↑

Properties

Label 416.2.bf.a
Level $416$
Weight $2$
Character orbit 416.bf
Analytic conductor $3.322$
Analytic rank $0$
Dimension $192$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [416,2,Mod(53,416)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(416, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 5, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("416.53");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 416 = 2^{5} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 416.bf (of order \(8\), 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: \(3.32177672409\)
Analytic rank: \(0\)
Dimension: \(192\)
Relative dimension: \(48\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 192 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 192 q - 16 q^{10} - 16 q^{12} - 40 q^{16} - 40 q^{18} - 8 q^{22} - 16 q^{23} + 24 q^{24} + 80 q^{30} - 48 q^{31} - 48 q^{35} + 80 q^{36} - 56 q^{38} - 64 q^{40} - 16 q^{43} - 80 q^{44} + 24 q^{50} + 32 q^{51} - 32 q^{53} - 24 q^{54} + 32 q^{55} + 16 q^{56} - 64 q^{58} - 56 q^{60} - 64 q^{61} + 24 q^{62} + 80 q^{63} + 72 q^{64} + 96 q^{66} + 80 q^{67} - 32 q^{68} - 64 q^{69} - 120 q^{72} - 32 q^{74} + 32 q^{75} + 64 q^{76} - 32 q^{77} - 56 q^{80} - 80 q^{82} - 80 q^{83} + 112 q^{84} - 112 q^{87} - 88 q^{88} - 120 q^{90} + 144 q^{92} + 40 q^{94} + 136 q^{96} + 56 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1 −1.41213 − 0.0766792i 2.98623 − 1.23694i 1.98824 + 0.216563i −1.24588 + 3.00782i −4.31180 + 1.51774i 2.81433 + 2.81433i −2.79105 − 0.458272i 5.26623 − 5.26623i 1.98999 − 4.15191i
53.2 −1.40600 + 0.152175i −0.435996 + 0.180596i 1.95369 − 0.427917i 0.326436 − 0.788085i 0.585530 − 0.320266i 1.89112 + 1.89112i −2.68177 + 0.898955i −1.96384 + 1.96384i −0.339042 + 1.15773i
53.3 −1.40466 − 0.164109i −1.71912 + 0.712084i 1.94614 + 0.461036i 1.47294 − 3.55599i 2.53164 − 0.718111i −1.70912 − 1.70912i −2.65800 − 0.966978i 0.327000 − 0.327000i −2.65255 + 4.75323i
53.4 −1.34738 + 0.429614i 2.33432 − 0.966908i 1.63086 − 1.15771i 1.29458 − 3.12540i −2.72982 + 2.30565i −0.532783 − 0.532783i −1.70002 + 2.26051i 2.39283 − 2.39283i −0.401578 + 4.76727i
53.5 −1.33922 − 0.454405i −3.06223 + 1.26842i 1.58703 + 1.21710i −0.807892 + 1.95042i 4.67738 − 0.307199i 2.62405 + 2.62405i −1.57233 − 2.35112i 5.64706 − 5.64706i 1.96823 − 2.24494i
53.6 −1.30913 + 0.534951i −2.18213 + 0.903868i 1.42765 − 1.40064i −0.245375 + 0.592388i 2.37317 − 2.35062i −0.0561050 − 0.0561050i −1.11971 + 2.59735i 1.82339 − 1.82339i 0.00432984 − 0.906777i
53.7 −1.28283 − 0.595266i −1.29613 + 0.536874i 1.29132 + 1.52725i −0.0161519 + 0.0389942i 1.98230 + 0.0828216i −1.32535 − 1.32535i −0.747423 − 2.72789i −0.729605 + 0.729605i 0.0439321 − 0.0404083i
53.8 −1.22753 − 0.702263i 0.795910 − 0.329677i 1.01365 + 1.72410i −0.746776 + 1.80288i −1.20852 − 0.154250i −0.218632 − 0.218632i −0.0335206 − 2.82823i −1.59653 + 1.59653i 2.18278 − 1.68865i
53.9 −1.19033 − 0.763623i 2.04342 − 0.846410i 0.833761 + 1.81792i 1.04561 − 2.52432i −3.07867 − 0.552892i −1.49027 − 1.49027i 0.395758 − 2.80060i 1.33781 − 1.33781i −3.17224 + 2.20631i
53.10 −1.13640 + 0.841781i −1.01781 + 0.421589i 0.582811 − 1.91320i −1.65824 + 4.00334i 0.801750 − 1.33586i 0.742191 + 0.742191i 0.948187 + 2.66476i −1.26313 + 1.26313i −1.48551 − 5.94527i
53.11 −1.07659 + 0.917037i 1.38221 − 0.572530i 0.318088 − 1.97454i −0.0209193 + 0.0505036i −0.963042 + 1.88392i 3.16119 + 3.16119i 1.46828 + 2.41747i −0.538605 + 0.538605i −0.0237922 − 0.0735553i
53.12 −0.948103 − 1.04933i −0.976242 + 0.404373i −0.202200 + 1.98975i −0.597830 + 1.44329i 1.34990 + 0.641016i 2.62460 + 2.62460i 2.27962 − 1.67432i −1.33179 + 1.33179i 2.08130 − 0.741064i
53.13 −0.914272 + 1.07894i −0.832888 + 0.344993i −0.328213 − 1.97289i 0.202993 − 0.490067i 0.389260 − 1.21405i −3.33390 − 3.33390i 2.42870 + 1.44963i −1.54664 + 1.54664i 0.343162 + 0.667071i
53.14 −0.894410 + 1.09546i 0.0294720 − 0.0122077i −0.400063 − 1.95958i 1.27084 − 3.06808i −0.0129870 + 0.0432041i −0.470428 − 0.470428i 2.50446 + 1.31441i −2.12060 + 2.12060i 2.22431 + 4.13628i
53.15 −0.884565 − 1.10342i −0.349851 + 0.144913i −0.435089 + 1.95210i 1.55977 − 3.76561i 0.469367 + 0.257849i 3.21782 + 3.21782i 2.53886 − 1.24667i −2.01992 + 2.01992i −5.53478 + 1.60984i
53.16 −0.738912 − 1.20582i 2.63272 − 1.09051i −0.908017 + 1.78199i −0.0873689 + 0.210927i −3.26031 − 2.36880i 0.671813 + 0.671813i 2.81971 − 0.221831i 3.62067 − 3.62067i 0.318899 − 0.0505053i
53.17 −0.706943 − 1.22484i −3.07212 + 1.27251i −1.00046 + 1.73178i 0.840287 − 2.02863i 3.73043 + 2.86325i −1.05731 − 1.05731i 2.82843 + 0.00113391i 5.69729 − 5.69729i −3.07878 + 0.404911i
53.18 −0.705052 + 1.22593i 3.06559 − 1.26981i −1.00580 − 1.72869i 0.112862 − 0.272473i −0.604704 + 4.65347i −1.80857 − 1.80857i 2.82839 − 0.0142285i 5.66408 − 5.66408i 0.254459 + 0.330468i
53.19 −0.561631 − 1.29791i −1.77289 + 0.734354i −1.36914 + 1.45789i −1.54177 + 3.72217i 1.94884 + 1.88861i −2.71316 − 2.71316i 2.66117 + 0.958222i 0.482536 − 0.482536i 5.69694 − 0.0894043i
53.20 −0.493432 + 1.32534i 1.01315 − 0.419661i −1.51305 − 1.30793i −1.26766 + 3.06041i 0.0562725 + 1.54984i −0.757044 − 0.757044i 2.48004 − 1.35993i −1.27096 + 1.27096i −3.43058 − 3.19019i
See next 80 embeddings (of 192 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
32.g even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 416.2.bf.a ✓ 192
32.g even 8 1 inner 416.2.bf.a ✓ 192
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
416.2.bf.a ✓ 192 1.a even 1 1 trivial
416.2.bf.a ✓ 192 32.g even 8 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(416, [\chi])\).