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Properties

Label 144.7.q.d
Level $144$
Weight $7$
Character orbit 144.q
Analytic conductor $33.128$
Analytic rank $0$
Dimension $36$
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] = [144,7,Mod(65,144)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(144, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("144.65");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 144.q (of order \(6\), degree \(2\), not 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: \(33.1277880413\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(18\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 10 q^{3} + 74 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 10 q^{3} + 74 q^{9} - 1350 q^{11} - 7912 q^{15} - 9540 q^{19} + 3828 q^{21} - 30888 q^{23} + 56250 q^{25} - 11392 q^{27} + 38556 q^{29} - 27720 q^{31} + 33514 q^{33} - 134068 q^{39} + 179226 q^{41} - 15930 q^{43} - 185620 q^{45} - 187596 q^{47} - 198774 q^{49} + 158098 q^{51} + 197064 q^{55} - 244990 q^{57} + 408618 q^{59} + 17136 q^{61} + 417048 q^{63} - 125712 q^{65} - 27090 q^{67} - 848504 q^{69} - 534060 q^{73} + 1405714 q^{75} + 48168 q^{77} - 172620 q^{79} + 349010 q^{81} - 1801980 q^{83} - 791568 q^{85} - 28500 q^{87} - 538560 q^{91} - 1116448 q^{93} - 1832652 q^{95} + 770706 q^{97} + 614260 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1 0 −26.9914 − 0.680128i 0 158.588 − 91.5607i 0 −99.1040 + 171.653i 0 728.075 + 36.7153i 0
65.2 0 −24.8361 + 10.5910i 0 −87.4624 + 50.4964i 0 21.1751 − 36.6764i 0 504.663 − 526.077i 0
65.3 0 −24.3041 − 11.7607i 0 76.0344 − 43.8985i 0 287.890 − 498.640i 0 452.374 + 571.663i 0
65.4 0 −22.2209 + 15.3372i 0 −93.2385 + 53.8313i 0 32.4834 − 56.2629i 0 258.541 − 681.614i 0
65.5 0 −22.0600 − 15.5677i 0 −91.6877 + 52.9359i 0 −238.433 + 412.978i 0 244.291 + 686.850i 0
65.6 0 −16.4370 − 21.4202i 0 −124.675 + 71.9811i 0 −53.2895 + 92.3001i 0 −188.649 + 704.168i 0
65.7 0 −12.6322 + 23.8627i 0 152.846 − 88.2455i 0 −152.534 + 264.197i 0 −409.853 − 602.878i 0
65.8 0 −4.90324 + 26.5511i 0 −9.93230 + 5.73442i 0 128.008 − 221.717i 0 −680.917 − 260.372i 0
65.9 0 −1.83005 − 26.9379i 0 86.3751 − 49.8687i 0 207.661 − 359.679i 0 −722.302 + 98.5953i 0
65.10 0 −0.709327 − 26.9907i 0 157.631 − 91.0083i 0 −199.766 + 346.005i 0 −727.994 + 38.2905i 0
65.11 0 11.5131 − 24.4223i 0 −113.268 + 65.3953i 0 143.592 − 248.709i 0 −463.898 − 562.352i 0
65.12 0 13.0560 + 23.6335i 0 49.0773 − 28.3348i 0 226.487 − 392.287i 0 −388.084 + 617.116i 0
65.13 0 13.2266 + 23.5384i 0 −180.740 + 104.350i 0 −173.716 + 300.885i 0 −379.115 + 622.666i 0
65.14 0 14.5191 + 22.7639i 0 26.1779 − 15.1138i 0 −301.407 + 522.052i 0 −307.391 + 661.023i 0
65.15 0 19.7087 − 18.4545i 0 −11.9109 + 6.87674i 0 −174.223 + 301.763i 0 47.8620 − 727.427i 0
65.16 0 26.4412 + 5.46471i 0 174.849 − 100.949i 0 193.914 − 335.869i 0 669.274 + 288.987i 0
65.17 0 26.5258 + 5.03801i 0 −212.009 + 122.404i 0 198.515 − 343.838i 0 678.237 + 267.274i 0
65.18 0 26.9340 − 1.88607i 0 43.3462 − 25.0260i 0 −47.2533 + 81.8451i 0 721.885 − 101.599i 0
113.1 0 −26.9914 + 0.680128i 0 158.588 + 91.5607i 0 −99.1040 − 171.653i 0 728.075 − 36.7153i 0
113.2 0 −24.8361 − 10.5910i 0 −87.4624 − 50.4964i 0 21.1751 + 36.6764i 0 504.663 + 526.077i 0
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.7.q.d 36
3.b odd 2 1 432.7.q.d 36
4.b odd 2 1 72.7.m.a ✓ 36
9.c even 3 1 432.7.q.d 36
9.d odd 6 1 inner 144.7.q.d 36
12.b even 2 1 216.7.m.a 36
36.f odd 6 1 216.7.m.a 36
36.f odd 6 1 648.7.e.c 36
36.h even 6 1 72.7.m.a ✓ 36
36.h even 6 1 648.7.e.c 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.7.m.a ✓ 36 4.b odd 2 1
72.7.m.a ✓ 36 36.h even 6 1
144.7.q.d 36 1.a even 1 1 trivial
144.7.q.d 36 9.d odd 6 1 inner
216.7.m.a 36 12.b even 2 1
216.7.m.a 36 36.f odd 6 1
432.7.q.d 36 3.b odd 2 1
432.7.q.d 36 9.c even 3 1
648.7.e.c 36 36.f odd 6 1
648.7.e.c 36 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{36} - 168750 T_{5}^{34} + 17139483549 T_{5}^{32} - 168062762628 T_{5}^{31} + \cdots + 24\!\cdots\!00 \) acting on \(S_{7}^{\mathrm{new}}(144, [\chi])\). Copy content Toggle raw display