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Properties

Label 847.2.f.o
Level $847$
Weight $2$
Character orbit 847.f
Analytic conductor $6.763$
Analytic rank $0$
Dimension $8$
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] = [847,2,Mod(148,847)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(847, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("847.148");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 847.f (of order \(5\), degree \(4\), 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: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.446265625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 4x^{6} - 7x^{5} + 19x^{4} + 21x^{3} + 36x^{2} + 27x + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} + (\beta_{6} + \beta_{5} + \beta_{2} - \beta_1) q^{3} + (\beta_{3} - \beta_1) q^{4} + (2 \beta_{6} + \beta_{4}) q^{5} + ( - \beta_{6} + 3 \beta_{4}) q^{6} - \beta_{3} q^{7} + ( - 3 \beta_{7} + 3 \beta_{4} + \cdots - 3) q^{8}+ \cdots + \beta_{5} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{2} + (\beta_{6} + \beta_{5} + \beta_{2} - \beta_1) q^{3} + (\beta_{3} - \beta_1) q^{4} + (2 \beta_{6} + \beta_{4}) q^{5} + ( - \beta_{6} + 3 \beta_{4}) q^{6} - \beta_{3} q^{7} + ( - 3 \beta_{7} + 3 \beta_{4} + \cdots - 3) q^{8}+ \cdots + \beta_{2} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - q^{2} + q^{3} - 3 q^{4} + 7 q^{6} + 2 q^{7} - 6 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - q^{2} + q^{3} - 3 q^{4} + 7 q^{6} + 2 q^{7} - 6 q^{8} - q^{9} + 52 q^{10} - 32 q^{12} - 6 q^{13} + q^{14} + 13 q^{15} + 3 q^{16} - 9 q^{17} - 7 q^{18} + 6 q^{19} - 13 q^{20} + 4 q^{21} - 36 q^{23} + 3 q^{24} - 16 q^{25} - 16 q^{26} + 4 q^{27} + 3 q^{28} - 13 q^{29} + 13 q^{30} - 2 q^{31} - 28 q^{32} - 8 q^{34} - 8 q^{36} - 4 q^{37} + 3 q^{38} + 16 q^{39} - 14 q^{41} - 7 q^{42} + 28 q^{43} + 52 q^{45} - 2 q^{46} - 7 q^{47} + 5 q^{48} - 2 q^{49} - 8 q^{50} - 2 q^{51} - 22 q^{52} + 15 q^{53} + 44 q^{54} - 24 q^{56} - 3 q^{57} - 17 q^{59} + 26 q^{60} + 5 q^{61} - q^{62} + q^{63} + 4 q^{64} + 104 q^{65} - 4 q^{67} - 7 q^{68} + 2 q^{69} + 13 q^{70} + 5 q^{71} - 3 q^{72} + 10 q^{73} + 24 q^{74} + 8 q^{75} - 36 q^{76} - 136 q^{78} + 13 q^{79} - 13 q^{80} + 14 q^{81} - 7 q^{82} - 6 q^{83} - 8 q^{84} + 13 q^{85} + 3 q^{86} + 12 q^{89} - 13 q^{90} + 6 q^{91} + 7 q^{92} + q^{93} + 16 q^{94} - 10 q^{96} + 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 4x^{6} - 7x^{5} + 19x^{4} + 21x^{3} + 36x^{2} + 27x + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} - 21 ) / 19 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} + 40\nu ) / 57 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} - 97\nu^{2} ) / 171 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} + 4\nu^{6} - 7\nu^{5} + 19\nu^{4} - 76\nu^{3} + 36\nu^{2} + 27\nu + 81 ) / 171 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} - 40\nu^{2} ) / 57 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4\nu^{7} - 16\nu^{6} + 28\nu^{5} - 76\nu^{4} + 133\nu^{3} - 144\nu^{2} - 108\nu - 324 ) / 513 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - 3\beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -3\beta_{7} - 4\beta_{5} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -12\beta_{7} - 7\beta_{6} - 7\beta_{5} + 12\beta_{4} - 12\beta_{3} - 7\beta_{2} + 7\beta _1 - 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -19\beta_{2} - 21 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 57\beta_{3} - 40\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -97\beta_{6} + 120\beta_{4} \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/847\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(365\)
\(\chi(n)\) \(1\) \(\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
148.1
−1.05397 + 0.765752i
1.86298 − 1.35354i
−0.711597 − 2.19007i
0.402580 + 1.23901i
−1.05397 − 0.765752i
1.86298 + 1.35354i
−0.711597 + 2.19007i
0.402580 − 1.23901i
−0.402580 − 1.23901i −1.05397 − 0.765752i 0.244951 − 0.177967i −1.11418 + 3.42908i −0.524471 + 1.61416i 0.809017 − 0.587785i −2.42705 − 1.76336i −0.402580 − 1.23901i 4.69722
148.2 0.711597 + 2.19007i 1.86298 + 1.35354i −2.67200 + 1.94132i 1.11418 − 3.42908i −1.63865 + 5.04324i 0.809017 − 0.587785i −2.42705 − 1.76336i 0.711597 + 2.19007i 8.30278
323.1 −1.86298 − 1.35354i −0.711597 + 2.19007i 1.02061 + 3.14113i −2.91695 + 2.11929i 4.29004 − 3.11689i −0.309017 − 0.951057i 0.927051 − 2.85317i −1.86298 − 1.35354i 8.30278
323.2 1.05397 + 0.765752i 0.402580 − 1.23901i −0.0935628 − 0.287957i 2.91695 − 2.11929i 1.37308 − 0.997603i −0.309017 − 0.951057i 0.927051 − 2.85317i 1.05397 + 0.765752i 4.69722
372.1 −0.402580 + 1.23901i −1.05397 + 0.765752i 0.244951 + 0.177967i −1.11418 − 3.42908i −0.524471 − 1.61416i 0.809017 + 0.587785i −2.42705 + 1.76336i −0.402580 + 1.23901i 4.69722
372.2 0.711597 − 2.19007i 1.86298 − 1.35354i −2.67200 − 1.94132i 1.11418 + 3.42908i −1.63865 − 5.04324i 0.809017 + 0.587785i −2.42705 + 1.76336i 0.711597 − 2.19007i 8.30278
729.1 −1.86298 + 1.35354i −0.711597 − 2.19007i 1.02061 − 3.14113i −2.91695 − 2.11929i 4.29004 + 3.11689i −0.309017 + 0.951057i 0.927051 + 2.85317i −1.86298 + 1.35354i 8.30278
729.2 1.05397 − 0.765752i 0.402580 + 1.23901i −0.0935628 + 0.287957i 2.91695 + 2.11929i 1.37308 + 0.997603i −0.309017 + 0.951057i 0.927051 + 2.85317i 1.05397 − 0.765752i 4.69722
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 148.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 847.2.f.o 8
11.b odd 2 1 847.2.f.r 8
11.c even 5 1 847.2.a.g yes 2
11.c even 5 3 inner 847.2.f.o 8
11.d odd 10 1 847.2.a.e ✓ 2
11.d odd 10 3 847.2.f.r 8
33.f even 10 1 7623.2.a.bs 2
33.h odd 10 1 7623.2.a.bc 2
77.j odd 10 1 5929.2.a.p 2
77.l even 10 1 5929.2.a.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
847.2.a.e ✓ 2 11.d odd 10 1
847.2.a.g yes 2 11.c even 5 1
847.2.f.o 8 1.a even 1 1 trivial
847.2.f.o 8 11.c even 5 3 inner
847.2.f.r 8 11.b odd 2 1
847.2.f.r 8 11.d odd 10 3
5929.2.a.k 2 77.l even 10 1
5929.2.a.p 2 77.j odd 10 1
7623.2.a.bc 2 33.h odd 10 1
7623.2.a.bs 2 33.f even 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(847, [\chi])\):

\( T_{2}^{8} + T_{2}^{7} + 4T_{2}^{6} + 7T_{2}^{5} + 19T_{2}^{4} - 21T_{2}^{3} + 36T_{2}^{2} - 27T_{2} + 81 \) Copy content Toggle raw display
\( T_{3}^{8} - T_{3}^{7} + 4T_{3}^{6} - 7T_{3}^{5} + 19T_{3}^{4} + 21T_{3}^{3} + 36T_{3}^{2} + 27T_{3} + 81 \) Copy content Toggle raw display
\( T_{13}^{8} + 6T_{13}^{7} + 40T_{13}^{6} + 264T_{13}^{5} + 1744T_{13}^{4} - 1056T_{13}^{3} + 640T_{13}^{2} - 384T_{13} + 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + T^{7} + \cdots + 81 \) Copy content Toggle raw display
$3$ \( T^{8} - T^{7} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( T^{8} + 13 T^{6} + \cdots + 28561 \) Copy content Toggle raw display
$7$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} + 6 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$17$ \( T^{8} + 9 T^{7} + \cdots + 83521 \) Copy content Toggle raw display
$19$ \( (T^{4} - 3 T^{3} + 9 T^{2} + \cdots + 81)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 9 T + 17)^{4} \) Copy content Toggle raw display
$29$ \( T^{8} + 13 T^{7} + \cdots + 2313441 \) Copy content Toggle raw display
$31$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 4 T^{7} + \cdots + 5308416 \) Copy content Toggle raw display
$41$ \( (T^{4} + 7 T^{3} + \cdots + 2401)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 7 T + 9)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} + 7 T^{7} + \cdots + 83521 \) Copy content Toggle raw display
$53$ \( T^{8} - 15 T^{7} + \cdots + 531441 \) Copy content Toggle raw display
$59$ \( T^{8} + 17 T^{7} + \cdots + 22667121 \) Copy content Toggle raw display
$61$ \( T^{8} - 5 T^{7} + \cdots + 81 \) Copy content Toggle raw display
$67$ \( (T^{2} + T - 81)^{4} \) Copy content Toggle raw display
$71$ \( T^{8} - 5 T^{7} + \cdots + 81 \) Copy content Toggle raw display
$73$ \( (T^{4} - 5 T^{3} + \cdots + 625)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} - 13 T^{7} + \cdots + 2313441 \) Copy content Toggle raw display
$83$ \( (T^{4} + 3 T^{3} + 9 T^{2} + \cdots + 81)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 3 T - 261)^{4} \) Copy content Toggle raw display
$97$ \( T^{8} + 13 T^{6} + \cdots + 28561 \) Copy content Toggle raw display
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