Properties

Label 3016.2.a.g
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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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: \( x^{10} - 2x^{9} - 21x^{8} + 40x^{7} + 138x^{6} - 243x^{5} - 318x^{4} + 448x^{3} + 312x^{2} - 240x - 128 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{9} q^{5} + \beta_{2} q^{7} + ( - \beta_{9} - \beta_{7} + \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{9} q^{5} + \beta_{2} q^{7} + ( - \beta_{9} - \beta_{7} + \beta_{2} + 1) q^{9} + (\beta_{7} + \beta_{4} + 1) q^{11} + q^{13} + (\beta_{8} + \beta_{3} + \beta_{2} + \beta_1 + 1) q^{15} + ( - \beta_{9} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2}) q^{17} + ( - 2 \beta_{9} - \beta_{8} - \beta_{6} - \beta_{5} + \beta_{3} - 1) q^{19} + ( - \beta_{9} - \beta_{8} - \beta_{4} + \beta_1 - 1) q^{21} + (\beta_{9} + \beta_{5} - \beta_{3} + 3) q^{23} + (\beta_{8} - \beta_{6} + \beta_{4} - \beta_{3} - \beta_{2} + 3) q^{25} + (\beta_{9} + \beta_{8} - \beta_{7} + 2 \beta_{5} + \beta_{3} + \beta_1 + 1) q^{27} - q^{29} + (\beta_{9} + \beta_{7} + 2 \beta_{6} + \beta_{4} - 1) q^{31} + (\beta_{7} + 2 \beta_{6} - \beta_{5} + \beta_{3} + \beta_{2} + 1) q^{33} + ( - \beta_{9} - \beta_{8} - \beta_{7} - \beta_{6} - 2 \beta_{5} - \beta_{4} + 2 \beta_1 - 3) q^{35} + ( - \beta_{7} + \beta_{5} - \beta_{3} - \beta_{2} - 1) q^{37} + \beta_1 q^{39} + ( - \beta_{8} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{41} + (2 \beta_{9} + \beta_{7} + 2 \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_1) q^{43} + ( - 2 \beta_{9} - 3 \beta_{7} - 2 \beta_{6} + \beta_{5} + \beta_{4} + 5) q^{45} + (2 \beta_{9} + \beta_{8} + \beta_{5} + 2 \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 3) q^{47} + (2 \beta_{9} + 2 \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} + 1) q^{49} + ( - \beta_{8} + 2 \beta_{7} - 2 \beta_{6} - \beta_{5} - 2 \beta_{4} - \beta_{3} - \beta_{2} + \cdots - 3) q^{51}+ \cdots + ( - 2 \beta_{8} + 2 \beta_{5} + \beta_{4} - \beta_{3} - 3 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 2 q^{3} + 5 q^{5} - 2 q^{7} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 2 q^{3} + 5 q^{5} - 2 q^{7} + 16 q^{9} + 4 q^{11} + 10 q^{13} + 8 q^{15} + 10 q^{17} - q^{19} + q^{21} + 23 q^{23} + 25 q^{25} + 2 q^{27} - 10 q^{29} - 13 q^{31} + 15 q^{33} - 12 q^{35} - 7 q^{37} + 2 q^{39} + 16 q^{41} - 12 q^{43} + 55 q^{45} + 11 q^{47} - 25 q^{51} + 11 q^{53} + 22 q^{55} - 6 q^{57} - 11 q^{59} + 34 q^{61} + 37 q^{63} + 5 q^{65} - 23 q^{67} + 2 q^{69} - 4 q^{71} + 39 q^{73} + 11 q^{75} + 32 q^{77} + 5 q^{79} + 38 q^{81} + 6 q^{83} + 45 q^{85} - 2 q^{87} - 24 q^{89} - 2 q^{91} + 13 q^{93} + 33 q^{95} + 19 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} - 21x^{8} + 40x^{7} + 138x^{6} - 243x^{5} - 318x^{4} + 448x^{3} + 312x^{2} - 240x - 128 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 309 \nu^{9} + 1962 \nu^{8} + 3405 \nu^{7} - 38696 \nu^{6} + 14730 \nu^{5} + 219863 \nu^{4} - 196674 \nu^{3} - 304604 \nu^{2} + 207928 \nu + 68544 ) / 25808 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 291 \nu^{9} + 1143 \nu^{8} + 5587 \nu^{7} - 23209 \nu^{6} - 29820 \nu^{5} + 142003 \nu^{4} + 34323 \nu^{3} - 261240 \nu^{2} - 17852 \nu + 120552 ) / 12904 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1045 \nu^{9} + 3190 \nu^{8} + 20285 \nu^{7} - 60436 \nu^{6} - 112174 \nu^{5} + 325151 \nu^{4} + 141498 \nu^{3} - 411660 \nu^{2} + 2696 \nu + 49648 ) / 25808 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 929 \nu^{9} - 1138 \nu^{8} - 20009 \nu^{7} + 21208 \nu^{6} + 134742 \nu^{5} - 110859 \nu^{4} - 299958 \nu^{3} + 113564 \nu^{2} + 166232 \nu + 10912 ) / 12904 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 539 \nu^{9} - 1136 \nu^{8} - 10293 \nu^{7} + 21698 \nu^{6} + 54768 \nu^{5} - 119693 \nu^{4} - 64256 \nu^{3} + 162242 \nu^{2} + 12668 \nu - 44896 ) / 6452 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2835 \nu^{9} - 2372 \nu^{8} - 62811 \nu^{7} + 41478 \nu^{6} + 446726 \nu^{5} - 189645 \nu^{4} - 1132320 \nu^{3} + 39100 \nu^{2} + 838608 \nu + 274656 ) / 25808 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2845 \nu^{9} - 4440 \nu^{8} - 60165 \nu^{7} + 83238 \nu^{6} + 399394 \nu^{5} - 439723 \nu^{4} - 910972 \nu^{3} + 476836 \nu^{2} + 659408 \nu + 66976 ) / 25808 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1572 \nu^{9} + 2167 \nu^{8} + 33108 \nu^{7} - 40087 \nu^{6} - 215998 \nu^{5} + 204754 \nu^{4} + 467823 \nu^{3} - 184756 \nu^{2} - 315340 \nu - 51440 ) / 12904 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{9} - \beta_{7} + \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} + \beta_{8} - \beta_{7} + 2\beta_{5} + \beta_{3} + 7\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -12\beta_{9} + \beta_{8} - 12\beta_{7} - 2\beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 8\beta_{2} - \beta _1 + 30 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 12\beta_{9} + 18\beta_{8} - 16\beta_{7} + 23\beta_{5} + 2\beta_{4} + 14\beta_{3} + 2\beta_{2} + 60\beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 122 \beta_{9} + 22 \beta_{8} - 131 \beta_{7} - 32 \beta_{6} + 25 \beta_{5} + 25 \beta_{4} + 13 \beta_{3} + 60 \beta_{2} - 14 \beta _1 + 278 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 127 \beta_{9} + 240 \beta_{8} - 209 \beta_{7} + 6 \beta_{6} + 243 \beta_{5} + 42 \beta_{4} + 172 \beta_{3} + 37 \beta_{2} + 556 \beta _1 + 158 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1223 \beta_{9} + 327 \beta_{8} - 1414 \beta_{7} - 396 \beta_{6} + 395 \beta_{5} + 395 \beta_{4} + 158 \beta_{3} + 460 \beta_{2} - 135 \beta _1 + 2777 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1295 \beta_{9} + 2899 \beta_{8} - 2555 \beta_{7} + 136 \beta_{6} + 2590 \beta_{5} + 647 \beta_{4} + 2013 \beta_{3} + 533 \beta_{2} + 5389 \beta _1 + 2120 \) Copy content Toggle raw display

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
−3.19372
−2.71446
−1.30047
−0.866798
−0.478552
1.09823
1.29348
2.22036
2.58509
3.35685
0 −3.19372 0 3.55069 0 −0.0377760 0 7.19984 0
1.2 0 −2.71446 0 −0.957650 0 1.45512 0 4.36832 0
1.3 0 −1.30047 0 −2.99209 0 3.88835 0 −1.30878 0
1.4 0 −0.866798 0 2.19814 0 −4.35947 0 −2.24866 0
1.5 0 −0.478552 0 −1.65690 0 −2.65348 0 −2.77099 0
1.6 0 1.09823 0 4.16520 0 −1.76426 0 −1.79389 0
1.7 0 1.29348 0 −3.79162 0 −2.99662 0 −1.32691 0
1.8 0 2.22036 0 −0.143149 0 1.44357 0 1.92998 0
1.9 0 2.58509 0 1.19710 0 3.50800 0 3.68268 0
1.10 0 3.35685 0 3.43029 0 −0.483428 0 8.26842 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.g 10
4.b odd 2 1 6032.2.a.ba 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3016.2.a.g 10 1.a even 1 1 trivial
6032.2.a.ba 10 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} - 2 T_{3}^{9} - 21 T_{3}^{8} + 40 T_{3}^{7} + 138 T_{3}^{6} - 243 T_{3}^{5} - 318 T_{3}^{4} + 448 T_{3}^{3} + 312 T_{3}^{2} - 240 T_{3} - 128 \) 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
$3$ \( T^{10} - 2 T^{9} - 21 T^{8} + 40 T^{7} + \cdots - 128 \) Copy content Toggle raw display
$5$ \( T^{10} - 5 T^{9} - 25 T^{8} + 139 T^{7} + \cdots - 344 \) Copy content Toggle raw display
$7$ \( T^{10} + 2 T^{9} - 33 T^{8} - 60 T^{7} + \cdots + 32 \) Copy content Toggle raw display
$11$ \( T^{10} - 4 T^{9} - 52 T^{8} + \cdots + 2312 \) Copy content Toggle raw display
$13$ \( (T - 1)^{10} \) Copy content Toggle raw display
$17$ \( T^{10} - 10 T^{9} - 86 T^{8} + \cdots + 628576 \) Copy content Toggle raw display
$19$ \( T^{10} + T^{9} - 120 T^{8} + \cdots - 455848 \) Copy content Toggle raw display
$23$ \( T^{10} - 23 T^{9} + 187 T^{8} + \cdots - 9728 \) Copy content Toggle raw display
$29$ \( (T + 1)^{10} \) Copy content Toggle raw display
$31$ \( T^{10} + 13 T^{9} - 63 T^{8} + \cdots - 29056 \) Copy content Toggle raw display
$37$ \( T^{10} + 7 T^{9} - 68 T^{8} + \cdots + 1045504 \) Copy content Toggle raw display
$41$ \( T^{10} - 16 T^{9} - 131 T^{8} + \cdots + 2453504 \) Copy content Toggle raw display
$43$ \( T^{10} + 12 T^{9} - 97 T^{8} + \cdots - 15872 \) Copy content Toggle raw display
$47$ \( T^{10} - 11 T^{9} - 143 T^{8} + \cdots + 172448 \) Copy content Toggle raw display
$53$ \( T^{10} - 11 T^{9} + \cdots + 148185504 \) Copy content Toggle raw display
$59$ \( T^{10} + 11 T^{9} - 272 T^{8} + \cdots - 79055104 \) Copy content Toggle raw display
$61$ \( T^{10} - 34 T^{9} + 383 T^{8} + \cdots - 584 \) Copy content Toggle raw display
$67$ \( T^{10} + 23 T^{9} - 95 T^{8} + \cdots + 19559552 \) Copy content Toggle raw display
$71$ \( T^{10} + 4 T^{9} + \cdots + 1890479104 \) Copy content Toggle raw display
$73$ \( T^{10} - 39 T^{9} + 509 T^{8} + \cdots + 55296 \) Copy content Toggle raw display
$79$ \( T^{10} - 5 T^{9} - 178 T^{8} + \cdots + 944 \) Copy content Toggle raw display
$83$ \( T^{10} - 6 T^{9} - 381 T^{8} + \cdots - 71552 \) Copy content Toggle raw display
$89$ \( T^{10} + 24 T^{9} + \cdots - 316752512 \) Copy content Toggle raw display
$97$ \( T^{10} - 19 T^{9} - 318 T^{8} + \cdots - 38534656 \) Copy content Toggle raw display
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