↑

Properties

Label 847.2.n.h
Level $847$
Weight $2$
Character orbit 847.n
Analytic conductor $6.763$
Analytic rank $0$
Dimension $40$
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(9,847)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(847, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([10, 18]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("847.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 847.n (of order \(15\), degree \(8\), not 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: \(40\)
Relative dimension: \(5\) over \(\Q(\zeta_{15})\)
Twist minimal: no (minimal twist has level 77)
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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 2 q^{2} + q^{3} + 12 q^{4} - 6 q^{5} - 34 q^{6} - 13 q^{7} - 32 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 2 q^{2} + q^{3} + 12 q^{4} - 6 q^{5} - 34 q^{6} - 13 q^{7} - 32 q^{8} + 2 q^{9} - 14 q^{10} - 18 q^{12} + 4 q^{13} + 22 q^{14} + 16 q^{15} + 20 q^{16} + 12 q^{17} + 41 q^{18} + 24 q^{19} + 40 q^{20} + 2 q^{21} - 14 q^{23} + 7 q^{24} - 29 q^{25} - 5 q^{26} + 4 q^{27} + 24 q^{28} + 30 q^{29} - 6 q^{30} + 3 q^{31} + 30 q^{32} + 48 q^{34} - 6 q^{35} - 46 q^{36} - 11 q^{37} + 12 q^{38} + 32 q^{39} + 20 q^{40} - 10 q^{41} + 45 q^{42} + 72 q^{43} - 16 q^{45} + 17 q^{46} + 3 q^{47} - 62 q^{48} + 35 q^{49} - 6 q^{50} - 28 q^{51} - 2 q^{52} - 42 q^{53} - 34 q^{54} + 24 q^{56} + 36 q^{57} + 8 q^{58} - 9 q^{59} + 27 q^{60} + 20 q^{61} - 128 q^{62} + 36 q^{63} - 36 q^{64} + 40 q^{65} - 38 q^{67} + 33 q^{68} + 106 q^{69} - 18 q^{70} - 50 q^{71} - 42 q^{72} + 14 q^{73} + q^{74} - 16 q^{75} + 96 q^{76} - 100 q^{78} - 11 q^{79} - 18 q^{80} + 12 q^{81} - 24 q^{82} - 104 q^{83} + 44 q^{84} - 32 q^{85} + 2 q^{86} - 48 q^{87} - 10 q^{89} - 42 q^{90} - 14 q^{91} + 80 q^{92} - 13 q^{93} + 18 q^{94} + 8 q^{95} - 7 q^{96} + 46 q^{97} - 116 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 −1.49691 − 1.66249i 1.21952 + 0.542967i −0.314069 + 2.98816i 0.847402 + 0.180121i −0.922843 − 2.84022i −2.23951 + 1.40876i 1.81822 − 1.32101i −0.814967 − 0.905113i −0.969038 − 1.67842i
9.2 −0.853189 − 0.947563i −0.127266 − 0.0566625i 0.0391138 − 0.372143i −1.15917 − 0.246390i 0.0548908 + 0.168936i 1.05400 − 2.42674i −2.44911 + 1.77938i −1.99441 − 2.21501i 0.755525 + 1.30861i
9.3 0.292556 + 0.324917i −2.36017 − 1.05082i 0.189075 − 1.79893i 1.18922 + 0.252776i −0.349055 − 1.07428i −2.48973 − 0.895120i 1.34725 − 0.978836i 2.45880 + 2.73077i 0.265782 + 0.460348i
9.4 1.50246 + 1.66865i −1.74404 − 0.776495i −0.317954 + 3.02513i −2.41117 − 0.512511i −1.32465 − 4.07684i 2.63136 − 0.275607i −1.89247 + 1.37496i 0.431324 + 0.479033i −2.76749 − 4.79344i
9.5 1.63776 + 1.81892i 2.66301 + 1.18565i −0.417144 + 3.96886i 0.324669 + 0.0690106i 2.20477 + 6.78559i −1.08484 − 2.41312i −3.94192 + 2.86397i 3.67845 + 4.08533i 0.406206 + 0.703570i
81.1 −0.193255 + 1.83870i 0.790915 + 0.878400i −1.38718 − 0.294855i −2.15843 − 0.960996i −1.76796 + 1.28450i −2.47553 − 0.933664i −0.332410 + 1.02305i 0.167545 − 1.59409i 2.18411 − 3.78299i
81.2 −0.0704484 + 0.670272i 1.43939 + 1.59861i 1.51199 + 0.321384i 1.54306 + 0.687016i −1.17290 + 0.852164i 1.55543 + 2.14024i −0.738465 + 2.27276i −0.170108 + 1.61847i −0.569194 + 0.985873i
81.3 −0.00794644 + 0.0756054i −1.57585 − 1.75016i 1.95064 + 0.414622i 2.35518 + 1.04860i 0.144844 − 0.105235i −1.83494 + 1.90604i −0.0938324 + 0.288786i −0.266166 + 2.53240i −0.0979948 + 0.169732i
81.4 0.0611782 − 0.582072i −0.925174 − 1.02751i 1.62123 + 0.344603i −3.62146 − 1.61238i −0.654686 + 0.475657i 2.53895 − 0.744116i 0.661490 − 2.03586i 0.113756 − 1.08231i −1.16007 + 2.00931i
81.5 0.275074 − 2.61716i −1.48109 − 1.64492i −4.81754 − 1.02400i −1.07466 − 0.478467i −4.71241 + 3.42377i −2.62673 − 0.316648i −2.37875 + 7.32103i −0.198538 + 1.88897i −1.54783 + 2.68093i
130.1 −2.39411 + 0.508884i −0.304703 − 2.89905i 3.64571 − 1.62317i −0.222100 + 0.246667i 2.20477 + 6.78559i −2.63024 − 0.286049i −3.94192 + 2.86397i −5.37722 + 1.14296i 0.406206 − 0.703570i
130.2 −2.19633 + 0.466843i 0.199554 + 1.89863i 2.77881 − 1.23721i 1.64943 − 1.83188i −1.32465 − 4.07684i 0.551016 + 2.58774i −1.89247 + 1.37496i −0.630517 + 0.134020i −2.76749 + 4.79344i
130.3 −0.427664 + 0.0909028i 0.270052 + 2.56938i −1.65246 + 0.735722i −0.813520 + 0.903506i −0.349055 − 1.07428i −1.62068 − 2.09127i 1.34725 − 0.978836i −3.59432 + 0.763996i 0.265782 − 0.460348i
130.4 1.24721 − 0.265102i 0.0145619 + 0.138547i −0.341843 + 0.152198i 0.792967 − 0.880679i 0.0548908 + 0.168936i −1.98226 + 1.75232i −2.44911 + 1.77938i 2.91546 − 0.619700i 0.755525 − 1.30861i
130.5 2.18822 − 0.465120i −0.139539 − 1.32762i 2.74486 − 1.22209i −0.579690 + 0.643811i −0.922843 − 2.84022i 0.647767 − 2.56523i 1.81822 − 1.32101i 1.19133 − 0.253226i −0.969038 + 1.67842i
366.1 −0.193255 − 1.83870i 0.790915 − 0.878400i −1.38718 + 0.294855i −2.15843 + 0.960996i −1.76796 − 1.28450i −2.47553 + 0.933664i −0.332410 − 1.02305i 0.167545 + 1.59409i 2.18411 + 3.78299i
366.2 −0.0704484 − 0.670272i 1.43939 − 1.59861i 1.51199 − 0.321384i 1.54306 − 0.687016i −1.17290 − 0.852164i 1.55543 − 2.14024i −0.738465 − 2.27276i −0.170108 − 1.61847i −0.569194 − 0.985873i
366.3 −0.00794644 − 0.0756054i −1.57585 + 1.75016i 1.95064 − 0.414622i 2.35518 − 1.04860i 0.144844 + 0.105235i −1.83494 − 1.90604i −0.0938324 − 0.288786i −0.266166 − 2.53240i −0.0979948 − 0.169732i
366.4 0.0611782 + 0.582072i −0.925174 + 1.02751i 1.62123 − 0.344603i −3.62146 + 1.61238i −0.654686 − 0.475657i 2.53895 + 0.744116i 0.661490 + 2.03586i 0.113756 + 1.08231i −1.16007 − 2.00931i
366.5 0.275074 + 2.61716i −1.48109 + 1.64492i −4.81754 + 1.02400i −1.07466 + 0.478467i −4.71241 − 3.42377i −2.62673 + 0.316648i −2.37875 − 7.32103i −0.198538 − 1.88897i −1.54783 − 2.68093i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
11.c even 5 1 inner
77.m even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 847.2.n.h 40
7.c even 3 1 inner 847.2.n.h 40
11.b odd 2 1 847.2.n.i 40
11.c even 5 1 847.2.e.h 20
11.c even 5 1 inner 847.2.n.h 40
11.c even 5 2 847.2.n.j 40
11.d odd 10 2 77.2.m.b ✓ 40
11.d odd 10 1 847.2.e.i 20
11.d odd 10 1 847.2.n.i 40
33.f even 10 2 693.2.by.b 40
77.h odd 6 1 847.2.n.i 40
77.l even 10 2 539.2.q.h 40
77.m even 15 1 847.2.e.h 20
77.m even 15 1 inner 847.2.n.h 40
77.m even 15 2 847.2.n.j 40
77.m even 15 1 5929.2.a.by 10
77.n even 30 2 539.2.f.g 20
77.n even 30 2 539.2.q.h 40
77.n even 30 1 5929.2.a.bx 10
77.o odd 30 2 77.2.m.b ✓ 40
77.o odd 30 2 539.2.f.h 20
77.o odd 30 1 847.2.e.i 20
77.o odd 30 1 847.2.n.i 40
77.o odd 30 1 5929.2.a.bw 10
77.p odd 30 1 5929.2.a.bz 10
231.be even 30 2 693.2.by.b 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.m.b ✓ 40 11.d odd 10 2
77.2.m.b ✓ 40 77.o odd 30 2
539.2.f.g 20 77.n even 30 2
539.2.f.h 20 77.o odd 30 2
539.2.q.h 40 77.l even 10 2
539.2.q.h 40 77.n even 30 2
693.2.by.b 40 33.f even 10 2
693.2.by.b 40 231.be even 30 2
847.2.e.h 20 11.c even 5 1
847.2.e.h 20 77.m even 15 1
847.2.e.i 20 11.d odd 10 1
847.2.e.i 20 77.o odd 30 1
847.2.n.h 40 1.a even 1 1 trivial
847.2.n.h 40 7.c even 3 1 inner
847.2.n.h 40 11.c even 5 1 inner
847.2.n.h 40 77.m even 15 1 inner
847.2.n.i 40 11.b odd 2 1
847.2.n.i 40 11.d odd 10 1
847.2.n.i 40 77.h odd 6 1
847.2.n.i 40 77.o odd 30 1
847.2.n.j 40 11.c even 5 2
847.2.n.j 40 77.m even 15 2
5929.2.a.bw 10 77.o odd 30 1
5929.2.a.bx 10 77.n even 30 1
5929.2.a.by 10 77.m even 15 1
5929.2.a.bz 10 77.p odd 30 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{40} + 2 T_{2}^{39} - 9 T_{2}^{38} - 2 T_{2}^{37} + 66 T_{2}^{36} - 90 T_{2}^{35} - 137 T_{2}^{34} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(847, [\chi])\). Copy content Toggle raw display