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Properties

Label 4018.2.a.bu
Level $4018$
Weight $2$
Character orbit 4018.a
Self dual yes
Analytic conductor $32.084$
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] = [4018,2,Mod(1,4018)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4018, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("4018.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4018 = 2 \cdot 7^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4018.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: \(32.0838915322\)
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: \( x^{10} - x^{9} - 23x^{8} + 19x^{7} + 181x^{6} - 109x^{5} - 579x^{4} + 231x^{3} + 608x^{2} - 204x - 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 574)
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.

\(f(q)\) \(=\) \( q + q^{2} + \beta_1 q^{3} + q^{4} + \beta_{4} q^{5} + \beta_1 q^{6} + q^{8} + ( - \beta_{4} + \beta_{3} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + \beta_1 q^{3} + q^{4} + \beta_{4} q^{5} + \beta_1 q^{6} + q^{8} + ( - \beta_{4} + \beta_{3} + 2) q^{9} + \beta_{4} q^{10} + (\beta_{7} + 1) q^{11} + \beta_1 q^{12} - \beta_{2} q^{13} + (\beta_{9} - \beta_{5} - \beta_1 + 1) q^{15} + q^{16} + (\beta_{3} + 1) q^{17} + ( - \beta_{4} + \beta_{3} + 2) q^{18} + ( - \beta_{6} + \beta_1) q^{19} + \beta_{4} q^{20} + (\beta_{7} + 1) q^{22} + (\beta_{7} + \beta_{6} - \beta_{3} + \beta_1) q^{23} + \beta_1 q^{24} + (\beta_{8} - \beta_{7} - \beta_{6} + \cdots + 3) q^{25}+ \cdots + (2 \beta_{8} + \beta_{7} - \beta_{5} + \cdots + 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 10 q^{2} + q^{3} + 10 q^{4} - 2 q^{5} + q^{6} + 10 q^{8} + 17 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 10 q^{2} + q^{3} + 10 q^{4} - 2 q^{5} + q^{6} + 10 q^{8} + 17 q^{9} - 2 q^{10} + 11 q^{11} + q^{12} + 4 q^{13} + 4 q^{15} + 10 q^{16} + 5 q^{17} + 17 q^{18} - q^{19} - 2 q^{20} + 11 q^{22} + 9 q^{23} + q^{24} + 24 q^{25} + 4 q^{26} + 7 q^{27} + 23 q^{29} + 4 q^{30} - 5 q^{31} + 10 q^{32} - 5 q^{33} + 5 q^{34} + 17 q^{36} + 16 q^{37} - q^{38} - 7 q^{39} - 2 q^{40} + 10 q^{41} + 20 q^{43} + 11 q^{44} - 42 q^{45} + 9 q^{46} - 16 q^{47} + q^{48} + 24 q^{50} + 13 q^{51} + 4 q^{52} + 26 q^{53} + 7 q^{54} + 7 q^{55} + 37 q^{57} + 23 q^{58} - 10 q^{59} + 4 q^{60} - 5 q^{62} + 10 q^{64} + 18 q^{65} - 5 q^{66} + 7 q^{67} + 5 q^{68} + 39 q^{69} + 5 q^{71} + 17 q^{72} - 13 q^{73} + 16 q^{74} + 19 q^{75} - q^{76} - 7 q^{78} - q^{79} - 2 q^{80} + 18 q^{81} + 10 q^{82} + 21 q^{83} + 34 q^{85} + 20 q^{86} + 2 q^{87} + 11 q^{88} + 6 q^{89} - 42 q^{90} + 9 q^{92} - 5 q^{93} - 16 q^{94} + 24 q^{95} + q^{96} + 29 q^{97} + 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} - 23x^{8} + 19x^{7} + 181x^{6} - 109x^{5} - 579x^{4} + 231x^{3} + 608x^{2} - 204x - 12 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 19 \nu^{9} + 132 \nu^{8} + 513 \nu^{7} - 2594 \nu^{6} - 4759 \nu^{5} + 15220 \nu^{4} + 17781 \nu^{3} + \cdots + 4512 ) / 1636 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 39 \nu^{9} + 314 \nu^{8} + 235 \nu^{7} - 5712 \nu^{6} + 4525 \nu^{5} + 29562 \nu^{4} + \cdots - 16484 ) / 3272 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 39 \nu^{9} + 314 \nu^{8} + 235 \nu^{7} - 5712 \nu^{6} + 4525 \nu^{5} + 29562 \nu^{4} - 33721 \nu^{3} + \cdots - 124 ) / 3272 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 42 \nu^{9} + 55 \nu^{8} + 725 \nu^{7} - 740 \nu^{6} - 3244 \nu^{5} + 1161 \nu^{4} + 2603 \nu^{3} + \cdots - 7936 ) / 1636 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 101 \nu^{9} + 142 \nu^{8} + 1909 \nu^{7} - 2208 \nu^{6} - 11521 \nu^{5} + 7846 \nu^{4} + \cdots - 4516 ) / 3272 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 61 \nu^{9} + 187 \nu^{8} + 1238 \nu^{7} - 3334 \nu^{6} - 8003 \nu^{5} + 16381 \nu^{4} + \cdots - 1788 ) / 1636 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 91 \nu^{9} - 51 \nu^{8} - 2048 \nu^{7} + 1058 \nu^{6} + 15345 \nu^{5} - 6401 \nu^{4} - 44154 \nu^{3} + \cdots - 256 ) / 1636 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 191 \nu^{9} - 552 \nu^{8} - 3521 \nu^{7} + 10104 \nu^{6} + 18823 \nu^{5} - 53980 \nu^{4} + \cdots - 19612 ) / 3272 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{4} + \beta_{3} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} + \beta_{5} + \beta_{2} + 7\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{9} + \beta_{8} - \beta_{5} - 12\beta_{4} + 10\beta_{3} - \beta_{2} + \beta _1 + 36 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{9} - 13\beta_{7} - 4\beta_{6} + 16\beta_{5} - \beta_{3} + 13\beta_{2} + 59\beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -30\beta_{9} + 18\beta_{8} + 4\beta_{7} - 12\beta_{5} - 131\beta_{4} + 97\beta_{3} - 16\beta_{2} + 10\beta _1 + 297 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 18 \beta_{9} + 2 \beta_{8} - 139 \beta_{7} - 70 \beta_{6} + 195 \beta_{5} - 2 \beta_{4} - 22 \beta_{3} + \cdots + 79 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 358 \beta_{9} + 239 \beta_{8} + 80 \beta_{7} - 2 \beta_{6} - 123 \beta_{5} - 1388 \beta_{4} + \cdots + 2670 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 229 \beta_{9} + 58 \beta_{8} - 1423 \beta_{7} - 902 \beta_{6} + 2176 \beta_{5} - 26 \beta_{4} + \cdots + 397 \) 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
−3.24091
−2.19478
−1.97715
−1.56572
−0.0511860
0.386194
1.31428
2.41684
2.71108
3.20135
1.00000 −3.24091 1.00000 −1.45353 −3.24091 0 1.00000 7.50351 −1.45353
1.2 1.00000 −2.19478 1.00000 −2.75416 −2.19478 0 1.00000 1.81704 −2.75416
1.3 1.00000 −1.97715 1.00000 3.36998 −1.97715 0 1.00000 0.909140 3.36998
1.4 1.00000 −1.56572 1.00000 −2.99546 −1.56572 0 1.00000 −0.548522 −2.99546
1.5 1.00000 −0.0511860 1.00000 −0.949433 −0.0511860 0 1.00000 −2.99738 −0.949433
1.6 1.00000 0.386194 1.00000 4.13931 0.386194 0 1.00000 −2.85085 4.13931
1.7 1.00000 1.31428 1.00000 −0.106745 1.31428 0 1.00000 −1.27267 −0.106745
1.8 1.00000 2.41684 1.00000 2.64605 2.41684 0 1.00000 2.84113 2.64605
1.9 1.00000 2.71108 1.00000 0.432605 2.71108 0 1.00000 4.34997 0.432605
1.10 1.00000 3.20135 1.00000 −4.32862 3.20135 0 1.00000 7.24863 −4.32862
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(-1\)
\(41\) \(-1\)

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 4018.2.a.bu 10
7.b odd 2 1 4018.2.a.bt 10
7.d odd 6 2 574.2.e.h ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
574.2.e.h ✓ 20 7.d odd 6 2
4018.2.a.bt 10 7.b odd 2 1
4018.2.a.bu 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4018))\):

\( T_{3}^{10} - T_{3}^{9} - 23 T_{3}^{8} + 19 T_{3}^{7} + 181 T_{3}^{6} - 109 T_{3}^{5} - 579 T_{3}^{4} + \cdots - 12 \) Copy content Toggle raw display
\( T_{5}^{10} + 2 T_{5}^{9} - 35 T_{5}^{8} - 71 T_{5}^{7} + 373 T_{5}^{6} + 808 T_{5}^{5} - 1137 T_{5}^{4} + \cdots + 84 \) Copy content Toggle raw display
\( T_{11}^{10} - 11 T_{11}^{9} - 6 T_{11}^{8} + 415 T_{11}^{7} - 845 T_{11}^{6} - 4429 T_{11}^{5} + \cdots - 2268 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{10} \) Copy content Toggle raw display
$3$ \( T^{10} - T^{9} + \cdots - 12 \) Copy content Toggle raw display
$5$ \( T^{10} + 2 T^{9} + \cdots + 84 \) Copy content Toggle raw display
$7$ \( T^{10} \) Copy content Toggle raw display
$11$ \( T^{10} - 11 T^{9} + \cdots - 2268 \) Copy content Toggle raw display
$13$ \( T^{10} - 4 T^{9} + \cdots - 2041200 \) Copy content Toggle raw display
$17$ \( T^{10} - 5 T^{9} + \cdots + 16128 \) Copy content Toggle raw display
$19$ \( T^{10} + T^{9} + \cdots + 31352 \) Copy content Toggle raw display
$23$ \( T^{10} - 9 T^{9} + \cdots + 112128 \) Copy content Toggle raw display
$29$ \( T^{10} - 23 T^{9} + \cdots - 20160 \) Copy content Toggle raw display
$31$ \( T^{10} + 5 T^{9} + \cdots - 768 \) Copy content Toggle raw display
$37$ \( T^{10} - 16 T^{9} + \cdots - 1792512 \) Copy content Toggle raw display
$41$ \( (T - 1)^{10} \) Copy content Toggle raw display
$43$ \( T^{10} - 20 T^{9} + \cdots - 64196608 \) Copy content Toggle raw display
$47$ \( T^{10} + 16 T^{9} + \cdots + 60345936 \) Copy content Toggle raw display
$53$ \( T^{10} - 26 T^{9} + \cdots - 7449408 \) Copy content Toggle raw display
$59$ \( T^{10} + 10 T^{9} + \cdots + 1402224 \) Copy content Toggle raw display
$61$ \( T^{10} - 285 T^{8} + \cdots - 29873160 \) Copy content Toggle raw display
$67$ \( T^{10} - 7 T^{9} + \cdots - 14672 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 1145259045 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots - 493665088 \) Copy content Toggle raw display
$79$ \( T^{10} + T^{9} + \cdots + 65105604 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots - 309985872 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots - 186610752 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots - 1048305664 \) Copy content Toggle raw display
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