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Properties

Label 464.2.bi.a
Level $464$
Weight $2$
Character orbit 464.bi
Analytic conductor $3.705$
Analytic rank $0$
Dimension $696$
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] = [464,2,Mod(45,464)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(464, base_ring=CyclotomicField(28))
 
chi = DirichletCharacter(H, H._module([0, 21, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("464.45");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 464 = 2^{4} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 464.bi (of order \(28\), degree \(12\), 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: \(3.70505865379\)
Analytic rank: \(0\)
Dimension: \(696\)
Relative dimension: \(58\) over \(\Q(\zeta_{28})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{28}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 696 q - 10 q^{2} - 10 q^{3} - 14 q^{4} - 10 q^{5} - 22 q^{6} - 10 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 696 q - 10 q^{2} - 10 q^{3} - 14 q^{4} - 10 q^{5} - 22 q^{6} - 10 q^{8} - 10 q^{10} + 10 q^{11} - 36 q^{12} - 10 q^{13} - 22 q^{14} - 20 q^{15} + 10 q^{16} - 48 q^{17} + 2 q^{18} - 10 q^{19} - 38 q^{20} - 22 q^{21} - 32 q^{22} - 10 q^{24} - 10 q^{26} + 2 q^{27} + 52 q^{28} - 20 q^{29} - 44 q^{30} - 20 q^{31} - 10 q^{32} - 20 q^{33} - 62 q^{34} - 30 q^{35} - 14 q^{36} - 26 q^{37} - 88 q^{38} + 22 q^{40} - 34 q^{42} - 2 q^{43} - 4 q^{44} - 18 q^{45} - 20 q^{46} - 20 q^{47} + 70 q^{48} + 72 q^{49} - 46 q^{50} - 40 q^{51} + 12 q^{52} + 6 q^{53} + 10 q^{54} - 38 q^{56} - 2 q^{58} - 48 q^{59} - 86 q^{60} - 10 q^{61} + 56 q^{62} - 36 q^{63} + 100 q^{64} - 36 q^{65} - 182 q^{66} + 30 q^{67} - 140 q^{68} - 22 q^{69} + 32 q^{70} + 104 q^{72} + 18 q^{74} - 72 q^{75} - 10 q^{76} - 22 q^{77} - 42 q^{78} - 20 q^{79} + 160 q^{80} + 64 q^{81} - 46 q^{82} - 10 q^{83} + 16 q^{84} - 60 q^{85} - 24 q^{86} - 4 q^{88} - 142 q^{90} - 14 q^{91} - 32 q^{92} + 122 q^{93} - 92 q^{94} - 20 q^{95} + 86 q^{96} - 20 q^{97} + 66 q^{98} - 88 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
45.1 −1.41391 − 0.0293361i −2.61729 + 0.294898i 1.99828 + 0.0829570i −2.65983 + 0.930713i 3.70927 − 0.340178i −1.67146 − 1.33295i −2.82295 − 0.175915i 3.83848 − 0.876108i 3.78805 − 1.23792i
45.2 −1.40631 − 0.149288i 2.62876 − 0.296190i 1.95543 + 0.419892i −3.61446 + 1.26476i −3.74107 + 0.0240920i 3.07386 + 2.45132i −2.68725 − 0.882422i 3.89786 − 0.889662i 5.27188 − 1.23904i
45.3 −1.38705 + 0.275866i −0.348389 + 0.0392540i 1.84780 − 0.765278i 1.44108 − 0.504254i 0.472402 − 0.150556i 3.07737 + 2.45412i −2.35186 + 1.57122i −2.80495 + 0.640212i −1.85973 + 1.09697i
45.4 −1.38457 + 0.288049i 2.07978 − 0.234335i 1.83406 − 0.797647i 1.38211 − 0.483622i −2.81209 + 0.923530i −1.99117 − 1.58791i −2.30961 + 1.63269i 1.34578 − 0.307165i −1.77432 + 1.06772i
45.5 −1.36571 − 0.367198i 2.09525 − 0.236078i 1.73033 + 1.00297i 1.98812 − 0.695673i −2.94819 − 0.446956i 1.19428 + 0.952403i −1.99484 − 2.00514i 1.40954 − 0.321718i −2.97064 + 0.220055i
45.6 −1.34664 + 0.431942i −0.448267 + 0.0505075i 1.62685 − 1.16334i −3.16794 + 1.10851i 0.581835 − 0.261640i 0.199697 + 0.159253i −1.68828 + 2.26929i −2.72639 + 0.622281i 3.78724 − 2.86112i
45.7 −1.34247 − 0.444723i 1.12398 − 0.126642i 1.60444 + 1.19405i −1.09870 + 0.384452i −1.56522 − 0.329846i −2.38073 − 1.89857i −1.62289 − 2.31651i −1.67750 + 0.382879i 1.64594 − 0.0274968i
45.8 −1.33132 − 0.477067i −1.83649 + 0.206923i 1.54481 + 1.27026i −0.342204 + 0.119742i 2.54367 + 0.600649i 3.16925 + 2.52739i −1.45064 − 2.42809i 0.405101 − 0.0924617i 0.512707 + 0.00383908i
45.9 −1.32389 − 0.497316i −1.22014 + 0.137477i 1.50535 + 1.31678i 2.91598 − 1.02035i 1.68369 + 0.424791i −1.30786 − 1.04298i −1.33806 − 2.49191i −1.45495 + 0.332082i −4.36787 − 0.0993424i
45.10 −1.24609 + 0.668775i −2.68906 + 0.302984i 1.10548 − 1.66671i 3.10533 − 1.08660i 3.14818 − 2.17592i −0.258392 − 0.206061i −0.262876 + 2.81618i 4.21446 − 0.961923i −3.14282 + 3.43077i
45.11 −1.19521 + 0.755956i −0.0156654 + 0.00176507i 0.857060 − 1.80706i 0.0497111 − 0.0173947i 0.0173892 − 0.0139520i −0.883781 − 0.704792i 0.341687 + 2.80771i −2.92454 + 0.667508i −0.0462657 + 0.0583698i
45.12 −1.09468 + 0.895364i 2.76884 − 0.311973i 0.396645 − 1.96027i 1.42447 − 0.498444i −2.75166 + 2.82063i 3.38379 + 2.69848i 1.32096 + 2.50101i 4.64435 − 1.06004i −1.11305 + 1.82106i
45.13 −1.07147 − 0.923017i −0.585312 + 0.0659488i 0.296078 + 1.97796i −2.40103 + 0.840157i 0.688013 + 0.469591i −1.50400 − 1.19940i 1.50846 − 2.39261i −2.58654 + 0.590362i 3.34810 + 1.31599i
45.14 −1.03064 − 0.968386i 3.03974 − 0.342497i 0.124457 + 1.99612i 2.32415 − 0.813256i −3.46457 − 2.59065i −0.773270 − 0.616663i 1.80475 − 2.17782i 6.19796 − 1.41464i −3.18292 − 1.41250i
45.15 −0.997205 − 1.00279i −3.28314 + 0.369921i −0.0111657 + 1.99997i 0.142662 − 0.0499196i 3.64491 + 2.92340i −1.80341 − 1.43817i 2.01668 − 1.98318i 7.71735 − 1.76144i −0.192322 − 0.0932796i
45.16 −0.869274 + 1.11551i 1.37784 − 0.155245i −0.488724 − 1.93937i 4.11345 − 1.43936i −1.02454 + 1.67194i −2.03236 − 1.62075i 2.58822 + 1.14067i −1.05045 + 0.239758i −1.97010 + 5.83979i
45.17 −0.843539 + 1.13510i −2.77399 + 0.312554i −0.576883 − 1.91500i −1.47276 + 0.515340i 1.98519 − 3.41240i 2.93175 + 2.33799i 2.66033 + 0.960557i 4.67257 − 1.06648i 0.657369 − 2.10643i
45.18 −0.783622 − 1.17726i 0.957676 − 0.107904i −0.771873 + 1.84505i −2.16233 + 0.756630i −0.877487 − 1.04288i 0.390747 + 0.311610i 2.77696 − 0.537129i −2.01928 + 0.460888i 2.58520 + 1.95270i
45.19 −0.778597 + 1.18059i −2.01029 + 0.226506i −0.787575 − 1.83840i −1.65248 + 0.578226i 1.29780 − 2.54968i −3.67966 − 2.93443i 2.78360 + 0.501574i 1.06519 − 0.243122i 0.603965 − 2.40110i
45.20 −0.772931 − 1.18430i −1.14283 + 0.128766i −0.805154 + 1.83077i 2.67243 − 0.935125i 1.03583 + 1.25393i 1.24551 + 0.993261i 2.79052 − 0.461513i −1.63531 + 0.373248i −3.17308 − 2.44219i
See next 80 embeddings (of 696 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 45.58
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner
29.d even 7 1 inner
464.bi even 28 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 464.2.bi.a ✓ 696
16.e even 4 1 inner 464.2.bi.a ✓ 696
29.d even 7 1 inner 464.2.bi.a ✓ 696
464.bi even 28 1 inner 464.2.bi.a ✓ 696
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
464.2.bi.a ✓ 696 1.a even 1 1 trivial
464.2.bi.a ✓ 696 16.e even 4 1 inner
464.2.bi.a ✓ 696 29.d even 7 1 inner
464.2.bi.a ✓ 696 464.bi even 28 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(464, [\chi])\).