Properties

Label 1815.4.a.h
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: yes
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)
 
\(f(q)\) \(=\) \( q + q^{2} + 3 q^{3} - 7 q^{4} + 5 q^{5} + 3 q^{6} + 9 q^{7} - 15 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + 3 q^{3} - 7 q^{4} + 5 q^{5} + 3 q^{6} + 9 q^{7} - 15 q^{8} + 9 q^{9} + 5 q^{10} - 21 q^{12} - 77 q^{13} + 9 q^{14} + 15 q^{15} + 41 q^{16} + 129 q^{17} + 9 q^{18} - 53 q^{19} - 35 q^{20} + 27 q^{21} - 124 q^{23} - 45 q^{24} + 25 q^{25} - 77 q^{26} + 27 q^{27} - 63 q^{28} + 153 q^{29} + 15 q^{30} - 2 q^{31} + 161 q^{32} + 129 q^{34} + 45 q^{35} - 63 q^{36} - 155 q^{37} - 53 q^{38} - 231 q^{39} - 75 q^{40} + 122 q^{41} + 27 q^{42} - 430 q^{43} + 45 q^{45} - 124 q^{46} - 30 q^{47} + 123 q^{48} - 262 q^{49} + 25 q^{50} + 387 q^{51} + 539 q^{52} + 476 q^{53} + 27 q^{54} - 135 q^{56} - 159 q^{57} + 153 q^{58} - 472 q^{59} - 105 q^{60} - 512 q^{61} - 2 q^{62} + 81 q^{63} - 167 q^{64} - 385 q^{65} + 686 q^{67} - 903 q^{68} - 372 q^{69} + 45 q^{70} + 211 q^{71} - 135 q^{72} - 434 q^{73} - 155 q^{74} + 75 q^{75} + 371 q^{76} - 231 q^{78} - 218 q^{79} + 205 q^{80} + 81 q^{81} + 122 q^{82} - 295 q^{83} - 189 q^{84} + 645 q^{85} - 430 q^{86} + 459 q^{87} - 616 q^{89} + 45 q^{90} - 693 q^{91} + 868 q^{92} - 6 q^{93} - 30 q^{94} - 265 q^{95} + 483 q^{96} - 1334 q^{97} - 262 q^{98}+O(q^{100}) \) 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 3.00000 −7.00000 5.00000 3.00000 9.00000 −15.0000 9.00000 5.00000
\(n\): 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 1815.4.a.h yes 1
11.b odd 2 1 1815.4.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1815.4.a.d 1 11.b odd 2 1
1815.4.a.h yes 1 1.a even 1 1 trivial

Hecke kernels

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

\( T_{2} - 1 \) Copy content Toggle raw display
\( T_{7} - 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 1 \) Copy content Toggle raw display
$3$ \( T - 3 \) Copy content Toggle raw display
$5$ \( T - 5 \) Copy content Toggle raw display
$7$ \( T - 9 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T + 77 \) Copy content Toggle raw display
$17$ \( T - 129 \) Copy content Toggle raw display
$19$ \( T + 53 \) Copy content Toggle raw display
$23$ \( T + 124 \) Copy content Toggle raw display
$29$ \( T - 153 \) Copy content Toggle raw display
$31$ \( T + 2 \) Copy content Toggle raw display
$37$ \( T + 155 \) Copy content Toggle raw display
$41$ \( T - 122 \) Copy content Toggle raw display
$43$ \( T + 430 \) Copy content Toggle raw display
$47$ \( T + 30 \) Copy content Toggle raw display
$53$ \( T - 476 \) Copy content Toggle raw display
$59$ \( T + 472 \) Copy content Toggle raw display
$61$ \( T + 512 \) Copy content Toggle raw display
$67$ \( T - 686 \) Copy content Toggle raw display
$71$ \( T - 211 \) Copy content Toggle raw display
$73$ \( T + 434 \) Copy content Toggle raw display
$79$ \( T + 218 \) Copy content Toggle raw display
$83$ \( T + 295 \) Copy content Toggle raw display
$89$ \( T + 616 \) Copy content Toggle raw display
$97$ \( T + 1334 \) Copy content Toggle raw display
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