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Properties

Label 3703.2.a.k
Level $3703$
Weight $2$
Character orbit 3703.a
Self dual yes
Analytic conductor $29.569$
Analytic rank $1$
Dimension $10$
CM no
Inner twists $1$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3703,2,Mod(1,3703)] 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("3703.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3703, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3703 = 7 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3703.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,-2,8,-8,-8,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(29.5686038685\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 14x^{8} - 2x^{7} + 63x^{6} + 16x^{5} - 97x^{4} - 28x^{3} + 35x^{2} + 14x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - \beta_{6} q^{3} + (\beta_{8} + \beta_{7} + \beta_1 + 1) q^{4} + ( - \beta_{3} - 1) q^{5} + ( - \beta_{9} - \beta_{7} + 2 \beta_{5} + \cdots - 1) q^{6} + q^{7} + (\beta_{9} + \beta_{7} - \beta_{6} + \cdots - \beta_1) q^{8}+ \cdots + ( - 5 \beta_{9} - 2 \beta_{8} + \cdots - 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 2 q^{3} + 8 q^{4} - 8 q^{5} - 8 q^{6} + 10 q^{7} - 6 q^{8} + 10 q^{9} + 8 q^{10} - 12 q^{11} - 20 q^{12} + 8 q^{13} + 8 q^{15} + 12 q^{16} - 16 q^{17} - 14 q^{18} - 8 q^{19} - 2 q^{21} - 16 q^{22}+ \cdots - 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 14x^{8} - 2x^{7} + 63x^{6} + 16x^{5} - 97x^{4} - 28x^{3} + 35x^{2} + 14x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{9} - 4\nu^{8} + 31\nu^{7} + 55\nu^{6} - 159\nu^{5} - 240\nu^{4} + 308\nu^{3} + 342\nu^{2} - 189\nu - 76 ) / 22 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 9\nu^{9} - 4\nu^{8} - 123\nu^{7} + 33\nu^{6} + 545\nu^{5} - 53\nu^{4} - 858\nu^{3} - 32\nu^{2} + 372\nu + 78 ) / 22 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -9\nu^{9} + 4\nu^{8} + 123\nu^{7} - 33\nu^{6} - 534\nu^{5} + 53\nu^{4} + 759\nu^{3} + 21\nu^{2} - 185\nu - 67 ) / 22 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{9} + 14\nu^{7} + 2\nu^{6} - 63\nu^{5} - 15\nu^{4} + 97\nu^{3} + 21\nu^{2} - 36\nu - 7 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 14 \nu^{9} + 5 \nu^{8} + 195 \nu^{7} - 44 \nu^{6} - 871 \nu^{5} + 113 \nu^{4} + 1320 \nu^{3} + \cdots - 37 ) / 22 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -15\nu^{9} + 3\nu^{8} + 205\nu^{7} - 11\nu^{6} - 890\nu^{5} - 51\nu^{4} + 1276\nu^{3} + 101\nu^{2} - 367\nu - 86 ) / 22 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 15\nu^{9} - 3\nu^{8} - 205\nu^{7} + 11\nu^{6} + 890\nu^{5} + 51\nu^{4} - 1276\nu^{3} - 79\nu^{2} + 345\nu + 20 ) / 22 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 19 \nu^{9} + 6 \nu^{8} + 267 \nu^{7} - 44 \nu^{6} - 1219 \nu^{5} + 52 \nu^{4} + 1947 \nu^{3} + \cdots - 106 ) / 22 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} + \beta_{7} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{9} - \beta_{7} + \beta_{6} + \beta_{5} - \beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 6\beta_{8} + 6\beta_{7} + \beta_{5} - \beta_{4} - \beta_{2} + 7\beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -9\beta_{9} + \beta_{8} - 8\beta_{7} + 9\beta_{6} + 9\beta_{5} + 2\beta_{4} - 7\beta_{3} + 29\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -\beta_{9} + 37\beta_{8} + 34\beta_{7} - \beta_{6} + 14\beta_{5} - 7\beta_{4} - \beta_{3} - 11\beta_{2} + 48\beta _1 + 86 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 64 \beta_{9} + 14 \beta_{8} - 56 \beta_{7} + 66 \beta_{6} + 69 \beta_{5} + 23 \beta_{4} - 42 \beta_{3} + \cdots + 28 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 15 \beta_{9} + 234 \beta_{8} + 189 \beta_{7} - 9 \beta_{6} + 133 \beta_{5} - 36 \beta_{4} + \cdots + 523 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 428 \beta_{9} + 138 \beta_{8} - 378 \beta_{7} + 452 \beta_{6} + 507 \beta_{5} + 197 \beta_{4} + \cdots + 269 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.60913
2.32797
1.39018
0.792781
−0.0952751
−0.316250
−0.619990
−1.61618
−2.00669
−2.46567
−2.60913 −2.82546 4.80756 −2.81923 7.37198 1.00000 −7.32529 4.98320 7.35574
1.2 −2.32797 0.887608 3.41946 1.22446 −2.06633 1.00000 −3.30446 −2.21215 −2.85051
1.3 −1.39018 3.12952 −0.0674011 −0.211606 −4.35060 1.00000 2.87406 6.79392 0.294171
1.4 −0.792781 1.51020 −1.37150 −3.70459 −1.19726 1.00000 2.67286 −0.719306 2.93693
1.5 0.0952751 −0.172502 −1.99092 −2.95457 −0.0164352 1.00000 −0.380236 −2.97024 −0.281497
1.6 0.316250 −2.45025 −1.89999 −0.186800 −0.774892 1.00000 −1.23337 3.00371 −0.0590756
1.7 0.619990 1.78132 −1.61561 −0.444493 1.10440 1.00000 −2.24164 0.173094 −0.275581
1.8 1.61618 −0.0489067 0.612049 3.02005 −0.0790423 1.00000 −2.24318 −2.99761 4.88095
1.9 2.00669 −3.06367 2.02682 −1.61423 −6.14785 1.00000 0.0538157 6.38609 −3.23926
1.10 2.46567 −0.747862 4.07953 −0.308987 −1.84398 1.00000 5.12745 −2.44070 −0.761859
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \( -1 \)
\(23\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3703.2.a.k ✓ 10
23.b odd 2 1 3703.2.a.l yes 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3703.2.a.k ✓ 10 1.a even 1 1 trivial
3703.2.a.l yes 10 23.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3703))\):

\( T_{2}^{10} - 14T_{2}^{8} + 2T_{2}^{7} + 63T_{2}^{6} - 16T_{2}^{5} - 97T_{2}^{4} + 28T_{2}^{3} + 35T_{2}^{2} - 14T_{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{10} + 8T_{5}^{9} + 10T_{5}^{8} - 66T_{5}^{7} - 199T_{5}^{6} - 90T_{5}^{5} + 214T_{5}^{4} + 240T_{5}^{3} + 94T_{5}^{2} + 16T_{5} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - 14 T^{8} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{10} + 2 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{10} + 8 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( (T - 1)^{10} \) Copy content Toggle raw display
$11$ \( T^{10} + 12 T^{9} + \cdots + 20401 \) Copy content Toggle raw display
$13$ \( T^{10} - 8 T^{9} + \cdots + 98569 \) Copy content Toggle raw display
$17$ \( T^{10} + 16 T^{9} + \cdots + 955837 \) Copy content Toggle raw display
$19$ \( T^{10} + 8 T^{9} + \cdots + 176461 \) Copy content Toggle raw display
$23$ \( T^{10} \) Copy content Toggle raw display
$29$ \( T^{10} + 16 T^{9} + \cdots + 56209 \) Copy content Toggle raw display
$31$ \( T^{10} - 4 T^{9} + \cdots - 101651 \) Copy content Toggle raw display
$37$ \( T^{10} - 214 T^{8} + \cdots + 4693957 \) Copy content Toggle raw display
$41$ \( T^{10} - 38 T^{9} + \cdots - 2903 \) Copy content Toggle raw display
$43$ \( T^{10} - 8 T^{9} + \cdots + 812929 \) Copy content Toggle raw display
$47$ \( T^{10} + 14 T^{9} + \cdots - 5761583 \) Copy content Toggle raw display
$53$ \( T^{10} + 36 T^{9} + \cdots - 28508051 \) Copy content Toggle raw display
$59$ \( T^{10} + 8 T^{9} + \cdots + 27074437 \) Copy content Toggle raw display
$61$ \( T^{10} + 26 T^{9} + \cdots + 1448233 \) Copy content Toggle raw display
$67$ \( T^{10} + 36 T^{9} + \cdots + 16525297 \) Copy content Toggle raw display
$71$ \( T^{10} + 14 T^{9} + \cdots - 11445551 \) Copy content Toggle raw display
$73$ \( T^{10} - 12 T^{9} + \cdots - 47127803 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 336190669 \) Copy content Toggle raw display
$83$ \( T^{10} + 18 T^{9} + \cdots + 1437601 \) Copy content Toggle raw display
$89$ \( T^{10} - 6 T^{9} + \cdots - 11142419 \) Copy content Toggle raw display
$97$ \( T^{10} + 46 T^{9} + \cdots + 48242473 \) Copy content Toggle raw display
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