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Properties

Label 28.11.c.a
Level $28$
Weight $11$
Character orbit 28.c
Analytic conductor $17.790$
Analytic rank $0$
Dimension $30$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [28,11,Mod(15,28)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(28, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("28.15");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 28 = 2^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 28.c (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(17.7900030749\)
Analytic rank: \(0\)
Dimension: \(30\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 30 q + 11 q^{2} + 593 q^{4} + 3116 q^{5} - 12354 q^{6} + 27923 q^{8} - 611698 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q + 11 q^{2} + 593 q^{4} + 3116 q^{5} - 12354 q^{6} + 27923 q^{8} - 611698 q^{9} - 318668 q^{10} + 1229286 q^{12} - 145668 q^{13} - 184877 q^{14} + 4508305 q^{16} + 342764 q^{17} - 8181881 q^{18} + 9408928 q^{20} + 12710180 q^{22} - 11973190 q^{24} + 54901290 q^{25} + 59777704 q^{26} - 17563315 q^{28} - 17020404 q^{29} + 106669376 q^{30} - 27524549 q^{32} + 85980560 q^{33} - 75916990 q^{34} - 130259 q^{36} - 209815348 q^{37} - 259315990 q^{38} + 82780520 q^{40} + 533813020 q^{41} - 130926530 q^{42} + 677088452 q^{44} - 972456708 q^{45} - 697788264 q^{46} - 315974498 q^{48} - 1210608210 q^{49} + 520621301 q^{50} - 391303324 q^{52} - 1812724468 q^{53} + 2434663396 q^{54} + 575891855 q^{56} + 2908394272 q^{57} - 825819710 q^{58} - 61952816 q^{60} - 4214280820 q^{61} + 4818295012 q^{62} - 2172659863 q^{64} + 6916393304 q^{65} - 4122267960 q^{66} - 9865573282 q^{68} - 4387104240 q^{69} - 2448242076 q^{70} + 11033271319 q^{72} + 5609934060 q^{73} - 11606873054 q^{74} + 10004661906 q^{76} + 771508528 q^{77} + 3388572248 q^{78} + 7498620792 q^{80} + 5391591742 q^{81} + 16147068914 q^{82} + 4845021118 q^{84} + 7869843448 q^{85} - 1085950100 q^{86} + 20876033980 q^{88} - 6731586228 q^{89} + 9221408452 q^{90} - 46446241064 q^{92} - 5869492320 q^{93} - 1318868196 q^{94} - 49417561750 q^{96} - 28798058772 q^{97} - 443889677 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
15.1 −31.7804 − 3.74219i − 34.6436i 995.992 + 237.857i −1947.47 −129.643 + 1100.99i − 6352.45i −30763.0 − 11286.4i 57848.8 61891.5 + 7287.80i
15.2 −31.7804 + 3.74219i 34.6436i 995.992 − 237.857i −1947.47 −129.643 − 1100.99i 6352.45i −30763.0 + 11286.4i 57848.8 61891.5 − 7287.80i
15.3 −31.2978 − 6.66711i − 439.071i 935.099 + 417.331i 1820.96 −2927.34 + 13742.0i 6352.45i −26484.1 − 19295.9i −133735. −56992.0 − 12140.5i
15.4 −31.2978 + 6.66711i 439.071i 935.099 − 417.331i 1820.96 −2927.34 − 13742.0i − 6352.45i −26484.1 + 19295.9i −133735. −56992.0 + 12140.5i
15.5 −28.1282 − 15.2579i 175.922i 558.395 + 858.353i 5750.03 2684.19 − 4948.38i 6352.45i −2610.04 − 32663.9i 28100.4 −161738. − 87733.2i
15.6 −28.1282 + 15.2579i − 175.922i 558.395 − 858.353i 5750.03 2684.19 + 4948.38i − 6352.45i −2610.04 + 32663.9i 28100.4 −161738. + 87733.2i
15.7 −24.0652 − 21.0918i − 213.154i 134.268 + 1015.16i 1000.38 −4495.81 + 5129.60i − 6352.45i 18180.4 − 27262.0i 13614.4 −24074.3 − 21099.8i
15.8 −24.0652 + 21.0918i 213.154i 134.268 − 1015.16i 1000.38 −4495.81 − 5129.60i 6352.45i 18180.4 + 27262.0i 13614.4 −24074.3 + 21099.8i
15.9 −17.2855 − 26.9298i − 233.298i −426.424 + 930.988i −5341.18 −6282.65 + 4032.67i 6352.45i 32442.2 − 4609.10i 4621.14 92325.0 + 143837.i
15.10 −17.2855 + 26.9298i 233.298i −426.424 − 930.988i −5341.18 −6282.65 − 4032.67i − 6352.45i 32442.2 + 4609.10i 4621.14 92325.0 − 143837.i
15.11 −10.4016 − 30.2623i 219.639i −807.614 + 629.552i 1313.63 6646.77 − 2284.59i − 6352.45i 27452.2 + 17891.9i 10807.8 −13663.8 − 39753.5i
15.12 −10.4016 + 30.2623i − 219.639i −807.614 − 629.552i 1313.63 6646.77 + 2284.59i 6352.45i 27452.2 − 17891.9i 10807.8 −13663.8 + 39753.5i
15.13 −6.41208 − 31.3510i − 116.479i −941.770 + 402.050i 2684.06 −3651.74 + 746.874i 6352.45i 18643.4 + 26947.5i 45481.6 −17210.4 − 84148.0i
15.14 −6.41208 + 31.3510i 116.479i −941.770 − 402.050i 2684.06 −3651.74 − 746.874i − 6352.45i 18643.4 − 26947.5i 45481.6 −17210.4 + 84148.0i
15.15 2.54618 − 31.8985i − 451.444i −1011.03 − 162.439i 1306.27 −14400.4 − 1149.46i − 6352.45i −7755.84 + 31836.9i −144753. 3325.99 − 41668.0i
15.16 2.54618 + 31.8985i 451.444i −1011.03 + 162.439i 1306.27 −14400.4 + 1149.46i 6352.45i −7755.84 − 31836.9i −144753. 3325.99 + 41668.0i
15.17 4.12208 − 31.7334i 317.343i −990.017 − 261.615i −1824.36 10070.4 + 1308.11i 6352.45i −12382.9 + 30338.2i −41657.6 −7520.18 + 57893.3i
15.18 4.12208 + 31.7334i − 317.343i −990.017 + 261.615i −1824.36 10070.4 − 1308.11i − 6352.45i −12382.9 − 30338.2i −41657.6 −7520.18 − 57893.3i
15.19 10.9508 − 30.0679i − 58.3633i −784.161 − 658.535i −4849.37 −1754.86 − 639.124i − 6352.45i −28388.0 + 16366.6i 55642.7 −53104.4 + 145811.i
15.20 10.9508 + 30.0679i 58.3633i −784.161 + 658.535i −4849.37 −1754.86 + 639.124i 6352.45i −28388.0 − 16366.6i 55642.7 −53104.4 − 145811.i
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 15.30
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 28.11.c.a ✓ 30
4.b odd 2 1 inner 28.11.c.a ✓ 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.11.c.a ✓ 30 1.a even 1 1 trivial
28.11.c.a ✓ 30 4.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{11}^{\mathrm{new}}(28, [\chi])\).