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Properties

Label 370.2.w.b
Level $370$
Weight $2$
Character orbit 370.w
Analytic conductor $2.954$
Analytic rank $0$
Dimension $48$
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(21,370)]
 
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, 11]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("370.21");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.w (of order \(18\), 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: \(48\)
Relative dimension: \(8\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 12 q^{7} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 12 q^{7} + 24 q^{9} - 24 q^{10} + 18 q^{11} + 24 q^{17} - 18 q^{21} + 12 q^{22} - 12 q^{26} + 24 q^{27} + 6 q^{28} - 18 q^{29} - 12 q^{33} - 12 q^{34} + 6 q^{35} - 84 q^{36} - 6 q^{37} + 12 q^{38} - 42 q^{39} - 6 q^{41} - 54 q^{42} + 6 q^{44} + 36 q^{45} + 36 q^{46} - 30 q^{47} + 6 q^{48} + 18 q^{49} + 18 q^{52} + 54 q^{54} - 6 q^{55} - 6 q^{56} - 36 q^{57} - 12 q^{58} + 42 q^{59} - 24 q^{61} + 54 q^{62} - 24 q^{63} + 24 q^{64} - 36 q^{66} + 60 q^{67} - 102 q^{69} - 6 q^{70} + 60 q^{71} - 48 q^{73} + 30 q^{74} + 12 q^{75} - 126 q^{77} + 30 q^{78} + 36 q^{81} + 90 q^{83} + 12 q^{84} + 6 q^{85} + 36 q^{86} - 30 q^{87} - 36 q^{88} - 6 q^{89} - 12 q^{90} + 84 q^{91} - 6 q^{92} - 54 q^{93} - 18 q^{94} - 24 q^{95} + 36 q^{97} - 24 q^{98} - 72 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
21.1 −0.342020 + 0.939693i −2.79539 + 1.01744i −0.766044 − 0.642788i 0.984808 + 0.173648i − 2.97480i 0.125624 − 0.712447i 0.866025 − 0.500000i 4.48091 − 3.75993i −0.500000 + 0.866025i
21.2 −0.342020 + 0.939693i −1.18324 + 0.430663i −0.766044 − 0.642788i 0.984808 + 0.173648i − 1.25917i 0.165700 − 0.939732i 0.866025 − 0.500000i −1.08356 + 0.909211i −0.500000 + 0.866025i
21.3 −0.342020 + 0.939693i 1.67405 − 0.609304i −0.766044 − 0.642788i 0.984808 + 0.173648i 1.78149i 0.521989 − 2.96035i 0.866025 − 0.500000i 0.133056 − 0.111647i −0.500000 + 0.866025i
21.4 −0.342020 + 0.939693i 3.24427 − 1.18082i −0.766044 − 0.642788i 0.984808 + 0.173648i 3.45248i −0.802595 + 4.55174i 0.866025 − 0.500000i 6.83285 − 5.73344i −0.500000 + 0.866025i
21.5 0.342020 − 0.939693i −2.52763 + 0.919981i −0.766044 − 0.642788i −0.984808 − 0.173648i 2.68984i 0.420839 − 2.38669i −0.866025 + 0.500000i 3.24439 − 2.72237i −0.500000 + 0.866025i
21.6 0.342020 − 0.939693i −0.384909 + 0.140095i −0.766044 − 0.642788i −0.984808 − 0.173648i 0.409612i −0.723927 + 4.10559i −0.866025 + 0.500000i −2.16961 + 1.82051i −0.500000 + 0.866025i
21.7 0.342020 − 0.939693i 0.815392 − 0.296779i −0.766044 − 0.642788i −0.984808 − 0.173648i − 0.867722i 0.629880 − 3.57223i −0.866025 + 0.500000i −1.72135 + 1.44438i −0.500000 + 0.866025i
21.8 0.342020 − 0.939693i 3.03684 − 1.10532i −0.766044 − 0.642788i −0.984808 − 0.173648i − 3.23173i 0.130401 − 0.739542i −0.866025 + 0.500000i 5.70251 − 4.78497i −0.500000 + 0.866025i
41.1 −0.984808 − 0.173648i −0.396937 − 2.25114i 0.939693 + 0.342020i 0.642788 + 0.766044i 2.28587i −1.58779 + 1.33231i −0.866025 − 0.500000i −2.09099 + 0.761058i −0.500000 − 0.866025i
41.2 −0.984808 − 0.173648i −0.240758 − 1.36541i 0.939693 + 0.342020i 0.642788 + 0.766044i 1.38647i 2.42202 − 2.03232i −0.866025 − 0.500000i 1.01271 − 0.368596i −0.500000 − 0.866025i
41.3 −0.984808 − 0.173648i 0.105668 + 0.599271i 0.939693 + 0.342020i 0.642788 + 0.766044i − 0.608515i −2.12489 + 1.78299i −0.866025 − 0.500000i 2.47112 − 0.899413i −0.500000 − 0.866025i
41.4 −0.984808 − 0.173648i 0.358379 + 2.03247i 0.939693 + 0.342020i 0.642788 + 0.766044i − 2.06382i 3.75435 − 3.15027i −0.866025 − 0.500000i −1.18341 + 0.430725i −0.500000 − 0.866025i
41.5 0.984808 + 0.173648i −0.568495 − 3.22410i 0.939693 + 0.342020i −0.642788 − 0.766044i − 3.27383i 2.50060 − 2.09825i 0.866025 + 0.500000i −7.25253 + 2.63970i −0.500000 − 0.866025i
41.6 0.984808 + 0.173648i −0.192543 − 1.09196i 0.939693 + 0.342020i −0.642788 − 0.766044i − 1.10881i −0.750891 + 0.630073i 0.866025 + 0.500000i 1.66377 − 0.605561i −0.500000 − 0.866025i
41.7 0.984808 + 0.173648i 0.168670 + 0.956577i 0.939693 + 0.342020i −0.642788 − 0.766044i 0.971334i 2.87596 − 2.41322i 0.866025 + 0.500000i 1.93249 − 0.703368i −0.500000 − 0.866025i
41.8 0.984808 + 0.173648i 0.418719 + 2.37467i 0.939693 + 0.342020i −0.642788 − 0.766044i 2.41131i −3.20998 + 2.69349i 0.866025 + 0.500000i −2.64467 + 0.962583i −0.500000 − 0.866025i
141.1 −0.342020 − 0.939693i −2.79539 − 1.01744i −0.766044 + 0.642788i 0.984808 − 0.173648i 2.97480i 0.125624 + 0.712447i 0.866025 + 0.500000i 4.48091 + 3.75993i −0.500000 − 0.866025i
141.2 −0.342020 − 0.939693i −1.18324 − 0.430663i −0.766044 + 0.642788i 0.984808 − 0.173648i 1.25917i 0.165700 + 0.939732i 0.866025 + 0.500000i −1.08356 − 0.909211i −0.500000 − 0.866025i
141.3 −0.342020 − 0.939693i 1.67405 + 0.609304i −0.766044 + 0.642788i 0.984808 − 0.173648i − 1.78149i 0.521989 + 2.96035i 0.866025 + 0.500000i 0.133056 + 0.111647i −0.500000 − 0.866025i
141.4 −0.342020 − 0.939693i 3.24427 + 1.18082i −0.766044 + 0.642788i 0.984808 − 0.173648i − 3.45248i −0.802595 − 4.55174i 0.866025 + 0.500000i 6.83285 + 5.73344i −0.500000 − 0.866025i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 21.8
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.h even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 370.2.w.b ✓ 48
37.h even 18 1 inner 370.2.w.b ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.w.b ✓ 48 1.a even 1 1 trivial
370.2.w.b ✓ 48 37.h even 18 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{48} - 12 T_{3}^{46} - 20 T_{3}^{45} + 90 T_{3}^{44} + 450 T_{3}^{43} + 1006 T_{3}^{42} + \cdots + 71464259584 \) acting on \(S_{2}^{\mathrm{new}}(370, [\chi])\). Copy content Toggle raw display