Properties

Label 1575.4.a.bm
Level $1575$
Weight $4$
Character orbit 1575.a
Self dual yes
Analytic conductor $92.928$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(4\)
Coefficient field: 4.4.26729725.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 18x^{2} + 19x + 44 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 525)
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.

\(f(q)\) \(=\) \( q + (\beta_1 + 1) q^{2} + (\beta_{2} + 3 \beta_1 + 3) q^{4} + 7 q^{7} + (\beta_{3} + 4 \beta_{2} + 5 \beta_1 + 23) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 1) q^{2} + (\beta_{2} + 3 \beta_1 + 3) q^{4} + 7 q^{7} + (\beta_{3} + 4 \beta_{2} + 5 \beta_1 + 23) q^{8} + (\beta_{3} - 5 \beta_{2} + 3 \beta_1 - 18) q^{11} + (3 \beta_{3} + \beta_{2} + 3 \beta_1 - 11) q^{13} + (7 \beta_1 + 7) q^{14} + (6 \beta_{3} + 6 \beta_{2} + 27 \beta_1 + 45) q^{16} + ( - 3 \beta_{3} + 3 \beta_{2} + \cdots + 17) q^{17}+ \cdots + (49 \beta_1 + 49) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{2} + 16 q^{4} + 28 q^{7} + 93 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6 q^{2} + 16 q^{4} + 28 q^{7} + 93 q^{8} - 57 q^{11} - 43 q^{13} + 42 q^{14} + 216 q^{16} + 99 q^{17} - 12 q^{19} + 41 q^{22} + 156 q^{23} + 81 q^{26} + 112 q^{28} - 378 q^{29} - 93 q^{31} + 690 q^{32} + 783 q^{34} - 81 q^{37} + 216 q^{38} + 465 q^{41} + 64 q^{43} - 681 q^{44} + 310 q^{46} + 744 q^{47} + 196 q^{49} + 727 q^{52} + 729 q^{53} + 651 q^{56} - 1172 q^{58} - 231 q^{59} - 1353 q^{61} - 165 q^{62} + 3107 q^{64} - 1487 q^{67} + 2577 q^{68} + 1725 q^{71} - 512 q^{73} + 1953 q^{74} - 3046 q^{76} - 399 q^{77} + 1629 q^{79} + 693 q^{82} + 321 q^{83} + 4542 q^{86} - 3482 q^{88} + 978 q^{89} - 301 q^{91} + 852 q^{92} + 2480 q^{94} - 2616 q^{97} + 294 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 18x^{2} + 19x + 44 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 14\nu + 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 15\beta _1 + 8 \) 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
−3.56826
−1.21734
2.21734
4.56826
−2.56826 0 −1.40404 0 0 7.00000 24.1520 0 0
1.2 −0.217342 0 −7.95276 0 0 7.00000 3.46721 0 0
1.3 3.21734 0 2.35129 0 0 7.00000 −18.1738 0 0
1.4 5.56826 0 23.0055 0 0 7.00000 83.5546 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

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 1575.4.a.bm 4
3.b odd 2 1 525.4.a.s 4
5.b even 2 1 1575.4.a.bf 4
15.d odd 2 1 525.4.a.v yes 4
15.e even 4 2 525.4.d.o 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.4.a.s 4 3.b odd 2 1
525.4.a.v yes 4 15.d odd 2 1
525.4.d.o 8 15.e even 4 2
1575.4.a.bf 4 5.b even 2 1
1575.4.a.bm 4 1.a even 1 1 trivial

Hecke kernels

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

\( T_{2}^{4} - 6T_{2}^{3} - 6T_{2}^{2} + 45T_{2} + 10 \) Copy content Toggle raw display
\( T_{11}^{4} + 57T_{11}^{3} - 1323T_{11}^{2} - 44829T_{11} + 99334 \) Copy content Toggle raw display
\( T_{13}^{4} + 43T_{13}^{3} - 4324T_{13}^{2} - 62320T_{13} + 3240760 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 6 T^{3} + \cdots + 10 \) 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
$11$ \( T^{4} + 57 T^{3} + \cdots + 99334 \) Copy content Toggle raw display
$13$ \( T^{4} + 43 T^{3} + \cdots + 3240760 \) Copy content Toggle raw display
$17$ \( T^{4} - 99 T^{3} + \cdots + 3462544 \) Copy content Toggle raw display
$19$ \( T^{4} + 12 T^{3} + \cdots + 329386624 \) Copy content Toggle raw display
$23$ \( T^{4} - 156 T^{3} + \cdots + 92278225 \) Copy content Toggle raw display
$29$ \( T^{4} + 378 T^{3} + \cdots + 29419291 \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 1187132760 \) Copy content Toggle raw display
$37$ \( T^{4} + 81 T^{3} + \cdots + 210866814 \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots - 7122182144 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 7838190085 \) Copy content Toggle raw display
$47$ \( T^{4} - 744 T^{3} + \cdots + 425590240 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 8629646344 \) Copy content Toggle raw display
$59$ \( T^{4} + 231 T^{3} + \cdots + 323902840 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots - 9759896064 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots - 109414277684 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots - 423829861100 \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 67091874784 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 28849899226 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots - 22647834344 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots - 108748692656 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots - 5353888496 \) Copy content Toggle raw display
show more
show less