Properties

Label 315.2.j.c
Level $315$
Weight $2$
Character orbit 315.j
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(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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12}) q^{2} + ( - 4 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 2 \zeta_{12} - 2) q^{4} + \zeta_{12}^{2} q^{5} + (3 \zeta_{12}^{3} - \zeta_{12}) q^{7} + (2 \zeta_{12}^{3} - 4 \zeta_{12} + 6) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12}) q^{2} + ( - 4 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 2 \zeta_{12} - 2) q^{4} + \zeta_{12}^{2} q^{5} + (3 \zeta_{12}^{3} - \zeta_{12}) q^{7} + (2 \zeta_{12}^{3} - 4 \zeta_{12} + 6) q^{8} + (2 \zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12} + 1) q^{10} + (2 \zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12} - 1) q^{11} + ( - \zeta_{12}^{3} + 2 \zeta_{12} + 4) q^{13} + ( - 2 \zeta_{12}^{3} + \zeta_{12}^{2} + 3 \zeta_{12} - 5) q^{14} + (4 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 4 \zeta_{12}) q^{16} + ( - 2 \zeta_{12}^{3} - 5 \zeta_{12}^{2} + \zeta_{12} + 5) q^{17} + (2 \zeta_{12}^{3} - \zeta_{12}^{2} + 2 \zeta_{12}) q^{19} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12} - 2) q^{20} - 2 q^{22} + (\zeta_{12}^{3} - 3 \zeta_{12}^{2} + \zeta_{12}) q^{23} + (\zeta_{12}^{2} - 1) q^{25} + (3 \zeta_{12}^{3} - \zeta_{12}^{2} + 3 \zeta_{12}) q^{26} + ( - 2 \zeta_{12}^{3} + 8 \zeta_{12}^{2} - 4 \zeta_{12} + 2) q^{28} + (3 \zeta_{12}^{3} - 6 \zeta_{12} - 1) q^{29} + (4 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 2 \zeta_{12} - 3) q^{31} + ( - 16 \zeta_{12}^{3} + 8 \zeta_{12}^{2} + 8 \zeta_{12} - 8) q^{32} + ( - 4 \zeta_{12}^{3} + 8 \zeta_{12} - 2) q^{34} + (2 \zeta_{12}^{3} - 3 \zeta_{12}) q^{35} + ( - 3 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 3 \zeta_{12}) q^{37} + ( - 6 \zeta_{12}^{3} + 7 \zeta_{12}^{2} + 3 \zeta_{12} - 7) q^{38} + ( - 2 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 2 \zeta_{12}) q^{40} + (\zeta_{12}^{3} - 2 \zeta_{12} - 1) q^{41} + (3 \zeta_{12}^{3} - 6 \zeta_{12} - 2) q^{43} + 4 \zeta_{12}^{2} q^{44} + ( - 8 \zeta_{12}^{3} + 6 \zeta_{12}^{2} + 4 \zeta_{12} - 6) q^{46} + 2 \zeta_{12}^{2} q^{47} + ( - 5 \zeta_{12}^{2} - 3) q^{49} + (\zeta_{12}^{3} - 2 \zeta_{12} + 1) q^{50} + ( - 12 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 6 \zeta_{12} - 2) q^{52} + (12 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 6 \zeta_{12} + 2) q^{53} + (\zeta_{12}^{3} - 2 \zeta_{12} - 1) q^{55} + (18 \zeta_{12}^{3} - 10 \zeta_{12}^{2} - 6 \zeta_{12} + 8) q^{56} + (2 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 2 \zeta_{12}) q^{58} + ( - 6 \zeta_{12}^{3} - 5 \zeta_{12}^{2} + 3 \zeta_{12} + 5) q^{59} - 4 \zeta_{12}^{2} q^{61} + (\zeta_{12}^{3} - 2 \zeta_{12} - 3) q^{62} + (8 \zeta_{12}^{3} - 16 \zeta_{12} + 16) q^{64} + (\zeta_{12}^{3} + 4 \zeta_{12}^{2} + \zeta_{12}) q^{65} + (10 \zeta_{12}^{3} - 6 \zeta_{12}^{2} - 5 \zeta_{12} + 6) q^{67} + ( - 8 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 8 \zeta_{12}) q^{68} + (\zeta_{12}^{3} - 4 \zeta_{12}^{2} + 2 \zeta_{12} - 1) q^{70} + ( - 3 \zeta_{12}^{3} + 6 \zeta_{12} - 1) q^{71} + ( - 10 \zeta_{12}^{3} + 4 \zeta_{12}^{2} + 5 \zeta_{12} - 4) q^{73} + (2 \zeta_{12}^{3} - 7 \zeta_{12}^{2} - \zeta_{12} + 7) q^{74} + (6 \zeta_{12}^{3} - 12 \zeta_{12} + 14) q^{76} + ( - \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 2 \zeta_{12} - 1) q^{77} + ( - 6 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 6 \zeta_{12}) q^{79} + (8 \zeta_{12}^{3} - 8 \zeta_{12}^{2} - 4 \zeta_{12} + 8) q^{80} - 2 \zeta_{12}^{2} q^{82} + ( - 7 \zeta_{12}^{3} + 14 \zeta_{12} - 3) q^{83} + ( - \zeta_{12}^{3} + 2 \zeta_{12} + 5) q^{85} + (\zeta_{12}^{3} - 7 \zeta_{12}^{2} + \zeta_{12}) q^{86} + (8 \zeta_{12}^{3} - 4 \zeta_{12}) q^{88} + ( - 7 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 7 \zeta_{12}) q^{89} + (12 \zeta_{12}^{3} + 5 \zeta_{12}^{2} - 4 \zeta_{12} - 4) q^{91} + (8 \zeta_{12}^{3} - 16 \zeta_{12} + 12) q^{92} + (4 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12} + 2) q^{94} + (4 \zeta_{12}^{3} - \zeta_{12}^{2} - 2 \zeta_{12} + 1) q^{95} + (4 \zeta_{12}^{3} - 8 \zeta_{12} + 8) q^{97} + ( - 13 \zeta_{12}^{3} + 8 \zeta_{12}^{2} + 2 \zeta_{12} - 5) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 4 q^{4} + 2 q^{5} + 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} - 4 q^{4} + 2 q^{5} + 24 q^{8} + 2 q^{10} - 2 q^{11} + 16 q^{13} - 18 q^{14} - 16 q^{16} + 10 q^{17} - 2 q^{19} - 8 q^{20} - 8 q^{22} - 6 q^{23} - 2 q^{25} - 2 q^{26} + 24 q^{28} - 4 q^{29} - 6 q^{31} - 16 q^{32} - 8 q^{34} - 4 q^{37} - 14 q^{38} + 12 q^{40} - 4 q^{41} - 8 q^{43} + 8 q^{44} - 12 q^{46} + 4 q^{47} - 22 q^{49} + 4 q^{50} - 4 q^{52} + 4 q^{53} - 4 q^{55} + 12 q^{56} - 16 q^{58} + 10 q^{59} - 8 q^{61} - 12 q^{62} + 64 q^{64} + 8 q^{65} + 12 q^{67} + 8 q^{68} - 12 q^{70} - 4 q^{71} - 8 q^{73} + 14 q^{74} + 56 q^{76} - 12 q^{77} - 6 q^{79} + 16 q^{80} - 4 q^{82} - 12 q^{83} + 20 q^{85} - 14 q^{86} + 6 q^{89} - 6 q^{91} + 48 q^{92} + 4 q^{94} + 2 q^{95} + 32 q^{97} - 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(136\) \(281\)
\(\chi(n)\) \(1\) \(-\zeta_{12}^{2}\) \(1\)

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
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
−1.36603 2.36603i 0 −2.73205 + 4.73205i 0.500000 + 0.866025i 0 0.866025 2.50000i 9.46410 0 1.36603 2.36603i
46.2 0.366025 + 0.633975i 0 0.732051 1.26795i 0.500000 + 0.866025i 0 −0.866025 + 2.50000i 2.53590 0 −0.366025 + 0.633975i
226.1 −1.36603 + 2.36603i 0 −2.73205 4.73205i 0.500000 0.866025i 0 0.866025 + 2.50000i 9.46410 0 1.36603 + 2.36603i
226.2 0.366025 0.633975i 0 0.732051 + 1.26795i 0.500000 0.866025i 0 −0.866025 2.50000i 2.53590 0 −0.366025 0.633975i
\(n\): 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.c 4
3.b odd 2 1 105.2.i.d 4
7.c even 3 1 inner 315.2.j.c 4
7.c even 3 1 2205.2.a.z 2
7.d odd 6 1 2205.2.a.ba 2
12.b even 2 1 1680.2.bg.o 4
15.d odd 2 1 525.2.i.f 4
15.e even 4 1 525.2.r.a 4
15.e even 4 1 525.2.r.f 4
21.c even 2 1 735.2.i.l 4
21.g even 6 1 735.2.a.h 2
21.g even 6 1 735.2.i.l 4
21.h odd 6 1 105.2.i.d 4
21.h odd 6 1 735.2.a.g 2
84.n even 6 1 1680.2.bg.o 4
105.o odd 6 1 525.2.i.f 4
105.o odd 6 1 3675.2.a.bg 2
105.p even 6 1 3675.2.a.be 2
105.x even 12 1 525.2.r.a 4
105.x even 12 1 525.2.r.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.i.d 4 3.b odd 2 1
105.2.i.d 4 21.h odd 6 1
315.2.j.c 4 1.a even 1 1 trivial
315.2.j.c 4 7.c even 3 1 inner
525.2.i.f 4 15.d odd 2 1
525.2.i.f 4 105.o odd 6 1
525.2.r.a 4 15.e even 4 1
525.2.r.a 4 105.x even 12 1
525.2.r.f 4 15.e even 4 1
525.2.r.f 4 105.x even 12 1
735.2.a.g 2 21.h odd 6 1
735.2.a.h 2 21.g even 6 1
735.2.i.l 4 21.c even 2 1
735.2.i.l 4 21.g even 6 1
1680.2.bg.o 4 12.b even 2 1
1680.2.bg.o 4 84.n even 6 1
2205.2.a.z 2 7.c even 3 1
2205.2.a.ba 2 7.d odd 6 1
3675.2.a.be 2 105.p even 6 1
3675.2.a.bg 2 105.o odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 2T_{2}^{3} + 6T_{2}^{2} - 4T_{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(315, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 2 T^{3} + 6 T^{2} - 4 T + 4 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 11T^{2} + 49 \) Copy content Toggle raw display
$11$ \( T^{4} + 2 T^{3} + 6 T^{2} - 4 T + 4 \) Copy content Toggle raw display
$13$ \( (T^{2} - 8 T + 13)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 10 T^{3} + 78 T^{2} + \cdots + 484 \) Copy content Toggle raw display
$19$ \( T^{4} + 2 T^{3} + 15 T^{2} - 22 T + 121 \) Copy content Toggle raw display
$23$ \( T^{4} + 6 T^{3} + 30 T^{2} + 36 T + 36 \) Copy content Toggle raw display
$29$ \( (T^{2} + 2 T - 26)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 6 T^{3} + 39 T^{2} - 18 T + 9 \) Copy content Toggle raw display
$37$ \( T^{4} + 4 T^{3} + 39 T^{2} - 92 T + 529 \) Copy content Toggle raw display
$41$ \( (T^{2} + 2 T - 2)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 4 T - 23)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} - 4 T^{3} + 120 T^{2} + \cdots + 10816 \) Copy content Toggle raw display
$59$ \( T^{4} - 10 T^{3} + 102 T^{2} + 20 T + 4 \) Copy content Toggle raw display
$61$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 12 T^{3} + 183 T^{2} + \cdots + 1521 \) Copy content Toggle raw display
$71$ \( (T^{2} + 2 T - 26)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 8 T^{3} + 123 T^{2} + \cdots + 3481 \) Copy content Toggle raw display
$79$ \( T^{4} + 6 T^{3} + 135 T^{2} + \cdots + 9801 \) Copy content Toggle raw display
$83$ \( (T^{2} + 6 T - 138)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 6 T^{3} + 174 T^{2} + \cdots + 19044 \) Copy content Toggle raw display
$97$ \( (T^{2} - 16 T + 16)^{2} \) Copy content Toggle raw display
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