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Properties

Label 847.2.f.t
Level $847$
Weight $2$
Character orbit 847.f
Analytic conductor $6.763$
Analytic rank $0$
Dimension $12$
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: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} + 7 x^{10} - 15 x^{9} + 59 x^{8} + 118 x^{7} + 266 x^{6} + 324 x^{5} + 1036 x^{4} + \cdots + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{10} + \beta_{8} + \beta_{7} + \cdots - 1) q^{2}+ \cdots + (\beta_{11} + \beta_{10} + \beta_{9} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{10} + \beta_{8} + \beta_{7} + \cdots - 1) q^{2}+ \cdots + (\beta_{2} + 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{2} + q^{3} - 8 q^{4} - q^{5} - 12 q^{6} - 3 q^{7} - 6 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 2 q^{2} + q^{3} - 8 q^{4} - q^{5} - 12 q^{6} - 3 q^{7} - 6 q^{8} - 4 q^{9} - 16 q^{10} + 8 q^{12} - 8 q^{13} - 2 q^{14} + 5 q^{15} - 10 q^{16} - 8 q^{17} + 18 q^{18} + 14 q^{20} - 4 q^{21} + 28 q^{23} - 20 q^{24} - 2 q^{25} + 12 q^{26} + 19 q^{27} - 8 q^{28} + 8 q^{30} + 13 q^{31} + 136 q^{32} - 48 q^{34} - q^{35} - 18 q^{36} + 17 q^{37} - 16 q^{38} + 20 q^{39} + 36 q^{40} - 16 q^{41} - 12 q^{42} + 16 q^{43} - 24 q^{45} - 4 q^{47} - 36 q^{48} - 3 q^{49} - 22 q^{50} + 20 q^{51} + 10 q^{53} + 32 q^{54} + 24 q^{56} - 16 q^{57} + 16 q^{58} - q^{59} + 30 q^{60} + 16 q^{61} - 4 q^{62} - 4 q^{63} - 34 q^{64} + 96 q^{65} - 12 q^{67} + 7 q^{69} + 4 q^{70} - 5 q^{71} - 2 q^{72} - 16 q^{73} + 32 q^{74} - 20 q^{75} - 96 q^{76} + 112 q^{78} - 28 q^{79} + 56 q^{80} - 15 q^{81} - 28 q^{82} - 8 q^{83} - 2 q^{84} - 24 q^{85} - 12 q^{86} - 64 q^{87} - 84 q^{89} + 20 q^{90} - 8 q^{91} - 2 q^{92} - 17 q^{93} - 20 q^{94} - 24 q^{95} - 20 q^{96} + 11 q^{97} + 8 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} + 7 x^{10} - 15 x^{9} + 59 x^{8} + 118 x^{7} + 266 x^{6} + 324 x^{5} + 1036 x^{4} + \cdots + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 15\nu^{10} + 5251\nu^{5} + 24224 ) / 66764 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -59\nu^{11} - 16203\nu^{6} + 367616\nu ) / 133528 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 37\nu^{10} + 10727\nu^{5} - 104932 ) / 33382 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 133\nu^{11} + 37657\nu^{6} - 577480\nu ) / 66764 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 59 \nu^{11} - 413 \nu^{10} + 885 \nu^{9} - 3481 \nu^{8} + 9241 \nu^{7} - 15694 \nu^{6} - 19116 \nu^{5} + \cdots - 944 ) / 133528 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 48 \nu^{11} - 336 \nu^{10} + 720 \nu^{9} - 2832 \nu^{8} + 7801 \nu^{7} - 12768 \nu^{6} - 15552 \nu^{5} + \cdots - 768 ) / 33382 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1093 \nu^{11} - 1093 \nu^{10} + 7563 \nu^{9} - 16395 \nu^{8} + 64487 \nu^{7} + 128974 \nu^{6} + \cdots + 52464 ) / 133528 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 325 \nu^{11} + 2275 \nu^{10} - 4875 \nu^{9} + 19175 \nu^{8} - 53167 \nu^{7} + 86450 \nu^{6} + \cdots + 5200 ) / 66764 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 1514 \nu^{11} - 1514 \nu^{10} + 10583 \nu^{9} - 22710 \nu^{8} + 89326 \nu^{7} + 178652 \nu^{6} + \cdots + 72672 ) / 66764 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 4771 \nu^{11} + 4771 \nu^{10} - 33471 \nu^{9} + 71565 \nu^{8} - 281489 \nu^{7} - 562978 \nu^{6} + \cdots - 229008 ) / 66764 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{9} - 4\beta_{7} + 2\beta_{6} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{11} - 6 \beta_{10} - \beta_{9} + 8 \beta_{8} - 6 \beta_{7} + 8 \beta_{6} + \beta_{5} + \cdots + 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -7\beta_{11} - 30\beta_{10} + 22\beta_{8} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -15\beta_{4} + 74\beta_{2} - 74 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 59\beta_{5} + 266\beta_{3} - 222\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 163\beta_{9} + 770\beta_{7} - 710\beta_{6} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 547 \beta_{11} + 2514 \beta_{10} + 547 \beta_{9} - 2190 \beta_{8} + 2514 \beta_{7} - 2190 \beta_{6} + \cdots - 2514 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1643\beta_{11} + 7666\beta_{10} - 6894\beta_{8} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 5251\beta_{4} - 21454\beta_{2} + 24290 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -16203\beta_{5} - 75314\beta_{3} + 67198\beta_1 \) 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
2.52809 − 1.83676i
−0.293939 + 0.213559i
−1.42513 + 1.03542i
0.544351 + 1.67534i
0.112275 + 0.345546i
−0.965643 − 2.97194i
2.52809 + 1.83676i
−0.293939 − 0.213559i
−1.42513 − 1.03542i
0.544351 − 1.67534i
0.112275 − 0.345546i
−0.965643 + 2.97194i
−0.656626 − 2.02089i 2.52809 + 1.83676i −2.03479 + 1.47836i 0.149831 − 0.461131i 2.05188 − 6.31504i −0.809017 + 0.587785i 0.885558 + 0.643395i 2.09047 + 6.43381i −1.03028
148.2 0.421292 + 1.29660i −0.293939 − 0.213559i 0.114343 − 0.0830753i 0.970726 − 2.98759i 0.153067 − 0.471092i −0.809017 + 0.587785i 2.36180 + 1.71595i −0.886258 − 2.72762i 4.28267
148.3 0.853368 + 2.62640i −1.42513 − 1.03542i −4.55169 + 3.30700i −0.811540 + 2.49766i 1.50326 − 4.62655i −0.809017 + 0.587785i −8.10146 − 5.88605i 0.0318546 + 0.0980384i −7.25240
323.1 −2.23415 − 1.62320i 0.544351 − 1.67534i 1.73859 + 5.35083i 2.12464 − 1.54364i −3.93558 + 2.85936i 0.309017 + 0.951057i 3.09448 − 9.52384i −0.0833965 − 0.0605911i −7.25240
323.2 −1.10296 − 0.801344i 0.112275 − 0.345546i −0.0436753 − 0.134419i −2.54139 + 1.84643i −0.400735 + 0.291151i 0.309017 + 0.951057i −0.902127 + 2.77646i 2.32025 + 1.68576i 4.28267
323.3 1.71907 + 1.24898i −0.965643 + 2.97194i 0.777220 + 2.39204i −0.392262 + 0.284995i −5.37189 + 3.90291i 0.309017 + 0.951057i −0.338253 + 1.04104i −5.47293 − 3.97631i −1.03028
372.1 −0.656626 + 2.02089i 2.52809 − 1.83676i −2.03479 − 1.47836i 0.149831 + 0.461131i 2.05188 + 6.31504i −0.809017 − 0.587785i 0.885558 − 0.643395i 2.09047 − 6.43381i −1.03028
372.2 0.421292 − 1.29660i −0.293939 + 0.213559i 0.114343 + 0.0830753i 0.970726 + 2.98759i 0.153067 + 0.471092i −0.809017 − 0.587785i 2.36180 − 1.71595i −0.886258 + 2.72762i 4.28267
372.3 0.853368 − 2.62640i −1.42513 + 1.03542i −4.55169 − 3.30700i −0.811540 − 2.49766i 1.50326 + 4.62655i −0.809017 − 0.587785i −8.10146 + 5.88605i 0.0318546 − 0.0980384i −7.25240
729.1 −2.23415 + 1.62320i 0.544351 + 1.67534i 1.73859 − 5.35083i 2.12464 + 1.54364i −3.93558 − 2.85936i 0.309017 − 0.951057i 3.09448 + 9.52384i −0.0833965 + 0.0605911i −7.25240
729.2 −1.10296 + 0.801344i 0.112275 + 0.345546i −0.0436753 + 0.134419i −2.54139 − 1.84643i −0.400735 − 0.291151i 0.309017 − 0.951057i −0.902127 − 2.77646i 2.32025 − 1.68576i 4.28267
729.3 1.71907 − 1.24898i −0.965643 − 2.97194i 0.777220 − 2.39204i −0.392262 − 0.284995i −5.37189 − 3.90291i 0.309017 − 0.951057i −0.338253 − 1.04104i −5.47293 + 3.97631i −1.03028
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 148.3
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.t 12
11.b odd 2 1 847.2.f.u 12
11.c even 5 1 847.2.a.j yes 3
11.c even 5 3 inner 847.2.f.t 12
11.d odd 10 1 847.2.a.i ✓ 3
11.d odd 10 3 847.2.f.u 12
33.f even 10 1 7623.2.a.ce 3
33.h odd 10 1 7623.2.a.bz 3
77.j odd 10 1 5929.2.a.y 3
77.l even 10 1 5929.2.a.t 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
847.2.a.i ✓ 3 11.d odd 10 1
847.2.a.j yes 3 11.c even 5 1
847.2.f.t 12 1.a even 1 1 trivial
847.2.f.t 12 11.c even 5 3 inner
847.2.f.u 12 11.b odd 2 1
847.2.f.u 12 11.d odd 10 3
5929.2.a.t 3 77.l even 10 1
5929.2.a.y 3 77.j odd 10 1
7623.2.a.bz 3 33.h odd 10 1
7623.2.a.ce 3 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 \(S_{2}^{\mathrm{new}}(847, [\chi])\):

\( T_{2}^{12} + 2 T_{2}^{11} + 9 T_{2}^{10} + 20 T_{2}^{9} + 69 T_{2}^{8} + 44 T_{2}^{7} + 273 T_{2}^{6} + \cdots + 4096 \) Copy content Toggle raw display
\( T_{3}^{12} - T_{3}^{11} + 7 T_{3}^{10} - 15 T_{3}^{9} + 59 T_{3}^{8} + 118 T_{3}^{7} + 266 T_{3}^{6} + \cdots + 16 \) Copy content Toggle raw display
\( T_{13}^{12} + 8 T_{13}^{11} + 62 T_{13}^{10} + 416 T_{13}^{9} + 2692 T_{13}^{8} + 8768 T_{13}^{7} + \cdots + 16777216 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 2 T^{11} + \cdots + 4096 \) Copy content Toggle raw display
$3$ \( T^{12} - T^{11} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{12} + T^{11} + \cdots + 256 \) Copy content Toggle raw display
$7$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{12} \) Copy content Toggle raw display
$13$ \( T^{12} + 8 T^{11} + \cdots + 16777216 \) Copy content Toggle raw display
$17$ \( T^{12} + 8 T^{11} + \cdots + 16777216 \) Copy content Toggle raw display
$19$ \( T^{12} + 40 T^{10} + \cdots + 16777216 \) Copy content Toggle raw display
$23$ \( (T^{3} - 7 T^{2} + 8 T + 8)^{4} \) Copy content Toggle raw display
$29$ \( T^{12} + 40 T^{10} + \cdots + 16777216 \) Copy content Toggle raw display
$31$ \( T^{12} - 13 T^{11} + \cdots + 11316496 \) Copy content Toggle raw display
$37$ \( T^{12} - 17 T^{11} + \cdots + 65536 \) Copy content Toggle raw display
$41$ \( T^{12} + 16 T^{11} + \cdots + 40960000 \) Copy content Toggle raw display
$43$ \( (T^{3} - 4 T^{2} - 28 T + 32)^{4} \) Copy content Toggle raw display
$47$ \( T^{12} + 4 T^{11} + \cdots + 4096 \) Copy content Toggle raw display
$53$ \( T^{12} - 10 T^{11} + \cdots + 16777216 \) Copy content Toggle raw display
$59$ \( T^{12} + T^{11} + \cdots + 16 \) Copy content Toggle raw display
$61$ \( T^{12} - 16 T^{11} + \cdots + 40960000 \) Copy content Toggle raw display
$67$ \( (T^{3} + 3 T^{2} + \cdots - 424)^{4} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 4797852160000 \) Copy content Toggle raw display
$73$ \( T^{12} + 16 T^{11} + \cdots + 40960000 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 68719476736 \) Copy content Toggle raw display
$83$ \( T^{12} + 8 T^{11} + \cdots + 16777216 \) Copy content Toggle raw display
$89$ \( (T^{3} + 21 T^{2} + \cdots + 100)^{4} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 41740124416 \) Copy content Toggle raw display
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