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Properties

Label 580.2.z.b.121.3
Level $580$
Weight $2$
Character 580.121
Analytic conductor $4.631$
Analytic rank $0$
Dimension $36$
Inner twists $2$

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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(121,580)] 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("580.121"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(580, base_ring=CyclotomicField(14)) chi = DirichletCharacter(H, H._module([0, 0, 11])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 580 = 2^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 580.z (of order \(14\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [36] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.63132331723\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(6\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label 121.3
Character \(\chi\) \(=\) 580.121
Dual form 580.2.z.b.441.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.706830 + 0.161329i) q^{3} +(-0.623490 - 0.781831i) q^{5} +(0.634892 + 2.78164i) q^{7} +(-2.22933 + 1.07359i) q^{9} +(1.12444 - 2.33492i) q^{11} +(-1.67780 - 0.807988i) q^{13} +(0.566834 + 0.452035i) q^{15} +6.69249i q^{17} +(-4.99274 - 1.13956i) q^{19} +(-0.897521 - 1.86372i) q^{21} +(-1.64803 + 2.06657i) q^{23} +(-0.222521 + 0.974928i) q^{25} +(3.10305 - 2.47460i) q^{27} +(-3.55921 + 4.04129i) q^{29} +(-5.41428 + 4.31774i) q^{31} +(-0.418095 + 1.83179i) q^{33} +(1.77893 - 2.23070i) q^{35} +(-1.58728 - 3.29601i) q^{37} +(1.31628 + 0.300431i) q^{39} +11.8302i q^{41} +(-0.382563 - 0.305084i) q^{43} +(2.22933 + 1.07359i) q^{45} +(1.32135 - 2.74381i) q^{47} +(-1.02766 + 0.494896i) q^{49} +(-1.07970 - 4.73045i) q^{51} +(7.42371 + 9.30903i) q^{53} +(-2.52658 + 0.576676i) q^{55} +3.71286 q^{57} -10.9862 q^{59} +(3.71041 - 0.846876i) q^{61} +(-4.40171 - 5.51957i) q^{63} +(0.414384 + 1.81553i) q^{65} +(1.13026 - 0.544304i) q^{67} +(0.831480 - 1.72659i) q^{69} +(-4.19414 - 2.01979i) q^{71} +(-9.41909 - 7.51148i) q^{73} -0.725007i q^{75} +(7.20879 + 1.64536i) q^{77} +(-6.60490 - 13.7152i) q^{79} +(2.83412 - 3.55387i) q^{81} +(-0.980872 + 4.29748i) q^{83} +(5.23240 - 4.17270i) q^{85} +(1.86378 - 3.43071i) q^{87} +(5.62941 - 4.48931i) q^{89} +(1.18231 - 5.18004i) q^{91} +(3.13040 - 3.92539i) q^{93} +(2.22198 + 4.61399i) q^{95} +(14.5248 + 3.31519i) q^{97} +6.41246i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 36 q + 6 q^{5} - 4 q^{7} + 6 q^{9} + 4 q^{13} + 7 q^{21} - 10 q^{23} - 6 q^{25} + 63 q^{27} + 5 q^{29} + 7 q^{31} + 6 q^{33} + 4 q^{35} + 49 q^{39} - 35 q^{43} - 6 q^{45} - 28 q^{47} - 46 q^{49} - 2 q^{51}+ \cdots + 63 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(117\) \(291\) \(321\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{11}{14}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.706830 + 0.161329i −0.408088 + 0.0931435i −0.421637 − 0.906765i \(-0.638544\pi\)
0.0135486 + 0.999908i \(0.495687\pi\)
\(4\) 0 0
\(5\) −0.623490 − 0.781831i −0.278833 − 0.349646i
\(6\) 0 0
\(7\) 0.634892 + 2.78164i 0.239966 + 1.05136i 0.941046 + 0.338279i \(0.109845\pi\)
−0.701079 + 0.713083i \(0.747298\pi\)
\(8\) 0 0
\(9\) −2.22933 + 1.07359i −0.743108 + 0.357862i
\(10\) 0 0
\(11\) 1.12444 − 2.33492i 0.339030 − 0.704003i −0.659846 − 0.751401i \(-0.729378\pi\)
0.998876 + 0.0473975i \(0.0150928\pi\)
\(12\) 0 0
\(13\) −1.67780 − 0.807988i −0.465339 − 0.224096i 0.186497 − 0.982456i \(-0.440287\pi\)
−0.651836 + 0.758360i \(0.726001\pi\)
\(14\) 0 0
\(15\) 0.566834 + 0.452035i 0.146356 + 0.116715i
\(16\) 0 0
\(17\) 6.69249i 1.62317i 0.584236 + 0.811584i \(0.301394\pi\)
−0.584236 + 0.811584i \(0.698606\pi\)
\(18\) 0 0
\(19\) −4.99274 − 1.13956i −1.14541 − 0.261433i −0.392627 − 0.919698i \(-0.628434\pi\)
−0.752787 + 0.658265i \(0.771291\pi\)
\(20\) 0 0
\(21\) −0.897521 − 1.86372i −0.195855 − 0.406697i
\(22\) 0 0
\(23\) −1.64803 + 2.06657i −0.343638 + 0.430909i −0.923377 − 0.383893i \(-0.874583\pi\)
0.579739 + 0.814802i \(0.303154\pi\)
\(24\) 0 0
\(25\) −0.222521 + 0.974928i −0.0445042 + 0.194986i
\(26\) 0 0
\(27\) 3.10305 − 2.47460i 0.597183 − 0.476238i
\(28\) 0 0
\(29\) −3.55921 + 4.04129i −0.660928 + 0.750449i
\(30\) 0 0
\(31\) −5.41428 + 4.31774i −0.972433 + 0.775489i −0.974472 − 0.224508i \(-0.927922\pi\)
0.00203911 + 0.999998i \(0.499351\pi\)
\(32\) 0 0
\(33\) −0.418095 + 1.83179i −0.0727809 + 0.318874i
\(34\) 0 0
\(35\) 1.77893 − 2.23070i 0.300694 − 0.377058i
\(36\) 0 0
\(37\) −1.58728 − 3.29601i −0.260947 − 0.541861i 0.728795 − 0.684732i \(-0.240081\pi\)
−0.989741 + 0.142871i \(0.954367\pi\)
\(38\) 0 0
\(39\) 1.31628 + 0.300431i 0.210773 + 0.0481075i
\(40\) 0 0
\(41\) 11.8302i 1.84757i 0.382916 + 0.923783i \(0.374920\pi\)
−0.382916 + 0.923783i \(0.625080\pi\)
\(42\) 0 0
\(43\) −0.382563 − 0.305084i −0.0583403 − 0.0465248i 0.593886 − 0.804550i \(-0.297593\pi\)
−0.652226 + 0.758025i \(0.726165\pi\)
\(44\) 0 0
\(45\) 2.22933 + 1.07359i 0.332328 + 0.160041i
\(46\) 0 0
\(47\) 1.32135 − 2.74381i 0.192738 − 0.400225i −0.782095 − 0.623159i \(-0.785849\pi\)
0.974834 + 0.222934i \(0.0715633\pi\)
\(48\) 0 0
\(49\) −1.02766 + 0.494896i −0.146809 + 0.0706995i
\(50\) 0 0
\(51\) −1.07970 − 4.73045i −0.151188 − 0.662396i
\(52\) 0 0
\(53\) 7.42371 + 9.30903i 1.01972 + 1.27869i 0.959858 + 0.280488i \(0.0904962\pi\)
0.0598667 + 0.998206i \(0.480932\pi\)
\(54\) 0 0
\(55\) −2.52658 + 0.576676i −0.340685 + 0.0777590i
\(56\) 0 0
\(57\) 3.71286 0.491781
\(58\) 0 0
\(59\) −10.9862 −1.43028 −0.715141 − 0.698980i \(-0.753638\pi\)
−0.715141 + 0.698980i \(0.753638\pi\)
\(60\) 0 0
\(61\) 3.71041 − 0.846876i 0.475069 − 0.108431i 0.0217192 − 0.999764i \(-0.493086\pi\)
0.453350 + 0.891333i \(0.350229\pi\)
\(62\) 0 0
\(63\) −4.40171 − 5.51957i −0.554564 − 0.695401i
\(64\) 0 0
\(65\) 0.414384 + 1.81553i 0.0513980 + 0.225189i
\(66\) 0 0
\(67\) 1.13026 − 0.544304i 0.138083 − 0.0664973i −0.363566 − 0.931568i \(-0.618441\pi\)
0.501649 + 0.865071i \(0.332727\pi\)
\(68\) 0 0
\(69\) 0.831480 − 1.72659i 0.100098 − 0.207857i
\(70\) 0 0
\(71\) −4.19414 − 2.01979i −0.497753 − 0.239705i 0.168125 − 0.985766i \(-0.446229\pi\)
−0.665878 + 0.746060i \(0.731943\pi\)
\(72\) 0 0
\(73\) −9.41909 − 7.51148i −1.10242 − 0.879152i −0.109043 − 0.994037i \(-0.534779\pi\)
−0.993379 + 0.114885i \(0.963350\pi\)
\(74\) 0 0
\(75\) − 0.725007i − 0.0837166i
\(76\) 0 0
\(77\) 7.20879 + 1.64536i 0.821518 + 0.187506i
\(78\) 0 0
\(79\) −6.60490 − 13.7152i −0.743110 − 1.54308i −0.836817 − 0.547483i \(-0.815586\pi\)
0.0937069 − 0.995600i \(-0.470128\pi\)
\(80\) 0 0
\(81\) 2.83412 − 3.55387i 0.314902 − 0.394874i
\(82\) 0 0
\(83\) −0.980872 + 4.29748i −0.107665 + 0.471710i 0.892136 + 0.451766i \(0.149206\pi\)
−0.999801 + 0.0199437i \(0.993651\pi\)
\(84\) 0 0
\(85\) 5.23240 − 4.17270i 0.567534 − 0.452593i
\(86\) 0 0
\(87\) 1.86378 − 3.43071i 0.199818 − 0.367811i
\(88\) 0 0
\(89\) 5.62941 − 4.48931i 0.596717 − 0.475866i −0.277947 − 0.960597i \(-0.589654\pi\)
0.874663 + 0.484731i \(0.161082\pi\)
\(90\) 0 0
\(91\) 1.18231 − 5.18004i 0.123940 − 0.543016i
\(92\) 0 0
\(93\) 3.13040 − 3.92539i 0.324607 − 0.407044i
\(94\) 0 0
\(95\) 2.22198 + 4.61399i 0.227970 + 0.473385i
\(96\) 0 0
\(97\) 14.5248 + 3.31519i 1.47477 + 0.336607i 0.882952 − 0.469464i \(-0.155553\pi\)
0.591819 + 0.806071i \(0.298410\pi\)
\(98\) 0 0
\(99\) 6.41246i 0.644477i
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.z.b.121.3 ✓ 36
29.6 even 14 inner 580.2.z.b.441.3 yes 36
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
580.2.z.b.121.3 ✓ 36 1.1 even 1 trivial
580.2.z.b.441.3 yes 36 29.6 even 14 inner