Properties

Label 760.2.w.d
Level $760$
Weight $2$
Character orbit 760.w
Analytic conductor $6.069$
Analytic rank $0$
Dimension $108$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [760,2,Mod(267,760)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("760.267"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(760, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 2, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 760 = 2^{3} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 760.w (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [108,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.06863055362\)
Analytic rank: \(0\)
Dimension: \(108\)
Relative dimension: \(54\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 108 q + 8 q^{6} + 8 q^{10} + 18 q^{12} + 16 q^{16} + 4 q^{17} - 36 q^{22} - 4 q^{25} - 40 q^{26} + 16 q^{28} - 12 q^{30} - 10 q^{32} + 24 q^{35} - 80 q^{36} + 28 q^{40} + 108 q^{42} - 32 q^{43} + 64 q^{46}+ \cdots + 76 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
267.1 −1.40857 + 0.126238i −1.36883 + 1.36883i 1.96813 − 0.355630i 0.375211 − 2.20436i 1.75529 − 2.10089i 2.05093 − 2.05093i −2.72735 + 0.749382i − 0.747394i −0.250235 + 3.15236i
267.2 −1.39849 + 0.210323i 1.23627 − 1.23627i 1.91153 − 0.588268i −1.98563 − 1.02824i −1.46890 + 1.98893i −2.63738 + 2.63738i −2.54952 + 1.22472i − 0.0567502i 2.99314 + 1.02036i
267.3 −1.39309 − 0.243545i −2.03277 + 2.03277i 1.88137 + 0.678557i −2.21833 − 0.281128i 3.32689 − 2.33675i −2.44756 + 2.44756i −2.45565 − 1.40349i − 5.26427i 3.02185 + 0.931897i
267.4 −1.39205 + 0.249374i −0.527859 + 0.527859i 1.87562 − 0.694285i −1.43303 + 1.71651i 0.603173 − 0.866442i 1.24713 − 1.24713i −2.43783 + 1.43421i 2.44273i 1.56680 − 2.74684i
267.5 −1.38724 − 0.274874i −1.68946 + 1.68946i 1.84889 + 0.762634i 1.63822 + 1.52192i 2.80809 − 1.87931i −1.48938 + 1.48938i −2.35523 − 1.56617i − 2.70857i −1.85427 − 2.56158i
267.6 −1.38164 + 0.301766i 2.05043 − 2.05043i 1.81787 − 0.833866i 1.90605 − 1.16918i −2.21421 + 3.45171i 1.97140 − 1.97140i −2.26002 + 1.70068i − 5.40854i −2.28066 + 2.19057i
267.7 −1.37681 + 0.323095i −0.336310 + 0.336310i 1.79122 − 0.889683i 1.81791 + 1.30200i 0.354375 − 0.571695i −0.881910 + 0.881910i −2.17872 + 1.80366i 2.77379i −2.92359 − 1.20525i
267.8 −1.31439 − 0.521896i 0.746757 − 0.746757i 1.45525 + 1.37195i −2.20913 − 0.346067i −1.37126 + 0.591801i 2.83517 − 2.83517i −1.19675 − 2.56277i 1.88471i 2.72305 + 1.60780i
267.9 −1.25201 − 0.657626i 2.08229 − 2.08229i 1.13506 + 1.64671i 0.127264 − 2.23244i −3.97642 + 1.23768i 0.638542 − 0.638542i −0.338182 − 2.80814i − 5.67187i −1.62745 + 2.71135i
267.10 −1.13965 − 0.837382i 0.654350 − 0.654350i 0.597584 + 1.90864i 0.582206 + 2.15894i −1.29367 + 0.197786i 0.531937 − 0.531937i 0.917224 − 2.67557i 2.14365i 1.14435 − 2.94796i
267.11 −1.09829 + 0.890935i 1.15450 − 1.15450i 0.412468 − 1.95701i −1.79430 + 1.33434i −0.239388 + 2.29656i −0.577109 + 0.577109i 1.29056 + 2.51684i 0.334254i 0.781847 − 3.06410i
267.12 −1.07153 + 0.922945i 0.0817665 − 0.0817665i 0.296346 − 1.97792i 1.80920 − 1.31408i −0.0121492 + 0.163081i 1.71397 − 1.71397i 1.50797 + 2.39291i 2.98663i −0.725784 + 3.07786i
267.13 −1.03664 + 0.961964i −1.93572 + 1.93572i 0.149250 − 1.99442i −2.23450 + 0.0837651i 0.144554 − 3.86873i 2.96042 − 2.96042i 1.76384 + 2.21108i − 4.49399i 2.23579 − 2.23634i
267.14 −1.02073 − 0.978826i −0.723718 + 0.723718i 0.0837985 + 1.99824i −2.17752 + 0.508321i 1.44712 − 0.0303299i 1.00583 − 1.00583i 1.87040 − 2.12170i 1.95246i 2.72023 + 1.61256i
267.15 −0.978826 − 1.02073i −0.723718 + 0.723718i −0.0837985 + 1.99824i 2.17752 − 0.508321i 1.44712 + 0.0303299i −1.00583 + 1.00583i 2.12170 − 1.87040i 1.95246i −2.65028 − 1.72512i
267.16 −0.848677 + 1.13126i −1.23252 + 1.23252i −0.559496 − 1.92015i −0.104679 + 2.23362i −0.348290 − 2.44032i −3.04449 + 3.04449i 2.64702 + 0.996648i − 0.0382290i −2.43796 − 2.01404i
267.17 −0.837382 − 1.13965i 0.654350 − 0.654350i −0.597584 + 1.90864i −0.582206 − 2.15894i −1.29367 − 0.197786i −0.531937 + 0.531937i 2.67557 − 0.917224i 2.14365i −1.97290 + 2.47137i
267.18 −0.679797 + 1.24011i 2.16703 − 2.16703i −1.07575 − 1.68605i 0.704730 + 2.12211i 1.21422 + 4.16050i 3.03658 − 3.03658i 2.82218 − 0.187884i − 6.39203i −3.11073 − 0.568660i
267.19 −0.657626 − 1.25201i 2.08229 − 2.08229i −1.13506 + 1.64671i −0.127264 + 2.23244i −3.97642 − 1.23768i −0.638542 + 0.638542i 2.80814 + 0.338182i − 5.67187i 2.87873 − 1.30878i
267.20 −0.634707 + 1.26378i −2.43641 + 2.43641i −1.19429 − 1.60426i −0.120344 − 2.23283i −1.53269 − 4.62549i −2.17928 + 2.17928i 2.78547 − 0.491093i − 8.87215i 2.89819 + 1.26510i
See next 80 embeddings (of 108 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 267.54
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
8.d odd 2 1 inner
40.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 760.2.w.d ✓ 108
5.c odd 4 1 inner 760.2.w.d ✓ 108
8.d odd 2 1 inner 760.2.w.d ✓ 108
40.k even 4 1 inner 760.2.w.d ✓ 108
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.w.d ✓ 108 1.a even 1 1 trivial
760.2.w.d ✓ 108 5.c odd 4 1 inner
760.2.w.d ✓ 108 8.d odd 2 1 inner
760.2.w.d ✓ 108 40.k even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(760, [\chi])\):

\( T_{3}^{54} + 376 T_{3}^{50} - 20 T_{3}^{49} + 320 T_{3}^{47} + 56862 T_{3}^{46} - 3220 T_{3}^{45} + \cdots + 131072 \) Copy content Toggle raw display
\( T_{7}^{108} + 2860 T_{7}^{104} + 3729602 T_{7}^{100} + 2938272260 T_{7}^{96} + 1562341888351 T_{7}^{92} + \cdots + 10\!\cdots\!24 \) Copy content Toggle raw display