Properties

Label 6045.2.a.s
Level $6045$
Weight $2$
Character orbit 6045.a
Self dual yes
Analytic conductor $48.270$
Analytic rank $1$
Dimension $5$
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] = [6045,2,Mod(1,6045)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6045, 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("6045.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6045 = 3 \cdot 5 \cdot 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6045.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: \(48.2695680219\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.230224.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 4x^{3} + 6x^{2} + 3x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 2x^{4} - 4x^{3} + 6x^{2} + 3x - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 3\nu + 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 2\nu^{3} - 3\nu^{2} + 4\nu + 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 4\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 2\beta_{3} + 5\beta_{2} + 7\beta _1 + 7 \) 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
0.424945
1.49592
−0.757366
2.49889
−1.66240
−2.24437 1.00000 3.03718 1.00000 −2.24437 3.68511 −2.32782 1.00000 −2.24437
1.2 −1.25814 1.00000 −0.417093 1.00000 −1.25814 −1.35765 3.04103 1.00000 −1.25814
1.3 −0.669031 1.00000 −1.55240 1.00000 −0.669031 4.35950 2.37666 1.00000 −0.669031
1.4 1.74558 1.00000 1.04704 1.00000 1.74558 −3.12693 −1.66347 1.00000 1.74558
1.5 2.42596 1.00000 3.88527 1.00000 2.42596 −4.56003 4.57359 1.00000 2.42596
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)
\(13\) \(1\)
\(31\) \(1\)

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 6045.2.a.s 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6045.2.a.s 5 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6045))\):

\( T_{2}^{5} - 8T_{2}^{3} - 2T_{2}^{2} + 14T_{2} + 8 \) Copy content Toggle raw display
\( T_{7}^{5} + T_{7}^{4} - 32T_{7}^{3} - 34T_{7}^{2} + 241T_{7} + 311 \) Copy content Toggle raw display
\( T_{11}^{5} + 9T_{11}^{4} + 12T_{11}^{3} - 48T_{11}^{2} - 64T_{11} - 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} - 8 T^{3} - 2 T^{2} + 14 T + 8 \) Copy content Toggle raw display
$3$ \( (T - 1)^{5} \) Copy content Toggle raw display
$5$ \( (T - 1)^{5} \) Copy content Toggle raw display
$7$ \( T^{5} + T^{4} - 32 T^{3} - 34 T^{2} + \cdots + 311 \) Copy content Toggle raw display
$11$ \( T^{5} + 9 T^{4} + 12 T^{3} - 48 T^{2} + \cdots - 16 \) Copy content Toggle raw display
$13$ \( (T + 1)^{5} \) Copy content Toggle raw display
$17$ \( T^{5} + 17 T^{4} + 78 T^{3} + \cdots - 3112 \) Copy content Toggle raw display
$19$ \( T^{5} + 2 T^{4} - 36 T^{3} - 10 T^{2} + \cdots - 244 \) Copy content Toggle raw display
$23$ \( T^{5} + 17 T^{4} + 48 T^{3} + \cdots - 1072 \) Copy content Toggle raw display
$29$ \( T^{5} + 6 T^{4} - 48 T^{2} - 61 T + 8 \) Copy content Toggle raw display
$31$ \( (T + 1)^{5} \) Copy content Toggle raw display
$37$ \( T^{5} + 3 T^{4} - 110 T^{3} + \cdots + 5912 \) Copy content Toggle raw display
$41$ \( T^{5} - 9 T^{4} - 14 T^{3} + 144 T^{2} + \cdots - 61 \) Copy content Toggle raw display
$43$ \( T^{5} - 8 T^{4} - 140 T^{3} + \cdots - 3326 \) Copy content Toggle raw display
$47$ \( T^{5} + 2 T^{4} - 304 T^{3} + \cdots - 49744 \) Copy content Toggle raw display
$53$ \( T^{5} + 51 T^{4} + 1000 T^{3} + \cdots + 50494 \) Copy content Toggle raw display
$59$ \( T^{5} + 6 T^{4} - 114 T^{3} + \cdots + 2132 \) Copy content Toggle raw display
$61$ \( T^{5} + 23 T^{4} + 170 T^{3} + \cdots - 118 \) Copy content Toggle raw display
$67$ \( T^{5} - 150 T^{3} - 850 T^{2} + \cdots + 2476 \) Copy content Toggle raw display
$71$ \( T^{5} - 9 T^{4} - 168 T^{3} + \cdots - 20992 \) Copy content Toggle raw display
$73$ \( T^{5} + 16 T^{4} - 142 T^{3} + \cdots + 80116 \) Copy content Toggle raw display
$79$ \( T^{5} - 21 T^{4} + 58 T^{3} + \cdots + 18754 \) Copy content Toggle raw display
$83$ \( T^{5} + 44 T^{4} + 660 T^{3} + \cdots - 47458 \) Copy content Toggle raw display
$89$ \( T^{5} + 65 T^{4} + 1626 T^{3} + \cdots + 176392 \) Copy content Toggle raw display
$97$ \( T^{5} - 11 T^{4} - 258 T^{3} + \cdots - 21523 \) Copy content Toggle raw display
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