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Properties

Label 1352.2.f.g
Level $1352$
Weight $2$
Character orbit 1352.f
Analytic conductor $10.796$
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] = [1352,2,Mod(337,1352)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1352.337"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1352, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1352 = 2^{3} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1352.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,6,0,0,0,0,0,18,0,0,0,0,0,0,0,-22] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.7957743533\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 21x^{10} + 166x^{8} + 623x^{6} + 1140x^{4} + 896x^{2} + 169 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( - \beta_{6} - \beta_{4} + 1) q^{3} + ( - \beta_{10} - \beta_{8} + \cdots - \beta_1) q^{5} + ( - \beta_{11} - \beta_{9} + \cdots + \beta_1) q^{7} + ( - 2 \beta_{6} + 2 \beta_{5} + \cdots + 3) q^{9}+ \cdots + ( - 6 \beta_{11} + 2 \beta_{10} + \cdots - 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{3} + 18 q^{9} - 22 q^{17} - 30 q^{23} - 32 q^{25} + 42 q^{27} + 18 q^{29} + 28 q^{35} - 20 q^{43} - 54 q^{49} - 10 q^{51} + 2 q^{53} + 28 q^{55} - 64 q^{69} - 80 q^{75} + 4 q^{77} + 26 q^{79}+ \cdots + 42 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 21x^{10} + 166x^{8} + 623x^{6} + 1140x^{4} + 896x^{2} + 169 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -23\nu^{10} - 432\nu^{8} - 2842\nu^{6} - 7783\nu^{4} - 7913\nu^{2} - 1235 ) / 208 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 23\nu^{10} + 432\nu^{8} + 2842\nu^{6} + 7783\nu^{4} + 8121\nu^{2} + 2067 ) / 208 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -6\nu^{10} - 111\nu^{8} - 725\nu^{6} - 2023\nu^{4} - 2244\nu^{2} - 533 ) / 52 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 9\nu^{10} + 186\nu^{8} + 1380\nu^{6} + 4367\nu^{4} + 5251\nu^{2} + 1183 ) / 104 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 15\nu^{10} + 284\nu^{8} + 1910\nu^{6} + 5467\nu^{4} + 5961\nu^{2} + 1183 ) / 104 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 11\nu^{11} + 196\nu^{9} + 1198\nu^{7} + 2951\nu^{5} + 2557\nu^{3} + 315\nu ) / 208 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -7\nu^{11} - 132\nu^{9} - 878\nu^{7} - 2451\nu^{5} - 2513\nu^{3} - 311\nu ) / 104 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 7\nu^{11} + 138\nu^{9} + 976\nu^{7} + 2981\nu^{5} + 3613\nu^{3} + 1021\nu ) / 104 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -23\nu^{11} - 420\nu^{9} - 2646\nu^{7} - 6723\nu^{5} - 5713\nu^{3} + 185\nu ) / 208 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 6\nu^{11} + 111\nu^{9} + 725\nu^{7} + 2023\nu^{5} + 2244\nu^{3} + 533\nu ) / 52 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} + \beta_{10} + \beta_{9} + 3\beta_{7} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -3\beta_{6} + \beta_{5} - 3\beta_{4} - 8\beta_{3} - 8\beta_{2} + 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 10\beta_{11} - 11\beta_{10} - 6\beta_{9} + 8\beta_{8} - 27\beta_{7} + 31\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 36\beta_{6} - 11\beta_{5} + 31\beta_{4} + 58\beta_{3} + 64\beta_{2} - 163 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -78\beta_{11} + 100\beta_{10} + 30\beta_{9} - 103\beta_{8} + 210\beta_{7} - 210\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -335\beta_{6} + 102\beta_{5} - 256\beta_{4} - 420\beta_{3} - 510\beta_{2} + 1172 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 574\beta_{11} - 845\beta_{10} - 126\beta_{9} + 993\beta_{8} - 1595\beta_{7} + 1490\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2859\beta_{6} - 895\beta_{5} + 1993\beta_{4} + 3085\beta_{3} + 4025\beta_{2} - 8671 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -4183\beta_{11} + 6884\beta_{10} + 355\beta_{9} - 8622\beta_{8} + 12114\beta_{7} - 10861\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1352\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(1015\) \(1185\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
1.83636i
− 1.83636i
− 1.52958i
1.52958i
2.31865i
− 2.31865i
1.39132i
− 1.39132i
0.516708i
− 0.516708i
− 2.77656i
2.77656i
0 −2.63830 0 − 1.20857i 0 4.34343i 0 3.96061 0
337.2 0 −2.63830 0 1.20857i 0 − 4.34343i 0 3.96061 0
337.3 0 −0.974624 0 − 0.130796i 0 − 2.76527i 0 −2.05011 0
337.4 0 −0.974624 0 0.130796i 0 2.76527i 0 −2.05011 0
337.5 0 −0.0716659 0 − 3.69476i 0 1.67432i 0 −2.99486 0
337.6 0 −0.0716659 0 3.69476i 0 − 1.67432i 0 −2.99486 0
337.7 0 0.589380 0 − 3.45555i 0 4.70032i 0 −2.65263 0
337.8 0 0.589380 0 3.45555i 0 − 4.70032i 0 −2.65263 0
337.9 0 2.76369 0 − 4.24972i 0 − 2.37460i 0 4.63797 0
337.10 0 2.76369 0 4.24972i 0 2.37460i 0 4.63797 0
337.11 0 3.33152 0 − 0.932734i 0 − 3.45729i 0 8.09903 0
337.12 0 3.33152 0 0.932734i 0 3.45729i 0 8.09903 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 337.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1352.2.f.g 12
4.b odd 2 1 2704.2.f.r 12
13.b even 2 1 inner 1352.2.f.g 12
13.c even 3 2 1352.2.o.h 24
13.d odd 4 1 1352.2.a.m ✓ 6
13.d odd 4 1 1352.2.a.n yes 6
13.e even 6 2 1352.2.o.h 24
13.f odd 12 2 1352.2.i.m 12
13.f odd 12 2 1352.2.i.n 12
52.b odd 2 1 2704.2.f.r 12
52.f even 4 1 2704.2.a.bf 6
52.f even 4 1 2704.2.a.bg 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1352.2.a.m ✓ 6 13.d odd 4 1
1352.2.a.n yes 6 13.d odd 4 1
1352.2.f.g 12 1.a even 1 1 trivial
1352.2.f.g 12 13.b even 2 1 inner
1352.2.i.m 12 13.f odd 12 2
1352.2.i.n 12 13.f odd 12 2
1352.2.o.h 24 13.c even 3 2
1352.2.o.h 24 13.e even 6 2
2704.2.a.bf 6 52.f even 4 1
2704.2.a.bg 6 52.f even 4 1
2704.2.f.r 12 4.b odd 2 1
2704.2.f.r 12 52.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1352, [\chi])\):

\( T_{3}^{6} - 3T_{3}^{5} - 9T_{3}^{4} + 23T_{3}^{3} + 15T_{3}^{2} - 13T_{3} - 1 \) Copy content Toggle raw display
\( T_{5}^{12} + 46T_{5}^{10} + 729T_{5}^{8} + 4469T_{5}^{6} + 7732T_{5}^{4} + 3872T_{5}^{2} + 64 \) Copy content Toggle raw display
\( T_{11}^{12} + 87T_{11}^{10} + 2801T_{11}^{8} + 40723T_{11}^{6} + 257709T_{11}^{4} + 512483T_{11}^{2} + 16129 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{6} - 3 T^{5} - 9 T^{4} + \cdots - 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} + 46 T^{10} + \cdots + 64 \) Copy content Toggle raw display
$7$ \( T^{12} + 69 T^{10} + \cdots + 602176 \) Copy content Toggle raw display
$11$ \( T^{12} + 87 T^{10} + \cdots + 16129 \) Copy content Toggle raw display
$13$ \( T^{12} \) Copy content Toggle raw display
$17$ \( (T^{6} + 11 T^{5} + 29 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + 143 T^{10} + \cdots + 49970761 \) Copy content Toggle raw display
$23$ \( (T^{6} + 15 T^{5} + 43 T^{4} + \cdots + 8)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} - 9 T^{5} + \cdots - 24184)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + 117 T^{10} + \cdots + 440896 \) Copy content Toggle raw display
$37$ \( T^{12} + 114 T^{10} + \cdots + 18181696 \) Copy content Toggle raw display
$41$ \( T^{12} + 248 T^{10} + \cdots + 28561 \) Copy content Toggle raw display
$43$ \( (T^{6} + 10 T^{5} + \cdots + 2197)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + 210 T^{10} + \cdots + 7268416 \) Copy content Toggle raw display
$53$ \( (T^{6} - T^{5} - 106 T^{4} + \cdots - 8632)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 71247887929 \) Copy content Toggle raw display
$61$ \( (T^{6} - 135 T^{4} + \cdots - 5312)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 4561246369 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 12592430656 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 79777437601 \) Copy content Toggle raw display
$79$ \( (T^{6} - 13 T^{5} + \cdots - 246616)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 5010233089 \) Copy content Toggle raw display
$89$ \( T^{12} + 138 T^{10} + \cdots + 1164241 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 15507471841 \) Copy content Toggle raw display
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