Properties

Label 3381.2.a.q
Level $3381$
Weight $2$
Character orbit 3381.a
Self dual yes
Analytic conductor $26.997$
Analytic rank $1$
Dimension $2$
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] = [3381,2,Mod(1,3381)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3381, 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("3381.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3381 = 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3381.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: \(26.9974209234\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 483)
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{2} - q^{3} + (\beta + 2) q^{4} + \beta q^{5} + \beta q^{6} + ( - \beta - 4) q^{8} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} - q^{3} + (\beta + 2) q^{4} + \beta q^{5} + \beta q^{6} + ( - \beta - 4) q^{8} + q^{9} + ( - \beta - 4) q^{10} - 2 q^{11} + ( - \beta - 2) q^{12} + ( - 2 \beta + 3) q^{13} - \beta q^{15} + 3 \beta q^{16} + ( - \beta + 6) q^{17} - \beta q^{18} + ( - \beta - 3) q^{19} + (3 \beta + 4) q^{20} + 2 \beta q^{22} + q^{23} + (\beta + 4) q^{24} + (\beta - 1) q^{25} + ( - \beta + 8) q^{26} - q^{27} + (2 \beta - 6) q^{29} + (\beta + 4) q^{30} + (3 \beta - 1) q^{31} + ( - \beta - 4) q^{32} + 2 q^{33} + ( - 5 \beta + 4) q^{34} + (\beta + 2) q^{36} + (\beta + 5) q^{37} + (4 \beta + 4) q^{38} + (2 \beta - 3) q^{39} + ( - 5 \beta - 4) q^{40} + ( - 4 \beta + 4) q^{41} + ( - 3 \beta - 5) q^{43} + ( - 2 \beta - 4) q^{44} + \beta q^{45} - \beta q^{46} + (3 \beta - 2) q^{47} - 3 \beta q^{48} - 4 q^{50} + (\beta - 6) q^{51} + ( - 3 \beta - 2) q^{52} + 5 \beta q^{53} + \beta q^{54} - 2 \beta q^{55} + (\beta + 3) q^{57} + (4 \beta - 8) q^{58} + (2 \beta - 6) q^{59} + ( - 3 \beta - 4) q^{60} - 6 q^{61} + ( - 2 \beta - 12) q^{62} + ( - \beta + 4) q^{64} + (\beta - 8) q^{65} - 2 \beta q^{66} + (2 \beta - 11) q^{67} + (3 \beta + 8) q^{68} - q^{69} + ( - 5 \beta + 6) q^{71} + ( - \beta - 4) q^{72} + ( - 2 \beta - 9) q^{73} + ( - 6 \beta - 4) q^{74} + ( - \beta + 1) q^{75} + ( - 6 \beta - 10) q^{76} + (\beta - 8) q^{78} + ( - \beta - 7) q^{79} + (3 \beta + 12) q^{80} + q^{81} + 16 q^{82} + ( - 2 \beta - 8) q^{83} + (5 \beta - 4) q^{85} + (8 \beta + 12) q^{86} + ( - 2 \beta + 6) q^{87} + (2 \beta + 8) q^{88} + 10 q^{89} + ( - \beta - 4) q^{90} + (\beta + 2) q^{92} + ( - 3 \beta + 1) q^{93} + ( - \beta - 12) q^{94} + ( - 4 \beta - 4) q^{95} + (\beta + 4) q^{96} + (6 \beta - 6) q^{97} - 2 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - 2 q^{3} + 5 q^{4} + q^{5} + q^{6} - 9 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - 2 q^{3} + 5 q^{4} + q^{5} + q^{6} - 9 q^{8} + 2 q^{9} - 9 q^{10} - 4 q^{11} - 5 q^{12} + 4 q^{13} - q^{15} + 3 q^{16} + 11 q^{17} - q^{18} - 7 q^{19} + 11 q^{20} + 2 q^{22} + 2 q^{23} + 9 q^{24} - q^{25} + 15 q^{26} - 2 q^{27} - 10 q^{29} + 9 q^{30} + q^{31} - 9 q^{32} + 4 q^{33} + 3 q^{34} + 5 q^{36} + 11 q^{37} + 12 q^{38} - 4 q^{39} - 13 q^{40} + 4 q^{41} - 13 q^{43} - 10 q^{44} + q^{45} - q^{46} - q^{47} - 3 q^{48} - 8 q^{50} - 11 q^{51} - 7 q^{52} + 5 q^{53} + q^{54} - 2 q^{55} + 7 q^{57} - 12 q^{58} - 10 q^{59} - 11 q^{60} - 12 q^{61} - 26 q^{62} + 7 q^{64} - 15 q^{65} - 2 q^{66} - 20 q^{67} + 19 q^{68} - 2 q^{69} + 7 q^{71} - 9 q^{72} - 20 q^{73} - 14 q^{74} + q^{75} - 26 q^{76} - 15 q^{78} - 15 q^{79} + 27 q^{80} + 2 q^{81} + 32 q^{82} - 18 q^{83} - 3 q^{85} + 32 q^{86} + 10 q^{87} + 18 q^{88} + 20 q^{89} - 9 q^{90} + 5 q^{92} - q^{93} - 25 q^{94} - 12 q^{95} + 9 q^{96} - 6 q^{97} - 4 q^{99}+O(q^{100}) \) 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
2.56155
−1.56155
−2.56155 −1.00000 4.56155 2.56155 2.56155 0 −6.56155 1.00000 −6.56155
1.2 1.56155 −1.00000 0.438447 −1.56155 −1.56155 0 −2.43845 1.00000 −2.43845
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(-1\)
\(23\) \(-1\)

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 3381.2.a.q 2
7.b odd 2 1 3381.2.a.s 2
7.d odd 6 2 483.2.i.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.i.e 4 7.d odd 6 2
3381.2.a.q 2 1.a even 1 1 trivial
3381.2.a.s 2 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3381))\):

\( T_{2}^{2} + T_{2} - 4 \) Copy content Toggle raw display
\( T_{5}^{2} - T_{5} - 4 \) Copy content Toggle raw display
\( T_{11} + 2 \) Copy content Toggle raw display
\( T_{13}^{2} - 4T_{13} - 13 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T - 4 \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T + 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 4T - 13 \) Copy content Toggle raw display
$17$ \( T^{2} - 11T + 26 \) Copy content Toggle raw display
$19$ \( T^{2} + 7T + 8 \) Copy content Toggle raw display
$23$ \( (T - 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 10T + 8 \) Copy content Toggle raw display
$31$ \( T^{2} - T - 38 \) Copy content Toggle raw display
$37$ \( T^{2} - 11T + 26 \) Copy content Toggle raw display
$41$ \( T^{2} - 4T - 64 \) Copy content Toggle raw display
$43$ \( T^{2} + 13T + 4 \) Copy content Toggle raw display
$47$ \( T^{2} + T - 38 \) Copy content Toggle raw display
$53$ \( T^{2} - 5T - 100 \) Copy content Toggle raw display
$59$ \( T^{2} + 10T + 8 \) Copy content Toggle raw display
$61$ \( (T + 6)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 20T + 83 \) Copy content Toggle raw display
$71$ \( T^{2} - 7T - 94 \) Copy content Toggle raw display
$73$ \( T^{2} + 20T + 83 \) Copy content Toggle raw display
$79$ \( T^{2} + 15T + 52 \) Copy content Toggle raw display
$83$ \( T^{2} + 18T + 64 \) Copy content Toggle raw display
$89$ \( (T - 10)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 6T - 144 \) Copy content Toggle raw display
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