Properties

Label 330.4.a.a
Level $330$
Weight $4$
Character orbit 330.a
Self dual yes
Analytic conductor $19.471$
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] = [330,4,Mod(1,330)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(330, 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("330.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 330 = 2 \cdot 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 330.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: \(19.4706303019\)
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)
 
\(f(q)\) \(=\) \( q - 2 q^{2} - 3 q^{3} + 4 q^{4} - 5 q^{5} + 6 q^{6} + 2 q^{7} - 8 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} - 3 q^{3} + 4 q^{4} - 5 q^{5} + 6 q^{6} + 2 q^{7} - 8 q^{8} + 9 q^{9} + 10 q^{10} + 11 q^{11} - 12 q^{12} - 16 q^{13} - 4 q^{14} + 15 q^{15} + 16 q^{16} + 96 q^{17} - 18 q^{18} - 112 q^{19} - 20 q^{20} - 6 q^{21} - 22 q^{22} + 180 q^{23} + 24 q^{24} + 25 q^{25} + 32 q^{26} - 27 q^{27} + 8 q^{28} - 102 q^{29} - 30 q^{30} - 208 q^{31} - 32 q^{32} - 33 q^{33} - 192 q^{34} - 10 q^{35} + 36 q^{36} + 110 q^{37} + 224 q^{38} + 48 q^{39} + 40 q^{40} - 90 q^{41} + 12 q^{42} - 10 q^{43} + 44 q^{44} - 45 q^{45} - 360 q^{46} - 180 q^{47} - 48 q^{48} - 339 q^{49} - 50 q^{50} - 288 q^{51} - 64 q^{52} - 618 q^{53} + 54 q^{54} - 55 q^{55} - 16 q^{56} + 336 q^{57} + 204 q^{58} - 36 q^{59} + 60 q^{60} - 286 q^{61} + 416 q^{62} + 18 q^{63} + 64 q^{64} + 80 q^{65} + 66 q^{66} - 928 q^{67} + 384 q^{68} - 540 q^{69} + 20 q^{70} + 48 q^{71} - 72 q^{72} - 520 q^{73} - 220 q^{74} - 75 q^{75} - 448 q^{76} + 22 q^{77} - 96 q^{78} - 412 q^{79} - 80 q^{80} + 81 q^{81} + 180 q^{82} - 618 q^{83} - 24 q^{84} - 480 q^{85} + 20 q^{86} + 306 q^{87} - 88 q^{88} - 234 q^{89} + 90 q^{90} - 32 q^{91} + 720 q^{92} + 624 q^{93} + 360 q^{94} + 560 q^{95} + 96 q^{96} + 422 q^{97} + 678 q^{98} + 99 q^{99}+O(q^{100}) \) 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
−2.00000 −3.00000 4.00000 −5.00000 6.00000 2.00000 −8.00000 9.00000 10.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(5\) \(1\)
\(11\) \(-1\)

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 330.4.a.a 1
3.b odd 2 1 990.4.a.t 1
5.b even 2 1 1650.4.a.n 1
5.c odd 4 2 1650.4.c.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
330.4.a.a 1 1.a even 1 1 trivial
990.4.a.t 1 3.b odd 2 1
1650.4.a.n 1 5.b even 2 1
1650.4.c.l 2 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(330))\):

\( T_{7} - 2 \) Copy content Toggle raw display
\( T_{13} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 2 \) Copy content Toggle raw display
$3$ \( T + 3 \) Copy content Toggle raw display
$5$ \( T + 5 \) Copy content Toggle raw display
$7$ \( T - 2 \) Copy content Toggle raw display
$11$ \( T - 11 \) Copy content Toggle raw display
$13$ \( T + 16 \) Copy content Toggle raw display
$17$ \( T - 96 \) Copy content Toggle raw display
$19$ \( T + 112 \) Copy content Toggle raw display
$23$ \( T - 180 \) Copy content Toggle raw display
$29$ \( T + 102 \) Copy content Toggle raw display
$31$ \( T + 208 \) Copy content Toggle raw display
$37$ \( T - 110 \) Copy content Toggle raw display
$41$ \( T + 90 \) Copy content Toggle raw display
$43$ \( T + 10 \) Copy content Toggle raw display
$47$ \( T + 180 \) Copy content Toggle raw display
$53$ \( T + 618 \) Copy content Toggle raw display
$59$ \( T + 36 \) Copy content Toggle raw display
$61$ \( T + 286 \) Copy content Toggle raw display
$67$ \( T + 928 \) Copy content Toggle raw display
$71$ \( T - 48 \) Copy content Toggle raw display
$73$ \( T + 520 \) Copy content Toggle raw display
$79$ \( T + 412 \) Copy content Toggle raw display
$83$ \( T + 618 \) Copy content Toggle raw display
$89$ \( T + 234 \) Copy content Toggle raw display
$97$ \( T - 422 \) Copy content Toggle raw display
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