Properties

Label 8016.2.a.o
Level 8016
Weight 2
Character orbit 8016.a
Self dual yes
Analytic conductor 64.008
Analytic rank 0
Dimension 4
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8016.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.0080822603\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.2777.1
Defining polynomial: \(x^{4} - x^{3} - 4 x^{2} + x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1002)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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

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

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

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
2.36234
−0.679643
0.825785
−1.50848
0 1.00000 0 −1.87806 0 3.28324 0 1.00000 0
1.2 0 1.00000 0 0.416566 0 −4.65960 0 1.00000 0
1.3 0 1.00000 0 2.77037 0 −1.86579 0 1.00000 0
1.4 0 1.00000 0 3.69113 0 2.24216 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(167\) \(-1\)

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 8016.2.a.o 4
4.b odd 2 1 1002.2.a.i 4
12.b even 2 1 3006.2.a.s 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1002.2.a.i 4 4.b odd 2 1
3006.2.a.s 4 12.b even 2 1
8016.2.a.o 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8016))\):

\( T_{5}^{4} - 5 T_{5}^{3} + 20 T_{5} - 8 \)
\( T_{7}^{4} + T_{7}^{3} - 20 T_{7}^{2} + 64 \)
\( T_{11} \)
\( T_{13}^{4} - 8 T_{13}^{3} + 4 T_{13}^{2} + 56 T_{13} - 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - T )^{4} \)
$5$ \( 1 - 5 T + 20 T^{2} - 55 T^{3} + 142 T^{4} - 275 T^{5} + 500 T^{6} - 625 T^{7} + 625 T^{8} \)
$7$ \( 1 + T + 8 T^{2} + 21 T^{3} + 78 T^{4} + 147 T^{5} + 392 T^{6} + 343 T^{7} + 2401 T^{8} \)
$11$ \( ( 1 + 11 T^{2} )^{4} \)
$13$ \( 1 - 8 T + 56 T^{2} - 256 T^{3} + 1102 T^{4} - 3328 T^{5} + 9464 T^{6} - 17576 T^{7} + 28561 T^{8} \)
$17$ \( 1 - 10 T + 88 T^{2} - 494 T^{3} + 2398 T^{4} - 8398 T^{5} + 25432 T^{6} - 49130 T^{7} + 83521 T^{8} \)
$19$ \( 1 - 2 T + 44 T^{2} - 82 T^{3} + 1078 T^{4} - 1558 T^{5} + 15884 T^{6} - 13718 T^{7} + 130321 T^{8} \)
$23$ \( 1 + 2 T + 60 T^{2} + 106 T^{3} + 1830 T^{4} + 2438 T^{5} + 31740 T^{6} + 24334 T^{7} + 279841 T^{8} \)
$29$ \( 1 - 14 T + 152 T^{2} - 1162 T^{3} + 7102 T^{4} - 33698 T^{5} + 127832 T^{6} - 341446 T^{7} + 707281 T^{8} \)
$31$ \( 1 + T + 88 T^{2} - 3 T^{3} + 3470 T^{4} - 93 T^{5} + 84568 T^{6} + 29791 T^{7} + 923521 T^{8} \)
$37$ \( 1 - 5 T + 82 T^{2} - 371 T^{3} + 3898 T^{4} - 13727 T^{5} + 112258 T^{6} - 253265 T^{7} + 1874161 T^{8} \)
$41$ \( 1 - 14 T + 152 T^{2} - 1242 T^{3} + 8030 T^{4} - 50922 T^{5} + 255512 T^{6} - 964894 T^{7} + 2825761 T^{8} \)
$43$ \( 1 + 2 T + 140 T^{2} + 298 T^{3} + 8374 T^{4} + 12814 T^{5} + 258860 T^{6} + 159014 T^{7} + 3418801 T^{8} \)
$47$ \( 1 + 7 T + 156 T^{2} + 779 T^{3} + 10182 T^{4} + 36613 T^{5} + 344604 T^{6} + 726761 T^{7} + 4879681 T^{8} \)
$53$ \( 1 - 3 T + 172 T^{2} - 441 T^{3} + 12622 T^{4} - 23373 T^{5} + 483148 T^{6} - 446631 T^{7} + 7890481 T^{8} \)
$59$ \( 1 - 5 T + 120 T^{2} - 921 T^{3} + 8006 T^{4} - 54339 T^{5} + 417720 T^{6} - 1026895 T^{7} + 12117361 T^{8} \)
$61$ \( 1 - 2 T + 208 T^{2} - 294 T^{3} + 18062 T^{4} - 17934 T^{5} + 773968 T^{6} - 453962 T^{7} + 13845841 T^{8} \)
$67$ \( 1 - 7 T + 82 T^{2} + 441 T^{3} - 1198 T^{4} + 29547 T^{5} + 368098 T^{6} - 2105341 T^{7} + 20151121 T^{8} \)
$71$ \( 1 + 2 T + 116 T^{2} + 298 T^{3} + 13174 T^{4} + 21158 T^{5} + 584756 T^{6} + 715822 T^{7} + 25411681 T^{8} \)
$73$ \( 1 - 2 T + 160 T^{2} - 366 T^{3} + 13214 T^{4} - 26718 T^{5} + 852640 T^{6} - 778034 T^{7} + 28398241 T^{8} \)
$79$ \( 1 - 10 T + 260 T^{2} - 1994 T^{3} + 30070 T^{4} - 157526 T^{5} + 1622660 T^{6} - 4930390 T^{7} + 38950081 T^{8} \)
$83$ \( 1 + 13 T + 236 T^{2} + 2465 T^{3} + 29470 T^{4} + 204595 T^{5} + 1625804 T^{6} + 7433231 T^{7} + 47458321 T^{8} \)
$89$ \( 1 - 13 T + 354 T^{2} - 2955 T^{3} + 45722 T^{4} - 262995 T^{5} + 2804034 T^{6} - 9164597 T^{7} + 62742241 T^{8} \)
$97$ \( 1 - 5 T + 242 T^{2} - 827 T^{3} + 31386 T^{4} - 80219 T^{5} + 2276978 T^{6} - 4563365 T^{7} + 88529281 T^{8} \)
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