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Properties

Label 369.2.h.b
Level $369$
Weight $2$
Character orbit 369.h
Analytic conductor $2.946$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [369,2,Mod(10,369)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(369, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("369.10");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 369 = 3^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 369.h (of order \(5\), degree \(4\), 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: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.13140625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 5x^{6} - 3x^{5} + 4x^{4} + 3x^{3} + 5x^{2} + 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 41)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{7} + \beta_{2}) q^{2} + (\beta_{6} + 2 \beta_{5} + \cdots - 2 \beta_1) q^{4}+ \cdots + ( - \beta_{7} + \beta_{6} + 2 \beta_{5} + \cdots - 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{7} + \beta_{2}) q^{2} + (\beta_{6} + 2 \beta_{5} + \cdots - 2 \beta_1) q^{4}+ \cdots + (\beta_{6} + \beta_{5} - 2 \beta_{4} + \cdots - 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{2} - 3 q^{4} - 3 q^{5} - 3 q^{7} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + q^{2} - 3 q^{4} - 3 q^{5} - 3 q^{7} - 4 q^{8} - 12 q^{10} + 11 q^{13} + 10 q^{14} + 13 q^{16} - 14 q^{17} - 4 q^{19} + 9 q^{20} + 15 q^{22} + 15 q^{23} + 19 q^{25} - 12 q^{26} + 2 q^{28} + 9 q^{29} + 6 q^{31} - 12 q^{32} - 8 q^{34} + 2 q^{35} - 36 q^{37} + 22 q^{38} + 24 q^{40} + 7 q^{41} + 13 q^{43} + 20 q^{44} - 24 q^{46} + 12 q^{47} + 13 q^{49} + 30 q^{50} - 2 q^{52} - 8 q^{53} + 20 q^{55} - 11 q^{56} + 26 q^{58} - 23 q^{59} - 15 q^{61} - 30 q^{62} - 36 q^{64} - 2 q^{65} - 10 q^{67} - 6 q^{68} - 11 q^{70} - 2 q^{71} - 4 q^{73} - 34 q^{74} - 21 q^{76} - 10 q^{77} - 42 q^{79} - 2 q^{80} - 41 q^{82} + 16 q^{83} - 6 q^{85} - 15 q^{86} - 20 q^{88} + 9 q^{89} - 30 q^{91} - 21 q^{92} + 10 q^{94} - 21 q^{95} - 8 q^{97} - 16 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 5x^{6} - 3x^{5} + 4x^{4} + 3x^{3} + 5x^{2} + 3x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} + 2\nu^{6} - 3\nu^{5} - 4\nu^{3} - 7\nu^{2} - 12\nu - 7 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - 7\nu^{5} + 20\nu^{4} - 16\nu^{3} + 19\nu^{2} + 6\nu + 9 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + 4\nu^{6} - 9\nu^{5} + 12\nu^{4} - 16\nu^{3} + 13\nu^{2} - 10\nu - 1 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{7} + 10\nu^{6} - 17\nu^{5} + 8\nu^{4} - 4\nu^{3} - 13\nu^{2} - 8\nu - 5 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3\nu^{7} - 12\nu^{6} + 23\nu^{5} - 20\nu^{4} + 16\nu^{3} + \nu^{2} + 6\nu - 1 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -5\nu^{7} + 18\nu^{6} - 35\nu^{5} + 32\nu^{4} - 28\nu^{3} - 11\nu^{2} - 12\nu - 7 ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{5} + \beta_{4} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 3\beta_{6} + 2\beta_{5} + \beta_{4} - 3\beta_{2} - 2\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{7} + 4\beta_{6} + \beta_{4} - \beta_{3} - 5\beta_{2} - 4\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{7} - 6\beta_{5} - 4\beta_{3} - 6\beta_{2} - 6\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -16\beta_{6} - 16\beta_{5} - 6\beta_{4} - 6\beta_{3} - 7\beta _1 - 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -16\beta_{7} - 51\beta_{6} - 29\beta_{5} - 23\beta_{4} + 29\beta_{2} - 16 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(83\) \(334\)
\(\chi(n)\) \(1\) \(-1 + \beta_{3} - \beta_{4} - \beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1
−0.386111 + 0.280526i
1.69513 − 1.23158i
−0.386111 − 0.280526i
1.69513 + 1.23158i
−0.227943 + 0.701538i
0.418926 − 1.28932i
−0.227943 − 0.701538i
0.418926 + 1.28932i
−1.19513 − 0.868312i 0 0.0563329 + 0.173375i 0.0563329 + 0.173375i 0 −1.69513 + 1.23158i −0.829779 + 2.55380i 0 0.0832183 − 0.256120i
10.2 0.886111 + 0.643798i 0 −0.247316 − 0.761160i −0.247316 − 0.761160i 0 0.386111 − 0.280526i 0.947813 − 2.91707i 0 0.270884 − 0.833694i
37.1 −1.19513 + 0.868312i 0 0.0563329 − 0.173375i 0.0563329 − 0.173375i 0 −1.69513 − 1.23158i −0.829779 − 2.55380i 0 0.0832183 + 0.256120i
37.2 0.886111 − 0.643798i 0 −0.247316 + 0.761160i −0.247316 + 0.761160i 0 0.386111 + 0.280526i 0.947813 + 2.91707i 0 0.270884 + 0.833694i
100.1 0.0810736 + 0.249519i 0 1.56235 − 1.13511i 1.56235 − 1.13511i 0 −0.418926 + 1.28932i 0.834404 + 0.606230i 0 0.409897 + 0.297808i
100.2 0.727943 + 2.24038i 0 −2.87136 + 2.08617i −2.87136 + 2.08617i 0 0.227943 − 0.701538i −2.95244 − 2.14507i 0 −6.76400 − 4.91433i
262.1 0.0810736 − 0.249519i 0 1.56235 + 1.13511i 1.56235 + 1.13511i 0 −0.418926 − 1.28932i 0.834404 − 0.606230i 0 0.409897 − 0.297808i
262.2 0.727943 − 2.24038i 0 −2.87136 − 2.08617i −2.87136 − 2.08617i 0 0.227943 + 0.701538i −2.95244 + 2.14507i 0 −6.76400 + 4.91433i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 369.2.h.b 8
3.b odd 2 1 41.2.d.a ✓ 8
12.b even 2 1 656.2.u.d 8
41.d even 5 1 inner 369.2.h.b 8
123.k odd 10 1 41.2.d.a ✓ 8
123.k odd 10 1 1681.2.a.e 4
123.l odd 10 1 1681.2.a.f 4
492.w even 10 1 656.2.u.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
41.2.d.a ✓ 8 3.b odd 2 1
41.2.d.a ✓ 8 123.k odd 10 1
369.2.h.b 8 1.a even 1 1 trivial
369.2.h.b 8 41.d even 5 1 inner
656.2.u.d 8 12.b even 2 1
656.2.u.d 8 492.w even 10 1
1681.2.a.e 4 123.k odd 10 1
1681.2.a.f 4 123.l odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - T_{2}^{7} + 4T_{2}^{6} + 3T_{2}^{5} - T_{2}^{4} - 9T_{2}^{3} + 16T_{2}^{2} - 3T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(369, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{7} + 4 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 3 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{8} + 3 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{8} + 15 T^{6} + \cdots + 625 \) Copy content Toggle raw display
$13$ \( T^{8} - 11 T^{7} + \cdots + 121 \) Copy content Toggle raw display
$17$ \( T^{8} + 14 T^{7} + \cdots + 841 \) Copy content Toggle raw display
$19$ \( T^{8} + 4 T^{7} + \cdots + 271441 \) Copy content Toggle raw display
$23$ \( T^{8} - 15 T^{7} + \cdots + 19321 \) Copy content Toggle raw display
$29$ \( T^{8} - 9 T^{7} + \cdots + 10201 \) Copy content Toggle raw display
$31$ \( T^{8} - 6 T^{7} + \cdots + 92416 \) Copy content Toggle raw display
$37$ \( T^{8} + 36 T^{7} + \cdots + 8755681 \) Copy content Toggle raw display
$41$ \( T^{8} - 7 T^{7} + \cdots + 2825761 \) Copy content Toggle raw display
$43$ \( T^{8} - 13 T^{7} + \cdots + 358801 \) Copy content Toggle raw display
$47$ \( T^{8} - 12 T^{7} + \cdots + 674041 \) Copy content Toggle raw display
$53$ \( T^{8} + 8 T^{7} + \cdots + 6195121 \) Copy content Toggle raw display
$59$ \( T^{8} + 23 T^{7} + \cdots + 9740641 \) Copy content Toggle raw display
$61$ \( T^{8} + 15 T^{7} + \cdots + 203401 \) Copy content Toggle raw display
$67$ \( (T^{4} + 5 T^{3} + \cdots + 3025)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 2 T^{7} + \cdots + 15768841 \) Copy content Toggle raw display
$73$ \( (T^{4} + 2 T^{3} + \cdots - 359)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 21 T^{3} + \cdots - 571)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 8 T^{3} + \cdots - 211)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - 9 T^{7} + \cdots + 6345361 \) Copy content Toggle raw display
$97$ \( T^{8} + 8 T^{7} + \cdots + 78961 \) Copy content Toggle raw display
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