Properties

Label 315.2.d.a
Level $315$
Weight $2$
Character orbit 315.d
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(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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

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

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
1.00000i
1.00000i
2.00000i 0 −2.00000 2.00000 + 1.00000i 0 1.00000i 0 0 2.00000 4.00000i
64.2 2.00000i 0 −2.00000 2.00000 1.00000i 0 1.00000i 0 0 2.00000 + 4.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.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.a 2
3.b odd 2 1 35.2.b.a 2
4.b odd 2 1 5040.2.t.p 2
5.b even 2 1 inner 315.2.d.a 2
5.c odd 4 1 1575.2.a.a 1
5.c odd 4 1 1575.2.a.k 1
7.b odd 2 1 2205.2.d.b 2
12.b even 2 1 560.2.g.b 2
15.d odd 2 1 35.2.b.a 2
15.e even 4 1 175.2.a.a 1
15.e even 4 1 175.2.a.c 1
20.d odd 2 1 5040.2.t.p 2
21.c even 2 1 245.2.b.a 2
21.g even 6 2 245.2.j.d 4
21.h odd 6 2 245.2.j.e 4
24.f even 2 1 2240.2.g.g 2
24.h odd 2 1 2240.2.g.h 2
35.c odd 2 1 2205.2.d.b 2
60.h even 2 1 560.2.g.b 2
60.l odd 4 1 2800.2.a.l 1
60.l odd 4 1 2800.2.a.w 1
105.g even 2 1 245.2.b.a 2
105.k odd 4 1 1225.2.a.a 1
105.k odd 4 1 1225.2.a.i 1
105.o odd 6 2 245.2.j.e 4
105.p even 6 2 245.2.j.d 4
120.i odd 2 1 2240.2.g.h 2
120.m even 2 1 2240.2.g.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.b.a 2 3.b odd 2 1
35.2.b.a 2 15.d odd 2 1
175.2.a.a 1 15.e even 4 1
175.2.a.c 1 15.e even 4 1
245.2.b.a 2 21.c even 2 1
245.2.b.a 2 105.g even 2 1
245.2.j.d 4 21.g even 6 2
245.2.j.d 4 105.p even 6 2
245.2.j.e 4 21.h odd 6 2
245.2.j.e 4 105.o odd 6 2
315.2.d.a 2 1.a even 1 1 trivial
315.2.d.a 2 5.b even 2 1 inner
560.2.g.b 2 12.b even 2 1
560.2.g.b 2 60.h even 2 1
1225.2.a.a 1 105.k odd 4 1
1225.2.a.i 1 105.k odd 4 1
1575.2.a.a 1 5.c odd 4 1
1575.2.a.k 1 5.c odd 4 1
2205.2.d.b 2 7.b odd 2 1
2205.2.d.b 2 35.c odd 2 1
2240.2.g.g 2 24.f even 2 1
2240.2.g.g 2 120.m even 2 1
2240.2.g.h 2 24.h odd 2 1
2240.2.g.h 2 120.i odd 2 1
2800.2.a.l 1 60.l odd 4 1
2800.2.a.w 1 60.l odd 4 1
5040.2.t.p 2 4.b odd 2 1
5040.2.t.p 2 20.d odd 2 1

Hecke kernels

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

\( T_{2}^{2} + 4 \) Copy content Toggle raw display
\( T_{11} - 3 \) Copy content Toggle raw display
\( T_{29} + 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 4 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 4T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} + 1 \) Copy content Toggle raw display
$11$ \( (T - 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 1 \) Copy content Toggle raw display
$17$ \( T^{2} + 49 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 36 \) Copy content Toggle raw display
$29$ \( (T + 5)^{2} \) Copy content Toggle raw display
$31$ \( (T - 2)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 4 \) Copy content Toggle raw display
$41$ \( (T + 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 16 \) Copy content Toggle raw display
$47$ \( T^{2} + 9 \) Copy content Toggle raw display
$53$ \( T^{2} + 36 \) Copy content Toggle raw display
$59$ \( (T - 10)^{2} \) Copy content Toggle raw display
$61$ \( (T + 8)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 4 \) Copy content Toggle raw display
$71$ \( (T - 8)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 36 \) Copy content Toggle raw display
$79$ \( (T - 5)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 16 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 49 \) Copy content Toggle raw display
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