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Properties

Label 441.2.bb.d
Level $441$
Weight $2$
Character orbit 441.bb
Analytic conductor $3.521$
Analytic rank $0$
Dimension $48$
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] = [441,2,Mod(37,441)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(441, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([0, 32]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.37");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 441.bb (of order \(21\), degree \(12\), 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: \(3.52140272914\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(4\) over \(\Q(\zeta_{21})\)
Twist minimal: no (minimal twist has level 49)
Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 13 q^{2} - 9 q^{4} + 14 q^{5} - 14 q^{7} + 20 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 13 q^{2} - 9 q^{4} + 14 q^{5} - 14 q^{7} + 20 q^{8} - 14 q^{10} + 3 q^{11} - 14 q^{13} - 21 q^{14} - 3 q^{16} + 7 q^{17} + 21 q^{19} - 14 q^{20} - 20 q^{22} - 15 q^{23} - 4 q^{25} + 28 q^{28} - 12 q^{29} + 35 q^{31} - 45 q^{32} + 70 q^{34} + 15 q^{37} + 28 q^{38} - 42 q^{40} + 42 q^{41} - 30 q^{43} + 50 q^{44} - 78 q^{46} - 21 q^{47} - 70 q^{49} - 40 q^{50} - 70 q^{52} - 11 q^{53} - 7 q^{55} + 28 q^{56} + 16 q^{58} + 28 q^{59} + 7 q^{61} + 28 q^{62} - 32 q^{64} - 14 q^{65} + 11 q^{67} - 77 q^{68} + 70 q^{70} - 19 q^{71} + 7 q^{73} - 34 q^{74} + 119 q^{76} - 7 q^{77} + 15 q^{79} - 70 q^{80} - 14 q^{82} - 26 q^{85} + 33 q^{86} - 77 q^{88} + 14 q^{89} + 84 q^{91} + 38 q^{92} + 14 q^{94} + 61 q^{95} + 161 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1 −1.52543 + 1.04002i 0 0.514606 − 1.31120i 1.32986 + 0.410207i 0 −1.73764 − 1.99515i −0.242977 − 1.06455i 0 −2.45523 + 0.757337i
37.2 −0.174575 + 0.119023i 0 −0.714372 + 1.82019i 3.75050 + 1.15688i 0 1.85388 + 1.88763i −0.185965 − 0.814768i 0 −0.792437 + 0.244435i
37.3 0.968018 − 0.659983i 0 −0.229202 + 0.583997i −3.03447 − 0.936009i 0 −2.50722 − 0.844891i 0.684966 + 3.00103i 0 −3.55517 + 1.09662i
37.4 2.09732 − 1.42993i 0 1.62338 − 4.13631i 2.38971 + 0.737129i 0 −1.40100 + 2.24437i −1.38019 − 6.04701i 0 6.06605 − 1.87113i
46.1 −0.940144 + 2.39545i 0 −3.38820 − 3.14379i 0.392378 + 0.267519i 0 −1.60453 + 2.10369i 6.07919 − 2.92758i 0 −1.00972 + 0.688415i
46.2 −0.0341744 + 0.0870750i 0 1.45969 + 1.35439i 2.09852 + 1.43075i 0 −0.301609 − 2.62850i −0.336373 + 0.161989i 0 −0.196298 + 0.133834i
46.3 0.354207 − 0.902506i 0 0.777050 + 0.720997i 0.605902 + 0.413097i 0 −0.310399 + 2.62748i 2.67297 − 1.28723i 0 0.587437 − 0.400508i
46.4 0.887059 − 2.26019i 0 −2.85548 − 2.64950i −1.29033 − 0.879733i 0 −2.25101 − 1.39030i −4.14619 + 1.99670i 0 −3.13296 + 2.13602i
100.1 −1.16258 − 0.358609i 0 −0.429482 − 0.292816i 2.56601 − 0.386764i 0 −0.339612 + 2.62386i 1.91142 + 2.39684i 0 −3.12189 − 0.470550i
100.2 −0.681025 − 0.210069i 0 −1.23281 − 0.840516i −2.65874 + 0.400741i 0 −0.518691 − 2.59441i 1.55172 + 1.94579i 0 1.89485 + 0.285603i
100.3 1.16438 + 0.359164i 0 −0.425690 − 0.290231i 0.478452 − 0.0721150i 0 1.90196 − 1.83918i −1.91089 − 2.39618i 0 0.583002 + 0.0878735i
100.4 2.50546 + 0.772833i 0 4.02760 + 2.74597i −2.04183 + 0.307757i 0 0.189789 + 2.63894i 4.69930 + 5.89274i 0 −5.35358 − 0.806922i
109.1 −1.51353 + 1.40435i 0 0.169112 − 2.25665i 0.0830372 − 0.211575i 0 1.07894 + 2.41576i 0.338532 + 0.424506i 0 0.171447 + 0.436839i
109.2 −0.168237 + 0.156102i 0 −0.145524 + 1.94188i −0.711168 + 1.81203i 0 −2.04196 − 1.68237i −0.564834 − 0.708279i 0 −0.163215 − 0.415865i
109.3 1.02480 − 0.950878i 0 −0.00340854 + 0.0454838i 1.11243 − 2.83442i 0 2.44418 − 1.01290i 1.78303 + 2.23585i 0 −1.55517 − 3.96251i
109.4 1.73170 − 1.60678i 0 0.267571 − 3.57049i 0.567937 − 1.44708i 0 −2.62161 − 0.356599i −2.32789 − 2.91908i 0 −1.34164 − 3.41845i
163.1 −0.940144 − 2.39545i 0 −3.38820 + 3.14379i 0.392378 − 0.267519i 0 −1.60453 − 2.10369i 6.07919 + 2.92758i 0 −1.00972 − 0.688415i
163.2 −0.0341744 − 0.0870750i 0 1.45969 − 1.35439i 2.09852 − 1.43075i 0 −0.301609 + 2.62850i −0.336373 − 0.161989i 0 −0.196298 − 0.133834i
163.3 0.354207 + 0.902506i 0 0.777050 − 0.720997i 0.605902 − 0.413097i 0 −0.310399 − 2.62748i 2.67297 + 1.28723i 0 0.587437 + 0.400508i
163.4 0.887059 + 2.26019i 0 −2.85548 + 2.64950i −1.29033 + 0.879733i 0 −2.25101 + 1.39030i −4.14619 − 1.99670i 0 −3.13296 − 2.13602i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.g even 21 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.2.bb.d 48
3.b odd 2 1 49.2.g.a ✓ 48
12.b even 2 1 784.2.bg.c 48
21.c even 2 1 343.2.g.g 48
21.g even 6 1 343.2.e.c 48
21.g even 6 1 343.2.g.h 48
21.h odd 6 1 343.2.e.d 48
21.h odd 6 1 343.2.g.i 48
49.g even 21 1 inner 441.2.bb.d 48
147.k even 14 1 343.2.g.h 48
147.l odd 14 1 343.2.g.i 48
147.n odd 42 1 49.2.g.a ✓ 48
147.n odd 42 1 343.2.e.d 48
147.n odd 42 1 2401.2.a.h 24
147.o even 42 1 343.2.e.c 48
147.o even 42 1 343.2.g.g 48
147.o even 42 1 2401.2.a.i 24
588.bb even 42 1 784.2.bg.c 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.2.g.a ✓ 48 3.b odd 2 1
49.2.g.a ✓ 48 147.n odd 42 1
343.2.e.c 48 21.g even 6 1
343.2.e.c 48 147.o even 42 1
343.2.e.d 48 21.h odd 6 1
343.2.e.d 48 147.n odd 42 1
343.2.g.g 48 21.c even 2 1
343.2.g.g 48 147.o even 42 1
343.2.g.h 48 21.g even 6 1
343.2.g.h 48 147.k even 14 1
343.2.g.i 48 21.h odd 6 1
343.2.g.i 48 147.l odd 14 1
441.2.bb.d 48 1.a even 1 1 trivial
441.2.bb.d 48 49.g even 21 1 inner
784.2.bg.c 48 12.b even 2 1
784.2.bg.c 48 588.bb even 42 1
2401.2.a.h 24 147.n odd 42 1
2401.2.a.i 24 147.o even 42 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{48} - 13 T_{2}^{47} + 85 T_{2}^{46} - 382 T_{2}^{45} + 1339 T_{2}^{44} - 3829 T_{2}^{43} + \cdots + 729 \) acting on \(S_{2}^{\mathrm{new}}(441, [\chi])\). Copy content Toggle raw display