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Properties

Label 1575.2.d.l
Level $1575$
Weight $2$
Character orbit 1575.d
Analytic conductor $12.576$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1575,2,Mod(1324,1575)]
 
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, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1575.1324");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1575.d (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: \(12.5764383184\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.30599805184.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 3x^{6} + 13x^{4} + 27x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (\beta_{6} - 2) q^{4} - \beta_{4} q^{7} + ( - \beta_{7} - 2 \beta_{2}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + (\beta_{6} - 2) q^{4} - \beta_{4} q^{7} + ( - \beta_{7} - 2 \beta_{2}) q^{8} - \beta_1 q^{11} + 2 \beta_{3} q^{13} + \beta_{5} q^{14} + ( - 3 \beta_{6} + 3) q^{16} + 2 \beta_{2} q^{17} + ( - 2 \beta_{6} - 4) q^{19} + (\beta_{4} + 3 \beta_{3}) q^{22} + \beta_{7} q^{23} + (4 \beta_{5} + 2 \beta_1) q^{26} + (2 \beta_{4} - \beta_{3}) q^{28} + (2 \beta_{5} - \beta_1) q^{29} + 6 q^{31} + (\beta_{7} + 5 \beta_{2}) q^{32} + (2 \beta_{6} - 8) q^{34} + 3 \beta_{4} q^{37} + 2 \beta_{7} q^{38} + 4 \beta_{5} q^{41} + (3 \beta_{4} - 2 \beta_{3}) q^{43} + (5 \beta_{5} + \beta_1) q^{44} + (3 \beta_{6} + 1) q^{46} + ( - 2 \beta_{7} - 2 \beta_{2}) q^{47} - q^{49} + (14 \beta_{4} - 6 \beta_{3}) q^{52} + 4 \beta_{2} q^{53} + ( - 2 \beta_{5} - \beta_1) q^{56} + (9 \beta_{4} + \beta_{3}) q^{58} + (2 \beta_{5} + 2 \beta_1) q^{59} + 6 \beta_{2} q^{62} + (2 \beta_{6} - 13) q^{64} + ( - 3 \beta_{4} + 2 \beta_{3}) q^{67} + ( - 2 \beta_{7} - 8 \beta_{2}) q^{68} + (4 \beta_{5} + \beta_1) q^{71} + (2 \beta_{4} + 2 \beta_{3}) q^{73} - 3 \beta_{5} q^{74} + (2 \beta_{6} - 6) q^{76} + \beta_{7} q^{77} + ( - 2 \beta_{6} - 1) q^{79} + (16 \beta_{4} - 4 \beta_{3}) q^{82} + ( - 2 \beta_{7} + 4 \beta_{2}) q^{83} + ( - 7 \beta_{5} - 2 \beta_1) q^{86} + (21 \beta_{4} - 2 \beta_{3}) q^{88} + (2 \beta_{5} - 2 \beta_1) q^{89} + 2 \beta_{6} q^{91} + ( - \beta_{7} - 5 \beta_{2}) q^{92} + ( - 8 \beta_{6} + 6) q^{94} + (2 \beta_{4} + 4 \beta_{3}) q^{97} - \beta_{2} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 20 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 20 q^{4} + 36 q^{16} - 24 q^{19} + 48 q^{31} - 72 q^{34} - 4 q^{46} - 8 q^{49} - 112 q^{64} - 56 q^{76} - 8 q^{91} + 80 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 3x^{6} + 13x^{4} + 27x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + 4\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{6} + 15\nu^{4} + 32\nu^{2} + 90 ) / 63 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 6\nu^{5} + 31\nu^{3} + 120\nu ) / 63 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4\nu^{7} + 3\nu^{5} - 2\nu^{3} + 18\nu ) / 189 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} + 3\nu^{5} + 13\nu^{3} ) / 27 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} + 3\nu^{4} + 4\nu^{2} + 9 ) / 9 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 8\nu^{6} + 6\nu^{4} + 122\nu^{2} + 99 ) / 63 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{5} + \beta_{4} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - \beta_{6} + \beta_{2} - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{5} - 6\beta_{4} + \beta_{3} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -2\beta_{7} + 3\beta_{6} + 5\beta_{2} - 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{5} - 2\beta_{4} - 2\beta_{3} + 3\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2\beta_{7} + 13\beta_{6} - 19\beta_{2} + 11 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 3\beta_{5} + 90\beta_{4} - \beta_{3} - 5\beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1575\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(1226\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1324.1
1.34095 − 1.09629i
−1.34095 + 1.09629i
0.672201 + 1.59629i
−0.672201 − 1.59629i
−0.672201 + 1.59629i
0.672201 − 1.59629i
−1.34095 − 1.09629i
1.34095 + 1.09629i
− 2.68190i 0 −5.19258 0 0 − 1.00000i 8.56218i 0 0
1324.2 − 2.68190i 0 −5.19258 0 0 1.00000i 8.56218i 0 0
1324.3 − 1.34440i 0 0.192582 0 0 − 1.00000i − 2.94771i 0 0
1324.4 − 1.34440i 0 0.192582 0 0 1.00000i − 2.94771i 0 0
1324.5 1.34440i 0 0.192582 0 0 − 1.00000i 2.94771i 0 0
1324.6 1.34440i 0 0.192582 0 0 1.00000i 2.94771i 0 0
1324.7 2.68190i 0 −5.19258 0 0 − 1.00000i − 8.56218i 0 0
1324.8 2.68190i 0 −5.19258 0 0 1.00000i − 8.56218i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1324.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1575.2.d.l 8
3.b odd 2 1 inner 1575.2.d.l 8
5.b even 2 1 inner 1575.2.d.l 8
5.c odd 4 1 1575.2.a.y ✓ 4
5.c odd 4 1 1575.2.a.z yes 4
15.d odd 2 1 inner 1575.2.d.l 8
15.e even 4 1 1575.2.a.y ✓ 4
15.e even 4 1 1575.2.a.z yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1575.2.a.y ✓ 4 5.c odd 4 1
1575.2.a.y ✓ 4 15.e even 4 1
1575.2.a.z yes 4 5.c odd 4 1
1575.2.a.z yes 4 15.e even 4 1
1575.2.d.l 8 1.a even 1 1 trivial
1575.2.d.l 8 3.b odd 2 1 inner
1575.2.d.l 8 5.b even 2 1 inner
1575.2.d.l 8 15.d 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 \(S_{2}^{\mathrm{new}}(1575, [\chi])\):

\( T_{2}^{4} + 9T_{2}^{2} + 13 \) Copy content Toggle raw display
\( T_{11}^{4} - 42T_{11}^{2} + 325 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 9 T^{2} + 13)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} - 42 T^{2} + 325)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 60 T^{2} + 784)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 36 T^{2} + 208)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 6 T - 20)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + 42 T^{2} + 325)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 74 T^{2} + 325)^{2} \) Copy content Toggle raw display
$31$ \( (T - 6)^{8} \) Copy content Toggle raw display
$37$ \( (T^{2} + 9)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 144 T^{2} + 3328)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 90 T^{2} + 169)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 212 T^{2} + 10192)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 144 T^{2} + 3328)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 212 T^{2} + 10192)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( (T^{4} + 90 T^{2} + 169)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 194 T^{2} + 13)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 60 T^{2} + 784)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 29)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 296 T^{2} + 5200)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 196 T^{2} + 208)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 116)^{4} \) Copy content Toggle raw display
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