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Properties

Label 760.2.f.b
Level $760$
Weight $2$
Character orbit 760.f
Analytic conductor $6.069$
Analytic rank $0$
Dimension $44$
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] = [760,2,Mod(381,760)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(760, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("760.381");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 760 = 2^{3} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 760.f (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: \(6.06863055362\)
Analytic rank: \(0\)
Dimension: \(44\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 44 q + 2 q^{2} - 2 q^{4} - 6 q^{6} + 4 q^{7} + 8 q^{8} - 60 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 44 q + 2 q^{2} - 2 q^{4} - 6 q^{6} + 4 q^{7} + 8 q^{8} - 60 q^{9} + 4 q^{12} + 4 q^{14} - 6 q^{16} + 24 q^{17} - 14 q^{18} - 4 q^{20} - 4 q^{22} + 4 q^{23} + 2 q^{24} - 44 q^{25} + 18 q^{26} - 14 q^{28} + 12 q^{32} + 40 q^{33} + 32 q^{34} - 28 q^{36} - 24 q^{39} + 6 q^{40} - 32 q^{41} + 46 q^{42} - 36 q^{44} - 16 q^{46} - 20 q^{47} + 64 q^{48} + 108 q^{49} - 2 q^{50} - 24 q^{52} - 26 q^{54} + 8 q^{55} + 60 q^{56} + 2 q^{58} - 14 q^{60} - 72 q^{62} + 20 q^{63} + 22 q^{64} + 12 q^{65} + 88 q^{66} - 62 q^{68} - 18 q^{70} - 8 q^{71} + 48 q^{72} - 88 q^{73} + 84 q^{74} - 4 q^{76} - 116 q^{78} + 40 q^{79} + 8 q^{80} + 116 q^{81} + 44 q^{82} - 108 q^{84} - 88 q^{86} + 48 q^{87} + 76 q^{88} - 64 q^{89} + 30 q^{90} - 62 q^{92} - 24 q^{94} - 44 q^{95} + 118 q^{96} + 116 q^{97} + 78 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
381.1 −1.38212 − 0.299590i − 2.26423i 1.82049 + 0.828136i − 1.00000i −0.678340 + 3.12943i −1.28214 −2.26803 − 1.68998i −2.12673 −0.299590 + 1.38212i
381.2 −1.38212 + 0.299590i 2.26423i 1.82049 − 0.828136i 1.00000i −0.678340 − 3.12943i −1.28214 −2.26803 + 1.68998i −2.12673 −0.299590 − 1.38212i
381.3 −1.30630 − 0.541819i 2.28177i 1.41287 + 1.41556i 1.00000i 1.23630 − 2.98068i 2.16697 −1.07866 − 2.61467i −2.20646 0.541819 − 1.30630i
381.4 −1.30630 + 0.541819i − 2.28177i 1.41287 − 1.41556i − 1.00000i 1.23630 + 2.98068i 2.16697 −1.07866 + 2.61467i −2.20646 0.541819 + 1.30630i
381.5 −1.29073 − 0.577950i 1.40980i 1.33195 + 1.49195i − 1.00000i 0.814797 − 1.81967i −1.07761 −0.856907 − 2.69550i 1.01245 −0.577950 + 1.29073i
381.6 −1.29073 + 0.577950i − 1.40980i 1.33195 − 1.49195i 1.00000i 0.814797 + 1.81967i −1.07761 −0.856907 + 2.69550i 1.01245 −0.577950 − 1.29073i
381.7 −1.25358 − 0.654621i − 2.81164i 1.14294 + 1.64124i 1.00000i −1.84056 + 3.52463i −3.31936 −0.358379 − 2.80563i −4.90534 0.654621 − 1.25358i
381.8 −1.25358 + 0.654621i 2.81164i 1.14294 − 1.64124i − 1.00000i −1.84056 − 3.52463i −3.31936 −0.358379 + 2.80563i −4.90534 0.654621 + 1.25358i
381.9 −1.01061 − 0.989281i − 0.530865i 0.0426469 + 1.99955i 1.00000i −0.525175 + 0.536495i 1.59876 1.93501 − 2.06294i 2.71818 0.989281 − 1.01061i
381.10 −1.01061 + 0.989281i 0.530865i 0.0426469 − 1.99955i − 1.00000i −0.525175 − 0.536495i 1.59876 1.93501 + 2.06294i 2.71818 0.989281 + 1.01061i
381.11 −0.862980 − 1.12039i − 3.06281i −0.510530 + 1.93374i − 1.00000i −3.43153 + 2.64314i −2.81229 2.60712 − 1.09679i −6.38080 −1.12039 + 0.862980i
381.12 −0.862980 + 1.12039i 3.06281i −0.510530 − 1.93374i 1.00000i −3.43153 − 2.64314i −2.81229 2.60712 + 1.09679i −6.38080 −1.12039 − 0.862980i
381.13 −0.750213 − 1.19882i 1.80124i −0.874362 + 1.79875i − 1.00000i 2.15937 − 1.35131i 4.97691 2.81234 − 0.301237i −0.244448 −1.19882 + 0.750213i
381.14 −0.750213 + 1.19882i − 1.80124i −0.874362 − 1.79875i 1.00000i 2.15937 + 1.35131i 4.97691 2.81234 + 0.301237i −0.244448 −1.19882 − 0.750213i
381.15 −0.676154 − 1.24210i 1.70163i −1.08563 + 1.67970i 1.00000i 2.11360 − 1.15057i 1.91794 2.82042 + 0.212725i 0.104446 1.24210 − 0.676154i
381.16 −0.676154 + 1.24210i − 1.70163i −1.08563 − 1.67970i − 1.00000i 2.11360 + 1.15057i 1.91794 2.82042 − 0.212725i 0.104446 1.24210 + 0.676154i
381.17 −0.477081 − 1.33131i − 1.22456i −1.54479 + 1.27029i 1.00000i −1.63027 + 0.584214i −4.59099 2.42814 + 1.45057i 1.50045 1.33131 − 0.477081i
381.18 −0.477081 + 1.33131i 1.22456i −1.54479 − 1.27029i − 1.00000i −1.63027 − 0.584214i −4.59099 2.42814 − 1.45057i 1.50045 1.33131 + 0.477081i
381.19 −0.188358 − 1.40161i − 0.0766253i −1.92904 + 0.528011i − 1.00000i −0.107399 + 0.0144330i 1.59941 1.10342 + 2.60432i 2.99413 −1.40161 + 0.188358i
381.20 −0.188358 + 1.40161i 0.0766253i −1.92904 − 0.528011i 1.00000i −0.107399 − 0.0144330i 1.59941 1.10342 − 2.60432i 2.99413 −1.40161 − 0.188358i
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 381.44
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 760.2.f.b ✓ 44
4.b odd 2 1 3040.2.f.b 44
8.b even 2 1 inner 760.2.f.b ✓ 44
8.d odd 2 1 3040.2.f.b 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.f.b ✓ 44 1.a even 1 1 trivial
760.2.f.b ✓ 44 8.b even 2 1 inner
3040.2.f.b 44 4.b odd 2 1
3040.2.f.b 44 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{44} + 96 T_{3}^{42} + 4246 T_{3}^{40} + 114860 T_{3}^{38} + 2127757 T_{3}^{36} + 28642108 T_{3}^{34} + \cdots + 8573184 \) acting on \(S_{2}^{\mathrm{new}}(760, [\chi])\). Copy content Toggle raw display