Properties

Label 55.2.a.b
Level 55
Weight 2
Character orbit 55.a
Self dual yes
Analytic conductor 0.439
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 55 = 5 \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 55.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.439177211117\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta ) q^{2} -2 \beta q^{3} + ( 1 + 2 \beta ) q^{4} - q^{5} + ( -4 - 2 \beta ) q^{6} -2 q^{7} + ( 3 + \beta ) q^{8} + 5 q^{9} +O(q^{10})\) \( q + ( 1 + \beta ) q^{2} -2 \beta q^{3} + ( 1 + 2 \beta ) q^{4} - q^{5} + ( -4 - 2 \beta ) q^{6} -2 q^{7} + ( 3 + \beta ) q^{8} + 5 q^{9} + ( -1 - \beta ) q^{10} + q^{11} + ( -8 - 2 \beta ) q^{12} + ( -4 + 2 \beta ) q^{13} + ( -2 - 2 \beta ) q^{14} + 2 \beta q^{15} + 3 q^{16} + ( 4 + 2 \beta ) q^{17} + ( 5 + 5 \beta ) q^{18} + ( -1 - 2 \beta ) q^{20} + 4 \beta q^{21} + ( 1 + \beta ) q^{22} -2 \beta q^{23} + ( -4 - 6 \beta ) q^{24} + q^{25} -2 \beta q^{26} -4 \beta q^{27} + ( -2 - 4 \beta ) q^{28} + ( 2 - 4 \beta ) q^{29} + ( 4 + 2 \beta ) q^{30} + ( -3 + \beta ) q^{32} -2 \beta q^{33} + ( 8 + 6 \beta ) q^{34} + 2 q^{35} + ( 5 + 10 \beta ) q^{36} + ( -2 - 4 \beta ) q^{37} + ( -8 + 8 \beta ) q^{39} + ( -3 - \beta ) q^{40} + 6 q^{41} + ( 8 + 4 \beta ) q^{42} -6 q^{43} + ( 1 + 2 \beta ) q^{44} -5 q^{45} + ( -4 - 2 \beta ) q^{46} + 2 \beta q^{47} -6 \beta q^{48} -3 q^{49} + ( 1 + \beta ) q^{50} + ( -8 - 8 \beta ) q^{51} + ( 4 - 6 \beta ) q^{52} + ( 6 + 4 \beta ) q^{53} + ( -8 - 4 \beta ) q^{54} - q^{55} + ( -6 - 2 \beta ) q^{56} + ( -6 - 2 \beta ) q^{58} + ( -4 + 4 \beta ) q^{59} + ( 8 + 2 \beta ) q^{60} + ( 2 - 8 \beta ) q^{61} -10 q^{63} + ( -7 - 2 \beta ) q^{64} + ( 4 - 2 \beta ) q^{65} + ( -4 - 2 \beta ) q^{66} + ( 4 + 6 \beta ) q^{67} + ( 12 + 10 \beta ) q^{68} + 8 q^{69} + ( 2 + 2 \beta ) q^{70} + 8 \beta q^{71} + ( 15 + 5 \beta ) q^{72} + ( -4 + 2 \beta ) q^{73} + ( -10 - 6 \beta ) q^{74} -2 \beta q^{75} -2 q^{77} + 8 q^{78} + 4 q^{79} -3 q^{80} + q^{81} + ( 6 + 6 \beta ) q^{82} -6 q^{83} + ( 16 + 4 \beta ) q^{84} + ( -4 - 2 \beta ) q^{85} + ( -6 - 6 \beta ) q^{86} + ( 16 - 4 \beta ) q^{87} + ( 3 + \beta ) q^{88} + ( -2 - 8 \beta ) q^{89} + ( -5 - 5 \beta ) q^{90} + ( 8 - 4 \beta ) q^{91} + ( -8 - 2 \beta ) q^{92} + ( 4 + 2 \beta ) q^{94} + ( -4 + 6 \beta ) q^{96} + ( -2 + 4 \beta ) q^{97} + ( -3 - 3 \beta ) q^{98} + 5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} - 2q^{5} - 8q^{6} - 4q^{7} + 6q^{8} + 10q^{9} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} - 2q^{5} - 8q^{6} - 4q^{7} + 6q^{8} + 10q^{9} - 2q^{10} + 2q^{11} - 16q^{12} - 8q^{13} - 4q^{14} + 6q^{16} + 8q^{17} + 10q^{18} - 2q^{20} + 2q^{22} - 8q^{24} + 2q^{25} - 4q^{28} + 4q^{29} + 8q^{30} - 6q^{32} + 16q^{34} + 4q^{35} + 10q^{36} - 4q^{37} - 16q^{39} - 6q^{40} + 12q^{41} + 16q^{42} - 12q^{43} + 2q^{44} - 10q^{45} - 8q^{46} - 6q^{49} + 2q^{50} - 16q^{51} + 8q^{52} + 12q^{53} - 16q^{54} - 2q^{55} - 12q^{56} - 12q^{58} - 8q^{59} + 16q^{60} + 4q^{61} - 20q^{63} - 14q^{64} + 8q^{65} - 8q^{66} + 8q^{67} + 24q^{68} + 16q^{69} + 4q^{70} + 30q^{72} - 8q^{73} - 20q^{74} - 4q^{77} + 16q^{78} + 8q^{79} - 6q^{80} + 2q^{81} + 12q^{82} - 12q^{83} + 32q^{84} - 8q^{85} - 12q^{86} + 32q^{87} + 6q^{88} - 4q^{89} - 10q^{90} + 16q^{91} - 16q^{92} + 8q^{94} - 8q^{96} - 4q^{97} - 6q^{98} + 10q^{99} + O(q^{100}) \)

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 −2.00000 1.58579 5.00000 0.414214
1.2 2.41421 −2.82843 3.82843 −1.00000 −6.82843 −2.00000 4.41421 5.00000 −2.41421
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 55.2.a.b 2
3.b odd 2 1 495.2.a.b 2
4.b odd 2 1 880.2.a.m 2
5.b even 2 1 275.2.a.c 2
5.c odd 4 2 275.2.b.d 4
7.b odd 2 1 2695.2.a.f 2
8.b even 2 1 3520.2.a.bn 2
8.d odd 2 1 3520.2.a.bo 2
11.b odd 2 1 605.2.a.d 2
11.c even 5 4 605.2.g.f 8
11.d odd 10 4 605.2.g.l 8
12.b even 2 1 7920.2.a.ch 2
13.b even 2 1 9295.2.a.g 2
15.d odd 2 1 2475.2.a.x 2
15.e even 4 2 2475.2.c.l 4
20.d odd 2 1 4400.2.a.bn 2
20.e even 4 2 4400.2.b.q 4
33.d even 2 1 5445.2.a.y 2
44.c even 2 1 9680.2.a.bn 2
55.d odd 2 1 3025.2.a.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.a.b 2 1.a even 1 1 trivial
275.2.a.c 2 5.b even 2 1
275.2.b.d 4 5.c odd 4 2
495.2.a.b 2 3.b odd 2 1
605.2.a.d 2 11.b odd 2 1
605.2.g.f 8 11.c even 5 4
605.2.g.l 8 11.d odd 10 4
880.2.a.m 2 4.b odd 2 1
2475.2.a.x 2 15.d odd 2 1
2475.2.c.l 4 15.e even 4 2
2695.2.a.f 2 7.b odd 2 1
3025.2.a.o 2 55.d odd 2 1
3520.2.a.bn 2 8.b even 2 1
3520.2.a.bo 2 8.d odd 2 1
4400.2.a.bn 2 20.d odd 2 1
4400.2.b.q 4 20.e even 4 2
5445.2.a.y 2 33.d even 2 1
7920.2.a.ch 2 12.b even 2 1
9295.2.a.g 2 13.b even 2 1
9680.2.a.bn 2 44.c even 2 1

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 2 T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(55))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T + 3 T^{2} - 4 T^{3} + 4 T^{4} \)
$3$ \( 1 - 2 T^{2} + 9 T^{4} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( ( 1 + 2 T + 7 T^{2} )^{2} \)
$11$ \( ( 1 - T )^{2} \)
$13$ \( 1 + 8 T + 34 T^{2} + 104 T^{3} + 169 T^{4} \)
$17$ \( 1 - 8 T + 42 T^{2} - 136 T^{3} + 289 T^{4} \)
$19$ \( ( 1 + 19 T^{2} )^{2} \)
$23$ \( 1 + 38 T^{2} + 529 T^{4} \)
$29$ \( 1 - 4 T + 30 T^{2} - 116 T^{3} + 841 T^{4} \)
$31$ \( ( 1 + 31 T^{2} )^{2} \)
$37$ \( 1 + 4 T + 46 T^{2} + 148 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 - 6 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 + 6 T + 43 T^{2} )^{2} \)
$47$ \( 1 + 86 T^{2} + 2209 T^{4} \)
$53$ \( 1 - 12 T + 110 T^{2} - 636 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 8 T + 102 T^{2} + 472 T^{3} + 3481 T^{4} \)
$61$ \( 1 - 4 T - 2 T^{2} - 244 T^{3} + 3721 T^{4} \)
$67$ \( 1 - 8 T + 78 T^{2} - 536 T^{3} + 4489 T^{4} \)
$71$ \( 1 + 14 T^{2} + 5041 T^{4} \)
$73$ \( 1 + 8 T + 154 T^{2} + 584 T^{3} + 5329 T^{4} \)
$79$ \( ( 1 - 4 T + 79 T^{2} )^{2} \)
$83$ \( ( 1 + 6 T + 83 T^{2} )^{2} \)
$89$ \( 1 + 4 T + 54 T^{2} + 356 T^{3} + 7921 T^{4} \)
$97$ \( 1 + 4 T + 166 T^{2} + 388 T^{3} + 9409 T^{4} \)
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