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Properties

Label 3016.2.a.h
Level $3016$
Weight $2$
Character orbit 3016.a
Self dual yes
Analytic conductor $24.083$
Analytic rank $0$
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] = [3016,2,Mod(1,3016)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3016, 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("3016.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3016 = 2^{3} \cdot 13 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3016.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: \(24.0828812496\)
Analytic rank: \(0\)
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: \( x^{10} - 3x^{9} - 17x^{8} + 47x^{7} + 104x^{6} - 235x^{5} - 283x^{4} + 364x^{3} + 330x^{2} + 12x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{2} q^{5} + \beta_{6} q^{7} + (\beta_{6} + \beta_{5} + \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{2} q^{5} + \beta_{6} q^{7} + (\beta_{6} + \beta_{5} + \beta_1 + 1) q^{9} + ( - \beta_{7} + \beta_{5} + 1) q^{11} - q^{13} + (\beta_{9} - \beta_{8} + \beta_{7} + \cdots + \beta_1) q^{15}+ \cdots + (3 \beta_{9} + 2 \beta_{8} + 3 \beta_{6} + \cdots + 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 3 q^{3} + 4 q^{5} - 3 q^{7} + 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 3 q^{3} + 4 q^{5} - 3 q^{7} + 13 q^{9} + 14 q^{11} - 10 q^{13} + 7 q^{15} + 5 q^{17} + 11 q^{19} + 7 q^{23} + 10 q^{25} + 21 q^{27} - 10 q^{29} + 5 q^{31} + 5 q^{33} + 11 q^{35} + 8 q^{37} - 3 q^{39} + 14 q^{41} + 35 q^{43} + 7 q^{45} - 7 q^{49} + 20 q^{51} - 11 q^{53} + 8 q^{55} + 4 q^{57} + 23 q^{59} - 8 q^{61} + 43 q^{63} - 4 q^{65} + 27 q^{67} + 10 q^{69} + 3 q^{71} + 7 q^{73} + 23 q^{75} + 2 q^{77} + 9 q^{79} - 6 q^{81} + 48 q^{83} - 6 q^{85} - 3 q^{87} + 20 q^{89} + 3 q^{91} - 11 q^{93} + 11 q^{95} + q^{97} + 54 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 3x^{9} - 17x^{8} + 47x^{7} + 104x^{6} - 235x^{5} - 283x^{4} + 364x^{3} + 330x^{2} + 12x - 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 13 \nu^{9} - 71 \nu^{8} - 185 \nu^{7} + 1243 \nu^{6} + 828 \nu^{5} - 6935 \nu^{4} - 1999 \nu^{3} + \cdots - 524 ) / 492 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 27 \nu^{9} + 97 \nu^{8} + 359 \nu^{7} - 1257 \nu^{6} - 1808 \nu^{5} + 5005 \nu^{4} + 4997 \nu^{3} + \cdots - 1132 ) / 492 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 33 \nu^{9} + 73 \nu^{8} + 539 \nu^{7} - 1017 \nu^{6} - 2966 \nu^{5} + 4459 \nu^{4} + 6317 \nu^{3} + \cdots + 1040 ) / 492 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 33 \nu^{9} + 73 \nu^{8} + 539 \nu^{7} - 1017 \nu^{6} - 2966 \nu^{5} + 4459 \nu^{4} + 5825 \nu^{3} + \cdots + 56 ) / 492 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 33 \nu^{9} - 73 \nu^{8} - 539 \nu^{7} + 1017 \nu^{6} + 2966 \nu^{5} - 4459 \nu^{4} - 5825 \nu^{3} + \cdots - 2024 ) / 492 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 4 \nu^{9} - 57 \nu^{8} + 79 \nu^{7} + 1103 \nu^{6} - 239 \nu^{5} - 6678 \nu^{4} - 2031 \nu^{3} + \cdots + 136 ) / 246 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 45 \nu^{9} - 107 \nu^{8} - 817 \nu^{7} + 1767 \nu^{6} + 5200 \nu^{5} - 9599 \nu^{4} - 13795 \nu^{3} + \cdots - 1612 ) / 492 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 59 \nu^{9} - 51 \nu^{8} - 1073 \nu^{7} + 551 \nu^{6} + 6262 \nu^{5} - 945 \nu^{4} - 11955 \nu^{3} + \cdots + 1684 ) / 492 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{5} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + \beta_{4} + 7\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} + 9\beta_{6} + 9\beta_{5} + 3\beta_{4} - \beta_{3} + \beta_{2} + 12\beta _1 + 25 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{9} + 2\beta_{8} - \beta_{7} + 14\beta_{6} + 6\beta_{5} + 16\beta_{4} - 3\beta_{3} + 2\beta_{2} + 57\beta _1 + 22 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 12 \beta_{9} + 4 \beta_{8} - 3 \beta_{7} + 80 \beta_{6} + 79 \beta_{5} + 48 \beta_{4} - 15 \beta_{3} + \cdots + 177 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 16 \beta_{9} + 33 \beta_{8} - 22 \beta_{7} + 158 \beta_{6} + 104 \beta_{5} + 194 \beta_{4} - 52 \beta_{3} + \cdots + 224 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 119 \beta_{9} + 85 \beta_{8} - 66 \beta_{7} + 720 \beta_{6} + 718 \beta_{5} + 593 \beta_{4} + \cdots + 1361 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 200 \beta_{9} + 424 \beta_{8} - 323 \beta_{7} + 1664 \beta_{6} + 1336 \beta_{5} + 2145 \beta_{4} + \cdots + 2257 \) 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
−2.55589
−2.43593
−1.58994
−0.696475
−0.206934
0.131336
2.02548
2.52246
2.63743
3.16847
0 −2.55589 0 −2.69451 0 −1.01002 0 3.53259 0
1.2 0 −2.43593 0 1.40458 0 1.75008 0 2.93376 0
1.3 0 −1.58994 0 3.10645 0 2.36044 0 −0.472088 0
1.4 0 −0.696475 0 −1.84500 0 0.816882 0 −2.51492 0
1.5 0 −0.206934 0 2.11442 0 −4.22910 0 −2.95718 0
1.6 0 0.131336 0 −0.686438 0 −3.43967 0 −2.98275 0
1.7 0 2.02548 0 −3.01540 0 −3.00262 0 1.10258 0
1.8 0 2.52246 0 3.65209 0 0.974037 0 3.36279 0
1.9 0 2.63743 0 3.03277 0 −0.746946 0 3.95604 0
1.10 0 3.16847 0 −1.06896 0 3.52691 0 7.03919 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
\(2\) \(-1\)
\(13\) \(1\)
\(29\) \(1\)

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 3016.2.a.h ✓ 10
4.b odd 2 1 6032.2.a.z 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3016.2.a.h ✓ 10 1.a even 1 1 trivial
6032.2.a.z 10 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} - 3 T_{3}^{9} - 17 T_{3}^{8} + 47 T_{3}^{7} + 104 T_{3}^{6} - 235 T_{3}^{5} - 283 T_{3}^{4} + \cdots - 8 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3016))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} - 3 T^{9} + \cdots - 8 \) Copy content Toggle raw display
$5$ \( T^{10} - 4 T^{9} + \cdots - 1124 \) Copy content Toggle raw display
$7$ \( T^{10} + 3 T^{9} + \cdots - 382 \) Copy content Toggle raw display
$11$ \( T^{10} - 14 T^{9} + \cdots + 50704 \) Copy content Toggle raw display
$13$ \( (T + 1)^{10} \) Copy content Toggle raw display
$17$ \( T^{10} - 5 T^{9} + \cdots + 80496 \) Copy content Toggle raw display
$19$ \( T^{10} - 11 T^{9} + \cdots + 1664 \) Copy content Toggle raw display
$23$ \( T^{10} - 7 T^{9} + \cdots - 24832 \) Copy content Toggle raw display
$29$ \( (T + 1)^{10} \) Copy content Toggle raw display
$31$ \( T^{10} - 5 T^{9} + \cdots - 8363008 \) Copy content Toggle raw display
$37$ \( T^{10} - 8 T^{9} + \cdots - 12327232 \) Copy content Toggle raw display
$41$ \( T^{10} - 14 T^{9} + \cdots - 87440576 \) Copy content Toggle raw display
$43$ \( T^{10} - 35 T^{9} + \cdots - 123392 \) Copy content Toggle raw display
$47$ \( T^{10} - 272 T^{8} + \cdots - 14090688 \) Copy content Toggle raw display
$53$ \( T^{10} + 11 T^{9} + \cdots - 129792 \) Copy content Toggle raw display
$59$ \( T^{10} - 23 T^{9} + \cdots + 2251248 \) Copy content Toggle raw display
$61$ \( T^{10} + 8 T^{9} + \cdots - 2452944 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 2441290512 \) Copy content Toggle raw display
$71$ \( T^{10} - 3 T^{9} + \cdots + 14402944 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 9094030808 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots - 509589984 \) Copy content Toggle raw display
$83$ \( T^{10} - 48 T^{9} + \cdots + 2453728 \) Copy content Toggle raw display
$89$ \( T^{10} - 20 T^{9} + \cdots - 41203424 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots - 1470315104 \) Copy content Toggle raw display
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