Properties

Label 8048.2.a.n
Level $8048$
Weight $2$
Character orbit 8048.a
Self dual yes
Analytic conductor $64.264$
Analytic rank $1$
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: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.36497.1
Defining polynomial: \(x^{5} - 2 x^{4} - 3 x^{3} + 5 x^{2} + x - 1\)
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.

\(f(q)\) \(=\) \( q + ( -\beta_{1} + \beta_{3} - \beta_{4} ) q^{3} + ( \beta_{2} + \beta_{3} ) q^{5} + ( \beta_{2} - \beta_{3} ) q^{7} + ( -1 - \beta_{2} - \beta_{3} - \beta_{4} ) q^{9} +O(q^{10})\) \( q + ( -\beta_{1} + \beta_{3} - \beta_{4} ) q^{3} + ( \beta_{2} + \beta_{3} ) q^{5} + ( \beta_{2} - \beta_{3} ) q^{7} + ( -1 - \beta_{2} - \beta_{3} - \beta_{4} ) q^{9} + ( 2 - \beta_{2} + \beta_{4} ) q^{11} + ( 2 - 2 \beta_{1} - \beta_{3} ) q^{13} + ( 1 + 3 \beta_{1} - 2 \beta_{3} + \beta_{4} ) q^{15} + ( -4 - \beta_{1} + \beta_{3} - \beta_{4} ) q^{17} + ( \beta_{1} + \beta_{3} - \beta_{4} ) q^{19} + ( -1 + \beta_{1} + \beta_{4} ) q^{21} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{23} + ( -2 + 3 \beta_{1} - 2 \beta_{3} ) q^{25} + ( -\beta_{3} + 2 \beta_{4} ) q^{27} + ( -2 - 3 \beta_{1} - 3 \beta_{2} + \beta_{3} - 3 \beta_{4} ) q^{29} + ( 1 + 2 \beta_{1} + \beta_{2} - \beta_{3} + 3 \beta_{4} ) q^{31} + ( -1 - 3 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} ) q^{33} + ( 1 - \beta_{1} ) q^{35} + ( -3 + \beta_{1} + 3 \beta_{3} ) q^{37} + ( -1 - 3 \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{39} + ( -1 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + 5 \beta_{4} ) q^{41} + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{43} + ( -3 - 2 \beta_{1} + 2 \beta_{3} ) q^{45} + ( -\beta_{1} - 5 \beta_{2} - \beta_{4} ) q^{47} + ( -\beta_{1} + 2 \beta_{3} + 4 \beta_{4} ) q^{49} + ( 2 + 4 \beta_{1} - \beta_{2} - 5 \beta_{3} + 3 \beta_{4} ) q^{51} + ( -5 + 6 \beta_{1} + \beta_{2} - 4 \beta_{3} + 3 \beta_{4} ) q^{53} + ( -2 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{55} + ( 2 + \beta_{2} + \beta_{3} - \beta_{4} ) q^{57} + ( 1 + 4 \beta_{1} - 2 \beta_{3} + \beta_{4} ) q^{59} + ( -4 + 2 \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{61} + ( -1 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{63} + ( -1 - 2 \beta_{1} - \beta_{3} + 2 \beta_{4} ) q^{65} + ( -3 - 2 \beta_{2} + 3 \beta_{4} ) q^{67} + ( -1 + 4 \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{69} + ( 7 - 6 \beta_{1} + \beta_{2} + \beta_{3} - 4 \beta_{4} ) q^{71} + ( -3 - 3 \beta_{1} - 5 \beta_{2} - 2 \beta_{3} - 6 \beta_{4} ) q^{73} + ( -2 + 3 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} ) q^{75} + ( -4 + 3 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} ) q^{77} + ( \beta_{1} - 2 \beta_{2} + 4 \beta_{3} - 5 \beta_{4} ) q^{79} + ( \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + 5 \beta_{4} ) q^{81} + ( 3 + \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} ) q^{83} + ( 1 + 3 \beta_{1} - 4 \beta_{2} - 6 \beta_{3} + \beta_{4} ) q^{85} + ( 4 - 6 \beta_{1} - 3 \beta_{2} - 4 \beta_{4} ) q^{87} + ( -3 + 10 \beta_{1} + \beta_{2} - 6 \beta_{3} + 2 \beta_{4} ) q^{89} + ( 7 + 4 \beta_{2} - \beta_{3} + 4 \beta_{4} ) q^{91} + ( -4 + 3 \beta_{1} + 2 \beta_{2} + 3 \beta_{4} ) q^{93} + ( 1 + 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{95} + ( 3 - 2 \beta_{1} + 3 \beta_{2} - \beta_{4} ) q^{97} + ( -2 + \beta_{1} - 4 \beta_{3} - 4 \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - q^{5} - 3q^{7} - 3q^{9} + O(q^{10}) \) \( 5q - q^{5} - 3q^{7} - 3q^{9} + 11q^{11} + 5q^{13} + 8q^{15} - 20q^{17} + 4q^{19} - 4q^{21} + 4q^{23} - 6q^{25} - 3q^{27} - 6q^{29} + 3q^{31} - 7q^{33} + 3q^{35} - 10q^{37} - 4q^{39} - 6q^{41} - 11q^{43} - 17q^{45} + 9q^{47} - 4q^{49} + 12q^{51} - 22q^{53} - 14q^{55} + 10q^{57} + 10q^{59} - 11q^{61} + 3q^{63} - 12q^{65} - 14q^{67} - 3q^{69} + 26q^{71} - 7q^{73} - 15q^{75} - 26q^{77} + 15q^{79} - 7q^{81} + 12q^{83} + 12q^{85} + 18q^{87} - 5q^{89} + 22q^{91} - 21q^{93} + 10q^{95} + 6q^{97} - 8q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - 2 x^{4} - 3 x^{3} + 5 x^{2} + x - 1\):

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

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
1.33419
−1.55629
0.410375
2.31801
−0.506287
0 −2.40769 0 −3.21230 0 0.104033 0 2.79697 0
1.2 0 −0.522539 0 0.813221 0 3.14341 0 −2.72695 0
1.3 0 −0.330424 0 −0.135598 0 −4.34834 0 −2.89082 0
1.4 0 1.12789 0 2.61447 0 −0.504123 0 −1.72786 0
1.5 0 2.13276 0 −1.07979 0 −1.39498 0 1.54867 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.n 5
4.b odd 2 1 1006.2.a.g 5
12.b even 2 1 9054.2.a.bb 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1006.2.a.g 5 4.b odd 2 1
8048.2.a.n 5 1.a even 1 1 trivial
9054.2.a.bb 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8048))\):

\( T_{3}^{5} - 6 T_{3}^{3} + T_{3}^{2} + 4 T_{3} + 1 \)
\( T_{5}^{5} + T_{5}^{4} - 9 T_{5}^{3} - 4 T_{5}^{2} + 7 T_{5} + 1 \)
\( T_{7}^{5} + 3 T_{7}^{4} - 11 T_{7}^{3} - 24 T_{7}^{2} - 7 T_{7} + 1 \)
\( T_{13}^{5} - 5 T_{13}^{4} - 21 T_{13}^{3} + 81 T_{13}^{2} + 14 T_{13} - 69 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 9 T^{2} + T^{3} + 40 T^{4} + 7 T^{5} + 120 T^{6} + 9 T^{7} + 243 T^{8} + 243 T^{10} \)
$5$ \( 1 + T + 16 T^{2} + 16 T^{3} + 122 T^{4} + 111 T^{5} + 610 T^{6} + 400 T^{7} + 2000 T^{8} + 625 T^{9} + 3125 T^{10} \)
$7$ \( 1 + 3 T + 24 T^{2} + 60 T^{3} + 252 T^{4} + 547 T^{5} + 1764 T^{6} + 2940 T^{7} + 8232 T^{8} + 7203 T^{9} + 16807 T^{10} \)
$11$ \( 1 - 11 T + 89 T^{2} - 507 T^{3} + 2301 T^{4} - 8463 T^{5} + 25311 T^{6} - 61347 T^{7} + 118459 T^{8} - 161051 T^{9} + 161051 T^{10} \)
$13$ \( 1 - 5 T + 44 T^{2} - 179 T^{3} + 885 T^{4} - 3033 T^{5} + 11505 T^{6} - 30251 T^{7} + 96668 T^{8} - 142805 T^{9} + 371293 T^{10} \)
$17$ \( 1 + 20 T + 239 T^{2} + 1929 T^{3} + 11748 T^{4} + 54699 T^{5} + 199716 T^{6} + 557481 T^{7} + 1174207 T^{8} + 1670420 T^{9} + 1419857 T^{10} \)
$19$ \( 1 - 4 T + 79 T^{2} - 267 T^{3} + 2678 T^{4} - 7255 T^{5} + 50882 T^{6} - 96387 T^{7} + 541861 T^{8} - 521284 T^{9} + 2476099 T^{10} \)
$23$ \( 1 - 4 T + 96 T^{2} - 310 T^{3} + 4041 T^{4} - 10201 T^{5} + 92943 T^{6} - 163990 T^{7} + 1168032 T^{8} - 1119364 T^{9} + 6436343 T^{10} \)
$29$ \( 1 + 6 T + 94 T^{2} + 323 T^{3} + 3466 T^{4} + 8479 T^{5} + 100514 T^{6} + 271643 T^{7} + 2292566 T^{8} + 4243686 T^{9} + 20511149 T^{10} \)
$31$ \( 1 - 3 T + 127 T^{2} - 249 T^{3} + 6921 T^{4} - 9755 T^{5} + 214551 T^{6} - 239289 T^{7} + 3783457 T^{8} - 2770563 T^{9} + 28629151 T^{10} \)
$37$ \( 1 + 10 T + 161 T^{2} + 1149 T^{3} + 11234 T^{4} + 60063 T^{5} + 415658 T^{6} + 1572981 T^{7} + 8155133 T^{8} + 18741610 T^{9} + 69343957 T^{10} \)
$41$ \( 1 + 6 T + 79 T^{2} + 31 T^{3} + 194 T^{4} - 17503 T^{5} + 7954 T^{6} + 52111 T^{7} + 5444759 T^{8} + 16954566 T^{9} + 115856201 T^{10} \)
$43$ \( 1 + 11 T + 250 T^{2} + 1923 T^{3} + 23015 T^{4} + 124701 T^{5} + 989645 T^{6} + 3555627 T^{7} + 19876750 T^{8} + 37606811 T^{9} + 147008443 T^{10} \)
$47$ \( 1 - 9 T + 120 T^{2} - 610 T^{3} + 8412 T^{4} - 44117 T^{5} + 395364 T^{6} - 1347490 T^{7} + 12458760 T^{8} - 43917129 T^{9} + 229345007 T^{10} \)
$53$ \( 1 + 22 T + 345 T^{2} + 3931 T^{3} + 36038 T^{4} + 286901 T^{5} + 1910014 T^{6} + 11042179 T^{7} + 51362565 T^{8} + 173590582 T^{9} + 418195493 T^{10} \)
$59$ \( 1 - 10 T + 284 T^{2} - 2204 T^{3} + 33061 T^{4} - 190573 T^{5} + 1950599 T^{6} - 7672124 T^{7} + 58327636 T^{8} - 121173610 T^{9} + 714924299 T^{10} \)
$61$ \( 1 + 11 T + 272 T^{2} + 2418 T^{3} + 31232 T^{4} + 213643 T^{5} + 1905152 T^{6} + 8997378 T^{7} + 61738832 T^{8} + 152304251 T^{9} + 844596301 T^{10} \)
$67$ \( 1 + 14 T + 324 T^{2} + 3022 T^{3} + 39869 T^{4} + 276469 T^{5} + 2671223 T^{6} + 13565758 T^{7} + 97447212 T^{8} + 282115694 T^{9} + 1350125107 T^{10} \)
$71$ \( 1 - 26 T + 478 T^{2} - 6623 T^{3} + 75008 T^{4} - 682865 T^{5} + 5325568 T^{6} - 33386543 T^{7} + 171081458 T^{8} - 660703706 T^{9} + 1804229351 T^{10} \)
$73$ \( 1 + 7 T + 140 T^{2} + 324 T^{3} + 13742 T^{4} + 47981 T^{5} + 1003166 T^{6} + 1726596 T^{7} + 54462380 T^{8} + 198787687 T^{9} + 2073071593 T^{10} \)
$79$ \( 1 - 15 T + 218 T^{2} - 1565 T^{3} + 10815 T^{4} - 54601 T^{5} + 854385 T^{6} - 9767165 T^{7} + 107482502 T^{8} - 584251215 T^{9} + 3077056399 T^{10} \)
$83$ \( 1 - 12 T + 345 T^{2} - 2417 T^{3} + 44256 T^{4} - 226225 T^{5} + 3673248 T^{6} - 16650713 T^{7} + 197266515 T^{8} - 569499852 T^{9} + 3939040643 T^{10} \)
$89$ \( 1 + 5 T + 106 T^{2} - 715 T^{3} + 6727 T^{4} - 64699 T^{5} + 598703 T^{6} - 5663515 T^{7} + 74726714 T^{8} + 313711205 T^{9} + 5584059449 T^{10} \)
$97$ \( 1 - 6 T + 405 T^{2} - 2139 T^{3} + 71686 T^{4} - 303715 T^{5} + 6953542 T^{6} - 20125851 T^{7} + 369632565 T^{8} - 531175686 T^{9} + 8587340257 T^{10} \)
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