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Properties

Label 1404.2.i.c
Level $1404$
Weight $2$
Character orbit 1404.i
Analytic conductor $11.211$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1404,2,Mod(469,1404)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1404, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2, 0])) 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("1404.469"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1404 = 2^{2} \cdot 3^{3} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1404.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.2109964438\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} - x^{9} - x^{8} + 28x^{7} + 13x^{6} - 150x^{5} - 68x^{4} + 177x^{3} + 141x^{2} + 6x + 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{5} \)
Twist minimal: no (minimal twist has level 468)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} - \beta_1) q^{5} + (\beta_{11} - \beta_{6} + \cdots - \beta_{3}) q^{7} + ( - \beta_{11} + \beta_{6} + \cdots - \beta_{2}) q^{11} + (\beta_{7} + 1) q^{13} + ( - \beta_{9} - \beta_{6} - \beta_{5} + \cdots - 1) q^{17}+ \cdots + ( - 2 \beta_{11} + 2 \beta_{10} + \cdots - 2 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + q^{5} - q^{11} + 6 q^{13} - 12 q^{17} + 6 q^{19} + 7 q^{23} - 3 q^{25} + 10 q^{29} + 12 q^{31} - 26 q^{35} - 18 q^{37} + 12 q^{41} - 3 q^{43} + 17 q^{47} - 12 q^{49} - 48 q^{53} + 6 q^{55} + q^{59}+ \cdots + 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} - x^{9} - x^{8} + 28x^{7} + 13x^{6} - 150x^{5} - 68x^{4} + 177x^{3} + 141x^{2} + 6x + 12 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 1707195 \nu^{11} - 65543020 \nu^{10} + 73078749 \nu^{9} - 121987447 \nu^{8} + \cdots + 163350932 ) / 2335240613 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 8301223 \nu^{11} + 26206810 \nu^{10} - 161897541 \nu^{9} + 158509263 \nu^{8} + \cdots - 192954168 ) / 2335240613 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 13348737 \nu^{11} + 16413168 \nu^{10} - 17361143 \nu^{9} + 35570136 \nu^{8} + \cdots + 1022391242 ) / 2335240613 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 13510451 \nu^{11} - 17996204 \nu^{10} + 33656706 \nu^{9} - 53174140 \nu^{8} + \cdots - 584902030 ) / 2335240613 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 38662444 \nu^{11} - 45963780 \nu^{10} + 23126480 \nu^{9} - 51209303 \nu^{8} - 24497093 \nu^{7} + \cdots + 310763750 ) / 2335240613 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 50526903 \nu^{11} - 116916676 \nu^{10} + 54642748 \nu^{9} - 76422597 \nu^{8} + \cdots - 11593778670 ) / 2335240613 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2044407 \nu^{11} - 2431399 \nu^{10} + 1558912 \nu^{9} - 3336933 \nu^{8} + 211767 \nu^{7} + \cdots - 23120684 ) / 76565266 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 76834630 \nu^{11} - 45705051 \nu^{10} + 128591682 \nu^{9} - 33843251 \nu^{8} + \cdots - 4387033262 ) / 2335240613 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 87187588 \nu^{11} - 165604978 \nu^{10} + 197397506 \nu^{9} - 168618281 \nu^{8} + \cdots - 225629292 ) / 2335240613 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 102274651 \nu^{11} + 154237880 \nu^{10} - 58985417 \nu^{9} + 62961913 \nu^{8} + \cdots + 6826509202 ) / 2335240613 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 194944319 \nu^{11} + 173508867 \nu^{10} + 55987581 \nu^{9} + 209577515 \nu^{8} + \cdots - 8084389638 ) / 2335240613 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{5} - 2\beta_{4} - \beta_{3} + \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} + \beta_{9} - 2\beta_{8} - 2\beta_{7} + \beta_{5} - \beta_{4} - 3\beta_{3} + \beta_{2} + 3\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - \beta_{11} + 2 \beta_{9} - 3 \beta_{8} - 12 \beta_{7} + 2 \beta_{6} + 4 \beta_{5} + 2 \beta_{4} + \cdots - 6 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4 \beta_{11} + 3 \beta_{10} - 3 \beta_{9} - \beta_{8} + 10 \beta_{7} + 2 \beta_{6} + 16 \beta_{5} + \cdots + 4 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 23 \beta_{10} + 6 \beta_{9} + 12 \beta_{8} + 28 \beta_{7} + 4 \beta_{6} + 12 \beta_{5} - 18 \beta_{4} + \cdots - 14 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 12 \beta_{11} + 22 \beta_{10} + 25 \beta_{9} + 8 \beta_{8} + 20 \beta_{7} + 3 \beta_{6} - 33 \beta_{5} + \cdots - 44 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 24 \beta_{11} - 25 \beta_{10} - 46 \beta_{9} + 32 \beta_{8} + 68 \beta_{7} + 35 \beta_{6} + \cdots + 212 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 32 \beta_{11} - 52 \beta_{10} - 119 \beta_{9} + 137 \beta_{8} + 410 \beta_{7} - 35 \beta_{6} + \cdots + 494 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 71 \beta_{11} - 66 \beta_{10} + 151 \beta_{9} - 57 \beta_{8} + 32 \beta_{7} - 184 \beta_{6} - 565 \beta_{5} + \cdots + 4 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 115 \beta_{11} - 727 \beta_{10} + 350 \beta_{9} - 942 \beta_{8} - 2708 \beta_{7} - 24 \beta_{6} + \cdots - 740 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 568 \beta_{11} - 1277 \beta_{10} - 1082 \beta_{9} - 1163 \beta_{8} - 2094 \beta_{7} + 284 \beta_{6} + \cdots + 1568 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1404\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(703\) \(1081\)
\(\chi(n)\) \(-1 - \beta_{7}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
469.1
−1.47015 + 0.709134i
−0.911883 − 0.374754i
−0.348630 − 2.22587i
0.0260437 − 0.275663i
1.59451 − 0.133579i
1.61011 + 1.43470i
−1.47015 − 0.709134i
−0.911883 + 0.374754i
−0.348630 + 2.22587i
0.0260437 + 0.275663i
1.59451 + 0.133579i
1.61011 − 1.43470i
0 0 0 −1.47015 + 2.54637i 0 −0.0944197 − 0.163540i 0 0 0
469.2 0 0 0 −0.911883 + 1.57943i 0 0.687845 + 1.19138i 0 0 0
469.3 0 0 0 −0.348630 + 0.603844i 0 −1.09840 − 1.90248i 0 0 0
469.4 0 0 0 0.0260437 − 0.0451090i 0 2.52118 + 4.36681i 0 0 0
469.5 0 0 0 1.59451 − 2.76177i 0 −2.31626 − 4.01189i 0 0 0
469.6 0 0 0 1.61011 − 2.78880i 0 0.300057 + 0.519714i 0 0 0
937.1 0 0 0 −1.47015 − 2.54637i 0 −0.0944197 + 0.163540i 0 0 0
937.2 0 0 0 −0.911883 − 1.57943i 0 0.687845 − 1.19138i 0 0 0
937.3 0 0 0 −0.348630 − 0.603844i 0 −1.09840 + 1.90248i 0 0 0
937.4 0 0 0 0.0260437 + 0.0451090i 0 2.52118 − 4.36681i 0 0 0
937.5 0 0 0 1.59451 + 2.76177i 0 −2.31626 + 4.01189i 0 0 0
937.6 0 0 0 1.61011 + 2.78880i 0 0.300057 − 0.519714i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 469.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1404.2.i.c 12
3.b odd 2 1 468.2.i.c ✓ 12
9.c even 3 1 inner 1404.2.i.c 12
9.c even 3 1 4212.2.a.k 6
9.d odd 6 1 468.2.i.c ✓ 12
9.d odd 6 1 4212.2.a.m 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
468.2.i.c ✓ 12 3.b odd 2 1
468.2.i.c ✓ 12 9.d odd 6 1
1404.2.i.c 12 1.a even 1 1 trivial
1404.2.i.c 12 9.c even 3 1 inner
4212.2.a.k 6 9.c even 3 1
4212.2.a.m 6 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} - T_{5}^{11} + 17 T_{5}^{10} + 6 T_{5}^{9} + 196 T_{5}^{8} + 85 T_{5}^{7} + 1096 T_{5}^{6} + \cdots + 4 \) acting on \(S_{2}^{\mathrm{new}}(1404, [\chi])\). 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
$5$ \( T^{12} - T^{11} + \cdots + 4 \) Copy content Toggle raw display
$7$ \( T^{12} + 27 T^{10} + \cdots + 64 \) Copy content Toggle raw display
$11$ \( T^{12} + T^{11} + \cdots + 29584 \) Copy content Toggle raw display
$13$ \( (T^{2} - T + 1)^{6} \) Copy content Toggle raw display
$17$ \( (T^{6} + 6 T^{5} + \cdots + 1917)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} - 3 T^{5} + \cdots + 216)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} - 7 T^{11} + \cdots + 22801 \) Copy content Toggle raw display
$29$ \( T^{12} - 10 T^{11} + \cdots + 21316 \) Copy content Toggle raw display
$31$ \( T^{12} - 12 T^{11} + \cdots + 32330596 \) Copy content Toggle raw display
$37$ \( (T^{6} + 9 T^{5} + \cdots - 8046)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} - 12 T^{11} + \cdots + 1822500 \) Copy content Toggle raw display
$43$ \( T^{12} + 3 T^{11} + \cdots + 27426169 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 578979844 \) Copy content Toggle raw display
$53$ \( (T^{6} + 24 T^{5} + \cdots + 63477)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} - T^{11} + \cdots + 12013156 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 10422163921 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 201288617104 \) Copy content Toggle raw display
$71$ \( (T^{6} + 17 T^{5} + \cdots + 268)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} - 3 T^{5} + \cdots - 15206)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 24341304289 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 19095370596 \) Copy content Toggle raw display
$89$ \( (T^{6} + 21 T^{5} + \cdots + 37584)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 88982890000 \) Copy content Toggle raw display
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