Properties

Label 966.2.g.e
Level $966$
Weight $2$
Character orbit 966.g
Analytic conductor $7.714$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [966,2,Mod(643,966)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(966, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("966.643");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 966.g (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: \(7.71354883526\)
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: \( x^{16} + 326x^{12} + 27081x^{8} + 96196x^{4} + 65536 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{7} \)
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.

\(f(q)\) \(=\) \( q + q^{2} + \beta_{2} q^{3} + q^{4} - \beta_{8} q^{5} + \beta_{2} q^{6} + \beta_{11} q^{7} + q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + \beta_{2} q^{3} + q^{4} - \beta_{8} q^{5} + \beta_{2} q^{6} + \beta_{11} q^{7} + q^{8} - q^{9} - \beta_{8} q^{10} + (\beta_{9} + \beta_{7} + \beta_{3}) q^{11} + \beta_{2} q^{12} - \beta_{6} q^{13} + \beta_{11} q^{14} + \beta_{7} q^{15} + q^{16} + (\beta_{11} + \beta_{9}) q^{17} - q^{18} + ( - \beta_{12} - \beta_{8} + \beta_{5}) q^{19} - \beta_{8} q^{20} - \beta_{10} q^{21} + (\beta_{9} + \beta_{7} + \beta_{3}) q^{22} + (\beta_{13} + \beta_{10} - \beta_{9} + \cdots - 1) q^{23}+ \cdots + ( - \beta_{9} - \beta_{7} - \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 16 q^{2} + 16 q^{4} + 16 q^{8} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{2} + 16 q^{4} + 16 q^{8} - 16 q^{9} + 16 q^{16} - 16 q^{18} - 8 q^{23} + 36 q^{25} + 20 q^{29} + 16 q^{32} - 16 q^{35} - 16 q^{36} - 4 q^{39} - 8 q^{46} + 36 q^{50} + 20 q^{58} + 16 q^{64} - 16 q^{70} - 48 q^{71} - 16 q^{72} + 20 q^{77} - 4 q^{78} + 16 q^{81} - 32 q^{85} - 8 q^{92} + 80 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 326x^{12} + 27081x^{8} + 96196x^{4} + 65536 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 30\nu^{12} + 10275\nu^{8} + 897461\nu^{4} + 2736336 ) / 364028 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 425\nu^{14} + 139062\nu^{10} + 11684785\nu^{6} + 59306340\nu^{2} ) / 46595584 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1255 \nu^{15} + 19584 \nu^{13} + 332330 \nu^{11} + 5875456 \nu^{9} + 14339167 \nu^{7} + \cdots - 1324973568 \nu ) / 745529344 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 731 \nu^{14} - 1328 \nu^{12} - 230866 \nu^{10} - 350832 \nu^{8} - 18392099 \nu^{6} + \cdots - 40917504 ) / 23297792 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1255 \nu^{15} - 19584 \nu^{13} + 332330 \nu^{11} - 5875456 \nu^{9} + 14339167 \nu^{7} + \cdots + 1324973568 \nu ) / 745529344 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -1211\nu^{14} - 395266\nu^{10} - 32751475\nu^{6} - 94033900\nu^{2} ) / 11648896 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1019 \nu^{15} - 3328 \nu^{13} - 329506 \nu^{11} - 1139840 \nu^{9} - 25842835 \nu^{7} + \cdots + 41256448 \nu ) / 186382336 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1019 \nu^{15} + 3328 \nu^{13} - 329506 \nu^{11} + 1139840 \nu^{9} - 25842835 \nu^{7} + \cdots - 41256448 \nu ) / 186382336 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 40007 \nu^{15} + 50304 \nu^{13} + 12980842 \nu^{11} + 16397056 \nu^{9} + 1062386367 \nu^{7} + \cdots + 2968093184 \nu ) / 745529344 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 40007 \nu^{15} - 50304 \nu^{13} + 12980842 \nu^{11} - 16397056 \nu^{9} + 1062386367 \nu^{7} + \cdots - 2968093184 \nu ) / 745529344 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 46807 \nu^{15} - 50304 \nu^{13} - 15205834 \nu^{11} - 16397056 \nu^{9} + \cdots - 2222563840 \nu ) / 745529344 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 46807 \nu^{15} - 50304 \nu^{13} + 15205834 \nu^{11} - 16397056 \nu^{9} + 1249342927 \nu^{7} + \cdots - 2222563840 \nu ) / 745529344 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 46807 \nu^{15} + 50304 \nu^{13} + 53248 \nu^{12} - 15205834 \nu^{11} + 16397056 \nu^{9} + \cdots + 830955520 ) / 745529344 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 46807 \nu^{15} + 23392 \nu^{14} + 50304 \nu^{13} - 157184 \nu^{12} - 15205834 \nu^{11} + \cdots - 6998802432 ) / 745529344 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 46807 \nu^{15} + 29904 \nu^{14} + 50304 \nu^{13} - 15205834 \nu^{11} + 9722080 \nu^{10} + \cdots + 2968093184 \nu ) / 745529344 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{12} + \beta_{11} - \beta_{10} + \beta_{9} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{15} + \beta_{12} - \beta_{11} + \beta_{10} - \beta_{9} + 2\beta_{6} + 14\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 9\beta_{12} - 9\beta_{11} - 11\beta_{10} - 11\beta_{9} - 4\beta_{8} - 4\beta_{7} + 2\beta_{5} + 2\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -30\beta_{13} - 15\beta_{12} + 15\beta_{11} - 15\beta_{10} + 15\beta_{9} + 26\beta _1 - 162 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 163 \beta_{12} - 163 \beta_{11} + 137 \beta_{10} - 137 \beta_{9} - 60 \beta_{8} + 60 \beta_{7} + \cdots - 26 \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 394 \beta_{15} + 8 \beta_{14} + 8 \beta_{13} - 189 \beta_{12} + 189 \beta_{11} - 189 \beta_{10} + \cdots - 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 1413 \beta_{12} + 1413 \beta_{11} + 1747 \beta_{10} + 1747 \beta_{9} + 828 \beta_{8} + \cdots - 302 \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 224 \beta_{14} + 5310 \beta_{13} + 2767 \beta_{12} - 2767 \beta_{11} + 2767 \beta_{10} - 2767 \beta_{9} + \cdots + 25698 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 25795 \beta_{12} + 25795 \beta_{11} - 21473 \beta_{10} + 21473 \beta_{9} + 11292 \beta_{8} + \cdots + 3426 \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 65530 \beta_{15} - 4440 \beta_{14} - 4440 \beta_{13} + 28325 \beta_{12} - 28325 \beta_{11} + 28325 \beta_{10} + \cdots + 4440 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 230717 \beta_{12} - 230717 \beta_{11} - 286747 \beta_{10} - 286747 \beta_{9} - 153260 \beta_{8} + \cdots + 38270 \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 76720 \beta_{14} - 921214 \beta_{13} - 498967 \beta_{12} + 498967 \beta_{11} - 498967 \beta_{10} + \cdots - 4137698 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 4098331 \beta_{12} - 4098331 \beta_{11} + 3371577 \beta_{10} - 3371577 \beta_{9} + \cdots - 419874 \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 10888330 \beta_{15} + 1232840 \beta_{14} + 1232840 \beta_{13} - 4211325 \beta_{12} + 4211325 \beta_{11} + \cdots - 1232840 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 37789477 \beta_{12} + 37789477 \beta_{11} + 47218979 \beta_{10} + 47218979 \beta_{9} + \cdots - 4498142 \beta_{3} ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/966\mathbb{Z}\right)^\times\).

\(n\) \(323\) \(829\) \(925\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
643.1
2.55383 2.55383i
−0.912244 + 0.912244i
−2.48351 + 2.48351i
−0.691340 + 0.691340i
0.691340 0.691340i
2.48351 2.48351i
0.912244 0.912244i
−2.55383 + 2.55383i
2.55383 + 2.55383i
−0.912244 0.912244i
−2.48351 2.48351i
−0.691340 0.691340i
0.691340 + 0.691340i
2.48351 + 2.48351i
0.912244 + 0.912244i
−2.55383 2.55383i
1.00000 1.00000i 1.00000 −4.19800 1.00000i 2.55383 + 0.691340i 1.00000 −1.00000 −4.19800
643.2 1.00000 1.00000i 1.00000 −2.93440 1.00000i −0.912244 + 2.48351i 1.00000 −1.00000 −2.93440
643.3 1.00000 1.00000i 1.00000 −1.36314 1.00000i −2.48351 + 0.912244i 1.00000 −1.00000 −1.36314
643.4 1.00000 1.00000i 1.00000 −0.952834 1.00000i −0.691340 2.55383i 1.00000 −1.00000 −0.952834
643.5 1.00000 1.00000i 1.00000 0.952834 1.00000i 0.691340 + 2.55383i 1.00000 −1.00000 0.952834
643.6 1.00000 1.00000i 1.00000 1.36314 1.00000i 2.48351 0.912244i 1.00000 −1.00000 1.36314
643.7 1.00000 1.00000i 1.00000 2.93440 1.00000i 0.912244 2.48351i 1.00000 −1.00000 2.93440
643.8 1.00000 1.00000i 1.00000 4.19800 1.00000i −2.55383 0.691340i 1.00000 −1.00000 4.19800
643.9 1.00000 1.00000i 1.00000 −4.19800 1.00000i 2.55383 0.691340i 1.00000 −1.00000 −4.19800
643.10 1.00000 1.00000i 1.00000 −2.93440 1.00000i −0.912244 2.48351i 1.00000 −1.00000 −2.93440
643.11 1.00000 1.00000i 1.00000 −1.36314 1.00000i −2.48351 0.912244i 1.00000 −1.00000 −1.36314
643.12 1.00000 1.00000i 1.00000 −0.952834 1.00000i −0.691340 + 2.55383i 1.00000 −1.00000 −0.952834
643.13 1.00000 1.00000i 1.00000 0.952834 1.00000i 0.691340 2.55383i 1.00000 −1.00000 0.952834
643.14 1.00000 1.00000i 1.00000 1.36314 1.00000i 2.48351 + 0.912244i 1.00000 −1.00000 1.36314
643.15 1.00000 1.00000i 1.00000 2.93440 1.00000i 0.912244 + 2.48351i 1.00000 −1.00000 2.93440
643.16 1.00000 1.00000i 1.00000 4.19800 1.00000i −2.55383 + 0.691340i 1.00000 −1.00000 4.19800
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 643.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
23.b odd 2 1 inner
161.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 966.2.g.e 16
3.b odd 2 1 2898.2.g.j 16
7.b odd 2 1 inner 966.2.g.e 16
21.c even 2 1 2898.2.g.j 16
23.b odd 2 1 inner 966.2.g.e 16
69.c even 2 1 2898.2.g.j 16
161.c even 2 1 inner 966.2.g.e 16
483.c odd 2 1 2898.2.g.j 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.g.e 16 1.a even 1 1 trivial
966.2.g.e 16 7.b odd 2 1 inner
966.2.g.e 16 23.b odd 2 1 inner
966.2.g.e 16 161.c even 2 1 inner
2898.2.g.j 16 3.b odd 2 1
2898.2.g.j 16 21.c even 2 1
2898.2.g.j 16 69.c even 2 1
2898.2.g.j 16 483.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(966, [\chi])\):

\( T_{5}^{8} - 29T_{5}^{6} + 226T_{5}^{4} - 464T_{5}^{2} + 256 \) Copy content Toggle raw display
\( T_{11}^{8} + 90T_{11}^{6} + 2680T_{11}^{4} + 27200T_{11}^{2} + 25600 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{16} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$5$ \( (T^{8} - 29 T^{6} + \cdots + 256)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} - 64 T^{12} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( (T^{8} + 90 T^{6} + \cdots + 25600)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 49 T^{6} + \cdots + 256)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 56 T^{6} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} - 90 T^{6} + \cdots + 25600)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 2 T^{3} + \cdots + 529)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} - 5 T^{3} + \cdots - 400)^{4} \) Copy content Toggle raw display
$31$ \( T^{16} \) Copy content Toggle raw display
$37$ \( (T^{8} + 101 T^{6} + \cdots + 256)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 105 T^{6} + \cdots + 25600)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 129 T^{6} + \cdots + 602176)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 181 T^{6} + \cdots + 65536)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 274 T^{6} + \cdots + 15241216)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 264 T^{6} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 126 T^{6} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 406 T^{6} + \cdots + 38738176)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 12 T^{3} + \cdots - 1024)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} + 244 T^{6} + \cdots + 1048576)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 244 T^{6} + \cdots + 2383936)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 194 T^{6} + \cdots + 65536)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 380 T^{6} + \cdots + 409600)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} - 711 T^{6} + \cdots + 157351936)^{2} \) Copy content Toggle raw display
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