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Properties

Label 595.2.bg.a
Level $595$
Weight $2$
Character orbit 595.bg
Analytic conductor $4.751$
Analytic rank $0$
Dimension $224$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [595,2,Mod(134,595)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(595, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([4, 0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("595.134");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 595 = 5 \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 595.bg (of order \(8\), 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: \(4.75109892027\)
Analytic rank: \(0\)
Dimension: \(224\)
Relative dimension: \(56\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 224 q - 16 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 224 q - 16 q^{6} + 24 q^{10} - 272 q^{16} + 32 q^{19} + 40 q^{20} - 16 q^{24} - 48 q^{25} - 48 q^{26} - 64 q^{29} + 32 q^{31} - 32 q^{34} + 64 q^{36} - 8 q^{39} + 56 q^{40} + 96 q^{41} - 32 q^{44} - 80 q^{45} - 16 q^{46} - 40 q^{50} - 8 q^{51} - 32 q^{54} + 64 q^{59} - 104 q^{60} - 64 q^{61} - 80 q^{65} + 208 q^{66} + 32 q^{69} + 16 q^{70} + 48 q^{74} + 32 q^{75} + 64 q^{76} + 80 q^{79} - 56 q^{80} + 96 q^{84} - 64 q^{85} + 112 q^{86} + 56 q^{90} + 24 q^{91} - 96 q^{94} + 96 q^{95} - 240 q^{96} + 24 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
134.1 −1.95991 + 1.95991i −1.34864 + 0.558627i − 5.68246i 1.06267 − 1.96742i 1.54836 − 3.73807i 0.382683 − 0.923880i 7.21726 + 7.21726i −0.614543 + 0.614543i 1.77321 + 5.93869i
134.2 −1.93823 + 1.93823i 2.77469 − 1.14932i − 5.51347i 0.0222950 − 2.23596i −3.15036 + 7.60563i −0.382683 + 0.923880i 6.80991 + 6.80991i 4.25668 − 4.25668i 4.29058 + 4.37701i
134.3 −1.81535 + 1.81535i −0.770146 + 0.319005i − 4.59096i −0.0745411 + 2.23483i 0.818977 − 1.97719i 0.382683 − 0.923880i 4.70349 + 4.70349i −1.62996 + 1.62996i −3.92166 − 4.19230i
134.4 −1.78066 + 1.78066i −2.00155 + 0.829070i − 4.34153i −2.16158 − 0.572321i 2.08780 − 5.04039i −0.382683 + 0.923880i 4.16948 + 4.16948i 1.19754 − 1.19754i 4.86817 − 2.82995i
134.5 −1.77164 + 1.77164i 1.62280 − 0.672186i − 4.27740i −2.15712 + 0.588922i −1.68415 + 4.06589i 0.382683 − 0.923880i 4.03474 + 4.03474i 0.0603286 − 0.0603286i 2.77828 − 4.86500i
134.6 −1.74383 + 1.74383i 0.232342 − 0.0962390i − 4.08191i 2.13680 + 0.658865i −0.237340 + 0.572990i −0.382683 + 0.923880i 3.63050 + 3.63050i −2.07660 + 2.07660i −4.87517 + 2.57727i
134.7 −1.58130 + 1.58130i −2.28947 + 0.948329i − 3.00103i −0.242037 + 2.22293i 2.12075 − 5.11993i −0.382683 + 0.923880i 1.58292 + 1.58292i 2.22102 − 2.22102i −3.13239 − 3.89786i
134.8 −1.53964 + 1.53964i 0.999030 − 0.413812i − 2.74099i −1.15145 − 1.91681i −0.901026 + 2.17527i 0.382683 − 0.923880i 1.14086 + 1.14086i −1.29450 + 1.29450i 4.72402 + 1.17839i
134.9 −1.37414 + 1.37414i −2.44498 + 1.01274i − 1.77650i 1.85728 + 1.24519i 1.96809 − 4.75138i 0.382683 − 0.923880i −0.307123 − 0.307123i 2.83096 − 2.83096i −4.26322 + 0.841094i
134.10 −1.36848 + 1.36848i 1.85857 − 0.769847i − 1.74550i −2.07441 + 0.834764i −1.48991 + 3.59696i −0.382683 + 0.923880i −0.348278 − 0.348278i 0.740316 − 0.740316i 1.69643 − 3.98116i
134.11 −1.36485 + 1.36485i −2.48015 + 1.02731i − 1.72562i −1.41367 − 1.73250i 1.98290 − 4.78715i 0.382683 − 0.923880i −0.374483 − 0.374483i 2.97445 − 2.97445i 4.29404 + 0.435158i
134.12 −1.33247 + 1.33247i 2.92568 − 1.21185i − 1.55097i −0.510602 + 2.17699i −2.28362 + 5.51315i −0.382683 + 0.923880i −0.598319 − 0.598319i 4.96967 − 4.96967i −2.22042 − 3.58115i
134.13 −1.30161 + 1.30161i −0.644794 + 0.267082i − 1.38837i 0.785115 − 2.09370i 0.491632 − 1.18691i −0.382683 + 0.923880i −0.796102 − 0.796102i −1.77689 + 1.77689i 1.70327 + 3.74710i
134.14 −1.21117 + 1.21117i 0.788614 − 0.326654i − 0.933872i −0.920991 − 2.03759i −0.559512 + 1.35078i −0.382683 + 0.923880i −1.29126 − 1.29126i −1.60611 + 1.60611i 3.58335 + 1.35239i
134.15 −1.15576 + 1.15576i 2.23402 − 0.925360i − 0.671563i 1.17257 + 1.90396i −1.51249 + 3.65148i 0.382683 − 0.923880i −1.53535 − 1.53535i 2.01322 − 2.01322i −3.55574 − 0.845312i
134.16 −0.999323 + 0.999323i −0.902843 + 0.373970i 0.00270520i 2.05999 − 0.869743i 0.528515 − 1.27595i 0.382683 − 0.923880i −2.00135 − 2.00135i −1.44605 + 1.44605i −1.18944 + 2.92775i
134.17 −0.820674 + 0.820674i 3.10367 − 1.28558i 0.652989i −0.903122 − 2.04557i −1.49205 + 3.60214i 0.382683 − 0.923880i −2.17724 − 2.17724i 5.85870 − 5.85870i 2.41992 + 0.937580i
134.18 −0.791704 + 0.791704i 0.438136 − 0.181482i 0.746411i 0.407598 + 2.19861i −0.203194 + 0.490554i 0.382683 − 0.923880i −2.17434 − 2.17434i −1.96229 + 1.96229i −2.06334 − 1.41795i
134.19 −0.692416 + 0.692416i −1.68292 + 0.697090i 1.04112i 2.09914 + 0.770475i 0.682607 − 1.64796i −0.382683 + 0.923880i −2.10572 − 2.10572i 0.224975 − 0.224975i −1.98696 + 0.919985i
134.20 −0.627033 + 0.627033i 2.24631 − 0.930452i 1.21366i 2.10166 − 0.763562i −0.825086 + 1.99193i −0.382683 + 0.923880i −2.01507 − 2.01507i 2.05885 − 2.05885i −0.839031 + 1.79659i
See next 80 embeddings (of 224 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 134.56
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
17.d even 8 1 inner
85.m even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 595.2.bg.a ✓ 224
5.b even 2 1 inner 595.2.bg.a ✓ 224
17.d even 8 1 inner 595.2.bg.a ✓ 224
85.m even 8 1 inner 595.2.bg.a ✓ 224
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
595.2.bg.a ✓ 224 1.a even 1 1 trivial
595.2.bg.a ✓ 224 5.b even 2 1 inner
595.2.bg.a ✓ 224 17.d even 8 1 inner
595.2.bg.a ✓ 224 85.m even 8 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(595, [\chi])\).