LMFDB - Embedded newform 176.15.h.e.65.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [176,15,Mod(65,176)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(176, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 15, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("176.65");

S:= CuspForms(chi, 15);

N := Newforms(S);

Level:	$ N $	$=$	$ 176 = 2^{4} \cdot 11 $
Weight:	$ k $	$=$	$ 15 $
Character orbit:	$[\chi]$	$=$	176.h (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$218.818983947$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 2 x^{13} - 38299509 x^{12} + 1255603312 x^{11} + 548839279225666 x^{10} + \cdots + 61\!\cdots\!00 $ x^14 - 2x^13 - 38299509x^12 + 1255603312x^11 + 548839279225666x^10 - 62614853719962708x^9 - 3647746096331338780406x^8 + 909709401522475794317032x^7 + 11341528896592004854718535645x^6 - 4608641836922062401823748151666x^5 - 14199433672677462467950659880576257x^4 + 7648786246750765008517429480227297240x^3 + 5468207905418692926727112276614665574080x^2 - 3973529610997293170696151713639263984312200*x + 613220922939245616829253347741252697812260300
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{64}\cdot 3^{10}\cdot 11^{7} $
Twist minimal:	no (minimal twist has level 22)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			65.4
Root			$-2599.55 + 1.41421i$ of defining polynomial
Character	$\chi$	$=$	176.65
Dual form			176.15.h.e.65.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2913.55 q^{3} -75323.6 q^{5} +867659. i q^{7} +3.70579e6 q^{9} +O(q^{10})$	\(q-2913.55 q^{3} -75323.6 q^{5} +867659. i q^{7} +3.70579e6 q^{9} +(1.56983e7 - 1.15461e7i) q^{11} +3.47214e7i q^{13} +2.19459e8 q^{15} +2.76380e8i q^{17} +8.30217e8i q^{19} -2.52797e9i q^{21} +5.71780e9 q^{23} -4.29878e8 q^{25} +3.13842e9 q^{27} +2.61317e10i q^{29} +1.69092e10 q^{31} +(-4.57378e10 + 3.36401e10i) q^{33} -6.53552e10i q^{35} +7.99862e10 q^{37} -1.01163e11i q^{39} -1.95749e11i q^{41} -2.34553e11i q^{43} -2.79133e11 q^{45} +6.41052e11 q^{47} -7.46096e10 q^{49} -8.05247e11i q^{51} -1.06181e12 q^{53} +(-1.18245e12 + 8.69693e11i) q^{55} -2.41888e12i q^{57} -3.40545e12 q^{59} -3.58361e12i q^{61} +3.21536e12i q^{63} -2.61534e12i q^{65} -4.80232e11 q^{67} -1.66591e13 q^{69} -4.31225e11 q^{71} -4.05979e12i q^{73} +1.25247e12 q^{75} +(1.00181e13 + 1.36208e13i) q^{77} +2.11530e13i q^{79} -2.68686e13 q^{81} +4.92388e12i q^{83} -2.08180e13i q^{85} -7.61359e13i q^{87} -7.53865e12 q^{89} -3.01264e13 q^{91} -4.92656e13 q^{93} -6.25349e13i q^{95} -4.89366e13 q^{97} +(5.81746e13 - 4.27874e13i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 4394 q^{3} + 69758 q^{5} + 11016572 q^{9}+O(q^{10}) $ 14 * q - 4394 * q^3 + 69758 * q^5 + 11016572 * q^9	$ 14 q - 4394 q^{3} + 69758 q^{5} + 11016572 q^{9} - 20143042 q^{11} + 1359602 q^{15} + 7305755542 q^{23} + 19291879452 q^{25} - 34093422830 q^{27} + 33569873942 q^{31} + 2885838062 q^{33} + 73167823966 q^{37} + 2000205168616 q^{45} + 1612717386124 q^{47} + 3424602524990 q^{49} - 3530064068164 q^{53} + 3715439610854 q^{55} + 818496564070 q^{59} - 16485465276922 q^{67} - 11394452631206 q^{69} + 19380879179878 q^{71} - 23016770893992 q^{75} + 60534793808304 q^{77} - 10394309810662 q^{81} - 117770741987650 q^{89} - 150621364097712 q^{91} + 27345122803162 q^{93} + 123398138843566 q^{97} - 118861332531788 q^{99}+O(q^{100}) $ 14 * q - 4394 * q^3 + 69758 * q^5 + 11016572 * q^9 - 20143042 * q^11 + 1359602 * q^15 + 7305755542 * q^23 + 19291879452 * q^25 - 34093422830 * q^27 + 33569873942 * q^31 + 2885838062 * q^33 + 73167823966 * q^37 + 2000205168616 * q^45 + 1612717386124 * q^47 + 3424602524990 * q^49 - 3530064068164 * q^53 + 3715439610854 * q^55 + 818496564070 * q^59 - 16485465276922 * q^67 - 11394452631206 * q^69 + 19380879179878 * q^71 - 23016770893992 * q^75 + 60534793808304 * q^77 - 10394309810662 * q^81 - 117770741987650 * q^89 - 150621364097712 * q^91 + 27345122803162 * q^93 + 123398138843566 * q^97 - 118861332531788 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/176\mathbb{Z}\right)^\times$.

$n$	$111$	$133$	$145$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	−2913.55			−1.33221			−0.666106	−	0.745857i	$-0.732040\pi$
$3$	−2913.55			−1.33221			−0.666106	+	0.745857i	$0.732040\pi$
$4$		0			0
$5$	−75323.6			−0.964141			−0.482071	−	0.876132i	$-0.660115\pi$
$5$	−75323.6			−0.964141			−0.482071	+	0.876132i	$0.660115\pi$
$6$		0			0
$7$			867659.i			1.05357i	0.849999	+	0.526784i	$0.176603\pi$
$7$			867659.i			1.05357i	−0.849999	+	0.526784i	$0.823397\pi$
$8$		0			0
$9$	3.70579e6			0.774788
$10$		0			0
$11$	1.56983e7	−	1.15461e7i	0.805572	−	0.592497i
$11$	1.56983e7	−	1.15461e7i	0.805572	−	0.592497i
$12$		0			0
$13$			3.47214e7i			0.553343i	0.960965	+	0.276671i	$0.0892314\pi$
$13$			3.47214e7i			0.553343i	−0.960965	+	0.276671i	$0.910769\pi$
$14$		0			0
$15$	2.19459e8			1.28444
$16$		0			0
$17$			2.76380e8i			0.673542i	0.941587	+	0.336771i	$0.109335\pi$
$17$			2.76380e8i			0.673542i	−0.941587	+	0.336771i	$0.890665\pi$
$18$		0			0
$19$			8.30217e8i			0.928787i	0.885629	+	0.464394i	$0.153728\pi$
$19$			8.30217e8i			0.928787i	−0.885629	+	0.464394i	$0.846272\pi$
$20$		0			0
$21$		−	2.52797e9i		−	1.40358i
$22$		0			0
$23$	5.71780e9			1.67932			0.839661	−	0.543111i	$-0.182754\pi$
$23$	5.71780e9			1.67932			0.839661	+	0.543111i	$0.182754\pi$
$24$		0			0
$25$	−4.29878e8			−0.0704312
$26$		0			0
$27$	3.13842e9			0.300030
$28$		0			0
$29$			2.61317e10i			1.51489i	0.652898	+	0.757446i	$0.273553\pi$
$29$			2.61317e10i			1.51489i	−0.652898	+	0.757446i	$0.726447\pi$
$30$		0			0
$31$	1.69092e10			0.614597			0.307298	−	0.951613i	$-0.400575\pi$
$31$	1.69092e10			0.614597			0.307298	+	0.951613i	$0.400575\pi$
$32$		0			0
$33$	−4.57378e10	+	3.36401e10i	−1.07319	+	0.789332i
$34$		0			0
$35$		−	6.53552e10i		−	1.01579i
$36$		0			0
$37$	7.99862e10			0.842564			0.421282	−	0.906930i	$-0.361580\pi$
$37$	7.99862e10			0.842564			0.421282	+	0.906930i	$0.361580\pi$
$38$		0			0
$39$		−	1.01163e11i		−	0.737170i
$40$		0			0
$41$		−	1.95749e11i		−	1.00511i	−0.864546	−	0.502553i	$-0.832394\pi$
$41$		−	1.95749e11i		−	1.00511i	0.864546	−	0.502553i	$-0.167606\pi$
$42$		0			0
$43$		−	2.34553e11i		−	0.862904i	−0.902136	−	0.431452i	$-0.858002\pi$
$43$		−	2.34553e11i		−	0.862904i	0.902136	−	0.431452i	$-0.141998\pi$
$44$		0			0
$45$	−2.79133e11			−0.747005
$46$		0			0
$47$	6.41052e11			1.26534			0.632671	−	0.774421i	$-0.281959\pi$
$47$	6.41052e11			1.26534			0.632671	+	0.774421i	$0.281959\pi$
$48$		0			0
$49$	−7.46096e10			−0.110008
$50$		0			0
$51$		−	8.05247e11i		−	0.897301i
$52$		0			0
$53$	−1.06181e12			−0.903891			−0.451946	−	0.892046i	$-0.649270\pi$
$53$	−1.06181e12			−0.903891			−0.451946	+	0.892046i	$0.649270\pi$
$54$		0			0
$55$	−1.18245e12	+	8.69693e11i	−0.776686	+	0.571251i
$56$		0			0
$57$		−	2.41888e12i		−	1.23734i
$58$		0			0
$59$	−3.40545e12			−1.36839			−0.684195	−	0.729299i	$-0.739846\pi$
$59$	−3.40545e12			−1.36839			−0.684195	+	0.729299i	$0.739846\pi$
$60$		0			0
$61$		−	3.58361e12i		−	1.14028i	−0.821547	−	0.570140i	$-0.806889\pi$
$61$		−	3.58361e12i		−	1.14028i	0.821547	−	0.570140i	$-0.193111\pi$
$62$		0			0
$63$			3.21536e12i			0.816292i
$64$		0			0
$65$		−	2.61534e12i		−	0.533501i
$66$		0			0
$67$	−4.80232e11			−0.0792370			−0.0396185	−	0.999215i	$-0.512614\pi$
$67$	−4.80232e11			−0.0792370			−0.0396185	+	0.999215i	$0.512614\pi$
$68$		0			0
$69$	−1.66591e13			−2.23721
$70$		0			0
$71$	−4.31225e11			−0.0474128			−0.0237064	−	0.999719i	$-0.507547\pi$
$71$	−4.31225e11			−0.0474128			−0.0237064	+	0.999719i	$0.507547\pi$
$72$		0			0
$73$		−	4.05979e12i		−	0.367488i	−0.982974	−	0.183744i	$-0.941178\pi$
$73$		−	4.05979e12i		−	0.367488i	0.982974	−	0.183744i	$-0.0588218\pi$
$74$		0			0
$75$	1.25247e12			0.0938293
$76$		0			0
$77$	1.00181e13	+	1.36208e13i	0.624237	+	0.848726i
$78$		0			0
$79$			2.11530e13i			1.10149i	0.834672	+	0.550747i	$0.185657\pi$
$79$			2.11530e13i			1.10149i	−0.834672	+	0.550747i	$0.814343\pi$
$80$		0			0
$81$	−2.68686e13			−1.17449
$82$		0			0
$83$			4.92388e12i			0.181452i	0.995876	+	0.0907258i	$0.0289187\pi$
$83$			4.92388e12i			0.181452i	−0.995876	+	0.0907258i	$0.971081\pi$
$84$		0			0
$85$		−	2.08180e13i		−	0.649390i
$86$		0			0
$87$		−	7.61359e13i		−	2.01816i
$88$		0			0
$89$	−7.53865e12			−0.170437			−0.0852185	−	0.996362i	$-0.527159\pi$
$89$	−7.53865e12			−0.170437			−0.0852185	+	0.996362i	$0.527159\pi$
$90$		0			0
$91$	−3.01264e13			−0.582985
$92$		0			0
$93$	−4.92656e13			−0.818773
$94$		0			0
$95$		−	6.25349e13i		−	0.895482i
$96$		0			0
$97$	−4.89366e13			−0.605664			−0.302832	−	0.953044i	$-0.597932\pi$
$97$	−4.89366e13			−0.605664			−0.302832	+	0.953044i	$0.597932\pi$
$98$		0			0
$99$	5.81746e13	−	4.27874e13i	0.624148	−	0.459060i
$100$		0			0
$101$		−	1.98660e14i		−	1.85294i	−0.376373	−	0.926468i	$-0.622829\pi$
$101$		−	1.98660e14i		−	1.85294i	0.376373	−	0.926468i	$-0.377171\pi$
$102$		0			0
$103$	1.35157e14			1.09895			0.549477	−	0.835509i	$-0.314827\pi$
$103$	1.35157e14			1.09895			0.549477	+	0.835509i	$0.314827\pi$
$104$		0			0
$105$			1.90415e14i			1.35325i
$106$		0			0
$107$			1.67405e14i			1.04251i	0.853400	+	0.521256i	$0.174536\pi$
$107$			1.67405e14i			1.04251i	−0.853400	+	0.521256i	$0.825464\pi$
$108$		0			0
$109$		−	1.43567e14i		−	0.785362i	−0.919675	−	0.392681i	$-0.871548\pi$
$109$		−	1.43567e14i		−	0.785362i	0.919675	−	0.392681i	$-0.128452\pi$
$110$		0			0
$111$	−2.33044e14			−1.12247
$112$		0			0
$113$	1.01260e14			0.430416			0.215208	−	0.976568i	$-0.430957\pi$
$113$	1.01260e14			0.430416			0.215208	+	0.976568i	$0.430957\pi$
$114$		0			0
$115$	−4.30685e14			−1.61910
$116$		0			0
$117$			1.28670e14i			0.428723i
$118$		0			0
$119$	−2.39804e14			−0.709623
$120$		0			0
$121$	1.13125e14	−	3.62509e14i	0.297893	−	0.954599i
$122$		0			0
$123$			5.70323e14i			1.33901i
$124$		0			0
$125$	4.92118e14			1.03205
$126$		0			0
$127$		−	1.04866e15i		−	1.96792i	−0.178381	−	0.983962i	$-0.557086\pi$
$127$		−	1.04866e15i		−	1.96792i	0.178381	−	0.983962i	$-0.442914\pi$
$128$		0			0
$129$			6.83382e14i			1.14957i
$130$		0			0
$131$		−	1.08808e15i		−	1.64347i	−0.569867	−	0.821737i	$-0.693005\pi$
$131$		−	1.08808e15i		−	1.64347i	0.569867	−	0.821737i	$-0.306995\pi$
$132$		0			0
$133$	−7.20345e14			−0.978541
$134$		0			0
$135$	−2.36397e14			−0.289272
$136$		0			0
$137$	9.50193e14			1.04898			0.524491	−	0.851416i	$-0.324256\pi$
$137$	9.50193e14			1.04898			0.524491	+	0.851416i	$0.324256\pi$
$138$		0			0
$139$			1.42444e15i			1.42082i	0.703788	+	0.710410i	$0.251491\pi$
$139$			1.42444e15i			1.42082i	−0.703788	+	0.710410i	$0.748509\pi$
$140$		0			0
$141$	−1.86773e15			−1.68570
$142$		0			0
$143$	4.00897e14	+	5.45069e14i	0.327854	+	0.445758i
$144$		0			0
$145$		−	1.96833e15i		−	1.46057i
$146$		0			0
$147$	2.17379e14			0.146553
$148$		0			0
$149$			2.48573e15i			1.52458i	0.647238	+	0.762288i	$0.275924\pi$
$149$			2.48573e15i			1.52458i	−0.647238	+	0.762288i	$0.724076\pi$
$150$		0			0
$151$		−	8.33993e14i		−	0.465933i	−0.972485	−	0.232967i	$-0.925157\pi$
$151$		−	8.33993e14i		−	0.465933i	0.972485	−	0.232967i	$-0.0748433\pi$
$152$		0			0
$153$			1.02421e15i			0.521852i
$154$		0			0
$155$	−1.27366e15			−0.592558
$156$		0			0
$157$	−2.83439e14			−0.120549			−0.0602744	−	0.998182i	$-0.519198\pi$
$157$	−2.83439e14			−0.120549			−0.0602744	+	0.998182i	$0.519198\pi$
$158$		0			0
$159$	3.09364e15			1.20417
$160$		0			0
$161$			4.96110e15i			1.76928i
$162$		0			0
$163$	4.94421e15			1.61727			0.808636	−	0.588309i	$-0.200206\pi$
$163$	4.94421e15			1.61727			0.808636	+	0.588309i	$0.200206\pi$
$164$		0			0
$165$	3.44513e15	−	2.53389e15i	1.03471	−	0.761028i
$166$		0			0
$167$		−	4.57889e13i		−	0.0126399i	−0.999980	−	0.00631997i	$-0.997988\pi$
$167$		−	4.57889e13i		−	0.0126399i	0.999980	−	0.00631997i	$-0.00201172\pi$
$168$		0			0
$169$	2.73180e15			0.693812
$170$		0			0
$171$			3.07661e15i			0.719613i
$172$		0			0
$173$		−	6.39265e15i		−	1.37835i	−0.724597	−	0.689173i	$-0.757974\pi$
$173$		−	6.39265e15i		−	1.37835i	0.724597	−	0.689173i	$-0.242026\pi$
$174$		0			0
$175$		−	3.72988e14i		−	0.0742042i
$176$		0			0
$177$	9.92192e15			1.82298
$178$		0			0
$179$	−2.80106e15			−0.475719			−0.237860	−	0.971300i	$-0.576446\pi$
$179$	−2.80106e15			−0.475719			−0.237860	+	0.971300i	$0.576446\pi$
$180$		0			0
$181$	6.02858e14			0.0947251			0.0473626	−	0.998878i	$-0.484918\pi$
$181$	6.02858e14			0.0947251			0.0473626	+	0.998878i	$0.484918\pi$
$182$		0			0
$183$			1.04410e16i			1.51909i
$184$		0			0
$185$	−6.02484e15			−0.812351
$186$		0			0
$187$	3.19112e15	+	4.33871e15i	0.399072	+	0.542587i
$188$		0			0
$189$			2.72308e15i			0.316103i
$190$		0			0
$191$	1.61123e16			1.73749			0.868747	−	0.495257i	$-0.164926\pi$
$191$	1.61123e16			1.73749			0.868747	+	0.495257i	$0.164926\pi$
$192$		0			0
$193$		−	3.93899e15i		−	0.394897i	−0.980313	−	0.197449i	$-0.936734\pi$
$193$		−	3.93899e15i		−	0.394897i	0.980313	−	0.197449i	$-0.0632656\pi$
$194$		0			0
$195$			7.61992e15i			0.710736i
$196$		0			0
$197$			4.74963e15i			0.412474i	0.978502	+	0.206237i	$0.0661218\pi$
$197$			4.74963e15i			0.412474i	−0.978502	+	0.206237i	$0.933878\pi$
$198$		0			0
$199$	1.59379e16			1.28962			0.644808	−	0.764345i	$-0.276937\pi$
$199$	1.59379e16			1.28962			0.644808	+	0.764345i	$0.276937\pi$
$200$		0			0
$201$	1.39918e15			0.105560
$202$		0			0
$203$	−2.26734e16			−1.59604
$204$		0			0
$205$			1.47445e16i			0.969065i
$206$		0			0
$207$	2.11889e16			1.30112
$208$		0			0
$209$	9.58576e15	+	1.30330e16i	0.550304	+	0.748205i
$210$		0			0
$211$		−	1.73324e16i		−	0.930854i	−0.885086	−	0.465427i	$-0.845901\pi$
$211$		−	1.73324e16i		−	0.930854i	0.885086	−	0.465427i	$-0.154099\pi$
$212$		0			0
$213$	1.25639e15			0.0631639
$214$		0			0
$215$			1.76674e16i			0.831961i
$216$		0			0
$217$			1.46714e16i			0.647520i
$218$		0			0
$219$			1.18284e16i			0.489572i
$220$		0			0
$221$	−9.59633e15			−0.372700
$222$		0			0
$223$	−3.42689e16			−1.24959			−0.624794	−	0.780790i	$-0.714817\pi$
$223$	−3.42689e16			−1.24959			−0.624794	+	0.780790i	$0.714817\pi$
$224$		0			0
$225$	−1.59304e15			−0.0545693
$226$		0			0
$227$		−	3.10536e16i		−	0.999840i	−0.866072	−	0.499920i	$-0.833363\pi$
$227$		−	3.10536e16i		−	0.999840i	0.866072	−	0.499920i	$-0.166637\pi$
$228$		0			0
$229$	−1.49288e16			−0.452039			−0.226019	−	0.974123i	$-0.572571\pi$
$229$	−1.49288e16			−0.452039			−0.226019	+	0.974123i	$0.572571\pi$
$230$		0			0
$231$	−2.91881e16	−	3.96848e16i	−0.831616	−	1.13068i
$232$		0			0
$233$			4.44002e16i			1.19095i	0.803374	+	0.595475i	$0.203036\pi$
$233$			4.44002e16i			1.19095i	−0.803374	+	0.595475i	$0.796964\pi$
$234$		0			0
$235$	−4.82863e16			−1.21997
$236$		0			0
$237$		−	6.16302e16i		−	1.46742i
$238$		0			0
$239$		−	5.76683e16i		−	1.29465i	−0.762214	−	0.647325i	$-0.775888\pi$
$239$		−	5.76683e16i		−	1.29465i	0.762214	−	0.647325i	$-0.224112\pi$
$240$		0			0
$241$		−	4.57373e16i		−	0.968614i	−0.874898	−	0.484307i	$-0.839072\pi$
$241$		−	4.57373e16i		−	0.968614i	0.874898	−	0.484307i	$-0.160928\pi$
$242$		0			0
$243$	6.32720e16			1.26464
$244$		0			0
$245$	5.61986e15			0.106063
$246$		0			0
$247$	−2.88263e16			−0.513938
$248$		0			0
$249$		−	1.43459e16i		−	0.241732i
$250$		0			0
$251$	7.06905e16			1.12628			0.563138	−	0.826363i	$-0.309594\pi$
$251$	7.06905e16			1.12628			0.563138	+	0.826363i	$0.309594\pi$
$252$		0			0
$253$	8.97599e16	−	6.60183e16i	1.35282	−	0.994994i
$254$		0			0
$255$			6.06541e16i			0.865125i
$256$		0			0
$257$	3.76880e16			0.508946			0.254473	−	0.967080i	$-0.418098\pi$
$257$	3.76880e16			0.508946			0.254473	+	0.967080i	$0.418098\pi$
$258$		0			0
$259$			6.94008e16i			0.887699i
$260$		0			0
$261$			9.68385e16i			1.17372i
$262$		0			0
$263$			1.52822e17i			1.75588i	0.478769	+	0.877941i	$0.341083\pi$
$263$			1.52822e17i			1.75588i	−0.478769	+	0.877941i	$0.658917\pi$
$264$		0			0
$265$	7.99794e16			0.871479
$266$		0			0
$267$	2.19642e16			0.227058
$268$		0			0
$269$	−1.05450e17			−1.03462			−0.517311	−	0.855797i	$-0.673067\pi$
$269$	−1.05450e17			−1.03462			−0.517311	+	0.855797i	$0.673067\pi$
$270$		0			0
$271$		−	4.33967e16i		−	0.404270i	−0.979358	−	0.202135i	$-0.935212\pi$
$271$		−	4.33967e16i		−	0.404270i	0.979358	−	0.202135i	$-0.0647880\pi$
$272$		0			0
$273$	8.77746e16			0.776659
$274$		0			0
$275$	−6.74837e15	+	4.96342e15i	−0.0567375	+	0.0417303i
$276$		0			0
$277$		−	2.21949e16i		−	0.177376i	−0.996059	−	0.0886879i	$-0.971733\pi$
$277$		−	2.21949e16i		−	0.177376i	0.996059	−	0.0886879i	$-0.0282674\pi$
$278$		0			0
$279$	6.26617e16			0.476182
$280$		0			0
$281$			2.54163e17i			1.83725i	0.395132	+	0.918624i	$0.370699\pi$
$281$			2.54163e17i			1.83725i	−0.395132	+	0.918624i	$0.629301\pi$
$282$		0			0
$283$			3.93577e16i			0.270723i	0.990796	+	0.135361i	$0.0432196\pi$
$283$			3.93577e16i			0.270723i	−0.990796	+	0.135361i	$0.956780\pi$
$284$		0			0
$285$			1.82198e17i			1.19297i
$286$		0			0
$287$	1.69843e17			1.05895
$288$		0			0
$289$	9.19917e16			0.546341
$290$		0			0
$291$	1.42579e17			0.806873
$292$		0			0
$293$			7.46259e16i			0.402547i	0.979535	+	0.201273i	$0.0645079\pi$
$293$			7.46259e16i			0.402547i	−0.979535	+	0.201273i	$0.935492\pi$
$294$		0			0
$295$	2.56510e17			1.31932
$296$		0			0
$297$	4.92680e16	−	3.62365e16i	0.241696	−	0.177767i
$298$		0			0
$299$			1.98530e17i			0.929241i
$300$		0			0
$301$	2.03512e17			0.909128
$302$		0			0
$303$			5.78805e17i			2.46850i
$304$		0			0
$305$			2.69930e17i			1.09939i
$306$		0			0
$307$		−	4.23515e17i		−	1.64778i	−0.566747	−	0.823892i	$-0.691798\pi$
$307$		−	4.23515e17i		−	1.64778i	0.566747	−	0.823892i	$-0.308202\pi$
$308$		0			0
$309$	−3.93787e17			−1.46404
$310$		0			0
$311$	5.34000e17			1.89766			0.948831	−	0.315785i	$-0.102268\pi$
$311$	5.34000e17			1.89766			0.948831	+	0.315785i	$0.102268\pi$
$312$		0			0
$313$	3.33634e17			1.13360			0.566801	−	0.823855i	$-0.308181\pi$
$313$	3.33634e17			1.13360			0.566801	+	0.823855i	$0.308181\pi$
$314$		0			0
$315$		−	2.42192e17i		−	0.787021i
$316$		0			0
$317$	3.20010e17			0.994829			0.497414	−	0.867513i	$-0.334283\pi$
$317$	3.20010e17			0.994829			0.497414	+	0.867513i	$0.334283\pi$
$318$		0			0
$319$	3.01719e17	+	4.10224e17i	0.897569	+	1.22035i
$320$		0			0
$321$		−	4.87741e17i		−	1.38885i
$322$		0			0
$323$	−2.29456e17			−0.625577
$324$		0			0
$325$		−	1.49260e16i		−	0.0389726i
$326$		0			0
$327$			4.18290e17i			1.04627i
$328$		0			0
$329$			5.56214e17i			1.33313i
$330$		0			0
$331$	2.12757e17			0.488751			0.244375	−	0.969681i	$-0.421417\pi$
$331$	2.12757e17			0.488751			0.244375	+	0.969681i	$0.421417\pi$
$332$		0			0
$333$	2.96412e17			0.652808
$334$		0			0
$335$	3.61728e16			0.0763956
$336$		0			0
$337$			3.14484e17i			0.637074i	0.947910	+	0.318537i	$0.103191\pi$
$337$			3.14484e17i			0.637074i	−0.947910	+	0.318537i	$0.896809\pi$
$338$		0			0
$339$	−2.95025e17			−0.573405
$340$		0			0
$341$	2.65446e17	−	1.95235e17i	0.495102	−	0.364147i
$342$		0			0
$343$			5.23731e17i			0.937668i
$344$		0			0
$345$	1.25482e18			2.15699
$346$		0			0
$347$		−	3.11559e17i		−	0.514321i	−0.966369	−	0.257161i	$-0.917213\pi$
$347$		−	3.11559e17i		−	0.514321i	0.966369	−	0.257161i	$-0.0827870\pi$
$348$		0			0
$349$		−	1.05194e16i		−	0.0166807i	−0.999965	−	0.00834036i	$-0.997345\pi$
$349$		−	1.05194e16i		−	0.0166807i	0.999965	−	0.00834036i	$-0.00265485\pi$
$350$		0			0
$351$			1.08971e17i			0.166020i
$352$		0			0
$353$	−8.03740e17			−1.17677			−0.588386	−	0.808580i	$-0.700237\pi$
$353$	−8.03740e17			−1.17677			−0.588386	+	0.808580i	$0.700237\pi$
$354$		0			0
$355$	3.24814e16			0.0457126
$356$		0			0
$357$	6.98680e17			0.945368
$358$		0			0
$359$		−	1.25415e18i		−	1.63189i	−0.578133	−	0.815943i	$-0.696218\pi$
$359$		−	1.25415e18i		−	1.63189i	0.578133	−	0.815943i	$-0.303782\pi$
$360$		0			0
$361$	1.09747e17			0.137354
$362$		0			0
$363$	−3.29595e17	+	1.05619e18i	−0.396857	+	1.27173i
$364$		0			0
$365$			3.05798e17i			0.354311i
$366$		0			0
$367$	9.57611e17			1.06789			0.533945	−	0.845519i	$-0.320709\pi$
$367$	9.57611e17			1.06789			0.533945	+	0.845519i	$0.320709\pi$
$368$		0			0
$369$		−	7.25403e17i		−	0.778744i
$370$		0			0
$371$		−	9.21290e17i		−	0.952312i
$372$		0			0
$373$		−	8.80613e17i		−	0.876644i	−0.898818	−	0.438322i	$-0.855573\pi$
$373$		−	8.80613e17i		−	0.876644i	0.898818	−	0.438322i	$-0.144427\pi$
$374$		0			0
$375$	−1.43381e18			−1.37491
$376$		0			0
$377$	−9.07330e17			−0.838254
$378$		0			0
$379$	−1.16292e18			−1.03532			−0.517661	−	0.855586i	$-0.673197\pi$
$379$	−1.16292e18			−1.03532			−0.517661	+	0.855586i	$0.673197\pi$
$380$		0			0
$381$			3.05532e18i			2.62169i
$382$		0			0
$383$	4.81244e17			0.398083			0.199042	−	0.979991i	$-0.436217\pi$
$383$	4.81244e17			0.398083			0.199042	+	0.979991i	$0.436217\pi$
$384$		0			0
$385$	−7.54597e17	−	1.02597e18i	−0.601853	−	0.818292i
$386$		0			0
$387$		−	8.69204e17i		−	0.668567i
$388$		0			0
$389$	−1.53981e18			−1.14240			−0.571202	−	0.820810i	$-0.693523\pi$
$389$	−1.53981e18			−1.14240			−0.571202	+	0.820810i	$0.693523\pi$
$390$		0			0
$391$			1.58029e18i			1.13109i
$392$		0			0
$393$			3.17018e18i			2.18945i
$394$		0			0
$395$		−	1.59332e18i		−	1.06200i
$396$		0			0
$397$	9.44294e17			0.607538			0.303769	−	0.952746i	$-0.401755\pi$
$397$	9.44294e17			0.607538			0.303769	+	0.952746i	$0.401755\pi$
$398$		0			0
$399$	2.09876e18			1.30362
$400$		0			0
$401$	−2.13296e18			−1.27930			−0.639651	−	0.768666i	$-0.720921\pi$
$401$	−2.13296e18			−1.27930			−0.639651	+	0.768666i	$0.720921\pi$
$402$		0			0
$403$			5.87111e17i			0.340083i
$404$		0			0
$405$	2.02384e18			1.13238
$406$		0			0
$407$	1.25565e18	−	9.23529e17i	0.678746	−	0.499217i
$408$		0			0
$409$		−	1.40285e18i		−	0.732739i	−0.930469	−	0.366369i	$-0.880601\pi$
$409$		−	1.40285e18i		−	0.732739i	0.930469	−	0.366369i	$-0.119399\pi$
$410$		0			0
$411$	−2.76843e18			−1.39747
$412$		0			0
$413$		−	2.95477e18i		−	1.44169i
$414$		0			0
$415$		−	3.70884e17i		−	0.174945i
$416$		0			0
$417$		−	4.15016e18i		−	1.89283i
$418$		0			0
$419$	−3.90588e18			−1.72275			−0.861373	−	0.507974i	$-0.830395\pi$
$419$	−3.90588e18			−1.72275			−0.861373	+	0.507974i	$0.830395\pi$
$420$		0			0
$421$	2.25755e17			0.0963080			0.0481540	−	0.998840i	$-0.484666\pi$
$421$	2.25755e17			0.0963080			0.0481540	+	0.998840i	$0.484666\pi$
$422$		0			0
$423$	2.37560e18			0.980372
$424$		0			0
$425$		−	1.18810e17i		−	0.0474384i
$426$		0			0
$427$	3.10935e18			1.20136
$428$		0			0
$429$	−1.16803e18	−	1.58808e18i	−0.436771	−	0.593844i
$430$		0			0
$431$			2.76525e18i			1.00091i	0.865763	+	0.500454i	$0.166833\pi$
$431$			2.76525e18i			1.00091i	−0.865763	+	0.500454i	$0.833167\pi$
$432$		0			0
$433$	−1.73403e18			−0.607633			−0.303817	−	0.952731i	$-0.598261\pi$
$433$	−1.73403e18			−0.607633			−0.303817	+	0.952731i	$0.598261\pi$
$434$		0			0
$435$			5.73483e18i			1.94579i
$436$		0			0
$437$			4.74701e18i			1.55973i
$438$		0			0
$439$			1.38152e18i			0.439648i	0.975540	+	0.219824i	$0.0705484\pi$
$439$			1.38152e18i			0.439648i	−0.975540	+	0.219824i	$0.929452\pi$
$440$		0			0
$441$	−2.76487e17			−0.0852325
$442$		0			0
$443$	−1.78840e18			−0.534121			−0.267060	−	0.963680i	$-0.586052\pi$
$443$	−1.78840e18			−0.534121			−0.267060	+	0.963680i	$0.586052\pi$
$444$		0			0
$445$	5.67838e17			0.164325
$446$		0			0
$447$		−	7.24228e18i		−	2.03106i
$448$		0			0
$449$	2.40865e18			0.654710			0.327355	−	0.944901i	$-0.393843\pi$
$449$	2.40865e18			0.654710			0.327355	+	0.944901i	$0.393843\pi$
$450$		0			0
$451$	−2.26014e18	−	3.07293e18i	−0.595523	−	0.809686i
$452$		0			0
$453$			2.42988e18i			0.620722i
$454$		0			0
$455$	2.26923e18			0.562080
$456$		0			0
$457$		−	4.07586e18i		−	0.979051i	−0.871989	−	0.489526i	$-0.837170\pi$
$457$		−	4.07586e18i		−	0.979051i	0.871989	−	0.489526i	$-0.162830\pi$
$458$		0			0
$459$			8.67398e17i			0.202083i
$460$		0			0
$461$		−	2.96254e18i		−	0.669511i	−0.942305	−	0.334755i	$-0.891346\pi$
$461$		−	2.96254e18i		−	0.669511i	0.942305	−	0.334755i	$-0.108654\pi$
$462$		0			0
$463$	4.04238e18			0.886278			0.443139	−	0.896453i	$-0.353865\pi$
$463$	4.04238e18			0.886278			0.443139	+	0.896453i	$0.353865\pi$
$464$		0			0
$465$	3.71086e18			0.789413
$466$		0			0
$467$	−5.69967e18			−1.17661			−0.588304	−	0.808640i	$-0.700204\pi$
$467$	−5.69967e18			−1.17661			−0.588304	+	0.808640i	$0.700204\pi$
$468$		0			0
$469$		−	4.16678e17i		−	0.0834816i
$470$		0			0
$471$	8.25814e17			0.160596
$472$		0			0
$473$	−2.70818e18	−	3.68209e18i	−0.511268	−	0.695131i
$474$		0			0
$475$		−	3.56892e17i		−	0.0654156i
$476$		0			0
$477$	−3.93484e18			−0.700324
$478$		0			0
$479$			7.40327e18i			1.27960i	0.768541	+	0.639801i	$0.220983\pi$
$479$			7.40327e18i			1.27960i	−0.768541	+	0.639801i	$0.779017\pi$
$480$		0			0
$481$			2.77724e18i			0.466227i
$482$		0			0
$483$		−	1.44544e19i		−	2.35706i
$484$		0			0
$485$	3.68608e18			0.583946
$486$		0			0
$487$	−9.76288e18			−1.50271			−0.751355	−	0.659898i	$-0.770599\pi$
$487$	−9.76288e18			−1.50271			−0.751355	+	0.659898i	$0.770599\pi$
$488$		0			0
$489$	−1.44052e19			−2.15455
$490$		0			0
$491$		−	6.09802e18i		−	0.886378i	−0.896428	−	0.443189i	$-0.853847\pi$
$491$		−	6.09802e18i		−	0.886378i	0.896428	−	0.443189i	$-0.146153\pi$
$492$		0			0
$493$	−7.22229e18			−1.02034
$494$		0			0
$495$	−4.38192e18	+	3.22290e18i	−0.601767	+	0.442599i
$496$		0			0
$497$		−	3.74156e17i		−	0.0499527i
$498$		0			0
$499$	1.40351e19			1.82185			0.910924	−	0.412574i	$-0.135370\pi$
$499$	1.40351e19			1.82185			0.910924	+	0.412574i	$0.135370\pi$
$500$		0			0
$501$			1.33408e17i			0.0168391i
$502$		0			0
$503$		−	5.18305e18i		−	0.636223i	−0.948053	−	0.318111i	$-0.896951\pi$
$503$		−	5.18305e18i		−	0.636223i	0.948053	−	0.318111i	$-0.103049\pi$
$504$		0			0
$505$			1.49638e19i			1.78649i
$506$		0			0
$507$	−7.95922e18			−0.924304
$508$		0			0
$509$	−1.31946e19			−1.49064			−0.745319	−	0.666708i	$-0.767703\pi$
$509$	−1.31946e19			−1.49064			−0.745319	+	0.666708i	$0.767703\pi$
$510$		0			0
$511$	3.52251e18			0.387174
$512$		0			0
$513$			2.60557e18i			0.278664i
$514$		0			0
$515$	−1.01805e19			−1.05955
$516$		0			0
$517$	1.00634e19	−	7.40165e18i	1.01932	−	0.749712i
$518$		0			0
$519$			1.86253e19i			1.83625i
$520$		0			0
$521$	1.02649e19			0.985124			0.492562	−	0.870277i	$-0.336060\pi$
$521$	1.02649e19			0.985124			0.492562	+	0.870277i	$0.336060\pi$
$522$		0			0
$523$		−	8.64600e18i		−	0.807799i	−0.914803	−	0.403900i	$-0.867655\pi$
$523$		−	8.64600e18i		−	0.807799i	0.914803	−	0.403900i	$-0.132345\pi$
$524$		0			0
$525$			1.08672e18i			0.0988557i
$526$		0			0
$527$			4.67336e18i			0.413957i
$528$		0			0
$529$	2.11004e19			1.82012
$530$		0			0
$531$	−1.26199e19			−1.06021
$532$		0			0
$533$	6.79668e18			0.556169
$534$		0			0
$535$		−	1.26095e19i		−	1.00513i
$536$		0			0
$537$	8.16101e18			0.633759
$538$		0			0
$539$	−1.17125e18	+	8.61450e17i	−0.0886190	+	0.0651792i
$540$		0			0
$541$		−	1.19057e19i		−	0.877757i	−0.898546	−	0.438879i	$-0.855376\pi$
$541$		−	1.19057e19i		−	0.877757i	0.898546	−	0.438879i	$-0.144624\pi$
$542$		0			0
$543$	−1.75646e18			−0.126194
$544$		0			0
$545$			1.08140e19i			0.757200i
$546$		0			0
$547$			1.01805e19i			0.694799i	0.937717	+	0.347400i	$0.112935\pi$
$547$			1.01805e19i			0.694799i	−0.937717	+	0.347400i	$0.887065\pi$
$548$		0			0
$549$		−	1.32801e19i		−	0.883475i
$550$		0			0
$551$	−2.16950e19			−1.40701
$552$		0			0
$553$	−1.83536e19			−1.16050
$554$		0			0
$555$	1.75537e19			1.08222
$556$		0			0
$557$		−	2.05299e19i		−	1.23424i	−0.786867	−	0.617122i	$-0.788298\pi$
$557$		−	2.05299e19i		−	1.23424i	0.786867	−	0.617122i	$-0.211702\pi$
$558$		0			0
$559$	8.14403e18			0.477482
$560$		0			0
$561$	−9.29746e18	−	1.26410e19i	−0.531648	−	0.722840i
$562$		0			0
$563$		−	2.56873e19i		−	1.43272i	−0.697733	−	0.716358i	$-0.745808\pi$
$563$		−	2.56873e19i		−	1.43272i	0.697733	−	0.716358i	$-0.254192\pi$
$564$		0			0
$565$	−7.62726e18			−0.414982
$566$		0			0
$567$		−	2.33128e19i		−	1.23741i
$568$		0			0
$569$		−	1.26575e19i		−	0.655485i	−0.944767	−	0.327742i	$-0.893712\pi$
$569$		−	1.26575e19i		−	0.655485i	0.944767	−	0.327742i	$-0.106288\pi$
$570$		0			0
$571$			3.03372e18i			0.153293i	0.997058	+	0.0766466i	$0.0244213\pi$
$571$			3.03372e18i			0.153293i	−0.997058	+	0.0766466i	$0.975579\pi$
$572$		0			0
$573$	−4.69439e19			−2.31471
$574$		0			0
$575$	−2.45796e18			−0.118277
$576$		0			0
$577$	−1.75043e19			−0.822082			−0.411041	−	0.911617i	$-0.634835\pi$
$577$	−1.75043e19			−0.822082			−0.411041	+	0.911617i	$0.634835\pi$
$578$		0			0
$579$			1.14764e19i			0.526087i
$580$		0			0
$581$	−4.27225e18			−0.191172
$582$		0			0
$583$	−1.66687e19	+	1.22598e19i	−0.728150	+	0.535553i
$584$		0			0
$585$		−	9.69190e18i		−	0.413350i
$586$		0			0
$587$	−2.01981e17			−0.00841092			−0.00420546	−	0.999991i	$-0.501339\pi$
$587$	−2.01981e17			−0.00841092			−0.00420546	+	0.999991i	$0.501339\pi$
$588$		0			0
$589$			1.40383e19i			0.570830i
$590$		0			0
$591$		−	1.38383e19i		−	0.549503i
$592$		0			0
$593$		−	3.53501e19i		−	1.37091i	−0.728116	−	0.685454i	$-0.759604\pi$
$593$		−	3.53501e19i		−	1.37091i	0.728116	−	0.685454i	$-0.240396\pi$
$594$		0			0
$595$	1.80629e19			0.684177
$596$		0			0
$597$	−4.64359e19			−1.71804
$598$		0			0
$599$	−3.03003e19			−1.09511			−0.547557	−	0.836768i	$-0.684442\pi$
$599$	−3.03003e19			−1.09511			−0.547557	+	0.836768i	$0.684442\pi$
$600$		0			0
$601$			2.23510e19i			0.789181i	0.918857	+	0.394590i	$0.129113\pi$
$601$			2.23510e19i			0.789181i	−0.918857	+	0.394590i	$0.870887\pi$
$602$		0			0
$603$	−1.77964e18			−0.0613918
$604$		0			0
$605$	−8.52098e18	+	2.73055e19i	−0.287211	+	0.920369i
$606$		0			0
$607$		−	6.83948e18i		−	0.225269i	−0.993636	−	0.112635i	$-0.964071\pi$
$607$		−	6.83948e18i		−	0.225269i	0.993636	−	0.112635i	$-0.0359290\pi$
$608$		0			0
$609$	6.60600e19			2.12627
$610$		0			0
$611$			2.22582e19i			0.700168i
$612$		0			0
$613$		−	7.18627e18i		−	0.220943i	−0.993879	−	0.110471i	$-0.964764\pi$
$613$		−	7.18627e18i		−	0.220943i	0.993879	−	0.110471i	$-0.0352361\pi$
$614$		0			0
$615$		−	4.29588e19i		−	1.29100i
$616$		0			0
$617$	−8.06287e18			−0.236861			−0.118431	−	0.992962i	$-0.537786\pi$
$617$	−8.06287e18			−0.236861			−0.118431	+	0.992962i	$0.537786\pi$
$618$		0			0
$619$	2.95921e19			0.849849			0.424924	−	0.905229i	$-0.360301\pi$
$619$	2.95921e19			0.849849			0.424924	+	0.905229i	$0.360301\pi$
$620$		0			0
$621$	1.79449e19			0.503847
$622$		0			0
$623$		−	6.54098e18i		−	0.179567i
$624$		0			0
$625$	−3.44443e19			−0.924608
$626$		0			0
$627$	−2.79286e19	−	3.79723e19i	−0.733122	−	0.996768i
$628$		0			0
$629$			2.21066e19i			0.567502i
$630$		0			0
$631$	−5.84818e18			−0.146830			−0.0734151	−	0.997301i	$-0.523390\pi$
$631$	−5.84818e18			−0.146830			−0.0734151	+	0.997301i	$0.523390\pi$
$632$		0			0
$633$			5.04988e19i			1.24009i
$634$		0			0
$635$			7.89887e19i			1.89736i
$636$		0			0
$637$		−	2.59055e18i		−	0.0608719i
$638$		0			0
$639$	−1.59803e18			−0.0367349
$640$		0			0
$641$	3.07064e19			0.690594			0.345297	−	0.938493i	$-0.387778\pi$
$641$	3.07064e19			0.690594			0.345297	+	0.938493i	$0.387778\pi$
$642$		0			0
$643$	3.34635e19			0.736366			0.368183	−	0.929753i	$-0.379980\pi$
$643$	3.34635e19			0.736366			0.368183	+	0.929753i	$0.379980\pi$
$644$		0			0
$645$		−	5.14747e19i		−	1.10835i
$646$		0			0
$647$	3.22302e19			0.679098			0.339549	−	0.940588i	$-0.389725\pi$
$647$	3.22302e19			0.679098			0.339549	+	0.940588i	$0.389725\pi$
$648$		0			0
$649$	−5.34598e19	+	3.93196e19i	−1.10234	+	0.810767i
$650$		0			0
$651$		−	4.27458e19i		−	0.862634i
$652$		0			0
$653$	8.05170e19			1.59036			0.795179	−	0.606374i	$-0.207377\pi$
$653$	8.05170e19			1.59036			0.795179	+	0.606374i	$0.207377\pi$
$654$		0			0
$655$			8.19582e19i			1.58454i
$656$		0			0
$657$		−	1.50447e19i		−	0.284726i
$658$		0			0
$659$			8.02662e19i			1.48708i	0.668690	+	0.743541i	$0.266855\pi$
$659$			8.02662e19i			1.48708i	−0.668690	+	0.743541i	$0.733145\pi$
$660$		0			0
$661$	4.55001e19			0.825282			0.412641	−	0.910894i	$-0.364606\pi$
$661$	4.55001e19			0.825282			0.412641	+	0.910894i	$0.364606\pi$
$662$		0			0
$663$	2.79593e19			0.496515
$664$		0			0
$665$	5.42590e19			0.943452
$666$		0			0
$667$			1.49416e20i			2.54399i
$668$		0			0
$669$	9.98441e19			1.66471
$670$		0			0
$671$	−4.13767e19	−	5.62567e19i	−0.675613	−	0.918578i
$672$		0			0
$673$		−	1.46349e19i		−	0.234038i	−0.993130	−	0.117019i	$-0.962666\pi$
$673$		−	1.46349e19i		−	0.234038i	0.993130	−	0.117019i	$-0.0373338\pi$
$674$		0			0
$675$	−1.34914e18			−0.0211315
$676$		0			0
$677$		−	6.09622e19i		−	0.935276i	−0.883920	−	0.467638i	$-0.845105\pi$
$677$		−	6.09622e19i		−	0.935276i	0.883920	−	0.467638i	$-0.154895\pi$
$678$		0			0
$679$		−	4.24603e19i		−	0.638109i
$680$		0			0
$681$			9.04760e19i			1.33200i
$682$		0			0
$683$	−5.87414e17			−0.00847226			−0.00423613	−	0.999991i	$-0.501348\pi$
$683$	−5.87414e17			−0.00847226			−0.00423613	+	0.999991i	$0.501348\pi$
$684$		0			0
$685$	−7.15719e19			−1.01137
$686$		0			0
$687$	4.34957e19			0.602211
$688$		0			0
$689$		−	3.68676e19i		−	0.500162i
$690$		0			0
$691$	1.07427e20			1.42812			0.714062	−	0.700083i	$-0.246853\pi$
$691$	1.07427e20			1.42812			0.714062	+	0.700083i	$0.246853\pi$
$692$		0			0
$693$	3.71249e19	+	5.04758e19i	0.483651	+	0.657583i
$694$		0			0
$695$		−	1.07294e20i		−	1.36987i
$696$		0			0
$697$	5.41011e19			0.676982
$698$		0			0
$699$		−	1.29362e20i		−	1.58660i
$700$		0			0
$701$			1.14615e20i			1.37789i	0.724812	+	0.688947i	$0.241927\pi$
$701$			1.14615e20i			1.37789i	−0.724812	+	0.688947i	$0.758073\pi$
$702$		0			0
$703$			6.64059e19i			0.782563i
$704$		0			0
$705$	1.40684e20			1.62526
$706$		0			0
$707$	1.72369e20			1.95220
$708$		0			0
$709$	4.19675e19			0.466003			0.233002	−	0.972476i	$-0.425145\pi$
$709$	4.19675e19			0.466003			0.233002	+	0.972476i	$0.425145\pi$
$710$		0			0
$711$			7.83885e19i			0.853424i
$712$		0			0
$713$	9.66832e19			1.03211
$714$		0			0
$715$	−3.01970e19	−	4.10565e19i	−0.316098	−	0.429773i
$716$		0			0
$717$			1.68019e20i			1.72475i
$718$		0			0
$719$	1.88410e18			0.0189671			0.00948356	−	0.999955i	$-0.496981\pi$
$719$	1.88410e18			0.0189671			0.00948356	+	0.999955i	$0.496981\pi$
$720$		0			0
$721$			1.17271e20i			1.15782i
$722$		0			0
$723$			1.33258e20i			1.29040i
$724$		0			0
$725$		−	1.12334e19i		−	0.106696i
$726$		0			0
$727$	2.64591e19			0.246510			0.123255	−	0.992375i	$-0.460667\pi$
$727$	2.64591e19			0.246510			0.123255	+	0.992375i	$0.460667\pi$
$728$		0			0
$729$	−5.58341e19			−0.510278
$730$		0			0
$731$	6.48259e19			0.581202
$732$		0			0
$733$			2.62616e19i			0.230990i	0.993308	+	0.115495i	$0.0368454\pi$
$733$			2.62616e19i			0.230990i	−0.993308	+	0.115495i	$0.963155\pi$
$734$		0			0
$735$	−1.63737e19			−0.141298
$736$		0			0
$737$	−7.53884e18	+	5.54481e18i	−0.0638311	+	0.0469477i
$738$		0			0
$739$			2.35064e20i			1.95288i	0.215800	+	0.976438i	$0.430764\pi$
$739$			2.35064e20i			1.95288i	−0.215800	+	0.976438i	$0.569236\pi$
$740$		0			0
$741$	8.39868e19			0.684674
$742$		0			0
$743$		−	2.17854e20i		−	1.74279i	−0.490584	−	0.871394i	$-0.663217\pi$
$743$		−	2.17854e20i		−	1.74279i	0.490584	−	0.871394i	$-0.336783\pi$
$744$		0			0
$745$		−	1.87234e20i		−	1.46991i
$746$		0			0
$747$			1.82468e19i			0.140586i
$748$		0			0
$749$	−1.45250e20			−1.09836
$750$		0			0
$751$	7.99894e19			0.593680			0.296840	−	0.954927i	$-0.404067\pi$
$751$	7.99894e19			0.593680			0.296840	+	0.954927i	$0.404067\pi$
$752$		0			0
$753$	−2.05960e20			−1.50044
$754$		0			0
$755$			6.28193e19i			0.449226i
$756$		0			0
$757$	−1.22053e20			−0.856794			−0.428397	−	0.903591i	$-0.640922\pi$
$757$	−1.22053e20			−0.856794			−0.428397	+	0.903591i	$0.640922\pi$
$758$		0			0
$759$	−2.61520e20	+	1.92347e20i	−1.80224	+	1.32554i
$760$		0			0
$761$			1.68403e20i			1.13935i	0.821871	+	0.569674i	$0.192930\pi$
$761$			1.68403e20i			1.13935i	−0.821871	+	0.569674i	$0.807070\pi$
$762$		0			0
$763$	1.24567e20			0.827433
$764$		0			0
$765$		−	7.71469e19i		−	0.503139i
$766$		0			0
$767$		−	1.18242e20i		−	0.757189i
$768$		0			0
$769$			4.56302e19i			0.286925i	0.989656	+	0.143462i	$0.0458236\pi$
$769$			4.56302e19i			0.286925i	−0.989656	+	0.143462i	$0.954176\pi$
$770$		0			0
$771$	−1.09806e20			−0.678024
$772$		0			0
$773$	3.35365e19			0.203358			0.101679	−	0.994817i	$-0.467579\pi$
$773$	3.35365e19			0.203358			0.101679	+	0.994817i	$0.467579\pi$
$774$		0			0
$775$	−7.26888e18			−0.0432868
$776$		0			0
$777$		−	2.02202e20i		−	1.18260i
$778$		0			0
$779$	1.62514e20			0.933530
$780$		0			0
$781$	−6.76951e18	+	4.97897e18i	−0.0381944	+	0.0280920i
$782$		0			0
$783$			8.20123e19i			0.454513i
$784$		0			0
$785$	2.13497e19			0.116226
$786$		0			0
$787$		−	2.81672e20i		−	1.50633i	−0.657831	−	0.753165i	$-0.728526\pi$
$787$		−	2.81672e20i		−	1.50633i	0.657831	−	0.753165i	$-0.271474\pi$
$788$		0			0
$789$		−	4.45254e20i		−	2.33921i
$790$		0			0
$791$			8.78591e19i			0.453473i
$792$		0			0
$793$	1.24428e20			0.630966
$794$		0			0
$795$	−2.33024e20			−1.16099
$796$		0			0
$797$	−2.50971e20			−1.22861			−0.614307	−	0.789067i	$-0.710564\pi$
$797$	−2.50971e20			−1.22861			−0.614307	+	0.789067i	$0.710564\pi$
$798$		0			0
$799$			1.77174e20i			0.852261i
$800$		0			0
$801$	−2.79366e19			−0.132052
$802$		0			0
$803$	−4.68747e19	−	6.37319e19i	−0.217736	−	0.296038i
$804$		0			0
$805$		−	3.73688e20i		−	1.70584i
$806$		0			0
$807$	3.07234e20			1.37834
$808$		0			0
$809$			1.67113e20i			0.736834i	0.929661	+	0.368417i	$0.120100\pi$
$809$			1.67113e20i			0.736834i	−0.929661	+	0.368417i	$0.879900\pi$
$810$		0			0
$811$		−	8.31549e19i		−	0.360364i	−0.983633	−	0.180182i	$-0.942331\pi$
$811$		−	8.31549e19i		−	0.360364i	0.983633	−	0.180182i	$-0.0576687\pi$
$812$		0			0
$813$			1.26438e20i			0.538573i
$814$		0			0
$815$	−3.72415e20			−1.55928
$816$		0			0
$817$	1.94730e20			0.801454
$818$		0			0
$819$	−1.11642e20			−0.451690
$820$		0			0
$821$			1.86972e20i			0.743660i	0.928301	+	0.371830i	$0.121270\pi$
$821$			1.86972e20i			0.743660i	−0.928301	+	0.371830i	$0.878730\pi$
$822$		0			0
$823$	−1.33342e20			−0.521397			−0.260698	−	0.965420i	$-0.583953\pi$
$823$	−1.33342e20			−0.521397			−0.260698	+	0.965420i	$0.583953\pi$
$824$		0			0
$825$	1.96617e19	−	1.44611e19i	0.0755863	−	0.0555936i
$826$		0			0
$827$		−	5.37762e19i		−	0.203260i	−0.994822	−	0.101630i	$-0.967594\pi$
$827$		−	5.37762e19i		−	0.203260i	0.994822	−	0.101630i	$-0.0324057\pi$
$828$		0			0
$829$	3.94236e20			1.46512			0.732562	−	0.680700i	$-0.238324\pi$
$829$	3.94236e20			1.46512			0.732562	+	0.680700i	$0.238324\pi$
$830$		0			0
$831$			6.46658e19i			0.236302i
$832$		0			0
$833$		−	2.06206e19i		−	0.0740947i
$834$		0			0
$835$			3.44898e18i			0.0121867i
$836$		0			0
$837$	5.30681e19			0.184398
$838$		0			0
$839$	−1.55689e19			−0.0532015			−0.0266007	−	0.999646i	$-0.508468\pi$
$839$	−1.55689e19			−0.0532015			−0.0266007	+	0.999646i	$0.508468\pi$
$840$		0			0
$841$	−3.85307e20			−1.29490
$842$		0			0
$843$		−	7.40515e20i		−	2.44760i
$844$		0			0
$845$	−2.05769e20			−0.668933
$846$		0			0
$847$	3.14534e20	+	9.81540e19i	1.00574	+	0.313851i
$848$		0			0
$849$		−	1.14671e20i		−	0.360660i
$850$		0			0
$851$	4.57345e20			1.41494
$852$		0			0
$853$			5.69284e19i			0.173255i	0.996241	+	0.0866274i	$0.0276090\pi$
$853$			5.69284e19i			0.173255i	−0.996241	+	0.0866274i	$0.972391\pi$
$854$		0			0
$855$		−	2.31741e20i		−	0.693809i
$856$		0			0
$857$			1.82987e20i			0.538958i	0.963006	+	0.269479i	$0.0868515\pi$
$857$			1.82987e20i			0.538958i	−0.963006	+	0.269479i	$0.913148\pi$
$858$		0			0
$859$	−3.81449e20			−1.10531			−0.552655	−	0.833410i	$-0.686385\pi$
$859$	−3.81449e20			−1.10531			−0.552655	+	0.833410i	$0.686385\pi$
$860$		0			0
$861$	−4.94846e20			−1.41074
$862$		0			0
$863$	1.41862e20			0.397916			0.198958	−	0.980008i	$-0.436244\pi$
$863$	1.41862e20			0.397916			0.198958	+	0.980008i	$0.436244\pi$
$864$		0			0
$865$			4.81517e20i			1.32892i
$866$		0			0
$867$	−2.68022e20			−0.727842
$868$		0			0
$869$	2.44235e20	+	3.32066e20i	0.652632	+	0.887333i
$870$		0			0
$871$		−	1.66744e19i		−	0.0438452i
$872$		0			0
$873$	−1.81349e20			−0.469261
$874$		0			0
$875$			4.26991e20i			1.08733i
$876$		0			0
$877$		−	4.00254e19i		−	0.100309i	−0.998741	−	0.0501544i	$-0.984029\pi$
$877$		−	4.00254e19i		−	0.100309i	0.998741	−	0.0501544i	$-0.0159713\pi$
$878$		0			0
$879$		−	2.17426e20i		−	0.536277i
$880$		0			0
$881$	7.93534e20			1.92634			0.963172	−	0.268885i	$-0.0866552\pi$
$881$	7.93534e20			1.92634			0.963172	+	0.268885i	$0.0866552\pi$
$882$		0			0
$883$	6.32658e20			1.51162			0.755812	−	0.654788i	$-0.227242\pi$
$883$	6.32658e20			1.51162			0.755812	+	0.654788i	$0.227242\pi$
$884$		0			0
$885$	−7.47355e20			−1.75762
$886$		0			0
$887$			2.94341e20i			0.681373i	0.940177	+	0.340686i	$0.110659\pi$
$887$			2.94341e20i			0.681373i	−0.940177	+	0.340686i	$0.889341\pi$
$888$		0			0
$889$	9.09878e20			2.07334
$890$		0			0
$891$	−4.21792e20	+	3.10228e20i	−0.946138	+	0.695883i
$892$		0			0
$893$			5.32212e20i			1.17523i
$894$		0			0
$895$	2.10986e20			0.458661
$896$		0			0
$897$		−	5.78427e20i		−	1.23795i
$898$		0			0
$899$			4.41865e20i			0.931047i
$900$		0			0
$901$		−	2.93464e20i		−	0.608809i
$902$		0			0
$903$	−5.92943e20			−1.21115
$904$		0			0
$905$	−4.54094e19			−0.0913284
$906$		0			0
$907$	−5.43443e20			−1.07623			−0.538113	−	0.842873i	$-0.680862\pi$
$907$	−5.43443e20			−1.07623			−0.538113	+	0.842873i	$0.680862\pi$
$908$		0			0
$909$		−	7.36191e20i		−	1.43563i
$910$		0			0
$911$	4.55809e20			0.875294			0.437647	−	0.899147i	$-0.355812\pi$
$911$	4.55809e20			0.875294			0.437647	+	0.899147i	$0.355812\pi$
$912$		0			0
$913$	5.68516e19	+	7.72966e19i	0.107510	+	0.146172i
$914$		0			0
$915$		−	7.86454e20i		−	1.46462i
$916$		0			0
$917$	9.44085e20			1.73151
$918$		0			0
$919$			5.46063e20i			0.986356i	0.869928	+	0.493178i	$0.164165\pi$
$919$			5.46063e20i			0.986356i	−0.869928	+	0.493178i	$0.835835\pi$
$920$		0			0
$921$			1.23393e21i			2.19520i
$922$		0			0
$923$		−	1.49728e19i		−	0.0262355i
$924$		0			0
$925$	−3.43843e19			−0.0593428
$926$		0			0
$927$	5.00865e20			0.851456
$928$		0			0
$929$	−2.23731e20			−0.374641			−0.187321	−	0.982299i	$-0.559980\pi$
$929$	−2.23731e20			−0.374641			−0.187321	+	0.982299i	$0.559980\pi$
$930$		0			0
$931$		−	6.19422e19i		−	0.102174i
$932$		0			0
$933$	−1.55583e21			−2.52809
$934$		0			0
$935$	−2.40366e20	−	3.26807e20i	−0.384762	−	0.523130i
$936$		0			0
$937$		−	8.55067e20i		−	1.34841i	−0.738543	−	0.674206i	$-0.764486\pi$
$937$		−	8.55067e20i		−	1.34841i	0.738543	−	0.674206i	$-0.235514\pi$
$938$		0			0
$939$	−9.72059e20			−1.51020
$940$		0			0
$941$		−	9.05540e20i		−	1.38605i	−0.720911	−	0.693027i	$-0.756277\pi$
$941$		−	9.05540e20i		−	1.38605i	0.720911	−	0.693027i	$-0.243723\pi$
$942$		0			0
$943$		−	1.11925e21i		−	1.68790i
$944$		0			0
$945$		−	2.05112e20i		−	0.304768i
$946$		0			0
$947$	6.85540e20			1.00365			0.501826	−	0.864969i	$-0.332662\pi$
$947$	6.85540e20			1.00365			0.501826	+	0.864969i	$0.332662\pi$
$948$		0			0
$949$	1.40962e20			0.203347
$950$		0			0
$951$	−9.32363e20			−1.32532
$952$		0			0
$953$			6.24466e20i			0.874698i	0.899292	+	0.437349i	$0.144083\pi$
$953$			6.24466e20i			0.874698i	−0.899292	+	0.437349i	$0.855917\pi$
$954$		0			0
$955$	−1.21363e21			−1.67519
$956$		0			0
$957$	−8.79073e20	−	1.19521e21i	−1.19575	−	1.62577i
$958$		0			0
$959$			8.24444e20i			1.10517i
$960$		0			0
$961$	−4.71024e20			−0.622271
$962$		0			0
$963$			6.20366e20i			0.807726i
$964$		0			0
$965$			2.96699e20i			0.380737i
$966$		0			0
$967$			1.81318e20i			0.229327i	0.993404	+	0.114664i	$0.0365790\pi$
$967$			1.81318e20i			0.229327i	−0.993404	+	0.114664i	$0.963421\pi$
$968$		0			0
$969$	6.68530e20			0.833401
$970$		0			0
$971$	5.19816e20			0.638726			0.319363	−	0.947632i	$-0.396531\pi$
$971$	5.19816e20			0.638726			0.319363	+	0.947632i	$0.396531\pi$
$972$		0			0
$973$	−1.23593e21			−1.49693
$974$		0			0
$975$			4.34876e19i			0.0519198i
$976$		0			0
$977$	3.78331e20			0.445257			0.222628	−	0.974903i	$-0.428536\pi$
$977$	3.78331e20			0.445257			0.222628	+	0.974903i	$0.428536\pi$
$978$		0			0
$979$	−1.18344e20	+	8.70420e19i	−0.137299	+	0.100983i
$980$		0			0
$981$		−	5.32030e20i		−	0.608489i
$982$		0			0
$983$	7.85688e20			0.885881			0.442941	−	0.896551i	$-0.353935\pi$
$983$	7.85688e20			0.885881			0.442941	+	0.896551i	$0.353935\pi$
$984$		0			0
$985$		−	3.57759e20i		−	0.397683i
$986$		0			0
$987$		−	1.62056e21i		−	1.77600i
$988$		0			0
$989$		−	1.34113e21i		−	1.44909i
$990$		0			0
$991$	1.09370e21			1.16515			0.582574	−	0.812778i	$-0.302046\pi$
$991$	1.09370e21			1.16515			0.582574	+	0.812778i	$0.302046\pi$
$992$		0			0
$993$	−6.19877e20			−0.651120
$994$		0			0
$995$	−1.20050e21			−1.24337
$996$		0			0
$997$			1.11125e21i			1.13487i	0.823418	+	0.567435i	$0.192064\pi$
$997$			1.11125e21i			1.13487i	−0.823418	+	0.567435i	$0.807936\pi$
$998$		0			0
$999$	2.51030e20			0.252795

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	176.15.h.e.65.4		14
4.3	odd	2		22.15.b.a.21.6	✓	14
11.10	odd	2	inner	176.15.h.e.65.3		14
44.43	even	2		22.15.b.a.21.13	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.15.b.a.21.6	✓	14	4.3	odd	2
22.15.b.a.21.13	yes	14	44.43	even	2
176.15.h.e.65.3		14	11.10	odd	2	inner
176.15.h.e.65.4		14	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 176 = 2^{4} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 15 \)
Character orbit:	\([\chi]\)	\(=\)	176.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2913.55 q^{3} -75323.6 q^{5} +867659. i q^{7} +3.70579e6 q^{9} +O(q^{10})\)	\(q-2913.55 q^{3} -75323.6 q^{5} +867659. i q^{7} +3.70579e6 q^{9} +(1.56983e7 - 1.15461e7i) q^{11} +3.47214e7i q^{13} +2.19459e8 q^{15} +2.76380e8i q^{17} +8.30217e8i q^{19} -2.52797e9i q^{21} +5.71780e9 q^{23} -4.29878e8 q^{25} +3.13842e9 q^{27} +2.61317e10i q^{29} +1.69092e10 q^{31} +(-4.57378e10 + 3.36401e10i) q^{33} -6.53552e10i q^{35} +7.99862e10 q^{37} -1.01163e11i q^{39} -1.95749e11i q^{41} -2.34553e11i q^{43} -2.79133e11 q^{45} +6.41052e11 q^{47} -7.46096e10 q^{49} -8.05247e11i q^{51} -1.06181e12 q^{53} +(-1.18245e12 + 8.69693e11i) q^{55} -2.41888e12i q^{57} -3.40545e12 q^{59} -3.58361e12i q^{61} +3.21536e12i q^{63} -2.61534e12i q^{65} -4.80232e11 q^{67} -1.66591e13 q^{69} -4.31225e11 q^{71} -4.05979e12i q^{73} +1.25247e12 q^{75} +(1.00181e13 + 1.36208e13i) q^{77} +2.11530e13i q^{79} -2.68686e13 q^{81} +4.92388e12i q^{83} -2.08180e13i q^{85} -7.61359e13i q^{87} -7.53865e12 q^{89} -3.01264e13 q^{91} -4.92656e13 q^{93} -6.25349e13i q^{95} -4.89366e13 q^{97} +(5.81746e13 - 4.27874e13i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 4394 q^{3} + 69758 q^{5} + 11016572 q^{9}+O(q^{10}) \) 14 * q - 4394 * q^3 + 69758 * q^5 + 11016572 * q^9	\( 14 q - 4394 q^{3} + 69758 q^{5} + 11016572 q^{9} - 20143042 q^{11} + 1359602 q^{15} + 7305755542 q^{23} + 19291879452 q^{25} - 34093422830 q^{27} + 33569873942 q^{31} + 2885838062 q^{33} + 73167823966 q^{37} + 2000205168616 q^{45} + 1612717386124 q^{47} + 3424602524990 q^{49} - 3530064068164 q^{53} + 3715439610854 q^{55} + 818496564070 q^{59} - 16485465276922 q^{67} - 11394452631206 q^{69} + 19380879179878 q^{71} - 23016770893992 q^{75} + 60534793808304 q^{77} - 10394309810662 q^{81} - 117770741987650 q^{89} - 150621364097712 q^{91} + 27345122803162 q^{93} + 123398138843566 q^{97} - 118861332531788 q^{99}+O(q^{100}) \) 14 * q - 4394 * q^3 + 69758 * q^5 + 11016572 * q^9 - 20143042 * q^11 + 1359602 * q^15 + 7305755542 * q^23 + 19291879452 * q^25 - 34093422830 * q^27 + 33569873942 * q^31 + 2885838062 * q^33 + 73167823966 * q^37 + 2000205168616 * q^45 + 1612717386124 * q^47 + 3424602524990 * q^49 - 3530064068164 * q^53 + 3715439610854 * q^55 + 818496564070 * q^59 - 16485465276922 * q^67 - 11394452631206 * q^69 + 19380879179878 * q^71 - 23016770893992 * q^75 + 60534793808304 * q^77 - 10394309810662 * q^81 - 117770741987650 * q^89 - 150621364097712 * q^91 + 27345122803162 * q^93 + 123398138843566 * q^97 - 118861332531788 * q^99

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