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Properties

Label 48.11.i.a
Level $48$
Weight $11$
Character orbit 48.i
Analytic conductor $30.497$
Analytic rank $0$
Dimension $156$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [48,11,Mod(5,48)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(48, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 2]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("48.5");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 48.i (of order \(4\), degree \(2\), 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: \(30.4971481283\)
Analytic rank: \(0\)
Dimension: \(156\)
Relative dimension: \(78\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 156 q - 2 q^{3} - 4 q^{4} + 17212 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 156 q - 2 q^{3} - 4 q^{4} + 17212 q^{6} - 62504 q^{10} - 609152 q^{12} - 4 q^{13} - 4 q^{15} + 6057200 q^{16} - 102300 q^{18} + 5107036 q^{19} + 118096 q^{21} - 499176 q^{22} - 794552 q^{24} + 25989646 q^{27} - 39131400 q^{28} - 21772956 q^{30} - 8 q^{31} - 4 q^{33} - 155222488 q^{34} - 325616636 q^{36} - 4 q^{37} - 357544872 q^{40} - 446785160 q^{42} - 15904068 q^{43} - 19531252 q^{45} + 1080624184 q^{46} - 1345105808 q^{48} - 5326676132 q^{49} + 300520416 q^{51} + 630061824 q^{52} + 2617981672 q^{54} + 616484408 q^{58} - 1111117056 q^{60} - 1673280164 q^{61} + 1129900992 q^{63} + 7171940888 q^{64} - 4511042924 q^{66} + 3393841788 q^{67} - 118100 q^{69} - 9204615528 q^{70} + 8135656392 q^{72} + 5737368178 q^{75} + 16820532680 q^{76} - 11356100692 q^{78} - 12448971528 q^{79} - 4 q^{81} + 18989054464 q^{82} + 23899300720 q^{84} + 4838562496 q^{85} - 43708597152 q^{88} + 36517328688 q^{90} + 14890002336 q^{91} - 15001853252 q^{93} + 37194302064 q^{94} - 53113702168 q^{96} - 8 q^{97} - 35045761444 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −31.9873 − 0.900229i 71.6047 − 232.211i 1022.38 + 57.5919i 3659.65 − 3659.65i −2499.49 + 7363.34i 12667.3i −32651.3 − 2762.59i −48794.5 − 33254.7i −120357. + 113768.i
5.2 −31.9776 + 1.19690i −125.440 − 208.120i 1021.13 − 76.5482i −2743.54 + 2743.54i 4260.36 + 6505.04i 29237.4i −32561.8 + 3670.03i −27578.7 + 52213.0i 84448.1 − 91015.6i
5.3 −31.9045 + 2.46996i −234.889 + 62.2574i 1011.80 − 157.606i −1558.23 + 1558.23i 7340.26 − 2566.46i − 7777.54i −31891.7 + 7527.45i 51297.0 − 29247.2i 45865.7 − 53563.2i
5.4 −31.8886 − 2.66771i −133.494 − 203.047i 1009.77 + 170.139i −51.4996 + 51.4996i 3715.28 + 6831.02i − 20727.7i −31746.2 − 8119.26i −23407.4 + 54211.4i 1779.63 − 1504.86i
5.5 −31.8685 + 2.89830i 87.4186 + 226.731i 1007.20 − 184.729i −4304.46 + 4304.46i −3443.03 − 6972.21i − 17207.4i −31562.5 + 8806.20i −43765.0 + 39641.0i 124701. − 149652.i
5.6 −31.4508 − 5.90308i −197.906 + 141.004i 954.307 + 371.313i 3850.19 − 3850.19i 7056.67 − 3266.42i − 19172.4i −27821.9 − 17311.4i 19284.9 − 55811.1i −143820. + 98363.6i
5.7 −31.4294 + 6.01596i 241.846 + 23.6513i 951.617 − 378.156i −274.668 + 274.668i −7743.37 + 711.590i − 18500.7i −27633.8 + 17610.1i 57930.2 + 11439.9i 6980.26 − 10285.0i
5.8 −30.5628 − 9.48230i 241.685 + 25.2455i 844.172 + 579.612i 489.531 − 489.531i −7147.19 − 3063.30i 11917.6i −20304.2 − 25719.3i 57774.3 + 12202.9i −19603.3 + 10319.6i
5.9 −30.5542 + 9.51018i 115.908 + 213.575i 843.113 − 581.151i 3230.24 − 3230.24i −5572.60 − 5423.31i 9381.17i −20233.8 + 25774.7i −32179.8 + 49510.1i −67977.1 + 129417.i
5.10 −29.7823 − 11.7052i −30.4704 + 241.082i 749.974 + 697.219i −322.044 + 322.044i 3729.40 − 6823.32i 12702.5i −14174.9 − 29543.4i −57192.1 − 14691.7i 13360.8 − 5821.62i
5.11 −28.7827 + 13.9841i −137.469 + 200.378i 632.891 − 805.000i −548.987 + 548.987i 1154.62 − 7689.80i 14626.3i −6959.14 + 32020.5i −21253.7 − 55091.4i 8124.26 − 23478.4i
5.12 −28.4208 + 14.7058i 211.750 − 119.210i 591.482 − 835.898i −2554.89 + 2554.89i −4265.04 + 6501.97i 26466.3i −4517.86 + 32455.1i 30627.2 − 50485.3i 35040.4 − 110184.i
5.13 −28.1458 − 15.2254i 124.389 − 208.750i 560.371 + 857.065i −2035.08 + 2035.08i −6679.33 + 3981.54i − 7192.59i −2722.91 − 32654.7i −28103.8 − 51932.3i 88263.8 − 26293.9i
5.14 −28.1318 + 15.2513i 137.043 − 200.669i 558.799 − 858.091i 505.821 − 505.821i −794.826 + 7735.27i − 18457.9i −2633.06 + 32662.0i −21487.2 − 55000.8i −6515.27 + 21944.1i
5.15 −27.0943 + 17.0265i −223.162 − 96.1657i 444.197 − 922.640i 1879.86 − 1879.86i 7683.77 − 1194.12i 2133.29i 3674.13 + 32561.4i 40553.3 + 42921.0i −18926.0 + 82941.0i
5.16 −26.6345 − 17.7370i −241.073 − 30.5385i 394.797 + 944.834i 1987.96 − 1987.96i 5879.22 + 5089.30i 33535.4i 6243.30 − 32167.7i 57183.8 + 14724.1i −88208.7 + 17687.9i
5.17 −25.2398 − 19.6712i −242.937 + 5.52323i 250.091 + 992.991i −2917.67 + 2917.67i 6240.33 + 4639.45i − 7520.61i 13221.0 − 29982.4i 58988.0 − 2683.60i 131035. − 16247.3i
5.18 −24.6932 − 20.3530i 149.105 + 191.877i 195.511 + 1005.16i 1745.93 − 1745.93i 223.377 − 7772.79i − 29296.5i 15630.3 − 28799.9i −14584.3 + 57219.6i −78647.5 + 7577.74i
5.19 −22.0556 + 23.1851i −60.4946 − 235.350i −51.0973 − 1022.72i −3857.18 + 3857.18i 6790.85 + 3788.21i − 18917.7i 24838.9 + 21372.1i −51729.8 + 28474.7i −4356.50 − 174502.i
5.20 −22.0040 − 23.2341i −133.632 − 202.957i −55.6449 + 1022.49i 1944.43 − 1944.43i −1775.07 + 7570.69i − 11713.7i 24981.0 − 21206.0i −23333.9 + 54243.1i −87962.5 − 2391.74i
See next 80 embeddings (of 156 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.78
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
16.e even 4 1 inner
48.i odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.11.i.a ✓ 156
3.b odd 2 1 inner 48.11.i.a ✓ 156
16.e even 4 1 inner 48.11.i.a ✓ 156
48.i odd 4 1 inner 48.11.i.a ✓ 156
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.11.i.a ✓ 156 1.a even 1 1 trivial
48.11.i.a ✓ 156 3.b odd 2 1 inner
48.11.i.a ✓ 156 16.e even 4 1 inner
48.11.i.a ✓ 156 48.i odd 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{11}^{\mathrm{new}}(48, [\chi])\).