LMFDB - Embedded newform 48.27.e.d.17.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,27,Mod(17,48)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(48, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 27, names="a")

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

chi := DirichletCharacter("48.17");

S:= CuspForms(chi, 27);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 27 $
Character orbit:	$[\chi]$	$=$	48.e (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:	$205.580601950$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 345470142396704 x^{6} + \cdots + 34\!\cdots\!00 $ x^8 + 345470142396704x^6 + 40918711054321233923976727125x^4 + 2002930760379199788283298503963129162531250*x^2 + 34160921082543983814440857408143040807600840705398437500
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{58}\cdot 3^{38}\cdot 5^{3} $
Twist minimal:	no (minimal twist has level 3)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			17.8
Root			$6.30303e6i$ of defining polynomial
Character	$\chi$	$=$	48.17
Dual form			48.27.e.d.17.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.57937e6 + 217880. i) q^{3} +7.56364e8i q^{5} +6.43856e10 q^{7} +(2.44692e12 + 6.88223e11i) q^{9} +O(q^{10})$	\(q+(1.57937e6 + 217880. i) q^{3} +7.56364e8i q^{5} +6.43856e10 q^{7} +(2.44692e12 + 6.88223e11i) q^{9} -4.53285e13i q^{11} -1.22189e14 q^{13} +(-1.64796e14 + 1.19458e15i) q^{15} -8.44454e14i q^{17} -1.38890e16 q^{19} +(1.01688e17 + 1.40283e16i) q^{21} +6.13136e17i q^{23} +9.18029e17 q^{25} +(3.71463e18 + 1.62009e18i) q^{27} -1.18072e19i q^{29} +2.12418e19 q^{31} +(9.87616e18 - 7.15902e19i) q^{33} +4.86990e19i q^{35} +2.40487e19 q^{37} +(-1.92980e20 - 2.66224e19i) q^{39} +2.62573e20i q^{41} +8.46691e20 q^{43} +(-5.20548e20 + 1.85076e21i) q^{45} -8.66938e21i q^{47} -5.24197e21 q^{49} +(1.83990e20 - 1.33370e21i) q^{51} -3.15687e22i q^{53} +3.42848e22 q^{55} +(-2.19357e22 - 3.02612e21i) q^{57} +4.46845e22i q^{59} -8.62736e22 q^{61} +(1.57547e23 + 4.43117e22i) q^{63} -9.24190e22i q^{65} +7.14670e23 q^{67} +(-1.33590e23 + 9.68365e23i) q^{69} -1.40204e24i q^{71} +1.49986e24 q^{73} +(1.44990e24 + 2.00020e23i) q^{75} -2.91850e24i q^{77} +1.64771e24 q^{79} +(5.51378e24 + 3.36806e24i) q^{81} +1.43170e25i q^{83} +6.38715e23 q^{85} +(2.57256e24 - 1.86480e25i) q^{87} -2.88024e25i q^{89} -7.86718e24 q^{91} +(3.35486e25 + 4.62816e24i) q^{93} -1.05051e25i q^{95} +9.43022e25 q^{97} +(3.11961e25 - 1.10915e26i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 1261080 q^{3} + 83723264240 q^{7} + 3535237525320 q^{9}+O(q^{10}) $ 8 * q + 1261080 * q^3 + 83723264240 * q^7 + 3535237525320 * q^9	$ 8 q + 1261080 q^{3} + 83723264240 q^{7} + 3535237525320 q^{9} - 397915354631600 q^{13} + 719893807027200 q^{15} - 16\!\cdots\!00 q^{19}+ \cdots - 12\!\cdots\!20 q^{99}+O(q^{100}) $ 8 * q + 1261080 * q^3 + 83723264240 * q^7 + 3535237525320 * q^9 - 397915354631600 * q^13 + 719893807027200 * q^15 - 16320401555362000 * q^19 + 194580840365988048 * q^21 + 1971388854053049800 * q^25 + 7894649015962792920 * q^27 + 9272518953250081904 * q^31 - 65168600178932490240 * q^33 + 879374467120740709840 * q^37 + 1582238541124790450928 * q^39 - 6748593942335676212560 * q^43 - 16831420952659929907200 * q^45 + 45841873858091621647512 * q^49 + 62114113806050924347392 * q^51 - 31541176921532063385600 * q^55 - 68601650728105742286960 * q^57 - 319050283467071524566704 * q^61 - 658789337835295522626960 * q^63 + 3016122636770760226629680 * q^67 + 2874694883813365556256768 * q^69 - 1734362529074066854728560 * q^73 - 551279759153288953055400 * q^75 - 5851464318766097144790160 * q^79 - 7387002303358032367832952 * q^81 + 36766362882842005714329600 * q^85 + 66105320167985812595481600 * q^87 - 57658591275235919126596256 * q^91 - 721851591602497641585840 * q^93 + 87088434574088023884109840 * q^97 - 126385397236902706353285120 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/48\mathbb{Z}\right)^\times$.

$n$	$17$	$31$	$37$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	1.57937e6	+	217880.i	0.990618	+	0.136660i
$3$	1.57937e6	+	217880.i	0.990618	+	0.136660i
$4$		0			0
$5$			7.56364e8i			0.619614i	0.950800	+	0.309807i	$0.100264\pi$
$5$			7.56364e8i			0.619614i	−0.950800	+	0.309807i	$0.899736\pi$
$6$		0			0
$7$	6.43856e10			0.664529			0.332265	−	0.943186i	$-0.392187\pi$
$7$	6.43856e10			0.664529			0.332265	+	0.943186i	$0.392187\pi$
$8$		0			0
$9$	2.44692e12	+	6.88223e11i	0.962648	+	0.270755i
$10$		0			0
$11$		−	4.53285e13i		−	1.31300i	−0.754324	−	0.656502i	$-0.772035\pi$
$11$		−	4.53285e13i		−	1.31300i	0.754324	−	0.656502i	$-0.227965\pi$
$12$		0			0
$13$	−1.22189e14			−0.403429			−0.201714	−	0.979444i	$-0.564651\pi$
$13$	−1.22189e14			−0.403429			−0.201714	+	0.979444i	$0.564651\pi$
$14$		0			0
$15$	−1.64796e14	+	1.19458e15i	−0.0846762	+	0.613800i
$16$		0			0
$17$		−	8.44454e14i		−	0.0852590i	−0.999091	−	0.0426295i	$-0.986426\pi$
$17$		−	8.44454e14i		−	0.0852590i	0.999091	−	0.0426295i	$-0.0135735\pi$
$18$		0			0
$19$	−1.38890e16			−0.330273			−0.165136	−	0.986271i	$-0.552806\pi$
$19$	−1.38890e16			−0.330273			−0.165136	+	0.986271i	$0.552806\pi$
$20$		0			0
$21$	1.01688e17	+	1.40283e16i	0.658295	+	0.0908144i
$22$		0			0
$23$			6.13136e17i			1.21645i	0.793764	+	0.608226i	$0.208118\pi$
$23$			6.13136e17i			1.21645i	−0.793764	+	0.608226i	$0.791882\pi$
$24$		0			0
$25$	9.18029e17			0.616079
$26$		0			0
$27$	3.71463e18	+	1.62009e18i	0.916615	+	0.399770i
$28$		0			0
$29$		−	1.18072e19i		−	1.15073i	−0.817896	−	0.575367i	$-0.804859\pi$
$29$		−	1.18072e19i		−	1.15073i	0.817896	−	0.575367i	$-0.195141\pi$
$30$		0			0
$31$	2.12418e19			0.869940			0.434970	−	0.900445i	$-0.356759\pi$
$31$	2.12418e19			0.869940			0.434970	+	0.900445i	$0.356759\pi$
$32$		0			0
$33$	9.87616e18	−	7.15902e19i	0.179435	−	1.30069i
$34$		0			0
$35$			4.86990e19i			0.411751i
$36$		0			0
$37$	2.40487e19			0.0987345			0.0493673	−	0.998781i	$-0.484280\pi$
$37$	2.40487e19			0.0987345			0.0493673	+	0.998781i	$0.484280\pi$
$38$		0			0
$39$	−1.92980e20	−	2.66224e19i	−0.399644	−	0.0551325i
$40$		0			0
$41$			2.62573e20i			0.283831i	0.989879	+	0.141916i	$0.0453262\pi$
$41$			2.62573e20i			0.283831i	−0.989879	+	0.141916i	$0.954674\pi$
$42$		0			0
$43$	8.46691e20			0.492759			0.246380	−	0.969173i	$-0.420759\pi$
$43$	8.46691e20			0.492759			0.246380	+	0.969173i	$0.420759\pi$
$44$		0			0
$45$	−5.20548e20	+	1.85076e21i	−0.167764	+	0.596470i
$46$		0			0
$47$		−	8.66938e21i		−	1.58751i	−0.608239	−	0.793754i	$-0.708124\pi$
$47$		−	8.66938e21i		−	1.58751i	0.608239	−	0.793754i	$-0.291876\pi$
$48$		0			0
$49$	−5.24197e21			−0.558401
$50$		0			0
$51$	1.83990e20	−	1.33370e21i	0.0116515	−	0.0844591i
$52$		0			0
$53$		−	3.15687e22i		−	1.21247i	−0.795286	−	0.606234i	$-0.792680\pi$
$53$		−	3.15687e22i		−	1.21247i	0.795286	−	0.606234i	$-0.207320\pi$
$54$		0			0
$55$	3.42848e22			0.813555
$56$		0			0
$57$	−2.19357e22	−	3.02612e21i	−0.327174	−	0.0451350i
$58$		0			0
$59$			4.46845e22i			0.425677i	0.977087	+	0.212839i	$0.0682709\pi$
$59$			4.46845e22i			0.425677i	−0.977087	+	0.212839i	$0.931729\pi$
$60$		0			0
$61$	−8.62736e22			−0.532831			−0.266416	−	0.963858i	$-0.585839\pi$
$61$	−8.62736e22			−0.532831			−0.266416	+	0.963858i	$0.585839\pi$
$62$		0			0
$63$	1.57547e23	+	4.43117e22i	0.639708	+	0.179925i
$64$		0			0
$65$		−	9.24190e22i		−	0.249970i
$66$		0			0
$67$	7.14670e23			1.30357			0.651783	−	0.758405i	$-0.274021\pi$
$67$	7.14670e23			1.30357			0.651783	+	0.758405i	$0.274021\pi$
$68$		0			0
$69$	−1.33590e23	+	9.68365e23i	−0.166240	+	1.20504i
$70$		0			0
$71$		−	1.40204e24i		−	1.20338i	−0.798730	−	0.601689i	$-0.794495\pi$
$71$		−	1.40204e24i		−	1.20338i	0.798730	−	0.601689i	$-0.205505\pi$
$72$		0			0
$73$	1.49986e24			0.897125			0.448563	−	0.893751i	$-0.351936\pi$
$73$	1.49986e24			0.897125			0.448563	+	0.893751i	$0.351936\pi$
$74$		0			0
$75$	1.44990e24	+	2.00020e23i	0.610299	+	0.0841932i
$76$		0			0
$77$		−	2.91850e24i		−	0.872530i
$78$		0			0
$79$	1.64771e24			0.352962			0.176481	−	0.984304i	$-0.443529\pi$
$79$	1.64771e24			0.352962			0.176481	+	0.984304i	$0.443529\pi$
$80$		0			0
$81$	5.51378e24	+	3.36806e24i	0.853383	+	0.521284i
$82$		0			0
$83$			1.43170e25i			1.61375i	0.590721	+	0.806876i	$0.298844\pi$
$83$			1.43170e25i			1.61375i	−0.590721	+	0.806876i	$0.701156\pi$
$84$		0			0
$85$	6.38715e23			0.0528276
$86$		0			0
$87$	2.57256e24	−	1.86480e25i	0.157259	−	1.13994i
$88$		0			0
$89$		−	2.88024e25i		−	1.31026i	−0.755515	−	0.655132i	$-0.772613\pi$
$89$		−	2.88024e25i		−	1.31026i	0.755515	−	0.655132i	$-0.227387\pi$
$90$		0			0
$91$	−7.86718e24			−0.268090
$92$		0			0
$93$	3.35486e25	+	4.62816e24i	0.861779	+	0.118886i
$94$		0			0
$95$		−	1.05051e25i		−	0.204642i
$96$		0			0
$97$	9.43022e25			1.40116			0.700582	−	0.713572i	$-0.252924\pi$
$97$	9.43022e25			1.40116			0.700582	+	0.713572i	$0.252924\pi$
$98$		0			0
$99$	3.11961e25	−	1.10915e26i	0.355503	−	1.26396i
$100$		0			0
$101$			2.88995e25i			0.253929i	0.991907	+	0.126965i	$0.0405235\pi$
$101$			2.88995e25i			0.253929i	−0.991907	+	0.126965i	$0.959477\pi$
$102$		0			0
$103$	2.36365e25			0.160953			0.0804767	−	0.996756i	$-0.474356\pi$
$103$	2.36365e25			0.160953			0.0804767	+	0.996756i	$0.474356\pi$
$104$		0			0
$105$	−1.06105e25	+	7.69134e25i	−0.0562699	+	0.407888i
$106$		0			0
$107$			3.73318e26i			1.54914i	0.632491	+	0.774568i	$0.282032\pi$
$107$			3.73318e26i			1.54914i	−0.632491	+	0.774568i	$0.717968\pi$
$108$		0			0
$109$	−5.14103e26			−1.67689			−0.838447	−	0.544983i	$-0.816536\pi$
$109$	−5.14103e26			−1.67689			−0.838447	+	0.544983i	$0.816536\pi$
$110$		0			0
$111$	3.79817e25	+	5.23972e24i	0.0978082	+	0.0134930i
$112$		0			0
$113$		−	1.57160e26i		−	0.320866i	−0.987047	−	0.160433i	$-0.948711\pi$
$113$		−	1.57160e26i		−	0.320866i	0.987047	−	0.160433i	$-0.0512890\pi$
$114$		0			0
$115$	−4.63754e26			−0.753730
$116$		0			0
$117$	−2.98986e26	−	8.40930e25i	−0.388360	−	0.109230i
$118$		0			0
$119$		−	5.43707e25i		−	0.0566571i
$120$		0			0
$121$	−8.62854e26			−0.723981
$122$		0			0
$123$	−5.72094e25	+	4.14699e26i	−0.0387883	+	0.281169i
$124$		0			0
$125$			1.82143e27i			1.00134i
$126$		0			0
$127$	8.92136e26			0.399009			0.199505	−	0.979897i	$-0.436067\pi$
$127$	8.92136e26			0.399009			0.199505	+	0.979897i	$0.436067\pi$
$128$		0			0
$129$	1.33723e27	+	1.84477e26i	0.488136	+	0.0673404i
$130$		0			0
$131$		−	4.35795e27i		−	1.30243i	−0.758892	−	0.651217i	$-0.774259\pi$
$131$		−	4.35795e27i		−	1.30243i	0.758892	−	0.651217i	$-0.225741\pi$
$132$		0			0
$133$	−8.94249e26			−0.219476
$134$		0			0
$135$	−1.22538e27	+	2.80962e27i	−0.247703	+	0.567947i
$136$		0			0
$137$			3.72366e25i			0.00621732i	0.999995	+	0.00310866i	$0.000989519\pi$
$137$			3.72366e25i			0.00621732i	−0.999995	+	0.00310866i	$0.999010\pi$
$138$		0			0
$139$	−6.17990e27			−0.854652			−0.427326	−	0.904098i	$-0.640544\pi$
$139$	−6.17990e27			−0.854652			−0.427326	+	0.904098i	$0.640544\pi$
$140$		0			0
$141$	1.88888e27	−	1.36921e28i	0.216948	−	1.57261i
$142$		0			0
$143$			5.53862e27i			0.529704i
$144$		0			0
$145$	8.93058e27			0.713010
$146$		0			0
$147$	−8.27899e27	−	1.14212e27i	−0.553162	−	0.0763109i
$148$		0			0
$149$			1.14194e28i			0.640064i	0.947407	+	0.320032i	$0.103694\pi$
$149$			1.14194e28i			0.640064i	−0.947407	+	0.320032i	$0.896306\pi$
$150$		0			0
$151$	2.99898e28			1.41343			0.706713	−	0.707501i	$-0.250177\pi$
$151$	2.99898e28			1.41343			0.706713	+	0.707501i	$0.250177\pi$
$152$		0			0
$153$	5.81173e26	−	2.06631e27i	0.0230843	−	0.0820744i
$154$		0			0
$155$			1.60665e28i			0.539027i
$156$		0			0
$157$	2.76145e28			0.784226			0.392113	−	0.919917i	$-0.371744\pi$
$157$	2.76145e28			0.784226			0.392113	+	0.919917i	$0.371744\pi$
$158$		0			0
$159$	6.87818e27	−	4.98585e28i	0.165696	−	1.20109i
$160$		0			0
$161$			3.94771e28i			0.808368i
$162$		0			0
$163$	8.18364e28			1.42728			0.713638	−	0.700515i	$-0.247046\pi$
$163$	8.18364e28			1.42728			0.713638	+	0.700515i	$0.247046\pi$
$164$		0			0
$165$	5.41483e28	+	7.46997e27i	0.805923	+	0.111180i
$166$		0			0
$167$			2.88328e27i			0.0366921i	0.999832	+	0.0183461i	$0.00584006\pi$
$167$			2.88328e27i			0.0366921i	−0.999832	+	0.0183461i	$0.994160\pi$
$168$		0			0
$169$	−7.68033e28			−0.837245
$170$		0			0
$171$	−3.39852e28	−	9.55871e27i	−0.317937	−	0.0894231i
$172$		0			0
$173$		−	1.95703e29i		−	1.57398i	−0.616968	−	0.786988i	$-0.711639\pi$
$173$		−	1.95703e29i		−	1.57398i	0.616968	−	0.786988i	$-0.288361\pi$
$174$		0			0
$175$	5.91079e28			0.409403
$176$		0			0
$177$	−9.73584e27	+	7.05731e28i	−0.0581730	+	0.421684i
$178$		0			0
$179$			1.31094e29i			0.676849i	0.940994	+	0.338424i	$0.109894\pi$
$179$			1.31094e29i			0.676849i	−0.940994	+	0.338424i	$0.890106\pi$
$180$		0			0
$181$	−1.90225e29			−0.850054			−0.425027	−	0.905181i	$-0.639735\pi$
$181$	−1.90225e29			−0.850054			−0.425027	+	0.905181i	$0.639735\pi$
$182$		0			0
$183$	−1.36257e29	−	1.87973e28i	−0.527832	−	0.0728166i
$184$		0			0
$185$			1.81896e28i			0.0611773i
$186$		0			0
$187$	−3.82778e28			−0.111945
$188$		0			0
$189$	2.39169e29	+	1.04311e29i	0.609118	+	0.265659i
$190$		0			0
$191$		−	1.16985e29i		−	0.259833i	−0.991525	−	0.129917i	$-0.958529\pi$
$191$		−	1.16985e29i		−	0.259833i	0.991525	−	0.129917i	$-0.0414710\pi$
$192$		0			0
$193$	2.85826e29			0.554443			0.277221	−	0.960806i	$-0.410586\pi$
$193$	2.85826e29			0.554443			0.277221	+	0.960806i	$0.410586\pi$
$194$		0			0
$195$	2.01362e28	−	1.45963e29i	0.0341608	−	0.247625i
$196$		0			0
$197$			7.75872e28i			0.115273i	0.998338	+	0.0576367i	$0.0183565\pi$
$197$			7.75872e28i			0.115273i	−0.998338	+	0.0576367i	$0.981643\pi$
$198$		0			0
$199$	−3.66878e28			−0.0478005			−0.0239002	−	0.999714i	$-0.507608\pi$
$199$	−3.66878e28			−0.0478005			−0.0239002	+	0.999714i	$0.507608\pi$
$200$		0			0
$201$	1.12872e30	+	1.55712e29i	1.29134	+	0.178145i
$202$		0			0
$203$		−	7.60217e29i		−	0.764696i
$204$		0			0
$205$	−1.98601e29			−0.175866
$206$		0			0
$207$	−4.21974e29	+	1.50030e30i	−0.329361	+	1.17101i
$208$		0			0
$209$			6.29566e29i			0.433650i
$210$		0			0
$211$	2.52317e30			1.53559			0.767793	−	0.640699i	$-0.221355\pi$
$211$	2.52317e30			1.53559			0.767793	+	0.640699i	$0.221355\pi$
$212$		0			0
$213$	3.05476e29	−	2.21433e30i	0.164453	−	1.19209i
$214$		0			0
$215$			6.40407e29i			0.305320i
$216$		0			0
$217$	1.36767e30			0.578101
$218$		0			0
$219$	2.36882e30	+	3.26789e29i	0.888709	+	0.122601i
$220$		0			0
$221$			1.03183e29i			0.0343959i
$222$		0			0
$223$	5.39481e30			1.59960			0.799802	−	0.600263i	$-0.204938\pi$
$223$	5.39481e30			1.59960			0.799802	+	0.600263i	$0.204938\pi$
$224$		0			0
$225$	2.24635e30	+	6.31809e29i	0.593067	+	0.166807i
$226$		0			0
$227$		−	4.37641e30i		−	1.02987i	−0.857229	−	0.514935i	$-0.827816\pi$
$227$		−	4.37641e30i		−	1.02987i	0.857229	−	0.514935i	$-0.172184\pi$
$228$		0			0
$229$	5.23211e30			1.09854			0.549270	−	0.835645i	$-0.314906\pi$
$229$	5.23211e30			1.09854			0.549270	+	0.835645i	$0.314906\pi$
$230$		0			0
$231$	6.35883e29	−	4.60938e30i	0.119240	−	0.864344i
$232$		0			0
$233$		−	4.10091e30i		−	0.687469i	−0.939067	−	0.343735i	$-0.888308\pi$
$233$		−	4.10091e30i		−	0.687469i	0.939067	−	0.343735i	$-0.111692\pi$
$234$		0			0
$235$	6.55721e30			0.983641
$236$		0			0
$237$	2.60233e30	+	3.59002e29i	0.349651	+	0.0482357i
$238$		0			0
$239$		−	6.18375e30i		−	0.744868i	−0.928059	−	0.372434i	$-0.878523\pi$
$239$		−	6.18375e30i		−	0.744868i	0.928059	−	0.372434i	$-0.121477\pi$
$240$		0			0
$241$	6.70018e30			0.724211			0.362105	−	0.932137i	$-0.382058\pi$
$241$	6.70018e30			0.724211			0.362105	+	0.932137i	$0.382058\pi$
$242$		0			0
$243$	7.97444e30	+	6.52074e30i	0.774138	+	0.633017i
$244$		0			0
$245$		−	3.96484e30i		−	0.345993i
$246$		0			0
$247$	1.69707e30			0.133242
$248$		0			0
$249$	−3.11939e30	+	2.26118e31i	−0.220535	+	1.59861i
$250$		0			0
$251$			2.03891e31i			1.29909i	0.760322	+	0.649547i	$0.225041\pi$
$251$			2.03891e31i			1.29909i	−0.760322	+	0.649547i	$0.774959\pi$
$252$		0			0
$253$	2.77925e31			1.59721
$254$		0			0
$255$	1.00876e30	+	1.39163e29i	0.0523320	+	0.00721941i
$256$		0			0
$257$		−	2.66641e31i		−	1.24967i	−0.780755	−	0.624837i	$-0.785165\pi$
$257$		−	2.66641e31i		−	1.24967i	0.780755	−	0.624837i	$-0.214835\pi$
$258$		0			0
$259$	1.54839e30			0.0656120
$260$		0			0
$261$	8.12603e30	−	2.88914e31i	0.311567	−	1.10775i
$262$		0			0
$263$			1.77891e31i			0.617631i	0.951122	+	0.308815i	$0.0999325\pi$
$263$			1.77891e31i			0.617631i	−0.951122	+	0.308815i	$0.900067\pi$
$264$		0			0
$265$	2.38774e31			0.751261
$266$		0			0
$267$	6.27546e30	−	4.54895e31i	0.179060	−	1.29797i
$268$		0			0
$269$			6.43690e31i			1.66685i	0.552631	+	0.833426i	$0.313624\pi$
$269$			6.43690e31i			1.66685i	−0.552631	+	0.833426i	$0.686376\pi$
$270$		0			0
$271$	1.23255e29			0.00289871			0.00144935	−	0.999999i	$-0.499539\pi$
$271$	1.23255e29			0.00289871			0.00144935	+	0.999999i	$0.499539\pi$
$272$		0			0
$273$	−1.24252e31	−	1.71410e30i	−0.265575	−	0.0366372i
$274$		0			0
$275$		−	4.16129e31i		−	0.808915i
$276$		0			0
$277$	−4.19607e30			−0.0742346			−0.0371173	−	0.999311i	$-0.511818\pi$
$277$	−4.19607e30			−0.0742346			−0.0371173	+	0.999311i	$0.511818\pi$
$278$		0			0
$279$	5.19771e31	+	1.46191e31i	0.837447	+	0.235541i
$280$		0			0
$281$		−	8.93137e30i		−	0.131140i	−0.997848	−	0.0655701i	$-0.979113\pi$
$281$		−	8.93137e30i		−	0.131140i	0.997848	−	0.0655701i	$-0.0208866\pi$
$282$		0			0
$283$	6.97738e31			0.934262			0.467131	−	0.884188i	$-0.345288\pi$
$283$	6.97738e31			0.934262			0.467131	+	0.884188i	$0.345288\pi$
$284$		0			0
$285$	2.28885e30	−	1.65914e31i	0.0279663	−	0.202722i
$286$		0			0
$287$			1.69059e31i			0.188614i
$288$		0			0
$289$	9.73876e31			0.992731
$290$		0			0
$291$	1.48938e32	+	2.05465e31i	1.38802	+	0.191483i
$292$		0			0
$293$			1.57236e31i			0.134052i	0.997751	+	0.0670259i	$0.0213510\pi$
$293$			1.57236e31i			0.134052i	−0.997751	+	0.0670259i	$0.978649\pi$
$294$		0			0
$295$	−3.37977e31			−0.263755
$296$		0			0
$297$	7.34363e31	−	1.68379e32i	0.524900	−	1.20352i
$298$		0			0
$299$		−	7.49181e31i		−	0.490751i
$300$		0			0
$301$	5.45147e31			0.327453
$302$		0			0
$303$	−6.29662e30	+	4.56429e31i	−0.0347019	+	0.251547i
$304$		0			0
$305$		−	6.52542e31i		−	0.330150i
$306$		0			0
$307$	2.66329e32			1.23771			0.618856	−	0.785505i	$-0.287597\pi$
$307$	2.66329e32			1.23771			0.618856	+	0.785505i	$0.287597\pi$
$308$		0			0
$309$	3.73307e31	+	5.14993e30i	0.159443	+	0.0219958i
$310$		0			0
$311$		−	8.37175e31i		−	0.328800i	−0.986394	−	0.164400i	$-0.947431\pi$
$311$		−	8.37175e31i		−	0.328800i	0.986394	−	0.164400i	$-0.0525687\pi$
$312$		0			0
$313$	5.08239e31			0.183650			0.0918251	−	0.995775i	$-0.470730\pi$
$313$	5.08239e31			0.183650			0.0918251	+	0.995775i	$0.470730\pi$
$314$		0			0
$315$	−3.35158e31	+	1.19163e32i	−0.111484	+	0.396372i
$316$		0			0
$317$			2.99132e32i			0.916417i	0.888845	+	0.458208i	$0.151509\pi$
$317$			2.99132e32i			0.916417i	−0.888845	+	0.458208i	$0.848491\pi$
$318$		0			0
$319$	−5.35205e32			−1.51092
$320$		0			0
$321$	−8.13384e31	+	5.89605e32i	−0.211704	+	1.53460i
$322$		0			0
$323$			1.17286e31i			0.0281587i
$324$		0			0
$325$	−1.12173e32			−0.248544
$326$		0			0
$327$	−8.11956e32	−	1.12013e32i	−1.66116	−	0.229164i
$328$		0			0
$329$		−	5.58183e32i		−	1.05495i
$330$		0			0
$331$	2.78561e32			0.486583			0.243292	−	0.969953i	$-0.421773\pi$
$331$	2.78561e32			0.486583			0.243292	+	0.969953i	$0.421773\pi$
$332$		0			0
$333$	5.88453e31	+	1.65509e31i	0.0950466	+	0.0267329i
$334$		0			0
$335$			5.40551e32i			0.807707i
$336$		0			0
$337$	−7.29911e32			−1.00944			−0.504721	−	0.863283i	$-0.668405\pi$
$337$	−7.29911e32			−1.00944			−0.504721	+	0.863283i	$0.668405\pi$
$338$		0			0
$339$	3.42421e31	−	2.48214e32i	0.0438494	−	0.317855i
$340$		0			0
$341$		−	9.62859e32i		−	1.14224i
$342$		0			0
$343$	−9.41926e32			−1.03560
$344$		0			0
$345$	−7.32437e32	−	1.01043e32i	−0.746658	−	0.103005i
$346$		0			0
$347$			1.30904e32i			0.123785i	0.998083	+	0.0618927i	$0.0197137\pi$
$347$			1.30904e32i			0.123785i	−0.998083	+	0.0618927i	$0.980286\pi$
$348$		0			0
$349$	−1.55440e33			−1.36406			−0.682030	−	0.731324i	$-0.738903\pi$
$349$	−1.55440e33			−1.36406			−0.682030	+	0.731324i	$0.738903\pi$
$350$		0			0
$351$	−4.53886e32	−	1.97957e32i	−0.369789	−	0.161279i
$352$		0			0
$353$			1.39321e33i			1.05426i	0.849785	+	0.527129i	$0.176732\pi$
$353$			1.39321e33i			1.05426i	−0.849785	+	0.527129i	$0.823268\pi$
$354$		0			0
$355$	1.06045e33			0.745629
$356$		0			0
$357$	1.18463e31	−	8.58712e31i	0.00774275	−	0.0561256i
$358$		0			0
$359$			4.86706e32i			0.295828i	0.989000	+	0.147914i	$0.0472559\pi$
$359$			4.86706e32i			0.295828i	−0.989000	+	0.147914i	$0.952744\pi$
$360$		0			0
$361$	−1.57555e33			−0.890920
$362$		0			0
$363$	−1.36276e33	−	1.87998e32i	−0.717189	−	0.0989391i
$364$		0			0
$365$			1.13444e33i			0.555871i
$366$		0			0
$367$	−2.49772e33			−1.13995			−0.569974	−	0.821663i	$-0.693047\pi$
$367$	−2.49772e33			−1.13995			−0.569974	+	0.821663i	$0.693047\pi$
$368$		0			0
$369$	−1.80709e32	+	6.42497e32i	−0.0768488	+	0.273230i
$370$		0			0
$371$		−	2.03257e33i		−	0.805721i
$372$		0			0
$373$	−2.22041e33			−0.820765			−0.410383	−	0.911913i	$-0.634605\pi$
$373$	−2.22041e33			−0.820765			−0.410383	+	0.911913i	$0.634605\pi$
$374$		0			0
$375$	−3.96854e32	+	2.87671e33i	−0.136843	+	0.991950i
$376$		0			0
$377$			1.44271e33i			0.464239i
$378$		0			0
$379$	−1.36618e33			−0.410392			−0.205196	−	0.978721i	$-0.565783\pi$
$379$	−1.36618e33			−0.410392			−0.205196	+	0.978721i	$0.565783\pi$
$380$		0			0
$381$	1.40901e33	+	1.94378e32i	0.395266	+	0.0545285i
$382$		0			0
$383$		−	6.09447e32i		−	0.159717i	−0.996806	−	0.0798586i	$-0.974553\pi$
$383$		−	6.09447e32i		−	0.159717i	0.996806	−	0.0798586i	$-0.0254469\pi$
$384$		0			0
$385$	2.20745e33			0.540632
$386$		0			0
$387$	2.07179e33	+	5.82713e32i	0.474354	+	0.133417i
$388$		0			0
$389$		−	5.78437e33i		−	1.23854i	−0.785177	−	0.619272i	$-0.787428\pi$
$389$		−	5.78437e33i		−	1.23854i	0.785177	−	0.619272i	$-0.212572\pi$
$390$		0			0
$391$	5.17765e32			0.103713
$392$		0			0
$393$	9.49510e32	−	6.88280e33i	0.177990	−	1.29021i
$394$		0			0
$395$			1.24627e33i			0.218700i
$396$		0			0
$397$	1.63919e33			0.269372			0.134686	−	0.990888i	$-0.456997\pi$
$397$	1.63919e33			0.269372			0.134686	+	0.990888i	$0.456997\pi$
$398$		0			0
$399$	−1.41235e33	−	1.94839e32i	−0.217417	−	0.0299936i
$400$		0			0
$401$			6.56795e33i			0.947443i	0.880675	+	0.473721i	$0.157090\pi$
$401$			6.56795e33i			0.947443i	−0.880675	+	0.473721i	$0.842910\pi$
$402$		0			0
$403$	−2.59551e33			−0.350959
$404$		0			0
$405$	−2.54748e33	+	4.17042e33i	−0.322995	+	0.528768i
$406$		0			0
$407$		−	1.09009e33i		−	0.129639i
$408$		0			0
$409$	2.39163e33			0.266864			0.133432	−	0.991058i	$-0.457400\pi$
$409$	2.39163e33			0.266864			0.133432	+	0.991058i	$0.457400\pi$
$410$		0			0
$411$	−8.11311e30	+	5.88103e31i	−0.000849657	+	0.00615899i
$412$		0			0
$413$			2.87704e33i			0.282875i
$414$		0			0
$415$	−1.08289e34			−0.999903
$416$		0			0
$417$	−9.76032e33	−	1.34648e33i	−0.846633	−	0.116796i
$418$		0			0
$419$			2.86768e33i			0.233748i	0.993147	+	0.116874i	$0.0372874\pi$
$419$			2.86768e33i			0.233748i	−0.993147	+	0.116874i	$0.962713\pi$
$420$		0			0
$421$	−1.43806e34			−1.10182			−0.550908	−	0.834566i	$-0.685718\pi$
$421$	−1.43806e34			−1.10182			−0.550908	+	0.834566i	$0.685718\pi$
$422$		0			0
$423$	5.96647e33	−	2.12133e34i	0.429826	−	1.52821i
$424$		0			0
$425$		−	7.75234e32i		−	0.0525263i
$426$		0			0
$427$	−5.55478e33			−0.354082
$428$		0			0
$429$	−1.20675e33	+	8.74750e33i	−0.0723892	+	0.524734i
$430$		0			0
$431$			1.54654e34i			0.873288i	0.899634	+	0.436644i	$0.143833\pi$
$431$			1.54654e34i			0.873288i	−0.899634	+	0.436644i	$0.856167\pi$
$432$		0			0
$433$	1.26710e34			0.673705			0.336852	−	0.941557i	$-0.390638\pi$
$433$	1.26710e34			0.673705			0.336852	+	0.941557i	$0.390638\pi$
$434$		0			0
$435$	1.41046e34	+	1.94579e33i	0.706321	+	0.0974398i
$436$		0			0
$437$		−	8.51582e33i		−	0.401761i
$438$		0			0
$439$	3.61999e34			1.60942			0.804709	−	0.593670i	$-0.202321\pi$
$439$	3.61999e34			1.60942			0.804709	+	0.593670i	$0.202321\pi$
$440$		0			0
$441$	−1.28267e34	−	3.60765e33i	−0.537543	−	0.151190i
$442$		0			0
$443$			1.20051e34i			0.474371i	0.971464	+	0.237186i	$0.0762250\pi$
$443$			1.20051e34i			0.474371i	−0.971464	+	0.237186i	$0.923775\pi$
$444$		0			0
$445$	2.17851e34			0.811857
$446$		0			0
$447$	−2.48806e33	+	1.80355e34i	−0.0874710	+	0.634059i
$448$		0			0
$449$		−	1.65389e34i		−	0.548662i	−0.961635	−	0.274331i	$-0.911544\pi$
$449$		−	1.65389e34i		−	0.548662i	0.961635	−	0.274331i	$-0.0884563\pi$
$450$		0			0
$451$	1.19020e34			0.372672
$452$		0			0
$453$	4.73648e34	+	6.53417e33i	1.40016	+	0.193158i
$454$		0			0
$455$		−	5.95045e33i		−	0.166112i
$456$		0			0
$457$	3.92198e34			1.03418			0.517088	−	0.855932i	$-0.327016\pi$
$457$	3.92198e34			1.03418			0.517088	+	0.855932i	$0.327016\pi$
$458$		0			0
$459$	1.36809e33	−	3.13684e33i	0.0340840	−	0.0781497i
$460$		0			0
$461$			1.19774e34i			0.282000i	0.990010	+	0.141000i	$0.0450318\pi$
$461$			1.19774e34i			0.282000i	−0.990010	+	0.141000i	$0.954968\pi$
$462$		0			0
$463$	6.00206e33			0.133582			0.0667909	−	0.997767i	$-0.478724\pi$
$463$	6.00206e33			0.133582			0.0667909	+	0.997767i	$0.478724\pi$
$464$		0			0
$465$	−3.50058e33	+	2.53749e34i	−0.0736633	+	0.533970i
$466$		0			0
$467$		−	7.71241e34i		−	1.53487i	−0.641128	−	0.767434i	$-0.721533\pi$
$467$		−	7.71241e34i		−	1.53487i	0.641128	−	0.767434i	$-0.278467\pi$
$468$		0			0
$469$	4.60145e34			0.866258
$470$		0			0
$471$	4.36133e34	+	6.01664e33i	0.776869	+	0.107172i
$472$		0			0
$473$		−	3.83792e34i		−	0.646995i
$474$		0			0
$475$	−1.27505e34			−0.203474
$476$		0			0
$477$	2.17263e34	−	7.72461e34i	0.328282	−	1.16718i
$478$		0			0
$479$		−	9.39293e34i		−	1.34412i	−0.740495	−	0.672062i	$-0.765409\pi$
$479$		−	9.39293e34i		−	1.34412i	0.740495	−	0.672062i	$-0.234591\pi$
$480$		0			0
$481$	−2.93847e33			−0.0398323
$482$		0			0
$483$	−8.60127e33	+	6.23488e34i	−0.110471	+	0.800784i
$484$		0			0
$485$			7.13268e34i			0.868181i
$486$		0			0
$487$	1.26082e35			1.45471			0.727356	−	0.686260i	$-0.240749\pi$
$487$	1.26082e35			1.45471			0.727356	+	0.686260i	$0.240749\pi$
$488$		0			0
$489$	1.29250e35	+	1.78305e34i	1.41388	+	0.195051i
$490$		0			0
$491$			4.80809e34i			0.498785i	0.968403	+	0.249392i	$0.0802309\pi$
$491$			4.80809e34i			0.498785i	−0.968403	+	0.249392i	$0.919769\pi$
$492$		0			0
$493$	−9.97068e33			−0.0981104
$494$		0			0
$495$	8.38923e34	+	2.35956e34i	0.783168	+	0.220274i
$496$		0			0
$497$		−	9.02712e34i		−	0.799680i
$498$		0			0
$499$	8.42930e32			0.00708736			0.00354368	−	0.999994i	$-0.498872\pi$
$499$	8.42930e32			0.00708736			0.00354368	+	0.999994i	$0.498872\pi$
$500$		0			0
$501$	−6.28209e32	+	4.55375e33i	−0.00501434	+	0.0363479i
$502$		0			0
$503$			1.44535e35i			1.09544i	0.836661	+	0.547722i	$0.184505\pi$
$503$			1.44535e35i			1.09544i	−0.836661	+	0.547722i	$0.815495\pi$
$504$		0			0
$505$	−2.18586e34			−0.157338
$506$		0			0
$507$	−1.21300e35	−	1.67339e34i	−0.829390	−	0.114418i
$508$		0			0
$509$			4.92853e33i			0.0320174i	0.999872	+	0.0160087i	$0.00509594\pi$
$509$			4.92853e33i			0.0320174i	−0.999872	+	0.0160087i	$0.994904\pi$
$510$		0			0
$511$	9.65693e34			0.596166
$512$		0			0
$513$	−5.15924e34	−	2.25014e34i	−0.302733	−	0.132033i
$514$		0			0
$515$			1.78778e34i			0.0997289i
$516$		0			0
$517$	−3.92970e35			−2.08440
$518$		0			0
$519$	4.26397e34	−	3.09087e35i	0.215099	−	1.55921i
$520$		0			0
$521$			2.96233e35i			1.42149i	0.703451	+	0.710744i	$0.251642\pi$
$521$			2.96233e35i			1.42149i	−0.703451	+	0.710744i	$0.748358\pi$
$522$		0			0
$523$	−2.24702e35			−1.02585			−0.512927	−	0.858432i	$-0.671439\pi$
$523$	−2.24702e35			−1.02585			−0.512927	+	0.858432i	$0.671439\pi$
$524$		0			0
$525$	9.33529e34	+	1.28784e34i	0.405562	+	0.0559489i
$526$		0			0
$527$		−	1.79377e34i		−	0.0741702i
$528$		0			0
$529$	−1.21883e35			−0.479754
$530$		0			0
$531$	−3.07529e34	+	1.09339e35i	−0.115254	+	0.409777i
$532$		0			0
$533$		−	3.20834e34i		−	0.114506i
$534$		0			0
$535$	−2.82364e35			−0.959865
$536$		0			0
$537$	−2.85626e34	+	2.07045e35i	−0.0924980	+	0.670499i
$538$		0			0
$539$			2.37611e35i			0.733182i
$540$		0			0
$541$	8.30656e34			0.244263			0.122131	−	0.992514i	$-0.461027\pi$
$541$	8.30656e34			0.244263			0.122131	+	0.992514i	$0.461027\pi$
$542$		0			0
$543$	−3.00435e35	−	4.14462e34i	−0.842079	−	0.116168i
$544$		0			0
$545$		−	3.88849e35i		−	1.03903i
$546$		0			0
$547$	−2.26776e35			−0.577779			−0.288890	−	0.957362i	$-0.593286\pi$
$547$	−2.26776e35			−0.577779			−0.288890	+	0.957362i	$0.593286\pi$
$548$		0			0
$549$	−2.11105e35	−	5.93755e34i	−0.512929	−	0.144267i
$550$		0			0
$551$			1.63990e35i			0.380056i
$552$		0			0
$553$	1.06089e35			0.234554
$554$		0			0
$555$	−3.96314e33	+	2.87280e34i	−0.00836047	+	0.0606033i
$556$		0			0
$557$		−	4.42912e35i		−	0.891662i	−0.895117	−	0.445831i	$-0.852908\pi$
$557$		−	4.42912e35i		−	0.891662i	0.895117	−	0.445831i	$-0.147092\pi$
$558$		0			0
$559$	−1.03456e35			−0.198793
$560$		0			0
$561$	−6.04547e34	−	8.33997e33i	−0.110895	−	0.0152984i
$562$		0			0
$563$		−	6.73030e35i		−	1.17876i	−0.807856	−	0.589380i	$-0.799372\pi$
$563$		−	6.73030e35i		−	1.17876i	0.807856	−	0.589380i	$-0.200628\pi$
$564$		0			0
$565$	1.18870e35			0.198813
$566$		0			0
$567$	3.55008e35	+	2.16855e35i	0.567098	+	0.346409i
$568$		0			0
$569$			3.02745e35i			0.461975i	0.972957	+	0.230987i	$0.0741956\pi$
$569$			3.02745e35i			0.461975i	−0.972957	+	0.230987i	$0.925804\pi$
$570$		0			0
$571$	−3.40373e34			−0.0496234			−0.0248117	−	0.999692i	$-0.507899\pi$
$571$	−3.40373e34			−0.0496234			−0.0248117	+	0.999692i	$0.507899\pi$
$572$		0			0
$573$	2.54886e34	−	1.84761e35i	0.0355087	−	0.257395i
$574$		0			0
$575$			5.62877e35i			0.749430i
$576$		0			0
$577$	−9.39165e35			−1.19524			−0.597621	−	0.801778i	$-0.703887\pi$
$577$	−9.39165e35			−1.19524			−0.597621	+	0.801778i	$0.703887\pi$
$578$		0			0
$579$	4.51423e35	+	6.22757e34i	0.549241	+	0.0757700i
$580$		0			0
$581$			9.21809e35i			1.07239i
$582$		0			0
$583$	−1.43096e36			−1.59198
$584$		0			0
$585$	6.36049e34	−	2.26142e35i	0.0676807	−	0.240633i
$586$		0			0
$587$		−	1.03983e36i		−	1.05844i	−0.848484	−	0.529222i	$-0.822484\pi$
$587$		−	1.03983e36i		−	1.05844i	0.848484	−	0.529222i	$-0.177516\pi$
$588$		0			0
$589$	−2.95027e35			−0.287318
$590$		0			0
$591$	−1.69047e34	+	1.22539e35i	−0.0157532	+	0.114192i
$592$		0			0
$593$			1.70525e36i			1.52082i	0.649445	+	0.760409i	$0.275001\pi$
$593$			1.70525e36i			1.52082i	−0.649445	+	0.760409i	$0.724999\pi$
$594$		0			0
$595$	4.11240e34			0.0351055
$596$		0			0
$597$	−5.79435e34	−	7.99354e33i	−0.0473520	−	0.00653240i
$598$		0			0
$599$			3.19697e35i			0.250144i	0.992148	+	0.125072i	$0.0399162\pi$
$599$			3.19697e35i			0.250144i	−0.992148	+	0.125072i	$0.960084\pi$
$600$		0			0
$601$	−8.59736e35			−0.644165			−0.322082	−	0.946712i	$-0.604383\pi$
$601$	−8.59736e35			−0.644165			−0.322082	+	0.946712i	$0.604383\pi$
$602$		0			0
$603$	1.74874e36	+	4.91853e35i	1.25488	+	0.352947i
$604$		0			0
$605$		−	6.52631e35i		−	0.448588i
$606$		0			0
$607$	−1.57024e36			−1.03398			−0.516990	−	0.855991i	$-0.672948\pi$
$607$	−1.57024e36			−1.03398			−0.516990	+	0.855991i	$0.672948\pi$
$608$		0			0
$609$	1.65636e35	−	1.20066e36i	0.104503	−	0.757522i
$610$		0			0
$611$			1.05930e36i			0.640446i
$612$		0			0
$613$	−1.50442e36			−0.871734			−0.435867	−	0.900011i	$-0.643558\pi$
$613$	−1.50442e36			−0.871734			−0.435867	+	0.900011i	$0.643558\pi$
$614$		0			0
$615$	−3.13664e35	−	4.32712e34i	−0.174216	−	0.0240338i
$616$		0			0
$617$		−	3.46870e36i		−	1.84697i	−0.383634	−	0.923485i	$-0.625328\pi$
$617$		−	3.46870e36i		−	1.84697i	0.383634	−	0.923485i	$-0.374672\pi$
$618$		0			0
$619$	1.86858e34			0.00953968			0.00476984	−	0.999989i	$-0.498482\pi$
$619$	1.86858e34			0.00953968			0.00476984	+	0.999989i	$0.498482\pi$
$620$		0			0
$621$	−9.93336e35	+	2.27757e36i	−0.486301	+	1.11502i
$622$		0			0
$623$		−	1.85446e36i		−	0.870709i
$624$		0			0
$625$	−9.69762e33			−0.00436742
$626$		0			0
$627$	−1.37170e35	+	9.94314e35i	−0.0592625	+	0.429581i
$628$		0			0
$629$		−	2.03080e34i		−	0.00841801i
$630$		0			0
$631$	−2.53086e36			−1.00667			−0.503334	−	0.864092i	$-0.667894\pi$
$631$	−2.53086e36			−1.00667			−0.503334	+	0.864092i	$0.667894\pi$
$632$		0			0
$633$	3.98500e36	+	5.49747e35i	1.52118	+	0.209853i
$634$		0			0
$635$			6.74780e35i			0.247231i
$636$		0			0
$637$	6.40509e35			0.225275
$638$		0			0
$639$	9.64917e35	−	3.43069e36i	0.325821	−	1.15843i
$640$		0			0
$641$			1.14525e36i			0.371318i	0.982614	+	0.185659i	$0.0594420\pi$
$641$			1.14525e36i			0.371318i	−0.982614	+	0.185659i	$0.940558\pi$
$642$		0			0
$643$	−1.31385e36			−0.409076			−0.204538	−	0.978859i	$-0.565569\pi$
$643$	−1.31385e36			−0.409076			−0.204538	+	0.978859i	$0.565569\pi$
$644$		0			0
$645$	−1.39532e35	+	1.01144e36i	−0.0417250	+	0.302456i
$646$		0			0
$647$			4.88372e36i			1.40280i	0.712768	+	0.701400i	$0.247441\pi$
$647$			4.88372e36i			1.40280i	−0.712768	+	0.701400i	$0.752559\pi$
$648$		0			0
$649$	2.02548e36			0.558916
$650$		0			0
$651$	2.16005e36	+	2.97987e35i	0.572677	+	0.0790032i
$652$		0			0
$653$		−	7.49065e35i		−	0.190831i	−0.995438	−	0.0954153i	$-0.969582\pi$
$653$		−	7.49065e35i		−	0.190831i	0.995438	−	0.0954153i	$-0.0304179\pi$
$654$		0			0
$655$	3.29620e36			0.807005
$656$		0			0
$657$	3.67004e36	+	1.03224e36i	0.863616	+	0.242901i
$658$		0			0
$659$		−	8.20579e36i		−	1.85614i	−0.372409	−	0.928069i	$-0.621468\pi$
$659$		−	8.20579e36i		−	1.85614i	0.372409	−	0.928069i	$-0.378532\pi$
$660$		0			0
$661$	−1.11276e35			−0.0241981			−0.0120990	−	0.999927i	$-0.503851\pi$
$661$	−1.11276e35			−0.0241981			−0.0120990	+	0.999927i	$0.503851\pi$
$662$		0			0
$663$	−2.24814e34	+	1.62963e35i	−0.00470054	+	0.0340732i
$664$		0			0
$665$		−	6.76378e35i		−	0.135990i
$666$		0			0
$667$	7.23945e36			1.39981
$668$		0			0
$669$	8.52037e36	+	1.17542e36i	1.58460	+	0.218602i
$670$		0			0
$671$			3.91065e36i			0.699610i
$672$		0			0
$673$	−3.71183e36			−0.638841			−0.319420	−	0.947613i	$-0.603488\pi$
$673$	−3.71183e36			−0.638841			−0.319420	+	0.947613i	$0.603488\pi$
$674$		0			0
$675$	3.41014e36	+	1.48729e36i	0.564708	+	0.246290i
$676$		0			0
$677$		−	4.08735e36i		−	0.651312i	−0.945488	−	0.325656i	$-0.894415\pi$
$677$		−	4.08735e36i		−	0.651312i	0.945488	−	0.325656i	$-0.105585\pi$
$678$		0			0
$679$	6.07170e36			0.931115
$680$		0			0
$681$	9.53532e35	−	6.91195e36i	0.140742	−	1.02021i
$682$		0			0
$683$			3.02236e36i			0.429416i	0.976678	+	0.214708i	$0.0688799\pi$
$683$			3.02236e36i			0.429416i	−0.976678	+	0.214708i	$0.931120\pi$
$684$		0			0
$685$	−2.81645e34			−0.00385233
$686$		0			0
$687$	8.26342e36	+	1.13997e36i	1.08823	+	0.150126i
$688$		0			0
$689$			3.85733e36i			0.489144i
$690$		0			0
$691$	−3.54345e36			−0.432725			−0.216362	−	0.976313i	$-0.569419\pi$
$691$	−3.54345e36			−0.432725			−0.216362	+	0.976313i	$0.569419\pi$
$692$		0			0
$693$	2.00858e36	−	7.14135e36i	0.236242	−	0.839940i
$694$		0			0
$695$		−	4.67426e36i		−	0.529554i
$696$		0			0
$697$	2.21731e35			0.0241992
$698$		0			0
$699$	8.93506e35	−	6.47684e36i	0.0939494	−	0.681020i
$700$		0			0
$701$			1.11647e37i			1.13113i	0.824704	+	0.565565i	$0.191342\pi$
$701$			1.11647e37i			1.13113i	−0.824704	+	0.565565i	$0.808658\pi$
$702$		0			0
$703$	−3.34011e35			−0.0326093
$704$		0			0
$705$	1.03562e37	+	1.42868e36i	0.974413	+	0.134424i
$706$		0			0
$707$			1.86071e36i			0.168743i
$708$		0			0
$709$	−3.51818e36			−0.307551			−0.153775	−	0.988106i	$-0.549143\pi$
$709$	−3.51818e36			−0.307551			−0.153775	+	0.988106i	$0.549143\pi$
$710$		0			0
$711$	4.03181e36	+	1.13399e36i	0.339778	+	0.0955663i
$712$		0			0
$713$			1.30241e37i			1.05824i
$714$		0			0
$715$	−4.18921e36			−0.328212
$716$		0			0
$717$	1.34731e36	−	9.76640e36i	0.101793	−	0.737880i
$718$		0			0
$719$		−	1.06482e37i		−	0.775890i	−0.921682	−	0.387945i	$-0.873185\pi$
$719$		−	1.06482e37i		−	0.775890i	0.921682	−	0.387945i	$-0.126815\pi$
$720$		0			0
$721$	1.52185e36			0.106958
$722$		0			0
$723$	1.05820e37	+	1.45983e36i	0.717416	+	0.0989705i
$724$		0			0
$725$		−	1.08394e37i		−	0.708943i
$726$		0			0
$727$	−1.97297e37			−1.24501			−0.622506	−	0.782615i	$-0.713885\pi$
$727$	−1.97297e37			−1.24501			−0.622506	+	0.782615i	$0.713885\pi$
$728$		0			0
$729$	1.11738e37	+	1.20361e37i	0.680367	+	0.732871i
$730$		0			0
$731$		−	7.14992e35i		−	0.0420122i
$732$		0			0
$733$	−2.92530e37			−1.65890			−0.829448	−	0.558584i	$-0.811345\pi$
$733$	−2.92530e37			−1.65890			−0.829448	+	0.558584i	$0.811345\pi$
$734$		0			0
$735$	8.63859e35	−	6.26193e36i	0.0472833	−	0.342746i
$736$		0			0
$737$		−	3.23949e37i		−	1.71159i
$738$		0			0
$739$	−3.22318e37			−1.64402			−0.822009	−	0.569474i	$-0.807147\pi$
$739$	−3.22318e37			−1.64402			−0.822009	+	0.569474i	$0.807147\pi$
$740$		0			0
$741$	2.68030e36	+	3.69758e35i	0.131992	+	0.0182088i
$742$		0			0
$743$			1.03473e37i			0.492009i	0.969269	+	0.246004i	$0.0791178\pi$
$743$			1.03473e37i			0.492009i	−0.969269	+	0.246004i	$0.920882\pi$
$744$		0			0
$745$	−8.63725e36			−0.396592
$746$		0			0
$747$	−9.85330e36	+	3.50326e37i	−0.436932	+	1.55348i
$748$		0			0
$749$			2.40363e37i			1.02945i
$750$		0			0
$751$	−3.01520e37			−1.24737			−0.623687	−	0.781674i	$-0.714366\pi$
$751$	−3.01520e37			−1.24737			−0.623687	+	0.781674i	$0.714366\pi$
$752$		0			0
$753$	−4.44238e36	+	3.22019e37i	−0.177534	+	1.28691i
$754$		0			0
$755$			2.26832e37i			0.875777i
$756$		0			0
$757$	−2.94589e37			−1.09893			−0.549465	−	0.835517i	$-0.685168\pi$
$757$	−2.94589e37			−1.09893			−0.549465	+	0.835517i	$0.685168\pi$
$758$		0			0
$759$	4.38945e37	+	6.05543e36i	1.58222	+	0.218274i
$760$		0			0
$761$		−	2.58816e37i		−	0.901550i	−0.892638	−	0.450775i	$-0.851148\pi$
$761$		−	2.58816e37i		−	0.901550i	0.892638	−	0.450775i	$-0.148852\pi$
$762$		0			0
$763$	−3.31008e37			−1.11435
$764$		0			0
$765$	1.56289e36	+	4.39579e35i	0.0508544	+	0.0143034i
$766$		0			0
$767$		−	5.45993e36i		−	0.171730i
$768$		0			0
$769$	1.64070e37			0.498869			0.249435	−	0.968392i	$-0.419755\pi$
$769$	1.64070e37			0.498869			0.249435	+	0.968392i	$0.419755\pi$
$770$		0			0
$771$	5.80958e36	−	4.21124e37i	0.170780	−	1.23795i
$772$		0			0
$773$			1.17016e37i			0.332592i	0.986076	+	0.166296i	$0.0531807\pi$
$773$			1.17016e37i			0.332592i	−0.986076	+	0.166296i	$0.946819\pi$
$774$		0			0
$775$	1.95006e37			0.535952
$776$		0			0
$777$	2.44547e36	+	3.37363e35i	0.0649964	+	0.00896652i
$778$		0			0
$779$		−	3.64687e36i		−	0.0937418i
$780$		0			0
$781$	−6.35524e37			−1.58004
$782$		0			0
$783$	1.91288e37	−	4.38596e37i	0.460029	−	1.05478i
$784$		0			0
$785$			2.08866e37i			0.485917i
$786$		0			0
$787$	−2.20060e37			−0.495302			−0.247651	−	0.968849i	$-0.579659\pi$
$787$	−2.20060e37			−0.495302			−0.247651	+	0.968849i	$0.579659\pi$
$788$		0			0
$789$	−3.87588e36	+	2.80955e37i	−0.0844053	+	0.611836i
$790$		0			0
$791$		−	1.01189e37i		−	0.213225i
$792$		0			0
$793$	1.05416e37			0.214960
$794$		0			0
$795$	3.77112e37	+	5.20241e36i	0.744213	+	0.102667i
$796$		0			0
$797$		−	6.59991e37i		−	1.26061i	−0.776349	−	0.630304i	$-0.782930\pi$
$797$		−	6.59991e37i		−	1.26061i	0.776349	−	0.630304i	$-0.217070\pi$
$798$		0			0
$799$	−7.32089e36			−0.135349
$800$		0			0
$801$	1.98225e37	−	7.04772e37i	0.354761	−	1.26132i
$802$		0			0
$803$		−	6.79863e37i		−	1.17793i
$804$		0			0
$805$	−2.98591e37			−0.500876
$806$		0			0
$807$	−1.40247e37	+	1.01662e38i	−0.227792	+	1.65121i
$808$		0			0
$809$			2.57683e37i			0.405280i	0.979253	+	0.202640i	$0.0649522\pi$
$809$			2.57683e37i			0.405280i	−0.979253	+	0.202640i	$0.935048\pi$
$810$		0			0
$811$	−1.07165e38			−1.63224			−0.816118	−	0.577885i	$-0.803878\pi$
$811$	−1.07165e38			−1.63224			−0.816118	+	0.577885i	$0.803878\pi$
$812$		0			0
$813$	1.94665e35	+	2.68548e34i	0.00287151	+	0.000396136i
$814$		0			0
$815$			6.18981e37i			0.884359i
$816$		0			0
$817$	−1.17597e37			−0.162745
$818$		0			0
$819$	−1.92504e37	−	5.41438e36i	−0.258077	−	0.0725868i
$820$		0			0
$821$			1.13885e38i			1.47913i	0.673084	+	0.739566i	$0.264969\pi$
$821$			1.13885e38i			1.47913i	−0.673084	+	0.739566i	$0.735031\pi$
$822$		0			0
$823$	−1.25928e37			−0.158462			−0.0792308	−	0.996856i	$-0.525246\pi$
$823$	−1.25928e37			−0.158462			−0.0792308	+	0.996856i	$0.525246\pi$
$824$		0			0
$825$	9.06661e36	−	6.57219e37i	0.110546	−	0.801326i
$826$		0			0
$827$			1.16233e38i			1.37328i	0.726999	+	0.686638i	$0.240914\pi$
$827$			1.16233e38i			1.37328i	−0.726999	+	0.686638i	$0.759086\pi$
$828$		0			0
$829$	−9.32435e37			−1.06760			−0.533802	−	0.845610i	$-0.679237\pi$
$829$	−9.32435e37			−1.06760			−0.533802	+	0.845610i	$0.679237\pi$
$830$		0			0
$831$	−6.62713e36	−	9.14240e35i	−0.0735381	−	0.0101449i
$832$		0			0
$833$			4.42661e36i			0.0476087i
$834$		0			0
$835$	−2.18081e36			−0.0227349
$836$		0			0
$837$	7.89056e37	+	3.44137e37i	0.797401	+	0.347776i
$838$		0			0
$839$		−	1.62318e38i		−	1.59023i	−0.606458	−	0.795116i	$-0.707410\pi$
$839$		−	1.62318e38i		−	1.59023i	0.606458	−	0.795116i	$-0.292590\pi$
$840$		0			0
$841$	−3.41306e37			−0.324188
$842$		0			0
$843$	1.94597e36	−	1.41059e37i	0.0179216	−	0.129910i
$844$		0			0
$845$		−	5.80913e37i		−	0.518768i
$846$		0			0
$847$	−5.55553e37			−0.481107
$848$		0			0
$849$	1.10198e38	+	1.52023e37i	0.925497	+	0.127676i
$850$		0			0
$851$			1.47451e37i			0.120106i
$852$		0			0
$853$	−6.26827e37			−0.495233			−0.247617	−	0.968858i	$-0.579647\pi$
$853$	−6.26827e37			−0.495233			−0.247617	+	0.968858i	$0.579647\pi$
$854$		0			0
$855$	7.22987e36	−	2.57052e37i	0.0554078	−	0.196998i
$856$		0			0
$857$			1.31009e38i			0.973980i	0.873407	+	0.486990i	$0.161905\pi$
$857$			1.31009e38i			0.973980i	−0.873407	+	0.486990i	$0.838095\pi$
$858$		0			0
$859$	1.51682e37			0.109402			0.0547008	−	0.998503i	$-0.482580\pi$
$859$	1.51682e37			0.109402			0.0547008	+	0.998503i	$0.482580\pi$
$860$		0			0
$861$	−3.68346e36	+	2.67007e37i	−0.0257760	+	0.186845i
$862$		0			0
$863$		−	8.19066e37i		−	0.556133i	−0.960562	−	0.278067i	$-0.910306\pi$
$863$		−	8.19066e37i		−	0.556133i	0.960562	−	0.278067i	$-0.0896936\pi$
$864$		0			0
$865$	1.48023e38			0.975257
$866$		0			0
$867$	1.53811e38	+	2.12188e37i	0.983417	+	0.135666i
$868$		0			0
$869$		−	7.46881e37i		−	0.463441i
$870$		0			0
$871$	−8.73245e37			−0.525896
$872$		0			0
$873$	2.30750e38	+	6.49010e37i	1.34883	+	0.379373i
$874$		0			0
$875$			1.17274e38i			0.665423i
$876$		0			0
$877$	−3.27529e38			−1.80408			−0.902039	−	0.431655i	$-0.857930\pi$
$877$	−3.27529e38			−1.80408			−0.902039	+	0.431655i	$0.857930\pi$
$878$		0			0
$879$	−3.42585e36	+	2.48333e37i	−0.0183195	+	0.132794i
$880$		0			0
$881$		−	2.58631e38i		−	1.34275i	−0.741119	−	0.671374i	$-0.765705\pi$
$881$		−	2.58631e38i		−	1.34275i	0.741119	−	0.671374i	$-0.234295\pi$
$882$		0			0
$883$	3.80240e38			1.91676			0.958381	−	0.285492i	$-0.0921570\pi$
$883$	3.80240e38			1.91676			0.958381	+	0.285492i	$0.0921570\pi$
$884$		0			0
$885$	−5.33790e37	−	7.36384e36i	−0.261281	−	0.0360447i
$886$		0			0
$887$			3.39306e38i			1.61281i	0.591361	+	0.806407i	$0.298591\pi$
$887$			3.39306e38i			1.61281i	−0.591361	+	0.806407i	$0.701409\pi$
$888$		0			0
$889$	5.74407e37			0.265153
$890$		0			0
$891$	1.52669e38	−	2.49931e38i	0.684448	−	1.12050i
$892$		0			0
$893$			1.20409e38i			0.524311i
$894$		0			0
$895$	−9.91544e37			−0.419385
$896$		0			0
$897$	1.63231e37	−	1.18323e38i	0.0670660	−	0.486147i
$898$		0			0
$899$		−	2.50807e38i		−	1.00107i
$900$		0			0
$901$	−2.66583e37			−0.103374
$902$		0			0
$903$	8.60986e37	+	1.18777e37i	0.324381	+	0.0447497i
$904$		0			0
$905$		−	1.43880e38i		−	0.526705i
$906$		0			0
$907$	2.48670e38			0.884562			0.442281	−	0.896877i	$-0.354169\pi$
$907$	2.48670e38			0.884562			0.442281	+	0.896877i	$0.354169\pi$
$908$		0			0
$909$	−1.98893e37	+	7.07149e37i	−0.0687527	+	0.244444i
$910$		0			0
$911$		−	1.82157e38i		−	0.611937i	−0.952042	−	0.305968i	$-0.901020\pi$
$911$		−	1.82157e38i		−	0.611937i	0.952042	−	0.305968i	$-0.0989802\pi$
$912$		0			0
$913$	6.48968e38			2.11886
$914$		0			0
$915$	1.42176e37	−	1.03060e38i	0.0451182	−	0.327052i
$916$		0			0
$917$		−	2.80589e38i		−	0.865506i
$918$		0			0
$919$	1.99483e38			0.598141			0.299070	−	0.954231i	$-0.403323\pi$
$919$	1.99483e38			0.598141			0.299070	+	0.954231i	$0.403323\pi$
$920$		0			0
$921$	4.20631e38	+	5.80277e37i	1.22610	+	0.169145i
$922$		0			0
$923$			1.71313e38i			0.485477i
$924$		0			0
$925$	2.20774e37			0.0608283
$926$		0			0
$927$	5.78368e37	+	1.62672e37i	0.154941	+	0.0435790i
$928$		0			0
$929$			6.60756e38i			1.72122i	0.509263	+	0.860611i	$0.329918\pi$
$929$			6.60756e38i			1.72122i	−0.509263	+	0.860611i	$0.670082\pi$
$930$		0			0
$931$	7.28056e37			0.184425
$932$		0			0
$933$	1.82403e37	−	1.32220e38i	0.0449337	−	0.325715i
$934$		0			0
$935$		−	2.89520e37i		−	0.0693629i
$936$		0			0
$937$	7.73152e38			1.80157			0.900783	−	0.434269i	$-0.142993\pi$
$937$	7.73152e38			1.80157			0.900783	+	0.434269i	$0.142993\pi$
$938$		0			0
$939$	8.02695e37	+	1.10735e37i	0.181927	+	0.0250976i
$940$		0			0
$941$			2.03187e38i			0.447952i	0.974595	+	0.223976i	$0.0719038\pi$
$941$			2.03187e38i			0.447952i	−0.974595	+	0.223976i	$0.928096\pi$
$942$		0			0
$943$	−1.60993e38			−0.345267
$944$		0			0
$945$	−7.88968e37	+	1.80899e38i	−0.164606	+	0.377418i
$946$		0			0
$947$		−	4.38067e38i		−	0.889183i	−0.895734	−	0.444591i	$-0.853349\pi$
$947$		−	4.38067e38i		−	0.889183i	0.895734	−	0.444591i	$-0.146651\pi$
$948$		0			0
$949$	−1.83265e38			−0.361926
$950$		0			0
$951$	−6.51749e37	+	4.72439e38i	−0.125237	+	0.907819i
$952$		0			0
$953$		−	7.13804e38i		−	1.33466i	−0.744761	−	0.667331i	$-0.767437\pi$
$953$		−	7.13804e38i		−	1.33466i	0.744761	−	0.667331i	$-0.232563\pi$
$954$		0			0
$955$	8.84829e37			0.160996
$956$		0			0
$957$	−8.45284e38	−	1.16610e38i	−1.49674	−	0.206482i
$958$		0			0
$959$			2.39750e36i			0.00413159i
$960$		0			0
$961$	−1.45002e38			−0.243204
$962$		0			0
$963$	−2.56926e38	+	9.13479e38i	−0.419436	+	1.49127i
$964$		0			0
$965$			2.16188e38i			0.343540i
$966$		0			0
$967$	−5.57724e38			−0.862730			−0.431365	−	0.902177i	$-0.641968\pi$
$967$	−5.57724e38			−0.862730			−0.431365	+	0.902177i	$0.641968\pi$
$968$		0			0
$969$	−2.55542e36	+	1.85237e37i	−0.00384817	+	0.0278946i
$970$		0			0
$971$			3.37407e38i			0.494656i	0.968932	+	0.247328i	$0.0795526\pi$
$971$			3.37407e38i			0.494656i	−0.968932	+	0.247328i	$0.920447\pi$
$972$		0			0
$973$	−3.97897e38			−0.567941
$974$		0			0
$975$	−1.77162e38	−	2.44402e37i	−0.246212	−	0.0339660i
$976$		0			0
$977$		−	3.47100e38i		−	0.469705i	−0.972031	−	0.234853i	$-0.924539\pi$
$977$		−	3.47100e38i		−	0.469705i	0.972031	−	0.234853i	$-0.0754608\pi$
$978$		0			0
$979$	−1.30557e39			−1.72038
$980$		0			0
$981$	−1.25797e39	−	3.53818e38i	−1.61426	−	0.454028i
$982$		0			0
$983$		−	9.14447e38i		−	1.14278i	−0.820679	−	0.571390i	$-0.806404\pi$
$983$		−	9.14447e38i		−	1.14278i	0.820679	−	0.571390i	$-0.193596\pi$
$984$		0			0
$985$	−5.86842e37			−0.0714250
$986$		0			0
$987$	1.21617e38	−	8.81575e38i	0.144169	−	1.04505i
$988$		0			0
$989$			5.19136e38i			0.599418i
$990$		0			0
$991$	8.31021e38			0.934662			0.467331	−	0.884083i	$-0.345216\pi$
$991$	8.31021e38			0.934662			0.467331	+	0.884083i	$0.345216\pi$
$992$		0			0
$993$	4.39950e38	+	6.06929e37i	0.482018	+	0.0664963i
$994$		0			0
$995$		−	2.77494e37i		−	0.0296178i
$996$		0			0
$997$	−7.04940e38			−0.733019			−0.366510	−	0.930414i	$-0.619447\pi$
$997$	−7.04940e38			−0.733019			−0.366510	+	0.930414i	$0.619447\pi$
$998$		0			0
$999$	8.93321e37	+	3.89611e37i	0.0905016	+	0.0394711i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.27.e.d.17.8		8
3.2	odd	2	inner	48.27.e.d.17.7		8
4.3	odd	2		3.27.b.a.2.8	yes	8
12.11	even	2		3.27.b.a.2.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3.27.b.a.2.1	✓	8	12.11	even	2
3.27.b.a.2.8	yes	8	4.3	odd	2
48.27.e.d.17.7		8	3.2	odd	2	inner
48.27.e.d.17.8		8	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:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 27 \)
Character orbit:	\([\chi]\)	\(=\)	48.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.57937e6 + 217880. i) q^{3} +7.56364e8i q^{5} +6.43856e10 q^{7} +(2.44692e12 + 6.88223e11i) q^{9} +O(q^{10})\)	\(q+(1.57937e6 + 217880. i) q^{3} +7.56364e8i q^{5} +6.43856e10 q^{7} +(2.44692e12 + 6.88223e11i) q^{9} -4.53285e13i q^{11} -1.22189e14 q^{13} +(-1.64796e14 + 1.19458e15i) q^{15} -8.44454e14i q^{17} -1.38890e16 q^{19} +(1.01688e17 + 1.40283e16i) q^{21} +6.13136e17i q^{23} +9.18029e17 q^{25} +(3.71463e18 + 1.62009e18i) q^{27} -1.18072e19i q^{29} +2.12418e19 q^{31} +(9.87616e18 - 7.15902e19i) q^{33} +4.86990e19i q^{35} +2.40487e19 q^{37} +(-1.92980e20 - 2.66224e19i) q^{39} +2.62573e20i q^{41} +8.46691e20 q^{43} +(-5.20548e20 + 1.85076e21i) q^{45} -8.66938e21i q^{47} -5.24197e21 q^{49} +(1.83990e20 - 1.33370e21i) q^{51} -3.15687e22i q^{53} +3.42848e22 q^{55} +(-2.19357e22 - 3.02612e21i) q^{57} +4.46845e22i q^{59} -8.62736e22 q^{61} +(1.57547e23 + 4.43117e22i) q^{63} -9.24190e22i q^{65} +7.14670e23 q^{67} +(-1.33590e23 + 9.68365e23i) q^{69} -1.40204e24i q^{71} +1.49986e24 q^{73} +(1.44990e24 + 2.00020e23i) q^{75} -2.91850e24i q^{77} +1.64771e24 q^{79} +(5.51378e24 + 3.36806e24i) q^{81} +1.43170e25i q^{83} +6.38715e23 q^{85} +(2.57256e24 - 1.86480e25i) q^{87} -2.88024e25i q^{89} -7.86718e24 q^{91} +(3.35486e25 + 4.62816e24i) q^{93} -1.05051e25i q^{95} +9.43022e25 q^{97} +(3.11961e25 - 1.10915e26i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 1261080 q^{3} + 83723264240 q^{7} + 3535237525320 q^{9}+O(q^{10}) \) 8 * q + 1261080 * q^3 + 83723264240 * q^7 + 3535237525320 * q^9	\( 8 q + 1261080 q^{3} + 83723264240 q^{7} + 3535237525320 q^{9} - 397915354631600 q^{13} + 719893807027200 q^{15} - 16\!\cdots\!00 q^{19}+ \cdots - 12\!\cdots\!20 q^{99}+O(q^{100}) \) 8 * q + 1261080 * q^3 + 83723264240 * q^7 + 3535237525320 * q^9 - 397915354631600 * q^13 + 719893807027200 * q^15 - 16320401555362000 * q^19 + 194580840365988048 * q^21 + 1971388854053049800 * q^25 + 7894649015962792920 * q^27 + 9272518953250081904 * q^31 - 65168600178932490240 * q^33 + 879374467120740709840 * q^37 + 1582238541124790450928 * q^39 - 6748593942335676212560 * q^43 - 16831420952659929907200 * q^45 + 45841873858091621647512 * q^49 + 62114113806050924347392 * q^51 - 31541176921532063385600 * q^55 - 68601650728105742286960 * q^57 - 319050283467071524566704 * q^61 - 658789337835295522626960 * q^63 + 3016122636770760226629680 * q^67 + 2874694883813365556256768 * q^69 - 1734362529074066854728560 * q^73 - 551279759153288953055400 * q^75 - 5851464318766097144790160 * q^79 - 7387002303358032367832952 * q^81 + 36766362882842005714329600 * q^85 + 66105320167985812595481600 * q^87 - 57658591275235919126596256 * q^91 - 721851591602497641585840 * q^93 + 87088434574088023884109840 * q^97 - 126385397236902706353285120 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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