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Properties

Label 3575.2.a.t
Level $3575$
Weight $2$
Character orbit 3575.a
Self dual yes
Analytic conductor $28.547$
Analytic rank $0$
Dimension $9$
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] = [3575,2,Mod(1,3575)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3575, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 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("3575.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 3575 = 5^{2} \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3575.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [9,-1,-2,15,0,-1,4,-3,23,0,-9,-6,9,16,0,15,-13] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(28.5465187226\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - x^{8} - 16x^{7} + 14x^{6} + 86x^{5} - 57x^{4} - 179x^{3} + 64x^{2} + 118x - 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 715)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - \beta_{3} q^{3} + (\beta_{2} + 2) q^{4} + ( - \beta_{8} + \beta_{4}) q^{6} + \beta_{7} q^{7} + (\beta_{7} - \beta_{6} - \beta_1 - 1) q^{8} + (\beta_{7} - \beta_{4} + \beta_{3} + \cdots + 1) q^{9}+ \cdots + ( - \beta_{7} + \beta_{4} - \beta_{3} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - q^{2} - 2 q^{3} + 15 q^{4} - q^{6} + 4 q^{7} - 3 q^{8} + 23 q^{9} - 9 q^{11} - 6 q^{12} + 9 q^{13} + 16 q^{14} + 15 q^{16} - 13 q^{17} - 3 q^{18} - 3 q^{19} + 14 q^{21} + q^{22} + 2 q^{24} - q^{26}+ \cdots - 23 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - x^{8} - 16x^{7} + 14x^{6} + 86x^{5} - 57x^{4} - 179x^{3} + 64x^{2} + 118x - 14 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{8} + \nu^{7} + 20\nu^{6} - 18\nu^{5} - 31\nu^{4} + 86\nu^{3} - 99\nu^{2} - 115\nu + 94 ) / 25 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -8\nu^{8} + 4\nu^{7} + 105\nu^{6} - 72\nu^{5} - 399\nu^{4} + 344\nu^{3} + 379\nu^{2} - 385\nu - 24 ) / 25 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -7\nu^{8} - 9\nu^{7} + 95\nu^{6} + 112\nu^{5} - 371\nu^{4} - 399\nu^{3} + 341\nu^{2} + 335\nu - 21 ) / 25 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -7\nu^{8} - 9\nu^{7} + 95\nu^{6} + 112\nu^{5} - 396\nu^{4} - 399\nu^{3} + 516\nu^{2} + 310\nu - 171 ) / 25 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -7\nu^{8} - 9\nu^{7} + 95\nu^{6} + 112\nu^{5} - 396\nu^{4} - 424\nu^{3} + 516\nu^{2} + 435\nu - 146 ) / 25 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -7\nu^{8} + 16\nu^{7} + 95\nu^{6} - 213\nu^{5} - 371\nu^{4} + 801\nu^{3} + 366\nu^{2} - 715\nu + 4 ) / 25 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} + \beta_{6} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{6} + \beta_{5} + 7\beta_{2} - \beta _1 + 22 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{8} - 8\beta_{7} + 9\beta_{6} - 2\beta_{4} + \beta_{3} - \beta_{2} + 30\beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -11\beta_{6} + 11\beta_{5} + \beta_{4} - 4\beta_{3} + 46\beta_{2} - 14\beta _1 + 134 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 14\beta_{8} - 56\beta_{7} + 69\beta_{6} - \beta_{5} - 26\beta_{4} + 13\beta_{3} - 14\beta_{2} + 192\beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -2\beta_{8} + \beta_{7} - 98\beta_{6} + 94\beta_{5} + 15\beta_{4} - 55\beta_{3} + 304\beta_{2} - 141\beta _1 + 852 \) 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
2.54892
2.34321
2.16069
1.13118
0.113915
−1.06779
−1.32285
−2.20826
−2.69902
−2.54892 1.71826 4.49699 0 −4.37972 −3.27678 −6.36464 −0.0475696 0
1.2 −2.34321 −2.48065 3.49065 0 5.81268 3.89108 −3.49290 3.15360 0
1.3 −2.16069 −1.29393 2.66858 0 2.79578 −2.72538 −1.44460 −1.32574 0
1.4 −1.13118 3.33877 −0.720430 0 −3.77675 4.42443 3.07730 8.14739 0
1.5 −0.113915 −3.18947 −1.98702 0 0.363329 −3.61768 0.454183 7.17269 0
1.6 1.06779 −0.343328 −0.859816 0 −0.366604 −1.31980 −3.05369 −2.88213 0
1.7 1.32285 2.67512 −0.250078 0 3.53878 0.156885 −2.97651 4.15628 0
1.8 2.20826 −3.17271 2.87640 0 −7.00615 1.59773 1.93531 7.06608 0
1.9 2.69902 0.747922 5.28473 0 2.01866 4.86951 8.86555 −2.44061 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( +1 \)
\(11\) \( +1 \)
\(13\) \( -1 \)

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 3575.2.a.t 9
5.b even 2 1 715.2.a.i ✓ 9
15.d odd 2 1 6435.2.a.bq 9
55.d odd 2 1 7865.2.a.w 9
65.d even 2 1 9295.2.a.t 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
715.2.a.i ✓ 9 5.b even 2 1
3575.2.a.t 9 1.a even 1 1 trivial
6435.2.a.bq 9 15.d odd 2 1
7865.2.a.w 9 55.d odd 2 1
9295.2.a.t 9 65.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3575))\):

\( T_{2}^{9} + T_{2}^{8} - 16T_{2}^{7} - 14T_{2}^{6} + 86T_{2}^{5} + 57T_{2}^{4} - 179T_{2}^{3} - 64T_{2}^{2} + 118T_{2} + 14 \) Copy content Toggle raw display
\( T_{3}^{9} + 2T_{3}^{8} - 23T_{3}^{7} - 42T_{3}^{6} + 169T_{3}^{5} + 260T_{3}^{4} - 434T_{3}^{3} - 466T_{3}^{2} + 272T_{3} + 128 \) Copy content Toggle raw display
\( T_{17}^{9} + 13 T_{17}^{8} - 39 T_{17}^{7} - 1089 T_{17}^{6} - 2318 T_{17}^{5} + 20468 T_{17}^{4} + \cdots + 3776 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} + T^{8} + \cdots + 14 \) Copy content Toggle raw display
$3$ \( T^{9} + 2 T^{8} + \cdots + 128 \) Copy content Toggle raw display
$5$ \( T^{9} \) Copy content Toggle raw display
$7$ \( T^{9} - 4 T^{8} + \cdots - 896 \) Copy content Toggle raw display
$11$ \( (T + 1)^{9} \) Copy content Toggle raw display
$13$ \( (T - 1)^{9} \) Copy content Toggle raw display
$17$ \( T^{9} + 13 T^{8} + \cdots + 3776 \) Copy content Toggle raw display
$19$ \( T^{9} + 3 T^{8} + \cdots + 1285360 \) Copy content Toggle raw display
$23$ \( T^{9} - 111 T^{7} + \cdots - 219520 \) Copy content Toggle raw display
$29$ \( T^{9} - 16 T^{8} + \cdots - 2929024 \) Copy content Toggle raw display
$31$ \( T^{9} - 12 T^{8} + \cdots + 131072 \) Copy content Toggle raw display
$37$ \( T^{9} + T^{8} + \cdots + 7082432 \) Copy content Toggle raw display
$41$ \( T^{9} - 5 T^{8} + \cdots - 155500 \) Copy content Toggle raw display
$43$ \( T^{9} - 3 T^{8} + \cdots + 256 \) Copy content Toggle raw display
$47$ \( T^{9} + 27 T^{8} + \cdots + 24153344 \) Copy content Toggle raw display
$53$ \( T^{9} + 34 T^{8} + \cdots - 283264 \) Copy content Toggle raw display
$59$ \( T^{9} + 4 T^{8} + \cdots - 17783296 \) Copy content Toggle raw display
$61$ \( T^{9} - 30 T^{8} + \cdots + 114329728 \) Copy content Toggle raw display
$67$ \( T^{9} - 11 T^{8} + \cdots - 1252352 \) Copy content Toggle raw display
$71$ \( T^{9} + 4 T^{8} + \cdots - 702464 \) Copy content Toggle raw display
$73$ \( T^{9} - 12 T^{8} + \cdots - 100765000 \) Copy content Toggle raw display
$79$ \( T^{9} + 16 T^{8} + \cdots + 49283072 \) Copy content Toggle raw display
$83$ \( T^{9} + 34 T^{8} + \cdots + 88272128 \) Copy content Toggle raw display
$89$ \( T^{9} + 10 T^{8} + \cdots - 560000 \) Copy content Toggle raw display
$97$ \( T^{9} - 15 T^{8} + \cdots - 1863872 \) Copy content Toggle raw display
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