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Properties

Label 2768.2.a.k
Level $2768$
Weight $2$
Character orbit 2768.a
Self dual yes
Analytic conductor $22.103$
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] = [2768,2,Mod(1,2768)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2768, 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("2768.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2768 = 2^{4} \cdot 173 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2768.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: \(22.1025912795\)
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: \( x^{10} - x^{9} - 16x^{8} + 16x^{7} + 85x^{6} - 80x^{5} - 175x^{4} + 136x^{3} + 138x^{2} - 71x - 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 173)
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.

\(f(q)\) \(=\) \( q + (\beta_{3} - 1) q^{3} + \beta_{9} q^{5} + ( - \beta_{5} - \beta_{4} - 1) q^{7} + ( - \beta_{8} - \beta_{6} + \beta_{5} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - 1) q^{3} + \beta_{9} q^{5} + ( - \beta_{5} - \beta_{4} - 1) q^{7} + ( - \beta_{8} - \beta_{6} + \beta_{5} + \cdots + 1) q^{9}+ \cdots + (2 \beta_{9} + 2 \beta_{8} + 2 \beta_{7} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 8 q^{3} + q^{5} - 11 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 8 q^{3} + q^{5} - 11 q^{7} + 12 q^{9} - 5 q^{11} + q^{13} + 4 q^{15} - 2 q^{17} - 7 q^{19} - 5 q^{21} - 5 q^{23} + 9 q^{25} - 23 q^{27} - 5 q^{29} - 3 q^{31} - 11 q^{33} + 7 q^{35} + 8 q^{37} + 14 q^{39} - 21 q^{41} - 48 q^{43} - 11 q^{45} + 18 q^{47} + 11 q^{49} + 6 q^{51} + 6 q^{53} + 6 q^{55} - 2 q^{57} + 2 q^{59} - 3 q^{61} - 7 q^{63} - 38 q^{65} - 62 q^{67} + 3 q^{69} - 6 q^{71} + 7 q^{73} - 43 q^{75} - 11 q^{77} - 17 q^{79} + 6 q^{81} - 4 q^{83} - 25 q^{85} + 6 q^{87} - 20 q^{89} - 32 q^{91} - 30 q^{93} - 17 q^{95} - q^{97} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} - 16x^{8} + 16x^{7} + 85x^{6} - 80x^{5} - 175x^{4} + 136x^{3} + 138x^{2} - 71x - 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 3 \nu^{9} - 12 \nu^{8} + 46 \nu^{7} + 182 \nu^{6} - 215 \nu^{5} - 835 \nu^{4} + 294 \nu^{3} + \cdots - 247 ) / 116 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 9 \nu^{9} + 22 \nu^{8} + 138 \nu^{7} - 324 \nu^{6} - 645 \nu^{5} + 1439 \nu^{4} + 940 \nu^{3} + \cdots + 419 ) / 116 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11 \nu^{9} - 14 \nu^{8} - 130 \nu^{7} + 222 \nu^{6} + 305 \nu^{5} - 1037 \nu^{4} + 604 \nu^{3} + \cdots - 467 ) / 116 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 13 \nu^{9} - 23 \nu^{8} + 209 \nu^{7} + 315 \nu^{6} - 1154 \nu^{5} - 1279 \nu^{4} + 2521 \nu^{3} + \cdots - 36 ) / 116 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 20 \nu^{9} + 7 \nu^{8} + 297 \nu^{7} - 111 \nu^{6} - 1385 \nu^{5} + 446 \nu^{4} + 2221 \nu^{3} + \cdots - 187 ) / 116 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 20 \nu^{9} - 7 \nu^{8} - 297 \nu^{7} + 111 \nu^{6} + 1385 \nu^{5} - 446 \nu^{4} - 2105 \nu^{3} + \cdots - 161 ) / 116 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 20 \nu^{9} + 7 \nu^{8} + 297 \nu^{7} - 111 \nu^{6} - 1385 \nu^{5} + 446 \nu^{4} + 2105 \nu^{3} + \cdots - 303 ) / 116 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 7 \nu^{9} + 30 \nu^{8} + 88 \nu^{7} - 426 \nu^{6} - 231 \nu^{5} + 1783 \nu^{4} - 358 \nu^{3} + \cdots + 371 ) / 58 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} + \beta_{7} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{8} + \beta_{6} + 5\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} + 7\beta_{8} + 8\beta_{7} - \beta_{6} + 2\beta_{5} + 22 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{9} - 9\beta_{8} - 2\beta_{7} + 8\beta_{6} - 3\beta_{5} - \beta_{4} + 3\beta_{3} + 3\beta_{2} + 28\beta _1 - 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 14 \beta_{9} + 47 \beta_{8} + 58 \beta_{7} - 10 \beta_{6} + 23 \beta_{5} + 3 \beta_{4} - 5 \beta_{3} + \cdots + 137 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 28 \beta_{9} - 70 \beta_{8} - 29 \beta_{7} + 56 \beta_{6} - 41 \beta_{5} - 11 \beta_{4} + 43 \beta_{3} + \cdots - 83 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 142 \beta_{9} + 320 \beta_{8} + 416 \beta_{7} - 83 \beta_{6} + 209 \beta_{5} + 45 \beta_{4} - 73 \beta_{3} + \cdots + 900 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 283 \beta_{9} - 530 \beta_{8} - 300 \beta_{7} + 387 \beta_{6} - 411 \beta_{5} - 95 \beta_{4} + \cdots - 655 \) 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
0.780301
−1.14366
−2.73030
2.26133
−1.25011
1.33077
1.72153
−2.31385
2.60667
−0.262676
0 −3.35054 0 1.01452 0 4.00811 0 8.22611 0
1.2 0 −2.99556 0 −3.35439 0 −4.18849 0 5.97338 0
1.3 0 −2.39046 0 2.43842 0 −3.51801 0 2.71432 0
1.4 0 −1.77282 0 −3.97878 0 1.30563 0 0.142887 0
1.5 0 −1.40688 0 3.06970 0 1.06630 0 −1.02068 0
1.6 0 −1.35503 0 0.350532 0 −3.19681 0 −1.16389 0
1.7 0 0.431830 0 3.14293 0 0.331344 0 −2.81352 0
1.8 0 0.815067 0 −2.17862 0 −1.18973 0 −2.33567 0
1.9 0 1.71290 0 −0.321822 0 −1.28506 0 −0.0659809 0
1.10 0 2.31150 0 0.817507 0 −4.33328 0 2.34304 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
\(2\) \( -1 \)
\(173\) \( -1 \)

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 2768.2.a.k 10
4.b odd 2 1 173.2.a.b ✓ 10
12.b even 2 1 1557.2.a.d 10
20.d odd 2 1 4325.2.a.e 10
28.d even 2 1 8477.2.a.n 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
173.2.a.b ✓ 10 4.b odd 2 1
1557.2.a.d 10 12.b even 2 1
2768.2.a.k 10 1.a even 1 1 trivial
4325.2.a.e 10 20.d odd 2 1
8477.2.a.n 10 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} + 8 T_{3}^{9} + 11 T_{3}^{8} - 59 T_{3}^{7} - 165 T_{3}^{6} + 55 T_{3}^{5} + 484 T_{3}^{4} + \cdots + 113 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2768))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + 8 T^{9} + \cdots + 113 \) Copy content Toggle raw display
$5$ \( T^{10} - T^{9} + \cdots + 64 \) Copy content Toggle raw display
$7$ \( T^{10} + 11 T^{9} + \cdots + 577 \) Copy content Toggle raw display
$11$ \( T^{10} + 5 T^{9} + \cdots - 25 \) Copy content Toggle raw display
$13$ \( T^{10} - T^{9} + \cdots + 5285 \) Copy content Toggle raw display
$17$ \( T^{10} + 2 T^{9} + \cdots + 6464 \) Copy content Toggle raw display
$19$ \( T^{10} + 7 T^{9} + \cdots + 7 \) Copy content Toggle raw display
$23$ \( T^{10} + 5 T^{9} + \cdots - 832 \) Copy content Toggle raw display
$29$ \( T^{10} + 5 T^{9} + \cdots + 4141 \) Copy content Toggle raw display
$31$ \( T^{10} + 3 T^{9} + \cdots - 583744 \) Copy content Toggle raw display
$37$ \( T^{10} - 8 T^{9} + \cdots + 2245561 \) Copy content Toggle raw display
$41$ \( T^{10} + 21 T^{9} + \cdots - 74423567 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 710485184 \) Copy content Toggle raw display
$47$ \( T^{10} - 18 T^{9} + \cdots + 1469120 \) Copy content Toggle raw display
$53$ \( T^{10} - 6 T^{9} + \cdots + 10368704 \) Copy content Toggle raw display
$59$ \( T^{10} - 2 T^{9} + \cdots + 22113776 \) Copy content Toggle raw display
$61$ \( T^{10} + 3 T^{9} + \cdots + 60224 \) Copy content Toggle raw display
$67$ \( T^{10} + 62 T^{9} + \cdots - 5737408 \) Copy content Toggle raw display
$71$ \( T^{10} + 6 T^{9} + \cdots + 2198771 \) Copy content Toggle raw display
$73$ \( T^{10} - 7 T^{9} + \cdots + 39229645 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 4525919897 \) Copy content Toggle raw display
$83$ \( T^{10} + 4 T^{9} + \cdots - 5440448 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 124693673 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots - 3716511808 \) Copy content Toggle raw display
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