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Properties

Label 370.2.q.f
Level $370$
Weight $2$
Character orbit 370.q
Analytic conductor $2.954$
Analytic rank $0$
Dimension $32$
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(97,370)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(370, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("370.97");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.q (of order \(12\), degree \(4\), 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: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 16 q^{2} - 2 q^{3} - 16 q^{4} + 6 q^{5} + 8 q^{6} - 10 q^{7} - 32 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 16 q^{2} - 2 q^{3} - 16 q^{4} + 6 q^{5} + 8 q^{6} - 10 q^{7} - 32 q^{8} + 12 q^{9} + 6 q^{10} + 10 q^{12} + 10 q^{13} - 8 q^{14} - 8 q^{15} - 16 q^{16} - 12 q^{17} + 12 q^{18} + 2 q^{19} - 54 q^{21} + 6 q^{22} + 12 q^{23} + 2 q^{24} - 16 q^{25} + 20 q^{26} + 40 q^{27} + 2 q^{28} + 6 q^{29} + 8 q^{30} - 4 q^{31} + 16 q^{32} + 26 q^{33} - 12 q^{34} - 12 q^{35} + 20 q^{37} - 26 q^{38} - 58 q^{39} - 6 q^{40} + 18 q^{41} - 54 q^{42} - 32 q^{43} + 6 q^{44} + 56 q^{45} + 6 q^{46} + 18 q^{47} - 8 q^{48} - 12 q^{49} - 14 q^{50} - 4 q^{51} + 10 q^{52} - 24 q^{53} + 20 q^{54} - 32 q^{55} + 10 q^{56} + 8 q^{57} + 36 q^{58} + 42 q^{59} + 16 q^{60} - 46 q^{61} - 14 q^{62} + 32 q^{64} - 18 q^{65} + 4 q^{66} + 50 q^{67} - 66 q^{69} + 12 q^{70} - 12 q^{71} - 12 q^{72} - 28 q^{73} + 16 q^{74} - 20 q^{75} - 28 q^{76} + 12 q^{77} - 26 q^{78} + 38 q^{79} - 6 q^{80} + 56 q^{81} - 8 q^{85} - 16 q^{86} + 18 q^{87} + 18 q^{89} + 4 q^{90} + 4 q^{91} - 6 q^{92} + 32 q^{93} + 30 q^{94} + 96 q^{95} - 10 q^{96} - 12 q^{98} - 26 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1 0.500000 − 0.866025i −0.875179 + 3.26621i −0.500000 − 0.866025i −0.0954522 + 2.23403i 2.39103 + 2.39103i −0.371808 + 1.38761i −1.00000 −7.30413 − 4.21704i 1.88700 + 1.19968i
97.2 0.500000 − 0.866025i −0.676906 + 2.52625i −0.500000 − 0.866025i 1.00515 − 1.99742i 1.84934 + 1.84934i 1.06481 − 3.97392i −1.00000 −3.32566 − 1.92007i −1.22724 − 1.86919i
97.3 0.500000 − 0.866025i −0.453882 + 1.69391i −0.500000 − 0.866025i 1.94207 − 1.10831i 1.24003 + 1.24003i −1.25240 + 4.67400i −1.00000 −0.0652541 − 0.0376745i 0.0112097 − 2.23604i
97.4 0.500000 − 0.866025i −0.418603 + 1.56225i −0.500000 − 0.866025i −2.09367 − 0.785216i 1.14364 + 1.14364i −0.298699 + 1.11476i −1.00000 0.332689 + 0.192078i −1.72685 + 1.42056i
97.5 0.500000 − 0.866025i −0.0267201 + 0.0997208i −0.500000 − 0.866025i 0.206483 + 2.22651i 0.0730007 + 0.0730007i −0.571833 + 2.13411i −1.00000 2.58885 + 1.49467i 2.03146 + 0.934438i
97.6 0.500000 − 0.866025i 0.0936541 − 0.349522i −0.500000 − 0.866025i −0.693098 − 2.12594i −0.255868 − 0.255868i 0.0100267 − 0.0374200i −1.00000 2.48468 + 1.43453i −2.18767 − 0.462729i
97.7 0.500000 − 0.866025i 0.304778 − 1.13745i −0.500000 − 0.866025i 2.21062 − 0.336359i −0.832670 − 0.832670i 0.993767 − 3.70879i −1.00000 1.39718 + 0.806661i 0.814017 − 2.08264i
97.8 0.500000 − 0.866025i 0.686833 − 2.56329i −0.500000 − 0.866025i −1.84814 + 1.25873i −1.87646 − 1.87646i 0.524210 − 1.95638i −1.00000 −3.50066 − 2.02111i 0.166021 + 2.22990i
103.1 0.500000 + 0.866025i −0.875179 − 3.26621i −0.500000 + 0.866025i −0.0954522 − 2.23403i 2.39103 − 2.39103i −0.371808 − 1.38761i −1.00000 −7.30413 + 4.21704i 1.88700 − 1.19968i
103.2 0.500000 + 0.866025i −0.676906 − 2.52625i −0.500000 + 0.866025i 1.00515 + 1.99742i 1.84934 − 1.84934i 1.06481 + 3.97392i −1.00000 −3.32566 + 1.92007i −1.22724 + 1.86919i
103.3 0.500000 + 0.866025i −0.453882 − 1.69391i −0.500000 + 0.866025i 1.94207 + 1.10831i 1.24003 − 1.24003i −1.25240 − 4.67400i −1.00000 −0.0652541 + 0.0376745i 0.0112097 + 2.23604i
103.4 0.500000 + 0.866025i −0.418603 − 1.56225i −0.500000 + 0.866025i −2.09367 + 0.785216i 1.14364 − 1.14364i −0.298699 − 1.11476i −1.00000 0.332689 − 0.192078i −1.72685 − 1.42056i
103.5 0.500000 + 0.866025i −0.0267201 − 0.0997208i −0.500000 + 0.866025i 0.206483 − 2.22651i 0.0730007 − 0.0730007i −0.571833 − 2.13411i −1.00000 2.58885 − 1.49467i 2.03146 − 0.934438i
103.6 0.500000 + 0.866025i 0.0936541 + 0.349522i −0.500000 + 0.866025i −0.693098 + 2.12594i −0.255868 + 0.255868i 0.0100267 + 0.0374200i −1.00000 2.48468 − 1.43453i −2.18767 + 0.462729i
103.7 0.500000 + 0.866025i 0.304778 + 1.13745i −0.500000 + 0.866025i 2.21062 + 0.336359i −0.832670 + 0.832670i 0.993767 + 3.70879i −1.00000 1.39718 − 0.806661i 0.814017 + 2.08264i
103.8 0.500000 + 0.866025i 0.686833 + 2.56329i −0.500000 + 0.866025i −1.84814 − 1.25873i −1.87646 + 1.87646i 0.524210 + 1.95638i −1.00000 −3.50066 + 2.02111i 0.166021 − 2.22990i
267.1 0.500000 + 0.866025i −2.85838 + 0.765901i −0.500000 + 0.866025i 2.19958 − 0.402277i −2.09248 − 2.09248i 3.43627 − 0.920746i −1.00000 4.98565 − 2.87847i 1.44817 + 1.70376i
267.2 0.500000 + 0.866025i −2.35325 + 0.630551i −0.500000 + 0.866025i −0.0214737 + 2.23596i −1.72270 − 1.72270i −1.71779 + 0.460281i −1.00000 2.54210 − 1.46768i −1.94714 + 1.09939i
267.3 0.500000 + 0.866025i −1.76952 + 0.474141i −0.500000 + 0.866025i −0.686657 − 2.12803i −1.29538 − 1.29538i 1.57494 − 0.422005i −1.00000 0.308315 − 0.178006i 1.49960 − 1.65868i
267.4 0.500000 + 0.866025i −0.243526 + 0.0652525i −0.500000 + 0.866025i 2.10570 + 0.752339i −0.178273 − 0.178273i −3.85162 + 1.03204i −1.00000 −2.54303 + 1.46822i 0.401306 + 2.19976i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.8
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
185.p even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 370.2.q.f ✓ 32
5.c odd 4 1 370.2.r.f yes 32
37.g odd 12 1 370.2.r.f yes 32
185.p even 12 1 inner 370.2.q.f ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.q.f ✓ 32 1.a even 1 1 trivial
370.2.q.f ✓ 32 185.p even 12 1 inner
370.2.r.f yes 32 5.c odd 4 1
370.2.r.f yes 32 37.g odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{32} + 2 T_{3}^{31} - 4 T_{3}^{30} - 40 T_{3}^{29} - 192 T_{3}^{28} + 16 T_{3}^{27} + 1600 T_{3}^{26} + \cdots + 1024 \) acting on \(S_{2}^{\mathrm{new}}(370, [\chi])\). Copy content Toggle raw display