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Properties

Label 675.2.b.h
Level $675$
Weight $2$
Character orbit 675.b
Analytic conductor $5.390$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [675,2,Mod(649,675)]
 
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, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("675.649");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 675.b (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(5.38990213644\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{13})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 7x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 135)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{3} - 2) q^{4} + ( - 2 \beta_{2} - 2 \beta_1) q^{7} + 3 \beta_{2} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{3} - 2) q^{4} + ( - 2 \beta_{2} - 2 \beta_1) q^{7} + 3 \beta_{2} q^{8} + (2 \beta_{3} - 2) q^{11} + (2 \beta_{2} - 2 \beta_1) q^{13} + 6 q^{14} + ( - \beta_{3} - 1) q^{16} + ( - 3 \beta_{2} - 2 \beta_1) q^{17} + (2 \beta_{3} - 1) q^{19} + (6 \beta_{2} - 2 \beta_1) q^{22} + 3 \beta_{2} q^{23} + ( - 4 \beta_{3} + 10) q^{26} + ( - 4 \beta_{2} + 2 \beta_1) q^{28} + (2 \beta_{3} + 4) q^{29} + (2 \beta_{3} - 3) q^{31} + (3 \beta_{2} - \beta_1) q^{32} + (\beta_{3} + 5) q^{34} - 2 \beta_{2} q^{37} + (6 \beta_{2} - \beta_1) q^{38} + ( - 2 \beta_{3} + 2) q^{41} + ( - 4 \beta_{2} - 2 \beta_1) q^{43} + ( - 4 \beta_{3} + 10) q^{44} + ( - 3 \beta_{3} + 3) q^{46} + 4 \beta_1 q^{47} + ( - 4 \beta_{3} - 5) q^{49} + ( - 8 \beta_{2} + 6 \beta_1) q^{52} + (3 \beta_{2} + 2 \beta_1) q^{53} + 6 \beta_{3} q^{56} + (6 \beta_{2} + 4 \beta_1) q^{58} + ( - 2 \beta_{3} - 4) q^{59} + (4 \beta_{3} + 1) q^{61} + (6 \beta_{2} - 3 \beta_1) q^{62} + ( - 6 \beta_{3} + 5) q^{64} + (10 \beta_{2} + 4 \beta_1) q^{67} + ( - 3 \beta_{2} + \beta_1) q^{68} + ( - 2 \beta_{3} - 10) q^{71} + (8 \beta_{2} - 2 \beta_1) q^{73} + (2 \beta_{3} - 2) q^{74} + ( - 3 \beta_{3} + 8) q^{76} - 12 \beta_{2} q^{77} + ( - 2 \beta_{3} + 9) q^{79} + ( - 6 \beta_{2} + 2 \beta_1) q^{82} - 3 \beta_{2} q^{83} + (2 \beta_{3} + 4) q^{86} + 6 \beta_1 q^{88} + (6 \beta_{3} - 6) q^{89} + (4 \beta_{3} - 12) q^{91} + ( - 3 \beta_{2} + 3 \beta_1) q^{92} + (4 \beta_{3} - 16) q^{94} - 8 \beta_{2} q^{97} + ( - 12 \beta_{2} - 5 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{4} - 4 q^{11} + 24 q^{14} - 6 q^{16} + 32 q^{26} + 20 q^{29} - 8 q^{31} + 22 q^{34} + 4 q^{41} + 32 q^{44} + 6 q^{46} - 28 q^{49} + 12 q^{56} - 20 q^{59} + 12 q^{61} + 8 q^{64} - 44 q^{71} - 4 q^{74} + 26 q^{76} + 32 q^{79} + 20 q^{86} - 12 q^{89} - 40 q^{91} - 56 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 7x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 4\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 4 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{2} - 4\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/675\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
649.1
− 2.30278i
− 1.30278i
1.30278i
2.30278i
− 2.30278i 0 −3.30278 0 0 2.60555i 3.00000i 0 0
649.2 − 1.30278i 0 0.302776 0 0 4.60555i − 3.00000i 0 0
649.3 1.30278i 0 0.302776 0 0 − 4.60555i 3.00000i 0 0
649.4 2.30278i 0 −3.30278 0 0 − 2.60555i − 3.00000i 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 675.2.b.h 4
3.b odd 2 1 675.2.b.i 4
5.b even 2 1 inner 675.2.b.h 4
5.c odd 4 1 135.2.a.d yes 2
5.c odd 4 1 675.2.a.k 2
15.d odd 2 1 675.2.b.i 4
15.e even 4 1 135.2.a.c ✓ 2
15.e even 4 1 675.2.a.p 2
20.e even 4 1 2160.2.a.y 2
35.f even 4 1 6615.2.a.v 2
40.i odd 4 1 8640.2.a.df 2
40.k even 4 1 8640.2.a.cy 2
45.k odd 12 2 405.2.e.j 4
45.l even 12 2 405.2.e.k 4
60.l odd 4 1 2160.2.a.ba 2
105.k odd 4 1 6615.2.a.p 2
120.q odd 4 1 8640.2.a.ck 2
120.w even 4 1 8640.2.a.cr 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.2.a.c ✓ 2 15.e even 4 1
135.2.a.d yes 2 5.c odd 4 1
405.2.e.j 4 45.k odd 12 2
405.2.e.k 4 45.l even 12 2
675.2.a.k 2 5.c odd 4 1
675.2.a.p 2 15.e even 4 1
675.2.b.h 4 1.a even 1 1 trivial
675.2.b.h 4 5.b even 2 1 inner
675.2.b.i 4 3.b odd 2 1
675.2.b.i 4 15.d odd 2 1
2160.2.a.y 2 20.e even 4 1
2160.2.a.ba 2 60.l odd 4 1
6615.2.a.p 2 105.k odd 4 1
6615.2.a.v 2 35.f even 4 1
8640.2.a.ck 2 120.q odd 4 1
8640.2.a.cr 2 120.w even 4 1
8640.2.a.cy 2 40.k even 4 1
8640.2.a.df 2 40.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(675, [\chi])\):

\( T_{2}^{4} + 7T_{2}^{2} + 9 \) Copy content Toggle raw display
\( T_{7}^{4} + 28T_{7}^{2} + 144 \) Copy content Toggle raw display
\( T_{11}^{2} + 2T_{11} - 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 7T^{2} + 9 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 28T^{2} + 144 \) Copy content Toggle raw display
$11$ \( (T^{2} + 2 T - 12)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 44T^{2} + 16 \) Copy content Toggle raw display
$17$ \( T^{4} + 34T^{2} + 81 \) Copy content Toggle raw display
$19$ \( (T^{2} - 13)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 10 T + 12)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 4 T - 9)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 2 T - 12)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 44T^{2} + 16 \) Copy content Toggle raw display
$47$ \( T^{4} + 112T^{2} + 2304 \) Copy content Toggle raw display
$53$ \( T^{4} + 34T^{2} + 81 \) Copy content Toggle raw display
$59$ \( (T^{2} + 10 T + 12)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 6 T - 43)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 232T^{2} + 144 \) Copy content Toggle raw display
$71$ \( (T^{2} + 22 T + 108)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 188T^{2} + 4624 \) Copy content Toggle raw display
$79$ \( (T^{2} - 16 T + 51)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 6 T - 108)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
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