Properties

Label 2695.2.a.f
Level $2695$
Weight $2$
Character orbit 2695.a
Self dual yes
Analytic conductor $21.520$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 55)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{2} + 2 \beta q^{3} + (2 \beta + 1) q^{4} + q^{5} + (2 \beta + 4) q^{6} + (\beta + 3) q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 1) q^{2} + 2 \beta q^{3} + (2 \beta + 1) q^{4} + q^{5} + (2 \beta + 4) q^{6} + (\beta + 3) q^{8} + 5 q^{9} + (\beta + 1) q^{10} + q^{11} + (2 \beta + 8) q^{12} + ( - 2 \beta + 4) q^{13} + 2 \beta q^{15} + 3 q^{16} + ( - 2 \beta - 4) q^{17} + (5 \beta + 5) q^{18} + (2 \beta + 1) q^{20} + (\beta + 1) q^{22} - 2 \beta q^{23} + (6 \beta + 4) q^{24} + q^{25} + 2 \beta q^{26} + 4 \beta q^{27} + ( - 4 \beta + 2) q^{29} + (2 \beta + 4) q^{30} + (\beta - 3) q^{32} + 2 \beta q^{33} + ( - 6 \beta - 8) q^{34} + (10 \beta + 5) q^{36} + ( - 4 \beta - 2) q^{37} + (8 \beta - 8) q^{39} + (\beta + 3) q^{40} - 6 q^{41} - 6 q^{43} + (2 \beta + 1) q^{44} + 5 q^{45} + ( - 2 \beta - 4) q^{46} - 2 \beta q^{47} + 6 \beta q^{48} + (\beta + 1) q^{50} + ( - 8 \beta - 8) q^{51} + (6 \beta - 4) q^{52} + (4 \beta + 6) q^{53} + (4 \beta + 8) q^{54} + q^{55} + ( - 2 \beta - 6) q^{58} + ( - 4 \beta + 4) q^{59} + (2 \beta + 8) q^{60} + (8 \beta - 2) q^{61} + ( - 2 \beta - 7) q^{64} + ( - 2 \beta + 4) q^{65} + (2 \beta + 4) q^{66} + (6 \beta + 4) q^{67} + ( - 10 \beta - 12) q^{68} - 8 q^{69} + 8 \beta q^{71} + (5 \beta + 15) q^{72} + ( - 2 \beta + 4) q^{73} + ( - 6 \beta - 10) q^{74} + 2 \beta q^{75} + 8 q^{78} + 4 q^{79} + 3 q^{80} + q^{81} + ( - 6 \beta - 6) q^{82} + 6 q^{83} + ( - 2 \beta - 4) q^{85} + ( - 6 \beta - 6) q^{86} + (4 \beta - 16) q^{87} + (\beta + 3) q^{88} + (8 \beta + 2) q^{89} + (5 \beta + 5) q^{90} + ( - 2 \beta - 8) q^{92} + ( - 2 \beta - 4) q^{94} + ( - 6 \beta + 4) q^{96} + ( - 4 \beta + 2) q^{97} + 5 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 8 q^{6} + 6 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 8 q^{6} + 6 q^{8} + 10 q^{9} + 2 q^{10} + 2 q^{11} + 16 q^{12} + 8 q^{13} + 6 q^{16} - 8 q^{17} + 10 q^{18} + 2 q^{20} + 2 q^{22} + 8 q^{24} + 2 q^{25} + 4 q^{29} + 8 q^{30} - 6 q^{32} - 16 q^{34} + 10 q^{36} - 4 q^{37} - 16 q^{39} + 6 q^{40} - 12 q^{41} - 12 q^{43} + 2 q^{44} + 10 q^{45} - 8 q^{46} + 2 q^{50} - 16 q^{51} - 8 q^{52} + 12 q^{53} + 16 q^{54} + 2 q^{55} - 12 q^{58} + 8 q^{59} + 16 q^{60} - 4 q^{61} - 14 q^{64} + 8 q^{65} + 8 q^{66} + 8 q^{67} - 24 q^{68} - 16 q^{69} + 30 q^{72} + 8 q^{73} - 20 q^{74} + 16 q^{78} + 8 q^{79} + 6 q^{80} + 2 q^{81} - 12 q^{82} + 12 q^{83} - 8 q^{85} - 12 q^{86} - 32 q^{87} + 6 q^{88} + 4 q^{89} + 10 q^{90} - 16 q^{92} - 8 q^{94} + 8 q^{96} + 4 q^{97} + 10 q^{99}+O(q^{100}) \) 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
−1.41421
1.41421
−0.414214 −2.82843 −1.82843 1.00000 1.17157 0 1.58579 5.00000 −0.414214
1.2 2.41421 2.82843 3.82843 1.00000 6.82843 0 4.41421 5.00000 2.41421
\(n\): e.g. 2-40 or 990-1000
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.f 2
7.b odd 2 1 55.2.a.b 2
21.c even 2 1 495.2.a.b 2
28.d even 2 1 880.2.a.m 2
35.c odd 2 1 275.2.a.c 2
35.f even 4 2 275.2.b.d 4
56.e even 2 1 3520.2.a.bo 2
56.h odd 2 1 3520.2.a.bn 2
77.b even 2 1 605.2.a.d 2
77.j odd 10 4 605.2.g.f 8
77.l even 10 4 605.2.g.l 8
84.h odd 2 1 7920.2.a.ch 2
91.b odd 2 1 9295.2.a.g 2
105.g even 2 1 2475.2.a.x 2
105.k odd 4 2 2475.2.c.l 4
140.c even 2 1 4400.2.a.bn 2
140.j odd 4 2 4400.2.b.q 4
231.h odd 2 1 5445.2.a.y 2
308.g odd 2 1 9680.2.a.bn 2
385.h even 2 1 3025.2.a.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.a.b 2 7.b odd 2 1
275.2.a.c 2 35.c odd 2 1
275.2.b.d 4 35.f even 4 2
495.2.a.b 2 21.c even 2 1
605.2.a.d 2 77.b even 2 1
605.2.g.f 8 77.j odd 10 4
605.2.g.l 8 77.l even 10 4
880.2.a.m 2 28.d even 2 1
2475.2.a.x 2 105.g even 2 1
2475.2.c.l 4 105.k odd 4 2
2695.2.a.f 2 1.a even 1 1 trivial
3025.2.a.o 2 385.h even 2 1
3520.2.a.bn 2 56.h odd 2 1
3520.2.a.bo 2 56.e even 2 1
4400.2.a.bn 2 140.c even 2 1
4400.2.b.q 4 140.j odd 4 2
5445.2.a.y 2 231.h odd 2 1
7920.2.a.ch 2 84.h odd 2 1
9295.2.a.g 2 91.b odd 2 1
9680.2.a.bn 2 308.g odd 2 1

Hecke kernels

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

\( T_{2}^{2} - 2T_{2} - 1 \) Copy content Toggle raw display
\( T_{3}^{2} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2T - 1 \) Copy content Toggle raw display
$3$ \( T^{2} - 8 \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T - 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 8T + 8 \) Copy content Toggle raw display
$17$ \( T^{2} + 8T + 8 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 8 \) Copy content Toggle raw display
$29$ \( T^{2} - 4T - 28 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 4T - 28 \) Copy content Toggle raw display
$41$ \( (T + 6)^{2} \) Copy content Toggle raw display
$43$ \( (T + 6)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 8 \) Copy content Toggle raw display
$53$ \( T^{2} - 12T + 4 \) Copy content Toggle raw display
$59$ \( T^{2} - 8T - 16 \) Copy content Toggle raw display
$61$ \( T^{2} + 4T - 124 \) Copy content Toggle raw display
$67$ \( T^{2} - 8T - 56 \) Copy content Toggle raw display
$71$ \( T^{2} - 128 \) Copy content Toggle raw display
$73$ \( T^{2} - 8T + 8 \) Copy content Toggle raw display
$79$ \( (T - 4)^{2} \) Copy content Toggle raw display
$83$ \( (T - 6)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 4T - 124 \) Copy content Toggle raw display
$97$ \( T^{2} - 4T - 28 \) Copy content Toggle raw display
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