Properties

Label 4016.2.a.e
Level 4016
Weight 2
Character orbit 4016.a
Self dual yes
Analytic conductor 32.068
Analytic rank 1
Dimension 5
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4016 = 2^{4} \cdot 251 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4016.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.0679214517\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.242773.1
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 502)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( -1 + \beta_{2} ) q^{5} + ( 1 - 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{7} + ( -\beta_{2} + \beta_{3} + \beta_{4} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( -1 + \beta_{2} ) q^{5} + ( 1 - 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{7} + ( -\beta_{2} + \beta_{3} + \beta_{4} ) q^{9} + ( \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{11} + ( -2 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} ) q^{13} + ( 2 - 3 \beta_{1} + \beta_{2} + \beta_{3} ) q^{15} + ( -2 + \beta_{1} - \beta_{2} - 2 \beta_{4} ) q^{17} + ( -1 - \beta_{2} - \beta_{3} ) q^{19} + ( -2 + \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{21} + ( 4 - \beta_{1} + 2 \beta_{3} ) q^{23} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{25} + ( \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{27} + ( -3 + \beta_{2} - \beta_{3} ) q^{29} + ( 2 \beta_{1} + \beta_{2} - 3 \beta_{4} ) q^{31} + ( -2 - \beta_{1} - \beta_{2} + \beta_{4} ) q^{33} + ( 5 \beta_{1} - 2 \beta_{3} - 3 \beta_{4} ) q^{35} + ( -1 + \beta_{2} - \beta_{3} - 4 \beta_{4} ) q^{37} + ( 1 - 3 \beta_{2} + \beta_{4} ) q^{39} + ( -2 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} ) q^{41} + ( -1 + 3 \beta_{1} - \beta_{3} - \beta_{4} ) q^{43} + ( -3 + \beta_{1} + \beta_{2} - 2 \beta_{3} - 3 \beta_{4} ) q^{45} + ( 5 - 3 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} ) q^{47} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{49} + ( -1 - 2 \beta_{1} + \beta_{4} ) q^{51} + ( -6 - 4 \beta_{1} + 2 \beta_{4} ) q^{53} + ( -3 - 2 \beta_{1} - \beta_{2} + \beta_{3} + 4 \beta_{4} ) q^{55} + ( -3 - \beta_{2} - \beta_{3} ) q^{57} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} ) q^{59} + ( -3 \beta_{1} - 3 \beta_{2} + 3 \beta_{4} ) q^{61} + ( 1 - 3 \beta_{1} - 2 \beta_{4} ) q^{63} + ( 1 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} ) q^{65} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{67} + ( -1 + 6 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} ) q^{69} + ( 3 - 6 \beta_{1} + 5 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} ) q^{71} + ( 2 - 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{73} + ( -4 + 3 \beta_{1} - 2 \beta_{3} - \beta_{4} ) q^{75} + ( -6 - \beta_{1} - \beta_{2} + 5 \beta_{4} ) q^{77} + ( -8 + 3 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{79} + ( -2 + 3 \beta_{1} - 4 \beta_{3} - 2 \beta_{4} ) q^{81} + ( -1 - 5 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} ) q^{83} + ( -2 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + 4 \beta_{4} ) q^{85} + ( 1 - 6 \beta_{1} + \beta_{2} + \beta_{3} ) q^{87} + ( 2 + 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} ) q^{89} + ( -5 + 4 \beta_{1} - 2 \beta_{2} - 5 \beta_{3} - 4 \beta_{4} ) q^{91} + ( 5 - 5 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} ) q^{93} + ( -3 + \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} ) q^{95} + ( \beta_{1} + 5 \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{97} + ( -4 - 2 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + q^{3} - 6q^{5} + q^{7} + O(q^{10}) \) \( 5q + q^{3} - 6q^{5} + q^{7} + 4q^{11} - 7q^{13} + 5q^{15} - 8q^{17} - 3q^{19} - 10q^{21} + 17q^{23} - q^{25} + 4q^{27} - 15q^{29} + q^{31} - 10q^{33} + 7q^{35} - 5q^{37} + 8q^{39} - 12q^{41} - q^{43} - 13q^{45} + 19q^{47} + 4q^{49} - 7q^{51} - 34q^{53} - 17q^{55} - 13q^{57} + 10q^{59} + 2q^{63} + 6q^{65} + 11q^{67} + q^{69} + q^{71} + 3q^{73} - 15q^{75} - 30q^{77} - 35q^{79} - 3q^{81} - 3q^{83} - 13q^{85} - 3q^{87} + 15q^{89} - 14q^{91} + 15q^{93} - 11q^{95} - 2q^{97} - 28q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - x^{4} - 7 x^{3} + 4 x^{2} + 6 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{4} - 2 \nu^{3} - 7 \nu^{2} + 11 \nu + 5 \)\()/2\)
\(\beta_{3}\)\(=\)\( -\nu^{4} + \nu^{3} + 7 \nu^{2} - 4 \nu - 5 \)
\(\beta_{4}\)\(=\)\((\)\( 3 \nu^{4} - 4 \nu^{3} - 19 \nu^{2} + 19 \nu + 9 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} + \beta_{3} - \beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(-\beta_{3} - 2 \beta_{2} + 7 \beta_{1}\)
\(\nu^{4}\)\(=\)\(7 \beta_{4} + 5 \beta_{3} - 9 \beta_{2} + 3 \beta_{1} + 16\)

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
−2.31766
−0.567497
−0.203945
1.37208
2.71702
0 −2.31766 0 −3.17136 0 3.66414 0 2.37155 0
1.2 0 −0.567497 0 −2.51380 0 −3.57054 0 −2.67795 0
1.3 0 −0.203945 0 0.242071 0 0.933626 0 −2.95841 0
1.4 0 1.37208 0 1.64633 0 2.43111 0 −1.11739 0
1.5 0 2.71702 0 −2.20324 0 −2.45833 0 4.38220 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

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 4016.2.a.e 5
4.b odd 2 1 502.2.a.c 5
12.b even 2 1 4518.2.a.v 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
502.2.a.c 5 4.b odd 2 1
4016.2.a.e 5 1.a even 1 1 trivial
4518.2.a.v 5 12.b even 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(251\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{5} - T_{3}^{4} - 7 T_{3}^{3} + 4 T_{3}^{2} + 6 T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4016))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( 1 - T + 8 T^{2} - 8 T^{3} + 33 T^{4} - 29 T^{5} + 99 T^{6} - 72 T^{7} + 216 T^{8} - 81 T^{9} + 243 T^{10} \)
$5$ \( 1 + 6 T + 31 T^{2} + 102 T^{3} + 315 T^{4} + 727 T^{5} + 1575 T^{6} + 2550 T^{7} + 3875 T^{8} + 3750 T^{9} + 3125 T^{10} \)
$7$ \( 1 - T + 16 T^{2} - 10 T^{3} + 169 T^{4} - 115 T^{5} + 1183 T^{6} - 490 T^{7} + 5488 T^{8} - 2401 T^{9} + 16807 T^{10} \)
$11$ \( 1 - 4 T + 26 T^{2} - 100 T^{3} + 413 T^{4} - 1168 T^{5} + 4543 T^{6} - 12100 T^{7} + 34606 T^{8} - 58564 T^{9} + 161051 T^{10} \)
$13$ \( 1 + 7 T + 58 T^{2} + 264 T^{3} + 1411 T^{4} + 4871 T^{5} + 18343 T^{6} + 44616 T^{7} + 127426 T^{8} + 199927 T^{9} + 371293 T^{10} \)
$17$ \( 1 + 8 T + 63 T^{2} + 302 T^{3} + 1267 T^{4} + 5331 T^{5} + 21539 T^{6} + 87278 T^{7} + 309519 T^{8} + 668168 T^{9} + 1419857 T^{10} \)
$19$ \( 1 + 3 T + 82 T^{2} + 230 T^{3} + 2885 T^{4} + 6566 T^{5} + 54815 T^{6} + 83030 T^{7} + 562438 T^{8} + 390963 T^{9} + 2476099 T^{10} \)
$23$ \( 1 - 17 T + 194 T^{2} - 1498 T^{3} + 9545 T^{4} - 48923 T^{5} + 219535 T^{6} - 792442 T^{7} + 2360398 T^{8} - 4757297 T^{9} + 6436343 T^{10} \)
$29$ \( 1 + 15 T + 214 T^{2} + 1788 T^{3} + 14073 T^{4} + 77938 T^{5} + 408117 T^{6} + 1503708 T^{7} + 5219246 T^{8} + 10609215 T^{9} + 20511149 T^{10} \)
$31$ \( 1 - T + 75 T^{2} - 139 T^{3} + 3758 T^{4} - 4900 T^{5} + 116498 T^{6} - 133579 T^{7} + 2234325 T^{8} - 923521 T^{9} + 28629151 T^{10} \)
$37$ \( 1 + 5 T + 54 T^{2} - 70 T^{3} - 1555 T^{4} - 17902 T^{5} - 57535 T^{6} - 95830 T^{7} + 2735262 T^{8} + 9370805 T^{9} + 69343957 T^{10} \)
$41$ \( 1 + 12 T + 161 T^{2} + 1212 T^{3} + 11153 T^{4} + 67159 T^{5} + 457273 T^{6} + 2037372 T^{7} + 11096281 T^{8} + 33909132 T^{9} + 115856201 T^{10} \)
$43$ \( 1 + T + 172 T^{2} + 286 T^{3} + 12855 T^{4} + 20906 T^{5} + 552765 T^{6} + 528814 T^{7} + 13675204 T^{8} + 3418801 T^{9} + 147008443 T^{10} \)
$47$ \( 1 - 19 T + 332 T^{2} - 3522 T^{3} + 34851 T^{4} - 247598 T^{5} + 1637997 T^{6} - 7780098 T^{7} + 34469236 T^{8} - 92713939 T^{9} + 229345007 T^{10} \)
$53$ \( 1 + 34 T + 625 T^{2} + 8072 T^{3} + 81586 T^{4} + 661196 T^{5} + 4324058 T^{6} + 22674248 T^{7} + 93048125 T^{8} + 268276354 T^{9} + 418195493 T^{10} \)
$59$ \( 1 - 10 T + 259 T^{2} - 1968 T^{3} + 27990 T^{4} - 163116 T^{5} + 1651410 T^{6} - 6850608 T^{7} + 53193161 T^{8} - 121173610 T^{9} + 714924299 T^{10} \)
$61$ \( 1 + 134 T^{2} + 54 T^{3} + 8185 T^{4} + 4644 T^{5} + 499285 T^{6} + 200934 T^{7} + 30415454 T^{8} + 844596301 T^{10} \)
$67$ \( 1 - 11 T + 328 T^{2} - 2684 T^{3} + 43483 T^{4} - 262229 T^{5} + 2913361 T^{6} - 12048476 T^{7} + 98650264 T^{8} - 221662331 T^{9} + 1350125107 T^{10} \)
$71$ \( 1 - T + 124 T^{2} + 10 T^{3} + 13175 T^{4} - 21722 T^{5} + 935425 T^{6} + 50410 T^{7} + 44380964 T^{8} - 25411681 T^{9} + 1804229351 T^{10} \)
$73$ \( 1 - 3 T + 283 T^{2} - 953 T^{3} + 35808 T^{4} - 107372 T^{5} + 2613984 T^{6} - 5078537 T^{7} + 110091811 T^{8} - 85194723 T^{9} + 2073071593 T^{10} \)
$79$ \( 1 + 35 T + 848 T^{2} + 13730 T^{3} + 176739 T^{4} + 1738679 T^{5} + 13962381 T^{6} + 85688930 T^{7} + 418097072 T^{8} + 1363252835 T^{9} + 3077056399 T^{10} \)
$83$ \( 1 + 3 T + 126 T^{2} + 320 T^{3} + 16065 T^{4} + 60618 T^{5} + 1333395 T^{6} + 2204480 T^{7} + 72045162 T^{8} + 142374963 T^{9} + 3939040643 T^{10} \)
$89$ \( 1 - 15 T + 450 T^{2} - 4826 T^{3} + 79871 T^{4} - 621321 T^{5} + 7108519 T^{6} - 38226746 T^{7} + 317236050 T^{8} - 941133615 T^{9} + 5584059449 T^{10} \)
$97$ \( 1 + 2 T + 132 T^{2} - 318 T^{3} + 18695 T^{4} + 9736 T^{5} + 1813415 T^{6} - 2992062 T^{7} + 120472836 T^{8} + 177058562 T^{9} + 8587340257 T^{10} \)
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