Properties

Label 4016.2.a.d
Level 4016
Weight 2
Character orbit 4016.a
Self dual yes
Analytic conductor 32.068
Analytic rank 0
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: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.138917.1
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{4} - 6 \nu^{2} - 2 \nu + 2 \)
\(\beta_{3}\)\(=\)\( -\nu^{4} + \nu^{3} + 5 \nu^{2} - 3 \nu - 1 \)
\(\beta_{4}\)\(=\)\( -\nu^{4} + \nu^{3} + 6 \nu^{2} - 3 \nu - 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} - \beta_{3} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{4} + \beta_{2} + 5 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(6 \beta_{4} - 6 \beta_{3} + \beta_{2} + 2 \beta_{1} + 16\)

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
0.729679
−0.331620
−2.11990
−0.779397
2.50123
0 −2.88943 0 0.781072 0 −2.79232 0 5.34878 0
1.2 0 −2.39387 0 3.18773 0 2.51166 0 2.73064 0
1.3 0 −0.398976 0 1.24476 0 −2.42122 0 −2.84082 0
1.4 0 1.14049 0 −2.02938 0 3.81607 0 −1.69928 0
1.5 0 2.54179 0 3.81582 0 −2.11419 0 3.46068 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.d 5
4.b odd 2 1 502.2.a.d 5
12.b even 2 1 4518.2.a.t 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
502.2.a.d 5 4.b odd 2 1
4016.2.a.d 5 1.a even 1 1 trivial
4518.2.a.t 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 \( T_{3}^{5} + 2 T_{3}^{4} - 9 T_{3}^{3} - 14 T_{3}^{2} + 16 T_{3} + 8 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4016))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( 1 + 2 T + 6 T^{2} + 10 T^{3} + 25 T^{4} + 32 T^{5} + 75 T^{6} + 90 T^{7} + 162 T^{8} + 162 T^{9} + 243 T^{10} \)
$5$ \( 1 - 7 T + 34 T^{2} - 116 T^{3} + 333 T^{4} - 786 T^{5} + 1665 T^{6} - 2900 T^{7} + 4250 T^{8} - 4375 T^{9} + 3125 T^{10} \)
$7$ \( 1 + T + 16 T^{2} + 171 T^{4} + 39 T^{5} + 1197 T^{6} + 5488 T^{8} + 2401 T^{9} + 16807 T^{10} \)
$11$ \( 1 + 7 T + 68 T^{2} + 308 T^{3} + 1629 T^{4} + 5083 T^{5} + 17919 T^{6} + 37268 T^{7} + 90508 T^{8} + 102487 T^{9} + 161051 T^{10} \)
$13$ \( 1 - 4 T + 42 T^{2} - 70 T^{3} + 593 T^{4} - 396 T^{5} + 7709 T^{6} - 11830 T^{7} + 92274 T^{8} - 114244 T^{9} + 371293 T^{10} \)
$17$ \( 1 - 6 T + 61 T^{2} - 250 T^{3} + 1707 T^{4} - 5563 T^{5} + 29019 T^{6} - 72250 T^{7} + 299693 T^{8} - 501126 T^{9} + 1419857 T^{10} \)
$19$ \( 1 - 10 T + 117 T^{2} - 724 T^{3} + 4703 T^{4} - 20163 T^{5} + 89357 T^{6} - 261364 T^{7} + 802503 T^{8} - 1303210 T^{9} + 2476099 T^{10} \)
$23$ \( 1 + 13 T + 112 T^{2} + 726 T^{3} + 4663 T^{4} + 24345 T^{5} + 107249 T^{6} + 384054 T^{7} + 1362704 T^{8} + 3637933 T^{9} + 6436343 T^{10} \)
$29$ \( 1 - 4 T + 121 T^{2} - 396 T^{3} + 6485 T^{4} - 16363 T^{5} + 188065 T^{6} - 333036 T^{7} + 2951069 T^{8} - 2829124 T^{9} + 20511149 T^{10} \)
$31$ \( 1 - 13 T + 171 T^{2} - 1223 T^{3} + 9378 T^{4} - 48892 T^{5} + 290718 T^{6} - 1175303 T^{7} + 5094261 T^{8} - 12005773 T^{9} + 28629151 T^{10} \)
$37$ \( 1 - 16 T + 163 T^{2} - 1254 T^{3} + 9303 T^{4} - 59013 T^{5} + 344211 T^{6} - 1716726 T^{7} + 8256439 T^{8} - 29986576 T^{9} + 69343957 T^{10} \)
$41$ \( 1 - 6 T + 123 T^{2} - 768 T^{3} + 8837 T^{4} - 40071 T^{5} + 362317 T^{6} - 1291008 T^{7} + 8477283 T^{8} - 16954566 T^{9} + 115856201 T^{10} \)
$43$ \( 1 - 8 T + 99 T^{2} - 568 T^{3} + 6133 T^{4} - 35897 T^{5} + 263719 T^{6} - 1050232 T^{7} + 7871193 T^{8} - 27350408 T^{9} + 147008443 T^{10} \)
$47$ \( 1 + 29 T + 530 T^{2} + 6754 T^{3} + 66085 T^{4} + 508202 T^{5} + 3105995 T^{6} + 14919586 T^{7} + 55026190 T^{8} + 141510749 T^{9} + 229345007 T^{10} \)
$53$ \( 1 - 25 T + 481 T^{2} - 5989 T^{3} + 62516 T^{4} - 492068 T^{5} + 3313348 T^{6} - 16823101 T^{7} + 71609837 T^{8} - 197262025 T^{9} + 418195493 T^{10} \)
$59$ \( 1 + 11 T + 293 T^{2} + 2373 T^{3} + 34336 T^{4} + 203496 T^{5} + 2025824 T^{6} + 8260413 T^{7} + 60176047 T^{8} + 133290971 T^{9} + 714924299 T^{10} \)
$61$ \( 1 - 11 T + 282 T^{2} - 2318 T^{3} + 33067 T^{4} - 201041 T^{5} + 2017087 T^{6} - 8625278 T^{7} + 64008642 T^{8} - 152304251 T^{9} + 844596301 T^{10} \)
$67$ \( 1 - 6 T + 32 T^{2} + 152 T^{3} + 5791 T^{4} - 64428 T^{5} + 387997 T^{6} + 682328 T^{7} + 9624416 T^{8} - 120906726 T^{9} + 1350125107 T^{10} \)
$71$ \( 1 + 15 T + 300 T^{2} + 2752 T^{3} + 33867 T^{4} + 239018 T^{5} + 2404557 T^{6} + 13872832 T^{7} + 107373300 T^{8} + 381175215 T^{9} + 1804229351 T^{10} \)
$73$ \( 1 + 9 T + 159 T^{2} + 1799 T^{3} + 21272 T^{4} + 138124 T^{5} + 1552856 T^{6} + 9586871 T^{7} + 61853703 T^{8} + 255584169 T^{9} + 2073071593 T^{10} \)
$79$ \( 1 - 29 T + 426 T^{2} - 2590 T^{3} - 4859 T^{4} + 200521 T^{5} - 383861 T^{6} - 16164190 T^{7} + 210034614 T^{8} - 1129552349 T^{9} + 3077056399 T^{10} \)
$83$ \( 1 - 9 T + 416 T^{2} - 2882 T^{3} + 69119 T^{4} - 354626 T^{5} + 5736877 T^{6} - 19854098 T^{7} + 237863392 T^{8} - 427124889 T^{9} + 3939040643 T^{10} \)
$89$ \( 1 + 9 T + 302 T^{2} + 1778 T^{3} + 37739 T^{4} + 172799 T^{5} + 3358771 T^{6} + 14083538 T^{7} + 212900638 T^{8} + 564680169 T^{9} + 5584059449 T^{10} \)
$97$ \( 1 + 6 T + 392 T^{2} + 1594 T^{3} + 66059 T^{4} + 196320 T^{5} + 6407723 T^{6} + 14997946 T^{7} + 357767816 T^{8} + 531175686 T^{9} + 8587340257 T^{10} \)
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