Properties

Label 3381.2.a.z
Level $3381$
Weight $2$
Character orbit 3381.a
Self dual yes
Analytic conductor $26.997$
Analytic rank $0$
Dimension $5$
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: \(5\)
Coefficient field: 5.5.3176240.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 9x^{3} + 7x^{2} + 19x - 9 \) 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - x^{4} - 9x^{3} + 7x^{2} + 19x - 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 6\nu^{2} - \nu + 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 6\beta_{2} + \beta _1 + 20 \) 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
−2.34113
−1.67261
0.443666
2.08269
2.48738
−2.34113 1.00000 3.48089 −1.12582 −2.34113 0 −3.46695 1.00000 2.63569
1.2 −1.67261 1.00000 0.797614 3.68373 −1.67261 0 2.01112 1.00000 −6.16143
1.3 0.443666 1.00000 −1.80316 −2.13100 0.443666 0 −1.68733 1.00000 −0.945453
1.4 2.08269 1.00000 2.33760 −1.37958 2.08269 0 0.703109 1.00000 −2.87324
1.5 2.48738 1.00000 4.18706 2.95267 2.48738 0 5.44006 1.00000 7.34443
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.z yes 5
7.b odd 2 1 3381.2.a.y 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3381.2.a.y 5 7.b odd 2 1
3381.2.a.z yes 5 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3381))\):

\( T_{2}^{5} - T_{2}^{4} - 9T_{2}^{3} + 7T_{2}^{2} + 19T_{2} - 9 \) Copy content Toggle raw display
\( T_{5}^{5} - 2T_{5}^{4} - 13T_{5}^{3} + 8T_{5}^{2} + 53T_{5} + 36 \) Copy content Toggle raw display
\( T_{11}^{5} + 2T_{11}^{4} - 16T_{11}^{3} - 26T_{11}^{2} + 7T_{11} + 4 \) Copy content Toggle raw display
\( T_{13}^{5} - 4T_{13}^{4} - 17T_{13}^{3} + 104T_{13}^{2} - 145T_{13} + 40 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} - T^{4} - 9 T^{3} + 7 T^{2} + 19 T - 9 \) Copy content Toggle raw display
$3$ \( (T - 1)^{5} \) Copy content Toggle raw display
$5$ \( T^{5} - 2 T^{4} - 13 T^{3} + 8 T^{2} + \cdots + 36 \) Copy content Toggle raw display
$7$ \( T^{5} \) Copy content Toggle raw display
$11$ \( T^{5} + 2 T^{4} - 16 T^{3} - 26 T^{2} + \cdots + 4 \) Copy content Toggle raw display
$13$ \( T^{5} - 4 T^{4} - 17 T^{3} + 104 T^{2} + \cdots + 40 \) Copy content Toggle raw display
$17$ \( T^{5} - 2 T^{4} - 38 T^{3} - 78 T^{2} + \cdots + 4 \) Copy content Toggle raw display
$19$ \( T^{5} - 8 T^{4} - 14 T^{3} + 192 T^{2} + \cdots - 336 \) Copy content Toggle raw display
$23$ \( (T - 1)^{5} \) Copy content Toggle raw display
$29$ \( T^{5} - 4 T^{4} - 58 T^{3} + \cdots - 1162 \) Copy content Toggle raw display
$31$ \( T^{5} - 12 T^{4} + 40 T^{3} + \cdots + 148 \) Copy content Toggle raw display
$37$ \( T^{5} + 6 T^{4} - 144 T^{3} + \cdots - 334 \) Copy content Toggle raw display
$41$ \( T^{5} - 6 T^{4} - 52 T^{3} + 426 T^{2} + \cdots + 448 \) Copy content Toggle raw display
$43$ \( T^{5} + 24 T^{4} + 207 T^{3} + \cdots - 1324 \) Copy content Toggle raw display
$47$ \( T^{5} - 24 T^{4} + 136 T^{3} + \cdots + 5376 \) Copy content Toggle raw display
$53$ \( T^{5} - 2 T^{4} - 123 T^{3} + \cdots - 7554 \) Copy content Toggle raw display
$59$ \( T^{5} - 16 T^{4} + 29 T^{3} + \cdots - 500 \) Copy content Toggle raw display
$61$ \( T^{5} - 22 T^{4} + 19 T^{3} + \cdots - 30564 \) Copy content Toggle raw display
$67$ \( T^{5} + 16 T^{4} - 83 T^{3} + \cdots - 1084 \) Copy content Toggle raw display
$71$ \( T^{5} - 16 T^{4} + 55 T^{3} + \cdots - 960 \) Copy content Toggle raw display
$73$ \( T^{5} - 328 T^{3} + 576 T^{2} + \cdots - 100600 \) Copy content Toggle raw display
$79$ \( T^{5} - 12 T^{4} - 388 T^{3} + \cdots - 435816 \) Copy content Toggle raw display
$83$ \( T^{5} - 10 T^{4} - 18 T^{3} + \cdots + 216 \) Copy content Toggle raw display
$89$ \( T^{5} + 16 T^{4} - 209 T^{3} + \cdots - 7388 \) Copy content Toggle raw display
$97$ \( T^{5} + 4 T^{4} - 102 T^{3} + 344 T^{2} + \cdots - 36 \) Copy content Toggle raw display
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