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Properties

Label 608.2.u.a
Level $608$
Weight $2$
Character orbit 608.u
Analytic conductor $4.855$
Analytic rank $0$
Dimension $312$
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] = [608,2,Mod(75,608)]
 
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([4, 5, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("608.75");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 608 = 2^{5} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 608.u (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: \(312\)
Relative dimension: \(78\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 312 q - 8 q^{4} - 8 q^{5} - 8 q^{6} - 8 q^{7} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 312 q - 8 q^{4} - 8 q^{5} - 8 q^{6} - 8 q^{7} - 8 q^{9} - 8 q^{11} - 8 q^{16} - 4 q^{19} - 8 q^{20} - 8 q^{23} - 24 q^{24} - 8 q^{25} - 48 q^{26} + 32 q^{28} + 8 q^{30} - 56 q^{35} - 8 q^{36} - 52 q^{38} - 8 q^{39} - 88 q^{42} - 8 q^{43} + 80 q^{44} + 16 q^{45} - 16 q^{47} - 32 q^{54} - 8 q^{55} - 4 q^{57} - 8 q^{58} - 72 q^{61} - 48 q^{62} - 80 q^{64} - 120 q^{66} - 8 q^{68} - 8 q^{73} - 8 q^{74} + 60 q^{76} - 8 q^{77} + 128 q^{80} - 88 q^{82} - 8 q^{83} + 32 q^{85} + 104 q^{87} - 8 q^{92} - 32 q^{93} + 96 q^{96} - 48 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
75.1 −1.41174 − 0.0836446i 1.47722 − 0.611884i 1.98601 + 0.236168i 1.61508 − 3.89916i −2.13662 + 0.740258i −1.84018 + 1.84018i −2.78397 − 0.499527i −0.313550 + 0.313550i −2.60622 + 5.36950i
75.2 −1.40604 + 0.151790i −1.14536 + 0.474423i 1.95392 − 0.426846i −1.37693 + 3.32420i 1.53841 − 0.840912i −3.28684 + 3.28684i −2.68251 + 0.896748i −1.03455 + 1.03455i 1.43144 − 4.88298i
75.3 −1.40431 + 0.167102i 0.0712862 − 0.0295277i 1.94415 − 0.469325i −0.710707 + 1.71580i −0.0951735 + 0.0533780i 3.39997 − 3.39997i −2.65176 + 0.983948i −2.11711 + 2.11711i 0.711337 − 2.52827i
75.4 −1.40386 − 0.170773i 1.38615 − 0.574163i 1.94167 + 0.479485i −0.365261 + 0.881818i −2.04402 + 0.569330i −1.07988 + 1.07988i −2.64396 − 1.00472i −0.529566 + 0.529566i 0.663368 − 1.17558i
75.5 −1.40316 + 0.176449i −1.96235 + 0.812830i 1.93773 − 0.495175i 1.13366 − 2.73691i 2.61007 − 1.48679i −0.635317 + 0.635317i −2.63158 + 1.03672i 1.06879 − 1.06879i −1.10779 + 4.04036i
75.6 −1.39471 + 0.234033i 3.07725 − 1.27464i 1.89046 − 0.652819i −0.0739347 + 0.178494i −3.99357 + 2.49793i 1.01289 − 1.01289i −2.48387 + 1.35293i 5.72343 − 5.72343i 0.0613442 − 0.266251i
75.7 −1.37655 − 0.324210i −0.182692 + 0.0756733i 1.78978 + 0.892582i 0.557106 − 1.34497i 0.276018 − 0.0449376i 1.37960 − 1.37960i −2.17433 − 1.80895i −2.09367 + 2.09367i −1.20294 + 1.67080i
75.8 −1.35671 − 0.399170i −2.48164 + 1.02793i 1.68133 + 1.08312i −1.02548 + 2.47573i 3.77719 − 0.404007i 0.525750 − 0.525750i −1.84873 − 2.14061i 2.98060 − 2.98060i 2.37952 − 2.94950i
75.9 −1.33823 + 0.457308i −1.25392 + 0.519390i 1.58174 − 1.22397i 0.374985 − 0.905295i 1.44052 − 1.26849i −0.701678 + 0.701678i −1.55701 + 2.36130i −0.818774 + 0.818774i −0.0878196 + 1.38298i
75.10 −1.22935 − 0.699064i −2.55712 + 1.05920i 1.02262 + 1.71879i 0.790516 − 1.90847i 3.88406 + 0.485467i −3.57633 + 3.57633i −0.0556168 − 2.82788i 3.29567 − 3.29567i −2.30597 + 1.79357i
75.11 −1.22438 − 0.707737i −0.0242344 + 0.0100382i 0.998216 + 1.73308i −0.610366 + 1.47355i 0.0367765 + 0.00486099i −0.609102 + 0.609102i 0.00436918 − 2.82842i −2.12083 + 2.12083i 1.79021 − 1.37221i
75.12 −1.22344 + 0.709358i −2.32996 + 0.965103i 0.993623 − 1.73572i −1.00287 + 2.42115i 2.16597 − 2.83353i 1.87199 − 1.87199i 0.0156030 + 2.82838i 2.37599 − 2.37599i −0.490503 − 3.67353i
75.13 −1.22075 + 0.713979i 1.37809 − 0.570822i 0.980469 − 1.74318i 1.47554 − 3.56228i −1.27475 + 1.68076i 2.93428 − 2.93428i 0.0476853 + 2.82803i −0.548036 + 0.548036i 0.742118 + 5.40216i
75.14 −1.21606 − 0.721946i 2.70346 − 1.11981i 0.957587 + 1.75586i 0.412246 − 0.995251i −4.09600 − 0.590001i 2.15423 − 2.15423i 0.103154 − 2.82655i 3.93340 − 3.93340i −1.21983 + 0.912662i
75.15 −1.18004 + 0.779420i 1.67980 − 0.695794i 0.785010 − 1.83950i 0.315547 − 0.761798i −1.43992 + 2.13033i −3.15837 + 3.15837i 0.507395 + 2.78254i 0.216264 − 0.216264i 0.221401 + 1.14490i
75.16 −1.15791 + 0.811942i 2.22453 − 0.921432i 0.681502 − 1.88031i −0.560773 + 1.35383i −1.82766 + 2.87312i −0.859362 + 0.859362i 0.737583 + 2.73056i 1.97819 − 1.97819i −0.449904 − 2.02292i
75.17 −1.13505 − 0.843595i −1.02267 + 0.423604i 0.576694 + 1.91505i 1.25329 − 3.02570i 1.51814 + 0.381907i 2.66524 − 2.66524i 0.960950 − 2.66018i −1.25490 + 1.25490i −3.97502 + 2.37707i
75.18 −1.10027 + 0.888487i −1.10037 + 0.455789i 0.421180 − 1.95515i 0.297137 − 0.717353i 0.805742 − 1.47916i 0.662472 − 0.662472i 1.27371 + 2.52540i −1.11824 + 1.11824i 0.310428 + 1.05328i
75.19 −1.07728 − 0.916221i 1.87406 − 0.776261i 0.321077 + 1.97406i −1.38614 + 3.34644i −2.73012 − 0.880801i 2.86665 − 2.86665i 1.46278 − 2.42080i 0.788198 − 0.788198i 4.55934 − 2.33505i
75.20 −1.06771 + 0.927355i 0.343014 − 0.142081i 0.280024 − 1.98030i −1.33603 + 3.22546i −0.234481 + 0.469798i −0.114753 + 0.114753i 1.53746 + 2.37407i −2.02385 + 2.02385i −1.56465 − 4.68284i
See next 80 embeddings (of 312 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 75.78
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 inner
32.h odd 8 1 inner
608.u 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.u.a ✓ 312
19.b odd 2 1 inner 608.2.u.a ✓ 312
32.h odd 8 1 inner 608.2.u.a ✓ 312
608.u even 8 1 inner 608.2.u.a ✓ 312
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
608.2.u.a ✓ 312 1.a even 1 1 trivial
608.2.u.a ✓ 312 19.b odd 2 1 inner
608.2.u.a ✓ 312 32.h odd 8 1 inner
608.2.u.a ✓ 312 608.u even 8 1 inner

Hecke kernels

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