Properties

Label 2695.2.a.u
Level $2695$
Weight $2$
Character orbit 2695.a
Self dual yes
Analytic conductor $21.520$
Analytic rank $1$
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: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \( x^{10} - 2x^{9} - 13x^{8} + 24x^{7} + 56x^{6} - 92x^{5} - 86x^{4} + 116x^{3} + 31x^{2} - 30x - 7 \) 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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + ( - \beta_{9} - 1) q^{3} + (\beta_{4} + \beta_{3} + 1) q^{4} + q^{5} + (\beta_{9} + \beta_{7} + \beta_{6} + 2 \beta_1 - 1) q^{6} + ( - \beta_{7} - \beta_{6} - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{8} + (\beta_{9} + \beta_{8} - \beta_{6} - \beta_{4} - \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + ( - \beta_{9} - 1) q^{3} + (\beta_{4} + \beta_{3} + 1) q^{4} + q^{5} + (\beta_{9} + \beta_{7} + \beta_{6} + 2 \beta_1 - 1) q^{6} + ( - \beta_{7} - \beta_{6} - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{8} + (\beta_{9} + \beta_{8} - \beta_{6} - \beta_{4} - \beta_{2} + 1) q^{9} - \beta_1 q^{10} + q^{11} + ( - \beta_{9} - 2 \beta_{7} - \beta_{5} - 2 \beta_{4} - \beta_{3} - 3) q^{12} + ( - \beta_{9} + \beta_{5} + \beta_{3} + 2 \beta_{2} - 1) q^{13} + ( - \beta_{9} - 1) q^{15} + (\beta_{9} + \beta_{7} + \beta_{5} + \beta_{2} + \beta_1 + 1) q^{16} + (\beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{2} + \beta_1 - 3) q^{17} + ( - \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - 2 \beta_1) q^{18} + (\beta_{8} + \beta_{7} - \beta_{3} - \beta_{2} + \beta_1) q^{19} + (\beta_{4} + \beta_{3} + 1) q^{20} - \beta_1 q^{22} + (\beta_{9} + 2 \beta_{6} + 2 \beta_{4} + \beta_{3} + \beta_1 - 2) q^{23} + (3 \beta_{7} + \beta_{6} + 2 \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2} + 3 \beta_1 + 1) q^{24} + q^{25} + (2 \beta_{9} - 2 \beta_{8} - \beta_{7} + \beta_{6} - 3 \beta_{3} - \beta_{2} + 3 \beta_1 - 1) q^{26} + ( - \beta_{9} - 2 \beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} + 2 \beta_{4} + 2 \beta_{2} + \cdots - 2) q^{27}+ \cdots + (\beta_{9} + \beta_{8} - \beta_{6} - \beta_{4} - \beta_{2} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 2 q^{2} - 8 q^{3} + 10 q^{4} + 10 q^{5} - 4 q^{6} - 6 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 2 q^{2} - 8 q^{3} + 10 q^{4} + 10 q^{5} - 4 q^{6} - 6 q^{8} + 10 q^{9} - 2 q^{10} + 10 q^{11} - 20 q^{12} - 8 q^{13} - 8 q^{15} + 6 q^{16} - 28 q^{17} - 14 q^{18} + 10 q^{20} - 2 q^{22} - 16 q^{23} + 8 q^{24} + 10 q^{25} - 20 q^{26} - 32 q^{27} - 4 q^{30} - 20 q^{31} - 14 q^{32} - 8 q^{33} + 4 q^{34} + 42 q^{36} - 36 q^{37} - 24 q^{38} + 24 q^{39} - 6 q^{40} - 36 q^{41} - 4 q^{43} + 10 q^{44} + 10 q^{45} - 4 q^{46} - 12 q^{47} - 40 q^{48} - 2 q^{50} + 20 q^{51} - 4 q^{52} - 16 q^{53} + 48 q^{54} + 10 q^{55} + 4 q^{57} + 16 q^{58} - 32 q^{59} - 20 q^{60} + 16 q^{61} + 4 q^{62} - 34 q^{64} - 8 q^{65} - 4 q^{66} - 20 q^{67} - 32 q^{68} - 28 q^{69} + 12 q^{71} - 2 q^{72} - 20 q^{73} + 32 q^{74} - 8 q^{75} - 12 q^{76} + 20 q^{78} + 12 q^{79} + 6 q^{80} + 42 q^{81} + 40 q^{82} - 8 q^{83} - 28 q^{85} - 4 q^{86} - 28 q^{87} - 6 q^{88} - 68 q^{89} - 14 q^{90} + 32 q^{92} - 32 q^{93} - 16 q^{94} + 80 q^{96} - 36 q^{97} + 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} - 13x^{8} + 24x^{7} + 56x^{6} - 92x^{5} - 86x^{4} + 116x^{3} + 31x^{2} - 30x - 7 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{9} - \nu^{8} + 10\nu^{7} + 17\nu^{6} - 27\nu^{5} - 66\nu^{4} + 20\nu^{3} + 32\nu^{2} - 34\nu + 16 ) / 11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{9} - \nu^{8} + 10\nu^{7} + 6\nu^{6} - 5\nu^{5} + 22\nu^{4} - 123\nu^{3} - 133\nu^{2} + 131\nu + 38 ) / 11 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{9} + \nu^{8} - 10\nu^{7} - 6\nu^{6} + 5\nu^{5} - 22\nu^{4} + 123\nu^{3} + 144\nu^{2} - 131\nu - 71 ) / 11 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{9} - \nu^{8} + 21\nu^{7} + 6\nu^{6} - 126\nu^{5} - 11\nu^{4} + 251\nu^{3} + 32\nu^{2} - 111\nu - 28 ) / 11 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5\nu^{9} - 6\nu^{8} - 61\nu^{7} + 58\nu^{6} + 245\nu^{5} - 176\nu^{4} - 342\nu^{3} + 170\nu^{2} + 71\nu + 8 ) / 11 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -4\nu^{9} + 7\nu^{8} + 51\nu^{7} - 75\nu^{6} - 218\nu^{5} + 242\nu^{4} + 333\nu^{3} - 213\nu^{2} - 92\nu + 9 ) / 11 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3\nu^{9} - 8\nu^{8} - 41\nu^{7} + 103\nu^{6} + 180\nu^{5} - 407\nu^{4} - 247\nu^{3} + 476\nu^{2} - 8\nu - 70 ) / 11 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 6\nu^{9} - 5\nu^{8} - 82\nu^{7} + 52\nu^{6} + 371\nu^{5} - 154\nu^{4} - 604\nu^{3} + 83\nu^{2} + 226\nu + 36 ) / 11 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} + \beta_{7} + \beta_{5} + 6\beta_{4} + 6\beta_{3} + \beta_{2} + \beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{9} + \beta_{8} + 9\beta_{7} + 7\beta_{6} + \beta_{5} + 7\beta_{4} + 8\beta_{3} + 6\beta_{2} + 27\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 10 \beta_{9} + 2 \beta_{8} + 13 \beta_{7} + \beta_{6} + 10 \beta_{5} + 34 \beta_{4} + 35 \beta_{3} + 8 \beta_{2} + 12 \beta _1 + 81 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 14 \beta_{9} + 11 \beta_{8} + 68 \beta_{7} + 43 \beta_{6} + 15 \beta_{5} + 46 \beta_{4} + 56 \beta_{3} + 35 \beta_{2} + 152 \beta _1 + 28 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 80 \beta_{9} + 25 \beta_{8} + 123 \beta_{7} + 17 \beta_{6} + 79 \beta_{5} + 198 \beta_{4} + 211 \beta_{3} + 56 \beta_{2} + 109 \beta _1 + 453 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 137 \beta_{9} + 92 \beta_{8} + 489 \beta_{7} + 261 \beta_{6} + 148 \beta_{5} + 307 \beta_{4} + 384 \beta_{3} + 211 \beta_{2} + 886 \beta _1 + 272 \) 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
2.62366
2.18262
2.13489
0.911764
0.777582
−0.231979
−0.480122
−1.45836
−2.20075
−2.25931
−2.62366 −3.16125 4.88361 1.00000 8.29406 0 −7.56562 6.99351 −2.62366
1.2 −2.18262 1.88048 2.76381 1.00000 −4.10437 0 −1.66711 0.536219 −2.18262
1.3 −2.13489 −0.317256 2.55774 1.00000 0.677306 0 −1.19071 −2.89935 −2.13489
1.4 −0.911764 1.93300 −1.16869 1.00000 −1.76244 0 2.88909 0.736495 −0.911764
1.5 −0.777582 −3.22419 −1.39537 1.00000 2.50707 0 2.64018 7.39539 −0.777582
1.6 0.231979 −0.535939 −1.94619 1.00000 −0.124327 0 −0.915432 −2.71277 0.231979
1.7 0.480122 −0.720208 −1.76948 1.00000 −0.345788 0 −1.80981 −2.48130 0.480122
1.8 1.45836 0.658575 0.126803 1.00000 0.960437 0 −2.73179 −2.56628 1.45836
1.9 2.20075 −1.61883 2.84330 1.00000 −3.56264 0 1.85588 −0.379385 2.20075
1.10 2.25931 −2.89439 3.10446 1.00000 −6.53930 0 2.49532 5.37747 2.25931
\(n\): 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.u 10
7.b odd 2 1 2695.2.a.v yes 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2695.2.a.u 10 1.a even 1 1 trivial
2695.2.a.v yes 10 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2695))\):

\( T_{2}^{10} + 2T_{2}^{9} - 13T_{2}^{8} - 24T_{2}^{7} + 56T_{2}^{6} + 92T_{2}^{5} - 86T_{2}^{4} - 116T_{2}^{3} + 31T_{2}^{2} + 30T_{2} - 7 \) Copy content Toggle raw display
\( T_{3}^{10} + 8T_{3}^{9} + 12T_{3}^{8} - 48T_{3}^{7} - 134T_{3}^{6} + 28T_{3}^{5} + 300T_{3}^{4} + 172T_{3}^{3} - 74T_{3}^{2} - 76T_{3} - 14 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 2 T^{9} - 13 T^{8} - 24 T^{7} + \cdots - 7 \) Copy content Toggle raw display
$3$ \( T^{10} + 8 T^{9} + 12 T^{8} - 48 T^{7} + \cdots - 14 \) 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
$13$ \( T^{10} + 8 T^{9} - 58 T^{8} + \cdots + 317506 \) Copy content Toggle raw display
$17$ \( T^{10} + 28 T^{9} + 298 T^{8} + \cdots + 47474 \) Copy content Toggle raw display
$19$ \( T^{10} - 82 T^{8} + 56 T^{7} + \cdots + 392 \) Copy content Toggle raw display
$23$ \( T^{10} + 16 T^{9} - 38 T^{8} + \cdots - 2653052 \) Copy content Toggle raw display
$29$ \( T^{10} - 220 T^{8} - 112 T^{7} + \cdots + 379792 \) Copy content Toggle raw display
$31$ \( T^{10} + 20 T^{9} - 18 T^{8} + \cdots + 49523026 \) Copy content Toggle raw display
$37$ \( T^{10} + 36 T^{9} + 344 T^{8} + \cdots + 16621444 \) Copy content Toggle raw display
$41$ \( T^{10} + 36 T^{9} + 424 T^{8} + \cdots - 4092238 \) Copy content Toggle raw display
$43$ \( T^{10} + 4 T^{9} - 204 T^{8} + \cdots + 20333636 \) Copy content Toggle raw display
$47$ \( T^{10} + 12 T^{9} - 166 T^{8} + \cdots + 3299858 \) Copy content Toggle raw display
$53$ \( T^{10} + 16 T^{9} - 248 T^{8} + \cdots - 658364 \) Copy content Toggle raw display
$59$ \( T^{10} + 32 T^{9} + 64 T^{8} + \cdots + 62775794 \) Copy content Toggle raw display
$61$ \( T^{10} - 16 T^{9} - 148 T^{8} + \cdots - 62213006 \) Copy content Toggle raw display
$67$ \( T^{10} + 20 T^{9} - 212 T^{8} + \cdots + 28861700 \) Copy content Toggle raw display
$71$ \( T^{10} - 12 T^{9} + \cdots + 159481232 \) Copy content Toggle raw display
$73$ \( T^{10} + 20 T^{9} + \cdots + 115334962 \) Copy content Toggle raw display
$79$ \( T^{10} - 12 T^{9} - 472 T^{8} + \cdots - 16197692 \) Copy content Toggle raw display
$83$ \( T^{10} + 8 T^{9} - 450 T^{8} + \cdots + 142234696 \) Copy content Toggle raw display
$89$ \( T^{10} + 68 T^{9} + \cdots - 317037112 \) Copy content Toggle raw display
$97$ \( T^{10} + 36 T^{9} + \cdots + 206918152 \) Copy content Toggle raw display
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