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Properties

Label 357.2.bl.a
Level $357$
Weight $2$
Character orbit 357.bl
Analytic conductor $2.851$
Analytic rank $0$
Dimension $384$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [357,2,Mod(10,357)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(357, base_ring=CyclotomicField(48)) chi = DirichletCharacter(H, H._module([0, 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("357.10"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 357 = 3 \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 357.bl (of order \(48\), degree \(16\), 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: \(2.85065935216\)
Analytic rank: \(0\)
Dimension: \(384\)
Relative dimension: \(24\) over \(\Q(\zeta_{48})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{48}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 384 q - 16 q^{11} + 32 q^{14} - 64 q^{22} - 16 q^{25} - 48 q^{28} - 64 q^{30} - 64 q^{35} - 32 q^{37} - 48 q^{42} - 64 q^{44} + 32 q^{46} - 64 q^{49} + 80 q^{56} - 32 q^{58} - 288 q^{61} - 528 q^{68} + 176 q^{70}+ \cdots + 32 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1 −2.71325 + 0.357206i 0.442289 + 0.896873i 5.30228 − 1.42074i 1.66378 − 1.45910i −1.52041 − 2.27545i −2.09469 + 1.61625i −8.82222 + 3.65428i −0.608761 + 0.793353i −3.99306 + 4.55322i
10.2 −2.58210 + 0.339940i −0.442289 − 0.896873i 4.61985 − 1.23788i −3.08620 + 2.70653i 1.44692 + 2.16547i −1.79393 + 1.94469i −6.69585 + 2.77351i −0.608761 + 0.793353i 7.04884 − 8.03766i
10.3 −2.43556 + 0.320648i 0.442289 + 0.896873i 3.89730 − 1.04428i −1.44985 + 1.27149i −1.36480 − 2.04257i 2.55244 − 0.696467i −4.61810 + 1.91288i −0.608761 + 0.793353i 3.12351 − 3.56168i
10.4 −2.23347 + 0.294042i −0.442289 − 0.896873i 2.97008 − 0.795831i 1.86915 − 1.63920i 1.25156 + 1.87309i 2.39791 + 1.11804i −2.23706 + 0.926619i −0.608761 + 0.793353i −3.69270 + 4.21072i
10.5 −1.79328 + 0.236090i 0.442289 + 0.896873i 1.22827 − 0.329115i 0.723589 − 0.634571i −1.00489 − 1.50393i −2.30150 − 1.30502i 1.21721 − 0.504184i −0.608761 + 0.793353i −1.14778 + 1.30880i
10.6 −1.52158 + 0.200320i −0.442289 − 0.896873i 0.343232 − 0.0919687i 0.535735 − 0.469827i 0.852640 + 1.27607i −0.930484 + 2.47673i 2.33194 − 0.965923i −0.608761 + 0.793353i −0.721050 + 0.822199i
10.7 −1.37138 + 0.180546i 0.442289 + 0.896873i −0.0837632 + 0.0224443i −0.803646 + 0.704779i −0.768473 − 1.15010i 0.896773 + 2.48914i 2.66667 − 1.10457i −0.608761 + 0.793353i 0.974860 − 1.11161i
10.8 −1.32179 + 0.174017i −0.442289 − 0.896873i −0.215000 + 0.0576090i −2.58490 + 2.26690i 0.740685 + 1.10851i 2.64573 + 0.0116518i 2.73759 − 1.13395i −0.608761 + 0.793353i 3.02223 − 3.44619i
10.9 −0.870777 + 0.114640i −0.442289 − 0.896873i −1.18674 + 0.317986i 3.24146 − 2.84269i 0.487952 + 0.730272i −2.13834 − 1.55805i 2.61980 − 1.08516i −0.608761 + 0.793353i −2.49670 + 2.84695i
10.10 −0.528748 + 0.0696109i 0.442289 + 0.896873i −1.65712 + 0.444025i 0.601022 − 0.527083i −0.296291 − 0.443431i 1.64107 − 2.07531i 1.83072 − 0.758309i −0.608761 + 0.793353i −0.281098 + 0.320531i
10.11 −0.466046 + 0.0613562i −0.442289 − 0.896873i −1.71842 + 0.460448i −1.35733 + 1.19035i 0.261156 + 0.390847i −1.84474 − 1.89656i 1.64118 − 0.679800i −0.608761 + 0.793353i 0.559543 − 0.638037i
10.12 0.0465376 − 0.00612679i 0.442289 + 0.896873i −1.92972 + 0.517068i 3.21502 − 2.81950i 0.0260780 + 0.0390285i −1.10998 + 2.40166i −0.173369 + 0.0718118i −0.608761 + 0.793353i 0.132345 − 0.150910i
10.13 0.0808615 − 0.0106456i 0.442289 + 0.896873i −1.92543 + 0.515916i 0.199390 − 0.174860i 0.0453119 + 0.0678141i −2.58704 − 0.554286i −0.300903 + 0.124638i −0.608761 + 0.793353i 0.0142615 − 0.0162621i
10.14 0.235499 − 0.0310041i −0.442289 − 0.896873i −1.87735 + 0.503035i −0.00654072 + 0.00573606i −0.131965 − 0.197500i 1.26448 + 2.32402i −0.865421 + 0.358469i −0.608761 + 0.793353i −0.00136250 + 0.00155363i
10.15 0.559246 − 0.0736262i −0.442289 − 0.896873i −1.62452 + 0.435288i 1.94651 − 1.70704i −0.313382 − 0.469009i 2.53814 − 0.746878i −1.91873 + 0.794762i −0.608761 + 0.793353i 0.962895 − 1.09797i
10.16 1.03243 − 0.135922i 0.442289 + 0.896873i −0.884410 + 0.236977i −2.20366 + 1.93256i 0.578538 + 0.865843i 2.58525 + 0.562582i −2.80503 + 1.16188i −0.608761 + 0.793353i −2.01245 + 2.29476i
10.17 1.06738 − 0.140524i −0.442289 − 0.896873i −0.812294 + 0.217654i −2.05370 + 1.80104i −0.598123 − 0.895154i −0.418160 − 2.61250i −2.82573 + 1.17045i −0.608761 + 0.793353i −1.93899 + 2.21099i
10.18 1.25234 − 0.164874i 0.442289 + 0.896873i −0.390673 + 0.104680i −1.76440 + 1.54734i 0.701768 + 1.05027i −2.47727 + 0.929057i −2.80599 + 1.16228i −0.608761 + 0.793353i −1.95452 + 2.22870i
10.19 1.77641 − 0.233869i 0.442289 + 0.896873i 1.16909 − 0.313256i 2.18661 − 1.91760i 0.995437 + 1.48978i 1.61314 − 2.09709i −1.30718 + 0.541453i −0.608761 + 0.793353i 3.43585 − 3.91783i
10.20 2.11462 − 0.278396i 0.442289 + 0.896873i 2.46228 − 0.659766i 0.750143 − 0.657858i 1.18496 + 1.77342i 0.498513 + 2.59836i 1.08209 − 0.448216i −0.608761 + 0.793353i 1.40313 − 1.59996i
See next 80 embeddings (of 384 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
17.e odd 16 1 inner
119.s even 48 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 357.2.bl.a ✓ 384
7.d odd 6 1 inner 357.2.bl.a ✓ 384
17.e odd 16 1 inner 357.2.bl.a ✓ 384
119.s even 48 1 inner 357.2.bl.a ✓ 384
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
357.2.bl.a ✓ 384 1.a even 1 1 trivial
357.2.bl.a ✓ 384 7.d odd 6 1 inner
357.2.bl.a ✓ 384 17.e odd 16 1 inner
357.2.bl.a ✓ 384 119.s even 48 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(357, [\chi])\).