LMFDB - Embedded newform 507.4.a.i.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [507,4,Mod(1,507)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, 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("507.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 507 = 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	507.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:	$29.9139683729$
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} - 2x^{3} - 25x^{2} + 24x + 78 $ x^4 - 2x^3 - 25x^2 + 24*x + 78
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-4.22605$ of defining polynomial
Character	$\chi$	$=$	507.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.22605 q^{2} -3.00000 q^{3} +9.85953 q^{4} +5.85953 q^{5} -12.6782 q^{6} +24.1254 q^{7} +7.85849 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+4.22605 q^{2} -3.00000 q^{3} +9.85953 q^{4} +5.85953 q^{5} -12.6782 q^{6} +24.1254 q^{7} +7.85849 q^{8} +9.00000 q^{9} +24.7627 q^{10} +33.8892 q^{11} -29.5786 q^{12} +101.955 q^{14} -17.5786 q^{15} -45.6659 q^{16} -49.3956 q^{17} +38.0345 q^{18} +76.8548 q^{19} +57.7723 q^{20} -72.3763 q^{21} +143.218 q^{22} +6.29163 q^{23} -23.5755 q^{24} -90.6659 q^{25} -27.0000 q^{27} +237.866 q^{28} +100.995 q^{29} -74.2881 q^{30} +307.580 q^{31} -255.854 q^{32} -101.668 q^{33} -208.749 q^{34} +141.364 q^{35} +88.7358 q^{36} +76.0189 q^{37} +324.793 q^{38} +46.0471 q^{40} +514.418 q^{41} -305.866 q^{42} -268.184 q^{43} +334.132 q^{44} +52.7358 q^{45} +26.5888 q^{46} +460.912 q^{47} +136.998 q^{48} +239.037 q^{49} -383.159 q^{50} +148.187 q^{51} +67.8057 q^{53} -114.103 q^{54} +198.575 q^{55} +189.589 q^{56} -230.564 q^{57} +426.812 q^{58} -25.2021 q^{59} -173.317 q^{60} -588.832 q^{61} +1299.85 q^{62} +217.129 q^{63} -715.927 q^{64} -429.653 q^{66} +1004.46 q^{67} -487.018 q^{68} -18.8749 q^{69} +597.411 q^{70} -895.481 q^{71} +70.7264 q^{72} -968.599 q^{73} +321.260 q^{74} +271.998 q^{75} +757.753 q^{76} +817.592 q^{77} -119.053 q^{79} -267.581 q^{80} +81.0000 q^{81} +2173.96 q^{82} -480.784 q^{83} -713.597 q^{84} -289.435 q^{85} -1133.36 q^{86} -302.986 q^{87} +266.318 q^{88} -1085.91 q^{89} +222.864 q^{90} +62.0325 q^{92} -922.740 q^{93} +1947.84 q^{94} +450.333 q^{95} +767.563 q^{96} +16.6552 q^{97} +1010.18 q^{98} +305.003 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} - 12 q^{3} + 22 q^{4} + 6 q^{5} + 6 q^{6} + 14 q^{7} - 54 q^{8} + 36 q^{9}+O(q^{10}) $ 4 * q - 2 * q^2 - 12 * q^3 + 22 * q^4 + 6 * q^5 + 6 * q^6 + 14 * q^7 - 54 * q^8 + 36 * q^9	\( 4 q - 2 q^{2} - 12 q^{3} + 22 q^{4} + 6 q^{5} + 6 q^{6} + 14 q^{7} - 54 q^{8} + 36 q^{9} - 62 q^{10} - 40 q^{11} - 66 q^{12} + 40 q^{14} - 18 q^{15} + 122 q^{16} + 98 q^{17} - 18 q^{18} - 124 q^{19} + 466 q^{20} - 42 q^{21} + 220 q^{22} + 104 q^{23} + 162 q^{24} - 58 q^{25} - 108 q^{27} + 144 q^{28} + 194 q^{29} + 186 q^{30} - 26 q^{31} - 654 q^{32} + 120 q^{33} - 1062 q^{34} + 88 q^{35} + 198 q^{36} - 102 q^{37} + 332 q^{38} - 998 q^{40} + 1054 q^{41} - 120 q^{42} + 450 q^{43} + 44 q^{44} + 54 q^{45} + 172 q^{46} + 96 q^{47} - 366 q^{48} + 1070 q^{49} - 996 q^{50} - 294 q^{51} + 262 q^{53} + 54 q^{54} + 204 q^{55} + 2164 q^{56} + 372 q^{57} - 722 q^{58} - 308 q^{59} - 1398 q^{60} - 928 q^{61} + 2780 q^{62} + 126 q^{63} + 1026 q^{64} - 660 q^{66} + 1134 q^{67} + 1786 q^{68} - 312 q^{69} + 2324 q^{70} - 1064 q^{71} - 486 q^{72} - 952 q^{73} + 1158 q^{74} + 174 q^{75} + 1708 q^{76} + 2508 q^{77} - 746 q^{79} + 2922 q^{80} + 324 q^{81} + 1734 q^{82} + 404 q^{83} - 432 q^{84} + 1394 q^{85} - 3168 q^{86} - 582 q^{87} + 3060 q^{88} - 1620 q^{89} - 558 q^{90} + 332 q^{92} + 78 q^{93} - 772 q^{94} + 2204 q^{95} + 1962 q^{96} - 2166 q^{97} + 1906 q^{98} - 360 q^{99}+O(q^{100}) \) 4 * q - 2 * q^2 - 12 * q^3 + 22 * q^4 + 6 * q^5 + 6 * q^6 + 14 * q^7 - 54 * q^8 + 36 * q^9 - 62 * q^10 - 40 * q^11 - 66 * q^12 + 40 * q^14 - 18 * q^15 + 122 * q^16 + 98 * q^17 - 18 * q^18 - 124 * q^19 + 466 * q^20 - 42 * q^21 + 220 * q^22 + 104 * q^23 + 162 * q^24 - 58 * q^25 - 108 * q^27 + 144 * q^28 + 194 * q^29 + 186 * q^30 - 26 * q^31 - 654 * q^32 + 120 * q^33 - 1062 * q^34 + 88 * q^35 + 198 * q^36 - 102 * q^37 + 332 * q^38 - 998 * q^40 + 1054 * q^41 - 120 * q^42 + 450 * q^43 + 44 * q^44 + 54 * q^45 + 172 * q^46 + 96 * q^47 - 366 * q^48 + 1070 * q^49 - 996 * q^50 - 294 * q^51 + 262 * q^53 + 54 * q^54 + 204 * q^55 + 2164 * q^56 + 372 * q^57 - 722 * q^58 - 308 * q^59 - 1398 * q^60 - 928 * q^61 + 2780 * q^62 + 126 * q^63 + 1026 * q^64 - 660 * q^66 + 1134 * q^67 + 1786 * q^68 - 312 * q^69 + 2324 * q^70 - 1064 * q^71 - 486 * q^72 - 952 * q^73 + 1158 * q^74 + 174 * q^75 + 1708 * q^76 + 2508 * q^77 - 746 * q^79 + 2922 * q^80 + 324 * q^81 + 1734 * q^82 + 404 * q^83 - 432 * q^84 + 1394 * q^85 - 3168 * q^86 - 582 * q^87 + 3060 * q^88 - 1620 * q^89 - 558 * q^90 + 332 * q^92 + 78 * q^93 - 772 * q^94 + 2204 * q^95 + 1962 * q^96 - 2166 * q^97 + 1906 * q^98 - 360 * 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 $

$2$	4.22605	1.49414	0.747068	−	0.664748i	$-0.231461\pi$
$2$	4.22605	1.49414	0.747068	+	0.664748i	$0.231461\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	9.85953	1.23244
$5$	5.85953	0.524093	0.262046	−	0.965055i	$-0.415603\pi$
$5$	5.85953	0.524093	0.262046	+	0.965055i	$0.415603\pi$
$6$	−12.6782	−0.862640
$7$	24.1254	1.30265	0.651326	−	0.758798i	$-0.274213\pi$
$7$	24.1254	1.30265	0.651326	+	0.758798i	$0.274213\pi$
$8$	7.85849	0.347299
$9$	9.00000	0.333333
$10$	24.7627	0.783065
$11$	33.8892	0.928907	0.464453	−	0.885598i	$-0.346251\pi$
$11$	33.8892	0.928907	0.464453	+	0.885598i	$0.346251\pi$
$12$	−29.5786	−0.711551
$13$	0	0
$13$	0	0
$14$	101.955	1.94634
$15$	−17.5786	−0.302585
$16$	−45.6659	−0.713529
$17$	−49.3956	−0.704718	−0.352359	−	0.935865i	$-0.614620\pi$
$17$	−49.3956	−0.704718	−0.352359	+	0.935865i	$0.614620\pi$
$18$	38.0345	0.498045
$19$	76.8548	0.927985	0.463992	−	0.885839i	$-0.346417\pi$
$19$	76.8548	0.927985	0.463992	+	0.885839i	$0.346417\pi$
$20$	57.7723	0.645913
$21$	−72.3763	−0.752086
$22$	143.218	1.38791
$23$	6.29163	0.0570389	0.0285195	−	0.999593i	$-0.490921\pi$
$23$	6.29163	0.0570389	0.0285195	+	0.999593i	$0.490921\pi$
$24$	−23.5755	−0.200513
$25$	−90.6659	−0.725327
$26$	0	0
$27$	−27.0000	−0.192450
$28$	237.866	1.60544
$29$	100.995	0.646702	0.323351	−	0.946279i	$-0.395191\pi$
$29$	100.995	0.646702	0.323351	+	0.946279i	$0.395191\pi$
$30$	−74.2881	−0.452103
$31$	307.580	1.78203	0.891016	−	0.453972i	$-0.149993\pi$
$31$	307.580	1.78203	0.891016	+	0.453972i	$0.149993\pi$
$32$	−255.854	−1.41341
$33$	−101.668	−0.536305
$34$	−208.749	−1.05294
$35$	141.364	0.682710
$36$	88.7358	0.410814
$37$	76.0189	0.337768	0.168884	−	0.985636i	$-0.445984\pi$
$37$	76.0189	0.337768	0.168884	+	0.985636i	$0.445984\pi$
$38$	324.793	1.38653
$39$	0	0
$40$	46.0471	0.182017
$41$	514.418	1.95948	0.979740	−	0.200274i	$-0.0641833\pi$
$41$	514.418	1.95948	0.979740	+	0.200274i	$0.0641833\pi$
$42$	−305.866	−1.12372
$43$	−268.184	−0.951108	−0.475554	−	0.879686i	$-0.657752\pi$
$43$	−268.184	−0.951108	−0.475554	+	0.879686i	$0.657752\pi$
$44$	334.132	1.14482
$45$	52.7358	0.174698
$46$	26.5888	0.0852239
$47$	460.912	1.43045	0.715223	−	0.698896i	$-0.246325\pi$
$47$	460.912	1.43045	0.715223	+	0.698896i	$0.246325\pi$
$48$	136.998	0.411956
$49$	239.037	0.696901
$50$	−383.159	−1.08374
$51$	148.187	0.406869
$52$	0	0
$53$	67.8057	0.175733	0.0878663	−	0.996132i	$-0.471995\pi$
$53$	67.8057	0.175733	0.0878663	+	0.996132i	$0.471995\pi$
$54$	−114.103	−0.287547
$55$	198.575	0.486833
$56$	189.589	0.452410
$57$	−230.564	−0.535772
$58$	426.812	0.966261
$59$	−25.2021	−0.0556107	−0.0278053	−	0.999613i	$-0.508852\pi$
$59$	−25.2021	−0.0556107	−0.0278053	+	0.999613i	$0.508852\pi$
$60$	−173.317	−0.372918
$61$	−588.832	−1.23594	−0.617969	−	0.786202i	$-0.712044\pi$
$61$	−588.832	−1.23594	−0.617969	+	0.786202i	$0.712044\pi$
$62$	1299.85	2.66260
$63$	217.129	0.434217
$64$	−715.927	−1.39830
$65$	0	0
$66$	−429.653	−0.801312
$67$	1004.46	1.83156	0.915778	−	0.401684i	$-0.131575\pi$
$67$	1004.46	1.83156	0.915778	+	0.401684i	$0.131575\pi$
$68$	−487.018	−0.868523
$69$	−18.8749	−0.0329315
$70$	597.411	1.02006
$71$	−895.481	−1.49682	−0.748408	−	0.663238i	$-0.769182\pi$
$71$	−895.481	−1.49682	−0.748408	+	0.663238i	$0.769182\pi$
$72$	70.7264	0.115766
$73$	−968.599	−1.55296	−0.776479	−	0.630143i	$-0.782996\pi$
$73$	−968.599	−1.55296	−0.776479	+	0.630143i	$0.782996\pi$
$74$	321.260	0.504672
$75$	271.998	0.418768
$76$	757.753	1.14369
$77$	817.592	1.21004
$78$	0	0
$79$	−119.053	−0.169551	−0.0847755	−	0.996400i	$-0.527017\pi$
$79$	−119.053	−0.169551	−0.0847755	+	0.996400i	$0.527017\pi$
$80$	−267.581	−0.373955
$81$	81.0000	0.111111
$82$	2173.96	2.92773
$83$	−480.784	−0.635818	−0.317909	−	0.948121i	$-0.602981\pi$
$83$	−480.784	−0.635818	−0.317909	+	0.948121i	$0.602981\pi$
$84$	−713.597	−0.926902
$85$	−289.435	−0.369337
$86$	−1133.36	−1.42109
$87$	−302.986	−0.373374
$88$	266.318	0.322609
$89$	−1085.91	−1.29333	−0.646663	−	0.762776i	$-0.723836\pi$
$89$	−1085.91	−1.29333	−0.646663	+	0.762776i	$0.723836\pi$
$90$	222.864	0.261022
$91$	0	0
$92$	62.0325	0.0702972
$93$	−922.740	−1.02886
$94$	1947.84	2.13728
$95$	450.333	0.486350
$96$	767.563	0.816032
$97$	16.6552	0.0174338	0.00871692	−	0.999962i	$-0.497225\pi$
$97$	16.6552	0.0174338	0.00871692	+	0.999962i	$0.497225\pi$
$98$	1010.18	1.04126
$99$	305.003	0.309636
$100$	−893.923	−0.893923
$101$	−958.004	−0.943811	−0.471906	−	0.881649i	$-0.656434\pi$
$101$	−958.004	−0.943811	−0.471906	+	0.881649i	$0.656434\pi$
$102$	626.246	0.607917
$103$	−2.70560	−0.00258826	−0.00129413	−	0.999999i	$-0.500412\pi$
$103$	−2.70560	−0.00258826	−0.00129413	+	0.999999i	$0.500412\pi$
$104$	0	0
$105$	−424.092	−0.394163
$106$	286.551	0.262568
$107$	1351.28	1.22087	0.610437	−	0.792065i	$-0.290994\pi$
$107$	1351.28	1.22087	0.610437	+	0.792065i	$0.290994\pi$
$108$	−266.207	−0.237184
$109$	−448.455	−0.394075	−0.197037	−	0.980396i	$-0.563132\pi$
$109$	−448.455	−0.394075	−0.197037	+	0.980396i	$0.563132\pi$
$110$	839.188	0.727395
$111$	−228.057	−0.195011
$112$	−1101.71	−0.929480
$113$	1398.85	1.16453	0.582267	−	0.812997i	$-0.302166\pi$
$113$	1398.85	1.16453	0.582267	+	0.812997i	$0.302166\pi$
$114$	−974.378	−0.800516
$115$	36.8660	0.0298937
$116$	995.767	0.797023
$117$	0	0
$118$	−106.505	−0.0830899
$119$	−1191.69	−0.918002
$120$	−138.141	−0.105088
$121$	−182.523	−0.137132
$122$	−2488.44	−1.84666
$123$	−1543.26	−1.13131
$124$	3032.59	2.19625
$125$	−1263.70	−0.904231
$126$	917.599	0.648779
$127$	−119.504	−0.0834985	−0.0417492	−	0.999128i	$-0.513293\pi$
$127$	−119.504	−0.0834985	−0.0417492	+	0.999128i	$0.513293\pi$
$128$	−978.713	−0.675834
$129$	804.552	0.549123
$130$	0	0
$131$	−2251.70	−1.50177	−0.750886	−	0.660432i	$-0.770373\pi$
$131$	−2251.70	−1.50177	−0.750886	+	0.660432i	$0.770373\pi$
$132$	−1002.39	−0.660964
$133$	1854.16	1.20884
$134$	4244.90	2.73659
$135$	−158.207	−0.100862
$136$	−388.175	−0.244748
$137$	−1130.62	−0.705076	−0.352538	−	0.935797i	$-0.614681\pi$
$137$	−1130.62	−0.705076	−0.352538	+	0.935797i	$0.614681\pi$
$138$	−79.7663	−0.0492041
$139$	−595.287	−0.363249	−0.181624	−	0.983368i	$-0.558136\pi$
$139$	−595.287	−0.363249	−0.181624	+	0.983368i	$0.558136\pi$
$140$	1393.78	0.841400
$141$	−1382.74	−0.825869
$142$	−3784.35	−2.23645
$143$	0	0
$144$	−410.993	−0.237843
$145$	591.786	0.338932
$146$	−4093.35	−2.32033
$147$	−717.111	−0.402356
$148$	749.511	0.416280
$149$	793.175	0.436103	0.218052	−	0.975937i	$-0.430030\pi$
$149$	793.175	0.436103	0.218052	+	0.975937i	$0.430030\pi$
$150$	1149.48	0.625696
$151$	134.213	0.0723317	0.0361659	−	0.999346i	$-0.488486\pi$
$151$	134.213	0.0723317	0.0361659	+	0.999346i	$0.488486\pi$
$152$	603.963	0.322288
$153$	−444.561	−0.234906
$154$	3455.19	1.80797
$155$	1802.28	0.933950
$156$	0	0
$157$	1509.07	0.767114	0.383557	−	0.923517i	$-0.374699\pi$
$157$	1509.07	0.767114	0.383557	+	0.923517i	$0.374699\pi$
$158$	−503.125	−0.253332
$159$	−203.417	−0.101459
$160$	−1499.19	−0.740757
$161$	151.788	0.0743019
$162$	342.310	0.166015
$163$	−1175.08	−0.564658	−0.282329	−	0.959318i	$-0.591107\pi$
$163$	−1175.08	−0.564658	−0.282329	+	0.959318i	$0.591107\pi$
$164$	5071.93	2.41494
$165$	−595.725	−0.281073
$166$	−2031.82	−0.949999
$167$	−1474.01	−0.683010	−0.341505	−	0.939880i	$-0.610937\pi$
$167$	−1474.01	−0.683010	−0.341505	+	0.939880i	$0.610937\pi$
$168$	−568.768	−0.261199
$169$	0	0
$170$	−1223.17	−0.551840
$171$	691.693	0.309328
$172$	−2644.17	−1.17219
$173$	−2328.31	−1.02322	−0.511612	−	0.859216i	$-0.670952\pi$
$173$	−2328.31	−1.02322	−0.511612	+	0.859216i	$0.670952\pi$
$174$	−1280.44	−0.557871
$175$	−2187.35	−0.944848
$176$	−1547.58	−0.662802
$177$	75.6062	0.0321069
$178$	−4589.10	−1.93240
$179$	−2133.85	−0.891015	−0.445508	−	0.895278i	$-0.646977\pi$
$179$	−2133.85	−0.891015	−0.445508	+	0.895278i	$0.646977\pi$
$180$	519.950	0.215304
$181$	−2485.41	−1.02066	−0.510329	−	0.859979i	$-0.670476\pi$
$181$	−2485.41	−1.02066	−0.510329	+	0.859979i	$0.670476\pi$
$182$	0	0
$183$	1766.50	0.713570
$184$	49.4427	0.0198096
$185$	445.435	0.177022
$186$	−3899.55	−1.53725
$187$	−1673.98	−0.654617
$188$	4544.38	1.76294
$189$	−651.387	−0.250695
$190$	1903.13	0.726673
$191$	−2325.07	−0.880816	−0.440408	−	0.897798i	$-0.645166\pi$
$191$	−2325.07	−0.880816	−0.440408	+	0.897798i	$0.645166\pi$
$192$	2147.78	0.807306
$193$	−3350.12	−1.24946	−0.624732	−	0.780839i	$-0.714792\pi$
$193$	−3350.12	−1.24946	−0.624732	+	0.780839i	$0.714792\pi$
$194$	70.3859	0.0260485
$195$	0	0
$196$	2356.79	0.858890
$197$	3859.30	1.39576	0.697878	−	0.716217i	$-0.254128\pi$
$197$	3859.30	1.39576	0.697878	+	0.716217i	$0.254128\pi$
$198$	1288.96	0.462638
$199$	−4083.60	−1.45467	−0.727333	−	0.686284i	$-0.759241\pi$
$199$	−4083.60	−1.45467	−0.727333	+	0.686284i	$0.759241\pi$
$200$	−712.497	−0.251906
$201$	−3013.38	−1.05745
$202$	−4048.58	−1.41018
$203$	2436.56	0.842428
$204$	1461.05	0.501442
$205$	3014.25	1.02695
$206$	−11.4340	−0.00386721
$207$	56.6247	0.0190130
$208$	0	0
$209$	2604.55	0.862011
$210$	−1792.23	−0.588933
$211$	3027.30	0.987714	0.493857	−	0.869543i	$-0.335587\pi$
$211$	3027.30	0.987714	0.493857	+	0.869543i	$0.335587\pi$
$212$	668.533	0.216580
$213$	2686.44	0.864188
$214$	5710.60	1.82415
$215$	−1571.43	−0.498469
$216$	−212.179	−0.0668378
$217$	7420.50	2.32137
$218$	−1895.19	−0.588801
$219$	2905.80	0.896601
$220$	1957.86	0.599994
$221$	0	0
$222$	−963.780	−0.291372
$223$	1724.76	0.517930	0.258965	−	0.965887i	$-0.416619\pi$
$223$	1724.76	0.517930	0.258965	+	0.965887i	$0.416619\pi$
$224$	−6172.60	−1.84118
$225$	−815.993	−0.241776
$226$	5911.60	1.73997
$227$	1923.27	0.562344	0.281172	−	0.959657i	$-0.409277\pi$
$227$	1923.27	0.562344	0.281172	+	0.959657i	$0.409277\pi$
$228$	−2273.26	−0.660308
$229$	−373.993	−0.107922	−0.0539610	−	0.998543i	$-0.517185\pi$
$229$	−373.993	−0.107922	−0.0539610	+	0.998543i	$0.517185\pi$
$230$	155.798	0.0446652
$231$	−2452.78	−0.698618
$232$	793.671	0.224599
$233$	3094.49	0.870073	0.435036	−	0.900413i	$-0.356735\pi$
$233$	3094.49	0.870073	0.435036	+	0.900413i	$0.356735\pi$
$234$	0	0
$235$	2700.73	0.749686
$236$	−248.481	−0.0685369
$237$	357.160	0.0978903
$238$	−5036.15	−1.37162
$239$	1221.18	0.330510	0.165255	−	0.986251i	$-0.447155\pi$
$239$	1221.18	0.330510	0.165255	+	0.986251i	$0.447155\pi$
$240$	802.742	0.215903
$241$	−145.401	−0.0388634	−0.0194317	−	0.999811i	$-0.506186\pi$
$241$	−145.401	−0.0388634	−0.0194317	+	0.999811i	$0.506186\pi$
$242$	−771.350	−0.204894
$243$	−243.000	−0.0641500
$244$	−5805.61	−1.52322
$245$	1400.65	0.365241
$246$	−6521.88	−1.69033
$247$	0	0
$248$	2417.11	0.618899
$249$	1442.35	0.367090
$250$	−5340.47	−1.35104
$251$	−985.670	−0.247868	−0.123934	−	0.992290i	$-0.539551\pi$
$251$	−985.670	−0.247868	−0.123934	+	0.992290i	$0.539551\pi$
$252$	2140.79	0.535147
$253$	213.218	0.0529839
$254$	−505.032	−0.124758
$255$	868.306	0.213237
$256$	1591.33	0.388507
$257$	2929.32	0.710995	0.355498	−	0.934677i	$-0.384311\pi$
$257$	2929.32	0.710995	0.355498	+	0.934677i	$0.384311\pi$
$258$	3400.08	0.820464
$259$	1833.99	0.439995
$260$	0	0
$261$	908.958	0.215567
$262$	−9515.82	−2.24385
$263$	2238.00	0.524719	0.262360	−	0.964970i	$-0.415499\pi$
$263$	2238.00	0.524719	0.262360	+	0.964970i	$0.415499\pi$
$264$	−798.953	−0.186258
$265$	397.310	0.0921002
$266$	7835.77	1.80617
$267$	3257.72	0.746702
$268$	9903.50	2.25729
$269$	1925.98	0.436540	0.218270	−	0.975888i	$-0.429959\pi$
$269$	1925.98	0.436540	0.218270	+	0.975888i	$0.429959\pi$
$270$	−668.593	−0.150701
$271$	3562.29	0.798500	0.399250	−	0.916842i	$-0.369271\pi$
$271$	3562.29	0.798500	0.399250	+	0.916842i	$0.369271\pi$
$272$	2255.69	0.502837
$273$	0	0
$274$	−4778.06	−1.05348
$275$	−3072.59	−0.673761
$276$	−186.098	−0.0405861
$277$	−1437.42	−0.311792	−0.155896	−	0.987773i	$-0.549827\pi$
$277$	−1437.42	−0.311792	−0.155896	+	0.987773i	$0.549827\pi$
$278$	−2515.72	−0.542743
$279$	2768.22	0.594011
$280$	1110.91	0.237105
$281$	3913.51	0.830820	0.415410	−	0.909634i	$-0.363638\pi$
$281$	3913.51	0.830820	0.415410	+	0.909634i	$0.363638\pi$
$282$	−5843.52	−1.23396
$283$	−3212.31	−0.674743	−0.337371	−	0.941372i	$-0.609538\pi$
$283$	−3212.31	−0.674743	−0.337371	+	0.941372i	$0.609538\pi$
$284$	−8829.02	−1.84474
$285$	−1351.00	−0.280794
$286$	0	0
$287$	12410.6	2.55252
$288$	−2302.69	−0.471136
$289$	−2473.07	−0.503373
$290$	2500.92	0.506410
$291$	−49.9657	−0.0100654
$292$	−9549.94	−1.91393
$293$	4901.77	0.977353	0.488677	−	0.872465i	$-0.337480\pi$
$293$	4901.77	0.977353	0.488677	+	0.872465i	$0.337480\pi$
$294$	−3030.55	−0.601175
$295$	−147.672	−0.0291452
$296$	597.394	0.117307
$297$	−915.008	−0.178768
$298$	3352.00	0.651598
$299$	0	0
$300$	2681.77	0.516107
$301$	−6470.06	−1.23896
$302$	567.191	0.108073
$303$	2874.01	0.544910
$304$	−3509.64	−0.662144
$305$	−3450.28	−0.647746
$306$	−1878.74	−0.350981
$307$	−5800.63	−1.07837	−0.539185	−	0.842188i	$-0.681267\pi$
$307$	−5800.63	−1.07837	−0.539185	+	0.842188i	$0.681267\pi$
$308$	8061.07	1.49131
$309$	8.11679	0.00149433
$310$	7616.51	1.39545
$311$	−4913.51	−0.895884	−0.447942	−	0.894063i	$-0.647843\pi$
$311$	−4913.51	−0.895884	−0.447942	+	0.894063i	$0.647843\pi$
$312$	0	0
$313$	−8104.97	−1.46364	−0.731822	−	0.681496i	$-0.761330\pi$
$313$	−8104.97	−1.46364	−0.731822	+	0.681496i	$0.761330\pi$
$314$	6377.41	1.14617
$315$	1272.27	0.227570
$316$	−1173.81	−0.208962
$317$	−5149.92	−0.912455	−0.456227	−	0.889863i	$-0.650800\pi$
$317$	−5149.92	−0.912455	−0.456227	+	0.889863i	$0.650800\pi$
$318$	−859.652	−0.151594
$319$	3422.65	0.600726
$320$	−4195.00	−0.732836
$321$	−4053.85	−0.704872
$322$	641.466	0.111017
$323$	−3796.29	−0.653967
$324$	798.622	0.136938
$325$	0	0
$326$	−4965.95	−0.843676
$327$	1345.36	0.227519
$328$	4042.55	0.680526
$329$	11119.7	1.86337
$330$	−2517.56	−0.419962
$331$	−6061.98	−1.00664	−0.503318	−	0.864101i	$-0.667888\pi$
$331$	−6061.98	−1.00664	−0.503318	+	0.864101i	$0.667888\pi$
$332$	−4740.31	−0.783609
$333$	684.170	0.112589
$334$	−6229.26	−1.02051
$335$	5885.67	0.959905
$336$	3305.13	0.536636
$337$	3743.50	0.605108	0.302554	−	0.953132i	$-0.402161\pi$
$337$	3743.50	0.605108	0.302554	+	0.953132i	$0.402161\pi$
$338$	0	0
$339$	−4196.54	−0.672344
$340$	−2853.70	−0.455187
$341$	10423.6	1.65534
$342$	2923.13	0.462178
$343$	−2508.15	−0.394832
$344$	−2107.52	−0.330319
$345$	−110.598	−0.0172591
$346$	−9839.55	−1.52884
$347$	−2520.41	−0.389921	−0.194961	−	0.980811i	$-0.562458\pi$
$347$	−2520.41	−0.389921	−0.194961	+	0.980811i	$0.562458\pi$
$348$	−2987.30	−0.460161
$349$	−10650.7	−1.63359	−0.816793	−	0.576931i	$-0.804250\pi$
$349$	−10650.7	−1.63359	−0.816793	+	0.576931i	$0.804250\pi$
$350$	−9243.88	−1.41173
$351$	0	0
$352$	−8670.70	−1.31293
$353$	9002.82	1.35743	0.678714	−	0.734403i	$-0.262538\pi$
$353$	9002.82	1.35743	0.678714	+	0.734403i	$0.262538\pi$
$354$	319.516	0.0479720
$355$	−5247.10	−0.784471
$356$	−10706.5	−1.59395
$357$	3575.08	0.530009
$358$	−9017.78	−1.33130
$359$	11360.9	1.67021	0.835106	−	0.550089i	$-0.185406\pi$
$359$	11360.9	1.67021	0.835106	+	0.550089i	$0.185406\pi$
$360$	414.424	0.0606723
$361$	−952.336	−0.138845
$362$	−10503.5	−1.52500
$363$	547.568	0.0791731
$364$	0	0
$365$	−5675.54	−0.813894
$366$	7465.31	1.06617
$367$	−13938.8	−1.98257	−0.991283	−	0.131753i	$-0.957939\pi$
$367$	−13938.8	−1.98257	−0.991283	+	0.131753i	$0.957939\pi$
$368$	−287.313	−0.0406990
$369$	4629.77	0.653160
$370$	1882.43	0.264495
$371$	1635.84	0.228918
$372$	−9097.78	−1.26801
$373$	1593.07	0.221142	0.110571	−	0.993868i	$-0.464732\pi$
$373$	1593.07	0.221142	0.110571	+	0.993868i	$0.464732\pi$
$374$	−7074.32	−0.978087
$375$	3791.10	0.522058
$376$	3622.07	0.496793
$377$	0	0
$378$	−2752.80	−0.374573
$379$	−9137.56	−1.23843	−0.619215	−	0.785221i	$-0.712549\pi$
$379$	−9137.56	−1.23843	−0.619215	+	0.785221i	$0.712549\pi$
$380$	4440.08	0.599398
$381$	358.513	0.0482079
$382$	−9825.86	−1.31606
$383$	−9551.07	−1.27425	−0.637124	−	0.770761i	$-0.719876\pi$
$383$	−9551.07	−1.27425	−0.637124	+	0.770761i	$0.719876\pi$
$384$	2936.14	0.390193
$385$	4790.71	0.634174
$386$	−14157.8	−1.86687
$387$	−2413.66	−0.317036
$388$	164.213	0.0214862
$389$	7366.50	0.960145	0.480072	−	0.877229i	$-0.340610\pi$
$389$	7366.50	0.960145	0.480072	+	0.877229i	$0.340610\pi$
$390$	0	0
$391$	−310.779	−0.0401964
$392$	1878.47	0.242033
$393$	6755.11	0.867049
$394$	16309.6	2.08545
$395$	−697.596	−0.0888604
$396$	3007.18	0.381608
$397$	11696.5	1.47866	0.739332	−	0.673342i	$-0.235142\pi$
$397$	11696.5	1.47866	0.739332	+	0.673342i	$0.235142\pi$
$398$	−17257.5	−2.17347
$399$	−5562.47	−0.697924
$400$	4140.34	0.517542
$401$	14167.6	1.76433	0.882167	−	0.470937i	$-0.156084\pi$
$401$	14167.6	1.76433	0.882167	+	0.470937i	$0.156084\pi$
$402$	−12734.7	−1.57997
$403$	0	0
$404$	−9445.47	−1.16319
$405$	474.622	0.0582325
$406$	10297.0	1.25870
$407$	2576.22	0.313755
$408$	1164.52	0.141305
$409$	−2703.69	−0.326868	−0.163434	−	0.986554i	$-0.552257\pi$
$409$	−2703.69	−0.326868	−0.163434	+	0.986554i	$0.552257\pi$
$410$	12738.4	1.53440
$411$	3391.86	0.407076
$412$	−26.6759	−0.00318988
$413$	−608.011	−0.0724414
$414$	239.299	0.0284080
$415$	−2817.17	−0.333228
$416$	0	0
$417$	1785.86	0.209722
$418$	11007.0	1.28796
$419$	7142.52	0.832781	0.416390	−	0.909186i	$-0.363295\pi$
$419$	7142.52	0.832781	0.416390	+	0.909186i	$0.363295\pi$
$420$	−4181.34	−0.485783
$421$	3406.45	0.394347	0.197174	−	0.980369i	$-0.436824\pi$
$421$	3406.45	0.394347	0.197174	+	0.980369i	$0.436824\pi$
$422$	12793.5	1.47578
$423$	4148.21	0.476815
$424$	532.850	0.0610318
$425$	4478.50	0.511151
$426$	11353.0	1.29121
$427$	−14205.8	−1.61000
$428$	13323.0	1.50466
$429$	0	0
$430$	−6640.96	−0.744780
$431$	−5172.97	−0.578128	−0.289064	−	0.957310i	$-0.593344\pi$
$431$	−5172.97	−0.578128	−0.289064	+	0.957310i	$0.593344\pi$
$432$	1232.98	0.137319
$433$	10955.0	1.21585	0.607924	−	0.793995i	$-0.292002\pi$
$433$	10955.0	1.21585	0.607924	+	0.793995i	$0.292002\pi$
$434$	31359.4	3.46844
$435$	−1775.36	−0.195682
$436$	−4421.55	−0.485674
$437$	483.542	0.0529313
$438$	12280.1	1.33964
$439$	11832.4	1.28640	0.643202	−	0.765696i	$-0.277605\pi$
$439$	11832.4	1.28640	0.643202	+	0.765696i	$0.277605\pi$
$440$	1560.50	0.169077
$441$	2151.33	0.232300
$442$	0	0
$443$	13479.8	1.44570	0.722852	−	0.691003i	$-0.242831\pi$
$443$	13479.8	1.44570	0.722852	+	0.691003i	$0.242831\pi$
$444$	−2248.53	−0.240339
$445$	−6362.91	−0.677822
$446$	7288.92	0.773857
$447$	−2379.52	−0.251784
$448$	−17272.1	−1.82149
$449$	6774.34	0.712028	0.356014	−	0.934481i	$-0.384135\pi$
$449$	6774.34	0.712028	0.356014	+	0.934481i	$0.384135\pi$
$450$	−3448.43	−0.361246
$451$	17433.2	1.82017
$452$	13792.0	1.43522
$453$	−402.639	−0.0417607
$454$	8127.86	0.840219
$455$	0	0
$456$	−1811.89	−0.186073
$457$	4642.36	0.475187	0.237594	−	0.971365i	$-0.423641\pi$
$457$	4642.36	0.475187	0.237594	+	0.971365i	$0.423641\pi$
$458$	−1580.51	−0.161250
$459$	1333.68	0.135623
$460$	363.482	0.0368422
$461$	−2460.37	−0.248570	−0.124285	−	0.992247i	$-0.539664\pi$
$461$	−2460.37	−0.248570	−0.124285	+	0.992247i	$0.539664\pi$
$462$	−10365.6	−1.04383
$463$	−4290.01	−0.430613	−0.215306	−	0.976547i	$-0.569075\pi$
$463$	−4290.01	−0.430613	−0.215306	+	0.976547i	$0.569075\pi$
$464$	−4612.04	−0.461441
$465$	−5406.83	−0.539216
$466$	13077.5	1.30001
$467$	−8798.99	−0.871882	−0.435941	−	0.899975i	$-0.643584\pi$
$467$	−8798.99	−0.871882	−0.435941	+	0.899975i	$0.643584\pi$
$468$	0	0
$469$	24233.0	2.38588
$470$	11413.4	1.12013
$471$	−4527.21	−0.442894
$472$	−198.050	−0.0193136
$473$	−9088.54	−0.883491
$474$	1509.38	0.146261
$475$	−6968.11	−0.673092
$476$	−11749.5	−1.13138
$477$	610.251	0.0585776
$478$	5160.79	0.493826
$479$	10973.4	1.04673	0.523367	−	0.852107i	$-0.324676\pi$
$479$	10973.4	1.04673	0.523367	+	0.852107i	$0.324676\pi$
$480$	4497.56	0.427676
$481$	0	0
$482$	−614.470	−0.0580671
$483$	−455.365	−0.0428982
$484$	−1799.59	−0.169007
$485$	97.5919	0.00913694
$486$	−1026.93	−0.0958489
$487$	5209.58	0.484740	0.242370	−	0.970184i	$-0.422075\pi$
$487$	5209.58	0.484740	0.242370	+	0.970184i	$0.422075\pi$
$488$	−4627.33	−0.429241
$489$	3525.24	0.326006
$490$	5919.20	0.545719
$491$	−8779.22	−0.806926	−0.403463	−	0.914996i	$-0.632194\pi$
$491$	−8779.22	−0.806926	−0.403463	+	0.914996i	$0.632194\pi$
$492$	−15215.8	−1.39427
$493$	−4988.73	−0.455742
$494$	0	0
$495$	1787.17	0.162278
$496$	−14045.9	−1.27153
$497$	−21603.9	−1.94983
$498$	6095.46	0.548482
$499$	15590.1	1.39861	0.699305	−	0.714823i	$-0.253493\pi$
$499$	15590.1	1.39861	0.699305	+	0.714823i	$0.253493\pi$
$500$	−12459.5	−1.11441
$501$	4422.04	0.394336
$502$	−4165.49	−0.370349
$503$	−64.3909	−0.00570785	−0.00285393	−	0.999996i	$-0.500908\pi$
$503$	−64.3909	−0.00570785	−0.00285393	+	0.999996i	$0.500908\pi$
$504$	1706.31	0.150803
$505$	−5613.45	−0.494644
$506$	901.072	0.0791651
$507$	0	0
$508$	−1178.26	−0.102907
$509$	−3214.15	−0.279891	−0.139946	−	0.990159i	$-0.544693\pi$
$509$	−3214.15	−0.279891	−0.139946	+	0.990159i	$0.544693\pi$
$510$	3669.51	0.318605
$511$	−23367.9	−2.02296
$512$	14554.7	1.25632
$513$	−2075.08	−0.178591
$514$	12379.5	1.06232
$515$	−15.8535	−0.00135649
$516$	7932.50	0.676762
$517$	15619.9	1.32875
$518$	7750.54	0.657412
$519$	6984.92	0.590759
$520$	0	0
$521$	−3053.01	−0.256727	−0.128363	−	0.991727i	$-0.540972\pi$
$521$	−3053.01	−0.256727	−0.128363	+	0.991727i	$0.540972\pi$
$522$	3841.31	0.322087
$523$	5096.02	0.426067	0.213034	−	0.977045i	$-0.431666\pi$
$523$	5096.02	0.426067	0.213034	+	0.977045i	$0.431666\pi$
$524$	−22200.7	−1.85085
$525$	6562.06	0.545508
$526$	9457.92	0.784002
$527$	−15193.1	−1.25583
$528$	4642.74	0.382669
$529$	−12127.4	−0.996747
$530$	1679.05	0.137610
$531$	−226.819	−0.0185369
$532$	18281.1	1.48983
$533$	0	0
$534$	13767.3	1.11567
$535$	7917.89	0.639851
$536$	7893.53	0.636098
$537$	6401.56	0.514428
$538$	8139.31	0.652250
$539$	8100.77	0.647356
$540$	−1559.85	−0.124306
$541$	−7861.99	−0.624793	−0.312397	−	0.949952i	$-0.601132\pi$
$541$	−7861.99	−0.624793	−0.312397	+	0.949952i	$0.601132\pi$
$542$	15054.4	1.19307
$543$	7456.23	0.589277
$544$	12638.1	0.996054
$545$	−2627.73	−0.206532
$546$	0	0
$547$	−6317.48	−0.493814	−0.246907	−	0.969039i	$-0.579414\pi$
$547$	−6317.48	−0.493814	−0.246907	+	0.969039i	$0.579414\pi$
$548$	−11147.4	−0.868965
$549$	−5299.49	−0.411980
$550$	−12984.9	−1.00669
$551$	7761.98	0.600130
$552$	−148.328	−0.0114371
$553$	−2872.21	−0.220866
$554$	−6074.63	−0.465860
$555$	−1336.31	−0.102204
$556$	−5869.25	−0.447683
$557$	971.234	0.0738824	0.0369412	−	0.999317i	$-0.488239\pi$
$557$	971.234	0.0738824	0.0369412	+	0.999317i	$0.488239\pi$
$558$	11698.6	0.887533
$559$	0	0
$560$	−6455.50	−0.487134
$561$	5021.93	0.377943
$562$	16538.7	1.24136
$563$	9328.49	0.698311	0.349155	−	0.937065i	$-0.386469\pi$
$563$	9328.49	0.698311	0.349155	+	0.937065i	$0.386469\pi$
$564$	−13633.1	−1.01783
$565$	8196.59	0.610324
$566$	−13575.4	−1.00816
$567$	1954.16	0.144739
$568$	−7037.12	−0.519843
$569$	17452.2	1.28582	0.642911	−	0.765941i	$-0.277727\pi$
$569$	17452.2	1.28582	0.642911	+	0.765941i	$0.277727\pi$
$570$	−5709.40	−0.419545
$571$	−20181.4	−1.47910	−0.739548	−	0.673103i	$-0.764961\pi$
$571$	−20181.4	−1.47910	−0.739548	+	0.673103i	$0.764961\pi$
$572$	0	0
$573$	6975.20	0.508540
$574$	52447.8	3.81381
$575$	−570.436	−0.0413719
$576$	−6443.35	−0.466099
$577$	−6382.72	−0.460513	−0.230257	−	0.973130i	$-0.573957\pi$
$577$	−6382.72	−0.460513	−0.230257	+	0.973130i	$0.573957\pi$
$578$	−10451.3	−0.752107
$579$	10050.3	0.721379
$580$	5834.73	0.417714
$581$	−11599.1	−0.828250
$582$	−211.158	−0.0150391
$583$	2297.88	0.163239
$584$	−7611.72	−0.539341
$585$	0	0
$586$	20715.2	1.46030
$587$	775.527	0.0545305	0.0272653	−	0.999628i	$-0.491320\pi$
$587$	775.527	0.0545305	0.0272653	+	0.999628i	$0.491320\pi$
$588$	−7070.38	−0.495880
$589$	23639.0	1.65370
$590$	−624.072	−0.0435468
$591$	−11577.9	−0.805840
$592$	−3471.47	−0.241008
$593$	17843.3	1.23564	0.617821	−	0.786319i	$-0.288016\pi$
$593$	17843.3	1.23564	0.617821	+	0.786319i	$0.288016\pi$
$594$	−3866.87	−0.267104
$595$	−6982.76	−0.481118
$596$	7820.33	0.537472
$597$	12250.8	0.839852
$598$	0	0
$599$	−24373.3	−1.66255	−0.831274	−	0.555863i	$-0.812388\pi$
$599$	−24373.3	−1.66255	−0.831274	+	0.555863i	$0.812388\pi$
$600$	2137.49	0.145438
$601$	−3526.99	−0.239383	−0.119691	−	0.992811i	$-0.538190\pi$
$601$	−3526.99	−0.239383	−0.119691	+	0.992811i	$0.538190\pi$
$602$	−27342.8	−1.85118
$603$	9040.14	0.610519
$604$	1323.28	0.0891446
$605$	−1069.50	−0.0718698
$606$	12145.7	0.814169
$607$	7991.55	0.534377	0.267189	−	0.963644i	$-0.413905\pi$
$607$	7991.55	0.534377	0.267189	+	0.963644i	$0.413905\pi$
$608$	−19663.6	−1.31162
$609$	−7309.67	−0.486376
$610$	−14581.1	−0.967821
$611$	0	0
$612$	−4383.16	−0.289508
$613$	16332.2	1.07610	0.538051	−	0.842912i	$-0.319161\pi$
$613$	16332.2	1.07610	0.538051	+	0.842912i	$0.319161\pi$
$614$	−24513.8	−1.61123
$615$	−9042.76	−0.592909
$616$	6425.04	0.420247
$617$	19353.6	1.26280	0.631401	−	0.775457i	$-0.282480\pi$
$617$	19353.6	1.26280	0.631401	+	0.775457i	$0.282480\pi$
$618$	34.3020	0.00223273
$619$	9982.52	0.648193	0.324096	−	0.946024i	$-0.394940\pi$
$619$	9982.52	0.648193	0.324096	+	0.946024i	$0.394940\pi$
$620$	17769.6	1.15104
$621$	−169.874	−0.0109772
$622$	−20764.8	−1.33857
$623$	−26198.0	−1.68475
$624$	0	0
$625$	3928.53	0.251426
$626$	−34252.1	−2.18688
$627$	−7813.64	−0.497682
$628$	14878.7	0.945423
$629$	−3755.00	−0.238031
$630$	5376.70	0.340020
$631$	575.775	0.0363253	0.0181626	−	0.999835i	$-0.494218\pi$
$631$	575.775	0.0363253	0.0181626	+	0.999835i	$0.494218\pi$
$632$	−935.578	−0.0588850
$633$	−9081.89	−0.570257
$634$	−21763.8	−1.36333
$635$	−700.240	−0.0437609
$636$	−2005.60	−0.125043
$637$	0	0
$638$	14464.3	0.897566
$639$	−8059.33	−0.498939
$640$	−5734.80	−0.354200
$641$	−24521.8	−1.51100	−0.755500	−	0.655149i	$-0.772606\pi$
$641$	−24521.8	−1.51100	−0.755500	+	0.655149i	$0.772606\pi$
$642$	−17131.8	−1.05317
$643$	22667.0	1.39020	0.695099	−	0.718914i	$-0.255360\pi$
$643$	22667.0	1.39020	0.695099	+	0.718914i	$0.255360\pi$
$644$	1496.56	0.0915727
$645$	4714.30	0.287791
$646$	−16043.3	−0.977116
$647$	−2397.45	−0.145678	−0.0728389	−	0.997344i	$-0.523206\pi$
$647$	−2397.45	−0.145678	−0.0728389	+	0.997344i	$0.523206\pi$
$648$	636.537	0.0385888
$649$	−854.078	−0.0516572
$650$	0	0
$651$	−22261.5	−1.34024
$652$	−11585.7	−0.695908
$653$	20002.1	1.19868	0.599342	−	0.800493i	$-0.295429\pi$
$653$	20002.1	1.19868	0.599342	+	0.800493i	$0.295429\pi$
$654$	5685.58	0.339945
$655$	−13193.9	−0.787068
$656$	−23491.4	−1.39815
$657$	−8717.39	−0.517653
$658$	46992.5	2.78413
$659$	−3517.96	−0.207952	−0.103976	−	0.994580i	$-0.533156\pi$
$659$	−3517.96	−0.207952	−0.103976	+	0.994580i	$0.533156\pi$
$660$	−5873.57	−0.346406
$661$	−13583.4	−0.799294	−0.399647	−	0.916669i	$-0.630867\pi$
$661$	−13583.4	−0.799294	−0.399647	+	0.916669i	$0.630867\pi$
$662$	−25618.2	−1.50405
$663$	0	0
$664$	−3778.24	−0.220819
$665$	10864.5	0.633544
$666$	2891.34	0.168224
$667$	635.425	0.0368872
$668$	−14533.1	−0.841770
$669$	−5174.27	−0.299027
$670$	24873.1	1.43423
$671$	−19955.1	−1.14807
$672$	18517.8	1.06301
$673$	−10895.8	−0.624077	−0.312038	−	0.950069i	$-0.601012\pi$
$673$	−10895.8	−0.624077	−0.312038	+	0.950069i	$0.601012\pi$
$674$	15820.2	0.904113
$675$	2447.98	0.139589
$676$	0	0
$677$	1449.03	0.0822609	0.0411305	−	0.999154i	$-0.486904\pi$
$677$	1449.03	0.0822609	0.0411305	+	0.999154i	$0.486904\pi$
$678$	−17734.8	−1.00457
$679$	401.815	0.0227102
$680$	−2274.52	−0.128271
$681$	−5769.82	−0.324670
$682$	44050.9	2.47331
$683$	15366.4	0.860878	0.430439	−	0.902620i	$-0.358359\pi$
$683$	15366.4	0.860878	0.430439	+	0.902620i	$0.358359\pi$
$684$	6819.77	0.381229
$685$	−6624.91	−0.369525
$686$	−10599.6	−0.589933
$687$	1121.98	0.0623088
$688$	12246.9	0.678644
$689$	0	0
$690$	−467.393	−0.0257875
$691$	2019.71	0.111191	0.0555957	−	0.998453i	$-0.482294\pi$
$691$	2019.71	0.111191	0.0555957	+	0.998453i	$0.482294\pi$
$692$	−22956.0	−1.26106
$693$	7358.33	0.403347
$694$	−10651.4	−0.582595
$695$	−3488.10	−0.190376
$696$	−2381.01	−0.129672
$697$	−25410.0	−1.38088
$698$	−45010.6	−2.44080
$699$	−9283.48	−0.502337
$700$	−21566.3	−1.16447
$701$	28031.6	1.51033	0.755164	−	0.655536i	$-0.227557\pi$
$701$	28031.6	1.51033	0.755164	+	0.655536i	$0.227557\pi$
$702$	0	0
$703$	5842.42	0.313444
$704$	−24262.2	−1.29889
$705$	−8102.19	−0.432832
$706$	38046.4	2.02818
$707$	−23112.3	−1.22946
$708$	745.442	0.0395698
$709$	−19604.4	−1.03845	−0.519224	−	0.854638i	$-0.673779\pi$
$709$	−19604.4	−1.03845	−0.519224	+	0.854638i	$0.673779\pi$
$710$	−22174.5	−1.17211
$711$	−1071.48	−0.0565170
$712$	−8533.59	−0.449171
$713$	1935.18	0.101645
$714$	15108.5	0.791905
$715$	0	0
$716$	−21038.8	−1.09812
$717$	−3663.55	−0.190820
$718$	48011.8	2.49552
$719$	−14727.4	−0.763894	−0.381947	−	0.924184i	$-0.624746\pi$
$719$	−14727.4	−0.763894	−0.381947	+	0.924184i	$0.624746\pi$
$720$	−2408.23	−0.124652
$721$	−65.2737	−0.00337160
$722$	−4024.62	−0.207453
$723$	436.202	0.0224378
$724$	−24505.0	−1.25790
$725$	−9156.83	−0.469071
$726$	2314.05	0.118295
$727$	−16890.5	−0.861668	−0.430834	−	0.902431i	$-0.641781\pi$
$727$	−16890.5	−0.861668	−0.430834	+	0.902431i	$0.641781\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	−23985.1	−1.21607
$731$	13247.1	0.670263
$732$	17416.8	0.879433
$733$	−12553.6	−0.632578	−0.316289	−	0.948663i	$-0.602437\pi$
$733$	−12553.6	−0.632578	−0.316289	+	0.948663i	$0.602437\pi$
$734$	−58906.3	−2.96222
$735$	−4201.94	−0.210872
$736$	−1609.74	−0.0806194
$737$	34040.3	1.70135
$738$	19565.6	0.975910
$739$	37874.8	1.88531	0.942656	−	0.333766i	$-0.108320\pi$
$739$	37874.8	1.88531	0.942656	+	0.333766i	$0.108320\pi$
$740$	4391.78	0.218169
$741$	0	0
$742$	6913.16	0.342035
$743$	−35882.6	−1.77174	−0.885872	−	0.463929i	$-0.846439\pi$
$743$	−35882.6	−1.77174	−0.885872	+	0.463929i	$0.846439\pi$
$744$	−7251.34	−0.357321
$745$	4647.63	0.228559
$746$	6732.40	0.330416
$747$	−4327.06	−0.211939
$748$	−16504.6	−0.806777
$749$	32600.3	1.59037
$750$	16021.4	0.780026
$751$	−181.689	−0.00882815	−0.00441407	−	0.999990i	$-0.501405\pi$
$751$	−181.689	−0.00882815	−0.00441407	+	0.999990i	$0.501405\pi$
$752$	−21048.0	−1.02067
$753$	2957.01	0.143107
$754$	0	0
$755$	786.425	0.0379085
$756$	−6422.37	−0.308967
$757$	−491.053	−0.0235768	−0.0117884	−	0.999931i	$-0.503752\pi$
$757$	−491.053	−0.0235768	−0.0117884	+	0.999931i	$0.503752\pi$
$758$	−38615.8	−1.85038
$759$	−639.655	−0.0305903
$760$	3538.94	0.168909
$761$	8113.01	0.386460	0.193230	−	0.981153i	$-0.438104\pi$
$761$	8113.01	0.386460	0.193230	+	0.981153i	$0.438104\pi$
$762$	1515.10	0.0720291
$763$	−10819.2	−0.513342
$764$	−22924.1	−1.08555
$765$	−2604.92	−0.123112
$766$	−40363.3	−1.90390
$767$	0	0
$768$	−4773.98	−0.224305
$769$	−19864.7	−0.931519	−0.465759	−	0.884911i	$-0.654219\pi$
$769$	−19864.7	−0.931519	−0.465759	+	0.884911i	$0.654219\pi$
$770$	20245.8	0.947542
$771$	−8787.95	−0.410493
$772$	−33030.6	−1.53989
$773$	−9047.42	−0.420974	−0.210487	−	0.977597i	$-0.567505\pi$
$773$	−9047.42	−0.420974	−0.210487	+	0.977597i	$0.567505\pi$
$774$	−10200.2	−0.473695
$775$	−27887.0	−1.29256
$776$	130.885	0.00605476
$777$	−5501.97	−0.254031
$778$	31131.2	1.43459
$779$	39535.5	1.81837
$780$	0	0
$781$	−30347.1	−1.39040
$782$	−1313.37	−0.0600588
$783$	−2726.87	−0.124458
$784$	−10915.8	−0.497259
$785$	8842.45	0.402039
$786$	28547.5	1.29549
$787$	15018.4	0.680240	0.340120	−	0.940382i	$-0.389532\pi$
$787$	15018.4	0.680240	0.340120	+	0.940382i	$0.389532\pi$
$788$	38050.9	1.72019
$789$	−6714.01	−0.302947
$790$	−2948.08	−0.132770
$791$	33747.8	1.51698
$792$	2396.86	0.107536
$793$	0	0
$794$	49429.9	2.20932
$795$	−1191.93	−0.0531741
$796$	−40262.4	−1.79279
$797$	31941.1	1.41959	0.709794	−	0.704409i	$-0.248788\pi$
$797$	31941.1	1.41959	0.709794	+	0.704409i	$0.248788\pi$
$798$	−23507.3	−1.04279
$799$	−22767.1	−1.00806
$800$	23197.3	1.02518
$801$	−9773.17	−0.431108
$802$	59873.2	2.63615
$803$	−32825.0	−1.44255
$804$	−29710.5	−1.30325
$805$	889.409	0.0389411
$806$	0	0
$807$	−5777.95	−0.252037
$808$	−7528.46	−0.327785
$809$	27260.1	1.18469	0.592344	−	0.805685i	$-0.298203\pi$
$809$	27260.1	1.18469	0.592344	+	0.805685i	$0.298203\pi$
$810$	2005.78	0.0870073
$811$	20707.8	0.896607	0.448303	−	0.893881i	$-0.352028\pi$
$811$	20707.8	0.896607	0.448303	+	0.893881i	$0.352028\pi$
$812$	24023.3	1.03824
$813$	−10686.9	−0.461014
$814$	10887.2	0.468793
$815$	−6885.42	−0.295933
$816$	−6767.08	−0.290313
$817$	−20611.2	−0.882614
$818$	−11426.0	−0.488385
$819$	0	0
$820$	29719.1	1.26565
$821$	−15658.6	−0.665636	−0.332818	−	0.942991i	$-0.608000\pi$
$821$	−15658.6	−0.665636	−0.332818	+	0.942991i	$0.608000\pi$
$822$	14334.2	0.608227
$823$	−4106.58	−0.173932	−0.0869662	−	0.996211i	$-0.527717\pi$
$823$	−4106.58	−0.173932	−0.0869662	+	0.996211i	$0.527717\pi$
$824$	−21.2619	−0.000898900 0
$825$	9217.78	0.388996
$826$	−2569.49	−0.108237
$827$	−16747.3	−0.704184	−0.352092	−	0.935965i	$-0.614530\pi$
$827$	−16747.3	−0.704184	−0.352092	+	0.935965i	$0.614530\pi$
$828$	558.293	0.0234324
$829$	−29157.1	−1.22155	−0.610776	−	0.791803i	$-0.709142\pi$
$829$	−29157.1	−1.22155	−0.610776	+	0.791803i	$0.709142\pi$
$830$	−11905.5	−0.497887
$831$	4312.27	0.180013
$832$	0	0
$833$	−11807.4	−0.491119
$834$	7547.15	0.313353
$835$	−8637.03	−0.357960
$836$	25679.6	1.06238
$837$	−8304.66	−0.342952
$838$	30184.7	1.24429
$839$	−45819.4	−1.88541	−0.942707	−	0.333621i	$-0.891729\pi$
$839$	−45819.4	−1.88541	−0.942707	+	0.333621i	$0.891729\pi$
$840$	−3332.72	−0.136892
$841$	−14188.9	−0.581776
$842$	14395.8	0.589209
$843$	−11740.5	−0.479674
$844$	29847.7	1.21730
$845$	0	0
$846$	17530.6	0.712427
$847$	−4403.44	−0.178635
$848$	−3096.41	−0.125390
$849$	9636.94	0.389563
$850$	18926.4	0.763729
$851$	478.283	0.0192660
$852$	26487.1	1.06506
$853$	−17351.1	−0.696471	−0.348235	−	0.937407i	$-0.613219\pi$
$853$	−17351.1	−0.696471	−0.348235	+	0.937407i	$0.613219\pi$
$854$	−60034.7	−2.40556
$855$	4053.00	0.162117
$856$	10619.0	0.424009
$857$	−21768.1	−0.867659	−0.433829	−	0.900995i	$-0.642838\pi$
$857$	−21768.1	−0.867659	−0.433829	+	0.900995i	$0.642838\pi$
$858$	0	0
$859$	−29878.4	−1.18677	−0.593387	−	0.804918i	$-0.702209\pi$
$859$	−29878.4	−1.18677	−0.593387	+	0.804918i	$0.702209\pi$
$860$	−15493.6	−0.614334
$861$	−37231.7	−1.47370
$862$	−21861.2	−0.863801
$863$	15067.7	0.594335	0.297168	−	0.954825i	$-0.403958\pi$
$863$	15067.7	0.594335	0.297168	+	0.954825i	$0.403958\pi$
$864$	6908.07	0.272011
$865$	−13642.8	−0.536265
$866$	46296.3	1.81664
$867$	7419.21	0.290622
$868$	73162.7	2.86095
$869$	−4034.62	−0.157497
$870$	−7502.75	−0.292376
$871$	0	0
$872$	−3524.17	−0.136862
$873$	149.897	0.00581128
$874$	2043.48	0.0790865
$875$	−30487.4	−1.17790
$876$	28649.8	1.10501
$877$	21119.5	0.813177	0.406588	−	0.913611i	$-0.366718\pi$
$877$	21119.5	0.813177	0.406588	+	0.913611i	$0.366718\pi$
$878$	50004.5	1.92206
$879$	−14705.3	−0.564275
$880$	−9068.09	−0.347370
$881$	31652.5	1.21044	0.605221	−	0.796057i	$-0.293085\pi$
$881$	31652.5	1.21044	0.605221	+	0.796057i	$0.293085\pi$
$882$	9091.65	0.347088
$883$	11701.6	0.445969	0.222984	−	0.974822i	$-0.428420\pi$
$883$	11701.6	0.445969	0.222984	+	0.974822i	$0.428420\pi$
$884$	0	0
$885$	443.017	0.0168270
$886$	56966.6	2.16008
$887$	22507.9	0.852020	0.426010	−	0.904719i	$-0.359919\pi$
$887$	22507.9	0.852020	0.426010	+	0.904719i	$0.359919\pi$
$888$	−1792.18	−0.0677271
$889$	−2883.10	−0.108769
$890$	−26890.0	−1.01276
$891$	2745.02	0.103212
$892$	17005.3	0.638318
$893$	35423.3	1.32743
$894$	−10056.0	−0.376200
$895$	−12503.4	−0.466974
$896$	−23611.9	−0.880377
$897$	0	0
$898$	28628.7	1.06387
$899$	31064.1	1.15244
$900$	−8045.31	−0.297974
$901$	−3349.31	−0.123842
$902$	73673.8	2.71959
$903$	19410.2	0.715316
$904$	10992.8	0.404442
$905$	−14563.3	−0.534919
$906$	−1701.57	−0.0623962
$907$	18718.6	0.685273	0.342636	−	0.939468i	$-0.388680\pi$
$907$	18718.6	0.685273	0.342636	+	0.939468i	$0.388680\pi$
$908$	18962.6	0.693057
$909$	−8622.03	−0.314604
$910$	0	0
$911$	18616.7	0.677057	0.338529	−	0.940956i	$-0.390071\pi$
$911$	18616.7	0.677057	0.338529	+	0.940956i	$0.390071\pi$
$912$	10528.9	0.382289
$913$	−16293.4	−0.590616
$914$	19618.9	0.709994
$915$	10350.8	0.373977
$916$	−3687.39	−0.133008
$917$	−54323.3	−1.95629
$918$	5636.21	0.202639
$919$	−54764.4	−1.96573	−0.982867	−	0.184316i	$-0.940993\pi$
$919$	−54764.4	−1.96573	−0.982867	+	0.184316i	$0.940993\pi$
$920$	289.711	0.0103821
$921$	17401.9	0.622597
$922$	−10397.6	−0.371397
$923$	0	0
$924$	−24183.2	−0.861006
$925$	−6892.32	−0.244993
$926$	−18129.8	−0.643394
$927$	−24.3504	−0.000862752 0
$928$	−25840.1	−0.914055
$929$	31832.1	1.12419	0.562097	−	0.827071i	$-0.309995\pi$
$929$	31832.1	1.12419	0.562097	+	0.827071i	$0.309995\pi$
$930$	−22849.5	−0.805662
$931$	18371.2	0.646713
$932$	30510.3	1.07231
$933$	14740.5	0.517239
$934$	−37185.0	−1.30271
$935$	−9808.73	−0.343080
$936$	0	0
$937$	−27408.7	−0.955607	−0.477803	−	0.878467i	$-0.658567\pi$
$937$	−27408.7	−0.955607	−0.477803	+	0.878467i	$0.658567\pi$
$938$	102410.	3.56483
$939$	24314.9	0.845035
$940$	26627.9	0.923945
$941$	−54837.8	−1.89975	−0.949874	−	0.312634i	$-0.898789\pi$
$941$	−54837.8	−1.89975	−0.949874	+	0.312634i	$0.898789\pi$
$942$	−19132.2	−0.661743
$943$	3236.53	0.111767
$944$	1150.87	0.0396799
$945$	−3816.82	−0.131388
$946$	−38408.6	−1.32006
$947$	39707.8	1.36255	0.681273	−	0.732030i	$-0.261427\pi$
$947$	39707.8	1.36255	0.681273	+	0.732030i	$0.261427\pi$
$948$	3521.43	0.120644
$949$	0	0
$950$	−29447.6	−1.00569
$951$	15449.8	0.526806
$952$	−9364.89	−0.318821
$953$	−17106.6	−0.581468	−0.290734	−	0.956804i	$-0.593899\pi$
$953$	−17106.6	−0.581468	−0.290734	+	0.956804i	$0.593899\pi$
$954$	2578.96	0.0875228
$955$	−13623.8	−0.461629
$956$	12040.3	0.407334
$957$	−10268.0	−0.346829
$958$	46374.0	1.56396
$959$	−27276.7	−0.918469
$960$	12585.0	0.423103
$961$	64814.4	2.17564
$962$	0	0
$963$	12161.6	0.406958
$964$	−1433.58	−0.0478968
$965$	−19630.1	−0.654835
$966$	−1924.40	−0.0640957
$967$	23417.5	0.778756	0.389378	−	0.921078i	$-0.372690\pi$
$967$	23417.5	0.778756	0.389378	+	0.921078i	$0.372690\pi$
$968$	−1434.35	−0.0476258
$969$	11388.9	0.377568
$970$	412.428	0.0136518
$971$	16430.6	0.543032	0.271516	−	0.962434i	$-0.412475\pi$
$971$	16430.6	0.543032	0.271516	+	0.962434i	$0.412475\pi$
$972$	−2395.87	−0.0790612
$973$	−14361.6	−0.473187
$974$	22015.9	0.724267
$975$	0	0
$976$	26889.5	0.881879
$977$	10554.6	0.345622	0.172811	−	0.984955i	$-0.444715\pi$
$977$	10554.6	0.345622	0.172811	+	0.984955i	$0.444715\pi$
$978$	14897.8	0.487097
$979$	−36800.5	−1.20138
$980$	13809.7	0.450138
$981$	−4036.09	−0.131358
$982$	−37101.5	−1.20566
$983$	−1534.33	−0.0497839	−0.0248919	−	0.999690i	$-0.507924\pi$
$983$	−1534.33	−0.0497839	−0.0248919	+	0.999690i	$0.507924\pi$
$984$	−12127.7	−0.392902
$985$	22613.7	0.731505
$986$	−21082.6	−0.680941
$987$	−33359.1	−1.07582
$988$	0	0
$989$	−1687.31	−0.0542502
$990$	7552.69	0.242465
$991$	−18018.2	−0.577563	−0.288782	−	0.957395i	$-0.593250\pi$
$991$	−18018.2	−0.577563	−0.288782	+	0.957395i	$0.593250\pi$
$992$	−78695.7	−2.51874
$993$	18185.9	0.581181
$994$	−91299.1	−2.91331
$995$	−23928.0	−0.762380
$996$	14220.9	0.452417
$997$	48287.7	1.53389	0.766944	−	0.641714i	$-0.221776\pi$
$997$	48287.7	1.53389	0.766944	+	0.641714i	$0.221776\pi$
$998$	65884.4	2.08971
$999$	−2052.51	−0.0650036

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.a.i.1.4		4
3.2	odd	2		1521.4.a.bb.1.1		4
13.4	even	6		39.4.e.c.16.4	✓	8
13.5	odd	4		507.4.b.h.337.2		8
13.8	odd	4		507.4.b.h.337.7		8
13.10	even	6		39.4.e.c.22.4	yes	8
13.12	even	2		507.4.a.m.1.1		4
39.17	odd	6		117.4.g.e.55.1		8
39.23	odd	6		117.4.g.e.100.1		8
39.38	odd	2		1521.4.a.v.1.4		4
52.23	odd	6		624.4.q.i.529.2		8
52.43	odd	6		624.4.q.i.289.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.e.c.16.4	✓	8	13.4	even	6
39.4.e.c.22.4	yes	8	13.10	even	6
117.4.g.e.55.1		8	39.17	odd	6
117.4.g.e.100.1		8	39.23	odd	6
507.4.a.i.1.4		4	1.1	even	1	trivial
507.4.a.m.1.1		4	13.12	even	2
507.4.b.h.337.2		8	13.5	odd	4
507.4.b.h.337.7		8	13.8	odd	4
624.4.q.i.289.2		8	52.43	odd	6
624.4.q.i.529.2		8	52.23	odd	6
1521.4.a.v.1.4		4	39.38	odd	2
1521.4.a.bb.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:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	507.a (trivial)

\(f(q)\)	\(=\)	\(q+4.22605 q^{2} -3.00000 q^{3} +9.85953 q^{4} +5.85953 q^{5} -12.6782 q^{6} +24.1254 q^{7} +7.85849 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+4.22605 q^{2} -3.00000 q^{3} +9.85953 q^{4} +5.85953 q^{5} -12.6782 q^{6} +24.1254 q^{7} +7.85849 q^{8} +9.00000 q^{9} +24.7627 q^{10} +33.8892 q^{11} -29.5786 q^{12} +101.955 q^{14} -17.5786 q^{15} -45.6659 q^{16} -49.3956 q^{17} +38.0345 q^{18} +76.8548 q^{19} +57.7723 q^{20} -72.3763 q^{21} +143.218 q^{22} +6.29163 q^{23} -23.5755 q^{24} -90.6659 q^{25} -27.0000 q^{27} +237.866 q^{28} +100.995 q^{29} -74.2881 q^{30} +307.580 q^{31} -255.854 q^{32} -101.668 q^{33} -208.749 q^{34} +141.364 q^{35} +88.7358 q^{36} +76.0189 q^{37} +324.793 q^{38} +46.0471 q^{40} +514.418 q^{41} -305.866 q^{42} -268.184 q^{43} +334.132 q^{44} +52.7358 q^{45} +26.5888 q^{46} +460.912 q^{47} +136.998 q^{48} +239.037 q^{49} -383.159 q^{50} +148.187 q^{51} +67.8057 q^{53} -114.103 q^{54} +198.575 q^{55} +189.589 q^{56} -230.564 q^{57} +426.812 q^{58} -25.2021 q^{59} -173.317 q^{60} -588.832 q^{61} +1299.85 q^{62} +217.129 q^{63} -715.927 q^{64} -429.653 q^{66} +1004.46 q^{67} -487.018 q^{68} -18.8749 q^{69} +597.411 q^{70} -895.481 q^{71} +70.7264 q^{72} -968.599 q^{73} +321.260 q^{74} +271.998 q^{75} +757.753 q^{76} +817.592 q^{77} -119.053 q^{79} -267.581 q^{80} +81.0000 q^{81} +2173.96 q^{82} -480.784 q^{83} -713.597 q^{84} -289.435 q^{85} -1133.36 q^{86} -302.986 q^{87} +266.318 q^{88} -1085.91 q^{89} +222.864 q^{90} +62.0325 q^{92} -922.740 q^{93} +1947.84 q^{94} +450.333 q^{95} +767.563 q^{96} +16.6552 q^{97} +1010.18 q^{98} +305.003 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} - 12 q^{3} + 22 q^{4} + 6 q^{5} + 6 q^{6} + 14 q^{7} - 54 q^{8} + 36 q^{9}+O(q^{10}) \) 4 * q - 2 * q^2 - 12 * q^3 + 22 * q^4 + 6 * q^5 + 6 * q^6 + 14 * q^7 - 54 * q^8 + 36 * q^9	\( 4 q - 2 q^{2} - 12 q^{3} + 22 q^{4} + 6 q^{5} + 6 q^{6} + 14 q^{7} - 54 q^{8} + 36 q^{9} - 62 q^{10} - 40 q^{11} - 66 q^{12} + 40 q^{14} - 18 q^{15} + 122 q^{16} + 98 q^{17} - 18 q^{18} - 124 q^{19} + 466 q^{20} - 42 q^{21} + 220 q^{22} + 104 q^{23} + 162 q^{24} - 58 q^{25} - 108 q^{27} + 144 q^{28} + 194 q^{29} + 186 q^{30} - 26 q^{31} - 654 q^{32} + 120 q^{33} - 1062 q^{34} + 88 q^{35} + 198 q^{36} - 102 q^{37} + 332 q^{38} - 998 q^{40} + 1054 q^{41} - 120 q^{42} + 450 q^{43} + 44 q^{44} + 54 q^{45} + 172 q^{46} + 96 q^{47} - 366 q^{48} + 1070 q^{49} - 996 q^{50} - 294 q^{51} + 262 q^{53} + 54 q^{54} + 204 q^{55} + 2164 q^{56} + 372 q^{57} - 722 q^{58} - 308 q^{59} - 1398 q^{60} - 928 q^{61} + 2780 q^{62} + 126 q^{63} + 1026 q^{64} - 660 q^{66} + 1134 q^{67} + 1786 q^{68} - 312 q^{69} + 2324 q^{70} - 1064 q^{71} - 486 q^{72} - 952 q^{73} + 1158 q^{74} + 174 q^{75} + 1708 q^{76} + 2508 q^{77} - 746 q^{79} + 2922 q^{80} + 324 q^{81} + 1734 q^{82} + 404 q^{83} - 432 q^{84} + 1394 q^{85} - 3168 q^{86} - 582 q^{87} + 3060 q^{88} - 1620 q^{89} - 558 q^{90} + 332 q^{92} + 78 q^{93} - 772 q^{94} + 2204 q^{95} + 1962 q^{96} - 2166 q^{97} + 1906 q^{98} - 360 q^{99}+O(q^{100}) \) 4 * q - 2 * q^2 - 12 * q^3 + 22 * q^4 + 6 * q^5 + 6 * q^6 + 14 * q^7 - 54 * q^8 + 36 * q^9 - 62 * q^10 - 40 * q^11 - 66 * q^12 + 40 * q^14 - 18 * q^15 + 122 * q^16 + 98 * q^17 - 18 * q^18 - 124 * q^19 + 466 * q^20 - 42 * q^21 + 220 * q^22 + 104 * q^23 + 162 * q^24 - 58 * q^25 - 108 * q^27 + 144 * q^28 + 194 * q^29 + 186 * q^30 - 26 * q^31 - 654 * q^32 + 120 * q^33 - 1062 * q^34 + 88 * q^35 + 198 * q^36 - 102 * q^37 + 332 * q^38 - 998 * q^40 + 1054 * q^41 - 120 * q^42 + 450 * q^43 + 44 * q^44 + 54 * q^45 + 172 * q^46 + 96 * q^47 - 366 * q^48 + 1070 * q^49 - 996 * q^50 - 294 * q^51 + 262 * q^53 + 54 * q^54 + 204 * q^55 + 2164 * q^56 + 372 * q^57 - 722 * q^58 - 308 * q^59 - 1398 * q^60 - 928 * q^61 + 2780 * q^62 + 126 * q^63 + 1026 * q^64 - 660 * q^66 + 1134 * q^67 + 1786 * q^68 - 312 * q^69 + 2324 * q^70 - 1064 * q^71 - 486 * q^72 - 952 * q^73 + 1158 * q^74 + 174 * q^75 + 1708 * q^76 + 2508 * q^77 - 746 * q^79 + 2922 * q^80 + 324 * q^81 + 1734 * q^82 + 404 * q^83 - 432 * q^84 + 1394 * q^85 - 3168 * q^86 - 582 * q^87 + 3060 * q^88 - 1620 * q^89 - 558 * q^90 + 332 * q^92 + 78 * q^93 - 772 * q^94 + 2204 * q^95 + 1962 * q^96 - 2166 * q^97 + 1906 * q^98 - 360 * q^99

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