LMFDB - Embedded newform 50.6.b.d.49.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [50,6,Mod(49,50)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(50, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("50.49");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 50 = 2 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	50.b (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:	$8.01919099065$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 10)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	50.49
Dual form			50.6.b.d.49.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000i q^{2} -26.0000i q^{3} -16.0000 q^{4} +104.000 q^{6} +22.0000i q^{7} -64.0000i q^{8} -433.000 q^{9} +O(q^{10})$	\(q+4.00000i q^{2} -26.0000i q^{3} -16.0000 q^{4} +104.000 q^{6} +22.0000i q^{7} -64.0000i q^{8} -433.000 q^{9} -768.000 q^{11} +416.000i q^{12} -46.0000i q^{13} -88.0000 q^{14} +256.000 q^{16} -378.000i q^{17} -1732.00i q^{18} -1100.00 q^{19} +572.000 q^{21} -3072.00i q^{22} -1986.00i q^{23} -1664.00 q^{24} +184.000 q^{26} +4940.00i q^{27} -352.000i q^{28} +5610.00 q^{29} -3988.00 q^{31} +1024.00i q^{32} +19968.0i q^{33} +1512.00 q^{34} +6928.00 q^{36} +142.000i q^{37} -4400.00i q^{38} -1196.00 q^{39} +1542.00 q^{41} +2288.00i q^{42} -5026.00i q^{43} +12288.0 q^{44} +7944.00 q^{46} -24738.0i q^{47} -6656.00i q^{48} +16323.0 q^{49} -9828.00 q^{51} +736.000i q^{52} -14166.0i q^{53} -19760.0 q^{54} +1408.00 q^{56} +28600.0i q^{57} +22440.0i q^{58} -28380.0 q^{59} +5522.00 q^{61} -15952.0i q^{62} -9526.00i q^{63} -4096.00 q^{64} -79872.0 q^{66} +24742.0i q^{67} +6048.00i q^{68} -51636.0 q^{69} +42372.0 q^{71} +27712.0i q^{72} -52126.0i q^{73} -568.000 q^{74} +17600.0 q^{76} -16896.0i q^{77} -4784.00i q^{78} +39640.0 q^{79} +23221.0 q^{81} +6168.00i q^{82} -59826.0i q^{83} -9152.00 q^{84} +20104.0 q^{86} -145860. i q^{87} +49152.0i q^{88} -57690.0 q^{89} +1012.00 q^{91} +31776.0i q^{92} +103688. i q^{93} +98952.0 q^{94} +26624.0 q^{96} +144382. i q^{97} +65292.0i q^{98} +332544. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 32 q^{4} + 208 q^{6} - 866 q^{9}+O(q^{10}) $ 2 * q - 32 * q^4 + 208 * q^6 - 866 * q^9	$ 2 q - 32 q^{4} + 208 q^{6} - 866 q^{9} - 1536 q^{11} - 176 q^{14} + 512 q^{16} - 2200 q^{19} + 1144 q^{21} - 3328 q^{24} + 368 q^{26} + 11220 q^{29} - 7976 q^{31} + 3024 q^{34} + 13856 q^{36} - 2392 q^{39} + 3084 q^{41} + 24576 q^{44} + 15888 q^{46} + 32646 q^{49} - 19656 q^{51} - 39520 q^{54} + 2816 q^{56} - 56760 q^{59} + 11044 q^{61} - 8192 q^{64} - 159744 q^{66} - 103272 q^{69} + 84744 q^{71} - 1136 q^{74} + 35200 q^{76} + 79280 q^{79} + 46442 q^{81} - 18304 q^{84} + 40208 q^{86} - 115380 q^{89} + 2024 q^{91} + 197904 q^{94} + 53248 q^{96} + 665088 q^{99}+O(q^{100}) $ 2 * q - 32 * q^4 + 208 * q^6 - 866 * q^9 - 1536 * q^11 - 176 * q^14 + 512 * q^16 - 2200 * q^19 + 1144 * q^21 - 3328 * q^24 + 368 * q^26 + 11220 * q^29 - 7976 * q^31 + 3024 * q^34 + 13856 * q^36 - 2392 * q^39 + 3084 * q^41 + 24576 * q^44 + 15888 * q^46 + 32646 * q^49 - 19656 * q^51 - 39520 * q^54 + 2816 * q^56 - 56760 * q^59 + 11044 * q^61 - 8192 * q^64 - 159744 * q^66 - 103272 * q^69 + 84744 * q^71 - 1136 * q^74 + 35200 * q^76 + 79280 * q^79 + 46442 * q^81 - 18304 * q^84 + 40208 * q^86 - 115380 * q^89 + 2024 * q^91 + 197904 * q^94 + 53248 * q^96 + 665088 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/50\mathbb{Z}\right)^\times$.

$n$	$27$
$\chi(n)$	$-1$

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 $

$2$	4.00000i	0.707107i
$2$	4.00000i	0.707107i
$3$	− 26.0000i	− 1.66790i	−0.551839	−	0.833950i	$-0.686074\pi$
$3$	− 26.0000i	− 1.66790i	0.551839	−	0.833950i	$-0.313926\pi$
$4$	−16.0000	−0.500000
$5$	0	0
$5$	0	0
$6$	104.000	1.17938
$7$	22.0000i	0.169698i	0.996394	+	0.0848492i	$0.0270408\pi$
$7$	22.0000i	0.169698i	−0.996394	+	0.0848492i	$0.972959\pi$
$8$	− 64.0000i	− 0.353553i
$9$	−433.000	−1.78189
$10$	0	0
$11$	−768.000	−1.91372	−0.956862	−	0.290541i	$-0.906165\pi$
$11$	−768.000	−1.91372	−0.956862	+	0.290541i	$0.906165\pi$
$12$	416.000i	0.833950i
$13$	− 46.0000i	− 0.0754917i	−0.999287	−	0.0377459i	$-0.987982\pi$
$13$	− 46.0000i	− 0.0754917i	0.999287	−	0.0377459i	$-0.0120177\pi$
$14$	−88.0000	−0.119995
$15$	0	0
$16$	256.000	0.250000
$17$	− 378.000i	− 0.317227i	−0.987341	−	0.158613i	$-0.949298\pi$
$17$	− 378.000i	− 0.317227i	0.987341	−	0.158613i	$-0.0507023\pi$
$18$	− 1732.00i	− 1.25999i
$19$	−1100.00	−0.699051	−0.349525	−	0.936927i	$-0.613657\pi$
$19$	−1100.00	−0.699051	−0.349525	+	0.936927i	$0.613657\pi$
$20$	0	0
$21$	572.000	0.283040
$22$	− 3072.00i	− 1.35321i
$23$	− 1986.00i	− 0.782816i	−0.920217	−	0.391408i	$-0.871988\pi$
$23$	− 1986.00i	− 0.782816i	0.920217	−	0.391408i	$-0.128012\pi$
$24$	−1664.00	−0.589692
$25$	0	0
$26$	184.000	0.0533807
$27$	4940.00i	1.30412i
$28$	− 352.000i	− 0.0848492i
$29$	5610.00	1.23870	0.619352	−	0.785113i	$-0.287395\pi$
$29$	5610.00	1.23870	0.619352	+	0.785113i	$0.287395\pi$
$30$	0	0
$31$	−3988.00	−0.745334	−0.372667	−	0.927965i	$-0.621557\pi$
$31$	−3988.00	−0.745334	−0.372667	+	0.927965i	$0.621557\pi$
$32$	1024.00i	0.176777i
$33$	19968.0i	3.19190i
$34$	1512.00	0.224313
$35$	0	0
$36$	6928.00	0.890947
$37$	142.000i	0.0170523i	0.999964	+	0.00852617i	$0.00271400\pi$
$37$	142.000i	0.0170523i	−0.999964	+	0.00852617i	$0.997286\pi$
$38$	− 4400.00i	− 0.494303i
$39$	−1196.00	−0.125913
$40$	0	0
$41$	1542.00	0.143260	0.0716300	−	0.997431i	$-0.477180\pi$
$41$	1542.00	0.143260	0.0716300	+	0.997431i	$0.477180\pi$
$42$	2288.00i	0.200139i
$43$	− 5026.00i	− 0.414526i	−0.978285	−	0.207263i	$-0.933544\pi$
$43$	− 5026.00i	− 0.414526i	0.978285	−	0.207263i	$-0.0664555\pi$
$44$	12288.0	0.956862
$45$	0	0
$46$	7944.00	0.553534
$47$	− 24738.0i	− 1.63350i	−0.576990	−	0.816752i	$-0.695773\pi$
$47$	− 24738.0i	− 1.63350i	0.576990	−	0.816752i	$-0.304227\pi$
$48$	− 6656.00i	− 0.416975i
$49$	16323.0	0.971202
$50$	0	0
$51$	−9828.00	−0.529102
$52$	736.000i	0.0377459i
$53$	− 14166.0i	− 0.692720i	−0.938102	−	0.346360i	$-0.887418\pi$
$53$	− 14166.0i	− 0.692720i	0.938102	−	0.346360i	$-0.112582\pi$
$54$	−19760.0	−0.922152
$55$	0	0
$56$	1408.00	0.0599974
$57$	28600.0i	1.16595i
$58$	22440.0i	0.875897i
$59$	−28380.0	−1.06141	−0.530704	−	0.847557i	$-0.678072\pi$
$59$	−28380.0	−1.06141	−0.530704	+	0.847557i	$0.678072\pi$
$60$	0	0
$61$	5522.00	0.190008	0.0950040	−	0.995477i	$-0.469714\pi$
$61$	5522.00	0.190008	0.0950040	+	0.995477i	$0.469714\pi$
$62$	− 15952.0i	− 0.527031i
$63$	− 9526.00i	− 0.302384i
$64$	−4096.00	−0.125000
$65$	0	0
$66$	−79872.0	−2.25702
$67$	24742.0i	0.673361i	0.941619	+	0.336680i	$0.109304\pi$
$67$	24742.0i	0.673361i	−0.941619	+	0.336680i	$0.890696\pi$
$68$	6048.00i	0.158613i
$69$	−51636.0	−1.30566
$70$	0	0
$71$	42372.0	0.997546	0.498773	−	0.866733i	$-0.333784\pi$
$71$	42372.0	0.997546	0.498773	+	0.866733i	$0.333784\pi$
$72$	27712.0i	0.629994i
$73$	− 52126.0i	− 1.14485i	−0.819958	−	0.572423i	$-0.806003\pi$
$73$	− 52126.0i	− 1.14485i	0.819958	−	0.572423i	$-0.193997\pi$
$74$	−568.000	−0.0120578
$75$	0	0
$76$	17600.0	0.349525
$77$	− 16896.0i	− 0.324756i
$78$	− 4784.00i	− 0.0890337i
$79$	39640.0	0.714605	0.357302	−	0.933989i	$-0.383697\pi$
$79$	39640.0	0.714605	0.357302	+	0.933989i	$0.383697\pi$
$80$	0	0
$81$	23221.0	0.393250
$82$	6168.00i	0.101300i
$83$	− 59826.0i	− 0.953223i	−0.879114	−	0.476612i	$-0.841865\pi$
$83$	− 59826.0i	− 0.953223i	0.879114	−	0.476612i	$-0.158135\pi$
$84$	−9152.00	−0.141520
$85$	0	0
$86$	20104.0	0.293114
$87$	− 145860.i	− 2.06604i
$88$	49152.0i	0.676604i
$89$	−57690.0	−0.772015	−0.386007	−	0.922496i	$-0.626146\pi$
$89$	−57690.0	−0.772015	−0.386007	+	0.922496i	$0.626146\pi$
$90$	0	0
$91$	1012.00	0.0128108
$92$	31776.0i	0.391408i
$93$	103688.i	1.24314i
$94$	98952.0	1.15506
$95$	0	0
$96$	26624.0	0.294846
$97$	144382.i	1.55806i	0.626988	+	0.779029i	$0.284288\pi$
$97$	144382.i	1.55806i	−0.626988	+	0.779029i	$0.715712\pi$
$98$	65292.0i	0.686744i
$99$	332544.	3.41005
$100$	0	0
$101$	−141258.	−1.37787	−0.688937	−	0.724821i	$-0.741922\pi$
$101$	−141258.	−1.37787	−0.688937	+	0.724821i	$0.741922\pi$
$102$	− 39312.0i	− 0.374132i
$103$	139814.i	1.29855i	0.760555	+	0.649273i	$0.224927\pi$
$103$	139814.i	1.29855i	−0.760555	+	0.649273i	$0.775073\pi$
$104$	−2944.00	−0.0266904
$105$	0	0
$106$	56664.0	0.489827
$107$	− 86418.0i	− 0.729701i	−0.931066	−	0.364850i	$-0.881120\pi$
$107$	− 86418.0i	− 0.729701i	0.931066	−	0.364850i	$-0.118880\pi$
$108$	− 79040.0i	− 0.652060i
$109$	−218450.	−1.76111	−0.880554	−	0.473947i	$-0.842829\pi$
$109$	−218450.	−1.76111	−0.880554	+	0.473947i	$0.842829\pi$
$110$	0	0
$111$	3692.00	0.0284416
$112$	5632.00i	0.0424246i
$113$	− 28806.0i	− 0.212220i	−0.994354	−	0.106110i	$-0.966160\pi$
$113$	− 28806.0i	− 0.212220i	0.994354	−	0.106110i	$-0.0338396\pi$
$114$	−114400.	−0.824449
$115$	0	0
$116$	−89760.0	−0.619352
$117$	19918.0i	0.134518i
$118$	− 113520.i	− 0.750529i
$119$	8316.00	0.0538328
$120$	0	0
$121$	428773.	2.66234
$122$	22088.0i	0.134356i
$123$	− 40092.0i	− 0.238943i
$124$	63808.0	0.372667
$125$	0	0
$126$	38104.0	0.213818
$127$	216502.i	1.19111i	0.803314	+	0.595556i	$0.203068\pi$
$127$	216502.i	1.19111i	−0.803314	+	0.595556i	$0.796932\pi$
$128$	− 16384.0i	− 0.0883883i
$129$	−130676.	−0.691388
$130$	0	0
$131$	−244608.	−1.24535	−0.622676	−	0.782479i	$-0.713955\pi$
$131$	−244608.	−1.24535	−0.622676	+	0.782479i	$0.713955\pi$
$132$	− 319488.i	− 1.59595i
$133$	− 24200.0i	− 0.118628i
$134$	−98968.0	−0.476138
$135$	0	0
$136$	−24192.0	−0.112157
$137$	239502.i	1.09020i	0.838370	+	0.545102i	$0.183509\pi$
$137$	239502.i	1.09020i	−0.838370	+	0.545102i	$0.816491\pi$
$138$	− 206544.i	− 0.923241i
$139$	−30860.0	−0.135475	−0.0677375	−	0.997703i	$-0.521578\pi$
$139$	−30860.0	−0.135475	−0.0677375	+	0.997703i	$0.521578\pi$
$140$	0	0
$141$	−643188.	−2.72452
$142$	169488.i	0.705372i
$143$	35328.0i	0.144470i
$144$	−110848.	−0.445473
$145$	0	0
$146$	208504.	0.809529
$147$	− 424398.i	− 1.61987i
$148$	− 2272.00i	− 0.00852617i
$149$	100950.	0.372512	0.186256	−	0.982501i	$-0.440365\pi$
$149$	100950.	0.372512	0.186256	+	0.982501i	$0.440365\pi$
$150$	0	0
$151$	12452.0	0.0444423	0.0222212	−	0.999753i	$-0.492926\pi$
$151$	12452.0	0.0444423	0.0222212	+	0.999753i	$0.492926\pi$
$152$	70400.0i	0.247152i
$153$	163674.i	0.565264i
$154$	67584.0	0.229637
$155$	0	0
$156$	19136.0	0.0629564
$157$	6022.00i	0.0194981i	0.999952	+	0.00974903i	$0.00310326\pi$
$157$	6022.00i	0.0194981i	−0.999952	+	0.00974903i	$0.996897\pi$
$158$	158560.i	0.505302i
$159$	−368316.	−1.15539
$160$	0	0
$161$	43692.0	0.132843
$162$	92884.0i	0.278070i
$163$	− 500866.i	− 1.47656i	−0.674492	−	0.738282i	$-0.735637\pi$
$163$	− 500866.i	− 1.47656i	0.674492	−	0.738282i	$-0.264363\pi$
$164$	−24672.0	−0.0716300
$165$	0	0
$166$	239304.	0.674031
$167$	− 555258.i	− 1.54065i	−0.637652	−	0.770324i	$-0.720094\pi$
$167$	− 555258.i	− 1.54065i	0.637652	−	0.770324i	$-0.279906\pi$
$168$	− 36608.0i	− 0.100070i
$169$	369177.	0.994301
$170$	0	0
$171$	476300.	1.24563
$172$	80416.0i	0.207263i
$173$	417354.i	1.06020i	0.847934	+	0.530102i	$0.177846\pi$
$173$	417354.i	1.06020i	−0.847934	+	0.530102i	$0.822154\pi$
$174$	583440.	1.46091
$175$	0	0
$176$	−196608.	−0.478431
$177$	737880.i	1.77032i
$178$	− 230760.i	− 0.545897i
$179$	52380.0	0.122189	0.0610946	−	0.998132i	$-0.480541\pi$
$179$	52380.0	0.122189	0.0610946	+	0.998132i	$0.480541\pi$
$180$	0	0
$181$	546662.	1.24029	0.620144	−	0.784488i	$-0.287074\pi$
$181$	546662.	1.24029	0.620144	+	0.784488i	$0.287074\pi$
$182$	4048.00i	0.00905862i
$183$	− 143572.i	− 0.316914i
$184$	−127104.	−0.276767
$185$	0	0
$186$	−414752.	−0.879035
$187$	290304.i	0.607084i
$188$	395808.i	0.816752i
$189$	−108680.	−0.221307
$190$	0	0
$191$	−452028.	−0.896565	−0.448283	−	0.893892i	$-0.647964\pi$
$191$	−452028.	−0.896565	−0.448283	+	0.893892i	$0.647964\pi$
$192$	106496.i	0.208488i
$193$	485594.i	0.938383i	0.883097	+	0.469191i	$0.155455\pi$
$193$	485594.i	0.938383i	−0.883097	+	0.469191i	$0.844545\pi$
$194$	−577528.	−1.10171
$195$	0	0
$196$	−261168.	−0.485601
$197$	− 1.01018e6i	− 1.85452i	−0.374414	−	0.927262i	$-0.622156\pi$
$197$	− 1.01018e6i	− 1.85452i	0.374414	−	0.927262i	$-0.377844\pi$
$198$	1.33018e6i	2.41127i
$199$	807640.	1.44572	0.722862	−	0.690993i	$-0.242826\pi$
$199$	807640.	1.44572	0.722862	+	0.690993i	$0.242826\pi$
$200$	0	0
$201$	643292.	1.12310
$202$	− 565032.i	− 0.974304i
$203$	123420.i	0.210206i
$204$	157248.	0.264551
$205$	0	0
$206$	−559256.	−0.918211
$207$	859938.i	1.39489i
$208$	− 11776.0i	− 0.0188729i
$209$	844800.	1.33779
$210$	0	0
$211$	149552.	0.231252	0.115626	−	0.993293i	$-0.463113\pi$
$211$	149552.	0.231252	0.115626	+	0.993293i	$0.463113\pi$
$212$	226656.i	0.346360i
$213$	− 1.10167e6i	− 1.66381i
$214$	345672.	0.515976
$215$	0	0
$216$	316160.	0.461076
$217$	− 87736.0i	− 0.126482i
$218$	− 873800.i	− 1.24529i
$219$	−1.35528e6	−1.90949
$220$	0	0
$221$	−17388.0	−0.0239480
$222$	14768.0i	0.0201113i
$223$	− 443506.i	− 0.597224i	−0.954375	−	0.298612i	$-0.903476\pi$
$223$	− 443506.i	− 0.597224i	0.954375	−	0.298612i	$-0.0965237\pi$
$224$	−22528.0	−0.0299987
$225$	0	0
$226$	115224.	0.150062
$227$	− 420018.i	− 0.541007i	−0.962719	−	0.270504i	$-0.912810\pi$
$227$	− 420018.i	− 0.541007i	0.962719	−	0.270504i	$-0.0871902\pi$
$228$	− 457600.i	− 0.582974i
$229$	−1.05875e6	−1.33415	−0.667075	−	0.744990i	$-0.732454\pi$
$229$	−1.05875e6	−1.33415	−0.667075	+	0.744990i	$0.732454\pi$
$230$	0	0
$231$	−439296.	−0.541661
$232$	− 359040.i	− 0.437948i
$233$	− 1.27345e6i	− 1.53671i	−0.640026	−	0.768353i	$-0.721077\pi$
$233$	− 1.27345e6i	− 1.53671i	0.640026	−	0.768353i	$-0.278923\pi$
$234$	−79672.0	−0.0951187
$235$	0	0
$236$	454080.	0.530704
$237$	− 1.03064e6i	− 1.19189i
$238$	33264.0i	0.0380655i
$239$	370680.	0.419763	0.209882	−	0.977727i	$-0.432692\pi$
$239$	370680.	0.419763	0.209882	+	0.977727i	$0.432692\pi$
$240$	0	0
$241$	−561298.	−0.622517	−0.311258	−	0.950325i	$-0.600750\pi$
$241$	−561298.	−0.622517	−0.311258	+	0.950325i	$0.600750\pi$
$242$	1.71509e6i	1.88256i
$243$	596674.i	0.648219i
$244$	−88352.0	−0.0950040
$245$	0	0
$246$	160368.	0.168958
$247$	50600.0i	0.0527726i
$248$	255232.i	0.263515i
$249$	−1.55548e6	−1.58988
$250$	0	0
$251$	577152.	0.578237	0.289119	−	0.957293i	$-0.406638\pi$
$251$	577152.	0.578237	0.289119	+	0.957293i	$0.406638\pi$
$252$	152416.i	0.151192i
$253$	1.52525e6i	1.49809i
$254$	−866008.	−0.842243
$255$	0	0
$256$	65536.0	0.0625000
$257$	651462.i	0.615257i	0.951507	+	0.307628i	$0.0995353\pi$
$257$	651462.i	0.615257i	−0.951507	+	0.307628i	$0.900465\pi$
$258$	− 522704.i	− 0.488885i
$259$	−3124.00	−0.00289375
$260$	0	0
$261$	−2.42913e6	−2.20724
$262$	− 978432.i	− 0.880597i
$263$	917574.i	0.817997i	0.912535	+	0.408999i	$0.134122\pi$
$263$	917574.i	0.817997i	−0.912535	+	0.408999i	$0.865878\pi$
$264$	1.27795e6	1.12851
$265$	0	0
$266$	96800.0	0.0838825
$267$	1.49994e6i	1.28764i
$268$	− 395872.i	− 0.336680i
$269$	735390.	0.619637	0.309818	−	0.950796i	$-0.399732\pi$
$269$	735390.	0.619637	0.309818	+	0.950796i	$0.399732\pi$
$270$	0	0
$271$	−1.12131e6	−0.927474	−0.463737	−	0.885973i	$-0.653492\pi$
$271$	−1.12131e6	−0.927474	−0.463737	+	0.885973i	$0.653492\pi$
$272$	− 96768.0i	− 0.0793066i
$273$	− 26312.0i	− 0.0213672i
$274$	−958008.	−0.770891
$275$	0	0
$276$	826176.	0.652830
$277$	1.66034e6i	1.30016i	0.759864	+	0.650082i	$0.225265\pi$
$277$	1.66034e6i	1.30016i	−0.759864	+	0.650082i	$0.774735\pi$
$278$	− 123440.i	− 0.0957952i
$279$	1.72680e6	1.32811
$280$	0	0
$281$	1.45210e6	1.09706	0.548531	−	0.836130i	$-0.315187\pi$
$281$	1.45210e6	1.09706	0.548531	+	0.836130i	$0.315187\pi$
$282$	− 2.57275e6i	− 1.92653i
$283$	309014.i	0.229357i	0.993403	+	0.114679i	$0.0365838\pi$
$283$	309014.i	0.229357i	−0.993403	+	0.114679i	$0.963416\pi$
$284$	−677952.	−0.498773
$285$	0	0
$286$	−141312.	−0.102156
$287$	33924.0i	0.0243110i
$288$	− 443392.i	− 0.314997i
$289$	1.27697e6	0.899367
$290$	0	0
$291$	3.75393e6	2.59869
$292$	834016.i	0.572423i
$293$	− 1.59301e6i	− 1.08405i	−0.840363	−	0.542024i	$-0.817658\pi$
$293$	− 1.59301e6i	− 1.08405i	0.840363	−	0.542024i	$-0.182342\pi$
$294$	1.69759e6	1.14542
$295$	0	0
$296$	9088.00	0.00602891
$297$	− 3.79392e6i	− 2.49573i
$298$	403800.i	0.263406i
$299$	−91356.0	−0.0590961
$300$	0	0
$301$	110572.	0.0703443
$302$	49808.0i	0.0314255i
$303$	3.67271e6i	2.29816i
$304$	−281600.	−0.174763
$305$	0	0
$306$	−654696.	−0.399702
$307$	− 1.24726e6i	− 0.755284i	−0.925952	−	0.377642i	$-0.876735\pi$
$307$	− 1.24726e6i	− 0.755284i	0.925952	−	0.377642i	$-0.123265\pi$
$308$	270336.i	0.162378i
$309$	3.63516e6	2.16585
$310$	0	0
$311$	−665988.	−0.390450	−0.195225	−	0.980758i	$-0.562544\pi$
$311$	−665988.	−0.390450	−0.195225	+	0.980758i	$0.562544\pi$
$312$	76544.0i	0.0445169i
$313$	− 591286.i	− 0.341143i	−0.985345	−	0.170572i	$-0.945439\pi$
$313$	− 591286.i	− 0.341143i	0.985345	−	0.170572i	$-0.0545614\pi$
$314$	−24088.0	−0.0137872
$315$	0	0
$316$	−634240.	−0.357302
$317$	516342.i	0.288595i	0.989534	+	0.144298i	$0.0460923\pi$
$317$	516342.i	0.288595i	−0.989534	+	0.144298i	$0.953908\pi$
$318$	− 1.47326e6i	− 0.816983i
$319$	−4.30848e6	−2.37054
$320$	0	0
$321$	−2.24687e6	−1.21707
$322$	174768.i	0.0939339i
$323$	415800.i	0.221757i
$324$	−371536.	−0.196625
$325$	0	0
$326$	2.00346e6	1.04409
$327$	5.67970e6i	2.93735i
$328$	− 98688.0i	− 0.0506500i
$329$	544236.	0.277203
$330$	0	0
$331$	−3.29577e6	−1.65343	−0.826717	−	0.562619i	$-0.809794\pi$
$331$	−3.29577e6	−1.65343	−0.826717	+	0.562619i	$0.809794\pi$
$332$	957216.i	0.476612i
$333$	− 61486.0i	− 0.0303854i
$334$	2.22103e6	1.08940
$335$	0	0
$336$	146432.	0.0707600
$337$	− 1.91098e6i	− 0.916602i	−0.888797	−	0.458301i	$-0.848458\pi$
$337$	− 1.91098e6i	− 0.916602i	0.888797	−	0.458301i	$-0.151542\pi$
$338$	1.47671e6i	0.703077i
$339$	−748956.	−0.353962
$340$	0	0
$341$	3.06278e6	1.42636
$342$	1.90520e6i	0.880796i
$343$	728860.i	0.334510i
$344$	−321664.	−0.146557
$345$	0	0
$346$	−1.66942e6	−0.749677
$347$	− 2.42006e6i	− 1.07895i	−0.842001	−	0.539476i	$-0.818622\pi$
$347$	− 2.42006e6i	− 1.07895i	0.842001	−	0.539476i	$-0.181378\pi$
$348$	2.33376e6i	1.03302i
$349$	−2.50727e6	−1.10189	−0.550944	−	0.834542i	$-0.685732\pi$
$349$	−2.50727e6	−1.10189	−0.550944	+	0.834542i	$0.685732\pi$
$350$	0	0
$351$	227240.	0.0984503
$352$	− 786432.i	− 0.338302i
$353$	− 413166.i	− 0.176477i	−0.996099	−	0.0882384i	$-0.971876\pi$
$353$	− 413166.i	− 0.176477i	0.996099	−	0.0882384i	$-0.0281237\pi$
$354$	−2.95152e6	−1.25181
$355$	0	0
$356$	923040.	0.386007
$357$	− 216216.i	− 0.0897878i
$358$	209520.i	0.0864008i
$359$	−1.73772e6	−0.711613	−0.355806	−	0.934560i	$-0.615794\pi$
$359$	−1.73772e6	−0.711613	−0.355806	+	0.934560i	$0.615794\pi$
$360$	0	0
$361$	−1.26610e6	−0.511328
$362$	2.18665e6i	0.877016i
$363$	− 1.11481e7i	− 4.44052i
$364$	−16192.0	−0.00640541
$365$	0	0
$366$	574288.	0.224092
$367$	− 1.16098e6i	− 0.449944i	−0.974365	−	0.224972i	$-0.927771\pi$
$367$	− 1.16098e6i	− 0.449944i	0.974365	−	0.224972i	$-0.0722291\pi$
$368$	− 508416.i	− 0.195704i
$369$	−667686.	−0.255274
$370$	0	0
$371$	311652.	0.117553
$372$	− 1.65901e6i	− 0.621572i
$373$	343754.i	0.127931i	0.997952	+	0.0639655i	$0.0203748\pi$
$373$	343754.i	0.127931i	−0.997952	+	0.0639655i	$0.979625\pi$
$374$	−1.16122e6	−0.429273
$375$	0	0
$376$	−1.58323e6	−0.577531
$377$	− 258060.i	− 0.0935120i
$378$	− 434720.i	− 0.156488i
$379$	−573140.	−0.204957	−0.102478	−	0.994735i	$-0.532677\pi$
$379$	−573140.	−0.204957	−0.102478	+	0.994735i	$0.532677\pi$
$380$	0	0
$381$	5.62905e6	1.98666
$382$	− 1.80811e6i	− 0.633967i
$383$	− 2.88055e6i	− 1.00341i	−0.865039	−	0.501704i	$-0.832707\pi$
$383$	− 2.88055e6i	− 1.00341i	0.865039	−	0.501704i	$-0.167293\pi$
$384$	−425984.	−0.147423
$385$	0	0
$386$	−1.94238e6	−0.663537
$387$	2.17626e6i	0.738640i
$388$	− 2.31011e6i	− 0.779029i
$389$	3.08559e6	1.03387	0.516933	−	0.856026i	$-0.327074\pi$
$389$	3.08559e6	1.03387	0.516933	+	0.856026i	$0.327074\pi$
$390$	0	0
$391$	−750708.	−0.248330
$392$	− 1.04467e6i	− 0.343372i
$393$	6.35981e6i	2.07712i
$394$	4.04071e6	1.31135
$395$	0	0
$396$	−5.32070e6	−1.70503
$397$	− 885458.i	− 0.281963i	−0.990012	−	0.140981i	$-0.954974\pi$
$397$	− 885458.i	− 0.281963i	0.990012	−	0.140981i	$-0.0450258\pi$
$398$	3.23056e6i	1.02228i
$399$	−629200.	−0.197859
$400$	0	0
$401$	−3.75344e6	−1.16565	−0.582825	−	0.812598i	$-0.698053\pi$
$401$	−3.75344e6	−1.16565	−0.582825	+	0.812598i	$0.698053\pi$
$402$	2.57317e6i	0.794151i
$403$	183448.i	0.0562666i
$404$	2.26013e6	0.688937
$405$	0	0
$406$	−493680.	−0.148638
$407$	− 109056.i	− 0.0326335i
$408$	628992.i	0.187066i
$409$	1.94653e6	0.575377	0.287689	−	0.957724i	$-0.407113\pi$
$409$	1.94653e6	0.575377	0.287689	+	0.957724i	$0.407113\pi$
$410$	0	0
$411$	6.22705e6	1.81835
$412$	− 2.23702e6i	− 0.649273i
$413$	− 624360.i	− 0.180119i
$414$	−3.43975e6	−0.986339
$415$	0	0
$416$	47104.0	0.0133452
$417$	802360.i	0.225959i
$418$	3.37920e6i	0.945961i
$419$	2.99166e6	0.832486	0.416243	−	0.909253i	$-0.363346\pi$
$419$	2.99166e6	0.832486	0.416243	+	0.909253i	$0.363346\pi$
$420$	0	0
$421$	3.96660e6	1.09072	0.545360	−	0.838202i	$-0.316393\pi$
$421$	3.96660e6	1.09072	0.545360	+	0.838202i	$0.316393\pi$
$422$	598208.i	0.163520i
$423$	1.07116e7i	2.91073i
$424$	−906624.	−0.244913
$425$	0	0
$426$	4.40669e6	1.17649
$427$	121484.i	0.0322440i
$428$	1.38269e6i	0.364850i
$429$	918528.	0.240962
$430$	0	0
$431$	−5.17115e6	−1.34089	−0.670446	−	0.741958i	$-0.733897\pi$
$431$	−5.17115e6	−1.34089	−0.670446	+	0.741958i	$0.733897\pi$
$432$	1.26464e6i	0.326030i
$433$	− 4.53485e6i	− 1.16237i	−0.813773	−	0.581183i	$-0.802590\pi$
$433$	− 4.53485e6i	− 1.16237i	0.813773	−	0.581183i	$-0.197410\pi$
$434$	350944.	0.0894362
$435$	0	0
$436$	3.49520e6	0.880554
$437$	2.18460e6i	0.547228i
$438$	− 5.42110e6i	− 1.35021i
$439$	1.08220e6	0.268007	0.134004	−	0.990981i	$-0.457217\pi$
$439$	1.08220e6	0.268007	0.134004	+	0.990981i	$0.457217\pi$
$440$	0	0
$441$	−7.06786e6	−1.73058
$442$	− 69552.0i	− 0.0169338i
$443$	− 1.08079e6i	− 0.261656i	−0.991405	−	0.130828i	$-0.958236\pi$
$443$	− 1.08079e6i	− 0.261656i	0.991405	−	0.130828i	$-0.0417635\pi$
$444$	−59072.0	−0.0142208
$445$	0	0
$446$	1.77402e6	0.422301
$447$	− 2.62470e6i	− 0.621314i
$448$	− 90112.0i	− 0.0212123i
$449$	−2.61783e6	−0.612810	−0.306405	−	0.951901i	$-0.599126\pi$
$449$	−2.61783e6	−0.612810	−0.306405	+	0.951901i	$0.599126\pi$
$450$	0	0
$451$	−1.18426e6	−0.274160
$452$	460896.i	0.106110i
$453$	− 323752.i	− 0.0741254i
$454$	1.68007e6	0.382550
$455$	0	0
$456$	1.83040e6	0.412225
$457$	− 1.59046e6i	− 0.356231i	−0.984010	−	0.178115i	$-0.943000\pi$
$457$	− 1.59046e6i	− 0.356231i	0.984010	−	0.178115i	$-0.0570001\pi$
$458$	− 4.23500e6i	− 0.943387i
$459$	1.86732e6	0.413701
$460$	0	0
$461$	4.25470e6	0.932431	0.466216	−	0.884671i	$-0.345617\pi$
$461$	4.25470e6	0.932431	0.466216	+	0.884671i	$0.345617\pi$
$462$	− 1.75718e6i	− 0.383012i
$463$	3.26605e6i	0.708061i	0.935234	+	0.354031i	$0.115189\pi$
$463$	3.26605e6i	0.708061i	−0.935234	+	0.354031i	$0.884811\pi$
$464$	1.43616e6	0.309676
$465$	0	0
$466$	5.09378e6	1.08662
$467$	601542.i	0.127636i	0.997962	+	0.0638181i	$0.0203277\pi$
$467$	601542.i	0.127636i	−0.997962	+	0.0638181i	$0.979672\pi$
$468$	− 318688.i	− 0.0672591i
$469$	−544324.	−0.114268
$470$	0	0
$471$	156572.	0.0325208
$472$	1.81632e6i	0.375264i
$473$	3.85997e6i	0.793288i
$474$	4.12256e6	0.842793
$475$	0	0
$476$	−133056.	−0.0269164
$477$	6.13388e6i	1.23435i
$478$	1.48272e6i	0.296817i
$479$	4.57932e6	0.911931	0.455966	−	0.889997i	$-0.349294\pi$
$479$	4.57932e6	0.911931	0.455966	+	0.889997i	$0.349294\pi$
$480$	0	0
$481$	6532.00	0.00128731
$482$	− 2.24519e6i	− 0.440186i
$483$	− 1.13599e6i	− 0.221568i
$484$	−6.86037e6	−1.33117
$485$	0	0
$486$	−2.38670e6	−0.458360
$487$	− 7.05226e6i	− 1.34743i	−0.738992	−	0.673714i	$-0.764698\pi$
$487$	− 7.05226e6i	− 1.34743i	0.738992	−	0.673714i	$-0.235302\pi$
$488$	− 353408.i	− 0.0671780i
$489$	−1.30225e7	−2.46276
$490$	0	0
$491$	−2.62349e6	−0.491106	−0.245553	−	0.969383i	$-0.578970\pi$
$491$	−2.62349e6	−0.491106	−0.245553	+	0.969383i	$0.578970\pi$
$492$	641472.i	0.119472i
$493$	− 2.12058e6i	− 0.392950i
$494$	−202400.	−0.0373158
$495$	0	0
$496$	−1.02093e6	−0.186333
$497$	932184.i	0.169282i
$498$	− 6.22190e6i	− 1.12422i
$499$	3.61234e6	0.649437	0.324719	−	0.945811i	$-0.394730\pi$
$499$	3.61234e6	0.649437	0.324719	+	0.945811i	$0.394730\pi$
$500$	0	0
$501$	−1.44367e7	−2.56965
$502$	2.30861e6i	0.408875i
$503$	9.15629e6i	1.61361i	0.590815	+	0.806807i	$0.298806\pi$
$503$	9.15629e6i	1.61361i	−0.590815	+	0.806807i	$0.701194\pi$
$504$	−609664.	−0.106909
$505$	0	0
$506$	−6.10099e6	−1.05931
$507$	− 9.59860e6i	− 1.65840i
$508$	− 3.46403e6i	− 0.595556i
$509$	−7.26159e6	−1.24233	−0.621165	−	0.783679i	$-0.713340\pi$
$509$	−7.26159e6	−1.24233	−0.621165	+	0.783679i	$0.713340\pi$
$510$	0	0
$511$	1.14677e6	0.194279
$512$	262144.i	0.0441942i
$513$	− 5.43400e6i	− 0.911646i
$514$	−2.60585e6	−0.435052
$515$	0	0
$516$	2.09082e6	0.345694
$517$	1.89988e7i	3.12608i
$518$	− 12496.0i	− 0.00204619i
$519$	1.08512e7	1.76831
$520$	0	0
$521$	5.81020e6	0.937771	0.468886	−	0.883259i	$-0.344656\pi$
$521$	5.81020e6	0.937771	0.468886	+	0.883259i	$0.344656\pi$
$522$	− 9.71652e6i	− 1.56075i
$523$	− 8.17067e6i	− 1.30618i	−0.757280	−	0.653090i	$-0.773472\pi$
$523$	− 8.17067e6i	− 1.30618i	0.757280	−	0.653090i	$-0.226528\pi$
$524$	3.91373e6	0.622676
$525$	0	0
$526$	−3.67030e6	−0.578411
$527$	1.50746e6i	0.236440i
$528$	5.11181e6i	0.797976i
$529$	2.49215e6	0.387199
$530$	0	0
$531$	1.22885e7	1.89132
$532$	387200.i	0.0593139i
$533$	− 70932.0i	− 0.0108149i
$534$	−5.99976e6	−0.910502
$535$	0	0
$536$	1.58349e6	0.238069
$537$	− 1.36188e6i	− 0.203800i
$538$	2.94156e6i	0.438149i
$539$	−1.25361e7	−1.85861
$540$	0	0
$541$	−817378.	−0.120069	−0.0600343	−	0.998196i	$-0.519121\pi$
$541$	−817378.	−0.120069	−0.0600343	+	0.998196i	$0.519121\pi$
$542$	− 4.48523e6i	− 0.655823i
$543$	− 1.42132e7i	− 2.06868i
$544$	387072.	0.0560783
$545$	0	0
$546$	105248.	0.0151089
$547$	3.50750e6i	0.501221i	0.968088	+	0.250611i	$0.0806314\pi$
$547$	3.50750e6i	0.501221i	−0.968088	+	0.250611i	$0.919369\pi$
$548$	− 3.83203e6i	− 0.545102i
$549$	−2.39103e6	−0.338574
$550$	0	0
$551$	−6.17100e6	−0.865918
$552$	3.30470e6i	0.461620i
$553$	872080.i	0.121267i
$554$	−6.64137e6	−0.919355
$555$	0	0
$556$	493760.	0.0677375
$557$	− 9.61490e6i	− 1.31313i	−0.754271	−	0.656563i	$-0.772009\pi$
$557$	− 9.61490e6i	− 1.31313i	0.754271	−	0.656563i	$-0.227991\pi$
$558$	6.90722e6i	0.939112i
$559$	−231196.	−0.0312933
$560$	0	0
$561$	7.54790e6	1.01256
$562$	5.80841e6i	0.775740i
$563$	2.01941e6i	0.268506i	0.990947	+	0.134253i	$0.0428635\pi$
$563$	2.01941e6i	0.268506i	−0.990947	+	0.134253i	$0.957136\pi$
$564$	1.02910e7	1.36226
$565$	0	0
$566$	−1.23606e6	−0.162180
$567$	510862.i	0.0667338i
$568$	− 2.71181e6i	− 0.352686i
$569$	−1.37859e6	−0.178507	−0.0892533	−	0.996009i	$-0.528448\pi$
$569$	−1.37859e6	−0.178507	−0.0892533	+	0.996009i	$0.528448\pi$
$570$	0	0
$571$	8.54295e6	1.09652	0.548261	−	0.836307i	$-0.315290\pi$
$571$	8.54295e6	1.09652	0.548261	+	0.836307i	$0.315290\pi$
$572$	− 565248.i	− 0.0722352i
$573$	1.17527e7i	1.49538i
$574$	−135696.	−0.0171905
$575$	0	0
$576$	1.77357e6	0.222737
$577$	2.31458e6i	0.289423i	0.989474	+	0.144711i	$0.0462254\pi$
$577$	2.31458e6i	0.289423i	−0.989474	+	0.144711i	$0.953775\pi$
$578$	5.10789e6i	0.635949i
$579$	1.26254e7	1.56513
$580$	0	0
$581$	1.31617e6	0.161760
$582$	1.50157e7i	1.83755i
$583$	1.08795e7i	1.32568i
$584$	−3.33606e6	−0.404764
$585$	0	0
$586$	6.37202e6	0.766537
$587$	− 928338.i	− 0.111202i	−0.998453	−	0.0556008i	$-0.982293\pi$
$587$	− 928338.i	− 0.111202i	0.998453	−	0.0556008i	$-0.0177074\pi$
$588$	6.79037e6i	0.809935i
$589$	4.38680e6	0.521026
$590$	0	0
$591$	−2.62646e7	−3.09316
$592$	36352.0i	0.00426309i
$593$	− 909486.i	− 0.106209i	−0.998589	−	0.0531043i	$-0.983088\pi$
$593$	− 909486.i	− 0.106209i	0.998589	−	0.0531043i	$-0.0169116\pi$
$594$	1.51757e7	1.76475
$595$	0	0
$596$	−1.61520e6	−0.186256
$597$	− 2.09986e7i	− 2.41132i
$598$	− 365424.i	− 0.0417873i
$599$	8.51136e6	0.969241	0.484621	−	0.874724i	$-0.338958\pi$
$599$	8.51136e6	0.969241	0.484621	+	0.874724i	$0.338958\pi$
$600$	0	0
$601$	6.12498e6	0.691701	0.345851	−	0.938290i	$-0.387590\pi$
$601$	6.12498e6	0.691701	0.345851	+	0.938290i	$0.387590\pi$
$602$	442288.i	0.0497409i
$603$	− 1.07133e7i	− 1.19986i
$604$	−199232.	−0.0222212
$605$	0	0
$606$	−1.46908e7	−1.62504
$607$	4.51646e6i	0.497538i	0.968563	+	0.248769i	$0.0800261\pi$
$607$	4.51646e6i	0.497538i	−0.968563	+	0.248769i	$0.919974\pi$
$608$	− 1.12640e6i	− 0.123576i
$609$	3.20892e6	0.350603
$610$	0	0
$611$	−1.13795e6	−0.123316
$612$	− 2.61878e6i	− 0.282632i
$613$	9.63979e6i	1.03614i	0.855340	+	0.518068i	$0.173349\pi$
$613$	9.63979e6i	1.03614i	−0.855340	+	0.518068i	$0.826651\pi$
$614$	4.98903e6	0.534067
$615$	0	0
$616$	−1.08134e6	−0.114819
$617$	9.92650e6i	1.04974i	0.851181	+	0.524872i	$0.175887\pi$
$617$	9.92650e6i	1.04974i	−0.851181	+	0.524872i	$0.824113\pi$
$618$	1.45407e7i	1.53149i
$619$	−7.63322e6	−0.800721	−0.400360	−	0.916358i	$-0.631115\pi$
$619$	−7.63322e6	−0.800721	−0.400360	+	0.916358i	$0.631115\pi$
$620$	0	0
$621$	9.81084e6	1.02089
$622$	− 2.66395e6i	− 0.276090i
$623$	− 1.26918e6i	− 0.131010i
$624$	−306176.	−0.0314782
$625$	0	0
$626$	2.36514e6	0.241225
$627$	− 2.19648e7i	− 2.23130i
$628$	− 96352.0i	− 0.00974903i
$629$	53676.0	0.00540946
$630$	0	0
$631$	1.80314e7	1.80284	0.901418	−	0.432949i	$-0.142527\pi$
$631$	1.80314e7	1.80284	0.901418	+	0.432949i	$0.142527\pi$
$632$	− 2.53696e6i	− 0.252651i
$633$	− 3.88835e6i	− 0.385706i
$634$	−2.06537e6	−0.204068
$635$	0	0
$636$	5.89306e6	0.577694
$637$	− 750858.i	− 0.0733178i
$638$	− 1.72339e7i	− 1.67623i
$639$	−1.83471e7	−1.77752
$640$	0	0
$641$	9.30190e6	0.894184	0.447092	−	0.894488i	$-0.352460\pi$
$641$	9.30190e6	0.894184	0.447092	+	0.894488i	$0.352460\pi$
$642$	− 8.98747e6i	− 0.860597i
$643$	− 1.38332e7i	− 1.31946i	−0.751503	−	0.659730i	$-0.770671\pi$
$643$	− 1.38332e7i	− 1.31946i	0.751503	−	0.659730i	$-0.229329\pi$
$644$	−699072.	−0.0664213
$645$	0	0
$646$	−1.66320e6	−0.156806
$647$	1.48997e7i	1.39932i	0.714478	+	0.699658i	$0.246664\pi$
$647$	1.48997e7i	1.39932i	−0.714478	+	0.699658i	$0.753336\pi$
$648$	− 1.48614e6i	− 0.139035i
$649$	2.17958e7	2.03124
$650$	0	0
$651$	−2.28114e6	−0.210959
$652$	8.01386e6i	0.738282i
$653$	− 1.93306e7i	− 1.77403i	−0.461738	−	0.887016i	$-0.652774\pi$
$653$	− 1.93306e7i	− 1.77403i	0.461738	−	0.887016i	$-0.347226\pi$
$654$	−2.27188e7	−2.07702
$655$	0	0
$656$	394752.	0.0358150
$657$	2.25706e7i	2.03999i
$658$	2.17694e6i	0.196012i
$659$	4.06110e6	0.364276	0.182138	−	0.983273i	$-0.441698\pi$
$659$	4.06110e6	0.364276	0.182138	+	0.983273i	$0.441698\pi$
$660$	0	0
$661$	−1.35152e7	−1.20315	−0.601575	−	0.798816i	$-0.705460\pi$
$661$	−1.35152e7	−1.20315	−0.601575	+	0.798816i	$0.705460\pi$
$662$	− 1.31831e7i	− 1.16915i
$663$	452088.i	0.0399429i
$664$	−3.82886e6	−0.337015
$665$	0	0
$666$	245944.	0.0214858
$667$	− 1.11415e7i	− 0.969678i
$668$	8.88413e6i	0.770324i
$669$	−1.15312e7	−0.996111
$670$	0	0
$671$	−4.24090e6	−0.363623
$672$	585728.i	0.0500349i
$673$	1.43520e7i	1.22144i	0.791845	+	0.610722i	$0.209121\pi$
$673$	1.43520e7i	1.22144i	−0.791845	+	0.610722i	$0.790879\pi$
$674$	7.64391e6	0.648136
$675$	0	0
$676$	−5.90683e6	−0.497150
$677$	− 1.89530e6i	− 0.158930i	−0.996838	−	0.0794650i	$-0.974679\pi$
$677$	− 1.89530e6i	− 0.158930i	0.996838	−	0.0794650i	$-0.0253212\pi$
$678$	− 2.99582e6i	− 0.250289i
$679$	−3.17640e6	−0.264400
$680$	0	0
$681$	−1.09205e7	−0.902347
$682$	1.22511e7i	1.00859i
$683$	2.91641e6i	0.239220i	0.992821	+	0.119610i	$0.0381644\pi$
$683$	2.91641e6i	0.239220i	−0.992821	+	0.119610i	$0.961836\pi$
$684$	−7.62080e6	−0.622817
$685$	0	0
$686$	−2.91544e6	−0.236534
$687$	2.75275e7i	2.22523i
$688$	− 1.28666e6i	− 0.103631i
$689$	−651636.	−0.0522946
$690$	0	0
$691$	1.44278e7	1.14949	0.574743	−	0.818334i	$-0.305102\pi$
$691$	1.44278e7	1.14949	0.574743	+	0.818334i	$0.305102\pi$
$692$	− 6.67766e6i	− 0.530102i
$693$	7.31597e6i	0.578680i
$694$	9.68023e6	0.762934
$695$	0	0
$696$	−9.33504e6	−0.730454
$697$	− 582876.i	− 0.0454458i
$698$	− 1.00291e7i	− 0.779153i
$699$	−3.31096e7	−2.56307
$700$	0	0
$701$	−1.58679e7	−1.21962	−0.609811	−	0.792547i	$-0.708754\pi$
$701$	−1.58679e7	−1.21962	−0.609811	+	0.792547i	$0.708754\pi$
$702$	908960.i	0.0696149i
$703$	− 156200.i	− 0.0119205i
$704$	3.14573e6	0.239216
$705$	0	0
$706$	1.65266e6	0.124788
$707$	− 3.10768e6i	− 0.233823i
$708$	− 1.18061e7i	− 0.885162i
$709$	301810.	0.0225485	0.0112743	−	0.999936i	$-0.496411\pi$
$709$	301810.	0.0225485	0.0112743	+	0.999936i	$0.496411\pi$
$710$	0	0
$711$	−1.71641e7	−1.27335
$712$	3.69216e6i	0.272948i
$713$	7.92017e6i	0.583459i
$714$	864864.	0.0634896
$715$	0	0
$716$	−838080.	−0.0610946
$717$	− 9.63768e6i	− 0.700123i
$718$	− 6.95088e6i	− 0.503186i
$719$	−2.12677e7	−1.53426	−0.767130	−	0.641492i	$-0.778316\pi$
$719$	−2.12677e7	−1.53426	−0.767130	+	0.641492i	$0.778316\pi$
$720$	0	0
$721$	−3.07591e6	−0.220361
$722$	− 5.06440e6i	− 0.361564i
$723$	1.45937e7i	1.03830i
$724$	−8.74659e6	−0.620144
$725$	0	0
$726$	4.45924e7	3.13992
$727$	− 1.55009e7i	− 1.08773i	−0.839174	−	0.543863i	$-0.816961\pi$
$727$	− 1.55009e7i	− 1.08773i	0.839174	−	0.543863i	$-0.183039\pi$
$728$	− 64768.0i	− 0.00452931i
$729$	2.11562e7	1.47441
$730$	0	0
$731$	−1.89983e6	−0.131499
$732$	2.29715e6i	0.158457i
$733$	− 1.21850e7i	− 0.837653i	−0.908066	−	0.418827i	$-0.862441\pi$
$733$	− 1.21850e7i	− 0.837653i	0.908066	−	0.418827i	$-0.137559\pi$
$734$	4.64391e6	0.318159
$735$	0	0
$736$	2.03366e6	0.138384
$737$	− 1.90019e7i	− 1.28863i
$738$	− 2.67074e6i	− 0.180506i
$739$	2.90282e7	1.95528	0.977641	−	0.210282i	$-0.0674382\pi$
$739$	2.90282e7	1.95528	0.977641	+	0.210282i	$0.0674382\pi$
$740$	0	0
$741$	1.31560e6	0.0880194
$742$	1.24661e6i	0.0831228i
$743$	1.61145e7i	1.07089i	0.844570	+	0.535445i	$0.179856\pi$
$743$	1.61145e7i	1.07089i	−0.844570	+	0.535445i	$0.820144\pi$
$744$	6.63603e6	0.439517
$745$	0	0
$746$	−1.37502e6	−0.0904609
$747$	2.59047e7i	1.69854i
$748$	− 4.64486e6i	− 0.303542i
$749$	1.90120e6	0.123829
$750$	0	0
$751$	−2.92431e6	−0.189201	−0.0946005	−	0.995515i	$-0.530157\pi$
$751$	−2.92431e6	−0.189201	−0.0946005	+	0.995515i	$0.530157\pi$
$752$	− 6.33293e6i	− 0.408376i
$753$	− 1.50060e7i	− 0.964442i
$754$	1.03224e6	0.0661230
$755$	0	0
$756$	1.73888e6	0.110653
$757$	− 2.60325e7i	− 1.65111i	−0.564319	−	0.825557i	$-0.690861\pi$
$757$	− 2.60325e7i	− 1.65111i	0.564319	−	0.825557i	$-0.309139\pi$
$758$	− 2.29256e6i	− 0.144926i
$759$	3.96564e7	2.49867
$760$	0	0
$761$	1.63263e7	1.02194	0.510970	−	0.859598i	$-0.329286\pi$
$761$	1.63263e7	1.02194	0.510970	+	0.859598i	$0.329286\pi$
$762$	2.25162e7i	1.40478i
$763$	− 4.80590e6i	− 0.298857i
$764$	7.23245e6	0.448283
$765$	0	0
$766$	1.15222e7	0.709517
$767$	1.30548e6i	0.0801275i
$768$	− 1.70394e6i	− 0.104244i
$769$	−2.58132e7	−1.57408	−0.787040	−	0.616902i	$-0.788388\pi$
$769$	−2.58132e7	−1.57408	−0.787040	+	0.616902i	$0.788388\pi$
$770$	0	0
$771$	1.69380e7	1.02619
$772$	− 7.76950e6i	− 0.469191i
$773$	− 1.90592e7i	− 1.14725i	−0.819119	−	0.573624i	$-0.805537\pi$
$773$	− 1.90592e7i	− 1.14725i	0.819119	−	0.573624i	$-0.194463\pi$
$774$	−8.70503e6	−0.522298
$775$	0	0
$776$	9.24045e6	0.550857
$777$	81224.0i	0.00482649i
$778$	1.23424e7i	0.731054i
$779$	−1.69620e6	−0.100146
$780$	0	0
$781$	−3.25417e7	−1.90903
$782$	− 3.00283e6i	− 0.175596i
$783$	2.77134e7i	1.61542i
$784$	4.17869e6	0.242801
$785$	0	0
$786$	−2.54392e7	−1.46875
$787$	1.73411e7i	0.998021i	0.866596	+	0.499011i	$0.166303\pi$
$787$	1.73411e7i	0.998021i	−0.866596	+	0.499011i	$0.833697\pi$
$788$	1.61628e7i	0.927262i
$789$	2.38569e7	1.36434
$790$	0	0
$791$	633732.	0.0360134
$792$	− 2.12828e7i	− 1.20564i
$793$	− 254012.i	− 0.0143440i
$794$	3.54183e6	0.199378
$795$	0	0
$796$	−1.29222e7	−0.722862
$797$	2.58169e7i	1.43965i	0.694153	+	0.719827i	$0.255779\pi$
$797$	2.58169e7i	1.43965i	−0.694153	+	0.719827i	$0.744221\pi$
$798$	− 2.51680e6i	− 0.139908i
$799$	−9.35096e6	−0.518190
$800$	0	0
$801$	2.49798e7	1.37565
$802$	− 1.50138e7i	− 0.824239i
$803$	4.00328e7i	2.19092i
$804$	−1.02927e7	−0.561549
$805$	0	0
$806$	−733792.	−0.0397865
$807$	− 1.91201e7i	− 1.03349i
$808$	9.04051e6i	0.487152i
$809$	−8.88489e6	−0.477288	−0.238644	−	0.971107i	$-0.576703\pi$
$809$	−8.88489e6	−0.477288	−0.238644	+	0.971107i	$0.576703\pi$
$810$	0	0
$811$	−2.46396e7	−1.31547	−0.657735	−	0.753249i	$-0.728485\pi$
$811$	−2.46396e7	−1.31547	−0.657735	+	0.753249i	$0.728485\pi$
$812$	− 1.97472e6i	− 0.105103i
$813$	2.91540e7i	1.54693i
$814$	436224.	0.0230754
$815$	0	0
$816$	−2.51597e6	−0.132276
$817$	5.52860e6i	0.289774i
$818$	7.78612e6i	0.406853i
$819$	−438196.	−0.0228275
$820$	0	0
$821$	1.13768e7	0.589062	0.294531	−	0.955642i	$-0.404837\pi$
$821$	1.13768e7	0.589062	0.294531	+	0.955642i	$0.404837\pi$
$822$	2.49082e7i	1.28577i
$823$	− 1.30783e7i	− 0.673057i	−0.941673	−	0.336529i	$-0.890747\pi$
$823$	− 1.30783e7i	− 0.673057i	0.941673	−	0.336529i	$-0.109253\pi$
$824$	8.94810e6	0.459106
$825$	0	0
$826$	2.49744e6	0.127363
$827$	3.57188e7i	1.81607i	0.418891	+	0.908037i	$0.362419\pi$
$827$	3.57188e7i	1.81607i	−0.418891	+	0.908037i	$0.637581\pi$
$828$	− 1.37590e7i	− 0.697447i
$829$	−1.61880e7	−0.818103	−0.409052	−	0.912511i	$-0.634140\pi$
$829$	−1.61880e7	−0.818103	−0.409052	+	0.912511i	$0.634140\pi$
$830$	0	0
$831$	4.31689e7	2.16854
$832$	188416.i	0.00943647i
$833$	− 6.17009e6i	− 0.308091i
$834$	−3.20944e6	−0.159777
$835$	0	0
$836$	−1.35168e7	−0.668895
$837$	− 1.97007e7i	− 0.972005i
$838$	1.19666e7i	0.588657i
$839$	2.55497e7	1.25309	0.626543	−	0.779387i	$-0.284469\pi$
$839$	2.55497e7	1.25309	0.626543	+	0.779387i	$0.284469\pi$
$840$	0	0
$841$	1.09610e7	0.534390
$842$	1.58664e7i	0.771256i
$843$	− 3.77547e7i	− 1.82979i
$844$	−2.39283e6	−0.115626
$845$	0	0
$846$	−4.28462e7	−2.05820
$847$	9.43301e6i	0.451795i
$848$	− 3.62650e6i	− 0.173180i
$849$	8.03436e6	0.382545
$850$	0	0
$851$	282012.	0.0133488
$852$	1.76268e7i	0.831904i
$853$	− 2.22953e7i	− 1.04916i	−0.851362	−	0.524579i	$-0.824223\pi$
$853$	− 2.22953e7i	− 1.04916i	0.851362	−	0.524579i	$-0.175777\pi$
$854$	−485936.	−0.0228000
$855$	0	0
$856$	−5.53075e6	−0.257988
$857$	− 1.96872e7i	− 0.915656i	−0.889041	−	0.457828i	$-0.848628\pi$
$857$	− 1.96872e7i	− 0.915656i	0.889041	−	0.457828i	$-0.151372\pi$
$858$	3.67411e6i	0.170386i
$859$	−6.77582e6	−0.313313	−0.156657	−	0.987653i	$-0.550072\pi$
$859$	−6.77582e6	−0.313313	−0.156657	+	0.987653i	$0.550072\pi$
$860$	0	0
$861$	882024.	0.0405483
$862$	− 2.06846e7i	− 0.948154i
$863$	− 2.63804e7i	− 1.20574i	−0.797839	−	0.602871i	$-0.794023\pi$
$863$	− 2.63804e7i	− 1.20574i	0.797839	−	0.602871i	$-0.205977\pi$
$864$	−5.05856e6	−0.230538
$865$	0	0
$866$	1.81394e7	0.821917
$867$	− 3.32013e7i	− 1.50006i
$868$	1.40378e6i	0.0632410i
$869$	−3.04435e7	−1.36756
$870$	0	0
$871$	1.13813e6	0.0508332
$872$	1.39808e7i	0.622645i
$873$	− 6.25174e7i	− 2.77629i
$874$	−8.73840e6	−0.386949
$875$	0	0
$876$	2.16844e7	0.954745
$877$	− 2.95161e7i	− 1.29587i	−0.761697	−	0.647934i	$-0.775633\pi$
$877$	− 2.95161e7i	− 1.29587i	0.761697	−	0.647934i	$-0.224367\pi$
$878$	4.32880e6i	0.189510i
$879$	−4.14182e7	−1.80808
$880$	0	0
$881$	−1.48565e7	−0.644877	−0.322438	−	0.946590i	$-0.604502\pi$
$881$	−1.48565e7	−0.644877	−0.322438	+	0.946590i	$0.604502\pi$
$882$	− 2.82714e7i	− 1.22370i
$883$	− 1.45340e7i	− 0.627313i	−0.949537	−	0.313656i	$-0.898446\pi$
$883$	− 1.45340e7i	− 0.627313i	0.949537	−	0.313656i	$-0.101554\pi$
$884$	278208.	0.0119740
$885$	0	0
$886$	4.32314e6	0.185019
$887$	1.72028e7i	0.734160i	0.930189	+	0.367080i	$0.119642\pi$
$887$	1.72028e7i	0.734160i	−0.930189	+	0.367080i	$0.880358\pi$
$888$	− 236288.i	− 0.0100556i
$889$	−4.76304e6	−0.202130
$890$	0	0
$891$	−1.78337e7	−0.752572
$892$	7.09610e6i	0.298612i
$893$	2.72118e7i	1.14190i
$894$	1.04988e7	0.439335
$895$	0	0
$896$	360448.	0.0149994
$897$	2.37526e6i	0.0985665i
$898$	− 1.04713e7i	− 0.433322i
$899$	−2.23727e7	−0.923249
$900$	0	0
$901$	−5.35475e6	−0.219749
$902$	− 4.73702e6i	− 0.193860i
$903$	− 2.87487e6i	− 0.117327i
$904$	−1.84358e6	−0.0750312
$905$	0	0
$906$	1.29501e6	0.0524146
$907$	3.44434e7i	1.39023i	0.718897	+	0.695116i	$0.244647\pi$
$907$	3.44434e7i	1.39023i	−0.718897	+	0.695116i	$0.755353\pi$
$908$	6.72029e6i	0.270504i
$909$	6.11647e7	2.45522
$910$	0	0
$911$	−983748.	−0.0392724	−0.0196362	−	0.999807i	$-0.506251\pi$
$911$	−983748.	−0.0392724	−0.0196362	+	0.999807i	$0.506251\pi$
$912$	7.32160e6i	0.291487i
$913$	4.59464e7i	1.82421i
$914$	6.36183e6	0.251893
$915$	0	0
$916$	1.69400e7	0.667075
$917$	− 5.38138e6i	− 0.211334i
$918$	7.46928e6i	0.292531i
$919$	−3.08857e7	−1.20634	−0.603168	−	0.797614i	$-0.706095\pi$
$919$	−3.08857e7	−1.20634	−0.603168	+	0.797614i	$0.706095\pi$
$920$	0	0
$921$	−3.24287e7	−1.25974
$922$	1.70188e7i	0.659328i
$923$	− 1.94911e6i	− 0.0753065i
$924$	7.02874e6	0.270830
$925$	0	0
$926$	−1.30642e7	−0.500675
$927$	− 6.05395e7i	− 2.31387i
$928$	5.74464e6i	0.218974i
$929$	3.20874e7	1.21982	0.609909	−	0.792472i	$-0.291206\pi$
$929$	3.20874e7	1.21982	0.609909	+	0.792472i	$0.291206\pi$
$930$	0	0
$931$	−1.79553e7	−0.678920
$932$	2.03751e7i	0.768353i
$933$	1.73157e7i	0.651232i
$934$	−2.40617e6	−0.0902524
$935$	0	0
$936$	1.27475e6	0.0475594
$937$	− 1.52520e7i	− 0.567515i	−0.958896	−	0.283757i	$-0.908419\pi$
$937$	− 1.52520e7i	− 0.567515i	0.958896	−	0.283757i	$-0.0915810\pi$
$938$	− 2.17730e6i	− 0.0807998i
$939$	−1.53734e7	−0.568993
$940$	0	0
$941$	3.48166e6	0.128178	0.0640889	−	0.997944i	$-0.479586\pi$
$941$	3.48166e6	0.128178	0.0640889	+	0.997944i	$0.479586\pi$
$942$	626288.i	0.0229957i
$943$	− 3.06241e6i	− 0.112146i
$944$	−7.26528e6	−0.265352
$945$	0	0
$946$	−1.54399e7	−0.560939
$947$	2.54010e7i	0.920398i	0.887816	+	0.460199i	$0.152222\pi$
$947$	2.54010e7i	0.920398i	−0.887816	+	0.460199i	$0.847778\pi$
$948$	1.64902e7i	0.595945i
$949$	−2.39780e6	−0.0864265
$950$	0	0
$951$	1.34249e7	0.481348
$952$	− 532224.i	− 0.0190328i
$953$	− 4.97352e7i	− 1.77391i	−0.461856	−	0.886955i	$-0.652816\pi$
$953$	− 4.97352e7i	− 1.77391i	0.461856	−	0.886955i	$-0.347184\pi$
$954$	−2.45355e7	−0.872819
$955$	0	0
$956$	−5.93088e6	−0.209882
$957$	1.12020e8i	3.95383i
$958$	1.83173e7i	0.644833i
$959$	−5.26904e6	−0.185006
$960$	0	0
$961$	−1.27250e7	−0.444477
$962$	26128.0i	0 0.000910266i
$963$	3.74190e7i	1.30025i
$964$	8.98077e6	0.311258
$965$	0	0
$966$	4.54397e6	0.156672
$967$	− 3.05173e7i	− 1.04949i	−0.851258	−	0.524747i	$-0.824160\pi$
$967$	− 3.05173e7i	− 1.04949i	0.851258	−	0.524747i	$-0.175840\pi$
$968$	− 2.74415e7i	− 0.941280i
$969$	1.08108e7	0.369869
$970$	0	0
$971$	3.19854e7	1.08869	0.544344	−	0.838862i	$-0.316779\pi$
$971$	3.19854e7	1.08869	0.544344	+	0.838862i	$0.316779\pi$
$972$	− 9.54678e6i	− 0.324109i
$973$	− 678920.i	− 0.0229899i
$974$	2.82090e7	0.952776
$975$	0	0
$976$	1.41363e6	0.0475020
$977$	− 2.90786e6i	− 0.0974623i	−0.998812	−	0.0487312i	$-0.984482\pi$
$977$	− 2.90786e6i	− 0.0974623i	0.998812	−	0.0487312i	$-0.0155178\pi$
$978$	− 5.20901e7i	− 1.74144i
$979$	4.43059e7	1.47742
$980$	0	0
$981$	9.45888e7	3.13810
$982$	− 1.04940e7i	− 0.347264i
$983$	3.49621e7i	1.15402i	0.816737	+	0.577010i	$0.195781\pi$
$983$	3.49621e7i	1.15402i	−0.816737	+	0.577010i	$0.804219\pi$
$984$	−2.56589e6	−0.0844792
$985$	0	0
$986$	8.48232e6	0.277858
$987$	− 1.41501e7i	− 0.462347i
$988$	− 809600.i	− 0.0263863i
$989$	−9.98164e6	−0.324497
$990$	0	0
$991$	3.00465e6	0.0971874	0.0485937	−	0.998819i	$-0.484526\pi$
$991$	3.00465e6	0.0971874	0.0485937	+	0.998819i	$0.484526\pi$
$992$	− 4.08371e6i	− 0.131758i
$993$	8.56900e7i	2.75776i
$994$	−3.72874e6	−0.119700
$995$	0	0
$996$	2.48876e7	0.794941
$997$	− 3.20789e7i	− 1.02207i	−0.859560	−	0.511035i	$-0.829262\pi$
$997$	− 3.20789e7i	− 1.02207i	0.859560	−	0.511035i	$-0.170738\pi$
$998$	1.44494e7i	0.459222i
$999$	−701480.	−0.0222383

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	50.6.b.d.49.2		2
3.2	odd	2		450.6.c.o.199.1		2
4.3	odd	2		400.6.c.a.49.2		2
5.2	odd	4		10.6.a.a.1.1	✓	1
5.3	odd	4		50.6.a.g.1.1		1
5.4	even	2	inner	50.6.b.d.49.1		2
15.2	even	4		90.6.a.f.1.1		1
15.8	even	4		450.6.a.h.1.1		1
15.14	odd	2		450.6.c.o.199.2		2
20.3	even	4		400.6.a.a.1.1		1
20.7	even	4		80.6.a.h.1.1		1
20.19	odd	2		400.6.c.a.49.1		2
35.27	even	4		490.6.a.j.1.1		1
40.27	even	4		320.6.a.a.1.1		1
40.37	odd	4		320.6.a.p.1.1		1
60.47	odd	4		720.6.a.r.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.6.a.a.1.1	✓	1	5.2	odd	4
50.6.a.g.1.1		1	5.3	odd	4
50.6.b.d.49.1		2	5.4	even	2	inner
50.6.b.d.49.2		2	1.1	even	1	trivial
80.6.a.h.1.1		1	20.7	even	4
90.6.a.f.1.1		1	15.2	even	4
320.6.a.a.1.1		1	40.27	even	4
320.6.a.p.1.1		1	40.37	odd	4
400.6.a.a.1.1		1	20.3	even	4
400.6.c.a.49.1		2	20.19	odd	2
400.6.c.a.49.2		2	4.3	odd	2
450.6.a.h.1.1		1	15.8	even	4
450.6.c.o.199.1		2	3.2	odd	2
450.6.c.o.199.2		2	15.14	odd	2
490.6.a.j.1.1		1	35.27	even	4
720.6.a.r.1.1		1	60.47	odd	4

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:	\( N \)	\(=\)	\( 50 = 2 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	50.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.00000i q^{2} -26.0000i q^{3} -16.0000 q^{4} +104.000 q^{6} +22.0000i q^{7} -64.0000i q^{8} -433.000 q^{9} +O(q^{10})\)	\(q+4.00000i q^{2} -26.0000i q^{3} -16.0000 q^{4} +104.000 q^{6} +22.0000i q^{7} -64.0000i q^{8} -433.000 q^{9} -768.000 q^{11} +416.000i q^{12} -46.0000i q^{13} -88.0000 q^{14} +256.000 q^{16} -378.000i q^{17} -1732.00i q^{18} -1100.00 q^{19} +572.000 q^{21} -3072.00i q^{22} -1986.00i q^{23} -1664.00 q^{24} +184.000 q^{26} +4940.00i q^{27} -352.000i q^{28} +5610.00 q^{29} -3988.00 q^{31} +1024.00i q^{32} +19968.0i q^{33} +1512.00 q^{34} +6928.00 q^{36} +142.000i q^{37} -4400.00i q^{38} -1196.00 q^{39} +1542.00 q^{41} +2288.00i q^{42} -5026.00i q^{43} +12288.0 q^{44} +7944.00 q^{46} -24738.0i q^{47} -6656.00i q^{48} +16323.0 q^{49} -9828.00 q^{51} +736.000i q^{52} -14166.0i q^{53} -19760.0 q^{54} +1408.00 q^{56} +28600.0i q^{57} +22440.0i q^{58} -28380.0 q^{59} +5522.00 q^{61} -15952.0i q^{62} -9526.00i q^{63} -4096.00 q^{64} -79872.0 q^{66} +24742.0i q^{67} +6048.00i q^{68} -51636.0 q^{69} +42372.0 q^{71} +27712.0i q^{72} -52126.0i q^{73} -568.000 q^{74} +17600.0 q^{76} -16896.0i q^{77} -4784.00i q^{78} +39640.0 q^{79} +23221.0 q^{81} +6168.00i q^{82} -59826.0i q^{83} -9152.00 q^{84} +20104.0 q^{86} -145860. i q^{87} +49152.0i q^{88} -57690.0 q^{89} +1012.00 q^{91} +31776.0i q^{92} +103688. i q^{93} +98952.0 q^{94} +26624.0 q^{96} +144382. i q^{97} +65292.0i q^{98} +332544. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 32 q^{4} + 208 q^{6} - 866 q^{9}+O(q^{10}) \) 2 * q - 32 * q^4 + 208 * q^6 - 866 * q^9	\( 2 q - 32 q^{4} + 208 q^{6} - 866 q^{9} - 1536 q^{11} - 176 q^{14} + 512 q^{16} - 2200 q^{19} + 1144 q^{21} - 3328 q^{24} + 368 q^{26} + 11220 q^{29} - 7976 q^{31} + 3024 q^{34} + 13856 q^{36} - 2392 q^{39} + 3084 q^{41} + 24576 q^{44} + 15888 q^{46} + 32646 q^{49} - 19656 q^{51} - 39520 q^{54} + 2816 q^{56} - 56760 q^{59} + 11044 q^{61} - 8192 q^{64} - 159744 q^{66} - 103272 q^{69} + 84744 q^{71} - 1136 q^{74} + 35200 q^{76} + 79280 q^{79} + 46442 q^{81} - 18304 q^{84} + 40208 q^{86} - 115380 q^{89} + 2024 q^{91} + 197904 q^{94} + 53248 q^{96} + 665088 q^{99}+O(q^{100}) \) 2 * q - 32 * q^4 + 208 * q^6 - 866 * q^9 - 1536 * q^11 - 176 * q^14 + 512 * q^16 - 2200 * q^19 + 1144 * q^21 - 3328 * q^24 + 368 * q^26 + 11220 * q^29 - 7976 * q^31 + 3024 * q^34 + 13856 * q^36 - 2392 * q^39 + 3084 * q^41 + 24576 * q^44 + 15888 * q^46 + 32646 * q^49 - 19656 * q^51 - 39520 * q^54 + 2816 * q^56 - 56760 * q^59 + 11044 * q^61 - 8192 * q^64 - 159744 * q^66 - 103272 * q^69 + 84744 * q^71 - 1136 * q^74 + 35200 * q^76 + 79280 * q^79 + 46442 * q^81 - 18304 * q^84 + 40208 * q^86 - 115380 * q^89 + 2024 * q^91 + 197904 * q^94 + 53248 * q^96 + 665088 * q^99

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