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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Newspace parameters

Level: \( N \) \(=\) \( 8048 = 2^{4} \cdot 503 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8048.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.2636035467\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - 2 x^{9} - 9 x^{8} + 14 x^{7} + 27 x^{6} - 27 x^{5} - 34 x^{4} + 14 x^{3} + 17 x^{2} + x - 1\)
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

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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( 3 \nu^{9} - 8 \nu^{8} - 22 \nu^{7} + 57 \nu^{6} + 46 \nu^{5} - 113 \nu^{4} - 34 \nu^{3} + 65 \nu^{2} + 12 \nu - 5 \)
\(\beta_{3}\)\(=\)\( 5 \nu^{9} - 13 \nu^{8} - 37 \nu^{7} + 92 \nu^{6} + 78 \nu^{5} - 181 \nu^{4} - 57 \nu^{3} + 104 \nu^{2} + 20 \nu - 8 \)
\(\beta_{4}\)\(=\)\( -6 \nu^{9} + 15 \nu^{8} + 47 \nu^{7} - 108 \nu^{6} - 113 \nu^{5} + 221 \nu^{4} + 108 \nu^{3} - 139 \nu^{2} - 44 \nu + 13 \)
\(\beta_{5}\)\(=\)\( 10 \nu^{9} - 25 \nu^{8} - 78 \nu^{7} + 180 \nu^{6} + 184 \nu^{5} - 368 \nu^{4} - 165 \nu^{3} + 230 \nu^{2} + 60 \nu - 21 \)
\(\beta_{6}\)\(=\)\( -11 \nu^{9} + 28 \nu^{8} + 84 \nu^{7} - 200 \nu^{6} - 191 \nu^{5} + 402 \nu^{4} + 165 \nu^{3} - 244 \nu^{2} - 63 \nu + 22 \)
\(\beta_{7}\)\(=\)\( -18 \nu^{9} + 45 \nu^{8} + 139 \nu^{7} - 321 \nu^{6} - 321 \nu^{5} + 645 \nu^{4} + 278 \nu^{3} - 394 \nu^{2} - 102 \nu + 38 \)
\(\beta_{8}\)\(=\)\( -21 \nu^{9} + 53 \nu^{8} + 162 \nu^{7} - 380 \nu^{6} - 375 \nu^{5} + 770 \nu^{4} + 331 \nu^{3} - 475 \nu^{2} - 127 \nu + 45 \)
\(\beta_{9}\)\(=\)\( -22 \nu^{9} + 55 \nu^{8} + 170 \nu^{7} - 392 \nu^{6} - 394 \nu^{5} + 785 \nu^{4} + 346 \nu^{3} - 473 \nu^{2} - 130 \nu + 42 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{6} + \beta_{4} - \beta_{3} + \beta_{1} + 1\)
\(\nu^{3}\)\(=\)\(\beta_{8} - 2 \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + 5 \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(-\beta_{9} + \beta_{8} + 2 \beta_{7} - 9 \beta_{6} + 2 \beta_{5} + 8 \beta_{4} - 6 \beta_{3} - 2 \beta_{2} + 10 \beta_{1} - 1\)
\(\nu^{5}\)\(=\)\(-\beta_{9} + 8 \beta_{8} + 3 \beta_{7} - 22 \beta_{6} + 9 \beta_{5} + 12 \beta_{4} - 9 \beta_{3} - 5 \beta_{2} + 34 \beta_{1} - 12\)
\(\nu^{6}\)\(=\)\(-8 \beta_{9} + 12 \beta_{8} + 19 \beta_{7} - 73 \beta_{6} + 22 \beta_{5} + 58 \beta_{4} - 37 \beta_{3} - 24 \beta_{2} + 83 \beta_{1} - 28\)
\(\nu^{7}\)\(=\)\(-12 \beta_{9} + 58 \beta_{8} + 37 \beta_{7} - 192 \beta_{6} + 73 \beta_{5} + 113 \beta_{4} - 71 \beta_{3} - 63 \beta_{2} + 252 \beta_{1} - 107\)
\(\nu^{8}\)\(=\)\(-58 \beta_{9} + 113 \beta_{8} + 152 \beta_{7} - 578 \beta_{6} + 192 \beta_{5} + 425 \beta_{4} - 247 \beta_{3} - 220 \beta_{2} + 661 \beta_{1} - 278\)
\(\nu^{9}\)\(=\)\(-113 \beta_{9} + 425 \beta_{8} + 345 \beta_{7} - 1565 \beta_{6} + 578 \beta_{5} + 967 \beta_{4} - 554 \beta_{3} - 591 \beta_{2} + 1920 \beta_{1} - 879\)

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.

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8048))\):

\(T_{3}^{10} - \cdots\)
\(T_{5}^{10} + \cdots\)
\(T_{7}^{10} - \cdots\)
\(T_{13}^{10} + \cdots\)