Properties

Label 144.7.o.c
Level $144$
Weight $7$
Character orbit 144.o
Analytic conductor $33.128$
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] = [144,7,Mod(31,144)]
 
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([3, 0, 2]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("144.31");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 144.o (of order \(6\), degree \(2\), 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: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 48 q^{3} - 72 q^{5} - 360 q^{7} - 1452 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 48 q^{3} - 72 q^{5} - 360 q^{7} - 1452 q^{9} - 864 q^{11} - 840 q^{13} - 11544 q^{15} - 12888 q^{17} + 21792 q^{21} + 60264 q^{23} - 42828 q^{25} + 39312 q^{27} - 5760 q^{29} - 18360 q^{31} + 54252 q^{33} + 49728 q^{37} + 31704 q^{39} - 52164 q^{41} + 283968 q^{45} + 104760 q^{47} + 236004 q^{49} + 305664 q^{51} + 134352 q^{53} - 325524 q^{57} + 280368 q^{59} + 76440 q^{61} + 266760 q^{63} - 22752 q^{65} + 1158048 q^{67} - 446904 q^{69} + 43800 q^{73} + 979632 q^{75} + 652104 q^{77} - 225576 q^{79} - 1391652 q^{81} - 306288 q^{83} - 414000 q^{85} - 1793928 q^{87} + 2486304 q^{89} - 1107576 q^{93} + 1538784 q^{95} - 365916 q^{97} - 1331280 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1 0 −26.9422 1.76641i 0 −22.4186 + 38.8302i 0 −44.0273 + 25.4192i 0 722.760 + 95.1818i 0
31.2 0 −18.8245 19.3555i 0 119.170 206.408i 0 −297.697 + 171.875i 0 −20.2728 + 728.718i 0
31.3 0 −18.5878 + 19.5830i 0 60.1651 104.209i 0 336.978 194.554i 0 −37.9871 728.010i 0
31.4 0 −12.3138 + 24.0285i 0 −111.515 + 193.150i 0 −258.486 + 149.237i 0 −425.738 591.767i 0
31.5 0 −8.92132 25.4835i 0 −39.8643 + 69.0470i 0 343.678 198.423i 0 −569.820 + 454.693i 0
31.6 0 0.0475887 + 27.0000i 0 44.4894 77.0579i 0 −295.893 + 170.834i 0 −728.995 + 2.56978i 0
31.7 0 3.22191 26.8071i 0 −37.9246 + 65.6873i 0 −492.804 + 284.521i 0 −708.239 172.740i 0
31.8 0 14.6976 + 22.6491i 0 −82.8778 + 143.549i 0 552.707 319.106i 0 −296.962 + 665.774i 0
31.9 0 19.4106 18.7678i 0 73.7563 127.750i 0 190.108 109.759i 0 24.5406 728.587i 0
31.10 0 21.2667 + 16.6352i 0 52.2018 90.4162i 0 145.321 83.9012i 0 175.541 + 707.549i 0
31.11 0 24.8124 10.6464i 0 −81.8187 + 141.714i 0 30.8728 17.8244i 0 502.310 528.324i 0
31.12 0 26.1330 + 6.78738i 0 −9.36320 + 16.2175i 0 −390.758 + 225.604i 0 636.863 + 354.748i 0
79.1 0 −26.9422 + 1.76641i 0 −22.4186 38.8302i 0 −44.0273 25.4192i 0 722.760 95.1818i 0
79.2 0 −18.8245 + 19.3555i 0 119.170 + 206.408i 0 −297.697 171.875i 0 −20.2728 728.718i 0
79.3 0 −18.5878 19.5830i 0 60.1651 + 104.209i 0 336.978 + 194.554i 0 −37.9871 + 728.010i 0
79.4 0 −12.3138 24.0285i 0 −111.515 193.150i 0 −258.486 149.237i 0 −425.738 + 591.767i 0
79.5 0 −8.92132 + 25.4835i 0 −39.8643 69.0470i 0 343.678 + 198.423i 0 −569.820 454.693i 0
79.6 0 0.0475887 27.0000i 0 44.4894 + 77.0579i 0 −295.893 170.834i 0 −728.995 2.56978i 0
79.7 0 3.22191 + 26.8071i 0 −37.9246 65.6873i 0 −492.804 284.521i 0 −708.239 + 172.740i 0
79.8 0 14.6976 22.6491i 0 −82.8778 143.549i 0 552.707 + 319.106i 0 −296.962 665.774i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
36.f 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.o.c yes 24
3.b odd 2 1 432.7.o.b 24
4.b odd 2 1 144.7.o.a 24
9.c even 3 1 144.7.o.a 24
9.d odd 6 1 432.7.o.c 24
12.b even 2 1 432.7.o.c 24
36.f odd 6 1 inner 144.7.o.c yes 24
36.h even 6 1 432.7.o.b 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.7.o.a 24 4.b odd 2 1
144.7.o.a 24 9.c even 3 1
144.7.o.c yes 24 1.a even 1 1 trivial
144.7.o.c yes 24 36.f odd 6 1 inner
432.7.o.b 24 3.b odd 2 1
432.7.o.b 24 36.h even 6 1
432.7.o.c 24 9.d odd 6 1
432.7.o.c 24 12.b even 2 1