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Properties

Label 60.7.c.a
Level $60$
Weight $7$
Character orbit 60.c
Analytic conductor $13.803$
Analytic rank $0$
Dimension $24$
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] = [60,7,Mod(31,60)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(60, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("60.31");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 60.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: \(13.8032450172\)
Analytic rank: \(0\)
Dimension: \(24\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 20 q^{2} - 246 q^{4} + 162 q^{6} - 340 q^{8} - 5832 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 20 q^{2} - 246 q^{4} + 162 q^{6} - 340 q^{8} - 5832 q^{9} - 750 q^{10} + 5040 q^{13} - 2596 q^{14} + 4194 q^{16} - 4860 q^{18} + 7000 q^{20} + 19440 q^{21} + 45780 q^{22} + 24786 q^{24} + 75000 q^{25} + 75852 q^{26} + 54300 q^{28} + 132800 q^{29} - 10700 q^{32} - 173484 q^{34} + 59778 q^{36} - 69840 q^{37} + 215800 q^{38} - 14250 q^{40} - 70448 q^{41} - 189540 q^{42} - 395668 q^{44} - 158760 q^{46} + 252720 q^{48} - 642984 q^{49} + 62500 q^{50} - 210240 q^{52} - 644320 q^{53} - 39366 q^{54} - 917708 q^{56} + 408240 q^{57} - 1345020 q^{58} + 222750 q^{60} - 222864 q^{61} + 1948520 q^{62} + 935922 q^{64} + 266000 q^{65} - 640548 q^{66} + 572680 q^{68} - 541728 q^{69} + 220500 q^{70} + 82620 q^{72} + 771120 q^{73} - 589164 q^{74} - 191544 q^{76} + 1383840 q^{77} - 693360 q^{78} - 946000 q^{80} + 1417176 q^{81} + 2672520 q^{82} + 1256796 q^{84} - 372000 q^{85} + 1781528 q^{86} + 956940 q^{88} - 1566224 q^{89} + 182250 q^{90} - 3040560 q^{92} + 1496880 q^{93} - 3788352 q^{94} - 413262 q^{96} - 1666800 q^{97} - 2709660 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1 −7.80071 − 1.77450i − 15.5885i 57.7023 + 27.6847i 55.9017 −27.6617 + 121.601i 264.480i −400.993 − 318.353i −243.000 −436.073 − 99.1975i
31.2 −7.80071 + 1.77450i 15.5885i 57.7023 − 27.6847i 55.9017 −27.6617 − 121.601i − 264.480i −400.993 + 318.353i −243.000 −436.073 + 99.1975i
31.3 −6.61106 − 4.50487i 15.5885i 23.4123 + 59.5639i −55.9017 70.2239 − 103.056i − 489.339i 113.547 − 499.250i −243.000 369.570 + 251.830i
31.4 −6.61106 + 4.50487i − 15.5885i 23.4123 − 59.5639i −55.9017 70.2239 + 103.056i 489.339i 113.547 + 499.250i −243.000 369.570 − 251.830i
31.5 −3.53904 − 7.17462i − 15.5885i −38.9505 + 50.7825i 55.9017 −111.841 + 55.1681i 99.9522i 502.192 + 99.7338i −243.000 −197.838 − 401.074i
31.6 −3.53904 + 7.17462i 15.5885i −38.9505 − 50.7825i 55.9017 −111.841 − 55.1681i − 99.9522i 502.192 − 99.7338i −243.000 −197.838 + 401.074i
31.7 −3.08457 − 7.38143i 15.5885i −44.9709 + 45.5370i 55.9017 115.065 − 48.0836i − 200.003i 474.844 + 191.487i −243.000 −172.433 − 412.634i
31.8 −3.08457 + 7.38143i − 15.5885i −44.9709 − 45.5370i 55.9017 115.065 + 48.0836i 200.003i 474.844 − 191.487i −243.000 −172.433 + 412.634i
31.9 −1.36400 − 7.88286i 15.5885i −60.2790 + 21.5045i −55.9017 122.882 − 21.2627i 86.8984i 251.737 + 445.839i −243.000 76.2500 + 440.665i
31.10 −1.36400 + 7.88286i − 15.5885i −60.2790 − 21.5045i −55.9017 122.882 + 21.2627i − 86.8984i 251.737 − 445.839i −243.000 76.2500 − 440.665i
31.11 0.331992 − 7.99311i − 15.5885i −63.7796 − 5.30729i −55.9017 −124.600 − 5.17524i − 552.003i −63.5960 + 508.035i −243.000 −18.5589 + 446.828i
31.12 0.331992 + 7.99311i 15.5885i −63.7796 + 5.30729i −55.9017 −124.600 + 5.17524i 552.003i −63.5960 − 508.035i −243.000 −18.5589 − 446.828i
31.13 2.59351 − 7.56794i − 15.5885i −50.5474 − 39.2551i −55.9017 −117.972 − 40.4289i 671.100i −428.176 + 280.731i −243.000 −144.982 + 423.061i
31.14 2.59351 + 7.56794i 15.5885i −50.5474 + 39.2551i −55.9017 −117.972 + 40.4289i − 671.100i −428.176 − 280.731i −243.000 −144.982 − 423.061i
31.15 3.43879 − 7.22321i 15.5885i −40.3494 − 49.6782i 55.9017 112.599 + 53.6055i 472.024i −497.589 + 120.619i −243.000 192.234 − 403.790i
31.16 3.43879 + 7.22321i − 15.5885i −40.3494 + 49.6782i 55.9017 112.599 − 53.6055i − 472.024i −497.589 − 120.619i −243.000 192.234 + 403.790i
31.17 5.21792 − 6.06410i − 15.5885i −9.54653 − 63.2840i 55.9017 −94.5299 − 81.3394i − 234.050i −433.573 − 272.320i −243.000 291.691 − 338.993i
31.18 5.21792 + 6.06410i 15.5885i −9.54653 + 63.2840i 55.9017 −94.5299 + 81.3394i 234.050i −433.573 + 272.320i −243.000 291.691 + 338.993i
31.19 5.40467 − 5.89827i 15.5885i −5.57916 − 63.7564i −55.9017 91.9449 + 84.2504i − 125.427i −406.206 − 311.674i −243.000 −302.130 + 329.723i
31.20 5.40467 + 5.89827i − 15.5885i −5.57916 + 63.7564i −55.9017 91.9449 − 84.2504i 125.427i −406.206 + 311.674i −243.000 −302.130 − 329.723i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.24
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 60.7.c.a ✓ 24
3.b odd 2 1 180.7.c.b 24
4.b odd 2 1 inner 60.7.c.a ✓ 24
12.b even 2 1 180.7.c.b 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.7.c.a ✓ 24 1.a even 1 1 trivial
60.7.c.a ✓ 24 4.b odd 2 1 inner
180.7.c.b 24 3.b odd 2 1
180.7.c.b 24 12.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(60, [\chi])\).