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Properties

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

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [680,2,Mod(523,680)] 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, 1])) 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("680.523"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 680 = 2^{3} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 680.t (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.

\(\operatorname{Tr}(f)(q) = \) \( 208 q - 8 q^{6} - 192 q^{9} + 2 q^{10} - 8 q^{11} + 8 q^{12} - 16 q^{14} - 16 q^{16} - 12 q^{18} - 10 q^{20} + 4 q^{22} + 16 q^{24} + 12 q^{28} - 24 q^{30} + 20 q^{32} - 8 q^{33} - 4 q^{34} - 8 q^{35} - 12 q^{38}+ \cdots + 24 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
523.1 −1.41408 − 0.0194916i 1.69876i 1.99924 + 0.0551253i 0.460305 + 2.18818i 0.0331116 − 2.40219i 2.40099 −2.82601 − 0.116920i 0.114199 −0.608257 − 3.10323i
523.2 −1.40960 − 0.114089i − 0.180096i 1.97397 + 0.321642i 2.23492 − 0.0714892i −0.0205471 + 0.253865i −2.81925 −2.74582 − 0.678597i 2.96757 −3.15852 − 0.154210i
523.3 −1.40941 − 0.116416i 1.89046i 1.97289 + 0.328156i −2.09461 − 0.782699i 0.220080 − 2.66444i 2.61042 −2.74242 − 0.692184i −0.573848 2.86105 + 1.34699i
523.4 −1.40702 − 0.142432i − 0.227143i 1.95943 + 0.400810i 0.753993 − 2.10511i −0.0323525 + 0.319596i 4.66544 −2.69987 − 0.843034i 2.94841 −1.36072 + 2.85455i
523.5 −1.40194 + 0.185930i − 2.65957i 1.93086 − 0.521326i −0.163750 − 2.23006i 0.494495 + 3.72855i 1.56006 −2.61002 + 1.08987i −4.07332 0.644204 + 3.09597i
523.6 −1.38711 − 0.275553i − 1.42706i 1.84814 + 0.764443i −1.79291 + 1.33622i −0.393230 + 1.97948i −0.420628 −2.35293 − 1.56963i 0.963506 2.85516 − 1.35944i
523.7 −1.38701 + 0.276036i 0.805848i 1.84761 − 0.765731i −1.83519 + 1.27753i −0.222443 − 1.11772i −1.35899 −2.35129 + 1.57209i 2.35061 2.19279 − 2.27853i
523.8 −1.37567 + 0.327922i − 1.51555i 1.78493 − 0.902224i −1.48529 − 1.67150i 0.496982 + 2.08490i −3.71705 −2.15962 + 1.82648i 0.703111 2.59139 + 1.81238i
523.9 −1.36150 + 0.382524i − 2.90882i 1.70735 − 1.04161i 0.926438 + 2.03512i 1.11269 + 3.96035i 1.64123 −1.92611 + 2.07126i −5.46123 −2.03983 − 2.41642i
523.10 −1.35867 + 0.392439i 1.94322i 1.69198 − 1.06639i 0.520493 − 2.17465i −0.762597 − 2.64020i −3.06337 −1.88036 + 2.11288i −0.776111 0.146238 + 3.15889i
523.11 −1.35436 − 0.407065i − 2.58701i 1.66860 + 1.10263i 0.524498 + 2.17368i −1.05308 + 3.50375i −2.06399 −1.81104 − 2.17259i −3.69261 0.174470 − 3.15746i
523.12 −1.33036 − 0.479733i 2.39666i 1.53971 + 1.27643i −1.22372 − 1.87150i 1.14976 − 3.18842i −2.01789 −1.43602 − 2.43677i −2.74399 0.730169 + 3.07683i
523.13 −1.29871 + 0.559787i 2.45050i 1.37328 − 1.45400i 2.23573 + 0.0391272i −1.37175 − 3.18248i 1.51507 −0.969557 + 2.65706i −3.00493 −2.92545 + 1.20071i
523.14 −1.25175 + 0.658124i − 0.113733i 1.13375 − 1.64761i 0.945534 + 2.02632i 0.0748502 + 0.142365i 3.07354 −0.334831 + 2.80854i 2.98706 −2.51714 − 1.91416i
523.15 −1.24750 − 0.666136i 3.02737i 1.11253 + 1.66201i 2.14004 − 0.648258i 2.01664 − 3.77665i 0.102339 −0.280755 − 2.81446i −6.16497 −3.10153 − 0.616851i
523.16 −1.24029 − 0.679470i − 0.936631i 1.07664 + 1.68548i −1.93873 − 1.11414i −0.636412 + 1.16170i 1.67544 −0.190117 − 2.82203i 2.12272 1.64757 + 2.69917i
523.17 −1.23139 − 0.695478i − 2.83257i 1.03262 + 1.71280i 1.41742 − 1.72943i −1.96999 + 3.48799i −3.96131 −0.0803370 − 2.82729i −5.02347 −2.94817 + 1.14381i
523.18 −1.18147 − 0.777262i 0.666399i 0.791726 + 1.83662i 2.03785 + 0.920425i 0.517967 − 0.787328i 1.83739 0.492137 − 2.78528i 2.55591 −1.69224 − 2.67139i
523.19 −1.17696 + 0.784070i − 2.07488i 0.770468 − 1.84564i −2.04113 + 0.913116i 1.62685 + 2.44204i 4.09851 0.540301 + 2.77634i −1.30511 1.68638 − 2.67509i
523.20 −1.17016 − 0.794181i 1.92180i 0.738553 + 1.85864i −0.280701 + 2.21838i 1.52625 − 2.24881i −4.67203 0.611869 − 2.76145i −0.693301 2.09026 − 2.37293i
See next 80 embeddings (of 208 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 523.104
Significant digits:
Format:

Inner twists

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

Hecke kernels

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