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Properties

Label 6045.2.a.bb
Level $6045$
Weight $2$
Character orbit 6045.a
Self dual yes
Analytic conductor $48.270$
Analytic rank $0$
Dimension $13$
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: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 3 x^{12} - 16 x^{11} + 49 x^{10} + 89 x^{9} - 282 x^{8} - 201 x^{7} + 683 x^{6} + 167 x^{5} + \cdots - 6 \) 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 \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{13} - 3 x^{12} - 16 x^{11} + 49 x^{10} + 89 x^{9} - 282 x^{8} - 201 x^{7} + 683 x^{6} + 167 x^{5} + \cdots - 6 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu - 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 9 \nu^{12} - 30 \nu^{11} - 134 \nu^{10} + 473 \nu^{9} + 656 \nu^{8} - 2554 \nu^{7} - 1097 \nu^{6} + \cdots - 200 ) / 38 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 9 \nu^{12} - 11 \nu^{11} - 172 \nu^{10} + 150 \nu^{9} + 1226 \nu^{8} - 597 \nu^{7} - 3947 \nu^{6} + \cdots + 47 ) / 19 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 13 \nu^{12} + 56 \nu^{11} + 164 \nu^{10} - 907 \nu^{9} - 462 \nu^{8} + 5112 \nu^{7} - 1143 \nu^{6} + \cdots + 120 ) / 38 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 13 \nu^{12} + 37 \nu^{11} + 202 \nu^{10} - 584 \nu^{9} - 1051 \nu^{8} + 3174 \nu^{7} + 1935 \nu^{6} + \cdots + 63 ) / 19 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 14 \nu^{12} + 15 \nu^{11} + 276 \nu^{10} - 208 \nu^{9} - 2038 \nu^{8} + 859 \nu^{7} + 6847 \nu^{6} + \cdots - 204 ) / 19 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 25 \nu^{12} - 20 \nu^{11} - 520 \nu^{10} + 271 \nu^{9} + 4060 \nu^{8} - 1006 \nu^{7} - 14443 \nu^{6} + \cdots + 348 ) / 38 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 37 \nu^{12} + 60 \nu^{11} + 686 \nu^{10} - 889 \nu^{9} - 4732 \nu^{8} + 4310 \nu^{7} + 14753 \nu^{6} + \cdots - 322 ) / 38 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 37 \nu^{12} + 60 \nu^{11} + 686 \nu^{10} - 889 \nu^{9} - 4732 \nu^{8} + 4272 \nu^{7} + 14791 \nu^{6} + \cdots - 398 ) / 38 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 45 \nu^{12} + 112 \nu^{11} + 746 \nu^{10} - 1757 \nu^{9} - 4344 \nu^{8} + 9388 \nu^{7} + 10273 \nu^{6} + \cdots + 354 ) / 38 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{11} + \beta_{8} - \beta_{4} + 7\beta_{2} + \beta _1 + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{12} + \beta_{11} + \beta_{9} - \beta_{7} - \beta_{6} + \beta_{5} + 9\beta_{3} + 2\beta_{2} + 30\beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -10\beta_{11} + \beta_{9} + 12\beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} - 10\beta_{4} + 46\beta_{2} + 15\beta _1 + 98 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 12 \beta_{12} + 11 \beta_{11} + \beta_{10} + 13 \beta_{9} + 2 \beta_{8} - 13 \beta_{7} - 11 \beta_{6} + \cdots + 98 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 3 \beta_{12} - 80 \beta_{11} + \beta_{10} + 16 \beta_{9} + 108 \beta_{8} - 17 \beta_{7} + 12 \beta_{6} + \cdots + 635 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 109 \beta_{12} + 88 \beta_{11} + 16 \beta_{10} + 126 \beta_{9} + 38 \beta_{8} - 128 \beta_{7} + \cdots + 792 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 54 \beta_{12} - 594 \beta_{11} + 21 \beta_{10} + 177 \beta_{9} + 875 \beta_{8} - 198 \beta_{7} + \cdots + 4247 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 896 \beta_{12} + 618 \beta_{11} + 181 \beta_{10} + 1088 \beta_{9} + 468 \beta_{8} - 1134 \beta_{7} + \cdots + 6100 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 649 \beta_{12} - 4267 \beta_{11} + 286 \beta_{10} + 1685 \beta_{9} + 6741 \beta_{8} - 1973 \beta_{7} + \cdots + 28996 \) 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
−2.56981
−2.06781
−1.99690
−0.892553
−0.559786
−0.444168
0.0818008
1.11779
1.20942
1.30813
2.44349
2.63964
2.73075
−2.56981 1.00000 4.60391 1.00000 −2.56981 −0.0753473 −6.69154 1.00000 −2.56981
1.2 −2.06781 1.00000 2.27583 1.00000 −2.06781 0.896679 −0.570357 1.00000 −2.06781
1.3 −1.99690 1.00000 1.98759 1.00000 −1.99690 −2.28275 0.0247719 1.00000 −1.99690
1.4 −0.892553 1.00000 −1.20335 1.00000 −0.892553 3.34043 2.85916 1.00000 −0.892553
1.5 −0.559786 1.00000 −1.68664 1.00000 −0.559786 0.0223322 2.06373 1.00000 −0.559786
1.6 −0.444168 1.00000 −1.80271 1.00000 −0.444168 −2.85747 1.68904 1.00000 −0.444168
1.7 0.0818008 1.00000 −1.99331 1.00000 0.0818008 2.84360 −0.326656 1.00000 0.0818008
1.8 1.11779 1.00000 −0.750553 1.00000 1.11779 −2.48876 −3.07453 1.00000 1.11779
1.9 1.20942 1.00000 −0.537299 1.00000 1.20942 −3.86285 −3.06866 1.00000 1.20942
1.10 1.30813 1.00000 −0.288796 1.00000 1.30813 4.16657 −2.99404 1.00000 1.30813
1.11 2.44349 1.00000 3.97062 1.00000 2.44349 4.25786 4.81519 1.00000 2.44349
1.12 2.63964 1.00000 4.96770 1.00000 2.63964 2.02015 7.83365 1.00000 2.63964
1.13 2.73075 1.00000 5.45701 1.00000 2.73075 −2.98044 9.44024 1.00000 2.73075
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
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.bb ✓ 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6045.2.a.bb ✓ 13 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6045))\):

\( T_{2}^{13} - 3 T_{2}^{12} - 16 T_{2}^{11} + 49 T_{2}^{10} + 89 T_{2}^{9} - 282 T_{2}^{8} - 201 T_{2}^{7} + \cdots - 6 \) Copy content Toggle raw display
\( T_{7}^{13} - 3 T_{7}^{12} - 47 T_{7}^{11} + 122 T_{7}^{10} + 864 T_{7}^{9} - 1849 T_{7}^{8} - 7756 T_{7}^{7} + \cdots - 96 \) Copy content Toggle raw display
\( T_{11}^{13} - 4 T_{11}^{12} - 52 T_{11}^{11} + 242 T_{11}^{10} + 749 T_{11}^{9} - 4536 T_{11}^{8} + \cdots + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{13} - 3 T^{12} + \cdots - 6 \) Copy content Toggle raw display
$3$ \( (T - 1)^{13} \) Copy content Toggle raw display
$5$ \( (T - 1)^{13} \) Copy content Toggle raw display
$7$ \( T^{13} - 3 T^{12} + \cdots - 96 \) Copy content Toggle raw display
$11$ \( T^{13} - 4 T^{12} + \cdots + 16 \) Copy content Toggle raw display
$13$ \( (T - 1)^{13} \) Copy content Toggle raw display
$17$ \( T^{13} - 5 T^{12} + \cdots - 9443488 \) Copy content Toggle raw display
$19$ \( T^{13} - 129 T^{11} + \cdots - 3218808 \) Copy content Toggle raw display
$23$ \( T^{13} - 15 T^{12} + \cdots - 4896 \) Copy content Toggle raw display
$29$ \( T^{13} - 11 T^{12} + \cdots - 20128386 \) Copy content Toggle raw display
$31$ \( (T + 1)^{13} \) Copy content Toggle raw display
$37$ \( T^{13} - 7 T^{12} + \cdots + 28659552 \) Copy content Toggle raw display
$41$ \( T^{13} + \cdots - 13972154298 \) Copy content Toggle raw display
$43$ \( T^{13} + 2 T^{12} + \cdots + 133616 \) Copy content Toggle raw display
$47$ \( T^{13} + \cdots - 255521824 \) Copy content Toggle raw display
$53$ \( T^{13} + \cdots + 6090030096 \) Copy content Toggle raw display
$59$ \( T^{13} - 28 T^{12} + \cdots + 1833072 \) Copy content Toggle raw display
$61$ \( T^{13} + \cdots + 850012128 \) Copy content Toggle raw display
$67$ \( T^{13} + \cdots + 12467501232 \) Copy content Toggle raw display
$71$ \( T^{13} - 9 T^{12} + \cdots - 90287616 \) Copy content Toggle raw display
$73$ \( T^{13} + \cdots - 1507745952 \) Copy content Toggle raw display
$79$ \( T^{13} + \cdots - 458681984 \) Copy content Toggle raw display
$83$ \( T^{13} + \cdots - 14698851632 \) Copy content Toggle raw display
$89$ \( T^{13} + \cdots - 2780059552 \) Copy content Toggle raw display
$97$ \( T^{13} + 17 T^{12} + \cdots + 2142432 \) Copy content Toggle raw display
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