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Properties

Label 369.2.f.d
Level $369$
Weight $2$
Character orbit 369.f
Analytic conductor $2.946$
Analytic rank $0$
Dimension $12$
Inner twists $4$

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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: \( x^{12} + 107x^{8} + 727x^{4} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - \beta_{10} q^{2} + (\beta_{3} - 2) q^{4} + ( - \beta_{10} + \beta_{7} - \beta_1) q^{5} + ( - \beta_{8} - \beta_{6} - \beta_{5} + \cdots + 1) q^{7}+ \cdots + (2 \beta_{10} - \beta_{9} - \beta_{2}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{10} q^{2} + (\beta_{3} - 2) q^{4} + ( - \beta_{10} + \beta_{7} - \beta_1) q^{5} + ( - \beta_{8} - \beta_{6} - \beta_{5} + \cdots + 1) q^{7}+ \cdots + ( - 7 \beta_{11} + 3 \beta_{7} + 3 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 24 q^{4} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 24 q^{4} + 8 q^{7} - 8 q^{10} - 4 q^{13} + 32 q^{16} - 28 q^{19} + 28 q^{22} - 12 q^{25} + 28 q^{28} - 4 q^{31} - 40 q^{34} + 4 q^{37} + 8 q^{40} - 28 q^{52} + 40 q^{55} - 84 q^{58} + 8 q^{64} - 8 q^{67} + 100 q^{70} + 100 q^{76} + 4 q^{82} - 4 q^{85} - 44 q^{88} - 32 q^{94} - 56 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 107x^{8} + 727x^{4} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{9} + 138\nu^{5} + 3035\nu ) / 1182 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -11\nu^{8} - 1124\nu^{4} - 3441 ) / 1182 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -8\nu^{10} + 12\nu^{8} - 907\nu^{6} + 1065\nu^{4} - 9899\nu^{2} - 3177 ) / 3546 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -8\nu^{10} - 12\nu^{8} - 907\nu^{6} - 1065\nu^{4} - 9899\nu^{2} + 3177 ) / 3546 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 13\nu^{10} + 1400\nu^{6} + 10693\nu^{2} ) / 3546 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -13\nu^{11} - 1400\nu^{7} - 10693\nu^{3} ) / 3546 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -49\nu^{10} - 5186\nu^{6} - 30121\nu^{2} ) / 3546 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 52\nu^{11} + 5600\nu^{7} + 40999\nu^{3} ) / 5319 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -145\nu^{11} - 54\nu^{9} - 15479\nu^{7} - 5679\nu^{5} - 102220\nu^{3} - 34461\nu ) / 10638 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 145\nu^{11} - 54\nu^{9} + 15479\nu^{7} - 5679\nu^{5} + 102220\nu^{3} - 34461\nu ) / 10638 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} + 5\beta_{6} + \beta_{5} + \beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -3\beta_{9} - 8\beta_{7} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 11\beta_{5} - 11\beta_{4} - 8\beta_{3} - 43 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3\beta_{11} + 3\beta_{10} + 36\beta_{2} - 73\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -73\beta_{8} - 413\beta_{6} - 112\beta_{5} - 112\beta_{4} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -39\beta_{11} + 39\beta_{10} + 375\beta_{9} + 710\beta_{7} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -1124\beta_{5} + 1124\beta_{4} + 710\beta_{3} + 4081 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -414\beta_{11} - 414\beta_{10} - 3786\beta_{2} + 7039\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 7039\beta_{8} + 40637\beta_{6} + 11239\beta_{5} + 11239\beta_{4} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 4200\beta_{11} - 4200\beta_{10} - 37917\beta_{9} - 70154\beta_{7} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(83\) \(334\)
\(\chi(n)\) \(1\) \(\beta_{6}\)

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
−0.410248 − 0.410248i
−2.23449 − 2.23449i
1.15705 + 1.15705i
−1.15705 − 1.15705i
2.23449 + 2.23449i
0.410248 + 0.410248i
0.410248 − 0.410248i
2.23449 − 2.23449i
−1.15705 + 1.15705i
1.15705 − 1.15705i
−2.23449 + 2.23449i
−0.410248 + 0.410248i
− 2.60836i 0 −4.80354 − 1.78786i 0 −2.73347 + 2.73347i 7.31265i 0 −4.66339
73.2 − 2.15043i 0 −2.62434 2.31855i 0 3.18077 − 3.18077i 1.34259i 0 4.98587
73.3 − 0.756387i 0 1.42788 − 3.07049i 0 1.55270 − 1.55270i − 2.59280i 0 −2.32247
73.4 0.756387i 0 1.42788 3.07049i 0 1.55270 − 1.55270i 2.59280i 0 −2.32247
73.5 2.15043i 0 −2.62434 − 2.31855i 0 3.18077 − 3.18077i − 1.34259i 0 4.98587
73.6 2.60836i 0 −4.80354 1.78786i 0 −2.73347 + 2.73347i − 7.31265i 0 −4.66339
91.1 − 2.60836i 0 −4.80354 − 1.78786i 0 −2.73347 − 2.73347i 7.31265i 0 −4.66339
91.2 − 2.15043i 0 −2.62434 2.31855i 0 3.18077 + 3.18077i 1.34259i 0 4.98587
91.3 − 0.756387i 0 1.42788 − 3.07049i 0 1.55270 + 1.55270i − 2.59280i 0 −2.32247
91.4 0.756387i 0 1.42788 3.07049i 0 1.55270 + 1.55270i 2.59280i 0 −2.32247
91.5 2.15043i 0 −2.62434 − 2.31855i 0 3.18077 + 3.18077i − 1.34259i 0 4.98587
91.6 2.60836i 0 −4.80354 1.78786i 0 −2.73347 − 2.73347i − 7.31265i 0 −4.66339
\(n\): 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
3.b odd 2 1 inner
41.c even 4 1 inner
123.f odd 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.d ✓ 12
3.b odd 2 1 inner 369.2.f.d ✓ 12
41.c even 4 1 inner 369.2.f.d ✓ 12
123.f odd 4 1 inner 369.2.f.d ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
369.2.f.d ✓ 12 1.a even 1 1 trivial
369.2.f.d ✓ 12 3.b odd 2 1 inner
369.2.f.d ✓ 12 41.c even 4 1 inner
369.2.f.d ✓ 12 123.f odd 4 1 inner

Hecke kernels

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

\( T_{2}^{6} + 12T_{2}^{4} + 38T_{2}^{2} + 18 \) Copy content Toggle raw display
\( T_{11}^{12} + 147T_{11}^{8} + 5007T_{11}^{4} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + 12 T^{4} + \cdots + 18)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{6} + 18 T^{4} + \cdots + 162)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} - 4 T^{5} + \cdots + 1458)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + 147 T^{8} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( (T^{6} + 2 T^{5} + \cdots + 450)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + 727 T^{8} + \cdots + 6561 \) Copy content Toggle raw display
$19$ \( (T^{6} + 14 T^{5} + \cdots + 18)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} - 44 T^{4} + \cdots - 450)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + 12387 T^{8} + \cdots + 25411681 \) Copy content Toggle raw display
$31$ \( (T^{3} + T^{2} - 61 T - 115)^{4} \) Copy content Toggle raw display
$37$ \( (T^{3} - T^{2} - 25 T - 29)^{4} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 4750104241 \) Copy content Toggle raw display
$43$ \( (T^{6} + 227 T^{4} + \cdots + 225)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 2750058481 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 8318169616 \) Copy content Toggle raw display
$59$ \( (T^{6} - 304 T^{4} + \cdots - 839808)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 227 T^{4} + \cdots + 47961)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + 4 T^{5} + \cdots + 23328)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 22430753361 \) Copy content Toggle raw display
$73$ \( (T^{6} + 155 T^{4} + \cdots + 729)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 192 T^{3} + \cdots + 18432)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} - 284 T^{4} + \cdots - 357858)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 126178406656 \) Copy content Toggle raw display
$97$ \( (T^{6} + 28 T^{5} + \cdots + 20402)^{2} \) Copy content Toggle raw display
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