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Properties

Label 784.2.bt.a
Level $784$
Weight $2$
Character orbit 784.bt
Analytic conductor $6.260$
Analytic rank $0$
Dimension $2640$
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] = [784,2,Mod(37,784)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(784, base_ring=CyclotomicField(84))
 
chi = DirichletCharacter(H, H._module([0, 21, 64]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("784.37");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 784.bt (of order \(84\), degree \(24\), 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: \(6.26027151847\)
Analytic rank: \(0\)
Dimension: \(2640\)
Relative dimension: \(110\) over \(\Q(\zeta_{84})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{84}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 2640 q - 26 q^{2} - 26 q^{3} - 26 q^{4} - 26 q^{5} - 20 q^{6} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 2640 q - 26 q^{2} - 26 q^{3} - 26 q^{4} - 26 q^{5} - 20 q^{6} - 8 q^{8} - 32 q^{10} - 26 q^{11} - 26 q^{12} - 20 q^{13} - 4 q^{14} - 40 q^{15} - 26 q^{16} - 52 q^{17} - 14 q^{18} - 12 q^{19} - 36 q^{20} - 18 q^{21} + 4 q^{22} - 46 q^{24} - 26 q^{26} - 8 q^{27} - 98 q^{28} - 20 q^{29} - 8 q^{30} + 120 q^{31} - 36 q^{32} - 52 q^{33} - 108 q^{34} - 26 q^{35} - 4 q^{36} - 26 q^{37} - 46 q^{38} + 34 q^{40} - 58 q^{42} - 20 q^{43} - 68 q^{44} - 28 q^{46} - 12 q^{47} - 96 q^{48} - 48 q^{49} - 52 q^{50} - 58 q^{51} - 32 q^{52} - 26 q^{53} - 110 q^{54} + 40 q^{56} - 24 q^{58} - 10 q^{59} - 242 q^{60} - 26 q^{61} - 28 q^{62} - 20 q^{63} - 80 q^{64} - 52 q^{65} - 170 q^{66} - 24 q^{67} - 38 q^{68} - 8 q^{69} - 104 q^{70} - 42 q^{72} - 4 q^{74} - 4 q^{75} - 170 q^{76} - 10 q^{77} - 112 q^{78} - 24 q^{79} - 4 q^{80} - 240 q^{81} + 130 q^{82} - 60 q^{83} + 212 q^{84} - 40 q^{85} - 20 q^{86} - 312 q^{88} - 8 q^{90} + 108 q^{91} - 68 q^{92} + 4 q^{93} - 142 q^{94} - 76 q^{95} - 628 q^{96} - 96 q^{97} + 42 q^{98} - 24 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1 −1.41296 + 0.0594596i 0.0220594 − 0.589550i 1.99293 − 0.168029i 0.642394 + 1.21547i 0.00388530 + 0.834324i 0.649672 − 2.56475i −2.80594 + 0.355917i 2.64453 + 0.198180i −0.979951 − 1.67922i
37.2 −1.41229 − 0.0738235i −0.110251 + 2.94651i 1.98910 + 0.208520i −1.19886 − 2.26835i 0.373227 − 4.15317i 2.62038 + 0.365500i −2.79378 − 0.441332i −5.67816 − 0.425519i 1.52567 + 3.29207i
37.3 −1.40957 + 0.114536i −0.0833711 + 2.22814i 1.97376 − 0.322891i −0.788648 − 1.49219i −0.137684 − 3.15026i 0.373356 − 2.61928i −2.74517 + 0.681204i −1.96604 − 0.147334i 1.28256 + 2.01302i
37.4 −1.40849 − 0.127103i 0.105684 − 2.82445i 1.96769 + 0.358047i 1.82699 + 3.45683i −0.507851 + 3.96478i 1.78407 + 1.95374i −2.72596 − 0.754405i −4.97474 − 0.372806i −2.13392 − 5.10113i
37.5 −1.40485 + 0.162487i −0.0480669 + 1.28462i 1.94720 − 0.456540i 1.74044 + 3.29306i −0.141207 − 1.81250i −2.36603 − 1.18403i −2.66133 + 0.957764i 1.34368 + 0.100695i −2.98013 − 4.34346i
37.6 −1.40414 − 0.168478i 0.127333 − 3.40303i 1.94323 + 0.473135i −1.50609 − 2.84967i −0.752131 + 4.75689i −2.62146 − 0.357677i −2.64886 − 0.991742i −8.57281 − 0.642443i 1.63466 + 4.25509i
37.7 −1.39250 + 0.246841i 0.0386629 − 1.03329i 1.87814 − 0.687455i −1.02536 − 1.94008i 0.201220 + 1.44840i 0.129125 + 2.64260i −2.44563 + 1.42089i 1.92542 + 0.144290i 1.90672 + 2.44847i
37.8 −1.38943 − 0.263620i −0.0353741 + 0.945392i 1.86101 + 0.732562i 0.486276 + 0.920079i 0.298374 − 1.30423i 2.40636 + 1.09974i −2.39262 − 1.50844i 2.09910 + 0.157306i −0.433093 − 1.40657i
37.9 −1.38731 + 0.274553i −0.0297465 + 0.794991i 1.84924 − 0.761779i −1.82338 − 3.45000i −0.177000 − 1.11106i −2.61181 − 0.422461i −2.35632 + 1.56454i 2.36049 + 0.176894i 3.47679 + 4.28559i
37.10 −1.36812 + 0.358129i 0.0917590 − 2.45231i 1.74349 − 0.979926i −0.159467 − 0.301727i 0.752708 + 3.38791i 2.31675 − 1.27776i −2.03435 + 1.96505i −3.01380 − 0.225853i 0.326227 + 0.355688i
37.11 −1.35258 + 0.412939i 0.0554548 − 1.48206i 1.65896 − 1.11707i 0.915972 + 1.73310i 0.536993 + 2.02751i −2.63734 + 0.210802i −1.78260 + 2.19598i 0.798184 + 0.0598156i −1.95459 − 1.96593i
37.12 −1.34308 − 0.442859i 0.0617173 − 1.64943i 1.60775 + 1.18959i −0.297187 − 0.562306i −0.813356 + 2.18799i −0.347153 + 2.62288i −1.63252 − 2.30973i 0.274805 + 0.0205938i 0.150126 + 0.886837i
37.13 −1.33028 − 0.479964i 0.0108582 − 0.290191i 1.53927 + 1.27697i −0.245961 − 0.465380i −0.153726 + 0.380823i −2.51609 − 0.818106i −1.43475 − 2.43752i 2.90752 + 0.217888i 0.103830 + 0.737137i
37.14 −1.31568 − 0.518629i 0.0779232 − 2.08254i 1.46205 + 1.36470i −1.38115 − 2.61326i −1.18259 + 2.69955i 2.35853 − 1.19889i −1.21582 − 2.55378i −1.33929 − 0.100366i 0.461843 + 4.15452i
37.15 −1.30849 + 0.536532i −0.116744 + 3.12006i 1.42427 − 1.40409i 1.92261 + 3.63775i −1.52126 − 4.14519i 2.60258 + 0.476006i −1.11029 + 2.60139i −6.72954 − 0.504309i −4.46747 − 3.72840i
37.16 −1.26186 − 0.638521i −0.118081 + 3.15579i 1.18458 + 1.61145i −0.153691 − 0.290798i 2.16404 − 3.90676i −2.50877 − 0.840273i −0.465831 − 2.78980i −6.95344 − 0.521088i 0.00825610 + 0.465081i
37.17 −1.25710 + 0.647833i −0.00846442 + 0.226216i 1.16062 − 1.62879i 1.00081 + 1.89363i −0.135910 − 0.289861i 1.53901 + 2.15208i −0.403845 + 2.79945i 2.94051 + 0.220361i −2.48488 − 1.73213i
37.18 −1.23370 + 0.691357i −0.0445768 + 1.19134i 1.04405 − 1.70586i −0.0886082 − 0.167655i −0.768647 − 1.50058i −0.576187 + 2.58225i −0.108690 + 2.82634i 1.57431 + 0.117978i 0.225226 + 0.145577i
37.19 −1.22290 − 0.710299i −0.0882130 + 2.35754i 0.990951 + 1.73724i 1.14404 + 2.16463i 1.78243 − 2.82037i 0.414136 − 2.61314i 0.0221326 − 2.82834i −2.55861 − 0.191741i 0.138491 − 3.45972i
37.20 −1.19640 − 0.754078i −0.0574716 + 1.53596i 0.862732 + 1.80435i −1.69643 − 3.20981i 1.22699 − 1.79428i −0.491120 + 2.59977i 0.328454 − 2.80929i 0.635735 + 0.0476418i −0.390838 + 5.11945i
See next 80 embeddings (of 2640 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.110
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner
49.g even 21 1 inner
784.bt even 84 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.2.bt.a ✓ 2640
16.e even 4 1 inner 784.2.bt.a ✓ 2640
49.g even 21 1 inner 784.2.bt.a ✓ 2640
784.bt even 84 1 inner 784.2.bt.a ✓ 2640
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
784.2.bt.a ✓ 2640 1.a even 1 1 trivial
784.2.bt.a ✓ 2640 16.e even 4 1 inner
784.2.bt.a ✓ 2640 49.g even 21 1 inner
784.2.bt.a ✓ 2640 784.bt even 84 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(784, [\chi])\).