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Properties

Label 153.5.p.a
Level $153$
Weight $5$
Character orbit 153.p
Analytic conductor $15.816$
Analytic rank $0$
Dimension $40$
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] = [153,5,Mod(10,153)]
 
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([0, 3]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("153.10");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 153 = 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 153.p (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: \(15.8156043518\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(5\) over \(\Q(\zeta_{16})\)
Twist minimal: no (minimal twist has level 17)
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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 8 q^{2} - 8 q^{4} + 8 q^{5} - 8 q^{7} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 8 q^{2} - 8 q^{4} + 8 q^{5} - 8 q^{7} + 8 q^{8} + 376 q^{10} - 112 q^{11} - 416 q^{13} + 776 q^{14} - 256 q^{17} + 688 q^{19} - 2680 q^{20} + 760 q^{22} + 176 q^{23} + 2600 q^{26} - 7448 q^{28} + 3368 q^{29} - 3720 q^{31} + 2400 q^{32} + 4280 q^{34} - 4208 q^{35} + 7416 q^{37} - 16720 q^{38} + 20280 q^{40} - 2656 q^{41} - 7512 q^{43} + 31592 q^{44} - 25752 q^{46} + 10208 q^{47} - 3112 q^{49} + 12784 q^{52} - 24424 q^{53} + 26648 q^{55} - 40432 q^{56} - 4336 q^{58} + 3176 q^{59} - 24600 q^{61} + 39248 q^{62} - 45560 q^{64} + 37928 q^{65} - 34912 q^{68} + 59536 q^{70} - 21736 q^{71} + 28592 q^{73} - 15976 q^{74} - 9280 q^{76} - 2392 q^{77} - 15912 q^{79} + 47640 q^{80} - 58368 q^{82} + 27296 q^{83} + 18872 q^{85} - 74336 q^{86} + 55608 q^{88} + 1232 q^{89} - 7800 q^{91} + 27032 q^{92} - 37096 q^{94} + 49640 q^{95} - 12392 q^{97} + 76304 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1 −2.24361 + 5.41655i 0 −12.9915 − 12.9915i 33.5479 − 6.67310i 0 −47.5881 − 9.46586i 12.8521 − 5.32353i 0 −39.1232 + 196.686i
10.2 −1.96744 + 4.74983i 0 −7.37633 − 7.37633i −12.5266 + 2.49169i 0 32.9912 + 6.56237i −26.4485 + 10.9553i 0 12.8102 − 64.4014i
10.3 0.308156 − 0.743954i 0 10.8552 + 10.8552i −11.5167 + 2.29082i 0 −14.9104 − 2.96587i 23.3241 − 9.66117i 0 −1.84468 + 9.27385i
10.4 1.66405 − 4.01738i 0 −2.05653 − 2.05653i 17.1588 − 3.41309i 0 −18.6444 − 3.70861i 52.5940 − 21.7851i 0 14.8414 − 74.6128i
10.5 2.53174 − 6.11215i 0 −19.6350 − 19.6350i 21.1399 − 4.20498i 0 70.6926 + 14.0616i −71.9284 + 29.7937i 0 27.8191 − 139.856i
28.1 −5.21362 − 2.15955i 0 11.2044 + 11.2044i 9.99058 + 6.67549i 0 −35.0400 + 23.4130i 0.333694 + 0.805609i 0 −37.6710 − 56.3786i
28.2 −1.59548 − 0.660869i 0 −9.20491 − 9.20491i −9.54385 − 6.37700i 0 68.4978 − 45.7688i 19.1769 + 46.2971i 0 11.0126 + 16.4816i
28.3 −0.347647 − 0.144000i 0 −11.2136 − 11.2136i −33.0442 − 22.0794i 0 6.11031 − 4.08278i 4.58761 + 11.0755i 0 8.30827 + 12.4342i
28.4 3.17127 + 1.31358i 0 −2.98223 − 2.98223i 30.6373 + 20.4712i 0 18.6713 − 12.4757i −26.5574 − 64.1153i 0 70.2687 + 105.164i
28.5 5.69258 + 2.35794i 0 15.5318 + 15.5318i −30.3835 − 20.3016i 0 −64.4884 + 43.0898i 14.0658 + 33.9579i 0 −125.090 − 187.211i
37.1 −6.53721 + 2.70780i 0 24.0892 − 24.0892i −16.7298 − 25.0379i 0 −31.8191 + 47.6207i −48.9226 + 118.110i 0 177.164 + 118.377i
37.2 −3.43536 + 1.42297i 0 −1.53684 + 1.53684i 8.81924 + 13.1989i 0 5.40438 − 8.08823i 25.8603 − 62.4323i 0 −49.0790 − 32.7936i
37.3 1.63167 − 0.675861i 0 −9.10814 + 9.10814i 6.20205 + 9.28203i 0 −12.6661 + 18.9562i −19.5194 + 47.1241i 0 16.3931 + 10.9535i
37.4 2.92749 − 1.21261i 0 −4.21393 + 4.21393i −21.2448 − 31.7951i 0 5.22138 − 7.81435i −26.6281 + 64.2859i 0 −100.749 − 67.3181i
37.5 7.12052 − 2.94942i 0 30.6890 − 30.6890i 14.8705 + 22.2553i 0 16.8953 − 25.2857i 80.8164 − 195.108i 0 171.526 + 114.610i
46.1 −2.24361 − 5.41655i 0 −12.9915 + 12.9915i 33.5479 + 6.67310i 0 −47.5881 + 9.46586i 12.8521 + 5.32353i 0 −39.1232 − 196.686i
46.2 −1.96744 − 4.74983i 0 −7.37633 + 7.37633i −12.5266 − 2.49169i 0 32.9912 − 6.56237i −26.4485 − 10.9553i 0 12.8102 + 64.4014i
46.3 0.308156 + 0.743954i 0 10.8552 − 10.8552i −11.5167 − 2.29082i 0 −14.9104 + 2.96587i 23.3241 + 9.66117i 0 −1.84468 − 9.27385i
46.4 1.66405 + 4.01738i 0 −2.05653 + 2.05653i 17.1588 + 3.41309i 0 −18.6444 + 3.70861i 52.5940 + 21.7851i 0 14.8414 + 74.6128i
46.5 2.53174 + 6.11215i 0 −19.6350 + 19.6350i 21.1399 + 4.20498i 0 70.6926 − 14.0616i −71.9284 − 29.7937i 0 27.8191 + 139.856i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.5
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.e odd 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 153.5.p.a 40
3.b odd 2 1 17.5.e.a ✓ 40
17.e odd 16 1 inner 153.5.p.a 40
51.i even 16 1 17.5.e.a ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.5.e.a ✓ 40 3.b odd 2 1
17.5.e.a ✓ 40 51.i even 16 1
153.5.p.a 40 1.a even 1 1 trivial
153.5.p.a 40 17.e odd 16 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{40} - 8 T_{2}^{39} + 36 T_{2}^{38} - 120 T_{2}^{37} + 328 T_{2}^{36} - 1152 T_{2}^{35} + \cdots + 14\!\cdots\!44 \) acting on \(S_{5}^{\mathrm{new}}(153, [\chi])\). Copy content Toggle raw display