Properties

Label 6030.2.a.bu
Level 6030
Weight 2
Character orbit 6030.a
Self dual yes
Analytic conductor 48.150
Analytic rank 0
Dimension 4
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 6030 = 2 \cdot 3^{2} \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6030.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.1497924188\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.70292.1
Defining polynomial: \(x^{4} - x^{3} - 8 x^{2} + 10 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 2010)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + q^{5} + \beta_{1} q^{7} + q^{8} +O(q^{10})\) \( q + q^{2} + q^{4} + q^{5} + \beta_{1} q^{7} + q^{8} + q^{10} + ( 1 + \beta_{1} - \beta_{2} ) q^{11} + ( 1 - \beta_{2} ) q^{13} + \beta_{1} q^{14} + q^{16} + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{17} + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{19} + q^{20} + ( 1 + \beta_{1} - \beta_{2} ) q^{22} + ( 3 + \beta_{1} - \beta_{3} ) q^{23} + q^{25} + ( 1 - \beta_{2} ) q^{26} + \beta_{1} q^{28} + ( -1 + \beta_{1} + \beta_{3} ) q^{29} + ( 3 - \beta_{2} ) q^{31} + q^{32} + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{34} + \beta_{1} q^{35} + ( 1 - \beta_{3} ) q^{37} + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{38} + q^{40} + ( -\beta_{1} + \beta_{2} - \beta_{3} ) q^{41} + ( -2 - 2 \beta_{1} + 2 \beta_{2} ) q^{43} + ( 1 + \beta_{1} - \beta_{2} ) q^{44} + ( 3 + \beta_{1} - \beta_{3} ) q^{46} + ( 5 - \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{47} + ( 4 + \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{49} + q^{50} + ( 1 - \beta_{2} ) q^{52} + ( 2 + 2 \beta_{1} ) q^{53} + ( 1 + \beta_{1} - \beta_{2} ) q^{55} + \beta_{1} q^{56} + ( -1 + \beta_{1} + \beta_{3} ) q^{58} + ( -\beta_{1} + \beta_{2} - \beta_{3} ) q^{59} + ( 2 + \beta_{1} ) q^{61} + ( 3 - \beta_{2} ) q^{62} + q^{64} + ( 1 - \beta_{2} ) q^{65} - q^{67} + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{68} + \beta_{1} q^{70} + ( 2 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{71} + ( -4 + 2 \beta_{3} ) q^{73} + ( 1 - \beta_{3} ) q^{74} + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{76} + ( 7 + 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} ) q^{77} + ( 3 + \beta_{2} - 2 \beta_{3} ) q^{79} + q^{80} + ( -\beta_{1} + \beta_{2} - \beta_{3} ) q^{82} + ( 5 - 2 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} ) q^{83} + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{85} + ( -2 - 2 \beta_{1} + 2 \beta_{2} ) q^{86} + ( 1 + \beta_{1} - \beta_{2} ) q^{88} + ( 3 - 3 \beta_{3} ) q^{89} + ( -4 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{91} + ( 3 + \beta_{1} - \beta_{3} ) q^{92} + ( 5 - \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{94} + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{95} + ( -1 - \beta_{1} + \beta_{2} ) q^{97} + ( 4 + \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{2} + 4q^{4} + 4q^{5} + q^{7} + 4q^{8} + O(q^{10}) \) \( 4q + 4q^{2} + 4q^{4} + 4q^{5} + q^{7} + 4q^{8} + 4q^{10} + 3q^{11} + 2q^{13} + q^{14} + 4q^{16} + 2q^{17} + 2q^{19} + 4q^{20} + 3q^{22} + 12q^{23} + 4q^{25} + 2q^{26} + q^{28} - 2q^{29} + 10q^{31} + 4q^{32} + 2q^{34} + q^{35} + 3q^{37} + 2q^{38} + 4q^{40} - 6q^{43} + 3q^{44} + 12q^{46} + 14q^{47} + 13q^{49} + 4q^{50} + 2q^{52} + 10q^{53} + 3q^{55} + q^{56} - 2q^{58} + 9q^{61} + 10q^{62} + 4q^{64} + 2q^{65} - 4q^{67} + 2q^{68} + q^{70} + 9q^{71} - 14q^{73} + 3q^{74} + 2q^{76} + 23q^{77} + 12q^{79} + 4q^{80} + 25q^{83} + 2q^{85} - 6q^{86} + 3q^{88} + 9q^{89} - 18q^{91} + 12q^{92} + 14q^{94} + 2q^{95} - 3q^{97} + 13q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 8 x^{2} + 10 x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{2}\)\(=\)\( \nu^{3} + \nu^{2} - 6 \nu - 1 \)
\(\beta_{3}\)\(=\)\( -\nu^{3} + 8 \nu - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{1} + 4\)
\(\nu^{3}\)\(=\)\(3 \beta_{3} + 4 \beta_{2} - 4 \beta_{1} - 3\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−0.175890
1.59676
2.46640
−2.88727
1.00000 0 1.00000 1.00000 0 −3.96906 1.00000 0 1.00000
1.2 1.00000 0 1.00000 1.00000 0 −1.45037 1.00000 0 1.00000
1.3 1.00000 0 1.00000 1.00000 0 2.08313 1.00000 0 1.00000
1.4 1.00000 0 1.00000 1.00000 0 4.33630 1.00000 0 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 6030.2.a.bu 4
3.b odd 2 1 2010.2.a.r 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2010.2.a.r 4 3.b odd 2 1
6030.2.a.bu 4 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(-1\)
\(67\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6030))\):

\( T_{7}^{4} - T_{7}^{3} - 20 T_{7}^{2} + 12 T_{7} + 52 \)
\( T_{11}^{4} - 3 T_{11}^{3} - 20 T_{11}^{2} + 32 T_{11} + 112 \)
\( T_{13}^{4} - 2 T_{13}^{3} - 20 T_{13}^{2} + 28 T_{13} - 8 \)
\( T_{17}^{4} - 2 T_{17}^{3} - 32 T_{17}^{2} + 80 T_{17} + 32 \)
\( T_{23}^{4} - 12 T_{23}^{3} + 8 T_{23}^{2} + 204 T_{23} - 448 \)
\( T_{29}^{4} + 2 T_{29}^{3} - 52 T_{29}^{2} + 148 T_{29} - 112 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{4} \)
$3$ 1
$5$ \( ( 1 - T )^{4} \)
$7$ \( 1 - T + 8 T^{2} - 9 T^{3} + 66 T^{4} - 63 T^{5} + 392 T^{6} - 343 T^{7} + 2401 T^{8} \)
$11$ \( 1 - 3 T + 24 T^{2} - 67 T^{3} + 398 T^{4} - 737 T^{5} + 2904 T^{6} - 3993 T^{7} + 14641 T^{8} \)
$13$ \( 1 - 2 T + 32 T^{2} - 50 T^{3} + 486 T^{4} - 650 T^{5} + 5408 T^{6} - 4394 T^{7} + 28561 T^{8} \)
$17$ \( 1 - 2 T + 36 T^{2} - 22 T^{3} + 678 T^{4} - 374 T^{5} + 10404 T^{6} - 9826 T^{7} + 83521 T^{8} \)
$19$ \( 1 - 2 T + 44 T^{2} - 34 T^{3} + 982 T^{4} - 646 T^{5} + 15884 T^{6} - 13718 T^{7} + 130321 T^{8} \)
$23$ \( 1 - 12 T + 100 T^{2} - 624 T^{3} + 3094 T^{4} - 14352 T^{5} + 52900 T^{6} - 146004 T^{7} + 279841 T^{8} \)
$29$ \( 1 + 2 T + 64 T^{2} + 322 T^{3} + 1918 T^{4} + 9338 T^{5} + 53824 T^{6} + 48778 T^{7} + 707281 T^{8} \)
$31$ \( 1 - 10 T + 140 T^{2} - 878 T^{3} + 6646 T^{4} - 27218 T^{5} + 134540 T^{6} - 297910 T^{7} + 923521 T^{8} \)
$37$ \( 1 - 3 T + 122 T^{2} - 273 T^{3} + 6346 T^{4} - 10101 T^{5} + 167018 T^{6} - 151959 T^{7} + 1874161 T^{8} \)
$41$ \( 1 + 92 T^{2} - 16 T^{3} + 4358 T^{4} - 656 T^{5} + 154652 T^{6} + 2825761 T^{8} \)
$43$ \( 1 + 6 T + 92 T^{2} + 518 T^{3} + 6006 T^{4} + 22274 T^{5} + 170108 T^{6} + 477042 T^{7} + 3418801 T^{8} \)
$47$ \( 1 - 14 T + 116 T^{2} - 762 T^{3} + 4862 T^{4} - 35814 T^{5} + 256244 T^{6} - 1453522 T^{7} + 4879681 T^{8} \)
$53$ \( 1 - 10 T + 168 T^{2} - 1230 T^{3} + 12542 T^{4} - 65190 T^{5} + 471912 T^{6} - 1488770 T^{7} + 7890481 T^{8} \)
$59$ \( 1 + 164 T^{2} - 16 T^{3} + 12566 T^{4} - 944 T^{5} + 570884 T^{6} + 12117361 T^{8} \)
$61$ \( 1 - 9 T + 254 T^{2} - 1599 T^{3} + 23518 T^{4} - 97539 T^{5} + 945134 T^{6} - 2042829 T^{7} + 13845841 T^{8} \)
$67$ \( ( 1 + T )^{4} \)
$71$ \( 1 - 9 T + 242 T^{2} - 1313 T^{3} + 22814 T^{4} - 93223 T^{5} + 1219922 T^{6} - 3221199 T^{7} + 25411681 T^{8} \)
$73$ \( 1 + 14 T + 248 T^{2} + 2274 T^{3} + 25134 T^{4} + 166002 T^{5} + 1321592 T^{6} + 5446238 T^{7} + 28398241 T^{8} \)
$79$ \( 1 - 12 T + 200 T^{2} - 992 T^{3} + 13686 T^{4} - 78368 T^{5} + 1248200 T^{6} - 5916468 T^{7} + 38950081 T^{8} \)
$83$ \( 1 - 25 T + 324 T^{2} - 3793 T^{3} + 40262 T^{4} - 314819 T^{5} + 2232036 T^{6} - 14294675 T^{7} + 47458321 T^{8} \)
$89$ \( 1 - 9 T + 122 T^{2} - 783 T^{3} + 10410 T^{4} - 69687 T^{5} + 966362 T^{6} - 6344721 T^{7} + 62742241 T^{8} \)
$97$ \( 1 + 3 T + 368 T^{2} + 841 T^{3} + 52686 T^{4} + 81577 T^{5} + 3462512 T^{6} + 2738019 T^{7} + 88529281 T^{8} \)
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