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Properties

Label 847.2.b.e
Level $847$
Weight $2$
Character orbit 847.b
Analytic conductor $6.763$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [847,2,Mod(846,847)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(847, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("847.846");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 847.b (of order \(2\), degree \(1\), 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: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} - 8 x^{14} + 48 x^{13} + 26 x^{12} - 288 x^{11} + 32 x^{10} + 968 x^{9} - 462 x^{8} + \cdots + 1252 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{10} - \beta_{3}) q^{2} + (\beta_{15} + \beta_{14} - \beta_{5}) q^{3} + ( - \beta_{6} - 1) q^{4} - \beta_{5} q^{5} + (\beta_{12} - \beta_{9} - 2 \beta_{2}) q^{6} + (\beta_{7} - \beta_{3}) q^{7} - \beta_{4} q^{8} + ( - 2 \beta_{11} - \beta_{8} + \cdots - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{10} - \beta_{3}) q^{2} + (\beta_{15} + \beta_{14} - \beta_{5}) q^{3} + ( - \beta_{6} - 1) q^{4} - \beta_{5} q^{5} + (\beta_{12} - \beta_{9} - 2 \beta_{2}) q^{6} + (\beta_{7} - \beta_{3}) q^{7} - \beta_{4} q^{8} + ( - 2 \beta_{11} - \beta_{8} + \cdots - 2) q^{9}+ \cdots + (2 \beta_{13} - 2 \beta_{12} + \cdots - 2 \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{4} - 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{4} - 40 q^{9} - 32 q^{14} - 16 q^{15} - 24 q^{16} + 32 q^{23} + 40 q^{25} + 24 q^{36} + 72 q^{37} - 40 q^{42} + 16 q^{49} - 8 q^{53} - 8 q^{56} + 96 q^{60} + 112 q^{64} + 24 q^{67} + 32 q^{70} - 56 q^{71} - 104 q^{78} + 48 q^{86} - 24 q^{91} + 8 q^{92} - 24 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} - 8 x^{14} + 48 x^{13} + 26 x^{12} - 288 x^{11} + 32 x^{10} + 968 x^{9} - 462 x^{8} + \cdots + 1252 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 1806236810424 \nu^{15} + 33412641339072 \nu^{14} - 21271288918721 \nu^{13} + \cdots + 45\!\cdots\!42 ) / 28\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 118561841142667 \nu^{15} - 258387126282021 \nu^{14} + 394285081798624 \nu^{13} + \cdots + 62\!\cdots\!98 ) / 10\!\cdots\!54 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 147861519361595 \nu^{15} + \cdots - 14\!\cdots\!66 ) / 10\!\cdots\!54 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 335015184264499 \nu^{15} + \cdots - 65\!\cdots\!54 ) / 10\!\cdots\!54 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 11845355524648 \nu^{15} - 8185822681324 \nu^{14} + 283299612034374 \nu^{13} + \cdots - 34\!\cdots\!42 ) / 28\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 11917111215328 \nu^{15} + 23807655997484 \nu^{14} - 453609661960810 \nu^{13} + \cdots - 16\!\cdots\!72 ) / 28\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 793929381913919 \nu^{15} + \cdots - 98\!\cdots\!92 ) / 10\!\cdots\!54 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 27011392152086 \nu^{15} - 94322220582592 \nu^{14} - 225696122558276 \nu^{13} + \cdots + 41\!\cdots\!60 ) / 28\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 10\!\cdots\!27 \nu^{15} + \cdots - 66\!\cdots\!80 ) / 10\!\cdots\!54 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 11\!\cdots\!43 \nu^{15} + \cdots + 10\!\cdots\!18 ) / 10\!\cdots\!54 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 27586196 \nu^{15} - 60484224 \nu^{14} - 429812678 \nu^{13} + 1098954297 \nu^{12} + \cdots - 30791568858 ) / 22992599138 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 15\!\cdots\!64 \nu^{15} + \cdots + 14\!\cdots\!30 ) / 10\!\cdots\!54 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 17\!\cdots\!59 \nu^{15} + \cdots + 81\!\cdots\!70 ) / 10\!\cdots\!54 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 65540689755620 \nu^{15} - 229996243725450 \nu^{14} - 657627068184028 \nu^{13} + \cdots - 53\!\cdots\!56 ) / 28\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 75710870805340 \nu^{15} + 246212044566059 \nu^{14} + 725706845851137 \nu^{13} + \cdots + 59\!\cdots\!66 ) / 28\!\cdots\!42 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{12} - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{4} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{10} + \beta_{9} + \beta_{3} - \beta_{2} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3 \beta_{15} + 2 \beta_{12} + \beta_{11} - 7 \beta_{10} + 3 \beta_{9} + 2 \beta_{8} + \beta_{7} + \cdots + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{15} + \beta_{11} - 6 \beta_{10} + 2 \beta_{9} + \beta_{8} + 2 \beta_{5} + 2 \beta_{4} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 10\beta_{15} - 2\beta_{12} + \beta_{11} - 17\beta_{10} - \beta_{9} - \beta_{8} + 17\beta_{4} + 5\beta_{3} + 10\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 16 \beta_{15} + 3 \beta_{14} + \beta_{12} - 29 \beta_{10} - 11 \beta_{9} + 10 \beta_{5} + 19 \beta_{4} + \cdots - 22 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 42 \beta_{15} + 7 \beta_{14} - 27 \beta_{12} - 12 \beta_{11} - 59 \beta_{10} - 39 \beta_{9} - 29 \beta_{8} + \cdots - 15 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 68 \beta_{15} + 24 \beta_{14} + 4 \beta_{13} - 4 \beta_{12} - 42 \beta_{11} - 92 \beta_{10} - 132 \beta_{9} + \cdots - 170 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 84 \beta_{15} + 48 \beta_{14} + 9 \beta_{13} - 142 \beta_{12} - 149 \beta_{11} - 89 \beta_{10} + \cdots - 213 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 68 \beta_{15} + 74 \beta_{14} + 30 \beta_{13} - 104 \beta_{12} - 400 \beta_{11} + 4 \beta_{10} + \cdots - 848 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 418 \beta_{15} + 77 \beta_{14} + 55 \beta_{13} - 521 \beta_{12} - 1008 \beta_{11} + 709 \beta_{10} + \cdots - 1356 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1520 \beta_{15} - 304 \beta_{14} + 12 \beta_{13} - 736 \beta_{12} - 2270 \beta_{11} + 2564 \beta_{10} + \cdots - 2998 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 5928 \beta_{15} - 1404 \beta_{14} - 130 \beta_{13} - 1168 \beta_{12} - 4444 \beta_{11} + 8314 \beta_{10} + \cdots - 5116 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 15354 \beta_{15} - 5886 \beta_{14} - 1422 \beta_{13} - 2548 \beta_{12} - 7476 \beta_{11} + \cdots - 4436 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 40066 \beta_{15} - 16672 \beta_{14} - 4230 \beta_{13} + 2008 \beta_{12} - 7726 \beta_{11} + \cdots - 3728 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(122\) \(365\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
846.1
−1.81128 − 0.707107i
−1.31120 − 0.707107i
2.06986 + 0.707107i
0.845980 + 0.707107i
0.160107 + 0.707107i
2.23033 + 0.707107i
−1.67036 − 0.707107i
1.48657 − 0.707107i
1.48657 + 0.707107i
−1.67036 + 0.707107i
2.23033 − 0.707107i
0.160107 − 0.707107i
0.845980 − 0.707107i
2.06986 − 0.707107i
−1.31120 + 0.707107i
−1.81128 + 0.707107i
− 2.20793i − 1.86190i −2.87496 − 2.47437i −4.11094 −2.03541 − 1.69029i 1.93185i −0.466655 −5.46323
846.2 − 2.20793i 1.86190i −2.87496 2.47437i 4.11094 2.03541 − 1.69029i 1.93185i −0.466655 5.46323
846.3 − 2.06181i − 1.78904i −2.25106 − 0.290104i −3.68867 2.64256 − 0.129958i 0.517638i −0.200678 −0.598140
846.4 − 2.06181i 1.78904i −2.25106 0.290104i 3.68867 −2.64256 − 0.129958i 0.517638i −0.200678 0.598140
846.5 − 1.69029i − 3.31624i −0.857091 − 0.780746i −5.60542 1.45775 − 2.20793i − 1.93185i −7.99745 −1.31969
846.6 − 1.69029i 3.31624i −0.857091 0.780746i 5.60542 −1.45775 − 2.20793i − 1.93185i −7.99745 1.31969
846.7 − 0.129958i − 2.08212i 1.98311 1.78432i −0.270589 1.65799 − 2.06181i − 0.517638i −1.33522 0.231887
846.8 − 0.129958i 2.08212i 1.98311 − 1.78432i 0.270589 −1.65799 − 2.06181i − 0.517638i −1.33522 −0.231887
846.9 0.129958i − 2.08212i 1.98311 1.78432i 0.270589 −1.65799 + 2.06181i 0.517638i −1.33522 −0.231887
846.10 0.129958i 2.08212i 1.98311 − 1.78432i −0.270589 1.65799 + 2.06181i 0.517638i −1.33522 0.231887
846.11 1.69029i − 3.31624i −0.857091 − 0.780746i 5.60542 −1.45775 + 2.20793i 1.93185i −7.99745 1.31969
846.12 1.69029i 3.31624i −0.857091 0.780746i −5.60542 1.45775 + 2.20793i 1.93185i −7.99745 −1.31969
846.13 2.06181i − 1.78904i −2.25106 − 0.290104i 3.68867 −2.64256 + 0.129958i − 0.517638i −0.200678 0.598140
846.14 2.06181i 1.78904i −2.25106 0.290104i −3.68867 2.64256 + 0.129958i − 0.517638i −0.200678 −0.598140
846.15 2.20793i − 1.86190i −2.87496 − 2.47437i 4.11094 2.03541 + 1.69029i − 1.93185i −0.466655 5.46323
846.16 2.20793i 1.86190i −2.87496 2.47437i −4.11094 −2.03541 + 1.69029i − 1.93185i −0.466655 −5.46323
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 846.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
11.b odd 2 1 inner
77.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 847.2.b.e ✓ 16
7.b odd 2 1 inner 847.2.b.e ✓ 16
11.b odd 2 1 inner 847.2.b.e ✓ 16
11.c even 5 4 847.2.l.m 64
11.d odd 10 4 847.2.l.m 64
77.b even 2 1 inner 847.2.b.e ✓ 16
77.j odd 10 4 847.2.l.m 64
77.l even 10 4 847.2.l.m 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
847.2.b.e ✓ 16 1.a even 1 1 trivial
847.2.b.e ✓ 16 7.b odd 2 1 inner
847.2.b.e ✓ 16 11.b odd 2 1 inner
847.2.b.e ✓ 16 77.b even 2 1 inner
847.2.l.m 64 11.c even 5 4
847.2.l.m 64 11.d odd 10 4
847.2.l.m 64 77.j odd 10 4
847.2.l.m 64 77.l even 10 4

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 12T_{2}^{6} + 47T_{2}^{4} + 60T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(847, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + 12 T^{6} + 47 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{8} + 22 T^{6} + \cdots + 529)^{2} \) Copy content Toggle raw display
$5$ \( (T^{8} + 10 T^{6} + 26 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} - 8 T^{14} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( (T^{8} - 52 T^{6} + \cdots + 21904)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 94 T^{6} + \cdots + 14641)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} - 94 T^{6} + \cdots + 121801)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 8 T^{3} + \cdots + 313)^{4} \) Copy content Toggle raw display
$29$ \( (T^{8} + 84 T^{6} + \cdots + 42849)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 94 T^{6} + \cdots + 36481)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 18 T^{3} + \cdots + 36)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} - 130 T^{6} + \cdots + 167281)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 108 T^{6} + \cdots + 157609)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 304 T^{6} + \cdots + 9665881)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 2 T^{3} + \cdots + 3289)^{4} \) Copy content Toggle raw display
$59$ \( (T^{8} + 160 T^{6} + \cdots + 829921)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 328 T^{6} + \cdots + 13935289)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 6 T^{3} + \cdots - 143)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 14 T^{3} + \cdots + 3373)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} - 314 T^{6} + \cdots + 10896601)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 360 T^{6} + \cdots + 42849)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 202 T^{6} + \cdots + 32761)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 526 T^{6} + \cdots + 45468049)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 274 T^{6} + \cdots + 966289)^{2} \) Copy content Toggle raw display
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