Properties

Label 3381.2.a.bf
Level $3381$
Weight $2$
Character orbit 3381.a
Self dual yes
Analytic conductor $26.997$
Analytic rank $0$
Dimension $8$
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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 15x^{6} + 11x^{5} + 75x^{4} - 35x^{3} - 141x^{2} + 37x + 80 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 2) q^{4} + (\beta_{7} + 1) q^{5} + \beta_1 q^{6} + (\beta_{5} - \beta_{4} + \beta_{2} + \beta_1 + 2) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 2) q^{4} + (\beta_{7} + 1) q^{5} + \beta_1 q^{6} + (\beta_{5} - \beta_{4} + \beta_{2} + \beta_1 + 2) q^{8} + q^{9} + (\beta_{7} + \beta_{6} - \beta_{5} - \beta_{3} + \beta_1) q^{10} + (\beta_{6} - \beta_{4} + \beta_1 + 2) q^{11} + (\beta_{2} + 2) q^{12} + (\beta_{4} - \beta_{3} - 1) q^{13} + (\beta_{7} + 1) q^{15} + ( - \beta_{4} + \beta_{3} + 2 \beta_{2} + \beta_1 + 2) q^{16} + ( - \beta_{7} + \beta_{4} - \beta_1 + 2) q^{17} + \beta_1 q^{18} + (\beta_{6} + \beta_{5} - \beta_{4} + 2) q^{19} + (\beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - 1) q^{20} + ( - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{2} + 2) q^{22} - q^{23} + (\beta_{5} - \beta_{4} + \beta_{2} + \beta_1 + 2) q^{24} + ( - 2 \beta_{2} + \beta_1 + 3) q^{25} + (2 \beta_{7} - 2 \beta_{5} - \beta_{2}) q^{26} + q^{27} + ( - \beta_{5} - \beta_{3} - \beta_1) q^{29} + (\beta_{7} + \beta_{6} - \beta_{5} - \beta_{3} + \beta_1) q^{30} + (\beta_{5} + \beta_{3} + \beta_{2} - \beta_1 - 1) q^{31} + ( - 2 \beta_{7} + 2 \beta_{5} + 2 \beta_{2} + \beta_1 + 4) q^{32} + (\beta_{6} - \beta_{4} + \beta_1 + 2) q^{33} + ( - \beta_{7} + \beta_{5} + \beta_{3} - 2 \beta_{2} + 3 \beta_1 - 3) q^{34} + (\beta_{2} + 2) q^{36} + ( - \beta_{5} - \beta_{4} - 2 \beta_{2} + 2 \beta_1 + 1) q^{37} + (\beta_{4} + 2 \beta_{3} + 2 \beta_{2} - \beta_1 - 1) q^{38} + (\beta_{4} - \beta_{3} - 1) q^{39} + (\beta_{7} - 2 \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2}) q^{40} + ( - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 + 2) q^{41} + ( - \beta_{6} - \beta_{4} - \beta_{3} - \beta_1 + 2) q^{43} + ( - 2 \beta_{7} + \beta_{6} + 2 \beta_{5} - \beta_{4} + 2 \beta_{2} + 3 \beta_1 + 4) q^{44} + (\beta_{7} + 1) q^{45} - \beta_1 q^{46} + (\beta_{5} + 2 \beta_{4} - \beta_{3} - 2 \beta_1) q^{47} + ( - \beta_{4} + \beta_{3} + 2 \beta_{2} + \beta_1 + 2) q^{48} + ( - 2 \beta_{5} + 2 \beta_{4} - \beta_{2} + \beta_1) q^{50} + ( - \beta_{7} + \beta_{4} - \beta_1 + 2) q^{51} + (2 \beta_{7} - \beta_{5} - \beta_{4} - 2 \beta_{3} - 3 \beta_{2} + \beta_1 - 2) q^{52} + (\beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + 1) q^{53} + \beta_1 q^{54} + (2 \beta_{7} - 2 \beta_{5} - \beta_{4} + \beta_{3} - 3 \beta_{2} + 3 \beta_1 - 2) q^{55} + (\beta_{6} + \beta_{5} - \beta_{4} + 2) q^{57} + (2 \beta_{7} - 2 \beta_{6} - \beta_{5} - \beta_{3} - 2 \beta_{2} + \beta_1 - 6) q^{58} + ( - 2 \beta_{7} - 2 \beta_{6} + \beta_{5} + \beta_{4} + \beta_{2} - 2 \beta_1 + 2) q^{59} + (\beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - 1) q^{60} + ( - \beta_{6} - \beta_{4} + 2 \beta_{3} - \beta_1 - 2) q^{61} + ( - 2 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta_1) q^{62} + ( - 2 \beta_{7} + 2 \beta_{5} + 2 \beta_{3} + \beta_{2} + 2 \beta_1 + 6) q^{64} + ( - \beta_{7} - \beta_{6} + \beta_{5} + 4 \beta_{4} - \beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{65} + ( - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{2} + 2) q^{66} + ( - \beta_{7} - 2 \beta_{6} + \beta_{5} + 2 \beta_{4} + \beta_1 + 3) q^{67} + ( - \beta_{7} + \beta_{6} + 2 \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 6) q^{68} - q^{69} + (2 \beta_{6} - 3 \beta_{5} + \beta_{4} - 2 \beta_{2}) q^{71} + (\beta_{5} - \beta_{4} + \beta_{2} + \beta_1 + 2) q^{72} + (2 \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - 3 \beta_1 + 1) q^{73} + ( - 2 \beta_{6} - \beta_{5} + 2 \beta_{4} - \beta_{3} - \beta_1 + 2) q^{74} + ( - 2 \beta_{2} + \beta_1 + 3) q^{75} + ( - 4 \beta_{7} + \beta_{6} + 4 \beta_{5} + 2 \beta_1 - 1) q^{76} + (2 \beta_{7} - 2 \beta_{5} - \beta_{2}) q^{78} + (\beta_{6} - \beta_{5} + \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 2) q^{79} + (\beta_{7} - 4 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 3) q^{80} + q^{81} + ( - 2 \beta_{7} + \beta_{5} + \beta_{4} + \beta_{2} + 2) q^{82} + (2 \beta_{7} - 2 \beta_{6} + 2 \beta_{2} - 2 \beta_1 + 4) q^{83} + (2 \beta_{7} + \beta_{5} + 3 \beta_{4} + 2 \beta_{2} - 2 \beta_1 - 4) q^{85} + (2 \beta_{7} - 2 \beta_{6} - 3 \beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_1 - 5) q^{86} + ( - \beta_{5} - \beta_{3} - \beta_1) q^{87} + ( - 2 \beta_{7} + \beta_{6} + \beta_{5} - 3 \beta_{4} + 3 \beta_{3} + 4 \beta_{2} + \cdots + 12) q^{88}+ \cdots + (\beta_{6} - \beta_{4} + \beta_1 + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{2} + 8 q^{3} + 15 q^{4} + 5 q^{5} + q^{6} + 9 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + q^{2} + 8 q^{3} + 15 q^{4} + 5 q^{5} + q^{6} + 9 q^{8} + 8 q^{9} - 3 q^{10} + 10 q^{11} + 15 q^{12} - 6 q^{13} + 5 q^{15} + 13 q^{16} + 21 q^{17} + q^{18} + 5 q^{19} - q^{20} + 18 q^{22} - 8 q^{23} + 9 q^{24} + 27 q^{25} + 3 q^{26} + 8 q^{27} + 2 q^{29} - 3 q^{30} - 13 q^{31} + 29 q^{32} + 10 q^{33} - 19 q^{34} + 15 q^{36} + 13 q^{37} - 6 q^{38} - 6 q^{39} + 7 q^{40} + 16 q^{41} + 15 q^{43} + 24 q^{44} + 5 q^{45} - q^{46} - q^{47} + 13 q^{48} + 16 q^{50} + 21 q^{51} - 19 q^{52} + 3 q^{53} + q^{54} - 10 q^{55} + 5 q^{57} - 40 q^{58} + 26 q^{59} - q^{60} - 14 q^{61} - 14 q^{62} + 49 q^{64} - 3 q^{65} + 18 q^{66} + 38 q^{67} + 43 q^{68} - 8 q^{69} + 9 q^{71} + 9 q^{72} - 6 q^{73} + 32 q^{74} + 27 q^{75} - 14 q^{76} + 3 q^{78} + 23 q^{79} - 17 q^{80} + 8 q^{81} + 20 q^{82} + 30 q^{83} - 37 q^{85} - 28 q^{86} + 2 q^{87} + 86 q^{88} + 12 q^{89} - 3 q^{90} - 15 q^{92} - 13 q^{93} - 45 q^{94} + 16 q^{95} + 29 q^{96} + 14 q^{97} + 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} - 15x^{6} + 11x^{5} + 75x^{4} - 35x^{3} - 141x^{2} + 37x + 80 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} - \nu^{5} - 10\nu^{4} + 8\nu^{3} + 25\nu^{2} - 13\nu - 14 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} - \nu^{5} - 12\nu^{4} + 8\nu^{3} + 41\nu^{2} - 11\nu - 34 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} - \nu^{5} - 12\nu^{4} + 10\nu^{3} + 39\nu^{2} - 21\nu - 30 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} - \nu^{6} - 12\nu^{5} + 8\nu^{4} + 41\nu^{3} - 9\nu^{2} - 36\nu - 10 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{6} - 2\nu^{5} - 12\nu^{4} + 18\nu^{3} + 41\nu^{2} - 32\nu - 34 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} - \beta_{4} + \beta_{2} + 5\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{4} + \beta_{3} + 8\beta_{2} + \beta _1 + 22 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{7} + 10\beta_{5} - 8\beta_{4} + 10\beta_{2} + 29\beta _1 + 20 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -2\beta_{7} + 2\beta_{5} - 10\beta_{4} + 12\beta_{3} + 57\beta_{2} + 12\beta _1 + 138 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -26\beta_{7} + 2\beta_{6} + 81\beta_{5} - 57\beta_{4} + 4\beta_{3} + 81\beta_{2} + 183\beta _1 + 166 \) 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
−2.55222
−1.87992
−1.67781
−0.818544
1.19194
1.47391
2.51679
2.74585
−2.55222 1.00000 4.51383 0.648159 −2.55222 0 −6.41586 1.00000 −1.65425
1.2 −1.87992 1.00000 1.53408 −2.65555 −1.87992 0 0.875886 1.00000 4.99220
1.3 −1.67781 1.00000 0.815035 2.94824 −1.67781 0 1.98814 1.00000 −4.94658
1.4 −0.818544 1.00000 −1.32999 3.72041 −0.818544 0 2.72574 1.00000 −3.04532
1.5 1.19194 1.00000 −0.579274 −3.78820 1.19194 0 −3.07435 1.00000 −4.51532
1.6 1.47391 1.00000 0.172415 3.62341 1.47391 0 −2.69370 1.00000 5.34058
1.7 2.51679 1.00000 4.33421 2.41834 2.51679 0 5.87470 1.00000 6.08644
1.8 2.74585 1.00000 5.53968 −1.91481 2.74585 0 9.71944 1.00000 −5.25777
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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.bf 8
7.b odd 2 1 3381.2.a.be 8
7.c even 3 2 483.2.i.g 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.i.g 16 7.c even 3 2
3381.2.a.be 8 7.b odd 2 1
3381.2.a.bf 8 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3381))\):

\( T_{2}^{8} - T_{2}^{7} - 15T_{2}^{6} + 11T_{2}^{5} + 75T_{2}^{4} - 35T_{2}^{3} - 141T_{2}^{2} + 37T_{2} + 80 \) Copy content Toggle raw display
\( T_{5}^{8} - 5T_{5}^{7} - 21T_{5}^{6} + 127T_{5}^{5} + 81T_{5}^{4} - 933T_{5}^{3} + 291T_{5}^{2} + 2013T_{5} - 1200 \) Copy content Toggle raw display
\( T_{11}^{8} - 10T_{11}^{7} - 4T_{11}^{6} + 256T_{11}^{5} - 304T_{11}^{4} - 1864T_{11}^{3} + 2704T_{11}^{2} + 3056T_{11} - 4384 \) Copy content Toggle raw display
\( T_{13}^{8} + 6T_{13}^{7} - 38T_{13}^{6} - 158T_{13}^{5} + 660T_{13}^{4} + 930T_{13}^{3} - 4602T_{13}^{2} + 2370T_{13} + 2127 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{7} - 15 T^{6} + 11 T^{5} + \cdots + 80 \) Copy content Toggle raw display
$3$ \( (T - 1)^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 5 T^{7} - 21 T^{6} + \cdots - 1200 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} - 10 T^{7} - 4 T^{6} + \cdots - 4384 \) Copy content Toggle raw display
$13$ \( T^{8} + 6 T^{7} - 38 T^{6} + \cdots + 2127 \) Copy content Toggle raw display
$17$ \( T^{8} - 21 T^{7} + 139 T^{6} + \cdots + 67182 \) Copy content Toggle raw display
$19$ \( T^{8} - 5 T^{7} - 94 T^{6} + \cdots - 99108 \) Copy content Toggle raw display
$23$ \( (T + 1)^{8} \) Copy content Toggle raw display
$29$ \( T^{8} - 2 T^{7} - 108 T^{6} + \cdots + 34688 \) Copy content Toggle raw display
$31$ \( T^{8} + 13 T^{7} - 22 T^{6} + \cdots + 10896 \) Copy content Toggle raw display
$37$ \( T^{8} - 13 T^{7} - 52 T^{6} + \cdots - 19776 \) Copy content Toggle raw display
$41$ \( T^{8} - 16 T^{7} + 40 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$43$ \( T^{8} - 15 T^{7} - 84 T^{6} + \cdots - 937796 \) Copy content Toggle raw display
$47$ \( T^{8} + T^{7} - 185 T^{6} + \cdots - 1290798 \) Copy content Toggle raw display
$53$ \( T^{8} - 3 T^{7} - 113 T^{6} + \cdots + 29772 \) Copy content Toggle raw display
$59$ \( T^{8} - 26 T^{7} + 100 T^{6} + \cdots + 1087200 \) Copy content Toggle raw display
$61$ \( T^{8} + 14 T^{7} - 152 T^{6} + \cdots + 146592 \) Copy content Toggle raw display
$67$ \( T^{8} - 38 T^{7} + 404 T^{6} + \cdots - 2418489 \) Copy content Toggle raw display
$71$ \( T^{8} - 9 T^{7} - 421 T^{6} + \cdots + 44639898 \) Copy content Toggle raw display
$73$ \( T^{8} + 6 T^{7} - 254 T^{6} + \cdots - 302829 \) Copy content Toggle raw display
$79$ \( T^{8} - 23 T^{7} - 178 T^{6} + \cdots - 14818292 \) Copy content Toggle raw display
$83$ \( T^{8} - 30 T^{7} + 36 T^{6} + \cdots + 9331200 \) Copy content Toggle raw display
$89$ \( T^{8} - 12 T^{7} + \cdots + 111729608 \) Copy content Toggle raw display
$97$ \( T^{8} - 14 T^{7} - 300 T^{6} + \cdots + 1304736 \) Copy content Toggle raw display
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