Properties

Label 2695.2.a.t
Level $2695$
Weight $2$
Character orbit 2695.a
Self dual yes
Analytic conductor $21.520$
Analytic rank $0$
Dimension $8$
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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - 3x^{7} - 8x^{6} + 26x^{5} + 15x^{4} - 60x^{3} - 2x^{2} + 37x - 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 385)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{5} q^{3} + (\beta_{2} + 1) q^{4} + q^{5} + (\beta_{6} + \beta_{5} + \beta_{4} - \beta_{2} + \beta_1 - 1) q^{6} + (\beta_{6} - \beta_{5} + \beta_{2} + 1) q^{8} + ( - \beta_{7} - \beta_{3} + \beta_1 + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{5} q^{3} + (\beta_{2} + 1) q^{4} + q^{5} + (\beta_{6} + \beta_{5} + \beta_{4} - \beta_{2} + \beta_1 - 1) q^{6} + (\beta_{6} - \beta_{5} + \beta_{2} + 1) q^{8} + ( - \beta_{7} - \beta_{3} + \beta_1 + 2) q^{9} + \beta_1 q^{10} - q^{11} + (\beta_{7} + 2 \beta_{5} + 2 \beta_{4} + \beta_{3} - \beta_1 + 1) q^{12} + ( - \beta_{4} - \beta_{3} + \beta_1 + 1) q^{13} + \beta_{5} q^{15} + (\beta_{7} - \beta_{5} + \beta_{2} + \beta_1) q^{16} + ( - \beta_{7} - \beta_{3} - \beta_1 + 1) q^{17} + ( - 2 \beta_{7} - 2 \beta_{4} - \beta_{3} + 3 \beta_1 + 3) q^{18} + (\beta_{7} - \beta_1) q^{19} + (\beta_{2} + 1) q^{20} - \beta_1 q^{22} + (\beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{2} - 1) q^{23} + (2 \beta_{7} + 2 \beta_{5} + 2 \beta_{4} + 3 \beta_{3} - 3) q^{24} + q^{25} + ( - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_1 + 2) q^{26} + ( - \beta_{7} - \beta_{6} + 4 \beta_{5} + \beta_{4} - 3 \beta_{2} + 2 \beta_1 - 1) q^{27} + ( - \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 + 4) q^{29} + (\beta_{6} + \beta_{5} + \beta_{4} - \beta_{2} + \beta_1 - 1) q^{30} + ( - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{31} + (\beta_{7} - \beta_{6} + \beta_{3} + \beta_{2} + 2) q^{32} - \beta_{5} q^{33} + ( - 2 \beta_{7} - 2 \beta_{4} - \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 3) q^{34} + ( - \beta_{7} - \beta_{6} - 2 \beta_{5} - 3 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 6) q^{36} + (\beta_{5} + 2 \beta_{4} + 3 \beta_{3} + 1) q^{37} + (\beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} - \beta_{2} - 4) q^{38} + (3 \beta_{5} + 2 \beta_{4} - 2 \beta_{2} + 2) q^{39} + (\beta_{6} - \beta_{5} + \beta_{2} + 1) q^{40} + (\beta_{6} + \beta_{5} - \beta_{4} + \beta_{2} + \beta_1 - 1) q^{41} + (\beta_{6} - 2 \beta_{5} - 2 \beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{43} + ( - \beta_{2} - 1) q^{44} + ( - \beta_{7} - \beta_{3} + \beta_1 + 2) q^{45} + (2 \beta_{7} + \beta_{6} + \beta_{4} + 2 \beta_{3} + 3 \beta_{2} - \beta_1 + 1) q^{46} + ( - \beta_{6} - \beta_{5} - 3 \beta_{4} + \beta_{2} - \beta_1 + 1) q^{47} + (3 \beta_{7} + \beta_{6} + 3 \beta_{4} + 2 \beta_{3} + \beta_{2} - 2 \beta_1 - 3) q^{48} + \beta_1 q^{50} + ( - \beta_{7} - 3 \beta_{6} + 4 \beta_{5} - \beta_{4} - \beta_{2} + 1) q^{51} + ( - \beta_{7} - \beta_{6} - \beta_{5} + 3 \beta_{2} + 5) q^{52} + (\beta_{6} - \beta_{4} - \beta_{2} + \beta_1 + 3) q^{53} + ( - 2 \beta_{7} + 7 \beta_{5} + 2 \beta_{4} - 6 \beta_{2}) q^{54} - q^{55} + (\beta_{7} - \beta_{6} - 3 \beta_{5} - \beta_{4} + 3 \beta_{2} - 2 \beta_1 + 3) q^{57} + ( - \beta_{5} + \beta_{3} + 4 \beta_1 - 1) q^{58} + (\beta_{7} - \beta_{6} - \beta_{4} - \beta_{3} - \beta_{2} - 3) q^{59} + (\beta_{7} + 2 \beta_{5} + 2 \beta_{4} + \beta_{3} - \beta_1 + 1) q^{60} + ( - \beta_{7} - 2 \beta_{6} + \beta_{5} + 2 \beta_{3} - \beta_1 + 2) q^{61} + ( - \beta_{7} - 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 2 \beta_1 - 4) q^{62} + ( - \beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} - \beta_{2} + 1) q^{64} + ( - \beta_{4} - \beta_{3} + \beta_1 + 1) q^{65} + ( - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{2} - \beta_1 + 1) q^{66} + (\beta_{7} - \beta_{6} - 2 \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - 2) q^{67} + ( - \beta_{7} - 3 \beta_{6} - 3 \beta_{4} - 2 \beta_{3} - \beta_{2} - 2 \beta_1 + 3) q^{68} + (3 \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + 3 \beta_{3} + 3 \beta_{2} - 2) q^{69} + ( - 2 \beta_{6} + \beta_{5} - 2 \beta_{4} + \beta_{3} - 2 \beta_{2}) q^{71} + (\beta_{6} - 8 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 3 \beta_{2} + 2 \beta_1 + 3) q^{72} + ( - 2 \beta_{7} + 2 \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - 3 \beta_1 + 3) q^{73} + (3 \beta_{7} - 2 \beta_{6} + 3 \beta_{5} + 4 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + \cdots + 2) q^{74}+ \cdots + (\beta_{7} + \beta_{3} - \beta_1 - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 3 q^{2} + q^{3} + 9 q^{4} + 8 q^{5} - 3 q^{6} + 9 q^{8} + 19 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 3 q^{2} + q^{3} + 9 q^{4} + 8 q^{5} - 3 q^{6} + 9 q^{8} + 19 q^{9} + 3 q^{10} - 8 q^{11} + 9 q^{12} + 14 q^{13} + q^{15} + 7 q^{16} + 5 q^{17} + 27 q^{18} + q^{19} + 9 q^{20} - 3 q^{22} - 2 q^{23} - 24 q^{24} + 8 q^{25} + 21 q^{26} - 5 q^{27} + 26 q^{29} - 3 q^{30} + 2 q^{31} + 16 q^{32} - q^{33} - 26 q^{34} + 54 q^{36} - q^{37} - 31 q^{38} + 19 q^{39} + 9 q^{40} - 3 q^{41} + 4 q^{43} - 9 q^{44} + 19 q^{45} + 10 q^{46} + q^{47} - 21 q^{48} + 3 q^{50} + 3 q^{51} + 37 q^{52} + 26 q^{53} - 5 q^{54} - 8 q^{55} + 20 q^{57} - q^{58} - 19 q^{59} + 9 q^{60} - 26 q^{62} + q^{64} + 14 q^{65} + 3 q^{66} - 13 q^{67} + 15 q^{68} - 14 q^{69} - 9 q^{71} + 32 q^{72} + 11 q^{73} + 24 q^{74} + q^{75} - 18 q^{76} - 33 q^{78} - 8 q^{79} + 7 q^{80} + 52 q^{81} + 41 q^{82} + 32 q^{83} + 5 q^{85} + 28 q^{86} - 16 q^{87} - 9 q^{88} + 5 q^{89} + 27 q^{90} + 30 q^{92} - 14 q^{93} - 5 q^{94} + q^{95} + q^{96} + 9 q^{97} - 19 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} - 8x^{6} + 26x^{5} + 15x^{4} - 60x^{3} - 2x^{2} + 37x - 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{5} - \nu^{4} - 7\nu^{3} + 5\nu^{2} + 9\nu - 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{6} - \nu^{5} - 8\nu^{4} + 6\nu^{3} + 14\nu^{2} - 6\nu - 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{7} - \nu^{6} - 9\nu^{5} + 7\nu^{4} + 21\nu^{3} - 11\nu^{2} - 12\nu + 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{7} - \nu^{6} - 9\nu^{5} + 7\nu^{4} + 22\nu^{3} - 12\nu^{2} - 16\nu + 6 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{7} - \nu^{6} - 9\nu^{5} + 8\nu^{4} + 21\nu^{3} - 18\nu^{2} - 13\nu + 11 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} - \beta_{5} + \beta_{2} + 4\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} - \beta_{5} + 7\beta_{2} + \beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{7} + 7\beta_{6} - 8\beta_{5} + \beta_{3} + 9\beta_{2} + 20\beta _1 + 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 9\beta_{7} + \beta_{6} - 10\beta_{5} + \beta_{4} + \beta_{3} + 45\beta_{2} + 10\beta _1 + 77 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 11\beta_{7} + 43\beta_{6} - 53\beta_{5} + \beta_{4} + 10\beta_{3} + 67\beta_{2} + 111\beta _1 + 77 \) 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
−2.31051
−1.47049
−1.06232
0.279211
0.840020
1.65515
2.51872
2.55022
−2.31051 2.43360 3.33845 1.00000 −5.62286 0 −3.09251 2.92243 −2.31051
1.2 −1.47049 0.718935 0.162341 1.00000 −1.05719 0 2.70226 −2.48313 −1.47049
1.3 −1.06232 −2.71455 −0.871473 1.00000 2.88373 0 3.05043 4.36879 −1.06232
1.4 0.279211 0.275960 −1.92204 1.00000 0.0770509 0 −1.09508 −2.92385 0.279211
1.5 0.840020 −1.72967 −1.29437 1.00000 −1.45296 0 −2.76733 −0.00824088 0.840020
1.6 1.65515 3.43210 0.739530 1.00000 5.68065 0 −2.08627 8.77931 1.65515
1.7 2.51872 −3.29098 4.34396 1.00000 −8.28906 0 5.90377 7.83055 2.51872
1.8 2.55022 1.87460 4.50360 1.00000 4.78064 0 6.38473 0.514136 2.55022
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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.t 8
7.b odd 2 1 2695.2.a.s 8
7.c even 3 2 385.2.i.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
385.2.i.c 16 7.c even 3 2
2695.2.a.s 8 7.b odd 2 1
2695.2.a.t 8 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2695))\):

\( T_{2}^{8} - 3T_{2}^{7} - 8T_{2}^{6} + 26T_{2}^{5} + 15T_{2}^{4} - 60T_{2}^{3} - 2T_{2}^{2} + 37T_{2} - 9 \) Copy content Toggle raw display
\( T_{3}^{8} - T_{3}^{7} - 21T_{3}^{6} + 21T_{3}^{5} + 129T_{3}^{4} - 136T_{3}^{3} - 212T_{3}^{2} + 240T_{3} - 48 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 3 T^{7} - 8 T^{6} + 26 T^{5} + \cdots - 9 \) Copy content Toggle raw display
$3$ \( T^{8} - T^{7} - 21 T^{6} + 21 T^{5} + \cdots - 48 \) Copy content Toggle raw display
$5$ \( (T - 1)^{8} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T + 1)^{8} \) Copy content Toggle raw display
$13$ \( T^{8} - 14 T^{7} + 44 T^{6} + \cdots - 5119 \) Copy content Toggle raw display
$17$ \( T^{8} - 5 T^{7} - 65 T^{6} + \cdots + 10512 \) Copy content Toggle raw display
$19$ \( T^{8} - T^{7} - 58 T^{6} + 20 T^{5} + \cdots + 16 \) Copy content Toggle raw display
$23$ \( T^{8} + 2 T^{7} - 78 T^{6} - 281 T^{5} + \cdots + 144 \) Copy content Toggle raw display
$29$ \( T^{8} - 26 T^{7} + 229 T^{6} + \cdots + 1296 \) Copy content Toggle raw display
$31$ \( T^{8} - 2 T^{7} - 96 T^{6} + \cdots + 48297 \) Copy content Toggle raw display
$37$ \( T^{8} + T^{7} - 250 T^{6} + \cdots + 354192 \) Copy content Toggle raw display
$41$ \( T^{8} + 3 T^{7} - 155 T^{6} + \cdots - 82944 \) Copy content Toggle raw display
$43$ \( T^{8} - 4 T^{7} - 137 T^{6} + \cdots + 37189 \) Copy content Toggle raw display
$47$ \( T^{8} - T^{7} - 243 T^{6} + \cdots + 1104624 \) Copy content Toggle raw display
$53$ \( T^{8} - 26 T^{7} + 187 T^{6} + \cdots - 20688 \) Copy content Toggle raw display
$59$ \( T^{8} + 19 T^{7} - 7 T^{6} + \cdots + 1208661 \) Copy content Toggle raw display
$61$ \( T^{8} - 180 T^{6} - 223 T^{5} + \cdots + 20400 \) Copy content Toggle raw display
$67$ \( T^{8} + 13 T^{7} - 138 T^{6} + \cdots - 43728 \) Copy content Toggle raw display
$71$ \( T^{8} + 9 T^{7} - 267 T^{6} + \cdots - 1485459 \) Copy content Toggle raw display
$73$ \( T^{8} - 11 T^{7} - 196 T^{6} + \cdots + 676121 \) Copy content Toggle raw display
$79$ \( T^{8} + 8 T^{7} - 346 T^{6} + \cdots - 864688 \) Copy content Toggle raw display
$83$ \( T^{8} - 32 T^{7} + 230 T^{6} + \cdots + 307773 \) Copy content Toggle raw display
$89$ \( T^{8} - 5 T^{7} - 413 T^{6} + \cdots - 3273441 \) Copy content Toggle raw display
$97$ \( T^{8} - 9 T^{7} - 526 T^{6} + \cdots + 554768 \) Copy content Toggle raw display
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