Properties

Label 1155.2.k.a
Level 1155
Weight 2
Character orbit 1155.k
Analytic conductor 9.223
Analytic rank 0
Dimension 48
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 1155 = 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1155.k (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.22272143346\)
Analytic rank: \(0\)
Dimension: \(48\)
Coefficient ring index: multiple of None
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48q - 48q^{3} + 48q^{4} + 4q^{5} + 48q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 48q - 48q^{3} + 48q^{4} + 4q^{5} + 48q^{9} - 48q^{12} - 4q^{15} + 40q^{16} + 18q^{20} + 20q^{25} - 48q^{27} + 48q^{36} + 20q^{38} - 16q^{44} + 4q^{45} - 8q^{47} - 40q^{48} + 24q^{49} + 8q^{55} - 8q^{56} - 18q^{60} - 4q^{64} - 14q^{70} - 32q^{71} - 20q^{75} + 32q^{77} + 46q^{80} + 48q^{81} + 32q^{82} - 16q^{86} - 68q^{97} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
769.1 −2.66471 −1.00000 5.10065 1.18527 1.89608i 2.66471 2.09935 + 1.61019i −8.26233 1.00000 −3.15840 + 5.05250i
769.2 −2.66471 −1.00000 5.10065 1.18527 + 1.89608i 2.66471 2.09935 1.61019i −8.26233 1.00000 −3.15840 5.05250i
769.3 −2.52930 −1.00000 4.39737 1.63381 1.52665i 2.52930 −2.48590 0.905718i −6.06368 1.00000 −4.13240 + 3.86136i
769.4 −2.52930 −1.00000 4.39737 1.63381 + 1.52665i 2.52930 −2.48590 + 0.905718i −6.06368 1.00000 −4.13240 3.86136i
769.5 −2.43104 −1.00000 3.90997 −2.21714 + 0.290295i 2.43104 1.43522 + 2.22265i −4.64322 1.00000 5.38997 0.705719i
769.6 −2.43104 −1.00000 3.90997 −2.21714 0.290295i 2.43104 1.43522 2.22265i −4.64322 1.00000 5.38997 + 0.705719i
769.7 −2.15215 −1.00000 2.63174 −1.03706 1.98104i 2.15215 −2.36665 1.18277i −1.35961 1.00000 2.23191 + 4.26348i
769.8 −2.15215 −1.00000 2.63174 −1.03706 + 1.98104i 2.15215 −2.36665 + 1.18277i −1.35961 1.00000 2.23191 4.26348i
769.9 −1.83264 −1.00000 1.35857 −0.538446 2.17027i 1.83264 −0.996474 + 2.45093i 1.17551 1.00000 0.986778 + 3.97732i
769.10 −1.83264 −1.00000 1.35857 −0.538446 + 2.17027i 1.83264 −0.996474 2.45093i 1.17551 1.00000 0.986778 3.97732i
769.11 −1.69546 −1.00000 0.874594 2.22547 0.217459i 1.69546 0.698105 2.55199i 1.90808 1.00000 −3.77320 + 0.368694i
769.12 −1.69546 −1.00000 0.874594 2.22547 + 0.217459i 1.69546 0.698105 + 2.55199i 1.90808 1.00000 −3.77320 0.368694i
769.13 −1.63529 −1.00000 0.674169 1.84877 1.25780i 1.63529 1.79271 1.94582i 2.16812 1.00000 −3.02327 + 2.05687i
769.14 −1.63529 −1.00000 0.674169 1.84877 + 1.25780i 1.63529 1.79271 + 1.94582i 2.16812 1.00000 −3.02327 2.05687i
769.15 −1.17317 −1.00000 −0.623669 −2.20931 0.344884i 1.17317 −1.71739 + 2.01260i 3.07801 1.00000 2.59190 + 0.404608i
769.16 −1.17317 −1.00000 −0.623669 −2.20931 + 0.344884i 1.17317 −1.71739 2.01260i 3.07801 1.00000 2.59190 0.404608i
769.17 −1.12050 −1.00000 −0.744486 −1.27295 + 1.83837i 1.12050 2.54893 0.709189i 3.07519 1.00000 1.42634 2.05989i
769.18 −1.12050 −1.00000 −0.744486 −1.27295 1.83837i 1.12050 2.54893 + 0.709189i 3.07519 1.00000 1.42634 + 2.05989i
769.19 −0.567201 −1.00000 −1.67828 2.10264 0.760858i 0.567201 −2.63586 + 0.228555i 2.08633 1.00000 −1.19262 + 0.431559i
769.20 −0.567201 −1.00000 −1.67828 2.10264 + 0.760858i 0.567201 −2.63586 0.228555i 2.08633 1.00000 −1.19262 0.431559i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 769.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner
35.c odd 2 1 inner
385.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1155.2.k.a 48
5.b even 2 1 1155.2.k.b yes 48
7.b odd 2 1 1155.2.k.b yes 48
11.b odd 2 1 inner 1155.2.k.a 48
35.c odd 2 1 inner 1155.2.k.a 48
55.d odd 2 1 1155.2.k.b yes 48
77.b even 2 1 1155.2.k.b yes 48
385.h even 2 1 inner 1155.2.k.a 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1155.2.k.a 48 1.a even 1 1 trivial
1155.2.k.a 48 11.b odd 2 1 inner
1155.2.k.a 48 35.c odd 2 1 inner
1155.2.k.a 48 385.h even 2 1 inner
1155.2.k.b yes 48 5.b even 2 1
1155.2.k.b yes 48 7.b odd 2 1
1155.2.k.b yes 48 55.d odd 2 1
1155.2.k.b yes 48 77.b even 2 1

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database