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Properties

Label 816.2.bs.a
Level $816$
Weight $2$
Character orbit 816.bs
Analytic conductor $6.516$
Analytic rank $0$
Dimension $560$
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] = [816,2,Mod(155,816)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(816, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([4, 2, 4, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("816.155");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 816 = 2^{4} \cdot 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 816.bs (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: \(6.51579280494\)
Analytic rank: \(0\)
Dimension: \(560\)
Relative dimension: \(140\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 560 q - 4 q^{3} - 12 q^{6} - 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 560 q - 4 q^{3} - 12 q^{6} - 16 q^{7} - 24 q^{10} - 24 q^{12} - 16 q^{16} - 8 q^{18} - 16 q^{22} - 12 q^{24} - 4 q^{27} + 16 q^{28} - 32 q^{31} - 16 q^{33} - 8 q^{34} - 20 q^{36} - 8 q^{37} - 8 q^{39} - 56 q^{40} - 32 q^{42} + 16 q^{45} + 48 q^{46} + 48 q^{48} - 16 q^{49} - 44 q^{51} - 16 q^{52} - 48 q^{54} + 24 q^{57} + 8 q^{58} - 72 q^{60} - 40 q^{61} + 80 q^{63} - 108 q^{66} - 16 q^{67} - 8 q^{69} - 56 q^{70} + 40 q^{72} - 36 q^{75} + 56 q^{76} - 48 q^{78} - 8 q^{82} - 8 q^{84} - 8 q^{85} - 8 q^{87} - 120 q^{88} + 64 q^{90} + 96 q^{91} - 32 q^{94} - 16 q^{96} - 16 q^{97} - 40 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
155.1 −1.41421 + 0.00299064i −1.72061 − 0.198751i 1.99998 − 0.00845879i 3.33076 + 1.37965i 2.43390 + 0.275930i −1.14350 + 2.76066i −2.82837 + 0.0179437i 2.92100 + 0.683946i −4.71452 − 1.94115i
155.2 −1.41411 + 0.0171549i 1.73165 + 0.0372076i 1.99941 − 0.0485177i −1.73967 − 0.720596i −2.44938 − 0.0229094i −0.259070 + 0.625450i −2.82655 + 0.102909i 2.99723 + 0.128861i 2.47245 + 0.989158i
155.3 −1.40802 − 0.132217i 1.57631 − 0.717816i 1.96504 + 0.372329i −0.900205 − 0.372877i −2.31438 + 0.802284i 1.72535 − 4.16537i −2.71758 − 0.784058i 1.96948 − 2.26299i 1.21821 + 0.644041i
155.4 −1.40313 + 0.176704i −0.162188 + 1.72444i 1.93755 − 0.495878i 0.178402 + 0.0738965i −0.0771456 − 2.44827i 0.534459 − 1.29030i −2.63101 + 1.03816i −2.94739 − 0.559366i −0.263379 − 0.0721620i
155.5 −1.40171 − 0.187658i 1.10091 − 1.33716i 1.92957 + 0.526084i −1.61306 − 0.668152i −1.79409 + 1.66771i −1.53343 + 3.70202i −2.60597 − 1.09952i −0.575980 − 2.94419i 2.13566 + 1.23926i
155.6 −1.39860 − 0.209561i −0.359646 − 1.69430i 1.91217 + 0.586184i 3.54234 + 1.46729i 0.147942 + 2.44502i −0.146006 + 0.352490i −2.55152 − 1.22055i −2.74131 + 1.21870i −4.64684 − 2.79448i
155.7 −1.39691 + 0.220572i −1.72889 − 0.104575i 1.90270 − 0.616237i −2.95322 − 1.22326i 2.43817 − 0.235264i −1.01364 + 2.44714i −2.52196 + 1.28051i 2.97813 + 0.361597i 4.39518 + 1.05739i
155.8 −1.38841 + 0.268925i −1.32857 − 1.11126i 1.85536 − 0.746756i 0.510924 + 0.211632i 2.14345 + 1.18559i 0.535190 − 1.29206i −2.37517 + 1.53576i 0.530218 + 2.95277i −0.766284 − 0.156431i
155.9 −1.38803 − 0.270891i −1.33281 − 1.10617i 1.85324 + 0.752007i −2.60702 − 1.07986i 1.55032 + 1.89644i 1.72779 − 4.17126i −2.36863 − 1.54583i 0.552766 + 2.94864i 3.32609 + 2.20510i
155.10 −1.38229 − 0.298792i −0.733767 + 1.56894i 1.82145 + 0.826034i −3.31839 − 1.37452i 1.48307 − 1.94949i −0.352147 + 0.850158i −2.27095 − 1.68605i −1.92317 − 2.30248i 4.17628 + 2.89149i
155.11 −1.37275 + 0.339952i 1.32705 + 1.11307i 1.76887 − 0.933335i 0.638599 + 0.264516i −2.20010 − 1.07683i −1.70880 + 4.12540i −2.11092 + 1.88256i 0.522134 + 2.95421i −0.966557 − 0.146021i
155.12 −1.36688 − 0.362814i 0.880032 + 1.49183i 1.73673 + 0.991847i 2.89472 + 1.19903i −0.661644 − 2.35844i 1.04743 − 2.52872i −2.01405 − 1.98585i −1.45109 + 2.62571i −3.52171 − 2.68918i
155.13 −1.36515 + 0.369294i −1.25977 + 1.18869i 1.72724 − 1.00828i 0.899303 + 0.372504i 1.28080 − 2.08796i −0.714806 + 1.72569i −1.98559 + 2.01431i 0.174048 − 2.99495i −1.36524 − 0.176415i
155.14 −1.35450 + 0.406618i 0.959860 − 1.44176i 1.66932 − 1.10152i 2.05158 + 0.849793i −0.713883 + 2.34315i 1.22683 − 2.96182i −1.81320 + 2.17079i −1.15734 − 2.76777i −3.12440 − 0.316833i
155.15 −1.35393 − 0.408505i 0.867148 + 1.49935i 1.66625 + 1.10617i 0.548548 + 0.227216i −0.561566 − 2.38425i −0.892230 + 2.15403i −1.80411 − 2.17835i −1.49611 + 2.60032i −0.649876 − 0.531719i
155.16 −1.34319 − 0.442529i −0.0225716 − 1.73190i 1.60834 + 1.18880i −0.0616062 − 0.0255181i −0.736099 + 2.33627i 0.428647 − 1.03485i −1.63423 − 2.30853i −2.99898 + 0.0781838i 0.0714565 + 0.0615383i
155.17 −1.32272 + 0.500422i 1.35682 − 1.07659i 1.49916 − 1.32383i 2.57032 + 1.06466i −1.25594 + 2.10300i −1.20678 + 2.91343i −1.32048 + 2.50127i 0.681913 − 2.92147i −3.93259 − 0.121999i
155.18 −1.31498 − 0.520422i −1.62382 + 0.602680i 1.45832 + 1.36869i −0.476036 − 0.197180i 2.44893 + 0.0525606i 0.559564 − 1.35091i −1.20536 − 2.55873i 2.27355 − 1.95728i 0.523358 + 0.507027i
155.19 −1.30751 − 0.538911i 1.73205 − 0.00385689i 1.41915 + 1.40926i 2.52228 + 1.04476i −2.26674 − 0.928376i 0.142024 − 0.342877i −1.09608 − 2.60741i 2.99997 − 0.0133606i −2.73487 − 2.72532i
155.20 −1.27357 + 0.614829i −0.131020 − 1.72709i 1.24397 − 1.56606i −3.29615 − 1.36531i 1.22873 + 2.11902i 0.172277 − 0.415913i −0.621427 + 2.75932i −2.96567 + 0.452565i 5.03731 − 0.287748i
See next 80 embeddings (of 560 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 155.140
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
272.x odd 8 1 inner
816.bs even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 816.2.bs.a ✓ 560
3.b odd 2 1 inner 816.2.bs.a ✓ 560
16.f odd 4 1 816.2.bw.a yes 560
17.d even 8 1 816.2.bw.a yes 560
48.k even 4 1 816.2.bw.a yes 560
51.g odd 8 1 816.2.bw.a yes 560
272.x odd 8 1 inner 816.2.bs.a ✓ 560
816.bs even 8 1 inner 816.2.bs.a ✓ 560
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
816.2.bs.a ✓ 560 1.a even 1 1 trivial
816.2.bs.a ✓ 560 3.b odd 2 1 inner
816.2.bs.a ✓ 560 272.x odd 8 1 inner
816.2.bs.a ✓ 560 816.bs even 8 1 inner
816.2.bw.a yes 560 16.f odd 4 1
816.2.bw.a yes 560 17.d even 8 1
816.2.bw.a yes 560 48.k even 4 1
816.2.bw.a yes 560 51.g odd 8 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(816, [\chi])\).