−	0.629968i	0.949097	−	0.314984i	$-0.101999\pi$
$884$		0			0
$885$			1.27395e13i			0.698082i
$886$		0			0
$887$	−1.98246e13			−1.07535			−0.537674	−	0.843153i	$-0.680697\pi$
$887$	−1.98246e13			−1.07535			−0.537674	+	0.843153i	$0.680697\pi$
$888$		0			0
$889$	1.58265e13	−	2.29351e12i	0.849822	−	0.123152i
$890$		0			0
$891$		−	3.30182e13i		−	1.75511i
$892$		0			0
$893$	−2.45729e12			−0.129308
$894$		0			0
$895$	5.25520e11			0.0273770
$896$		0			0
$897$	3.76847e10			0.00194357
$898$		0			0
$899$	−3.80139e13			−1.94099
$900$		0			0
$901$		−	2.33436e11i		−	0.0118007i
$902$		0			0
$903$	3.48335e13	−	5.04792e12i	1.74342	−	0.252649i
$904$		0			0
$905$	−8.79170e12			−0.435667
$906$		0			0
$907$			1.83615e13i			0.900898i	0.892802	+	0.450449i	$0.148736\pi$
$907$			1.83615e13i			0.900898i	−0.892802	+	0.450449i	$0.851264\pi$
$908$		0			0
$909$			6.46514e12i			0.314080i
$910$		0			0
$911$			1.96288e12i			0.0944193i	0.998885	+	0.0472097i	$0.0150329\pi$
$911$			1.96288e12i			0.0944193i	−0.998885	+	0.0472097i	$0.984967\pi$
$912$		0			0
$913$			3.23098e13i			1.53892i
$914$		0			0
$915$	1.52170e13			0.717684
$916$		0			0
$917$	−5.49379e12	−	3.79103e13i	−0.256573	−	1.77050i
$918$		0			0
$919$		−	3.37030e13i		−	1.55865i	−0.626620	−	0.779325i	$-0.715562\pi$
$919$		−	3.37030e13i		−	1.55865i	0.626620	−	0.779325i	$-0.284438\pi$
$920$		0			0
$921$	3.58254e13			1.64067
$922$		0			0
$923$	−8.34490e10			−0.00378454
$924$		0			0
$925$	9.30605e12			0.417953
$926$		0			0
$927$	−1.57134e11			−0.00698892
$928$		0			0
$929$			2.36893e12i			0.104348i	0.998638	+	0.0521738i	$0.0166150\pi$
$929$			2.36893e12i			0.104348i	−0.998638	+	0.0521738i	$0.983385\pi$
$930$		0			0
$931$	−3.70330e13	+	1.09635e13i	−1.61553	+	0.478275i
$932$		0			0
$933$	2.67694e13			1.15657
$934$		0			0
$935$		−	1.40282e13i		−	0.600274i
$936$		0			0
$937$		−	3.48576e13i		−	1.47730i	−0.674088	−	0.738651i	$-0.735463\pi$
$937$		−	3.48576e13i		−	1.47730i	0.674088	−	0.738651i	$-0.264537\pi$
$938$		0			0
$939$		−	1.00741e13i		−	0.422873i
$940$		0			0
$941$			5.16797e12i			0.214865i	0.994212	+	0.107433i	$0.0342630\pi$
$941$			5.16797e12i			0.214865i	−0.994212	+	0.107433i	$0.965737\pi$
$942$		0			0
$943$	−1.36104e12			−0.0560489
$944$		0			0
$945$	−7.68575e12	+	1.11378e12i	−0.313504	+	0.0454316i
$946$		0			0
$947$			2.25617e13i			0.911583i	0.890087	+	0.455791i	$0.150644\pi$
$947$			2.25617e13i			0.911583i	−0.890087	+	0.455791i	$0.849356\pi$
$948$		0			0
$949$	1.89756e11			0.00759448
$950$		0			0
$951$	−4.61397e13			−1.82920
$952$		0			0
$953$	1.59596e13			0.626763			0.313382	−	0.949627i	$-0.398538\pi$
$953$	1.59596e13			0.626763			0.313382	+	0.949627i	$0.398538\pi$
$954$		0			0
$955$	1.58862e13			0.618025
$956$		0			0
$957$			5.14608e13i			1.98323i
$958$		0			0
$959$	1.70096e12	+	1.17376e13i	0.0649396	+	0.448120i
$960$		0			0
$961$	4.75750e13			1.79938
$962$		0			0
$963$			1.26112e13i			0.472541i
$964$		0			0
$965$			7.89847e12i			0.293204i
$966$		0			0
$967$			1.34087e13i			0.493138i	0.969125	+	0.246569i	$0.0793032\pi$
$967$			1.34087e13i			0.493138i	−0.969125	+	0.246569i	$0.920697\pi$
$968$		0			0
$969$		−	4.79161e13i		−	1.74592i
$970$		0			0
$971$	2.96980e13			1.07211			0.536057	−	0.844182i	$-0.319913\pi$
$971$	2.96980e13			1.07211			0.536057	+	0.844182i	$0.319913\pi$
$972$		0			0
$973$	4.17670e12	+	2.88216e13i	0.149391	+	1.03089i
$974$		0			0
$975$		−	1.59592e11i		−	0.00565574i
$976$		0			0
$977$	6.06605e11			0.0213000			0.0106500	−	0.999943i	$-0.496610\pi$
$977$	6.06605e11			0.0213000			0.0106500	+	0.999943i	$0.496610\pi$
$978$		0			0
$979$	6.00041e13			2.08766
$980$		0			0
$981$	−1.48024e13			−0.510296
$982$		0			0
$983$	4.86971e13			1.66346			0.831731	−	0.555179i	$-0.187350\pi$
$983$	4.86971e13			1.66346			0.831731	+	0.555179i	$0.187350\pi$
$984$		0			0
$985$			6.40387e11i			0.0216760i
$986$		0			0
$987$	3.99452e11	+	2.75644e12i	0.0133979	+	0.0924532i
$988$		0			0
$989$	−1.11922e13			−0.371990
$990$		0			0
$991$		−	4.02951e13i		−	1.32715i	−0.748108	−	0.663577i	$-0.769038\pi$
$991$		−	4.02951e13i		−	1.32715i	0.748108	−	0.663577i	$-0.230962\pi$
$992$		0			0
$993$		−	2.16701e13i		−	0.707277i
$994$		0			0
$995$		−	9.68900e12i		−	0.313383i
$996$		0			0
$997$		−	2.94016e13i		−	0.942418i	−0.882022	−	0.471209i	$-0.843818\pi$
$997$		−	2.94016e13i		−	0.942418i	0.882022	−	0.471209i	$-0.156182\pi$
$998$		0			0
$999$	1.10998e13			0.352589

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.10.f.b.111.4	yes	24
4.3	odd	2	inner	112.10.f.b.111.22	yes	24
7.6	odd	2	inner	112.10.f.b.111.21	yes	24
28.27	even	2	inner	112.10.f.b.111.3	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.10.f.b.111.3	✓	24	28.27	even	2	inner
112.10.f.b.111.4	yes	24	1.1	even	1	trivial
112.10.f.b.111.21	yes	24	7.6	odd	2	inner
112.10.f.b.111.22	yes	24	4.3	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)

\(f(q)\)	\(=\)	\(q-170.771 q^{3} +701.626i q^{5} +(911.052 + 6286.78i) q^{7} +9479.76 q^{9} +68199.0i q^{11} +639.722i q^{13} -119817. i q^{15} +293169. i q^{17} +957083. q^{19} +(-155581. - 1.07360e6i) q^{21} +344952. i q^{23} +1.46085e6 q^{25} +1.74242e6 q^{27} -4.41859e6 q^{29} +8.60318e6 q^{31} -1.16464e7i q^{33} +(-4.41097e6 + 639217. i) q^{35} +6.37031e6 q^{37} -109246. i q^{39} +3.94558e6i q^{41} +3.24456e7i q^{43} +6.65124e6i q^{45} -2.56748e6 q^{47} +(-3.86936e7 + 1.14552e7i) q^{49} -5.00647e7i q^{51} -796252. q^{53} -4.78502e7 q^{55} -1.63442e8 q^{57} -1.06324e8 q^{59} +1.27001e8i q^{61} +(8.63655e6 + 5.95972e7i) q^{63} -448846. q^{65} +1.97960e8i q^{67} -5.89079e7i q^{69} +1.30446e8i q^{71} -2.96623e8i q^{73} -2.49470e8 q^{75} +(-4.28752e8 + 6.21328e7i) q^{77} +1.96821e8i q^{79} -4.84145e8 q^{81} +4.73757e8 q^{83} -2.05695e8 q^{85} +7.54568e8 q^{87} -8.79838e8i q^{89} +(-4.02179e6 + 582820. i) q^{91} -1.46917e9 q^{93} +6.71514e8i q^{95} +6.60272e7i q^{97} +6.46510e8i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 155768 q^{9} + 220672 q^{21} - 9187656 q^{25} + 14881104 q^{29} + 2829456 q^{37} - 214802472 q^{49} + 327087120 q^{53} + 238245440 q^{57} - 495797952 q^{65} + 347010000 q^{77} + 1816013720 q^{81}+ \cdots + 288442240 q^{93}+O(q^{100}) \) 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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