LMFDB - Embedded newform 272.10.b.c.33.12

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [272,10,Mod(33,272)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("272.33"); S:= CuspForms(chi, 10); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(272, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 10, names="a")

Level:	$ N $	$=$	$ 272 = 2^{4} \cdot 17 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	272.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,0,0,-9184] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$140.089747437$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 122690 x^{10} + 5157152560 x^{8} + 87983684680032 x^{6} + \cdots + 20\!\cdots\!28 $ x^12 + 122690x^10 + 5157152560x^8 + 87983684680032x^6 + 612743619071665152x^4 + 1335826553351738886144*x^2 + 203949399568932198678528
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{28}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 17)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			33.12
Root			$225.146i$ of defining polynomial
Character	$\chi$	$=$	272.33
Dual form			272.10.b.c.33.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+225.146i q^{3} -96.4328i q^{5} +1385.77i q^{7} -31007.8 q^{9} +33833.7i q^{11} +138558. q^{13} +21711.5 q^{15} +(177982. - 294805. i) q^{17} -429726. q^{19} -312001. q^{21} -352831. i q^{23} +1.94383e6 q^{25} -2.54973e6i q^{27} +508224. i q^{29} -8.73026e6i q^{31} -7.61753e6 q^{33} +133634. q^{35} -1.62218e7i q^{37} +3.11958e7i q^{39} +1.12435e7i q^{41} +2.43795e7 q^{43} +2.99017e6i q^{45} +1.11673e7 q^{47} +3.84332e7 q^{49} +(6.63743e7 + 4.00720e7i) q^{51} +3.54297e7 q^{53} +3.26268e6 q^{55} -9.67510e7i q^{57} -5.03437e7 q^{59} -1.53033e8i q^{61} -4.29697e7i q^{63} -1.33615e7i q^{65} +2.44023e8 q^{67} +7.94386e7 q^{69} -3.78713e8i q^{71} -1.38738e7i q^{73} +4.37645e8i q^{75} -4.68858e7 q^{77} -5.65164e8i q^{79} -3.62648e7 q^{81} +5.68408e8 q^{83} +(-2.84289e7 - 1.71633e7i) q^{85} -1.14425e8 q^{87} +2.61194e8 q^{89} +1.92010e8i q^{91} +1.96558e9 q^{93} +4.14396e7i q^{95} +1.04492e9i q^{97} -1.04911e9i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 9184 q^{9} - 63204 q^{13} + 243480 q^{15} - 105960 q^{17} - 1110672 q^{19} - 172580 q^{21} - 4441796 q^{25} - 6557404 q^{33} - 3519864 q^{35} - 10004616 q^{43} + 112552440 q^{47} + 121354720 q^{49}+ \cdots + 1635779524 q^{93}+O(q^{100}) $ 12 * q - 9184 * q^9 - 63204 * q^13 + 243480 * q^15 - 105960 * q^17 - 1110672 * q^19 - 172580 * q^21 - 4441796 * q^25 - 6557404 * q^33 - 3519864 * q^35 - 10004616 * q^43 + 112552440 * q^47 + 121354720 * q^49 + 52506472 * q^51 + 76804272 * q^53 - 300732568 * q^55 - 11618904 * q^59 + 304208752 * q^67 - 211308236 * q^69 + 138357828 * q^77 - 363335792 * q^81 + 845042136 * q^83 - 388949632 * q^85 - 610860648 * q^87 - 938223804 * q^89 + 1635779524 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/272\mathbb{Z}\right)^\times$.

$n$	$69$	$239$	$241$
$\chi(n)$	$1$	$1$	$-1$

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
$3$			225.146i			1.60479i	0.596792	+	0.802396i	$0.296442\pi$
$3$			225.146i			1.60479i	−0.596792	+	0.802396i	$0.703558\pi$
$4$		0			0
$5$		−	96.4328i		−	0.0690017i	−0.999405	−	0.0345009i	$-0.989016\pi$
$5$		−	96.4328i		−	0.0690017i	0.999405	−	0.0345009i	$-0.0109841\pi$
$6$		0			0
$7$			1385.77i			0.218148i	0.994034	+	0.109074i	$0.0347885\pi$
$7$			1385.77i			0.218148i	−0.994034	+	0.109074i	$0.965211\pi$
$8$		0			0
$9$	−31007.8			−1.57536
$10$		0			0
$11$			33833.7i			0.696759i	0.937354	+	0.348379i	$0.113268\pi$
$11$			33833.7i			0.696759i	−0.937354	+	0.348379i	$0.886732\pi$
$12$		0			0
$13$	138558.			1.34551			0.672755	−	0.739866i	$-0.265111\pi$
$13$	138558.			1.34551			0.672755	+	0.739866i	$0.265111\pi$
$14$		0			0
$15$	21711.5			0.110733
$16$		0			0
$17$	177982.	−	294805.i	0.516840	−	0.856082i
$17$	177982.	−	294805.i	0.516840	−	0.856082i
$18$		0			0
$19$	−429726.			−0.756484			−0.378242	−	0.925707i	$-0.623471\pi$
$19$	−429726.			−0.756484			−0.378242	+	0.925707i	$0.623471\pi$
$20$		0			0
$21$	−312001.			−0.350082
$22$		0			0
$23$		−	352831.i		−	0.262901i	−0.991323	−	0.131450i	$-0.958037\pi$
$23$		−	352831.i		−	0.262901i	0.991323	−	0.131450i	$-0.0419634\pi$
$24$		0			0
$25$	1.94383e6			0.995239
$26$		0			0
$27$		−	2.54973e6i		−	0.923330i
$28$		0			0
$29$			508224.i			0.133433i	0.997772	+	0.0667166i	$0.0212523\pi$
$29$			508224.i			0.133433i	−0.997772	+	0.0667166i	$0.978748\pi$
$30$		0			0
$31$		−	8.73026e6i		−	1.69785i	−0.528513	−	0.848925i	$-0.677250\pi$
$31$		−	8.73026e6i		−	1.69785i	0.528513	−	0.848925i	$-0.322750\pi$
$32$		0			0
$33$	−7.61753e6			−1.11815
$34$		0			0
$35$	133634.			0.0150526
$36$		0			0
$37$		−	1.62218e7i		−	1.42295i	−0.702710	−	0.711476i	$-0.748027\pi$
$37$		−	1.62218e7i		−	1.42295i	0.702710	−	0.711476i	$-0.251973\pi$
$38$		0			0
$39$			3.11958e7i			2.15926i
$40$		0			0
$41$			1.12435e7i			0.621406i	0.950507	+	0.310703i	$0.100564\pi$
$41$			1.12435e7i			0.621406i	−0.950507	+	0.310703i	$0.899436\pi$
$42$		0			0
$43$	2.43795e7			1.08747			0.543735	−	0.839257i	$-0.317010\pi$
$43$	2.43795e7			1.08747			0.543735	+	0.839257i	$0.317010\pi$
$44$		0			0
$45$			2.99017e6i			0.108702i
$46$		0			0
$47$	1.11673e7			0.333817			0.166909	−	0.985972i	$-0.446621\pi$
$47$	1.11673e7			0.333817			0.166909	+	0.985972i	$0.446621\pi$
$48$		0			0
$49$	3.84332e7			0.952412
$50$		0			0
$51$	6.63743e7	+	4.00720e7i	1.37383	+	0.829421i
$52$		0			0
$53$	3.54297e7			0.616774			0.308387	−	0.951261i	$-0.400211\pi$
$53$	3.54297e7			0.616774			0.308387	+	0.951261i	$0.400211\pi$
$54$		0			0
$55$	3.26268e6			0.0480775
$56$		0			0
$57$		−	9.67510e7i		−	1.21400i
$58$		0			0
$59$	−5.03437e7			−0.540893			−0.270446	−	0.962735i	$-0.587171\pi$
$59$	−5.03437e7			−0.540893			−0.270446	+	0.962735i	$0.587171\pi$
$60$		0			0
$61$		−	1.53033e8i		−	1.41515i	−0.706639	−	0.707574i	$-0.749789\pi$
$61$		−	1.53033e8i		−	1.41515i	0.706639	−	0.707574i	$-0.250211\pi$
$62$		0			0
$63$		−	4.29697e7i		−	0.343661i
$64$		0			0
$65$		−	1.33615e7i		−	0.0928425i
$66$		0			0
$67$	2.44023e8			1.47943			0.739713	−	0.672922i	$-0.234961\pi$
$67$	2.44023e8			1.47943			0.739713	+	0.672922i	$0.234961\pi$
$68$		0			0
$69$	7.94386e7			0.421901
$70$		0			0
$71$		−	3.78713e8i		−	1.76867i	−0.466850	−	0.884337i	$-0.654611\pi$
$71$		−	3.78713e8i		−	1.76867i	0.466850	−	0.884337i	$-0.345389\pi$
$72$		0			0
$73$		−	1.38738e7i		−	0.0571798i	−0.999591	−	0.0285899i	$-0.990898\pi$
$73$		−	1.38738e7i		−	0.0571798i	0.999591	−	0.0285899i	$-0.00910169\pi$
$74$		0			0
$75$			4.37645e8i			1.59715i
$76$		0			0
$77$	−4.68858e7			−0.151996
$78$		0			0
$79$		−	5.65164e8i		−	1.63250i	−0.577700	−	0.816250i	$-0.696049\pi$
$79$		−	5.65164e8i		−	1.63250i	0.577700	−	0.816250i	$-0.303951\pi$
$80$		0			0
$81$	−3.62648e7			−0.0936057
$82$		0			0
$83$	5.68408e8			1.31465			0.657323	−	0.753609i	$-0.271689\pi$
$83$	5.68408e8			1.31465			0.657323	+	0.753609i	$0.271689\pi$
$84$		0			0
$85$	−2.84289e7	−	1.71633e7i	−0.0590711	−	0.0356629i
$86$		0			0
$87$	−1.14425e8			−0.214133
$88$		0			0
$89$	2.61194e8			0.441275			0.220637	−	0.975356i	$-0.429186\pi$
$89$	2.61194e8			0.441275			0.220637	+	0.975356i	$0.429186\pi$
$90$		0			0
$91$			1.92010e8i			0.293520i
$92$		0			0
$93$	1.96558e9			2.72470
$94$		0			0
$95$			4.14396e7i			0.0521987i
$96$		0			0
$97$			1.04492e9i			1.19842i	0.800590	+	0.599212i	$0.204519\pi$
$97$			1.04492e9i			1.19842i	−0.800590	+	0.599212i	$0.795481\pi$
$98$		0			0
$99$		−	1.04911e9i		−	1.09764i
$100$		0			0
$101$	2.97630e7			0.0284597			0.0142299	−	0.999899i	$-0.495470\pi$
$101$	2.97630e7			0.0284597			0.0142299	+	0.999899i	$0.495470\pi$
$102$		0			0
$103$	−1.60523e9			−1.40530			−0.702650	−	0.711536i	$-0.748000\pi$
$103$	−1.60523e9			−1.40530			−0.702650	+	0.711536i	$0.748000\pi$
$104$		0			0
$105$			3.00872e7i			0.0241562i
$106$		0			0
$107$			2.35483e9i			1.73673i	0.495925	+	0.868365i	$0.334829\pi$
$107$			2.35483e9i			1.73673i	−0.495925	+	0.868365i	$0.665171\pi$
$108$		0			0
$109$			1.68886e9i			1.14597i	0.819566	+	0.572985i	$0.194215\pi$
$109$			1.68886e9i			1.14597i	−0.819566	+	0.572985i	$0.805785\pi$
$110$		0			0
$111$	3.65227e9			2.28354
$112$		0			0
$113$		−	2.39661e9i		−	1.38275i	−0.722496	−	0.691375i	$-0.757005\pi$
$113$		−	2.39661e9i		−	1.38275i	0.722496	−	0.691375i	$-0.242995\pi$
$114$		0			0
$115$	−3.40245e7			−0.0181406
$116$		0			0
$117$	−4.29638e9			−2.11966
$118$		0			0
$119$	4.08533e8	+	2.46643e8i	0.186752	+	0.112748i
$120$		0			0
$121$	1.21323e9			0.514527
$122$		0			0
$123$	−2.53144e9			−0.997227
$124$		0			0
$125$		−	3.75794e8i		−	0.137675i
$126$		0			0
$127$	−4.20715e9			−1.43506			−0.717532	−	0.696526i	$-0.754728\pi$
$127$	−4.20715e9			−1.43506			−0.717532	+	0.696526i	$0.754728\pi$
$128$		0			0
$129$			5.48896e9i			1.74516i
$130$		0			0
$131$		−	1.46332e9i		−	0.434129i	−0.976157	−	0.217064i	$-0.930352\pi$
$131$		−	1.46332e9i		−	0.434129i	0.976157	−	0.217064i	$-0.0696481\pi$
$132$		0			0
$133$		−	5.95502e8i		−	0.165025i
$134$		0			0
$135$	−2.45877e8			−0.0637113
$136$		0			0
$137$	−4.13890e9			−1.00379			−0.501894	−	0.864929i	$-0.667363\pi$
$137$	−4.13890e9			−1.00379			−0.501894	+	0.864929i	$0.667363\pi$
$138$		0			0
$139$		−	1.74238e9i		−	0.395892i	−0.980213	−	0.197946i	$-0.936573\pi$
$139$		−	1.74238e9i		−	0.395892i	0.980213	−	0.197946i	$-0.0634271\pi$
$140$		0			0
$141$			2.51428e9i			0.535708i
$142$		0			0
$143$			4.68793e9i			0.937496i
$144$		0			0
$145$	4.90094e7			0.00920712
$146$		0			0
$147$			8.65309e9i			1.52842i
$148$		0			0
$149$	4.80220e9			0.798183			0.399092	−	0.916911i	$-0.369326\pi$
$149$	4.80220e9			0.798183			0.399092	+	0.916911i	$0.369326\pi$
$150$		0			0
$151$	−1.22756e9			−0.192152			−0.0960762	−	0.995374i	$-0.530629\pi$
$151$	−1.22756e9			−0.192152			−0.0960762	+	0.995374i	$0.530629\pi$
$152$		0			0
$153$	−5.51883e9	+	9.14126e9i	−0.814208	+	1.34864i
$154$		0			0
$155$	−8.41883e8			−0.117155
$156$		0			0
$157$	1.72654e9			0.226792			0.113396	−	0.993550i	$-0.463827\pi$
$157$	1.72654e9			0.226792			0.113396	+	0.993550i	$0.463827\pi$
$158$		0			0
$159$			7.97686e9i			0.989794i
$160$		0			0
$161$	4.88944e8			0.0573512
$162$		0			0
$163$		−	2.27451e9i		−	0.252374i	−0.992006	−	0.126187i	$-0.959726\pi$
$163$		−	2.27451e9i		−	0.252374i	0.992006	−	0.126187i	$-0.0402739\pi$
$164$		0			0
$165$			7.34580e8i			0.0771545i
$166$		0			0
$167$			1.42586e10i			1.41857i	0.704921	+	0.709286i	$0.250982\pi$
$167$			1.42586e10i			1.41857i	−0.704921	+	0.709286i	$0.749018\pi$
$168$		0			0
$169$	8.59384e9			0.810396
$170$		0			0
$171$	1.33248e10			1.19173
$172$		0			0
$173$			6.92744e9i			0.587984i	0.955808	+	0.293992i	$0.0949839\pi$
$173$			6.92744e9i			0.587984i	−0.955808	+	0.293992i	$0.905016\pi$
$174$		0			0
$175$			2.69370e9i			0.217109i
$176$		0			0
$177$		−	1.13347e10i		−	0.868020i
$178$		0			0
$179$	−1.32700e10			−0.966120			−0.483060	−	0.875587i	$-0.660475\pi$
$179$	−1.32700e10			−0.966120			−0.483060	+	0.875587i	$0.660475\pi$
$180$		0			0
$181$		−	8.95066e9i		−	0.619871i	−0.950758	−	0.309936i	$-0.899692\pi$
$181$		−	8.95066e9i		−	0.619871i	0.950758	−	0.309936i	$-0.100308\pi$
$182$		0			0
$183$	3.44549e10			2.27102
$184$		0			0
$185$	−1.56431e9			−0.0981861
$186$		0			0
$187$	9.97436e9	+	6.02180e9i	0.596483	+	0.360113i
$188$		0			0
$189$	3.53334e9			0.201422
$190$		0			0
$191$	2.10560e9			0.114479			0.0572394	−	0.998360i	$-0.481770\pi$
$191$	2.10560e9			0.114479			0.0572394	+	0.998360i	$0.481770\pi$
$192$		0			0
$193$		−	5.10730e9i		−	0.264962i	−0.991186	−	0.132481i	$-0.957706\pi$
$193$		−	5.10730e9i		−	0.264962i	0.991186	−	0.132481i	$-0.0422944\pi$
$194$		0			0
$195$	3.00830e9			0.148993
$196$		0			0
$197$			1.85517e10i			0.877578i	0.898590	+	0.438789i	$0.144592\pi$
$197$			1.85517e10i			0.877578i	−0.898590	+	0.438789i	$0.855408\pi$
$198$		0			0
$199$		−	1.37208e10i		−	0.620211i	−0.950702	−	0.310105i	$-0.899636\pi$
$199$		−	1.37208e10i		−	0.620211i	0.950702	−	0.310105i	$-0.100364\pi$
$200$		0			0
$201$			5.49407e10i			2.37417i
$202$		0			0
$203$	−7.04282e8			−0.0291082
$204$		0			0
$205$	1.08425e9			0.0428781
$206$		0			0
$207$			1.09405e10i			0.414163i
$208$		0			0
$209$		−	1.45392e10i		−	0.527087i
$210$		0			0
$211$		−	6.12032e9i		−	0.212570i	−0.994336	−	0.106285i	$-0.966104\pi$
$211$		−	6.12032e9i		−	0.212570i	0.994336	−	0.106285i	$-0.0338957\pi$
$212$		0			0
$213$	8.52658e10			2.83835
$214$		0			0
$215$		−	2.35099e9i		−	0.0750373i
$216$		0			0
$217$	1.20982e10			0.370382
$218$		0			0
$219$	3.12363e9			0.0917617
$220$		0			0
$221$	2.46609e10	−	4.08477e10i	0.695414	−	1.15187i
$222$		0			0
$223$	6.31228e9			0.170928			0.0854642	−	0.996341i	$-0.472763\pi$
$223$	6.31228e9			0.170928			0.0854642	+	0.996341i	$0.472763\pi$
$224$		0			0
$225$	−6.02737e10			−1.56786
$226$		0			0
$227$			5.43032e10i			1.35740i	0.734413	+	0.678702i	$0.237457\pi$
$227$			5.43032e10i			1.35740i	−0.734413	+	0.678702i	$0.762543\pi$
$228$		0			0
$229$	−1.03751e10			−0.249306			−0.124653	−	0.992200i	$-0.539782\pi$
$229$	−1.03751e10			−0.249306			−0.124653	+	0.992200i	$0.539782\pi$
$230$		0			0
$231$		−	1.05562e10i		−	0.243923i
$232$		0			0
$233$			5.93158e10i			1.31846i	0.751939	+	0.659232i	$0.229119\pi$
$233$			5.93158e10i			1.31846i	−0.751939	+	0.659232i	$0.770881\pi$
$234$		0			0
$235$		−	1.07690e9i		−	0.0230340i
$236$		0			0
$237$	1.27245e11			2.61982
$238$		0			0
$239$	6.57143e10			1.30278			0.651388	−	0.758745i	$-0.274187\pi$
$239$	6.57143e10			1.30278			0.651388	+	0.758745i	$0.274187\pi$
$240$		0			0
$241$			2.02970e10i			0.387574i	0.981044	+	0.193787i	$0.0620771\pi$
$241$			2.02970e10i			0.387574i	−0.981044	+	0.193787i	$0.937923\pi$
$242$		0			0
$243$		−	5.83511e10i		−	1.07355i
$244$		0			0
$245$		−	3.70623e9i		−	0.0657180i
$246$		0			0
$247$	−5.95420e10			−1.01786
$248$		0			0
$249$			1.27975e11i			2.10974i
$250$		0			0
$251$	8.71654e10			1.38616			0.693078	−	0.720863i	$-0.256254\pi$
$251$	8.71654e10			1.38616			0.693078	+	0.720863i	$0.256254\pi$
$252$		0			0
$253$	1.19376e10			0.183178
$254$		0			0
$255$	3.86426e9	−	6.40066e9i	0.0572315	−	0.0947968i
$256$		0			0
$257$	−1.20306e10			−0.172023			−0.0860115	−	0.996294i	$-0.527412\pi$
$257$	−1.20306e10			−0.172023			−0.0860115	+	0.996294i	$0.527412\pi$
$258$		0			0
$259$	2.24797e10			0.310414
$260$		0			0
$261$		−	1.57589e10i		−	0.210205i
$262$		0			0
$263$	−3.51365e10			−0.452853			−0.226426	−	0.974028i	$-0.572704\pi$
$263$	−3.51365e10			−0.452853			−0.226426	+	0.974028i	$0.572704\pi$
$264$		0			0
$265$		−	3.41659e9i		−	0.0425585i
$266$		0			0
$267$			5.88069e10i			0.708154i
$268$		0			0
$269$		−	8.30055e10i		−	0.966544i	−0.875470	−	0.483272i	$-0.839448\pi$
$269$		−	8.30055e10i		−	0.966544i	0.875470	−	0.483272i	$-0.160552\pi$
$270$		0			0
$271$	5.57473e10			0.627858			0.313929	−	0.949446i	$-0.398355\pi$
$271$	5.57473e10			0.627858			0.313929	+	0.949446i	$0.398355\pi$
$272$		0			0
$273$	−4.32303e10			−0.471038
$274$		0			0
$275$			6.57668e10i			0.693441i
$276$		0			0
$277$			1.84969e11i			1.88773i	0.330333	+	0.943864i	$0.392839\pi$
$277$			1.84969e11i			1.88773i	−0.330333	+	0.943864i	$0.607161\pi$
$278$		0			0
$279$			2.70706e11i			2.67472i
$280$		0			0
$281$	−1.35823e11			−1.29956			−0.649778	−	0.760124i	$-0.725138\pi$
$281$	−1.35823e11			−1.29956			−0.649778	+	0.760124i	$0.725138\pi$
$282$		0			0
$283$			5.85987e10i			0.543062i	0.962430	+	0.271531i	$0.0875299\pi$
$283$			5.85987e10i			0.543062i	−0.962430	+	0.271531i	$0.912470\pi$
$284$		0			0
$285$	−9.32997e9			−0.0837681
$286$		0			0
$287$	−1.55810e10			−0.135558
$288$		0			0
$289$	−5.52326e10	−	1.04940e11i	−0.465752	−	0.884915i
$290$		0			0
$291$	−2.35260e11			−1.92322
$292$		0			0
$293$	−5.48232e10			−0.434571			−0.217285	−	0.976108i	$-0.569720\pi$
$293$	−5.48232e10			−0.434571			−0.217285	+	0.976108i	$0.569720\pi$
$294$		0			0
$295$			4.85478e9i			0.0373225i
$296$		0			0
$297$	8.62667e10			0.643338
$298$		0			0
$299$		−	4.88876e10i		−	0.353735i
$300$		0			0
$301$			3.37845e10i			0.237229i
$302$		0			0
$303$			6.70103e9i			0.0456720i
$304$		0			0
$305$	−1.47574e10			−0.0976477
$306$		0			0
$307$	1.88055e11			1.20826			0.604132	−	0.796884i	$-0.293520\pi$
$307$	1.88055e11			1.20826			0.604132	+	0.796884i	$0.293520\pi$
$308$		0			0
$309$		−	3.61411e11i		−	2.25521i
$310$		0			0
$311$		−	1.81717e11i		−	1.10147i	−0.834679	−	0.550737i	$-0.814347\pi$
$311$		−	1.81717e11i		−	1.10147i	0.834679	−	0.550737i	$-0.185653\pi$
$312$		0			0
$313$			2.21549e11i			1.30473i	0.757905	+	0.652364i	$0.226223\pi$
$313$			2.21549e11i			1.30473i	−0.757905	+	0.652364i	$0.773777\pi$
$314$		0			0
$315$	−4.14369e9			−0.0237132
$316$		0			0
$317$		−	2.66841e11i		−	1.48418i	−0.670301	−	0.742089i	$-0.733835\pi$
$317$		−	2.66841e11i		−	1.48418i	0.670301	−	0.742089i	$-0.266165\pi$
$318$		0			0
$319$	−1.71951e10			−0.0929708
$320$		0			0
$321$	−5.30181e11			−2.78709
$322$		0			0
$323$	−7.64835e10	+	1.26685e11i	−0.390982	+	0.647613i
$324$		0			0
$325$	2.69333e11			1.33910
$326$		0			0
$327$	−3.80239e11			−1.83904
$328$		0			0
$329$			1.54754e10i			0.0728215i
$330$		0			0
$331$	−3.46276e11			−1.58561			−0.792804	−	0.609476i	$-0.791380\pi$
$331$	−3.46276e11			−1.58561			−0.792804	+	0.609476i	$0.791380\pi$
$332$		0			0
$333$			5.03001e11i			2.24166i
$334$		0			0
$335$		−	2.35318e10i		−	0.102083i
$336$		0			0
$337$			2.14440e11i			0.905671i	0.891594	+	0.452836i	$0.149588\pi$
$337$			2.14440e11i			0.905671i	−0.891594	+	0.452836i	$0.850412\pi$
$338$		0			0
$339$	5.39586e11			2.21903
$340$		0			0
$341$	2.95377e11			1.18299
$342$		0			0
$343$			1.09181e11i			0.425914i
$344$		0			0
$345$		−	7.66048e9i		−	0.0291119i
$346$		0			0
$347$		−	2.25675e11i		−	0.835603i	−0.908538	−	0.417801i	$-0.862801\pi$
$347$		−	2.25675e11i		−	0.835603i	0.908538	−	0.417801i	$-0.137199\pi$
$348$		0			0
$349$	−3.13210e10			−0.113011			−0.0565055	−	0.998402i	$-0.517996\pi$
$349$	−3.13210e10			−0.113011			−0.0565055	+	0.998402i	$0.517996\pi$
$350$		0			0
$351$		−	3.53285e11i		−	1.24235i
$352$		0			0
$353$	1.67054e11			0.572625			0.286312	−	0.958136i	$-0.407571\pi$
$353$	1.67054e11			0.572625			0.286312	+	0.958136i	$0.407571\pi$
$354$		0			0
$355$	−3.65204e10			−0.122041
$356$		0			0
$357$	−5.55307e10	+	9.19797e10i	−0.180936	+	0.299699i
$358$		0			0
$359$	5.25130e11			1.66856			0.834280	−	0.551341i	$-0.185884\pi$
$359$	5.25130e11			1.66856			0.834280	+	0.551341i	$0.185884\pi$
$360$		0			0
$361$	−1.38024e11			−0.427731
$362$		0			0
$363$			2.73154e11i			0.825709i
$364$		0			0
$365$	−1.33789e9			−0.00394551
$366$		0			0
$367$		−	9.23982e10i		−	0.265868i	−0.991125	−	0.132934i	$-0.957560\pi$
$367$		−	9.23982e10i		−	0.265868i	0.991125	−	0.132934i	$-0.0424399\pi$
$368$		0			0
$369$		−	3.48637e11i		−	0.978936i
$370$		0			0
$371$			4.90975e10i			0.134548i
$372$		0			0
$373$	1.27208e11			0.340272			0.170136	−	0.985421i	$-0.445579\pi$
$373$	1.27208e11			0.340272			0.170136	+	0.985421i	$0.445579\pi$
$374$		0			0
$375$	8.46085e10			0.220940
$376$		0			0
$377$			7.04185e10i			0.179536i
$378$		0			0
$379$		−	4.08652e11i		−	1.01737i	−0.860954	−	0.508683i	$-0.830132\pi$
$379$		−	4.08652e11i		−	1.01737i	0.860954	−	0.508683i	$-0.169868\pi$
$380$		0			0
$381$		−	9.47223e11i		−	2.30298i
$382$		0			0
$383$	−2.61140e11			−0.620123			−0.310062	−	0.950716i	$-0.600350\pi$
$383$	−2.61140e11			−0.620123			−0.310062	+	0.950716i	$0.600350\pi$
$384$		0			0
$385$			4.52133e9i			0.0104880i
$386$		0			0
$387$	−7.55955e11			−1.71316
$388$		0			0
$389$	−3.51283e11			−0.777829			−0.388914	−	0.921274i	$-0.627150\pi$
$389$	−3.51283e11			−0.777829			−0.388914	+	0.921274i	$0.627150\pi$
$390$		0			0
$391$	−1.04017e11	−	6.27977e10i	−0.225064	−	0.135878i
$392$		0			0
$393$	3.29461e11			0.696686
$394$		0			0
$395$	−5.45004e10			−0.112645
$396$		0			0
$397$		−	4.17307e11i		−	0.843137i	−0.906797	−	0.421569i	$-0.861480\pi$
$397$		−	4.17307e11i		−	0.843137i	0.906797	−	0.421569i	$-0.138520\pi$
$398$		0			0
$399$	1.34075e11			0.264831
$400$		0			0
$401$		−	4.93060e11i		−	0.952248i	−0.879378	−	0.476124i	$-0.842041\pi$
$401$		−	4.93060e11i		−	0.952248i	0.879378	−	0.476124i	$-0.157959\pi$
$402$		0			0
$403$		−	1.20965e12i		−	2.28447i
$404$		0			0
$405$			3.49711e9i			0.00645895i
$406$		0			0
$407$	5.48842e11			0.991454
$408$		0			0
$409$	2.79374e11			0.493663			0.246831	−	0.969058i	$-0.420611\pi$
$409$	2.79374e11			0.493663			0.246831	+	0.969058i	$0.420611\pi$
$410$		0			0
$411$		−	9.31857e11i		−	1.61087i
$412$		0			0
$413$		−	6.97649e10i		−	0.117995i
$414$		0			0
$415$		−	5.48132e10i		−	0.0907129i
$416$		0			0
$417$	3.92291e11			0.635325
$418$		0			0
$419$		−	4.89354e10i		−	0.0775640i	−0.999248	−	0.0387820i	$-0.987652\pi$
$419$		−	4.89354e10i		−	0.0775640i	0.999248	−	0.0387820i	$-0.0123478\pi$
$420$		0			0
$421$	9.73572e11			1.51042			0.755211	−	0.655481i	$-0.227534\pi$
$421$	9.73572e11			1.51042			0.755211	+	0.655481i	$0.227534\pi$
$422$		0			0
$423$	−3.46274e11			−0.525882
$424$		0			0
$425$	3.45966e11	−	5.73050e11i	0.514380	−	0.852006i
$426$		0			0
$427$	2.12070e11			0.308712
$428$		0			0
$429$	−1.05547e12			−1.50449
$430$		0			0
$431$			3.60887e11i			0.503760i	0.967758	+	0.251880i	$0.0810489\pi$
$431$			3.60887e11i			0.503760i	−0.967758	+	0.251880i	$0.918951\pi$
$432$		0			0
$433$	−1.32088e12			−1.80579			−0.902895	−	0.429862i	$-0.858562\pi$
$433$	−1.32088e12			−1.80579			−0.902895	+	0.429862i	$0.858562\pi$
$434$		0			0
$435$			1.10343e10i			0.0147755i
$436$		0			0
$437$			1.51621e11i			0.198880i
$438$		0			0
$439$		−	4.29288e11i		−	0.551643i	−0.961209	−	0.275822i	$-0.911050\pi$
$439$		−	4.29288e11i		−	0.551643i	0.961209	−	0.275822i	$-0.0889499\pi$
$440$		0			0
$441$	−1.19173e12			−1.50039
$442$		0			0
$443$	9.88207e11			1.21908			0.609538	−	0.792757i	$-0.291355\pi$
$443$	9.88207e11			1.21908			0.609538	+	0.792757i	$0.291355\pi$
$444$		0			0
$445$		−	2.51877e10i		−	0.0304487i
$446$		0			0
$447$			1.08120e12i			1.28092i
$448$		0			0
$449$			5.27610e11i			0.612639i	0.951929	+	0.306319i	$0.0990975\pi$
$449$			5.27610e11i			0.612639i	−0.951929	+	0.306319i	$0.900902\pi$
$450$		0			0
$451$	−3.80410e11			−0.432970
$452$		0			0
$453$		−	2.76380e11i		−	0.308365i
$454$		0			0
$455$	1.85161e10			0.0202534
$456$		0			0
$457$	1.13024e11			0.121213			0.0606064	−	0.998162i	$-0.480697\pi$
$457$	1.13024e11			0.121213			0.0606064	+	0.998162i	$0.480697\pi$
$458$		0			0
$459$	−7.51673e11	−	4.53806e11i	−0.790446	−	0.477214i
$460$		0			0
$461$	1.11660e12			1.15145			0.575725	−	0.817643i	$-0.304720\pi$
$461$	1.11660e12			1.15145			0.575725	+	0.817643i	$0.304720\pi$
$462$		0			0
$463$	3.92248e11			0.396685			0.198343	−	0.980133i	$-0.436444\pi$
$463$	3.92248e11			0.396685			0.198343	+	0.980133i	$0.436444\pi$
$464$		0			0
$465$		−	1.89547e11i		−	0.188009i
$466$		0			0
$467$	−1.59510e11			−0.155189			−0.0775946	−	0.996985i	$-0.524724\pi$
$467$	−1.59510e11			−0.155189			−0.0775946	+	0.996985i	$0.524724\pi$
$468$		0			0
$469$			3.38160e11i			0.322734i
$470$		0			0
$471$			3.88724e11i			0.363955i
$472$		0			0
$473$			8.24850e11i			0.757705i
$474$		0			0
$475$	−8.35312e11			−0.752883
$476$		0			0
$477$	−1.09860e12			−0.971640
$478$		0			0
$479$			4.37040e11i			0.379325i	0.981849	+	0.189662i	$0.0607393\pi$
$479$			4.37040e11i			0.379325i	−0.981849	+	0.189662i	$0.939261\pi$
$480$		0			0
$481$		−	2.24766e12i		−	1.91460i
$482$		0			0
$483$			1.10084e11i			0.0920367i
$484$		0			0
$485$	1.00765e11			0.0826933
$486$		0			0
$487$			1.05048e12i			0.846270i	0.906067	+	0.423135i	$0.139070\pi$
$487$			1.05048e12i			0.846270i	−0.906067	+	0.423135i	$0.860930\pi$
$488$		0			0
$489$	5.12097e11			0.405007
$490$		0			0
$491$	−1.77945e12			−1.38172			−0.690859	−	0.722990i	$-0.742767\pi$
$491$	−1.77945e12			−1.38172			−0.690859	+	0.722990i	$0.742767\pi$
$492$		0			0
$493$	1.49827e11	+	9.04548e10i	0.114230	+	0.0689637i
$494$		0			0
$495$	−1.01168e11			−0.0757393
$496$		0			0
$497$	5.24810e11			0.385832
$498$		0			0
$499$		−	1.14978e12i		−	0.830162i	−0.909785	−	0.415081i	$-0.863753\pi$
$499$		−	1.14978e12i		−	0.830162i	0.909785	−	0.415081i	$-0.136247\pi$
$500$		0			0
$501$	−3.21026e12			−2.27651
$502$		0			0
$503$		−	1.33344e12i		−	0.928788i	−0.885629	−	0.464394i	$-0.846272\pi$
$503$		−	1.33344e12i		−	0.928788i	0.885629	−	0.464394i	$-0.153728\pi$
$504$		0			0
$505$		−	2.87013e9i		−	0.00196377i
$506$		0			0
$507$			1.93487e12i			1.30052i
$508$		0			0
$509$	−2.18589e12			−1.44344			−0.721721	−	0.692185i	$-0.756648\pi$
$509$	−2.18589e12			−1.44344			−0.721721	+	0.692185i	$0.756648\pi$
$510$		0			0
$511$	1.92259e10			0.0124737
$512$		0			0
$513$			1.09568e12i			0.698484i
$514$		0			0
$515$			1.54797e11i			0.0969681i
$516$		0			0
$517$			3.77832e11i			0.232590i
$518$		0			0
$519$	−1.55969e12			−0.943592
$520$		0			0
$521$		−	9.81356e11i		−	0.583521i	−0.956491	−	0.291761i	$-0.905759\pi$
$521$		−	9.81356e11i		−	0.583521i	0.956491	−	0.291761i	$-0.0942411\pi$
$522$		0			0
$523$	−1.01562e11			−0.0593572			−0.0296786	−	0.999559i	$-0.509448\pi$
$523$	−1.01562e11			−0.0593572			−0.0296786	+	0.999559i	$0.509448\pi$
$524$		0			0
$525$	−6.06476e11			−0.348415
$526$		0			0
$527$	−2.57373e12	−	1.55383e12i	−1.45350	−	0.877518i
$528$		0			0
$529$	1.67666e12			0.930883
$530$		0			0
$531$	1.56105e12			0.852099
$532$		0			0
$533$			1.55788e12i			0.836107i
$534$		0			0
$535$	2.27083e11			0.119837
$536$		0			0
$537$		−	2.98768e12i		−	1.55042i
$538$		0			0
$539$			1.30034e12i			0.663601i
$540$		0			0
$541$		−	2.29392e12i		−	1.15131i	−0.817694	−	0.575653i	$-0.804748\pi$
$541$		−	2.29392e12i		−	1.15131i	0.817694	−	0.575653i	$-0.195252\pi$
$542$		0			0
$543$	2.01521e12			0.994765
$544$		0			0
$545$	1.62861e11			0.0790739
$546$		0			0
$547$			4.31625e10i			0.0206140i	0.999947	+	0.0103070i	$0.00328088\pi$
$547$			4.31625e10i			0.0206140i	−0.999947	+	0.0103070i	$0.996719\pi$
$548$		0			0
$549$			4.74522e12i			2.22937i
$550$		0			0
$551$		−	2.18397e11i		−	0.100940i
$552$		0			0
$553$	7.83189e11			0.356126
$554$		0			0
$555$		−	3.52198e11i		−	0.157568i
$556$		0			0
$557$	−1.26017e12			−0.554730			−0.277365	−	0.960765i	$-0.589461\pi$
$557$	−1.26017e12			−0.554730			−0.277365	+	0.960765i	$0.589461\pi$
$558$		0			0
$559$	3.37798e12			1.46320
$560$		0			0
$561$	−1.35578e12	+	2.24569e12i	−0.577907	+	0.957231i
$562$		0			0
$563$	−1.57680e12			−0.661436			−0.330718	−	0.943730i	$-0.607291\pi$
$563$	−1.57680e12			−0.661436			−0.330718	+	0.943730i	$0.607291\pi$
$564$		0			0
$565$	−2.31111e11			−0.0954121
$566$		0			0
$567$		−	5.02547e10i		−	0.0204199i
$568$		0			0
$569$	4.44023e12			1.77583			0.887913	−	0.460012i	$-0.152155\pi$
$569$	4.44023e12			1.77583			0.887913	+	0.460012i	$0.152155\pi$
$570$		0			0
$571$			2.59026e11i			0.101972i	0.998699	+	0.0509860i	$0.0162364\pi$
$571$			2.59026e11i			0.101972i	−0.998699	+	0.0509860i	$0.983764\pi$
$572$		0			0
$573$			4.74067e11i			0.183715i
$574$		0			0
$575$		−	6.85842e11i		−	0.261649i
$576$		0			0
$577$	2.66713e12			1.00173			0.500867	−	0.865524i	$-0.333015\pi$
$577$	2.66713e12			1.00173			0.500867	+	0.865524i	$0.333015\pi$
$578$		0			0
$579$	1.14989e12			0.425209
$580$		0			0
$581$			7.87685e11i			0.286787i
$582$		0			0
$583$			1.19872e12i			0.429743i
$584$		0			0
$585$			4.14312e11i			0.146260i
$586$		0			0
$587$	4.82084e12			1.67591			0.837956	−	0.545738i	$-0.183751\pi$
$587$	4.82084e12			1.67591			0.837956	+	0.545738i	$0.183751\pi$
$588$		0			0
$589$			3.75162e12i			1.28440i
$590$		0			0
$591$	−4.17684e12			−1.40833
$592$		0			0
$593$	2.49924e12			0.829968			0.414984	−	0.909829i	$-0.363787\pi$
$593$	2.49924e12			0.829968			0.414984	+	0.909829i	$0.363787\pi$
$594$		0			0
$595$	2.37845e10	−	3.93960e10i	0.00777977	−	0.0128862i
$596$		0			0
$597$	3.08918e12			0.995310
$598$		0			0
$599$	3.23334e12			1.02620			0.513098	−	0.858330i	$-0.328498\pi$
$599$	3.23334e12			1.02620			0.513098	+	0.858330i	$0.328498\pi$
$600$		0			0
$601$			5.50244e12i			1.72036i	0.509988	+	0.860181i	$0.329650\pi$
$601$			5.50244e12i			1.72036i	−0.509988	+	0.860181i	$0.670350\pi$
$602$		0			0
$603$	−7.56660e12			−2.33063
$604$		0			0
$605$		−	1.16995e11i		−	0.0355032i
$606$		0			0
$607$			3.65674e12i			1.09331i	0.837357	+	0.546657i	$0.184100\pi$
$607$			3.65674e12i			1.09331i	−0.837357	+	0.546657i	$0.815900\pi$
$608$		0			0
$609$		−	1.58566e11i		−	0.0467125i
$610$		0			0
$611$	1.54732e12			0.449155
$612$		0			0
$613$	1.17726e12			0.336743			0.168372	−	0.985724i	$-0.446149\pi$
$613$	1.17726e12			0.336743			0.168372	+	0.985724i	$0.446149\pi$
$614$		0			0
$615$			2.44114e11i			0.0688104i
$616$		0			0
$617$		−	3.74523e12i		−	1.04039i	−0.854048	−	0.520194i	$-0.825859\pi$
$617$		−	3.74523e12i		−	1.04039i	0.854048	−	0.520194i	$-0.174141\pi$
$618$		0			0
$619$		−	6.72874e12i		−	1.84215i	−0.389381	−	0.921077i	$-0.627311\pi$
$619$		−	6.72874e12i		−	1.84215i	0.389381	−	0.921077i	$-0.372689\pi$
$620$		0			0
$621$	−8.99623e11			−0.242744
$622$		0			0
$623$			3.61956e11i			0.0962631i
$624$		0			0
$625$	3.76030e12			0.985739
$626$		0			0
$627$	3.27345e12			0.845865
$628$		0			0
$629$	−4.78226e12	−	2.88718e12i	−1.21816	−	0.735439i
$630$		0			0
$631$	4.61439e12			1.15873			0.579365	−	0.815068i	$-0.303300\pi$
$631$	4.61439e12			1.15873			0.579365	+	0.815068i	$0.303300\pi$
$632$		0			0
$633$	1.37797e12			0.341131
$634$		0			0
$635$			4.05707e11i			0.0990218i
$636$		0			0
$637$	5.32524e12			1.28148
$638$		0			0
$639$			1.17430e13i			2.78629i
$640$		0			0
$641$		−	7.42031e12i		−	1.73605i	−0.496524	−	0.868023i	$-0.665391\pi$
$641$		−	7.42031e12i		−	1.73605i	0.496524	−	0.868023i	$-0.334609\pi$
$642$		0			0
$643$		−	6.05336e12i		−	1.39652i	−0.715845	−	0.698260i	$-0.753958\pi$
$643$		−	6.05336e12i		−	1.39652i	0.715845	−	0.698260i	$-0.246042\pi$
$644$		0			0
$645$	5.29316e11			0.120419
$646$		0			0
$647$	4.28077e12			0.960402			0.480201	−	0.877159i	$-0.340564\pi$
$647$	4.28077e12			0.960402			0.480201	+	0.877159i	$0.340564\pi$
$648$		0			0
$649$		−	1.70331e12i		−	0.376872i
$650$		0			0
$651$			2.72385e12i			0.594387i
$652$		0			0
$653$		−	2.02322e12i		−	0.435445i	−0.976011	−	0.217723i	$-0.930137\pi$
$653$		−	2.02322e12i		−	0.435445i	0.976011	−	0.217723i	$-0.0698629\pi$
$654$		0			0
$655$	−1.41112e11			−0.0299556
$656$		0			0
$657$			4.30196e11i			0.0900787i
$658$		0			0
$659$	2.09148e12			0.431985			0.215993	−	0.976395i	$-0.430701\pi$
$659$	2.09148e12			0.431985			0.215993	+	0.976395i	$0.430701\pi$
$660$		0			0
$661$	−3.66578e12			−0.746895			−0.373448	−	0.927651i	$-0.621824\pi$
$661$	−3.66578e12			−0.746895			−0.373448	+	0.927651i	$0.621824\pi$
$662$		0			0
$663$	9.19669e12	+	5.55230e12i	1.84851	+	1.11599i
$664$		0			0
$665$	−5.74259e10			−0.0113870
$666$		0			0
$667$	1.79317e11			0.0350797
$668$		0			0
$669$			1.42118e12i			0.274305i
$670$		0			0
$671$	5.17769e12			0.986017
$672$		0			0
$673$			4.12814e12i			0.775686i	0.921725	+	0.387843i	$0.126780\pi$
$673$			4.12814e12i			0.775686i	−0.921725	+	0.387843i	$0.873220\pi$
$674$		0			0
$675$		−	4.95622e12i		−	0.918933i
$676$		0			0
$677$			3.86753e12i			0.707594i	0.935322	+	0.353797i	$0.115110\pi$
$677$			3.86753e12i			0.707594i	−0.935322	+	0.353797i	$0.884890\pi$
$678$		0			0
$679$	−1.44802e12			−0.261434
$680$		0			0
$681$	−1.22262e13			−2.17835
$682$		0			0
$683$			6.60671e12i			1.16170i	0.814012	+	0.580848i	$0.197279\pi$
$683$			6.60671e12i			1.16170i	−0.814012	+	0.580848i	$0.802721\pi$
$684$		0			0
$685$			3.99126e11i			0.0692631i
$686$		0			0
$687$		−	2.33591e12i		−	0.400084i
$688$		0			0
$689$	4.90907e12			0.829875
$690$		0			0
$691$		−	8.12202e12i		−	1.35523i	−0.735417	−	0.677615i	$-0.763014\pi$
$691$		−	8.12202e12i		−	1.35523i	0.735417	−	0.677615i	$-0.236986\pi$
$692$		0			0
$693$	1.45382e12			0.239449
$694$		0			0
$695$	−1.68023e11			−0.0273172
$696$		0			0
$697$	3.31465e12	+	2.00115e12i	0.531974	+	0.321168i
$698$		0			0
$699$	−1.33547e13			−2.11586
$700$		0			0
$701$	6.09018e12			0.952574			0.476287	−	0.879290i	$-0.341982\pi$
$701$	6.09018e12			0.952574			0.476287	+	0.879290i	$0.341982\pi$
$702$		0			0
$703$			6.97091e12i			1.07644i
$704$		0			0
$705$	2.42459e11			0.0369647
$706$		0			0
$707$			4.12448e10i			0.00620843i
$708$		0			0
$709$		−	1.24847e12i		−	0.185553i	−0.995687	−	0.0927767i	$-0.970426\pi$
$709$		−	1.24847e12i		−	0.185553i	0.995687	−	0.0927767i	$-0.0295743\pi$
$710$		0			0
$711$			1.75245e13i			2.57177i
$712$		0			0
$713$	−3.08031e12			−0.446366
$714$		0			0
$715$	4.52071e11			0.0646888
$716$		0			0
$717$			1.47953e13i			2.09068i
$718$		0			0
$719$			4.85534e12i			0.677547i	0.940868	+	0.338774i	$0.110012\pi$
$719$			4.85534e12i			0.677547i	−0.940868	+	0.338774i	$0.889988\pi$
$720$		0			0
$721$		−	2.22448e12i		−	0.306563i
$722$		0			0
$723$	−4.56979e12			−0.621976
$724$		0			0
$725$			9.87898e11i			0.132798i
$726$		0			0
$727$	−3.36183e12			−0.446345			−0.223173	−	0.974779i	$-0.571641\pi$
$727$	−3.36183e12			−0.446345			−0.223173	+	0.974779i	$0.571641\pi$
$728$		0			0
$729$	1.24237e13			1.62921
$730$		0			0
$731$	4.33912e12	−	7.18722e12i	0.562049	−	0.930964i
$732$		0			0
$733$	−1.02897e13			−1.31654			−0.658271	−	0.752781i	$-0.728712\pi$
$733$	−1.02897e13			−1.31654			−0.658271	+	0.752781i	$0.728712\pi$
$734$		0			0
$735$	8.34442e11			0.105464
$736$		0			0
$737$			8.25619e12i			1.03080i
$738$		0			0
$739$	1.10714e12			0.136553			0.0682766	−	0.997666i	$-0.478250\pi$
$739$	1.10714e12			0.136553			0.0682766	+	0.997666i	$0.478250\pi$
$740$		0			0
$741$		−	1.34056e13i		−	1.63345i
$742$		0			0
$743$		−	5.80468e12i		−	0.698761i	−0.936981	−	0.349380i	$-0.886392\pi$
$743$		−	5.80468e12i		−	0.698761i	0.936981	−	0.349380i	$-0.113608\pi$
$744$		0			0
$745$		−	4.63090e11i		−	0.0550760i
$746$		0			0
$747$	−1.76251e13			−2.07104
$748$		0			0
$749$	−3.26326e12			−0.378864
$750$		0			0
$751$			8.01566e12i			0.919517i	0.888044	+	0.459758i	$0.152064\pi$
$751$			8.01566e12i			0.919517i	−0.888044	+	0.459758i	$0.847936\pi$
$752$		0			0
$753$			1.96249e13i			2.22449i
$754$		0			0
$755$			1.18377e11i			0.0132588i
$756$		0			0
$757$	−4.40521e12			−0.487568			−0.243784	−	0.969829i	$-0.578389\pi$
$757$	−4.40521e12			−0.487568			−0.243784	+	0.969829i	$0.578389\pi$
$758$		0			0
$759$			2.68770e12i			0.293963i
$760$		0			0
$761$	−1.23860e13			−1.33875			−0.669377	−	0.742923i	$-0.733439\pi$
$761$	−1.23860e13			−1.33875			−0.669377	+	0.742923i	$0.733439\pi$
$762$		0			0
$763$	−2.34037e12			−0.249991
$764$		0			0
$765$	8.81517e11	+	5.32196e11i	0.0930581	+	0.0561818i
$766$		0			0
$767$	−6.97553e12			−0.727776
$768$		0			0
$769$	−1.34212e13			−1.38396			−0.691978	−	0.721919i	$-0.743260\pi$
$769$	−1.34212e13			−1.38396			−0.691978	+	0.721919i	$0.743260\pi$
$770$		0			0
$771$		−	2.70863e12i		−	0.276061i
$772$		0			0
$773$	3.10743e12			0.313036			0.156518	−	0.987675i	$-0.449973\pi$
$773$	3.10743e12			0.313036			0.156518	+	0.987675i	$0.449973\pi$
$774$		0			0
$775$		−	1.69701e13i		−	1.68977i
$776$		0			0
$777$			5.06121e12i			0.498150i
$778$		0			0
$779$		−	4.83163e12i		−	0.470084i
$780$		0			0
$781$	1.28133e13			1.23234
$782$		0			0
$783$	1.29583e12			0.123203
$784$		0			0
$785$		−	1.66495e11i		−	0.0156491i
$786$		0			0
$787$			4.60263e12i			0.427681i	0.976869	+	0.213840i	$0.0685973\pi$
$787$			4.60263e12i			0.427681i	−0.976869	+	0.213840i	$0.931403\pi$
$788$		0			0
$789$		−	7.91084e12i		−	0.726735i
$790$		0			0
$791$	3.32115e12			0.301644
$792$		0			0
$793$		−	2.12040e13i		−	1.90410i
$794$		0			0
$795$	7.69231e11			0.0682975
$796$		0			0
$797$	1.39540e13			1.22500			0.612502	−	0.790469i	$-0.290163\pi$
$797$	1.39540e13			1.22500			0.612502	+	0.790469i	$0.290163\pi$
$798$		0			0
$799$	1.98759e12	−	3.29219e12i	0.172530	−	0.285775i
$800$		0			0
$801$	−8.09906e12			−0.695165
$802$		0			0
$803$	4.69402e11			0.0398405
$804$		0			0
$805$		−	4.71502e10i		−	0.00395733i
$806$		0			0
$807$	1.86884e13			1.55110
$808$		0			0
$809$			2.10896e13i			1.73101i	0.500896	+	0.865507i	$0.333004\pi$
$809$			2.10896e13i			1.73101i	−0.500896	+	0.865507i	$0.666996\pi$
$810$		0			0
$811$			3.64082e12i			0.295533i	0.989022	+	0.147766i	$0.0472084\pi$
$811$			3.64082e12i			0.295533i	−0.989022	+	0.147766i	$0.952792\pi$
$812$		0			0
$813$			1.25513e13i			1.00758i
$814$		0			0
$815$	−2.19338e11			−0.0174142
$816$		0			0
$817$	−1.04765e13			−0.822655
$818$		0			0
$819$		−	5.95380e12i		−	0.462399i
$820$		0			0
$821$		−	1.79752e13i		−	1.38080i	−0.723429	−	0.690399i	$-0.757435\pi$
$821$		−	1.79752e13i		−	1.38080i	0.723429	−	0.690399i	$-0.242565\pi$
$822$		0			0
$823$		−	4.92718e12i		−	0.374368i	−0.982325	−	0.187184i	$-0.940064\pi$
$823$		−	4.92718e12i		−	0.374368i	0.982325	−	0.187184i	$-0.0599361\pi$
$824$		0			0
$825$	−1.48071e13			−1.11283
$826$		0			0
$827$			5.12459e12i			0.380964i	0.981691	+	0.190482i	$0.0610051\pi$
$827$			5.12459e12i			0.380964i	−0.981691	+	0.190482i	$0.938995\pi$
$828$		0			0
$829$	1.91602e13			1.40898			0.704490	−	0.709714i	$-0.251176\pi$
$829$	1.91602e13			1.40898			0.704490	+	0.709714i	$0.251176\pi$
$830$		0			0
$831$	−4.16450e13			−3.02941
$832$		0			0
$833$	6.84043e12	−	1.13303e13i	0.492245	−	0.815342i
$834$		0			0
$835$	1.37499e12			0.0978839
$836$		0			0
$837$	−2.22598e13			−1.56768
$838$		0			0
$839$			9.27849e12i			0.646470i	0.946319	+	0.323235i	$0.104770\pi$
$839$			9.27849e12i			0.646470i	−0.946319	+	0.323235i	$0.895230\pi$
$840$		0			0
$841$	1.42489e13			0.982196
$842$		0			0
$843$		−	3.05800e13i		−	2.08552i
$844$		0			0
$845$		−	8.28729e11i		−	0.0559187i
$846$		0			0
$847$			1.68126e12i			0.112243i
$848$		0			0
$849$	−1.31933e13			−0.871501
$850$		0			0
$851$	−5.72354e12			−0.374095
$852$		0			0
$853$		−	1.45073e13i		−	0.938244i	−0.883133	−	0.469122i	$-0.844571\pi$
$853$		−	1.45073e13i		−	0.938244i	0.883133	−	0.469122i	$-0.155429\pi$
$854$		0			0
$855$		−	1.28495e12i		−	0.0822317i
$856$		0			0
$857$		−	6.75756e12i		−	0.427933i	−0.976841	−	0.213967i	$-0.931362\pi$
$857$		−	6.75756e12i		−	0.427933i	0.976841	−	0.213967i	$-0.0686384\pi$
$858$		0			0
$859$	6.20315e11			0.0388726			0.0194363	−	0.999811i	$-0.493813\pi$
$859$	6.20315e11			0.0388726			0.0194363	+	0.999811i	$0.493813\pi$
$860$		0			0
$861$		−	3.50800e12i		−	0.217543i
$862$		0			0
$863$	−1.36639e12			−0.0838542			−0.0419271	−	0.999121i	$-0.513350\pi$
$863$	−1.36639e12			−0.0838542			−0.0419271	+	0.999121i	$0.513350\pi$
$864$		0			0
$865$	6.68033e11			0.0405719
$866$		0			0
$867$	2.36269e13	−	1.24354e13i	1.42010	−	0.747435i
$868$		0			0
$869$	1.91216e13			1.13746
$870$		0			0
$871$	3.38113e13			1.99058
$872$		0			0
$873$		−	3.24007e13i		−	1.88795i
$874$		0			0
$875$	5.20765e11			0.0300335
$876$		0			0
$877$		−	1.47910e13i		−	0.844304i	−0.906525	−	0.422152i	$-0.861275\pi$
$877$		−	1.47910e13i		−	0.844304i	0.906525	−	0.422152i	$-0.138725\pi$
$878$		0			0
$879$		−	1.23432e13i		−	0.697396i
$880$		0			0
$881$		−	1.90695e12i		−	0.106647i	−0.998577	−	0.0533234i	$-0.983019\pi$
$881$		−	1.90695e12i		−	0.106647i	0.998577	−	0.0533234i	$-0.0169814\pi$
$882$		0			0
$883$	−9.34392e12			−0.517256			−0.258628	−	0.965977i	$-0.583270\pi$
$883$	−9.34392e12			−0.517256			−0.258628	+	0.965977i	$0.583270\pi$
$884$		0			0
$885$	−1.09304e12			−0.0598949
$886$		0			0
$887$			6.67058e12i			0.361832i	0.983499	+	0.180916i	$0.0579063\pi$
$887$			6.67058e12i			0.361832i	−0.983499	+	0.180916i	$0.942094\pi$
$888$		0			0
$889$		−	5.83015e12i		−	0.313056i
$890$		0			0
$891$		−	1.22697e12i		−	0.0652206i
$892$		0			0
$893$	−4.79889e12			−0.252528
$894$		0			0
$895$			1.27966e12i			0.0666639i
$896$		0			0
$897$	1.10069e13			0.567672
$898$		0			0
$899$	4.43692e12			0.226550
$900$		0			0
$901$	6.30586e12	−	1.04449e13i	0.318774	−	0.528009i
$902$		0			0
$903$	−7.60645e12			−0.380704
$904$		0			0
$905$	−8.63138e11			−0.0427722
$906$		0			0
$907$			2.25953e13i			1.10863i	0.832308	+	0.554313i	$0.187019\pi$
$907$			2.25953e13i			1.10863i	−0.832308	+	0.554313i	$0.812981\pi$
$908$		0			0
$909$	−9.22885e11			−0.0448343
$910$		0			0
$911$			2.70620e13i			1.30175i	0.759186	+	0.650873i	$0.225597\pi$
$911$			2.70620e13i			1.30175i	−0.759186	+	0.650873i	$0.774403\pi$
$912$		0			0
$913$			1.92314e13i			0.915992i
$914$		0			0
$915$		−	3.32258e12i		−	0.156704i
$916$		0			0
$917$	2.02783e12			0.0947042
$918$		0			0
$919$	−1.59897e13			−0.739468			−0.369734	−	0.929138i	$-0.620551\pi$
$919$	−1.59897e13			−0.739468			−0.369734	+	0.929138i	$0.620551\pi$
$920$		0			0
$921$			4.23398e13i			1.93901i
$922$		0			0
$923$		−	5.24738e13i		−	2.37977i
$924$		0			0
$925$		−	3.15323e13i		−	1.41618i
$926$		0			0
$927$	4.97745e13			2.21385
$928$		0			0
$929$		−	9.00625e12i		−	0.396710i	−0.980130	−	0.198355i	$-0.936440\pi$
$929$		−	9.00625e12i		−	0.396710i	0.980130	−	0.198355i	$-0.0635599\pi$
$930$		0			0
$931$	−1.65157e13			−0.720485
$932$		0			0
$933$	4.09129e13			1.76764
$934$		0			0
$935$	5.80699e11	−	9.61855e11i	0.0248484	−	0.0411583i
$936$		0			0
$937$	−4.00157e13			−1.69591			−0.847954	−	0.530071i	$-0.822165\pi$
$937$	−4.00157e13			−1.69591			−0.847954	+	0.530071i	$0.822165\pi$
$938$		0			0
$939$	−4.98809e13			−2.09382
$940$		0			0
$941$			2.42260e13i			1.00723i	0.863928	+	0.503615i	$0.167997\pi$
$941$			2.42260e13i			1.00723i	−0.863928	+	0.503615i	$0.832003\pi$
$942$		0			0
$943$	3.96707e12			0.163368
$944$		0			0
$945$		−	3.40730e11i		−	0.0138985i
$946$		0			0
$947$		−	3.52100e13i		−	1.42263i	−0.702875	−	0.711313i	$-0.748101\pi$
$947$		−	3.52100e13i		−	1.42263i	0.702875	−	0.711313i	$-0.251899\pi$
$948$		0			0
$949$		−	1.92233e12i		−	0.0769360i
$950$		0			0
$951$	6.00782e13			2.38180
$952$		0			0
$953$	9.99226e12			0.392415			0.196208	−	0.980562i	$-0.437137\pi$
$953$	9.99226e12			0.392415			0.196208	+	0.980562i	$0.437137\pi$
$954$		0			0
$955$		−	2.03049e11i		−	0.00789923i
$956$		0			0
$957$		−	3.87141e12i		−	0.149199i
$958$		0			0
$959$		−	5.73557e12i		−	0.218974i
$960$		0			0
$961$	−4.97778e13			−1.88270
$962$		0			0
$963$		−	7.30180e13i		−	2.73597i
$964$		0			0
$965$	−4.92511e11			−0.0182828
$966$		0			0
$967$	−3.17277e12			−0.116686			−0.0583431	−	0.998297i	$-0.518582\pi$
$967$	−3.17277e12			−0.116686			−0.0583431	+	0.998297i	$0.518582\pi$
$968$		0			0
$969$	−2.85227e13	−	1.72200e13i	−1.03928	−	0.627444i
$970$		0			0
$971$	−2.55138e13			−0.921061			−0.460531	−	0.887644i	$-0.652341\pi$
$971$	−2.55138e13			−0.921061			−0.460531	+	0.887644i	$0.652341\pi$
$972$		0			0
$973$	2.41455e12			0.0863630
$974$		0			0
$975$			6.06392e13i			2.14898i
$976$		0			0
$977$	−4.02976e12			−0.141499			−0.0707496	−	0.997494i	$-0.522539\pi$
$977$	−4.02976e12			−0.141499			−0.0707496	+	0.997494i	$0.522539\pi$
$978$		0			0
$979$			8.83718e12i			0.307462i
$980$		0			0
$981$		−	5.23676e13i		−	1.80531i
$982$		0			0
$983$			9.84380e12i			0.336258i	0.985765	+	0.168129i	$0.0537724\pi$
$983$			9.84380e12i			0.336258i	−0.985765	+	0.168129i	$0.946228\pi$
$984$		0			0
$985$	1.78899e12			0.0605544
$986$		0			0
$987$	−3.48422e12			−0.116863
$988$		0			0
$989$		−	8.60186e12i		−	0.285897i
$990$		0			0
$991$			7.10324e12i			0.233951i	0.993135	+	0.116976i	$0.0373199\pi$
$991$			7.10324e12i			0.233951i	−0.993135	+	0.116976i	$0.962680\pi$
$992$		0			0
$993$		−	7.79626e13i		−	2.54457i
$994$		0			0
$995$	−1.32313e12			−0.0427956
$996$		0			0
$997$		−	3.04471e13i		−	0.975928i	−0.872864	−	0.487964i	$-0.837740\pi$
$997$		−	3.04471e13i		−	0.975928i	0.872864	−	0.487964i	$-0.162260\pi$
$998$		0			0
$999$	−4.13611e13			−1.31385

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	272.10.b.c.33.12		12
4.3	odd	2		17.10.b.a.16.9	✓	12
12.11	even	2		153.10.d.b.118.4		12
17.16	even	2	inner	272.10.b.c.33.1		12
68.47	odd	4		289.10.a.c.1.4		12
68.55	odd	4		289.10.a.c.1.3		12
68.67	odd	2		17.10.b.a.16.10	yes	12
204.203	even	2		153.10.d.b.118.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.10.b.a.16.9	✓	12	4.3	odd	2
17.10.b.a.16.10	yes	12	68.67	odd	2
153.10.d.b.118.3		12	204.203	even	2
153.10.d.b.118.4		12	12.11	even	2
272.10.b.c.33.1		12	17.16	even	2	inner
272.10.b.c.33.12		12	1.1	even	1	trivial
289.10.a.c.1.3		12	68.55	odd	4
289.10.a.c.1.4		12	68.47	odd	4

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:	\( N \)	\(=\)	\( 272 = 2^{4} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	272.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+225.146i q^{3} -96.4328i q^{5} +1385.77i q^{7} -31007.8 q^{9} +33833.7i q^{11} +138558. q^{13} +21711.5 q^{15} +(177982. - 294805. i) q^{17} -429726. q^{19} -312001. q^{21} -352831. i q^{23} +1.94383e6 q^{25} -2.54973e6i q^{27} +508224. i q^{29} -8.73026e6i q^{31} -7.61753e6 q^{33} +133634. q^{35} -1.62218e7i q^{37} +3.11958e7i q^{39} +1.12435e7i q^{41} +2.43795e7 q^{43} +2.99017e6i q^{45} +1.11673e7 q^{47} +3.84332e7 q^{49} +(6.63743e7 + 4.00720e7i) q^{51} +3.54297e7 q^{53} +3.26268e6 q^{55} -9.67510e7i q^{57} -5.03437e7 q^{59} -1.53033e8i q^{61} -4.29697e7i q^{63} -1.33615e7i q^{65} +2.44023e8 q^{67} +7.94386e7 q^{69} -3.78713e8i q^{71} -1.38738e7i q^{73} +4.37645e8i q^{75} -4.68858e7 q^{77} -5.65164e8i q^{79} -3.62648e7 q^{81} +5.68408e8 q^{83} +(-2.84289e7 - 1.71633e7i) q^{85} -1.14425e8 q^{87} +2.61194e8 q^{89} +1.92010e8i q^{91} +1.96558e9 q^{93} +4.14396e7i q^{95} +1.04492e9i q^{97} -1.04911e9i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 9184 q^{9} - 63204 q^{13} + 243480 q^{15} - 105960 q^{17} - 1110672 q^{19} - 172580 q^{21} - 4441796 q^{25} - 6557404 q^{33} - 3519864 q^{35} - 10004616 q^{43} + 112552440 q^{47} + 121354720 q^{49}+ \cdots + 1635779524 q^{93}+O(q^{100}) \) 12 * q - 9184 * q^9 - 63204 * q^13 + 243480 * q^15 - 105960 * q^17 - 1110672 * q^19 - 172580 * q^21 - 4441796 * q^25 - 6557404 * q^33 - 3519864 * q^35 - 10004616 * q^43 + 112552440 * q^47 + 121354720 * q^49 + 52506472 * q^51 + 76804272 * q^53 - 300732568 * q^55 - 11618904 * q^59 + 304208752 * q^67 - 211308236 * q^69 + 138357828 * q^77 - 363335792 * q^81 + 845042136 * q^83 - 388949632 * q^85 - 610860648 * q^87 - 938223804 * q^89 + 1635779524 * q^93

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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