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Properties

Label 124.4.f.a
Level $124$
Weight $4$
Character orbit 124.f
Analytic conductor $7.316$
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] = [124,4,Mod(33,124)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(124, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 8]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("124.33");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 124 = 2^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 124.f (of order \(5\), degree \(4\), 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: \(7.31623684071\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 2 q^{3} + 8 q^{5} - 34 q^{7} - 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 2 q^{3} + 8 q^{5} - 34 q^{7} - 40 q^{9} + 52 q^{11} + 106 q^{13} - 190 q^{15} + 78 q^{17} - 64 q^{19} + 58 q^{21} - 348 q^{23} + 748 q^{25} - 166 q^{27} + 186 q^{29} + 440 q^{31} - 390 q^{33} + 148 q^{35} - 1044 q^{37} - 2 q^{39} - 172 q^{41} + 798 q^{43} - 530 q^{45} - 390 q^{47} + 1008 q^{49} + 1574 q^{51} + 272 q^{53} + 1116 q^{55} - 3660 q^{57} + 860 q^{59} - 2160 q^{61} + 2072 q^{63} - 3134 q^{65} - 660 q^{67} - 1234 q^{69} - 1368 q^{71} + 486 q^{73} + 2520 q^{75} + 1736 q^{77} - 1392 q^{79} - 2220 q^{81} - 1034 q^{83} + 1216 q^{85} + 3540 q^{87} + 2378 q^{89} + 1422 q^{91} + 4054 q^{93} + 3406 q^{95} - 1400 q^{97} - 3408 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
33.1 0 −6.50048 + 4.72287i 0 −12.3463 0 1.31444 + 4.04544i 0 11.6072 − 35.7234i 0
33.2 0 −5.31146 + 3.85900i 0 18.5822 0 −5.82471 − 17.9266i 0 4.97627 − 15.3154i 0
33.3 0 −4.02575 + 2.92488i 0 4.83677 0 3.45244 + 10.6255i 0 −0.691709 + 2.12886i 0
33.4 0 0.891385 − 0.647629i 0 0.734265 0 −10.4203 − 32.0705i 0 −7.96831 + 24.5240i 0
33.5 0 1.36862 − 0.994358i 0 −11.9252 0 1.24456 + 3.83037i 0 −7.45910 + 22.9567i 0
33.6 0 2.70721 − 1.96690i 0 10.1972 0 8.36053 + 25.7311i 0 −4.88319 + 15.0289i 0
33.7 0 6.93521 − 5.03872i 0 14.5615 0 −3.88859 − 11.9679i 0 14.3649 − 44.2107i 0
33.8 0 7.78938 − 5.65931i 0 −18.1683 0 0.615771 + 1.89515i 0 20.3031 − 62.4866i 0
97.1 0 −2.70767 + 8.33336i 0 2.46797 0 −28.2119 + 20.4972i 0 −40.2699 − 29.2578i 0
97.2 0 −2.20223 + 6.77778i 0 14.4214 0 26.8980 − 19.5425i 0 −19.2450 − 13.9823i 0
97.3 0 −2.03738 + 6.27041i 0 −8.85060 0 3.57653 − 2.59850i 0 −13.3236 − 9.68020i 0
97.4 0 −0.339973 + 1.04633i 0 −16.5589 0 13.3202 − 9.67768i 0 20.8642 + 15.1588i 0
97.5 0 −0.123140 + 0.378986i 0 −5.10927 0 −4.64807 + 3.37702i 0 21.7150 + 15.7769i 0
97.6 0 0.0889905 − 0.273885i 0 16.1523 0 −15.2462 + 11.0770i 0 21.7764 + 15.8215i 0
97.7 0 2.00131 − 6.15941i 0 7.45090 0 12.5346 − 9.10689i 0 −12.0896 − 8.78358i 0
97.8 0 2.46599 − 7.58955i 0 −12.4459 0 −20.0772 + 14.5869i 0 −29.6767 − 21.5614i 0
101.1 0 −2.70767 − 8.33336i 0 2.46797 0 −28.2119 − 20.4972i 0 −40.2699 + 29.2578i 0
101.2 0 −2.20223 − 6.77778i 0 14.4214 0 26.8980 + 19.5425i 0 −19.2450 + 13.9823i 0
101.3 0 −2.03738 − 6.27041i 0 −8.85060 0 3.57653 + 2.59850i 0 −13.3236 + 9.68020i 0
101.4 0 −0.339973 − 1.04633i 0 −16.5589 0 13.3202 + 9.67768i 0 20.8642 − 15.1588i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 33.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 124.4.f.a ✓ 32
31.d even 5 1 inner 124.4.f.a ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
124.4.f.a ✓ 32 1.a even 1 1 trivial
124.4.f.a ✓ 32 31.d even 5 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(124, [\chi])\).