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Properties

Label 369.2.f.e
Level $369$
Weight $2$
Character orbit 369.f
Analytic conductor $2.946$
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] = [369,2,Mod(73,369)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(369, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([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("369.73");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 369 = 3^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 369.f (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: \(2.94647983459\)
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: \( x^{12} + 16x^{10} + 92x^{8} + 236x^{6} + 260x^{4} + 88x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 123)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{11} + \beta_{10} - \beta_{7} + \cdots - 1) q^{4}+ \cdots + ( - \beta_{7} - \beta_{6} + \cdots - 2 \beta_{2}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{11} + \beta_{10} - \beta_{7} + \cdots - 1) q^{4}+ \cdots + (2 \beta_{11} + \beta_{10} - \beta_{9} + \cdots - 8) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 8 q^{4} - 24 q^{10} + 12 q^{11} + 4 q^{13} - 20 q^{14} - 8 q^{17} + 4 q^{19} + 12 q^{22} + 8 q^{23} - 28 q^{25} + 20 q^{26} - 44 q^{28} - 12 q^{29} + 12 q^{31} + 8 q^{34} - 36 q^{35} + 4 q^{37} - 20 q^{38} + 56 q^{40} + 20 q^{41} - 20 q^{44} + 40 q^{47} + 20 q^{52} + 12 q^{53} - 16 q^{55} + 40 q^{56} + 36 q^{58} + 16 q^{59} - 8 q^{64} - 4 q^{65} + 16 q^{67} - 4 q^{68} - 36 q^{70} - 8 q^{71} - 52 q^{76} - 20 q^{82} - 40 q^{83} + 36 q^{85} - 48 q^{86} + 36 q^{88} + 24 q^{89} + 72 q^{92} - 16 q^{94} - 28 q^{95} + 40 q^{97} - 80 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 16x^{10} + 92x^{8} + 236x^{6} + 260x^{4} + 88x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{11} - 14\nu^{9} - 66\nu^{7} - 130\nu^{5} - 108\nu^{3} - 40\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{11} - 14\nu^{9} - 64\nu^{7} - 106\nu^{5} - 24\nu^{3} + 44\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{10} + \nu^{9} + 14\nu^{8} + 12\nu^{7} + 64\nu^{6} + 42\nu^{5} + 110\nu^{4} + 46\nu^{3} + 56\nu^{2} + 8\nu + 4 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{11} + 15\nu^{9} + \nu^{8} + 78\nu^{7} + 12\nu^{6} + 172\nu^{5} + 42\nu^{4} + 150\nu^{3} + 46\nu^{2} + 32\nu + 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{10} - \nu^{9} + 14\nu^{8} - 12\nu^{7} + 64\nu^{6} - 42\nu^{5} + 110\nu^{4} - 46\nu^{3} + 56\nu^{2} - 8\nu + 4 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{11} + 15\nu^{9} - \nu^{8} + 78\nu^{7} - 12\nu^{6} + 172\nu^{5} - 42\nu^{4} + 150\nu^{3} - 46\nu^{2} + 32\nu - 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{11} + 16\nu^{9} + \nu^{8} + 92\nu^{7} + 14\nu^{6} + 234\nu^{5} + 62\nu^{4} + 240\nu^{3} + 90\nu^{2} + 44\nu + 12 ) / 8 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - \nu^{11} - 16 \nu^{9} + \nu^{8} - 92 \nu^{7} + 14 \nu^{6} - 234 \nu^{5} + 62 \nu^{4} - 240 \nu^{3} + \cdots + 12 ) / 8 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{10} + 13\nu^{8} + 53\nu^{6} + 76\nu^{4} + 24\nu^{2} + 2 ) / 4 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -\nu^{10} - 14\nu^{8} - 65\nu^{6} - 118\nu^{4} - 66\nu^{2} + 2 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} + \beta_{10} - \beta_{7} + \beta_{5} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - 2\beta_{2} - 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -4\beta_{11} - 6\beta_{10} + \beta_{9} + \beta_{8} + 7\beta_{7} + 2\beta_{6} - 7\beta_{5} + 2\beta_{4} + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{9} - \beta_{8} + 11\beta_{7} + 9\beta_{6} + 11\beta_{5} - 9\beta_{4} + 2\beta_{3} + 18\beta_{2} + 20\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 18\beta_{11} + 38\beta_{10} - 8\beta_{9} - 8\beta_{8} - 46\beta_{7} - 20\beta_{6} + 46\beta_{5} - 20\beta_{4} - 76 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 12 \beta_{9} + 12 \beta_{8} - 90 \beta_{7} - 66 \beta_{6} - 90 \beta_{5} + 66 \beta_{4} + \cdots - 114 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 94 \beta_{11} - 250 \beta_{10} + 54 \beta_{9} + 54 \beta_{8} + 300 \beta_{7} + 156 \beta_{6} + \cdots + 454 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 102 \beta_{9} - 102 \beta_{8} + 664 \beta_{7} + 456 \beta_{6} + 664 \beta_{5} - 456 \beta_{4} + \cdots + 704 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 548 \beta_{11} + 1672 \beta_{10} - 354 \beta_{9} - 354 \beta_{8} - 1970 \beta_{7} - 1120 \beta_{6} + \cdots - 2868 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 766 \beta_{9} + 766 \beta_{8} - 4678 \beta_{7} - 3090 \beta_{6} - 4678 \beta_{5} + 3090 \beta_{4} + \cdots - 4540 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/369\mathbb{Z}\right)^\times\).

\(n\) \(83\) \(334\)
\(\chi(n)\) \(1\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
73.1
− 2.59428i
− 1.71117i
0.231328i
0.691736i
1.47985i
1.90254i
− 1.90254i
− 1.47985i
− 0.691736i
− 0.231328i
1.71117i
2.59428i
− 2.59428i 0 −4.73027 − 2.98774i 0 3.21989 − 3.21989i 7.08307i 0 −7.75102
73.2 − 1.71117i 0 −0.928117 − 1.18955i 0 0.374411 − 0.374411i − 1.83418i 0 −2.03553
73.3 0.231328i 0 1.94649 2.07114i 0 −3.15929 + 3.15929i 0.912934i 0 −0.479114
73.4 0.691736i 0 1.52150 − 3.89594i 0 −0.934769 + 0.934769i 2.43595i 0 2.69496
73.5 1.47985i 0 −0.189944 − 1.46738i 0 1.37066 − 1.37066i 2.67860i 0 2.17150
73.6 1.90254i 0 −1.61966 3.46947i 0 −0.870912 + 0.870912i 0.723616i 0 −6.60080
91.1 − 1.90254i 0 −1.61966 − 3.46947i 0 −0.870912 − 0.870912i − 0.723616i 0 −6.60080
91.2 − 1.47985i 0 −0.189944 1.46738i 0 1.37066 + 1.37066i − 2.67860i 0 2.17150
91.3 − 0.691736i 0 1.52150 3.89594i 0 −0.934769 − 0.934769i − 2.43595i 0 2.69496
91.4 − 0.231328i 0 1.94649 − 2.07114i 0 −3.15929 − 3.15929i − 0.912934i 0 −0.479114
91.5 1.71117i 0 −0.928117 1.18955i 0 0.374411 + 0.374411i 1.83418i 0 −2.03553
91.6 2.59428i 0 −4.73027 2.98774i 0 3.21989 + 3.21989i − 7.08307i 0 −7.75102
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 73.6
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 369.2.f.e 12
3.b odd 2 1 123.2.e.a ✓ 12
12.b even 2 1 1968.2.t.d 12
41.c even 4 1 inner 369.2.f.e 12
123.f odd 4 1 123.2.e.a ✓ 12
123.i even 8 1 5043.2.a.s 6
123.i even 8 1 5043.2.a.t 6
492.l even 4 1 1968.2.t.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
123.2.e.a ✓ 12 3.b odd 2 1
123.2.e.a ✓ 12 123.f odd 4 1
369.2.f.e 12 1.a even 1 1 trivial
369.2.f.e 12 41.c even 4 1 inner
1968.2.t.d 12 12.b even 2 1
1968.2.t.d 12 492.l even 4 1
5043.2.a.s 6 123.i even 8 1
5043.2.a.t 6 123.i even 8 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(369, [\chi])\):

\( T_{2}^{12} + 16T_{2}^{10} + 92T_{2}^{8} + 236T_{2}^{6} + 260T_{2}^{4} + 88T_{2}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{12} - 12 T_{11}^{11} + 72 T_{11}^{10} - 160 T_{11}^{9} + 203 T_{11}^{8} - 208 T_{11}^{7} + \cdots + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 16 T^{10} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 44 T^{10} + \cdots + 21316 \) Copy content Toggle raw display
$7$ \( T^{12} + 4 T^{9} + \cdots + 1156 \) Copy content Toggle raw display
$11$ \( T^{12} - 12 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{12} - 4 T^{11} + \cdots + 749956 \) Copy content Toggle raw display
$17$ \( T^{12} + 8 T^{11} + \cdots + 22801 \) Copy content Toggle raw display
$19$ \( T^{12} - 4 T^{11} + \cdots + 27730756 \) Copy content Toggle raw display
$23$ \( (T^{6} - 4 T^{5} + \cdots - 526)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 296907361 \) Copy content Toggle raw display
$31$ \( (T^{6} - 6 T^{5} - 25 T^{4} + \cdots - 79)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} - 2 T^{5} + \cdots - 2663)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 4750104241 \) Copy content Toggle raw display
$43$ \( T^{12} + 134 T^{10} + \cdots + 37249 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 1505983249 \) Copy content Toggle raw display
$53$ \( T^{12} - 12 T^{11} + \cdots + 56490256 \) Copy content Toggle raw display
$59$ \( (T^{6} - 8 T^{5} + \cdots + 21376)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 1911525841 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 15423652864 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 124523281 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 23298664321 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 233967616 \) Copy content Toggle raw display
$83$ \( (T^{6} + 20 T^{5} + \cdots - 7454)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} - 24 T^{11} + \cdots + 5992704 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 274695484996 \) Copy content Toggle raw display
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