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Properties

Label 1368.2.e.e
Level $1368$
Weight $2$
Character orbit 1368.e
Analytic conductor $10.924$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1368,2,Mod(379,1368)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1368, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 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("1368.379");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1368.e (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: \(10.9235349965\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: 12.0.319794774016000000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2x^{10} + 2x^{8} + 8x^{4} - 32x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 152)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{2} q^{4} + ( - \beta_{8} + \beta_{5}) q^{5} - \beta_{8} q^{7} + ( - \beta_{4} - \beta_{3}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + \beta_{2} q^{4} + ( - \beta_{8} + \beta_{5}) q^{5} - \beta_{8} q^{7} + ( - \beta_{4} - \beta_{3}) q^{8} + (\beta_{9} + \beta_{7} + \cdots - \beta_{4}) q^{10}+ \cdots - 2 \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 4 q^{4} - 12 q^{17} - 24 q^{19} - 4 q^{20} - 44 q^{25} + 44 q^{26} - 20 q^{28} - 40 q^{35} - 4 q^{38} - 24 q^{43} + 4 q^{44} + 24 q^{49} - 4 q^{58} + 8 q^{62} - 8 q^{64} - 12 q^{68} + 4 q^{73} - 48 q^{74} - 12 q^{76} - 32 q^{80} - 8 q^{82} - 24 q^{83} - 8 q^{92}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2x^{10} + 2x^{8} + 8x^{4} - 32x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{11} + 2\nu^{7} + 4\nu^{5} + 16\nu^{3} ) / 32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{11} - 2\nu^{7} - 4\nu^{5} + 16\nu^{3} ) / 32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{6} + 2\nu^{4} - 2\nu^{2} ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{11} - 2\nu^{7} + 12\nu^{5} - 16\nu^{3} ) / 32 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{11} + 6\nu^{7} - 4\nu^{5} ) / 32 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{8} + 2\nu^{4} - 4\nu^{2} + 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -\nu^{11} + 4\nu^{9} - 2\nu^{7} + 4\nu^{5} + 32\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -\nu^{10} + 2\nu^{6} + 4\nu^{4} - 8\nu^{2} + 16 ) / 16 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -\nu^{10} + 2\nu^{8} - 2\nu^{6} - 8\nu^{2} + 32 ) / 16 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{11} + \beta_{10} + \beta_{8} + \beta_{5} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{6} + 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -2\beta_{11} + 2\beta_{10} + 2\beta_{8} - 2\beta_{5} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 4\beta_{7} - 2\beta_{4} + 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2\beta_{11} - 2\beta_{10} + 6\beta_{8} - 2\beta_{5} + 2\beta_{2} - 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 8\beta_{9} - 4\beta_{6} - 4\beta_{4} - 8\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -8\beta_{11} - 8\beta_{10} + 8\beta_{8} - 4\beta_{2} + 16 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -8\beta_{7} - 8\beta_{6} - 12\beta_{4} + 4\beta_{3} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1368\mathbb{Z}\right)^\times\).

\(n\) \(343\) \(685\) \(1009\) \(1217\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
379.1
1.37364 + 0.336338i
1.37364 − 0.336338i
1.17117 + 0.792696i
1.17117 − 0.792696i
0.491416 + 1.32609i
0.491416 − 1.32609i
−0.491416 + 1.32609i
−0.491416 − 1.32609i
−1.17117 + 0.792696i
−1.17117 − 0.792696i
−1.37364 + 0.336338i
−1.37364 − 0.336338i
−1.37364 − 0.336338i 0 1.77375 + 0.924013i − 3.04222i 0 − 2.23607i −2.12571 − 1.86584i 0 −1.02321 + 4.17890i
379.2 −1.37364 + 0.336338i 0 1.77375 − 0.924013i 3.04222i 0 2.23607i −2.12571 + 1.86584i 0 −1.02321 − 4.17890i
379.3 −1.17117 − 0.792696i 0 0.743268 + 1.85676i 3.51876i 0 2.23607i 0.601353 − 2.76376i 0 2.78930 − 4.12105i
379.4 −1.17117 + 0.792696i 0 0.743268 − 1.85676i − 3.51876i 0 − 2.23607i 0.601353 + 2.76376i 0 2.78930 + 4.12105i
379.5 −0.491416 − 1.32609i 0 −1.51702 + 1.30332i − 2.08884i 0 2.23607i 2.47381 + 1.37123i 0 −2.76999 + 1.02649i
379.6 −0.491416 + 1.32609i 0 −1.51702 − 1.30332i 2.08884i 0 − 2.23607i 2.47381 − 1.37123i 0 −2.76999 − 1.02649i
379.7 0.491416 − 1.32609i 0 −1.51702 − 1.30332i 2.08884i 0 − 2.23607i −2.47381 + 1.37123i 0 2.76999 + 1.02649i
379.8 0.491416 + 1.32609i 0 −1.51702 + 1.30332i − 2.08884i 0 2.23607i −2.47381 − 1.37123i 0 2.76999 − 1.02649i
379.9 1.17117 − 0.792696i 0 0.743268 − 1.85676i − 3.51876i 0 − 2.23607i −0.601353 − 2.76376i 0 −2.78930 − 4.12105i
379.10 1.17117 + 0.792696i 0 0.743268 + 1.85676i 3.51876i 0 2.23607i −0.601353 + 2.76376i 0 −2.78930 + 4.12105i
379.11 1.37364 − 0.336338i 0 1.77375 − 0.924013i 3.04222i 0 2.23607i 2.12571 − 1.86584i 0 1.02321 + 4.17890i
379.12 1.37364 + 0.336338i 0 1.77375 + 0.924013i − 3.04222i 0 − 2.23607i 2.12571 + 1.86584i 0 1.02321 − 4.17890i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 379.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
19.b odd 2 1 inner
152.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1368.2.e.e 12
3.b odd 2 1 152.2.b.c ✓ 12
4.b odd 2 1 5472.2.e.e 12
8.b even 2 1 5472.2.e.e 12
8.d odd 2 1 inner 1368.2.e.e 12
12.b even 2 1 608.2.b.c 12
19.b odd 2 1 inner 1368.2.e.e 12
24.f even 2 1 152.2.b.c ✓ 12
24.h odd 2 1 608.2.b.c 12
57.d even 2 1 152.2.b.c ✓ 12
76.d even 2 1 5472.2.e.e 12
152.b even 2 1 inner 1368.2.e.e 12
152.g odd 2 1 5472.2.e.e 12
228.b odd 2 1 608.2.b.c 12
456.l odd 2 1 152.2.b.c ✓ 12
456.p even 2 1 608.2.b.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.2.b.c ✓ 12 3.b odd 2 1
152.2.b.c ✓ 12 24.f even 2 1
152.2.b.c ✓ 12 57.d even 2 1
152.2.b.c ✓ 12 456.l odd 2 1
608.2.b.c 12 12.b even 2 1
608.2.b.c 12 24.h odd 2 1
608.2.b.c 12 228.b odd 2 1
608.2.b.c 12 456.p even 2 1
1368.2.e.e 12 1.a even 1 1 trivial
1368.2.e.e 12 8.d odd 2 1 inner
1368.2.e.e 12 19.b odd 2 1 inner
1368.2.e.e 12 152.b even 2 1 inner
5472.2.e.e 12 4.b odd 2 1
5472.2.e.e 12 8.b even 2 1
5472.2.e.e 12 76.d even 2 1
5472.2.e.e 12 152.g odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1368, [\chi])\):

\( T_{5}^{6} + 26T_{5}^{4} + 209T_{5}^{2} + 500 \) Copy content Toggle raw display
\( T_{7}^{2} + 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 2 T^{10} + \cdots + 64 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{6} + 26 T^{4} + \cdots + 500)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 5)^{6} \) Copy content Toggle raw display
$11$ \( (T^{3} - 7 T + 4)^{4} \) Copy content Toggle raw display
$13$ \( (T^{6} - 51 T^{4} + \cdots - 640)^{2} \) Copy content Toggle raw display
$17$ \( (T^{3} + 3 T^{2} - 25 T - 59)^{4} \) Copy content Toggle raw display
$19$ \( (T^{6} + 12 T^{5} + \cdots + 6859)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + 69 T^{4} + \cdots + 320)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} - 71 T^{4} + \cdots - 40)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 44 T^{4} + \cdots - 640)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} - 84 T^{4} + \cdots - 2560)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 136 T^{4} + \cdots + 70688)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} + 6 T^{2} - 51 T - 10)^{4} \) Copy content Toggle raw display
$47$ \( (T^{6} + 106 T^{4} + \cdots + 80)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} - 191 T^{4} + \cdots - 112360)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 103 T^{4} + \cdots + 12800)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 214 T^{4} + \cdots + 50000)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + 155 T^{4} + \cdots + 75272)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 476 T^{4} + \cdots - 3504640)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} - T^{2} - 65 T + 97)^{4} \) Copy content Toggle raw display
$79$ \( (T^{6} - 424 T^{4} + \cdots - 1697440)^{2} \) Copy content Toggle raw display
$83$ \( (T + 2)^{12} \) Copy content Toggle raw display
$89$ \( (T^{6} + 232 T^{4} + \cdots + 231200)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 316 T^{4} + \cdots + 2048)^{2} \) Copy content Toggle raw display
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