LMFDB - Embedded newform 550.6.b.b.199.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [550,6,Mod(199,550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(550, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("550.199");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 550 = 2 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	550.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:	$88.2111008971$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 110)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	550.199
Dual form			550.6.b.b.199.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.00000i q^{2} -12.0000i q^{3} -16.0000 q^{4} -48.0000 q^{6} +54.0000i q^{7} +64.0000i q^{8} +99.0000 q^{9} +O(q^{10})$	\(q-4.00000i q^{2} -12.0000i q^{3} -16.0000 q^{4} -48.0000 q^{6} +54.0000i q^{7} +64.0000i q^{8} +99.0000 q^{9} -121.000 q^{11} +192.000i q^{12} +540.000i q^{13} +216.000 q^{14} +256.000 q^{16} +340.000i q^{17} -396.000i q^{18} +952.000 q^{19} +648.000 q^{21} +484.000i q^{22} -1092.00i q^{23} +768.000 q^{24} +2160.00 q^{26} -4104.00i q^{27} -864.000i q^{28} +62.0000 q^{29} -7560.00 q^{31} -1024.00i q^{32} +1452.00i q^{33} +1360.00 q^{34} -1584.00 q^{36} -9186.00i q^{37} -3808.00i q^{38} +6480.00 q^{39} -6818.00 q^{41} -2592.00i q^{42} +13310.0i q^{43} +1936.00 q^{44} -4368.00 q^{46} -22420.0i q^{47} -3072.00i q^{48} +13891.0 q^{49} +4080.00 q^{51} -8640.00i q^{52} -19654.0i q^{53} -16416.0 q^{54} -3456.00 q^{56} -11424.0i q^{57} -248.000i q^{58} -48292.0 q^{59} +17530.0 q^{61} +30240.0i q^{62} +5346.00i q^{63} -4096.00 q^{64} +5808.00 q^{66} -35344.0i q^{67} -5440.00i q^{68} -13104.0 q^{69} -22912.0 q^{71} +6336.00i q^{72} -47852.0i q^{73} -36744.0 q^{74} -15232.0 q^{76} -6534.00i q^{77} -25920.0i q^{78} -52396.0 q^{79} -25191.0 q^{81} +27272.0i q^{82} -7890.00i q^{83} -10368.0 q^{84} +53240.0 q^{86} -744.000i q^{87} -7744.00i q^{88} -41958.0 q^{89} -29160.0 q^{91} +17472.0i q^{92} +90720.0i q^{93} -89680.0 q^{94} -12288.0 q^{96} -37602.0i q^{97} -55564.0i q^{98} -11979.0 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 32 q^{4} - 96 q^{6} + 198 q^{9}+O(q^{10}) $ 2 * q - 32 * q^4 - 96 * q^6 + 198 * q^9	$ 2 q - 32 q^{4} - 96 q^{6} + 198 q^{9} - 242 q^{11} + 432 q^{14} + 512 q^{16} + 1904 q^{19} + 1296 q^{21} + 1536 q^{24} + 4320 q^{26} + 124 q^{29} - 15120 q^{31} + 2720 q^{34} - 3168 q^{36} + 12960 q^{39} - 13636 q^{41} + 3872 q^{44} - 8736 q^{46} + 27782 q^{49} + 8160 q^{51} - 32832 q^{54} - 6912 q^{56} - 96584 q^{59} + 35060 q^{61} - 8192 q^{64} + 11616 q^{66} - 26208 q^{69} - 45824 q^{71} - 73488 q^{74} - 30464 q^{76} - 104792 q^{79} - 50382 q^{81} - 20736 q^{84} + 106480 q^{86} - 83916 q^{89} - 58320 q^{91} - 179360 q^{94} - 24576 q^{96} - 23958 q^{99}+O(q^{100}) $ 2 * q - 32 * q^4 - 96 * q^6 + 198 * q^9 - 242 * q^11 + 432 * q^14 + 512 * q^16 + 1904 * q^19 + 1296 * q^21 + 1536 * q^24 + 4320 * q^26 + 124 * q^29 - 15120 * q^31 + 2720 * q^34 - 3168 * q^36 + 12960 * q^39 - 13636 * q^41 + 3872 * q^44 - 8736 * q^46 + 27782 * q^49 + 8160 * q^51 - 32832 * q^54 - 6912 * q^56 - 96584 * q^59 + 35060 * q^61 - 8192 * q^64 + 11616 * q^66 - 26208 * q^69 - 45824 * q^71 - 73488 * q^74 - 30464 * q^76 - 104792 * q^79 - 50382 * q^81 - 20736 * q^84 + 106480 * q^86 - 83916 * q^89 - 58320 * q^91 - 179360 * q^94 - 24576 * q^96 - 23958 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/550\mathbb{Z}\right)^\times$.

$n$	$101$	$177$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 4.00000i	− 0.707107i
$2$	− 4.00000i	− 0.707107i
$3$	− 12.0000i	− 0.769800i	−0.922958	−	0.384900i	$-0.874236\pi$
$3$	− 12.0000i	− 0.769800i	0.922958	−	0.384900i	$-0.125764\pi$
$4$	−16.0000	−0.500000
$5$	0	0
$5$	0	0
$6$	−48.0000	−0.544331
$7$	54.0000i	0.416532i	0.978072	+	0.208266i	$0.0667820\pi$
$7$	54.0000i	0.416532i	−0.978072	+	0.208266i	$0.933218\pi$
$8$	64.0000i	0.353553i
$9$	99.0000	0.407407
$10$	0	0
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	192.000i	0.384900i
$13$	540.000i	0.886207i	0.896470	+	0.443104i	$0.146123\pi$
$13$	540.000i	0.886207i	−0.896470	+	0.443104i	$0.853877\pi$
$14$	216.000	0.294533
$15$	0	0
$16$	256.000	0.250000
$17$	340.000i	0.285336i	0.989771	+	0.142668i	$0.0455681\pi$
$17$	340.000i	0.285336i	−0.989771	+	0.142668i	$0.954432\pi$
$18$	− 396.000i	− 0.288081i
$19$	952.000	0.604997	0.302498	−	0.953150i	$-0.402179\pi$
$19$	952.000	0.604997	0.302498	+	0.953150i	$0.402179\pi$
$20$	0	0
$21$	648.000	0.320647
$22$	484.000i	0.213201i
$23$	− 1092.00i	− 0.430431i	−0.976567	−	0.215215i	$-0.930955\pi$
$23$	− 1092.00i	− 0.430431i	0.976567	−	0.215215i	$-0.0690453\pi$
$24$	768.000	0.272166
$25$	0	0
$26$	2160.00	0.626643
$27$	− 4104.00i	− 1.08342i
$28$	− 864.000i	− 0.208266i
$29$	62.0000	0.0136898	0.00684489	−	0.999977i	$-0.497821\pi$
$29$	62.0000	0.0136898	0.00684489	+	0.999977i	$0.497821\pi$
$30$	0	0
$31$	−7560.00	−1.41292	−0.706460	−	0.707753i	$-0.749709\pi$
$31$	−7560.00	−1.41292	−0.706460	+	0.707753i	$0.749709\pi$
$32$	− 1024.00i	− 0.176777i
$33$	1452.00i	0.232104i
$34$	1360.00	0.201763
$35$	0	0
$36$	−1584.00	−0.203704
$37$	− 9186.00i	− 1.10312i	−0.834136	−	0.551559i	$-0.814033\pi$
$37$	− 9186.00i	− 1.10312i	0.834136	−	0.551559i	$-0.185967\pi$
$38$	− 3808.00i	− 0.427797i
$39$	6480.00	0.682203
$40$	0	0
$41$	−6818.00	−0.633428	−0.316714	−	0.948521i	$-0.602580\pi$
$41$	−6818.00	−0.633428	−0.316714	+	0.948521i	$0.602580\pi$
$42$	− 2592.00i	− 0.226731i
$43$	13310.0i	1.09776i	0.835902	+	0.548879i	$0.184945\pi$
$43$	13310.0i	1.09776i	−0.835902	+	0.548879i	$0.815055\pi$
$44$	1936.00	0.150756
$45$	0	0
$46$	−4368.00	−0.304360
$47$	− 22420.0i	− 1.48044i	−0.672364	−	0.740220i	$-0.734721\pi$
$47$	− 22420.0i	− 1.48044i	0.672364	−	0.740220i	$-0.265279\pi$
$48$	− 3072.00i	− 0.192450i
$49$	13891.0	0.826501
$50$	0	0
$51$	4080.00	0.219652
$52$	− 8640.00i	− 0.443104i
$53$	− 19654.0i	− 0.961084i	−0.876972	−	0.480542i	$-0.840440\pi$
$53$	− 19654.0i	− 0.961084i	0.876972	−	0.480542i	$-0.159560\pi$
$54$	−16416.0	−0.766096
$55$	0	0
$56$	−3456.00	−0.147266
$57$	− 11424.0i	− 0.465727i
$58$	− 248.000i	− 0.00968014i
$59$	−48292.0	−1.80611	−0.903057	−	0.429521i	$-0.858683\pi$
$59$	−48292.0	−1.80611	−0.903057	+	0.429521i	$0.858683\pi$
$60$	0	0
$61$	17530.0	0.603194	0.301597	−	0.953435i	$-0.402480\pi$
$61$	17530.0	0.603194	0.301597	+	0.953435i	$0.402480\pi$
$62$	30240.0i	0.999085i
$63$	5346.00i	0.169698i
$64$	−4096.00	−0.125000
$65$	0	0
$66$	5808.00	0.164122
$67$	− 35344.0i	− 0.961897i	−0.876749	−	0.480949i	$-0.840292\pi$
$67$	− 35344.0i	− 0.961897i	0.876749	−	0.480949i	$-0.159708\pi$
$68$	− 5440.00i	− 0.142668i
$69$	−13104.0	−0.331346
$70$	0	0
$71$	−22912.0	−0.539408	−0.269704	−	0.962943i	$-0.586926\pi$
$71$	−22912.0	−0.539408	−0.269704	+	0.962943i	$0.586926\pi$
$72$	6336.00i	0.144040i
$73$	− 47852.0i	− 1.05098i	−0.850801	−	0.525488i	$-0.823883\pi$
$73$	− 47852.0i	− 1.05098i	0.850801	−	0.525488i	$-0.176117\pi$
$74$	−36744.0	−0.780023
$75$	0	0
$76$	−15232.0	−0.302498
$77$	− 6534.00i	− 0.125589i
$78$	− 25920.0i	− 0.482390i
$79$	−52396.0	−0.944562	−0.472281	−	0.881448i	$-0.656569\pi$
$79$	−52396.0	−0.944562	−0.472281	+	0.881448i	$0.656569\pi$
$80$	0	0
$81$	−25191.0	−0.426612
$82$	27272.0i	0.447901i
$83$	− 7890.00i	− 0.125713i	−0.998023	−	0.0628567i	$-0.979979\pi$
$83$	− 7890.00i	− 0.125713i	0.998023	−	0.0628567i	$-0.0200211\pi$
$84$	−10368.0	−0.160323
$85$	0	0
$86$	53240.0	0.776233
$87$	− 744.000i	− 0.0105384i
$88$	− 7744.00i	− 0.106600i
$89$	−41958.0	−0.561487	−0.280744	−	0.959783i	$-0.590581\pi$
$89$	−41958.0	−0.561487	−0.280744	+	0.959783i	$0.590581\pi$
$90$	0	0
$91$	−29160.0	−0.369134
$92$	17472.0i	0.215215i
$93$	90720.0i	1.08767i
$94$	−89680.0	−1.04683
$95$	0	0
$96$	−12288.0	−0.136083
$97$	− 37602.0i	− 0.405772i	−0.979202	−	0.202886i	$-0.934968\pi$
$97$	− 37602.0i	− 0.405772i	0.979202	−	0.202886i	$-0.0650320\pi$
$98$	− 55564.0i	− 0.584424i
$99$	−11979.0	−0.122838
$100$	0	0
$101$	57406.0	0.559956	0.279978	−	0.960006i	$-0.409673\pi$
$101$	57406.0	0.559956	0.279978	+	0.960006i	$0.409673\pi$
$102$	− 16320.0i	− 0.155317i
$103$	50528.0i	0.469288i	0.972081	+	0.234644i	$0.0753924\pi$
$103$	50528.0i	0.469288i	−0.972081	+	0.234644i	$0.924608\pi$
$104$	−34560.0	−0.313322
$105$	0	0
$106$	−78616.0	−0.679589
$107$	− 78350.0i	− 0.661576i	−0.943705	−	0.330788i	$-0.892686\pi$
$107$	− 78350.0i	− 0.661576i	0.943705	−	0.330788i	$-0.107314\pi$
$108$	65664.0i	0.541711i
$109$	−59990.0	−0.483629	−0.241815	−	0.970322i	$-0.577743\pi$
$109$	−59990.0	−0.483629	−0.241815	+	0.970322i	$0.577743\pi$
$110$	0	0
$111$	−110232.	−0.849181
$112$	13824.0i	0.104133i
$113$	60194.0i	0.443463i	0.975108	+	0.221731i	$0.0711708\pi$
$113$	60194.0i	0.443463i	−0.975108	+	0.221731i	$0.928829\pi$
$114$	−45696.0	−0.329318
$115$	0	0
$116$	−992.000	−0.00684489
$117$	53460.0i	0.361047i
$118$	193168.i	1.27712i
$119$	−18360.0	−0.118852
$120$	0	0
$121$	14641.0	0.0909091
$122$	− 70120.0i	− 0.426523i
$123$	81816.0i	0.487613i
$124$	120960.	0.706460
$125$	0	0
$126$	21384.0	0.119995
$127$	115466.i	0.635250i	0.948216	+	0.317625i	$0.102885\pi$
$127$	115466.i	0.635250i	−0.948216	+	0.317625i	$0.897115\pi$
$128$	16384.0i	0.0883883i
$129$	159720.	0.845055
$130$	0	0
$131$	302896.	1.54211	0.771055	−	0.636769i	$-0.219729\pi$
$131$	302896.	1.54211	0.771055	+	0.636769i	$0.219729\pi$
$132$	− 23232.0i	− 0.116052i
$133$	51408.0i	0.252001i
$134$	−141376.	−0.680164
$135$	0	0
$136$	−21760.0	−0.100882
$137$	− 131658.i	− 0.599302i	−0.954049	−	0.299651i	$-0.903130\pi$
$137$	− 131658.i	− 0.599302i	0.954049	−	0.299651i	$-0.0968703\pi$
$138$	52416.0i	0.234297i
$139$	−129052.	−0.566536	−0.283268	−	0.959041i	$-0.591419\pi$
$139$	−129052.	−0.566536	−0.283268	+	0.959041i	$0.591419\pi$
$140$	0	0
$141$	−269040.	−1.13964
$142$	91648.0i	0.381419i
$143$	− 65340.0i	− 0.267202i
$144$	25344.0	0.101852
$145$	0	0
$146$	−191408.	−0.743153
$147$	− 166692.i	− 0.636241i
$148$	146976.i	0.551559i
$149$	−494814.	−1.82590	−0.912949	−	0.408075i	$-0.866200\pi$
$149$	−494814.	−1.82590	−0.912949	+	0.408075i	$0.866200\pi$
$150$	0	0
$151$	27044.0	0.0965225	0.0482612	−	0.998835i	$-0.484632\pi$
$151$	27044.0	0.0965225	0.0482612	+	0.998835i	$0.484632\pi$
$152$	60928.0i	0.213899i
$153$	33660.0i	0.116248i
$154$	−26136.0	−0.0888050
$155$	0	0
$156$	−103680.	−0.341101
$157$	− 252414.i	− 0.817268i	−0.912698	−	0.408634i	$-0.866005\pi$
$157$	− 252414.i	− 0.817268i	0.912698	−	0.408634i	$-0.133995\pi$
$158$	209584.i	0.667906i
$159$	−235848.	−0.739843
$160$	0	0
$161$	58968.0	0.179288
$162$	100764.i	0.301660i
$163$	325724.i	0.960242i	0.877202	+	0.480121i	$0.159407\pi$
$163$	325724.i	0.960242i	−0.877202	+	0.480121i	$0.840593\pi$
$164$	109088.	0.316714
$165$	0	0
$166$	−31560.0	−0.0888928
$167$	− 134214.i	− 0.372397i	−0.982512	−	0.186199i	$-0.940383\pi$
$167$	− 134214.i	− 0.372397i	0.982512	−	0.186199i	$-0.0596168\pi$
$168$	41472.0i	0.113366i
$169$	79693.0	0.214636
$170$	0	0
$171$	94248.0	0.246480
$172$	− 212960.i	− 0.548879i
$173$	359936.i	0.914345i	0.889378	+	0.457172i	$0.151138\pi$
$173$	359936.i	0.914345i	−0.889378	+	0.457172i	$0.848862\pi$
$174$	−2976.00	−0.00745178
$175$	0	0
$176$	−30976.0	−0.0753778
$177$	579504.i	1.39035i
$178$	167832.i	0.397031i
$179$	−413652.	−0.964945	−0.482472	−	0.875911i	$-0.660261\pi$
$179$	−413652.	−0.964945	−0.482472	+	0.875911i	$0.660261\pi$
$180$	0	0
$181$	−112030.	−0.254178	−0.127089	−	0.991891i	$-0.540563\pi$
$181$	−112030.	−0.254178	−0.127089	+	0.991891i	$0.540563\pi$
$182$	116640.i	0.261017i
$183$	− 210360.i	− 0.464339i
$184$	69888.0	0.152180
$185$	0	0
$186$	362880.	0.769096
$187$	− 41140.0i	− 0.0860320i
$188$	358720.i	0.740220i
$189$	221616.	0.451281
$190$	0	0
$191$	−317280.	−0.629302	−0.314651	−	0.949207i	$-0.601888\pi$
$191$	−317280.	−0.629302	−0.314651	+	0.949207i	$0.601888\pi$
$192$	49152.0i	0.0962250i
$193$	494996.i	0.956552i	0.878210	+	0.478276i	$0.158738\pi$
$193$	494996.i	0.956552i	−0.878210	+	0.478276i	$0.841262\pi$
$194$	−150408.	−0.286924
$195$	0	0
$196$	−222256.	−0.413250
$197$	202032.i	0.370898i	0.982654	+	0.185449i	$0.0593740\pi$
$197$	202032.i	0.370898i	−0.982654	+	0.185449i	$0.940626\pi$
$198$	47916.0i	0.0868596i
$199$	−471896.	−0.844722	−0.422361	−	0.906428i	$-0.638798\pi$
$199$	−471896.	−0.844722	−0.422361	+	0.906428i	$0.638798\pi$
$200$	0	0
$201$	−424128.	−0.740469
$202$	− 229624.i	− 0.395949i
$203$	3348.00i	0.00570224i
$204$	−65280.0	−0.109826
$205$	0	0
$206$	202112.	0.331836
$207$	− 108108.i	− 0.175361i
$208$	138240.i	0.221552i
$209$	−115192.	−0.182413
$210$	0	0
$211$	171136.	0.264628	0.132314	−	0.991208i	$-0.457759\pi$
$211$	171136.	0.264628	0.132314	+	0.991208i	$0.457759\pi$
$212$	314464.i	0.480542i
$213$	274944.i	0.415236i
$214$	−313400.	−0.467805
$215$	0	0
$216$	262656.	0.383048
$217$	− 408240.i	− 0.588527i
$218$	239960.i	0.341978i
$219$	−574224.	−0.809042
$220$	0	0
$221$	−183600.	−0.252867
$222$	440928.i	0.600462i
$223$	− 832572.i	− 1.12114i	−0.828107	−	0.560570i	$-0.810582\pi$
$223$	− 832572.i	− 1.12114i	0.828107	−	0.560570i	$-0.189418\pi$
$224$	55296.0	0.0736332
$225$	0	0
$226$	240776.	0.313575
$227$	1.20700e6i	1.55468i	0.629079	+	0.777342i	$0.283432\pi$
$227$	1.20700e6i	1.55468i	−0.629079	+	0.777342i	$0.716568\pi$
$228$	182784.i	0.232863i
$229$	1.02502e6	1.29164	0.645822	−	0.763488i	$-0.276515\pi$
$229$	1.02502e6	1.29164	0.645822	+	0.763488i	$0.276515\pi$
$230$	0	0
$231$	−78408.0	−0.0966786
$232$	3968.00i	0.00484007i
$233$	− 722808.i	− 0.872234i	−0.899890	−	0.436117i	$-0.856353\pi$
$233$	− 722808.i	− 0.872234i	0.899890	−	0.436117i	$-0.143647\pi$
$234$	213840.	0.255299
$235$	0	0
$236$	772672.	0.903057
$237$	628752.i	0.727124i
$238$	73440.0i	0.0840408i
$239$	51556.0	0.0583827	0.0291914	−	0.999574i	$-0.490707\pi$
$239$	51556.0	0.0583827	0.0291914	+	0.999574i	$0.490707\pi$
$240$	0	0
$241$	−955890.	−1.06015	−0.530073	−	0.847952i	$-0.677835\pi$
$241$	−955890.	−1.06015	−0.530073	+	0.847952i	$0.677835\pi$
$242$	− 58564.0i	− 0.0642824i
$243$	− 694980.i	− 0.755017i
$244$	−280480.	−0.301597
$245$	0	0
$246$	327264.	0.344795
$247$	514080.i	0.536152i
$248$	− 483840.i	− 0.499543i
$249$	−94680.0	−0.0967743
$250$	0	0
$251$	−1.80948e6	−1.81288	−0.906439	−	0.422337i	$-0.861210\pi$
$251$	−1.80948e6	−1.81288	−0.906439	+	0.422337i	$0.861210\pi$
$252$	− 85536.0i	− 0.0848492i
$253$	132132.i	0.129780i
$254$	461864.	0.449190
$255$	0	0
$256$	65536.0	0.0625000
$257$	− 977806.i	− 0.923464i	−0.887020	−	0.461732i	$-0.847228\pi$
$257$	− 977806.i	− 0.923464i	0.887020	−	0.461732i	$-0.152772\pi$
$258$	− 638880.i	− 0.597544i
$259$	496044.	0.459484
$260$	0	0
$261$	6138.00	0.00557732
$262$	− 1.21158e6i	− 1.09044i
$263$	− 1.39455e6i	− 1.24321i	−0.783331	−	0.621605i	$-0.786481\pi$
$263$	− 1.39455e6i	− 1.24321i	0.783331	−	0.621605i	$-0.213519\pi$
$264$	−92928.0	−0.0820610
$265$	0	0
$266$	205632.	0.178191
$267$	503496.i	0.432233i
$268$	565504.i	0.480949i
$269$	−1.18931e6	−1.00211	−0.501054	−	0.865416i	$-0.667054\pi$
$269$	−1.18931e6	−1.00211	−0.501054	+	0.865416i	$0.667054\pi$
$270$	0	0
$271$	510412.	0.422180	0.211090	−	0.977467i	$-0.432299\pi$
$271$	510412.	0.422180	0.211090	+	0.977467i	$0.432299\pi$
$272$	87040.0i	0.0713340i
$273$	349920.i	0.284159i
$274$	−526632.	−0.423771
$275$	0	0
$276$	209664.	0.165673
$277$	− 1.99828e6i	− 1.56479i	−0.622783	−	0.782395i	$-0.713998\pi$
$277$	− 1.99828e6i	− 1.56479i	0.622783	−	0.782395i	$-0.286002\pi$
$278$	516208.i	0.400602i
$279$	−748440.	−0.575634
$280$	0	0
$281$	−2.61405e6	−1.97491	−0.987457	−	0.157890i	$-0.949531\pi$
$281$	−2.61405e6	−1.97491	−0.987457	+	0.157890i	$0.949531\pi$
$282$	1.07616e6i	0.805850i
$283$	1.80037e6i	1.33627i	0.744039	+	0.668136i	$0.232907\pi$
$283$	1.80037e6i	1.33627i	−0.744039	+	0.668136i	$0.767093\pi$
$284$	366592.	0.269704
$285$	0	0
$286$	−261360.	−0.188940
$287$	− 368172.i	− 0.263843i
$288$	− 101376.i	− 0.0720201i
$289$	1.30426e6	0.918583
$290$	0	0
$291$	−451224.	−0.312363
$292$	765632.i	0.525488i
$293$	− 867228.i	− 0.590152i	−0.955474	−	0.295076i	$-0.904655\pi$
$293$	− 867228.i	− 0.590152i	0.955474	−	0.295076i	$-0.0953451\pi$
$294$	−666768.	−0.449890
$295$	0	0
$296$	587904.	0.390011
$297$	496584.i	0.326664i
$298$	1.97926e6i	1.29110i
$299$	589680.	0.381451
$300$	0	0
$301$	−718740.	−0.457252
$302$	− 108176.i	− 0.0682517i
$303$	− 688872.i	− 0.431054i
$304$	243712.	0.151249
$305$	0	0
$306$	134640.	0.0821998
$307$	− 735734.i	− 0.445528i	−0.974872	−	0.222764i	$-0.928492\pi$
$307$	− 735734.i	− 0.445528i	0.974872	−	0.222764i	$-0.0715079\pi$
$308$	104544.i	0.0627946i
$309$	606336.	0.361258
$310$	0	0
$311$	1.55405e6	0.911095	0.455547	−	0.890212i	$-0.349444\pi$
$311$	1.55405e6	0.911095	0.455547	+	0.890212i	$0.349444\pi$
$312$	414720.i	0.241195i
$313$	− 2.04743e6i	− 1.18127i	−0.806939	−	0.590635i	$-0.798877\pi$
$313$	− 2.04743e6i	− 1.18127i	0.806939	−	0.590635i	$-0.201123\pi$
$314$	−1.00966e6	−0.577895
$315$	0	0
$316$	838336.	0.472281
$317$	− 1.11066e6i	− 0.620774i	−0.950610	−	0.310387i	$-0.899541\pi$
$317$	− 1.11066e6i	− 0.620774i	0.950610	−	0.310387i	$-0.100459\pi$
$318$	943392.i	0.523148i
$319$	−7502.00	−0.00412763
$320$	0	0
$321$	−940200.	−0.509281
$322$	− 235872.i	− 0.126776i
$323$	323680.i	0.172627i
$324$	403056.	0.213306
$325$	0	0
$326$	1.30290e6	0.678994
$327$	719880.i	0.372298i
$328$	− 436352.i	− 0.223951i
$329$	1.21068e6	0.616651
$330$	0	0
$331$	−1.29735e6	−0.650858	−0.325429	−	0.945566i	$-0.605509\pi$
$331$	−1.29735e6	−0.650858	−0.325429	+	0.945566i	$0.605509\pi$
$332$	126240.i	0.0628567i
$333$	− 909414.i	− 0.449419i
$334$	−536856.	−0.263325
$335$	0	0
$336$	165888.	0.0801617
$337$	− 146684.i	− 0.0703571i	−0.999381	−	0.0351786i	$-0.988800\pi$
$337$	− 146684.i	− 0.0703571i	0.999381	−	0.0351786i	$-0.0112000\pi$
$338$	− 318772.i	− 0.151771i
$339$	722328.	0.341378
$340$	0	0
$341$	914760.	0.426011
$342$	− 376992.i	− 0.174288i
$343$	1.65769e6i	0.760797i
$344$	−851840.	−0.388116
$345$	0	0
$346$	1.43974e6	0.646539
$347$	− 3.18236e6i	− 1.41881i	−0.704799	−	0.709407i	$-0.748963\pi$
$347$	− 3.18236e6i	− 1.41881i	0.704799	−	0.709407i	$-0.251037\pi$
$348$	11904.0i	0.00526920i
$349$	2.14613e6	0.943177	0.471589	−	0.881819i	$-0.343681\pi$
$349$	2.14613e6	0.943177	0.471589	+	0.881819i	$0.343681\pi$
$350$	0	0
$351$	2.21616e6	0.960137
$352$	123904.i	0.0533002i
$353$	3.65609e6i	1.56164i	0.624759	+	0.780818i	$0.285197\pi$
$353$	3.65609e6i	1.56164i	−0.624759	+	0.780818i	$0.714803\pi$
$354$	2.31802e6	0.983124
$355$	0	0
$356$	671328.	0.280744
$357$	220320.i	0.0914921i
$358$	1.65461e6i	0.682319i
$359$	−751316.	−0.307671	−0.153835	−	0.988096i	$-0.549163\pi$
$359$	−751316.	−0.307671	−0.153835	+	0.988096i	$0.549163\pi$
$360$	0	0
$361$	−1.56980e6	−0.633979
$362$	448120.i	0.179731i
$363$	− 175692.i	− 0.0699819i
$364$	466560.	0.184567
$365$	0	0
$366$	−841440.	−0.328337
$367$	− 2.30709e6i	− 0.894128i	−0.894502	−	0.447064i	$-0.852470\pi$
$367$	− 2.30709e6i	− 0.894128i	0.894502	−	0.447064i	$-0.147530\pi$
$368$	− 279552.i	− 0.107608i
$369$	−674982.	−0.258063
$370$	0	0
$371$	1.06132e6	0.400322
$372$	− 1.45152e6i	− 0.543833i
$373$	− 4.95593e6i	− 1.84439i	−0.386724	−	0.922195i	$-0.626394\pi$
$373$	− 4.95593e6i	− 1.84439i	0.386724	−	0.922195i	$-0.373606\pi$
$374$	−164560.	−0.0608338
$375$	0	0
$376$	1.43488e6	0.523415
$377$	33480.0i	0.0121320i
$378$	− 886464.i	− 0.319104i
$379$	−3.64597e6	−1.30381	−0.651906	−	0.758299i	$-0.726031\pi$
$379$	−3.64597e6	−1.30381	−0.651906	+	0.758299i	$0.726031\pi$
$380$	0	0
$381$	1.38559e6	0.489016
$382$	1.26912e6i	0.444984i
$383$	− 3.82201e6i	− 1.33136i	−0.746238	−	0.665679i	$-0.768142\pi$
$383$	− 3.82201e6i	− 1.33136i	0.746238	−	0.665679i	$-0.231858\pi$
$384$	196608.	0.0680414
$385$	0	0
$386$	1.97998e6	0.676384
$387$	1.31769e6i	0.447235i
$388$	601632.i	0.202886i
$389$	2.36807e6	0.793453	0.396727	−	0.917937i	$-0.370146\pi$
$389$	2.36807e6	0.793453	0.396727	+	0.917937i	$0.370146\pi$
$390$	0	0
$391$	371280.	0.122817
$392$	889024.i	0.292212i
$393$	− 3.63475e6i	− 1.18712i
$394$	808128.	0.262265
$395$	0	0
$396$	191664.	0.0614190
$397$	890626.i	0.283608i	0.989895	+	0.141804i	$0.0452903\pi$
$397$	890626.i	0.283608i	−0.989895	+	0.141804i	$0.954710\pi$
$398$	1.88758e6i	0.597308i
$399$	616896.	0.193990
$400$	0	0
$401$	3.58277e6	1.11265	0.556324	−	0.830966i	$-0.312211\pi$
$401$	3.58277e6	1.11265	0.556324	+	0.830966i	$0.312211\pi$
$402$	1.69651e6i	0.523591i
$403$	− 4.08240e6i	− 1.25214i
$404$	−918496.	−0.279978
$405$	0	0
$406$	13392.0	0.00403209
$407$	1.11151e6i	0.332603i
$408$	261120.i	0.0776586i
$409$	4.07749e6	1.20527	0.602636	−	0.798016i	$-0.294117\pi$
$409$	4.07749e6	1.20527	0.602636	+	0.798016i	$0.294117\pi$
$410$	0	0
$411$	−1.57990e6	−0.461343
$412$	− 808448.i	− 0.234644i
$413$	− 2.60777e6i	− 0.752305i
$414$	−432432.	−0.123999
$415$	0	0
$416$	552960.	0.156661
$417$	1.54862e6i	0.436120i
$418$	460768.i	0.128986i
$419$	20820.0	0.00579356	0.00289678	−	0.999996i	$-0.499078\pi$
$419$	20820.0	0.00579356	0.00289678	+	0.999996i	$0.499078\pi$
$420$	0	0
$421$	4.93515e6	1.35705	0.678523	−	0.734579i	$-0.262620\pi$
$421$	4.93515e6	1.35705	0.678523	+	0.734579i	$0.262620\pi$
$422$	− 684544.i	− 0.187120i
$423$	− 2.21958e6i	− 0.603142i
$424$	1.25786e6	0.339794
$425$	0	0
$426$	1.09978e6	0.293616
$427$	946620.i	0.251250i
$428$	1.25360e6i	0.330788i
$429$	−784080.	−0.205692
$430$	0	0
$431$	−1.27881e6	−0.331598	−0.165799	−	0.986160i	$-0.553020\pi$
$431$	−1.27881e6	−0.331598	−0.165799	+	0.986160i	$0.553020\pi$
$432$	− 1.05062e6i	− 0.270856i
$433$	− 3.04729e6i	− 0.781078i	−0.920586	−	0.390539i	$-0.872289\pi$
$433$	− 3.04729e6i	− 0.781078i	0.920586	−	0.390539i	$-0.127711\pi$
$434$	−1.63296e6	−0.416151
$435$	0	0
$436$	959840.	0.241815
$437$	− 1.03958e6i	− 0.260409i
$438$	2.29690e6i	0.572079i
$439$	−3.55304e6	−0.879911	−0.439956	−	0.898020i	$-0.645006\pi$
$439$	−3.55304e6	−0.879911	−0.439956	+	0.898020i	$0.645006\pi$
$440$	0	0
$441$	1.37521e6	0.336723
$442$	734400.i	0.178804i
$443$	− 3.87688e6i	− 0.938585i	−0.883043	−	0.469292i	$-0.844509\pi$
$443$	− 3.87688e6i	− 0.938585i	0.883043	−	0.469292i	$-0.155491\pi$
$444$	1.76371e6	0.424590
$445$	0	0
$446$	−3.33029e6	−0.792765
$447$	5.93777e6i	1.40558i
$448$	− 221184.i	− 0.0520665i
$449$	2.29420e6	0.537051	0.268526	−	0.963273i	$-0.413464\pi$
$449$	2.29420e6	0.537051	0.268526	+	0.963273i	$0.413464\pi$
$450$	0	0
$451$	824978.	0.190986
$452$	− 963104.i	− 0.221731i
$453$	− 324528.i	− 0.0743031i
$454$	4.82799e6	1.09933
$455$	0	0
$456$	731136.	0.164659
$457$	391368.i	0.0876587i	0.999039	+	0.0438293i	$0.0139558\pi$
$457$	391368.i	0.0876587i	−0.999039	+	0.0438293i	$0.986044\pi$
$458$	− 4.10007e6i	− 0.913330i
$459$	1.39536e6	0.309140
$460$	0	0
$461$	869814.	0.190622	0.0953112	−	0.995448i	$-0.469615\pi$
$461$	869814.	0.190622	0.0953112	+	0.995448i	$0.469615\pi$
$462$	313632.i	0.0683621i
$463$	5.75085e6i	1.24675i	0.781923	+	0.623375i	$0.214239\pi$
$463$	5.75085e6i	1.24675i	−0.781923	+	0.623375i	$0.785761\pi$
$464$	15872.0	0.00342245
$465$	0	0
$466$	−2.89123e6	−0.616763
$467$	− 4.56445e6i	− 0.968492i	−0.874932	−	0.484246i	$-0.839094\pi$
$467$	− 4.56445e6i	− 0.968492i	0.874932	−	0.484246i	$-0.160906\pi$
$468$	− 855360.i	− 0.180524i
$469$	1.90858e6	0.400661
$470$	0	0
$471$	−3.02897e6	−0.629133
$472$	− 3.09069e6i	− 0.638558i
$473$	− 1.61051e6i	− 0.330987i
$474$	2.51501e6	0.514154
$475$	0	0
$476$	293760.	0.0594258
$477$	− 1.94575e6i	− 0.391553i
$478$	− 206224.i	− 0.0412828i
$479$	−3.15604e6	−0.628497	−0.314248	−	0.949341i	$-0.601753\pi$
$479$	−3.15604e6	−0.628497	−0.314248	+	0.949341i	$0.601753\pi$
$480$	0	0
$481$	4.96044e6	0.977592
$482$	3.82356e6i	0.749636i
$483$	− 707616.i	− 0.138016i
$484$	−234256.	−0.0454545
$485$	0	0
$486$	−2.77992e6	−0.533878
$487$	− 3.14120e6i	− 0.600169i	−0.953913	−	0.300085i	$-0.902985\pi$
$487$	− 3.14120e6i	− 0.600169i	0.953913	−	0.300085i	$-0.0970149\pi$
$488$	1.12192e6i	0.213261i
$489$	3.90869e6	0.739195
$490$	0	0
$491$	−7.23388e6	−1.35415	−0.677076	−	0.735913i	$-0.736753\pi$
$491$	−7.23388e6	−1.35415	−0.677076	+	0.735913i	$0.736753\pi$
$492$	− 1.30906e6i	− 0.243807i
$493$	21080.0i	0.00390619i
$494$	2.05632e6	0.379117
$495$	0	0
$496$	−1.93536e6	−0.353230
$497$	− 1.23725e6i	− 0.224681i
$498$	378720.i	0.0684297i
$499$	7.84402e6	1.41022	0.705111	−	0.709097i	$-0.250897\pi$
$499$	7.84402e6	1.41022	0.705111	+	0.709097i	$0.250897\pi$
$500$	0	0
$501$	−1.61057e6	−0.286672
$502$	7.23790e6i	1.28190i
$503$	5.94665e6i	1.04798i	0.851725	+	0.523990i	$0.175557\pi$
$503$	5.94665e6i	1.04798i	−0.851725	+	0.523990i	$0.824443\pi$
$504$	−342144.	−0.0599974
$505$	0	0
$506$	528528.	0.0917681
$507$	− 956316.i	− 0.165227i
$508$	− 1.84746e6i	− 0.317625i
$509$	6.20971e6	1.06237	0.531187	−	0.847255i	$-0.321746\pi$
$509$	6.20971e6	1.06237	0.531187	+	0.847255i	$0.321746\pi$
$510$	0	0
$511$	2.58401e6	0.437766
$512$	− 262144.i	− 0.0441942i
$513$	− 3.90701e6i	− 0.655467i
$514$	−3.91122e6	−0.652988
$515$	0	0
$516$	−2.55552e6	−0.422528
$517$	2.71282e6i	0.446370i
$518$	− 1.98418e6i	− 0.324905i
$519$	4.31923e6	0.703863
$520$	0	0
$521$	−8.89086e6	−1.43499	−0.717496	−	0.696563i	$-0.754712\pi$
$521$	−8.89086e6	−1.43499	−0.717496	+	0.696563i	$0.754712\pi$
$522$	− 24552.0i	− 0.00394376i
$523$	− 2.63425e6i	− 0.421118i	−0.977581	−	0.210559i	$-0.932472\pi$
$523$	− 2.63425e6i	− 0.421118i	0.977581	−	0.210559i	$-0.0675283\pi$
$524$	−4.84634e6	−0.771055
$525$	0	0
$526$	−5.57820e6	−0.879083
$527$	− 2.57040e6i	− 0.403157i
$528$	371712.i	0.0580259i
$529$	5.24388e6	0.814730
$530$	0	0
$531$	−4.78091e6	−0.735824
$532$	− 822528.i	− 0.126000i
$533$	− 3.68172e6i	− 0.561349i
$534$	2.01398e6	0.305635
$535$	0	0
$536$	2.26202e6	0.340082
$537$	4.96382e6i	0.742815i
$538$	4.75724e6i	0.708597i
$539$	−1.68081e6	−0.249199
$540$	0	0
$541$	505174.	0.0742075	0.0371038	−	0.999311i	$-0.488187\pi$
$541$	505174.	0.0742075	0.0371038	+	0.999311i	$0.488187\pi$
$542$	− 2.04165e6i	− 0.298526i
$543$	1.34436e6i	0.195666i
$544$	348160.	0.0504408
$545$	0	0
$546$	1.39968e6	0.200931
$547$	− 2.53143e6i	− 0.361741i	−0.983507	−	0.180871i	$-0.942108\pi$
$547$	− 2.53143e6i	− 0.361741i	0.983507	−	0.180871i	$-0.0578916\pi$
$548$	2.10653e6i	0.299651i
$549$	1.73547e6	0.245746
$550$	0	0
$551$	59024.0	0.00828227
$552$	− 838656.i	− 0.117148i
$553$	− 2.82938e6i	− 0.393441i
$554$	−7.99310e6	−1.10647
$555$	0	0
$556$	2.06483e6	0.283268
$557$	− 7.02739e6i	− 0.959746i	−0.877338	−	0.479873i	$-0.840683\pi$
$557$	− 7.02739e6i	− 0.959746i	0.877338	−	0.479873i	$-0.159317\pi$
$558$	2.99376e6i	0.407035i
$559$	−7.18740e6	−0.972842
$560$	0	0
$561$	−493680.	−0.0662275
$562$	1.04562e7i	1.39647i
$563$	2.21500e6i	0.294512i	0.989098	+	0.147256i	$0.0470441\pi$
$563$	2.21500e6i	0.294512i	−0.989098	+	0.147256i	$0.952956\pi$
$564$	4.30464e6	0.569822
$565$	0	0
$566$	7.20146e6	0.944887
$567$	− 1.36031e6i	− 0.177698i
$568$	− 1.46637e6i	− 0.190709i
$569$	−3.70077e6	−0.479194	−0.239597	−	0.970872i	$-0.577015\pi$
$569$	−3.70077e6	−0.479194	−0.239597	+	0.970872i	$0.577015\pi$
$570$	0	0
$571$	1.18319e6	0.151867	0.0759335	−	0.997113i	$-0.475806\pi$
$571$	1.18319e6	0.151867	0.0759335	+	0.997113i	$0.475806\pi$
$572$	1.04544e6i	0.133601i
$573$	3.80736e6i	0.484437i
$574$	−1.47269e6	−0.186565
$575$	0	0
$576$	−405504.	−0.0509259
$577$	1.33023e7i	1.66336i	0.555255	+	0.831680i	$0.312621\pi$
$577$	1.33023e7i	1.66336i	−0.555255	+	0.831680i	$0.687379\pi$
$578$	− 5.21703e6i	− 0.649537i
$579$	5.93995e6	0.736354
$580$	0	0
$581$	426060.	0.0523637
$582$	1.80490e6i	0.220874i
$583$	2.37813e6i	0.289778i
$584$	3.06253e6	0.371576
$585$	0	0
$586$	−3.46891e6	−0.417301
$587$	− 1.83924e6i	− 0.220315i	−0.993914	−	0.110157i	$-0.964865\pi$
$587$	− 1.83924e6i	− 0.220315i	0.993914	−	0.110157i	$-0.0351355\pi$
$588$	2.66707e6i	0.318120i
$589$	−7.19712e6	−0.854812
$590$	0	0
$591$	2.42438e6	0.285517
$592$	− 2.35162e6i	− 0.275780i
$593$	1.35225e7i	1.57914i	0.613663	+	0.789568i	$0.289696\pi$
$593$	1.35225e7i	1.57914i	−0.613663	+	0.789568i	$0.710304\pi$
$594$	1.98634e6	0.230987
$595$	0	0
$596$	7.91702e6	0.912949
$597$	5.66275e6i	0.650267i
$598$	− 2.35872e6i	− 0.269726i
$599$	169296.	0.0192788	0.00963939	−	0.999954i	$-0.496932\pi$
$599$	169296.	0.0192788	0.00963939	+	0.999954i	$0.496932\pi$
$600$	0	0
$601$	−5.30071e6	−0.598615	−0.299307	−	0.954157i	$-0.596756\pi$
$601$	−5.30071e6	−0.598615	−0.299307	+	0.954157i	$0.596756\pi$
$602$	2.87496e6i	0.323326i
$603$	− 3.49906e6i	− 0.391884i
$604$	−432704.	−0.0482612
$605$	0	0
$606$	−2.75549e6	−0.304801
$607$	− 26534.0i	− 0.00292301i	−0.999999	−	0.00146151i	$-0.999535\pi$
$607$	− 26534.0i	− 0.00292301i	0.999999	−	0.00146151i	$-0.000465212\pi$
$608$	− 974848.i	− 0.106949i
$609$	40176.0	0.00438959
$610$	0	0
$611$	1.21068e7	1.31198
$612$	− 538560.i	− 0.0581240i
$613$	8.54910e6i	0.918902i	0.888203	+	0.459451i	$0.151954\pi$
$613$	8.54910e6i	0.918902i	−0.888203	+	0.459451i	$0.848046\pi$
$614$	−2.94294e6	−0.315036
$615$	0	0
$616$	418176.	0.0444025
$617$	1.80098e6i	0.190457i	0.995455	+	0.0952284i	$0.0303581\pi$
$617$	1.80098e6i	0.190457i	−0.995455	+	0.0952284i	$0.969642\pi$
$618$	− 2.42534e6i	− 0.255448i
$619$	−7.27982e6	−0.763649	−0.381825	−	0.924235i	$-0.624704\pi$
$619$	−7.27982e6	−0.763649	−0.381825	+	0.924235i	$0.624704\pi$
$620$	0	0
$621$	−4.48157e6	−0.466338
$622$	− 6.21619e6i	− 0.644241i
$623$	− 2.26573e6i	− 0.233877i
$624$	1.65888e6	0.170551
$625$	0	0
$626$	−8.18974e6	−0.835284
$627$	1.38230e6i	0.140422i
$628$	4.03862e6i	0.408634i
$629$	3.12324e6	0.314759
$630$	0	0
$631$	−5.31298e6	−0.531208	−0.265604	−	0.964082i	$-0.585571\pi$
$631$	−5.31298e6	−0.531208	−0.265604	+	0.964082i	$0.585571\pi$
$632$	− 3.35334e6i	− 0.333953i
$633$	− 2.05363e6i	− 0.203710i
$634$	−4.44265e6	−0.438954
$635$	0	0
$636$	3.77357e6	0.369921
$637$	7.50114e6i	0.732451i
$638$	30008.0i	0.00291867i
$639$	−2.26829e6	−0.219759
$640$	0	0
$641$	−1.03744e6	−0.0997284	−0.0498642	−	0.998756i	$-0.515879\pi$
$641$	−1.03744e6	−0.0997284	−0.0498642	+	0.998756i	$0.515879\pi$
$642$	3.76080e6i	0.360116i
$643$	1.48514e7i	1.41658i	0.705922	+	0.708290i	$0.250533\pi$
$643$	1.48514e7i	1.41658i	−0.705922	+	0.708290i	$0.749467\pi$
$644$	−943488.	−0.0896441
$645$	0	0
$646$	1.29472e6	0.122066
$647$	− 1.00954e7i	− 0.948115i	−0.880494	−	0.474058i	$-0.842789\pi$
$647$	− 1.00954e7i	− 0.948115i	0.880494	−	0.474058i	$-0.157211\pi$
$648$	− 1.61222e6i	− 0.150830i
$649$	5.84333e6	0.544564
$650$	0	0
$651$	−4.89888e6	−0.453048
$652$	− 5.21158e6i	− 0.480121i
$653$	− 1.01545e7i	− 0.931911i	−0.884808	−	0.465955i	$-0.845711\pi$
$653$	− 1.01545e7i	− 0.931911i	0.884808	−	0.465955i	$-0.154289\pi$
$654$	2.87952e6	0.263254
$655$	0	0
$656$	−1.74541e6	−0.158357
$657$	− 4.73735e6i	− 0.428176i
$658$	− 4.84272e6i	− 0.436038i
$659$	−1.81794e7	−1.63067	−0.815334	−	0.578990i	$-0.803447\pi$
$659$	−1.81794e7	−1.63067	−0.815334	+	0.578990i	$0.803447\pi$
$660$	0	0
$661$	1.13115e7	1.00697	0.503483	−	0.864005i	$-0.332052\pi$
$661$	1.13115e7	1.00697	0.503483	+	0.864005i	$0.332052\pi$
$662$	5.18939e6i	0.460226i
$663$	2.20320e6i	0.194657i
$664$	504960.	0.0444464
$665$	0	0
$666$	−3.63766e6	−0.317787
$667$	− 67704.0i	− 0.00589250i
$668$	2.14742e6i	0.186199i
$669$	−9.99086e6	−0.863054
$670$	0	0
$671$	−2.12113e6	−0.181870
$672$	− 663552.i	− 0.0566829i
$673$	7.56284e6i	0.643646i	0.946800	+	0.321823i	$0.104296\pi$
$673$	7.56284e6i	0.643646i	−0.946800	+	0.321823i	$0.895704\pi$
$674$	−586736.	−0.0497500
$675$	0	0
$676$	−1.27509e6	−0.107318
$677$	7.84728e6i	0.658032i	0.944324	+	0.329016i	$0.106717\pi$
$677$	7.84728e6i	0.658032i	−0.944324	+	0.329016i	$0.893283\pi$
$678$	− 2.88931e6i	− 0.241391i
$679$	2.03051e6	0.169017
$680$	0	0
$681$	1.44840e7	1.19680
$682$	− 3.65904e6i	− 0.301236i
$683$	1.38512e7i	1.13615i	0.822977	+	0.568075i	$0.192312\pi$
$683$	1.38512e7i	1.13615i	−0.822977	+	0.568075i	$0.807688\pi$
$684$	−1.50797e6	−0.123240
$685$	0	0
$686$	6.63077e6	0.537964
$687$	− 1.23002e7i	− 0.994308i
$688$	3.40736e6i	0.274440i
$689$	1.06132e7	0.851720
$690$	0	0
$691$	−5.21179e6	−0.415233	−0.207616	−	0.978210i	$-0.566571\pi$
$691$	−5.21179e6	−0.415233	−0.207616	+	0.978210i	$0.566571\pi$
$692$	− 5.75898e6i	− 0.457172i
$693$	− 646866.i	− 0.0511660i
$694$	−1.27294e7	−1.00325
$695$	0	0
$696$	47616.0	0.00372589
$697$	− 2.31812e6i	− 0.180740i
$698$	− 8.58454e6i	− 0.666927i
$699$	−8.67370e6	−0.671446
$700$	0	0
$701$	1.45479e7	1.11816	0.559081	−	0.829113i	$-0.311154\pi$
$701$	1.45479e7	1.11816	0.559081	+	0.829113i	$0.311154\pi$
$702$	− 8.86464e6i	− 0.678920i
$703$	− 8.74507e6i	− 0.667383i
$704$	495616.	0.0376889
$705$	0	0
$706$	1.46243e7	1.10424
$707$	3.09992e6i	0.233240i
$708$	− 9.27206e6i	− 0.695174i
$709$	−2.51026e7	−1.87544	−0.937721	−	0.347389i	$-0.887068\pi$
$709$	−2.51026e7	−1.87544	−0.937721	+	0.347389i	$0.887068\pi$
$710$	0	0
$711$	−5.18720e6	−0.384821
$712$	− 2.68531e6i	− 0.198516i
$713$	8.25552e6i	0.608164i
$714$	881280.	0.0646947
$715$	0	0
$716$	6.61843e6	0.482472
$717$	− 618672.i	− 0.0449431i
$718$	3.00526e6i	0.217556i
$719$	1.65769e7	1.19586	0.597932	−	0.801547i	$-0.295989\pi$
$719$	1.65769e7	1.19586	0.597932	+	0.801547i	$0.295989\pi$
$720$	0	0
$721$	−2.72851e6	−0.195473
$722$	6.27918e6i	0.448291i
$723$	1.14707e7i	0.816100i
$724$	1.79248e6	0.127089
$725$	0	0
$726$	−702768.	−0.0494846
$727$	1.62103e7i	1.13751i	0.822508	+	0.568754i	$0.192574\pi$
$727$	1.62103e7i	1.13751i	−0.822508	+	0.568754i	$0.807426\pi$
$728$	− 1.86624e6i	− 0.130509i
$729$	−1.44612e7	−1.00782
$730$	0	0
$731$	−4.52540e6	−0.313230
$732$	3.36576e6i	0.232170i
$733$	2.48702e7i	1.70970i	0.518875	+	0.854850i	$0.326351\pi$
$733$	2.48702e7i	1.70970i	−0.518875	+	0.854850i	$0.673649\pi$
$734$	−9.22837e6	−0.632244
$735$	0	0
$736$	−1.11821e6	−0.0760901
$737$	4.27662e6i	0.290023i
$738$	2.69993e6i	0.182478i
$739$	−2.67918e7	−1.80464	−0.902321	−	0.431066i	$-0.858138\pi$
$739$	−2.67918e7	−1.80464	−0.902321	+	0.431066i	$0.858138\pi$
$740$	0	0
$741$	6.16896e6	0.412730
$742$	− 4.24526e6i	− 0.283071i
$743$	5.52863e6i	0.367406i	0.982982	+	0.183703i	$0.0588084\pi$
$743$	5.52863e6i	0.367406i	−0.982982	+	0.183703i	$0.941192\pi$
$744$	−5.80608e6	−0.384548
$745$	0	0
$746$	−1.98237e7	−1.30418
$747$	− 781110.i	− 0.0512166i
$748$	658240.i	0.0430160i
$749$	4.23090e6	0.275568
$750$	0	0
$751$	2.01222e7	1.30189	0.650946	−	0.759124i	$-0.274373\pi$
$751$	2.01222e7	1.30189	0.650946	+	0.759124i	$0.274373\pi$
$752$	− 5.73952e6i	− 0.370110i
$753$	2.17137e7i	1.39555i
$754$	133920.	0.00857861
$755$	0	0
$756$	−3.54586e6	−0.225640
$757$	2.05783e7i	1.30518i	0.757711	+	0.652590i	$0.226317\pi$
$757$	2.05783e7i	1.30518i	−0.757711	+	0.652590i	$0.773683\pi$
$758$	1.45839e7i	0.921935i
$759$	1.58558e6	0.0999044
$760$	0	0
$761$	−9.53120e6	−0.596604	−0.298302	−	0.954472i	$-0.596420\pi$
$761$	−9.53120e6	−0.596604	−0.298302	+	0.954472i	$0.596420\pi$
$762$	− 5.54237e6i	− 0.345786i
$763$	− 3.23946e6i	− 0.201447i
$764$	5.07648e6	0.314651
$765$	0	0
$766$	−1.52880e7	−0.941412
$767$	− 2.60777e7i	− 1.60059i
$768$	− 786432.i	− 0.0481125i
$769$	−2.13743e7	−1.30340	−0.651698	−	0.758479i	$-0.725943\pi$
$769$	−2.13743e7	−1.30340	−0.651698	+	0.758479i	$0.725943\pi$
$770$	0	0
$771$	−1.17337e7	−0.710883
$772$	− 7.91994e6i	− 0.478276i
$773$	− 1.27480e7i	− 0.767352i	−0.923468	−	0.383676i	$-0.874658\pi$
$773$	− 1.27480e7i	− 0.767352i	0.923468	−	0.383676i	$-0.125342\pi$
$774$	5.27076e6	0.316243
$775$	0	0
$776$	2.40653e6	0.143462
$777$	− 5.95253e6i	− 0.353711i
$778$	− 9.47230e6i	− 0.561056i
$779$	−6.49074e6	−0.383222
$780$	0	0
$781$	2.77235e6	0.162638
$782$	− 1.48512e6i	− 0.0868450i
$783$	− 254448.i	− 0.0148318i
$784$	3.55610e6	0.206625
$785$	0	0
$786$	−1.45390e7	−0.839418
$787$	− 3.14391e7i	− 1.80939i	−0.426055	−	0.904697i	$-0.640097\pi$
$787$	− 3.14391e7i	− 1.80939i	0.426055	−	0.904697i	$-0.359903\pi$
$788$	− 3.23251e6i	− 0.185449i
$789$	−1.67346e7	−0.957024
$790$	0	0
$791$	−3.25048e6	−0.184717
$792$	− 766656.i	− 0.0434298i
$793$	9.46620e6i	0.534555i
$794$	3.56250e6	0.200541
$795$	0	0
$796$	7.55034e6	0.422361
$797$	− 1.77995e7i	− 0.992573i	−0.868159	−	0.496286i	$-0.834697\pi$
$797$	− 1.77995e7i	− 0.992573i	0.868159	−	0.496286i	$-0.165303\pi$
$798$	− 2.46758e6i	− 0.137172i
$799$	7.62280e6	0.422423
$800$	0	0
$801$	−4.15384e6	−0.228754
$802$	− 1.43311e7i	− 0.786760i
$803$	5.79009e6i	0.316881i
$804$	6.78605e6	0.370234
$805$	0	0
$806$	−1.63296e7	−0.885397
$807$	1.42717e7i	0.771423i
$808$	3.67398e6i	0.197974i
$809$	1.99550e6	0.107196	0.0535982	−	0.998563i	$-0.482931\pi$
$809$	1.99550e6	0.107196	0.0535982	+	0.998563i	$0.482931\pi$
$810$	0	0
$811$	9.09296e6	0.485459	0.242730	−	0.970094i	$-0.421957\pi$
$811$	9.09296e6	0.485459	0.242730	+	0.970094i	$0.421957\pi$
$812$	− 53568.0i	− 0.00285112i
$813$	− 6.12494e6i	− 0.324994i
$814$	4.44602e6	0.235186
$815$	0	0
$816$	1.04448e6	0.0549129
$817$	1.26711e7i	0.664140i
$818$	− 1.63100e7i	− 0.852256i
$819$	−2.88684e6	−0.150388
$820$	0	0
$821$	−2.45679e7	−1.27207	−0.636034	−	0.771661i	$-0.719426\pi$
$821$	−2.45679e7	−1.27207	−0.636034	+	0.771661i	$0.719426\pi$
$822$	6.31958e6i	0.326219i
$823$	6.20308e6i	0.319233i	0.987179	+	0.159617i	$0.0510258\pi$
$823$	6.20308e6i	0.319233i	−0.987179	+	0.159617i	$0.948974\pi$
$824$	−3.23379e6	−0.165918
$825$	0	0
$826$	−1.04311e7	−0.531960
$827$	2.88374e7i	1.46620i	0.680122	+	0.733099i	$0.261927\pi$
$827$	2.88374e7i	1.46620i	−0.680122	+	0.733099i	$0.738073\pi$
$828$	1.72973e6i	0.0876803i
$829$	−2.07058e7	−1.04642	−0.523209	−	0.852204i	$-0.675265\pi$
$829$	−2.07058e7	−1.04642	−0.523209	+	0.852204i	$0.675265\pi$
$830$	0	0
$831$	−2.39793e7	−1.20458
$832$	− 2.21184e6i	− 0.110776i
$833$	4.72294e6i	0.235830i
$834$	6.19450e6	0.308383
$835$	0	0
$836$	1.84307e6	0.0912067
$837$	3.10262e7i	1.53079i
$838$	− 83280.0i	− 0.00409667i
$839$	1.68022e7	0.824066	0.412033	−	0.911169i	$-0.364819\pi$
$839$	1.68022e7	0.824066	0.412033	+	0.911169i	$0.364819\pi$
$840$	0	0
$841$	−2.05073e7	−0.999813
$842$	− 1.97406e7i	− 0.959577i
$843$	3.13686e7i	1.52029i
$844$	−2.73818e6	−0.132314
$845$	0	0
$846$	−8.87832e6	−0.426486
$847$	790614.i	0.0378666i
$848$	− 5.03142e6i	− 0.240271i
$849$	2.16044e7	1.02866
$850$	0	0
$851$	−1.00311e7	−0.474816
$852$	− 4.39910e6i	− 0.207618i
$853$	− 2.06202e7i	− 0.970334i	−0.874422	−	0.485167i	$-0.838759\pi$
$853$	− 2.06202e7i	− 0.970334i	0.874422	−	0.485167i	$-0.161241\pi$
$854$	3.78648e6	0.177661
$855$	0	0
$856$	5.01440e6	0.233902
$857$	− 2.32986e7i	− 1.08362i	−0.840501	−	0.541810i	$-0.817739\pi$
$857$	− 2.32986e7i	− 1.08362i	0.840501	−	0.541810i	$-0.182261\pi$
$858$	3.13632e6i	0.145446i
$859$	−2.64808e7	−1.22447	−0.612235	−	0.790676i	$-0.709729\pi$
$859$	−2.64808e7	−1.22447	−0.612235	+	0.790676i	$0.709729\pi$
$860$	0	0
$861$	−4.41806e6	−0.203107
$862$	5.11523e6i	0.234475i
$863$	− 7.80878e6i	− 0.356908i	−0.983948	−	0.178454i	$-0.942890\pi$
$863$	− 7.80878e6i	− 0.356908i	0.983948	−	0.178454i	$-0.0571096\pi$
$864$	−4.20250e6	−0.191524
$865$	0	0
$866$	−1.21892e7	−0.552306
$867$	− 1.56511e7i	− 0.707126i
$868$	6.53184e6i	0.294263i
$869$	6.33992e6	0.284796
$870$	0	0
$871$	1.90858e7	0.852440
$872$	− 3.83936e6i	− 0.170989i
$873$	− 3.72260e6i	− 0.165314i
$874$	−4.15834e6	−0.184137
$875$	0	0
$876$	9.18758e6	0.404521
$877$	2.30969e7i	1.01404i	0.861934	+	0.507020i	$0.169253\pi$
$877$	2.30969e7i	1.01404i	−0.861934	+	0.507020i	$0.830747\pi$
$878$	1.42122e7i	0.622191i
$879$	−1.04067e7	−0.454300
$880$	0	0
$881$	2.48521e7	1.07875	0.539377	−	0.842064i	$-0.318660\pi$
$881$	2.48521e7	1.07875	0.539377	+	0.842064i	$0.318660\pi$
$882$	− 5.50084e6i	− 0.238099i
$883$	− 3.39308e7i	− 1.46451i	−0.681032	−	0.732254i	$-0.738468\pi$
$883$	− 3.39308e7i	− 1.46451i	0.681032	−	0.732254i	$-0.261532\pi$
$884$	2.93760e6	0.126433
$885$	0	0
$886$	−1.55075e7	−0.663680
$887$	− 3.99879e7i	− 1.70655i	−0.521459	−	0.853277i	$-0.674612\pi$
$887$	− 3.99879e7i	− 1.70655i	0.521459	−	0.853277i	$-0.325388\pi$
$888$	− 7.05485e6i	− 0.300231i
$889$	−6.23516e6	−0.264602
$890$	0	0
$891$	3.04811e6	0.128628
$892$	1.33212e7i	0.560570i
$893$	− 2.13438e7i	− 0.895662i
$894$	2.37511e7	0.993893
$895$	0	0
$896$	−884736.	−0.0368166
$897$	− 7.07616e6i	− 0.293641i
$898$	− 9.17681e6i	− 0.379753i
$899$	−468720.	−0.0193426
$900$	0	0
$901$	6.68236e6	0.274232
$902$	− 3.29991e6i	− 0.135047i
$903$	8.62488e6i	0.351993i
$904$	−3.85242e6	−0.156788
$905$	0	0
$906$	−1.29811e6	−0.0525402
$907$	− 4.52230e6i	− 0.182533i	−0.995827	−	0.0912664i	$-0.970909\pi$
$907$	− 4.52230e6i	− 0.182533i	0.995827	−	0.0912664i	$-0.0290915\pi$
$908$	− 1.93120e7i	− 0.777342i
$909$	5.68319e6	0.228130
$910$	0	0
$911$	2.90952e7	1.16151	0.580757	−	0.814077i	$-0.302757\pi$
$911$	2.90952e7	1.16151	0.580757	+	0.814077i	$0.302757\pi$
$912$	− 2.92454e6i	− 0.116432i
$913$	954690.i	0.0379040i
$914$	1.56547e6	0.0619840
$915$	0	0
$916$	−1.64003e7	−0.645822
$917$	1.63564e7i	0.642339i
$918$	− 5.58144e6i	− 0.218595i
$919$	2.51315e7	0.981588	0.490794	−	0.871276i	$-0.336707\pi$
$919$	2.51315e7	0.981588	0.490794	+	0.871276i	$0.336707\pi$
$920$	0	0
$921$	−8.82881e6	−0.342968
$922$	− 3.47926e6i	− 0.134790i
$923$	− 1.23725e7i	− 0.478027i
$924$	1.25453e6	0.0483393
$925$	0	0
$926$	2.30034e7	0.881586
$927$	5.00227e6i	0.191191i
$928$	− 63488.0i	− 0.00242004i
$929$	1.93302e7	0.734849	0.367424	−	0.930053i	$-0.380240\pi$
$929$	1.93302e7	0.734849	0.367424	+	0.930053i	$0.380240\pi$
$930$	0	0
$931$	1.32242e7	0.500030
$932$	1.15649e7i	0.436117i
$933$	− 1.86486e7i	− 0.701361i
$934$	−1.82578e7	−0.684827
$935$	0	0
$936$	−3.42144e6	−0.127650
$937$	3.95931e7i	1.47323i	0.676311	+	0.736616i	$0.263577\pi$
$937$	3.95931e7i	1.47323i	−0.676311	+	0.736616i	$0.736423\pi$
$938$	− 7.63430e6i	− 0.283310i
$939$	−2.45692e7	−0.909342
$940$	0	0
$941$	4.49734e7	1.65570	0.827850	−	0.560949i	$-0.189564\pi$
$941$	4.49734e7	1.65570	0.827850	+	0.560949i	$0.189564\pi$
$942$	1.21159e7i	0.444864i
$943$	7.44526e6i	0.272647i
$944$	−1.23628e7	−0.451529
$945$	0	0
$946$	−6.44204e6	−0.234043
$947$	3.57695e7i	1.29610i	0.761599	+	0.648049i	$0.224415\pi$
$947$	3.57695e7i	1.29610i	−0.761599	+	0.648049i	$0.775585\pi$
$948$	− 1.00600e7i	− 0.363562i
$949$	2.58401e7	0.931383
$950$	0	0
$951$	−1.33279e7	−0.477872
$952$	− 1.17504e6i	− 0.0420204i
$953$	3.83107e7i	1.36643i	0.730217	+	0.683216i	$0.239419\pi$
$953$	3.83107e7i	1.36643i	−0.730217	+	0.683216i	$0.760581\pi$
$954$	−7.78298e6	−0.276870
$955$	0	0
$956$	−824896.	−0.0291914
$957$	90024.0i	0.00317745i
$958$	1.26241e7i	0.444414i
$959$	7.10953e6	0.249629
$960$	0	0
$961$	2.85244e7	0.996343
$962$	− 1.98418e7i	− 0.691262i
$963$	− 7.75665e6i	− 0.269531i
$964$	1.52942e7	0.530073
$965$	0	0
$966$	−2.83046e6	−0.0975921
$967$	− 1.19488e7i	− 0.410920i	−0.978666	−	0.205460i	$-0.934131\pi$
$967$	− 1.19488e7i	− 0.410920i	0.978666	−	0.205460i	$-0.0658690\pi$
$968$	937024.i	0.0321412i
$969$	3.88416e6	0.132889
$970$	0	0
$971$	2.74434e7	0.934091	0.467046	−	0.884233i	$-0.345318\pi$
$971$	2.74434e7	0.934091	0.467046	+	0.884233i	$0.345318\pi$
$972$	1.11197e7i	0.377508i
$973$	− 6.96881e6i	− 0.235981i
$974$	−1.25648e7	−0.424384
$975$	0	0
$976$	4.48768e6	0.150799
$977$	− 2.74269e7i	− 0.919263i	−0.888110	−	0.459631i	$-0.847982\pi$
$977$	− 2.74269e7i	− 0.919263i	0.888110	−	0.459631i	$-0.152018\pi$
$978$	− 1.56348e7i	− 0.522690i
$979$	5.07692e6	0.169295
$980$	0	0
$981$	−5.93901e6	−0.197034
$982$	2.89355e7i	0.957530i
$983$	− 1.32192e7i	− 0.436336i	−0.975911	−	0.218168i	$-0.929992\pi$
$983$	− 1.32192e7i	− 0.436336i	0.975911	−	0.218168i	$-0.0700080\pi$
$984$	−5.23622e6	−0.172397
$985$	0	0
$986$	84320.0	0.00276209
$987$	− 1.45282e7i	− 0.474698i
$988$	− 8.22528e6i	− 0.268076i
$989$	1.45345e7	0.472509
$990$	0	0
$991$	3.20180e7	1.03564	0.517822	−	0.855488i	$-0.326743\pi$
$991$	3.20180e7	1.03564	0.517822	+	0.855488i	$0.326743\pi$
$992$	7.74144e6i	0.249771i
$993$	1.55682e7i	0.501031i
$994$	−4.94899e6	−0.158873
$995$	0	0
$996$	1.51488e6	0.0483871
$997$	4.78223e7i	1.52368i	0.647768	+	0.761838i	$0.275703\pi$
$997$	4.78223e7i	1.52368i	−0.647768	+	0.761838i	$0.724297\pi$
$998$	− 3.13761e7i	− 0.997177i
$999$	−3.76993e7	−1.19514

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	550.6.b.b.199.1		2
5.2	odd	4		550.6.a.d.1.1		1
5.3	odd	4		110.6.a.b.1.1	✓	1
5.4	even	2	inner	550.6.b.b.199.2		2
15.8	even	4		990.6.a.e.1.1		1
20.3	even	4		880.6.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
110.6.a.b.1.1	✓	1	5.3	odd	4
550.6.a.d.1.1		1	5.2	odd	4
550.6.b.b.199.1		2	1.1	even	1	trivial
550.6.b.b.199.2		2	5.4	even	2	inner
880.6.a.b.1.1		1	20.3	even	4
990.6.a.e.1.1		1	15.8	even	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:	\( N \)	\(=\)	\( 550 = 2 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	550.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.00000i q^{2} -12.0000i q^{3} -16.0000 q^{4} -48.0000 q^{6} +54.0000i q^{7} +64.0000i q^{8} +99.0000 q^{9} +O(q^{10})\)	\(q-4.00000i q^{2} -12.0000i q^{3} -16.0000 q^{4} -48.0000 q^{6} +54.0000i q^{7} +64.0000i q^{8} +99.0000 q^{9} -121.000 q^{11} +192.000i q^{12} +540.000i q^{13} +216.000 q^{14} +256.000 q^{16} +340.000i q^{17} -396.000i q^{18} +952.000 q^{19} +648.000 q^{21} +484.000i q^{22} -1092.00i q^{23} +768.000 q^{24} +2160.00 q^{26} -4104.00i q^{27} -864.000i q^{28} +62.0000 q^{29} -7560.00 q^{31} -1024.00i q^{32} +1452.00i q^{33} +1360.00 q^{34} -1584.00 q^{36} -9186.00i q^{37} -3808.00i q^{38} +6480.00 q^{39} -6818.00 q^{41} -2592.00i q^{42} +13310.0i q^{43} +1936.00 q^{44} -4368.00 q^{46} -22420.0i q^{47} -3072.00i q^{48} +13891.0 q^{49} +4080.00 q^{51} -8640.00i q^{52} -19654.0i q^{53} -16416.0 q^{54} -3456.00 q^{56} -11424.0i q^{57} -248.000i q^{58} -48292.0 q^{59} +17530.0 q^{61} +30240.0i q^{62} +5346.00i q^{63} -4096.00 q^{64} +5808.00 q^{66} -35344.0i q^{67} -5440.00i q^{68} -13104.0 q^{69} -22912.0 q^{71} +6336.00i q^{72} -47852.0i q^{73} -36744.0 q^{74} -15232.0 q^{76} -6534.00i q^{77} -25920.0i q^{78} -52396.0 q^{79} -25191.0 q^{81} +27272.0i q^{82} -7890.00i q^{83} -10368.0 q^{84} +53240.0 q^{86} -744.000i q^{87} -7744.00i q^{88} -41958.0 q^{89} -29160.0 q^{91} +17472.0i q^{92} +90720.0i q^{93} -89680.0 q^{94} -12288.0 q^{96} -37602.0i q^{97} -55564.0i q^{98} -11979.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 32 q^{4} - 96 q^{6} + 198 q^{9}+O(q^{10}) \) 2 * q - 32 * q^4 - 96 * q^6 + 198 * q^9	\( 2 q - 32 q^{4} - 96 q^{6} + 198 q^{9} - 242 q^{11} + 432 q^{14} + 512 q^{16} + 1904 q^{19} + 1296 q^{21} + 1536 q^{24} + 4320 q^{26} + 124 q^{29} - 15120 q^{31} + 2720 q^{34} - 3168 q^{36} + 12960 q^{39} - 13636 q^{41} + 3872 q^{44} - 8736 q^{46} + 27782 q^{49} + 8160 q^{51} - 32832 q^{54} - 6912 q^{56} - 96584 q^{59} + 35060 q^{61} - 8192 q^{64} + 11616 q^{66} - 26208 q^{69} - 45824 q^{71} - 73488 q^{74} - 30464 q^{76} - 104792 q^{79} - 50382 q^{81} - 20736 q^{84} + 106480 q^{86} - 83916 q^{89} - 58320 q^{91} - 179360 q^{94} - 24576 q^{96} - 23958 q^{99}+O(q^{100}) \) 2 * q - 32 * q^4 - 96 * q^6 + 198 * q^9 - 242 * q^11 + 432 * q^14 + 512 * q^16 + 1904 * q^19 + 1296 * q^21 + 1536 * q^24 + 4320 * q^26 + 124 * q^29 - 15120 * q^31 + 2720 * q^34 - 3168 * q^36 + 12960 * q^39 - 13636 * q^41 + 3872 * q^44 - 8736 * q^46 + 27782 * q^49 + 8160 * q^51 - 32832 * q^54 - 6912 * q^56 - 96584 * q^59 + 35060 * q^61 - 8192 * q^64 + 11616 * q^66 - 26208 * q^69 - 45824 * q^71 - 73488 * q^74 - 30464 * q^76 - 104792 * q^79 - 50382 * q^81 - 20736 * q^84 + 106480 * q^86 - 83916 * q^89 - 58320 * q^91 - 179360 * q^94 - 24576 * q^96 - 23958 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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