Properties

Label 1575.4.a.bi
Level $1575$
Weight $4$
Character orbit 1575.a
Self dual yes
Analytic conductor $92.928$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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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: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.3030748.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 21x^{2} + 28 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{3} + 3) q^{4} + 7 q^{7} + (2 \beta_{2} + 3 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{3} + 3) q^{4} + 7 q^{7} + (2 \beta_{2} + 3 \beta_1) q^{8} + ( - \beta_{2} - 5 \beta_1) q^{11} + (\beta_{3} - 5) q^{13} + 7 \beta_1 q^{14} + ( - 3 \beta_{3} + 3) q^{16} - 13 \beta_1 q^{17} + ( - 2 \beta_{3} - 34) q^{19} + ( - 6 \beta_{3} - 52) q^{22} + ( - 17 \beta_{2} - 10 \beta_1) q^{23} + (2 \beta_{2} + 3 \beta_1) q^{26} + (7 \beta_{3} + 21) q^{28} + (19 \beta_{2} + 4 \beta_1) q^{29} + ( - 15 \beta_{3} - 39) q^{31} + ( - 22 \beta_{2} - 45 \beta_1) q^{32} + ( - 13 \beta_{3} - 143) q^{34} + ( - 15 \beta_{3} - 96) q^{37} + ( - 4 \beta_{2} - 50 \beta_1) q^{38} + ( - 30 \beta_{2} + 53 \beta_1) q^{41} + (14 \beta_{3} - 61) q^{43} + ( - 4 \beta_{2} - 60 \beta_1) q^{44} + ( - 27 \beta_{3} - 59) q^{46} + (44 \beta_{2} - 22 \beta_1) q^{47} + 49 q^{49} + ( - 3 \beta_{3} + 67) q^{52} + (30 \beta_{2} - \beta_1) q^{53} + (14 \beta_{2} + 21 \beta_1) q^{56} + (23 \beta_{3} - 13) q^{58} + ( - 38 \beta_{2} - 41 \beta_1) q^{59} + ( - 45 \beta_{3} - 315) q^{61} + ( - 30 \beta_{2} - 159 \beta_1) q^{62} + ( - 43 \beta_{3} - 453) q^{64} + ( - 49 \beta_{3} - 34) q^{67} + ( - 26 \beta_{2} - 143 \beta_1) q^{68} + (61 \beta_{2} + 193 \beta_1) q^{71} + (46 \beta_{3} - 24) q^{73} + ( - 30 \beta_{2} - 216 \beta_1) q^{74} + ( - 38 \beta_{3} - 266) q^{76} + ( - 7 \beta_{2} - 35 \beta_1) q^{77} + (13 \beta_{3} - 166) q^{79} + (23 \beta_{3} + 673) q^{82} + (64 \beta_{2} + 79 \beta_1) q^{83} + (28 \beta_{2} + 51 \beta_1) q^{86} + ( - 16 \beta_{3} - 232) q^{88} + (68 \beta_{2} - 186 \beta_1) q^{89} + (7 \beta_{3} - 35) q^{91} + (82 \beta_{2} - 195 \beta_1) q^{92} + (22 \beta_{3} - 374) q^{94} + ( - 38 \beta_{3} + 476) q^{97} + 49 \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 10 q^{4} + 28 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 10 q^{4} + 28 q^{7} - 22 q^{13} + 18 q^{16} - 132 q^{19} - 196 q^{22} + 70 q^{28} - 126 q^{31} - 546 q^{34} - 354 q^{37} - 272 q^{43} - 182 q^{46} + 196 q^{49} + 274 q^{52} - 98 q^{58} - 1170 q^{61} - 1726 q^{64} - 38 q^{67} - 188 q^{73} - 988 q^{76} - 690 q^{79} + 2646 q^{82} - 896 q^{88} - 154 q^{91} - 1540 q^{94} + 1980 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 21x^{2} + 28 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 19\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - 11 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{2} + 19\beta_1 \) 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
−4.42371
−1.19617
1.19617
4.42371
−4.42371 0 11.5692 0 0 7.00000 −15.7890 0 0
1.2 −1.19617 0 −6.56918 0 0 7.00000 17.4272 0 0
1.3 1.19617 0 −6.56918 0 0 7.00000 −17.4272 0 0
1.4 4.42371 0 11.5692 0 0 7.00000 15.7890 0 0
\(n\): e.g. 2-40 or 990-1000
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.bi yes 4
3.b odd 2 1 inner 1575.4.a.bi yes 4
5.b even 2 1 1575.4.a.bh 4
15.d odd 2 1 1575.4.a.bh 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1575.4.a.bh 4 5.b even 2 1
1575.4.a.bh 4 15.d odd 2 1
1575.4.a.bi yes 4 1.a even 1 1 trivial
1575.4.a.bi yes 4 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1575))\):

\( T_{2}^{4} - 21T_{2}^{2} + 28 \) Copy content Toggle raw display
\( T_{11}^{4} - 567T_{11}^{2} + 11200 \) Copy content Toggle raw display
\( T_{13}^{2} + 11T_{13} - 52 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 21T^{2} + 28 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T - 7)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 567 T^{2} + 11200 \) Copy content Toggle raw display
$13$ \( (T^{2} + 11 T - 52)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 3549 T^{2} + 799708 \) Copy content Toggle raw display
$19$ \( (T^{2} + 66 T + 760)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 32088 T^{2} + \cdots + 119687575 \) Copy content Toggle raw display
$29$ \( T^{4} - 39704 T^{2} + \cdots + 65759575 \) Copy content Toggle raw display
$31$ \( (T^{2} + 63 T - 17514)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 177 T - 10674)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 182049 T^{2} + \cdots + 5546169328 \) Copy content Toggle raw display
$43$ \( (T^{2} + 136 T - 11497)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 240548 T^{2} + \cdots + 419786752 \) Copy content Toggle raw display
$53$ \( T^{4} - 101241 T^{2} + \cdots + 111329008 \) Copy content Toggle raw display
$59$ \( T^{4} - 175217 T^{2} + \cdots + 6445308352 \) Copy content Toggle raw display
$61$ \( (T^{2} + 585 T - 81000)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 19 T - 197392)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} - 1034159 T^{2} + \cdots + 145652398528 \) Copy content Toggle raw display
$73$ \( (T^{2} + 94 T - 171832)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 345 T + 15856)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 519029 T^{2} + \cdots + 61786217500 \) Copy content Toggle raw display
$89$ \( T^{4} - 1421476 T^{2} + \cdots + 477168606208 \) Copy content Toggle raw display
$97$ \( (T^{2} - 990 T + 126256)^{2} \) Copy content Toggle raw display
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