Properties

Label 3381.2.a.bb
Level $3381$
Weight $2$
Character orbit 3381.a
Self dual yes
Analytic conductor $26.997$
Analytic rank $1$
Dimension $6$
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: \(6\)
Coefficient field: 6.6.62622704.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} - 5x^{4} + 13x^{3} + 9x^{2} - 5x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + q^{3} + (\beta_{2} - \beta_1 + 2) q^{4} + ( - \beta_{3} - \beta_1) q^{5} + (\beta_1 - 1) q^{6} + (\beta_{3} - 2 \beta_{2} + 2 \beta_1 - 3) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} + q^{3} + (\beta_{2} - \beta_1 + 2) q^{4} + ( - \beta_{3} - \beta_1) q^{5} + (\beta_1 - 1) q^{6} + (\beta_{3} - 2 \beta_{2} + 2 \beta_1 - 3) q^{8} + q^{9} + ( - \beta_{4} - \beta_{2} - 2) q^{10} + ( - \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_1 - 1) q^{11} + (\beta_{2} - \beta_1 + 2) q^{12} + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{13} + ( - \beta_{3} - \beta_1) q^{15} + (\beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 5 \beta_1 + 4) q^{16} + (\beta_{5} - 2 \beta_{2} - 2) q^{17} + (\beta_1 - 1) q^{18} + (\beta_{5} - 2 \beta_{4} - \beta_{3} - 4) q^{19} + ( - \beta_{5} - \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 2) q^{20} + (2 \beta_{5} + \beta_{3} + \beta_{2} - \beta_1 + 5) q^{22} - q^{23} + (\beta_{3} - 2 \beta_{2} + 2 \beta_1 - 3) q^{24} + (\beta_{5} + \beta_{4} + \beta_{3} + \beta_1) q^{25} + (\beta_{4} + \beta_{3} - 2 \beta_{2} + 3 \beta_1 - 5) q^{26} + q^{27} + ( - 3 \beta_{5} - 2) q^{29} + ( - \beta_{4} - \beta_{2} - 2) q^{30} + (\beta_{5} + \beta_{4} + 2 \beta_{3} + 1) q^{31} + (\beta_{5} - 2 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} + 5 \beta_1 - 11) q^{32} + ( - \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_1 - 1) q^{33} + ( - \beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_{2} - 5 \beta_1 + 3) q^{34} + (\beta_{2} - \beta_1 + 2) q^{36} + ( - \beta_{5} + \beta_{4} + 2 \beta_{3} - 1) q^{37} + ( - 3 \beta_{5} - 3 \beta_{3} + 2 \beta_{2} - 5 \beta_1 + 6) q^{38} + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{39} + (\beta_{5} + \beta_{3} - 3 \beta_{2} + 5 \beta_1 - 7) q^{40} + (3 \beta_{5} - \beta_{4} - \beta_{3} - 4 \beta_1 + 1) q^{41} + ( - 2 \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 - 1) q^{43} + (\beta_{4} + \beta_{3} - 2 \beta_{2} + 5 \beta_1 - 5) q^{44} + ( - \beta_{3} - \beta_1) q^{45} + ( - \beta_1 + 1) q^{46} + (\beta_{5} + \beta_{4} - 3 \beta_{3} + \beta_{2} - 2 \beta_1) q^{47} + (\beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 5 \beta_1 + 4) q^{48} + (2 \beta_{4} + 3 \beta_{3} + 2 \beta_1 + 3) q^{50} + (\beta_{5} - 2 \beta_{2} - 2) q^{51} + (\beta_{5} + \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 6 \beta_1 + 11) q^{52} + (\beta_{5} - 2 \beta_{4} - 2 \beta_{3} + \beta_{2} - \beta_1 - 1) q^{53} + (\beta_1 - 1) q^{54} + (\beta_{5} - 3 \beta_{4} - \beta_{3} - \beta_{2} - 3 \beta_1 - 4) q^{55} + (\beta_{5} - 2 \beta_{4} - \beta_{3} - 4) q^{57} + (3 \beta_{5} - 3 \beta_{4} - 3 \beta_{3} - 5 \beta_1 - 1) q^{58} + ( - \beta_{5} - 3 \beta_{2} + \beta_1 - 5) q^{59} + ( - \beta_{5} - \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 2) q^{60} + ( - 4 \beta_{5} + \beta_{2} + \beta_1 - 5) q^{61} + (3 \beta_{4} + 3 \beta_{3} - \beta_{2} + 3 \beta_1 - 2) q^{62} + ( - 3 \beta_{5} + \beta_{4} - 3 \beta_{3} + 7 \beta_{2} - 10 \beta_1 + 17) q^{64} + ( - 2 \beta_{5} + 3 \beta_{2} - 2 \beta_1 - 1) q^{65} + (2 \beta_{5} + \beta_{3} + \beta_{2} - \beta_1 + 5) q^{66} + ( - \beta_{5} - \beta_{4} - 3 \beta_{3} - \beta_{2} - \beta_1 + 2) q^{67} + ( - 2 \beta_{4} + 3 \beta_{3} - 4 \beta_{2} + 7 \beta_1 - 14) q^{68} - q^{69} + (5 \beta_{5} + \beta_{3} - 2 \beta_{2} + \beta_1 - 2) q^{71} + (\beta_{3} - 2 \beta_{2} + 2 \beta_1 - 3) q^{72} + ( - 3 \beta_{5} + \beta_{4} + 2 \beta_{3} + 2 \beta_1 + 1) q^{73} + (2 \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 - 2) q^{74} + (\beta_{5} + \beta_{4} + \beta_{3} + \beta_1) q^{75} + (\beta_{5} - 2 \beta_{4} + \beta_{3} - 7 \beta_{2} + 7 \beta_1 - 13) q^{76} + (\beta_{4} + \beta_{3} - 2 \beta_{2} + 3 \beta_1 - 5) q^{78} + ( - \beta_{5} - 3 \beta_{4} - 3) q^{79} + (\beta_{5} + 2 \beta_{4} + 4 \beta_{2} - 6 \beta_1 + 18) q^{80} + q^{81} + ( - 4 \beta_{5} + 2 \beta_{4} + \beta_{3} - 3 \beta_{2} + 3 \beta_1 - 9) q^{82} + ( - 3 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 2 \beta_1 - 2) q^{83} + (\beta_{5} + 2 \beta_{4} + \beta_{3} - 3 \beta_{2} + 5 \beta_1 - 1) q^{85} + ( - 2 \beta_{5} + \beta_{4} - 3 \beta_{3} + 2 \beta_{2} - \beta_1 + 3) q^{86} + ( - 3 \beta_{5} - 2) q^{87} + ( - 3 \beta_{5} + \beta_{4} - 2 \beta_{3} + 4 \beta_{2} - 6 \beta_1 + 9) q^{88} + (\beta_{5} - \beta_{3} + 2 \beta_{2} - \beta_1 - 2) q^{89} + ( - \beta_{4} - \beta_{2} - 2) q^{90} + ( - \beta_{2} + \beta_1 - 2) q^{92} + (\beta_{5} + \beta_{4} + 2 \beta_{3} + 1) q^{93} + ( - 2 \beta_{4} + 4 \beta_{3} - 4 \beta_{2} + 4 \beta_1 - 2) q^{94} + ( - 2 \beta_{5} + 2 \beta_{4} + 5 \beta_{3} + 9 \beta_1 - 2) q^{95} + (\beta_{5} - 2 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} + 5 \beta_1 - 11) q^{96} + ( - 2 \beta_{5} + \beta_{4} + \beta_{2} - 2 \beta_1 - 2) q^{97} + ( - \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_1 - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{2} + 6 q^{3} + 7 q^{4} - 2 q^{5} - 3 q^{6} - 9 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{2} + 6 q^{3} + 7 q^{4} - 2 q^{5} - 3 q^{6} - 9 q^{8} + 6 q^{9} - 8 q^{10} - 3 q^{11} + 7 q^{12} - 2 q^{15} + 5 q^{16} - 8 q^{17} - 3 q^{18} - 19 q^{19} + 24 q^{22} - 6 q^{23} - 9 q^{24} - 20 q^{26} + 6 q^{27} - 12 q^{29} - 8 q^{30} + 2 q^{31} - 41 q^{32} - 3 q^{33} - 2 q^{34} + 7 q^{36} - 10 q^{37} + 20 q^{38} - 22 q^{40} - 3 q^{41} - 2 q^{43} - 14 q^{44} - 2 q^{45} + 3 q^{46} - 7 q^{47} + 5 q^{48} + 17 q^{50} - 8 q^{51} + 44 q^{52} - 5 q^{53} - 3 q^{54} - 24 q^{55} - 19 q^{57} - 12 q^{58} - 21 q^{59} - 29 q^{61} - 10 q^{62} + 59 q^{64} - 18 q^{65} + 24 q^{66} + 16 q^{67} - 54 q^{68} - 6 q^{69} - 6 q^{71} - 9 q^{72} + 8 q^{73} - 16 q^{74} - 40 q^{76} - 20 q^{78} - 12 q^{79} + 78 q^{80} + 6 q^{81} - 44 q^{82} - 24 q^{83} + 10 q^{85} + 12 q^{86} - 12 q^{87} + 28 q^{88} - 18 q^{89} - 8 q^{90} - 7 q^{92} + 2 q^{93} + 8 q^{94} + 6 q^{95} - 41 q^{96} - 22 q^{97} - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} - 5x^{4} + 13x^{3} + 9x^{2} - 5x - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 5\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 2\nu^{3} - 4\nu^{2} + 5\nu + 1 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} - 3\nu^{4} - 4\nu^{3} + 12\nu^{2} + 4\nu - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 6\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 2\beta_{3} + 6\beta_{2} + 11\beta _1 + 17 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} + 3\beta_{4} + 10\beta_{3} + 10\beta_{2} + 41\beta _1 + 31 \) 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
−1.82095
−0.835848
−0.311423
0.584512
2.52648
2.85723
−2.82095 1.00000 5.95777 2.07008 −2.82095 0 −11.1647 1.00000 −5.83961
1.2 −1.83585 1.00000 1.37034 −2.06079 −1.83585 0 1.15597 1.00000 3.78330
1.3 −1.31142 1.00000 −0.280171 −1.11850 −1.31142 0 2.99027 1.00000 1.46683
1.4 −0.415488 1.00000 −1.82737 2.48000 −0.415488 0 1.59023 1.00000 −1.03041
1.5 1.52648 1.00000 0.330154 0.362203 1.52648 0 −2.54899 1.00000 0.552896
1.6 1.85723 1.00000 1.44929 −3.73299 1.85723 0 −1.02280 1.00000 −6.93301
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.bb yes 6
7.b odd 2 1 3381.2.a.ba 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3381.2.a.ba 6 7.b odd 2 1
3381.2.a.bb yes 6 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3381))\):

\( T_{2}^{6} + 3T_{2}^{5} - 5T_{2}^{4} - 17T_{2}^{3} + 3T_{2}^{2} + 23T_{2} + 8 \) Copy content Toggle raw display
\( T_{5}^{6} + 2T_{5}^{5} - 13T_{5}^{4} - 16T_{5}^{3} + 41T_{5}^{2} + 32T_{5} - 16 \) Copy content Toggle raw display
\( T_{11}^{6} + 3T_{11}^{5} - 38T_{11}^{4} - 122T_{11}^{3} + 181T_{11}^{2} + 827T_{11} + 596 \) Copy content Toggle raw display
\( T_{13}^{6} - 41T_{13}^{4} + 36T_{13}^{3} + 347T_{13}^{2} - 480T_{13} + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 3 T^{5} - 5 T^{4} - 17 T^{3} + \cdots + 8 \) Copy content Toggle raw display
$3$ \( (T - 1)^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + 2 T^{5} - 13 T^{4} - 16 T^{3} + \cdots - 16 \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} + 3 T^{5} - 38 T^{4} - 122 T^{3} + \cdots + 596 \) Copy content Toggle raw display
$13$ \( T^{6} - 41 T^{4} + 36 T^{3} + 347 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$17$ \( T^{6} + 8 T^{5} - 30 T^{4} + \cdots + 1480 \) Copy content Toggle raw display
$19$ \( T^{6} + 19 T^{5} + 70 T^{4} + \cdots + 26816 \) Copy content Toggle raw display
$23$ \( (T + 1)^{6} \) Copy content Toggle raw display
$29$ \( T^{6} + 12 T^{5} - 66 T^{4} + \cdots + 4744 \) Copy content Toggle raw display
$31$ \( T^{6} - 2 T^{5} - 84 T^{4} + 248 T^{3} + \cdots - 824 \) Copy content Toggle raw display
$37$ \( T^{6} + 10 T^{5} - 24 T^{4} + \cdots - 128 \) Copy content Toggle raw display
$41$ \( T^{6} + 3 T^{5} - 162 T^{4} + \cdots - 81584 \) Copy content Toggle raw display
$43$ \( T^{6} + 2 T^{5} - 125 T^{4} + \cdots + 392 \) Copy content Toggle raw display
$47$ \( T^{6} + 7 T^{5} - 168 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$53$ \( T^{6} + 5 T^{5} - 85 T^{4} - 433 T^{3} + \cdots + 674 \) Copy content Toggle raw display
$59$ \( T^{6} + 21 T^{5} + 29 T^{4} + \cdots + 44476 \) Copy content Toggle raw display
$61$ \( T^{6} + 29 T^{5} + 153 T^{4} + \cdots + 130124 \) Copy content Toggle raw display
$67$ \( T^{6} - 16 T^{5} - 71 T^{4} + \cdots + 12304 \) Copy content Toggle raw display
$71$ \( T^{6} + 6 T^{5} - 345 T^{4} + \cdots - 1394528 \) Copy content Toggle raw display
$73$ \( T^{6} - 8 T^{5} - 100 T^{4} + 56 T^{3} + \cdots - 64 \) Copy content Toggle raw display
$79$ \( T^{6} + 12 T^{5} - 188 T^{4} + \cdots - 297056 \) Copy content Toggle raw display
$83$ \( T^{6} + 24 T^{5} - 34 T^{4} + \cdots + 612464 \) Copy content Toggle raw display
$89$ \( T^{6} + 18 T^{5} + 51 T^{4} + \cdots + 1864 \) Copy content Toggle raw display
$97$ \( T^{6} + 22 T^{5} + 94 T^{4} + \cdots + 71272 \) Copy content Toggle raw display
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