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Properties

Label 495.2.bg.a
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.

\(\operatorname{Tr}(f)(q) = \) \( 192 q + 2 q^{2} - 4 q^{3} + 24 q^{4} - 24 q^{5} + 9 q^{6} - 8 q^{7} - 12 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 192 q + 2 q^{2} - 4 q^{3} + 24 q^{4} - 24 q^{5} + 9 q^{6} - 8 q^{7} - 12 q^{8} + 12 q^{9} - 16 q^{10} - 2 q^{11} + 8 q^{12} - 6 q^{13} + q^{15} + 32 q^{16} + 14 q^{17} + 26 q^{18} + 14 q^{19} - 24 q^{20} + 8 q^{21} - 31 q^{22} - 6 q^{23} - 49 q^{24} + 24 q^{25} - 8 q^{26} - 55 q^{27} + 40 q^{28} + 13 q^{29} - 18 q^{30} - 12 q^{31} - 56 q^{32} - 15 q^{33} - 6 q^{34} - 16 q^{35} + 7 q^{36} - 12 q^{38} - 35 q^{39} - 6 q^{40} - 32 q^{41} - 55 q^{42} + 116 q^{43} - 34 q^{44} - 8 q^{45} - 8 q^{46} + 18 q^{47} + 120 q^{48} + 36 q^{49} + 2 q^{50} - 97 q^{51} - 36 q^{52} - 24 q^{53} - 72 q^{54} + 6 q^{55} - 94 q^{56} + 84 q^{57} + 22 q^{58} - 19 q^{59} - 4 q^{60} - 16 q^{61} - 120 q^{62} + q^{63} + 28 q^{64} - 34 q^{65} - 69 q^{66} - 30 q^{67} + 13 q^{68} - 18 q^{69} - 32 q^{71} + 103 q^{72} + 36 q^{73} - 98 q^{74} - 5 q^{75} + 146 q^{76} - 2 q^{77} + 46 q^{78} - 26 q^{79} + 24 q^{80} - 76 q^{81} - 86 q^{82} + 51 q^{83} - 6 q^{84} - 23 q^{85} + 47 q^{86} + 106 q^{87} - 71 q^{88} + 20 q^{89} - 6 q^{90} - 74 q^{91} + 26 q^{92} + 96 q^{93} - 20 q^{94} + 7 q^{95} + 25 q^{96} + 27 q^{97} + 104 q^{98} + 86 q^{99}+O(q^{100}) \) 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 −2.72939 − 0.580149i −0.598473 − 1.62537i 5.28589 + 2.35343i 0.978148 − 0.207912i 0.690508 + 4.78347i 0.168950 + 1.60746i −8.54700 − 6.20976i −2.28366 + 1.94548i −2.79036
16.2 −2.49631 − 0.530607i 1.22253 + 1.22695i 4.12293 + 1.83565i 0.978148 − 0.207912i −2.40080 − 3.71154i 0.297322 + 2.82883i −5.18876 − 3.76985i −0.0108176 + 2.99998i −2.55208
16.3 −2.37432 − 0.504677i −1.72558 + 0.149556i 3.55560 + 1.58306i 0.978148 − 0.207912i 4.17256 + 0.515768i −0.338101 − 3.21682i −3.71565 − 2.69958i 2.95527 − 0.516143i −2.42736
16.4 −1.96670 − 0.418034i 1.63157 + 0.581346i 1.86606 + 0.830821i 0.978148 − 0.207912i −2.96579 − 1.82539i −0.393108 − 3.74017i −0.0693785 − 0.0504065i 2.32407 + 1.89702i −2.01063
16.5 −1.92235 − 0.408608i −1.11495 + 1.32548i 1.70138 + 0.757503i 0.978148 − 0.207912i 2.68492 − 2.09245i 0.428654 + 4.07837i 0.218790 + 0.158960i −0.513777 − 2.95568i −1.96530
16.6 −1.81071 − 0.384878i 1.12161 − 1.31985i 1.30344 + 0.580330i 0.978148 − 0.207912i −2.53888 + 1.95818i −0.0219404 − 0.208749i 0.858443 + 0.623695i −0.483994 − 2.96070i −1.85116
16.7 −1.65293 − 0.351340i −0.115849 + 1.72817i 0.781630 + 0.348004i 0.978148 − 0.207912i 0.798665 − 2.81584i −0.0667737 − 0.635309i 1.56453 + 1.13670i −2.97316 − 0.400413i −1.68985
16.8 −1.47567 − 0.313664i −0.636706 − 1.61078i 0.252131 + 0.112256i 0.978148 − 0.207912i 0.434327 + 2.57669i −0.0974165 − 0.926856i 2.10418 + 1.52877i −2.18921 + 2.05119i −1.50864
16.9 −0.888058 − 0.188763i −1.06408 − 1.36665i −1.07408 − 0.478209i 0.978148 − 0.207912i 0.686993 + 1.41452i 0.131391 + 1.25010i 2.33258 + 1.69472i −0.735464 + 2.90845i −0.907898
16.10 −0.876170 − 0.186236i 1.42059 − 0.990916i −1.09410 − 0.487125i 0.978148 − 0.207912i −1.42923 + 0.603646i 0.539248 + 5.13060i 2.31724 + 1.68358i 1.03617 − 2.81538i −0.895744
16.11 −0.383704 − 0.0815589i 0.385722 + 1.68856i −1.68651 − 0.750884i 0.978148 − 0.207912i −0.0102866 − 0.679365i −0.150167 − 1.42874i 1.22060 + 0.886816i −2.70244 + 1.30263i −0.392277
16.12 −0.342647 − 0.0728318i 1.47396 + 0.909638i −1.71499 − 0.763562i 0.978148 − 0.207912i −0.438797 − 0.419036i 0.0718226 + 0.683347i 1.09882 + 0.798342i 1.34512 + 2.68154i −0.350302
16.13 −0.215352 − 0.0457746i −1.73109 + 0.0576392i −1.78281 − 0.793758i 0.978148 − 0.207912i 0.375433 + 0.0668272i 0.153346 + 1.45899i 0.703831 + 0.511363i 2.99336 − 0.199558i −0.220164
16.14 0.630325 + 0.133980i −0.0704490 − 1.73062i −1.44773 − 0.644572i 0.978148 − 0.207912i 0.187462 − 1.10029i −0.403995 − 3.84375i −1.86885 − 1.35780i −2.99007 + 0.243841i 0.644407
16.15 0.718693 + 0.152763i 0.757488 − 1.55763i −1.33391 − 0.593894i 0.978148 − 0.207912i 0.782350 − 1.00374i 0.170095 + 1.61835i −2.05679 − 1.49435i −1.85242 − 2.35977i 0.734749
16.16 0.913942 + 0.194264i −1.72020 − 0.202288i −1.02954 − 0.458380i 0.978148 − 0.207912i −1.53286 − 0.519053i 0.383998 + 3.65350i −2.36372 − 1.71734i 2.91816 + 0.695952i 0.934360
16.17 0.942650 + 0.200366i 1.70535 − 0.302971i −0.978649 − 0.435723i 0.978148 − 0.207912i 1.66825 + 0.0560991i −0.163790 − 1.55835i −2.39453 − 1.73973i 2.81642 − 1.03334i 0.963709
16.18 1.14509 + 0.243397i −1.05245 + 1.37562i −0.575091 − 0.256047i 0.978148 − 0.207912i −1.53998 + 1.31906i −0.380295 − 3.61827i −2.49041 − 1.80939i −0.784683 − 2.89556i 1.17068
16.19 1.61631 + 0.343558i 0.573135 + 1.63448i 0.667348 + 0.297123i 0.978148 − 0.207912i 0.364828 + 2.83873i 0.469349 + 4.46556i −1.69711 − 1.23302i −2.34303 + 1.87355i 1.65242
16.20 1.96168 + 0.416967i −1.58993 − 0.687120i 1.84722 + 0.822436i 0.978148 − 0.207912i −2.83241 − 2.01085i −0.294077 − 2.79795i 0.0357529 + 0.0259760i 2.05573 + 2.18494i 2.00550
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.a ✓ 192
9.c even 3 1 inner 495.2.bg.a ✓ 192
11.c even 5 1 inner 495.2.bg.a ✓ 192
99.m even 15 1 inner 495.2.bg.a ✓ 192
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
495.2.bg.a ✓ 192 1.a even 1 1 trivial
495.2.bg.a ✓ 192 9.c even 3 1 inner
495.2.bg.a ✓ 192 11.c even 5 1 inner
495.2.bg.a ✓ 192 99.m even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{192} - 2 T_{2}^{191} - 34 T_{2}^{190} + 80 T_{2}^{189} + 466 T_{2}^{188} - 1300 T_{2}^{187} + \cdots + 34297420960000 \) acting on \(S_{2}^{\mathrm{new}}(495, [\chi])\). Copy content Toggle raw display