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Properties

Label 952.2.q.c
Level $952$
Weight $2$
Character orbit 952.q
Analytic conductor $7.602$
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] = [952,2,Mod(137,952)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(952, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("952.137"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 952 = 2^{3} \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 952.q (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,-1] 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: \(7.60175827243\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
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: \( x^{12} - x^{11} + 10 x^{10} - 11 x^{9} + 73 x^{8} - 77 x^{7} + 243 x^{6} - 236 x^{5} + 572 x^{4} + \cdots + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - \beta_{11} q^{5} + ( - \beta_{9} - \beta_{6} + \beta_{2}) q^{7} + (\beta_{9} + \beta_{4}) q^{9} + ( - \beta_{11} - \beta_{10} + \cdots - \beta_1) q^{11} + ( - \beta_{7} + \beta_{5} + \beta_{4}) q^{13}+ \cdots + ( - \beta_{7} + 2 \beta_{5} + \cdots + 4 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - q^{3} - q^{5} + 3 q^{7} - q^{9} + 6 q^{11} - 4 q^{13} + 6 q^{17} - 2 q^{19} + 14 q^{21} + 5 q^{23} - q^{25} - 16 q^{27} - 30 q^{29} - 8 q^{31} - 11 q^{33} + 7 q^{35} + 16 q^{37} + 2 q^{39} + 8 q^{41}+ \cdots + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} + 10 x^{10} - 11 x^{9} + 73 x^{8} - 77 x^{7} + 243 x^{6} - 236 x^{5} + 572 x^{4} + \cdots + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 14290269 \nu^{11} + 594547151 \nu^{10} + 953976395 \nu^{9} + 5580755644 \nu^{8} + \cdots - 19343310930 ) / 21130868701 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 23459088 \nu^{11} - 28890745 \nu^{10} - 190246065 \nu^{9} - 201644976 \nu^{8} + \cdots + 1913729518 ) / 21130868701 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 52349833 \nu^{11} - 44344815 \nu^{10} + 459694944 \nu^{9} - 340552273 \nu^{8} + \cdots + 63298769751 ) / 21130868701 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 257870635 \nu^{11} - 265752455 \nu^{10} - 2399231901 \nu^{9} - 1840052477 \nu^{8} + \cdots - 59327961246 ) / 21130868701 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 360135604 \nu^{11} - 265470224 \nu^{10} - 2260600454 \nu^{9} - 545641141 \nu^{8} + \cdots - 420518644 ) / 21130868701 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 375166075 \nu^{11} - 410206180 \nu^{10} - 3350462226 \nu^{9} - 2848277357 \nu^{8} + \cdots - 70890182357 ) / 21130868701 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 956864759 \nu^{11} - 909946583 \nu^{10} + 9626429080 \nu^{9} - 10145020219 \nu^{8} + \cdots - 3162064816 ) / 42261737402 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 2975293943 \nu^{11} + 2818529379 \nu^{10} - 29798677128 \nu^{9} + 31116165203 \nu^{8} + \cdots + 9673867152 ) / 42261737402 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1544721230 \nu^{11} + 1685045763 \nu^{10} - 14897292195 \nu^{9} + 18266973422 \nu^{8} + \cdots + 64293926046 ) / 21130868701 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 3795706303 \nu^{11} - 4441338787 \nu^{10} + 37047961040 \nu^{9} - 47534303897 \nu^{8} + \cdots - 154455560466 ) / 42261737402 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + 3\beta_{8} + \beta_{4} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - \beta_{5} - 5\beta_{3} + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{11} + \beta_{10} - 7\beta_{9} - 14\beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} + \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -8\beta_{11} - 9\beta_{10} + \beta_{9} + 7\beta_{8} + \beta_{4} + 27\beta_{3} - 27\beta _1 - 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -7\beta_{7} + 9\beta_{6} - 10\beta_{5} - 44\beta_{4} - 9\beta_{3} - 9\beta_{2} + 73 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 51\beta_{11} + 63\beta_{10} - 12\beta_{9} - 43\beta_{8} - 51\beta_{7} - \beta_{6} + 63\beta_{5} + 152\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 39 \beta_{11} - 76 \beta_{10} + 266 \beta_{9} + 403 \beta_{8} + 266 \beta_{4} + 66 \beta_{3} + \cdots - 403 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 305\beta_{7} + 14\beta_{6} - 404\beta_{5} - 103\beta_{4} - 873\beta_{3} - 14\beta_{2} + 264 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 202 \beta_{11} + 521 \beta_{10} - 1582 \beta_{9} - 2290 \beta_{8} + 202 \beta_{7} - 390 \beta_{6} + \cdots + 456 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1784 \beta_{11} - 2493 \beta_{10} + 775 \beta_{9} + 1649 \beta_{8} + 775 \beta_{4} + 5064 \beta_{3} + \cdots - 1649 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/952\mathbb{Z}\right)^\times\).

\(n\) \(239\) \(409\) \(477\) \(785\)
\(\chi(n)\) \(1\) \(-1 + \beta_{8}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
137.1
1.18253 − 2.04820i
0.747056 − 1.29394i
0.667223 − 1.15566i
0.0471199 − 0.0816140i
−0.917183 + 1.58861i
−1.22674 + 2.12478i
1.18253 + 2.04820i
0.747056 + 1.29394i
0.667223 + 1.15566i
0.0471199 + 0.0816140i
−0.917183 − 1.58861i
−1.22674 − 2.12478i
0 −1.18253 + 2.04820i 0 0.949958 + 1.64538i 0 −1.39834 + 2.24603i 0 −1.29675 − 2.24603i 0
137.2 0 −0.747056 + 1.29394i 0 −0.776774 − 1.34541i 0 2.56087 − 0.664785i 0 0.383814 + 0.664785i 0
137.3 0 −0.667223 + 1.15566i 0 −1.82280 − 3.15717i 0 −2.42592 − 1.05590i 0 0.609627 + 1.05590i 0
137.4 0 −0.0471199 + 0.0816140i 0 1.65116 + 2.85990i 0 0.538429 − 2.59038i 0 1.49556 + 2.59038i 0
137.5 0 0.917183 − 1.58861i 0 −0.0643710 − 0.111494i 0 2.62681 + 0.316014i 0 −0.182451 − 0.316014i 0
137.6 0 1.22674 − 2.12478i 0 −0.437182 − 0.757221i 0 −0.401860 + 2.61505i 0 −1.50980 − 2.61505i 0
681.1 0 −1.18253 − 2.04820i 0 0.949958 − 1.64538i 0 −1.39834 − 2.24603i 0 −1.29675 + 2.24603i 0
681.2 0 −0.747056 − 1.29394i 0 −0.776774 + 1.34541i 0 2.56087 + 0.664785i 0 0.383814 − 0.664785i 0
681.3 0 −0.667223 − 1.15566i 0 −1.82280 + 3.15717i 0 −2.42592 + 1.05590i 0 0.609627 − 1.05590i 0
681.4 0 −0.0471199 − 0.0816140i 0 1.65116 − 2.85990i 0 0.538429 + 2.59038i 0 1.49556 − 2.59038i 0
681.5 0 0.917183 + 1.58861i 0 −0.0643710 + 0.111494i 0 2.62681 − 0.316014i 0 −0.182451 + 0.316014i 0
681.6 0 1.22674 + 2.12478i 0 −0.437182 + 0.757221i 0 −0.401860 − 2.61505i 0 −1.50980 + 2.61505i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 137.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 952.2.q.c ✓ 12
7.c even 3 1 inner 952.2.q.c ✓ 12
7.c even 3 1 6664.2.a.s 6
7.d odd 6 1 6664.2.a.r 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
952.2.q.c ✓ 12 1.a even 1 1 trivial
952.2.q.c ✓ 12 7.c even 3 1 inner
6664.2.a.r 6 7.d odd 6 1
6664.2.a.s 6 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + T_{3}^{11} + 10 T_{3}^{10} + 11 T_{3}^{9} + 73 T_{3}^{8} + 77 T_{3}^{7} + 243 T_{3}^{6} + \cdots + 4 \) acting on \(S_{2}^{\mathrm{new}}(952, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + T^{11} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( T^{12} + T^{11} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( T^{12} - 3 T^{11} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( T^{12} - 6 T^{11} + \cdots + 4096 \) Copy content Toggle raw display
$13$ \( (T^{6} + 2 T^{5} - 26 T^{4} + \cdots + 54)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - T + 1)^{6} \) Copy content Toggle raw display
$19$ \( T^{12} + 2 T^{11} + \cdots + 425104 \) Copy content Toggle raw display
$23$ \( T^{12} - 5 T^{11} + \cdots + 1024 \) Copy content Toggle raw display
$29$ \( (T^{6} + 15 T^{5} + \cdots + 968)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 1439140096 \) Copy content Toggle raw display
$37$ \( T^{12} - 16 T^{11} + \cdots + 1048576 \) Copy content Toggle raw display
$41$ \( (T^{6} - 4 T^{5} + \cdots + 10552)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} - 24 T^{5} + \cdots + 1312)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 122589184 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 912402436 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 107827456 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 33527074816 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 3208542736 \) Copy content Toggle raw display
$71$ \( (T^{6} + 12 T^{5} + \cdots - 3107)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 225480256 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 1198267456 \) Copy content Toggle raw display
$83$ \( (T^{6} + 20 T^{5} + \cdots + 245696)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} - 33 T^{11} + \cdots + 4096 \) Copy content Toggle raw display
$97$ \( (T^{6} + 9 T^{5} + \cdots + 31328)^{2} \) Copy content Toggle raw display
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