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Properties

Label 297.3.m.d
Level $297$
Weight $3$
Character orbit 297.m
Analytic conductor $8.093$
Analytic rank $0$
Dimension $64$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [297,3,Mod(26,297)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(297, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("297.26");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 297 = 3^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 297.m (of order \(10\), degree \(4\), 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: \(8.09266385150\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(16\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 32 q^{4} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q + 32 q^{4} - 8 q^{7} + 4 q^{13} - 168 q^{16} - 20 q^{19} - 120 q^{22} + 110 q^{25} - 300 q^{28} - 34 q^{31} + 52 q^{34} + 174 q^{37} + 384 q^{40} + 92 q^{43} - 560 q^{46} + 104 q^{49} + 388 q^{52} + 530 q^{55} + 532 q^{58} + 48 q^{61} + 712 q^{64} - 240 q^{67} - 340 q^{70} - 64 q^{73} - 1564 q^{76} + 120 q^{79} - 394 q^{82} - 604 q^{85} - 658 q^{88} - 280 q^{91} + 220 q^{94} - 144 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1 −2.29100 + 3.15330i 0 −3.45851 − 10.6442i −2.95599 − 4.06857i 0 −2.24734 − 6.91659i 26.6601 + 8.66240i 0 19.6016
26.2 −2.08238 + 2.86616i 0 −2.64246 − 8.13265i 4.37297 + 6.01887i 0 −0.485607 − 1.49455i 15.3346 + 4.98251i 0 −26.3572
26.3 −1.65860 + 2.28287i 0 −1.22446 − 3.76850i −2.27992 − 3.13804i 0 2.26451 + 6.96944i −0.100805 − 0.0327536i 0 10.9452
26.4 −1.44692 + 1.99151i 0 −0.636483 − 1.95889i 3.95879 + 5.44880i 0 0.0906837 + 0.279096i −4.54255 − 1.47597i 0 −16.5794
26.5 −1.23319 + 1.69734i 0 −0.124145 − 0.382080i −1.64989 − 2.27088i 0 −0.0179354 − 0.0551993i −7.17978 − 2.33285i 0 5.88909
26.6 −0.694768 + 0.956266i 0 0.804326 + 2.47546i 1.63280 + 2.24736i 0 −3.92120 − 12.0682i −7.42265 − 2.41177i 0 −3.28350
26.7 −0.327936 + 0.451365i 0 1.13988 + 3.50819i −3.54394 − 4.87782i 0 −0.345286 − 1.06268i −4.07973 − 1.32558i 0 3.36386
26.8 −0.207428 + 0.285500i 0 1.19758 + 3.68578i 4.20380 + 5.78603i 0 3.66217 + 11.2710i −2.64320 − 0.858829i 0 −2.52390
26.9 0.207428 − 0.285500i 0 1.19758 + 3.68578i −4.20380 − 5.78603i 0 3.66217 + 11.2710i 2.64320 + 0.858829i 0 −2.52390
26.10 0.327936 − 0.451365i 0 1.13988 + 3.50819i 3.54394 + 4.87782i 0 −0.345286 − 1.06268i 4.07973 + 1.32558i 0 3.36386
26.11 0.694768 − 0.956266i 0 0.804326 + 2.47546i −1.63280 − 2.24736i 0 −3.92120 − 12.0682i 7.42265 + 2.41177i 0 −3.28350
26.12 1.23319 − 1.69734i 0 −0.124145 − 0.382080i 1.64989 + 2.27088i 0 −0.0179354 − 0.0551993i 7.17978 + 2.33285i 0 5.88909
26.13 1.44692 − 1.99151i 0 −0.636483 − 1.95889i −3.95879 − 5.44880i 0 0.0906837 + 0.279096i 4.54255 + 1.47597i 0 −16.5794
26.14 1.65860 − 2.28287i 0 −1.22446 − 3.76850i 2.27992 + 3.13804i 0 2.26451 + 6.96944i 0.100805 + 0.0327536i 0 10.9452
26.15 2.08238 − 2.86616i 0 −2.64246 − 8.13265i −4.37297 − 6.01887i 0 −0.485607 − 1.49455i −15.3346 − 4.98251i 0 −26.3572
26.16 2.29100 − 3.15330i 0 −3.45851 − 10.6442i 2.95599 + 4.06857i 0 −2.24734 − 6.91659i −26.6601 − 8.66240i 0 19.6016
53.1 −3.48569 − 1.13257i 0 7.63128 + 5.54445i −6.70573 + 2.17882i 0 −7.56448 − 5.49592i −11.7037 − 16.1088i 0 25.8418
53.2 −3.37486 − 1.09656i 0 6.95116 + 5.05031i −0.207237 + 0.0673353i 0 6.32644 + 4.59643i −9.57811 − 13.1831i 0 0.773231
53.3 −2.61191 − 0.848662i 0 2.86579 + 2.08212i 1.61531 − 0.524845i 0 8.70612 + 6.32537i 0.738817 + 1.01689i 0 −4.66446
53.4 −2.56884 − 0.834665i 0 2.66618 + 1.93710i 7.07792 − 2.29975i 0 −9.48618 − 6.89212i 1.11835 + 1.53928i 0 −20.1015
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.16
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.c even 5 1 inner
33.h odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 297.3.m.d ✓ 64
3.b odd 2 1 inner 297.3.m.d ✓ 64
11.c even 5 1 inner 297.3.m.d ✓ 64
33.h odd 10 1 inner 297.3.m.d ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
297.3.m.d ✓ 64 1.a even 1 1 trivial
297.3.m.d ✓ 64 3.b odd 2 1 inner
297.3.m.d ✓ 64 11.c even 5 1 inner
297.3.m.d ✓ 64 33.h odd 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{64} - 48 T_{2}^{62} + 1418 T_{2}^{60} - 33500 T_{2}^{58} + 686363 T_{2}^{56} - 11823272 T_{2}^{54} + \cdots + 10485760000 \) acting on \(S_{3}^{\mathrm{new}}(297, [\chi])\). Copy content Toggle raw display