LMFDB - Embedded newform 242.5.b.e.241.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [242,5,Mod(241,242)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(242, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("242.241");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 242 = 2 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	242.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$25.0155310663$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} + 138 x^{14} - 428 x^{13} + 7783 x^{12} - 18620 x^{11} + 235604 x^{10} + \cdots + 1499670491 $ x^16 - 4x^15 + 138x^14 - 428x^13 + 7783x^12 - 18620x^11 + 235604x^10 - 425164x^9 + 4199998x^8 - 5446172x^7 + 45313276x^6 - 38408204x^5 + 290282498x^4 - 132888724x^3 + 1016893202x^2 - 153602264*x + 1499670491
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{16}\cdot 11^{10} $
Twist minimal:	no (minimal twist has level 22)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			241.5
Root			$0.809017 - 5.77971i$ of defining polynomial
Character	$\chi$	$=$	242.241
Dual form			242.5.b.e.241.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843i q^{2} +3.95773 q^{3} -8.00000 q^{4} +43.3108 q^{5} -11.1941i q^{6} -13.2707i q^{7} +22.6274i q^{8} -65.3364 q^{9} +O(q^{10})$	\(q-2.82843i q^{2} +3.95773 q^{3} -8.00000 q^{4} +43.3108 q^{5} -11.1941i q^{6} -13.2707i q^{7} +22.6274i q^{8} -65.3364 q^{9} -122.501i q^{10} -31.6618 q^{12} -179.176i q^{13} -37.5351 q^{14} +171.412 q^{15} +64.0000 q^{16} -176.541i q^{17} +184.799i q^{18} -362.786i q^{19} -346.486 q^{20} -52.5217i q^{21} +564.946 q^{23} +89.5531i q^{24} +1250.82 q^{25} -506.787 q^{26} -579.159 q^{27} +106.165i q^{28} +860.276i q^{29} -484.827i q^{30} +893.611 q^{31} -181.019i q^{32} -499.334 q^{34} -574.763i q^{35} +522.691 q^{36} +203.232 q^{37} -1026.11 q^{38} -709.131i q^{39} +980.011i q^{40} -2620.22i q^{41} -148.554 q^{42} -2603.62i q^{43} -2829.77 q^{45} -1597.91i q^{46} +1546.16 q^{47} +253.295 q^{48} +2224.89 q^{49} -3537.86i q^{50} -698.703i q^{51} +1433.41i q^{52} -1470.96 q^{53} +1638.11i q^{54} +300.281 q^{56} -1435.81i q^{57} +2433.23 q^{58} -4916.25 q^{59} -1371.30 q^{60} +1586.39i q^{61} -2527.51i q^{62} +867.057i q^{63} -512.000 q^{64} -7760.26i q^{65} -3004.99 q^{67} +1412.33i q^{68} +2235.90 q^{69} -1625.67 q^{70} -8497.57 q^{71} -1478.39i q^{72} +4550.26i q^{73} -574.828i q^{74} +4950.41 q^{75} +2902.29i q^{76} -2005.72 q^{78} +2113.32i q^{79} +2771.89 q^{80} +3000.09 q^{81} -7411.09 q^{82} +9457.56i q^{83} +420.173i q^{84} -7646.14i q^{85} -7364.14 q^{86} +3404.74i q^{87} +14557.2 q^{89} +8003.80i q^{90} -2377.79 q^{91} -4519.56 q^{92} +3536.67 q^{93} -4373.21i q^{94} -15712.5i q^{95} -716.425i q^{96} +6786.44 q^{97} -6292.94i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 48 q^{3} - 128 q^{4} + 60 q^{5} - 40 q^{9}+O(q^{10}) $ 16 * q + 48 * q^3 - 128 * q^4 + 60 * q^5 - 40 * q^9	$ 16 q + 48 q^{3} - 128 q^{4} + 60 q^{5} - 40 q^{9} - 384 q^{12} + 384 q^{14} + 1352 q^{15} + 1024 q^{16} - 480 q^{20} - 2424 q^{23} + 5692 q^{25} + 4344 q^{27} - 428 q^{31} - 3904 q^{34} + 320 q^{36} - 228 q^{37} + 1440 q^{38} - 5568 q^{42} + 12152 q^{45} - 9228 q^{47} + 3072 q^{48} - 11956 q^{49} + 12468 q^{53} - 3072 q^{56} + 2944 q^{58} - 22320 q^{59} - 10816 q^{60} - 8192 q^{64} + 21524 q^{67} - 24704 q^{69} + 16960 q^{70} + 8868 q^{71} + 19716 q^{75} - 39424 q^{78} + 3840 q^{80} + 1136 q^{81} - 20992 q^{82} - 35616 q^{86} + 46596 q^{89} - 35600 q^{91} + 19392 q^{92} + 14832 q^{93} + 69448 q^{97}+O(q^{100}) $ 16 * q + 48 * q^3 - 128 * q^4 + 60 * q^5 - 40 * q^9 - 384 * q^12 + 384 * q^14 + 1352 * q^15 + 1024 * q^16 - 480 * q^20 - 2424 * q^23 + 5692 * q^25 + 4344 * q^27 - 428 * q^31 - 3904 * q^34 + 320 * q^36 - 228 * q^37 + 1440 * q^38 - 5568 * q^42 + 12152 * q^45 - 9228 * q^47 + 3072 * q^48 - 11956 * q^49 + 12468 * q^53 - 3072 * q^56 + 2944 * q^58 - 22320 * q^59 - 10816 * q^60 - 8192 * q^64 + 21524 * q^67 - 24704 * q^69 + 16960 * q^70 + 8868 * q^71 + 19716 * q^75 - 39424 * q^78 + 3840 * q^80 + 1136 * q^81 - 20992 * q^82 - 35616 * q^86 + 46596 * q^89 - 35600 * q^91 + 19392 * q^92 + 14832 * q^93 + 69448 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/242\mathbb{Z}\right)^\times$.

$n$	$123$
$\chi(n)$	$-1$

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 $

$2$	− 2.82843i	− 0.707107i
$2$	− 2.82843i	− 0.707107i
$3$	3.95773	0.439747	0.219874	−	0.975528i	$-0.429435\pi$
$3$	3.95773	0.439747	0.219874	+	0.975528i	$0.429435\pi$
$4$	−8.00000	−0.500000
$5$	43.3108	1.73243	0.866215	−	0.499671i	$-0.166546\pi$
$5$	43.3108	1.73243	0.866215	+	0.499671i	$0.166546\pi$
$6$	− 11.1941i	− 0.310948i
$7$	− 13.2707i	− 0.270830i	−0.990789	−	0.135415i	$-0.956763\pi$
$7$	− 13.2707i	− 0.270830i	0.990789	−	0.135415i	$-0.0432367\pi$
$8$	22.6274i	0.353553i
$9$	−65.3364	−0.806622
$10$	− 122.501i	− 1.22501i
$11$	0	0
$11$	0	0
$12$	−31.6618	−0.219874
$13$	− 179.176i	− 1.06021i	−0.847931	−	0.530107i	$-0.822152\pi$
$13$	− 179.176i	− 1.06021i	0.847931	−	0.530107i	$-0.177848\pi$
$14$	−37.5351	−0.191506
$15$	171.412	0.761832
$16$	64.0000	0.250000
$17$	− 176.541i	− 0.610870i	−0.952213	−	0.305435i	$-0.901198\pi$
$17$	− 176.541i	− 0.610870i	0.952213	−	0.305435i	$-0.0988019\pi$
$18$	184.799i	0.570368i
$19$	− 362.786i	− 1.00495i	−0.864593	−	0.502474i	$-0.832423\pi$
$19$	− 362.786i	− 1.00495i	0.864593	−	0.502474i	$-0.167577\pi$
$20$	−346.486	−0.866215
$21$	− 52.5217i	− 0.119097i
$22$	0	0
$23$	564.946	1.06795	0.533975	−	0.845500i	$-0.320698\pi$
$23$	564.946	1.06795	0.533975	+	0.845500i	$0.320698\pi$
$24$	89.5531i	0.155474i
$25$	1250.82	2.00132
$26$	−506.787	−0.749685
$27$	−579.159	−0.794457
$28$	106.165i	0.135415i
$29$	860.276i	1.02292i	0.859307	+	0.511460i	$0.170895\pi$
$29$	860.276i	1.02292i	−0.859307	+	0.511460i	$0.829105\pi$
$30$	− 484.827i	− 0.538696i
$31$	893.611	0.929876	0.464938	−	0.885343i	$-0.346077\pi$
$31$	893.611	0.929876	0.464938	+	0.885343i	$0.346077\pi$
$32$	− 181.019i	− 0.176777i
$33$	0	0
$34$	−499.334	−0.431950
$35$	− 574.763i	− 0.469194i
$36$	522.691	0.403311
$37$	203.232	0.148453	0.0742265	−	0.997241i	$-0.476351\pi$
$37$	203.232	0.148453	0.0742265	+	0.997241i	$0.476351\pi$
$38$	−1026.11	−0.710605
$39$	− 709.131i	− 0.466227i
$40$	980.011i	0.612507i
$41$	− 2620.22i	− 1.55872i	−0.626573	−	0.779362i	$-0.715543\pi$
$41$	− 2620.22i	− 1.55872i	0.626573	−	0.779362i	$-0.284457\pi$
$42$	−148.554	−0.0842141
$43$	− 2603.62i	− 1.40812i	−0.710140	−	0.704061i	$-0.751368\pi$
$43$	− 2603.62i	− 1.40812i	0.710140	−	0.704061i	$-0.248632\pi$
$44$	0	0
$45$	−2829.77	−1.39742
$46$	− 1597.91i	− 0.755155i
$47$	1546.16	0.699938	0.349969	−	0.936761i	$-0.386192\pi$
$47$	1546.16	0.699938	0.349969	+	0.936761i	$0.386192\pi$
$48$	253.295	0.109937
$49$	2224.89	0.926651
$50$	− 3537.86i	− 1.41514i
$51$	− 698.703i	− 0.268628i
$52$	1433.41i	0.530107i
$53$	−1470.96	−0.523659	−0.261830	−	0.965114i	$-0.584326\pi$
$53$	−1470.96	−0.523659	−0.261830	+	0.965114i	$0.584326\pi$
$54$	1638.11i	0.561766i
$55$	0	0
$56$	300.281	0.0957528
$57$	− 1435.81i	− 0.441923i
$58$	2433.23	0.723314
$59$	−4916.25	−1.41231	−0.706155	−	0.708057i	$-0.749572\pi$
$59$	−4916.25	−1.41231	−0.706155	+	0.708057i	$0.749572\pi$
$60$	−1371.30	−0.380916
$61$	1586.39i	0.426334i	0.977016	+	0.213167i	$0.0683779\pi$
$61$	1586.39i	0.426334i	−0.977016	+	0.213167i	$0.931622\pi$
$62$	− 2527.51i	− 0.657522i
$63$	867.057i	0.218457i
$64$	−512.000	−0.125000
$65$	− 7760.26i	− 1.83675i
$66$	0	0
$67$	−3004.99	−0.669413	−0.334706	−	0.942323i	$-0.608637\pi$
$67$	−3004.99	−0.669413	−0.334706	+	0.942323i	$0.608637\pi$
$68$	1412.33i	0.305435i
$69$	2235.90	0.469628
$70$	−1625.67	−0.331770
$71$	−8497.57	−1.68569	−0.842846	−	0.538155i	$-0.819121\pi$
$71$	−8497.57	−1.68569	−0.842846	+	0.538155i	$0.819121\pi$
$72$	− 1478.39i	− 0.285184i
$73$	4550.26i	0.853868i	0.904283	+	0.426934i	$0.140406\pi$
$73$	4550.26i	0.853868i	−0.904283	+	0.426934i	$0.859594\pi$
$74$	− 574.828i	− 0.104972i
$75$	4950.41	0.880073
$76$	2902.29i	0.502474i
$77$	0	0
$78$	−2005.72	−0.329672
$79$	2113.32i	0.338618i	0.985563	+	0.169309i	$0.0541536\pi$
$79$	2113.32i	0.338618i	−0.985563	+	0.169309i	$0.945846\pi$
$80$	2771.89	0.433108
$81$	3000.09	0.457262
$82$	−7411.09	−1.10219
$83$	9457.56i	1.37285i	0.727201	+	0.686424i	$0.240821\pi$
$83$	9457.56i	1.37285i	−0.727201	+	0.686424i	$0.759179\pi$
$84$	420.173i	0.0595484i
$85$	− 7646.14i	− 1.05829i
$86$	−7364.14	−0.995693
$87$	3404.74i	0.449827i
$88$	0	0
$89$	14557.2	1.83779	0.918896	−	0.394499i	$-0.129082\pi$
$89$	14557.2	1.83779	0.918896	+	0.394499i	$0.129082\pi$
$90$	8003.80i	0.988123i
$91$	−2377.79	−0.287138
$92$	−4519.56	−0.533975
$93$	3536.67	0.408911
$94$	− 4373.21i	− 0.494931i
$95$	− 15712.5i	− 1.74100i
$96$	− 716.425i	− 0.0777371i
$97$	6786.44	0.721271	0.360635	−	0.932707i	$-0.382560\pi$
$97$	6786.44	0.721271	0.360635	+	0.932707i	$0.382560\pi$
$98$	− 6292.94i	− 0.655241i
$99$	0	0
$100$	−10006.6	−1.00066
$101$	− 13666.1i	− 1.33968i	−0.742504	−	0.669842i	$-0.766362\pi$
$101$	− 13666.1i	− 1.33968i	0.742504	−	0.669842i	$-0.233638\pi$
$102$	−1976.23	−0.189949
$103$	1230.09	0.115948	0.0579739	−	0.998318i	$-0.481536\pi$
$103$	1230.09	0.115948	0.0579739	+	0.998318i	$0.481536\pi$
$104$	4054.30	0.374842
$105$	− 2274.75i	− 0.206327i
$106$	4160.50i	0.370283i
$107$	8463.30i	0.739218i	0.929187	+	0.369609i	$0.120508\pi$
$107$	8463.30i	0.739218i	−0.929187	+	0.369609i	$0.879492\pi$
$108$	4633.28	0.397229
$109$	7648.65i	0.643771i	0.946779	+	0.321886i	$0.104317\pi$
$109$	7648.65i	0.643771i	−0.946779	+	0.321886i	$0.895683\pi$
$110$	0	0
$111$	804.338	0.0652819
$112$	− 849.322i	− 0.0677075i
$113$	−9079.67	−0.711071	−0.355536	−	0.934663i	$-0.615702\pi$
$113$	−9079.67	−0.711071	−0.355536	+	0.934663i	$0.615702\pi$
$114$	−4061.08	−0.312487
$115$	24468.2	1.85015
$116$	− 6882.21i	− 0.511460i
$117$	11706.7i	0.855192i
$118$	13905.3i	0.998654i
$119$	−2342.82	−0.165442
$120$	3878.61i	0.269348i
$121$	0	0
$122$	4486.99	0.301464
$123$	− 10370.1i	− 0.685445i
$124$	−7148.89	−0.464938
$125$	27104.8	1.73471
$126$	2452.41	0.154473
$127$	6887.90i	0.427051i	0.976938	+	0.213525i	$0.0684946\pi$
$127$	6887.90i	0.427051i	−0.976938	+	0.213525i	$0.931505\pi$
$128$	1448.15i	0.0883883i
$129$	− 10304.4i	− 0.619218i
$130$	−21949.3	−1.29878
$131$	− 1471.01i	− 0.0857182i	−0.999081	−	0.0428591i	$-0.986353\pi$
$131$	− 1471.01i	− 0.0857182i	0.999081	−	0.0428591i	$-0.0136467\pi$
$132$	0	0
$133$	−4814.41	−0.272170
$134$	8499.40i	0.473346i
$135$	−25083.8	−1.37634
$136$	3994.68	0.215975
$137$	18123.8	0.965622	0.482811	−	0.875725i	$-0.339616\pi$
$137$	18123.8	0.965622	0.482811	+	0.875725i	$0.339616\pi$
$138$	− 6324.08i	− 0.332077i
$139$	10181.4i	0.526963i	0.964664	+	0.263481i	$0.0848708\pi$
$139$	10181.4i	0.526963i	−0.964664	+	0.263481i	$0.915129\pi$
$140$	4598.10i	0.234597i
$141$	6119.29	0.307796
$142$	24034.8i	1.19196i
$143$	0	0
$144$	−4181.53	−0.201656
$145$	37259.2i	1.77214i
$146$	12870.1	0.603776
$147$	8805.50	0.407492
$148$	−1625.86	−0.0742265
$149$	26550.5i	1.19592i	0.801528	+	0.597958i	$0.204021\pi$
$149$	26550.5i	1.19592i	−0.801528	+	0.597958i	$0.795979\pi$
$150$	− 14001.9i	− 0.622306i
$151$	18988.5i	0.832793i	0.909183	+	0.416397i	$0.136707\pi$
$151$	18988.5i	0.832793i	−0.909183	+	0.416397i	$0.863293\pi$
$152$	8208.91	0.355303
$153$	11534.6i	0.492741i
$154$	0	0
$155$	38703.0	1.61095
$156$	5673.04i	0.233113i
$157$	−3405.68	−0.138167	−0.0690834	−	0.997611i	$-0.522007\pi$
$157$	−3405.68	−0.138167	−0.0690834	+	0.997611i	$0.522007\pi$
$158$	5977.36	0.239439
$159$	−5821.65	−0.230278
$160$	− 7840.09i	− 0.306253i
$161$	− 7497.20i	− 0.289233i
$162$	− 8485.55i	− 0.323333i
$163$	8519.83	0.320668	0.160334	−	0.987063i	$-0.448743\pi$
$163$	8519.83	0.320668	0.160334	+	0.987063i	$0.448743\pi$
$164$	20961.7i	0.779362i
$165$	0	0
$166$	26750.0	0.970751
$167$	13260.3i	0.475467i	0.971330	+	0.237733i	$0.0764044\pi$
$167$	13260.3i	0.475467i	−0.971330	+	0.237733i	$0.923596\pi$
$168$	1188.43	0.0421071
$169$	−3543.12	−0.124055
$170$	−21626.6	−0.748324
$171$	23703.1i	0.810613i
$172$	20828.9i	0.704061i
$173$	22090.3i	0.738091i	0.929411	+	0.369045i	$0.120315\pi$
$173$	22090.3i	0.738091i	−0.929411	+	0.369045i	$0.879685\pi$
$174$	9630.05	0.318075
$175$	− 16599.2i	− 0.542016i
$176$	0	0
$177$	−19457.2	−0.621060
$178$	− 41173.9i	− 1.29952i
$179$	−8356.88	−0.260818	−0.130409	−	0.991460i	$-0.541629\pi$
$179$	−8356.88	−0.260818	−0.130409	+	0.991460i	$0.541629\pi$
$180$	22638.2	0.698708
$181$	10070.0	0.307377	0.153689	−	0.988119i	$-0.450885\pi$
$181$	10070.0	0.307377	0.153689	+	0.988119i	$0.450885\pi$
$182$	6725.40i	0.203037i
$183$	6278.50i	0.187479i
$184$	12783.3i	0.377577i
$185$	8802.14	0.257185
$186$	− 10003.2i	− 0.289144i
$187$	0	0
$188$	−12369.3	−0.349969
$189$	7685.83i	0.215163i
$190$	−44441.8	−1.23107
$191$	−35313.9	−0.968007	−0.484003	−	0.875066i	$-0.660818\pi$
$191$	−35313.9	−0.968007	−0.484003	+	0.875066i	$0.660818\pi$
$192$	−2026.36	−0.0549684
$193$	7824.13i	0.210049i	0.994470	+	0.105025i	$0.0334922\pi$
$193$	7824.13i	0.210049i	−0.994470	+	0.105025i	$0.966508\pi$
$194$	− 19194.9i	− 0.510015i
$195$	− 30713.0i	− 0.807705i
$196$	−17799.1	−0.463326
$197$	− 1473.23i	− 0.0379610i	−0.999820	−	0.0189805i	$-0.993958\pi$
$197$	− 1473.23i	− 0.0379610i	0.999820	−	0.0189805i	$-0.00604205\pi$
$198$	0	0
$199$	−26096.6	−0.658987	−0.329494	−	0.944158i	$-0.606878\pi$
$199$	−26096.6	−0.658987	−0.329494	+	0.944158i	$0.606878\pi$
$200$	28302.9i	0.707572i
$201$	−11892.9	−0.294372
$202$	−38653.6	−0.947299
$203$	11416.4	0.277037
$204$	5589.62i	0.134314i
$205$	− 113484.i	− 2.70038i
$206$	− 3479.22i	− 0.0819875i
$207$	−36911.5	−0.861432
$208$	− 11467.3i	− 0.265054i
$209$	0	0
$210$	−6433.97	−0.145895
$211$	− 72402.6i	− 1.62626i	−0.582083	−	0.813129i	$-0.697762\pi$
$211$	− 72402.6i	− 1.62626i	0.582083	−	0.813129i	$-0.302238\pi$
$212$	11767.7	0.261830
$213$	−33631.1	−0.741279
$214$	23937.8	0.522706
$215$	− 112765.i	− 2.43947i
$216$	− 13104.9i	− 0.280883i
$217$	− 11858.8i	− 0.251838i
$218$	21633.6	0.455215
$219$	18008.7i	0.375486i
$220$	0	0
$221$	−31632.0	−0.647653
$222$	− 2275.01i	− 0.0461612i
$223$	28290.6	0.568895	0.284448	−	0.958692i	$-0.408190\pi$
$223$	28290.6	0.568895	0.284448	+	0.958692i	$0.408190\pi$
$224$	−2402.25	−0.0478764
$225$	−81724.2	−1.61431
$226$	25681.2i	0.502803i
$227$	72054.2i	1.39832i	0.714964	+	0.699161i	$0.246443\pi$
$227$	72054.2i	1.39832i	−0.714964	+	0.699161i	$0.753557\pi$
$228$	11486.5i	0.220961i
$229$	8938.31	0.170445	0.0852226	−	0.996362i	$-0.472840\pi$
$229$	8938.31	0.170445	0.0852226	+	0.996362i	$0.472840\pi$
$230$	− 69206.6i	− 1.30825i
$231$	0	0
$232$	−19465.8	−0.361657
$233$	48416.6i	0.891831i	0.895075	+	0.445915i	$0.147122\pi$
$233$	48416.6i	0.891831i	−0.895075	+	0.445915i	$0.852878\pi$
$234$	33111.6	0.604712
$235$	66965.5	1.21259
$236$	39330.0	0.706155
$237$	8363.93i	0.148906i
$238$	6626.50i	0.116985i
$239$	54155.2i	0.948078i	0.880504	+	0.474039i	$0.157204\pi$
$239$	54155.2i	0.948078i	−0.880504	+	0.474039i	$0.842796\pi$
$240$	10970.4	0.190458
$241$	− 54166.9i	− 0.932610i	−0.884624	−	0.466305i	$-0.845585\pi$
$241$	− 54166.9i	− 0.932610i	0.884624	−	0.466305i	$-0.154415\pi$
$242$	0	0
$243$	58785.5	0.995537
$244$	− 12691.1i	− 0.213167i
$245$	96361.7	1.60536
$246$	−29331.1	−0.484683
$247$	−65002.6	−1.06546
$248$	20220.1i	0.328761i
$249$	37430.4i	0.603707i
$250$	− 76664.0i	− 1.22662i
$251$	14291.8	0.226851	0.113425	−	0.993547i	$-0.463818\pi$
$251$	14291.8	0.226851	0.113425	+	0.993547i	$0.463818\pi$
$252$	− 6936.46i	− 0.109229i
$253$	0	0
$254$	19481.9	0.301970
$255$	− 30261.3i	− 0.465380i
$256$	4096.00	0.0625000
$257$	−38483.3	−0.582648	−0.291324	−	0.956624i	$-0.594096\pi$
$257$	−38483.3	−0.582648	−0.291324	+	0.956624i	$0.594096\pi$
$258$	−29145.3	−0.437853
$259$	− 2697.03i	− 0.0402055i
$260$	62082.1i	0.918374i
$261$	− 56207.3i	− 0.825110i
$262$	−4160.65	−0.0606119
$263$	130054.i	1.88024i	0.340841	+	0.940121i	$0.389288\pi$
$263$	130054.i	1.88024i	−0.340841	+	0.940121i	$0.610712\pi$
$264$	0	0
$265$	−63708.4	−0.907203
$266$	13617.2i	0.192453i
$267$	57613.2	0.808164
$268$	24039.9	0.334706
$269$	−70648.7	−0.976337	−0.488169	−	0.872749i	$-0.662335\pi$
$269$	−70648.7	−0.976337	−0.488169	+	0.872749i	$0.662335\pi$
$270$	70947.8i	0.973221i
$271$	79735.1i	1.08570i	0.839829	+	0.542851i	$0.182655\pi$
$271$	79735.1i	1.08570i	−0.839829	+	0.542851i	$0.817345\pi$
$272$	− 11298.6i	− 0.152717i
$273$	−9410.63	−0.126268
$274$	− 51261.7i	− 0.682798i
$275$	0	0
$276$	−17887.2	−0.234814
$277$	44311.8i	0.577511i	0.957403	+	0.288755i	$0.0932414\pi$
$277$	44311.8i	0.577511i	−0.957403	+	0.288755i	$0.906759\pi$
$278$	28797.5	0.372619
$279$	−58385.3	−0.750059
$280$	13005.4	0.165885
$281$	− 78270.6i	− 0.991256i	−0.868535	−	0.495628i	$-0.834938\pi$
$281$	− 78270.6i	− 0.991256i	0.868535	−	0.495628i	$-0.165062\pi$
$282$	− 17308.0i	− 0.217645i
$283$	− 89798.3i	− 1.12123i	−0.828076	−	0.560616i	$-0.810564\pi$
$283$	− 89798.3i	− 1.12123i	0.828076	−	0.560616i	$-0.189436\pi$
$284$	67980.6	0.842846
$285$	− 62185.9i	− 0.765601i
$286$	0	0
$287$	−34772.0	−0.422149
$288$	11827.2i	0.142592i
$289$	52354.1	0.626838
$290$	105385.	1.25309
$291$	26858.9	0.317177
$292$	− 36402.1i	− 0.426934i
$293$	20515.8i	0.238976i	0.992836	+	0.119488i	$0.0381252\pi$
$293$	20515.8i	0.238976i	−0.992836	+	0.119488i	$0.961875\pi$
$294$	− 24905.7i	− 0.288141i
$295$	−212927.	−2.44673
$296$	4598.62i	0.0524861i
$297$	0	0
$298$	75096.2	0.845640
$299$	− 101225.i	− 1.13226i
$300$	−39603.3	−0.440037
$301$	−34551.7	−0.381361
$302$	53707.6	0.588874
$303$	− 54086.7i	− 0.589122i
$304$	− 23218.3i	− 0.251237i
$305$	68707.7i	0.738594i
$306$	32624.7	0.348421
$307$	− 133825.i	− 1.41991i	−0.704248	−	0.709954i	$-0.748716\pi$
$307$	− 133825.i	− 1.41991i	0.704248	−	0.709954i	$-0.251284\pi$
$308$	0	0
$309$	4868.36	0.0509878
$310$	− 109469.i	− 1.13911i
$311$	42282.8	0.437162	0.218581	−	0.975819i	$-0.429857\pi$
$311$	42282.8	0.437162	0.218581	+	0.975819i	$0.429857\pi$
$312$	16045.8	0.164836
$313$	172787.	1.76369	0.881846	−	0.471538i	$-0.156301\pi$
$313$	172787.	1.76369	0.881846	+	0.471538i	$0.156301\pi$
$314$	9632.70i	0.0976987i
$315$	37552.9i	0.378462i
$316$	− 16906.5i	− 0.169309i
$317$	84331.1	0.839208	0.419604	−	0.907707i	$-0.362169\pi$
$317$	84331.1	0.839208	0.419604	+	0.907707i	$0.362169\pi$
$318$	16466.1i	0.162831i
$319$	0	0
$320$	−22175.1	−0.216554
$321$	33495.4i	0.325069i
$322$	−21205.3	−0.204518
$323$	−64046.7	−0.613892
$324$	−24000.7	−0.228631
$325$	− 224118.i	− 2.12182i
$326$	− 24097.7i	− 0.226746i
$327$	30271.3i	0.283097i
$328$	59288.7	0.551093
$329$	− 20518.6i	− 0.189564i
$330$	0	0
$331$	−130067.	−1.18717	−0.593583	−	0.804773i	$-0.702287\pi$
$331$	−130067.	−1.18717	−0.593583	+	0.804773i	$0.702287\pi$
$332$	− 75660.4i	− 0.686424i
$333$	−13278.5	−0.119746
$334$	37505.8	0.336206
$335$	−130149.	−1.15971
$336$	− 3361.39i	− 0.0297742i
$337$	− 81806.9i	− 0.720327i	−0.932889	−	0.360164i	$-0.882721\pi$
$337$	− 81806.9i	− 0.720327i	0.932889	−	0.360164i	$-0.117279\pi$
$338$	10021.5i	0.0877198i
$339$	−35934.8	−0.312692
$340$	61169.1i	0.529145i
$341$	0	0
$342$	67042.6	0.573190
$343$	− 61388.6i	− 0.521795i
$344$	58913.1	0.497846
$345$	96838.5	0.813598
$346$	62480.8	0.521909
$347$	231071.i	1.91905i	0.281622	+	0.959526i	$0.409128\pi$
$347$	231071.i	1.91905i	−0.281622	+	0.959526i	$0.590872\pi$
$348$	− 27237.9i	− 0.224913i
$349$	12937.7i	0.106220i	0.998589	+	0.0531102i	$0.0169134\pi$
$349$	12937.7i	0.106220i	−0.998589	+	0.0531102i	$0.983087\pi$
$350$	−46949.7	−0.383263
$351$	103772.i	0.842295i
$352$	0	0
$353$	59097.3	0.474262	0.237131	−	0.971478i	$-0.423793\pi$
$353$	59097.3	0.474262	0.237131	+	0.971478i	$0.423793\pi$
$354$	55033.2i	0.439156i
$355$	−368036.	−2.92034
$356$	−116457.	−0.918896
$357$	−9272.25	−0.0727526
$358$	23636.8i	0.184426i
$359$	− 55851.2i	− 0.433355i	−0.976243	−	0.216677i	$-0.930478\pi$
$359$	− 55851.2i	− 0.433355i	0.976243	−	0.216677i	$-0.0695220\pi$
$360$	− 64030.4i	− 0.494061i
$361$	−1292.66	−0.00991908
$362$	− 28482.2i	− 0.217349i
$363$	0	0
$364$	19022.3	0.143569
$365$	197075.i	1.47927i
$366$	17758.3	0.132568
$367$	−158293.	−1.17525	−0.587624	−	0.809134i	$-0.699937\pi$
$367$	−158293.	−1.17525	−0.587624	+	0.809134i	$0.699937\pi$
$368$	36156.5	0.266987
$369$	171196.i	1.25730i
$370$	− 24896.2i	− 0.181857i
$371$	19520.6i	0.141823i
$372$	−28293.3	−0.204455
$373$	133744.i	0.961297i	0.876913	+	0.480649i	$0.159599\pi$
$373$	133744.i	0.961297i	−0.876913	+	0.480649i	$0.840401\pi$
$374$	0	0
$375$	107274.	0.762834
$376$	34985.7i	0.247466i
$377$	154141.	1.08451
$378$	21738.8	0.152143
$379$	−53897.4	−0.375223	−0.187611	−	0.982243i	$-0.560075\pi$
$379$	−53897.4	−0.375223	−0.187611	+	0.982243i	$0.560075\pi$
$380$	125700.i	0.870501i
$381$	27260.4i	0.187794i
$382$	99882.7i	0.684484i
$383$	48838.4	0.332939	0.166469	−	0.986047i	$-0.446763\pi$
$383$	48838.4	0.332939	0.166469	+	0.986047i	$0.446763\pi$
$384$	5731.40i	0.0388685i
$385$	0	0
$386$	22130.0	0.148527
$387$	170111.i	1.13582i
$388$	−54291.5	−0.360635
$389$	274094.	1.81134	0.905670	−	0.423984i	$-0.139369\pi$
$389$	274094.	1.81134	0.905670	+	0.423984i	$0.139369\pi$
$390$	−86869.4	−0.571134
$391$	− 99736.3i	− 0.652378i
$392$	50343.5i	0.327621i
$393$	− 5821.86i	− 0.0376944i
$394$	−4166.92	−0.0268425
$395$	91529.3i	0.586632i
$396$	0	0
$397$	49313.3	0.312884	0.156442	−	0.987687i	$-0.449998\pi$
$397$	49313.3	0.312884	0.156442	+	0.987687i	$0.449998\pi$
$398$	73812.2i	0.465974i
$399$	−19054.1	−0.119686
$400$	80052.6	0.500329
$401$	7577.67	0.0471245	0.0235623	−	0.999722i	$-0.492499\pi$
$401$	7577.67	0.0471245	0.0235623	+	0.999722i	$0.492499\pi$
$402$	33638.3i	0.208153i
$403$	− 160114.i	− 0.985868i
$404$	109329.i	0.669842i
$405$	129936.	0.792174
$406$	− 32290.6i	− 0.195895i
$407$	0	0
$408$	15809.8	0.0949745
$409$	− 14430.3i	− 0.0862637i	−0.999069	−	0.0431319i	$-0.986266\pi$
$409$	− 14430.3i	− 0.0862637i	0.999069	−	0.0431319i	$-0.0137336\pi$
$410$	−320980.	−1.90946
$411$	71728.9	0.424630
$412$	−9840.73	−0.0579739
$413$	65241.9i	0.382496i
$414$	104402.i	0.609125i
$415$	409614.i	2.37837i
$416$	−32434.4	−0.187421
$417$	40295.4i	0.231731i
$418$	0	0
$419$	229500.	1.30724	0.653619	−	0.756824i	$-0.273250\pi$
$419$	229500.	1.30724	0.653619	+	0.756824i	$0.273250\pi$
$420$	18198.0i	0.103163i
$421$	12180.4	0.0687222	0.0343611	−	0.999409i	$-0.489060\pi$
$421$	12180.4	0.0687222	0.0343611	+	0.999409i	$0.489060\pi$
$422$	−204786.	−1.14994
$423$	−101021.	−0.564586
$424$	− 33284.0i	− 0.185142i
$425$	− 220822.i	− 1.22254i
$426$	95123.0i	0.524163i
$427$	21052.4	0.115464
$428$	− 67706.4i	− 0.369609i
$429$	0	0
$430$	−318947.	−1.72497
$431$	− 42248.5i	− 0.227435i	−0.993513	−	0.113717i	$-0.963724\pi$
$431$	− 42248.5i	− 0.227435i	0.993513	−	0.113717i	$-0.0362759\pi$
$432$	−37066.2	−0.198614
$433$	66292.6	0.353581	0.176791	−	0.984248i	$-0.443428\pi$
$433$	66292.6	0.353581	0.176791	+	0.984248i	$0.443428\pi$
$434$	−33541.8	−0.178077
$435$	147462.i	0.779293i
$436$	− 61189.2i	− 0.321886i
$437$	− 204954.i	− 1.07323i
$438$	50936.3	0.265509
$439$	212489.i	1.10257i	0.834316	+	0.551287i	$0.185863\pi$
$439$	212489.i	1.10257i	−0.834316	+	0.551287i	$0.814137\pi$
$440$	0	0
$441$	−145366.	−0.747457
$442$	89468.9i	0.457960i
$443$	−357555.	−1.82195	−0.910974	−	0.412465i	$-0.864668\pi$
$443$	−357555.	−1.82195	−0.910974	+	0.412465i	$0.864668\pi$
$444$	−6434.70	−0.0326409
$445$	630481.	3.18385
$446$	− 80017.9i	− 0.402270i
$447$	105080.i	0.525901i
$448$	6794.58i	0.0338537i
$449$	36147.2	0.179301	0.0896503	−	0.995973i	$-0.471425\pi$
$449$	36147.2	0.179301	0.0896503	+	0.995973i	$0.471425\pi$
$450$	231151.i	1.14149i
$451$	0	0
$452$	72637.3	0.355536
$453$	75151.3i	0.366219i
$454$	203800.	0.988764
$455$	−102984.	−0.497446
$456$	32488.6	0.156243
$457$	− 24559.6i	− 0.117595i	−0.998270	−	0.0587976i	$-0.981273\pi$
$457$	− 24559.6i	− 0.117595i	0.998270	−	0.0587976i	$-0.0187266\pi$
$458$	− 25281.4i	− 0.120523i
$459$	102246.i	0.485310i
$460$	−195746.	−0.925075
$461$	− 41038.8i	− 0.193105i	−0.995328	−	0.0965523i	$-0.969218\pi$
$461$	− 41038.8i	− 0.193105i	0.995328	−	0.0965523i	$-0.0307815\pi$
$462$	0	0
$463$	−263948.	−1.23128	−0.615640	−	0.788027i	$-0.711103\pi$
$463$	−263948.	−1.23128	−0.615640	+	0.788027i	$0.711103\pi$
$464$	55057.7i	0.255730i
$465$	153176.	0.708409
$466$	136943.	0.630620
$467$	−22661.1	−0.103907	−0.0519537	−	0.998649i	$-0.516545\pi$
$467$	−22661.1	−0.103907	−0.0519537	+	0.998649i	$0.516545\pi$
$468$	− 93653.8i	− 0.427596i
$469$	39878.2i	0.181297i
$470$	− 189407.i	− 0.857434i
$471$	−13478.7	−0.0607585
$472$	− 111242.i	− 0.499327i
$473$	0	0
$474$	23656.8	0.105293
$475$	− 453781.i	− 2.01122i
$476$	18742.6	0.0827209
$477$	96107.2	0.422395
$478$	153174.	0.670393
$479$	− 66459.2i	− 0.289657i	−0.989457	−	0.144828i	$-0.953737\pi$
$479$	− 66459.2i	− 0.289657i	0.989457	−	0.144828i	$-0.0462630\pi$
$480$	− 31028.9i	− 0.134674i
$481$	− 36414.4i	− 0.157392i
$482$	−153207.	−0.659455
$483$	− 29671.9i	− 0.127189i
$484$	0	0
$485$	293926.	1.24955
$486$	− 166270.i	− 0.703951i
$487$	−407424.	−1.71786	−0.858931	−	0.512091i	$-0.828871\pi$
$487$	−407424.	−1.71786	−0.858931	+	0.512091i	$0.828871\pi$
$488$	−35895.9	−0.150732
$489$	33719.1	0.141013
$490$	− 272552.i	− 1.13516i
$491$	35332.2i	0.146557i	0.997312	+	0.0732786i	$0.0233462\pi$
$491$	35332.2i	0.146557i	−0.997312	+	0.0732786i	$0.976654\pi$
$492$	82960.8i	0.342723i
$493$	151874.	0.624871
$494$	183855.i	0.753394i
$495$	0	0
$496$	57191.1	0.232469
$497$	112768.i	0.456536i
$498$	105869.	0.426885
$499$	30656.5	0.123118	0.0615591	−	0.998103i	$-0.480393\pi$
$499$	30656.5	0.123118	0.0615591	+	0.998103i	$0.480393\pi$
$500$	−216839.	−0.867355
$501$	52480.6i	0.209085i
$502$	− 40423.4i	− 0.160408i
$503$	− 279687.i	− 1.10544i	−0.833366	−	0.552722i	$-0.813589\pi$
$503$	− 279687.i	− 1.10544i	0.833366	−	0.552722i	$-0.186411\pi$
$504$	−19619.3	−0.0772363
$505$	− 591890.i	− 2.32091i
$506$	0	0
$507$	−14022.7	−0.0545527
$508$	− 55103.2i	− 0.213525i
$509$	151802.	0.585925	0.292963	−	0.956124i	$-0.405359\pi$
$509$	151802.	0.585925	0.292963	+	0.956124i	$0.405359\pi$
$510$	−85592.0	−0.329073
$511$	60385.0	0.231253
$512$	− 11585.2i	− 0.0441942i
$513$	210111.i	0.798388i
$514$	108847.i	0.411994i
$515$	53276.2	0.200872
$516$	82435.2i	0.309609i
$517$	0	0
$518$	−7628.34	−0.0284296
$519$	87427.4i	0.324573i
$520$	175595.	0.649388
$521$	−2088.46	−0.00769397	−0.00384698	−	0.999993i	$-0.501225\pi$
$521$	−2088.46	−0.00769397	−0.00384698	+	0.999993i	$0.501225\pi$
$522$	−158978.	−0.583441
$523$	305638.i	1.11739i	0.829374	+	0.558695i	$0.188698\pi$
$523$	305638.i	1.11739i	−0.829374	+	0.558695i	$0.811302\pi$
$524$	11768.1i	0.0428591i
$525$	− 65695.2i	− 0.238350i
$526$	367850.	1.32953
$527$	− 157759.i	− 0.568033i
$528$	0	0
$529$	39322.4	0.140517
$530$	180194.i	0.641490i
$531$	321210.	1.13920
$532$	38515.3	0.136085
$533$	−469481.	−1.65258
$534$	− 162955.i	− 0.571459i
$535$	366552.i	1.28064i
$536$	− 67995.2i	− 0.236673i
$537$	−33074.2	−0.114694
$538$	199825.i	0.690375i
$539$	0	0
$540$	200671.	0.688171
$541$	− 126541.i	− 0.432353i	−0.976354	−	0.216176i	$-0.930641\pi$
$541$	− 126541.i	− 0.432353i	0.976354	−	0.216176i	$-0.0693586\pi$
$542$	225525.	0.767707
$543$	39854.3	0.135168
$544$	−31957.4	−0.107988
$545$	331269.i	1.11529i
$546$	26617.3i	0.0892850i
$547$	− 354275.i	− 1.18404i	−0.805923	−	0.592020i	$-0.798331\pi$
$547$	− 354275.i	− 1.18404i	0.805923	−	0.592020i	$-0.201669\pi$
$548$	−144990.	−0.482811
$549$	− 103649.i	− 0.343891i
$550$	0	0
$551$	312096.	1.02798
$552$	50592.6i	0.166039i
$553$	28045.1	0.0917079
$554$	125333.	0.408362
$555$	34836.5	0.113096
$556$	− 81451.6i	− 0.263481i
$557$	30932.9i	0.0997034i	0.998757	+	0.0498517i	$0.0158749\pi$
$557$	30932.9i	0.0997034i	−0.998757	+	0.0498517i	$0.984125\pi$
$558$	165139.i	0.530372i
$559$	−466506.	−1.49291
$560$	− 36784.8i	− 0.117298i
$561$	0	0
$562$	−221383.	−0.700924
$563$	− 355669.i	− 1.12210i	−0.827784	−	0.561048i	$-0.810398\pi$
$563$	− 355669.i	− 1.12210i	0.827784	−	0.561048i	$-0.189602\pi$
$564$	−48954.3	−0.153898
$565$	−393247.	−1.23188
$566$	−253988.	−0.792830
$567$	− 39813.2i	− 0.123840i
$568$	− 192278.i	− 0.595982i
$569$	− 437475.i	− 1.35123i	−0.737256	−	0.675614i	$-0.763879\pi$
$569$	− 437475.i	− 1.35123i	0.737256	−	0.675614i	$-0.236121\pi$
$570$	−175888.	−0.541362
$571$	− 1830.95i	− 0.00561570i	−0.999996	−	0.00280785i	$-0.999106\pi$
$571$	− 1830.95i	− 0.00561570i	0.999996	−	0.00280785i	$-0.000893767\pi$
$572$	0	0
$573$	−139763.	−0.425679
$574$	98350.1i	0.298505i
$575$	706646.	2.13730
$576$	33452.2	0.100828
$577$	231875.	0.696468	0.348234	−	0.937408i	$-0.386781\pi$
$577$	231875.	0.696468	0.348234	+	0.937408i	$0.386781\pi$
$578$	− 148080.i	− 0.443241i
$579$	30965.8i	0.0923687i
$580$	− 298074.i	− 0.886069i
$581$	125508.	0.371808
$582$	− 75968.3i	− 0.224278i
$583$	0	0
$584$	−102961.	−0.301888
$585$	507027.i	1.48156i
$586$	58027.5	0.168981
$587$	51842.7	0.150457	0.0752283	−	0.997166i	$-0.476031\pi$
$587$	51842.7	0.150457	0.0752283	+	0.997166i	$0.476031\pi$
$588$	−70444.0	−0.203746
$589$	− 324190.i	− 0.934477i
$590$	602248.i	1.73010i
$591$	− 5830.64i	− 0.0166933i
$592$	13006.9	0.0371133
$593$	94653.9i	0.269171i	0.990902	+	0.134586i	$0.0429704\pi$
$593$	94653.9i	0.269171i	−0.990902	+	0.134586i	$0.957030\pi$
$594$	0	0
$595$	−101469.	−0.286616
$596$	− 212404.i	− 0.597958i
$597$	−103283.	−0.289788
$598$	−286307.	−0.800626
$599$	114529.	0.319200	0.159600	−	0.987182i	$-0.448980\pi$
$599$	114529.	0.319200	0.159600	+	0.987182i	$0.448980\pi$
$600$	112015.i	0.311153i
$601$	− 324147.i	− 0.897413i	−0.893679	−	0.448707i	$-0.851885\pi$
$601$	− 324147.i	− 0.897413i	0.893679	−	0.448707i	$-0.148115\pi$
$602$	97727.0i	0.269663i
$603$	196335.	0.539963
$604$	− 151908.i	− 0.416397i
$605$	0	0
$606$	−152980.	−0.416572
$607$	− 700511.i	− 1.90124i	−0.310351	−	0.950622i	$-0.600447\pi$
$607$	− 700511.i	− 1.90124i	0.310351	−	0.950622i	$-0.399553\pi$
$608$	−65671.3	−0.177651
$609$	45183.1	0.121826
$610$	194335.	0.522265
$611$	− 277036.i	− 0.742085i
$612$	− 92276.6i	− 0.246371i
$613$	597737.i	1.59070i	0.606148	+	0.795352i	$0.292714\pi$
$613$	597737.i	1.59070i	−0.606148	+	0.795352i	$0.707286\pi$
$614$	−378514.	−1.00403
$615$	− 449137.i	− 1.18749i
$616$	0	0
$617$	−704195.	−1.84979	−0.924896	−	0.380221i	$-0.875848\pi$
$617$	−704195.	−1.84979	−0.924896	+	0.380221i	$0.875848\pi$
$618$	− 13769.8i	− 0.0360538i
$619$	38982.5	0.101739	0.0508696	−	0.998705i	$-0.483801\pi$
$619$	38982.5	0.101739	0.0508696	+	0.998705i	$0.483801\pi$
$620$	−309624.	−0.805473
$621$	−327194.	−0.848441
$622$	− 119594.i	− 0.309121i
$623$	− 193183.i	− 0.497729i
$624$	− 45384.4i	− 0.116557i
$625$	392167.	1.00395
$626$	− 488716.i	− 1.24712i
$627$	0	0
$628$	27245.4	0.0690834
$629$	− 35878.9i	− 0.0906855i
$630$	106216.	0.267613
$631$	−19130.3	−0.0480467	−0.0240233	−	0.999711i	$-0.507648\pi$
$631$	−19130.3	−0.0480467	−0.0240233	+	0.999711i	$0.507648\pi$
$632$	−47818.9	−0.119720
$633$	− 286550.i	− 0.715143i
$634$	− 238524.i	− 0.593409i
$635$	298320.i	0.739836i
$636$	46573.2	0.115139
$637$	− 398647.i	− 0.982449i
$638$	0	0
$639$	555201.	1.35972
$640$	62720.7i	0.153127i
$641$	−343560.	−0.836154	−0.418077	−	0.908412i	$-0.637296\pi$
$641$	−343560.	−0.836154	−0.418077	+	0.908412i	$0.637296\pi$
$642$	94739.4	0.229858
$643$	−71246.1	−0.172321	−0.0861607	−	0.996281i	$-0.527460\pi$
$643$	−71246.1	−0.172321	−0.0861607	+	0.996281i	$0.527460\pi$
$644$	59977.6i	0.144616i
$645$	− 446292.i	− 1.07275i
$646$	181152.i	0.434087i
$647$	136791.	0.326775	0.163387	−	0.986562i	$-0.447758\pi$
$647$	136791.	0.326775	0.163387	+	0.986562i	$0.447758\pi$
$648$	67884.4i	0.161666i
$649$	0	0
$650$	−633900.	−1.50036
$651$	− 46933.9i	− 0.110745i
$652$	−68158.6	−0.160334
$653$	542958.	1.27333	0.636663	−	0.771142i	$-0.280314\pi$
$653$	542958.	1.27333	0.636663	+	0.771142i	$0.280314\pi$
$654$	85620.0	0.200180
$655$	− 63710.6i	− 0.148501i
$656$	− 167694.i	− 0.389681i
$657$	− 297298.i	− 0.688749i
$658$	−58035.4	−0.134042
$659$	702060.i	1.61660i	0.588768	+	0.808302i	$0.299613\pi$
$659$	702060.i	1.61660i	−0.588768	+	0.808302i	$0.700387\pi$
$660$	0	0
$661$	−226372.	−0.518108	−0.259054	−	0.965863i	$-0.583411\pi$
$661$	−226372.	−0.518108	−0.259054	+	0.965863i	$0.583411\pi$
$662$	367885.i	0.839452i
$663$	−125191.	−0.284804
$664$	−214000.	−0.485375
$665$	−208516.	−0.471515
$666$	37557.2i	0.0846729i
$667$	486009.i	1.09243i
$668$	− 106082.i	− 0.237733i
$669$	111966.	0.250170
$670$	368116.i	0.820039i
$671$	0	0
$672$	−9507.44	−0.0210535
$673$	− 43692.4i	− 0.0964664i	−0.998836	−	0.0482332i	$-0.984641\pi$
$673$	− 43692.4i	− 0.0964664i	0.998836	−	0.0482332i	$-0.0153591\pi$
$674$	−231385.	−0.509348
$675$	−724425.	−1.58996
$676$	28345.0	0.0620273
$677$	513609.i	1.12061i	0.828286	+	0.560306i	$0.189317\pi$
$677$	513609.i	1.12061i	−0.828286	+	0.560306i	$0.810683\pi$
$678$	101639.i	0.221106i
$679$	− 90060.5i	− 0.195342i
$680$	173012.	0.374162
$681$	285171.i	0.614909i
$682$	0	0
$683$	−289751.	−0.621131	−0.310566	−	0.950552i	$-0.600518\pi$
$683$	−289751.	−0.621131	−0.310566	+	0.950552i	$0.600518\pi$
$684$	− 189625.i	− 0.405306i
$685$	784954.	1.67287
$686$	−173633.	−0.368965
$687$	35375.4	0.0749528
$688$	− 166632.i	− 0.352030i
$689$	263561.i	0.555191i
$690$	− 273901.i	− 0.575301i
$691$	−479558.	−1.00435	−0.502175	−	0.864766i	$-0.667467\pi$
$691$	−479558.	−1.00435	−0.502175	+	0.864766i	$0.667467\pi$
$692$	− 176723.i	− 0.369045i
$693$	0	0
$694$	653568.	1.35697
$695$	440966.i	0.912926i
$696$	−77040.4	−0.159038
$697$	−462577.	−0.952178
$698$	36593.5	0.0751091
$699$	191620.i	0.392180i
$700$	132794.i	0.271008i
$701$	− 676849.i	− 1.37739i	−0.725053	−	0.688693i	$-0.758185\pi$
$701$	− 676849.i	− 1.37739i	0.725053	−	0.688693i	$-0.241815\pi$
$702$	293510.	0.595593
$703$	− 73729.8i	− 0.149188i
$704$	0	0
$705$	265031.	0.533235
$706$	− 167153.i	− 0.335354i
$707$	−181358.	−0.362826
$708$	155657.	0.310530
$709$	−225756.	−0.449104	−0.224552	−	0.974462i	$-0.572092\pi$
$709$	−225756.	−0.449104	−0.224552	+	0.974462i	$0.572092\pi$
$710$	1.04096e6i	2.06499i
$711$	− 138076.i	− 0.273137i
$712$	329391.i	0.649758i
$713$	504842.	0.993061
$714$	26225.9i	0.0514439i
$715$	0	0
$716$	66855.0	0.130409
$717$	214331.i	0.416915i
$718$	−157971.	−0.306428
$719$	930146.	1.79926	0.899629	−	0.436655i	$-0.143837\pi$
$719$	930146.	1.79926	0.899629	+	0.436655i	$0.143837\pi$
$720$	−181105.	−0.349354
$721$	− 16324.1i	− 0.0314021i
$722$	3656.21i	0.00701385i
$723$	− 214378.i	− 0.410113i
$724$	−80559.9	−0.153689
$725$	1.07605e6i	2.04719i
$726$	0	0
$727$	141940.	0.268556	0.134278	−	0.990944i	$-0.457128\pi$
$727$	141940.	0.268556	0.134278	+	0.990944i	$0.457128\pi$
$728$	− 53803.2i	− 0.101519i
$729$	−10350.8	−0.0194768
$730$	557413.	1.04600
$731$	−459646.	−0.860179
$732$	− 50228.0i	− 0.0937397i
$733$	− 1.01944e6i	− 1.89738i	−0.316203	−	0.948691i	$-0.602408\pi$
$733$	− 1.01944e6i	− 1.89738i	0.316203	−	0.948691i	$-0.397592\pi$
$734$	447720.i	0.831026i
$735$	381373.	0.705952
$736$	− 102266.i	− 0.188789i
$737$	0	0
$738$	484214.	0.889047
$739$	949192.i	1.73806i	0.494758	+	0.869031i	$0.335257\pi$
$739$	949192.i	1.73806i	−0.494758	+	0.869031i	$0.664743\pi$
$740$	−70417.2	−0.128592
$741$	−257263.	−0.468533
$742$	55212.6	0.100284
$743$	− 454366.i	− 0.823054i	−0.911398	−	0.411527i	$-0.864996\pi$
$743$	− 454366.i	− 0.823054i	0.911398	−	0.411527i	$-0.135004\pi$
$744$	80025.7i	0.144572i
$745$	1.14992e6i	2.07184i
$746$	378286.	0.679740
$747$	− 617923.i	− 1.10737i
$748$	0	0
$749$	112314.	0.200202
$750$	− 303415.i	− 0.539405i
$751$	−417159.	−0.739642	−0.369821	−	0.929103i	$-0.620581\pi$
$751$	−417159.	−0.739642	−0.369821	+	0.929103i	$0.620581\pi$
$752$	98954.5	0.174985
$753$	56563.2	0.0997571
$754$	− 435977.i	− 0.766868i
$755$	822407.i	1.44276i
$756$	− 61486.6i	− 0.107581i
$757$	−1.11344e6	−1.94302	−0.971508	−	0.237008i	$-0.923833\pi$
$757$	−1.11344e6	−1.94302	−0.971508	+	0.237008i	$0.923833\pi$
$758$	152445.i	0.265322i
$759$	0	0
$760$	355534.	0.615537
$761$	242415.i	0.418591i	0.977852	+	0.209295i	$0.0671170\pi$
$761$	242415.i	0.418591i	−0.977852	+	0.209295i	$0.932883\pi$
$762$	77104.1	0.132791
$763$	101503.	0.174352
$764$	282511.	0.484003
$765$	499571.i	0.853640i
$766$	− 138136.i	− 0.235423i
$767$	880876.i	1.49735i
$768$	16210.8	0.0274842
$769$	− 502205.i	− 0.849235i	−0.905373	−	0.424618i	$-0.860409\pi$
$769$	− 502205.i	− 0.849235i	0.905373	−	0.424618i	$-0.139591\pi$
$770$	0	0
$771$	−152306.	−0.256218
$772$	− 62593.1i	− 0.105025i
$773$	−635668.	−1.06383	−0.531914	−	0.846799i	$-0.678527\pi$
$773$	−635668.	−1.06383	−0.531914	+	0.846799i	$0.678527\pi$
$774$	481147.	0.803148
$775$	1.11775e6	1.86098
$776$	153560.i	0.255008i
$777$	− 10674.1i	− 0.0176803i
$778$	− 775254.i	− 1.28081i
$779$	−950578.	−1.56644
$780$	245704.i	0.403853i
$781$	0	0
$782$	−282097.	−0.461301
$783$	− 498237.i	− 0.812667i
$784$	142393.	0.231663
$785$	−147502.	−0.239364
$786$	−16466.7	−0.0266539
$787$	− 603258.i	− 0.973988i	−0.873405	−	0.486994i	$-0.838093\pi$
$787$	− 603258.i	− 0.973988i	0.873405	−	0.486994i	$-0.161907\pi$
$788$	11785.8i	0.0189805i
$789$	514720.i	0.826832i
$790$	258884.	0.414812
$791$	120493.i	0.192579i
$792$	0	0
$793$	284243.	0.452006
$794$	− 139479.i	− 0.221242i
$795$	−252140.	−0.398940
$796$	208772.	0.329494
$797$	869443.	1.36875	0.684375	−	0.729130i	$-0.260075\pi$
$797$	869443.	1.36875	0.684375	+	0.729130i	$0.260075\pi$
$798$	53893.2i	0.0846307i
$799$	− 272962.i	− 0.427571i
$800$	− 226423.i	− 0.353786i
$801$	−951112.	−1.48240
$802$	− 21432.9i	− 0.0333221i
$803$	0	0
$804$	95143.5	0.147186
$805$	− 324710.i	− 0.501076i
$806$	−452870.	−0.697114
$807$	−279608.	−0.429342
$808$	309229.	0.473650
$809$	− 361998.i	− 0.553107i	−0.960998	−	0.276554i	$-0.910808\pi$
$809$	− 361998.i	− 0.553107i	0.960998	−	0.276554i	$-0.0891924\pi$
$810$	− 367515.i	− 0.560152i
$811$	− 566141.i	− 0.860762i	−0.902647	−	0.430381i	$-0.858379\pi$
$811$	− 566141.i	− 0.860762i	0.902647	−	0.430381i	$-0.141621\pi$
$812$	−91331.5	−0.138519
$813$	315570.i	0.477435i
$814$	0	0
$815$	369000.	0.555535
$816$	− 44717.0i	− 0.0671571i
$817$	−944556.	−1.41509
$818$	−40815.0	−0.0609976
$819$	155356.	0.231612
$820$	907869.i	1.35019i
$821$	725449.i	1.07627i	0.842859	+	0.538134i	$0.180871\pi$
$821$	725449.i	1.07627i	−0.842859	+	0.538134i	$0.819129\pi$
$822$	− 202880.i	− 0.300259i
$823$	942525.	1.39153	0.695766	−	0.718269i	$-0.255065\pi$
$823$	942525.	1.39153	0.695766	+	0.718269i	$0.255065\pi$
$824$	27833.8i	0.0409938i
$825$	0	0
$826$	184532.	0.270465
$827$	318771.i	0.466088i	0.972466	+	0.233044i	$0.0748686\pi$
$827$	318771.i	0.466088i	−0.972466	+	0.233044i	$0.925131\pi$
$828$	295292.	0.430716
$829$	980360.	1.42652	0.713258	−	0.700902i	$-0.247219\pi$
$829$	980360.	1.42652	0.713258	+	0.700902i	$0.247219\pi$
$830$	1.15856e6	1.68176
$831$	175374.i	0.253959i
$832$	91738.2i	0.132527i
$833$	− 392785.i	− 0.566063i
$834$	113973.	0.163858
$835$	574313.i	0.823713i
$836$	0	0
$837$	−517543.	−0.738747
$838$	− 649124.i	− 0.924357i
$839$	523401.	0.743551	0.371776	−	0.928323i	$-0.378749\pi$
$839$	523401.	0.743551	0.371776	+	0.928323i	$0.378749\pi$
$840$	51471.8	0.0729475
$841$	−32794.0	−0.0463662
$842$	− 34451.3i	− 0.0485939i
$843$	− 309773.i	− 0.435902i
$844$	579221.i	0.813129i
$845$	−153455.	−0.214916
$846$	285730.i	0.399222i
$847$	0	0
$848$	−94141.4	−0.130915
$849$	− 355397.i	− 0.493059i
$850$	−624579.	−0.864469
$851$	114815.	0.158540
$852$	269049.	0.370639
$853$	− 345002.i	− 0.474158i	−0.971490	−	0.237079i	$-0.923810\pi$
$853$	− 345002.i	− 0.474158i	0.971490	−	0.237079i	$-0.0761901\pi$
$854$	− 59545.3i	− 0.0816454i
$855$	1.02660e6i	1.40433i
$856$	−191503.	−0.261353
$857$	652071.i	0.887837i	0.896067	+	0.443919i	$0.146412\pi$
$857$	652071.i	0.887837i	−0.896067	+	0.443919i	$0.853588\pi$
$858$	0	0
$859$	−559418.	−0.758142	−0.379071	−	0.925368i	$-0.623756\pi$
$859$	−559418.	−0.758142	−0.379071	+	0.925368i	$0.623756\pi$
$860$	902117.i	1.21974i
$861$	−137618.	−0.185639
$862$	−119497.	−0.160821
$863$	−378358.	−0.508021	−0.254011	−	0.967201i	$-0.581750\pi$
$863$	−378358.	−0.508021	−0.254011	+	0.967201i	$0.581750\pi$
$864$	104839.i	0.140442i
$865$	956748.i	1.27869i
$866$	− 187504.i	− 0.250020i
$867$	207203.	0.275650
$868$	94870.5i	0.125919i
$869$	0	0
$870$	417085.	0.551044
$871$	538423.i	0.709721i
$872$	−173069.	−0.227608
$873$	−443401.	−0.581793
$874$	−579698.	−0.758891
$875$	− 359699.i	− 0.469811i
$876$	− 144069.i	− 0.187743i
$877$	− 227306.i	− 0.295537i	−0.989022	−	0.147768i	$-0.952791\pi$
$877$	− 227306.i	− 0.295537i	0.989022	−	0.147768i	$-0.0472090\pi$
$878$	601010.	0.779637
$879$	81196.0i	0.105089i
$880$	0	0
$881$	721351.	0.929383	0.464692	−	0.885473i	$-0.346165\pi$
$881$	721351.	0.929383	0.464692	+	0.885473i	$0.346165\pi$
$882$	411158.i	0.528532i
$883$	214061.	0.274546	0.137273	−	0.990533i	$-0.456166\pi$
$883$	214061.	0.274546	0.137273	+	0.990533i	$0.456166\pi$
$884$	253056.	0.323827
$885$	−842706.	−1.07594
$886$	1.01132e6i	1.28831i
$887$	− 1.22543e6i	− 1.55755i	−0.627304	−	0.778774i	$-0.715842\pi$
$887$	− 1.22543e6i	− 1.55755i	0.627304	−	0.778774i	$-0.284158\pi$
$888$	18200.1i	0.0230806i
$889$	91407.0	0.115658
$890$	− 1.78327e6i	− 2.25132i
$891$	0	0
$892$	−226325.	−0.284448
$893$	− 560926.i	− 0.703401i
$894$	297210.	0.371868
$895$	−361943.	−0.451850
$896$	19218.0	0.0239382
$897$	− 400620.i	− 0.497907i
$898$	− 102240.i	− 0.126785i
$899$	768752.i	0.951189i
$900$	653794.	0.807153
$901$	259685.i	0.319888i
$902$	0	0
$903$	−136746.	−0.167703
$904$	− 205449.i	− 0.251402i
$905$	436139.	0.532510
$906$	212560.	0.258956
$907$	−1.01473e6	−1.23349	−0.616746	−	0.787162i	$-0.711550\pi$
$907$	−1.01473e6	−1.23349	−0.616746	+	0.787162i	$0.711550\pi$
$908$	− 576433.i	− 0.699161i
$909$	892895.i	1.08062i
$910$	291282.i	0.351748i
$911$	1.19460e6	1.43941	0.719705	−	0.694280i	$-0.244277\pi$
$911$	1.19460e6	1.43941	0.719705	+	0.694280i	$0.244277\pi$
$912$	− 91891.7i	− 0.110481i
$913$	0	0
$914$	−69465.1	−0.0831523
$915$	271926.i	0.324795i
$916$	−71506.5	−0.0852226
$917$	−19521.3	−0.0232151
$918$	289194.	0.343166
$919$	1.49415e6i	1.76914i	0.466405	+	0.884571i	$0.345549\pi$
$919$	1.49415e6i	1.76914i	−0.466405	+	0.884571i	$0.654451\pi$
$920$	553653.i	0.654126i
$921$	− 529643.i	− 0.624401i
$922$	−116075.	−0.136546
$923$	1.52256e6i	1.78719i
$924$	0	0
$925$	254207.	0.297101
$926$	746559.i	0.870647i
$927$	−80369.7	−0.0935261
$928$	155727.	0.180828
$929$	−475837.	−0.551349	−0.275674	−	0.961251i	$-0.588901\pi$
$929$	−475837.	−0.551349	−0.275674	+	0.961251i	$0.588901\pi$
$930$	− 433247.i	− 0.500921i
$931$	− 807159.i	− 0.931236i
$932$	− 387333.i	− 0.445915i
$933$	167344.	0.192241
$934$	64095.2i	0.0734737i
$935$	0	0
$936$	−264893.	−0.302356
$937$	914722.i	1.04186i	0.853599	+	0.520930i	$0.174415\pi$
$937$	914722.i	1.04186i	−0.853599	+	0.520930i	$0.825585\pi$
$938$	112793.	0.128196
$939$	683844.	0.775579
$940$	−535724.	−0.606297
$941$	− 830845.i	− 0.938298i	−0.883119	−	0.469149i	$-0.844561\pi$
$941$	− 830845.i	− 0.938298i	0.883119	−	0.469149i	$-0.155439\pi$
$942$	38123.6i	0.0429628i
$943$	− 1.48028e6i	− 1.66464i
$944$	−314640.	−0.353078
$945$	332879.i	0.372755i
$946$	0	0
$947$	−322279.	−0.359362	−0.179681	−	0.983725i	$-0.557506\pi$
$947$	−322279.	−0.359362	−0.179681	+	0.983725i	$0.557506\pi$
$948$	− 66911.4i	− 0.0744532i
$949$	815299.	0.905283
$950$	−1.28349e6	−1.42214
$951$	333760.	0.369039
$952$	− 53012.0i	− 0.0584925i
$953$	− 311258.i	− 0.342716i	−0.985209	−	0.171358i	$-0.945184\pi$
$953$	− 311258.i	− 0.342716i	0.985209	−	0.171358i	$-0.0548155\pi$
$954$	− 271832.i	− 0.298679i
$955$	−1.52947e6	−1.67700
$956$	− 433241.i	− 0.474039i
$957$	0	0
$958$	−187975.	−0.204818
$959$	− 240514.i	− 0.261519i
$960$	−87763.0	−0.0952290
$961$	−124980.	−0.135330
$962$	−102995.	−0.111293
$963$	− 552962.i	− 0.596269i
$964$	433335.i	0.466305i
$965$	338869.i	0.363896i
$966$	−83924.7	−0.0899364
$967$	− 852868.i	− 0.912072i	−0.889961	−	0.456036i	$-0.849269\pi$
$967$	− 852868.i	− 0.912072i	0.889961	−	0.456036i	$-0.150731\pi$
$968$	0	0
$969$	−253480.	−0.269957
$970$	− 831348.i	− 0.883566i
$971$	460906.	0.488848	0.244424	−	0.969668i	$-0.421401\pi$
$971$	460906.	0.488848	0.244424	+	0.969668i	$0.421401\pi$
$972$	−470284.	−0.497769
$973$	135115.	0.142717
$974$	1.15237e6i	1.21471i
$975$	− 886996.i	− 0.933066i
$976$	101529.i	0.106584i
$977$	−816357.	−0.855246	−0.427623	−	0.903957i	$-0.640649\pi$
$977$	−816357.	−0.855246	−0.427623	+	0.903957i	$0.640649\pi$
$978$	− 95372.1i	− 0.0997112i
$979$	0	0
$980$	−770893.	−0.802679
$981$	− 499735.i	− 0.519280i
$982$	99934.4	0.103632
$983$	−1.62854e6	−1.68536	−0.842678	−	0.538418i	$-0.819022\pi$
$983$	−1.62854e6	−1.68536	−0.842678	+	0.538418i	$0.819022\pi$
$984$	234649.	0.242341
$985$	− 63806.7i	− 0.0657648i
$986$	− 429566.i	− 0.441851i
$987$	− 81207.1i	− 0.0833603i
$988$	520021.	0.532730
$989$	− 1.47090e6i	− 1.50380i
$990$	0	0
$991$	−552727.	−0.562812	−0.281406	−	0.959589i	$-0.590801\pi$
$991$	−552727.	−0.562812	−0.281406	+	0.959589i	$0.590801\pi$
$992$	− 161761.i	− 0.164380i
$993$	−514770.	−0.522053
$994$	318957.	0.322819
$995$	−1.13026e6	−1.14165
$996$	− 299443.i	− 0.301853i
$997$	1.19345e6i	1.20065i	0.799757	+	0.600323i	$0.204961\pi$
$997$	1.19345e6i	1.20065i	−0.799757	+	0.600323i	$0.795039\pi$
$998$	− 86709.8i	− 0.0870576i
$999$	−117704.	−0.117940

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	242.5.b.e.241.5		16
11.3	even	5		22.5.d.a.13.1	✓	16
11.7	odd	10		22.5.d.a.17.1	yes	16
11.10	odd	2	inner	242.5.b.e.241.13		16
33.14	odd	10		198.5.j.a.145.3		16
33.29	even	10		198.5.j.a.127.3		16
44.3	odd	10		176.5.n.c.145.3		16
44.7	even	10		176.5.n.c.17.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.5.d.a.13.1	✓	16	11.3	even	5
22.5.d.a.17.1	yes	16	11.7	odd	10
176.5.n.c.17.3		16	44.7	even	10
176.5.n.c.145.3		16	44.3	odd	10
198.5.j.a.127.3		16	33.29	even	10
198.5.j.a.145.3		16	33.14	odd	10
242.5.b.e.241.5		16	1.1	even	1	trivial
242.5.b.e.241.13		16	11.10	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 242 = 2 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	242.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.82843i q^{2} +3.95773 q^{3} -8.00000 q^{4} +43.3108 q^{5} -11.1941i q^{6} -13.2707i q^{7} +22.6274i q^{8} -65.3364 q^{9} +O(q^{10})\)	\(q-2.82843i q^{2} +3.95773 q^{3} -8.00000 q^{4} +43.3108 q^{5} -11.1941i q^{6} -13.2707i q^{7} +22.6274i q^{8} -65.3364 q^{9} -122.501i q^{10} -31.6618 q^{12} -179.176i q^{13} -37.5351 q^{14} +171.412 q^{15} +64.0000 q^{16} -176.541i q^{17} +184.799i q^{18} -362.786i q^{19} -346.486 q^{20} -52.5217i q^{21} +564.946 q^{23} +89.5531i q^{24} +1250.82 q^{25} -506.787 q^{26} -579.159 q^{27} +106.165i q^{28} +860.276i q^{29} -484.827i q^{30} +893.611 q^{31} -181.019i q^{32} -499.334 q^{34} -574.763i q^{35} +522.691 q^{36} +203.232 q^{37} -1026.11 q^{38} -709.131i q^{39} +980.011i q^{40} -2620.22i q^{41} -148.554 q^{42} -2603.62i q^{43} -2829.77 q^{45} -1597.91i q^{46} +1546.16 q^{47} +253.295 q^{48} +2224.89 q^{49} -3537.86i q^{50} -698.703i q^{51} +1433.41i q^{52} -1470.96 q^{53} +1638.11i q^{54} +300.281 q^{56} -1435.81i q^{57} +2433.23 q^{58} -4916.25 q^{59} -1371.30 q^{60} +1586.39i q^{61} -2527.51i q^{62} +867.057i q^{63} -512.000 q^{64} -7760.26i q^{65} -3004.99 q^{67} +1412.33i q^{68} +2235.90 q^{69} -1625.67 q^{70} -8497.57 q^{71} -1478.39i q^{72} +4550.26i q^{73} -574.828i q^{74} +4950.41 q^{75} +2902.29i q^{76} -2005.72 q^{78} +2113.32i q^{79} +2771.89 q^{80} +3000.09 q^{81} -7411.09 q^{82} +9457.56i q^{83} +420.173i q^{84} -7646.14i q^{85} -7364.14 q^{86} +3404.74i q^{87} +14557.2 q^{89} +8003.80i q^{90} -2377.79 q^{91} -4519.56 q^{92} +3536.67 q^{93} -4373.21i q^{94} -15712.5i q^{95} -716.425i q^{96} +6786.44 q^{97} -6292.94i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 48 q^{3} - 128 q^{4} + 60 q^{5} - 40 q^{9}+O(q^{10}) \) 16 * q + 48 * q^3 - 128 * q^4 + 60 * q^5 - 40 * q^9	\( 16 q + 48 q^{3} - 128 q^{4} + 60 q^{5} - 40 q^{9} - 384 q^{12} + 384 q^{14} + 1352 q^{15} + 1024 q^{16} - 480 q^{20} - 2424 q^{23} + 5692 q^{25} + 4344 q^{27} - 428 q^{31} - 3904 q^{34} + 320 q^{36} - 228 q^{37} + 1440 q^{38} - 5568 q^{42} + 12152 q^{45} - 9228 q^{47} + 3072 q^{48} - 11956 q^{49} + 12468 q^{53} - 3072 q^{56} + 2944 q^{58} - 22320 q^{59} - 10816 q^{60} - 8192 q^{64} + 21524 q^{67} - 24704 q^{69} + 16960 q^{70} + 8868 q^{71} + 19716 q^{75} - 39424 q^{78} + 3840 q^{80} + 1136 q^{81} - 20992 q^{82} - 35616 q^{86} + 46596 q^{89} - 35600 q^{91} + 19392 q^{92} + 14832 q^{93} + 69448 q^{97}+O(q^{100}) \) 16 * q + 48 * q^3 - 128 * q^4 + 60 * q^5 - 40 * q^9 - 384 * q^12 + 384 * q^14 + 1352 * q^15 + 1024 * q^16 - 480 * q^20 - 2424 * q^23 + 5692 * q^25 + 4344 * q^27 - 428 * q^31 - 3904 * q^34 + 320 * q^36 - 228 * q^37 + 1440 * q^38 - 5568 * q^42 + 12152 * q^45 - 9228 * q^47 + 3072 * q^48 - 11956 * q^49 + 12468 * q^53 - 3072 * q^56 + 2944 * q^58 - 22320 * q^59 - 10816 * q^60 - 8192 * q^64 + 21524 * q^67 - 24704 * q^69 + 16960 * q^70 + 8868 * q^71 + 19716 * q^75 - 39424 * q^78 + 3840 * q^80 + 1136 * q^81 - 20992 * q^82 - 35616 * q^86 + 46596 * q^89 - 35600 * q^91 + 19392 * q^92 + 14832 * q^93 + 69448 * q^97

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