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Properties

Label 1064.2.r.j
Level $1064$
Weight $2$
Character orbit 1064.r
Analytic conductor $8.496$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1064,2,Mod(505,1064)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1064, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("1064.505");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1064 = 2^{3} \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1064.r (of order \(3\), degree \(2\), 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: \(8.49608277506\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 3 x^{15} + 22 x^{14} - 49 x^{13} + 285 x^{12} - 583 x^{11} + 1859 x^{10} - 2088 x^{9} + \cdots + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - \beta_1) q^{3} + ( - \beta_{13} + \beta_{7}) q^{5} + q^{7} + ( - \beta_{15} + \beta_{11} + \beta_{10} + \cdots - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - \beta_1) q^{3} + ( - \beta_{13} + \beta_{7}) q^{5} + q^{7} + ( - \beta_{15} + \beta_{11} + \beta_{10} + \cdots - 2) q^{9}+ \cdots + ( - 3 \beta_{15} - \beta_{14} + \cdots - 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 3 q^{3} - 5 q^{5} + 16 q^{7} - 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 3 q^{3} - 5 q^{5} + 16 q^{7} - 11 q^{9} + 4 q^{11} + 2 q^{13} - 15 q^{15} - 8 q^{17} + 7 q^{19} + 3 q^{21} - 10 q^{23} - 9 q^{25} + 12 q^{27} - 3 q^{29} - 10 q^{31} - 3 q^{33} - 5 q^{35} - 26 q^{37} + 36 q^{39} + 5 q^{41} - 23 q^{43} + 74 q^{45} + 3 q^{47} + 16 q^{49} - 8 q^{51} + 8 q^{53} - 25 q^{55} + 15 q^{57} + 11 q^{59} + 7 q^{61} - 11 q^{63} + 32 q^{65} - 7 q^{67} - 28 q^{69} - 6 q^{71} + 3 q^{73} + 40 q^{75} + 4 q^{77} + 16 q^{79} - 52 q^{81} - 32 q^{83} - 27 q^{85} - 62 q^{87} - q^{89} + 2 q^{91} + 6 q^{93} + 20 q^{95} + 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 3 x^{15} + 22 x^{14} - 49 x^{13} + 285 x^{12} - 583 x^{11} + 1859 x^{10} - 2088 x^{9} + \cdots + 144 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 219334901456575 \nu^{15} + 262815868572823 \nu^{14} + \cdots - 98\!\cdots\!68 ) / 32\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 39\!\cdots\!14 \nu^{15} + \cdots - 30\!\cdots\!88 ) / 98\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 67\!\cdots\!35 \nu^{15} + \cdots + 42\!\cdots\!40 ) / 98\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 68\!\cdots\!93 \nu^{15} + \cdots - 31\!\cdots\!44 ) / 49\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 70\!\cdots\!54 \nu^{15} + \cdots + 24\!\cdots\!20 ) / 49\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 68\!\cdots\!22 \nu^{15} + \cdots - 29\!\cdots\!64 ) / 32\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 25\!\cdots\!49 \nu^{15} + \cdots + 65\!\cdots\!36 ) / 98\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 40\!\cdots\!97 \nu^{15} + \cdots + 93\!\cdots\!92 ) / 98\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 53\!\cdots\!95 \nu^{15} + \cdots - 16\!\cdots\!56 ) / 98\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 64\!\cdots\!15 \nu^{15} + \cdots - 28\!\cdots\!12 ) / 98\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 35\!\cdots\!32 \nu^{15} + \cdots - 28\!\cdots\!20 ) / 49\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 36\!\cdots\!78 \nu^{15} + \cdots - 19\!\cdots\!04 ) / 49\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 12\!\cdots\!53 \nu^{15} + \cdots + 71\!\cdots\!76 ) / 98\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 13\!\cdots\!15 \nu^{15} + \cdots - 90\!\cdots\!48 ) / 98\!\cdots\!84 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{15} - \beta_{10} + \beta_{8} + 5\beta_{7} + \beta_{3} - \beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} - \beta_{10} - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + 9\beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 12 \beta_{15} - 2 \beta_{14} + \beta_{13} + 11 \beta_{11} + 12 \beta_{10} + 12 \beta_{9} - 47 \beta_{7} + \cdots - 47 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{15} - 14 \beta_{14} + 15 \beta_{13} + 16 \beta_{12} + 12 \beta_{11} - 4 \beta_{10} + \cdots - 96 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 134 \beta_{11} - 16 \beta_{10} - 118 \beta_{9} - 135 \beta_{8} + \beta_{6} - 29 \beta_{5} + \cdots + 494 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 31 \beta_{15} + 165 \beta_{14} - 187 \beta_{13} - 197 \beta_{12} + 77 \beta_{11} + 211 \beta_{10} + \cdots - 566 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1496 \beta_{15} + 331 \beta_{14} - 226 \beta_{13} - 7 \beta_{12} + 211 \beta_{11} - 1285 \beta_{10} + \cdots - 2320 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 2637 \beta_{11} - 1496 \beta_{10} - 1141 \beta_{9} - 599 \beta_{8} - 2264 \beta_{6} + 1834 \beta_{5} + \cdots + 7437 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 16544 \beta_{15} - 3499 \beta_{14} + 2463 \beta_{13} - 133 \beta_{12} + 14168 \beta_{11} + \cdots - 59966 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 9548 \beta_{15} - 19910 \beta_{14} + 24235 \beta_{13} + 25404 \beta_{12} + 16805 \beta_{11} + \cdots - 137767 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 189732 \beta_{11} - 32228 \beta_{10} - 157504 \beta_{9} - 183289 \beta_{8} - 4888 \beta_{6} + \cdots + 672227 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 137931 \beta_{15} + 214167 \beta_{14} - 266754 \beta_{13} - 282567 \beta_{12} + 199482 \beta_{11} + \cdots - 1155354 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 2036335 \beta_{15} + 358803 \beta_{14} - 237129 \beta_{13} + 98025 \beta_{12} + 389214 \beta_{11} + \cdots - 3895354 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 4665604 \beta_{11} - 2149831 \beta_{10} - 2515773 \beta_{9} - 1881252 \beta_{8} - 3134977 \beta_{6} + \cdots + 14072360 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1064\mathbb{Z}\right)^\times\).

\(n\) \(533\) \(799\) \(913\) \(1009\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1 - \beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
505.1
−1.70215 − 2.94821i
−0.749091 − 1.29746i
−0.360310 − 0.624075i
0.147334 + 0.255190i
0.357488 + 0.619188i
1.05052 + 1.81955i
1.14326 + 1.98018i
1.61295 + 2.79371i
−1.70215 + 2.94821i
−0.749091 + 1.29746i
−0.360310 + 0.624075i
0.147334 − 0.255190i
0.357488 − 0.619188i
1.05052 − 1.81955i
1.14326 − 1.98018i
1.61295 − 2.79371i
0 −1.70215 + 2.94821i 0 −1.86887 + 3.23698i 0 1.00000 0 −4.29461 − 7.43848i 0
505.2 0 −0.749091 + 1.29746i 0 −0.366767 + 0.635259i 0 1.00000 0 0.377726 + 0.654241i 0
505.3 0 −0.360310 + 0.624075i 0 −1.35310 + 2.34365i 0 1.00000 0 1.24035 + 2.14836i 0
505.4 0 0.147334 − 0.255190i 0 1.28751 − 2.23003i 0 1.00000 0 1.45659 + 2.52288i 0
505.5 0 0.357488 − 0.619188i 0 0.0300010 − 0.0519632i 0 1.00000 0 1.24440 + 2.15537i 0
505.6 0 1.05052 − 1.81955i 0 −1.38197 + 2.39365i 0 1.00000 0 −0.707184 − 1.22488i 0
505.7 0 1.14326 − 1.98018i 0 1.70765 − 2.95773i 0 1.00000 0 −1.11407 − 1.92963i 0
505.8 0 1.61295 − 2.79371i 0 −0.554442 + 0.960321i 0 1.00000 0 −3.70321 − 6.41414i 0
729.1 0 −1.70215 − 2.94821i 0 −1.86887 − 3.23698i 0 1.00000 0 −4.29461 + 7.43848i 0
729.2 0 −0.749091 − 1.29746i 0 −0.366767 − 0.635259i 0 1.00000 0 0.377726 − 0.654241i 0
729.3 0 −0.360310 − 0.624075i 0 −1.35310 − 2.34365i 0 1.00000 0 1.24035 − 2.14836i 0
729.4 0 0.147334 + 0.255190i 0 1.28751 + 2.23003i 0 1.00000 0 1.45659 − 2.52288i 0
729.5 0 0.357488 + 0.619188i 0 0.0300010 + 0.0519632i 0 1.00000 0 1.24440 − 2.15537i 0
729.6 0 1.05052 + 1.81955i 0 −1.38197 − 2.39365i 0 1.00000 0 −0.707184 + 1.22488i 0
729.7 0 1.14326 + 1.98018i 0 1.70765 + 2.95773i 0 1.00000 0 −1.11407 + 1.92963i 0
729.8 0 1.61295 + 2.79371i 0 −0.554442 − 0.960321i 0 1.00000 0 −3.70321 + 6.41414i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 505.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1064.2.r.j ✓ 16
19.c even 3 1 inner 1064.2.r.j ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1064.2.r.j ✓ 16 1.a even 1 1 trivial
1064.2.r.j ✓ 16 19.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1064, [\chi])\):

\( T_{3}^{16} - 3 T_{3}^{15} + 22 T_{3}^{14} - 49 T_{3}^{13} + 285 T_{3}^{12} - 583 T_{3}^{11} + 1859 T_{3}^{10} + \cdots + 144 \) Copy content Toggle raw display
\( T_{11}^{8} - 2T_{11}^{7} - 65T_{11}^{6} + 97T_{11}^{5} + 1159T_{11}^{4} - 716T_{11}^{3} - 5508T_{11}^{2} + 684T_{11} + 48 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} - 3 T^{15} + \cdots + 144 \) Copy content Toggle raw display
$5$ \( T^{16} + 5 T^{15} + \cdots + 144 \) Copy content Toggle raw display
$7$ \( (T - 1)^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} - 2 T^{7} - 65 T^{6} + \cdots + 48)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} - 2 T^{15} + \cdots + 1454436 \) Copy content Toggle raw display
$17$ \( T^{16} + 8 T^{15} + \cdots + 42302016 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 16983563041 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 1279206756 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 7407890401536 \) Copy content Toggle raw display
$31$ \( (T^{8} + 5 T^{7} + \cdots + 12256)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 13 T^{7} + \cdots + 137664)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} - 5 T^{15} + \cdots + 36144144 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 85142571264 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 272646144 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 201821573657664 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 92895920892009 \) Copy content Toggle raw display
$61$ \( T^{16} - 7 T^{15} + \cdots + 6718464 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 931116923136 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 452980149444 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 33629691456 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 2530492416 \) Copy content Toggle raw display
$83$ \( (T^{8} + 16 T^{7} + \cdots - 26390472)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 20994850816 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 25\!\cdots\!96 \) Copy content Toggle raw display
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