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Properties

Label 465.2.bg.d
Level $465$
Weight $2$
Character orbit 465.bg
Analytic conductor $3.713$
Analytic rank $0$
Dimension $56$
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: \(56\)
Relative dimension: \(7\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 56 q - 2 q^{2} - 7 q^{3} - 22 q^{4} - 28 q^{5} + 4 q^{6} + 7 q^{7} - 6 q^{8} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q - 2 q^{2} - 7 q^{3} - 22 q^{4} - 28 q^{5} + 4 q^{6} + 7 q^{7} - 6 q^{8} + 7 q^{9} + q^{10} - 5 q^{11} - 21 q^{12} - 2 q^{13} - q^{14} + 14 q^{15} - 10 q^{16} - 12 q^{17} + q^{18} - q^{19} + 11 q^{20} + 3 q^{21} - 13 q^{22} + 22 q^{23} - 3 q^{24} - 28 q^{25} - 9 q^{26} + 14 q^{27} + 49 q^{28} + 2 q^{29} - 8 q^{30} - 5 q^{31} + 96 q^{32} - 10 q^{33} + 46 q^{34} - 14 q^{35} - 34 q^{36} + 5 q^{37} + 14 q^{38} - 4 q^{39} + 3 q^{40} + 20 q^{41} + q^{42} - 22 q^{43} + 58 q^{44} + 7 q^{45} + 55 q^{46} - 9 q^{47} - 65 q^{48} - 74 q^{49} + q^{50} + 12 q^{51} + 42 q^{52} - 53 q^{53} + 2 q^{54} - 5 q^{55} + 17 q^{56} + 26 q^{57} - 65 q^{58} - 16 q^{59} + 12 q^{60} - 26 q^{61} - 130 q^{62} + 16 q^{63} - 92 q^{64} - 2 q^{65} - 11 q^{66} + 10 q^{67} - 22 q^{68} + 11 q^{69} + 32 q^{70} + 96 q^{71} + 3 q^{72} - 56 q^{73} - 162 q^{74} - 7 q^{75} + 8 q^{76} - 26 q^{77} + 12 q^{78} + 29 q^{79} + 35 q^{80} + 7 q^{81} + 148 q^{82} + 31 q^{83} + 21 q^{84} - 6 q^{85} + 28 q^{86} - 14 q^{87} + 31 q^{88} + 50 q^{89} + q^{90} - 16 q^{91} + 44 q^{92} - 17 q^{93} + 56 q^{94} - 13 q^{95} + 38 q^{96} - 20 q^{97} + 63 q^{98} + 20 q^{99}+O(q^{100}) \) 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 −0.781082 + 2.40392i −0.669131 + 0.743145i −3.55072 − 2.57975i −0.500000 + 0.866025i −1.26382 − 2.18900i −0.519328 + 4.94107i 4.88513 − 3.54925i −0.104528 − 0.994522i −1.69132 − 1.87840i
76.2 −0.448406 + 1.38005i −0.669131 + 0.743145i −0.0854415 − 0.0620769i −0.500000 + 0.866025i −0.725536 − 1.25667i 0.515687 − 4.90644i −2.22390 + 1.61576i −0.104528 − 0.994522i −0.970957 − 1.07836i
76.3 −0.192445 + 0.592285i −0.669131 + 0.743145i 1.30427 + 0.947606i −0.500000 + 0.866025i −0.311382 − 0.539330i −0.488045 + 4.64344i −1.81991 + 1.32224i −0.104528 − 0.994522i −0.416711 − 0.462804i
76.4 0.0873208 − 0.268746i −0.669131 + 0.743145i 1.55343 + 1.12864i −0.500000 + 0.866025i 0.141288 + 0.244718i −0.0721897 + 0.686839i 0.896181 − 0.651114i −0.104528 − 0.994522i 0.189080 + 0.209995i
76.5 0.217591 − 0.669677i −0.669131 + 0.743145i 1.21691 + 0.884139i −0.500000 + 0.866025i 0.352070 + 0.609803i 0.457081 − 4.34883i 1.99620 − 1.45032i −0.104528 − 0.994522i 0.471161 + 0.523278i
76.6 0.611126 − 1.88085i −0.669131 + 0.743145i −1.54609 − 1.12330i −0.500000 + 0.866025i 0.988822 + 1.71269i −0.0307499 + 0.292566i 0.142270 − 0.103365i −0.104528 − 0.994522i 1.32330 + 1.46968i
76.7 0.814912 − 2.50804i −0.669131 + 0.743145i −4.00816 − 2.91210i −0.500000 + 0.866025i 1.31856 + 2.28381i −0.305245 + 2.90421i −6.30302 + 4.57941i −0.104528 − 0.994522i 1.76457 + 1.95976i
121.1 −2.26713 + 1.64716i 0.104528 − 0.994522i 1.80868 − 5.56655i −0.500000 + 0.866025i 1.40116 + 2.42688i −1.72120 + 0.365853i 3.33658 + 10.2689i −0.978148 − 0.207912i −0.292923 − 2.78697i
121.2 −1.48492 + 1.07886i 0.104528 − 0.994522i 0.423023 − 1.30193i −0.500000 + 0.866025i 0.917732 + 1.58956i 1.57210 − 0.334160i −0.357936 − 1.10161i −0.978148 − 0.207912i −0.191858 − 1.82541i
121.3 −1.11654 + 0.811217i 0.104528 − 0.994522i −0.0294351 + 0.0905918i −0.500000 + 0.866025i 0.690063 + 1.19522i −4.39382 + 0.933935i −0.893588 − 2.75018i −0.978148 − 0.207912i −0.144262 − 1.37256i
121.4 −0.00679230 + 0.00493489i 0.104528 − 0.994522i −0.618012 + 1.90205i −0.500000 + 0.866025i 0.00419787 + 0.00727093i −1.73416 + 0.368606i −0.0103775 − 0.0319387i −0.978148 − 0.207912i −0.000877594 − 0.00834975i
121.5 0.622426 − 0.452219i 0.104528 − 0.994522i −0.435122 + 1.33917i −0.500000 + 0.866025i −0.384680 − 0.666286i 2.78632 − 0.592250i 0.810257 + 2.49371i −0.978148 − 0.207912i 0.0804201 + 0.765146i
121.6 1.26481 − 0.918939i 0.104528 − 0.994522i 0.137264 − 0.422456i −0.500000 + 0.866025i −0.781696 − 1.35394i 3.80533 − 0.808847i 0.751632 + 2.31328i −0.978148 − 0.207912i 0.163419 + 1.55483i
121.7 2.17913 − 1.58323i 0.104528 − 0.994522i 1.62396 − 4.99803i −0.500000 + 0.866025i −1.34678 − 2.33269i −0.0836582 + 0.0177821i −2.70951 − 8.33902i −0.978148 − 0.207912i 0.281553 + 2.67880i
196.1 −2.26713 − 1.64716i 0.104528 + 0.994522i 1.80868 + 5.56655i −0.500000 − 0.866025i 1.40116 − 2.42688i −1.72120 − 0.365853i 3.33658 − 10.2689i −0.978148 + 0.207912i −0.292923 + 2.78697i
196.2 −1.48492 − 1.07886i 0.104528 + 0.994522i 0.423023 + 1.30193i −0.500000 − 0.866025i 0.917732 − 1.58956i 1.57210 + 0.334160i −0.357936 + 1.10161i −0.978148 + 0.207912i −0.191858 + 1.82541i
196.3 −1.11654 − 0.811217i 0.104528 + 0.994522i −0.0294351 − 0.0905918i −0.500000 − 0.866025i 0.690063 − 1.19522i −4.39382 − 0.933935i −0.893588 + 2.75018i −0.978148 + 0.207912i −0.144262 + 1.37256i
196.4 −0.00679230 − 0.00493489i 0.104528 + 0.994522i −0.618012 − 1.90205i −0.500000 − 0.866025i 0.00419787 − 0.00727093i −1.73416 − 0.368606i −0.0103775 + 0.0319387i −0.978148 + 0.207912i −0.000877594 0.00834975i
196.5 0.622426 + 0.452219i 0.104528 + 0.994522i −0.435122 − 1.33917i −0.500000 − 0.866025i −0.384680 + 0.666286i 2.78632 + 0.592250i 0.810257 − 2.49371i −0.978148 + 0.207912i 0.0804201 − 0.765146i
196.6 1.26481 + 0.918939i 0.104528 + 0.994522i 0.137264 + 0.422456i −0.500000 − 0.866025i −0.781696 + 1.35394i 3.80533 + 0.808847i 0.751632 − 2.31328i −0.978148 + 0.207912i 0.163419 − 1.55483i
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 76.7
Significant digits:
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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.d ✓ 56
31.g even 15 1 inner 465.2.bg.d ✓ 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
465.2.bg.d ✓ 56 1.a even 1 1 trivial
465.2.bg.d ✓ 56 31.g even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{56} + 2 T_{2}^{55} + 27 T_{2}^{54} + 56 T_{2}^{53} + 421 T_{2}^{52} + 792 T_{2}^{51} + 4824 T_{2}^{50} + \cdots + 256 \) acting on \(S_{2}^{\mathrm{new}}(465, [\chi])\). Copy content Toggle raw display