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Properties

Label 8016.2.a.bd
Level $8016$
Weight $2$
Character orbit 8016.a
Self dual yes
Analytic conductor $64.008$
Analytic rank $1$
Dimension $10$
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: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 26x^{8} - 3x^{7} + 220x^{6} + 42x^{5} - 675x^{4} - 67x^{3} + 628x^{2} - 48x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 4008)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + ( - \beta_1 - 1) q^{5} + ( - \beta_{7} - \beta_{4} - \beta_{2}) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + ( - \beta_1 - 1) q^{5} + ( - \beta_{7} - \beta_{4} - \beta_{2}) q^{7} + q^{9} + (\beta_{7} + \beta_{4}) q^{11} + (\beta_{8} + \beta_1 - 1) q^{13} + ( - \beta_1 - 1) q^{15} + ( - \beta_{8} + \beta_{3}) q^{17} + (\beta_{9} + \beta_{8} - \beta_{6} + \cdots - 1) q^{19}+ \cdots + (\beta_{7} + \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 10 q^{3} - 10 q^{5} - q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 10 q^{3} - 10 q^{5} - q^{7} + 10 q^{9} + q^{11} - 6 q^{13} - 10 q^{15} - 9 q^{17} - 2 q^{19} - q^{21} - 7 q^{23} + 12 q^{25} + 10 q^{27} - 13 q^{29} - 23 q^{31} + q^{33} - q^{35} - 6 q^{37} - 6 q^{39} - 12 q^{41} - 10 q^{45} - 10 q^{47} + 7 q^{49} - 9 q^{51} - 26 q^{53} - 11 q^{55} - 2 q^{57} + 10 q^{59} - 10 q^{61} - q^{63} - 22 q^{65} + 5 q^{67} - 7 q^{69} - 25 q^{71} - 8 q^{73} + 12 q^{75} - 46 q^{77} - 26 q^{79} + 10 q^{81} + 14 q^{83} + 9 q^{85} - 13 q^{87} - 31 q^{89} + 3 q^{91} - 23 q^{93} + 5 q^{95} - 32 q^{97} + q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 26x^{8} - 3x^{7} + 220x^{6} + 42x^{5} - 675x^{4} - 67x^{3} + 628x^{2} - 48x - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 31274 \nu^{9} + 188403 \nu^{8} - 799085 \nu^{7} - 4445172 \nu^{6} + 5651462 \nu^{5} + \cdots + 17959625 ) / 6386707 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 45076 \nu^{9} - 26608 \nu^{8} - 1096194 \nu^{7} + 535650 \nu^{6} + 9006576 \nu^{5} + \cdots + 3445799 ) / 6386707 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 140074 \nu^{9} - 197290 \nu^{8} + 4018521 \nu^{7} + 4959639 \nu^{6} - 37660725 \nu^{5} + \cdots - 3074005 ) / 6386707 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 252127 \nu^{9} - 49534 \nu^{8} + 6474592 \nu^{7} + 2278935 \nu^{6} - 53041455 \nu^{5} + \cdots - 20051371 ) / 6386707 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 400908 \nu^{9} + 156741 \nu^{8} + 10079448 \nu^{7} - 2045962 \nu^{6} - 81378952 \nu^{5} + \cdots + 15210925 ) / 6386707 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 446372 \nu^{9} + 13553 \nu^{8} + 11479221 \nu^{7} + 1022842 \nu^{6} - 95483893 \nu^{5} + \cdots + 3313559 ) / 6386707 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 581351 \nu^{9} + 4817 \nu^{8} + 14786968 \nu^{7} + 1678767 \nu^{6} - 120466276 \nu^{5} + \cdots + 24062029 ) / 6386707 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 676598 \nu^{9} + 130521 \nu^{8} - 17690130 \nu^{7} - 4911181 \nu^{6} + 151157770 \nu^{5} + \cdots - 18894784 ) / 6386707 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + \beta_{7} + \beta_{4} - 2\beta_{3} - \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2} + 9\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 12\beta_{9} + \beta_{8} + 9\beta_{7} + 3\beta_{6} - 3\beta_{5} + 14\beta_{4} - 24\beta_{3} - \beta_{2} - 6\beta _1 + 33 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4 \beta_{9} + 18 \beta_{8} - 11 \beta_{7} - 18 \beta_{6} + 16 \beta_{5} + 2 \beta_{4} - 13 \beta_{3} + \cdots + 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 132 \beta_{9} + 6 \beta_{8} + 81 \beta_{7} + 59 \beta_{6} - 40 \beta_{5} + 167 \beta_{4} - 268 \beta_{3} + \cdots + 341 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 81 \beta_{9} + 246 \beta_{8} - 103 \beta_{7} - 253 \beta_{6} + 212 \beta_{5} + 51 \beta_{4} - 139 \beta_{3} + \cdots + 45 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1477 \beta_{9} - 11 \beta_{8} + 784 \beta_{7} + 876 \beta_{6} - 447 \beta_{5} + 1957 \beta_{4} + \cdots + 3816 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1188 \beta_{9} + 3084 \beta_{8} - 1004 \beta_{7} - 3245 \beta_{6} + 2675 \beta_{5} + 861 \beta_{4} + \cdots + 459 \) 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
3.44741
3.20371
1.76045
1.15141
0.108417
−0.0299246
−1.39234
−2.25873
−2.53098
−3.45943
0 1.00000 0 −4.44741 0 −2.42708 0 1.00000 0
1.2 0 1.00000 0 −4.20371 0 3.93441 0 1.00000 0
1.3 0 1.00000 0 −2.76045 0 −3.28542 0 1.00000 0
1.4 0 1.00000 0 −2.15141 0 2.22742 0 1.00000 0
1.5 0 1.00000 0 −1.10842 0 3.58606 0 1.00000 0
1.6 0 1.00000 0 −0.970075 0 −4.67245 0 1.00000 0
1.7 0 1.00000 0 0.392343 0 −0.0476950 0 1.00000 0
1.8 0 1.00000 0 1.25873 0 0.924269 0 1.00000 0
1.9 0 1.00000 0 1.53098 0 −1.95354 0 1.00000 0
1.10 0 1.00000 0 2.45943 0 0.714036 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
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.bd 10
4.b odd 2 1 4008.2.a.j ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4008.2.a.j ✓ 10 4.b odd 2 1
8016.2.a.bd 10 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8016))\):

\( T_{5}^{10} + 10 T_{5}^{9} + 19 T_{5}^{8} - 85 T_{5}^{7} - 277 T_{5}^{6} + 137 T_{5}^{5} + 910 T_{5}^{4} + \cdots + 222 \) Copy content Toggle raw display
\( T_{7}^{10} + T_{7}^{9} - 38 T_{7}^{8} - 25 T_{7}^{7} + 467 T_{7}^{6} + 186 T_{7}^{5} - 2125 T_{7}^{4} + \cdots - 72 \) Copy content Toggle raw display
\( T_{11}^{10} - T_{11}^{9} - 53 T_{11}^{8} + 76 T_{11}^{7} + 781 T_{11}^{6} - 1395 T_{11}^{5} + \cdots + 4656 \) Copy content Toggle raw display
\( T_{13}^{10} + 6 T_{13}^{9} - 47 T_{13}^{8} - 224 T_{13}^{7} + 920 T_{13}^{6} + 2443 T_{13}^{5} + \cdots - 96 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( (T - 1)^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + 10 T^{9} + \cdots + 222 \) Copy content Toggle raw display
$7$ \( T^{10} + T^{9} + \cdots - 72 \) Copy content Toggle raw display
$11$ \( T^{10} - T^{9} + \cdots + 4656 \) Copy content Toggle raw display
$13$ \( T^{10} + 6 T^{9} + \cdots - 96 \) Copy content Toggle raw display
$17$ \( T^{10} + 9 T^{9} + \cdots - 3336 \) Copy content Toggle raw display
$19$ \( T^{10} + 2 T^{9} + \cdots + 184480 \) Copy content Toggle raw display
$23$ \( T^{10} + 7 T^{9} + \cdots + 1580032 \) Copy content Toggle raw display
$29$ \( T^{10} + 13 T^{9} + \cdots - 15617040 \) Copy content Toggle raw display
$31$ \( T^{10} + 23 T^{9} + \cdots + 1637000 \) Copy content Toggle raw display
$37$ \( T^{10} + 6 T^{9} + \cdots + 4012928 \) Copy content Toggle raw display
$41$ \( T^{10} + 12 T^{9} + \cdots - 317520 \) Copy content Toggle raw display
$43$ \( T^{10} - 178 T^{8} + \cdots - 7185208 \) Copy content Toggle raw display
$47$ \( T^{10} + 10 T^{9} + \cdots + 780 \) Copy content Toggle raw display
$53$ \( T^{10} + 26 T^{9} + \cdots + 8344626 \) Copy content Toggle raw display
$59$ \( T^{10} - 10 T^{9} + \cdots - 13097044 \) Copy content Toggle raw display
$61$ \( T^{10} + 10 T^{9} + \cdots - 51200 \) Copy content Toggle raw display
$67$ \( T^{10} - 5 T^{9} + \cdots - 11149482 \) Copy content Toggle raw display
$71$ \( T^{10} + 25 T^{9} + \cdots + 424928 \) Copy content Toggle raw display
$73$ \( T^{10} + 8 T^{9} + \cdots - 2097200 \) Copy content Toggle raw display
$79$ \( T^{10} + 26 T^{9} + \cdots - 7804352 \) Copy content Toggle raw display
$83$ \( T^{10} - 14 T^{9} + \cdots - 374292 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots - 195650252 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots - 476686580 \) Copy content Toggle raw display
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