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Properties

Label 1682.2.b.i
Level $1682$
Weight $2$
Character orbit 1682.b
Analytic conductor $13.431$
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] = [1682,2,Mod(1681,1682)] 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("1682.1681"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1682, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1682 = 2 \cdot 29^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1682.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,-12,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.4308376200\)
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: \( x^{12} + 30x^{10} + 341x^{8} + 1897x^{6} + 5456x^{4} + 7680x^{2} + 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 58)
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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} + \beta_1 q^{3} - q^{4} + (\beta_{11} + \beta_{4}) q^{5} + \beta_{11} q^{6} + \beta_{8} q^{7} - \beta_{6} q^{8} + (\beta_{10} + \beta_{8} - \beta_{4} - 1) q^{9} + (\beta_{2} - \beta_1) q^{10}+ \cdots + ( - \beta_{9} + 4 \beta_{6} + \cdots - 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{4} - 4 q^{6} - 6 q^{7} - 24 q^{9} + 6 q^{13} + 12 q^{16} + 2 q^{22} - 28 q^{23} + 4 q^{24} + 8 q^{25} + 6 q^{28} + 54 q^{30} - 40 q^{33} + 12 q^{34} + 18 q^{35} + 24 q^{36} - 36 q^{38} + 20 q^{42}+ \cdots - 4 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 30x^{10} + 341x^{8} + 1897x^{6} + 5456x^{4} + 7680x^{2} + 4096 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 15\nu^{11} + 386\nu^{9} + 3451\nu^{7} + 13287\nu^{5} + 21104\nu^{3} + 9984\nu ) / 1024 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{11} + 26\nu^{9} + 237\nu^{7} + 949\nu^{5} + 1660\nu^{3} + 1040\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{10} - 28\nu^{8} - 281\nu^{6} - 1247\nu^{4} - 2366\nu^{2} - 1472 ) / 32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -47\nu^{11} - 1282\nu^{9} - 12443\nu^{7} - 53191\nu^{5} - 96816\nu^{3} - 57088\nu ) / 1024 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -45\nu^{11} - 1206\nu^{9} - 11537\nu^{7} - 49573\nu^{5} - 94720\nu^{3} - 63232\nu ) / 1024 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 9\nu^{10} + 254\nu^{8} + 2589\nu^{6} + 11873\nu^{4} + 24128\nu^{2} + 16896 ) / 256 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{10} + 26\nu^{8} + 237\nu^{6} + 949\nu^{4} + 1660\nu^{2} + 1024 ) / 16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -17\nu^{11} - 478\nu^{9} - 4837\nu^{7} - 21849\nu^{5} - 43056\nu^{3} - 28928\nu ) / 256 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -3\nu^{10} - 80\nu^{8} - 755\nu^{6} - 3145\nu^{4} - 5654\nu^{2} - 3392 ) / 32 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 9\nu^{10} + 238\nu^{8} + 2237\nu^{6} + 9425\nu^{4} + 17648\nu^{2} + 11520 ) / 64 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} + \beta_{8} - \beta_{4} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} - 2\beta_{5} + 4\beta_{3} - 6\beta_{2} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{11} - 13\beta_{10} - 11\beta_{8} + 4\beta_{7} + 17\beta_{4} + 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -18\beta_{9} + 4\beta_{6} + 35\beta_{5} - 44\beta_{3} + 87\beta_{2} + 31\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 22\beta_{11} + 153\beta_{10} + 110\beta_{8} - 72\beta_{7} - 221\beta_{4} - 196 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 247\beta_{9} - 88\beta_{6} - 468\beta_{5} + 440\beta_{3} - 1080\beta_{2} - 234\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -335\beta_{11} - 1782\beta_{10} - 1142\beta_{8} + 988\beta_{7} + 2666\beta_{4} + 1924 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -3105\beta_{9} + 1340\beta_{6} + 5771\beta_{5} - 4568\beta_{3} + 12899\beta_{2} + 2078\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 4445\beta_{11} + 20748\beta_{10} + 12417\beta_{8} - 12420\beta_{7} - 31412\beta_{4} - 20732 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 37613\beta_{9} - 17780\beta_{6} - 69025\beta_{5} + 49668\beta_{3} - 152017\beta_{2} - 20729\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1682\mathbb{Z}\right)^\times\).

\(n\) \(843\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1681.1
− 3.41744i
− 2.29664i
− 1.63883i
1.17047i
1.74168i
2.44077i
− 2.44077i
− 1.74168i
− 1.17047i
1.63883i
2.29664i
3.41744i
− 1.00000i − 3.41744i −1.00000 −1.61551 −3.41744 −1.52091 1.00000i −8.67893 1.61551i
1681.2 − 1.00000i − 2.29664i −1.00000 −3.54362 −2.29664 −4.13840 1.00000i −2.27454 3.54362i
1681.3 − 1.00000i − 1.63883i −1.00000 −1.19379 −1.63883 2.04359 1.00000i 0.314239 1.19379i
1681.4 − 1.00000i 1.17047i −1.00000 2.97240 1.17047 0.520906 1.00000i 1.63001 − 2.97240i
1681.5 − 1.00000i 1.74168i −1.00000 0.494698 1.74168 3.13840 1.00000i −0.0334417 − 0.494698i
1681.6 − 1.00000i 2.44077i −1.00000 2.88581 2.44077 −3.04359 1.00000i −2.95734 − 2.88581i
1681.7 1.00000i − 2.44077i −1.00000 2.88581 2.44077 −3.04359 − 1.00000i −2.95734 2.88581i
1681.8 1.00000i − 1.74168i −1.00000 0.494698 1.74168 3.13840 − 1.00000i −0.0334417 0.494698i
1681.9 1.00000i − 1.17047i −1.00000 2.97240 1.17047 0.520906 − 1.00000i 1.63001 2.97240i
1681.10 1.00000i 1.63883i −1.00000 −1.19379 −1.63883 2.04359 − 1.00000i 0.314239 − 1.19379i
1681.11 1.00000i 2.29664i −1.00000 −3.54362 −2.29664 −4.13840 − 1.00000i −2.27454 − 3.54362i
1681.12 1.00000i 3.41744i −1.00000 −1.61551 −3.41744 −1.52091 − 1.00000i −8.67893 − 1.61551i
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1681.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1682.2.b.i 12
29.b even 2 1 inner 1682.2.b.i 12
29.c odd 4 1 1682.2.a.q 6
29.c odd 4 1 1682.2.a.t 6
29.f odd 28 2 58.2.d.b ✓ 12
87.k even 28 2 522.2.k.h 12
116.l even 28 2 464.2.u.h 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
58.2.d.b ✓ 12 29.f odd 28 2
464.2.u.h 12 116.l even 28 2
522.2.k.h 12 87.k even 28 2
1682.2.a.q 6 29.c odd 4 1
1682.2.a.t 6 29.c odd 4 1
1682.2.b.i 12 1.a even 1 1 trivial
1682.2.b.i 12 29.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1682, [\chi])\):

\( T_{3}^{12} + 30T_{3}^{10} + 341T_{3}^{8} + 1897T_{3}^{6} + 5456T_{3}^{4} + 7680T_{3}^{2} + 4096 \) Copy content Toggle raw display
\( T_{5}^{6} - 17T_{5}^{4} + 66T_{5}^{2} + 28T_{5} - 29 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$3$ \( T^{12} + 30 T^{10} + \cdots + 4096 \) Copy content Toggle raw display
$5$ \( (T^{6} - 17 T^{4} + \cdots - 29)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} + 3 T^{5} - 17 T^{4} + \cdots - 64)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + 53 T^{10} + \cdots + 4096 \) Copy content Toggle raw display
$13$ \( (T^{6} - 3 T^{5} - 25 T^{4} + \cdots - 7)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + 142 T^{10} + \cdots + 1585081 \) Copy content Toggle raw display
$19$ \( T^{12} + 122 T^{10} + \cdots + 12845056 \) Copy content Toggle raw display
$23$ \( (T^{6} + 14 T^{5} + \cdots - 2752)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} \) Copy content Toggle raw display
$31$ \( T^{12} + 173 T^{10} + \cdots + 3444736 \) Copy content Toggle raw display
$37$ \( T^{12} + 198 T^{10} + \cdots + 68442529 \) Copy content Toggle raw display
$41$ \( T^{12} + 187 T^{10} + \cdots + 93180409 \) Copy content Toggle raw display
$43$ \( T^{12} + 53 T^{10} + \cdots + 4096 \) Copy content Toggle raw display
$47$ \( T^{12} + 230 T^{10} + \cdots + 7573504 \) Copy content Toggle raw display
$53$ \( (T^{6} - 3 T^{5} + \cdots - 5977)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} - 19 T^{5} + \cdots - 3584)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + 251 T^{10} + \cdots + 97160449 \) Copy content Toggle raw display
$67$ \( (T^{6} + 21 T^{5} + \cdots - 64)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 9 T^{5} + \cdots + 60992)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 490724067289 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 113844158464 \) Copy content Toggle raw display
$83$ \( (T^{6} + 2 T^{5} + \cdots - 448)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 126405049 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 36316162624 \) Copy content Toggle raw display
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