LMFDB - Embedded newform 48.25.e.d.17.3

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			17.3
Root			$3.80484e7i$ of defining polynomial
Character	$\chi$	$=$	48.17
Dual form			48.25.e.d.17.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-269557. - 458005. i) q^{3} +4.56581e8i q^{5} -1.81935e9 q^{7} +(-1.37107e11 + 2.46917e11i) q^{9} +O(q^{10})$	\(q+(-269557. - 458005. i) q^{3} +4.56581e8i q^{5} -1.81935e9 q^{7} +(-1.37107e11 + 2.46917e11i) q^{9} -3.36073e12i q^{11} +3.10150e13 q^{13} +(2.09116e14 - 1.23075e14i) q^{15} -1.14524e14i q^{17} -9.25261e14 q^{19} +(4.90420e14 + 8.33273e14i) q^{21} +2.51730e16i q^{23} -1.48862e17 q^{25} +(1.50047e17 - 3.76256e15i) q^{27} -5.38831e17i q^{29} -4.67868e17 q^{31} +(-1.53923e18 + 9.05909e17i) q^{33} -8.30683e17i q^{35} +3.11155e18 q^{37} +(-8.36032e18 - 1.42050e19i) q^{39} +1.15365e19i q^{41} -5.41569e19 q^{43} +(-1.12738e20 - 6.26006e19i) q^{45} -1.41317e20i q^{47} -1.88271e20 q^{49} +(-5.24524e19 + 3.08707e19i) q^{51} +5.85803e20i q^{53} +1.53445e21 q^{55} +(2.49411e20 + 4.23774e20i) q^{57} +1.22231e21i q^{59} +3.16259e21 q^{61} +(2.49447e20 - 4.49230e20i) q^{63} +1.41609e22i q^{65} +2.88702e21 q^{67} +(1.15294e22 - 6.78557e21i) q^{69} +5.86562e21i q^{71} -4.89426e21 q^{73} +(4.01268e22 + 6.81794e22i) q^{75} +6.11435e21i q^{77} -4.71039e22 q^{79} +(-4.21697e22 - 6.77082e22i) q^{81} -1.25799e23i q^{83} +5.22894e22 q^{85} +(-2.46787e23 + 1.45246e23i) q^{87} +4.49583e22i q^{89} -5.64273e22 q^{91} +(1.26117e23 + 2.14286e23i) q^{93} -4.22457e23i q^{95} -2.08004e23 q^{97} +(8.29821e23 + 4.60780e23i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−269557.	−	458005.i	−0.507220	−	0.861817i
$3$	−269557.	−	458005.i	−0.507220	−	0.861817i
$4$		0			0
$5$			4.56581e8i			1.87016i	0.354441	+	0.935078i	$0.384671\pi$
$5$			4.56581e8i			1.87016i	−0.354441	+	0.935078i	$0.615329\pi$
$6$		0			0
$7$	−1.81935e9			−0.131444			−0.0657220	−	0.997838i	$-0.520935\pi$
$7$	−1.81935e9			−0.131444			−0.0657220	+	0.997838i	$0.520935\pi$
$8$		0			0
$9$	−1.37107e11	+	2.46917e11i	−0.485456	+	0.874261i
$10$		0			0
$11$		−	3.36073e12i		−	1.07083i	−0.844589	−	0.535416i	$-0.820155\pi$
$11$		−	3.36073e12i		−	1.07083i	0.844589	−	0.535416i	$-0.179845\pi$
$12$		0			0
$13$	3.10150e13			1.33123			0.665613	−	0.746297i	$-0.268170\pi$
$13$	3.10150e13			1.33123			0.665613	+	0.746297i	$0.268170\pi$
$14$		0			0
$15$	2.09116e14	−	1.23075e14i	1.61173	−	0.948580i
$16$		0			0
$17$		−	1.14524e14i		−	0.196566i	−0.995159	−	0.0982830i	$-0.968665\pi$
$17$		−	1.14524e14i		−	0.196566i	0.995159	−	0.0982830i	$-0.0313350\pi$
$18$		0			0
$19$	−9.25261e14			−0.418043			−0.209022	−	0.977911i	$-0.567028\pi$
$19$	−9.25261e14			−0.418043			−0.209022	+	0.977911i	$0.567028\pi$
$20$		0			0
$21$	4.90420e14	+	8.33273e14i	0.0666710	+	0.113281i
$22$		0			0
$23$			2.51730e16i			1.14869i	0.818615	+	0.574343i	$0.194742\pi$
$23$			2.51730e16i			1.14869i	−0.818615	+	0.574343i	$0.805258\pi$
$24$		0			0
$25$	−1.48862e17			−2.49749
$26$		0			0
$27$	1.50047e17	−	3.76256e15i	0.999686	−	0.0250679i
$28$		0			0
$29$		−	5.38831e17i		−	1.52292i	−0.648213	−	0.761459i	$-0.724483\pi$
$29$		−	5.38831e17i		−	1.52292i	0.648213	−	0.761459i	$-0.275517\pi$
$30$		0			0
$31$	−4.67868e17			−0.593995			−0.296998	−	0.954878i	$-0.595985\pi$
$31$	−4.67868e17			−0.593995			−0.296998	+	0.954878i	$0.595985\pi$
$32$		0			0
$33$	−1.53923e18	+	9.05909e17i	−0.922860	+	0.543147i
$34$		0			0
$35$		−	8.30683e17i		−	0.245821i
$36$		0			0
$37$	3.11155e18			0.472668			0.236334	−	0.971672i	$-0.424054\pi$
$37$	3.11155e18			0.472668			0.236334	+	0.971672i	$0.424054\pi$
$38$		0			0
$39$	−8.36032e18	−	1.42050e19i	−0.675224	−	1.14727i
$40$		0			0
$41$			1.15365e19i			0.511291i	0.966771	+	0.255646i	$0.0822881\pi$
$41$			1.15365e19i			0.511291i	−0.966771	+	0.255646i	$0.917712\pi$
$42$		0			0
$43$	−5.41569e19			−1.35529			−0.677645	−	0.735390i	$-0.736999\pi$
$43$	−5.41569e19			−1.35529			−0.677645	+	0.735390i	$0.736999\pi$
$44$		0			0
$45$	−1.12738e20	−	6.26006e19i	−1.63500	−	0.907879i
$46$		0			0
$47$		−	1.41317e20i		−	1.21624i	−0.793844	−	0.608122i	$-0.791923\pi$
$47$		−	1.41317e20i		−	1.21624i	0.793844	−	0.608122i	$-0.208077\pi$
$48$		0			0
$49$	−1.88271e20			−0.982722
$50$		0			0
$51$	−5.24524e19	+	3.08707e19i	−0.169404	+	0.0997022i
$52$		0			0
$53$			5.85803e20i			1.19245i	0.802816	+	0.596226i	$0.203334\pi$
$53$			5.85803e20i			1.19245i	−0.802816	+	0.596226i	$0.796666\pi$
$54$		0			0
$55$	1.53445e21			2.00262
$56$		0			0
$57$	2.49411e20	+	4.23774e20i	0.212040	+	0.360277i
$58$		0			0
$59$			1.22231e21i			0.686998i	0.939153	+	0.343499i	$0.111612\pi$
$59$			1.22231e21i			0.686998i	−0.939153	+	0.343499i	$0.888388\pi$
$60$		0			0
$61$	3.16259e21			1.19148			0.595738	−	0.803179i	$-0.296860\pi$
$61$	3.16259e21			1.19148			0.595738	+	0.803179i	$0.296860\pi$
$62$		0			0
$63$	2.49447e20	−	4.49230e20i	0.0638103	−	0.114916i
$64$		0			0
$65$			1.41609e22i			2.48960i
$66$		0			0
$67$	2.88702e21			0.352820			0.176410	−	0.984317i	$-0.443552\pi$
$67$	2.88702e21			0.352820			0.176410	+	0.984317i	$0.443552\pi$
$68$		0			0
$69$	1.15294e22	−	6.78557e21i	0.989956	−	0.582636i
$70$		0			0
$71$			5.86562e21i			0.357449i	0.983899	+	0.178724i	$0.0571970\pi$
$71$			5.86562e21i			0.357449i	−0.983899	+	0.178724i	$0.942803\pi$
$72$		0			0
$73$	−4.89426e21			−0.213704			−0.106852	−	0.994275i	$-0.534077\pi$
$73$	−4.89426e21			−0.213704			−0.106852	+	0.994275i	$0.534077\pi$
$74$		0			0
$75$	4.01268e22	+	6.81794e22i	1.26677	+	2.15238i
$76$		0			0
$77$			6.11435e21i			0.140754i
$78$		0			0
$79$	−4.71039e22			−0.797135			−0.398567	−	0.917139i	$-0.630493\pi$
$79$	−4.71039e22			−0.797135			−0.398567	+	0.917139i	$0.630493\pi$
$80$		0			0
$81$	−4.21697e22	−	6.77082e22i	−0.528664	−	0.848831i
$82$		0			0
$83$		−	1.25799e23i		−	1.17690i	−0.808534	−	0.588449i	$-0.799739\pi$
$83$		−	1.25799e23i		−	1.17690i	0.808534	−	0.588449i	$-0.200261\pi$
$84$		0			0
$85$	5.22894e22			0.367609
$86$		0			0
$87$	−2.46787e23	+	1.45246e23i	−1.31248	+	0.772454i
$88$		0			0
$89$			4.49583e22i			0.182024i	0.995850	+	0.0910122i	$0.0290102\pi$
$89$			4.49583e22i			0.182024i	−0.995850	+	0.0910122i	$0.970990\pi$
$90$		0			0
$91$	−5.64273e22			−0.174982
$92$		0			0
$93$	1.26117e23	+	2.14286e23i	0.301286	+	0.511915i
$94$		0			0
$95$		−	4.22457e23i		−	0.781806i
$96$		0			0
$97$	−2.08004e23			−0.299786			−0.149893	−	0.988702i	$-0.547893\pi$
$97$	−2.08004e23			−0.299786			−0.149893	+	0.988702i	$0.547893\pi$
$98$		0			0
$99$	8.29821e23	+	4.60780e23i	0.936186	+	0.519842i
$100$		0			0
$101$		−	4.79708e23i		−	0.425717i	−0.977083	−	0.212858i	$-0.931723\pi$
$101$		−	4.79708e23i		−	0.425717i	0.977083	−	0.212858i	$-0.0682773\pi$
$102$		0			0
$103$	2.07015e24			1.45196			0.725982	−	0.687714i	$-0.241386\pi$
$103$	2.07015e24			1.45196			0.725982	+	0.687714i	$0.241386\pi$
$104$		0			0
$105$	−3.80457e23	+	2.23917e23i	−0.211853	+	0.124685i
$106$		0			0
$107$		−	2.26189e24i		−	1.00430i	−0.864779	−	0.502152i	$-0.832542\pi$
$107$		−	2.26189e24i		−	1.00430i	0.864779	−	0.502152i	$-0.167458\pi$
$108$		0			0
$109$	4.92671e23			0.175162			0.0875808	−	0.996157i	$-0.472086\pi$
$109$	4.92671e23			0.175162			0.0875808	+	0.996157i	$0.472086\pi$
$110$		0			0
$111$	−8.38741e23	−	1.42511e24i	−0.239747	−	0.407353i
$112$		0			0
$113$		−	3.60667e24i		−	0.832080i	−0.909346	−	0.416040i	$-0.863418\pi$
$113$		−	3.60667e24i		−	0.832080i	0.909346	−	0.416040i	$-0.136582\pi$
$114$		0			0
$115$	−1.14935e25			−2.14822
$116$		0			0
$117$	−4.25238e24	+	7.65813e24i	−0.646252	+	1.16384i
$118$		0			0
$119$			2.08359e23i			0.0258374i
$120$		0			0
$121$	−1.44476e24			−0.146680
$122$		0			0
$123$	5.28378e24	−	3.10975e24i	0.440640	−	0.259337i
$124$		0			0
$125$		−	4.07531e25i		−	2.80053i
$126$		0			0
$127$	−1.55723e24			−0.0884519			−0.0442259	−	0.999022i	$-0.514082\pi$
$127$	−1.55723e24			−0.0884519			−0.0442259	+	0.999022i	$0.514082\pi$
$128$		0			0
$129$	1.45984e25	+	2.48041e25i	0.687429	+	1.16801i
$130$		0			0
$131$			3.80638e24i			0.149024i	0.997220	+	0.0745121i	$0.0237399\pi$
$131$			3.80638e24i			0.149024i	−0.997220	+	0.0745121i	$0.976260\pi$
$132$		0			0
$133$	1.68338e24			0.0549493
$134$		0			0
$135$	1.71792e24	+	6.85089e25i	0.0468810	+	1.86957i
$136$		0			0
$137$			2.25473e25i			0.515760i	0.966177	+	0.257880i	$0.0830239\pi$
$137$			2.25473e25i			0.515760i	−0.966177	+	0.257880i	$0.916976\pi$
$138$		0			0
$139$	7.62088e25			1.46497			0.732483	−	0.680785i	$-0.238361\pi$
$139$	7.62088e25			1.46497			0.732483	+	0.680785i	$0.238361\pi$
$140$		0			0
$141$	−6.47239e25	+	3.80931e25i	−1.04818	+	0.616903i
$142$		0			0
$143$		−	1.04233e26i		−	1.42552i
$144$		0			0
$145$	2.46020e26			2.84810
$146$		0			0
$147$	5.07499e25	+	8.62291e25i	0.498456	+	0.846927i
$148$		0			0
$149$			2.27765e25i			0.190218i	0.995467	+	0.0951089i	$0.0303199\pi$
$149$			2.27765e25i			0.190218i	−0.995467	+	0.0951089i	$0.969680\pi$
$150$		0			0
$151$	−1.16110e26			−0.826314			−0.413157	−	0.910660i	$-0.635574\pi$
$151$	−1.16110e26			−0.826314			−0.413157	+	0.910660i	$0.635574\pi$
$152$		0			0
$153$	2.82779e25	+	1.57020e25i	0.171850	+	0.0954242i
$154$		0			0
$155$		−	2.13620e26i		−	1.11086i
$156$		0			0
$157$	−3.85801e26			−1.72015			−0.860077	−	0.510164i	$-0.829585\pi$
$157$	−3.85801e26			−1.72015			−0.860077	+	0.510164i	$0.829585\pi$
$158$		0			0
$159$	2.68301e26	−	1.57908e26i	1.02768	−	0.604836i
$160$		0			0
$161$		−	4.57986e25i		−	0.150988i
$162$		0			0
$163$	−3.94400e26			−1.12121			−0.560604	−	0.828084i	$-0.689431\pi$
$163$	−3.94400e26			−1.12121			−0.560604	+	0.828084i	$0.689431\pi$
$164$		0			0
$165$	−4.13621e26	−	7.02783e26i	−1.01577	−	1.72589i
$166$		0			0
$167$		−	2.99402e26i		−	0.636292i	−0.948042	−	0.318146i	$-0.896940\pi$
$167$		−	2.99402e26i		−	0.636292i	0.948042	−	0.318146i	$-0.103060\pi$
$168$		0			0
$169$	4.19130e26			0.772161
$170$		0			0
$171$	1.26860e26	−	2.28463e26i	0.202942	−	0.365479i
$172$		0			0
$173$		−	1.10874e27i		−	1.54268i	−0.636424	−	0.771339i	$-0.719587\pi$
$173$		−	1.10874e27i		−	1.54268i	0.636424	−	0.771339i	$-0.280413\pi$
$174$		0			0
$175$	2.70832e26			0.328280
$176$		0			0
$177$	5.59822e26	−	3.29481e26i	0.592067	−	0.348459i
$178$		0			0
$179$			2.47530e26i			0.228766i	0.993437	+	0.114383i	$0.0364891\pi$
$179$			2.47530e26i			0.228766i	−0.993437	+	0.114383i	$0.963511\pi$
$180$		0			0
$181$	9.06907e26			0.733533			0.366767	−	0.930313i	$-0.380465\pi$
$181$	9.06907e26			0.733533			0.366767	+	0.930313i	$0.380465\pi$
$182$		0			0
$183$	−8.52500e26	−	1.44848e27i	−0.604340	−	1.02683i
$184$		0			0
$185$			1.42068e27i			0.883963i
$186$		0			0
$187$	−3.84883e26			−0.210489
$188$		0			0
$189$	−2.72990e26	+	6.84543e24i	−0.131403	+	0.00329503i
$190$		0			0
$191$			2.37087e27i			1.00579i	0.864348	+	0.502895i	$0.167732\pi$
$191$			2.37087e27i			1.00579i	−0.864348	+	0.502895i	$0.832268\pi$
$192$		0			0
$193$	5.09004e27			1.90561			0.952804	−	0.303585i	$-0.0981835\pi$
$193$	5.09004e27			1.90561			0.952804	+	0.303585i	$0.0981835\pi$
$194$		0			0
$195$	6.48575e27	−	3.81717e27i	2.14558	−	1.26277i
$196$		0			0
$197$		−	4.72370e27i		−	1.38257i	−0.722582	−	0.691285i	$-0.757045\pi$
$197$		−	4.72370e27i		−	1.38257i	0.722582	−	0.691285i	$-0.242955\pi$
$198$		0			0
$199$	7.31715e27			1.89716			0.948582	−	0.316530i	$-0.102518\pi$
$199$	7.31715e27			1.89716			0.948582	+	0.316530i	$0.102518\pi$
$200$		0			0
$201$	−7.78219e26	−	1.32227e27i	−0.178957	−	0.304066i
$202$		0			0
$203$			9.80325e26i			0.200179i
$204$		0			0
$205$	−5.26736e27			−0.956195
$206$		0			0
$207$	−6.21564e27	−	3.45140e27i	−1.00425	−	0.557636i
$208$		0			0
$209$			3.10955e27i			0.447654i
$210$		0			0
$211$	−5.52494e27			−0.709476			−0.354738	−	0.934966i	$-0.615430\pi$
$211$	−5.52494e27			−0.709476			−0.354738	+	0.934966i	$0.615430\pi$
$212$		0			0
$213$	2.68648e27	−	1.58112e27i	0.308055	−	0.181305i
$214$		0			0
$215$		−	2.47270e28i		−	2.53460i
$216$		0			0
$217$	8.51218e26			0.0780771
$218$		0			0
$219$	1.31928e27	+	2.24160e27i	0.108395	+	0.184174i
$220$		0			0
$221$		−	3.55195e27i		−	0.261674i
$222$		0			0
$223$	2.65651e27			0.175652			0.0878261	−	0.996136i	$-0.472008\pi$
$223$	2.65651e27			0.175652			0.0878261	+	0.996136i	$0.472008\pi$
$224$		0			0
$225$	2.04100e28	−	3.67565e28i	1.21242	−	2.18345i
$226$		0			0
$227$		−	3.79716e27i		−	0.202837i	−0.994844	−	0.101419i	$-0.967662\pi$
$227$		−	3.79716e27i		−	0.202837i	0.994844	−	0.101419i	$-0.0323382\pi$
$228$		0			0
$229$	1.19916e28			0.576570			0.288285	−	0.957545i	$-0.406915\pi$
$229$	1.19916e28			0.576570			0.288285	+	0.957545i	$0.406915\pi$
$230$		0			0
$231$	2.80040e27	−	1.64817e27i	0.121304	−	0.0713934i
$232$		0			0
$233$		−	2.52886e28i		−	0.987766i	−0.869528	−	0.493883i	$-0.835577\pi$
$233$		−	2.52886e28i		−	0.987766i	0.869528	−	0.493883i	$-0.164423\pi$
$234$		0			0
$235$	6.45227e28			2.27457
$236$		0			0
$237$	1.26972e28	+	2.15738e28i	0.404323	+	0.686984i
$238$		0			0
$239$		−	2.31937e28i		−	0.667720i	−0.942623	−	0.333860i	$-0.891649\pi$
$239$		−	2.31937e28i		−	0.667720i	0.942623	−	0.333860i	$-0.108351\pi$
$240$		0			0
$241$	−9.28349e27			−0.241828			−0.120914	−	0.992663i	$-0.538583\pi$
$241$	−9.28349e27			−0.241828			−0.120914	+	0.992663i	$0.538583\pi$
$242$		0			0
$243$	−1.96435e28	+	3.75652e28i	−0.463388	+	0.886156i
$244$		0			0
$245$		−	8.59611e28i		−	1.83785i
$246$		0			0
$247$	−2.86970e28			−0.556510
$248$		0			0
$249$	−5.76164e28	+	3.39100e28i	−1.01427	+	0.596946i
$250$		0			0
$251$		−	1.06381e29i		−	1.70129i	−0.525742	−	0.850644i	$-0.676212\pi$
$251$		−	1.06381e29i		−	1.70129i	0.525742	−	0.850644i	$-0.323788\pi$
$252$		0			0
$253$	8.45996e28			1.23005
$254$		0			0
$255$	−1.40950e28	−	2.39488e28i	−0.186459	−	0.316812i
$256$		0			0
$257$		−	7.43493e28i		−	0.895528i	−0.894152	−	0.447764i	$-0.852220\pi$
$257$		−	7.43493e28i		−	0.895528i	0.894152	−	0.447764i	$-0.147780\pi$
$258$		0			0
$259$	−5.66101e27			−0.0621294
$260$		0			0
$261$	1.33047e29	+	7.38776e28i	1.33143	+	0.739310i
$262$		0			0
$263$		−	1.08384e29i		−	0.989685i	−0.868983	−	0.494843i	$-0.835226\pi$
$263$		−	1.08384e29i		−	0.989685i	0.868983	−	0.494843i	$-0.164774\pi$
$264$		0			0
$265$	−2.67467e29			−2.23007
$266$		0			0
$267$	2.05911e28	−	1.21188e28i	0.156872	−	0.0923264i
$268$		0			0
$269$			1.29861e29i			0.904591i	0.891868	+	0.452296i	$0.149395\pi$
$269$			1.29861e29i			0.904591i	−0.891868	+	0.452296i	$0.850605\pi$
$270$		0			0
$271$	4.00692e28			0.255375			0.127688	−	0.991814i	$-0.459244\pi$
$271$	4.00692e28			0.255375			0.127688	+	0.991814i	$0.459244\pi$
$272$		0			0
$273$	1.52104e28	+	2.58440e28i	0.0887541	+	0.150802i
$274$		0			0
$275$			5.00284e29i			2.67439i
$276$		0			0
$277$	1.01057e29			0.495232			0.247616	−	0.968858i	$-0.420353\pi$
$277$	1.01057e29			0.495232			0.247616	+	0.968858i	$0.420353\pi$
$278$		0			0
$279$	6.41481e28	−	1.15525e29i	0.288359	−	0.519307i
$280$		0			0
$281$		−	5.95802e28i		−	0.245825i	−0.992418	−	0.122912i	$-0.960777\pi$
$281$		−	5.95802e28i		−	0.245825i	0.992418	−	0.122912i	$-0.0392234\pi$
$282$		0			0
$283$	1.57986e29			0.598663			0.299332	−	0.954149i	$-0.403236\pi$
$283$	1.57986e29			0.598663			0.299332	+	0.954149i	$0.403236\pi$
$284$		0			0
$285$	−1.93487e29	+	1.13876e29i	−0.673774	+	0.396548i
$286$		0			0
$287$		−	2.09890e28i		−	0.0672062i
$288$		0			0
$289$	3.26333e29			0.961362
$290$		0			0
$291$	5.60690e28	+	9.52668e28i	0.152057	+	0.258360i
$292$		0			0
$293$			3.62941e29i			0.906618i	0.891354	+	0.453309i	$0.149756\pi$
$293$			3.62941e29i			0.906618i	−0.891354	+	0.453309i	$0.850244\pi$
$294$		0			0
$295$	−5.58082e29			−1.28479
$296$		0			0
$297$	−1.26449e28	−	5.04269e29i	−0.0268435	−	1.07049i
$298$		0			0
$299$			7.80741e29i			1.52916i
$300$		0			0
$301$	9.85305e28			0.178145
$302$		0			0
$303$	−2.19709e29	+	1.29309e29i	−0.366890	+	0.215932i
$304$		0			0
$305$			1.44398e30i			2.22825i
$306$		0			0
$307$	−1.61139e29			−0.229901			−0.114950	−	0.993371i	$-0.536671\pi$
$307$	−1.61139e29			−0.229901			−0.114950	+	0.993371i	$0.536671\pi$
$308$		0			0
$309$	−5.58025e29	−	9.48140e29i	−0.736465	−	1.25133i
$310$		0			0
$311$		−	7.23373e29i		−	0.883563i	−0.897123	−	0.441781i	$-0.854347\pi$
$311$		−	7.23373e29i		−	0.883563i	0.897123	−	0.441781i	$-0.145653\pi$
$312$		0			0
$313$	5.06999e29			0.573423			0.286711	−	0.958017i	$-0.407438\pi$
$313$	5.06999e29			0.573423			0.286711	+	0.958017i	$0.407438\pi$
$314$		0			0
$315$	2.05110e29	+	1.13893e29i	0.214912	+	0.119335i
$316$		0			0
$317$		−	3.48880e29i		−	0.338817i	−0.985546	−	0.169409i	$-0.945814\pi$
$317$		−	3.48880e29i		−	0.338817i	0.985546	−	0.169409i	$-0.0541857\pi$
$318$		0			0
$319$	−1.81086e30			−1.63079
$320$		0			0
$321$	−1.03595e30	+	6.09708e29i	−0.865526	+	0.509403i
$322$		0			0
$323$			1.05964e29i			0.0821731i
$324$		0			0
$325$	−4.61695e30			−3.32472
$326$		0			0
$327$	−1.32803e29	−	2.25646e29i	−0.0888454	−	0.150957i
$328$		0			0
$329$			2.57106e29i			0.159868i
$330$		0			0
$331$	1.05153e30			0.607976			0.303988	−	0.952676i	$-0.401682\pi$
$331$	1.05153e30			0.607976			0.303988	+	0.952676i	$0.401682\pi$
$332$		0			0
$333$	−4.26616e29	+	7.68295e29i	−0.229460	+	0.413235i
$334$		0			0
$335$			1.31816e30i			0.659828i
$336$		0			0
$337$	2.48164e30			1.15659			0.578295	−	0.815828i	$-0.303718\pi$
$337$	2.48164e30			1.15659			0.578295	+	0.815828i	$0.303718\pi$
$338$		0			0
$339$	−1.65187e30	+	9.72204e29i	−0.717100	+	0.422047i
$340$		0			0
$341$			1.57238e30i			0.636069i
$342$		0			0
$343$	6.91086e29			0.260617
$344$		0			0
$345$	3.09816e30	+	5.26409e30i	1.08962	+	1.85137i
$346$		0			0
$347$			3.73656e30i			1.22608i	0.790052	+	0.613040i	$0.210053\pi$
$347$			3.73656e30i			1.22608i	−0.790052	+	0.613040i	$0.789947\pi$
$348$		0			0
$349$	5.31023e30			1.62633			0.813164	−	0.582035i	$-0.197743\pi$
$349$	5.31023e30			1.62633			0.813164	+	0.582035i	$0.197743\pi$
$350$		0			0
$351$	4.65372e30	−	1.16696e29i	1.33081	−	0.0333711i
$352$		0			0
$353$			1.35177e28i			0.00361082i	0.999998	+	0.00180541i	$0.000574680\pi$
$353$			1.35177e28i			0.00361082i	−0.999998	+	0.00180541i	$0.999425\pi$
$354$		0			0
$355$	−2.67813e30			−0.668485
$356$		0			0
$357$	9.54296e28	−	5.61648e28i	0.0222671	−	0.0131053i
$358$		0			0
$359$			1.08720e30i			0.237234i	0.992940	+	0.118617i	$0.0378460\pi$
$359$			1.08720e30i			0.237234i	−0.992940	+	0.118617i	$0.962154\pi$
$360$		0			0
$361$	−4.04265e30			−0.825240
$362$		0			0
$363$	3.89444e29	+	6.61705e29i	0.0743988	+	0.126411i
$364$		0			0
$365$		−	2.23463e30i		−	0.399660i
$366$		0			0
$367$	4.97294e30			0.832952			0.416476	−	0.909147i	$-0.363265\pi$
$367$	4.97294e30			0.832952			0.416476	+	0.909147i	$0.363265\pi$
$368$		0			0
$369$	−2.84856e30	−	1.58174e30i	−0.447002	−	0.248210i
$370$		0			0
$371$		−	1.06578e30i		−	0.156741i
$372$		0			0
$373$	−4.16953e30			−0.574885			−0.287443	−	0.957798i	$-0.592805\pi$
$373$	−4.16953e30			−0.574885			−0.287443	+	0.957798i	$0.592805\pi$
$374$		0			0
$375$	−1.86651e31	+	1.09853e31i	−2.41355	+	1.42049i
$376$		0			0
$377$		−	1.67118e31i		−	2.02735i
$378$		0			0
$379$	5.25196e30			0.597931			0.298966	−	0.954264i	$-0.403358\pi$
$379$	5.25196e30			0.597931			0.298966	+	0.954264i	$0.403358\pi$
$380$		0			0
$381$	4.19762e29	+	7.13217e29i	0.0448645	+	0.0762293i
$382$		0			0
$383$		−	1.79845e31i		−	1.80515i	−0.430532	−	0.902575i	$-0.641674\pi$
$383$		−	1.79845e31i		−	1.80515i	0.430532	−	0.902575i	$-0.358326\pi$
$384$		0			0
$385$	−2.79170e30			−0.263233
$386$		0			0
$387$	7.42530e30	−	1.33723e31i	0.657934	−	1.18488i
$388$		0			0
$389$			5.39514e30i			0.449374i	0.974431	+	0.224687i	$0.0721359\pi$
$389$			5.39514e30i			0.449374i	−0.974431	+	0.224687i	$0.927864\pi$
$390$		0			0
$391$	2.88291e30			0.225793
$392$		0			0
$393$	1.74334e30	−	1.02604e30i	0.128432	−	0.0755880i
$394$		0			0
$395$		−	2.15068e31i		−	1.49077i
$396$		0			0
$397$	1.06217e31			0.692958			0.346479	−	0.938058i	$-0.387377\pi$
$397$	1.06217e31			0.692958			0.346479	+	0.938058i	$0.387377\pi$
$398$		0			0
$399$	−4.53767e29	−	7.70995e29i	−0.0278714	−	0.0473562i
$400$		0			0
$401$			3.13328e30i			0.181245i	0.995885	+	0.0906226i	$0.0288857\pi$
$401$			3.13328e30i			0.181245i	−0.995885	+	0.0906226i	$0.971114\pi$
$402$		0			0
$403$	−1.45109e31			−0.790742
$404$		0			0
$405$	3.09143e31	−	1.92539e31i	1.58745	−	0.988685i
$406$		0			0
$407$		−	1.04571e31i		−	0.506148i
$408$		0			0
$409$	−2.02665e31			−0.924909			−0.462454	−	0.886643i	$-0.653031\pi$
$409$	−2.02665e31			−0.924909			−0.462454	+	0.886643i	$0.653031\pi$
$410$		0			0
$411$	1.03268e31	−	6.07779e30i	0.444491	−	0.261604i
$412$		0			0
$413$		−	2.22381e30i		−	0.0903018i
$414$		0			0
$415$	5.74373e31			2.20098
$416$		0			0
$417$	−2.05426e31	−	3.49040e31i	−0.743060	−	1.26253i
$418$		0			0
$419$			1.73826e31i			0.593674i	0.954928	+	0.296837i	$0.0959317\pi$
$419$			1.73826e31i			0.593674i	−0.954928	+	0.296837i	$0.904068\pi$
$420$		0			0
$421$	3.94616e31			1.27289			0.636443	−	0.771324i	$-0.280405\pi$
$421$	3.94616e31			1.27289			0.636443	+	0.771324i	$0.280405\pi$
$422$		0			0
$423$	3.48936e31	+	1.93756e31i	1.06331	+	0.590433i
$424$		0			0
$425$			1.70482e31i			0.490921i
$426$		0			0
$427$	−5.75388e30			−0.156612
$428$		0			0
$429$	−4.77392e31	+	2.80968e31i	−1.22854	+	0.723051i
$430$		0			0
$431$			5.04053e31i			1.22673i	0.789799	+	0.613365i	$0.210185\pi$
$431$			5.04053e31i			1.22673i	−0.789799	+	0.613365i	$0.789815\pi$
$432$		0			0
$433$	7.89926e31			1.81858			0.909289	−	0.416166i	$-0.136627\pi$
$433$	7.89926e31			1.81858			0.909289	+	0.416166i	$0.136627\pi$
$434$		0			0
$435$	−6.63165e31	−	1.12678e32i	−1.44461	−	2.45454i
$436$		0			0
$437$		−	2.32916e31i		−	0.480200i
$438$		0			0
$439$	−2.86248e31			−0.558687			−0.279344	−	0.960191i	$-0.590117\pi$
$439$	−2.86248e31			−0.558687			−0.279344	+	0.960191i	$0.590117\pi$
$440$		0			0
$441$	2.58133e31	−	4.64874e31i	0.477069	−	0.859156i
$442$		0			0
$443$			4.49140e31i			0.786206i	0.919494	+	0.393103i	$0.128599\pi$
$443$			4.49140e31i			0.786206i	−0.919494	+	0.393103i	$0.871401\pi$
$444$		0			0
$445$	−2.05271e31			−0.340414
$446$		0			0
$447$	1.04317e31	−	6.13956e30i	0.163933	−	0.0964822i
$448$		0			0
$449$		−	5.39778e31i		−	0.804006i	−0.915638	−	0.402003i	$-0.868314\pi$
$449$		−	5.39778e31i		−	0.804006i	0.915638	−	0.402003i	$-0.131686\pi$
$450$		0			0
$451$	3.87711e31			0.547507
$452$		0			0
$453$	3.12982e31	+	5.31788e31i	0.419123	+	0.712132i
$454$		0			0
$455$		−	2.57636e31i		−	0.327243i
$456$		0			0
$457$	−2.97204e31			−0.358146			−0.179073	−	0.983836i	$-0.557310\pi$
$457$	−2.97204e31			−0.358146			−0.179073	+	0.983836i	$0.557310\pi$
$458$		0			0
$459$	−4.30903e29	−	1.71840e31i	−0.00492750	−	0.196504i
$460$		0			0
$461$		−	2.21817e31i		−	0.240759i	−0.992728	−	0.120380i	$-0.961589\pi$
$461$		−	2.21817e31i		−	0.240759i	0.992728	−	0.120380i	$-0.0384112\pi$
$462$		0			0
$463$	−1.34906e32			−1.39014			−0.695072	−	0.718940i	$-0.744628\pi$
$463$	−1.34906e32			−1.39014			−0.695072	+	0.718940i	$0.744628\pi$
$464$		0			0
$465$	−9.78389e31	+	5.75828e31i	−0.957361	+	0.563452i
$466$		0			0
$467$		−	2.20131e31i		−	0.204587i	−0.994754	−	0.102294i	$-0.967382\pi$
$467$		−	2.20131e31i		−	0.204587i	0.994754	−	0.102294i	$-0.0326181\pi$
$468$		0			0
$469$	−5.25252e30			−0.0463760
$470$		0			0
$471$	1.03995e32	+	1.76699e32i	0.872496	+	1.48246i
$472$		0			0
$473$			1.82006e32i			1.45129i
$474$		0			0
$475$	1.37736e32			1.04406
$476$		0			0
$477$	−1.44645e32	−	8.03178e31i	−1.04251	−	0.578884i
$478$		0			0
$479$		−	1.84865e32i		−	1.26715i	−0.773681	−	0.633575i	$-0.781587\pi$
$479$		−	1.84865e32i		−	1.26715i	0.773681	−	0.633575i	$-0.218413\pi$
$480$		0			0
$481$	9.65047e31			0.629228
$482$		0			0
$483$	−2.09760e31	+	1.23454e31i	−0.130124	+	0.0765840i
$484$		0			0
$485$		−	9.49707e31i		−	0.560646i
$486$		0			0
$487$	8.47310e31			0.476096			0.238048	−	0.971253i	$-0.423492\pi$
$487$	8.47310e31			0.476096			0.238048	+	0.971253i	$0.423492\pi$
$488$		0			0
$489$	1.06314e32	+	1.80637e32i	0.568699	+	0.966275i
$490$		0			0
$491$			8.67400e31i			0.441816i	0.975295	+	0.220908i	$0.0709021\pi$
$491$			8.67400e31i			0.441816i	−0.975295	+	0.220908i	$0.929098\pi$
$492$		0			0
$493$	−6.17090e31			−0.299354
$494$		0			0
$495$	−2.10383e32	+	3.78881e32i	−0.972186	+	1.75081i
$496$		0			0
$497$		−	1.06716e31i		−	0.0469845i
$498$		0			0
$499$	−2.49415e32			−1.04645			−0.523223	−	0.852196i	$-0.675270\pi$
$499$	−2.49415e32			−1.04645			−0.523223	+	0.852196i	$0.675270\pi$
$500$		0			0
$501$	−1.37127e32	+	8.07059e31i	−0.548367	+	0.322740i
$502$		0			0
$503$		−	4.71984e32i		−	1.79933i	−0.436579	−	0.899666i	$-0.643810\pi$
$503$		−	4.71984e32i		−	1.79933i	0.436579	−	0.899666i	$-0.356190\pi$
$504$		0			0
$505$	2.19026e32			0.796157
$506$		0			0
$507$	−1.12979e32	−	1.91963e32i	−0.391655	−	0.665461i
$508$		0			0
$509$			1.88162e32i			0.622183i	0.950380	+	0.311091i	$0.100694\pi$
$509$			1.88162e32i			0.622183i	−0.950380	+	0.311091i	$0.899306\pi$
$510$		0			0
$511$	8.90440e30			0.0280901
$512$		0			0
$513$	−1.38833e32	+	3.48135e30i	−0.417912	+	0.0104795i
$514$		0			0
$515$			9.45193e32i			2.71540i
$516$		0			0
$517$	−4.74928e32			−1.30239
$518$		0			0
$519$	−5.07807e32	+	2.98868e32i	−1.32951	+	0.782477i
$520$		0			0
$521$		−	6.65852e31i		−	0.166466i	−0.996530	−	0.0832329i	$-0.973475\pi$
$521$		−	6.65852e31i		−	0.166466i	0.996530	−	0.0832329i	$-0.0265245\pi$
$522$		0			0
$523$	7.70218e30			0.0183905			0.00919525	−	0.999958i	$-0.497073\pi$
$523$	7.70218e30			0.0183905			0.00919525	+	0.999958i	$0.497073\pi$
$524$		0			0
$525$	−7.30049e31	−	1.24043e32i	−0.166510	−	0.282917i
$526$		0			0
$527$			5.35820e31i			0.116759i
$528$		0			0
$529$	−1.53429e32			−0.319477
$530$		0			0
$531$	−3.01808e32	−	1.67587e32i	−0.600616	−	0.333508i
$532$		0			0
$533$			3.57805e32i			0.680644i
$534$		0			0
$535$	1.03273e33			1.87821
$536$		0			0
$537$	1.13370e32	−	6.67235e31i	0.197154	−	0.116035i
$538$		0			0
$539$			6.32728e32i			1.05233i
$540$		0			0
$541$	6.06284e32			0.964514			0.482257	−	0.876030i	$-0.339817\pi$
$541$	6.06284e32			0.964514			0.482257	+	0.876030i	$0.339817\pi$
$542$		0			0
$543$	−2.44463e32	−	4.15368e32i	−0.372062	−	0.632171i
$544$		0			0
$545$			2.24944e32i			0.327580i
$546$		0			0
$547$	8.04943e32			1.12180			0.560902	−	0.827882i	$-0.310454\pi$
$547$	8.04943e32			1.12180			0.560902	+	0.827882i	$0.310454\pi$
$548$		0			0
$549$	−4.33614e32	+	7.80898e32i	−0.578409	+	1.04166i
$550$		0			0
$551$			4.98559e32i			0.636646i
$552$		0			0
$553$	8.56987e31			0.104779
$554$		0			0
$555$	6.50676e32	−	3.82954e32i	0.761814	−	0.448364i
$556$		0			0
$557$			9.36098e32i			1.04968i	0.851200	+	0.524842i	$0.175876\pi$
$557$			9.36098e32i			1.04968i	−0.851200	+	0.524842i	$0.824124\pi$
$558$		0			0
$559$	−1.67967e33			−1.80420
$560$		0			0
$561$	1.03748e32	+	1.76278e32i	0.106764	+	0.181403i
$562$		0			0
$563$			3.63395e32i			0.358326i	0.983819	+	0.179163i	$0.0573389\pi$
$563$			3.63395e32i			0.358326i	−0.983819	+	0.179163i	$0.942661\pi$
$564$		0			0
$565$	1.64674e33			1.55612
$566$		0			0
$567$	7.67216e31	+	1.23185e32i	0.0694898	+	0.111574i
$568$		0			0
$569$		−	1.56580e33i		−	1.35953i	−0.733431	−	0.679764i	$-0.762082\pi$
$569$		−	1.56580e33i		−	1.35953i	0.733431	−	0.679764i	$-0.237918\pi$
$570$		0			0
$571$	1.35659e33			1.12932			0.564659	−	0.825324i	$-0.309008\pi$
$571$	1.35659e33			1.12932			0.564659	+	0.825324i	$0.309008\pi$
$572$		0			0
$573$	1.08587e33	−	6.39085e32i	0.866806	−	0.510156i
$574$		0			0
$575$		−	3.74730e33i		−	2.86883i
$576$		0			0
$577$	−1.34896e33			−0.990578			−0.495289	−	0.868728i	$-0.664938\pi$
$577$	−1.34896e33			−0.990578			−0.495289	+	0.868728i	$0.664938\pi$
$578$		0			0
$579$	−1.37206e33	−	2.33126e33i	−0.966562	−	1.64229i
$580$		0			0
$581$			2.28872e32i			0.154696i
$582$		0			0
$583$	1.96872e33			1.27692
$584$		0			0
$585$	−3.49656e33	−	1.94156e33i	−2.17656	−	1.20859i
$586$		0			0
$587$			3.28919e33i			1.96531i	0.185437	+	0.982656i	$0.440630\pi$
$587$			3.28919e33i			1.96531i	−0.185437	+	0.982656i	$0.559370\pi$
$588$		0			0
$589$	4.32900e32			0.248316
$590$		0			0
$591$	−2.16348e33	+	1.27331e33i	−1.19152	+	0.701267i
$592$		0			0
$593$		−	1.50339e33i		−	0.795088i	−0.917583	−	0.397544i	$-0.869863\pi$
$593$		−	1.50339e33i		−	0.795088i	0.917583	−	0.397544i	$-0.130137\pi$
$594$		0			0
$595$	−9.51330e31			−0.0483201
$596$		0			0
$597$	−1.97239e33	−	3.35129e33i	−0.962279	−	1.63501i
$598$		0			0
$599$		−	2.92781e33i		−	1.37221i	−0.727501	−	0.686107i	$-0.759318\pi$
$599$		−	2.92781e33i		−	1.37221i	0.727501	−	0.686107i	$-0.240682\pi$
$600$		0			0
$601$	1.78506e33			0.803821			0.401910	−	0.915679i	$-0.368346\pi$
$601$	1.78506e33			0.803821			0.401910	+	0.915679i	$0.368346\pi$
$602$		0			0
$603$	−3.95832e32	+	7.12856e32i	−0.171279	+	0.308456i
$604$		0			0
$605$		−	6.59648e32i		−	0.274314i
$606$		0			0
$607$	−2.30435e32			−0.0921050			−0.0460525	−	0.998939i	$-0.514664\pi$
$607$	−2.30435e32			−0.0921050			−0.0460525	+	0.998939i	$0.514664\pi$
$608$		0			0
$609$	4.48993e32	−	2.64254e32i	0.172517	−	0.101534i
$610$		0			0
$611$		−	4.38295e33i		−	1.61909i
$612$		0			0
$613$	4.46344e33			1.58542			0.792710	−	0.609599i	$-0.208670\pi$
$613$	4.46344e33			1.58542			0.792710	+	0.609599i	$0.208670\pi$
$614$		0			0
$615$	1.41986e33	+	2.41248e33i	0.485001	+	0.824065i
$616$		0			0
$617$			3.92178e32i			0.128843i	0.997923	+	0.0644216i	$0.0205203\pi$
$617$			3.92178e32i			0.128843i	−0.997923	+	0.0644216i	$0.979480\pi$
$618$		0			0
$619$	2.74576e32			0.0867710			0.0433855	−	0.999058i	$-0.486186\pi$
$619$	2.74576e32			0.0867710			0.0433855	+	0.999058i	$0.486186\pi$
$620$		0			0
$621$	9.47150e31	+	3.77715e33i	0.0287952	+	1.14832i
$622$		0			0
$623$		−	8.17950e31i		−	0.0239260i
$624$		0			0
$625$	9.73426e33			2.73995
$626$		0			0
$627$	1.42419e33	−	8.38202e32i	0.385796	−	0.227059i
$628$		0			0
$629$		−	3.56346e32i		−	0.0929105i
$630$		0			0
$631$	7.59300e33			1.90573			0.952864	−	0.303397i	$-0.0981210\pi$
$631$	7.59300e33			1.90573			0.952864	+	0.303397i	$0.0981210\pi$
$632$		0			0
$633$	1.48929e33	+	2.53045e33i	0.359860	+	0.611439i
$634$		0			0
$635$		−	7.11000e32i		−	0.165419i
$636$		0			0
$637$	−5.83923e33			−1.30823
$638$		0			0
$639$	−1.44832e33	−	8.04219e32i	−0.312503	−	0.173526i
$640$		0			0
$641$		−	5.38294e33i		−	1.11872i	−0.828924	−	0.559362i	$-0.811046\pi$
$641$		−	5.38294e33i		−	1.11872i	0.828924	−	0.559362i	$-0.188954\pi$
$642$		0			0
$643$	5.58872e32			0.111887			0.0559436	−	0.998434i	$-0.482183\pi$
$643$	5.58872e32			0.111887			0.0559436	+	0.998434i	$0.482183\pi$
$644$		0			0
$645$	−1.13251e34	+	6.66535e33i	−2.18436	+	1.28560i
$646$		0			0
$647$			9.97296e33i			1.85342i	0.375783	+	0.926708i	$0.377374\pi$
$647$			9.97296e33i			1.85342i	−0.375783	+	0.926708i	$0.622626\pi$
$648$		0			0
$649$	4.10784e33			0.735659
$650$		0			0
$651$	−2.29452e32	−	3.89862e32i	−0.0396023	−	0.0672882i
$652$		0			0
$653$		−	6.38820e32i		−	0.106272i	−0.998587	−	0.0531362i	$-0.983078\pi$
$653$		−	6.38820e32i		−	0.106272i	0.998587	−	0.0531362i	$-0.0169217\pi$
$654$		0			0
$655$	−1.73792e33			−0.278699
$656$		0			0
$657$	6.71039e32	−	1.20848e33i	0.103744	−	0.186833i
$658$		0			0
$659$		−	6.90599e33i		−	1.02944i	−0.857359	−	0.514720i	$-0.827896\pi$
$659$		−	6.90599e33i		−	1.02944i	0.857359	−	0.514720i	$-0.172104\pi$
$660$		0			0
$661$	−1.70554e33			−0.245156			−0.122578	−	0.992459i	$-0.539116\pi$
$661$	−1.70554e33			−0.245156			−0.122578	+	0.992459i	$0.539116\pi$
$662$		0			0
$663$	−1.62681e33	+	9.57455e32i	−0.225515	+	0.132726i
$664$		0			0
$665$			7.68599e32i			0.102764i
$666$		0			0
$667$	1.35640e34			1.74935
$668$		0			0
$669$	−7.16083e32	−	1.21670e33i	−0.0890943	−	0.151380i
$670$		0			0
$671$		−	1.06286e34i		−	1.27587i
$672$		0			0
$673$	5.28039e33			0.611625			0.305813	−	0.952092i	$-0.401072\pi$
$673$	5.28039e33			0.611625			0.305813	+	0.952092i	$0.401072\pi$
$674$		0			0
$675$	−2.23363e34	+	5.60102e32i	−2.49670	+	0.0626068i
$676$		0			0
$677$		−	1.06855e34i		−	1.15274i	−0.817188	−	0.576371i	$-0.804468\pi$
$677$		−	1.06855e34i		−	1.15274i	0.817188	−	0.576371i	$-0.195532\pi$
$678$		0			0
$679$	3.78433e32			0.0394050
$680$		0			0
$681$	−1.73912e33	+	1.02355e33i	−0.174809	+	0.102883i
$682$		0			0
$683$			6.35426e32i			0.0616618i	0.999525	+	0.0308309i	$0.00981534\pi$
$683$			6.35426e32i			0.0616618i	−0.999525	+	0.0308309i	$0.990185\pi$
$684$		0			0
$685$	−1.02947e34			−0.964552
$686$		0			0
$687$	−3.23243e33	−	5.49222e33i	−0.292448	−	0.496898i
$688$		0			0
$689$			1.81687e34i			1.58742i
$690$		0			0
$691$	−2.02979e34			−1.71283			−0.856417	−	0.516284i	$-0.827315\pi$
$691$	−2.02979e34			−1.71283			−0.856417	+	0.516284i	$0.827315\pi$
$692$		0			0
$693$	−1.50974e33	−	8.38322e32i	−0.123056	−	0.0683301i
$694$		0			0
$695$			3.47955e34i			2.73972i
$696$		0			0
$697$	1.32121e33			0.100503
$698$		0			0
$699$	−1.15823e34	+	6.81673e33i	−0.851274	+	0.501014i
$700$		0			0
$701$			1.31882e34i			0.936633i	0.883561	+	0.468317i	$0.155139\pi$
$701$			1.31882e34i			0.936633i	−0.883561	+	0.468317i	$0.844861\pi$
$702$		0			0
$703$	−2.87900e33			−0.197596
$704$		0			0
$705$	−1.73926e34	−	2.95517e34i	−1.15370	−	1.96026i
$706$		0			0
$707$			8.72759e32i			0.0559579i
$708$		0			0
$709$	1.70214e34			1.05497			0.527487	−	0.849563i	$-0.323134\pi$
$709$	1.70214e34			1.05497			0.527487	+	0.849563i	$0.323134\pi$
$710$		0			0
$711$	6.45829e33	−	1.16308e34i	0.386974	−	0.696904i
$712$		0			0
$713$		−	1.17776e34i		−	0.682313i
$714$		0			0
$715$	4.75908e34			2.66594
$716$		0			0
$717$	−1.06228e34	+	6.25202e33i	−0.575453	+	0.338681i
$718$		0			0
$719$			9.84142e33i			0.515598i	0.966199	+	0.257799i	$0.0829972\pi$
$719$			9.84142e33i			0.515598i	−0.966199	+	0.257799i	$0.917003\pi$
$720$		0			0
$721$	−3.76634e33			−0.190852
$722$		0			0
$723$	2.50243e33	+	4.25188e33i	0.122660	+	0.208411i
$724$		0			0
$725$			8.02114e34i			3.80347i
$726$		0			0
$727$	3.38827e34			1.55441			0.777205	−	0.629247i	$-0.216637\pi$
$727$	3.38827e34			1.55441			0.777205	+	0.629247i	$0.216637\pi$
$728$		0			0
$729$	2.25001e34	−	1.12913e33i	0.998743	−	0.0501201i
$730$		0			0
$731$			6.20225e33i			0.266404i
$732$		0			0
$733$	6.94273e33			0.288591			0.144295	−	0.989535i	$-0.453908\pi$
$733$	6.94273e33			0.288591			0.144295	+	0.989535i	$0.453908\pi$
$734$		0			0
$735$	−3.93706e34	+	2.31714e34i	−1.58389	+	0.932191i
$736$		0			0
$737$		−	9.70250e33i		−	0.377810i
$738$		0			0
$739$	−4.80426e34			−1.81090			−0.905448	−	0.424458i	$-0.860465\pi$
$739$	−4.80426e34			−1.81090			−0.905448	+	0.424458i	$0.860465\pi$
$740$		0			0
$741$	7.73548e33	+	1.31434e34i	0.282273	+	0.479609i
$742$		0			0
$743$			5.57616e33i			0.197001i	0.995137	+	0.0985007i	$0.0314047\pi$
$743$			5.57616e33i			0.197001i	−0.995137	+	0.0985007i	$0.968595\pi$
$744$		0			0
$745$	−1.03993e34			−0.355737
$746$		0			0
$747$	3.10618e34	+	1.72479e34i	1.02892	+	0.571333i
$748$		0			0
$749$			4.11517e33i			0.132010i
$750$		0			0
$751$	−2.09656e34			−0.651371			−0.325685	−	0.945478i	$-0.605595\pi$
$751$	−2.09656e34			−0.651371			−0.325685	+	0.945478i	$0.605595\pi$
$752$		0			0
$753$	−4.87228e34	+	2.86757e34i	−1.46620	+	0.862927i
$754$		0			0
$755$		−	5.30135e34i		−	1.54534i
$756$		0			0
$757$	−2.59094e34			−0.731657			−0.365829	−	0.930682i	$-0.619214\pi$
$757$	−2.59094e34			−0.731657			−0.365829	+	0.930682i	$0.619214\pi$
$758$		0			0
$759$	−2.28044e34	−	3.87470e34i	−0.623905	−	1.06008i
$760$		0			0
$761$		−	6.37561e34i		−	1.69008i	−0.534707	−	0.845038i	$-0.679578\pi$
$761$		−	6.37561e34i		−	1.69008i	0.534707	−	0.845038i	$-0.320422\pi$
$762$		0			0
$763$	−8.96343e32			−0.0230239
$764$		0			0
$765$	−7.16925e33	+	1.29111e34i	−0.178458	+	0.321387i
$766$		0			0
$767$			3.79098e34i			0.914549i
$768$		0			0
$769$	4.52884e34			1.05894			0.529470	−	0.848329i	$-0.322391\pi$
$769$	4.52884e34			1.05894			0.529470	+	0.848329i	$0.322391\pi$
$770$		0			0
$771$	−3.40523e34	+	2.00414e34i	−0.771781	+	0.454230i
$772$		0			0
$773$			2.41056e34i			0.529619i	0.964301	+	0.264810i	$0.0853091\pi$
$773$			2.41056e34i			0.529619i	−0.964301	+	0.264810i	$0.914691\pi$
$774$		0			0
$775$	6.96477e34			1.48350
$776$		0			0
$777$	1.52597e33	+	2.59277e33i	0.0315132	+	0.0535441i
$778$		0			0
$779$		−	1.06743e34i		−	0.213742i
$780$		0			0
$781$	1.97127e34			0.382767
$782$		0			0
$783$	−2.02739e33	−	8.08502e34i	−0.0381764	−	1.52244i
$784$		0			0
$785$		−	1.76149e35i		−	3.21696i
$786$		0			0
$787$	2.11726e33			0.0375040			0.0187520	−	0.999824i	$-0.494031\pi$
$787$	2.11726e33			0.0375040			0.0187520	+	0.999824i	$0.494031\pi$
$788$		0			0
$789$	−4.96405e34	+	2.92158e34i	−0.852927	+	0.501988i
$790$		0			0
$791$			6.56181e33i			0.109372i
$792$		0			0
$793$	9.80878e34			1.58612
$794$		0			0
$795$	7.20976e34	+	1.22501e35i	1.13114	+	1.92192i
$796$		0			0
$797$		−	1.31847e34i		−	0.200711i	−0.994952	−	0.100355i	$-0.968002\pi$
$797$		−	1.31847e34i		−	0.200711i	0.994952	−	0.100355i	$-0.0319979\pi$
$798$		0			0
$799$	−1.61842e34			−0.239072
$800$		0			0
$801$	−1.11010e34	−	6.16410e33i	−0.159137	−	0.0883649i
$802$		0			0
$803$			1.64483e34i			0.228841i
$804$		0			0
$805$	2.09108e34			0.282371
$806$		0			0
$807$	5.94771e34	−	3.50051e34i	0.779592	−	0.458826i
$808$		0			0
$809$			5.48800e34i			0.698284i	0.937070	+	0.349142i	$0.113527\pi$
$809$			5.48800e34i			0.698284i	−0.937070	+	0.349142i	$0.886473\pi$
$810$		0			0
$811$	2.02114e34			0.249659			0.124829	−	0.992178i	$-0.460162\pi$
$811$	2.02114e34			0.249659			0.124829	+	0.992178i	$0.460162\pi$
$812$		0			0
$813$	−1.08009e34	−	1.83519e34i	−0.129531	−	0.220087i
$814$		0			0
$815$		−	1.80076e35i		−	2.09683i
$816$		0			0
$817$	5.01092e34			0.566569
$818$		0			0
$819$	7.73659e33	−	1.39329e34i	0.0849459	−	0.152980i
$820$		0			0
$821$		−	4.91287e34i		−	0.523862i	−0.965087	−	0.261931i	$-0.915641\pi$
$821$		−	4.91287e34i		−	0.523862i	0.965087	−	0.261931i	$-0.0843593\pi$
$822$		0			0
$823$	−1.85256e35			−1.91856			−0.959279	−	0.282461i	$-0.908849\pi$
$823$	−1.85256e35			−1.91856			−0.959279	+	0.282461i	$0.908849\pi$
$824$		0			0
$825$	2.29132e35	−	1.34855e35i	2.30483	−	1.35650i
$826$		0			0
$827$		−	1.63343e35i		−	1.59600i	−0.602654	−	0.798002i	$-0.705890\pi$
$827$		−	1.63343e35i		−	1.59600i	0.602654	−	0.798002i	$-0.294110\pi$
$828$		0			0
$829$	−1.13394e35			−1.07630			−0.538152	−	0.842848i	$-0.680877\pi$
$829$	−1.13394e35			−1.07630			−0.538152	+	0.842848i	$0.680877\pi$
$830$		0			0
$831$	−2.72406e34	−	4.62845e34i	−0.251191	−	0.426799i
$832$		0			0
$833$			2.15615e34i			0.193170i
$834$		0			0
$835$	1.36701e35			1.18997
$836$		0			0
$837$	−7.02024e34	+	1.76038e33i	−0.593809	+	0.0148902i
$838$		0			0
$839$			2.05511e34i			0.168924i	0.996427	+	0.0844622i	$0.0269172\pi$
$839$			2.05511e34i			0.168924i	−0.996427	+	0.0844622i	$0.973083\pi$
$840$		0			0
$841$	−1.65154e35			−1.31928
$842$		0			0
$843$	−2.72880e34	+	1.60603e34i	−0.211856	+	0.124687i
$844$		0			0
$845$			1.91367e35i			1.44406i
$846$		0			0
$847$	2.62852e33			0.0192802
$848$		0			0
$849$	−4.25864e34	−	7.23585e34i	−0.303654	−	0.515938i
$850$		0			0
$851$			7.83271e34i			0.542947i
$852$		0			0
$853$	−1.23579e35			−0.832832			−0.416416	−	0.909174i	$-0.636714\pi$
$853$	−1.23579e35			−0.832832			−0.416416	+	0.909174i	$0.636714\pi$
$854$		0			0
$855$	1.04312e35	+	5.79219e34i	0.683503	+	0.379533i
$856$		0			0
$857$		−	1.56805e35i		−	0.999055i	−0.866298	−	0.499527i	$-0.833507\pi$
$857$		−	1.56805e35i		−	0.999055i	0.866298	−	0.499527i	$-0.166493\pi$
$858$		0			0
$859$	1.59845e35			0.990330			0.495165	−	0.868799i	$-0.335108\pi$
$859$	1.59845e35			0.990330			0.495165	+	0.868799i	$0.335108\pi$
$860$		0			0
$861$	−9.61307e33	+	5.65774e33i	−0.0579194	+	0.0340883i
$862$		0			0
$863$			6.37895e34i			0.373783i	0.982381	+	0.186892i	$0.0598413\pi$
$863$			6.37895e34i			0.373783i	−0.982381	+	0.186892i	$0.940159\pi$
$864$		0			0
$865$	5.06229e35			2.88505
$866$		0			0
$867$	−8.79655e34	−	1.49462e35i	−0.487622	−	0.828518i
$868$		0			0
$869$			1.58303e35i			0.853597i
$870$		0			0
$871$	8.95411e34			0.469682
$872$		0			0
$873$	2.85188e34	−	5.13597e34i	0.145533	−	0.262091i
$874$		0			0
$875$			7.41444e34i			0.368114i
$876$		0			0
$877$	−3.55972e35			−1.71957			−0.859787	−	0.510652i	$-0.829404\pi$
$877$	−3.55972e35			−1.71957			−0.859787	+	0.510652i	$0.829404\pi$
$878$		0			0
$879$	1.66229e35	−	9.78333e34i	0.781338	−	0.459854i
$880$		0			0
$881$		−	1.83672e35i		−	0.840101i	−0.907501	−	0.420050i	$-0.862012\pi$
$881$		−	1.83672e35i		−	0.840101i	0.907501	−	0.420050i	$-0.137988\pi$
$882$		0			0
$883$	−1.85716e35			−0.826647			−0.413324	−	0.910584i	$-0.635632\pi$
$883$	−1.85716e35			−0.826647			−0.413324	+	0.910584i	$0.635632\pi$
$884$		0			0
$885$	1.50435e35	+	2.55604e35i	0.651673	+	1.10726i
$886$		0			0
$887$			2.36928e35i			0.998926i	0.866335	+	0.499463i	$0.166469\pi$
$887$			2.36928e35i			0.998926i	−0.866335	+	0.499463i	$0.833531\pi$
$888$		0			0
$889$	2.83315e33			0.0116265
$890$		0			0
$891$	−2.27549e35	+	1.41721e35i	−0.908955	+	0.566110i
$892$		0			0
$893$			1.30755e35i			0.508442i
$894$		0			0
$895$	−1.13018e35			−0.427828
$896$		0			0
$897$	3.57583e35	−	2.10454e35i	1.31785	−	0.775619i
$898$		0			0
$899$			2.52102e35i			0.904606i
$900$		0			0
$901$	6.70884e34			0.234396
$902$		0			0
$903$	−2.65596e34	−	4.51274e34i	−0.0903585	−	0.153528i
$904$		0			0
$905$			4.14077e35i			1.37182i
$906$		0			0
$907$	−5.44825e35			−1.75780			−0.878901	−	0.477004i	$-0.841723\pi$
$907$	−5.44825e35			−1.75780			−0.878901	+	0.477004i	$0.841723\pi$
$908$		0			0
$909$	1.18448e35	+	6.57714e34i	0.372187	+	0.206667i
$910$		0			0
$911$			9.49061e34i			0.290452i	0.989399	+	0.145226i	$0.0463908\pi$
$911$			9.49061e34i			0.290452i	−0.989399	+	0.145226i	$0.953609\pi$
$912$		0			0
$913$	−4.22775e35			−1.26026
$914$		0			0
$915$	6.61350e35	−	3.89235e35i	1.92034	−	1.13021i
$916$		0			0
$917$		−	6.92516e33i		−	0.0195883i
$918$		0			0
$919$	2.17698e35			0.599885			0.299943	−	0.953957i	$-0.403032\pi$
$919$	2.17698e35			0.599885			0.299943	+	0.953957i	$0.403032\pi$
$920$		0			0
$921$	4.34362e34	+	7.38025e34i	0.116610	+	0.198132i
$922$		0			0
$923$			1.81922e35i			0.475845i
$924$		0			0
$925$	−4.63191e35			−1.18048
$926$		0			0
$927$	−2.83833e35	+	5.11156e35i	−0.704865	+	1.26940i
$928$		0			0
$929$		−	7.35473e35i		−	1.77983i	−0.456127	−	0.889915i	$-0.650764\pi$
$929$		−	7.35473e35i		−	1.77983i	0.456127	−	0.889915i	$-0.349236\pi$
$930$		0			0
$931$	1.74200e35			0.410820
$932$		0			0
$933$	−3.31308e35	+	1.94990e35i	−0.761469	+	0.448160i
$934$		0			0
$935$		−	1.75730e35i		−	0.393648i
$936$		0			0
$937$	−6.48186e35			−1.41522			−0.707611	−	0.706602i	$-0.750227\pi$
$937$	−6.48186e35			−1.41522			−0.707611	+	0.706602i	$0.750227\pi$
$938$		0			0
$939$	−1.36665e35	−	2.32208e35i	−0.290851	−	0.494185i
$940$		0			0
$941$			3.85253e35i			0.799229i	0.916683	+	0.399614i	$0.130856\pi$
$941$			3.85253e35i			0.799229i	−0.916683	+	0.399614i	$0.869144\pi$
$942$		0			0
$943$	−2.90409e35			−0.587313
$944$		0			0
$945$	−3.12550e33	−	1.24642e35i	−0.00616222	−	0.245744i
$946$		0			0
$947$		−	4.72280e35i		−	0.907820i	−0.891048	−	0.453910i	$-0.850029\pi$
$947$		−	4.72280e35i		−	0.907820i	0.891048	−	0.453910i	$-0.149971\pi$
$948$		0			0
$949$	−1.51796e35			−0.284488
$950$		0			0
$951$	−1.59789e35	+	9.40432e34i	−0.291998	+	0.171855i
$952$		0			0
$953$		−	1.06604e36i		−	1.89959i	−0.312877	−	0.949794i	$-0.601293\pi$
$953$		−	1.06604e36i		−	1.89959i	0.312877	−	0.949794i	$-0.398707\pi$
$954$		0			0
$955$	−1.08249e36			−1.88098
$956$		0			0
$957$	4.88132e35	+	8.29384e35i	0.827168	+	1.40544i
$958$		0			0
$959$		−	4.10215e34i		−	0.0677936i
$960$		0			0
$961$	−4.01512e35			−0.647170
$962$		0			0
$963$	5.58498e35	+	3.10121e35i	0.878024	+	0.487546i
$964$		0			0
$965$			2.32402e36i			3.56379i
$966$		0			0
$967$	−1.18623e36			−1.77439			−0.887197	−	0.461391i	$-0.847351\pi$
$967$	−1.18623e36			−1.77439			−0.887197	+	0.461391i	$0.847351\pi$
$968$		0			0
$969$	4.85322e34	−	2.85635e34i	0.0708182	−	0.0416798i
$970$		0			0
$971$		−	9.47421e35i		−	1.34869i	−0.738416	−	0.674345i	$-0.764426\pi$
$971$		−	9.47421e35i		−	1.34869i	0.738416	−	0.674345i	$-0.235574\pi$
$972$		0			0
$973$	−1.38651e35			−0.192561
$974$		0			0
$975$	1.24453e36	+	2.11458e36i	1.68636	+	2.86530i
$976$		0			0
$977$			3.91558e35i			0.517679i	0.965920	+	0.258840i	$0.0833401\pi$
$977$			3.91558e35i			0.517679i	−0.965920	+	0.258840i	$0.916660\pi$
$978$		0			0
$979$	1.51093e35			0.194917
$980$		0			0
$981$	−6.75487e34	+	1.21649e35i	−0.0850333	+	0.153137i
$982$		0			0
$983$		−	8.43619e35i		−	1.03634i	−0.855276	−	0.518172i	$-0.826613\pi$
$983$		−	8.43619e35i		−	1.03634i	0.855276	−	0.518172i	$-0.173387\pi$
$984$		0			0
$985$	2.15675e36			2.58562
$986$		0			0
$987$	1.17756e35	−	6.93048e34i	0.137777	−	0.0810882i
$988$		0			0
$989$		−	1.36329e36i		−	1.55680i
$990$		0			0
$991$	1.42451e36			1.58775			0.793876	−	0.608079i	$-0.208060\pi$
$991$	1.42451e36			1.58775			0.793876	+	0.608079i	$0.208060\pi$
$992$		0			0
$993$	−2.83448e35	−	4.81606e35i	−0.308377	−	0.523964i
$994$		0			0
$995$			3.34087e36i			3.54800i
$996$		0			0
$997$	8.85938e35			0.918463			0.459231	−	0.888317i	$-0.348125\pi$
$997$	8.85938e35			0.918463			0.459231	+	0.888317i	$0.348125\pi$
$998$		0			0
$999$	4.66880e35	−	1.17074e34i	0.472519	−	0.0118488i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.25.e.d.17.3		8
3.2	odd	2	inner	48.25.e.d.17.4		8
4.3	odd	2		6.25.b.a.5.7	yes	8
12.11	even	2		6.25.b.a.5.3	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6.25.b.a.5.3	✓	8	12.11	even	2
6.25.b.a.5.7	yes	8	4.3	odd	2
48.25.e.d.17.3		8	1.1	even	1	trivial
48.25.e.d.17.4		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)

\(f(q)\)	\(=\)	\(q+(-269557. - 458005. i) q^{3} +4.56581e8i q^{5} -1.81935e9 q^{7} +(-1.37107e11 + 2.46917e11i) q^{9} +O(q^{10})\)	\(q+(-269557. - 458005. i) q^{3} +4.56581e8i q^{5} -1.81935e9 q^{7} +(-1.37107e11 + 2.46917e11i) q^{9} -3.36073e12i q^{11} +3.10150e13 q^{13} +(2.09116e14 - 1.23075e14i) q^{15} -1.14524e14i q^{17} -9.25261e14 q^{19} +(4.90420e14 + 8.33273e14i) q^{21} +2.51730e16i q^{23} -1.48862e17 q^{25} +(1.50047e17 - 3.76256e15i) q^{27} -5.38831e17i q^{29} -4.67868e17 q^{31} +(-1.53923e18 + 9.05909e17i) q^{33} -8.30683e17i q^{35} +3.11155e18 q^{37} +(-8.36032e18 - 1.42050e19i) q^{39} +1.15365e19i q^{41} -5.41569e19 q^{43} +(-1.12738e20 - 6.26006e19i) q^{45} -1.41317e20i q^{47} -1.88271e20 q^{49} +(-5.24524e19 + 3.08707e19i) q^{51} +5.85803e20i q^{53} +1.53445e21 q^{55} +(2.49411e20 + 4.23774e20i) q^{57} +1.22231e21i q^{59} +3.16259e21 q^{61} +(2.49447e20 - 4.49230e20i) q^{63} +1.41609e22i q^{65} +2.88702e21 q^{67} +(1.15294e22 - 6.78557e21i) q^{69} +5.86562e21i q^{71} -4.89426e21 q^{73} +(4.01268e22 + 6.81794e22i) q^{75} +6.11435e21i q^{77} -4.71039e22 q^{79} +(-4.21697e22 - 6.77082e22i) q^{81} -1.25799e23i q^{83} +5.22894e22 q^{85} +(-2.46787e23 + 1.45246e23i) q^{87} +4.49583e22i q^{89} -5.64273e22 q^{91} +(1.26117e23 + 2.14286e23i) q^{93} -4.22457e23i q^{95} -2.08004e23 q^{97} +(8.29821e23 + 4.60780e23i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 131880 q^{3} + 10160794640 q^{7} + 295169053896 q^{9}+O(q^{10}) \) 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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Database

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