Properties

Label 4864.2.a.bs
Level $4864$
Weight $2$
Character orbit 4864.a
Self dual yes
Analytic conductor $38.839$
Analytic rank $1$
Dimension $10$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4864,2,Mod(1,4864)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4864, 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("4864.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4864 = 2^{8} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4864.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: \(38.8392355432\)
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: \( x^{10} - 2x^{9} - 23x^{8} + 44x^{7} + 167x^{6} - 266x^{5} - 491x^{4} + 460x^{3} + 546x^{2} + 56x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 2432)
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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} - \beta_1 q^{5} + ( - \beta_{6} + \beta_1) q^{7} + ( - \beta_{7} - \beta_{4} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{3} - \beta_1 q^{5} + ( - \beta_{6} + \beta_1) q^{7} + ( - \beta_{7} - \beta_{4} + 1) q^{9} + ( - \beta_{5} - 2) q^{11} + \beta_{6} q^{13} + ( - \beta_{8} + \beta_{3}) q^{15} + (\beta_{7} + \beta_{5} - \beta_{4}) q^{17} - q^{19} + (\beta_{8} + \beta_{6} - \beta_{3} - \beta_{2}) q^{21} + (\beta_{8} + \beta_{2} + \beta_1) q^{23} + (\beta_{5} + 1) q^{25} + ( - \beta_{9} + 2 \beta_{7} + \beta_{5} + 2 \beta_{4} - 2) q^{27} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{29} + (\beta_{6} - \beta_{3} - \beta_{2} + \beta_1) q^{31} + (\beta_{9} - \beta_{5} - 3 \beta_{4} - 2) q^{33} + (\beta_{9} + \beta_{7} - \beta_{4} - 4) q^{35} + ( - \beta_{8} + \beta_{3} - \beta_{2}) q^{37} + ( - \beta_{6} + \beta_{2}) q^{39} + (\beta_{7} - 2 \beta_{4} - 2) q^{41} + (\beta_{9} - \beta_{7} - \beta_{4}) q^{43} + (\beta_{8} + \beta_{6} + 3 \beta_{2} - 2 \beta_1) q^{45} + ( - 2 \beta_{3} - \beta_{2} - \beta_1) q^{47} + ( - 2 \beta_{9} + \beta_{7} + \beta_{5} + \beta_{4} + 3) q^{49} + ( - \beta_{4} - 4) q^{51} + (2 \beta_{6} + \beta_{3} - \beta_{2} - \beta_1) q^{53} + (\beta_{8} + \beta_{6} - \beta_{2} + 4 \beta_1) q^{55} - \beta_{4} q^{57} + (\beta_{9} + \beta_{5} - 2 \beta_{4} - 6) q^{59} + ( - \beta_{8} - \beta_{6} + 2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{61} + ( - \beta_{2} + 3 \beta_1) q^{63} + ( - \beta_{9} - \beta_{7} - \beta_{5} + \beta_{4} - 2) q^{65} + ( - 2 \beta_{9} + 3 \beta_{4}) q^{67} + ( - \beta_{8} - 4 \beta_{2} + 3 \beta_1) q^{69} + (\beta_{8} + \beta_{6} + \beta_{2} - \beta_1) q^{71} + ( - \beta_{7} - \beta_{5} - 3 \beta_{4}) q^{73} + ( - \beta_{9} + \beta_{5} + 2 \beta_{4} + 2) q^{75} + ( - \beta_{8} + \beta_{6} + 2 \beta_{3} + 3 \beta_{2} - 2 \beta_1) q^{77} + ( - \beta_{8} + 2 \beta_{6} + 3 \beta_{3}) q^{79} + (2 \beta_{9} - 2 \beta_{7} - 6 \beta_{4} + 1) q^{81} + ( - \beta_{9} - \beta_{7} + \beta_{5} - \beta_{4} - 2) q^{83} + ( - 2 \beta_{6} - 2 \beta_{3} - 2 \beta_{2} - \beta_1) q^{85} + ( - 2 \beta_{8} + \beta_{6} - \beta_{2}) q^{87} + (\beta_{7} - 2 \beta_{5} - 2) q^{89} + (\beta_{9} - 2 \beta_{7} - \beta_{5} - 6) q^{91} + (2 \beta_{8} - 2 \beta_{6} - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{93} + \beta_1 q^{95} + (\beta_{9} - \beta_{7} - \beta_{5} - 3 \beta_{4} - 4) q^{97} + (4 \beta_{7} + \beta_{5} - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 4 q^{3} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 4 q^{3} + 14 q^{9} - 20 q^{11} + 4 q^{17} - 10 q^{19} + 10 q^{25} - 28 q^{27} - 8 q^{33} - 36 q^{35} - 12 q^{41} + 4 q^{43} + 26 q^{49} - 36 q^{51} + 4 q^{57} - 52 q^{59} - 24 q^{65} - 12 q^{67} + 12 q^{73} + 12 q^{75} + 34 q^{81} - 16 q^{83} - 20 q^{89} - 60 q^{91} - 28 q^{97} - 60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} - 23x^{8} + 44x^{7} + 167x^{6} - 266x^{5} - 491x^{4} + 460x^{3} + 546x^{2} + 56x - 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{9} - 623 \nu^{8} + 315 \nu^{7} + 13464 \nu^{6} - 7337 \nu^{5} - 85439 \nu^{4} + 32563 \nu^{3} + 168722 \nu^{2} - 9026 \nu - 27588 ) / 10670 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{9} - 28 \nu^{8} - 55 \nu^{7} + 634 \nu^{6} + 803 \nu^{5} - 4424 \nu^{4} - 4707 \nu^{3} + 9522 \nu^{2} + 9854 \nu + 412 ) / 440 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 169 \nu^{9} + 443 \nu^{8} - 3995 \nu^{7} - 9419 \nu^{6} + 29777 \nu^{5} + 61339 \nu^{4} - 64653 \nu^{3} - 121837 \nu^{2} - 14134 \nu - 4432 ) / 10670 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 34 \nu^{9} - 36 \nu^{8} - 736 \nu^{7} + 809 \nu^{6} + 4488 \nu^{5} - 4820 \nu^{4} - 6418 \nu^{3} + 9377 \nu^{2} - 8900 \nu - 6404 ) / 2134 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 66 \nu^{9} - 281 \nu^{8} - 1520 \nu^{7} + 6118 \nu^{6} + 10846 \nu^{5} - 38291 \nu^{4} - 32372 \nu^{3} + 72802 \nu^{2} + 37112 \nu - 2252 ) / 2134 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2371 \nu^{9} + 4188 \nu^{8} + 51445 \nu^{7} - 92594 \nu^{6} - 337513 \nu^{5} + 555424 \nu^{4} + 855817 \nu^{3} - 999322 \nu^{2} - 816234 \nu + 17628 ) / 42680 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 79 \nu^{9} + 135 \nu^{8} + 1790 \nu^{7} - 3058 \nu^{6} - 12562 \nu^{5} + 19045 \nu^{4} + 33137 \nu^{3} - 34218 \nu^{2} - 27444 \nu - 1496 ) / 1067 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 941 \nu^{9} - 1333 \nu^{8} - 20775 \nu^{7} + 28099 \nu^{6} + 138083 \nu^{5} - 149029 \nu^{4} - 329347 \nu^{3} + 189377 \nu^{2} + 226834 \nu + 64792 ) / 10670 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 254 \nu^{9} - 423 \nu^{8} - 5738 \nu^{7} + 9045 \nu^{6} + 39930 \nu^{5} - 51009 \nu^{4} - 104754 \nu^{3} + 81201 \nu^{2} + 84284 \nu - 5116 ) / 2134 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{4} + \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{9} - \beta_{8} - \beta_{5} + \beta _1 + 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} + 7\beta_{5} - 19\beta_{4} + 18\beta_{3} - 5\beta_{2} + 13\beta _1 - 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 12 \beta_{9} - 10 \beta_{8} - 5 \beta_{7} + 5 \beta_{6} - 16 \beta_{5} - \beta_{4} - 8 \beta_{3} - \beta_{2} + 10 \beta _1 + 84 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 19 \beta_{9} - 15 \beta_{8} - 11 \beta_{7} + 3 \beta_{6} + 117 \beta_{5} - 217 \beta_{4} + 186 \beta_{3} - 79 \beta_{2} + 95 \beta _1 - 102 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 137 \beta_{9} - 109 \beta_{8} - 90 \beta_{7} + 78 \beta_{6} - 221 \beta_{5} + 14 \beta_{4} - 152 \beta_{3} - 6 \beta_{2} + 97 \beta _1 + 834 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 323 \beta_{9} - 151 \beta_{8} - 87 \beta_{7} - 93 \beta_{6} + 1613 \beta_{5} - 2541 \beta_{4} + 2086 \beta_{3} - 1039 \beta_{2} + 787 \beta _1 - 1990 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 1586 \beta_{9} - 1232 \beta_{8} - 1243 \beta_{7} + 983 \beta_{6} - 2934 \beta_{5} + 569 \beta_{4} - 2280 \beta_{3} + 69 \beta_{2} + 932 \beta _1 + 9144 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 4983 \beta_{9} - 1183 \beta_{8} - 387 \beta_{7} - 2433 \beta_{6} + 21113 \beta_{5} - 29965 \beta_{4} + 24662 \beta_{3} - 12859 \beta_{2} + 7115 \beta _1 - 31262 ) / 4 \) 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
3.22490
0.0787429
2.07657
−0.217867
2.76111
−2.30294
−3.58141
2.42960
−0.719482
−1.74923
0 −3.30364 0 −2.55295 0 1.32515 0 7.91406 0
1.2 0 −3.30364 0 2.55295 0 −1.32515 0 7.91406 0
1.3 0 −1.85871 0 −1.69983 0 4.99365 0 0.454788 0
1.4 0 −1.85871 0 1.69983 0 −4.99365 0 0.454788 0
1.5 0 −0.458177 0 −3.00397 0 −2.21624 0 −2.79007 0
1.6 0 −0.458177 0 3.00397 0 2.21624 0 −2.79007 0
1.7 0 1.15181 0 −1.07587 0 −0.238565 0 −1.67333 0
1.8 0 1.15181 0 1.07587 0 0.238565 0 −1.67333 0
1.9 0 2.46871 0 −3.22672 0 4.04213 0 3.09455 0
1.10 0 2.46871 0 3.22672 0 −4.04213 0 3.09455 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
\(2\) \(1\)
\(19\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4864.2.a.bs 10
4.b odd 2 1 4864.2.a.bt 10
8.b even 2 1 4864.2.a.bt 10
8.d odd 2 1 inner 4864.2.a.bs 10
16.e even 4 2 2432.2.c.j 20
16.f odd 4 2 2432.2.c.j 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2432.2.c.j 20 16.e even 4 2
2432.2.c.j 20 16.f odd 4 2
4864.2.a.bs 10 1.a even 1 1 trivial
4864.2.a.bs 10 8.d odd 2 1 inner
4864.2.a.bt 10 4.b odd 2 1
4864.2.a.bt 10 8.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4864))\):

\( T_{3}^{5} + 2T_{3}^{4} - 9T_{3}^{3} - 12T_{3}^{2} + 14T_{3} + 8 \) Copy content Toggle raw display
\( T_{5}^{10} - 30T_{5}^{8} + 329T_{5}^{6} - 1592T_{5}^{4} + 3216T_{5}^{2} - 2048 \) Copy content Toggle raw display
\( T_{7}^{10} - 48T_{7}^{8} + 694T_{7}^{6} - 3112T_{7}^{4} + 3689T_{7}^{2} - 200 \) Copy content Toggle raw display
\( T_{11}^{5} + 10T_{11}^{4} + 9T_{11}^{3} - 116T_{11}^{2} - 128T_{11} + 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( (T^{5} + 2 T^{4} - 9 T^{3} - 12 T^{2} + 14 T + 8)^{2} \) Copy content Toggle raw display
$5$ \( T^{10} - 30 T^{8} + 329 T^{6} + \cdots - 2048 \) Copy content Toggle raw display
$7$ \( T^{10} - 48 T^{8} + 694 T^{6} + \cdots - 200 \) Copy content Toggle raw display
$11$ \( (T^{5} + 10 T^{4} + 9 T^{3} - 116 T^{2} + \cdots + 256)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} - 42 T^{8} + 449 T^{6} + \cdots - 512 \) Copy content Toggle raw display
$17$ \( (T^{5} - 2 T^{4} - 42 T^{3} - 32 T^{2} + \cdots - 22)^{2} \) Copy content Toggle raw display
$19$ \( (T + 1)^{10} \) Copy content Toggle raw display
$23$ \( T^{10} - 154 T^{8} + 8049 T^{6} + \cdots - 1548800 \) Copy content Toggle raw display
$29$ \( T^{10} - 134 T^{8} + 5165 T^{6} + \cdots - 59168 \) Copy content Toggle raw display
$31$ \( T^{10} - 172 T^{8} + \cdots - 24780800 \) Copy content Toggle raw display
$37$ \( T^{10} - 172 T^{8} + 6260 T^{6} + \cdots - 12800 \) Copy content Toggle raw display
$41$ \( (T^{5} + 6 T^{4} - 66 T^{3} + 12 T^{2} + \cdots - 416)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} - 2 T^{4} - 75 T^{3} + 256 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} - 254 T^{8} + 21129 T^{6} + \cdots - 4524032 \) Copy content Toggle raw display
$53$ \( T^{10} - 254 T^{8} + \cdots - 67931168 \) Copy content Toggle raw display
$59$ \( (T^{5} + 26 T^{4} + 149 T^{3} - 592 T^{2} + \cdots + 8912)^{2} \) Copy content Toggle raw display
$61$ \( T^{10} - 382 T^{8} + \cdots - 633110528 \) Copy content Toggle raw display
$67$ \( (T^{5} + 6 T^{4} - 285 T^{3} - 1496 T^{2} + \cdots + 91880)^{2} \) Copy content Toggle raw display
$71$ \( T^{10} - 208 T^{8} + 10464 T^{6} + \cdots - 991232 \) Copy content Toggle raw display
$73$ \( (T^{5} - 6 T^{4} - 130 T^{3} + 232 T^{2} + \cdots + 470)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} - 764 T^{8} + \cdots - 5939628032 \) Copy content Toggle raw display
$83$ \( (T^{5} + 8 T^{4} - 132 T^{3} - 336 T^{2} + \cdots - 10624)^{2} \) Copy content Toggle raw display
$89$ \( (T^{5} + 10 T^{4} - 162 T^{3} + \cdots + 23696)^{2} \) Copy content Toggle raw display
$97$ \( (T^{5} + 14 T^{4} - 68 T^{3} - 1304 T^{2} + \cdots - 832)^{2} \) Copy content Toggle raw display
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