LMFDB - Embedded newform 48.25.e.d.17.5

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			17.5
Root			$-5.64115e6i$ of defining polynomial
Character	$\chi$	$=$	48.17
Dual form			48.25.e.d.17.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(317945. - 425841. i) q^{3} -6.76938e7i q^{5} +2.41596e10 q^{7} +(-8.02515e10 - 2.70788e11i) q^{9} +O(q^{10})$	\(q+(317945. - 425841. i) q^{3} -6.76938e7i q^{5} +2.41596e10 q^{7} +(-8.02515e10 - 2.70788e11i) q^{9} -4.00004e12i q^{11} +2.67023e13 q^{13} +(-2.88268e13 - 2.15229e13i) q^{15} -6.60084e14i q^{17} +3.05285e14 q^{19} +(7.68142e15 - 1.02881e16i) q^{21} +7.66067e14i q^{23} +5.50222e16 q^{25} +(-1.40828e17 - 5.19213e16i) q^{27} -4.00191e17i q^{29} -1.08492e18 q^{31} +(-1.70338e18 - 1.27179e18i) q^{33} -1.63546e18i q^{35} +2.67466e18 q^{37} +(8.48986e18 - 1.13709e19i) q^{39} +3.10777e19i q^{41} +2.43678e19 q^{43} +(-1.83307e19 + 5.43253e18i) q^{45} +3.79896e19i q^{47} +3.92105e20 q^{49} +(-2.81091e20 - 2.09870e20i) q^{51} -8.82597e19i q^{53} -2.70778e20 q^{55} +(9.70638e19 - 1.30003e20i) q^{57} -1.88092e21i q^{59} -1.92240e21 q^{61} +(-1.93884e21 - 6.54213e21i) q^{63} -1.80758e21i q^{65} +7.96234e21 q^{67} +(3.26223e20 + 2.43567e20i) q^{69} +2.04327e22i q^{71} +3.28296e22 q^{73} +(1.74940e22 - 2.34307e22i) q^{75} -9.66394e22i q^{77} -5.13170e21 q^{79} +(-6.68858e22 + 4.34623e22i) q^{81} -1.14963e21i q^{83} -4.46836e22 q^{85} +(-1.70418e23 - 1.27239e23i) q^{87} -3.69857e23i q^{89} +6.45117e23 q^{91} +(-3.44945e23 + 4.62003e23i) q^{93} -2.06659e22i q^{95} +1.10819e24 q^{97} +(-1.08316e24 + 3.21010e23i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	317945.	−	425841.i	0.598270	−	0.801295i
$3$	317945.	−	425841.i	0.598270	−	0.801295i
$4$		0			0
$5$		−	6.76938e7i		−	0.277274i	−0.990343	−	0.138637i	$-0.955728\pi$
$5$		−	6.76938e7i		−	0.277274i	0.990343	−	0.138637i	$-0.0442721\pi$
$6$		0			0
$7$	2.41596e10			1.74547			0.872736	−	0.488192i	$-0.162343\pi$
$7$	2.41596e10			1.74547			0.872736	+	0.488192i	$0.162343\pi$
$8$		0			0
$9$	−8.02515e10	−	2.70788e11i	−0.284147	−	0.958781i
$10$		0			0
$11$		−	4.00004e12i		−	1.27454i	−0.770642	−	0.637268i	$-0.780064\pi$
$11$		−	4.00004e12i		−	1.27454i	0.770642	−	0.637268i	$-0.219936\pi$
$12$		0			0
$13$	2.67023e13			1.14612			0.573058	−	0.819515i	$-0.305757\pi$
$13$	2.67023e13			1.14612			0.573058	+	0.819515i	$0.305757\pi$
$14$		0			0
$15$	−2.88268e13	−	2.15229e13i	−0.222178	−	0.165885i
$16$		0			0
$17$		−	6.60084e14i		−	1.13295i	−0.824078	−	0.566477i	$-0.808306\pi$
$17$		−	6.60084e14i		−	1.13295i	0.824078	−	0.566477i	$-0.191694\pi$
$18$		0			0
$19$	3.05285e14			0.137931			0.0689656	−	0.997619i	$-0.478030\pi$
$19$	3.05285e14			0.137931			0.0689656	+	0.997619i	$0.478030\pi$
$20$		0			0
$21$	7.68142e15	−	1.02881e16i	1.04426	−	1.39864i
$22$		0			0
$23$			7.66067e14i			0.0349569i	0.999847	+	0.0174784i	$0.00556384\pi$
$23$			7.66067e14i			0.0349569i	−0.999847	+	0.0174784i	$0.994436\pi$
$24$		0			0
$25$	5.50222e16			0.923119
$26$		0			0
$27$	−1.40828e17	−	5.19213e16i	−0.938263	−	0.345924i
$28$		0			0
$29$		−	4.00191e17i		−	1.13107i	−0.824723	−	0.565537i	$-0.808669\pi$
$29$		−	4.00191e17i		−	1.13107i	0.824723	−	0.565537i	$-0.191331\pi$
$30$		0			0
$31$	−1.08492e18			−1.37739			−0.688696	−	0.725051i	$-0.741816\pi$
$31$	−1.08492e18			−1.37739			−0.688696	+	0.725051i	$0.741816\pi$
$32$		0			0
$33$	−1.70338e18	−	1.27179e18i	−1.02128	−	0.762517i
$34$		0			0
$35$		−	1.63546e18i		−	0.483974i
$36$		0			0
$37$	2.67466e18			0.406302			0.203151	−	0.979147i	$-0.434882\pi$
$37$	2.67466e18			0.406302			0.203151	+	0.979147i	$0.434882\pi$
$38$		0			0
$39$	8.48986e18	−	1.13709e19i	0.685686	−	0.918377i
$40$		0			0
$41$			3.10777e19i			1.37734i	0.725074	+	0.688671i	$0.241806\pi$
$41$			3.10777e19i			1.37734i	−0.725074	+	0.688671i	$0.758194\pi$
$42$		0			0
$43$	2.43678e19			0.609810			0.304905	−	0.952383i	$-0.401375\pi$
$43$	2.43678e19			0.609810			0.304905	+	0.952383i	$0.401375\pi$
$44$		0			0
$45$	−1.83307e19	+	5.43253e18i	−0.265845	+	0.0787866i
$46$		0			0
$47$			3.79896e19i			0.326957i	0.986547	+	0.163478i	$0.0522714\pi$
$47$			3.79896e19i			0.326957i	−0.986547	+	0.163478i	$0.947729\pi$
$48$		0			0
$49$	3.92105e20			2.04668
$50$		0			0
$51$	−2.81091e20	−	2.09870e20i	−0.907830	−	0.677811i
$52$		0			0
$53$		−	8.82597e19i		−	0.179660i	−0.995957	−	0.0898301i	$-0.971368\pi$
$53$		−	8.82597e19i		−	0.179660i	0.995957	−	0.0898301i	$-0.0286324\pi$
$54$		0			0
$55$	−2.70778e20			−0.353396
$56$		0			0
$57$	9.70638e19	−	1.30003e20i	0.0825200	−	0.110524i
$58$		0			0
$59$		−	1.88092e21i		−	1.05718i	−0.848879	−	0.528588i	$-0.822722\pi$
$59$		−	1.88092e21i		−	1.05718i	0.848879	−	0.528588i	$-0.177278\pi$
$60$		0			0
$61$	−1.92240e21			−0.724244			−0.362122	−	0.932131i	$-0.617948\pi$
$61$	−1.92240e21			−0.724244			−0.362122	+	0.932131i	$0.617948\pi$
$62$		0			0
$63$	−1.93884e21	−	6.54213e21i	−0.495971	−	1.67353i
$64$		0			0
$65$		−	1.80758e21i		−	0.317788i
$66$		0			0
$67$	7.96234e21			0.973068			0.486534	−	0.873662i	$-0.338261\pi$
$67$	7.96234e21			0.973068			0.486534	+	0.873662i	$0.338261\pi$
$68$		0			0
$69$	3.26223e20	+	2.43567e20i	0.0280108	+	0.0209136i
$70$		0			0
$71$			2.04327e22i			1.24516i	0.782557	+	0.622579i	$0.213915\pi$
$71$			2.04327e22i			1.24516i	−0.782557	+	0.622579i	$0.786085\pi$
$72$		0			0
$73$	3.28296e22			1.43348			0.716740	−	0.697340i	$-0.245633\pi$
$73$	3.28296e22			1.43348			0.716740	+	0.697340i	$0.245633\pi$
$74$		0			0
$75$	1.74940e22	−	2.34307e22i	0.552274	−	0.739691i
$76$		0			0
$77$		−	9.66394e22i		−	2.22467i
$78$		0			0
$79$	−5.13170e21			−0.0868433			−0.0434216	−	0.999057i	$-0.513826\pi$
$79$	−5.13170e21			−0.0868433			−0.0434216	+	0.999057i	$0.513826\pi$
$80$		0			0
$81$	−6.68858e22	+	4.34623e22i	−0.838521	+	0.544870i
$82$		0			0
$83$		−	1.14963e21i		−	0.0107552i	−0.999986	−	0.00537761i	$-0.998288\pi$
$83$		−	1.14963e21i		−	0.0107552i	0.999986	−	0.00537761i	$-0.00171175\pi$
$84$		0			0
$85$	−4.46836e22			−0.314138
$86$		0			0
$87$	−1.70418e23	−	1.27239e23i	−0.906324	−	0.676688i
$88$		0			0
$89$		−	3.69857e23i		−	1.49745i	−0.662879	−	0.748727i	$-0.730665\pi$
$89$		−	3.69857e23i		−	1.49745i	0.662879	−	0.748727i	$-0.269335\pi$
$90$		0			0
$91$	6.45117e23			2.00051
$92$		0			0
$93$	−3.44945e23	+	4.62003e23i	−0.824051	+	1.10370i
$94$		0			0
$95$		−	2.06659e22i		−	0.0382447i
$96$		0			0
$97$	1.10819e24			1.59718			0.798592	−	0.601873i	$-0.205579\pi$
$97$	1.10819e24			1.59718			0.798592	+	0.601873i	$0.205579\pi$
$98$		0			0
$99$	−1.08316e24	+	3.21010e23i	−1.22200	+	0.362156i
$100$		0			0
$101$			1.86377e24i			1.65400i	0.562202	+	0.827000i	$0.309954\pi$
$101$			1.86377e24i			1.65400i	−0.562202	+	0.827000i	$0.690046\pi$
$102$		0			0
$103$	−1.22843e24			−0.861594			−0.430797	−	0.902449i	$-0.641768\pi$
$103$	−1.22843e24			−0.861594			−0.430797	+	0.902449i	$0.641768\pi$
$104$		0			0
$105$	−6.96444e23	−	5.19985e23i	−0.387806	−	0.289547i
$106$		0			0
$107$			3.55456e24i			1.57827i	0.614220	+	0.789134i	$0.289471\pi$
$107$			3.55456e24i			1.57827i	−0.614220	+	0.789134i	$0.710529\pi$
$108$		0			0
$109$	2.18543e23			0.0776995			0.0388497	−	0.999245i	$-0.487631\pi$
$109$	2.18543e23			0.0776995			0.0388497	+	0.999245i	$0.487631\pi$
$110$		0			0
$111$	8.50396e23	−	1.13898e24i	0.243078	−	0.325567i
$112$		0			0
$113$			5.66320e24i			1.30653i	0.757128	+	0.653267i	$0.226602\pi$
$113$			5.66320e24i			1.30653i	−0.757128	+	0.653267i	$0.773398\pi$
$114$		0			0
$115$	5.18580e22			0.00969263
$116$		0			0
$117$	−2.14290e24	−	7.23066e24i	−0.325666	−	1.09887i
$118$		0			0
$119$		−	1.59474e25i		−	1.97754i
$120$		0			0
$121$	−6.15061e24			−0.624444
$122$		0			0
$123$	1.32341e25	+	9.88098e24i	1.10366	+	0.824022i
$124$		0			0
$125$		−	7.75953e24i		−	0.533231i
$126$		0			0
$127$	−1.31873e25			−0.749051			−0.374526	−	0.927217i	$-0.622194\pi$
$127$	−1.31873e25			−0.749051			−0.374526	+	0.927217i	$0.622194\pi$
$128$		0			0
$129$	7.74761e24	−	1.03768e25i	0.364831	−	0.488638i
$130$		0			0
$131$			2.72828e25i			1.06815i	0.845436	+	0.534077i	$0.179341\pi$
$131$			2.72828e25i			1.06815i	−0.845436	+	0.534077i	$0.820659\pi$
$132$		0			0
$133$	7.37556e24			0.240755
$134$		0			0
$135$	−3.51475e24	+	9.53320e24i	−0.0959156	+	0.260156i
$136$		0			0
$137$		−	1.88032e25i		−	0.430115i	−0.976601	−	0.215058i	$-0.931006\pi$
$137$		−	1.88032e25i		−	0.430115i	0.976601	−	0.215058i	$-0.0689939\pi$
$138$		0			0
$139$	−4.16312e24			−0.0800279			−0.0400140	−	0.999199i	$-0.512740\pi$
$139$	−4.16312e24			−0.0800279			−0.0400140	+	0.999199i	$0.512740\pi$
$140$		0			0
$141$	1.61775e25	+	1.20786e25i	0.261989	+	0.195608i
$142$		0			0
$143$		−	1.06810e26i		−	1.46077i
$144$		0			0
$145$	−2.70905e25			−0.313618
$146$		0			0
$147$	1.24668e26	−	1.66974e26i	1.22446	−	1.63999i
$148$		0			0
$149$		−	9.17379e25i		−	0.766149i	−0.923717	−	0.383075i	$-0.874865\pi$
$149$		−	9.17379e25i		−	0.766149i	0.923717	−	0.383075i	$-0.125135\pi$
$150$		0			0
$151$	−1.86184e26			−1.32501			−0.662505	−	0.749058i	$-0.730507\pi$
$151$	−1.86184e26			−1.32501			−0.662505	+	0.749058i	$0.730507\pi$
$152$		0			0
$153$	−1.78743e26	+	5.29727e25i	−1.08625	+	0.321925i
$154$		0			0
$155$			7.34424e25i			0.381915i
$156$		0			0
$157$	1.58923e26			0.708584			0.354292	−	0.935135i	$-0.384722\pi$
$157$	1.58923e26			0.708584			0.354292	+	0.935135i	$0.384722\pi$
$158$		0			0
$159$	−3.75846e25	−	2.80617e25i	−0.143961	−	0.107485i
$160$		0			0
$161$			1.85079e25i			0.0610163i
$162$		0			0
$163$	−4.86804e26			−1.38390			−0.691948	−	0.721948i	$-0.743247\pi$
$163$	−4.86804e26			−1.38390			−0.691948	+	0.721948i	$0.743247\pi$
$164$		0			0
$165$	−8.60926e25	+	1.15308e26i	−0.211426	+	0.283174i
$166$		0			0
$167$			6.29000e26i			1.33676i	0.743821	+	0.668379i	$0.233011\pi$
$167$			6.29000e26i			1.33676i	−0.743821	+	0.668379i	$0.766989\pi$
$168$		0			0
$169$	1.70212e26			0.313582
$170$		0			0
$171$	−2.44996e25	−	8.26675e25i	−0.0391927	−	0.132246i
$172$		0			0
$173$		−	1.85146e26i		−	0.257609i	−0.991670	−	0.128805i	$-0.958886\pi$
$173$		−	1.85146e26i		−	0.257609i	0.991670	−	0.128805i	$-0.0411140\pi$
$174$		0			0
$175$	1.32931e27			1.61128
$176$		0			0
$177$	−8.00974e26	−	5.98030e26i	−0.847109	−	0.632476i
$178$		0			0
$179$		−	1.47704e27i		−	1.36507i	−0.730853	−	0.682535i	$-0.760877\pi$
$179$		−	1.47704e27i		−	1.36507i	0.730853	−	0.682535i	$-0.239123\pi$
$180$		0			0
$181$	−6.65463e26			−0.538246			−0.269123	−	0.963106i	$-0.586734\pi$
$181$	−6.65463e26			−0.538246			−0.269123	+	0.963106i	$0.586734\pi$
$182$		0			0
$183$	−6.11216e26	+	8.18635e26i	−0.433293	+	0.580333i
$184$		0			0
$185$		−	1.81058e26i		−	0.112657i
$186$		0			0
$187$	−2.64036e27			−1.44399
$188$		0			0
$189$	−3.40235e27	−	1.25440e27i	−1.63771	−	0.603800i
$190$		0			0
$191$		−	2.76451e27i		−	1.17279i	−0.810027	−	0.586393i	$-0.800548\pi$
$191$		−	2.76451e27i		−	1.17279i	0.810027	−	0.586393i	$-0.199452\pi$
$192$		0			0
$193$	7.06698e26			0.264573			0.132287	−	0.991212i	$-0.457768\pi$
$193$	7.06698e26			0.264573			0.132287	+	0.991212i	$0.457768\pi$
$194$		0			0
$195$	−7.69742e26	−	5.74711e26i	−0.254642	−	0.190123i
$196$		0			0
$197$		−	4.71662e27i		−	1.38050i	−0.723571	−	0.690250i	$-0.757501\pi$
$197$		−	4.71662e27i		−	1.38050i	0.723571	−	0.690250i	$-0.242499\pi$
$198$		0			0
$199$	−2.25704e27			−0.585197			−0.292598	−	0.956235i	$-0.594520\pi$
$199$	−2.25704e27			−0.585197			−0.292598	+	0.956235i	$0.594520\pi$
$200$		0			0
$201$	2.53159e27	−	3.39069e27i	0.582157	−	0.779714i
$202$		0			0
$203$		−	9.66845e27i		−	1.97426i
$204$		0			0
$205$	2.10377e27			0.381901
$206$		0			0
$207$	2.07442e26	−	6.14780e25i	0.0335160	−	0.00993289i
$208$		0			0
$209$		−	1.22115e27i		−	0.175798i
$210$		0			0
$211$	−3.37907e27			−0.433918			−0.216959	−	0.976181i	$-0.569614\pi$
$211$	−3.37907e27			−0.433918			−0.216959	+	0.976181i	$0.569614\pi$
$212$		0			0
$213$	8.70106e27	+	6.49646e27i	0.997739	+	0.744940i
$214$		0			0
$215$		−	1.64955e27i		−	0.169084i
$216$		0			0
$217$	−2.62112e28			−2.40420
$218$		0			0
$219$	1.04380e28	−	1.39802e28i	0.857608	−	1.14864i
$220$		0			0
$221$		−	1.76258e28i		−	1.29850i
$222$		0			0
$223$	−1.36332e28			−0.901443			−0.450722	−	0.892665i	$-0.648833\pi$
$223$	−1.36332e28			−0.901443			−0.450722	+	0.892665i	$0.648833\pi$
$224$		0			0
$225$	−4.41562e27	−	1.48993e28i	−0.262302	−	0.885069i
$226$		0			0
$227$			6.82027e27i			0.364327i	0.983268	+	0.182164i	$0.0583100\pi$
$227$			6.82027e27i			0.364327i	−0.983268	+	0.182164i	$0.941690\pi$
$228$		0			0
$229$	−2.90439e28			−1.39646			−0.698231	−	0.715872i	$-0.746029\pi$
$229$	−2.90439e28			−1.39646			−0.698231	+	0.715872i	$0.746029\pi$
$230$		0			0
$231$	−4.11530e28	−	3.07260e28i	−1.78262	−	1.33095i
$232$		0			0
$233$		−	8.69231e27i		−	0.339519i	−0.985486	−	0.169760i	$-0.945701\pi$
$233$		−	8.69231e27i		−	0.339519i	0.985486	−	0.169760i	$-0.0542991\pi$
$234$		0			0
$235$	2.57166e27			0.0906566
$236$		0			0
$237$	−1.63160e27	+	2.18529e27i	−0.0519557	+	0.0695871i
$238$		0			0
$239$			3.86579e28i			1.11292i	0.830875	+	0.556459i	$0.187840\pi$
$239$			3.86579e28i			1.11292i	−0.830875	+	0.556459i	$0.812160\pi$
$240$		0			0
$241$	1.20605e28			0.314168			0.157084	−	0.987585i	$-0.449791\pi$
$241$	1.20605e28			0.314168			0.157084	+	0.987585i	$0.449791\pi$
$242$		0			0
$243$	−2.75798e27	+	4.23013e28i	−0.0650603	+	0.997881i
$244$		0			0
$245$		−	2.65431e28i		−	0.567490i
$246$		0			0
$247$	8.15181e27			0.158085
$248$		0			0
$249$	−4.89557e26	−	3.65518e26i	−0.00861810	−	0.00643452i
$250$		0			0
$251$		−	8.43194e28i		−	1.34848i	−0.738515	−	0.674238i	$-0.764472\pi$
$251$		−	8.43194e28i		−	1.34848i	0.738515	−	0.674238i	$-0.235528\pi$
$252$		0			0
$253$	3.06430e27			0.0445538
$254$		0			0
$255$	−1.42069e28	+	1.90281e28i	−0.187939	+	0.251717i
$256$		0			0
$257$			5.82077e28i			0.701105i	0.936543	+	0.350553i	$0.114006\pi$
$257$			5.82077e28i			0.701105i	−0.936543	+	0.350553i	$0.885994\pi$
$258$		0			0
$259$	6.46188e28			0.709189
$260$		0			0
$261$	−1.08367e29	+	3.21159e28i	−1.08445	+	0.321392i
$262$		0			0
$263$		−	1.33775e29i		−	1.22153i	−0.791810	−	0.610767i	$-0.790861\pi$
$263$		−	1.33775e29i		−	1.22153i	0.791810	−	0.610767i	$-0.209139\pi$
$264$		0			0
$265$	−5.97464e27			−0.0498151
$266$		0			0
$267$	−1.57500e29	−	1.17594e29i	−1.19990	−	0.895881i
$268$		0			0
$269$		−	4.31823e28i		−	0.300800i	−0.988625	−	0.150400i	$-0.951944\pi$
$269$		−	4.31823e28i		−	0.300800i	0.988625	−	0.150400i	$-0.0480562\pi$
$270$		0			0
$271$	−1.25855e28			−0.0802119			−0.0401059	−	0.999195i	$-0.512770\pi$
$271$	−1.25855e28			−0.0802119			−0.0401059	+	0.999195i	$0.512770\pi$
$272$		0			0
$273$	2.05112e29	−	2.74717e29i	1.19685	−	1.60300i
$274$		0			0
$275$		−	2.20091e29i		−	1.17655i
$276$		0			0
$277$	9.17464e28			0.449606			0.224803	−	0.974404i	$-0.427826\pi$
$277$	9.17464e28			0.449606			0.224803	+	0.974404i	$0.427826\pi$
$278$		0			0
$279$	8.70665e28	+	2.93783e29i	0.391382	+	1.32062i
$280$		0			0
$281$		−	2.21314e28i		−	0.0913131i	−0.998957	−	0.0456566i	$-0.985462\pi$
$281$		−	2.21314e28i		−	0.0913131i	0.998957	−	0.0456566i	$-0.0145380\pi$
$282$		0			0
$283$	−3.26573e28			−0.123749			−0.0618747	−	0.998084i	$-0.519708\pi$
$283$	−3.26573e28			−0.123749			−0.0618747	+	0.998084i	$0.519708\pi$
$284$		0			0
$285$	−8.80039e27	−	6.57062e27i	−0.0306453	−	0.0228806i
$286$		0			0
$287$			7.50824e29i			2.40411i
$288$		0			0
$289$	−9.62619e28			−0.283583
$290$		0			0
$291$	3.52344e29	−	4.71914e29i	0.955546	−	1.27981i
$292$		0			0
$293$			3.37056e29i			0.841958i	0.907070	+	0.420979i	$0.138313\pi$
$293$			3.37056e29i			0.841958i	−0.907070	+	0.420979i	$0.861687\pi$
$294$		0			0
$295$	−1.27327e29			−0.293127
$296$		0			0
$297$	−2.07687e29	+	5.63319e29i	−0.440892	+	1.19585i
$298$		0			0
$299$			2.04557e28i			0.0400646i
$300$		0			0
$301$	5.88716e29			1.06441
$302$		0			0
$303$	7.93669e29	+	5.92576e29i	1.32534	+	0.989538i
$304$		0			0
$305$			1.30134e29i			0.200814i
$306$		0			0
$307$	6.58441e29			0.939412			0.469706	−	0.882823i	$-0.344360\pi$
$307$	6.58441e29			0.939412			0.469706	+	0.882823i	$0.344360\pi$
$308$		0			0
$309$	−3.90572e29	+	5.23115e29i	−0.515466	+	0.690391i
$310$		0			0
$311$			1.25703e30i			1.53540i	0.640810	+	0.767700i	$0.278599\pi$
$311$			1.25703e30i			1.53540i	−0.640810	+	0.767700i	$0.721401\pi$
$312$		0			0
$313$	−4.01699e29			−0.454328			−0.227164	−	0.973857i	$-0.572945\pi$
$313$	−4.01699e29			−0.454328			−0.227164	+	0.973857i	$0.572945\pi$
$314$		0			0
$315$	−4.42862e29	+	1.31248e29i	−0.464025	+	0.137520i
$316$		0			0
$317$		−	4.83365e29i		−	0.469422i	−0.972065	−	0.234711i	$-0.924586\pi$
$317$		−	4.83365e29i		−	0.469422i	0.972065	−	0.234711i	$-0.0754144\pi$
$318$		0			0
$319$	−1.60078e30			−1.44160
$320$		0			0
$321$	1.51368e30	+	1.13016e30i	1.26466	+	0.944230i
$322$		0			0
$323$		−	2.01514e29i		−	0.156270i
$324$		0			0
$325$	1.46922e30			1.05800
$326$		0			0
$327$	6.94845e28	−	9.30644e28i	0.0464852	−	0.0622602i
$328$		0			0
$329$			9.17814e29i			0.570694i
$330$		0			0
$331$	1.33834e29			0.0773802			0.0386901	−	0.999251i	$-0.487681\pi$
$331$	1.33834e29			0.0773802			0.0386901	+	0.999251i	$0.487681\pi$
$332$		0			0
$333$	−2.14646e29	−	7.24267e29i	−0.115449	−	0.389554i
$334$		0			0
$335$		−	5.39001e29i		−	0.269806i
$336$		0			0
$337$	2.49698e26			0.000116374			5.81870e−5	−	1.00000i	$-0.499981\pi$
$337$	2.49698e26			0.000116374			5.81870e−5		1.00000i	$0.499981\pi$
$338$		0			0
$339$	2.41162e30	+	1.80059e30i	1.04692	+	0.781660i
$340$		0			0
$341$			4.33973e30i			1.75554i
$342$		0			0
$343$	4.84457e30			1.82694
$344$		0			0
$345$	1.64880e28	−	2.20833e28i	0.00579880	−	0.00776665i
$346$		0			0
$347$			1.01133e30i			0.331847i	0.986139	+	0.165923i	$0.0530605\pi$
$347$			1.01133e30i			0.331847i	−0.986139	+	0.165923i	$0.946940\pi$
$348$		0			0
$349$	4.79831e30			1.46955			0.734773	−	0.678314i	$-0.237289\pi$
$349$	4.79831e30			1.46955			0.734773	+	0.678314i	$0.237289\pi$
$350$		0			0
$351$	−3.76044e30	−	1.38642e30i	−1.07536	−	0.396469i
$352$		0			0
$353$		−	4.36735e29i		−	0.116660i	−0.998297	−	0.0583300i	$-0.981422\pi$
$353$		−	4.36735e29i		−	0.116660i	0.998297	−	0.0583300i	$-0.0185776\pi$
$354$		0			0
$355$	1.38316e30			0.345250
$356$		0			0
$357$	−6.79104e30	−	5.07038e30i	−1.58459	−	1.18310i
$358$		0			0
$359$		−	1.36292e30i		−	0.297397i	−0.988883	−	0.148699i	$-0.952492\pi$
$359$		−	1.36292e30i		−	0.297397i	0.988883	−	0.148699i	$-0.0475084\pi$
$360$		0			0
$361$	−4.80556e30			−0.980975
$362$		0			0
$363$	−1.95556e30	+	2.61918e30i	−0.373586	+	0.500364i
$364$		0			0
$365$		−	2.22236e30i		−	0.397467i
$366$		0			0
$367$	−1.44710e30			−0.242384			−0.121192	−	0.992629i	$-0.538672\pi$
$367$	−1.44710e30			−0.242384			−0.121192	+	0.992629i	$0.538672\pi$
$368$		0			0
$369$	8.41546e30	−	2.49403e30i	1.32057	−	0.391368i
$370$		0			0
$371$		−	2.13232e30i		−	0.313592i
$372$		0			0
$373$	2.12953e30			0.293614			0.146807	−	0.989165i	$-0.453100\pi$
$373$	2.12953e30			0.293614			0.146807	+	0.989165i	$0.453100\pi$
$374$		0			0
$375$	−3.30433e30	−	2.46710e30i	−0.427275	−	0.319016i
$376$		0			0
$377$		−	1.06860e31i		−	1.29634i
$378$		0			0
$379$	4.51756e30			0.514320			0.257160	−	0.966369i	$-0.417213\pi$
$379$	4.51756e30			0.514320			0.257160	+	0.966369i	$0.417213\pi$
$380$		0			0
$381$	−4.19284e30	+	5.61570e30i	−0.448135	+	0.600211i
$382$		0			0
$383$			5.00583e30i			0.502448i	0.967929	+	0.251224i	$0.0808330\pi$
$383$			5.00583e30i			0.502448i	−0.967929	+	0.251224i	$0.919167\pi$
$384$		0			0
$385$	−6.54189e30			−0.616843
$386$		0			0
$387$	−1.95555e30	−	6.59850e30i	−0.173276	−	0.584674i
$388$		0			0
$389$			7.79334e30i			0.649125i	0.945864	+	0.324563i	$0.105217\pi$
$389$			7.79334e30i			0.649125i	−0.945864	+	0.324563i	$0.894783\pi$
$390$		0			0
$391$	5.05668e29			0.0396045
$392$		0			0
$393$	1.16181e31	+	8.67443e30i	0.855906	+	0.639044i
$394$		0			0
$395$			3.47384e29i			0.0240794i
$396$		0			0
$397$	−1.68507e31			−1.09934			−0.549669	−	0.835383i	$-0.685246\pi$
$397$	−1.68507e31			−1.09934			−0.549669	+	0.835383i	$0.685246\pi$
$398$		0			0
$399$	2.34502e30	−	3.14082e30i	0.144036	−	0.192916i
$400$		0			0
$401$			1.09723e31i			0.634693i	0.948310	+	0.317346i	$0.102792\pi$
$401$			1.09723e31i			0.634693i	−0.948310	+	0.317346i	$0.897208\pi$
$402$		0			0
$403$	−2.89699e31			−1.57865
$404$		0			0
$405$	2.94213e30	+	4.52776e30i	0.151078	+	0.232500i
$406$		0			0
$407$		−	1.06988e31i		−	0.517846i
$408$		0			0
$409$	−3.46568e31			−1.58164			−0.790822	−	0.612046i	$-0.790347\pi$
$409$	−3.46568e31			−1.58164			−0.790822	+	0.612046i	$0.790347\pi$
$410$		0			0
$411$	−8.00717e30	−	5.97838e30i	−0.344649	−	0.257325i
$412$		0			0
$413$		−	4.54423e31i		−	1.84527i
$414$		0			0
$415$	−7.78225e28			−0.00298214
$416$		0			0
$417$	−1.32364e30	+	1.77283e30i	−0.0478783	+	0.0641260i
$418$		0			0
$419$		−	4.32637e30i		−	0.147759i	−0.997267	−	0.0738797i	$-0.976462\pi$
$419$		−	4.32637e30i		−	0.147759i	0.997267	−	0.0738797i	$-0.0235381\pi$
$420$		0			0
$421$	5.35766e31			1.72818			0.864092	−	0.503334i	$-0.167893\pi$
$421$	5.35766e31			1.72818			0.864092	+	0.503334i	$0.167893\pi$
$422$		0			0
$423$	1.02871e31	−	3.04872e30i	0.313480	−	0.0929039i
$424$		0			0
$425$		−	3.63193e31i		−	1.04585i
$426$		0			0
$427$	−4.64443e31			−1.26415
$428$		0			0
$429$	−4.54842e31	−	3.39598e31i	−1.17051	−	0.873932i
$430$		0			0
$431$		−	7.38282e31i		−	1.79678i	−0.439197	−	0.898391i	$-0.644737\pi$
$431$		−	7.38282e31i		−	1.79678i	0.439197	−	0.898391i	$-0.355263\pi$
$432$		0			0
$433$	2.54475e30			0.0585856			0.0292928	−	0.999571i	$-0.490674\pi$
$433$	2.54475e30			0.0585856			0.0292928	+	0.999571i	$0.490674\pi$
$434$		0			0
$435$	−8.61327e30	+	1.15362e31i	−0.187628	+	0.251300i
$436$		0			0
$437$			2.33869e29i			0.00482164i
$438$		0			0
$439$	6.51383e31			1.27134			0.635671	−	0.771960i	$-0.280724\pi$
$439$	6.51383e31			1.27134			0.635671	+	0.771960i	$0.280724\pi$
$440$		0			0
$441$	−3.14670e31	−	1.06177e32i	−0.581557	−	1.96231i
$442$		0			0
$443$			3.26967e31i			0.572347i	0.958178	+	0.286173i	$0.0923833\pi$
$443$			3.26967e31i			0.572347i	−0.958178	+	0.286173i	$0.907617\pi$
$444$		0			0
$445$	−2.50370e31			−0.415205
$446$		0			0
$447$	−3.90657e31	−	2.91676e31i	−0.613912	−	0.458364i
$448$		0			0
$449$			7.64915e31i			1.13935i	0.821870	+	0.569675i	$0.192931\pi$
$449$			7.64915e31i			1.13935i	−0.821870	+	0.569675i	$0.807069\pi$
$450$		0			0
$451$	1.24312e32			1.75547
$452$		0			0
$453$	−5.91963e31	+	7.92848e31i	−0.792713	+	1.06172i
$454$		0			0
$455$		−	4.36704e31i		−	0.554690i
$456$		0			0
$457$	5.89189e31			0.710003			0.355001	−	0.934866i	$-0.384480\pi$
$457$	5.89189e31			0.710003			0.355001	+	0.934866i	$0.384480\pi$
$458$		0			0
$459$	−3.42724e31	+	9.29584e31i	−0.391915	+	1.06301i
$460$		0			0
$461$			4.35731e31i			0.472941i	0.971639	+	0.236470i	$0.0759907\pi$
$461$			4.35731e31i			0.472941i	−0.971639	+	0.236470i	$0.924009\pi$
$462$		0			0
$463$	−1.14041e32			−1.17514			−0.587569	−	0.809174i	$-0.699915\pi$
$463$	−1.14041e32			−1.17514			−0.587569	+	0.809174i	$0.699915\pi$
$464$		0			0
$465$	3.12748e31	+	2.33506e31i	0.306026	+	0.228488i
$466$		0			0
$467$			2.25380e31i			0.209466i	0.994500	+	0.104733i	$0.0333988\pi$
$467$			2.25380e31i			0.209466i	−0.994500	+	0.104733i	$0.966601\pi$
$468$		0			0
$469$	1.92367e32			1.69846
$470$		0			0
$471$	5.05288e31	−	6.76760e31i	0.423924	−	0.567785i
$472$		0			0
$473$		−	9.74722e31i		−	0.777225i
$474$		0			0
$475$	1.67975e31			0.127327
$476$		0			0
$477$	−2.38997e31	+	7.08298e30i	−0.172255	+	0.0510499i
$478$		0			0
$479$			8.30835e31i			0.569493i	0.958603	+	0.284747i	$0.0919095\pi$
$479$			8.30835e31i			0.569493i	−0.958603	+	0.284747i	$0.908090\pi$
$480$		0			0
$481$	7.14197e31			0.465669
$482$		0			0
$483$	7.88140e30	+	5.88448e30i	0.0488920	+	0.0365042i
$484$		0			0
$485$		−	7.50178e31i		−	0.442857i
$486$		0			0
$487$	−2.38724e32			−1.34137			−0.670686	−	0.741741i	$-0.734000\pi$
$487$	−2.38724e32			−1.34137			−0.670686	+	0.741741i	$0.734000\pi$
$488$		0			0
$489$	−1.54777e32	+	2.07301e32i	−0.827942	+	1.10891i
$490$		0			0
$491$			1.30787e32i			0.666173i	0.942896	+	0.333087i	$0.108090\pi$
$491$			1.30787e32i			0.666173i	−0.942896	+	0.333087i	$0.891910\pi$
$492$		0			0
$493$	−2.64160e32			−1.28145
$494$		0			0
$495$	2.17304e31	+	7.33235e31i	0.100416	+	0.338829i
$496$		0			0
$497$			4.93645e32i			2.17339i
$498$		0			0
$499$	2.33308e32			0.978868			0.489434	−	0.872040i	$-0.337203\pi$
$499$	2.33308e32			0.978868			0.489434	+	0.872040i	$0.337203\pi$
$500$		0			0
$501$	2.67854e32	+	1.99987e32i	1.07114	+	0.799741i
$502$		0			0
$503$			3.12030e32i			1.18954i	0.803895	+	0.594772i	$0.202758\pi$
$503$			3.12030e32i			1.18954i	−0.803895	+	0.594772i	$0.797242\pi$
$504$		0			0
$505$	1.26166e32			0.458611
$506$		0			0
$507$	5.41182e31	−	7.24834e31i	0.187606	−	0.251271i
$508$		0			0
$509$			7.26199e31i			0.240128i	0.992766	+	0.120064i	$0.0383099\pi$
$509$			7.26199e31i			0.240128i	−0.992766	+	0.120064i	$0.961690\pi$
$510$		0			0
$511$	7.93151e32			2.50210
$512$		0			0
$513$	−4.29927e31	−	1.58508e31i	−0.129416	−	0.0477136i
$514$		0			0
$515$			8.31569e31i			0.238898i
$516$		0			0
$517$	1.51960e32			0.416719
$518$		0			0
$519$	−7.88429e31	−	5.88663e31i	−0.206421	−	0.154120i
$520$		0			0
$521$		−	6.64395e32i		−	1.66102i	−0.557007	−	0.830508i	$-0.688050\pi$
$521$		−	6.64395e32i		−	1.66102i	0.557007	−	0.830508i	$-0.311950\pi$
$522$		0			0
$523$	5.25543e31			0.125484			0.0627420	−	0.998030i	$-0.480015\pi$
$523$	5.25543e31			0.125484			0.0627420	+	0.998030i	$0.480015\pi$
$524$		0			0
$525$	4.22649e32	−	5.66076e32i	0.963980	−	1.29111i
$526$		0			0
$527$			7.16138e32i			1.56052i
$528$		0			0
$529$	4.79664e32			0.998778
$530$		0			0
$531$	−5.09332e32	+	1.50947e32i	−1.01360	+	0.300393i
$532$		0			0
$533$			8.29845e32i			1.57859i
$534$		0			0
$535$	2.40622e32			0.437613
$536$		0			0
$537$	−6.28983e32	−	4.69616e32i	−1.09382	−	0.816680i
$538$		0			0
$539$		−	1.56844e33i		−	2.60856i
$540$		0			0
$541$	−1.00798e32			−0.160356			−0.0801778	−	0.996781i	$-0.525549\pi$
$541$	−1.00798e32			−0.160356			−0.0801778	+	0.996781i	$0.525549\pi$
$542$		0			0
$543$	−2.11581e32	+	2.83381e32i	−0.322016	+	0.431294i
$544$		0			0
$545$		−	1.47940e31i		−	0.0215440i
$546$		0			0
$547$	−2.11119e32			−0.294225			−0.147113	−	0.989120i	$-0.546998\pi$
$547$	−2.11119e32			−0.294225			−0.147113	+	0.989120i	$0.546998\pi$
$548$		0			0
$549$	1.54275e32	+	5.20562e32i	0.205792	+	0.694391i
$550$		0			0
$551$		−	1.22172e32i		−	0.156010i
$552$		0			0
$553$	−1.23980e32			−0.151583
$554$		0			0
$555$	−7.71020e31	−	5.75666e31i	−0.0902714	−	0.0673992i
$556$		0			0
$557$		−	1.33948e32i		−	0.150201i	−0.997176	−	0.0751005i	$-0.976072\pi$
$557$		−	1.33948e32i		−	0.150201i	0.997176	−	0.0751005i	$-0.0239278\pi$
$558$		0			0
$559$	6.50676e32			0.698913
$560$		0			0
$561$	−8.39490e32	+	1.12437e33i	−0.863896	+	1.15706i
$562$		0			0
$563$		−	6.33412e32i		−	0.624576i	−0.949987	−	0.312288i	$-0.898905\pi$
$563$		−	6.33412e32i		−	0.624576i	0.949987	−	0.312288i	$-0.101095\pi$
$564$		0			0
$565$	3.83364e32			0.362268
$566$		0			0
$567$	−1.61593e33	+	1.05003e33i	−1.46362	+	0.951055i
$568$		0			0
$569$			3.30678e32i			0.287116i	0.989642	+	0.143558i	$0.0458544\pi$
$569$			3.30678e32i			0.287116i	−0.989642	+	0.143558i	$0.954146\pi$
$570$		0			0
$571$	1.57897e33			1.31444			0.657218	−	0.753700i	$-0.271733\pi$
$571$	1.57897e33			1.31444			0.657218	+	0.753700i	$0.271733\pi$
$572$		0			0
$573$	−1.17724e33	−	8.78964e32i	−0.939747	−	0.701642i
$574$		0			0
$575$			4.21507e31i			0.0322694i
$576$		0			0
$577$	2.68744e33			1.97346			0.986730	−	0.162371i	$-0.0519140\pi$
$577$	2.68744e33			1.97346			0.986730	+	0.162371i	$0.0519140\pi$
$578$		0			0
$579$	2.24691e32	−	3.00941e32i	0.158286	−	0.212001i
$580$		0			0
$581$		−	2.77745e31i		−	0.0187729i
$582$		0			0
$583$	−3.53043e32			−0.228984
$584$		0			0
$585$	−4.89471e32	+	1.45061e32i	−0.304689	+	0.0902986i
$586$		0			0
$587$			1.59247e33i			0.951510i	0.879578	+	0.475755i	$0.157825\pi$
$587$			1.59247e33i			0.951510i	−0.879578	+	0.475755i	$0.842175\pi$
$588$		0			0
$589$	−3.31210e32			−0.189985
$590$		0			0
$591$	−2.00853e33	−	1.49963e33i	−1.10619	−	0.825911i
$592$		0			0
$593$		−	2.89338e33i		−	1.53020i	−0.643910	−	0.765102i	$-0.722689\pi$
$593$		−	2.89338e33i		−	1.53020i	0.643910	−	0.765102i	$-0.277311\pi$
$594$		0			0
$595$	−1.07954e33			−0.548320
$596$		0			0
$597$	−7.17614e32	+	9.61139e32i	−0.350105	+	0.468915i
$598$		0			0
$599$			1.27146e33i			0.595911i	0.954580	+	0.297955i	$0.0963047\pi$
$599$			1.27146e33i			0.595911i	−0.954580	+	0.297955i	$0.903695\pi$
$600$		0			0
$601$	−1.15108e33			−0.518337			−0.259169	−	0.965832i	$-0.583449\pi$
$601$	−1.15108e33			−0.518337			−0.259169	+	0.965832i	$0.583449\pi$
$602$		0			0
$603$	−6.38990e32	−	2.15611e33i	−0.276494	−	0.932959i
$604$		0			0
$605$			4.16358e32i			0.173142i
$606$		0			0
$607$	−4.18715e32			−0.167361			−0.0836805	−	0.996493i	$-0.526668\pi$
$607$	−4.18715e32			−0.167361			−0.0836805	+	0.996493i	$0.526668\pi$
$608$		0			0
$609$	−4.11722e33	−	3.07404e33i	−1.58196	−	1.18114i
$610$		0			0
$611$			1.01441e33i			0.374731i
$612$		0			0
$613$	−3.03008e33			−1.07629			−0.538144	−	0.842853i	$-0.680874\pi$
$613$	−3.03008e33			−1.07629			−0.538144	+	0.842853i	$0.680874\pi$
$614$		0			0
$615$	6.68882e32	−	8.95870e32i	0.228480	−	0.306015i
$616$		0			0
$617$		−	3.78856e33i		−	1.24466i	−0.782753	−	0.622332i	$-0.786185\pi$
$617$		−	3.78856e33i		−	1.24466i	0.782753	−	0.622332i	$-0.213815\pi$
$618$		0			0
$619$	2.01070e33			0.635418			0.317709	−	0.948188i	$-0.397087\pi$
$619$	2.01070e33			0.635418			0.317709	+	0.948188i	$0.397087\pi$
$620$		0			0
$621$	3.97752e31	−	1.07884e32i	0.0120924	−	0.0327987i
$622$		0			0
$623$		−	8.93559e33i		−	2.61377i
$624$		0			0
$625$	2.75431e33			0.775268
$626$		0			0
$627$	−5.20017e32	−	3.88260e32i	−0.140866	−	0.105175i
$628$		0			0
$629$		−	1.76550e33i		−	0.460321i
$630$		0			0
$631$	−2.99733e33			−0.752284			−0.376142	−	0.926562i	$-0.622750\pi$
$631$	−2.99733e33			−0.752284			−0.376142	+	0.926562i	$0.622750\pi$
$632$		0			0
$633$	−1.07436e33	+	1.43895e33i	−0.259600	+	0.347696i
$634$		0			0
$635$			8.92699e32i			0.207692i
$636$		0			0
$637$	1.04701e34			2.34573
$638$		0			0
$639$	5.53292e33	−	1.63975e33i	1.19383	−	0.353808i
$640$		0			0
$641$			6.59215e33i			1.37003i	0.728528	+	0.685016i	$0.240205\pi$
$641$			6.59215e33i			1.37003i	−0.728528	+	0.685016i	$0.759795\pi$
$642$		0			0
$643$	−1.70251e32			−0.0340845			−0.0170423	−	0.999855i	$-0.505425\pi$
$643$	−1.70251e32			−0.0340845			−0.0170423	+	0.999855i	$0.505425\pi$
$644$		0			0
$645$	−7.02445e32	−	5.24466e32i	−0.135486	−	0.101158i
$646$		0			0
$647$			2.44965e33i			0.455253i	0.973749	+	0.227626i	$0.0730965\pi$
$647$			2.44965e33i			0.455253i	−0.973749	+	0.227626i	$0.926904\pi$
$648$		0			0
$649$	−7.52378e33			−1.34741
$650$		0			0
$651$	−8.33373e33	+	1.11618e34i	−1.43836	+	1.92647i
$652$		0			0
$653$		−	1.61675e32i		−	0.0268958i	−0.999910	−	0.0134479i	$-0.995719\pi$
$653$		−	1.61675e32i		−	0.0268958i	0.999910	−	0.0134479i	$-0.00428073\pi$
$654$		0			0
$655$	1.84688e33			0.296171
$656$		0			0
$657$	−2.63463e33	−	8.88987e33i	−0.407319	−	1.37439i
$658$		0			0
$659$		−	1.23809e34i		−	1.84556i	−0.385333	−	0.922778i	$-0.625913\pi$
$659$		−	1.23809e34i		−	1.84556i	0.385333	−	0.922778i	$-0.374087\pi$
$660$		0			0
$661$	1.28758e34			1.85079			0.925395	−	0.379003i	$-0.123733\pi$
$661$	1.28758e34			1.85079			0.925395	+	0.379003i	$0.123733\pi$
$662$		0			0
$663$	−7.50577e33	−	5.60402e33i	−1.04048	−	0.776850i
$664$		0			0
$665$		−	4.99280e32i		−	0.0667551i
$666$		0			0
$667$	3.06573e32			0.0395388
$668$		0			0
$669$	−4.33460e33	+	5.80556e33i	−0.539306	+	0.722322i
$670$		0			0
$671$			7.68967e33i			0.923076i
$672$		0			0
$673$	−4.97150e33			−0.575847			−0.287923	−	0.957653i	$-0.592965\pi$
$673$	−4.97150e33			−0.575847			−0.287923	+	0.957653i	$0.592965\pi$
$674$		0			0
$675$	−7.74868e33	−	2.85682e33i	−0.866128	−	0.319329i
$676$		0			0
$677$			3.16889e33i			0.341856i	0.985283	+	0.170928i	$0.0546766\pi$
$677$			3.16889e33i			0.341856i	−0.985283	+	0.170928i	$0.945323\pi$
$678$		0			0
$679$	2.67735e34			2.78784
$680$		0			0
$681$	2.90435e33	+	2.16847e33i	0.291933	+	0.217966i
$682$		0			0
$683$			6.57986e33i			0.638510i	0.947669	+	0.319255i	$0.103433\pi$
$683$			6.57986e33i			0.638510i	−0.947669	+	0.319255i	$0.896567\pi$
$684$		0			0
$685$	−1.27286e33			−0.119260
$686$		0			0
$687$	−9.23436e33	+	1.23681e34i	−0.835461	+	1.11898i
$688$		0			0
$689$		−	2.35674e33i		−	0.205911i
$690$		0			0
$691$	2.28870e34			1.93131			0.965657	−	0.259819i	$-0.0836629\pi$
$691$	2.28870e34			1.93131			0.965657	+	0.259819i	$0.0836629\pi$
$692$		0			0
$693$	−2.61688e34	+	7.75546e33i	−2.13297	+	0.632133i
$694$		0			0
$695$			2.81818e32i			0.0221897i
$696$		0			0
$697$	2.05139e34			1.56046
$698$		0			0
$699$	−3.70154e33	−	2.76368e33i	−0.272055	−	0.203124i
$700$		0			0
$701$			2.42814e34i			1.72448i	0.506504	+	0.862238i	$0.330938\pi$
$701$			2.42814e34i			1.72448i	−0.506504	+	0.862238i	$0.669062\pi$
$702$		0			0
$703$	8.16535e32			0.0560417
$704$		0			0
$705$	8.17647e32	−	1.09512e33i	0.0542371	−	0.0726427i
$706$		0			0
$707$			4.50279e34i			2.88701i
$708$		0			0
$709$	3.13216e34			1.94129			0.970643	−	0.240526i	$-0.0773201\pi$
$709$	3.13216e34			1.94129			0.970643	+	0.240526i	$0.0773201\pi$
$710$		0			0
$711$	4.11827e32	+	1.38960e33i	0.0246763	+	0.0832636i
$712$		0			0
$713$		−	8.31121e32i		−	0.0481493i
$714$		0			0
$715$	−7.23040e33			−0.405033
$716$		0			0
$717$	1.64621e34	+	1.22911e34i	0.891775	+	0.665825i
$718$		0			0
$719$			6.69265e33i			0.350632i	0.984512	+	0.175316i	$0.0560947\pi$
$719$			6.69265e33i			0.350632i	−0.984512	+	0.175316i	$0.943905\pi$
$720$		0			0
$721$	−2.96783e34			−1.50389
$722$		0			0
$723$	3.83458e33	−	5.13586e33i	0.187957	−	0.251741i
$724$		0			0
$725$		−	2.20194e34i		−	1.04412i
$726$		0			0
$727$	3.17481e32			0.0145648			0.00728241	−	0.999973i	$-0.497682\pi$
$727$	3.17481e32			0.0145648			0.00728241	+	0.999973i	$0.497682\pi$
$728$		0			0
$729$	1.71368e34	+	1.46240e34i	0.760674	+	0.649135i
$730$		0			0
$731$		−	1.60848e34i		−	0.690886i
$732$		0			0
$733$	9.81929e33			0.408162			0.204081	−	0.978954i	$-0.434579\pi$
$733$	9.81929e33			0.408162			0.204081	+	0.978954i	$0.434579\pi$
$734$		0			0
$735$	−1.13031e34	−	8.43923e33i	−0.454727	−	0.339512i
$736$		0			0
$737$		−	3.18497e34i		−	1.24021i
$738$		0			0
$739$	9.87359e33			0.372171			0.186085	−	0.982534i	$-0.440420\pi$
$739$	9.87359e33			0.372171			0.186085	+	0.982534i	$0.440420\pi$
$740$		0			0
$741$	2.59183e33	−	3.47138e33i	0.0945775	−	0.126673i
$742$		0			0
$743$		−	4.57424e34i		−	1.61604i	−0.589153	−	0.808022i	$-0.700538\pi$
$743$		−	4.57424e34i		−	1.61604i	0.589153	−	0.808022i	$-0.299462\pi$
$744$		0			0
$745$	−6.21009e33			−0.212433
$746$		0			0
$747$	−3.11305e32	+	9.22592e31i	−0.0103119	+	0.00305606i
$748$		0			0
$749$			8.58768e34i			2.75483i
$750$		0			0
$751$	2.95235e34			0.917253			0.458626	−	0.888629i	$-0.348342\pi$
$751$	2.95235e34			0.917253			0.458626	+	0.888629i	$0.348342\pi$
$752$		0			0
$753$	−3.59067e34	−	2.68089e34i	−1.08053	−	0.806752i
$754$		0			0
$755$			1.26035e34i			0.367391i
$756$		0			0
$757$	2.19612e34			0.620163			0.310082	−	0.950710i	$-0.399644\pi$
$757$	2.19612e34			0.620163			0.310082	+	0.950710i	$0.399644\pi$
$758$		0			0
$759$	9.74278e32	−	1.30490e33i	0.0266552	−	0.0357007i
$760$		0			0
$761$		−	4.02417e34i		−	1.06675i	−0.845880	−	0.533373i	$-0.820924\pi$
$761$		−	4.02417e34i		−	1.06675i	0.845880	−	0.533373i	$-0.179076\pi$
$762$		0			0
$763$	5.27990e33			0.135622
$764$		0			0
$765$	3.58593e33	+	1.20998e34i	0.0892615	+	0.301190i
$766$		0			0
$767$		−	5.02250e34i		−	1.21165i
$768$		0			0
$769$	−1.32227e34			−0.309175			−0.154587	−	0.987979i	$-0.549405\pi$
$769$	−1.32227e34			−0.309175			−0.154587	+	0.987979i	$0.549405\pi$
$770$		0			0
$771$	2.47872e34	+	1.85069e34i	0.561792	+	0.419450i
$772$		0			0
$773$			2.79581e33i			0.0614261i	0.999528	+	0.0307131i	$0.00977781\pi$
$773$			2.79581e33i			0.0614261i	−0.999528	+	0.0307131i	$0.990222\pi$
$774$		0			0
$775$	−5.96947e34			−1.27150
$776$		0			0
$777$	2.05452e34	−	2.75173e34i	0.424286	−	0.568269i
$778$		0			0
$779$			9.48754e33i			0.189978i
$780$		0			0
$781$	8.17315e34			1.58700
$782$		0			0
$783$	−2.07784e34	+	5.63582e34i	−0.391266	+	1.06125i
$784$		0			0
$785$		−	1.07581e34i		−	0.196472i
$786$		0			0
$787$	1.02458e33			0.0181489			0.00907443	−	0.999959i	$-0.497111\pi$
$787$	1.02458e33			0.0181489			0.00907443	+	0.999959i	$0.497111\pi$
$788$		0			0
$789$	−5.69668e34	−	4.25331e34i	−0.978809	−	0.730807i
$790$		0			0
$791$			1.36821e35i			2.28052i
$792$		0			0
$793$	−5.13324e34			−0.830068
$794$		0			0
$795$	−1.89961e33	+	2.54425e33i	−0.0298029	+	0.0399166i
$796$		0			0
$797$			1.28416e34i			0.195487i	0.995212	+	0.0977437i	$0.0311625\pi$
$797$			1.28416e34i			0.195487i	−0.995212	+	0.0977437i	$0.968837\pi$
$798$		0			0
$799$	2.50763e34			0.370427
$800$		0			0
$801$	−1.00153e35	+	2.96816e34i	−1.43573	+	0.425497i
$802$		0			0
$803$		−	1.31320e35i		−	1.82702i
$804$		0			0
$805$	1.25287e33			0.0169182
$806$		0			0
$807$	−1.83888e34	−	1.37296e34i	−0.241030	−	0.179960i
$808$		0			0
$809$		−	2.52757e34i		−	0.321604i	−0.986987	−	0.160802i	$-0.948592\pi$
$809$		−	2.52757e34i		−	0.321604i	0.986987	−	0.160802i	$-0.0514081\pi$
$810$		0			0
$811$	−6.11893e34			−0.755832			−0.377916	−	0.925840i	$-0.623359\pi$
$811$	−6.11893e34			−0.755832			−0.377916	+	0.925840i	$0.623359\pi$
$812$		0			0
$813$	−4.00149e33	+	5.35942e33i	−0.0479883	+	0.0642734i
$814$		0			0
$815$			3.29536e34i			0.383718i
$816$		0			0
$817$	7.43912e33			0.0841118
$818$		0			0
$819$	−5.17716e34	−	1.74690e35i	−0.568440	−	1.91805i
$820$		0			0
$821$			1.06023e35i			1.13053i	0.824910	+	0.565264i	$0.191226\pi$
$821$			1.06023e35i			1.13053i	−0.824910	+	0.565264i	$0.808774\pi$
$822$		0			0
$823$	−2.36678e34			−0.245110			−0.122555	−	0.992462i	$-0.539109\pi$
$823$	−2.36678e34			−0.245110			−0.122555	+	0.992462i	$0.539109\pi$
$824$		0			0
$825$	−9.37238e34	−	6.99769e34i	−0.942763	−	0.703894i
$826$		0			0
$827$		−	8.95156e34i		−	0.874647i	−0.899304	−	0.437324i	$-0.855926\pi$
$827$		−	8.95156e34i		−	0.874647i	0.899304	−	0.437324i	$-0.144074\pi$
$828$		0			0
$829$	−9.98135e34			−0.947404			−0.473702	−	0.880685i	$-0.657083\pi$
$829$	−9.98135e34			−0.947404			−0.473702	+	0.880685i	$0.657083\pi$
$830$		0			0
$831$	2.91703e34	−	3.90694e34i	0.268985	−	0.360267i
$832$		0			0
$833$		−	2.58822e35i		−	2.31879i
$834$		0			0
$835$	4.25794e34			0.370648
$836$		0			0
$837$	1.52787e35	+	5.63304e34i	1.29235	+	0.476472i
$838$		0			0
$839$			6.41910e34i			0.527631i	0.964573	+	0.263816i	$0.0849811\pi$
$839$			6.41910e34i			0.527631i	−0.964573	+	0.263816i	$0.915019\pi$
$840$		0			0
$841$	−3.49679e34			−0.279330
$842$		0			0
$843$	−9.42446e33	−	7.03657e33i	−0.0731687	−	0.0546299i
$844$		0			0
$845$		−	1.15223e34i		−	0.0869480i
$846$		0			0
$847$	−1.48596e35			−1.08995
$848$		0			0
$849$	−1.03832e34	+	1.39068e34i	−0.0740355	+	0.0991598i
$850$		0			0
$851$			2.04897e33i			0.0142030i
$852$		0			0
$853$	1.28735e35			0.867577			0.433788	−	0.901015i	$-0.357177\pi$
$853$	1.28735e35			0.867577			0.433788	+	0.901015i	$0.357177\pi$
$854$		0			0
$855$	−5.59608e33	+	1.65847e33i	−0.0366683	+	0.0108671i
$856$		0			0
$857$		−	1.19549e35i		−	0.761686i	−0.924640	−	0.380843i	$-0.875634\pi$
$857$		−	1.19549e35i		−	0.761686i	0.924640	−	0.380843i	$-0.124366\pi$
$858$		0			0
$859$	−3.64373e33			−0.0225750			−0.0112875	−	0.999936i	$-0.503593\pi$
$859$	−3.64373e33			−0.0225750			−0.0112875	+	0.999936i	$0.503593\pi$
$860$		0			0
$861$	3.19731e35	+	2.38721e35i	1.92640	+	1.43831i
$862$		0			0
$863$		−	1.13114e35i		−	0.662808i	−0.943489	−	0.331404i	$-0.892478\pi$
$863$		−	1.13114e35i		−	0.662808i	0.943489	−	0.331404i	$-0.107522\pi$
$864$		0			0
$865$	−1.25333e34			−0.0714284
$866$		0			0
$867$	−3.06060e34	+	4.09923e34i	−0.169659	+	0.227234i
$868$		0			0
$869$			2.05270e34i			0.110685i
$870$		0			0
$871$	2.12613e35			1.11525
$872$		0			0
$873$	−8.89342e34	−	3.00085e35i	−0.453835	−	1.53135i
$874$		0			0
$875$		−	1.87467e35i		−	0.930740i
$876$		0			0
$877$	−2.74651e35			−1.32674			−0.663371	−	0.748291i	$-0.730875\pi$
$877$	−2.74651e35			−1.32674			−0.663371	+	0.748291i	$0.730875\pi$
$878$		0			0
$879$	1.43532e35	+	1.07165e35i	0.674657	+	0.503718i
$880$		0			0
$881$			3.92641e33i			0.0179591i	0.999960	+	0.00897954i	$0.00285832\pi$
$881$			3.92641e33i			0.0179591i	−0.999960	+	0.00897954i	$0.997142\pi$
$882$		0			0
$883$	−3.14107e35			−1.39814			−0.699068	−	0.715055i	$-0.746402\pi$
$883$	−3.14107e35			−1.39814			−0.699068	+	0.715055i	$0.746402\pi$
$884$		0			0
$885$	−4.04830e34	+	5.42210e34i	−0.175369	+	0.234881i
$886$		0			0
$887$		−	1.36842e35i		−	0.576949i	−0.957488	−	0.288474i	$-0.906852\pi$
$887$		−	1.36842e35i		−	0.576949i	0.957488	−	0.288474i	$-0.0931480\pi$
$888$		0			0
$889$	−3.18600e35			−1.30745
$890$		0			0
$891$	1.73851e35	+	2.67546e35i	0.694456	+	1.06873i
$892$		0			0
$893$			1.15977e34i			0.0450975i
$894$		0			0
$895$	−9.99863e34			−0.378498
$896$		0			0
$897$	8.71089e33	+	6.50380e33i	0.0321036	+	0.0239694i
$898$		0			0
$899$			4.34175e35i			1.55793i
$900$		0			0
$901$	−5.82588e34			−0.203547
$902$		0			0
$903$	1.87179e35	−	2.50699e35i	0.636802	−	0.852904i
$904$		0			0
$905$			4.50477e34i			0.149242i
$906$		0			0
$907$	4.89113e35			1.57805			0.789027	−	0.614359i	$-0.210585\pi$
$907$	4.89113e35			1.57805			0.789027	+	0.614359i	$0.210585\pi$
$908$		0			0
$909$	5.04686e35	−	1.49570e35i	1.58582	−	0.469979i
$910$		0			0
$911$		−	1.37776e35i		−	0.421652i	−0.977524	−	0.210826i	$-0.932385\pi$
$911$		−	1.37776e35i		−	0.421652i	0.977524	−	0.210826i	$-0.0676153\pi$
$912$		0			0
$913$	−4.59855e33			−0.0137079
$914$		0			0
$915$	5.54166e34	+	4.13756e34i	0.160911	+	0.120141i
$916$		0			0
$917$			6.59142e35i			1.86443i
$918$		0			0
$919$	−5.77554e35			−1.59150			−0.795749	−	0.605626i	$-0.792923\pi$
$919$	−5.77554e35			−1.59150			−0.795749	+	0.605626i	$0.792923\pi$
$920$		0			0
$921$	2.09348e35	−	2.80391e35i	0.562022	−	0.752746i
$922$		0			0
$923$			5.45599e35i			1.42710i
$924$		0			0
$925$	1.47166e35			0.375065
$926$		0			0
$927$	9.85832e34	+	3.32643e35i	0.244820	+	0.826080i
$928$		0			0
$929$			4.96289e35i			1.20101i	0.799621	+	0.600504i	$0.205033\pi$
$929$			4.96289e35i			1.20101i	−0.799621	+	0.600504i	$0.794967\pi$
$930$		0			0
$931$	1.19704e35			0.282300
$932$		0			0
$933$	5.35295e35	+	3.99667e35i	1.23031	+	0.918583i
$934$		0			0
$935$			1.78736e35i			0.400381i
$936$		0			0
$937$	−6.65386e35			−1.45278			−0.726388	−	0.687284i	$-0.758803\pi$
$937$	−6.65386e35			−1.45278			−0.726388	+	0.687284i	$0.758803\pi$
$938$		0			0
$939$	−1.27718e35	+	1.71060e35i	−0.271810	+	0.364050i
$940$		0			0
$941$			5.63970e35i			1.16999i	0.811038	+	0.584993i	$0.198903\pi$
$941$			5.63970e35i			1.16999i	−0.811038	+	0.584993i	$0.801097\pi$
$942$		0			0
$943$	−2.38076e34			−0.0481476
$944$		0			0
$945$	−8.49149e34	+	2.30318e35i	−0.167418	+	0.454095i
$946$		0			0
$947$			7.06990e35i			1.35898i	0.733684	+	0.679491i	$0.237799\pi$
$947$			7.06990e35i			1.35898i	−0.733684	+	0.679491i	$0.762201\pi$
$948$		0			0
$949$	8.76627e35			1.64294
$950$		0			0
$951$	−2.05836e35	−	1.53683e35i	−0.376146	−	0.280841i
$952$		0			0
$953$		−	1.01676e36i		−	1.81177i	−0.423527	−	0.905883i	$-0.639208\pi$
$953$		−	1.01676e36i		−	1.81177i	0.423527	−	0.905883i	$-0.360792\pi$
$954$		0			0
$955$	−1.87141e35			−0.325183
$956$		0			0
$957$	−5.08960e35	+	6.81678e35i	−0.862463	+	1.15514i
$958$		0			0
$959$		−	4.54278e35i		−	0.750755i
$960$		0			0
$961$	5.56638e35			0.897207
$962$		0			0
$963$	9.62533e35	−	2.85259e35i	1.51321	−	0.448461i
$964$		0			0
$965$		−	4.78391e34i		−	0.0733593i
$966$		0			0
$967$	9.91788e35			1.48355			0.741773	−	0.670651i	$-0.233985\pi$
$967$	9.91788e35			1.48355			0.741773	+	0.670651i	$0.233985\pi$
$968$		0			0
$969$	−8.58128e34	−	6.40703e34i	−0.125218	−	0.0934913i
$970$		0			0
$971$		−	1.14482e36i		−	1.62970i	−0.579675	−	0.814848i	$-0.696820\pi$
$971$		−	1.14482e36i		−	1.62970i	0.579675	−	0.814848i	$-0.303180\pi$
$972$		0			0
$973$	−1.00579e35			−0.139687
$974$		0			0
$975$	4.67131e35	−	6.25654e35i	0.632970	−	0.847771i
$976$		0			0
$977$		−	1.52947e35i		−	0.202212i	−0.994876	−	0.101106i	$-0.967762\pi$
$977$		−	1.52947e35i		−	0.202212i	0.994876	−	0.101106i	$-0.0322380\pi$
$978$		0			0
$979$	−1.47944e36			−1.90856
$980$		0			0
$981$	−1.75384e34	−	5.91787e34i	−0.0220781	−	0.0744967i
$982$		0			0
$983$			8.83352e35i			1.08515i	0.840006	+	0.542577i	$0.182552\pi$
$983$			8.83352e35i			1.08515i	−0.840006	+	0.542577i	$0.817448\pi$
$984$		0			0
$985$	−3.19286e35			−0.382777
$986$		0			0
$987$	3.90843e35	+	2.91814e35i	0.457295	+	0.341429i
$988$		0			0
$989$			1.86673e34i			0.0213171i
$990$		0			0
$991$	3.62104e35			0.403598			0.201799	−	0.979427i	$-0.435321\pi$
$991$	3.62104e35			0.403598			0.201799	+	0.979427i	$0.435321\pi$
$992$		0			0
$993$	4.25518e34	−	5.69919e34i	0.0462942	−	0.0620043i
$994$		0			0
$995$			1.52788e35i			0.162260i
$996$		0			0
$997$	1.31697e35			0.136532			0.0682662	−	0.997667i	$-0.478253\pi$
$997$	1.31697e35			0.136532			0.0682662	+	0.997667i	$0.478253\pi$
$998$		0			0
$999$	−3.76668e35	−	1.38872e35i	−0.381218	−	0.140549i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.25.e.d.17.5		8
3.2	odd	2	inner	48.25.e.d.17.6		8
4.3	odd	2		6.25.b.a.5.2	✓	8
12.11	even	2		6.25.b.a.5.6	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6.25.b.a.5.2	✓	8	4.3	odd	2
6.25.b.a.5.6	yes	8	12.11	even	2
48.25.e.d.17.5		8	1.1	even	1	trivial
48.25.e.d.17.6		8	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 25 \)
Character orbit:	\([\chi]\)	\(=\)	48.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(317945. - 425841. i) q^{3} -6.76938e7i q^{5} +2.41596e10 q^{7} +(-8.02515e10 - 2.70788e11i) q^{9} +O(q^{10})\)	\(q+(317945. - 425841. i) q^{3} -6.76938e7i q^{5} +2.41596e10 q^{7} +(-8.02515e10 - 2.70788e11i) q^{9} -4.00004e12i q^{11} +2.67023e13 q^{13} +(-2.88268e13 - 2.15229e13i) q^{15} -6.60084e14i q^{17} +3.05285e14 q^{19} +(7.68142e15 - 1.02881e16i) q^{21} +7.66067e14i q^{23} +5.50222e16 q^{25} +(-1.40828e17 - 5.19213e16i) q^{27} -4.00191e17i q^{29} -1.08492e18 q^{31} +(-1.70338e18 - 1.27179e18i) q^{33} -1.63546e18i q^{35} +2.67466e18 q^{37} +(8.48986e18 - 1.13709e19i) q^{39} +3.10777e19i q^{41} +2.43678e19 q^{43} +(-1.83307e19 + 5.43253e18i) q^{45} +3.79896e19i q^{47} +3.92105e20 q^{49} +(-2.81091e20 - 2.09870e20i) q^{51} -8.82597e19i q^{53} -2.70778e20 q^{55} +(9.70638e19 - 1.30003e20i) q^{57} -1.88092e21i q^{59} -1.92240e21 q^{61} +(-1.93884e21 - 6.54213e21i) q^{63} -1.80758e21i q^{65} +7.96234e21 q^{67} +(3.26223e20 + 2.43567e20i) q^{69} +2.04327e22i q^{71} +3.28296e22 q^{73} +(1.74940e22 - 2.34307e22i) q^{75} -9.66394e22i q^{77} -5.13170e21 q^{79} +(-6.68858e22 + 4.34623e22i) q^{81} -1.14963e21i q^{83} -4.46836e22 q^{85} +(-1.70418e23 - 1.27239e23i) q^{87} -3.69857e23i q^{89} +6.45117e23 q^{91} +(-3.44945e23 + 4.62003e23i) q^{93} -2.06659e22i q^{95} +1.10819e24 q^{97} +(-1.08316e24 + 3.21010e23i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 131880 q^{3} + 10160794640 q^{7} + 295169053896 q^{9}+O(q^{10}) \) 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

Fields

Representations

Groups

Database

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