↑

Properties

Label 847.2.be.a
Level $847$
Weight $2$
Character orbit 847.be
Analytic conductor $6.763$
Analytic rank $0$
Dimension $6880$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [847,2,Mod(17,847)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(847, base_ring=CyclotomicField(330))
 
chi = DirichletCharacter(H, H._module([55, 147]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("847.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 847.be (of order \(330\), degree \(80\), 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: \(6.76332905120\)
Analytic rank: \(0\)
Dimension: \(6880\)
Relative dimension: \(86\) over \(\Q(\zeta_{330})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{330}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 6880 q - 39 q^{2} - 90 q^{3} + 45 q^{4} - 117 q^{5} - 83 q^{7} - 176 q^{8} - 844 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 6880 q - 39 q^{2} - 90 q^{3} + 45 q^{4} - 117 q^{5} - 83 q^{7} - 176 q^{8} - 844 q^{9} - 198 q^{10} - 32 q^{11} - 120 q^{12} - 110 q^{14} - 154 q^{15} - 101 q^{16} - 147 q^{17} - 141 q^{18} - 117 q^{19} - 55 q^{21} - 188 q^{22} - 32 q^{23} - 108 q^{24} + 35 q^{25} - 159 q^{26} - 48 q^{28} - 136 q^{29} - 3 q^{30} - 141 q^{31} - 44 q^{32} - 174 q^{33} - 93 q^{35} - 20 q^{36} - 72 q^{37} - 135 q^{38} - 32 q^{39} - 240 q^{40} - 137 q^{42} - 176 q^{43} - 85 q^{44} + 51 q^{45} - 24 q^{46} - 129 q^{47} - 53 q^{49} - 206 q^{50} + 55 q^{51} + 81 q^{52} - 21 q^{53} - 858 q^{54} - 56 q^{56} - 170 q^{57} - 19 q^{58} - 153 q^{59} - 74 q^{60} - 102 q^{61} - 37 q^{63} - 560 q^{64} - 44 q^{65} + 60 q^{66} - 20 q^{67} - 57 q^{68} - 61 q^{70} - 180 q^{71} + 170 q^{72} - 72 q^{73} - 89 q^{74} - 51 q^{75} + 176 q^{77} - 90 q^{78} + 70 q^{79} - 81 q^{80} + 628 q^{81} - 27 q^{82} + 70 q^{84} - 406 q^{85} + 30 q^{86} - 33 q^{87} - 41 q^{88} + 135 q^{89} - 59 q^{91} + 258 q^{92} + 131 q^{93} - 270 q^{94} + 14 q^{95} + 189 q^{96} - 186 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 −2.08791 + 1.74070i −1.20069 − 2.69680i 0.969529 − 5.30150i −1.44070 + 1.06783i 7.20128 + 3.54063i −0.641783 + 2.56673i 4.53065 + 8.02273i −3.82369 + 4.24664i 1.14928 − 4.73738i
17.2 −2.05985 + 1.71731i 0.130199 + 0.292432i 0.934033 − 5.10740i 2.69195 − 1.99524i −0.770386 − 0.378773i −2.55328 + 0.693363i 4.20954 + 7.45413i 1.93883 − 2.15329i −2.11855 + 8.73279i
17.3 −2.04726 + 1.70682i 0.307526 + 0.690714i 0.918275 − 5.02123i −0.820089 + 0.607841i −1.80851 − 0.889184i 1.21954 + 2.34792i 4.06902 + 7.20529i 1.62488 − 1.80461i 0.641465 − 2.64415i
17.4 −2.02092 + 1.68486i −0.275222 − 0.618158i 0.885599 − 4.84256i 0.706785 − 0.523862i 1.59771 + 0.785541i 2.64537 − 0.0446952i 3.78166 + 6.69645i 1.70102 − 1.88917i −0.545727 + 2.24952i
17.5 −2.00976 + 1.67555i 0.842269 + 1.89177i 0.871880 − 4.76754i −0.510600 + 0.378452i −4.86251 − 2.39074i 0.0860555 − 2.64435i 3.66264 + 6.48569i −0.861977 + 0.957322i 0.392070 − 1.61613i
17.6 −1.86476 + 1.55466i 0.916760 + 2.05908i 0.700564 − 3.83076i −0.591258 + 0.438234i −4.91070 − 2.41443i −2.58033 + 0.584716i 2.26148 + 4.00457i −1.39196 + 1.54592i 0.421248 − 1.73641i
17.7 −1.85077 + 1.54300i 1.17420 + 2.63731i 0.684718 − 3.74412i −2.93036 + 2.17195i −6.24255 − 3.06926i 2.22866 + 1.42586i 2.14015 + 3.78972i −3.56924 + 3.96404i 2.07210 − 8.54133i
17.8 −1.83365 + 1.52873i −0.138547 − 0.311183i 0.665481 − 3.63892i −3.22431 + 2.38982i 0.729760 + 0.358799i −2.57491 − 0.608149i 1.99482 + 3.53236i 1.92975 − 2.14321i 2.25887 − 9.31119i
17.9 −1.78226 + 1.48588i −1.08301 − 2.43248i 0.608813 − 3.32906i 3.08248 − 2.28470i 5.54456 + 2.72608i −0.149837 + 2.64151i 1.57948 + 2.79689i −2.73664 + 3.03935i −2.09898 + 8.65212i
17.10 −1.77561 + 1.48033i −0.509664 − 1.14472i 0.601599 − 3.28961i 1.79345 − 1.32928i 2.59954 + 1.27811i −0.744238 − 2.53892i 1.52800 + 2.70573i 0.956755 − 1.06258i −1.21667 + 5.01519i
17.11 −1.73569 + 1.44706i −0.981739 − 2.20502i 0.558861 − 3.05592i −2.62084 + 1.94254i 4.89479 + 2.40660i 2.62119 − 0.359701i 1.22966 + 2.17744i −1.89092 + 2.10008i 1.73800 − 7.16415i
17.12 −1.63343 + 1.36180i −1.34163 − 3.01334i 0.453798 − 2.48142i 0.470532 − 0.348753i 6.29501 + 3.09505i −0.710835 − 2.54847i 0.546473 + 0.967679i −5.27288 + 5.85613i −0.293647 + 1.21043i
17.13 −1.62401 + 1.35395i −0.309931 − 0.696116i 0.444447 − 2.43029i −1.48332 + 1.09942i 1.44584 + 0.710870i 1.15240 − 2.38159i 0.489280 + 0.866402i 1.61887 − 1.79794i 0.920366 − 3.79380i
17.14 −1.57573 + 1.31370i −0.784235 − 1.76142i 0.397338 − 2.17269i 1.15753 − 0.857946i 3.54972 + 1.74528i −2.60659 + 0.453528i 0.210563 + 0.372858i −0.480188 + 0.533303i −0.696868 + 2.87253i
17.15 −1.56186 + 1.30213i 0.309898 + 0.696043i 0.384068 − 2.10013i 2.58460 − 1.91568i −1.39036 − 0.683593i 2.59551 − 0.513166i 0.134946 + 0.238958i 1.61895 − 1.79803i −1.54232 + 6.35753i
17.16 −1.50090 + 1.25131i 0.812051 + 1.82390i 0.327137 − 1.78882i 1.60533 − 1.18985i −3.50107 − 1.72136i 0.153479 + 2.64130i −0.174414 − 0.308846i −0.659782 + 0.732762i −0.920565 + 3.79462i
17.17 −1.49075 + 1.24284i 0.979694 + 2.20043i 0.317872 − 1.73816i −0.0234651 + 0.0173921i −4.19527 − 2.06267i 1.67794 − 2.04561i −0.222384 − 0.393791i −1.87469 + 2.08206i 0.0133649 − 0.0550907i
17.18 −1.39089 + 1.15959i −0.0705075 − 0.158362i 0.230127 − 1.25836i −0.770439 + 0.571041i 0.281704 + 0.138505i −0.0295781 + 2.64559i −0.641820 − 1.13652i 1.98728 − 2.20710i 0.409420 − 1.68765i
17.19 −1.34720 + 1.12317i −0.810888 − 1.82128i 0.193649 − 1.05889i −0.159322 + 0.118088i 3.13804 + 1.54287i 1.40647 + 2.24095i −0.796549 − 1.41051i −0.652142 + 0.724277i 0.0820060 − 0.338034i
17.20 −1.29646 + 1.08087i 1.35842 + 3.05106i 0.152745 − 0.835226i 1.93078 − 1.43107i −5.05892 − 2.48730i −1.99539 − 1.73736i −0.955273 − 1.69157i −5.45626 + 6.05979i −0.956379 + 3.94225i
See next 80 embeddings (of 6880 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.86
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
121.h odd 110 1 inner
847.be even 330 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 847.2.be.a ✓ 6880
7.d odd 6 1 inner 847.2.be.a ✓ 6880
121.h odd 110 1 inner 847.2.be.a ✓ 6880
847.be even 330 1 inner 847.2.be.a ✓ 6880
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
847.2.be.a ✓ 6880 1.a even 1 1 trivial
847.2.be.a ✓ 6880 7.d odd 6 1 inner
847.2.be.a ✓ 6880 121.h odd 110 1 inner
847.2.be.a ✓ 6880 847.be even 330 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(847, [\chi])\).