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Properties

Label 198.2.m.b
Level $198$
Weight $2$
Character orbit 198.m
Analytic conductor $1.581$
Analytic rank $0$
Dimension $56$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [198,2,Mod(25,198)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(198, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([20, 24]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("198.25");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 198 = 2 \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 198.m (of order \(15\), degree \(8\), 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: \(1.58103796002\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(7\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 7 q^{2} + 3 q^{3} + 7 q^{4} - q^{5} - 5 q^{6} - 14 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q + 7 q^{2} + 3 q^{3} + 7 q^{4} - q^{5} - 5 q^{6} - 14 q^{8} + 9 q^{9} - 8 q^{10} - 4 q^{11} - 8 q^{12} - 4 q^{13} + 11 q^{15} + 7 q^{16} + 15 q^{18} - q^{20} + 2 q^{21} - 4 q^{22} - 14 q^{23} - 2 q^{24} + 22 q^{25} - 12 q^{26} - 48 q^{27} + 14 q^{29} + 2 q^{30} + 5 q^{31} - 28 q^{32} - 44 q^{33} - 20 q^{34} - 20 q^{35} - 9 q^{36} + 2 q^{37} + 5 q^{38} - 34 q^{39} - q^{40} - 14 q^{41} + 4 q^{42} + 6 q^{43} - 12 q^{44} + 8 q^{46} + 11 q^{47} + 35 q^{49} + 2 q^{50} - 69 q^{51} + 6 q^{52} - 50 q^{53} + 4 q^{54} + 6 q^{55} + 36 q^{57} + 14 q^{58} + 10 q^{59} + 12 q^{60} - 24 q^{61} - 10 q^{62} - 16 q^{63} - 14 q^{64} - 42 q^{65} - 6 q^{66} - 56 q^{67} + 70 q^{69} - 6 q^{71} + 9 q^{72} + 68 q^{73} - q^{74} - 92 q^{75} - 10 q^{76} + 10 q^{77} + 60 q^{78} - 12 q^{79} + 2 q^{80} - 67 q^{81} - 52 q^{82} - 25 q^{83} + 4 q^{84} - 7 q^{85} + 11 q^{86} + 86 q^{87} - 4 q^{88} + 104 q^{89} + 25 q^{90} - 158 q^{91} + 11 q^{92} + 95 q^{93} - 19 q^{94} + 68 q^{95} - 3 q^{96} + 32 q^{97} + 120 q^{98} + 155 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1 −0.104528 − 0.994522i −1.71075 + 0.270798i −0.978148 + 0.207912i −0.104099 + 0.990431i 0.448137 + 1.67307i 1.84731 + 2.05165i 0.309017 + 0.951057i 2.85334 − 0.926537i 0.995887
25.2 −0.104528 − 0.994522i −1.39978 − 1.02011i −0.978148 + 0.207912i 0.0870823 − 0.828533i −0.868200 + 1.49874i −2.21319 − 2.45799i 0.309017 + 0.951057i 0.918771 + 2.85585i −0.833096
25.3 −0.104528 − 0.994522i 0.156905 + 1.72493i −0.978148 + 0.207912i −0.109789 + 1.04457i 1.69908 − 0.336350i 1.12247 + 1.24663i 0.309017 + 0.951057i −2.95076 + 0.541300i 1.05032
25.4 −0.104528 − 0.994522i 0.730501 − 1.57047i −0.978148 + 0.207912i −0.0332930 + 0.316762i −1.63822 − 0.562341i −2.24672 − 2.49524i 0.309017 + 0.951057i −1.93274 − 2.29446i 0.318507
25.5 −0.104528 − 0.994522i 1.33058 + 1.10885i −0.978148 + 0.207912i 0.391167 − 3.72171i 0.963697 − 1.43920i −2.41346 − 2.68042i 0.309017 + 0.951057i 0.540882 + 2.95084i −3.74221
25.6 −0.104528 − 0.994522i 1.41412 − 1.00013i −0.978148 + 0.207912i 0.305557 − 2.90718i −1.14247 − 1.30183i 3.46795 + 3.85155i 0.309017 + 0.951057i 0.999479 − 2.82861i −2.92320
25.7 −0.104528 − 0.994522i 1.73023 + 0.0793834i −0.978148 + 0.207912i −0.432098 + 4.11114i −0.101910 − 1.72905i 0.435640 + 0.483827i 0.309017 + 0.951057i 2.98740 + 0.274703i 4.13379
31.1 −0.978148 + 0.207912i −1.61110 + 0.635883i 0.913545 − 0.406737i 1.60660 + 0.341493i 1.44369 − 0.956955i 0.156191 − 1.48606i −0.809017 + 0.587785i 2.19130 − 2.04895i −1.64249
31.2 −0.978148 + 0.207912i −1.31310 − 1.12949i 0.913545 − 0.406737i −2.36093 − 0.501830i 1.51925 + 0.831802i −0.355834 + 3.38554i −0.809017 + 0.587785i 0.448487 + 2.96629i 2.41367
31.3 −0.978148 + 0.207912i −0.150983 − 1.72546i 0.913545 − 0.406737i 1.11581 + 0.237173i 0.506427 + 1.65636i 0.329162 − 3.13177i −0.809017 + 0.587785i −2.95441 + 0.521031i −1.14074
31.4 −0.978148 + 0.207912i −0.0531070 + 1.73124i 0.913545 − 0.406737i −2.56708 − 0.545650i −0.307998 − 1.70445i −0.160692 + 1.52888i −0.809017 + 0.587785i −2.99436 − 0.183881i 2.62443
31.5 −0.978148 + 0.207912i 0.678322 + 1.59370i 0.913545 − 0.406737i 3.70361 + 0.787228i −0.994848 − 1.41784i 0.181247 − 1.72445i −0.809017 + 0.587785i −2.07976 + 2.16208i −3.78636
31.6 −0.978148 + 0.207912i 1.36498 − 1.06622i 0.913545 − 0.406737i 2.85496 + 0.606841i −1.11347 + 1.32672i −0.443991 + 4.22430i −0.809017 + 0.587785i 0.726337 − 2.91074i −2.91874
31.7 −0.978148 + 0.207912i 1.54507 − 0.782790i 0.913545 − 0.406737i −3.37483 − 0.717342i −1.34855 + 1.08692i 0.293917 − 2.79644i −0.809017 + 0.587785i 1.77448 − 2.41893i 3.45023
49.1 0.669131 − 0.743145i −1.66291 + 0.484474i −0.104528 − 0.994522i 1.75609 + 1.95033i −0.752673 + 1.55996i 1.40440 + 0.625278i −0.809017 − 0.587785i 2.53057 − 1.61128i 2.62443
49.2 0.669131 − 0.743145i −1.30609 + 1.13760i −0.104528 − 0.994522i −2.53357 − 2.81381i −0.0285376 + 1.73182i −1.58404 − 0.705260i −0.809017 − 0.587785i 0.411717 − 2.97161i −3.78636
49.3 0.669131 − 0.743145i −1.10262 − 1.33575i −0.104528 − 0.994522i −1.09904 − 1.22061i −1.73045 − 0.0743863i −1.36506 − 0.607764i −0.809017 − 0.587785i −0.568462 + 2.94565i −1.64249
49.4 0.669131 − 0.743145i 0.668441 − 1.59787i −0.104528 − 0.994522i 1.61506 + 1.79371i −0.740174 − 1.56593i 3.10988 + 1.38461i −0.809017 − 0.587785i −2.10637 − 2.13616i 2.41367
49.5 0.669131 − 0.743145i 1.22193 + 1.22755i −0.104528 − 0.994522i 2.30865 + 2.56402i 1.72988 − 0.0866776i −2.56874 − 1.14368i −0.809017 − 0.587785i −0.0137736 + 2.99997i 3.45023
49.6 0.669131 − 0.743145i 1.43584 + 0.968692i −0.104528 − 0.994522i −1.95302 − 2.16905i 1.68064 − 0.418856i 3.88034 + 1.72764i −0.809017 − 0.587785i 1.12327 + 2.78177i −2.91874
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
11.c even 5 1 inner
99.m even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 198.2.m.b ✓ 56
3.b odd 2 1 594.2.n.b 56
9.c even 3 1 inner 198.2.m.b ✓ 56
9.d odd 6 1 594.2.n.b 56
11.c even 5 1 inner 198.2.m.b ✓ 56
33.h odd 10 1 594.2.n.b 56
99.m even 15 1 inner 198.2.m.b ✓ 56
99.n odd 30 1 594.2.n.b 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
198.2.m.b ✓ 56 1.a even 1 1 trivial
198.2.m.b ✓ 56 9.c even 3 1 inner
198.2.m.b ✓ 56 11.c even 5 1 inner
198.2.m.b ✓ 56 99.m even 15 1 inner
594.2.n.b 56 3.b odd 2 1
594.2.n.b 56 9.d odd 6 1
594.2.n.b 56 33.h odd 10 1
594.2.n.b 56 99.n odd 30 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{56} + T_{5}^{55} - 28 T_{5}^{54} - 39 T_{5}^{53} + 232 T_{5}^{52} + 532 T_{5}^{51} + \cdots + 10\!\cdots\!00 \) acting on \(S_{2}^{\mathrm{new}}(198, [\chi])\). Copy content Toggle raw display