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Properties

Label 6018.2.a.y
Level $6018$
Weight $2$
Character orbit 6018.a
Self dual yes
Analytic conductor $48.054$
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] = [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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} - 33x^{8} + 53x^{7} + 356x^{6} - 433x^{5} - 1296x^{4} + 1135x^{3} + 930x^{2} - 186x - 104 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
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.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} - 33x^{8} + 53x^{7} + 356x^{6} - 433x^{5} - 1296x^{4} + 1135x^{3} + 930x^{2} - 186x - 104 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 43951 \nu^{9} - 76913 \nu^{8} + 988626 \nu^{7} + 8141512 \nu^{6} - 20812677 \nu^{5} + \cdots - 232805875 ) / 51474769 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 214570 \nu^{9} - 1275880 \nu^{8} - 6472015 \nu^{7} + 39620633 \nu^{6} + 61025321 \nu^{5} + \cdots - 459680139 ) / 51474769 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 249465 \nu^{9} + 1418601 \nu^{8} + 4174377 \nu^{7} - 28056148 \nu^{6} - 13181246 \nu^{5} + \cdots - 47082704 ) / 51474769 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 414280 \nu^{9} - 956844 \nu^{8} - 14645292 \nu^{7} + 33222269 \nu^{6} + 160178219 \nu^{5} + \cdots - 154245468 ) / 51474769 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 417990 \nu^{9} - 662240 \nu^{8} - 15536862 \nu^{7} + 20221183 \nu^{6} + 181938909 \nu^{5} + \cdots - 87407927 ) / 51474769 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 743180 \nu^{9} - 1756250 \nu^{8} - 21953329 \nu^{7} + 43199010 \nu^{6} + 209998564 \nu^{5} + \cdots - 191137786 ) / 51474769 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1112920 \nu^{9} + 3475010 \nu^{8} + 31167323 \nu^{7} - 82936201 \nu^{6} - 285693609 \nu^{5} + \cdots + 114968772 ) / 51474769 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1849040 \nu^{9} - 4265675 \nu^{8} - 59471294 \nu^{7} + 114940606 \nu^{6} + 618418870 \nu^{5} + \cdots - 338810025 ) / 51474769 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{8} - 2\beta_{5} + 2\beta_{4} + \beta_{3} + 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{9} - 2\beta_{8} - 3\beta_{7} - 3\beta_{6} - 4\beta_{5} + 4\beta_{4} + \beta_{3} + 10\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -21\beta_{8} - 5\beta_{7} - 2\beta_{6} - 31\beta_{5} + 36\beta_{4} + 14\beta_{3} + 2\beta _1 + 90 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 38 \beta_{9} - 46 \beta_{8} - 59 \beta_{7} - 60 \beta_{6} - 86 \beta_{5} + 91 \beta_{4} + 26 \beta_{3} + \cdots + 67 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 14 \beta_{9} - 349 \beta_{8} - 117 \beta_{7} - 70 \beta_{6} - 475 \beta_{5} + 585 \beta_{4} + \cdots + 1256 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 596 \beta_{9} - 866 \beta_{8} - 952 \beta_{7} - 1005 \beta_{6} - 1522 \beta_{5} + 1682 \beta_{4} + \cdots + 1552 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 501 \beta_{9} - 5491 \beta_{8} - 2178 \beta_{7} - 1690 \beta_{6} - 7398 \beta_{5} + 9322 \beta_{4} + \cdots + 18164 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 9007 \beta_{9} - 15231 \beta_{8} - 14597 \beta_{7} - 16198 \beta_{6} - 25570 \beta_{5} + 29253 \beta_{4} + \cdots + 30910 \) 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
4.05993
3.94189
2.38622
1.40255
0.390474
−0.313984
−0.556815
−2.34533
−3.46791
−3.49703
−1.00000 1.00000 1.00000 −4.05993 −1.00000 4.32251 −1.00000 1.00000 4.05993
1.2 −1.00000 1.00000 1.00000 −3.94189 −1.00000 −4.15692 −1.00000 1.00000 3.94189
1.3 −1.00000 1.00000 1.00000 −2.38622 −1.00000 0.0691424 −1.00000 1.00000 2.38622
1.4 −1.00000 1.00000 1.00000 −1.40255 −1.00000 0.699671 −1.00000 1.00000 1.40255
1.5 −1.00000 1.00000 1.00000 −0.390474 −1.00000 −3.35881 −1.00000 1.00000 0.390474
1.6 −1.00000 1.00000 1.00000 0.313984 −1.00000 2.10092 −1.00000 1.00000 −0.313984
1.7 −1.00000 1.00000 1.00000 0.556815 −1.00000 0.727556 −1.00000 1.00000 −0.556815
1.8 −1.00000 1.00000 1.00000 2.34533 −1.00000 −0.244087 −1.00000 1.00000 −2.34533
1.9 −1.00000 1.00000 1.00000 3.46791 −1.00000 −1.61634 −1.00000 1.00000 −3.46791
1.10 −1.00000 1.00000 1.00000 3.49703 −1.00000 −4.54363 −1.00000 1.00000 −3.49703
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

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

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.y ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6018.2.a.y ✓ 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6018))\):

\( T_{5}^{10} + 2 T_{5}^{9} - 33 T_{5}^{8} - 53 T_{5}^{7} + 356 T_{5}^{6} + 433 T_{5}^{5} - 1296 T_{5}^{4} + \cdots - 104 \) Copy content Toggle raw display
\( T_{7}^{10} + 6 T_{7}^{9} - 20 T_{7}^{8} - 153 T_{7}^{7} + 19 T_{7}^{6} + 820 T_{7}^{5} + 85 T_{7}^{4} + \cdots - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{10} \) Copy content Toggle raw display
$3$ \( (T - 1)^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + 2 T^{9} + \cdots - 104 \) Copy content Toggle raw display
$7$ \( T^{10} + 6 T^{9} + \cdots - 8 \) Copy content Toggle raw display
$11$ \( T^{10} + 3 T^{9} + \cdots + 208 \) Copy content Toggle raw display
$13$ \( T^{10} + 10 T^{9} + \cdots - 10528 \) Copy content Toggle raw display
$17$ \( (T + 1)^{10} \) Copy content Toggle raw display
$19$ \( T^{10} - 8 T^{9} + \cdots - 1642112 \) Copy content Toggle raw display
$23$ \( T^{10} + 9 T^{9} + \cdots - 8 \) Copy content Toggle raw display
$29$ \( T^{10} + 24 T^{9} + \cdots + 3291832 \) Copy content Toggle raw display
$31$ \( T^{10} + 7 T^{9} + \cdots + 4886272 \) Copy content Toggle raw display
$37$ \( T^{10} + 4 T^{9} + \cdots + 8617616 \) Copy content Toggle raw display
$41$ \( T^{10} + 9 T^{9} + \cdots + 4078472 \) Copy content Toggle raw display
$43$ \( T^{10} + 11 T^{9} + \cdots - 10479872 \) Copy content Toggle raw display
$47$ \( T^{10} + 18 T^{9} + \cdots + 6313984 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots - 144005728 \) Copy content Toggle raw display
$59$ \( (T - 1)^{10} \) Copy content Toggle raw display
$61$ \( T^{10} + 25 T^{9} + \cdots + 40422224 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 164788352 \) Copy content Toggle raw display
$71$ \( T^{10} + 30 T^{9} + \cdots - 16426144 \) Copy content Toggle raw display
$73$ \( T^{10} + 11 T^{9} + \cdots - 90242696 \) Copy content Toggle raw display
$79$ \( T^{10} - 3 T^{9} + \cdots - 231752 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 144187904 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 161541328 \) Copy content Toggle raw display
$97$ \( T^{10} + 10 T^{9} + \cdots + 12864448 \) Copy content Toggle raw display
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