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Properties

Label 60.12.i.a
Level $60$
Weight $12$
Character orbit 60.i
Analytic conductor $46.101$
Analytic rank $0$
Dimension $44$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [60,12,Mod(17,60)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(60, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 2, 1]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("60.17");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 60.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: \(46.1005908336\)
Analytic rank: \(0\)
Dimension: \(44\)
Relative dimension: \(22\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 44 q + 506 q^{3} - 31508 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 44 q + 506 q^{3} - 31508 q^{7} + 3569724 q^{13} + 4045190 q^{15} - 58590844 q^{21} - 37491380 q^{25} + 37434038 q^{27} - 14255824 q^{31} + 155475980 q^{33} + 803791356 q^{37} + 1630282884 q^{43} - 984426620 q^{45} - 3819070820 q^{51} - 4345914000 q^{55} + 9776627492 q^{57} + 16883428536 q^{61} + 6520377704 q^{63} + 17783240332 q^{67} - 50848391668 q^{73} + 28006771810 q^{75} - 27998403956 q^{81} + 79641662620 q^{85} + 33539593220 q^{87} + 84318294264 q^{91} - 17716290136 q^{93} - 107136772308 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 0 −420.864 + 4.56531i 0 −1727.82 − 6770.73i 0 12208.3 + 12208.3i 0 177105. − 3842.75i 0
17.2 0 −407.948 − 103.564i 0 −2623.93 + 6476.35i 0 −33585.9 − 33585.9i 0 155696. + 84497.7i 0
17.3 0 −380.780 + 179.315i 0 −5107.99 + 4768.28i 0 46364.1 + 46364.1i 0 112839. − 136559.i 0
17.4 0 −377.508 + 186.104i 0 6757.95 + 1777.13i 0 −29650.5 − 29650.5i 0 107878. − 140511.i 0
17.5 0 −292.455 − 302.683i 0 5061.01 + 4818.12i 0 47707.1 + 47707.1i 0 −6086.72 + 177042.i 0
17.6 0 −288.925 − 306.054i 0 4824.77 − 5054.67i 0 −11294.7 − 11294.7i 0 −10191.4 + 176854.i 0
17.7 0 −247.726 − 340.263i 0 −6465.86 − 2649.68i 0 −21842.9 − 21842.9i 0 −54410.9 + 168584.i 0
17.8 0 −186.104 + 377.508i 0 −6757.95 − 1777.13i 0 −29650.5 − 29650.5i 0 −107878. − 140511.i 0
17.9 0 −179.315 + 380.780i 0 5107.99 − 4768.28i 0 46364.1 + 46364.1i 0 −112839. − 136559.i 0
17.10 0 −23.7965 − 420.215i 0 −6454.24 + 2677.85i 0 24221.8 + 24221.8i 0 −176014. + 19999.3i 0
17.11 0 −4.56531 + 420.864i 0 1727.82 + 6770.73i 0 12208.3 + 12208.3i 0 −177105. − 3842.75i 0
17.12 0 76.3853 − 413.899i 0 2459.31 + 6540.64i 0 −60188.9 − 60188.9i 0 −165478. − 63231.5i 0
17.13 0 103.564 + 407.948i 0 2623.93 − 6476.35i 0 −33585.9 − 33585.9i 0 −155696. + 84497.7i 0
17.14 0 107.415 − 406.951i 0 5767.22 − 3945.55i 0 −7954.92 − 7954.92i 0 −154071. − 87425.3i 0
17.15 0 239.351 − 346.205i 0 −3251.24 − 6185.27i 0 26139.5 + 26139.5i 0 −62569.2 − 165729.i 0
17.16 0 302.683 + 292.455i 0 −5061.01 − 4818.12i 0 47707.1 + 47707.1i 0 6086.72 + 177042.i 0
17.17 0 306.054 + 288.925i 0 −4824.77 + 5054.67i 0 −11294.7 − 11294.7i 0 10191.4 + 176854.i 0
17.18 0 340.263 + 247.726i 0 6465.86 + 2649.68i 0 −21842.9 − 21842.9i 0 54410.9 + 168584.i 0
17.19 0 346.205 − 239.351i 0 3251.24 + 6185.27i 0 26139.5 + 26139.5i 0 62569.2 − 165729.i 0
17.20 0 406.951 − 107.415i 0 −5767.22 + 3945.55i 0 −7954.92 − 7954.92i 0 154071. − 87425.3i 0
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 60.12.i.a ✓ 44
3.b odd 2 1 inner 60.12.i.a ✓ 44
5.c odd 4 1 inner 60.12.i.a ✓ 44
15.e even 4 1 inner 60.12.i.a ✓ 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.12.i.a ✓ 44 1.a even 1 1 trivial
60.12.i.a ✓ 44 3.b odd 2 1 inner
60.12.i.a ✓ 44 5.c odd 4 1 inner
60.12.i.a ✓ 44 15.e even 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{12}^{\mathrm{new}}(60, [\chi])\).