↑

Properties

Label 936.2.db.a
Level $936$
Weight $2$
Character orbit 936.db
Analytic conductor $7.474$
Analytic rank $0$
Dimension $328$
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] = [936,2,Mod(61,936)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(936, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 4, 4])) 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("936.61"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 936 = 2^{3} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 936.db (of order \(6\), 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: \(7.47399762919\)
Analytic rank: \(0\)
Dimension: \(328\)
Relative dimension: \(164\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 328 q + q^{2} + q^{4} + q^{6} - 4 q^{7} - 2 q^{8} - 2 q^{9} - 4 q^{10} + 5 q^{12} - 6 q^{14} + 4 q^{15} + q^{16} - 4 q^{17} - 12 q^{18} - 18 q^{20} - 7 q^{22} + 44 q^{23} - 6 q^{24} + 144 q^{25} - 16 q^{26}+ \cdots + 39 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
61.1 −1.41383 − 0.0328320i 1.19534 − 1.25346i 1.99784 + 0.0928379i 1.07343 − 0.619746i −1.73116 + 1.73294i −3.62566 −2.82157 − 0.196850i −0.142340 − 2.99662i −1.53800 + 0.840974i
61.2 −1.41382 + 0.0331869i 1.68403 − 0.405016i 1.99780 − 0.0938410i −0.203142 + 0.117284i −2.36748 + 0.628510i 2.52349 −2.82142 + 0.198975i 2.67192 − 1.36412i 0.283314 − 0.172560i
61.3 −1.41074 + 0.0991241i −1.62386 + 0.602548i 1.98035 − 0.279676i −2.94424 + 1.69986i 2.23112 − 1.01100i −2.86012 −2.76603 + 0.590849i 2.27387 − 1.95691i 3.98505 − 2.68990i
61.4 −1.40766 + 0.135961i −0.195465 + 1.72099i 1.96303 − 0.382776i 0.951819 − 0.549533i 0.0411610 − 2.44914i −2.60565 −2.71124 + 0.805715i −2.92359 − 0.672785i −1.26512 + 0.902967i
61.5 −1.40487 − 0.162318i 0.958745 + 1.44250i 1.94731 + 0.456070i −0.947786 + 0.547205i −1.11277 − 2.18214i −2.57743 −2.66168 − 0.956801i −1.16162 + 2.76598i 1.42034 − 0.614907i
61.6 −1.40448 + 0.165649i 1.30381 − 1.14021i 1.94512 − 0.465300i 2.42054 − 1.39750i −1.64230 + 1.81738i 2.26644 −2.65480 + 0.975710i 0.399837 − 2.97324i −3.16810 + 2.36372i
61.7 −1.40331 + 0.175255i −1.70212 − 0.320630i 1.93857 − 0.491875i 2.82831 − 1.63292i 2.44479 + 0.151640i 0.761796 −2.63422 + 1.03000i 2.79439 + 1.09150i −3.68282 + 2.78718i
61.8 −1.40283 + 0.179043i −1.20910 + 1.24019i 1.93589 − 0.502335i −0.00979680 + 0.00565618i 1.47412 − 1.95627i 3.11386 −2.62579 + 1.05130i −0.0761639 − 2.99903i 0.0127306 − 0.00968873i
61.9 −1.40265 − 0.180518i −1.62706 − 0.593865i 1.93483 + 0.506405i −2.31463 + 1.33635i 2.17498 + 1.12669i 3.16045 −2.62246 − 1.05958i 2.29465 + 1.93251i 3.48784 − 1.45660i
61.10 −1.39677 + 0.221459i −1.10721 − 1.33195i 1.90191 − 0.618653i 2.08463 − 1.20356i 1.84148 + 1.61522i 0.223466 −2.51952 + 1.28531i −0.548181 + 2.94949i −2.64520 + 2.14276i
61.11 −1.38780 − 0.272025i 1.10140 − 1.33676i 1.85201 + 0.755035i −3.53261 + 2.03955i −1.89216 + 1.55555i −2.22614 −2.36483 − 1.55163i −0.573846 − 2.94461i 5.45738 − 1.86954i
61.12 −1.37196 + 0.343128i −0.241474 − 1.71514i 1.76453 − 0.941512i −0.309123 + 0.178472i 0.919802 + 2.27023i −3.61607 −2.09779 + 1.89717i −2.88338 + 0.828320i 0.362864 − 0.350925i
61.13 −1.36693 − 0.362617i −0.347433 − 1.69685i 1.73702 + 0.991346i −0.513997 + 0.296756i −0.140387 + 2.44546i 3.23696 −2.01491 − 1.98498i −2.75858 + 1.17908i 0.810208 − 0.219262i
61.14 −1.36024 + 0.386956i 0.0294737 + 1.73180i 1.70053 − 1.05271i −3.14960 + 1.81842i −0.710221 − 2.34427i 3.48447 −1.90579 + 2.08997i −2.99826 + 0.102085i 3.58058 − 3.69226i
61.15 −1.35749 − 0.396522i 1.70060 + 0.328583i 1.68554 + 1.07655i 2.88728 − 1.66697i −2.17825 − 1.12037i 1.57859 −1.86123 − 2.12975i 2.78407 + 1.11757i −4.58044 + 1.11803i
61.16 −1.33779 + 0.458590i 1.73180 + 0.0293872i 1.57939 − 1.22700i −2.62102 + 1.51325i −2.33027 + 0.754873i 0.0194906 −1.55021 + 2.36577i 2.99827 + 0.101786i 2.81243 − 3.22639i
61.17 −1.33515 − 0.466242i −1.42357 − 0.986631i 1.56524 + 1.24500i 0.715331 − 0.412997i 1.44067 + 1.98103i −2.35753 −1.50935 − 2.39204i 1.05312 + 2.80908i −1.14763 + 0.217894i
61.18 −1.33314 − 0.471953i 1.60244 + 0.657405i 1.55452 + 1.25836i −0.823961 + 0.475714i −1.82601 − 1.63269i −1.67702 −1.47851 − 2.41123i 2.13564 + 2.10691i 1.32297 − 0.245323i
61.19 −1.31791 − 0.512942i 0.259180 + 1.71255i 1.47378 + 1.35202i 3.37624 − 1.94927i 0.536863 − 2.38993i 0.690243 −1.24880 − 2.53781i −2.86565 + 0.887716i −5.44944 + 0.837152i
61.20 −1.31196 + 0.527970i 0.650684 − 1.60518i 1.44250 − 1.38535i −0.862801 + 0.498139i −0.00618455 + 2.44948i 4.66240 −1.16107 + 2.57913i −2.15322 − 2.08893i 0.868961 − 1.10907i
See next 80 embeddings (of 328 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 61.164
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
117.f even 3 1 inner
936.db even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 936.2.db.a yes 328
8.b even 2 1 inner 936.2.db.a yes 328
9.c even 3 1 936.2.cj.a ✓ 328
13.c even 3 1 936.2.cj.a ✓ 328
72.n even 6 1 936.2.cj.a ✓ 328
104.r even 6 1 936.2.cj.a ✓ 328
117.f even 3 1 inner 936.2.db.a yes 328
936.db even 6 1 inner 936.2.db.a yes 328
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
936.2.cj.a ✓ 328 9.c even 3 1
936.2.cj.a ✓ 328 13.c even 3 1
936.2.cj.a ✓ 328 72.n even 6 1
936.2.cj.a ✓ 328 104.r even 6 1
936.2.db.a yes 328 1.a even 1 1 trivial
936.2.db.a yes 328 8.b even 2 1 inner
936.2.db.a yes 328 117.f even 3 1 inner
936.2.db.a yes 328 936.db even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(936, [\chi])\).