LMFDB - Embedded newform 1875.2.b.h.1249.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1875,2,Mod(1249,1875)]

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, 1]))

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

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

chi := DirichletCharacter("1875.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1875 = 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1875.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:	$14.9719503790$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 20x^{14} + 156x^{12} + 610x^{10} + 1286x^{8} + 1440x^{6} + 761x^{4} + 130x^{2} + 1 $ x^16 + 20x^14 + 156x^12 + 610x^10 + 1286x^8 + 1440x^6 + 761x^4 + 130*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 75)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.10
Root			$-1.53655i$ of defining polynomial
Character	$\chi$	$=$	1875.1249
Dual form			1875.2.b.h.1249.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.536547i q^{2} -1.00000i q^{3} +1.71212 q^{4} +0.536547 q^{6} +2.57318i q^{7} +1.99173i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+0.536547i q^{2} -1.00000i q^{3} +1.71212 q^{4} +0.536547 q^{6} +2.57318i q^{7} +1.99173i q^{8} -1.00000 q^{9} -5.14369 q^{11} -1.71212i q^{12} +1.47492i q^{13} -1.38063 q^{14} +2.35558 q^{16} +0.687859i q^{17} -0.536547i q^{18} +8.09265 q^{19} +2.57318 q^{21} -2.75983i q^{22} -0.372750i q^{23} +1.99173 q^{24} -0.791365 q^{26} +1.00000i q^{27} +4.40559i q^{28} -0.0356723 q^{29} -4.48902 q^{31} +5.24733i q^{32} +5.14369i q^{33} -0.369069 q^{34} -1.71212 q^{36} +1.90935i q^{37} +4.34209i q^{38} +1.47492 q^{39} -5.16925 q^{41} +1.38063i q^{42} +11.4506i q^{43} -8.80660 q^{44} +0.199998 q^{46} +8.52114i q^{47} -2.35558i q^{48} +0.378747 q^{49} +0.687859 q^{51} +2.52524i q^{52} +9.12317i q^{53} -0.536547 q^{54} -5.12507 q^{56} -8.09265i q^{57} -0.0191399i q^{58} +0.176190 q^{59} -6.41152 q^{61} -2.40857i q^{62} -2.57318i q^{63} +1.89572 q^{64} -2.75983 q^{66} +0.0834377i q^{67} +1.17770i q^{68} -0.372750 q^{69} +12.1578 q^{71} -1.99173i q^{72} +12.0518i q^{73} -1.02446 q^{74} +13.8556 q^{76} -13.2356i q^{77} +0.791365i q^{78} -4.95687 q^{79} +1.00000 q^{81} -2.77354i q^{82} -9.36322i q^{83} +4.40559 q^{84} -6.14378 q^{86} +0.0356723i q^{87} -10.2448i q^{88} +0.0123190 q^{89} -3.79524 q^{91} -0.638192i q^{92} +4.48902i q^{93} -4.57199 q^{94} +5.24733 q^{96} -7.62041i q^{97} +0.203215i q^{98} +5.14369 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 8 q^{4} - 8 q^{6} - 16 q^{9}+O(q^{10}) $ 16 * q - 8 * q^4 - 8 * q^6 - 16 * q^9	$ 16 q - 8 q^{4} - 8 q^{6} - 16 q^{9} + 4 q^{11} - 12 q^{14} + 28 q^{19} - 16 q^{21} + 24 q^{24} + 12 q^{26} - 4 q^{29} - 44 q^{31} + 24 q^{34} + 8 q^{36} + 32 q^{39} + 16 q^{41} - 44 q^{44} - 4 q^{46} - 32 q^{51} + 8 q^{54} + 60 q^{56} - 28 q^{59} - 40 q^{61} - 12 q^{64} - 24 q^{66} + 28 q^{69} + 32 q^{71} - 52 q^{74} - 32 q^{76} + 60 q^{79} + 16 q^{81} + 32 q^{84} + 64 q^{86} - 32 q^{89} - 24 q^{91} - 28 q^{94} + 4 q^{96} - 4 q^{99}+O(q^{100}) $ 16 * q - 8 * q^4 - 8 * q^6 - 16 * q^9 + 4 * q^11 - 12 * q^14 + 28 * q^19 - 16 * q^21 + 24 * q^24 + 12 * q^26 - 4 * q^29 - 44 * q^31 + 24 * q^34 + 8 * q^36 + 32 * q^39 + 16 * q^41 - 44 * q^44 - 4 * q^46 - 32 * q^51 + 8 * q^54 + 60 * q^56 - 28 * q^59 - 40 * q^61 - 12 * q^64 - 24 * q^66 + 28 * q^69 + 32 * q^71 - 52 * q^74 - 32 * q^76 + 60 * q^79 + 16 * q^81 + 32 * q^84 + 64 * q^86 - 32 * q^89 - 24 * q^91 - 28 * q^94 + 4 * q^96 - 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1875\mathbb{Z}\right)^\times$.

$n$	$626$	$1252$
$\chi(n)$	$1$	$-1$

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 $

$2$	0.536547i	0.379396i	0.981842	+	0.189698i	$0.0607509\pi$
$2$	0.536547i	0.379396i	−0.981842	+	0.189698i	$0.939249\pi$
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	1.71212	0.856059
$5$	0	0
$5$	0	0
$6$	0.536547	0.219044
$7$	2.57318i	0.972570i	0.873800	+	0.486285i	$0.161648\pi$
$7$	2.57318i	0.972570i	−0.873800	+	0.486285i	$0.838352\pi$
$8$	1.99173i	0.704181i
$9$	−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	$0.782472\pi$
$12$	− 1.71212i	− 0.494246i
$13$	1.47492i	0.409070i	0.978859	+	0.204535i	$0.0655682\pi$
$13$	1.47492i	0.409070i	−0.978859	+	0.204535i	$0.934432\pi$
$14$	−1.38063	−0.368989
$15$	0	0
$16$	2.35558	0.588895
$17$	0.687859i	0.166830i	0.996515	+	0.0834152i	$0.0265828\pi$
$17$	0.687859i	0.166830i	−0.996515	+	0.0834152i	$0.973417\pi$
$18$	− 0.536547i	− 0.126465i
$19$	8.09265	1.85658	0.928291	−	0.371855i	$-0.121278\pi$
$19$	8.09265	1.85658	0.928291	+	0.371855i	$0.121278\pi$
$20$	0	0
$21$	2.57318	0.561514
$22$	− 2.75983i	− 0.588398i
$23$	− 0.372750i	− 0.0777237i	−0.999245	−	0.0388619i	$-0.987627\pi$
$23$	− 0.372750i	− 0.0777237i	0.999245	−	0.0388619i	$-0.0123732\pi$
$24$	1.99173	0.406559
$25$	0	0
$26$	−0.791365	−0.155200
$27$	1.00000i	0.192450i
$28$	4.40559i	0.832577i
$29$	−0.0356723	−0.00662418	−0.00331209	−	0.999995i	$-0.501054\pi$
$29$	−0.0356723	−0.00662418	−0.00331209	+	0.999995i	$0.501054\pi$
$30$	0	0
$31$	−4.48902	−0.806252	−0.403126	−	0.915144i	$-0.632076\pi$
$31$	−4.48902	−0.806252	−0.403126	+	0.915144i	$0.632076\pi$
$32$	5.24733i	0.927606i
$33$	5.14369i	0.895401i
$34$	−0.369069	−0.0632948
$35$	0	0
$36$	−1.71212	−0.285353
$37$	1.90935i	0.313896i	0.987607	+	0.156948i	$0.0501654\pi$
$37$	1.90935i	0.313896i	−0.987607	+	0.156948i	$0.949835\pi$
$38$	4.34209i	0.704380i
$39$	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	$0.632259\pi$
$42$	1.38063i	0.213036i
$43$	11.4506i	1.74620i	0.487543	+	0.873099i	$0.337893\pi$
$43$	11.4506i	1.74620i	−0.487543	+	0.873099i	$0.662107\pi$
$44$	−8.80660	−1.32764
$45$	0	0
$46$	0.199998	0.0294881
$47$	8.52114i	1.24294i	0.783440	+	0.621468i	$0.213463\pi$
$47$	8.52114i	1.24294i	−0.783440	+	0.621468i	$0.786537\pi$
$48$	− 2.35558i	− 0.339999i
$49$	0.378747	0.0541067
$50$	0	0
$51$	0.687859	0.0963196
$52$	2.52524i	0.350188i
$53$	9.12317i	1.25316i	0.779355	+	0.626582i	$0.215547\pi$
$53$	9.12317i	1.25316i	−0.779355	+	0.626582i	$0.784453\pi$
$54$	−0.536547	−0.0730148
$55$	0	0
$56$	−5.12507	−0.684866
$57$	− 8.09265i	− 1.07190i
$58$	− 0.0191399i	− 0.00251319i
$59$	0.176190	0.0229380	0.0114690	−	0.999934i	$-0.496349\pi$
$59$	0.176190	0.0229380	0.0114690	+	0.999934i	$0.496349\pi$
$60$	0	0
$61$	−6.41152	−0.820911	−0.410456	−	0.911881i	$-0.634630\pi$
$61$	−6.41152	−0.820911	−0.410456	+	0.911881i	$0.634630\pi$
$62$	− 2.40857i	− 0.305889i
$63$	− 2.57318i	− 0.324190i
$64$	1.89572	0.236965
$65$	0	0
$66$	−2.75983	−0.339712
$67$	0.0834377i	0.0101935i	0.999987	+	0.00509677i	$0.00162236\pi$
$67$	0.0834377i	0.0101935i	−0.999987	+	0.00509677i	$0.998378\pi$
$68$	1.17770i	0.142817i
$69$	−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	$0.243483\pi$
$72$	− 1.99173i	− 0.234727i
$73$	12.0518i	1.41056i	0.708930	+	0.705279i	$0.249178\pi$
$73$	12.0518i	1.41056i	−0.708930	+	0.705279i	$0.750822\pi$
$74$	−1.02446	−0.119091
$75$	0	0
$76$	13.8556	1.58934
$77$	− 13.2356i	− 1.50834i
$78$	0.791365i	0.0896045i
$79$	−4.95687	−0.557691	−0.278846	−	0.960336i	$-0.589952\pi$
$79$	−4.95687	−0.557691	−0.278846	+	0.960336i	$0.589952\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 2.77354i	− 0.306287i
$83$	− 9.36322i	− 1.02775i	−0.857866	−	0.513874i	$-0.828210\pi$
$83$	− 9.36322i	− 1.02775i	0.857866	−	0.513874i	$-0.171790\pi$
$84$	4.40559	0.480689
$85$	0	0
$86$	−6.14378	−0.662500
$87$	0.0356723i	0.00382447i
$88$	− 10.2448i	− 1.09210i
$89$	0.0123190	0.00130581	0.000652905	−	1.00000i	$-0.499792\pi$
$89$	0.0123190	0.00130581	0.000652905		1.00000i	$0.499792\pi$
$90$	0	0
$91$	−3.79524	−0.397850
$92$	− 0.638192i	− 0.0665361i
$93$	4.48902i	0.465490i
$94$	−4.57199	−0.471565
$95$	0	0
$96$	5.24733	0.535553
$97$	− 7.62041i	− 0.773736i	−0.922135	−	0.386868i	$-0.873557\pi$
$97$	− 7.62041i	− 0.773736i	0.922135	−	0.386868i	$-0.126443\pi$
$98$	0.203215i	0.0205278i
$99$	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	$0.364933\pi$
$102$	0.369069i	0.0365433i
$103$	− 10.3860i	− 1.02336i	−0.859175	−	0.511682i	$-0.829022\pi$
$103$	− 10.3860i	− 1.02336i	0.859175	−	0.511682i	$-0.170978\pi$
$104$	−2.93764	−0.288059
$105$	0	0
$106$	−4.89501	−0.475445
$107$	− 12.2737i	− 1.18655i	−0.805001	−	0.593274i	$-0.797835\pi$
$107$	− 12.2737i	− 1.18655i	0.805001	−	0.593274i	$-0.202165\pi$
$108$	1.71212i	0.164749i
$109$	4.22902	0.405066	0.202533	−	0.979275i	$-0.435083\pi$
$109$	4.22902	0.405066	0.202533	+	0.979275i	$0.435083\pi$
$110$	0	0
$111$	1.90935	0.181228
$112$	6.06133i	0.572742i
$113$	18.3687i	1.72798i	0.503511	+	0.863989i	$0.332041\pi$
$113$	18.3687i	1.72798i	−0.503511	+	0.863989i	$0.667959\pi$
$114$	4.34209	0.406674
$115$	0	0
$116$	−0.0610751	−0.00567068
$117$	− 1.47492i	− 0.136357i
$118$	0.0945341i	0.00870257i
$119$	−1.76999	−0.162254
$120$	0	0
$121$	15.4575	1.40523
$122$	− 3.44008i	− 0.311450i
$123$	5.16925i	0.466095i
$124$	−7.68573	−0.690199
$125$	0	0
$126$	1.38063	0.122996
$127$	0.345501i	0.0306582i	0.999883	+	0.0153291i	$0.00487960\pi$
$127$	0.345501i	0.0306582i	−0.999883	+	0.0153291i	$0.995120\pi$
$128$	11.5118i	1.01751i
$129$	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	$0.563928\pi$
$132$	8.80660i	0.766516i
$133$	20.8238i	1.80566i
$134$	−0.0447683	−0.00386739
$135$	0	0
$136$	−1.37003	−0.117479
$137$	− 4.81753i	− 0.411589i	−0.978595	−	0.205795i	$-0.934022\pi$
$137$	− 4.81753i	− 0.411589i	0.978595	−	0.205795i	$-0.0659779\pi$
$138$	− 0.199998i	− 0.0170249i
$139$	3.74075	0.317286	0.158643	−	0.987336i	$-0.449288\pi$
$139$	3.74075	0.317286	0.158643	+	0.987336i	$0.449288\pi$
$140$	0	0
$141$	8.52114	0.717609
$142$	6.52325i	0.547419i
$143$	− 7.58655i	− 0.634419i
$144$	−2.35558	−0.196298
$145$	0	0
$146$	−6.46636	−0.535160
$147$	− 0.378747i	− 0.0312385i
$148$	3.26904i	0.268713i
$149$	−8.64621	−0.708325	−0.354163	−	0.935184i	$-0.615234\pi$
$149$	−8.64621	−0.708325	−0.354163	+	0.935184i	$0.615234\pi$
$150$	0	0
$151$	−1.24898	−0.101641	−0.0508205	−	0.998708i	$-0.516184\pi$
$151$	−1.24898	−0.101641	−0.0508205	+	0.998708i	$0.516184\pi$
$152$	16.1183i	1.30737i
$153$	− 0.687859i	− 0.0556101i
$154$	7.10154	0.572258
$155$	0	0
$156$	2.52524	0.202181
$157$	3.86574i	0.308520i	0.988030	+	0.154260i	$0.0492993\pi$
$157$	3.86574i	0.308520i	−0.988030	+	0.154260i	$0.950701\pi$
$158$	− 2.65959i	− 0.211586i
$159$	9.12317	0.723515
$160$	0	0
$161$	0.959153	0.0755918
$162$	0.536547i	0.0421551i
$163$	6.97411i	0.546254i	0.961978	+	0.273127i	$0.0880580\pi$
$163$	6.97411i	0.546254i	−0.961978	+	0.273127i	$0.911942\pi$
$164$	−8.85036	−0.691097
$165$	0	0
$166$	5.02381	0.389923
$167$	− 15.9594i	− 1.23497i	−0.786581	−	0.617487i	$-0.788151\pi$
$167$	− 15.9594i	− 1.23497i	0.786581	−	0.617487i	$-0.211849\pi$
$168$	5.12507i	0.395407i
$169$	10.8246	0.832662
$170$	0	0
$171$	−8.09265	−0.618861
$172$	19.6047i	1.49485i
$173$	7.67330i	0.583390i	0.956511	+	0.291695i	$0.0942193\pi$
$173$	7.67330i	0.583390i	−0.956511	+	0.291695i	$0.905781\pi$
$174$	−0.0191399	−0.00145099
$175$	0	0
$176$	−12.1164	−0.913306
$177$	− 0.176190i	− 0.0132432i
$178$	0.00660971i	0 0.000495419i
$179$	−10.6093	−0.792977	−0.396489	−	0.918040i	$-0.629771\pi$
$179$	−10.6093	−0.792977	−0.396489	+	0.918040i	$0.629771\pi$
$180$	0	0
$181$	15.1076	1.12294	0.561471	−	0.827496i	$-0.310236\pi$
$181$	15.1076	1.12294	0.561471	+	0.827496i	$0.310236\pi$
$182$	− 2.03633i	− 0.150942i
$183$	6.41152i	0.473953i
$184$	0.742415	0.0547316
$185$	0	0
$186$	−2.40857	−0.176605
$187$	− 3.53813i	− 0.258734i
$188$	14.5892i	1.06403i
$189$	−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	$0.353925\pi$
$192$	− 1.89572i	− 0.136812i
$193$	− 14.2421i	− 1.02517i	−0.858637	−	0.512585i	$-0.828688\pi$
$193$	− 14.2421i	− 1.02517i	0.858637	−	0.512585i	$-0.171312\pi$
$194$	4.08871	0.293552
$195$	0	0
$196$	0.648459	0.0463185
$197$	− 17.7488i	− 1.26455i	−0.774743	−	0.632276i	$-0.782121\pi$
$197$	− 17.7488i	− 1.26455i	0.774743	−	0.632276i	$-0.217879\pi$
$198$	2.75983i	0.196133i
$199$	18.9550	1.34369	0.671843	−	0.740693i	$-0.265503\pi$
$199$	18.9550	1.34369	0.671843	+	0.740693i	$0.265503\pi$
$200$	0	0
$201$	0.0834377	0.00588524
$202$	4.44002i	0.312399i
$203$	− 0.0917912i	− 0.00644248i
$204$	1.17770	0.0824552
$205$	0	0
$206$	5.57259	0.388260
$207$	0.372750i	0.0259079i
$208$	3.47430i	0.240899i
$209$	−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	$0.620969\pi$
$212$	15.6199i	1.07278i
$213$	− 12.1578i	− 0.833041i
$214$	6.58544	0.450171
$215$	0	0
$216$	−1.99173	−0.135520
$217$	− 11.5511i	− 0.784137i
$218$	2.26907i	0.153681i
$219$	12.0518	0.814386
$220$	0	0
$221$	−1.01454	−0.0682453
$222$	1.02446i	0.0687571i
$223$	− 27.5707i	− 1.84627i	−0.384472	−	0.923137i	$-0.625617\pi$
$223$	− 27.5707i	− 1.84627i	0.384472	−	0.923137i	$-0.374383\pi$
$224$	−13.5023	−0.902162
$225$	0	0
$226$	−9.85565	−0.655588
$227$	− 3.98221i	− 0.264309i	−0.991229	−	0.132154i	$-0.957811\pi$
$227$	− 3.98221i	− 0.264309i	0.991229	−	0.132154i	$-0.0421894\pi$
$228$	− 13.8556i	− 0.917608i
$229$	−11.9697	−0.790983	−0.395492	−	0.918470i	$-0.629426\pi$
$229$	−11.9697	−0.790983	−0.395492	+	0.918470i	$0.629426\pi$
$230$	0	0
$231$	−13.2356	−0.870841
$232$	− 0.0710494i	− 0.00466462i
$233$	− 21.1181i	− 1.38349i	−0.722140	−	0.691747i	$-0.756841\pi$
$233$	− 21.1181i	− 1.38349i	0.722140	−	0.691747i	$-0.243159\pi$
$234$	0.791365	0.0517332
$235$	0	0
$236$	0.301658	0.0196362
$237$	4.95687i	0.321983i
$238$	− 0.949680i	− 0.0615586i
$239$	−26.2050	−1.69506	−0.847529	−	0.530749i	$-0.821911\pi$
$239$	−26.2050	−1.69506	−0.847529	+	0.530749i	$0.821911\pi$
$240$	0	0
$241$	−8.17452	−0.526567	−0.263284	−	0.964719i	$-0.584805\pi$
$241$	−8.17452	−0.526567	−0.263284	+	0.964719i	$0.584805\pi$
$242$	8.29369i	0.533139i
$243$	− 1.00000i	− 0.0641500i
$244$	−10.9773	−0.702748
$245$	0	0
$246$	−2.77354	−0.176835
$247$	11.9360i	0.759472i
$248$	− 8.94090i	− 0.567748i
$249$	−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	$0.213618\pi$
$252$	− 4.40559i	− 0.277526i
$253$	1.91731i	0.120540i
$254$	−0.185377	−0.0116316
$255$	0	0
$256$	−2.38518	−0.149074
$257$	24.2995i	1.51576i	0.652392	+	0.757882i	$0.273766\pi$
$257$	24.2995i	1.51576i	−0.652392	+	0.757882i	$0.726234\pi$
$258$	6.14378i	0.382495i
$259$	−4.91311	−0.305286
$260$	0	0
$261$	0.0356723	0.00220806
$262$	− 2.45015i	− 0.151371i
$263$	− 8.52269i	− 0.525532i	−0.964860	−	0.262766i	$-0.915365\pi$
$263$	− 8.52269i	− 0.525532i	0.964860	−	0.262766i	$-0.0846347\pi$
$264$	−10.2448	−0.630525
$265$	0	0
$266$	−11.1730	−0.685059
$267$	− 0.0123190i	0 0.000753909i
$268$	0.142855i	0.00872627i
$269$	−2.58312	−0.157496	−0.0787479	−	0.996895i	$-0.525092\pi$
$269$	−2.58312	−0.157496	−0.0787479	+	0.996895i	$0.525092\pi$
$270$	0	0
$271$	−13.5893	−0.825491	−0.412746	−	0.910846i	$-0.635430\pi$
$271$	−13.5893	−0.825491	−0.412746	+	0.910846i	$0.635430\pi$
$272$	1.62031i	0.0982456i
$273$	3.79524i	0.229699i
$274$	2.58483	0.156155
$275$	0	0
$276$	−0.638192	−0.0384146
$277$	9.13975i	0.549154i	0.961565	+	0.274577i	$0.0885379\pi$
$277$	9.13975i	0.549154i	−0.961565	+	0.274577i	$0.911462\pi$
$278$	2.00709i	0.120377i
$279$	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	$0.399501\pi$
$282$	4.57199i	0.272258i
$283$	− 4.81586i	− 0.286273i	−0.989703	−	0.143137i	$-0.954281\pi$
$283$	− 4.81586i	− 0.286273i	0.989703	−	0.143137i	$-0.0457188\pi$
$284$	20.8156	1.23518
$285$	0	0
$286$	4.07054	0.240696
$287$	− 13.3014i	− 0.785157i
$288$	− 5.24733i	− 0.309202i
$289$	16.5268	0.972168
$290$	0	0
$291$	−7.62041	−0.446716
$292$	20.6341i	1.20752i
$293$	− 24.4506i	− 1.42842i	−0.699932	−	0.714210i	$-0.746786\pi$
$293$	− 24.4506i	− 1.42842i	0.699932	−	0.714210i	$-0.253214\pi$
$294$	0.203215	0.0118518
$295$	0	0
$296$	−3.80291	−0.221039
$297$	− 5.14369i	− 0.298467i
$298$	− 4.63910i	− 0.268736i
$299$	0.549778	0.0317945
$300$	0	0
$301$	−29.4644	−1.69830
$302$	− 0.670139i	− 0.0385622i
$303$	− 8.27518i	− 0.475397i
$304$	19.0629	1.09333
$305$	0	0
$306$	0.369069	0.0210983
$307$	− 9.51655i	− 0.543138i	−0.962419	−	0.271569i	$-0.912457\pi$
$307$	− 9.51655i	− 0.543138i	0.962419	−	0.271569i	$-0.0875426\pi$
$308$	− 22.6610i	− 1.29123i
$309$	−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	$0.249471\pi$
$312$	2.93764i	0.166311i
$313$	− 9.88161i	− 0.558542i	−0.960212	−	0.279271i	$-0.909907\pi$
$313$	− 9.88161i	− 0.558542i	0.960212	−	0.279271i	$-0.0900927\pi$
$314$	−2.07415	−0.117051
$315$	0	0
$316$	−8.48674	−0.477416
$317$	24.7310i	1.38903i	0.719477	+	0.694516i	$0.244382\pi$
$317$	24.7310i	1.38903i	−0.719477	+	0.694516i	$0.755618\pi$
$318$	4.89501i	0.274499i
$319$	0.183487	0.0102733
$320$	0	0
$321$	−12.2737	−0.685054
$322$	0.514630i	0.0286792i
$323$	5.56661i	0.309734i
$324$	1.71212	0.0951176
$325$	0	0
$326$	−3.74194	−0.207247
$327$	− 4.22902i	− 0.233865i
$328$	− 10.2957i	− 0.568486i
$329$	−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	$0.490566\pi$
$332$	− 16.0309i	− 0.879812i
$333$	− 1.90935i	− 0.104632i
$334$	8.56296	0.468544
$335$	0	0
$336$	6.06133	0.330673
$337$	− 14.0609i	− 0.765948i	−0.923759	−	0.382974i	$-0.874900\pi$
$337$	− 14.0609i	− 0.765948i	0.923759	−	0.382974i	$-0.125100\pi$
$338$	5.80791i	0.315908i
$339$	18.3687	0.997648
$340$	0	0
$341$	23.0901	1.25040
$342$	− 4.34209i	− 0.234793i
$343$	18.9868i	1.02519i
$344$	−22.8064	−1.22964
$345$	0	0
$346$	−4.11709	−0.221336
$347$	− 5.69584i	− 0.305769i	−0.988244	−	0.152884i	$-0.951144\pi$
$347$	− 5.69584i	− 0.305769i	0.988244	−	0.152884i	$-0.0488562\pi$
$348$	0.0610751i	0.00327397i
$349$	28.4950	1.52530	0.762651	−	0.646810i	$-0.223897\pi$
$349$	28.4950	1.52530	0.762651	+	0.646810i	$0.223897\pi$
$350$	0	0
$351$	−1.47492	−0.0787256
$352$	− 26.9906i	− 1.43861i
$353$	12.7656i	0.679443i	0.940526	+	0.339721i	$0.110333\pi$
$353$	12.7656i	0.679443i	−0.940526	+	0.339721i	$0.889667\pi$
$354$	0.0945341	0.00502443
$355$	0	0
$356$	0.0210915	0.00111785
$357$	1.76999i	0.0936776i
$358$	− 5.69240i	− 0.300852i
$359$	4.51821	0.238462	0.119231	−	0.992867i	$-0.461957\pi$
$359$	4.51821	0.238462	0.119231	+	0.992867i	$0.461957\pi$
$360$	0	0
$361$	46.4910	2.44690
$362$	8.10596i	0.426040i
$363$	− 15.4575i	− 0.811310i
$364$	−6.49790	−0.340583
$365$	0	0
$366$	−3.44008	−0.179816
$367$	− 3.03446i	− 0.158398i	−0.996859	−	0.0791989i	$-0.974764\pi$
$367$	− 3.03446i	− 0.158398i	0.996859	−	0.0791989i	$-0.0252362\pi$
$368$	− 0.878043i	− 0.0457711i
$369$	5.16925	0.269100
$370$	0	0
$371$	−23.4756	−1.21879
$372$	7.68573i	0.398487i
$373$	15.3659i	0.795614i	0.917469	+	0.397807i	$0.130229\pi$
$373$	15.3659i	0.795614i	−0.917469	+	0.397807i	$0.869771\pi$
$374$	1.89838	0.0981626
$375$	0	0
$376$	−16.9718	−0.875252
$377$	− 0.0526139i	− 0.00270975i
$378$	− 1.38063i	− 0.0710120i
$379$	−10.3597	−0.532142	−0.266071	−	0.963953i	$-0.585725\pi$
$379$	−10.3597	−0.532142	−0.266071	+	0.963953i	$0.585725\pi$
$380$	0	0
$381$	0.345501	0.0177005
$382$	6.56946i	0.336123i
$383$	− 15.4866i	− 0.791327i	−0.918396	−	0.395663i	$-0.870515\pi$
$383$	− 15.4866i	− 0.791327i	0.918396	−	0.395663i	$-0.129485\pi$
$384$	11.5118	0.587459
$385$	0	0
$386$	7.64156	0.388945
$387$	− 11.4506i	− 0.582066i
$388$	− 13.0470i	− 0.662363i
$389$	21.3949	1.08476	0.542382	−	0.840132i	$-0.317522\pi$
$389$	21.3949	1.08476	0.542382	+	0.840132i	$0.317522\pi$
$390$	0	0
$391$	0.256400	0.0129667
$392$	0.754359i	0.0381009i
$393$	4.56651i	0.230350i
$394$	9.52309	0.479766
$395$	0	0
$396$	8.80660	0.442548
$397$	− 12.5016i	− 0.627439i	−0.949516	−	0.313719i	$-0.898425\pi$
$397$	− 12.5016i	− 0.627439i	0.949516	−	0.313719i	$-0.101575\pi$
$398$	10.1703i	0.509789i
$399$	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	$0.808917\pi$
$402$	0.0447683i	0.00223284i
$403$	− 6.62096i	− 0.329814i
$404$	14.1681	0.704888
$405$	0	0
$406$	0.0492503	0.00244425
$407$	− 9.82111i	− 0.486815i
$408$	1.37003i	0.0678264i
$409$	−3.01739	−0.149200	−0.0746002	−	0.997214i	$-0.523768\pi$
$409$	−3.01739	−0.149200	−0.0746002	+	0.997214i	$0.523768\pi$
$410$	0	0
$411$	−4.81753	−0.237631
$412$	− 17.7821i	− 0.876060i
$413$	0.453368i	0.0223088i
$414$	−0.199998	−0.00982936
$415$	0	0
$416$	−7.73941	−0.379456
$417$	− 3.74075i	− 0.183185i
$418$	− 22.3343i	− 1.09241i
$419$	−7.11371	−0.347528	−0.173764	−	0.984787i	$-0.555593\pi$
$419$	−7.11371	−0.347528	−0.173764	+	0.984787i	$0.555593\pi$
$420$	0	0
$421$	−21.8569	−1.06524	−0.532620	−	0.846354i	$-0.678793\pi$
$421$	−21.8569	−1.06524	−0.532620	+	0.846354i	$0.678793\pi$
$422$	− 5.78227i	− 0.281477i
$423$	− 8.52114i	− 0.414312i
$424$	−18.1709	−0.882455
$425$	0	0
$426$	6.52325	0.316052
$427$	− 16.4980i	− 0.798394i
$428$	− 21.0141i	− 1.01575i
$429$	−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	$0.469334\pi$
$432$	2.35558i	0.113333i
$433$	25.1913i	1.21062i	0.795991	+	0.605309i	$0.206950\pi$
$433$	25.1913i	1.21062i	−0.795991	+	0.605309i	$0.793050\pi$
$434$	6.19769	0.297498
$435$	0	0
$436$	7.24057	0.346761
$437$	− 3.01654i	− 0.144300i
$438$	6.46636i	0.308975i
$439$	1.71510	0.0818575	0.0409287	−	0.999162i	$-0.486968\pi$
$439$	1.71510	0.0818575	0.0409287	+	0.999162i	$0.486968\pi$
$440$	0	0
$441$	−0.378747	−0.0180356
$442$	− 0.544348i	− 0.0258920i
$443$	− 11.7475i	− 0.558140i	−0.960271	−	0.279070i	$-0.909974\pi$
$443$	− 11.7475i	− 0.558140i	0.960271	−	0.279070i	$-0.0900261\pi$
$444$	3.26904	0.155142
$445$	0	0
$446$	14.7930	0.700469
$447$	8.64621i	0.408952i
$448$	4.87804i	0.230466i
$449$	−12.8415	−0.606030	−0.303015	−	0.952986i	$-0.597993\pi$
$449$	−12.8415	−0.606030	−0.303015	+	0.952986i	$0.597993\pi$
$450$	0	0
$451$	26.5890	1.25203
$452$	31.4493i	1.47925i
$453$	1.24898i	0.0586824i
$454$	2.13664	0.100278
$455$	0	0
$456$	16.1183	0.754810
$457$	− 13.6882i	− 0.640309i	−0.947365	−	0.320155i	$-0.896265\pi$
$457$	− 13.6882i	− 0.640309i	0.947365	−	0.320155i	$-0.103735\pi$
$458$	− 6.42233i	− 0.300096i
$459$	−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	$0.427370\pi$
$462$	− 7.10154i	− 0.330393i
$463$	− 21.2740i	− 0.988687i	−0.869267	−	0.494344i	$-0.835408\pi$
$463$	− 21.2740i	− 0.988687i	0.869267	−	0.494344i	$-0.164592\pi$
$464$	−0.0840289	−0.00390095
$465$	0	0
$466$	11.3309	0.524892
$467$	− 13.9014i	− 0.643282i	−0.946862	−	0.321641i	$-0.895766\pi$
$467$	− 13.9014i	− 0.643282i	0.946862	−	0.321641i	$-0.104234\pi$
$468$	− 2.52524i	− 0.116729i
$469$	−0.214700	−0.00991394
$470$	0	0
$471$	3.86574	0.178124
$472$	0.350922i	0.0161525i
$473$	− 58.8982i	− 2.70814i
$474$	−2.65959	−0.122159
$475$	0	0
$476$	−3.03042	−0.138899
$477$	− 9.12317i	− 0.417721i
$478$	− 14.0602i	− 0.643098i
$479$	12.3677	0.565096	0.282548	−	0.959253i	$-0.408820\pi$
$479$	12.3677	0.565096	0.282548	+	0.959253i	$0.408820\pi$
$480$	0	0
$481$	−2.81615	−0.128405
$482$	− 4.38601i	− 0.199777i
$483$	− 0.959153i	− 0.0436430i
$484$	26.4651	1.20296
$485$	0	0
$486$	0.536547	0.0243383
$487$	35.3280i	1.60087i	0.599422	+	0.800433i	$0.295397\pi$
$487$	35.3280i	1.60087i	−0.599422	+	0.800433i	$0.704603\pi$
$488$	− 12.7700i	− 0.578070i
$489$	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	$0.662825\pi$
$492$	8.85036i	0.399005i
$493$	− 0.0245375i	− 0.00110511i
$494$	−6.40425	−0.288141
$495$	0	0
$496$	−10.5743	−0.474798
$497$	31.2843i	1.40329i
$498$	− 5.02381i	− 0.225122i
$499$	−38.7869	−1.73634	−0.868171	−	0.496265i	$-0.834704\pi$
$499$	−38.7869	−1.73634	−0.868171	+	0.496265i	$0.834704\pi$
$500$	0	0
$501$	−15.9594	−0.713013
$502$	13.3141i	0.594239i
$503$	6.41964i	0.286238i	0.989706	+	0.143119i	$0.0457131\pi$
$503$	6.41964i	0.286238i	−0.989706	+	0.143119i	$0.954287\pi$
$504$	5.12507	0.228289
$505$	0	0
$506$	−1.02873	−0.0457325
$507$	− 10.8246i	− 0.480737i
$508$	0.591538i	0.0262452i
$509$	7.19516	0.318920	0.159460	−	0.987204i	$-0.449025\pi$
$509$	7.19516	0.318920	0.159460	+	0.987204i	$0.449025\pi$
$510$	0	0
$511$	−31.0115	−1.37187
$512$	21.7438i	0.960951i
$513$	8.09265i	0.357299i
$514$	−13.0378	−0.575074
$515$	0	0
$516$	19.6047	0.863051
$517$	− 43.8301i	− 1.92764i
$518$	− 2.63611i	− 0.115824i
$519$	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	$0.574021\pi$
$522$	0.0191399i	0 0.000837729i
$523$	− 11.3242i	− 0.495174i	−0.968866	−	0.247587i	$-0.920362\pi$
$523$	− 11.3242i	− 0.495174i	0.968866	−	0.247587i	$-0.0796375\pi$
$524$	−7.81840	−0.341548
$525$	0	0
$526$	4.57282	0.199385
$527$	− 3.08782i	− 0.134507i
$528$	12.1164i	0.527297i
$529$	22.8611	0.993959
$530$	0	0
$531$	−0.176190	−0.00764599
$532$	35.6529i	1.54575i
$533$	− 7.62424i	− 0.330243i
$534$	0.00660971	0.000286030 0
$535$	0	0
$536$	−0.166185	−0.00717810
$537$	10.6093i	0.457826i
$538$	− 1.38597i	− 0.0597533i
$539$	−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	$0.381495\pi$
$542$	− 7.29130i	− 0.313188i
$543$	− 15.1076i	− 0.648331i
$544$	−3.60943	−0.154753
$545$	0	0
$546$	−2.03633	−0.0871467
$547$	− 44.1367i	− 1.88715i	−0.331160	−	0.943575i	$-0.607440\pi$
$547$	− 44.1367i	− 1.88715i	0.331160	−	0.943575i	$-0.392560\pi$
$548$	− 8.24818i	− 0.352345i
$549$	6.41152	0.273637
$550$	0	0
$551$	−0.288683	−0.0122983
$552$	− 0.742415i	− 0.0315993i
$553$	− 12.7549i	− 0.542394i
$554$	−4.90390	−0.208347
$555$	0	0
$556$	6.40460	0.271615
$557$	17.8472i	0.756210i	0.925763	+	0.378105i	$0.123424\pi$
$557$	17.8472i	0.756210i	−0.925763	+	0.378105i	$0.876576\pi$
$558$	2.40857i	0.101963i
$559$	−16.8887	−0.714317
$560$	0	0
$561$	−3.53813	−0.149380
$562$	5.58551i	0.235610i
$563$	21.4063i	0.902167i	0.892482	+	0.451083i	$0.148962\pi$
$563$	21.4063i	0.902167i	−0.892482	+	0.451083i	$0.851038\pi$
$564$	14.5892	0.614316
$565$	0	0
$566$	2.58394	0.108611
$567$	2.57318i	0.108063i
$568$	24.2151i	1.01604i
$569$	26.0723	1.09301	0.546504	−	0.837456i	$-0.315958\pi$
$569$	26.0723	1.09301	0.546504	+	0.837456i	$0.315958\pi$
$570$	0	0
$571$	−9.53879	−0.399186	−0.199593	−	0.979879i	$-0.563962\pi$
$571$	−9.53879	−0.399186	−0.199593	+	0.979879i	$0.563962\pi$
$572$	− 12.9891i	− 0.543100i
$573$	− 12.2440i	− 0.511498i
$574$	7.13683	0.297885
$575$	0	0
$576$	−1.89572	−0.0789885
$577$	9.10504i	0.379048i	0.981876	+	0.189524i	$0.0606944\pi$
$577$	9.10504i	0.379048i	−0.981876	+	0.189524i	$0.939306\pi$
$578$	8.86743i	0.368836i
$579$	−14.2421	−0.591882
$580$	0	0
$581$	24.0933	0.999557
$582$	− 4.08871i	− 0.169482i
$583$	− 46.9268i	− 1.94351i
$584$	−24.0039	−0.993288
$585$	0	0
$586$	13.1189	0.541936
$587$	− 29.4485i	− 1.21547i	−0.794140	−	0.607734i	$-0.792079\pi$
$587$	− 29.4485i	− 1.21547i	0.794140	−	0.607734i	$-0.207921\pi$
$588$	− 0.648459i	− 0.0267420i
$589$	−36.3281	−1.49687
$590$	0	0
$591$	−17.7488	−0.730090
$592$	4.49763i	0.184852i
$593$	33.7757i	1.38700i	0.720456	+	0.693501i	$0.243933\pi$
$593$	33.7757i	1.38700i	−0.720456	+	0.693501i	$0.756067\pi$
$594$	2.75983	0.113237
$595$	0	0
$596$	−14.8033	−0.606368
$597$	− 18.9550i	− 0.775778i
$598$	0.294981i	0.0120627i
$599$	12.0575	0.492656	0.246328	−	0.969187i	$-0.420776\pi$
$599$	12.0575	0.492656	0.246328	+	0.969187i	$0.420776\pi$
$600$	0	0
$601$	0.0653240	0.00266462	0.00133231	−	0.999999i	$-0.499576\pi$
$601$	0.0653240	0.00266462	0.00133231	+	0.999999i	$0.499576\pi$
$602$	− 15.8090i	− 0.644328i
$603$	− 0.0834377i	− 0.00339785i
$604$	−2.13841	−0.0870106
$605$	0	0
$606$	4.44002	0.180364
$607$	− 25.9556i	− 1.05350i	−0.850019	−	0.526752i	$-0.823410\pi$
$607$	− 25.9556i	− 1.05350i	0.850019	−	0.526752i	$-0.176590\pi$
$608$	42.4648i	1.72218i
$609$	−0.0917912	−0.00371957
$610$	0	0
$611$	−12.5680	−0.508448
$612$	− 1.17770i	− 0.0476055i
$613$	− 12.5712i	− 0.507747i	−0.967237	−	0.253873i	$-0.918295\pi$
$613$	− 12.5712i	− 0.507747i	0.967237	−	0.253873i	$-0.0817047\pi$
$614$	5.10608	0.206064
$615$	0	0
$616$	26.3617	1.06214
$617$	− 1.85723i	− 0.0747693i	−0.999301	−	0.0373846i	$-0.988097\pi$
$617$	− 1.85723i	− 0.0747693i	0.999301	−	0.0373846i	$-0.0119027\pi$
$618$	− 5.57259i	− 0.224162i
$619$	18.0236	0.724431	0.362216	−	0.932094i	$-0.382020\pi$
$619$	18.0236	0.724431	0.362216	+	0.932094i	$0.382020\pi$
$620$	0	0
$621$	0.372750	0.0149579
$622$	13.4037i	0.537438i
$623$	0.0316990i	0.00126999i
$624$	3.47430	0.139083
$625$	0	0
$626$	5.30195	0.211908
$627$	41.6261i	1.66239i
$628$	6.61860i	0.264111i
$629$	−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	$0.359498\pi$
$632$	− 9.87272i	− 0.392716i
$633$	10.7768i	0.428340i
$634$	−13.2694	−0.526993
$635$	0	0
$636$	15.6199	0.619371
$637$	0.558622i	0.0221334i
$638$	0.0984494i	0.00389765i
$639$	−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	$0.213266\pi$
$642$	− 6.58544i	− 0.259907i
$643$	1.26211i	0.0497729i	0.999690	+	0.0248864i	$0.00792242\pi$
$643$	1.26211i	0.0497729i	−0.999690	+	0.0248864i	$0.992078\pi$
$644$	1.64218	0.0647110
$645$	0	0
$646$	−2.98675	−0.117512
$647$	28.9276i	1.13726i	0.822593	+	0.568631i	$0.192527\pi$
$647$	28.9276i	1.13726i	−0.822593	+	0.568631i	$0.807473\pi$
$648$	1.99173i	0.0782424i
$649$	−0.906266	−0.0355740
$650$	0	0
$651$	−11.5511	−0.452722
$652$	11.9405i	0.467626i
$653$	17.4796i	0.684031i	0.939694	+	0.342016i	$0.111110\pi$
$653$	17.4796i	0.684031i	−0.939694	+	0.342016i	$0.888890\pi$
$654$	2.26907	0.0887275
$655$	0	0
$656$	−12.1766	−0.475416
$657$	− 12.0518i	− 0.470186i
$658$	− 11.7645i	− 0.458630i
$659$	−11.3394	−0.441720	−0.220860	−	0.975306i	$-0.570886\pi$
$659$	−11.3394	−0.441720	−0.220860	+	0.975306i	$0.570886\pi$
$660$	0	0
$661$	4.63889	0.180432	0.0902160	−	0.995922i	$-0.471244\pi$
$661$	4.63889	0.180432	0.0902160	+	0.995922i	$0.471244\pi$
$662$	0.578541i	0.0224856i
$663$	1.01454i	0.0394015i
$664$	18.6490	0.723720
$665$	0	0
$666$	1.02446	0.0396969
$667$	0.0132968i	0 0.000514856i
$668$	− 27.3243i	− 1.05721i
$669$	−27.5707	−1.06595
$670$	0	0
$671$	32.9789	1.27314
$672$	13.5023i	0.520863i
$673$	7.53164i	0.290323i	0.989408	+	0.145162i	$0.0463702\pi$
$673$	7.53164i	0.290323i	−0.989408	+	0.145162i	$0.953630\pi$
$674$	7.54435	0.290598
$675$	0	0
$676$	18.5330	0.712807
$677$	− 31.6182i	− 1.21518i	−0.794249	−	0.607592i	$-0.792136\pi$
$677$	− 31.6182i	− 1.21518i	0.794249	−	0.607592i	$-0.207864\pi$
$678$	9.85565i	0.378504i
$679$	19.6087	0.752512
$680$	0	0
$681$	−3.98221	−0.152599
$682$	12.3889i	0.474397i
$683$	30.6601i	1.17317i	0.809886	+	0.586587i	$0.199529\pi$
$683$	30.6601i	1.17317i	−0.809886	+	0.586587i	$0.800471\pi$
$684$	−13.8556	−0.529781
$685$	0	0
$686$	−10.1873	−0.388954
$687$	11.9697i	0.456674i
$688$	26.9728i	1.02833i
$689$	−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	$0.729729\pi$
$692$	13.1376i	0.499416i
$693$	13.2356i	0.502780i
$694$	3.05609	0.116008
$695$	0	0
$696$	−0.0710494	−0.00269312
$697$	− 3.55571i	− 0.134682i
$698$	15.2889i	0.578694i
$699$	−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	$0.620326\pi$
$702$	− 0.791365i	− 0.0298682i
$703$	15.4517i	0.582773i
$704$	−9.75101	−0.367505
$705$	0	0
$706$	−6.84933	−0.257778
$707$	21.2935i	0.800825i
$708$	− 0.301658i	− 0.0113370i
$709$	25.3106	0.950558	0.475279	−	0.879835i	$-0.342347\pi$
$709$	25.3106	0.950558	0.475279	+	0.879835i	$0.342347\pi$
$710$	0	0
$711$	4.95687	0.185897
$712$	0.0245360i	0 0.000919526i
$713$	1.67328i	0.0626649i
$714$	−0.949680	−0.0355409
$715$	0	0
$716$	−18.1644	−0.678835
$717$	26.2050i	0.978643i
$718$	2.42423i	0.0904715i
$719$	9.19065	0.342754	0.171377	−	0.985206i	$-0.445178\pi$
$719$	9.19065	0.342754	0.171377	+	0.985206i	$0.445178\pi$
$720$	0	0
$721$	26.7251	0.995294
$722$	24.9446i	0.928342i
$723$	8.17452i	0.304014i
$724$	25.8661	0.961305
$725$	0	0
$726$	8.29369	0.307808
$727$	13.0649i	0.484549i	0.970208	+	0.242274i	$0.0778934\pi$
$727$	13.0649i	0.484549i	−0.970208	+	0.242274i	$0.922107\pi$
$728$	− 7.55908i	− 0.280158i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−7.87639	−0.291319
$732$	10.9773i	0.405732i
$733$	− 9.68091i	− 0.357573i	−0.983888	−	0.178786i	$-0.942783\pi$
$733$	− 9.68091i	− 0.357573i	0.983888	−	0.178786i	$-0.0572171\pi$
$734$	1.62813	0.0600955
$735$	0	0
$736$	1.95594	0.0720970
$737$	− 0.429178i	− 0.0158090i
$738$	2.77354i	0.102096i
$739$	−9.13413	−0.336004	−0.168002	−	0.985787i	$-0.553732\pi$
$739$	−9.13413	−0.336004	−0.168002	+	0.985787i	$0.553732\pi$
$740$	0	0
$741$	11.9360	0.438481
$742$	− 12.5957i	− 0.462404i
$743$	− 6.35760i	− 0.233238i	−0.993177	−	0.116619i	$-0.962794\pi$
$743$	− 6.35760i	− 0.233238i	0.993177	−	0.116619i	$-0.0372056\pi$
$744$	−8.94090	−0.327789
$745$	0	0
$746$	−8.24450	−0.301853
$747$	9.36322i	0.342582i
$748$	− 6.05770i	− 0.221492i
$749$	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	$0.279414\pi$
$752$	20.0722i	0.731959i
$753$	− 24.8145i	− 0.904291i
$754$	0.0282298	0.00102807
$755$	0	0
$756$	−4.40559	−0.160230
$757$	− 11.5175i	− 0.418609i	−0.977850	−	0.209305i	$-0.932880\pi$
$757$	− 11.5175i	− 0.418609i	0.977850	−	0.209305i	$-0.0671200\pi$
$758$	− 5.55846i	− 0.201892i
$759$	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	$0.232706\pi$
$762$	0.185377i	0.00671551i
$763$	10.8820i	0.393956i
$764$	20.9631	0.758418
$765$	0	0
$766$	8.30927	0.300226
$767$	0.259867i	0.00938324i
$768$	2.38518i	0.0860677i
$769$	−17.9468	−0.647178	−0.323589	−	0.946198i	$-0.604889\pi$
$769$	−17.9468	−0.647178	−0.323589	+	0.946198i	$0.604889\pi$
$770$	0	0
$771$	24.2995	0.875126
$772$	− 24.3842i	− 0.877606i
$773$	52.0238i	1.87117i	0.353107	+	0.935583i	$0.385125\pi$
$773$	52.0238i	1.87117i	−0.353107	+	0.935583i	$0.614875\pi$
$774$	6.14378	0.220833
$775$	0	0
$776$	15.1778	0.544850
$777$	4.91311i	0.176257i
$778$	11.4794i	0.411555i
$779$	−41.8329	−1.49882
$780$	0	0
$781$	−62.5361	−2.23772
$782$	0.137570i	0.00491951i
$783$	− 0.0356723i	− 0.00127482i
$784$	0.892168	0.0318632
$785$	0	0
$786$	−2.45015	−0.0873938
$787$	12.5243i	0.446442i	0.974768	+	0.223221i	$0.0716572\pi$
$787$	12.5243i	0.446442i	−0.974768	+	0.223221i	$0.928343\pi$
$788$	− 30.3881i	− 1.08253i
$789$	−8.52269	−0.303416
$790$	0	0
$791$	−47.2659	−1.68058
$792$	10.2448i	0.364034i
$793$	− 9.45650i	− 0.335810i
$794$	6.70771	0.238048
$795$	0	0
$796$	32.4532	1.15027
$797$	− 25.1417i	− 0.890563i	−0.895391	−	0.445282i	$-0.853104\pi$
$797$	− 25.1417i	− 0.890563i	0.895391	−	0.445282i	$-0.146896\pi$
$798$	11.1730i	0.395519i
$799$	−5.86134	−0.207359
$800$	0	0
$801$	−0.0123190	−0.000435270 0
$802$	− 17.7317i	− 0.626128i
$803$	− 61.9907i	− 2.18761i
$804$	0.142855	0.00503811
$805$	0	0
$806$	3.55246	0.125130
$807$	2.58312i	0.0909302i
$808$	16.4819i	0.579831i
$809$	4.82683	0.169702	0.0848512	−	0.996394i	$-0.472959\pi$
$809$	4.82683	0.169702	0.0848512	+	0.996394i	$0.472959\pi$
$810$	0	0
$811$	−36.8583	−1.29427	−0.647135	−	0.762375i	$-0.724033\pi$
$811$	−36.8583	−1.29427	−0.647135	+	0.762375i	$0.724033\pi$
$812$	− 0.157157i	− 0.00551514i
$813$	13.5893i	0.476598i
$814$	5.26949	0.184695
$815$	0	0
$816$	1.62031	0.0567221
$817$	92.6656i	3.24196i
$818$	− 1.61897i	− 0.0566061i
$819$	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	$0.420372\pi$
$822$	− 2.58483i	− 0.0901564i
$823$	25.9934i	0.906074i	0.891492	+	0.453037i	$0.149660\pi$
$823$	25.9934i	0.906074i	−0.891492	+	0.453037i	$0.850340\pi$
$824$	20.6861	0.720634
$825$	0	0
$826$	−0.243253	−0.00846386
$827$	− 42.9339i	− 1.49296i	−0.665408	−	0.746480i	$-0.731743\pi$
$827$	− 42.9339i	− 1.49296i	0.665408	−	0.746480i	$-0.268257\pi$
$828$	0.638192i	0.0221787i
$829$	5.63771	0.195806	0.0979029	−	0.995196i	$-0.468787\pi$
$829$	5.63771	0.195806	0.0979029	+	0.995196i	$0.468787\pi$
$830$	0	0
$831$	9.13975	0.317054
$832$	2.79605i	0.0969355i
$833$	0.260524i	0.00902664i
$834$	2.00709	0.0694997
$835$	0	0
$836$	−71.2687	−2.46488
$837$	− 4.48902i	− 0.155163i
$838$	− 3.81684i	− 0.131851i
$839$	23.0630	0.796225	0.398112	−	0.917337i	$-0.369665\pi$
$839$	23.0630	0.796225	0.398112	+	0.917337i	$0.369665\pi$
$840$	0	0
$841$	−28.9987	−0.999956
$842$	− 11.7273i	− 0.404148i
$843$	− 10.4101i	− 0.358543i
$844$	−18.4512	−0.635116
$845$	0	0
$846$	4.57199	0.157188
$847$	39.7750i	1.36669i
$848$	21.4904i	0.737982i
$849$	−4.81586	−0.165280
$850$	0	0
$851$	0.711711	0.0243971
$852$	− 20.8156i	− 0.713132i
$853$	39.0456i	1.33690i	0.743758	+	0.668448i	$0.233041\pi$
$853$	39.0456i	1.33690i	−0.743758	+	0.668448i	$0.766959\pi$
$854$	8.85195	0.302908
$855$	0	0
$856$	24.4459	0.835544
$857$	− 12.7315i	− 0.434899i	−0.976072	−	0.217450i	$-0.930226\pi$
$857$	− 12.7315i	− 0.434899i	0.976072	−	0.217450i	$-0.0697738\pi$
$858$	− 4.07054i	− 0.138966i
$859$	−14.5866	−0.497690	−0.248845	−	0.968543i	$-0.580051\pi$
$859$	−14.5866	−0.497690	−0.248845	+	0.968543i	$0.580051\pi$
$860$	0	0
$861$	−13.3014	−0.453310
$862$	2.14293i	0.0729884i
$863$	− 17.5145i	− 0.596201i	−0.954534	−	0.298101i	$-0.903647\pi$
$863$	− 17.5145i	− 0.596201i	0.954534	−	0.298101i	$-0.0963531\pi$
$864$	−5.24733	−0.178518
$865$	0	0
$866$	−13.5163	−0.459303
$867$	− 16.5268i	− 0.561281i
$868$	− 19.7768i	− 0.671267i
$869$	25.4966	0.864912
$870$	0	0
$871$	−0.123064	−0.00416987
$872$	8.42304i	0.285240i
$873$	7.62041i	0.257912i
$874$	1.61851	0.0547470
$875$	0	0
$876$	20.6341	0.697162
$877$	− 9.10744i	− 0.307536i	−0.988107	−	0.153768i	$-0.950859\pi$
$877$	− 9.10744i	− 0.307536i	0.988107	−	0.153768i	$-0.0491409\pi$
$878$	0.920234i	0.0310564i
$879$	−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	$0.775564\pi$
$882$	− 0.203215i	− 0.00684262i
$883$	− 4.24226i	− 0.142763i	−0.997449	−	0.0713817i	$-0.977259\pi$
$883$	− 4.24226i	− 0.142763i	0.997449	−	0.0713817i	$-0.0227408\pi$
$884$	−1.73701	−0.0584220
$885$	0	0
$886$	6.30307	0.211756
$887$	12.7865i	0.429328i	0.976688	+	0.214664i	$0.0688657\pi$
$887$	12.7865i	0.429328i	−0.976688	+	0.214664i	$0.931134\pi$
$888$	3.80291i	0.127617i
$889$	−0.889035	−0.0298173
$890$	0	0
$891$	−5.14369	−0.172320
$892$	− 47.2043i	− 1.58052i
$893$	68.9586i	2.30761i
$894$	−4.63910	−0.155155
$895$	0	0
$896$	−29.6219	−0.989600
$897$	− 0.549778i	− 0.0183565i
$898$	− 6.89009i	− 0.229925i
$899$	0.160134	0.00534076
$900$	0	0
$901$	−6.27546	−0.209066
$902$	14.2662i	0.475014i
$903$	29.4644i	0.980514i
$904$	−36.5853	−1.21681
$905$	0	0
$906$	−0.670139	−0.0222639
$907$	− 9.51928i	− 0.316082i	−0.987433	−	0.158041i	$-0.949482\pi$
$907$	− 9.51928i	− 0.316082i	0.987433	−	0.158041i	$-0.0505179\pi$
$908$	− 6.81801i	− 0.226264i
$909$	−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	$0.740966\pi$
$912$	− 19.0629i	− 0.631236i
$913$	48.1615i	1.59391i
$914$	7.34438	0.242931
$915$	0	0
$916$	−20.4936	−0.677128
$917$	− 11.7504i	− 0.388034i
$918$	− 0.369069i	− 0.0121811i
$919$	−15.8676	−0.523424	−0.261712	−	0.965146i	$-0.584287\pi$
$919$	−15.8676	−0.523424	−0.261712	+	0.965146i	$0.584287\pi$
$920$	0	0
$921$	−9.51655	−0.313581
$922$	5.21169i	0.171638i
$923$	17.9319i	0.590235i
$924$	−22.6610	−0.745491
$925$	0	0
$926$	11.4145	0.375104
$927$	10.3860i	0.341122i
$928$	− 0.187184i	− 0.00614462i
$929$	29.4108	0.964936	0.482468	−	0.875914i	$-0.339740\pi$
$929$	29.4108	0.964936	0.482468	+	0.875914i	$0.339740\pi$
$930$	0	0
$931$	3.06506	0.100453
$932$	− 36.1567i	− 1.18435i
$933$	− 24.9814i	− 0.817853i
$934$	7.45878	0.244059
$935$	0	0
$936$	2.93764	0.0960198
$937$	− 58.1131i	− 1.89847i	−0.314565	−	0.949236i	$-0.601859\pi$
$937$	− 58.1131i	− 1.89847i	0.314565	−	0.949236i	$-0.398141\pi$
$938$	− 0.115197i	− 0.00376131i
$939$	−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	$0.311591\pi$
$942$	2.07415i	0.0675795i
$943$	1.92684i	0.0627464i
$944$	0.415029	0.0135081
$945$	0	0
$946$	31.6017	1.02746
$947$	− 38.0104i	− 1.23517i	−0.786503	−	0.617586i	$-0.788111\pi$
$947$	− 38.0104i	− 1.23517i	0.786503	−	0.617586i	$-0.211889\pi$
$948$	8.48674i	0.275636i
$949$	−17.7755	−0.577017
$950$	0	0
$951$	24.7310	0.801958
$952$	− 3.52533i	− 0.114256i
$953$	16.9149i	0.547929i	0.961740	+	0.273964i	$0.0883350\pi$
$953$	16.9149i	0.547929i	−0.961740	+	0.273964i	$0.911665\pi$
$954$	4.89501	0.158482
$955$	0	0
$956$	−44.8660	−1.45107
$957$	− 0.183487i	− 0.00593130i
$958$	6.63587i	0.214395i
$959$	12.3964	0.400300
$960$	0	0
$961$	−10.8487	−0.349958
$962$	− 1.51100i	− 0.0487165i
$963$	12.2737i	0.395516i
$964$	−13.9957	−0.450772
$965$	0	0
$966$	0.514630	0.0165580
$967$	22.7845i	0.732700i	0.930477	+	0.366350i	$0.119393\pi$
$967$	22.7845i	0.732700i	−0.930477	+	0.366350i	$0.880607\pi$
$968$	30.7872i	0.989537i
$969$	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	$0.250598\pi$
$972$	− 1.71212i	− 0.0549162i
$973$	9.62561i	0.308583i
$974$	−18.9552	−0.607362
$975$	0	0
$976$	−15.1029	−0.483431
$977$	18.6331i	0.596125i	0.954546	+	0.298062i	$0.0963404\pi$
$977$	18.6331i	0.596125i	−0.954546	+	0.298062i	$0.903660\pi$
$978$	3.74194i	0.119654i
$979$	−0.0633650	−0.00202515
$980$	0	0
$981$	−4.22902	−0.135022
$982$	− 11.6397i	− 0.371438i
$983$	54.5076i	1.73852i	0.494354	+	0.869261i	$0.335405\pi$
$983$	54.5076i	1.73852i	−0.494354	+	0.869261i	$0.664595\pi$
$984$	−10.2957	−0.328216
$985$	0	0
$986$	0.0131655	0.000419276 0
$987$	21.9264i	0.697925i
$988$	20.4359i	0.650153i
$989$	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	$0.304981\pi$
$992$	− 23.5554i	− 0.747884i
$993$	− 1.07827i	− 0.0342178i
$994$	−16.7855	−0.532403
$995$	0	0
$996$	−16.0309	−0.507960
$997$	7.19225i	0.227781i	0.993493	+	0.113890i	$0.0363313\pi$
$997$	7.19225i	0.227781i	−0.993493	+	0.113890i	$0.963669\pi$
$998$	− 20.8110i	− 0.658761i
$999$	−1.90935	−0.0604092

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.2.i.a.4.2	✓	16	25.14	even	10
75.2.i.a.19.2	yes	16	25.16	even	5
225.2.m.b.19.3		16	75.41	odd	10
225.2.m.b.154.3		16	75.14	odd	10
375.2.g.d.151.2		16	25.12	odd	20
375.2.g.d.226.2		16	25.2	odd	20
375.2.g.e.151.3		16	25.13	odd	20
375.2.g.e.226.3		16	25.23	odd	20
375.2.i.c.274.3		16	25.11	even	5
375.2.i.c.349.3		16	25.9	even	10
1875.2.a.m.1.6		8	5.3	odd	4
1875.2.a.p.1.3		8	5.2	odd	4
1875.2.b.h.1249.7		16	5.4	even	2	inner
1875.2.b.h.1249.10		16	1.1	even	1	trivial
5625.2.a.t.1.6		8	15.2	even	4
5625.2.a.bd.1.3		8	15.8	even	4

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.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.536547i q^{2} -1.00000i q^{3} +1.71212 q^{4} +0.536547 q^{6} +2.57318i q^{7} +1.99173i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+0.536547i q^{2} -1.00000i q^{3} +1.71212 q^{4} +0.536547 q^{6} +2.57318i q^{7} +1.99173i q^{8} -1.00000 q^{9} -5.14369 q^{11} -1.71212i q^{12} +1.47492i q^{13} -1.38063 q^{14} +2.35558 q^{16} +0.687859i q^{17} -0.536547i q^{18} +8.09265 q^{19} +2.57318 q^{21} -2.75983i q^{22} -0.372750i q^{23} +1.99173 q^{24} -0.791365 q^{26} +1.00000i q^{27} +4.40559i q^{28} -0.0356723 q^{29} -4.48902 q^{31} +5.24733i q^{32} +5.14369i q^{33} -0.369069 q^{34} -1.71212 q^{36} +1.90935i q^{37} +4.34209i q^{38} +1.47492 q^{39} -5.16925 q^{41} +1.38063i q^{42} +11.4506i q^{43} -8.80660 q^{44} +0.199998 q^{46} +8.52114i q^{47} -2.35558i q^{48} +0.378747 q^{49} +0.687859 q^{51} +2.52524i q^{52} +9.12317i q^{53} -0.536547 q^{54} -5.12507 q^{56} -8.09265i q^{57} -0.0191399i q^{58} +0.176190 q^{59} -6.41152 q^{61} -2.40857i q^{62} -2.57318i q^{63} +1.89572 q^{64} -2.75983 q^{66} +0.0834377i q^{67} +1.17770i q^{68} -0.372750 q^{69} +12.1578 q^{71} -1.99173i q^{72} +12.0518i q^{73} -1.02446 q^{74} +13.8556 q^{76} -13.2356i q^{77} +0.791365i q^{78} -4.95687 q^{79} +1.00000 q^{81} -2.77354i q^{82} -9.36322i q^{83} +4.40559 q^{84} -6.14378 q^{86} +0.0356723i q^{87} -10.2448i q^{88} +0.0123190 q^{89} -3.79524 q^{91} -0.638192i q^{92} +4.48902i q^{93} -4.57199 q^{94} +5.24733 q^{96} -7.62041i q^{97} +0.203215i q^{98} +5.14369 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{4} - 8 q^{6} - 16 q^{9}+O(q^{10}) \) 16 * q - 8 * q^4 - 8 * q^6 - 16 * q^9	\( 16 q - 8 q^{4} - 8 q^{6} - 16 q^{9} + 4 q^{11} - 12 q^{14} + 28 q^{19} - 16 q^{21} + 24 q^{24} + 12 q^{26} - 4 q^{29} - 44 q^{31} + 24 q^{34} + 8 q^{36} + 32 q^{39} + 16 q^{41} - 44 q^{44} - 4 q^{46} - 32 q^{51} + 8 q^{54} + 60 q^{56} - 28 q^{59} - 40 q^{61} - 12 q^{64} - 24 q^{66} + 28 q^{69} + 32 q^{71} - 52 q^{74} - 32 q^{76} + 60 q^{79} + 16 q^{81} + 32 q^{84} + 64 q^{86} - 32 q^{89} - 24 q^{91} - 28 q^{94} + 4 q^{96} - 4 q^{99}+O(q^{100}) \) 16 * q - 8 * q^4 - 8 * q^6 - 16 * q^9 + 4 * q^11 - 12 * q^14 + 28 * q^19 - 16 * q^21 + 24 * q^24 + 12 * q^26 - 4 * q^29 - 44 * q^31 + 24 * q^34 + 8 * q^36 + 32 * q^39 + 16 * q^41 - 44 * q^44 - 4 * q^46 - 32 * q^51 + 8 * q^54 + 60 * q^56 - 28 * q^59 - 40 * q^61 - 12 * q^64 - 24 * q^66 + 28 * q^69 + 32 * q^71 - 52 * q^74 - 32 * q^76 + 60 * q^79 + 16 * q^81 + 32 * q^84 + 64 * q^86 - 32 * q^89 - 24 * q^91 - 28 * q^94 + 4 * q^96 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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