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Properties

Label 256.2.m.a
Level $256$
Weight $2$
Character orbit 256.m
Analytic conductor $2.044$
Analytic rank $0$
Dimension $992$
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] = [256,2,Mod(5,256)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(256, base_ring=CyclotomicField(64))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("256.5");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 256.m (of order \(64\), degree \(32\), 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: \(2.04417029174\)
Analytic rank: \(0\)
Dimension: \(992\)
Relative dimension: \(31\) over \(\Q(\zeta_{64})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{64}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 992 q - 32 q^{2} - 32 q^{3} - 32 q^{4} - 32 q^{5} - 32 q^{6} - 32 q^{7} - 32 q^{8} - 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 992 q - 32 q^{2} - 32 q^{3} - 32 q^{4} - 32 q^{5} - 32 q^{6} - 32 q^{7} - 32 q^{8} - 32 q^{9} - 32 q^{10} - 32 q^{11} - 32 q^{12} - 32 q^{13} - 32 q^{14} - 32 q^{15} - 32 q^{16} - 32 q^{17} - 32 q^{18} - 32 q^{19} - 32 q^{20} - 32 q^{21} - 32 q^{22} - 32 q^{23} - 32 q^{24} - 32 q^{25} - 32 q^{26} - 32 q^{27} - 32 q^{28} - 32 q^{29} - 32 q^{30} - 32 q^{31} - 32 q^{32} - 32 q^{33} - 32 q^{34} - 32 q^{35} - 32 q^{36} - 32 q^{37} - 32 q^{38} - 32 q^{39} - 32 q^{40} - 32 q^{41} - 32 q^{42} - 32 q^{43} - 32 q^{44} - 32 q^{45} - 32 q^{46} - 32 q^{47} - 32 q^{48} - 32 q^{49} - 32 q^{50} - 32 q^{51} - 32 q^{52} - 32 q^{53} - 32 q^{54} - 32 q^{55} - 32 q^{56} - 32 q^{57} - 32 q^{58} - 32 q^{59} - 32 q^{60} - 32 q^{61} - 32 q^{62} - 32 q^{63} - 32 q^{64} - 32 q^{65} - 32 q^{66} - 32 q^{67} - 32 q^{68} - 32 q^{69} - 32 q^{70} - 32 q^{71} - 32 q^{72} - 32 q^{73} - 32 q^{74} - 32 q^{75} - 32 q^{76} - 32 q^{77} - 32 q^{78} - 32 q^{79} - 32 q^{80} - 32 q^{81} - 32 q^{82} - 32 q^{83} - 32 q^{84} - 32 q^{85} - 32 q^{86} - 32 q^{87} - 32 q^{88} - 32 q^{89} - 32 q^{90} - 32 q^{91} - 32 q^{92} - 32 q^{93} - 32 q^{94} - 32 q^{95} - 32 q^{96} - 32 q^{97} - 32 q^{98} - 32 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −1.41268 + 0.0658638i −0.355192 + 2.39451i 1.99132 − 0.186089i −0.744117 − 0.0365561i 0.344061 − 3.40607i −4.42442 − 2.36490i −2.80084 + 0.394040i −2.73669 − 0.830167i 1.05361 + 0.00263169i
5.2 −1.38616 + 0.280272i 0.180713 − 1.21827i 1.84290 − 0.777004i −1.95340 − 0.0959644i 0.0909482 + 1.73936i 2.82372 + 1.50931i −2.33678 + 1.59357i 1.41931 + 0.430542i 2.73463 − 0.414460i
5.3 −1.36541 + 0.368314i 0.116030 − 0.782214i 1.72869 − 1.00580i 4.03087 + 0.198024i 0.129671 + 1.11078i −0.803953 − 0.429722i −1.98992 + 2.01003i 2.27243 + 0.689333i −5.57673 + 1.21424i
5.4 −1.33300 − 0.472334i −0.379489 + 2.55831i 1.55380 + 1.25925i 2.31738 + 0.113846i 1.71424 − 3.23099i 4.12682 + 2.20583i −1.47644 − 2.41249i −3.53011 − 1.07085i −3.03530 − 1.24633i
5.5 −1.32340 − 0.498599i −0.0465769 + 0.313996i 1.50280 + 1.31970i −2.22226 − 0.109173i 0.218198 − 0.392320i −0.0426083 − 0.0227746i −1.33081 − 2.49578i 2.77440 + 0.841604i 2.88652 + 1.25250i
5.6 −1.15776 + 0.812153i −0.322964 + 2.17724i 0.680816 − 1.88056i 0.378649 + 0.0186018i −1.39434 − 2.78302i 1.23593 + 0.660620i 0.739076 + 2.73016i −1.76526 − 0.535486i −0.453492 + 0.285984i
5.7 −1.14085 − 0.835738i 0.284707 − 1.91934i 0.603086 + 1.90691i 2.56674 + 0.126096i −1.92887 + 1.95174i 2.08571 + 1.11483i 0.905641 − 2.67952i −0.731981 − 0.222044i −2.82289 − 2.28898i
5.8 −1.00909 − 0.990829i 0.123137 − 0.830120i 0.0365159 + 1.99967i 0.948935 + 0.0466182i −0.946762 + 0.715656i −3.17633 − 1.69778i 1.94448 − 2.05402i 2.19689 + 0.666418i −0.911368 − 0.987274i
5.9 −0.843136 + 1.13540i −0.177658 + 1.19767i −0.578244 − 1.91458i −3.58155 − 0.175950i −1.21004 − 1.21151i −0.919470 − 0.491467i 2.66135 + 0.957719i 1.46797 + 0.445304i 3.21951 − 3.91812i
5.10 −0.813538 + 1.15679i 0.498412 − 3.36002i −0.676313 − 1.88218i 1.91052 + 0.0938577i 3.48135 + 3.31006i 3.27945 + 1.75290i 2.72749 + 0.748874i −8.17052 − 2.47850i −1.66285 + 2.13370i
5.11 −0.731994 − 1.21004i 0.434101 − 2.92647i −0.928370 + 1.77148i −4.11288 − 0.202053i −3.85889 + 1.61688i 2.18104 + 1.16579i 2.82311 − 0.173349i −5.50497 − 1.66991i 2.76611 + 5.12463i
5.12 −0.563821 − 1.29696i −0.373278 + 2.51644i −1.36421 + 1.46251i 3.52543 + 0.173193i 3.47418 − 0.934695i −3.07133 − 1.64166i 2.66599 + 0.944734i −3.32231 − 1.00781i −1.76309 − 4.66999i
5.13 −0.550381 + 1.30272i 0.0413256 − 0.278595i −1.39416 − 1.43398i 2.13467 + 0.104869i 0.340186 + 0.207169i −3.02795 − 1.61847i 2.63540 − 1.02697i 2.79491 + 0.847828i −1.31149 + 2.72316i
5.14 −0.408395 + 1.35396i −0.0625001 + 0.421342i −1.66643 − 1.10590i 1.29990 + 0.0638600i −0.544956 − 0.256697i 3.11690 + 1.66602i 2.17791 − 1.80463i 2.69720 + 0.818186i −0.617338 + 1.73394i
5.15 −0.369587 − 1.36507i −0.111501 + 0.751680i −1.72681 + 1.00902i −1.56967 − 0.0771129i 1.06730 − 0.125605i 1.56830 + 0.838272i 2.01559 + 1.98429i 2.31823 + 0.703228i 0.474865 + 2.17120i
5.16 −0.00498777 + 1.41420i −0.488058 + 3.29022i −1.99995 − 0.0141075i −0.957916 − 0.0470594i −4.65061 − 0.706624i 1.14204 + 0.610433i 0.0299261 − 2.82827i −7.71651 − 2.34078i 0.0713295 − 1.35445i
5.17 0.101501 + 1.41057i 0.345303 − 2.32784i −1.97939 + 0.286349i −1.81333 − 0.0890833i 3.31862 + 0.250793i −2.21013 − 1.18134i −0.604825 − 2.76300i −2.42879 − 0.736767i −0.0583977 − 2.56687i
5.18 0.104160 − 1.41037i 0.381661 − 2.57295i −1.97830 − 0.293808i 2.12912 + 0.104597i −3.58906 − 0.806282i −1.14227 − 0.610554i −0.620438 + 2.75954i −3.60358 − 1.09313i 0.369290 − 2.99196i
5.19 0.247919 − 1.39231i 0.0918072 − 0.618914i −1.87707 − 0.690363i −2.56691 − 0.126104i −0.838961 − 0.281265i −3.84053 − 2.05281i −1.42656 + 2.44232i 2.49620 + 0.757213i −0.811964 + 3.54268i
5.20 0.251229 − 1.39172i −0.448258 + 3.02191i −1.87377 − 0.699281i −2.12455 − 0.104373i 4.09304 + 1.38304i 0.433462 + 0.231690i −1.44395 + 2.43208i −6.06019 − 1.83834i −0.679007 + 2.93056i
See next 80 embeddings (of 992 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.31
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
256.m even 64 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 256.2.m.a ✓ 992
256.m even 64 1 inner 256.2.m.a ✓ 992
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
256.2.m.a ✓ 992 1.a even 1 1 trivial
256.2.m.a ✓ 992 256.m even 64 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(256, [\chi])\).