Properties

Label 75.22.a.i
Level $75$
Weight $22$
Character orbit 75.a
Self dual yes
Analytic conductor $209.608$
Analytic rank $1$
Dimension $6$
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: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 530373x^{4} - 23850203x^{3} + 59940607490x^{2} + 2888057920800x - 684850637484000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{18}\cdot 3^{6}\cdot 5^{7}\cdot 7^{2} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 15) q^{2} + 59049 q^{3} + (\beta_{2} + 243 \beta_1 + 731822) q^{4} + (59049 \beta_1 - 885735) q^{6} + ( - \beta_{3} - 49 \beta_{2} - 19683 \beta_1 - 221197120) q^{7} + (\beta_{4} + \beta_{3} + 406 \beta_{2} + 663939 \beta_1 + 708933081) q^{8} + 3486784401 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 15) q^{2} + 59049 q^{3} + (\beta_{2} + 243 \beta_1 + 731822) q^{4} + (59049 \beta_1 - 885735) q^{6} + ( - \beta_{3} - 49 \beta_{2} - 19683 \beta_1 - 221197120) q^{7} + (\beta_{4} + \beta_{3} + 406 \beta_{2} + 663939 \beta_1 + 708933081) q^{8} + 3486784401 q^{9} + (5 \beta_{5} - 22 \beta_{4} - 7 \beta_{3} + 856 \beta_{2} + \cdots - 13051322369) q^{11}+ \cdots + (17433922005 \beta_{5} - 76709256822 \beta_{4} + \cdots - 45\!\cdots\!69) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 92 q^{2} + 354294 q^{3} + 4390448 q^{4} - 5432508 q^{6} - 1327143454 q^{7} + 4252271424 q^{8} + 20920706406 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 92 q^{2} + 354294 q^{3} + 4390448 q^{4} - 5432508 q^{6} - 1327143454 q^{7} + 4252271424 q^{8} + 20920706406 q^{9} - 78310112516 q^{11} + 259251563952 q^{12} - 116029746338 q^{13} - 313825730148 q^{14} + 1996701612800 q^{16} - 5382900513068 q^{17} - 320784164892 q^{18} + 6969638997622 q^{19} - 78366493815246 q^{21} + 19701817271864 q^{22} - 295754895311052 q^{23} + 251092375315776 q^{24} - 274314326263628 q^{26} + 12\!\cdots\!94 q^{27}+ \cdots - 27\!\cdots\!16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 530373x^{4} - 23850203x^{3} + 59940607490x^{2} + 2888057920800x - 684850637484000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 4\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 16\nu^{2} - 1100\nu - 2828475 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} - 157\nu^{4} - 456377\nu^{3} + 37318593\nu^{2} + 36666498918\nu - 909462022508 ) / 4004 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} + 157\nu^{4} + 712633\nu^{3} - 66403649\nu^{2} - 112675632118\nu + 2940276712412 ) / 4004 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} + 571\nu^{4} - 544465\nu^{3} - 249005991\nu^{2} + 48958029442\nu + 11600260895639 ) / 1001 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + 275\beta _1 + 2828750 ) / 16 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{4} + \beta_{3} + 454\beta_{2} + 4870675\beta _1 + 781801849 ) / 64 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 88\beta_{5} + 121\beta_{4} - 231\beta_{3} + 1628146\beta_{2} + 751841447\beta _1 + 3444795464265 ) / 64 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 6908 \beta_{5} + 237687 \beta_{4} + 338183 \beta_{3} + 156769854 \beta_{2} + 856594358333 \beta _1 + 266494147996751 ) / 32 \) Copy content Toggle raw display

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
−571.187
−388.716
−144.571
89.2274
370.009
646.238
−2300.75 59049.0 3.19629e6 0 −1.35857e8 2.10374e8 −2.52885e9 3.48678e9 0
1.2 −1570.86 59049.0 370456. 0 −9.27578e7 −1.39580e9 2.71240e9 3.48678e9 0
1.3 −594.286 59049.0 −1.74398e6 0 −3.50920e7 9.49356e8 2.28273e9 3.48678e9 0
1.4 340.910 59049.0 −1.98093e6 0 2.01304e7 −6.73159e8 −1.39026e9 3.48678e9 0
1.5 1464.04 59049.0 46252.4 0 8.64499e7 1.40408e8 −3.00259e9 3.48678e9 0
1.6 2568.95 59049.0 4.50235e6 0 1.51694e8 −5.58323e8 6.17885e9 3.48678e9 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.i 6
5.b even 2 1 75.22.a.j yes 6
5.c odd 4 2 75.22.b.i 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.22.a.i 6 1.a even 1 1 trivial
75.22.a.j yes 6 5.b even 2 1
75.22.b.i 12 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 92 T_{2}^{5} - 8482448 T_{2}^{4} - 2069443264 T_{2}^{3} + 15258494066176 T_{2}^{2} + \cdots - 27\!\cdots\!28 \) acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(75))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 92 T^{5} + \cdots - 27\!\cdots\!28 \) Copy content Toggle raw display
$3$ \( (T - 59049)^{6} \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 1327143454 T^{5} + \cdots - 14\!\cdots\!75 \) Copy content Toggle raw display
$11$ \( T^{6} + 78310112516 T^{5} + \cdots - 99\!\cdots\!52 \) Copy content Toggle raw display
$13$ \( T^{6} + 116029746338 T^{5} + \cdots + 45\!\cdots\!97 \) Copy content Toggle raw display
$17$ \( T^{6} + 5382900513068 T^{5} + \cdots - 85\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( T^{6} - 6969638997622 T^{5} + \cdots - 69\!\cdots\!75 \) Copy content Toggle raw display
$23$ \( T^{6} + 295754895311052 T^{5} + \cdots - 72\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots - 38\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{6} - 492586859482706 T^{5} + \cdots + 43\!\cdots\!25 \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots - 26\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots - 29\!\cdots\!31 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots - 14\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 61\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots - 36\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 14\!\cdots\!41 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots - 96\!\cdots\!51 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 54\!\cdots\!84 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots - 37\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots - 16\!\cdots\!32 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots - 85\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots - 80\!\cdots\!91 \) Copy content Toggle raw display
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