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Properties

Label 3432.2.g.d
Level $3432$
Weight $2$
Character orbit 3432.g
Analytic conductor $27.405$
Analytic rank $0$
Dimension $14$
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] = [3432,2,Mod(1585,3432)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3432, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("3432.1585");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3432 = 2^{3} \cdot 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3432.g (of order \(2\), degree \(1\), 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: \(27.4046579737\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 25x^{12} + 236x^{10} + 1040x^{8} + 2124x^{6} + 1676x^{4} + 340x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + \beta_1 q^{5} - \beta_{7} q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + \beta_1 q^{5} - \beta_{7} q^{7} + q^{9} + \beta_{8} q^{11} + (\beta_{13} - \beta_{11} + \cdots - \beta_{7}) q^{13}+ \cdots + \beta_{8} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 14 q^{3} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 14 q^{3} + 14 q^{9} - 4 q^{13} - 4 q^{17} + 2 q^{23} + 20 q^{25} + 14 q^{27} - 10 q^{29} - 14 q^{35} - 4 q^{39} + 22 q^{43} + 8 q^{49} - 4 q^{51} - 20 q^{53} + 2 q^{55} - 2 q^{61} - 20 q^{65} + 2 q^{69} + 20 q^{75} + 2 q^{77} - 40 q^{79} + 14 q^{81} - 10 q^{87} - 44 q^{91} + 4 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 25x^{12} + 236x^{10} + 1040x^{8} + 2124x^{6} + 1676x^{4} + 340x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 17\nu^{12} + 608\nu^{10} + 7687\nu^{8} + 42498\nu^{6} + 97322\nu^{4} + 62514\nu^{2} + 3332 ) / 3614 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -6\nu^{12} - 2\nu^{10} + 1645\nu^{8} + 15826\nu^{6} + 51643\nu^{4} + 54468\nu^{2} + 6052 ) / 1807 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 29\nu^{13} + 612\nu^{11} + 4397\nu^{9} + 10846\nu^{7} - 5964\nu^{5} - 50036\nu^{3} - 23228\nu ) / 3614 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 4\nu^{12} + 94\nu^{10} + 803\nu^{8} + 3025\nu^{6} + 4862\nu^{4} + 2608\nu^{2} + 228 ) / 278 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -193\nu^{13} - 4883\nu^{11} - 46772\nu^{9} - 209514\nu^{7} - 431624\nu^{5} - 318768\nu^{3} - 23372\nu ) / 7228 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -57\nu^{13} - 1409\nu^{11} - 13076\nu^{9} - 56068\nu^{7} - 108968\nu^{5} - 76084\nu^{3} - 8948\nu ) / 1112 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 274\nu^{12} + 6717\nu^{10} + 61608\nu^{8} + 259685\nu^{6} + 492480\nu^{4} + 335162\nu^{2} + 42862 ) / 3614 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 833 \nu^{13} + 386 \nu^{12} + 20757 \nu^{11} + 9766 \nu^{10} + 194156 \nu^{9} + 93544 \nu^{8} + \cdots + 46744 ) / 14456 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 833 \nu^{13} - 386 \nu^{12} + 20757 \nu^{11} - 9766 \nu^{10} + 194156 \nu^{9} - 93544 \nu^{8} + \cdots - 46744 ) / 14456 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1899 \nu^{13} + 47615 \nu^{11} + 450620 \nu^{9} + 1985968 \nu^{7} + 4020784 \nu^{5} + \cdots + 531220 \nu ) / 14456 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 53\nu^{13} + 1315\nu^{11} + 12273\nu^{9} + 53043\nu^{7} + 104106\nu^{5} + 73476\nu^{3} + 8720\nu ) / 278 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{13} + \beta_{12} - \beta_{8} + \beta_{5} - 7\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} - \beta_{10} + \beta_{9} - \beta_{6} + \beta_{4} - \beta_{3} - 8\beta_{2} + 25 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 10\beta_{13} - 9\beta_{12} + 11\beta_{8} + 3\beta_{7} - 10\beta_{5} + 51\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -13\beta_{11} + 13\beta_{10} - 13\beta_{9} + 15\beta_{6} - 10\beta_{4} + 9\beta_{3} + 64\beta_{2} - 173 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -88\beta_{13} + 73\beta_{12} - \beta_{11} - \beta_{10} - 109\beta_{8} - 39\beta_{7} + 84\beta_{5} - 381\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 123\beta_{11} - 123\beta_{10} + 127\beta_{9} - 171\beta_{6} + 80\beta_{4} - 71\beta_{3} - 508\beta_{2} + 1259 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 750 \beta_{13} - 579 \beta_{12} + 9 \beta_{11} + 9 \beta_{10} + 1027 \beta_{8} + 373 \beta_{7} + \cdots + 2887 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1041 \beta_{11} + 1041 \beta_{10} - 1123 \beta_{9} + 1735 \beta_{6} - 586 \beta_{4} + 561 \beta_{3} + \cdots - 9431 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 6306 \beta_{13} + 4571 \beta_{12} - 25 \beta_{11} - 25 \beta_{10} - 9315 \beta_{8} + \cdots - 22071 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 8387 \beta_{11} - 8387 \beta_{10} + 9511 \beta_{9} - 16503 \beta_{6} + 4058 \beta_{4} - 4521 \beta_{3} + \cdots + 71879 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 52606 \beta_{13} - 36103 \beta_{12} - 463 \beta_{11} - 463 \beta_{10} + 82167 \beta_{8} + \cdots + 169751 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3432\mathbb{Z}\right)^\times\).

\(n\) \(937\) \(1145\) \(1717\) \(2575\) \(2641\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(1\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1585.1
− 2.83259i
− 2.68836i
− 2.15782i
− 1.90876i
− 1.08554i
− 0.450909i
− 0.260548i
0.260548i
0.450909i
1.08554i
1.90876i
2.15782i
2.68836i
2.83259i
0 1.00000 0 − 2.83259i 0 2.34008i 0 1.00000 0
1585.2 0 1.00000 0 − 2.68836i 0 − 0.456068i 0 1.00000 0
1585.3 0 1.00000 0 − 2.15782i 0 − 3.35467i 0 1.00000 0
1585.4 0 1.00000 0 − 1.90876i 0 − 4.43738i 0 1.00000 0
1585.5 0 1.00000 0 − 1.08554i 0 2.42229i 0 1.00000 0
1585.6 0 1.00000 0 − 0.450909i 0 1.57703i 0 1.00000 0
1585.7 0 1.00000 0 − 0.260548i 0 − 0.131822i 0 1.00000 0
1585.8 0 1.00000 0 0.260548i 0 0.131822i 0 1.00000 0
1585.9 0 1.00000 0 0.450909i 0 − 1.57703i 0 1.00000 0
1585.10 0 1.00000 0 1.08554i 0 − 2.42229i 0 1.00000 0
1585.11 0 1.00000 0 1.90876i 0 4.43738i 0 1.00000 0
1585.12 0 1.00000 0 2.15782i 0 3.35467i 0 1.00000 0
1585.13 0 1.00000 0 2.68836i 0 0.456068i 0 1.00000 0
1585.14 0 1.00000 0 2.83259i 0 − 2.34008i 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1585.14
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3432.2.g.d ✓ 14
13.b even 2 1 inner 3432.2.g.d ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3432.2.g.d ✓ 14 1.a even 1 1 trivial
3432.2.g.d ✓ 14 13.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 \(S_{2}^{\mathrm{new}}(3432, [\chi])\):

\( T_{5}^{14} + 25T_{5}^{12} + 236T_{5}^{10} + 1040T_{5}^{8} + 2124T_{5}^{6} + 1676T_{5}^{4} + 340T_{5}^{2} + 16 \) Copy content Toggle raw display
\( T_{17}^{7} + 2T_{17}^{6} - 46T_{17}^{5} - 88T_{17}^{4} + 474T_{17}^{3} + 822T_{17}^{2} - 564T_{17} - 944 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( (T - 1)^{14} \) Copy content Toggle raw display
$5$ \( T^{14} + 25 T^{12} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( T^{14} + 45 T^{12} + \cdots + 64 \) Copy content Toggle raw display
$11$ \( (T^{2} + 1)^{7} \) Copy content Toggle raw display
$13$ \( T^{14} + 4 T^{13} + \cdots + 62748517 \) Copy content Toggle raw display
$17$ \( (T^{7} + 2 T^{6} + \cdots - 944)^{2} \) Copy content Toggle raw display
$19$ \( T^{14} + 160 T^{12} + \cdots + 4096 \) Copy content Toggle raw display
$23$ \( (T^{7} - T^{6} - 76 T^{5} + \cdots - 3056)^{2} \) Copy content Toggle raw display
$29$ \( (T^{7} + 5 T^{6} + \cdots + 596)^{2} \) Copy content Toggle raw display
$31$ \( T^{14} + 176 T^{12} + \cdots + 30118144 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 15162474496 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 3411494464 \) Copy content Toggle raw display
$43$ \( (T^{7} - 11 T^{6} + \cdots + 46712)^{2} \) Copy content Toggle raw display
$47$ \( T^{14} + 208 T^{12} + \cdots + 262144 \) Copy content Toggle raw display
$53$ \( (T^{7} + 10 T^{6} + \cdots - 135616)^{2} \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 34733031424 \) Copy content Toggle raw display
$61$ \( (T^{7} + T^{6} - 124 T^{5} + \cdots + 1424)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 1309568964496 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 6679465984 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 1072533639424 \) Copy content Toggle raw display
$79$ \( (T^{7} + 20 T^{6} + \cdots + 785792)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 7180189696 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 105932022784 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 158466606075904 \) Copy content Toggle raw display
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