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Properties

Label 680.2.co.a
Level $680$
Weight $2$
Character orbit 680.co
Analytic conductor $5.430$
Analytic rank $0$
Dimension $832$
Inner twists $8$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [680,2,Mod(99,680)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(680, base_ring=CyclotomicField(16)) chi = DirichletCharacter(H, H._module([8, 8, 8, 9])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("680.99"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 680 = 2^{3} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 680.co (of order \(16\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.42982733745\)
Analytic rank: \(0\)
Dimension: \(832\)
Relative dimension: \(104\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 832 q - 16 q^{4} - 16 q^{6} - 32 q^{9} - 8 q^{10} - 32 q^{11} - 16 q^{14} - 32 q^{19} - 8 q^{20} - 32 q^{24} - 16 q^{25} - 48 q^{26} + 56 q^{30} + 48 q^{34} - 32 q^{35} + 48 q^{36} - 72 q^{40} - 32 q^{41}+ \cdots - 128 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1 −1.41418 + 0.0101729i −0.640662 − 0.127436i 1.99979 − 0.0287725i 1.56955 + 1.59264i 0.907306 + 0.173699i −3.42068 − 2.28563i −2.82777 + 0.0610331i −2.37743 − 0.984764i −2.23583 − 2.23631i
99.2 −1.41120 + 0.0922559i −2.87525 − 0.571923i 1.98298 − 0.260383i −1.43338 − 1.71622i 4.11032 + 0.541840i −0.212560 − 0.142028i −2.77436 + 0.550394i 5.16834 + 2.14080i 2.18112 + 2.28970i
99.3 −1.40993 − 0.110028i 1.23327 + 0.245314i 1.97579 + 0.310262i 2.01756 − 0.964082i −1.71184 − 0.481569i 2.31563 + 1.54725i −2.75158 − 0.654838i −1.31085 − 0.542972i −2.95069 + 1.13730i
99.4 −1.40789 − 0.133562i −0.261892 − 0.0520936i 1.96432 + 0.376082i −2.19746 − 0.413747i 0.361758 + 0.108321i 3.74174 + 2.50015i −2.71532 − 0.791841i −2.70576 − 1.12076i 3.03852 + 0.876008i
99.5 −1.40534 + 0.158195i 1.68688 + 0.335540i 1.94995 − 0.444636i −1.31805 + 1.80630i −2.42371 − 0.204691i −1.88168 − 1.25730i −2.67000 + 0.933337i −0.0386776 − 0.0160208i 1.56656 − 2.74698i
99.6 −1.40033 + 0.197690i 2.34414 + 0.466278i 1.92184 − 0.553661i 0.931769 − 2.03268i −3.37474 − 0.189530i −3.26252 − 2.17994i −2.58175 + 1.15523i 2.50594 + 1.03799i −0.902942 + 3.03063i
99.7 −1.39450 − 0.235336i −1.73290 − 0.344695i 1.88923 + 0.656351i −0.781951 + 2.09489i 2.33540 + 0.888490i 1.67205 + 1.11723i −2.48006 − 1.35988i 0.112490 + 0.0465948i 1.58343 − 2.73729i
99.8 −1.38718 − 0.275197i 2.51872 + 0.501004i 1.84853 + 0.763494i 0.942122 + 2.02791i −3.35604 − 1.38812i 2.23913 + 1.49614i −2.35414 − 1.56781i 3.32129 + 1.37572i −0.748819 − 3.07234i
99.9 −1.37562 + 0.328119i −1.63264 − 0.324753i 1.78468 − 0.902737i 0.608763 − 2.15161i 2.35246 − 0.0889645i −0.693159 − 0.463154i −2.15883 + 1.82741i −0.211574 − 0.0876370i −0.131444 + 3.15954i
99.10 −1.36678 − 0.363200i −0.582212 − 0.115809i 1.73617 + 0.992829i 2.18602 − 0.470433i 0.753693 + 0.369745i −0.120434 − 0.0804716i −2.01237 − 1.98756i −2.44608 − 1.01320i −3.15867 − 0.150986i
99.11 −1.35478 + 0.405659i −3.02940 − 0.602586i 1.67088 − 1.09916i 2.03379 + 0.929359i 4.34863 − 0.412531i 3.84724 + 2.57064i −1.81780 + 2.16693i 6.04253 + 2.50290i −3.13235 − 0.434057i
99.12 −1.32441 + 0.495923i 1.03954 + 0.206777i 1.50812 − 1.31361i −2.15293 − 0.604079i −1.47932 + 0.241673i 0.290647 + 0.194204i −1.34592 + 2.48767i −1.73376 − 0.718147i 3.15093 − 0.267637i
99.13 −1.31885 − 0.510535i −0.821426 − 0.163392i 1.47871 + 1.34663i −1.70342 − 1.44857i 0.999917 + 0.634856i −3.51029 − 2.34550i −1.26268 − 2.53094i −2.12359 − 0.879621i 1.50701 + 2.78010i
99.14 −1.30228 − 0.551433i 3.30415 + 0.657237i 1.39184 + 1.43624i −2.22931 − 0.173723i −3.94050 − 2.67792i −0.135632 − 0.0906264i −1.02058 − 2.63788i 7.71383 + 3.19517i 2.80738 + 1.45555i
99.15 −1.28717 + 0.585828i −1.03954 − 0.206777i 1.31361 − 1.50812i −0.265792 + 2.22021i 1.45919 − 0.342833i 0.290647 + 0.194204i −0.807340 + 2.71076i −1.73376 − 0.718147i −0.958545 − 3.01350i
99.16 −1.24482 + 0.671133i 3.02940 + 0.602586i 1.09916 − 1.67088i −0.0803191 − 2.23462i −4.17548 + 1.28302i 3.84724 + 2.57064i −0.246876 + 2.81763i 6.04253 + 2.50290i 1.59971 + 2.72780i
99.17 −1.20473 + 0.740697i 1.63264 + 0.324753i 0.902737 − 1.78468i 2.22079 + 0.260961i −2.20744 + 0.818055i −0.693159 − 0.463154i 0.234351 + 2.81870i −0.211574 − 0.0876370i −2.86874 + 1.33054i
99.18 −1.18089 − 0.778143i −2.76972 − 0.550931i 0.788988 + 1.83780i 1.55179 − 1.60995i 2.84202 + 2.80582i −0.826983 − 0.552572i 0.498362 − 2.78418i 4.59617 + 1.90380i −3.08526 + 0.693654i
99.19 −1.17976 − 0.779844i 0.420865 + 0.0837152i 0.783687 + 1.84006i −0.334927 − 2.21084i −0.431236 − 0.426973i 0.582822 + 0.389429i 0.510396 − 2.78199i −2.60152 − 1.07758i −1.32898 + 2.86946i
99.20 −1.16464 − 0.802257i −1.96701 − 0.391262i 0.712767 + 1.86868i 0.939232 + 2.02925i 1.97696 + 2.03373i 1.80528 + 1.20625i 0.669045 − 2.74816i 0.944397 + 0.391182i 0.534111 − 3.11685i
See next 80 embeddings (of 832 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 99.104
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
8.d odd 2 1 inner
17.e odd 16 1 inner
40.e odd 2 1 inner
85.p odd 16 1 inner
136.s even 16 1 inner
680.co even 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 680.2.co.a ✓ 832
5.b even 2 1 inner 680.2.co.a ✓ 832
8.d odd 2 1 inner 680.2.co.a ✓ 832
17.e odd 16 1 inner 680.2.co.a ✓ 832
40.e odd 2 1 inner 680.2.co.a ✓ 832
85.p odd 16 1 inner 680.2.co.a ✓ 832
136.s even 16 1 inner 680.2.co.a ✓ 832
680.co even 16 1 inner 680.2.co.a ✓ 832
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
680.2.co.a ✓ 832 1.a even 1 1 trivial
680.2.co.a ✓ 832 5.b even 2 1 inner
680.2.co.a ✓ 832 8.d odd 2 1 inner
680.2.co.a ✓ 832 17.e odd 16 1 inner
680.2.co.a ✓ 832 40.e odd 2 1 inner
680.2.co.a ✓ 832 85.p odd 16 1 inner
680.2.co.a ✓ 832 136.s even 16 1 inner
680.2.co.a ✓ 832 680.co even 16 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(680, [\chi])\).