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Properties

Label 847.2.m.a
Level $847$
Weight $2$
Character orbit 847.m
Analytic conductor $6.763$
Analytic rank $0$
Dimension $320$
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] = [847,2,Mod(78,847)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(847, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 16]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("847.78");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 847.m (of order \(11\), degree \(10\), 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: \(320\)
Relative dimension: \(32\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 320 q - q^{2} - 20 q^{3} - 27 q^{4} + 4 q^{5} - 2 q^{6} + 32 q^{7} + 3 q^{8} + 272 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 320 q - q^{2} - 20 q^{3} - 27 q^{4} + 4 q^{5} - 2 q^{6} + 32 q^{7} + 3 q^{8} + 272 q^{9} - 4 q^{10} + 33 q^{12} + q^{14} + 17 q^{15} - 33 q^{16} - 6 q^{17} - 5 q^{18} + 6 q^{19} - 2 q^{21} + 21 q^{22} - 21 q^{23} + 88 q^{24} - 22 q^{25} - 2 q^{26} - 68 q^{27} + 27 q^{28} - 52 q^{29} + 4 q^{30} + 16 q^{31} - 5 q^{32} - 27 q^{33} - 92 q^{34} - 4 q^{35} - 13 q^{36} - 30 q^{37} + 68 q^{38} - 24 q^{39} - 18 q^{40} - 2 q^{41} + 2 q^{42} - 52 q^{43} - 78 q^{44} + 22 q^{45} + 4 q^{46} + 8 q^{47} + 24 q^{48} - 32 q^{49} + 129 q^{50} + 107 q^{51} + 78 q^{52} - 78 q^{54} + 89 q^{55} + 30 q^{56} - 11 q^{57} + 3 q^{58} - 4 q^{59} - 154 q^{60} - 10 q^{61} + 41 q^{62} + 3 q^{63} + 21 q^{64} - 55 q^{65} - 13 q^{66} + 31 q^{67} - 8 q^{68} + 8 q^{69} + 26 q^{70} - 4 q^{71} - 9 q^{72} + 16 q^{73} - 86 q^{74} + 118 q^{76} + 112 q^{78} + 26 q^{79} + 187 q^{80} + 160 q^{81} + 50 q^{82} - 126 q^{83} - 33 q^{84} + 57 q^{85} - 84 q^{86} - 206 q^{87} - 250 q^{88} - 34 q^{89} + 66 q^{90} + 6 q^{92} - 46 q^{93} - 242 q^{94} + 26 q^{95} - 38 q^{96} - 9 q^{97} - q^{98} + 5 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
78.1 −1.80730 + 2.08573i −2.69195 −0.799327 − 5.55944i −1.89481 + 1.21772i 4.86516 − 5.61470i 0.959493 + 0.281733i 8.39672 + 5.39624i 4.24662 0.884649 − 6.15287i
78.2 −1.68586 + 1.94559i 2.65187 −0.658555 − 4.58035i −1.73874 + 1.11742i −4.47069 + 5.15945i 0.959493 + 0.281733i 5.69031 + 3.65694i 4.03242 0.757237 − 5.26670i
78.3 −1.66103 + 1.91693i −1.41881 −0.630969 − 4.38849i 2.55163 − 1.63983i 2.35669 − 2.71976i 0.959493 + 0.281733i 5.19286 + 3.33725i −0.986967 −1.09489 + 7.61510i
78.4 −1.57261 + 1.81489i 0.142432 −0.536089 − 3.72858i −2.64146 + 1.69757i −0.223990 + 0.258498i 0.959493 + 0.281733i 3.56957 + 2.29402i −2.97971 1.07310 − 7.46357i
78.5 −1.33091 + 1.53595i 0.0770141 −0.303195 − 2.10876i 0.390078 − 0.250688i −0.102499 + 0.118290i 0.959493 + 0.281733i 0.223032 + 0.143334i −2.99407 −0.134114 + 0.932781i
78.6 −1.25388 + 1.44706i −0.679774 −0.237124 − 1.64923i −2.76617 + 1.77771i 0.852356 − 0.983671i 0.959493 + 0.281733i −0.537687 − 0.345550i −2.53791 0.896004 − 6.23184i
78.7 −1.20245 + 1.38770i −2.84131 −0.195195 − 1.35761i 0.410635 − 0.263899i 3.41652 − 3.94288i 0.959493 + 0.281733i −0.970727 − 0.623849i 5.07305 −0.127555 + 0.887161i
78.8 −1.17036 + 1.35067i 3.10681 −0.169934 − 1.18192i 1.35758 − 0.872467i −3.63610 + 4.19628i 0.959493 + 0.281733i −1.21170 − 0.778715i 6.65225 −0.410452 + 2.85476i
78.9 −0.935315 + 1.07941i 0.911730 −0.00568467 − 0.0395377i 1.71713 − 1.10353i −0.852755 + 0.984131i 0.959493 + 0.281733i −2.35507 − 1.51351i −2.16875 −0.414891 + 2.88563i
78.10 −0.934952 + 1.07899i −1.92414 −0.00545909 − 0.0379688i 3.63497 − 2.33606i 1.79898 − 2.07613i 0.959493 + 0.281733i −2.35606 − 1.51415i 0.702307 −0.877940 + 6.10621i
78.11 −0.768590 + 0.887000i 1.62462 0.0885911 + 0.616165i −1.04212 + 0.669731i −1.24867 + 1.44104i 0.959493 + 0.281733i −2.58933 − 1.66406i −0.360597 0.206913 − 1.43911i
78.12 −0.494205 + 0.570342i −1.78860 0.203577 + 1.41591i −0.756631 + 0.486257i 0.883934 − 1.02011i 0.959493 + 0.281733i −2.17790 − 1.39965i 0.199086 0.0965974 − 0.671849i
78.13 −0.349719 + 0.403597i −0.521588 0.244042 + 1.69735i 1.18521 − 0.761688i 0.182409 − 0.210512i 0.959493 + 0.281733i −1.66891 − 1.07254i −2.72795 −0.107075 + 0.744724i
78.14 −0.295977 + 0.341576i −2.78253 0.255558 + 1.77745i −0.630721 + 0.405340i 0.823566 − 0.950445i 0.959493 + 0.281733i −1.44322 − 0.927498i 4.74246 0.0482248 − 0.335411i
78.15 −0.156723 + 0.180869i 1.51489 0.276479 + 1.92295i −1.31500 + 0.845097i −0.237418 + 0.273995i 0.959493 + 0.281733i −0.793795 − 0.510141i −0.705115 0.0532394 − 0.370288i
78.16 −0.131857 + 0.152171i 2.82693 0.278860 + 1.93951i 0.658679 − 0.423307i −0.372749 + 0.430176i 0.959493 + 0.281733i −0.670680 − 0.431020i 4.99153 −0.0224362 + 0.156047i
78.17 0.0397379 − 0.0458600i −1.33973 0.284106 + 1.97600i −3.45187 + 2.21838i −0.0532379 + 0.0614398i 0.959493 + 0.281733i 0.204006 + 0.131107i −1.20513 −0.0354351 + 0.246457i
78.18 0.0841831 − 0.0971525i 1.61774 0.282278 + 1.96329i 2.40779 − 1.54739i 0.136187 − 0.157168i 0.959493 + 0.281733i 0.430790 + 0.276852i −0.382909 0.0523623 − 0.364188i
78.19 0.117914 − 0.136080i 1.55513 0.280016 + 1.94755i −3.32095 + 2.13424i 0.183372 − 0.211622i 0.959493 + 0.281733i 0.600994 + 0.386236i −0.581584 −0.101159 + 0.703575i
78.20 0.573326 − 0.661654i −2.56991 0.175547 + 1.22096i 0.531277 − 0.341431i −1.47340 + 1.70039i 0.959493 + 0.281733i 2.38152 + 1.53051i 3.60446 0.0786859 − 0.547273i
See next 80 embeddings (of 320 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 78.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
121.e even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 847.2.m.a ✓ 320
121.e even 11 1 inner 847.2.m.a ✓ 320
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
847.2.m.a ✓ 320 1.a even 1 1 trivial
847.2.m.a ✓ 320 121.e even 11 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{320} + T_{2}^{319} + 46 T_{2}^{318} + 46 T_{2}^{317} + 1171 T_{2}^{316} + 1171 T_{2}^{315} + \cdots + 12\!\cdots\!01 \) acting on \(S_{2}^{\mathrm{new}}(847, [\chi])\). Copy content Toggle raw display