LMFDB - Embedded newform 242.4.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [242,4,Mod(1,242)]

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([0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("242.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 242 = 2 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	242.a (trivial)

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:	yes
Analytic conductor:	$14.2784622214$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.978025.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 99x^{2} + 100x + 2420 $ x^4 - 2x^3 - 99x^2 + 100*x + 2420
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 11 $
Twist minimal:	no (minimal twist has level 22)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$8.19378$ of defining polynomial
Character	$\chi$	$=$	242.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} -7.81182 q^{3} +4.00000 q^{4} +14.9181 q^{5} -15.6236 q^{6} -21.7679 q^{7} +8.00000 q^{8} +34.0245 q^{9} +O(q^{10})$	\(q+2.00000 q^{2} -7.81182 q^{3} +4.00000 q^{4} +14.9181 q^{5} -15.6236 q^{6} -21.7679 q^{7} +8.00000 q^{8} +34.0245 q^{9} +29.8363 q^{10} -31.2473 q^{12} +44.0286 q^{13} -43.5357 q^{14} -116.538 q^{15} +16.0000 q^{16} +24.9145 q^{17} +68.0489 q^{18} -21.9573 q^{19} +59.6725 q^{20} +170.046 q^{21} +177.749 q^{23} -62.4945 q^{24} +97.5508 q^{25} +88.0571 q^{26} -54.8738 q^{27} -87.0714 q^{28} +149.396 q^{29} -233.075 q^{30} +75.1436 q^{31} +32.0000 q^{32} +49.8290 q^{34} -324.736 q^{35} +136.098 q^{36} +222.336 q^{37} -43.9145 q^{38} -343.943 q^{39} +119.345 q^{40} +253.121 q^{41} +340.093 q^{42} -130.623 q^{43} +507.582 q^{45} +355.498 q^{46} +499.093 q^{47} -124.989 q^{48} +130.839 q^{49} +195.102 q^{50} -194.627 q^{51} +176.114 q^{52} +12.9421 q^{53} -109.748 q^{54} -174.143 q^{56} +171.526 q^{57} +298.793 q^{58} +35.5614 q^{59} -466.151 q^{60} -538.343 q^{61} +150.287 q^{62} -740.639 q^{63} +64.0000 q^{64} +656.824 q^{65} -519.621 q^{67} +99.6580 q^{68} -1388.54 q^{69} -649.472 q^{70} +78.4486 q^{71} +272.196 q^{72} -1144.07 q^{73} +444.673 q^{74} -762.049 q^{75} -87.8290 q^{76} -687.886 q^{78} -772.546 q^{79} +238.690 q^{80} -489.997 q^{81} +506.243 q^{82} -537.242 q^{83} +680.186 q^{84} +371.678 q^{85} -261.247 q^{86} -1167.06 q^{87} +667.089 q^{89} +1015.16 q^{90} -958.407 q^{91} +710.996 q^{92} -587.008 q^{93} +998.187 q^{94} -327.561 q^{95} -249.978 q^{96} -179.654 q^{97} +261.679 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{2} + 4 q^{3} + 16 q^{4} + 25 q^{5} + 8 q^{6} + 3 q^{7} + 32 q^{8} + 102 q^{9}+O(q^{10}) $ 4 * q + 8 * q^2 + 4 * q^3 + 16 * q^4 + 25 * q^5 + 8 * q^6 + 3 * q^7 + 32 * q^8 + 102 * q^9	\( 4 q + 8 q^{2} + 4 q^{3} + 16 q^{4} + 25 q^{5} + 8 q^{6} + 3 q^{7} + 32 q^{8} + 102 q^{9} + 50 q^{10} + 16 q^{12} - 41 q^{13} + 6 q^{14} + 68 q^{15} + 64 q^{16} + 52 q^{17} + 204 q^{18} + 16 q^{19} + 100 q^{20} + 25 q^{21} + 314 q^{23} + 32 q^{24} - 21 q^{25} - 82 q^{26} + 286 q^{27} + 12 q^{28} + 561 q^{29} + 136 q^{30} + 199 q^{31} + 128 q^{32} + 104 q^{34} - 714 q^{35} + 408 q^{36} + 357 q^{37} + 32 q^{38} - 1038 q^{39} + 200 q^{40} + 32 q^{41} + 50 q^{42} - 721 q^{43} + 1326 q^{45} + 628 q^{46} + 403 q^{47} + 64 q^{48} + 823 q^{49} - 42 q^{50} - 174 q^{51} - 164 q^{52} - 133 q^{53} + 572 q^{54} + 24 q^{56} - 1031 q^{57} + 1122 q^{58} + 1016 q^{59} + 272 q^{60} - 919 q^{61} + 398 q^{62} - 1367 q^{63} + 256 q^{64} + 69 q^{65} + 289 q^{67} + 208 q^{68} - 1620 q^{69} - 1428 q^{70} - 1205 q^{71} + 816 q^{72} - 1234 q^{73} + 714 q^{74} - 911 q^{75} + 64 q^{76} - 2076 q^{78} - 603 q^{79} + 400 q^{80} - 1400 q^{81} + 64 q^{82} + 1514 q^{83} + 100 q^{84} + 717 q^{85} - 1442 q^{86} - 1061 q^{87} - 1101 q^{89} + 2652 q^{90} - 2306 q^{91} + 1256 q^{92} - 2298 q^{93} + 806 q^{94} - 1766 q^{95} + 128 q^{96} + 2116 q^{97} + 1646 q^{98}+O(q^{100}) \) 4 * q + 8 * q^2 + 4 * q^3 + 16 * q^4 + 25 * q^5 + 8 * q^6 + 3 * q^7 + 32 * q^8 + 102 * q^9 + 50 * q^10 + 16 * q^12 - 41 * q^13 + 6 * q^14 + 68 * q^15 + 64 * q^16 + 52 * q^17 + 204 * q^18 + 16 * q^19 + 100 * q^20 + 25 * q^21 + 314 * q^23 + 32 * q^24 - 21 * q^25 - 82 * q^26 + 286 * q^27 + 12 * q^28 + 561 * q^29 + 136 * q^30 + 199 * q^31 + 128 * q^32 + 104 * q^34 - 714 * q^35 + 408 * q^36 + 357 * q^37 + 32 * q^38 - 1038 * q^39 + 200 * q^40 + 32 * q^41 + 50 * q^42 - 721 * q^43 + 1326 * q^45 + 628 * q^46 + 403 * q^47 + 64 * q^48 + 823 * q^49 - 42 * q^50 - 174 * q^51 - 164 * q^52 - 133 * q^53 + 572 * q^54 + 24 * q^56 - 1031 * q^57 + 1122 * q^58 + 1016 * q^59 + 272 * q^60 - 919 * q^61 + 398 * q^62 - 1367 * q^63 + 256 * q^64 + 69 * q^65 + 289 * q^67 + 208 * q^68 - 1620 * q^69 - 1428 * q^70 - 1205 * q^71 + 816 * q^72 - 1234 * q^73 + 714 * q^74 - 911 * q^75 + 64 * q^76 - 2076 * q^78 - 603 * q^79 + 400 * q^80 - 1400 * q^81 + 64 * q^82 + 1514 * q^83 + 100 * q^84 + 717 * q^85 - 1442 * q^86 - 1061 * q^87 - 1101 * q^89 + 2652 * q^90 - 2306 * q^91 + 1256 * q^92 - 2298 * q^93 + 806 * q^94 - 1766 * q^95 + 128 * q^96 + 2116 * q^97 + 1646 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	−7.81182	−1.50338	−0.751692	−	0.659514i	$-0.770762\pi$
$3$	−7.81182	−1.50338	−0.751692	+	0.659514i	$0.770762\pi$
$4$	4.00000	0.500000
$5$	14.9181	1.33432	0.667159	−	0.744915i	$-0.267510\pi$
$5$	14.9181	1.33432	0.667159	+	0.744915i	$0.267510\pi$
$6$	−15.6236	−1.06305
$7$	−21.7679	−1.17535	−0.587677	−	0.809096i	$-0.699957\pi$
$7$	−21.7679	−1.17535	−0.587677	+	0.809096i	$0.699957\pi$
$8$	8.00000	0.353553
$9$	34.0245	1.26017
$10$	29.8363	0.943506
$11$	0	0
$11$	0	0
$12$	−31.2473	−0.751692
$13$	44.0286	0.939333	0.469666	−	0.882844i	$-0.344374\pi$
$13$	44.0286	0.939333	0.469666	+	0.882844i	$0.344374\pi$
$14$	−43.5357	−0.831101
$15$	−116.538	−2.00599
$16$	16.0000	0.250000
$17$	24.9145	0.355450	0.177725	−	0.984080i	$-0.443126\pi$
$17$	24.9145	0.355450	0.177725	+	0.984080i	$0.443126\pi$
$18$	68.0489	0.891071
$19$	−21.9573	−0.265123	−0.132562	−	0.991175i	$-0.542320\pi$
$19$	−21.9573	−0.265123	−0.132562	+	0.991175i	$0.542320\pi$
$20$	59.6725	0.667159
$21$	170.046	1.76701
$22$	0	0
$23$	177.749	1.61145	0.805723	−	0.592293i	$-0.201777\pi$
$23$	177.749	1.61145	0.805723	+	0.592293i	$0.201777\pi$
$24$	−62.4945	−0.531527
$25$	97.5508	0.780407
$26$	88.0571	0.664208
$27$	−54.8738	−0.391128
$28$	−87.0714	−0.587677
$29$	149.396	0.956628	0.478314	−	0.878189i	$-0.341248\pi$
$29$	149.396	0.956628	0.478314	+	0.878189i	$0.341248\pi$
$30$	−233.075	−1.41845
$31$	75.1436	0.435361	0.217681	−	0.976020i	$-0.430151\pi$
$31$	75.1436	0.435361	0.217681	+	0.976020i	$0.430151\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	49.8290	0.251341
$35$	−324.736	−1.56830
$36$	136.098	0.630083
$37$	222.336	0.987889	0.493944	−	0.869493i	$-0.335555\pi$
$37$	222.336	0.987889	0.493944	+	0.869493i	$0.335555\pi$
$38$	−43.9145	−0.187470
$39$	−343.943	−1.41218
$40$	119.345	0.471753
$41$	253.121	0.964168	0.482084	−	0.876125i	$-0.339880\pi$
$41$	253.121	0.964168	0.482084	+	0.876125i	$0.339880\pi$
$42$	340.093	1.24946
$43$	−130.623	−0.463253	−0.231626	−	0.972805i	$-0.574405\pi$
$43$	−130.623	−0.463253	−0.231626	+	0.972805i	$0.574405\pi$
$44$	0	0
$45$	507.582	1.68146
$46$	355.498	1.13946
$47$	499.093	1.54894	0.774471	−	0.632610i	$-0.218016\pi$
$47$	499.093	1.54894	0.774471	+	0.632610i	$0.218016\pi$
$48$	−124.989	−0.375846
$49$	130.839	0.381456
$50$	195.102	0.551831
$51$	−194.627	−0.534378
$52$	176.114	0.469666
$53$	12.9421	0.0335422	0.0167711	−	0.999859i	$-0.494661\pi$
$53$	12.9421	0.0335422	0.0167711	+	0.999859i	$0.494661\pi$
$54$	−109.748	−0.276569
$55$	0	0
$56$	−174.143	−0.415550
$57$	171.526	0.398582
$58$	298.793	0.676438
$59$	35.5614	0.0784694	0.0392347	−	0.999230i	$-0.487508\pi$
$59$	35.5614	0.0784694	0.0392347	+	0.999230i	$0.487508\pi$
$60$	−466.151	−1.00300
$61$	−538.343	−1.12996	−0.564982	−	0.825103i	$-0.691117\pi$
$61$	−538.343	−1.12996	−0.564982	+	0.825103i	$0.691117\pi$
$62$	150.287	0.307847
$63$	−740.639	−1.48114
$64$	64.0000	0.125000
$65$	656.824	1.25337
$66$	0	0
$67$	−519.621	−0.947491	−0.473745	−	0.880662i	$-0.657098\pi$
$67$	−519.621	−0.947491	−0.473745	+	0.880662i	$0.657098\pi$
$68$	99.6580	0.177725
$69$	−1388.54	−2.42262
$70$	−649.472	−1.10895
$71$	78.4486	0.131129	0.0655644	−	0.997848i	$-0.479115\pi$
$71$	78.4486	0.131129	0.0655644	+	0.997848i	$0.479115\pi$
$72$	272.196	0.445536
$73$	−1144.07	−1.83429	−0.917145	−	0.398554i	$-0.869512\pi$
$73$	−1144.07	−1.83429	−0.917145	+	0.398554i	$0.869512\pi$
$74$	444.673	0.698543
$75$	−762.049	−1.17325
$76$	−87.8290	−0.132562
$77$	0	0
$78$	−687.886	−0.998561
$79$	−772.546	−1.10023	−0.550115	−	0.835089i	$-0.685416\pi$
$79$	−772.546	−1.10023	−0.550115	+	0.835089i	$0.685416\pi$
$80$	238.690	0.333580
$81$	−489.997	−0.672149
$82$	506.243	0.681770
$83$	−537.242	−0.710481	−0.355241	−	0.934775i	$-0.615601\pi$
$83$	−537.242	−0.710481	−0.355241	+	0.934775i	$0.615601\pi$
$84$	680.186	0.883504
$85$	371.678	0.474284
$86$	−261.247	−0.327569
$87$	−1167.06	−1.43818
$88$	0	0
$89$	667.089	0.794509	0.397255	−	0.917708i	$-0.369963\pi$
$89$	667.089	0.794509	0.397255	+	0.917708i	$0.369963\pi$
$90$	1015.16	1.18897
$91$	−958.407	−1.10405
$92$	710.996	0.805723
$93$	−587.008	−0.654515
$94$	998.187	1.09527
$95$	−327.561	−0.353759
$96$	−249.978	−0.265763
$97$	−179.654	−0.188052	−0.0940262	−	0.995570i	$-0.529974\pi$
$97$	−179.654	−0.188052	−0.0940262	+	0.995570i	$0.529974\pi$
$98$	261.679	0.269730
$99$	0	0
$100$	390.203	0.390203
$101$	410.736	0.404651	0.202325	−	0.979318i	$-0.435150\pi$
$101$	410.736	0.404651	0.202325	+	0.979318i	$0.435150\pi$
$102$	−389.255	−0.377863
$103$	1367.66	1.30834	0.654172	−	0.756345i	$-0.273017\pi$
$103$	1367.66	1.30834	0.654172	+	0.756345i	$0.273017\pi$
$104$	352.228	0.332104
$105$	2536.78	2.35775
$106$	25.8842	0.0237179
$107$	395.241	0.357097	0.178548	−	0.983931i	$-0.442860\pi$
$107$	395.241	0.357097	0.178548	+	0.983931i	$0.442860\pi$
$108$	−219.495	−0.195564
$109$	−505.826	−0.444490	−0.222245	−	0.974991i	$-0.571338\pi$
$109$	−505.826	−0.444490	−0.222245	+	0.974991i	$0.571338\pi$
$110$	0	0
$111$	−1736.85	−1.48518
$112$	−348.286	−0.293838
$113$	1537.62	1.28007	0.640033	−	0.768347i	$-0.278921\pi$
$113$	1537.62	1.28007	0.640033	+	0.768347i	$0.278921\pi$
$114$	343.052	0.281840
$115$	2651.68	2.15018
$116$	597.585	0.478314
$117$	1498.05	1.18371
$118$	71.1227	0.0554862
$119$	−542.335	−0.417780
$120$	−932.302	−0.709226
$121$	0	0
$122$	−1076.69	−0.799005
$123$	−1977.34	−1.44952
$124$	300.574	0.217681
$125$	−409.491	−0.293008
$126$	−1481.28	−1.04732
$127$	697.332	0.487230	0.243615	−	0.969872i	$-0.421667\pi$
$127$	697.332	0.487230	0.243615	+	0.969872i	$0.421667\pi$
$128$	128.000	0.0883883
$129$	1020.40	0.696447
$130$	1313.65	0.886266
$131$	259.910	0.173347	0.0866735	−	0.996237i	$-0.472376\pi$
$131$	259.910	0.173347	0.0866735	+	0.996237i	$0.472376\pi$
$132$	0	0
$133$	477.962	0.311613
$134$	−1039.24	−0.669977
$135$	−818.615	−0.521890
$136$	199.316	0.125671
$137$	−2083.56	−1.29935	−0.649674	−	0.760213i	$-0.725094\pi$
$137$	−2083.56	−1.29935	−0.649674	+	0.760213i	$0.725094\pi$
$138$	−2777.09	−1.71305
$139$	−173.429	−0.105828	−0.0529138	−	0.998599i	$-0.516851\pi$
$139$	−173.429	−0.105828	−0.0529138	+	0.998599i	$0.516851\pi$
$140$	−1298.94	−0.784148
$141$	−3898.83	−2.32866
$142$	156.897	0.0927220
$143$	0	0
$144$	544.391	0.315041
$145$	2228.71	1.27645
$146$	−2288.14	−1.29704
$147$	−1022.09	−0.573475
$148$	889.346	0.493944
$149$	449.055	0.246899	0.123450	−	0.992351i	$-0.460604\pi$
$149$	449.055	0.246899	0.123450	+	0.992351i	$0.460604\pi$
$150$	−1524.10	−0.829614
$151$	−28.5358	−0.0153789	−0.00768944	−	0.999970i	$-0.502448\pi$
$151$	−28.5358	−0.0153789	−0.00768944	+	0.999970i	$0.502448\pi$
$152$	−175.658	−0.0937352
$153$	847.702	0.447926
$154$	0	0
$155$	1121.00	0.580910
$156$	−1375.77	−0.706089
$157$	1723.86	0.876300	0.438150	−	0.898902i	$-0.355634\pi$
$157$	1723.86	0.876300	0.438150	+	0.898902i	$0.355634\pi$
$158$	−1545.09	−0.777980
$159$	−101.101	−0.0504268
$160$	477.380	0.235876
$161$	−3869.22	−1.89402
$162$	−979.993	−0.475281
$163$	3318.22	1.59450	0.797249	−	0.603651i	$-0.206288\pi$
$163$	3318.22	1.59450	0.797249	+	0.603651i	$0.206288\pi$
$164$	1012.49	0.482084
$165$	0	0
$166$	−1074.48	−0.502386
$167$	3015.60	1.39733	0.698665	−	0.715449i	$-0.253778\pi$
$167$	3015.60	1.39733	0.698665	+	0.715449i	$0.253778\pi$
$168$	1360.37	0.624732
$169$	−258.486	−0.117654
$170$	743.356	0.335369
$171$	−747.084	−0.334099
$172$	−522.493	−0.231626
$173$	2209.41	0.970973	0.485486	−	0.874244i	$-0.338643\pi$
$173$	2209.41	0.970973	0.485486	+	0.874244i	$0.338643\pi$
$174$	−2334.11	−1.01695
$175$	−2123.47	−0.917254
$176$	0	0
$177$	−277.799	−0.117970
$178$	1334.18	0.561803
$179$	−2271.83	−0.948629	−0.474315	−	0.880355i	$-0.657304\pi$
$179$	−2271.83	−0.948629	−0.474315	+	0.880355i	$0.657304\pi$
$180$	2030.33	0.840731
$181$	−624.435	−0.256430	−0.128215	−	0.991746i	$-0.540925\pi$
$181$	−624.435	−0.256430	−0.128215	+	0.991746i	$0.540925\pi$
$182$	−1916.81	−0.780680
$183$	4205.44	1.69877
$184$	1421.99	0.569732
$185$	3316.85	1.31816
$186$	−1174.02	−0.462812
$187$	0	0
$188$	1996.37	0.774471
$189$	1194.48	0.459714
$190$	−655.123	−0.250145
$191$	−1344.20	−0.509229	−0.254615	−	0.967043i	$-0.581949\pi$
$191$	−1344.20	−0.509229	−0.254615	+	0.967043i	$0.581949\pi$
$192$	−499.956	−0.187923
$193$	−4623.76	−1.72449	−0.862243	−	0.506496i	$-0.830941\pi$
$193$	−4623.76	−1.72449	−0.862243	+	0.506496i	$0.830941\pi$
$194$	−359.308	−0.132973
$195$	−5130.99	−1.88430
$196$	523.358	0.190728
$197$	−664.691	−0.240392	−0.120196	−	0.992750i	$-0.538352\pi$
$197$	−664.691	−0.240392	−0.120196	+	0.992750i	$0.538352\pi$
$198$	0	0
$199$	−3042.82	−1.08392	−0.541959	−	0.840405i	$-0.682317\pi$
$199$	−3042.82	−1.08392	−0.541959	+	0.840405i	$0.682317\pi$
$200$	780.407	0.275915
$201$	4059.19	1.42444
$202$	821.471	0.286131
$203$	−3252.04	−1.12438
$204$	−778.510	−0.267189
$205$	3776.10	1.28651
$206$	2735.32	0.925139
$207$	6047.82	2.03069
$208$	704.457	0.234833
$209$	0	0
$210$	5073.55	1.66718
$211$	−2591.61	−0.845563	−0.422781	−	0.906232i	$-0.638946\pi$
$211$	−2591.61	−0.845563	−0.422781	+	0.906232i	$0.638946\pi$
$212$	51.7684	0.0167711
$213$	−612.826	−0.197137
$214$	790.481	0.252506
$215$	−1948.66	−0.618127
$216$	−438.990	−0.138285
$217$	−1635.72	−0.511703
$218$	−1011.65	−0.314302
$219$	8937.26	2.75764
$220$	0	0
$221$	1096.95	0.333886
$222$	−3473.70	−1.05018
$223$	2637.20	0.791929	0.395965	−	0.918266i	$-0.370410\pi$
$223$	2637.20	0.791929	0.395965	+	0.918266i	$0.370410\pi$
$224$	−696.571	−0.207775
$225$	3319.11	0.983441
$226$	3075.25	0.905143
$227$	−250.670	−0.0732932	−0.0366466	−	0.999328i	$-0.511668\pi$
$227$	−250.670	−0.0732932	−0.0366466	+	0.999328i	$0.511668\pi$
$228$	686.104	0.199291
$229$	1799.31	0.519222	0.259611	−	0.965713i	$-0.416406\pi$
$229$	1799.31	0.519222	0.259611	+	0.965713i	$0.416406\pi$
$230$	5303.37	1.52041
$231$	0	0
$232$	1195.17	0.338219
$233$	−3180.48	−0.894248	−0.447124	−	0.894472i	$-0.647552\pi$
$233$	−3180.48	−0.894248	−0.447124	+	0.894472i	$0.647552\pi$
$234$	2996.10	0.837012
$235$	7445.54	2.06678
$236$	142.245	0.0392347
$237$	6034.98	1.65407
$238$	−1084.67	−0.295415
$239$	−5013.80	−1.35697	−0.678485	−	0.734614i	$-0.737363\pi$
$239$	−5013.80	−1.35697	−0.678485	+	0.734614i	$0.737363\pi$
$240$	−1864.60	−0.501499
$241$	−6074.13	−1.62352	−0.811761	−	0.583990i	$-0.801491\pi$
$241$	−6074.13	−1.62352	−0.811761	+	0.583990i	$0.801491\pi$
$242$	0	0
$243$	5309.35	1.40163
$244$	−2153.37	−0.564982
$245$	1951.88	0.508984
$246$	−3954.67	−1.02496
$247$	−966.746	−0.249039
$248$	601.149	0.153923
$249$	4196.83	1.06813
$250$	−818.981	−0.207188
$251$	−5569.69	−1.40062	−0.700310	−	0.713838i	$-0.746955\pi$
$251$	−5569.69	−1.40062	−0.700310	+	0.713838i	$0.746955\pi$
$252$	−2962.56	−0.740570
$253$	0	0
$254$	1394.66	0.344524
$255$	−2903.48	−0.713031
$256$	256.000	0.0625000
$257$	−5907.05	−1.43374	−0.716871	−	0.697206i	$-0.754426\pi$
$257$	−5907.05	−1.43374	−0.716871	+	0.697206i	$0.754426\pi$
$258$	2040.81	0.492462
$259$	−4839.79	−1.16112
$260$	2627.30	0.626685
$261$	5083.13	1.20551
$262$	519.820	0.122575
$263$	6853.74	1.60692	0.803459	−	0.595360i	$-0.202991\pi$
$263$	6853.74	1.60692	0.803459	+	0.595360i	$0.202991\pi$
$264$	0	0
$265$	193.072	0.0447559
$266$	955.924	0.220344
$267$	−5211.18	−1.19445
$268$	−2078.49	−0.473745
$269$	1101.36	0.249633	0.124817	−	0.992180i	$-0.460166\pi$
$269$	1101.36	0.249633	0.124817	+	0.992180i	$0.460166\pi$
$270$	−1637.23	−0.369032
$271$	2571.69	0.576455	0.288227	−	0.957562i	$-0.406934\pi$
$271$	2571.69	0.576455	0.288227	+	0.957562i	$0.406934\pi$
$272$	398.632	0.0888626
$273$	7486.90	1.65981
$274$	−4167.12	−0.918777
$275$	0	0
$276$	−5554.17	−1.21131
$277$	7340.10	1.59214	0.796072	−	0.605202i	$-0.206908\pi$
$277$	7340.10	1.59214	0.796072	+	0.605202i	$0.206908\pi$
$278$	−346.857	−0.0748314
$279$	2556.72	0.548627
$280$	−2597.89	−0.554477
$281$	7492.94	1.59072	0.795359	−	0.606139i	$-0.207282\pi$
$281$	7492.94	1.59072	0.795359	+	0.606139i	$0.207282\pi$
$282$	−7797.65	−1.64661
$283$	6494.10	1.36408	0.682039	−	0.731316i	$-0.261093\pi$
$283$	6494.10	1.36408	0.682039	+	0.731316i	$0.261093\pi$
$284$	313.795	0.0655644
$285$	2558.85	0.531835
$286$	0	0
$287$	−5509.91	−1.13324
$288$	1088.78	0.222768
$289$	−4292.27	−0.873655
$290$	4457.43	0.902584
$291$	1403.42	0.282715
$292$	−4576.28	−0.917145
$293$	2684.87	0.535331	0.267666	−	0.963512i	$-0.413748\pi$
$293$	2684.87	0.535331	0.267666	+	0.963512i	$0.413748\pi$
$294$	−2044.19	−0.405508
$295$	530.509	0.104703
$296$	1778.69	0.349271
$297$	0	0
$298$	898.110	0.174584
$299$	7826.03	1.51368
$300$	−3048.20	−0.586626
$301$	2843.39	0.544486
$302$	−57.0716	−0.0108745
$303$	−3208.59	−0.608345
$304$	−351.316	−0.0662808
$305$	−8031.08	−1.50773
$306$	1695.40	0.316732
$307$	−8331.66	−1.54890	−0.774451	−	0.632633i	$-0.781974\pi$
$307$	−8331.66	−1.54890	−0.774451	+	0.632633i	$0.781974\pi$
$308$	0	0
$309$	−10683.9	−1.96695
$310$	2242.01	0.410766
$311$	−5020.35	−0.915363	−0.457682	−	0.889116i	$-0.651320\pi$
$311$	−5020.35	−0.915363	−0.457682	+	0.889116i	$0.651320\pi$
$312$	−2751.54	−0.499280
$313$	3022.08	0.545744	0.272872	−	0.962050i	$-0.412026\pi$
$313$	3022.08	0.545744	0.272872	+	0.962050i	$0.412026\pi$
$314$	3447.72	0.619637
$315$	−11049.0	−1.97631
$316$	−3090.18	−0.550115
$317$	−10540.1	−1.86749	−0.933744	−	0.357942i	$-0.883478\pi$
$317$	−10540.1	−1.86749	−0.933744	+	0.357942i	$0.883478\pi$
$318$	−202.203	−0.0356571
$319$	0	0
$320$	954.761	0.166790
$321$	−3087.55	−0.536854
$322$	−7738.43	−1.33927
$323$	−547.054	−0.0942381
$324$	−1959.99	−0.336074
$325$	4295.02	0.733061
$326$	6636.44	1.12748
$327$	3951.42	0.668239
$328$	2024.97	0.340885
$329$	−10864.2	−1.82055
$330$	0	0
$331$	309.871	0.0514563	0.0257281	−	0.999669i	$-0.491810\pi$
$331$	309.871	0.0514563	0.0257281	+	0.999669i	$0.491810\pi$
$332$	−2148.97	−0.355241
$333$	7564.88	1.24490
$334$	6031.20	0.988061
$335$	−7751.78	−1.26425
$336$	2720.74	0.441752
$337$	−11164.7	−1.80468	−0.902340	−	0.431024i	$-0.858152\pi$
$337$	−11164.7	−1.80468	−0.902340	+	0.431024i	$0.858152\pi$
$338$	−516.972	−0.0831941
$339$	−12011.6	−1.92443
$340$	1486.71	0.237142
$341$	0	0
$342$	−1494.17	−0.236244
$343$	4618.28	0.727008
$344$	−1044.99	−0.163785
$345$	−20714.5	−3.23255
$346$	4418.82	0.686581
$347$	−2515.48	−0.389159	−0.194580	−	0.980887i	$-0.562334\pi$
$347$	−2515.48	−0.389159	−0.194580	+	0.980887i	$0.562334\pi$
$348$	−4668.23	−0.719090
$349$	11376.6	1.74491	0.872457	−	0.488692i	$-0.162526\pi$
$349$	11376.6	1.74491	0.872457	+	0.488692i	$0.162526\pi$
$350$	−4246.94	−0.648596
$351$	−2416.01	−0.367400
$352$	0	0
$353$	6210.08	0.936343	0.468172	−	0.883638i	$-0.344913\pi$
$353$	6210.08	0.936343	0.468172	+	0.883638i	$0.344913\pi$
$354$	−555.598	−0.0834172
$355$	1170.31	0.174968
$356$	2668.36	0.397255
$357$	4236.62	0.628084
$358$	−4543.66	−0.670782
$359$	2487.51	0.365698	0.182849	−	0.983141i	$-0.441468\pi$
$359$	2487.51	0.365698	0.182849	+	0.983141i	$0.441468\pi$
$360$	4060.65	0.594487
$361$	−6376.88	−0.929710
$362$	−1248.87	−0.181324
$363$	0	0
$364$	−3833.63	−0.552024
$365$	−17067.4	−2.44753
$366$	8410.88	1.20121
$367$	6719.53	0.955739	0.477870	−	0.878431i	$-0.341409\pi$
$367$	6719.53	0.955739	0.477870	+	0.878431i	$0.341409\pi$
$368$	2843.98	0.402861
$369$	8612.31	1.21501
$370$	6633.69	0.932079
$371$	−281.722	−0.0394239
$372$	−2348.03	−0.327258
$373$	8614.37	1.19580	0.597902	−	0.801569i	$-0.296001\pi$
$373$	8614.37	1.19580	0.597902	+	0.801569i	$0.296001\pi$
$374$	0	0
$375$	3198.87	0.440503
$376$	3992.75	0.547634
$377$	6577.70	0.898592
$378$	2388.97	0.325067
$379$	−8009.10	−1.08549	−0.542744	−	0.839898i	$-0.682614\pi$
$379$	−8009.10	−1.08549	−0.542744	+	0.839898i	$0.682614\pi$
$380$	−1310.25	−0.176879
$381$	−5447.43	−0.732494
$382$	−2688.40	−0.360079
$383$	8011.95	1.06891	0.534454	−	0.845198i	$-0.320517\pi$
$383$	8011.95	1.06891	0.534454	+	0.845198i	$0.320517\pi$
$384$	−999.912	−0.132882
$385$	0	0
$386$	−9247.52	−1.21940
$387$	−4444.39	−0.583775
$388$	−718.615	−0.0940262
$389$	8029.60	1.04657	0.523287	−	0.852157i	$-0.324706\pi$
$389$	8029.60	1.04657	0.523287	+	0.852157i	$0.324706\pi$
$390$	−10262.0	−1.33240
$391$	4428.53	0.572789
$392$	1046.72	0.134865
$393$	−2030.37	−0.260607
$394$	−1329.38	−0.169983
$395$	−11524.9	−1.46806
$396$	0	0
$397$	4435.00	0.560670	0.280335	−	0.959902i	$-0.409554\pi$
$397$	4435.00	0.560670	0.280335	+	0.959902i	$0.409554\pi$
$398$	−6085.64	−0.766446
$399$	−3733.75	−0.468475
$400$	1560.81	0.195102
$401$	−3929.21	−0.489315	−0.244657	−	0.969610i	$-0.578675\pi$
$401$	−3929.21	−0.489315	−0.244657	+	0.969610i	$0.578675\pi$
$402$	8118.37	1.00723
$403$	3308.46	0.408949
$404$	1642.94	0.202325
$405$	−7309.84	−0.896861
$406$	−6504.07	−0.795054
$407$	0	0
$408$	−1557.02	−0.188931
$409$	14564.2	1.76077	0.880384	−	0.474261i	$-0.157285\pi$
$409$	14564.2	1.76077	0.880384	+	0.474261i	$0.157285\pi$
$410$	7552.20	0.909698
$411$	16276.4	1.95342
$412$	5470.64	0.654172
$413$	−774.094	−0.0922293
$414$	12095.6	1.43591
$415$	−8014.65	−0.948008
$416$	1408.91	0.166052
$417$	1354.79	0.159099
$418$	0	0
$419$	4028.77	0.469734	0.234867	−	0.972028i	$-0.424535\pi$
$419$	4028.77	0.469734	0.234867	+	0.972028i	$0.424535\pi$
$420$	10147.1	1.17888
$421$	−8527.60	−0.987197	−0.493599	−	0.869690i	$-0.664319\pi$
$421$	−8527.60	−0.987197	−0.493599	+	0.869690i	$0.664319\pi$
$422$	−5183.22	−0.597903
$423$	16981.4	1.95192
$424$	103.537	0.0118589
$425$	2430.43	0.277396
$426$	−1225.65	−0.139397
$427$	11718.6	1.32811
$428$	1580.96	0.178548
$429$	0	0
$430$	−3897.31	−0.437082
$431$	−13415.2	−1.49928	−0.749639	−	0.661847i	$-0.769773\pi$
$431$	−13415.2	−1.49928	−0.749639	+	0.661847i	$0.769773\pi$
$432$	−877.980	−0.0977821
$433$	4132.31	0.458628	0.229314	−	0.973352i	$-0.426352\pi$
$433$	4132.31	0.458628	0.229314	+	0.973352i	$0.426352\pi$
$434$	−3271.43	−0.361829
$435$	−17410.3	−1.91899
$436$	−2023.30	−0.222245
$437$	−3902.88	−0.427231
$438$	17874.5	1.94995
$439$	−3358.46	−0.365126	−0.182563	−	0.983194i	$-0.558439\pi$
$439$	−3358.46	−0.365126	−0.182563	+	0.983194i	$0.558439\pi$
$440$	0	0
$441$	4451.74	0.480698
$442$	2193.90	0.236093
$443$	−441.749	−0.0473773	−0.0236886	−	0.999719i	$-0.507541\pi$
$443$	−441.749	−0.0473773	−0.0236886	+	0.999719i	$0.507541\pi$
$444$	−6947.41	−0.742588
$445$	9951.73	1.06013
$446$	5274.41	0.559979
$447$	−3507.93	−0.371185
$448$	−1393.14	−0.146919
$449$	−408.478	−0.0429338	−0.0214669	−	0.999770i	$-0.506834\pi$
$449$	−408.478	−0.0429338	−0.0214669	+	0.999770i	$0.506834\pi$
$450$	6638.23	0.695398
$451$	0	0
$452$	6150.49	0.640033
$453$	222.916	0.0231204
$454$	−501.340	−0.0518261
$455$	−14297.6	−1.47315
$456$	1372.21	0.140920
$457$	−1522.85	−0.155877	−0.0779384	−	0.996958i	$-0.524834\pi$
$457$	−1522.85	−0.155877	−0.0779384	+	0.996958i	$0.524834\pi$
$458$	3598.63	0.367146
$459$	−1367.15	−0.139027
$460$	10606.7	1.07509
$461$	13861.4	1.40041	0.700203	−	0.713944i	$-0.253093\pi$
$461$	13861.4	1.40041	0.700203	+	0.713944i	$0.253093\pi$
$462$	0	0
$463$	−6502.26	−0.652669	−0.326334	−	0.945254i	$-0.605814\pi$
$463$	−6502.26	−0.652669	−0.326334	+	0.945254i	$0.605814\pi$
$464$	2390.34	0.239157
$465$	−8757.07	−0.873332
$466$	−6360.95	−0.632329
$467$	423.973	0.0420110	0.0210055	−	0.999779i	$-0.493313\pi$
$467$	423.973	0.0420110	0.0210055	+	0.999779i	$0.493313\pi$
$468$	5992.19	0.591857
$469$	11311.0	1.11364
$470$	14891.1	1.46144
$471$	−13466.5	−1.31742
$472$	284.491	0.0277431
$473$	0	0
$474$	12070.0	1.16960
$475$	−2141.95	−0.206904
$476$	−2169.34	−0.208890
$477$	440.348	0.0422687
$478$	−10027.6	−0.959523
$479$	−4557.34	−0.434718	−0.217359	−	0.976092i	$-0.569744\pi$
$479$	−4557.34	−0.434718	−0.217359	+	0.976092i	$0.569744\pi$
$480$	−3729.21	−0.354613
$481$	9789.15	0.927956
$482$	−12148.3	−1.14800
$483$	30225.6	2.84744
$484$	0	0
$485$	−2680.10	−0.250922
$486$	10618.7	0.991100
$487$	−6708.92	−0.624251	−0.312126	−	0.950041i	$-0.601041\pi$
$487$	−6708.92	−0.624251	−0.312126	+	0.950041i	$0.601041\pi$
$488$	−4306.75	−0.399503
$489$	−25921.3	−2.39714
$490$	3903.76	0.359906
$491$	−14527.7	−1.33528	−0.667641	−	0.744483i	$-0.732696\pi$
$491$	−14527.7	−1.33528	−0.667641	+	0.744483i	$0.732696\pi$
$492$	−7909.35	−0.724758
$493$	3722.13	0.340034
$494$	−1933.49	−0.176097
$495$	0	0
$496$	1202.30	0.108840
$497$	−1707.66	−0.154123
$498$	8393.67	0.755279
$499$	−9699.53	−0.870162	−0.435081	−	0.900391i	$-0.643280\pi$
$499$	−9699.53	−0.870162	−0.435081	+	0.900391i	$0.643280\pi$
$500$	−1637.96	−0.146504
$501$	−23557.3	−2.10072
$502$	−11139.4	−0.990388
$503$	−15707.1	−1.39234	−0.696170	−	0.717877i	$-0.745114\pi$
$503$	−15707.1	−1.39234	−0.696170	+	0.717877i	$0.745114\pi$
$504$	−5925.12	−0.523662
$505$	6127.41	0.539933
$506$	0	0
$507$	2019.25	0.176879
$508$	2789.33	0.243615
$509$	−1474.08	−0.128364	−0.0641822	−	0.997938i	$-0.520444\pi$
$509$	−1474.08	−0.128364	−0.0641822	+	0.997938i	$0.520444\pi$
$510$	−5806.96	−0.504189
$511$	24903.9	2.15594
$512$	512.000	0.0441942
$513$	1204.88	0.103697
$514$	−11814.1	−1.01381
$515$	20402.9	1.74575
$516$	4081.62	0.348223
$517$	0	0
$518$	−9679.57	−0.821035
$519$	−17259.5	−1.45975
$520$	5254.59	0.443133
$521$	−9272.84	−0.779752	−0.389876	−	0.920867i	$-0.627482\pi$
$521$	−9272.84	−0.779752	−0.389876	+	0.920867i	$0.627482\pi$
$522$	10166.3	0.852423
$523$	4242.50	0.354707	0.177353	−	0.984147i	$-0.443246\pi$
$523$	4242.50	0.354707	0.177353	+	0.984147i	$0.443246\pi$
$524$	1039.64	0.0866735
$525$	16588.2	1.37899
$526$	13707.5	1.13626
$527$	1872.17	0.154749
$528$	0	0
$529$	19427.7	1.59676
$530$	386.144	0.0316472
$531$	1209.96	0.0988844
$532$	1911.85	0.155807
$533$	11144.6	0.905675
$534$	−10422.4	−0.844606
$535$	5896.25	0.476481
$536$	−4156.97	−0.334988
$537$	17747.1	1.42615
$538$	2202.73	0.176517
$539$	0	0
$540$	−3274.46	−0.260945
$541$	6520.25	0.518166	0.259083	−	0.965855i	$-0.416580\pi$
$541$	6520.25	0.518166	0.259083	+	0.965855i	$0.416580\pi$
$542$	5143.39	0.407615
$543$	4877.97	0.385513
$544$	797.264	0.0628353
$545$	−7545.98	−0.593091
$546$	14973.8	1.17366
$547$	−6370.78	−0.497980	−0.248990	−	0.968506i	$-0.580099\pi$
$547$	−6370.78	−0.497980	−0.248990	+	0.968506i	$0.580099\pi$
$548$	−8334.24	−0.649674
$549$	−18316.8	−1.42394
$550$	0	0
$551$	−3280.33	−0.253624
$552$	−11108.3	−0.856526
$553$	16816.7	1.29316
$554$	14680.2	1.12582
$555$	−25910.6	−1.98170
$556$	−693.715	−0.0529138
$557$	−6103.99	−0.464335	−0.232167	−	0.972676i	$-0.574582\pi$
$557$	−6103.99	−0.464335	−0.232167	+	0.972676i	$0.574582\pi$
$558$	5113.44	0.387938
$559$	−5751.15	−0.435148
$560$	−5195.77	−0.392074
$561$	0	0
$562$	14985.9	1.12481
$563$	13592.6	1.01751	0.508756	−	0.860911i	$-0.330106\pi$
$563$	13592.6	1.01751	0.508756	+	0.860911i	$0.330106\pi$
$564$	−15595.3	−1.16433
$565$	22938.5	1.70802
$566$	12988.2	0.964549
$567$	10666.2	0.790013
$568$	627.589	0.0463610
$569$	−17197.6	−1.26706	−0.633532	−	0.773717i	$-0.718395\pi$
$569$	−17197.6	−1.26706	−0.633532	+	0.773717i	$0.718395\pi$
$570$	5117.70	0.376064
$571$	−2475.65	−0.181441	−0.0907203	−	0.995876i	$-0.528917\pi$
$571$	−2475.65	−0.181441	−0.0907203	+	0.995876i	$0.528917\pi$
$572$	0	0
$573$	10500.6	0.765567
$574$	−11019.8	−0.801321
$575$	17339.6	1.25758
$576$	2177.57	0.157521
$577$	−20339.7	−1.46751	−0.733755	−	0.679414i	$-0.762234\pi$
$577$	−20339.7	−1.46751	−0.733755	+	0.679414i	$0.762234\pi$
$578$	−8584.54	−0.617767
$579$	36120.0	2.59256
$580$	8914.86	0.638223
$581$	11694.6	0.835067
$582$	2806.85	0.199910
$583$	0	0
$584$	−9152.55	−0.648519
$585$	22348.1	1.57945
$586$	5369.75	0.378536
$587$	−14818.4	−1.04195	−0.520973	−	0.853573i	$-0.674431\pi$
$587$	−14818.4	−1.04195	−0.520973	+	0.853573i	$0.674431\pi$
$588$	−4088.37	−0.286738
$589$	−1649.95	−0.115424
$590$	1061.02	0.0740363
$591$	5192.44	0.361402
$592$	3557.38	0.246972
$593$	−11123.2	−0.770281	−0.385140	−	0.922858i	$-0.625847\pi$
$593$	−11123.2	−0.770281	−0.385140	+	0.922858i	$0.625847\pi$
$594$	0	0
$595$	−8090.63	−0.557451
$596$	1796.22	0.123450
$597$	23770.0	1.62955
$598$	15652.1	1.07034
$599$	8285.18	0.565147	0.282574	−	0.959246i	$-0.408812\pi$
$599$	8285.18	0.565147	0.282574	+	0.959246i	$0.408812\pi$
$600$	−6096.39	−0.414807
$601$	−27318.3	−1.85414	−0.927069	−	0.374890i	$-0.877680\pi$
$601$	−27318.3	−1.85414	−0.927069	+	0.374890i	$0.877680\pi$
$602$	5686.78	0.385009
$603$	−17679.8	−1.19399
$604$	−114.143	−0.00768944
$605$	0	0
$606$	−6417.18	−0.430165
$607$	−16327.7	−1.09180	−0.545898	−	0.837851i	$-0.683812\pi$
$607$	−16327.7	−1.09180	−0.545898	+	0.837851i	$0.683812\pi$
$608$	−702.632	−0.0468676
$609$	25404.3	1.69037
$610$	−16062.2	−1.06613
$611$	21974.4	1.45497
$612$	3390.81	0.223963
$613$	21025.3	1.38533	0.692663	−	0.721262i	$-0.256437\pi$
$613$	21025.3	1.38533	0.692663	+	0.721262i	$0.256437\pi$
$614$	−16663.3	−1.09524
$615$	−29498.2	−1.93412
$616$	0	0
$617$	871.824	0.0568854	0.0284427	−	0.999595i	$-0.490945\pi$
$617$	871.824	0.0568854	0.0284427	+	0.999595i	$0.490945\pi$
$618$	−21367.8	−1.39084
$619$	9397.52	0.610207	0.305104	−	0.952319i	$-0.401309\pi$
$619$	9397.52	0.610207	0.305104	+	0.952319i	$0.401309\pi$
$620$	4484.01	0.290455
$621$	−9753.76	−0.630282
$622$	−10040.7	−0.647260
$623$	−14521.1	−0.933829
$624$	−5503.09	−0.353045
$625$	−18302.7	−1.17137
$626$	6044.15	0.385899
$627$	0	0
$628$	6895.44	0.438150
$629$	5539.40	0.351145
$630$	−22097.9	−1.39746
$631$	−14535.9	−0.917062	−0.458531	−	0.888678i	$-0.651624\pi$
$631$	−14535.9	−0.917062	−0.458531	+	0.888678i	$0.651624\pi$
$632$	−6180.37	−0.388990
$633$	20245.2	1.27121
$634$	−21080.3	−1.32051
$635$	10402.9	0.650120
$636$	−404.405	−0.0252134
$637$	5760.67	0.358314
$638$	0	0
$639$	2669.17	0.165244
$640$	1909.52	0.117938
$641$	−11419.7	−0.703666	−0.351833	−	0.936063i	$-0.614441\pi$
$641$	−11419.7	−0.703666	−0.351833	+	0.936063i	$0.614441\pi$
$642$	−6175.09	−0.379613
$643$	23969.9	1.47011	0.735055	−	0.678007i	$-0.237156\pi$
$643$	23969.9	1.47011	0.735055	+	0.678007i	$0.237156\pi$
$644$	−15476.9	−0.947009
$645$	15222.5	0.929282
$646$	−1094.11	−0.0666364
$647$	8314.25	0.505204	0.252602	−	0.967570i	$-0.418714\pi$
$647$	8314.25	0.505204	0.252602	+	0.967570i	$0.418714\pi$
$648$	−3919.97	−0.237641
$649$	0	0
$650$	8590.04	0.518353
$651$	12777.9	0.769287
$652$	13272.9	0.797249
$653$	−30079.3	−1.80260	−0.901298	−	0.433200i	$-0.857384\pi$
$653$	−30079.3	−1.80260	−0.901298	+	0.433200i	$0.857384\pi$
$654$	7902.84	0.472516
$655$	3877.38	0.231300
$656$	4049.94	0.241042
$657$	−38926.3	−2.31151
$658$	−21728.4	−1.28733
$659$	10041.6	0.593572	0.296786	−	0.954944i	$-0.404085\pi$
$659$	10041.6	0.593572	0.296786	+	0.954944i	$0.404085\pi$
$660$	0	0
$661$	1402.50	0.0825281	0.0412640	−	0.999148i	$-0.486862\pi$
$661$	1402.50	0.0825281	0.0412640	+	0.999148i	$0.486862\pi$
$662$	619.741	0.0363851
$663$	−8569.17	−0.501959
$664$	−4297.93	−0.251193
$665$	7130.31	0.415792
$666$	15129.8	0.880279
$667$	26555.1	1.54155
$668$	12062.4	0.698665
$669$	−20601.4	−1.19057
$670$	−15503.6	−0.893963
$671$	0	0
$672$	5441.49	0.312366
$673$	−20385.4	−1.16761	−0.583803	−	0.811895i	$-0.698436\pi$
$673$	−20385.4	−1.16761	−0.583803	+	0.811895i	$0.698436\pi$
$674$	−22329.3	−1.27610
$675$	−5352.98	−0.305239
$676$	−1033.94	−0.0588271
$677$	7385.00	0.419245	0.209622	−	0.977782i	$-0.432777\pi$
$677$	7385.00	0.419245	0.209622	+	0.977782i	$0.432777\pi$
$678$	−24023.3	−1.36078
$679$	3910.68	0.221028
$680$	2973.42	0.167685
$681$	1958.19	0.110188
$682$	0	0
$683$	−25844.0	−1.44787	−0.723935	−	0.689868i	$-0.757668\pi$
$683$	−25844.0	−1.44787	−0.723935	+	0.689868i	$0.757668\pi$
$684$	−2988.33	−0.167049
$685$	−31082.8	−1.73374
$686$	9236.56	0.514072
$687$	−14055.9	−0.780591
$688$	−2089.97	−0.115813
$689$	569.822	0.0315073
$690$	−41428.9	−2.28576
$691$	8438.80	0.464583	0.232292	−	0.972646i	$-0.425378\pi$
$691$	8438.80	0.464583	0.232292	+	0.972646i	$0.425378\pi$
$692$	8837.64	0.485486
$693$	0	0
$694$	−5030.97	−0.275177
$695$	−2587.23	−0.141208
$696$	−9336.45	−0.508473
$697$	6306.39	0.342714
$698$	22753.2	1.23384
$699$	24845.3	1.34440
$700$	−8493.89	−0.458627
$701$	12983.4	0.699536	0.349768	−	0.936836i	$-0.386260\pi$
$701$	12983.4	0.699536	0.349768	+	0.936836i	$0.386260\pi$
$702$	−4832.03	−0.259791
$703$	−4881.90	−0.261912
$704$	0	0
$705$	−58163.2	−3.10717
$706$	12420.2	0.662095
$707$	−8940.83	−0.475608
$708$	−1111.20	−0.0589848
$709$	−17918.6	−0.949150	−0.474575	−	0.880215i	$-0.657398\pi$
$709$	−17918.6	−0.949150	−0.474575	+	0.880215i	$0.657398\pi$
$710$	2340.61	0.123721
$711$	−26285.4	−1.38647
$712$	5336.71	0.280901
$713$	13356.7	0.701560
$714$	8473.24	0.444122
$715$	0	0
$716$	−9087.33	−0.474315
$717$	39166.9	2.04005
$718$	4975.01	0.258587
$719$	3517.35	0.182441	0.0912204	−	0.995831i	$-0.470923\pi$
$719$	3517.35	0.182441	0.0912204	+	0.995831i	$0.470923\pi$
$720$	8121.30	0.420366
$721$	−29771.0	−1.53777
$722$	−12753.8	−0.657404
$723$	47449.9	2.44078
$724$	−2497.74	−0.128215
$725$	14573.7	0.746559
$726$	0	0
$727$	−29438.9	−1.50183	−0.750913	−	0.660401i	$-0.770386\pi$
$727$	−29438.9	−1.50183	−0.750913	+	0.660401i	$0.770386\pi$
$728$	−7667.26	−0.390340
$729$	−28245.8	−1.43503
$730$	−34134.8	−1.73066
$731$	−3254.41	−0.164663
$732$	16821.8	0.849385
$733$	−3435.90	−0.173135	−0.0865673	−	0.996246i	$-0.527590\pi$
$733$	−3435.90	−0.173135	−0.0865673	+	0.996246i	$0.527590\pi$
$734$	13439.1	0.675810
$735$	−15247.7	−0.765199
$736$	5687.97	0.284866
$737$	0	0
$738$	17224.6	0.859143
$739$	33601.8	1.67261	0.836307	−	0.548262i	$-0.184710\pi$
$739$	33601.8	1.67261	0.836307	+	0.548262i	$0.184710\pi$
$740$	13267.4	0.659079
$741$	7552.04	0.374401
$742$	−563.444	−0.0278769
$743$	1193.70	0.0589404	0.0294702	−	0.999566i	$-0.490618\pi$
$743$	1193.70	0.0589404	0.0294702	+	0.999566i	$0.490618\pi$
$744$	−4696.06	−0.231406
$745$	6699.06	0.329443
$746$	17228.7	0.845562
$747$	−18279.4	−0.895324
$748$	0	0
$749$	−8603.54	−0.419715
$750$	6397.73	0.311483
$751$	24509.4	1.19089	0.595446	−	0.803396i	$-0.296976\pi$
$751$	24509.4	1.19089	0.595446	+	0.803396i	$0.296976\pi$
$752$	7985.50	0.387235
$753$	43509.4	2.10567
$754$	13155.4	0.635400
$755$	−425.701	−0.0205203
$756$	4777.94	0.229857
$757$	10803.8	0.518721	0.259360	−	0.965781i	$-0.416488\pi$
$757$	10803.8	0.518721	0.259360	+	0.965781i	$0.416488\pi$
$758$	−16018.2	−0.767555
$759$	0	0
$760$	−2620.49	−0.125073
$761$	−8175.91	−0.389457	−0.194728	−	0.980857i	$-0.562383\pi$
$761$	−8175.91	−0.389457	−0.194728	+	0.980857i	$0.562383\pi$
$762$	−10894.9	−0.517951
$763$	11010.8	0.522432
$764$	−5376.79	−0.254615
$765$	12646.1	0.597676
$766$	16023.9	0.755831
$767$	1565.72	0.0737089
$768$	−1999.82	−0.0939615
$769$	28895.9	1.35502	0.677511	−	0.735513i	$-0.263059\pi$
$769$	28895.9	1.35502	0.677511	+	0.735513i	$0.263059\pi$
$770$	0	0
$771$	46144.8	2.15547
$772$	−18495.0	−0.862243
$773$	−17996.8	−0.837387	−0.418693	−	0.908128i	$-0.637512\pi$
$773$	−17996.8	−0.837387	−0.418693	+	0.908128i	$0.637512\pi$
$774$	−8888.77	−0.412791
$775$	7330.32	0.339759
$776$	−1437.23	−0.0664866
$777$	37807.5	1.74561
$778$	16059.2	0.740039
$779$	−5557.85	−0.255623
$780$	−20524.0	−0.942148
$781$	0	0
$782$	8857.06	0.405023
$783$	−8197.94	−0.374164
$784$	2093.43	0.0953640
$785$	25716.8	1.16926
$786$	−4060.74	−0.184277
$787$	15876.7	0.719114	0.359557	−	0.933123i	$-0.382928\pi$
$787$	15876.7	0.719114	0.359557	+	0.933123i	$0.382928\pi$
$788$	−2658.76	−0.120196
$789$	−53540.1	−2.41582
$790$	−23049.9	−1.03807
$791$	−33470.8	−1.50453
$792$	0	0
$793$	−23702.5	−1.06141
$794$	8869.99	0.396454
$795$	−1508.24	−0.0672854
$796$	−12171.3	−0.541959
$797$	16497.7	0.733223	0.366611	−	0.930374i	$-0.380518\pi$
$797$	16497.7	0.733223	0.366611	+	0.930374i	$0.380518\pi$
$798$	−7467.51	−0.331262
$799$	12434.7	0.550572
$800$	3121.63	0.137958
$801$	22697.3	1.00121
$802$	−7858.41	−0.345998
$803$	0	0
$804$	16236.7	0.712221
$805$	−57721.5	−2.52722
$806$	6616.93	0.289170
$807$	−8603.65	−0.375295
$808$	3285.88	0.143066
$809$	9102.95	0.395603	0.197801	−	0.980242i	$-0.436620\pi$
$809$	9102.95	0.395603	0.197801	+	0.980242i	$0.436620\pi$
$810$	−14619.7	−0.634176
$811$	−36576.2	−1.58368	−0.791841	−	0.610728i	$-0.790877\pi$
$811$	−36576.2	−1.58368	−0.791841	+	0.610728i	$0.790877\pi$
$812$	−13008.1	−0.562188
$813$	−20089.6	−0.866633
$814$	0	0
$815$	49501.7	2.12757
$816$	−3114.04	−0.133595
$817$	2868.13	0.122819
$818$	29128.4	1.24505
$819$	−32609.3	−1.39128
$820$	15104.4	0.643254
$821$	3644.21	0.154913	0.0774567	−	0.996996i	$-0.475320\pi$
$821$	3644.21	0.154913	0.0774567	+	0.996996i	$0.475320\pi$
$822$	32552.8	1.38128
$823$	−16895.9	−0.715618	−0.357809	−	0.933795i	$-0.616476\pi$
$823$	−16895.9	−0.715618	−0.357809	+	0.933795i	$0.616476\pi$
$824$	10941.3	0.462570
$825$	0	0
$826$	−1548.19	−0.0652160
$827$	38778.1	1.63053	0.815263	−	0.579090i	$-0.196592\pi$
$827$	38778.1	1.63053	0.815263	+	0.579090i	$0.196592\pi$
$828$	24191.3	1.01534
$829$	−12394.3	−0.519267	−0.259634	−	0.965707i	$-0.583602\pi$
$829$	−12394.3	−0.519267	−0.259634	+	0.965707i	$0.583602\pi$
$830$	−16029.3	−0.670343
$831$	−57339.5	−2.39361
$832$	2817.83	0.117417
$833$	3259.80	0.135589
$834$	2709.59	0.112500
$835$	44987.1	1.86448
$836$	0	0
$837$	−4123.41	−0.170282
$838$	8057.55	0.332152
$839$	24563.8	1.01077	0.505386	−	0.862893i	$-0.331350\pi$
$839$	24563.8	1.01077	0.505386	+	0.862893i	$0.331350\pi$
$840$	20294.2	0.833591
$841$	−2069.74	−0.0848635
$842$	−17055.2	−0.698054
$843$	−58533.5	−2.39146
$844$	−10366.4	−0.422781
$845$	−3856.13	−0.156988
$846$	33962.8	1.38022
$847$	0	0
$848$	207.074	0.00838554
$849$	−50730.7	−2.05073
$850$	4860.86	0.196148
$851$	39520.1	1.59193
$852$	−2451.30	−0.0985684
$853$	12207.6	0.490013	0.245007	−	0.969521i	$-0.421210\pi$
$853$	12207.6	0.490013	0.245007	+	0.969521i	$0.421210\pi$
$854$	23437.2	0.939114
$855$	−11145.1	−0.445794
$856$	3161.93	0.126253
$857$	−7281.72	−0.290244	−0.145122	−	0.989414i	$-0.546357\pi$
$857$	−7281.72	−0.290244	−0.145122	+	0.989414i	$0.546357\pi$
$858$	0	0
$859$	5927.39	0.235437	0.117718	−	0.993047i	$-0.462442\pi$
$859$	5927.39	0.235437	0.117718	+	0.993047i	$0.462442\pi$
$860$	−7794.62	−0.309063
$861$	43042.4	1.70369
$862$	−26830.4	−1.06015
$863$	−38561.9	−1.52104	−0.760522	−	0.649312i	$-0.775057\pi$
$863$	−38561.9	−1.52104	−0.760522	+	0.649312i	$0.775057\pi$
$864$	−1755.96	−0.0691424
$865$	32960.3	1.29559
$866$	8264.62	0.324299
$867$	33530.4	1.31344
$868$	−6542.86	−0.255852
$869$	0	0
$870$	−34820.6	−1.35693
$871$	−22878.2	−0.890009
$872$	−4046.61	−0.157151
$873$	−6112.63	−0.236977
$874$	−7805.76	−0.302098
$875$	8913.73	0.344388
$876$	35749.0	1.37882
$877$	−28512.7	−1.09784	−0.548919	−	0.835875i	$-0.684961\pi$
$877$	−28512.7	−1.09784	−0.548919	+	0.835875i	$0.684961\pi$
$878$	−6716.91	−0.258183
$879$	−20973.7	−0.804809
$880$	0	0
$881$	40747.6	1.55826	0.779128	−	0.626865i	$-0.215662\pi$
$881$	40747.6	1.55826	0.779128	+	0.626865i	$0.215662\pi$
$882$	8903.48	0.339905
$883$	−3595.59	−0.137034	−0.0685171	−	0.997650i	$-0.521827\pi$
$883$	−3595.59	−0.137034	−0.0685171	+	0.997650i	$0.521827\pi$
$884$	4387.80	0.166943
$885$	−4144.24	−0.157409
$886$	−883.499	−0.0335008
$887$	7715.30	0.292057	0.146028	−	0.989280i	$-0.453351\pi$
$887$	7715.30	0.292057	0.146028	+	0.989280i	$0.453351\pi$
$888$	−13894.8	−0.525089
$889$	−15179.4	−0.572667
$890$	19903.5	0.749624
$891$	0	0
$892$	10548.8	0.395965
$893$	−10958.7	−0.410660
$894$	−7015.87	−0.262467
$895$	−33891.5	−1.26577
$896$	−2786.29	−0.103888
$897$	−61135.5	−2.27565
$898$	−816.957	−0.0303588
$899$	11226.2	0.416478
$900$	13276.5	0.491721
$901$	322.446	0.0119226
$902$	0	0
$903$	−22212.0	−0.818571
$904$	12301.0	0.452572
$905$	−9315.41	−0.342160
$906$	445.833	0.0163486
$907$	−33713.7	−1.23423	−0.617114	−	0.786874i	$-0.711698\pi$
$907$	−33713.7	−1.23423	−0.617114	+	0.786874i	$0.711698\pi$
$908$	−1002.68	−0.0366466
$909$	13975.1	0.509927
$910$	−28595.3	−1.04168
$911$	15030.7	0.546642	0.273321	−	0.961923i	$-0.411878\pi$
$911$	15030.7	0.546642	0.273321	+	0.961923i	$0.411878\pi$
$912$	2744.42	0.0996455
$913$	0	0
$914$	−3045.69	−0.110222
$915$	62737.3	2.26670
$916$	7197.25	0.259611
$917$	−5657.69	−0.203744
$918$	−2734.31	−0.0983067
$919$	17090.7	0.613460	0.306730	−	0.951797i	$-0.400765\pi$
$919$	17090.7	0.613460	0.306730	+	0.951797i	$0.400765\pi$
$920$	21213.5	0.760204
$921$	65085.4	2.32860
$922$	27722.7	0.990237
$923$	3453.98	0.123173
$924$	0	0
$925$	21689.1	0.770955
$926$	−13004.5	−0.461506
$927$	46533.9	1.64873
$928$	4780.68	0.169109
$929$	13160.3	0.464773	0.232386	−	0.972624i	$-0.425347\pi$
$929$	13160.3	0.464773	0.232386	+	0.972624i	$0.425347\pi$
$930$	−17514.1	−0.617539
$931$	−2872.87	−0.101133
$932$	−12721.9	−0.447124
$933$	39218.1	1.37614
$934$	847.946	0.0297063
$935$	0	0
$936$	11984.4	0.418506
$937$	−4056.29	−0.141423	−0.0707114	−	0.997497i	$-0.522527\pi$
$937$	−4056.29	−0.141423	−0.0707114	+	0.997497i	$0.522527\pi$
$938$	22622.1	0.787460
$939$	−23607.9	−0.820463
$940$	29782.2	1.03339
$941$	−33397.6	−1.15699	−0.578497	−	0.815684i	$-0.696361\pi$
$941$	−33397.6	−1.15699	−0.578497	+	0.815684i	$0.696361\pi$
$942$	−26933.0	−0.931553
$943$	44992.1	1.55370
$944$	568.982	0.0196173
$945$	17819.5	0.613405
$946$	0	0
$947$	25660.8	0.880533	0.440267	−	0.897867i	$-0.354884\pi$
$947$	25660.8	0.880533	0.440267	+	0.897867i	$0.354884\pi$
$948$	24139.9	0.827034
$949$	−50371.7	−1.72301
$950$	−4283.90	−0.146303
$951$	82337.7	2.80755
$952$	−4338.68	−0.147707
$953$	−48852.8	−1.66054	−0.830271	−	0.557359i	$-0.811815\pi$
$953$	−48852.8	−1.66054	−0.830271	+	0.557359i	$0.811815\pi$
$954$	880.696	0.0298885
$955$	−20052.9	−0.679474
$956$	−20055.2	−0.678485
$957$	0	0
$958$	−9114.68	−0.307392
$959$	45354.6	1.52719
$960$	−7458.42	−0.250749
$961$	−24144.4	−0.810461
$962$	19578.3	0.656164
$963$	13447.8	0.450001
$964$	−24296.5	−0.811761
$965$	−68977.9	−2.30101
$966$	60451.2	2.01344
$967$	48861.8	1.62491	0.812456	−	0.583023i	$-0.198130\pi$
$967$	48861.8	1.62491	0.812456	+	0.583023i	$0.198130\pi$
$968$	0	0
$969$	4273.48	0.141676
$970$	−5360.20	−0.177429
$971$	34887.5	1.15303	0.576516	−	0.817086i	$-0.304412\pi$
$971$	34887.5	1.15303	0.576516	+	0.817086i	$0.304412\pi$
$972$	21237.4	0.700813
$973$	3775.17	0.124385
$974$	−13417.8	−0.441412
$975$	−33551.9	−1.10207
$976$	−8613.49	−0.282491
$977$	−19012.4	−0.622580	−0.311290	−	0.950315i	$-0.600761\pi$
$977$	−19012.4	−0.622580	−0.311290	+	0.950315i	$0.600761\pi$
$978$	−51842.7	−1.69504
$979$	0	0
$980$	7807.52	0.254492
$981$	−17210.5	−0.560130
$982$	−29055.3	−0.944187
$983$	−7447.89	−0.241659	−0.120829	−	0.992673i	$-0.538555\pi$
$983$	−7447.89	−0.241659	−0.120829	+	0.992673i	$0.538555\pi$
$984$	−15818.7	−0.512481
$985$	−9915.95	−0.320760
$986$	7444.27	0.240440
$987$	84869.1	2.73699
$988$	−3866.98	−0.124519
$989$	−23218.2	−0.746506
$990$	0	0
$991$	5313.37	0.170318	0.0851588	−	0.996367i	$-0.472860\pi$
$991$	5313.37	0.170318	0.0851588	+	0.996367i	$0.472860\pi$
$992$	2404.60	0.0769617
$993$	−2420.65	−0.0773586
$994$	−3415.32	−0.108981
$995$	−45393.2	−1.44629
$996$	16787.3	0.534063
$997$	−53523.9	−1.70022	−0.850110	−	0.526604i	$-0.823465\pi$
$997$	−53523.9	−1.70022	−0.850110	+	0.526604i	$0.823465\pi$
$998$	−19399.1	−0.615297
$999$	−12200.4	−0.386391

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	242.4.a.o.1.1		4
3.2	odd	2		2178.4.a.bt.1.1		4
4.3	odd	2		1936.4.a.bm.1.4		4
11.2	odd	10		242.4.c.r.81.1		8
11.3	even	5		242.4.c.q.9.2		8
11.4	even	5		242.4.c.q.27.2		8
11.5	even	5		242.4.c.n.3.1		8
11.6	odd	10		242.4.c.r.3.1		8
11.7	odd	10		22.4.c.b.5.2	✓	8
11.8	odd	10		22.4.c.b.9.2	yes	8
11.9	even	5		242.4.c.n.81.1		8
11.10	odd	2		242.4.a.n.1.1		4
33.8	even	10		198.4.f.d.163.1		8
33.29	even	10		198.4.f.d.181.1		8
33.32	even	2		2178.4.a.by.1.1		4
44.7	even	10		176.4.m.b.49.1		8
44.19	even	10		176.4.m.b.97.1		8
44.43	even	2		1936.4.a.bn.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.4.c.b.5.2	✓	8	11.7	odd	10
22.4.c.b.9.2	yes	8	11.8	odd	10
176.4.m.b.49.1		8	44.7	even	10
176.4.m.b.97.1		8	44.19	even	10
198.4.f.d.163.1		8	33.8	even	10
198.4.f.d.181.1		8	33.29	even	10
242.4.a.n.1.1		4	11.10	odd	2
242.4.a.o.1.1		4	1.1	even	1	trivial
242.4.c.n.3.1		8	11.5	even	5
242.4.c.n.81.1		8	11.9	even	5
242.4.c.q.9.2		8	11.3	even	5
242.4.c.q.27.2		8	11.4	even	5
242.4.c.r.3.1		8	11.6	odd	10
242.4.c.r.81.1		8	11.2	odd	10
1936.4.a.bm.1.4		4	4.3	odd	2
1936.4.a.bn.1.4		4	44.43	even	2
2178.4.a.bt.1.1		4	3.2	odd	2
2178.4.a.by.1.1		4	33.32	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 242 = 2 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	242.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} -7.81182 q^{3} +4.00000 q^{4} +14.9181 q^{5} -15.6236 q^{6} -21.7679 q^{7} +8.00000 q^{8} +34.0245 q^{9} +O(q^{10})\)	\(q+2.00000 q^{2} -7.81182 q^{3} +4.00000 q^{4} +14.9181 q^{5} -15.6236 q^{6} -21.7679 q^{7} +8.00000 q^{8} +34.0245 q^{9} +29.8363 q^{10} -31.2473 q^{12} +44.0286 q^{13} -43.5357 q^{14} -116.538 q^{15} +16.0000 q^{16} +24.9145 q^{17} +68.0489 q^{18} -21.9573 q^{19} +59.6725 q^{20} +170.046 q^{21} +177.749 q^{23} -62.4945 q^{24} +97.5508 q^{25} +88.0571 q^{26} -54.8738 q^{27} -87.0714 q^{28} +149.396 q^{29} -233.075 q^{30} +75.1436 q^{31} +32.0000 q^{32} +49.8290 q^{34} -324.736 q^{35} +136.098 q^{36} +222.336 q^{37} -43.9145 q^{38} -343.943 q^{39} +119.345 q^{40} +253.121 q^{41} +340.093 q^{42} -130.623 q^{43} +507.582 q^{45} +355.498 q^{46} +499.093 q^{47} -124.989 q^{48} +130.839 q^{49} +195.102 q^{50} -194.627 q^{51} +176.114 q^{52} +12.9421 q^{53} -109.748 q^{54} -174.143 q^{56} +171.526 q^{57} +298.793 q^{58} +35.5614 q^{59} -466.151 q^{60} -538.343 q^{61} +150.287 q^{62} -740.639 q^{63} +64.0000 q^{64} +656.824 q^{65} -519.621 q^{67} +99.6580 q^{68} -1388.54 q^{69} -649.472 q^{70} +78.4486 q^{71} +272.196 q^{72} -1144.07 q^{73} +444.673 q^{74} -762.049 q^{75} -87.8290 q^{76} -687.886 q^{78} -772.546 q^{79} +238.690 q^{80} -489.997 q^{81} +506.243 q^{82} -537.242 q^{83} +680.186 q^{84} +371.678 q^{85} -261.247 q^{86} -1167.06 q^{87} +667.089 q^{89} +1015.16 q^{90} -958.407 q^{91} +710.996 q^{92} -587.008 q^{93} +998.187 q^{94} -327.561 q^{95} -249.978 q^{96} -179.654 q^{97} +261.679 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{2} + 4 q^{3} + 16 q^{4} + 25 q^{5} + 8 q^{6} + 3 q^{7} + 32 q^{8} + 102 q^{9}+O(q^{10}) \) 4 * q + 8 * q^2 + 4 * q^3 + 16 * q^4 + 25 * q^5 + 8 * q^6 + 3 * q^7 + 32 * q^8 + 102 * q^9	\( 4 q + 8 q^{2} + 4 q^{3} + 16 q^{4} + 25 q^{5} + 8 q^{6} + 3 q^{7} + 32 q^{8} + 102 q^{9} + 50 q^{10} + 16 q^{12} - 41 q^{13} + 6 q^{14} + 68 q^{15} + 64 q^{16} + 52 q^{17} + 204 q^{18} + 16 q^{19} + 100 q^{20} + 25 q^{21} + 314 q^{23} + 32 q^{24} - 21 q^{25} - 82 q^{26} + 286 q^{27} + 12 q^{28} + 561 q^{29} + 136 q^{30} + 199 q^{31} + 128 q^{32} + 104 q^{34} - 714 q^{35} + 408 q^{36} + 357 q^{37} + 32 q^{38} - 1038 q^{39} + 200 q^{40} + 32 q^{41} + 50 q^{42} - 721 q^{43} + 1326 q^{45} + 628 q^{46} + 403 q^{47} + 64 q^{48} + 823 q^{49} - 42 q^{50} - 174 q^{51} - 164 q^{52} - 133 q^{53} + 572 q^{54} + 24 q^{56} - 1031 q^{57} + 1122 q^{58} + 1016 q^{59} + 272 q^{60} - 919 q^{61} + 398 q^{62} - 1367 q^{63} + 256 q^{64} + 69 q^{65} + 289 q^{67} + 208 q^{68} - 1620 q^{69} - 1428 q^{70} - 1205 q^{71} + 816 q^{72} - 1234 q^{73} + 714 q^{74} - 911 q^{75} + 64 q^{76} - 2076 q^{78} - 603 q^{79} + 400 q^{80} - 1400 q^{81} + 64 q^{82} + 1514 q^{83} + 100 q^{84} + 717 q^{85} - 1442 q^{86} - 1061 q^{87} - 1101 q^{89} + 2652 q^{90} - 2306 q^{91} + 1256 q^{92} - 2298 q^{93} + 806 q^{94} - 1766 q^{95} + 128 q^{96} + 2116 q^{97} + 1646 q^{98}+O(q^{100}) \) 4 * q + 8 * q^2 + 4 * q^3 + 16 * q^4 + 25 * q^5 + 8 * q^6 + 3 * q^7 + 32 * q^8 + 102 * q^9 + 50 * q^10 + 16 * q^12 - 41 * q^13 + 6 * q^14 + 68 * q^15 + 64 * q^16 + 52 * q^17 + 204 * q^18 + 16 * q^19 + 100 * q^20 + 25 * q^21 + 314 * q^23 + 32 * q^24 - 21 * q^25 - 82 * q^26 + 286 * q^27 + 12 * q^28 + 561 * q^29 + 136 * q^30 + 199 * q^31 + 128 * q^32 + 104 * q^34 - 714 * q^35 + 408 * q^36 + 357 * q^37 + 32 * q^38 - 1038 * q^39 + 200 * q^40 + 32 * q^41 + 50 * q^42 - 721 * q^43 + 1326 * q^45 + 628 * q^46 + 403 * q^47 + 64 * q^48 + 823 * q^49 - 42 * q^50 - 174 * q^51 - 164 * q^52 - 133 * q^53 + 572 * q^54 + 24 * q^56 - 1031 * q^57 + 1122 * q^58 + 1016 * q^59 + 272 * q^60 - 919 * q^61 + 398 * q^62 - 1367 * q^63 + 256 * q^64 + 69 * q^65 + 289 * q^67 + 208 * q^68 - 1620 * q^69 - 1428 * q^70 - 1205 * q^71 + 816 * q^72 - 1234 * q^73 + 714 * q^74 - 911 * q^75 + 64 * q^76 - 2076 * q^78 - 603 * q^79 + 400 * q^80 - 1400 * q^81 + 64 * q^82 + 1514 * q^83 + 100 * q^84 + 717 * q^85 - 1442 * q^86 - 1061 * q^87 - 1101 * q^89 + 2652 * q^90 - 2306 * q^91 + 1256 * q^92 - 2298 * q^93 + 806 * q^94 - 1766 * q^95 + 128 * q^96 + 2116 * q^97 + 1646 * q^98

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