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Properties

Label 357.2.d.b
Level $357$
Weight $2$
Character orbit 357.d
Analytic conductor $2.851$
Analytic rank $0$
Dimension $22$
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] = [357,2,Mod(188,357)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(357, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("357.188");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 357 = 3 \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 357.d (of order \(2\), degree \(1\), 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: \(2.85065935216\)
Analytic rank: \(0\)
Dimension: \(22\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 22 q - 24 q^{4} + 5 q^{6} - 2 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q - 24 q^{4} + 5 q^{6} - 2 q^{7} - 4 q^{9} - 8 q^{14} - 4 q^{15} + 20 q^{16} + 22 q^{17} + 8 q^{18} - 30 q^{20} - 4 q^{21} - 12 q^{22} - 44 q^{24} + 14 q^{25} - 24 q^{26} + 6 q^{27} + 8 q^{28} + 5 q^{30} + 28 q^{33} + 10 q^{35} - 3 q^{36} - 16 q^{37} + 88 q^{38} - 14 q^{39} - 16 q^{41} + 19 q^{42} - 24 q^{43} - 46 q^{45} + 4 q^{46} - 16 q^{47} + 25 q^{48} + 6 q^{49} + 36 q^{54} - 40 q^{56} - 6 q^{57} + 24 q^{58} + 24 q^{59} - 21 q^{60} - 20 q^{62} - 6 q^{63} - 20 q^{64} - 116 q^{66} + 8 q^{67} - 24 q^{68} + 6 q^{69} + 4 q^{70} - 7 q^{72} + 54 q^{75} + 6 q^{77} + 2 q^{78} + 16 q^{79} + 128 q^{80} - 4 q^{81} + 8 q^{83} + 42 q^{84} - 48 q^{87} + 32 q^{88} - 100 q^{89} + 47 q^{90} + 18 q^{91} + 20 q^{93} + 88 q^{96} - 8 q^{98} + 18 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
188.1 − 2.67583i 1.42596 + 0.983178i −5.16008 2.78395 2.63082 − 3.81563i 1.08035 + 2.41513i 8.45583i 1.06672 + 2.80394i − 7.44937i
188.2 − 2.62830i −0.685367 + 1.59068i −4.90796 0.419096 4.18079 + 1.80135i −2.38080 − 1.15403i 7.64298i −2.06054 − 2.18040i − 1.10151i
188.3 − 2.17326i −0.647242 − 1.60657i −2.72304 3.66218 −3.49150 + 1.40662i 1.14868 − 2.38339i 1.57136i −2.16216 + 2.07968i − 7.95885i
188.4 − 2.04055i −1.36093 + 1.07138i −2.16383 −0.263720 2.18621 + 2.77705i 2.64172 − 0.145918i 0.334296i 0.704276 − 2.91616i 0.538133i
188.5 − 1.99552i 1.72460 − 0.160529i −1.98210 −3.32746 −0.320339 − 3.44146i −2.59132 − 0.533926i − 0.0357248i 2.94846 − 0.553695i 6.64000i
188.6 − 1.84940i −1.59187 − 0.682594i −1.42026 −2.40027 −1.26239 + 2.94401i −1.26533 + 2.32356i − 1.07216i 2.06813 + 2.17321i 4.43904i
188.7 − 1.22862i 0.136369 − 1.72667i 0.490482 −3.00418 −2.12143 − 0.167546i 0.249788 − 2.63393i − 3.05987i −2.96281 − 0.470928i 3.69101i
188.8 − 0.889073i 0.351208 + 1.69607i 1.20955 1.21532 1.50793 − 0.312249i 1.43665 − 2.22172i − 2.85352i −2.75331 + 1.19135i − 1.08051i
188.9 − 0.886197i 0.730545 − 1.57045i 1.21466 2.83371 −1.39173 − 0.647406i −2.45773 + 0.979562i − 2.84882i −1.93261 − 2.29456i − 2.51122i
188.10 − 0.735767i 1.46476 + 0.924372i 1.45865 −2.26067 0.680123 − 1.07773i 2.53573 − 0.755015i − 2.54476i 1.29107 + 2.70798i 1.66333i
188.11 − 0.126768i −1.54802 − 0.776930i 1.98393 0.342053 −0.0984896 + 0.196239i −1.39775 − 2.24640i − 0.505033i 1.79276 + 2.40541i − 0.0433612i
188.12 0.126768i −1.54802 + 0.776930i 1.98393 0.342053 −0.0984896 − 0.196239i −1.39775 + 2.24640i 0.505033i 1.79276 − 2.40541i 0.0433612i
188.13 0.735767i 1.46476 − 0.924372i 1.45865 −2.26067 0.680123 + 1.07773i 2.53573 + 0.755015i 2.54476i 1.29107 − 2.70798i − 1.66333i
188.14 0.886197i 0.730545 + 1.57045i 1.21466 2.83371 −1.39173 + 0.647406i −2.45773 − 0.979562i 2.84882i −1.93261 + 2.29456i 2.51122i
188.15 0.889073i 0.351208 − 1.69607i 1.20955 1.21532 1.50793 + 0.312249i 1.43665 + 2.22172i 2.85352i −2.75331 − 1.19135i 1.08051i
188.16 1.22862i 0.136369 + 1.72667i 0.490482 −3.00418 −2.12143 + 0.167546i 0.249788 + 2.63393i 3.05987i −2.96281 + 0.470928i − 3.69101i
188.17 1.84940i −1.59187 + 0.682594i −1.42026 −2.40027 −1.26239 − 2.94401i −1.26533 − 2.32356i 1.07216i 2.06813 − 2.17321i − 4.43904i
188.18 1.99552i 1.72460 + 0.160529i −1.98210 −3.32746 −0.320339 + 3.44146i −2.59132 + 0.533926i 0.0357248i 2.94846 + 0.553695i − 6.64000i
188.19 2.04055i −1.36093 − 1.07138i −2.16383 −0.263720 2.18621 − 2.77705i 2.64172 + 0.145918i − 0.334296i 0.704276 + 2.91616i − 0.538133i
188.20 2.17326i −0.647242 + 1.60657i −2.72304 3.66218 −3.49150 − 1.40662i 1.14868 + 2.38339i − 1.57136i −2.16216 − 2.07968i 7.95885i
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 188.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 357.2.d.b yes 22
3.b odd 2 1 357.2.d.a ✓ 22
7.b odd 2 1 357.2.d.a ✓ 22
21.c even 2 1 inner 357.2.d.b yes 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
357.2.d.a ✓ 22 3.b odd 2 1
357.2.d.a ✓ 22 7.b odd 2 1
357.2.d.b yes 22 1.a even 1 1 trivial
357.2.d.b yes 22 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{11} - 31 T_{5}^{9} - 2 T_{5}^{8} + 338 T_{5}^{7} + 18 T_{5}^{6} - 1515 T_{5}^{5} + 100 T_{5}^{4} + \cdots + 72 \) acting on \(S_{2}^{\mathrm{new}}(357, [\chi])\). Copy content Toggle raw display