Properties

Label 1815.4.a.a
Level $1815$
Weight $4$
Character orbit 1815.a
Self dual yes
Analytic conductor $107.088$
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] = [1815,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1815.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
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: \(107.088466660\)
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)
 
\(f(q)\) \(=\) \( q - 3 q^{2} - 3 q^{3} + q^{4} - 5 q^{5} + 9 q^{6} - 20 q^{7} + 21 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{2} - 3 q^{3} + q^{4} - 5 q^{5} + 9 q^{6} - 20 q^{7} + 21 q^{8} + 9 q^{9} + 15 q^{10} - 3 q^{12} - 74 q^{13} + 60 q^{14} + 15 q^{15} - 71 q^{16} - 54 q^{17} - 27 q^{18} + 124 q^{19} - 5 q^{20} + 60 q^{21} - 120 q^{23} - 63 q^{24} + 25 q^{25} + 222 q^{26} - 27 q^{27} - 20 q^{28} + 78 q^{29} - 45 q^{30} + 200 q^{31} + 45 q^{32} + 162 q^{34} + 100 q^{35} + 9 q^{36} - 70 q^{37} - 372 q^{38} + 222 q^{39} - 105 q^{40} - 330 q^{41} - 180 q^{42} - 92 q^{43} - 45 q^{45} + 360 q^{46} - 24 q^{47} + 213 q^{48} + 57 q^{49} - 75 q^{50} + 162 q^{51} - 74 q^{52} + 450 q^{53} + 81 q^{54} - 420 q^{56} - 372 q^{57} - 234 q^{58} + 24 q^{59} + 15 q^{60} + 322 q^{61} - 600 q^{62} - 180 q^{63} + 433 q^{64} + 370 q^{65} - 196 q^{67} - 54 q^{68} + 360 q^{69} - 300 q^{70} - 288 q^{71} + 189 q^{72} + 430 q^{73} + 210 q^{74} - 75 q^{75} + 124 q^{76} - 666 q^{78} + 520 q^{79} + 355 q^{80} + 81 q^{81} + 990 q^{82} - 156 q^{83} + 60 q^{84} + 270 q^{85} + 276 q^{86} - 234 q^{87} + 1026 q^{89} + 135 q^{90} + 1480 q^{91} - 120 q^{92} - 600 q^{93} + 72 q^{94} - 620 q^{95} - 135 q^{96} - 286 q^{97} - 171 q^{98}+O(q^{100}) \) 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
−3.00000 −3.00000 1.00000 −5.00000 9.00000 −20.0000 21.0000 9.00000 15.0000
\(n\): 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 1815.4.a.a 1
11.b odd 2 1 15.4.a.b 1
33.d even 2 1 45.4.a.b 1
44.c even 2 1 240.4.a.f 1
55.d odd 2 1 75.4.a.a 1
55.e even 4 2 75.4.b.a 2
77.b even 2 1 735.4.a.i 1
88.b odd 2 1 960.4.a.bi 1
88.g even 2 1 960.4.a.l 1
99.g even 6 2 405.4.e.k 2
99.h odd 6 2 405.4.e.d 2
132.d odd 2 1 720.4.a.r 1
165.d even 2 1 225.4.a.g 1
165.l odd 4 2 225.4.b.d 2
220.g even 2 1 1200.4.a.o 1
220.i odd 4 2 1200.4.f.m 2
231.h odd 2 1 2205.4.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.a.b 1 11.b odd 2 1
45.4.a.b 1 33.d even 2 1
75.4.a.a 1 55.d odd 2 1
75.4.b.a 2 55.e even 4 2
225.4.a.g 1 165.d even 2 1
225.4.b.d 2 165.l odd 4 2
240.4.a.f 1 44.c even 2 1
405.4.e.d 2 99.h odd 6 2
405.4.e.k 2 99.g even 6 2
720.4.a.r 1 132.d odd 2 1
735.4.a.i 1 77.b even 2 1
960.4.a.l 1 88.g even 2 1
960.4.a.bi 1 88.b odd 2 1
1200.4.a.o 1 220.g even 2 1
1200.4.f.m 2 220.i odd 4 2
1815.4.a.a 1 1.a even 1 1 trivial
2205.4.a.c 1 231.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1815))\):

\( T_{2} + 3 \) Copy content Toggle raw display
\( T_{7} + 20 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 3 \) Copy content Toggle raw display
$3$ \( T + 3 \) Copy content Toggle raw display
$5$ \( T + 5 \) Copy content Toggle raw display
$7$ \( T + 20 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T + 74 \) Copy content Toggle raw display
$17$ \( T + 54 \) Copy content Toggle raw display
$19$ \( T - 124 \) Copy content Toggle raw display
$23$ \( T + 120 \) Copy content Toggle raw display
$29$ \( T - 78 \) Copy content Toggle raw display
$31$ \( T - 200 \) Copy content Toggle raw display
$37$ \( T + 70 \) Copy content Toggle raw display
$41$ \( T + 330 \) Copy content Toggle raw display
$43$ \( T + 92 \) Copy content Toggle raw display
$47$ \( T + 24 \) Copy content Toggle raw display
$53$ \( T - 450 \) Copy content Toggle raw display
$59$ \( T - 24 \) Copy content Toggle raw display
$61$ \( T - 322 \) Copy content Toggle raw display
$67$ \( T + 196 \) Copy content Toggle raw display
$71$ \( T + 288 \) Copy content Toggle raw display
$73$ \( T - 430 \) Copy content Toggle raw display
$79$ \( T - 520 \) Copy content Toggle raw display
$83$ \( T + 156 \) Copy content Toggle raw display
$89$ \( T - 1026 \) Copy content Toggle raw display
$97$ \( T + 286 \) Copy content Toggle raw display
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