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Properties

Label 156.2.w.d
Level $156$
Weight $2$
Character orbit 156.w
Analytic conductor $1.246$
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] = [156,2,Mod(7,156)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(156, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 0, 11]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("156.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 156 = 2^{2} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 156.w (of order \(12\), degree \(4\), 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: \(1.24566627153\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 2 q^{2} - 4 q^{4} + 2 q^{5} + 4 q^{6} + 2 q^{7} + 4 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 2 q^{2} - 4 q^{4} + 2 q^{5} + 4 q^{6} + 2 q^{7} + 4 q^{8} + 12 q^{9} - 4 q^{11} + 4 q^{12} - 22 q^{14} + 4 q^{15} - 12 q^{16} + 2 q^{18} + 22 q^{19} - 56 q^{20} + 10 q^{21} + 26 q^{22} + 2 q^{24} - 12 q^{26} - 26 q^{28} + 8 q^{29} + 4 q^{30} + 2 q^{31} - 22 q^{32} + 4 q^{33} - 44 q^{34} - 24 q^{35} + 4 q^{36} - 12 q^{37} + 12 q^{38} + 6 q^{39} + 28 q^{40} - 36 q^{41} - 10 q^{43} + 12 q^{44} + 4 q^{45} + 26 q^{46} - 20 q^{47} - 16 q^{48} - 54 q^{49} + 34 q^{50} - 36 q^{51} + 40 q^{52} - 36 q^{53} + 2 q^{54} + 24 q^{55} + 62 q^{56} - 32 q^{57} + 20 q^{58} + 36 q^{59} + 20 q^{60} - 2 q^{61} + 2 q^{62} - 8 q^{63} + 8 q^{64} + 16 q^{65} + 20 q^{66} + 16 q^{67} + 2 q^{68} + 12 q^{69} + 8 q^{70} - 36 q^{71} + 2 q^{72} - 40 q^{73} + 54 q^{74} + 8 q^{75} - 58 q^{76} - 6 q^{78} + 60 q^{80} - 12 q^{81} - 6 q^{82} + 24 q^{83} + 14 q^{84} + 18 q^{85} - 34 q^{86} - 6 q^{87} + 18 q^{88} + 66 q^{89} - 12 q^{90} - 42 q^{91} + 36 q^{92} - 10 q^{93} + 42 q^{94} + 28 q^{95} - 32 q^{96} + 4 q^{97} + 80 q^{98} + 4 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −1.31456 + 0.521479i −0.866025 − 0.500000i 1.45612 − 1.37103i −1.90811 − 1.90811i 1.39918 + 0.205665i −1.18978 + 4.44032i −1.19919 + 2.56163i 0.500000 + 0.866025i 3.50335 + 1.51328i
7.2 −0.880389 + 1.10676i −0.866025 − 0.500000i −0.449829 − 1.94876i 0.713774 + 0.713774i 1.31582 − 0.518286i 0.872392 − 3.25581i 2.55283 + 1.21781i 0.500000 + 0.866025i −1.41837 + 0.161577i
7.3 −0.793527 − 1.17060i −0.866025 − 0.500000i −0.740630 + 1.85781i 2.43250 + 2.43250i 0.101912 + 1.41054i −0.522652 + 1.95057i 2.76247 − 0.607239i 0.500000 + 0.866025i 0.917241 − 4.77774i
7.4 0.117097 − 1.40936i −0.866025 − 0.500000i −1.97258 − 0.330063i −0.922964 − 0.922964i −0.806088 + 1.16199i 0.757268 − 2.82616i −0.696159 + 2.74142i 0.500000 + 0.866025i −1.40886 + 1.19271i
7.5 0.331729 + 1.37476i −0.866025 − 0.500000i −1.77991 + 0.912093i 1.23779 + 1.23779i 0.400093 − 1.35644i −1.04473 + 3.89897i −1.84435 − 2.14438i 0.500000 + 0.866025i −1.29105 + 2.11227i
7.6 1.17362 − 0.789057i −0.866025 − 0.500000i 0.754777 − 1.85211i −1.91901 − 1.91901i −1.41092 + 0.0965326i −0.104554 + 0.390202i −0.575598 − 2.76924i 0.500000 + 0.866025i −3.76641 − 0.737985i
19.1 −1.40179 − 0.187026i 0.866025 + 0.500000i 1.93004 + 0.524344i 0.756294 − 0.756294i −1.12047 − 0.862866i 0.936058 + 0.250816i −2.60745 − 1.09599i 0.500000 + 0.866025i −1.20161 + 0.918720i
19.2 −0.719617 − 1.21744i 0.866025 + 0.500000i −0.964302 + 1.75218i −2.26006 + 2.26006i −0.0144886 − 1.41414i 3.65215 + 0.978592i 2.82709 − 0.0869195i 0.500000 + 0.866025i 4.37786 + 1.12510i
19.3 −0.616993 + 1.27253i 0.866025 + 0.500000i −1.23864 − 1.57028i 2.78626 − 2.78626i −1.17059 + 0.793543i 0.826736 + 0.221523i 2.76245 − 0.607351i 0.500000 + 0.866025i 1.82648 + 5.26468i
19.4 0.595366 − 1.28279i 0.866025 + 0.500000i −1.29108 − 1.52745i 1.41026 − 1.41026i 1.15700 − 0.813242i −0.844432 − 0.226265i −2.72806 + 0.746782i 0.500000 + 0.866025i −0.969439 − 2.64868i
19.5 1.24187 + 0.676576i 0.866025 + 0.500000i 1.08449 + 1.68044i 0.218120 − 0.218120i 0.737204 + 1.20687i −3.40856 − 0.913320i 0.209852 + 2.82063i 0.500000 + 0.866025i 0.418452 − 0.123303i
19.6 1.26719 − 0.627878i 0.866025 + 0.500000i 1.21154 − 1.59128i −1.54484 + 1.54484i 1.41136 + 0.0898367i 1.07009 + 0.286731i 0.536120 − 2.77715i 0.500000 + 0.866025i −0.987636 + 2.92758i
67.1 −1.31456 − 0.521479i −0.866025 + 0.500000i 1.45612 + 1.37103i −1.90811 + 1.90811i 1.39918 − 0.205665i −1.18978 − 4.44032i −1.19919 − 2.56163i 0.500000 − 0.866025i 3.50335 − 1.51328i
67.2 −0.880389 − 1.10676i −0.866025 + 0.500000i −0.449829 + 1.94876i 0.713774 − 0.713774i 1.31582 + 0.518286i 0.872392 + 3.25581i 2.55283 − 1.21781i 0.500000 − 0.866025i −1.41837 − 0.161577i
67.3 −0.793527 + 1.17060i −0.866025 + 0.500000i −0.740630 − 1.85781i 2.43250 − 2.43250i 0.101912 − 1.41054i −0.522652 − 1.95057i 2.76247 + 0.607239i 0.500000 − 0.866025i 0.917241 + 4.77774i
67.4 0.117097 + 1.40936i −0.866025 + 0.500000i −1.97258 + 0.330063i −0.922964 + 0.922964i −0.806088 − 1.16199i 0.757268 + 2.82616i −0.696159 − 2.74142i 0.500000 − 0.866025i −1.40886 − 1.19271i
67.5 0.331729 − 1.37476i −0.866025 + 0.500000i −1.77991 − 0.912093i 1.23779 − 1.23779i 0.400093 + 1.35644i −1.04473 − 3.89897i −1.84435 + 2.14438i 0.500000 − 0.866025i −1.29105 − 2.11227i
67.6 1.17362 + 0.789057i −0.866025 + 0.500000i 0.754777 + 1.85211i −1.91901 + 1.91901i −1.41092 − 0.0965326i −0.104554 − 0.390202i −0.575598 + 2.76924i 0.500000 − 0.866025i −3.76641 + 0.737985i
115.1 −1.40179 + 0.187026i 0.866025 − 0.500000i 1.93004 − 0.524344i 0.756294 + 0.756294i −1.12047 + 0.862866i 0.936058 − 0.250816i −2.60745 + 1.09599i 0.500000 − 0.866025i −1.20161 − 0.918720i
115.2 −0.719617 + 1.21744i 0.866025 − 0.500000i −0.964302 − 1.75218i −2.26006 − 2.26006i −0.0144886 + 1.41414i 3.65215 − 0.978592i 2.82709 + 0.0869195i 0.500000 − 0.866025i 4.37786 − 1.12510i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
52.l even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 156.2.w.d yes 24
3.b odd 2 1 468.2.cb.g 24
4.b odd 2 1 156.2.w.c ✓ 24
12.b even 2 1 468.2.cb.h 24
13.f odd 12 1 156.2.w.c ✓ 24
39.k even 12 1 468.2.cb.h 24
52.l even 12 1 inner 156.2.w.d yes 24
156.v odd 12 1 468.2.cb.g 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.2.w.c ✓ 24 4.b odd 2 1
156.2.w.c ✓ 24 13.f odd 12 1
156.2.w.d yes 24 1.a even 1 1 trivial
156.2.w.d yes 24 52.l even 12 1 inner
468.2.cb.g 24 3.b odd 2 1
468.2.cb.g 24 156.v odd 12 1
468.2.cb.h 24 12.b even 2 1
468.2.cb.h 24 39.k even 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(156, [\chi])\):

\( T_{5}^{24} - 2 T_{5}^{23} + 2 T_{5}^{22} + 14 T_{5}^{21} + 293 T_{5}^{20} - 388 T_{5}^{19} + \cdots + 1106704 \) Copy content Toggle raw display
\( T_{7}^{24} - 2 T_{7}^{23} + 29 T_{7}^{22} - 112 T_{7}^{21} + 209 T_{7}^{20} - 816 T_{7}^{19} + \cdots + 2560000 \) Copy content Toggle raw display