Properties

Label 8048.2.a.o
Level $8048$
Weight $2$
Character orbit 8048.a
Self dual yes
Analytic conductor $64.264$
Analytic rank $0$
Dimension $5$
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: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.205225.1
Defining polynomial: \(x^{5} - x^{4} - 6 x^{3} + 3 x^{2} + 7 x - 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1006)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} ) q^{3} + ( -1 - \beta_{1} - \beta_{3} - \beta_{4} ) q^{5} + ( 1 + \beta_{1} - \beta_{3} - \beta_{4} ) q^{7} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( 1 - \beta_{1} ) q^{3} + ( -1 - \beta_{1} - \beta_{3} - \beta_{4} ) q^{5} + ( 1 + \beta_{1} - \beta_{3} - \beta_{4} ) q^{7} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{9} + ( 4 + 2 \beta_{2} + \beta_{3} ) q^{11} + ( \beta_{1} + \beta_{2} + 2 \beta_{4} ) q^{13} + ( \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{15} + ( -1 + \beta_{1} + 2 \beta_{3} ) q^{17} + ( -3 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{19} + ( -4 - 2 \beta_{1} - \beta_{2} - 4 \beta_{3} - 2 \beta_{4} ) q^{21} + ( 4 + \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{23} + ( 2 - 2 \beta_{1} + 2 \beta_{2} - \beta_{4} ) q^{25} + ( 2 + \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{27} + ( -2 + 3 \beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{4} ) q^{29} + ( 7 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{31} + ( 5 - 6 \beta_{1} - \beta_{4} ) q^{33} + ( 3 - 6 \beta_{1} - \beta_{4} ) q^{35} + ( -1 - \beta_{1} - \beta_{2} - 4 \beta_{3} - \beta_{4} ) q^{37} + ( -1 + \beta_{1} - \beta_{2} + \beta_{4} ) q^{39} + ( -3 - 2 \beta_{2} + \beta_{3} ) q^{41} + ( 3 - 3 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{43} + ( -1 - 4 \beta_{3} - 2 \beta_{4} ) q^{45} + ( 4 + 4 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} + 5 \beta_{4} ) q^{47} + ( 8 + 2 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} - \beta_{4} ) q^{49} + ( -2 + \beta_{1} - \beta_{2} + 3 \beta_{3} + 2 \beta_{4} ) q^{51} + ( -1 + 2 \beta_{1} + \beta_{3} - 2 \beta_{4} ) q^{53} + ( -3 - 6 \beta_{1} - \beta_{2} - 7 \beta_{3} - 5 \beta_{4} ) q^{55} + ( -2 + 5 \beta_{1} + \beta_{2} - \beta_{3} ) q^{57} + ( -4 + 2 \beta_{1} + \beta_{2} - \beta_{4} ) q^{59} + ( -1 + 3 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} ) q^{61} + ( -7 + 2 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} ) q^{63} + ( -3 + \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{65} + ( -4 - \beta_{2} - 2 \beta_{3} - 5 \beta_{4} ) q^{67} + ( 4 - 4 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} ) q^{69} + ( 6 - 3 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} ) q^{71} + ( 2 + 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{73} + ( 7 - 5 \beta_{1} + 2 \beta_{2} - \beta_{3} - 3 \beta_{4} ) q^{75} + ( 3 + 6 \beta_{1} + 7 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} ) q^{77} + ( 3 - 3 \beta_{2} ) q^{79} + ( -2 - 4 \beta_{2} - \beta_{3} + \beta_{4} ) q^{81} + ( -1 - \beta_{1} - 5 \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{83} + ( -6 + 4 \beta_{1} - 3 \beta_{2} + 4 \beta_{4} ) q^{85} + ( -9 - 3 \beta_{2} - 3 \beta_{3} - \beta_{4} ) q^{87} + ( 2 - 4 \beta_{1} - 5 \beta_{2} - \beta_{3} + \beta_{4} ) q^{89} + ( -1 + 3 \beta_{1} + 5 \beta_{2} + 2 \beta_{3} + 6 \beta_{4} ) q^{91} + ( 5 - 8 \beta_{1} - \beta_{2} ) q^{93} + ( 6 + 4 \beta_{1} + 3 \beta_{2} + 6 \beta_{3} + 2 \beta_{4} ) q^{95} + ( 3 - 2 \beta_{1} - 4 \beta_{2} - 5 \beta_{3} ) q^{97} + ( 10 - 6 \beta_{1} + 2 \beta_{3} - \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 4q^{3} - 3q^{5} + 9q^{7} + q^{9} + O(q^{10}) \) \( 5q + 4q^{3} - 3q^{5} + 9q^{7} + q^{9} + 15q^{11} - 5q^{13} + 4q^{15} - 6q^{17} - 10q^{19} - 12q^{21} + 12q^{23} + 6q^{25} + 7q^{27} - 16q^{29} + 33q^{31} + 21q^{33} + 11q^{35} + 2q^{37} - 4q^{39} - 12q^{41} + 17q^{43} + 3q^{45} + 7q^{47} + 36q^{49} - 14q^{51} - 2q^{55} - 6q^{57} - 18q^{59} + q^{61} - 31q^{63} - 14q^{65} - 6q^{67} + 15q^{69} + 26q^{71} + 9q^{73} + 33q^{75} + 16q^{77} + 21q^{79} - 3q^{81} - 28q^{85} - 34q^{87} + 15q^{89} - 26q^{91} + 19q^{93} + 18q^{95} + 26q^{97} + 44q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - x^{4} - 6 x^{3} + 3 x^{2} + 7 x - 3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{3} - \nu^{2} - 4 \nu + 1 \)
\(\beta_{3}\)\(=\)\( -\nu^{3} + 2 \nu^{2} + 3 \nu - 4 \)
\(\beta_{4}\)\(=\)\( \nu^{4} - \nu^{3} - 5 \nu^{2} + \nu + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{2} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 2 \beta_{2} + 5 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(\beta_{4} + 6 \beta_{3} + 7 \beta_{2} + 9 \beta_{1} + 14\)

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
2.54180
1.17073
0.418933
−1.35347
−1.77799
0 −1.54180 0 −2.22418 0 4.85942 0 −0.622849 0
1.2 0 −0.170728 0 −0.411173 0 3.93028 0 −2.97085 0
1.3 0 0.581067 0 −1.45190 0 1.38596 0 −2.66236 0
1.4 0 2.35347 0 3.94839 0 3.24145 0 2.53883 0
1.5 0 2.77799 0 −2.86114 0 −4.41712 0 4.71723 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.o 5
4.b odd 2 1 1006.2.a.h 5
12.b even 2 1 9054.2.a.ba 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1006.2.a.h 5 4.b odd 2 1
8048.2.a.o 5 1.a even 1 1 trivial
9054.2.a.ba 5 12.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8048))\):

\( T_{3}^{5} - 4 T_{3}^{4} + 11 T_{3}^{2} - 4 T_{3} - 1 \)
\( T_{5}^{5} + 3 T_{5}^{4} - 11 T_{5}^{3} - 50 T_{5}^{2} - 55 T_{5} - 15 \)
\( T_{7}^{5} - 9 T_{7}^{4} + 5 T_{7}^{3} + 156 T_{7}^{2} - 479 T_{7} + 379 \)
\( T_{13}^{5} + 5 T_{13}^{4} - 27 T_{13}^{3} - 85 T_{13}^{2} + 10 T_{13} + 15 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 4 T + 15 T^{2} - 37 T^{3} + 86 T^{4} - 151 T^{5} + 258 T^{6} - 333 T^{7} + 405 T^{8} - 324 T^{9} + 243 T^{10} \)
$5$ \( 1 + 3 T + 14 T^{2} + 10 T^{3} + 30 T^{4} - 65 T^{5} + 150 T^{6} + 250 T^{7} + 1750 T^{8} + 1875 T^{9} + 3125 T^{10} \)
$7$ \( 1 - 9 T + 40 T^{2} - 96 T^{3} + 116 T^{4} - 83 T^{5} + 812 T^{6} - 4704 T^{7} + 13720 T^{8} - 21609 T^{9} + 16807 T^{10} \)
$11$ \( 1 - 15 T + 115 T^{2} - 625 T^{3} + 2765 T^{4} - 10129 T^{5} + 30415 T^{6} - 75625 T^{7} + 153065 T^{8} - 219615 T^{9} + 161051 T^{10} \)
$13$ \( 1 + 5 T + 38 T^{2} + 175 T^{3} + 647 T^{4} + 2875 T^{5} + 8411 T^{6} + 29575 T^{7} + 83486 T^{8} + 142805 T^{9} + 371293 T^{10} \)
$17$ \( 1 + 6 T + 63 T^{2} + 331 T^{3} + 2054 T^{4} + 7567 T^{5} + 34918 T^{6} + 95659 T^{7} + 309519 T^{8} + 501126 T^{9} + 1419857 T^{10} \)
$19$ \( 1 + 10 T + 97 T^{2} + 617 T^{3} + 3658 T^{4} + 16559 T^{5} + 69502 T^{6} + 222737 T^{7} + 665323 T^{8} + 1303210 T^{9} + 2476099 T^{10} \)
$23$ \( 1 - 12 T + 124 T^{2} - 730 T^{3} + 4391 T^{4} - 19387 T^{5} + 100993 T^{6} - 386170 T^{7} + 1508708 T^{8} - 3358092 T^{9} + 6436343 T^{10} \)
$29$ \( 1 + 16 T + 168 T^{2} + 1239 T^{3} + 7332 T^{4} + 40831 T^{5} + 212628 T^{6} + 1041999 T^{7} + 4097352 T^{8} + 11316496 T^{9} + 20511149 T^{10} \)
$31$ \( 1 - 33 T + 579 T^{2} - 6751 T^{3} + 57197 T^{4} - 364933 T^{5} + 1773107 T^{6} - 6487711 T^{7} + 17248989 T^{8} - 30476193 T^{9} + 28629151 T^{10} \)
$37$ \( 1 - 2 T + 79 T^{2} + 19 T^{3} + 3444 T^{4} + 4987 T^{5} + 127428 T^{6} + 26011 T^{7} + 4001587 T^{8} - 3748322 T^{9} + 69343957 T^{10} \)
$41$ \( 1 + 12 T + 207 T^{2} + 1687 T^{3} + 16586 T^{4} + 98677 T^{5} + 680026 T^{6} + 2835847 T^{7} + 14266647 T^{8} + 33909132 T^{9} + 115856201 T^{10} \)
$43$ \( 1 - 17 T + 270 T^{2} - 2417 T^{3} + 21761 T^{4} - 137929 T^{5} + 935723 T^{6} - 4469033 T^{7} + 21466890 T^{8} - 58119617 T^{9} + 147008443 T^{10} \)
$47$ \( 1 - 7 T + 42 T^{2} + 124 T^{3} + 3434 T^{4} - 23499 T^{5} + 161398 T^{6} + 273916 T^{7} + 4360566 T^{8} - 34157767 T^{9} + 229345007 T^{10} \)
$53$ \( 1 + 177 T^{2} - 51 T^{3} + 15884 T^{4} - 2059 T^{5} + 841852 T^{6} - 143259 T^{7} + 26351229 T^{8} + 418195493 T^{10} \)
$59$ \( 1 + 18 T + 380 T^{2} + 4190 T^{3} + 48801 T^{4} + 367541 T^{5} + 2879259 T^{6} + 14585390 T^{7} + 78044020 T^{8} + 218112498 T^{9} + 714924299 T^{10} \)
$61$ \( 1 - T + 194 T^{2} + 4 T^{3} + 19824 T^{4} - 933 T^{5} + 1209264 T^{6} + 14884 T^{7} + 44034314 T^{8} - 13845841 T^{9} + 844596301 T^{10} \)
$67$ \( 1 + 6 T + 128 T^{2} + 1136 T^{3} + 14413 T^{4} + 78367 T^{5} + 965671 T^{6} + 5099504 T^{7} + 38497664 T^{8} + 120906726 T^{9} + 1350125107 T^{10} \)
$71$ \( 1 - 26 T + 402 T^{2} - 4177 T^{3} + 35480 T^{4} - 279111 T^{5} + 2519080 T^{6} - 21056257 T^{7} + 143880222 T^{8} - 660703706 T^{9} + 1804229351 T^{10} \)
$73$ \( 1 - 9 T + 372 T^{2} - 2620 T^{3} + 54822 T^{4} - 286599 T^{5} + 4002006 T^{6} - 13961980 T^{7} + 144714324 T^{8} - 255584169 T^{9} + 2073071593 T^{10} \)
$79$ \( 1 - 21 T + 494 T^{2} - 6501 T^{3} + 84577 T^{4} - 764307 T^{5} + 6681583 T^{6} - 40572741 T^{7} + 243561266 T^{8} - 817951701 T^{9} + 3077056399 T^{10} \)
$83$ \( 1 + 103 T^{2} + 223 T^{3} + 10958 T^{4} + 36289 T^{5} + 909514 T^{6} + 1536247 T^{7} + 58894061 T^{8} + 3939040643 T^{10} \)
$89$ \( 1 - 15 T + 292 T^{2} - 2249 T^{3} + 32285 T^{4} - 207719 T^{5} + 2873365 T^{6} - 17814329 T^{7} + 205850948 T^{8} - 941133615 T^{9} + 5584059449 T^{10} \)
$97$ \( 1 - 26 T + 557 T^{2} - 8411 T^{3} + 110746 T^{4} - 1172915 T^{5} + 10742362 T^{6} - 79139099 T^{7} + 508358861 T^{8} - 2301761306 T^{9} + 8587340257 T^{10} \)
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