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Properties

Label 175.10.a.l
Level $175$
Weight $10$
Character orbit 175.a
Self dual yes
Analytic conductor $90.131$
Analytic rank $1$
Dimension $13$
CM no
Inner twists $1$

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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] = [175,10,Mod(1,175)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(175, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 10, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("175.1"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 175.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [13,-32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(90.1312713287\)
Analytic rank: \(1\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 6 x^{12} - 4637 x^{11} + 24436 x^{10} + 8207010 x^{9} - 36480204 x^{8} - 6997218824 x^{7} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{2}\cdot 5^{10} \)
Twist minimal: no (minimal twist has level 35)
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 \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 2) q^{2} + (\beta_{3} - 12) q^{3} + (\beta_{2} + 5 \beta_1 + 208) q^{4} + (\beta_{4} - 6 \beta_{3} + 19 \beta_1 + 227) q^{6} + 2401 q^{7} + ( - \beta_{5} - \beta_{4} + \cdots - 2545) q^{8}+ \cdots + (10136 \beta_{12} + 38122 \beta_{11} + \cdots + 26840102) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q - 32 q^{2} - 158 q^{3} + 2730 q^{4} + 3078 q^{6} + 31213 q^{7} - 33642 q^{8} + 92723 q^{9} - 66270 q^{11} - 120320 q^{12} - 203722 q^{13} - 76832 q^{14} - 444134 q^{16} - 426118 q^{17} - 1689266 q^{18}+ \cdots + 371812524 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{13} - 6 x^{12} - 4637 x^{11} + 24436 x^{10} + 8207010 x^{9} - 36480204 x^{8} - 6997218824 x^{7} + \cdots + 14\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 716 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 32\!\cdots\!87 \nu^{12} + \cdots - 20\!\cdots\!00 ) / 44\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 29\!\cdots\!61 \nu^{12} + \cdots + 19\!\cdots\!00 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 20\!\cdots\!53 \nu^{12} + \cdots - 13\!\cdots\!00 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 85\!\cdots\!63 \nu^{12} + \cdots + 53\!\cdots\!00 ) / 44\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 10\!\cdots\!97 \nu^{12} + \cdots - 65\!\cdots\!00 ) / 44\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 19\!\cdots\!61 \nu^{12} + \cdots - 11\!\cdots\!00 ) / 55\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 35\!\cdots\!19 \nu^{12} + \cdots + 23\!\cdots\!00 ) / 44\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 42\!\cdots\!83 \nu^{12} + \cdots + 27\!\cdots\!00 ) / 44\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 16\!\cdots\!09 \nu^{12} + \cdots - 10\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 69\!\cdots\!39 \nu^{12} + \cdots + 44\!\cdots\!00 ) / 44\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 716 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} - 2\beta_{3} + 1101\beta _1 + 289 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7 \beta_{11} + 9 \beta_{10} + \beta_{8} + \beta_{7} + 2 \beta_{6} + 4 \beta_{5} - 3 \beta_{4} + \cdots + 788790 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 60 \beta_{12} - 39 \beta_{11} - 69 \beta_{10} - 96 \beta_{9} + 107 \beta_{8} + 83 \beta_{7} + \cdots + 522846 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 748 \beta_{12} + 16239 \beta_{11} + 18861 \beta_{10} + 2480 \beta_{9} + 1725 \beta_{8} + \cdots + 1014530792 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 115788 \beta_{12} - 86157 \beta_{11} - 146447 \beta_{10} - 250288 \beta_{9} + 252241 \beta_{8} + \cdots + 1408094260 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1522852 \beta_{12} + 29449667 \beta_{11} + 30364433 \beta_{10} + 7865104 \beta_{9} + \cdots + 1396141277400 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 142174844 \beta_{12} - 145771969 \beta_{11} - 227295051 \beta_{10} - 470130064 \beta_{9} + \cdots + 3184401834232 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1863807412 \beta_{12} + 49529905339 \beta_{11} + 44864816937 \beta_{10} + 17541066064 \beta_{9} + \cdots + 19\!\cdots\!68 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 113237063516 \beta_{12} - 225682612081 \beta_{11} - 310294825595 \beta_{10} - 787420557072 \beta_{9} + \cdots + 62\!\cdots\!24 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1292079455412 \beta_{12} + 80758373183779 \beta_{11} + 64025636098881 \beta_{10} + \cdots + 28\!\cdots\!56 \) 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
39.1148
37.5568
25.4175
22.6718
22.6549
14.1576
−3.80576
−9.74202
−15.6862
−23.7968
−27.4620
−36.7267
−38.3540
−41.1148 −238.422 1178.43 0 9802.66 2401.00 −27400.1 37161.8 0
1.2 −39.5568 124.465 1052.74 0 −4923.44 2401.00 −21389.9 −4191.43 0
1.3 −27.4175 243.480 239.721 0 −6675.61 2401.00 7465.23 39599.4 0
1.4 −24.6718 −212.283 96.6987 0 5237.41 2401.00 10246.2 25381.1 0
1.5 −24.6549 −17.6734 95.8645 0 435.737 2401.00 10259.8 −19370.6 0
1.6 −16.1576 17.2974 −250.933 0 −279.484 2401.00 12327.1 −19383.8 0
1.7 1.80576 −23.8664 −508.739 0 −43.0971 2401.00 −1843.21 −19113.4 0
1.8 7.74202 −216.721 −452.061 0 −1677.86 2401.00 −7463.78 27284.9 0
1.9 13.6862 223.464 −324.689 0 3058.37 2401.00 −11451.1 30253.3 0
1.10 21.7968 −114.276 −36.8999 0 −2490.84 2401.00 −11964.3 −6624.06 0
1.11 25.4620 111.752 136.314 0 2845.44 2401.00 −9565.72 −7194.45 0
1.12 34.7267 125.299 693.945 0 4351.22 2401.00 6318.35 −3983.15 0
1.13 36.3540 −180.517 809.610 0 −6562.51 2401.00 10819.3 12903.4 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( -1 \)
\(7\) \( -1 \)

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 175.10.a.l 13
5.b even 2 1 175.10.a.m 13
5.c odd 4 2 35.10.b.a ✓ 26
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.10.b.a ✓ 26 5.c odd 4 2
175.10.a.l 13 1.a even 1 1 trivial
175.10.a.m 13 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{13} + 32 T_{2}^{12} - 4181 T_{2}^{11} - 122578 T_{2}^{10} + 6720150 T_{2}^{9} + \cdots - 58\!\cdots\!00 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(175))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{13} + \cdots - 58\!\cdots\!00 \) Copy content Toggle raw display
$3$ \( T^{13} + \cdots + 15\!\cdots\!96 \) Copy content Toggle raw display
$5$ \( T^{13} \) Copy content Toggle raw display
$7$ \( (T - 2401)^{13} \) Copy content Toggle raw display
$11$ \( T^{13} + \cdots + 32\!\cdots\!08 \) Copy content Toggle raw display
$13$ \( T^{13} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{13} + \cdots + 61\!\cdots\!28 \) Copy content Toggle raw display
$19$ \( T^{13} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{13} + \cdots + 16\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( T^{13} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{13} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{13} + \cdots - 73\!\cdots\!68 \) Copy content Toggle raw display
$41$ \( T^{13} + \cdots - 44\!\cdots\!32 \) Copy content Toggle raw display
$43$ \( T^{13} + \cdots - 36\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{13} + \cdots + 10\!\cdots\!12 \) Copy content Toggle raw display
$53$ \( T^{13} + \cdots - 60\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{13} + \cdots + 50\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{13} + \cdots - 36\!\cdots\!72 \) Copy content Toggle raw display
$67$ \( T^{13} + \cdots - 98\!\cdots\!72 \) Copy content Toggle raw display
$71$ \( T^{13} + \cdots - 32\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{13} + \cdots + 10\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( T^{13} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{13} + \cdots + 48\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{13} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{13} + \cdots + 38\!\cdots\!08 \) Copy content Toggle raw display
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