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Properties

Label 465.2.bg.a
Level $465$
Weight $2$
Character orbit 465.bg
Analytic conductor $3.713$
Analytic rank $0$
Dimension $32$
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] = [465,2,Mod(76,465)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(465, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([0, 0, 22]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("465.76");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 465 = 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 465.bg (of order \(15\), degree \(8\), 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: \(3.71304369399\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(4\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 4 q^{3} - 12 q^{4} - 16 q^{5} + 5 q^{6} - 3 q^{7} + 16 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 4 q^{3} - 12 q^{4} - 16 q^{5} + 5 q^{6} - 3 q^{7} + 16 q^{8} + 4 q^{9} - 9 q^{11} + 6 q^{12} + 7 q^{13} + 21 q^{14} - 8 q^{15} + 2 q^{16} + 8 q^{17} + 14 q^{19} + 6 q^{20} + 2 q^{21} + 15 q^{22} - 14 q^{23} - 23 q^{24} - 16 q^{25} + 5 q^{26} - 8 q^{27} - 59 q^{28} + 18 q^{29} - 10 q^{30} + 15 q^{31} + 12 q^{32} - 2 q^{33} - 31 q^{34} + 6 q^{35} - 9 q^{36} + 4 q^{37} + 16 q^{38} + 21 q^{39} - 23 q^{40} + 2 q^{41} + 21 q^{42} + 34 q^{43} - 26 q^{44} + 4 q^{45} + 36 q^{46} + 17 q^{47} - 26 q^{48} - 93 q^{49} + 8 q^{51} - 56 q^{52} + 15 q^{53} + 5 q^{54} - 9 q^{55} - 15 q^{56} + 29 q^{57} - 7 q^{58} + 10 q^{59} + 3 q^{60} + 14 q^{61} - 9 q^{62} - 4 q^{63} + 20 q^{64} + 7 q^{65} - 15 q^{66} - 28 q^{67} - 7 q^{68} + 7 q^{69} - 12 q^{70} - 16 q^{71} + 2 q^{72} - 35 q^{73} + 12 q^{74} + 4 q^{75} - 56 q^{76} + 14 q^{77} - 20 q^{78} - 10 q^{79} + 14 q^{80} + 4 q^{81} - 6 q^{82} - 41 q^{83} + 41 q^{84} - 16 q^{85} - 86 q^{86} + 6 q^{87} + 53 q^{88} + 50 q^{89} + 53 q^{91} - 44 q^{92} + 38 q^{93} - 68 q^{94} + 17 q^{95} + 49 q^{96} + 77 q^{97} + 28 q^{98} + 16 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
76.1 −0.682397 + 2.10020i 0.669131 − 0.743145i −2.32715 − 1.69078i −0.500000 + 0.866025i 1.10414 + 1.91243i 0.498617 − 4.74402i 1.56594 − 1.13772i −0.104528 − 0.994522i −1.47763 − 1.64107i
76.2 −0.378553 + 1.16507i 0.669131 − 0.743145i 0.403957 + 0.293492i −0.500000 + 0.866025i 0.612512 + 1.06090i −0.273951 + 2.60647i −2.47699 + 1.79964i −0.104528 − 0.994522i −0.819700 − 0.910370i
76.3 0.268094 − 0.825108i 0.669131 − 0.743145i 1.00911 + 0.733158i −0.500000 + 0.866025i −0.433785 − 0.751338i −0.344963 + 3.28210i 2.27923 − 1.65596i −0.104528 − 0.994522i 0.580518 + 0.644730i
76.4 0.548442 − 1.68793i 0.669131 − 0.743145i −0.930285 − 0.675891i −0.500000 + 0.866025i −0.887397 − 1.53702i 0.289427 − 2.75372i 1.22061 − 0.886828i −0.104528 − 0.994522i 1.18757 + 1.31893i
121.1 −1.81280 + 1.31708i −0.104528 + 0.994522i 0.933522 − 2.87309i −0.500000 + 0.866025i −1.12037 − 1.94054i 0.806709 − 0.171471i 0.706929 + 2.17570i −0.978148 − 0.207912i −0.234222 − 2.22847i
121.2 −0.833057 + 0.605252i −0.104528 + 0.994522i −0.290379 + 0.893694i −0.500000 + 0.866025i −0.514858 − 0.891760i −1.60913 + 0.342032i −0.935407 − 2.87889i −0.978148 − 0.207912i −0.107635 − 1.02407i
121.3 0.414273 − 0.300987i −0.104528 + 0.994522i −0.537005 + 1.65273i −0.500000 + 0.866025i 0.256035 + 0.443466i −1.40118 + 0.297830i 0.591461 + 1.82033i −0.978148 − 0.207912i 0.0535259 + 0.509265i
121.4 1.45793 − 1.05925i −0.104528 + 0.994522i 0.385514 − 1.18649i −0.500000 + 0.866025i 0.901048 + 1.56066i 1.59908 − 0.339894i 0.419023 + 1.28962i −0.978148 − 0.207912i 0.188370 + 1.79222i
196.1 −1.81280 − 1.31708i −0.104528 − 0.994522i 0.933522 + 2.87309i −0.500000 − 0.866025i −1.12037 + 1.94054i 0.806709 + 0.171471i 0.706929 − 2.17570i −0.978148 + 0.207912i −0.234222 + 2.22847i
196.2 −0.833057 − 0.605252i −0.104528 − 0.994522i −0.290379 − 0.893694i −0.500000 − 0.866025i −0.514858 + 0.891760i −1.60913 − 0.342032i −0.935407 + 2.87889i −0.978148 + 0.207912i −0.107635 + 1.02407i
196.3 0.414273 + 0.300987i −0.104528 − 0.994522i −0.537005 − 1.65273i −0.500000 − 0.866025i 0.256035 − 0.443466i −1.40118 − 0.297830i 0.591461 − 1.82033i −0.978148 + 0.207912i 0.0535259 − 0.509265i
196.4 1.45793 + 1.05925i −0.104528 − 0.994522i 0.385514 + 1.18649i −0.500000 − 0.866025i 0.901048 − 1.56066i 1.59908 + 0.339894i 0.419023 − 1.28962i −0.978148 + 0.207912i 0.188370 − 1.79222i
226.1 −0.804615 + 2.47635i −0.978148 − 0.207912i −3.86687 − 2.80945i −0.500000 − 0.866025i 1.30189 − 2.25495i 1.29368 − 0.575985i 5.85550 − 4.25427i 0.913545 + 0.406737i 2.54689 − 0.541358i
226.2 −0.582736 + 1.79348i −0.978148 − 0.207912i −1.25894 − 0.914674i −0.500000 − 0.866025i 0.942886 − 1.63313i −1.03235 + 0.459634i −0.677169 + 0.491992i 0.913545 + 0.406737i 1.84456 − 0.392074i
226.3 −0.0408649 + 0.125769i −0.978148 − 0.207912i 1.60389 + 1.16529i −0.500000 − 0.866025i 0.0661207 − 0.114524i −2.92676 + 1.30308i −0.426071 + 0.309559i 0.913545 + 0.406737i 0.129352 − 0.0274945i
226.4 0.554596 − 1.70687i −0.978148 − 0.207912i −0.987801 − 0.717679i −0.500000 − 0.866025i −0.897356 + 1.55427i 1.18729 − 0.528614i 1.13109 − 0.821783i 0.913545 + 0.406737i −1.75549 + 0.373141i
286.1 −0.804615 − 2.47635i −0.978148 + 0.207912i −3.86687 + 2.80945i −0.500000 + 0.866025i 1.30189 + 2.25495i 1.29368 + 0.575985i 5.85550 + 4.25427i 0.913545 − 0.406737i 2.54689 + 0.541358i
286.2 −0.582736 − 1.79348i −0.978148 + 0.207912i −1.25894 + 0.914674i −0.500000 + 0.866025i 0.942886 + 1.63313i −1.03235 − 0.459634i −0.677169 − 0.491992i 0.913545 − 0.406737i 1.84456 + 0.392074i
286.3 −0.0408649 − 0.125769i −0.978148 + 0.207912i 1.60389 − 1.16529i −0.500000 + 0.866025i 0.0661207 + 0.114524i −2.92676 − 1.30308i −0.426071 − 0.309559i 0.913545 − 0.406737i 0.129352 + 0.0274945i
286.4 0.554596 + 1.70687i −0.978148 + 0.207912i −0.987801 + 0.717679i −0.500000 + 0.866025i −0.897356 − 1.55427i 1.18729 + 0.528614i 1.13109 + 0.821783i 0.913545 − 0.406737i −1.75549 − 0.373141i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 76.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.g even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 465.2.bg.a ✓ 32
31.g even 15 1 inner 465.2.bg.a ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
465.2.bg.a ✓ 32 1.a even 1 1 trivial
465.2.bg.a ✓ 32 31.g even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} + 14 T_{2}^{30} - 12 T_{2}^{29} + 109 T_{2}^{28} - 92 T_{2}^{27} + 712 T_{2}^{26} - 474 T_{2}^{25} + \cdots + 81 \) acting on \(S_{2}^{\mathrm{new}}(465, [\chi])\). Copy content Toggle raw display