Properties

Label 3381.2.a.t
Level $3381$
Weight $2$
Character orbit 3381.a
Self dual yes
Analytic conductor $26.997$
Analytic rank $0$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 69)
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 \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{2} + q^{3} + 3 q^{4} + ( - \beta + 1) q^{5} - \beta q^{6} - \beta q^{8} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} + q^{3} + 3 q^{4} + ( - \beta + 1) q^{5} - \beta q^{6} - \beta q^{8} + q^{9} + ( - \beta + 5) q^{10} + 4 q^{11} + 3 q^{12} + 2 \beta q^{13} + ( - \beta + 1) q^{15} - q^{16} + ( - \beta + 5) q^{17} - \beta q^{18} + ( - \beta - 5) q^{19} + ( - 3 \beta + 3) q^{20} - 4 \beta q^{22} + q^{23} - \beta q^{24} + ( - 2 \beta + 1) q^{25} - 10 q^{26} + q^{27} + 2 \beta q^{29} + ( - \beta + 5) q^{30} + (2 \beta + 2) q^{31} + 3 \beta q^{32} + 4 q^{33} + ( - 5 \beta + 5) q^{34} + 3 q^{36} + 2 \beta q^{37} + (5 \beta + 5) q^{38} + 2 \beta q^{39} + ( - \beta + 5) q^{40} + (4 \beta + 2) q^{41} + ( - 3 \beta + 1) q^{43} + 12 q^{44} + ( - \beta + 1) q^{45} - \beta q^{46} + 4 q^{47} - q^{48} + ( - \beta + 10) q^{50} + ( - \beta + 5) q^{51} + 6 \beta q^{52} + ( - \beta - 3) q^{53} - \beta q^{54} + ( - 4 \beta + 4) q^{55} + ( - \beta - 5) q^{57} - 10 q^{58} + (4 \beta - 4) q^{59} + ( - 3 \beta + 3) q^{60} - 2 \beta q^{61} + ( - 2 \beta - 10) q^{62} - 13 q^{64} + (2 \beta - 10) q^{65} - 4 \beta q^{66} + ( - \beta + 3) q^{67} + ( - 3 \beta + 15) q^{68} + q^{69} - 8 q^{71} - \beta q^{72} + ( - 4 \beta + 2) q^{73} - 10 q^{74} + ( - 2 \beta + 1) q^{75} + ( - 3 \beta - 15) q^{76} - 10 q^{78} + (3 \beta + 3) q^{79} + (\beta - 1) q^{80} + q^{81} + ( - 2 \beta - 20) q^{82} - 4 q^{83} + ( - 6 \beta + 10) q^{85} + ( - \beta + 15) q^{86} + 2 \beta q^{87} - 4 \beta q^{88} + (\beta - 1) q^{89} + ( - \beta + 5) q^{90} + 3 q^{92} + (2 \beta + 2) q^{93} - 4 \beta q^{94} + 4 \beta q^{95} + 3 \beta q^{96} + ( - 2 \beta - 4) q^{97} + 4 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 6 q^{4} + 2 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 6 q^{4} + 2 q^{5} + 2 q^{9} + 10 q^{10} + 8 q^{11} + 6 q^{12} + 2 q^{15} - 2 q^{16} + 10 q^{17} - 10 q^{19} + 6 q^{20} + 2 q^{23} + 2 q^{25} - 20 q^{26} + 2 q^{27} + 10 q^{30} + 4 q^{31} + 8 q^{33} + 10 q^{34} + 6 q^{36} + 10 q^{38} + 10 q^{40} + 4 q^{41} + 2 q^{43} + 24 q^{44} + 2 q^{45} + 8 q^{47} - 2 q^{48} + 20 q^{50} + 10 q^{51} - 6 q^{53} + 8 q^{55} - 10 q^{57} - 20 q^{58} - 8 q^{59} + 6 q^{60} - 20 q^{62} - 26 q^{64} - 20 q^{65} + 6 q^{67} + 30 q^{68} + 2 q^{69} - 16 q^{71} + 4 q^{73} - 20 q^{74} + 2 q^{75} - 30 q^{76} - 20 q^{78} + 6 q^{79} - 2 q^{80} + 2 q^{81} - 40 q^{82} - 8 q^{83} + 20 q^{85} + 30 q^{86} - 2 q^{89} + 10 q^{90} + 6 q^{92} + 4 q^{93} - 8 q^{97} + 8 q^{99}+O(q^{100}) \) 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
1.61803
−0.618034
−2.23607 1.00000 3.00000 −1.23607 −2.23607 0 −2.23607 1.00000 2.76393
1.2 2.23607 1.00000 3.00000 3.23607 2.23607 0 2.23607 1.00000 7.23607
\(n\): 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.t 2
7.b odd 2 1 69.2.a.b 2
21.c even 2 1 207.2.a.c 2
28.d even 2 1 1104.2.a.m 2
35.c odd 2 1 1725.2.a.ba 2
35.f even 4 2 1725.2.b.o 4
56.e even 2 1 4416.2.a.bg 2
56.h odd 2 1 4416.2.a.bm 2
77.b even 2 1 8349.2.a.i 2
84.h odd 2 1 3312.2.a.bb 2
105.g even 2 1 5175.2.a.bk 2
161.c even 2 1 1587.2.a.i 2
483.c odd 2 1 4761.2.a.v 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.2.a.b 2 7.b odd 2 1
207.2.a.c 2 21.c even 2 1
1104.2.a.m 2 28.d even 2 1
1587.2.a.i 2 161.c even 2 1
1725.2.a.ba 2 35.c odd 2 1
1725.2.b.o 4 35.f even 4 2
3312.2.a.bb 2 84.h odd 2 1
3381.2.a.t 2 1.a even 1 1 trivial
4416.2.a.bg 2 56.e even 2 1
4416.2.a.bm 2 56.h odd 2 1
4761.2.a.v 2 483.c odd 2 1
5175.2.a.bk 2 105.g even 2 1
8349.2.a.i 2 77.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3381))\):

\( T_{2}^{2} - 5 \) Copy content Toggle raw display
\( T_{5}^{2} - 2T_{5} - 4 \) Copy content Toggle raw display
\( T_{11} - 4 \) Copy content Toggle raw display
\( T_{13}^{2} - 20 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 5 \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 2T - 4 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T - 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 20 \) Copy content Toggle raw display
$17$ \( T^{2} - 10T + 20 \) Copy content Toggle raw display
$19$ \( T^{2} + 10T + 20 \) Copy content Toggle raw display
$23$ \( (T - 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 20 \) Copy content Toggle raw display
$31$ \( T^{2} - 4T - 16 \) Copy content Toggle raw display
$37$ \( T^{2} - 20 \) Copy content Toggle raw display
$41$ \( T^{2} - 4T - 76 \) Copy content Toggle raw display
$43$ \( T^{2} - 2T - 44 \) Copy content Toggle raw display
$47$ \( (T - 4)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 6T + 4 \) Copy content Toggle raw display
$59$ \( T^{2} + 8T - 64 \) Copy content Toggle raw display
$61$ \( T^{2} - 20 \) Copy content Toggle raw display
$67$ \( T^{2} - 6T + 4 \) Copy content Toggle raw display
$71$ \( (T + 8)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 4T - 76 \) Copy content Toggle raw display
$79$ \( T^{2} - 6T - 36 \) Copy content Toggle raw display
$83$ \( (T + 4)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 2T - 4 \) Copy content Toggle raw display
$97$ \( T^{2} + 8T - 4 \) Copy content Toggle raw display
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