LMFDB - Embedded newform 354.4.a.i.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [354,4,Mod(1,354)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(354, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("354.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 354 = 2 \cdot 3 \cdot 59 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	354.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:	$20.8866761420$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 492x^{4} + 3376x^{3} + 13255x^{2} - 108942x + 106740 $ x^6 - 2x^5 - 492x^4 + 3376x^3 + 13255x^2 - 108942*x + 106740
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2\cdot 3^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-5.84847$ of defining polynomial
Character	$\chi$	$=$	354.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -2.84847 q^{5} +6.00000 q^{6} -2.00843 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -2.84847 q^{5} +6.00000 q^{6} -2.00843 q^{7} +8.00000 q^{8} +9.00000 q^{9} -5.69694 q^{10} +71.0516 q^{11} +12.0000 q^{12} -79.7024 q^{13} -4.01685 q^{14} -8.54541 q^{15} +16.0000 q^{16} +138.146 q^{17} +18.0000 q^{18} +47.8573 q^{19} -11.3939 q^{20} -6.02528 q^{21} +142.103 q^{22} +77.8494 q^{23} +24.0000 q^{24} -116.886 q^{25} -159.405 q^{26} +27.0000 q^{27} -8.03370 q^{28} -110.084 q^{29} -17.0908 q^{30} +280.840 q^{31} +32.0000 q^{32} +213.155 q^{33} +276.292 q^{34} +5.72094 q^{35} +36.0000 q^{36} +99.8653 q^{37} +95.7146 q^{38} -239.107 q^{39} -22.7878 q^{40} +87.4915 q^{41} -12.0506 q^{42} +417.970 q^{43} +284.206 q^{44} -25.6362 q^{45} +155.699 q^{46} -476.698 q^{47} +48.0000 q^{48} -338.966 q^{49} -233.772 q^{50} +414.438 q^{51} -318.809 q^{52} -517.275 q^{53} +54.0000 q^{54} -202.388 q^{55} -16.0674 q^{56} +143.572 q^{57} -220.168 q^{58} -59.0000 q^{59} -34.1817 q^{60} -169.495 q^{61} +561.680 q^{62} -18.0758 q^{63} +64.0000 q^{64} +227.030 q^{65} +426.309 q^{66} +386.562 q^{67} +552.584 q^{68} +233.548 q^{69} +11.4419 q^{70} -65.3562 q^{71} +72.0000 q^{72} +341.558 q^{73} +199.731 q^{74} -350.659 q^{75} +191.429 q^{76} -142.702 q^{77} -478.214 q^{78} -686.296 q^{79} -45.5755 q^{80} +81.0000 q^{81} +174.983 q^{82} +464.254 q^{83} -24.1011 q^{84} -393.505 q^{85} +835.940 q^{86} -330.252 q^{87} +568.412 q^{88} -1373.62 q^{89} -51.2725 q^{90} +160.076 q^{91} +311.397 q^{92} +842.519 q^{93} -953.397 q^{94} -136.320 q^{95} +96.0000 q^{96} -117.863 q^{97} -677.932 q^{98} +639.464 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 12 q^{2} + 18 q^{3} + 24 q^{4} + 20 q^{5} + 36 q^{6} + 26 q^{7} + 48 q^{8} + 54 q^{9}+O(q^{10}) $ 6 * q + 12 * q^2 + 18 * q^3 + 24 * q^4 + 20 * q^5 + 36 * q^6 + 26 * q^7 + 48 * q^8 + 54 * q^9	\( 6 q + 12 q^{2} + 18 q^{3} + 24 q^{4} + 20 q^{5} + 36 q^{6} + 26 q^{7} + 48 q^{8} + 54 q^{9} + 40 q^{10} + 63 q^{11} + 72 q^{12} + 93 q^{13} + 52 q^{14} + 60 q^{15} + 96 q^{16} + 230 q^{17} + 108 q^{18} + 89 q^{19} + 80 q^{20} + 78 q^{21} + 126 q^{22} + 81 q^{23} + 144 q^{24} + 304 q^{25} + 186 q^{26} + 162 q^{27} + 104 q^{28} + 131 q^{29} + 120 q^{30} + 51 q^{31} + 192 q^{32} + 189 q^{33} + 460 q^{34} - 87 q^{35} + 216 q^{36} - 16 q^{37} + 178 q^{38} + 279 q^{39} + 160 q^{40} + 176 q^{41} + 156 q^{42} + 375 q^{43} + 252 q^{44} + 180 q^{45} + 162 q^{46} - 255 q^{47} + 288 q^{48} - 290 q^{49} + 608 q^{50} + 690 q^{51} + 372 q^{52} - 256 q^{53} + 324 q^{54} - 184 q^{55} + 208 q^{56} + 267 q^{57} + 262 q^{58} - 354 q^{59} + 240 q^{60} + 39 q^{61} + 102 q^{62} + 234 q^{63} + 384 q^{64} - 954 q^{65} + 378 q^{66} + 86 q^{67} + 920 q^{68} + 243 q^{69} - 174 q^{70} - 895 q^{71} + 432 q^{72} + 155 q^{73} - 32 q^{74} + 912 q^{75} + 356 q^{76} - 498 q^{77} + 558 q^{78} - 565 q^{79} + 320 q^{80} + 486 q^{81} + 352 q^{82} + 140 q^{83} + 312 q^{84} + q^{85} + 750 q^{86} + 393 q^{87} + 504 q^{88} - 1769 q^{89} + 360 q^{90} - 422 q^{91} + 324 q^{92} + 153 q^{93} - 510 q^{94} - 2209 q^{95} + 576 q^{96} + 48 q^{97} - 580 q^{98} + 567 q^{99}+O(q^{100}) \) 6 * q + 12 * q^2 + 18 * q^3 + 24 * q^4 + 20 * q^5 + 36 * q^6 + 26 * q^7 + 48 * q^8 + 54 * q^9 + 40 * q^10 + 63 * q^11 + 72 * q^12 + 93 * q^13 + 52 * q^14 + 60 * q^15 + 96 * q^16 + 230 * q^17 + 108 * q^18 + 89 * q^19 + 80 * q^20 + 78 * q^21 + 126 * q^22 + 81 * q^23 + 144 * q^24 + 304 * q^25 + 186 * q^26 + 162 * q^27 + 104 * q^28 + 131 * q^29 + 120 * q^30 + 51 * q^31 + 192 * q^32 + 189 * q^33 + 460 * q^34 - 87 * q^35 + 216 * q^36 - 16 * q^37 + 178 * q^38 + 279 * q^39 + 160 * q^40 + 176 * q^41 + 156 * q^42 + 375 * q^43 + 252 * q^44 + 180 * q^45 + 162 * q^46 - 255 * q^47 + 288 * q^48 - 290 * q^49 + 608 * q^50 + 690 * q^51 + 372 * q^52 - 256 * q^53 + 324 * q^54 - 184 * q^55 + 208 * q^56 + 267 * q^57 + 262 * q^58 - 354 * q^59 + 240 * q^60 + 39 * q^61 + 102 * q^62 + 234 * q^63 + 384 * q^64 - 954 * q^65 + 378 * q^66 + 86 * q^67 + 920 * q^68 + 243 * q^69 - 174 * q^70 - 895 * q^71 + 432 * q^72 + 155 * q^73 - 32 * q^74 + 912 * q^75 + 356 * q^76 - 498 * q^77 + 558 * q^78 - 565 * q^79 + 320 * q^80 + 486 * q^81 + 352 * q^82 + 140 * q^83 + 312 * q^84 + q^85 + 750 * q^86 + 393 * q^87 + 504 * q^88 - 1769 * q^89 + 360 * q^90 - 422 * q^91 + 324 * q^92 + 153 * q^93 - 510 * q^94 - 2209 * q^95 + 576 * q^96 + 48 * q^97 - 580 * q^98 + 567 * q^99

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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	4.00000	0.500000
$5$	−2.84847	−0.254775	−0.127388	−	0.991853i	$-0.540659\pi$
$5$	−2.84847	−0.254775	−0.127388	+	0.991853i	$0.540659\pi$
$6$	6.00000	0.408248
$7$	−2.00843	−0.108445	−0.0542224	−	0.998529i	$-0.517268\pi$
$7$	−2.00843	−0.108445	−0.0542224	+	0.998529i	$0.517268\pi$
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	−5.69694	−0.180153
$11$	71.0516	1.94753	0.973766	−	0.227552i	$-0.0730723\pi$
$11$	71.0516	1.94753	0.973766	+	0.227552i	$0.0730723\pi$
$12$	12.0000	0.288675
$13$	−79.7024	−1.70042	−0.850210	−	0.526444i	$-0.823525\pi$
$13$	−79.7024	−1.70042	−0.850210	+	0.526444i	$0.823525\pi$
$14$	−4.01685	−0.0766821
$15$	−8.54541	−0.147094
$16$	16.0000	0.250000
$17$	138.146	1.97090	0.985450	−	0.169965i	$-0.0543653\pi$
$17$	138.146	1.97090	0.985450	+	0.169965i	$0.0543653\pi$
$18$	18.0000	0.235702
$19$	47.8573	0.577853	0.288927	−	0.957351i	$-0.406702\pi$
$19$	47.8573	0.577853	0.288927	+	0.957351i	$0.406702\pi$
$20$	−11.3939	−0.127388
$21$	−6.02528	−0.0626106
$22$	142.103	1.37711
$23$	77.8494	0.705770	0.352885	−	0.935667i	$-0.385201\pi$
$23$	77.8494	0.705770	0.352885	+	0.935667i	$0.385201\pi$
$24$	24.0000	0.204124
$25$	−116.886	−0.935090
$26$	−159.405	−1.20238
$27$	27.0000	0.192450
$28$	−8.03370	−0.0542224
$29$	−110.084	−0.704899	−0.352450	−	0.935831i	$-0.614651\pi$
$29$	−110.084	−0.704899	−0.352450	+	0.935831i	$0.614651\pi$
$30$	−17.0908	−0.104011
$31$	280.840	1.62711	0.813553	−	0.581490i	$-0.197530\pi$
$31$	280.840	1.62711	0.813553	+	0.581490i	$0.197530\pi$
$32$	32.0000	0.176777
$33$	213.155	1.12441
$34$	276.292	1.39364
$35$	5.72094	0.0276290
$36$	36.0000	0.166667
$37$	99.8653	0.443723	0.221861	−	0.975078i	$-0.428787\pi$
$37$	99.8653	0.443723	0.221861	+	0.975078i	$0.428787\pi$
$38$	95.7146	0.408604
$39$	−239.107	−0.981738
$40$	−22.7878	−0.0900766
$41$	87.4915	0.333265	0.166633	−	0.986019i	$-0.446711\pi$
$41$	87.4915	0.333265	0.166633	+	0.986019i	$0.446711\pi$
$42$	−12.0506	−0.0442724
$43$	417.970	1.48232	0.741161	−	0.671328i	$-0.234276\pi$
$43$	417.970	1.48232	0.741161	+	0.671328i	$0.234276\pi$
$44$	284.206	0.973766
$45$	−25.6362	−0.0849250
$46$	155.699	0.499055
$47$	−476.698	−1.47944	−0.739719	−	0.672916i	$-0.765042\pi$
$47$	−476.698	−1.47944	−0.739719	+	0.672916i	$0.765042\pi$
$48$	48.0000	0.144338
$49$	−338.966	−0.988240
$50$	−233.772	−0.661208
$51$	414.438	1.13790
$52$	−318.809	−0.850210
$53$	−517.275	−1.34063	−0.670313	−	0.742078i	$-0.733840\pi$
$53$	−517.275	−1.34063	−0.670313	+	0.742078i	$0.733840\pi$
$54$	54.0000	0.136083
$55$	−202.388	−0.496182
$56$	−16.0674	−0.0383410
$57$	143.572	0.333624
$58$	−220.168	−0.498439
$59$	−59.0000	−0.130189
$59$	−59.0000	−0.130189
$60$	−34.1817	−0.0735472
$61$	−169.495	−0.355764	−0.177882	−	0.984052i	$-0.556925\pi$
$61$	−169.495	−0.355764	−0.177882	+	0.984052i	$0.556925\pi$
$62$	561.680	1.15054
$63$	−18.0758	−0.0361483
$64$	64.0000	0.125000
$65$	227.030	0.433225
$66$	426.309	0.795077
$67$	386.562	0.704866	0.352433	−	0.935837i	$-0.385354\pi$
$67$	386.562	0.704866	0.352433	+	0.935837i	$0.385354\pi$
$68$	552.584	0.985450
$69$	233.548	0.407477
$70$	11.4419	0.0195367
$71$	−65.3562	−0.109244	−0.0546222	−	0.998507i	$-0.517395\pi$
$71$	−65.3562	−0.109244	−0.0546222	+	0.998507i	$0.517395\pi$
$72$	72.0000	0.117851
$73$	341.558	0.547621	0.273811	−	0.961784i	$-0.411716\pi$
$73$	341.558	0.547621	0.273811	+	0.961784i	$0.411716\pi$
$74$	199.731	0.313759
$75$	−350.659	−0.539874
$76$	191.429	0.288927
$77$	−142.702	−0.211200
$78$	−478.214	−0.694194
$79$	−686.296	−0.977397	−0.488698	−	0.872453i	$-0.662528\pi$
$79$	−686.296	−0.977397	−0.488698	+	0.872453i	$0.662528\pi$
$80$	−45.5755	−0.0636938
$81$	81.0000	0.111111
$82$	174.983	0.235654
$83$	464.254	0.613958	0.306979	−	0.951716i	$-0.400682\pi$
$83$	464.254	0.613958	0.306979	+	0.951716i	$0.400682\pi$
$84$	−24.1011	−0.0313053
$85$	−393.505	−0.502136
$86$	835.940	1.04816
$87$	−330.252	−0.406974
$88$	568.412	0.688556
$89$	−1373.62	−1.63600	−0.817999	−	0.575220i	$-0.804916\pi$
$89$	−1373.62	−1.63600	−0.817999	+	0.575220i	$0.804916\pi$
$90$	−51.2725	−0.0600510
$91$	160.076	0.184402
$92$	311.397	0.352885
$93$	842.519	0.939411
$94$	−953.397	−1.04612
$95$	−136.320	−0.147223
$96$	96.0000	0.102062
$97$	−117.863	−0.123373	−0.0616866	−	0.998096i	$-0.519648\pi$
$97$	−117.863	−0.123373	−0.0616866	+	0.998096i	$0.519648\pi$
$98$	−677.932	−0.698791
$99$	639.464	0.649177
$100$	−467.545	−0.467545
$101$	1181.26	1.16376	0.581882	−	0.813273i	$-0.302316\pi$
$101$	1181.26	1.16376	0.581882	+	0.813273i	$0.302316\pi$
$102$	828.875	0.804617
$103$	311.644	0.298128	0.149064	−	0.988828i	$-0.452374\pi$
$103$	311.644	0.298128	0.149064	+	0.988828i	$0.452374\pi$
$104$	−637.619	−0.601189
$105$	17.1628	0.0159516
$106$	−1034.55	−0.947966
$107$	−857.803	−0.775018	−0.387509	−	0.921866i	$-0.626664\pi$
$107$	−857.803	−0.775018	−0.387509	+	0.921866i	$0.626664\pi$
$108$	108.000	0.0962250
$109$	−1424.40	−1.25168	−0.625840	−	0.779952i	$-0.715244\pi$
$109$	−1424.40	−1.25168	−0.625840	+	0.779952i	$0.715244\pi$
$110$	−404.777	−0.350854
$111$	299.596	0.256184
$112$	−32.1348	−0.0271112
$113$	−1284.72	−1.06953	−0.534764	−	0.845002i	$-0.679599\pi$
$113$	−1284.72	−1.06953	−0.534764	+	0.845002i	$0.679599\pi$
$114$	287.144	0.235908
$115$	−221.752	−0.179813
$116$	−440.336	−0.352450
$117$	−717.321	−0.566807
$118$	−118.000	−0.0920575
$119$	−277.456	−0.213734
$120$	−68.3633	−0.0520057
$121$	3717.32	2.79288
$122$	−338.990	−0.251563
$123$	262.474	0.192411
$124$	1123.36	0.813553
$125$	689.006	0.493012
$126$	−36.1517	−0.0255607
$127$	−786.239	−0.549350	−0.274675	−	0.961537i	$-0.588570\pi$
$127$	−786.239	−0.549350	−0.274675	+	0.961537i	$0.588570\pi$
$128$	128.000	0.0883883
$129$	1253.91	0.855819
$130$	454.060	0.306336
$131$	1705.49	1.13748	0.568739	−	0.822518i	$-0.307431\pi$
$131$	1705.49	1.13748	0.568739	+	0.822518i	$0.307431\pi$
$132$	852.619	0.562204
$133$	−96.1178	−0.0626652
$134$	773.123	0.498415
$135$	−76.9087	−0.0490315
$136$	1105.17	0.696819
$137$	−2818.96	−1.75795	−0.878977	−	0.476863i	$-0.841774\pi$
$137$	−2818.96	−1.75795	−0.878977	+	0.476863i	$0.841774\pi$
$138$	467.096	0.288129
$139$	308.341	0.188152	0.0940760	−	0.995565i	$-0.470010\pi$
$139$	308.341	0.188152	0.0940760	+	0.995565i	$0.470010\pi$
$140$	22.8838	0.0138145
$141$	−1430.09	−0.854154
$142$	−130.712	−0.0772474
$143$	−5662.98	−3.31162
$144$	144.000	0.0833333
$145$	313.571	0.179591
$146$	683.116	0.387227
$147$	−1016.90	−0.570560
$148$	399.461	0.221861
$149$	1059.90	0.582755	0.291377	−	0.956608i	$-0.405886\pi$
$149$	1059.90	0.582755	0.291377	+	0.956608i	$0.405886\pi$
$150$	−701.317	−0.381749
$151$	−2929.27	−1.57868	−0.789339	−	0.613957i	$-0.789577\pi$
$151$	−2929.27	−1.57868	−0.789339	+	0.613957i	$0.789577\pi$
$152$	382.858	0.204302
$153$	1243.31	0.656967
$154$	−285.404	−0.149341
$155$	−799.964	−0.414546
$156$	−956.428	−0.490869
$157$	−3264.12	−1.65927	−0.829634	−	0.558307i	$-0.811451\pi$
$157$	−3264.12	−1.65927	−0.829634	+	0.558307i	$0.811451\pi$
$158$	−1372.59	−0.691124
$159$	−1551.82	−0.774011
$160$	−91.1511	−0.0450383
$161$	−156.355	−0.0765371
$162$	162.000	0.0785674
$163$	−2875.17	−1.38160	−0.690799	−	0.723047i	$-0.742741\pi$
$163$	−2875.17	−1.38160	−0.690799	+	0.723047i	$0.742741\pi$
$164$	349.966	0.166633
$165$	−607.165	−0.286471
$166$	928.509	0.434134
$167$	1256.13	0.582048	0.291024	−	0.956716i	$-0.406004\pi$
$167$	1256.13	0.582048	0.291024	+	0.956716i	$0.406004\pi$
$168$	−48.2022	−0.0221362
$169$	4155.47	1.89143
$170$	−787.009	−0.355064
$171$	430.715	0.192618
$172$	1671.88	0.741161
$173$	−1669.01	−0.733481	−0.366741	−	0.930323i	$-0.619526\pi$
$173$	−1669.01	−0.733481	−0.366741	+	0.930323i	$0.619526\pi$
$174$	−660.504	−0.287774
$175$	234.757	0.101406
$176$	1136.82	0.486883
$177$	−177.000	−0.0751646
$178$	−2747.25	−1.15683
$179$	−1884.48	−0.786885	−0.393442	−	0.919349i	$-0.628716\pi$
$179$	−1884.48	−0.786885	−0.393442	+	0.919349i	$0.628716\pi$
$180$	−102.545	−0.0424625
$181$	−1259.87	−0.517378	−0.258689	−	0.965961i	$-0.583291\pi$
$181$	−1259.87	−0.517378	−0.258689	+	0.965961i	$0.583291\pi$
$182$	320.153	0.130392
$183$	−508.486	−0.205401
$184$	622.795	0.249527
$185$	−284.463	−0.113049
$186$	1685.04	0.664264
$187$	9815.48	3.83839
$188$	−1906.79	−0.739719
$189$	−54.2275	−0.0208702
$190$	−272.640	−0.104102
$191$	3250.82	1.23152	0.615762	−	0.787932i	$-0.288848\pi$
$191$	3250.82	1.23152	0.615762	+	0.787932i	$0.288848\pi$
$192$	192.000	0.0721688
$193$	1019.79	0.380342	0.190171	−	0.981751i	$-0.439096\pi$
$193$	1019.79	0.380342	0.190171	+	0.981751i	$0.439096\pi$
$194$	−235.726	−0.0872380
$195$	681.090	0.250122
$196$	−1355.86	−0.494120
$197$	−4308.83	−1.55833	−0.779166	−	0.626818i	$-0.784357\pi$
$197$	−4308.83	−1.55833	−0.779166	+	0.626818i	$0.784357\pi$
$198$	1278.93	0.459038
$199$	−872.377	−0.310760	−0.155380	−	0.987855i	$-0.549660\pi$
$199$	−872.377	−0.310760	−0.155380	+	0.987855i	$0.549660\pi$
$200$	−935.090	−0.330604
$201$	1159.68	0.406955
$202$	2362.53	0.822906
$203$	221.095	0.0764426
$204$	1657.75	0.568950
$205$	−249.217	−0.0849076
$206$	623.288	0.210808
$207$	700.644	0.235257
$208$	−1275.24	−0.425105
$209$	3400.33	1.12539
$210$	34.3257	0.0112795
$211$	−2897.61	−0.945401	−0.472701	−	0.881223i	$-0.656721\pi$
$211$	−2897.61	−0.945401	−0.472701	+	0.881223i	$0.656721\pi$
$212$	−2069.10	−0.670313
$213$	−196.069	−0.0630723
$214$	−1715.61	−0.548020
$215$	−1190.58	−0.377658
$216$	216.000	0.0680414
$217$	−564.046	−0.176451
$218$	−2848.81	−0.885071
$219$	1024.67	0.316169
$220$	−809.553	−0.248091
$221$	−11010.6	−3.35136
$222$	599.192	0.181149
$223$	2456.61	0.737697	0.368849	−	0.929489i	$-0.379752\pi$
$223$	2456.61	0.737697	0.368849	+	0.929489i	$0.379752\pi$
$224$	−64.2696	−0.0191705
$225$	−1051.98	−0.311697
$226$	−2569.45	−0.756270
$227$	913.585	0.267122	0.133561	−	0.991041i	$-0.457359\pi$
$227$	913.585	0.267122	0.133561	+	0.991041i	$0.457359\pi$
$228$	574.287	0.166812
$229$	6652.80	1.91978	0.959889	−	0.280379i	$-0.0904600\pi$
$229$	6652.80	1.91978	0.959889	+	0.280379i	$0.0904600\pi$
$230$	−443.503	−0.127147
$231$	−428.105	−0.121936
$232$	−880.671	−0.249219
$233$	−4180.44	−1.17541	−0.587703	−	0.809077i	$-0.699968\pi$
$233$	−4180.44	−1.17541	−0.587703	+	0.809077i	$0.699968\pi$
$234$	−1434.64	−0.400793
$235$	1357.86	0.376924
$236$	−236.000	−0.0650945
$237$	−2058.89	−0.564300
$238$	−554.912	−0.151133
$239$	3971.70	1.07493	0.537464	−	0.843287i	$-0.319382\pi$
$239$	3971.70	1.07493	0.537464	+	0.843287i	$0.319382\pi$
$240$	−136.727	−0.0367736
$241$	−2047.02	−0.547137	−0.273569	−	0.961852i	$-0.588204\pi$
$241$	−2047.02	−0.547137	−0.273569	+	0.961852i	$0.588204\pi$
$242$	7434.65	1.97486
$243$	243.000	0.0641500
$244$	−677.981	−0.177882
$245$	965.536	0.251779
$246$	524.949	0.136055
$247$	−3814.34	−0.982593
$248$	2246.72	0.575269
$249$	1392.76	0.354469
$250$	1378.01	0.348612
$251$	461.573	0.116073	0.0580363	−	0.998314i	$-0.481516\pi$
$251$	461.573	0.116073	0.0580363	+	0.998314i	$0.481516\pi$
$252$	−72.3033	−0.0180741
$253$	5531.32	1.37451
$254$	−1572.48	−0.388449
$255$	−1180.51	−0.289908
$256$	256.000	0.0625000
$257$	−1788.31	−0.434052	−0.217026	−	0.976166i	$-0.569636\pi$
$257$	−1788.31	−0.434052	−0.217026	+	0.976166i	$0.569636\pi$
$258$	2507.82	0.605155
$259$	−200.572	−0.0481194
$260$	908.120	0.216612
$261$	−990.755	−0.234966
$262$	3410.99	0.804319
$263$	−2125.74	−0.498399	−0.249199	−	0.968452i	$-0.580167\pi$
$263$	−2125.74	−0.498399	−0.249199	+	0.968452i	$0.580167\pi$
$264$	1705.24	0.397538
$265$	1473.44	0.341558
$266$	−192.236	−0.0443110
$267$	−4120.87	−0.944544
$268$	1546.25	0.352433
$269$	−1965.60	−0.445521	−0.222760	−	0.974873i	$-0.571507\pi$
$269$	−1965.60	−0.445521	−0.222760	+	0.974873i	$0.571507\pi$
$270$	−153.817	−0.0346705
$271$	5640.86	1.26442	0.632210	−	0.774797i	$-0.282148\pi$
$271$	5640.86	1.26442	0.632210	+	0.774797i	$0.282148\pi$
$272$	2210.33	0.492725
$273$	480.229	0.106464
$274$	−5637.92	−1.24306
$275$	−8304.95	−1.82112
$276$	934.192	0.203738
$277$	4032.62	0.874718	0.437359	−	0.899287i	$-0.355914\pi$
$277$	4032.62	0.874718	0.437359	+	0.899287i	$0.355914\pi$
$278$	616.682	0.133044
$279$	2527.56	0.542369
$280$	45.7675	0.00976833
$281$	6537.31	1.38784	0.693920	−	0.720052i	$-0.255882\pi$
$281$	6537.31	1.38784	0.693920	+	0.720052i	$0.255882\pi$
$282$	−2860.19	−0.603978
$283$	−926.420	−0.194593	−0.0972967	−	0.995255i	$-0.531020\pi$
$283$	−926.420	−0.194593	−0.0972967	+	0.995255i	$0.531020\pi$
$284$	−261.425	−0.0546222
$285$	−408.960	−0.0849990
$286$	−11326.0	−2.34167
$287$	−175.720	−0.0361409
$288$	288.000	0.0589256
$289$	14171.3	2.88445
$290$	627.142	0.126990
$291$	−353.590	−0.0712295
$292$	1366.23	0.273811
$293$	3494.78	0.696817	0.348408	−	0.937343i	$-0.386722\pi$
$293$	3494.78	0.696817	0.348408	+	0.937343i	$0.386722\pi$
$294$	−2033.80	−0.403447
$295$	168.060	0.0331689
$296$	798.922	0.156880
$297$	1918.39	0.374803
$298$	2119.80	0.412070
$299$	−6204.78	−1.20011
$300$	−1402.63	−0.269937
$301$	−839.461	−0.160750
$302$	−5858.54	−1.11629
$303$	3543.79	0.671900
$304$	765.716	0.144463
$305$	482.802	0.0906399
$306$	2486.63	0.464546
$307$	4556.80	0.847135	0.423568	−	0.905865i	$-0.360778\pi$
$307$	4556.80	0.847135	0.423568	+	0.905865i	$0.360778\pi$
$308$	−570.807	−0.105600
$309$	934.931	0.172124
$310$	−1599.93	−0.293128
$311$	6587.15	1.20104	0.600519	−	0.799610i	$-0.294961\pi$
$311$	6587.15	1.20104	0.600519	+	0.799610i	$0.294961\pi$
$312$	−1912.86	−0.347097
$313$	4662.62	0.842002	0.421001	−	0.907060i	$-0.361679\pi$
$313$	4662.62	0.842002	0.421001	+	0.907060i	$0.361679\pi$
$314$	−6528.24	−1.17328
$315$	51.4885	0.00920967
$316$	−2745.18	−0.488698
$317$	9912.19	1.75623	0.878114	−	0.478451i	$-0.158802\pi$
$317$	9912.19	1.75623	0.878114	+	0.478451i	$0.158802\pi$
$318$	−3103.65	−0.547308
$319$	−7821.63	−1.37281
$320$	−182.302	−0.0318469
$321$	−2573.41	−0.447457
$322$	−312.709	−0.0541199
$323$	6611.29	1.13889
$324$	324.000	0.0555556
$325$	9316.11	1.59005
$326$	−5750.33	−0.976937
$327$	−4273.21	−0.722657
$328$	699.932	0.117827
$329$	957.413	0.160437
$330$	−1214.33	−0.202566
$331$	−8243.05	−1.36882	−0.684410	−	0.729098i	$-0.739940\pi$
$331$	−8243.05	−1.36882	−0.684410	+	0.729098i	$0.739940\pi$
$332$	1857.02	0.306979
$333$	898.787	0.147908
$334$	2512.25	0.411570
$335$	−1101.11	−0.179582
$336$	−96.4044	−0.0156527
$337$	−5322.49	−0.860340	−0.430170	−	0.902748i	$-0.641546\pi$
$337$	−5322.49	−0.860340	−0.430170	+	0.902748i	$0.641546\pi$
$338$	8310.94	1.33744
$339$	−3854.17	−0.617492
$340$	−1574.02	−0.251068
$341$	19954.1	3.16884
$342$	861.431	0.136201
$343$	1369.68	0.215614
$344$	3343.76	0.524080
$345$	−665.255	−0.103815
$346$	−3338.01	−0.518649
$347$	−639.651	−0.0989575	−0.0494787	−	0.998775i	$-0.515756\pi$
$347$	−639.651	−0.0989575	−0.0494787	+	0.998775i	$0.515756\pi$
$348$	−1321.01	−0.203487
$349$	6739.89	1.03375	0.516874	−	0.856062i	$-0.327096\pi$
$349$	6739.89	1.03375	0.516874	+	0.856062i	$0.327096\pi$
$350$	469.515	0.0717046
$351$	−2151.96	−0.327246
$352$	2273.65	0.344278
$353$	7368.69	1.11104	0.555518	−	0.831504i	$-0.312520\pi$
$353$	7368.69	1.11104	0.555518	+	0.831504i	$0.312520\pi$
$354$	−354.000	−0.0531494
$355$	186.165	0.0278327
$356$	−5494.49	−0.817999
$357$	−832.367	−0.123399
$358$	−3768.95	−0.556412
$359$	−12876.5	−1.89303	−0.946514	−	0.322664i	$-0.895422\pi$
$359$	−12876.5	−1.89303	−0.946514	+	0.322664i	$0.895422\pi$
$360$	−205.090	−0.0300255
$361$	−4568.68	−0.666086
$362$	−2519.74	−0.365842
$363$	11152.0	1.61247
$364$	640.305	0.0922009
$365$	−972.919	−0.139520
$366$	−1016.97	−0.145240
$367$	−5897.53	−0.838824	−0.419412	−	0.907796i	$-0.637764\pi$
$367$	−5897.53	−0.838824	−0.419412	+	0.907796i	$0.637764\pi$
$368$	1245.59	0.176443
$369$	787.423	0.111088
$370$	−568.927	−0.0799381
$371$	1038.91	0.145384
$372$	3370.08	0.469705
$373$	1165.60	0.161803	0.0809013	−	0.996722i	$-0.474220\pi$
$373$	1165.60	0.161803	0.0809013	+	0.996722i	$0.474220\pi$
$374$	19631.0	2.71415
$375$	2067.02	0.284641
$376$	−3813.59	−0.523060
$377$	8773.95	1.19862
$378$	−108.455	−0.0147575
$379$	2434.11	0.329899	0.164949	−	0.986302i	$-0.447254\pi$
$379$	2434.11	0.329899	0.164949	+	0.986302i	$0.447254\pi$
$380$	−545.280	−0.0736113
$381$	−2358.72	−0.317167
$382$	6501.64	0.870819
$383$	2889.55	0.385506	0.192753	−	0.981247i	$-0.438258\pi$
$383$	2889.55	0.385506	0.192753	+	0.981247i	$0.438258\pi$
$384$	384.000	0.0510310
$385$	406.482	0.0538084
$386$	2039.58	0.268942
$387$	3761.73	0.494107
$388$	−471.453	−0.0616866
$389$	14306.5	1.86471	0.932353	−	0.361548i	$-0.117752\pi$
$389$	14306.5	1.86471	0.932353	+	0.361548i	$0.117752\pi$
$390$	1362.18	0.176863
$391$	10754.6	1.39100
$392$	−2711.73	−0.349396
$393$	5116.48	0.656723
$394$	−8617.66	−1.10191
$395$	1954.89	0.249016
$396$	2557.86	0.324589
$397$	4441.92	0.561545	0.280773	−	0.959774i	$-0.409409\pi$
$397$	4441.92	0.561545	0.280773	+	0.959774i	$0.409409\pi$
$398$	−1744.75	−0.219740
$399$	−288.353	−0.0361798
$400$	−1870.18	−0.233772
$401$	−7566.57	−0.942285	−0.471143	−	0.882057i	$-0.656158\pi$
$401$	−7566.57	−0.942285	−0.471143	+	0.882057i	$0.656158\pi$
$402$	2319.37	0.287760
$403$	−22383.6	−2.76677
$404$	4725.06	0.581882
$405$	−230.726	−0.0283083
$406$	442.191	0.0540531
$407$	7095.58	0.864164
$408$	3315.50	0.402308
$409$	−11926.2	−1.44184	−0.720921	−	0.693018i	$-0.756281\pi$
$409$	−11926.2	−1.44184	−0.720921	+	0.693018i	$0.756281\pi$
$410$	−498.434	−0.0600388
$411$	−8456.87	−1.01496
$412$	1246.58	0.149064
$413$	118.497	0.0141183
$414$	1401.29	0.166352
$415$	−1322.42	−0.156421
$416$	−2550.48	−0.300595
$417$	925.023	0.108630
$418$	6800.67	0.795769
$419$	−8841.44	−1.03086	−0.515432	−	0.856930i	$-0.672369\pi$
$419$	−8841.44	−1.03086	−0.515432	+	0.856930i	$0.672369\pi$
$420$	68.6513	0.00797581
$421$	15343.1	1.77619	0.888095	−	0.459660i	$-0.152029\pi$
$421$	15343.1	1.77619	0.888095	+	0.459660i	$0.152029\pi$
$422$	−5795.22	−0.668500
$423$	−4290.28	−0.493146
$424$	−4138.20	−0.473983
$425$	−16147.4	−1.84297
$426$	−392.137	−0.0445988
$427$	340.418	0.0385808
$428$	−3431.21	−0.387509
$429$	−16988.9	−1.91197
$430$	−2381.15	−0.267045
$431$	−1750.61	−0.195647	−0.0978235	−	0.995204i	$-0.531188\pi$
$431$	−1750.61	−0.195647	−0.0978235	+	0.995204i	$0.531188\pi$
$432$	432.000	0.0481125
$433$	−9153.10	−1.01587	−0.507933	−	0.861397i	$-0.669590\pi$
$433$	−9153.10	−1.01587	−0.507933	+	0.861397i	$0.669590\pi$
$434$	−1128.09	−0.124770
$435$	940.713	0.103687
$436$	−5697.61	−0.625840
$437$	3725.66	0.407832
$438$	2049.35	0.223566
$439$	−4986.14	−0.542085	−0.271042	−	0.962567i	$-0.587368\pi$
$439$	−4986.14	−0.542085	−0.271042	+	0.962567i	$0.587368\pi$
$440$	−1619.11	−0.175427
$441$	−3050.70	−0.329413
$442$	−22021.1	−2.36977
$443$	−8062.57	−0.864705	−0.432352	−	0.901705i	$-0.642316\pi$
$443$	−8062.57	−0.864705	−0.432352	+	0.901705i	$0.642316\pi$
$444$	1198.38	0.128092
$445$	3912.73	0.416811
$446$	4913.21	0.521631
$447$	3179.70	0.336454
$448$	−128.539	−0.0135556
$449$	8007.17	0.841608	0.420804	−	0.907152i	$-0.361748\pi$
$449$	8007.17	0.841608	0.420804	+	0.907152i	$0.361748\pi$
$450$	−2103.95	−0.220403
$451$	6216.40	0.649044
$452$	−5138.89	−0.534764
$453$	−8787.80	−0.911450
$454$	1827.17	0.188884
$455$	−455.973	−0.0469809
$456$	1148.57	0.117954
$457$	12839.0	1.31419	0.657095	−	0.753808i	$-0.271785\pi$
$457$	12839.0	1.31419	0.657095	+	0.753808i	$0.271785\pi$
$458$	13305.6	1.35749
$459$	3729.94	0.379300
$460$	−887.007	−0.0899063
$461$	−7671.41	−0.775039	−0.387520	−	0.921861i	$-0.626668\pi$
$461$	−7671.41	−0.775039	−0.387520	+	0.921861i	$0.626668\pi$
$462$	−856.211	−0.0862219
$463$	12016.6	1.20618	0.603088	−	0.797674i	$-0.293937\pi$
$463$	12016.6	1.20618	0.603088	+	0.797674i	$0.293937\pi$
$464$	−1761.34	−0.176225
$465$	−2399.89	−0.239338
$466$	−8360.88	−0.831138
$467$	−11359.5	−1.12560	−0.562798	−	0.826594i	$-0.690275\pi$
$467$	−11359.5	−1.12560	−0.562798	+	0.826594i	$0.690275\pi$
$468$	−2869.29	−0.283403
$469$	−776.380	−0.0764390
$470$	2715.72	0.266525
$471$	−9792.36	−0.957979
$472$	−472.000	−0.0460287
$473$	29697.4	2.88687
$474$	−4117.78	−0.399021
$475$	−5593.86	−0.540345
$476$	−1109.82	−0.106867
$477$	−4655.47	−0.446875
$478$	7943.40	0.760089
$479$	−8727.94	−0.832546	−0.416273	−	0.909240i	$-0.636664\pi$
$479$	−8727.94	−0.832546	−0.416273	+	0.909240i	$0.636664\pi$
$480$	−273.453	−0.0260029
$481$	−7959.50	−0.754515
$482$	−4094.04	−0.386885
$483$	−469.064	−0.0441887
$484$	14869.3	1.39644
$485$	335.730	0.0314324
$486$	486.000	0.0453609
$487$	−4589.15	−0.427011	−0.213505	−	0.976942i	$-0.568488\pi$
$487$	−4589.15	−0.427011	−0.213505	+	0.976942i	$0.568488\pi$
$488$	−1355.96	−0.125782
$489$	−8625.50	−0.797666
$490$	1931.07	0.178034
$491$	8385.48	0.770736	0.385368	−	0.922763i	$-0.374074\pi$
$491$	8385.48	0.770736	0.385368	+	0.922763i	$0.374074\pi$
$492$	1049.90	0.0962054
$493$	−15207.6	−1.38929
$494$	−7628.68	−0.694798
$495$	−1821.49	−0.165394
$496$	4493.44	0.406777
$497$	131.263	0.0118470
$498$	2785.53	0.250647
$499$	11493.4	1.03109	0.515545	−	0.856863i	$-0.327590\pi$
$499$	11493.4	1.03109	0.515545	+	0.856863i	$0.327590\pi$
$500$	2756.02	0.246506
$501$	3768.38	0.336045
$502$	923.146	0.0820757
$503$	4816.15	0.426922	0.213461	−	0.976952i	$-0.431526\pi$
$503$	4816.15	0.426922	0.213461	+	0.976952i	$0.431526\pi$
$504$	−144.607	−0.0127803
$505$	−3364.80	−0.296498
$506$	11062.6	0.971925
$507$	12466.4	1.09202
$508$	−3144.96	−0.274675
$509$	−8180.77	−0.712390	−0.356195	−	0.934412i	$-0.615926\pi$
$509$	−8180.77	−0.712390	−0.356195	+	0.934412i	$0.615926\pi$
$510$	−2361.03	−0.204996
$511$	−685.994	−0.0593867
$512$	512.000	0.0441942
$513$	1292.15	0.111208
$514$	−3576.61	−0.306921
$515$	−887.708	−0.0759555
$516$	5015.64	0.427909
$517$	−33870.2	−2.88125
$518$	−401.144	−0.0340256
$519$	−5007.02	−0.423475
$520$	1816.24	0.153168
$521$	1793.90	0.150849	0.0754244	−	0.997152i	$-0.475969\pi$
$521$	1793.90	0.150849	0.0754244	+	0.997152i	$0.475969\pi$
$522$	−1981.51	−0.166146
$523$	−4341.44	−0.362979	−0.181489	−	0.983393i	$-0.558092\pi$
$523$	−4341.44	−0.362979	−0.181489	+	0.983393i	$0.558092\pi$
$524$	6821.97	0.568739
$525$	704.272	0.0585466
$526$	−4251.48	−0.352421
$527$	38796.9	3.20687
$528$	3410.47	0.281102
$529$	−6106.48	−0.501888
$530$	2946.89	0.241518
$531$	−531.000	−0.0433963
$532$	−384.471	−0.0313326
$533$	−6973.28	−0.566691
$534$	−8241.74	−0.667893
$535$	2443.43	0.197455
$536$	3092.49	0.249208
$537$	−5653.43	−0.454308
$538$	−3931.21	−0.315031
$539$	−24084.1	−1.92463
$540$	−307.635	−0.0245157
$541$	−8578.71	−0.681751	−0.340876	−	0.940108i	$-0.610724\pi$
$541$	−8578.71	−0.681751	−0.340876	+	0.940108i	$0.610724\pi$
$542$	11281.7	0.894080
$543$	−3779.61	−0.298709
$544$	4420.67	0.348409
$545$	4057.37	0.318897
$546$	960.458	0.0752817
$547$	−11810.3	−0.923165	−0.461583	−	0.887097i	$-0.652718\pi$
$547$	−11810.3	−0.923165	−0.461583	+	0.887097i	$0.652718\pi$
$548$	−11275.8	−0.878977
$549$	−1525.46	−0.118588
$550$	−16609.9	−1.28772
$551$	−5268.32	−0.407328
$552$	1868.38	0.144065
$553$	1378.37	0.105994
$554$	8065.25	0.618519
$555$	−853.390	−0.0652692
$556$	1233.36	0.0940760
$557$	−9746.05	−0.741389	−0.370694	−	0.928755i	$-0.620880\pi$
$557$	−9746.05	−0.741389	−0.370694	+	0.928755i	$0.620880\pi$
$558$	5055.12	0.383513
$559$	−33313.2	−2.52057
$560$	91.5351	0.00690726
$561$	29446.4	2.21610
$562$	13074.6	0.981352
$563$	−9640.70	−0.721682	−0.360841	−	0.932627i	$-0.617510\pi$
$563$	−9640.70	−0.721682	−0.360841	+	0.932627i	$0.617510\pi$
$564$	−5720.38	−0.427077
$565$	3659.50	0.272489
$566$	−1852.84	−0.137598
$567$	−162.682	−0.0120494
$568$	−522.849	−0.0386237
$569$	−12080.0	−0.890017	−0.445009	−	0.895526i	$-0.646799\pi$
$569$	−12080.0	−0.890017	−0.445009	+	0.895526i	$0.646799\pi$
$570$	−817.920	−0.0601034
$571$	696.360	0.0510364	0.0255182	−	0.999674i	$-0.491876\pi$
$571$	696.360	0.0510364	0.0255182	+	0.999674i	$0.491876\pi$
$572$	−22651.9	−1.65581
$573$	9752.46	0.711021
$574$	−351.440	−0.0255555
$575$	−9099.52	−0.659958
$576$	576.000	0.0416667
$577$	−12572.2	−0.907084	−0.453542	−	0.891235i	$-0.649840\pi$
$577$	−12572.2	−0.907084	−0.453542	+	0.891235i	$0.649840\pi$
$578$	28342.6	2.03961
$579$	3059.37	0.219591
$580$	1254.28	0.0897953
$581$	−932.420	−0.0665806
$582$	−707.179	−0.0503669
$583$	−36753.2	−2.61091
$584$	2732.47	0.193613
$585$	2043.27	0.144408
$586$	6989.56	0.492724
$587$	8308.56	0.584209	0.292105	−	0.956386i	$-0.405644\pi$
$587$	8308.56	0.584209	0.292105	+	0.956386i	$0.405644\pi$
$588$	−4067.59	−0.285280
$589$	13440.2	0.940229
$590$	336.120	0.0234539
$591$	−12926.5	−0.899703
$592$	1597.84	0.110931
$593$	−25381.2	−1.75764	−0.878820	−	0.477154i	$-0.841668\pi$
$593$	−25381.2	−1.75764	−0.878820	+	0.477154i	$0.841668\pi$
$594$	3836.78	0.265026
$595$	790.325	0.0544541
$596$	4239.60	0.291377
$597$	−2617.13	−0.179417
$598$	−12409.6	−0.848603
$599$	−24456.8	−1.66824	−0.834121	−	0.551582i	$-0.814025\pi$
$599$	−24456.8	−1.66824	−0.834121	+	0.551582i	$0.814025\pi$
$600$	−2805.27	−0.190874
$601$	26481.3	1.79733	0.898664	−	0.438637i	$-0.144539\pi$
$601$	26481.3	1.79733	0.898664	+	0.438637i	$0.144539\pi$
$602$	−1678.92	−0.113667
$603$	3479.05	0.234955
$604$	−11717.1	−0.789339
$605$	−10588.7	−0.711556
$606$	7087.59	0.475105
$607$	−743.818	−0.0497374	−0.0248687	−	0.999691i	$-0.507917\pi$
$607$	−743.818	−0.0497374	−0.0248687	+	0.999691i	$0.507917\pi$
$608$	1531.43	0.102151
$609$	663.286	0.0441342
$610$	965.604	0.0640921
$611$	37994.0	2.51567
$612$	4973.25	0.328483
$613$	5487.68	0.361574	0.180787	−	0.983522i	$-0.442135\pi$
$613$	5487.68	0.361574	0.180787	+	0.983522i	$0.442135\pi$
$614$	9113.61	0.599015
$615$	−747.651	−0.0490214
$616$	−1141.61	−0.0746704
$617$	−20063.7	−1.30913	−0.654567	−	0.756004i	$-0.727149\pi$
$617$	−20063.7	−1.30913	−0.654567	+	0.756004i	$0.727149\pi$
$618$	1869.86	0.121710
$619$	12496.4	0.811426	0.405713	−	0.914000i	$-0.367023\pi$
$619$	12496.4	0.811426	0.405713	+	0.914000i	$0.367023\pi$
$620$	−3199.86	−0.207273
$621$	2101.93	0.135826
$622$	13174.3	0.849263
$623$	2758.82	0.177415
$624$	−3825.71	−0.245434
$625$	12648.2	0.809482
$626$	9325.23	0.595386
$627$	10201.0	0.649743
$628$	−13056.5	−0.829634
$629$	13796.0	0.874534
$630$	102.977	0.00651222
$631$	−3812.97	−0.240557	−0.120279	−	0.992740i	$-0.538379\pi$
$631$	−3812.97	−0.240557	−0.120279	+	0.992740i	$0.538379\pi$
$632$	−5490.37	−0.345562
$633$	−8692.83	−0.545828
$634$	19824.4	1.24184
$635$	2239.58	0.139961
$636$	−6207.30	−0.387005
$637$	27016.4	1.68042
$638$	−15643.3	−0.970726
$639$	−588.206	−0.0364148
$640$	−364.604	−0.0225191
$641$	13373.6	0.824062	0.412031	−	0.911170i	$-0.364819\pi$
$641$	13373.6	0.824062	0.412031	+	0.911170i	$0.364819\pi$
$642$	−5146.82	−0.316400
$643$	12973.7	0.795696	0.397848	−	0.917451i	$-0.369757\pi$
$643$	12973.7	0.795696	0.397848	+	0.917451i	$0.369757\pi$
$644$	−625.419	−0.0382686
$645$	−3571.73	−0.218041
$646$	13222.6	0.805318
$647$	16868.6	1.02500	0.512498	−	0.858688i	$-0.328720\pi$
$647$	16868.6	1.02500	0.512498	+	0.858688i	$0.328720\pi$
$648$	648.000	0.0392837
$649$	−4192.04	−0.253547
$650$	18632.2	1.12433
$651$	−1692.14	−0.101874
$652$	−11500.7	−0.690799
$653$	184.471	0.0110550	0.00552750	−	0.999985i	$-0.498241\pi$
$653$	184.471	0.0110550	0.00552750	+	0.999985i	$0.498241\pi$
$654$	−8546.42	−0.510996
$655$	−4858.05	−0.289801
$656$	1399.86	0.0833163
$657$	3074.02	0.182540
$658$	1914.83	0.113446
$659$	12858.1	0.760063	0.380031	−	0.924974i	$-0.375913\pi$
$659$	12858.1	0.760063	0.380031	+	0.924974i	$0.375913\pi$
$660$	−2428.66	−0.143236
$661$	15270.4	0.898565	0.449282	−	0.893390i	$-0.351680\pi$
$661$	15270.4	0.898565	0.449282	+	0.893390i	$0.351680\pi$
$662$	−16486.1	−0.967902
$663$	−33031.7	−1.93491
$664$	3714.03	0.217067
$665$	273.789	0.0159655
$666$	1797.57	0.104586
$667$	−8569.96	−0.497497
$668$	5024.50	0.291024
$669$	7369.82	0.425910
$670$	−2202.22	−0.126984
$671$	−12042.9	−0.692863
$672$	−192.809	−0.0110681
$673$	−4568.30	−0.261657	−0.130828	−	0.991405i	$-0.541764\pi$
$673$	−4568.30	−0.261657	−0.130828	+	0.991405i	$0.541764\pi$
$674$	−10645.0	−0.608352
$675$	−3155.93	−0.179958
$676$	16621.9	0.945714
$677$	3438.74	0.195216	0.0976082	−	0.995225i	$-0.468881\pi$
$677$	3438.74	0.195216	0.0976082	+	0.995225i	$0.468881\pi$
$678$	−7708.34	−0.436633
$679$	236.720	0.0133792
$680$	−3148.04	−0.177532
$681$	2740.76	0.154223
$682$	39908.2	2.24071
$683$	−30459.3	−1.70643	−0.853216	−	0.521558i	$-0.825351\pi$
$683$	−30459.3	−1.70643	−0.853216	+	0.521558i	$0.825351\pi$
$684$	1722.86	0.0963089
$685$	8029.72	0.447883
$686$	2739.36	0.152462
$687$	19958.4	1.10838
$688$	6687.52	0.370580
$689$	41228.0	2.27963
$690$	−1330.51	−0.0734082
$691$	−29401.5	−1.61865	−0.809323	−	0.587364i	$-0.800166\pi$
$691$	−29401.5	−1.61865	−0.809323	+	0.587364i	$0.800166\pi$
$692$	−6676.03	−0.366741
$693$	−1284.32	−0.0703999
$694$	−1279.30	−0.0699735
$695$	−878.300	−0.0479364
$696$	−2642.01	−0.143887
$697$	12086.6	0.656832
$698$	13479.8	0.730970
$699$	−12541.3	−0.678621
$700$	939.029	0.0507028
$701$	−13025.9	−0.701828	−0.350914	−	0.936408i	$-0.614129\pi$
$701$	−13025.9	−0.701828	−0.350914	+	0.936408i	$0.614129\pi$
$702$	−4303.93	−0.231398
$703$	4779.28	0.256407
$704$	4547.30	0.243441
$705$	4073.58	0.217617
$706$	14737.4	0.785622
$707$	−2372.48	−0.126204
$708$	−708.000	−0.0375823
$709$	−7900.40	−0.418485	−0.209243	−	0.977864i	$-0.567100\pi$
$709$	−7900.40	−0.418485	−0.209243	+	0.977864i	$0.567100\pi$
$710$	372.330	0.0196807
$711$	−6176.66	−0.325799
$712$	−10989.0	−0.578413
$713$	21863.2	1.14836
$714$	−1664.73	−0.0872565
$715$	16130.8	0.843718
$716$	−7537.91	−0.393442
$717$	11915.1	0.620610
$718$	−25753.0	−1.33857
$719$	−624.747	−0.0324049	−0.0162025	−	0.999869i	$-0.505158\pi$
$719$	−624.747	−0.0324049	−0.0162025	+	0.999869i	$0.505158\pi$
$720$	−410.180	−0.0212313
$721$	−625.913	−0.0323304
$722$	−9137.36	−0.470994
$723$	−6141.06	−0.315890
$724$	−5039.48	−0.258689
$725$	12867.3	0.659144
$726$	22303.9	1.14019
$727$	4454.34	0.227238	0.113619	−	0.993524i	$-0.463756\pi$
$727$	4454.34	0.227238	0.113619	+	0.993524i	$0.463756\pi$
$728$	1280.61	0.0651958
$729$	729.000	0.0370370
$730$	−1945.84	−0.0986557
$731$	57740.8	2.92151
$732$	−2033.94	−0.102700
$733$	10744.3	0.541403	0.270702	−	0.962663i	$-0.412744\pi$
$733$	10744.3	0.541403	0.270702	+	0.962663i	$0.412744\pi$
$734$	−11795.1	−0.593138
$735$	2896.61	0.145365
$736$	2491.18	0.124764
$737$	27465.8	1.37275
$738$	1574.85	0.0785513
$739$	23326.6	1.16114	0.580571	−	0.814210i	$-0.302830\pi$
$739$	23326.6	1.16114	0.580571	+	0.814210i	$0.302830\pi$
$740$	−1137.85	−0.0565247
$741$	−11443.0	−0.567300
$742$	2077.82	0.102802
$743$	−14424.3	−0.712214	−0.356107	−	0.934445i	$-0.615896\pi$
$743$	−14424.3	−0.712214	−0.356107	+	0.934445i	$0.615896\pi$
$744$	6740.15	0.332132
$745$	−3019.10	−0.148471
$746$	2331.20	0.114412
$747$	4178.29	0.204653
$748$	39261.9	1.91920
$749$	1722.83	0.0840467
$750$	4134.04	0.201272
$751$	17242.2	0.837787	0.418893	−	0.908035i	$-0.362418\pi$
$751$	17242.2	0.837787	0.418893	+	0.908035i	$0.362418\pi$
$752$	−7627.17	−0.369860
$753$	1384.72	0.0670145
$754$	17547.9	0.847556
$755$	8343.94	0.402208
$756$	−216.910	−0.0104351
$757$	10821.8	0.519586	0.259793	−	0.965664i	$-0.416346\pi$
$757$	10821.8	0.519586	0.259793	+	0.965664i	$0.416346\pi$
$758$	4868.21	0.233274
$759$	16594.0	0.793574
$760$	−1090.56	−0.0520510
$761$	−24071.2	−1.14662	−0.573311	−	0.819338i	$-0.694341\pi$
$761$	−24071.2	−1.14662	−0.573311	+	0.819338i	$0.694341\pi$
$762$	−4717.43	−0.224271
$763$	2860.81	0.135738
$764$	13003.3	0.615762
$765$	−3541.54	−0.167379
$766$	5779.09	0.272594
$767$	4702.44	0.221376
$768$	768.000	0.0360844
$769$	−4563.64	−0.214004	−0.107002	−	0.994259i	$-0.534125\pi$
$769$	−4563.64	−0.214004	−0.107002	+	0.994259i	$0.534125\pi$
$770$	812.964	0.0380483
$771$	−5364.92	−0.250600
$772$	4079.16	0.190171
$773$	−23735.8	−1.10442	−0.552210	−	0.833705i	$-0.686215\pi$
$773$	−23735.8	−1.10442	−0.552210	+	0.833705i	$0.686215\pi$
$774$	7523.46	0.349386
$775$	−32826.3	−1.52149
$776$	−942.906	−0.0436190
$777$	−601.716	−0.0277818
$778$	28613.1	1.31855
$779$	4187.10	0.192578
$780$	2724.36	0.125061
$781$	−4643.66	−0.212757
$782$	21509.1	0.983587
$783$	−2972.27	−0.135658
$784$	−5423.46	−0.247060
$785$	9297.75	0.422740
$786$	10233.0	0.464374
$787$	−404.236	−0.0183093	−0.00915466	−	0.999958i	$-0.502914\pi$
$787$	−404.236	−0.0183093	−0.00915466	+	0.999958i	$0.502914\pi$
$788$	−17235.3	−0.779166
$789$	−6377.23	−0.287751
$790$	3909.79	0.176081
$791$	2580.27	0.115985
$792$	5115.71	0.229519
$793$	13509.2	0.604949
$794$	8883.84	0.397073
$795$	4420.33	0.197199
$796$	−3489.51	−0.155380
$797$	−9647.60	−0.428777	−0.214389	−	0.976748i	$-0.568776\pi$
$797$	−9647.60	−0.428777	−0.214389	+	0.976748i	$0.568776\pi$
$798$	−576.707	−0.0255830
$799$	−65853.9	−2.91583
$800$	−3740.36	−0.165302
$801$	−12362.6	−0.545333
$802$	−15133.1	−0.666296
$803$	24268.2	1.06651
$804$	4638.74	0.203477
$805$	445.372	0.0194997
$806$	−44767.2	−1.95640
$807$	−5896.81	−0.257221
$808$	9450.12	0.411453
$809$	−35450.8	−1.54065	−0.770325	−	0.637652i	$-0.779906\pi$
$809$	−35450.8	−1.54065	−0.770325	+	0.637652i	$0.779906\pi$
$810$	−461.452	−0.0200170
$811$	25953.6	1.12374	0.561871	−	0.827225i	$-0.310082\pi$
$811$	25953.6	1.12374	0.561871	+	0.827225i	$0.310082\pi$
$812$	884.382	0.0382213
$813$	16922.6	0.730013
$814$	14191.2	0.611056
$815$	8189.83	0.351997
$816$	6631.00	0.284475
$817$	20002.9	0.856564
$818$	−23852.4	−1.01954
$819$	1440.69	0.0614672
$820$	−996.868	−0.0424538
$821$	−18775.6	−0.798140	−0.399070	−	0.916921i	$-0.630667\pi$
$821$	−18775.6	−0.798140	−0.399070	+	0.916921i	$0.630667\pi$
$822$	−16913.7	−0.717682
$823$	31898.8	1.35106	0.675530	−	0.737332i	$-0.263915\pi$
$823$	31898.8	1.35106	0.675530	+	0.737332i	$0.263915\pi$
$824$	2493.15	0.105404
$825$	−24914.8	−1.05142
$826$	236.994	0.00998315
$827$	−14629.4	−0.615130	−0.307565	−	0.951527i	$-0.599514\pi$
$827$	−14629.4	−0.615130	−0.307565	+	0.951527i	$0.599514\pi$
$828$	2802.58	0.117628
$829$	13235.9	0.554525	0.277263	−	0.960794i	$-0.410573\pi$
$829$	13235.9	0.554525	0.277263	+	0.960794i	$0.410573\pi$
$830$	−2644.83	−0.110607
$831$	12097.9	0.505019
$832$	−5100.95	−0.212552
$833$	−46826.8	−1.94772
$834$	1850.05	0.0768127
$835$	−3578.04	−0.148291
$836$	13601.3	0.562694
$837$	7582.67	0.313137
$838$	−17682.9	−0.728932
$839$	27229.3	1.12045	0.560226	−	0.828340i	$-0.310714\pi$
$839$	27229.3	1.12045	0.560226	+	0.828340i	$0.310714\pi$
$840$	137.303	0.00563975
$841$	−12270.5	−0.503117
$842$	30686.2	1.25596
$843$	19611.9	0.801270
$844$	−11590.4	−0.472701
$845$	−11836.7	−0.481889
$846$	−8580.57	−0.348707
$847$	−7465.97	−0.302873
$848$	−8276.40	−0.335157
$849$	−2779.26	−0.112349
$850$	−32294.7	−1.30318
$851$	7774.45	0.313166
$852$	−784.274	−0.0315361
$853$	−8119.58	−0.325919	−0.162960	−	0.986633i	$-0.552104\pi$
$853$	−8119.58	−0.325919	−0.162960	+	0.986633i	$0.552104\pi$
$854$	680.837	0.0272807
$855$	−1226.88	−0.0490742
$856$	−6862.42	−0.274010
$857$	−6527.03	−0.260162	−0.130081	−	0.991503i	$-0.541524\pi$
$857$	−6527.03	−0.260162	−0.130081	+	0.991503i	$0.541524\pi$
$858$	−33977.9	−1.35196
$859$	20395.4	0.810106	0.405053	−	0.914293i	$-0.367253\pi$
$859$	20395.4	0.810106	0.405053	+	0.914293i	$0.367253\pi$
$860$	−4762.30	−0.188829
$861$	−527.160	−0.0208659
$862$	−3501.22	−0.138343
$863$	−12988.5	−0.512322	−0.256161	−	0.966634i	$-0.582458\pi$
$863$	−12988.5	−0.512322	−0.256161	+	0.966634i	$0.582458\pi$
$864$	864.000	0.0340207
$865$	4754.12	0.186873
$866$	−18306.2	−0.718326
$867$	42513.9	1.66534
$868$	−2256.18	−0.0882256
$869$	−48762.4	−1.90351
$870$	1881.43	0.0733176
$871$	−30809.9	−1.19857
$872$	−11395.2	−0.442535
$873$	−1060.77	−0.0411244
$874$	7451.32	0.288381
$875$	−1383.82	−0.0534646
$876$	4098.70	0.158085
$877$	43771.1	1.68534	0.842671	−	0.538428i	$-0.180982\pi$
$877$	43771.1	1.68534	0.842671	+	0.538428i	$0.180982\pi$
$878$	−9972.27	−0.383312
$879$	10484.3	0.402307
$880$	−3238.21	−0.124046
$881$	11248.5	0.430161	0.215081	−	0.976596i	$-0.430999\pi$
$881$	11248.5	0.430161	0.215081	+	0.976596i	$0.430999\pi$
$882$	−6101.39	−0.232930
$883$	11462.5	0.436856	0.218428	−	0.975853i	$-0.429907\pi$
$883$	11462.5	0.436856	0.218428	+	0.975853i	$0.429907\pi$
$884$	−44042.2	−1.67568
$885$	504.179	0.0191501
$886$	−16125.1	−0.611439
$887$	15176.2	0.574484	0.287242	−	0.957858i	$-0.407262\pi$
$887$	15176.2	0.574484	0.287242	+	0.957858i	$0.407262\pi$
$888$	2396.77	0.0905746
$889$	1579.10	0.0595741
$890$	7825.45	0.294730
$891$	5755.18	0.216392
$892$	9826.42	0.368849
$893$	−22813.5	−0.854898
$894$	6359.40	0.237909
$895$	5367.88	0.200479
$896$	−257.078	−0.00958526
$897$	−18614.3	−0.692881
$898$	16014.3	0.595107
$899$	−30915.9	−1.14695
$900$	−4207.90	−0.155848
$901$	−71459.4	−2.64224
$902$	12432.8	0.458944
$903$	−2518.38	−0.0928091
$904$	−10277.8	−0.378135
$905$	3588.71	0.131815
$906$	−17575.6	−0.644493
$907$	14286.1	0.523000	0.261500	−	0.965203i	$-0.415783\pi$
$907$	14286.1	0.523000	0.261500	+	0.965203i	$0.415783\pi$
$908$	3654.34	0.133561
$909$	10631.4	0.387922
$910$	−911.945	−0.0332205
$911$	35953.3	1.30756	0.653779	−	0.756685i	$-0.273182\pi$
$911$	35953.3	1.30756	0.653779	+	0.756685i	$0.273182\pi$
$912$	2297.15	0.0834059
$913$	32986.0	1.19570
$914$	25678.1	0.929273
$915$	1448.41	0.0523310
$916$	26611.2	0.959889
$917$	−3425.36	−0.123354
$918$	7459.88	0.268206
$919$	−28087.3	−1.00818	−0.504088	−	0.863652i	$-0.668171\pi$
$919$	−28087.3	−1.00818	−0.504088	+	0.863652i	$0.668171\pi$
$920$	−1774.01	−0.0635734
$921$	13670.4	0.489094
$922$	−15342.8	−0.548036
$923$	5209.04	0.185761
$924$	−1712.42	−0.0609681
$925$	−11672.9	−0.414921
$926$	24033.2	0.852896
$927$	2804.79	0.0993760
$928$	−3522.69	−0.124610
$929$	26420.2	0.933067	0.466533	−	0.884504i	$-0.345503\pi$
$929$	26420.2	0.933067	0.466533	+	0.884504i	$0.345503\pi$
$930$	−4799.78	−0.169238
$931$	−16222.0	−0.571058
$932$	−16721.8	−0.587703
$933$	19761.5	0.693420
$934$	−22718.9	−0.795917
$935$	−27959.1	−0.977926
$936$	−5738.57	−0.200396
$937$	14001.8	0.488173	0.244087	−	0.969753i	$-0.421512\pi$
$937$	14001.8	0.488173	0.244087	+	0.969753i	$0.421512\pi$
$938$	−1552.76	−0.0540506
$939$	13987.9	0.486130
$940$	5431.45	0.188462
$941$	−45633.7	−1.58089	−0.790444	−	0.612534i	$-0.790150\pi$
$941$	−45633.7	−1.58089	−0.790444	+	0.612534i	$0.790150\pi$
$942$	−19584.7	−0.677394
$943$	6811.15	0.235209
$944$	−944.000	−0.0325472
$945$	154.465	0.00531721
$946$	59394.8	2.04132
$947$	−52543.2	−1.80298	−0.901492	−	0.432795i	$-0.857527\pi$
$947$	−52543.2	−1.80298	−0.901492	+	0.432795i	$0.857527\pi$
$948$	−8235.55	−0.282150
$949$	−27223.0	−0.931186
$950$	−11187.7	−0.382081
$951$	29736.6	1.01396
$952$	−2219.65	−0.0755663
$953$	−9136.96	−0.310572	−0.155286	−	0.987870i	$-0.549630\pi$
$953$	−9136.96	−0.310572	−0.155286	+	0.987870i	$0.549630\pi$
$954$	−9310.95	−0.315989
$955$	−9259.87	−0.313762
$956$	15886.8	0.537464
$957$	−23464.9	−0.792594
$958$	−17455.9	−0.588699
$959$	5661.67	0.190641
$960$	−546.906	−0.0183868
$961$	49080.0	1.64748
$962$	−15919.0	−0.533523
$963$	−7720.22	−0.258339
$964$	−8188.08	−0.273569
$965$	−2904.84	−0.0969017
$966$	−938.128	−0.0312461
$967$	35685.3	1.18672	0.593361	−	0.804936i	$-0.297801\pi$
$967$	35685.3	1.18672	0.593361	+	0.804936i	$0.297801\pi$
$968$	29738.6	0.987432
$969$	19833.9	0.657539
$970$	671.460	0.0222261
$971$	−23931.1	−0.790922	−0.395461	−	0.918483i	$-0.629415\pi$
$971$	−23931.1	−0.790922	−0.395461	+	0.918483i	$0.629415\pi$
$972$	972.000	0.0320750
$973$	−619.280	−0.0204041
$974$	−9178.30	−0.301942
$975$	27948.3	0.918013
$976$	−2711.92	−0.0889411
$977$	−13960.1	−0.457137	−0.228568	−	0.973528i	$-0.573404\pi$
$977$	−13960.1	−0.457137	−0.228568	+	0.973528i	$0.573404\pi$
$978$	−17251.0	−0.564035
$979$	−97598.1	−3.18616
$980$	3862.14	0.125889
$981$	−12819.6	−0.417226
$982$	16771.0	0.544993
$983$	−16070.9	−0.521447	−0.260723	−	0.965414i	$-0.583961\pi$
$983$	−16070.9	−0.521447	−0.260723	+	0.965414i	$0.583961\pi$
$984$	2099.80	0.0680275
$985$	12273.6	0.397024
$986$	−30415.3	−0.982373
$987$	2872.24	0.0926286
$988$	−15257.4	−0.491297
$989$	32538.7	1.04618
$990$	−3642.99	−0.116951
$991$	9225.17	0.295708	0.147854	−	0.989009i	$-0.452763\pi$
$991$	9225.17	0.295708	0.147854	+	0.989009i	$0.452763\pi$
$992$	8986.87	0.287635
$993$	−24729.2	−0.790288
$994$	262.526	0.00837708
$995$	2484.94	0.0791738
$996$	5571.05	0.177234
$997$	−11031.9	−0.350436	−0.175218	−	0.984530i	$-0.556063\pi$
$997$	−11031.9	−0.350436	−0.175218	+	0.984530i	$0.556063\pi$
$998$	22986.7	0.729090
$999$	2696.36	0.0853945

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	354.4.a.i.1.2	✓	6
3.2	odd	2		1062.4.a.s.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.4.a.i.1.2	✓	6	1.1	even	1	trivial
1062.4.a.s.1.5		6	3.2	odd	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:	\( N \)	\(=\)	\( 354 = 2 \cdot 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	354.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -2.84847 q^{5} +6.00000 q^{6} -2.00843 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -2.84847 q^{5} +6.00000 q^{6} -2.00843 q^{7} +8.00000 q^{8} +9.00000 q^{9} -5.69694 q^{10} +71.0516 q^{11} +12.0000 q^{12} -79.7024 q^{13} -4.01685 q^{14} -8.54541 q^{15} +16.0000 q^{16} +138.146 q^{17} +18.0000 q^{18} +47.8573 q^{19} -11.3939 q^{20} -6.02528 q^{21} +142.103 q^{22} +77.8494 q^{23} +24.0000 q^{24} -116.886 q^{25} -159.405 q^{26} +27.0000 q^{27} -8.03370 q^{28} -110.084 q^{29} -17.0908 q^{30} +280.840 q^{31} +32.0000 q^{32} +213.155 q^{33} +276.292 q^{34} +5.72094 q^{35} +36.0000 q^{36} +99.8653 q^{37} +95.7146 q^{38} -239.107 q^{39} -22.7878 q^{40} +87.4915 q^{41} -12.0506 q^{42} +417.970 q^{43} +284.206 q^{44} -25.6362 q^{45} +155.699 q^{46} -476.698 q^{47} +48.0000 q^{48} -338.966 q^{49} -233.772 q^{50} +414.438 q^{51} -318.809 q^{52} -517.275 q^{53} +54.0000 q^{54} -202.388 q^{55} -16.0674 q^{56} +143.572 q^{57} -220.168 q^{58} -59.0000 q^{59} -34.1817 q^{60} -169.495 q^{61} +561.680 q^{62} -18.0758 q^{63} +64.0000 q^{64} +227.030 q^{65} +426.309 q^{66} +386.562 q^{67} +552.584 q^{68} +233.548 q^{69} +11.4419 q^{70} -65.3562 q^{71} +72.0000 q^{72} +341.558 q^{73} +199.731 q^{74} -350.659 q^{75} +191.429 q^{76} -142.702 q^{77} -478.214 q^{78} -686.296 q^{79} -45.5755 q^{80} +81.0000 q^{81} +174.983 q^{82} +464.254 q^{83} -24.1011 q^{84} -393.505 q^{85} +835.940 q^{86} -330.252 q^{87} +568.412 q^{88} -1373.62 q^{89} -51.2725 q^{90} +160.076 q^{91} +311.397 q^{92} +842.519 q^{93} -953.397 q^{94} -136.320 q^{95} +96.0000 q^{96} -117.863 q^{97} -677.932 q^{98} +639.464 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 12 q^{2} + 18 q^{3} + 24 q^{4} + 20 q^{5} + 36 q^{6} + 26 q^{7} + 48 q^{8} + 54 q^{9}+O(q^{10}) \) 6 * q + 12 * q^2 + 18 * q^3 + 24 * q^4 + 20 * q^5 + 36 * q^6 + 26 * q^7 + 48 * q^8 + 54 * q^9	\( 6 q + 12 q^{2} + 18 q^{3} + 24 q^{4} + 20 q^{5} + 36 q^{6} + 26 q^{7} + 48 q^{8} + 54 q^{9} + 40 q^{10} + 63 q^{11} + 72 q^{12} + 93 q^{13} + 52 q^{14} + 60 q^{15} + 96 q^{16} + 230 q^{17} + 108 q^{18} + 89 q^{19} + 80 q^{20} + 78 q^{21} + 126 q^{22} + 81 q^{23} + 144 q^{24} + 304 q^{25} + 186 q^{26} + 162 q^{27} + 104 q^{28} + 131 q^{29} + 120 q^{30} + 51 q^{31} + 192 q^{32} + 189 q^{33} + 460 q^{34} - 87 q^{35} + 216 q^{36} - 16 q^{37} + 178 q^{38} + 279 q^{39} + 160 q^{40} + 176 q^{41} + 156 q^{42} + 375 q^{43} + 252 q^{44} + 180 q^{45} + 162 q^{46} - 255 q^{47} + 288 q^{48} - 290 q^{49} + 608 q^{50} + 690 q^{51} + 372 q^{52} - 256 q^{53} + 324 q^{54} - 184 q^{55} + 208 q^{56} + 267 q^{57} + 262 q^{58} - 354 q^{59} + 240 q^{60} + 39 q^{61} + 102 q^{62} + 234 q^{63} + 384 q^{64} - 954 q^{65} + 378 q^{66} + 86 q^{67} + 920 q^{68} + 243 q^{69} - 174 q^{70} - 895 q^{71} + 432 q^{72} + 155 q^{73} - 32 q^{74} + 912 q^{75} + 356 q^{76} - 498 q^{77} + 558 q^{78} - 565 q^{79} + 320 q^{80} + 486 q^{81} + 352 q^{82} + 140 q^{83} + 312 q^{84} + q^{85} + 750 q^{86} + 393 q^{87} + 504 q^{88} - 1769 q^{89} + 360 q^{90} - 422 q^{91} + 324 q^{92} + 153 q^{93} - 510 q^{94} - 2209 q^{95} + 576 q^{96} + 48 q^{97} - 580 q^{98} + 567 q^{99}+O(q^{100}) \) 6 * q + 12 * q^2 + 18 * q^3 + 24 * q^4 + 20 * q^5 + 36 * q^6 + 26 * q^7 + 48 * q^8 + 54 * q^9 + 40 * q^10 + 63 * q^11 + 72 * q^12 + 93 * q^13 + 52 * q^14 + 60 * q^15 + 96 * q^16 + 230 * q^17 + 108 * q^18 + 89 * q^19 + 80 * q^20 + 78 * q^21 + 126 * q^22 + 81 * q^23 + 144 * q^24 + 304 * q^25 + 186 * q^26 + 162 * q^27 + 104 * q^28 + 131 * q^29 + 120 * q^30 + 51 * q^31 + 192 * q^32 + 189 * q^33 + 460 * q^34 - 87 * q^35 + 216 * q^36 - 16 * q^37 + 178 * q^38 + 279 * q^39 + 160 * q^40 + 176 * q^41 + 156 * q^42 + 375 * q^43 + 252 * q^44 + 180 * q^45 + 162 * q^46 - 255 * q^47 + 288 * q^48 - 290 * q^49 + 608 * q^50 + 690 * q^51 + 372 * q^52 - 256 * q^53 + 324 * q^54 - 184 * q^55 + 208 * q^56 + 267 * q^57 + 262 * q^58 - 354 * q^59 + 240 * q^60 + 39 * q^61 + 102 * q^62 + 234 * q^63 + 384 * q^64 - 954 * q^65 + 378 * q^66 + 86 * q^67 + 920 * q^68 + 243 * q^69 - 174 * q^70 - 895 * q^71 + 432 * q^72 + 155 * q^73 - 32 * q^74 + 912 * q^75 + 356 * q^76 - 498 * q^77 + 558 * q^78 - 565 * q^79 + 320 * q^80 + 486 * q^81 + 352 * q^82 + 140 * q^83 + 312 * q^84 + q^85 + 750 * q^86 + 393 * q^87 + 504 * q^88 - 1769 * q^89 + 360 * q^90 - 422 * q^91 + 324 * q^92 + 153 * q^93 - 510 * q^94 - 2209 * q^95 + 576 * q^96 + 48 * q^97 - 580 * q^98 + 567 * q^99

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