LMFDB - Embedded newform 165.4.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [165,4,Mod(1,165)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(165, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("165.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 165 = 3 \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	165.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:	$9.73531515095$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.47528.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 26x - 22 $ x^3 - x^2 - 26*x - 22
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.906392$ of defining polynomial
Character	$\chi$	$=$	165.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.90639 q^{2} +3.00000 q^{3} -4.36567 q^{4} -5.00000 q^{5} -5.71918 q^{6} -22.9186 q^{7} +23.5738 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-1.90639 q^{2} +3.00000 q^{3} -4.36567 q^{4} -5.00000 q^{5} -5.71918 q^{6} -22.9186 q^{7} +23.5738 q^{8} +9.00000 q^{9} +9.53196 q^{10} +11.0000 q^{11} -13.0970 q^{12} +66.7313 q^{13} +43.6917 q^{14} -15.0000 q^{15} -10.0156 q^{16} -3.45588 q^{17} -17.1575 q^{18} +78.2359 q^{19} +21.8283 q^{20} -68.7557 q^{21} -20.9703 q^{22} +12.2907 q^{23} +70.7214 q^{24} +25.0000 q^{25} -127.216 q^{26} +27.0000 q^{27} +100.055 q^{28} -31.1827 q^{29} +28.5959 q^{30} +247.181 q^{31} -169.497 q^{32} +33.0000 q^{33} +6.58825 q^{34} +114.593 q^{35} -39.2910 q^{36} +304.128 q^{37} -149.148 q^{38} +200.194 q^{39} -117.869 q^{40} -29.8219 q^{41} +131.075 q^{42} +269.060 q^{43} -48.0224 q^{44} -45.0000 q^{45} -23.4309 q^{46} +225.463 q^{47} -30.0467 q^{48} +182.260 q^{49} -47.6598 q^{50} -10.3676 q^{51} -291.327 q^{52} -16.8371 q^{53} -51.4726 q^{54} -55.0000 q^{55} -540.278 q^{56} +234.708 q^{57} +59.4464 q^{58} +28.0701 q^{59} +65.4850 q^{60} -853.742 q^{61} -471.224 q^{62} -206.267 q^{63} +403.252 q^{64} -333.657 q^{65} -62.9109 q^{66} -36.7885 q^{67} +15.0872 q^{68} +36.8722 q^{69} -218.459 q^{70} +23.8552 q^{71} +212.164 q^{72} +707.265 q^{73} -579.787 q^{74} +75.0000 q^{75} -341.552 q^{76} -252.104 q^{77} -381.648 q^{78} -412.126 q^{79} +50.0778 q^{80} +81.0000 q^{81} +56.8522 q^{82} -552.596 q^{83} +300.165 q^{84} +17.2794 q^{85} -512.934 q^{86} -93.5480 q^{87} +259.312 q^{88} -1495.26 q^{89} +85.7876 q^{90} -1529.39 q^{91} -53.6573 q^{92} +741.543 q^{93} -429.820 q^{94} -391.179 q^{95} -508.491 q^{96} +1199.45 q^{97} -347.459 q^{98} +99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{2} + 9 q^{3} + 30 q^{4} - 15 q^{5} - 6 q^{6} + 10 q^{7} + 18 q^{8} + 27 q^{9}+O(q^{10}) $ 3 * q - 2 * q^2 + 9 * q^3 + 30 * q^4 - 15 * q^5 - 6 * q^6 + 10 * q^7 + 18 * q^8 + 27 * q^9	\( 3 q - 2 q^{2} + 9 q^{3} + 30 q^{4} - 15 q^{5} - 6 q^{6} + 10 q^{7} + 18 q^{8} + 27 q^{9} + 10 q^{10} + 33 q^{11} + 90 q^{12} + 114 q^{13} - 68 q^{14} - 45 q^{15} + 178 q^{16} - 104 q^{17} - 18 q^{18} - 58 q^{19} - 150 q^{20} + 30 q^{21} - 22 q^{22} + 120 q^{23} + 54 q^{24} + 75 q^{25} - 120 q^{26} + 81 q^{27} + 676 q^{28} - 220 q^{29} + 30 q^{30} + 248 q^{31} - 258 q^{32} + 99 q^{33} - 80 q^{34} - 50 q^{35} + 270 q^{36} + 838 q^{37} + 600 q^{38} + 342 q^{39} - 90 q^{40} + 156 q^{41} - 204 q^{42} + 122 q^{43} + 330 q^{44} - 135 q^{45} - 1256 q^{46} + 504 q^{47} + 534 q^{48} + 279 q^{49} - 50 q^{50} - 312 q^{51} + 520 q^{52} + 282 q^{53} - 54 q^{54} - 165 q^{55} - 1644 q^{56} - 174 q^{57} - 1644 q^{58} + 548 q^{59} - 450 q^{60} + 414 q^{61} - 2448 q^{62} + 90 q^{63} - 58 q^{64} - 570 q^{65} - 66 q^{66} - 428 q^{67} - 1704 q^{68} + 360 q^{69} + 340 q^{70} - 912 q^{71} + 162 q^{72} + 618 q^{73} - 1612 q^{74} + 225 q^{75} - 2752 q^{76} + 110 q^{77} - 360 q^{78} - 542 q^{79} - 890 q^{80} + 243 q^{81} + 3372 q^{82} + 2028 q^{84} + 520 q^{85} - 1548 q^{86} - 660 q^{87} + 198 q^{88} + 790 q^{89} + 90 q^{90} - 772 q^{91} + 1912 q^{92} + 744 q^{93} - 424 q^{94} + 290 q^{95} - 774 q^{96} + 2074 q^{97} - 3978 q^{98} + 297 q^{99}+O(q^{100}) \) 3 * q - 2 * q^2 + 9 * q^3 + 30 * q^4 - 15 * q^5 - 6 * q^6 + 10 * q^7 + 18 * q^8 + 27 * q^9 + 10 * q^10 + 33 * q^11 + 90 * q^12 + 114 * q^13 - 68 * q^14 - 45 * q^15 + 178 * q^16 - 104 * q^17 - 18 * q^18 - 58 * q^19 - 150 * q^20 + 30 * q^21 - 22 * q^22 + 120 * q^23 + 54 * q^24 + 75 * q^25 - 120 * q^26 + 81 * q^27 + 676 * q^28 - 220 * q^29 + 30 * q^30 + 248 * q^31 - 258 * q^32 + 99 * q^33 - 80 * q^34 - 50 * q^35 + 270 * q^36 + 838 * q^37 + 600 * q^38 + 342 * q^39 - 90 * q^40 + 156 * q^41 - 204 * q^42 + 122 * q^43 + 330 * q^44 - 135 * q^45 - 1256 * q^46 + 504 * q^47 + 534 * q^48 + 279 * q^49 - 50 * q^50 - 312 * q^51 + 520 * q^52 + 282 * q^53 - 54 * q^54 - 165 * q^55 - 1644 * q^56 - 174 * q^57 - 1644 * q^58 + 548 * q^59 - 450 * q^60 + 414 * q^61 - 2448 * q^62 + 90 * q^63 - 58 * q^64 - 570 * q^65 - 66 * q^66 - 428 * q^67 - 1704 * q^68 + 360 * q^69 + 340 * q^70 - 912 * q^71 + 162 * q^72 + 618 * q^73 - 1612 * q^74 + 225 * q^75 - 2752 * q^76 + 110 * q^77 - 360 * q^78 - 542 * q^79 - 890 * q^80 + 243 * q^81 + 3372 * q^82 + 2028 * q^84 + 520 * q^85 - 1548 * q^86 - 660 * q^87 + 198 * q^88 + 790 * q^89 + 90 * q^90 - 772 * q^91 + 1912 * q^92 + 744 * q^93 - 424 * q^94 + 290 * q^95 - 774 * q^96 + 2074 * q^97 - 3978 * q^98 + 297 * 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 $

$2$	−1.90639	−0.674011	−0.337006	−	0.941503i	$-0.609414\pi$
$2$	−1.90639	−0.674011	−0.337006	+	0.941503i	$0.609414\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−4.36567	−0.545709
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	−5.71918	−0.389141
$7$	−22.9186	−1.23749	−0.618743	−	0.785594i	$-0.712358\pi$
$7$	−22.9186	−1.23749	−0.618743	+	0.785594i	$0.712358\pi$
$8$	23.5738	1.04183
$9$	9.00000	0.333333
$10$	9.53196	0.301427
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	−13.0970	−0.315065
$13$	66.7313	1.42369	0.711844	−	0.702338i	$-0.247860\pi$
$13$	66.7313	1.42369	0.711844	+	0.702338i	$0.247860\pi$
$14$	43.6917	0.834079
$15$	−15.0000	−0.258199
$16$	−10.0156	−0.156493
$17$	−3.45588	−0.0493043	−0.0246522	−	0.999696i	$-0.507848\pi$
$17$	−3.45588	−0.0493043	−0.0246522	+	0.999696i	$0.507848\pi$
$18$	−17.1575	−0.224670
$19$	78.2359	0.944660	0.472330	−	0.881422i	$-0.343413\pi$
$19$	78.2359	0.944660	0.472330	+	0.881422i	$0.343413\pi$
$20$	21.8283	0.244048
$21$	−68.7557	−0.714463
$22$	−20.9703	−0.203222
$23$	12.2907	0.111426	0.0557129	−	0.998447i	$-0.482257\pi$
$23$	12.2907	0.111426	0.0557129	+	0.998447i	$0.482257\pi$
$24$	70.7214	0.601498
$25$	25.0000	0.200000
$26$	−127.216	−0.959582
$27$	27.0000	0.192450
$28$	100.055	0.675307
$29$	−31.1827	−0.199672	−0.0998358	−	0.995004i	$-0.531832\pi$
$29$	−31.1827	−0.199672	−0.0998358	+	0.995004i	$0.531832\pi$
$30$	28.5959	0.174029
$31$	247.181	1.43210	0.716049	−	0.698050i	$-0.245949\pi$
$31$	247.181	1.43210	0.716049	+	0.698050i	$0.245949\pi$
$32$	−169.497	−0.936347
$33$	33.0000	0.174078
$34$	6.58825	0.0332317
$35$	114.593	0.553420
$36$	−39.2910	−0.181903
$37$	304.128	1.35131	0.675653	−	0.737220i	$-0.263862\pi$
$37$	304.128	1.35131	0.675653	+	0.737220i	$0.263862\pi$
$38$	−149.148	−0.636712
$39$	200.194	0.821967
$40$	−117.869	−0.465918
$41$	−29.8219	−0.113595	−0.0567975	−	0.998386i	$-0.518089\pi$
$41$	−29.8219	−0.113595	−0.0567975	+	0.998386i	$0.518089\pi$
$42$	131.075	0.481556
$43$	269.060	0.954215	0.477108	−	0.878845i	$-0.341685\pi$
$43$	269.060	0.954215	0.477108	+	0.878845i	$0.341685\pi$
$44$	−48.0224	−0.164537
$45$	−45.0000	−0.149071
$46$	−23.4309	−0.0751023
$47$	225.463	0.699726	0.349863	−	0.936801i	$-0.386228\pi$
$47$	225.463	0.699726	0.349863	+	0.936801i	$0.386228\pi$
$48$	−30.0467	−0.0903514
$49$	182.260	0.531371
$50$	−47.6598	−0.134802
$51$	−10.3676	−0.0284659
$52$	−291.327	−0.776919
$53$	−16.8371	−0.0436369	−0.0218184	−	0.999762i	$-0.506946\pi$
$53$	−16.8371	−0.0436369	−0.0218184	+	0.999762i	$0.506946\pi$
$54$	−51.4726	−0.129714
$55$	−55.0000	−0.134840
$56$	−540.278	−1.28924
$57$	234.708	0.545400
$58$	59.4464	0.134581
$59$	28.0701	0.0619392	0.0309696	−	0.999520i	$-0.490140\pi$
$59$	28.0701	0.0619392	0.0309696	+	0.999520i	$0.490140\pi$
$60$	65.4850	0.140901
$61$	−853.742	−1.79198	−0.895988	−	0.444079i	$-0.853531\pi$
$61$	−853.742	−1.79198	−0.895988	+	0.444079i	$0.853531\pi$
$62$	−471.224	−0.965250
$63$	−206.267	−0.412495
$64$	403.252	0.787602
$65$	−333.657	−0.636693
$66$	−62.9109	−0.117330
$67$	−36.7885	−0.0670810	−0.0335405	−	0.999437i	$-0.510678\pi$
$67$	−36.7885	−0.0670810	−0.0335405	+	0.999437i	$0.510678\pi$
$68$	15.0872	0.0269058
$69$	36.8722	0.0643317
$70$	−218.459	−0.373012
$71$	23.8552	0.0398746	0.0199373	−	0.999801i	$-0.493653\pi$
$71$	23.8552	0.0398746	0.0199373	+	0.999801i	$0.493653\pi$
$72$	212.164	0.347275
$73$	707.265	1.13396	0.566980	−	0.823731i	$-0.308111\pi$
$73$	707.265	1.13396	0.566980	+	0.823731i	$0.308111\pi$
$74$	−579.787	−0.910795
$75$	75.0000	0.115470
$76$	−341.552	−0.515509
$77$	−252.104	−0.373116
$78$	−381.648	−0.554015
$79$	−412.126	−0.586934	−0.293467	−	0.955969i	$-0.594809\pi$
$79$	−412.126	−0.586934	−0.293467	+	0.955969i	$0.594809\pi$
$80$	50.0778	0.0699859
$81$	81.0000	0.111111
$82$	56.8522	0.0765643
$83$	−552.596	−0.730786	−0.365393	−	0.930853i	$-0.619065\pi$
$83$	−552.596	−0.730786	−0.365393	+	0.930853i	$0.619065\pi$
$84$	300.165	0.389889
$85$	17.2794	0.0220496
$86$	−512.934	−0.643152
$87$	−93.5480	−0.115280
$88$	259.312	0.314122
$89$	−1495.26	−1.78087	−0.890434	−	0.455113i	$-0.849599\pi$
$89$	−1495.26	−1.78087	−0.890434	+	0.455113i	$0.849599\pi$
$90$	85.7876	0.100476
$91$	−1529.39	−1.76179
$92$	−53.6573	−0.0608060
$93$	741.543	0.826822
$94$	−429.820	−0.471623
$95$	−391.179	−0.422465
$96$	−508.491	−0.540600
$97$	1199.45	1.25552	0.627761	−	0.778406i	$-0.283971\pi$
$97$	1199.45	1.25552	0.627761	+	0.778406i	$0.283971\pi$
$98$	−347.459	−0.358150
$99$	99.0000	0.100504
$100$	−109.142	−0.109142
$101$	−1009.66	−0.994701	−0.497351	−	0.867550i	$-0.665694\pi$
$101$	−1009.66	−0.994701	−0.497351	+	0.867550i	$0.665694\pi$
$102$	19.7648	0.0191863
$103$	1156.70	1.10653	0.553266	−	0.833004i	$-0.313381\pi$
$103$	1156.70	1.10653	0.553266	+	0.833004i	$0.313381\pi$
$104$	1573.11	1.48323
$105$	343.778	0.319517
$106$	32.0981	0.0294117
$107$	491.857	0.444389	0.222194	−	0.975002i	$-0.428678\pi$
$107$	491.857	0.444389	0.222194	+	0.975002i	$0.428678\pi$
$108$	−117.873	−0.105022
$109$	1340.77	1.17819	0.589093	−	0.808066i	$-0.299485\pi$
$109$	1340.77	1.17819	0.589093	+	0.808066i	$0.299485\pi$
$110$	104.852	0.0908837
$111$	912.384	0.780177
$112$	229.542	0.193658
$113$	1849.21	1.53946	0.769729	−	0.638371i	$-0.220391\pi$
$113$	1849.21	1.53946	0.769729	+	0.638371i	$0.220391\pi$
$114$	−447.445	−0.367606
$115$	−61.4536	−0.0498311
$116$	136.133	0.108963
$117$	600.582	0.474563
$118$	−53.5126	−0.0417478
$119$	79.2037	0.0610134
$120$	−353.607	−0.268998
$121$	121.000	0.0909091
$122$	1627.57	1.20781
$123$	−89.4656	−0.0655841
$124$	−1079.11	−0.781508
$125$	−125.000	−0.0894427
$126$	393.226	0.278026
$127$	−1020.24	−0.712850	−0.356425	−	0.934324i	$-0.616005\pi$
$127$	−1020.24	−0.712850	−0.356425	+	0.934324i	$0.616005\pi$
$128$	587.219	0.405495
$129$	807.180	0.550917
$130$	636.080	0.429138
$131$	1003.30	0.669147	0.334574	−	0.942370i	$-0.391408\pi$
$131$	1003.30	0.669147	0.334574	+	0.942370i	$0.391408\pi$
$132$	−144.067	−0.0949957
$133$	−1793.05	−1.16900
$134$	70.1332	0.0452134
$135$	−135.000	−0.0860663
$136$	−81.4682	−0.0513665
$137$	2665.14	1.66203	0.831016	−	0.556249i	$-0.187760\pi$
$137$	2665.14	1.66203	0.831016	+	0.556249i	$0.187760\pi$
$138$	−70.2928	−0.0433603
$139$	2557.64	1.56069	0.780347	−	0.625347i	$-0.215043\pi$
$139$	2557.64	1.56069	0.780347	+	0.625347i	$0.215043\pi$
$140$	−500.274	−0.302006
$141$	676.388	0.403987
$142$	−45.4774	−0.0268759
$143$	734.045	0.429258
$144$	−90.1401	−0.0521644
$145$	155.913	0.0892959
$146$	−1348.32	−0.764303
$147$	546.781	0.306787
$148$	−1327.72	−0.737419
$149$	−2644.66	−1.45409	−0.727044	−	0.686591i	$-0.759106\pi$
$149$	−2644.66	−1.45409	−0.727044	+	0.686591i	$0.759106\pi$
$150$	−142.979	−0.0778281
$151$	−2871.44	−1.54751	−0.773757	−	0.633482i	$-0.781625\pi$
$151$	−2871.44	−1.54751	−0.773757	+	0.633482i	$0.781625\pi$
$152$	1844.32	0.984171
$153$	−31.1029	−0.0164348
$154$	480.609	0.251484
$155$	−1235.91	−0.640454
$156$	−873.981	−0.448554
$157$	1048.51	0.532996	0.266498	−	0.963835i	$-0.414133\pi$
$157$	1048.51	0.532996	0.266498	+	0.963835i	$0.414133\pi$
$158$	785.674	0.395600
$159$	−50.5113	−0.0251937
$160$	847.485	0.418747
$161$	−281.686	−0.137888
$162$	−154.418	−0.0748901
$163$	−1953.90	−0.938905	−0.469452	−	0.882958i	$-0.655549\pi$
$163$	−1953.90	−0.938905	−0.469452	+	0.882958i	$0.655549\pi$
$164$	130.192	0.0619898
$165$	−165.000	−0.0778499
$166$	1053.46	0.492558
$167$	2085.79	0.966488	0.483244	−	0.875486i	$-0.339458\pi$
$167$	2085.79	0.966488	0.483244	+	0.875486i	$0.339458\pi$
$168$	−1620.83	−0.744345
$169$	2256.07	1.02689
$170$	−32.9413	−0.0148616
$171$	704.123	0.314887
$172$	−1174.63	−0.520724
$173$	−3999.28	−1.75757	−0.878785	−	0.477218i	$-0.841645\pi$
$173$	−3999.28	−1.75757	−0.878785	+	0.477218i	$0.841645\pi$
$174$	178.339	0.0777003
$175$	−572.964	−0.247497
$176$	−110.171	−0.0471845
$177$	84.2103	0.0357606
$178$	2850.55	1.20032
$179$	2046.60	0.854580	0.427290	−	0.904115i	$-0.359468\pi$
$179$	2046.60	0.854580	0.427290	+	0.904115i	$0.359468\pi$
$180$	196.455	0.0813495
$181$	64.2973	0.0264043	0.0132022	−	0.999913i	$-0.495797\pi$
$181$	64.2973	0.0264043	0.0132022	+	0.999913i	$0.495797\pi$
$182$	2915.61	1.18747
$183$	−2561.23	−1.03460
$184$	289.739	0.116086
$185$	−1520.64	−0.604322
$186$	−1413.67	−0.557287
$187$	−38.0146	−0.0148658
$188$	−984.296	−0.381846
$189$	−618.801	−0.238154
$190$	745.741	0.284746
$191$	4816.60	1.82470	0.912348	−	0.409416i	$-0.134268\pi$
$191$	4816.60	1.82470	0.912348	+	0.409416i	$0.134268\pi$
$192$	1209.76	0.454722
$193$	−295.804	−0.110323	−0.0551617	−	0.998477i	$-0.517567\pi$
$193$	−295.804	−0.110323	−0.0551617	+	0.998477i	$0.517567\pi$
$194$	−2286.62	−0.846237
$195$	−1000.97	−0.367595
$196$	−795.688	−0.289974
$197$	−2147.97	−0.776835	−0.388418	−	0.921483i	$-0.626978\pi$
$197$	−2147.97	−0.776835	−0.388418	+	0.921483i	$0.626978\pi$
$198$	−188.733	−0.0677407
$199$	−876.260	−0.312143	−0.156071	−	0.987746i	$-0.549883\pi$
$199$	−876.260	−0.312143	−0.156071	+	0.987746i	$0.549883\pi$
$200$	589.345	0.208365
$201$	−110.365	−0.0387292
$202$	1924.81	0.670440
$203$	714.662	0.247091
$204$	45.2616	0.0155341
$205$	149.109	0.0508012
$206$	−2205.12	−0.745815
$207$	110.617	0.0371419
$208$	−668.352	−0.222798
$209$	860.595	0.284826
$210$	−655.376	−0.215358
$211$	1413.99	0.461341	0.230670	−	0.973032i	$-0.425908\pi$
$211$	1413.99	0.461341	0.230670	+	0.973032i	$0.425908\pi$
$212$	73.5052	0.0238130
$213$	71.5657	0.0230216
$214$	−937.672	−0.299523
$215$	−1345.30	−0.426738
$216$	636.493	0.200499
$217$	−5665.03	−1.77220
$218$	−2556.03	−0.794110
$219$	2121.80	0.654693
$220$	240.112	0.0735834
$221$	−230.615	−0.0701939
$222$	−1739.36	−0.525848
$223$	3365.53	1.01064	0.505320	−	0.862932i	$-0.331374\pi$
$223$	3365.53	1.01064	0.505320	+	0.862932i	$0.331374\pi$
$224$	3884.62	1.15872
$225$	225.000	0.0666667
$226$	−3525.31	−1.03761
$227$	−5724.31	−1.67373	−0.836864	−	0.547411i	$-0.815613\pi$
$227$	−5724.31	−1.67373	−0.836864	+	0.547411i	$0.815613\pi$
$228$	−1024.66	−0.297629
$229$	2586.74	0.746447	0.373224	−	0.927741i	$-0.378252\pi$
$229$	2586.74	0.746447	0.373224	+	0.927741i	$0.378252\pi$
$230$	117.155	0.0335867
$231$	−756.312	−0.215419
$232$	−735.095	−0.208023
$233$	5571.63	1.56656	0.783282	−	0.621666i	$-0.213544\pi$
$233$	5571.63	1.56656	0.783282	+	0.621666i	$0.213544\pi$
$234$	−1144.94	−0.319861
$235$	−1127.31	−0.312927
$236$	−122.545	−0.0338008
$237$	−1236.38	−0.338867
$238$	−150.993	−0.0411237
$239$	−1822.92	−0.493369	−0.246685	−	0.969096i	$-0.579341\pi$
$239$	−1822.92	−0.493369	−0.246685	+	0.969096i	$0.579341\pi$
$240$	150.233	0.0404064
$241$	−2226.66	−0.595152	−0.297576	−	0.954698i	$-0.596178\pi$
$241$	−2226.66	−0.595152	−0.297576	+	0.954698i	$0.596178\pi$
$242$	−230.673	−0.0612738
$243$	243.000	0.0641500
$244$	3727.16	0.977897
$245$	−911.301	−0.237636
$246$	170.556	0.0442044
$247$	5220.78	1.34490
$248$	5827.00	1.49200
$249$	−1657.79	−0.421920
$250$	238.299	0.0602854
$251$	7888.22	1.98367	0.991833	−	0.127543i	$-0.0407092\pi$
$251$	7888.22	1.98367	0.991833	+	0.127543i	$0.0407092\pi$
$252$	900.494	0.225102
$253$	135.198	0.0335961
$254$	1944.98	0.480469
$255$	51.8381	0.0127303
$256$	−4345.49	−1.06091
$257$	−4755.32	−1.15420	−0.577099	−	0.816674i	$-0.695815\pi$
$257$	−4755.32	−1.15420	−0.577099	+	0.816674i	$0.695815\pi$
$258$	−1538.80	−0.371324
$259$	−6970.17	−1.67222
$260$	1456.64	0.347449
$261$	−280.644	−0.0665572
$262$	−1912.67	−0.451013
$263$	−103.437	−0.0242517	−0.0121258	−	0.999926i	$-0.503860\pi$
$263$	−103.437	−0.0242517	−0.0121258	+	0.999926i	$0.503860\pi$
$264$	777.936	0.181358
$265$	84.1855	0.0195150
$266$	3418.26	0.787921
$267$	−4485.78	−1.02818
$268$	160.606	0.0366067
$269$	−2179.50	−0.494002	−0.247001	−	0.969015i	$-0.579445\pi$
$269$	−2179.50	−0.494002	−0.247001	+	0.969015i	$0.579445\pi$
$270$	257.363	0.0580097
$271$	−3688.54	−0.826800	−0.413400	−	0.910550i	$-0.635659\pi$
$271$	−3688.54	−0.826800	−0.413400	+	0.910550i	$0.635659\pi$
$272$	34.6126	0.00771579
$273$	−4588.16	−1.01717
$274$	−5080.80	−1.12023
$275$	275.000	0.0603023
$276$	−160.972	−0.0351064
$277$	−3087.18	−0.669641	−0.334821	−	0.942282i	$-0.608676\pi$
$277$	−3087.18	−0.669641	−0.334821	+	0.942282i	$0.608676\pi$
$278$	−4875.87	−1.05193
$279$	2224.63	0.477366
$280$	2701.39	0.576567
$281$	3338.91	0.708836	0.354418	−	0.935087i	$-0.384679\pi$
$281$	3338.91	0.708836	0.354418	+	0.935087i	$0.384679\pi$
$282$	−1289.46	−0.272292
$283$	1483.75	0.311661	0.155830	−	0.987784i	$-0.450195\pi$
$283$	1483.75	0.311661	0.155830	+	0.987784i	$0.450195\pi$
$284$	−104.144	−0.0217599
$285$	−1173.54	−0.243910
$286$	−1399.38	−0.289325
$287$	683.474	0.140572
$288$	−1525.47	−0.312116
$289$	−4901.06	−0.997569
$290$	−297.232	−0.0601864
$291$	3598.35	0.724877
$292$	−3087.69	−0.618812
$293$	−1590.55	−0.317137	−0.158569	−	0.987348i	$-0.550688\pi$
$293$	−1590.55	−0.317137	−0.158569	+	0.987348i	$0.550688\pi$
$294$	−1042.38	−0.206778
$295$	−140.350	−0.0277001
$296$	7169.45	1.40782
$297$	297.000	0.0580259
$298$	5041.76	0.980071
$299$	820.177	0.158636
$300$	−327.425	−0.0630130
$301$	−6166.47	−1.18083
$302$	5474.10	1.04304
$303$	−3028.98	−0.574291
$304$	−783.577	−0.147833
$305$	4268.71	0.801396
$306$	59.2943	0.0110772
$307$	10175.3	1.89164	0.945818	−	0.324697i	$-0.105262\pi$
$307$	10175.3	1.89164	0.945818	+	0.324697i	$0.105262\pi$
$308$	1100.60	0.203613
$309$	3470.09	0.638857
$310$	2356.12	0.431673
$311$	−6258.24	−1.14107	−0.570535	−	0.821274i	$-0.693264\pi$
$311$	−6258.24	−1.14107	−0.570535	+	0.821274i	$0.693264\pi$
$312$	4719.34	0.856346
$313$	9592.24	1.73222	0.866111	−	0.499852i	$-0.166612\pi$
$313$	9592.24	1.73222	0.866111	+	0.499852i	$0.166612\pi$
$314$	−1998.88	−0.359245
$315$	1031.34	0.184473
$316$	1799.21	0.320295
$317$	−10632.3	−1.88381	−0.941907	−	0.335874i	$-0.890969\pi$
$317$	−10632.3	−1.88381	−0.941907	+	0.335874i	$0.890969\pi$
$318$	96.2943	0.0169809
$319$	−343.009	−0.0602033
$320$	−2016.26	−0.352226
$321$	1475.57	0.256568
$322$	537.003	0.0929380
$323$	−270.373	−0.0465758
$324$	−353.619	−0.0606343
$325$	1668.28	0.284738
$326$	3724.90	0.632832
$327$	4022.30	0.680226
$328$	−703.015	−0.118346
$329$	−5167.28	−0.865901
$330$	314.555	0.0524717
$331$	8222.61	1.36542	0.682712	−	0.730687i	$-0.260800\pi$
$331$	8222.61	1.36542	0.682712	+	0.730687i	$0.260800\pi$
$332$	2412.45	0.398796
$333$	2737.15	0.450435
$334$	−3976.34	−0.651424
$335$	183.942	0.0299995
$336$	688.627	0.111809
$337$	−2947.77	−0.476484	−0.238242	−	0.971206i	$-0.576571\pi$
$337$	−2947.77	−0.476484	−0.238242	+	0.971206i	$0.576571\pi$
$338$	−4300.96	−0.692134
$339$	5547.62	0.888806
$340$	−75.4361	−0.0120326
$341$	2718.99	0.431794
$342$	−1342.33	−0.212237
$343$	3683.92	0.579922
$344$	6342.77	0.994126
$345$	−184.361	−0.0287700
$346$	7624.20	1.18462
$347$	3322.43	0.513999	0.256999	−	0.966412i	$-0.417266\pi$
$347$	3322.43	0.513999	0.256999	+	0.966412i	$0.417266\pi$
$348$	408.400	0.0629096
$349$	−9199.67	−1.41102	−0.705511	−	0.708699i	$-0.749283\pi$
$349$	−9199.67	−1.41102	−0.705511	+	0.708699i	$0.749283\pi$
$350$	1092.29	0.166816
$351$	1801.75	0.273989
$352$	−1864.47	−0.282319
$353$	−10105.2	−1.52363	−0.761817	−	0.647792i	$-0.775693\pi$
$353$	−10105.2	−1.52363	−0.761817	+	0.647792i	$0.775693\pi$
$354$	−160.538	−0.0241031
$355$	−119.276	−0.0178325
$356$	6527.81	0.971835
$357$	237.611	0.0352261
$358$	−3901.62	−0.575997
$359$	−5236.42	−0.769826	−0.384913	−	0.922953i	$-0.625769\pi$
$359$	−5236.42	−0.769826	−0.384913	+	0.922953i	$0.625769\pi$
$360$	−1060.82	−0.155306
$361$	−738.148	−0.107617
$362$	−122.576	−0.0177968
$363$	363.000	0.0524864
$364$	6676.79	0.961426
$365$	−3536.33	−0.507123
$366$	4882.70	0.697330
$367$	−9337.95	−1.32817	−0.664083	−	0.747659i	$-0.731178\pi$
$367$	−9337.95	−1.32817	−0.664083	+	0.747659i	$0.731178\pi$
$368$	−123.099	−0.0174374
$369$	−268.397	−0.0378650
$370$	2898.93	0.407320
$371$	385.882	0.0540000
$372$	−3237.33	−0.451204
$373$	14256.1	1.97896	0.989482	−	0.144659i	$-0.0462086\pi$
$373$	14256.1	1.97896	0.989482	+	0.144659i	$0.0462086\pi$
$374$	72.4708	0.0100197
$375$	−375.000	−0.0516398
$376$	5315.02	0.728992
$377$	−2080.86	−0.284270
$378$	1179.68	0.160519
$379$	1911.19	0.259027	0.129514	−	0.991578i	$-0.458658\pi$
$379$	1911.19	0.259027	0.129514	+	0.991578i	$0.458658\pi$
$380$	1707.76	0.230543
$381$	−3060.73	−0.411564
$382$	−9182.32	−1.22987
$383$	1743.91	0.232662	0.116331	−	0.993210i	$-0.462887\pi$
$383$	1743.91	0.232662	0.116331	+	0.993210i	$0.462887\pi$
$384$	1761.66	0.234112
$385$	1260.52	0.166863
$386$	563.917	0.0743592
$387$	2421.54	0.318072
$388$	−5236.40	−0.685150
$389$	−2734.88	−0.356462	−0.178231	−	0.983989i	$-0.557037\pi$
$389$	−2734.88	−0.356462	−0.178231	+	0.983989i	$0.557037\pi$
$390$	1908.24	0.247763
$391$	−42.4752	−0.00549377
$392$	4296.57	0.553596
$393$	3009.89	0.386332
$394$	4094.87	0.523596
$395$	2060.63	0.262485
$396$	−432.201	−0.0548458
$397$	−14879.5	−1.88106	−0.940530	−	0.339709i	$-0.889671\pi$
$397$	−14879.5	−1.88106	−0.940530	+	0.339709i	$0.889671\pi$
$398$	1670.50	0.210388
$399$	−5379.16	−0.674924
$400$	−250.389	−0.0312986
$401$	−11638.2	−1.44934	−0.724668	−	0.689098i	$-0.758007\pi$
$401$	−11638.2	−1.44934	−0.724668	+	0.689098i	$0.758007\pi$
$402$	210.400	0.0261039
$403$	16494.7	2.03886
$404$	4407.84	0.542817
$405$	−405.000	−0.0496904
$406$	−1362.43	−0.166542
$407$	3345.41	0.407434
$408$	−244.405	−0.0296564
$409$	936.594	0.113231	0.0566157	−	0.998396i	$-0.481969\pi$
$409$	936.594	0.113231	0.0566157	+	0.998396i	$0.481969\pi$
$410$	−284.261	−0.0342406
$411$	7995.42	0.959574
$412$	−5049.76	−0.603845
$413$	−643.326	−0.0766489
$414$	−210.878	−0.0250341
$415$	2762.98	0.326818
$416$	−11310.8	−1.33307
$417$	7672.93	0.901067
$418$	−1640.63	−0.191976
$419$	128.037	0.0149285	0.00746425	−	0.999972i	$-0.497624\pi$
$419$	128.037	0.0149285	0.00746425	+	0.999972i	$0.497624\pi$
$420$	−1500.82	−0.174363
$421$	−6628.97	−0.767402	−0.383701	−	0.923457i	$-0.625351\pi$
$421$	−6628.97	−0.767402	−0.383701	+	0.923457i	$0.625351\pi$
$422$	−2695.61	−0.310949
$423$	2029.16	0.233242
$424$	−396.915	−0.0454620
$425$	−86.3969	−0.00986086
$426$	−136.432	−0.0155168
$427$	19566.5	2.21754
$428$	−2147.29	−0.242507
$429$	2202.13	0.247832
$430$	2564.67	0.287626
$431$	14677.8	1.64038	0.820192	−	0.572089i	$-0.193867\pi$
$431$	14677.8	1.64038	0.820192	+	0.572089i	$0.193867\pi$
$432$	−270.420	−0.0301171
$433$	−1150.36	−0.127674	−0.0638369	−	0.997960i	$-0.520334\pi$
$433$	−1150.36	−0.127674	−0.0638369	+	0.997960i	$0.520334\pi$
$434$	10799.8	1.19448
$435$	467.740	0.0515550
$436$	−5853.35	−0.642946
$437$	961.576	0.105259
$438$	−4044.97	−0.441270
$439$	−9308.30	−1.01198	−0.505992	−	0.862538i	$-0.668873\pi$
$439$	−9308.30	−1.01198	−0.505992	+	0.862538i	$0.668873\pi$
$440$	−1296.56	−0.140480
$441$	1640.34	0.177124
$442$	439.643	0.0473115
$443$	−5689.21	−0.610164	−0.305082	−	0.952326i	$-0.598684\pi$
$443$	−5689.21	−0.610164	−0.305082	+	0.952326i	$0.598684\pi$
$444$	−3983.17	−0.425749
$445$	7476.30	0.796428
$446$	−6416.02	−0.681183
$447$	−7933.98	−0.839518
$448$	−9241.96	−0.974646
$449$	−6644.91	−0.698425	−0.349212	−	0.937044i	$-0.613551\pi$
$449$	−6644.91	−0.698425	−0.349212	+	0.937044i	$0.613551\pi$
$450$	−428.938	−0.0449341
$451$	−328.041	−0.0342502
$452$	−8073.02	−0.840095
$453$	−8614.33	−0.893458
$454$	10912.8	1.12811
$455$	7646.93	0.787898
$456$	5532.95	0.568211
$457$	−8355.11	−0.855220	−0.427610	−	0.903963i	$-0.640644\pi$
$457$	−8355.11	−0.855220	−0.427610	+	0.903963i	$0.640644\pi$
$458$	−4931.33	−0.503114
$459$	−93.3087	−0.00948862
$460$	268.286	0.0271933
$461$	−15580.5	−1.57409	−0.787044	−	0.616897i	$-0.788389\pi$
$461$	−15580.5	−1.57409	−0.787044	+	0.616897i	$0.788389\pi$
$462$	1441.83	0.145195
$463$	7139.16	0.716598	0.358299	−	0.933607i	$-0.383357\pi$
$463$	7139.16	0.716598	0.358299	+	0.933607i	$0.383357\pi$
$464$	312.312	0.0312473
$465$	−3707.72	−0.369766
$466$	−10621.7	−1.05588
$467$	−12415.1	−1.23019	−0.615097	−	0.788451i	$-0.710883\pi$
$467$	−12415.1	−1.23019	−0.615097	+	0.788451i	$0.710883\pi$
$468$	−2621.94	−0.258973
$469$	843.139	0.0830118
$470$	2149.10	0.210916
$471$	3145.54	0.307726
$472$	661.719	0.0645299
$473$	2959.66	0.287707
$474$	2357.02	0.228400
$475$	1955.90	0.188932
$476$	−345.777	−0.0332955
$477$	−151.534	−0.0145456
$478$	3475.21	0.332536
$479$	−14151.0	−1.34985	−0.674925	−	0.737887i	$-0.735824\pi$
$479$	−14151.0	−1.34985	−0.674925	+	0.737887i	$0.735824\pi$
$480$	2542.45	0.241764
$481$	20294.9	1.92384
$482$	4244.88	0.401139
$483$	−845.057	−0.0796096
$484$	−528.246	−0.0496099
$485$	−5997.25	−0.561487
$486$	−463.253	−0.0432378
$487$	−76.6358	−0.00713080	−0.00356540	−	0.999994i	$-0.501135\pi$
$487$	−76.6358	−0.00713080	−0.00356540	+	0.999994i	$0.501135\pi$
$488$	−20126.0	−1.86693
$489$	−5861.71	−0.542077
$490$	1737.30	0.160170
$491$	−1707.54	−0.156945	−0.0784725	−	0.996916i	$-0.525004\pi$
$491$	−1707.54	−0.156945	−0.0784725	+	0.996916i	$0.525004\pi$
$492$	390.577	0.0357898
$493$	107.763	0.00984467
$494$	−9952.86	−0.906479
$495$	−495.000	−0.0449467
$496$	−2475.66	−0.224114
$497$	−546.728	−0.0493443
$498$	3160.39	0.284379
$499$	−13168.5	−1.18137	−0.590686	−	0.806902i	$-0.701143\pi$
$499$	−13168.5	−1.18137	−0.590686	+	0.806902i	$0.701143\pi$
$500$	545.709	0.0488097
$501$	6257.38	0.558002
$502$	−15038.0	−1.33701
$503$	−1525.32	−0.135210	−0.0676052	−	0.997712i	$-0.521536\pi$
$503$	−1525.32	−0.135210	−0.0676052	+	0.997712i	$0.521536\pi$
$504$	−4862.50	−0.429748
$505$	5048.29	0.444844
$506$	−257.740	−0.0226442
$507$	6768.22	0.592874
$508$	4454.05	0.389009
$509$	9006.01	0.784253	0.392126	−	0.919911i	$-0.371740\pi$
$509$	9006.01	0.784253	0.392126	+	0.919911i	$0.371740\pi$
$510$	−98.8238	−0.00858038
$511$	−16209.5	−1.40326
$512$	3586.45	0.309571
$513$	2112.37	0.181800
$514$	9065.51	0.777942
$515$	−5783.49	−0.494856
$516$	−3523.88	−0.300640
$517$	2480.09	0.210975
$518$	13287.9	1.12710
$519$	−11997.8	−1.01473
$520$	−7865.56	−0.663322
$521$	21707.8	1.82541	0.912703	−	0.408623i	$-0.133991\pi$
$521$	21707.8	1.82541	0.912703	+	0.408623i	$0.133991\pi$
$522$	535.018	0.0448603
$523$	−14015.0	−1.17177	−0.585883	−	0.810396i	$-0.699252\pi$
$523$	−14015.0	−1.17177	−0.585883	+	0.810396i	$0.699252\pi$
$524$	−4380.06	−0.365160
$525$	−1718.89	−0.142893
$526$	197.191	0.0163459
$527$	−854.227	−0.0706086
$528$	−330.514	−0.0272420
$529$	−12015.9	−0.987584
$530$	−160.491	−0.0131533
$531$	252.631	0.0206464
$532$	7827.88	0.637935
$533$	−1990.05	−0.161724
$534$	8551.65	0.693008
$535$	−2459.29	−0.198737
$536$	−867.244	−0.0698867
$537$	6139.79	0.493392
$538$	4154.98	0.332963
$539$	2004.86	0.160214
$540$	589.365	0.0469671
$541$	14444.9	1.14793	0.573967	−	0.818878i	$-0.305404\pi$
$541$	14444.9	1.14793	0.573967	+	0.818878i	$0.305404\pi$
$542$	7031.80	0.557273
$543$	192.892	0.0152445
$544$	585.760	0.0461659
$545$	−6703.83	−0.526900
$546$	8746.83	0.685585
$547$	20286.4	1.58571	0.792854	−	0.609412i	$-0.208594\pi$
$547$	20286.4	1.58571	0.792854	+	0.609412i	$0.208594\pi$
$548$	−11635.1	−0.906985
$549$	−7683.68	−0.597325
$550$	−524.258	−0.0406444
$551$	−2439.60	−0.188622
$552$	869.218	0.0670224
$553$	9445.34	0.726323
$554$	5885.38	0.451346
$555$	−4561.92	−0.348906
$556$	−11165.8	−0.851684
$557$	−24482.4	−1.86239	−0.931196	−	0.364520i	$-0.881233\pi$
$557$	−24482.4	−1.86239	−0.931196	+	0.364520i	$0.881233\pi$
$558$	−4241.02	−0.321750
$559$	17954.7	1.35851
$560$	−1147.71	−0.0866065
$561$	−114.044	−0.00858278
$562$	−6365.27	−0.477763
$563$	−20383.5	−1.52587	−0.762933	−	0.646478i	$-0.776241\pi$
$563$	−20383.5	−1.52587	−0.762933	+	0.646478i	$0.776241\pi$
$564$	−2952.89	−0.220459
$565$	−9246.03	−0.688466
$566$	−2828.62	−0.210063
$567$	−1856.40	−0.137498
$568$	562.359	0.0415424
$569$	3297.74	0.242967	0.121484	−	0.992593i	$-0.461235\pi$
$569$	3297.74	0.242967	0.121484	+	0.992593i	$0.461235\pi$
$570$	2237.22	0.164398
$571$	−7213.03	−0.528644	−0.264322	−	0.964434i	$-0.585148\pi$
$571$	−7213.03	−0.528644	−0.264322	+	0.964434i	$0.585148\pi$
$572$	−3204.60	−0.234250
$573$	14449.8	1.05349
$574$	−1302.97	−0.0947472
$575$	307.268	0.0222852
$576$	3629.27	0.262534
$577$	−18080.4	−1.30450	−0.652251	−	0.758003i	$-0.726175\pi$
$577$	−18080.4	−1.30450	−0.652251	+	0.758003i	$0.726175\pi$
$578$	9343.34	0.672373
$579$	−887.411	−0.0636952
$580$	−680.666	−0.0487295
$581$	12664.7	0.904337
$582$	−6859.87	−0.488575
$583$	−185.208	−0.0131570
$584$	16672.9	1.18139
$585$	−3002.91	−0.212231
$586$	3032.22	0.213754
$587$	13457.5	0.946250	0.473125	−	0.880995i	$-0.343126\pi$
$587$	13457.5	0.946250	0.473125	+	0.880995i	$0.343126\pi$
$588$	−2387.06	−0.167416
$589$	19338.4	1.35285
$590$	267.563	0.0186702
$591$	−6443.91	−0.448506
$592$	−3046.01	−0.211470
$593$	−10262.3	−0.710665	−0.355332	−	0.934740i	$-0.615632\pi$
$593$	−10262.3	−0.710665	−0.355332	+	0.934740i	$0.615632\pi$
$594$	−566.198	−0.0391101
$595$	−396.018	−0.0272860
$596$	11545.7	0.793508
$597$	−2628.78	−0.180216
$598$	−1563.58	−0.106922
$599$	23153.3	1.57933	0.789665	−	0.613538i	$-0.210254\pi$
$599$	23153.3	1.57933	0.789665	+	0.613538i	$0.210254\pi$
$600$	1768.04	0.120300
$601$	14414.8	0.978357	0.489178	−	0.872184i	$-0.337297\pi$
$601$	14414.8	0.978357	0.489178	+	0.872184i	$0.337297\pi$
$602$	11755.7	0.795891
$603$	−331.096	−0.0223603
$604$	12535.8	0.844492
$605$	−605.000	−0.0406558
$606$	5774.42	0.387079
$607$	10808.8	0.722757	0.361379	−	0.932419i	$-0.382306\pi$
$607$	10808.8	0.722757	0.361379	+	0.932419i	$0.382306\pi$
$608$	−13260.7	−0.884530
$609$	2143.99	0.142658
$610$	−8137.84	−0.540150
$611$	15045.4	0.996191
$612$	135.785	0.00896860
$613$	12072.8	0.795455	0.397727	−	0.917504i	$-0.369799\pi$
$613$	12072.8	0.795455	0.397727	+	0.917504i	$0.369799\pi$
$614$	−19398.0	−1.27498
$615$	447.328	0.0293301
$616$	−5943.06	−0.388722
$617$	11593.5	0.756462	0.378231	−	0.925711i	$-0.376532\pi$
$617$	11593.5	0.756462	0.378231	+	0.925711i	$0.376532\pi$
$618$	−6615.36	−0.430597
$619$	4037.96	0.262196	0.131098	−	0.991369i	$-0.458150\pi$
$619$	4037.96	0.262196	0.131098	+	0.991369i	$0.458150\pi$
$620$	5395.55	0.349501
$621$	331.850	0.0214439
$622$	11930.7	0.769094
$623$	34269.2	2.20380
$624$	−2005.06	−0.128632
$625$	625.000	0.0400000
$626$	−18286.6	−1.16754
$627$	2581.78	0.164444
$628$	−4577.46	−0.290861
$629$	−1051.03	−0.0666252
$630$	−1966.13	−0.124337
$631$	2896.22	0.182721	0.0913604	−	0.995818i	$-0.470878\pi$
$631$	2896.22	0.182721	0.0913604	+	0.995818i	$0.470878\pi$
$632$	−9715.39	−0.611483
$633$	4241.96	0.266355
$634$	20269.3	1.26971
$635$	5101.22	0.318796
$636$	220.516	0.0137484
$637$	12162.5	0.756506
$638$	653.910	0.0405777
$639$	214.697	0.0132915
$640$	−2936.09	−0.181343
$641$	13371.2	0.823916	0.411958	−	0.911203i	$-0.364845\pi$
$641$	13371.2	0.823916	0.411958	+	0.911203i	$0.364845\pi$
$642$	−2813.02	−0.172930
$643$	14063.6	0.862544	0.431272	−	0.902222i	$-0.358065\pi$
$643$	14063.6	0.862544	0.431272	+	0.902222i	$0.358065\pi$
$644$	1229.75	0.0752466
$645$	−4035.90	−0.246377
$646$	515.438	0.0313926
$647$	20899.7	1.26994	0.634971	−	0.772536i	$-0.281012\pi$
$647$	20899.7	1.26994	0.634971	+	0.772536i	$0.281012\pi$
$648$	1909.48	0.115758
$649$	308.771	0.0186754
$650$	−3180.40	−0.191916
$651$	−16995.1	−1.02318
$652$	8530.09	0.512368
$653$	−1607.80	−0.0963523	−0.0481761	−	0.998839i	$-0.515341\pi$
$653$	−1607.80	−0.0963523	−0.0481761	+	0.998839i	$0.515341\pi$
$654$	−7668.08	−0.458480
$655$	−5016.48	−0.299252
$656$	298.683	0.0177768
$657$	6365.39	0.377987
$658$	9850.86	0.583627
$659$	14733.1	0.870896	0.435448	−	0.900214i	$-0.356590\pi$
$659$	14733.1	0.870896	0.435448	+	0.900214i	$0.356590\pi$
$660$	720.336	0.0424834
$661$	−16442.2	−0.967518	−0.483759	−	0.875201i	$-0.660729\pi$
$661$	−16442.2	−0.967518	−0.483759	+	0.875201i	$0.660729\pi$
$662$	−15675.5	−0.920312
$663$	−691.846	−0.0405265
$664$	−13026.8	−0.761351
$665$	8965.27	0.522794
$666$	−5218.08	−0.303598
$667$	−383.258	−0.0222486
$668$	−9105.88	−0.527421
$669$	10096.6	0.583493
$670$	−350.666	−0.0202200
$671$	−9391.16	−0.540301
$672$	11653.9	0.668985
$673$	−25246.6	−1.44604	−0.723021	−	0.690826i	$-0.757247\pi$
$673$	−25246.6	−1.44604	−0.723021	+	0.690826i	$0.757247\pi$
$674$	5619.60	0.321156
$675$	675.000	0.0384900
$676$	−9849.26	−0.560381
$677$	24582.6	1.39555	0.697774	−	0.716318i	$-0.254174\pi$
$677$	24582.6	1.39555	0.697774	+	0.716318i	$0.254174\pi$
$678$	−10575.9	−0.599065
$679$	−27489.7	−1.55369
$680$	407.341	0.0229718
$681$	−17172.9	−0.966327
$682$	−5183.46	−0.291034
$683$	11459.5	0.642000	0.321000	−	0.947079i	$-0.395981\pi$
$683$	11459.5	0.642000	0.321000	+	0.947079i	$0.395981\pi$
$684$	−3073.97	−0.171836
$685$	−13325.7	−0.743283
$686$	−7023.00	−0.390874
$687$	7760.21	0.430961
$688$	−2694.79	−0.149328
$689$	−1123.56	−0.0621253
$690$	351.464	0.0193913
$691$	−15683.8	−0.863445	−0.431723	−	0.902006i	$-0.642094\pi$
$691$	−15683.8	−0.863445	−0.431723	+	0.902006i	$0.642094\pi$
$692$	17459.5	0.959121
$693$	−2268.94	−0.124372
$694$	−6333.86	−0.346441
$695$	−12788.2	−0.697963
$696$	−2205.28	−0.120102
$697$	103.061	0.00560072
$698$	17538.2	0.951045
$699$	16714.9	0.904457
$700$	2501.37	0.135061
$701$	6185.65	0.333279	0.166640	−	0.986018i	$-0.446708\pi$
$701$	6185.65	0.333279	0.166640	+	0.986018i	$0.446708\pi$
$702$	−3434.83	−0.184672
$703$	23793.7	1.27652
$704$	4435.77	0.237471
$705$	−3381.94	−0.180668
$706$	19264.4	1.02695
$707$	23139.9	1.23093
$708$	−367.634	−0.0195149
$709$	−28031.2	−1.48481	−0.742407	−	0.669950i	$-0.766316\pi$
$709$	−28031.2	−1.48481	−0.742407	+	0.669950i	$0.766316\pi$
$710$	227.387	0.0120193
$711$	−3709.14	−0.195645
$712$	−35249.0	−1.85535
$713$	3038.03	0.159573
$714$	−452.980	−0.0237428
$715$	−3670.22	−0.191970
$716$	−8934.77	−0.466352
$717$	−5468.77	−0.284847
$718$	9982.67	0.518872
$719$	23313.0	1.20922	0.604610	−	0.796522i	$-0.293329\pi$
$719$	23313.0	1.20922	0.604610	+	0.796522i	$0.293329\pi$
$720$	450.700	0.0233286
$721$	−26509.9	−1.36932
$722$	1407.20	0.0725354
$723$	−6679.98	−0.343611
$724$	−280.701	−0.0144091
$725$	−779.567	−0.0399343
$726$	−692.020	−0.0353764
$727$	−69.6810	−0.00355478	−0.00177739	−	0.999998i	$-0.500566\pi$
$727$	−69.6810	−0.00355478	−0.00177739	+	0.999998i	$0.500566\pi$
$728$	−36053.5	−1.83548
$729$	729.000	0.0370370
$730$	6741.62	0.341806
$731$	−929.838	−0.0470469
$732$	11181.5	0.564589
$733$	15152.9	0.763552	0.381776	−	0.924255i	$-0.375313\pi$
$733$	15152.9	0.763552	0.381776	+	0.924255i	$0.375313\pi$
$734$	17801.8	0.895199
$735$	−2733.90	−0.137199
$736$	−2083.24	−0.104333
$737$	−404.673	−0.0202257
$738$	511.669	0.0255214
$739$	−31557.9	−1.57087	−0.785437	−	0.618941i	$-0.787562\pi$
$739$	−31557.9	−1.57087	−0.785437	+	0.618941i	$0.787562\pi$
$740$	6638.61	0.329784
$741$	15662.4	0.776479
$742$	−735.642	−0.0363966
$743$	6925.00	0.341929	0.170965	−	0.985277i	$-0.445312\pi$
$743$	6925.00	0.341929	0.170965	+	0.985277i	$0.445312\pi$
$744$	17481.0	0.861404
$745$	13223.3	0.650288
$746$	−27177.7	−1.33384
$747$	−4973.36	−0.243595
$748$	165.959	0.00811240
$749$	−11272.7	−0.549925
$750$	714.897	0.0348058
$751$	−22202.1	−1.07878	−0.539392	−	0.842055i	$-0.681346\pi$
$751$	−22202.1	−1.07878	−0.539392	+	0.842055i	$0.681346\pi$
$752$	−2258.14	−0.109502
$753$	23664.7	1.14527
$754$	3966.94	0.191601
$755$	14357.2	0.692070
$756$	2701.48	0.129963
$757$	18928.6	0.908813	0.454406	−	0.890795i	$-0.349851\pi$
$757$	18928.6	0.908813	0.454406	+	0.890795i	$0.349851\pi$
$758$	−3643.48	−0.174587
$759$	405.594	0.0193967
$760$	−9221.59	−0.440134
$761$	−22883.3	−1.09004	−0.545019	−	0.838423i	$-0.683478\pi$
$761$	−22883.3	−1.09004	−0.545019	+	0.838423i	$0.683478\pi$
$762$	5834.95	0.277399
$763$	−30728.4	−1.45799
$764$	−21027.7	−0.995752
$765$	155.514	0.00734985
$766$	−3324.57	−0.156817
$767$	1873.16	0.0881822
$768$	−13036.5	−0.612516
$769$	8903.82	0.417529	0.208765	−	0.977966i	$-0.433056\pi$
$769$	8903.82	0.417529	0.208765	+	0.977966i	$0.433056\pi$
$770$	−2403.05	−0.112467
$771$	−14266.0	−0.666376
$772$	1291.38	0.0602044
$773$	24691.9	1.14891	0.574454	−	0.818537i	$-0.305214\pi$
$773$	24691.9	1.14891	0.574454	+	0.818537i	$0.305214\pi$
$774$	−4616.40	−0.214384
$775$	6179.53	0.286420
$776$	28275.6	1.30804
$777$	−20910.5	−0.965457
$778$	5213.75	0.240259
$779$	−2333.14	−0.107309
$780$	4369.91	0.200600
$781$	262.408	0.0120226
$782$	80.9744	0.00370286
$783$	−841.932	−0.0384268
$784$	−1825.44	−0.0831559
$785$	−5242.56	−0.238363
$786$	−5738.02	−0.260392
$787$	−7418.83	−0.336026	−0.168013	−	0.985785i	$-0.553735\pi$
$787$	−7418.83	−0.336026	−0.168013	+	0.985785i	$0.553735\pi$
$788$	9377.33	0.423926
$789$	−310.311	−0.0140017
$790$	−3928.37	−0.176918
$791$	−42381.1	−1.90506
$792$	2333.81	0.104707
$793$	−56971.4	−2.55121
$794$	28366.2	1.26786
$795$	252.556	0.0112670
$796$	3825.46	0.170339
$797$	−27971.2	−1.24315	−0.621576	−	0.783354i	$-0.713507\pi$
$797$	−27971.2	−1.24315	−0.621576	+	0.783354i	$0.713507\pi$
$798$	10254.8	0.454907
$799$	−779.171	−0.0344995
$800$	−4237.42	−0.187269
$801$	−13457.3	−0.593622
$802$	22187.0	0.976869
$803$	7779.92	0.341902
$804$	481.819	0.0211349
$805$	1408.43	0.0616653
$806$	−31445.4	−1.37421
$807$	−6538.51	−0.285212
$808$	−23801.5	−1.03630
$809$	−10670.0	−0.463704	−0.231852	−	0.972751i	$-0.574479\pi$
$809$	−10670.0	−0.463704	−0.231852	+	0.972751i	$0.574479\pi$
$810$	772.089	0.0334919
$811$	40618.3	1.75869	0.879346	−	0.476182i	$-0.157980\pi$
$811$	40618.3	1.75869	0.879346	+	0.476182i	$0.157980\pi$
$812$	−3119.98	−0.134840
$813$	−11065.6	−0.477353
$814$	−6377.66	−0.274615
$815$	9769.51	0.419891
$816$	103.838	0.00445471
$817$	21050.1	0.901409
$818$	−1785.52	−0.0763192
$819$	−13764.5	−0.587265
$820$	−650.962	−0.0277227
$821$	16710.5	0.710355	0.355178	−	0.934799i	$-0.384420\pi$
$821$	16710.5	0.710355	0.355178	+	0.934799i	$0.384420\pi$
$822$	−15242.4	−0.646764
$823$	−16920.5	−0.716661	−0.358331	−	0.933595i	$-0.616654\pi$
$823$	−16920.5	−0.716661	−0.358331	+	0.933595i	$0.616654\pi$
$824$	27267.8	1.15281
$825$	825.000	0.0348155
$826$	1226.43	0.0516623
$827$	−9478.39	−0.398544	−0.199272	−	0.979944i	$-0.563858\pi$
$827$	−9478.39	−0.398544	−0.199272	+	0.979944i	$0.563858\pi$
$828$	−482.915	−0.0202687
$829$	−31908.1	−1.33681	−0.668405	−	0.743798i	$-0.733023\pi$
$829$	−31908.1	−1.33681	−0.668405	+	0.743798i	$0.733023\pi$
$830$	−5267.32	−0.220279
$831$	−9261.54	−0.386618
$832$	26909.6	1.12130
$833$	−629.869	−0.0261989
$834$	−14627.6	−0.607329
$835$	−10429.0	−0.432226
$836$	−3757.07	−0.155432
$837$	6673.89	0.275607
$838$	−244.090	−0.0100620
$839$	25195.7	1.03677	0.518386	−	0.855147i	$-0.326533\pi$
$839$	25195.7	1.03677	0.518386	+	0.855147i	$0.326533\pi$
$840$	8104.17	0.332881
$841$	−23416.6	−0.960131
$842$	12637.4	0.517238
$843$	10016.7	0.409246
$844$	−6173.00	−0.251758
$845$	−11280.4	−0.459238
$846$	−3868.38	−0.157208
$847$	−2773.15	−0.112499
$848$	168.633	0.00682887
$849$	4451.26	0.179938
$850$	164.706	0.00664633
$851$	3737.95	0.150570
$852$	−312.432	−0.0125631
$853$	8239.19	0.330720	0.165360	−	0.986233i	$-0.447121\pi$
$853$	8239.19	0.330720	0.165360	+	0.986233i	$0.447121\pi$
$854$	−37301.5	−1.49465
$855$	−3520.61	−0.140822
$856$	11594.9	0.462976
$857$	12912.8	0.514694	0.257347	−	0.966319i	$-0.417152\pi$
$857$	12912.8	0.514694	0.257347	+	0.966319i	$0.417152\pi$
$858$	−4198.13	−0.167042
$859$	18534.6	0.736196	0.368098	−	0.929787i	$-0.380009\pi$
$859$	18534.6	0.736196	0.368098	+	0.929787i	$0.380009\pi$
$860$	5873.14	0.232875
$861$	2050.42	0.0811594
$862$	−27981.7	−1.10564
$863$	−18743.4	−0.739318	−0.369659	−	0.929168i	$-0.620525\pi$
$863$	−18743.4	−0.739318	−0.369659	+	0.929168i	$0.620525\pi$
$864$	−4576.42	−0.180200
$865$	19996.4	0.786009
$866$	2193.04	0.0860536
$867$	−14703.2	−0.575947
$868$	24731.7	0.967105
$869$	−4533.39	−0.176967
$870$	−891.696	−0.0347487
$871$	−2454.94	−0.0955024
$872$	31607.0	1.22746
$873$	10795.1	0.418508
$874$	−1833.14	−0.0709461
$875$	2864.82	0.110684
$876$	−9263.06	−0.357272
$877$	25629.7	0.986833	0.493417	−	0.869793i	$-0.335748\pi$
$877$	25629.7	0.986833	0.493417	+	0.869793i	$0.335748\pi$
$878$	17745.3	0.682088
$879$	−4771.66	−0.183099
$880$	550.856	0.0211015
$881$	14336.5	0.548250	0.274125	−	0.961694i	$-0.411612\pi$
$881$	14336.5	0.548250	0.274125	+	0.961694i	$0.411612\pi$
$882$	−3127.13	−0.119383
$883$	1012.36	0.0385827	0.0192914	−	0.999814i	$-0.493859\pi$
$883$	1012.36	0.0385827	0.0192914	+	0.999814i	$0.493859\pi$
$884$	1006.79	0.0383054
$885$	−421.051	−0.0159926
$886$	10845.9	0.411258
$887$	−2586.40	−0.0979061	−0.0489530	−	0.998801i	$-0.515588\pi$
$887$	−2586.40	−0.0979061	−0.0489530	+	0.998801i	$0.515588\pi$
$888$	21508.4	0.812808
$889$	23382.5	0.882142
$890$	−14252.8	−0.536801
$891$	891.000	0.0335013
$892$	−14692.8	−0.551515
$893$	17639.3	0.661003
$894$	15125.3	0.565844
$895$	−10233.0	−0.382180
$896$	−13458.2	−0.501794
$897$	2460.53	0.0915883
$898$	12667.8	0.470746
$899$	−7707.77	−0.285949
$900$	−982.276	−0.0363806
$901$	58.1869	0.00215148
$902$	625.374	0.0230850
$903$	−18499.4	−0.681751
$904$	43592.8	1.60385
$905$	−321.487	−0.0118084
$906$	16422.3	0.602201
$907$	43346.1	1.58686	0.793431	−	0.608660i	$-0.208293\pi$
$907$	43346.1	1.58686	0.793431	+	0.608660i	$0.208293\pi$
$908$	24990.5	0.913368
$909$	−9086.93	−0.331567
$910$	−14578.0	−0.531052
$911$	−5218.71	−0.189796	−0.0948978	−	0.995487i	$-0.530252\pi$
$911$	−5218.71	−0.189796	−0.0948978	+	0.995487i	$0.530252\pi$
$912$	−2350.73	−0.0853514
$913$	−6078.55	−0.220340
$914$	15928.1	0.576428
$915$	12806.1	0.462686
$916$	−11292.8	−0.407343
$917$	−22994.1	−0.828060
$918$	177.883	0.00639544
$919$	−29209.5	−1.04846	−0.524230	−	0.851577i	$-0.675647\pi$
$919$	−29209.5	−1.04846	−0.524230	+	0.851577i	$0.675647\pi$
$920$	−1448.70	−0.0519153
$921$	30525.8	1.09214
$922$	29702.5	1.06095
$923$	1591.89	0.0567690
$924$	3301.81	0.117556
$925$	7603.20	0.270261
$926$	−13610.0	−0.482995
$927$	10410.3	0.368844
$928$	5285.37	0.186962
$929$	−39401.0	−1.39150	−0.695751	−	0.718283i	$-0.744928\pi$
$929$	−39401.0	−1.39150	−0.695751	+	0.718283i	$0.744928\pi$
$930$	7068.36	0.249226
$931$	14259.3	0.501965
$932$	−24323.9	−0.854888
$933$	−18774.7	−0.658797
$934$	23668.0	0.829165
$935$	190.073	0.00664819
$936$	14158.0	0.494411
$937$	6099.66	0.212665	0.106332	−	0.994331i	$-0.466089\pi$
$937$	6099.66	0.212665	0.106332	+	0.994331i	$0.466089\pi$
$938$	−1607.35	−0.0559509
$939$	28776.7	1.00010
$940$	4921.48	0.170767
$941$	11534.4	0.399587	0.199793	−	0.979838i	$-0.435973\pi$
$941$	11534.4	0.399587	0.199793	+	0.979838i	$0.435973\pi$
$942$	−5996.63	−0.207410
$943$	−366.532	−0.0126574
$944$	−281.138	−0.00969307
$945$	3094.01	0.106506
$946$	−5642.27	−0.193918
$947$	−16680.4	−0.572376	−0.286188	−	0.958174i	$-0.592388\pi$
$947$	−16680.4	−0.572376	−0.286188	+	0.958174i	$0.592388\pi$
$948$	5397.62	0.184923
$949$	47196.8	1.61441
$950$	−3728.71	−0.127342
$951$	−31896.9	−1.08762
$952$	1867.13	0.0635653
$953$	−17114.4	−0.581733	−0.290866	−	0.956764i	$-0.593943\pi$
$953$	−17114.4	−0.581733	−0.290866	+	0.956764i	$0.593943\pi$
$954$	288.883	0.00980391
$955$	−24083.0	−0.816029
$956$	7958.29	0.269236
$957$	−1029.03	−0.0347584
$958$	26977.4	0.909814
$959$	−61081.2	−2.05674
$960$	−6048.78	−0.203358
$961$	31307.5	1.05090
$962$	−38690.0	−1.29669
$963$	4426.71	0.148130
$964$	9720.86	0.324780
$965$	1479.02	0.0493381
$966$	1611.01	0.0536578
$967$	−35200.4	−1.17060	−0.585299	−	0.810817i	$-0.699023\pi$
$967$	−35200.4	−1.17060	−0.585299	+	0.810817i	$0.699023\pi$
$968$	2852.43	0.0947114
$969$	−811.120	−0.0268906
$970$	11433.1	0.378449
$971$	−43502.6	−1.43776	−0.718881	−	0.695134i	$-0.755345\pi$
$971$	−43502.6	−1.43776	−0.718881	+	0.695134i	$0.755345\pi$
$972$	−1060.86	−0.0350072
$973$	−58617.5	−1.93134
$974$	146.098	0.00480624
$975$	5004.85	0.164393
$976$	8550.71	0.280432
$977$	−5365.50	−0.175699	−0.0878494	−	0.996134i	$-0.527999\pi$
$977$	−5365.50	−0.175699	−0.0878494	+	0.996134i	$0.527999\pi$
$978$	11174.7	0.365366
$979$	−16447.9	−0.536952
$980$	3978.44	0.129680
$981$	12066.9	0.392728
$982$	3255.23	0.105783
$983$	42459.0	1.37765	0.688826	−	0.724926i	$-0.258126\pi$
$983$	42459.0	1.37765	0.688826	+	0.724926i	$0.258126\pi$
$984$	−2109.05	−0.0683271
$985$	10739.8	0.347411
$986$	−205.439	−0.00663542
$987$	−15501.8	−0.499928
$988$	−22792.2	−0.733924
$989$	3306.94	0.106324
$990$	943.664	0.0302946
$991$	−6881.29	−0.220576	−0.110288	−	0.993900i	$-0.535177\pi$
$991$	−6881.29	−0.220576	−0.110288	+	0.993900i	$0.535177\pi$
$992$	−41896.4	−1.34094
$993$	24667.8	0.788328
$994$	1042.28	0.0332586
$995$	4381.30	0.139595
$996$	7237.35	0.230245
$997$	−3166.87	−0.100597	−0.0502987	−	0.998734i	$-0.516017\pi$
$997$	−3166.87	−0.100597	−0.0502987	+	0.998734i	$0.516017\pi$
$998$	25104.4	0.796258
$999$	8211.45	0.260059

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	165.4.a.e.1.2	✓	3
3.2	odd	2		495.4.a.k.1.2		3
5.2	odd	4		825.4.c.k.199.3		6
5.3	odd	4		825.4.c.k.199.4		6
5.4	even	2		825.4.a.r.1.2		3
11.10	odd	2		1815.4.a.r.1.2		3
15.14	odd	2		2475.4.a.t.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.4.a.e.1.2	✓	3	1.1	even	1	trivial
495.4.a.k.1.2		3	3.2	odd	2
825.4.a.r.1.2		3	5.4	even	2
825.4.c.k.199.3		6	5.2	odd	4
825.4.c.k.199.4		6	5.3	odd	4
1815.4.a.r.1.2		3	11.10	odd	2
2475.4.a.t.1.2		3	15.14	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:	\( N \)	\(=\)	\( 165 = 3 \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	165.a (trivial)

\(f(q)\)	\(=\)	\(q-1.90639 q^{2} +3.00000 q^{3} -4.36567 q^{4} -5.00000 q^{5} -5.71918 q^{6} -22.9186 q^{7} +23.5738 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-1.90639 q^{2} +3.00000 q^{3} -4.36567 q^{4} -5.00000 q^{5} -5.71918 q^{6} -22.9186 q^{7} +23.5738 q^{8} +9.00000 q^{9} +9.53196 q^{10} +11.0000 q^{11} -13.0970 q^{12} +66.7313 q^{13} +43.6917 q^{14} -15.0000 q^{15} -10.0156 q^{16} -3.45588 q^{17} -17.1575 q^{18} +78.2359 q^{19} +21.8283 q^{20} -68.7557 q^{21} -20.9703 q^{22} +12.2907 q^{23} +70.7214 q^{24} +25.0000 q^{25} -127.216 q^{26} +27.0000 q^{27} +100.055 q^{28} -31.1827 q^{29} +28.5959 q^{30} +247.181 q^{31} -169.497 q^{32} +33.0000 q^{33} +6.58825 q^{34} +114.593 q^{35} -39.2910 q^{36} +304.128 q^{37} -149.148 q^{38} +200.194 q^{39} -117.869 q^{40} -29.8219 q^{41} +131.075 q^{42} +269.060 q^{43} -48.0224 q^{44} -45.0000 q^{45} -23.4309 q^{46} +225.463 q^{47} -30.0467 q^{48} +182.260 q^{49} -47.6598 q^{50} -10.3676 q^{51} -291.327 q^{52} -16.8371 q^{53} -51.4726 q^{54} -55.0000 q^{55} -540.278 q^{56} +234.708 q^{57} +59.4464 q^{58} +28.0701 q^{59} +65.4850 q^{60} -853.742 q^{61} -471.224 q^{62} -206.267 q^{63} +403.252 q^{64} -333.657 q^{65} -62.9109 q^{66} -36.7885 q^{67} +15.0872 q^{68} +36.8722 q^{69} -218.459 q^{70} +23.8552 q^{71} +212.164 q^{72} +707.265 q^{73} -579.787 q^{74} +75.0000 q^{75} -341.552 q^{76} -252.104 q^{77} -381.648 q^{78} -412.126 q^{79} +50.0778 q^{80} +81.0000 q^{81} +56.8522 q^{82} -552.596 q^{83} +300.165 q^{84} +17.2794 q^{85} -512.934 q^{86} -93.5480 q^{87} +259.312 q^{88} -1495.26 q^{89} +85.7876 q^{90} -1529.39 q^{91} -53.6573 q^{92} +741.543 q^{93} -429.820 q^{94} -391.179 q^{95} -508.491 q^{96} +1199.45 q^{97} -347.459 q^{98} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{2} + 9 q^{3} + 30 q^{4} - 15 q^{5} - 6 q^{6} + 10 q^{7} + 18 q^{8} + 27 q^{9}+O(q^{10}) \) 3 * q - 2 * q^2 + 9 * q^3 + 30 * q^4 - 15 * q^5 - 6 * q^6 + 10 * q^7 + 18 * q^8 + 27 * q^9	\( 3 q - 2 q^{2} + 9 q^{3} + 30 q^{4} - 15 q^{5} - 6 q^{6} + 10 q^{7} + 18 q^{8} + 27 q^{9} + 10 q^{10} + 33 q^{11} + 90 q^{12} + 114 q^{13} - 68 q^{14} - 45 q^{15} + 178 q^{16} - 104 q^{17} - 18 q^{18} - 58 q^{19} - 150 q^{20} + 30 q^{21} - 22 q^{22} + 120 q^{23} + 54 q^{24} + 75 q^{25} - 120 q^{26} + 81 q^{27} + 676 q^{28} - 220 q^{29} + 30 q^{30} + 248 q^{31} - 258 q^{32} + 99 q^{33} - 80 q^{34} - 50 q^{35} + 270 q^{36} + 838 q^{37} + 600 q^{38} + 342 q^{39} - 90 q^{40} + 156 q^{41} - 204 q^{42} + 122 q^{43} + 330 q^{44} - 135 q^{45} - 1256 q^{46} + 504 q^{47} + 534 q^{48} + 279 q^{49} - 50 q^{50} - 312 q^{51} + 520 q^{52} + 282 q^{53} - 54 q^{54} - 165 q^{55} - 1644 q^{56} - 174 q^{57} - 1644 q^{58} + 548 q^{59} - 450 q^{60} + 414 q^{61} - 2448 q^{62} + 90 q^{63} - 58 q^{64} - 570 q^{65} - 66 q^{66} - 428 q^{67} - 1704 q^{68} + 360 q^{69} + 340 q^{70} - 912 q^{71} + 162 q^{72} + 618 q^{73} - 1612 q^{74} + 225 q^{75} - 2752 q^{76} + 110 q^{77} - 360 q^{78} - 542 q^{79} - 890 q^{80} + 243 q^{81} + 3372 q^{82} + 2028 q^{84} + 520 q^{85} - 1548 q^{86} - 660 q^{87} + 198 q^{88} + 790 q^{89} + 90 q^{90} - 772 q^{91} + 1912 q^{92} + 744 q^{93} - 424 q^{94} + 290 q^{95} - 774 q^{96} + 2074 q^{97} - 3978 q^{98} + 297 q^{99}+O(q^{100}) \) 3 * q - 2 * q^2 + 9 * q^3 + 30 * q^4 - 15 * q^5 - 6 * q^6 + 10 * q^7 + 18 * q^8 + 27 * q^9 + 10 * q^10 + 33 * q^11 + 90 * q^12 + 114 * q^13 - 68 * q^14 - 45 * q^15 + 178 * q^16 - 104 * q^17 - 18 * q^18 - 58 * q^19 - 150 * q^20 + 30 * q^21 - 22 * q^22 + 120 * q^23 + 54 * q^24 + 75 * q^25 - 120 * q^26 + 81 * q^27 + 676 * q^28 - 220 * q^29 + 30 * q^30 + 248 * q^31 - 258 * q^32 + 99 * q^33 - 80 * q^34 - 50 * q^35 + 270 * q^36 + 838 * q^37 + 600 * q^38 + 342 * q^39 - 90 * q^40 + 156 * q^41 - 204 * q^42 + 122 * q^43 + 330 * q^44 - 135 * q^45 - 1256 * q^46 + 504 * q^47 + 534 * q^48 + 279 * q^49 - 50 * q^50 - 312 * q^51 + 520 * q^52 + 282 * q^53 - 54 * q^54 - 165 * q^55 - 1644 * q^56 - 174 * q^57 - 1644 * q^58 + 548 * q^59 - 450 * q^60 + 414 * q^61 - 2448 * q^62 + 90 * q^63 - 58 * q^64 - 570 * q^65 - 66 * q^66 - 428 * q^67 - 1704 * q^68 + 360 * q^69 + 340 * q^70 - 912 * q^71 + 162 * q^72 + 618 * q^73 - 1612 * q^74 + 225 * q^75 - 2752 * q^76 + 110 * q^77 - 360 * q^78 - 542 * q^79 - 890 * q^80 + 243 * q^81 + 3372 * q^82 + 2028 * q^84 + 520 * q^85 - 1548 * q^86 - 660 * q^87 + 198 * q^88 + 790 * q^89 + 90 * q^90 - 772 * q^91 + 1912 * q^92 + 744 * q^93 - 424 * q^94 + 290 * q^95 - 774 * q^96 + 2074 * q^97 - 3978 * q^98 + 297 * q^99

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