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Properties

Label 805.2.i.e
Level $805$
Weight $2$
Character orbit 805.i
Analytic conductor $6.428$
Analytic rank $0$
Dimension $30$
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] = [805,2,Mod(116,805)]
 
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([0, 4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("805.116");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 805 = 5 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 805.i (of order \(3\), 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: \(30\)
Relative dimension: \(15\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 30 q - 5 q^{2} - 19 q^{4} + 15 q^{5} - 16 q^{6} - 3 q^{7} + 30 q^{8} - 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q - 5 q^{2} - 19 q^{4} + 15 q^{5} - 16 q^{6} - 3 q^{7} + 30 q^{8} - 27 q^{9} + 5 q^{10} - 11 q^{11} + 6 q^{13} - 5 q^{14} - 43 q^{16} + q^{17} + 13 q^{18} + 17 q^{19} - 38 q^{20} - 3 q^{21} + 50 q^{22} - 15 q^{23} + 4 q^{24} - 15 q^{25} + 14 q^{26} - 6 q^{27} - q^{28} + 62 q^{29} - 8 q^{30} + 23 q^{31} - 30 q^{32} - 28 q^{33} - 20 q^{34} + 6 q^{35} + 30 q^{36} - 20 q^{37} - 12 q^{38} - 4 q^{39} + 15 q^{40} - 16 q^{42} + 8 q^{43} - 33 q^{44} + 27 q^{45} - 5 q^{46} + q^{47} + 52 q^{48} + 33 q^{49} + 10 q^{50} - 27 q^{51} - 11 q^{52} - 36 q^{53} - 12 q^{54} - 22 q^{55} + 18 q^{56} + 56 q^{57} - 14 q^{58} + 10 q^{59} + 10 q^{61} + 28 q^{62} - 51 q^{63} + 18 q^{64} + 3 q^{65} + 17 q^{66} - 13 q^{67} - 3 q^{68} - q^{70} - 14 q^{71} - 35 q^{72} - 9 q^{73} - 62 q^{74} + 10 q^{76} - 26 q^{77} - 34 q^{78} - 4 q^{79} + 43 q^{80} - 43 q^{81} - 32 q^{82} + 58 q^{83} - 78 q^{84} + 2 q^{85} - 100 q^{86} + 22 q^{87} - 24 q^{88} + 6 q^{89} + 26 q^{90} - 6 q^{91} + 38 q^{92} - q^{93} + 62 q^{94} - 17 q^{95} + 63 q^{96} + 72 q^{97} + q^{98} + 26 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
116.1 −1.39458 − 2.41548i −0.706998 + 1.22456i −2.88969 + 5.00510i 0.500000 + 0.866025i 3.94385 0.884352 + 2.49358i 10.5413 0.500308 + 0.866558i 1.39458 − 2.41548i
116.2 −1.28220 − 2.22083i 1.25773 − 2.17845i −2.28806 + 3.96303i 0.500000 + 0.866025i −6.45063 2.53323 − 0.763369i 6.60618 −1.66377 − 2.88173i 1.28220 − 2.22083i
116.3 −1.27808 − 2.21371i 0.868481 − 1.50425i −2.26700 + 3.92656i 0.500000 + 0.866025i −4.43997 −2.61172 + 0.422972i 6.47732 −0.00851900 − 0.0147553i 1.27808 − 2.21371i
116.4 −1.13720 − 1.96969i −1.33307 + 2.30894i −1.58644 + 2.74779i 0.500000 + 0.866025i 6.06385 −0.155013 − 2.64121i 2.66759 −2.05414 − 3.55788i 1.13720 − 1.96969i
116.5 −0.831747 − 1.44063i 0.400424 − 0.693555i −0.383605 + 0.664423i 0.500000 + 0.866025i −1.33221 −2.64565 − 0.0226953i −2.05074 1.17932 + 2.04264i 0.831747 − 1.44063i
116.6 −0.702969 − 1.21758i 0.262875 − 0.455313i 0.0116691 − 0.0202115i 0.500000 + 0.866025i −0.739172 1.05563 − 2.42604i −2.84469 1.36179 + 2.35870i 0.702969 − 1.21758i
116.7 −0.297748 − 0.515714i 1.64829 − 2.85492i 0.822693 − 1.42495i 0.500000 + 0.866025i −1.96309 1.62883 − 2.08492i −2.17081 −3.93370 − 6.81337i 0.297748 − 0.515714i
116.8 −0.110541 − 0.191463i −0.787100 + 1.36330i 0.975561 − 1.68972i 0.500000 + 0.866025i 0.348029 2.03078 + 1.69586i −0.873525 0.260946 + 0.451972i 0.110541 − 0.191463i
116.9 0.110311 + 0.191064i 0.0971305 − 0.168235i 0.975663 − 1.68990i 0.500000 + 0.866025i 0.0428582 −1.68540 + 2.03947i 0.871748 1.48113 + 2.56539i −0.110311 + 0.191064i
116.10 0.152095 + 0.263437i −1.43958 + 2.49343i 0.953734 − 1.65192i 0.500000 + 0.866025i −0.875815 −2.56033 − 0.666849i 1.18861 −2.64480 − 4.58093i −0.152095 + 0.263437i
116.11 0.270268 + 0.468117i 1.08279 − 1.87545i 0.853911 − 1.47902i 0.500000 + 0.866025i 1.17058 −1.51572 − 2.16855i 2.00421 −0.844883 − 1.46338i −0.270268 + 0.468117i
116.12 0.525095 + 0.909491i −1.55910 + 2.70044i 0.448550 − 0.776912i 0.500000 + 0.866025i −3.27470 2.64484 + 0.0692983i 3.04251 −3.36157 − 5.82241i −0.525095 + 0.909491i
116.13 1.01640 + 1.76045i 1.58149 − 2.73922i −1.06613 + 1.84660i 0.500000 + 0.866025i 6.42970 −1.55536 + 2.14029i −0.268867 −3.50223 − 6.06604i −1.01640 + 1.76045i
116.14 1.19562 + 2.07088i −0.0985182 + 0.170638i −1.85902 + 3.21991i 0.500000 + 0.866025i −0.471161 2.52848 − 0.778970i −4.10824 1.48059 + 2.56445i −1.19562 + 2.07088i
116.15 1.26527 + 2.19152i −1.27485 + 2.20810i −2.20184 + 3.81369i 0.500000 + 0.866025i −6.45212 −2.07695 − 1.63899i −6.08260 −1.75047 − 3.03190i −1.26527 + 2.19152i
576.1 −1.39458 + 2.41548i −0.706998 − 1.22456i −2.88969 − 5.00510i 0.500000 − 0.866025i 3.94385 0.884352 − 2.49358i 10.5413 0.500308 − 0.866558i 1.39458 + 2.41548i
576.2 −1.28220 + 2.22083i 1.25773 + 2.17845i −2.28806 − 3.96303i 0.500000 − 0.866025i −6.45063 2.53323 + 0.763369i 6.60618 −1.66377 + 2.88173i 1.28220 + 2.22083i
576.3 −1.27808 + 2.21371i 0.868481 + 1.50425i −2.26700 − 3.92656i 0.500000 − 0.866025i −4.43997 −2.61172 − 0.422972i 6.47732 −0.00851900 + 0.0147553i 1.27808 + 2.21371i
576.4 −1.13720 + 1.96969i −1.33307 − 2.30894i −1.58644 − 2.74779i 0.500000 − 0.866025i 6.06385 −0.155013 + 2.64121i 2.66759 −2.05414 + 3.55788i 1.13720 + 1.96969i
576.5 −0.831747 + 1.44063i 0.400424 + 0.693555i −0.383605 − 0.664423i 0.500000 − 0.866025i −1.33221 −2.64565 + 0.0226953i −2.05074 1.17932 − 2.04264i 0.831747 + 1.44063i
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 116.15
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 805.2.i.e ✓ 30
7.c even 3 1 inner 805.2.i.e ✓ 30
7.c even 3 1 5635.2.a.bi 15
7.d odd 6 1 5635.2.a.bj 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
805.2.i.e ✓ 30 1.a even 1 1 trivial
805.2.i.e ✓ 30 7.c even 3 1 inner
5635.2.a.bi 15 7.c even 3 1
5635.2.a.bj 15 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{30} + 5 T_{2}^{29} + 37 T_{2}^{28} + 120 T_{2}^{27} + 585 T_{2}^{26} + 1570 T_{2}^{25} + \cdots + 36 \) acting on \(S_{2}^{\mathrm{new}}(805, [\chi])\). Copy content Toggle raw display