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Properties

Label 1344.3.m.e
Level $1344$
Weight $3$
Character orbit 1344.m
Analytic conductor $36.621$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1344,3,Mod(127,1344)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1344, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0, 0])) N = Newforms(chi, 3, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1344.127"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1344.m (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(36.6213475300\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: 12.0.489494783471841.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} + 7 x^{10} - 11 x^{9} + 18 x^{8} - 22 x^{7} + 33 x^{6} - 44 x^{5} + 72 x^{4} + \cdots + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{28} \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{3} + ( - \beta_{8} - 1) q^{5} + \beta_1 q^{7} - 3 q^{9} + ( - \beta_{6} - 3 \beta_1) q^{11} + (\beta_{4} + 2) q^{13} + (\beta_{11} - \beta_{5}) q^{15} + ( - \beta_{10} + \beta_{8} - 2 \beta_{7} + \cdots - 3) q^{17}+ \cdots + (3 \beta_{6} + 9 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 8 q^{5} - 36 q^{9} + 24 q^{13} - 40 q^{17} + 180 q^{25} - 72 q^{29} + 88 q^{37} - 200 q^{41} + 24 q^{45} - 84 q^{49} - 104 q^{53} - 104 q^{61} + 176 q^{65} + 192 q^{69} + 312 q^{73} + 224 q^{77}+ \cdots - 264 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} + 7 x^{10} - 11 x^{9} + 18 x^{8} - 22 x^{7} + 33 x^{6} - 44 x^{5} + 72 x^{4} + \cdots + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 31 \nu^{11} + 23 \nu^{10} + 89 \nu^{9} + 43 \nu^{8} - 66 \nu^{7} + 386 \nu^{6} + 551 \nu^{5} + \cdots + 2608 ) / 1264 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 15 \nu^{11} + 136 \nu^{10} - 250 \nu^{9} + 426 \nu^{8} - 325 \nu^{7} + 620 \nu^{6} - 503 \nu^{5} + \cdots + 1792 ) / 316 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{11} - \nu^{10} - \nu^{9} - 7\nu^{8} - 16\nu^{6} - 5\nu^{5} - 28\nu^{4} + 2\nu^{3} - 56\nu^{2} + 8\nu - 88 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{11} - 13 \nu^{10} + 9 \nu^{9} - 37 \nu^{8} + 20 \nu^{7} - 78 \nu^{6} + 29 \nu^{5} - 158 \nu^{4} + \cdots - 320 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 191 \nu^{11} + 317 \nu^{10} - 793 \nu^{9} + 1101 \nu^{8} - 1734 \nu^{7} + 1954 \nu^{6} + \cdots + 6928 ) / 1264 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 285 \nu^{11} - 813 \nu^{10} + 1649 \nu^{9} - 2413 \nu^{8} + 4016 \nu^{7} - 4494 \nu^{6} + \cdots - 15248 ) / 1264 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2 \nu^{11} - \nu^{10} + 5 \nu^{9} - 5 \nu^{8} + 15 \nu^{7} + 4 \nu^{6} + 24 \nu^{5} - 15 \nu^{4} + \cdots + 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2 \nu^{11} - 5 \nu^{10} + 11 \nu^{9} - 15 \nu^{8} + 25 \nu^{7} - 26 \nu^{6} + 44 \nu^{5} - 55 \nu^{4} + \cdots - 88 ) / 8 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 423 \nu^{11} - 1205 \nu^{10} + 2509 \nu^{9} - 3929 \nu^{8} + 5450 \nu^{7} - 7638 \nu^{6} + \cdots - 33648 ) / 1264 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 3 \nu^{11} + 9 \nu^{10} - 11 \nu^{9} + 27 \nu^{8} - 32 \nu^{7} + 52 \nu^{6} - 63 \nu^{5} + 128 \nu^{4} + \cdots + 200 ) / 8 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1795 \nu^{11} + 3291 \nu^{10} - 7187 \nu^{9} + 9615 \nu^{8} - 15276 \nu^{7} + 17078 \nu^{6} + \cdots + 58160 ) / 1264 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} + \beta_{10} + 2 \beta_{9} - 3 \beta_{8} + \beta_{7} + 3 \beta_{6} - 3 \beta_{5} + \beta_{4} + \cdots + 7 ) / 32 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{11} - \beta_{10} - 2 \beta_{9} - 5 \beta_{8} - \beta_{7} + 5 \beta_{6} + 3 \beta_{5} + \cdots - 15 ) / 32 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 3 \beta_{11} + \beta_{10} - 2 \beta_{9} + 3 \beta_{8} - \beta_{7} + 3 \beta_{6} + 25 \beta_{5} + \cdots - 7 ) / 32 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3 \beta_{11} + 3 \beta_{10} - 6 \beta_{9} + 11 \beta_{8} - \beta_{7} + \beta_{6} - 25 \beta_{5} + \cdots - 31 ) / 32 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 3 \beta_{11} - 5 \beta_{10} + 5 \beta_{8} - 3 \beta_{7} - 3 \beta_{6} - 13 \beta_{5} - \beta_{4} + \cdots - 21 ) / 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2\beta_{9} + 5\beta_{8} + 9\beta_{7} - 6\beta_{6} + 20\beta_{5} + \beta_{4} - \beta_{3} - 3\beta_{2} + 4\beta _1 - 45 ) / 16 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 2 \beta_{10} - 2 \beta_{9} + 9 \beta_{8} + 5 \beta_{7} - 2 \beta_{6} + 8 \beta_{5} - 3 \beta_{4} + \cdots + 35 ) / 16 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - \beta_{11} - 5 \beta_{10} + 34 \beta_{9} - 5 \beta_{8} - 25 \beta_{7} - 3 \beta_{6} + 67 \beta_{5} + \cdots + 49 ) / 32 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 3 \beta_{11} + 11 \beta_{10} + 10 \beta_{9} - 24 \beta_{8} - 8 \beta_{7} - 3 \beta_{6} - 35 \beta_{5} + \cdots - 36 ) / 16 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 7 \beta_{11} + 3 \beta_{10} + 4 \beta_{9} - 35 \beta_{8} - 3 \beta_{7} - 9 \beta_{6} - 79 \beta_{5} + \cdots + 155 ) / 16 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 29 \beta_{11} + 39 \beta_{10} - 38 \beta_{9} - 103 \beta_{8} + 29 \beta_{7} - 11 \beta_{6} + \cdots + 3 ) / 32 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(127\) \(449\) \(577\) \(1093\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
−0.434363 + 1.34586i
0.0311486 + 1.41387i
−1.15503 − 0.816025i
0.754714 − 1.19600i
1.10978 + 0.876576i
1.19375 − 0.758257i
−0.434363 − 1.34586i
0.0311486 − 1.41387i
−1.15503 + 0.816025i
0.754714 + 1.19600i
1.10978 − 0.876576i
1.19375 + 0.758257i
0 − 1.73205i 0 −8.22808 0 − 2.64575i 0 −3.00000 0
127.2 0 − 1.73205i 0 −7.86764 0 2.64575i 0 −3.00000 0
127.3 0 − 1.73205i 0 −1.94731 0 − 2.64575i 0 −3.00000 0
127.4 0 − 1.73205i 0 −0.424396 0 2.64575i 0 −3.00000 0
127.5 0 − 1.73205i 0 6.29204 0 2.64575i 0 −3.00000 0
127.6 0 − 1.73205i 0 8.17539 0 − 2.64575i 0 −3.00000 0
127.7 0 1.73205i 0 −8.22808 0 2.64575i 0 −3.00000 0
127.8 0 1.73205i 0 −7.86764 0 − 2.64575i 0 −3.00000 0
127.9 0 1.73205i 0 −1.94731 0 2.64575i 0 −3.00000 0
127.10 0 1.73205i 0 −0.424396 0 − 2.64575i 0 −3.00000 0
127.11 0 1.73205i 0 6.29204 0 − 2.64575i 0 −3.00000 0
127.12 0 1.73205i 0 8.17539 0 2.64575i 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1344.3.m.e 12
4.b odd 2 1 inner 1344.3.m.e 12
8.b even 2 1 84.3.g.a ✓ 12
8.d odd 2 1 84.3.g.a ✓ 12
24.f even 2 1 252.3.g.b 12
24.h odd 2 1 252.3.g.b 12
56.e even 2 1 588.3.g.d 12
56.h odd 2 1 588.3.g.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.3.g.a ✓ 12 8.b even 2 1
84.3.g.a ✓ 12 8.d odd 2 1
252.3.g.b 12 24.f even 2 1
252.3.g.b 12 24.h odd 2 1
588.3.g.d 12 56.e even 2 1
588.3.g.d 12 56.h odd 2 1
1344.3.m.e 12 1.a even 1 1 trivial
1344.3.m.e 12 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 4T_{5}^{5} - 112T_{5}^{4} - 384T_{5}^{3} + 2976T_{5}^{2} + 7808T_{5} + 2752 \) acting on \(S_{3}^{\mathrm{new}}(1344, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3)^{6} \) Copy content Toggle raw display
$5$ \( (T^{6} + 4 T^{5} + \cdots + 2752)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 7)^{6} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 8305770496 \) Copy content Toggle raw display
$13$ \( (T^{6} - 12 T^{5} + \cdots - 1102016)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + 20 T^{5} + \cdots + 39448000)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 4294967296 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 12\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( (T^{6} + 36 T^{5} + \cdots - 12566720)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 128544076201984 \) Copy content Toggle raw display
$37$ \( (T^{6} - 44 T^{5} + \cdots - 155225024)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 100 T^{5} + \cdots + 8374720)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 49\!\cdots\!64 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 47\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{6} + 52 T^{5} + \cdots - 11239563200)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 90\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( (T^{6} + 52 T^{5} + \cdots + 44445760)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 13\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{6} - 156 T^{5} + \cdots + 11256899392)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 84\!\cdots\!44 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{6} + 276 T^{5} + \cdots + 733352728000)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 132 T^{5} + \cdots - 290193606848)^{2} \) Copy content Toggle raw display
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