Properties

Label 315.2.p.c
Level $315$
Weight $2$
Character orbit 315.p
Analytic conductor $2.515$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [315,2,Mod(118,315)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(315, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("315.118");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 315.p (of order \(4\), 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 \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_{3} \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(136\) \(281\)
\(\chi(n)\) \(\beta_{2}\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
118.1
−1.58114 + 1.58114i
1.58114 1.58114i
−1.58114 1.58114i
1.58114 + 1.58114i
1.00000 1.00000i 0 0 −1.58114 + 1.58114i 0 2.58114 + 0.581139i 2.00000 + 2.00000i 0 3.16228i
118.2 1.00000 1.00000i 0 0 1.58114 1.58114i 0 −0.581139 2.58114i 2.00000 + 2.00000i 0 3.16228i
307.1 1.00000 + 1.00000i 0 0 −1.58114 1.58114i 0 2.58114 0.581139i 2.00000 2.00000i 0 3.16228i
307.2 1.00000 + 1.00000i 0 0 1.58114 + 1.58114i 0 −0.581139 + 2.58114i 2.00000 2.00000i 0 3.16228i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.b odd 2 1 inner
35.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.p.c 4
3.b odd 2 1 35.2.f.a 4
5.c odd 4 1 inner 315.2.p.c 4
7.b odd 2 1 inner 315.2.p.c 4
12.b even 2 1 560.2.bj.a 4
15.d odd 2 1 175.2.f.c 4
15.e even 4 1 35.2.f.a 4
15.e even 4 1 175.2.f.c 4
21.c even 2 1 35.2.f.a 4
21.g even 6 2 245.2.l.c 8
21.h odd 6 2 245.2.l.c 8
35.f even 4 1 inner 315.2.p.c 4
60.l odd 4 1 560.2.bj.a 4
84.h odd 2 1 560.2.bj.a 4
105.g even 2 1 175.2.f.c 4
105.k odd 4 1 35.2.f.a 4
105.k odd 4 1 175.2.f.c 4
105.w odd 12 2 245.2.l.c 8
105.x even 12 2 245.2.l.c 8
420.w even 4 1 560.2.bj.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.f.a 4 3.b odd 2 1
35.2.f.a 4 15.e even 4 1
35.2.f.a 4 21.c even 2 1
35.2.f.a 4 105.k odd 4 1
175.2.f.c 4 15.d odd 2 1
175.2.f.c 4 15.e even 4 1
175.2.f.c 4 105.g even 2 1
175.2.f.c 4 105.k odd 4 1
245.2.l.c 8 21.g even 6 2
245.2.l.c 8 21.h odd 6 2
245.2.l.c 8 105.w odd 12 2
245.2.l.c 8 105.x even 12 2
315.2.p.c 4 1.a even 1 1 trivial
315.2.p.c 4 5.c odd 4 1 inner
315.2.p.c 4 7.b odd 2 1 inner
315.2.p.c 4 35.f even 4 1 inner
560.2.bj.a 4 12.b even 2 1
560.2.bj.a 4 60.l odd 4 1
560.2.bj.a 4 84.h odd 2 1
560.2.bj.a 4 420.w even 4 1

Hecke kernels

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

\( T_{2}^{2} - 2T_{2} + 2 \) Copy content Toggle raw display
\( T_{17}^{4} + 25 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 25 \) Copy content Toggle raw display
$7$ \( T^{4} - 4 T^{3} + 8 T^{2} - 28 T + 49 \) Copy content Toggle raw display
$11$ \( (T - 1)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 25 \) Copy content Toggle raw display
$17$ \( T^{4} + 25 \) Copy content Toggle raw display
$19$ \( (T^{2} - 10)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 4 T + 8)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 10)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 12 T + 72)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 90)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 6 T + 18)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 2025 \) Copy content Toggle raw display
$53$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 90)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 40)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$71$ \( (T - 6)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 169)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 400 \) Copy content Toggle raw display
$89$ \( (T^{2} - 40)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 25 \) Copy content Toggle raw display
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