LMFDB - Embedded newform 230.4.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [230,4,Mod(1,230)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(230, 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("230.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 230 = 2 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	230.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:	$13.5704393013$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 84x^{2} - 11x + 1242 $ x^4 - 84x^2 - 11x + 1242
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$8.16920$ of defining polynomial
Character	$\chi$	$=$	230.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} -7.16920 q^{3} +4.00000 q^{4} -5.00000 q^{5} -14.3384 q^{6} -25.3945 q^{7} +8.00000 q^{8} +24.3974 q^{9} +O(q^{10})$	\(q+2.00000 q^{2} -7.16920 q^{3} +4.00000 q^{4} -5.00000 q^{5} -14.3384 q^{6} -25.3945 q^{7} +8.00000 q^{8} +24.3974 q^{9} -10.0000 q^{10} +67.0072 q^{11} -28.6768 q^{12} -2.41842 q^{13} -50.7891 q^{14} +35.8460 q^{15} +16.0000 q^{16} -12.5097 q^{17} +48.7949 q^{18} +104.553 q^{19} -20.0000 q^{20} +182.058 q^{21} +134.014 q^{22} +23.0000 q^{23} -57.3536 q^{24} +25.0000 q^{25} -4.83684 q^{26} +18.6584 q^{27} -101.578 q^{28} +221.014 q^{29} +71.6920 q^{30} -102.786 q^{31} +32.0000 q^{32} -480.388 q^{33} -25.0194 q^{34} +126.973 q^{35} +97.5897 q^{36} +2.56360 q^{37} +209.106 q^{38} +17.3382 q^{39} -40.0000 q^{40} -89.4364 q^{41} +364.117 q^{42} +5.94181 q^{43} +268.029 q^{44} -121.987 q^{45} +46.0000 q^{46} +549.164 q^{47} -114.707 q^{48} +301.882 q^{49} +50.0000 q^{50} +89.6845 q^{51} -9.67369 q^{52} +159.714 q^{53} +37.3167 q^{54} -335.036 q^{55} -203.156 q^{56} -749.560 q^{57} +442.029 q^{58} +593.563 q^{59} +143.384 q^{60} -894.952 q^{61} -205.572 q^{62} -619.561 q^{63} +64.0000 q^{64} +12.0921 q^{65} -960.776 q^{66} +525.060 q^{67} -50.0388 q^{68} -164.892 q^{69} +253.945 q^{70} -57.4296 q^{71} +195.179 q^{72} -870.328 q^{73} +5.12720 q^{74} -179.230 q^{75} +418.211 q^{76} -1701.62 q^{77} +34.6763 q^{78} +578.999 q^{79} -80.0000 q^{80} -792.496 q^{81} -178.873 q^{82} -345.704 q^{83} +728.234 q^{84} +62.5485 q^{85} +11.8836 q^{86} -1584.50 q^{87} +536.058 q^{88} +311.282 q^{89} -243.974 q^{90} +61.4147 q^{91} +92.0000 q^{92} +736.894 q^{93} +1098.33 q^{94} -522.764 q^{95} -229.414 q^{96} +1815.23 q^{97} +603.764 q^{98} +1634.80 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{2} + 4 q^{3} + 16 q^{4} - 20 q^{5} + 8 q^{6} + 26 q^{7} + 32 q^{8} + 64 q^{9}+O(q^{10}) $ 4 * q + 8 * q^2 + 4 * q^3 + 16 * q^4 - 20 * q^5 + 8 * q^6 + 26 * q^7 + 32 * q^8 + 64 * q^9	\( 4 q + 8 q^{2} + 4 q^{3} + 16 q^{4} - 20 q^{5} + 8 q^{6} + 26 q^{7} + 32 q^{8} + 64 q^{9} - 40 q^{10} + 93 q^{11} + 16 q^{12} + 32 q^{13} + 52 q^{14} - 20 q^{15} + 64 q^{16} + 108 q^{17} + 128 q^{18} + 185 q^{19} - 80 q^{20} + 302 q^{21} + 186 q^{22} + 92 q^{23} + 32 q^{24} + 100 q^{25} + 64 q^{26} + 259 q^{27} + 104 q^{28} + 294 q^{29} - 40 q^{30} - 211 q^{31} + 128 q^{32} - 237 q^{33} + 216 q^{34} - 130 q^{35} + 256 q^{36} + 5 q^{37} + 370 q^{38} - 421 q^{39} - 160 q^{40} - 369 q^{41} + 604 q^{42} - 100 q^{43} + 372 q^{44} - 320 q^{45} + 184 q^{46} - 363 q^{47} + 64 q^{48} + 600 q^{49} + 200 q^{50} - 315 q^{51} + 128 q^{52} + 21 q^{53} + 518 q^{54} - 465 q^{55} + 208 q^{56} - 160 q^{57} + 588 q^{58} - 33 q^{59} - 80 q^{60} - 307 q^{61} - 422 q^{62} + 167 q^{63} + 256 q^{64} - 160 q^{65} - 474 q^{66} + 725 q^{67} + 432 q^{68} + 92 q^{69} - 260 q^{70} - 1257 q^{71} + 512 q^{72} + 509 q^{73} + 10 q^{74} + 100 q^{75} + 740 q^{76} - 1962 q^{77} - 842 q^{78} + 1202 q^{79} - 320 q^{80} - 992 q^{81} - 738 q^{82} - 1377 q^{83} + 1208 q^{84} - 540 q^{85} - 200 q^{86} - 2829 q^{87} + 744 q^{88} - 984 q^{89} - 640 q^{90} - 995 q^{91} + 368 q^{92} - 1843 q^{93} - 726 q^{94} - 925 q^{95} + 128 q^{96} + 137 q^{97} + 1200 q^{98} + 2013 q^{99}+O(q^{100}) \) 4 * q + 8 * q^2 + 4 * q^3 + 16 * q^4 - 20 * q^5 + 8 * q^6 + 26 * q^7 + 32 * q^8 + 64 * q^9 - 40 * q^10 + 93 * q^11 + 16 * q^12 + 32 * q^13 + 52 * q^14 - 20 * q^15 + 64 * q^16 + 108 * q^17 + 128 * q^18 + 185 * q^19 - 80 * q^20 + 302 * q^21 + 186 * q^22 + 92 * q^23 + 32 * q^24 + 100 * q^25 + 64 * q^26 + 259 * q^27 + 104 * q^28 + 294 * q^29 - 40 * q^30 - 211 * q^31 + 128 * q^32 - 237 * q^33 + 216 * q^34 - 130 * q^35 + 256 * q^36 + 5 * q^37 + 370 * q^38 - 421 * q^39 - 160 * q^40 - 369 * q^41 + 604 * q^42 - 100 * q^43 + 372 * q^44 - 320 * q^45 + 184 * q^46 - 363 * q^47 + 64 * q^48 + 600 * q^49 + 200 * q^50 - 315 * q^51 + 128 * q^52 + 21 * q^53 + 518 * q^54 - 465 * q^55 + 208 * q^56 - 160 * q^57 + 588 * q^58 - 33 * q^59 - 80 * q^60 - 307 * q^61 - 422 * q^62 + 167 * q^63 + 256 * q^64 - 160 * q^65 - 474 * q^66 + 725 * q^67 + 432 * q^68 + 92 * q^69 - 260 * q^70 - 1257 * q^71 + 512 * q^72 + 509 * q^73 + 10 * q^74 + 100 * q^75 + 740 * q^76 - 1962 * q^77 - 842 * q^78 + 1202 * q^79 - 320 * q^80 - 992 * q^81 - 738 * q^82 - 1377 * q^83 + 1208 * q^84 - 540 * q^85 - 200 * q^86 - 2829 * q^87 + 744 * q^88 - 984 * q^89 - 640 * q^90 - 995 * q^91 + 368 * q^92 - 1843 * q^93 - 726 * q^94 - 925 * q^95 + 128 * q^96 + 137 * q^97 + 1200 * q^98 + 2013 * 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
$3$	−7.16920	−1.37971	−0.689857	−	0.723946i	$-0.742326\pi$
$3$	−7.16920	−1.37971	−0.689857	+	0.723946i	$0.742326\pi$
$4$	4.00000	0.500000
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	−14.3384	−0.975605
$7$	−25.3945	−1.37118	−0.685588	−	0.727990i	$-0.740455\pi$
$7$	−25.3945	−1.37118	−0.685588	+	0.727990i	$0.740455\pi$
$8$	8.00000	0.353553
$9$	24.3974	0.903608
$10$	−10.0000	−0.316228
$11$	67.0072	1.83668	0.918338	−	0.395798i	$-0.129532\pi$
$11$	67.0072	1.83668	0.918338	+	0.395798i	$0.129532\pi$
$12$	−28.6768	−0.689857
$13$	−2.41842	−0.0515961	−0.0257981	−	0.999667i	$-0.508213\pi$
$13$	−2.41842	−0.0515961	−0.0257981	+	0.999667i	$0.508213\pi$
$14$	−50.7891	−0.969568
$15$	35.8460	0.617026
$16$	16.0000	0.250000
$17$	−12.5097	−0.178473	−0.0892366	−	0.996010i	$-0.528443\pi$
$17$	−12.5097	−0.178473	−0.0892366	+	0.996010i	$0.528443\pi$
$18$	48.7949	0.638948
$19$	104.553	1.26242	0.631212	−	0.775610i	$-0.282558\pi$
$19$	104.553	1.26242	0.631212	+	0.775610i	$0.282558\pi$
$20$	−20.0000	−0.223607
$21$	182.058	1.89183
$22$	134.014	1.29873
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	−57.3536	−0.487802
$25$	25.0000	0.200000
$26$	−4.83684	−0.0364840
$27$	18.6584	0.132993
$28$	−101.578	−0.685588
$29$	221.014	1.41522	0.707609	−	0.706604i	$-0.249774\pi$
$29$	221.014	1.41522	0.707609	+	0.706604i	$0.249774\pi$
$30$	71.6920	0.436304
$31$	−102.786	−0.595514	−0.297757	−	0.954642i	$-0.596238\pi$
$31$	−102.786	−0.595514	−0.297757	+	0.954642i	$0.596238\pi$
$32$	32.0000	0.176777
$33$	−480.388	−2.53409
$34$	−25.0194	−0.126200
$35$	126.973	0.613208
$36$	97.5897	0.451804
$37$	2.56360	0.0113906	0.00569531	−	0.999984i	$-0.498187\pi$
$37$	2.56360	0.0113906	0.00569531	+	0.999984i	$0.498187\pi$
$38$	209.106	0.892669
$39$	17.3382	0.0711879
$40$	−40.0000	−0.158114
$41$	−89.4364	−0.340674	−0.170337	−	0.985386i	$-0.554486\pi$
$41$	−89.4364	−0.340674	−0.170337	+	0.985386i	$0.554486\pi$
$42$	364.117	1.33773
$43$	5.94181	0.0210725	0.0105363	−	0.999944i	$-0.496646\pi$
$43$	5.94181	0.0210725	0.0105363	+	0.999944i	$0.496646\pi$
$44$	268.029	0.918338
$45$	−121.987	−0.404106
$46$	46.0000	0.147442
$47$	549.164	1.70434	0.852168	−	0.523269i	$-0.175288\pi$
$47$	549.164	1.70434	0.852168	+	0.523269i	$0.175288\pi$
$48$	−114.707	−0.344928
$49$	301.882	0.880123
$50$	50.0000	0.141421
$51$	89.6845	0.246242
$52$	−9.67369	−0.0257981
$53$	159.714	0.413931	0.206966	−	0.978348i	$-0.433641\pi$
$53$	159.714	0.413931	0.206966	+	0.978348i	$0.433641\pi$
$54$	37.3167	0.0940400
$55$	−335.036	−0.821386
$56$	−203.156	−0.484784
$57$	−749.560	−1.74178
$58$	442.029	1.00071
$59$	593.563	1.30975	0.654876	−	0.755737i	$-0.272721\pi$
$59$	593.563	1.30975	0.654876	+	0.755737i	$0.272721\pi$
$60$	143.384	0.308513
$61$	−894.952	−1.87847	−0.939237	−	0.343270i	$-0.888465\pi$
$61$	−894.952	−1.87847	−0.939237	+	0.343270i	$0.888465\pi$
$62$	−205.572	−0.421092
$63$	−619.561	−1.23901
$64$	64.0000	0.125000
$65$	12.0921	0.0230745
$66$	−960.776	−1.79187
$67$	525.060	0.957407	0.478703	−	0.877977i	$-0.341107\pi$
$67$	525.060	0.957407	0.478703	+	0.877977i	$0.341107\pi$
$68$	−50.0388	−0.0892366
$69$	−164.892	−0.287690
$70$	253.945	0.433604
$71$	−57.4296	−0.0959950	−0.0479975	−	0.998847i	$-0.515284\pi$
$71$	−57.4296	−0.0959950	−0.0479975	+	0.998847i	$0.515284\pi$
$72$	195.179	0.319474
$73$	−870.328	−1.39540	−0.697700	−	0.716390i	$-0.745793\pi$
$73$	−870.328	−1.39540	−0.697700	+	0.716390i	$0.745793\pi$
$74$	5.12720	0.00805439
$75$	−179.230	−0.275943
$76$	418.211	0.631212
$77$	−1701.62	−2.51840
$78$	34.6763	0.0503374
$79$	578.999	0.824589	0.412294	−	0.911051i	$-0.364728\pi$
$79$	578.999	0.824589	0.412294	+	0.911051i	$0.364728\pi$
$80$	−80.0000	−0.111803
$81$	−792.496	−1.08710
$82$	−178.873	−0.240893
$83$	−345.704	−0.457180	−0.228590	−	0.973523i	$-0.573411\pi$
$83$	−345.704	−0.457180	−0.228590	+	0.973523i	$0.573411\pi$
$84$	728.234	0.945915
$85$	62.5485	0.0798157
$86$	11.8836	0.0149005
$87$	−1584.50	−1.95260
$88$	536.058	0.649363
$89$	311.282	0.370740	0.185370	−	0.982669i	$-0.440652\pi$
$89$	311.282	0.370740	0.185370	+	0.982669i	$0.440652\pi$
$90$	−243.974	−0.285746
$91$	61.4147	0.0707474
$92$	92.0000	0.104257
$93$	736.894	0.821638
$94$	1098.33	1.20515
$95$	−522.764	−0.564573
$96$	−229.414	−0.243901
$97$	1815.23	1.90009	0.950043	−	0.312120i	$-0.101039\pi$
$97$	1815.23	1.90009	0.950043	+	0.312120i	$0.101039\pi$
$98$	603.764	0.622341
$99$	1634.80	1.65964
$100$	100.000	0.100000
$101$	−1492.17	−1.47006	−0.735032	−	0.678033i	$-0.762833\pi$
$101$	−1492.17	−1.47006	−0.735032	+	0.678033i	$0.762833\pi$
$102$	179.369	0.174119
$103$	−250.888	−0.240007	−0.120004	−	0.992773i	$-0.538291\pi$
$103$	−250.888	−0.240007	−0.120004	+	0.992773i	$0.538291\pi$
$104$	−19.3474	−0.0182420
$105$	−910.292	−0.846052
$106$	319.427	0.292693
$107$	−1035.24	−0.935331	−0.467665	−	0.883906i	$-0.654905\pi$
$107$	−1035.24	−0.935331	−0.467665	+	0.883906i	$0.654905\pi$
$108$	74.6334	0.0664963
$109$	354.522	0.311533	0.155766	−	0.987794i	$-0.450215\pi$
$109$	354.522	0.311533	0.155766	+	0.987794i	$0.450215\pi$
$110$	−670.072	−0.580808
$111$	−18.3790	−0.0157158
$112$	−406.312	−0.342794
$113$	1102.86	0.918131	0.459065	−	0.888402i	$-0.348184\pi$
$113$	1102.86	0.918131	0.459065	+	0.888402i	$0.348184\pi$
$114$	−1499.12	−1.23163
$115$	−115.000	−0.0932505
$116$	884.058	0.707609
$117$	−59.0033	−0.0466227
$118$	1187.13	0.926134
$119$	317.678	0.244718
$120$	286.768	0.218152
$121$	3158.96	2.37338
$122$	−1789.90	−1.32828
$123$	641.187	0.470032
$124$	−411.144	−0.297757
$125$	−125.000	−0.0894427
$126$	−1239.12	−0.876110
$127$	503.148	0.351553	0.175776	−	0.984430i	$-0.443756\pi$
$127$	503.148	0.351553	0.175776	+	0.984430i	$0.443756\pi$
$128$	128.000	0.0883883
$129$	−42.5981	−0.0290740
$130$	24.1842	0.0163161
$131$	2880.53	1.92117	0.960586	−	0.277985i	$-0.0896664\pi$
$131$	2880.53	1.92117	0.960586	+	0.277985i	$0.0896664\pi$
$132$	−1921.55	−1.26704
$133$	−2655.07	−1.73101
$134$	1050.12	0.676989
$135$	−93.2918	−0.0594761
$136$	−100.078	−0.0630998
$137$	1888.31	1.17758	0.588792	−	0.808285i	$-0.299604\pi$
$137$	1888.31	1.17758	0.588792	+	0.808285i	$0.299604\pi$
$138$	−329.783	−0.203428
$139$	−1429.79	−0.872472	−0.436236	−	0.899832i	$-0.643689\pi$
$139$	−1429.79	−0.872472	−0.436236	+	0.899832i	$0.643689\pi$
$140$	507.891	0.306604
$141$	−3937.06	−2.35149
$142$	−114.859	−0.0678787
$143$	−162.052	−0.0947653
$144$	390.359	0.225902
$145$	−1105.07	−0.632905
$146$	−1740.66	−0.986697
$147$	−2164.25	−1.21432
$148$	10.2544	0.00569531
$149$	−1268.34	−0.697357	−0.348679	−	0.937242i	$-0.613370\pi$
$149$	−1268.34	−0.697357	−0.348679	+	0.937242i	$0.613370\pi$
$150$	−358.460	−0.195121
$151$	2335.87	1.25888	0.629438	−	0.777050i	$-0.283285\pi$
$151$	2335.87	1.25888	0.629438	+	0.777050i	$0.283285\pi$
$152$	836.423	0.446334
$153$	−305.204	−0.161270
$154$	−3403.23	−1.78078
$155$	513.930	0.266322
$156$	69.3526	0.0355939
$157$	2989.36	1.51960	0.759799	−	0.650158i	$-0.225297\pi$
$157$	2989.36	1.51960	0.759799	+	0.650158i	$0.225297\pi$
$158$	1158.00	0.583072
$159$	−1145.02	−0.571106
$160$	−160.000	−0.0790569
$161$	−584.074	−0.285910
$162$	−1584.99	−0.768696
$163$	700.254	0.336492	0.168246	−	0.985745i	$-0.446190\pi$
$163$	700.254	0.336492	0.168246	+	0.985745i	$0.446190\pi$
$164$	−357.746	−0.170337
$165$	2401.94	1.13328
$166$	−691.408	−0.323275
$167$	1.69236	0.000784185 0	0.000392093	−	1.00000i	$-0.499875\pi$
$167$	1.69236	0.000784185 0	0.000392093		1.00000i	$0.499875\pi$
$168$	1456.47	0.668863
$169$	−2191.15	−0.997338
$170$	125.097	0.0564382
$171$	2550.82	1.14074
$172$	23.7673	0.0105363
$173$	−781.543	−0.343466	−0.171733	−	0.985144i	$-0.554937\pi$
$173$	−781.543	−0.343466	−0.171733	+	0.985144i	$0.554937\pi$
$174$	−3168.99	−1.38069
$175$	−634.863	−0.274235
$176$	1072.12	0.459169
$177$	−4255.37	−1.80708
$178$	622.564	0.262152
$179$	2670.22	1.11498	0.557490	−	0.830184i	$-0.311765\pi$
$179$	2670.22	1.11498	0.557490	+	0.830184i	$0.311765\pi$
$180$	−487.949	−0.202053
$181$	−1230.09	−0.505149	−0.252574	−	0.967577i	$-0.581277\pi$
$181$	−1230.09	−0.505149	−0.252574	+	0.967577i	$0.581277\pi$
$182$	122.829	0.0500259
$183$	6416.09	2.59175
$184$	184.000	0.0737210
$185$	−12.8180	−0.00509404
$186$	1473.79	0.580986
$187$	−838.239	−0.327797
$188$	2196.65	0.852168
$189$	−473.820	−0.182356
$190$	−1045.53	−0.399214
$191$	−1433.23	−0.542959	−0.271480	−	0.962444i	$-0.587513\pi$
$191$	−1433.23	−0.542959	−0.271480	+	0.962444i	$0.587513\pi$
$192$	−458.829	−0.172464
$193$	4348.12	1.62168	0.810841	−	0.585266i	$-0.199010\pi$
$193$	4348.12	1.62168	0.810841	+	0.585266i	$0.199010\pi$
$194$	3630.45	1.34356
$195$	−86.6908	−0.0318362
$196$	1207.53	0.440062
$197$	−2551.15	−0.922649	−0.461324	−	0.887232i	$-0.652626\pi$
$197$	−2551.15	−0.922649	−0.461324	+	0.887232i	$0.652626\pi$
$198$	3269.61	1.17354
$199$	1230.65	0.438386	0.219193	−	0.975682i	$-0.429658\pi$
$199$	1230.65	0.438386	0.219193	+	0.975682i	$0.429658\pi$
$200$	200.000	0.0707107
$201$	−3764.26	−1.32095
$202$	−2984.34	−1.03949
$203$	−5612.56	−1.94051
$204$	358.738	0.123121
$205$	447.182	0.152354
$206$	−501.777	−0.169711
$207$	561.141	0.188415
$208$	−38.6948	−0.0128990
$209$	7005.79	2.31866
$210$	−1820.58	−0.598249
$211$	2834.05	0.924663	0.462332	−	0.886707i	$-0.347013\pi$
$211$	2834.05	0.924663	0.462332	+	0.886707i	$0.347013\pi$
$212$	638.854	0.206966
$213$	411.724	0.132446
$214$	−2070.48	−0.661379
$215$	−29.7091	−0.00942392
$216$	149.267	0.0470200
$217$	2610.20	0.816554
$218$	709.045	0.220287
$219$	6239.56	1.92525
$220$	−1340.14	−0.410693
$221$	30.2537	0.00920853
$222$	−36.7579	−0.0111127
$223$	−391.251	−0.117489	−0.0587446	−	0.998273i	$-0.518710\pi$
$223$	−391.251	−0.117489	−0.0587446	+	0.998273i	$0.518710\pi$
$224$	−812.625	−0.242392
$225$	609.936	0.180722
$226$	2205.73	0.649216
$227$	1245.79	0.364256	0.182128	−	0.983275i	$-0.441702\pi$
$227$	1245.79	0.364256	0.182128	+	0.983275i	$0.441702\pi$
$228$	−2998.24	−0.870892
$229$	6497.87	1.87507	0.937535	−	0.347890i	$-0.113102\pi$
$229$	6497.87	1.87507	0.937535	+	0.347890i	$0.113102\pi$
$230$	−230.000	−0.0659380
$231$	12199.2	3.47468
$232$	1768.12	0.500355
$233$	−3076.87	−0.865118	−0.432559	−	0.901606i	$-0.642389\pi$
$233$	−3076.87	−0.865118	−0.432559	+	0.901606i	$0.642389\pi$
$234$	−118.007	−0.0329672
$235$	−2745.82	−0.762202
$236$	2374.25	0.654876
$237$	−4150.96	−1.13770
$238$	635.355	0.173042
$239$	532.313	0.144069	0.0720344	−	0.997402i	$-0.477051\pi$
$239$	532.313	0.144069	0.0720344	+	0.997402i	$0.477051\pi$
$240$	573.536	0.154257
$241$	−830.944	−0.222099	−0.111049	−	0.993815i	$-0.535421\pi$
$241$	−830.944	−0.222099	−0.111049	+	0.993815i	$0.535421\pi$
$242$	6317.93	1.67823
$243$	5177.79	1.36689
$244$	−3579.81	−0.939237
$245$	−1509.41	−0.393603
$246$	1282.37	0.332363
$247$	−252.853	−0.0651362
$248$	−822.288	−0.210546
$249$	2478.42	0.630777
$250$	−250.000	−0.0632456
$251$	3729.71	0.937916	0.468958	−	0.883220i	$-0.344629\pi$
$251$	3729.71	0.937916	0.468958	+	0.883220i	$0.344629\pi$
$252$	−2478.25	−0.619503
$253$	1541.17	0.382973
$254$	1006.30	0.248585
$255$	−448.422	−0.110123
$256$	256.000	0.0625000
$257$	−7399.44	−1.79597	−0.897985	−	0.440027i	$-0.854969\pi$
$257$	−7399.44	−1.79597	−0.897985	+	0.440027i	$0.854969\pi$
$258$	−85.1961	−0.0205584
$259$	−65.1014	−0.0156185
$260$	48.3684	0.0115372
$261$	5392.18	1.27880
$262$	5761.07	1.35847
$263$	−1907.88	−0.447319	−0.223659	−	0.974667i	$-0.571800\pi$
$263$	−1907.88	−0.447319	−0.223659	+	0.974667i	$0.571800\pi$
$264$	−3843.10	−0.895934
$265$	−798.568	−0.185116
$266$	−5310.14	−1.22401
$267$	−2231.64	−0.511514
$268$	2100.24	0.478703
$269$	−1291.74	−0.292784	−0.146392	−	0.989227i	$-0.546766\pi$
$269$	−1291.74	−0.292784	−0.146392	+	0.989227i	$0.546766\pi$
$270$	−186.584	−0.0420560
$271$	−5361.16	−1.20172	−0.600862	−	0.799353i	$-0.705176\pi$
$271$	−5361.16	−1.20172	−0.600862	+	0.799353i	$0.705176\pi$
$272$	−200.155	−0.0446183
$273$	−440.294	−0.0976111
$274$	3776.61	0.832677
$275$	1675.18	0.367335
$276$	−659.566	−0.143845
$277$	−839.923	−0.182188	−0.0910940	−	0.995842i	$-0.529036\pi$
$277$	−839.923	−0.182188	−0.0910940	+	0.995842i	$0.529036\pi$
$278$	−2859.59	−0.616931
$279$	−2507.72	−0.538111
$280$	1015.78	0.216802
$281$	−610.965	−0.129705	−0.0648525	−	0.997895i	$-0.520658\pi$
$281$	−610.965	−0.129705	−0.0648525	+	0.997895i	$0.520658\pi$
$282$	−7874.13	−1.66276
$283$	−8568.90	−1.79989	−0.899944	−	0.436006i	$-0.856393\pi$
$283$	−8568.90	−1.79989	−0.899944	+	0.436006i	$0.856393\pi$
$284$	−229.718	−0.0479975
$285$	3747.80	0.778949
$286$	−324.103	−0.0670092
$287$	2271.20	0.467123
$288$	780.718	0.159737
$289$	−4756.51	−0.968147
$290$	−2210.14	−0.447531
$291$	−13013.7	−2.62157
$292$	−3481.31	−0.697700
$293$	−9728.79	−1.93980	−0.969901	−	0.243498i	$-0.921705\pi$
$293$	−9728.79	−1.93980	−0.969901	+	0.243498i	$0.921705\pi$
$294$	−4328.51	−0.858652
$295$	−2967.82	−0.585739
$296$	20.5088	0.00402719
$297$	1250.24	0.244264
$298$	−2536.67	−0.493106
$299$	−55.6237	−0.0107585
$300$	−716.920	−0.137971
$301$	−150.890	−0.0288941
$302$	4671.74	0.890160
$303$	10697.7	2.02827
$304$	1672.85	0.315606
$305$	4474.76	0.840079
$306$	−610.409	−0.114035
$307$	−8115.58	−1.50873	−0.754366	−	0.656454i	$-0.772055\pi$
$307$	−8115.58	−1.50873	−0.754366	+	0.656454i	$0.772055\pi$
$308$	−6806.47	−1.25920
$309$	1798.67	0.331141
$310$	1027.86	0.188318
$311$	−6433.83	−1.17308	−0.586542	−	0.809919i	$-0.699511\pi$
$311$	−6433.83	−1.17308	−0.586542	+	0.809919i	$0.699511\pi$
$312$	138.705	0.0251687
$313$	2674.76	0.483023	0.241512	−	0.970398i	$-0.422357\pi$
$313$	2674.76	0.483023	0.241512	+	0.970398i	$0.422357\pi$
$314$	5978.72	1.07452
$315$	3097.81	0.554100
$316$	2316.00	0.412294
$317$	−8720.03	−1.54500	−0.772501	−	0.635013i	$-0.780995\pi$
$317$	−8720.03	−1.54500	−0.772501	+	0.635013i	$0.780995\pi$
$318$	−2290.04	−0.403833
$319$	14809.6	2.59930
$320$	−320.000	−0.0559017
$321$	7421.84	1.29049
$322$	−1168.15	−0.202169
$323$	−1307.92	−0.225309
$324$	−3169.98	−0.543550
$325$	−60.4606	−0.0103192
$326$	1400.51	0.237936
$327$	−2541.64	−0.429826
$328$	−715.491	−0.120446
$329$	−13945.8	−2.33694
$330$	4803.88	0.801348
$331$	−3384.40	−0.562005	−0.281002	−	0.959707i	$-0.590667\pi$
$331$	−3384.40	−0.562005	−0.281002	+	0.959707i	$0.590667\pi$
$332$	−1382.82	−0.228590
$333$	62.5452	0.0102927
$334$	3.38472	0.000554503 0
$335$	−2625.30	−0.428165
$336$	2912.94	0.472957
$337$	9334.10	1.50879	0.754393	−	0.656423i	$-0.227931\pi$
$337$	9334.10	1.50879	0.754393	+	0.656423i	$0.227931\pi$
$338$	−4382.30	−0.705224
$339$	−7906.66	−1.26676
$340$	250.194	0.0399078
$341$	−6887.40	−1.09377
$342$	5101.64	0.806623
$343$	1044.17	0.164372
$344$	47.5345	0.00745026
$345$	824.458	0.128659
$346$	−1563.09	−0.242867
$347$	−1943.02	−0.300595	−0.150298	−	0.988641i	$-0.548023\pi$
$347$	−1943.02	−0.300595	−0.150298	+	0.988641i	$0.548023\pi$
$348$	−6337.99	−0.976298
$349$	6978.55	1.07035	0.535177	−	0.844740i	$-0.320245\pi$
$349$	6978.55	1.07035	0.535177	+	0.844740i	$0.320245\pi$
$350$	−1269.73	−0.193914
$351$	−45.1238	−0.00686191
$352$	2144.23	0.324681
$353$	6854.18	1.03346	0.516730	−	0.856149i	$-0.327149\pi$
$353$	6854.18	1.03346	0.516730	+	0.856149i	$0.327149\pi$
$354$	−8510.75	−1.27780
$355$	287.148	0.0429302
$356$	1245.13	0.185370
$357$	−2277.50	−0.337641
$358$	5340.43	0.788410
$359$	10618.3	1.56104	0.780522	−	0.625128i	$-0.214953\pi$
$359$	10618.3	1.56104	0.780522	+	0.625128i	$0.214953\pi$
$360$	−975.897	−0.142873
$361$	4072.29	0.593715
$362$	−2460.18	−0.357194
$363$	−22647.2	−3.27458
$364$	245.659	0.0353737
$365$	4351.64	0.624042
$366$	12832.2	1.83265
$367$	−9250.18	−1.31568	−0.657841	−	0.753157i	$-0.728530\pi$
$367$	−9250.18	−1.31568	−0.657841	+	0.753157i	$0.728530\pi$
$368$	368.000	0.0521286
$369$	−2182.02	−0.307836
$370$	−25.6360	−0.00360203
$371$	−4055.85	−0.567572
$372$	2947.57	0.410819
$373$	8505.24	1.18066	0.590328	−	0.807163i	$-0.298998\pi$
$373$	8505.24	1.18066	0.590328	+	0.807163i	$0.298998\pi$
$374$	−1676.48	−0.231788
$375$	896.150	0.123405
$376$	4393.31	0.602573
$377$	−534.506	−0.0730198
$378$	−947.640	−0.128945
$379$	1047.43	0.141960	0.0709798	−	0.997478i	$-0.477387\pi$
$379$	1047.43	0.141960	0.0709798	+	0.997478i	$0.477387\pi$
$380$	−2091.06	−0.282287
$381$	−3607.17	−0.485042
$382$	−2866.47	−0.383930
$383$	9072.64	1.21042	0.605209	−	0.796066i	$-0.293089\pi$
$383$	9072.64	1.21042	0.605209	+	0.796066i	$0.293089\pi$
$384$	−917.658	−0.121951
$385$	8508.08	1.12626
$386$	8696.25	1.14670
$387$	144.965	0.0190413
$388$	7260.90	0.950043
$389$	−6880.63	−0.896817	−0.448408	−	0.893829i	$-0.648009\pi$
$389$	−6880.63	−0.896817	−0.448408	+	0.893829i	$0.648009\pi$
$390$	−173.382	−0.0225116
$391$	−287.723	−0.0372143
$392$	2415.06	0.311170
$393$	−20651.1	−2.65067
$394$	−5102.30	−0.652411
$395$	−2895.00	−0.368767
$396$	6539.21	0.829818
$397$	9297.23	1.17535	0.587676	−	0.809096i	$-0.300043\pi$
$397$	9297.23	1.17535	0.587676	+	0.809096i	$0.300043\pi$
$398$	2461.31	0.309986
$399$	19034.7	2.38829
$400$	400.000	0.0500000
$401$	−4904.30	−0.610745	−0.305373	−	0.952233i	$-0.598781\pi$
$401$	−4904.30	−0.610745	−0.305373	+	0.952233i	$0.598781\pi$
$402$	−7528.52	−0.934051
$403$	248.580	0.0307262
$404$	−5968.68	−0.735032
$405$	3962.48	0.486166
$406$	−11225.1	−1.37215
$407$	171.780	0.0209209
$408$	717.476	0.0870597
$409$	−11856.7	−1.43344	−0.716719	−	0.697362i	$-0.754357\pi$
$409$	−11856.7	−1.43344	−0.716719	+	0.697362i	$0.754357\pi$
$410$	894.364	0.107730
$411$	−13537.6	−1.62473
$412$	−1003.55	−0.120004
$413$	−15073.3	−1.79590
$414$	1122.28	0.133230
$415$	1728.52	0.204457
$416$	−77.3895	−0.00912099
$417$	10250.5	1.20376
$418$	14011.6	1.63954
$419$	12497.2	1.45711	0.728554	−	0.684988i	$-0.240193\pi$
$419$	12497.2	1.45711	0.728554	+	0.684988i	$0.240193\pi$
$420$	−3641.17	−0.423026
$421$	−2341.63	−0.271079	−0.135539	−	0.990772i	$-0.543277\pi$
$421$	−2341.63	−0.271079	−0.135539	+	0.990772i	$0.543277\pi$
$422$	5668.10	0.653836
$423$	13398.2	1.54005
$424$	1277.71	0.146347
$425$	−312.742	−0.0356947
$426$	823.449	0.0936531
$427$	22726.9	2.57572
$428$	−4140.96	−0.467665
$429$	1161.78	0.130749
$430$	−59.4181	−0.00666372
$431$	−14047.2	−1.56990	−0.784951	−	0.619558i	$-0.787312\pi$
$431$	−14047.2	−1.56990	−0.784951	+	0.619558i	$0.787312\pi$
$432$	298.534	0.0332482
$433$	9877.43	1.09626	0.548128	−	0.836394i	$-0.315341\pi$
$433$	9877.43	1.09626	0.548128	+	0.836394i	$0.315341\pi$
$434$	5220.41	0.577391
$435$	7922.48	0.873228
$436$	1418.09	0.155766
$437$	2404.72	0.263234
$438$	12479.1	1.36136
$439$	−7028.36	−0.764113	−0.382056	−	0.924139i	$-0.624784\pi$
$439$	−7028.36	−0.764113	−0.382056	+	0.924139i	$0.624784\pi$
$440$	−2680.29	−0.290404
$441$	7365.15	0.795287
$442$	60.5074	0.00651141
$443$	−13486.7	−1.44644	−0.723219	−	0.690618i	$-0.757338\pi$
$443$	−13486.7	−1.44644	−0.723219	+	0.690618i	$0.757338\pi$
$444$	−73.5158	−0.00785790
$445$	−1556.41	−0.165800
$446$	−782.502	−0.0830774
$447$	9092.96	0.962153
$448$	−1625.25	−0.171397
$449$	−6435.76	−0.676441	−0.338221	−	0.941067i	$-0.609825\pi$
$449$	−6435.76	−0.676441	−0.338221	+	0.941067i	$0.609825\pi$
$450$	1219.87	0.127790
$451$	−5992.88	−0.625707
$452$	4411.46	0.459065
$453$	−16746.3	−1.73689
$454$	2491.58	0.257568
$455$	−307.074	−0.0316392
$456$	−5996.48	−0.615814
$457$	−14325.5	−1.46634	−0.733172	−	0.680043i	$-0.761961\pi$
$457$	−14325.5	−1.46634	−0.733172	+	0.680043i	$0.761961\pi$
$458$	12995.7	1.32588
$459$	−233.410	−0.0237356
$460$	−460.000	−0.0466252
$461$	9966.13	1.00687	0.503437	−	0.864032i	$-0.332069\pi$
$461$	9966.13	1.00687	0.503437	+	0.864032i	$0.332069\pi$
$462$	24398.5	2.45697
$463$	3629.06	0.364270	0.182135	−	0.983274i	$-0.441699\pi$
$463$	3629.06	0.364270	0.182135	+	0.983274i	$0.441699\pi$
$464$	3536.23	0.353805
$465$	−3684.47	−0.367448
$466$	−6153.74	−0.611731
$467$	−4453.81	−0.441323	−0.220661	−	0.975350i	$-0.570822\pi$
$467$	−4453.81	−0.441323	−0.220661	+	0.975350i	$0.570822\pi$
$468$	−236.013	−0.0233113
$469$	−13333.6	−1.31277
$470$	−5491.64	−0.538958
$471$	−21431.3	−2.09661
$472$	4748.50	0.463067
$473$	398.144	0.0387034
$474$	−8301.92	−0.804473
$475$	2613.82	0.252485
$476$	1270.71	0.122359
$477$	3896.60	0.374032
$478$	1064.63	0.101872
$479$	3196.91	0.304949	0.152475	−	0.988307i	$-0.451276\pi$
$479$	3196.91	0.304949	0.152475	+	0.988307i	$0.451276\pi$
$480$	1147.07	0.109076
$481$	−6.19986	−0.000587712 0
$482$	−1661.89	−0.157047
$483$	4187.34	0.394474
$484$	12635.9	1.18669
$485$	−9076.13	−0.849744
$486$	10355.6	0.966540
$487$	9506.29	0.884540	0.442270	−	0.896882i	$-0.354173\pi$
$487$	9506.29	0.884540	0.442270	+	0.896882i	$0.354173\pi$
$488$	−7159.62	−0.664141
$489$	−5020.26	−0.464262
$490$	−3018.82	−0.278319
$491$	−9924.98	−0.912236	−0.456118	−	0.889919i	$-0.650761\pi$
$491$	−9924.98	−0.912236	−0.456118	+	0.889919i	$0.650761\pi$
$492$	2564.75	0.235016
$493$	−2764.82	−0.252579
$494$	−505.706	−0.0460583
$495$	−8174.02	−0.742212
$496$	−1644.58	−0.148878
$497$	1458.40	0.131626
$498$	4956.84	0.446027
$499$	3186.71	0.285885	0.142942	−	0.989731i	$-0.454344\pi$
$499$	3186.71	0.285885	0.142942	+	0.989731i	$0.454344\pi$
$500$	−500.000	−0.0447214
$501$	−12.1329	−0.00108195
$502$	7459.41	0.663207
$503$	1222.87	0.108399	0.0541997	−	0.998530i	$-0.482739\pi$
$503$	1222.87	0.108399	0.0541997	+	0.998530i	$0.482739\pi$
$504$	−4956.49	−0.438055
$505$	7460.85	0.657432
$506$	3082.33	0.270803
$507$	15708.8	1.37604
$508$	2012.59	0.175776
$509$	−19858.4	−1.72929	−0.864644	−	0.502385i	$-0.832456\pi$
$509$	−19858.4	−1.72929	−0.864644	+	0.502385i	$0.832456\pi$
$510$	−896.845	−0.0778685
$511$	22101.6	1.91334
$512$	512.000	0.0441942
$513$	1950.78	0.167893
$514$	−14798.9	−1.26994
$515$	1254.44	0.107335
$516$	−170.392	−0.0145370
$517$	36797.9	3.13031
$518$	−130.203	−0.0110440
$519$	5603.04	0.473884
$520$	96.7369	0.00815806
$521$	−1723.35	−0.144916	−0.0724581	−	0.997371i	$-0.523084\pi$
$521$	−1723.35	−0.144916	−0.0724581	+	0.997371i	$0.523084\pi$
$522$	10784.4	0.904251
$523$	21592.8	1.80533	0.902663	−	0.430347i	$-0.141609\pi$
$523$	21592.8	1.80533	0.902663	+	0.430347i	$0.141609\pi$
$524$	11522.1	0.960586
$525$	4551.46	0.378366
$526$	−3815.76	−0.316302
$527$	1285.82	0.106283
$528$	−7686.21	−0.633521
$529$	529.000	0.0434783
$530$	−1597.14	−0.130897
$531$	14481.4	1.18350
$532$	−10620.3	−0.865503
$533$	216.295	0.0175774
$534$	−4463.29	−0.361695
$535$	5176.20	0.418293
$536$	4200.48	0.338494
$537$	−19143.3	−1.53835
$538$	−2583.49	−0.207030
$539$	20228.3	1.61650
$540$	−373.167	−0.0297381
$541$	−20473.8	−1.62706	−0.813528	−	0.581525i	$-0.802456\pi$
$541$	−20473.8	−1.62706	−0.813528	+	0.581525i	$0.802456\pi$
$542$	−10722.3	−0.849747
$543$	8818.77	0.696960
$544$	−400.310	−0.0315499
$545$	−1772.61	−0.139322
$546$	−880.589	−0.0690214
$547$	−3212.07	−0.251075	−0.125538	−	0.992089i	$-0.540066\pi$
$547$	−3212.07	−0.251075	−0.125538	+	0.992089i	$0.540066\pi$
$548$	7553.23	0.588792
$549$	−21834.5	−1.69740
$550$	3350.36	0.259745
$551$	23107.7	1.78661
$552$	−1319.13	−0.101714
$553$	−14703.4	−1.13066
$554$	−1679.85	−0.128826
$555$	91.8948	0.00702832
$556$	−5719.18	−0.436236
$557$	14789.3	1.12503	0.562515	−	0.826787i	$-0.309834\pi$
$557$	14789.3	1.12503	0.562515	+	0.826787i	$0.309834\pi$
$558$	−5015.43	−0.380502
$559$	−14.3698	−0.00108726
$560$	2031.56	0.153302
$561$	6009.51	0.452267
$562$	−1221.93	−0.0917153
$563$	−24632.1	−1.84391	−0.921953	−	0.387302i	$-0.873407\pi$
$563$	−24632.1	−1.84391	−0.921953	+	0.387302i	$0.873407\pi$
$564$	−15748.3	−1.17575
$565$	−5514.32	−0.410601
$566$	−17137.8	−1.27271
$567$	20125.1	1.49061
$568$	−459.437	−0.0339393
$569$	−16924.5	−1.24695	−0.623473	−	0.781845i	$-0.714279\pi$
$569$	−16924.5	−1.24695	−0.623473	+	0.781845i	$0.714279\pi$
$570$	7495.60	0.550800
$571$	−907.889	−0.0665394	−0.0332697	−	0.999446i	$-0.510592\pi$
$571$	−907.889	−0.0665394	−0.0332697	+	0.999446i	$0.510592\pi$
$572$	−648.207	−0.0473827
$573$	10275.1	0.749128
$574$	4542.39	0.330306
$575$	575.000	0.0417029
$576$	1561.44	0.112951
$577$	21160.0	1.52669	0.763346	−	0.645990i	$-0.223555\pi$
$577$	21160.0	1.52669	0.763346	+	0.645990i	$0.223555\pi$
$578$	−9513.02	−0.684584
$579$	−31172.6	−2.23746
$580$	−4420.29	−0.316453
$581$	8778.99	0.626874
$582$	−26027.4	−1.85373
$583$	10702.0	0.760257
$584$	−6962.62	−0.493348
$585$	295.016	0.0208503
$586$	−19457.6	−1.37165
$587$	10085.6	0.709159	0.354579	−	0.935026i	$-0.384624\pi$
$587$	10085.6	0.709159	0.354579	+	0.935026i	$0.384624\pi$
$588$	−8657.02	−0.607159
$589$	−10746.6	−0.751791
$590$	−5935.63	−0.414180
$591$	18289.7	1.27299
$592$	41.0176	0.00284766
$593$	24405.1	1.69004	0.845022	−	0.534731i	$-0.179587\pi$
$593$	24405.1	1.69004	0.845022	+	0.534731i	$0.179587\pi$
$594$	2500.49	0.172721
$595$	−1588.39	−0.109441
$596$	−5073.35	−0.348679
$597$	−8822.81	−0.604847
$598$	−111.247	−0.00760743
$599$	−5361.08	−0.365689	−0.182845	−	0.983142i	$-0.558531\pi$
$599$	−5361.08	−0.365689	−0.182845	+	0.983142i	$0.558531\pi$
$600$	−1433.84	−0.0975605
$601$	15225.4	1.03338	0.516688	−	0.856174i	$-0.327165\pi$
$601$	15225.4	1.03338	0.516688	+	0.856174i	$0.327165\pi$
$602$	−301.779	−0.0204312
$603$	12810.1	0.865121
$604$	9343.48	0.629438
$605$	−15794.8	−1.06141
$606$	21395.3	1.43420
$607$	−11891.3	−0.795148	−0.397574	−	0.917570i	$-0.630148\pi$
$607$	−11891.3	−0.795148	−0.397574	+	0.917570i	$0.630148\pi$
$608$	3345.69	0.223167
$609$	40237.5	2.67735
$610$	8949.52	0.594025
$611$	−1328.11	−0.0879371
$612$	−1220.82	−0.0806350
$613$	155.039	0.0102153	0.00510765	−	0.999987i	$-0.498374\pi$
$613$	155.039	0.0102153	0.00510765	+	0.999987i	$0.498374\pi$
$614$	−16231.2	−1.06683
$615$	−3205.94	−0.210205
$616$	−13612.9	−0.890391
$617$	6028.99	0.393384	0.196692	−	0.980465i	$-0.436980\pi$
$617$	6028.99	0.393384	0.196692	+	0.980465i	$0.436980\pi$
$618$	3597.34	0.234152
$619$	−23232.7	−1.50856	−0.754282	−	0.656550i	$-0.772015\pi$
$619$	−23232.7	−1.50856	−0.754282	+	0.656550i	$0.772015\pi$
$620$	2055.72	0.133161
$621$	429.142	0.0277309
$622$	−12867.7	−0.829496
$623$	−7904.86	−0.508349
$624$	277.410	0.0177970
$625$	625.000	0.0400000
$626$	5349.52	0.341549
$627$	−50225.9	−3.19909
$628$	11957.4	0.759799
$629$	−32.0698	−0.00203292
$630$	6195.61	0.391808
$631$	14910.2	0.940673	0.470337	−	0.882487i	$-0.344132\pi$
$631$	14910.2	0.940673	0.470337	+	0.882487i	$0.344132\pi$
$632$	4631.99	0.291536
$633$	−20317.9	−1.27577
$634$	−17440.1	−1.09248
$635$	−2515.74	−0.157219
$636$	−4580.07	−0.285553
$637$	−730.079	−0.0454109
$638$	29619.1	1.83798
$639$	−1401.14	−0.0867419
$640$	−640.000	−0.0395285
$641$	22653.9	1.39591	0.697953	−	0.716144i	$-0.254095\pi$
$641$	22653.9	1.39591	0.697953	+	0.716144i	$0.254095\pi$
$642$	14843.7	0.912513
$643$	−9024.61	−0.553493	−0.276746	−	0.960943i	$-0.589256\pi$
$643$	−9024.61	−0.553493	−0.276746	+	0.960943i	$0.589256\pi$
$644$	−2336.30	−0.142955
$645$	212.990	0.0130023
$646$	−2615.85	−0.159318
$647$	−6904.07	−0.419516	−0.209758	−	0.977753i	$-0.567268\pi$
$647$	−6904.07	−0.419516	−0.209758	+	0.977753i	$0.567268\pi$
$648$	−6339.97	−0.384348
$649$	39773.0	2.40559
$650$	−120.921	−0.00729679
$651$	−18713.1	−1.12661
$652$	2801.02	0.168246
$653$	−6242.60	−0.374107	−0.187054	−	0.982350i	$-0.559894\pi$
$653$	−6242.60	−0.374107	−0.187054	+	0.982350i	$0.559894\pi$
$654$	−5083.28	−0.303933
$655$	−14402.7	−0.859174
$656$	−1430.98	−0.0851684
$657$	−21233.8	−1.26090
$658$	−27891.5	−1.65247
$659$	30150.4	1.78224	0.891118	−	0.453773i	$-0.149922\pi$
$659$	30150.4	1.78224	0.891118	+	0.453773i	$0.149922\pi$
$660$	9607.76	0.566639
$661$	−7567.21	−0.445280	−0.222640	−	0.974901i	$-0.571468\pi$
$661$	−7567.21	−0.445280	−0.222640	+	0.974901i	$0.571468\pi$
$662$	−6768.80	−0.397397
$663$	−216.895	−0.0127051
$664$	−2765.63	−0.161637
$665$	13275.4	0.774129
$666$	125.090	0.00727801
$667$	5083.33	0.295094
$668$	6.76945	0.000392093 0
$669$	2804.96	0.162101
$670$	−5250.60	−0.302759
$671$	−59968.2	−3.45015
$672$	5825.87	0.334431
$673$	−941.239	−0.0539110	−0.0269555	−	0.999637i	$-0.508581\pi$
$673$	−941.239	−0.0539110	−0.0269555	+	0.999637i	$0.508581\pi$
$674$	18668.2	1.06687
$675$	466.459	0.0265985
$676$	−8764.60	−0.498669
$677$	20383.3	1.15716	0.578578	−	0.815627i	$-0.303608\pi$
$677$	20383.3	1.15716	0.578578	+	0.815627i	$0.303608\pi$
$678$	−15813.3	−0.895733
$679$	−46096.8	−2.60535
$680$	500.388	0.0282191
$681$	−8931.33	−0.502569
$682$	−13774.8	−0.773409
$683$	−2076.87	−0.116353	−0.0581765	−	0.998306i	$-0.518529\pi$
$683$	−2076.87	−0.116353	−0.0581765	+	0.998306i	$0.518529\pi$
$684$	10203.3	0.570369
$685$	−9441.53	−0.526631
$686$	2088.33	0.116229
$687$	−46584.5	−2.58706
$688$	95.0690	0.00526813
$689$	−386.255	−0.0213572
$690$	1648.92	0.0909756
$691$	17273.6	0.950969	0.475484	−	0.879724i	$-0.342273\pi$
$691$	17273.6	0.950969	0.475484	+	0.879724i	$0.342273\pi$
$692$	−3126.17	−0.171733
$693$	−41515.1	−2.27565
$694$	−3886.03	−0.212553
$695$	7148.97	0.390181
$696$	−12676.0	−0.690347
$697$	1118.82	0.0608011
$698$	13957.1	0.756854
$699$	22058.7	1.19361
$700$	−2539.45	−0.137118
$701$	23844.5	1.28473	0.642364	−	0.766399i	$-0.277954\pi$
$701$	23844.5	1.28473	0.642364	+	0.766399i	$0.277954\pi$
$702$	−90.2476	−0.00485210
$703$	268.031	0.0143798
$704$	4288.46	0.229584
$705$	19685.3	1.05162
$706$	13708.4	0.730766
$707$	37892.9	2.01572
$708$	−17021.5	−0.903541
$709$	−9773.89	−0.517724	−0.258862	−	0.965914i	$-0.583347\pi$
$709$	−9773.89	−0.517724	−0.258862	+	0.965914i	$0.583347\pi$
$710$	574.296	0.0303563
$711$	14126.1	0.745105
$712$	2490.26	0.131076
$713$	−2364.08	−0.124173
$714$	−4554.99	−0.238748
$715$	810.259	0.0423803
$716$	10680.9	0.557490
$717$	−3816.26	−0.198774
$718$	21236.7	1.10382
$719$	−15782.2	−0.818605	−0.409303	−	0.912399i	$-0.634228\pi$
$719$	−15782.2	−0.818605	−0.409303	+	0.912399i	$0.634228\pi$
$720$	−1951.79	−0.101026
$721$	6371.19	0.329092
$722$	8144.59	0.419820
$723$	5957.20	0.306432
$724$	−4920.36	−0.252574
$725$	5525.36	0.283044
$726$	−45294.5	−2.31548
$727$	9655.80	0.492591	0.246295	−	0.969195i	$-0.420787\pi$
$727$	9655.80	0.492591	0.246295	+	0.969195i	$0.420787\pi$
$728$	491.318	0.0250130
$729$	−15723.2	−0.798821
$730$	8703.28	0.441264
$731$	−74.3303	−0.00376088
$732$	25664.4	1.29588
$733$	−7159.73	−0.360779	−0.180389	−	0.983595i	$-0.557736\pi$
$733$	−7159.73	−0.360779	−0.180389	+	0.983595i	$0.557736\pi$
$734$	−18500.4	−0.930328
$735$	10821.3	0.543059
$736$	736.000	0.0368605
$737$	35182.8	1.75845
$738$	−4364.04	−0.217673
$739$	−3017.01	−0.150179	−0.0750897	−	0.997177i	$-0.523924\pi$
$739$	−3017.01	−0.150179	−0.0750897	+	0.997177i	$0.523924\pi$
$740$	−51.2720	−0.00254702
$741$	1812.75	0.0898693
$742$	−8111.70	−0.401334
$743$	36291.7	1.79194	0.895971	−	0.444112i	$-0.146481\pi$
$743$	36291.7	1.79194	0.895971	+	0.444112i	$0.146481\pi$
$744$	5895.15	0.290493
$745$	6341.69	0.311868
$746$	17010.5	0.834850
$747$	−8434.29	−0.413112
$748$	−3352.96	−0.163899
$749$	26289.4	1.28250
$750$	1792.30	0.0872607
$751$	−22258.2	−1.08151	−0.540754	−	0.841181i	$-0.681861\pi$
$751$	−22258.2	−1.08151	−0.540754	+	0.841181i	$0.681861\pi$
$752$	8786.62	0.426084
$753$	−26739.0	−1.29406
$754$	−1069.01	−0.0516328
$755$	−11679.3	−0.562987
$756$	−1895.28	−0.0911782
$757$	13222.9	0.634868	0.317434	−	0.948280i	$-0.397179\pi$
$757$	13222.9	0.634868	0.317434	+	0.948280i	$0.397179\pi$
$758$	2094.85	0.100381
$759$	−11048.9	−0.528393
$760$	−4182.11	−0.199607
$761$	−1297.65	−0.0618129	−0.0309065	−	0.999522i	$-0.509839\pi$
$761$	−1297.65	−0.0618129	−0.0309065	+	0.999522i	$0.509839\pi$
$762$	−7214.34	−0.342976
$763$	−9002.93	−0.427166
$764$	−5732.94	−0.271480
$765$	1526.02	0.0721221
$766$	18145.3	0.855895
$767$	−1435.49	−0.0675781
$768$	−1835.32	−0.0862321
$769$	19892.1	0.932803	0.466402	−	0.884573i	$-0.345550\pi$
$769$	19892.1	0.932803	0.466402	+	0.884573i	$0.345550\pi$
$770$	17016.2	0.796390
$771$	53048.0	2.47792
$772$	17392.5	0.810841
$773$	−7926.68	−0.368826	−0.184413	−	0.982849i	$-0.559038\pi$
$773$	−7926.68	−0.368826	−0.184413	+	0.982849i	$0.559038\pi$
$774$	289.930	0.0134642
$775$	−2569.65	−0.119103
$776$	14521.8	0.671782
$777$	466.725	0.0215491
$778$	−13761.3	−0.634145
$779$	−9350.83	−0.430075
$780$	−346.763	−0.0159181
$781$	−3848.20	−0.176312
$782$	−575.446	−0.0263144
$783$	4123.77	0.188214
$784$	4830.12	0.220031
$785$	−14946.8	−0.679585
$786$	−41302.2	−1.87430
$787$	−2543.20	−0.115191	−0.0575955	−	0.998340i	$-0.518343\pi$
$787$	−2543.20	−0.115191	−0.0575955	+	0.998340i	$0.518343\pi$
$788$	−10204.6	−0.461324
$789$	13678.0	0.617171
$790$	−5789.99	−0.260758
$791$	−28006.7	−1.25892
$792$	13078.4	0.586770
$793$	2164.37	0.0969220
$794$	18594.5	0.831099
$795$	5725.09	0.255406
$796$	4922.62	0.219193
$797$	6813.74	0.302829	0.151415	−	0.988470i	$-0.451617\pi$
$797$	6813.74	0.302829	0.151415	+	0.988470i	$0.451617\pi$
$798$	38069.5	1.68878
$799$	−6869.87	−0.304178
$800$	800.000	0.0353553
$801$	7594.48	0.335003
$802$	−9808.59	−0.431862
$803$	−58318.2	−2.56290
$804$	−15057.0	−0.660473
$805$	2920.37	0.127863
$806$	497.160	0.0217267
$807$	9260.77	0.403958
$808$	−11937.4	−0.519746
$809$	−14588.3	−0.633991	−0.316996	−	0.948427i	$-0.602674\pi$
$809$	−14588.3	−0.633991	−0.316996	+	0.948427i	$0.602674\pi$
$810$	7924.96	0.343771
$811$	−22659.3	−0.981106	−0.490553	−	0.871411i	$-0.663205\pi$
$811$	−22659.3	−0.981106	−0.490553	+	0.871411i	$0.663205\pi$
$812$	−22450.2	−0.970257
$813$	38435.2	1.65803
$814$	343.559	0.0147933
$815$	−3501.27	−0.150484
$816$	1434.95	0.0615605
$817$	621.234	0.0266025
$818$	−23713.4	−1.01359
$819$	1498.36	0.0639279
$820$	1788.73	0.0761769
$821$	−15236.4	−0.647689	−0.323845	−	0.946110i	$-0.604975\pi$
$821$	−15236.4	−0.647689	−0.323845	+	0.946110i	$0.604975\pi$
$822$	−27075.3	−1.14886
$823$	−11201.4	−0.474430	−0.237215	−	0.971457i	$-0.576235\pi$
$823$	−11201.4	−0.474430	−0.237215	+	0.971457i	$0.576235\pi$
$824$	−2007.11	−0.0848554
$825$	−12009.7	−0.506817
$826$	−30146.5	−1.26989
$827$	23651.0	0.994468	0.497234	−	0.867617i	$-0.334349\pi$
$827$	23651.0	0.994468	0.497234	+	0.867617i	$0.334349\pi$
$828$	2244.56	0.0942077
$829$	−13484.2	−0.564929	−0.282465	−	0.959278i	$-0.591152\pi$
$829$	−13484.2	−0.564929	−0.282465	+	0.959278i	$0.591152\pi$
$830$	3457.04	0.144573
$831$	6021.57	0.251367
$832$	−154.779	−0.00644952
$833$	−3776.45	−0.157078
$834$	20501.0	0.851187
$835$	−8.46181	−0.000350698 0
$836$	28023.2	1.15933
$837$	−1917.82	−0.0791989
$838$	24994.4	1.03033
$839$	13389.9	0.550978	0.275489	−	0.961304i	$-0.411160\pi$
$839$	13389.9	0.550978	0.275489	+	0.961304i	$0.411160\pi$
$840$	−7282.34	−0.299124
$841$	24458.4	1.00284
$842$	−4683.26	−0.191681
$843$	4380.13	0.178956
$844$	11336.2	0.462332
$845$	10955.8	0.446023
$846$	26796.4	1.08898
$847$	−80220.4	−3.25432
$848$	2555.42	0.103483
$849$	61432.1	2.48333
$850$	−625.485	−0.0252399
$851$	58.9628	0.00237511
$852$	1646.90	0.0662228
$853$	−5212.82	−0.209242	−0.104621	−	0.994512i	$-0.533363\pi$
$853$	−5212.82	−0.209242	−0.104621	+	0.994512i	$0.533363\pi$
$854$	45453.8	1.82131
$855$	−12754.1	−0.510153
$856$	−8281.92	−0.330689
$857$	−29998.6	−1.19572	−0.597861	−	0.801600i	$-0.703983\pi$
$857$	−29998.6	−1.19572	−0.597861	+	0.801600i	$0.703983\pi$
$858$	2323.56	0.0924535
$859$	36624.0	1.45471	0.727354	−	0.686263i	$-0.240750\pi$
$859$	36624.0	1.45471	0.727354	+	0.686263i	$0.240750\pi$
$860$	−118.836	−0.00471196
$861$	−16282.7	−0.644496
$862$	−28094.3	−1.11009
$863$	−16979.4	−0.669740	−0.334870	−	0.942264i	$-0.608692\pi$
$863$	−16979.4	−0.669740	−0.334870	+	0.942264i	$0.608692\pi$
$864$	597.067	0.0235100
$865$	3907.71	0.153603
$866$	19754.9	0.775170
$867$	34100.4	1.33577
$868$	10440.8	0.408277
$869$	38797.1	1.51450
$870$	15845.0	0.617465
$871$	−1269.82	−0.0493985
$872$	2836.18	0.110144
$873$	44286.8	1.71693
$874$	4809.43	0.186134
$875$	3174.32	0.122642
$876$	24958.2	0.962626
$877$	17155.7	0.660554	0.330277	−	0.943884i	$-0.392858\pi$
$877$	17155.7	0.660554	0.330277	+	0.943884i	$0.392858\pi$
$878$	−14056.7	−0.540309
$879$	69747.7	2.67637
$880$	−5360.58	−0.205347
$881$	37047.2	1.41675	0.708373	−	0.705839i	$-0.249430\pi$
$881$	37047.2	1.41675	0.708373	+	0.705839i	$0.249430\pi$
$882$	14730.3	0.562353
$883$	36958.1	1.40854	0.704270	−	0.709932i	$-0.251275\pi$
$883$	36958.1	1.40854	0.704270	+	0.709932i	$0.251275\pi$
$884$	121.015	0.00460427
$885$	21276.9	0.808151
$886$	−26973.4	−1.02279
$887$	−40755.4	−1.54277	−0.771383	−	0.636372i	$-0.780434\pi$
$887$	−40755.4	−1.54277	−0.771383	+	0.636372i	$0.780434\pi$
$888$	−147.032	−0.00555637
$889$	−12777.2	−0.482040
$890$	−3112.82	−0.117238
$891$	−53102.9	−1.99665
$892$	−1565.00	−0.0587446
$893$	57416.6	2.15159
$894$	18185.9	0.680345
$895$	−13351.1	−0.498634
$896$	−3250.50	−0.121196
$897$	398.778	0.0148437
$898$	−12871.5	−0.478316
$899$	−22717.2	−0.842782
$900$	2439.74	0.0903608
$901$	−1997.97	−0.0738756
$902$	−11985.8	−0.442442
$903$	1081.76	0.0398656
$904$	8822.92	0.324608
$905$	6150.45	0.225909
$906$	−33492.6	−1.22817
$907$	−28701.3	−1.05073	−0.525365	−	0.850877i	$-0.676071\pi$
$907$	−28701.3	−1.05073	−0.525365	+	0.850877i	$0.676071\pi$
$908$	4983.16	0.182128
$909$	−36405.1	−1.32836
$910$	−614.147	−0.0223723
$911$	−30965.9	−1.12618	−0.563088	−	0.826397i	$-0.690387\pi$
$911$	−30965.9	−1.12618	−0.563088	+	0.826397i	$0.690387\pi$
$912$	−11993.0	−0.435446
$913$	−23164.6	−0.839691
$914$	−28651.0	−1.03686
$915$	−32080.5	−1.15907
$916$	25991.5	0.937535
$917$	−73149.8	−2.63426
$918$	−466.821	−0.0167836
$919$	−12189.9	−0.437550	−0.218775	−	0.975775i	$-0.570206\pi$
$919$	−12189.9	−0.437550	−0.218775	+	0.975775i	$0.570206\pi$
$920$	−920.000	−0.0329690
$921$	58182.2	2.08162
$922$	19932.3	0.711968
$923$	138.889	0.00495297
$924$	48796.9	1.73734
$925$	64.0900	0.00227812
$926$	7258.12	0.257577
$927$	−6121.03	−0.216873
$928$	7072.46	0.250178
$929$	−47633.5	−1.68224	−0.841121	−	0.540846i	$-0.818104\pi$
$929$	−47633.5	−1.68224	−0.841121	+	0.540846i	$0.818104\pi$
$930$	−7368.94	−0.259825
$931$	31562.6	1.11109
$932$	−12307.5	−0.432559
$933$	46125.4	1.61852
$934$	−8907.62	−0.312062
$935$	4191.20	0.146595
$936$	−472.026	−0.0164836
$937$	17375.8	0.605809	0.302904	−	0.953021i	$-0.402044\pi$
$937$	17375.8	0.605809	0.302904	+	0.953021i	$0.402044\pi$
$938$	−26667.3	−0.928271
$939$	−19175.9	−0.666434
$940$	−10983.3	−0.381101
$941$	27101.7	0.938883	0.469442	−	0.882964i	$-0.344455\pi$
$941$	27101.7	0.938883	0.469442	+	0.882964i	$0.344455\pi$
$942$	−42862.6	−1.48253
$943$	−2057.04	−0.0710354
$944$	9497.01	0.327438
$945$	2369.10	0.0815522
$946$	796.289	0.0273674
$947$	−15867.2	−0.544472	−0.272236	−	0.962230i	$-0.587763\pi$
$947$	−15867.2	−0.544472	−0.272236	+	0.962230i	$0.587763\pi$
$948$	−16603.8	−0.568848
$949$	2104.82	0.0719972
$950$	5227.64	0.178534
$951$	62515.7	2.13166
$952$	2541.42	0.0865210
$953$	5010.38	0.170306	0.0851532	−	0.996368i	$-0.472862\pi$
$953$	5010.38	0.170306	0.0851532	+	0.996368i	$0.472862\pi$
$954$	7793.20	0.264480
$955$	7166.17	0.242819
$956$	2129.25	0.0720344
$957$	−106173.	−3.58629
$958$	6393.82	0.215632
$959$	−47952.7	−1.61467
$960$	2294.14	0.0771283
$961$	−19226.0	−0.645364
$962$	−12.3997	−0.000415575 0
$963$	−25257.2	−0.845173
$964$	−3323.77	−0.111049
$965$	−21740.6	−0.725239
$966$	8374.69	0.278935
$967$	−35041.9	−1.16533	−0.582663	−	0.812714i	$-0.697989\pi$
$967$	−35041.9	−1.16533	−0.582663	+	0.812714i	$0.697989\pi$
$968$	25271.7	0.839115
$969$	9376.77	0.310862
$970$	−18152.3	−0.600860
$971$	−17750.1	−0.586641	−0.293321	−	0.956014i	$-0.594760\pi$
$971$	−17750.1	−0.586641	−0.293321	+	0.956014i	$0.594760\pi$
$972$	20711.1	0.683447
$973$	36309.0	1.19631
$974$	19012.6	0.625464
$975$	433.454	0.0142376
$976$	−14319.2	−0.469618
$977$	46981.1	1.53844	0.769221	−	0.638983i	$-0.220644\pi$
$977$	46981.1	1.53844	0.769221	+	0.638983i	$0.220644\pi$
$978$	−10040.5	−0.328283
$979$	20858.1	0.680928
$980$	−6037.64	−0.196802
$981$	8649.43	0.281504
$982$	−19850.0	−0.645049
$983$	3552.85	0.115278	0.0576390	−	0.998337i	$-0.481643\pi$
$983$	3552.85	0.115278	0.0576390	+	0.998337i	$0.481643\pi$
$984$	5129.50	0.166181
$985$	12755.7	0.412621
$986$	−5529.64	−0.178600
$987$	99979.9	3.22431
$988$	−1011.41	−0.0325681
$989$	136.662	0.00439392
$990$	−16348.0	−0.524823
$991$	−29989.4	−0.961295	−0.480648	−	0.876914i	$-0.659598\pi$
$991$	−29989.4	−0.961295	−0.480648	+	0.876914i	$0.659598\pi$
$992$	−3289.15	−0.105273
$993$	24263.5	0.775405
$994$	2916.80	0.0930736
$995$	−6153.27	−0.196052
$996$	9913.68	0.315389
$997$	−59911.2	−1.90312	−0.951559	−	0.307467i	$-0.900519\pi$
$997$	−59911.2	−1.90312	−0.951559	+	0.307467i	$0.900519\pi$
$998$	6373.41	0.202151
$999$	47.8325	0.00151487

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	230.4.a.i.1.1	✓	4
3.2	odd	2		2070.4.a.bi.1.1		4
4.3	odd	2		1840.4.a.l.1.4		4
5.2	odd	4		1150.4.b.m.599.8		8
5.3	odd	4		1150.4.b.m.599.1		8
5.4	even	2		1150.4.a.o.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.4.a.i.1.1	✓	4	1.1	even	1	trivial
1150.4.a.o.1.4		4	5.4	even	2
1150.4.b.m.599.1		8	5.3	odd	4
1150.4.b.m.599.8		8	5.2	odd	4
1840.4.a.l.1.4		4	4.3	odd	2
2070.4.a.bi.1.1		4	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:	\( N \)	\(=\)	\( 230 = 2 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	230.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} -7.16920 q^{3} +4.00000 q^{4} -5.00000 q^{5} -14.3384 q^{6} -25.3945 q^{7} +8.00000 q^{8} +24.3974 q^{9} +O(q^{10})\)	\(q+2.00000 q^{2} -7.16920 q^{3} +4.00000 q^{4} -5.00000 q^{5} -14.3384 q^{6} -25.3945 q^{7} +8.00000 q^{8} +24.3974 q^{9} -10.0000 q^{10} +67.0072 q^{11} -28.6768 q^{12} -2.41842 q^{13} -50.7891 q^{14} +35.8460 q^{15} +16.0000 q^{16} -12.5097 q^{17} +48.7949 q^{18} +104.553 q^{19} -20.0000 q^{20} +182.058 q^{21} +134.014 q^{22} +23.0000 q^{23} -57.3536 q^{24} +25.0000 q^{25} -4.83684 q^{26} +18.6584 q^{27} -101.578 q^{28} +221.014 q^{29} +71.6920 q^{30} -102.786 q^{31} +32.0000 q^{32} -480.388 q^{33} -25.0194 q^{34} +126.973 q^{35} +97.5897 q^{36} +2.56360 q^{37} +209.106 q^{38} +17.3382 q^{39} -40.0000 q^{40} -89.4364 q^{41} +364.117 q^{42} +5.94181 q^{43} +268.029 q^{44} -121.987 q^{45} +46.0000 q^{46} +549.164 q^{47} -114.707 q^{48} +301.882 q^{49} +50.0000 q^{50} +89.6845 q^{51} -9.67369 q^{52} +159.714 q^{53} +37.3167 q^{54} -335.036 q^{55} -203.156 q^{56} -749.560 q^{57} +442.029 q^{58} +593.563 q^{59} +143.384 q^{60} -894.952 q^{61} -205.572 q^{62} -619.561 q^{63} +64.0000 q^{64} +12.0921 q^{65} -960.776 q^{66} +525.060 q^{67} -50.0388 q^{68} -164.892 q^{69} +253.945 q^{70} -57.4296 q^{71} +195.179 q^{72} -870.328 q^{73} +5.12720 q^{74} -179.230 q^{75} +418.211 q^{76} -1701.62 q^{77} +34.6763 q^{78} +578.999 q^{79} -80.0000 q^{80} -792.496 q^{81} -178.873 q^{82} -345.704 q^{83} +728.234 q^{84} +62.5485 q^{85} +11.8836 q^{86} -1584.50 q^{87} +536.058 q^{88} +311.282 q^{89} -243.974 q^{90} +61.4147 q^{91} +92.0000 q^{92} +736.894 q^{93} +1098.33 q^{94} -522.764 q^{95} -229.414 q^{96} +1815.23 q^{97} +603.764 q^{98} +1634.80 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{2} + 4 q^{3} + 16 q^{4} - 20 q^{5} + 8 q^{6} + 26 q^{7} + 32 q^{8} + 64 q^{9}+O(q^{10}) \) 4 * q + 8 * q^2 + 4 * q^3 + 16 * q^4 - 20 * q^5 + 8 * q^6 + 26 * q^7 + 32 * q^8 + 64 * q^9	\( 4 q + 8 q^{2} + 4 q^{3} + 16 q^{4} - 20 q^{5} + 8 q^{6} + 26 q^{7} + 32 q^{8} + 64 q^{9} - 40 q^{10} + 93 q^{11} + 16 q^{12} + 32 q^{13} + 52 q^{14} - 20 q^{15} + 64 q^{16} + 108 q^{17} + 128 q^{18} + 185 q^{19} - 80 q^{20} + 302 q^{21} + 186 q^{22} + 92 q^{23} + 32 q^{24} + 100 q^{25} + 64 q^{26} + 259 q^{27} + 104 q^{28} + 294 q^{29} - 40 q^{30} - 211 q^{31} + 128 q^{32} - 237 q^{33} + 216 q^{34} - 130 q^{35} + 256 q^{36} + 5 q^{37} + 370 q^{38} - 421 q^{39} - 160 q^{40} - 369 q^{41} + 604 q^{42} - 100 q^{43} + 372 q^{44} - 320 q^{45} + 184 q^{46} - 363 q^{47} + 64 q^{48} + 600 q^{49} + 200 q^{50} - 315 q^{51} + 128 q^{52} + 21 q^{53} + 518 q^{54} - 465 q^{55} + 208 q^{56} - 160 q^{57} + 588 q^{58} - 33 q^{59} - 80 q^{60} - 307 q^{61} - 422 q^{62} + 167 q^{63} + 256 q^{64} - 160 q^{65} - 474 q^{66} + 725 q^{67} + 432 q^{68} + 92 q^{69} - 260 q^{70} - 1257 q^{71} + 512 q^{72} + 509 q^{73} + 10 q^{74} + 100 q^{75} + 740 q^{76} - 1962 q^{77} - 842 q^{78} + 1202 q^{79} - 320 q^{80} - 992 q^{81} - 738 q^{82} - 1377 q^{83} + 1208 q^{84} - 540 q^{85} - 200 q^{86} - 2829 q^{87} + 744 q^{88} - 984 q^{89} - 640 q^{90} - 995 q^{91} + 368 q^{92} - 1843 q^{93} - 726 q^{94} - 925 q^{95} + 128 q^{96} + 137 q^{97} + 1200 q^{98} + 2013 q^{99}+O(q^{100}) \) 4 * q + 8 * q^2 + 4 * q^3 + 16 * q^4 - 20 * q^5 + 8 * q^6 + 26 * q^7 + 32 * q^8 + 64 * q^9 - 40 * q^10 + 93 * q^11 + 16 * q^12 + 32 * q^13 + 52 * q^14 - 20 * q^15 + 64 * q^16 + 108 * q^17 + 128 * q^18 + 185 * q^19 - 80 * q^20 + 302 * q^21 + 186 * q^22 + 92 * q^23 + 32 * q^24 + 100 * q^25 + 64 * q^26 + 259 * q^27 + 104 * q^28 + 294 * q^29 - 40 * q^30 - 211 * q^31 + 128 * q^32 - 237 * q^33 + 216 * q^34 - 130 * q^35 + 256 * q^36 + 5 * q^37 + 370 * q^38 - 421 * q^39 - 160 * q^40 - 369 * q^41 + 604 * q^42 - 100 * q^43 + 372 * q^44 - 320 * q^45 + 184 * q^46 - 363 * q^47 + 64 * q^48 + 600 * q^49 + 200 * q^50 - 315 * q^51 + 128 * q^52 + 21 * q^53 + 518 * q^54 - 465 * q^55 + 208 * q^56 - 160 * q^57 + 588 * q^58 - 33 * q^59 - 80 * q^60 - 307 * q^61 - 422 * q^62 + 167 * q^63 + 256 * q^64 - 160 * q^65 - 474 * q^66 + 725 * q^67 + 432 * q^68 + 92 * q^69 - 260 * q^70 - 1257 * q^71 + 512 * q^72 + 509 * q^73 + 10 * q^74 + 100 * q^75 + 740 * q^76 - 1962 * q^77 - 842 * q^78 + 1202 * q^79 - 320 * q^80 - 992 * q^81 - 738 * q^82 - 1377 * q^83 + 1208 * q^84 - 540 * q^85 - 200 * q^86 - 2829 * q^87 + 744 * q^88 - 984 * q^89 - 640 * q^90 - 995 * q^91 + 368 * q^92 - 1843 * q^93 - 726 * q^94 - 925 * q^95 + 128 * q^96 + 137 * q^97 + 1200 * q^98 + 2013 * q^99

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