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Properties

Label 448.2.bc.c
Level $448$
Weight $2$
Character orbit 448.bc
Analytic conductor $3.577$
Analytic rank $0$
Dimension $192$
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] = [448,2,Mod(29,448)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(448, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([0, 11, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("448.29");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 448.bc (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: \(3.57729801055\)
Analytic rank: \(0\)
Dimension: \(192\)
Relative dimension: \(24\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 192 q - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 192 q - 16 q^{9} - 16 q^{11} + 72 q^{12} - 8 q^{13} + 8 q^{14} - 16 q^{17} - 16 q^{19} - 56 q^{22} + 8 q^{23} - 56 q^{26} - 16 q^{29} - 40 q^{30} + 40 q^{32} - 8 q^{36} - 16 q^{37} + 16 q^{39} - 40 q^{40} - 24 q^{43} - 64 q^{44} + 80 q^{45} + 16 q^{47} + 40 q^{50} - 16 q^{52} - 16 q^{53} + 192 q^{54} - 64 q^{55} + 32 q^{56} - 16 q^{57} + 64 q^{58} + 48 q^{60} + 24 q^{61} + 32 q^{62} - 224 q^{63} - 96 q^{64} - 160 q^{65} - 96 q^{66} - 56 q^{67} + 16 q^{68} + 8 q^{69} + 48 q^{70} + 16 q^{71} + 88 q^{72} + 8 q^{73} + 104 q^{74} - 64 q^{75} + 272 q^{76} + 8 q^{77} + 8 q^{78} - 64 q^{80} - 32 q^{81} + 16 q^{83} - 16 q^{84} + 128 q^{85} - 256 q^{86} + 8 q^{87} - 192 q^{88} + 8 q^{89} + 8 q^{91} - 136 q^{92} + 32 q^{93} - 80 q^{94} + 16 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1 −1.39476 + 0.233735i −0.236983 − 0.0471389i 1.89074 − 0.652011i 0.319548 − 0.478237i 0.341554 + 0.0103563i −0.923880 + 0.382683i −2.48473 + 1.35133i −2.71770 − 1.12571i −0.333913 + 0.741718i
29.2 −1.38246 − 0.297985i 1.89636 + 0.377209i 1.82241 + 0.823907i −0.996693 + 1.49166i −2.50924 − 1.08656i −0.923880 + 0.382683i −2.27390 − 1.68207i 0.682251 + 0.282598i 1.82238 − 1.76516i
29.3 −1.28980 + 0.580024i −3.05344 − 0.607368i 1.32715 − 1.49622i 0.545184 − 0.815926i 4.29061 − 0.987689i −0.923880 + 0.382683i −0.843901 + 2.69960i 6.18298 + 2.56107i −0.229920 + 1.36860i
29.4 −1.20537 − 0.739651i 2.99753 + 0.596246i 0.905832 + 1.78311i 1.05714 − 1.58212i −3.17212 − 2.93583i −0.923880 + 0.382683i 0.227015 − 2.81930i 5.85805 + 2.42648i −2.44446 + 1.12513i
29.5 −1.17944 + 0.780334i 1.88144 + 0.374242i 0.782156 − 1.84071i 1.91333 − 2.86350i −2.51108 + 1.02676i −0.923880 + 0.382683i 0.513867 + 2.78136i 0.628132 + 0.260181i −0.0221691 + 4.87036i
29.6 −1.14123 − 0.835216i −1.09732 − 0.218270i 0.604828 + 1.90635i −0.430262 + 0.643932i 1.06999 + 1.16559i −0.923880 + 0.382683i 0.901967 − 2.68076i −1.61518 − 0.669028i 1.02885 − 0.375516i
29.7 −0.937879 + 1.05848i −1.82564 − 0.363143i −0.240766 − 1.98546i −0.371764 + 0.556384i 2.09661 − 1.59183i −0.923880 + 0.382683i 2.32738 + 1.60727i 0.429465 + 0.177890i −0.240253 − 0.915326i
29.8 −0.712695 − 1.22150i 1.03851 + 0.206572i −0.984133 + 1.74112i −1.88927 + 2.82750i −0.487812 − 1.41576i −0.923880 + 0.382683i 2.82816 − 0.0387639i −1.73581 − 0.718996i 4.80027 + 0.292608i
29.9 −0.698297 + 1.22979i 1.87538 + 0.373036i −1.02476 − 1.71752i −1.41571 + 2.11876i −1.76832 + 2.04583i −0.923880 + 0.382683i 2.82777 − 0.0609068i 0.606244 + 0.251115i −1.61704 − 3.22055i
29.10 −0.612242 − 1.27482i −2.37538 − 0.472493i −1.25032 + 1.56099i 1.93055 − 2.88927i 0.851966 + 3.31746i −0.923880 + 0.382683i 2.75548 + 0.638226i 2.64756 + 1.09666i −4.86526 − 0.692167i
29.11 −0.220091 − 1.39698i −0.814253 − 0.161965i −1.90312 + 0.614927i 0.149441 − 0.223655i −0.0470523 + 1.17314i −0.923880 + 0.382683i 1.27790 + 2.52328i −2.13486 − 0.884290i −0.345332 − 0.159542i
29.12 −0.0915647 + 1.41125i −1.92546 − 0.382997i −1.98323 − 0.258441i −1.97675 + 2.95841i 0.716807 − 2.68222i −0.923880 + 0.382683i 0.546317 − 2.77516i 0.789052 + 0.326836i −3.99405 − 3.06056i
29.13 −0.0876261 + 1.41150i 0.121491 + 0.0241660i −1.98464 − 0.247368i 1.52076 − 2.27597i −0.0447559 + 0.169366i −0.923880 + 0.382683i 0.523065 − 2.77964i −2.75746 − 1.14218i 3.07927 + 2.34598i
29.14 0.243227 − 1.39314i 2.85266 + 0.567428i −1.88168 − 0.677698i 2.04405 − 3.05913i 1.48435 − 3.83614i −0.923880 + 0.382683i −1.40180 + 2.45661i 5.04403 + 2.08930i −3.76464 − 3.59171i
29.15 0.538967 + 1.30748i −2.28548 − 0.454610i −1.41903 + 1.40938i 1.69360 − 2.53466i −0.637403 − 3.23324i −0.923880 + 0.382683i −2.60755 − 1.09575i 2.24510 + 0.929949i 4.22682 + 0.848263i
29.16 0.616390 − 1.27282i −0.110539 − 0.0219876i −1.24013 − 1.56910i −0.0609872 + 0.0912738i −0.0961213 + 0.127143i −0.923880 + 0.382683i −2.76158 + 0.611276i −2.75990 − 1.14319i 0.0785830 + 0.133886i
29.17 0.829550 + 1.14536i −0.122986 − 0.0244635i −0.623695 + 1.90026i −1.18457 + 1.77283i −0.0740038 − 0.161157i −0.923880 + 0.382683i −2.69387 + 0.862008i −2.75711 − 1.14203i −3.01318 + 0.113895i
29.18 0.984463 − 1.01530i −2.18132 − 0.433891i −0.0616659 − 1.99905i −1.23353 + 1.84611i −2.58796 + 1.78754i −0.923880 + 0.382683i −2.09034 − 1.90538i 1.79825 + 0.744859i 0.659988 + 3.06983i
29.19 1.05979 + 0.936398i −3.22016 − 0.640531i 0.246319 + 1.98477i −1.07951 + 1.61561i −2.81291 − 3.69418i −0.923880 + 0.382683i −1.59749 + 2.33410i 7.18754 + 2.97718i −2.65691 + 0.701353i
29.20 1.17213 − 0.791272i 3.30175 + 0.656758i 0.747778 − 1.85495i −2.19915 + 3.29126i 4.38975 − 1.84277i −0.923880 + 0.382683i −0.591274 − 2.76593i 7.69856 + 3.18885i 0.0265918 + 5.59791i
See next 80 embeddings (of 192 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
64.i even 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 448.2.bc.c ✓ 192
64.i even 16 1 inner 448.2.bc.c ✓ 192
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
448.2.bc.c ✓ 192 1.a even 1 1 trivial
448.2.bc.c ✓ 192 64.i even 16 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{192} + 8 T_{3}^{190} + 16 T_{3}^{189} + 40 T_{3}^{188} + 48 T_{3}^{187} + 208 T_{3}^{186} + \cdots + 98\!\cdots\!36 \) acting on \(S_{2}^{\mathrm{new}}(448, [\chi])\). Copy content Toggle raw display