Properties

Label 8281.2.a.cp
Level $8281$
Weight $2$
Character orbit 8281.a
Self dual yes
Analytic conductor $66.124$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 8281 = 7^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8281.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(66.1241179138\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \( x^{12} - 16x^{10} + 88x^{8} - 197x^{6} + 172x^{4} - 36x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{3} q^{3} + (\beta_{9} - \beta_{8} + 1) q^{4} + (\beta_{11} + \beta_1) q^{5} - \beta_{4} q^{6} + ( - \beta_{11} + \beta_{10} + \beta_{7} + \beta_{5} + \beta_{4} + \beta_1) q^{8} + ( - \beta_{3} + \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{3} q^{3} + (\beta_{9} - \beta_{8} + 1) q^{4} + (\beta_{11} + \beta_1) q^{5} - \beta_{4} q^{6} + ( - \beta_{11} + \beta_{10} + \beta_{7} + \beta_{5} + \beta_{4} + \beta_1) q^{8} + ( - \beta_{3} + \beta_{2}) q^{9} + ( - \beta_{8} + 2) q^{10} + ( - \beta_{5} - \beta_{4} + \beta_1) q^{11} + ( - \beta_{9} - \beta_{6}) q^{12} + ( - \beta_{7} + \beta_{5} - \beta_{4}) q^{15} + ( - \beta_{6} + \beta_{3} - \beta_{2} + 2) q^{16} + ( - \beta_{9} - \beta_{8} + \beta_{3} + 3) q^{17} + (\beta_{11} - 2 \beta_{10} - \beta_{5} - \beta_{4} - \beta_1) q^{18} + ( - \beta_{11} - \beta_{5} - \beta_{4} + \beta_1) q^{19} + ( - 2 \beta_{11} + \beta_{10} + \beta_{5} + \beta_1) q^{20} + ( - \beta_{6} - 2 \beta_{3} + 2) q^{22} + ( - 2 \beta_{9} + \beta_{8} - \beta_{6} - \beta_{2}) q^{23} + ( - \beta_{11} + 2 \beta_{10} + \beta_{7} + \beta_{5} + \beta_{4} - \beta_1) q^{24} + ( - 2 \beta_{9} - \beta_{8} + \beta_{3} - 1) q^{25} + (\beta_{9} + 2 \beta_{2} + 2) q^{27} + ( - \beta_{9} - \beta_{8} + \beta_{3} + 2 \beta_{2} + 1) q^{29} + ( - \beta_{9} - \beta_{8} - \beta_{3} + 1) q^{30} + ( - 2 \beta_{11} + 2 \beta_{7} + \beta_{5} + \beta_{4} + 2 \beta_1) q^{31} + ( - \beta_{11} + 2 \beta_{10} - \beta_{4} + \beta_1) q^{32} + ( - \beta_{11} - 2 \beta_{10} - \beta_{5} - 2 \beta_{4} + \beta_1) q^{33} + (\beta_{11} + \beta_{10} - \beta_{7} + \beta_{5} + 3 \beta_1) q^{34} + ( - \beta_{9} + 2 \beta_{8} + \beta_{6} - 3) q^{36} + (\beta_{11} + \beta_{7} - \beta_{5} + \beta_{4} + 3 \beta_1) q^{37} + (\beta_{9} - \beta_{6} - 2 \beta_{3} + 3) q^{38} + (2 \beta_{9} - \beta_{6} - \beta_{2} + 1) q^{40} + (4 \beta_{11} - 3 \beta_{10} + \beta_{7} + 2 \beta_{4} + \beta_1) q^{41} + (2 \beta_{9} + 2 \beta_{6} - \beta_{3} + 1) q^{43} + ( - 2 \beta_{11} + 2 \beta_{10} + 2 \beta_{7} + 3 \beta_{5}) q^{44} + (\beta_{11} - \beta_{10} - \beta_{7} - \beta_{5} - \beta_{4}) q^{45} + ( - \beta_{11} + 3 \beta_{10} + \beta_{5} - 2 \beta_{4} - 2 \beta_1) q^{46} + ( - \beta_{11} + \beta_{10} - \beta_{7}) q^{47} + (\beta_{9} + \beta_{3} - 2 \beta_{2} - 3) q^{48} + (2 \beta_{11} + \beta_{10} - 2 \beta_{7} + \beta_{5} - \beta_{4} - 2 \beta_1) q^{50} + ( - \beta_{9} - 2 \beta_{8} + \beta_{6} - \beta_{3} - \beta_{2} - 1) q^{51} + (2 \beta_{9} - \beta_{8} - 2 \beta_{3} + 1) q^{53} + (\beta_{11} - 4 \beta_{10} + \beta_{7} - 2 \beta_{5} + \beta_{4} + \beta_1) q^{54} + ( - \beta_{9} - \beta_{8} - \beta_{6} + \beta_{2} + 3) q^{55} + ( - \beta_{11} - 2 \beta_{10} + \beta_{7} - 2 \beta_{5} - 2 \beta_{4} + \beta_1) q^{57} + (3 \beta_{11} - 3 \beta_{10} - \beta_{7} - \beta_{5} - \beta_1) q^{58} + (\beta_{11} + 4 \beta_{10} - 3 \beta_{7} + 4 \beta_1) q^{59} + (\beta_{11} + \beta_{10} + \beta_{7} - \beta_{5} + \beta_1) q^{60} + (2 \beta_{9} - \beta_{3} + 2 \beta_{2}) q^{61} + (5 \beta_{9} - 3 \beta_{8} - \beta_{6} + 9) q^{62} + ( - \beta_{9} - \beta_{8} - \beta_{6} - 4 \beta_{3} - 2) q^{64} + (2 \beta_{9} - 4 \beta_{3} + 2 \beta_{2} + 5) q^{66} + (5 \beta_{11} - \beta_{10} - \beta_{7} - \beta_{5} + \beta_{4} - \beta_1) q^{67} + (3 \beta_{9} - 2 \beta_{8} - \beta_{3} - \beta_{2} + 2) q^{68} + (2 \beta_{9} - \beta_{8} + 2 \beta_{6} + \beta_{3} - \beta_{2} + 1) q^{69} + (2 \beta_{11} - 3 \beta_{10} - \beta_{7} + 3 \beta_1) q^{71} + (\beta_{11} - 3 \beta_{7} - \beta_{5} + \beta_{4} - 4 \beta_1) q^{72} + ( - 4 \beta_{11} + 2 \beta_{10} - 2 \beta_{7} + 2 \beta_{5} + \beta_1) q^{73} + (3 \beta_{9} - 2 \beta_{8} + \beta_{3} + 7) q^{74} + ( - \beta_{9} - 3 \beta_{8} + 2 \beta_{6} + 3 \beta_{3} - \beta_{2}) q^{75} + ( - \beta_{11} + 2 \beta_{10} + 3 \beta_{7} + 3 \beta_{5} + \beta_{4} + 2 \beta_1) q^{76} + (\beta_{9} + 3 \beta_{8} + \beta_{6} + \beta_{2} + 6) q^{79} + ( - \beta_{11} + 2 \beta_{10} + 4 \beta_{7} + 2 \beta_{4} + 2 \beta_1) q^{80} + (2 \beta_{9} + \beta_{8} - \beta_{6} - 3 \beta_{3} - \beta_{2} - 3) q^{81} + (2 \beta_{9} - \beta_{8} + 4 \beta_{6} + 3 \beta_{3} + 3 \beta_{2} + 2) q^{82} + (4 \beta_{11} - \beta_{10} + \beta_{7} - 2 \beta_{5} + 3 \beta_{4} + 3 \beta_1) q^{83} + (6 \beta_{11} + \beta_{10} - 3 \beta_{7} - \beta_{4} + 5 \beta_1) q^{85} + (2 \beta_{11} - 4 \beta_{10} - 2 \beta_{7} - 2 \beta_{5} + \beta_{4} + 3 \beta_1) q^{86} + (\beta_{9} - 2 \beta_{8} + \beta_{6} - 3 \beta_{3} + \beta_{2} - 3) q^{87} + ( - 3 \beta_{8} - 2 \beta_{6} + 2 \beta_{3} - 2 \beta_{2} - 1) q^{88} + ( - 3 \beta_{11} + \beta_{10} + 2 \beta_{7} + \beta_{5} - 2 \beta_{4} - 4 \beta_1) q^{89} + ( - \beta_{9} + \beta_{8} + \beta_{6} - \beta_{3} + \beta_{2} - 1) q^{90} + ( - 2 \beta_{9} - \beta_{8} - 3 \beta_{6} - 4 \beta_{3} - \beta_{2} - 7) q^{92} + ( - \beta_{11} + 4 \beta_{7} - \beta_{5} - \beta_{4} - \beta_1) q^{93} + (\beta_{3} - \beta_{2}) q^{94} + (\beta_{9} - \beta_{8} - \beta_{6} - \beta_{3} + \beta_{2} + 1) q^{95} + ( - \beta_{11} - \beta_{7} + 2 \beta_1) q^{96} + ( - 2 \beta_{11} + 4 \beta_{10} + 2 \beta_{5} + \beta_{4} + 2 \beta_1) q^{97} + (2 \beta_{11} - 6 \beta_{10} + 3 \beta_{7} - 2 \beta_{5} - 2 \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{3} + 8 q^{4} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{3} + 8 q^{4} + 2 q^{9} + 24 q^{10} - 2 q^{12} + 16 q^{16} + 34 q^{17} + 30 q^{22} + 6 q^{23} - 10 q^{25} + 12 q^{27} + 2 q^{29} + 22 q^{30} - 26 q^{36} + 38 q^{38} + 2 q^{40} + 22 q^{43} - 38 q^{48} + 8 q^{51} + 16 q^{53} + 30 q^{55} - 10 q^{61} + 82 q^{62} - 2 q^{64} + 68 q^{66} + 22 q^{68} + 14 q^{69} + 66 q^{74} + 2 q^{75} + 70 q^{79} - 28 q^{81} + 10 q^{82} - 20 q^{87} - 28 q^{88} - 66 q^{92} - 2 q^{94} + 4 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 16x^{10} + 88x^{8} - 197x^{6} + 172x^{4} - 36x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5\nu^{10} - 66\nu^{8} + 242\nu^{6} - 127\nu^{4} - 248\nu^{2} + 32 ) / 11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{10} + 44\nu^{8} - 209\nu^{6} + 360\nu^{4} - 223\nu^{2} + 27 ) / 11 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{11} + 44\nu^{9} - 209\nu^{7} + 360\nu^{5} - 223\nu^{3} + 27\nu ) / 11 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2\nu^{11} + 22\nu^{9} - 33\nu^{7} - 244\nu^{5} + 559\nu^{3} - 191\nu ) / 11 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -8\nu^{10} + 110\nu^{8} - 451\nu^{6} + 476\nu^{4} + 91\nu^{2} - 27 ) / 11 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{9} - 14\nu^{7} + 60\nu^{5} - 76\nu^{3} + 12\nu \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 10\nu^{10} - 143\nu^{8} + 638\nu^{6} - 903\nu^{4} + 263\nu^{2} + 9 ) / 11 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 10\nu^{10} - 143\nu^{8} + 638\nu^{6} - 903\nu^{4} + 274\nu^{2} - 24 ) / 11 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -8\nu^{11} + 121\nu^{9} - 605\nu^{7} + 1147\nu^{5} - 822\nu^{3} + 171\nu ) / 11 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -13\nu^{11} + 198\nu^{9} - 1001\nu^{7} + 1923\nu^{5} - 1333\nu^{3} + 194\nu ) / 11 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} - \beta_{8} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} + \beta_{10} + \beta_{7} + \beta_{5} + \beta_{4} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 6\beta_{9} - 6\beta_{8} - \beta_{6} + \beta_{3} - \beta_{2} + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -9\beta_{11} + 10\beta_{10} + 8\beta_{7} + 8\beta_{5} + 7\beta_{4} + 29\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 35\beta_{9} - 37\beta_{8} - 11\beta_{6} + 6\beta_{3} - 10\beta_{2} + 94 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -67\beta_{11} + 79\beta_{10} + 57\beta_{7} + 58\beta_{5} + 41\beta_{4} + 176\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 205\beta_{9} - 234\beta_{8} - 95\beta_{6} + 25\beta_{3} - 79\beta_{2} + 574 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -474\beta_{11} + 582\beta_{10} + 395\beta_{7} + 408\beta_{5} + 230\beta_{4} + 1092\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1214\beta_{9} - 1500\beta_{8} - 747\beta_{6} + 65\beta_{3} - 582\beta_{2} + 3576 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -3290\beta_{11} + 4158\beta_{10} + 2708\beta_{7} + 2829\beta_{5} + 1279\beta_{4} + 6872\beta_1 \) 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
−2.58860
−2.30327
−1.37905
−1.34523
−0.499987
−0.180824
0.180824
0.499987
1.34523
1.37905
2.30327
2.58860
−2.58860 0.518466 4.70085 −1.61205 −1.34210 0 −6.99143 −2.73119 4.17296
1.2 −2.30327 −1.47336 3.30504 −0.847292 3.39354 0 −3.00585 −0.829208 1.95154
1.3 −1.37905 2.88120 −0.0982074 −0.805948 −3.97334 0 2.89354 5.30133 1.11145
1.4 −1.34523 2.05010 −0.190366 −3.56778 −2.75785 0 2.94654 1.20292 4.79947
1.5 −0.499987 0.849601 −1.75001 1.04248 −0.424789 0 1.87496 −2.27818 −0.521224
1.6 −0.180824 −1.82601 −1.96730 −2.68664 0.330186 0 0.717383 0.334323 0.485809
1.7 0.180824 −1.82601 −1.96730 2.68664 −0.330186 0 −0.717383 0.334323 0.485809
1.8 0.499987 0.849601 −1.75001 −1.04248 0.424789 0 −1.87496 −2.27818 −0.521224
1.9 1.34523 2.05010 −0.190366 3.56778 2.75785 0 −2.94654 1.20292 4.79947
1.10 1.37905 2.88120 −0.0982074 0.805948 3.97334 0 −2.89354 5.30133 1.11145
1.11 2.30327 −1.47336 3.30504 0.847292 −3.39354 0 3.00585 −0.829208 1.95154
1.12 2.58860 0.518466 4.70085 1.61205 1.34210 0 6.99143 −2.73119 4.17296
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(1\)
\(13\) \(-1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8281.2.a.cp 12
7.b odd 2 1 8281.2.a.co 12
7.c even 3 2 1183.2.e.j 24
13.b even 2 1 inner 8281.2.a.cp 12
13.f odd 12 2 637.2.q.g 12
91.b odd 2 1 8281.2.a.co 12
91.r even 6 2 1183.2.e.j 24
91.w even 12 2 637.2.u.g 12
91.x odd 12 2 91.2.k.b 12
91.ba even 12 2 637.2.k.i 12
91.bc even 12 2 637.2.q.i 12
91.bd odd 12 2 91.2.u.b yes 12
273.bv even 12 2 819.2.bm.f 12
273.bw even 12 2 819.2.do.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.k.b 12 91.x odd 12 2
91.2.u.b yes 12 91.bd odd 12 2
637.2.k.i 12 91.ba even 12 2
637.2.q.g 12 13.f odd 12 2
637.2.q.i 12 91.bc even 12 2
637.2.u.g 12 91.w even 12 2
819.2.bm.f 12 273.bv even 12 2
819.2.do.e 12 273.bw even 12 2
1183.2.e.j 24 7.c even 3 2
1183.2.e.j 24 91.r even 6 2
8281.2.a.co 12 7.b odd 2 1
8281.2.a.co 12 91.b odd 2 1
8281.2.a.cp 12 1.a even 1 1 trivial
8281.2.a.cp 12 13.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8281))\):

\( T_{2}^{12} - 16T_{2}^{10} + 88T_{2}^{8} - 197T_{2}^{6} + 172T_{2}^{4} - 36T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{6} - 3T_{3}^{5} - 5T_{3}^{4} + 16T_{3}^{3} + 4T_{3}^{2} - 19T_{3} + 7 \) Copy content Toggle raw display
\( T_{5}^{12} - 25T_{5}^{10} + 201T_{5}^{8} - 636T_{5}^{6} + 878T_{5}^{4} - 539T_{5}^{2} + 121 \) Copy content Toggle raw display
\( T_{11}^{12} - 62T_{11}^{10} + 1355T_{11}^{8} - 13284T_{11}^{6} + 61227T_{11}^{4} - 122593T_{11}^{2} + 85849 \) Copy content Toggle raw display
\( T_{17}^{6} - 17T_{17}^{5} + 96T_{17}^{4} - 198T_{17}^{3} + 56T_{17}^{2} + 146T_{17} + 19 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 16 T^{10} + 88 T^{8} - 197 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T^{6} - 3 T^{5} - 5 T^{4} + 16 T^{3} + 4 T^{2} + \cdots + 7)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} - 25 T^{10} + 201 T^{8} + \cdots + 121 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( T^{12} - 62 T^{10} + 1355 T^{8} + \cdots + 85849 \) Copy content Toggle raw display
$13$ \( T^{12} \) Copy content Toggle raw display
$17$ \( (T^{6} - 17 T^{5} + 96 T^{4} - 198 T^{3} + \cdots + 19)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} - 79 T^{10} + 1984 T^{8} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( (T^{6} - 3 T^{5} - 50 T^{4} + 259 T^{3} + \cdots + 793)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} - T^{5} - 86 T^{4} + 190 T^{3} + \cdots + 4009)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} - 232 T^{10} + \cdots + 241274089 \) Copy content Toggle raw display
$37$ \( T^{12} - 147 T^{10} + 7164 T^{8} + \cdots + 123201 \) Copy content Toggle raw display
$41$ \( T^{12} - 354 T^{10} + \cdots + 389707081 \) Copy content Toggle raw display
$43$ \( (T^{6} - 11 T^{5} - 49 T^{4} + 816 T^{3} + \cdots + 20467)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} - 41 T^{10} + 509 T^{8} + \cdots + 121 \) Copy content Toggle raw display
$53$ \( (T^{6} - 8 T^{5} - 38 T^{4} + 404 T^{3} + \cdots - 17)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} - 553 T^{10} + \cdots + 35582408689 \) Copy content Toggle raw display
$61$ \( (T^{6} + 5 T^{5} - 75 T^{4} - 354 T^{3} + \cdots + 1777)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} - 439 T^{10} + \cdots + 5708255809 \) Copy content Toggle raw display
$71$ \( T^{12} - 446 T^{10} + \cdots + 639230089 \) Copy content Toggle raw display
$73$ \( T^{12} - 400 T^{10} + \cdots + 484396081 \) Copy content Toggle raw display
$79$ \( (T^{6} - 35 T^{5} + 344 T^{4} + \cdots + 255121)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} - 463 T^{10} + \cdots + 402363481 \) Copy content Toggle raw display
$89$ \( T^{12} - 430 T^{10} + \cdots + 145033849 \) Copy content Toggle raw display
$97$ \( T^{12} - 361 T^{10} + 29979 T^{8} + \cdots + 1681 \) Copy content Toggle raw display
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