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Properties

Label 768.2.ba.a
Level $768$
Weight $2$
Character orbit 768.ba
Analytic conductor $6.133$
Analytic rank $0$
Dimension $4032$
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] = [768,2,Mod(11,768)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(768, base_ring=CyclotomicField(64))
 
chi = DirichletCharacter(H, H._module([32, 21, 32]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("768.11");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 768.ba (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: \(6.13251087523\)
Analytic rank: \(0\)
Dimension: \(4032\)
Relative dimension: \(126\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 4032 q - 32 q^{3} - 64 q^{4} - 32 q^{6} - 64 q^{7} - 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 4032 q - 32 q^{3} - 64 q^{4} - 32 q^{6} - 64 q^{7} - 32 q^{9} - 64 q^{10} - 32 q^{12} - 64 q^{13} - 32 q^{15} - 64 q^{16} - 32 q^{18} - 64 q^{19} - 32 q^{21} - 64 q^{22} - 32 q^{24} - 64 q^{25} - 32 q^{27} - 64 q^{28} - 32 q^{30} - 64 q^{31} - 32 q^{33} - 64 q^{34} - 32 q^{36} - 64 q^{37} - 32 q^{39} - 64 q^{40} - 32 q^{42} - 64 q^{43} - 32 q^{45} - 64 q^{46} - 32 q^{48} - 64 q^{49} - 32 q^{51} - 64 q^{52} - 32 q^{54} - 64 q^{55} - 32 q^{57} - 64 q^{58} - 32 q^{60} - 64 q^{61} - 32 q^{63} - 64 q^{64} - 32 q^{66} - 64 q^{67} - 32 q^{69} - 64 q^{70} - 32 q^{72} - 64 q^{73} - 32 q^{75} - 64 q^{76} - 32 q^{78} - 64 q^{79} - 32 q^{81} - 64 q^{82} - 32 q^{84} - 64 q^{85} - 32 q^{87} - 64 q^{88} - 32 q^{90} - 64 q^{91} - 32 q^{93} - 64 q^{94} - 32 q^{96} - 64 q^{97} - 32 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −1.41393 + 0.0283381i 1.46154 − 0.929459i 1.99839 − 0.0801361i 0.632155 − 0.378899i −2.04018 + 1.35561i −3.85375 + 3.16269i −2.82332 + 0.169937i 1.27221 − 2.71689i −0.883085 + 0.553651i
11.2 −1.41378 + 0.0350901i 0.707806 − 1.58083i 1.99754 − 0.0992192i −3.22012 + 1.93007i −0.945210 + 2.25977i −0.205950 + 0.169019i −2.82059 + 0.210368i −1.99802 − 2.23784i 4.48481 − 2.84168i
11.3 −1.41083 + 0.0977288i 0.00333359 − 1.73205i 1.98090 − 0.275758i 0.864841 − 0.518366i 0.164568 + 2.44396i 3.55130 − 2.91448i −2.76777 + 0.582639i −2.99998 − 0.0115479i −1.16949 + 0.815847i
11.4 −1.40992 − 0.110058i −0.334326 + 1.69948i 1.97577 + 0.310348i 2.77454 − 1.66300i 0.658416 − 2.35934i −2.09647 + 1.72053i −2.75154 − 0.655018i −2.77645 − 1.13636i −4.09492 + 2.03934i
11.5 −1.40491 + 0.161984i −1.70309 − 0.315411i 1.94752 − 0.455144i 1.48089 − 0.887614i 2.44377 + 0.167250i 3.29286 − 2.70238i −2.66236 + 0.954901i 2.80103 + 1.07435i −1.93674 + 1.48689i
11.6 −1.40403 + 0.169375i −1.37122 − 1.05818i 1.94262 − 0.475616i −2.37779 + 1.42519i 2.10447 + 1.25347i −0.947059 + 0.777231i −2.64695 + 0.996813i 0.760502 + 2.90201i 3.09711 − 2.40376i
11.7 −1.39561 − 0.228612i 1.67652 + 0.435082i 1.89547 + 0.638107i 2.61657 − 1.56831i −2.24030 − 0.990477i 0.446891 − 0.366754i −2.49947 − 1.32388i 2.62141 + 1.45884i −4.01026 + 1.59058i
11.8 −1.39285 + 0.244884i −0.688083 + 1.58951i 1.88006 − 0.682174i −2.04953 + 1.22844i 0.569151 − 2.38245i 2.38358 − 1.95615i −2.45159 + 1.41056i −2.05308 − 2.18743i 2.55386 − 2.21293i
11.9 −1.38676 − 0.277316i 1.71662 + 0.230687i 1.84619 + 0.769140i −0.810541 + 0.485819i −2.31656 − 0.795953i 1.45334 − 1.19273i −2.34692 − 1.57859i 2.89357 + 0.792004i 1.25875 − 0.448938i
11.10 −1.37112 + 0.346466i 0.719322 + 1.57562i 1.75992 − 0.950090i 0.362382 − 0.217204i −1.53217 − 1.91114i −0.120578 + 0.0989554i −2.08389 + 1.91244i −1.96515 + 2.26676i −0.421615 + 0.423364i
11.11 −1.36554 + 0.367825i 0.126056 − 1.72746i 1.72941 − 1.00456i 2.30232 − 1.37996i 0.463266 + 2.40528i −1.00396 + 0.823928i −1.99208 + 2.00789i −2.96822 − 0.435514i −2.63633 + 2.73124i
11.12 −1.36029 − 0.386792i −1.20916 + 1.24013i 1.70078 + 1.05230i −0.969042 + 0.580822i 2.12449 − 1.21925i −2.55598 + 2.09764i −1.90654 − 2.08928i −0.0758553 − 2.99904i 1.54284 − 0.415268i
11.13 −1.35941 + 0.389873i −1.38254 + 1.04335i 1.69600 − 1.06000i 3.60659 − 2.16171i 1.47266 − 1.95736i 0.163213 − 0.133946i −1.89230 + 2.10219i 0.822823 − 2.88495i −4.06005 + 4.34477i
11.14 −1.33876 − 0.455771i 1.09155 + 1.34481i 1.58455 + 1.22033i −3.09820 + 1.85699i −0.848390 − 2.29788i −3.19893 + 2.62530i −1.56513 − 2.35592i −0.617048 + 2.93586i 4.99411 − 1.07399i
11.15 −1.33864 + 0.456107i 1.20836 + 1.24091i 1.58393 − 1.22113i −0.914426 + 0.548086i −2.18355 − 1.11000i −0.525404 + 0.431188i −1.56336 + 2.35710i −0.0797320 + 2.99894i 0.974105 − 1.15077i
11.16 −1.31852 − 0.511384i −1.70461 − 0.307073i 1.47697 + 1.34854i 0.501425 − 0.300543i 2.09053 + 1.27659i 0.143066 − 0.117411i −1.25779 − 2.53337i 2.81141 + 1.04688i −0.814831 + 0.139850i
11.17 −1.31794 − 0.512874i −0.455235 − 1.67116i 1.47392 + 1.35187i 0.298426 − 0.178870i −0.257120 + 2.43596i −2.69673 + 2.21315i −1.24920 − 2.53762i −2.58552 + 1.52154i −0.485045 + 0.0826844i
11.18 −1.28974 − 0.580139i −0.155618 + 1.72505i 1.32688 + 1.49646i 0.489502 − 0.293396i 1.20147 − 2.13459i 1.55553 − 1.27659i −0.843178 − 2.69982i −2.95157 − 0.536897i −0.801543 + 0.0944269i
11.19 −1.25398 + 0.653863i −1.73144 + 0.0461259i 1.14493 − 1.63986i 0.274560 − 0.164565i 2.14103 − 1.18996i −2.77076 + 2.27390i −0.363472 + 2.80498i 2.99574 − 0.159728i −0.236690 + 0.385886i
11.20 −1.22593 + 0.705049i 1.66563 − 0.475038i 1.00581 − 1.72868i −1.27245 + 0.762680i −1.70703 + 1.75672i −1.35342 + 1.11072i −0.0142530 + 2.82839i 2.54868 − 1.58248i 1.02222 − 1.83214i
See next 80 embeddings (of 4032 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.126
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
256.n odd 64 1 inner
768.ba even 64 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 768.2.ba.a ✓ 4032
3.b odd 2 1 inner 768.2.ba.a ✓ 4032
256.n odd 64 1 inner 768.2.ba.a ✓ 4032
768.ba even 64 1 inner 768.2.ba.a ✓ 4032
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
768.2.ba.a ✓ 4032 1.a even 1 1 trivial
768.2.ba.a ✓ 4032 3.b odd 2 1 inner
768.2.ba.a ✓ 4032 256.n odd 64 1 inner
768.2.ba.a ✓ 4032 768.ba even 64 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(768, [\chi])\).