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Properties

Label 5220.2.b.d
Level $5220$
Weight $2$
Character orbit 5220.b
Analytic conductor $41.682$
Analytic rank $0$
Dimension $14$
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] = [5220,2,Mod(289,5220)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5220, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5220.289"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 5220 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5220.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [14,0,0,0,-2,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,-6,0,0,0,-8, 0,0,0,0,0,18,0,-14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(37)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(41.6819098551\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 2 x^{13} + 5 x^{12} - 8 x^{11} + 11 x^{10} - 30 x^{9} - 49 x^{8} + 176 x^{7} - 245 x^{6} + \cdots + 78125 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 1740)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{5} - \beta_{9} q^{7} + \beta_{5} q^{11} - \beta_{6} q^{13} - \beta_{11} q^{17} + \beta_{8} q^{19} + (\beta_{9} + \beta_{8} - \beta_{7} + \cdots + \beta_1) q^{23} + \beta_{2} q^{25} + ( - \beta_{4} + \beta_{3} - 1) q^{29}+ \cdots + (\beta_{13} - \beta_{12} + \beta_{11} + \cdots + 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 2 q^{5} + 4 q^{17} - 6 q^{25} - 8 q^{29} + 18 q^{35} - 14 q^{37} + 6 q^{43} - 16 q^{47} - 14 q^{49} - 8 q^{55} - 12 q^{59} - 10 q^{65} - 4 q^{71} - 14 q^{73} - 24 q^{77} - 14 q^{85} + 16 q^{91} - 8 q^{95}+ \cdots + 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 2 x^{13} + 5 x^{12} - 8 x^{11} + 11 x^{10} - 30 x^{9} - 49 x^{8} + 176 x^{7} - 245 x^{6} + \cdots + 78125 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 11 \nu^{13} + 29 \nu^{12} + 68 \nu^{11} + 112 \nu^{10} + 1813 \nu^{9} - 1689 \nu^{8} + \cdots - 2021875 ) / 700000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 21 \nu^{13} + 113 \nu^{12} - 155 \nu^{11} + 132 \nu^{10} - 634 \nu^{9} - 3075 \nu^{8} + \cdots + 671875 ) / 875000 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 53 \nu^{13} + 99 \nu^{12} - 320 \nu^{11} - 424 \nu^{10} - 1057 \nu^{9} + 2065 \nu^{8} + \cdots + 2546875 ) / 1000000 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 87 \nu^{13} + 1371 \nu^{12} - 1580 \nu^{11} - 2996 \nu^{10} - 903 \nu^{9} + 3285 \nu^{8} + \cdots + 13484375 ) / 3500000 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - \nu^{13} + 2 \nu^{12} - 5 \nu^{11} + 8 \nu^{10} - 11 \nu^{9} + 30 \nu^{8} + 49 \nu^{7} + \cdots + 31250 ) / 15625 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 261 \nu^{13} + 2643 \nu^{12} - 1800 \nu^{11} - 4088 \nu^{10} - 15449 \nu^{9} + 11185 \nu^{8} + \cdots + 45046875 ) / 7000000 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 131 \nu^{13} - 171 \nu^{12} - 204 \nu^{11} + 588 \nu^{10} + 1323 \nu^{9} - 5013 \nu^{8} + \cdots - 3384375 ) / 1400000 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2 \nu^{13} + \nu^{12} + 9 \nu^{10} - 18 \nu^{9} - 5 \nu^{8} - 248 \nu^{7} + 107 \nu^{6} + \cdots + 15625 ) / 15625 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 63 \nu^{13} - 176 \nu^{12} + 15 \nu^{11} - 454 \nu^{10} + 718 \nu^{9} + 135 \nu^{8} + \cdots - 1765625 ) / 437500 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 84 \nu^{13} + 127 \nu^{12} + 62 \nu^{11} - 8 \nu^{10} - 746 \nu^{9} + 369 \nu^{8} + \cdots + 1371875 ) / 350000 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 231 \nu^{13} + 537 \nu^{12} - 355 \nu^{11} + 1948 \nu^{10} - 4766 \nu^{9} - 1095 \nu^{8} + \cdots + 6859375 ) / 875000 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{13} + 2\beta_{11} + \beta_{8} - \beta_{5} - \beta_{4} + \beta_{3} - \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{11} - \beta_{10} - 4 \beta_{9} - 2 \beta_{8} + 4 \beta_{7} + 2 \beta_{6} - 6 \beta_{5} + \cdots + 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 3 \beta_{13} + 8 \beta_{12} + 6 \beta_{11} + 4 \beta_{10} + 3 \beta_{8} + 11 \beta_{7} - 8 \beta_{6} + \cdots + 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8 \beta_{13} + 8 \beta_{12} + 16 \beta_{11} + 7 \beta_{10} - 8 \beta_{9} - 20 \beta_{8} - 4 \beta_{7} + \cdots + 36 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 20 \beta_{13} - 24 \beta_{12} - 4 \beta_{11} - 52 \beta_{10} - 32 \beta_{9} + 4 \beta_{8} - 27 \beta_{7} + \cdots - 12 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 24 \beta_{13} + 16 \beta_{12} + 8 \beta_{11} + 4 \beta_{10} - 120 \beta_{9} - 36 \beta_{8} + \cdots - 175 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 52 \beta_{13} - 72 \beta_{12} - 44 \beta_{11} - 100 \beta_{10} - 64 \beta_{9} + 4 \beta_{8} + \cdots + 844 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 72 \beta_{13} - 264 \beta_{12} - 160 \beta_{11} + 108 \beta_{10} - 392 \beta_{9} - 244 \beta_{8} + \cdots + 588 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 295 \beta_{13} + 856 \beta_{12} - 330 \beta_{11} + 652 \beta_{10} - 608 \beta_{9} - 71 \beta_{8} + \cdots + 1436 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1168 \beta_{13} + 544 \beta_{12} + 1298 \beta_{11} + 879 \beta_{10} - 780 \beta_{9} + 378 \beta_{8} + \cdots - 5581 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 1701 \beta_{13} - 4656 \beta_{12} + 898 \beta_{11} - 2968 \beta_{10} - 7712 \beta_{9} - 389 \beta_{8} + \cdots - 616 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5220\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(2611\) \(4061\) \(4177\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
2.21984 + 0.268918i
2.21984 − 0.268918i
1.72817 + 1.41895i
1.72817 − 1.41895i
0.798478 + 2.08864i
0.798478 − 2.08864i
0.591682 + 2.15637i
0.591682 − 2.15637i
−0.886253 + 2.05294i
−0.886253 − 2.05294i
−1.30477 + 1.81592i
−1.30477 − 1.81592i
−2.14714 + 0.624326i
−2.14714 − 0.624326i
0 0 0 −2.21984 − 0.268918i 0 4.12381i 0 0 0
289.2 0 0 0 −2.21984 + 0.268918i 0 − 4.12381i 0 0 0
289.3 0 0 0 −1.72817 − 1.41895i 0 − 0.510239i 0 0 0
289.4 0 0 0 −1.72817 + 1.41895i 0 0.510239i 0 0 0
289.5 0 0 0 −0.798478 − 2.08864i 0 3.91876i 0 0 0
289.6 0 0 0 −0.798478 + 2.08864i 0 − 3.91876i 0 0 0
289.7 0 0 0 −0.591682 − 2.15637i 0 − 2.69732i 0 0 0
289.8 0 0 0 −0.591682 + 2.15637i 0 2.69732i 0 0 0
289.9 0 0 0 0.886253 − 2.05294i 0 3.83675i 0 0 0
289.10 0 0 0 0.886253 + 2.05294i 0 − 3.83675i 0 0 0
289.11 0 0 0 1.30477 − 1.81592i 0 − 0.538567i 0 0 0
289.12 0 0 0 1.30477 + 1.81592i 0 0.538567i 0 0 0
289.13 0 0 0 2.14714 − 0.624326i 0 − 1.04444i 0 0 0
289.14 0 0 0 2.14714 + 0.624326i 0 1.04444i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
145.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5220.2.b.d 14
3.b odd 2 1 1740.2.b.a ✓ 14
5.b even 2 1 5220.2.b.c 14
15.d odd 2 1 1740.2.b.b yes 14
15.e even 4 2 8700.2.l.j 28
29.b even 2 1 5220.2.b.c 14
87.d odd 2 1 1740.2.b.b yes 14
145.d even 2 1 inner 5220.2.b.d 14
435.b odd 2 1 1740.2.b.a ✓ 14
435.p even 4 2 8700.2.l.j 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1740.2.b.a ✓ 14 3.b odd 2 1
1740.2.b.a ✓ 14 435.b odd 2 1
1740.2.b.b yes 14 15.d odd 2 1
1740.2.b.b yes 14 87.d odd 2 1
5220.2.b.c 14 5.b even 2 1
5220.2.b.c 14 29.b even 2 1
5220.2.b.d 14 1.a even 1 1 trivial
5220.2.b.d 14 145.d even 2 1 inner
8700.2.l.j 28 15.e even 4 2
8700.2.l.j 28 435.p even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(5220, [\chi])\):

\( T_{7}^{14} + 56T_{7}^{12} + 1170T_{7}^{10} + 11020T_{7}^{8} + 43821T_{7}^{6} + 52220T_{7}^{4} + 19664T_{7}^{2} + 2304 \) Copy content Toggle raw display
\( T_{17}^{7} - 2T_{17}^{6} - 50T_{17}^{5} + 16T_{17}^{4} + 717T_{17}^{3} + 690T_{17}^{2} - 1472T_{17} - 1536 \) Copy content Toggle raw display
\( T_{37}^{7} + 7T_{37}^{6} - 80T_{37}^{5} - 286T_{37}^{4} + 2024T_{37}^{3} + 2196T_{37}^{2} - 14400T_{37} + 3776 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} \) Copy content Toggle raw display
$5$ \( T^{14} + 2 T^{13} + \cdots + 78125 \) Copy content Toggle raw display
$7$ \( T^{14} + 56 T^{12} + \cdots + 2304 \) Copy content Toggle raw display
$11$ \( T^{14} + 81 T^{12} + \cdots + 38416 \) Copy content Toggle raw display
$13$ \( T^{14} + 112 T^{12} + \cdots + 746496 \) Copy content Toggle raw display
$17$ \( (T^{7} - 2 T^{6} + \cdots - 1536)^{2} \) Copy content Toggle raw display
$19$ \( T^{14} + 116 T^{12} + \cdots + 36864 \) Copy content Toggle raw display
$23$ \( T^{14} + 157 T^{12} + \cdots + 12544 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 17249876309 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 257025024 \) Copy content Toggle raw display
$37$ \( (T^{7} + 7 T^{6} + \cdots + 3776)^{2} \) Copy content Toggle raw display
$41$ \( T^{14} + 161 T^{12} + \cdots + 65536 \) Copy content Toggle raw display
$43$ \( (T^{7} - 3 T^{6} + \cdots - 263744)^{2} \) Copy content Toggle raw display
$47$ \( (T^{7} + 8 T^{6} + \cdots + 31104)^{2} \) Copy content Toggle raw display
$53$ \( T^{14} + 153 T^{12} + \cdots + 65536 \) Copy content Toggle raw display
$59$ \( (T^{7} + 6 T^{6} + \cdots - 86016)^{2} \) Copy content Toggle raw display
$61$ \( T^{14} + 380 T^{12} + \cdots + 11943936 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 116046510336 \) Copy content Toggle raw display
$71$ \( (T^{7} + 2 T^{6} + \cdots - 539136)^{2} \) Copy content Toggle raw display
$73$ \( (T^{7} + 7 T^{6} + \cdots + 183296)^{2} \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 1904135049216 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 420381063424 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 44835521536 \) Copy content Toggle raw display
$97$ \( (T^{7} - 3 T^{6} + \cdots + 5888)^{2} \) Copy content Toggle raw display
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