↑

Properties

Label 580.2.z.b.241.3
Level $580$
Weight $2$
Character 580.241
Analytic conductor $4.631$
Analytic rank $0$
Dimension $36$
Inner twists $2$

Related objects

Downloads

Learn more

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 241.3
Character \(\chi\) \(=\) 580.241
Dual form 580.2.z.b.361.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.520764 + 0.415295i) q^{3} +(0.900969 - 0.433884i) q^{5} +(-2.54025 - 3.18538i) q^{7} +(-0.568838 + 2.49224i) q^{9} +(3.70125 - 0.844786i) q^{11} +(-0.333273 - 1.46016i) q^{13} +(-0.289002 + 0.600119i) q^{15} -6.23772i q^{17} +(-2.56269 - 2.04368i) q^{19} +(2.64574 + 0.603874i) q^{21} +(-0.605736 - 0.291707i) q^{23} +(0.623490 - 0.781831i) q^{25} +(-1.60579 - 3.33446i) q^{27} +(5.20422 - 1.38424i) q^{29} +(-2.57535 - 5.34777i) q^{31} +(-1.57664 + 1.97705i) q^{33} +(-3.67077 - 1.76775i) q^{35} +(4.34364 + 0.991406i) q^{37} +(0.779956 + 0.621994i) q^{39} +3.79410i q^{41} +(3.25837 - 6.76608i) q^{43} +(0.568838 + 2.49224i) q^{45} +(1.71482 - 0.391396i) q^{47} +(-2.13609 + 9.35883i) q^{49} +(2.59050 + 3.24838i) q^{51} +(6.88879 - 3.31747i) q^{53} +(2.96817 - 2.36704i) q^{55} +2.18329 q^{57} -5.57161 q^{59} +(-4.54693 + 3.62605i) q^{61} +(9.38372 - 4.51896i) q^{63} +(-0.933810 - 1.17096i) q^{65} +(0.658547 - 2.88528i) q^{67} +(0.436590 - 0.0996488i) q^{69} +(-3.32636 - 14.5737i) q^{71} +(-5.76930 + 11.9801i) q^{73} +0.666082i q^{75} +(-12.0931 - 9.64390i) q^{77} +(8.01494 + 1.82936i) q^{79} +(-4.68851 - 2.25787i) q^{81} +(-5.46774 + 6.85633i) q^{83} +(-2.70644 - 5.61999i) q^{85} +(-2.13530 + 2.88215i) q^{87} +(2.95212 + 6.13015i) q^{89} +(-3.80457 + 4.77078i) q^{91} +(3.56205 + 1.71539i) q^{93} +(-3.19562 - 0.729380i) q^{95} +(-1.05550 - 0.841734i) q^{97} +9.70496i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 36 q + 6 q^{5} - 4 q^{7} + 6 q^{9} + 4 q^{13} + 7 q^{21} - 10 q^{23} - 6 q^{25} + 63 q^{27} + 5 q^{29} + 7 q^{31} + 6 q^{33} + 4 q^{35} + 49 q^{39} - 35 q^{43} - 6 q^{45} - 28 q^{47} - 46 q^{49} - 2 q^{51}+ \cdots + 63 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(117\) \(291\) \(321\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{5}{14}\right)\)

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
\(3\) −0.520764 + 0.415295i −0.300663 + 0.239771i −0.762187 − 0.647357i \(-0.775874\pi\)
0.461524 + 0.887128i \(0.347303\pi\)
\(4\) 0 0
\(5\) 0.900969 − 0.433884i 0.402926 − 0.194039i
\(6\) 0 0
\(7\) −2.54025 − 3.18538i −0.960125 − 1.20396i −0.978942 − 0.204136i \(-0.934561\pi\)
0.0188171 − 0.999823i \(-0.494010\pi\)
\(8\) 0 0
\(9\) −0.568838 + 2.49224i −0.189613 + 0.830747i
\(10\) 0 0
\(11\) 3.70125 − 0.844786i 1.11597 − 0.254713i 0.375513 − 0.926817i \(-0.377467\pi\)
0.740456 + 0.672105i \(0.234610\pi\)
\(12\) 0 0
\(13\) −0.333273 − 1.46016i −0.0924332 − 0.404977i 0.907452 − 0.420157i \(-0.138025\pi\)
−0.999885 + 0.0151803i \(0.995168\pi\)
\(14\) 0 0
\(15\) −0.289002 + 0.600119i −0.0746200 + 0.154950i
\(16\) 0 0
\(17\) − 6.23772i − 1.51287i −0.654070 − 0.756434i \(-0.726940\pi\)
0.654070 − 0.756434i \(-0.273060\pi\)
\(18\) 0 0
\(19\) −2.56269 − 2.04368i −0.587922 − 0.468852i 0.283780 − 0.958890i \(-0.408412\pi\)
−0.871701 + 0.490038i \(0.836983\pi\)
\(20\) 0 0
\(21\) 2.64574 + 0.603874i 0.577349 + 0.131776i
\(22\) 0 0
\(23\) −0.605736 − 0.291707i −0.126305 − 0.0608251i 0.369664 − 0.929165i \(-0.379473\pi\)
−0.495969 + 0.868340i \(0.665187\pi\)
\(24\) 0 0
\(25\) 0.623490 − 0.781831i 0.124698 − 0.156366i
\(26\) 0 0
\(27\) −1.60579 − 3.33446i −0.309035 − 0.641718i
\(28\) 0 0
\(29\) 5.20422 − 1.38424i 0.966399 − 0.257046i
\(30\) 0 0
\(31\) −2.57535 − 5.34777i −0.462546 − 0.960487i −0.993579 − 0.113136i \(-0.963910\pi\)
0.531033 − 0.847351i \(-0.321804\pi\)
\(32\) 0 0
\(33\) −1.57664 + 1.97705i −0.274458 + 0.344159i
\(34\) 0 0
\(35\) −3.67077 − 1.76775i −0.620474 − 0.298804i
\(36\) 0 0
\(37\) 4.34364 + 0.991406i 0.714089 + 0.162986i 0.564106 − 0.825702i \(-0.309221\pi\)
0.149983 + 0.988689i \(0.452078\pi\)
\(38\) 0 0
\(39\) 0.779956 + 0.621994i 0.124893 + 0.0995987i
\(40\) 0 0
\(41\) 3.79410i 0.592539i 0.955104 + 0.296270i \(0.0957427\pi\)
−0.955104 + 0.296270i \(0.904257\pi\)
\(42\) 0 0
\(43\) 3.25837 − 6.76608i 0.496897 − 1.03182i −0.490187 − 0.871617i \(-0.663072\pi\)
0.987085 − 0.160200i \(-0.0512140\pi\)
\(44\) 0 0
\(45\) 0.568838 + 2.49224i 0.0847974 + 0.371522i
\(46\) 0 0
\(47\) 1.71482 − 0.391396i 0.250132 − 0.0570909i −0.0956165 − 0.995418i \(-0.530482\pi\)
0.345748 + 0.938327i \(0.387625\pi\)
\(48\) 0 0
\(49\) −2.13609 + 9.35883i −0.305156 + 1.33698i
\(50\) 0 0
\(51\) 2.59050 + 3.24838i 0.362742 + 0.454864i
\(52\) 0 0
\(53\) 6.88879 − 3.31747i 0.946248 − 0.455689i 0.103879 − 0.994590i \(-0.466875\pi\)
0.842369 + 0.538901i \(0.181160\pi\)
\(54\) 0 0
\(55\) 2.96817 − 2.36704i 0.400228 − 0.319171i
\(56\) 0 0
\(57\) 2.18329 0.289183
\(58\) 0 0
\(59\) −5.57161 −0.725361 −0.362681 − 0.931913i \(-0.618138\pi\)
−0.362681 + 0.931913i \(0.618138\pi\)
\(60\) 0 0
\(61\) −4.54693 + 3.62605i −0.582174 + 0.464268i −0.869752 − 0.493489i \(-0.835721\pi\)
0.287578 + 0.957757i \(0.407150\pi\)
\(62\) 0 0
\(63\) 9.38372 − 4.51896i 1.18224 − 0.569336i
\(64\) 0 0
\(65\) −0.933810 − 1.17096i −0.115825 − 0.145240i
\(66\) 0 0
\(67\) 0.658547 − 2.88528i 0.0804544 − 0.352494i −0.918638 − 0.395101i \(-0.870709\pi\)
0.999092 + 0.0426078i \(0.0135666\pi\)
\(68\) 0 0
\(69\) 0.436590 − 0.0996488i 0.0525593 − 0.0119963i
\(70\) 0 0
\(71\) −3.32636 − 14.5737i −0.394767 − 1.72959i −0.647514 − 0.762054i \(-0.724191\pi\)
0.252747 − 0.967532i \(-0.418666\pi\)
\(72\) 0 0
\(73\) −5.76930 + 11.9801i −0.675245 + 1.40216i 0.228267 + 0.973598i \(0.426694\pi\)
−0.903512 + 0.428562i \(0.859020\pi\)
\(74\) 0 0
\(75\) 0.666082i 0.0769125i
\(76\) 0 0
\(77\) −12.0931 − 9.64390i −1.37813 − 1.09902i
\(78\) 0 0
\(79\) 8.01494 + 1.82936i 0.901751 + 0.205819i 0.648170 − 0.761495i \(-0.275534\pi\)
0.253581 + 0.967314i \(0.418392\pi\)
\(80\) 0 0
\(81\) −4.68851 − 2.25787i −0.520945 − 0.250874i
\(82\) 0 0
\(83\) −5.46774 + 6.85633i −0.600163 + 0.752580i −0.985403 − 0.170236i \(-0.945547\pi\)
0.385240 + 0.922816i \(0.374118\pi\)
\(84\) 0 0
\(85\) −2.70644 − 5.61999i −0.293555 − 0.609573i
\(86\) 0 0
\(87\) −2.13530 + 2.88215i −0.228928 + 0.308999i
\(88\) 0 0
\(89\) 2.95212 + 6.13015i 0.312924 + 0.649794i 0.996811 − 0.0797937i \(-0.0254262\pi\)
−0.683887 + 0.729588i \(0.739712\pi\)
\(90\) 0 0
\(91\) −3.80457 + 4.77078i −0.398828 + 0.500114i
\(92\) 0 0
\(93\) 3.56205 + 1.71539i 0.369368 + 0.177878i
\(94\) 0 0
\(95\) −3.19562 − 0.729380i −0.327864 − 0.0748328i
\(96\) 0 0
\(97\) −1.05550 − 0.841734i −0.107170 − 0.0854651i 0.568439 − 0.822726i \(-0.307548\pi\)
−0.675609 + 0.737260i \(0.736119\pi\)
\(98\) 0 0
\(99\) 9.70496i 0.975385i
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.241.3 ✓ 36
29.13 even 14 inner 580.2.z.b.361.3 yes 36
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
580.2.z.b.241.3 ✓ 36 1.1 even 1 trivial
580.2.z.b.361.3 yes 36 29.13 even 14 inner