Properties

Label 675.4.a.a
Level $675$
Weight $4$
Character orbit 675.a
Self dual yes
Analytic conductor $39.826$
Analytic rank $1$
Dimension $1$
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] = [675,4,Mod(1,675)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(675, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("675.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 675.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: \(39.8262892539\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 27)
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)
 
\(f(q)\) \(=\) \( q - 3 q^{2} + q^{4} + 25 q^{7} + 21 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{2} + q^{4} + 25 q^{7} + 21 q^{8} - 15 q^{11} - 20 q^{13} - 75 q^{14} - 71 q^{16} - 72 q^{17} + 2 q^{19} + 45 q^{22} - 114 q^{23} + 60 q^{26} + 25 q^{28} + 30 q^{29} + 101 q^{31} + 45 q^{32} + 216 q^{34} + 430 q^{37} - 6 q^{38} - 30 q^{41} - 110 q^{43} - 15 q^{44} + 342 q^{46} + 330 q^{47} + 282 q^{49} - 20 q^{52} - 621 q^{53} + 525 q^{56} - 90 q^{58} - 660 q^{59} - 376 q^{61} - 303 q^{62} + 433 q^{64} + 250 q^{67} - 72 q^{68} - 360 q^{71} - 785 q^{73} - 1290 q^{74} + 2 q^{76} - 375 q^{77} + 488 q^{79} + 90 q^{82} - 489 q^{83} + 330 q^{86} - 315 q^{88} - 450 q^{89} - 500 q^{91} - 114 q^{92} - 990 q^{94} + 1105 q^{97} - 846 q^{98}+O(q^{100}) \) 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
0
−3.00000 0 1.00000 0 0 25.0000 21.0000 0 0
\(n\): e.g. 2-40 or 990-1000
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 675.4.a.a 1
3.b odd 2 1 675.4.a.j 1
5.b even 2 1 27.4.a.b yes 1
5.c odd 4 2 675.4.b.a 2
15.d odd 2 1 27.4.a.a 1
15.e even 4 2 675.4.b.b 2
20.d odd 2 1 432.4.a.n 1
35.c odd 2 1 1323.4.a.k 1
40.e odd 2 1 1728.4.a.d 1
40.f even 2 1 1728.4.a.c 1
45.h odd 6 2 81.4.c.c 2
45.j even 6 2 81.4.c.a 2
60.h even 2 1 432.4.a.a 1
105.g even 2 1 1323.4.a.d 1
120.i odd 2 1 1728.4.a.bc 1
120.m even 2 1 1728.4.a.bd 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.4.a.a 1 15.d odd 2 1
27.4.a.b yes 1 5.b even 2 1
81.4.c.a 2 45.j even 6 2
81.4.c.c 2 45.h odd 6 2
432.4.a.a 1 60.h even 2 1
432.4.a.n 1 20.d odd 2 1
675.4.a.a 1 1.a even 1 1 trivial
675.4.a.j 1 3.b odd 2 1
675.4.b.a 2 5.c odd 4 2
675.4.b.b 2 15.e even 4 2
1323.4.a.d 1 105.g even 2 1
1323.4.a.k 1 35.c odd 2 1
1728.4.a.c 1 40.f even 2 1
1728.4.a.d 1 40.e odd 2 1
1728.4.a.bc 1 120.i odd 2 1
1728.4.a.bd 1 120.m even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(675))\):

\( T_{2} + 3 \) Copy content Toggle raw display
\( T_{7} - 25 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 3 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 25 \) Copy content Toggle raw display
$11$ \( T + 15 \) Copy content Toggle raw display
$13$ \( T + 20 \) Copy content Toggle raw display
$17$ \( T + 72 \) Copy content Toggle raw display
$19$ \( T - 2 \) Copy content Toggle raw display
$23$ \( T + 114 \) Copy content Toggle raw display
$29$ \( T - 30 \) Copy content Toggle raw display
$31$ \( T - 101 \) Copy content Toggle raw display
$37$ \( T - 430 \) Copy content Toggle raw display
$41$ \( T + 30 \) Copy content Toggle raw display
$43$ \( T + 110 \) Copy content Toggle raw display
$47$ \( T - 330 \) Copy content Toggle raw display
$53$ \( T + 621 \) Copy content Toggle raw display
$59$ \( T + 660 \) Copy content Toggle raw display
$61$ \( T + 376 \) Copy content Toggle raw display
$67$ \( T - 250 \) Copy content Toggle raw display
$71$ \( T + 360 \) Copy content Toggle raw display
$73$ \( T + 785 \) Copy content Toggle raw display
$79$ \( T - 488 \) Copy content Toggle raw display
$83$ \( T + 489 \) Copy content Toggle raw display
$89$ \( T + 450 \) Copy content Toggle raw display
$97$ \( T - 1105 \) Copy content Toggle raw display
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