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Properties

Label 7440.2.a.bx.1.1
Level $7440$
Weight $2$
Character 7440.1
Self dual yes
Analytic conductor $59.409$
Analytic rank $0$
Dimension $4$
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] = [7440,2,Mod(1,7440)] 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("7440.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7440, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7440.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,4,0,-4,0,-1,0,4,0,-1,0,-2,0,-4,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(59.4086991038\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.78292.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 10x^{2} + 8x + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3720)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.68461\) of defining polynomial
Character \(\chi\) \(=\) 7440.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.00000 q^{5} -2.68461 q^{7} +1.00000 q^{9} -3.24072 q^{11} -4.97040 q^{13} -1.00000 q^{15} +0.921359 q^{17} -1.24072 q^{19} -2.68461 q^{21} -6.57637 q^{23} +1.00000 q^{25} +1.00000 q^{27} -3.89176 q^{29} -1.00000 q^{31} -3.24072 q^{33} +2.68461 q^{35} +2.97040 q^{37} -4.97040 q^{39} +4.00000 q^{41} -5.65501 q^{43} -1.00000 q^{45} +8.86216 q^{47} +0.207146 q^{49} +0.921359 q^{51} +0.792854 q^{53} +3.24072 q^{55} -1.24072 q^{57} +7.00397 q^{59} +10.8957 q^{61} -2.68461 q^{63} +4.97040 q^{65} +12.3396 q^{67} -6.57637 q^{69} -15.5468 q^{71} +7.09891 q^{73} +1.00000 q^{75} +8.70008 q^{77} +1.04986 q^{79} +1.00000 q^{81} +16.3887 q^{83} -0.921359 q^{85} -3.89176 q^{87} -3.13248 q^{89} +13.3436 q^{91} -1.00000 q^{93} +1.24072 q^{95} -4.57157 q^{97} -3.24072 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 4 q^{5} - q^{7} + 4 q^{9} - q^{11} - 2 q^{13} - 4 q^{15} + 4 q^{17} + 7 q^{19} - q^{21} + q^{23} + 4 q^{25} + 4 q^{27} + 2 q^{29} - 4 q^{31} - q^{33} + q^{35} - 6 q^{37} - 2 q^{39}+ \cdots - q^{99}+O(q^{100}) \) 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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −2.68461 −1.01469 −0.507344 − 0.861744i \(-0.669373\pi\)
−0.507344 + 0.861744i \(0.669373\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.24072 −0.977114 −0.488557 − 0.872532i \(-0.662477\pi\)
−0.488557 + 0.872532i \(0.662477\pi\)
\(12\) 0 0
\(13\) −4.97040 −1.37854 −0.689270 − 0.724504i \(-0.742069\pi\)
−0.689270 + 0.724504i \(0.742069\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 0.921359 0.223462 0.111731 − 0.993738i \(-0.464360\pi\)
0.111731 + 0.993738i \(0.464360\pi\)
\(18\) 0 0
\(19\) −1.24072 −0.284641 −0.142320 − 0.989821i \(-0.545456\pi\)
−0.142320 + 0.989821i \(0.545456\pi\)
\(20\) 0 0
\(21\) −2.68461 −0.585831
\(22\) 0 0
\(23\) −6.57637 −1.37127 −0.685634 − 0.727946i \(-0.740475\pi\)
−0.685634 + 0.727946i \(0.740475\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −3.89176 −0.722682 −0.361341 − 0.932434i \(-0.617681\pi\)
−0.361341 + 0.932434i \(0.617681\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) 0 0
\(33\) −3.24072 −0.564137
\(34\) 0 0
\(35\) 2.68461 0.453782
\(36\) 0 0
\(37\) 2.97040 0.488331 0.244165 − 0.969734i \(-0.421486\pi\)
0.244165 + 0.969734i \(0.421486\pi\)
\(38\) 0 0
\(39\) −4.97040 −0.795901
\(40\) 0 0
\(41\) 4.00000 0.624695 0.312348 − 0.949968i \(-0.398885\pi\)
0.312348 + 0.949968i \(0.398885\pi\)
\(42\) 0 0
\(43\) −5.65501 −0.862381 −0.431191 − 0.902261i \(-0.641906\pi\)
−0.431191 + 0.902261i \(0.641906\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 8.86216 1.29268 0.646339 − 0.763050i \(-0.276299\pi\)
0.646339 + 0.763050i \(0.276299\pi\)
\(48\) 0 0
\(49\) 0.207146 0.0295923
\(50\) 0 0
\(51\) 0.921359 0.129016
\(52\) 0 0
\(53\) 0.792854 0.108907 0.0544534 − 0.998516i \(-0.482658\pi\)
0.0544534 + 0.998516i \(0.482658\pi\)
\(54\) 0 0
\(55\) 3.24072 0.436979
\(56\) 0 0
\(57\) −1.24072 −0.164337
\(58\) 0 0
\(59\) 7.00397 0.911840 0.455920 − 0.890021i \(-0.349310\pi\)
0.455920 + 0.890021i \(0.349310\pi\)
\(60\) 0 0
\(61\) 10.8957 1.39506 0.697528 − 0.716558i \(-0.254283\pi\)
0.697528 + 0.716558i \(0.254283\pi\)
\(62\) 0 0
\(63\) −2.68461 −0.338229
\(64\) 0 0
\(65\) 4.97040 0.616502
\(66\) 0 0
\(67\) 12.3396 1.50753 0.753763 − 0.657147i \(-0.228237\pi\)
0.753763 + 0.657147i \(0.228237\pi\)
\(68\) 0 0
\(69\) −6.57637 −0.791702
\(70\) 0 0
\(71\) −15.5468 −1.84506 −0.922531 − 0.385923i \(-0.873883\pi\)
−0.922531 + 0.385923i \(0.873883\pi\)
\(72\) 0 0
\(73\) 7.09891 0.830864 0.415432 − 0.909624i \(-0.363630\pi\)
0.415432 + 0.909624i \(0.363630\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 8.70008 0.991466
\(78\) 0 0
\(79\) 1.04986 0.118119 0.0590595 − 0.998254i \(-0.481190\pi\)
0.0590595 + 0.998254i \(0.481190\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 16.3887 1.79889 0.899445 − 0.437034i \(-0.143971\pi\)
0.899445 + 0.437034i \(0.143971\pi\)
\(84\) 0 0
\(85\) −0.921359 −0.0999354
\(86\) 0 0
\(87\) −3.89176 −0.417240
\(88\) 0 0
\(89\) −3.13248 −0.332042 −0.166021 − 0.986122i \(-0.553092\pi\)
−0.166021 + 0.986122i \(0.553092\pi\)
\(90\) 0 0
\(91\) 13.3436 1.39879
\(92\) 0 0
\(93\) −1.00000 −0.103695
\(94\) 0 0
\(95\) 1.24072 0.127295
\(96\) 0 0
\(97\) −4.57157 −0.464173 −0.232087 − 0.972695i \(-0.574555\pi\)
−0.232087 + 0.972695i \(0.574555\pi\)
\(98\) 0 0
\(99\) −3.24072 −0.325705
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 7440.2.a.bx.1.1 4
4.3 odd 2 3720.2.a.r.1.4 ✓ 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3720.2.a.r.1.4 ✓ 4 4.3 odd 2
7440.2.a.bx.1.1 4 1.1 even 1 trivial