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Properties

Label 8016.2.a.bc
Level $8016$
Weight $2$
Character orbit 8016.a
Self dual yes
Analytic conductor $64.008$
Analytic rank $0$
Dimension $9$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8016,2,Mod(1,8016)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8016, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8016.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8016.a (trivial)

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: yes
Analytic conductor: \(64.0080822603\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - x^{8} - 31x^{7} + 24x^{6} + 293x^{5} - 101x^{4} - 864x^{3} - 278x^{2} + 24x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2004)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + \beta_1 q^{5} + (\beta_{6} + \beta_{3}) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + \beta_1 q^{5} + (\beta_{6} + \beta_{3}) q^{7} + q^{9} + (\beta_{7} + 1) q^{11} + ( - \beta_{4} - \beta_{3} + \cdots + \beta_1) q^{13}+ \cdots + (\beta_{7} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 9 q^{3} + q^{5} - 2 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 9 q^{3} + q^{5} - 2 q^{7} + 9 q^{9} + 9 q^{11} + 10 q^{13} + q^{15} + 7 q^{17} + 2 q^{19} - 2 q^{21} + 3 q^{23} + 18 q^{25} + 9 q^{27} + 5 q^{29} - 12 q^{31} + 9 q^{33} + 6 q^{35} + 15 q^{37} + 10 q^{39} + 14 q^{41} - 6 q^{43} + q^{45} + 3 q^{47} + 27 q^{49} + 7 q^{51} + 9 q^{53} - 19 q^{55} + 2 q^{57} + 9 q^{59} + 30 q^{61} - 2 q^{63} + 28 q^{65} - 16 q^{67} + 3 q^{69} + 3 q^{71} + 32 q^{73} + 18 q^{75} + 18 q^{77} - 24 q^{79} + 9 q^{81} + 3 q^{83} + 37 q^{85} + 5 q^{87} + 46 q^{89} - 33 q^{91} - 12 q^{93} - 11 q^{95} + 43 q^{97} + 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - x^{8} - 31x^{7} + 24x^{6} + 293x^{5} - 101x^{4} - 864x^{3} - 278x^{2} + 24x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 454 \nu^{8} + 294 \nu^{7} - 14998 \nu^{6} - 10080 \nu^{5} + 154841 \nu^{4} + 125150 \nu^{3} + \cdots - 14306 ) / 27853 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 575 \nu^{8} + 924 \nu^{7} + 16824 \nu^{6} - 25452 \nu^{5} - 142113 \nu^{4} + 175571 \nu^{3} + \cdots - 38190 ) / 27853 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 2488 \nu^{8} + 2646 \nu^{7} + 78500 \nu^{6} - 64078 \nu^{5} - 762703 \nu^{4} + 284359 \nu^{3} + \cdots - 147092 ) / 27853 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3597 \nu^{8} - 3731 \nu^{7} - 111199 \nu^{6} + 90552 \nu^{5} + 1044226 \nu^{4} - 410322 \nu^{3} + \cdots + 6960 ) / 27853 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 577 \nu^{8} + 611 \nu^{7} + 18018 \nu^{6} - 14597 \nu^{5} - 172292 \nu^{4} + 60797 \nu^{3} + \cdots - 3088 ) / 3979 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 5148 \nu^{8} + 5362 \nu^{7} + 158825 \nu^{6} - 128653 \nu^{5} - 1487567 \nu^{4} + 551542 \nu^{3} + \cdots - 76455 ) / 27853 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 5294 \nu^{8} + 4830 \nu^{7} + 163120 \nu^{6} - 116641 \nu^{5} - 1527615 \nu^{4} + 492437 \nu^{3} + \cdots - 110393 ) / 27853 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{6} - 2\beta_{5} + \beta_{4} - 2\beta_{3} + \beta_{2} + 13\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -4\beta_{8} + 16\beta_{7} - 16\beta_{6} + 10\beta_{5} + 14\beta_{4} + 4\beta_{3} - 5\beta_{2} + 2\beta _1 + 90 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -3\beta_{8} + 6\beta_{7} - 42\beta_{6} - 42\beta_{5} + 17\beta_{4} - 50\beta_{3} + 22\beta_{2} + 187\beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 101 \beta_{8} + 246 \beta_{7} - 255 \beta_{6} + 99 \beta_{5} + 218 \beta_{4} + 96 \beta_{3} + \cdots + 1271 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 91 \beta_{8} + 150 \beta_{7} - 711 \beta_{6} - 746 \beta_{5} + 253 \beta_{4} - 943 \beta_{3} + \cdots - 57 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1980 \beta_{8} + 3806 \beta_{7} - 3988 \beta_{6} + 1062 \beta_{5} + 3465 \beta_{4} + 1859 \beta_{3} + \cdots + 18844 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−4.06168
−2.94765
−1.74256
−0.419033
−0.0546093
0.111665
2.94172
3.25977
3.91239
0 1.00000 0 −4.06168 0 1.52198 0 1.00000 0
1.2 0 1.00000 0 −2.94765 0 −4.65088 0 1.00000 0
1.3 0 1.00000 0 −1.74256 0 −2.35372 0 1.00000 0
1.4 0 1.00000 0 −0.419033 0 4.23221 0 1.00000 0
1.5 0 1.00000 0 −0.0546093 0 −3.75281 0 1.00000 0
1.6 0 1.00000 0 0.111665 0 3.78430 0 1.00000 0
1.7 0 1.00000 0 2.94172 0 2.78729 0 1.00000 0
1.8 0 1.00000 0 3.25977 0 −1.95900 0 1.00000 0
1.9 0 1.00000 0 3.91239 0 −1.60938 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(167\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8016.2.a.bc 9
4.b odd 2 1 2004.2.a.c ✓ 9
12.b even 2 1 6012.2.a.i 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2004.2.a.c ✓ 9 4.b odd 2 1
6012.2.a.i 9 12.b even 2 1
8016.2.a.bc 9 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8016))\):

\( T_{5}^{9} - T_{5}^{8} - 31T_{5}^{7} + 24T_{5}^{6} + 293T_{5}^{5} - 101T_{5}^{4} - 864T_{5}^{3} - 278T_{5}^{2} + 24T_{5} + 2 \) Copy content Toggle raw display
\( T_{7}^{9} + 2T_{7}^{8} - 43T_{7}^{7} - 81T_{7}^{6} + 611T_{7}^{5} + 1125T_{7}^{4} - 3192T_{7}^{3} - 5976T_{7}^{2} + 4592T_{7} + 8800 \) Copy content Toggle raw display
\( T_{11}^{9} - 9 T_{11}^{8} - 43 T_{11}^{7} + 492 T_{11}^{6} + 583 T_{11}^{5} - 9565 T_{11}^{4} + \cdots - 170748 \) Copy content Toggle raw display
\( T_{13}^{9} - 10 T_{13}^{8} - 25 T_{13}^{7} + 416 T_{13}^{6} - 250 T_{13}^{5} - 2669 T_{13}^{4} + \cdots + 272 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} \) Copy content Toggle raw display
$3$ \( (T - 1)^{9} \) Copy content Toggle raw display
$5$ \( T^{9} - T^{8} - 31 T^{7} + \cdots + 2 \) Copy content Toggle raw display
$7$ \( T^{9} + 2 T^{8} + \cdots + 8800 \) Copy content Toggle raw display
$11$ \( T^{9} - 9 T^{8} + \cdots - 170748 \) Copy content Toggle raw display
$13$ \( T^{9} - 10 T^{8} + \cdots + 272 \) Copy content Toggle raw display
$17$ \( T^{9} - 7 T^{8} + \cdots - 190 \) Copy content Toggle raw display
$19$ \( T^{9} - 2 T^{8} + \cdots - 147008 \) Copy content Toggle raw display
$23$ \( T^{9} - 3 T^{8} + \cdots + 256 \) Copy content Toggle raw display
$29$ \( T^{9} - 5 T^{8} + \cdots + 16 \) Copy content Toggle raw display
$31$ \( T^{9} + 12 T^{8} + \cdots + 2138048 \) Copy content Toggle raw display
$37$ \( T^{9} - 15 T^{8} + \cdots + 1936 \) Copy content Toggle raw display
$41$ \( T^{9} - 14 T^{8} + \cdots + 1455166 \) Copy content Toggle raw display
$43$ \( T^{9} + 6 T^{8} + \cdots - 4286 \) Copy content Toggle raw display
$47$ \( T^{9} - 3 T^{8} + \cdots + 97276 \) Copy content Toggle raw display
$53$ \( T^{9} - 9 T^{8} + \cdots + 11282146 \) Copy content Toggle raw display
$59$ \( T^{9} - 9 T^{8} + \cdots - 43725824 \) Copy content Toggle raw display
$61$ \( T^{9} - 30 T^{8} + \cdots - 1198252 \) Copy content Toggle raw display
$67$ \( T^{9} + 16 T^{8} + \cdots - 789266 \) Copy content Toggle raw display
$71$ \( T^{9} - 3 T^{8} + \cdots - 82528 \) Copy content Toggle raw display
$73$ \( T^{9} - 32 T^{8} + \cdots + 127893904 \) Copy content Toggle raw display
$79$ \( T^{9} + 24 T^{8} + \cdots + 405486 \) Copy content Toggle raw display
$83$ \( T^{9} - 3 T^{8} + \cdots + 6165280 \) Copy content Toggle raw display
$89$ \( T^{9} + \cdots + 1336363088 \) Copy content Toggle raw display
$97$ \( T^{9} - 43 T^{8} + \cdots + 68517472 \) Copy content Toggle raw display
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