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Properties

Label 580.2.be.d.39.1
Level $580$
Weight $2$
Character 580.39
Analytic conductor $4.631$
Analytic rank $0$
Dimension $24$
CM discriminant -20
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [580,2,Mod(19,580)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("580.19"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(580, base_ring=CyclotomicField(28)) chi = DirichletCharacter(H, H._module([14, 14, 9])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 580 = 2^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 580.be (of order \(28\), degree \(12\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.63132331723\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(2\) over \(\Q(\zeta_{28})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{28}]$

Embedding invariants

Embedding label 39.1
Character \(\chi\) \(=\) 580.39
Dual form 580.2.be.d.119.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.19745 - 0.752407i) q^{2} +(-3.20702 - 1.12218i) q^{3} +(0.867767 - 1.80194i) q^{4} +(-2.18001 - 0.497572i) q^{5} +(-4.68458 + 1.06922i) q^{6} +(-2.66698 + 1.28435i) q^{7} +(-0.316683 - 2.81064i) q^{8} +(6.68017 + 5.32726i) q^{9} +(-2.98482 + 1.04443i) q^{10} +(-4.80505 + 4.80505i) q^{12} +(-2.22722 + 3.54460i) q^{14} +(6.43295 + 4.04209i) q^{15} +(-2.49396 - 3.12733i) q^{16} +(12.0074 + 1.35291i) q^{18} +(-2.78833 + 3.49646i) q^{20} +(9.99433 - 1.12609i) q^{21} +(0.856672 + 3.75333i) q^{23} +(-2.13845 + 9.36916i) q^{24} +(4.50484 + 2.16942i) q^{25} +(-10.0222 - 15.9503i) q^{27} +5.92025i q^{28} +(4.64329 + 2.72760i) q^{29} +10.7444 q^{30} +(-5.33941 - 1.86834i) q^{32} +(6.45308 - 1.47287i) q^{35} +(15.3962 - 7.41443i) q^{36} +(-0.708126 + 6.28479i) q^{40} +(-8.78599 + 8.78599i) q^{41} +(11.1204 - 8.86824i) q^{42} +(-6.34328 + 10.0953i) q^{43} +(-11.9121 - 14.9373i) q^{45} +(3.84985 + 3.84985i) q^{46} +(4.22537 + 0.476085i) q^{47} +(4.48874 + 12.8281i) q^{48} +(1.09880 - 1.37785i) q^{49} +(7.02661 - 0.791708i) q^{50} +(-24.0022 - 11.5589i) q^{54} +(4.45444 + 7.08919i) q^{56} +(7.61237 - 0.227480i) q^{58} +(12.8659 - 8.08418i) q^{60} +(-14.2939 - 5.00164i) q^{61} +(-24.6579 - 5.62802i) q^{63} +(-7.79942 + 1.78017i) q^{64} +(-6.39565 - 5.10036i) q^{67} +(1.46456 - 12.9983i) q^{69} +(6.61904 - 6.61904i) q^{70} +(12.8575 - 20.4626i) q^{72} +(-12.0126 - 12.0126i) q^{75} +(3.88077 + 8.05851i) q^{80} +(8.53849 + 37.4096i) q^{81} +(-3.91013 + 17.1314i) q^{82} +(12.9707 + 6.24634i) q^{83} +(6.64361 - 18.9863i) q^{84} +16.8613i q^{86} +(-11.8303 - 13.9581i) q^{87} +(-15.9746 + 10.0375i) q^{89} +(-25.5031 - 8.92392i) q^{90} +(7.50665 + 1.71334i) q^{92} +(5.41788 - 2.60911i) q^{94} +(15.0270 + 11.9836i) q^{96} +(0.279051 - 2.47664i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 4 q^{2} + 4 q^{3} - 30 q^{7} - 8 q^{8} - 8 q^{12} - 12 q^{14} + 70 q^{15} + 16 q^{16} + 4 q^{18} - 12 q^{21} - 10 q^{23} - 40 q^{24} + 20 q^{25} + 58 q^{27} + 12 q^{29} - 16 q^{32} + 104 q^{36} + 24 q^{41}+ \cdots + 84 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/580\mathbb{Z}\right)^\times\).

\(n\) \(117\) \(291\) \(321\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{23}{28}\right)\)

Coefficient data

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



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.19745 − 0.752407i 0.846724 − 0.532032i
\(3\) −3.20702 − 1.12218i −1.85157 − 0.647893i −0.990151 − 0.140003i \(-0.955289\pi\)
−0.861422 − 0.507890i \(-0.830425\pi\)
\(4\) 0.867767 − 1.80194i 0.433884 − 0.900969i
\(5\) −2.18001 − 0.497572i −0.974928 − 0.222521i
\(6\) −4.68458 + 1.06922i −1.91247 + 0.436509i
\(7\) −2.66698 + 1.28435i −1.00802 + 0.485439i −0.863654 − 0.504085i \(-0.831830\pi\)
−0.144370 + 0.989524i \(0.546115\pi\)
\(8\) −0.316683 − 2.81064i −0.111964 − 0.993712i
\(9\) 6.68017 + 5.32726i 2.22672 + 1.77575i
\(10\) −2.98482 + 1.04443i −0.943883 + 0.330279i
\(11\) 0 0 0.111964 − 0.993712i \(-0.464286\pi\)
−0.111964 + 0.993712i \(0.535714\pi\)
\(12\) −4.80505 + 4.80505i −1.38710 + 1.38710i
\(13\) 0 0 0.781831 − 0.623490i \(-0.214286\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(14\) −2.22722 + 3.54460i −0.595249 + 0.947333i
\(15\) 6.43295 + 4.04209i 1.66098 + 1.04366i
\(16\) −2.49396 − 3.12733i −0.623490 − 0.781831i
\(17\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(18\) 12.0074 + 1.35291i 2.83018 + 0.318885i
\(19\) 0 0 −0.330279 − 0.943883i \(-0.607143\pi\)
0.330279 + 0.943883i \(0.392857\pi\)
\(20\) −2.78833 + 3.49646i −0.623490 + 0.781831i
\(21\) 9.99433 − 1.12609i 2.18094 − 0.245733i
\(22\) 0 0
\(23\) 0.856672 + 3.75333i 0.178629 + 0.782623i 0.982264 + 0.187501i \(0.0600389\pi\)
−0.803636 + 0.595121i \(0.797104\pi\)
\(24\) −2.13845 + 9.36916i −0.436509 + 1.91247i
\(25\) 4.50484 + 2.16942i 0.900969 + 0.433884i
\(26\) 0 0
\(27\) −10.0222 − 15.9503i −1.92878 − 3.06964i
\(28\) 5.92025i 1.11882i
\(29\) 4.64329 + 2.72760i 0.862238 + 0.506503i
\(30\) 10.7444 1.96165
\(31\) 0 0 0.846724 − 0.532032i \(-0.178571\pi\)
−0.846724 + 0.532032i \(0.821429\pi\)
\(32\) −5.33941 − 1.86834i −0.943883 − 0.330279i
\(33\) 0 0
\(34\) 0 0
\(35\) 6.45308 − 1.47287i 1.09077 − 0.248961i
\(36\) 15.3962 − 7.41443i 2.56604 − 1.23574i
\(37\) 0 0 −0.111964 − 0.993712i \(-0.535714\pi\)
0.111964 + 0.993712i \(0.464286\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −0.708126 + 6.28479i −0.111964 + 0.993712i
\(41\) −8.78599 + 8.78599i −1.37214 + 1.37214i −0.514877 + 0.857264i \(0.672163\pi\)
−0.857264 + 0.514877i \(0.827837\pi\)
\(42\) 11.1204 − 8.86824i 1.71592 − 1.36840i
\(43\) −6.34328 + 10.0953i −0.967341 + 1.53951i −0.130944 + 0.991390i \(0.541801\pi\)
−0.836397 + 0.548125i \(0.815342\pi\)
\(44\) 0 0
\(45\) −11.9121 − 14.9373i −1.77575 − 2.22672i
\(46\) 3.84985 + 3.84985i 0.567630 + 0.567630i
\(47\) 4.22537 + 0.476085i 0.616334 + 0.0694442i 0.414612 − 0.909998i \(-0.363917\pi\)
0.201723 + 0.979443i \(0.435346\pi\)
\(48\) 4.48874 + 12.8281i 0.647893 + 1.85157i
\(49\) 1.09880 − 1.37785i 0.156971 − 0.196835i
\(50\) 7.02661 − 0.791708i 0.993712 − 0.111964i
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 0.222521 − 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(54\) −24.0022 − 11.5589i −3.26629 − 1.57296i
\(55\) 0 0
\(56\) 4.45444 + 7.08919i 0.595249 + 0.947333i
\(57\) 0 0
\(58\) 7.61237 − 0.227480i 0.999554 − 0.0298696i
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 12.8659 − 8.08418i 1.66098 − 1.04366i
\(61\) −14.2939 − 5.00164i −1.83014 − 0.640395i −0.996052 − 0.0887673i \(-0.971707\pi\)
−0.834091 − 0.551627i \(-0.814007\pi\)
\(62\) 0 0
\(63\) −24.6579 − 5.62802i −3.10661 − 0.709063i
\(64\) −7.79942 + 1.78017i −0.974928 + 0.222521i
\(65\) 0 0
\(66\) 0 0
\(67\) −6.39565 − 5.10036i −0.781353 − 0.623108i 0.149392 − 0.988778i \(-0.452268\pi\)
−0.930745 + 0.365670i \(0.880840\pi\)
\(68\) 0 0
\(69\) 1.46456 − 12.9983i 0.176312 − 1.56482i
\(70\) 6.61904 − 6.61904i 0.791126 − 0.791126i
\(71\) 0 0 0.781831 − 0.623490i \(-0.214286\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(72\) 12.8575 − 20.4626i 1.51527 − 2.41154i
\(73\) 0 0 −0.846724 − 0.532032i \(-0.821429\pi\)
0.846724 + 0.532032i \(0.178571\pi\)
\(74\) 0 0
\(75\) −12.0126 − 12.0126i −1.38710 − 1.38710i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.993712 − 0.111964i \(-0.0357143\pi\)
−0.993712 + 0.111964i \(0.964286\pi\)
\(80\) 3.88077 + 8.05851i 0.433884 + 0.900969i
\(81\) 8.53849 + 37.4096i 0.948721 + 4.15662i
\(82\) −3.91013 + 17.1314i −0.431802 + 1.89185i
\(83\) 12.9707 + 6.24634i 1.42371 + 0.685625i 0.977817 − 0.209459i \(-0.0671703\pi\)
0.445897 + 0.895084i \(0.352885\pi\)
\(84\) 6.64361 − 18.9863i 0.724877 − 2.07158i
\(85\) 0 0
\(86\) 16.8613i 1.81820i
\(87\) −11.8303 − 13.9581i −1.26834 − 1.49647i
\(88\) 0 0
\(89\) −15.9746 + 10.0375i −1.69331 + 1.06397i −0.839747 + 0.542978i \(0.817297\pi\)
−0.853559 + 0.520997i \(0.825560\pi\)
\(90\) −25.5031 − 8.92392i −2.68826 − 0.940664i
\(91\) 0 0
\(92\) 7.50665 + 1.71334i 0.782623 + 0.178629i
\(93\) 0 0
\(94\) 5.41788 − 2.60911i 0.558812 − 0.269110i
\(95\) 0 0
\(96\) 15.0270 + 11.9836i 1.53368 + 1.22307i
\(97\) 0 0 0.943883 − 0.330279i \(-0.107143\pi\)
−0.943883 + 0.330279i \(0.892857\pi\)
\(98\) 0.279051 − 2.47664i 0.0281884 − 0.250179i
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 580.2.be.d.39.1 yes 24
4.3 odd 2 580.2.be.c.39.2 ✓ 24
5.4 even 2 580.2.be.c.39.2 ✓ 24
20.19 odd 2 CM 580.2.be.d.39.1 yes 24
29.3 odd 28 inner 580.2.be.d.119.1 yes 24
116.3 even 28 580.2.be.c.119.2 yes 24
145.119 odd 28 580.2.be.c.119.2 yes 24
580.119 even 28 inner 580.2.be.d.119.1 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
580.2.be.c.39.2 ✓ 24 4.3 odd 2
580.2.be.c.39.2 ✓ 24 5.4 even 2
580.2.be.c.119.2 yes 24 116.3 even 28
580.2.be.c.119.2 yes 24 145.119 odd 28
580.2.be.d.39.1 yes 24 1.1 even 1 trivial
580.2.be.d.39.1 yes 24 20.19 odd 2 CM
580.2.be.d.119.1 yes 24 29.3 odd 28 inner
580.2.be.d.119.1 yes 24 580.119 even 28 inner