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Properties

Label 1183.2.c.h
Level $1183$
Weight $2$
Character orbit 1183.c
Analytic conductor $9.446$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1183,2,Mod(337,1183)] 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("1183.337"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1183, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1183 = 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1183.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.44630255912\)
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: \( x^{12} + 16x^{10} + 96x^{8} + 266x^{6} + 332x^{4} + 141x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{11} + \beta_{10} - \beta_{8}) q^{2} + ( - \beta_{4} - 1) q^{3} + (\beta_{7} + \beta_{5} - \beta_{4} - 1) q^{4} + \beta_{11} q^{5} + (\beta_{11} - 2 \beta_{10} + \cdots + \beta_1) q^{6}+ \cdots + ( - 2 \beta_{11} - \beta_{9} + \cdots + 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 8 q^{3} - 16 q^{4} + 28 q^{10} + 46 q^{12} - 4 q^{14} + 46 q^{17} - 8 q^{22} + 36 q^{23} + 20 q^{25} - 20 q^{27} - 30 q^{29} - 28 q^{30} - 4 q^{35} - 44 q^{36} + 22 q^{38} - 28 q^{40} + 16 q^{42}+ \cdots + 14 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 16x^{10} + 96x^{8} + 266x^{6} + 332x^{4} + 141x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{10} + 7\nu^{8} + 33\nu^{6} + 218\nu^{4} + 528\nu^{2} + 203 ) / 83 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{10} + 14\nu^{8} - 17\nu^{6} - 228\nu^{4} - 189\nu^{2} + 157 ) / 83 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{10} - 14\nu^{8} + 17\nu^{6} + 228\nu^{4} + 272\nu^{2} + 9 ) / 83 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 14\nu^{10} + 181\nu^{8} + 794\nu^{6} + 1309\nu^{4} + 503\nu^{2} - 146 ) / 83 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 12\nu^{11} + 167\nu^{9} + 811\nu^{7} + 1620\nu^{5} + 1273\nu^{3} + 444\nu ) / 83 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 12\nu^{10} + 167\nu^{8} + 811\nu^{6} + 1620\nu^{4} + 1190\nu^{2} + 112 ) / 83 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 9\nu^{11} + 146\nu^{9} + 878\nu^{7} + 2377\nu^{5} + 2760\nu^{3} + 997\nu ) / 83 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -6\nu^{11} - 42\nu^{9} + 134\nu^{7} + 1597\nu^{5} + 3555\nu^{3} + 1936\nu ) / 83 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 8\nu^{11} + 139\nu^{9} + 928\nu^{7} + 2906\nu^{5} + 4058\nu^{3} + 1790\nu ) / 83 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -21\nu^{11} - 313\nu^{9} - 1689\nu^{7} - 3997\nu^{5} - 3950\nu^{3} - 1109\nu ) / 83 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + \beta_{8} + \beta_{6} - 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} - \beta_{5} - 6\beta_{4} - 5\beta_{3} + 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -7\beta_{11} + \beta_{9} - 9\beta_{8} - 5\beta_{6} + 18\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -8\beta_{7} + 8\beta_{5} + 33\beta_{4} + 24\beta_{3} + 2\beta_{2} - 29 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 41\beta_{11} + 3\beta_{10} - 10\beta_{9} + 57\beta_{8} + 22\beta_{6} - 86\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 53\beta_{7} - 52\beta_{5} - 175\beta_{4} - 118\beta_{3} - 22\beta_{2} + 133 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -228\beta_{11} - 39\beta_{10} + 74\beta_{9} - 320\beta_{8} - 96\beta_{6} + 426\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -325\beta_{7} + 318\beta_{5} + 916\beta_{4} + 596\beta_{3} + 171\beta_{2} - 647 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1241\beta_{11} + 340\beta_{10} - 489\beta_{9} + 1710\beta_{8} + 425\beta_{6} - 2159\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1183\mathbb{Z}\right)^\times\).

\(n\) \(339\) \(1016\)
\(\chi(n)\) \(1\) \(-1\)

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
1.71083i
0.908891i
1.54570i
− 0.0849355i
− 2.33192i
− 2.10066i
2.10066i
2.33192i
0.0849355i
− 1.54570i
− 0.908891i
− 1.71083i
− 2.63777i −2.71083 −4.95781 2.08281i 7.15053i − 1.00000i 7.80201i 4.34860 5.49396
337.2 − 2.08281i −0.0911085 −2.33809 2.63777i 0.189762i 1.00000i 0.704173i −2.99170 5.49396
337.3 − 1.93488i −2.54570 −1.74376 − 0.312100i 4.92562i − 1.00000i − 0.495793i 3.48058 −0.603875
337.4 − 1.90785i −1.08494 −1.63989 1.10591i 2.06989i 1.00000i − 0.687029i −1.82292 2.10992
337.5 − 1.10591i 1.33192 0.776957 1.90785i − 1.47298i − 1.00000i − 3.07107i −1.22600 2.10992
337.6 − 0.312100i 1.10066 1.90259 − 1.93488i − 0.343514i − 1.00000i − 1.21800i −1.78856 −0.603875
337.7 0.312100i 1.10066 1.90259 1.93488i 0.343514i 1.00000i 1.21800i −1.78856 −0.603875
337.8 1.10591i 1.33192 0.776957 − 1.90785i 1.47298i 1.00000i 3.07107i −1.22600 2.10992
337.9 1.90785i −1.08494 −1.63989 − 1.10591i − 2.06989i − 1.00000i 0.687029i −1.82292 2.10992
337.10 1.93488i −2.54570 −1.74376 0.312100i − 4.92562i 1.00000i 0.495793i 3.48058 −0.603875
337.11 2.08281i −0.0911085 −2.33809 − 2.63777i − 0.189762i − 1.00000i − 0.704173i −2.99170 5.49396
337.12 2.63777i −2.71083 −4.95781 − 2.08281i − 7.15053i 1.00000i − 7.80201i 4.34860 5.49396
\(n\): 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 1183.2.c.h 12
13.b even 2 1 inner 1183.2.c.h 12
13.d odd 4 1 1183.2.a.n ✓ 6
13.d odd 4 1 1183.2.a.o yes 6
91.i even 4 1 8281.2.a.cb 6
91.i even 4 1 8281.2.a.cg 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1183.2.a.n ✓ 6 13.d odd 4 1
1183.2.a.o yes 6 13.d odd 4 1
1183.2.c.h 12 1.a even 1 1 trivial
1183.2.c.h 12 13.b even 2 1 inner
8281.2.a.cb 6 91.i even 4 1
8281.2.a.cg 6 91.i even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 20T_{2}^{10} + 152T_{2}^{8} + 547T_{2}^{6} + 924T_{2}^{4} + 588T_{2}^{2} + 49 \) acting on \(S_{2}^{\mathrm{new}}(1183, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 20 T^{10} + \cdots + 49 \) Copy content Toggle raw display
$3$ \( (T^{6} + 4 T^{5} - T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} + 20 T^{10} + \cdots + 49 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$11$ \( T^{12} + 96 T^{10} + \cdots + 790321 \) Copy content Toggle raw display
$13$ \( T^{12} \) Copy content Toggle raw display
$17$ \( (T^{6} - 23 T^{5} + \cdots - 7351)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + 75 T^{10} + \cdots + 337561 \) Copy content Toggle raw display
$23$ \( (T^{6} - 18 T^{5} + \cdots - 587)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 15 T^{5} + \cdots + 3569)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 460660369 \) Copy content Toggle raw display
$37$ \( T^{12} + 159 T^{10} + \cdots + 49 \) Copy content Toggle raw display
$41$ \( T^{12} + 246 T^{10} + \cdots + 253009 \) Copy content Toggle raw display
$43$ \( (T^{6} - 18 T^{5} + \cdots + 181)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 914034289 \) Copy content Toggle raw display
$53$ \( (T^{6} + 25 T^{5} + \cdots + 24193)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 8632453921 \) Copy content Toggle raw display
$61$ \( (T^{6} - 16 T^{5} + \cdots - 12979)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 715402009 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 317982082201 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 2058164689 \) Copy content Toggle raw display
$79$ \( (T^{6} - 2 T^{5} + \cdots - 10277)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 1697687209 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 49398174049 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 1929932761 \) Copy content Toggle raw display
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