Properties

Label 495.2.bc.d
Level $495$
Weight $2$
Character orbit 495.bc
Analytic conductor $3.953$
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] = [495,2,Mod(23,495)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(495, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([10, 9, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("495.23");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 495.bc (of order \(12\), degree \(4\), 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: \(3.95259490005\)
Analytic rank: \(0\)
Dimension: \(116\)
Relative dimension: \(29\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 116 q - 2 q^{2} + 6 q^{4} - 2 q^{5} - 2 q^{6} - 10 q^{7} + 2 q^{8} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 116 q - 2 q^{2} + 6 q^{4} - 2 q^{5} - 2 q^{6} - 10 q^{7} + 2 q^{8} + 16 q^{9} + 6 q^{10} - 8 q^{12} + 12 q^{13} - 10 q^{14} + 20 q^{15} + 62 q^{16} - 8 q^{17} - 54 q^{18} + 6 q^{20} - 10 q^{21} + 2 q^{22} - 14 q^{23} - 62 q^{24} - 12 q^{25} + 30 q^{27} + 18 q^{28} - 2 q^{29} - 18 q^{30} - 2 q^{31} - 48 q^{32} + 4 q^{33} - 24 q^{34} + 2 q^{35} + 24 q^{36} - 14 q^{37} - 6 q^{38} + 4 q^{39} + 98 q^{40} + 6 q^{41} - 44 q^{42} + 26 q^{43} - 120 q^{44} - 18 q^{45} - 44 q^{46} - 2 q^{47} - 20 q^{48} - 18 q^{49} - 20 q^{50} - 8 q^{51} + 102 q^{52} - 44 q^{53} + 28 q^{54} + 2 q^{55} + 42 q^{56} - 48 q^{57} - 16 q^{58} + 22 q^{59} - 8 q^{60} - 10 q^{61} - 16 q^{62} - 26 q^{63} - 108 q^{65} + 6 q^{66} - 36 q^{67} - 72 q^{68} - 76 q^{69} - 134 q^{70} + 30 q^{72} + 12 q^{73} - 8 q^{74} + 20 q^{75} - 6 q^{76} - 10 q^{77} + 210 q^{78} - 6 q^{79} + 4 q^{80} + 44 q^{81} - 50 q^{82} + 24 q^{83} + 222 q^{84} + 54 q^{85} + 90 q^{86} - 32 q^{87} - 4 q^{88} + 8 q^{89} - 74 q^{90} + 72 q^{91} + 18 q^{92} - 98 q^{93} + 42 q^{94} + 54 q^{95} + 68 q^{96} + 18 q^{97} + 16 q^{98} - 4 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1 −0.709061 + 2.64625i −1.30748 1.13600i −4.76783 2.75271i −1.96868 + 1.06033i 3.93323 2.65442i −4.33523 1.16162i 6.79066 6.79066i 0.419003 + 2.97060i −1.41000 5.96146i
23.2 −0.698485 + 2.60678i 1.03342 + 1.38998i −4.57538 2.64160i 2.20386 + 0.378161i −4.34521 + 1.72301i 3.64058 + 0.975490i 6.26532 6.26532i −0.864101 + 2.87286i −2.52515 + 5.48084i
23.3 −0.650441 + 2.42748i 1.72765 0.123451i −3.73753 2.15786i −0.602018 2.15350i −0.824056 + 4.27412i −0.582153 0.155987i 4.11513 4.11513i 2.96952 0.426560i 5.61916 0.0606604i
23.4 −0.545055 + 2.03417i −0.900940 + 1.47929i −2.10872 1.21747i −1.22843 + 1.86841i −2.51807 2.63896i 0.996328 + 0.266965i 0.647682 0.647682i −1.37661 2.66551i −3.13111 3.51722i
23.5 −0.535684 + 1.99920i −0.364559 1.69325i −1.97779 1.14188i −1.74294 1.40077i 3.58043 + 0.178220i 3.07868 + 0.824929i 0.415285 0.415285i −2.73419 + 1.23458i 3.73408 2.73412i
23.6 −0.505862 + 1.88790i −0.470036 1.66705i −1.57622 0.910033i 1.73643 + 1.40884i 3.38500 0.0440830i −0.132167 0.0354142i −0.248675 + 0.248675i −2.55813 + 1.56715i −3.53813 + 2.56553i
23.7 −0.463461 + 1.72966i 0.570504 + 1.63540i −1.04488 0.603259i −0.410017 2.19816i −3.09309 + 0.228835i −3.55584 0.952785i −1.00470 + 1.00470i −2.34905 + 1.86600i 3.99209 + 0.309569i
23.8 −0.414074 + 1.54534i 1.59144 + 0.683613i −0.484580 0.279772i −0.948906 + 2.02474i −1.71539 + 2.17625i 2.44300 + 0.654599i −1.62954 + 1.62954i 2.06535 + 2.17586i −2.73600 2.30478i
23.9 −0.386428 + 1.44217i −1.14470 + 1.29987i −0.198472 0.114588i 0.866165 2.06149i −1.43229 2.15316i 3.55920 + 0.953684i −1.86953 + 1.86953i −0.379319 2.97592i 2.63831 + 2.04577i
23.10 −0.347430 + 1.29663i −1.56587 0.740309i 0.171519 + 0.0990263i 1.33394 1.79461i 1.50393 1.77314i −3.42972 0.918992i −2.08638 + 2.08638i 1.90388 + 2.31845i 1.86349 + 2.35312i
23.11 −0.190036 + 0.709226i 0.275872 + 1.70994i 1.26516 + 0.730443i 2.05237 + 0.887562i −1.26516 0.129295i 0.703210 + 0.188424i −1.79685 + 1.79685i −2.84779 + 0.943450i −1.01951 + 1.28693i
23.12 −0.154094 + 0.575087i −1.72092 0.196064i 1.42507 + 0.822765i −2.07959 0.821775i 0.377937 0.959465i 2.02666 + 0.543042i −1.53474 + 1.53474i 2.92312 + 0.674820i 0.793044 1.06931i
23.13 −0.106714 + 0.398263i 1.48878 0.885180i 1.58483 + 0.914999i 2.22335 + 0.238187i 0.193661 + 0.687386i −4.49290 1.20387i −1.11663 + 1.11663i 1.43291 2.63567i −0.332124 + 0.860058i
23.14 −0.0652104 + 0.243368i 1.72698 + 0.132396i 1.67708 + 0.968260i 0.657560 2.13720i −0.144838 + 0.411660i 1.29753 + 0.347673i −0.701322 + 0.701322i 2.96494 + 0.457291i 0.477247 + 0.299397i
23.15 −0.0644467 + 0.240518i 1.51359 + 0.842050i 1.67836 + 0.968999i −1.56789 + 1.59428i −0.300074 + 0.309778i −3.86022 1.03434i −0.693369 + 0.693369i 1.58190 + 2.54904i −0.282409 0.479851i
23.16 −0.0360136 + 0.134405i −0.347592 1.69681i 1.71528 + 0.990319i −2.01237 + 0.974876i 0.240578 + 0.0143905i 0.855905 + 0.229339i −0.391659 + 0.391659i −2.75836 + 1.17960i −0.0585552 0.305580i
23.17 −0.00736284 + 0.0274785i 0.724814 1.57310i 1.73135 + 0.999595i 0.674857 + 2.13180i 0.0378897 + 0.0314993i 3.09816 + 0.830150i −0.0804463 + 0.0804463i −1.94929 2.28041i −0.0635475 + 0.00284797i
23.18 0.150659 0.562267i −1.69955 + 0.333984i 1.43860 + 0.830579i 2.15548 0.594897i −0.0682637 + 1.00592i 0.0901346 + 0.0241515i 1.50696 1.50696i 2.77691 1.13524i −0.00974836 1.30158i
23.19 0.241566 0.901536i −1.57517 + 0.720305i 0.977638 + 0.564439i −1.68402 + 1.47107i 0.268873 + 1.59407i −3.99403 1.07020i 2.06497 2.06497i 1.96232 2.26920i 0.919424 + 1.87357i
23.20 0.260420 0.971901i 1.09089 + 1.34535i 0.855277 + 0.493795i −1.65525 1.50337i 1.59163 0.709883i 0.704826 + 0.188858i 2.12561 2.12561i −0.619915 + 2.93525i −1.89219 + 1.21723i
See next 80 embeddings (of 116 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.29
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
45.l even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 495.2.bc.d yes 116
5.c odd 4 1 495.2.bc.c 116
9.d odd 6 1 495.2.bc.c 116
45.l even 12 1 inner 495.2.bc.d yes 116
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
495.2.bc.c 116 5.c odd 4 1
495.2.bc.c 116 9.d odd 6 1
495.2.bc.d yes 116 1.a even 1 1 trivial
495.2.bc.d yes 116 45.l even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{116} + 2 T_{2}^{115} - T_{2}^{114} - 8 T_{2}^{113} - 229 T_{2}^{112} - 412 T_{2}^{111} + 290 T_{2}^{110} + 1618 T_{2}^{109} + 29728 T_{2}^{108} + 50246 T_{2}^{107} - 43046 T_{2}^{106} - 200096 T_{2}^{105} - 2589232 T_{2}^{104} + \cdots + 44302336 \) acting on \(S_{2}^{\mathrm{new}}(495, [\chi])\). Copy content Toggle raw display