LMFDB - Embedded newform 48.7.e.d.17.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,7,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, 7, names="a")

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

chi := DirichletCharacter("48.17");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 7 $
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:	$11.0425960138$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.1173604352.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 14x^{4} - 12x^{3} + 112x^{2} + 192x + 324 $ x^6 - 14x^4 - 12x^3 + 112x^2 + 192x + 324
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{22}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 24)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			17.5
Root			$3.42788 + 1.41421i$ of defining polynomial
Character	$\chi$	$=$	48.17
Dual form			48.7.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)

$f(q)$	$=$	$q+(22.0014 - 15.6505i) q^{3} -161.417i q^{5} -562.144 q^{7} +(239.124 - 688.666i) q^{9} +O(q^{10})$	\(q+(22.0014 - 15.6505i) q^{3} -161.417i q^{5} -562.144 q^{7} +(239.124 - 688.666i) q^{9} +1453.91i q^{11} -1097.86 q^{13} +(-2526.25 - 3551.39i) q^{15} -4056.19i q^{17} -1575.09 q^{19} +(-12368.0 + 8797.83i) q^{21} -11564.2i q^{23} -10430.3 q^{25} +(-5516.88 - 18894.0i) q^{27} -37282.8i q^{29} +2341.99 q^{31} +(22754.4 + 31988.1i) q^{33} +90739.4i q^{35} +68041.9 q^{37} +(-24154.6 + 17182.1i) q^{39} +36036.8i q^{41} +101347. q^{43} +(-111162. - 38598.7i) q^{45} -5036.59i q^{47} +198357. q^{49} +(-63481.3 - 89241.9i) q^{51} +109585. i q^{53} +234686. q^{55} +(-34654.3 + 24651.0i) q^{57} -250830. i q^{59} -318142. q^{61} +(-134422. + 387129. i) q^{63} +177214. i q^{65} -226749. q^{67} +(-180985. - 254428. i) q^{69} -235944. i q^{71} +694381. q^{73} +(-229482. + 163240. i) q^{75} -817308. i q^{77} -205769. q^{79} +(-417080. - 329354. i) q^{81} +126777. i q^{83} -654736. q^{85} +(-583495. - 820275. i) q^{87} +1.03431e6i q^{89} +617158. q^{91} +(51527.0 - 36653.2i) q^{93} +254246. i q^{95} +317833. q^{97} +(1.00126e6 + 347666. i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 10 q^{3} - 156 q^{7} - 74 q^{9}+O(q^{10}) $ 6 * q + 10 * q^3 - 156 * q^7 - 74 * q^9	$ 6 q + 10 q^{3} - 156 q^{7} - 74 q^{9} + 156 q^{13} - 2912 q^{15} + 4500 q^{19} - 15108 q^{21} + 21366 q^{25} - 37574 q^{27} + 74244 q^{31} - 83104 q^{33} + 171132 q^{37} - 200444 q^{39} + 291060 q^{43} - 355136 q^{45} + 517746 q^{49} - 452224 q^{51} + 748224 q^{55} - 650420 q^{57} + 592092 q^{61} - 1009788 q^{63} + 570900 q^{67} - 981184 q^{69} + 1119660 q^{73} - 521446 q^{75} + 1053636 q^{79} - 742874 q^{81} + 197376 q^{85} - 1251360 q^{87} - 839640 q^{91} + 354652 q^{93} - 798516 q^{97} + 2849600 q^{99}+O(q^{100}) $ 6 * q + 10 * q^3 - 156 * q^7 - 74 * q^9 + 156 * q^13 - 2912 * q^15 + 4500 * q^19 - 15108 * q^21 + 21366 * q^25 - 37574 * q^27 + 74244 * q^31 - 83104 * q^33 + 171132 * q^37 - 200444 * q^39 + 291060 * q^43 - 355136 * q^45 + 517746 * q^49 - 452224 * q^51 + 748224 * q^55 - 650420 * q^57 + 592092 * q^61 - 1009788 * q^63 + 570900 * q^67 - 981184 * q^69 + 1119660 * q^73 - 521446 * q^75 + 1053636 * q^79 - 742874 * q^81 + 197376 * q^85 - 1251360 * q^87 - 839640 * q^91 + 354652 * q^93 - 798516 * q^97 + 2849600 * 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
$3$	22.0014	−	15.6505i	0.814867	−	0.579648i
$3$	22.0014	−	15.6505i	0.814867	−	0.579648i
$4$		0			0
$5$		−	161.417i		−	1.29133i	−0.763620	−	0.645666i	$-0.776580\pi$
$5$		−	161.417i		−	1.29133i	0.763620	−	0.645666i	$-0.223420\pi$
$6$		0			0
$7$	−562.144			−1.63890			−0.819452	−	0.573147i	$-0.805722\pi$
$7$	−562.144			−1.63890			−0.819452	+	0.573147i	$0.805722\pi$
$8$		0			0
$9$	239.124	−	688.666i	0.328017	−	0.944672i
$10$		0			0
$11$			1453.91i			1.09235i	0.837673	+	0.546173i	$0.183916\pi$
$11$			1453.91i			1.09235i	−0.837673	+	0.546173i	$0.816084\pi$
$12$		0			0
$13$	−1097.86			−0.499711			−0.249855	−	0.968283i	$-0.580383\pi$
$13$	−1097.86			−0.499711			−0.249855	+	0.968283i	$0.580383\pi$
$14$		0			0
$15$	−2526.25	−	3551.39i	−0.748518	−	1.05226i
$16$		0			0
$17$		−	4056.19i		−	0.825603i	−0.910821	−	0.412801i	$-0.864550\pi$
$17$		−	4056.19i		−	0.825603i	0.910821	−	0.412801i	$-0.135450\pi$
$18$		0			0
$19$	−1575.09			−0.229639			−0.114819	−	0.993386i	$-0.536629\pi$
$19$	−1575.09			−0.229639			−0.114819	+	0.993386i	$0.536629\pi$
$20$		0			0
$21$	−12368.0	+	8797.83i	−1.33549	+	0.949987i
$22$		0			0
$23$		−	11564.2i		−	0.950455i	−0.879863	−	0.475227i	$-0.842366\pi$
$23$		−	11564.2i		−	0.950455i	0.879863	−	0.475227i	$-0.157634\pi$
$24$		0			0
$25$	−10430.3			−0.667541
$26$		0			0
$27$	−5516.88	−	18894.0i	−0.280286	−	0.959916i
$28$		0			0
$29$		−	37282.8i		−	1.52867i	−0.644817	−	0.764337i	$-0.723066\pi$
$29$		−	37282.8i		−	1.52867i	0.644817	−	0.764337i	$-0.276934\pi$
$30$		0			0
$31$	2341.99			0.0786139			0.0393069	−	0.999227i	$-0.487485\pi$
$31$	2341.99			0.0786139			0.0393069	+	0.999227i	$0.487485\pi$
$32$		0			0
$33$	22754.4	+	31988.1i	0.633176	+	0.890116i
$34$		0			0
$35$			90739.4i			2.11637i
$36$		0			0
$37$	68041.9			1.34330			0.671648	−	0.740871i	$-0.265587\pi$
$37$	68041.9			1.34330			0.671648	+	0.740871i	$0.265587\pi$
$38$		0			0
$39$	−24154.6	+	17182.1i	−0.407198	+	0.289656i
$40$		0			0
$41$			36036.8i			0.522871i	0.965221	+	0.261435i	$0.0841958\pi$
$41$			36036.8i			0.522871i	−0.965221	+	0.261435i	$0.915804\pi$
$42$		0			0
$43$	101347.			1.27469			0.637344	−	0.770579i	$-0.280033\pi$
$43$	101347.			1.27469			0.637344	+	0.770579i	$0.280033\pi$
$44$		0			0
$45$	−111162.	−	38598.7i	−1.21989	−	0.423579i
$46$		0			0
$47$		−	5036.59i		−	0.0485113i	−0.999706	−	0.0242557i	$-0.992278\pi$
$47$		−	5036.59i		−	0.0485113i	0.999706	−	0.0242557i	$-0.00772157\pi$
$48$		0			0
$49$	198357.			1.68601
$50$		0			0
$51$	−63481.3	−	89241.9i	−0.478559	−	0.672757i
$52$		0			0
$53$			109585.i			0.736076i	0.929811	+	0.368038i	$0.119970\pi$
$53$			109585.i			0.736076i	−0.929811	+	0.368038i	$0.880030\pi$
$54$		0			0
$55$	234686.			1.41058
$56$		0			0
$57$	−34654.3	+	24651.0i	−0.187125	+	0.133110i
$58$		0			0
$59$		−	250830.i		−	1.22130i	−0.791899	−	0.610652i	$-0.790908\pi$
$59$		−	250830.i		−	1.22130i	0.791899	−	0.610652i	$-0.209092\pi$
$60$		0			0
$61$	−318142.			−1.40162			−0.700812	−	0.713346i	$-0.747179\pi$
$61$	−318142.			−1.40162			−0.700812	+	0.713346i	$0.747179\pi$
$62$		0			0
$63$	−134422.	+	387129.i	−0.537589	+	1.54823i
$64$		0			0
$65$			177214.i			0.645293i
$66$		0			0
$67$	−226749.			−0.753914			−0.376957	−	0.926231i	$-0.623030\pi$
$67$	−226749.			−0.753914			−0.376957	+	0.926231i	$0.623030\pi$
$68$		0			0
$69$	−180985.	−	254428.i	−0.550929	−	0.774494i
$70$		0			0
$71$		−	235944.i		−	0.659227i	−0.944116	−	0.329613i	$-0.893082\pi$
$71$		−	235944.i		−	0.659227i	0.944116	−	0.329613i	$-0.106918\pi$
$72$		0			0
$73$	694381.			1.78496			0.892481	−	0.451085i	$-0.148963\pi$
$73$	694381.			1.78496			0.892481	+	0.451085i	$0.148963\pi$
$74$		0			0
$75$	−229482.	+	163240.i	−0.543957	+	0.386939i
$76$		0			0
$77$		−	817308.i		−	1.79025i
$78$		0			0
$79$	−205769.			−0.417348			−0.208674	−	0.977985i	$-0.566915\pi$
$79$	−205769.			−0.417348			−0.208674	+	0.977985i	$0.566915\pi$
$80$		0			0
$81$	−417080.	−	329354.i	−0.784810	−	0.619737i
$82$		0			0
$83$			126777.i			0.221721i	0.993836	+	0.110860i	$0.0353606\pi$
$83$			126777.i			0.221721i	−0.993836	+	0.110860i	$0.964639\pi$
$84$		0			0
$85$	−654736.			−1.06613
$86$		0			0
$87$	−583495.	−	820275.i	−0.886092	−	1.24567i
$88$		0			0
$89$			1.03431e6i			1.46717i	0.679596	+	0.733587i	$0.262155\pi$
$89$			1.03431e6i			1.46717i	−0.679596	+	0.733587i	$0.737845\pi$
$90$		0			0
$91$	617158.			0.818978
$92$		0			0
$93$	51527.0	−	36653.2i	0.0640599	−	0.0455684i
$94$		0			0
$95$			254246.i			0.296540i
$96$		0			0
$97$	317833.			0.348244			0.174122	−	0.984724i	$-0.444291\pi$
$97$	317833.			0.348244			0.174122	+	0.984724i	$0.444291\pi$
$98$		0			0
$99$	1.00126e6	+	347666.i	1.03191	+	0.358308i
$100$		0			0
$101$			239045.i			0.232014i	0.993248	+	0.116007i	$0.0370096\pi$
$101$			239045.i			0.232014i	−0.993248	+	0.116007i	$0.962990\pi$
$102$		0			0
$103$	669879.			0.613034			0.306517	−	0.951865i	$-0.400836\pi$
$103$	669879.			0.613034			0.306517	+	0.951865i	$0.400836\pi$
$104$		0			0
$105$	1.42012e6	+	1.99640e6i	1.22675	+	1.72456i
$106$		0			0
$107$			503132.i			0.410706i	0.978688	+	0.205353i	$0.0658342\pi$
$107$			503132.i			0.410706i	−0.978688	+	0.205353i	$0.934166\pi$
$108$		0			0
$109$	−87225.3			−0.0673539			−0.0336770	−	0.999433i	$-0.510722\pi$
$109$	−87225.3			−0.0673539			−0.0336770	+	0.999433i	$0.510722\pi$
$110$		0			0
$111$	1.49702e6	−	1.06489e6i	1.09461	−	0.778638i
$112$		0			0
$113$		−	1.55492e6i		−	1.07764i	−0.842421	−	0.538820i	$-0.818870\pi$
$113$		−	1.55492e6i		−	1.07764i	0.842421	−	0.538820i	$-0.181130\pi$
$114$		0			0
$115$	−1.86665e6			−1.22735
$116$		0			0
$117$	−262526.	+	756061.i	−0.163914	+	0.472063i
$118$		0			0
$119$			2.28016e6i			1.35308i
$120$		0			0
$121$	−342299.			−0.193219
$122$		0			0
$123$	563993.	+	792860.i	0.303081	+	0.426070i
$124$		0			0
$125$		−	838507.i		−	0.429315i
$126$		0			0
$127$	−125890.			−0.0614581			−0.0307290	−	0.999528i	$-0.509783\pi$
$127$	−125890.			−0.0614581			−0.0307290	+	0.999528i	$0.509783\pi$
$128$		0			0
$129$	2.22977e6	−	1.58612e6i	1.03870	−	0.738870i
$130$		0			0
$131$		−	1.44014e6i		−	0.640606i	−0.947315	−	0.320303i	$-0.896215\pi$
$131$		−	1.44014e6i		−	0.640606i	0.947315	−	0.320303i	$-0.103785\pi$
$132$		0			0
$133$	885429.			0.376356
$134$		0			0
$135$	−3.04981e6	+	890516.i	−1.23957	+	0.361943i
$136$		0			0
$137$		−	1.68509e6i		−	0.655333i	−0.944793	−	0.327667i	$-0.893738\pi$
$137$		−	1.68509e6i		−	0.655333i	0.944793	−	0.327667i	$-0.106262\pi$
$138$		0			0
$139$	−1.50853e6			−0.561705			−0.280853	−	0.959751i	$-0.590617\pi$
$139$	−1.50853e6			−0.561705			−0.280853	+	0.959751i	$0.590617\pi$
$140$		0			0
$141$	−78825.1	−	110812.i	−0.0281195	−	0.0395303i
$142$		0			0
$143$		−	1.59620e6i		−	0.545857i
$144$		0			0
$145$	−6.01807e6			−1.97403
$146$		0			0
$147$	4.36414e6	−	3.10439e6i	1.37387	−	0.977291i
$148$		0			0
$149$			3.79005e6i			1.14574i	0.819646	+	0.572870i	$0.194170\pi$
$149$			3.79005e6i			1.14574i	−0.819646	+	0.572870i	$0.805830\pi$
$150$		0			0
$151$	−242876.			−0.0705429			−0.0352714	−	0.999378i	$-0.511230\pi$
$151$	−242876.			−0.0705429			−0.0352714	+	0.999378i	$0.511230\pi$
$152$		0			0
$153$	−2.79336e6	−	969934.i	−0.779924	−	0.270812i
$154$		0			0
$155$		−	378036.i		−	0.101517i
$156$		0			0
$157$	−658703.			−0.170212			−0.0851062	−	0.996372i	$-0.527123\pi$
$157$	−658703.			−0.170212			−0.0851062	+	0.996372i	$0.527123\pi$
$158$		0			0
$159$	1.71506e6	+	2.41102e6i	0.426665	+	0.599804i
$160$		0			0
$161$			6.50074e6i			1.55770i
$162$		0			0
$163$	−7.86756e6			−1.81668			−0.908338	−	0.418237i	$-0.862648\pi$
$163$	−7.86756e6			−1.81668			−0.908338	+	0.418237i	$0.862648\pi$
$164$		0			0
$165$	5.16341e6	−	3.67294e6i	1.14944	−	0.817640i
$166$		0			0
$167$		−	5.45397e6i		−	1.17102i	−0.810666	−	0.585509i	$-0.800895\pi$
$167$		−	5.45397e6i		−	1.17102i	0.810666	−	0.585509i	$-0.199105\pi$
$168$		0			0
$169$	−3.62150e6			−0.750289
$170$		0			0
$171$	−376643.	+	1.08471e6i	−0.0753254	+	0.216933i
$172$		0			0
$173$		−	2.88712e6i		−	0.557606i	−0.960348	−	0.278803i	$-0.910062\pi$
$173$		−	2.88712e6i		−	0.557606i	0.960348	−	0.278803i	$-0.0899376\pi$
$174$		0			0
$175$	5.86335e6			1.09404
$176$		0			0
$177$	−3.92561e6	−	5.51862e6i	−0.707926	−	0.995200i
$178$		0			0
$179$		−	3.43541e6i		−	0.598991i	−0.954098	−	0.299495i	$-0.903182\pi$
$179$		−	3.43541e6i		−	0.598991i	0.954098	−	0.299495i	$-0.0968183\pi$
$180$		0			0
$181$	5.27169e6			0.889025			0.444512	−	0.895773i	$-0.353377\pi$
$181$	5.27169e6			0.889025			0.444512	+	0.895773i	$0.353377\pi$
$182$		0			0
$183$	−6.99957e6	+	4.97908e6i	−1.14214	+	0.812448i
$184$		0			0
$185$		−	1.09831e7i		−	1.73464i
$186$		0			0
$187$	5.89734e6			0.901844
$188$		0			0
$189$	3.10128e6	+	1.06212e7i	0.459363	+	1.57321i
$190$		0			0
$191$			5.29632e6i			0.760106i	0.924965	+	0.380053i	$0.124094\pi$
$191$			5.29632e6i			0.760106i	−0.924965	+	0.380053i	$0.875906\pi$
$192$		0			0
$193$	1.23035e7			1.71143			0.855713	−	0.517451i	$-0.173119\pi$
$193$	1.23035e7			1.71143			0.855713	+	0.517451i	$0.173119\pi$
$194$		0			0
$195$	2.77348e6	+	3.89895e6i	0.374042	+	0.525828i
$196$		0			0
$197$			1.18358e7i			1.54810i	0.633126	+	0.774049i	$0.281772\pi$
$197$			1.18358e7i			1.54810i	−0.633126	+	0.774049i	$0.718228\pi$
$198$		0			0
$199$	5.84495e6			0.741688			0.370844	−	0.928695i	$-0.379068\pi$
$199$	5.84495e6			0.741688			0.370844	+	0.928695i	$0.379068\pi$
$200$		0			0
$201$	−4.98881e6	+	3.54874e6i	−0.614340	+	0.437005i
$202$		0			0
$203$			2.09583e7i			2.50535i
$204$		0			0
$205$	5.81694e6			0.675200
$206$		0			0
$207$	−7.96386e6	−	2.76528e6i	−0.897868	−	0.311765i
$208$		0			0
$209$		−	2.29004e6i		−	0.250845i
$210$		0			0
$211$	−3.33581e6			−0.355103			−0.177551	−	0.984112i	$-0.556818\pi$
$211$	−3.33581e6			−0.355103			−0.177551	+	0.984112i	$0.556818\pi$
$212$		0			0
$213$	−3.69265e6	−	5.19111e6i	−0.382119	−	0.537182i
$214$		0			0
$215$		−	1.63590e7i		−	1.64605i
$216$		0			0
$217$	−1.31653e6			−0.128841
$218$		0			0
$219$	1.52774e7	−	1.08674e7i	1.45451	−	1.03465i
$220$		0			0
$221$			4.45314e6i			0.412563i
$222$		0			0
$223$	1.36715e7			1.23282			0.616411	−	0.787425i	$-0.288586\pi$
$223$	1.36715e7			1.23282			0.616411	+	0.787425i	$0.288586\pi$
$224$		0			0
$225$	−2.49415e6	+	7.18301e6i	−0.218965	+	0.630607i
$226$		0			0
$227$		−	1.85519e6i		−	0.158603i	−0.996851	−	0.0793014i	$-0.974731\pi$
$227$		−	1.85519e6i		−	0.158603i	0.996851	−	0.0793014i	$-0.0252690\pi$
$228$		0			0
$229$	−4.40573e6			−0.366870			−0.183435	−	0.983032i	$-0.558722\pi$
$229$	−4.40573e6			−0.366870			−0.183435	+	0.983032i	$0.558722\pi$
$230$		0			0
$231$	−1.27913e7	−	1.79819e7i	−1.03771	−	1.45882i
$232$		0			0
$233$		−	1.57817e6i		−	0.124763i	−0.998052	−	0.0623816i	$-0.980130\pi$
$233$		−	1.57817e6i		−	0.124763i	0.998052	−	0.0623816i	$-0.0198696\pi$
$234$		0			0
$235$	−812989.			−0.0626443
$236$		0			0
$237$	−4.52721e6	+	3.22038e6i	−0.340083	+	0.241915i
$238$		0			0
$239$			2.47475e7i			1.81275i	0.422472	+	0.906376i	$0.361163\pi$
$239$			2.47475e7i			1.81275i	−0.422472	+	0.906376i	$0.638837\pi$
$240$		0			0
$241$	−1.59319e7			−1.13819			−0.569097	−	0.822271i	$-0.692707\pi$
$241$	−1.59319e7			−1.13819			−0.569097	+	0.822271i	$0.692707\pi$
$242$		0			0
$243$	−1.43309e7	−	718741.i	−0.998745	−	0.0500903i
$244$		0			0
$245$		−	3.20181e7i		−	2.17720i
$246$		0			0
$247$	1.72924e6			0.114753
$248$		0			0
$249$	1.98412e6	+	2.78927e6i	0.128520	+	0.180673i
$250$		0			0
$251$		−	3.00814e7i		−	1.90229i	−0.308747	−	0.951144i	$-0.599910\pi$
$251$		−	3.00814e7i		−	1.90229i	0.308747	−	0.951144i	$-0.400090\pi$
$252$		0			0
$253$	1.68133e7			1.03822
$254$		0			0
$255$	−1.44051e7	+	1.02469e7i	−0.868753	+	0.617979i
$256$		0			0
$257$			9.13866e6i			0.538373i	0.963088	+	0.269186i	$0.0867547\pi$
$257$			9.13866e6i			0.538373i	−0.963088	+	0.269186i	$0.913245\pi$
$258$		0			0
$259$	−3.82494e7			−2.20153
$260$		0			0
$261$	−2.56754e7	−	8.91524e6i	−1.44410	−	0.501431i
$262$		0			0
$263$			2.15883e7i			1.18673i	0.804934	+	0.593365i	$0.202201\pi$
$263$			2.15883e7i			1.18673i	−0.804934	+	0.593365i	$0.797799\pi$
$264$		0			0
$265$	1.76888e7			0.950519
$266$		0			0
$267$	1.61875e7	+	2.27563e7i	0.850444	+	1.19555i
$268$		0			0
$269$		−	2.61709e6i		−	0.134450i	−0.997738	−	0.0672250i	$-0.978585\pi$
$269$		−	2.61709e6i		−	0.134450i	0.997738	−	0.0672250i	$-0.0214145\pi$
$270$		0			0
$271$	3.23455e7			1.62520			0.812599	−	0.582823i	$-0.198052\pi$
$271$	3.23455e7			1.62520			0.812599	+	0.582823i	$0.198052\pi$
$272$		0			0
$273$	1.35784e7	−	9.65882e6i	0.667358	−	0.474719i
$274$		0			0
$275$		−	1.51648e7i		−	0.729185i
$276$		0			0
$277$	−1.38413e7			−0.651236			−0.325618	−	0.945501i	$-0.605572\pi$
$277$	−1.38413e7			−0.651236			−0.325618	+	0.945501i	$0.605572\pi$
$278$		0			0
$279$	560026.	−	1.61285e6i	0.0257867	−	0.0742643i
$280$		0			0
$281$			2.54444e6i			0.114676i	0.998355	+	0.0573382i	$0.0182613\pi$
$281$			2.54444e6i			0.114676i	−0.998355	+	0.0573382i	$0.981739\pi$
$282$		0			0
$283$	4.09696e6			0.180760			0.0903800	−	0.995907i	$-0.471192\pi$
$283$	4.09696e6			0.180760			0.0903800	+	0.995907i	$0.471192\pi$
$284$		0			0
$285$	3.97907e6	+	5.59377e6i	0.171889	+	0.241641i
$286$		0			0
$287$		−	2.02579e7i		−	0.856935i
$288$		0			0
$289$	7.68491e6			0.318380
$290$		0			0
$291$	6.99278e6	−	4.97424e6i	0.283773	−	0.201859i
$292$		0			0
$293$			9.97335e6i			0.396496i	0.980152	+	0.198248i	$0.0635251\pi$
$293$			9.97335e6i			0.396496i	−0.980152	+	0.198248i	$0.936475\pi$
$294$		0			0
$295$	−4.04881e7			−1.57711
$296$		0			0
$297$	2.74703e7	−	8.02105e6i	1.04856	−	0.306170i
$298$		0			0
$299$			1.26959e7i			0.474952i
$300$		0			0
$301$	−5.69714e7			−2.08909
$302$		0			0
$303$	3.74117e6	+	5.25932e6i	0.134487	+	0.189061i
$304$		0			0
$305$			5.13534e7i			1.80996i
$306$		0			0
$307$	1.67597e7			0.579229			0.289614	−	0.957143i	$-0.406473\pi$
$307$	1.67597e7			0.579229			0.289614	+	0.957143i	$0.406473\pi$
$308$		0			0
$309$	1.47383e7	−	1.04839e7i	0.499541	−	0.355344i
$310$		0			0
$311$		−	3.43802e7i		−	1.14295i	−0.820620	−	0.571475i	$-0.806372\pi$
$311$		−	3.43802e7i		−	1.14295i	0.820620	−	0.571475i	$-0.193628\pi$
$312$		0			0
$313$	7.32819e6			0.238981			0.119491	−	0.992835i	$-0.461874\pi$
$313$	7.32819e6			0.238981			0.119491	+	0.992835i	$0.461874\pi$
$314$		0			0
$315$	6.24891e7	+	2.16980e7i	1.99928	+	0.694206i
$316$		0			0
$317$			2.21528e7i			0.695425i	0.937601	+	0.347712i	$0.113041\pi$
$317$			2.21528e7i			0.695425i	−0.937601	+	0.347712i	$0.886959\pi$
$318$		0			0
$319$	5.42060e7			1.66984
$320$		0			0
$321$	7.87426e6	+	1.10696e7i	0.238065	+	0.334671i
$322$		0			0
$323$			6.38887e6i			0.189590i
$324$		0			0
$325$	1.14511e7			0.333577
$326$		0			0
$327$	−1.91908e6	+	1.36512e6i	−0.0548845	+	0.0390415i
$328$		0			0
$329$			2.83129e6i			0.0795054i
$330$		0			0
$331$	−3.71104e7			−1.02332			−0.511660	−	0.859188i	$-0.670969\pi$
$331$	−3.71104e7			−1.02332			−0.511660	+	0.859188i	$0.670969\pi$
$332$		0			0
$333$	1.62705e7	−	4.68582e7i	0.440624	−	1.26897i
$334$		0			0
$335$			3.66011e7i			0.973554i
$336$		0			0
$337$	6.41291e7			1.67558			0.837790	−	0.545992i	$-0.183847\pi$
$337$	6.41291e7			1.67558			0.837790	+	0.545992i	$0.183847\pi$
$338$		0			0
$339$	−2.43353e7	−	3.42105e7i	−0.624651	−	0.878133i
$340$		0			0
$341$			3.40504e6i			0.0858735i
$342$		0			0
$343$	−4.53696e7			−1.12430
$344$		0			0
$345$	−4.10690e7	+	2.92140e7i	−1.00013	+	0.711433i
$346$		0			0
$347$			8.19630e7i			1.96169i	0.194799	+	0.980843i	$0.437595\pi$
$347$			8.19630e7i			1.96169i	−0.194799	+	0.980843i	$0.562405\pi$
$348$		0			0
$349$	−4.31801e7			−1.01580			−0.507899	−	0.861417i	$-0.669578\pi$
$349$	−4.31801e7			−1.01580			−0.507899	+	0.861417i	$0.669578\pi$
$350$		0			0
$351$	6.05678e6	+	2.07431e7i	0.140062	+	0.479680i
$352$		0			0
$353$		−	7.86856e7i		−	1.78884i	−0.447229	−	0.894420i	$-0.647589\pi$
$353$		−	7.86856e7i		−	1.78884i	0.447229	−	0.894420i	$-0.352411\pi$
$354$		0			0
$355$	−3.80854e7			−0.851281
$356$		0			0
$357$	3.56857e7	+	5.01668e7i	0.784312	+	1.10258i
$358$		0			0
$359$		−	4.06169e7i		−	0.877857i	−0.898522	−	0.438929i	$-0.855358\pi$
$359$		−	4.06169e7i		−	0.877857i	0.898522	−	0.438929i	$-0.144642\pi$
$360$		0			0
$361$	−4.45650e7			−0.947266
$362$		0			0
$363$	−7.53105e6	+	5.35714e6i	−0.157448	+	0.111999i
$364$		0			0
$365$		−	1.12085e8i		−	2.30498i
$366$		0			0
$367$	7.10068e6			0.143649			0.0718243	−	0.997417i	$-0.477118\pi$
$367$	7.10068e6			0.143649			0.0718243	+	0.997417i	$0.477118\pi$
$368$		0			0
$369$	2.48173e7	+	8.61728e6i	0.493941	+	0.171511i
$370$		0			0
$371$		−	6.16025e7i		−	1.20636i
$372$		0			0
$373$	−6.93868e6			−0.133706			−0.0668529	−	0.997763i	$-0.521296\pi$
$373$	−6.93868e6			−0.133706			−0.0668529	+	0.997763i	$0.521296\pi$
$374$		0			0
$375$	−1.31230e7	−	1.84483e7i	−0.248852	−	0.349835i
$376$		0			0
$377$			4.09315e7i			0.763895i
$378$		0			0
$379$	1.37811e7			0.253143			0.126572	−	0.991957i	$-0.459603\pi$
$379$	1.37811e7			0.253143			0.126572	+	0.991957i	$0.459603\pi$
$380$		0			0
$381$	−2.76975e6	+	1.97024e6i	−0.0500802	+	0.0356240i
$382$		0			0
$383$		−	5.63601e7i		−	1.00317i	−0.865108	−	0.501586i	$-0.832750\pi$
$383$		−	5.63601e7i		−	1.00317i	0.865108	−	0.501586i	$-0.167250\pi$
$384$		0			0
$385$	−1.31927e8			−2.31181
$386$		0			0
$387$	2.42345e7	−	6.97939e7i	0.418119	−	1.20416i
$388$		0			0
$389$			1.13847e7i			0.193408i	0.995313	+	0.0967038i	$0.0308300\pi$
$389$			1.13847e7i			0.193408i	−0.995313	+	0.0967038i	$0.969170\pi$
$390$		0			0
$391$	−4.69065e7			−0.784698
$392$		0			0
$393$	−2.25389e7	−	3.16852e7i	−0.371326	−	0.522009i
$394$		0			0
$395$			3.32145e7i			0.538935i
$396$		0			0
$397$	4.77423e7			0.763013			0.381507	−	0.924366i	$-0.375405\pi$
$397$	4.77423e7			0.763013			0.381507	+	0.924366i	$0.375405\pi$
$398$		0			0
$399$	1.94807e7	−	1.38574e7i	0.306680	−	0.218154i
$400$		0			0
$401$		−	5.80283e7i		−	0.899926i	−0.893047	−	0.449963i	$-0.851437\pi$
$401$		−	5.80283e7i		−	0.899926i	0.893047	−	0.449963i	$-0.148563\pi$
$402$		0			0
$403$	−2.57118e6			−0.0392842
$404$		0			0
$405$	−5.31632e7	+	6.73236e7i	−0.800287	+	1.01345i
$406$		0			0
$407$			9.89270e7i			1.46734i
$408$		0			0
$409$	8.86645e7			1.29592			0.647962	−	0.761672i	$-0.275621\pi$
$409$	8.86645e7			1.29592			0.647962	+	0.761672i	$0.275621\pi$
$410$		0			0
$411$	−2.63725e7	−	3.70744e7i	−0.379862	−	0.534009i
$412$		0			0
$413$			1.41003e8i			2.00160i
$414$		0			0
$415$	2.04639e7			0.286315
$416$		0			0
$417$	−3.31897e7	+	2.36092e7i	−0.457715	+	0.325591i
$418$		0			0
$419$			3.24299e6i			0.0440863i	0.999757	+	0.0220431i	$0.00701712\pi$
$419$			3.24299e6i			0.0440863i	−0.999757	+	0.0220431i	$0.992983\pi$
$420$		0			0
$421$	7.38808e7			0.990114			0.495057	−	0.868861i	$-0.335147\pi$
$421$	7.38808e7			0.990114			0.495057	+	0.868861i	$0.335147\pi$
$422$		0			0
$423$	−3.46853e6	−	1.20437e6i	−0.0458273	−	0.0159125i
$424$		0			0
$425$			4.23074e7i			0.551124i
$426$		0			0
$427$	1.78842e8			2.29713
$428$		0			0
$429$	−2.49813e7	−	3.51186e7i	−0.316405	−	0.444801i
$430$		0			0
$431$			8.59791e7i			1.07389i	0.843616	+	0.536946i	$0.180422\pi$
$431$			8.59791e7i			1.07389i	−0.843616	+	0.536946i	$0.819578\pi$
$432$		0			0
$433$	−3.96565e7			−0.488485			−0.244242	−	0.969714i	$-0.578539\pi$
$433$	−3.96565e7			−0.488485			−0.244242	+	0.969714i	$0.578539\pi$
$434$		0			0
$435$	−1.32406e8	+	9.41857e7i	−1.60857	+	1.14424i
$436$		0			0
$437$			1.82147e7i			0.218261i
$438$		0			0
$439$	4.24613e7			0.501879			0.250940	−	0.968003i	$-0.419260\pi$
$439$	4.24613e7			0.501879			0.250940	+	0.968003i	$0.419260\pi$
$440$		0			0
$441$	4.74321e7	−	1.36602e8i	0.553039	−	1.59272i
$442$		0			0
$443$		−	1.03857e8i		−	1.19460i	−0.802017	−	0.597301i	$-0.796240\pi$
$443$		−	1.03857e8i		−	1.19460i	0.802017	−	0.597301i	$-0.203760\pi$
$444$		0			0
$445$	1.66955e8			1.89461
$446$		0			0
$447$	5.93161e7	+	8.33865e7i	0.664126	+	0.933626i
$448$		0			0
$449$		−	1.60326e8i		−	1.77118i	−0.464464	−	0.885592i	$-0.653753\pi$
$449$		−	1.60326e8i		−	1.77118i	0.464464	−	0.885592i	$-0.346247\pi$
$450$		0			0
$451$	−5.23943e7			−0.571156
$452$		0			0
$453$	−5.34361e6	+	3.80112e6i	−0.0574831	+	0.0408900i
$454$		0			0
$455$		−	9.96196e7i		−	1.05757i
$456$		0			0
$457$	−1.33118e7			−0.139472			−0.0697361	−	0.997565i	$-0.522216\pi$
$457$	−1.33118e7			−0.139472			−0.0697361	+	0.997565i	$0.522216\pi$
$458$		0			0
$459$	−7.66377e7	+	2.23775e7i	−0.792510	+	0.231405i
$460$		0			0
$461$			4.73010e7i			0.482800i	0.970426	+	0.241400i	$0.0776066\pi$
$461$			4.73010e7i			0.482800i	−0.970426	+	0.241400i	$0.922393\pi$
$462$		0			0
$463$	1.88140e7			0.189556			0.0947779	−	0.995498i	$-0.469786\pi$
$463$	1.88140e7			0.189556			0.0947779	+	0.995498i	$0.469786\pi$
$464$		0			0
$465$	−5.91644e6	−	8.31732e6i	−0.0588439	−	0.0827226i
$466$		0			0
$467$			3.39635e7i			0.333474i	0.986001	+	0.166737i	$0.0533231\pi$
$467$			3.39635e7i			0.333474i	−0.986001	+	0.166737i	$0.946677\pi$
$468$		0			0
$469$	1.27466e8			1.23559
$470$		0			0
$471$	−1.44924e7	+	1.03090e7i	−0.138700	+	0.0986632i
$472$		0			0
$473$			1.47349e8i			1.39240i
$474$		0			0
$475$	1.64287e7			0.153293
$476$		0			0
$477$	7.54673e7	+	2.62044e7i	0.695350	+	0.241446i
$478$		0			0
$479$			7.85728e7i			0.714934i	0.933926	+	0.357467i	$0.116359\pi$
$479$			7.85728e7i			0.714934i	−0.933926	+	0.357467i	$0.883641\pi$
$480$		0			0
$481$	−7.47008e7			−0.671259
$482$		0			0
$483$	1.01740e8	+	1.43025e8i	0.902920	+	1.26932i
$484$		0			0
$485$		−	5.13035e7i		−	0.449699i
$486$		0			0
$487$	2.12816e7			0.184254			0.0921269	−	0.995747i	$-0.470633\pi$
$487$	2.12816e7			0.184254			0.0921269	+	0.995747i	$0.470633\pi$
$488$		0			0
$489$	−1.73098e8	+	1.23131e8i	−1.48035	+	1.05303i
$490$		0			0
$491$		−	8.04568e7i		−	0.679701i	−0.940479	−	0.339851i	$-0.889623\pi$
$491$		−	8.04568e7i		−	0.679701i	0.940479	−	0.339851i	$-0.110377\pi$
$492$		0			0
$493$	−1.51226e8			−1.26208
$494$		0			0
$495$	5.61191e7	−	1.61620e8i	0.462695	−	1.33254i
$496$		0			0
$497$			1.32635e8i			1.08041i
$498$		0			0
$499$	1.60696e8			1.29331			0.646656	−	0.762782i	$-0.276167\pi$
$499$	1.60696e8			1.29331			0.646656	+	0.762782i	$0.276167\pi$
$500$		0			0
$501$	−8.53574e7	−	1.19995e8i	−0.678778	−	0.954224i
$502$		0			0
$503$		−	1.56985e8i		−	1.23354i	−0.787142	−	0.616771i	$-0.788440\pi$
$503$		−	1.56985e8i		−	1.23354i	0.787142	−	0.616771i	$-0.211560\pi$
$504$		0			0
$505$	3.85858e7			0.299608
$506$		0			0
$507$	−7.96782e7	+	5.66783e7i	−0.611386	+	0.434903i
$508$		0			0
$509$			1.05504e8i			0.800050i	0.916504	+	0.400025i	$0.130999\pi$
$509$			1.05504e8i			0.800050i	−0.916504	+	0.400025i	$0.869001\pi$
$510$		0			0
$511$	−3.90342e8			−2.92538
$512$		0			0
$513$	8.68959e6	+	2.97598e7i	0.0643646	+	0.220434i
$514$		0			0
$515$		−	1.08130e8i		−	0.791631i
$516$		0			0
$517$	7.32276e6			0.0529911
$518$		0			0
$519$	−4.51849e7	−	6.35208e7i	−0.323215	−	0.454375i
$520$		0			0
$521$		−	2.70219e8i		−	1.91075i	−0.295400	−	0.955374i	$-0.595453\pi$
$521$		−	2.70219e8i		−	1.91075i	0.295400	−	0.955374i	$-0.404547\pi$
$522$		0			0
$523$	2.43442e8			1.70173			0.850863	−	0.525387i	$-0.176079\pi$
$523$	2.43442e8			1.70173			0.850863	+	0.525387i	$0.176079\pi$
$524$		0			0
$525$	1.29002e8	−	9.17643e7i	0.891494	−	0.634155i
$526$		0			0
$527$		−	9.49954e6i		−	0.0649039i
$528$		0			0
$529$	1.43056e7			0.0966358
$530$		0			0
$531$	−1.72738e8	−	5.99796e7i	−1.15373	−	0.400608i
$532$		0			0
$533$		−	3.95635e7i		−	0.261284i
$534$		0			0
$535$	8.12139e7			0.530358
$536$		0			0
$537$	−5.37659e7	−	7.55840e7i	−0.347203	−	0.488098i
$538$		0			0
$539$			2.88394e8i			1.84170i
$540$		0			0
$541$	−2.43819e8			−1.53984			−0.769920	−	0.638140i	$-0.779704\pi$
$541$	−2.43819e8			−1.53984			−0.769920	+	0.638140i	$0.779704\pi$
$542$		0			0
$543$	1.15985e8	−	8.25044e7i	0.724437	−	0.515321i
$544$		0			0
$545$			1.40796e7i			0.0869763i
$546$		0			0
$547$	−2.50520e8			−1.53067			−0.765333	−	0.643634i	$-0.777426\pi$
$547$	−2.50520e8			−1.53067			−0.765333	+	0.643634i	$0.777426\pi$
$548$		0			0
$549$	−7.60755e7	+	2.19094e8i	−0.459757	+	1.32407i
$550$		0			0
$551$			5.87239e7i			0.351043i
$552$		0			0
$553$	1.15672e8			0.683994
$554$		0			0
$555$	−1.71891e8	−	2.41644e8i	−1.00548	−	1.41350i
$556$		0			0
$557$		−	2.69462e8i		−	1.55931i	−0.626210	−	0.779654i	$-0.715395\pi$
$557$		−	2.69462e8i		−	1.55931i	0.626210	−	0.779654i	$-0.284605\pi$
$558$		0			0
$559$	−1.11265e8			−0.636975
$560$		0			0
$561$	1.29750e8	−	9.22962e7i	0.734883	−	0.522752i
$562$		0			0
$563$			1.85501e8i			1.03949i	0.854322	+	0.519745i	$0.173973\pi$
$563$			1.85501e8i			1.03949i	−0.854322	+	0.519745i	$0.826027\pi$
$564$		0			0
$565$	−2.50990e8			−1.39159
$566$		0			0
$567$	2.34459e8	+	1.85144e8i	1.28623	+	1.01569i
$568$		0			0
$569$		−	7.09949e7i		−	0.385381i	−0.981260	−	0.192691i	$-0.938279\pi$
$569$		−	7.09949e7i		−	0.385381i	0.981260	−	0.192691i	$-0.0617214\pi$
$570$		0			0
$571$	−2.08845e8			−1.12180			−0.560900	−	0.827883i	$-0.689545\pi$
$571$	−2.08845e8			−1.12180			−0.560900	+	0.827883i	$0.689545\pi$
$572$		0			0
$573$	8.28900e7	+	1.16527e8i	0.440594	+	0.619385i
$574$		0			0
$575$			1.20618e8i			0.634467i
$576$		0			0
$577$	3.43247e8			1.78681			0.893407	−	0.449249i	$-0.148308\pi$
$577$	3.43247e8			1.78681			0.893407	+	0.449249i	$0.148308\pi$
$578$		0			0
$579$	2.70695e8	−	1.92556e8i	1.39458	−	0.992024i
$580$		0			0
$581$		−	7.12670e7i		−	0.363379i
$582$		0			0
$583$	−1.59327e8			−0.804049
$584$		0			0
$585$	1.22041e8	+	4.23761e7i	0.609590	+	0.211667i
$586$		0			0
$587$		−	2.80621e8i		−	1.38741i	−0.720257	−	0.693707i	$-0.755976\pi$
$587$		−	2.80621e8i		−	1.38741i	0.720257	−	0.693707i	$-0.244024\pi$
$588$		0			0
$589$	−3.68884e6			−0.0180528
$590$		0			0
$591$	1.85236e8	+	2.60404e8i	0.897351	+	1.26149i
$592$		0			0
$593$			3.30407e8i			1.58448i	0.610213	+	0.792238i	$0.291084\pi$
$593$			3.30407e8i			1.58448i	−0.610213	+	0.792238i	$0.708916\pi$
$594$		0			0
$595$	3.68056e8			1.74728
$596$		0			0
$597$	1.28597e8	−	9.14763e7i	0.604377	−	0.429918i
$598$		0			0
$599$			2.96086e8i			1.37764i	0.724930	+	0.688822i	$0.241872\pi$
$599$			2.96086e8i			1.37764i	−0.724930	+	0.688822i	$0.758128\pi$
$600$		0			0
$601$	−2.23246e8			−1.02839			−0.514197	−	0.857672i	$-0.671910\pi$
$601$	−2.23246e8			−1.02839			−0.514197	+	0.857672i	$0.671910\pi$
$602$		0			0
$603$	−5.42213e7	+	1.56155e8i	−0.247297	+	0.712201i
$604$		0			0
$605$			5.52527e7i			0.249510i
$606$		0			0
$607$	9.25728e7			0.413921			0.206960	−	0.978349i	$-0.433643\pi$
$607$	9.25728e7			0.413921			0.206960	+	0.978349i	$0.433643\pi$
$608$		0			0
$609$	3.28008e8	+	4.61113e8i	1.45222	+	2.04153i
$610$		0			0
$611$			5.52949e6i			0.0242416i
$612$		0			0
$613$	9.84939e7			0.427590			0.213795	−	0.976878i	$-0.431417\pi$
$613$	9.84939e7			0.427590			0.213795	+	0.976878i	$0.431417\pi$
$614$		0			0
$615$	1.27981e8	−	9.10379e7i	0.550199	−	0.391378i
$616$		0			0
$617$			1.19884e8i			0.510395i	0.966889	+	0.255197i	$0.0821404\pi$
$617$			1.19884e8i			0.510395i	−0.966889	+	0.255197i	$0.917860\pi$
$618$		0			0
$619$	−2.87961e7			−0.121412			−0.0607061	−	0.998156i	$-0.519335\pi$
$619$	−2.87961e7			−0.121412			−0.0607061	+	0.998156i	$0.519335\pi$
$620$		0			0
$621$	−2.18494e8	+	6.37982e7i	−0.912357	+	0.266400i
$622$		0			0
$623$		−	5.81432e8i		−	2.40456i
$624$		0			0
$625$	−2.98323e8			−1.22193
$626$		0			0
$627$	−3.58403e7	−	5.03842e7i	−0.145402	−	0.204405i
$628$		0			0
$629$		−	2.75991e8i		−	1.10903i
$630$		0			0
$631$	7.14913e7			0.284554			0.142277	−	0.989827i	$-0.454558\pi$
$631$	7.14913e7			0.284554			0.142277	+	0.989827i	$0.454558\pi$
$632$		0			0
$633$	−7.33925e7	+	5.22071e7i	−0.289362	+	0.205834i
$634$		0			0
$635$			2.03207e7i			0.0793629i
$636$		0			0
$637$	−2.17769e8			−0.842516
$638$		0			0
$639$	−1.62487e8	−	5.64201e7i	−0.622753	−	0.216238i
$640$		0			0
$641$			2.08226e8i			0.790607i	0.918551	+	0.395304i	$0.129361\pi$
$641$			2.08226e8i			0.790607i	−0.918551	+	0.395304i	$0.870639\pi$
$642$		0			0
$643$	3.03410e8			1.14129			0.570646	−	0.821196i	$-0.306693\pi$
$643$	3.03410e8			1.14129			0.570646	+	0.821196i	$0.306693\pi$
$644$		0			0
$645$	−2.56027e8	−	3.59922e8i	−0.954127	−	1.34131i
$646$		0			0
$647$			2.26989e8i			0.838094i	0.907965	+	0.419047i	$0.137636\pi$
$647$			2.26989e8i			0.838094i	−0.907965	+	0.419047i	$0.862364\pi$
$648$		0			0
$649$	3.64685e8			1.33408
$650$		0			0
$651$	−2.89656e7	+	2.06044e7i	−0.104988	+	0.0746822i
$652$		0			0
$653$			2.83435e8i			1.01792i	0.860790	+	0.508960i	$0.169970\pi$
$653$			2.83435e8i			1.01792i	−0.860790	+	0.508960i	$0.830030\pi$
$654$		0			0
$655$	−2.32463e8			−0.827236
$656$		0			0
$657$	1.66043e8	−	4.78196e8i	0.585498	−	1.68620i
$658$		0			0
$659$			2.42021e8i			0.845662i	0.906209	+	0.422831i	$0.138964\pi$
$659$			2.42021e8i			0.845662i	−0.906209	+	0.422831i	$0.861036\pi$
$660$		0			0
$661$	4.96002e8			1.71743			0.858715	−	0.512454i	$-0.171263\pi$
$661$	4.96002e8			1.71743			0.858715	+	0.512454i	$0.171263\pi$
$662$		0			0
$663$	6.96939e7	+	9.79754e7i	0.239141	+	0.336184i
$664$		0			0
$665$		−	1.42923e8i		−	0.486001i
$666$		0			0
$667$	−4.31145e8			−1.45294
$668$		0			0
$669$	3.00791e8	−	2.13965e8i	1.00459	−	0.714602i
$670$		0			0
$671$		−	4.62550e8i		−	1.53106i
$672$		0			0
$673$	−2.38665e8			−0.782968			−0.391484	−	0.920185i	$-0.628038\pi$
$673$	−2.38665e8			−0.782968			−0.391484	+	0.920185i	$0.628038\pi$
$674$		0			0
$675$	5.75428e7	+	1.97071e8i	0.187103	+	0.640783i
$676$		0			0
$677$			2.78503e8i			0.897561i	0.893642	+	0.448781i	$0.148142\pi$
$677$			2.78503e8i			0.897561i	−0.893642	+	0.448781i	$0.851858\pi$
$678$		0			0
$679$	−1.78668e8			−0.570739
$680$		0			0
$681$	−2.90346e7	−	4.08168e7i	−0.0919337	−	0.129240i
$682$		0			0
$683$		−	7.48322e7i		−	0.234869i	−0.993081	−	0.117435i	$-0.962533\pi$
$683$		−	7.48322e7i		−	0.234869i	0.993081	−	0.117435i	$-0.0374671\pi$
$684$		0			0
$685$	−2.72002e8			−0.846253
$686$		0			0
$687$	−9.69324e7	+	6.89519e7i	−0.298950	+	0.212655i
$688$		0			0
$689$		−	1.20309e8i		−	0.367825i
$690$		0			0
$691$	6.49944e7			0.196989			0.0984944	−	0.995138i	$-0.468597\pi$
$691$	6.49944e7			0.196989			0.0984944	+	0.995138i	$0.468597\pi$
$692$		0			0
$693$	−5.62852e8	−	1.95438e8i	−1.69120	−	0.587233i
$694$		0			0
$695$			2.43501e8i			0.725349i
$696$		0			0
$697$	1.46172e8			0.431684
$698$		0			0
$699$	−2.46992e7	−	3.47220e7i	−0.0723188	−	0.101666i
$700$		0			0
$701$		−	3.56669e8i		−	1.03541i	−0.855560	−	0.517704i	$-0.826787\pi$
$701$		−	3.56669e8i		−	1.03541i	0.855560	−	0.517704i	$-0.173213\pi$
$702$		0			0
$703$	−1.07172e8			−0.308473
$704$		0			0
$705$	−1.78869e7	+	1.27237e7i	−0.0510468	+	0.0363116i
$706$		0			0
$707$		−	1.34378e8i		−	0.380249i
$708$		0			0
$709$	−3.60514e8			−1.01154			−0.505770	−	0.862668i	$-0.668792\pi$
$709$	−3.60514e8			−1.01154			−0.505770	+	0.862668i	$0.668792\pi$
$710$		0			0
$711$	−4.92044e7	+	1.41706e8i	−0.136897	+	0.394257i
$712$		0			0
$713$		−	2.70832e7i		−	0.0747189i
$714$		0			0
$715$	−2.57653e8			−0.704883
$716$		0			0
$717$	3.87311e8	+	5.44481e8i	1.05076	+	1.47715i
$718$		0			0
$719$			1.38066e8i			0.371450i	0.982602	+	0.185725i	$0.0594633\pi$
$719$			1.38066e8i			0.371450i	−0.982602	+	0.185725i	$0.940537\pi$
$720$		0			0
$721$	−3.76569e8			−1.00470
$722$		0			0
$723$	−3.50524e8	+	2.49342e8i	−0.927477	+	0.659751i
$724$		0			0
$725$			3.88872e8i			1.02045i
$726$		0			0
$727$	5.51729e8			1.43589			0.717947	−	0.696097i	$-0.245082\pi$
$727$	5.51729e8			1.43589			0.717947	+	0.696097i	$0.245082\pi$
$728$		0			0
$729$	−3.26549e8	+	2.08472e8i	−0.842879	+	0.538103i
$730$		0			0
$731$		−	4.11081e8i		−	1.05239i
$732$		0			0
$733$	3.81707e8			0.969212			0.484606	−	0.874733i	$-0.338963\pi$
$733$	3.81707e8			0.969212			0.484606	+	0.874733i	$0.338963\pi$
$734$		0			0
$735$	−5.01100e8	−	7.04444e8i	−1.26201	−	1.77413i
$736$		0			0
$737$		−	3.29674e8i		−	0.823535i
$738$		0			0
$739$	4.31468e8			1.06909			0.534546	−	0.845139i	$-0.320483\pi$
$739$	4.31468e8			1.06909			0.534546	+	0.845139i	$0.320483\pi$
$740$		0			0
$741$	3.80457e7	−	2.70634e7i	0.0935084	−	0.0665163i
$742$		0			0
$743$		−	5.13797e8i		−	1.25264i	−0.779567	−	0.626318i	$-0.784561\pi$
$743$		−	5.13797e8i		−	1.25264i	0.779567	−	0.626318i	$-0.215439\pi$
$744$		0			0
$745$	6.11777e8			1.47953
$746$		0			0
$747$	8.73070e7	+	3.03155e7i	0.209453	+	0.0727282i
$748$		0			0
$749$		−	2.82833e8i		−	0.673107i
$750$		0			0
$751$	3.82215e8			0.902378			0.451189	−	0.892428i	$-0.351000\pi$
$751$	3.82215e8			0.902378			0.451189	+	0.892428i	$0.351000\pi$
$752$		0			0
$753$	−4.70788e8	−	6.61833e8i	−1.10266	−	1.55011i
$754$		0			0
$755$			3.92042e7i			0.0910944i
$756$		0			0
$757$	1.57293e8			0.362596			0.181298	−	0.983428i	$-0.441970\pi$
$757$	1.57293e8			0.362596			0.181298	+	0.983428i	$0.441970\pi$
$758$		0			0
$759$	3.69916e8	−	2.63136e8i	0.846015	−	0.601805i
$760$		0			0
$761$		−	1.17082e7i		−	0.0265666i	−0.999912	−	0.0132833i	$-0.995772\pi$
$761$		−	1.17082e7i		−	0.0265666i	0.999912	−	0.0132833i	$-0.00422833\pi$
$762$		0			0
$763$	4.90332e7			0.110387
$764$		0			0
$765$	−1.56563e8	+	4.50894e8i	−0.349708	+	1.00714i
$766$		0			0
$767$			2.75377e8i			0.610298i
$768$		0			0
$769$	−2.32400e8			−0.511043			−0.255522	−	0.966803i	$-0.582247\pi$
$769$	−2.32400e8			−0.511043			−0.255522	+	0.966803i	$0.582247\pi$
$770$		0			0
$771$	1.43024e8	+	2.01063e8i	0.312066	+	0.438702i
$772$		0			0
$773$			4.42964e8i			0.959025i	0.877535	+	0.479512i	$0.159186\pi$
$773$			4.42964e8i			0.959025i	−0.877535	+	0.479512i	$0.840814\pi$
$774$		0			0
$775$	−2.44277e7			−0.0524780
$776$		0			0
$777$	−8.41541e8	+	5.98622e8i	−1.79396	+	1.27611i
$778$		0			0
$779$		−	5.67612e7i		−	0.120071i
$780$		0			0
$781$	3.43042e8			0.720103
$782$		0			0
$783$	−7.04423e8	+	2.05685e8i	−1.46740	+	0.428467i
$784$		0			0
$785$			1.06326e8i			0.219801i
$786$		0			0
$787$	4.74577e8			0.973604			0.486802	−	0.873512i	$-0.338163\pi$
$787$	4.74577e8			0.973604			0.486802	+	0.873512i	$0.338163\pi$
$788$		0			0
$789$	3.37868e8	+	4.74974e8i	0.687885	+	0.967027i
$790$		0			0
$791$			8.74091e8i			1.76615i
$792$		0			0
$793$	3.49277e8			0.700406
$794$		0			0
$795$	3.89179e8	−	2.76838e8i	0.774547	−	0.550966i
$796$		0			0
$797$		−	5.64334e8i		−	1.11471i	−0.830275	−	0.557354i	$-0.811817\pi$
$797$		−	5.64334e8i		−	1.11471i	0.830275	−	0.557354i	$-0.188183\pi$
$798$		0			0
$799$	−2.04294e7			−0.0400511
$800$		0			0
$801$	7.12295e8	+	2.47329e8i	1.38600	+	0.481258i
$802$		0			0
$803$			1.00957e9i			1.94980i
$804$		0			0
$805$	1.04933e9			2.01152
$806$		0			0
$807$	−4.09587e7	−	5.75796e7i	−0.0779337	−	0.109559i
$808$		0			0
$809$			9.33422e6i			0.0176292i	0.999961	+	0.00881460i	$0.00280581\pi$
$809$			9.33422e6i			0.0176292i	−0.999961	+	0.00881460i	$0.997194\pi$
$810$		0			0
$811$	−5.39372e8			−1.01117			−0.505587	−	0.862776i	$-0.668724\pi$
$811$	−5.39372e8			−1.01117			−0.505587	+	0.862776i	$0.668724\pi$
$812$		0			0
$813$	7.11647e8	−	5.06223e8i	1.32432	−	0.942042i
$814$		0			0
$815$			1.26996e9i			2.34593i
$816$		0			0
$817$	−1.59630e8			−0.292718
$818$		0			0
$819$	1.47578e8	−	4.25016e8i	0.268639	−	0.773665i
$820$		0			0
$821$			2.94011e8i			0.531293i	0.964070	+	0.265647i	$0.0855854\pi$
$821$			2.94011e8i			0.531293i	−0.964070	+	0.265647i	$0.914415\pi$
$822$		0			0
$823$	9.74855e8			1.74880			0.874401	−	0.485204i	$-0.161255\pi$
$823$	9.74855e8			1.74880			0.874401	+	0.485204i	$0.161255\pi$
$824$		0			0
$825$	−2.37336e8	−	3.33646e8i	−0.422671	−	0.594189i
$826$		0			0
$827$		−	5.04607e8i		−	0.892148i	−0.894996	−	0.446074i	$-0.852822\pi$
$827$		−	5.04607e8i		−	0.892148i	0.894996	−	0.446074i	$-0.147178\pi$
$828$		0			0
$829$	−6.41820e8			−1.12655			−0.563274	−	0.826270i	$-0.690459\pi$
$829$	−6.41820e8			−1.12655			−0.563274	+	0.826270i	$0.690459\pi$
$830$		0			0
$831$	−3.04529e8	+	2.16624e8i	−0.530671	+	0.377488i
$832$		0			0
$833$		−	8.04574e8i		−	1.39197i
$834$		0			0
$835$	−8.80362e8			−1.51217
$836$		0			0
$837$	−1.29205e7	−	4.42496e7i	−0.0220344	−	0.0754628i
$838$		0			0
$839$		−	5.67669e8i		−	0.961190i	−0.876943	−	0.480595i	$-0.840421\pi$
$839$		−	5.67669e8i		−	0.961190i	0.876943	−	0.480595i	$-0.159579\pi$
$840$		0			0
$841$	−7.95186e8			−1.33684
$842$		0			0
$843$	3.98218e7	+	5.59813e7i	0.0664719	+	0.0934460i
$844$		0			0
$845$			5.84571e8i			0.968873i
$846$		0			0
$847$	1.92421e8			0.316667
$848$		0			0
$849$	9.01389e7	−	6.41194e7i	0.147295	−	0.104777i
$850$		0			0
$851$		−	7.86849e8i		−	1.27674i
$852$		0			0
$853$	8.51887e8			1.37257			0.686286	−	0.727332i	$-0.259240\pi$
$853$	8.51887e8			1.37257			0.686286	+	0.727332i	$0.259240\pi$
$854$		0			0
$855$	1.75091e8	+	6.07964e7i	0.280133	+	0.0972702i
$856$		0			0
$857$		−	1.76222e7i		−	0.0279973i	−0.999902	−	0.0139987i	$-0.995544\pi$
$857$		−	1.76222e7i		−	0.0279973i	0.999902	−	0.0139987i	$-0.00445606\pi$
$858$		0			0
$859$	−6.56246e8			−1.03535			−0.517675	−	0.855577i	$-0.673202\pi$
$859$	−6.56246e8			−1.03535			−0.517675	+	0.855577i	$0.673202\pi$
$860$		0			0
$861$	−3.17046e8	−	4.45702e8i	−0.496721	−	0.698288i
$862$		0			0
$863$		−	1.02971e9i		−	1.60207i	−0.598617	−	0.801036i	$-0.704283\pi$
$863$		−	1.02971e9i		−	1.60207i	0.598617	−	0.801036i	$-0.295717\pi$
$864$		0			0
$865$	−4.66030e8			−0.720055
$866$		0			0
$867$	1.69079e8	−	1.20273e8i	0.259437	−	0.184548i
$868$		0			0
$869$		−	2.99170e8i		−	0.455888i
$870$		0			0
$871$	2.48940e8			0.376739
$872$		0			0
$873$	7.60017e7	−	2.18881e8i	0.114230	−	0.328976i
$874$		0			0
$875$			4.71362e8i			0.703607i
$876$		0			0
$877$	−1.26282e9			−1.87216			−0.936082	−	0.351781i	$-0.885576\pi$
$877$	−1.26282e9			−1.87216			−0.936082	+	0.351781i	$0.885576\pi$
$878$		0			0
$879$	1.56088e8	+	2.19428e8i	0.229828	+	0.323091i
$880$		0			0
$881$			9.22205e8i			1.34865i	0.738434	+	0.674326i	$0.235566\pi$
$881$			9.22205e8i			1.34865i	−0.738434	+	0.674326i	$0.764434\pi$
$882$		0			0
$883$	1.47782e7			0.0214655			0.0107327	−	0.999942i	$-0.496584\pi$
$883$	1.47782e7			0.0214655			0.0107327	+	0.999942i	$0.496584\pi$
$884$		0			0
$885$	−8.90796e8	+	6.33659e8i	−1.28513	+	0.914168i
$886$		0			0
$887$		−	4.02774e8i		−	0.577153i	−0.957457	−	0.288576i	$-0.906818\pi$
$887$		−	4.02774e8i		−	0.577153i	0.957457	−	0.288576i	$-0.0931819\pi$
$888$		0			0
$889$	7.07682e7			0.100724
$890$		0			0
$891$	4.78851e8	−	6.06398e8i	0.676967	−	0.857283i
$892$		0			0
$893$			7.93309e6i			0.0111401i
$894$		0			0
$895$	−5.54533e8			−0.773496
$896$		0			0
$897$	1.98697e8	+	2.79328e8i	0.275305	+	0.387023i
$898$		0			0
$899$		−	8.73159e7i		−	0.120175i
$900$		0			0
$901$	4.44496e8			0.607707
$902$		0			0
$903$	−1.25345e9	+	8.91630e8i	−1.70233	+	1.21094i
$904$		0			0
$905$		−	8.50938e8i		−	1.14803i
$906$		0			0
$907$	−3.20659e8			−0.429755			−0.214878	−	0.976641i	$-0.568935\pi$
$907$	−3.20659e8			−0.429755			−0.214878	+	0.976641i	$0.568935\pi$
$908$		0			0
$909$	1.64622e8	+	5.71614e7i	0.219177	+	0.0761047i
$910$		0			0
$911$		−	1.48274e9i		−	1.96115i	−0.196139	−	0.980576i	$-0.562840\pi$
$911$		−	1.48274e9i		−	1.96115i	0.196139	−	0.980576i	$-0.437160\pi$
$912$		0			0
$913$	−1.84323e8			−0.242196
$914$		0			0
$915$	8.03706e8	+	1.12985e9i	1.04914	+	1.47488i
$916$		0			0
$917$			8.09567e8i			1.04989i
$918$		0			0
$919$	−3.73389e8			−0.481077			−0.240539	−	0.970640i	$-0.577324\pi$
$919$	−3.73389e8			−0.481077			−0.240539	+	0.970640i	$0.577324\pi$
$920$		0			0
$921$	3.68736e8	−	2.62297e8i	0.471994	−	0.335748i
$922$		0			0
$923$			2.59035e8i			0.329423i
$924$		0			0
$925$	−7.09700e8			−0.896705
$926$		0			0
$927$	1.60184e8	−	4.61323e8i	0.201086	−	0.579116i
$928$		0			0
$929$		−	7.60909e8i		−	0.949043i	−0.880244	−	0.474521i	$-0.842621\pi$
$929$		−	7.60909e8i		−	0.949043i	0.880244	−	0.474521i	$-0.157379\pi$
$930$		0			0
$931$	−3.12431e8			−0.387173
$932$		0			0
$933$	−5.38067e8	−	7.56413e8i	−0.662508	−	0.931352i
$934$		0			0
$935$		−	9.51929e8i		−	1.16458i
$936$		0			0
$937$	−8.75257e7			−0.106394			−0.0531969	−	0.998584i	$-0.516941\pi$
$937$	−8.75257e7			−0.106394			−0.0531969	+	0.998584i	$0.516941\pi$
$938$		0			0
$939$	1.61231e8	−	1.14690e8i	0.194738	−	0.138525i
$940$		0			0
$941$			7.56838e8i			0.908310i	0.890923	+	0.454155i	$0.150059\pi$
$941$			7.56838e8i			0.908310i	−0.890923	+	0.454155i	$0.849941\pi$
$942$		0			0
$943$	4.16736e8			0.496965
$944$		0			0
$945$	1.71443e9	−	5.00598e8i	2.03154	−	0.593190i
$946$		0			0
$947$			8.47068e8i			0.997397i	0.866775	+	0.498699i	$0.166189\pi$
$947$			8.47068e8i			0.997397i	−0.866775	+	0.498699i	$0.833811\pi$
$948$		0			0
$949$	−7.62336e8			−0.891965
$950$		0			0
$951$	3.46702e8	+	4.87392e8i	0.403101	+	0.566679i
$952$		0			0
$953$			1.31295e9i			1.51694i	0.651708	+	0.758470i	$0.274053\pi$
$953$			1.31295e9i			1.51694i	−0.651708	+	0.758470i	$0.725947\pi$
$954$		0			0
$955$	8.54914e8			0.981550
$956$		0			0
$957$	1.19261e9	−	8.48350e8i	1.36070	−	0.967919i
$958$		0			0
$959$			9.47265e8i			1.07403i
$960$		0			0
$961$	−8.82019e8			−0.993820
$962$		0			0
$963$	3.46490e8	+	1.20311e8i	0.387982	+	0.134718i
$964$		0			0
$965$		−	1.98600e9i		−	2.21002i
$966$		0			0
$967$	−1.58569e9			−1.75364			−0.876819	−	0.480821i	$-0.840339\pi$
$967$	−1.58569e9			−1.75364			−0.876819	+	0.480821i	$0.840339\pi$
$968$		0			0
$969$	9.99889e7	+	1.40564e8i	0.109896	+	0.154491i
$970$		0			0
$971$			7.56916e8i			0.826781i	0.910554	+	0.413390i	$0.135655\pi$
$971$			7.56916e8i			0.826781i	−0.910554	+	0.413390i	$0.864345\pi$
$972$		0			0
$973$	8.48010e8			0.920582
$974$		0			0
$975$	2.51940e8	−	1.79215e8i	0.271821	−	0.193357i
$976$		0			0
$977$			1.63453e9i			1.75270i	0.481671	+	0.876352i	$0.340030\pi$
$977$			1.63453e9i			1.75270i	−0.481671	+	0.876352i	$0.659970\pi$
$978$		0			0
$979$	−1.50380e9			−1.60266
$980$		0			0
$981$	−2.08577e7	+	6.00691e7i	−0.0220932	+	0.0636274i
$982$		0			0
$983$			1.64466e9i			1.73148i	0.500498	+	0.865738i	$0.333150\pi$
$983$			1.64466e9i			1.73148i	−0.500498	+	0.865738i	$0.666850\pi$
$984$		0			0
$985$	1.91049e9			1.99911
$986$		0			0
$987$	4.43111e7	+	6.22924e7i	0.0460851	+	0.0647864i
$988$		0			0
$989$		−	1.17199e9i		−	1.21153i
$990$		0			0
$991$	4.41128e8			0.453256			0.226628	−	0.973981i	$-0.427230\pi$
$991$	4.41128e8			0.453256			0.226628	+	0.973981i	$0.427230\pi$
$992$		0			0
$993$	−8.16481e8	+	5.80796e8i	−0.833870	+	0.593165i
$994$		0			0
$995$		−	9.43472e8i		−	0.957766i
$996$		0			0
$997$	−4.30176e8			−0.434071			−0.217036	−	0.976164i	$-0.569639\pi$
$997$	−4.30176e8			−0.434071			−0.217036	+	0.976164i	$0.569639\pi$
$998$		0			0
$999$	−3.75379e8	−	1.28559e9i	−0.376508	−	1.28945i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.7.e.d.17.5		6
3.2	odd	2	inner	48.7.e.d.17.6		6
4.3	odd	2		24.7.e.a.17.2	yes	6
8.3	odd	2		192.7.e.h.65.5		6
8.5	even	2		192.7.e.g.65.2		6
12.11	even	2		24.7.e.a.17.1	✓	6
24.5	odd	2		192.7.e.g.65.1		6
24.11	even	2		192.7.e.h.65.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.7.e.a.17.1	✓	6	12.11	even	2
24.7.e.a.17.2	yes	6	4.3	odd	2
48.7.e.d.17.5		6	1.1	even	1	trivial
48.7.e.d.17.6		6	3.2	odd	2	inner
192.7.e.g.65.1		6	24.5	odd	2
192.7.e.g.65.2		6	8.5	even	2
192.7.e.h.65.5		6	8.3	odd	2
192.7.e.h.65.6		6	24.11	even	2

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:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	48.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(22.0014 - 15.6505i) q^{3} -161.417i q^{5} -562.144 q^{7} +(239.124 - 688.666i) q^{9} +O(q^{10})\)	\(q+(22.0014 - 15.6505i) q^{3} -161.417i q^{5} -562.144 q^{7} +(239.124 - 688.666i) q^{9} +1453.91i q^{11} -1097.86 q^{13} +(-2526.25 - 3551.39i) q^{15} -4056.19i q^{17} -1575.09 q^{19} +(-12368.0 + 8797.83i) q^{21} -11564.2i q^{23} -10430.3 q^{25} +(-5516.88 - 18894.0i) q^{27} -37282.8i q^{29} +2341.99 q^{31} +(22754.4 + 31988.1i) q^{33} +90739.4i q^{35} +68041.9 q^{37} +(-24154.6 + 17182.1i) q^{39} +36036.8i q^{41} +101347. q^{43} +(-111162. - 38598.7i) q^{45} -5036.59i q^{47} +198357. q^{49} +(-63481.3 - 89241.9i) q^{51} +109585. i q^{53} +234686. q^{55} +(-34654.3 + 24651.0i) q^{57} -250830. i q^{59} -318142. q^{61} +(-134422. + 387129. i) q^{63} +177214. i q^{65} -226749. q^{67} +(-180985. - 254428. i) q^{69} -235944. i q^{71} +694381. q^{73} +(-229482. + 163240. i) q^{75} -817308. i q^{77} -205769. q^{79} +(-417080. - 329354. i) q^{81} +126777. i q^{83} -654736. q^{85} +(-583495. - 820275. i) q^{87} +1.03431e6i q^{89} +617158. q^{91} +(51527.0 - 36653.2i) q^{93} +254246. i q^{95} +317833. q^{97} +(1.00126e6 + 347666. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 10 q^{3} - 156 q^{7} - 74 q^{9}+O(q^{10}) \) 6 * q + 10 * q^3 - 156 * q^7 - 74 * q^9	\( 6 q + 10 q^{3} - 156 q^{7} - 74 q^{9} + 156 q^{13} - 2912 q^{15} + 4500 q^{19} - 15108 q^{21} + 21366 q^{25} - 37574 q^{27} + 74244 q^{31} - 83104 q^{33} + 171132 q^{37} - 200444 q^{39} + 291060 q^{43} - 355136 q^{45} + 517746 q^{49} - 452224 q^{51} + 748224 q^{55} - 650420 q^{57} + 592092 q^{61} - 1009788 q^{63} + 570900 q^{67} - 981184 q^{69} + 1119660 q^{73} - 521446 q^{75} + 1053636 q^{79} - 742874 q^{81} + 197376 q^{85} - 1251360 q^{87} - 839640 q^{91} + 354652 q^{93} - 798516 q^{97} + 2849600 q^{99}+O(q^{100}) \) 6 * q + 10 * q^3 - 156 * q^7 - 74 * q^9 + 156 * q^13 - 2912 * q^15 + 4500 * q^19 - 15108 * q^21 + 21366 * q^25 - 37574 * q^27 + 74244 * q^31 - 83104 * q^33 + 171132 * q^37 - 200444 * q^39 + 291060 * q^43 - 355136 * q^45 + 517746 * q^49 - 452224 * q^51 + 748224 * q^55 - 650420 * q^57 + 592092 * q^61 - 1009788 * q^63 + 570900 * q^67 - 981184 * q^69 + 1119660 * q^73 - 521446 * q^75 + 1053636 * q^79 - 742874 * q^81 + 197376 * q^85 - 1251360 * q^87 - 839640 * q^91 + 354652 * q^93 - 798516 * q^97 + 2849600 * q^99

Introduction

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