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Properties

Label 3204.2.k.c
Level $3204$
Weight $2$
Character orbit 3204.k
Analytic conductor $25.584$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3204,2,Mod(1369,3204)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3204, base_ring=CyclotomicField(4))
 
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("3204.1369");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3204 = 2^{2} \cdot 3^{2} \cdot 89 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3204.k (of order \(4\), degree \(2\), 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: \(25.5840688076\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} + 8 x^{10} + 4 x^{9} + 43 x^{8} - 158 x^{7} + 296 x^{6} + 166 x^{5} + 559 x^{4} + \cdots + 121 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 356)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{11} + \beta_{4}) q^{5} + (\beta_{9} - \beta_{4} - \beta_1 + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{11} + \beta_{4}) q^{5} + (\beta_{9} - \beta_{4} - \beta_1 + 1) q^{7} + ( - \beta_{9} + \beta_{8} + \beta_{5}) q^{11} - \beta_{7} q^{13} + ( - \beta_{6} + 2 \beta_{4}) q^{17} + (\beta_{8} + \beta_{6} + \cdots + \beta_{2}) q^{19}+ \cdots + ( - \beta_{10} + \beta_{9} + \cdots - \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 4 q^{7} + 12 q^{11} + 2 q^{13} + 12 q^{19} + 8 q^{23} - 16 q^{25} + 2 q^{29} + 10 q^{31} - 24 q^{35} + 8 q^{37} - 18 q^{41} + 2 q^{43} - 12 q^{59} - 14 q^{61} + 10 q^{65} - 24 q^{67} - 56 q^{73} - 36 q^{77} + 4 q^{83} - 52 q^{85} - 38 q^{89} - 16 q^{91} + 2 q^{95} - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} + 8 x^{10} + 4 x^{9} + 43 x^{8} - 158 x^{7} + 296 x^{6} + 166 x^{5} + 559 x^{4} + \cdots + 121 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 108707635 \nu^{11} - 488891325 \nu^{10} + 1372998479 \nu^{9} - 1386963369 \nu^{8} + \cdots - 1238317091626 ) / 279624055389 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1212837851 \nu^{11} + 3964636458 \nu^{10} - 7204740325 \nu^{9} - 7937096292 \nu^{8} + \cdots + 506215990181 ) / 838872166167 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 46019635471 \nu^{11} + 170737325523 \nu^{10} - 324546082730 \nu^{9} + \cdots + 14196859994746 ) / 9227593827837 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 48257743841 \nu^{11} + 223068383028 \nu^{10} - 464182192972 \nu^{9} + \cdots + 35248818936122 ) / 9227593827837 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 63796547354 \nu^{11} + 278444479053 \nu^{10} - 513551838610 \nu^{9} + \cdots + 13975348028774 ) / 9227593827837 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 105340427650 \nu^{11} - 468039369837 \nu^{10} + 1005956058251 \nu^{9} + \cdots - 77360076766291 ) / 9227593827837 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 10546857976 \nu^{11} - 30747570969 \nu^{10} + 42661666526 \nu^{9} + 117193534650 \nu^{8} + \cdots + 2966916894911 ) / 838872166167 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 15141655612 \nu^{11} + 55066871378 \nu^{10} - 101502116577 \nu^{9} - 96746753237 \nu^{8} + \cdots + 1704652009698 ) / 1025288203093 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 251393273050 \nu^{11} + 884382892740 \nu^{10} - 1535199514685 \nu^{9} + \cdots + 29657210089672 ) / 9227593827837 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 261216305599 \nu^{11} + 1005004782183 \nu^{10} - 1890988313105 \nu^{9} + \cdots + 76331163899749 ) / 9227593827837 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} - \beta_{6} - 4\beta_{4} - \beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} - \beta_{6} - \beta_{5} - \beta_{4} - 8\beta_{3} - \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} - \beta_{5} - 11\beta_{3} - 8\beta_{2} - 11\beta _1 - 25 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -14\beta_{11} + 3\beta_{10} + 12\beta_{9} + 3\beta_{8} + 11\beta_{6} + 19\beta_{4} - 11\beta_{2} - 65\beta _1 - 19 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 86 \beta_{11} + 15 \beta_{10} + 17 \beta_{9} - 15 \beta_{7} + 65 \beta_{6} + 17 \beta_{5} + \cdots - 109 \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 162 \beta_{11} - 53 \beta_{8} - 55 \beta_{7} + 109 \beta_{6} + 125 \beta_{5} + 224 \beta_{4} + \cdots + 224 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 178 \beta_{10} - 219 \beta_{9} - 286 \beta_{8} - 178 \beta_{7} + 219 \beta_{5} + 1057 \beta_{3} + \cdots + 1461 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1740 \beta_{11} - 724 \beta_{10} - 1272 \beta_{9} - 683 \beta_{8} - 1057 \beta_{6} - 2329 \beta_{4} + \cdots + 2329 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 8413 \beta_{11} - 1955 \beta_{10} - 2505 \beta_{9} + 1955 \beta_{7} - 5051 \beta_{6} + \cdots + 10218 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 18040 \beta_{11} + 7822 \beta_{8} + 8372 \beta_{7} - 10218 \beta_{6} - 12871 \beta_{5} - 23214 \beta_{4} + \cdots - 23214 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3204\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(713\) \(1603\)
\(\chi(n)\) \(-\beta_{4}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1369.1
2.22655 + 2.22655i
−1.30017 − 1.30017i
0.573552 + 0.573552i
−1.61091 − 1.61091i
1.83007 + 1.83007i
0.280905 + 0.280905i
0.280905 − 0.280905i
1.83007 − 1.83007i
−1.61091 + 1.61091i
0.573552 − 0.573552i
−1.30017 + 1.30017i
2.22655 − 2.22655i
0 0 0 − 3.61347i 0 −3.02857 − 3.02857i 0 0 0
1369.2 0 0 0 − 2.90663i 0 −0.724251 − 0.724251i 0 0 0
1369.3 0 0 0 − 2.16872i 0 1.14833 + 1.14833i 0 0 0
1369.4 0 0 0 − 0.0875617i 0 3.49617 + 3.49617i 0 0 0
1369.5 0 0 0 1.95432i 0 1.59031 + 1.59031i 0 0 0
1369.6 0 0 0 2.82205i 0 −0.481988 − 0.481988i 0 0 0
1657.1 0 0 0 − 2.82205i 0 −0.481988 + 0.481988i 0 0 0
1657.2 0 0 0 − 1.95432i 0 1.59031 − 1.59031i 0 0 0
1657.3 0 0 0 0.0875617i 0 3.49617 − 3.49617i 0 0 0
1657.4 0 0 0 2.16872i 0 1.14833 − 1.14833i 0 0 0
1657.5 0 0 0 2.90663i 0 −0.724251 + 0.724251i 0 0 0
1657.6 0 0 0 3.61347i 0 −3.02857 + 3.02857i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1369.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
89.c even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3204.2.k.c 12
3.b odd 2 1 356.2.f.b ✓ 12
12.b even 2 1 1424.2.l.g 12
89.c even 4 1 inner 3204.2.k.c 12
267.e odd 4 1 356.2.f.b ✓ 12
1068.l even 4 1 1424.2.l.g 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
356.2.f.b ✓ 12 3.b odd 2 1
356.2.f.b ✓ 12 267.e odd 4 1
1424.2.l.g 12 12.b even 2 1
1424.2.l.g 12 1068.l even 4 1
3204.2.k.c 12 1.a even 1 1 trivial
3204.2.k.c 12 89.c even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 38T_{5}^{10} + 551T_{5}^{8} + 3812T_{5}^{6} + 12575T_{5}^{4} + 15878T_{5}^{2} + 121 \) acting on \(S_{2}^{\mathrm{new}}(3204, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 38 T^{10} + \cdots + 121 \) Copy content Toggle raw display
$7$ \( T^{12} - 4 T^{11} + \cdots + 2916 \) Copy content Toggle raw display
$11$ \( (T^{6} - 6 T^{5} + \cdots - 108)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} - 2 T^{11} + \cdots + 810000 \) Copy content Toggle raw display
$17$ \( T^{12} + 90 T^{10} + \cdots + 9 \) Copy content Toggle raw display
$19$ \( T^{12} - 12 T^{11} + \cdots + 245025 \) Copy content Toggle raw display
$23$ \( T^{12} - 8 T^{11} + \cdots + 841 \) Copy content Toggle raw display
$29$ \( T^{12} - 2 T^{11} + \cdots + 82664464 \) Copy content Toggle raw display
$31$ \( T^{12} - 10 T^{11} + \cdots + 4566769 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 247118400 \) Copy content Toggle raw display
$41$ \( T^{12} + 18 T^{11} + \cdots + 725904 \) Copy content Toggle raw display
$43$ \( T^{12} - 2 T^{11} + \cdots + 10387729 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 309900816 \) Copy content Toggle raw display
$53$ \( T^{12} + 178 T^{10} + \cdots + 729 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 14921599716 \) Copy content Toggle raw display
$61$ \( T^{12} + 14 T^{11} + \cdots + 1106704 \) Copy content Toggle raw display
$67$ \( (T^{6} + 12 T^{5} + \cdots - 89172)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 119439360000 \) Copy content Toggle raw display
$73$ \( (T^{6} + 28 T^{5} + \cdots + 31879)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + 316 T^{10} + \cdots + 10969344 \) Copy content Toggle raw display
$83$ \( T^{12} - 4 T^{11} + \cdots + 62188996 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 496981290961 \) Copy content Toggle raw display
$97$ \( (T^{6} + 6 T^{5} + \cdots - 94491)^{2} \) Copy content Toggle raw display
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