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Properties

Label 75.22.a.m
Level $75$
Weight $22$
Character orbit 75.a
Self dual yes
Analytic conductor $209.608$
Analytic rank $1$
Dimension $10$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [75,22,Mod(1,75)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(75, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.1");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 75.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(209.608008215\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 5 x^{9} - 15486550 x^{8} - 930225020 x^{7} + 80619202368510 x^{6} + \cdots + 47\!\cdots\!18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{24}\cdot 3^{14}\cdot 5^{24}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
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.

\(f(q)\) \(=\) \( q + (\beta_1 - 65) q^{2} + 59049 q^{3} + (\beta_{2} - 33 \beta_1 + 1004337) q^{4} + (59049 \beta_1 - 3838185) q^{6} + ( - \beta_{3} - 22 \beta_{2} + 51583 \beta_1 - 11561521) q^{7} + (\beta_{4} + \beta_{3} - 517 \beta_{2} + 875124 \beta_1 - 31851835) q^{8} + 3486784401 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 65) q^{2} + 59049 q^{3} + (\beta_{2} - 33 \beta_1 + 1004337) q^{4} + (59049 \beta_1 - 3838185) q^{6} + ( - \beta_{3} - 22 \beta_{2} + 51583 \beta_1 - 11561521) q^{7} + (\beta_{4} + \beta_{3} - 517 \beta_{2} + 875124 \beta_1 - 31851835) q^{8} + 3486784401 q^{9} + ( - 2 \beta_{9} - \beta_{8} - 2 \beta_{7} - 14 \beta_{6} + \cdots + 605070536) q^{11}+ \cdots + ( - 6973568802 \beta_{9} + \cdots + 21\!\cdots\!36) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 645 q^{2} + 590490 q^{3} + 10043205 q^{4} - 38086605 q^{6} - 115357290 q^{7} - 314142735 q^{8} + 34867844010 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 645 q^{2} + 590490 q^{3} + 10043205 q^{4} - 38086605 q^{6} - 115357290 q^{7} - 314142735 q^{8} + 34867844010 q^{9} + 5976221790 q^{11} + 593041212045 q^{12} - 842570747430 q^{13} + 1605059874570 q^{14} + 6061861547825 q^{16} - 12910340404230 q^{17} - 2248975938645 q^{18} - 76155422176280 q^{19} - 6811732617210 q^{21} - 461780887241010 q^{22} - 82640920915920 q^{23} - 18549814359015 q^{24} + 286555331159670 q^{26} + 20\!\cdots\!90 q^{27}+ \cdots + 20\!\cdots\!90 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 5 x^{9} - 15486550 x^{8} - 930225020 x^{7} + 80619202368510 x^{6} + \cdots + 47\!\cdots\!18 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 97\nu - 3097264 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 26\!\cdots\!11 \nu^{9} + \cdots + 18\!\cdots\!30 ) / 94\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 26\!\cdots\!11 \nu^{9} + \cdots - 30\!\cdots\!82 ) / 94\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 42\!\cdots\!43 \nu^{9} + \cdots + 38\!\cdots\!10 ) / 11\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 13\!\cdots\!05 \nu^{9} + \cdots + 29\!\cdots\!94 ) / 23\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 80\!\cdots\!57 \nu^{9} + \cdots - 45\!\cdots\!06 ) / 59\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 12\!\cdots\!93 \nu^{9} + \cdots - 27\!\cdots\!14 ) / 47\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 42\!\cdots\!51 \nu^{9} + \cdots + 89\!\cdots\!34 ) / 94\!\cdots\!64 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 97\beta _1 + 3097264 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + \beta_{3} - 322\beta_{2} + 5075668\beta _1 + 299759510 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 128 \beta_{9} - 205 \beta_{8} + 85 \beta_{7} + 190 \beta_{6} + 171 \beta_{5} - 84 \beta_{4} - 1130 \beta_{3} + 6755618 \beta_{2} + 157434970 \beta _1 + 15720901830703 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 285232 \beta_{9} + 247348 \beta_{8} + 310540 \beta_{7} - 203224 \beta_{6} - 130188 \beta_{5} + 8507436 \beta_{4} + 7308884 \beta_{3} - 3013169458 \beta_{2} + \cdots + 483057304801666 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 1393440912 \beta_{9} - 2236243116 \beta_{8} + 385164396 \beta_{7} + 846881256 \beta_{6} + 1739173140 \beta_{5} - 2139425954 \beta_{4} + \cdots + 89\!\cdots\!24 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 3466587403696 \beta_{9} + 3256823931146 \beta_{8} + 3738315792070 \beta_{7} - 1249106068220 \beta_{6} - 1738420673094 \beta_{5} + \cdots - 82\!\cdots\!68 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 11\!\cdots\!32 \beta_{9} + \cdots + 54\!\cdots\!39 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 31\!\cdots\!52 \beta_{9} + \cdots - 12\!\cdots\!48 \) 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.

comment: embeddings in the coefficient field
 
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
−2641.03
−2176.21
−1403.87
−1018.03
−294.921
−173.651
1063.71
1793.04
2403.38
2452.58
−2706.03 59049.0 5.22544e6 0 −1.59788e8 9.90426e7 −8.46523e9 3.48678e9 0
1.2 −2241.21 59049.0 2.92587e6 0 −1.32341e8 −1.14262e8 −1.85733e9 3.48678e9 0
1.3 −1468.87 59049.0 60422.4 0 −8.67352e7 −1.30137e9 2.99169e9 3.48678e9 0
1.4 −1083.03 59049.0 −924202. 0 −6.39517e7 7.89823e8 3.27221e9 3.48678e9 0
1.5 −359.921 59049.0 −1.96761e6 0 −2.12530e7 2.25801e8 1.46299e9 3.48678e9 0
1.6 −238.651 59049.0 −2.04020e6 0 −1.40921e7 −6.18210e8 9.87381e8 3.48678e9 0
1.7 998.710 59049.0 −1.09973e6 0 5.89728e7 1.20991e9 −3.19276e9 3.48678e9 0
1.8 1728.04 59049.0 888953. 0 1.02039e8 −3.15759e8 −2.08781e9 3.48678e9 0
1.9 2338.38 59049.0 3.37088e6 0 1.38079e8 −9.66595e8 2.97845e9 3.48678e9 0
1.10 2387.58 59049.0 3.60339e6 0 1.40984e8 8.76258e8 3.59626e9 3.48678e9 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)

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 75.22.a.m 10
5.b even 2 1 75.22.a.n 10
5.c odd 4 2 15.22.b.a ✓ 20
15.e even 4 2 45.22.b.d 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.22.b.a ✓ 20 5.c odd 4 2
45.22.b.d 20 15.e even 4 2
75.22.a.m 10 1.a even 1 1 trivial
75.22.a.n 10 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} + 645 T_{2}^{9} - 15299350 T_{2}^{8} - 8951036520 T_{2}^{7} + 78367524408160 T_{2}^{6} + \cdots + 79\!\cdots\!68 \) acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(75))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 645 T^{9} + \cdots + 79\!\cdots\!68 \) Copy content Toggle raw display
$3$ \( (T - 59049)^{10} \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} + 115357290 T^{9} + \cdots - 52\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{10} - 5976221790 T^{9} + \cdots - 61\!\cdots\!68 \) Copy content Toggle raw display
$13$ \( T^{10} + 842570747430 T^{9} + \cdots - 32\!\cdots\!32 \) Copy content Toggle raw display
$17$ \( T^{10} + 12910340404230 T^{9} + \cdots - 11\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{10} + 76155422176280 T^{9} + \cdots - 47\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{10} + 82640920915920 T^{9} + \cdots - 16\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots - 11\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots - 79\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 48\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 82\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots - 10\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 11\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 71\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots - 78\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 17\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots - 12\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots - 32\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots - 57\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 31\!\cdots\!32 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 39\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots - 61\!\cdots\!76 \) Copy content Toggle raw display
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