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Properties

Label 4840.2.a.bh.1.5
Level $4840$
Weight $2$
Character 4840.1
Self dual yes
Analytic conductor $38.648$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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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] = [4840,2,Mod(1,4840)] 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("4840.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4840, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4840 = 2^{3} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4840.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,1,0,8,0,6,0,19,0,0,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(38.6475945783\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 21x^{6} + 15x^{5} + 140x^{4} - 60x^{3} - 295x^{2} + 50x + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 440)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(0.713352\) of defining polynomial
Character \(\chi\) \(=\) 4840.1

$q$-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)
 
\(f(q)\) \(=\) \(q+0.713352 q^{3} +1.00000 q^{5} -0.376172 q^{7} -2.49113 q^{9} +2.82831 q^{13} +0.713352 q^{15} +4.83768 q^{17} -2.92990 q^{19} -0.268343 q^{21} +3.77226 q^{23} +1.00000 q^{25} -3.91711 q^{27} -8.44396 q^{29} +8.04011 q^{31} -0.376172 q^{35} +2.83990 q^{37} +2.01758 q^{39} +3.72034 q^{41} +6.48484 q^{43} -2.49113 q^{45} +2.58897 q^{47} -6.85849 q^{49} +3.45097 q^{51} -0.0487626 q^{53} -2.09005 q^{57} -1.64929 q^{59} -8.69705 q^{61} +0.937093 q^{63} +2.82831 q^{65} +11.4395 q^{67} +2.69095 q^{69} +14.1216 q^{71} -0.513225 q^{73} +0.713352 q^{75} -12.9823 q^{79} +4.67911 q^{81} -2.73687 q^{83} +4.83768 q^{85} -6.02351 q^{87} +12.0195 q^{89} -1.06393 q^{91} +5.73543 q^{93} -2.92990 q^{95} +16.0745 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{3} + 8 q^{5} + 6 q^{7} + 19 q^{9} - 12 q^{13} + q^{15} - 2 q^{17} + 6 q^{19} - 6 q^{21} + 10 q^{23} + 8 q^{25} + 13 q^{27} - 8 q^{29} + 19 q^{31} + 6 q^{35} + 12 q^{37} + 21 q^{39} + 3 q^{41}+ \cdots + 21 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 0.713352 0.411854 0.205927 − 0.978567i \(-0.433979\pi\)
0.205927 + 0.978567i \(0.433979\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −0.376172 −0.142180 −0.0710899 − 0.997470i \(-0.522648\pi\)
−0.0710899 + 0.997470i \(0.522648\pi\)
\(8\) 0 0
\(9\) −2.49113 −0.830376
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) 2.82831 0.784432 0.392216 − 0.919873i \(-0.371709\pi\)
0.392216 + 0.919873i \(0.371709\pi\)
\(14\) 0 0
\(15\) 0.713352 0.184187
\(16\) 0 0
\(17\) 4.83768 1.17331 0.586655 − 0.809837i \(-0.300444\pi\)
0.586655 + 0.809837i \(0.300444\pi\)
\(18\) 0 0
\(19\) −2.92990 −0.672165 −0.336082 − 0.941833i \(-0.609102\pi\)
−0.336082 + 0.941833i \(0.609102\pi\)
\(20\) 0 0
\(21\) −0.268343 −0.0585573
\(22\) 0 0
\(23\) 3.77226 0.786571 0.393285 − 0.919416i \(-0.371338\pi\)
0.393285 + 0.919416i \(0.371338\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −3.91711 −0.753848
\(28\) 0 0
\(29\) −8.44396 −1.56800 −0.784002 − 0.620759i \(-0.786825\pi\)
−0.784002 + 0.620759i \(0.786825\pi\)
\(30\) 0 0
\(31\) 8.04011 1.44405 0.722023 − 0.691869i \(-0.243213\pi\)
0.722023 + 0.691869i \(0.243213\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.376172 −0.0635847
\(36\) 0 0
\(37\) 2.83990 0.466877 0.233438 − 0.972372i \(-0.425002\pi\)
0.233438 + 0.972372i \(0.425002\pi\)
\(38\) 0 0
\(39\) 2.01758 0.323071
\(40\) 0 0
\(41\) 3.72034 0.581020 0.290510 − 0.956872i \(-0.406175\pi\)
0.290510 + 0.956872i \(0.406175\pi\)
\(42\) 0 0
\(43\) 6.48484 0.988929 0.494465 − 0.869198i \(-0.335364\pi\)
0.494465 + 0.869198i \(0.335364\pi\)
\(44\) 0 0
\(45\) −2.49113 −0.371356
\(46\) 0 0
\(47\) 2.58897 0.377639 0.188820 − 0.982012i \(-0.439534\pi\)
0.188820 + 0.982012i \(0.439534\pi\)
\(48\) 0 0
\(49\) −6.85849 −0.979785
\(50\) 0 0
\(51\) 3.45097 0.483233
\(52\) 0 0
\(53\) −0.0487626 −0.00669805 −0.00334903 − 0.999994i \(-0.501066\pi\)
−0.00334903 + 0.999994i \(0.501066\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −2.09005 −0.276834
\(58\) 0 0
\(59\) −1.64929 −0.214720 −0.107360 − 0.994220i \(-0.534240\pi\)
−0.107360 + 0.994220i \(0.534240\pi\)
\(60\) 0 0
\(61\) −8.69705 −1.11354 −0.556772 − 0.830666i \(-0.687960\pi\)
−0.556772 + 0.830666i \(0.687960\pi\)
\(62\) 0 0
\(63\) 0.937093 0.118063
\(64\) 0 0
\(65\) 2.82831 0.350809
\(66\) 0 0
\(67\) 11.4395 1.39756 0.698779 − 0.715338i \(-0.253727\pi\)
0.698779 + 0.715338i \(0.253727\pi\)
\(68\) 0 0
\(69\) 2.69095 0.323952
\(70\) 0 0
\(71\) 14.1216 1.67592 0.837962 − 0.545728i \(-0.183747\pi\)
0.837962 + 0.545728i \(0.183747\pi\)
\(72\) 0 0
\(73\) −0.513225 −0.0600684 −0.0300342 − 0.999549i \(-0.509562\pi\)
−0.0300342 + 0.999549i \(0.509562\pi\)
\(74\) 0 0
\(75\) 0.713352 0.0823708
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −12.9823 −1.46063 −0.730313 − 0.683113i \(-0.760626\pi\)
−0.730313 + 0.683113i \(0.760626\pi\)
\(80\) 0 0
\(81\) 4.67911 0.519901
\(82\) 0 0
\(83\) −2.73687 −0.300411 −0.150205 − 0.988655i \(-0.547993\pi\)
−0.150205 + 0.988655i \(0.547993\pi\)
\(84\) 0 0
\(85\) 4.83768 0.524720
\(86\) 0 0
\(87\) −6.02351 −0.645789
\(88\) 0 0
\(89\) 12.0195 1.27407 0.637034 − 0.770835i \(-0.280161\pi\)
0.637034 + 0.770835i \(0.280161\pi\)
\(90\) 0 0
\(91\) −1.06393 −0.111530
\(92\) 0 0
\(93\) 5.73543 0.594736
\(94\) 0 0
\(95\) −2.92990 −0.300601
\(96\) 0 0
\(97\) 16.0745 1.63212 0.816060 − 0.577967i \(-0.196154\pi\)
0.816060 + 0.577967i \(0.196154\pi\)
\(98\) 0 0
\(99\) 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 4840.2.a.bh.1.5 8
4.3 odd 2 9680.2.a.de.1.4 8
11.2 odd 10 440.2.y.d.81.3 ✓ 16
11.6 odd 10 440.2.y.d.201.3 yes 16
11.10 odd 2 4840.2.a.bg.1.5 8
44.35 even 10 880.2.bo.k.81.2 16
44.39 even 10 880.2.bo.k.641.2 16
44.43 even 2 9680.2.a.df.1.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.y.d.81.3 ✓ 16 11.2 odd 10
440.2.y.d.201.3 yes 16 11.6 odd 10
880.2.bo.k.81.2 16 44.35 even 10
880.2.bo.k.641.2 16 44.39 even 10
4840.2.a.bg.1.5 8 11.10 odd 2
4840.2.a.bh.1.5 8 1.1 even 1 trivial
9680.2.a.de.1.4 8 4.3 odd 2
9680.2.a.df.1.4 8 44.43 even 2