Properties

Label 1344.2.bb.e
Level $1344$
Weight $2$
Character orbit 1344.bb
Analytic conductor $10.732$
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] = [1344,2,Mod(31,1344)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1344, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 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("1344.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1344.bb (of order \(6\), 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: \(10.7318940317\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 15x^{10} + 84x^{8} - 187x^{6} + 141x^{4} + 108x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{9} + \beta_{3}) q^{3} + (\beta_{10} - \beta_{4} + \beta_1) q^{5} + (\beta_{5} + \beta_{3}) q^{7} + \beta_{4} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{9} + \beta_{3}) q^{3} + (\beta_{10} - \beta_{4} + \beta_1) q^{5} + (\beta_{5} + \beta_{3}) q^{7} + \beta_{4} q^{9} + (\beta_{11} - \beta_{5} - \beta_{3}) q^{11} + ( - \beta_{10} + 2 \beta_{8} + \beta_{6} + \beta_{4} + 2 \beta_1) q^{13} + ( - \beta_{7} - \beta_{3}) q^{15} + (\beta_{10} - \beta_{8} - \beta_{4} + \beta_1 - 1) q^{17} - \beta_{11} q^{19} + ( - \beta_{10} + \beta_{8} + \beta_{6} + \beta_{4} - 1) q^{21} + ( - 2 \beta_{11} + \beta_{9} - \beta_{7} + 2 \beta_{5} + \beta_{3} + \beta_{2}) q^{23} + ( - 2 \beta_{10} + \beta_{8} + 2 \beta_{6} + 3 \beta_{4} + \beta_1 - 3) q^{25} + \beta_{3} q^{27} + ( - \beta_{10} + \beta_{6} + 5 \beta_{4} - 3) q^{29} + ( - 2 \beta_{9} - \beta_{7} + \beta_{5} + 3 \beta_{3}) q^{31} + (\beta_{10} - \beta_{8} - \beta_{6} - \beta_{4} - \beta_1 + 1) q^{33} + (4 \beta_{11} - \beta_{9} + 2 \beta_{7} - 2 \beta_{5} - 4 \beta_{3} - \beta_{2}) q^{35} + (\beta_{10} + \beta_{8} + \beta_{6} - \beta_1 + 1) q^{37} + ( - 3 \beta_{11} + \beta_{9} - 2 \beta_{7} + \beta_{5} + 2 \beta_{2}) q^{39} + (2 \beta_{10} + 2 \beta_{6} + 2 \beta_{4} + 2 \beta_1 - 2) q^{41} + ( - 2 \beta_{11} - \beta_{7} + 2 \beta_{5} + 2 \beta_{2}) q^{43} + (\beta_{10} - \beta_{4} + 1) q^{45} + ( - 2 \beta_{11} + \beta_{9} - 3 \beta_{7} + \beta_{3} + \beta_{2}) q^{47} + ( - \beta_{10} + \beta_{6} - 4 \beta_{4} + 2 \beta_1) q^{49} + ( - \beta_{9} - \beta_{7} - \beta_{3} - \beta_{2}) q^{51} + (\beta_{10} - 2 \beta_{8} - 3 \beta_{4} - 3) q^{53} + ( - 6 \beta_{11} - 4 \beta_{9} - 3 \beta_{7} + 3 \beta_{5} - 2 \beta_{3} + 3 \beta_{2}) q^{55} + \beta_1 q^{57} + ( - \beta_{11} - \beta_{9} + \beta_{7} + 2 \beta_{5} - 3 \beta_{3} + 4 \beta_{2}) q^{59} + (2 \beta_{10} - 4 \beta_{4} + 2 \beta_1) q^{61} + ( - \beta_{11} + \beta_{5} + \beta_{2}) q^{63} + ( - 2 \beta_{10} - 3 \beta_{8} + 3 \beta_{6} + 8 \beta_{4} - 2 \beta_1 - 3) q^{65} + ( - \beta_{11} - 2 \beta_{9} + \beta_{7} + 2 \beta_{3}) q^{67} + ( - \beta_{10} + 2 \beta_{8} + \beta_{6} + \beta_{4} + \beta_1) q^{69} + (\beta_{5} - 5 \beta_{3} - \beta_{2}) q^{71} + (2 \beta_{10} - 3 \beta_{8} - 3 \beta_{4} + \beta_1 - 3) q^{73} + ( - 3 \beta_{11} - \beta_{9} - \beta_{7} + 2 \beta_{5} + \beta_{3} + \beta_{2}) q^{75} + ( - \beta_{10} - \beta_{8} - \beta_{6} + 3 \beta_{4} - 3 \beta_1 + 3) q^{77} + (2 \beta_{11} + 6 \beta_{9} - \beta_{7} + 2 \beta_{5} + \beta_{3} + \beta_{2}) q^{79} + (\beta_{4} - 1) q^{81} + (3 \beta_{7} - 2 \beta_{5} + \beta_{3} + 2 \beta_{2}) q^{83} + (8 \beta_{4} - 4) q^{85} + ( - \beta_{11} - 2 \beta_{9} + \beta_{5} + 3 \beta_{3}) q^{87} + ( - 4 \beta_{10} + 4 \beta_{8} + 4 \beta_{6} + 4 \beta_1 + 4) q^{89} + ( - \beta_{11} + 2 \beta_{9} - 4 \beta_{7} + 2 \beta_{5} - 8 \beta_{3} + 2 \beta_{2}) q^{91} + (\beta_{8} + \beta_{6} + 3 \beta_{4} - 5) q^{93} + (2 \beta_{11} + 5 \beta_{9} + \beta_{7} - \beta_{5} + 6 \beta_{3} - 2 \beta_{2}) q^{95} + ( - 3 \beta_{10} - \beta_{6} + 7 \beta_{4} - 2 \beta_1 - 3) q^{97} + (2 \beta_{11} + \beta_{7} - \beta_{5} - \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{5} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 6 q^{5} + 6 q^{9} - 12 q^{17} - 6 q^{21} - 12 q^{25} + 6 q^{33} + 12 q^{37} + 6 q^{45} - 18 q^{49} - 42 q^{53} - 24 q^{61} + 48 q^{65} - 36 q^{73} + 54 q^{77} - 6 q^{81} + 48 q^{89} - 42 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 15x^{10} + 84x^{8} - 187x^{6} + 141x^{4} + 108x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -67\nu^{10} + 870\nu^{8} - 4056\nu^{6} + 5925\nu^{4} + 3698\nu^{2} - 21684 ) / 8084 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 241\nu^{11} - 3974\nu^{9} + 24604\nu^{7} - 61491\nu^{5} + 38098\nu^{3} + 65932\nu ) / 16168 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -321\nu^{11} + 4681\nu^{9} - 25224\nu^{7} + 51915\nu^{5} - 33411\nu^{3} - 43440\nu ) / 16168 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -19\nu^{10} + 286\nu^{8} - 1616\nu^{6} + 3737\nu^{4} - 3410\nu^{2} - 888 ) / 376 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -562\nu^{11} + 8655\nu^{9} - 49828\nu^{7} + 113406\nu^{5} - 71509\nu^{3} - 77036\nu ) / 16168 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -483\nu^{10} + 7780\nu^{8} - 47700\nu^{6} + 124277\nu^{4} - 122292\nu^{2} - 22752 ) / 8084 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -829\nu^{11} + 12303\nu^{9} - 67560\nu^{7} + 143895\nu^{5} - 91461\nu^{3} - 119288\nu ) / 16168 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 463\nu^{10} - 7098\nu^{8} + 41482\nu^{6} - 102419\nu^{4} + 101738\nu^{2} + 26354 ) / 4042 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -1367\nu^{11} + 20948\nu^{9} - 121244\nu^{7} + 290893\nu^{5} - 273308\nu^{3} - 66452\nu ) / 16168 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2185\nu^{10} - 33078\nu^{8} + 189224\nu^{6} - 444795\nu^{4} + 401362\nu^{2} + 149120 ) / 16168 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -3727\nu^{11} + 56962\nu^{9} - 329508\nu^{7} + 792549\nu^{5} - 754478\nu^{3} - 183044\nu ) / 16168 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} - \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{10} + \beta_{6} + \beta_{4} + 3\beta _1 + 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{11} - 6\beta_{9} + 4\beta_{7} + 3\beta_{5} - 11\beta_{3} + 3\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 13\beta_{10} - 4\beta_{8} + 5\beta_{6} + 17\beta_{4} + 17\beta _1 + 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 12\beta_{11} - 34\beta_{9} + 36\beta_{7} + 3\beta_{5} - 83\beta_{3} + 13\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 97\beta_{10} - 18\beta_{8} + 33\beta_{6} + 165\beta_{4} + 89\beta _1 - 56 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 28\beta_{11} - 78\beta_{9} + 238\beta_{7} - 41\beta_{5} - 493\beta_{3} + 57\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 583\beta_{10} - 4\beta_{8} + 243\beta_{6} + 1199\beta_{4} + 389\beta _1 - 792 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -186\beta_{11} + 534\beta_{9} + 1284\beta_{7} - 495\beta_{5} - 2449\beta_{3} + 153\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 2903\beta_{10} + 684\beta_{8} + 1655\beta_{6} + 7139\beta_{4} + 1091\beta _1 - 6790 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( -3180\beta_{11} + 9042\beta_{9} + 5428\beta_{7} - 3959\beta_{5} - 9217\beta_{3} - 593\beta_{2} ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(449\) \(577\) \(1093\)
\(\chi(n)\) \(-1\) \(1\) \(\beta_{4}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
−0.0585812 + 0.500000i
−1.51496 + 0.500000i
2.43956 + 0.500000i
0.0585812 0.500000i
1.51496 0.500000i
−2.43956 0.500000i
−0.0585812 0.500000i
−1.51496 0.500000i
2.43956 0.500000i
0.0585812 + 0.500000i
1.51496 + 0.500000i
−2.43956 + 0.500000i
0 −0.866025 0.500000i 0 −1.87328 3.24462i 0 −0.0585812 2.64510i 0 0.500000 + 0.866025i 0
31.2 0 −0.866025 0.500000i 0 −0.727452 1.25998i 0 −1.51496 + 2.16908i 0 0.500000 + 0.866025i 0
31.3 0 −0.866025 0.500000i 0 1.10074 + 1.90653i 0 2.43956 1.02398i 0 0.500000 + 0.866025i 0
31.4 0 0.866025 + 0.500000i 0 −1.87328 3.24462i 0 0.0585812 + 2.64510i 0 0.500000 + 0.866025i 0
31.5 0 0.866025 + 0.500000i 0 −0.727452 1.25998i 0 1.51496 2.16908i 0 0.500000 + 0.866025i 0
31.6 0 0.866025 + 0.500000i 0 1.10074 + 1.90653i 0 −2.43956 + 1.02398i 0 0.500000 + 0.866025i 0
607.1 0 −0.866025 + 0.500000i 0 −1.87328 + 3.24462i 0 −0.0585812 + 2.64510i 0 0.500000 0.866025i 0
607.2 0 −0.866025 + 0.500000i 0 −0.727452 + 1.25998i 0 −1.51496 2.16908i 0 0.500000 0.866025i 0
607.3 0 −0.866025 + 0.500000i 0 1.10074 1.90653i 0 2.43956 + 1.02398i 0 0.500000 0.866025i 0
607.4 0 0.866025 0.500000i 0 −1.87328 + 3.24462i 0 0.0585812 2.64510i 0 0.500000 0.866025i 0
607.5 0 0.866025 0.500000i 0 −0.727452 + 1.25998i 0 1.51496 + 2.16908i 0 0.500000 0.866025i 0
607.6 0 0.866025 0.500000i 0 1.10074 1.90653i 0 −2.43956 1.02398i 0 0.500000 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
56.j odd 6 1 inner
56.m even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1344.2.bb.e 12
4.b odd 2 1 inner 1344.2.bb.e 12
7.d odd 6 1 1344.2.bb.h yes 12
8.b even 2 1 1344.2.bb.h yes 12
8.d odd 2 1 1344.2.bb.h yes 12
28.f even 6 1 1344.2.bb.h yes 12
56.j odd 6 1 inner 1344.2.bb.e 12
56.m even 6 1 inner 1344.2.bb.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1344.2.bb.e 12 1.a even 1 1 trivial
1344.2.bb.e 12 4.b odd 2 1 inner
1344.2.bb.e 12 56.j odd 6 1 inner
1344.2.bb.e 12 56.m even 6 1 inner
1344.2.bb.h yes 12 7.d odd 6 1
1344.2.bb.h yes 12 8.b even 2 1
1344.2.bb.h yes 12 8.d odd 2 1
1344.2.bb.h yes 12 28.f even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1344, [\chi])\):

\( T_{5}^{6} + 3T_{5}^{5} + 15T_{5}^{4} + 6T_{5}^{3} + 72T_{5}^{2} + 72T_{5} + 144 \) Copy content Toggle raw display
\( T_{13}^{3} - 39T_{13} + 74 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{2} + 1)^{3} \) Copy content Toggle raw display
$5$ \( (T^{6} + 3 T^{5} + 15 T^{4} + 6 T^{3} + \cdots + 144)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} + 9 T^{10} + 30 T^{8} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( T^{12} + 27 T^{10} + 609 T^{8} + \cdots + 2304 \) Copy content Toggle raw display
$13$ \( (T^{3} - 39 T + 74)^{4} \) Copy content Toggle raw display
$17$ \( (T^{6} + 6 T^{5} - 72 T^{3} + 48 T^{2} + \cdots + 768)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} - 18 T^{10} + 243 T^{8} + \cdots + 256 \) Copy content Toggle raw display
$23$ \( T^{12} - 48 T^{10} + 1728 T^{8} + \cdots + 331776 \) Copy content Toggle raw display
$29$ \( (T^{6} + 75 T^{4} + 624 T^{2} + 768)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + 81 T^{10} + 5766 T^{8} + \cdots + 21609 \) Copy content Toggle raw display
$37$ \( (T^{6} - 6 T^{5} - 57 T^{4} + 414 T^{3} + \cdots + 101568)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 132 T^{4} + 3504 T^{2} + \cdots + 192)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} - 66 T^{4} + 1425 T^{2} + \cdots - 10092)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + 132 T^{10} + \cdots + 1416167424 \) Copy content Toggle raw display
$53$ \( (T^{6} + 21 T^{5} + 165 T^{4} + 378 T^{3} + \cdots + 48)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} - 309 T^{10} + \cdots + 38862602496 \) Copy content Toggle raw display
$61$ \( (T^{6} + 12 T^{5} + 132 T^{4} + \cdots + 12544)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + 90 T^{10} + 6507 T^{8} + \cdots + 28005264 \) Copy content Toggle raw display
$71$ \( (T^{6} + 156 T^{4} + 3600 T^{2} + \cdots + 9216)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 18 T^{5} + 63 T^{4} - 810 T^{3} + \cdots + 1728)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} - 363 T^{10} + \cdots + 2512481776561 \) Copy content Toggle raw display
$83$ \( (T^{6} + 309 T^{4} + 19260 T^{2} + \cdots + 121104)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} - 24 T^{5} + 48 T^{4} + \cdots + 3145728)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 267 T^{4} + 19200 T^{2} + \cdots + 196608)^{2} \) Copy content Toggle raw display
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