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Properties

Label 370.2.r.f
Level $370$
Weight $2$
Character orbit 370.r
Analytic conductor $2.954$
Analytic rank $0$
Dimension $32$
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(23,370)]
 
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([9, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("370.23");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.r (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 algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 10 q^{3} + 16 q^{4} + 6 q^{5} + 8 q^{6} + 2 q^{7} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 10 q^{3} + 16 q^{4} + 6 q^{5} + 8 q^{6} + 2 q^{7} - 12 q^{9} + 6 q^{10} - 2 q^{12} + 12 q^{13} + 8 q^{14} - 8 q^{15} - 16 q^{16} - 6 q^{17} - 24 q^{18} - 2 q^{19} + 6 q^{20} - 54 q^{21} + 12 q^{22} - 2 q^{24} - 14 q^{25} + 20 q^{26} - 40 q^{27} + 10 q^{28} - 6 q^{29} - 8 q^{30} - 4 q^{31} - 22 q^{33} + 12 q^{34} + 24 q^{37} - 26 q^{38} + 58 q^{39} + 18 q^{41} - 2 q^{42} - 6 q^{44} - 18 q^{45} + 6 q^{46} + 18 q^{47} + 8 q^{48} + 12 q^{49} + 24 q^{50} - 4 q^{51} + 12 q^{52} - 20 q^{54} + 42 q^{55} + 10 q^{56} + 36 q^{57} + 30 q^{58} - 42 q^{59} - 16 q^{60} - 46 q^{61} - 10 q^{62} - 32 q^{64} - 18 q^{65} + 4 q^{66} + 46 q^{67} - 12 q^{68} + 66 q^{69} + 12 q^{70} - 12 q^{71} + 24 q^{72} + 28 q^{73} - 16 q^{74} - 20 q^{75} - 28 q^{76} + 24 q^{77} - 58 q^{78} - 38 q^{79} + 56 q^{81} - 48 q^{82} + 12 q^{83} + 8 q^{85} - 16 q^{86} - 2 q^{87} + 24 q^{88} - 18 q^{89} + 52 q^{90} + 4 q^{91} - 12 q^{92} + 36 q^{93} - 30 q^{94} + 30 q^{95} - 10 q^{96} - 12 q^{97} - 16 q^{98} + 26 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1 −0.866025 − 0.500000i −3.26621 − 0.875179i 0.500000 + 0.866025i −1.03435 + 1.98245i 2.39103 + 2.39103i 1.38761 + 0.371808i − 1.00000i 7.30413 + 4.21704i 1.88700 − 1.19968i
23.2 −0.866025 − 0.500000i −2.52625 − 0.676906i 0.500000 + 0.866025i 0.128227 − 2.23239i 1.84934 + 1.84934i −3.97392 − 1.06481i − 1.00000i 3.32566 + 1.92007i −1.22724 + 1.86919i
23.3 −0.866025 − 0.500000i −1.69391 − 0.453882i 0.500000 + 0.866025i −1.12773 − 1.93086i 1.24003 + 1.24003i 4.67400 + 1.25240i − 1.00000i 0.0652541 + 0.0376745i 0.0112097 + 2.23604i
23.4 −0.866025 − 0.500000i −1.56225 − 0.418603i 0.500000 + 0.866025i 2.20578 + 0.366816i 1.14364 + 1.14364i 1.11476 + 0.298699i − 1.00000i −0.332689 − 0.192078i −1.72685 − 1.42056i
23.5 −0.866025 − 0.500000i −0.0997208 − 0.0267201i 0.500000 + 0.866025i −1.29208 + 1.82498i 0.0730007 + 0.0730007i 2.13411 + 0.571833i − 1.00000i −2.58885 − 1.49467i 2.03146 − 0.934438i
23.6 −0.866025 − 0.500000i 0.349522 + 0.0936541i 0.500000 + 0.866025i 1.66321 − 1.49457i −0.255868 − 0.255868i −0.0374200 − 0.0100267i − 1.00000i −2.48468 − 1.43453i −2.18767 + 0.462729i
23.7 −0.866025 − 0.500000i 1.13745 + 0.304778i 0.500000 + 0.866025i −1.74628 − 1.39661i −0.832670 − 0.832670i −3.70879 − 0.993767i − 1.00000i −1.39718 − 0.806661i 0.814017 + 2.08264i
23.8 −0.866025 − 0.500000i 2.56329 + 0.686833i 0.500000 + 0.866025i 0.971170 + 2.01416i −1.87646 − 1.87646i −1.95638 − 0.524210i − 1.00000i 3.50066 + 2.02111i 0.166021 − 2.22990i
177.1 −0.866025 + 0.500000i −3.26621 + 0.875179i 0.500000 − 0.866025i −1.03435 − 1.98245i 2.39103 − 2.39103i 1.38761 − 0.371808i 1.00000i 7.30413 − 4.21704i 1.88700 + 1.19968i
177.2 −0.866025 + 0.500000i −2.52625 + 0.676906i 0.500000 − 0.866025i 0.128227 + 2.23239i 1.84934 − 1.84934i −3.97392 + 1.06481i 1.00000i 3.32566 − 1.92007i −1.22724 − 1.86919i
177.3 −0.866025 + 0.500000i −1.69391 + 0.453882i 0.500000 − 0.866025i −1.12773 + 1.93086i 1.24003 − 1.24003i 4.67400 − 1.25240i 1.00000i 0.0652541 − 0.0376745i 0.0112097 − 2.23604i
177.4 −0.866025 + 0.500000i −1.56225 + 0.418603i 0.500000 − 0.866025i 2.20578 − 0.366816i 1.14364 − 1.14364i 1.11476 − 0.298699i 1.00000i −0.332689 + 0.192078i −1.72685 + 1.42056i
177.5 −0.866025 + 0.500000i −0.0997208 + 0.0267201i 0.500000 − 0.866025i −1.29208 − 1.82498i 0.0730007 − 0.0730007i 2.13411 − 0.571833i 1.00000i −2.58885 + 1.49467i 2.03146 + 0.934438i
177.6 −0.866025 + 0.500000i 0.349522 − 0.0936541i 0.500000 − 0.866025i 1.66321 + 1.49457i −0.255868 + 0.255868i −0.0374200 + 0.0100267i 1.00000i −2.48468 + 1.43453i −2.18767 − 0.462729i
177.7 −0.866025 + 0.500000i 1.13745 − 0.304778i 0.500000 − 0.866025i −1.74628 + 1.39661i −0.832670 + 0.832670i −3.70879 + 0.993767i 1.00000i −1.39718 + 0.806661i 0.814017 − 2.08264i
177.8 −0.866025 + 0.500000i 2.56329 − 0.686833i 0.500000 − 0.866025i 0.971170 − 2.01416i −1.87646 + 1.87646i −1.95638 + 0.524210i 1.00000i 3.50066 − 2.02111i 0.166021 + 2.22990i
193.1 0.866025 − 0.500000i −0.765901 − 2.85838i 0.500000 − 0.866025i 2.10603 − 0.751410i −2.09248 − 2.09248i −0.920746 − 3.43627i − 1.00000i −4.98565 + 2.87847i 1.44817 − 1.70376i
193.2 0.866025 − 0.500000i −0.630551 − 2.35325i 0.500000 − 0.866025i −1.13658 − 1.92567i −1.72270 − 1.72270i 0.460281 + 1.71779i − 1.00000i −2.54210 + 1.46768i −1.94714 − 1.09939i
193.3 0.866025 − 0.500000i −0.474141 − 1.76952i 0.500000 − 0.866025i 0.469352 + 2.18625i −1.29538 − 1.29538i −0.422005 − 1.57494i − 1.00000i −0.308315 + 0.178006i 1.49960 + 1.65868i
193.4 0.866025 − 0.500000i −0.0652525 − 0.243526i 0.500000 − 0.866025i 1.44742 − 1.70440i −0.178273 − 0.178273i 1.03204 + 3.85162i − 1.00000i 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 23.8
Significant digits:
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Inner twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{32} + 10 T_{3}^{31} + 56 T_{3}^{30} + 256 T_{3}^{29} + 888 T_{3}^{28} + 2204 T_{3}^{27} + \cdots + 1024 \) acting on \(S_{2}^{\mathrm{new}}(370, [\chi])\). Copy content Toggle raw display