Properties

Label 153.2.n.a
Level $153$
Weight $2$
Character orbit 153.n
Analytic conductor $1.222$
Analytic rank $0$
Dimension $64$
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] = [153,2,Mod(4,153)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(153, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([4, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("153.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 153 = 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 153.n (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: \(1.22171115093\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 64 q - 6 q^{3} + 24 q^{4} - 2 q^{5} - 10 q^{6} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q - 6 q^{3} + 24 q^{4} - 2 q^{5} - 10 q^{6} - 2 q^{7} - 16 q^{10} - 24 q^{12} - 4 q^{13} - 16 q^{16} - 8 q^{17} - 8 q^{18} + 18 q^{20} - 16 q^{21} - 4 q^{22} - 8 q^{23} - 2 q^{24} - 10 q^{29} - 36 q^{30} - 2 q^{31} + 12 q^{33} + 20 q^{34} - 128 q^{35} - 8 q^{37} - 24 q^{38} + 34 q^{39} - 20 q^{40} + 32 q^{41} + 20 q^{44} + 20 q^{45} - 40 q^{46} - 64 q^{47} + 62 q^{48} + 48 q^{50} + 40 q^{51} + 36 q^{52} - 46 q^{54} - 16 q^{55} + 12 q^{56} + 72 q^{57} - 10 q^{58} - 2 q^{61} - 28 q^{62} + 64 q^{63} - 8 q^{64} + 8 q^{65} - 4 q^{67} - 60 q^{68} - 24 q^{69} - 84 q^{71} + 72 q^{72} - 44 q^{73} - 14 q^{74} + 46 q^{75} - 56 q^{78} + 10 q^{79} + 204 q^{80} + 44 q^{81} - 52 q^{82} - 60 q^{84} + 22 q^{85} + 32 q^{86} + 16 q^{88} + 128 q^{89} - 66 q^{90} + 44 q^{91} + 136 q^{92} + 4 q^{95} - 2 q^{96} - 44 q^{97} + 208 q^{98} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −2.25660 + 1.30285i −1.06362 + 1.36701i 2.39483 4.14796i 0.847469 + 3.16280i 0.619154 4.47053i −0.996589 + 3.71932i 7.26900i −0.737432 2.90795i −6.03304 6.03304i
4.2 −2.07111 + 1.19576i 1.60904 0.641087i 1.85966 3.22103i 0.229055 + 0.854845i −2.56591 + 3.25178i 0.639316 2.38596i 4.11179i 2.17801 2.06307i −1.49658 1.49658i
4.3 −1.84575 + 1.06565i 0.00129728 + 1.73205i 1.27121 2.20180i −0.945773 3.52967i −1.84815 3.19556i 0.990725 3.69744i 1.15604i −3.00000 + 0.00449391i 5.50705 + 5.50705i
4.4 −1.64395 + 0.949132i −1.09575 1.34139i 0.801704 1.38859i 0.583765 + 2.17864i 3.07451 + 1.16516i 0.516120 1.92618i 0.752835i −0.598665 + 2.93966i −3.02750 3.02750i
4.5 −1.25944 + 0.727138i −1.72945 0.0948539i 0.0574603 0.0995242i −0.384824 1.43618i 2.24711 1.13809i −0.382826 + 1.42872i 2.74143i 2.98201 + 0.328090i 1.52897 + 1.52897i
4.6 −1.15506 + 0.666877i 1.50110 + 0.864115i −0.110551 + 0.191481i −0.0545182 0.203465i −2.31013 + 0.00294161i −0.965572 + 3.60357i 2.96240i 1.50661 + 2.59425i 0.198658 + 0.198658i
4.7 −0.429699 + 0.248087i −0.0790082 1.73025i −0.876906 + 1.51885i −0.533502 1.99106i 0.463201 + 0.723884i 0.631823 2.35799i 1.86254i −2.98752 + 0.273408i 0.723200 + 0.723200i
4.8 −0.218953 + 0.126413i 0.963283 1.43947i −0.968040 + 1.67669i 1.02659 + 3.83129i −0.0289461 + 0.436949i −0.551719 + 2.05904i 0.995142i −1.14417 2.77324i −0.709099 0.709099i
4.9 0.145499 0.0840042i −0.814046 + 1.52883i −0.985887 + 1.70761i −0.454855 1.69754i 0.00998502 + 0.290828i −0.714807 + 2.66770i 0.667291i −1.67466 2.48908i −0.208782 0.208782i
4.10 0.268917 0.155259i 1.64363 + 0.546341i −0.951789 + 1.64855i 0.0951356 + 0.355051i 0.526823 0.108268i 0.984025 3.67243i 1.21213i 2.40302 + 1.79596i 0.0807084 + 0.0807084i
4.11 0.793288 0.458005i −1.72440 0.162580i −0.580463 + 1.00539i 0.915201 + 3.41558i −1.44241 + 0.660813i 0.0913582 0.340953i 2.89544i 2.94714 + 0.560706i 2.29037 + 2.29037i
4.12 1.11815 0.645565i 1.43984 0.962729i −0.166493 + 0.288374i −0.684695 2.55532i 0.988459 2.00599i −0.585596 + 2.18547i 3.01219i 1.14630 2.77236i −2.41521 2.41521i
4.13 1.19294 0.688745i 0.266702 + 1.71139i −0.0512619 + 0.0887882i 0.193813 + 0.723321i 1.49687 + 1.85790i 0.137184 0.511976i 2.89620i −2.85774 + 0.912864i 0.729391 + 0.729391i
4.14 1.41154 0.814955i −1.69753 0.344088i 0.328304 0.568639i −0.870254 3.24783i −2.67655 + 0.897715i 1.08515 4.04983i 2.18961i 2.76321 + 1.16820i −3.87524 3.87524i
4.15 1.92090 1.10903i −0.421083 1.68009i 1.45992 2.52865i 0.182375 + 0.680632i −2.67213 2.76029i −0.556256 + 2.07597i 2.04026i −2.64538 + 1.41491i 1.10517 + 1.10517i
4.16 2.29727 1.32633i −1.16603 + 1.28077i 2.51830 4.36183i 0.221043 + 0.824944i −0.979965 + 4.48882i 0.0436914 0.163059i 8.05510i −0.280745 2.98683i 1.60194 + 1.60194i
13.1 −2.29727 + 1.32633i −1.28077 1.16603i 2.51830 4.36183i −0.824944 + 0.221043i 4.48882 + 0.979965i −0.163059 0.0436914i 8.05510i 0.280745 + 2.98683i 1.60194 1.60194i
13.2 −1.92090 + 1.10903i 1.68009 0.421083i 1.45992 2.52865i −0.680632 + 0.182375i −2.76029 + 2.67213i 2.07597 + 0.556256i 2.04026i 2.64538 1.41491i 1.10517 1.10517i
13.3 −1.41154 + 0.814955i 0.344088 1.69753i 0.328304 0.568639i 3.24783 0.870254i 0.897715 + 2.67655i −4.04983 1.08515i 2.18961i −2.76321 1.16820i −3.87524 + 3.87524i
13.4 −1.19294 + 0.688745i −1.71139 + 0.266702i −0.0512619 + 0.0887882i −0.723321 + 0.193813i 1.85790 1.49687i −0.511976 0.137184i 2.89620i 2.85774 0.912864i 0.729391 0.729391i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
17.c even 4 1 inner
153.n even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 153.2.n.a 64
3.b odd 2 1 459.2.o.a 64
9.c even 3 1 inner 153.2.n.a 64
9.d odd 6 1 459.2.o.a 64
17.c even 4 1 inner 153.2.n.a 64
51.f odd 4 1 459.2.o.a 64
153.m odd 12 1 459.2.o.a 64
153.n even 12 1 inner 153.2.n.a 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
153.2.n.a 64 1.a even 1 1 trivial
153.2.n.a 64 9.c even 3 1 inner
153.2.n.a 64 17.c even 4 1 inner
153.2.n.a 64 153.n even 12 1 inner
459.2.o.a 64 3.b odd 2 1
459.2.o.a 64 9.d odd 6 1
459.2.o.a 64 51.f odd 4 1
459.2.o.a 64 153.m odd 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(153, [\chi])\).