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Properties

Label 806.2.bl.b
Level $806$
Weight $2$
Character orbit 806.bl
Analytic conductor $6.436$
Analytic rank $0$
Dimension $144$
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] = [806,2,Mod(35,806)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(806, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([20, 18]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("806.35");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 806 = 2 \cdot 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 806.bl (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: \(6.43594240292\)
Analytic rank: \(0\)
Dimension: \(144\)
Relative dimension: \(18\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 144 q + 18 q^{2} + 18 q^{4} + q^{7} - 36 q^{8} + 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 144 q + 18 q^{2} + 18 q^{4} + q^{7} - 36 q^{8} + 22 q^{9} + 7 q^{11} - 19 q^{13} - 2 q^{14} + 4 q^{15} + 18 q^{16} + 7 q^{17} - 24 q^{18} - 12 q^{19} - 60 q^{21} + 7 q^{22} + 7 q^{23} + 104 q^{25} - 6 q^{26} + 6 q^{27} - 4 q^{28} + 3 q^{29} - 16 q^{30} + 30 q^{31} - 72 q^{32} + q^{33} + 6 q^{34} + 16 q^{35} - 68 q^{36} - 66 q^{37} - 46 q^{38} - 23 q^{39} - 16 q^{41} - 10 q^{42} - 5 q^{43} + 6 q^{44} - q^{45} - 3 q^{46} + 12 q^{47} + 25 q^{49} + 18 q^{50} + 114 q^{51} + 5 q^{52} + 52 q^{53} - 3 q^{54} - 19 q^{55} + 6 q^{56} - 16 q^{57} + 13 q^{58} - 24 q^{59} - 8 q^{60} - 18 q^{61} + 108 q^{63} - 36 q^{64} - 24 q^{65} - 12 q^{66} - 8 q^{67} - 8 q^{68} + 28 q^{69} - 32 q^{70} + 35 q^{71} + 12 q^{72} - 20 q^{73} + 39 q^{74} - 64 q^{75} + 23 q^{76} - 92 q^{77} + 20 q^{78} + 6 q^{79} + 60 q^{81} + 9 q^{82} + 18 q^{83} - 10 q^{84} + 69 q^{85} - 40 q^{86} + 42 q^{87} - 8 q^{88} - 25 q^{89} - 28 q^{90} - 23 q^{91} + 16 q^{92} + 29 q^{93} - 36 q^{94} + 13 q^{95} - 56 q^{97} - 90 q^{98} - 12 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
35.1 −0.978148 − 0.207912i −1.98251 − 2.20180i 0.913545 + 0.406737i −3.47209 1.48141 + 2.56587i 1.86260 + 0.829283i −0.809017 − 0.587785i −0.603991 + 5.74659i 3.39621 + 0.721888i
35.2 −0.978148 − 0.207912i −1.81739 − 2.01842i 0.913545 + 0.406737i 1.40451 1.35803 + 2.35217i 0.956404 + 0.425819i −0.809017 − 0.587785i −0.457514 + 4.35296i −1.37382 − 0.292014i
35.3 −0.978148 − 0.207912i −1.78086 − 1.97785i 0.913545 + 0.406737i 0.386298 1.33073 + 2.30489i 2.26473 + 1.00832i −0.809017 − 0.587785i −0.426825 + 4.06097i −0.377856 − 0.0803158i
35.4 −0.978148 − 0.207912i −1.69218 − 1.87935i 0.913545 + 0.406737i −0.606455 1.26446 + 2.19011i −4.40140 − 1.95963i −0.809017 − 0.587785i −0.354918 + 3.37682i 0.593202 + 0.126089i
35.5 −0.978148 − 0.207912i −1.50942 − 1.67638i 0.913545 + 0.406737i 3.31839 1.12789 + 1.95357i −3.66279 − 1.63078i −0.809017 − 0.587785i −0.218316 + 2.07714i −3.24587 − 0.689931i
35.6 −0.978148 − 0.207912i −0.826120 − 0.917499i 0.913545 + 0.406737i 3.35513 0.617308 + 1.06921i 1.61993 + 0.721239i −0.809017 − 0.587785i 0.154255 − 1.46764i −3.28181 − 0.697570i
35.7 −0.978148 − 0.207912i −0.522904 − 0.580744i 0.913545 + 0.406737i −2.58108 0.390734 + 0.676771i 2.13762 + 0.951731i −0.809017 − 0.587785i 0.249751 − 2.37622i 2.52468 + 0.536637i
35.8 −0.978148 − 0.207912i −0.507661 − 0.563814i 0.913545 + 0.406737i −3.19987 0.379343 + 0.657042i −2.37350 − 1.05675i −0.809017 − 0.587785i 0.253418 − 2.41111i 3.12994 + 0.665290i
35.9 −0.978148 − 0.207912i −0.191803 − 0.213019i 0.913545 + 0.406737i −1.05453 0.143322 + 0.248242i −1.58292 − 0.704761i −0.809017 − 0.587785i 0.304997 − 2.90185i 1.03149 + 0.219249i
35.10 −0.978148 − 0.207912i 0.392307 + 0.435701i 0.913545 + 0.406737i −0.400170 −0.293147 − 0.507746i −0.398260 − 0.177317i −0.809017 − 0.587785i 0.277655 − 2.64171i 0.391425 + 0.0832000i
35.11 −0.978148 − 0.207912i 0.394457 + 0.438089i 0.913545 + 0.406737i 0.975885 −0.294753 − 0.510528i 4.63316 + 2.06281i −0.809017 − 0.587785i 0.277260 − 2.63795i −0.954559 − 0.202898i
35.12 −0.978148 − 0.207912i 0.913067 + 1.01406i 0.913545 + 0.406737i 1.43581 −0.682279 − 1.18174i −3.93255 − 1.75088i −0.809017 − 0.587785i 0.118952 − 1.13175i −1.40444 − 0.298522i
35.13 −0.978148 − 0.207912i 0.913919 + 1.01501i 0.913545 + 0.406737i −0.273606 −0.682915 − 1.18284i 3.68001 + 1.63845i −0.809017 − 0.587785i 0.118588 − 1.12829i 0.267627 + 0.0568858i
35.14 −0.978148 − 0.207912i 1.26332 + 1.40306i 0.913545 + 0.406737i −3.60391 −0.944000 − 1.63506i 0.172979 + 0.0770153i −0.809017 − 0.587785i −0.0590109 + 0.561452i 3.52515 + 0.749295i
35.15 −0.978148 − 0.207912i 1.39440 + 1.54864i 0.913545 + 0.406737i 2.92978 −1.04195 − 1.80471i −0.542048 − 0.241336i −0.809017 − 0.587785i −0.140346 + 1.33530i −2.86576 − 0.609136i
35.16 −0.978148 − 0.207912i 1.57332 + 1.74735i 0.913545 + 0.406737i −0.176876 −1.17565 − 2.03628i 1.13977 + 0.507458i −0.809017 − 0.587785i −0.264310 + 2.51474i 0.173011 + 0.0367746i
35.17 −0.978148 − 0.207912i 1.82705 + 2.02914i 0.913545 + 0.406737i 3.93290 −1.36524 − 2.36467i 1.41168 + 0.628519i −0.809017 − 0.587785i −0.465731 + 4.43113i −3.84696 − 0.817696i
35.18 −0.978148 − 0.207912i 2.15899 + 2.39780i 0.913545 + 0.406737i −2.37012 −1.61328 − 2.79429i −3.33436 − 1.48455i −0.809017 − 0.587785i −0.774630 + 7.37011i 2.31833 + 0.492777i
159.1 0.669131 − 0.743145i −3.15606 + 0.670840i −0.104528 − 0.994522i −2.37012 −1.61328 + 2.79429i 0.381520 + 3.62992i −0.809017 − 0.587785i 6.77002 − 3.01421i −1.58592 + 1.76135i
159.2 0.669131 − 0.743145i −2.67082 + 0.567699i −0.104528 − 0.994522i 3.93290 −1.36524 + 2.36467i −0.161525 − 1.53681i −0.809017 − 0.587785i 4.07034 − 1.81223i 2.63162 − 2.92271i
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 35.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner
31.d even 5 1 inner
403.bl even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 806.2.bl.b ✓ 144
13.c even 3 1 inner 806.2.bl.b ✓ 144
31.d even 5 1 inner 806.2.bl.b ✓ 144
403.bl even 15 1 inner 806.2.bl.b ✓ 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
806.2.bl.b ✓ 144 1.a even 1 1 trivial
806.2.bl.b ✓ 144 13.c even 3 1 inner
806.2.bl.b ✓ 144 31.d even 5 1 inner
806.2.bl.b ✓ 144 403.bl even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{144} - 38 T_{3}^{142} - 2 T_{3}^{141} + 599 T_{3}^{140} - 14 T_{3}^{139} - 2997 T_{3}^{138} + \cdots + 33\!\cdots\!25 \) acting on \(S_{2}^{\mathrm{new}}(806, [\chi])\). Copy content Toggle raw display