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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [28,5,Mod(11,28)] 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("28.11"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(28, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 4])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 28 = 2^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 28.g (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: \(2.89435896635\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 2 q^{2} - 4 q^{4} - 2 q^{5} - 36 q^{6} - 184 q^{8} + 268 q^{9} + 130 q^{10} - 48 q^{12} + 344 q^{13} - 474 q^{14} - 432 q^{16} - 2 q^{17} - 568 q^{18} + 664 q^{20} + 426 q^{21} + 196 q^{22} - 692 q^{24}+ \cdots - 9682 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −3.98249 + 0.373846i −10.4527 − 6.03487i 15.7205 − 2.97768i 13.8853 + 24.0501i 43.8839 + 20.1261i 34.1259 + 35.1628i −61.4935 + 17.7356i 32.3394 + 56.0135i −64.2891 − 90.5882i
11.2 −3.86347 − 1.03614i 1.18377 + 0.683453i 13.8528 + 8.00618i −16.2684 − 28.1777i −3.86533 − 3.86706i −47.8742 − 10.4432i −45.2245 − 45.2851i −39.5658 − 68.5299i 33.6565 + 125.720i
11.3 −3.31594 + 2.23708i 12.8404 + 7.41339i 5.99095 − 14.8361i 0.212899 + 0.368751i −59.1623 + 4.14255i 13.4191 + 47.1267i 13.3238 + 62.5977i 69.4168 + 120.233i −1.53089 − 0.746487i
11.4 −2.68081 − 2.96871i 8.26641 + 4.77262i −1.62652 + 15.9171i 8.72165 + 15.1063i −7.99215 − 37.3351i 48.2343 − 8.62858i 51.6137 − 37.8421i 5.05573 + 8.75678i 21.4653 − 66.3893i
11.5 −2.34212 + 3.24261i −3.98530 − 2.30092i −5.02899 − 15.1891i −5.96978 − 10.3400i 16.7950 − 7.53376i 8.20764 − 48.3077i 61.0308 + 19.2677i −29.9116 − 51.8084i 47.5103 + 4.85975i
11.6 −1.23058 − 3.80601i −8.26641 − 4.77262i −12.9714 + 9.36716i 8.72165 + 15.1063i −7.99215 + 37.3351i −48.2343 + 8.62858i 51.6137 + 37.8421i 5.05573 + 8.75678i 46.7622 − 51.7842i
11.7 0.123509 + 3.99809i 2.78459 + 1.60768i −15.9695 + 0.987599i 16.3700 + 28.3536i −6.08374 + 11.3316i −41.5540 + 25.9667i −5.92089 − 63.7255i −35.3307 − 61.1946i −111.339 + 68.9506i
11.8 1.03441 − 3.86393i −1.18377 − 0.683453i −13.8600 − 7.99382i −16.2684 − 28.1777i −3.86533 + 3.86706i 47.8742 + 10.4432i −45.2245 + 45.2851i −39.5658 − 68.5299i −125.705 + 33.7126i
11.9 1.54095 + 3.69127i −12.5795 − 7.26279i −11.2510 + 11.3761i −17.4517 − 30.2272i 7.42457 − 57.6260i 1.32995 + 48.9819i −59.3294 − 24.0005i 64.9961 + 112.577i 84.6846 − 110.997i
11.10 2.31501 − 3.26202i 10.4527 + 6.03487i −5.28149 − 15.1032i 13.8853 + 24.0501i 43.8839 − 20.1261i −34.1259 − 35.1628i −61.4935 − 17.7356i 32.3394 + 56.0135i 110.596 + 10.3819i
11.11 2.42626 + 3.18013i 12.5795 + 7.26279i −4.22650 + 15.4317i −17.4517 − 30.2272i 7.42457 + 57.6260i −1.32995 − 48.9819i −59.3294 + 24.0005i 64.9961 + 112.577i 53.7841 − 128.838i
11.12 3.40070 + 2.10601i −2.78459 − 1.60768i 7.12946 + 14.3238i 16.3700 + 28.3536i −6.08374 − 11.3316i 41.5540 − 25.9667i −5.92089 + 63.7255i −35.3307 − 61.1946i −4.04367 + 130.897i
11.13 3.59534 − 1.75315i −12.8404 − 7.41339i 9.85292 − 12.6063i 0.212899 + 0.368751i −59.1623 − 4.14255i −13.4191 − 47.1267i 13.3238 − 62.5977i 69.4168 + 120.233i 1.41192 + 0.952543i
11.14 3.97924 − 0.407029i 3.98530 + 2.30092i 15.6687 − 3.23933i −5.96978 − 10.3400i 16.7950 + 7.53376i −8.20764 + 48.3077i 61.0308 − 19.2677i −29.9116 − 51.8084i −27.9638 − 38.7153i
23.1 −3.98249 − 0.373846i −10.4527 + 6.03487i 15.7205 + 2.97768i 13.8853 − 24.0501i 43.8839 − 20.1261i 34.1259 − 35.1628i −61.4935 − 17.7356i 32.3394 − 56.0135i −64.2891 + 90.5882i
23.2 −3.86347 + 1.03614i 1.18377 − 0.683453i 13.8528 − 8.00618i −16.2684 + 28.1777i −3.86533 + 3.86706i −47.8742 + 10.4432i −45.2245 + 45.2851i −39.5658 + 68.5299i 33.6565 − 125.720i
23.3 −3.31594 − 2.23708i 12.8404 − 7.41339i 5.99095 + 14.8361i 0.212899 − 0.368751i −59.1623 − 4.14255i 13.4191 − 47.1267i 13.3238 − 62.5977i 69.4168 − 120.233i −1.53089 + 0.746487i
23.4 −2.68081 + 2.96871i 8.26641 − 4.77262i −1.62652 − 15.9171i 8.72165 − 15.1063i −7.99215 + 37.3351i 48.2343 + 8.62858i 51.6137 + 37.8421i 5.05573 − 8.75678i 21.4653 + 66.3893i
23.5 −2.34212 − 3.24261i −3.98530 + 2.30092i −5.02899 + 15.1891i −5.96978 + 10.3400i 16.7950 + 7.53376i 8.20764 + 48.3077i 61.0308 − 19.2677i −29.9116 + 51.8084i 47.5103 − 4.85975i
23.6 −1.23058 + 3.80601i −8.26641 + 4.77262i −12.9714 − 9.36716i 8.72165 − 15.1063i −7.99215 − 37.3351i −48.2343 − 8.62858i 51.6137 − 37.8421i 5.05573 − 8.75678i 46.7622 + 51.7842i
See all 28 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 11.14
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.c even 3 1 inner
28.g odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 28.5.g.a ✓ 28
4.b odd 2 1 inner 28.5.g.a ✓ 28
7.c even 3 1 inner 28.5.g.a ✓ 28
7.c even 3 1 196.5.c.h 14
7.d odd 6 1 196.5.c.g 14
28.f even 6 1 196.5.c.g 14
28.g odd 6 1 inner 28.5.g.a ✓ 28
28.g odd 6 1 196.5.c.h 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.5.g.a ✓ 28 1.a even 1 1 trivial
28.5.g.a ✓ 28 4.b odd 2 1 inner
28.5.g.a ✓ 28 7.c even 3 1 inner
28.5.g.a ✓ 28 28.g odd 6 1 inner
196.5.c.g 14 7.d odd 6 1
196.5.c.g 14 28.f even 6 1
196.5.c.h 14 7.c even 3 1
196.5.c.h 14 28.g odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(28, [\chi])\).