Properties

Label 465.2.bg.c
Level $465$
Weight $2$
Character orbit 465.bg
Analytic conductor $3.713$
Analytic rank $0$
Dimension $48$
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: \(48\)
Relative dimension: \(6\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 2 q^{2} + 6 q^{3} - 10 q^{4} + 24 q^{5} - 4 q^{6} - 5 q^{7} - 6 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 2 q^{2} + 6 q^{3} - 10 q^{4} + 24 q^{5} - 4 q^{6} - 5 q^{7} - 6 q^{8} + 6 q^{9} - q^{10} + 17 q^{11} - 5 q^{12} + 7 q^{13} - 9 q^{14} + 12 q^{15} - 26 q^{16} + 24 q^{17} + q^{18} - 2 q^{19} - 5 q^{20} + 10 q^{21} + q^{22} - 6 q^{23} + 3 q^{24} - 24 q^{25} - 15 q^{26} - 12 q^{27} + 95 q^{28} - 32 q^{29} - 8 q^{30} - 13 q^{31} + 56 q^{32} - 14 q^{33} - 66 q^{34} - 10 q^{35} - 30 q^{36} + 14 q^{37} - 4 q^{38} + 21 q^{39} - 3 q^{40} - 20 q^{41} - 9 q^{42} - 28 q^{43} + 74 q^{44} - 6 q^{45} - 53 q^{46} - 45 q^{47} + 53 q^{48} + 61 q^{49} + q^{50} + 24 q^{51} - 78 q^{52} + 7 q^{53} - 2 q^{54} + 13 q^{55} - 11 q^{56} - 7 q^{57} - 73 q^{58} + 56 q^{59} + 20 q^{60} + 26 q^{61} + 26 q^{62} + 20 q^{63} - 80 q^{64} - 7 q^{65} + 13 q^{66} - 46 q^{67} - 66 q^{68} + 3 q^{69} - 48 q^{70} + 72 q^{71} + 3 q^{72} + 23 q^{73} - 64 q^{74} + 6 q^{75} - 74 q^{76} - 60 q^{77} - 20 q^{78} + 34 q^{79} + 17 q^{80} + 6 q^{81} + 108 q^{82} - 25 q^{83} + 15 q^{84} + 18 q^{85} + 130 q^{86} - 4 q^{87} - 25 q^{88} - 18 q^{89} - q^{90} + 33 q^{91} + 52 q^{92} - 22 q^{93} - 72 q^{94} + 11 q^{95} + 32 q^{96} - 79 q^{97} - 37 q^{98} - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −0.791724 + 2.43667i 0.669131 0.743145i −3.69252 2.68277i 0.500000 0.866025i 1.28104 + 2.21882i −0.323047 + 3.07359i 5.31499 3.86156i −0.104528 0.994522i 1.71436 + 1.90399i
76.2 −0.547008 + 1.68352i 0.669131 0.743145i −0.916984 0.666228i 0.500000 0.866025i 0.885078 + 1.53300i 0.289063 2.75025i −1.24097 + 0.901616i −0.104528 0.994522i 1.18447 + 1.31548i
76.3 −0.0126485 + 0.0389281i 0.669131 0.743145i 1.61668 + 1.17459i 0.500000 0.866025i 0.0204657 + 0.0354476i −0.375369 + 3.57140i −0.132401 + 0.0961952i −0.104528 0.994522i 0.0273885 + 0.0304180i
76.4 0.197542 0.607972i 0.669131 0.743145i 1.28743 + 0.935370i 0.500000 0.866025i −0.319630 0.553615i 0.264793 2.51934i 1.85734 1.34944i −0.104528 0.994522i −0.427748 0.475063i
76.5 0.599664 1.84558i 0.669131 0.743145i −1.42852 1.03788i 0.500000 0.866025i −0.970277 1.68057i 0.0159459 0.151715i 0.367763 0.267195i −0.104528 0.994522i −1.29848 1.44211i
76.6 0.863191 2.65663i 0.669131 0.743145i −4.69455 3.41079i 0.500000 0.866025i −1.39667 2.41911i 0.217892 2.07310i −8.59378 + 6.24374i −0.104528 0.994522i −1.86911 2.07586i
121.1 −2.17121 + 1.57748i −0.104528 + 0.994522i 1.60769 4.94796i 0.500000 0.866025i −1.34188 2.32421i −3.36862 + 0.716022i 2.65600 + 8.17434i −0.978148 0.207912i 0.280530 + 2.66906i
121.2 −1.42167 + 1.03290i −0.104528 + 0.994522i 0.336215 1.03476i 0.500000 0.866025i −0.878637 1.52184i 1.53605 0.326498i −0.495233 1.52417i −0.978148 0.207912i 0.183685 + 1.74765i
121.3 −0.491640 + 0.357197i −0.104528 + 0.994522i −0.503914 + 1.55089i 0.500000 0.866025i −0.303850 0.526284i −3.62443 + 0.770396i −0.681808 2.09839i −0.978148 0.207912i 0.0635220 + 0.604371i
121.4 0.278085 0.202041i −0.104528 + 0.994522i −0.581523 + 1.78974i 0.500000 0.866025i 0.171866 + 0.297681i 2.80882 0.597033i 0.412326 + 1.26901i −0.978148 0.207912i −0.0359298 0.341849i
121.5 0.943990 0.685849i −0.104528 + 0.994522i −0.197306 + 0.607244i 0.500000 0.866025i 0.583418 + 1.01051i −4.63922 + 0.986096i 0.951367 + 2.92801i −0.978148 0.207912i −0.121968 1.16044i
121.6 2.05342 1.49190i −0.104528 + 0.994522i 1.37275 4.22490i 0.500000 0.866025i 1.26909 + 2.19812i 1.56122 0.331847i −1.91561 5.89563i −0.978148 0.207912i −0.265311 2.52427i
196.1 −2.17121 1.57748i −0.104528 0.994522i 1.60769 + 4.94796i 0.500000 + 0.866025i −1.34188 + 2.32421i −3.36862 0.716022i 2.65600 8.17434i −0.978148 + 0.207912i 0.280530 2.66906i
196.2 −1.42167 1.03290i −0.104528 0.994522i 0.336215 + 1.03476i 0.500000 + 0.866025i −0.878637 + 1.52184i 1.53605 + 0.326498i −0.495233 + 1.52417i −0.978148 + 0.207912i 0.183685 1.74765i
196.3 −0.491640 0.357197i −0.104528 0.994522i −0.503914 1.55089i 0.500000 + 0.866025i −0.303850 + 0.526284i −3.62443 0.770396i −0.681808 + 2.09839i −0.978148 + 0.207912i 0.0635220 0.604371i
196.4 0.278085 + 0.202041i −0.104528 0.994522i −0.581523 1.78974i 0.500000 + 0.866025i 0.171866 0.297681i 2.80882 + 0.597033i 0.412326 1.26901i −0.978148 + 0.207912i −0.0359298 + 0.341849i
196.5 0.943990 + 0.685849i −0.104528 0.994522i −0.197306 0.607244i 0.500000 + 0.866025i 0.583418 1.01051i −4.63922 0.986096i 0.951367 2.92801i −0.978148 + 0.207912i −0.121968 + 1.16044i
196.6 2.05342 + 1.49190i −0.104528 0.994522i 1.37275 + 4.22490i 0.500000 + 0.866025i 1.26909 2.19812i 1.56122 + 0.331847i −1.91561 + 5.89563i −0.978148 + 0.207912i −0.265311 + 2.52427i
226.1 −0.652567 + 2.00839i −0.978148 0.207912i −1.98977 1.44565i 0.500000 + 0.866025i 1.05588 1.82883i 2.63550 1.17340i 0.785009 0.570342i 0.913545 + 0.406737i −2.06560 + 0.439058i
226.2 −0.403889 + 1.24304i −0.978148 0.207912i 0.236004 + 0.171467i 0.500000 + 0.866025i 0.653506 1.13191i −2.27510 + 1.01294i −2.42325 + 1.76060i 0.913545 + 0.406737i −1.27845 + 0.271743i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 76.6
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.c 48
31.g even 15 1 inner 465.2.bg.c 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
465.2.bg.c 48 1.a even 1 1 trivial
465.2.bg.c 48 31.g even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{48} + 2 T_{2}^{47} + 19 T_{2}^{46} + 40 T_{2}^{45} + 249 T_{2}^{44} + 440 T_{2}^{43} + 2538 T_{2}^{42} + 4226 T_{2}^{41} + 22794 T_{2}^{40} + 36407 T_{2}^{39} + 166979 T_{2}^{38} + 248093 T_{2}^{37} + 1056614 T_{2}^{36} + \cdots + 2025 \) acting on \(S_{2}^{\mathrm{new}}(465, [\chi])\). Copy content Toggle raw display