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Properties

Label 8384.2.a.bv
Level $8384$
Weight $2$
Character orbit 8384.a
Self dual yes
Analytic conductor $66.947$
Analytic rank $1$
Dimension $10$
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] = [8384,2,Mod(1,8384)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8384, 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("8384.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8384 = 2^{6} \cdot 131 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8384.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: \(66.9465770546\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 3x^{9} - 13x^{8} + 45x^{7} + 26x^{6} - 155x^{5} + 21x^{4} + 168x^{3} - 55x^{2} - 30x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 4192)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{9} q^{5} + (\beta_{8} - \beta_{5} - \beta_{3} + \cdots - 1) q^{7}+ \cdots + (\beta_{3} + \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{9} q^{5} + (\beta_{8} - \beta_{5} - \beta_{3} + \cdots - 1) q^{7}+ \cdots + (\beta_{6} - \beta_{5} + 2 \beta_{3} + \cdots + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 3 q^{3} - 2 q^{5} - 11 q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 3 q^{3} - 2 q^{5} - 11 q^{7} + 5 q^{9} + 12 q^{11} - 5 q^{13} - 8 q^{15} + 14 q^{17} + 14 q^{19} - 19 q^{21} - 20 q^{23} + 14 q^{25} - 9 q^{27} + 2 q^{29} - 24 q^{31} - 27 q^{33} + 7 q^{35} - 8 q^{37} + 6 q^{39} + 3 q^{41} - 7 q^{43} + 7 q^{45} - 30 q^{47} + 3 q^{49} - 10 q^{51} + 2 q^{53} + 2 q^{55} - 26 q^{57} - 11 q^{59} + 3 q^{61} - 49 q^{63} - 2 q^{65} - 4 q^{67} + 4 q^{69} - 40 q^{71} + 4 q^{73} + 40 q^{75} - 18 q^{77} - 26 q^{79} + 18 q^{81} + 36 q^{83} + 10 q^{85} + 6 q^{89} - q^{91} + 24 q^{93} - 56 q^{95} - 6 q^{97} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 3x^{9} - 13x^{8} + 45x^{7} + 26x^{6} - 155x^{5} + 21x^{4} + 168x^{3} - 55x^{2} - 30x - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 53 \nu^{9} + 183 \nu^{8} + 539 \nu^{7} - 2506 \nu^{6} + 540 \nu^{5} + 6460 \nu^{4} - 4911 \nu^{3} + \cdots - 13 ) / 593 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 53 \nu^{9} - 183 \nu^{8} - 539 \nu^{7} + 2506 \nu^{6} - 540 \nu^{5} - 6460 \nu^{4} + 4911 \nu^{3} + \cdots - 2359 ) / 593 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 71 \nu^{9} + \nu^{8} - 1371 \nu^{7} + 90 \nu^{6} + 8373 \nu^{5} - 710 \nu^{4} - 18439 \nu^{3} + \cdots - 363 ) / 593 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 152 \nu^{9} - 357 \nu^{8} - 2150 \nu^{7} + 5229 \nu^{6} + 6675 \nu^{5} - 16345 \nu^{4} - 6693 \nu^{3} + \cdots + 317 ) / 593 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 181 \nu^{9} + 390 \nu^{8} + 2568 \nu^{7} - 5817 \nu^{6} - 7890 \nu^{5} + 19007 \nu^{4} + 5438 \nu^{3} + \cdots + 750 ) / 593 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 268 \nu^{9} + 489 \nu^{8} + 3822 \nu^{7} - 6988 \nu^{6} - 12128 \nu^{5} + 19284 \nu^{4} + \cdots - 1386 ) / 593 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 290 \nu^{9} - 923 \nu^{8} - 3587 \nu^{7} + 13589 \nu^{6} + 5034 \nu^{5} - 44410 \nu^{4} + 12550 \nu^{3} + \cdots - 4740 ) / 593 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 362 \nu^{9} + 780 \nu^{8} + 5136 \nu^{7} - 11634 \nu^{6} - 15780 \nu^{5} + 38014 \nu^{4} + \cdots + 2686 ) / 593 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} - 2\beta_{6} + 7\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{9} - \beta_{8} + \beta_{6} + 2\beta_{5} - 2\beta_{4} + 9\beta_{3} + 10\beta_{2} - 3\beta _1 + 29 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 10\beta_{9} - 2\beta_{8} + 2\beta_{7} - 24\beta_{6} + 2\beta_{5} - \beta_{4} - \beta_{3} - 4\beta_{2} + 63\beta _1 - 32 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 18 \beta_{9} - 15 \beta_{8} + 3 \beta_{7} + 15 \beta_{6} + 25 \beta_{5} - 27 \beta_{4} + 85 \beta_{3} + \cdots + 260 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 92 \beta_{9} - 30 \beta_{8} + 30 \beta_{7} - 249 \beta_{6} + 23 \beta_{5} - 15 \beta_{4} - 27 \beta_{3} + \cdots - 402 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 232 \beta_{9} - 172 \beta_{8} + 45 \beta_{7} + 194 \beta_{6} + 254 \beta_{5} - 294 \beta_{4} + \cdots + 2514 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 873 \beta_{9} - 332 \beta_{8} + 339 \beta_{7} - 2509 \beta_{6} + 193 \beta_{5} - 145 \beta_{4} + \cdots - 4659 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
−3.23498
−1.51422
−1.34480
−0.235492
−0.0820520
0.896916
1.47958
1.65681
2.39758
2.98067
0 −3.23498 0 2.21911 0 −1.88686 0 7.46512 0
1.2 0 −1.51422 0 −1.41941 0 −1.84437 0 −0.707153 0
1.3 0 −1.34480 0 −0.932160 0 2.42597 0 −1.19151 0
1.4 0 −0.235492 0 0.752425 0 3.53863 0 −2.94454 0
1.5 0 −0.0820520 0 −3.46538 0 −2.77305 0 −2.99327 0
1.6 0 0.896916 0 3.48072 0 −2.59945 0 −2.19554 0
1.7 0 1.47958 0 −1.51000 0 0.687606 0 −0.810850 0
1.8 0 1.65681 0 3.29512 0 −0.949290 0 −0.254991 0
1.9 0 2.39758 0 −4.28307 0 −2.58795 0 2.74837 0
1.10 0 2.98067 0 −0.137350 0 −5.01123 0 5.88438 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(131\) \( -1 \)

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 8384.2.a.bv 10
4.b odd 2 1 8384.2.a.bs 10
8.b even 2 1 4192.2.a.k ✓ 10
8.d odd 2 1 4192.2.a.l yes 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4192.2.a.k ✓ 10 8.b even 2 1
4192.2.a.l yes 10 8.d odd 2 1
8384.2.a.bs 10 4.b odd 2 1
8384.2.a.bv 10 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8384))\):

\( T_{3}^{10} - 3T_{3}^{9} - 13T_{3}^{8} + 45T_{3}^{7} + 26T_{3}^{6} - 155T_{3}^{5} + 21T_{3}^{4} + 168T_{3}^{3} - 55T_{3}^{2} - 30T_{3} - 2 \) Copy content Toggle raw display
\( T_{5}^{10} + 2 T_{5}^{9} - 30 T_{5}^{8} - 50 T_{5}^{7} + 279 T_{5}^{6} + 412 T_{5}^{5} - 772 T_{5}^{4} + \cdots + 78 \) Copy content Toggle raw display
\( T_{7}^{10} + 11 T_{7}^{9} + 24 T_{7}^{8} - 134 T_{7}^{7} - 679 T_{7}^{6} - 480 T_{7}^{5} + 2579 T_{7}^{4} + \cdots - 1823 \) Copy content Toggle raw display
\( T_{11}^{10} - 12 T_{11}^{9} + 26 T_{11}^{8} + 220 T_{11}^{7} - 1189 T_{11}^{6} + 820 T_{11}^{5} + \cdots + 996 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} - 3 T^{9} + \cdots - 2 \) Copy content Toggle raw display
$5$ \( T^{10} + 2 T^{9} + \cdots + 78 \) Copy content Toggle raw display
$7$ \( T^{10} + 11 T^{9} + \cdots - 1823 \) Copy content Toggle raw display
$11$ \( T^{10} - 12 T^{9} + \cdots + 996 \) Copy content Toggle raw display
$13$ \( T^{10} + 5 T^{9} + \cdots + 1782 \) Copy content Toggle raw display
$17$ \( T^{10} - 14 T^{9} + \cdots + 90112 \) Copy content Toggle raw display
$19$ \( T^{10} - 14 T^{9} + \cdots + 256 \) Copy content Toggle raw display
$23$ \( T^{10} + 20 T^{9} + \cdots - 264704 \) Copy content Toggle raw display
$29$ \( T^{10} - 2 T^{9} + \cdots + 205056 \) Copy content Toggle raw display
$31$ \( T^{10} + 24 T^{9} + \cdots - 26502656 \) Copy content Toggle raw display
$37$ \( T^{10} + 8 T^{9} + \cdots - 585728 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots - 301397791 \) Copy content Toggle raw display
$43$ \( T^{10} + 7 T^{9} + \cdots - 29320224 \) Copy content Toggle raw display
$47$ \( T^{10} + 30 T^{9} + \cdots - 98304 \) Copy content Toggle raw display
$53$ \( T^{10} - 2 T^{9} + \cdots - 31457696 \) Copy content Toggle raw display
$59$ \( T^{10} + 11 T^{9} + \cdots + 155502 \) Copy content Toggle raw display
$61$ \( T^{10} - 3 T^{9} + \cdots - 419136 \) Copy content Toggle raw display
$67$ \( T^{10} + 4 T^{9} + \cdots - 196352 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 1377656832 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots - 115268096 \) Copy content Toggle raw display
$79$ \( T^{10} + 26 T^{9} + \cdots + 5325312 \) Copy content Toggle raw display
$83$ \( T^{10} - 36 T^{9} + \cdots + 49318656 \) Copy content Toggle raw display
$89$ \( T^{10} - 6 T^{9} + \cdots + 1907408 \) Copy content Toggle raw display
$97$ \( T^{10} + 6 T^{9} + \cdots - 51355648 \) Copy content Toggle raw display
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