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Properties

Label 847.2.n.f
Level $847$
Weight $2$
Character orbit 847.n
Analytic conductor $6.763$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $8$

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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: \(24\)
Relative dimension: \(3\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 3 q^{3} + 6 q^{5} - 6 q^{6} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 3 q^{3} + 6 q^{5} - 6 q^{6} + 6 q^{8} + 12 q^{10} - 12 q^{12} + 6 q^{13} + 12 q^{14} - 18 q^{15} - 6 q^{16} + 3 q^{17} + 12 q^{18} - 9 q^{19} - 12 q^{20} + 48 q^{21} - 6 q^{24} + 3 q^{25} + 9 q^{26} - 12 q^{27} + 3 q^{28} - 6 q^{29} + 6 q^{30} + 9 q^{31} - 36 q^{32} - 48 q^{34} - 15 q^{35} - 6 q^{36} - 3 q^{39} - 3 q^{40} + 18 q^{41} + 18 q^{42} - 12 q^{45} + 24 q^{46} - 3 q^{47} + 36 q^{48} + 30 q^{50} + 18 q^{51} - 9 q^{52} + 9 q^{53} - 72 q^{54} + 12 q^{56} + 15 q^{58} + 6 q^{60} - 12 q^{61} - 6 q^{62} + 6 q^{63} + 6 q^{64} + 60 q^{65} + 21 q^{68} - 42 q^{69} - 45 q^{70} + 18 q^{71} - 12 q^{72} - 6 q^{73} - 18 q^{74} + 9 q^{75} + 72 q^{76} + 48 q^{78} + 3 q^{79} - 27 q^{80} + 15 q^{81} - 3 q^{82} - 30 q^{83} - 54 q^{85} - 9 q^{86} - 96 q^{87} + 60 q^{89} - 72 q^{90} - 9 q^{91} + 6 q^{92} + 6 q^{93} - 9 q^{95} - 3 q^{96} - 90 q^{97} + 24 q^{98}+O(q^{100}) \) 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 −1.25755 − 1.39666i 0.596274 + 0.265478i −0.160147 + 1.52370i −3.45490 − 0.734363i −0.379065 − 1.16664i −1.19676 + 2.35961i −0.711438 + 0.516890i −1.72233 − 1.91284i 3.31908 + 5.74881i
9.2 0.232387 + 0.258091i −0.486087 − 0.216420i 0.196449 − 1.86909i −0.117979 − 0.0250772i −0.0571040 − 0.175748i −1.44511 − 2.21623i 1.08999 − 0.791921i −1.81795 − 2.01904i −0.0209445 − 0.0362770i
9.3 1.02517 + 1.13856i 2.63045 + 1.17115i −0.0363024 + 0.345394i −2.29600 − 0.488030i 1.36322 + 4.19556i 2.64186 − 0.143384i 2.04850 − 1.48832i 3.54028 + 3.93187i −1.79813 − 3.11446i
81.1 −0.160147 + 1.52370i 1.92668 + 2.13980i −0.339707 − 0.0722070i 2.14436 + 0.954731i −3.56896 + 2.59300i 0.952747 − 2.46825i −0.782458 + 2.40816i −0.553045 + 5.26188i −1.79813 + 3.11446i
81.2 −0.0363024 + 0.345394i −0.356037 − 0.395419i 1.83832 + 0.390746i 0.110187 + 0.0490584i 0.149500 − 0.108618i 1.66120 + 2.05923i −0.416337 + 1.28135i 0.283991 − 2.70200i −0.0209445 + 0.0362770i
81.3 0.196449 − 1.86909i 0.436744 + 0.485053i −1.49861 − 0.318539i 3.22672 + 1.43663i 0.992406 − 0.721025i −2.61394 + 0.409024i 0.271745 − 0.836345i 0.269054 − 2.55988i 3.31908 − 5.74881i
130.1 −1.49861 + 0.318539i −0.300978 − 2.86361i 0.317271 − 0.141258i 1.57065 − 1.74438i 1.36322 + 4.19556i 0.680015 + 2.55687i 2.04850 − 1.48832i −5.17524 + 1.10003i −1.79813 + 3.11446i
130.2 −0.339707 + 0.0722070i 0.0556184 + 0.529174i −1.71690 + 0.764415i 0.0807070 − 0.0896342i −0.0571040 − 0.175748i −2.55432 − 0.689525i 1.08999 − 0.791921i 2.65751 − 0.564871i −0.0209445 + 0.0362770i
130.3 1.83832 − 0.390746i −0.0682261 − 0.649128i 1.39963 − 0.623157i 2.36343 − 2.62485i −0.379065 − 1.16664i 1.87431 − 1.86734i −0.711438 + 0.516890i 2.51773 − 0.535160i 3.31908 − 5.74881i
366.1 −0.160147 − 1.52370i 1.92668 − 2.13980i −0.339707 + 0.0722070i 2.14436 − 0.954731i −3.56896 − 2.59300i 0.952747 + 2.46825i −0.782458 − 2.40816i −0.553045 − 5.26188i −1.79813 − 3.11446i
366.2 −0.0363024 − 0.345394i −0.356037 + 0.395419i 1.83832 − 0.390746i 0.110187 − 0.0490584i 0.149500 + 0.108618i 1.66120 − 2.05923i −0.416337 − 1.28135i 0.283991 + 2.70200i −0.0209445 − 0.0362770i
366.3 0.196449 + 1.86909i 0.436744 − 0.485053i −1.49861 + 0.318539i 3.22672 − 1.43663i 0.992406 + 0.721025i −2.61394 − 0.409024i 0.271745 + 0.836345i 0.269054 + 2.55988i 3.31908 + 5.74881i
487.1 −1.71690 − 0.764415i −0.638441 − 0.135705i 1.02517 + 1.13856i −0.369204 + 3.51274i 0.992406 + 0.721025i 2.35514 + 1.20553i 0.271745 + 0.836345i −2.35145 − 1.04693i 3.31908 − 5.74881i
487.2 0.317271 + 0.141258i 0.520461 + 0.110628i −1.25755 − 1.39666i −0.0126077 + 0.119954i 0.149500 + 0.108618i −0.133552 − 2.64238i −0.416337 − 1.28135i −2.48199 − 1.10506i −0.0209445 + 0.0362770i
487.3 1.39963 + 0.623157i −2.81646 − 0.598658i 0.232387 + 0.258091i −0.245359 + 2.33444i −3.56896 − 2.59300i −2.22159 + 1.43685i −0.782458 − 2.40816i 4.83344 + 2.15199i −1.79813 + 3.11446i
632.1 −1.49861 − 0.318539i −0.300978 + 2.86361i 0.317271 + 0.141258i 1.57065 + 1.74438i 1.36322 − 4.19556i 0.680015 − 2.55687i 2.04850 + 1.48832i −5.17524 − 1.10003i −1.79813 − 3.11446i
632.2 −0.339707 − 0.0722070i 0.0556184 − 0.529174i −1.71690 − 0.764415i 0.0807070 + 0.0896342i −0.0571040 + 0.175748i −2.55432 + 0.689525i 1.08999 + 0.791921i 2.65751 + 0.564871i −0.0209445 − 0.0362770i
632.3 1.83832 + 0.390746i −0.0682261 + 0.649128i 1.39963 + 0.623157i 2.36343 + 2.62485i −0.379065 + 1.16664i 1.87431 + 1.86734i −0.711438 − 0.516890i 2.51773 + 0.535160i 3.31908 + 5.74881i
753.1 −1.25755 + 1.39666i 0.596274 − 0.265478i −0.160147 − 1.52370i −3.45490 + 0.734363i −0.379065 + 1.16664i −1.19676 − 2.35961i −0.711438 − 0.516890i −1.72233 + 1.91284i 3.31908 − 5.74881i
753.2 0.232387 − 0.258091i −0.486087 + 0.216420i 0.196449 + 1.86909i −0.117979 + 0.0250772i −0.0571040 + 0.175748i −1.44511 + 2.21623i 1.08999 + 0.791921i −1.81795 + 2.01904i −0.0209445 + 0.0362770i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.3
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 3 inner
77.m even 15 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 847.2.n.f 24
7.c even 3 1 inner 847.2.n.f 24
11.b odd 2 1 847.2.n.g 24
11.c even 5 1 847.2.e.c 6
11.c even 5 3 inner 847.2.n.f 24
11.d odd 10 1 77.2.e.a ✓ 6
11.d odd 10 3 847.2.n.g 24
33.f even 10 1 693.2.i.h 6
44.g even 10 1 1232.2.q.m 6
77.h odd 6 1 847.2.n.g 24
77.l even 10 1 539.2.e.m 6
77.m even 15 1 847.2.e.c 6
77.m even 15 3 inner 847.2.n.f 24
77.m even 15 1 5929.2.a.x 3
77.n even 30 1 539.2.a.g 3
77.n even 30 1 539.2.e.m 6
77.o odd 30 1 77.2.e.a ✓ 6
77.o odd 30 1 539.2.a.j 3
77.o odd 30 3 847.2.n.g 24
77.p odd 30 1 5929.2.a.u 3
231.be even 30 1 693.2.i.h 6
231.be even 30 1 4851.2.a.bj 3
231.bf odd 30 1 4851.2.a.bk 3
308.bc even 30 1 1232.2.q.m 6
308.bc even 30 1 8624.2.a.ch 3
308.bd odd 30 1 8624.2.a.co 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.e.a ✓ 6 11.d odd 10 1
77.2.e.a ✓ 6 77.o odd 30 1
539.2.a.g 3 77.n even 30 1
539.2.a.j 3 77.o odd 30 1
539.2.e.m 6 77.l even 10 1
539.2.e.m 6 77.n even 30 1
693.2.i.h 6 33.f even 10 1
693.2.i.h 6 231.be even 30 1
847.2.e.c 6 11.c even 5 1
847.2.e.c 6 77.m even 15 1
847.2.n.f 24 1.a even 1 1 trivial
847.2.n.f 24 7.c even 3 1 inner
847.2.n.f 24 11.c even 5 3 inner
847.2.n.f 24 77.m even 15 3 inner
847.2.n.g 24 11.b odd 2 1
847.2.n.g 24 11.d odd 10 3
847.2.n.g 24 77.h odd 6 1
847.2.n.g 24 77.o odd 30 3
1232.2.q.m 6 44.g even 10 1
1232.2.q.m 6 308.bc even 30 1
4851.2.a.bj 3 231.be even 30 1
4851.2.a.bk 3 231.bf odd 30 1
5929.2.a.u 3 77.p odd 30 1
5929.2.a.x 3 77.m even 15 1
8624.2.a.ch 3 308.bc even 30 1
8624.2.a.co 3 308.bd odd 30 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} - 3 T_{2}^{22} - 2 T_{2}^{21} - 6 T_{2}^{19} + 30 T_{2}^{18} + 45 T_{2}^{17} - 69 T_{2}^{16} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(847, [\chi])\). Copy content Toggle raw display