Properties

Label 370.2.o.d
Level $370$
Weight $2$
Character orbit 370.o
Analytic conductor $2.954$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.o (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.95446487479\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(4\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24q + 6q^{6} - 9q^{7} - 12q^{8} + 12q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 24q + 6q^{6} - 9q^{7} - 12q^{8} + 12q^{9} + 12q^{10} - 3q^{11} - 3q^{14} + 6q^{17} - 6q^{18} - 18q^{19} + 18q^{21} + 3q^{22} + 15q^{23} - 6q^{26} - 12q^{27} + 9q^{28} - 18q^{29} - 6q^{31} - 45q^{33} - 3q^{34} + 30q^{36} - 18q^{37} - 24q^{38} + 12q^{39} + 3q^{41} + 18q^{42} + 3q^{44} + 15q^{45} + 9q^{46} - 36q^{47} - 3q^{48} - 15q^{49} + 54q^{51} - 9q^{52} - 6q^{53} + 45q^{54} - 3q^{55} + 3q^{57} - 15q^{58} - 9q^{59} + 3q^{60} - 24q^{61} + 81q^{62} - 45q^{63} - 12q^{64} - 3q^{66} + 12q^{67} + 12q^{68} + 60q^{69} - 9q^{70} + 42q^{71} - 6q^{72} - 42q^{73} + 12q^{74} + 6q^{75} - 18q^{76} + 9q^{77} + 30q^{78} - 6q^{79} - 24q^{80} - 66q^{81} - 24q^{82} - 3q^{83} - 42q^{84} + 6q^{85} - 18q^{86} - 84q^{87} - 3q^{88} + 27q^{89} + 6q^{90} + 153q^{91} - 9q^{92} - 54q^{93} - 21q^{94} - 9q^{95} - 9q^{97} - 24q^{98} + 165q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
71.1 0.766044 + 0.642788i −1.72501 + 1.44746i 0.173648 + 0.984808i 0.939693 + 0.342020i −2.25184 −2.62554 0.955619i −0.500000 + 0.866025i 0.359591 2.03934i 0.500000 + 0.866025i
71.2 0.766044 + 0.642788i −0.599525 + 0.503061i 0.173648 + 0.984808i 0.939693 + 0.342020i −0.782624 2.34566 + 0.853751i −0.500000 + 0.866025i −0.414585 + 2.35123i 0.500000 + 0.866025i
71.3 0.766044 + 0.642788i 0.479302 0.402182i 0.173648 + 0.984808i 0.939693 + 0.342020i 0.625684 0.117038 + 0.0425983i −0.500000 + 0.866025i −0.452965 + 2.56889i 0.500000 + 0.866025i
71.4 0.766044 + 0.642788i 2.61128 2.19112i 0.173648 + 0.984808i 0.939693 + 0.342020i 3.40878 −2.86925 1.04432i −0.500000 + 0.866025i 1.49681 8.48886i 0.500000 + 0.866025i
81.1 −0.939693 + 0.342020i −3.13308 1.14035i 0.766044 0.642788i −0.173648 0.984808i 3.33416 0.599161 + 3.39801i −0.500000 + 0.866025i 6.21768 + 5.21726i 0.500000 + 0.866025i
81.2 −0.939693 + 0.342020i −0.436963 0.159041i 0.766044 0.642788i −0.173648 0.984808i 0.465006 0.593088 + 3.36357i −0.500000 + 0.866025i −2.13249 1.78937i 0.500000 + 0.866025i
81.3 −0.939693 + 0.342020i −0.142661 0.0519243i 0.766044 0.642788i −0.173648 0.984808i 0.151817 −0.282147 1.60013i −0.500000 + 0.866025i −2.28048 1.91355i 0.500000 + 0.866025i
81.4 −0.939693 + 0.342020i 2.77301 + 1.00929i 0.766044 0.642788i −0.173648 0.984808i −2.95098 −0.530717 3.00985i −0.500000 + 0.866025i 4.37280 + 3.66922i 0.500000 + 0.866025i
181.1 0.173648 0.984808i −0.471123 2.67187i −0.939693 0.342020i −0.766044 + 0.642788i −2.71309 −2.35481 + 1.97592i −0.500000 + 0.866025i −4.09785 + 1.49149i 0.500000 + 0.866025i
181.2 0.173648 0.984808i 0.0797653 + 0.452371i −0.939693 0.342020i −0.766044 + 0.642788i 0.459350 −0.742184 + 0.622767i −0.500000 + 0.866025i 2.62080 0.953893i 0.500000 + 0.866025i
181.3 0.173648 0.984808i 0.207212 + 1.17516i −0.939693 0.342020i −0.766044 + 0.642788i 1.19329 −2.54999 + 2.13970i −0.500000 + 0.866025i 1.48102 0.539047i 0.500000 + 0.866025i
181.4 0.173648 0.984808i 0.357794 + 2.02915i −0.939693 0.342020i −0.766044 + 0.642788i 2.06045 3.79969 3.18832i −0.500000 + 0.866025i −1.17035 + 0.425971i 0.500000 + 0.866025i
201.1 −0.939693 0.342020i −3.13308 + 1.14035i 0.766044 + 0.642788i −0.173648 + 0.984808i 3.33416 0.599161 3.39801i −0.500000 0.866025i 6.21768 5.21726i 0.500000 0.866025i
201.2 −0.939693 0.342020i −0.436963 + 0.159041i 0.766044 + 0.642788i −0.173648 + 0.984808i 0.465006 0.593088 3.36357i −0.500000 0.866025i −2.13249 + 1.78937i 0.500000 0.866025i
201.3 −0.939693 0.342020i −0.142661 + 0.0519243i 0.766044 + 0.642788i −0.173648 + 0.984808i 0.151817 −0.282147 + 1.60013i −0.500000 0.866025i −2.28048 + 1.91355i 0.500000 0.866025i
201.4 −0.939693 0.342020i 2.77301 1.00929i 0.766044 + 0.642788i −0.173648 + 0.984808i −2.95098 −0.530717 + 3.00985i −0.500000 0.866025i 4.37280 3.66922i 0.500000 0.866025i
231.1 0.173648 + 0.984808i −0.471123 + 2.67187i −0.939693 + 0.342020i −0.766044 0.642788i −2.71309 −2.35481 1.97592i −0.500000 0.866025i −4.09785 1.49149i 0.500000 0.866025i
231.2 0.173648 + 0.984808i 0.0797653 0.452371i −0.939693 + 0.342020i −0.766044 0.642788i 0.459350 −0.742184 0.622767i −0.500000 0.866025i 2.62080 + 0.953893i 0.500000 0.866025i
231.3 0.173648 + 0.984808i 0.207212 1.17516i −0.939693 + 0.342020i −0.766044 0.642788i 1.19329 −2.54999 2.13970i −0.500000 0.866025i 1.48102 + 0.539047i 0.500000 0.866025i
231.4 0.173648 + 0.984808i 0.357794 2.02915i −0.939693 + 0.342020i −0.766044 0.642788i 2.06045 3.79969 + 3.18832i −0.500000 0.866025i −1.17035 0.425971i 0.500000 0.866025i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 271.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.f even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 370.2.o.d 24
37.f even 9 1 inner 370.2.o.d 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.o.d 24 1.a even 1 1 trivial
370.2.o.d 24 37.f even 9 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{24} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(370, [\chi])\).