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Properties

Label 370.2.ba.b
Level $370$
Weight $2$
Character orbit 370.ba
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(17,370)]
 
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([9, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("370.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.ba (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.

\(\operatorname{Tr}(f)(q) = \) \( 120 q - 6 q^{3} - 6 q^{5} + 60 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 120 q - 6 q^{3} - 6 q^{5} + 60 q^{8} + 12 q^{10} - 36 q^{11} - 6 q^{12} - 6 q^{13} + 12 q^{14} - 24 q^{15} - 12 q^{19} + 12 q^{20} + 42 q^{21} + 6 q^{22} + 6 q^{24} + 6 q^{25} + 6 q^{26} + 6 q^{27} - 36 q^{30} - 18 q^{33} - 30 q^{35} + 12 q^{37} - 48 q^{38} - 12 q^{40} - 30 q^{41} + 42 q^{42} + 6 q^{44} - 30 q^{45} - 6 q^{46} + 12 q^{47} - 60 q^{49} - 48 q^{50} + 12 q^{51} - 6 q^{52} + 12 q^{53} + 18 q^{54} + 36 q^{57} + 6 q^{58} - 24 q^{59} + 54 q^{60} - 72 q^{61} + 6 q^{62} + 96 q^{63} - 60 q^{64} - 18 q^{65} - 42 q^{67} - 96 q^{69} - 12 q^{70} - 90 q^{73} + 24 q^{74} - 60 q^{75} + 18 q^{76} + 6 q^{77} + 24 q^{78} - 18 q^{79} + 6 q^{80} - 108 q^{81} - 36 q^{82} + 48 q^{83} + 36 q^{85} + 24 q^{86} + 108 q^{87} - 36 q^{88} + 54 q^{89} - 6 q^{90} + 42 q^{91} - 12 q^{92} - 12 q^{93} - 18 q^{94} - 198 q^{95} - 12 q^{96} - 72 q^{97} + 48 q^{98} + 42 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 −0.173648 − 0.984808i −2.77903 + 1.94590i −0.939693 + 0.342020i −0.881644 + 2.05492i 2.39891 + 2.39891i 0.205490 + 2.34876i 0.500000 + 0.866025i 2.91044 − 7.99636i 2.17680 + 0.511416i
17.2 −0.173648 − 0.984808i −1.47946 + 1.03593i −0.939693 + 0.342020i 1.60255 − 1.55943i 1.27710 + 1.27710i −0.151990 − 1.73726i 0.500000 + 0.866025i 0.0895920 − 0.246152i −1.81402 − 1.30741i
17.3 −0.173648 − 0.984808i −1.28340 + 0.898643i −0.939693 + 0.342020i −2.22534 − 0.218791i 1.10785 + 1.10785i −0.0448989 − 0.513197i 0.500000 + 0.866025i −0.186516 + 0.512448i 0.170958 + 2.22952i
17.4 −0.173648 − 0.984808i −0.809902 + 0.567099i −0.939693 + 0.342020i 0.168535 + 2.22971i 0.699122 + 0.699122i −0.288350 − 3.29585i 0.500000 + 0.866025i −0.691721 + 1.90049i 2.16657 − 0.553159i
17.5 −0.173648 − 0.984808i −0.124693 + 0.0873112i −0.939693 + 0.342020i 2.15046 + 0.612816i 0.107638 + 0.107638i 0.230601 + 2.63579i 0.500000 + 0.866025i −1.01814 + 2.79730i 0.230083 − 2.22420i
17.6 −0.173648 − 0.984808i 0.810570 − 0.567567i −0.939693 + 0.342020i −2.22842 + 0.184728i −0.699698 − 0.699698i 0.406884 + 4.65071i 0.500000 + 0.866025i −0.691169 + 1.89897i 0.568884 + 2.16249i
17.7 −0.173648 − 0.984808i 1.21087 − 0.847863i −0.939693 + 0.342020i −0.415110 − 2.19720i −1.04525 − 1.04525i −0.203881 − 2.33037i 0.500000 + 0.866025i −0.278717 + 0.765769i −2.09174 + 0.790343i
17.8 −0.173648 − 0.984808i 1.32542 − 0.928067i −0.939693 + 0.342020i 0.101493 + 2.23376i −1.14412 − 1.14412i 0.111534 + 1.27484i 0.500000 + 0.866025i −0.130639 + 0.358928i 2.18220 − 0.487840i
17.9 −0.173648 − 0.984808i 2.30212 − 1.61196i −0.939693 + 0.342020i −2.13489 + 0.665008i −1.98723 − 1.98723i −0.396149 − 4.52800i 0.500000 + 0.866025i 1.67527 − 4.60277i 1.02563 + 1.98698i
17.10 −0.173648 − 0.984808i 2.60236 − 1.82219i −0.939693 + 0.342020i 1.31462 − 1.80881i −2.24641 − 2.24641i 0.299132 + 3.41909i 0.500000 + 0.866025i 2.42584 − 6.66494i −2.00961 − 0.980548i
87.1 −0.173648 + 0.984808i −1.92441 + 2.74835i −0.939693 − 0.342020i 2.17626 − 0.513687i −2.37242 − 2.37242i −3.50361 − 0.306526i 0.500000 − 0.866025i −2.82399 − 7.75885i 0.127979 + 2.23240i
87.2 −0.173648 + 0.984808i −1.51194 + 2.15928i −0.939693 − 0.342020i −1.73937 − 1.40520i −1.86393 − 1.86393i −0.0300439 − 0.00262850i 0.500000 − 0.866025i −1.35045 − 3.71033i 1.68589 − 1.46893i
87.3 −0.173648 + 0.984808i −0.880716 + 1.25779i −0.939693 − 0.342020i −1.51276 + 1.64668i −1.08575 − 1.08575i −0.477859 − 0.0418072i 0.500000 − 0.866025i 0.219679 + 0.603563i −1.35898 − 1.77572i
87.4 −0.173648 + 0.984808i −0.673462 + 0.961804i −0.939693 − 0.342020i −1.35496 − 1.77879i −0.830246 − 0.830246i 4.61019 + 0.403339i 0.500000 − 0.866025i 0.554545 + 1.52360i 1.98705 − 1.02549i
87.5 −0.173648 + 0.984808i −0.572741 + 0.817959i −0.939693 − 0.342020i 1.67937 + 1.47639i −0.706077 − 0.706077i 1.51235 + 0.132313i 0.500000 − 0.866025i 0.685035 + 1.88212i −1.74558 + 1.39748i
87.6 −0.173648 + 0.984808i −0.308015 + 0.439891i −0.939693 − 0.342020i 1.13091 − 1.92900i −0.379722 − 0.379722i −2.53970 − 0.222195i 0.500000 − 0.866025i 0.927429 + 2.54809i 1.70331 + 1.44870i
87.7 −0.173648 + 0.984808i 0.681558 − 0.973365i −0.939693 − 0.342020i −1.46658 − 1.68794i 0.840226 + 0.840226i −4.08083 − 0.357026i 0.500000 − 0.866025i 0.543142 + 1.49227i 1.91696 − 1.15120i
87.8 −0.173648 + 0.984808i 0.995829 − 1.42219i −0.939693 − 0.342020i 1.96796 − 1.06167i 1.22766 + 1.22766i 2.05370 + 0.179675i 0.500000 − 0.866025i −0.00489254 − 0.0134421i 0.703804 + 2.12242i
87.9 −0.173648 + 0.984808i 1.04818 − 1.49696i −0.939693 − 0.342020i −0.716196 + 2.11827i 1.29220 + 1.29220i 0.497240 + 0.0435029i 0.500000 − 0.866025i −0.116144 − 0.319104i −1.96172 − 1.07315i
87.10 −0.173648 + 0.984808i 1.90295 − 2.71770i −0.939693 − 0.342020i −2.02835 − 0.941159i 2.34597 + 2.34597i 1.44290 + 0.126237i 0.500000 − 0.866025i −2.73859 − 7.52423i 1.27908 − 1.83411i
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.10
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{120} + 6 T_{3}^{119} + 18 T_{3}^{118} + 34 T_{3}^{117} + 69 T_{3}^{116} + 348 T_{3}^{115} + \cdots + 17\!\cdots\!36 \) acting on \(S_{2}^{\mathrm{new}}(370, [\chi])\). Copy content Toggle raw display