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Properties

Label 495.2.bg.b
Level $495$
Weight $2$
Character orbit 495.bg
Analytic conductor $3.953$
Analytic rank $0$
Dimension $192$
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] = [495,2,Mod(16,495)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(495, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([20, 0, 12]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("495.16");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 495.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: \(3.95259490005\)
Analytic rank: \(0\)
Dimension: \(192\)
Relative dimension: \(24\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 192 q + 2 q^{2} + 8 q^{3} + 24 q^{4} + 24 q^{5} - 15 q^{6} + 8 q^{7} - 12 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 192 q + 2 q^{2} + 8 q^{3} + 24 q^{4} + 24 q^{5} - 15 q^{6} + 8 q^{7} - 12 q^{8} + 12 q^{9} + 16 q^{10} - 2 q^{11} - 32 q^{12} + 6 q^{13} - 7 q^{15} + 16 q^{16} - 22 q^{17} + 2 q^{18} - 50 q^{19} + 24 q^{20} - 24 q^{21} + 25 q^{22} - 6 q^{23} + 47 q^{24} + 24 q^{25} - 8 q^{26} + 5 q^{27} - 40 q^{28} + 13 q^{29} - 14 q^{30} + 12 q^{31} - 56 q^{32} - 35 q^{33} - 6 q^{34} - 16 q^{35} - 97 q^{36} - 12 q^{38} + 69 q^{39} + 6 q^{40} - 32 q^{41} - 15 q^{42} - 128 q^{43} + 14 q^{44} + 8 q^{45} + 8 q^{46} + 6 q^{47} + 60 q^{48} + 12 q^{49} + 2 q^{50} + 39 q^{51} + 36 q^{52} - 24 q^{53} - 6 q^{55} - 94 q^{56} + 12 q^{57} - 22 q^{58} + 5 q^{59} - 22 q^{60} + 16 q^{61} - 120 q^{62} - 47 q^{63} - 124 q^{64} - 34 q^{65} - 73 q^{66} + 18 q^{67} + 85 q^{68} - 30 q^{69} - 32 q^{71} - 67 q^{72} + 36 q^{73} - 2 q^{74} - q^{75} - 182 q^{76} - 8 q^{77} - 70 q^{78} + 14 q^{79} - 72 q^{80} + 76 q^{81} + 50 q^{82} + 3 q^{83} - 30 q^{84} + 11 q^{85} + 47 q^{86} + 154 q^{87} + 53 q^{88} - 28 q^{89} - 26 q^{90} + 38 q^{91} + 26 q^{92} + 10 q^{93} + 20 q^{94} + 25 q^{95} - 115 q^{96} + 27 q^{97} + 272 q^{98} - 22 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −2.67800 − 0.569227i 1.25831 − 1.19023i 5.02059 + 2.23531i −0.978148 + 0.207912i −4.04727 + 2.47118i −0.454966 − 4.32871i −7.74285 − 5.62551i 0.166690 − 2.99537i 2.73783
16.2 −2.55580 − 0.543252i 0.698889 + 1.58479i 4.40989 + 1.96341i −0.978148 + 0.207912i −0.925280 − 4.43007i −0.0480355 − 0.457027i −5.97642 − 4.34212i −2.02311 + 2.21518i 2.61290
16.3 −2.46381 − 0.523699i −1.56611 + 0.739801i 3.96901 + 1.76712i −0.978148 + 0.207912i 4.24603 − 1.00256i 0.137120 + 1.30461i −4.77786 − 3.47132i 1.90539 − 2.31722i 2.51885
16.4 −2.13134 − 0.453030i 0.184346 − 1.72221i 2.51028 + 1.11765i −0.978148 + 0.207912i −1.17312 + 3.58710i 0.479789 + 4.56489i −1.31830 − 0.957804i −2.93203 − 0.634967i 2.17895
16.5 −1.96704 − 0.418108i −1.04251 − 1.38318i 1.86736 + 0.831402i −0.978148 + 0.207912i 1.47235 + 3.15665i −0.540152 − 5.13920i −0.0717126 − 0.0521023i −0.826346 + 2.88395i 2.01099
16.6 −1.69167 − 0.359577i −0.636978 + 1.61067i 0.905378 + 0.403100i −0.978148 + 0.207912i 1.65672 − 2.49569i −0.144981 − 1.37940i 1.41168 + 1.02564i −2.18852 − 2.05192i 1.72947
16.7 −1.54155 − 0.327666i 1.71170 + 0.264713i 0.441913 + 0.196753i −0.978148 + 0.207912i −2.55193 − 0.968935i 0.0131496 + 0.125110i 1.93324 + 1.40458i 2.85985 + 0.906221i 1.57599
16.8 −1.33064 − 0.282836i 0.815919 + 1.52783i −0.136487 − 0.0607679i −0.978148 + 0.207912i −0.653566 − 2.26377i 0.449101 + 4.27291i 2.36555 + 1.71867i −1.66855 + 2.49318i 1.36037
16.9 −0.953180 − 0.202605i −1.60122 − 0.660382i −0.959588 − 0.427236i −0.978148 + 0.207912i 1.39245 + 0.953877i 0.137460 + 1.30785i 2.40483 + 1.74721i 2.12779 + 2.11483i 0.974475
16.10 −0.843761 − 0.179347i 1.56156 − 0.749351i −1.14732 − 0.510822i −0.978148 + 0.207912i −1.45198 + 0.352212i −0.0800491 − 0.761616i 2.27219 + 1.65084i 1.87695 − 2.34032i 0.862611
16.11 −0.427951 − 0.0909639i 0.292512 − 1.70717i −1.65222 − 0.735617i −0.978148 + 0.207912i −0.280472 + 0.703978i −0.00894732 − 0.0851280i 1.34807 + 0.979427i −2.82887 − 0.998737i 0.437512
16.12 −0.0433194 − 0.00920783i −0.705673 + 1.58178i −1.82530 − 0.812676i −0.978148 + 0.207912i 0.0451341 − 0.0620241i −0.119308 − 1.13514i 0.143246 + 0.104074i −2.00405 − 2.23244i 0.0442872
16.13 0.0180674 + 0.00384034i 1.02531 + 1.39597i −1.82678 − 0.813335i −0.978148 + 0.207912i 0.0131636 + 0.0291591i −0.387663 − 3.68837i −0.0597683 − 0.0434242i −0.897491 + 2.86261i −0.0184710
16.14 0.289537 + 0.0615429i −0.792955 − 1.53988i −1.74705 − 0.777835i −0.978148 + 0.207912i −0.134821 − 0.494652i 0.340789 + 3.24240i −0.936911 − 0.680705i −1.74245 + 2.44211i −0.296005
16.15 0.325445 + 0.0691755i −1.71152 − 0.265921i −1.72596 − 0.768448i −0.978148 + 0.207912i −0.538609 − 0.204938i −0.454229 − 4.32170i −1.04689 − 0.760612i 2.85857 + 0.910256i −0.332716
16.16 0.795868 + 0.169167i 1.41192 + 1.00323i −1.22230 − 0.544204i −0.978148 + 0.207912i 0.953991 + 1.03729i 0.349628 + 3.32649i −2.19724 − 1.59639i 0.987053 + 2.83297i −0.813648
16.17 1.21964 + 0.259242i 1.66315 − 0.483681i −0.406778 − 0.181109i −0.978148 + 0.207912i 2.15383 − 0.158758i −0.445871 − 4.24218i −2.46668 − 1.79214i 2.53211 − 1.60886i −1.24689
16.18 1.34347 + 0.285564i −1.70842 + 0.285118i −0.103721 − 0.0461794i −0.978148 + 0.207912i −2.37664 − 0.104816i 0.105174 + 1.00066i −2.34851 − 1.70629i 2.83742 − 0.974203i −1.37349
16.19 1.56438 + 0.332520i −1.31788 + 1.12392i 0.509635 + 0.226904i −0.978148 + 0.207912i −2.43539 + 1.32003i 0.203744 + 1.93850i −1.86596 − 1.35570i 0.473593 − 2.96238i −1.59933
16.20 1.57949 + 0.335730i −0.584364 − 1.63050i 0.554971 + 0.247089i −0.978148 + 0.207912i −0.375587 − 2.77154i −0.223635 − 2.12774i −1.81914 − 1.32169i −2.31704 + 1.90561i −1.61477
See next 80 embeddings (of 192 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
11.c even 5 1 inner
99.m even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 495.2.bg.b ✓ 192
9.c even 3 1 inner 495.2.bg.b ✓ 192
11.c even 5 1 inner 495.2.bg.b ✓ 192
99.m even 15 1 inner 495.2.bg.b ✓ 192
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
495.2.bg.b ✓ 192 1.a even 1 1 trivial
495.2.bg.b ✓ 192 9.c even 3 1 inner
495.2.bg.b ✓ 192 11.c even 5 1 inner
495.2.bg.b ✓ 192 99.m even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{192} - 2 T_{2}^{191} - 34 T_{2}^{190} + 80 T_{2}^{189} + 470 T_{2}^{188} - 1308 T_{2}^{187} + \cdots + 113164960000 \) acting on \(S_{2}^{\mathrm{new}}(495, [\chi])\). Copy content Toggle raw display