Properties

Label 2695.2.a.x
Level $2695$
Weight $2$
Character orbit 2695.a
Self dual yes
Analytic conductor $21.520$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2695 = 5 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2695.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(21.5196833447\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \( x^{10} - 2x^{9} - 13x^{8} + 24x^{7} + 56x^{6} - 92x^{5} - 86x^{4} + 116x^{3} + 31x^{2} - 22x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{8} q^{3} + (\beta_{2} + 1) q^{4} + q^{5} + (\beta_{4} + \beta_{2}) q^{6} + (\beta_{9} + \beta_{8} + \beta_{3} + \beta_1) q^{8} + (\beta_{9} - \beta_{6} + \beta_{5} - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{8} q^{3} + (\beta_{2} + 1) q^{4} + q^{5} + (\beta_{4} + \beta_{2}) q^{6} + (\beta_{9} + \beta_{8} + \beta_{3} + \beta_1) q^{8} + (\beta_{9} - \beta_{6} + \beta_{5} - \beta_{2}) q^{9} + \beta_1 q^{10} - q^{11} + (\beta_{9} + \beta_{7} + \beta_1) q^{12} + (\beta_{9} - \beta_{7} + \beta_1) q^{13} + \beta_{8} q^{15} + (\beta_{9} + \beta_{8} + \beta_{6} + \beta_{4} + \beta_1) q^{16} + (\beta_{9} + \beta_{5} + \beta_{3} - \beta_{2} + \beta_1 + 2) q^{17} + ( - \beta_{9} + \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_{2} - \beta_1) q^{18} + (\beta_{9} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{2} + \beta_1) q^{19} + (\beta_{2} + 1) q^{20} - \beta_1 q^{22} + ( - \beta_{9} - \beta_{7} - \beta_{6} - 1) q^{23} + (\beta_{7} - \beta_{6} + \beta_{3} + \beta_{2} + 3) q^{24} + q^{25} + (2 \beta_{9} + 2 \beta_{8} - \beta_{7} + \beta_{6} - 2 \beta_{4} - \beta_{3} - \beta_{2} + 2 \beta_1 + 1) q^{26} + ( - 2 \beta_{9} - \beta_{6} - \beta_{4} + \beta_{2} - \beta_1 + 1) q^{27} + ( - \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5}) q^{29} + (\beta_{4} + \beta_{2}) q^{30} + ( - 2 \beta_{9} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{31} + (\beta_{5} + \beta_{4} + \beta_{2} - \beta_1 + 1) q^{32} - \beta_{8} q^{33} + (2 \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 + 2) q^{34} + ( - 2 \beta_{9} + \beta_{7} - 2 \beta_{6} - \beta_{5} + \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + \cdots - 2) q^{36}+ \cdots + ( - \beta_{9} + \beta_{6} - \beta_{5} + \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 2 q^{2} + 10 q^{4} + 10 q^{5} + 4 q^{6} + 6 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 2 q^{2} + 10 q^{4} + 10 q^{5} + 4 q^{6} + 6 q^{8} + 10 q^{9} + 2 q^{10} - 10 q^{11} + 4 q^{12} + 8 q^{13} + 6 q^{16} + 28 q^{17} - 10 q^{18} + 8 q^{19} + 10 q^{20} - 2 q^{22} - 8 q^{23} + 32 q^{24} + 10 q^{25} + 12 q^{26} - 8 q^{29} + 4 q^{30} - 4 q^{31} + 14 q^{32} + 20 q^{34} - 22 q^{36} + 28 q^{37} + 24 q^{38} - 24 q^{39} + 6 q^{40} + 44 q^{41} + 20 q^{43} - 10 q^{44} + 10 q^{45} - 12 q^{46} + 12 q^{47} + 16 q^{48} + 2 q^{50} - 4 q^{51} + 36 q^{52} - 8 q^{54} - 10 q^{55} + 12 q^{57} - 8 q^{58} + 16 q^{59} + 4 q^{60} + 16 q^{61} + 36 q^{62} - 34 q^{64} + 8 q^{65} - 4 q^{66} + 20 q^{67} + 8 q^{68} + 4 q^{69} - 4 q^{71} + 10 q^{72} + 20 q^{73} - 16 q^{74} + 4 q^{76} + 52 q^{78} - 20 q^{79} + 6 q^{80} + 10 q^{81} - 32 q^{82} + 16 q^{83} + 28 q^{85} - 20 q^{86} + 20 q^{87} - 6 q^{88} + 44 q^{89} - 10 q^{90} - 24 q^{92} + 16 q^{93} + 24 q^{94} + 8 q^{95} - 4 q^{97} - 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} - 13x^{8} + 24x^{7} + 56x^{6} - 92x^{5} - 86x^{4} + 116x^{3} + 31x^{2} - 22x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{9} - 3\nu^{8} + 27\nu^{7} + 35\nu^{6} - 116\nu^{5} - 130\nu^{4} + 177\nu^{3} + 169\nu^{2} - 80\nu - 29 ) / 23 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5\nu^{9} - 4\nu^{8} - 56\nu^{7} + 39\nu^{6} + 198\nu^{5} - 112\nu^{4} - 247\nu^{3} + 49\nu^{2} + 85\nu + 61 ) / 23 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5\nu^{9} + 4\nu^{8} + 56\nu^{7} - 39\nu^{6} - 175\nu^{5} + 112\nu^{4} + 63\nu^{3} - 72\nu^{2} + 214\nu - 15 ) / 23 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -7\nu^{9} + \nu^{8} + 83\nu^{7} - 4\nu^{6} - 314\nu^{5} + 5\nu^{4} + 401\nu^{3} - 18\nu^{2} - 73\nu + 2 ) / 23 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -4\nu^{9} + 17\nu^{8} + 54\nu^{7} - 183\nu^{6} - 255\nu^{5} + 591\nu^{4} + 446\nu^{3} - 582\nu^{2} - 206\nu + 57 ) / 23 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 8\nu^{9} - 11\nu^{8} - 108\nu^{7} + 136\nu^{6} + 487\nu^{5} - 538\nu^{4} - 800\nu^{3} + 681\nu^{2} + 320\nu - 91 ) / 23 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 6 \nu^{9} + 14 \nu^{8} + 81 \nu^{7} - 171 \nu^{6} - 371 \nu^{5} + 668 \nu^{4} + 646 \nu^{3} - 850 \nu^{2} - 355 \nu + 120 ) / 23 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} + \beta_{8} + \beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} + \beta_{8} + \beta_{6} + \beta_{4} + 6\beta_{2} + \beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8\beta_{9} + 8\beta_{8} + \beta_{5} + \beta_{4} + 8\beta_{3} + \beta_{2} + 27\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 9\beta_{9} + 10\beta_{8} + \beta_{7} + 9\beta_{6} + 9\beta_{4} + 2\beta_{3} + 35\beta_{2} + 11\beta _1 + 73 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 52 \beta_{9} + 52 \beta_{8} + \beta_{7} + \beta_{6} + 9 \beta_{5} + 12 \beta_{4} + 54 \beta_{3} + 14 \beta_{2} + 151 \beta _1 + 16 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 66 \beta_{9} + 77 \beta_{8} + 12 \beta_{7} + 62 \beta_{6} + \beta_{5} + 63 \beta_{4} + 24 \beta_{3} + 204 \beta_{2} + 93 \beta _1 + 401 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 320 \beta_{9} + 321 \beta_{8} + 13 \beta_{7} + 13 \beta_{6} + 62 \beta_{5} + 102 \beta_{4} + 341 \beta_{3} + 132 \beta_{2} + 863 \beta _1 + 163 \) 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
−2.29554
−2.15225
−1.47696
−0.568445
0.0495267
0.330732
1.29753
1.81378
2.48590
2.51572
−2.29554 −0.293572 3.26952 1.00000 0.673908 0 −2.91425 −2.91382 −2.29554
1.2 −2.15225 −2.37643 2.63217 1.00000 5.11466 0 −1.36058 2.64742 −2.15225
1.3 −1.47696 2.93275 0.181403 1.00000 −4.33154 0 2.68599 5.60100 −1.47696
1.4 −0.568445 0.674153 −1.67687 1.00000 −0.383219 0 2.09010 −2.54552 −0.568445
1.5 0.0495267 −3.19919 −1.99755 1.00000 −0.158445 0 −0.197985 7.23480 0.0495267
1.6 0.330732 2.43493 −1.89062 1.00000 0.805311 0 −1.28675 2.92889 0.330732
1.7 1.29753 −1.54034 −0.316404 1.00000 −1.99865 0 −3.00561 −0.627346 1.29753
1.8 1.81378 −1.20588 1.28980 1.00000 −2.18721 0 −1.28815 −1.54584 1.81378
1.9 2.48590 0.309874 4.17970 1.00000 0.770315 0 5.41853 −2.90398 2.48590
1.10 2.51572 2.26371 4.32884 1.00000 5.69486 0 5.85871 2.12439 2.51572
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(7\) \(1\)
\(11\) \(1\)

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 2695.2.a.x yes 10
7.b odd 2 1 2695.2.a.w 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2695.2.a.w 10 7.b odd 2 1
2695.2.a.x yes 10 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2695))\):

\( T_{2}^{10} - 2T_{2}^{9} - 13T_{2}^{8} + 24T_{2}^{7} + 56T_{2}^{6} - 92T_{2}^{5} - 86T_{2}^{4} + 116T_{2}^{3} + 31T_{2}^{2} - 22T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{10} - 20T_{3}^{8} + 130T_{3}^{6} + 12T_{3}^{5} - 308T_{3}^{4} - 68T_{3}^{3} + 182T_{3}^{2} + 4T_{3} - 14 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - 2 T^{9} - 13 T^{8} + 24 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{10} - 20 T^{8} + 130 T^{6} + \cdots - 14 \) Copy content Toggle raw display
$5$ \( (T - 1)^{10} \) Copy content Toggle raw display
$7$ \( T^{10} \) Copy content Toggle raw display
$11$ \( (T + 1)^{10} \) Copy content Toggle raw display
$13$ \( T^{10} - 8 T^{9} - 42 T^{8} + 448 T^{7} + \cdots + 34 \) Copy content Toggle raw display
$17$ \( T^{10} - 28 T^{9} + 298 T^{8} + \cdots - 9486 \) Copy content Toggle raw display
$19$ \( T^{10} - 8 T^{9} - 42 T^{8} + \cdots + 14536 \) Copy content Toggle raw display
$23$ \( T^{10} + 8 T^{9} - 70 T^{8} + \cdots + 58052 \) Copy content Toggle raw display
$29$ \( T^{10} + 8 T^{9} - 100 T^{8} + \cdots - 17648 \) Copy content Toggle raw display
$31$ \( T^{10} + 4 T^{9} - 130 T^{8} + \cdots + 106642 \) Copy content Toggle raw display
$37$ \( T^{10} - 28 T^{9} + 280 T^{8} + \cdots - 65596 \) Copy content Toggle raw display
$41$ \( T^{10} - 44 T^{9} + 704 T^{8} + \cdots - 1704254 \) Copy content Toggle raw display
$43$ \( T^{10} - 20 T^{9} - 36 T^{8} + \cdots + 7268 \) Copy content Toggle raw display
$47$ \( T^{10} - 12 T^{9} - 134 T^{8} + \cdots + 223634 \) Copy content Toggle raw display
$53$ \( T^{10} - 360 T^{8} + 500 T^{7} + \cdots - 889916 \) Copy content Toggle raw display
$59$ \( T^{10} - 16 T^{9} - 128 T^{8} + \cdots + 21298 \) Copy content Toggle raw display
$61$ \( T^{10} - 16 T^{9} + \cdots + 127321954 \) Copy content Toggle raw display
$67$ \( T^{10} - 20 T^{9} + 44 T^{8} + \cdots - 485852 \) Copy content Toggle raw display
$71$ \( T^{10} + 4 T^{9} - 124 T^{8} + \cdots - 349808 \) Copy content Toggle raw display
$73$ \( T^{10} - 20 T^{9} - 200 T^{8} + \cdots - 29120686 \) Copy content Toggle raw display
$79$ \( T^{10} + 20 T^{9} - 232 T^{8} + \cdots - 7623452 \) Copy content Toggle raw display
$83$ \( T^{10} - 16 T^{9} + \cdots + 3533457544 \) Copy content Toggle raw display
$89$ \( T^{10} - 44 T^{9} + \cdots + 344547208 \) Copy content Toggle raw display
$97$ \( T^{10} + 4 T^{9} - 480 T^{8} + \cdots - 706936 \) Copy content Toggle raw display
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