LMFDB - Embedded newform 8030.2.a.bl.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8030,2,Mod(1,8030)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8030, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8030.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8030 = 2 \cdot 5 \cdot 11 \cdot 73 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8030.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:	$64.1198728231$
Analytic rank:	$0$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 9 x^{18} - x^{17} + 200 x^{16} - 263 x^{15} - 1900 x^{14} + 3165 x^{13} + 10217 x^{12} + \cdots + 1388 $ x^19 - 9x^18 - x^17 + 200x^16 - 263x^15 - 1900x^14 + 3165x^13 + 10217x^12 - 16393x^11 - 33875x^10 + 44043x^9 + 68537x^8 - 63620x^7 - 79890x^6 + 48388x^5 + 49353x^4 - 17806x^3 - 14287x^2 + 2436*x + 1388
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Root			$-2.08616$ of defining polynomial
Character	$\chi$	$=$	8030.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +3.08616 q^{3} +1.00000 q^{4} +1.00000 q^{5} +3.08616 q^{6} -0.0642466 q^{7} +1.00000 q^{8} +6.52440 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +3.08616 q^{3} +1.00000 q^{4} +1.00000 q^{5} +3.08616 q^{6} -0.0642466 q^{7} +1.00000 q^{8} +6.52440 q^{9} +1.00000 q^{10} +1.00000 q^{11} +3.08616 q^{12} +5.56214 q^{13} -0.0642466 q^{14} +3.08616 q^{15} +1.00000 q^{16} -7.49466 q^{17} +6.52440 q^{18} +2.42333 q^{19} +1.00000 q^{20} -0.198276 q^{21} +1.00000 q^{22} +4.92705 q^{23} +3.08616 q^{24} +1.00000 q^{25} +5.56214 q^{26} +10.8769 q^{27} -0.0642466 q^{28} -4.58607 q^{29} +3.08616 q^{30} -7.10237 q^{31} +1.00000 q^{32} +3.08616 q^{33} -7.49466 q^{34} -0.0642466 q^{35} +6.52440 q^{36} +3.80161 q^{37} +2.42333 q^{38} +17.1657 q^{39} +1.00000 q^{40} +0.755066 q^{41} -0.198276 q^{42} +0.634695 q^{43} +1.00000 q^{44} +6.52440 q^{45} +4.92705 q^{46} -0.525114 q^{47} +3.08616 q^{48} -6.99587 q^{49} +1.00000 q^{50} -23.1297 q^{51} +5.56214 q^{52} -2.63826 q^{53} +10.8769 q^{54} +1.00000 q^{55} -0.0642466 q^{56} +7.47880 q^{57} -4.58607 q^{58} +13.9386 q^{59} +3.08616 q^{60} -4.27434 q^{61} -7.10237 q^{62} -0.419171 q^{63} +1.00000 q^{64} +5.56214 q^{65} +3.08616 q^{66} -1.45417 q^{67} -7.49466 q^{68} +15.2057 q^{69} -0.0642466 q^{70} +4.99291 q^{71} +6.52440 q^{72} -1.00000 q^{73} +3.80161 q^{74} +3.08616 q^{75} +2.42333 q^{76} -0.0642466 q^{77} +17.1657 q^{78} -4.68171 q^{79} +1.00000 q^{80} +13.9946 q^{81} +0.755066 q^{82} -4.73371 q^{83} -0.198276 q^{84} -7.49466 q^{85} +0.634695 q^{86} -14.1534 q^{87} +1.00000 q^{88} +12.4173 q^{89} +6.52440 q^{90} -0.357349 q^{91} +4.92705 q^{92} -21.9191 q^{93} -0.525114 q^{94} +2.42333 q^{95} +3.08616 q^{96} +10.4291 q^{97} -6.99587 q^{98} +6.52440 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q + 19 q^{2} + 10 q^{3} + 19 q^{4} + 19 q^{5} + 10 q^{6} + 8 q^{7} + 19 q^{8} + 27 q^{9}+O(q^{10}) $ 19 * q + 19 * q^2 + 10 * q^3 + 19 * q^4 + 19 * q^5 + 10 * q^6 + 8 * q^7 + 19 * q^8 + 27 * q^9	\( 19 q + 19 q^{2} + 10 q^{3} + 19 q^{4} + 19 q^{5} + 10 q^{6} + 8 q^{7} + 19 q^{8} + 27 q^{9} + 19 q^{10} + 19 q^{11} + 10 q^{12} + 16 q^{13} + 8 q^{14} + 10 q^{15} + 19 q^{16} + 12 q^{17} + 27 q^{18} + 12 q^{19} + 19 q^{20} + 3 q^{21} + 19 q^{22} + 26 q^{23} + 10 q^{24} + 19 q^{25} + 16 q^{26} + 25 q^{27} + 8 q^{28} + q^{29} + 10 q^{30} + 24 q^{31} + 19 q^{32} + 10 q^{33} + 12 q^{34} + 8 q^{35} + 27 q^{36} + 23 q^{37} + 12 q^{38} - 5 q^{39} + 19 q^{40} + 3 q^{42} + 8 q^{43} + 19 q^{44} + 27 q^{45} + 26 q^{46} + 34 q^{47} + 10 q^{48} + 27 q^{49} + 19 q^{50} + 15 q^{51} + 16 q^{52} + 25 q^{53} + 25 q^{54} + 19 q^{55} + 8 q^{56} + q^{57} + q^{58} + 24 q^{59} + 10 q^{60} + 31 q^{61} + 24 q^{62} + 15 q^{63} + 19 q^{64} + 16 q^{65} + 10 q^{66} + 24 q^{67} + 12 q^{68} + q^{69} + 8 q^{70} + 5 q^{71} + 27 q^{72} - 19 q^{73} + 23 q^{74} + 10 q^{75} + 12 q^{76} + 8 q^{77} - 5 q^{78} + 18 q^{79} + 19 q^{80} + 11 q^{81} + 12 q^{83} + 3 q^{84} + 12 q^{85} + 8 q^{86} + 12 q^{87} + 19 q^{88} + 27 q^{90} + 23 q^{91} + 26 q^{92} + 18 q^{93} + 34 q^{94} + 12 q^{95} + 10 q^{96} + 15 q^{97} + 27 q^{98} + 27 q^{99}+O(q^{100}) \) 19 * q + 19 * q^2 + 10 * q^3 + 19 * q^4 + 19 * q^5 + 10 * q^6 + 8 * q^7 + 19 * q^8 + 27 * q^9 + 19 * q^10 + 19 * q^11 + 10 * q^12 + 16 * q^13 + 8 * q^14 + 10 * q^15 + 19 * q^16 + 12 * q^17 + 27 * q^18 + 12 * q^19 + 19 * q^20 + 3 * q^21 + 19 * q^22 + 26 * q^23 + 10 * q^24 + 19 * q^25 + 16 * q^26 + 25 * q^27 + 8 * q^28 + q^29 + 10 * q^30 + 24 * q^31 + 19 * q^32 + 10 * q^33 + 12 * q^34 + 8 * q^35 + 27 * q^36 + 23 * q^37 + 12 * q^38 - 5 * q^39 + 19 * q^40 + 3 * q^42 + 8 * q^43 + 19 * q^44 + 27 * q^45 + 26 * q^46 + 34 * q^47 + 10 * q^48 + 27 * q^49 + 19 * q^50 + 15 * q^51 + 16 * q^52 + 25 * q^53 + 25 * q^54 + 19 * q^55 + 8 * q^56 + q^57 + q^58 + 24 * q^59 + 10 * q^60 + 31 * q^61 + 24 * q^62 + 15 * q^63 + 19 * q^64 + 16 * q^65 + 10 * q^66 + 24 * q^67 + 12 * q^68 + q^69 + 8 * q^70 + 5 * q^71 + 27 * q^72 - 19 * q^73 + 23 * q^74 + 10 * q^75 + 12 * q^76 + 8 * q^77 - 5 * q^78 + 18 * q^79 + 19 * q^80 + 11 * q^81 + 12 * q^83 + 3 * q^84 + 12 * q^85 + 8 * q^86 + 12 * q^87 + 19 * q^88 + 27 * q^90 + 23 * q^91 + 26 * q^92 + 18 * q^93 + 34 * q^94 + 12 * q^95 + 10 * q^96 + 15 * q^97 + 27 * q^98 + 27 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	3.08616	1.78180	0.890899	−	0.454202i	$-0.150076\pi$
$3$	3.08616	1.78180	0.890899	+	0.454202i	$0.150076\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	3.08616	1.25992
$7$	−0.0642466	−0.0242829	−0.0121415	−	0.999926i	$-0.503865\pi$
$7$	−0.0642466	−0.0242829	−0.0121415	+	0.999926i	$0.503865\pi$
$8$	1.00000	0.353553
$9$	6.52440	2.17480
$10$	1.00000	0.316228
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	3.08616	0.890899
$13$	5.56214	1.54266	0.771330	−	0.636436i	$-0.219592\pi$
$13$	5.56214	1.54266	0.771330	+	0.636436i	$0.219592\pi$
$14$	−0.0642466	−0.0171706
$15$	3.08616	0.796844
$16$	1.00000	0.250000
$17$	−7.49466	−1.81772	−0.908861	−	0.417100i	$-0.863046\pi$
$17$	−7.49466	−1.81772	−0.908861	+	0.417100i	$0.863046\pi$
$18$	6.52440	1.53782
$19$	2.42333	0.555950	0.277975	−	0.960588i	$-0.410337\pi$
$19$	2.42333	0.555950	0.277975	+	0.960588i	$0.410337\pi$
$20$	1.00000	0.223607
$21$	−0.198276	−0.0432673
$22$	1.00000	0.213201
$23$	4.92705	1.02736	0.513681	−	0.857981i	$-0.328282\pi$
$23$	4.92705	1.02736	0.513681	+	0.857981i	$0.328282\pi$
$24$	3.08616	0.629960
$25$	1.00000	0.200000
$26$	5.56214	1.09083
$27$	10.8769	2.09326
$28$	−0.0642466	−0.0121415
$29$	−4.58607	−0.851611	−0.425806	−	0.904815i	$-0.640009\pi$
$29$	−4.58607	−0.851611	−0.425806	+	0.904815i	$0.640009\pi$
$30$	3.08616	0.563454
$31$	−7.10237	−1.27562	−0.637811	−	0.770193i	$-0.720160\pi$
$31$	−7.10237	−1.27562	−0.637811	+	0.770193i	$0.720160\pi$
$32$	1.00000	0.176777
$33$	3.08616	0.537232
$34$	−7.49466	−1.28532
$35$	−0.0642466	−0.0108597
$36$	6.52440	1.08740
$37$	3.80161	0.624980	0.312490	−	0.949921i	$-0.398837\pi$
$37$	3.80161	0.624980	0.312490	+	0.949921i	$0.398837\pi$
$38$	2.42333	0.393116
$39$	17.1657	2.74871
$40$	1.00000	0.158114
$41$	0.755066	0.117922	0.0589608	−	0.998260i	$-0.481221\pi$
$41$	0.755066	0.117922	0.0589608	+	0.998260i	$0.481221\pi$
$42$	−0.198276	−0.0305946
$43$	0.634695	0.0967901	0.0483950	−	0.998828i	$-0.484589\pi$
$43$	0.634695	0.0967901	0.0483950	+	0.998828i	$0.484589\pi$
$44$	1.00000	0.150756
$45$	6.52440	0.972601
$46$	4.92705	0.726454
$47$	−0.525114	−0.0765958	−0.0382979	−	0.999266i	$-0.512194\pi$
$47$	−0.525114	−0.0765958	−0.0382979	+	0.999266i	$0.512194\pi$
$48$	3.08616	0.445449
$49$	−6.99587	−0.999410
$50$	1.00000	0.141421
$51$	−23.1297	−3.23881
$52$	5.56214	0.771330
$53$	−2.63826	−0.362392	−0.181196	−	0.983447i	$-0.557997\pi$
$53$	−2.63826	−0.362392	−0.181196	+	0.983447i	$0.557997\pi$
$54$	10.8769	1.48016
$55$	1.00000	0.134840
$56$	−0.0642466	−0.00858532
$57$	7.47880	0.990591
$58$	−4.58607	−0.602180
$59$	13.9386	1.81465	0.907323	−	0.420434i	$-0.138122\pi$
$59$	13.9386	1.81465	0.907323	+	0.420434i	$0.138122\pi$
$60$	3.08616	0.398422
$61$	−4.27434	−0.547274	−0.273637	−	0.961833i	$-0.588227\pi$
$61$	−4.27434	−0.547274	−0.273637	+	0.961833i	$0.588227\pi$
$62$	−7.10237	−0.902001
$63$	−0.419171	−0.0528106
$64$	1.00000	0.125000
$65$	5.56214	0.689898
$66$	3.08616	0.379880
$67$	−1.45417	−0.177655	−0.0888276	−	0.996047i	$-0.528312\pi$
$67$	−1.45417	−0.177655	−0.0888276	+	0.996047i	$0.528312\pi$
$68$	−7.49466	−0.908861
$69$	15.2057	1.83055
$70$	−0.0642466	−0.00767894
$71$	4.99291	0.592549	0.296275	−	0.955103i	$-0.404256\pi$
$71$	4.99291	0.592549	0.296275	+	0.955103i	$0.404256\pi$
$72$	6.52440	0.768908
$73$	−1.00000	−0.117041
$73$	−1.00000	−0.117041
$74$	3.80161	0.441928
$75$	3.08616	0.356359
$76$	2.42333	0.277975
$77$	−0.0642466	−0.00732158
$78$	17.1657	1.94363
$79$	−4.68171	−0.526734	−0.263367	−	0.964696i	$-0.584833\pi$
$79$	−4.68171	−0.526734	−0.263367	+	0.964696i	$0.584833\pi$
$80$	1.00000	0.111803
$81$	13.9946	1.55496
$82$	0.755066	0.0833831
$83$	−4.73371	−0.519592	−0.259796	−	0.965664i	$-0.583655\pi$
$83$	−4.73371	−0.519592	−0.259796	+	0.965664i	$0.583655\pi$
$84$	−0.198276	−0.0216336
$85$	−7.49466	−0.812910
$86$	0.634695	0.0684409
$87$	−14.1534	−1.51740
$88$	1.00000	0.106600
$89$	12.4173	1.31624	0.658118	−	0.752915i	$-0.271353\pi$
$89$	12.4173	1.31624	0.658118	+	0.752915i	$0.271353\pi$
$90$	6.52440	0.687733
$91$	−0.357349	−0.0374603
$92$	4.92705	0.513681
$93$	−21.9191	−2.27290
$94$	−0.525114	−0.0541614
$95$	2.42333	0.248629
$96$	3.08616	0.314980
$97$	10.4291	1.05891	0.529457	−	0.848337i	$-0.322396\pi$
$97$	10.4291	1.05891	0.529457	+	0.848337i	$0.322396\pi$
$98$	−6.99587	−0.706690
$99$	6.52440	0.655727
$100$	1.00000	0.100000
$101$	−2.58736	−0.257452	−0.128726	−	0.991680i	$-0.541089\pi$
$101$	−2.58736	−0.257452	−0.128726	+	0.991680i	$0.541089\pi$
$102$	−23.1297	−2.29018
$103$	−11.1056	−1.09427	−0.547136	−	0.837044i	$-0.684282\pi$
$103$	−11.1056	−1.09427	−0.547136	+	0.837044i	$0.684282\pi$
$104$	5.56214	0.545413
$105$	−0.198276	−0.0193497
$106$	−2.63826	−0.256250
$107$	−5.80496	−0.561187	−0.280593	−	0.959827i	$-0.590531\pi$
$107$	−5.80496	−0.561187	−0.280593	+	0.959827i	$0.590531\pi$
$108$	10.8769	1.04663
$109$	−2.45195	−0.234854	−0.117427	−	0.993082i	$-0.537465\pi$
$109$	−2.45195	−0.234854	−0.117427	+	0.993082i	$0.537465\pi$
$110$	1.00000	0.0953463
$111$	11.7324	1.11359
$112$	−0.0642466	−0.00607074
$113$	−15.6199	−1.46940	−0.734700	−	0.678392i	$-0.762677\pi$
$113$	−15.6199	−1.46940	−0.734700	+	0.678392i	$0.762677\pi$
$114$	7.47880	0.700453
$115$	4.92705	0.459450
$116$	−4.58607	−0.425806
$117$	36.2896	3.35498
$118$	13.9386	1.28315
$119$	0.481506	0.0441396
$120$	3.08616	0.281727
$121$	1.00000	0.0909091
$122$	−4.27434	−0.386981
$123$	2.33026	0.210112
$124$	−7.10237	−0.637811
$125$	1.00000	0.0894427
$126$	−0.419171	−0.0373427
$127$	−0.939599	−0.0833759	−0.0416880	−	0.999131i	$-0.513274\pi$
$127$	−0.939599	−0.0833759	−0.0416880	+	0.999131i	$0.513274\pi$
$128$	1.00000	0.0883883
$129$	1.95877	0.172460
$130$	5.56214	0.487832
$131$	−10.0525	−0.878295	−0.439147	−	0.898415i	$-0.644719\pi$
$131$	−10.0525	−0.878295	−0.439147	+	0.898415i	$0.644719\pi$
$132$	3.08616	0.268616
$133$	−0.155691	−0.0135001
$134$	−1.45417	−0.125621
$135$	10.8769	0.936133
$136$	−7.49466	−0.642661
$137$	−0.182257	−0.0155713	−0.00778564	−	0.999970i	$-0.502478\pi$
$137$	−0.182257	−0.0155713	−0.00778564	+	0.999970i	$0.502478\pi$
$138$	15.2057	1.29439
$139$	−9.70087	−0.822817	−0.411409	−	0.911451i	$-0.634963\pi$
$139$	−9.70087	−0.822817	−0.411409	+	0.911451i	$0.634963\pi$
$140$	−0.0642466	−0.00542983
$141$	−1.62059	−0.136478
$142$	4.99291	0.418996
$143$	5.56214	0.465129
$144$	6.52440	0.543700
$145$	−4.58607	−0.380852
$146$	−1.00000	−0.0827606
$147$	−21.5904	−1.78075
$148$	3.80161	0.312490
$149$	15.4733	1.26763	0.633813	−	0.773486i	$-0.281489\pi$
$149$	15.4733	1.26763	0.633813	+	0.773486i	$0.281489\pi$
$150$	3.08616	0.251984
$151$	−3.86321	−0.314384	−0.157192	−	0.987568i	$-0.550244\pi$
$151$	−3.86321	−0.314384	−0.157192	+	0.987568i	$0.550244\pi$
$152$	2.42333	0.196558
$153$	−48.8982	−3.95318
$154$	−0.0642466	−0.00517714
$155$	−7.10237	−0.570476
$156$	17.1657	1.37435
$157$	20.1569	1.60869	0.804347	−	0.594160i	$-0.202515\pi$
$157$	20.1569	1.60869	0.804347	+	0.594160i	$0.202515\pi$
$158$	−4.68171	−0.372457
$159$	−8.14209	−0.645710
$160$	1.00000	0.0790569
$161$	−0.316547	−0.0249474
$162$	13.9946	1.09952
$163$	10.6386	0.833277	0.416639	−	0.909072i	$-0.363208\pi$
$163$	10.6386	0.833277	0.416639	+	0.909072i	$0.363208\pi$
$164$	0.755066	0.0589608
$165$	3.08616	0.240257
$166$	−4.73371	−0.367407
$167$	−15.3624	−1.18878	−0.594390	−	0.804177i	$-0.702606\pi$
$167$	−15.3624	−1.18878	−0.594390	+	0.804177i	$0.702606\pi$
$168$	−0.198276	−0.0152973
$169$	17.9374	1.37980
$170$	−7.49466	−0.574814
$171$	15.8108	1.20908
$172$	0.634695	0.0483950
$173$	5.00209	0.380302	0.190151	−	0.981755i	$-0.439102\pi$
$173$	5.00209	0.380302	0.190151	+	0.981755i	$0.439102\pi$
$174$	−14.1534	−1.07296
$175$	−0.0642466	−0.00485659
$176$	1.00000	0.0753778
$177$	43.0167	3.23333
$178$	12.4173	0.930719
$179$	−4.72451	−0.353126	−0.176563	−	0.984289i	$-0.556498\pi$
$179$	−4.72451	−0.353126	−0.176563	+	0.984289i	$0.556498\pi$
$180$	6.52440	0.486300
$181$	−3.47598	−0.258367	−0.129184	−	0.991621i	$-0.541236\pi$
$181$	−3.47598	−0.258367	−0.129184	+	0.991621i	$0.541236\pi$
$182$	−0.357349	−0.0264884
$183$	−13.1913	−0.975131
$184$	4.92705	0.363227
$185$	3.80161	0.279500
$186$	−21.9191	−1.60718
$187$	−7.49466	−0.548064
$188$	−0.525114	−0.0382979
$189$	−0.698803	−0.0508305
$190$	2.42333	0.175807
$191$	−12.2341	−0.885227	−0.442613	−	0.896713i	$-0.645949\pi$
$191$	−12.2341	−0.885227	−0.442613	+	0.896713i	$0.645949\pi$
$192$	3.08616	0.222725
$193$	−15.4090	−1.10917	−0.554583	−	0.832128i	$-0.687122\pi$
$193$	−15.4090	−1.10917	−0.554583	+	0.832128i	$0.687122\pi$
$194$	10.4291	0.748765
$195$	17.1657	1.22926
$196$	−6.99587	−0.499705
$197$	11.8426	0.843753	0.421876	−	0.906653i	$-0.361372\pi$
$197$	11.8426	0.843753	0.421876	+	0.906653i	$0.361372\pi$
$198$	6.52440	0.463669
$199$	1.50306	0.106549	0.0532747	−	0.998580i	$-0.483034\pi$
$199$	1.50306	0.106549	0.0532747	+	0.998580i	$0.483034\pi$
$200$	1.00000	0.0707107
$201$	−4.48781	−0.316546
$202$	−2.58736	−0.182046
$203$	0.294639	0.0206796
$204$	−23.1297	−1.61941
$205$	0.755066	0.0527361
$206$	−11.1056	−0.773767
$207$	32.1461	2.23431
$208$	5.56214	0.385665
$209$	2.42333	0.167625
$210$	−0.198276	−0.0136823
$211$	9.37065	0.645102	0.322551	−	0.946552i	$-0.395460\pi$
$211$	9.37065	0.645102	0.322551	+	0.946552i	$0.395460\pi$
$212$	−2.63826	−0.181196
$213$	15.4089	1.05580
$214$	−5.80496	−0.396819
$215$	0.634695	0.0432858
$216$	10.8769	0.740078
$217$	0.456303	0.0309759
$218$	−2.45195	−0.166067
$219$	−3.08616	−0.208544
$220$	1.00000	0.0674200
$221$	−41.6863	−2.80412
$222$	11.7324	0.787426
$223$	−22.5220	−1.50818	−0.754092	−	0.656769i	$-0.771923\pi$
$223$	−22.5220	−1.50818	−0.754092	+	0.656769i	$0.771923\pi$
$224$	−0.0642466	−0.00429266
$225$	6.52440	0.434960
$226$	−15.6199	−1.03902
$227$	−21.8609	−1.45096	−0.725479	−	0.688245i	$-0.758382\pi$
$227$	−21.8609	−1.45096	−0.725479	+	0.688245i	$0.758382\pi$
$228$	7.47880	0.495295
$229$	−7.86222	−0.519550	−0.259775	−	0.965669i	$-0.583648\pi$
$229$	−7.86222	−0.519550	−0.259775	+	0.965669i	$0.583648\pi$
$230$	4.92705	0.324880
$231$	−0.198276	−0.0130456
$232$	−4.58607	−0.301090
$233$	−20.3966	−1.33622	−0.668111	−	0.744061i	$-0.732897\pi$
$233$	−20.3966	−1.33622	−0.668111	+	0.744061i	$0.732897\pi$
$234$	36.2896	2.37233
$235$	−0.525114	−0.0342547
$236$	13.9386	0.907323
$237$	−14.4485	−0.938533
$238$	0.481506	0.0312114
$239$	18.9265	1.22426	0.612128	−	0.790759i	$-0.290314\pi$
$239$	18.9265	1.22426	0.612128	+	0.790759i	$0.290314\pi$
$240$	3.08616	0.199211
$241$	8.25488	0.531744	0.265872	−	0.964008i	$-0.414340\pi$
$241$	8.25488	0.531744	0.265872	+	0.964008i	$0.414340\pi$
$242$	1.00000	0.0642824
$243$	10.5591	0.677364
$244$	−4.27434	−0.273637
$245$	−6.99587	−0.446950
$246$	2.33026	0.148572
$247$	13.4789	0.857642
$248$	−7.10237	−0.451001
$249$	−14.6090	−0.925807
$250$	1.00000	0.0632456
$251$	−15.0182	−0.947943	−0.473972	−	0.880540i	$-0.657180\pi$
$251$	−15.0182	−0.947943	−0.473972	+	0.880540i	$0.657180\pi$
$252$	−0.419171	−0.0264053
$253$	4.92705	0.309761
$254$	−0.939599	−0.0589557
$255$	−23.1297	−1.44844
$256$	1.00000	0.0625000
$257$	−25.2141	−1.57282	−0.786408	−	0.617708i	$-0.788061\pi$
$257$	−25.2141	−1.57282	−0.786408	+	0.617708i	$0.788061\pi$
$258$	1.95877	0.121948
$259$	−0.244240	−0.0151764
$260$	5.56214	0.344949
$261$	−29.9214	−1.85209
$262$	−10.0525	−0.621048
$263$	23.1840	1.42959	0.714794	−	0.699335i	$-0.246520\pi$
$263$	23.1840	1.42959	0.714794	+	0.699335i	$0.246520\pi$
$264$	3.08616	0.189940
$265$	−2.63826	−0.162067
$266$	−0.155691	−0.00954602
$267$	38.3220	2.34527
$268$	−1.45417	−0.0888276
$269$	−25.3005	−1.54260	−0.771299	−	0.636473i	$-0.780393\pi$
$269$	−25.3005	−1.54260	−0.771299	+	0.636473i	$0.780393\pi$
$270$	10.8769	0.661946
$271$	−22.2007	−1.34860	−0.674299	−	0.738458i	$-0.735554\pi$
$271$	−22.2007	−1.34860	−0.674299	+	0.738458i	$0.735554\pi$
$272$	−7.49466	−0.454430
$273$	−1.10284	−0.0667467
$274$	−0.182257	−0.0110106
$275$	1.00000	0.0603023
$276$	15.2057	0.915275
$277$	−13.0807	−0.785942	−0.392971	−	0.919551i	$-0.628553\pi$
$277$	−13.0807	−0.785942	−0.392971	+	0.919551i	$0.628553\pi$
$278$	−9.70087	−0.581820
$279$	−46.3387	−2.77423
$280$	−0.0642466	−0.00383947
$281$	0.591860	0.0353074	0.0176537	−	0.999844i	$-0.494380\pi$
$281$	0.591860	0.0353074	0.0176537	+	0.999844i	$0.494380\pi$
$282$	−1.62059	−0.0965046
$283$	22.5080	1.33796	0.668980	−	0.743280i	$-0.266731\pi$
$283$	22.5080	1.33796	0.668980	+	0.743280i	$0.266731\pi$
$284$	4.99291	0.296275
$285$	7.47880	0.443006
$286$	5.56214	0.328896
$287$	−0.0485105	−0.00286348
$288$	6.52440	0.384454
$289$	39.1699	2.30411
$290$	−4.58607	−0.269303
$291$	32.1859	1.88677
$292$	−1.00000	−0.0585206
$293$	26.8131	1.56644	0.783220	−	0.621745i	$-0.213576\pi$
$293$	26.8131	1.56644	0.783220	+	0.621745i	$0.213576\pi$
$294$	−21.5904	−1.25918
$295$	13.9386	0.811534
$296$	3.80161	0.220964
$297$	10.8769	0.631141
$298$	15.4733	0.896347
$299$	27.4050	1.58487
$300$	3.08616	0.178180
$301$	−0.0407770	−0.00235035
$302$	−3.86321	−0.222303
$303$	−7.98503	−0.458728
$304$	2.42333	0.138988
$305$	−4.27434	−0.244748
$306$	−48.8982	−2.79532
$307$	18.0313	1.02910	0.514551	−	0.857460i	$-0.327959\pi$
$307$	18.0313	1.02910	0.514551	+	0.857460i	$0.327959\pi$
$308$	−0.0642466	−0.00366079
$309$	−34.2738	−1.94977
$310$	−7.10237	−0.403387
$311$	6.41047	0.363505	0.181752	−	0.983344i	$-0.441823\pi$
$311$	6.41047	0.363505	0.181752	+	0.983344i	$0.441823\pi$
$312$	17.1657	0.971814
$313$	26.8853	1.51964	0.759822	−	0.650131i	$-0.225286\pi$
$313$	26.8853	1.51964	0.759822	+	0.650131i	$0.225286\pi$
$314$	20.1569	1.13752
$315$	−0.419171	−0.0236176
$316$	−4.68171	−0.263367
$317$	6.62012	0.371823	0.185911	−	0.982567i	$-0.440476\pi$
$317$	6.62012	0.371823	0.185911	+	0.982567i	$0.440476\pi$
$318$	−8.14209	−0.456586
$319$	−4.58607	−0.256770
$320$	1.00000	0.0559017
$321$	−17.9151	−0.999921
$322$	−0.316547	−0.0176405
$323$	−18.1620	−1.01056
$324$	13.9946	0.777480
$325$	5.56214	0.308532
$326$	10.6386	0.589216
$327$	−7.56711	−0.418462
$328$	0.755066	0.0416916
$329$	0.0337368	0.00185997
$330$	3.08616	0.169888
$331$	17.5582	0.965086	0.482543	−	0.875872i	$-0.339713\pi$
$331$	17.5582	0.965086	0.482543	+	0.875872i	$0.339713\pi$
$332$	−4.73371	−0.259796
$333$	24.8032	1.35921
$334$	−15.3624	−0.840594
$335$	−1.45417	−0.0794498
$336$	−0.198276	−0.0108168
$337$	21.4259	1.16714	0.583571	−	0.812062i	$-0.301655\pi$
$337$	21.4259	1.16714	0.583571	+	0.812062i	$0.301655\pi$
$338$	17.9374	0.975665
$339$	−48.2057	−2.61817
$340$	−7.49466	−0.406455
$341$	−7.10237	−0.384615
$342$	15.8108	0.854950
$343$	0.899188	0.0485516
$344$	0.634695	0.0342205
$345$	15.2057	0.818647
$346$	5.00209	0.268914
$347$	−23.8923	−1.28261	−0.641303	−	0.767288i	$-0.721606\pi$
$347$	−23.8923	−1.28261	−0.641303	+	0.767288i	$0.721606\pi$
$348$	−14.1534	−0.758699
$349$	29.7403	1.59196	0.795980	−	0.605322i	$-0.206956\pi$
$349$	29.7403	1.59196	0.795980	+	0.605322i	$0.206956\pi$
$350$	−0.0642466	−0.00343413
$351$	60.4987	3.22918
$352$	1.00000	0.0533002
$353$	10.2490	0.545497	0.272749	−	0.962085i	$-0.412067\pi$
$353$	10.2490	0.545497	0.272749	+	0.962085i	$0.412067\pi$
$354$	43.0167	2.28631
$355$	4.99291	0.264996
$356$	12.4173	0.658118
$357$	1.48601	0.0786479
$358$	−4.72451	−0.249698
$359$	−22.5144	−1.18827	−0.594133	−	0.804367i	$-0.702505\pi$
$359$	−22.5144	−1.18827	−0.594133	+	0.804367i	$0.702505\pi$
$360$	6.52440	0.343866
$361$	−13.1275	−0.690919
$362$	−3.47598	−0.182693
$363$	3.08616	0.161982
$364$	−0.357349	−0.0187302
$365$	−1.00000	−0.0523424
$366$	−13.1913	−0.689522
$367$	−3.76698	−0.196635	−0.0983174	−	0.995155i	$-0.531346\pi$
$367$	−3.76698	−0.196635	−0.0983174	+	0.995155i	$0.531346\pi$
$368$	4.92705	0.256840
$369$	4.92636	0.256456
$370$	3.80161	0.197636
$371$	0.169499	0.00879996
$372$	−21.9191	−1.13645
$373$	16.7422	0.866878	0.433439	−	0.901183i	$-0.357300\pi$
$373$	16.7422	0.866878	0.433439	+	0.901183i	$0.357300\pi$
$374$	−7.49466	−0.387539
$375$	3.08616	0.159369
$376$	−0.525114	−0.0270807
$377$	−25.5083	−1.31375
$378$	−0.698803	−0.0359426
$379$	14.3990	0.739629	0.369814	−	0.929106i	$-0.379421\pi$
$379$	14.3990	0.739629	0.369814	+	0.929106i	$0.379421\pi$
$380$	2.42333	0.124314
$381$	−2.89976	−0.148559
$382$	−12.2341	−0.625950
$383$	−10.9828	−0.561193	−0.280597	−	0.959826i	$-0.590532\pi$
$383$	−10.9828	−0.561193	−0.280597	+	0.959826i	$0.590532\pi$
$384$	3.08616	0.157490
$385$	−0.0642466	−0.00327431
$386$	−15.4090	−0.784299
$387$	4.14101	0.210499
$388$	10.4291	0.529457
$389$	−27.8581	−1.41246	−0.706230	−	0.707983i	$-0.749605\pi$
$389$	−27.8581	−1.41246	−0.706230	+	0.707983i	$0.749605\pi$
$390$	17.1657	0.869217
$391$	−36.9266	−1.86746
$392$	−6.99587	−0.353345
$393$	−31.0238	−1.56494
$394$	11.8426	0.596623
$395$	−4.68171	−0.235563
$396$	6.52440	0.327864
$397$	−2.31045	−0.115958	−0.0579790	−	0.998318i	$-0.518466\pi$
$397$	−2.31045	−0.115958	−0.0579790	+	0.998318i	$0.518466\pi$
$398$	1.50306	0.0753418
$399$	−0.480487	−0.0240545
$400$	1.00000	0.0500000
$401$	22.2471	1.11097	0.555483	−	0.831528i	$-0.312533\pi$
$401$	22.2471	1.11097	0.555483	+	0.831528i	$0.312533\pi$
$402$	−4.48781	−0.223832
$403$	−39.5043	−1.96785
$404$	−2.58736	−0.128726
$405$	13.9946	0.695399
$406$	0.294639	0.0146227
$407$	3.80161	0.188439
$408$	−23.1297	−1.14509
$409$	−27.6451	−1.36696	−0.683481	−	0.729969i	$-0.739535\pi$
$409$	−27.6451	−1.36696	−0.683481	+	0.729969i	$0.739535\pi$
$410$	0.755066	0.0372901
$411$	−0.562475	−0.0277449
$412$	−11.1056	−0.547136
$413$	−0.895506	−0.0440650
$414$	32.1461	1.57989
$415$	−4.73371	−0.232369
$416$	5.56214	0.272706
$417$	−29.9385	−1.46609
$418$	2.42333	0.118529
$419$	16.7963	0.820553	0.410277	−	0.911961i	$-0.365432\pi$
$419$	16.7963	0.820553	0.410277	+	0.911961i	$0.365432\pi$
$420$	−0.198276	−0.00967486
$421$	10.5644	0.514877	0.257439	−	0.966295i	$-0.417121\pi$
$421$	10.5644	0.514877	0.257439	+	0.966295i	$0.417121\pi$
$422$	9.37065	0.456156
$423$	−3.42606	−0.166581
$424$	−2.63826	−0.128125
$425$	−7.49466	−0.363544
$426$	15.4089	0.746565
$427$	0.274612	0.0132894
$428$	−5.80496	−0.280593
$429$	17.1657	0.828766
$430$	0.634695	0.0306077
$431$	24.3828	1.17448	0.587239	−	0.809413i	$-0.300215\pi$
$431$	24.3828	1.17448	0.587239	+	0.809413i	$0.300215\pi$
$432$	10.8769	0.523314
$433$	5.47148	0.262943	0.131471	−	0.991320i	$-0.458030\pi$
$433$	5.47148	0.262943	0.131471	+	0.991320i	$0.458030\pi$
$434$	0.456303	0.0219033
$435$	−14.1534	−0.678601
$436$	−2.45195	−0.117427
$437$	11.9399	0.571162
$438$	−3.08616	−0.147463
$439$	10.0675	0.480494	0.240247	−	0.970712i	$-0.422772\pi$
$439$	10.0675	0.480494	0.240247	+	0.970712i	$0.422772\pi$
$440$	1.00000	0.0476731
$441$	−45.6439	−2.17352
$442$	−41.6863	−1.98282
$443$	−26.2135	−1.24544	−0.622721	−	0.782444i	$-0.713973\pi$
$443$	−26.2135	−1.24544	−0.622721	+	0.782444i	$0.713973\pi$
$444$	11.7324	0.556794
$445$	12.4173	0.588639
$446$	−22.5220	−1.06645
$447$	47.7533	2.25865
$448$	−0.0642466	−0.00303537
$449$	4.10283	0.193624	0.0968122	−	0.995303i	$-0.469135\pi$
$449$	4.10283	0.193624	0.0968122	+	0.995303i	$0.469135\pi$
$450$	6.52440	0.307563
$451$	0.755066	0.0355547
$452$	−15.6199	−0.734700
$453$	−11.9225	−0.560168
$454$	−21.8609	−1.02598
$455$	−0.357349	−0.0167528
$456$	7.47880	0.350227
$457$	−25.9553	−1.21414	−0.607069	−	0.794649i	$-0.707655\pi$
$457$	−25.9553	−1.21414	−0.607069	+	0.794649i	$0.707655\pi$
$458$	−7.86222	−0.367377
$459$	−81.5185	−3.80496
$460$	4.92705	0.229725
$461$	8.50836	0.396274	0.198137	−	0.980174i	$-0.436511\pi$
$461$	8.50836	0.396274	0.198137	+	0.980174i	$0.436511\pi$
$462$	−0.198276	−0.00922462
$463$	12.9267	0.600755	0.300377	−	0.953820i	$-0.402887\pi$
$463$	12.9267	0.600755	0.300377	+	0.953820i	$0.402887\pi$
$464$	−4.58607	−0.212903
$465$	−21.9191	−1.01647
$466$	−20.3966	−0.944852
$467$	13.2177	0.611643	0.305822	−	0.952089i	$-0.401069\pi$
$467$	13.2177	0.611643	0.305822	+	0.952089i	$0.401069\pi$
$468$	36.2896	1.67749
$469$	0.0934256	0.00431399
$470$	−0.525114	−0.0242217
$471$	62.2074	2.86637
$472$	13.9386	0.641574
$473$	0.634695	0.0291833
$474$	−14.4485	−0.663643
$475$	2.42333	0.111190
$476$	0.481506	0.0220698
$477$	−17.2131	−0.788132
$478$	18.9265	0.865680
$479$	34.2977	1.56710	0.783551	−	0.621327i	$-0.213406\pi$
$479$	34.2977	1.56710	0.783551	+	0.621327i	$0.213406\pi$
$480$	3.08616	0.140863
$481$	21.1451	0.964132
$482$	8.25488	0.376000
$483$	−0.976915	−0.0444512
$484$	1.00000	0.0454545
$485$	10.4291	0.473561
$486$	10.5591	0.478969
$487$	8.85802	0.401395	0.200698	−	0.979653i	$-0.435679\pi$
$487$	8.85802	0.401395	0.200698	+	0.979653i	$0.435679\pi$
$488$	−4.27434	−0.193490
$489$	32.8324	1.48473
$490$	−6.99587	−0.316041
$491$	−11.8647	−0.535447	−0.267724	−	0.963496i	$-0.586271\pi$
$491$	−11.8647	−0.535447	−0.267724	+	0.963496i	$0.586271\pi$
$492$	2.33026	0.105056
$493$	34.3710	1.54799
$494$	13.4789	0.606445
$495$	6.52440	0.293250
$496$	−7.10237	−0.318906
$497$	−0.320778	−0.0143888
$498$	−14.6090	−0.654645
$499$	−0.998958	−0.0447195	−0.0223597	−	0.999750i	$-0.507118\pi$
$499$	−0.998958	−0.0447195	−0.0223597	+	0.999750i	$0.507118\pi$
$500$	1.00000	0.0447214
$501$	−47.4109	−2.11816
$502$	−15.0182	−0.670297
$503$	12.2411	0.545804	0.272902	−	0.962042i	$-0.412016\pi$
$503$	12.2411	0.545804	0.272902	+	0.962042i	$0.412016\pi$
$504$	−0.419171	−0.0186714
$505$	−2.58736	−0.115136
$506$	4.92705	0.219034
$507$	55.3577	2.45852
$508$	−0.939599	−0.0416880
$509$	22.5847	1.00105	0.500524	−	0.865723i	$-0.333141\pi$
$509$	22.5847	1.00105	0.500524	+	0.865723i	$0.333141\pi$
$510$	−23.1297	−1.02420
$511$	0.0642466	0.00284210
$512$	1.00000	0.0441942
$513$	26.3583	1.16375
$514$	−25.2141	−1.11215
$515$	−11.1056	−0.489373
$516$	1.95877	0.0862301
$517$	−0.525114	−0.0230945
$518$	−0.244240	−0.0107313
$519$	15.4373	0.677621
$520$	5.56214	0.243916
$521$	−16.6119	−0.727783	−0.363891	−	0.931441i	$-0.618552\pi$
$521$	−16.6119	−0.727783	−0.363891	+	0.931441i	$0.618552\pi$
$522$	−29.9214	−1.30962
$523$	36.0479	1.57626	0.788132	−	0.615506i	$-0.211048\pi$
$523$	36.0479	1.57626	0.788132	+	0.615506i	$0.211048\pi$
$524$	−10.0525	−0.439147
$525$	−0.198276	−0.00865346
$526$	23.1840	1.01087
$527$	53.2298	2.31873
$528$	3.08616	0.134308
$529$	1.27586	0.0554722
$530$	−2.63826	−0.114599
$531$	90.9408	3.94649
$532$	−0.155691	−0.00675006
$533$	4.19978	0.181913
$534$	38.3220	1.65835
$535$	−5.80496	−0.250970
$536$	−1.45417	−0.0628106
$537$	−14.5806	−0.629199
$538$	−25.3005	−1.09078
$539$	−6.99587	−0.301334
$540$	10.8769	0.468067
$541$	−35.8332	−1.54059	−0.770296	−	0.637687i	$-0.779891\pi$
$541$	−35.8332	−1.54059	−0.770296	+	0.637687i	$0.779891\pi$
$542$	−22.2007	−0.953603
$543$	−10.7274	−0.460358
$544$	−7.49466	−0.321331
$545$	−2.45195	−0.105030
$546$	−1.10284	−0.0471970
$547$	−9.61245	−0.410998	−0.205499	−	0.978657i	$-0.565882\pi$
$547$	−9.61245	−0.410998	−0.205499	+	0.978657i	$0.565882\pi$
$548$	−0.182257	−0.00778564
$549$	−27.8875	−1.19021
$550$	1.00000	0.0426401
$551$	−11.1136	−0.473454
$552$	15.2057	0.647197
$553$	0.300784	0.0127906
$554$	−13.0807	−0.555745
$555$	11.7324	0.498012
$556$	−9.70087	−0.411409
$557$	−40.8700	−1.73172	−0.865859	−	0.500287i	$-0.833228\pi$
$557$	−40.8700	−1.73172	−0.865859	+	0.500287i	$0.833228\pi$
$558$	−46.3387	−1.96167
$559$	3.53026	0.149314
$560$	−0.0642466	−0.00271492
$561$	−23.1297	−0.976538
$562$	0.591860	0.0249661
$563$	−40.6654	−1.71384	−0.856921	−	0.515448i	$-0.827626\pi$
$563$	−40.6654	−1.71384	−0.856921	+	0.515448i	$0.827626\pi$
$564$	−1.62059	−0.0682390
$565$	−15.6199	−0.657136
$566$	22.5080	0.946081
$567$	−0.899108	−0.0377590
$568$	4.99291	0.209498
$569$	8.31259	0.348482	0.174241	−	0.984703i	$-0.444253\pi$
$569$	8.31259	0.348482	0.174241	+	0.984703i	$0.444253\pi$
$570$	7.47880	0.313252
$571$	17.7425	0.742501	0.371251	−	0.928533i	$-0.378929\pi$
$571$	17.7425	0.742501	0.371251	+	0.928533i	$0.378929\pi$
$572$	5.56214	0.232565
$573$	−37.7564	−1.57729
$574$	−0.0485105	−0.00202479
$575$	4.92705	0.205472
$576$	6.52440	0.271850
$577$	18.9154	0.787458	0.393729	−	0.919226i	$-0.371185\pi$
$577$	18.9154	0.787458	0.393729	+	0.919226i	$0.371185\pi$
$578$	39.1699	1.62925
$579$	−47.5548	−1.97631
$580$	−4.58607	−0.190426
$581$	0.304125	0.0126172
$582$	32.1859	1.33415
$583$	−2.63826	−0.109265
$584$	−1.00000	−0.0413803
$585$	36.2896	1.50039
$586$	26.8131	1.10764
$587$	4.90833	0.202589	0.101294	−	0.994857i	$-0.467702\pi$
$587$	4.90833	0.202589	0.101294	+	0.994857i	$0.467702\pi$
$588$	−21.5904	−0.890373
$589$	−17.2114	−0.709183
$590$	13.9386	0.573841
$591$	36.5483	1.50340
$592$	3.80161	0.156245
$593$	−44.3829	−1.82259	−0.911294	−	0.411756i	$-0.864915\pi$
$593$	−44.3829	−1.82259	−0.911294	+	0.411756i	$0.864915\pi$
$594$	10.8769	0.446284
$595$	0.481506	0.0197398
$596$	15.4733	0.633813
$597$	4.63870	0.189849
$598$	27.4050	1.12067
$599$	−31.9662	−1.30610	−0.653052	−	0.757313i	$-0.726512\pi$
$599$	−31.9662	−1.30610	−0.653052	+	0.757313i	$0.726512\pi$
$600$	3.08616	0.125992
$601$	−28.2137	−1.15086	−0.575430	−	0.817851i	$-0.695165\pi$
$601$	−28.2137	−1.15086	−0.575430	+	0.817851i	$0.695165\pi$
$602$	−0.0407770	−0.00166195
$603$	−9.48760	−0.386365
$604$	−3.86321	−0.157192
$605$	1.00000	0.0406558
$606$	−7.98503	−0.324370
$607$	−35.6574	−1.44729	−0.723644	−	0.690174i	$-0.757534\pi$
$607$	−35.6574	−1.44729	−0.723644	+	0.690174i	$0.757534\pi$
$608$	2.42333	0.0982791
$609$	0.909305	0.0368469
$610$	−4.27434	−0.173063
$611$	−2.92076	−0.118161
$612$	−48.8982	−1.97659
$613$	−12.5804	−0.508119	−0.254060	−	0.967189i	$-0.581766\pi$
$613$	−12.5804	−0.508119	−0.254060	+	0.967189i	$0.581766\pi$
$614$	18.0313	0.727685
$615$	2.33026	0.0939651
$616$	−0.0642466	−0.00258857
$617$	17.3219	0.697354	0.348677	−	0.937243i	$-0.386631\pi$
$617$	17.3219	0.697354	0.348677	+	0.937243i	$0.386631\pi$
$618$	−34.2738	−1.37870
$619$	27.6056	1.10956	0.554782	−	0.831996i	$-0.312802\pi$
$619$	27.6056	1.10956	0.554782	+	0.831996i	$0.312802\pi$
$620$	−7.10237	−0.285238
$621$	53.5910	2.15053
$622$	6.41047	0.257037
$623$	−0.797773	−0.0319621
$624$	17.1657	0.687177
$625$	1.00000	0.0400000
$626$	26.8853	1.07455
$627$	7.47880	0.298674
$628$	20.1569	0.804347
$629$	−28.4917	−1.13604
$630$	−0.419171	−0.0167002
$631$	10.1026	0.402178	0.201089	−	0.979573i	$-0.435552\pi$
$631$	10.1026	0.402178	0.201089	+	0.979573i	$0.435552\pi$
$632$	−4.68171	−0.186229
$633$	28.9194	1.14944
$634$	6.62012	0.262918
$635$	−0.939599	−0.0372869
$636$	−8.14209	−0.322855
$637$	−38.9120	−1.54175
$638$	−4.58607	−0.181564
$639$	32.5758	1.28868
$640$	1.00000	0.0395285
$641$	−29.2640	−1.15586	−0.577930	−	0.816087i	$-0.696139\pi$
$641$	−29.2640	−1.15586	−0.577930	+	0.816087i	$0.696139\pi$
$642$	−17.9151	−0.707051
$643$	−10.9387	−0.431382	−0.215691	−	0.976462i	$-0.569200\pi$
$643$	−10.9387	−0.431382	−0.215691	+	0.976462i	$0.569200\pi$
$644$	−0.316547	−0.0124737
$645$	1.95877	0.0771266
$646$	−18.1620	−0.714576
$647$	39.8444	1.56644	0.783222	−	0.621742i	$-0.213575\pi$
$647$	39.8444	1.56644	0.783222	+	0.621742i	$0.213575\pi$
$648$	13.9946	0.549761
$649$	13.9386	0.547136
$650$	5.56214	0.218165
$651$	1.40823	0.0551927
$652$	10.6386	0.416639
$653$	19.4138	0.759719	0.379860	−	0.925044i	$-0.375972\pi$
$653$	19.4138	0.759719	0.379860	+	0.925044i	$0.375972\pi$
$654$	−7.56711	−0.295897
$655$	−10.0525	−0.392785
$656$	0.755066	0.0294804
$657$	−6.52440	−0.254541
$658$	0.0337368	0.00131520
$659$	−35.2355	−1.37258	−0.686290	−	0.727328i	$-0.740762\pi$
$659$	−35.2355	−1.37258	−0.686290	+	0.727328i	$0.740762\pi$
$660$	3.08616	0.120129
$661$	34.2255	1.33122	0.665609	−	0.746301i	$-0.268172\pi$
$661$	34.2255	1.33122	0.665609	+	0.746301i	$0.268172\pi$
$662$	17.5582	0.682419
$663$	−128.651	−4.99638
$664$	−4.73371	−0.183703
$665$	−0.155691	−0.00603743
$666$	24.8032	0.961105
$667$	−22.5958	−0.874913
$668$	−15.3624	−0.594390
$669$	−69.5065	−2.68728
$670$	−1.45417	−0.0561795
$671$	−4.27434	−0.165009
$672$	−0.198276	−0.00764865
$673$	2.55472	0.0984771	0.0492385	−	0.998787i	$-0.484321\pi$
$673$	2.55472	0.0984771	0.0492385	+	0.998787i	$0.484321\pi$
$674$	21.4259	0.825294
$675$	10.8769	0.418652
$676$	17.9374	0.689899
$677$	21.0147	0.807660	0.403830	−	0.914834i	$-0.367679\pi$
$677$	21.0147	0.807660	0.403830	+	0.914834i	$0.367679\pi$
$678$	−48.2057	−1.85133
$679$	−0.670034	−0.0257135
$680$	−7.49466	−0.287407
$681$	−67.4662	−2.58531
$682$	−7.10237	−0.271964
$683$	−25.1656	−0.962936	−0.481468	−	0.876464i	$-0.659896\pi$
$683$	−25.1656	−0.962936	−0.481468	+	0.876464i	$0.659896\pi$
$684$	15.8108	0.604541
$685$	−0.182257	−0.00696369
$686$	0.899188	0.0343311
$687$	−24.2641	−0.925733
$688$	0.634695	0.0241975
$689$	−14.6743	−0.559048
$690$	15.2057	0.578871
$691$	35.3884	1.34624	0.673119	−	0.739534i	$-0.264954\pi$
$691$	35.3884	1.34624	0.673119	+	0.739534i	$0.264954\pi$
$692$	5.00209	0.190151
$693$	−0.419171	−0.0159230
$694$	−23.8923	−0.906940
$695$	−9.70087	−0.367975
$696$	−14.1534	−0.536481
$697$	−5.65896	−0.214348
$698$	29.7403	1.12569
$699$	−62.9471	−2.38088
$700$	−0.0642466	−0.00242829
$701$	−45.9497	−1.73550	−0.867749	−	0.497003i	$-0.834434\pi$
$701$	−45.9497	−1.73550	−0.867749	+	0.497003i	$0.834434\pi$
$702$	60.4987	2.28338
$703$	9.21255	0.347458
$704$	1.00000	0.0376889
$705$	−1.62059	−0.0610349
$706$	10.2490	0.385725
$707$	0.166229	0.00625170
$708$	43.0167	1.61667
$709$	−36.6004	−1.37456	−0.687279	−	0.726393i	$-0.741195\pi$
$709$	−36.6004	−1.37456	−0.687279	+	0.726393i	$0.741195\pi$
$710$	4.99291	0.187381
$711$	−30.5454	−1.14554
$712$	12.4173	0.465360
$713$	−34.9937	−1.31053
$714$	1.48601	0.0556124
$715$	5.56214	0.208012
$716$	−4.72451	−0.176563
$717$	58.4104	2.18138
$718$	−22.5144	−0.840231
$719$	21.9124	0.817196	0.408598	−	0.912714i	$-0.366018\pi$
$719$	21.9124	0.817196	0.408598	+	0.912714i	$0.366018\pi$
$720$	6.52440	0.243150
$721$	0.713501	0.0265722
$722$	−13.1275	−0.488554
$723$	25.4759	0.947460
$724$	−3.47598	−0.129184
$725$	−4.58607	−0.170322
$726$	3.08616	0.114538
$727$	−0.503577	−0.0186766	−0.00933831	−	0.999956i	$-0.502973\pi$
$727$	−0.503577	−0.0186766	−0.00933831	+	0.999956i	$0.502973\pi$
$728$	−0.357349	−0.0132442
$729$	−9.39690	−0.348033
$730$	−1.00000	−0.0370117
$731$	−4.75682	−0.175937
$732$	−13.1913	−0.487565
$733$	−4.58646	−0.169405	−0.0847023	−	0.996406i	$-0.526994\pi$
$733$	−4.58646	−0.169405	−0.0847023	+	0.996406i	$0.526994\pi$
$734$	−3.76698	−0.139042
$735$	−21.5904	−0.796374
$736$	4.92705	0.181614
$737$	−1.45417	−0.0535651
$738$	4.92636	0.181342
$739$	−24.3354	−0.895193	−0.447596	−	0.894236i	$-0.647720\pi$
$739$	−24.3354	−0.895193	−0.447596	+	0.894236i	$0.647720\pi$
$740$	3.80161	0.139750
$741$	41.5981	1.52814
$742$	0.169499	0.00622251
$743$	13.8615	0.508529	0.254264	−	0.967135i	$-0.418167\pi$
$743$	13.8615	0.508529	0.254264	+	0.967135i	$0.418167\pi$
$744$	−21.9191	−0.803592
$745$	15.4733	0.566900
$746$	16.7422	0.612975
$747$	−30.8846	−1.13001
$748$	−7.49466	−0.274032
$749$	0.372949	0.0136273
$750$	3.08616	0.112691
$751$	21.8464	0.797186	0.398593	−	0.917128i	$-0.369499\pi$
$751$	21.8464	0.797186	0.398593	+	0.917128i	$0.369499\pi$
$752$	−0.525114	−0.0191489
$753$	−46.3487	−1.68904
$754$	−25.5083	−0.928959
$755$	−3.86321	−0.140597
$756$	−0.698803	−0.0254152
$757$	52.0340	1.89121	0.945603	−	0.325322i	$-0.105473\pi$
$757$	52.0340	1.89121	0.945603	+	0.325322i	$0.105473\pi$
$758$	14.3990	0.522996
$759$	15.2057	0.551932
$760$	2.42333	0.0879035
$761$	0.770007	0.0279127	0.0139564	−	0.999903i	$-0.495557\pi$
$761$	0.770007	0.0279127	0.0139564	+	0.999903i	$0.495557\pi$
$762$	−2.89976	−0.105047
$763$	0.157529	0.00570295
$764$	−12.2341	−0.442613
$765$	−48.8982	−1.76792
$766$	−10.9828	−0.396824
$767$	77.5282	2.79938
$768$	3.08616	0.111362
$769$	−6.14635	−0.221643	−0.110822	−	0.993840i	$-0.535348\pi$
$769$	−6.14635	−0.221643	−0.110822	+	0.993840i	$0.535348\pi$
$770$	−0.0642466	−0.00231529
$771$	−77.8150	−2.80244
$772$	−15.4090	−0.554583
$773$	−35.4610	−1.27544	−0.637722	−	0.770266i	$-0.720123\pi$
$773$	−35.4610	−1.27544	−0.637722	+	0.770266i	$0.720123\pi$
$774$	4.14101	0.148845
$775$	−7.10237	−0.255125
$776$	10.4291	0.374383
$777$	−0.753766	−0.0270412
$778$	−27.8581	−0.998760
$779$	1.82978	0.0655585
$780$	17.1657	0.614629
$781$	4.99291	0.178660
$782$	−36.9266	−1.32049
$783$	−49.8821	−1.78264
$784$	−6.99587	−0.249853
$785$	20.1569	0.719430
$786$	−31.0238	−1.10658
$787$	−30.0773	−1.07214	−0.536070	−	0.844174i	$-0.680092\pi$
$787$	−30.0773	−1.07214	−0.536070	+	0.844174i	$0.680092\pi$
$788$	11.8426	0.421876
$789$	71.5497	2.54724
$790$	−4.68171	−0.166568
$791$	1.00353	0.0356814
$792$	6.52440	0.231835
$793$	−23.7745	−0.844257
$794$	−2.31045	−0.0819947
$795$	−8.14209	−0.288770
$796$	1.50306	0.0532747
$797$	−15.1355	−0.536127	−0.268063	−	0.963401i	$-0.586384\pi$
$797$	−15.1355	−0.536127	−0.268063	+	0.963401i	$0.586384\pi$
$798$	−0.480487	−0.0170091
$799$	3.93555	0.139230
$800$	1.00000	0.0353553
$801$	81.0158	2.86255
$802$	22.2471	0.785572
$803$	−1.00000	−0.0352892
$804$	−4.48781	−0.158273
$805$	−0.316547	−0.0111568
$806$	−39.5043	−1.39148
$807$	−78.0814	−2.74860
$808$	−2.58736	−0.0910232
$809$	−33.8382	−1.18969	−0.594843	−	0.803842i	$-0.702786\pi$
$809$	−33.8382	−1.18969	−0.594843	+	0.803842i	$0.702786\pi$
$810$	13.9946	0.491721
$811$	−4.99958	−0.175559	−0.0877795	−	0.996140i	$-0.527977\pi$
$811$	−4.99958	−0.175559	−0.0877795	+	0.996140i	$0.527977\pi$
$812$	0.294639	0.0103398
$813$	−68.5151	−2.40293
$814$	3.80161	0.133246
$815$	10.6386	0.372653
$816$	−23.1297	−0.809703
$817$	1.53808	0.0538105
$818$	−27.6451	−0.966587
$819$	−2.33149	−0.0814687
$820$	0.755066	0.0263681
$821$	−40.2370	−1.40428	−0.702141	−	0.712038i	$-0.747773\pi$
$821$	−40.2370	−1.40428	−0.702141	+	0.712038i	$0.747773\pi$
$822$	−0.562475	−0.0196186
$823$	−14.5331	−0.506594	−0.253297	−	0.967389i	$-0.581515\pi$
$823$	−14.5331	−0.506594	−0.253297	+	0.967389i	$0.581515\pi$
$824$	−11.1056	−0.386884
$825$	3.08616	0.107446
$826$	−0.895506	−0.0311586
$827$	40.6035	1.41192	0.705960	−	0.708251i	$-0.250516\pi$
$827$	40.6035	1.41192	0.705960	+	0.708251i	$0.250516\pi$
$828$	32.1461	1.11715
$829$	0.00576945	0.000200381 0	0.000100191	−	1.00000i	$-0.499968\pi$
$829$	0.00576945	0.000200381 0	0.000100191		1.00000i	$0.499968\pi$
$830$	−4.73371	−0.164309
$831$	−40.3691	−1.40039
$832$	5.56214	0.192832
$833$	52.4317	1.81665
$834$	−29.9385	−1.03668
$835$	−15.3624	−0.531638
$836$	2.42333	0.0838127
$837$	−77.2516	−2.67021
$838$	16.7963	0.580219
$839$	−23.6465	−0.816367	−0.408183	−	0.912900i	$-0.633838\pi$
$839$	−23.6465	−0.816367	−0.408183	+	0.912900i	$0.633838\pi$
$840$	−0.198276	−0.00684116
$841$	−7.96798	−0.274758
$842$	10.5644	0.364073
$843$	1.82658	0.0629106
$844$	9.37065	0.322551
$845$	17.9374	0.617065
$846$	−3.42606	−0.117790
$847$	−0.0642466	−0.00220754
$848$	−2.63826	−0.0905981
$849$	69.4633	2.38397
$850$	−7.49466	−0.257065
$851$	18.7307	0.642081
$852$	15.4089	0.527901
$853$	−37.1827	−1.27311	−0.636555	−	0.771231i	$-0.719641\pi$
$853$	−37.1827	−1.27311	−0.636555	+	0.771231i	$0.719641\pi$
$854$	0.274612	0.00939704
$855$	15.8108	0.540718
$856$	−5.80496	−0.198409
$857$	−27.8404	−0.951009	−0.475505	−	0.879713i	$-0.657735\pi$
$857$	−27.8404	−0.951009	−0.475505	+	0.879713i	$0.657735\pi$
$858$	17.1657	0.586026
$859$	−16.6332	−0.567516	−0.283758	−	0.958896i	$-0.591581\pi$
$859$	−16.6332	−0.567516	−0.283758	+	0.958896i	$0.591581\pi$
$860$	0.634695	0.0216429
$861$	−0.149711	−0.00510214
$862$	24.3828	0.830482
$863$	−46.9955	−1.59975	−0.799873	−	0.600169i	$-0.795100\pi$
$863$	−46.9955	−1.59975	−0.799873	+	0.600169i	$0.795100\pi$
$864$	10.8769	0.370039
$865$	5.00209	0.170076
$866$	5.47148	0.185929
$867$	120.885	4.10546
$868$	0.456303	0.0154879
$869$	−4.68171	−0.158816
$870$	−14.1534	−0.479844
$871$	−8.08830	−0.274062
$872$	−2.45195	−0.0830334
$873$	68.0436	2.30293
$874$	11.9399	0.403873
$875$	−0.0642466	−0.00217193
$876$	−3.08616	−0.104272
$877$	−11.8045	−0.398609	−0.199304	−	0.979938i	$-0.563868\pi$
$877$	−11.8045	−0.398609	−0.199304	+	0.979938i	$0.563868\pi$
$878$	10.0675	0.339761
$879$	82.7497	2.79108
$880$	1.00000	0.0337100
$881$	48.9749	1.65001	0.825003	−	0.565128i	$-0.191173\pi$
$881$	48.9749	1.65001	0.825003	+	0.565128i	$0.191173\pi$
$882$	−45.6439	−1.53691
$883$	−34.6150	−1.16489	−0.582443	−	0.812871i	$-0.697903\pi$
$883$	−34.6150	−1.16489	−0.582443	+	0.812871i	$0.697903\pi$
$884$	−41.6863	−1.40206
$885$	43.0167	1.44599
$886$	−26.2135	−0.880660
$887$	−16.1556	−0.542453	−0.271227	−	0.962515i	$-0.587429\pi$
$887$	−16.1556	−0.542453	−0.271227	+	0.962515i	$0.587429\pi$
$888$	11.7324	0.393713
$889$	0.0603661	0.00202461
$890$	12.4173	0.416230
$891$	13.9946	0.468838
$892$	−22.5220	−0.754092
$893$	−1.27253	−0.0425834
$894$	47.7533	1.59711
$895$	−4.72451	−0.157923
$896$	−0.0642466	−0.00214633
$897$	84.5762	2.82392
$898$	4.10283	0.136913
$899$	32.5719	1.08633
$900$	6.52440	0.217480
$901$	19.7728	0.658728
$902$	0.755066	0.0251410
$903$	−0.125845	−0.00418784
$904$	−15.6199	−0.519512
$905$	−3.47598	−0.115545
$906$	−11.9225	−0.396098
$907$	−24.3001	−0.806870	−0.403435	−	0.915008i	$-0.632184\pi$
$907$	−24.3001	−0.806870	−0.403435	+	0.915008i	$0.632184\pi$
$908$	−21.8609	−0.725479
$909$	−16.8810	−0.559908
$910$	−0.357349	−0.0118460
$911$	8.97903	0.297488	0.148744	−	0.988876i	$-0.452477\pi$
$911$	8.97903	0.297488	0.148744	+	0.988876i	$0.452477\pi$
$912$	7.47880	0.247648
$913$	−4.73371	−0.156663
$914$	−25.9553	−0.858526
$915$	−13.1913	−0.436092
$916$	−7.86222	−0.259775
$917$	0.645842	0.0213276
$918$	−81.5185	−2.69051
$919$	−6.23671	−0.205730	−0.102865	−	0.994695i	$-0.532801\pi$
$919$	−6.23671	−0.205730	−0.102865	+	0.994695i	$0.532801\pi$
$920$	4.92705	0.162440
$921$	55.6476	1.83365
$922$	8.50836	0.280208
$923$	27.7712	0.914102
$924$	−0.198276	−0.00652279
$925$	3.80161	0.124996
$926$	12.9267	0.424798
$927$	−72.4577	−2.37982
$928$	−4.58607	−0.150545
$929$	−4.09326	−0.134296	−0.0671478	−	0.997743i	$-0.521390\pi$
$929$	−4.09326	−0.134296	−0.0671478	+	0.997743i	$0.521390\pi$
$930$	−21.9191	−0.718754
$931$	−16.9533	−0.555623
$932$	−20.3966	−0.668111
$933$	19.7838	0.647692
$934$	13.2177	0.432497
$935$	−7.49466	−0.245101
$936$	36.2896	1.18616
$937$	19.1836	0.626699	0.313350	−	0.949638i	$-0.398549\pi$
$937$	19.1836	0.626699	0.313350	+	0.949638i	$0.398549\pi$
$938$	0.0934256	0.00305045
$939$	82.9723	2.70770
$940$	−0.525114	−0.0171273
$941$	−53.9251	−1.75791	−0.878954	−	0.476907i	$-0.841758\pi$
$941$	−53.9251	−1.75791	−0.878954	+	0.476907i	$0.841758\pi$
$942$	62.2074	2.02683
$943$	3.72025	0.121148
$944$	13.9386	0.453661
$945$	−0.698803	−0.0227321
$946$	0.634695	0.0206357
$947$	34.3995	1.11783	0.558917	−	0.829223i	$-0.311217\pi$
$947$	34.3995	1.11783	0.558917	+	0.829223i	$0.311217\pi$
$948$	−14.4485	−0.469266
$949$	−5.56214	−0.180555
$950$	2.42333	0.0786233
$951$	20.4308	0.662513
$952$	0.481506	0.0156057
$953$	17.9796	0.582416	0.291208	−	0.956660i	$-0.405943\pi$
$953$	17.9796	0.582416	0.291208	+	0.956660i	$0.405943\pi$
$954$	−17.2131	−0.557293
$955$	−12.2341	−0.395885
$956$	18.9265	0.612128
$957$	−14.1534	−0.457513
$958$	34.2977	1.10811
$959$	0.0117094	0.000378116 0
$960$	3.08616	0.0996055
$961$	19.4436	0.627213
$962$	21.1451	0.681744
$963$	−37.8739	−1.22047
$964$	8.25488	0.265872
$965$	−15.4090	−0.496034
$966$	−0.976915	−0.0314317
$967$	−49.0756	−1.57816	−0.789082	−	0.614288i	$-0.789443\pi$
$967$	−49.0756	−1.57816	−0.789082	+	0.614288i	$0.789443\pi$
$968$	1.00000	0.0321412
$969$	−56.0510	−1.80062
$970$	10.4291	0.334858
$971$	1.22258	0.0392345	0.0196172	−	0.999808i	$-0.493755\pi$
$971$	1.22258	0.0392345	0.0196172	+	0.999808i	$0.493755\pi$
$972$	10.5591	0.338682
$973$	0.623248	0.0199804
$974$	8.85802	0.283829
$975$	17.1657	0.549741
$976$	−4.27434	−0.136818
$977$	−35.6419	−1.14029	−0.570143	−	0.821545i	$-0.693112\pi$
$977$	−35.6419	−1.14029	−0.570143	+	0.821545i	$0.693112\pi$
$978$	32.8324	1.04986
$979$	12.4173	0.396860
$980$	−6.99587	−0.223475
$981$	−15.9975	−0.510761
$982$	−11.8647	−0.378618
$983$	17.8781	0.570222	0.285111	−	0.958495i	$-0.407970\pi$
$983$	17.8781	0.570222	0.285111	+	0.958495i	$0.407970\pi$
$984$	2.33026	0.0742859
$985$	11.8426	0.377338
$986$	34.3710	1.09460
$987$	0.104117	0.00331409
$988$	13.4789	0.428821
$989$	3.12718	0.0994384
$990$	6.52440	0.207359
$991$	45.1752	1.43504	0.717519	−	0.696538i	$-0.245277\pi$
$991$	45.1752	1.43504	0.717519	+	0.696538i	$0.245277\pi$
$992$	−7.10237	−0.225500
$993$	54.1875	1.71959
$994$	−0.320778	−0.0101744
$995$	1.50306	0.0476503
$996$	−14.6090	−0.462904
$997$	35.1455	1.11307	0.556535	−	0.830824i	$-0.312131\pi$
$997$	35.1455	1.11307	0.556535	+	0.830824i	$0.312131\pi$
$998$	−0.998958	−0.0316214
$999$	41.3496	1.30824

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8030.2.a.bl.1.18	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8030.2.a.bl.1.18	✓	19	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8030 = 2 \cdot 5 \cdot 11 \cdot 73 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8030.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +3.08616 q^{3} +1.00000 q^{4} +1.00000 q^{5} +3.08616 q^{6} -0.0642466 q^{7} +1.00000 q^{8} +6.52440 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +3.08616 q^{3} +1.00000 q^{4} +1.00000 q^{5} +3.08616 q^{6} -0.0642466 q^{7} +1.00000 q^{8} +6.52440 q^{9} +1.00000 q^{10} +1.00000 q^{11} +3.08616 q^{12} +5.56214 q^{13} -0.0642466 q^{14} +3.08616 q^{15} +1.00000 q^{16} -7.49466 q^{17} +6.52440 q^{18} +2.42333 q^{19} +1.00000 q^{20} -0.198276 q^{21} +1.00000 q^{22} +4.92705 q^{23} +3.08616 q^{24} +1.00000 q^{25} +5.56214 q^{26} +10.8769 q^{27} -0.0642466 q^{28} -4.58607 q^{29} +3.08616 q^{30} -7.10237 q^{31} +1.00000 q^{32} +3.08616 q^{33} -7.49466 q^{34} -0.0642466 q^{35} +6.52440 q^{36} +3.80161 q^{37} +2.42333 q^{38} +17.1657 q^{39} +1.00000 q^{40} +0.755066 q^{41} -0.198276 q^{42} +0.634695 q^{43} +1.00000 q^{44} +6.52440 q^{45} +4.92705 q^{46} -0.525114 q^{47} +3.08616 q^{48} -6.99587 q^{49} +1.00000 q^{50} -23.1297 q^{51} +5.56214 q^{52} -2.63826 q^{53} +10.8769 q^{54} +1.00000 q^{55} -0.0642466 q^{56} +7.47880 q^{57} -4.58607 q^{58} +13.9386 q^{59} +3.08616 q^{60} -4.27434 q^{61} -7.10237 q^{62} -0.419171 q^{63} +1.00000 q^{64} +5.56214 q^{65} +3.08616 q^{66} -1.45417 q^{67} -7.49466 q^{68} +15.2057 q^{69} -0.0642466 q^{70} +4.99291 q^{71} +6.52440 q^{72} -1.00000 q^{73} +3.80161 q^{74} +3.08616 q^{75} +2.42333 q^{76} -0.0642466 q^{77} +17.1657 q^{78} -4.68171 q^{79} +1.00000 q^{80} +13.9946 q^{81} +0.755066 q^{82} -4.73371 q^{83} -0.198276 q^{84} -7.49466 q^{85} +0.634695 q^{86} -14.1534 q^{87} +1.00000 q^{88} +12.4173 q^{89} +6.52440 q^{90} -0.357349 q^{91} +4.92705 q^{92} -21.9191 q^{93} -0.525114 q^{94} +2.42333 q^{95} +3.08616 q^{96} +10.4291 q^{97} -6.99587 q^{98} +6.52440 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q + 19 q^{2} + 10 q^{3} + 19 q^{4} + 19 q^{5} + 10 q^{6} + 8 q^{7} + 19 q^{8} + 27 q^{9}+O(q^{10}) \) 19 * q + 19 * q^2 + 10 * q^3 + 19 * q^4 + 19 * q^5 + 10 * q^6 + 8 * q^7 + 19 * q^8 + 27 * q^9	\( 19 q + 19 q^{2} + 10 q^{3} + 19 q^{4} + 19 q^{5} + 10 q^{6} + 8 q^{7} + 19 q^{8} + 27 q^{9} + 19 q^{10} + 19 q^{11} + 10 q^{12} + 16 q^{13} + 8 q^{14} + 10 q^{15} + 19 q^{16} + 12 q^{17} + 27 q^{18} + 12 q^{19} + 19 q^{20} + 3 q^{21} + 19 q^{22} + 26 q^{23} + 10 q^{24} + 19 q^{25} + 16 q^{26} + 25 q^{27} + 8 q^{28} + q^{29} + 10 q^{30} + 24 q^{31} + 19 q^{32} + 10 q^{33} + 12 q^{34} + 8 q^{35} + 27 q^{36} + 23 q^{37} + 12 q^{38} - 5 q^{39} + 19 q^{40} + 3 q^{42} + 8 q^{43} + 19 q^{44} + 27 q^{45} + 26 q^{46} + 34 q^{47} + 10 q^{48} + 27 q^{49} + 19 q^{50} + 15 q^{51} + 16 q^{52} + 25 q^{53} + 25 q^{54} + 19 q^{55} + 8 q^{56} + q^{57} + q^{58} + 24 q^{59} + 10 q^{60} + 31 q^{61} + 24 q^{62} + 15 q^{63} + 19 q^{64} + 16 q^{65} + 10 q^{66} + 24 q^{67} + 12 q^{68} + q^{69} + 8 q^{70} + 5 q^{71} + 27 q^{72} - 19 q^{73} + 23 q^{74} + 10 q^{75} + 12 q^{76} + 8 q^{77} - 5 q^{78} + 18 q^{79} + 19 q^{80} + 11 q^{81} + 12 q^{83} + 3 q^{84} + 12 q^{85} + 8 q^{86} + 12 q^{87} + 19 q^{88} + 27 q^{90} + 23 q^{91} + 26 q^{92} + 18 q^{93} + 34 q^{94} + 12 q^{95} + 10 q^{96} + 15 q^{97} + 27 q^{98} + 27 q^{99}+O(q^{100}) \) 19 * q + 19 * q^2 + 10 * q^3 + 19 * q^4 + 19 * q^5 + 10 * q^6 + 8 * q^7 + 19 * q^8 + 27 * q^9 + 19 * q^10 + 19 * q^11 + 10 * q^12 + 16 * q^13 + 8 * q^14 + 10 * q^15 + 19 * q^16 + 12 * q^17 + 27 * q^18 + 12 * q^19 + 19 * q^20 + 3 * q^21 + 19 * q^22 + 26 * q^23 + 10 * q^24 + 19 * q^25 + 16 * q^26 + 25 * q^27 + 8 * q^28 + q^29 + 10 * q^30 + 24 * q^31 + 19 * q^32 + 10 * q^33 + 12 * q^34 + 8 * q^35 + 27 * q^36 + 23 * q^37 + 12 * q^38 - 5 * q^39 + 19 * q^40 + 3 * q^42 + 8 * q^43 + 19 * q^44 + 27 * q^45 + 26 * q^46 + 34 * q^47 + 10 * q^48 + 27 * q^49 + 19 * q^50 + 15 * q^51 + 16 * q^52 + 25 * q^53 + 25 * q^54 + 19 * q^55 + 8 * q^56 + q^57 + q^58 + 24 * q^59 + 10 * q^60 + 31 * q^61 + 24 * q^62 + 15 * q^63 + 19 * q^64 + 16 * q^65 + 10 * q^66 + 24 * q^67 + 12 * q^68 + q^69 + 8 * q^70 + 5 * q^71 + 27 * q^72 - 19 * q^73 + 23 * q^74 + 10 * q^75 + 12 * q^76 + 8 * q^77 - 5 * q^78 + 18 * q^79 + 19 * q^80 + 11 * q^81 + 12 * q^83 + 3 * q^84 + 12 * q^85 + 8 * q^86 + 12 * q^87 + 19 * q^88 + 27 * q^90 + 23 * q^91 + 26 * q^92 + 18 * q^93 + 34 * q^94 + 12 * q^95 + 10 * q^96 + 15 * q^97 + 27 * q^98 + 27 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more