Properties

Label 7168.2.a.bb
Level $7168$
Weight $2$
Character orbit 7168.a
Self dual yes
Analytic conductor $57.237$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7168 = 2^{10} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7168.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(57.2367681689\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: 8.8.9433055232.1
Defining polynomial: \(x^{8} - 4 x^{7} - 6 x^{6} + 32 x^{5} + 9 x^{4} - 76 x^{3} - 4 x^{2} + 48 x - 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1792)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{3} + ( -1 - \beta_{2} ) q^{5} + q^{7} + ( 2 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{6} ) q^{9} +O(q^{10})\) \( q + \beta_{5} q^{3} + ( -1 - \beta_{2} ) q^{5} + q^{7} + ( 2 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{6} ) q^{9} + ( -1 - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{11} + ( -2 - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{13} + ( -1 - \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{15} + ( \beta_{2} - \beta_{4} - 2 \beta_{6} + \beta_{7} ) q^{17} + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{19} + \beta_{5} q^{21} + ( 1 + \beta_{1} - 2 \beta_{2} + \beta_{3} + 3 \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{23} + ( 1 + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{7} ) q^{25} + ( 2 + \beta_{1} + \beta_{2} + 3 \beta_{3} + \beta_{4} + 2 \beta_{6} - \beta_{7} ) q^{27} + ( -1 - 2 \beta_{1} - \beta_{2} + 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{29} + ( -1 - \beta_{1} + 3 \beta_{4} - \beta_{5} - \beta_{6} ) q^{31} + ( -\beta_{1} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{7} ) q^{33} + ( -1 - \beta_{2} ) q^{35} + ( -2 - \beta_{1} - 2 \beta_{4} - 2 \beta_{7} ) q^{37} + ( -2 + \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} - 3 \beta_{5} ) q^{39} + ( -2 - \beta_{1} + \beta_{2} - 3 \beta_{3} - \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{41} + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{43} + ( -8 - 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - 3 \beta_{6} ) q^{45} + ( 3 - \beta_{1} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{47} + q^{49} + ( -5 - 3 \beta_{1} - 3 \beta_{2} + 4 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{51} + ( -6 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{6} ) q^{53} + ( 2 + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{7} ) q^{55} + ( -2 \beta_{1} - \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{7} ) q^{57} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - 3 \beta_{5} + \beta_{7} ) q^{59} + ( 1 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{61} + ( 2 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{6} ) q^{63} + ( 5 - 3 \beta_{1} + 3 \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{65} + ( -6 + \beta_{1} + \beta_{2} + 3 \beta_{3} - \beta_{4} - 3 \beta_{6} - 2 \beta_{7} ) q^{67} + ( -1 + 2 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} - 5 \beta_{4} + \beta_{5} - \beta_{6} ) q^{69} + ( -1 - \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{71} + ( 2 + 3 \beta_{1} - 3 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{73} + ( 6 + 3 \beta_{1} - \beta_{2} - 3 \beta_{3} + 3 \beta_{4} + \beta_{5} + 4 \beta_{6} + \beta_{7} ) q^{75} + ( -1 - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{77} + ( 2 + \beta_{1} + \beta_{3} - 2 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} ) q^{79} + ( 5 + 4 \beta_{1} + 5 \beta_{3} + 2 \beta_{4} + \beta_{5} + 4 \beta_{6} + \beta_{7} ) q^{81} + ( -2 \beta_{1} - 2 \beta_{3} - 4 \beta_{4} + \beta_{5} - 2 \beta_{7} ) q^{83} + ( -1 + 3 \beta_{1} - \beta_{2} + 6 \beta_{4} - \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{85} + ( 3 - 3 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} - 3 \beta_{4} + \beta_{5} - 3 \beta_{6} + 4 \beta_{7} ) q^{87} + ( 2 + \beta_{1} + \beta_{3} + 2 \beta_{4} ) q^{89} + ( -2 - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{91} + ( -1 - 3 \beta_{1} + \beta_{2} - 2 \beta_{3} - 4 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} + 5 \beta_{7} ) q^{93} + ( -4 + \beta_{1} - \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} + 2 \beta_{6} ) q^{95} + ( -2 + \beta_{2} - 3 \beta_{4} + 2 \beta_{6} + 3 \beta_{7} ) q^{97} + ( 1 - \beta_{3} - 3 \beta_{4} + \beta_{5} - 2 \beta_{6} + 3 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{5} + 8q^{7} + 12q^{9} + O(q^{10}) \) \( 8q - 8q^{5} + 8q^{7} + 12q^{9} - 12q^{11} - 20q^{13} + 4q^{17} - 4q^{19} + 8q^{23} + 12q^{25} + 12q^{27} - 8q^{29} - 4q^{31} + 8q^{33} - 8q^{35} - 8q^{37} - 16q^{39} - 12q^{41} + 4q^{43} - 52q^{45} + 20q^{47} + 8q^{49} - 32q^{51} - 40q^{53} + 24q^{55} - 4q^{57} - 4q^{59} + 8q^{61} + 12q^{63} + 36q^{65} - 28q^{67} - 4q^{69} - 16q^{71} + 16q^{73} + 28q^{75} - 12q^{77} + 20q^{81} + 8q^{83} - 16q^{85} + 20q^{87} + 16q^{89} - 20q^{91} - 16q^{93} - 40q^{95} - 36q^{97} + 4q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 4 x^{7} - 6 x^{6} + 32 x^{5} + 9 x^{4} - 76 x^{3} - 4 x^{2} + 48 x - 8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{4} - 2 \nu^{3} - 5 \nu^{2} + 6 \nu + 4 \)\()/2\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{7} - 4 \nu^{6} - 2 \nu^{5} + 20 \nu^{4} - 15 \nu^{3} - 8 \nu^{2} + 32 \nu - 20 \)\()/4\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{6} - 3 \nu^{5} - 6 \nu^{4} + 17 \nu^{3} + 9 \nu^{2} - 18 \nu \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{6} + 3 \nu^{5} + 6 \nu^{4} - 17 \nu^{3} - 7 \nu^{2} + 16 \nu - 6 \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} - 4 \nu^{6} - 4 \nu^{5} + 26 \nu^{4} - \nu^{3} - 42 \nu^{2} + 8 \)\()/4\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{7} + 4 \nu^{6} + 4 \nu^{5} - 26 \nu^{4} + 5 \nu^{3} + 38 \nu^{2} - 16 \nu - 8 \)\()/4\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{7} + 4 \nu^{6} + 6 \nu^{5} - 30 \nu^{4} - 13 \nu^{3} + 62 \nu^{2} + 20 \nu - 24 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{7} + \beta_{6} - \beta_{5} + \beta_{2} + \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{7} + \beta_{6} - \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + \beta_{2} + \beta_{1} + 7\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-5 \beta_{7} + 7 \beta_{6} - 3 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 5 \beta_{2} + 5 \beta_{1} + 11\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-9 \beta_{7} + 13 \beta_{6} - 5 \beta_{5} + 14 \beta_{4} + 14 \beta_{3} + 9 \beta_{2} + 13 \beta_{1} + 43\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-29 \beta_{7} + 55 \beta_{6} - 7 \beta_{5} + 22 \beta_{4} + 22 \beta_{3} + 33 \beta_{2} + 41 \beta_{1} + 99\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(-65 \beta_{7} + 133 \beta_{6} - 9 \beta_{5} + 98 \beta_{4} + 102 \beta_{3} + 77 \beta_{2} + 125 \beta_{1} + 323\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-189 \beta_{7} + 463 \beta_{6} + 29 \beta_{5} + 202 \beta_{4} + 218 \beta_{3} + 253 \beta_{2} + 373 \beta_{1} + 859\)\()/2\)

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
0.822080
2.38887
2.21236
−1.21236
−2.02908
0.177920
−1.38887
3.02908
0 −2.88693 0 −0.992279 0 1.00000 0 5.33435 0
1.2 0 −2.29954 0 −1.65321 0 1.00000 0 2.28788 0
1.3 0 −1.78579 0 −4.18248 0 1.00000 0 0.189043 0
1.4 0 −0.463900 0 1.98268 0 1.00000 0 −2.78480 0
1.5 0 0.242103 0 0.379610 0 1.00000 0 −2.94139 0
1.6 0 1.67251 0 2.65618 0 1.00000 0 −0.202696 0
1.7 0 2.09974 0 −2.59647 0 1.00000 0 1.40890 0
1.8 0 3.42180 0 −3.59402 0 1.00000 0 8.70871 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(-1\)

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 7168.2.a.bb 8
4.b odd 2 1 7168.2.a.ba 8
8.b even 2 1 7168.2.a.bf 8
8.d odd 2 1 7168.2.a.be 8
32.g even 8 2 1792.2.m.e 16
32.g even 8 2 1792.2.m.h yes 16
32.h odd 8 2 1792.2.m.f yes 16
32.h odd 8 2 1792.2.m.g yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1792.2.m.e 16 32.g even 8 2
1792.2.m.f yes 16 32.h odd 8 2
1792.2.m.g yes 16 32.h odd 8 2
1792.2.m.h yes 16 32.g even 8 2
7168.2.a.ba 8 4.b odd 2 1
7168.2.a.bb 8 1.a even 1 1 trivial
7168.2.a.be 8 8.d odd 2 1
7168.2.a.bf 8 8.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7168))\):

\( T_{3}^{8} - 18 T_{3}^{6} - 4 T_{3}^{5} + 94 T_{3}^{4} + 24 T_{3}^{3} - 152 T_{3}^{2} - 32 T_{3} + 16 \)
\(T_{5}^{8} + \cdots\)
\(T_{11}^{8} + \cdots\)
\( T_{13}^{8} + 20 T_{13}^{7} + 138 T_{13}^{6} + 284 T_{13}^{5} - 882 T_{13}^{4} - 4960 T_{13}^{3} - 6784 T_{13}^{2} + 4096 \)
\(T_{17}^{8} - \cdots\)
\(T_{23}^{8} - \cdots\)
\(T_{31}^{8} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( 16 - 32 T - 152 T^{2} + 24 T^{3} + 94 T^{4} - 4 T^{5} - 18 T^{6} + T^{8} \)
$5$ \( -128 + 128 T + 512 T^{2} + 176 T^{3} - 162 T^{4} - 84 T^{5} + 6 T^{6} + 8 T^{7} + T^{8} \)
$7$ \( ( -1 + T )^{8} \)
$11$ \( 64 + 128 T - 192 T^{2} - 640 T^{3} - 492 T^{4} - 88 T^{5} + 32 T^{6} + 12 T^{7} + T^{8} \)
$13$ \( 4096 - 6784 T^{2} - 4960 T^{3} - 882 T^{4} + 284 T^{5} + 138 T^{6} + 20 T^{7} + T^{8} \)
$17$ \( 116608 + 15872 T - 46272 T^{2} - 4992 T^{3} + 3528 T^{4} + 272 T^{5} - 100 T^{6} - 4 T^{7} + T^{8} \)
$19$ \( 16 - 160 T - 1528 T^{2} + 584 T^{3} + 526 T^{4} - 140 T^{5} - 46 T^{6} + 4 T^{7} + T^{8} \)
$23$ \( -70592 + 56064 T + 12800 T^{2} - 15680 T^{3} + 1428 T^{4} + 688 T^{5} - 84 T^{6} - 8 T^{7} + T^{8} \)
$29$ \( 91792 - 112704 T + 2816 T^{2} + 21728 T^{3} + 1560 T^{4} - 880 T^{5} - 96 T^{6} + 8 T^{7} + T^{8} \)
$31$ \( -251648 - 366848 T - 59584 T^{2} + 25792 T^{3} + 5304 T^{4} - 576 T^{5} - 132 T^{6} + 4 T^{7} + T^{8} \)
$37$ \( 695056 - 197056 T - 172960 T^{2} + 40544 T^{3} + 9240 T^{4} - 1104 T^{5} - 168 T^{6} + 8 T^{7} + T^{8} \)
$41$ \( -9344 - 2048 T + 15360 T^{2} + 2560 T^{3} - 4440 T^{4} - 1424 T^{5} - 76 T^{6} + 12 T^{7} + T^{8} \)
$43$ \( 45952 + 51840 T - 22624 T^{2} - 12064 T^{3} + 3012 T^{4} + 536 T^{5} - 120 T^{6} - 4 T^{7} + T^{8} \)
$47$ \( -18176 + 24832 T + 32832 T^{2} - 2880 T^{3} - 4680 T^{4} + 784 T^{5} + 68 T^{6} - 20 T^{7} + T^{8} \)
$53$ \( 365584 + 127808 T - 504800 T^{2} - 228896 T^{3} - 28392 T^{4} + 1136 T^{5} + 520 T^{6} + 40 T^{7} + T^{8} \)
$59$ \( 407824 - 240928 T - 107960 T^{2} + 46072 T^{3} + 10254 T^{4} - 964 T^{5} - 206 T^{6} + 4 T^{7} + T^{8} \)
$61$ \( -98432 + 84992 T + 16480 T^{2} - 21344 T^{3} + 990 T^{4} + 1124 T^{5} - 122 T^{6} - 8 T^{7} + T^{8} \)
$67$ \( -51056384 - 6458368 T + 2259520 T^{2} + 323776 T^{3} - 27900 T^{4} - 5272 T^{5} + 16 T^{6} + 28 T^{7} + T^{8} \)
$71$ \( -8192 - 45056 T + 35328 T^{2} + 14592 T^{3} - 2544 T^{4} - 1088 T^{5} - 16 T^{6} + 16 T^{7} + T^{8} \)
$73$ \( 7618816 + 2791424 T - 384640 T^{2} - 162304 T^{3} + 10656 T^{4} + 2880 T^{5} - 168 T^{6} - 16 T^{7} + T^{8} \)
$79$ \( -4822784 - 3951616 T - 633344 T^{2} + 107520 T^{3} + 25504 T^{4} - 704 T^{5} - 288 T^{6} + T^{8} \)
$83$ \( -20041712 + 10520992 T - 569112 T^{2} - 398808 T^{3} + 41022 T^{4} + 3652 T^{5} - 418 T^{6} - 8 T^{7} + T^{8} \)
$89$ \( ( 16 + 32 T - 16 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$97$ \( -2571392 + 3808512 T + 3034816 T^{2} + 99840 T^{3} - 84856 T^{4} - 8256 T^{5} + 116 T^{6} + 36 T^{7} + T^{8} \)
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