LMFDB - Embedded newform 1875.2.a.p.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1875,2,Mod(1,1875)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1875, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1875.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1875 = 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1875.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:	$14.9719503790$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.5444000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 2x^{6} + 20x^{5} - 4x^{4} - 30x^{3} + 7x^{2} + 12x + 1 $ x^8 - 4x^7 - 2x^6 + 20x^5 - 4x^4 - 30x^3 + 7x^2 + 12*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 75)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.536547$ of defining polynomial
Character	$\chi$	$=$	1875.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.536547 q^{2} -1.00000 q^{3} -1.71212 q^{4} +0.536547 q^{6} -2.57318 q^{7} +1.99173 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.536547 q^{2} -1.00000 q^{3} -1.71212 q^{4} +0.536547 q^{6} -2.57318 q^{7} +1.99173 q^{8} +1.00000 q^{9} -5.14369 q^{11} +1.71212 q^{12} +1.47492 q^{13} +1.38063 q^{14} +2.35558 q^{16} -0.687859 q^{17} -0.536547 q^{18} -8.09265 q^{19} +2.57318 q^{21} +2.75983 q^{22} -0.372750 q^{23} -1.99173 q^{24} -0.791365 q^{26} -1.00000 q^{27} +4.40559 q^{28} +0.0356723 q^{29} -4.48902 q^{31} -5.24733 q^{32} +5.14369 q^{33} +0.369069 q^{34} -1.71212 q^{36} -1.90935 q^{37} +4.34209 q^{38} -1.47492 q^{39} -5.16925 q^{41} -1.38063 q^{42} +11.4506 q^{43} +8.80660 q^{44} +0.199998 q^{46} -8.52114 q^{47} -2.35558 q^{48} -0.378747 q^{49} +0.687859 q^{51} -2.52524 q^{52} +9.12317 q^{53} +0.536547 q^{54} -5.12507 q^{56} +8.09265 q^{57} -0.0191399 q^{58} -0.176190 q^{59} -6.41152 q^{61} +2.40857 q^{62} -2.57318 q^{63} -1.89572 q^{64} -2.75983 q^{66} -0.0834377 q^{67} +1.17770 q^{68} +0.372750 q^{69} +12.1578 q^{71} +1.99173 q^{72} +12.0518 q^{73} +1.02446 q^{74} +13.8556 q^{76} +13.2356 q^{77} +0.791365 q^{78} +4.95687 q^{79} +1.00000 q^{81} +2.77354 q^{82} -9.36322 q^{83} -4.40559 q^{84} -6.14378 q^{86} -0.0356723 q^{87} -10.2448 q^{88} -0.0123190 q^{89} -3.79524 q^{91} +0.638192 q^{92} +4.48902 q^{93} +4.57199 q^{94} +5.24733 q^{96} +7.62041 q^{97} +0.203215 q^{98} -5.14369 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{2} - 8 q^{3} + 4 q^{4} - 4 q^{6} + 8 q^{7} + 12 q^{8} + 8 q^{9}+O(q^{10}) $ 8 * q + 4 * q^2 - 8 * q^3 + 4 * q^4 - 4 * q^6 + 8 * q^7 + 12 * q^8 + 8 * q^9	$ 8 q + 4 q^{2} - 8 q^{3} + 4 q^{4} - 4 q^{6} + 8 q^{7} + 12 q^{8} + 8 q^{9} + 2 q^{11} - 4 q^{12} + 16 q^{13} + 6 q^{14} + 16 q^{17} + 4 q^{18} - 14 q^{19} - 8 q^{21} + 12 q^{22} + 14 q^{23} - 12 q^{24} + 6 q^{26} - 8 q^{27} + 16 q^{28} + 2 q^{29} - 22 q^{31} - 2 q^{32} - 2 q^{33} - 12 q^{34} + 4 q^{36} + 28 q^{37} - 16 q^{38} - 16 q^{39} + 8 q^{41} - 6 q^{42} + 20 q^{43} + 22 q^{44} - 2 q^{46} + 10 q^{47} - 16 q^{51} + 16 q^{52} + 44 q^{53} - 4 q^{54} + 30 q^{56} + 14 q^{57} + 8 q^{58} + 14 q^{59} - 20 q^{61} + 16 q^{62} + 8 q^{63} + 6 q^{64} - 12 q^{66} + 16 q^{67} - 2 q^{68} - 14 q^{69} + 16 q^{71} + 12 q^{72} + 24 q^{73} + 26 q^{74} - 16 q^{76} + 46 q^{77} - 6 q^{78} - 30 q^{79} + 8 q^{81} + 16 q^{82} + 12 q^{83} - 16 q^{84} + 32 q^{86} - 2 q^{87} + 32 q^{88} + 16 q^{89} - 12 q^{91} - 2 q^{92} + 22 q^{93} + 14 q^{94} + 2 q^{96} + 16 q^{97} + 4 q^{98} + 2 q^{99}+O(q^{100}) $ 8 * q + 4 * q^2 - 8 * q^3 + 4 * q^4 - 4 * q^6 + 8 * q^7 + 12 * q^8 + 8 * q^9 + 2 * q^11 - 4 * q^12 + 16 * q^13 + 6 * q^14 + 16 * q^17 + 4 * q^18 - 14 * q^19 - 8 * q^21 + 12 * q^22 + 14 * q^23 - 12 * q^24 + 6 * q^26 - 8 * q^27 + 16 * q^28 + 2 * q^29 - 22 * q^31 - 2 * q^32 - 2 * q^33 - 12 * q^34 + 4 * q^36 + 28 * q^37 - 16 * q^38 - 16 * q^39 + 8 * q^41 - 6 * q^42 + 20 * q^43 + 22 * q^44 - 2 * q^46 + 10 * q^47 - 16 * q^51 + 16 * q^52 + 44 * q^53 - 4 * q^54 + 30 * q^56 + 14 * q^57 + 8 * q^58 + 14 * q^59 - 20 * q^61 + 16 * q^62 + 8 * q^63 + 6 * q^64 - 12 * q^66 + 16 * q^67 - 2 * q^68 - 14 * q^69 + 16 * q^71 + 12 * q^72 + 24 * q^73 + 26 * q^74 - 16 * q^76 + 46 * q^77 - 6 * q^78 - 30 * q^79 + 8 * q^81 + 16 * q^82 + 12 * q^83 - 16 * q^84 + 32 * q^86 - 2 * q^87 + 32 * q^88 + 16 * q^89 - 12 * q^91 - 2 * q^92 + 22 * q^93 + 14 * q^94 + 2 * q^96 + 16 * q^97 + 4 * q^98 + 2 * 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 $

$2$	−0.536547	−0.379396	−0.189698	−	0.981842i	$-0.560751\pi$
$2$	−0.536547	−0.379396	−0.189698	+	0.981842i	$0.560751\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.71212	−0.856059
$5$	0	0
$5$	0	0
$6$	0.536547	0.219044
$7$	−2.57318	−0.972570	−0.486285	−	0.873800i	$-0.661648\pi$
$7$	−2.57318	−0.972570	−0.486285	+	0.873800i	$0.661648\pi$
$8$	1.99173	0.704181
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.14369	−1.55088	−0.775440	−	0.631421i	$-0.782472\pi$
$11$	−5.14369	−1.55088	−0.775440	+	0.631421i	$0.782472\pi$
$12$	1.71212	0.494246
$13$	1.47492	0.409070	0.204535	−	0.978859i	$-0.434432\pi$
$13$	1.47492	0.409070	0.204535	+	0.978859i	$0.434432\pi$
$14$	1.38063	0.368989
$15$	0	0
$16$	2.35558	0.588895
$17$	−0.687859	−0.166830	−0.0834152	−	0.996515i	$-0.526583\pi$
$17$	−0.687859	−0.166830	−0.0834152	+	0.996515i	$0.526583\pi$
$18$	−0.536547	−0.126465
$19$	−8.09265	−1.85658	−0.928291	−	0.371855i	$-0.878722\pi$
$19$	−8.09265	−1.85658	−0.928291	+	0.371855i	$0.878722\pi$
$20$	0	0
$21$	2.57318	0.561514
$22$	2.75983	0.588398
$23$	−0.372750	−0.0777237	−0.0388619	−	0.999245i	$-0.512373\pi$
$23$	−0.372750	−0.0777237	−0.0388619	+	0.999245i	$0.512373\pi$
$24$	−1.99173	−0.406559
$25$	0	0
$26$	−0.791365	−0.155200
$27$	−1.00000	−0.192450
$28$	4.40559	0.832577
$29$	0.0356723	0.00662418	0.00331209	−	0.999995i	$-0.498946\pi$
$29$	0.0356723	0.00662418	0.00331209	+	0.999995i	$0.498946\pi$
$30$	0	0
$31$	−4.48902	−0.806252	−0.403126	−	0.915144i	$-0.632076\pi$
$31$	−4.48902	−0.806252	−0.403126	+	0.915144i	$0.632076\pi$
$32$	−5.24733	−0.927606
$33$	5.14369	0.895401
$34$	0.369069	0.0632948
$35$	0	0
$36$	−1.71212	−0.285353
$37$	−1.90935	−0.313896	−0.156948	−	0.987607i	$-0.550165\pi$
$37$	−1.90935	−0.313896	−0.156948	+	0.987607i	$0.550165\pi$
$38$	4.34209	0.704380
$39$	−1.47492	−0.236177
$40$	0	0
$41$	−5.16925	−0.807301	−0.403650	−	0.914913i	$-0.632259\pi$
$41$	−5.16925	−0.807301	−0.403650	+	0.914913i	$0.632259\pi$
$42$	−1.38063	−0.213036
$43$	11.4506	1.74620	0.873099	−	0.487543i	$-0.162107\pi$
$43$	11.4506	1.74620	0.873099	+	0.487543i	$0.162107\pi$
$44$	8.80660	1.32764
$45$	0	0
$46$	0.199998	0.0294881
$47$	−8.52114	−1.24294	−0.621468	−	0.783440i	$-0.713463\pi$
$47$	−8.52114	−1.24294	−0.621468	+	0.783440i	$0.713463\pi$
$48$	−2.35558	−0.339999
$49$	−0.378747	−0.0541067
$50$	0	0
$51$	0.687859	0.0963196
$52$	−2.52524	−0.350188
$53$	9.12317	1.25316	0.626582	−	0.779355i	$-0.284453\pi$
$53$	9.12317	1.25316	0.626582	+	0.779355i	$0.284453\pi$
$54$	0.536547	0.0730148
$55$	0	0
$56$	−5.12507	−0.684866
$57$	8.09265	1.07190
$58$	−0.0191399	−0.00251319
$59$	−0.176190	−0.0229380	−0.0114690	−	0.999934i	$-0.503651\pi$
$59$	−0.176190	−0.0229380	−0.0114690	+	0.999934i	$0.503651\pi$
$60$	0	0
$61$	−6.41152	−0.820911	−0.410456	−	0.911881i	$-0.634630\pi$
$61$	−6.41152	−0.820911	−0.410456	+	0.911881i	$0.634630\pi$
$62$	2.40857	0.305889
$63$	−2.57318	−0.324190
$64$	−1.89572	−0.236965
$65$	0	0
$66$	−2.75983	−0.339712
$67$	−0.0834377	−0.0101935	−0.00509677	−	0.999987i	$-0.501622\pi$
$67$	−0.0834377	−0.0101935	−0.00509677	+	0.999987i	$0.501622\pi$
$68$	1.17770	0.142817
$69$	0.372750	0.0448738
$70$	0	0
$71$	12.1578	1.44287	0.721434	−	0.692483i	$-0.243483\pi$
$71$	12.1578	1.44287	0.721434	+	0.692483i	$0.243483\pi$
$72$	1.99173	0.234727
$73$	12.0518	1.41056	0.705279	−	0.708930i	$-0.250822\pi$
$73$	12.0518	1.41056	0.705279	+	0.708930i	$0.250822\pi$
$74$	1.02446	0.119091
$75$	0	0
$76$	13.8556	1.58934
$77$	13.2356	1.50834
$78$	0.791365	0.0896045
$79$	4.95687	0.557691	0.278846	−	0.960336i	$-0.410048\pi$
$79$	4.95687	0.557691	0.278846	+	0.960336i	$0.410048\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	2.77354	0.306287
$83$	−9.36322	−1.02775	−0.513874	−	0.857866i	$-0.671790\pi$
$83$	−9.36322	−1.02775	−0.513874	+	0.857866i	$0.671790\pi$
$84$	−4.40559	−0.480689
$85$	0	0
$86$	−6.14378	−0.662500
$87$	−0.0356723	−0.00382447
$88$	−10.2448	−1.09210
$89$	−0.0123190	−0.00130581	−0.000652905	−	1.00000i	$-0.500208\pi$
$89$	−0.0123190	−0.00130581	−0.000652905		1.00000i	$0.500208\pi$
$90$	0	0
$91$	−3.79524	−0.397850
$92$	0.638192	0.0665361
$93$	4.48902	0.465490
$94$	4.57199	0.471565
$95$	0	0
$96$	5.24733	0.535553
$97$	7.62041	0.773736	0.386868	−	0.922135i	$-0.373557\pi$
$97$	7.62041	0.773736	0.386868	+	0.922135i	$0.373557\pi$
$98$	0.203215	0.0205278
$99$	−5.14369	−0.516960
$100$	0	0
$101$	8.27518	0.823411	0.411706	−	0.911317i	$-0.364933\pi$
$101$	8.27518	0.823411	0.411706	+	0.911317i	$0.364933\pi$
$102$	−0.369069	−0.0365433
$103$	−10.3860	−1.02336	−0.511682	−	0.859175i	$-0.670978\pi$
$103$	−10.3860	−1.02336	−0.511682	+	0.859175i	$0.670978\pi$
$104$	2.93764	0.288059
$105$	0	0
$106$	−4.89501	−0.475445
$107$	12.2737	1.18655	0.593274	−	0.805001i	$-0.297835\pi$
$107$	12.2737	1.18655	0.593274	+	0.805001i	$0.297835\pi$
$108$	1.71212	0.164749
$109$	−4.22902	−0.405066	−0.202533	−	0.979275i	$-0.564917\pi$
$109$	−4.22902	−0.405066	−0.202533	+	0.979275i	$0.564917\pi$
$110$	0	0
$111$	1.90935	0.181228
$112$	−6.06133	−0.572742
$113$	18.3687	1.72798	0.863989	−	0.503511i	$-0.167959\pi$
$113$	18.3687	1.72798	0.863989	+	0.503511i	$0.167959\pi$
$114$	−4.34209	−0.406674
$115$	0	0
$116$	−0.0610751	−0.00567068
$117$	1.47492	0.136357
$118$	0.0945341	0.00870257
$119$	1.76999	0.162254
$120$	0	0
$121$	15.4575	1.40523
$122$	3.44008	0.311450
$123$	5.16925	0.466095
$124$	7.68573	0.690199
$125$	0	0
$126$	1.38063	0.122996
$127$	−0.345501	−0.0306582	−0.0153291	−	0.999883i	$-0.504880\pi$
$127$	−0.345501	−0.0306582	−0.0153291	+	0.999883i	$0.504880\pi$
$128$	11.5118	1.01751
$129$	−11.4506	−1.00817
$130$	0	0
$131$	−4.56651	−0.398978	−0.199489	−	0.979900i	$-0.563928\pi$
$131$	−4.56651	−0.398978	−0.199489	+	0.979900i	$0.563928\pi$
$132$	−8.80660	−0.766516
$133$	20.8238	1.80566
$134$	0.0447683	0.00386739
$135$	0	0
$136$	−1.37003	−0.117479
$137$	4.81753	0.411589	0.205795	−	0.978595i	$-0.434022\pi$
$137$	4.81753	0.411589	0.205795	+	0.978595i	$0.434022\pi$
$138$	−0.199998	−0.0170249
$139$	−3.74075	−0.317286	−0.158643	−	0.987336i	$-0.550712\pi$
$139$	−3.74075	−0.317286	−0.158643	+	0.987336i	$0.550712\pi$
$140$	0	0
$141$	8.52114	0.717609
$142$	−6.52325	−0.547419
$143$	−7.58655	−0.634419
$144$	2.35558	0.196298
$145$	0	0
$146$	−6.46636	−0.535160
$147$	0.378747	0.0312385
$148$	3.26904	0.268713
$149$	8.64621	0.708325	0.354163	−	0.935184i	$-0.384766\pi$
$149$	8.64621	0.708325	0.354163	+	0.935184i	$0.384766\pi$
$150$	0	0
$151$	−1.24898	−0.101641	−0.0508205	−	0.998708i	$-0.516184\pi$
$151$	−1.24898	−0.101641	−0.0508205	+	0.998708i	$0.516184\pi$
$152$	−16.1183	−1.30737
$153$	−0.687859	−0.0556101
$154$	−7.10154	−0.572258
$155$	0	0
$156$	2.52524	0.202181
$157$	−3.86574	−0.308520	−0.154260	−	0.988030i	$-0.549299\pi$
$157$	−3.86574	−0.308520	−0.154260	+	0.988030i	$0.549299\pi$
$158$	−2.65959	−0.211586
$159$	−9.12317	−0.723515
$160$	0	0
$161$	0.959153	0.0755918
$162$	−0.536547	−0.0421551
$163$	6.97411	0.546254	0.273127	−	0.961978i	$-0.411942\pi$
$163$	6.97411	0.546254	0.273127	+	0.961978i	$0.411942\pi$
$164$	8.85036	0.691097
$165$	0	0
$166$	5.02381	0.389923
$167$	15.9594	1.23497	0.617487	−	0.786581i	$-0.288151\pi$
$167$	15.9594	1.23497	0.617487	+	0.786581i	$0.288151\pi$
$168$	5.12507	0.395407
$169$	−10.8246	−0.832662
$170$	0	0
$171$	−8.09265	−0.618861
$172$	−19.6047	−1.49485
$173$	7.67330	0.583390	0.291695	−	0.956511i	$-0.405781\pi$
$173$	7.67330	0.583390	0.291695	+	0.956511i	$0.405781\pi$
$174$	0.0191399	0.00145099
$175$	0	0
$176$	−12.1164	−0.913306
$177$	0.176190	0.0132432
$178$	0.00660971	0.000495419 0
$179$	10.6093	0.792977	0.396489	−	0.918040i	$-0.370229\pi$
$179$	10.6093	0.792977	0.396489	+	0.918040i	$0.370229\pi$
$180$	0	0
$181$	15.1076	1.12294	0.561471	−	0.827496i	$-0.310236\pi$
$181$	15.1076	1.12294	0.561471	+	0.827496i	$0.310236\pi$
$182$	2.03633	0.150942
$183$	6.41152	0.473953
$184$	−0.742415	−0.0547316
$185$	0	0
$186$	−2.40857	−0.176605
$187$	3.53813	0.258734
$188$	14.5892	1.06403
$189$	2.57318	0.187171
$190$	0	0
$191$	12.2440	0.885941	0.442971	−	0.896536i	$-0.353925\pi$
$191$	12.2440	0.885941	0.442971	+	0.896536i	$0.353925\pi$
$192$	1.89572	0.136812
$193$	−14.2421	−1.02517	−0.512585	−	0.858637i	$-0.671312\pi$
$193$	−14.2421	−1.02517	−0.512585	+	0.858637i	$0.671312\pi$
$194$	−4.08871	−0.293552
$195$	0	0
$196$	0.648459	0.0463185
$197$	17.7488	1.26455	0.632276	−	0.774743i	$-0.282121\pi$
$197$	17.7488	1.26455	0.632276	+	0.774743i	$0.282121\pi$
$198$	2.75983	0.196133
$199$	−18.9550	−1.34369	−0.671843	−	0.740693i	$-0.734497\pi$
$199$	−18.9550	−1.34369	−0.671843	+	0.740693i	$0.734497\pi$
$200$	0	0
$201$	0.0834377	0.00588524
$202$	−4.44002	−0.312399
$203$	−0.0917912	−0.00644248
$204$	−1.17770	−0.0824552
$205$	0	0
$206$	5.57259	0.388260
$207$	−0.372750	−0.0259079
$208$	3.47430	0.240899
$209$	41.6261	2.87934
$210$	0	0
$211$	−10.7768	−0.741907	−0.370954	−	0.928651i	$-0.620969\pi$
$211$	−10.7768	−0.741907	−0.370954	+	0.928651i	$0.620969\pi$
$212$	−15.6199	−1.07278
$213$	−12.1578	−0.833041
$214$	−6.58544	−0.450171
$215$	0	0
$216$	−1.99173	−0.135520
$217$	11.5511	0.784137
$218$	2.26907	0.153681
$219$	−12.0518	−0.814386
$220$	0	0
$221$	−1.01454	−0.0682453
$222$	−1.02446	−0.0687571
$223$	−27.5707	−1.84627	−0.923137	−	0.384472i	$-0.874383\pi$
$223$	−27.5707	−1.84627	−0.923137	+	0.384472i	$0.874383\pi$
$224$	13.5023	0.902162
$225$	0	0
$226$	−9.85565	−0.655588
$227$	3.98221	0.264309	0.132154	−	0.991229i	$-0.457811\pi$
$227$	3.98221	0.264309	0.132154	+	0.991229i	$0.457811\pi$
$228$	−13.8556	−0.917608
$229$	11.9697	0.790983	0.395492	−	0.918470i	$-0.370574\pi$
$229$	11.9697	0.790983	0.395492	+	0.918470i	$0.370574\pi$
$230$	0	0
$231$	−13.2356	−0.870841
$232$	0.0710494	0.00466462
$233$	−21.1181	−1.38349	−0.691747	−	0.722140i	$-0.743159\pi$
$233$	−21.1181	−1.38349	−0.691747	+	0.722140i	$0.743159\pi$
$234$	−0.791365	−0.0517332
$235$	0	0
$236$	0.301658	0.0196362
$237$	−4.95687	−0.321983
$238$	−0.949680	−0.0615586
$239$	26.2050	1.69506	0.847529	−	0.530749i	$-0.178089\pi$
$239$	26.2050	1.69506	0.847529	+	0.530749i	$0.178089\pi$
$240$	0	0
$241$	−8.17452	−0.526567	−0.263284	−	0.964719i	$-0.584805\pi$
$241$	−8.17452	−0.526567	−0.263284	+	0.964719i	$0.584805\pi$
$242$	−8.29369	−0.533139
$243$	−1.00000	−0.0641500
$244$	10.9773	0.702748
$245$	0	0
$246$	−2.77354	−0.176835
$247$	−11.9360	−0.759472
$248$	−8.94090	−0.567748
$249$	9.36322	0.593370
$250$	0	0
$251$	24.8145	1.56628	0.783139	−	0.621847i	$-0.213618\pi$
$251$	24.8145	1.56628	0.783139	+	0.621847i	$0.213618\pi$
$252$	4.40559	0.277526
$253$	1.91731	0.120540
$254$	0.185377	0.0116316
$255$	0	0
$256$	−2.38518	−0.149074
$257$	−24.2995	−1.51576	−0.757882	−	0.652392i	$-0.773766\pi$
$257$	−24.2995	−1.51576	−0.757882	+	0.652392i	$0.773766\pi$
$258$	6.14378	0.382495
$259$	4.91311	0.305286
$260$	0	0
$261$	0.0356723	0.00220806
$262$	2.45015	0.151371
$263$	−8.52269	−0.525532	−0.262766	−	0.964860i	$-0.584635\pi$
$263$	−8.52269	−0.525532	−0.262766	+	0.964860i	$0.584635\pi$
$264$	10.2448	0.630525
$265$	0	0
$266$	−11.1730	−0.685059
$267$	0.0123190	0.000753909 0
$268$	0.142855	0.00872627
$269$	2.58312	0.157496	0.0787479	−	0.996895i	$-0.474908\pi$
$269$	2.58312	0.157496	0.0787479	+	0.996895i	$0.474908\pi$
$270$	0	0
$271$	−13.5893	−0.825491	−0.412746	−	0.910846i	$-0.635430\pi$
$271$	−13.5893	−0.825491	−0.412746	+	0.910846i	$0.635430\pi$
$272$	−1.62031	−0.0982456
$273$	3.79524	0.229699
$274$	−2.58483	−0.156155
$275$	0	0
$276$	−0.638192	−0.0384146
$277$	−9.13975	−0.549154	−0.274577	−	0.961565i	$-0.588538\pi$
$277$	−9.13975	−0.549154	−0.274577	+	0.961565i	$0.588538\pi$
$278$	2.00709	0.120377
$279$	−4.48902	−0.268751
$280$	0	0
$281$	10.4101	0.621014	0.310507	−	0.950571i	$-0.399501\pi$
$281$	10.4101	0.621014	0.310507	+	0.950571i	$0.399501\pi$
$282$	−4.57199	−0.272258
$283$	−4.81586	−0.286273	−0.143137	−	0.989703i	$-0.545719\pi$
$283$	−4.81586	−0.286273	−0.143137	+	0.989703i	$0.545719\pi$
$284$	−20.8156	−1.23518
$285$	0	0
$286$	4.07054	0.240696
$287$	13.3014	0.785157
$288$	−5.24733	−0.309202
$289$	−16.5268	−0.972168
$290$	0	0
$291$	−7.62041	−0.446716
$292$	−20.6341	−1.20752
$293$	−24.4506	−1.42842	−0.714210	−	0.699932i	$-0.753214\pi$
$293$	−24.4506	−1.42842	−0.714210	+	0.699932i	$0.753214\pi$
$294$	−0.203215	−0.0118518
$295$	0	0
$296$	−3.80291	−0.221039
$297$	5.14369	0.298467
$298$	−4.63910	−0.268736
$299$	−0.549778	−0.0317945
$300$	0	0
$301$	−29.4644	−1.69830
$302$	0.670139	0.0385622
$303$	−8.27518	−0.475397
$304$	−19.0629	−1.09333
$305$	0	0
$306$	0.369069	0.0210983
$307$	9.51655	0.543138	0.271569	−	0.962419i	$-0.412457\pi$
$307$	9.51655	0.543138	0.271569	+	0.962419i	$0.412457\pi$
$308$	−22.6610	−1.29123
$309$	10.3860	0.590840
$310$	0	0
$311$	24.9814	1.41656	0.708282	−	0.705930i	$-0.249471\pi$
$311$	24.9814	1.41656	0.708282	+	0.705930i	$0.249471\pi$
$312$	−2.93764	−0.166311
$313$	−9.88161	−0.558542	−0.279271	−	0.960212i	$-0.590093\pi$
$313$	−9.88161	−0.558542	−0.279271	+	0.960212i	$0.590093\pi$
$314$	2.07415	0.117051
$315$	0	0
$316$	−8.48674	−0.477416
$317$	−24.7310	−1.38903	−0.694516	−	0.719477i	$-0.744382\pi$
$317$	−24.7310	−1.38903	−0.694516	+	0.719477i	$0.744382\pi$
$318$	4.89501	0.274499
$319$	−0.183487	−0.0102733
$320$	0	0
$321$	−12.2737	−0.685054
$322$	−0.514630	−0.0286792
$323$	5.56661	0.309734
$324$	−1.71212	−0.0951176
$325$	0	0
$326$	−3.74194	−0.207247
$327$	4.22902	0.233865
$328$	−10.2957	−0.568486
$329$	21.9264	1.20884
$330$	0	0
$331$	1.07827	0.0592669	0.0296335	−	0.999561i	$-0.490566\pi$
$331$	1.07827	0.0592669	0.0296335	+	0.999561i	$0.490566\pi$
$332$	16.0309	0.879812
$333$	−1.90935	−0.104632
$334$	−8.56296	−0.468544
$335$	0	0
$336$	6.06133	0.330673
$337$	14.0609	0.765948	0.382974	−	0.923759i	$-0.374900\pi$
$337$	14.0609	0.765948	0.382974	+	0.923759i	$0.374900\pi$
$338$	5.80791	0.315908
$339$	−18.3687	−0.997648
$340$	0	0
$341$	23.0901	1.25040
$342$	4.34209	0.234793
$343$	18.9868	1.02519
$344$	22.8064	1.22964
$345$	0	0
$346$	−4.11709	−0.221336
$347$	5.69584	0.305769	0.152884	−	0.988244i	$-0.451144\pi$
$347$	5.69584	0.305769	0.152884	+	0.988244i	$0.451144\pi$
$348$	0.0610751	0.00327397
$349$	−28.4950	−1.52530	−0.762651	−	0.646810i	$-0.776103\pi$
$349$	−28.4950	−1.52530	−0.762651	+	0.646810i	$0.776103\pi$
$350$	0	0
$351$	−1.47492	−0.0787256
$352$	26.9906	1.43861
$353$	12.7656	0.679443	0.339721	−	0.940526i	$-0.389667\pi$
$353$	12.7656	0.679443	0.339721	+	0.940526i	$0.389667\pi$
$354$	−0.0945341	−0.00502443
$355$	0	0
$356$	0.0210915	0.00111785
$357$	−1.76999	−0.0936776
$358$	−5.69240	−0.300852
$359$	−4.51821	−0.238462	−0.119231	−	0.992867i	$-0.538043\pi$
$359$	−4.51821	−0.238462	−0.119231	+	0.992867i	$0.538043\pi$
$360$	0	0
$361$	46.4910	2.44690
$362$	−8.10596	−0.426040
$363$	−15.4575	−0.811310
$364$	6.49790	0.340583
$365$	0	0
$366$	−3.44008	−0.179816
$367$	3.03446	0.158398	0.0791989	−	0.996859i	$-0.474764\pi$
$367$	3.03446	0.158398	0.0791989	+	0.996859i	$0.474764\pi$
$368$	−0.878043	−0.0457711
$369$	−5.16925	−0.269100
$370$	0	0
$371$	−23.4756	−1.21879
$372$	−7.68573	−0.398487
$373$	15.3659	0.795614	0.397807	−	0.917469i	$-0.369771\pi$
$373$	15.3659	0.795614	0.397807	+	0.917469i	$0.369771\pi$
$374$	−1.89838	−0.0981626
$375$	0	0
$376$	−16.9718	−0.875252
$377$	0.0526139	0.00270975
$378$	−1.38063	−0.0710120
$379$	10.3597	0.532142	0.266071	−	0.963953i	$-0.414275\pi$
$379$	10.3597	0.532142	0.266071	+	0.963953i	$0.414275\pi$
$380$	0	0
$381$	0.345501	0.0177005
$382$	−6.56946	−0.336123
$383$	−15.4866	−0.791327	−0.395663	−	0.918396i	$-0.629485\pi$
$383$	−15.4866	−0.791327	−0.395663	+	0.918396i	$0.629485\pi$
$384$	−11.5118	−0.587459
$385$	0	0
$386$	7.64156	0.388945
$387$	11.4506	0.582066
$388$	−13.0470	−0.662363
$389$	−21.3949	−1.08476	−0.542382	−	0.840132i	$-0.682478\pi$
$389$	−21.3949	−1.08476	−0.542382	+	0.840132i	$0.682478\pi$
$390$	0	0
$391$	0.256400	0.0129667
$392$	−0.754359	−0.0381009
$393$	4.56651	0.230350
$394$	−9.52309	−0.479766
$395$	0	0
$396$	8.80660	0.442548
$397$	12.5016	0.627439	0.313719	−	0.949516i	$-0.398425\pi$
$397$	12.5016	0.627439	0.313719	+	0.949516i	$0.398425\pi$
$398$	10.1703	0.509789
$399$	−20.8238	−1.04250
$400$	0	0
$401$	−33.0478	−1.65033	−0.825164	−	0.564893i	$-0.808917\pi$
$401$	−33.0478	−1.65033	−0.825164	+	0.564893i	$0.808917\pi$
$402$	−0.0447683	−0.00223284
$403$	−6.62096	−0.329814
$404$	−14.1681	−0.704888
$405$	0	0
$406$	0.0492503	0.00244425
$407$	9.82111	0.486815
$408$	1.37003	0.0678264
$409$	3.01739	0.149200	0.0746002	−	0.997214i	$-0.476232\pi$
$409$	3.01739	0.149200	0.0746002	+	0.997214i	$0.476232\pi$
$410$	0	0
$411$	−4.81753	−0.237631
$412$	17.7821	0.876060
$413$	0.453368	0.0223088
$414$	0.199998	0.00982936
$415$	0	0
$416$	−7.73941	−0.379456
$417$	3.74075	0.183185
$418$	−22.3343	−1.09241
$419$	7.11371	0.347528	0.173764	−	0.984787i	$-0.444407\pi$
$419$	7.11371	0.347528	0.173764	+	0.984787i	$0.444407\pi$
$420$	0	0
$421$	−21.8569	−1.06524	−0.532620	−	0.846354i	$-0.678793\pi$
$421$	−21.8569	−1.06524	−0.532620	+	0.846354i	$0.678793\pi$
$422$	5.78227	0.281477
$423$	−8.52114	−0.414312
$424$	18.1709	0.882455
$425$	0	0
$426$	6.52325	0.316052
$427$	16.4980	0.798394
$428$	−21.0141	−1.01575
$429$	7.58655	0.366282
$430$	0	0
$431$	3.99392	0.192381	0.0961903	−	0.995363i	$-0.469334\pi$
$431$	3.99392	0.192381	0.0961903	+	0.995363i	$0.469334\pi$
$432$	−2.35558	−0.113333
$433$	25.1913	1.21062	0.605309	−	0.795991i	$-0.293050\pi$
$433$	25.1913	1.21062	0.605309	+	0.795991i	$0.293050\pi$
$434$	−6.19769	−0.297498
$435$	0	0
$436$	7.24057	0.346761
$437$	3.01654	0.144300
$438$	6.46636	0.308975
$439$	−1.71510	−0.0818575	−0.0409287	−	0.999162i	$-0.513032\pi$
$439$	−1.71510	−0.0818575	−0.0409287	+	0.999162i	$0.513032\pi$
$440$	0	0
$441$	−0.378747	−0.0180356
$442$	0.544348	0.0258920
$443$	−11.7475	−0.558140	−0.279070	−	0.960271i	$-0.590026\pi$
$443$	−11.7475	−0.558140	−0.279070	+	0.960271i	$0.590026\pi$
$444$	−3.26904	−0.155142
$445$	0	0
$446$	14.7930	0.700469
$447$	−8.64621	−0.408952
$448$	4.87804	0.230466
$449$	12.8415	0.606030	0.303015	−	0.952986i	$-0.402007\pi$
$449$	12.8415	0.606030	0.303015	+	0.952986i	$0.402007\pi$
$450$	0	0
$451$	26.5890	1.25203
$452$	−31.4493	−1.47925
$453$	1.24898	0.0586824
$454$	−2.13664	−0.100278
$455$	0	0
$456$	16.1183	0.754810
$457$	13.6882	0.640309	0.320155	−	0.947365i	$-0.396265\pi$
$457$	13.6882	0.640309	0.320155	+	0.947365i	$0.396265\pi$
$458$	−6.42233	−0.300096
$459$	0.687859	0.0321065
$460$	0	0
$461$	9.71340	0.452398	0.226199	−	0.974081i	$-0.427370\pi$
$461$	9.71340	0.452398	0.226199	+	0.974081i	$0.427370\pi$
$462$	7.10154	0.330393
$463$	−21.2740	−0.988687	−0.494344	−	0.869267i	$-0.664592\pi$
$463$	−21.2740	−0.988687	−0.494344	+	0.869267i	$0.664592\pi$
$464$	0.0840289	0.00390095
$465$	0	0
$466$	11.3309	0.524892
$467$	13.9014	0.643282	0.321641	−	0.946862i	$-0.395766\pi$
$467$	13.9014	0.643282	0.321641	+	0.946862i	$0.395766\pi$
$468$	−2.52524	−0.116729
$469$	0.214700	0.00991394
$470$	0	0
$471$	3.86574	0.178124
$472$	−0.350922	−0.0161525
$473$	−58.8982	−2.70814
$474$	2.65959	0.122159
$475$	0	0
$476$	−3.03042	−0.138899
$477$	9.12317	0.417721
$478$	−14.0602	−0.643098
$479$	−12.3677	−0.565096	−0.282548	−	0.959253i	$-0.591180\pi$
$479$	−12.3677	−0.565096	−0.282548	+	0.959253i	$0.591180\pi$
$480$	0	0
$481$	−2.81615	−0.128405
$482$	4.38601	0.199777
$483$	−0.959153	−0.0436430
$484$	−26.4651	−1.20296
$485$	0	0
$486$	0.536547	0.0243383
$487$	−35.3280	−1.60087	−0.800433	−	0.599422i	$-0.795397\pi$
$487$	−35.3280	−1.60087	−0.800433	+	0.599422i	$0.795397\pi$
$488$	−12.7700	−0.578070
$489$	−6.97411	−0.315380
$490$	0	0
$491$	−21.6938	−0.979026	−0.489513	−	0.871996i	$-0.662825\pi$
$491$	−21.6938	−0.979026	−0.489513	+	0.871996i	$0.662825\pi$
$492$	−8.85036	−0.399005
$493$	−0.0245375	−0.00110511
$494$	6.40425	0.288141
$495$	0	0
$496$	−10.5743	−0.474798
$497$	−31.2843	−1.40329
$498$	−5.02381	−0.225122
$499$	38.7869	1.73634	0.868171	−	0.496265i	$-0.165296\pi$
$499$	38.7869	1.73634	0.868171	+	0.496265i	$0.165296\pi$
$500$	0	0
$501$	−15.9594	−0.713013
$502$	−13.3141	−0.594239
$503$	6.41964	0.286238	0.143119	−	0.989706i	$-0.454287\pi$
$503$	6.41964	0.286238	0.143119	+	0.989706i	$0.454287\pi$
$504$	−5.12507	−0.228289
$505$	0	0
$506$	−1.02873	−0.0457325
$507$	10.8246	0.480737
$508$	0.591538	0.0262452
$509$	−7.19516	−0.318920	−0.159460	−	0.987204i	$-0.550975\pi$
$509$	−7.19516	−0.318920	−0.159460	+	0.987204i	$0.550975\pi$
$510$	0	0
$511$	−31.0115	−1.37187
$512$	−21.7438	−0.960951
$513$	8.09265	0.357299
$514$	13.0378	0.575074
$515$	0	0
$516$	19.6047	0.863051
$517$	43.8301	1.92764
$518$	−2.63611	−0.115824
$519$	−7.67330	−0.336820
$520$	0	0
$521$	−10.5204	−0.460908	−0.230454	−	0.973083i	$-0.574021\pi$
$521$	−10.5204	−0.460908	−0.230454	+	0.973083i	$0.574021\pi$
$522$	−0.0191399	−0.000837729 0
$523$	−11.3242	−0.495174	−0.247587	−	0.968866i	$-0.579638\pi$
$523$	−11.3242	−0.495174	−0.247587	+	0.968866i	$0.579638\pi$
$524$	7.81840	0.341548
$525$	0	0
$526$	4.57282	0.199385
$527$	3.08782	0.134507
$528$	12.1164	0.527297
$529$	−22.8611	−0.993959
$530$	0	0
$531$	−0.176190	−0.00764599
$532$	−35.6529	−1.54575
$533$	−7.62424	−0.330243
$534$	−0.00660971	−0.000286030 0
$535$	0	0
$536$	−0.166185	−0.00717810
$537$	−10.6093	−0.457826
$538$	−1.38597	−0.0597533
$539$	1.94815	0.0839130
$540$	0	0
$541$	16.9213	0.727505	0.363752	−	0.931496i	$-0.381495\pi$
$541$	16.9213	0.727505	0.363752	+	0.931496i	$0.381495\pi$
$542$	7.29130	0.313188
$543$	−15.1076	−0.648331
$544$	3.60943	0.154753
$545$	0	0
$546$	−2.03633	−0.0871467
$547$	44.1367	1.88715	0.943575	−	0.331160i	$-0.107440\pi$
$547$	44.1367	1.88715	0.943575	+	0.331160i	$0.107440\pi$
$548$	−8.24818	−0.352345
$549$	−6.41152	−0.273637
$550$	0	0
$551$	−0.288683	−0.0122983
$552$	0.742415	0.0315993
$553$	−12.7549	−0.542394
$554$	4.90390	0.208347
$555$	0	0
$556$	6.40460	0.271615
$557$	−17.8472	−0.756210	−0.378105	−	0.925763i	$-0.623424\pi$
$557$	−17.8472	−0.756210	−0.378105	+	0.925763i	$0.623424\pi$
$558$	2.40857	0.101963
$559$	16.8887	0.714317
$560$	0	0
$561$	−3.53813	−0.149380
$562$	−5.58551	−0.235610
$563$	21.4063	0.902167	0.451083	−	0.892482i	$-0.351038\pi$
$563$	21.4063	0.902167	0.451083	+	0.892482i	$0.351038\pi$
$564$	−14.5892	−0.614316
$565$	0	0
$566$	2.58394	0.108611
$567$	−2.57318	−0.108063
$568$	24.2151	1.01604
$569$	−26.0723	−1.09301	−0.546504	−	0.837456i	$-0.684042\pi$
$569$	−26.0723	−1.09301	−0.546504	+	0.837456i	$0.684042\pi$
$570$	0	0
$571$	−9.53879	−0.399186	−0.199593	−	0.979879i	$-0.563962\pi$
$571$	−9.53879	−0.399186	−0.199593	+	0.979879i	$0.563962\pi$
$572$	12.9891	0.543100
$573$	−12.2440	−0.511498
$574$	−7.13683	−0.297885
$575$	0	0
$576$	−1.89572	−0.0789885
$577$	−9.10504	−0.379048	−0.189524	−	0.981876i	$-0.560694\pi$
$577$	−9.10504	−0.379048	−0.189524	+	0.981876i	$0.560694\pi$
$578$	8.86743	0.368836
$579$	14.2421	0.591882
$580$	0	0
$581$	24.0933	0.999557
$582$	4.08871	0.169482
$583$	−46.9268	−1.94351
$584$	24.0039	0.993288
$585$	0	0
$586$	13.1189	0.541936
$587$	29.4485	1.21547	0.607734	−	0.794140i	$-0.292079\pi$
$587$	29.4485	1.21547	0.607734	+	0.794140i	$0.292079\pi$
$588$	−0.648459	−0.0267420
$589$	36.3281	1.49687
$590$	0	0
$591$	−17.7488	−0.730090
$592$	−4.49763	−0.184852
$593$	33.7757	1.38700	0.693501	−	0.720456i	$-0.256067\pi$
$593$	33.7757	1.38700	0.693501	+	0.720456i	$0.256067\pi$
$594$	−2.75983	−0.113237
$595$	0	0
$596$	−14.8033	−0.606368
$597$	18.9550	0.775778
$598$	0.294981	0.0120627
$599$	−12.0575	−0.492656	−0.246328	−	0.969187i	$-0.579224\pi$
$599$	−12.0575	−0.492656	−0.246328	+	0.969187i	$0.579224\pi$
$600$	0	0
$601$	0.0653240	0.00266462	0.00133231	−	0.999999i	$-0.499576\pi$
$601$	0.0653240	0.00266462	0.00133231	+	0.999999i	$0.499576\pi$
$602$	15.8090	0.644328
$603$	−0.0834377	−0.00339785
$604$	2.13841	0.0870106
$605$	0	0
$606$	4.44002	0.180364
$607$	25.9556	1.05350	0.526752	−	0.850019i	$-0.323410\pi$
$607$	25.9556	1.05350	0.526752	+	0.850019i	$0.323410\pi$
$608$	42.4648	1.72218
$609$	0.0917912	0.00371957
$610$	0	0
$611$	−12.5680	−0.508448
$612$	1.17770	0.0476055
$613$	−12.5712	−0.507747	−0.253873	−	0.967237i	$-0.581705\pi$
$613$	−12.5712	−0.507747	−0.253873	+	0.967237i	$0.581705\pi$
$614$	−5.10608	−0.206064
$615$	0	0
$616$	26.3617	1.06214
$617$	1.85723	0.0747693	0.0373846	−	0.999301i	$-0.488097\pi$
$617$	1.85723	0.0747693	0.0373846	+	0.999301i	$0.488097\pi$
$618$	−5.57259	−0.224162
$619$	−18.0236	−0.724431	−0.362216	−	0.932094i	$-0.617980\pi$
$619$	−18.0236	−0.724431	−0.362216	+	0.932094i	$0.617980\pi$
$620$	0	0
$621$	0.372750	0.0149579
$622$	−13.4037	−0.537438
$623$	0.0316990	0.00126999
$624$	−3.47430	−0.139083
$625$	0	0
$626$	5.30195	0.211908
$627$	−41.6261	−1.66239
$628$	6.61860	0.264111
$629$	1.31337	0.0523673
$630$	0	0
$631$	21.4626	0.854411	0.427206	−	0.904155i	$-0.359498\pi$
$631$	21.4626	0.854411	0.427206	+	0.904155i	$0.359498\pi$
$632$	9.87272	0.392716
$633$	10.7768	0.428340
$634$	13.2694	0.526993
$635$	0	0
$636$	15.6199	0.619371
$637$	−0.558622	−0.0221334
$638$	0.0984494	0.00389765
$639$	12.1578	0.480956
$640$	0	0
$641$	39.6897	1.56765	0.783824	−	0.620983i	$-0.213266\pi$
$641$	39.6897	1.56765	0.783824	+	0.620983i	$0.213266\pi$
$642$	6.58544	0.259907
$643$	1.26211	0.0497729	0.0248864	−	0.999690i	$-0.492078\pi$
$643$	1.26211	0.0497729	0.0248864	+	0.999690i	$0.492078\pi$
$644$	−1.64218	−0.0647110
$645$	0	0
$646$	−2.98675	−0.117512
$647$	−28.9276	−1.13726	−0.568631	−	0.822593i	$-0.692527\pi$
$647$	−28.9276	−1.13726	−0.568631	+	0.822593i	$0.692527\pi$
$648$	1.99173	0.0782424
$649$	0.906266	0.0355740
$650$	0	0
$651$	−11.5511	−0.452722
$652$	−11.9405	−0.467626
$653$	17.4796	0.684031	0.342016	−	0.939694i	$-0.388890\pi$
$653$	17.4796	0.684031	0.342016	+	0.939694i	$0.388890\pi$
$654$	−2.26907	−0.0887275
$655$	0	0
$656$	−12.1766	−0.475416
$657$	12.0518	0.470186
$658$	−11.7645	−0.458630
$659$	11.3394	0.441720	0.220860	−	0.975306i	$-0.429114\pi$
$659$	11.3394	0.441720	0.220860	+	0.975306i	$0.429114\pi$
$660$	0	0
$661$	4.63889	0.180432	0.0902160	−	0.995922i	$-0.471244\pi$
$661$	4.63889	0.180432	0.0902160	+	0.995922i	$0.471244\pi$
$662$	−0.578541	−0.0224856
$663$	1.01454	0.0394015
$664$	−18.6490	−0.723720
$665$	0	0
$666$	1.02446	0.0396969
$667$	−0.0132968	−0.000514856 0
$668$	−27.3243	−1.05721
$669$	27.5707	1.06595
$670$	0	0
$671$	32.9789	1.27314
$672$	−13.5023	−0.520863
$673$	7.53164	0.290323	0.145162	−	0.989408i	$-0.453630\pi$
$673$	7.53164	0.290323	0.145162	+	0.989408i	$0.453630\pi$
$674$	−7.54435	−0.290598
$675$	0	0
$676$	18.5330	0.712807
$677$	31.6182	1.21518	0.607592	−	0.794249i	$-0.292136\pi$
$677$	31.6182	1.21518	0.607592	+	0.794249i	$0.292136\pi$
$678$	9.85565	0.378504
$679$	−19.6087	−0.752512
$680$	0	0
$681$	−3.98221	−0.152599
$682$	−12.3889	−0.474397
$683$	30.6601	1.17317	0.586587	−	0.809886i	$-0.300471\pi$
$683$	30.6601	1.17317	0.586587	+	0.809886i	$0.300471\pi$
$684$	13.8556	0.529781
$685$	0	0
$686$	−10.1873	−0.388954
$687$	−11.9697	−0.456674
$688$	26.9728	1.02833
$689$	13.4560	0.512632
$690$	0	0
$691$	−34.7341	−1.32135	−0.660673	−	0.750674i	$-0.729729\pi$
$691$	−34.7341	−1.32135	−0.660673	+	0.750674i	$0.729729\pi$
$692$	−13.1376	−0.499416
$693$	13.2356	0.502780
$694$	−3.05609	−0.116008
$695$	0	0
$696$	−0.0710494	−0.00269312
$697$	3.55571	0.134682
$698$	15.2889	0.578694
$699$	21.1181	0.798761
$700$	0	0
$701$	−19.5437	−0.738154	−0.369077	−	0.929399i	$-0.620326\pi$
$701$	−19.5437	−0.738154	−0.369077	+	0.929399i	$0.620326\pi$
$702$	0.791365	0.0298682
$703$	15.4517	0.582773
$704$	9.75101	0.367505
$705$	0	0
$706$	−6.84933	−0.257778
$707$	−21.2935	−0.800825
$708$	−0.301658	−0.0113370
$709$	−25.3106	−0.950558	−0.475279	−	0.879835i	$-0.657653\pi$
$709$	−25.3106	−0.950558	−0.475279	+	0.879835i	$0.657653\pi$
$710$	0	0
$711$	4.95687	0.185897
$712$	−0.0245360	−0.000919526 0
$713$	1.67328	0.0626649
$714$	0.949680	0.0355409
$715$	0	0
$716$	−18.1644	−0.678835
$717$	−26.2050	−0.978643
$718$	2.42423	0.0904715
$719$	−9.19065	−0.342754	−0.171377	−	0.985206i	$-0.554822\pi$
$719$	−9.19065	−0.342754	−0.171377	+	0.985206i	$0.554822\pi$
$720$	0	0
$721$	26.7251	0.995294
$722$	−24.9446	−0.928342
$723$	8.17452	0.304014
$724$	−25.8661	−0.961305
$725$	0	0
$726$	8.29369	0.307808
$727$	−13.0649	−0.484549	−0.242274	−	0.970208i	$-0.577893\pi$
$727$	−13.0649	−0.484549	−0.242274	+	0.970208i	$0.577893\pi$
$728$	−7.55908	−0.280158
$729$	1.00000	0.0370370
$730$	0	0
$731$	−7.87639	−0.291319
$732$	−10.9773	−0.405732
$733$	−9.68091	−0.357573	−0.178786	−	0.983888i	$-0.557217\pi$
$733$	−9.68091	−0.357573	−0.178786	+	0.983888i	$0.557217\pi$
$734$	−1.62813	−0.0600955
$735$	0	0
$736$	1.95594	0.0720970
$737$	0.429178	0.0158090
$738$	2.77354	0.102096
$739$	9.13413	0.336004	0.168002	−	0.985787i	$-0.446268\pi$
$739$	9.13413	0.336004	0.168002	+	0.985787i	$0.446268\pi$
$740$	0	0
$741$	11.9360	0.438481
$742$	12.5957	0.462404
$743$	−6.35760	−0.233238	−0.116619	−	0.993177i	$-0.537206\pi$
$743$	−6.35760	−0.233238	−0.116619	+	0.993177i	$0.537206\pi$
$744$	8.94090	0.327789
$745$	0	0
$746$	−8.24450	−0.301853
$747$	−9.36322	−0.342582
$748$	−6.05770	−0.221492
$749$	−31.5825	−1.15400
$750$	0	0
$751$	35.0142	1.27768	0.638842	−	0.769338i	$-0.279414\pi$
$751$	35.0142	1.27768	0.638842	+	0.769338i	$0.279414\pi$
$752$	−20.0722	−0.731959
$753$	−24.8145	−0.904291
$754$	−0.0282298	−0.00102807
$755$	0	0
$756$	−4.40559	−0.160230
$757$	11.5175	0.418609	0.209305	−	0.977850i	$-0.432880\pi$
$757$	11.5175	0.418609	0.209305	+	0.977850i	$0.432880\pi$
$758$	−5.55846	−0.201892
$759$	−1.91731	−0.0695939
$760$	0	0
$761$	41.0738	1.48892	0.744461	−	0.667665i	$-0.232706\pi$
$761$	41.0738	1.48892	0.744461	+	0.667665i	$0.232706\pi$
$762$	−0.185377	−0.00671551
$763$	10.8820	0.393956
$764$	−20.9631	−0.758418
$765$	0	0
$766$	8.30927	0.300226
$767$	−0.259867	−0.00938324
$768$	2.38518	0.0860677
$769$	17.9468	0.647178	0.323589	−	0.946198i	$-0.395111\pi$
$769$	17.9468	0.647178	0.323589	+	0.946198i	$0.395111\pi$
$770$	0	0
$771$	24.2995	0.875126
$772$	24.3842	0.877606
$773$	52.0238	1.87117	0.935583	−	0.353107i	$-0.114875\pi$
$773$	52.0238	1.87117	0.935583	+	0.353107i	$0.114875\pi$
$774$	−6.14378	−0.220833
$775$	0	0
$776$	15.1778	0.544850
$777$	−4.91311	−0.176257
$778$	11.4794	0.411555
$779$	41.8329	1.49882
$780$	0	0
$781$	−62.5361	−2.23772
$782$	−0.137570	−0.00491951
$783$	−0.0356723	−0.00127482
$784$	−0.892168	−0.0318632
$785$	0	0
$786$	−2.45015	−0.0873938
$787$	−12.5243	−0.446442	−0.223221	−	0.974768i	$-0.571657\pi$
$787$	−12.5243	−0.446442	−0.223221	+	0.974768i	$0.571657\pi$
$788$	−30.3881	−1.08253
$789$	8.52269	0.303416
$790$	0	0
$791$	−47.2659	−1.68058
$792$	−10.2448	−0.364034
$793$	−9.45650	−0.335810
$794$	−6.70771	−0.238048
$795$	0	0
$796$	32.4532	1.15027
$797$	25.1417	0.890563	0.445282	−	0.895391i	$-0.353104\pi$
$797$	25.1417	0.890563	0.445282	+	0.895391i	$0.353104\pi$
$798$	11.1730	0.395519
$799$	5.86134	0.207359
$800$	0	0
$801$	−0.0123190	−0.000435270 0
$802$	17.7317	0.626128
$803$	−61.9907	−2.18761
$804$	−0.142855	−0.00503811
$805$	0	0
$806$	3.55246	0.125130
$807$	−2.58312	−0.0909302
$808$	16.4819	0.579831
$809$	−4.82683	−0.169702	−0.0848512	−	0.996394i	$-0.527041\pi$
$809$	−4.82683	−0.169702	−0.0848512	+	0.996394i	$0.527041\pi$
$810$	0	0
$811$	−36.8583	−1.29427	−0.647135	−	0.762375i	$-0.724033\pi$
$811$	−36.8583	−1.29427	−0.647135	+	0.762375i	$0.724033\pi$
$812$	0.157157	0.00551514
$813$	13.5893	0.476598
$814$	−5.26949	−0.184695
$815$	0	0
$816$	1.62031	0.0567221
$817$	−92.6656	−3.24196
$818$	−1.61897	−0.0566061
$819$	−3.79524	−0.132617
$820$	0	0
$821$	14.1866	0.495116	0.247558	−	0.968873i	$-0.420372\pi$
$821$	14.1866	0.495116	0.247558	+	0.968873i	$0.420372\pi$
$822$	2.58483	0.0901564
$823$	25.9934	0.906074	0.453037	−	0.891492i	$-0.350340\pi$
$823$	25.9934	0.906074	0.453037	+	0.891492i	$0.350340\pi$
$824$	−20.6861	−0.720634
$825$	0	0
$826$	−0.243253	−0.00846386
$827$	42.9339	1.49296	0.746480	−	0.665408i	$-0.231743\pi$
$827$	42.9339	1.49296	0.746480	+	0.665408i	$0.231743\pi$
$828$	0.638192	0.0221787
$829$	−5.63771	−0.195806	−0.0979029	−	0.995196i	$-0.531213\pi$
$829$	−5.63771	−0.195806	−0.0979029	+	0.995196i	$0.531213\pi$
$830$	0	0
$831$	9.13975	0.317054
$832$	−2.79605	−0.0969355
$833$	0.260524	0.00902664
$834$	−2.00709	−0.0694997
$835$	0	0
$836$	−71.2687	−2.46488
$837$	4.48902	0.155163
$838$	−3.81684	−0.131851
$839$	−23.0630	−0.796225	−0.398112	−	0.917337i	$-0.630335\pi$
$839$	−23.0630	−0.796225	−0.398112	+	0.917337i	$0.630335\pi$
$840$	0	0
$841$	−28.9987	−0.999956
$842$	11.7273	0.404148
$843$	−10.4101	−0.358543
$844$	18.4512	0.635116
$845$	0	0
$846$	4.57199	0.157188
$847$	−39.7750	−1.36669
$848$	21.4904	0.737982
$849$	4.81586	0.165280
$850$	0	0
$851$	0.711711	0.0243971
$852$	20.8156	0.713132
$853$	39.0456	1.33690	0.668448	−	0.743758i	$-0.266959\pi$
$853$	39.0456	1.33690	0.668448	+	0.743758i	$0.266959\pi$
$854$	−8.85195	−0.302908
$855$	0	0
$856$	24.4459	0.835544
$857$	12.7315	0.434899	0.217450	−	0.976072i	$-0.430226\pi$
$857$	12.7315	0.434899	0.217450	+	0.976072i	$0.430226\pi$
$858$	−4.07054	−0.138966
$859$	14.5866	0.497690	0.248845	−	0.968543i	$-0.419949\pi$
$859$	14.5866	0.497690	0.248845	+	0.968543i	$0.419949\pi$
$860$	0	0
$861$	−13.3014	−0.453310
$862$	−2.14293	−0.0729884
$863$	−17.5145	−0.596201	−0.298101	−	0.954534i	$-0.596353\pi$
$863$	−17.5145	−0.596201	−0.298101	+	0.954534i	$0.596353\pi$
$864$	5.24733	0.178518
$865$	0	0
$866$	−13.5163	−0.459303
$867$	16.5268	0.561281
$868$	−19.7768	−0.671267
$869$	−25.4966	−0.864912
$870$	0	0
$871$	−0.123064	−0.00416987
$872$	−8.42304	−0.285240
$873$	7.62041	0.257912
$874$	−1.61851	−0.0547470
$875$	0	0
$876$	20.6341	0.697162
$877$	9.10744	0.307536	0.153768	−	0.988107i	$-0.450859\pi$
$877$	9.10744	0.307536	0.153768	+	0.988107i	$0.450859\pi$
$878$	0.920234	0.0310564
$879$	24.4506	0.824698
$880$	0	0
$881$	−45.2084	−1.52311	−0.761555	−	0.648101i	$-0.775564\pi$
$881$	−45.2084	−1.52311	−0.761555	+	0.648101i	$0.775564\pi$
$882$	0.203215	0.00684262
$883$	−4.24226	−0.142763	−0.0713817	−	0.997449i	$-0.522741\pi$
$883$	−4.24226	−0.142763	−0.0713817	+	0.997449i	$0.522741\pi$
$884$	1.73701	0.0584220
$885$	0	0
$886$	6.30307	0.211756
$887$	−12.7865	−0.429328	−0.214664	−	0.976688i	$-0.568866\pi$
$887$	−12.7865	−0.429328	−0.214664	+	0.976688i	$0.568866\pi$
$888$	3.80291	0.127617
$889$	0.889035	0.0298173
$890$	0	0
$891$	−5.14369	−0.172320
$892$	47.2043	1.58052
$893$	68.9586	2.30761
$894$	4.63910	0.155155
$895$	0	0
$896$	−29.6219	−0.989600
$897$	0.549778	0.0183565
$898$	−6.89009	−0.229925
$899$	−0.160134	−0.00534076
$900$	0	0
$901$	−6.27546	−0.209066
$902$	−14.2662	−0.475014
$903$	29.4644	0.980514
$904$	36.5853	1.21681
$905$	0	0
$906$	−0.670139	−0.0222639
$907$	9.51928	0.316082	0.158041	−	0.987433i	$-0.449482\pi$
$907$	9.51928	0.316082	0.158041	+	0.987433i	$0.449482\pi$
$908$	−6.81801	−0.226264
$909$	8.27518	0.274470
$910$	0	0
$911$	−41.4564	−1.37351	−0.686755	−	0.726889i	$-0.740966\pi$
$911$	−41.4564	−1.37351	−0.686755	+	0.726889i	$0.740966\pi$
$912$	19.0629	0.631236
$913$	48.1615	1.59391
$914$	−7.34438	−0.242931
$915$	0	0
$916$	−20.4936	−0.677128
$917$	11.7504	0.388034
$918$	−0.369069	−0.0121811
$919$	15.8676	0.523424	0.261712	−	0.965146i	$-0.415713\pi$
$919$	15.8676	0.523424	0.261712	+	0.965146i	$0.415713\pi$
$920$	0	0
$921$	−9.51655	−0.313581
$922$	−5.21169	−0.171638
$923$	17.9319	0.590235
$924$	22.6610	0.745491
$925$	0	0
$926$	11.4145	0.375104
$927$	−10.3860	−0.341122
$928$	−0.187184	−0.00614462
$929$	−29.4108	−0.964936	−0.482468	−	0.875914i	$-0.660260\pi$
$929$	−29.4108	−0.964936	−0.482468	+	0.875914i	$0.660260\pi$
$930$	0	0
$931$	3.06506	0.100453
$932$	36.1567	1.18435
$933$	−24.9814	−0.817853
$934$	−7.45878	−0.244059
$935$	0	0
$936$	2.93764	0.0960198
$937$	58.1131	1.89847	0.949236	−	0.314565i	$-0.101859\pi$
$937$	58.1131	1.89847	0.949236	+	0.314565i	$0.101859\pi$
$938$	−0.115197	−0.00376131
$939$	9.88161	0.322474
$940$	0	0
$941$	34.2306	1.11589	0.557944	−	0.829879i	$-0.311591\pi$
$941$	34.2306	1.11589	0.557944	+	0.829879i	$0.311591\pi$
$942$	−2.07415	−0.0675795
$943$	1.92684	0.0627464
$944$	−0.415029	−0.0135081
$945$	0	0
$946$	31.6017	1.02746
$947$	38.0104	1.23517	0.617586	−	0.786503i	$-0.288111\pi$
$947$	38.0104	1.23517	0.617586	+	0.786503i	$0.288111\pi$
$948$	8.48674	0.275636
$949$	17.7755	0.577017
$950$	0	0
$951$	24.7310	0.801958
$952$	3.52533	0.114256
$953$	16.9149	0.547929	0.273964	−	0.961740i	$-0.411665\pi$
$953$	16.9149	0.547929	0.273964	+	0.961740i	$0.411665\pi$
$954$	−4.89501	−0.158482
$955$	0	0
$956$	−44.8660	−1.45107
$957$	0.183487	0.00593130
$958$	6.63587	0.214395
$959$	−12.3964	−0.400300
$960$	0	0
$961$	−10.8487	−0.349958
$962$	1.51100	0.0487165
$963$	12.2737	0.395516
$964$	13.9957	0.450772
$965$	0	0
$966$	0.514630	0.0165580
$967$	−22.7845	−0.732700	−0.366350	−	0.930477i	$-0.619393\pi$
$967$	−22.7845	−0.732700	−0.366350	+	0.930477i	$0.619393\pi$
$968$	30.7872	0.989537
$969$	−5.56661	−0.178825
$970$	0	0
$971$	43.9853	1.41156	0.705778	−	0.708433i	$-0.250598\pi$
$971$	43.9853	1.41156	0.705778	+	0.708433i	$0.250598\pi$
$972$	1.71212	0.0549162
$973$	9.62561	0.308583
$974$	18.9552	0.607362
$975$	0	0
$976$	−15.1029	−0.483431
$977$	−18.6331	−0.596125	−0.298062	−	0.954546i	$-0.596340\pi$
$977$	−18.6331	−0.596125	−0.298062	+	0.954546i	$0.596340\pi$
$978$	3.74194	0.119654
$979$	0.0633650	0.00202515
$980$	0	0
$981$	−4.22902	−0.135022
$982$	11.6397	0.371438
$983$	54.5076	1.73852	0.869261	−	0.494354i	$-0.164595\pi$
$983$	54.5076	1.73852	0.869261	+	0.494354i	$0.164595\pi$
$984$	10.2957	0.328216
$985$	0	0
$986$	0.0131655	0.000419276 0
$987$	−21.9264	−0.697925
$988$	20.4359	0.650153
$989$	−4.26820	−0.135721
$990$	0	0
$991$	36.2056	1.15011	0.575054	−	0.818116i	$-0.304981\pi$
$991$	36.2056	1.15011	0.575054	+	0.818116i	$0.304981\pi$
$992$	23.5554	0.747884
$993$	−1.07827	−0.0342178
$994$	16.7855	0.532403
$995$	0	0
$996$	−16.0309	−0.507960
$997$	−7.19225	−0.227781	−0.113890	−	0.993493i	$-0.536331\pi$
$997$	−7.19225	−0.227781	−0.113890	+	0.993493i	$0.536331\pi$
$998$	−20.8110	−0.658761
$999$	1.90935	0.0604092

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1875.2.a.p.1.3		8
3.2	odd	2		5625.2.a.t.1.6		8
5.2	odd	4		1875.2.b.h.1249.7		16
5.3	odd	4		1875.2.b.h.1249.10		16
5.4	even	2		1875.2.a.m.1.6		8
15.14	odd	2		5625.2.a.bd.1.3		8
25.2	odd	20		75.2.i.a.4.2	✓	16
25.9	even	10		375.2.g.e.151.3		16
25.11	even	5		375.2.g.d.226.2		16
25.12	odd	20		375.2.i.c.349.3		16
25.13	odd	20		75.2.i.a.19.2	yes	16
25.14	even	10		375.2.g.e.226.3		16
25.16	even	5		375.2.g.d.151.2		16
25.23	odd	20		375.2.i.c.274.3		16
75.2	even	20		225.2.m.b.154.3		16
75.38	even	20		225.2.m.b.19.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.2.i.a.4.2	✓	16	25.2	odd	20
75.2.i.a.19.2	yes	16	25.13	odd	20
225.2.m.b.19.3		16	75.38	even	20
225.2.m.b.154.3		16	75.2	even	20
375.2.g.d.151.2		16	25.16	even	5
375.2.g.d.226.2		16	25.11	even	5
375.2.g.e.151.3		16	25.9	even	10
375.2.g.e.226.3		16	25.14	even	10
375.2.i.c.274.3		16	25.23	odd	20
375.2.i.c.349.3		16	25.12	odd	20
1875.2.a.m.1.6		8	5.4	even	2
1875.2.a.p.1.3		8	1.1	even	1	trivial
1875.2.b.h.1249.7		16	5.2	odd	4
1875.2.b.h.1249.10		16	5.3	odd	4
5625.2.a.t.1.6		8	3.2	odd	2
5625.2.a.bd.1.3		8	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:	\( N \)	\(=\)	\( 1875 = 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1875.a (trivial)

\(f(q)\)	\(=\)	\(q-0.536547 q^{2} -1.00000 q^{3} -1.71212 q^{4} +0.536547 q^{6} -2.57318 q^{7} +1.99173 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.536547 q^{2} -1.00000 q^{3} -1.71212 q^{4} +0.536547 q^{6} -2.57318 q^{7} +1.99173 q^{8} +1.00000 q^{9} -5.14369 q^{11} +1.71212 q^{12} +1.47492 q^{13} +1.38063 q^{14} +2.35558 q^{16} -0.687859 q^{17} -0.536547 q^{18} -8.09265 q^{19} +2.57318 q^{21} +2.75983 q^{22} -0.372750 q^{23} -1.99173 q^{24} -0.791365 q^{26} -1.00000 q^{27} +4.40559 q^{28} +0.0356723 q^{29} -4.48902 q^{31} -5.24733 q^{32} +5.14369 q^{33} +0.369069 q^{34} -1.71212 q^{36} -1.90935 q^{37} +4.34209 q^{38} -1.47492 q^{39} -5.16925 q^{41} -1.38063 q^{42} +11.4506 q^{43} +8.80660 q^{44} +0.199998 q^{46} -8.52114 q^{47} -2.35558 q^{48} -0.378747 q^{49} +0.687859 q^{51} -2.52524 q^{52} +9.12317 q^{53} +0.536547 q^{54} -5.12507 q^{56} +8.09265 q^{57} -0.0191399 q^{58} -0.176190 q^{59} -6.41152 q^{61} +2.40857 q^{62} -2.57318 q^{63} -1.89572 q^{64} -2.75983 q^{66} -0.0834377 q^{67} +1.17770 q^{68} +0.372750 q^{69} +12.1578 q^{71} +1.99173 q^{72} +12.0518 q^{73} +1.02446 q^{74} +13.8556 q^{76} +13.2356 q^{77} +0.791365 q^{78} +4.95687 q^{79} +1.00000 q^{81} +2.77354 q^{82} -9.36322 q^{83} -4.40559 q^{84} -6.14378 q^{86} -0.0356723 q^{87} -10.2448 q^{88} -0.0123190 q^{89} -3.79524 q^{91} +0.638192 q^{92} +4.48902 q^{93} +4.57199 q^{94} +5.24733 q^{96} +7.62041 q^{97} +0.203215 q^{98} -5.14369 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{2} - 8 q^{3} + 4 q^{4} - 4 q^{6} + 8 q^{7} + 12 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q + 4 * q^2 - 8 * q^3 + 4 * q^4 - 4 * q^6 + 8 * q^7 + 12 * q^8 + 8 * q^9	\( 8 q + 4 q^{2} - 8 q^{3} + 4 q^{4} - 4 q^{6} + 8 q^{7} + 12 q^{8} + 8 q^{9} + 2 q^{11} - 4 q^{12} + 16 q^{13} + 6 q^{14} + 16 q^{17} + 4 q^{18} - 14 q^{19} - 8 q^{21} + 12 q^{22} + 14 q^{23} - 12 q^{24} + 6 q^{26} - 8 q^{27} + 16 q^{28} + 2 q^{29} - 22 q^{31} - 2 q^{32} - 2 q^{33} - 12 q^{34} + 4 q^{36} + 28 q^{37} - 16 q^{38} - 16 q^{39} + 8 q^{41} - 6 q^{42} + 20 q^{43} + 22 q^{44} - 2 q^{46} + 10 q^{47} - 16 q^{51} + 16 q^{52} + 44 q^{53} - 4 q^{54} + 30 q^{56} + 14 q^{57} + 8 q^{58} + 14 q^{59} - 20 q^{61} + 16 q^{62} + 8 q^{63} + 6 q^{64} - 12 q^{66} + 16 q^{67} - 2 q^{68} - 14 q^{69} + 16 q^{71} + 12 q^{72} + 24 q^{73} + 26 q^{74} - 16 q^{76} + 46 q^{77} - 6 q^{78} - 30 q^{79} + 8 q^{81} + 16 q^{82} + 12 q^{83} - 16 q^{84} + 32 q^{86} - 2 q^{87} + 32 q^{88} + 16 q^{89} - 12 q^{91} - 2 q^{92} + 22 q^{93} + 14 q^{94} + 2 q^{96} + 16 q^{97} + 4 q^{98} + 2 q^{99}+O(q^{100}) \) 8 * q + 4 * q^2 - 8 * q^3 + 4 * q^4 - 4 * q^6 + 8 * q^7 + 12 * q^8 + 8 * q^9 + 2 * q^11 - 4 * q^12 + 16 * q^13 + 6 * q^14 + 16 * q^17 + 4 * q^18 - 14 * q^19 - 8 * q^21 + 12 * q^22 + 14 * q^23 - 12 * q^24 + 6 * q^26 - 8 * q^27 + 16 * q^28 + 2 * q^29 - 22 * q^31 - 2 * q^32 - 2 * q^33 - 12 * q^34 + 4 * q^36 + 28 * q^37 - 16 * q^38 - 16 * q^39 + 8 * q^41 - 6 * q^42 + 20 * q^43 + 22 * q^44 - 2 * q^46 + 10 * q^47 - 16 * q^51 + 16 * q^52 + 44 * q^53 - 4 * q^54 + 30 * q^56 + 14 * q^57 + 8 * q^58 + 14 * q^59 - 20 * q^61 + 16 * q^62 + 8 * q^63 + 6 * q^64 - 12 * q^66 + 16 * q^67 - 2 * q^68 - 14 * q^69 + 16 * q^71 + 12 * q^72 + 24 * q^73 + 26 * q^74 - 16 * q^76 + 46 * q^77 - 6 * q^78 - 30 * q^79 + 8 * q^81 + 16 * q^82 + 12 * q^83 - 16 * q^84 + 32 * q^86 - 2 * q^87 + 32 * q^88 + 16 * q^89 - 12 * q^91 - 2 * q^92 + 22 * q^93 + 14 * q^94 + 2 * q^96 + 16 * q^97 + 4 * q^98 + 2 * q^99

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