Properties

Label 1110.2.ba.b
Level $1110$
Weight $2$
Character orbit 1110.ba
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.ba (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.86339462436\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(18\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36q + 18q^{2} - 18q^{4} + 4q^{5} - 36q^{8} + 18q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 36q + 18q^{2} - 18q^{4} + 4q^{5} - 36q^{8} + 18q^{9} + 2q^{10} + 4q^{11} + 14q^{13} + 2q^{15} - 18q^{16} - 18q^{18} + 6q^{19} - 2q^{20} + 2q^{22} + 20q^{23} - 2q^{25} + 28q^{26} - 2q^{30} + 18q^{32} + 6q^{33} - 20q^{35} - 36q^{36} - 20q^{37} + 6q^{39} - 4q^{40} + 10q^{41} - 2q^{44} + 2q^{45} + 10q^{46} + 10q^{49} - 4q^{50} + 14q^{52} + 12q^{53} + 40q^{55} - 8q^{57} - 30q^{58} + 18q^{59} - 4q^{60} - 6q^{61} + 12q^{62} + 36q^{64} - 32q^{65} - 36q^{67} + 12q^{69} - 40q^{70} - 24q^{71} - 18q^{72} - 34q^{74} + 8q^{75} - 6q^{76} + 24q^{77} + 6q^{78} - 2q^{80} - 18q^{81} + 20q^{82} - 36q^{83} + 26q^{85} + 10q^{87} - 4q^{88} - 2q^{90} - 36q^{91} - 10q^{92} - 12q^{93} + 12q^{94} + 18q^{95} - 52q^{97} - 10q^{98} + 2q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
529.1 0.500000 + 0.866025i −0.866025 0.500000i −0.500000 + 0.866025i −1.62555 1.53544i 1.00000i 2.75229 + 1.58904i −1.00000 0.500000 + 0.866025i 0.516959 2.17549i
529.2 0.500000 + 0.866025i −0.866025 0.500000i −0.500000 + 0.866025i −2.16062 + 0.575954i 1.00000i −1.29297 0.746494i −1.00000 0.500000 + 0.866025i −1.57910 1.58317i
529.3 0.500000 + 0.866025i −0.866025 0.500000i −0.500000 + 0.866025i −1.56260 + 1.59947i 1.00000i −0.998396 0.576424i −1.00000 0.500000 + 0.866025i −2.16648 0.553515i
529.4 0.500000 + 0.866025i −0.866025 0.500000i −0.500000 + 0.866025i −0.696189 2.12493i 1.00000i 1.07219 + 0.619032i −1.00000 0.500000 + 0.866025i 1.49215 1.66538i
529.5 0.500000 + 0.866025i −0.866025 0.500000i −0.500000 + 0.866025i 1.57405 + 1.58819i 1.00000i −1.99912 1.15419i −1.00000 0.500000 + 0.866025i −0.588388 + 2.15727i
529.6 0.500000 + 0.866025i −0.866025 0.500000i −0.500000 + 0.866025i 2.18223 0.487744i 1.00000i −2.13280 1.23137i −1.00000 0.500000 + 0.866025i 1.51351 + 1.64599i
529.7 0.500000 + 0.866025i −0.866025 0.500000i −0.500000 + 0.866025i −0.465711 2.18703i 1.00000i −3.89483 2.24868i −1.00000 0.500000 + 0.866025i 1.66117 1.49683i
529.8 0.500000 + 0.866025i −0.866025 0.500000i −0.500000 + 0.866025i 0.653296 + 2.13851i 1.00000i 3.20005 + 1.84755i −1.00000 0.500000 + 0.866025i −1.52535 + 1.63502i
529.9 0.500000 + 0.866025i −0.866025 0.500000i −0.500000 + 0.866025i 2.23506 0.0669680i 1.00000i 3.29356 + 1.90154i −1.00000 0.500000 + 0.866025i 1.17553 + 1.90214i
529.10 0.500000 + 0.866025i 0.866025 + 0.500000i −0.500000 + 0.866025i 2.02746 + 0.943089i 1.00000i 3.31268 + 1.91258i −1.00000 0.500000 + 0.866025i 0.196990 + 2.22737i
529.11 0.500000 + 0.866025i 0.866025 + 0.500000i −0.500000 + 0.866025i −1.19962 + 1.88704i 1.00000i 3.17295 + 1.83190i −1.00000 0.500000 + 0.866025i −2.23403 0.0953865i
529.12 0.500000 + 0.866025i 0.866025 + 0.500000i −0.500000 + 0.866025i 1.93300 1.12406i 1.00000i −3.60632 2.08211i −1.00000 0.500000 + 0.866025i 1.93996 + 1.11200i
529.13 0.500000 + 0.866025i 0.866025 + 0.500000i −0.500000 + 0.866025i −1.38147 1.75828i 1.00000i 1.17143 + 0.676327i −1.00000 0.500000 + 0.866025i 0.831981 2.07553i
529.14 0.500000 + 0.866025i 0.866025 + 0.500000i −0.500000 + 0.866025i −0.672968 + 2.13240i 1.00000i −0.0701099 0.0404780i −1.00000 0.500000 + 0.866025i −2.18319 + 0.483391i
529.15 0.500000 + 0.866025i 0.866025 + 0.500000i −0.500000 + 0.866025i 1.50247 + 1.65608i 1.00000i −0.827955 0.478020i −1.00000 0.500000 + 0.866025i −0.682971 + 2.12921i
529.16 0.500000 + 0.866025i 0.866025 + 0.500000i −0.500000 + 0.866025i 1.64331 1.51642i 1.00000i 0.916644 + 0.529225i −1.00000 0.500000 + 0.866025i 2.13491 + 0.664945i
529.17 0.500000 + 0.866025i 0.866025 + 0.500000i −0.500000 + 0.866025i −2.17760 + 0.508014i 1.00000i −1.49882 0.865342i −1.00000 0.500000 + 0.866025i −1.52875 1.63185i
529.18 0.500000 + 0.866025i 0.866025 + 0.500000i −0.500000 + 0.866025i 0.191439 2.22786i 1.00000i −2.57051 1.48408i −1.00000 0.500000 + 0.866025i 2.02510 0.948138i
619.1 0.500000 0.866025i −0.866025 + 0.500000i −0.500000 0.866025i −1.62555 + 1.53544i 1.00000i 2.75229 1.58904i −1.00000 0.500000 0.866025i 0.516959 + 2.17549i
619.2 0.500000 0.866025i −0.866025 + 0.500000i −0.500000 0.866025i −2.16062 0.575954i 1.00000i −1.29297 + 0.746494i −1.00000 0.500000 0.866025i −1.57910 + 1.58317i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 619.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
185.l even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1110.2.ba.b yes 36
5.b even 2 1 1110.2.ba.a 36
37.e even 6 1 1110.2.ba.a 36
185.l even 6 1 inner 1110.2.ba.b yes 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.ba.a 36 5.b even 2 1
1110.2.ba.a 36 37.e even 6 1
1110.2.ba.b yes 36 1.a even 1 1 trivial
1110.2.ba.b yes 36 185.l even 6 1 inner