Properties

Label 112.5.u.a
Level $112$
Weight $5$
Character orbit 112.u
Analytic conductor $11.577$
Analytic rank $0$
Dimension $248$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [112,5,Mod(11,112)] 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("112.11"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([6, 3, 8])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 112.u (of order \(12\), degree \(4\), 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: \(11.5774358654\)
Analytic rank: \(0\)
Dimension: \(248\)
Relative dimension: \(62\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 248 q - 2 q^{2} - 2 q^{3} + 4 q^{4} - 2 q^{5} - 8 q^{6} - 8 q^{7} + 172 q^{8} - 200 q^{10} - 98 q^{11} - 2 q^{12} - 8 q^{13} - 404 q^{14} - 312 q^{16} - 4 q^{17} - 326 q^{18} - 2 q^{19} - 1208 q^{20} + 158 q^{21}+ \cdots - 8 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −3.99982 − 0.0380779i 4.15411 − 1.11309i 15.9971 + 0.304610i −11.8085 + 44.0700i −16.6581 + 4.29398i 34.1359 + 35.1531i −63.9739 − 1.82752i −54.1304 + 31.2522i 48.9100 − 175.822i
11.2 −3.94789 − 0.643555i 16.6481 − 4.46086i 15.1717 + 5.08137i −4.17374 + 15.5766i −68.5958 + 6.89697i −18.7190 − 45.2835i −56.6260 − 29.8245i 187.113 − 108.030i 26.5019 − 58.8087i
11.3 −3.93629 + 0.711056i 7.14869 − 1.91549i 14.9888 − 5.59785i 1.72495 − 6.43760i −26.7773 + 12.6230i −39.7813 + 28.6085i −55.0199 + 32.6926i −22.7133 + 13.1136i −2.21241 + 26.5668i
11.4 −3.92357 − 0.778199i −8.75692 + 2.34641i 14.7888 + 6.10663i −0.230543 + 0.860400i 36.1844 − 2.39168i −46.0910 − 16.6319i −53.2728 − 35.4684i 1.02994 − 0.594639i 1.57412 − 3.19643i
11.5 −3.87462 + 0.993639i −4.73462 + 1.26864i 14.0254 − 7.69995i 2.48767 − 9.28412i 17.0843 − 9.61999i 48.3831 − 7.75082i −46.6920 + 43.7705i −49.3409 + 28.4870i −0.413720 + 38.4443i
11.6 −3.86957 − 1.01312i 5.34494 − 1.43217i 13.9472 + 7.84070i 9.64041 − 35.9785i −22.1336 + 0.126813i 15.1048 − 46.6138i −46.0260 − 44.4703i −43.6308 + 25.1903i −73.7549 + 129.454i
11.7 −3.75643 − 1.37450i −14.1007 + 3.77828i 12.2215 + 10.3264i −6.83589 + 25.5119i 58.1617 + 5.18869i 48.4315 − 7.44235i −31.7153 − 55.5890i 114.407 − 66.0532i 60.7447 − 86.4375i
11.8 −3.73010 + 1.44442i −15.0108 + 4.02214i 11.8273 − 10.7756i 9.79558 − 36.5576i 50.1822 − 36.6848i −9.74301 + 48.0216i −28.5526 + 57.2778i 138.999 − 80.2511i 16.2659 + 150.512i
11.9 −3.66878 + 1.59375i −9.56936 + 2.56410i 10.9199 − 11.6942i −4.67861 + 17.4608i 31.0214 − 24.6583i −18.1801 − 45.5026i −21.4251 + 60.3073i 14.8500 − 8.57365i −10.6634 − 71.5164i
11.10 −3.64656 − 1.64396i −4.74805 + 1.27224i 10.5948 + 11.9896i 6.16458 − 23.0065i 19.4056 + 3.16632i 13.8215 + 47.0103i −18.9241 − 61.1382i −49.2226 + 28.4187i −60.3013 + 73.7603i
11.11 −3.38093 + 2.13760i 8.86803 − 2.37618i 6.86137 − 14.4541i −3.92219 + 14.6378i −24.9029 + 26.9900i 34.4621 − 34.8334i 7.69925 + 63.5352i 2.84764 − 1.64409i −18.0291 − 57.8735i
11.12 −3.33618 + 2.20679i 14.6928 − 3.93692i 6.26015 − 14.7245i 10.5620 − 39.4181i −40.3298 + 45.5582i 39.6689 + 28.7642i 11.6089 + 62.9383i 130.231 − 75.1886i 51.7506 + 154.814i
11.13 −3.26156 − 2.31565i 13.0548 − 3.49803i 5.27551 + 15.1053i −1.18861 + 4.43594i −50.6793 − 18.8214i 19.6026 + 44.9081i 17.7722 − 61.4829i 88.0445 − 50.8325i 14.1488 − 11.7157i
11.14 −3.13236 − 2.48763i 0.211826 − 0.0567586i 3.62335 + 15.5843i −8.20558 + 30.6236i −0.804710 − 0.349157i −48.9979 − 0.458956i 27.4185 − 57.8293i −70.1064 + 40.4760i 101.883 − 75.5118i
11.15 −2.98688 + 2.66056i −8.74431 + 2.34303i 1.84287 − 15.8935i −11.0546 + 41.2562i 19.8844 − 30.2631i −21.5198 + 44.0216i 36.7812 + 52.3750i 0.825047 − 0.476341i −76.7459 − 152.639i
11.16 −2.72739 − 2.92597i 3.49997 − 0.937814i −1.12264 + 15.9606i −3.81718 + 14.2459i −12.2898 − 7.68302i 27.8887 − 40.2892i 49.7621 − 40.2460i −58.7778 + 33.9354i 52.0942 − 27.6853i
11.17 −2.60847 − 3.03247i −15.8659 + 4.25126i −2.39179 + 15.8202i 8.60652 − 32.1200i 54.2776 + 37.0237i −11.7953 − 47.5591i 54.2133 − 34.0135i 163.506 − 94.4003i −119.853 + 57.6849i
11.18 −2.60098 + 3.03890i −3.66465 + 0.981940i −2.46983 − 15.8082i 11.8650 − 44.2809i 6.54765 − 13.6905i −24.2313 − 42.5893i 54.4636 + 33.6112i −57.6826 + 33.3031i 103.705 + 151.230i
11.19 −2.54471 + 3.08617i 8.71948 − 2.33638i −3.04888 − 15.7068i −4.81991 + 17.9882i −14.9781 + 32.8552i −40.5163 − 27.5577i 56.2325 + 30.5600i 0.422671 − 0.244029i −43.2492 − 60.6498i
11.20 −2.31488 + 3.26211i 0.106082 − 0.0284245i −5.28270 − 15.1028i 1.43166 − 5.34302i −0.152842 + 0.411849i −10.6601 + 47.8264i 61.4956 + 17.7283i −70.1376 + 40.4940i 14.1154 + 17.0387i
See next 80 embeddings (of 248 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 11.62
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
16.f odd 4 1 inner
112.u odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 112.5.u.a ✓ 248
7.c even 3 1 inner 112.5.u.a ✓ 248
16.f odd 4 1 inner 112.5.u.a ✓ 248
112.u odd 12 1 inner 112.5.u.a ✓ 248
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.5.u.a ✓ 248 1.a even 1 1 trivial
112.5.u.a ✓ 248 7.c even 3 1 inner
112.5.u.a ✓ 248 16.f odd 4 1 inner
112.5.u.a ✓ 248 112.u odd 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(112, [\chi])\).