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Properties

Label 315.2.d.e
Level $315$
Weight $2$
Character orbit 315.d
Analytic conductor $2.515$
Analytic rank $0$
Dimension $6$
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(64,315)]
 
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([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("315.64");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 315.d (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: \(6\)
Coefficient field: 6.0.350464.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 105)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{5} + \beta_{3} - 1) q^{4} + (\beta_{3} - \beta_1) q^{5} + \beta_{4} q^{7} + ( - \beta_{5} - 2 \beta_{4} + \cdots + \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{5} + \beta_{3} - 1) q^{4} + (\beta_{3} - \beta_1) q^{5} + \beta_{4} q^{7} + ( - \beta_{5} - 2 \beta_{4} + \cdots + \beta_1) q^{8}+ \cdots + \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 10 q^{4} - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 10 q^{4} - 2 q^{5} - 12 q^{10} - 12 q^{11} - 2 q^{14} + 26 q^{16} - 12 q^{19} + 30 q^{20} - 2 q^{25} - 20 q^{26} + 4 q^{29} + 4 q^{31} + 24 q^{34} + 4 q^{40} - 4 q^{41} + 20 q^{44} - 16 q^{46} - 6 q^{49} + 16 q^{50} + 4 q^{55} + 18 q^{56} + 32 q^{59} - 12 q^{61} - 26 q^{64} - 32 q^{65} + 10 q^{70} - 12 q^{71} - 88 q^{74} + 4 q^{76} - 24 q^{79} - 46 q^{80} + 32 q^{85} + 8 q^{86} + 28 q^{89} + 12 q^{91} + 32 q^{94} - 4 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -3\nu^{5} + \nu^{4} + 11\nu^{3} - 26\nu^{2} + 6\nu - 1 ) / 23 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -4\nu^{5} + 9\nu^{4} - 16\nu^{3} - 4\nu^{2} + 8\nu - 9 ) / 23 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 6\nu^{5} - 2\nu^{4} + \nu^{3} + 6\nu^{2} + 80\nu + 2 ) / 23 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7\nu^{5} - 10\nu^{4} + 5\nu^{3} + 30\nu^{2} + 32\nu - 13 ) / 23 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -16\nu^{5} + 36\nu^{4} - 41\nu^{3} - 16\nu^{2} - 60\nu + 56 ) / 23 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} + \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + 4\beta_{4} - \beta_{3} + 2\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + 2\beta_{4} - 2\beta_{2} + 2\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} + 2\beta_{3} - 5\beta_{2} - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -9\beta_{4} + 5\beta_{3} - 8\beta_{2} - 8\beta _1 - 9 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1
−0.854638 + 0.854638i
1.45161 + 1.45161i
0.403032 − 0.403032i
0.403032 + 0.403032i
1.45161 − 1.45161i
−0.854638 − 0.854638i
− 2.70928i 0 −5.34017 −2.17009 − 0.539189i 0 − 1.00000i 9.04945i 0 −1.46081 + 5.87936i
64.2 − 1.90321i 0 −1.62222 −0.311108 − 2.21432i 0 1.00000i − 0.719004i 0 −4.21432 + 0.592104i
64.3 − 0.193937i 0 1.96239 1.48119 − 1.67513i 0 − 1.00000i − 0.768452i 0 −0.324869 − 0.287258i
64.4 0.193937i 0 1.96239 1.48119 + 1.67513i 0 1.00000i 0.768452i 0 −0.324869 + 0.287258i
64.5 1.90321i 0 −1.62222 −0.311108 + 2.21432i 0 − 1.00000i 0.719004i 0 −4.21432 − 0.592104i
64.6 2.70928i 0 −5.34017 −2.17009 + 0.539189i 0 1.00000i − 9.04945i 0 −1.46081 − 5.87936i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 64.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b 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.d.e 6
3.b odd 2 1 105.2.d.b ✓ 6
4.b odd 2 1 5040.2.t.v 6
5.b even 2 1 inner 315.2.d.e 6
5.c odd 4 1 1575.2.a.w 3
5.c odd 4 1 1575.2.a.x 3
7.b odd 2 1 2205.2.d.l 6
12.b even 2 1 1680.2.t.k 6
15.d odd 2 1 105.2.d.b ✓ 6
15.e even 4 1 525.2.a.j 3
15.e even 4 1 525.2.a.k 3
20.d odd 2 1 5040.2.t.v 6
21.c even 2 1 735.2.d.b 6
21.g even 6 2 735.2.q.f 12
21.h odd 6 2 735.2.q.e 12
35.c odd 2 1 2205.2.d.l 6
60.h even 2 1 1680.2.t.k 6
60.l odd 4 1 8400.2.a.dg 3
60.l odd 4 1 8400.2.a.dj 3
105.g even 2 1 735.2.d.b 6
105.k odd 4 1 3675.2.a.bi 3
105.k odd 4 1 3675.2.a.bj 3
105.o odd 6 2 735.2.q.e 12
105.p even 6 2 735.2.q.f 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.d.b ✓ 6 3.b odd 2 1
105.2.d.b ✓ 6 15.d odd 2 1
315.2.d.e 6 1.a even 1 1 trivial
315.2.d.e 6 5.b even 2 1 inner
525.2.a.j 3 15.e even 4 1
525.2.a.k 3 15.e even 4 1
735.2.d.b 6 21.c even 2 1
735.2.d.b 6 105.g even 2 1
735.2.q.e 12 21.h odd 6 2
735.2.q.e 12 105.o odd 6 2
735.2.q.f 12 21.g even 6 2
735.2.q.f 12 105.p even 6 2
1575.2.a.w 3 5.c odd 4 1
1575.2.a.x 3 5.c odd 4 1
1680.2.t.k 6 12.b even 2 1
1680.2.t.k 6 60.h even 2 1
2205.2.d.l 6 7.b odd 2 1
2205.2.d.l 6 35.c odd 2 1
3675.2.a.bi 3 105.k odd 4 1
3675.2.a.bj 3 105.k odd 4 1
5040.2.t.v 6 4.b odd 2 1
5040.2.t.v 6 20.d odd 2 1
8400.2.a.dg 3 60.l odd 4 1
8400.2.a.dj 3 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(315, [\chi])\):

\( T_{2}^{6} + 11T_{2}^{4} + 27T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{11} + 2 \) Copy content Toggle raw display
\( T_{29}^{3} - 2T_{29}^{2} - 52T_{29} + 40 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 11 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + 2 T^{5} + \cdots + 125 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$11$ \( (T + 2)^{6} \) Copy content Toggle raw display
$13$ \( T^{6} + 44 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$17$ \( T^{6} + 32 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$19$ \( (T^{3} + 6 T^{2} - 4 T - 40)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 32 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$29$ \( (T^{3} - 2 T^{2} - 52 T + 40)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 2 T^{2} + \cdots + 184)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 176 T^{4} + \cdots + 4096 \) Copy content Toggle raw display
$41$ \( (T^{3} + 2 T^{2} + \cdots - 200)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 304 T^{4} + \cdots + 692224 \) Copy content Toggle raw display
$47$ \( T^{6} + 128 T^{4} + \cdots + 16384 \) Copy content Toggle raw display
$53$ \( T^{6} + 172 T^{4} + \cdots + 87616 \) Copy content Toggle raw display
$59$ \( (T^{3} - 16 T^{2} + \cdots + 1280)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 6 T^{2} + \cdots - 248)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 128 T^{4} + \cdots + 16384 \) Copy content Toggle raw display
$71$ \( (T + 2)^{6} \) Copy content Toggle raw display
$73$ \( T^{6} + 140 T^{4} + \cdots + 10816 \) Copy content Toggle raw display
$79$ \( (T^{3} + 12 T^{2} + \cdots - 320)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 192 T^{4} + \cdots + 65536 \) Copy content Toggle raw display
$89$ \( (T^{3} - 14 T^{2} + \cdots - 40)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 556 T^{4} + \cdots + 3474496 \) Copy content Toggle raw display
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