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Properties

Label 495.2.bc.c
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.

\(\operatorname{Tr}(f)(q) = \) \( 116 q - 4 q^{2} + 8 q^{3} - 6 q^{4} + 2 q^{5} - 2 q^{6} + 8 q^{7} - 2 q^{8} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 116 q - 4 q^{2} + 8 q^{3} - 6 q^{4} + 2 q^{5} - 2 q^{6} + 8 q^{7} - 2 q^{8} - 16 q^{9} + 6 q^{10} + 6 q^{12} - 12 q^{13} + 10 q^{14} - 18 q^{15} + 62 q^{16} + 8 q^{17} - 12 q^{18} - 18 q^{20} - 10 q^{21} - 4 q^{22} - 40 q^{23} + 62 q^{24} - 6 q^{25} - 58 q^{27} + 18 q^{28} + 2 q^{29} - 70 q^{30} - 2 q^{31} - 66 q^{32} + 24 q^{34} - 2 q^{35} + 24 q^{36} - 14 q^{37} - 6 q^{38} - 4 q^{39} - 100 q^{40} + 6 q^{41} - 30 q^{42} - 22 q^{43} + 120 q^{44} + 94 q^{45} - 44 q^{46} + 32 q^{47} + 108 q^{48} + 18 q^{49} - 22 q^{50} - 8 q^{51} - 126 q^{52} + 44 q^{53} - 28 q^{54} + 2 q^{55} + 42 q^{56} + 20 q^{57} + 2 q^{58} - 22 q^{59} - 68 q^{60} - 10 q^{61} + 16 q^{62} - 8 q^{63} - 60 q^{65} + 6 q^{66} + 36 q^{67} - 48 q^{68} + 76 q^{69} + 154 q^{70} - 228 q^{72} + 12 q^{73} + 8 q^{74} - 72 q^{75} - 6 q^{76} - 8 q^{77} - 110 q^{78} + 6 q^{79} - 4 q^{80} + 44 q^{81} - 50 q^{82} + 30 q^{83} - 222 q^{84} - 126 q^{85} + 90 q^{86} + 112 q^{87} + 2 q^{88} - 8 q^{89} + 72 q^{90} + 72 q^{91} - 132 q^{92} + 112 q^{93} - 42 q^{94} + 78 q^{95} + 68 q^{96} - 72 q^{97} - 16 q^{98} + 4 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1 −0.695692 + 2.59636i −0.566483 + 1.63679i −4.52503 − 2.61253i −1.67397 − 1.48250i −3.85561 − 2.60950i 0.811799 + 0.217521i 6.12975 − 6.12975i −2.35819 − 1.85443i 5.01368 − 3.31486i
23.2 −0.681865 + 2.54475i 0.951142 − 1.44753i −4.27878 − 2.47035i −0.953606 + 2.02253i 3.03504 + 3.40744i 4.20241 + 1.12603i 5.47821 − 5.47821i −1.19066 − 2.75360i −4.49661 − 3.80579i
23.3 −0.635820 + 2.37291i 0.561069 − 1.63866i −3.49439 − 2.01749i 1.80540 − 1.31929i 3.53165 + 2.37326i −2.41257 − 0.646447i 3.53494 − 3.53494i −2.37040 − 1.83880i 1.98264 + 5.12289i
23.4 −0.629065 + 2.34770i −1.69619 − 0.350640i −3.38393 − 1.95371i 0.435925 − 2.19316i 1.89021 − 3.76157i 1.65959 + 0.444685i 3.27818 − 3.27818i 2.75410 + 1.18950i 4.87467 + 2.40306i
23.5 −0.592711 + 2.21203i 0.632659 + 1.61237i −2.80970 − 1.62218i 0.479462 + 2.18406i −3.94159 + 0.443788i −4.33405 − 1.16130i 2.01502 − 2.01502i −2.19949 + 2.04016i −5.11538 − 0.233934i
23.6 −0.463281 + 1.72899i 1.62694 + 0.594200i −1.04272 − 0.602013i 1.97387 − 1.05065i −1.78109 + 2.53767i −0.348917 − 0.0934920i −1.00747 + 1.00747i 2.29385 + 1.93345i 0.902100 + 3.89953i
23.7 −0.411545 + 1.53591i −1.60491 + 0.651360i −0.457586 − 0.264187i −2.22322 − 0.239387i −0.339936 − 2.73305i −2.74190 − 0.734690i −1.65464 + 1.65464i 2.15146 − 2.09075i 1.28263 − 3.31613i
23.8 −0.370900 + 1.38422i 1.66253 − 0.485784i −0.0464383 − 0.0268111i 1.21782 + 1.87535i 0.0557980 + 2.48148i 0.848332 + 0.227310i −1.97230 + 1.97230i 2.52803 − 1.61526i −3.04757 + 0.990155i
23.9 −0.347994 + 1.29873i 0.901715 + 1.47882i 0.166444 + 0.0960964i −1.90139 − 1.17674i −2.23438 + 0.656466i 4.07458 + 1.09178i −2.08420 + 2.08420i −1.37382 + 2.66695i 2.18994 − 2.05990i
23.10 −0.304903 + 1.13792i 0.00993826 − 1.73202i 0.530166 + 0.306091i −0.569442 + 2.16234i 1.96786 + 0.539409i −3.43669 − 0.920859i −2.17598 + 2.17598i −2.99980 − 0.0344266i −2.28694 − 1.30728i
23.11 −0.276848 + 1.03321i −1.25805 + 1.19051i 0.741174 + 0.427917i 2.18073 + 0.494391i −0.881759 − 1.62942i −2.83799 − 0.760438i −2.16004 + 2.16004i 0.165369 − 2.99544i −1.11454 + 2.11628i
23.12 −0.209862 + 0.783216i −1.07336 − 1.35937i 1.16267 + 0.671265i 2.18873 − 0.457674i 1.28994 − 0.555395i 3.89193 + 1.04284i −1.91645 + 1.91645i −0.695781 + 2.91820i −0.100874 + 1.81030i
23.13 −0.180284 + 0.672828i 0.985916 − 1.42407i 1.31186 + 0.757400i −0.276235 − 2.21894i 0.780409 + 0.920088i 1.66403 + 0.445875i −1.73120 + 1.73120i −1.05594 − 2.80802i 1.54277 + 0.214180i
23.14 −0.0890303 + 0.332266i −0.00850249 + 1.73203i 1.62958 + 0.940837i −1.71831 + 1.43087i −0.574737 − 0.157028i −0.368808 − 0.0988218i −0.944160 + 0.944160i −2.99986 − 0.0294531i −0.322448 − 0.698327i
23.15 −0.0366050 + 0.136612i −1.52067 + 0.829200i 1.71473 + 0.989999i 0.0142926 + 2.23602i −0.0576145 − 0.238094i 4.20296 + 1.12618i −0.398027 + 0.398027i 1.62486 − 2.52187i −0.305990 − 0.0798971i
23.16 −0.0293022 + 0.109357i −0.995458 − 1.41741i 1.72095 + 0.993591i −1.42043 − 1.72696i 0.184174 − 0.0673273i −3.03685 − 0.813722i −0.319194 + 0.319194i −1.01812 + 2.82195i 0.230477 − 0.104731i
23.17 0.141596 − 0.528444i 1.68879 − 0.384686i 1.47285 + 0.850349i −2.20790 + 0.353784i 0.0358412 − 0.946902i 2.19488 + 0.588115i 1.43161 − 1.43161i 2.70403 − 1.29931i −0.125675 + 1.21685i
23.18 0.223068 − 0.832501i 0.620970 − 1.61691i 1.08875 + 0.628591i 2.22213 − 0.249258i −1.20756 − 0.877639i −0.328457 − 0.0880097i 1.98504 − 1.98504i −2.22879 − 2.00811i 0.288179 − 1.90553i
23.19 0.233159 − 0.870160i 1.43635 + 0.967942i 1.02923 + 0.594229i 1.56117 + 1.60086i 1.17716 − 1.02417i −1.44124 − 0.386179i 2.03105 − 2.03105i 1.12618 + 2.78060i 1.75700 − 0.985214i
23.20 0.310475 − 1.15871i −0.103237 + 1.72897i 0.485839 + 0.280500i 1.58680 − 1.57546i 1.97132 + 0.656424i 4.78549 + 1.28227i 2.17233 − 2.17233i −2.97868 − 0.356986i −1.33283 − 2.32778i
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.c ✓ 116
5.c odd 4 1 495.2.bc.d yes 116
9.d odd 6 1 495.2.bc.d yes 116
45.l even 12 1 inner 495.2.bc.c ✓ 116
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
495.2.bc.c ✓ 116 1.a even 1 1 trivial
495.2.bc.c ✓ 116 45.l even 12 1 inner
495.2.bc.d yes 116 5.c odd 4 1
495.2.bc.d yes 116 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{116} + 4 T_{2}^{115} + 11 T_{2}^{114} + 26 T_{2}^{113} - 169 T_{2}^{112} - 776 T_{2}^{111} + \cdots + 44302336 \) acting on \(S_{2}^{\mathrm{new}}(495, [\chi])\). Copy content Toggle raw display