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Properties

Label 462.2.bc.a
Level $462$
Weight $2$
Character orbit 462.bc
Analytic conductor $3.689$
Analytic rank $0$
Dimension $128$
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] = [462,2,Mod(95,462)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(462, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([15, 20, 21]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("462.95");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 462.bc (of order \(30\), 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: \(3.68908857338\)
Analytic rank: \(0\)
Dimension: \(128\)
Relative dimension: \(16\) over \(\Q(\zeta_{30})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 128 q - 16 q^{2} + 16 q^{4} + 32 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 128 q - 16 q^{2} + 16 q^{4} + 32 q^{8} - 4 q^{9} + 6 q^{11} - 12 q^{15} + 16 q^{16} + 2 q^{17} + 4 q^{18} + 2 q^{22} - 12 q^{25} - 18 q^{27} - 5 q^{28} - 38 q^{29} - 6 q^{30} - 3 q^{31} + 64 q^{32} - 4 q^{33} - 16 q^{34} + 31 q^{35} + 8 q^{36} + 2 q^{37} + 22 q^{39} + 5 q^{40} - 16 q^{41} + 17 q^{42} + q^{44} + 28 q^{45} + 38 q^{49} - 34 q^{50} + 16 q^{51} - 25 q^{53} + 6 q^{54} - 42 q^{55} + 20 q^{57} - 19 q^{58} - 40 q^{59} - 4 q^{60} + 40 q^{61} + 4 q^{62} + 6 q^{63} - 32 q^{64} - 20 q^{65} - 41 q^{66} + 16 q^{67} + 2 q^{68} - 68 q^{69} - 21 q^{70} - 80 q^{71} - 16 q^{72} + 10 q^{73} - 2 q^{74} - 14 q^{75} - q^{77} - 16 q^{78} + 5 q^{80} - 88 q^{81} - 8 q^{82} + 92 q^{83} - 48 q^{84} - 100 q^{85} + 40 q^{86} + 38 q^{87} - q^{88} - 164 q^{90} + 12 q^{91} + 20 q^{92} + 47 q^{93} + 40 q^{94} - 38 q^{95} - 16 q^{97} - 18 q^{98} - 138 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
95.1 0.104528 − 0.994522i −1.64414 + 0.544800i −0.978148 − 0.207912i 1.25353 − 2.81547i 0.369957 + 1.69208i 1.28350 + 2.31358i −0.309017 + 0.951057i 2.40638 − 1.79146i −2.66902 − 1.54096i
95.2 0.104528 − 0.994522i −1.63004 + 0.585637i −0.978148 − 0.207912i 0.112013 − 0.251585i 0.412043 + 1.68233i −2.57741 − 0.597448i −0.309017 + 0.951057i 2.31406 − 1.90922i −0.238498 − 0.137697i
95.3 0.104528 − 0.994522i −1.45000 − 0.947367i −0.978148 − 0.207912i 0.625478 − 1.40485i −1.09374 + 1.34303i −1.00131 − 2.44895i −0.309017 + 0.951057i 1.20499 + 2.74736i −1.33177 − 0.768898i
95.4 0.104528 − 0.994522i −1.41798 − 0.994648i −0.978148 − 0.207912i −1.12093 + 2.51765i −1.13742 + 1.30625i 2.40939 − 1.09308i −0.309017 + 0.951057i 1.02135 + 2.82079i 2.38669 + 1.37795i
95.5 0.104528 − 0.994522i −1.18732 − 1.26106i −0.978148 − 0.207912i −0.157991 + 0.354854i −1.37826 + 1.04900i 0.106147 + 2.64362i −0.309017 + 0.951057i −0.180521 + 2.99456i 0.336396 + 0.194218i
95.6 0.104528 − 0.994522i −0.612442 + 1.62016i −0.978148 − 0.207912i −1.31655 + 2.95701i 1.54727 + 0.778440i −1.86309 − 1.87854i −0.309017 + 0.951057i −2.24983 − 1.98451i 2.80319 + 1.61843i
95.7 0.104528 − 0.994522i −0.611589 + 1.62048i −0.978148 − 0.207912i 0.684905 − 1.53832i 1.54768 + 0.777625i 1.54934 − 2.14465i −0.309017 + 0.951057i −2.25192 − 1.98214i −1.45830 − 0.841952i
95.8 0.104528 − 0.994522i 0.0618530 − 1.73095i −0.978148 − 0.207912i 1.30191 − 2.92414i −1.71500 − 0.242447i −2.04556 + 1.67800i −0.309017 + 0.951057i −2.99235 − 0.214128i −2.77204 − 1.60044i
95.9 0.104528 − 0.994522i 0.333518 + 1.69964i −0.978148 − 0.207912i 0.0756668 − 0.169950i 1.72519 − 0.154030i 1.63703 + 2.07849i −0.309017 + 0.951057i −2.77753 + 1.13372i −0.161110 − 0.0930169i
95.10 0.104528 − 0.994522i 0.398305 − 1.68563i −0.978148 − 0.207912i −1.53261 + 3.44230i −1.63476 − 0.572320i −2.01818 + 1.71083i −0.309017 + 0.951057i −2.68271 − 1.34279i 3.26324 + 1.88403i
95.11 0.104528 − 0.994522i 0.812842 − 1.52947i −0.978148 − 0.207912i 0.565716 − 1.27062i −1.43613 − 0.968262i 0.339893 − 2.62383i −0.309017 + 0.951057i −1.67858 − 2.48644i −1.20453 − 0.695433i
95.12 0.104528 − 0.994522i 0.891795 + 1.48482i −0.978148 − 0.207912i −0.333330 + 0.748672i 1.56991 − 0.731703i −2.01015 + 1.72026i −0.309017 + 0.951057i −1.40940 + 2.64832i 0.709728 + 0.409762i
95.13 0.104528 − 0.994522i 1.28109 − 1.16568i −0.978148 − 0.207912i −0.572012 + 1.28476i −1.02538 − 1.39592i 2.51605 + 0.818231i −0.309017 + 0.951057i 0.282395 − 2.98668i 1.21793 + 0.703172i
95.14 0.104528 − 0.994522i 1.39020 + 1.03313i −0.978148 − 0.207912i −1.35789 + 3.04988i 1.17278 − 1.27459i 1.07443 − 2.41777i −0.309017 + 0.951057i 0.865302 + 2.87250i 2.89123 + 1.66925i
95.15 0.104528 − 0.994522i 1.67644 + 0.435353i −0.978148 − 0.207912i 0.789465 − 1.77317i 0.608205 − 1.62175i −2.28945 + 1.32606i −0.309017 + 0.951057i 2.62093 + 1.45969i −1.68093 − 0.970486i
95.16 0.104528 − 0.994522i 1.70747 + 0.290781i −0.978148 − 0.207912i 1.75629 − 3.94468i 0.467667 − 1.66772i 2.64495 + 0.0650804i −0.309017 + 0.951057i 2.83089 + 0.992999i −3.73949 − 2.15900i
107.1 0.104528 + 0.994522i −1.64414 − 0.544800i −0.978148 + 0.207912i 1.25353 + 2.81547i 0.369957 − 1.69208i 1.28350 − 2.31358i −0.309017 − 0.951057i 2.40638 + 1.79146i −2.66902 + 1.54096i
107.2 0.104528 + 0.994522i −1.63004 − 0.585637i −0.978148 + 0.207912i 0.112013 + 0.251585i 0.412043 − 1.68233i −2.57741 + 0.597448i −0.309017 − 0.951057i 2.31406 + 1.90922i −0.238498 + 0.137697i
107.3 0.104528 + 0.994522i −1.45000 + 0.947367i −0.978148 + 0.207912i 0.625478 + 1.40485i −1.09374 − 1.34303i −1.00131 + 2.44895i −0.309017 − 0.951057i 1.20499 − 2.74736i −1.33177 + 0.768898i
107.4 0.104528 + 0.994522i −1.41798 + 0.994648i −0.978148 + 0.207912i −1.12093 − 2.51765i −1.13742 − 1.30625i 2.40939 + 1.09308i −0.309017 − 0.951057i 1.02135 − 2.82079i 2.38669 − 1.37795i
See next 80 embeddings (of 128 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 95.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
33.f even 10 1 inner
231.be even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 462.2.bc.a ✓ 128
3.b odd 2 1 462.2.bc.b yes 128
7.c even 3 1 inner 462.2.bc.a ✓ 128
11.d odd 10 1 462.2.bc.b yes 128
21.h odd 6 1 462.2.bc.b yes 128
33.f even 10 1 inner 462.2.bc.a ✓ 128
77.o odd 30 1 462.2.bc.b yes 128
231.be even 30 1 inner 462.2.bc.a ✓ 128
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.bc.a ✓ 128 1.a even 1 1 trivial
462.2.bc.a ✓ 128 7.c even 3 1 inner
462.2.bc.a ✓ 128 33.f even 10 1 inner
462.2.bc.a ✓ 128 231.be even 30 1 inner
462.2.bc.b yes 128 3.b odd 2 1
462.2.bc.b yes 128 11.d odd 10 1
462.2.bc.b yes 128 21.h odd 6 1
462.2.bc.b yes 128 77.o odd 30 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{128} + 46 T_{5}^{126} + 50 T_{5}^{125} + 762 T_{5}^{124} + 2300 T_{5}^{123} + \cdots + 21\!\cdots\!81 \) acting on \(S_{2}^{\mathrm{new}}(462, [\chi])\). Copy content Toggle raw display