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Properties

Label 464.2.j.a
Level $464$
Weight $2$
Character orbit 464.j
Analytic conductor $3.705$
Analytic rank $0$
Dimension $116$
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] = [464,2,Mod(307,464)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(464, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("464.307");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 464 = 2^{4} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 464.j (of order \(4\), degree \(2\), 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: \(116\)
Relative dimension: \(58\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 116 q + 8 q^{6} - 8 q^{7} + 6 q^{8} - 108 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 116 q + 8 q^{6} - 8 q^{7} + 6 q^{8} - 108 q^{9} + 2 q^{10} - 12 q^{11} - 2 q^{12} - 4 q^{14} + 12 q^{15} - 24 q^{16} - 4 q^{17} - 30 q^{18} - 4 q^{20} - 8 q^{22} - 8 q^{23} - 4 q^{24} - 10 q^{26} - 20 q^{28} - 10 q^{29} + 24 q^{30} - 10 q^{32} + 12 q^{34} - 8 q^{36} - 20 q^{37} + 8 q^{38} - 28 q^{39} - 14 q^{40} - 36 q^{42} - 12 q^{43} - 2 q^{44} - 24 q^{45} - 28 q^{46} + 2 q^{48} + 84 q^{49} - 26 q^{50} + 12 q^{51} + 40 q^{52} - 20 q^{53} - 28 q^{54} - 4 q^{55} - 8 q^{56} + 24 q^{57} - 12 q^{58} - 4 q^{59} - 34 q^{60} - 4 q^{61} + 24 q^{63} + 36 q^{64} - 8 q^{65} - 74 q^{66} - 40 q^{67} + 16 q^{68} + 80 q^{70} + 12 q^{72} + 8 q^{73} - 4 q^{74} + 44 q^{75} + 68 q^{76} - 16 q^{77} + 56 q^{78} + 40 q^{79} - 64 q^{80} + 76 q^{81} - 24 q^{82} - 4 q^{83} + 48 q^{84} + 16 q^{85} + 112 q^{86} - 4 q^{87} + 36 q^{88} - 8 q^{89} - 32 q^{90} + 24 q^{91} - 20 q^{92} - 36 q^{93} + 8 q^{94} + 20 q^{95} + 12 q^{96} - 4 q^{97} + 48 q^{98} + 12 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1 −1.40590 + 0.153120i 1.47445i 1.95311 − 0.430544i −1.30331 − 1.30331i −0.225769 − 2.07293i 0.643539 −2.67995 + 0.904362i 0.825985 2.03189 + 1.63276i
307.2 −1.39107 + 0.254782i − 0.957398i 1.87017 − 0.708842i 0.495500 + 0.495500i 0.243928 + 1.33181i −4.02592 −2.42095 + 1.46254i 2.08339 −0.815522 − 0.563033i
307.3 −1.39027 + 0.259126i 1.86792i 1.86571 − 0.720511i 0.724991 + 0.724991i −0.484027 − 2.59691i 1.97950 −2.40714 + 1.48516i −0.489118 −1.19580 − 0.820070i
307.4 −1.38898 − 0.265979i − 0.565427i 1.85851 + 0.738878i −2.30504 − 2.30504i −0.150392 + 0.785364i 2.76445 −2.38490 − 1.52061i 2.68029 2.58855 + 3.81474i
307.5 −1.38098 − 0.304763i 2.76327i 1.81424 + 0.841747i 1.91033 + 1.91033i 0.842141 − 3.81603i −3.53093 −2.24890 − 1.71535i −4.63563 −2.05594 − 3.22034i
307.6 −1.34509 − 0.436718i − 1.06938i 1.61855 + 1.17485i 2.35358 + 2.35358i −0.467017 + 1.43841i −0.405342 −1.66403 − 2.28714i 1.85643 −2.13794 − 4.19364i
307.7 −1.32415 − 0.496625i − 2.62170i 1.50673 + 1.31521i 0.308473 + 0.308473i −1.30200 + 3.47152i 3.67230 −1.34196 − 2.48981i −3.87332 −0.255268 − 0.561658i
307.8 −1.32269 + 0.500493i − 2.70451i 1.49901 − 1.32399i −2.72729 − 2.72729i 1.35359 + 3.57722i −1.27999 −1.32008 + 2.50148i −4.31435 4.97235 + 2.24237i
307.9 −1.21445 − 0.724641i 2.05571i 0.949791 + 1.76008i −2.15664 − 2.15664i 1.48965 − 2.49656i −2.04484 0.121954 − 2.82580i −1.22594 1.05635 + 4.18194i
307.10 −1.20643 + 0.737913i − 0.475725i 0.910970 − 1.78049i 2.68100 + 2.68100i 0.351043 + 0.573931i 3.31259 0.214817 + 2.82026i 2.77369 −5.21280 − 1.25611i
307.11 −1.13989 − 0.837051i 0.0707663i 0.598691 + 1.90829i −0.122049 − 0.122049i 0.0592350 − 0.0806657i −2.86673 0.914895 − 2.67637i 2.99499 0.0369611 + 0.241284i
307.12 −1.13604 − 0.842261i − 3.14073i 0.581193 + 1.91369i −0.813373 − 0.813373i −2.64531 + 3.56801i −2.78953 0.951566 − 2.66355i −6.86417 0.238956 + 1.60910i
307.13 −1.04841 + 0.949119i − 2.44381i 0.198348 − 1.99014i 0.0104209 + 0.0104209i 2.31946 + 2.56212i 3.06495 1.68093 + 2.27475i −2.97219 −0.0208160 − 0.00103475i
307.14 −1.04659 + 0.951132i 0.691461i 0.190696 − 1.99089i −1.76409 − 1.76409i −0.657671 − 0.723675i −2.67588 1.69402 + 2.26502i 2.52188 3.52417 + 0.168394i
307.15 −1.03227 + 0.966648i 0.0538569i 0.131183 − 1.99569i −0.695345 − 0.695345i −0.0520607 − 0.0555951i −0.265456 1.79372 + 2.18691i 2.99710 1.38994 + 0.0456334i
307.16 −0.911676 + 1.08113i 2.88181i −0.337693 − 1.97128i 1.46472 + 1.46472i −3.11562 − 2.62728i 1.66671 2.43909 + 1.43208i −5.30485 −2.91891 + 0.248206i
307.17 −0.906177 − 1.08575i 0.749495i −0.357686 + 1.96776i −0.0423063 − 0.0423063i 0.813761 − 0.679175i 2.50708 2.46061 − 1.39478i 2.43826 −0.00759688 + 0.0842710i
307.18 −0.845131 + 1.13391i − 3.02225i −0.571506 − 1.91661i 2.21820 + 2.21820i 3.42696 + 2.55420i −4.61929 2.65626 + 0.971748i −6.13402 −4.38991 + 0.640570i
307.19 −0.740668 − 1.20475i 2.39616i −0.902823 + 1.78463i 2.35528 + 2.35528i 2.88677 − 1.77476i 1.43567 2.81872 − 0.234147i −2.74161 1.09303 − 4.58199i
307.20 −0.716772 − 1.21911i − 1.31108i −0.972475 + 1.74765i −1.08552 − 1.08552i −1.59835 + 0.939745i 3.08418 2.82763 − 0.0671134i 1.28107 −0.545299 + 2.10143i
See next 80 embeddings (of 116 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 307.58
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
464.j even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 464.2.j.a ✓ 116
16.f odd 4 1 464.2.t.a yes 116
29.c odd 4 1 464.2.t.a yes 116
464.j even 4 1 inner 464.2.j.a ✓ 116
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
464.2.j.a ✓ 116 1.a even 1 1 trivial
464.2.j.a ✓ 116 464.j even 4 1 inner
464.2.t.a yes 116 16.f odd 4 1
464.2.t.a yes 116 29.c odd 4 1

Hecke kernels

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