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Properties

Label 190.2.r.a
Level $190$
Weight $2$
Character orbit 190.r
Analytic conductor $1.517$
Analytic rank $0$
Dimension $120$
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] = [190,2,Mod(3,190)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(190, base_ring=CyclotomicField(36))
 
chi = DirichletCharacter(H, H._module([27, 26]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("190.3");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 190 = 2 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 190.r (of order \(36\), degree \(12\), 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: \(1.51715763840\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(10\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 120 q - 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 120 q - 12 q^{7} - 36 q^{15} + 36 q^{17} - 96 q^{21} + 24 q^{22} - 24 q^{25} - 12 q^{26} - 96 q^{33} + 36 q^{35} - 12 q^{41} - 72 q^{43} - 36 q^{45} - 24 q^{47} - 144 q^{50} + 24 q^{51} + 36 q^{53} - 72 q^{55} + 84 q^{57} - 24 q^{60} + 48 q^{61} - 24 q^{62} + 36 q^{63} - 180 q^{65} - 24 q^{66} - 96 q^{67} - 12 q^{68} - 48 q^{70} - 36 q^{73} + 12 q^{76} + 96 q^{78} + 144 q^{81} + 48 q^{82} + 24 q^{83} + 48 q^{85} + 48 q^{86} + 72 q^{87} + 168 q^{90} + 72 q^{91} + 72 q^{92} + 156 q^{93} + 24 q^{95} + 120 q^{97} + 24 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −0.0871557 − 0.996195i −0.772202 + 1.65599i −0.984808 + 0.173648i 1.15159 − 1.91673i 1.71699 + 0.624934i 2.34531 + 0.628423i 0.258819 + 0.965926i −0.217652 − 0.259388i −2.00980 − 0.980156i
3.2 −0.0871557 − 0.996195i −0.715886 + 1.53522i −0.984808 + 0.173648i 0.627647 + 2.14617i 1.59177 + 0.579359i −4.29599 − 1.15111i 0.258819 + 0.965926i 0.0839459 + 0.100043i 2.08330 − 0.812310i
3.3 −0.0871557 − 0.996195i −0.202610 + 0.434499i −0.984808 + 0.173648i −1.29189 + 1.82511i 0.450504 + 0.163970i 3.80940 + 1.02072i 0.258819 + 0.965926i 1.78062 + 2.12207i 1.93076 + 1.12790i
3.4 −0.0871557 − 0.996195i 0.602383 − 1.29181i −0.984808 + 0.173648i 1.49498 − 1.66284i −1.33940 − 0.487501i −1.39748 − 0.374454i 0.258819 + 0.965926i 0.622444 + 0.741800i −1.78681 − 1.34437i
3.5 −0.0871557 − 0.996195i 1.08832 − 2.33390i −0.984808 + 0.173648i −2.15598 + 0.593089i −2.41987 − 0.880761i −3.72976 − 0.999386i 0.258819 + 0.965926i −2.33430 − 2.78191i 0.778739 + 2.09608i
3.6 0.0871557 + 0.996195i −1.36202 + 2.92085i −0.984808 + 0.173648i 1.16760 + 1.90702i −3.02844 − 1.10226i 1.04490 + 0.279979i −0.258819 − 0.965926i −4.74793 − 5.65836i −1.79800 + 1.32936i
3.7 0.0871557 + 0.996195i −0.684861 + 1.46869i −0.984808 + 0.173648i −2.14169 − 0.642793i −1.52279 − 0.554250i −3.13506 − 0.840038i −0.258819 − 0.965926i 0.240351 + 0.286439i 0.453687 − 2.18956i
3.8 0.0871557 + 0.996195i 0.186197 − 0.399301i −0.984808 + 0.173648i −0.832474 + 2.07533i 0.414010 + 0.150687i 1.97694 + 0.529719i −0.258819 − 0.965926i 1.80359 + 2.14944i −2.13999 − 0.648429i
3.9 0.0871557 + 0.996195i 0.507694 − 1.08875i −0.984808 + 0.173648i −0.594274 − 2.15565i 1.12886 + 0.410871i 2.08332 + 0.558224i −0.258819 − 0.965926i 1.00073 + 1.19263i 2.09566 − 0.779890i
3.10 0.0871557 + 0.996195i 1.35298 − 2.90149i −0.984808 + 0.173648i 2.22719 − 0.199092i 3.00836 + 1.09496i −1.43361 − 0.384135i −0.258819 − 0.965926i −4.65969 − 5.55320i 0.392446 + 2.20136i
13.1 −0.996195 − 0.0871557i −2.90149 + 1.35298i 0.984808 + 0.173648i −2.02478 + 0.948828i 3.00836 − 1.09496i 0.384135 + 1.43361i −0.965926 − 0.258819i 4.65969 − 5.55320i 2.09977 − 0.768746i
13.2 −0.996195 − 0.0871557i −1.08875 + 0.507694i 0.984808 + 0.173648i 1.29571 + 1.82240i 1.12886 − 0.410871i −0.558224 − 2.08332i −0.965926 − 0.258819i −1.00073 + 1.19263i −1.13195 − 1.92839i
13.3 −0.996195 − 0.0871557i −0.399301 + 0.186197i 0.984808 + 0.173648i 0.0724654 − 2.23489i 0.414010 − 0.150687i −0.529719 − 1.97694i −0.965926 − 0.258819i −1.80359 + 2.14944i −0.266973 + 2.22007i
13.4 −0.996195 − 0.0871557i 1.46869 − 0.684861i 0.984808 + 0.173648i 2.23237 − 0.128472i −1.52279 + 0.554250i 0.840038 + 3.13506i −0.965926 − 0.258819i −0.240351 + 0.286439i −2.23508 − 0.0665816i
13.5 −0.996195 − 0.0871557i 2.92085 − 1.36202i 0.984808 + 0.173648i −1.74942 − 1.39267i −3.02844 + 1.10226i −0.279979 − 1.04490i −0.965926 − 0.258819i 4.74793 − 5.65836i 1.62139 + 1.53984i
13.6 0.996195 + 0.0871557i −2.33390 + 1.08832i 0.984808 + 0.173648i 1.82311 − 1.29471i −2.41987 + 0.880761i 0.999386 + 3.72976i 0.965926 + 0.258819i 2.33430 − 2.78191i 1.92901 − 1.13089i
13.7 0.996195 + 0.0871557i −1.29181 + 0.602383i 0.984808 + 0.173648i −0.836098 + 2.07387i −1.33940 + 0.487501i 0.374454 + 1.39748i 0.965926 + 0.258819i −0.622444 + 0.741800i −1.01367 + 1.99311i
13.8 0.996195 + 0.0871557i 0.434499 − 0.202610i 0.984808 + 0.173648i 0.589754 − 2.15689i 0.450504 − 0.163970i −1.02072 − 3.80940i 0.965926 + 0.258819i −1.78062 + 2.12207i 0.775496 − 2.09729i
13.9 0.996195 + 0.0871557i 1.53522 − 0.715886i 0.984808 + 0.173648i −1.32383 − 1.80208i 1.59177 − 0.579359i 1.15111 + 4.29599i 0.965926 + 0.258819i −0.0839459 + 0.100043i −1.16173 − 1.91060i
13.10 0.996195 + 0.0871557i 1.65599 − 0.772202i 0.984808 + 0.173648i −0.426584 + 2.19500i 1.71699 − 0.624934i −0.628423 − 2.34531i 0.965926 + 0.258819i 0.217652 − 0.259388i −0.616267 + 2.14947i
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
19.f odd 18 1 inner
95.r even 36 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 190.2.r.a ✓ 120
5.b even 2 1 950.2.bb.e 120
5.c odd 4 1 inner 190.2.r.a ✓ 120
5.c odd 4 1 950.2.bb.e 120
19.f odd 18 1 inner 190.2.r.a ✓ 120
95.o odd 18 1 950.2.bb.e 120
95.r even 36 1 inner 190.2.r.a ✓ 120
95.r even 36 1 950.2.bb.e 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.r.a ✓ 120 1.a even 1 1 trivial
190.2.r.a ✓ 120 5.c odd 4 1 inner
190.2.r.a ✓ 120 19.f odd 18 1 inner
190.2.r.a ✓ 120 95.r even 36 1 inner
950.2.bb.e 120 5.b even 2 1
950.2.bb.e 120 5.c odd 4 1
950.2.bb.e 120 95.o odd 18 1
950.2.bb.e 120 95.r even 36 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(190, [\chi])\).