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Properties

Label 5040.2.f.c.881.1
Level $5040$
Weight $2$
Character 5040.881
Analytic conductor $40.245$
Analytic rank $0$
Dimension $4$
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] = [5040,2,Mod(881,5040)] 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("5040.881"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5040, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5040.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,4,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(40.2446026187\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 315)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 881.1
Root \(-0.517638i\) of defining polynomial
Character \(\chi\) \(=\) 5040.881
Dual form 5040.2.f.c.881.4

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{5} +(-1.00000 - 2.44949i) q^{7} -0.378937i q^{11} +1.79315i q^{13} -3.46410 q^{17} +1.79315i q^{19} +1.41421i q^{23} +1.00000 q^{25} -1.41421i q^{29} +6.69213i q^{31} +(-1.00000 - 2.44949i) q^{35} -1.46410 q^{37} +10.3923 q^{41} +4.92820 q^{43} +9.46410 q^{47} +(-5.00000 + 4.89898i) q^{49} +10.1769i q^{53} -0.378937i q^{55} +9.46410 q^{59} -13.3843i q^{61} +1.79315i q^{65} +10.9282 q^{67} -15.0759i q^{71} -1.79315i q^{73} +(-0.928203 + 0.378937i) q^{77} +10.9282 q^{79} -9.46410 q^{83} -3.46410 q^{85} -12.9282 q^{89} +(4.39230 - 1.79315i) q^{91} +1.79315i q^{95} +15.1774i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{5} - 4 q^{7} + 4 q^{25} - 4 q^{35} + 8 q^{37} - 8 q^{43} + 24 q^{47} - 20 q^{49} + 24 q^{59} + 16 q^{67} + 24 q^{77} + 16 q^{79} - 24 q^{83} - 24 q^{89} - 24 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5040\mathbb{Z}\right)^\times\).

\(n\) \(2017\) \(2801\) \(3151\) \(3601\) \(3781\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\) \(1\)

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 0
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −1.00000 − 2.44949i −0.377964 − 0.925820i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 0.378937i − 0.114254i −0.998367 − 0.0571270i \(-0.981806\pi\)
0.998367 − 0.0571270i \(-0.0181940\pi\)
\(12\) 0 0
\(13\) 1.79315i 0.497331i 0.968589 + 0.248665i \(0.0799919\pi\)
−0.968589 + 0.248665i \(0.920008\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.46410 −0.840168 −0.420084 − 0.907485i \(-0.637999\pi\)
−0.420084 + 0.907485i \(0.637999\pi\)
\(18\) 0 0
\(19\) 1.79315i 0.411377i 0.978618 + 0.205689i \(0.0659434\pi\)
−0.978618 + 0.205689i \(0.934057\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.41421i 0.294884i 0.989071 + 0.147442i \(0.0471040\pi\)
−0.989071 + 0.147442i \(0.952896\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 1.41421i − 0.262613i −0.991342 − 0.131306i \(-0.958083\pi\)
0.991342 − 0.131306i \(-0.0419172\pi\)
\(30\) 0 0
\(31\) 6.69213i 1.20194i 0.799271 + 0.600971i \(0.205219\pi\)
−0.799271 + 0.600971i \(0.794781\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.00000 − 2.44949i −0.169031 − 0.414039i
\(36\) 0 0
\(37\) −1.46410 −0.240697 −0.120348 − 0.992732i \(-0.538401\pi\)
−0.120348 + 0.992732i \(0.538401\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 10.3923 1.62301 0.811503 − 0.584349i \(-0.198650\pi\)
0.811503 + 0.584349i \(0.198650\pi\)
\(42\) 0 0
\(43\) 4.92820 0.751544 0.375772 − 0.926712i \(-0.377378\pi\)
0.375772 + 0.926712i \(0.377378\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 9.46410 1.38048 0.690241 − 0.723580i \(-0.257505\pi\)
0.690241 + 0.723580i \(0.257505\pi\)
\(48\) 0 0
\(49\) −5.00000 + 4.89898i −0.714286 + 0.699854i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 10.1769i 1.39790i 0.715168 + 0.698952i \(0.246350\pi\)
−0.715168 + 0.698952i \(0.753650\pi\)
\(54\) 0 0
\(55\) − 0.378937i − 0.0510959i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 9.46410 1.23212 0.616061 − 0.787699i \(-0.288728\pi\)
0.616061 + 0.787699i \(0.288728\pi\)
\(60\) 0 0
\(61\) − 13.3843i − 1.71368i −0.515583 − 0.856840i \(-0.672425\pi\)
0.515583 − 0.856840i \(-0.327575\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.79315i 0.222413i
\(66\) 0 0
\(67\) 10.9282 1.33509 0.667546 − 0.744568i \(-0.267345\pi\)
0.667546 + 0.744568i \(0.267345\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 15.0759i − 1.78918i −0.446891 − 0.894589i \(-0.647469\pi\)
0.446891 − 0.894589i \(-0.352531\pi\)
\(72\) 0 0
\(73\) − 1.79315i − 0.209872i −0.994479 − 0.104936i \(-0.966536\pi\)
0.994479 − 0.104936i \(-0.0334638\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −0.928203 + 0.378937i −0.105779 + 0.0431839i
\(78\) 0 0
\(79\) 10.9282 1.22952 0.614759 − 0.788715i \(-0.289253\pi\)
0.614759 + 0.788715i \(0.289253\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −9.46410 −1.03882 −0.519410 − 0.854525i \(-0.673848\pi\)
−0.519410 + 0.854525i \(0.673848\pi\)
\(84\) 0 0
\(85\) −3.46410 −0.375735
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −12.9282 −1.37039 −0.685193 − 0.728361i \(-0.740282\pi\)
−0.685193 + 0.728361i \(0.740282\pi\)
\(90\) 0 0
\(91\) 4.39230 − 1.79315i 0.460439 − 0.187973i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.79315i 0.183973i
\(96\) 0 0
\(97\) 15.1774i 1.54103i 0.637420 + 0.770516i \(0.280002\pi\)
−0.637420 + 0.770516i \(0.719998\pi\)
\(98\) 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 5040.2.f.c.881.1 4
3.2 odd 2 5040.2.f.a.881.2 4
4.3 odd 2 315.2.b.b.251.2 yes 4
7.6 odd 2 5040.2.f.a.881.3 4
12.11 even 2 315.2.b.a.251.3 yes 4
20.3 even 4 1575.2.g.a.1574.3 8
20.7 even 4 1575.2.g.a.1574.6 8
20.19 odd 2 1575.2.b.c.251.3 4
21.20 even 2 inner 5040.2.f.c.881.4 4
28.27 even 2 315.2.b.a.251.2 ✓ 4
60.23 odd 4 1575.2.g.c.1574.5 8
60.47 odd 4 1575.2.g.c.1574.4 8
60.59 even 2 1575.2.b.b.251.2 4
84.83 odd 2 315.2.b.b.251.3 yes 4
140.27 odd 4 1575.2.g.c.1574.6 8
140.83 odd 4 1575.2.g.c.1574.3 8
140.139 even 2 1575.2.b.b.251.3 4
420.83 even 4 1575.2.g.a.1574.5 8
420.167 even 4 1575.2.g.a.1574.4 8
420.419 odd 2 1575.2.b.c.251.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
315.2.b.a.251.2 ✓ 4 28.27 even 2
315.2.b.a.251.3 yes 4 12.11 even 2
315.2.b.b.251.2 yes 4 4.3 odd 2
315.2.b.b.251.3 yes 4 84.83 odd 2
1575.2.b.b.251.2 4 60.59 even 2
1575.2.b.b.251.3 4 140.139 even 2
1575.2.b.c.251.2 4 420.419 odd 2
1575.2.b.c.251.3 4 20.19 odd 2
1575.2.g.a.1574.3 8 20.3 even 4
1575.2.g.a.1574.4 8 420.167 even 4
1575.2.g.a.1574.5 8 420.83 even 4
1575.2.g.a.1574.6 8 20.7 even 4
1575.2.g.c.1574.3 8 140.83 odd 4
1575.2.g.c.1574.4 8 60.47 odd 4
1575.2.g.c.1574.5 8 60.23 odd 4
1575.2.g.c.1574.6 8 140.27 odd 4
5040.2.f.a.881.2 4 3.2 odd 2
5040.2.f.a.881.3 4 7.6 odd 2
5040.2.f.c.881.1 4 1.1 even 1 trivial
5040.2.f.c.881.4 4 21.20 even 2 inner