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Properties

Label 760.2.w.c
Level $760$
Weight $2$
Character orbit 760.w
Analytic conductor $6.069$
Analytic rank $0$
Dimension $104$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [760,2,Mod(267,760)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(760, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 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("760.267");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 760 = 2^{3} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 760.w (of order \(4\), degree \(2\), 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: \(104\)
Relative dimension: \(52\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 104 q - 8 q^{3} - 8 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 104 q - 8 q^{3} - 8 q^{6} + 4 q^{10} + 16 q^{11} - 18 q^{12} - 16 q^{16} + 20 q^{22} + 8 q^{25} - 24 q^{26} + 16 q^{27} + 16 q^{28} - 44 q^{30} + 50 q^{32} + 32 q^{33} - 36 q^{35} + 136 q^{36} + 4 q^{40} + 8 q^{41} + 20 q^{42} - 44 q^{43} - 56 q^{46} + 38 q^{48} + 24 q^{50} - 16 q^{51} - 94 q^{52} - 28 q^{56} + 8 q^{57} + 8 q^{58} - 2 q^{60} - 20 q^{62} + 44 q^{65} + 52 q^{66} - 40 q^{67} - 56 q^{68} - 32 q^{70} + 88 q^{72} - 48 q^{73} + 56 q^{75} + 8 q^{76} + 52 q^{78} - 80 q^{80} - 152 q^{81} - 100 q^{82} - 44 q^{83} - 104 q^{86} + 20 q^{88} + 72 q^{90} - 64 q^{91} - 64 q^{92} - 52 q^{96} + 20 q^{97} - 100 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
267.1 −1.40565 − 0.155404i 2.13203 − 2.13203i 1.95170 + 0.436886i −1.55346 + 1.60834i −3.32821 + 2.66556i 0.758554 − 0.758554i −2.67551 − 0.917409i − 6.09111i 2.43357 − 2.01934i
267.2 −1.38362 − 0.292576i 0.685526 − 0.685526i 1.82880 + 0.809627i 2.17301 + 0.527280i −1.14907 + 0.747938i 1.03632 − 1.03632i −2.29348 − 1.65528i 2.06011i −2.85235 − 1.36532i
267.3 −1.37178 + 0.343844i 0.929890 − 0.929890i 1.76354 − 0.943354i −1.47116 − 1.68395i −0.955864 + 1.59534i 3.32714 − 3.32714i −2.09482 + 1.90045i 1.27061i 2.59712 + 1.80415i
267.4 −1.34678 + 0.431504i −2.17446 + 2.17446i 1.62761 − 1.16228i 1.96091 − 1.07464i 1.99022 − 3.86680i 0.483709 − 0.483709i −1.69050 + 2.26765i − 6.45655i −2.17719 + 2.29344i
267.5 −1.34262 − 0.444279i −1.69762 + 1.69762i 1.60523 + 1.19299i 0.143240 + 2.23148i 3.03347 − 1.52503i 3.45743 − 3.45743i −1.62519 − 2.31490i − 2.76384i 0.799082 − 3.05965i
267.6 −1.30931 + 0.534525i 1.64962 − 1.64962i 1.42856 − 1.39972i 1.08792 + 1.95357i −1.27809 + 3.04162i −1.60504 + 1.60504i −1.12225 + 2.59626i − 2.44250i −2.46865 − 1.97630i
267.7 −1.29709 + 0.563528i −1.05307 + 1.05307i 1.36487 − 1.46189i −1.60536 − 1.55654i 0.772487 − 1.95935i −1.06078 + 1.06078i −0.946544 + 2.66534i 0.782105i 2.95945 + 1.11430i
267.8 −1.29605 − 0.565915i −0.899454 + 0.899454i 1.35948 + 1.46691i −0.710484 − 2.12019i 1.67475 − 0.656720i 0.152896 − 0.152896i −0.931806 − 2.67053i 1.38197i −0.279027 + 3.14994i
267.9 −1.24129 − 0.677637i 0.0295384 − 0.0295384i 1.08162 + 1.68229i −0.945240 + 2.02646i −0.0566821 + 0.0166495i −3.50400 + 3.50400i −0.202618 − 2.82116i 2.99825i 2.54652 − 1.87490i
267.10 −1.18702 + 0.768758i −0.696739 + 0.696739i 0.818024 − 1.82506i 2.11169 − 0.735369i 0.291418 − 1.36267i −3.48083 + 3.48083i 0.432019 + 2.79524i 2.02911i −1.94129 + 2.49627i
267.11 −1.14719 + 0.827016i 0.0947206 − 0.0947206i 0.632090 − 1.89749i 0.772974 + 2.09822i −0.0303271 + 0.186998i 2.71510 − 2.71510i 0.844126 + 2.69953i 2.98206i −2.62201 − 1.76779i
267.12 −1.08661 − 0.905136i −1.87646 + 1.87646i 0.361456 + 1.96707i 2.04208 − 0.911004i 3.73743 − 0.340533i −0.531458 + 0.531458i 1.38770 − 2.46461i − 4.04218i −3.04353 − 0.858448i
267.13 −1.03366 − 0.965164i 1.76063 − 1.76063i 0.136915 + 1.99531i −2.15452 − 0.598366i −3.51920 + 0.120599i −2.14797 + 2.14797i 1.78428 − 2.19462i − 3.19965i 1.64952 + 2.69798i
267.14 −0.965164 − 1.03366i 1.76063 − 1.76063i −0.136915 + 1.99531i 2.15452 + 0.598366i −3.51920 − 0.120599i 2.14797 − 2.14797i 2.19462 − 1.78428i − 3.19965i −1.46096 − 2.80457i
267.15 −0.905136 − 1.08661i −1.87646 + 1.87646i −0.361456 + 1.96707i −2.04208 + 0.911004i 3.73743 + 0.340533i 0.531458 − 0.531458i 2.46461 − 1.38770i − 4.04218i 2.83827 + 1.39436i
267.16 −0.897815 + 1.09267i 1.75587 − 1.75587i −0.387856 − 1.96203i −0.899297 − 2.04726i 0.342141 + 3.49504i 1.30028 − 1.30028i 2.49208 + 1.33774i − 3.16619i 3.04438 + 0.855424i
267.17 −0.881214 + 1.10610i −0.667159 + 0.667159i −0.446923 − 1.94943i −2.22363 − 0.235482i −0.150036 − 1.32586i −0.840214 + 0.840214i 2.55010 + 1.22352i 2.10980i 2.21997 − 2.25206i
267.18 −0.677637 − 1.24129i 0.0295384 − 0.0295384i −1.08162 + 1.68229i 0.945240 − 2.02646i −0.0566821 − 0.0166495i 3.50400 − 3.50400i 2.82116 + 0.202618i 2.99825i −3.15595 + 0.199882i
267.19 −0.598568 + 1.28129i −1.10517 + 1.10517i −1.28343 − 1.53388i 1.15323 − 1.91574i −0.754530 − 2.07757i 1.04961 − 1.04961i 2.73358 − 0.726323i 0.557193i 1.76434 + 2.62433i
267.20 −0.565915 − 1.29605i −0.899454 + 0.899454i −1.35948 + 1.46691i 0.710484 + 2.12019i 1.67475 + 0.656720i −0.152896 + 0.152896i 2.67053 + 0.931806i 1.38197i 2.34580 − 2.12067i
See next 80 embeddings (of 104 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 267.52
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
8.d odd 2 1 inner
40.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 760.2.w.c ✓ 104
5.c odd 4 1 inner 760.2.w.c ✓ 104
8.d odd 2 1 inner 760.2.w.c ✓ 104
40.k even 4 1 inner 760.2.w.c ✓ 104
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.w.c ✓ 104 1.a even 1 1 trivial
760.2.w.c ✓ 104 5.c odd 4 1 inner
760.2.w.c ✓ 104 8.d odd 2 1 inner
760.2.w.c ✓ 104 40.k even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(760, [\chi])\):

\( T_{3}^{52} + 4 T_{3}^{51} + 8 T_{3}^{50} + 320 T_{3}^{48} + 1276 T_{3}^{47} + 2544 T_{3}^{46} + \cdots + 102400 \) Copy content Toggle raw display
\( T_{7}^{104} + 3304 T_{7}^{100} + 4690210 T_{7}^{96} + 3747630204 T_{7}^{92} + 1859385007855 T_{7}^{88} + \cdots + 46\!\cdots\!56 \) Copy content Toggle raw display