Properties

Label 6069.2.a.b
Level 6069
Weight 2
Character orbit 6069.a
Self dual yes
Analytic conductor 48.461
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 6069 = 3 \cdot 7 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6069.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.4612089867\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} - q^{3} - q^{4} + 2q^{5} + q^{6} + q^{7} + 3q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} - q^{3} - q^{4} + 2q^{5} + q^{6} + q^{7} + 3q^{8} + q^{9} - 2q^{10} - 4q^{11} + q^{12} - 2q^{13} - q^{14} - 2q^{15} - q^{16} - q^{18} + 4q^{19} - 2q^{20} - q^{21} + 4q^{22} - 3q^{24} - q^{25} + 2q^{26} - q^{27} - q^{28} + 2q^{29} + 2q^{30} - 5q^{32} + 4q^{33} + 2q^{35} - q^{36} - 6q^{37} - 4q^{38} + 2q^{39} + 6q^{40} - 2q^{41} + q^{42} - 4q^{43} + 4q^{44} + 2q^{45} + q^{48} + q^{49} + q^{50} + 2q^{52} + 6q^{53} + q^{54} - 8q^{55} + 3q^{56} - 4q^{57} - 2q^{58} + 12q^{59} + 2q^{60} + 2q^{61} + q^{63} + 7q^{64} - 4q^{65} - 4q^{66} + 4q^{67} - 2q^{70} + 3q^{72} + 6q^{73} + 6q^{74} + q^{75} - 4q^{76} - 4q^{77} - 2q^{78} + 16q^{79} - 2q^{80} + q^{81} + 2q^{82} - 12q^{83} + q^{84} + 4q^{86} - 2q^{87} - 12q^{88} - 14q^{89} - 2q^{90} - 2q^{91} + 8q^{95} + 5q^{96} - 18q^{97} - q^{98} - 4q^{99} + O(q^{100}) \)

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
−1.00000 −1.00000 −1.00000 2.00000 1.00000 1.00000 3.00000 1.00000 −2.00000
\(n\): 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 6069.2.a.b 1
17.b even 2 1 21.2.a.a 1
51.c odd 2 1 63.2.a.a 1
68.d odd 2 1 336.2.a.a 1
85.c even 2 1 525.2.a.d 1
85.g odd 4 2 525.2.d.a 2
119.d odd 2 1 147.2.a.a 1
119.h odd 6 2 147.2.e.c 2
119.j even 6 2 147.2.e.b 2
136.e odd 2 1 1344.2.a.s 1
136.h even 2 1 1344.2.a.g 1
153.h even 6 2 567.2.f.g 2
153.i odd 6 2 567.2.f.b 2
187.b odd 2 1 2541.2.a.j 1
204.h even 2 1 1008.2.a.l 1
221.b even 2 1 3549.2.a.c 1
255.h odd 2 1 1575.2.a.c 1
255.o even 4 2 1575.2.d.a 2
272.k odd 4 2 5376.2.c.l 2
272.r even 4 2 5376.2.c.r 2
323.c odd 2 1 7581.2.a.d 1
340.d odd 2 1 8400.2.a.bn 1
357.c even 2 1 441.2.a.f 1
357.q odd 6 2 441.2.e.a 2
357.s even 6 2 441.2.e.b 2
408.b odd 2 1 4032.2.a.h 1
408.h even 2 1 4032.2.a.k 1
476.e even 2 1 2352.2.a.v 1
476.o odd 6 2 2352.2.q.x 2
476.q even 6 2 2352.2.q.e 2
561.h even 2 1 7623.2.a.g 1
595.b odd 2 1 3675.2.a.n 1
952.e odd 2 1 9408.2.a.bv 1
952.k even 2 1 9408.2.a.m 1
1428.b odd 2 1 7056.2.a.p 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.2.a.a 1 17.b even 2 1
63.2.a.a 1 51.c odd 2 1
147.2.a.a 1 119.d odd 2 1
147.2.e.b 2 119.j even 6 2
147.2.e.c 2 119.h odd 6 2
336.2.a.a 1 68.d odd 2 1
441.2.a.f 1 357.c even 2 1
441.2.e.a 2 357.q odd 6 2
441.2.e.b 2 357.s even 6 2
525.2.a.d 1 85.c even 2 1
525.2.d.a 2 85.g odd 4 2
567.2.f.b 2 153.i odd 6 2
567.2.f.g 2 153.h even 6 2
1008.2.a.l 1 204.h even 2 1
1344.2.a.g 1 136.h even 2 1
1344.2.a.s 1 136.e odd 2 1
1575.2.a.c 1 255.h odd 2 1
1575.2.d.a 2 255.o even 4 2
2352.2.a.v 1 476.e even 2 1
2352.2.q.e 2 476.q even 6 2
2352.2.q.x 2 476.o odd 6 2
2541.2.a.j 1 187.b odd 2 1
3549.2.a.c 1 221.b even 2 1
3675.2.a.n 1 595.b odd 2 1
4032.2.a.h 1 408.b odd 2 1
4032.2.a.k 1 408.h even 2 1
5376.2.c.l 2 272.k odd 4 2
5376.2.c.r 2 272.r even 4 2
6069.2.a.b 1 1.a even 1 1 trivial
7056.2.a.p 1 1428.b odd 2 1
7581.2.a.d 1 323.c odd 2 1
7623.2.a.g 1 561.h even 2 1
8400.2.a.bn 1 340.d odd 2 1
9408.2.a.m 1 952.k even 2 1
9408.2.a.bv 1 952.e odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(-1\)
\(17\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6069))\):

\( T_{2} + 1 \)
\( T_{5} - 2 \)
\( T_{11} + 4 \)
\( T_{23} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + 2 T^{2} \)
$3$ \( 1 + T \)
$5$ \( 1 - 2 T + 5 T^{2} \)
$7$ \( 1 - T \)
$11$ \( 1 + 4 T + 11 T^{2} \)
$13$ \( 1 + 2 T + 13 T^{2} \)
$17$ 1
$19$ \( 1 - 4 T + 19 T^{2} \)
$23$ \( 1 + 23 T^{2} \)
$29$ \( 1 - 2 T + 29 T^{2} \)
$31$ \( 1 + 31 T^{2} \)
$37$ \( 1 + 6 T + 37 T^{2} \)
$41$ \( 1 + 2 T + 41 T^{2} \)
$43$ \( 1 + 4 T + 43 T^{2} \)
$47$ \( 1 + 47 T^{2} \)
$53$ \( 1 - 6 T + 53 T^{2} \)
$59$ \( 1 - 12 T + 59 T^{2} \)
$61$ \( 1 - 2 T + 61 T^{2} \)
$67$ \( 1 - 4 T + 67 T^{2} \)
$71$ \( 1 + 71 T^{2} \)
$73$ \( 1 - 6 T + 73 T^{2} \)
$79$ \( 1 - 16 T + 79 T^{2} \)
$83$ \( 1 + 12 T + 83 T^{2} \)
$89$ \( 1 + 14 T + 89 T^{2} \)
$97$ \( 1 + 18 T + 97 T^{2} \)
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