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Properties

Label 180.7.c.b
Level $180$
Weight $7$
Character orbit 180.c
Analytic conductor $41.410$
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] = [180,7,Mod(91,180)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(180, 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("180.91");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 180.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: \(41.4097350516\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: no (minimal twist has level 60)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 20 q^{2} - 246 q^{4} + 340 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 20 q^{2} - 246 q^{4} + 340 q^{8} - 750 q^{10} + 5040 q^{13} + 2596 q^{14} + 4194 q^{16} - 7000 q^{20} + 45780 q^{22} + 75000 q^{25} - 75852 q^{26} + 54300 q^{28} - 132800 q^{29} + 10700 q^{32} - 173484 q^{34} - 69840 q^{37} - 215800 q^{38} - 14250 q^{40} + 70448 q^{41} + 395668 q^{44} - 158760 q^{46} - 642984 q^{49} - 62500 q^{50} - 210240 q^{52} + 644320 q^{53} + 917708 q^{56} - 1345020 q^{58} - 222864 q^{61} - 1948520 q^{62} + 935922 q^{64} - 266000 q^{65} - 572680 q^{68} + 220500 q^{70} + 771120 q^{73} + 589164 q^{74} - 191544 q^{76} - 1383840 q^{77} + 946000 q^{80} + 2672520 q^{82} - 372000 q^{85} - 1781528 q^{86} + 956940 q^{88} + 1566224 q^{89} + 3040560 q^{92} - 3788352 q^{94} - 1666800 q^{97} + 2709660 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
91.1 −7.99899 − 0.126876i 0 63.9678 + 2.02976i 55.9017 0 335.195i −511.421 − 24.3521i 0 −447.157 − 7.09260i
91.2 −7.99899 + 0.126876i 0 63.9678 − 2.02976i 55.9017 0 − 335.195i −511.421 + 24.3521i 0 −447.157 + 7.09260i
91.3 −7.41350 − 3.00666i 0 45.9200 + 44.5798i −55.9017 0 453.408i −206.391 − 468.558i 0 414.427 + 168.078i
91.4 −7.41350 + 3.00666i 0 45.9200 − 44.5798i −55.9017 0 − 453.408i −206.391 + 468.558i 0 414.427 − 168.078i
91.5 −5.40467 − 5.89827i 0 −5.57916 + 63.7564i 55.9017 0 125.427i 406.206 − 311.674i 0 −302.130 − 329.723i
91.6 −5.40467 + 5.89827i 0 −5.57916 − 63.7564i 55.9017 0 − 125.427i 406.206 + 311.674i 0 −302.130 + 329.723i
91.7 −5.21792 − 6.06410i 0 −9.54653 + 63.2840i −55.9017 0 234.050i 433.573 − 272.320i 0 291.691 + 338.993i
91.8 −5.21792 + 6.06410i 0 −9.54653 − 63.2840i −55.9017 0 − 234.050i 433.573 + 272.320i 0 291.691 − 338.993i
91.9 −3.43879 − 7.22321i 0 −40.3494 + 49.6782i −55.9017 0 − 472.024i 497.589 + 120.619i 0 192.234 + 403.790i
91.10 −3.43879 + 7.22321i 0 −40.3494 − 49.6782i −55.9017 0 472.024i 497.589 − 120.619i 0 192.234 − 403.790i
91.11 −2.59351 − 7.56794i 0 −50.5474 + 39.2551i 55.9017 0 − 671.100i 428.176 + 280.731i 0 −144.982 − 423.061i
91.12 −2.59351 + 7.56794i 0 −50.5474 − 39.2551i 55.9017 0 671.100i 428.176 − 280.731i 0 −144.982 + 423.061i
91.13 −0.331992 − 7.99311i 0 −63.7796 + 5.30729i 55.9017 0 552.003i 63.5960 + 508.035i 0 −18.5589 − 446.828i
91.14 −0.331992 + 7.99311i 0 −63.7796 − 5.30729i 55.9017 0 − 552.003i 63.5960 − 508.035i 0 −18.5589 + 446.828i
91.15 1.36400 − 7.88286i 0 −60.2790 − 21.5045i 55.9017 0 − 86.8984i −251.737 + 445.839i 0 76.2500 − 440.665i
91.16 1.36400 + 7.88286i 0 −60.2790 + 21.5045i 55.9017 0 86.8984i −251.737 − 445.839i 0 76.2500 + 440.665i
91.17 3.08457 − 7.38143i 0 −44.9709 − 45.5370i −55.9017 0 200.003i −474.844 + 191.487i 0 −172.433 + 412.634i
91.18 3.08457 + 7.38143i 0 −44.9709 + 45.5370i −55.9017 0 − 200.003i −474.844 − 191.487i 0 −172.433 − 412.634i
91.19 3.53904 − 7.17462i 0 −38.9505 − 50.7825i −55.9017 0 − 99.9522i −502.192 + 99.7338i 0 −197.838 + 401.074i
91.20 3.53904 + 7.17462i 0 −38.9505 + 50.7825i −55.9017 0 99.9522i −502.192 − 99.7338i 0 −197.838 − 401.074i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 91.24
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 180.7.c.b 24
3.b odd 2 1 60.7.c.a ✓ 24
4.b odd 2 1 inner 180.7.c.b 24
12.b even 2 1 60.7.c.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.7.c.a ✓ 24 3.b odd 2 1
60.7.c.a ✓ 24 12.b even 2 1
180.7.c.b 24 1.a even 1 1 trivial
180.7.c.b 24 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{24} + 1733280 T_{7}^{22} + 1268405172480 T_{7}^{20} + \cdots + 30\!\cdots\!16 \) acting on \(S_{7}^{\mathrm{new}}(180, [\chi])\). Copy content Toggle raw display