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Properties

Label 48.5.l.a
Level $48$
Weight $5$
Character orbit 48.l
Analytic conductor $4.962$
Analytic rank $0$
Dimension $32$
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] = [48,5,Mod(19,48)]
 
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([2, 3, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("48.19");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 48.l (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: \(4.96175822802\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 12 q^{4} + 180 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 12 q^{4} + 180 q^{8} + 296 q^{10} - 192 q^{11} + 360 q^{12} - 156 q^{14} + 352 q^{16} - 324 q^{18} + 704 q^{19} - 1200 q^{20} - 1568 q^{22} - 2304 q^{23} + 1188 q^{24} + 2700 q^{26} + 4680 q^{28} - 1728 q^{29} + 1512 q^{30} - 3360 q^{32} - 9312 q^{34} - 5184 q^{35} - 756 q^{36} + 3648 q^{37} - 5880 q^{38} + 5232 q^{40} + 4500 q^{42} + 1088 q^{43} + 18840 q^{44} + 680 q^{46} + 2160 q^{48} + 10976 q^{49} - 25884 q^{50} - 4032 q^{51} - 25584 q^{52} + 960 q^{53} + 972 q^{54} + 11776 q^{55} + 15456 q^{56} + 12624 q^{58} + 13056 q^{59} + 7992 q^{60} + 3776 q^{61} + 21852 q^{62} - 8664 q^{64} + 4032 q^{65} - 8856 q^{66} - 896 q^{67} - 17280 q^{68} - 9792 q^{69} - 18240 q^{70} - 39936 q^{71} + 4860 q^{72} + 24204 q^{74} - 1152 q^{75} + 16776 q^{76} + 9408 q^{77} - 3780 q^{78} - 14232 q^{80} - 23328 q^{81} - 33800 q^{82} + 24000 q^{83} - 11448 q^{84} - 11200 q^{85} - 1200 q^{86} - 11424 q^{88} + 4104 q^{90} + 30528 q^{91} - 11664 q^{92} - 8040 q^{94} + 10080 q^{96} + 52968 q^{98} - 5184 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −3.98297 − 0.368670i −3.67423 + 3.67423i 15.7282 + 2.93680i −9.21870 + 9.21870i 15.9890 − 13.2798i 17.0639 −61.5622 − 17.4957i − 27.0000i 40.1165 − 33.3192i
19.2 −3.94725 − 0.647479i 3.67423 − 3.67423i 15.1615 + 5.11152i −21.2428 + 21.2428i −16.8821 + 12.1241i 35.6927 −56.5368 − 29.9932i − 27.0000i 97.6049 − 70.0964i
19.3 −3.15376 + 2.46044i −3.67423 + 3.67423i 3.89246 − 15.5193i 18.4189 − 18.4189i 2.54743 − 20.6279i 4.95460 25.9084 + 58.5214i − 27.0000i −12.7702 + 103.407i
19.4 −3.09682 + 2.53174i 3.67423 − 3.67423i 3.18056 − 15.6807i −7.37831 + 7.37831i −2.07622 + 20.6807i −75.1984 29.8498 + 56.6126i − 27.0000i 4.16931 − 41.5293i
19.5 −2.82809 − 2.82877i 3.67423 − 3.67423i −0.00384203 + 16.0000i 21.6601 − 21.6601i −20.7846 − 0.00249547i 15.5177 45.2711 − 45.2385i − 27.0000i −122.528 − 0.0147111i
19.6 −1.62589 − 3.65465i −3.67423 + 3.67423i −10.7130 + 11.8841i −2.10268 + 2.10268i 19.4020 + 7.45415i 84.8276 60.8504 + 19.8299i − 27.0000i 11.1033 + 4.26583i
19.7 −0.922521 + 3.89217i 3.67423 − 3.67423i −14.2979 − 7.18121i 17.5812 − 17.5812i 10.9112 + 17.6903i 50.9945 41.1406 − 49.0250i − 27.0000i 52.2099 + 84.6479i
19.8 −0.827018 − 3.91357i 3.67423 − 3.67423i −14.6321 + 6.47318i −25.9147 + 25.9147i −17.4180 − 11.3407i −48.6273 37.4343 + 51.9103i − 27.0000i 122.851 + 79.9872i
19.9 −0.149438 + 3.99721i −3.67423 + 3.67423i −15.9553 − 1.19467i −0.419719 + 0.419719i −14.1376 − 15.2357i −40.4181 7.15966 − 63.5983i − 27.0000i −1.61498 − 1.74043i
19.10 0.896210 − 3.89831i −3.67423 + 3.67423i −14.3936 − 6.98741i 30.1272 − 30.1272i 11.0304 + 17.6162i −80.6454 −40.1388 + 49.8486i − 27.0000i −90.4449 − 144.446i
19.11 2.05307 − 3.43291i −3.67423 + 3.67423i −7.56978 − 14.0960i −34.5052 + 34.5052i 5.06985 + 20.1568i −15.7115 −63.9318 − 2.95383i − 27.0000i 47.6116 + 189.295i
19.12 2.69875 − 2.95241i 3.67423 − 3.67423i −1.43350 − 15.9357i 5.17564 − 5.17564i −0.932024 − 20.7637i 6.77541 −50.9173 − 38.7741i − 27.0000i −1.31288 − 29.2484i
19.13 3.43325 + 2.05251i −3.67423 + 3.67423i 7.57441 + 14.0936i −15.8310 + 15.8310i −20.1560 + 5.07316i −14.0988 −2.92234 + 63.9332i − 27.0000i −86.8452 + 21.8585i
19.14 3.75428 − 1.38036i −3.67423 + 3.67423i 12.1892 − 10.3645i 13.5312 − 13.5312i −8.72233 + 18.8659i 44.0276 31.4549 − 55.7368i − 27.0000i 32.1220 − 69.4780i
19.15 3.79804 + 1.25496i 3.67423 − 3.67423i 12.8502 + 9.53276i 27.4836 − 27.4836i 18.5659 − 9.34386i −74.9802 36.8422 + 52.3322i − 27.0000i 138.874 − 69.8928i
19.16 3.90016 + 0.888110i 3.67423 − 3.67423i 14.4225 + 6.92754i −17.3646 + 17.3646i 17.5932 − 11.0670i 89.8255 50.0978 + 39.8273i − 27.0000i −83.1466 + 52.3032i
43.1 −3.98297 + 0.368670i −3.67423 − 3.67423i 15.7282 − 2.93680i −9.21870 − 9.21870i 15.9890 + 13.2798i 17.0639 −61.5622 + 17.4957i 27.0000i 40.1165 + 33.3192i
43.2 −3.94725 + 0.647479i 3.67423 + 3.67423i 15.1615 − 5.11152i −21.2428 − 21.2428i −16.8821 − 12.1241i 35.6927 −56.5368 + 29.9932i 27.0000i 97.6049 + 70.0964i
43.3 −3.15376 − 2.46044i −3.67423 − 3.67423i 3.89246 + 15.5193i 18.4189 + 18.4189i 2.54743 + 20.6279i 4.95460 25.9084 − 58.5214i 27.0000i −12.7702 − 103.407i
43.4 −3.09682 − 2.53174i 3.67423 + 3.67423i 3.18056 + 15.6807i −7.37831 − 7.37831i −2.07622 − 20.6807i −75.1984 29.8498 − 56.6126i 27.0000i 4.16931 + 41.5293i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.f odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.5.l.a ✓ 32
3.b odd 2 1 144.5.m.c 32
4.b odd 2 1 192.5.l.a 32
8.b even 2 1 384.5.l.b 32
8.d odd 2 1 384.5.l.a 32
12.b even 2 1 576.5.m.b 32
16.e even 4 1 192.5.l.a 32
16.e even 4 1 384.5.l.a 32
16.f odd 4 1 inner 48.5.l.a ✓ 32
16.f odd 4 1 384.5.l.b 32
48.i odd 4 1 576.5.m.b 32
48.k even 4 1 144.5.m.c 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.5.l.a ✓ 32 1.a even 1 1 trivial
48.5.l.a ✓ 32 16.f odd 4 1 inner
144.5.m.c 32 3.b odd 2 1
144.5.m.c 32 48.k even 4 1
192.5.l.a 32 4.b odd 2 1
192.5.l.a 32 16.e even 4 1
384.5.l.a 32 8.d odd 2 1
384.5.l.a 32 16.e even 4 1
384.5.l.b 32 8.b even 2 1
384.5.l.b 32 16.f odd 4 1
576.5.m.b 32 12.b even 2 1
576.5.m.b 32 48.i odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(48, [\chi])\).