↑

Properties

Label 680.2.bl.a
Level $680$
Weight $2$
Character orbit 680.bl
Analytic conductor $5.430$
Analytic rank $0$
Dimension $208$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [680,2,Mod(123,680)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(680, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 2, 3, 3])) 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("680.123"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 680 = 2^{3} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 680.bl (of order \(4\), degree \(2\), 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: \(208\)
Relative dimension: \(104\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 208 q - 8 q^{3} - 8 q^{6} + 192 q^{9} - 6 q^{10} - 8 q^{11} + 16 q^{14} - 16 q^{16} - 12 q^{18} - 10 q^{20} - 16 q^{24} - 32 q^{27} - 24 q^{30} - 20 q^{32} - 8 q^{33} + 4 q^{34} - 8 q^{35} - 12 q^{38} + 38 q^{40}+ \cdots - 24 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
123.1 −1.41208 + 0.0777098i 2.77161 1.98792 − 0.219464i −2.02542 − 0.947459i −3.91373 + 0.215381i 2.38616i −2.79004 + 0.464382i 4.68182 2.93367 + 1.18049i
123.2 −1.41175 − 0.0834622i −0.942276 1.98607 + 0.235655i −1.58440 − 1.57787i 1.33026 + 0.0786444i 0.802214i −2.78416 − 0.498447i −2.11212 2.10508 + 2.35980i
123.3 −1.41161 + 0.0857726i −3.33428 1.98529 − 0.242155i 1.35641 − 1.77768i 4.70670 − 0.285990i 0.378310i −2.78168 + 0.512112i 8.11740 −1.76225 + 2.62574i
123.4 −1.40232 − 0.183044i 1.40881 1.93299 + 0.513372i 2.20458 − 0.373952i −1.97560 − 0.257874i − 0.664990i −2.61670 − 1.07373i −1.01525 −3.15997 + 0.120865i
123.5 −1.39133 − 0.253361i 0.719242 1.87162 + 0.705020i 0.825729 + 2.07802i −1.00071 − 0.182228i − 1.97447i −2.42542 − 1.45511i −2.48269 −0.622374 − 3.10043i
123.6 −1.38955 + 0.262950i 0.114761 1.86171 − 0.730766i −1.62562 + 1.53537i −0.159467 + 0.0301765i 0.891454i −2.39480 + 1.50498i −2.98683 1.85516 − 2.56093i
123.7 −1.38536 − 0.284198i −1.66268 1.83846 + 0.787434i 1.75053 + 1.39127i 2.30342 + 0.472530i − 0.572159i −2.32315 − 1.61337i −0.235488 −2.02973 − 2.42491i
123.8 −1.37806 + 0.317714i 3.29489 1.79812 − 0.875660i 2.03374 + 0.929456i −4.54056 + 1.04683i 2.70735i −2.19971 + 1.77800i 7.85628 −3.09793 − 0.634700i
123.9 −1.36847 − 0.356773i 2.93734 1.74543 + 0.976468i −1.10520 + 1.94384i −4.01967 − 1.04797i − 5.06704i −2.04019 − 1.95899i 5.62799 2.20595 − 2.26579i
123.10 −1.34516 + 0.436527i −2.03744 1.61889 − 1.17439i −0.397855 + 2.20039i 2.74068 − 0.889398i − 3.62989i −1.66500 + 2.28643i 1.15116 −0.425353 − 3.13354i
123.11 −1.34353 + 0.441514i 1.08019 1.61013 − 1.18637i 1.65925 − 1.49896i −1.45127 + 0.476920i − 1.54588i −1.63946 + 2.30482i −1.83318 −1.56744 + 2.74648i
123.12 −1.34098 − 0.449197i −2.30800 1.59644 + 1.20473i −1.25162 + 1.85296i 3.09497 + 1.03675i 4.47457i −1.59963 − 2.33263i 2.32684 2.51074 − 1.92255i
123.13 −1.32841 + 0.485099i 1.01540 1.52936 − 1.28882i −1.74286 − 1.40087i −1.34887 + 0.492569i − 4.93845i −1.40641 + 2.45398i −1.96896 2.99480 + 1.01547i
123.14 −1.32444 − 0.495854i −1.10496 1.50826 + 1.31345i 0.236605 − 2.22351i 1.46345 + 0.547901i − 2.48520i −1.34631 − 2.48746i −1.77905 −1.41591 + 2.82758i
123.15 −1.29592 + 0.566210i −2.40143 1.35881 − 1.46752i −2.19168 − 0.443317i 3.11207 − 1.35972i 0.998624i −0.929985 + 2.67117i 2.76689 3.09125 − 0.666449i
123.16 −1.24114 + 0.677911i 1.66129 1.08087 − 1.68277i 0.0270834 + 2.23590i −2.06190 + 1.12621i 2.02798i −0.200753 + 2.82129i −0.240104 −1.54936 − 2.75672i
123.17 −1.23794 − 0.683748i 1.23752 1.06498 + 1.69287i 1.08259 + 1.95653i −1.53197 − 0.846148i 4.38346i −0.160877 − 2.82385i −1.46856 −0.00241194 − 3.16228i
123.18 −1.20585 − 0.738875i 2.74346 0.908129 + 1.78194i 0.584289 − 2.15838i −3.30818 − 2.02707i − 1.18693i 0.221566 − 2.81974i 4.52655 −2.29933 + 2.17096i
123.19 −1.19288 − 0.759626i −0.380697 0.845937 + 1.81229i −2.14321 + 0.637679i 0.454127 + 0.289188i − 1.82524i 0.367558 − 2.80444i −2.85507 3.04100 + 0.867365i
123.20 −1.17522 − 0.786672i 1.61017 0.762294 + 1.84903i −2.21833 − 0.281104i −1.89231 − 1.26668i 2.13601i 0.558715 − 2.77270i −0.407346 2.38589 + 2.07546i
See next 80 embeddings (of 208 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 123.104
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
85.i odd 4 1 inner
680.bl even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 680.2.bl.a yes 208
5.c odd 4 1 680.2.t.a ✓ 208
8.d odd 2 1 inner 680.2.bl.a yes 208
17.c even 4 1 680.2.t.a ✓ 208
40.k even 4 1 680.2.t.a ✓ 208
85.i odd 4 1 inner 680.2.bl.a yes 208
136.j odd 4 1 680.2.t.a ✓ 208
680.bl even 4 1 inner 680.2.bl.a yes 208
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
680.2.t.a ✓ 208 5.c odd 4 1
680.2.t.a ✓ 208 17.c even 4 1
680.2.t.a ✓ 208 40.k even 4 1
680.2.t.a ✓ 208 136.j odd 4 1
680.2.bl.a yes 208 1.a even 1 1 trivial
680.2.bl.a yes 208 8.d odd 2 1 inner
680.2.bl.a yes 208 85.i odd 4 1 inner
680.2.bl.a yes 208 680.bl even 4 1 inner

Hecke kernels

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