Properties

Label 1155.2.a.w
Level 1155
Weight 2
Character orbit 1155.a
Self dual yes
Analytic conductor 9.223
Analytic rank 0
Dimension 5
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: \(5\)
Coefficient field: 5.5.352076.1
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{3} \)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - x^{4} - 8 x^{3} + 3 x^{2} + 8 x - 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{4} - \nu^{3} - 8 \nu^{2} + 3 \nu + 6 \)\()/2\)
\(\beta_{2}\)\(=\)\( \nu^{4} - 9 \nu^{2} - 5 \nu + 8 \)
\(\beta_{3}\)\(=\)\( \nu^{4} + \nu^{3} - 10 \nu^{2} - 9 \nu + 9 \)
\(\beta_{4}\)\(=\)\( 2 \nu^{4} - \nu^{3} - 15 \nu^{2} - 4 \nu + 8 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} - 2 \beta_{2} + 2 \beta_{1} + 1\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{4} + \beta_{3} - 4 \beta_{2} - 2 \beta_{1} + 13\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(2 \beta_{4} + 9 \beta_{3} - 16 \beta_{2} + 6 \beta_{1} + 13\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(9 \beta_{4} + 7 \beta_{3} - 21 \beta_{2} - 4 \beta_{1} + 45\)\()/2\)

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
0.567739
0.844040
3.06028
−2.27097
−1.20109
−2.52275 1.00000 4.36426 −1.00000 −2.52275 1.00000 −5.96443 1.00000 2.52275
1.2 −1.36955 1.00000 −0.124319 −1.00000 −1.36955 1.00000 2.90937 1.00000 1.36955
1.3 0.346466 1.00000 −1.87996 −1.00000 0.346466 1.00000 −1.34427 1.00000 −0.346466
1.4 1.88068 1.00000 1.53695 −1.00000 1.88068 1.00000 −0.870842 1.00000 −1.88068
1.5 2.66516 1.00000 5.10307 −1.00000 2.66516 1.00000 8.27017 1.00000 −2.66516
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.w 5
3.b odd 2 1 3465.2.a.bm 5
5.b even 2 1 5775.2.a.cg 5
7.b odd 2 1 8085.2.a.bv 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1155.2.a.w 5 1.a even 1 1 trivial
3465.2.a.bm 5 3.b odd 2 1
5775.2.a.cg 5 5.b even 2 1
8085.2.a.bv 5 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1155))\):

\( T_{2}^{5} - T_{2}^{4} - 9 T_{2}^{3} + 7 T_{2}^{2} + 16 T_{2} - 6 \)
\( T_{13}^{5} - 8 T_{13}^{4} - 10 T_{13}^{3} + 164 T_{13}^{2} - 40 T_{13} - 784 \)
\( T_{17}^{5} - 63 T_{17}^{3} + 70 T_{17}^{2} + 380 T_{17} - 504 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} - T^{3} + 2 T^{4} - 2 T^{5} + 4 T^{6} - 4 T^{7} + 8 T^{8} - 16 T^{9} + 32 T^{10} \)
$3$ \( ( 1 - T )^{5} \)
$5$ \( ( 1 + T )^{5} \)
$7$ \( ( 1 - T )^{5} \)
$11$ \( ( 1 - T )^{5} \)
$13$ \( 1 - 8 T + 55 T^{2} - 252 T^{3} + 1260 T^{4} - 4632 T^{5} + 16380 T^{6} - 42588 T^{7} + 120835 T^{8} - 228488 T^{9} + 371293 T^{10} \)
$17$ \( 1 + 22 T^{2} + 70 T^{3} + 57 T^{4} + 1876 T^{5} + 969 T^{6} + 20230 T^{7} + 108086 T^{8} + 1419857 T^{10} \)
$19$ \( 1 - 6 T + 92 T^{2} - 412 T^{3} + 3439 T^{4} - 11388 T^{5} + 65341 T^{6} - 148732 T^{7} + 631028 T^{8} - 781926 T^{9} + 2476099 T^{10} \)
$23$ \( 1 + 2 T + 68 T^{2} + 260 T^{3} + 2175 T^{4} + 9652 T^{5} + 50025 T^{6} + 137540 T^{7} + 827356 T^{8} + 559682 T^{9} + 6436343 T^{10} \)
$29$ \( 1 - 6 T + 58 T^{2} - 142 T^{3} + 1365 T^{4} - 3016 T^{5} + 39585 T^{6} - 119422 T^{7} + 1414562 T^{8} - 4243686 T^{9} + 20511149 T^{10} \)
$31$ \( 1 - 10 T + 145 T^{2} - 1040 T^{3} + 8424 T^{4} - 45388 T^{5} + 261144 T^{6} - 999440 T^{7} + 4319695 T^{8} - 9235210 T^{9} + 28629151 T^{10} \)
$37$ \( 1 - 4 T + 71 T^{2} - 204 T^{3} + 1316 T^{4} - 4928 T^{5} + 48692 T^{6} - 279276 T^{7} + 3596363 T^{8} - 7496644 T^{9} + 69343957 T^{10} \)
$41$ \( 1 + 83 T^{2} - 260 T^{3} + 4964 T^{4} - 11192 T^{5} + 203524 T^{6} - 437060 T^{7} + 5720443 T^{8} + 115856201 T^{10} \)
$43$ \( 1 + 48 T^{2} + 12 T^{3} + 2563 T^{4} + 12104 T^{5} + 110209 T^{6} + 22188 T^{7} + 3816336 T^{8} + 147008443 T^{10} \)
$47$ \( 1 + 2 T + 97 T^{2} + 328 T^{3} + 6984 T^{4} + 12780 T^{5} + 328248 T^{6} + 724552 T^{7} + 10070831 T^{8} + 9759362 T^{9} + 229345007 T^{10} \)
$53$ \( 1 + 8 T + 194 T^{2} + 1338 T^{3} + 17981 T^{4} + 100460 T^{5} + 952993 T^{6} + 3758442 T^{7} + 28882138 T^{8} + 63123848 T^{9} + 418195493 T^{10} \)
$59$ \( 1 + 4 T + 164 T^{2} + 404 T^{3} + 15719 T^{4} + 36528 T^{5} + 927421 T^{6} + 1406324 T^{7} + 33682156 T^{8} + 48469444 T^{9} + 714924299 T^{10} \)
$61$ \( 1 - 16 T + 246 T^{2} - 2642 T^{3} + 28105 T^{4} - 220396 T^{5} + 1714405 T^{6} - 9830882 T^{7} + 55837326 T^{8} - 221533456 T^{9} + 844596301 T^{10} \)
$67$ \( 1 - 16 T + 323 T^{2} - 3792 T^{3} + 43246 T^{4} - 364224 T^{5} + 2897482 T^{6} - 17022288 T^{7} + 97146449 T^{8} - 322417936 T^{9} + 1350125107 T^{10} \)
$71$ \( 1 + 6 T + 161 T^{2} + 392 T^{3} + 13184 T^{4} + 19748 T^{5} + 936064 T^{6} + 1976072 T^{7} + 57623671 T^{8} + 152470086 T^{9} + 1804229351 T^{10} \)
$73$ \( 1 - 2 T + 121 T^{2} + 240 T^{3} + 13998 T^{4} - 10908 T^{5} + 1021854 T^{6} + 1278960 T^{7} + 47071057 T^{8} - 56796482 T^{9} + 2073071593 T^{10} \)
$79$ \( 1 - 42 T + 1065 T^{2} - 18336 T^{3} + 239440 T^{4} - 2398060 T^{5} + 18915760 T^{6} - 114434976 T^{7} + 525086535 T^{8} - 1635903402 T^{9} + 3077056399 T^{10} \)
$83$ \( 1 + 22 T + 432 T^{2} + 5200 T^{3} + 64115 T^{4} + 577172 T^{5} + 5321545 T^{6} + 35822800 T^{7} + 247011984 T^{8} + 1044083062 T^{9} + 3939040643 T^{10} \)
$89$ \( 1 + 10 T + 202 T^{2} + 1822 T^{3} + 26405 T^{4} + 244592 T^{5} + 2350045 T^{6} + 14432062 T^{7} + 142403738 T^{8} + 627422410 T^{9} + 5584059449 T^{10} \)
$97$ \( 1 + 2 T + 58 T^{2} + 502 T^{3} + 12309 T^{4} + 560 T^{5} + 1193973 T^{6} + 4723318 T^{7} + 52935034 T^{8} + 177058562 T^{9} + 8587340257 T^{10} \)
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