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Properties

Label 75.22.a.l
Level $75$
Weight $22$
Character orbit 75.a
Self dual yes
Analytic conductor $209.608$
Analytic rank $0$
Dimension $8$
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: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2 x^{7} - 13701836 x^{6} + 201589162 x^{5} + 55078074399586 x^{4} + \cdots + 23\!\cdots\!25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{21}\cdot 3^{10}\cdot 5^{14}\cdot 7^{2} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 83) q^{2} - 59049 q^{3} + (\beta_{2} + 147 \beta_1 + 1335201) q^{4} + ( - 59049 \beta_1 - 4901067) q^{6} + (\beta_{3} + 56 \beta_{2} + 148059 \beta_1 + 16717280) q^{7} + (\beta_{4} + 3 \beta_{3} + 279 \beta_{2} + 1483913 \beta_1 + 438649955) q^{8} + 3486784401 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 83) q^{2} - 59049 q^{3} + (\beta_{2} + 147 \beta_1 + 1335201) q^{4} + ( - 59049 \beta_1 - 4901067) q^{6} + (\beta_{3} + 56 \beta_{2} + 148059 \beta_1 + 16717280) q^{7} + (\beta_{4} + 3 \beta_{3} + 279 \beta_{2} + 1483913 \beta_1 + 438649955) q^{8} + 3486784401 q^{9} + ( - \beta_{5} - 12 \beta_{4} - 30 \beta_{3} - 5002 \beta_{2} + \cdots - 1797385509) q^{11}+ \cdots + ( - 3486784401 \beta_{5} + \cdots - 62\!\cdots\!09) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 666 q^{2} - 472392 q^{3} + 10681904 q^{4} - 39326634 q^{6} + 134034472 q^{7} + 3512168028 q^{8} + 27894275208 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 666 q^{2} - 472392 q^{3} + 10681904 q^{4} - 39326634 q^{6} + 134034472 q^{7} + 3512168028 q^{8} + 27894275208 q^{9} - 14410165968 q^{11} - 630755749296 q^{12} - 328226481176 q^{13} + 4068036669186 q^{14} + 18553054308920 q^{16} - 5718214953936 q^{17} + 2322198411066 q^{18} + 75919698170296 q^{19} - 7914601537128 q^{21} - 426897542691372 q^{22} + 49712781537936 q^{23} - 207390009885372 q^{24} - 126056679642054 q^{26} - 16\!\cdots\!92 q^{27}+ \cdots - 50\!\cdots\!68 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2 x^{7} - 13701836 x^{6} + 201589162 x^{5} + 55078074399586 x^{4} + \cdots + 23\!\cdots\!25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 19\nu - 3425464 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 321202317 \nu^{7} - 186534335407 \nu^{6} + \cdots + 67\!\cdots\!41 ) / 39\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 963606951 \nu^{7} + 559603006221 \nu^{6} + \cdots - 13\!\cdots\!35 ) / 39\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 133748537493 \nu^{7} + 730347015969159 \nu^{6} + \cdots - 18\!\cdots\!69 ) / 43\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 393603917171 \nu^{7} + 320153027486417 \nu^{6} + \cdots - 71\!\cdots\!79 ) / 43\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 302835281351 \nu^{7} + 246792224013677 \nu^{6} + \cdots + 97\!\cdots\!05 ) / 21\!\cdots\!64 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 19\beta _1 + 3425464 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + 3\beta_{3} + 30\beta_{2} + 5662281\beta _1 - 66735136 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 10 \beta_{7} - 44 \beta_{6} + 4 \beta_{5} + 828 \beta_{4} - 552 \beta_{3} + 7635372 \beta_{2} - 104015606 \beta _1 + 19395890152827 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 42158 \beta_{7} - 10252 \beta_{6} - 7724 \beta_{5} + 9444344 \beta_{4} + 34125348 \beta_{3} + 1294083360 \beta_{2} + 36473237814027 \beta _1 - 368911807691694 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 46884322 \beta_{7} - 474024380 \beta_{6} + 122383924 \beta_{5} + 10330386844 \beta_{4} - 9145742360 \beta_{3} + 56368731735085 \beta_{2} + \cdots + 12\!\cdots\!62 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 600833936150 \beta_{7} - 102434672348 \beta_{6} - 154577254140 \beta_{5} + 75193348500649 \beta_{4} + 299109028605959 \beta_{3} + \cdots + 63\!\cdots\!58 \) 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
−2688.59
−2210.46
−1237.86
−249.679
155.916
1642.33
1827.63
2762.71
−2605.59 −59049.0 4.69194e6 0 1.53857e8 −1.24132e9 −6.76095e9 3.48678e9 0
1.2 −2127.46 −59049.0 2.42895e6 0 1.25625e8 1.04187e9 −7.05898e8 3.48678e9 0
1.3 −1154.86 −59049.0 −763454. 0 6.81932e7 −5.42271e7 3.30359e9 3.48678e9 0
1.4 −166.679 −59049.0 −2.06937e6 0 9.84223e6 −1.02476e9 6.94472e8 3.48678e9 0
1.5 238.916 −59049.0 −2.04007e6 0 −1.41077e7 6.22955e8 −9.88448e8 3.48678e9 0
1.6 1725.33 −59049.0 879617. 0 −1.01879e8 −5.00000e8 −2.10065e9 3.48678e9 0
1.7 1910.63 −59049.0 1.55335e6 0 −1.12821e8 1.48601e8 −1.03900e9 3.48678e9 0
1.8 2845.71 −59049.0 6.00093e6 0 −1.68037e8 1.14091e9 1.11090e10 3.48678e9 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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.l yes 8
5.b even 2 1 75.22.a.k ✓ 8
5.c odd 4 2 75.22.b.j 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.22.a.k ✓ 8 5.b even 2 1
75.22.a.l yes 8 1.a even 1 1 trivial
75.22.b.j 16 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 666 T_{2}^{7} - 13507782 T_{2}^{6} + 6992794080 T_{2}^{5} + 53581897781856 T_{2}^{4} + \cdots + 23\!\cdots\!76 \) acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(75))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 666 T^{7} + \cdots + 23\!\cdots\!76 \) Copy content Toggle raw display
$3$ \( (T + 59049)^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 134034472 T^{7} + \cdots + 37\!\cdots\!25 \) Copy content Toggle raw display
$11$ \( T^{8} + 14410165968 T^{7} + \cdots - 32\!\cdots\!04 \) Copy content Toggle raw display
$13$ \( T^{8} + 328226481176 T^{7} + \cdots - 51\!\cdots\!79 \) Copy content Toggle raw display
$17$ \( T^{8} + 5718214953936 T^{7} + \cdots - 13\!\cdots\!84 \) Copy content Toggle raw display
$19$ \( T^{8} - 75919698170296 T^{7} + \cdots + 10\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( T^{8} - 49712781537936 T^{7} + \cdots + 99\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots - 81\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 65\!\cdots\!25 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 39\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 55\!\cdots\!01 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots - 53\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 72\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots - 14\!\cdots\!39 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 55\!\cdots\!81 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots - 43\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 55\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots - 13\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 29\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots - 35\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 95\!\cdots\!41 \) Copy content Toggle raw display
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