Properties

Label 315.2.j.a
Level $315$
Weight $2$
Character orbit 315.j
Analytic conductor $2.515$
Analytic rank $0$
Dimension $2$
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(46,315)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(315, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("315.46");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 315.j (of order \(3\), degree \(2\), 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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 \zeta_{6} q^{2} + (2 \zeta_{6} - 2) q^{4} + \zeta_{6} q^{5} + (3 \zeta_{6} - 2) q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q - 2 \zeta_{6} q^{2} + (2 \zeta_{6} - 2) q^{4} + \zeta_{6} q^{5} + (3 \zeta_{6} - 2) q^{7} + ( - 2 \zeta_{6} + 2) q^{10} + (6 \zeta_{6} - 6) q^{11} - 3 q^{13} + ( - 2 \zeta_{6} + 6) q^{14} + 4 \zeta_{6} q^{16} + (4 \zeta_{6} - 4) q^{17} - \zeta_{6} q^{19} - 2 q^{20} + 12 q^{22} - 4 \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{25} + 6 \zeta_{6} q^{26} + ( - 4 \zeta_{6} - 2) q^{28} + 8 q^{29} + (\zeta_{6} - 1) q^{31} + ( - 8 \zeta_{6} + 8) q^{32} + 8 q^{34} + (\zeta_{6} - 3) q^{35} - 7 \zeta_{6} q^{37} + (2 \zeta_{6} - 2) q^{38} + 6 q^{41} + q^{43} - 12 \zeta_{6} q^{44} + (8 \zeta_{6} - 8) q^{46} + 2 \zeta_{6} q^{47} + ( - 3 \zeta_{6} - 5) q^{49} + 2 q^{50} + ( - 6 \zeta_{6} + 6) q^{52} + ( - 4 \zeta_{6} + 4) q^{53} - 6 q^{55} - 16 \zeta_{6} q^{58} + (8 \zeta_{6} - 8) q^{59} + 14 \zeta_{6} q^{61} + 2 q^{62} - 8 q^{64} - 3 \zeta_{6} q^{65} + (7 \zeta_{6} - 7) q^{67} - 8 \zeta_{6} q^{68} + (4 \zeta_{6} + 2) q^{70} - 6 q^{71} + (\zeta_{6} - 1) q^{73} + (14 \zeta_{6} - 14) q^{74} + 2 q^{76} + ( - 12 \zeta_{6} - 6) q^{77} + \zeta_{6} q^{79} + (4 \zeta_{6} - 4) q^{80} - 12 \zeta_{6} q^{82} - 2 q^{83} - 4 q^{85} - 2 \zeta_{6} q^{86} - 12 \zeta_{6} q^{89} + ( - 9 \zeta_{6} + 6) q^{91} + 8 q^{92} + ( - 4 \zeta_{6} + 4) q^{94} + ( - \zeta_{6} + 1) q^{95} - 6 q^{97} + (16 \zeta_{6} - 6) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{4} + q^{5} - q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 2 q^{4} + q^{5} - q^{7} + 2 q^{10} - 6 q^{11} - 6 q^{13} + 10 q^{14} + 4 q^{16} - 4 q^{17} - q^{19} - 4 q^{20} + 24 q^{22} - 4 q^{23} - q^{25} + 6 q^{26} - 8 q^{28} + 16 q^{29} - q^{31} + 8 q^{32} + 16 q^{34} - 5 q^{35} - 7 q^{37} - 2 q^{38} + 12 q^{41} + 2 q^{43} - 12 q^{44} - 8 q^{46} + 2 q^{47} - 13 q^{49} + 4 q^{50} + 6 q^{52} + 4 q^{53} - 12 q^{55} - 16 q^{58} - 8 q^{59} + 14 q^{61} + 4 q^{62} - 16 q^{64} - 3 q^{65} - 7 q^{67} - 8 q^{68} + 8 q^{70} - 12 q^{71} - q^{73} - 14 q^{74} + 4 q^{76} - 24 q^{77} + q^{79} - 4 q^{80} - 12 q^{82} - 4 q^{83} - 8 q^{85} - 2 q^{86} - 12 q^{89} + 3 q^{91} + 16 q^{92} + 4 q^{94} + q^{95} - 12 q^{97} + 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(136\) \(281\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
46.1
0.500000 + 0.866025i
0.500000 0.866025i
−1.00000 1.73205i 0 −1.00000 + 1.73205i 0.500000 + 0.866025i 0 −0.500000 + 2.59808i 0 0 1.00000 1.73205i
226.1 −1.00000 + 1.73205i 0 −1.00000 1.73205i 0.500000 0.866025i 0 −0.500000 2.59808i 0 0 1.00000 + 1.73205i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.j.a 2
3.b odd 2 1 105.2.i.b 2
7.c even 3 1 inner 315.2.j.a 2
7.c even 3 1 2205.2.a.k 1
7.d odd 6 1 2205.2.a.m 1
12.b even 2 1 1680.2.bg.l 2
15.d odd 2 1 525.2.i.a 2
15.e even 4 2 525.2.r.d 4
21.c even 2 1 735.2.i.f 2
21.g even 6 1 735.2.a.a 1
21.g even 6 1 735.2.i.f 2
21.h odd 6 1 105.2.i.b 2
21.h odd 6 1 735.2.a.b 1
84.n even 6 1 1680.2.bg.l 2
105.o odd 6 1 525.2.i.a 2
105.o odd 6 1 3675.2.a.o 1
105.p even 6 1 3675.2.a.p 1
105.x even 12 2 525.2.r.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.i.b 2 3.b odd 2 1
105.2.i.b 2 21.h odd 6 1
315.2.j.a 2 1.a even 1 1 trivial
315.2.j.a 2 7.c even 3 1 inner
525.2.i.a 2 15.d odd 2 1
525.2.i.a 2 105.o odd 6 1
525.2.r.d 4 15.e even 4 2
525.2.r.d 4 105.x even 12 2
735.2.a.a 1 21.g even 6 1
735.2.a.b 1 21.h odd 6 1
735.2.i.f 2 21.c even 2 1
735.2.i.f 2 21.g even 6 1
1680.2.bg.l 2 12.b even 2 1
1680.2.bg.l 2 84.n even 6 1
2205.2.a.k 1 7.c even 3 1
2205.2.a.m 1 7.d odd 6 1
3675.2.a.o 1 105.o odd 6 1
3675.2.a.p 1 105.p even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 2T_{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(315, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} + T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$13$ \( (T + 3)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$19$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$23$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$29$ \( (T - 8)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$37$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$41$ \( (T - 6)^{2} \) Copy content Toggle raw display
$43$ \( (T - 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$53$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$59$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$61$ \( T^{2} - 14T + 196 \) Copy content Toggle raw display
$67$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$71$ \( (T + 6)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$79$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$83$ \( (T + 2)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$97$ \( (T + 6)^{2} \) Copy content Toggle raw display
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