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Properties

Label 153.4.o.a
Level $153$
Weight $4$
Character orbit 153.o
Analytic conductor $9.027$
Analytic rank $0$
Dimension $72$
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] = [153,4,Mod(44,153)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(153, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([8, 3]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("153.44");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 153 = 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 153.o (of order \(16\), 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: \(9.02729223088\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(9\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 56 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q - 56 q^{5} + 80 q^{11} + 176 q^{17} + 672 q^{20} + 208 q^{23} - 312 q^{25} + 144 q^{28} - 1008 q^{29} + 864 q^{31} - 1520 q^{32} + 528 q^{34} + 672 q^{37} - 704 q^{38} + 1344 q^{40} - 640 q^{41} + 96 q^{43} - 528 q^{44} - 2928 q^{46} + 736 q^{47} - 1536 q^{49} + 2112 q^{50} - 3072 q^{52} - 544 q^{53} - 960 q^{55} + 2208 q^{56} + 192 q^{58} + 1152 q^{59} + 960 q^{61} + 5072 q^{62} + 4224 q^{64} - 3704 q^{65} + 2240 q^{68} - 4032 q^{70} + 672 q^{71} - 768 q^{73} + 9968 q^{74} + 768 q^{76} - 7840 q^{77} + 1728 q^{79} - 10288 q^{80} + 624 q^{82} - 8896 q^{83} + 9504 q^{85} + 12096 q^{88} - 3616 q^{89} + 576 q^{91} - 4720 q^{92} - 4704 q^{94} + 976 q^{95} - 4896 q^{97} + 15920 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
44.1 −4.48163 − 1.85635i 0 10.9821 + 10.9821i 3.41142 + 17.1504i 0 31.4046 + 6.24676i −13.9803 − 33.7513i 0 16.5484 − 83.1944i
44.2 −4.29531 − 1.77918i 0 9.62739 + 9.62739i −2.87670 − 14.4621i 0 −2.36058 − 0.469549i −9.99042 − 24.1190i 0 −13.3744 + 67.2376i
44.3 −2.35694 − 0.976278i 0 −1.05479 − 1.05479i 1.51925 + 7.63778i 0 −15.0068 − 2.98504i 9.26654 + 22.3714i 0 3.87581 − 19.4850i
44.4 −2.17931 − 0.902698i 0 −1.72235 − 1.72235i 0.199045 + 1.00067i 0 −2.62755 − 0.522652i 9.42034 + 22.7427i 0 0.469519 − 2.36043i
44.5 −0.196335 − 0.0813246i 0 −5.62492 − 5.62492i −2.15730 − 10.8455i 0 6.42946 + 1.27890i 1.29752 + 3.13249i 0 −0.458452 + 2.30479i
44.6 1.83136 + 0.758575i 0 −2.87840 − 2.87840i 3.53114 + 17.7523i 0 −27.1882 − 5.40806i −9.15651 − 22.1058i 0 −6.99962 + 35.1895i
44.7 1.83755 + 0.761138i 0 −2.85960 − 2.85960i 0.401922 + 2.02060i 0 24.8201 + 4.93703i −9.16721 − 22.1316i 0 −0.799403 + 4.01887i
44.8 3.39939 + 1.40807i 0 3.91632 + 3.91632i −3.67803 − 18.4907i 0 −22.3855 − 4.45275i −3.46595 − 8.36754i 0 13.5332 − 68.0360i
44.9 4.59347 + 1.90268i 0 11.8229 + 11.8229i 1.07774 + 5.41818i 0 6.91445 + 1.37537i 16.5916 + 40.0555i 0 −5.35847 + 26.9388i
62.1 −2.15562 + 5.20414i 0 −16.7795 − 16.7795i −6.36507 + 9.52600i 0 23.1077 − 15.4401i 81.8597 − 33.9074i 0 −35.8539 − 53.6591i
62.2 −1.62146 + 3.91456i 0 −7.03778 − 7.03778i 6.41879 − 9.60639i 0 −25.6497 + 17.1386i 7.64483 − 3.16659i 0 27.1970 + 40.7031i
62.3 −1.27371 + 3.07502i 0 −2.17654 − 2.17654i 2.23963 − 3.35185i 0 8.79750 − 5.87830i −15.1350 + 6.26910i 0 7.45435 + 11.1562i
62.4 −0.731822 + 1.76677i 0 3.07093 + 3.07093i −11.0055 + 16.4709i 0 −4.54532 + 3.03709i −21.8072 + 9.03284i 0 −21.0462 − 31.4979i
62.5 −0.148033 + 0.357382i 0 5.55105 + 5.55105i 4.86951 − 7.28774i 0 14.6978 − 9.82073i −5.66464 + 2.34637i 0 1.88366 + 2.81910i
62.6 0.478458 − 1.15510i 0 4.55152 + 4.55152i −1.55350 + 2.32498i 0 −23.1820 + 15.4897i 16.6760 − 6.90742i 0 1.94230 + 2.90686i
62.7 1.11689 − 2.69641i 0 −0.366345 − 0.366345i −5.24577 + 7.85085i 0 12.0057 − 8.02195i 20.1743 − 8.35648i 0 15.3102 + 22.9133i
62.8 1.52465 − 3.68082i 0 −5.56706 − 5.56706i 7.73665 − 11.5787i 0 0.239191 − 0.159822i 0.467442 − 0.193621i 0 −30.8235 − 46.1307i
62.9 2.04529 − 4.93777i 0 −14.5415 − 14.5415i −8.60356 + 12.8761i 0 −5.47083 + 3.65549i −62.0423 + 25.6988i 0 45.9827 + 68.8179i
71.1 −1.93084 − 4.66146i 0 −12.3442 + 12.3442i −0.894668 + 0.597798i 0 −9.53722 + 14.2735i 44.0853 + 18.2607i 0 4.51408 + 3.01621i
71.2 −1.32460 − 3.19787i 0 −2.81496 + 2.81496i −4.85395 + 3.24331i 0 12.0864 − 18.0885i −12.8524 − 5.32364i 0 16.8012 + 11.2262i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 44.9
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
51.i even 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 153.4.o.a ✓ 72
3.b odd 2 1 153.4.o.b yes 72
17.e odd 16 1 153.4.o.b yes 72
51.i even 16 1 inner 153.4.o.a ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
153.4.o.a ✓ 72 1.a even 1 1 trivial
153.4.o.a ✓ 72 51.i even 16 1 inner
153.4.o.b yes 72 3.b odd 2 1
153.4.o.b yes 72 17.e odd 16 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{72} + 304 T_{2}^{67} - 4224 T_{2}^{66} + 10176 T_{2}^{65} + 1301136 T_{2}^{64} + \cdots + 18\!\cdots\!96 \) acting on \(S_{4}^{\mathrm{new}}(153, [\chi])\). Copy content Toggle raw display