Properties

Label 5780.2.c.g
Level $5780$
Weight $2$
Character orbit 5780.c
Analytic conductor $46.154$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5780,2,Mod(5201,5780)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5780, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5780.5201");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5780 = 2^{2} \cdot 5 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5780.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(46.1535323683\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: 12.0.851059918206111744.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 33x^{10} + 360x^{8} + 1423x^{6} + 1269x^{4} + 234x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{11} q^{3} - \beta_{3} q^{5} - \beta_{6} q^{7} + \beta_{8} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{11} q^{3} - \beta_{3} q^{5} - \beta_{6} q^{7} + \beta_{8} q^{9} + ( - \beta_{9} + \beta_{7} + \beta_{6}) q^{11} + (\beta_{8} + \beta_{5} - \beta_{4} - 2) q^{13} - \beta_{8} q^{15} + (\beta_{8} - \beta_{2} + \beta_1 + 1) q^{19} + ( - \beta_{5} + \beta_{4} - \beta_{2} + \beta_1) q^{21} + (\beta_{10} + \beta_{9} - \beta_{7} + 2 \beta_{6}) q^{23} - q^{25} + (2 \beta_{11} - 3 \beta_{3}) q^{27} + ( - 2 \beta_{11} - \beta_{9} + \beta_{7} + \beta_{6} + 2 \beta_{3}) q^{29} + ( - 2 \beta_{11} + \beta_{10} - 2 \beta_{9} + \beta_{7} + \beta_{6} + 2 \beta_{3}) q^{31} + ( - 3 \beta_{5} + 3 \beta_{4}) q^{33} - \beta_{4} q^{35} + ( - \beta_{11} + 2 \beta_{10} - \beta_{9} + 2 \beta_{7} + \beta_{6} + \beta_{3}) q^{37} + ( - 3 \beta_{11} + \beta_{7} - 3 \beta_{3}) q^{39} + (\beta_{11} - \beta_{9} + \beta_{7} + \beta_{6} - \beta_{3}) q^{41} + ( - \beta_{8} - \beta_{5} - \beta_{4} + 2 \beta_{2} - \beta_1 - 1) q^{43} + \beta_{11} q^{45} + (\beta_{8} + 2 \beta_1 - 2) q^{47} + (\beta_{5} + 5) q^{49} + ( - 2 \beta_{8} - 2 \beta_{5} - 2 \beta_{2} + 2 \beta_1 + 1) q^{53} - \beta_1 q^{55} + (\beta_{10} + \beta_{7} + 4 \beta_{6} - 3 \beta_{3}) q^{57} + (2 \beta_{8} + \beta_{5} - 3 \beta_{4} + \beta_{2} + 2) q^{59} + (\beta_{11} + 2 \beta_{10} - 2 \beta_{9} + 3 \beta_{6} - \beta_{3}) q^{61} + (\beta_{10} + \beta_{6}) q^{63} + (\beta_{11} - \beta_{9} + \beta_{6} + 2 \beta_{3}) q^{65} + ( - 2 \beta_{8} - 5 \beta_{5} + 2 \beta_{4} + 2 \beta_1 - 2) q^{67} + (6 \beta_{5} - 9 \beta_{4} + 3 \beta_{2} - 3 \beta_1) q^{69} + ( - 2 \beta_{11} + 2 \beta_{10} - 2 \beta_{9} + 2 \beta_{7} + 5 \beta_{6} + 2 \beta_{3}) q^{71} + (\beta_{11} - 4 \beta_{10} - 3 \beta_{9} - 3 \beta_{7} + \beta_{3}) q^{73} - \beta_{11} q^{75} + (\beta_{8} - \beta_1 + 1) q^{77} + (\beta_{11} + \beta_{10} - 5 \beta_{9} + 2 \beta_{7} + 5 \beta_{6} + 6 \beta_{3}) q^{79} + (2 \beta_{8} - 6) q^{81} + ( - \beta_{8} + 2 \beta_{5} + 3 \beta_{4} + 2 \beta_{2} + \beta_1 + 2) q^{83} + ( - 3 \beta_{5} + 3 \beta_{4} + 6) q^{87} + ( - 2 \beta_{8} + 2 \beta_{5} - 3 \beta_{4} - \beta_{2} - 3 \beta_1 - 2) q^{89} + (\beta_{10} - \beta_{9} + 4 \beta_{6} + \beta_{3}) q^{91} + ( - 3 \beta_{5} - \beta_{2} + 6) q^{93} + (\beta_{11} - \beta_{10} - \beta_{9} - \beta_{3}) q^{95} + (\beta_{10} + 6 \beta_{9} - 2 \beta_{7} - 3 \beta_{6} + 3 \beta_{3}) q^{97} + ( - 3 \beta_{9} + 3 \beta_{6}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 6 q^{9} - 30 q^{13} + 6 q^{15} + 6 q^{19} - 12 q^{25} - 6 q^{43} - 30 q^{47} + 60 q^{49} + 24 q^{53} + 12 q^{59} - 12 q^{67} + 6 q^{77} - 84 q^{81} + 30 q^{83} + 72 q^{87} - 12 q^{89} + 72 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 33x^{10} + 360x^{8} + 1423x^{6} + 1269x^{4} + 234x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 3760\nu^{10} + 126642\nu^{8} + 1422495\nu^{6} + 5912210\nu^{4} + 6470694\nu^{2} + 2460930 ) / 605663 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 15553\nu^{10} + 526746\nu^{8} + 6002454\nu^{6} + 25752178\nu^{4} + 29356755\nu^{2} + 5922120 ) / 1816989 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 157\nu^{11} + 5255\nu^{9} + 58845\nu^{7} + 246955\nu^{5} + 280457\nu^{3} + 76464\nu ) / 14307 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -1628\nu^{10} - 53409\nu^{8} - 573690\nu^{6} - 2156651\nu^{4} - 1369446\nu^{2} - 32919 ) / 95631 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1682\nu^{10} + 55533\nu^{8} + 606582\nu^{6} + 2409932\nu^{4} + 2213283\nu^{2} + 289536 ) / 95631 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -39591\nu^{11} - 1302230\nu^{9} - 14105940\nu^{7} - 54603024\nu^{5} - 42225431\nu^{3} + 680379\nu ) / 1816989 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 62603\nu^{11} + 2037358\nu^{9} + 21584478\nu^{7} + 78641531\nu^{5} + 39061204\nu^{3} - 13975590\nu ) / 1816989 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 96\nu^{10} + 3141\nu^{8} + 33752\nu^{6} + 128925\nu^{4} + 97101\nu^{2} + 7898 ) / 2413 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 155117\nu^{11} + 5102456\nu^{9} + 55313739\nu^{7} + 215208887\nu^{5} + 176806943\nu^{3} + 23601756\nu ) / 1816989 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -55370\nu^{11} - 1816614\nu^{9} - 19580178\nu^{7} - 74940657\nu^{5} - 55736126\nu^{3} - 4838763\nu ) / 605663 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 194708 \nu^{11} - 6404686 \nu^{9} - 69419679 \nu^{7} - 269811911 \nu^{5} - 219032374 \nu^{3} - 21104388 \nu ) / 1816989 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_{11} + \beta_{9} - \beta_{6} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} - 2\beta_{5} + \beta_{4} + 2\beta _1 - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -10\beta_{11} - 3\beta_{10} - 12\beta_{9} - 3\beta_{7} + 9\beta_{6} - 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -15\beta_{8} + 33\beta_{5} - 12\beta_{4} - 6\beta_{2} - 22\beta _1 + 54 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 119\beta_{11} + 55\beta_{10} + 159\beta_{9} + 60\beta_{7} - 75\beta_{6} + 46\beta_{3} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 230\beta_{8} - 504\beta_{5} + 180\beta_{4} + 144\beta_{2} + 249\beta _1 - 646 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -1518\beta_{11} - 861\beta_{10} - 2149\beta_{9} - 1022\beta_{7} + 601\beta_{6} - 876\beta_{3} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -3577\beta_{8} + 7609\beta_{5} - 2877\beta_{4} - 2570\beta_{2} - 2980\beta _1 + 8165 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 20272\beta_{11} + 12852\beta_{10} + 29547\beta_{9} + 16326\beta_{7} - 4356\beta_{6} + 15257\beta_{3} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 55329\beta_{8} - 114054\beta_{5} + 45951\beta_{4} + 41517\beta_{2} + 37480\beta _1 - 107571 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -279377\beta_{11} - 188617\beta_{10} - 412665\beta_{9} - 252417\beta_{7} + 22923\beta_{6} - 251032\beta_{3} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5780\mathbb{Z}\right)^\times\).

\(n\) \(581\) \(1157\) \(2891\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5201.1
2.65007i
3.83486i
0.423390i
3.18216i
0.229313i
0.955479i
0.955479i
0.229313i
3.18216i
0.423390i
3.83486i
2.65007i
0 2.30278i 0 1.00000i 0 1.87939i 0 −2.30278 0
5201.2 0 2.30278i 0 1.00000i 0 0.347296i 0 −2.30278 0
5201.3 0 2.30278i 0 1.00000i 0 1.53209i 0 −2.30278 0
5201.4 0 1.30278i 0 1.00000i 0 1.53209i 0 1.30278 0
5201.5 0 1.30278i 0 1.00000i 0 0.347296i 0 1.30278 0
5201.6 0 1.30278i 0 1.00000i 0 1.87939i 0 1.30278 0
5201.7 0 1.30278i 0 1.00000i 0 1.87939i 0 1.30278 0
5201.8 0 1.30278i 0 1.00000i 0 0.347296i 0 1.30278 0
5201.9 0 1.30278i 0 1.00000i 0 1.53209i 0 1.30278 0
5201.10 0 2.30278i 0 1.00000i 0 1.53209i 0 −2.30278 0
5201.11 0 2.30278i 0 1.00000i 0 0.347296i 0 −2.30278 0
5201.12 0 2.30278i 0 1.00000i 0 1.87939i 0 −2.30278 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5201.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5780.2.c.g 12
17.b even 2 1 inner 5780.2.c.g 12
17.c even 4 1 5780.2.a.l 6
17.c even 4 1 5780.2.a.o yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5780.2.a.l 6 17.c even 4 1
5780.2.a.o yes 6 17.c even 4 1
5780.2.c.g 12 1.a even 1 1 trivial
5780.2.c.g 12 17.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(5780, [\chi])\):

\( T_{3}^{4} + 7T_{3}^{2} + 9 \) Copy content Toggle raw display
\( T_{7}^{6} + 6T_{7}^{4} + 9T_{7}^{2} + 1 \) Copy content Toggle raw display
\( T_{11}^{12} + 42T_{11}^{10} + 603T_{11}^{8} + 3556T_{11}^{6} + 8235T_{11}^{4} + 5103T_{11}^{2} + 729 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{4} + 7 T^{2} + 9)^{3} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$7$ \( (T^{6} + 6 T^{4} + 9 T^{2} + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + 42 T^{10} + 603 T^{8} + \cdots + 729 \) Copy content Toggle raw display
$13$ \( (T^{6} + 15 T^{5} + 78 T^{4} + 157 T^{3} + \cdots - 51)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} \) Copy content Toggle raw display
$19$ \( (T^{6} - 3 T^{5} - 30 T^{4} + 92 T^{3} + \cdots + 51)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + 180 T^{10} + 10044 T^{8} + \cdots + 59049 \) Copy content Toggle raw display
$29$ \( T^{12} + 126 T^{10} + 4311 T^{8} + \cdots + 59049 \) Copy content Toggle raw display
$31$ \( T^{12} + 150 T^{10} + 6255 T^{8} + \cdots + 2277081 \) Copy content Toggle raw display
$37$ \( T^{12} + 189 T^{10} + \cdots + 96255721 \) Copy content Toggle raw display
$41$ \( T^{12} + 63 T^{10} + 1287 T^{8} + \cdots + 59049 \) Copy content Toggle raw display
$43$ \( (T^{6} + 3 T^{5} - 111 T^{4} - 778 T^{3} + \cdots - 51)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 15 T^{5} - 701 T^{3} - 1548 T^{2} + \cdots - 1377)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} - 12 T^{5} - 63 T^{4} + 1056 T^{3} + \cdots + 8667)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} - 6 T^{5} - 90 T^{4} + 483 T^{3} + \cdots - 8667)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + 225 T^{10} + \cdots + 11444689 \) Copy content Toggle raw display
$67$ \( (T^{6} + 6 T^{5} - 180 T^{4} - 1206 T^{3} + \cdots - 32941)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + 324 T^{10} + \cdots + 75116889 \) Copy content Toggle raw display
$73$ \( T^{12} + 741 T^{10} + \cdots + 357602804001 \) Copy content Toggle raw display
$79$ \( T^{12} + 561 T^{10} + \cdots + 155102856561 \) Copy content Toggle raw display
$83$ \( (T^{6} - 15 T^{5} - 126 T^{4} + 1968 T^{3} + \cdots - 4131)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + 6 T^{5} - 288 T^{4} - 1563 T^{3} + \cdots - 65529)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + 552 T^{10} + \cdots + 39167972281 \) Copy content Toggle raw display
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