Properties

Label 1155.2.a.p
Level 1155
Weight 2
Character orbit 1155.a
Self dual yes
Analytic conductor 9.223
Analytic rank 0
Dimension 2
CM no
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 1155 = 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1155.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.22272143346\)
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.

\(f(q)\) \(=\) \( q + \beta q^{2} - q^{3} - q^{5} -\beta q^{6} - q^{7} -2 \beta q^{8} + q^{9} +O(q^{10})\) \( q + \beta q^{2} - q^{3} - q^{5} -\beta q^{6} - q^{7} -2 \beta q^{8} + q^{9} -\beta q^{10} + q^{11} + ( 2 + \beta ) q^{13} -\beta q^{14} + q^{15} -4 q^{16} + ( 1 + 2 \beta ) q^{17} + \beta q^{18} + ( -1 + 2 \beta ) q^{19} + q^{21} + \beta q^{22} + ( -1 + 2 \beta ) q^{23} + 2 \beta q^{24} + q^{25} + ( 2 + 2 \beta ) q^{26} - q^{27} + ( 3 + 3 \beta ) q^{29} + \beta q^{30} + ( 2 + \beta ) q^{31} - q^{33} + ( 4 + \beta ) q^{34} + q^{35} + ( -2 + 3 \beta ) q^{37} + ( 4 - \beta ) q^{38} + ( -2 - \beta ) q^{39} + 2 \beta q^{40} + ( 4 - 5 \beta ) q^{41} + \beta q^{42} + ( -5 + \beta ) q^{43} - q^{45} + ( 4 - \beta ) q^{46} + ( 6 + 3 \beta ) q^{47} + 4 q^{48} + q^{49} + \beta q^{50} + ( -1 - 2 \beta ) q^{51} + ( 3 - 4 \beta ) q^{53} -\beta q^{54} - q^{55} + 2 \beta q^{56} + ( 1 - 2 \beta ) q^{57} + ( 6 + 3 \beta ) q^{58} + ( 13 - \beta ) q^{59} + ( 5 - 4 \beta ) q^{61} + ( 2 + 2 \beta ) q^{62} - q^{63} + 8 q^{64} + ( -2 - \beta ) q^{65} -\beta q^{66} -6 q^{67} + ( 1 - 2 \beta ) q^{69} + \beta q^{70} + ( -4 - 5 \beta ) q^{71} -2 \beta q^{72} + 6 \beta q^{73} + ( 6 - 2 \beta ) q^{74} - q^{75} - q^{77} + ( -2 - 2 \beta ) q^{78} + ( -4 + 3 \beta ) q^{79} + 4 q^{80} + q^{81} + ( -10 + 4 \beta ) q^{82} + ( 3 - 10 \beta ) q^{83} + ( -1 - 2 \beta ) q^{85} + ( 2 - 5 \beta ) q^{86} + ( -3 - 3 \beta ) q^{87} -2 \beta q^{88} + ( 7 + 5 \beta ) q^{89} -\beta q^{90} + ( -2 - \beta ) q^{91} + ( -2 - \beta ) q^{93} + ( 6 + 6 \beta ) q^{94} + ( 1 - 2 \beta ) q^{95} + ( -3 + 3 \beta ) q^{97} + \beta q^{98} + q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} - 2q^{5} - 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} - 2q^{5} - 2q^{7} + 2q^{9} + 2q^{11} + 4q^{13} + 2q^{15} - 8q^{16} + 2q^{17} - 2q^{19} + 2q^{21} - 2q^{23} + 2q^{25} + 4q^{26} - 2q^{27} + 6q^{29} + 4q^{31} - 2q^{33} + 8q^{34} + 2q^{35} - 4q^{37} + 8q^{38} - 4q^{39} + 8q^{41} - 10q^{43} - 2q^{45} + 8q^{46} + 12q^{47} + 8q^{48} + 2q^{49} - 2q^{51} + 6q^{53} - 2q^{55} + 2q^{57} + 12q^{58} + 26q^{59} + 10q^{61} + 4q^{62} - 2q^{63} + 16q^{64} - 4q^{65} - 12q^{67} + 2q^{69} - 8q^{71} + 12q^{74} - 2q^{75} - 2q^{77} - 4q^{78} - 8q^{79} + 8q^{80} + 2q^{81} - 20q^{82} + 6q^{83} - 2q^{85} + 4q^{86} - 6q^{87} + 14q^{89} - 4q^{91} - 4q^{93} + 12q^{94} + 2q^{95} - 6q^{97} + 2q^{99} + O(q^{100}) \)

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
−1.41421 −1.00000 0 −1.00000 1.41421 −1.00000 2.82843 1.00000 1.41421
1.2 1.41421 −1.00000 0 −1.00000 −1.41421 −1.00000 −2.82843 1.00000 −1.41421
\(n\): 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 1155.2.a.p 2
3.b odd 2 1 3465.2.a.w 2
5.b even 2 1 5775.2.a.bi 2
7.b odd 2 1 8085.2.a.bf 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1155.2.a.p 2 1.a even 1 1 trivial
3465.2.a.w 2 3.b odd 2 1
5775.2.a.bi 2 5.b even 2 1
8085.2.a.bf 2 7.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(1\)
\(7\) \(1\)
\(11\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1155))\):

\( T_{2}^{2} - 2 \)
\( T_{13}^{2} - 4 T_{13} + 2 \)
\( T_{17}^{2} - 2 T_{17} - 7 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T^{2} + 4 T^{4} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( ( 1 - T )^{2} \)
$13$ \( 1 - 4 T + 28 T^{2} - 52 T^{3} + 169 T^{4} \)
$17$ \( 1 - 2 T + 27 T^{2} - 34 T^{3} + 289 T^{4} \)
$19$ \( 1 + 2 T + 31 T^{2} + 38 T^{3} + 361 T^{4} \)
$23$ \( 1 + 2 T + 39 T^{2} + 46 T^{3} + 529 T^{4} \)
$29$ \( 1 - 6 T + 49 T^{2} - 174 T^{3} + 841 T^{4} \)
$31$ \( 1 - 4 T + 64 T^{2} - 124 T^{3} + 961 T^{4} \)
$37$ \( 1 + 4 T + 60 T^{2} + 148 T^{3} + 1369 T^{4} \)
$41$ \( 1 - 8 T + 48 T^{2} - 328 T^{3} + 1681 T^{4} \)
$43$ \( 1 + 10 T + 109 T^{2} + 430 T^{3} + 1849 T^{4} \)
$47$ \( 1 - 12 T + 112 T^{2} - 564 T^{3} + 2209 T^{4} \)
$53$ \( 1 - 6 T + 83 T^{2} - 318 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 26 T + 285 T^{2} - 1534 T^{3} + 3481 T^{4} \)
$61$ \( 1 - 10 T + 115 T^{2} - 610 T^{3} + 3721 T^{4} \)
$67$ \( ( 1 + 6 T + 67 T^{2} )^{2} \)
$71$ \( 1 + 8 T + 108 T^{2} + 568 T^{3} + 5041 T^{4} \)
$73$ \( 1 + 74 T^{2} + 5329 T^{4} \)
$79$ \( 1 + 8 T + 156 T^{2} + 632 T^{3} + 6241 T^{4} \)
$83$ \( 1 - 6 T - 25 T^{2} - 498 T^{3} + 6889 T^{4} \)
$89$ \( 1 - 14 T + 177 T^{2} - 1246 T^{3} + 7921 T^{4} \)
$97$ \( 1 + 6 T + 185 T^{2} + 582 T^{3} + 9409 T^{4} \)
show more
show less