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Properties

Label 370.2.bd.b
Level $370$
Weight $2$
Character orbit 370.bd
Analytic conductor $2.954$
Analytic rank $0$
Dimension $120$
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(13,370)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(370, base_ring=CyclotomicField(36))
 
chi = DirichletCharacter(H, H._module([27, 11]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("370.13");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.bd (of order \(36\), degree \(12\), 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: \(120\)
Relative dimension: \(10\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 120 q + 6 q^{3}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 120 q + 6 q^{3} - 6 q^{10} - 36 q^{11} - 6 q^{12} - 12 q^{14} + 6 q^{17} + 12 q^{19} + 42 q^{21} + 36 q^{23} - 6 q^{24} + 24 q^{25} + 6 q^{26} - 6 q^{27} - 24 q^{30} + 6 q^{33} - 66 q^{35} - 24 q^{37} - 48 q^{38} - 24 q^{40} - 30 q^{41} + 18 q^{42} - 6 q^{44} + 42 q^{45} - 6 q^{46} - 24 q^{47} + 60 q^{49} + 12 q^{50} + 12 q^{51} + 12 q^{53} - 18 q^{54} - 72 q^{57} - 48 q^{58} + 24 q^{59} - 72 q^{61} - 30 q^{62} - 102 q^{63} + 60 q^{64} - 18 q^{65} - 30 q^{67} + 96 q^{69} + 12 q^{70} + 90 q^{73} - 24 q^{74} - 60 q^{75} + 18 q^{76} + 24 q^{77} + 36 q^{78} + 18 q^{79} - 6 q^{80} - 108 q^{81} - 6 q^{82} - 36 q^{83} + 18 q^{85} + 24 q^{86} - 48 q^{87} + 12 q^{88} - 54 q^{89} - 12 q^{90} + 42 q^{91} - 6 q^{92} + 18 q^{94} + 102 q^{95} - 12 q^{96} + 60 q^{97} + 36 q^{98} - 42 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 0.984808 + 0.173648i −2.74835 − 1.92441i 0.939693 + 0.342020i 2.05400 + 0.883787i −2.37242 − 2.37242i −0.306526 + 3.50361i 0.866025 + 0.500000i 2.82399 + 7.75885i 1.86933 + 1.22703i
13.2 0.984808 + 0.173648i −2.15928 − 1.51194i 0.939693 + 0.342020i −1.95696 + 1.08182i −1.86393 − 1.86393i −0.00262850 + 0.0300439i 0.866025 + 0.500000i 1.35045 + 3.71033i −2.11508 + 0.725559i
13.3 0.984808 + 0.173648i −1.25779 − 0.880716i 0.939693 + 0.342020i −1.20384 − 1.88435i −1.08575 − 1.08575i −0.0418072 + 0.477859i 0.866025 + 0.500000i −0.219679 − 0.603563i −0.858332 − 2.06477i
13.4 0.984808 + 0.173648i −0.961804 − 0.673462i 0.939693 + 0.342020i −1.64326 + 1.51648i −0.830246 − 0.830246i 0.403339 − 4.61019i 0.866025 + 0.500000i −0.554545 − 1.52360i −1.88163 + 1.20809i
13.5 0.984808 + 0.173648i −0.817959 − 0.572741i 0.939693 + 0.342020i 1.91023 − 1.16234i −0.706077 − 0.706077i 0.132313 − 1.51235i 0.866025 + 0.500000i −0.685035 − 1.88212i 2.08304 − 0.812974i
13.6 0.984808 + 0.173648i −0.439891 − 0.308015i 0.939693 + 0.342020i 0.778763 + 2.09607i −0.379722 − 0.379722i −0.222195 + 2.53970i 0.866025 + 0.500000i −0.927429 − 2.54809i 0.402953 + 2.19946i
13.7 0.984808 + 0.173648i 0.973365 + 0.681558i 0.939693 + 0.342020i −1.73741 + 1.40762i 0.840226 + 0.840226i −0.357026 + 4.08083i 0.866025 + 0.500000i −0.543142 − 1.49227i −1.95545 + 1.08454i
13.8 0.984808 + 0.173648i 1.42219 + 0.995829i 0.939693 + 0.342020i 1.75371 + 1.38727i 1.22766 + 1.22766i 0.179675 − 2.05370i 0.866025 + 0.500000i 0.00489254 + 0.0134421i 1.48617 + 1.67072i
13.9 0.984808 + 0.173648i 1.49696 + 1.04818i 0.939693 + 0.342020i −0.337481 − 2.21045i 1.29220 + 1.29220i 0.0435029 − 0.497240i 0.866025 + 0.500000i 0.116144 + 0.319104i 0.0514870 − 2.23548i
13.10 0.984808 + 0.173648i 2.71770 + 1.90295i 0.939693 + 0.342020i −2.16097 + 0.574640i 2.34597 + 2.34597i 0.126237 − 1.44290i 0.866025 + 0.500000i 2.73859 + 7.52423i −2.22792 + 0.190662i
57.1 0.984808 − 0.173648i −2.74835 + 1.92441i 0.939693 − 0.342020i 2.05400 − 0.883787i −2.37242 + 2.37242i −0.306526 − 3.50361i 0.866025 − 0.500000i 2.82399 − 7.75885i 1.86933 − 1.22703i
57.2 0.984808 − 0.173648i −2.15928 + 1.51194i 0.939693 − 0.342020i −1.95696 − 1.08182i −1.86393 + 1.86393i −0.00262850 − 0.0300439i 0.866025 − 0.500000i 1.35045 − 3.71033i −2.11508 − 0.725559i
57.3 0.984808 − 0.173648i −1.25779 + 0.880716i 0.939693 − 0.342020i −1.20384 + 1.88435i −1.08575 + 1.08575i −0.0418072 − 0.477859i 0.866025 − 0.500000i −0.219679 + 0.603563i −0.858332 + 2.06477i
57.4 0.984808 − 0.173648i −0.961804 + 0.673462i 0.939693 − 0.342020i −1.64326 − 1.51648i −0.830246 + 0.830246i 0.403339 + 4.61019i 0.866025 − 0.500000i −0.554545 + 1.52360i −1.88163 − 1.20809i
57.5 0.984808 − 0.173648i −0.817959 + 0.572741i 0.939693 − 0.342020i 1.91023 + 1.16234i −0.706077 + 0.706077i 0.132313 + 1.51235i 0.866025 − 0.500000i −0.685035 + 1.88212i 2.08304 + 0.812974i
57.6 0.984808 − 0.173648i −0.439891 + 0.308015i 0.939693 − 0.342020i 0.778763 − 2.09607i −0.379722 + 0.379722i −0.222195 − 2.53970i 0.866025 − 0.500000i −0.927429 + 2.54809i 0.402953 − 2.19946i
57.7 0.984808 − 0.173648i 0.973365 − 0.681558i 0.939693 − 0.342020i −1.73741 − 1.40762i 0.840226 − 0.840226i −0.357026 − 4.08083i 0.866025 − 0.500000i −0.543142 + 1.49227i −1.95545 − 1.08454i
57.8 0.984808 − 0.173648i 1.42219 − 0.995829i 0.939693 − 0.342020i 1.75371 − 1.38727i 1.22766 − 1.22766i 0.179675 + 2.05370i 0.866025 − 0.500000i 0.00489254 − 0.0134421i 1.48617 − 1.67072i
57.9 0.984808 − 0.173648i 1.49696 − 1.04818i 0.939693 − 0.342020i −0.337481 + 2.21045i 1.29220 − 1.29220i 0.0435029 + 0.497240i 0.866025 − 0.500000i 0.116144 − 0.319104i 0.0514870 + 2.23548i
57.10 0.984808 − 0.173648i 2.71770 − 1.90295i 0.939693 − 0.342020i −2.16097 − 0.574640i 2.34597 − 2.34597i 0.126237 + 1.44290i 0.866025 − 0.500000i 2.73859 − 7.52423i −2.22792 − 0.190662i
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.10
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
185.bc even 36 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 370.2.bd.b yes 120
5.c odd 4 1 370.2.ba.b ✓ 120
37.i odd 36 1 370.2.ba.b ✓ 120
185.bc even 36 1 inner 370.2.bd.b yes 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.ba.b ✓ 120 5.c odd 4 1
370.2.ba.b ✓ 120 37.i odd 36 1
370.2.bd.b yes 120 1.a even 1 1 trivial
370.2.bd.b yes 120 185.bc even 36 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{120} - 6 T_{3}^{119} + 18 T_{3}^{118} - 34 T_{3}^{117} + 69 T_{3}^{116} - 336 T_{3}^{115} + \cdots + 17\!\cdots\!36 \) acting on \(S_{2}^{\mathrm{new}}(370, [\chi])\). Copy content Toggle raw display