LMFDB - Embedded newform 2031.2.a.g.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2031,2,Mod(1,2031)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2031, 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("2031.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2031 = 3 \cdot 677 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2031.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:	$16.2176166505$
Analytic rank:	$1$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 7 x^{17} + 99 x^{15} - 148 x^{14} - 514 x^{13} + 1204 x^{12} + 1143 x^{11} - 4191 x^{10} + \cdots + 48 $ x^18 - 7x^17 + 99x^15 - 148x^14 - 514x^13 + 1204x^12 + 1143x^11 - 4191x^10 - 668x^9 + 7531x^8 - 1457x^7 - 7108x^6 + 2569x^5 + 3238x^4 - 1325x^3 - 571x^2 + 160x + 48
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Root			$-1.46392$ of defining polynomial
Character	$\chi$	$=$	2031.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.46392 q^{2} +1.00000 q^{3} +0.143048 q^{4} +0.387918 q^{5} +1.46392 q^{6} -2.28879 q^{7} -2.71842 q^{8} +1.00000 q^{9} +0.567880 q^{10} -1.78909 q^{11} +0.143048 q^{12} +0.983519 q^{13} -3.35060 q^{14} +0.387918 q^{15} -4.26563 q^{16} -1.60300 q^{17} +1.46392 q^{18} -4.74512 q^{19} +0.0554910 q^{20} -2.28879 q^{21} -2.61907 q^{22} -3.45899 q^{23} -2.71842 q^{24} -4.84952 q^{25} +1.43979 q^{26} +1.00000 q^{27} -0.327407 q^{28} -9.04599 q^{29} +0.567880 q^{30} +2.62121 q^{31} -0.807685 q^{32} -1.78909 q^{33} -2.34665 q^{34} -0.887865 q^{35} +0.143048 q^{36} -0.148493 q^{37} -6.94645 q^{38} +0.983519 q^{39} -1.05453 q^{40} +5.04274 q^{41} -3.35060 q^{42} +6.65939 q^{43} -0.255925 q^{44} +0.387918 q^{45} -5.06368 q^{46} +7.92475 q^{47} -4.26563 q^{48} -1.76143 q^{49} -7.09929 q^{50} -1.60300 q^{51} +0.140690 q^{52} -1.72374 q^{53} +1.46392 q^{54} -0.694020 q^{55} +6.22190 q^{56} -4.74512 q^{57} -13.2426 q^{58} -0.0905729 q^{59} +0.0554910 q^{60} +7.66859 q^{61} +3.83723 q^{62} -2.28879 q^{63} +7.34888 q^{64} +0.381525 q^{65} -2.61907 q^{66} -9.42400 q^{67} -0.229306 q^{68} -3.45899 q^{69} -1.29976 q^{70} -12.4803 q^{71} -2.71842 q^{72} -4.58440 q^{73} -0.217382 q^{74} -4.84952 q^{75} -0.678780 q^{76} +4.09485 q^{77} +1.43979 q^{78} -3.84230 q^{79} -1.65472 q^{80} +1.00000 q^{81} +7.38214 q^{82} -9.85407 q^{83} -0.327407 q^{84} -0.621833 q^{85} +9.74879 q^{86} -9.04599 q^{87} +4.86349 q^{88} +5.42797 q^{89} +0.567880 q^{90} -2.25107 q^{91} -0.494802 q^{92} +2.62121 q^{93} +11.6012 q^{94} -1.84072 q^{95} -0.807685 q^{96} +0.0506965 q^{97} -2.57858 q^{98} -1.78909 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 7 q^{2} + 18 q^{3} + 13 q^{4} - 13 q^{5} - 7 q^{6} - 3 q^{7} - 18 q^{8} + 18 q^{9} - 6 q^{10} - 26 q^{11} + 13 q^{12} - 11 q^{13} - 11 q^{14} - 13 q^{15} - q^{16} - 34 q^{17} - 7 q^{18} - 2 q^{19}+ \cdots - 26 q^{99}+O(q^{100}) $ 18 * q - 7 * q^2 + 18 * q^3 + 13 * q^4 - 13 * q^5 - 7 * q^6 - 3 * q^7 - 18 * q^8 + 18 * q^9 - 6 * q^10 - 26 * q^11 + 13 * q^12 - 11 * q^13 - 11 * q^14 - 13 * q^15 - q^16 - 34 * q^17 - 7 * q^18 - 2 * q^19 - 10 * q^20 - 3 * q^21 + 8 * q^22 - 35 * q^23 - 18 * q^24 - 5 * q^25 - 17 * q^26 + 18 * q^27 - 5 * q^28 - 25 * q^29 - 6 * q^30 - 11 * q^31 - 18 * q^32 - 26 * q^33 - 3 * q^34 - 27 * q^35 + 13 * q^36 - q^37 - 18 * q^38 - 11 * q^39 - 8 * q^40 - 44 * q^41 - 11 * q^42 + 5 * q^43 - 46 * q^44 - 13 * q^45 + 20 * q^46 - 51 * q^47 - q^48 - 33 * q^49 + 11 * q^50 - 34 * q^51 - 2 * q^52 - 28 * q^53 - 7 * q^54 + q^55 - q^56 - 2 * q^57 + 22 * q^58 - 44 * q^59 - 10 * q^60 - 7 * q^61 + q^62 - 3 * q^63 + 8 * q^64 - 11 * q^65 + 8 * q^66 + 2 * q^67 - 11 * q^68 - 35 * q^69 + 16 * q^70 - 23 * q^71 - 18 * q^72 + q^73 - 9 * q^74 - 5 * q^75 + 3 * q^76 - 23 * q^77 - 17 * q^78 + 9 * q^79 - 15 * q^80 + 18 * q^81 + 19 * q^82 - 45 * q^83 - 5 * q^84 + 9 * q^85 - 4 * q^86 - 25 * q^87 + 51 * q^88 - 90 * q^89 - 6 * q^90 - 24 * q^91 - 57 * q^92 - 11 * q^93 + 24 * q^94 - 9 * q^95 - 18 * q^96 - 17 * q^97 + 12 * q^98 - 26 * q^99

Coefficient data

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

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

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

$2$	1.46392	1.03514	0.517572	−	0.855640i	$-0.326836\pi$
$2$	1.46392	1.03514	0.517572	+	0.855640i	$0.326836\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0.143048	0.0715240
$5$	0.387918	0.173482	0.0867412	−	0.996231i	$-0.472355\pi$
$5$	0.387918	0.173482	0.0867412	+	0.996231i	$0.472355\pi$
$6$	1.46392	0.597641
$7$	−2.28879	−0.865082	−0.432541	−	0.901614i	$-0.642383\pi$
$7$	−2.28879	−0.865082	−0.432541	+	0.901614i	$0.642383\pi$
$8$	−2.71842	−0.961107
$9$	1.00000	0.333333
$10$	0.567880	0.179579
$11$	−1.78909	−0.539430	−0.269715	−	0.962940i	$-0.586930\pi$
$11$	−1.78909	−0.539430	−0.269715	+	0.962940i	$0.586930\pi$
$12$	0.143048	0.0412944
$13$	0.983519	0.272779	0.136389	−	0.990655i	$-0.456450\pi$
$13$	0.983519	0.272779	0.136389	+	0.990655i	$0.456450\pi$
$14$	−3.35060	−0.895485
$15$	0.387918	0.100160
$16$	−4.26563	−1.06641
$17$	−1.60300	−0.388784	−0.194392	−	0.980924i	$-0.562273\pi$
$17$	−1.60300	−0.388784	−0.194392	+	0.980924i	$0.562273\pi$
$18$	1.46392	0.345048
$19$	−4.74512	−1.08860	−0.544302	−	0.838889i	$-0.683206\pi$
$19$	−4.74512	−1.08860	−0.544302	+	0.838889i	$0.683206\pi$
$20$	0.0554910	0.0124082
$21$	−2.28879	−0.499455
$22$	−2.61907	−0.558388
$23$	−3.45899	−0.721250	−0.360625	−	0.932711i	$-0.617437\pi$
$23$	−3.45899	−0.721250	−0.360625	+	0.932711i	$0.617437\pi$
$24$	−2.71842	−0.554895
$25$	−4.84952	−0.969904
$26$	1.43979	0.282366
$27$	1.00000	0.192450
$28$	−0.327407	−0.0618742
$29$	−9.04599	−1.67980	−0.839899	−	0.542743i	$-0.817386\pi$
$29$	−9.04599	−1.67980	−0.839899	+	0.542743i	$0.817386\pi$
$30$	0.567880	0.103680
$31$	2.62121	0.470783	0.235391	−	0.971901i	$-0.424363\pi$
$31$	2.62121	0.470783	0.235391	+	0.971901i	$0.424363\pi$
$32$	−0.807685	−0.142780
$33$	−1.78909	−0.311440
$34$	−2.34665	−0.402448
$35$	−0.887865	−0.150077
$36$	0.143048	0.0238413
$37$	−0.148493	−0.0244122	−0.0122061	−	0.999926i	$-0.503885\pi$
$37$	−0.148493	−0.0244122	−0.0122061	+	0.999926i	$0.503885\pi$
$38$	−6.94645	−1.12686
$39$	0.983519	0.157489
$40$	−1.05453	−0.166735
$41$	5.04274	0.787543	0.393772	−	0.919208i	$-0.371170\pi$
$41$	5.04274	0.787543	0.393772	+	0.919208i	$0.371170\pi$
$42$	−3.35060	−0.517009
$43$	6.65939	1.01555	0.507774	−	0.861490i	$-0.330468\pi$
$43$	6.65939	1.01555	0.507774	+	0.861490i	$0.330468\pi$
$44$	−0.255925	−0.0385822
$45$	0.387918	0.0578275
$46$	−5.06368	−0.746598
$47$	7.92475	1.15594	0.577972	−	0.816057i	$-0.303844\pi$
$47$	7.92475	1.15594	0.577972	+	0.816057i	$0.303844\pi$
$48$	−4.26563	−0.615691
$49$	−1.76143	−0.251633
$50$	−7.09929	−1.00399
$51$	−1.60300	−0.224465
$52$	0.140690	0.0195102
$53$	−1.72374	−0.236774	−0.118387	−	0.992968i	$-0.537772\pi$
$53$	−1.72374	−0.236774	−0.118387	+	0.992968i	$0.537772\pi$
$54$	1.46392	0.199214
$55$	−0.694020	−0.0935817
$56$	6.22190	0.831436
$57$	−4.74512	−0.628506
$58$	−13.2426	−1.73883
$59$	−0.0905729	−0.0117916	−0.00589579	−	0.999983i	$-0.501877\pi$
$59$	−0.0905729	−0.0117916	−0.00589579	+	0.999983i	$0.501877\pi$
$60$	0.0554910	0.00716385
$61$	7.66859	0.981863	0.490931	−	0.871198i	$-0.336657\pi$
$61$	7.66859	0.981863	0.490931	+	0.871198i	$0.336657\pi$
$62$	3.83723	0.487328
$63$	−2.28879	−0.288361
$64$	7.34888	0.918611
$65$	0.381525	0.0473224
$66$	−2.61907	−0.322386
$67$	−9.42400	−1.15132	−0.575662	−	0.817688i	$-0.695256\pi$
$67$	−9.42400	−1.15132	−0.575662	+	0.817688i	$0.695256\pi$
$68$	−0.229306	−0.0278074
$69$	−3.45899	−0.416414
$70$	−1.29976	−0.155351
$71$	−12.4803	−1.48114	−0.740572	−	0.671977i	$-0.765445\pi$
$71$	−12.4803	−1.48114	−0.740572	+	0.671977i	$0.765445\pi$
$72$	−2.71842	−0.320369
$73$	−4.58440	−0.536563	−0.268282	−	0.963341i	$-0.586456\pi$
$73$	−4.58440	−0.536563	−0.268282	+	0.963341i	$0.586456\pi$
$74$	−0.217382	−0.0252701
$75$	−4.84952	−0.559974
$76$	−0.678780	−0.0778614
$77$	4.09485	0.466651
$78$	1.43979	0.163024
$79$	−3.84230	−0.432293	−0.216146	−	0.976361i	$-0.569349\pi$
$79$	−3.84230	−0.432293	−0.216146	+	0.976361i	$0.569349\pi$
$80$	−1.65472	−0.185003
$81$	1.00000	0.111111
$82$	7.38214	0.815221
$83$	−9.85407	−1.08162	−0.540812	−	0.841143i	$-0.681883\pi$
$83$	−9.85407	−1.08162	−0.540812	+	0.841143i	$0.681883\pi$
$84$	−0.327407	−0.0357231
$85$	−0.621833	−0.0674472
$86$	9.74879	1.05124
$87$	−9.04599	−0.969831
$88$	4.86349	0.518450
$89$	5.42797	0.575364	0.287682	−	0.957726i	$-0.407115\pi$
$89$	5.42797	0.575364	0.287682	+	0.957726i	$0.407115\pi$
$90$	0.567880	0.0598598
$91$	−2.25107	−0.235976
$92$	−0.494802	−0.0515867
$93$	2.62121	0.271807
$94$	11.6012	1.19657
$95$	−1.84072	−0.188854
$96$	−0.807685	−0.0824340
$97$	0.0506965	0.00514745	0.00257372	−	0.999997i	$-0.499181\pi$
$97$	0.0506965	0.00514745	0.00257372	+	0.999997i	$0.499181\pi$
$98$	−2.57858	−0.260476
$99$	−1.78909	−0.179810
$100$	−0.693714	−0.0693714
$101$	−8.03677	−0.799689	−0.399844	−	0.916583i	$-0.630936\pi$
$101$	−8.03677	−0.799689	−0.399844	+	0.916583i	$0.630936\pi$
$102$	−2.34665	−0.232353
$103$	9.35634	0.921907	0.460954	−	0.887424i	$-0.347507\pi$
$103$	9.35634	0.921907	0.460954	+	0.887424i	$0.347507\pi$
$104$	−2.67362	−0.262170
$105$	−0.887865	−0.0866467
$106$	−2.52341	−0.245096
$107$	9.29217	0.898308	0.449154	−	0.893454i	$-0.351725\pi$
$107$	9.29217	0.898308	0.449154	+	0.893454i	$0.351725\pi$
$108$	0.143048	0.0137648
$109$	−17.3534	−1.66216	−0.831079	−	0.556154i	$-0.812277\pi$
$109$	−17.3534	−1.66216	−0.831079	+	0.556154i	$0.812277\pi$
$110$	−1.01599	−0.0968705
$111$	−0.148493	−0.0140944
$112$	9.76315	0.922531
$113$	−10.7244	−1.00886	−0.504432	−	0.863452i	$-0.668298\pi$
$113$	−10.7244	−1.00886	−0.504432	+	0.863452i	$0.668298\pi$
$114$	−6.94645	−0.650595
$115$	−1.34181	−0.125124
$116$	−1.29401	−0.120146
$117$	0.983519	0.0909263
$118$	−0.132591	−0.0122060
$119$	3.66893	0.336330
$120$	−1.05453	−0.0962646
$121$	−7.79917	−0.709015
$122$	11.2262	1.01637
$123$	5.04274	0.454688
$124$	0.374959	0.0336723
$125$	−3.82081	−0.341744
$126$	−3.35060	−0.298495
$127$	0.558833	0.0495884	0.0247942	−	0.999693i	$-0.492107\pi$
$127$	0.558833	0.0495884	0.0247942	+	0.999693i	$0.492107\pi$
$128$	12.3735	1.09367
$129$	6.65939	0.586327
$130$	0.558520	0.0489855
$131$	21.6051	1.88764	0.943822	−	0.330453i	$-0.107202\pi$
$131$	21.6051	1.88764	0.943822	+	0.330453i	$0.107202\pi$
$132$	−0.255925	−0.0222755
$133$	10.8606	0.941733
$134$	−13.7959	−1.19179
$135$	0.387918	0.0333867
$136$	4.35762	0.373663
$137$	9.15886	0.782494	0.391247	−	0.920286i	$-0.372044\pi$
$137$	9.15886	0.782494	0.391247	+	0.920286i	$0.372044\pi$
$138$	−5.06368	−0.431049
$139$	9.19295	0.779736	0.389868	−	0.920871i	$-0.372521\pi$
$139$	9.19295	0.779736	0.389868	+	0.920871i	$0.372521\pi$
$140$	−0.127007	−0.0107341
$141$	7.92475	0.667384
$142$	−18.2702	−1.53320
$143$	−1.75960	−0.147145
$144$	−4.26563	−0.355469
$145$	−3.50910	−0.291415
$146$	−6.71117	−0.555421
$147$	−1.76143	−0.145280
$148$	−0.0212417	−0.00174606
$149$	15.8387	1.29756	0.648778	−	0.760978i	$-0.275280\pi$
$149$	15.8387	1.29756	0.648778	+	0.760978i	$0.275280\pi$
$150$	−7.09929	−0.579654
$151$	9.02723	0.734625	0.367313	−	0.930098i	$-0.380278\pi$
$151$	9.02723	0.734625	0.367313	+	0.930098i	$0.380278\pi$
$152$	12.8992	1.04627
$153$	−1.60300	−0.129595
$154$	5.99451	0.483052
$155$	1.01681	0.0816725
$156$	0.140690	0.0112642
$157$	−4.12933	−0.329557	−0.164778	−	0.986331i	$-0.552691\pi$
$157$	−4.12933	−0.329557	−0.164778	+	0.986331i	$0.552691\pi$
$158$	−5.62480	−0.447485
$159$	−1.72374	−0.136702
$160$	−0.313316	−0.0247698
$161$	7.91692	0.623941
$162$	1.46392	0.115016
$163$	−8.78697	−0.688249	−0.344124	−	0.938924i	$-0.611824\pi$
$163$	−8.78697	−0.688249	−0.344124	+	0.938924i	$0.611824\pi$
$164$	0.721354	0.0563283
$165$	−0.694020	−0.0540294
$166$	−14.4255	−1.11964
$167$	5.89653	0.456287	0.228144	−	0.973627i	$-0.426734\pi$
$167$	5.89653	0.456287	0.228144	+	0.973627i	$0.426734\pi$
$168$	6.22190	0.480030
$169$	−12.0327	−0.925592
$170$	−0.910310	−0.0698176
$171$	−4.74512	−0.362868
$172$	0.952613	0.0726361
$173$	−23.0978	−1.75609	−0.878045	−	0.478577i	$-0.841153\pi$
$173$	−23.0978	−1.75609	−0.878045	+	0.478577i	$0.841153\pi$
$174$	−13.2426	−1.00392
$175$	11.0995	0.839047
$176$	7.63159	0.575253
$177$	−0.0905729	−0.00680788
$178$	7.94609	0.595585
$179$	21.6632	1.61918	0.809591	−	0.586995i	$-0.199689\pi$
$179$	21.6632	1.61918	0.809591	+	0.586995i	$0.199689\pi$
$180$	0.0554910	0.00413605
$181$	23.3481	1.73545	0.867726	−	0.497044i	$-0.165581\pi$
$181$	23.3481	1.73545	0.867726	+	0.497044i	$0.165581\pi$
$182$	−3.29538	−0.244270
$183$	7.66859	0.566879
$184$	9.40300	0.693198
$185$	−0.0576033	−0.00423508
$186$	3.83723	0.281359
$187$	2.86790	0.209722
$188$	1.13362	0.0826777
$189$	−2.28879	−0.166485
$190$	−2.69466	−0.195491
$191$	−24.6992	−1.78717	−0.893587	−	0.448890i	$-0.851819\pi$
$191$	−24.6992	−1.78717	−0.893587	+	0.448890i	$0.851819\pi$
$192$	7.34888	0.530360
$193$	−11.8669	−0.854197	−0.427098	−	0.904205i	$-0.640464\pi$
$193$	−11.8669	−0.854197	−0.427098	+	0.904205i	$0.640464\pi$
$194$	0.0742154	0.00532835
$195$	0.381525	0.0273216
$196$	−0.251969	−0.0179978
$197$	−2.04200	−0.145486	−0.0727432	−	0.997351i	$-0.523175\pi$
$197$	−2.04200	−0.145486	−0.0727432	+	0.997351i	$0.523175\pi$
$198$	−2.61907	−0.186129
$199$	21.1109	1.49651	0.748256	−	0.663410i	$-0.230891\pi$
$199$	21.1109	1.49651	0.748256	+	0.663410i	$0.230891\pi$
$200$	13.1830	0.932181
$201$	−9.42400	−0.664718
$202$	−11.7652	−0.827793
$203$	20.7044	1.45316
$204$	−0.229306	−0.0160546
$205$	1.95617	0.136625
$206$	13.6969	0.954307
$207$	−3.45899	−0.240417
$208$	−4.19533	−0.290894
$209$	8.48943	0.587226
$210$	−1.29976	−0.0896919
$211$	18.5988	1.28040	0.640198	−	0.768210i	$-0.278852\pi$
$211$	18.5988	1.28040	0.640198	+	0.768210i	$0.278852\pi$
$212$	−0.246578	−0.0169351
$213$	−12.4803	−0.855139
$214$	13.6030	0.929879
$215$	2.58330	0.176180
$216$	−2.71842	−0.184965
$217$	−5.99940	−0.407266
$218$	−25.4040	−1.72057
$219$	−4.58440	−0.309785
$220$	−0.0992782	−0.00669334
$221$	−1.57658	−0.106052
$222$	−0.217382	−0.0145897
$223$	−0.939854	−0.0629372	−0.0314686	−	0.999505i	$-0.510018\pi$
$223$	−0.939854	−0.0629372	−0.0314686	+	0.999505i	$0.510018\pi$
$224$	1.84862	0.123516
$225$	−4.84952	−0.323301
$226$	−15.6996	−1.04432
$227$	−8.16364	−0.541840	−0.270920	−	0.962602i	$-0.587328\pi$
$227$	−8.16364	−0.541840	−0.270920	+	0.962602i	$0.587328\pi$
$228$	−0.678780	−0.0449533
$229$	−11.5793	−0.765179	−0.382589	−	0.923918i	$-0.624968\pi$
$229$	−11.5793	−0.765179	−0.382589	+	0.923918i	$0.624968\pi$
$230$	−1.96429	−0.129522
$231$	4.09485	0.269421
$232$	24.5908	1.61446
$233$	−7.90222	−0.517692	−0.258846	−	0.965919i	$-0.583342\pi$
$233$	−7.90222	−0.517692	−0.258846	+	0.965919i	$0.583342\pi$
$234$	1.43979	0.0941219
$235$	3.07416	0.200536
$236$	−0.0129563	−0.000843382 0
$237$	−3.84230	−0.249584
$238$	5.37100	0.348150
$239$	16.0618	1.03895	0.519476	−	0.854485i	$-0.326127\pi$
$239$	16.0618	1.03895	0.519476	+	0.854485i	$0.326127\pi$
$240$	−1.65472	−0.106812
$241$	9.31539	0.600057	0.300029	−	0.953930i	$-0.403004\pi$
$241$	9.31539	0.600057	0.300029	+	0.953930i	$0.403004\pi$
$242$	−11.4173	−0.733933
$243$	1.00000	0.0641500
$244$	1.09698	0.0702268
$245$	−0.683291	−0.0436539
$246$	7.38214	0.470668
$247$	−4.66691	−0.296948
$248$	−7.12555	−0.452473
$249$	−9.85407	−0.624476
$250$	−5.59334	−0.353754
$251$	−20.5549	−1.29742	−0.648708	−	0.761038i	$-0.724690\pi$
$251$	−20.5549	−1.29742	−0.648708	+	0.761038i	$0.724690\pi$
$252$	−0.327407	−0.0206247
$253$	6.18844	0.389064
$254$	0.818084	0.0513312
$255$	−0.621833	−0.0389407
$256$	3.41601	0.213501
$257$	−28.7839	−1.79549	−0.897747	−	0.440512i	$-0.854797\pi$
$257$	−28.7839	−1.79549	−0.897747	+	0.440512i	$0.854797\pi$
$258$	9.74879	0.606933
$259$	0.339871	0.0211185
$260$	0.0545764	0.00338468
$261$	−9.04599	−0.559932
$262$	31.6280	1.95398
$263$	−14.3614	−0.885565	−0.442782	−	0.896629i	$-0.646009\pi$
$263$	−14.3614	−0.885565	−0.442782	+	0.896629i	$0.646009\pi$
$264$	4.86349	0.299327
$265$	−0.668672	−0.0410762
$266$	15.8990	0.974829
$267$	5.42797	0.332187
$268$	−1.34808	−0.0823474
$269$	0.286312	0.0174568	0.00872839	−	0.999962i	$-0.497222\pi$
$269$	0.286312	0.0174568	0.00872839	+	0.999962i	$0.497222\pi$
$270$	0.567880	0.0345601
$271$	23.1392	1.40561	0.702804	−	0.711384i	$-0.251931\pi$
$271$	23.1392	1.40561	0.702804	+	0.711384i	$0.251931\pi$
$272$	6.83780	0.414603
$273$	−2.25107	−0.136241
$274$	13.4078	0.809994
$275$	8.67621	0.523195
$276$	−0.494802	−0.0297836
$277$	−12.8797	−0.773865	−0.386932	−	0.922108i	$-0.626465\pi$
$277$	−12.8797	−0.773865	−0.386932	+	0.922108i	$0.626465\pi$
$278$	13.4577	0.807139
$279$	2.62121	0.156928
$280$	2.41359	0.144240
$281$	−16.5883	−0.989578	−0.494789	−	0.869013i	$-0.664755\pi$
$281$	−16.5883	−0.989578	−0.494789	+	0.869013i	$0.664755\pi$
$282$	11.6012	0.690839
$283$	−0.328687	−0.0195384	−0.00976922	−	0.999952i	$-0.503110\pi$
$283$	−0.328687	−0.0195384	−0.00976922	+	0.999952i	$0.503110\pi$
$284$	−1.78529	−0.105937
$285$	−1.84072	−0.109035
$286$	−2.57591	−0.152317
$287$	−11.5418	−0.681290
$288$	−0.807685	−0.0475933
$289$	−14.4304	−0.848847
$290$	−5.13703	−0.301657
$291$	0.0506965	0.00297188
$292$	−0.655789	−0.0383772
$293$	10.3647	0.605510	0.302755	−	0.953068i	$-0.402094\pi$
$293$	10.3647	0.605510	0.302755	+	0.953068i	$0.402094\pi$
$294$	−2.57858	−0.150386
$295$	−0.0351349	−0.00204563
$296$	0.403667	0.0234627
$297$	−1.78909	−0.103813
$298$	23.1865	1.34316
$299$	−3.40199	−0.196742
$300$	−0.693714	−0.0400516
$301$	−15.2420	−0.878533
$302$	13.2151	0.760443
$303$	−8.03677	−0.461700
$304$	20.2409	1.16090
$305$	2.97479	0.170336
$306$	−2.34665	−0.134149
$307$	−5.02595	−0.286846	−0.143423	−	0.989661i	$-0.545811\pi$
$307$	−5.02595	−0.286846	−0.143423	+	0.989661i	$0.545811\pi$
$308$	0.585760	0.0333768
$309$	9.35634	0.532264
$310$	1.48853	0.0845429
$311$	3.63333	0.206027	0.103014	−	0.994680i	$-0.467151\pi$
$311$	3.63333	0.206027	0.103014	+	0.994680i	$0.467151\pi$
$312$	−2.67362	−0.151364
$313$	23.0916	1.30521	0.652606	−	0.757697i	$-0.273676\pi$
$313$	23.0916	1.30521	0.652606	+	0.757697i	$0.273676\pi$
$314$	−6.04499	−0.341139
$315$	−0.887865	−0.0500255
$316$	−0.549634	−0.0309193
$317$	−4.51754	−0.253730	−0.126865	−	0.991920i	$-0.540492\pi$
$317$	−4.51754	−0.253730	−0.126865	+	0.991920i	$0.540492\pi$
$318$	−2.52341	−0.141506
$319$	16.1841	0.906133
$320$	2.85077	0.159363
$321$	9.29217	0.518638
$322$	11.5897	0.645869
$323$	7.60642	0.423232
$324$	0.143048	0.00794711
$325$	−4.76959	−0.264569
$326$	−12.8634	−0.712437
$327$	−17.3534	−0.959648
$328$	−13.7083	−0.756913
$329$	−18.1381	−0.999986
$330$	−1.01599	−0.0559282
$331$	6.57073	0.361160	0.180580	−	0.983560i	$-0.442203\pi$
$331$	6.57073	0.361160	0.180580	+	0.983560i	$0.442203\pi$
$332$	−1.40961	−0.0773622
$333$	−0.148493	−0.00813739
$334$	8.63202	0.472323
$335$	−3.65574	−0.199735
$336$	9.76315	0.532623
$337$	21.6310	1.17831	0.589157	−	0.808019i	$-0.299460\pi$
$337$	21.6310	1.17831	0.589157	+	0.808019i	$0.299460\pi$
$338$	−17.6148	−0.958121
$339$	−10.7244	−0.582468
$340$	−0.0889519	−0.00482410
$341$	−4.68957	−0.253954
$342$	−6.94645	−0.375621
$343$	20.0531	1.08277
$344$	−18.1030	−0.976050
$345$	−1.34181	−0.0722405
$346$	−33.8132	−1.81781
$347$	−1.99669	−0.107188	−0.0535941	−	0.998563i	$-0.517068\pi$
$347$	−1.99669	−0.107188	−0.0535941	+	0.998563i	$0.517068\pi$
$348$	−1.29401	−0.0693662
$349$	−17.5651	−0.940240	−0.470120	−	0.882602i	$-0.655789\pi$
$349$	−17.5651	−0.940240	−0.470120	+	0.882602i	$0.655789\pi$
$350$	16.2488	0.868534
$351$	0.983519	0.0524963
$352$	1.44502	0.0770198
$353$	20.0470	1.06700	0.533498	−	0.845802i	$-0.320877\pi$
$353$	20.0470	1.06700	0.533498	+	0.845802i	$0.320877\pi$
$354$	−0.132591	−0.00704713
$355$	−4.84136	−0.256952
$356$	0.776461	0.0411523
$357$	3.66893	0.194180
$358$	31.7131	1.67609
$359$	−33.8830	−1.78828	−0.894138	−	0.447791i	$-0.852211\pi$
$359$	−33.8830	−1.78828	−0.894138	+	0.447791i	$0.852211\pi$
$360$	−1.05453	−0.0555784
$361$	3.51614	0.185060
$362$	34.1797	1.79644
$363$	−7.79917	−0.409350
$364$	−0.322011	−0.0168780
$365$	−1.77837	−0.0930843
$366$	11.2262	0.586801
$367$	−1.55099	−0.0809610	−0.0404805	−	0.999180i	$-0.512889\pi$
$367$	−1.55099	−0.0809610	−0.0404805	+	0.999180i	$0.512889\pi$
$368$	14.7548	0.769147
$369$	5.04274	0.262514
$370$	−0.0843264	−0.00438392
$371$	3.94529	0.204829
$372$	0.374959	0.0194407
$373$	5.59834	0.289871	0.144936	−	0.989441i	$-0.453702\pi$
$373$	5.59834	0.289871	0.144936	+	0.989441i	$0.453702\pi$
$374$	4.19837	0.217092
$375$	−3.82081	−0.197306
$376$	−21.5428	−1.11099
$377$	−8.89690	−0.458213
$378$	−3.35060	−0.172336
$379$	−3.49594	−0.179574	−0.0897872	−	0.995961i	$-0.528619\pi$
$379$	−3.49594	−0.179574	−0.0897872	+	0.995961i	$0.528619\pi$
$380$	−0.263311	−0.0135076
$381$	0.558833	0.0286299
$382$	−36.1576	−1.84998
$383$	23.6240	1.20713	0.603565	−	0.797314i	$-0.293747\pi$
$383$	23.6240	1.20713	0.603565	+	0.797314i	$0.293747\pi$
$384$	12.3735	0.631433
$385$	1.58847	0.0809558
$386$	−17.3721	−0.884217
$387$	6.65939	0.338516
$388$	0.00725203	0.000368166 0
$389$	15.2090	0.771126	0.385563	−	0.922681i	$-0.374007\pi$
$389$	15.2090	0.771126	0.385563	+	0.922681i	$0.374007\pi$
$390$	0.558520	0.0282818
$391$	5.54476	0.280411
$392$	4.78831	0.241846
$393$	21.6051	1.08983
$394$	−2.98931	−0.150599
$395$	−1.49050	−0.0749952
$396$	−0.255925	−0.0128607
$397$	12.0872	0.606637	0.303318	−	0.952889i	$-0.401905\pi$
$397$	12.0872	0.606637	0.303318	+	0.952889i	$0.401905\pi$
$398$	30.9046	1.54911
$399$	10.8606	0.543710
$400$	20.6863	1.03431
$401$	6.11032	0.305135	0.152567	−	0.988293i	$-0.451246\pi$
$401$	6.11032	0.305135	0.152567	+	0.988293i	$0.451246\pi$
$402$	−13.7959	−0.688079
$403$	2.57801	0.128420
$404$	−1.14964	−0.0571969
$405$	0.387918	0.0192758
$406$	30.3095	1.50423
$407$	0.265668	0.0131687
$408$	4.35762	0.215734
$409$	13.3281	0.659033	0.329516	−	0.944150i	$-0.393114\pi$
$409$	13.3281	0.659033	0.329516	+	0.944150i	$0.393114\pi$
$410$	2.86367	0.141427
$411$	9.15886	0.451773
$412$	1.33841	0.0659385
$413$	0.207303	0.0102007
$414$	−5.06368	−0.248866
$415$	−3.82258	−0.187643
$416$	−0.794373	−0.0389474
$417$	9.19295	0.450181
$418$	12.4278	0.607864
$419$	−18.0114	−0.879917	−0.439958	−	0.898018i	$-0.645007\pi$
$419$	−18.0114	−0.879917	−0.439958	+	0.898018i	$0.645007\pi$
$420$	−0.127007	−0.00619732
$421$	14.0404	0.684290	0.342145	−	0.939647i	$-0.388847\pi$
$421$	14.0404	0.684290	0.342145	+	0.939647i	$0.388847\pi$
$422$	27.2271	1.32539
$423$	7.92475	0.385315
$424$	4.68586	0.227565
$425$	7.77377	0.377083
$426$	−18.2702	−0.885192
$427$	−17.5518	−0.849392
$428$	1.32923	0.0642506
$429$	−1.75960	−0.0849543
$430$	3.78174	0.182371
$431$	−38.7591	−1.86696	−0.933479	−	0.358632i	$-0.883243\pi$
$431$	−38.7591	−1.86696	−0.933479	+	0.358632i	$0.883243\pi$
$432$	−4.26563	−0.205230
$433$	−29.1583	−1.40126	−0.700629	−	0.713526i	$-0.747097\pi$
$433$	−29.1583	−1.40126	−0.700629	+	0.713526i	$0.747097\pi$
$434$	−8.78261	−0.421579
$435$	−3.50910	−0.168249
$436$	−2.48238	−0.118884
$437$	16.4133	0.785156
$438$	−6.71117	−0.320672
$439$	−9.37853	−0.447613	−0.223807	−	0.974634i	$-0.571848\pi$
$439$	−9.37853	−0.447613	−0.223807	+	0.974634i	$0.571848\pi$
$440$	1.88664	0.0899420
$441$	−1.76143	−0.0838776
$442$	−2.30798	−0.109779
$443$	−10.6502	−0.506009	−0.253004	−	0.967465i	$-0.581419\pi$
$443$	−10.6502	−0.506009	−0.253004	+	0.967465i	$0.581419\pi$
$444$	−0.0212417	−0.00100809
$445$	2.10561	0.0998155
$446$	−1.37587	−0.0651491
$447$	15.8387	0.749145
$448$	−16.8201	−0.794674
$449$	−22.5384	−1.06365	−0.531826	−	0.846853i	$-0.678494\pi$
$449$	−22.5384	−1.06365	−0.531826	+	0.846853i	$0.678494\pi$
$450$	−7.09929	−0.334664
$451$	−9.02190	−0.424825
$452$	−1.53410	−0.0721580
$453$	9.02723	0.424136
$454$	−11.9509	−0.560882
$455$	−0.873232	−0.0409377
$456$	12.8992	0.604062
$457$	34.9180	1.63340	0.816698	−	0.577066i	$-0.195802\pi$
$457$	34.9180	1.63340	0.816698	+	0.577066i	$0.195802\pi$
$458$	−16.9511	−0.792071
$459$	−1.60300	−0.0748215
$460$	−0.191943	−0.00894939
$461$	28.2073	1.31374	0.656872	−	0.754002i	$-0.271879\pi$
$461$	28.2073	1.31374	0.656872	+	0.754002i	$0.271879\pi$
$462$	5.99451	0.278890
$463$	−17.6647	−0.820946	−0.410473	−	0.911873i	$-0.634636\pi$
$463$	−17.6647	−0.820946	−0.410473	+	0.911873i	$0.634636\pi$
$464$	38.5869	1.79135
$465$	1.01681	0.0471537
$466$	−11.5682	−0.535886
$467$	−18.6212	−0.861688	−0.430844	−	0.902426i	$-0.641784\pi$
$467$	−18.6212	−0.861688	−0.430844	+	0.902426i	$0.641784\pi$
$468$	0.140690	0.00650342
$469$	21.5696	0.995990
$470$	4.50031	0.207584
$471$	−4.12933	−0.190270
$472$	0.246215	0.0113330
$473$	−11.9142	−0.547817
$474$	−5.62480	−0.258356
$475$	23.0115	1.05584
$476$	0.524833	0.0240557
$477$	−1.72374	−0.0789248
$478$	23.5131	1.07547
$479$	2.54315	0.116200	0.0580998	−	0.998311i	$-0.481496\pi$
$479$	2.54315	0.116200	0.0580998	+	0.998311i	$0.481496\pi$
$480$	−0.313316	−0.0143009
$481$	−0.146046	−0.00665913
$482$	13.6369	0.621146
$483$	7.91692	0.360232
$484$	−1.11566	−0.0507116
$485$	0.0196661	0.000892992 0
$486$	1.46392	0.0664045
$487$	−1.14047	−0.0516797	−0.0258398	−	0.999666i	$-0.508226\pi$
$487$	−1.14047	−0.0516797	−0.0258398	+	0.999666i	$0.508226\pi$
$488$	−20.8465	−0.943675
$489$	−8.78697	−0.397360
$490$	−1.00028	−0.0451881
$491$	−27.6703	−1.24874	−0.624372	−	0.781127i	$-0.714645\pi$
$491$	−27.6703	−1.24874	−0.624372	+	0.781127i	$0.714645\pi$
$492$	0.721354	0.0325211
$493$	14.5007	0.653079
$494$	−6.83196	−0.307385
$495$	−0.694020	−0.0311939
$496$	−11.1811	−0.502047
$497$	28.5649	1.28131
$498$	−14.4255	−0.646423
$499$	−10.2687	−0.459689	−0.229844	−	0.973227i	$-0.573822\pi$
$499$	−10.2687	−0.459689	−0.229844	+	0.973227i	$0.573822\pi$
$500$	−0.546559	−0.0244429
$501$	5.89653	0.263438
$502$	−30.0907	−1.34301
$503$	−1.81344	−0.0808574	−0.0404287	−	0.999182i	$-0.512872\pi$
$503$	−1.81344	−0.0808574	−0.0404287	+	0.999182i	$0.512872\pi$
$504$	6.22190	0.277145
$505$	−3.11761	−0.138732
$506$	9.05936	0.402738
$507$	−12.0327	−0.534391
$508$	0.0799400	0.00354676
$509$	−22.5991	−1.00169	−0.500844	−	0.865538i	$-0.666977\pi$
$509$	−22.5991	−1.00169	−0.500844	+	0.865538i	$0.666977\pi$
$510$	−0.910310	−0.0403092
$511$	10.4927	0.464171
$512$	−19.7463	−0.872671
$513$	−4.74512	−0.209502
$514$	−42.1373	−1.85859
$515$	3.62950	0.159935
$516$	0.952613	0.0419365
$517$	−14.1781	−0.623551
$518$	0.497542	0.0218607
$519$	−23.0978	−1.01388
$520$	−1.03715	−0.0454818
$521$	−7.28990	−0.319377	−0.159688	−	0.987167i	$-0.551049\pi$
$521$	−7.28990	−0.319377	−0.159688	+	0.987167i	$0.551049\pi$
$522$	−13.2426	−0.579611
$523$	24.5365	1.07291	0.536453	−	0.843930i	$-0.319764\pi$
$523$	24.5365	1.07291	0.536453	+	0.843930i	$0.319764\pi$
$524$	3.09056	0.135012
$525$	11.0995	0.484424
$526$	−21.0239	−0.916688
$527$	−4.20179	−0.183033
$528$	7.63159	0.332122
$529$	−11.0354	−0.479798
$530$	−0.978879	−0.0425198
$531$	−0.0905729	−0.00393053
$532$	1.55359	0.0673565
$533$	4.95963	0.214825
$534$	7.94609	0.343861
$535$	3.60460	0.155841
$536$	25.6184	1.10655
$537$	21.6632	0.934835
$538$	0.419137	0.0180703
$539$	3.15135	0.135738
$540$	0.0554910	0.00238795
$541$	−27.8290	−1.19646	−0.598231	−	0.801324i	$-0.704130\pi$
$541$	−27.8290	−1.19646	−0.598231	+	0.801324i	$0.704130\pi$
$542$	33.8738	1.45501
$543$	23.3481	1.00196
$544$	1.29472	0.0555106
$545$	−6.73172	−0.288355
$546$	−3.29538	−0.141029
$547$	33.7277	1.44209	0.721046	−	0.692887i	$-0.243662\pi$
$547$	33.7277	1.44209	0.721046	+	0.692887i	$0.243662\pi$
$548$	1.31016	0.0559671
$549$	7.66859	0.327288
$550$	12.7012	0.541583
$551$	42.9243	1.82864
$552$	9.40300	0.400218
$553$	8.79423	0.373969
$554$	−18.8548	−0.801062
$555$	−0.0576033	−0.00244513
$556$	1.31503	0.0557698
$557$	−31.3138	−1.32681	−0.663404	−	0.748261i	$-0.730889\pi$
$557$	−31.3138	−1.32681	−0.663404	+	0.748261i	$0.730889\pi$
$558$	3.83723	0.162443
$559$	6.54964	0.277020
$560$	3.78731	0.160043
$561$	2.86790	0.121083
$562$	−24.2839	−1.02436
$563$	−32.2048	−1.35727	−0.678635	−	0.734475i	$-0.737428\pi$
$563$	−32.2048	−1.35727	−0.678635	+	0.734475i	$0.737428\pi$
$564$	1.13362	0.0477340
$565$	−4.16018	−0.175020
$566$	−0.481171	−0.0202251
$567$	−2.28879	−0.0961202
$568$	33.9268	1.42354
$569$	−45.0406	−1.88820	−0.944100	−	0.329658i	$-0.893066\pi$
$569$	−45.0406	−1.88820	−0.944100	+	0.329658i	$0.893066\pi$
$570$	−2.69466	−0.112867
$571$	−27.9975	−1.17166	−0.585829	−	0.810435i	$-0.699231\pi$
$571$	−27.9975	−1.17166	−0.585829	+	0.810435i	$0.699231\pi$
$572$	−0.251707	−0.0105244
$573$	−24.6992	−1.03183
$574$	−16.8962	−0.705233
$575$	16.7745	0.699543
$576$	7.34888	0.306204
$577$	−11.4961	−0.478591	−0.239295	−	0.970947i	$-0.576916\pi$
$577$	−11.4961	−0.478591	−0.239295	+	0.970947i	$0.576916\pi$
$578$	−21.1249	−0.878679
$579$	−11.8669	−0.493171
$580$	−0.501971	−0.0208432
$581$	22.5539	0.935694
$582$	0.0742154	0.00307633
$583$	3.08393	0.127723
$584$	12.4623	0.515695
$585$	0.381525	0.0157741
$586$	15.1730	0.626790
$587$	−30.6640	−1.26564	−0.632819	−	0.774300i	$-0.718102\pi$
$587$	−30.6640	−1.26564	−0.632819	+	0.774300i	$0.718102\pi$
$588$	−0.251969	−0.0103910
$589$	−12.4379	−0.512496
$590$	−0.0514345	−0.00211753
$591$	−2.04200	−0.0839966
$592$	0.633418	0.0260333
$593$	−41.7369	−1.71393	−0.856965	−	0.515374i	$-0.827653\pi$
$593$	−41.7369	−1.71393	−0.856965	+	0.515374i	$0.827653\pi$
$594$	−2.61907	−0.107462
$595$	1.42325	0.0583474
$596$	2.26569	0.0928065
$597$	21.1109	0.864012
$598$	−4.98022	−0.203656
$599$	−22.8597	−0.934021	−0.467010	−	0.884252i	$-0.654669\pi$
$599$	−22.8597	−0.934021	−0.467010	+	0.884252i	$0.654669\pi$
$600$	13.1830	0.538195
$601$	−16.8840	−0.688711	−0.344356	−	0.938839i	$-0.611903\pi$
$601$	−16.8840	−0.688711	−0.344356	+	0.938839i	$0.611903\pi$
$602$	−22.3130	−0.909408
$603$	−9.42400	−0.383775
$604$	1.29133	0.0525433
$605$	−3.02544	−0.123002
$606$	−11.7652	−0.477927
$607$	2.63279	0.106862	0.0534309	−	0.998572i	$-0.482984\pi$
$607$	2.63279	0.106862	0.0534309	+	0.998572i	$0.482984\pi$
$608$	3.83256	0.155431
$609$	20.7044	0.838984
$610$	4.35484	0.176322
$611$	7.79414	0.315317
$612$	−0.229306	−0.00926913
$613$	32.8318	1.32606	0.663032	−	0.748591i	$-0.269269\pi$
$613$	32.8318	1.32606	0.663032	+	0.748591i	$0.269269\pi$
$614$	−7.35756	−0.296927
$615$	1.95617	0.0788804
$616$	−11.1315	−0.448502
$617$	9.15201	0.368446	0.184223	−	0.982884i	$-0.441023\pi$
$617$	9.15201	0.368446	0.184223	+	0.982884i	$0.441023\pi$
$618$	13.6969	0.550970
$619$	11.0552	0.444345	0.222172	−	0.975007i	$-0.428685\pi$
$619$	11.0552	0.444345	0.222172	+	0.975007i	$0.428685\pi$
$620$	0.145453	0.00584155
$621$	−3.45899	−0.138805
$622$	5.31889	0.213268
$623$	−12.4235	−0.497737
$624$	−4.19533	−0.167948
$625$	22.7654	0.910617
$626$	33.8041	1.35108
$627$	8.48943	0.339035
$628$	−0.590693	−0.0235712
$629$	0.238035	0.00949106
$630$	−1.29976	−0.0517836
$631$	31.4339	1.25136	0.625681	−	0.780079i	$-0.284821\pi$
$631$	31.4339	1.25136	0.625681	+	0.780079i	$0.284821\pi$
$632$	10.4450	0.415479
$633$	18.5988	0.739237
$634$	−6.61329	−0.262647
$635$	0.216782	0.00860272
$636$	−0.246578	−0.00977746
$637$	−1.73240	−0.0686401
$638$	23.6921	0.937979
$639$	−12.4803	−0.493715
$640$	4.79991	0.189733
$641$	−35.0632	−1.38491	−0.692457	−	0.721459i	$-0.743472\pi$
$641$	−35.0632	−1.38491	−0.692457	+	0.721459i	$0.743472\pi$
$642$	13.6030	0.536866
$643$	−28.5069	−1.12420	−0.562101	−	0.827069i	$-0.690007\pi$
$643$	−28.5069	−1.12420	−0.562101	+	0.827069i	$0.690007\pi$
$644$	1.13250	0.0446267
$645$	2.58330	0.101717
$646$	11.1351	0.438107
$647$	5.49463	0.216016	0.108008	−	0.994150i	$-0.465553\pi$
$647$	5.49463	0.216016	0.108008	+	0.994150i	$0.465553\pi$
$648$	−2.71842	−0.106790
$649$	0.162043	0.00636074
$650$	−6.98228	−0.273868
$651$	−5.99940	−0.235135
$652$	−1.25696	−0.0492263
$653$	−37.8811	−1.48240	−0.741202	−	0.671283i	$-0.765744\pi$
$653$	−37.8811	−1.48240	−0.741202	+	0.671283i	$0.765744\pi$
$654$	−25.4040	−0.993374
$655$	8.38101	0.327473
$656$	−21.5105	−0.839843
$657$	−4.58440	−0.178854
$658$	−26.5527	−1.03513
$659$	32.9824	1.28481	0.642406	−	0.766364i	$-0.277936\pi$
$659$	32.9824	1.28481	0.642406	+	0.766364i	$0.277936\pi$
$660$	−0.0992782	−0.00386440
$661$	3.70894	0.144261	0.0721306	−	0.997395i	$-0.477020\pi$
$661$	3.70894	0.144261	0.0721306	+	0.997395i	$0.477020\pi$
$662$	9.61899	0.373853
$663$	−1.57658	−0.0612292
$664$	26.7875	1.03956
$665$	4.21302	0.163374
$666$	−0.217382	−0.00842337
$667$	31.2900	1.21155
$668$	0.843487	0.0326355
$669$	−0.939854	−0.0363368
$670$	−5.35170	−0.206754
$671$	−13.7198	−0.529646
$672$	1.84862	0.0713122
$673$	−13.4252	−0.517502	−0.258751	−	0.965944i	$-0.583311\pi$
$673$	−13.4252	−0.517502	−0.258751	+	0.965944i	$0.583311\pi$
$674$	31.6659	1.21972
$675$	−4.84952	−0.186658
$676$	−1.72125	−0.0662020
$677$	1.00000	0.0384331
$677$	1.00000	0.0384331
$678$	−15.6996	−0.602938
$679$	−0.116034	−0.00445297
$680$	1.69040	0.0648240
$681$	−8.16364	−0.312831
$682$	−6.86513	−0.262880
$683$	11.9448	0.457054	0.228527	−	0.973538i	$-0.426609\pi$
$683$	11.9448	0.457054	0.228527	+	0.973538i	$0.426609\pi$
$684$	−0.678780	−0.0259538
$685$	3.55289	0.135749
$686$	29.3560	1.12082
$687$	−11.5793	−0.441776
$688$	−28.4065	−1.08299
$689$	−1.69533	−0.0645871
$690$	−1.96429	−0.0747794
$691$	41.1418	1.56511	0.782554	−	0.622583i	$-0.213917\pi$
$691$	41.1418	1.56511	0.782554	+	0.622583i	$0.213917\pi$
$692$	−3.30409	−0.125603
$693$	4.09485	0.155550
$694$	−2.92299	−0.110955
$695$	3.56611	0.135270
$696$	24.5908	0.932112
$697$	−8.08350	−0.306184
$698$	−25.7139	−0.973285
$699$	−7.90222	−0.298889
$700$	1.58777	0.0600120
$701$	−34.8834	−1.31753	−0.658765	−	0.752349i	$-0.728921\pi$
$701$	−34.8834	−1.31753	−0.658765	+	0.752349i	$0.728921\pi$
$702$	1.43979	0.0543413
$703$	0.704619	0.0265752
$704$	−13.1478	−0.495526
$705$	3.07416	0.115779
$706$	29.3471	1.10449
$707$	18.3945	0.691796
$708$	−0.0129563	−0.000486927 0
$709$	−9.10734	−0.342033	−0.171017	−	0.985268i	$-0.554705\pi$
$709$	−9.10734	−0.342033	−0.171017	+	0.985268i	$0.554705\pi$
$710$	−7.08733	−0.265983
$711$	−3.84230	−0.144098
$712$	−14.7555	−0.552986
$713$	−9.06674	−0.339552
$714$	5.37100	0.201005
$715$	−0.682582	−0.0255271
$716$	3.09887	0.115810
$717$	16.0618	0.599840
$718$	−49.6018	−1.85112
$719$	8.80298	0.328296	0.164148	−	0.986436i	$-0.447513\pi$
$719$	8.80298	0.328296	0.164148	+	0.986436i	$0.447513\pi$
$720$	−1.65472	−0.0616677
$721$	−21.4147	−0.797526
$722$	5.14733	0.191564
$723$	9.31539	0.346443
$724$	3.33990	0.124126
$725$	43.8687	1.62924
$726$	−11.4173	−0.423736
$727$	41.2941	1.53151	0.765757	−	0.643130i	$-0.222364\pi$
$727$	41.2941	1.53151	0.765757	+	0.643130i	$0.222364\pi$
$728$	6.11935	0.226798
$729$	1.00000	0.0370370
$730$	−2.60339	−0.0963557
$731$	−10.6750	−0.394829
$732$	1.09698	0.0405454
$733$	19.0753	0.704564	0.352282	−	0.935894i	$-0.385406\pi$
$733$	19.0753	0.704564	0.352282	+	0.935894i	$0.385406\pi$
$734$	−2.27052	−0.0838063
$735$	−0.683291	−0.0252036
$736$	2.79378	0.102980
$737$	16.8604	0.621059
$738$	7.38214	0.271740
$739$	34.9890	1.28709	0.643546	−	0.765407i	$-0.277463\pi$
$739$	34.9890	1.28709	0.643546	+	0.765407i	$0.277463\pi$
$740$	−0.00824004	−0.000302910 0
$741$	−4.66691	−0.171443
$742$	5.77557	0.212028
$743$	−9.84477	−0.361170	−0.180585	−	0.983559i	$-0.557799\pi$
$743$	−9.84477	−0.361170	−0.180585	+	0.983559i	$0.557799\pi$
$744$	−7.12555	−0.261235
$745$	6.14412	0.225103
$746$	8.19550	0.300059
$747$	−9.85407	−0.360542
$748$	0.410248	0.0150002
$749$	−21.2679	−0.777110
$750$	−5.59334	−0.204240
$751$	2.32732	0.0849250	0.0424625	−	0.999098i	$-0.486480\pi$
$751$	2.32732	0.0849250	0.0424625	+	0.999098i	$0.486480\pi$
$752$	−33.8041	−1.23271
$753$	−20.5549	−0.749063
$754$	−13.0243	−0.474317
$755$	3.50183	0.127445
$756$	−0.327407	−0.0119077
$757$	−17.2815	−0.628108	−0.314054	−	0.949405i	$-0.601687\pi$
$757$	−17.2815	−0.628108	−0.314054	+	0.949405i	$0.601687\pi$
$758$	−5.11776	−0.185885
$759$	6.18844	0.224626
$760$	5.00385	0.181509
$761$	15.8739	0.575428	0.287714	−	0.957716i	$-0.407105\pi$
$761$	15.8739	0.575428	0.287714	+	0.957716i	$0.407105\pi$
$762$	0.818084	0.0296361
$763$	39.7184	1.43790
$764$	−3.53318	−0.127826
$765$	−0.621833	−0.0224824
$766$	34.5835	1.24955
$767$	−0.0890801	−0.00321650
$768$	3.41601	0.123265
$769$	15.3085	0.552040	0.276020	−	0.961152i	$-0.410984\pi$
$769$	15.3085	0.552040	0.276020	+	0.961152i	$0.410984\pi$
$770$	2.32538	0.0838010
$771$	−28.7839	−1.03663
$772$	−1.69753	−0.0610956
$773$	15.3167	0.550905	0.275453	−	0.961315i	$-0.411172\pi$
$773$	15.3167	0.550905	0.275453	+	0.961315i	$0.411172\pi$
$774$	9.74879	0.350413
$775$	−12.7116	−0.456614
$776$	−0.137814	−0.00494725
$777$	0.339871	0.0121928
$778$	22.2647	0.798227
$779$	−23.9284	−0.857323
$780$	0.0545764	0.00195415
$781$	22.3284	0.798974
$782$	8.11706	0.290266
$783$	−9.04599	−0.323277
$784$	7.51361	0.268343
$785$	−1.60184	−0.0571723
$786$	31.6280	1.12813
$787$	−55.7092	−1.98582	−0.992909	−	0.118875i	$-0.962071\pi$
$787$	−55.7092	−1.98582	−0.992909	+	0.118875i	$0.962071\pi$
$788$	−0.292104	−0.0104058
$789$	−14.3614	−0.511281
$790$	−2.18197	−0.0776308
$791$	24.5459	0.872750
$792$	4.86349	0.172817
$793$	7.54220	0.267832
$794$	17.6946	0.627957
$795$	−0.668672	−0.0237153
$796$	3.01987	0.107037
$797$	2.22098	0.0786710	0.0393355	−	0.999226i	$-0.487476\pi$
$797$	2.22098	0.0786710	0.0393355	+	0.999226i	$0.487476\pi$
$798$	15.8990	0.562818
$799$	−12.7034	−0.449413
$800$	3.91688	0.138483
$801$	5.42797	0.191788
$802$	8.94500	0.315859
$803$	8.20189	0.289438
$804$	−1.34808	−0.0475433
$805$	3.07112	0.108243
$806$	3.77398	0.132933
$807$	0.286312	0.0100787
$808$	21.8473	0.768586
$809$	4.24804	0.149353	0.0746765	−	0.997208i	$-0.476208\pi$
$809$	4.24804	0.149353	0.0746765	+	0.997208i	$0.476208\pi$
$810$	0.567880	0.0199533
$811$	−33.5125	−1.17678	−0.588392	−	0.808576i	$-0.700239\pi$
$811$	−33.5125	−1.17678	−0.588392	+	0.808576i	$0.700239\pi$
$812$	2.96172	0.103936
$813$	23.1392	0.811528
$814$	0.388915	0.0136315
$815$	−3.40863	−0.119399
$816$	6.83780	0.239371
$817$	−31.5996	−1.10553
$818$	19.5112	0.682194
$819$	−2.25107	−0.0786587
$820$	0.279826	0.00977196
$821$	18.8922	0.659341	0.329670	−	0.944096i	$-0.393062\pi$
$821$	18.8922	0.659341	0.329670	+	0.944096i	$0.393062\pi$
$822$	13.4078	0.467650
$823$	−0.253189	−0.00882561	−0.00441280	−	0.999990i	$-0.501405\pi$
$823$	−0.253189	−0.00882561	−0.00441280	+	0.999990i	$0.501405\pi$
$824$	−25.4345	−0.886051
$825$	8.67621	0.302067
$826$	0.303473	0.0105592
$827$	−26.9985	−0.938830	−0.469415	−	0.882978i	$-0.655535\pi$
$827$	−26.9985	−0.938830	−0.469415	+	0.882978i	$0.655535\pi$
$828$	−0.494802	−0.0171956
$829$	−57.0561	−1.98164	−0.990821	−	0.135183i	$-0.956838\pi$
$829$	−57.0561	−1.98164	−0.990821	+	0.135183i	$0.956838\pi$
$830$	−5.59593	−0.194237
$831$	−12.8797	−0.446791
$832$	7.22776	0.250578
$833$	2.82357	0.0978308
$834$	13.4577	0.466002
$835$	2.28737	0.0791578
$836$	1.21440	0.0420008
$837$	2.62121	0.0906022
$838$	−26.3672	−0.910841
$839$	−38.4221	−1.32648	−0.663240	−	0.748407i	$-0.730819\pi$
$839$	−38.4221	−1.32648	−0.663240	+	0.748407i	$0.730819\pi$
$840$	2.41359	0.0832768
$841$	52.8299	1.82172
$842$	20.5540	0.708338
$843$	−16.5883	−0.571333
$844$	2.66052	0.0915790
$845$	−4.66770	−0.160574
$846$	11.6012	0.398856
$847$	17.8507	0.613356
$848$	7.35286	0.252498
$849$	−0.328687	−0.0112805
$850$	11.3801	0.390336
$851$	0.513638	0.0176073
$852$	−1.78529	−0.0611630
$853$	−10.3269	−0.353587	−0.176793	−	0.984248i	$-0.556572\pi$
$853$	−10.3269	−0.353587	−0.176793	+	0.984248i	$0.556572\pi$
$854$	−25.6944	−0.879243
$855$	−1.84072	−0.0629513
$856$	−25.2600	−0.863370
$857$	11.3429	0.387467	0.193734	−	0.981054i	$-0.437940\pi$
$857$	11.3429	0.387467	0.193734	+	0.981054i	$0.437940\pi$
$858$	−2.57591	−0.0879400
$859$	−8.77050	−0.299245	−0.149623	−	0.988743i	$-0.547806\pi$
$859$	−8.77050	−0.299245	−0.149623	+	0.988743i	$0.547806\pi$
$860$	0.369536	0.0126011
$861$	−11.5418	−0.393343
$862$	−56.7400	−1.93257
$863$	−32.5706	−1.10872	−0.554359	−	0.832278i	$-0.687036\pi$
$863$	−32.5706	−1.10872	−0.554359	+	0.832278i	$0.687036\pi$
$864$	−0.807685	−0.0274780
$865$	−8.96005	−0.304651
$866$	−42.6853	−1.45050
$867$	−14.4304	−0.490082
$868$	−0.858202	−0.0291293
$869$	6.87421	0.233192
$870$	−5.13703	−0.174162
$871$	−9.26868	−0.314057
$872$	47.1740	1.59751
$873$	0.0506965	0.00171582
$874$	24.0277	0.812750
$875$	8.74504	0.295636
$876$	−0.655789	−0.0221571
$877$	52.5225	1.77356	0.886780	−	0.462192i	$-0.152937\pi$
$877$	52.5225	1.77356	0.886780	+	0.462192i	$0.152937\pi$
$878$	−13.7294	−0.463344
$879$	10.3647	0.349591
$880$	2.96044	0.0997963
$881$	39.2658	1.32290	0.661450	−	0.749989i	$-0.269941\pi$
$881$	39.2658	1.32290	0.661450	+	0.749989i	$0.269941\pi$
$882$	−2.57858	−0.0868254
$883$	−23.1714	−0.779780	−0.389890	−	0.920862i	$-0.627487\pi$
$883$	−23.1714	−0.779780	−0.389890	+	0.920862i	$0.627487\pi$
$884$	−0.225526	−0.00758528
$885$	−0.0351349	−0.00118105
$886$	−15.5911	−0.523792
$887$	32.1376	1.07907	0.539537	−	0.841962i	$-0.318599\pi$
$887$	32.1376	1.07907	0.539537	+	0.841962i	$0.318599\pi$
$888$	0.403667	0.0135462
$889$	−1.27905	−0.0428980
$890$	3.08244	0.103323
$891$	−1.78909	−0.0599367
$892$	−0.134444	−0.00450152
$893$	−37.6039	−1.25837
$894$	23.1865	0.775473
$895$	8.40354	0.280900
$896$	−28.3204	−0.946118
$897$	−3.40199	−0.113589
$898$	−32.9943	−1.10103
$899$	−23.7114	−0.790820
$900$	−0.693714	−0.0231238
$901$	2.76316	0.0920541
$902$	−13.2073	−0.439755
$903$	−15.2420	−0.507221
$904$	29.1533	0.969626
$905$	9.05716	0.301070
$906$	13.2151	0.439042
$907$	14.0634	0.466967	0.233483	−	0.972361i	$-0.424988\pi$
$907$	14.0634	0.466967	0.233483	+	0.972361i	$0.424988\pi$
$908$	−1.16779	−0.0387545
$909$	−8.03677	−0.266563
$910$	−1.27834	−0.0423765
$911$	−5.55457	−0.184031	−0.0920155	−	0.995758i	$-0.529331\pi$
$911$	−5.55457	−0.184031	−0.0920155	+	0.995758i	$0.529331\pi$
$912$	20.2409	0.670244
$913$	17.6298	0.583461
$914$	51.1170	1.69080
$915$	2.97479	0.0983435
$916$	−1.65639	−0.0547287
$917$	−49.4495	−1.63297
$918$	−2.34665	−0.0774511
$919$	21.0669	0.694933	0.347466	−	0.937692i	$-0.387042\pi$
$919$	21.0669	0.694933	0.347466	+	0.937692i	$0.387042\pi$
$920$	3.64760	0.120258
$921$	−5.02595	−0.165611
$922$	41.2931	1.35992
$923$	−12.2747	−0.404025
$924$	0.585760	0.0192701
$925$	0.720122	0.0236775
$926$	−25.8596	−0.849798
$927$	9.35634	0.307302
$928$	7.30631	0.239841
$929$	−9.97010	−0.327108	−0.163554	−	0.986534i	$-0.552296\pi$
$929$	−9.97010	−0.327108	−0.163554	+	0.986534i	$0.552296\pi$
$930$	1.48853	0.0488109
$931$	8.35819	0.273929
$932$	−1.13040	−0.0370274
$933$	3.63333	0.118950
$934$	−27.2599	−0.891972
$935$	1.11251	0.0363831
$936$	−2.67362	−0.0873899
$937$	24.7208	0.807594	0.403797	−	0.914849i	$-0.367690\pi$
$937$	24.7208	0.807594	0.403797	+	0.914849i	$0.367690\pi$
$938$	31.5760	1.03099
$939$	23.0916	0.753565
$940$	0.439752	0.0143431
$941$	40.7105	1.32712	0.663562	−	0.748122i	$-0.269044\pi$
$941$	40.7105	1.32712	0.663562	+	0.748122i	$0.269044\pi$
$942$	−6.04499	−0.196957
$943$	−17.4428	−0.568016
$944$	0.386351	0.0125746
$945$	−0.887865	−0.0288822
$946$	−17.4414	−0.567070
$947$	18.8055	0.611096	0.305548	−	0.952177i	$-0.401160\pi$
$947$	18.8055	0.611096	0.305548	+	0.952177i	$0.401160\pi$
$948$	−0.549634	−0.0178513
$949$	−4.50884	−0.146363
$950$	33.6869	1.09295
$951$	−4.51754	−0.146491
$952$	−9.97369	−0.323249
$953$	−14.7613	−0.478166	−0.239083	−	0.970999i	$-0.576847\pi$
$953$	−14.7613	−0.478166	−0.239083	+	0.970999i	$0.576847\pi$
$954$	−2.52341	−0.0816986
$955$	−9.58129	−0.310043
$956$	2.29761	0.0743101
$957$	16.1841	0.523156
$958$	3.72296	0.120283
$959$	−20.9627	−0.676922
$960$	2.85077	0.0920081
$961$	−24.1293	−0.778364
$962$	−0.213799	−0.00689316
$963$	9.29217	0.299436
$964$	1.33255	0.0429185
$965$	−4.60338	−0.148188
$966$	11.5897	0.372893
$967$	34.4322	1.10727	0.553633	−	0.832761i	$-0.313241\pi$
$967$	34.4322	1.10727	0.553633	+	0.832761i	$0.313241\pi$
$968$	21.2014	0.681439
$969$	7.60642	0.244353
$970$	0.0287895	0.000924375 0
$971$	28.1893	0.904639	0.452319	−	0.891856i	$-0.350597\pi$
$971$	28.1893	0.904639	0.452319	+	0.891856i	$0.350597\pi$
$972$	0.143048	0.00458827
$973$	−21.0407	−0.674535
$974$	−1.66955	−0.0534959
$975$	−4.76959	−0.152749
$976$	−32.7114	−1.04707
$977$	−31.4468	−1.00607	−0.503036	−	0.864265i	$-0.667784\pi$
$977$	−31.4468	−1.00607	−0.503036	+	0.864265i	$0.667784\pi$
$978$	−12.8634	−0.411326
$979$	−9.71112	−0.310369
$980$	−0.0977434	−0.00312230
$981$	−17.3534	−0.554053
$982$	−40.5070	−1.29263
$983$	−6.88201	−0.219502	−0.109751	−	0.993959i	$-0.535005\pi$
$983$	−6.88201	−0.219502	−0.109751	+	0.993959i	$0.535005\pi$
$984$	−13.7083	−0.437004
$985$	−0.792129	−0.0252393
$986$	21.2278	0.676031
$987$	−18.1381	−0.577342
$988$	−0.667593	−0.0212389
$989$	−23.0348	−0.732464
$990$	−1.01599	−0.0322902
$991$	21.3698	0.678833	0.339416	−	0.940636i	$-0.389770\pi$
$991$	21.3698	0.678833	0.339416	+	0.940636i	$0.389770\pi$
$992$	−2.11711	−0.0672183
$993$	6.57073	0.208516
$994$	41.8166	1.32634
$995$	8.18931	0.259619
$996$	−1.40961	−0.0446651
$997$	15.2851	0.484083	0.242041	−	0.970266i	$-0.422183\pi$
$997$	15.2851	0.484083	0.242041	+	0.970266i	$0.422183\pi$
$998$	−15.0325	−0.475844
$999$	−0.148493	−0.00469812

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2031.2.a.g.1.16	✓	18
3.2	odd	2		6093.2.a.l.1.3		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2031.2.a.g.1.16	✓	18	1.1	even	1	trivial
6093.2.a.l.1.3		18	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2031 = 3 \cdot 677 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2031.a (trivial)

\(f(q)\)	\(=\)	\(q+1.46392 q^{2} +1.00000 q^{3} +0.143048 q^{4} +0.387918 q^{5} +1.46392 q^{6} -2.28879 q^{7} -2.71842 q^{8} +1.00000 q^{9} +0.567880 q^{10} -1.78909 q^{11} +0.143048 q^{12} +0.983519 q^{13} -3.35060 q^{14} +0.387918 q^{15} -4.26563 q^{16} -1.60300 q^{17} +1.46392 q^{18} -4.74512 q^{19} +0.0554910 q^{20} -2.28879 q^{21} -2.61907 q^{22} -3.45899 q^{23} -2.71842 q^{24} -4.84952 q^{25} +1.43979 q^{26} +1.00000 q^{27} -0.327407 q^{28} -9.04599 q^{29} +0.567880 q^{30} +2.62121 q^{31} -0.807685 q^{32} -1.78909 q^{33} -2.34665 q^{34} -0.887865 q^{35} +0.143048 q^{36} -0.148493 q^{37} -6.94645 q^{38} +0.983519 q^{39} -1.05453 q^{40} +5.04274 q^{41} -3.35060 q^{42} +6.65939 q^{43} -0.255925 q^{44} +0.387918 q^{45} -5.06368 q^{46} +7.92475 q^{47} -4.26563 q^{48} -1.76143 q^{49} -7.09929 q^{50} -1.60300 q^{51} +0.140690 q^{52} -1.72374 q^{53} +1.46392 q^{54} -0.694020 q^{55} +6.22190 q^{56} -4.74512 q^{57} -13.2426 q^{58} -0.0905729 q^{59} +0.0554910 q^{60} +7.66859 q^{61} +3.83723 q^{62} -2.28879 q^{63} +7.34888 q^{64} +0.381525 q^{65} -2.61907 q^{66} -9.42400 q^{67} -0.229306 q^{68} -3.45899 q^{69} -1.29976 q^{70} -12.4803 q^{71} -2.71842 q^{72} -4.58440 q^{73} -0.217382 q^{74} -4.84952 q^{75} -0.678780 q^{76} +4.09485 q^{77} +1.43979 q^{78} -3.84230 q^{79} -1.65472 q^{80} +1.00000 q^{81} +7.38214 q^{82} -9.85407 q^{83} -0.327407 q^{84} -0.621833 q^{85} +9.74879 q^{86} -9.04599 q^{87} +4.86349 q^{88} +5.42797 q^{89} +0.567880 q^{90} -2.25107 q^{91} -0.494802 q^{92} +2.62121 q^{93} +11.6012 q^{94} -1.84072 q^{95} -0.807685 q^{96} +0.0506965 q^{97} -2.57858 q^{98} -1.78909 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 7 q^{2} + 18 q^{3} + 13 q^{4} - 13 q^{5} - 7 q^{6} - 3 q^{7} - 18 q^{8} + 18 q^{9} - 6 q^{10} - 26 q^{11} + 13 q^{12} - 11 q^{13} - 11 q^{14} - 13 q^{15} - q^{16} - 34 q^{17} - 7 q^{18} - 2 q^{19}+ \cdots - 26 q^{99}+O(q^{100}) \) 18 * q - 7 * q^2 + 18 * q^3 + 13 * q^4 - 13 * q^5 - 7 * q^6 - 3 * q^7 - 18 * q^8 + 18 * q^9 - 6 * q^10 - 26 * q^11 + 13 * q^12 - 11 * q^13 - 11 * q^14 - 13 * q^15 - q^16 - 34 * q^17 - 7 * q^18 - 2 * q^19 - 10 * q^20 - 3 * q^21 + 8 * q^22 - 35 * q^23 - 18 * q^24 - 5 * q^25 - 17 * q^26 + 18 * q^27 - 5 * q^28 - 25 * q^29 - 6 * q^30 - 11 * q^31 - 18 * q^32 - 26 * q^33 - 3 * q^34 - 27 * q^35 + 13 * q^36 - q^37 - 18 * q^38 - 11 * q^39 - 8 * q^40 - 44 * q^41 - 11 * q^42 + 5 * q^43 - 46 * q^44 - 13 * q^45 + 20 * q^46 - 51 * q^47 - q^48 - 33 * q^49 + 11 * q^50 - 34 * q^51 - 2 * q^52 - 28 * q^53 - 7 * q^54 + q^55 - q^56 - 2 * q^57 + 22 * q^58 - 44 * q^59 - 10 * q^60 - 7 * q^61 + q^62 - 3 * q^63 + 8 * q^64 - 11 * q^65 + 8 * q^66 + 2 * q^67 - 11 * q^68 - 35 * q^69 + 16 * q^70 - 23 * q^71 - 18 * q^72 + q^73 - 9 * q^74 - 5 * q^75 + 3 * q^76 - 23 * q^77 - 17 * q^78 + 9 * q^79 - 15 * q^80 + 18 * q^81 + 19 * q^82 - 45 * q^83 - 5 * q^84 + 9 * q^85 - 4 * q^86 - 25 * q^87 + 51 * q^88 - 90 * q^89 - 6 * q^90 - 24 * q^91 - 57 * q^92 - 11 * q^93 + 24 * q^94 - 9 * q^95 - 18 * q^96 - 17 * q^97 + 12 * q^98 - 26 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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