Properties

Label 1110.2.l.a
Level $1110$
Weight $2$
Character orbit 1110.l
Analytic conductor $8.863$
Analytic rank $0$
Dimension $36$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1110.l (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.86339462436\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(18\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36q - 36q^{4} + 4q^{7} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 36q - 36q^{4} + 4q^{7} - 4q^{10} + 4q^{14} + 36q^{16} - 32q^{17} + 36q^{18} + 4q^{19} + 8q^{22} - 4q^{25} + 8q^{26} - 4q^{28} + 36q^{29} - 4q^{31} + 4q^{33} - 12q^{35} - 4q^{37} + 4q^{38} + 4q^{39} + 4q^{40} - 16q^{42} + 4q^{45} + 16q^{47} - 16q^{50} - 8q^{53} + 16q^{55} - 4q^{56} - 36q^{58} - 4q^{59} - 4q^{61} - 4q^{62} - 4q^{63} - 36q^{64} + 52q^{65} - 4q^{66} + 16q^{67} + 32q^{68} - 8q^{69} - 28q^{70} - 8q^{71} - 36q^{72} - 4q^{73} + 28q^{74} + 16q^{75} - 4q^{76} + 8q^{77} - 4q^{78} - 12q^{79} - 36q^{81} - 8q^{82} + 8q^{83} + 8q^{85} + 32q^{86} - 8q^{87} - 8q^{88} - 24q^{89} + 56q^{91} + 16q^{94} - 20q^{95} + 40q^{97} - 12q^{98} - 8q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 1.00000i −0.707107 + 0.707107i −1.00000 −1.51129 1.64803i −0.707107 0.707107i 2.75974 2.75974i 1.00000i 1.00000i 1.64803 1.51129i
43.2 1.00000i −0.707107 + 0.707107i −1.00000 −0.887844 + 2.05225i −0.707107 0.707107i −1.84414 + 1.84414i 1.00000i 1.00000i −2.05225 0.887844i
43.3 1.00000i −0.707107 + 0.707107i −1.00000 2.04060 + 0.914315i −0.707107 0.707107i 0.867388 0.867388i 1.00000i 1.00000i −0.914315 + 2.04060i
43.4 1.00000i −0.707107 + 0.707107i −1.00000 1.11117 + 1.94044i −0.707107 0.707107i 0.636201 0.636201i 1.00000i 1.00000i −1.94044 + 1.11117i
43.5 1.00000i −0.707107 + 0.707107i −1.00000 −0.943349 2.02734i −0.707107 0.707107i −0.516238 + 0.516238i 1.00000i 1.00000i 2.02734 0.943349i
43.6 1.00000i −0.707107 + 0.707107i −1.00000 1.50038 1.65797i −0.707107 0.707107i −1.06172 + 1.06172i 1.00000i 1.00000i 1.65797 + 1.50038i
43.7 1.00000i −0.707107 + 0.707107i −1.00000 −2.19864 + 0.407420i −0.707107 0.707107i 1.64494 1.64494i 1.00000i 1.00000i −0.407420 2.19864i
43.8 1.00000i −0.707107 + 0.707107i −1.00000 2.05054 0.891783i −0.707107 0.707107i −1.56460 + 1.56460i 1.00000i 1.00000i 0.891783 + 2.05054i
43.9 1.00000i −0.707107 + 0.707107i −1.00000 −1.16157 + 1.91070i −0.707107 0.707107i 2.90687 2.90687i 1.00000i 1.00000i −1.91070 1.16157i
43.10 1.00000i 0.707107 0.707107i −1.00000 1.87641 + 1.21617i 0.707107 + 0.707107i −2.24767 + 2.24767i 1.00000i 1.00000i −1.21617 + 1.87641i
43.11 1.00000i 0.707107 0.707107i −1.00000 −0.0221544 + 2.23596i 0.707107 + 0.707107i −0.281427 + 0.281427i 1.00000i 1.00000i −2.23596 0.0221544i
43.12 1.00000i 0.707107 0.707107i −1.00000 −2.06140 0.866398i 0.707107 + 0.707107i −0.662100 + 0.662100i 1.00000i 1.00000i 0.866398 2.06140i
43.13 1.00000i 0.707107 0.707107i −1.00000 1.45560 + 1.69741i 0.707107 + 0.707107i −0.764053 + 0.764053i 1.00000i 1.00000i −1.69741 + 1.45560i
43.14 1.00000i 0.707107 0.707107i −1.00000 −1.52512 + 1.63524i 0.707107 + 0.707107i 1.84299 1.84299i 1.00000i 1.00000i −1.63524 1.52512i
43.15 1.00000i 0.707107 0.707107i −1.00000 −2.17211 0.530977i 0.707107 + 0.707107i −1.93388 + 1.93388i 1.00000i 1.00000i 0.530977 2.17211i
43.16 1.00000i 0.707107 0.707107i −1.00000 2.23501 + 0.0687789i 0.707107 + 0.707107i 2.13605 2.13605i 1.00000i 1.00000i −0.0687789 + 2.23501i
43.17 1.00000i 0.707107 0.707107i −1.00000 −0.0484605 2.23554i 0.707107 + 0.707107i 2.77693 2.77693i 1.00000i 1.00000i 2.23554 0.0484605i
43.18 1.00000i 0.707107 0.707107i −1.00000 0.262217 2.22064i 0.707107 + 0.707107i −2.69527 + 2.69527i 1.00000i 1.00000i 2.22064 + 0.262217i
697.1 1.00000i −0.707107 0.707107i −1.00000 −1.51129 + 1.64803i −0.707107 + 0.707107i 2.75974 + 2.75974i 1.00000i 1.00000i 1.64803 + 1.51129i
697.2 1.00000i −0.707107 0.707107i −1.00000 −0.887844 2.05225i −0.707107 + 0.707107i −1.84414 1.84414i 1.00000i 1.00000i −2.05225 + 0.887844i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 697.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
185.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1110.2.l.a 36
5.c odd 4 1 1110.2.o.a yes 36
37.d odd 4 1 1110.2.o.a yes 36
185.f even 4 1 inner 1110.2.l.a 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.l.a 36 1.a even 1 1 trivial
1110.2.l.a 36 185.f even 4 1 inner
1110.2.o.a yes 36 5.c odd 4 1
1110.2.o.a yes 36 37.d odd 4 1