Properties

Label 8048.2.a.p
Level $8048$
Weight $2$
Character orbit 8048.a
Self dual yes
Analytic conductor $64.264$
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] = [8048,2,Mod(1,8048)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8048, 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("8048.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8048 = 2^{4} \cdot 503 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8048.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: \(64.2636035467\)
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: \( x^{10} - 2x^{9} - 9x^{8} + 14x^{7} + 27x^{6} - 27x^{5} - 34x^{4} + 14x^{3} + 17x^{2} + x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 503)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{3} - \beta_{2} q^{5} + ( - \beta_{9} + \beta_{5} + \beta_{4} - \beta_{2} + \beta_1) q^{7} + ( - \beta_{6} + \beta_{4} - \beta_{3} - \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{3} - \beta_{2} q^{5} + ( - \beta_{9} + \beta_{5} + \beta_{4} - \beta_{2} + \beta_1) q^{7} + ( - \beta_{6} + \beta_{4} - \beta_{3} - \beta_1 - 1) q^{9} + ( - \beta_{9} - \beta_{8} + \beta_{7} - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 + 1) q^{11} + (\beta_{7} + \beta_{3} - 2) q^{13} - \beta_{3} q^{15} + (2 \beta_{9} - \beta_{8} + \beta_{6} - 2 \beta_{4} + 2 \beta_{3} + \beta_{2} - \beta_1) q^{17} + (\beta_{9} + \beta_{8} - \beta_{5} - \beta_{4} + 2 \beta_{2}) q^{19} + ( - \beta_{9} - \beta_{8} + 2 \beta_{6} + \beta_{5} + \beta_{2} + 1) q^{21} + (3 \beta_{9} + \beta_{8} - 3 \beta_{7} - \beta_{6} - 2 \beta_{5} - \beta_{4} + 2 \beta_{2} - 2 \beta_1 + 1) q^{23} + ( - \beta_{7} + \beta_{6} + \beta_{2} - 2) q^{25} + ( - \beta_{8} - \beta_{6} - \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + \beta_1 - 1) q^{27} + (\beta_{9} + 2 \beta_{8} - 2 \beta_{7} - 2 \beta_{6} - \beta_{3} + \beta_{2} - 2) q^{29} + ( - \beta_{9} + \beta_{8} + \beta_{7} - \beta_{4} - \beta_{3} + \beta_1 + 2) q^{31} + ( - 2 \beta_{9} - \beta_{4} + \beta_1) q^{33} + (\beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{3} + 1) q^{35} + ( - 3 \beta_{9} + \beta_{8} + 2 \beta_{7} - \beta_{6} + \beta_{5} + 3 \beta_{4} - \beta_{3} + \cdots - 6) q^{37}+ \cdots + (\beta_{9} + 4 \beta_{8} - 4 \beta_{7} - \beta_{6} - 2 \beta_{3} + 3 \beta_{2} - \beta_1 - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 8 q^{3} - q^{5} + 5 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 8 q^{3} - q^{5} + 5 q^{7} - 2 q^{9} + 3 q^{11} - 18 q^{13} + 2 q^{15} - 11 q^{17} + q^{21} + 2 q^{23} - 27 q^{25} + 2 q^{27} - 9 q^{29} + 22 q^{31} - 10 q^{33} + 6 q^{35} - 35 q^{37} - 8 q^{39} - 4 q^{41} + 20 q^{43} + 2 q^{45} - 7 q^{47} - 27 q^{49} - 9 q^{51} - 24 q^{53} + 11 q^{55} - 23 q^{57} - 17 q^{59} - 4 q^{61} - 10 q^{63} - 16 q^{65} + 6 q^{67} - 2 q^{69} + q^{71} - 31 q^{73} - 30 q^{75} + 3 q^{77} + 10 q^{79} - 6 q^{81} - 22 q^{83} - 6 q^{85} - 25 q^{87} + q^{89} - 10 q^{91} - 6 q^{93} - 39 q^{95} - 57 q^{97} - 35 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} - 9x^{8} + 14x^{7} + 27x^{6} - 27x^{5} - 34x^{4} + 14x^{3} + 17x^{2} + x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 3\nu^{9} - 8\nu^{8} - 22\nu^{7} + 57\nu^{6} + 46\nu^{5} - 113\nu^{4} - 34\nu^{3} + 65\nu^{2} + 12\nu - 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 5\nu^{9} - 13\nu^{8} - 37\nu^{7} + 92\nu^{6} + 78\nu^{5} - 181\nu^{4} - 57\nu^{3} + 104\nu^{2} + 20\nu - 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -6\nu^{9} + 15\nu^{8} + 47\nu^{7} - 108\nu^{6} - 113\nu^{5} + 221\nu^{4} + 108\nu^{3} - 139\nu^{2} - 44\nu + 13 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 10\nu^{9} - 25\nu^{8} - 78\nu^{7} + 180\nu^{6} + 184\nu^{5} - 368\nu^{4} - 165\nu^{3} + 230\nu^{2} + 60\nu - 21 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -11\nu^{9} + 28\nu^{8} + 84\nu^{7} - 200\nu^{6} - 191\nu^{5} + 402\nu^{4} + 165\nu^{3} - 244\nu^{2} - 63\nu + 22 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -18\nu^{9} + 45\nu^{8} + 139\nu^{7} - 321\nu^{6} - 321\nu^{5} + 645\nu^{4} + 278\nu^{3} - 394\nu^{2} - 102\nu + 38 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( -21\nu^{9} + 53\nu^{8} + 162\nu^{7} - 380\nu^{6} - 375\nu^{5} + 770\nu^{4} + 331\nu^{3} - 475\nu^{2} - 127\nu + 45 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( -22\nu^{9} + 55\nu^{8} + 170\nu^{7} - 392\nu^{6} - 394\nu^{5} + 785\nu^{4} + 346\nu^{3} - 473\nu^{2} - 130\nu + 42 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{6} + \beta_{4} - \beta_{3} + \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} - 2\beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + 5\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{9} + \beta_{8} + 2\beta_{7} - 9\beta_{6} + 2\beta_{5} + 8\beta_{4} - 6\beta_{3} - 2\beta_{2} + 10\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{9} + 8 \beta_{8} + 3 \beta_{7} - 22 \beta_{6} + 9 \beta_{5} + 12 \beta_{4} - 9 \beta_{3} - 5 \beta_{2} + 34 \beta _1 - 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 8 \beta_{9} + 12 \beta_{8} + 19 \beta_{7} - 73 \beta_{6} + 22 \beta_{5} + 58 \beta_{4} - 37 \beta_{3} - 24 \beta_{2} + 83 \beta _1 - 28 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 12 \beta_{9} + 58 \beta_{8} + 37 \beta_{7} - 192 \beta_{6} + 73 \beta_{5} + 113 \beta_{4} - 71 \beta_{3} - 63 \beta_{2} + 252 \beta _1 - 107 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 58 \beta_{9} + 113 \beta_{8} + 152 \beta_{7} - 578 \beta_{6} + 192 \beta_{5} + 425 \beta_{4} - 247 \beta_{3} - 220 \beta_{2} + 661 \beta _1 - 278 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 113 \beta_{9} + 425 \beta_{8} + 345 \beta_{7} - 1565 \beta_{6} + 578 \beta_{5} + 967 \beta_{4} - 554 \beta_{3} - 591 \beta_{2} + 1920 \beta _1 - 879 \) 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
2.78533
1.95007
1.31567
1.07636
0.208270
−0.489003
−0.510671
−0.858231
−1.40552
−2.07227
0 −1.78533 0 −0.701114 0 2.02991 0 0.187388 0
1.2 0 −0.950069 0 −2.28693 0 −2.71022 0 −2.09737 0
1.3 0 −0.315672 0 2.25024 0 3.20647 0 −2.90035 0
1.4 0 −0.0763625 0 1.17276 0 −0.469303 0 −2.99417 0
1.5 0 0.791730 0 0.178789 0 0.0809018 0 −2.37316 0
1.6 0 1.48900 0 −1.79865 0 −0.552233 0 −0.782869 0
1.7 0 1.51067 0 −2.23445 0 3.60329 0 −0.717874 0
1.8 0 1.85823 0 1.44291 0 1.96509 0 0.453023 0
1.9 0 2.40552 0 0.590303 0 −1.95900 0 2.78655 0
1.10 0 3.07227 0 0.386144 0 −0.194914 0 6.43884 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
\(2\) \(-1\)
\(503\) \(-1\)

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 8048.2.a.p 10
4.b odd 2 1 503.2.a.e 10
12.b even 2 1 4527.2.a.k 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
503.2.a.e 10 4.b odd 2 1
4527.2.a.k 10 12.b even 2 1
8048.2.a.p 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8048))\):

\( T_{3}^{10} - 8T_{3}^{9} + 18T_{3}^{8} + 10T_{3}^{7} - 85T_{3}^{6} + 75T_{3}^{5} + 54T_{3}^{4} - 86T_{3}^{3} + 5T_{3}^{2} + 14T_{3} + 1 \) Copy content Toggle raw display
\( T_{5}^{10} + T_{5}^{9} - 11T_{5}^{8} - 7T_{5}^{7} + 41T_{5}^{6} + 7T_{5}^{5} - 59T_{5}^{4} + 16T_{5}^{3} + 18T_{5}^{2} - 9T_{5} + 1 \) Copy content Toggle raw display
\( T_{7}^{10} - 5T_{7}^{9} - 9T_{7}^{8} + 64T_{7}^{7} + 8T_{7}^{6} - 228T_{7}^{5} + 23T_{7}^{4} + 214T_{7}^{3} + 86T_{7}^{2} + 4T_{7} - 1 \) Copy content Toggle raw display
\( T_{13}^{10} + 18 T_{13}^{9} + 120 T_{13}^{8} + 307 T_{13}^{7} - 212 T_{13}^{6} - 2777 T_{13}^{5} - 4882 T_{13}^{4} - 481 T_{13}^{3} + 5985 T_{13}^{2} + 4107 T_{13} - 19 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} - 8 T^{9} + 18 T^{8} + 10 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{10} + T^{9} - 11 T^{8} - 7 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{10} - 5 T^{9} - 9 T^{8} + 64 T^{7} + \cdots - 1 \) Copy content Toggle raw display
$11$ \( T^{10} - 3 T^{9} - 41 T^{8} + 80 T^{7} + \cdots + 311 \) Copy content Toggle raw display
$13$ \( T^{10} + 18 T^{9} + 120 T^{8} + \cdots - 19 \) Copy content Toggle raw display
$17$ \( T^{10} + 11 T^{9} + 2 T^{8} + \cdots + 30151 \) Copy content Toggle raw display
$19$ \( T^{10} - 73 T^{8} - 144 T^{7} + \cdots + 8863 \) Copy content Toggle raw display
$23$ \( T^{10} - 2 T^{9} - 102 T^{8} + \cdots - 2281 \) Copy content Toggle raw display
$29$ \( T^{10} + 9 T^{9} - 65 T^{8} + \cdots - 1397 \) Copy content Toggle raw display
$31$ \( T^{10} - 22 T^{9} + 127 T^{8} + \cdots + 8207 \) Copy content Toggle raw display
$37$ \( T^{10} + 35 T^{9} + 437 T^{8} + \cdots - 3774629 \) Copy content Toggle raw display
$41$ \( T^{10} + 4 T^{9} - 150 T^{8} + \cdots - 17357 \) Copy content Toggle raw display
$43$ \( T^{10} - 20 T^{9} + 5 T^{8} + \cdots - 147629 \) Copy content Toggle raw display
$47$ \( T^{10} + 7 T^{9} - 148 T^{8} + \cdots + 34183 \) Copy content Toggle raw display
$53$ \( T^{10} + 24 T^{9} + 40 T^{8} + \cdots + 30585517 \) Copy content Toggle raw display
$59$ \( T^{10} + 17 T^{9} - 123 T^{8} + \cdots - 3373 \) Copy content Toggle raw display
$61$ \( T^{10} + 4 T^{9} - 358 T^{8} + \cdots + 160395869 \) Copy content Toggle raw display
$67$ \( T^{10} - 6 T^{9} - 352 T^{8} + \cdots - 52161527 \) Copy content Toggle raw display
$71$ \( T^{10} - T^{9} - 448 T^{8} + \cdots + 14183807 \) Copy content Toggle raw display
$73$ \( T^{10} + 31 T^{9} + 127 T^{8} + \cdots - 3955559 \) Copy content Toggle raw display
$79$ \( T^{10} - 10 T^{9} - 542 T^{8} + \cdots + 8912581 \) Copy content Toggle raw display
$83$ \( T^{10} + 22 T^{9} - 161 T^{8} + \cdots + 40035623 \) Copy content Toggle raw display
$89$ \( T^{10} - T^{9} - 380 T^{8} + \cdots - 789547 \) Copy content Toggle raw display
$97$ \( T^{10} + 57 T^{9} + \cdots + 3229338523 \) Copy content Toggle raw display
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