LMFDB - Embedded newform 99.5.c.c.10.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [99,5,Mod(10,99)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(99, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("99.10");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 99 = 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	99.c (of order $2$, degree $1$, 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:	$10.2336263453$
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} + 102x^{6} + 2913x^{4} + 23292x^{2} + 41364 $ x^8 + 102x^6 + 2913x^4 + 23292*x^2 + 41364
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{3} $
Twist minimal:	no (minimal twist has level 33)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			10.8
Root			$7.70102i$ of defining polynomial
Character	$\chi$	$=$	99.10
Dual form			99.5.c.c.10.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+7.70102i q^{2} -43.3057 q^{4} +12.5296 q^{5} -63.6540i q^{7} -210.281i q^{8} +O(q^{10})$	\(q+7.70102i q^{2} -43.3057 q^{4} +12.5296 q^{5} -63.6540i q^{7} -210.281i q^{8} +96.4909i q^{10} +(-119.745 - 17.3819i) q^{11} -194.814i q^{13} +490.200 q^{14} +926.491 q^{16} +108.192i q^{17} +69.0344i q^{19} -542.604 q^{20} +(133.858 - 922.159i) q^{22} -576.600 q^{23} -468.008 q^{25} +1500.26 q^{26} +2756.58i q^{28} -382.039i q^{29} -36.6327 q^{31} +3770.42i q^{32} -833.186 q^{34} -797.560i q^{35} +1791.57 q^{37} -531.635 q^{38} -2634.75i q^{40} -2882.25i q^{41} +319.351i q^{43} +(5185.64 + 752.735i) q^{44} -4440.41i q^{46} -2299.80 q^{47} -1650.83 q^{49} -3604.14i q^{50} +8436.53i q^{52} -857.015 q^{53} +(-1500.36 - 217.789i) q^{55} -13385.2 q^{56} +2942.09 q^{58} +2149.26 q^{59} -4966.14i q^{61} -282.109i q^{62} -14212.2 q^{64} -2440.94i q^{65} -5369.49 q^{67} -4685.32i q^{68} +6142.03 q^{70} +4954.38 q^{71} -3583.14i q^{73} +13796.9i q^{74} -2989.58i q^{76} +(-1106.43 + 7622.24i) q^{77} +7143.06i q^{79} +11608.6 q^{80} +22196.2 q^{82} +156.911i q^{83} +1355.60i q^{85} -2459.33 q^{86} +(-3655.09 + 25180.2i) q^{88} -7181.21 q^{89} -12400.7 q^{91} +24970.1 q^{92} -17710.8i q^{94} +864.975i q^{95} +2419.65 q^{97} -12713.0i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 76 q^{4} + 36 q^{5}+O(q^{10}) $ 8 * q - 76 * q^4 + 36 * q^5	$ 8 q - 76 q^{4} + 36 q^{5} - 36 q^{11} + 1140 q^{14} + 1412 q^{16} - 2532 q^{20} - 780 q^{22} - 516 q^{23} - 2280 q^{25} + 1524 q^{26} + 2752 q^{31} - 4920 q^{34} + 5296 q^{37} - 696 q^{38} + 6540 q^{44} - 420 q^{47} - 6832 q^{49} - 3540 q^{53} + 3784 q^{55} - 17964 q^{56} + 21624 q^{58} + 16632 q^{59} - 27508 q^{64} - 3656 q^{67} + 3312 q^{70} + 13212 q^{71} - 23268 q^{77} + 4476 q^{80} + 17088 q^{82} - 19896 q^{86} - 12516 q^{88} - 15528 q^{89} - 19752 q^{91} + 81180 q^{92} + 7624 q^{97}+O(q^{100}) $ 8 * q - 76 * q^4 + 36 * q^5 - 36 * q^11 + 1140 * q^14 + 1412 * q^16 - 2532 * q^20 - 780 * q^22 - 516 * q^23 - 2280 * q^25 + 1524 * q^26 + 2752 * q^31 - 4920 * q^34 + 5296 * q^37 - 696 * q^38 + 6540 * q^44 - 420 * q^47 - 6832 * q^49 - 3540 * q^53 + 3784 * q^55 - 17964 * q^56 + 21624 * q^58 + 16632 * q^59 - 27508 * q^64 - 3656 * q^67 + 3312 * q^70 + 13212 * q^71 - 23268 * q^77 + 4476 * q^80 + 17088 * q^82 - 19896 * q^86 - 12516 * q^88 - 15528 * q^89 - 19752 * q^91 + 81180 * q^92 + 7624 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/99\mathbb{Z}\right)^\times$.

$n$	$46$	$56$
$\chi(n)$	$-1$	$1$

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 $

$2$			7.70102i			1.92525i	0.270830	+	0.962627i	$0.412702\pi$
$2$			7.70102i			1.92525i	−0.270830	+	0.962627i	$0.587298\pi$
$3$		0			0
$3$		0			0
$4$	−43.3057			−2.70660
$5$	12.5296			0.501185			0.250593	−	0.968093i	$-0.419375\pi$
$5$	12.5296			0.501185			0.250593	+	0.968093i	$0.419375\pi$
$6$		0			0
$7$		−	63.6540i		−	1.29906i	−0.760336	−	0.649530i	$-0.774966\pi$
$7$		−	63.6540i		−	1.29906i	0.760336	−	0.649530i	$-0.225034\pi$
$8$		−	210.281i		−	3.28565i
$9$		0			0
$10$			96.4909i			0.964909i
$11$	−119.745	−	17.3819i	−0.989628	−	0.143652i
$11$	−119.745	−	17.3819i	−0.989628	−	0.143652i
$12$		0			0
$13$		−	194.814i		−	1.15274i	−0.817188	−	0.576371i	$-0.804468\pi$
$13$		−	194.814i		−	1.15274i	0.817188	−	0.576371i	$-0.195532\pi$
$14$	490.200			2.50102
$15$		0			0
$16$	926.491			3.61910
$17$			108.192i			0.374366i	0.982325	+	0.187183i	$0.0599357\pi$
$17$			108.192i			0.374366i	−0.982325	+	0.187183i	$0.940064\pi$
$18$		0			0
$19$			69.0344i			0.191231i	0.995418	+	0.0956155i	$0.0304819\pi$
$19$			69.0344i			0.191231i	−0.995418	+	0.0956155i	$0.969518\pi$
$20$	−542.604			−1.35651
$21$		0			0
$22$	133.858	−	922.159i	0.276567	−	1.90529i
$23$	−576.600			−1.08998			−0.544991	−	0.838442i	$-0.683467\pi$
$23$	−576.600			−1.08998			−0.544991	+	0.838442i	$0.683467\pi$
$24$		0			0
$25$	−468.008			−0.748813
$26$	1500.26			2.21932
$27$		0			0
$28$			2756.58i			3.51604i
$29$		−	382.039i		−	0.454268i	−0.973864	−	0.227134i	$-0.927064\pi$
$29$		−	382.039i		−	0.454268i	0.973864	−	0.227134i	$-0.0729356\pi$
$30$		0			0
$31$	−36.6327			−0.0381193			−0.0190597	−	0.999818i	$-0.506067\pi$
$31$	−36.6327			−0.0381193			−0.0190597	+	0.999818i	$0.506067\pi$
$32$			3770.42i			3.68205i
$33$		0			0
$34$	−833.186			−0.720749
$35$		−	797.560i		−	0.651070i
$36$		0			0
$37$	1791.57			1.30867			0.654334	−	0.756205i	$-0.272949\pi$
$37$	1791.57			1.30867			0.654334	+	0.756205i	$0.272949\pi$
$38$	−531.635			−0.368168
$39$		0			0
$40$		−	2634.75i		−	1.64672i
$41$		−	2882.25i		−	1.71460i	−0.514815	−	0.857301i	$-0.672139\pi$
$41$		−	2882.25i		−	1.71460i	0.514815	−	0.857301i	$-0.327861\pi$
$42$		0			0
$43$			319.351i			0.172716i	0.996264	+	0.0863579i	$0.0275228\pi$
$43$			319.351i			0.172716i	−0.996264	+	0.0863579i	$0.972477\pi$
$44$	5185.64	+	752.735i	2.67853	+	0.388809i
$45$		0			0
$46$		−	4440.41i		−	2.09849i
$47$	−2299.80			−1.04110			−0.520552	−	0.853830i	$-0.674274\pi$
$47$	−2299.80			−1.04110			−0.520552	+	0.853830i	$0.674274\pi$
$48$		0			0
$49$	−1650.83			−0.687558
$50$		−	3604.14i		−	1.44166i
$51$		0			0
$52$			8436.53i			3.12002i
$53$	−857.015			−0.305096			−0.152548	−	0.988296i	$-0.548748\pi$
$53$	−857.015			−0.305096			−0.152548	+	0.988296i	$0.548748\pi$
$54$		0			0
$55$	−1500.36	−	217.789i	−0.495987	−	0.0719963i
$56$	−13385.2			−4.26826
$57$		0			0
$58$	2942.09			0.874582
$59$	2149.26			0.617427			0.308714	−	0.951155i	$-0.400102\pi$
$59$	2149.26			0.617427			0.308714	+	0.951155i	$0.400102\pi$
$60$		0			0
$61$		−	4966.14i		−	1.33462i	−0.744778	−	0.667312i	$-0.767445\pi$
$61$		−	4966.14i		−	1.33462i	0.744778	−	0.667312i	$-0.232555\pi$
$62$		−	282.109i		−	0.0733894i
$63$		0			0
$64$	−14212.2			−3.46978
$65$		−	2440.94i		−	0.577738i
$66$		0			0
$67$	−5369.49			−1.19614			−0.598072	−	0.801443i	$-0.704066\pi$
$67$	−5369.49			−1.19614			−0.598072	+	0.801443i	$0.704066\pi$
$68$		−	4685.32i		−	1.01326i
$69$		0			0
$70$	6142.03			1.25347
$71$	4954.38			0.982817			0.491408	−	0.870929i	$-0.336482\pi$
$71$	4954.38			0.982817			0.491408	+	0.870929i	$0.336482\pi$
$72$		0			0
$73$		−	3583.14i		−	0.672386i	−0.941793	−	0.336193i	$-0.890861\pi$
$73$		−	3583.14i		−	0.672386i	0.941793	−	0.336193i	$-0.109139\pi$
$74$			13796.9i			2.51952i
$75$		0			0
$76$		−	2989.58i		−	0.517587i
$77$	−1106.43	+	7622.24i	−0.186613	+	1.28559i
$78$		0			0
$79$			7143.06i			1.14454i	0.820066	+	0.572269i	$0.193937\pi$
$79$			7143.06i			1.14454i	−0.820066	+	0.572269i	$0.806063\pi$
$80$	11608.6			1.81384
$81$		0			0
$82$	22196.2			3.30105
$83$			156.911i			0.0227771i	0.999935	+	0.0113885i	$0.00362516\pi$
$83$			156.911i			0.0227771i	−0.999935	+	0.0113885i	$0.996375\pi$
$84$		0			0
$85$			1355.60i			0.187627i
$86$	−2459.33			−0.332522
$87$		0			0
$88$	−3655.09	+	25180.2i	−0.471990	+	3.25157i
$89$	−7181.21			−0.906604			−0.453302	−	0.891357i	$-0.649754\pi$
$89$	−7181.21			−0.906604			−0.453302	+	0.891357i	$0.649754\pi$
$90$		0			0
$91$	−12400.7			−1.49748
$92$	24970.1			2.95015
$93$		0			0
$94$		−	17710.8i		−	2.00439i
$95$			864.975i			0.0958421i
$96$		0			0
$97$	2419.65			0.257164			0.128582	−	0.991699i	$-0.458958\pi$
$97$	2419.65			0.257164			0.128582	+	0.991699i	$0.458958\pi$
$98$		−	12713.0i		−	1.32372i
$99$		0			0
$100$	20267.4			2.02674
$101$			3839.47i			0.376381i	0.982133	+	0.188191i	$0.0602623\pi$
$101$			3839.47i			0.376381i	−0.982133	+	0.188191i	$0.939738\pi$
$102$		0			0
$103$	−6804.82			−0.641420			−0.320710	−	0.947177i	$-0.603921\pi$
$103$	−6804.82			−0.641420			−0.320710	+	0.947177i	$0.603921\pi$
$104$	−40965.7			−3.78751
$105$		0			0
$106$		−	6599.89i		−	0.587388i
$107$			5960.40i			0.520604i	0.965527	+	0.260302i	$0.0838222\pi$
$107$			5960.40i			0.520604i	−0.965527	+	0.260302i	$0.916178\pi$
$108$		0			0
$109$			21808.4i			1.83557i	0.397080	+	0.917784i	$0.370024\pi$
$109$			21808.4i			1.83557i	−0.397080	+	0.917784i	$0.629976\pi$
$110$	1677.20	−	11554.3i	0.138611	−	0.954901i
$111$		0			0
$112$		−	58974.8i		−	4.70144i
$113$	−2293.56			−0.179619			−0.0898097	−	0.995959i	$-0.528626\pi$
$113$	−2293.56			−0.179619			−0.0898097	+	0.995959i	$0.528626\pi$
$114$		0			0
$115$	−7224.59			−0.546283
$116$			16544.5i			1.22952i
$117$		0			0
$118$			16551.5i			1.18870i
$119$	6886.83			0.486324
$120$		0			0
$121$	14036.7	+	4162.79i	0.958728	+	0.284324i
$122$	38244.3			2.56949
$123$		0			0
$124$	1586.40			0.103174
$125$	−13695.0			−0.876479
$126$		0			0
$127$			8755.61i			0.542849i	0.962460	+	0.271424i	$0.0874947\pi$
$127$			8755.61i			0.542849i	−0.962460	+	0.271424i	$0.912505\pi$
$128$		−	49121.7i		−	2.99815i
$129$		0			0
$130$	18797.7			1.11229
$131$		−	13196.5i		−	0.768981i	−0.923129	−	0.384490i	$-0.874377\pi$
$131$		−	13196.5i		−	0.768981i	0.923129	−	0.384490i	$-0.125623\pi$
$132$		0			0
$133$	4394.31			0.248421
$134$		−	41350.5i		−	2.30288i
$135$		0			0
$136$	22750.7			1.23003
$137$	2034.69			0.108407			0.0542034	−	0.998530i	$-0.482738\pi$
$137$	2034.69			0.108407			0.0542034	+	0.998530i	$0.482738\pi$
$138$		0			0
$139$		−	24766.1i		−	1.28183i	−0.767614	−	0.640913i	$-0.778556\pi$
$139$		−	24766.1i		−	1.28183i	0.767614	−	0.640913i	$-0.221444\pi$
$140$			34538.9i			1.76219i
$141$		0			0
$142$			38153.8i			1.89217i
$143$	−3386.23	+	23327.9i	−0.165594	+	1.14079i
$144$		0			0
$145$		−	4786.81i		−	0.227672i
$146$	27593.9			1.29451
$147$		0			0
$148$	−77585.0			−3.54205
$149$			20475.5i			0.922277i	0.887328	+	0.461138i	$0.152559\pi$
$149$			20475.5i			0.922277i	−0.887328	+	0.461138i	$0.847441\pi$
$150$		0			0
$151$		−	17123.5i		−	0.750996i	−0.926823	−	0.375498i	$-0.877472\pi$
$151$		−	17123.5i		−	0.750996i	0.926823	−	0.375498i	$-0.122528\pi$
$152$	14516.7			0.628318
$153$		0			0
$154$	−58699.0	−	8520.61i	−2.47508	−	0.359277i
$155$	−458.994			−0.0191048
$156$		0			0
$157$	42755.1			1.73456			0.867279	−	0.497823i	$-0.165867\pi$
$157$	42755.1			1.73456			0.867279	+	0.497823i	$0.165867\pi$
$158$	−55008.8			−2.20353
$159$		0			0
$160$			47241.9i			1.84539i
$161$			36702.9i			1.41595i
$162$		0			0
$163$	−32922.2			−1.23912			−0.619561	−	0.784949i	$-0.712689\pi$
$163$	−32922.2			−1.23912			−0.619561	+	0.784949i	$0.712689\pi$
$164$			124818.i			4.64075i
$165$		0			0
$166$	−1208.38			−0.0438516
$167$		−	22705.4i		−	0.814136i	−0.913398	−	0.407068i	$-0.866551\pi$
$167$		−	22705.4i		−	0.814136i	0.913398	−	0.407068i	$-0.133449\pi$
$168$		0			0
$169$	−9391.31			−0.328816
$170$	−10439.5			−0.361229
$171$		0			0
$172$		−	13829.7i		−	0.467473i
$173$		−	32607.5i		−	1.08949i	−0.838600	−	0.544747i	$-0.816626\pi$
$173$		−	32607.5i		−	1.08949i	0.838600	−	0.544747i	$-0.183374\pi$
$174$		0			0
$175$			29790.6i			0.972754i
$176$	−110943.	−	16104.2i	−3.58157	−	0.519892i
$177$		0			0
$178$		−	55302.6i		−	1.74544i
$179$	−4293.51			−0.134001			−0.0670003	−	0.997753i	$-0.521343\pi$
$179$	−4293.51			−0.134001			−0.0670003	+	0.997753i	$0.521343\pi$
$180$		0			0
$181$	24070.9			0.734743			0.367371	−	0.930074i	$-0.380258\pi$
$181$	24070.9			0.734743			0.367371	+	0.930074i	$0.380258\pi$
$182$		−	95497.6i		−	2.88303i
$183$		0			0
$184$			121248.i			3.58130i
$185$	22447.7			0.655885
$186$		0			0
$187$	1880.58	−	12955.4i	0.0537784	−	0.370483i
$188$	99594.4			2.81786
$189$		0			0
$190$	−6661.19			−0.184520
$191$	15866.1			0.434913			0.217457	−	0.976070i	$-0.430224\pi$
$191$	15866.1			0.434913			0.217457	+	0.976070i	$0.430224\pi$
$192$		0			0
$193$		−	29252.3i		−	0.785319i	−0.919684	−	0.392659i	$-0.871555\pi$
$193$		−	29252.3i		−	0.785319i	0.919684	−	0.392659i	$-0.128445\pi$
$194$			18633.8i			0.495105i
$195$		0			0
$196$	71490.1			1.86095
$197$		−	54943.2i		−	1.41573i	−0.706346	−	0.707867i	$-0.749658\pi$
$197$		−	54943.2i		−	1.41573i	0.706346	−	0.707867i	$-0.250342\pi$
$198$		0			0
$199$	31818.1			0.803468			0.401734	−	0.915756i	$-0.368408\pi$
$199$	31818.1			0.803468			0.401734	+	0.915756i	$0.368408\pi$
$200$			98413.5i			2.46034i
$201$		0			0
$202$	−29567.8			−0.724630
$203$	−24318.3			−0.590122
$204$		0			0
$205$		−	36113.5i		−	0.859333i
$206$		−	52404.1i		−	1.23490i
$207$		0			0
$208$		−	180493.i		−	4.17190i
$209$	1199.95	−	8266.52i	0.0274707	−	0.189248i
$210$		0			0
$211$			59860.2i			1.34454i	0.740306	+	0.672270i	$0.234680\pi$
$211$			59860.2i			1.34454i	−0.740306	+	0.672270i	$0.765320\pi$
$212$	37113.6			0.825774
$213$		0			0
$214$	−45901.1			−1.00230
$215$			4001.35i			0.0865626i
$216$		0			0
$217$			2331.81i			0.0495193i
$218$	−167947.			−3.53394
$219$		0			0
$220$	64974.1	+	9431.49i	1.34244	+	0.194866i
$221$	21077.2			0.431547
$222$		0			0
$223$	−8345.72			−0.167824			−0.0839120	−	0.996473i	$-0.526741\pi$
$223$	−8345.72			−0.167824			−0.0839120	+	0.996473i	$0.526741\pi$
$224$	240002.			4.78320
$225$		0			0
$226$		−	17662.8i		−	0.345813i
$227$		−	59279.0i		−	1.15040i	−0.818013	−	0.575200i	$-0.804924\pi$
$227$		−	59279.0i		−	1.15040i	0.818013	−	0.575200i	$-0.195076\pi$
$228$		0			0
$229$	−45919.1			−0.875634			−0.437817	−	0.899064i	$-0.644248\pi$
$229$	−45919.1			−0.875634			−0.437817	+	0.899064i	$0.644248\pi$
$230$		−	55636.7i		−	1.05173i
$231$		0			0
$232$	−80335.8			−1.49257
$233$		−	69926.9i		−	1.28805i	−0.765005	−	0.644025i	$-0.777263\pi$
$233$		−	69926.9i		−	1.28805i	0.765005	−	0.644025i	$-0.222737\pi$
$234$		0			0
$235$	−28815.7			−0.521786
$236$	−93075.4			−1.67113
$237$		0			0
$238$			53035.6i			0.936297i
$239$		−	16074.1i		−	0.281404i	−0.990052	−	0.140702i	$-0.955064\pi$
$239$		−	16074.1i		−	0.281404i	0.990052	−	0.140702i	$-0.0449359\pi$
$240$		0			0
$241$			6183.67i			0.106466i	0.998582	+	0.0532332i	$0.0169527\pi$
$241$			6183.67i			0.106466i	−0.998582	+	0.0532332i	$0.983047\pi$
$242$	−32057.7	+	108097.i	−0.547397	+	1.84580i
$243$		0			0
$244$			215062.i			3.61230i
$245$	−20684.2			−0.344594
$246$		0			0
$247$	13448.8			0.220440
$248$			7703.17i			0.125247i
$249$		0			0
$250$		−	105465.i		−	1.68745i
$251$	50585.6			0.802933			0.401467	−	0.915874i	$-0.368501\pi$
$251$	50585.6			0.802933			0.401467	+	0.915874i	$0.368501\pi$
$252$		0			0
$253$	69045.0	+	10022.4i	1.07868	+	0.156578i
$254$	−67427.1			−1.04512
$255$		0			0
$256$	150892.			2.30243
$257$	26033.8			0.394159			0.197080	−	0.980387i	$-0.436854\pi$
$257$	26033.8			0.394159			0.197080	+	0.980387i	$0.436854\pi$
$258$		0			0
$259$		−	114040.i		−	1.70004i
$260$			105707.i			1.56371i
$261$		0			0
$262$	101626.			1.48048
$263$			79564.2i			1.15029i	0.818053	+	0.575143i	$0.195054\pi$
$263$			79564.2i			1.15029i	−0.818053	+	0.575143i	$0.804946\pi$
$264$		0			0
$265$	−10738.1			−0.152910
$266$			33840.7i			0.478273i
$267$		0			0
$268$	232529.			3.23749
$269$	101837.			1.40734			0.703671	−	0.710526i	$-0.251543\pi$
$269$	101837.			1.40734			0.703671	+	0.710526i	$0.251543\pi$
$270$		0			0
$271$			84386.2i			1.14903i	0.818493	+	0.574517i	$0.194810\pi$
$271$			84386.2i			1.14903i	−0.818493	+	0.574517i	$0.805190\pi$
$272$			100239.i			1.35487i
$273$		0			0
$274$			15669.2i			0.208711i
$275$	56041.7	+	8134.88i	0.741047	+	0.107569i
$276$		0			0
$277$		−	77513.5i		−	1.01022i	−0.863054	−	0.505112i	$-0.831451\pi$
$277$		−	77513.5i		−	1.01022i	0.863054	−	0.505112i	$-0.168549\pi$
$278$	190725.			2.46784
$279$		0			0
$280$	−167712.			−2.13919
$281$			31892.6i			0.403903i	0.979395	+	0.201952i	$0.0647284\pi$
$281$			31892.6i			0.403903i	−0.979395	+	0.201952i	$0.935272\pi$
$282$		0			0
$283$		−	91416.2i		−	1.14143i	−0.821147	−	0.570716i	$-0.806666\pi$
$283$		−	91416.2i		−	1.14143i	0.821147	−	0.570716i	$-0.193334\pi$
$284$	−214553.			−2.66010
$285$		0			0
$286$	−179649.	−	26077.4i	−2.19630	−	0.318810i
$287$	−183466.			−2.22737
$288$		0			0
$289$	71815.6			0.859850
$290$	36863.3			0.438327
$291$		0			0
$292$			155170.i			1.81988i
$293$			98100.1i			1.14271i	0.820704	+	0.571353i	$0.193581\pi$
$293$			98100.1i			1.14271i	−0.820704	+	0.571353i	$0.806419\pi$
$294$		0			0
$295$	26929.5			0.309445
$296$		−	376733.i		−	4.29982i
$297$		0			0
$298$	−157682.			−1.77562
$299$			112330.i			1.25647i
$300$		0			0
$301$	20328.0			0.224368
$302$	131868.			1.44586
$303$		0			0
$304$			63959.7i			0.692085i
$305$		−	62223.9i		−	0.668894i
$306$		0			0
$307$			142750.i			1.51461i	0.653063	+	0.757304i	$0.273484\pi$
$307$			142750.i			1.51461i	−0.653063	+	0.757304i	$0.726516\pi$
$308$	47914.6	−	330086.i	0.505087	−	3.47958i
$309$		0			0
$310$		−	3534.72i		−	0.0367817i
$311$	−165392.			−1.70999			−0.854996	−	0.518635i	$-0.826440\pi$
$311$	−165392.			−1.70999			−0.854996	+	0.518635i	$0.826440\pi$
$312$		0			0
$313$	73167.5			0.746843			0.373422	−	0.927662i	$-0.378185\pi$
$313$	73167.5			0.746843			0.373422	+	0.927662i	$0.378185\pi$
$314$			329258.i			3.33946i
$315$		0			0
$316$		−	309335.i		−	3.09781i
$317$	112606.			1.12058			0.560292	−	0.828295i	$-0.310689\pi$
$317$	112606.			1.12058			0.560292	+	0.828295i	$0.310689\pi$
$318$		0			0
$319$	−6640.57	+	45747.3i	−0.0652566	+	0.449557i
$320$	−178074.			−1.73900
$321$		0			0
$322$	−282650.			−2.72607
$323$	−7468.95			−0.0715903
$324$		0			0
$325$			91174.4i			0.863189i
$326$		−	253535.i		−	2.38562i
$327$		0			0
$328$	−606083.			−5.63358
$329$			146391.i			1.35246i
$330$		0			0
$331$	−72976.7			−0.666083			−0.333041	−	0.942912i	$-0.608075\pi$
$331$	−72976.7			−0.666083			−0.333041	+	0.942912i	$0.608075\pi$
$332$		−	6795.14i		−	0.0616485i
$333$		0			0
$334$	174855.			1.56742
$335$	−67277.7			−0.599489
$336$		0			0
$337$			196562.i			1.73077i	0.501108	+	0.865384i	$0.332926\pi$
$337$			196562.i			1.73077i	−0.501108	+	0.865384i	$0.667074\pi$
$338$		−	72322.6i		−	0.633054i
$339$		0			0
$340$		−	58705.3i		−	0.507831i
$341$	4386.58	+	636.745i	0.0377240	+	0.00547592i
$342$		0			0
$343$		−	47751.5i		−	0.405881i
$344$	67153.7			0.567483
$345$		0			0
$346$	251111.			2.09755
$347$		−	206637.i		−	1.71612i	−0.513546	−	0.858062i	$-0.671668\pi$
$347$		−	206637.i		−	1.71612i	0.513546	−	0.858062i	$-0.328332\pi$
$348$		0			0
$349$		−	210273.i		−	1.72636i	−0.504894	−	0.863181i	$-0.668468\pi$
$349$		−	210273.i		−	1.72636i	0.504894	−	0.863181i	$-0.331532\pi$
$350$	−229418.			−1.87280
$351$		0			0
$352$	65537.0	−	451489.i	0.528934	−	3.64386i
$353$	140022.			1.12369			0.561847	−	0.827241i	$-0.310091\pi$
$353$	140022.			1.12369			0.561847	+	0.827241i	$0.310091\pi$
$354$		0			0
$355$	62076.5			0.492573
$356$	310987.			2.45382
$357$		0			0
$358$		−	33064.4i		−	0.257985i
$359$			90549.6i			0.702583i	0.936266	+	0.351292i	$0.114257\pi$
$359$			90549.6i			0.702583i	−0.936266	+	0.351292i	$0.885743\pi$
$360$		0			0
$361$	125555.			0.963431
$362$			185370.i			1.41457i
$363$		0			0
$364$	537019.			4.05309
$365$		−	44895.5i		−	0.336990i
$366$		0			0
$367$	−60682.5			−0.450538			−0.225269	−	0.974297i	$-0.572326\pi$
$367$	−60682.5			−0.450538			−0.225269	+	0.974297i	$0.572326\pi$
$368$	−534215.			−3.94476
$369$		0			0
$370$			172870.i			1.26275i
$371$			54552.4i			0.396338i
$372$		0			0
$373$			100208.i			0.720252i	0.932904	+	0.360126i	$0.117266\pi$
$373$			100208.i			0.720252i	−0.932904	+	0.360126i	$0.882734\pi$
$374$	99769.9	+	14482.4i	0.713274	+	0.103537i
$375$		0			0
$376$			483606.i			3.42070i
$377$	−74426.5			−0.523654
$378$		0			0
$379$	−228068.			−1.58777			−0.793883	−	0.608070i	$-0.791944\pi$
$379$	−228068.			−1.58777			−0.793883	+	0.608070i	$0.791944\pi$
$380$		−	37458.3i		−	0.259407i
$381$		0			0
$382$			122185.i			0.837319i
$383$	13819.4			0.0942091			0.0471045	−	0.998890i	$-0.485001\pi$
$383$	13819.4			0.0942091			0.0471045	+	0.998890i	$0.485001\pi$
$384$		0			0
$385$	−13863.1	+	95503.9i	−0.0935275	+	0.644317i
$386$	225273.			1.51194
$387$		0			0
$388$	−104785.			−0.696040
$389$	159107.			1.05145			0.525726	−	0.850654i	$-0.323794\pi$
$389$	159107.			1.05145			0.525726	+	0.850654i	$0.323794\pi$
$390$		0			0
$391$		−	62383.4i		−	0.408052i
$392$			347138.i			2.25907i
$393$		0			0
$394$	423118.			2.72565
$395$			89499.9i			0.573625i
$396$		0			0
$397$	−76802.1			−0.487295			−0.243648	−	0.969864i	$-0.578344\pi$
$397$	−76802.1			−0.487295			−0.243648	+	0.969864i	$0.578344\pi$
$398$			245032.i			1.54688i
$399$		0			0
$400$	−433605.			−2.71003
$401$	−121201.			−0.753732			−0.376866	−	0.926268i	$-0.622998\pi$
$401$	−121201.			−0.753732			−0.376866	+	0.926268i	$0.622998\pi$
$402$		0			0
$403$			7136.54i			0.0439418i
$404$		−	166271.i		−	1.01872i
$405$		0			0
$406$		−	187276.i		−	1.13613i
$407$	−214531.	−	31140.8i	−1.29510	−	0.187993i
$408$		0			0
$409$		−	81034.4i		−	0.484421i	−0.970224	−	0.242210i	$-0.922128\pi$
$409$		−	81034.4i		−	0.484421i	0.970224	−	0.242210i	$-0.0778724\pi$
$410$	278111.			1.65444
$411$		0			0
$412$	294687.			1.73607
$413$		−	136809.i		−	0.802075i
$414$		0			0
$415$			1966.04i			0.0114155i
$416$	734528.			4.24445
$417$		0			0
$418$	63660.6	+	9240.83i	0.364350	+	0.0528881i
$419$	−160750.			−0.915635			−0.457817	−	0.889046i	$-0.651369\pi$
$419$	−160750.			−0.915635			−0.457817	+	0.889046i	$0.651369\pi$
$420$		0			0
$421$	−92831.1			−0.523756			−0.261878	−	0.965101i	$-0.584342\pi$
$421$	−92831.1			−0.523756			−0.261878	+	0.965101i	$0.584342\pi$
$422$	−460985.			−2.58858
$423$		0			0
$424$			180214.i			1.00244i
$425$		−	50634.6i		−	0.280330i
$426$		0			0
$427$	−316114.			−1.73376
$428$		−	258119.i		−	1.40907i
$429$		0			0
$430$	−30814.5			−0.166655
$431$			213351.i			1.14853i	0.818671	+	0.574263i	$0.194711\pi$
$431$			213351.i			1.14853i	−0.818671	+	0.574263i	$0.805289\pi$
$432$		0			0
$433$	−33161.8			−0.176873			−0.0884366	−	0.996082i	$-0.528187\pi$
$433$	−33161.8			−0.176873			−0.0884366	+	0.996082i	$0.528187\pi$
$434$	−17957.3			−0.0953372
$435$		0			0
$436$		−	944427.i		−	4.96816i
$437$		−	39805.3i		−	0.208438i
$438$		0			0
$439$		−	293728.i		−	1.52411i	−0.647513	−	0.762055i	$-0.724191\pi$
$439$		−	293728.i		−	1.52411i	0.647513	−	0.762055i	$-0.275809\pi$
$440$	−45797.0	+	315498.i	−0.236555	+	1.62964i
$441$		0			0
$442$			162316.i			0.830839i
$443$	274048.			1.39643			0.698215	−	0.715888i	$-0.253978\pi$
$443$	274048.			1.39643			0.698215	+	0.715888i	$0.253978\pi$
$444$		0			0
$445$	−89977.9			−0.454376
$446$		−	64270.6i		−	0.323104i
$447$		0			0
$448$			904663.i			4.50745i
$449$	231379.			1.14771			0.573854	−	0.818958i	$-0.305448\pi$
$449$	231379.			1.14771			0.573854	+	0.818958i	$0.305448\pi$
$450$		0			0
$451$	−50098.9	+	345135.i	−0.246306	+	1.69682i
$452$	99324.2			0.486159
$453$		0			0
$454$	456509.			2.21481
$455$	−155376.			−0.750516
$456$		0			0
$457$		−	271864.i		−	1.30172i	−0.759196	−	0.650862i	$-0.774407\pi$
$457$		−	271864.i		−	1.30172i	0.759196	−	0.650862i	$-0.225593\pi$
$458$		−	353624.i		−	1.68582i
$459$		0			0
$460$	312866.			1.47857
$461$			264865.i			1.24630i	0.782102	+	0.623150i	$0.214148\pi$
$461$			264865.i			1.24630i	−0.782102	+	0.623150i	$0.785852\pi$
$462$		0			0
$463$	255889.			1.19369			0.596843	−	0.802358i	$-0.296422\pi$
$463$	255889.			1.19369			0.596843	+	0.802358i	$0.296422\pi$
$464$		−	353956.i		−	1.64404i
$465$		0			0
$466$	538508.			2.47982
$467$	184125.			0.844264			0.422132	−	0.906534i	$-0.361282\pi$
$467$	184125.			0.844264			0.422132	+	0.906534i	$0.361282\pi$
$468$		0			0
$469$			341789.i			1.55386i
$470$		−	221910.i		−	1.00457i
$471$		0			0
$472$		−	451951.i		−	2.02865i
$473$	5550.93	−	38240.7i	0.0248110	−	0.170924i
$474$		0			0
$475$		−	32308.7i		−	0.143196i
$476$	−298239.			−1.31629
$477$		0			0
$478$	123787.			0.541774
$479$		−	452588.i		−	1.97257i	−0.165062	−	0.986283i	$-0.552782\pi$
$479$		−	452588.i		−	1.97257i	0.165062	−	0.986283i	$-0.447218\pi$
$480$		0			0
$481$		−	349022.i		−	1.50856i
$482$	−47620.6			−0.204975
$483$		0			0
$484$	−607870.	−	180273.i	−2.59490	−	0.769554i
$485$	30317.3			0.128887
$486$		0			0
$487$	10896.3			0.0459432			0.0229716	−	0.999736i	$-0.492687\pi$
$487$	10896.3			0.0459432			0.0229716	+	0.999736i	$0.492687\pi$
$488$	−1.04429e6			−4.38511
$489$		0			0
$490$		−	159290.i		−	0.663431i
$491$		−	178944.i		−	0.742256i	−0.928582	−	0.371128i	$-0.878971\pi$
$491$		−	178944.i		−	0.742256i	0.928582	−	0.371128i	$-0.121029\pi$
$492$		0			0
$493$	41333.5			0.170062
$494$			103570.i			0.424403i
$495$		0			0
$496$	−33939.8			−0.137958
$497$		−	315366.i		−	1.27674i
$498$		0			0
$499$	−402187.			−1.61520			−0.807601	−	0.589729i	$-0.799235\pi$
$499$	−402187.			−1.61520			−0.807601	+	0.589729i	$0.799235\pi$
$500$	593071.			2.37228
$501$		0			0
$502$			389561.i			1.54585i
$503$		−	211540.i		−	0.836096i	−0.908425	−	0.418048i	$-0.862714\pi$
$503$		−	211540.i		−	0.836096i	0.908425	−	0.418048i	$-0.137286\pi$
$504$		0			0
$505$			48107.1i			0.188637i
$506$	−77182.8	+	531717.i	−0.301453	+	2.07673i
$507$		0			0
$508$		−	379167.i		−	1.46928i
$509$	−185877.			−0.717446			−0.358723	−	0.933444i	$-0.616788\pi$
$509$	−185877.			−0.717446			−0.358723	+	0.933444i	$0.616788\pi$
$510$		0			0
$511$	−228081.			−0.873470
$512$			376076.i			1.43461i
$513$		0			0
$514$			200487.i			0.758857i
$515$	−85261.9			−0.321470
$516$		0			0
$517$	275390.	+	39974.9i	1.03031	+	0.149557i
$518$	878227.			3.27301
$519$		0			0
$520$	−513285.			−1.89824
$521$	−324045.			−1.19380			−0.596898	−	0.802317i	$-0.703600\pi$
$521$	−324045.			−1.19380			−0.596898	+	0.802317i	$0.703600\pi$
$522$		0			0
$523$		−	159328.i		−	0.582492i	−0.956648	−	0.291246i	$-0.905930\pi$
$523$		−	159328.i		−	0.582492i	0.956648	−	0.291246i	$-0.0940698\pi$
$524$			571482.i			2.08133i
$525$		0			0
$526$	−612725.			−2.21459
$527$		−	3963.35i		−	0.0142706i
$528$		0			0
$529$	52627.1			0.188061
$530$		−	82694.1i		−	0.294390i
$531$		0			0
$532$	−190299.			−0.672376
$533$	−561501.			−1.97650
$534$		0			0
$535$			74681.6i			0.260919i
$536$			1.12910e6i			3.93011i
$537$		0			0
$538$			784246.i			2.70949i
$539$	197678.	+	28694.5i	0.680426	+	0.0987691i
$540$		0			0
$541$		−	94598.0i		−	0.323212i	−0.986855	−	0.161606i	$-0.948333\pi$
$541$		−	94598.0i		−	0.323212i	0.986855	−	0.161606i	$-0.0516674\pi$
$542$	−649860.			−2.21218
$543$		0			0
$544$	−407928.			−1.37843
$545$			273251.i			0.919960i
$546$		0			0
$547$		−	65315.6i		−	0.218294i	−0.994026	−	0.109147i	$-0.965188\pi$
$547$		−	65315.6i		−	0.218294i	0.994026	−	0.109147i	$-0.0348119\pi$
$548$	−88113.5			−0.293414
$549$		0			0
$550$	−62646.8	+	431578.i	−0.207097	+	1.42670i
$551$	26373.9			0.0868701
$552$		0			0
$553$	454684.			1.48682
$554$	596933.			1.94494
$555$		0			0
$556$			1.07251e6i			3.46939i
$557$		−	334291.i		−	1.07749i	−0.842468	−	0.538747i	$-0.818898\pi$
$557$		−	334291.i		−	1.07749i	0.842468	−	0.538747i	$-0.181102\pi$
$558$		0			0
$559$	62214.0			0.199097
$560$		−	738932.i		−	2.35629i
$561$		0			0
$562$	−245606.			−0.777617
$563$			296644.i			0.935876i	0.883761	+	0.467938i	$0.155003\pi$
$563$			296644.i			0.935876i	−0.883761	+	0.467938i	$0.844997\pi$
$564$		0			0
$565$	−28737.5			−0.0900226
$566$	703998.			2.19755
$567$		0			0
$568$		−	1.04181e6i		−	3.22919i
$569$		−	87509.9i		−	0.270292i	−0.990826	−	0.135146i	$-0.956850\pi$
$569$		−	87509.9i		−	0.270292i	0.990826	−	0.135146i	$-0.0431503\pi$
$570$		0			0
$571$			615448.i			1.88764i	0.330462	+	0.943819i	$0.392795\pi$
$571$			615448.i			1.88764i	−0.330462	+	0.943819i	$0.607205\pi$
$572$	146643.	−	1.01023e6i	0.448197	−	3.08766i
$573$		0			0
$574$		−	1.41288e6i		−	4.28826i
$575$	269854.			0.816193
$576$		0			0
$577$	516552.			1.55154			0.775770	−	0.631016i	$-0.217362\pi$
$577$	516552.			1.55154			0.775770	+	0.631016i	$0.217362\pi$
$578$			553053.i			1.65543i
$579$		0			0
$580$			207296.i			0.616219i
$581$	9988.01			0.0295888
$582$		0			0
$583$	102623.	+	14896.5i	0.301932	+	0.0438277i
$584$	−753469.			−2.20922
$585$		0			0
$586$	−755471.			−2.20000
$587$	523368.			1.51891			0.759453	−	0.650562i	$-0.225467\pi$
$587$	523368.			1.51891			0.759453	+	0.650562i	$0.225467\pi$
$588$		0			0
$589$		−	2528.91i		−	0.00728960i
$590$			207384.i			0.595761i
$591$		0			0
$592$	1.65987e6			4.73621
$593$			221167.i			0.628942i	0.949267	+	0.314471i	$0.101827\pi$
$593$			221167.i			0.628942i	−0.949267	+	0.314471i	$0.898173\pi$
$594$		0			0
$595$	86289.4			0.243738
$596$		−	886704.i		−	2.49624i
$597$		0			0
$598$	−865052.			−2.41902
$599$	−293216.			−0.817210			−0.408605	−	0.912711i	$-0.633985\pi$
$599$	−293216.			−0.817210			−0.408605	+	0.912711i	$0.633985\pi$
$600$		0			0
$601$		−	256959.i		−	0.711401i	−0.934600	−	0.355701i	$-0.884242\pi$
$601$		−	256959.i		−	0.711401i	0.934600	−	0.355701i	$-0.115758\pi$
$602$			156546.i			0.431966i
$603$		0			0
$604$			741543.i			2.03265i
$605$	175875.	+	52158.2i	0.480500	+	0.142499i
$606$		0			0
$607$		−	182631.i		−	0.495676i	−0.968802	−	0.247838i	$-0.920280\pi$
$607$		−	182631.i		−	0.495676i	0.968802	−	0.247838i	$-0.0797200\pi$
$608$	−260288.			−0.704122
$609$		0			0
$610$	479187.			1.28779
$611$			448032.i			1.20013i
$612$		0			0
$613$		−	199000.i		−	0.529580i	−0.964306	−	0.264790i	$-0.914697\pi$
$613$		−	199000.i		−	0.529580i	0.964306	−	0.264790i	$-0.0853026\pi$
$614$	−1.09932e6			−2.91601
$615$		0			0
$616$	1.60282e6	+	232661.i	4.22399	+	0.613144i
$617$	−385966.			−1.01386			−0.506931	−	0.861987i	$-0.669220\pi$
$617$	−385966.			−1.01386			−0.506931	+	0.861987i	$0.669220\pi$
$618$		0			0
$619$	−287681.			−0.750809			−0.375404	−	0.926861i	$-0.622496\pi$
$619$	−287681.			−0.750809			−0.375404	+	0.926861i	$0.622496\pi$
$620$	19877.0			0.0517092
$621$		0			0
$622$		−	1.27369e6i		−	3.29217i
$623$			457112.i			1.17773i
$624$		0			0
$625$	120912.			0.309535
$626$			563464.i			1.43786i
$627$		0			0
$628$	−1.85154e6			−4.69476
$629$			193833.i			0.489921i
$630$		0			0
$631$	505556.			1.26973			0.634863	−	0.772625i	$-0.281056\pi$
$631$	505556.			1.26973			0.634863	+	0.772625i	$0.281056\pi$
$632$	1.50205e6			3.76055
$633$		0			0
$634$			867184.i			2.15741i
$635$			109704.i			0.272068i
$636$		0			0
$637$			321603.i			0.792577i
$638$	−352301.	−	51139.2i	−0.865511	−	0.125635i
$639$		0			0
$640$		−	615477.i		−	1.50263i
$641$	729128.			1.77455			0.887274	−	0.461242i	$-0.152596\pi$
$641$	729128.			1.77455			0.887274	+	0.461242i	$0.152596\pi$
$642$		0			0
$643$	−548088.			−1.32565			−0.662824	−	0.748775i	$-0.730642\pi$
$643$	−548088.			−1.32565			−0.662824	+	0.748775i	$0.730642\pi$
$644$		−	1.58944e6i		−	3.83242i
$645$		0			0
$646$		−	57518.5i		−	0.137830i
$647$	301562.			0.720390			0.360195	−	0.932877i	$-0.382710\pi$
$647$	301562.			0.720390			0.360195	+	0.932877i	$0.382710\pi$
$648$		0			0
$649$	−257364.	−	37358.3i	−0.611024	−	0.0886947i
$650$	−702135.			−1.66186
$651$		0			0
$652$	1.42572e6			3.35381
$653$	−184146.			−0.431853			−0.215926	−	0.976410i	$-0.569277\pi$
$653$	−184146.			−0.431853			−0.215926	+	0.976410i	$0.569277\pi$
$654$		0			0
$655$		−	165347.i		−	0.385402i
$656$		−	2.67038e6i		−	6.20533i
$657$		0			0
$658$	−1.12736e6			−2.60383
$659$		−	608904.i		−	1.40210i	−0.713114	−	0.701048i	$-0.752716\pi$
$659$		−	608904.i		−	1.40210i	0.713114	−	0.701048i	$-0.247284\pi$
$660$		0			0
$661$	−211140.			−0.483246			−0.241623	−	0.970370i	$-0.577680\pi$
$661$	−211140.			−0.483246			−0.241623	+	0.970370i	$0.577680\pi$
$662$		−	561995.i		−	1.28238i
$663$		0			0
$664$	32995.5			0.0748374
$665$	55059.1			0.124505
$666$		0			0
$667$			220284.i			0.495144i
$668$			983274.i			2.20354i
$669$		0			0
$670$		−	518107.i		−	1.15417i
$671$	−86320.9	+	594670.i	−0.191722	+	1.32078i
$672$		0			0
$673$			207518.i			0.458169i	0.973406	+	0.229085i	$0.0735733\pi$
$673$			207518.i			0.458169i	−0.973406	+	0.229085i	$0.926427\pi$
$674$	−1.51373e6			−3.33217
$675$		0			0
$676$	406697.			0.889974
$677$			419679.i			0.915673i	0.889037	+	0.457836i	$0.151375\pi$
$677$			419679.i			0.915673i	−0.889037	+	0.457836i	$0.848625\pi$
$678$		0			0
$679$		−	154020.i		−	0.334071i
$680$	285058.			0.616475
$681$		0			0
$682$	−4903.59	+	33781.1i	−0.0105425	+	0.0726282i
$683$	−266045.			−0.570313			−0.285156	−	0.958481i	$-0.592046\pi$
$683$	−266045.			−0.570313			−0.285156	+	0.958481i	$0.592046\pi$
$684$		0			0
$685$	25493.9			0.0543319
$686$	367736.			0.781425
$687$		0			0
$688$			295876.i			0.625076i
$689$			166958.i			0.351697i
$690$		0			0
$691$	594449.			1.24497			0.622485	−	0.782632i	$-0.286123\pi$
$691$	594449.			1.24497			0.622485	+	0.782632i	$0.286123\pi$
$692$			1.41209e6i			2.94883i
$693$		0			0
$694$	1.59131e6			3.30398
$695$		−	310311.i		−	0.642432i
$696$		0			0
$697$	311835.			0.641889
$698$	1.61931e6			3.32369
$699$		0			0
$700$		−	1.29010e6i		−	2.63286i
$701$		−	329130.i		−	0.669779i	−0.942257	−	0.334890i	$-0.891301\pi$
$701$		−	329130.i		−	0.669779i	0.942257	−	0.334890i	$-0.108699\pi$
$702$		0			0
$703$			123680.i			0.250258i
$704$	1.70184e6	+	247035.i	3.43379	+	0.498441i
$705$		0			0
$706$			1.07832e6i			2.16340i
$707$	244397.			0.488942
$708$		0			0
$709$	−148161.			−0.294742			−0.147371	−	0.989081i	$-0.547081\pi$
$709$	−148161.			−0.294742			−0.147371	+	0.989081i	$0.547081\pi$
$710$			478053.i			0.948329i
$711$		0			0
$712$			1.51008e6i			2.97878i
$713$	21122.4			0.0415494
$714$		0			0
$715$	−42428.2	+	292291.i	−0.0829932	+	0.571745i
$716$	185933.			0.362687
$717$		0			0
$718$	−697324.			−1.35265
$719$	140295.			0.271384			0.135692	−	0.990751i	$-0.456674\pi$
$719$	140295.			0.271384			0.135692	+	0.990751i	$0.456674\pi$
$720$		0			0
$721$			433154.i			0.833243i
$722$			966903.i			1.85485i
$723$		0			0
$724$	−1.04241e6			−1.98866
$725$			178798.i			0.340162i
$726$		0			0
$727$	−116812.			−0.221013			−0.110507	−	0.993875i	$-0.535247\pi$
$727$	−116812.			−0.221013			−0.110507	+	0.993875i	$0.535247\pi$
$728$			2.60763e6i			4.92020i
$729$		0			0
$730$	345741.			0.648791
$731$	−34551.2			−0.0646589
$732$		0			0
$733$		−	645954.i		−	1.20225i	−0.799156	−	0.601123i	$-0.794720\pi$
$733$		−	645954.i		−	1.20225i	0.799156	−	0.601123i	$-0.205280\pi$
$734$		−	467317.i		−	0.867400i
$735$		0			0
$736$		−	2.17402e6i		−	4.01337i
$737$	642969.	+	93331.9i	1.18374	+	0.171828i
$738$		0			0
$739$			72867.2i			0.133427i	0.997772	+	0.0667134i	$0.0212513\pi$
$739$			72867.2i			0.133427i	−0.997772	+	0.0667134i	$0.978749\pi$
$740$	−972112.			−1.77522
$741$		0			0
$742$	−420109.			−0.763052
$743$		−	675603.i		−	1.22381i	−0.790932	−	0.611905i	$-0.790404\pi$
$743$		−	675603.i		−	1.22381i	0.790932	−	0.611905i	$-0.209596\pi$
$744$		0			0
$745$			256550.i			0.462231i
$746$	−771703.			−1.38667
$747$		0			0
$748$	−81439.7	+	561043.i	−0.145557	+	1.00275i
$749$	379403.			0.676296
$750$		0			0
$751$	−168392.			−0.298567			−0.149283	−	0.988794i	$-0.547697\pi$
$751$	−168392.			−0.298567			−0.149283	+	0.988794i	$0.547697\pi$
$752$	−2.13074e6			−3.76787
$753$		0			0
$754$		−	573159.i		−	1.00817i
$755$		−	214551.i		−	0.376388i
$756$		0			0
$757$	−63422.1			−0.110675			−0.0553374	−	0.998468i	$-0.517623\pi$
$757$	−63422.1			−0.110675			−0.0553374	+	0.998468i	$0.517623\pi$
$758$		−	1.75636e6i		−	3.05685i
$759$		0			0
$760$	181888.			0.314904
$761$		−	377856.i		−	0.652464i	−0.945290	−	0.326232i	$-0.894221\pi$
$761$		−	377856.i		−	0.652464i	0.945290	−	0.326232i	$-0.105779\pi$
$762$		0			0
$763$	1.38819e6			2.38451
$764$	−687091.			−1.17714
$765$		0			0
$766$			106424.i			0.181376i
$767$		−	418706.i		−	0.711735i
$768$		0			0
$769$			795673.i			1.34549i	0.739872	+	0.672747i	$0.234886\pi$
$769$			795673.i			1.34549i	−0.739872	+	0.672747i	$0.765114\pi$
$770$	−735477.	−	106760.i	−1.24047	−	0.180064i
$771$		0			0
$772$			1.26679e6i			2.12555i
$773$	−929744.			−1.55598			−0.777991	−	0.628276i	$-0.783761\pi$
$773$	−929744.			−1.55598			−0.777991	+	0.628276i	$0.783761\pi$
$774$		0			0
$775$	17144.4			0.0285443
$776$		−	508808.i		−	0.844949i
$777$		0			0
$778$			1.22528e6i			2.02431i
$779$	198974.			0.327885
$780$		0			0
$781$	−593262.	−	86116.5i	−0.972623	−	0.141184i
$782$	480416.			0.785604
$783$		0			0
$784$	−1.52947e6			−2.48834
$785$	535706.			0.869334
$786$		0			0
$787$		−	852028.i		−	1.37564i	−0.725882	−	0.687819i	$-0.758568\pi$
$787$		−	852028.i		−	1.37564i	0.725882	−	0.687819i	$-0.241432\pi$
$788$			2.37935e6i			3.83183i
$789$		0			0
$790$	−689240.			−1.10437
$791$			145994.i			0.233337i
$792$		0			0
$793$	−967471.			−1.53848
$794$		−	591454.i		−	0.938167i
$795$		0			0
$796$	−1.37791e6			−2.17467
$797$	−691392.			−1.08845			−0.544224	−	0.838940i	$-0.683176\pi$
$797$	−691392.			−1.08845			−0.544224	+	0.838940i	$0.683176\pi$
$798$		0			0
$799$		−	248819.i		−	0.389754i
$800$		−	1.76459e6i		−	2.75717i
$801$		0			0
$802$		−	933370.i		−	1.45113i
$803$	−62281.9	+	429064.i	−0.0965896	+	0.665412i
$804$		0			0
$805$			459874.i			0.709654i
$806$	−54958.6			−0.0845991
$807$		0			0
$808$	807369.			1.23666
$809$		−	227949.i		−	0.348289i	−0.984720	−	0.174145i	$-0.944284\pi$
$809$		−	227949.i		−	0.348289i	0.984720	−	0.174145i	$-0.0557161\pi$
$810$		0			0
$811$		−	1.02649e6i		−	1.56068i	−0.625356	−	0.780340i	$-0.715046\pi$
$811$		−	1.02649e6i		−	1.56068i	0.625356	−	0.780340i	$-0.284954\pi$
$812$	1.05312e6			1.59723
$813$		0			0
$814$	239816.	−	1.65211e6i	0.361934	−	2.49339i
$815$	−412503.			−0.621029
$816$		0			0
$817$	−22046.2			−0.0330286
$818$	624047.			0.932633
$819$		0			0
$820$			1.56392e6i			2.32588i
$821$			714226.i			1.05962i	0.848117	+	0.529809i	$0.177736\pi$
$821$			714226.i			1.05962i	−0.848117	+	0.529809i	$0.822264\pi$
$822$		0			0
$823$	1.17206e6			1.73042			0.865210	−	0.501409i	$-0.167185\pi$
$823$	1.17206e6			1.73042			0.865210	+	0.501409i	$0.167185\pi$
$824$			1.43093e6i			2.10748i
$825$		0			0
$826$	1.05357e6			1.54420
$827$		−	788589.i		−	1.15303i	−0.817087	−	0.576514i	$-0.804413\pi$
$827$		−	788589.i		−	1.15303i	0.817087	−	0.576514i	$-0.195587\pi$
$828$		0			0
$829$	−1.30745e6			−1.90246			−0.951229	−	0.308484i	$-0.900178\pi$
$829$	−1.30745e6			−1.90246			−0.951229	+	0.308484i	$0.900178\pi$
$830$	−15140.5			−0.0219778
$831$		0			0
$832$			2.76873e6i			3.99976i
$833$		−	178606.i		−	0.257398i
$834$		0			0
$835$		−	284491.i		−	0.408033i
$836$	−51964.6	+	357987.i	−0.0743524	+	0.512218i
$837$		0			0
$838$		−	1.23794e6i		−	1.76283i
$839$	872858.			1.23999			0.619997	−	0.784604i	$-0.287134\pi$
$839$	872858.			1.23999			0.619997	+	0.784604i	$0.287134\pi$
$840$		0			0
$841$	561327.			0.793640
$842$		−	714894.i		−	1.00836i
$843$		0			0
$844$		−	2.59229e6i		−	3.63914i
$845$	−117670.			−0.164798
$846$		0			0
$847$	264978.	−	893494.i	0.369354	−	1.24545i
$848$	−794016.			−1.10417
$849$		0			0
$850$	389938.			0.539707
$851$	−1.03302e6			−1.42643
$852$		0			0
$853$		−	62382.2i		−	0.0857359i	−0.999081	−	0.0428680i	$-0.986351\pi$
$853$		−	62382.2i		−	0.0857359i	0.999081	−	0.0428680i	$-0.0136495\pi$
$854$		−	2.43440e6i		−	3.33793i
$855$		0			0
$856$	1.25336e6			1.71052
$857$		−	378766.i		−	0.515714i	−0.966183	−	0.257857i	$-0.916984\pi$
$857$		−	378766.i		−	0.515714i	0.966183	−	0.257857i	$-0.0830163\pi$
$858$		0			0
$859$	128702.			0.174421			0.0872106	−	0.996190i	$-0.472205\pi$
$859$	128702.			0.174421			0.0872106	+	0.996190i	$0.472205\pi$
$860$		−	173281.i		−	0.234291i
$861$		0			0
$862$	−1.64302e6			−2.21120
$863$	−692449.			−0.929750			−0.464875	−	0.885376i	$-0.653901\pi$
$863$	−692449.			−0.929750			−0.464875	+	0.885376i	$0.653901\pi$
$864$		0			0
$865$		−	408560.i		−	0.546038i
$866$		−	255379.i		−	0.340526i
$867$		0			0
$868$		−	100981.i		−	0.134029i
$869$	124160.	−	855346.i	0.164415	−	1.13267i
$870$		0			0
$871$			1.04605e6i			1.37885i
$872$	4.58590e6			6.03103
$873$		0			0
$874$	306541.			0.401297
$875$			871740.i			1.13860i
$876$		0			0
$877$		−	202045.i		−	0.262694i	−0.991336	−	0.131347i	$-0.958070\pi$
$877$		−	202045.i		−	0.262694i	0.991336	−	0.131347i	$-0.0419302\pi$
$878$	2.26200e6			2.93430
$879$		0			0
$880$	−1.39007e6	−	201779.i	−1.79503	−	0.260562i
$881$	777238.			1.00139			0.500694	−	0.865624i	$-0.333078\pi$
$881$	777238.			1.00139			0.500694	+	0.865624i	$0.333078\pi$
$882$		0			0
$883$	−138486.			−0.177617			−0.0888083	−	0.996049i	$-0.528306\pi$
$883$	−138486.			−0.177617			−0.0888083	+	0.996049i	$0.528306\pi$
$884$	−912763.			−1.16803
$885$		0			0
$886$			2.11045e6i			2.68848i
$887$			637104.i			0.809773i	0.914367	+	0.404886i	$0.132689\pi$
$887$			637104.i			0.809773i	−0.914367	+	0.404886i	$0.867311\pi$
$888$		0			0
$889$	557329.			0.705193
$890$		−	692921.i		−	0.874790i
$891$		0			0
$892$	361417.			0.454233
$893$		−	158765.i		−	0.199092i
$894$		0			0
$895$	−53796.1			−0.0671591
$896$	−3.12679e6			−3.89478
$897$		0			0
$898$			1.78185e6i			2.20963i
$899$			13995.1i			0.0173164i
$900$		0			0
$901$		−	92721.9i		−	0.114218i
$902$	−2.65789e6	−	385813.i	−3.26681	−	0.474202i
$903$		0			0
$904$			482293.i			0.590166i
$905$	301600.			0.368242
$906$		0			0
$907$	−249854.			−0.303719			−0.151859	−	0.988402i	$-0.548526\pi$
$907$	−249854.			−0.303719			−0.151859	+	0.988402i	$0.548526\pi$
$908$			2.56712e6i			3.11368i
$909$		0			0
$910$		−	1.19655e6i		−	1.44493i
$911$	−96364.5			−0.116113			−0.0580564	−	0.998313i	$-0.518490\pi$
$911$	−96364.5			−0.116113			−0.0580564	+	0.998313i	$0.518490\pi$
$912$		0			0
$913$	2727.41	−	18789.3i	0.00327197	−	0.0225408i
$914$	2.09363e6			2.50615
$915$		0			0
$916$	1.98856e6			2.36999
$917$	−840008.			−0.998952
$918$		0			0
$919$			586637.i			0.694606i	0.937753	+	0.347303i	$0.112902\pi$
$919$			586637.i			0.694606i	−0.937753	+	0.347303i	$0.887098\pi$
$920$			1.51920e6i			1.79489i
$921$		0			0
$922$	−2.03973e6			−2.39945
$923$		−	965180.i		−	1.13294i
$924$		0			0
$925$	−838469.			−0.979949
$926$			1.97061e6i			2.29815i
$927$		0			0
$928$	1.44045e6			1.67264
$929$	−702941.			−0.814494			−0.407247	−	0.913318i	$-0.633511\pi$
$929$	−702941.			−0.814494			−0.407247	+	0.913318i	$0.633511\pi$
$930$		0			0
$931$		−	113964.i		−	0.131482i
$932$			3.02823e6i			3.48624i
$933$		0			0
$934$			1.41795e6i			1.62542i
$935$	23562.9	−	162327.i	0.0269530	−	0.185681i
$936$		0			0
$937$		−	1.34733e6i		−	1.53459i	−0.641292	−	0.767297i	$-0.721601\pi$
$937$		−	1.34733e6i		−	1.53459i	0.641292	−	0.767297i	$-0.278399\pi$
$938$	−2.63212e6			−2.99158
$939$		0			0
$940$	1.24788e6			1.41227
$941$			146016.i			0.164900i	0.996595	+	0.0824501i	$0.0262745\pi$
$941$			146016.i			0.164900i	−0.996595	+	0.0824501i	$0.973725\pi$
$942$		0			0
$943$			1.66190e6i			1.86889i
$944$	1.99127e6			2.23453
$945$		0			0
$946$	294493.	+	42747.8i	0.329073	+	0.0477674i
$947$	−688104.			−0.767280			−0.383640	−	0.923483i	$-0.625330\pi$
$947$	−688104.			−0.767280			−0.383640	+	0.923483i	$0.625330\pi$
$948$		0			0
$949$	−698045.			−0.775088
$950$	248810.			0.275689
$951$		0			0
$952$		−	1.44817e6i		−	1.59789i
$953$		−	93659.4i		−	0.103125i	−0.998670	−	0.0515627i	$-0.983580\pi$
$953$		−	93659.4i		−	0.103125i	0.998670	−	0.0515627i	$-0.0164202\pi$
$954$		0			0
$955$	198796.			0.217972
$956$			696098.i			0.761649i
$957$		0			0
$958$	3.48539e6			3.79769
$959$		−	129516.i		−	0.140827i
$960$		0			0
$961$	−922179.			−0.998547
$962$	2.68782e6			2.90436
$963$		0			0
$964$		−	267788.i		−	0.288162i
$965$		−	366521.i		−	0.393590i
$966$		0			0
$967$		−	4474.08i		−	0.00478465i	−0.999997	−	0.00239233i	$-0.999238\pi$
$967$		−	4474.08i		−	0.00478465i	0.999997	−	0.00239233i	$-0.000761502\pi$
$968$	875358.	−	2.95167e6i	0.934190	−	3.15004i
$969$		0			0
$970$			233474.i			0.248139i
$971$	344326.			0.365200			0.182600	−	0.983187i	$-0.441549\pi$
$971$	344326.			0.365200			0.182600	+	0.983187i	$0.441549\pi$
$972$		0			0
$973$	−1.57646e6			−1.66517
$974$			83912.7i			0.0884524i
$975$		0			0
$976$		−	4.60108e6i		−	4.83015i
$977$	−1.02530e6			−1.07414			−0.537069	−	0.843539i	$-0.680468\pi$
$977$	−1.02530e6			−1.07414			−0.537069	+	0.843539i	$0.680468\pi$
$978$		0			0
$979$	859914.	+	124823.i	0.897201	+	0.130235i
$980$	895745.			0.932679
$981$		0			0
$982$	1.37805e6			1.42903
$983$	−230734.			−0.238784			−0.119392	−	0.992847i	$-0.538095\pi$
$983$	−230734.			−0.238784			−0.119392	+	0.992847i	$0.538095\pi$
$984$		0			0
$985$		−	688418.i		−	0.709544i
$986$			318310.i			0.327414i
$987$		0			0
$988$	−582411.			−0.596644
$989$		−	184138.i		−	0.188257i
$990$		0			0
$991$	378202.			0.385103			0.192551	−	0.981287i	$-0.438324\pi$
$991$	378202.			0.385103			0.192551	+	0.981287i	$0.438324\pi$
$992$		−	138120.i		−	0.140357i
$993$		0			0
$994$	2.42864e6			2.45805
$995$	398669.			0.402686
$996$		0			0
$997$			1.02115e6i			1.02731i	0.857997	+	0.513655i	$0.171709\pi$
$997$			1.02115e6i			1.02731i	−0.857997	+	0.513655i	$0.828291\pi$
$998$		−	3.09725e6i		−	3.10968i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	99.5.c.c.10.8		8
3.2	odd	2		33.5.c.a.10.1	✓	8
11.10	odd	2	inner	99.5.c.c.10.1		8
12.11	even	2		528.5.j.a.241.2		8
33.32	even	2		33.5.c.a.10.8	yes	8
132.131	odd	2		528.5.j.a.241.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.5.c.a.10.1	✓	8	3.2	odd	2
33.5.c.a.10.8	yes	8	33.32	even	2
99.5.c.c.10.1		8	11.10	odd	2	inner
99.5.c.c.10.8		8	1.1	even	1	trivial
528.5.j.a.241.1		8	132.131	odd	2
528.5.j.a.241.2		8	12.11	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:	\( N \)	\(=\)	\( 99 = 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	99.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+7.70102i q^{2} -43.3057 q^{4} +12.5296 q^{5} -63.6540i q^{7} -210.281i q^{8} +O(q^{10})\)	\(q+7.70102i q^{2} -43.3057 q^{4} +12.5296 q^{5} -63.6540i q^{7} -210.281i q^{8} +96.4909i q^{10} +(-119.745 - 17.3819i) q^{11} -194.814i q^{13} +490.200 q^{14} +926.491 q^{16} +108.192i q^{17} +69.0344i q^{19} -542.604 q^{20} +(133.858 - 922.159i) q^{22} -576.600 q^{23} -468.008 q^{25} +1500.26 q^{26} +2756.58i q^{28} -382.039i q^{29} -36.6327 q^{31} +3770.42i q^{32} -833.186 q^{34} -797.560i q^{35} +1791.57 q^{37} -531.635 q^{38} -2634.75i q^{40} -2882.25i q^{41} +319.351i q^{43} +(5185.64 + 752.735i) q^{44} -4440.41i q^{46} -2299.80 q^{47} -1650.83 q^{49} -3604.14i q^{50} +8436.53i q^{52} -857.015 q^{53} +(-1500.36 - 217.789i) q^{55} -13385.2 q^{56} +2942.09 q^{58} +2149.26 q^{59} -4966.14i q^{61} -282.109i q^{62} -14212.2 q^{64} -2440.94i q^{65} -5369.49 q^{67} -4685.32i q^{68} +6142.03 q^{70} +4954.38 q^{71} -3583.14i q^{73} +13796.9i q^{74} -2989.58i q^{76} +(-1106.43 + 7622.24i) q^{77} +7143.06i q^{79} +11608.6 q^{80} +22196.2 q^{82} +156.911i q^{83} +1355.60i q^{85} -2459.33 q^{86} +(-3655.09 + 25180.2i) q^{88} -7181.21 q^{89} -12400.7 q^{91} +24970.1 q^{92} -17710.8i q^{94} +864.975i q^{95} +2419.65 q^{97} -12713.0i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 76 q^{4} + 36 q^{5}+O(q^{10}) \) 8 * q - 76 * q^4 + 36 * q^5	\( 8 q - 76 q^{4} + 36 q^{5} - 36 q^{11} + 1140 q^{14} + 1412 q^{16} - 2532 q^{20} - 780 q^{22} - 516 q^{23} - 2280 q^{25} + 1524 q^{26} + 2752 q^{31} - 4920 q^{34} + 5296 q^{37} - 696 q^{38} + 6540 q^{44} - 420 q^{47} - 6832 q^{49} - 3540 q^{53} + 3784 q^{55} - 17964 q^{56} + 21624 q^{58} + 16632 q^{59} - 27508 q^{64} - 3656 q^{67} + 3312 q^{70} + 13212 q^{71} - 23268 q^{77} + 4476 q^{80} + 17088 q^{82} - 19896 q^{86} - 12516 q^{88} - 15528 q^{89} - 19752 q^{91} + 81180 q^{92} + 7624 q^{97}+O(q^{100}) \) 8 * q - 76 * q^4 + 36 * q^5 - 36 * q^11 + 1140 * q^14 + 1412 * q^16 - 2532 * q^20 - 780 * q^22 - 516 * q^23 - 2280 * q^25 + 1524 * q^26 + 2752 * q^31 - 4920 * q^34 + 5296 * q^37 - 696 * q^38 + 6540 * q^44 - 420 * q^47 - 6832 * q^49 - 3540 * q^53 + 3784 * q^55 - 17964 * q^56 + 21624 * q^58 + 16632 * q^59 - 27508 * q^64 - 3656 * q^67 + 3312 * q^70 + 13212 * q^71 - 23268 * q^77 + 4476 * q^80 + 17088 * q^82 - 19896 * q^86 - 12516 * q^88 - 15528 * q^89 - 19752 * q^91 + 81180 * q^92 + 7624 * q^97

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