Properties

Label 315.2.b.b
Level $315$
Weight $2$
Character orbit 315.b
Analytic conductor $2.515$
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] = [315,2,Mod(251,315)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(315, 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("315.251");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 315.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: \(2.51528766367\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{2} q^{4} + q^{5} + ( - \beta_{3} - \beta_1 + 1) q^{7} + \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{2} q^{4} + q^{5} + ( - \beta_{3} - \beta_1 + 1) q^{7} + \beta_{3} q^{8} + \beta_1 q^{10} + ( - \beta_{3} + 3 \beta_1) q^{11} + ( - 2 \beta_{3} + 4 \beta_1) q^{13} + ( - \beta_{2} + \beta_1 + 3) q^{14} + (2 \beta_{2} - 1) q^{16} - 2 \beta_{2} q^{17} + (2 \beta_{3} - 4 \beta_1) q^{19} + \beta_{2} q^{20} + (3 \beta_{2} - 5) q^{22} + (\beta_{3} - \beta_1) q^{23} + q^{25} + (4 \beta_{2} - 6) q^{26} + ( - 3 \beta_{3} + \beta_{2} + 3 \beta_1) q^{28} + (\beta_{3} - \beta_1) q^{29} + (4 \beta_{3} - 2 \beta_1) q^{31} + (4 \beta_{3} - 5 \beta_1) q^{32} + ( - 2 \beta_{3} + 4 \beta_1) q^{34} + ( - \beta_{3} - \beta_1 + 1) q^{35} + ( - 2 \beta_{2} + 2) q^{37} + ( - 4 \beta_{2} + 6) q^{38} + \beta_{3} q^{40} + 6 \beta_{2} q^{41} + ( - 4 \beta_{2} + 2) q^{43} + (\beta_{3} - 5 \beta_1) q^{44} + ( - \beta_{2} + 1) q^{46} + ( - 2 \beta_{2} - 6) q^{47} + ( - 2 \beta_{3} - 2 \beta_1 - 5) q^{49} + \beta_1 q^{50} - 6 \beta_1 q^{52} + ( - 5 \beta_{3} - \beta_1) q^{53} + ( - \beta_{3} + 3 \beta_1) q^{55} + (\beta_{3} + \beta_{2} + 3) q^{56} + ( - \beta_{2} + 1) q^{58} + ( - 2 \beta_{2} - 6) q^{59} + (8 \beta_{3} - 4 \beta_1) q^{61} - 2 \beta_{2} q^{62} + ( - \beta_{2} + 4) q^{64} + ( - 2 \beta_{3} + 4 \beta_1) q^{65} + ( - 4 \beta_{2} - 4) q^{67} - 6 q^{68} + ( - \beta_{2} + \beta_1 + 3) q^{70} + ( - 7 \beta_{3} - 3 \beta_1) q^{71} + (2 \beta_{3} - 4 \beta_1) q^{73} + ( - 2 \beta_{3} + 6 \beta_1) q^{74} + 6 \beta_1 q^{76} + ( - \beta_{3} - 4 \beta_{2} + 3 \beta_1 + 6) q^{77} + ( - 4 \beta_{2} - 4) q^{79} + (2 \beta_{2} - 1) q^{80} + (6 \beta_{3} - 12 \beta_1) q^{82} + (2 \beta_{2} + 6) q^{83} - 2 \beta_{2} q^{85} + ( - 4 \beta_{3} + 10 \beta_1) q^{86} + (\beta_{2} - 1) q^{88} + ( - 4 \beta_{2} - 6) q^{89} + ( - 2 \beta_{3} - 6 \beta_{2} + 4 \beta_1 + 6) q^{91} + (\beta_{3} + \beta_1) q^{92} + ( - 2 \beta_{3} - 2 \beta_1) q^{94} + (2 \beta_{3} - 4 \beta_1) q^{95} + ( - 10 \beta_{3} + 8 \beta_1) q^{97} + ( - 2 \beta_{2} - 5 \beta_1 + 6) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{5} + 4 q^{7} + 12 q^{14} - 4 q^{16} - 20 q^{22} + 4 q^{25} - 24 q^{26} + 4 q^{35} + 8 q^{37} + 24 q^{38} + 8 q^{43} + 4 q^{46} - 24 q^{47} - 20 q^{49} + 12 q^{56} + 4 q^{58} - 24 q^{59} + 16 q^{64} - 16 q^{67} - 24 q^{68} + 12 q^{70} + 24 q^{77} - 16 q^{79} - 4 q^{80} + 24 q^{83} - 4 q^{88} - 24 q^{89} + 24 q^{91} + 24 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 4x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 4\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 4\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(136\) \(281\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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
1.93185i
0.517638i
0.517638i
1.93185i
1.93185i 0 −1.73205 1.00000 0 1.00000 + 2.44949i 0.517638i 0 1.93185i
251.2 0.517638i 0 1.73205 1.00000 0 1.00000 + 2.44949i 1.93185i 0 0.517638i
251.3 0.517638i 0 1.73205 1.00000 0 1.00000 2.44949i 1.93185i 0 0.517638i
251.4 1.93185i 0 −1.73205 1.00000 0 1.00000 2.44949i 0.517638i 0 1.93185i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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 315.2.b.b yes 4
3.b odd 2 1 315.2.b.a 4
4.b odd 2 1 5040.2.f.c 4
5.b even 2 1 1575.2.b.c 4
5.c odd 4 2 1575.2.g.a 8
7.b odd 2 1 315.2.b.a 4
12.b even 2 1 5040.2.f.a 4
15.d odd 2 1 1575.2.b.b 4
15.e even 4 2 1575.2.g.c 8
21.c even 2 1 inner 315.2.b.b yes 4
28.d even 2 1 5040.2.f.a 4
35.c odd 2 1 1575.2.b.b 4
35.f even 4 2 1575.2.g.c 8
84.h odd 2 1 5040.2.f.c 4
105.g even 2 1 1575.2.b.c 4
105.k odd 4 2 1575.2.g.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.b.a 4 3.b odd 2 1
315.2.b.a 4 7.b odd 2 1
315.2.b.b yes 4 1.a even 1 1 trivial
315.2.b.b yes 4 21.c even 2 1 inner
1575.2.b.b 4 15.d odd 2 1
1575.2.b.b 4 35.c odd 2 1
1575.2.b.c 4 5.b even 2 1
1575.2.b.c 4 105.g even 2 1
1575.2.g.a 8 5.c odd 4 2
1575.2.g.a 8 105.k odd 4 2
1575.2.g.c 8 15.e even 4 2
1575.2.g.c 8 35.f even 4 2
5040.2.f.a 4 12.b even 2 1
5040.2.f.a 4 28.d even 2 1
5040.2.f.c 4 4.b odd 2 1
5040.2.f.c 4 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{47}^{2} + 12T_{47} + 24 \) acting on \(S_{2}^{\mathrm{new}}(315, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 4T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T - 1)^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} - 2 T + 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 28T^{2} + 4 \) Copy content Toggle raw display
$13$ \( T^{4} + 48T^{2} + 144 \) Copy content Toggle raw display
$17$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 48T^{2} + 144 \) Copy content Toggle raw display
$23$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 48T^{2} + 144 \) Copy content Toggle raw display
$37$ \( (T^{2} - 4 T - 8)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 108)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 4 T - 44)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 12 T + 24)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 124T^{2} + 2116 \) Copy content Toggle raw display
$59$ \( (T^{2} + 12 T + 24)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 192T^{2} + 2304 \) Copy content Toggle raw display
$67$ \( (T^{2} + 8 T - 32)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 316 T^{2} + 20164 \) Copy content Toggle raw display
$73$ \( T^{4} + 48T^{2} + 144 \) Copy content Toggle raw display
$79$ \( (T^{2} + 8 T - 32)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 12 T + 24)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 12 T - 12)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 336 T^{2} + 24336 \) Copy content Toggle raw display
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