Properties

Label 1155.2.a.v
Level 1155
Weight 2
Character orbit 1155.a
Self dual yes
Analytic conductor 9.223
Analytic rank 0
Dimension 4
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: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.7232.1
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} - 5 x^{2} + 4 x + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} - 2 \nu^{2} - \nu \)\()/2\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} - 2 \nu^{2} - 5 \nu + 2 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 4 \nu^{2} + \nu - 8 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{2} + \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{1} + 4\)
\(\nu^{3}\)\(=\)\((\)\(4 \beta_{3} - \beta_{2} + 9 \beta_{1} + 17\)\()/2\)

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
−0.652223
−1.63640
3.06644
1.22219
−2.06644 −1.00000 2.27016 1.00000 2.06644 1.00000 −0.558268 1.00000 −2.06644
1.2 −0.222191 −1.00000 −1.95063 1.00000 0.222191 1.00000 0.877796 1.00000 −0.222191
1.3 1.65222 −1.00000 0.729840 1.00000 −1.65222 1.00000 −2.09859 1.00000 1.65222
1.4 2.63640 −1.00000 4.95063 1.00000 −2.63640 1.00000 7.77906 1.00000 2.63640
\(n\): 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.v 4
3.b odd 2 1 3465.2.a.bj 4
5.b even 2 1 5775.2.a.by 4
7.b odd 2 1 8085.2.a.bq 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1155.2.a.v 4 1.a even 1 1 trivial
3465.2.a.bj 4 3.b odd 2 1
5775.2.a.by 4 5.b even 2 1
8085.2.a.bq 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1155))\):

\( T_{2}^{4} - 2 T_{2}^{3} - 5 T_{2}^{2} + 8 T_{2} + 2 \)
\( T_{13}^{4} - 4 T_{13}^{3} - 14 T_{13}^{2} + 24 T_{13} - 8 \)
\( T_{17}^{4} - 6 T_{17}^{3} - 11 T_{17}^{2} + 4 T_{17} + 4 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T + 3 T^{2} - 4 T^{3} + 6 T^{4} - 8 T^{5} + 12 T^{6} - 16 T^{7} + 16 T^{8} \)
$3$ \( ( 1 + T )^{4} \)
$5$ \( ( 1 - T )^{4} \)
$7$ \( ( 1 - T )^{4} \)
$11$ \( ( 1 - T )^{4} \)
$13$ \( 1 - 4 T + 38 T^{2} - 132 T^{3} + 642 T^{4} - 1716 T^{5} + 6422 T^{6} - 8788 T^{7} + 28561 T^{8} \)
$17$ \( 1 - 6 T + 57 T^{2} - 302 T^{3} + 1364 T^{4} - 5134 T^{5} + 16473 T^{6} - 29478 T^{7} + 83521 T^{8} \)
$19$ \( 1 + 6 T + 77 T^{2} + 318 T^{3} + 2188 T^{4} + 6042 T^{5} + 27797 T^{6} + 41154 T^{7} + 130321 T^{8} \)
$23$ \( 1 - 10 T + 53 T^{2} - 242 T^{3} + 1252 T^{4} - 5566 T^{5} + 28037 T^{6} - 121670 T^{7} + 279841 T^{8} \)
$29$ \( 1 - 6 T + 71 T^{2} - 422 T^{3} + 2720 T^{4} - 12238 T^{5} + 59711 T^{6} - 146334 T^{7} + 707281 T^{8} \)
$31$ \( 1 + 8 T + 106 T^{2} + 552 T^{3} + 4394 T^{4} + 17112 T^{5} + 101866 T^{6} + 238328 T^{7} + 923521 T^{8} \)
$37$ \( 1 - 12 T + 118 T^{2} - 988 T^{3} + 6818 T^{4} - 36556 T^{5} + 161542 T^{6} - 607836 T^{7} + 1874161 T^{8} \)
$41$ \( 1 + 58 T^{2} + 264 T^{3} + 2362 T^{4} + 10824 T^{5} + 97498 T^{6} + 2825761 T^{8} \)
$43$ \( 1 - 6 T + 11 T^{2} - 238 T^{3} + 3504 T^{4} - 10234 T^{5} + 20339 T^{6} - 477042 T^{7} + 3418801 T^{8} \)
$47$ \( 1 + 42 T^{2} - 352 T^{3} + 1706 T^{4} - 16544 T^{5} + 92778 T^{6} + 4879681 T^{8} \)
$53$ \( 1 - 14 T + 233 T^{2} - 2118 T^{3} + 19148 T^{4} - 112254 T^{5} + 654497 T^{6} - 2084278 T^{7} + 7890481 T^{8} \)
$59$ \( 1 - 2 T + 99 T^{2} - 266 T^{3} + 9056 T^{4} - 15694 T^{5} + 344619 T^{6} - 410758 T^{7} + 12117361 T^{8} \)
$61$ \( 1 - 2 T + 193 T^{2} - 226 T^{3} + 16300 T^{4} - 13786 T^{5} + 718153 T^{6} - 453962 T^{7} + 13845841 T^{8} \)
$67$ \( 1 + 12 T + 96 T^{2} - 180 T^{3} - 2706 T^{4} - 12060 T^{5} + 430944 T^{6} + 3609156 T^{7} + 20151121 T^{8} \)
$71$ \( 1 - 4 T + 190 T^{2} - 692 T^{3} + 19074 T^{4} - 49132 T^{5} + 957790 T^{6} - 1431644 T^{7} + 25411681 T^{8} \)
$73$ \( 1 - 20 T + 416 T^{2} - 4604 T^{3} + 50046 T^{4} - 336092 T^{5} + 2216864 T^{6} - 7780340 T^{7} + 28398241 T^{8} \)
$79$ \( 1 + 4 T + 206 T^{2} + 596 T^{3} + 21154 T^{4} + 47084 T^{5} + 1285646 T^{6} + 1972156 T^{7} + 38950081 T^{8} \)
$83$ \( 1 - 14 T + 285 T^{2} - 3094 T^{3} + 34668 T^{4} - 256802 T^{5} + 1963365 T^{6} - 8005018 T^{7} + 47458321 T^{8} \)
$89$ \( 1 + 6 T + 231 T^{2} + 1758 T^{3} + 25464 T^{4} + 156462 T^{5} + 1829751 T^{6} + 4229814 T^{7} + 62742241 T^{8} \)
$97$ \( 1 + 14 T + 447 T^{2} + 4150 T^{3} + 67896 T^{4} + 402550 T^{5} + 4205823 T^{6} + 12777422 T^{7} + 88529281 T^{8} \)
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