Properties

Label 7168.2.a.bj
Level $7168$
Weight $2$
Character orbit 7168.a
Self dual yes
Analytic conductor $57.237$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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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: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \( x^{12} - 24x^{10} + 221x^{8} - 968x^{6} + 2008x^{4} - 1640x^{2} + 196 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 112)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{4} + \beta_{2} + \beta_1) q^{5} + q^{7} + (\beta_{9} + \beta_{8} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{4} + \beta_{2} + \beta_1) q^{5} + q^{7} + (\beta_{9} + \beta_{8} + 1) q^{9} + \beta_{5} q^{11} + \beta_{6} q^{13} + (\beta_{8} + \beta_{3} + 2) q^{15} + (\beta_{10} + 1) q^{17} + (\beta_{5} - \beta_{2} + \beta_1) q^{19} + \beta_1 q^{21} + (\beta_{10} + \beta_{7} + 2) q^{23} + (\beta_{10} + \beta_{7} + \beta_{3} + 2) q^{25} + (\beta_{11} + \beta_{4} + 2 \beta_1) q^{27} + (\beta_{6} + \beta_{5} - \beta_1) q^{29} + (\beta_{3} - 1) q^{31} + ( - \beta_{10} + \beta_{8} - \beta_{7} - \beta_{3}) q^{33} + (\beta_{4} + \beta_{2} + \beta_1) q^{35} + ( - \beta_{4} + 2 \beta_{2}) q^{37} + ( - \beta_{10} + \beta_{8} - 2 \beta_{3} + 2) q^{39} + (\beta_{9} + \beta_{3} + 2) q^{41} + ( - \beta_{11} + \beta_{6} + \beta_{5} - 2 \beta_{4} - \beta_{2} - \beta_1) q^{43} + (\beta_{11} - \beta_{6} - \beta_{5} + 5 \beta_{4} + \beta_{2} + 4 \beta_1) q^{45} + ( - \beta_{9} - \beta_{8} - \beta_{7} + \beta_{3} - 2) q^{47} + q^{49} + ( - \beta_{6} - \beta_{5} - \beta_{4} + \beta_1) q^{51} + ( - \beta_{11} - \beta_{5} + \beta_{2}) q^{53} + (\beta_{9} - \beta_{8} - \beta_{7} + 1) q^{55} + ( - \beta_{10} + 2 \beta_{9} + 2 \beta_{8} - \beta_{7} - \beta_{3} + 5) q^{57} + ( - \beta_{11} - \beta_{4} + \beta_1) q^{59} + ( - \beta_{11} + \beta_{5} + 2 \beta_{4} + \beta_1) q^{61} + (\beta_{9} + \beta_{8} + 1) q^{63} + ( - \beta_{9} - \beta_{8} + 2 \beta_{7} - 2 \beta_{3} + 4) q^{65} + ( - \beta_{6} - \beta_{4} + 2 \beta_{2} + \beta_1) q^{67} + ( - \beta_{6} - 2 \beta_{5} + \beta_{4} + \beta_{2} + 3 \beta_1) q^{69} + ( - \beta_{10} - 2 \beta_{8} - \beta_{3}) q^{71} + ( - \beta_{10} + \beta_{9} - 2 \beta_{8} - \beta_{3} - 1) q^{73} + ( - 2 \beta_{6} - 3 \beta_{5} + 6 \beta_{4} + 3 \beta_{2} + 5 \beta_1) q^{75} + \beta_{5} q^{77} + (\beta_{10} - \beta_{8} + \beta_{7} + \beta_{3} - 2) q^{79} + ( - \beta_{10} + 2 \beta_{8} - \beta_{7} - \beta_{3} + 4) q^{81} + ( - \beta_{11} + 2 \beta_{6} + \beta_{5} - 3 \beta_{4} - 3 \beta_{2} - \beta_1) q^{83} + (\beta_{11} - \beta_{6} + 2 \beta_{4} + 3 \beta_{2} + \beta_1) q^{85} + ( - 2 \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} - 3 \beta_{3} - 2) q^{87} + ( - \beta_{10} - \beta_{8} - \beta_{7} + 3 \beta_{3}) q^{89} + \beta_{6} q^{91} + ( - \beta_{6} - \beta_{5} + 5 \beta_{4} + 2 \beta_{2} + \beta_1) q^{93} + ( - \beta_{10} + 2 \beta_{9} + \beta_{8} - 2 \beta_{7} + 2 \beta_{3} - 2) q^{95} + (\beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} + 2 \beta_{3} + 4) q^{97} + (\beta_{11} + 2 \beta_{6} - 3 \beta_{4} - \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{7} + 12 q^{9} + 24 q^{15} + 8 q^{17} + 16 q^{23} + 20 q^{25} - 8 q^{31} + 16 q^{39} + 32 q^{41} - 16 q^{47} + 12 q^{49} + 24 q^{55} + 64 q^{57} + 12 q^{63} + 32 q^{65} + 8 q^{71} - 24 q^{79} + 44 q^{81} - 32 q^{87} + 24 q^{89} + 48 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 24x^{10} + 221x^{8} - 968x^{6} + 2008x^{4} - 1640x^{2} + 196 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{11} + 17\nu^{9} - 95\nu^{7} + 177\nu^{5} - 6\nu^{3} - 68\nu ) / 42 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{11} + 17\nu^{9} - 95\nu^{7} + 177\nu^{5} + 36\nu^{3} - 278\nu ) / 42 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{8} - 16\nu^{6} + 79\nu^{4} - 116\nu^{2} + 16 ) / 6 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{11} + 24\nu^{9} - 207\nu^{7} + 772\nu^{5} - 1154\nu^{3} + 464\nu ) / 84 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{11} - 41\nu^{9} + 316\nu^{7} - 1117\nu^{5} + 1720\nu^{3} - 830\nu ) / 42 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{11} + 18\nu^{9} - 115\nu^{7} + 322\nu^{5} - 426\nu^{3} + 264\nu ) / 12 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{10} + 18\nu^{8} - 113\nu^{6} + 286\nu^{4} - 232\nu^{2} - 26 ) / 6 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{10} + 18\nu^{8} - 113\nu^{6} + 286\nu^{4} - 244\nu^{2} + 22 ) / 6 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{10} - 18\nu^{8} + 115\nu^{6} - 310\nu^{4} + 318\nu^{2} - 54 ) / 6 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -\nu^{10} + 19\nu^{8} - 127\nu^{6} + 353\nu^{4} - 370\nu^{2} + 72 ) / 6 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 15\nu^{11} - 276\nu^{9} + 1789\nu^{7} - 4776\nu^{5} + 4402\nu^{3} + 152\nu ) / 84 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{11} - \beta_{5} + \beta_{4} + 3\beta_{2} + 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{8} + \beta_{7} + 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{11} - 5\beta_{5} + 5\beta_{4} + 19\beta_{2} + 6\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{10} - \beta_{9} - 9\beta_{8} + 7\beta_{7} - \beta_{3} + 45 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 29\beta_{11} + 4\beta_{6} - 25\beta_{5} + 33\beta_{4} + 119\beta_{2} + 18\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 12\beta_{10} - 6\beta_{9} - 65\beta_{8} + 47\beta_{7} - 12\beta_{3} + 276 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 179\beta_{11} + 48\beta_{6} - 119\beta_{5} + 251\beta_{4} + 761\beta_{2} + 50\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 113\beta_{10} - 17\beta_{9} - 445\beta_{8} + 315\beta_{7} - 101\beta_{3} + 1757 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 1143\beta_{11} + 428\beta_{6} - 523\beta_{5} + 2003\beta_{4} + 4949\beta_{2} + 78\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 964\beta_{10} + 86\beta_{9} - 3007\beta_{8} + 2117\beta_{7} - 748\beta_{3} + 11400 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 7461\beta_{11} + 3424\beta_{6} - 1913\beta_{5} + 15949\beta_{4} + 32583\beta_{2} - 578\beta_1 ) / 4 \) Copy content Toggle raw display

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
−1.33191
−1.75510
2.66376
−2.48658
2.39251
0.377920
−0.377920
−2.39251
2.48658
−2.66376
1.75510
1.33191
0 −3.13347 0 −0.555963 0 1.00000 0 6.81864 0
1.2 0 −2.90620 0 −3.85750 0 1.00000 0 5.44602 0
1.3 0 −1.96750 0 3.06146 0 1.00000 0 0.871066 0
1.4 0 −0.892634 0 −3.31293 0 1.00000 0 −2.20320 0
1.5 0 −0.848497 0 −1.37882 0 1.00000 0 −2.28005 0
1.6 0 −0.589521 0 −1.60045 0 1.00000 0 −2.65247 0
1.7 0 0.589521 0 1.60045 0 1.00000 0 −2.65247 0
1.8 0 0.848497 0 1.37882 0 1.00000 0 −2.28005 0
1.9 0 0.892634 0 3.31293 0 1.00000 0 −2.20320 0
1.10 0 1.96750 0 −3.06146 0 1.00000 0 0.871066 0
1.11 0 2.90620 0 3.85750 0 1.00000 0 5.44602 0
1.12 0 3.13347 0 0.555963 0 1.00000 0 6.81864 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7168.2.a.bj 12
4.b odd 2 1 7168.2.a.bi 12
8.b even 2 1 inner 7168.2.a.bj 12
8.d odd 2 1 7168.2.a.bi 12
32.g even 8 2 112.2.m.d 12
32.g even 8 2 896.2.m.g 12
32.h odd 8 2 448.2.m.d 12
32.h odd 8 2 896.2.m.h 12
224.v odd 8 2 784.2.m.h 12
224.bc odd 24 4 784.2.x.m 24
224.bd even 24 4 784.2.x.l 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.m.d 12 32.g even 8 2
448.2.m.d 12 32.h odd 8 2
784.2.m.h 12 224.v odd 8 2
784.2.x.l 24 224.bd even 24 4
784.2.x.m 24 224.bc odd 24 4
896.2.m.g 12 32.g even 8 2
896.2.m.h 12 32.h odd 8 2
7168.2.a.bi 12 4.b odd 2 1
7168.2.a.bi 12 8.d odd 2 1
7168.2.a.bj 12 1.a even 1 1 trivial
7168.2.a.bj 12 8.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7168))\):

\( T_{3}^{12} - 24T_{3}^{10} + 196T_{3}^{8} - 632T_{3}^{6} + 772T_{3}^{4} - 384T_{3}^{2} + 64 \) Copy content Toggle raw display
\( T_{5}^{12} - 40T_{5}^{10} + 580T_{5}^{8} - 3688T_{5}^{6} + 9892T_{5}^{4} - 10176T_{5}^{2} + 2304 \) Copy content Toggle raw display
\( T_{11}^{12} - 76T_{11}^{10} + 2148T_{11}^{8} - 28736T_{11}^{6} + 190080T_{11}^{4} - 569344T_{11}^{2} + 541696 \) Copy content Toggle raw display
\( T_{13}^{12} - 112T_{13}^{10} + 4548T_{13}^{8} - 82856T_{13}^{6} + 694020T_{13}^{4} - 2563072T_{13}^{2} + 3211264 \) Copy content Toggle raw display
\( T_{17}^{6} - 4T_{17}^{5} - 44T_{17}^{4} + 200T_{17}^{3} + 40T_{17}^{2} - 288T_{17} - 96 \) Copy content Toggle raw display
\( T_{23}^{6} - 8T_{23}^{5} - 36T_{23}^{4} + 320T_{23}^{3} + 340T_{23}^{2} - 2656T_{23} - 3184 \) Copy content Toggle raw display
\( T_{31}^{6} + 4T_{31}^{5} - 16T_{31}^{4} - 72T_{31}^{3} - 24T_{31}^{2} + 96T_{31} + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - 24 T^{10} + 196 T^{8} + \cdots + 64 \) Copy content Toggle raw display
$5$ \( T^{12} - 40 T^{10} + 580 T^{8} + \cdots + 2304 \) Copy content Toggle raw display
$7$ \( (T - 1)^{12} \) Copy content Toggle raw display
$11$ \( T^{12} - 76 T^{10} + 2148 T^{8} + \cdots + 541696 \) Copy content Toggle raw display
$13$ \( T^{12} - 112 T^{10} + 4548 T^{8} + \cdots + 3211264 \) Copy content Toggle raw display
$17$ \( (T^{6} - 4 T^{5} - 44 T^{4} + 200 T^{3} + \cdots - 96)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} - 128 T^{10} + 5748 T^{8} + \cdots + 2849344 \) Copy content Toggle raw display
$23$ \( (T^{6} - 8 T^{5} - 36 T^{4} + 320 T^{3} + \cdots - 3184)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} - 156 T^{10} + 8188 T^{8} + \cdots + 8620096 \) Copy content Toggle raw display
$31$ \( (T^{6} + 4 T^{5} - 16 T^{4} - 72 T^{3} + \cdots + 64)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} - 172 T^{10} + 9980 T^{8} + \cdots + 5053504 \) Copy content Toggle raw display
$41$ \( (T^{6} - 16 T^{5} + 68 T^{4} + 40 T^{3} + \cdots + 160)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} - 220 T^{10} + \cdots + 23040000 \) Copy content Toggle raw display
$47$ \( (T^{6} + 8 T^{5} - 104 T^{4} - 1032 T^{3} + \cdots + 19776)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} - 348 T^{10} + 40156 T^{8} + \cdots + 78400 \) Copy content Toggle raw display
$59$ \( T^{12} - 240 T^{10} + \cdots + 119596096 \) Copy content Toggle raw display
$61$ \( T^{12} - 264 T^{10} + 16100 T^{8} + \cdots + 256 \) Copy content Toggle raw display
$67$ \( T^{12} - 236 T^{10} + 10980 T^{8} + \cdots + 3686400 \) Copy content Toggle raw display
$71$ \( (T^{6} - 4 T^{5} - 208 T^{4} + 880 T^{3} + \cdots + 2560)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} - 308 T^{4} - 384 T^{3} + \cdots - 55232)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 12 T^{5} - 44 T^{4} - 576 T^{3} + \cdots - 2240)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} - 568 T^{10} + \cdots + 320093429824 \) Copy content Toggle raw display
$89$ \( (T^{6} - 12 T^{5} - 204 T^{4} + \cdots - 188352)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} - 24 T^{5} - 60 T^{4} + 2568 T^{3} + \cdots - 39008)^{2} \) Copy content Toggle raw display
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