LMFDB - Embedded newform 75.6.b.f.49.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,6,Mod(49,75)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("75.49");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	75.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$12.0287864860$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{31})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 15x^{2} + 64 $ x^4 - 15*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.2
Root			$2.78388 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	75.49
Dual form			75.6.b.f.49.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.56776i q^{2} -9.00000i q^{3} +25.4066 q^{4} -23.1099 q^{6} -82.6263i q^{7} -147.407i q^{8} -81.0000 q^{9} +O(q^{10})$	\(q-2.56776i q^{2} -9.00000i q^{3} +25.4066 q^{4} -23.1099 q^{6} -82.6263i q^{7} -147.407i q^{8} -81.0000 q^{9} +483.963 q^{11} -228.659i q^{12} +7.50538i q^{13} -212.165 q^{14} +434.505 q^{16} -1080.71i q^{17} +207.989i q^{18} -3042.38 q^{19} -743.637 q^{21} -1242.70i q^{22} -3385.91i q^{23} -1326.66 q^{24} +19.2720 q^{26} +729.000i q^{27} -2099.25i q^{28} -2345.79 q^{29} +3912.43 q^{31} -5832.72i q^{32} -4355.67i q^{33} -2775.01 q^{34} -2057.93 q^{36} +12094.3i q^{37} +7812.13i q^{38} +67.5484 q^{39} +7264.06 q^{41} +1909.48i q^{42} +3022.36i q^{43} +12295.9 q^{44} -8694.22 q^{46} -17299.5i q^{47} -3910.55i q^{48} +9979.89 q^{49} -9726.40 q^{51} +190.686i q^{52} +31151.2i q^{53} +1871.90 q^{54} -12179.7 q^{56} +27381.5i q^{57} +6023.45i q^{58} +48709.8 q^{59} -1957.31 q^{61} -10046.2i q^{62} +6692.73i q^{63} -1072.87 q^{64} -11184.3 q^{66} +59596.7i q^{67} -27457.2i q^{68} -30473.2 q^{69} +83672.1 q^{71} +11939.9i q^{72} -5692.94i q^{73} +31055.4 q^{74} -77296.6 q^{76} -39988.1i q^{77} -173.448i q^{78} -31309.9 q^{79} +6561.00 q^{81} -18652.4i q^{82} +37162.0i q^{83} -18893.3 q^{84} +7760.72 q^{86} +21112.2i q^{87} -71339.4i q^{88} -69190.7 q^{89} +620.142 q^{91} -86024.4i q^{92} -35211.8i q^{93} -44421.1 q^{94} -52494.5 q^{96} -88249.2i q^{97} -25626.0i q^{98} -39201.0 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 32 q^{4} + 108 q^{6} - 324 q^{9}+O(q^{10}) $ 4 * q - 32 * q^4 + 108 * q^6 - 324 * q^9	$ 4 q - 32 q^{4} + 108 q^{6} - 324 q^{9} - 24 q^{11} - 3588 q^{14} - 400 q^{16} - 8428 q^{19} + 1836 q^{21} - 4104 q^{24} + 18228 q^{26} - 8136 q^{29} - 5196 q^{31} + 5336 q^{34} + 2592 q^{36} - 18972 q^{39} + 22464 q^{41} + 65664 q^{44} - 83016 q^{46} - 14600 q^{49} - 30888 q^{51} - 8748 q^{54} + 5400 q^{56} + 127848 q^{59} + 14620 q^{61} + 94592 q^{64} - 98856 q^{66} + 7992 q^{69} + 196608 q^{71} + 193752 q^{74} - 57568 q^{76} - 168000 q^{79} + 26244 q^{81} - 175392 q^{84} - 77484 q^{86} - 206208 q^{89} + 393204 q^{91} - 427432 q^{94} - 249264 q^{96} + 1944 q^{99}+O(q^{100}) $ 4 * q - 32 * q^4 + 108 * q^6 - 324 * q^9 - 24 * q^11 - 3588 * q^14 - 400 * q^16 - 8428 * q^19 + 1836 * q^21 - 4104 * q^24 + 18228 * q^26 - 8136 * q^29 - 5196 * q^31 + 5336 * q^34 + 2592 * q^36 - 18972 * q^39 + 22464 * q^41 + 65664 * q^44 - 83016 * q^46 - 14600 * q^49 - 30888 * q^51 - 8748 * q^54 + 5400 * q^56 + 127848 * q^59 + 14620 * q^61 + 94592 * q^64 - 98856 * q^66 + 7992 * q^69 + 196608 * q^71 + 193752 * q^74 - 57568 * q^76 - 168000 * q^79 + 26244 * q^81 - 175392 * q^84 - 77484 * q^86 - 206208 * q^89 + 393204 * q^91 - 427432 * q^94 - 249264 * q^96 + 1944 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$.

$n$	$26$	$52$
$\chi(n)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	− 2.56776i	− 0.453921i	−0.973904	−	0.226960i	$-0.927121\pi$
$2$	− 2.56776i	− 0.453921i	0.973904	−	0.226960i	$-0.0728788\pi$
$3$	− 9.00000i	− 0.577350i
$3$	− 9.00000i	− 0.577350i
$4$	25.4066	0.793956
$5$	0	0
$5$	0	0
$6$	−23.1099	−0.262071
$7$	− 82.6263i	− 0.637343i	−0.947865	−	0.318672i	$-0.896763\pi$
$7$	− 82.6263i	− 0.637343i	0.947865	−	0.318672i	$-0.103237\pi$
$8$	− 147.407i	− 0.814314i
$9$	−81.0000	−0.333333
$10$	0	0
$11$	483.963	1.20595	0.602977	−	0.797759i	$-0.293981\pi$
$11$	483.963	1.20595	0.602977	+	0.797759i	$0.293981\pi$
$12$	− 228.659i	− 0.458391i
$13$	7.50538i	0.0123173i	0.999981	+	0.00615863i	$0.00196037\pi$
$13$	7.50538i	0.0123173i	−0.999981	+	0.00615863i	$0.998040\pi$
$14$	−212.165	−0.289303
$15$	0	0
$16$	434.505	0.424322
$17$	− 1080.71i	− 0.906958i	−0.891267	−	0.453479i	$-0.850183\pi$
$17$	− 1080.71i	− 0.906958i	0.891267	−	0.453479i	$-0.149817\pi$
$18$	207.989i	0.151307i
$19$	−3042.38	−1.93344	−0.966719	−	0.255842i	$-0.917647\pi$
$19$	−3042.38	−1.93344	−0.966719	+	0.255842i	$0.917647\pi$
$20$	0	0
$21$	−743.637	−0.367970
$22$	− 1242.70i	− 0.547408i
$23$	− 3385.91i	− 1.33461i	−0.744782	−	0.667307i	$-0.767447\pi$
$23$	− 3385.91i	− 1.33461i	0.744782	−	0.667307i	$-0.232553\pi$
$24$	−1326.66	−0.470144
$25$	0	0
$26$	19.2720	0.00559106
$27$	729.000i	0.192450i
$28$	− 2099.25i	− 0.506022i
$29$	−2345.79	−0.517959	−0.258979	−	0.965883i	$-0.583386\pi$
$29$	−2345.79	−0.517959	−0.258979	+	0.965883i	$0.583386\pi$
$30$	0	0
$31$	3912.43	0.731210	0.365605	−	0.930770i	$-0.380862\pi$
$31$	3912.43	0.731210	0.365605	+	0.930770i	$0.380862\pi$
$32$	− 5832.72i	− 1.00692i
$33$	− 4355.67i	− 0.696258i
$34$	−2775.01	−0.411687
$35$	0	0
$36$	−2057.93	−0.264652
$37$	12094.3i	1.45237i	0.687498	+	0.726187i	$0.258709\pi$
$37$	12094.3i	1.45237i	−0.687498	+	0.726187i	$0.741291\pi$
$38$	7812.13i	0.877628i
$39$	67.5484	0.00711138
$40$	0	0
$41$	7264.06	0.674869	0.337435	−	0.941349i	$-0.390441\pi$
$41$	7264.06	0.674869	0.337435	+	0.941349i	$0.390441\pi$
$42$	1909.48i	0.167029i
$43$	3022.36i	0.249273i	0.992202	+	0.124637i	$0.0397765\pi$
$43$	3022.36i	0.249273i	−0.992202	+	0.124637i	$0.960224\pi$
$44$	12295.9	0.957474
$45$	0	0
$46$	−8694.22	−0.605810
$47$	− 17299.5i	− 1.14232i	−0.820837	−	0.571162i	$-0.806493\pi$
$47$	− 17299.5i	− 1.14232i	0.820837	−	0.571162i	$-0.193507\pi$
$48$	− 3910.55i	− 0.244982i
$49$	9979.89	0.593793
$50$	0	0
$51$	−9726.40	−0.523632
$52$	190.686i	0.00977936i
$53$	31151.2i	1.52330i	0.647989	+	0.761649i	$0.275610\pi$
$53$	31151.2i	1.52330i	−0.647989	+	0.761649i	$0.724390\pi$
$54$	1871.90	0.0873571
$55$	0	0
$56$	−12179.7	−0.518998
$57$	27381.5i	1.11627i
$58$	6023.45i	0.235112i
$59$	48709.8	1.82174	0.910870	−	0.412692i	$-0.135412\pi$
$59$	48709.8	1.82174	0.910870	+	0.412692i	$0.135412\pi$
$60$	0	0
$61$	−1957.31	−0.0673495	−0.0336747	−	0.999433i	$-0.510721\pi$
$61$	−1957.31	−0.0673495	−0.0336747	+	0.999433i	$0.510721\pi$
$62$	− 10046.2i	− 0.331911i
$63$	6692.73i	0.212448i
$64$	−1072.87	−0.0327415
$65$	0	0
$66$	−11184.3	−0.316046
$67$	59596.7i	1.62194i	0.585087	+	0.810970i	$0.301060\pi$
$67$	59596.7i	1.62194i	−0.585087	+	0.810970i	$0.698940\pi$
$68$	− 27457.2i	− 0.720084i
$69$	−30473.2	−0.770540
$70$	0	0
$71$	83672.1	1.96986	0.984929	−	0.172958i	$-0.0553325\pi$
$71$	83672.1	1.96986	0.984929	+	0.172958i	$0.0553325\pi$
$72$	11939.9i	0.271438i
$73$	− 5692.94i	− 0.125034i	−0.998044	−	0.0625172i	$-0.980087\pi$
$73$	− 5692.94i	− 0.125034i	0.998044	−	0.0625172i	$-0.0199128\pi$
$74$	31055.4	0.659263
$75$	0	0
$76$	−77296.6	−1.53506
$77$	− 39988.1i	− 0.768607i
$78$	− 173.448i	− 0.00322800i
$79$	−31309.9	−0.564435	−0.282217	−	0.959350i	$-0.591070\pi$
$79$	−31309.9	−0.564435	−0.282217	+	0.959350i	$0.591070\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	− 18652.4i	− 0.306337i
$83$	37162.0i	0.592113i	0.955170	+	0.296056i	$0.0956716\pi$
$83$	37162.0i	0.592113i	−0.955170	+	0.296056i	$0.904328\pi$
$84$	−18893.3	−0.292152
$85$	0	0
$86$	7760.72	0.113150
$87$	21112.2i	0.299044i
$88$	− 71339.4i	− 0.982025i
$89$	−69190.7	−0.925918	−0.462959	−	0.886380i	$-0.653212\pi$
$89$	−69190.7	−0.925918	−0.462959	+	0.886380i	$0.653212\pi$
$90$	0	0
$91$	620.142	0.00785033
$92$	− 86024.4i	− 1.05963i
$93$	− 35211.8i	− 0.422164i
$94$	−44421.1	−0.518525
$95$	0	0
$96$	−52494.5	−0.581347
$97$	− 88249.2i	− 0.952317i	−0.879359	−	0.476159i	$-0.842029\pi$
$97$	− 88249.2i	− 0.952317i	0.879359	−	0.476159i	$-0.157971\pi$
$98$	− 25626.0i	− 0.269535i
$99$	−39201.0	−0.401985
$100$	0	0
$101$	89217.2	0.870252	0.435126	−	0.900370i	$-0.356704\pi$
$101$	89217.2	0.870252	0.435126	+	0.900370i	$0.356704\pi$
$102$	24975.1i	0.237688i
$103$	− 9687.70i	− 0.0899762i	−0.998988	−	0.0449881i	$-0.985675\pi$
$103$	− 9687.70i	− 0.0899762i	0.998988	−	0.0449881i	$-0.0143250\pi$
$104$	1106.34	0.0100301
$105$	0	0
$106$	79988.9	0.691457
$107$	− 59011.6i	− 0.498285i	−0.968467	−	0.249143i	$-0.919851\pi$
$107$	− 59011.6i	− 0.498285i	0.968467	−	0.249143i	$-0.0801488\pi$
$108$	18521.4i	0.152797i
$109$	−85229.1	−0.687103	−0.343551	−	0.939134i	$-0.611630\pi$
$109$	−85229.1	−0.687103	−0.343551	+	0.939134i	$0.611630\pi$
$110$	0	0
$111$	108849.	0.838528
$112$	− 35901.6i	− 0.270439i
$113$	113652.i	0.837297i	0.908148	+	0.418649i	$0.137496\pi$
$113$	113652.i	0.837297i	−0.908148	+	0.418649i	$0.862504\pi$
$114$	70309.1	0.506699
$115$	0	0
$116$	−59598.6	−0.411236
$117$	− 607.936i	− 0.00410575i
$118$	− 125075.i	− 0.826926i
$119$	−89295.2	−0.578044
$120$	0	0
$121$	73169.4	0.454325
$122$	5025.90i	0.0305713i
$123$	− 65376.5i	− 0.389636i
$124$	99401.4	0.580548
$125$	0	0
$126$	17185.4	0.0964345
$127$	295771.i	1.62722i	0.581410	+	0.813611i	$0.302501\pi$
$127$	295771.i	1.62722i	−0.581410	+	0.813611i	$0.697499\pi$
$128$	− 183892.i	− 0.992060i
$129$	27201.3	0.143918
$130$	0	0
$131$	−250130.	−1.27347	−0.636734	−	0.771084i	$-0.719715\pi$
$131$	−250130.	−1.27347	−0.636734	+	0.771084i	$0.719715\pi$
$132$	− 110663.i	− 0.552798i
$133$	251381.i	1.23226i
$134$	153030.	0.736233
$135$	0	0
$136$	−159304.	−0.738548
$137$	269811.i	1.22817i	0.789241	+	0.614084i	$0.210474\pi$
$137$	269811.i	1.22817i	−0.789241	+	0.614084i	$0.789526\pi$
$138$	78248.0i	0.349764i
$139$	262877.	1.15403	0.577013	−	0.816735i	$-0.304218\pi$
$139$	262877.	1.15403	0.577013	+	0.816735i	$0.304218\pi$
$140$	0	0
$141$	−155696.	−0.659521
$142$	− 214850.i	− 0.894160i
$143$	3632.33i	0.0148541i
$144$	−35194.9	−0.141441
$145$	0	0
$146$	−14618.1	−0.0567557
$147$	− 89819.0i	− 0.342827i
$148$	307276.i	1.15312i
$149$	−131202.	−0.484144	−0.242072	−	0.970258i	$-0.577827\pi$
$149$	−131202.	−0.484144	−0.242072	+	0.970258i	$0.577827\pi$
$150$	0	0
$151$	175144.	0.625104	0.312552	−	0.949901i	$-0.398816\pi$
$151$	175144.	0.625104	0.312552	+	0.949901i	$0.398816\pi$
$152$	448468.i	1.57443i
$153$	87537.6i	0.302319i
$154$	−102680.	−0.348887
$155$	0	0
$156$	1716.17	0.00564612
$157$	− 47706.6i	− 0.154465i	−0.997013	−	0.0772323i	$-0.975392\pi$
$157$	− 47706.6i	− 0.154465i	0.997013	−	0.0772323i	$-0.0246083\pi$
$158$	80396.4i	0.256209i
$159$	280361.	0.879477
$160$	0	0
$161$	−279765.	−0.850608
$162$	− 16847.1i	− 0.0504357i
$163$	− 22618.8i	− 0.0666809i	−0.999444	−	0.0333404i	$-0.989385\pi$
$163$	− 22618.8i	− 0.0666809i	0.999444	−	0.0333404i	$-0.0106146\pi$
$164$	184555.	0.535816
$165$	0	0
$166$	95423.4	0.268772
$167$	339093.i	0.940866i	0.882436	+	0.470433i	$0.155902\pi$
$167$	339093.i	0.940866i	−0.882436	+	0.470433i	$0.844098\pi$
$168$	109617.i	0.299643i
$169$	371237.	0.999848
$170$	0	0
$171$	246433.	0.644479
$172$	76787.9i	0.197912i
$173$	− 269075.i	− 0.683530i	−0.939785	−	0.341765i	$-0.888975\pi$
$173$	− 269075.i	− 0.683530i	0.939785	−	0.341765i	$-0.111025\pi$
$174$	54211.0	0.135742
$175$	0	0
$176$	210285.	0.511712
$177$	− 438389.i	− 1.05178i
$178$	177665.i	0.420294i
$179$	−193640.	−0.451714	−0.225857	−	0.974160i	$-0.572518\pi$
$179$	−193640.	−0.451714	−0.225857	+	0.974160i	$0.572518\pi$
$180$	0	0
$181$	165755.	0.376071	0.188035	−	0.982162i	$-0.439788\pi$
$181$	165755.	0.376071	0.188035	+	0.982162i	$0.439788\pi$
$182$	− 1592.38i	− 0.00356343i
$183$	17615.8i	0.0388842i
$184$	−499106.	−1.08680
$185$	0	0
$186$	−90415.7	−0.191629
$187$	− 523024.i	− 1.09375i
$188$	− 439522.i	− 0.906955i
$189$	60234.6	0.122657
$190$	0	0
$191$	−522342.	−1.03603	−0.518014	−	0.855372i	$-0.673329\pi$
$191$	−522342.	−1.03603	−0.518014	+	0.855372i	$0.673329\pi$
$192$	9655.86i	0.0189033i
$193$	− 579738.i	− 1.12031i	−0.828387	−	0.560156i	$-0.810741\pi$
$193$	− 579738.i	− 1.12031i	0.828387	−	0.560156i	$-0.189259\pi$
$194$	−226603.	−0.432277
$195$	0	0
$196$	253555.	0.471446
$197$	− 576819.i	− 1.05895i	−0.848327	−	0.529473i	$-0.822390\pi$
$197$	− 576819.i	− 1.05895i	0.848327	−	0.529473i	$-0.177610\pi$
$198$	100659.i	0.182469i
$199$	493665.	0.883690	0.441845	−	0.897091i	$-0.354324\pi$
$199$	493665.	0.883690	0.441845	+	0.897091i	$0.354324\pi$
$200$	0	0
$201$	536370.	0.936428
$202$	− 229089.i	− 0.395026i
$203$	193824.i	0.330117i
$204$	−247114.	−0.415741
$205$	0	0
$206$	−24875.7	−0.0408421
$207$	274259.i	0.444872i
$208$	3261.13i	0.00522648i
$209$	−1.47240e6	−2.33164
$210$	0	0
$211$	−383876.	−0.593588	−0.296794	−	0.954942i	$-0.595917\pi$
$211$	−383876.	−0.593588	−0.296794	+	0.954942i	$0.595917\pi$
$212$	791446.i	1.20943i
$213$	− 753049.i	− 1.13730i
$214$	−151528.	−0.226182
$215$	0	0
$216$	107459.	0.156715
$217$	− 323270.i	− 0.466032i
$218$	218848.i	0.311890i
$219$	−51236.5	−0.0721886
$220$	0	0
$221$	8111.14	0.0111712
$222$	− 279499.i	− 0.380625i
$223$	− 321593.i	− 0.433056i	−0.976276	−	0.216528i	$-0.930527\pi$
$223$	− 321593.i	− 0.433056i	0.976276	−	0.216528i	$-0.0694733\pi$
$224$	−481936.	−0.641755
$225$	0	0
$226$	291831.	0.380067
$227$	− 331402.i	− 0.426865i	−0.976958	−	0.213432i	$-0.931536\pi$
$227$	− 331402.i	− 0.426865i	0.976958	−	0.213432i	$-0.0684643\pi$
$228$	695669.i	0.886269i
$229$	−979958.	−1.23486	−0.617432	−	0.786625i	$-0.711827\pi$
$229$	−979958.	−1.23486	−0.617432	+	0.786625i	$0.711827\pi$
$230$	0	0
$231$	−359893.	−0.443755
$232$	345786.i	0.421781i
$233$	309466.i	0.373442i	0.982413	+	0.186721i	$0.0597861\pi$
$233$	309466.i	0.373442i	−0.982413	+	0.186721i	$0.940214\pi$
$234$	−1561.04	−0.00186369
$235$	0	0
$236$	1.23755e6	1.44638
$237$	281789.i	0.325877i
$238$	229289.i	0.262386i
$239$	1.52544e6	1.72743	0.863714	−	0.503983i	$-0.168133\pi$
$239$	1.52544e6	1.72743	0.863714	+	0.503983i	$0.168133\pi$
$240$	0	0
$241$	−772397.	−0.856639	−0.428320	−	0.903627i	$-0.640894\pi$
$241$	−772397.	−0.856639	−0.428320	+	0.903627i	$0.640894\pi$
$242$	− 187882.i	− 0.206227i
$243$	− 59049.0i	− 0.0641500i
$244$	−49728.5	−0.0534725
$245$	0	0
$246$	−167872.	−0.176864
$247$	− 22834.2i	− 0.0238147i
$248$	− 576718.i	− 0.595434i
$249$	334458.	0.341856
$250$	0	0
$251$	−401992.	−0.402747	−0.201374	−	0.979514i	$-0.564541\pi$
$251$	−401992.	−0.402747	−0.201374	+	0.979514i	$0.564541\pi$
$252$	170040.i	0.168674i
$253$	− 1.63866e6i	− 1.60948i
$254$	759471.	0.738630
$255$	0	0
$256$	−506524.	−0.483058
$257$	− 484514.i	− 0.457587i	−0.973475	−	0.228793i	$-0.926522\pi$
$257$	− 484514.i	− 0.457587i	0.973475	−	0.228793i	$-0.0734780\pi$
$258$	− 69846.4i	− 0.0653273i
$259$	999312.	0.925660
$260$	0	0
$261$	190009.	0.172653
$262$	642276.i	0.578054i
$263$	979486.i	0.873190i	0.899658	+	0.436595i	$0.143816\pi$
$263$	979486.i	0.873190i	−0.899658	+	0.436595i	$0.856184\pi$
$264$	−642054.	−0.566972
$265$	0	0
$266$	645487.	0.559350
$267$	622716.i	0.534579i
$268$	1.51415e6i	1.28775i
$269$	995409.	0.838727	0.419364	−	0.907818i	$-0.362253\pi$
$269$	995409.	0.838727	0.419364	+	0.907818i	$0.362253\pi$
$270$	0	0
$271$	46175.9	0.0381937	0.0190969	−	0.999818i	$-0.493921\pi$
$271$	46175.9	0.0381937	0.0190969	+	0.999818i	$0.493921\pi$
$272$	− 469575.i	− 0.384842i
$273$	− 5581.28i	− 0.00453239i
$274$	692810.	0.557491
$275$	0	0
$276$	−774220.	−0.611775
$277$	844717.i	0.661473i	0.943723	+	0.330736i	$0.107297\pi$
$277$	844717.i	0.661473i	−0.943723	+	0.330736i	$0.892703\pi$
$278$	− 675006.i	− 0.523837i
$279$	−316907.	−0.243737
$280$	0	0
$281$	−453572.	−0.342674	−0.171337	−	0.985213i	$-0.554809\pi$
$281$	−453572.	−0.342674	−0.171337	+	0.985213i	$0.554809\pi$
$282$	399790.i	0.299370i
$283$	2.37359e6i	1.76173i	0.473363	+	0.880867i	$0.343040\pi$
$283$	2.37359e6i	1.76173i	−0.473363	+	0.880867i	$0.656960\pi$
$284$	2.12582e6	1.56398
$285$	0	0
$286$	9326.96	0.00674256
$287$	− 600203.i	− 0.430123i
$288$	472450.i	0.335641i
$289$	251922.	0.177427
$290$	0	0
$291$	−794243.	−0.549821
$292$	− 144638.i	− 0.0992718i
$293$	1.71847e6i	1.16943i	0.811240	+	0.584714i	$0.198793\pi$
$293$	1.71847e6i	1.16943i	−0.811240	+	0.584714i	$0.801207\pi$
$294$	−230634.	−0.155616
$295$	0	0
$296$	1.78279e6	1.18269
$297$	352809.i	0.232086i
$298$	336896.i	0.219763i
$299$	25412.5	0.0164388
$300$	0	0
$301$	249727.	0.158873
$302$	− 449728.i	− 0.283748i
$303$	− 802955.i	− 0.502440i
$304$	−1.32193e6	−0.820399
$305$	0	0
$306$	224776.	0.137229
$307$	2.89249e6i	1.75156i	0.482709	+	0.875781i	$0.339653\pi$
$307$	2.89249e6i	1.75156i	−0.482709	+	0.875781i	$0.660347\pi$
$308$	− 1.01596e6i	− 0.610240i
$309$	−87189.3	−0.0519478
$310$	0	0
$311$	−1.28455e6	−0.753097	−0.376548	−	0.926397i	$-0.622889\pi$
$311$	−1.28455e6	−0.753097	−0.376548	+	0.926397i	$0.622889\pi$
$312$	− 9957.08i	− 0.00579089i
$313$	− 2.32830e6i	− 1.34332i	−0.740860	−	0.671659i	$-0.765582\pi$
$313$	− 2.32830e6i	− 1.34332i	0.740860	−	0.671659i	$-0.234418\pi$
$314$	−122499.	−0.0701147
$315$	0	0
$316$	−795477.	−0.448136
$317$	− 2.86466e6i	− 1.60112i	−0.599251	−	0.800561i	$-0.704535\pi$
$317$	− 2.86466e6i	− 1.60112i	0.599251	−	0.800561i	$-0.295465\pi$
$318$	− 719900.i	− 0.399213i
$319$	−1.13528e6	−0.624634
$320$	0	0
$321$	−531105.	−0.287685
$322$	718372.i	0.386109i
$323$	3.28794e6i	1.75355i
$324$	166693.	0.0882173
$325$	0	0
$326$	−58079.8	−0.0302678
$327$	767062.i	0.396699i
$328$	− 1.07077e6i	− 0.549556i
$329$	−1.42940e6	−0.728053
$330$	0	0
$331$	33623.6	0.0168684	0.00843422	−	0.999964i	$-0.497315\pi$
$331$	33623.6	0.0168684	0.00843422	+	0.999964i	$0.497315\pi$
$332$	944161.i	0.470111i
$333$	− 979642.i	− 0.484124i
$334$	870711.	0.427079
$335$	0	0
$336$	−323114.	−0.156138
$337$	− 1.18330e6i	− 0.567571i	−0.958888	−	0.283786i	$-0.908410\pi$
$337$	− 1.18330e6i	− 0.567571i	0.958888	−	0.283786i	$-0.0915904\pi$
$338$	− 953248.i	− 0.453852i
$339$	1.02286e6	0.483414
$340$	0	0
$341$	1.89347e6	0.881805
$342$	− 632782.i	− 0.292543i
$343$	− 2.21330e6i	− 1.01579i
$344$	445516.	0.202987
$345$	0	0
$346$	−690921.	−0.310269
$347$	− 982641.i	− 0.438098i	−0.975714	−	0.219049i	$-0.929705\pi$
$347$	− 982641.i	− 0.438098i	0.975714	−	0.219049i	$-0.0702954\pi$
$348$	536388.i	0.237427i
$349$	−3.86491e6	−1.69854	−0.849270	−	0.527958i	$-0.822958\pi$
$349$	−3.86491e6	−1.69854	−0.849270	+	0.527958i	$0.822958\pi$
$350$	0	0
$351$	−5471.42	−0.00237046
$352$	− 2.82282e6i	− 1.21430i
$353$	− 3.08371e6i	− 1.31715i	−0.752513	−	0.658577i	$-0.771159\pi$
$353$	− 3.08371e6i	− 1.31715i	0.752513	−	0.658577i	$-0.228841\pi$
$354$	−1.12568e6	−0.477426
$355$	0	0
$356$	−1.75790e6	−0.735138
$357$	803656.i	0.333734i
$358$	497223.i	0.205042i
$359$	−1.39479e6	−0.571179	−0.285589	−	0.958352i	$-0.592189\pi$
$359$	−1.39479e6	−0.571179	−0.285589	+	0.958352i	$0.592189\pi$
$360$	0	0
$361$	6.78000e6	2.73818
$362$	− 425619.i	− 0.170706i
$363$	− 658525.i	− 0.262304i
$364$	15755.7	0.00623281
$365$	0	0
$366$	45233.1	0.0176504
$367$	201558.i	0.0781152i	0.999237	+	0.0390576i	$0.0124356\pi$
$367$	201558.i	0.0781152i	−0.999237	+	0.0390576i	$0.987564\pi$
$368$	− 1.47120e6i	− 0.566306i
$369$	−588389.	−0.224956
$370$	0	0
$371$	2.57391e6	0.970864
$372$	− 894613.i	− 0.335180i
$373$	3.49422e6i	1.30040i	0.759761	+	0.650202i	$0.225316\pi$
$373$	3.49422e6i	1.30040i	−0.759761	+	0.650202i	$0.774684\pi$
$374$	−1.34300e6	−0.496476
$375$	0	0
$376$	−2.55006e6	−0.930211
$377$	− 17606.1i	− 0.00637983i
$378$	− 154668.i	− 0.0556765i
$379$	3.36760e6	1.20427	0.602133	−	0.798396i	$-0.294318\pi$
$379$	3.36760e6	1.20427	0.602133	+	0.798396i	$0.294318\pi$
$380$	0	0
$381$	2.66194e6	0.939477
$382$	1.34125e6i	0.470275i
$383$	3.53112e6i	1.23003i	0.788515	+	0.615015i	$0.210850\pi$
$383$	3.53112e6i	1.23003i	−0.788515	+	0.615015i	$0.789150\pi$
$384$	−1.65503e6	−0.572766
$385$	0	0
$386$	−1.48863e6	−0.508533
$387$	− 244811.i	− 0.0830910i
$388$	− 2.24211e6i	− 0.756098i
$389$	−3.49889e6	−1.17235	−0.586173	−	0.810186i	$-0.699366\pi$
$389$	−3.49889e6	−1.17235	−0.586173	+	0.810186i	$0.699366\pi$
$390$	0	0
$391$	−3.65919e6	−1.21044
$392$	− 1.47110e6i	− 0.483534i
$393$	2.25117e6i	0.735237i
$394$	−1.48113e6	−0.480678
$395$	0	0
$396$	−995964.	−0.319158
$397$	− 4.17547e6i	− 1.32963i	−0.747010	−	0.664813i	$-0.768511\pi$
$397$	− 4.17547e6i	− 1.32963i	0.747010	−	0.664813i	$-0.231489\pi$
$398$	− 1.26762e6i	− 0.401125i
$399$	2.26243e6	0.711448
$400$	0	0
$401$	−893371.	−0.277441	−0.138721	−	0.990332i	$-0.544299\pi$
$401$	−893371.	−0.277441	−0.138721	+	0.990332i	$0.544299\pi$
$402$	− 1.37727e6i	− 0.425064i
$403$	29364.3i	0.00900651i
$404$	2.26670e6	0.690942
$405$	0	0
$406$	497696.	0.149847
$407$	5.85322e6i	1.75149i
$408$	1.43373e6i	0.426401i
$409$	−495248.	−0.146391	−0.0731955	−	0.997318i	$-0.523320\pi$
$409$	−495248.	−0.146391	−0.0731955	+	0.997318i	$0.523320\pi$
$410$	0	0
$411$	2.42829e6	0.709083
$412$	− 246131.i	− 0.0714372i
$413$	− 4.02472e6i	− 1.16107i
$414$	704232.	0.201937
$415$	0	0
$416$	43776.8	0.0124025
$417$	− 2.36589e6i	− 0.666277i
$418$	3.78078e6i	1.05838i
$419$	4.96965e6	1.38290	0.691450	−	0.722425i	$-0.256972\pi$
$419$	4.96965e6	1.38290	0.691450	+	0.722425i	$0.256972\pi$
$420$	0	0
$421$	−932548.	−0.256428	−0.128214	−	0.991747i	$-0.540924\pi$
$421$	−932548.	−0.256428	−0.128214	+	0.991747i	$0.540924\pi$
$422$	985703.i	0.269442i
$423$	1.40126e6i	0.380775i
$424$	4.59189e6	1.24044
$425$	0	0
$426$	−1.93365e6	−0.516243
$427$	161725.i	0.0429247i
$428$	− 1.49928e6i	− 0.395617i
$429$	32690.9	0.00857599
$430$	0	0
$431$	−1.81726e6	−0.471221	−0.235611	−	0.971848i	$-0.575709\pi$
$431$	−1.81726e6	−0.471221	−0.235611	+	0.971848i	$0.575709\pi$
$432$	316754.i	0.0816607i
$433$	− 2.61506e6i	− 0.670288i	−0.942167	−	0.335144i	$-0.891215\pi$
$433$	− 2.61506e6i	− 0.670288i	0.942167	−	0.335144i	$-0.108785\pi$
$434$	−830080.	−0.211542
$435$	0	0
$436$	−2.16538e6	−0.545529
$437$	1.03012e7i	2.58039i
$438$	131563.i	0.0327679i
$439$	3.03139e6	0.750723	0.375362	−	0.926878i	$-0.377519\pi$
$439$	3.03139e6	0.750723	0.375362	+	0.926878i	$0.377519\pi$
$440$	0	0
$441$	−808371.	−0.197931
$442$	− 20827.5i	− 0.00507086i
$443$	3.47567e6i	0.841452i	0.907188	+	0.420726i	$0.138225\pi$
$443$	3.47567e6i	0.841452i	−0.907188	+	0.420726i	$0.861775\pi$
$444$	2.76549e6	0.665754
$445$	0	0
$446$	−825775.	−0.196573
$447$	1.18082e6i	0.279521i
$448$	88647.6i	0.0208676i
$449$	−5.53025e6	−1.29458	−0.647290	−	0.762244i	$-0.724098\pi$
$449$	−5.53025e6	−1.29458	−0.647290	+	0.762244i	$0.724098\pi$
$450$	0	0
$451$	3.51554e6	0.813861
$452$	2.88750e6i	0.664777i
$453$	− 1.57629e6i	− 0.360904i
$454$	−850962.	−0.193763
$455$	0	0
$456$	4.03621e6	0.908995
$457$	1.42548e6i	0.319279i	0.987175	+	0.159639i	$0.0510331\pi$
$457$	1.42548e6i	0.319279i	−0.987175	+	0.159639i	$0.948967\pi$
$458$	2.51630e6i	0.560530i
$459$	787838.	0.174544
$460$	0	0
$461$	−4.21687e6	−0.924141	−0.462070	−	0.886843i	$-0.652893\pi$
$461$	−4.21687e6	−0.924141	−0.462070	+	0.886843i	$0.652893\pi$
$462$	924121.i	0.201430i
$463$	4.18428e6i	0.907128i	0.891224	+	0.453564i	$0.149848\pi$
$463$	4.18428e6i	0.907128i	−0.891224	+	0.453564i	$0.850152\pi$
$464$	−1.01926e6	−0.219781
$465$	0	0
$466$	794637.	0.169513
$467$	1.36779e6i	0.290219i	0.989416	+	0.145110i	$0.0463535\pi$
$467$	1.36779e6i	0.290219i	−0.989416	+	0.145110i	$0.953646\pi$
$468$	− 15445.6i	− 0.00325979i
$469$	4.92425e6	1.03373
$470$	0	0
$471$	−429359.	−0.0891802
$472$	− 7.18015e6i	− 1.48347i
$473$	1.46271e6i	0.300612i
$474$	723568.	0.147922
$475$	0	0
$476$	−2.26869e6	−0.458941
$477$	− 2.52325e6i	− 0.507766i
$478$	− 3.91697e6i	− 0.784115i
$479$	2.01693e6	0.401654	0.200827	−	0.979627i	$-0.435637\pi$
$479$	2.01693e6	0.401654	0.200827	+	0.979627i	$0.435637\pi$
$480$	0	0
$481$	−90772.7	−0.0178893
$482$	1.98333e6i	0.388846i
$483$	2.51789e6i	0.491099i
$484$	1.85899e6	0.360714
$485$	0	0
$486$	−151624.	−0.0291190
$487$	8.07935e6i	1.54367i	0.635824	+	0.771834i	$0.280660\pi$
$487$	8.07935e6i	1.54367i	−0.635824	+	0.771834i	$0.719340\pi$
$488$	288520.i	0.0548436i
$489$	−203570.	−0.0384982
$490$	0	0
$491$	1.09450e6	0.204885	0.102443	−	0.994739i	$-0.467334\pi$
$491$	1.09450e6	0.204885	0.102443	+	0.994739i	$0.467334\pi$
$492$	− 1.66099e6i	− 0.309354i
$493$	2.53513e6i	0.469767i
$494$	−58633.0	−0.0108100
$495$	0	0
$496$	1.69997e6	0.310268
$497$	− 6.91352e6i	− 1.25548i
$498$	− 858810.i	− 0.155176i
$499$	−3.24830e6	−0.583989	−0.291994	−	0.956420i	$-0.594319\pi$
$499$	−3.24830e6	−0.583989	−0.291994	+	0.956420i	$0.594319\pi$
$500$	0	0
$501$	3.05184e6	0.543209
$502$	1.03222e6i	0.182815i
$503$	− 5.19359e6i	− 0.915267i	−0.889141	−	0.457633i	$-0.848697\pi$
$503$	− 5.19359e6i	− 0.915267i	0.889141	−	0.457633i	$-0.151303\pi$
$504$	986553.	0.172999
$505$	0	0
$506$	−4.20768e6	−0.730578
$507$	− 3.34113e6i	− 0.577263i
$508$	7.51454e6i	1.29194i
$509$	−4.95131e6	−0.847083	−0.423541	−	0.905877i	$-0.639213\pi$
$509$	−4.95131e6	−0.847083	−0.423541	+	0.905877i	$0.639213\pi$
$510$	0	0
$511$	−470387.	−0.0796898
$512$	− 4.58391e6i	− 0.772790i
$513$	− 2.21790e6i	− 0.372090i
$514$	−1.24412e6	−0.207708
$515$	0	0
$516$	691091.	0.114264
$517$	− 8.37233e6i	− 1.37759i
$518$	− 2.56600e6i	− 0.420177i
$519$	−2.42167e6	−0.394636
$520$	0	0
$521$	−5.43792e6	−0.877685	−0.438843	−	0.898564i	$-0.644611\pi$
$521$	−5.43792e6	−0.877685	−0.438843	+	0.898564i	$0.644611\pi$
$522$	− 487899.i	− 0.0783707i
$523$	− 9.04176e6i	− 1.44544i	−0.691143	−	0.722718i	$-0.742893\pi$
$523$	− 9.04176e6i	− 1.44544i	0.691143	−	0.722718i	$-0.257107\pi$
$524$	−6.35496e6	−1.01108
$525$	0	0
$526$	2.51509e6	0.396359
$527$	− 4.22820e6i	− 0.663177i
$528$	− 1.89256e6i	− 0.295437i
$529$	−5.02805e6	−0.781197
$530$	0	0
$531$	−3.94550e6	−0.607247
$532$	6.38674e6i	0.978363i
$533$	54519.5i	0.00831254i
$534$	1.59899e6	0.242657
$535$	0	0
$536$	8.78494e6	1.32077
$537$	1.74276e6i	0.260797i
$538$	− 2.55598e6i	− 0.380716i
$539$	4.82990e6	0.716088
$540$	0	0
$541$	5.97997e6	0.878427	0.439213	−	0.898383i	$-0.355257\pi$
$541$	5.97997e6	0.878427	0.439213	+	0.898383i	$0.355257\pi$
$542$	− 118569.i	− 0.0173369i
$543$	− 1.49179e6i	− 0.217125i
$544$	−6.30348e6	−0.913236
$545$	0	0
$546$	−14331.4	−0.00205735
$547$	− 9.52731e6i	− 1.36145i	−0.732538	−	0.680726i	$-0.761665\pi$
$547$	− 9.52731e6i	− 1.36145i	0.732538	−	0.680726i	$-0.238335\pi$
$548$	6.85496e6i	0.975110i
$549$	158542.	0.0224498
$550$	0	0
$551$	7.13681e6	1.00144
$552$	4.49195e6i	0.627462i
$553$	2.58702e6i	0.359739i
$554$	2.16903e6	0.300256
$555$	0	0
$556$	6.67881e6	0.916246
$557$	3.80921e6i	0.520232i	0.965577	+	0.260116i	$0.0837609\pi$
$557$	3.80921e6i	0.520232i	−0.965577	+	0.260116i	$0.916239\pi$
$558$	813742.i	0.110637i
$559$	−22684.0	−0.00307036
$560$	0	0
$561$	−4.70722e6	−0.631476
$562$	1.16467e6i	0.155547i
$563$	− 1.42604e7i	− 1.89609i	−0.318131	−	0.948047i	$-0.603055\pi$
$563$	− 1.42604e7i	− 1.89609i	0.318131	−	0.948047i	$-0.396945\pi$
$564$	−3.95570e6	−0.523631
$565$	0	0
$566$	6.09483e6	0.799688
$567$	− 542111.i	− 0.0708159i
$568$	− 1.23338e7i	− 1.60408i
$569$	1.07341e6	0.138990	0.0694951	−	0.997582i	$-0.477861\pi$
$569$	1.07341e6	0.138990	0.0694951	+	0.997582i	$0.477861\pi$
$570$	0	0
$571$	−9.35129e6	−1.20028	−0.600138	−	0.799896i	$-0.704888\pi$
$571$	−9.35129e6	−1.20028	−0.600138	+	0.799896i	$0.704888\pi$
$572$	92285.0i	0.0117935i
$573$	4.70108e6i	0.598151i
$574$	−1.54118e6	−0.195242
$575$	0	0
$576$	86902.8	0.0109138
$577$	3.64162e6i	0.455361i	0.973736	+	0.227680i	$0.0731141\pi$
$577$	3.64162e6i	0.455361i	−0.973736	+	0.227680i	$0.926886\pi$
$578$	− 646875.i	− 0.0805380i
$579$	−5.21765e6	−0.646812
$580$	0	0
$581$	3.07056e6	0.377379
$582$	2.03943e6i	0.249575i
$583$	1.50760e7i	1.83703i
$584$	−839177.	−0.101817
$585$	0	0
$586$	4.41263e6	0.530828
$587$	− 7.66557e6i	− 0.918225i	−0.888378	−	0.459113i	$-0.848167\pi$
$587$	− 7.66557e6i	− 0.918225i	0.888378	−	0.459113i	$-0.151833\pi$
$588$	− 2.28199e6i	− 0.272189i
$589$	−1.19031e7	−1.41375
$590$	0	0
$591$	−5.19137e6	−0.611383
$592$	5.25506e6i	0.616273i
$593$	− 3.74383e6i	− 0.437199i	−0.975815	−	0.218599i	$-0.929851\pi$
$593$	− 3.74383e6i	− 0.437199i	0.975815	−	0.218599i	$-0.0701488\pi$
$594$	905931.	0.105349
$595$	0	0
$596$	−3.33339e6	−0.384389
$597$	− 4.44299e6i	− 0.510199i
$598$	− 65253.4i	− 0.00746192i
$599$	3.50682e6	0.399343	0.199672	−	0.979863i	$-0.436012\pi$
$599$	3.50682e6	0.399343	0.199672	+	0.979863i	$0.436012\pi$
$600$	0	0
$601$	−1.63306e7	−1.84423	−0.922116	−	0.386914i	$-0.873541\pi$
$601$	−1.63306e7	−1.84423	−0.922116	+	0.386914i	$0.873541\pi$
$602$	− 641240.i	− 0.0721156i
$603$	− 4.82733e6i	− 0.540647i
$604$	4.44980e6	0.496305
$605$	0	0
$606$	−2.06180e6	−0.228068
$607$	1.00001e7i	1.10163i	0.834628	+	0.550814i	$0.185683\pi$
$607$	1.00001e7i	1.10163i	−0.834628	+	0.550814i	$0.814317\pi$
$608$	1.77454e7i	1.94682i
$609$	1.74442e6	0.190593
$610$	0	0
$611$	129839.	0.0140703
$612$	2.22403e6i	0.240028i
$613$	− 7.99681e6i	− 0.859539i	−0.902939	−	0.429770i	$-0.858595\pi$
$613$	− 7.99681e6i	− 0.859539i	0.902939	−	0.429770i	$-0.141405\pi$
$614$	7.42722e6	0.795070
$615$	0	0
$616$	−5.89451e6	−0.625887
$617$	− 1.55620e7i	− 1.64571i	−0.568255	−	0.822853i	$-0.692381\pi$
$617$	− 1.55620e7i	− 1.64571i	0.568255	−	0.822853i	$-0.307619\pi$
$618$	223882.i	0.0235802i
$619$	9.45241e6	0.991553	0.495777	−	0.868450i	$-0.334883\pi$
$619$	9.45241e6	0.991553	0.495777	+	0.868450i	$0.334883\pi$
$620$	0	0
$621$	2.46833e6	0.256847
$622$	3.29843e6i	0.341846i
$623$	5.71697e6i	0.590128i
$624$	29350.1	0.00301751
$625$	0	0
$626$	−5.97854e6	−0.609760
$627$	1.32516e7i	1.34617i
$628$	− 1.21206e6i	− 0.122638i
$629$	1.30705e7	1.31724
$630$	0	0
$631$	−3.53523e6	−0.353464	−0.176732	−	0.984259i	$-0.556553\pi$
$631$	−3.53523e6	−0.353464	−0.176732	+	0.984259i	$0.556553\pi$
$632$	4.61528e6i	0.459627i
$633$	3.45488e6i	0.342708i
$634$	−7.35577e6	−0.726783
$635$	0	0
$636$	7.12301e6	0.698266
$637$	74902.8i	0.00731391i
$638$	2.91513e6i	0.283534i
$639$	−6.77744e6	−0.656619
$640$	0	0
$641$	1.19103e7	1.14493	0.572465	−	0.819929i	$-0.305987\pi$
$641$	1.19103e7	1.14493	0.572465	+	0.819929i	$0.305987\pi$
$642$	1.36375e6i	0.130586i
$643$	8.67256e6i	0.827218i	0.910455	+	0.413609i	$0.135732\pi$
$643$	8.67256e6i	0.827218i	−0.910455	+	0.413609i	$0.864268\pi$
$644$	−7.10789e6	−0.675345
$645$	0	0
$646$	8.44265e6	0.795971
$647$	− 1.21563e7i	− 1.14167i	−0.821063	−	0.570837i	$-0.806619\pi$
$647$	− 1.21563e7i	− 1.14167i	0.821063	−	0.570837i	$-0.193381\pi$
$648$	− 967135.i	− 0.0904793i
$649$	2.35738e7	2.19694
$650$	0	0
$651$	−2.90943e6	−0.269064
$652$	− 574667.i	− 0.0529417i
$653$	− 475958.i	− 0.0436803i	−0.999761	−	0.0218401i	$-0.993048\pi$
$653$	− 475958.i	− 0.0436803i	0.999761	−	0.0218401i	$-0.00695249\pi$
$654$	1.96963e6	0.180070
$655$	0	0
$656$	3.15627e6	0.286362
$657$	461128.i	0.0416781i
$658$	3.67035e6i	0.330478i
$659$	1.95724e7	1.75562	0.877810	−	0.479008i	$-0.159004\pi$
$659$	1.95724e7	1.75562	0.877810	+	0.479008i	$0.159004\pi$
$660$	0	0
$661$	−1.23639e7	−1.10066	−0.550328	−	0.834948i	$-0.685497\pi$
$661$	−1.23639e7	−1.10066	−0.550328	+	0.834948i	$0.685497\pi$
$662$	− 86337.6i	− 0.00765693i
$663$	− 73000.3i	− 0.00644972i
$664$	5.47793e6	0.482166
$665$	0	0
$666$	−2.51549e6	−0.219754
$667$	7.94265e6i	0.691275i
$668$	8.61520e6i	0.747006i
$669$	−2.89434e6	−0.250025
$670$	0	0
$671$	−947264.	−0.0812204
$672$	4.33743e6i	0.370518i
$673$	− 1.85745e6i	− 0.158081i	−0.996871	−	0.0790403i	$-0.974814\pi$
$673$	− 1.85745e6i	− 0.158081i	0.996871	−	0.0790403i	$-0.0251856\pi$
$674$	−3.03844e6	−0.257632
$675$	0	0
$676$	9.43186e6	0.793835
$677$	1.96586e7i	1.64847i	0.566246	+	0.824236i	$0.308395\pi$
$677$	1.96586e7i	1.64847i	−0.566246	+	0.824236i	$0.691605\pi$
$678$	− 2.62648e6i	− 0.219432i
$679$	−7.29171e6	−0.606953
$680$	0	0
$681$	−2.98262e6	−0.246450
$682$	− 4.86199e6i	− 0.400270i
$683$	1.00629e7i	0.825411i	0.910864	+	0.412706i	$0.135416\pi$
$683$	1.00629e7i	0.825411i	−0.910864	+	0.412706i	$0.864584\pi$
$684$	6.26102e6	0.511688
$685$	0	0
$686$	−5.68324e6	−0.461090
$687$	8.81962e6i	0.712949i
$688$	1.31323e6i	0.105772i
$689$	−233802.	−0.0187629
$690$	0	0
$691$	5.94278e6	0.473472	0.236736	−	0.971574i	$-0.423922\pi$
$691$	5.94278e6	0.473472	0.236736	+	0.971574i	$0.423922\pi$
$692$	− 6.83627e6i	− 0.542693i
$693$	3.23904e6i	0.256202i
$694$	−2.52319e6	−0.198862
$695$	0	0
$696$	3.11207e6	0.243515
$697$	− 7.85034e6i	− 0.612078i
$698$	9.92418e6i	0.771003i
$699$	2.78520e6	0.215607
$700$	0	0
$701$	9.79494e6	0.752847	0.376423	−	0.926448i	$-0.377154\pi$
$701$	9.79494e6	0.752847	0.376423	+	0.926448i	$0.377154\pi$
$702$	14049.3i	0.00107600i
$703$	− 3.67957e7i	− 2.80807i
$704$	−519231.	−0.0394847
$705$	0	0
$706$	−7.91824e6	−0.597884
$707$	− 7.37169e6i	− 0.554649i
$708$	− 1.11380e7i	− 0.835069i
$709$	−3.95802e6	−0.295707	−0.147854	−	0.989009i	$-0.547236\pi$
$709$	−3.95802e6	−0.295707	−0.147854	+	0.989009i	$0.547236\pi$
$710$	0	0
$711$	2.53610e6	0.188145
$712$	1.01992e7i	0.753988i
$713$	− 1.32471e7i	− 0.975884i
$714$	2.06360e6	0.151489
$715$	0	0
$716$	−4.91974e6	−0.358641
$717$	− 1.37289e7i	− 0.997331i
$718$	3.58149e6i	0.259270i
$719$	−1.16721e7	−0.842029	−0.421014	−	0.907054i	$-0.638326\pi$
$719$	−1.16721e7	−0.842029	−0.421014	+	0.907054i	$0.638326\pi$
$720$	0	0
$721$	−800460.	−0.0573458
$722$	− 1.74095e7i	− 1.24292i
$723$	6.95157e6i	0.494581i
$724$	4.21127e6	0.298584
$725$	0	0
$726$	−1.69094e6	−0.119065
$727$	− 1.05723e7i	− 0.741879i	−0.928657	−	0.370940i	$-0.879036\pi$
$727$	− 1.05723e7i	− 0.741879i	0.928657	−	0.370940i	$-0.120964\pi$
$728$	− 91413.0i	− 0.00639263i
$729$	−531441.	−0.0370370
$730$	0	0
$731$	3.26630e6	0.226080
$732$	447556.i	0.0308724i
$733$	2.80641e7i	1.92927i	0.263598	+	0.964633i	$0.415091\pi$
$733$	2.80641e7i	1.92927i	−0.263598	+	0.964633i	$0.584909\pi$
$734$	517554.	0.0354581
$735$	0	0
$736$	−1.97491e7	−1.34385
$737$	2.88426e7i	1.95599i
$738$	1.51084e6i	0.102112i
$739$	−2.17470e7	−1.46483	−0.732417	−	0.680857i	$-0.761608\pi$
$739$	−2.17470e7	−1.46483	−0.732417	+	0.680857i	$0.761608\pi$
$740$	0	0
$741$	−205508.	−0.0137494
$742$	− 6.60919e6i	− 0.440696i
$743$	− 1.68996e7i	− 1.12306i	−0.827456	−	0.561531i	$-0.810213\pi$
$743$	− 1.68996e7i	− 1.12306i	0.827456	−	0.561531i	$-0.189787\pi$
$744$	−5.19046e6	−0.343774
$745$	0	0
$746$	8.97234e6	0.590281
$747$	− 3.01013e6i	− 0.197371i
$748$	− 1.32883e7i	− 0.868389i
$749$	−4.87591e6	−0.317579
$750$	0	0
$751$	−3.68515e6	−0.238427	−0.119213	−	0.992869i	$-0.538037\pi$
$751$	−3.68515e6	−0.238427	−0.119213	+	0.992869i	$0.538037\pi$
$752$	− 7.51673e6i	− 0.484713i
$753$	3.61792e6i	0.232526i
$754$	−45208.3	−0.00289594
$755$	0	0
$756$	1.53036e6	0.0973841
$757$	1.58441e7i	1.00491i	0.864603	+	0.502455i	$0.167570\pi$
$757$	1.58441e7i	1.00491i	−0.864603	+	0.502455i	$0.832430\pi$
$758$	− 8.64720e6i	− 0.546641i
$759$	−1.47479e7	−0.929236
$760$	0	0
$761$	−1.08409e7	−0.678584	−0.339292	−	0.940681i	$-0.610187\pi$
$761$	−1.08409e7	−0.678584	−0.339292	+	0.940681i	$0.610187\pi$
$762$	− 6.83524e6i	− 0.426448i
$763$	7.04217e6i	0.437920i
$764$	−1.32709e7	−0.822561
$765$	0	0
$766$	9.06709e6	0.558337
$767$	365586.i	0.0224389i
$768$	4.55871e6i	0.278894i
$769$	−4.56486e6	−0.278363	−0.139182	−	0.990267i	$-0.544447\pi$
$769$	−4.56486e6	−0.278363	−0.139182	+	0.990267i	$0.544447\pi$
$770$	0	0
$771$	−4.36062e6	−0.264188
$772$	− 1.47292e7i	− 0.889478i
$773$	− 1.39415e7i	− 0.839189i	−0.907712	−	0.419595i	$-0.862172\pi$
$773$	− 1.39415e7i	− 0.839189i	0.907712	−	0.419595i	$-0.137828\pi$
$774$	−628618.	−0.0377168
$775$	0	0
$776$	−1.30085e7	−0.775485
$777$	− 8.99381e6i	− 0.534430i
$778$	8.98432e6i	0.532152i
$779$	−2.21001e7	−1.30482
$780$	0	0
$781$	4.04942e7	2.37556
$782$	9.39594e6i	0.549444i
$783$	− 1.71008e6i	− 0.0996812i
$784$	4.33631e6	0.251959
$785$	0	0
$786$	5.78048e6	0.333739
$787$	− 1.64075e7i	− 0.944290i	−0.881521	−	0.472145i	$-0.843480\pi$
$787$	− 1.64075e7i	− 0.944290i	0.881521	−	0.472145i	$-0.156520\pi$
$788$	− 1.46550e7i	− 0.840756i
$789$	8.81537e6	0.504136
$790$	0	0
$791$	9.39062e6	0.533646
$792$	5.77849e6i	0.327342i
$793$	− 14690.3i	0 0.000829561i
$794$	−1.07216e7	−0.603545
$795$	0	0
$796$	1.25424e7	0.701611
$797$	1.72151e7i	0.959981i	0.877274	+	0.479990i	$0.159360\pi$
$797$	1.72151e7i	0.959981i	−0.877274	+	0.479990i	$0.840640\pi$
$798$	− 5.80939e6i	− 0.322941i
$799$	−1.86958e7	−1.03604
$800$	0	0
$801$	5.60444e6	0.308639
$802$	2.29397e6i	0.125936i
$803$	− 2.75517e6i	− 0.150786i
$804$	1.36273e7	0.743482
$805$	0	0
$806$	75400.5	0.00408824
$807$	− 8.95868e6i	− 0.484240i
$808$	− 1.31512e7i	− 0.708658i
$809$	1.64138e7	0.881736	0.440868	−	0.897572i	$-0.354671\pi$
$809$	1.64138e7	0.881736	0.440868	+	0.897572i	$0.354671\pi$
$810$	0	0
$811$	1.74140e7	0.929706	0.464853	−	0.885388i	$-0.346107\pi$
$811$	1.74140e7	0.929706	0.464853	+	0.885388i	$0.346107\pi$
$812$	4.92442e6i	0.262099i
$813$	− 415583.i	− 0.0220512i
$814$	1.50297e7	0.795040
$815$	0	0
$816$	−4.22617e6	−0.222189
$817$	− 9.19519e6i	− 0.481954i
$818$	1.27168e6i	0.0664500i
$819$	−50231.5	−0.00261678
$820$	0	0
$821$	−1.90041e7	−0.983986	−0.491993	−	0.870599i	$-0.663731\pi$
$821$	−1.90041e7	−0.983986	−0.491993	+	0.870599i	$0.663731\pi$
$822$	− 6.23529e6i	− 0.321867i
$823$	2.26008e7i	1.16312i	0.813505	+	0.581559i	$0.197557\pi$
$823$	2.26008e7i	1.16312i	−0.813505	+	0.581559i	$0.802443\pi$
$824$	−1.42803e6	−0.0732689
$825$	0	0
$826$	−1.03345e7	−0.527036
$827$	− 1.71443e7i	− 0.871679i	−0.900024	−	0.435840i	$-0.856452\pi$
$827$	− 1.71443e7i	− 0.871679i	0.900024	−	0.435840i	$-0.143548\pi$
$828$	6.96798e6i	0.353208i
$829$	2.99057e7	1.51136	0.755680	−	0.654941i	$-0.227307\pi$
$829$	2.99057e7	1.51136	0.755680	+	0.654941i	$0.227307\pi$
$830$	0	0
$831$	7.60245e6	0.381901
$832$	− 8052.32i	0 0.000403286i
$833$	− 1.07854e7i	− 0.538546i
$834$	−6.07506e6	−0.302437
$835$	0	0
$836$	−3.74087e7	−1.85122
$837$	2.85216e6i	0.140721i
$838$	− 1.27609e7i	− 0.627727i
$839$	1.35881e7	0.666431	0.333215	−	0.942851i	$-0.391866\pi$
$839$	1.35881e7	0.666431	0.333215	+	0.942851i	$0.391866\pi$
$840$	0	0
$841$	−1.50084e7	−0.731719
$842$	2.39456e6i	0.116398i
$843$	4.08215e6i	0.197843i
$844$	−9.75298e6	−0.471282
$845$	0	0
$846$	3.59811e6	0.172842
$847$	− 6.04572e6i	− 0.289561i
$848$	1.35354e7i	0.646369i
$849$	2.13624e7	1.01714
$850$	0	0
$851$	4.09504e7	1.93836
$852$	− 1.91324e7i	− 0.902965i
$853$	9.45702e6i	0.445022i	0.974930	+	0.222511i	$0.0714253\pi$
$853$	9.45702e6i	0.445022i	−0.974930	+	0.222511i	$0.928575\pi$
$854$	415272.	0.0194844
$855$	0	0
$856$	−8.69870e6	−0.405761
$857$	4.10899e7i	1.91110i	0.294832	+	0.955549i	$0.404736\pi$
$857$	4.10899e7i	1.91110i	−0.294832	+	0.955549i	$0.595264\pi$
$858$	− 83942.7i	− 0.00389282i
$859$	−2.17103e7	−1.00388	−0.501941	−	0.864902i	$-0.667380\pi$
$859$	−2.17103e7	−1.00388	−0.501941	+	0.864902i	$0.667380\pi$
$860$	0	0
$861$	−5.40182e6	−0.248332
$862$	4.66631e6i	0.213897i
$863$	1.59005e7i	0.726750i	0.931643	+	0.363375i	$0.118376\pi$
$863$	1.59005e7i	0.726750i	−0.931643	+	0.363375i	$0.881624\pi$
$864$	4.25205e6	0.193782
$865$	0	0
$866$	−6.71485e6	−0.304258
$867$	− 2.26729e6i	− 0.102438i
$868$	− 8.21318e6i	− 0.370009i
$869$	−1.51528e7	−0.680682
$870$	0	0
$871$	−447296.	−0.0199779
$872$	1.25633e7i	0.559517i
$873$	7.14819e6i	0.317439i
$874$	2.64512e7	1.17129
$875$	0	0
$876$	−1.30174e6	−0.0573146
$877$	1.04121e7i	0.457129i	0.973529	+	0.228564i	$0.0734031\pi$
$877$	1.04121e7i	0.457129i	−0.973529	+	0.228564i	$0.926597\pi$
$878$	− 7.78388e6i	− 0.340769i
$879$	1.54662e7	0.675169
$880$	0	0
$881$	−4.04001e7	−1.75365	−0.876824	−	0.480811i	$-0.840342\pi$
$881$	−4.04001e7	−1.75365	−0.876824	+	0.480811i	$0.840342\pi$
$882$	2.07571e6i	0.0898451i
$883$	6.22610e6i	0.268729i	0.990932	+	0.134365i	$0.0428993\pi$
$883$	6.22610e6i	0.268729i	−0.990932	+	0.134365i	$0.957101\pi$
$884$	206076.	0.00886947
$885$	0	0
$886$	8.92470e6	0.381953
$887$	8.02230e6i	0.342365i	0.985239	+	0.171183i	$0.0547588\pi$
$887$	8.02230e6i	0.342365i	−0.985239	+	0.171183i	$0.945241\pi$
$888$	− 1.60451e7i	− 0.682825i
$889$	2.44385e7	1.03710
$890$	0	0
$891$	3.17528e6	0.133995
$892$	− 8.17058e6i	− 0.343828i
$893$	5.26318e7i	2.20861i
$894$	3.03206e6	0.126880
$895$	0	0
$896$	−1.51943e7	−0.632283
$897$	− 228713.i	− 0.00949095i
$898$	1.42004e7i	0.587637i
$899$	−9.17775e6	−0.378736
$900$	0	0
$901$	3.36654e7	1.38157
$902$	− 9.02707e6i	− 0.369429i
$903$	− 2.24754e6i	− 0.0917251i
$904$	1.67530e7	0.681823
$905$	0	0
$906$	−4.04755e6	−0.163822
$907$	3.16239e7i	1.27643i	0.769858	+	0.638215i	$0.220327\pi$
$907$	3.16239e7i	1.27643i	−0.769858	+	0.638215i	$0.779673\pi$
$908$	− 8.41979e6i	− 0.338912i
$909$	−7.22659e6	−0.290084
$910$	0	0
$911$	469497.	0.0187429	0.00937146	−	0.999956i	$-0.497017\pi$
$911$	469497.	0.0187429	0.00937146	+	0.999956i	$0.497017\pi$
$912$	1.18974e7i	0.473658i
$913$	1.79851e7i	0.714061i
$914$	3.66029e6	0.144927
$915$	0	0
$916$	−2.48974e7	−0.980427
$917$	2.06674e7i	0.811636i
$918$	− 2.02298e6i	− 0.0792292i
$919$	1.27281e7	0.497137	0.248568	−	0.968614i	$-0.420040\pi$
$919$	1.27281e7	0.497137	0.248568	+	0.968614i	$0.420040\pi$
$920$	0	0
$921$	2.60324e7	1.01126
$922$	1.08279e7i	0.419487i
$923$	627991.i	0.0242633i
$924$	−9.14365e6	−0.352322
$925$	0	0
$926$	1.07443e7	0.411764
$927$	784704.i	0.0299921i
$928$	1.36824e7i	0.521544i
$929$	−2.23196e7	−0.848492	−0.424246	−	0.905547i	$-0.639461\pi$
$929$	−2.23196e7	−0.848492	−0.424246	+	0.905547i	$0.639461\pi$
$930$	0	0
$931$	−3.03627e7	−1.14806
$932$	7.86248e6i	0.296497i
$933$	1.15610e7i	0.434801i
$934$	3.51216e6	0.131737
$935$	0	0
$936$	−89613.7	−0.00334337
$937$	− 4.56779e7i	− 1.69964i	−0.527072	−	0.849820i	$-0.676710\pi$
$937$	− 4.56779e7i	− 1.69964i	0.527072	−	0.849820i	$-0.323290\pi$
$938$	− 1.26443e7i	− 0.469233i
$939$	−2.09547e7	−0.775565
$940$	0	0
$941$	−6.91771e6	−0.254676	−0.127338	−	0.991859i	$-0.540643\pi$
$941$	−6.91771e6	−0.254676	−0.127338	+	0.991859i	$0.540643\pi$
$942$	1.10249e6i	0.0404808i
$943$	− 2.45955e7i	− 0.900691i
$944$	2.11647e7	0.773004
$945$	0	0
$946$	3.75590e6	0.136454
$947$	− 7.12916e6i	− 0.258323i	−0.991624	−	0.129162i	$-0.958771\pi$
$947$	− 7.12916e6i	− 0.258323i	0.991624	−	0.129162i	$-0.0412286\pi$
$948$	7.15930e6i	0.258732i
$949$	42727.7	0.00154008
$950$	0	0
$951$	−2.57819e7	−0.924408
$952$	1.31627e7i	0.470709i
$953$	− 2.68417e7i	− 0.957365i	−0.877988	−	0.478683i	$-0.841114\pi$
$953$	− 2.68417e7i	− 0.957365i	0.877988	−	0.478683i	$-0.158886\pi$
$954$	−6.47910e6	−0.230486
$955$	0	0
$956$	3.87562e7	1.37150
$957$	1.02175e7i	0.360633i
$958$	− 5.17900e6i	− 0.182319i
$959$	2.22935e7	0.782764
$960$	0	0
$961$	−1.33221e7	−0.465332
$962$	233083.i	0.00812031i
$963$	4.77994e6i	0.166095i
$964$	−1.96240e7	−0.680134
$965$	0	0
$966$	6.46535e6	0.222920
$967$	1.01997e7i	0.350770i	0.984500	+	0.175385i	$0.0561169\pi$
$967$	1.01997e7i	0.350770i	−0.984500	+	0.175385i	$0.943883\pi$
$968$	− 1.07857e7i	− 0.369963i
$969$	2.95914e7	1.01241
$970$	0	0
$971$	−3.54899e6	−0.120797	−0.0603986	−	0.998174i	$-0.519237\pi$
$971$	−3.54899e6	−0.120797	−0.0603986	+	0.998174i	$0.519237\pi$
$972$	− 1.50023e6i	− 0.0509323i
$973$	− 2.17206e7i	− 0.735511i
$974$	2.07459e7	0.700703
$975$	0	0
$976$	−850460.	−0.0285778
$977$	− 94269.8i	− 0.00315963i	−0.999999	−	0.00157981i	$-0.999497\pi$
$977$	− 94269.8i	− 0.00315963i	0.999999	−	0.00157981i	$-0.000502871\pi$
$978$	522719.i	0.0174751i
$979$	−3.34857e7	−1.11661
$980$	0	0
$981$	6.90356e6	0.229034
$982$	− 2.81041e6i	− 0.0930018i
$983$	− 2.17884e7i	− 0.719185i	−0.933109	−	0.359593i	$-0.882916\pi$
$983$	− 2.17884e7i	− 0.719185i	0.933109	−	0.359593i	$-0.117084\pi$
$984$	−9.63693e6	−0.317286
$985$	0	0
$986$	6.50960e6	0.213237
$987$	1.28646e7i	0.420341i
$988$	− 580140.i	− 0.0189078i
$989$	1.02335e7	0.332684
$990$	0	0
$991$	−672678.	−0.0217582	−0.0108791	−	0.999941i	$-0.503463\pi$
$991$	−672678.	−0.0217582	−0.0108791	+	0.999941i	$0.503463\pi$
$992$	− 2.28201e7i	− 0.736272i
$993$	− 302613.i	− 0.00973899i
$994$	−1.77523e7	−0.569887
$995$	0	0
$996$	8.49745e6	0.271419
$997$	− 391424.i	− 0.0124712i	−0.999981	−	0.00623562i	$-0.998015\pi$
$997$	− 391424.i	− 0.0124712i	0.999981	−	0.00623562i	$-0.00198487\pi$
$998$	8.34086e6i	0.265085i
$999$	−8.81678e6	−0.279509

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.6.b.f.49.2		4
3.2	odd	2		225.6.b.l.199.3		4
5.2	odd	4		75.6.a.g.1.2	✓	2
5.3	odd	4		75.6.a.i.1.1	yes	2
5.4	even	2	inner	75.6.b.f.49.3		4
15.2	even	4		225.6.a.t.1.1		2
15.8	even	4		225.6.a.j.1.2		2
15.14	odd	2		225.6.b.l.199.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.6.a.g.1.2	✓	2	5.2	odd	4
75.6.a.i.1.1	yes	2	5.3	odd	4
75.6.b.f.49.2		4	1.1	even	1	trivial
75.6.b.f.49.3		4	5.4	even	2	inner
225.6.a.j.1.2		2	15.8	even	4
225.6.a.t.1.1		2	15.2	even	4
225.6.b.l.199.2		4	15.14	odd	2
225.6.b.l.199.3		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:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	75.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.56776i q^{2} -9.00000i q^{3} +25.4066 q^{4} -23.1099 q^{6} -82.6263i q^{7} -147.407i q^{8} -81.0000 q^{9} +O(q^{10})\)	\(q-2.56776i q^{2} -9.00000i q^{3} +25.4066 q^{4} -23.1099 q^{6} -82.6263i q^{7} -147.407i q^{8} -81.0000 q^{9} +483.963 q^{11} -228.659i q^{12} +7.50538i q^{13} -212.165 q^{14} +434.505 q^{16} -1080.71i q^{17} +207.989i q^{18} -3042.38 q^{19} -743.637 q^{21} -1242.70i q^{22} -3385.91i q^{23} -1326.66 q^{24} +19.2720 q^{26} +729.000i q^{27} -2099.25i q^{28} -2345.79 q^{29} +3912.43 q^{31} -5832.72i q^{32} -4355.67i q^{33} -2775.01 q^{34} -2057.93 q^{36} +12094.3i q^{37} +7812.13i q^{38} +67.5484 q^{39} +7264.06 q^{41} +1909.48i q^{42} +3022.36i q^{43} +12295.9 q^{44} -8694.22 q^{46} -17299.5i q^{47} -3910.55i q^{48} +9979.89 q^{49} -9726.40 q^{51} +190.686i q^{52} +31151.2i q^{53} +1871.90 q^{54} -12179.7 q^{56} +27381.5i q^{57} +6023.45i q^{58} +48709.8 q^{59} -1957.31 q^{61} -10046.2i q^{62} +6692.73i q^{63} -1072.87 q^{64} -11184.3 q^{66} +59596.7i q^{67} -27457.2i q^{68} -30473.2 q^{69} +83672.1 q^{71} +11939.9i q^{72} -5692.94i q^{73} +31055.4 q^{74} -77296.6 q^{76} -39988.1i q^{77} -173.448i q^{78} -31309.9 q^{79} +6561.00 q^{81} -18652.4i q^{82} +37162.0i q^{83} -18893.3 q^{84} +7760.72 q^{86} +21112.2i q^{87} -71339.4i q^{88} -69190.7 q^{89} +620.142 q^{91} -86024.4i q^{92} -35211.8i q^{93} -44421.1 q^{94} -52494.5 q^{96} -88249.2i q^{97} -25626.0i q^{98} -39201.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 32 q^{4} + 108 q^{6} - 324 q^{9}+O(q^{10}) \) 4 * q - 32 * q^4 + 108 * q^6 - 324 * q^9	\( 4 q - 32 q^{4} + 108 q^{6} - 324 q^{9} - 24 q^{11} - 3588 q^{14} - 400 q^{16} - 8428 q^{19} + 1836 q^{21} - 4104 q^{24} + 18228 q^{26} - 8136 q^{29} - 5196 q^{31} + 5336 q^{34} + 2592 q^{36} - 18972 q^{39} + 22464 q^{41} + 65664 q^{44} - 83016 q^{46} - 14600 q^{49} - 30888 q^{51} - 8748 q^{54} + 5400 q^{56} + 127848 q^{59} + 14620 q^{61} + 94592 q^{64} - 98856 q^{66} + 7992 q^{69} + 196608 q^{71} + 193752 q^{74} - 57568 q^{76} - 168000 q^{79} + 26244 q^{81} - 175392 q^{84} - 77484 q^{86} - 206208 q^{89} + 393204 q^{91} - 427432 q^{94} - 249264 q^{96} + 1944 q^{99}+O(q^{100}) \) 4 * q - 32 * q^4 + 108 * q^6 - 324 * q^9 - 24 * q^11 - 3588 * q^14 - 400 * q^16 - 8428 * q^19 + 1836 * q^21 - 4104 * q^24 + 18228 * q^26 - 8136 * q^29 - 5196 * q^31 + 5336 * q^34 + 2592 * q^36 - 18972 * q^39 + 22464 * q^41 + 65664 * q^44 - 83016 * q^46 - 14600 * q^49 - 30888 * q^51 - 8748 * q^54 + 5400 * q^56 + 127848 * q^59 + 14620 * q^61 + 94592 * q^64 - 98856 * q^66 + 7992 * q^69 + 196608 * q^71 + 193752 * q^74 - 57568 * q^76 - 168000 * q^79 + 26244 * q^81 - 175392 * q^84 - 77484 * q^86 - 206208 * q^89 + 393204 * q^91 - 427432 * q^94 - 249264 * q^96 + 1944 * q^99

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