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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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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [370,2,Mod(71,370)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(370, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("370.71");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.o (of order \(9\), degree \(6\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
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.

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

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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 71.4
Significant digits:
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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 \( T_{3}^{24} - 6 T_{3}^{22} + 10 T_{3}^{21} + 48 T_{3}^{20} + 111 T_{3}^{19} + 536 T_{3}^{18} - 1620 T_{3}^{17} + \cdots + 64 \) acting on \(S_{2}^{\mathrm{new}}(370, [\chi])\). Copy content Toggle raw display