LMFDB - Embedded newform 1003.2.a.i.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1003,2,Mod(1,1003)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1003, 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("1003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1003 = 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1003.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:	$8.00899532273$
Analytic rank:	$0$
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} - 5 x^{17} - 16 x^{16} + 112 x^{15} + 52 x^{14} - 984 x^{13} + 431 x^{12} + 4304 x^{11} + \cdots + 28 $ x^18 - 5x^17 - 16x^16 + 112x^15 + 52x^14 - 984x^13 + 431x^12 + 4304x^11 - 3825x^10 - 9826x^9 + 11533x^8 + 11182x^7 - 15697x^6 - 5604x^5 + 9201x^4 + 1189x^3 - 1952x^2 - 100*x + 28
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Root			$2.66948$ of defining polynomial
Character	$\chi$	$=$	1003.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.66948 q^{2} +0.955769 q^{3} +5.12613 q^{4} +1.71138 q^{5} +2.55141 q^{6} -0.402139 q^{7} +8.34515 q^{8} -2.08651 q^{9} +O(q^{10})$	\(q+2.66948 q^{2} +0.955769 q^{3} +5.12613 q^{4} +1.71138 q^{5} +2.55141 q^{6} -0.402139 q^{7} +8.34515 q^{8} -2.08651 q^{9} +4.56850 q^{10} -2.27003 q^{11} +4.89940 q^{12} -6.19463 q^{13} -1.07350 q^{14} +1.63568 q^{15} +12.0250 q^{16} +1.00000 q^{17} -5.56989 q^{18} +6.85815 q^{19} +8.77276 q^{20} -0.384352 q^{21} -6.05980 q^{22} +3.93346 q^{23} +7.97604 q^{24} -2.07118 q^{25} -16.5364 q^{26} -4.86153 q^{27} -2.06142 q^{28} -5.15524 q^{29} +4.36643 q^{30} -1.68879 q^{31} +15.4101 q^{32} -2.16962 q^{33} +2.66948 q^{34} -0.688213 q^{35} -10.6957 q^{36} +3.48377 q^{37} +18.3077 q^{38} -5.92064 q^{39} +14.2817 q^{40} +1.92264 q^{41} -1.02602 q^{42} -2.39274 q^{43} -11.6365 q^{44} -3.57080 q^{45} +10.5003 q^{46} -5.55801 q^{47} +11.4931 q^{48} -6.83828 q^{49} -5.52897 q^{50} +0.955769 q^{51} -31.7545 q^{52} -3.14326 q^{53} -12.9778 q^{54} -3.88488 q^{55} -3.35591 q^{56} +6.55481 q^{57} -13.7618 q^{58} -1.00000 q^{59} +8.38473 q^{60} +2.10564 q^{61} -4.50819 q^{62} +0.839066 q^{63} +17.0871 q^{64} -10.6014 q^{65} -5.79177 q^{66} +8.91869 q^{67} +5.12613 q^{68} +3.75948 q^{69} -1.83717 q^{70} +13.5829 q^{71} -17.4122 q^{72} +2.16087 q^{73} +9.29986 q^{74} -1.97957 q^{75} +35.1558 q^{76} +0.912867 q^{77} -15.8050 q^{78} +2.30217 q^{79} +20.5793 q^{80} +1.61302 q^{81} +5.13244 q^{82} +2.12347 q^{83} -1.97024 q^{84} +1.71138 q^{85} -6.38738 q^{86} -4.92722 q^{87} -18.9437 q^{88} -4.12968 q^{89} -9.53219 q^{90} +2.49110 q^{91} +20.1634 q^{92} -1.61409 q^{93} -14.8370 q^{94} +11.7369 q^{95} +14.7285 q^{96} +1.87743 q^{97} -18.2547 q^{98} +4.73642 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 5 q^{2} + q^{3} + 21 q^{4} + 21 q^{5} + 5 q^{6} - q^{7} + 9 q^{8} + 17 q^{9}+O(q^{10}) $ 18 * q + 5 * q^2 + q^3 + 21 * q^4 + 21 * q^5 + 5 * q^6 - q^7 + 9 * q^8 + 17 * q^9	\( 18 q + 5 q^{2} + q^{3} + 21 q^{4} + 21 q^{5} + 5 q^{6} - q^{7} + 9 q^{8} + 17 q^{9} + 10 q^{10} + 6 q^{11} + 20 q^{12} + 12 q^{13} + 9 q^{14} - q^{15} + 19 q^{16} + 18 q^{17} + 10 q^{18} - 14 q^{19} + 17 q^{20} - 14 q^{21} - 4 q^{22} + 7 q^{23} - 3 q^{24} + 35 q^{25} + 3 q^{26} + 13 q^{27} + q^{28} + 23 q^{29} + 17 q^{30} - q^{31} + 13 q^{32} + 7 q^{33} + 5 q^{34} + 5 q^{35} - 16 q^{36} + 36 q^{37} - 19 q^{38} + 8 q^{39} - 4 q^{40} + 37 q^{41} + 9 q^{42} - 15 q^{43} - 9 q^{44} + 17 q^{45} + q^{46} + 23 q^{47} + 17 q^{48} + 15 q^{49} + 32 q^{50} + q^{51} - 11 q^{52} + 35 q^{53} - 17 q^{54} - 6 q^{55} + 37 q^{56} - 16 q^{57} + 2 q^{58} - 18 q^{59} - 6 q^{60} + 39 q^{61} - q^{62} + 15 q^{63} + 5 q^{64} + 15 q^{65} - 36 q^{66} + 14 q^{67} + 21 q^{68} + 16 q^{69} - 45 q^{70} + 3 q^{71} - 71 q^{72} + 56 q^{73} - 27 q^{74} + 16 q^{75} - q^{76} + 39 q^{77} - 71 q^{78} - 22 q^{79} + 45 q^{80} - 18 q^{81} + 5 q^{82} + 12 q^{83} + 2 q^{84} + 21 q^{85} + 4 q^{86} + 8 q^{87} - 18 q^{88} + 34 q^{89} + 48 q^{90} + 3 q^{91} + 67 q^{92} - 19 q^{93} - 38 q^{94} - 72 q^{95} + 6 q^{96} + 31 q^{97} + 19 q^{98} + 44 q^{99}+O(q^{100}) \) 18 * q + 5 * q^2 + q^3 + 21 * q^4 + 21 * q^5 + 5 * q^6 - q^7 + 9 * q^8 + 17 * q^9 + 10 * q^10 + 6 * q^11 + 20 * q^12 + 12 * q^13 + 9 * q^14 - q^15 + 19 * q^16 + 18 * q^17 + 10 * q^18 - 14 * q^19 + 17 * q^20 - 14 * q^21 - 4 * q^22 + 7 * q^23 - 3 * q^24 + 35 * q^25 + 3 * q^26 + 13 * q^27 + q^28 + 23 * q^29 + 17 * q^30 - q^31 + 13 * q^32 + 7 * q^33 + 5 * q^34 + 5 * q^35 - 16 * q^36 + 36 * q^37 - 19 * q^38 + 8 * q^39 - 4 * q^40 + 37 * q^41 + 9 * q^42 - 15 * q^43 - 9 * q^44 + 17 * q^45 + q^46 + 23 * q^47 + 17 * q^48 + 15 * q^49 + 32 * q^50 + q^51 - 11 * q^52 + 35 * q^53 - 17 * q^54 - 6 * q^55 + 37 * q^56 - 16 * q^57 + 2 * q^58 - 18 * q^59 - 6 * q^60 + 39 * q^61 - q^62 + 15 * q^63 + 5 * q^64 + 15 * q^65 - 36 * q^66 + 14 * q^67 + 21 * q^68 + 16 * q^69 - 45 * q^70 + 3 * q^71 - 71 * q^72 + 56 * q^73 - 27 * q^74 + 16 * q^75 - q^76 + 39 * q^77 - 71 * q^78 - 22 * q^79 + 45 * q^80 - 18 * q^81 + 5 * q^82 + 12 * q^83 + 2 * q^84 + 21 * q^85 + 4 * q^86 + 8 * q^87 - 18 * q^88 + 34 * q^89 + 48 * q^90 + 3 * q^91 + 67 * q^92 - 19 * q^93 - 38 * q^94 - 72 * q^95 + 6 * q^96 + 31 * q^97 + 19 * q^98 + 44 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	2.66948	1.88761	0.943804	−	0.330505i	$-0.107219\pi$
$2$	2.66948	1.88761	0.943804	+	0.330505i	$0.107219\pi$
$3$	0.955769	0.551814	0.275907	−	0.961184i	$-0.411022\pi$
$3$	0.955769	0.551814	0.275907	+	0.961184i	$0.411022\pi$
$4$	5.12613	2.56307
$5$	1.71138	0.765353	0.382676	−	0.923882i	$-0.375002\pi$
$5$	1.71138	0.765353	0.382676	+	0.923882i	$0.375002\pi$
$6$	2.55141	1.04161
$7$	−0.402139	−0.151994	−0.0759972	−	0.997108i	$-0.524214\pi$
$7$	−0.402139	−0.151994	−0.0759972	+	0.997108i	$0.524214\pi$
$8$	8.34515	2.95046
$9$	−2.08651	−0.695502
$10$	4.56850	1.44469
$11$	−2.27003	−0.684439	−0.342220	−	0.939620i	$-0.611179\pi$
$11$	−2.27003	−0.684439	−0.342220	+	0.939620i	$0.611179\pi$
$12$	4.89940	1.41433
$13$	−6.19463	−1.71808	−0.859040	−	0.511908i	$-0.828939\pi$
$13$	−6.19463	−1.71808	−0.859040	+	0.511908i	$0.828939\pi$
$14$	−1.07350	−0.286906
$15$	1.63568	0.422332
$16$	12.0250	3.00624
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−5.56989	−1.31283
$19$	6.85815	1.57337	0.786684	−	0.617356i	$-0.211796\pi$
$19$	6.85815	1.57337	0.786684	+	0.617356i	$0.211796\pi$
$20$	8.77276	1.96165
$21$	−0.384352	−0.0838726
$22$	−6.05980	−1.29195
$23$	3.93346	0.820184	0.410092	−	0.912044i	$-0.365497\pi$
$23$	3.93346	0.820184	0.410092	+	0.912044i	$0.365497\pi$
$24$	7.97604	1.62810
$25$	−2.07118	−0.414235
$26$	−16.5364	−3.24306
$27$	−4.86153	−0.935601
$28$	−2.06142	−0.389572
$29$	−5.15524	−0.957303	−0.478652	−	0.878005i	$-0.658874\pi$
$29$	−5.15524	−0.957303	−0.478652	+	0.878005i	$0.658874\pi$
$30$	4.36643	0.797197
$31$	−1.68879	−0.303315	−0.151658	−	0.988433i	$-0.548461\pi$
$31$	−1.68879	−0.303315	−0.151658	+	0.988433i	$0.548461\pi$
$32$	15.4101	2.72415
$33$	−2.16962	−0.377683
$34$	2.66948	0.457812
$35$	−0.688213	−0.116329
$36$	−10.6957	−1.78262
$37$	3.48377	0.572729	0.286364	−	0.958121i	$-0.407553\pi$
$37$	3.48377	0.572729	0.286364	+	0.958121i	$0.407553\pi$
$38$	18.3077	2.96990
$39$	−5.92064	−0.948061
$40$	14.2817	2.25814
$41$	1.92264	0.300265	0.150133	−	0.988666i	$-0.452030\pi$
$41$	1.92264	0.300265	0.150133	+	0.988666i	$0.452030\pi$
$42$	−1.02602	−0.158319
$43$	−2.39274	−0.364890	−0.182445	−	0.983216i	$-0.558401\pi$
$43$	−2.39274	−0.364890	−0.182445	+	0.983216i	$0.558401\pi$
$44$	−11.6365	−1.75426
$45$	−3.57080	−0.532304
$46$	10.5003	1.54819
$47$	−5.55801	−0.810719	−0.405360	−	0.914157i	$-0.632854\pi$
$47$	−5.55801	−0.810719	−0.405360	+	0.914157i	$0.632854\pi$
$48$	11.4931	1.65888
$49$	−6.83828	−0.976898
$50$	−5.52897	−0.781914
$51$	0.955769	0.133834
$52$	−31.7545	−4.40355
$53$	−3.14326	−0.431760	−0.215880	−	0.976420i	$-0.569262\pi$
$53$	−3.14326	−0.431760	−0.215880	+	0.976420i	$0.569262\pi$
$54$	−12.9778	−1.76605
$55$	−3.88488	−0.523837
$56$	−3.35591	−0.448453
$57$	6.55481	0.868206
$58$	−13.7618	−1.80701
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	8.38473	1.08246
$61$	2.10564	0.269600	0.134800	−	0.990873i	$-0.456961\pi$
$61$	2.10564	0.269600	0.134800	+	0.990873i	$0.456961\pi$
$62$	−4.50819	−0.572540
$63$	0.839066	0.105712
$64$	17.0871	2.13588
$65$	−10.6014	−1.31494
$66$	−5.79177	−0.712917
$67$	8.91869	1.08959	0.544796	−	0.838569i	$-0.316607\pi$
$67$	8.91869	1.08959	0.544796	+	0.838569i	$0.316607\pi$
$68$	5.12613	0.621635
$69$	3.75948	0.452589
$70$	−1.83717	−0.219584
$71$	13.5829	1.61200	0.805999	−	0.591916i	$-0.201628\pi$
$71$	13.5829	1.61200	0.805999	+	0.591916i	$0.201628\pi$
$72$	−17.4122	−2.05205
$73$	2.16087	0.252910	0.126455	−	0.991972i	$-0.459640\pi$
$73$	2.16087	0.252910	0.126455	+	0.991972i	$0.459640\pi$
$74$	9.29986	1.08109
$75$	−1.97957	−0.228581
$76$	35.1558	4.03264
$77$	0.912867	0.104031
$78$	−15.8050	−1.78957
$79$	2.30217	0.259015	0.129507	−	0.991578i	$-0.458660\pi$
$79$	2.30217	0.259015	0.129507	+	0.991578i	$0.458660\pi$
$80$	20.5793	2.30083
$81$	1.61302	0.179224
$82$	5.13244	0.566783
$83$	2.12347	0.233081	0.116540	−	0.993186i	$-0.462820\pi$
$83$	2.12347	0.233081	0.116540	+	0.993186i	$0.462820\pi$
$84$	−1.97024	−0.214971
$85$	1.71138	0.185625
$86$	−6.38738	−0.688769
$87$	−4.92722	−0.528253
$88$	−18.9437	−2.01941
$89$	−4.12968	−0.437745	−0.218873	−	0.975753i	$-0.570238\pi$
$89$	−4.12968	−0.437745	−0.218873	+	0.975753i	$0.570238\pi$
$90$	−9.53219	−1.00478
$91$	2.49110	0.261139
$92$	20.1634	2.10218
$93$	−1.61409	−0.167373
$94$	−14.8370	−1.53032
$95$	11.7369	1.20418
$96$	14.7285	1.50322
$97$	1.87743	0.190624	0.0953122	−	0.995447i	$-0.469615\pi$
$97$	1.87743	0.190624	0.0953122	+	0.995447i	$0.469615\pi$
$98$	−18.2547	−1.84400
$99$	4.73642	0.476029
$100$	−10.6171	−1.06171
$101$	8.86160	0.881762	0.440881	−	0.897565i	$-0.354666\pi$
$101$	8.86160	0.881762	0.440881	+	0.897565i	$0.354666\pi$
$102$	2.55141	0.252627
$103$	12.5115	1.23279	0.616397	−	0.787436i	$-0.288592\pi$
$103$	12.5115	1.23279	0.616397	+	0.787436i	$0.288592\pi$
$104$	−51.6951	−5.06912
$105$	−0.657773	−0.0641921
$106$	−8.39087	−0.814993
$107$	−10.9546	−1.05902	−0.529510	−	0.848304i	$-0.677624\pi$
$107$	−10.9546	−1.05902	−0.529510	+	0.848304i	$0.677624\pi$
$108$	−24.9208	−2.39801
$109$	15.9861	1.53119	0.765596	−	0.643322i	$-0.222444\pi$
$109$	15.9861	1.53119	0.765596	+	0.643322i	$0.222444\pi$
$110$	−10.3706	−0.988800
$111$	3.32968	0.316039
$112$	−4.83571	−0.456931
$113$	14.7084	1.38365	0.691823	−	0.722067i	$-0.256808\pi$
$113$	14.7084	1.38365	0.691823	+	0.722067i	$0.256808\pi$
$114$	17.4979	1.63883
$115$	6.73165	0.627730
$116$	−26.4264	−2.45363
$117$	12.9251	1.19493
$118$	−2.66948	−0.245746
$119$	−0.402139	−0.0368641
$120$	13.6500	1.24607
$121$	−5.84697	−0.531543
$122$	5.62098	0.508899
$123$	1.83760	0.165691
$124$	−8.65694	−0.777417
$125$	−12.1015	−1.08239
$126$	2.23987	0.199544
$127$	−11.7750	−1.04486	−0.522430	−	0.852682i	$-0.674974\pi$
$127$	−11.7750	−1.04486	−0.522430	+	0.852682i	$0.674974\pi$
$128$	14.7934	1.30756
$129$	−2.28691	−0.201351
$130$	−28.3001	−2.48209
$131$	−14.2679	−1.24659	−0.623296	−	0.781986i	$-0.714207\pi$
$131$	−14.2679	−1.24659	−0.623296	+	0.781986i	$0.714207\pi$
$132$	−11.1218	−0.968026
$133$	−2.75793	−0.239143
$134$	23.8083	2.05672
$135$	−8.31992	−0.716065
$136$	8.34515	0.715591
$137$	2.94337	0.251469	0.125735	−	0.992064i	$-0.459871\pi$
$137$	2.94337	0.251469	0.125735	+	0.992064i	$0.459871\pi$
$138$	10.0359	0.854310
$139$	3.55298	0.301359	0.150680	−	0.988583i	$-0.451854\pi$
$139$	3.55298	0.301359	0.150680	+	0.988583i	$0.451854\pi$
$140$	−3.52787	−0.298160
$141$	−5.31218	−0.447366
$142$	36.2594	3.04282
$143$	14.0620	1.17592
$144$	−25.0901	−2.09084
$145$	−8.82257	−0.732675
$146$	5.76840	0.477396
$147$	−6.53582	−0.539066
$148$	17.8583	1.46794
$149$	−15.1233	−1.23895	−0.619476	−	0.785016i	$-0.712655\pi$
$149$	−15.1233	−1.23895	−0.619476	+	0.785016i	$0.712655\pi$
$150$	−5.28442	−0.431471
$151$	−19.5629	−1.59200	−0.796001	−	0.605295i	$-0.793055\pi$
$151$	−19.5629	−1.59200	−0.796001	+	0.605295i	$0.793055\pi$
$152$	57.2323	4.64215
$153$	−2.08651	−0.168684
$154$	2.43688	0.196370
$155$	−2.89016	−0.232143
$156$	−30.3500	−2.42994
$157$	−0.756111	−0.0603442	−0.0301721	−	0.999545i	$-0.509606\pi$
$157$	−0.756111	−0.0603442	−0.0301721	+	0.999545i	$0.509606\pi$
$158$	6.14561	0.488918
$159$	−3.00423	−0.238251
$160$	26.3725	2.08493
$161$	−1.58180	−0.124663
$162$	4.30592	0.338305
$163$	−6.93526	−0.543211	−0.271606	−	0.962409i	$-0.587555\pi$
$163$	−6.93526	−0.543211	−0.271606	+	0.962409i	$0.587555\pi$
$164$	9.85568	0.769600
$165$	−3.71305	−0.289061
$166$	5.66856	0.439966
$167$	9.55082	0.739065	0.369532	−	0.929218i	$-0.379518\pi$
$167$	9.55082	0.739065	0.369532	+	0.929218i	$0.379518\pi$
$168$	−3.20748	−0.247462
$169$	25.3734	1.95180
$170$	4.56850	0.350388
$171$	−14.3096	−1.09428
$172$	−12.2655	−0.935237
$173$	22.1478	1.68387	0.841933	−	0.539582i	$-0.181418\pi$
$173$	22.1478	1.68387	0.841933	+	0.539582i	$0.181418\pi$
$174$	−13.1531	−0.997135
$175$	0.832902	0.0629615
$176$	−27.2970	−2.05759
$177$	−0.955769	−0.0718400
$178$	−11.0241	−0.826292
$179$	−20.9040	−1.56244	−0.781219	−	0.624257i	$-0.785402\pi$
$179$	−20.9040	−1.56244	−0.781219	+	0.624257i	$0.785402\pi$
$180$	−18.3044	−1.36433
$181$	26.0738	1.93805	0.969025	−	0.246962i	$-0.0794324\pi$
$181$	26.0738	1.93805	0.969025	+	0.246962i	$0.0794324\pi$
$182$	6.64996	0.492927
$183$	2.01251	0.148769
$184$	32.8253	2.41992
$185$	5.96206	0.438339
$186$	−4.30879	−0.315936
$187$	−2.27003	−0.166001
$188$	−28.4911	−2.07793
$189$	1.95501	0.142206
$190$	31.3314	2.27302
$191$	−8.93558	−0.646556	−0.323278	−	0.946304i	$-0.604785\pi$
$191$	−8.93558	−0.646556	−0.323278	+	0.946304i	$0.604785\pi$
$192$	16.3313	1.17861
$193$	−12.7615	−0.918595	−0.459298	−	0.888282i	$-0.651899\pi$
$193$	−12.7615	−0.918595	−0.459298	+	0.888282i	$0.651899\pi$
$194$	5.01177	0.359824
$195$	−10.1325	−0.725601
$196$	−35.0539	−2.50385
$197$	−13.8771	−0.988704	−0.494352	−	0.869262i	$-0.664595\pi$
$197$	−13.8771	−0.988704	−0.494352	+	0.869262i	$0.664595\pi$
$198$	12.6438	0.898555
$199$	−13.7873	−0.977353	−0.488676	−	0.872465i	$-0.662520\pi$
$199$	−13.7873	−0.977353	−0.488676	+	0.872465i	$0.662520\pi$
$200$	−17.2843	−1.22218
$201$	8.52421	0.601252
$202$	23.6559	1.66442
$203$	2.07312	0.145505
$204$	4.89940	0.343027
$205$	3.29036	0.229809
$206$	33.3992	2.32703
$207$	−8.20719	−0.570439
$208$	−74.4901	−5.16496
$209$	−15.5682	−1.07687
$210$	−1.75591	−0.121170
$211$	20.3654	1.40201	0.701005	−	0.713156i	$-0.252735\pi$
$211$	20.3654	1.40201	0.701005	+	0.713156i	$0.252735\pi$
$212$	−16.1127	−1.10663
$213$	12.9822	0.889523
$214$	−29.2431	−1.99901
$215$	−4.09489	−0.279269
$216$	−40.5701	−2.76045
$217$	0.679128	0.0461022
$218$	42.6746	2.89029
$219$	2.06529	0.139559
$220$	−19.9144	−1.34263
$221$	−6.19463	−0.416696
$222$	8.88852	0.596559
$223$	−3.31822	−0.222204	−0.111102	−	0.993809i	$-0.535438\pi$
$223$	−3.31822	−0.222204	−0.111102	+	0.993809i	$0.535438\pi$
$224$	−6.19701	−0.414055
$225$	4.32152	0.288101
$226$	39.2637	2.61178
$227$	6.34226	0.420951	0.210475	−	0.977599i	$-0.432499\pi$
$227$	6.34226	0.420951	0.210475	+	0.977599i	$0.432499\pi$
$228$	33.6008	2.22527
$229$	22.9989	1.51981	0.759906	−	0.650033i	$-0.225245\pi$
$229$	22.9989	1.51981	0.759906	+	0.650033i	$0.225245\pi$
$230$	17.9700	1.18491
$231$	0.872491	0.0574057
$232$	−43.0212	−2.82448
$233$	7.35635	0.481930	0.240965	−	0.970534i	$-0.422536\pi$
$233$	7.35635	0.481930	0.240965	+	0.970534i	$0.422536\pi$
$234$	34.5034	2.25556
$235$	−9.51187	−0.620486
$236$	−5.12613	−0.333683
$237$	2.20035	0.142928
$238$	−1.07350	−0.0695849
$239$	14.8980	0.963672	0.481836	−	0.876261i	$-0.339970\pi$
$239$	14.8980	0.963672	0.481836	+	0.876261i	$0.339970\pi$
$240$	19.6690	1.26963
$241$	−17.5623	−1.13129	−0.565643	−	0.824651i	$-0.691372\pi$
$241$	−17.5623	−1.13129	−0.565643	+	0.824651i	$0.691372\pi$
$242$	−15.6084	−1.00335
$243$	16.1262	1.03450
$244$	10.7938	0.691003
$245$	−11.7029	−0.747671
$246$	4.90543	0.312759
$247$	−42.4837	−2.70317
$248$	−14.0932	−0.894918
$249$	2.02955	0.128617
$250$	−32.3047	−2.04313
$251$	6.93812	0.437930	0.218965	−	0.975733i	$-0.429732\pi$
$251$	6.93812	0.437930	0.218965	+	0.975733i	$0.429732\pi$
$252$	4.30116	0.270948
$253$	−8.92907	−0.561366
$254$	−31.4331	−1.97229
$255$	1.63568	0.102431
$256$	5.31656	0.332285
$257$	22.4829	1.40245	0.701223	−	0.712942i	$-0.252638\pi$
$257$	22.4829	1.40245	0.701223	+	0.712942i	$0.252638\pi$
$258$	−6.10486	−0.380072
$259$	−1.40096	−0.0870515
$260$	−54.3440	−3.37027
$261$	10.7564	0.665806
$262$	−38.0879	−2.35308
$263$	−10.9030	−0.672307	−0.336154	−	0.941807i	$-0.609126\pi$
$263$	−10.9030	−0.672307	−0.336154	+	0.941807i	$0.609126\pi$
$264$	−18.1058	−1.11434
$265$	−5.37931	−0.330448
$266$	−7.36225	−0.451408
$267$	−3.94702	−0.241554
$268$	45.7184	2.79270
$269$	20.4457	1.24660	0.623299	−	0.781984i	$-0.285792\pi$
$269$	20.4457	1.24660	0.623299	+	0.781984i	$0.285792\pi$
$270$	−22.2099	−1.35165
$271$	−31.9921	−1.94338	−0.971691	−	0.236254i	$-0.924080\pi$
$271$	−31.9921	−1.94338	−0.971691	+	0.236254i	$0.924080\pi$
$272$	12.0250	0.729120
$273$	2.38092	0.144100
$274$	7.85728	0.474676
$275$	4.70163	0.283519
$276$	19.2716	1.16001
$277$	5.47368	0.328881	0.164441	−	0.986387i	$-0.447418\pi$
$277$	5.47368	0.328881	0.164441	+	0.986387i	$0.447418\pi$
$278$	9.48460	0.568849
$279$	3.52366	0.210956
$280$	−5.74324	−0.343224
$281$	0.277630	0.0165620	0.00828102	−	0.999966i	$-0.497364\pi$
$281$	0.277630	0.0165620	0.00828102	+	0.999966i	$0.497364\pi$
$282$	−14.1808	−0.844452
$283$	15.4609	0.919054	0.459527	−	0.888164i	$-0.348019\pi$
$283$	15.4609	0.919054	0.459527	+	0.888164i	$0.348019\pi$
$284$	69.6279	4.13166
$285$	11.2178	0.664483
$286$	37.5382	2.21968
$287$	−0.773168	−0.0456386
$288$	−32.1532	−1.89465
$289$	1.00000	0.0588235
$290$	−23.5517	−1.38300
$291$	1.79439	0.105189
$292$	11.0769	0.648226
$293$	13.6009	0.794573	0.397287	−	0.917695i	$-0.369952\pi$
$293$	13.6009	0.794573	0.397287	+	0.917695i	$0.369952\pi$
$294$	−17.4473	−1.01754
$295$	−1.71138	−0.0996404
$296$	29.0726	1.68981
$297$	11.0358	0.640362
$298$	−40.3715	−2.33866
$299$	−24.3663	−1.40914
$300$	−10.1475	−0.585868
$301$	0.962216	0.0554612
$302$	−52.2227	−3.00508
$303$	8.46965	0.486569
$304$	82.4689	4.72992
$305$	3.60356	0.206339
$306$	−5.56989	−0.318409
$307$	11.7486	0.670529	0.335264	−	0.942124i	$-0.391174\pi$
$307$	11.7486	0.670529	0.335264	+	0.942124i	$0.391174\pi$
$308$	4.67948	0.266638
$309$	11.9581	0.680272
$310$	−7.71522	−0.438195
$311$	35.1763	1.99466	0.997332	−	0.0729992i	$-0.0232570\pi$
$311$	35.1763	1.99466	0.997332	+	0.0729992i	$0.0232570\pi$
$312$	−49.4086	−2.79721
$313$	−14.9915	−0.847371	−0.423686	−	0.905809i	$-0.639264\pi$
$313$	−14.9915	−0.847371	−0.423686	+	0.905809i	$0.639264\pi$
$314$	−2.01842	−0.113906
$315$	1.43596	0.0809072
$316$	11.8012	0.663871
$317$	−25.1570	−1.41296	−0.706479	−	0.707734i	$-0.749717\pi$
$317$	−25.1570	−1.41296	−0.706479	+	0.707734i	$0.749717\pi$
$318$	−8.01973	−0.449724
$319$	11.7025	0.655216
$320$	29.2425	1.63470
$321$	−10.4701	−0.584381
$322$	−4.22259	−0.235316
$323$	6.85815	0.381598
$324$	8.26854	0.459363
$325$	12.8302	0.711690
$326$	−18.5135	−1.02537
$327$	15.2790	0.844933
$328$	16.0447	0.885919
$329$	2.23510	0.123225
$330$	−9.91192	−0.545633
$331$	−8.29217	−0.455779	−0.227890	−	0.973687i	$-0.573183\pi$
$331$	−8.29217	−0.455779	−0.227890	+	0.973687i	$0.573183\pi$
$332$	10.8852	0.597402
$333$	−7.26891	−0.398334
$334$	25.4957	1.39506
$335$	15.2633	0.833922
$336$	−4.62182	−0.252141
$337$	21.3030	1.16045	0.580225	−	0.814456i	$-0.302965\pi$
$337$	21.3030	1.16045	0.580225	+	0.814456i	$0.302965\pi$
$338$	67.7339	3.68424
$339$	14.0578	0.763515
$340$	8.77276	0.475770
$341$	3.83359	0.207601
$342$	−38.1991	−2.06557
$343$	5.56492	0.300477
$344$	−19.9678	−1.07659
$345$	6.43391	0.346390
$346$	59.1231	3.17848
$347$	24.0512	1.29114	0.645568	−	0.763702i	$-0.276621\pi$
$347$	24.0512	1.29114	0.645568	+	0.763702i	$0.276621\pi$
$348$	−25.2576	−1.35395
$349$	−7.63163	−0.408512	−0.204256	−	0.978918i	$-0.565477\pi$
$349$	−7.63163	−0.408512	−0.204256	+	0.978918i	$0.565477\pi$
$350$	2.22342	0.118847
$351$	30.1153	1.60744
$352$	−34.9813	−1.86451
$353$	−33.0235	−1.75767	−0.878833	−	0.477129i	$-0.841677\pi$
$353$	−33.0235	−1.75767	−0.878833	+	0.477129i	$0.841677\pi$
$354$	−2.55141	−0.135606
$355$	23.2456	1.23375
$356$	−21.1693	−1.12197
$357$	−0.384352	−0.0203421
$358$	−55.8029	−2.94927
$359$	−30.7407	−1.62243	−0.811215	−	0.584748i	$-0.801193\pi$
$359$	−30.7407	−1.62243	−0.811215	+	0.584748i	$0.801193\pi$
$360$	−29.7989	−1.57054
$361$	28.0342	1.47549
$362$	69.6035	3.65828
$363$	−5.58836	−0.293313
$364$	12.7697	0.669315
$365$	3.69807	0.193566
$366$	5.37236	0.280818
$367$	14.0604	0.733946	0.366973	−	0.930232i	$-0.380394\pi$
$367$	14.0604	0.733946	0.366973	+	0.930232i	$0.380394\pi$
$368$	47.2997	2.46567
$369$	−4.01159	−0.208835
$370$	15.9156	0.827413
$371$	1.26403	0.0656250
$372$	−8.27404	−0.428989
$373$	−33.7998	−1.75009	−0.875044	−	0.484043i	$-0.839168\pi$
$373$	−33.7998	−1.75009	−0.875044	+	0.484043i	$0.839168\pi$
$374$	−6.05980	−0.313345
$375$	−11.5662	−0.597277
$376$	−46.3824	−2.39199
$377$	31.9348	1.64472
$378$	5.21886	0.268429
$379$	−33.1973	−1.70523	−0.852616	−	0.522538i	$-0.824985\pi$
$379$	−33.1973	−1.70523	−0.852616	+	0.522538i	$0.824985\pi$
$380$	60.1649	3.08639
$381$	−11.2542	−0.576568
$382$	−23.8534	−1.22044
$383$	16.0360	0.819401	0.409701	−	0.912220i	$-0.365633\pi$
$383$	16.0360	0.819401	0.409701	+	0.912220i	$0.365633\pi$
$384$	14.1391	0.721531
$385$	1.56226	0.0796203
$386$	−34.0667	−1.73395
$387$	4.99247	0.253782
$388$	9.62397	0.488583
$389$	−5.77352	−0.292729	−0.146365	−	0.989231i	$-0.546757\pi$
$389$	−5.77352	−0.292729	−0.146365	+	0.989231i	$0.546757\pi$
$390$	−27.0484	−1.36965
$391$	3.93346	0.198924
$392$	−57.0665	−2.88229
$393$	−13.6368	−0.687887
$394$	−37.0447	−1.86629
$395$	3.93989	0.198237
$396$	24.2795	1.22009
$397$	−28.2912	−1.41989	−0.709946	−	0.704256i	$-0.751281\pi$
$397$	−28.2912	−1.41989	−0.709946	+	0.704256i	$0.751281\pi$
$398$	−36.8048	−1.84486
$399$	−2.63595	−0.131962
$400$	−24.9058	−1.24529
$401$	32.9444	1.64516	0.822581	−	0.568648i	$-0.192533\pi$
$401$	32.9444	1.64516	0.822581	+	0.568648i	$0.192533\pi$
$402$	22.7552	1.13493
$403$	10.4614	0.521120
$404$	45.4257	2.26001
$405$	2.76049	0.137170
$406$	5.53416	0.274656
$407$	−7.90826	−0.391998
$408$	7.97604	0.394873
$409$	13.0670	0.646121	0.323061	−	0.946378i	$-0.395288\pi$
$409$	13.0670	0.646121	0.323061	+	0.946378i	$0.395288\pi$
$410$	8.78356	0.433789
$411$	2.81319	0.138764
$412$	64.1355	3.15973
$413$	0.402139	0.0197880
$414$	−21.9089	−1.07677
$415$	3.63406	0.178389
$416$	−95.4598	−4.68030
$417$	3.39582	0.166294
$418$	−41.5590	−2.03272
$419$	−30.1050	−1.47073	−0.735363	−	0.677673i	$-0.762988\pi$
$419$	−30.1050	−1.47073	−0.735363	+	0.677673i	$0.762988\pi$
$420$	−3.37183	−0.164529
$421$	30.0148	1.46283	0.731415	−	0.681933i	$-0.238860\pi$
$421$	30.0148	1.46283	0.731415	+	0.681933i	$0.238860\pi$
$422$	54.3650	2.64645
$423$	11.5968	0.563857
$424$	−26.2309	−1.27389
$425$	−2.07118	−0.100467
$426$	34.6556	1.67907
$427$	−0.846762	−0.0409777
$428$	−56.1546	−2.71434
$429$	13.4400	0.648890
$430$	−10.9312	−0.527151
$431$	14.9526	0.720242	0.360121	−	0.932906i	$-0.382735\pi$
$431$	14.9526	0.720242	0.360121	+	0.932906i	$0.382735\pi$
$432$	−58.4596	−2.81264
$433$	−2.62240	−0.126025	−0.0630123	−	0.998013i	$-0.520071\pi$
$433$	−2.62240	−0.126025	−0.0630123	+	0.998013i	$0.520071\pi$
$434$	1.81292	0.0870229
$435$	−8.43234	−0.404300
$436$	81.9469	3.92455
$437$	26.9763	1.29045
$438$	5.51326	0.263434
$439$	7.63571	0.364432	0.182216	−	0.983258i	$-0.441673\pi$
$439$	7.63571	0.364432	0.182216	+	0.983258i	$0.441673\pi$
$440$	−32.4199	−1.54556
$441$	14.2681	0.679434
$442$	−16.5364	−0.786559
$443$	−14.7048	−0.698645	−0.349322	−	0.937003i	$-0.613588\pi$
$443$	−14.7048	−0.698645	−0.349322	+	0.937003i	$0.613588\pi$
$444$	17.0684	0.810030
$445$	−7.06745	−0.335029
$446$	−8.85792	−0.419434
$447$	−14.4544	−0.683671
$448$	−6.87138	−0.324642
$449$	−31.1956	−1.47221	−0.736106	−	0.676867i	$-0.763337\pi$
$449$	−31.1956	−1.47221	−0.736106	+	0.676867i	$0.763337\pi$
$450$	11.5362	0.543823
$451$	−4.36444	−0.205513
$452$	75.3970	3.54638
$453$	−18.6976	−0.878489
$454$	16.9306	0.794590
$455$	4.26323	0.199863
$456$	54.7009	2.56160
$457$	33.2818	1.55686	0.778428	−	0.627734i	$-0.216018\pi$
$457$	33.2818	1.55686	0.778428	+	0.627734i	$0.216018\pi$
$458$	61.3952	2.86881
$459$	−4.86153	−0.226917
$460$	34.5073	1.60891
$461$	−6.19607	−0.288580	−0.144290	−	0.989535i	$-0.546090\pi$
$461$	−6.19607	−0.288580	−0.144290	+	0.989535i	$0.546090\pi$
$462$	2.32910	0.108359
$463$	−17.1281	−0.796010	−0.398005	−	0.917383i	$-0.630297\pi$
$463$	−17.1281	−0.796010	−0.398005	+	0.917383i	$0.630297\pi$
$464$	−61.9915	−2.87788
$465$	−2.76232	−0.128100
$466$	19.6376	0.909696
$467$	−13.2940	−0.615174	−0.307587	−	0.951520i	$-0.599522\pi$
$467$	−13.2940	−0.615174	−0.307587	+	0.951520i	$0.599522\pi$
$468$	66.2559	3.06268
$469$	−3.58656	−0.165612
$470$	−25.3918	−1.17123
$471$	−0.722668	−0.0332988
$472$	−8.34515	−0.384117
$473$	5.43159	0.249745
$474$	5.87378	0.269792
$475$	−14.2044	−0.651745
$476$	−2.06142	−0.0944850
$477$	6.55842	0.300290
$478$	39.7700	1.81904
$479$	11.3581	0.518964	0.259482	−	0.965748i	$-0.416448\pi$
$479$	11.3581	0.518964	0.259482	+	0.965748i	$0.416448\pi$
$480$	25.2061	1.15049
$481$	−21.5807	−0.983994
$482$	−46.8821	−2.13542
$483$	−1.51184	−0.0687909
$484$	−29.9724	−1.36238
$485$	3.21300	0.145895
$486$	43.0487	1.95273
$487$	−26.4855	−1.20017	−0.600085	−	0.799936i	$-0.704867\pi$
$487$	−26.4855	−1.20017	−0.600085	+	0.799936i	$0.704867\pi$
$488$	17.5719	0.795443
$489$	−6.62851	−0.299751
$490$	−31.2407	−1.41131
$491$	26.4299	1.19276	0.596382	−	0.802701i	$-0.296604\pi$
$491$	26.4299	1.19276	0.596382	+	0.802701i	$0.296604\pi$
$492$	9.41976	0.424676
$493$	−5.15524	−0.232180
$494$	−113.409	−5.10253
$495$	8.10582	0.364330
$496$	−20.3076	−0.911838
$497$	−5.46223	−0.245015
$498$	5.41784	0.242779
$499$	4.50411	0.201632	0.100816	−	0.994905i	$-0.467855\pi$
$499$	4.50411	0.201632	0.100816	+	0.994905i	$0.467855\pi$
$500$	−62.0337	−2.77423
$501$	9.12838	0.407826
$502$	18.5212	0.826640
$503$	−29.3183	−1.30724	−0.653620	−	0.756823i	$-0.726750\pi$
$503$	−29.3183	−1.30724	−0.653620	+	0.756823i	$0.726750\pi$
$504$	7.00213	0.311900
$505$	15.1656	0.674859
$506$	−23.8360	−1.05964
$507$	24.2511	1.07703
$508$	−60.3600	−2.67804
$509$	−10.4293	−0.462269	−0.231135	−	0.972922i	$-0.574244\pi$
$509$	−10.4293	−0.462269	−0.231135	+	0.972922i	$0.574244\pi$
$510$	4.36643	0.193349
$511$	−0.868970	−0.0384410
$512$	−15.3943	−0.680339
$513$	−33.3411	−1.47204
$514$	60.0178	2.64727
$515$	21.4119	0.943521
$516$	−11.7230	−0.516076
$517$	12.6168	0.554888
$518$	−3.73984	−0.164319
$519$	21.1682	0.929180
$520$	−88.4700	−3.87966
$521$	−35.7866	−1.56784	−0.783919	−	0.620863i	$-0.786782\pi$
$521$	−35.7866	−1.56784	−0.783919	+	0.620863i	$0.786782\pi$
$522$	28.7141	1.25678
$523$	−1.06849	−0.0467220	−0.0233610	−	0.999727i	$-0.507437\pi$
$523$	−1.06849	−0.0467220	−0.0233610	+	0.999727i	$0.507437\pi$
$524$	−73.1391	−3.19510
$525$	0.796062	0.0347430
$526$	−29.1053	−1.26905
$527$	−1.68879	−0.0735647
$528$	−26.0896	−1.13540
$529$	−7.52787	−0.327299
$530$	−14.3600	−0.623757
$531$	2.08651	0.0905466
$532$	−14.1375	−0.612939
$533$	−11.9100	−0.515880
$534$	−10.5365	−0.455959
$535$	−18.7475	−0.810523
$536$	74.4278	3.21479
$537$	−19.9794	−0.862175
$538$	54.5795	2.35309
$539$	15.5231	0.668627
$540$	−42.6490	−1.83532
$541$	6.89581	0.296474	0.148237	−	0.988952i	$-0.452640\pi$
$541$	6.89581	0.296474	0.148237	+	0.988952i	$0.452640\pi$
$542$	−85.4024	−3.66835
$543$	24.9205	1.06944
$544$	15.4101	0.660702
$545$	27.3583	1.17190
$546$	6.35582	0.272004
$547$	−23.1277	−0.988869	−0.494435	−	0.869215i	$-0.664625\pi$
$547$	−23.1277	−0.988869	−0.494435	+	0.869215i	$0.664625\pi$
$548$	15.0881	0.644532
$549$	−4.39344	−0.187507
$550$	12.5509	0.535173
$551$	−35.3554	−1.50619
$552$	31.3734	1.33534
$553$	−0.925794	−0.0393688
$554$	14.6119	0.620799
$555$	5.69835	0.241882
$556$	18.2130	0.772404
$557$	2.34880	0.0995219	0.0497610	−	0.998761i	$-0.484154\pi$
$557$	2.34880	0.0995219	0.0497610	+	0.998761i	$0.484154\pi$
$558$	9.40635	0.398203
$559$	14.8222	0.626910
$560$	−8.27573	−0.349714
$561$	−2.16962	−0.0916016
$562$	0.741129	0.0312627
$563$	33.9226	1.42967	0.714833	−	0.699295i	$-0.246502\pi$
$563$	33.9226	1.42967	0.714833	+	0.699295i	$0.246502\pi$
$564$	−27.2309	−1.14663
$565$	25.1716	1.05898
$566$	41.2725	1.73481
$567$	−0.648658	−0.0272411
$568$	113.352	4.75613
$569$	−43.0900	−1.80643	−0.903214	−	0.429191i	$-0.858799\pi$
$569$	−43.0900	−1.80643	−0.903214	+	0.429191i	$0.858799\pi$
$570$	29.9456	1.25428
$571$	−30.4983	−1.27631	−0.638156	−	0.769907i	$-0.720303\pi$
$571$	−30.4983	−1.27631	−0.638156	+	0.769907i	$0.720303\pi$
$572$	72.0835	3.01396
$573$	−8.54035	−0.356778
$574$	−2.06396	−0.0861479
$575$	−8.14690	−0.339749
$576$	−35.6522	−1.48551
$577$	21.9348	0.913156	0.456578	−	0.889683i	$-0.349075\pi$
$577$	21.9348	0.913156	0.456578	+	0.889683i	$0.349075\pi$
$578$	2.66948	0.111036
$579$	−12.1971	−0.506893
$580$	−45.2257	−1.87789
$581$	−0.853930	−0.0354270
$582$	4.79010	0.198556
$583$	7.13528	0.295513
$584$	18.0328	0.746201
$585$	22.1198	0.914541
$586$	36.3074	1.49984
$587$	−2.32011	−0.0957610	−0.0478805	−	0.998853i	$-0.515247\pi$
$587$	−2.32011	−0.0957610	−0.0478805	+	0.998853i	$0.515247\pi$
$588$	−33.5035	−1.38166
$589$	−11.5820	−0.477226
$590$	−4.56850	−0.188082
$591$	−13.2633	−0.545581
$592$	41.8922	1.72176
$593$	11.7607	0.482955	0.241478	−	0.970406i	$-0.422368\pi$
$593$	11.7607	0.482955	0.241478	+	0.970406i	$0.422368\pi$
$594$	29.4599	1.20875
$595$	−0.688213	−0.0282140
$596$	−77.5242	−3.17552
$597$	−13.1774	−0.539317
$598$	−65.0455	−2.65991
$599$	31.0951	1.27051	0.635256	−	0.772302i	$-0.280895\pi$
$599$	31.0951	1.27051	0.635256	+	0.772302i	$0.280895\pi$
$600$	−16.5198	−0.674417
$601$	14.6399	0.597176	0.298588	−	0.954382i	$-0.403484\pi$
$601$	14.6399	0.597176	0.298588	+	0.954382i	$0.403484\pi$
$602$	2.56862	0.104689
$603$	−18.6089	−0.757813
$604$	−100.282	−4.08041
$605$	−10.0064	−0.406818
$606$	22.6096	0.918451
$607$	34.7210	1.40928	0.704641	−	0.709564i	$-0.251108\pi$
$607$	34.7210	1.40928	0.704641	+	0.709564i	$0.251108\pi$
$608$	105.685	4.28608
$609$	1.98143	0.0802915
$610$	9.61963	0.389487
$611$	34.4298	1.39288
$612$	−10.6957	−0.432348
$613$	−11.4751	−0.463475	−0.231738	−	0.972778i	$-0.574441\pi$
$613$	−11.4751	−0.463475	−0.231738	+	0.972778i	$0.574441\pi$
$614$	31.3627	1.26570
$615$	3.14483	0.126812
$616$	7.61801	0.306939
$617$	39.1772	1.57722	0.788608	−	0.614896i	$-0.210802\pi$
$617$	39.1772	1.57722	0.788608	+	0.614896i	$0.210802\pi$
$618$	31.9219	1.28409
$619$	−25.3103	−1.01731	−0.508654	−	0.860971i	$-0.669857\pi$
$619$	−25.3103	−1.01731	−0.508654	+	0.860971i	$0.669857\pi$
$620$	−14.8153	−0.594998
$621$	−19.1226	−0.767365
$622$	93.9024	3.76514
$623$	1.66071	0.0665348
$624$	−71.1954	−2.85010
$625$	−10.3543	−0.414173
$626$	−40.0196	−1.59950
$627$	−14.8796	−0.594234
$628$	−3.87592	−0.154666
$629$	3.48377	0.138907
$630$	3.83327	0.152721
$631$	−24.5231	−0.976250	−0.488125	−	0.872774i	$-0.662319\pi$
$631$	−24.5231	−0.976250	−0.488125	+	0.872774i	$0.662319\pi$
$632$	19.2120	0.764211
$633$	19.4646	0.773649
$634$	−67.1561	−2.66711
$635$	−20.1515	−0.799686
$636$	−15.4001	−0.610653
$637$	42.3606	1.67839
$638$	31.2397	1.23679
$639$	−28.3409	−1.12115
$640$	25.3171	1.00075
$641$	−11.5726	−0.457089	−0.228545	−	0.973533i	$-0.573397\pi$
$641$	−11.5726	−0.457089	−0.228545	+	0.973533i	$0.573397\pi$
$642$	−27.9496	−1.10308
$643$	−33.4677	−1.31984	−0.659919	−	0.751337i	$-0.729409\pi$
$643$	−33.4677	−1.31984	−0.659919	+	0.751337i	$0.729409\pi$
$644$	−8.10852	−0.319520
$645$	−3.91377	−0.154105
$646$	18.3077	0.720307
$647$	29.1738	1.14694	0.573470	−	0.819227i	$-0.305597\pi$
$647$	29.1738	1.14694	0.573470	+	0.819227i	$0.305597\pi$
$648$	13.4609	0.528793
$649$	2.27003	0.0891064
$650$	34.2499	1.34339
$651$	0.649089	0.0254398
$652$	−35.5510	−1.39229
$653$	−9.05572	−0.354378	−0.177189	−	0.984177i	$-0.556700\pi$
$653$	−9.05572	−0.354378	−0.177189	+	0.984177i	$0.556700\pi$
$654$	40.7871	1.59490
$655$	−24.4178	−0.954082
$656$	23.1196	0.902669
$657$	−4.50866	−0.175900
$658$	5.96655	0.232600
$659$	−21.1753	−0.824872	−0.412436	−	0.910987i	$-0.635322\pi$
$659$	−21.1753	−0.824872	−0.412436	+	0.910987i	$0.635322\pi$
$660$	−19.0336	−0.740881
$661$	1.91385	0.0744401	0.0372201	−	0.999307i	$-0.488150\pi$
$661$	1.91385	0.0744401	0.0372201	+	0.999307i	$0.488150\pi$
$662$	−22.1358	−0.860332
$663$	−5.92064	−0.229938
$664$	17.7207	0.687695
$665$	−4.71987	−0.183029
$666$	−19.4042	−0.751898
$667$	−20.2779	−0.785165
$668$	48.9587	1.89427
$669$	−3.17145	−0.122615
$670$	40.7450	1.57412
$671$	−4.77987	−0.184525
$672$	−5.92291	−0.228481
$673$	27.1207	1.04543	0.522713	−	0.852508i	$-0.324920\pi$
$673$	27.1207	1.04543	0.522713	+	0.852508i	$0.324920\pi$
$674$	56.8681	2.19048
$675$	10.0691	0.387559
$676$	130.068	5.00260
$677$	4.17307	0.160384	0.0801921	−	0.996779i	$-0.474447\pi$
$677$	4.17307	0.160384	0.0801921	+	0.996779i	$0.474447\pi$
$678$	37.5270	1.44122
$679$	−0.754990	−0.0289738
$680$	14.2817	0.547679
$681$	6.06174	0.232286
$682$	10.2337	0.391869
$683$	0.714482	0.0273389	0.0136694	−	0.999907i	$-0.495649\pi$
$683$	0.714482	0.0273389	0.0136694	+	0.999907i	$0.495649\pi$
$684$	−73.3527	−2.80471
$685$	5.03723	0.192463
$686$	14.8554	0.567184
$687$	21.9817	0.838653
$688$	−28.7726	−1.09695
$689$	19.4713	0.741798
$690$	17.1752	0.653848
$691$	23.8345	0.906709	0.453354	−	0.891330i	$-0.350227\pi$
$691$	23.8345	0.906709	0.453354	+	0.891330i	$0.350227\pi$
$692$	113.533	4.31586
$693$	−1.90470	−0.0723537
$694$	64.2043	2.43716
$695$	6.08049	0.230646
$696$	−41.1184	−1.55859
$697$	1.92264	0.0728250
$698$	−20.3725	−0.771110
$699$	7.03097	0.265936
$700$	4.26956	0.161374
$701$	35.8760	1.35502	0.677508	−	0.735515i	$-0.263060\pi$
$701$	35.8760	1.35502	0.677508	+	0.735515i	$0.263060\pi$
$702$	80.3924	3.03421
$703$	23.8922	0.901112
$704$	−38.7881	−1.46188
$705$	−9.09116	−0.342393
$706$	−88.1557	−3.31779
$707$	−3.56360	−0.134023
$708$	−4.89940	−0.184131
$709$	2.04948	0.0769697	0.0384848	−	0.999259i	$-0.487747\pi$
$709$	2.04948	0.0769697	0.0384848	+	0.999259i	$0.487747\pi$
$710$	62.0536	2.32883
$711$	−4.80349	−0.180145
$712$	−34.4628	−1.29155
$713$	−6.64278	−0.248774
$714$	−1.02602	−0.0383979
$715$	24.0654	0.899995
$716$	−107.157	−4.00463
$717$	14.2391	0.531768
$718$	−82.0616	−3.06251
$719$	6.97472	0.260113	0.130057	−	0.991507i	$-0.458484\pi$
$719$	6.97472	0.260113	0.130057	+	0.991507i	$0.458484\pi$
$720$	−42.9388	−1.60023
$721$	−5.03136	−0.187378
$722$	74.8368	2.78514
$723$	−16.7855	−0.624259
$724$	133.658	4.96735
$725$	10.6774	0.396549
$726$	−14.9180	−0.553660
$727$	7.10387	0.263468	0.131734	−	0.991285i	$-0.457946\pi$
$727$	7.10387	0.263468	0.131734	+	0.991285i	$0.457946\pi$
$728$	20.7886	0.770478
$729$	10.5739	0.391627
$730$	9.87192	0.365376
$731$	−2.39274	−0.0884988
$732$	10.3164	0.381305
$733$	−20.5811	−0.760180	−0.380090	−	0.924950i	$-0.624107\pi$
$733$	−20.5811	−0.760180	−0.380090	+	0.924950i	$0.624107\pi$
$734$	37.5339	1.38540
$735$	−11.1853	−0.412575
$736$	60.6151	2.23430
$737$	−20.2457	−0.745759
$738$	−10.7089	−0.394199
$739$	−18.8562	−0.693637	−0.346818	−	0.937932i	$-0.612738\pi$
$739$	−18.8562	−0.693637	−0.346818	+	0.937932i	$0.612738\pi$
$740$	30.5623	1.12349
$741$	−40.6046	−1.49165
$742$	3.37430	0.123874
$743$	−19.2037	−0.704517	−0.352258	−	0.935903i	$-0.614586\pi$
$743$	−19.2037	−0.704517	−0.352258	+	0.935903i	$0.614586\pi$
$744$	−13.4698	−0.493828
$745$	−25.8818	−0.948235
$746$	−90.2280	−3.30348
$747$	−4.43063	−0.162108
$748$	−11.6365	−0.425471
$749$	4.40527	0.160965
$750$	−30.8758	−1.12742
$751$	17.3536	0.633240	0.316620	−	0.948552i	$-0.397452\pi$
$751$	17.3536	0.633240	0.316620	+	0.948552i	$0.397452\pi$
$752$	−66.8348	−2.43722
$753$	6.63124	0.241656
$754$	85.2493	3.10460
$755$	−33.4795	−1.21844
$756$	10.0216	0.364484
$757$	44.0655	1.60159	0.800794	−	0.598940i	$-0.204411\pi$
$757$	44.0655	1.60159	0.800794	+	0.598940i	$0.204411\pi$
$758$	−88.6196	−3.21881
$759$	−8.53413	−0.309769
$760$	97.9462	3.55288
$761$	−13.1307	−0.475987	−0.237994	−	0.971267i	$-0.576490\pi$
$761$	−13.1307	−0.475987	−0.237994	+	0.971267i	$0.576490\pi$
$762$	−30.0428	−1.08833
$763$	−6.42865	−0.232733
$764$	−45.8050	−1.65717
$765$	−3.57080	−0.129103
$766$	42.8078	1.54671
$767$	6.19463	0.223675
$768$	5.08140	0.183359
$769$	16.1911	0.583867	0.291934	−	0.956439i	$-0.405701\pi$
$769$	16.1911	0.583867	0.291934	+	0.956439i	$0.405701\pi$
$770$	4.17043	0.150292
$771$	21.4885	0.773889
$772$	−65.4173	−2.35442
$773$	30.6654	1.10296	0.551479	−	0.834189i	$-0.314064\pi$
$773$	30.6654	1.10296	0.551479	+	0.834189i	$0.314064\pi$
$774$	13.3273	0.479040
$775$	3.49778	0.125644
$776$	15.6675	0.562429
$777$	−1.33900	−0.0480362
$778$	−15.4123	−0.552558
$779$	13.1857	0.472428
$780$	−51.9403	−1.85976
$781$	−30.8337	−1.10331
$782$	10.5003	0.375490
$783$	25.0623	0.895654
$784$	−82.2301	−2.93679
$785$	−1.29399	−0.0461846
$786$	−36.4032	−1.29846
$787$	−31.5733	−1.12547	−0.562733	−	0.826639i	$-0.690250\pi$
$787$	−31.5733	−1.12547	−0.562733	+	0.826639i	$0.690250\pi$
$788$	−71.1360	−2.53411
$789$	−10.4207	−0.370988
$790$	10.5175	0.374195
$791$	−5.91481	−0.210306
$792$	39.5262	1.40450
$793$	−13.0437	−0.463195
$794$	−75.5227	−2.68020
$795$	−5.14138	−0.182346
$796$	−70.6753	−2.50502
$797$	44.5086	1.57658	0.788288	−	0.615306i	$-0.210968\pi$
$797$	44.5086	1.57658	0.788288	+	0.615306i	$0.210968\pi$
$798$	−7.03661	−0.249093
$799$	−5.55801	−0.196628
$800$	−31.9170	−1.12844
$801$	8.61660	0.304453
$802$	87.9443	3.10542
$803$	−4.90523	−0.173102
$804$	43.6962	1.54105
$805$	−2.70706	−0.0954114
$806$	27.9265	0.983670
$807$	19.5414	0.687890
$808$	73.9514	2.60160
$809$	−14.5108	−0.510173	−0.255087	−	0.966918i	$-0.582104\pi$
$809$	−14.5108	−0.510173	−0.255087	+	0.966918i	$0.582104\pi$
$810$	7.36907	0.258923
$811$	−25.5717	−0.897945	−0.448972	−	0.893546i	$-0.648210\pi$
$811$	−25.5717	−0.897945	−0.448972	+	0.893546i	$0.648210\pi$
$812$	10.6271	0.372938
$813$	−30.5771	−1.07239
$814$	−21.1109	−0.739938
$815$	−11.8689	−0.415748
$816$	11.4931	0.402338
$817$	−16.4098	−0.574106
$818$	34.8821	1.21962
$819$	−5.19770	−0.181622
$820$	16.8668	0.589015
$821$	20.7209	0.723164	0.361582	−	0.932340i	$-0.382237\pi$
$821$	20.7209	0.723164	0.361582	+	0.932340i	$0.382237\pi$
$822$	7.50975	0.261933
$823$	−7.40677	−0.258184	−0.129092	−	0.991633i	$-0.541206\pi$
$823$	−7.40677	−0.258184	−0.129092	+	0.991633i	$0.541206\pi$
$824$	104.410	3.63730
$825$	4.49367	0.156450
$826$	1.07350	0.0373520
$827$	−12.0512	−0.419060	−0.209530	−	0.977802i	$-0.567193\pi$
$827$	−12.0512	−0.419060	−0.209530	+	0.977802i	$0.567193\pi$
$828$	−42.0711	−1.46207
$829$	−8.71266	−0.302603	−0.151302	−	0.988488i	$-0.548346\pi$
$829$	−8.71266	−0.302603	−0.151302	+	0.988488i	$0.548346\pi$
$830$	9.70106	0.336729
$831$	5.23157	0.181481
$832$	−105.848	−3.66962
$833$	−6.83828	−0.236932
$834$	9.06509	0.313898
$835$	16.3451	0.565645
$836$	−79.8046	−2.76010
$837$	8.21008	0.283782
$838$	−80.3648	−2.77615
$839$	36.1802	1.24908	0.624539	−	0.780994i	$-0.285287\pi$
$839$	36.1802	1.24908	0.624539	+	0.780994i	$0.285287\pi$
$840$	−5.48921	−0.189396
$841$	−2.42353	−0.0835700
$842$	80.1238	2.76125
$843$	0.265351	0.00913916
$844$	104.396	3.59344
$845$	43.4236	1.49382
$846$	30.9575	1.06434
$847$	2.35130	0.0807916
$848$	−37.7975	−1.29797
$849$	14.7770	0.507146
$850$	−5.52897	−0.189642
$851$	13.7033	0.469743
$852$	66.5482	2.27991
$853$	−44.5064	−1.52387	−0.761935	−	0.647654i	$-0.775750\pi$
$853$	−44.5064	−1.52387	−0.761935	+	0.647654i	$0.775750\pi$
$854$	−2.26042	−0.0773498
$855$	−24.4891	−0.837510
$856$	−91.4176	−3.12459
$857$	−25.9690	−0.887084	−0.443542	−	0.896254i	$-0.646278\pi$
$857$	−25.9690	−0.887084	−0.443542	+	0.896254i	$0.646278\pi$
$858$	35.8779	1.22485
$859$	4.45931	0.152150	0.0760749	−	0.997102i	$-0.475761\pi$
$859$	4.45931	0.152150	0.0760749	+	0.997102i	$0.475761\pi$
$860$	−20.9910	−0.715786
$861$	−0.738970	−0.0251840
$862$	39.9157	1.35953
$863$	33.9315	1.15504	0.577521	−	0.816376i	$-0.304020\pi$
$863$	33.9315	1.15504	0.577521	+	0.816376i	$0.304020\pi$
$864$	−74.9166	−2.54871
$865$	37.9033	1.28875
$866$	−7.00046	−0.237885
$867$	0.955769	0.0324596
$868$	3.48130	0.118163
$869$	−5.22599	−0.177280
$870$	−22.5100	−0.763160
$871$	−55.2480	−1.87201
$872$	133.406	4.51771
$873$	−3.91727	−0.132580
$874$	72.0127	2.43587
$875$	4.86648	0.164517
$876$	10.5870	0.357700
$877$	−36.3182	−1.22638	−0.613189	−	0.789936i	$-0.710114\pi$
$877$	−36.3182	−1.22638	−0.613189	+	0.789936i	$0.710114\pi$
$878$	20.3834	0.687906
$879$	12.9993	0.438456
$880$	−46.7155	−1.57478
$881$	−24.4924	−0.825170	−0.412585	−	0.910919i	$-0.635374\pi$
$881$	−24.4924	−0.825170	−0.412585	+	0.910919i	$0.635374\pi$
$882$	38.0885	1.28251
$883$	−38.0395	−1.28013	−0.640065	−	0.768321i	$-0.721093\pi$
$883$	−38.0395	−1.28013	−0.640065	+	0.768321i	$0.721093\pi$
$884$	−31.7545	−1.06802
$885$	−1.63568	−0.0549829
$886$	−39.2541	−1.31877
$887$	36.0917	1.21184	0.605921	−	0.795525i	$-0.292805\pi$
$887$	36.0917	1.21184	0.605921	+	0.795525i	$0.292805\pi$
$888$	27.7867	0.932460
$889$	4.73518	0.158813
$890$	−18.8664	−0.632404
$891$	−3.66160	−0.122668
$892$	−17.0096	−0.569524
$893$	−38.1177	−1.27556
$894$	−38.5858	−1.29050
$895$	−35.7747	−1.19582
$896$	−5.94900	−0.198742
$897$	−23.2886	−0.777584
$898$	−83.2761	−2.77896
$899$	8.70610	0.290365
$900$	22.1527	0.738423
$901$	−3.14326	−0.104717
$902$	−11.6508	−0.387929
$903$	0.919657	0.0306043
$904$	122.743	4.08239
$905$	44.6222	1.48329
$906$	−49.9128	−1.65824
$907$	5.34996	0.177642	0.0888212	−	0.996048i	$-0.471690\pi$
$907$	5.34996	0.177642	0.0888212	+	0.996048i	$0.471690\pi$
$908$	32.5113	1.07892
$909$	−18.4898	−0.613267
$910$	11.3806	0.377263
$911$	50.1375	1.66113	0.830565	−	0.556922i	$-0.188018\pi$
$911$	50.1375	1.66113	0.830565	+	0.556922i	$0.188018\pi$
$912$	78.8213	2.61003
$913$	−4.82033	−0.159530
$914$	88.8451	2.93873
$915$	3.44417	0.113861
$916$	117.895	3.89538
$917$	5.73768	0.189475
$918$	−12.9778	−0.428330
$919$	−15.2554	−0.503229	−0.251615	−	0.967827i	$-0.580962\pi$
$919$	−15.2554	−0.503229	−0.251615	+	0.967827i	$0.580962\pi$
$920$	56.1766	1.85209
$921$	11.2290	0.370007
$922$	−16.5403	−0.544726
$923$	−84.1413	−2.76954
$924$	4.47250	0.147134
$925$	−7.21551	−0.237244
$926$	−45.7231	−1.50256
$927$	−26.1053	−0.857410
$928$	−79.4427	−2.60783
$929$	59.9295	1.96622	0.983111	−	0.183011i	$-0.0585844\pi$
$929$	59.9295	1.96622	0.983111	+	0.183011i	$0.0585844\pi$
$930$	−7.37397	−0.241802
$931$	−46.8980	−1.53702
$932$	37.7096	1.23522
$933$	33.6204	1.10068
$934$	−35.4882	−1.16121
$935$	−3.88488	−0.127049
$936$	107.862	3.52558
$937$	−21.9672	−0.717636	−0.358818	−	0.933408i	$-0.616820\pi$
$937$	−21.9672	−0.717636	−0.358818	+	0.933408i	$0.616820\pi$
$938$	−9.57425	−0.312610
$939$	−14.3284	−0.467591
$940$	−48.7591	−1.59035
$941$	11.3928	0.371395	0.185698	−	0.982607i	$-0.440545\pi$
$941$	11.3928	0.371395	0.185698	+	0.982607i	$0.440545\pi$
$942$	−1.92915	−0.0628550
$943$	7.56262	0.246273
$944$	−12.0250	−0.391379
$945$	3.34577	0.108838
$946$	14.4995	0.471421
$947$	−16.9808	−0.551802	−0.275901	−	0.961186i	$-0.588976\pi$
$947$	−16.9808	−0.551802	−0.275901	+	0.961186i	$0.588976\pi$
$948$	11.2793	0.366333
$949$	−13.3858	−0.434521
$950$	−37.9185	−1.23024
$951$	−24.0443	−0.779689
$952$	−3.35591	−0.108766
$953$	13.5191	0.437926	0.218963	−	0.975733i	$-0.429733\pi$
$953$	13.5191	0.437926	0.218963	+	0.975733i	$0.429733\pi$
$954$	17.5076	0.566829
$955$	−15.2922	−0.494843
$956$	76.3692	2.46995
$957$	11.1849	0.361557
$958$	30.3202	0.979600
$959$	−1.18365	−0.0382219
$960$	27.9490	0.902051
$961$	−28.1480	−0.908000
$962$	−57.6092	−1.85740
$963$	22.8568	0.736550
$964$	−90.0265	−2.89956
$965$	−21.8398	−0.703049
$966$	−4.03582	−0.129850
$967$	−52.2334	−1.67971	−0.839856	−	0.542809i	$-0.817361\pi$
$967$	−52.2334	−1.67971	−0.839856	+	0.542809i	$0.817361\pi$
$968$	−48.7939	−1.56829
$969$	6.55481	0.210571
$970$	8.57705	0.275392
$971$	47.2896	1.51759	0.758797	−	0.651327i	$-0.225787\pi$
$971$	47.2896	1.51759	0.758797	+	0.651327i	$0.225787\pi$
$972$	82.6653	2.65149
$973$	−1.42879	−0.0458049
$974$	−70.7024	−2.26545
$975$	12.2627	0.392720
$976$	25.3203	0.810482
$977$	8.60965	0.275447	0.137724	−	0.990471i	$-0.456021\pi$
$977$	8.60965	0.275447	0.137724	+	0.990471i	$0.456021\pi$
$978$	−17.6947	−0.565813
$979$	9.37449	0.299610
$980$	−59.9906	−1.91633
$981$	−33.3551	−1.06495
$982$	70.5541	2.25147
$983$	−30.3365	−0.967585	−0.483793	−	0.875183i	$-0.660741\pi$
$983$	−30.3365	−0.967585	−0.483793	+	0.875183i	$0.660741\pi$
$984$	15.3350	0.488862
$985$	−23.7490	−0.756707
$986$	−13.7618	−0.438265
$987$	2.13624	0.0679971
$988$	−217.777	−6.92841
$989$	−9.41177	−0.299277
$990$	21.6383	0.687712
$991$	−31.0581	−0.986592	−0.493296	−	0.869861i	$-0.664208\pi$
$991$	−31.0581	−0.986592	−0.493296	+	0.869861i	$0.664208\pi$
$992$	−26.0244	−0.826275
$993$	−7.92540	−0.251505
$994$	−14.5813	−0.462492
$995$	−23.5952	−0.748020
$996$	10.4037	0.329654
$997$	44.1078	1.39691	0.698454	−	0.715655i	$-0.253872\pi$
$997$	44.1078	1.39691	0.698454	+	0.715655i	$0.253872\pi$
$998$	12.0236	0.380602
$999$	−16.9364	−0.535845

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1003.2.a.i.1.18	✓	18
3.2	odd	2		9027.2.a.q.1.1		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1003.2.a.i.1.18	✓	18	1.1	even	1	trivial
9027.2.a.q.1.1		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:	\( N \)	\(=\)	\( 1003 = 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1003.a (trivial)

\(f(q)\)	\(=\)	\(q+2.66948 q^{2} +0.955769 q^{3} +5.12613 q^{4} +1.71138 q^{5} +2.55141 q^{6} -0.402139 q^{7} +8.34515 q^{8} -2.08651 q^{9} +O(q^{10})\)	\(q+2.66948 q^{2} +0.955769 q^{3} +5.12613 q^{4} +1.71138 q^{5} +2.55141 q^{6} -0.402139 q^{7} +8.34515 q^{8} -2.08651 q^{9} +4.56850 q^{10} -2.27003 q^{11} +4.89940 q^{12} -6.19463 q^{13} -1.07350 q^{14} +1.63568 q^{15} +12.0250 q^{16} +1.00000 q^{17} -5.56989 q^{18} +6.85815 q^{19} +8.77276 q^{20} -0.384352 q^{21} -6.05980 q^{22} +3.93346 q^{23} +7.97604 q^{24} -2.07118 q^{25} -16.5364 q^{26} -4.86153 q^{27} -2.06142 q^{28} -5.15524 q^{29} +4.36643 q^{30} -1.68879 q^{31} +15.4101 q^{32} -2.16962 q^{33} +2.66948 q^{34} -0.688213 q^{35} -10.6957 q^{36} +3.48377 q^{37} +18.3077 q^{38} -5.92064 q^{39} +14.2817 q^{40} +1.92264 q^{41} -1.02602 q^{42} -2.39274 q^{43} -11.6365 q^{44} -3.57080 q^{45} +10.5003 q^{46} -5.55801 q^{47} +11.4931 q^{48} -6.83828 q^{49} -5.52897 q^{50} +0.955769 q^{51} -31.7545 q^{52} -3.14326 q^{53} -12.9778 q^{54} -3.88488 q^{55} -3.35591 q^{56} +6.55481 q^{57} -13.7618 q^{58} -1.00000 q^{59} +8.38473 q^{60} +2.10564 q^{61} -4.50819 q^{62} +0.839066 q^{63} +17.0871 q^{64} -10.6014 q^{65} -5.79177 q^{66} +8.91869 q^{67} +5.12613 q^{68} +3.75948 q^{69} -1.83717 q^{70} +13.5829 q^{71} -17.4122 q^{72} +2.16087 q^{73} +9.29986 q^{74} -1.97957 q^{75} +35.1558 q^{76} +0.912867 q^{77} -15.8050 q^{78} +2.30217 q^{79} +20.5793 q^{80} +1.61302 q^{81} +5.13244 q^{82} +2.12347 q^{83} -1.97024 q^{84} +1.71138 q^{85} -6.38738 q^{86} -4.92722 q^{87} -18.9437 q^{88} -4.12968 q^{89} -9.53219 q^{90} +2.49110 q^{91} +20.1634 q^{92} -1.61409 q^{93} -14.8370 q^{94} +11.7369 q^{95} +14.7285 q^{96} +1.87743 q^{97} -18.2547 q^{98} +4.73642 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 5 q^{2} + q^{3} + 21 q^{4} + 21 q^{5} + 5 q^{6} - q^{7} + 9 q^{8} + 17 q^{9}+O(q^{10}) \) 18 * q + 5 * q^2 + q^3 + 21 * q^4 + 21 * q^5 + 5 * q^6 - q^7 + 9 * q^8 + 17 * q^9	\( 18 q + 5 q^{2} + q^{3} + 21 q^{4} + 21 q^{5} + 5 q^{6} - q^{7} + 9 q^{8} + 17 q^{9} + 10 q^{10} + 6 q^{11} + 20 q^{12} + 12 q^{13} + 9 q^{14} - q^{15} + 19 q^{16} + 18 q^{17} + 10 q^{18} - 14 q^{19} + 17 q^{20} - 14 q^{21} - 4 q^{22} + 7 q^{23} - 3 q^{24} + 35 q^{25} + 3 q^{26} + 13 q^{27} + q^{28} + 23 q^{29} + 17 q^{30} - q^{31} + 13 q^{32} + 7 q^{33} + 5 q^{34} + 5 q^{35} - 16 q^{36} + 36 q^{37} - 19 q^{38} + 8 q^{39} - 4 q^{40} + 37 q^{41} + 9 q^{42} - 15 q^{43} - 9 q^{44} + 17 q^{45} + q^{46} + 23 q^{47} + 17 q^{48} + 15 q^{49} + 32 q^{50} + q^{51} - 11 q^{52} + 35 q^{53} - 17 q^{54} - 6 q^{55} + 37 q^{56} - 16 q^{57} + 2 q^{58} - 18 q^{59} - 6 q^{60} + 39 q^{61} - q^{62} + 15 q^{63} + 5 q^{64} + 15 q^{65} - 36 q^{66} + 14 q^{67} + 21 q^{68} + 16 q^{69} - 45 q^{70} + 3 q^{71} - 71 q^{72} + 56 q^{73} - 27 q^{74} + 16 q^{75} - q^{76} + 39 q^{77} - 71 q^{78} - 22 q^{79} + 45 q^{80} - 18 q^{81} + 5 q^{82} + 12 q^{83} + 2 q^{84} + 21 q^{85} + 4 q^{86} + 8 q^{87} - 18 q^{88} + 34 q^{89} + 48 q^{90} + 3 q^{91} + 67 q^{92} - 19 q^{93} - 38 q^{94} - 72 q^{95} + 6 q^{96} + 31 q^{97} + 19 q^{98} + 44 q^{99}+O(q^{100}) \) 18 * q + 5 * q^2 + q^3 + 21 * q^4 + 21 * q^5 + 5 * q^6 - q^7 + 9 * q^8 + 17 * q^9 + 10 * q^10 + 6 * q^11 + 20 * q^12 + 12 * q^13 + 9 * q^14 - q^15 + 19 * q^16 + 18 * q^17 + 10 * q^18 - 14 * q^19 + 17 * q^20 - 14 * q^21 - 4 * q^22 + 7 * q^23 - 3 * q^24 + 35 * q^25 + 3 * q^26 + 13 * q^27 + q^28 + 23 * q^29 + 17 * q^30 - q^31 + 13 * q^32 + 7 * q^33 + 5 * q^34 + 5 * q^35 - 16 * q^36 + 36 * q^37 - 19 * q^38 + 8 * q^39 - 4 * q^40 + 37 * q^41 + 9 * q^42 - 15 * q^43 - 9 * q^44 + 17 * q^45 + q^46 + 23 * q^47 + 17 * q^48 + 15 * q^49 + 32 * q^50 + q^51 - 11 * q^52 + 35 * q^53 - 17 * q^54 - 6 * q^55 + 37 * q^56 - 16 * q^57 + 2 * q^58 - 18 * q^59 - 6 * q^60 + 39 * q^61 - q^62 + 15 * q^63 + 5 * q^64 + 15 * q^65 - 36 * q^66 + 14 * q^67 + 21 * q^68 + 16 * q^69 - 45 * q^70 + 3 * q^71 - 71 * q^72 + 56 * q^73 - 27 * q^74 + 16 * q^75 - q^76 + 39 * q^77 - 71 * q^78 - 22 * q^79 + 45 * q^80 - 18 * q^81 + 5 * q^82 + 12 * q^83 + 2 * q^84 + 21 * q^85 + 4 * q^86 + 8 * q^87 - 18 * q^88 + 34 * q^89 + 48 * q^90 + 3 * q^91 + 67 * q^92 - 19 * q^93 - 38 * q^94 - 72 * q^95 + 6 * q^96 + 31 * q^97 + 19 * q^98 + 44 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more