Properties

Label 1815.2.a.d
Level $1815$
Weight $2$
Character orbit 1815.a
Self dual yes
Analytic conductor $14.493$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1815,2,Mod(1,1815)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1815, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1815.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1815.a (trivial)

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: yes
Analytic conductor: \(14.4928479669\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + q^{2} - q^{3} - q^{4} + q^{5} - q^{6} - 3 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - q^{3} - q^{4} + q^{5} - q^{6} - 3 q^{8} + q^{9} + q^{10} + q^{12} + 2 q^{13} - q^{15} - q^{16} - 2 q^{17} + q^{18} - 4 q^{19} - q^{20} + 3 q^{24} + q^{25} + 2 q^{26} - q^{27} + 2 q^{29} - q^{30} + 5 q^{32} - 2 q^{34} - q^{36} - 10 q^{37} - 4 q^{38} - 2 q^{39} - 3 q^{40} - 10 q^{41} - 4 q^{43} + q^{45} + 8 q^{47} + q^{48} - 7 q^{49} + q^{50} + 2 q^{51} - 2 q^{52} - 10 q^{53} - q^{54} + 4 q^{57} + 2 q^{58} - 4 q^{59} + q^{60} + 2 q^{61} + 7 q^{64} + 2 q^{65} + 12 q^{67} + 2 q^{68} - 8 q^{71} - 3 q^{72} - 10 q^{73} - 10 q^{74} - q^{75} + 4 q^{76} - 2 q^{78} - q^{80} + q^{81} - 10 q^{82} - 12 q^{83} - 2 q^{85} - 4 q^{86} - 2 q^{87} - 6 q^{89} + q^{90} + 8 q^{94} - 4 q^{95} - 5 q^{96} + 2 q^{97} - 7 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
1.00000 −1.00000 −1.00000 1.00000 −1.00000 0 −3.00000 1.00000 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(-1\)
\(11\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1815.2.a.d 1
3.b odd 2 1 5445.2.a.c 1
5.b even 2 1 9075.2.a.g 1
11.b odd 2 1 15.2.a.a 1
33.d even 2 1 45.2.a.a 1
44.c even 2 1 240.2.a.d 1
55.d odd 2 1 75.2.a.b 1
55.e even 4 2 75.2.b.b 2
77.b even 2 1 735.2.a.c 1
77.h odd 6 2 735.2.i.e 2
77.i even 6 2 735.2.i.d 2
88.b odd 2 1 960.2.a.l 1
88.g even 2 1 960.2.a.a 1
99.g even 6 2 405.2.e.c 2
99.h odd 6 2 405.2.e.f 2
132.d odd 2 1 720.2.a.c 1
143.d odd 2 1 2535.2.a.j 1
165.d even 2 1 225.2.a.b 1
165.l odd 4 2 225.2.b.b 2
176.i even 4 2 3840.2.k.r 2
176.l odd 4 2 3840.2.k.m 2
187.b odd 2 1 4335.2.a.c 1
209.d even 2 1 5415.2.a.j 1
220.g even 2 1 1200.2.a.e 1
220.i odd 4 2 1200.2.f.h 2
231.h odd 2 1 2205.2.a.i 1
253.b even 2 1 7935.2.a.d 1
264.m even 2 1 2880.2.a.y 1
264.p odd 2 1 2880.2.a.bc 1
385.h even 2 1 3675.2.a.j 1
429.e even 2 1 7605.2.a.g 1
440.c even 2 1 4800.2.a.bz 1
440.o odd 2 1 4800.2.a.t 1
440.t even 4 2 4800.2.f.bf 2
440.w odd 4 2 4800.2.f.c 2
660.g odd 2 1 3600.2.a.u 1
660.q even 4 2 3600.2.f.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.2.a.a 1 11.b odd 2 1
45.2.a.a 1 33.d even 2 1
75.2.a.b 1 55.d odd 2 1
75.2.b.b 2 55.e even 4 2
225.2.a.b 1 165.d even 2 1
225.2.b.b 2 165.l odd 4 2
240.2.a.d 1 44.c even 2 1
405.2.e.c 2 99.g even 6 2
405.2.e.f 2 99.h odd 6 2
720.2.a.c 1 132.d odd 2 1
735.2.a.c 1 77.b even 2 1
735.2.i.d 2 77.i even 6 2
735.2.i.e 2 77.h odd 6 2
960.2.a.a 1 88.g even 2 1
960.2.a.l 1 88.b odd 2 1
1200.2.a.e 1 220.g even 2 1
1200.2.f.h 2 220.i odd 4 2
1815.2.a.d 1 1.a even 1 1 trivial
2205.2.a.i 1 231.h odd 2 1
2535.2.a.j 1 143.d odd 2 1
2880.2.a.y 1 264.m even 2 1
2880.2.a.bc 1 264.p odd 2 1
3600.2.a.u 1 660.g odd 2 1
3600.2.f.e 2 660.q even 4 2
3675.2.a.j 1 385.h even 2 1
3840.2.k.m 2 176.l odd 4 2
3840.2.k.r 2 176.i even 4 2
4335.2.a.c 1 187.b odd 2 1
4800.2.a.t 1 440.o odd 2 1
4800.2.a.bz 1 440.c even 2 1
4800.2.f.c 2 440.w odd 4 2
4800.2.f.bf 2 440.t even 4 2
5415.2.a.j 1 209.d even 2 1
5445.2.a.c 1 3.b odd 2 1
7605.2.a.g 1 429.e even 2 1
7935.2.a.d 1 253.b even 2 1
9075.2.a.g 1 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1815))\):

\( T_{2} - 1 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 1 \) Copy content Toggle raw display
$3$ \( T + 1 \) Copy content Toggle raw display
$5$ \( T - 1 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T - 2 \) Copy content Toggle raw display
$17$ \( T + 2 \) Copy content Toggle raw display
$19$ \( T + 4 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T - 2 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T + 10 \) Copy content Toggle raw display
$41$ \( T + 10 \) Copy content Toggle raw display
$43$ \( T + 4 \) Copy content Toggle raw display
$47$ \( T - 8 \) Copy content Toggle raw display
$53$ \( T + 10 \) Copy content Toggle raw display
$59$ \( T + 4 \) Copy content Toggle raw display
$61$ \( T - 2 \) Copy content Toggle raw display
$67$ \( T - 12 \) Copy content Toggle raw display
$71$ \( T + 8 \) Copy content Toggle raw display
$73$ \( T + 10 \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T + 12 \) Copy content Toggle raw display
$89$ \( T + 6 \) Copy content Toggle raw display
$97$ \( T - 2 \) Copy content Toggle raw display
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