LMFDB - Embedded newform 48.25.e.d.17.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,25,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, 25, names="a")

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

chi := DirichletCharacter("48.17");

S:= CuspForms(chi, 25);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 25 $
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:	$175.184233084$
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} + \cdots + 21\!\cdots\!00 $ x^8 + 1629189373042052x^6 + 272106194286045879281514214500x^4 + 13671756443267173943059351340058372000500000*x^2 + 210991722585849839909329798386550369753038400000000000000
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{60}\cdot 3^{34}\cdot 17^{2} $
Twist minimal:	no (minimal twist has level 6)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			17.8
Root			$1.03344e7i$ of defining polynomial
Character	$\chi$	$=$	48.17
Dual form			48.25.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)

$f(q)$	$=$	$q+(490828. + 203759. i) q^{3} +1.24013e8i q^{5} -1.50611e10 q^{7} +(1.99394e11 + 2.00021e11i) q^{9} +O(q^{10})$	\(q+(490828. + 203759. i) q^{3} +1.24013e8i q^{5} -1.50611e10 q^{7} +(1.99394e11 + 2.00021e11i) q^{9} +2.25660e12i q^{11} +4.78761e12 q^{13} +(-2.52688e13 + 6.08691e13i) q^{15} +5.31661e14i q^{17} +2.04629e15 q^{19} +(-7.39242e15 - 3.06885e15i) q^{21} +3.16079e16i q^{23} +4.42254e16 q^{25} +(5.71119e16 + 1.38804e17i) q^{27} -6.82228e17i q^{29} +6.83154e17 q^{31} +(-4.59804e17 + 1.10760e18i) q^{33} -1.86778e18i q^{35} +9.74472e18 q^{37} +(2.34989e18 + 9.75520e17i) q^{39} +1.91167e19i q^{41} -6.07042e19 q^{43} +(-2.48053e19 + 2.47275e19i) q^{45} +1.14305e20i q^{47} +3.52568e19 q^{49} +(-1.08331e20 + 2.60954e20i) q^{51} -5.21591e20i q^{53} -2.79848e20 q^{55} +(1.00438e21 + 4.16951e20i) q^{57} +5.99992e20i q^{59} +7.42263e19 q^{61} +(-3.00310e21 - 3.01255e21i) q^{63} +5.93727e20i q^{65} -8.52972e21 q^{67} +(-6.44041e21 + 1.55140e22i) q^{69} +2.05121e22i q^{71} -3.58499e22 q^{73} +(2.17070e22 + 9.01133e21i) q^{75} -3.39870e22i q^{77} -2.61248e22 q^{79} +(-2.50559e20 + 7.97660e22i) q^{81} +1.17460e23i q^{83} -6.59329e22 q^{85} +(1.39010e23 - 3.34856e23i) q^{87} +1.60364e23i q^{89} -7.21069e22 q^{91} +(3.35311e23 + 1.39199e23i) q^{93} +2.53767e23i q^{95} -8.32244e22 q^{97} +(-4.51369e23 + 4.49953e23i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 131880 q^{3} + 10160794640 q^{7} + 295169053896 q^{9}+O(q^{10}) $ 8 * q + 131880 * q^3 + 10160794640 * q^7 + 295169053896 * q^9	$ 8 q + 131880 q^{3} + 10160794640 q^{7} + 295169053896 q^{9} + 50568363679120 q^{13} + 348034956760512 q^{15} + 978083631341264 q^{19} + 36\!\cdots\!36 q^{21}+ \cdots - 69\!\cdots\!44 q^{99}+O(q^{100}) $ 8 * q + 131880 * q^3 + 10160794640 * q^7 + 295169053896 * q^9 + 50568363679120 * q^13 + 348034956760512 * q^15 + 978083631341264 * q^19 + 3640012304241936 * q^21 + 7630618767014024 * q^25 + 86594528606057640 * q^27 - 3092119786822709104 * q^31 - 8158685952668529600 * q^33 + 22590293223992782480 * q^37 + 40190176581881465040 * q^39 - 226487466371803896880 * q^43 - 347709996757177504128 * q^45 + 104686700473616731800 * q^49 - 558091874936566543104 * q^51 + 2212687664250467338368 * q^55 + 3588879995640760725840 * q^57 - 6507010783838092385648 * q^61 - 10112982777612899380080 * q^63 + 7042120118150060144720 * q^67 - 4323335967368731345536 * q^69 + 36259820324758576687120 * q^73 + 108081978448612908655272 * q^75 - 316807052777330015315824 * q^79 - 268518660504396776813304 * q^81 - 222512862545524200678912 * q^85 - 449461577201807342128320 * q^87 + 1196839602335690774367520 * q^91 + 873237537450828783745680 * q^93 + 816517576248201265716880 * q^97 - 695858883367208922313344 * 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
$3$	490828.	+	203759.i	0.923579	+	0.383409i
$3$	490828.	+	203759.i	0.923579	+	0.383409i
$4$		0			0
$5$			1.24013e8i			0.507958i	0.967210	+	0.253979i	$0.0817394\pi$
$5$			1.24013e8i			0.507958i	−0.967210	+	0.253979i	$0.918261\pi$
$6$		0			0
$7$	−1.50611e10			−1.08813			−0.544066	−	0.839043i	$-0.683116\pi$
$7$	−1.50611e10			−1.08813			−0.544066	+	0.839043i	$0.683116\pi$
$8$		0			0
$9$	1.99394e11	+	2.00021e11i	0.705995	+	0.708216i
$10$		0			0
$11$			2.25660e12i			0.719023i	0.933141	+	0.359512i	$0.117057\pi$
$11$			2.25660e12i			0.719023i	−0.933141	+	0.359512i	$0.882943\pi$
$12$		0			0
$13$	4.78761e12			0.205494			0.102747	−	0.994708i	$-0.467237\pi$
$13$	4.78761e12			0.205494			0.102747	+	0.994708i	$0.467237\pi$
$14$		0			0
$15$	−2.52688e13	+	6.08691e13i	−0.194756	+	0.469139i
$16$		0			0
$17$			5.31661e14i			0.912531i	0.889844	+	0.456265i	$0.150813\pi$
$17$			5.31661e14i			0.912531i	−0.889844	+	0.456265i	$0.849187\pi$
$18$		0			0
$19$	2.04629e15			0.924537			0.462269	−	0.886740i	$-0.347036\pi$
$19$	2.04629e15			0.924537			0.462269	+	0.886740i	$0.347036\pi$
$20$		0			0
$21$	−7.39242e15	−	3.06885e15i	−1.00498	−	0.417199i
$22$		0			0
$23$			3.16079e16i			1.44232i	0.692768	+	0.721161i	$0.256391\pi$
$23$			3.16079e16i			1.44232i	−0.692768	+	0.721161i	$0.743609\pi$
$24$		0			0
$25$	4.42254e16			0.741979
$26$		0			0
$27$	5.71119e16	+	1.38804e17i	0.380506	+	0.924779i
$28$		0			0
$29$		−	6.82228e17i		−	1.92821i	−0.265527	−	0.964103i	$-0.585546\pi$
$29$		−	6.82228e17i		−	1.92821i	0.265527	−	0.964103i	$-0.414454\pi$
$30$		0			0
$31$	6.83154e17			0.867318			0.433659	−	0.901077i	$-0.357222\pi$
$31$	6.83154e17			0.867318			0.433659	+	0.901077i	$0.357222\pi$
$32$		0			0
$33$	−4.59804e17	+	1.10760e18i	−0.275680	+	0.664075i
$34$		0			0
$35$		−	1.86778e18i		−	0.552725i
$36$		0			0
$37$	9.74472e18			1.48030			0.740148	−	0.672444i	$-0.234755\pi$
$37$	9.74472e18			1.48030			0.740148	+	0.672444i	$0.234755\pi$
$38$		0			0
$39$	2.34989e18	+	9.75520e17i	0.189790	+	0.0787882i
$40$		0			0
$41$			1.91167e19i			0.847242i	0.905840	+	0.423621i	$0.139241\pi$
$41$			1.91167e19i			0.847242i	−0.905840	+	0.423621i	$0.860759\pi$
$42$		0			0
$43$	−6.07042e19			−1.51914			−0.759569	−	0.650426i	$-0.774590\pi$
$43$	−6.07042e19			−1.51914			−0.759569	+	0.650426i	$0.774590\pi$
$44$		0			0
$45$	−2.48053e19	+	2.47275e19i	−0.359744	+	0.358616i
$46$		0			0
$47$			1.14305e20i			0.983766i	0.870661	+	0.491883i	$0.163691\pi$
$47$			1.14305e20i			0.983766i	−0.870661	+	0.491883i	$0.836309\pi$
$48$		0			0
$49$	3.52568e19			0.184030
$50$		0			0
$51$	−1.08331e20	+	2.60954e20i	−0.349872	+	0.842794i
$52$		0			0
$53$		−	5.21591e20i		−	1.06174i	−0.847452	−	0.530872i	$-0.821864\pi$
$53$		−	5.21591e20i		−	1.06174i	0.847452	−	0.530872i	$-0.178136\pi$
$54$		0			0
$55$	−2.79848e20			−0.365233
$56$		0			0
$57$	1.00438e21	+	4.16951e20i	0.853883	+	0.354476i
$58$		0			0
$59$			5.99992e20i			0.337226i	0.985682	+	0.168613i	$0.0539289\pi$
$59$			5.99992e20i			0.337226i	−0.985682	+	0.168613i	$0.946071\pi$
$60$		0			0
$61$	7.42263e19			0.0279640			0.0139820	−	0.999902i	$-0.495549\pi$
$61$	7.42263e19			0.0279640			0.0139820	+	0.999902i	$0.495549\pi$
$62$		0			0
$63$	−3.00310e21	−	3.01255e21i	−0.768216	−	0.770633i
$64$		0			0
$65$			5.93727e20i			0.104382i
$66$		0			0
$67$	−8.52972e21			−1.04241			−0.521203	−	0.853433i	$-0.674517\pi$
$67$	−8.52972e21			−1.04241			−0.521203	+	0.853433i	$0.674517\pi$
$68$		0			0
$69$	−6.44041e21	+	1.55140e22i	−0.552999	+	1.33210i
$70$		0			0
$71$			2.05121e22i			1.25000i	0.780625	+	0.624999i	$0.214901\pi$
$71$			2.05121e22i			1.25000i	−0.780625	+	0.624999i	$0.785099\pi$
$72$		0			0
$73$	−3.58499e22			−1.56536			−0.782679	−	0.622425i	$-0.786148\pi$
$73$	−3.58499e22			−1.56536			−0.782679	+	0.622425i	$0.786148\pi$
$74$		0			0
$75$	2.17070e22	+	9.01133e21i	0.685276	+	0.284481i
$76$		0			0
$77$		−	3.39870e22i		−	0.782392i
$78$		0			0
$79$	−2.61248e22			−0.442108			−0.221054	−	0.975262i	$-0.570950\pi$
$79$	−2.61248e22			−0.442108			−0.221054	+	0.975262i	$0.570950\pi$
$80$		0			0
$81$	−2.50559e20	+	7.97660e22i	−0.00314115	+	0.999995i
$82$		0			0
$83$			1.17460e23i			1.09889i	0.835530	+	0.549445i	$0.185161\pi$
$83$			1.17460e23i			1.09889i	−0.835530	+	0.549445i	$0.814839\pi$
$84$		0			0
$85$	−6.59329e22			−0.463527
$86$		0			0
$87$	1.39010e23	−	3.34856e23i	0.739291	−	1.78085i
$88$		0			0
$89$			1.60364e23i			0.649273i	0.945839	+	0.324637i	$0.105242\pi$
$89$			1.60364e23i			0.649273i	−0.945839	+	0.324637i	$0.894758\pi$
$90$		0			0
$91$	−7.21069e22			−0.223604
$92$		0			0
$93$	3.35311e23	+	1.39199e23i	0.801036	+	0.332537i
$94$		0			0
$95$			2.53767e23i			0.469626i
$96$		0			0
$97$	−8.32244e22			−0.119947			−0.0599736	−	0.998200i	$-0.519102\pi$
$97$	−8.32244e22			−0.119947			−0.0599736	+	0.998200i	$0.519102\pi$
$98$		0			0
$99$	−4.51369e23	+	4.49953e23i	−0.509224	+	0.507627i
$100$		0			0
$101$		−	2.20951e24i		−	1.96083i	−0.196945	−	0.980415i	$-0.563102\pi$
$101$		−	2.20951e24i		−	1.96083i	0.196945	−	0.980415i	$-0.436898\pi$
$102$		0			0
$103$	−3.85825e23			−0.270610			−0.135305	−	0.990804i	$-0.543201\pi$
$103$	−3.85825e23			−0.270610			−0.135305	+	0.990804i	$0.543201\pi$
$104$		0			0
$105$	3.80577e23	−	9.16758e23i	0.211920	−	0.510485i
$106$		0			0
$107$		−	3.07071e24i		−	1.36343i	−0.731617	−	0.681716i	$-0.761234\pi$
$107$		−	3.07071e24i		−	1.36343i	0.731617	−	0.681716i	$-0.238766\pi$
$108$		0			0
$109$	1.69282e24			0.601857			0.300929	−	0.953647i	$-0.402703\pi$
$109$	1.69282e24			0.601857			0.300929	+	0.953647i	$0.402703\pi$
$110$		0			0
$111$	4.78298e24	+	1.98558e24i	1.36717	+	0.567558i
$112$		0			0
$113$			2.70784e24i			0.624715i	0.949965	+	0.312358i	$0.101119\pi$
$113$			2.70784e24i			0.624715i	−0.949965	+	0.312358i	$0.898881\pi$
$114$		0			0
$115$	−3.91980e24			−0.732638
$116$		0			0
$117$	9.54621e23	+	9.57624e23i	0.145078	+	0.145534i
$118$		0			0
$119$		−	8.00742e24i		−	0.992954i
$120$		0			0
$121$	4.75748e24			0.483006
$122$		0			0
$123$	−3.89521e24	+	9.38302e24i	−0.324840	+	0.782495i
$124$		0			0
$125$			1.28763e25i			0.884852i
$126$		0			0
$127$	2.15801e25			1.22577			0.612886	−	0.790172i	$-0.290009\pi$
$127$	2.15801e25			1.22577			0.612886	+	0.790172i	$0.290009\pi$
$128$		0			0
$129$	−2.97953e25	−	1.23690e25i	−1.40304	−	0.582451i
$130$		0			0
$131$			1.13650e25i			0.444954i	0.974938	+	0.222477i	$0.0714143\pi$
$131$			1.13650e25i			0.444954i	−0.974938	+	0.222477i	$0.928586\pi$
$132$		0			0
$133$	−3.08195e25			−1.00602
$134$		0			0
$135$	−1.72136e25	+	7.08262e24i	−0.469748	+	0.193281i
$136$		0			0
$137$		−	1.38297e25i		−	0.316349i	−0.987411	−	0.158175i	$-0.949439\pi$
$137$		−	1.38297e25i		−	0.316349i	0.987411	−	0.158175i	$-0.0505609\pi$
$138$		0			0
$139$	−6.91088e25			−1.32848			−0.664241	−	0.747518i	$-0.731245\pi$
$139$	−6.91088e25			−1.32848			−0.664241	+	0.747518i	$0.731245\pi$
$140$		0			0
$141$	−2.32907e25	+	5.61042e25i	−0.377185	+	0.908586i
$142$		0			0
$143$			1.08037e25i			0.147755i
$144$		0			0
$145$	8.46052e25			0.979448
$146$		0			0
$147$	1.73050e25	+	7.18389e24i	0.169967	+	0.0705589i
$148$		0			0
$149$			1.45823e25i			0.121785i	0.998144	+	0.0608923i	$0.0193946\pi$
$149$			1.45823e25i			0.121785i	−0.998144	+	0.0608923i	$0.980605\pi$
$150$		0			0
$151$	−1.28532e26			−0.914720			−0.457360	−	0.889282i	$-0.651205\pi$
$151$	−1.28532e26			−0.914720			−0.457360	+	0.889282i	$0.651205\pi$
$152$		0			0
$153$	−1.06343e26	+	1.06010e26i	−0.646269	+	0.644242i
$154$		0			0
$155$			8.47200e25i			0.440561i
$156$		0			0
$157$	1.95529e26			0.871796			0.435898	−	0.899996i	$-0.356431\pi$
$157$	1.95529e26			0.871796			0.435898	+	0.899996i	$0.356431\pi$
$158$		0			0
$159$	1.06279e26	−	2.56011e26i	0.407082	−	0.980604i
$160$		0			0
$161$		−	4.76052e26i		−	1.56944i
$162$		0			0
$163$	−1.44368e26			−0.410413			−0.205206	−	0.978719i	$-0.565787\pi$
$163$	−1.44368e26			−0.410413			−0.205206	+	0.978719i	$0.565787\pi$
$164$		0			0
$165$	−1.37357e26	−	5.70217e25i	−0.337322	−	0.140034i
$166$		0			0
$167$			4.74707e26i			1.00885i	0.863455	+	0.504426i	$0.168296\pi$
$167$			4.74707e26i			1.00885i	−0.863455	+	0.504426i	$0.831704\pi$
$168$		0			0
$169$	−5.19880e26			−0.957772
$170$		0			0
$171$	4.08018e26	+	4.09302e26i	0.652719	+	0.654772i
$172$		0			0
$173$			7.88219e26i			1.09671i	0.836244	+	0.548357i	$0.184747\pi$
$173$			7.88219e26i			1.09671i	−0.836244	+	0.548357i	$0.815253\pi$
$174$		0			0
$175$	−6.66085e26			−0.807371
$176$		0			0
$177$	−1.22254e26	+	2.94493e26i	−0.129296	+	0.311455i
$178$		0			0
$179$		−	1.71671e27i		−	1.58657i	−0.608847	−	0.793287i	$-0.708368\pi$
$179$		−	1.71671e27i		−	1.58657i	0.608847	−	0.793287i	$-0.291632\pi$
$180$		0			0
$181$	−6.59970e26			−0.533804			−0.266902	−	0.963724i	$-0.586000\pi$
$181$	−6.59970e26			−0.533804			−0.266902	+	0.963724i	$0.586000\pi$
$182$		0			0
$183$	3.64323e25	+	1.51243e25i	0.0258270	+	0.0107217i
$184$		0			0
$185$			1.20847e27i			0.751928i
$186$		0			0
$187$	−1.19975e27			−0.656131
$188$		0			0
$189$	−8.60170e26	−	2.09055e27i	−0.414040	−	1.00628i
$190$		0			0
$191$			8.80733e26i			0.373632i	0.982395	+	0.186816i	$0.0598168\pi$
$191$			8.80733e26i			0.373632i	−0.982395	+	0.186816i	$0.940183\pi$
$192$		0			0
$193$	−1.65998e27			−0.621461			−0.310730	−	0.950498i	$-0.600574\pi$
$193$	−1.65998e27			−0.621461			−0.310730	+	0.950498i	$0.600574\pi$
$194$		0			0
$195$	−1.20977e26	+	2.91418e26i	−0.0400211	+	0.0964052i
$196$		0			0
$197$			2.80575e27i			0.821209i	0.911814	+	0.410604i	$0.134682\pi$
$197$			2.80575e27i			0.821209i	−0.911814	+	0.410604i	$0.865318\pi$
$198$		0			0
$199$	−6.04163e27			−1.56645			−0.783226	−	0.621737i	$-0.786427\pi$
$199$	−6.04163e27			−1.56645			−0.783226	+	0.621737i	$0.786427\pi$
$200$		0			0
$201$	−4.18662e27	−	1.73801e27i	−0.962745	−	0.399668i
$202$		0			0
$203$			1.02751e28i			2.09814i
$204$		0			0
$205$	−2.37073e27			−0.430363
$206$		0			0
$207$	−6.32226e27	+	6.30243e27i	−1.02148	+	1.01827i
$208$		0			0
$209$			4.61767e27i			0.664764i
$210$		0			0
$211$	−9.62629e27			−1.23615			−0.618073	−	0.786121i	$-0.712086\pi$
$211$	−9.62629e27			−1.23615			−0.618073	+	0.786121i	$0.712086\pi$
$212$		0			0
$213$	−4.17952e27	+	1.00679e28i	−0.479260	+	1.15447i
$214$		0			0
$215$		−	7.52812e27i		−	0.771658i
$216$		0			0
$217$	−1.02891e28			−0.943756
$218$		0			0
$219$	−1.75961e28	−	7.30475e27i	−1.44573	−	0.600172i
$220$		0			0
$221$			2.54539e27i			0.187519i
$222$		0			0
$223$	1.86863e28			1.23556			0.617782	−	0.786349i	$-0.288031\pi$
$223$	1.86863e28			1.23556			0.617782	+	0.786349i	$0.288031\pi$
$224$		0			0
$225$	8.81827e27	+	8.84602e27i	0.523834	+	0.525482i
$226$		0			0
$227$			1.80444e28i			0.963901i	0.876198	+	0.481951i	$0.160072\pi$
$227$			1.80444e28i			0.963901i	−0.876198	+	0.481951i	$0.839928\pi$
$228$		0			0
$229$	−1.96660e28			−0.945562			−0.472781	−	0.881180i	$-0.656750\pi$
$229$	−1.96660e28			−0.945562			−0.472781	+	0.881180i	$0.656750\pi$
$230$		0			0
$231$	6.92517e27	−	1.66818e28i	0.299976	−	0.722601i
$232$		0			0
$233$		−	3.90415e28i		−	1.52495i	−0.647018	−	0.762474i	$-0.723984\pi$
$233$		−	3.90415e28i		−	1.52495i	0.647018	−	0.762474i	$-0.276016\pi$
$234$		0			0
$235$	−1.41754e28			−0.499712
$236$		0			0
$237$	−1.28228e28	−	5.32317e27i	−0.408321	−	0.169508i
$238$		0			0
$239$		−	1.14900e28i		−	0.330785i	−0.986228	−	0.165392i	$-0.947111\pi$
$239$		−	1.14900e28i		−	0.330785i	0.986228	−	0.165392i	$-0.0528891\pi$
$240$		0			0
$241$	−2.26729e28			−0.590612			−0.295306	−	0.955403i	$-0.595422\pi$
$241$	−2.26729e28			−0.590612			−0.295306	+	0.955403i	$0.595422\pi$
$242$		0			0
$243$	−1.63760e28	+	3.91003e28i	−0.386308	+	0.922370i
$244$		0			0
$245$			4.37231e27i			0.0934797i
$246$		0			0
$247$	9.79685e27			0.189987
$248$		0			0
$249$	−2.39336e28	+	5.76528e28i	−0.421324	+	1.01491i
$250$		0			0
$251$		−	2.74981e27i		−	0.0439762i	−0.999758	−	0.0219881i	$-0.993000\pi$
$251$		−	2.74981e27i		−	0.0439762i	0.999758	−	0.0219881i	$-0.00699960\pi$
$252$		0			0
$253$	−7.13265e28			−1.03706
$254$		0			0
$255$	−3.23617e28	−	1.34344e28i	−0.428104	−	0.177720i
$256$		0			0
$257$			1.14943e29i			1.38447i	0.721671	+	0.692236i	$0.243374\pi$
$257$			1.14943e29i			1.38447i	−0.721671	+	0.692236i	$0.756626\pi$
$258$		0			0
$259$	−1.46767e29			−1.61076
$260$		0			0
$261$	1.36460e29	−	1.36032e29i	1.36559	−	1.36130i
$262$		0			0
$263$		−	1.67988e28i		−	0.153394i	−0.997054	−	0.0766970i	$-0.975563\pi$
$263$		−	1.67988e28i		−	0.153394i	0.997054	−	0.0766970i	$-0.0244374\pi$
$264$		0			0
$265$	6.46842e28			0.539321
$266$		0			0
$267$	−3.26757e28	+	7.87112e28i	−0.248937	+	0.599655i
$268$		0			0
$269$			7.11629e28i			0.495708i	0.968797	+	0.247854i	$0.0797254\pi$
$269$			7.11629e28i			0.495708i	−0.968797	+	0.247854i	$0.920275\pi$
$270$		0			0
$271$	8.35305e28			0.532370			0.266185	−	0.963922i	$-0.414237\pi$
$271$	8.35305e28			0.532370			0.266185	+	0.963922i	$0.414237\pi$
$272$		0			0
$273$	−3.53921e28	−	1.46924e28i	−0.206516	−	0.0857319i
$274$		0			0
$275$			9.97991e28i			0.533500i
$276$		0			0
$277$	−1.50267e29			−0.736389			−0.368194	−	0.929749i	$-0.620024\pi$
$277$	−1.50267e29			−0.736389			−0.368194	+	0.929749i	$0.620024\pi$
$278$		0			0
$279$	1.36217e29	+	1.36645e29i	0.612322	+	0.614249i
$280$		0			0
$281$		−	1.37645e29i		−	0.567918i	−0.958836	−	0.283959i	$-0.908352\pi$
$281$		−	1.37645e29i		−	0.567918i	0.958836	−	0.283959i	$-0.0916479\pi$
$282$		0			0
$283$	4.27935e28			0.162159			0.0810794	−	0.996708i	$-0.474163\pi$
$283$	4.27935e28			0.162159			0.0810794	+	0.996708i	$0.474163\pi$
$284$		0			0
$285$	−5.17074e28	+	1.24556e29i	−0.180059	+	0.433737i
$286$		0			0
$287$		−	2.87920e29i		−	0.921911i
$288$		0			0
$289$	5.67855e28			0.167288
$290$		0			0
$291$	−4.08489e28	−	1.69577e28i	−0.110781	−	0.0459888i
$292$		0			0
$293$		−	1.71498e29i		−	0.428397i	−0.976790	−	0.214199i	$-0.931286\pi$
$293$		−	1.71498e29i		−	0.428397i	0.976790	−	0.214199i	$-0.0687140\pi$
$294$		0			0
$295$	−7.44069e28			−0.171297
$296$		0			0
$297$	−3.13226e29	+	1.28879e29i	−0.664937	+	0.273593i
$298$		0			0
$299$			1.51327e29i			0.296388i
$300$		0			0
$301$	9.14275e29			1.65302
$302$		0			0
$303$	4.50208e29	−	1.08449e30i	0.751799	−	1.81098i
$304$		0			0
$305$			9.20503e27i			0.0142045i
$306$		0			0
$307$	9.60156e29			1.36988			0.684938	−	0.728601i	$-0.259829\pi$
$307$	9.60156e29			1.36988			0.684938	+	0.728601i	$0.259829\pi$
$308$		0			0
$309$	−1.89374e29	−	7.86154e28i	−0.249930	−	0.103754i
$310$		0			0
$311$		−	1.83314e29i		−	0.223909i	−0.993713	−	0.111955i	$-0.964289\pi$
$311$		−	1.83314e29i		−	0.223909i	0.993713	−	0.111955i	$-0.0357111\pi$
$312$		0			0
$313$	1.66337e30			1.88130			0.940648	−	0.339385i	$-0.110219\pi$
$313$	1.66337e30			1.88130			0.940648	+	0.339385i	$0.110219\pi$
$314$		0			0
$315$	3.73596e29	−	3.72424e29i	0.391449	−	0.390221i
$316$		0			0
$317$		−	5.50864e29i		−	0.534975i	−0.963561	−	0.267487i	$-0.913807\pi$
$317$		−	5.50864e29i		−	0.534975i	0.963561	−	0.267487i	$-0.0861934\pi$
$318$		0			0
$319$	1.53952e30			1.38643
$320$		0			0
$321$	6.25685e29	−	1.50719e30i	0.522752	−	1.25924i
$322$		0			0
$323$			1.08793e30i			0.843669i
$324$		0			0
$325$	2.11734e29			0.152472
$326$		0			0
$327$	8.30884e29	+	3.44928e29i	0.555862	+	0.230757i
$328$		0			0
$329$		−	1.72157e30i		−	1.07047i
$330$		0			0
$331$	−2.52531e30			−1.46009			−0.730045	−	0.683399i	$-0.760501\pi$
$331$	−2.52531e30			−1.46009			−0.730045	+	0.683399i	$0.760501\pi$
$332$		0			0
$333$	1.94304e30	+	1.94915e30i	1.04508	+	1.04837i
$334$		0			0
$335$		−	1.05780e30i		−	0.529499i
$336$		0			0
$337$	6.18579e29			0.288295			0.144147	−	0.989556i	$-0.453956\pi$
$337$	6.18579e29			0.288295			0.144147	+	0.989556i	$0.453956\pi$
$338$		0			0
$339$	−5.51748e29	+	1.32908e30i	−0.239521	+	0.576974i
$340$		0			0
$341$			1.54161e30i			0.623621i
$342$		0			0
$343$	2.35442e30			0.887882
$344$		0			0
$345$	−1.92395e30	−	7.98695e29i	−0.676649	−	0.280900i
$346$		0			0
$347$		−	3.27882e30i		−	1.07588i	−0.842983	−	0.537940i	$-0.819203\pi$
$347$		−	3.27882e30i		−	1.07588i	0.842983	−	0.537940i	$-0.180797\pi$
$348$		0			0
$349$	−1.23269e30			−0.377527			−0.188764	−	0.982023i	$-0.560448\pi$
$349$	−1.23269e30			−0.377527			−0.188764	+	0.982023i	$0.560448\pi$
$350$		0			0
$351$	2.73430e29	+	6.64541e29i	0.0781916	+	0.190036i
$352$		0			0
$353$			5.25121e30i			1.40269i	0.712820	+	0.701347i	$0.247418\pi$
$353$			5.25121e30i			1.40269i	−0.712820	+	0.701347i	$0.752582\pi$
$354$		0			0
$355$	−2.54377e30			−0.634946
$356$		0			0
$357$	1.63158e30	−	3.93026e30i	0.380707	−	0.917071i
$358$		0			0
$359$		−	3.10562e30i		−	0.677664i	−0.940847	−	0.338832i	$-0.889968\pi$
$359$		−	3.10562e30i		−	0.677664i	0.940847	−	0.338832i	$-0.110032\pi$
$360$		0			0
$361$	−7.11452e29			−0.145231
$362$		0			0
$363$	2.33510e30	+	9.69379e29i	0.446094	+	0.185189i
$364$		0			0
$365$		−	4.44586e30i		−	0.795136i
$366$		0			0
$367$	1.54480e30			0.258749			0.129374	−	0.991596i	$-0.458703\pi$
$367$	1.54480e30			0.258749			0.129374	+	0.991596i	$0.458703\pi$
$368$		0			0
$369$	−3.82375e30	+	3.81176e30i	−0.600031	+	0.598149i
$370$		0			0
$371$			7.85576e30i			1.15532i
$372$		0			0
$373$	−8.17121e30			−1.12663			−0.563314	−	0.826243i	$-0.690474\pi$
$373$	−8.17121e30			−1.12663			−0.563314	+	0.826243i	$0.690474\pi$
$374$		0			0
$375$	−2.62366e30	+	6.32004e30i	−0.339260	+	0.817230i
$376$		0			0
$377$		−	3.26624e30i		−	0.396235i
$378$		0			0
$379$	1.36135e31			1.54989			0.774944	−	0.632029i	$-0.217778\pi$
$379$	1.36135e31			1.54989			0.774944	+	0.632029i	$0.217778\pi$
$380$		0			0
$381$	1.05921e31	+	4.39715e30i	1.13210	+	0.469971i
$382$		0			0
$383$		−	2.14155e29i		−	0.0214953i	−0.999942	−	0.0107477i	$-0.996579\pi$
$383$		−	2.14155e29i		−	0.0214953i	0.999942	−	0.0107477i	$-0.00342115\pi$
$384$		0			0
$385$	4.21484e30			0.397422
$386$		0			0
$387$	−1.21041e31	−	1.21421e31i	−1.07250	−	1.07588i
$388$		0			0
$389$			2.46642e30i			0.205433i	0.994711	+	0.102717i	$0.0327535\pi$
$389$			2.46642e30i			0.205433i	−0.994711	+	0.102717i	$0.967246\pi$
$390$		0			0
$391$	−1.68047e31			−1.31616
$392$		0			0
$393$	−2.31573e30	+	5.57828e30i	−0.170599	+	0.410950i
$394$		0			0
$395$		−	3.23982e30i		−	0.224572i
$396$		0			0
$397$	−1.20449e31			−0.785809			−0.392904	−	0.919579i	$-0.628530\pi$
$397$	−1.20449e31			−0.785809			−0.392904	+	0.919579i	$0.628530\pi$
$398$		0			0
$399$	−1.51271e31	−	6.27976e30i	−0.929137	−	0.385716i
$400$		0			0
$401$		−	4.88143e30i		−	0.282367i	−0.989983	−	0.141184i	$-0.954909\pi$
$401$		−	4.88143e30i		−	0.282367i	0.989983	−	0.141184i	$-0.0450908\pi$
$402$		0			0
$403$	3.27068e30			0.178228
$404$		0			0
$405$	−9.89204e30	−	3.10726e28i	−0.507955	−	0.00159557i
$406$		0			0
$407$			2.19900e31i			1.06437i
$408$		0			0
$409$	2.14098e31			0.977085			0.488543	−	0.872540i	$-0.337529\pi$
$409$	2.14098e31			0.977085			0.488543	+	0.872540i	$0.337529\pi$
$410$		0			0
$411$	2.81793e30	−	6.78801e30i	0.121291	−	0.292173i
$412$		0			0
$413$		−	9.03657e30i		−	0.366947i
$414$		0			0
$415$	−1.45666e31			−0.558190
$416$		0			0
$417$	−3.39205e31	−	1.40816e31i	−1.22696	−	0.509352i
$418$		0			0
$419$			9.90044e28i			0.00338132i	0.999999	+	0.00169066i	$0.000538154\pi$
$419$			9.90044e28i			0.00338132i	−0.999999	+	0.00169066i	$0.999462\pi$
$420$		0			0
$421$	−1.38801e31			−0.447720			−0.223860	−	0.974621i	$-0.571866\pi$
$421$	−1.38801e31			−0.447720			−0.223860	+	0.974621i	$0.571866\pi$
$422$		0			0
$423$	−2.28635e31	+	2.27918e31i	−0.696720	+	0.694535i
$424$		0			0
$425$			2.35129e31i			0.677079i
$426$		0			0
$427$	−1.11793e30			−0.0304285
$428$		0			0
$429$	−2.20136e30	+	5.30277e30i	−0.0566505	+	0.136463i
$430$		0			0
$431$			4.58411e31i			1.11565i	0.829959	+	0.557824i	$0.188364\pi$
$431$			4.58411e31i			1.11565i	−0.829959	+	0.557824i	$0.811636\pi$
$432$		0			0
$433$	−6.04562e31			−1.39183			−0.695915	−	0.718124i	$-0.745001\pi$
$433$	−6.04562e31			−1.39183			−0.695915	+	0.718124i	$0.745001\pi$
$434$		0			0
$435$	4.15266e31	+	1.72391e31i	0.904597	+	0.375529i
$436$		0			0
$437$			6.46790e31i			1.33348i
$438$		0			0
$439$	−4.04585e31			−0.789652			−0.394826	−	0.918756i	$-0.629195\pi$
$439$	−4.04585e31			−0.789652			−0.394826	+	0.918756i	$0.629195\pi$
$440$		0			0
$441$	7.02999e30	+	7.05211e30i	0.129925	+	0.130333i
$442$		0			0
$443$		−	1.03672e32i		−	1.81474i	−0.420329	−	0.907372i	$-0.638085\pi$
$443$		−	1.03672e32i		−	1.81474i	0.420329	−	0.907372i	$-0.361915\pi$
$444$		0			0
$445$	−1.98873e31			−0.329803
$446$		0			0
$447$	−2.97129e30	+	7.15742e30i	−0.0466933	+	0.112478i
$448$		0			0
$449$			5.40229e31i			0.804677i	0.915491	+	0.402338i	$0.131803\pi$
$449$			5.40229e31i			0.804677i	−0.915491	+	0.402338i	$0.868197\pi$
$450$		0			0
$451$	−4.31389e31			−0.609187
$452$		0			0
$453$	−6.30871e31	−	2.61896e31i	−0.844816	−	0.350712i
$454$		0			0
$455$		−	8.94221e30i		−	0.113582i
$456$		0			0
$457$	5.79920e31			0.698834			0.349417	−	0.936967i	$-0.386380\pi$
$457$	5.79920e31			0.698834			0.349417	+	0.936967i	$0.386380\pi$
$458$		0			0
$459$	−7.37968e31	+	3.03641e31i	−0.843889	+	0.347223i
$460$		0			0
$461$		−	5.22421e31i		−	0.567034i	−0.958967	−	0.283517i	$-0.908499\pi$
$461$		−	5.22421e31i		−	0.567034i	0.958967	−	0.283517i	$-0.0915013\pi$
$462$		0			0
$463$	2.10907e31			0.217330			0.108665	−	0.994078i	$-0.465342\pi$
$463$	2.10907e31			0.217330			0.108665	+	0.994078i	$0.465342\pi$
$464$		0			0
$465$	−1.72625e31	+	4.15829e31i	−0.168915	+	0.406893i
$466$		0			0
$467$			1.62891e32i			1.51389i	0.653479	+	0.756945i	$0.273309\pi$
$467$			1.62891e32i			1.51389i	−0.653479	+	0.756945i	$0.726691\pi$
$468$		0			0
$469$	1.28467e32			1.13428
$470$		0			0
$471$	9.59709e31	+	3.98408e31i	0.805172	+	0.334254i
$472$		0			0
$473$		−	1.36985e32i		−	1.09230i
$474$		0			0
$475$	9.04981e31			0.685987
$476$		0			0
$477$	1.04329e32	−	1.04002e32i	0.751945	−	0.749587i
$478$		0			0
$479$		−	1.75158e32i		−	1.20062i	−0.799769	−	0.600308i	$-0.795045\pi$
$479$		−	1.75158e32i		−	1.20062i	0.799769	−	0.600308i	$-0.204955\pi$
$480$		0			0
$481$	4.66539e31			0.304192
$482$		0			0
$483$	9.69999e31	−	2.33659e32i	0.601735	−	1.44950i
$484$		0			0
$485$		−	1.03209e31i		−	0.0609281i
$486$		0			0
$487$	7.23439e31			0.406494			0.203247	−	0.979127i	$-0.434851\pi$
$487$	7.23439e31			0.406494			0.203247	+	0.979127i	$0.434851\pi$
$488$		0			0
$489$	−7.08600e31	−	2.94164e31i	−0.379049	−	0.157356i
$490$		0			0
$491$			1.04300e32i			0.531260i	0.964075	+	0.265630i	$0.0855799\pi$
$491$			1.04300e32i			0.531260i	−0.964075	+	0.265630i	$0.914420\pi$
$492$		0			0
$493$	3.62714e32			1.75955
$494$		0			0
$495$	−5.58001e31	−	5.59756e31i	−0.257853	−	0.258664i
$496$		0			0
$497$		−	3.08935e32i		−	1.36016i
$498$		0			0
$499$	1.57053e32			0.658931			0.329466	−	0.944168i	$-0.393131\pi$
$499$	1.57053e32			0.658931			0.329466	+	0.944168i	$0.393131\pi$
$500$		0			0
$501$	−9.67258e31	+	2.32999e32i	−0.386803	+	0.931755i
$502$		0			0
$503$		−	2.71086e32i		−	1.03345i	−0.856151	−	0.516726i	$-0.827150\pi$
$503$		−	2.71086e32i		−	1.03345i	0.856151	−	0.516726i	$-0.172850\pi$
$504$		0			0
$505$	2.74008e32			0.996019
$506$		0			0
$507$	−2.55171e32	−	1.05930e32i	−0.884578	−	0.367218i
$508$		0			0
$509$		−	2.32358e31i		−	0.0768324i	−0.999262	−	0.0384162i	$-0.987769\pi$
$509$		−	2.32358e31i		−	0.0768324i	0.999262	−	0.0384162i	$-0.0122313\pi$
$510$		0			0
$511$	5.39941e32			1.70332
$512$		0			0
$513$	1.16868e32	+	2.84034e32i	0.351792	+	0.854992i
$514$		0			0
$515$		−	4.78474e31i		−	0.137458i
$516$		0			0
$517$	−2.57942e32			−0.707351
$518$		0			0
$519$	−1.60607e32	+	3.86879e32i	−0.420490	+	1.01290i
$520$		0			0
$521$			2.99413e32i			0.748546i	0.927319	+	0.374273i	$0.122108\pi$
$521$			2.99413e32i			0.748546i	−0.927319	+	0.374273i	$0.877892\pi$
$522$		0			0
$523$	5.02363e32			1.19949			0.599746	−	0.800190i	$-0.295268\pi$
$523$	5.02363e32			1.19949			0.599746	+	0.800190i	$0.295268\pi$
$524$		0			0
$525$	−3.26933e32	−	1.35721e32i	−0.745670	−	0.309553i
$526$		0			0
$527$			3.63206e32i			0.791454i
$528$		0			0
$529$	−5.18810e32			−1.08029
$530$		0			0
$531$	−1.20011e32	+	1.19635e32i	−0.238829	+	0.238080i
$532$		0			0
$533$			9.15235e31i			0.174103i
$534$		0			0
$535$	3.80808e32			0.692566
$536$		0			0
$537$	3.49795e32	−	8.42609e32i	0.608307	−	1.46533i
$538$		0			0
$539$			7.95606e31i			0.132322i
$540$		0			0
$541$	1.14877e33			1.82754			0.913768	−	0.406236i	$-0.133159\pi$
$541$	1.14877e33			1.82754			0.913768	+	0.406236i	$0.133159\pi$
$542$		0			0
$543$	−3.23932e32	−	1.34475e32i	−0.493010	−	0.204665i
$544$		0			0
$545$			2.09932e32i			0.305718i
$546$		0			0
$547$	−2.72421e32			−0.379658			−0.189829	−	0.981817i	$-0.560793\pi$
$547$	−2.72421e32			−0.379658			−0.189829	+	0.981817i	$0.560793\pi$
$548$		0			0
$549$	1.48003e31	+	1.48468e31i	0.0197425	+	0.0198046i
$550$		0			0
$551$		−	1.39604e33i		−	1.78270i
$552$		0			0
$553$	3.93470e32			0.481072
$554$		0			0
$555$	−2.46237e32	+	5.93152e32i	−0.288296	+	0.694465i
$556$		0			0
$557$		−	3.73854e32i		−	0.419218i	−0.977785	−	0.209609i	$-0.932781\pi$
$557$		−	3.73854e32i		−	0.419218i	0.977785	−	0.209609i	$-0.0672191\pi$
$558$		0			0
$559$	−2.90628e32			−0.312174
$560$		0			0
$561$	−5.88869e32	−	2.44459e32i	−0.605988	−	0.251566i
$562$		0			0
$563$			3.96278e32i			0.390751i	0.980729	+	0.195375i	$0.0625925\pi$
$563$			3.96278e32i			0.390751i	−0.980729	+	0.195375i	$0.937408\pi$
$564$		0			0
$565$	−3.35808e32			−0.317329
$566$		0			0
$567$	3.77370e30	−	1.20137e33i	0.00341799	−	1.08813i
$568$		0			0
$569$			1.91705e33i			1.66451i	0.554395	+	0.832254i	$0.312950\pi$
$569$			1.91705e33i			1.66451i	−0.554395	+	0.832254i	$0.687050\pi$
$570$		0			0
$571$	−7.07277e32			−0.588784			−0.294392	−	0.955685i	$-0.595117\pi$
$571$	−7.07277e32			−0.588784			−0.294392	+	0.955685i	$0.595117\pi$
$572$		0			0
$573$	−1.79457e32	+	4.32288e32i	−0.143254	+	0.345078i
$574$		0			0
$575$			1.39787e33i			1.07017i
$576$		0			0
$577$	−5.60843e32			−0.411843			−0.205921	−	0.978569i	$-0.566019\pi$
$577$	−5.60843e32			−0.411843			−0.205921	+	0.978569i	$0.566019\pi$
$578$		0			0
$579$	−8.14762e32	−	3.38235e32i	−0.573968	−	0.238274i
$580$		0			0
$581$		−	1.76909e33i		−	1.19574i
$582$		0			0
$583$	1.17702e33			0.763419
$584$		0			0
$585$	−1.18758e32	+	1.18386e32i	−0.0739252	+	0.0736934i
$586$		0			0
$587$			1.55484e33i			0.929027i	0.885566	+	0.464514i	$0.153771\pi$
$587$			1.55484e33i			0.929027i	−0.885566	+	0.464514i	$0.846229\pi$
$588$		0			0
$589$	1.39793e33			0.801867
$590$		0			0
$591$	−5.71696e32	+	1.37714e33i	−0.314859	+	0.758451i
$592$		0			0
$593$		−	8.57869e32i		−	0.453696i	−0.973930	−	0.226848i	$-0.927158\pi$
$593$		−	8.57869e32i		−	0.453696i	0.973930	−	0.226848i	$-0.0728420\pi$
$594$		0			0
$595$	9.93025e32			0.504379
$596$		0			0
$597$	−2.96540e33	−	1.23104e33i	−1.44674	−	0.600592i
$598$		0			0
$599$		−	1.41388e33i		−	0.662658i	−0.943515	−	0.331329i	$-0.892503\pi$
$599$		−	1.41388e33i		−	0.662658i	0.943515	−	0.331329i	$-0.107497\pi$
$600$		0			0
$601$	−1.65022e33			−0.743103			−0.371552	−	0.928412i	$-0.621174\pi$
$601$	−1.65022e33			−0.743103			−0.371552	+	0.928412i	$0.621174\pi$
$602$		0			0
$603$	−1.70077e33	−	1.70613e33i	−0.735934	−	0.738250i
$604$		0			0
$605$			5.89990e32i			0.245346i
$606$		0			0
$607$	2.77181e33			1.10790			0.553948	−	0.832552i	$-0.313121\pi$
$607$	2.77181e33			1.10790			0.553948	+	0.832552i	$0.313121\pi$
$608$		0			0
$609$	−2.09365e33	+	5.04332e33i	−0.804446	+	1.93780i
$610$		0			0
$611$			5.47249e32i			0.202158i
$612$		0			0
$613$	3.92257e33			1.39330			0.696651	−	0.717410i	$-0.254673\pi$
$613$	3.92257e33			1.39330			0.696651	+	0.717410i	$0.254673\pi$
$614$		0			0
$615$	−1.16362e33	−	4.83057e32i	−0.397474	−	0.165005i
$616$		0			0
$617$		−	2.49852e33i		−	0.820844i	−0.911896	−	0.410422i	$-0.865381\pi$
$617$		−	2.49852e33i		−	0.820844i	0.911896	−	0.410422i	$-0.134619\pi$
$618$		0			0
$619$	−3.36013e33			−1.06186			−0.530931	−	0.847415i	$-0.678158\pi$
$619$	−3.36013e33			−1.06186			−0.530931	+	0.847415i	$0.678158\pi$
$620$		0			0
$621$	−4.38732e33	+	1.80519e33i	−1.33383	+	0.548812i
$622$		0			0
$623$		−	2.41527e33i		−	0.706495i
$624$		0			0
$625$	1.03921e33			0.292511
$626$		0			0
$627$	−9.40892e32	+	2.26648e33i	−0.254876	+	0.613962i
$628$		0			0
$629$			5.18088e33i			1.35082i
$630$		0			0
$631$	−4.71285e33			−1.18285			−0.591427	−	0.806359i	$-0.701435\pi$
$631$	−4.71285e33			−1.18285			−0.591427	+	0.806359i	$0.701435\pi$
$632$		0			0
$633$	−4.72485e33	−	1.96145e33i	−1.14168	−	0.473949i
$634$		0			0
$635$			2.67622e33i			0.622640i
$636$		0			0
$637$	1.68796e32			0.0378171
$638$		0			0
$639$	−4.10285e33	+	4.08998e33i	−0.885269	+	0.882493i
$640$		0			0
$641$			4.12036e33i			0.856326i	0.903702	+	0.428163i	$0.140839\pi$
$641$			4.12036e33i			0.856326i	−0.903702	+	0.428163i	$0.859161\pi$
$642$		0			0
$643$	1.45104e33			0.290501			0.145250	−	0.989395i	$-0.453601\pi$
$643$	1.45104e33			0.290501			0.145250	+	0.989395i	$0.453601\pi$
$644$		0			0
$645$	1.53392e33	−	3.69501e33i	0.295861	−	0.712687i
$646$		0			0
$647$			9.20942e33i			1.71152i	0.517377	+	0.855758i	$0.326909\pi$
$647$			9.20942e33i			1.71152i	−0.517377	+	0.855758i	$0.673091\pi$
$648$		0			0
$649$	−1.35394e33			−0.242474
$650$		0			0
$651$	−5.05016e33	−	2.09649e33i	−0.871633	−	0.361844i
$652$		0			0
$653$		−	3.83686e33i		−	0.638289i	−0.947706	−	0.319144i	$-0.896604\pi$
$653$		−	3.83686e33i		−	0.638289i	0.947706	−	0.319144i	$-0.103396\pi$
$654$		0			0
$655$	−1.40941e33			−0.226018
$656$		0			0
$657$	−7.14826e33	−	7.17075e33i	−1.10514	−	1.10861i
$658$		0			0
$659$		−	1.22388e34i		−	1.82437i	−0.409774	−	0.912187i	$-0.634392\pi$
$659$		−	1.22388e34i		−	1.82437i	0.409774	−	0.912187i	$-0.365608\pi$
$660$		0			0
$661$	4.50499e33			0.647555			0.323777	−	0.946133i	$-0.395047\pi$
$661$	4.50499e33			0.647555			0.323777	+	0.946133i	$0.395047\pi$
$662$		0			0
$663$	−5.18646e32	+	1.24935e33i	−0.0718966	+	0.173189i
$664$		0			0
$665$		−	3.82202e33i		−	0.511015i
$666$		0			0
$667$	2.15638e34			2.78109
$668$		0			0
$669$	9.17176e33	+	3.80751e33i	1.14114	+	0.473726i
$670$		0			0
$671$			1.67499e32i			0.0201068i
$672$		0			0
$673$	1.47939e34			1.71356			0.856782	−	0.515678i	$-0.172460\pi$
$673$	1.47939e34			1.71356			0.856782	+	0.515678i	$0.172460\pi$
$674$		0			0
$675$	2.52580e33	+	6.13867e33i	0.282327	+	0.686166i
$676$		0			0
$677$			9.47154e33i			1.02178i	0.859647	+	0.510889i	$0.170684\pi$
$677$			9.47154e33i			1.02178i	−0.859647	+	0.510889i	$0.829316\pi$
$678$		0			0
$679$	1.25346e33			0.130518
$680$		0			0
$681$	−3.67672e33	+	8.85670e33i	−0.369568	+	0.890239i
$682$		0			0
$683$		−	1.32859e34i		−	1.28927i	−0.764492	−	0.644633i	$-0.777010\pi$
$683$		−	1.32859e34i		−	1.28927i	0.764492	−	0.644633i	$-0.222990\pi$
$684$		0			0
$685$	1.71507e33			0.160692
$686$		0			0
$687$	−9.65261e33	−	4.00713e33i	−0.873301	−	0.362537i
$688$		0			0
$689$		−	2.49718e33i		−	0.218182i
$690$		0			0
$691$	1.22829e34			1.03649			0.518246	−	0.855232i	$-0.326585\pi$
$691$	1.22829e34			1.03649			0.518246	+	0.855232i	$0.326585\pi$
$692$		0			0
$693$	6.79813e33	−	6.77681e33i	0.554103	−	0.552365i
$694$		0			0
$695$		−	8.57040e33i		−	0.674813i
$696$		0			0
$697$	−1.01636e34			−0.773134
$698$		0			0
$699$	7.95506e33	−	1.91626e34i	0.584679	−	1.40841i
$700$		0			0
$701$		−	2.57975e34i		−	1.83215i	−0.401007	−	0.916075i	$-0.631340\pi$
$701$		−	2.57975e34i		−	1.83215i	0.401007	−	0.916075i	$-0.368660\pi$
$702$		0			0
$703$	1.99405e34			1.36859
$704$		0			0
$705$	−6.95766e33	−	2.88836e33i	−0.461523	−	0.191594i
$706$		0			0
$707$			3.32778e34i			2.13364i
$708$		0			0
$709$	1.67711e34			1.03946			0.519728	−	0.854332i	$-0.326033\pi$
$709$	1.67711e34			1.03946			0.519728	+	0.854332i	$0.326033\pi$
$710$		0			0
$711$	−5.20913e33	−	5.22552e33i	−0.312126	−	0.313108i
$712$		0			0
$713$			2.15931e34i			1.25095i
$714$		0			0
$715$	−1.33981e33			−0.0750532
$716$		0			0
$717$	2.34120e33	−	5.63961e33i	0.126826	−	0.305506i
$718$		0			0
$719$			1.42987e34i			0.749115i	0.927204	+	0.374558i	$0.122205\pi$
$719$			1.42987e34i			0.749115i	−0.927204	+	0.374558i	$0.877795\pi$
$720$		0			0
$721$	5.81097e33			0.294459
$722$		0			0
$723$	−1.11285e34	−	4.61981e33i	−0.545477	−	0.226446i
$724$		0			0
$725$		−	3.01718e34i		−	1.43069i
$726$		0			0
$727$	1.19847e34			0.549813			0.274907	−	0.961471i	$-0.411353\pi$
$727$	1.19847e34			0.549813			0.274907	+	0.961471i	$0.411353\pi$
$728$		0			0
$729$	−1.60049e34	+	1.58547e34i	−0.710431	+	0.703767i
$730$		0			0
$731$		−	3.22740e34i		−	1.38626i
$732$		0			0
$733$	3.29541e34			1.36981			0.684906	−	0.728631i	$-0.259843\pi$
$733$	3.29541e34			1.36981			0.684906	+	0.728631i	$0.259843\pi$
$734$		0			0
$735$	−8.90897e32	+	2.14605e33i	−0.0358410	+	0.0863359i
$736$		0			0
$737$		−	1.92482e34i		−	0.749515i
$738$		0			0
$739$	1.97092e34			0.742909			0.371455	−	0.928451i	$-0.378859\pi$
$739$	1.97092e34			0.742909			0.371455	+	0.928451i	$0.378859\pi$
$740$		0			0
$741$	4.80857e33	+	1.99620e33i	0.175468	+	0.0728426i
$742$		0			0
$743$			1.02403e34i			0.361780i	0.983503	+	0.180890i	$0.0578979\pi$
$743$			1.02403e34i			0.361780i	−0.983503	+	0.180890i	$0.942102\pi$
$744$		0			0
$745$	−1.80840e33			−0.0618614
$746$		0			0
$747$	−2.34946e34	+	2.34209e34i	−0.778252	+	0.775811i
$748$		0			0
$749$			4.62484e34i			1.48359i
$750$		0			0
$751$	1.34386e34			0.417519			0.208759	−	0.977967i	$-0.433057\pi$
$751$	1.34386e34			0.417519			0.208759	+	0.977967i	$0.433057\pi$
$752$		0			0
$753$	5.60299e32	−	1.34968e33i	0.0168609	−	0.0406155i
$754$		0			0
$755$		−	1.59397e34i		−	0.464639i
$756$		0			0
$757$	5.51041e34			1.55608			0.778042	−	0.628212i	$-0.216213\pi$
$757$	5.51041e34			1.55608			0.778042	+	0.628212i	$0.216213\pi$
$758$		0			0
$759$	−3.50090e34	−	1.45334e34i	−0.957809	−	0.397619i
$760$		0			0
$761$			4.30641e34i			1.14156i	0.821102	+	0.570781i	$0.193360\pi$
$761$			4.30641e34i			1.14156i	−0.821102	+	0.570781i	$0.806640\pi$
$762$		0			0
$763$	−2.54958e34			−0.654900
$764$		0			0
$765$	−1.31466e34	−	1.31880e34i	−0.327248	−	0.328278i
$766$		0			0
$767$			2.87253e33i			0.0692980i
$768$		0			0
$769$	−3.03466e34			−0.709567			−0.354784	−	0.934948i	$-0.615445\pi$
$769$	−3.03466e34			−0.709567			−0.354784	+	0.934948i	$0.615445\pi$
$770$		0			0
$771$	−2.34207e34	+	5.64171e34i	−0.530819	+	1.27867i
$772$		0			0
$773$		−	1.67755e34i		−	0.368571i	−0.982873	−	0.184286i	$-0.941003\pi$
$773$		−	1.67755e34i		−	0.368571i	0.982873	−	0.184286i	$-0.0589972\pi$
$774$		0			0
$775$	3.02127e34			0.643531
$776$		0			0
$777$	−7.20371e34	−	2.99050e34i	−1.48766	−	0.617578i
$778$		0			0
$779$			3.91184e34i			0.783307i
$780$		0			0
$781$	−4.62876e34			−0.898778
$782$		0			0
$783$	9.46962e34	−	3.89633e34i	1.78316	−	0.733694i
$784$		0			0
$785$			2.42481e34i			0.442835i
$786$		0			0
$787$	−3.01739e34			−0.534484			−0.267242	−	0.963629i	$-0.586112\pi$
$787$	−3.01739e34			−0.534484			−0.267242	+	0.963629i	$0.586112\pi$
$788$		0			0
$789$	3.42290e33	−	8.24530e33i	0.0588126	−	0.141671i
$790$		0			0
$791$		−	4.07832e34i		−	0.679773i
$792$		0			0
$793$	3.55367e32			0.00574644
$794$		0			0
$795$	3.17488e34	+	1.31800e34i	0.498106	+	0.206781i
$796$		0			0
$797$		−	7.59723e34i		−	1.15653i	−0.815850	−	0.578263i	$-0.803731\pi$
$797$		−	7.59723e34i		−	1.15653i	0.815850	−	0.578263i	$-0.196269\pi$
$798$		0			0
$799$	−6.07716e34			−0.897717
$800$		0			0
$801$	−3.20763e34	+	3.19757e34i	−0.459826	+	0.458384i
$802$		0			0
$803$		−	8.08990e34i		−	1.12553i
$804$		0			0
$805$	5.90366e34			0.797207
$806$		0			0
$807$	−1.45001e34	+	3.49287e34i	−0.190059	+	0.457826i
$808$		0			0
$809$			5.59731e33i			0.0712192i	0.999366	+	0.0356096i	$0.0113373\pi$
$809$			5.59731e33i			0.0712192i	−0.999366	+	0.0356096i	$0.988663\pi$
$810$		0			0
$811$	−3.95151e34			−0.488104			−0.244052	−	0.969762i	$-0.578477\pi$
$811$	−3.95151e34			−0.488104			−0.244052	+	0.969762i	$0.578477\pi$
$812$		0			0
$813$	4.09991e34	+	1.70201e34i	0.491686	+	0.204115i
$814$		0			0
$815$		−	1.79036e34i		−	0.208472i
$816$		0			0
$817$	−1.24219e35			−1.40450
$818$		0			0
$819$	−1.43777e34	−	1.44229e34i	−0.157864	−	0.158360i
$820$		0			0
$821$			9.13276e33i			0.0973832i	0.998814	+	0.0486916i	$0.0155051\pi$
$821$			9.13276e33i			0.0973832i	−0.998814	+	0.0486916i	$0.984495\pi$
$822$		0			0
$823$	−1.85735e34			−0.192352			−0.0961760	−	0.995364i	$-0.530661\pi$
$823$	−1.85735e34			−0.192352			−0.0961760	+	0.995364i	$0.530661\pi$
$824$		0			0
$825$	−2.03350e34	+	4.89842e34i	−0.204549	+	0.492729i
$826$		0			0
$827$			1.06740e35i			1.04295i	0.853267	+	0.521474i	$0.174618\pi$
$827$			1.06740e35i			1.04295i	−0.853267	+	0.521474i	$0.825382\pi$
$828$		0			0
$829$	1.82567e35			1.73288			0.866440	−	0.499281i	$-0.166403\pi$
$829$	1.82567e35			1.73288			0.866440	+	0.499281i	$0.166403\pi$
$830$		0			0
$831$	−7.37553e34	−	3.06183e34i	−0.680113	−	0.282338i
$832$		0			0
$833$			1.87447e34i			0.167933i
$834$		0			0
$835$	−5.88699e34			−0.512454
$836$		0			0
$837$	3.90162e34	+	9.48247e34i	0.330019	+	0.802077i
$838$		0			0
$839$		−	1.68970e34i		−	0.138888i	−0.997586	−	0.0694442i	$-0.977877\pi$
$839$		−	1.68970e34i		−	0.138888i	0.997586	−	0.0694442i	$-0.0221226\pi$
$840$		0			0
$841$	−3.40250e35			−2.71798
$842$		0			0
$843$	2.80465e34	−	6.75601e34i	0.217745	−	0.524517i
$844$		0			0
$845$		−	6.44719e34i		−	0.486508i
$846$		0			0
$847$	−7.16530e34			−0.525574
$848$		0			0
$849$	2.10042e34	+	8.71957e33i	0.149766	+	0.0621731i
$850$		0			0
$851$			3.08010e35i			2.13506i
$852$		0			0
$853$	−5.01848e34			−0.338208			−0.169104	−	0.985598i	$-0.554087\pi$
$853$	−5.01848e34			−0.338208			−0.169104	+	0.985598i	$0.554087\pi$
$854$		0			0
$855$	−5.07588e34	+	5.05996e34i	−0.332597	+	0.331554i
$856$		0			0
$857$			1.02720e35i			0.654465i	0.944944	+	0.327233i	$0.106116\pi$
$857$			1.02720e35i			0.654465i	−0.944944	+	0.327233i	$0.893884\pi$
$858$		0			0
$859$	−1.27721e35			−0.791304			−0.395652	−	0.918400i	$-0.629481\pi$
$859$	−1.27721e35			−0.791304			−0.395652	+	0.918400i	$0.629481\pi$
$860$		0			0
$861$	5.86663e34	−	1.41319e35i	0.353469	−	0.851457i
$862$		0			0
$863$			2.40446e35i			1.40893i	0.709740	+	0.704463i	$0.248812\pi$
$863$			2.40446e35i			1.40893i	−0.709740	+	0.704463i	$0.751188\pi$
$864$		0			0
$865$	−9.77495e34			−0.557085
$866$		0			0
$867$	2.78719e34	+	1.15706e34i	0.154503	+	0.0641395i
$868$		0			0
$869$		−	5.89533e34i		−	0.317886i
$870$		0			0
$871$	−4.08370e34			−0.214208
$872$		0			0
$873$	−1.65944e34	−	1.66467e34i	−0.0846822	−	0.0849486i
$874$		0			0
$875$		−	1.93932e35i		−	0.962835i
$876$		0			0
$877$	1.99000e35			0.961300			0.480650	−	0.876912i	$-0.340401\pi$
$877$	1.99000e35			0.961300			0.480650	+	0.876912i	$0.340401\pi$
$878$		0			0
$879$	3.49442e34	−	8.41758e34i	0.164251	−	0.395659i
$880$		0			0
$881$			1.97792e35i			0.904684i	0.891845	+	0.452342i	$0.149411\pi$
$881$			1.97792e35i			0.904684i	−0.891845	+	0.452342i	$0.850589\pi$
$882$		0			0
$883$	1.50399e35			0.669446			0.334723	−	0.942317i	$-0.391357\pi$
$883$	1.50399e35			0.669446			0.334723	+	0.942317i	$0.391357\pi$
$884$		0			0
$885$	−3.65210e34	−	1.51611e34i	−0.158206	−	0.0656767i
$886$		0			0
$887$		−	3.11037e35i		−	1.31138i	−0.755031	−	0.655689i	$-0.772378\pi$
$887$		−	3.11037e35i		−	1.31138i	0.755031	−	0.655689i	$-0.227622\pi$
$888$		0			0
$889$	−3.25021e35			−1.33380
$890$		0			0
$891$	−1.80000e35	−	5.65411e32i	−0.719020	−	0.00225856i
$892$		0			0
$893$			2.33902e35i			0.909529i
$894$		0			0
$895$	2.12895e35			0.805913
$896$		0			0
$897$	−3.08342e34	+	7.42752e34i	−0.113638	+	0.273738i
$898$		0			0
$899$		−	4.66067e35i		−	1.67237i
$900$		0			0
$901$	2.77310e35			0.968874
$902$		0			0
$903$	4.48751e35	+	1.86292e35i	1.52670	+	0.633783i
$904$		0			0
$905$		−	8.18450e34i		−	0.271150i
$906$		0			0
$907$	1.62862e35			0.525450			0.262725	−	0.964871i	$-0.415379\pi$
$907$	1.62862e35			0.525450			0.262725	+	0.964871i	$0.415379\pi$
$908$		0			0
$909$	4.41949e35	−	4.40563e35i	1.38869	−	1.38434i
$910$		0			0
$911$			1.31582e35i			0.402695i	0.979520	+	0.201347i	$0.0645320\pi$
$911$			1.31582e35i			0.402695i	−0.979520	+	0.201347i	$0.935468\pi$
$912$		0			0
$913$	−2.65061e35			−0.790128
$914$		0			0
$915$	−1.87561e33	+	4.51809e33i	−0.00544615	+	0.0131190i
$916$		0			0
$917$		−	1.71171e35i		−	0.484169i
$918$		0			0
$919$	1.38423e35			0.381435			0.190718	−	0.981645i	$-0.438919\pi$
$919$	1.38423e35			0.381435			0.190718	+	0.981645i	$0.438919\pi$
$920$		0			0
$921$	4.71271e35	+	1.95641e35i	1.26519	+	0.525223i
$922$		0			0
$923$			9.82039e34i			0.256867i
$924$		0			0
$925$	4.30964e35			1.09835
$926$		0			0
$927$	−7.69312e34	−	7.71732e34i	−0.191049	−	0.191650i
$928$		0			0
$929$		−	5.79667e35i		−	1.40278i	−0.712776	−	0.701391i	$-0.752563\pi$
$929$		−	5.79667e35i		−	1.40278i	0.712776	−	0.701391i	$-0.247437\pi$
$930$		0			0
$931$	7.21457e34			0.170143
$932$		0			0
$933$	3.73520e34	−	8.99758e34i	0.0858488	−	0.206798i
$934$		0			0
$935$		−	1.48784e35i		−	0.333287i
$936$		0			0
$937$	−2.61380e35			−0.570685			−0.285343	−	0.958426i	$-0.592107\pi$
$937$	−2.61380e35			−0.570685			−0.285343	+	0.958426i	$0.592107\pi$
$938$		0			0
$939$	8.16428e35	+	3.38927e35i	1.73752	+	0.721305i
$940$		0			0
$941$		−	2.84502e35i		−	0.590215i	−0.955464	−	0.295107i	$-0.904645\pi$
$941$		−	2.84502e35i		−	0.590215i	0.955464	−	0.295107i	$-0.0953554\pi$
$942$		0			0
$943$	−6.04240e35			−1.22200
$944$		0			0
$945$	2.59256e35	−	1.06672e35i	0.511148	−	0.210315i
$946$		0			0
$947$		−	2.03824e35i		−	0.391792i	−0.980625	−	0.195896i	$-0.937239\pi$
$947$		−	2.03824e35i		−	0.391792i	0.980625	−	0.195896i	$-0.0627615\pi$
$948$		0			0
$949$	−1.71636e35			−0.321672
$950$		0			0
$951$	1.12244e35	−	2.70379e35i	0.205114	−	0.494091i
$952$		0			0
$953$			1.97399e35i			0.351746i	0.984413	+	0.175873i	$0.0562748\pi$
$953$			1.97399e35i			0.351746i	−0.984413	+	0.175873i	$0.943725\pi$
$954$		0			0
$955$	−1.09222e35			−0.189789
$956$		0			0
$957$	7.55638e35	+	3.13691e35i	1.28047	+	0.531568i
$958$		0			0
$959$			2.08291e35i			0.344230i
$960$		0			0
$961$	−1.53714e35			−0.247760
$962$		0			0
$963$	6.14207e35	−	6.12281e35i	0.965605	−	0.962576i
$964$		0			0
$965$		−	2.05859e35i		−	0.315676i
$966$		0			0
$967$	−6.81298e35			−1.01911			−0.509553	−	0.860439i	$-0.670189\pi$
$967$	−6.81298e35			−1.01911			−0.509553	+	0.860439i	$0.670189\pi$
$968$		0			0
$969$	−2.21676e35	+	5.33988e35i	−0.323470	+	0.779194i
$970$		0			0
$971$			4.00910e35i			0.570711i	0.958422	+	0.285355i	$0.0921117\pi$
$971$			4.00910e35i			0.570711i	−0.958422	+	0.285355i	$0.907888\pi$
$972$		0			0
$973$	1.04086e36			1.44556
$974$		0			0
$975$	1.03925e35	+	4.31428e34i	0.140820	+	0.0584591i
$976$		0			0
$977$			1.43289e36i			1.89443i	0.320600	+	0.947215i	$0.396116\pi$
$977$			1.43289e36i			1.89443i	−0.320600	+	0.947215i	$0.603884\pi$
$978$		0			0
$979$	−3.61878e35			−0.466843
$980$		0			0
$981$	3.37539e35	+	3.38600e35i	0.424908	+	0.426245i
$982$		0			0
$983$			8.32748e35i			1.02299i	0.859286	+	0.511495i	$0.170908\pi$
$983$			8.32748e35i			1.02299i	−0.859286	+	0.511495i	$0.829092\pi$
$984$		0			0
$985$	−3.47949e35			−0.417139
$986$		0			0
$987$	3.50785e35	−	8.44993e35i	0.410427	−	0.988661i
$988$		0			0
$989$		−	1.91873e36i		−	2.19109i
$990$		0			0
$991$	2.31096e35			0.257578			0.128789	−	0.991672i	$-0.458891\pi$
$991$	2.31096e35			0.257578			0.128789	+	0.991672i	$0.458891\pi$
$992$		0			0
$993$	−1.23949e36	−	5.14556e35i	−1.34851	−	0.559811i
$994$		0			0
$995$		−	7.49241e35i		−	0.795692i
$996$		0			0
$997$	−8.99382e35			−0.932400			−0.466200	−	0.884679i	$-0.654377\pi$
$997$	−8.99382e35			−0.932400			−0.466200	+	0.884679i	$0.654377\pi$
$998$		0			0
$999$	5.56539e35	+	1.35261e36i	0.563261	+	1.36895i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.25.e.d.17.8		8
3.2	odd	2	inner	48.25.e.d.17.7		8
4.3	odd	2		6.25.b.a.5.1	✓	8
12.11	even	2		6.25.b.a.5.5	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6.25.b.a.5.1	✓	8	4.3	odd	2
6.25.b.a.5.5	yes	8	12.11	even	2
48.25.e.d.17.7		8	3.2	odd	2	inner
48.25.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:	\( k \)	\(=\)	\( 25 \)
Character orbit:	\([\chi]\)	\(=\)	48.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(490828. + 203759. i) q^{3} +1.24013e8i q^{5} -1.50611e10 q^{7} +(1.99394e11 + 2.00021e11i) q^{9} +O(q^{10})\)	\(q+(490828. + 203759. i) q^{3} +1.24013e8i q^{5} -1.50611e10 q^{7} +(1.99394e11 + 2.00021e11i) q^{9} +2.25660e12i q^{11} +4.78761e12 q^{13} +(-2.52688e13 + 6.08691e13i) q^{15} +5.31661e14i q^{17} +2.04629e15 q^{19} +(-7.39242e15 - 3.06885e15i) q^{21} +3.16079e16i q^{23} +4.42254e16 q^{25} +(5.71119e16 + 1.38804e17i) q^{27} -6.82228e17i q^{29} +6.83154e17 q^{31} +(-4.59804e17 + 1.10760e18i) q^{33} -1.86778e18i q^{35} +9.74472e18 q^{37} +(2.34989e18 + 9.75520e17i) q^{39} +1.91167e19i q^{41} -6.07042e19 q^{43} +(-2.48053e19 + 2.47275e19i) q^{45} +1.14305e20i q^{47} +3.52568e19 q^{49} +(-1.08331e20 + 2.60954e20i) q^{51} -5.21591e20i q^{53} -2.79848e20 q^{55} +(1.00438e21 + 4.16951e20i) q^{57} +5.99992e20i q^{59} +7.42263e19 q^{61} +(-3.00310e21 - 3.01255e21i) q^{63} +5.93727e20i q^{65} -8.52972e21 q^{67} +(-6.44041e21 + 1.55140e22i) q^{69} +2.05121e22i q^{71} -3.58499e22 q^{73} +(2.17070e22 + 9.01133e21i) q^{75} -3.39870e22i q^{77} -2.61248e22 q^{79} +(-2.50559e20 + 7.97660e22i) q^{81} +1.17460e23i q^{83} -6.59329e22 q^{85} +(1.39010e23 - 3.34856e23i) q^{87} +1.60364e23i q^{89} -7.21069e22 q^{91} +(3.35311e23 + 1.39199e23i) q^{93} +2.53767e23i q^{95} -8.32244e22 q^{97} +(-4.51369e23 + 4.49953e23i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 131880 q^{3} + 10160794640 q^{7} + 295169053896 q^{9}+O(q^{10}) \) 8 * q + 131880 * q^3 + 10160794640 * q^7 + 295169053896 * q^9	\( 8 q + 131880 q^{3} + 10160794640 q^{7} + 295169053896 q^{9} + 50568363679120 q^{13} + 348034956760512 q^{15} + 978083631341264 q^{19} + 36\!\cdots\!36 q^{21}+ \cdots - 69\!\cdots\!44 q^{99}+O(q^{100}) \) 8 * q + 131880 * q^3 + 10160794640 * q^7 + 295169053896 * q^9 + 50568363679120 * q^13 + 348034956760512 * q^15 + 978083631341264 * q^19 + 3640012304241936 * q^21 + 7630618767014024 * q^25 + 86594528606057640 * q^27 - 3092119786822709104 * q^31 - 8158685952668529600 * q^33 + 22590293223992782480 * q^37 + 40190176581881465040 * q^39 - 226487466371803896880 * q^43 - 347709996757177504128 * q^45 + 104686700473616731800 * q^49 - 558091874936566543104 * q^51 + 2212687664250467338368 * q^55 + 3588879995640760725840 * q^57 - 6507010783838092385648 * q^61 - 10112982777612899380080 * q^63 + 7042120118150060144720 * q^67 - 4323335967368731345536 * q^69 + 36259820324758576687120 * q^73 + 108081978448612908655272 * q^75 - 316807052777330015315824 * q^79 - 268518660504396776813304 * q^81 - 222512862545524200678912 * q^85 - 449461577201807342128320 * q^87 + 1196839602335690774367520 * q^91 + 873237537450828783745680 * q^93 + 816517576248201265716880 * q^97 - 695858883367208922313344 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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