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Properties

Label 608.2.v.a
Level $608$
Weight $2$
Character orbit 608.v
Analytic conductor $4.855$
Analytic rank $0$
Dimension $288$
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] = [608,2,Mod(77,608)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(608, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 7, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("608.77");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 608 = 2^{5} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 608.v (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: \(4.85490444289\)
Analytic rank: \(0\)
Dimension: \(288\)
Relative dimension: \(72\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 288 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 288 q - 32 q^{12} - 32 q^{14} - 40 q^{16} - 40 q^{18} - 32 q^{20} - 24 q^{22} - 16 q^{23} + 40 q^{24} - 48 q^{27} + 80 q^{30} + 56 q^{36} - 48 q^{39} - 64 q^{40} + 40 q^{42} - 16 q^{43} + 8 q^{44} - 104 q^{48} - 104 q^{50} + 48 q^{52} - 32 q^{53} + 64 q^{55} + 56 q^{56} + 8 q^{58} + 64 q^{59} + 80 q^{63} - 32 q^{66} + 80 q^{67} + 24 q^{68} - 72 q^{70} + 64 q^{71} - 48 q^{72} - 32 q^{74} + 64 q^{75} - 32 q^{77} - 80 q^{78} + 32 q^{80} + 80 q^{82} - 80 q^{83} - 40 q^{84} + 104 q^{86} - 88 q^{88} - 120 q^{90} - 96 q^{91} + 72 q^{92} - 48 q^{94} - 64 q^{95} - 112 q^{96} - 64 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
77.1 −1.40874 − 0.124258i −0.941483 + 2.27294i 1.96912 + 0.350097i 3.07366 − 1.27315i 1.60874 − 3.08501i −1.31033 + 1.31033i −2.73048 − 0.737877i −2.15855 − 2.15855i −4.48820 + 1.41162i
77.2 −1.40612 + 0.151091i 0.243701 − 0.588346i 1.95434 − 0.424904i 2.00967 − 0.832433i −0.253779 + 0.864105i 2.22704 − 2.22704i −2.68384 + 0.892749i 1.83456 + 1.83456i −2.70007 + 1.47414i
77.3 −1.40373 − 0.171839i −0.728168 + 1.75795i 1.94094 + 0.482434i −2.88942 + 1.19684i 1.32424 − 2.34257i 2.61523 − 2.61523i −2.64167 − 1.01074i −0.438847 − 0.438847i 4.26165 − 1.18353i
77.4 −1.40257 − 0.181130i 0.624289 − 1.50717i 1.93438 + 0.508095i 3.62445 − 1.50130i −1.14860 + 2.00082i −2.35896 + 2.35896i −2.62107 − 1.06301i 0.239506 + 0.239506i −5.35547 + 1.44917i
77.5 −1.38573 − 0.282425i −0.519263 + 1.25361i 1.84047 + 0.782726i 0.936806 − 0.388038i 1.07361 − 1.59051i 1.26790 − 1.26790i −2.32933 − 1.60444i 0.819412 + 0.819412i −1.40775 + 0.273137i
77.6 −1.38421 + 0.289741i −0.0719190 + 0.173628i 1.83210 − 0.802128i −1.34738 + 0.558105i 0.0492442 − 0.261176i −2.66791 + 2.66791i −2.30361 + 1.64115i 2.09635 + 2.09635i 1.70336 − 1.16293i
77.7 −1.35375 + 0.409112i 1.11053 − 2.68105i 1.66525 − 1.10767i 2.25659 − 0.934711i −0.406521 + 4.08379i 0.0254620 − 0.0254620i −1.80117 + 2.18078i −3.83343 − 3.83343i −2.67245 + 2.18856i
77.8 −1.35114 + 0.417633i 1.28418 − 3.10029i 1.65117 − 1.12856i −3.49151 + 1.44623i −0.440328 + 4.72525i −1.92387 + 1.92387i −1.75963 + 2.21443i −5.84135 − 5.84135i 4.11353 − 3.41223i
77.9 −1.33124 + 0.477278i 0.0312328 − 0.0754026i 1.54441 − 1.27074i −0.899731 + 0.372681i −0.00559042 + 0.115286i 0.0398824 − 0.0398824i −1.44949 + 2.42878i 2.11661 + 2.11661i 1.01989 − 0.925550i
77.10 −1.31256 − 0.526486i −0.608594 + 1.46928i 1.44562 + 1.38209i −2.17974 + 0.902877i 1.57237 − 1.60809i −2.27930 + 2.27930i −1.16982 − 2.57517i 0.332937 + 0.332937i 3.33639 − 0.0374780i
77.11 −1.30428 − 0.546681i 0.643529 − 1.55362i 1.40228 + 1.42605i −0.394530 + 0.163420i −1.68867 + 1.67454i −2.18717 + 2.18717i −1.04937 − 2.62656i 0.121727 + 0.121727i 0.603915 + 0.00253738i
77.12 −1.30009 − 0.556560i 1.01788 − 2.45737i 1.38048 + 1.44716i −0.498812 + 0.206615i −2.69101 + 2.62830i 2.06448 − 2.06448i −0.989324 − 2.64976i −2.88129 − 2.88129i 0.763496 + 0.00900054i
77.13 −1.10999 + 0.876313i −0.894164 + 2.15870i 0.464151 − 1.94540i −3.92557 + 1.62602i −0.899187 − 3.17970i −1.35942 + 1.35942i 1.18957 + 2.56611i −1.73914 − 1.73914i 2.93243 − 5.24489i
77.14 −1.10395 + 0.883912i 0.453125 − 1.09394i 0.437400 − 1.95158i −2.85861 + 1.18408i 0.466721 + 1.60818i 3.23352 − 3.23352i 1.24216 + 2.54107i 1.12994 + 1.12994i 2.10914 − 3.83392i
77.15 −1.03720 + 0.961361i −0.132400 + 0.319643i 0.151572 − 1.99425i 3.78637 − 1.56837i −0.169966 − 0.458819i 1.56439 − 1.56439i 1.75998 + 2.21415i 2.03668 + 2.03668i −2.41946 + 5.26678i
77.16 −1.03348 − 0.965360i 0.149647 − 0.361280i 0.136162 + 1.99536i 2.10296 − 0.871073i −0.503422 + 0.228912i 1.99434 − 1.99434i 1.78552 − 2.19361i 2.01319 + 2.01319i −3.01426 − 1.12987i
77.17 −1.03289 − 0.965996i −1.30279 + 3.14522i 0.133704 + 1.99553i −1.13763 + 0.471224i 4.38390 − 1.99016i 0.675439 − 0.675439i 1.78957 − 2.19031i −6.07381 − 6.07381i 1.63025 + 0.612230i
77.18 −0.989454 − 1.01044i −0.0959338 + 0.231605i −0.0419608 + 1.99956i −2.76602 + 1.14572i 0.328944 − 0.132227i 2.00945 − 2.00945i 2.06194 − 1.93607i 2.07688 + 2.07688i 3.89453 + 1.66125i
77.19 −0.966072 + 1.03281i −0.824523 + 1.99057i −0.133411 − 1.99555i 0.514088 − 0.212942i −1.25935 − 2.77462i −0.990048 + 0.990048i 2.18991 + 1.79005i −1.16123 − 1.16123i −0.276716 + 0.736675i
77.20 −0.947840 − 1.04957i 0.444287 − 1.07260i −0.203201 + 1.98965i −3.67221 + 1.52108i −1.54689 + 0.550345i −1.34350 + 1.34350i 2.28088 − 1.67260i 1.16823 + 1.16823i 5.07715 + 2.41251i
See next 80 embeddings (of 288 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 77.72
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
32.g even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 608.2.v.a ✓ 288
32.g even 8 1 inner 608.2.v.a ✓ 288
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
608.2.v.a ✓ 288 1.a even 1 1 trivial
608.2.v.a ✓ 288 32.g even 8 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(608, [\chi])\).