↑

Properties

Label 6018.2.a.u
Level $6018$
Weight $2$
Character orbit 6018.a
Self dual yes
Analytic conductor $48.054$
Analytic rank $1$
Dimension $9$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6018,2,Mod(1,6018)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6018, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("6018.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6018 = 2 \cdot 3 \cdot 17 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6018.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: \(48.0539719364\)
Analytic rank: \(1\)
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: \( x^{9} - 4x^{8} - 16x^{7} + 37x^{6} + 97x^{5} - 72x^{4} - 182x^{3} + 24x^{2} + 70x - 19 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} - \beta_{6} q^{5} + q^{6} + (\beta_{7} + \beta_1 - 1) q^{7} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - q^{3} + q^{4} - \beta_{6} q^{5} + q^{6} + (\beta_{7} + \beta_1 - 1) q^{7} - q^{8} + q^{9} + \beta_{6} q^{10} - \beta_{8} q^{11} - q^{12} + (\beta_{5} + \beta_{2}) q^{13} + ( - \beta_{7} - \beta_1 + 1) q^{14} + \beta_{6} q^{15} + q^{16} + q^{17} - q^{18} + ( - \beta_{8} - \beta_{6} + \cdots - \beta_1) q^{19}+ \cdots - \beta_{8} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 9 q^{2} - 9 q^{3} + 9 q^{4} + 2 q^{5} + 9 q^{6} - 5 q^{7} - 9 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 9 q^{2} - 9 q^{3} + 9 q^{4} + 2 q^{5} + 9 q^{6} - 5 q^{7} - 9 q^{8} + 9 q^{9} - 2 q^{10} - q^{11} - 9 q^{12} - 4 q^{13} + 5 q^{14} - 2 q^{15} + 9 q^{16} + 9 q^{17} - 9 q^{18} - 7 q^{19} + 2 q^{20} + 5 q^{21} + q^{22} - 8 q^{23} + 9 q^{24} + 5 q^{25} + 4 q^{26} - 9 q^{27} - 5 q^{28} + 6 q^{29} + 2 q^{30} - 17 q^{31} - 9 q^{32} + q^{33} - 9 q^{34} + 10 q^{35} + 9 q^{36} + 2 q^{37} + 7 q^{38} + 4 q^{39} - 2 q^{40} + 14 q^{41} - 5 q^{42} - 27 q^{43} - q^{44} + 2 q^{45} + 8 q^{46} - 18 q^{47} - 9 q^{48} + 18 q^{49} - 5 q^{50} - 9 q^{51} - 4 q^{52} + 4 q^{53} + 9 q^{54} - 27 q^{55} + 5 q^{56} + 7 q^{57} - 6 q^{58} + 9 q^{59} - 2 q^{60} + 5 q^{61} + 17 q^{62} - 5 q^{63} + 9 q^{64} + 2 q^{65} - q^{66} - 22 q^{67} + 9 q^{68} + 8 q^{69} - 10 q^{70} + 16 q^{71} - 9 q^{72} - 12 q^{73} - 2 q^{74} - 5 q^{75} - 7 q^{76} + 6 q^{77} - 4 q^{78} - 9 q^{79} + 2 q^{80} + 9 q^{81} - 14 q^{82} + 10 q^{83} + 5 q^{84} + 2 q^{85} + 27 q^{86} - 6 q^{87} + q^{88} + 15 q^{89} - 2 q^{90} + 3 q^{91} - 8 q^{92} + 17 q^{93} + 18 q^{94} - 9 q^{95} + 9 q^{96} - 33 q^{97} - 18 q^{98} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 4x^{8} - 16x^{7} + 37x^{6} + 97x^{5} - 72x^{4} - 182x^{3} + 24x^{2} + 70x - 19 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 7 \nu^{8} - 595 \nu^{7} + 3093 \nu^{6} + 7114 \nu^{5} - 28461 \nu^{4} - 34541 \nu^{3} + \cdots - 30285 ) / 1472 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 77 \nu^{8} - 79 \nu^{7} - 2375 \nu^{6} + 498 \nu^{5} + 17199 \nu^{4} + 6063 \nu^{3} - 33031 \nu^{2} + \cdots + 10767 ) / 736 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 163 \nu^{8} - 865 \nu^{7} - 1577 \nu^{6} + 8462 \nu^{5} + 6433 \nu^{4} - 23583 \nu^{3} - 7401 \nu^{2} + \cdots - 3455 ) / 1472 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 211 \nu^{8} + 271 \nu^{7} - 8697 \nu^{6} - 5202 \nu^{5} + 68273 \nu^{4} + 47985 \nu^{3} + \cdots + 49233 ) / 1472 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 205 \nu^{8} + 975 \nu^{7} + 2471 \nu^{6} - 9202 \nu^{5} - 11599 \nu^{4} + 20945 \nu^{3} + \cdots + 2065 ) / 736 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 705 \nu^{8} - 3371 \nu^{7} - 8067 \nu^{6} + 29402 \nu^{5} + 39979 \nu^{4} - 56341 \nu^{3} + \cdots + 3723 ) / 1472 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 511 \nu^{8} + 2197 \nu^{7} + 7197 \nu^{6} - 19430 \nu^{5} - 40405 \nu^{4} + 33899 \nu^{3} + \cdots - 13109 ) / 736 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} + \beta_{7} + \beta_{4} + \beta_{3} + 3\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{8} + 4\beta_{7} - \beta_{6} - 2\beta_{5} + 3\beta_{4} + 5\beta_{3} - 3\beta_{2} + 17\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 24\beta_{8} + 23\beta_{7} - 7\beta_{6} - 10\beta_{5} + 21\beta_{4} + 26\beta_{3} - 18\beta_{2} + 79\beta _1 + 22 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 120 \beta_{8} + 113 \beta_{7} - 48 \beta_{6} - 63 \beta_{5} + 94 \beta_{4} + 133 \beta_{3} - 112 \beta_{2} + \cdots + 64 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 635 \beta_{8} + 592 \beta_{7} - 268 \beta_{6} - 326 \beta_{5} + 506 \beta_{4} + 674 \beta_{3} + \cdots + 328 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 3264 \beta_{8} + 3020 \beta_{7} - 1482 \beta_{6} - 1747 \beta_{5} + 2529 \beta_{4} + 3459 \beta_{3} + \cdots + 1431 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 16912 \beta_{8} + 15604 \beta_{7} - 7834 \beta_{6} - 9049 \beta_{5} + 13096 \beta_{4} + 17715 \beta_{3} + \cdots + 7168 \) 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
−1.12576
0.443000
2.85528
0.342793
−2.14979
−1.32747
1.53430
−1.72522
5.15287
−1.00000 −1.00000 1.00000 −3.14251 1.00000 −1.86106 −1.00000 1.00000 3.14251
1.2 −1.00000 −1.00000 1.00000 −2.03963 1.00000 −4.77480 −1.00000 1.00000 2.03963
1.3 −1.00000 −1.00000 1.00000 −1.56975 1.00000 1.56796 −1.00000 1.00000 1.56975
1.4 −1.00000 −1.00000 1.00000 −1.11819 1.00000 −1.49540 −1.00000 1.00000 1.11819
1.5 −1.00000 −1.00000 1.00000 −1.03642 1.00000 2.58814 −1.00000 1.00000 1.03642
1.6 −1.00000 −1.00000 1.00000 2.06239 1.00000 1.71252 −1.00000 1.00000 −2.06239
1.7 −1.00000 −1.00000 1.00000 2.21366 1.00000 −2.01100 −1.00000 1.00000 −2.21366
1.8 −1.00000 −1.00000 1.00000 3.17259 1.00000 −4.61193 −1.00000 1.00000 −3.17259
1.9 −1.00000 −1.00000 1.00000 3.45787 1.00000 3.88559 −1.00000 1.00000 −3.45787
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(17\) \(-1\)
\(59\) \(-1\)

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 6018.2.a.u ✓ 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6018.2.a.u ✓ 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6018))\):

\( T_{5}^{9} - 2T_{5}^{8} - 23T_{5}^{7} + 33T_{5}^{6} + 192T_{5}^{5} - 137T_{5}^{4} - 720T_{5}^{3} + 3T_{5}^{2} + 1030T_{5} + 584 \) Copy content Toggle raw display
\( T_{7}^{9} + 5T_{7}^{8} - 28T_{7}^{7} - 135T_{7}^{6} + 250T_{7}^{5} + 1081T_{7}^{4} - 815T_{7}^{3} - 3274T_{7}^{2} + 879T_{7} + 3328 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{9} \) Copy content Toggle raw display
$3$ \( (T + 1)^{9} \) Copy content Toggle raw display
$5$ \( T^{9} - 2 T^{8} + \cdots + 584 \) Copy content Toggle raw display
$7$ \( T^{9} + 5 T^{8} + \cdots + 3328 \) Copy content Toggle raw display
$11$ \( T^{9} + T^{8} + \cdots + 1112 \) Copy content Toggle raw display
$13$ \( T^{9} + 4 T^{8} + \cdots - 288 \) Copy content Toggle raw display
$17$ \( (T - 1)^{9} \) Copy content Toggle raw display
$19$ \( T^{9} + 7 T^{8} + \cdots + 10592 \) Copy content Toggle raw display
$23$ \( T^{9} + 8 T^{8} + \cdots + 145576 \) Copy content Toggle raw display
$29$ \( T^{9} - 6 T^{8} + \cdots + 18388 \) Copy content Toggle raw display
$31$ \( T^{9} + 17 T^{8} + \cdots + 507904 \) Copy content Toggle raw display
$37$ \( T^{9} - 2 T^{8} + \cdots + 16732412 \) Copy content Toggle raw display
$41$ \( T^{9} - 14 T^{8} + \cdots + 18646 \) Copy content Toggle raw display
$43$ \( T^{9} + 27 T^{8} + \cdots - 5696 \) Copy content Toggle raw display
$47$ \( T^{9} + 18 T^{8} + \cdots - 9216 \) Copy content Toggle raw display
$53$ \( T^{9} - 4 T^{8} + \cdots - 15056 \) Copy content Toggle raw display
$59$ \( (T - 1)^{9} \) Copy content Toggle raw display
$61$ \( T^{9} - 5 T^{8} + \cdots + 23036 \) Copy content Toggle raw display
$67$ \( T^{9} + 22 T^{8} + \cdots - 141184 \) Copy content Toggle raw display
$71$ \( T^{9} - 16 T^{8} + \cdots - 19456 \) Copy content Toggle raw display
$73$ \( T^{9} + 12 T^{8} + \cdots - 802 \) Copy content Toggle raw display
$79$ \( T^{9} + 9 T^{8} + \cdots + 1038848 \) Copy content Toggle raw display
$83$ \( T^{9} - 10 T^{8} + \cdots - 12184864 \) Copy content Toggle raw display
$89$ \( T^{9} - 15 T^{8} + \cdots + 7890034 \) Copy content Toggle raw display
$97$ \( T^{9} + 33 T^{8} + \cdots + 233424 \) Copy content Toggle raw display
show more
show less