Properties

Label 1575.4.a.bs
Level $1575$
Weight $4$
Character orbit 1575.a
Self dual yes
Analytic conductor $92.928$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1575,4,Mod(1,1575)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1575, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1575.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1575.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: \(92.9280082590\)
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: \( x^{8} - 50x^{6} + 698x^{4} - 2653x^{2} + 2268 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2\cdot 3 \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 5) q^{4} + 7 q^{7} + (\beta_{3} + 6 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 5) q^{4} + 7 q^{7} + (\beta_{3} + 6 \beta_1) q^{8} + ( - \beta_{5} - \beta_{3} + \beta_1) q^{11} + ( - \beta_{7} - 5) q^{13} + 7 \beta_1 q^{14} + (\beta_{7} + \beta_{6} + 6 \beta_{2} + 43) q^{16} + ( - \beta_{5} - \beta_{4} + \cdots + \beta_1) q^{17}+ \cdots + 49 \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 36 q^{4} + 56 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 36 q^{4} + 56 q^{7} - 38 q^{13} + 320 q^{16} + 68 q^{19} + 110 q^{22} + 252 q^{28} + 534 q^{31} + 118 q^{34} + 14 q^{37} + 816 q^{43} + 1588 q^{46} + 392 q^{49} - 266 q^{52} + 716 q^{58} + 874 q^{61} + 2502 q^{64} + 254 q^{67} - 1532 q^{73} + 4304 q^{76} + 5190 q^{79} + 1158 q^{82} - 4300 q^{88} - 266 q^{91} + 6236 q^{94} + 428 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 50x^{6} + 698x^{4} - 2653x^{2} + 2268 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 22\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} - 42\nu^{5} + 380\nu^{3} + 27\nu ) / 18 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} - 48\nu^{5} + 596\nu^{3} - 1353\nu ) / 18 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} - 42\nu^{4} + 407\nu^{2} - 558 ) / 9 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{6} + 51\nu^{4} - 677\nu^{2} + 1449 ) / 9 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 22\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} + \beta_{6} + 30\beta_{2} + 291 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -3\beta_{5} + 3\beta_{4} + 36\beta_{3} + 562\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 42\beta_{7} + 51\beta_{6} + 853\beta_{2} + 7489 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -126\beta_{5} + 144\beta_{4} + 1132\beta_{3} + 15217\beta_1 \) 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
−5.38335
−3.95827
−2.03648
−1.09744
1.09744
2.03648
3.95827
5.38335
−5.38335 0 20.9805 0 0 7.00000 −69.8783 0 0
1.2 −3.95827 0 7.66789 0 0 7.00000 1.31458 0 0
1.3 −2.03648 0 −3.85273 0 0 7.00000 24.1379 0 0
1.4 −1.09744 0 −6.79562 0 0 7.00000 16.2374 0 0
1.5 1.09744 0 −6.79562 0 0 7.00000 −16.2374 0 0
1.6 2.03648 0 −3.85273 0 0 7.00000 −24.1379 0 0
1.7 3.95827 0 7.66789 0 0 7.00000 −1.31458 0 0
1.8 5.38335 0 20.9805 0 0 7.00000 69.8783 0 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
\(3\) \( +1 \)
\(5\) \( -1 \)
\(7\) \( -1 \)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1575.4.a.bs yes 8
3.b odd 2 1 inner 1575.4.a.bs yes 8
5.b even 2 1 1575.4.a.br 8
15.d odd 2 1 1575.4.a.br 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1575.4.a.br 8 5.b even 2 1
1575.4.a.br 8 15.d odd 2 1
1575.4.a.bs yes 8 1.a even 1 1 trivial
1575.4.a.bs yes 8 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1575))\):

\( T_{2}^{8} - 50T_{2}^{6} + 698T_{2}^{4} - 2653T_{2}^{2} + 2268 \) Copy content Toggle raw display
\( T_{11}^{8} - 5061T_{11}^{6} + 2770339T_{11}^{4} - 405181287T_{11}^{2} + 3104665200 \) Copy content Toggle raw display
\( T_{13}^{4} + 19T_{13}^{3} - 6920T_{13}^{2} - 16232T_{13} + 9397752 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 50 T^{6} + \cdots + 2268 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T - 7)^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 3104665200 \) Copy content Toggle raw display
$13$ \( (T^{4} + 19 T^{3} + \cdots + 9397752)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 708112006463232 \) Copy content Toggle raw display
$19$ \( (T^{4} - 34 T^{3} + \cdots + 91746160)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 15\!\cdots\!75 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 51\!\cdots\!75 \) Copy content Toggle raw display
$31$ \( (T^{4} - 267 T^{3} + \cdots + 91463288)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 7 T^{3} + \cdots + 442781074)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 11\!\cdots\!92 \) Copy content Toggle raw display
$43$ \( (T^{4} - 408 T^{3} + \cdots - 1610375707)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 22\!\cdots\!32 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 68\!\cdots\!28 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 70\!\cdots\!52 \) Copy content Toggle raw display
$61$ \( (T^{4} - 437 T^{3} + \cdots - 43605452800)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 127 T^{3} + \cdots + 35816837576)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 65\!\cdots\!52 \) Copy content Toggle raw display
$73$ \( (T^{4} + 766 T^{3} + \cdots - 6635401472)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 2595 T^{3} + \cdots - 201947393092)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 29\!\cdots\!32 \) Copy content Toggle raw display
$97$ \( (T^{4} - 214 T^{3} + \cdots + 96747398592)^{2} \) Copy content Toggle raw display
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