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Properties

Label 805.2.s.d
Level $805$
Weight $2$
Character orbit 805.s
Analytic conductor $6.428$
Analytic rank $0$
Dimension $164$
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] = [805,2,Mod(254,805)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(805, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("805.254");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 805 = 5 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 805.s (of order \(6\), degree \(2\), 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: \(6.42795736271\)
Analytic rank: \(0\)
Dimension: \(164\)
Relative dimension: \(82\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 164 q + 90 q^{4} - 4 q^{5} - 8 q^{6} + 86 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 164 q + 90 q^{4} - 4 q^{5} - 8 q^{6} + 86 q^{9} - 2 q^{14} - 28 q^{15} - 82 q^{16} - 12 q^{19} - 48 q^{20} + 4 q^{21} - 12 q^{24} + 6 q^{25} - 8 q^{26} - 4 q^{29} + 6 q^{31} - 4 q^{34} + 20 q^{35} + 132 q^{36} + 56 q^{39} - 6 q^{40} + 16 q^{44} + 46 q^{45} - 6 q^{46} - 62 q^{49} + 4 q^{50} - 20 q^{51} - 68 q^{54} + 20 q^{55} - 10 q^{56} - 74 q^{59} - 44 q^{60} + 4 q^{61} - 60 q^{64} + 18 q^{65} - 64 q^{66} + 24 q^{69} + 110 q^{70} + 20 q^{71} + 30 q^{74} - 98 q^{75} + 72 q^{76} + 28 q^{79} + 12 q^{80} - 186 q^{81} - 300 q^{84} - 16 q^{85} + 110 q^{86} + 20 q^{89} - 136 q^{90} - 20 q^{91} - 4 q^{94} + 28 q^{95} + 64 q^{96} + 40 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
254.1 −2.34962 − 1.35655i 1.52884 − 0.882678i 2.68047 + 4.64271i −2.22008 − 0.266883i −4.78960 −2.09400 − 1.61714i − 9.11859i 0.0582402 − 0.100875i 4.85431 + 3.63873i
254.2 −2.34606 − 1.35450i −0.862797 + 0.498136i 2.66932 + 4.62339i 0.589205 − 2.15704i 2.69889 1.92466 + 1.81540i − 9.04434i −1.00372 + 1.73850i −4.30401 + 4.26247i
254.3 −2.32323 − 1.34132i −0.934200 + 0.539361i 2.59827 + 4.50034i 1.15754 + 1.91314i 2.89382 1.32899 − 2.28775i − 8.57517i −0.918180 + 1.59033i −0.123093 − 5.99730i
254.4 −2.29358 − 1.32420i 2.64117 − 1.52488i 2.50700 + 4.34225i 0.222932 + 2.22493i −8.07697 2.54692 + 0.716394i − 7.98226i 3.15052 − 5.45686i 2.43493 − 5.39825i
254.5 −2.25827 − 1.30381i −2.08101 + 1.20147i 2.39985 + 4.15667i −1.80584 + 1.31869i 6.26597 0.234775 + 2.63531i − 7.30058i 1.38706 − 2.40246i 5.79740 − 0.623473i
254.6 −2.12981 − 1.22965i 1.01774 − 0.587591i 2.02406 + 3.50577i −2.18499 − 0.475202i −2.89011 1.81846 + 1.92177i − 5.03690i −0.809474 + 1.40205i 4.06928 + 3.69885i
254.7 −2.10088 − 1.21295i 2.51781 − 1.45366i 1.94247 + 3.36446i −0.226321 − 2.22459i −7.05285 −2.14018 + 1.55552i − 4.57268i 2.72626 − 4.72202i −2.22283 + 4.94811i
254.8 −2.09080 − 1.20713i 0.331556 − 0.191424i 1.91431 + 3.31568i −0.502897 + 2.17878i −0.924291 −2.33169 + 1.25028i − 4.41473i −1.42671 + 2.47114i 3.68152 − 3.94835i
254.9 −2.08758 − 1.20527i −1.24436 + 0.718433i 1.90533 + 3.30013i 0.167575 − 2.22978i 3.46361 −1.65704 − 2.06258i − 4.36466i −0.467709 + 0.810095i −3.03730 + 4.45287i
254.10 −2.07349 − 1.19713i 0.831152 − 0.479866i 1.86623 + 3.23240i 2.23527 + 0.0598395i −2.29784 −1.41892 − 2.23308i − 4.14795i −1.03946 + 1.80039i −4.56316 − 2.79998i
254.11 −1.98996 − 1.14891i −2.89968 + 1.67413i 1.63997 + 2.84051i 2.11939 − 0.712882i 7.69369 2.58900 − 0.545071i − 2.94107i 4.10544 − 7.11083i −5.03654 − 1.01637i
254.12 −1.96708 − 1.13569i −0.885722 + 0.511372i 1.57960 + 2.73595i 2.00031 − 0.999384i 2.32305 −1.97103 + 1.76495i − 2.63301i −0.976997 + 1.69221i −5.06976 − 0.305869i
254.13 −1.87711 − 1.08375i −2.17720 + 1.25701i 1.34904 + 2.33660i −2.23540 + 0.0546493i 5.44914 1.82906 − 1.91169i − 1.51307i 1.66014 − 2.87545i 4.25533 + 2.32004i
254.14 −1.78815 − 1.03239i 0.0714794 − 0.0412687i 1.13166 + 1.96009i −1.54359 − 1.61781i −0.170421 1.65426 − 2.06481i − 0.543696i −1.49659 + 2.59218i 1.08996 + 4.48649i
254.15 −1.78428 − 1.03015i 2.79336 − 1.61275i 1.12243 + 1.94411i 2.14854 + 0.619490i −6.64552 −2.38947 − 1.13597i − 0.504500i 3.70192 − 6.41191i −3.19543 − 3.31867i
254.16 −1.76316 − 1.01796i 0.184549 − 0.106550i 1.07250 + 1.85762i 1.39419 + 1.74821i −0.433854 2.63704 + 0.214558i − 0.295200i −1.47729 + 2.55875i −0.678571 − 4.50161i
254.17 −1.71210 − 0.988481i −2.27360 + 1.31266i 0.954190 + 1.65271i −0.396060 + 2.20071i 5.19017 −1.31579 − 2.29536i 0.181129i 1.94617 − 3.37086i 2.85346 − 3.37634i
254.18 −1.65169 − 0.953606i 1.86296 − 1.07558i 0.818728 + 1.41808i −1.63575 + 1.52457i −4.10272 0.647840 − 2.56521i 0.691447i 0.813750 − 1.40946i 4.15560 − 0.958258i
254.19 −1.58596 − 0.915656i 1.52212 − 0.878797i 0.676851 + 1.17234i −0.324174 − 2.21244i −3.21870 −0.636677 + 2.56800i 1.18357i 0.0445677 − 0.0771935i −1.51171 + 3.80569i
254.20 −1.54737 − 0.893372i −1.55577 + 0.898224i 0.596227 + 1.03270i 1.91158 + 1.16011i 3.20979 0.303683 + 2.62826i 1.44288i 0.113612 − 0.196782i −1.92151 − 3.50287i
See next 80 embeddings (of 164 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 254.82
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 805.2.s.d ✓ 164
5.b even 2 1 inner 805.2.s.d ✓ 164
7.c even 3 1 inner 805.2.s.d ✓ 164
35.j even 6 1 inner 805.2.s.d ✓ 164
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
805.2.s.d ✓ 164 1.a even 1 1 trivial
805.2.s.d ✓ 164 5.b even 2 1 inner
805.2.s.d ✓ 164 7.c even 3 1 inner
805.2.s.d ✓ 164 35.j even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{164} - 127 T_{2}^{162} + 8384 T_{2}^{160} - 379517 T_{2}^{158} + 13160679 T_{2}^{156} + \cdots + 11\!\cdots\!00 \) acting on \(S_{2}^{\mathrm{new}}(805, [\chi])\). Copy content Toggle raw display