Properties

Label 3150.2.b.a
Level $3150$
Weight $2$
Character orbit 3150.b
Analytic conductor $25.153$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3150,2,Mod(251,3150)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3150, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("3150.251");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3150.b (of order \(2\), degree \(1\), 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: \(25.1528766367\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.40960000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 7x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - q^{4} + (\beta_{5} - \beta_{3}) q^{7} - \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - q^{4} + (\beta_{5} - \beta_{3}) q^{7} - \beta_1 q^{8} - \beta_{4} q^{11} + (\beta_{6} - \beta_{5}) q^{13} + ( - \beta_{4} - \beta_{2}) q^{14} + q^{16} + \beta_{2} q^{17} + (\beta_{6} + 2 \beta_{5}) q^{19} + \beta_{3} q^{22} - \beta_1 q^{23} + ( - \beta_{7} + \beta_{2}) q^{26} + ( - \beta_{5} + \beta_{3}) q^{28} + (3 \beta_{4} + 5 \beta_1) q^{29} + ( - 2 \beta_{6} + \beta_{5}) q^{31} + \beta_1 q^{32} + \beta_{5} q^{34} + 2 \beta_{3} q^{37} + ( - \beta_{7} - 2 \beta_{2}) q^{38} + (2 \beta_{7} + \beta_{2}) q^{41} + ( - \beta_{3} - 5) q^{43} + \beta_{4} q^{44} + q^{46} + (\beta_{7} - 2 \beta_{2}) q^{47} + (2 \beta_{6} - 3) q^{49} + ( - \beta_{6} + \beta_{5}) q^{52} + (5 \beta_{4} - \beta_1) q^{53} + (\beta_{4} + \beta_{2}) q^{56} + ( - 3 \beta_{3} - 5) q^{58} + (3 \beta_{7} - \beta_{2}) q^{59} + ( - \beta_{6} + \beta_{5}) q^{61} + (2 \beta_{7} - \beta_{2}) q^{62} - q^{64} - \beta_{2} q^{68} + ( - 4 \beta_{4} + 10 \beta_1) q^{71} + (3 \beta_{6} + 2 \beta_{5}) q^{73} + 2 \beta_{4} q^{74} + ( - \beta_{6} - 2 \beta_{5}) q^{76} + ( - \beta_{7} + 2 \beta_1) q^{77} + (5 \beta_{3} - 6) q^{79} + (2 \beta_{6} + \beta_{5}) q^{82} + ( - \beta_{7} - 5 \beta_{2}) q^{83} + ( - \beta_{4} - 5 \beta_1) q^{86} - \beta_{3} q^{88} + (2 \beta_{7} - 4 \beta_{2}) q^{89} + ( - \beta_{6} + 2 \beta_{5} + 5 \beta_{3} + 5) q^{91} + \beta_1 q^{92} + (\beta_{6} - 2 \beta_{5}) q^{94} + (3 \beta_{6} - 4 \beta_{5}) q^{97} + ( - 2 \beta_{7} - 3 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{4} + 8 q^{16} - 40 q^{43} + 8 q^{46} - 24 q^{49} - 40 q^{58} - 8 q^{64} - 48 q^{79} + 40 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 7x^{4} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} + 8\nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{4} + 7 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{7} - \nu^{5} + 13\nu^{3} - 5\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{7} - \nu^{5} - 13\nu^{3} - 5\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{6} + 6\nu^{2} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 4\nu^{7} + \nu^{5} + 29\nu^{3} + 11\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -4\nu^{7} + \nu^{5} - 29\nu^{3} + 11\nu ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{5} + 3\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{7} + \beta_{6} + 2\beta_{4} - 2\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{2} - 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -5\beta_{7} - 5\beta_{6} - 11\beta_{4} - 11\beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4\beta_{5} - 9\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 13\beta_{7} - 13\beta_{6} - 29\beta_{4} + 29\beta_{3} ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(2801\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1
−0.437016 + 0.437016i
1.14412 1.14412i
0.437016 0.437016i
−1.14412 + 1.14412i
1.14412 + 1.14412i
−0.437016 0.437016i
−1.14412 1.14412i
0.437016 + 0.437016i
1.00000i 0 −1.00000 0 0 −1.41421 2.23607i 1.00000i 0 0
251.2 1.00000i 0 −1.00000 0 0 −1.41421 + 2.23607i 1.00000i 0 0
251.3 1.00000i 0 −1.00000 0 0 1.41421 2.23607i 1.00000i 0 0
251.4 1.00000i 0 −1.00000 0 0 1.41421 + 2.23607i 1.00000i 0 0
251.5 1.00000i 0 −1.00000 0 0 −1.41421 2.23607i 1.00000i 0 0
251.6 1.00000i 0 −1.00000 0 0 −1.41421 + 2.23607i 1.00000i 0 0
251.7 1.00000i 0 −1.00000 0 0 1.41421 2.23607i 1.00000i 0 0
251.8 1.00000i 0 −1.00000 0 0 1.41421 + 2.23607i 1.00000i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 251.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3150.2.b.a 8
3.b odd 2 1 inner 3150.2.b.a 8
5.b even 2 1 3150.2.b.d yes 8
5.c odd 4 1 3150.2.d.b 8
5.c odd 4 1 3150.2.d.e 8
7.b odd 2 1 inner 3150.2.b.a 8
15.d odd 2 1 3150.2.b.d yes 8
15.e even 4 1 3150.2.d.b 8
15.e even 4 1 3150.2.d.e 8
21.c even 2 1 inner 3150.2.b.a 8
35.c odd 2 1 3150.2.b.d yes 8
35.f even 4 1 3150.2.d.b 8
35.f even 4 1 3150.2.d.e 8
105.g even 2 1 3150.2.b.d yes 8
105.k odd 4 1 3150.2.d.b 8
105.k odd 4 1 3150.2.d.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3150.2.b.a 8 1.a even 1 1 trivial
3150.2.b.a 8 3.b odd 2 1 inner
3150.2.b.a 8 7.b odd 2 1 inner
3150.2.b.a 8 21.c even 2 1 inner
3150.2.b.d yes 8 5.b even 2 1
3150.2.b.d yes 8 15.d odd 2 1
3150.2.b.d yes 8 35.c odd 2 1
3150.2.b.d yes 8 105.g even 2 1
3150.2.d.b 8 5.c odd 4 1
3150.2.d.b 8 15.e even 4 1
3150.2.d.b 8 35.f even 4 1
3150.2.d.b 8 105.k odd 4 1
3150.2.d.e 8 5.c odd 4 1
3150.2.d.e 8 15.e even 4 1
3150.2.d.e 8 35.f even 4 1
3150.2.d.e 8 105.k odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3150, [\chi])\):

\( T_{11}^{2} + 2 \) Copy content Toggle raw display
\( T_{13}^{4} + 30T_{13}^{2} + 25 \) Copy content Toggle raw display
\( T_{17}^{2} - 5 \) Copy content Toggle raw display
\( T_{43}^{2} + 10T_{43} + 23 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 6 T^{2} + 49)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 30 T^{2} + 25)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 5)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} + 60 T^{2} + 100)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} + 86 T^{2} + 49)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 90 T^{2} + 1225)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 8)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 90 T^{2} + 1225)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 10 T + 23)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} - 60 T^{2} + 100)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 102 T^{2} + 2401)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 190 T^{2} + 7225)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 30 T^{2} + 25)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( (T^{4} + 264 T^{2} + 4624)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 220 T^{2} + 4900)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 12 T - 14)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 270 T^{2} + 13225)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 240 T^{2} + 1600)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 340 T^{2} + 100)^{2} \) Copy content Toggle raw display
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