Properties

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

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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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{2} - q^{3} + ( 2 + \beta ) q^{4} + q^{5} + \beta q^{6} - q^{7} + ( -4 - \beta ) q^{8} + q^{9} +O(q^{10})\) \( q -\beta q^{2} - q^{3} + ( 2 + \beta ) q^{4} + q^{5} + \beta q^{6} - q^{7} + ( -4 - \beta ) q^{8} + q^{9} -\beta q^{10} + q^{11} + ( -2 - \beta ) q^{12} + ( -4 + 2 \beta ) q^{13} + \beta q^{14} - q^{15} + 3 \beta q^{16} + ( -1 - \beta ) q^{17} -\beta q^{18} + ( -1 + \beta ) q^{19} + ( 2 + \beta ) q^{20} + q^{21} -\beta q^{22} + ( -3 - \beta ) q^{23} + ( 4 + \beta ) q^{24} + q^{25} + ( -8 + 2 \beta ) q^{26} - q^{27} + ( -2 - \beta ) q^{28} + ( -3 - 3 \beta ) q^{29} + \beta q^{30} + ( -2 + 2 \beta ) q^{31} + ( -4 - \beta ) q^{32} - q^{33} + ( 4 + 2 \beta ) q^{34} - q^{35} + ( 2 + \beta ) q^{36} + ( -4 + 2 \beta ) q^{37} -4 q^{38} + ( 4 - 2 \beta ) q^{39} + ( -4 - \beta ) q^{40} + 2 q^{41} -\beta q^{42} + ( 1 - \beta ) q^{43} + ( 2 + \beta ) q^{44} + q^{45} + ( 4 + 4 \beta ) q^{46} + ( -2 + 2 \beta ) q^{47} -3 \beta q^{48} + q^{49} -\beta q^{50} + ( 1 + \beta ) q^{51} + 2 \beta q^{52} + ( -9 + 3 \beta ) q^{53} + \beta q^{54} + q^{55} + ( 4 + \beta ) q^{56} + ( 1 - \beta ) q^{57} + ( 12 + 6 \beta ) q^{58} + ( 9 - \beta ) q^{59} + ( -2 - \beta ) q^{60} + ( -5 - \beta ) q^{61} -8 q^{62} - q^{63} + ( 4 - \beta ) q^{64} + ( -4 + 2 \beta ) q^{65} + \beta q^{66} + ( 10 + 2 \beta ) q^{67} + ( -6 - 4 \beta ) q^{68} + ( 3 + \beta ) q^{69} + \beta q^{70} + ( 2 - 2 \beta ) q^{71} + ( -4 - \beta ) q^{72} + ( -8 + 2 \beta ) q^{73} + ( -8 + 2 \beta ) q^{74} - q^{75} + ( 2 + 2 \beta ) q^{76} - q^{77} + ( 8 - 2 \beta ) q^{78} + ( -4 + 4 \beta ) q^{79} + 3 \beta q^{80} + q^{81} -2 \beta q^{82} + ( -7 - \beta ) q^{83} + ( 2 + \beta ) q^{84} + ( -1 - \beta ) q^{85} + 4 q^{86} + ( 3 + 3 \beta ) q^{87} + ( -4 - \beta ) q^{88} + ( -7 - 3 \beta ) q^{89} -\beta q^{90} + ( 4 - 2 \beta ) q^{91} + ( -10 - 6 \beta ) q^{92} + ( 2 - 2 \beta ) q^{93} -8 q^{94} + ( -1 + \beta ) q^{95} + ( 4 + \beta ) q^{96} + ( -5 - 5 \beta ) q^{97} -\beta q^{98} + q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} - 2q^{3} + 5q^{4} + 2q^{5} + q^{6} - 2q^{7} - 9q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - q^{2} - 2q^{3} + 5q^{4} + 2q^{5} + q^{6} - 2q^{7} - 9q^{8} + 2q^{9} - q^{10} + 2q^{11} - 5q^{12} - 6q^{13} + q^{14} - 2q^{15} + 3q^{16} - 3q^{17} - q^{18} - q^{19} + 5q^{20} + 2q^{21} - q^{22} - 7q^{23} + 9q^{24} + 2q^{25} - 14q^{26} - 2q^{27} - 5q^{28} - 9q^{29} + q^{30} - 2q^{31} - 9q^{32} - 2q^{33} + 10q^{34} - 2q^{35} + 5q^{36} - 6q^{37} - 8q^{38} + 6q^{39} - 9q^{40} + 4q^{41} - q^{42} + q^{43} + 5q^{44} + 2q^{45} + 12q^{46} - 2q^{47} - 3q^{48} + 2q^{49} - q^{50} + 3q^{51} + 2q^{52} - 15q^{53} + q^{54} + 2q^{55} + 9q^{56} + q^{57} + 30q^{58} + 17q^{59} - 5q^{60} - 11q^{61} - 16q^{62} - 2q^{63} + 7q^{64} - 6q^{65} + q^{66} + 22q^{67} - 16q^{68} + 7q^{69} + q^{70} + 2q^{71} - 9q^{72} - 14q^{73} - 14q^{74} - 2q^{75} + 6q^{76} - 2q^{77} + 14q^{78} - 4q^{79} + 3q^{80} + 2q^{81} - 2q^{82} - 15q^{83} + 5q^{84} - 3q^{85} + 8q^{86} + 9q^{87} - 9q^{88} - 17q^{89} - q^{90} + 6q^{91} - 26q^{92} + 2q^{93} - 16q^{94} - q^{95} + 9q^{96} - 15q^{97} - q^{98} + 2q^{99} + O(q^{100}) \)

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
2.56155
−1.56155
−2.56155 −1.00000 4.56155 1.00000 2.56155 −1.00000 −6.56155 1.00000 −2.56155
1.2 1.56155 −1.00000 0.438447 1.00000 −1.56155 −1.00000 −2.43845 1.00000 1.56155
\(n\): 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.o 2
3.b odd 2 1 3465.2.a.z 2
5.b even 2 1 5775.2.a.bm 2
7.b odd 2 1 8085.2.a.bb 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1155.2.a.o 2 1.a even 1 1 trivial
3465.2.a.z 2 3.b odd 2 1
5775.2.a.bm 2 5.b even 2 1
8085.2.a.bb 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1155))\):

\( T_{2}^{2} + T_{2} - 4 \)
\( T_{13}^{2} + 6 T_{13} - 8 \)
\( T_{17}^{2} + 3 T_{17} - 2 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + 2 T^{3} + 4 T^{4} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( ( 1 - T )^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( ( 1 - T )^{2} \)
$13$ \( 1 + 6 T + 18 T^{2} + 78 T^{3} + 169 T^{4} \)
$17$ \( 1 + 3 T + 32 T^{2} + 51 T^{3} + 289 T^{4} \)
$19$ \( 1 + T + 34 T^{2} + 19 T^{3} + 361 T^{4} \)
$23$ \( 1 + 7 T + 54 T^{2} + 161 T^{3} + 529 T^{4} \)
$29$ \( 1 + 9 T + 40 T^{2} + 261 T^{3} + 841 T^{4} \)
$31$ \( 1 + 2 T + 46 T^{2} + 62 T^{3} + 961 T^{4} \)
$37$ \( 1 + 6 T + 66 T^{2} + 222 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 - 2 T + 41 T^{2} )^{2} \)
$43$ \( 1 - T + 82 T^{2} - 43 T^{3} + 1849 T^{4} \)
$47$ \( 1 + 2 T + 78 T^{2} + 94 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 15 T + 124 T^{2} + 795 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 17 T + 186 T^{2} - 1003 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 11 T + 148 T^{2} + 671 T^{3} + 3721 T^{4} \)
$67$ \( 1 - 22 T + 238 T^{2} - 1474 T^{3} + 4489 T^{4} \)
$71$ \( 1 - 2 T + 126 T^{2} - 142 T^{3} + 5041 T^{4} \)
$73$ \( 1 + 14 T + 178 T^{2} + 1022 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 4 T + 94 T^{2} + 316 T^{3} + 6241 T^{4} \)
$83$ \( 1 + 15 T + 218 T^{2} + 1245 T^{3} + 6889 T^{4} \)
$89$ \( 1 + 17 T + 212 T^{2} + 1513 T^{3} + 7921 T^{4} \)
$97$ \( 1 + 15 T + 144 T^{2} + 1455 T^{3} + 9409 T^{4} \)
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