Properties

Label 4200.2.a.t
Level $4200$
Weight $2$
Character orbit 4200.a
Self dual yes
Analytic conductor $33.537$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4200 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4200.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(33.5371688489\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 168)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{3} - q^{7} + q^{9} + O(q^{10}) \) \( q + q^{3} - q^{7} + q^{9} - 6q^{13} + 2q^{17} + 4q^{19} - q^{21} + 4q^{23} + q^{27} - 10q^{29} - 8q^{31} - 6q^{37} - 6q^{39} - 2q^{41} + 4q^{43} - 8q^{47} + q^{49} + 2q^{51} + 10q^{53} + 4q^{57} + 12q^{59} - 2q^{61} - q^{63} - 12q^{67} + 4q^{69} - 12q^{71} + 14q^{73} - 8q^{79} + q^{81} - 12q^{83} - 10q^{87} - 2q^{89} + 6q^{91} - 8q^{93} - 10q^{97} + O(q^{100}) \)

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
0 1.00000 0 0 0 −1.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(1\)
\(7\) \(1\)

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 4200.2.a.t 1
4.b odd 2 1 8400.2.a.y 1
5.b even 2 1 168.2.a.a 1
5.c odd 4 2 4200.2.t.j 2
15.d odd 2 1 504.2.a.e 1
20.d odd 2 1 336.2.a.e 1
35.c odd 2 1 1176.2.a.f 1
35.i odd 6 2 1176.2.q.d 2
35.j even 6 2 1176.2.q.f 2
40.e odd 2 1 1344.2.a.b 1
40.f even 2 1 1344.2.a.m 1
60.h even 2 1 1008.2.a.b 1
80.k odd 4 2 5376.2.c.bb 2
80.q even 4 2 5376.2.c.d 2
105.g even 2 1 3528.2.a.v 1
105.o odd 6 2 3528.2.s.w 2
105.p even 6 2 3528.2.s.g 2
120.i odd 2 1 4032.2.a.bh 1
120.m even 2 1 4032.2.a.bc 1
140.c even 2 1 2352.2.a.c 1
140.p odd 6 2 2352.2.q.d 2
140.s even 6 2 2352.2.q.w 2
280.c odd 2 1 9408.2.a.be 1
280.n even 2 1 9408.2.a.da 1
420.o odd 2 1 7056.2.a.bq 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.a.a 1 5.b even 2 1
336.2.a.e 1 20.d odd 2 1
504.2.a.e 1 15.d odd 2 1
1008.2.a.b 1 60.h even 2 1
1176.2.a.f 1 35.c odd 2 1
1176.2.q.d 2 35.i odd 6 2
1176.2.q.f 2 35.j even 6 2
1344.2.a.b 1 40.e odd 2 1
1344.2.a.m 1 40.f even 2 1
2352.2.a.c 1 140.c even 2 1
2352.2.q.d 2 140.p odd 6 2
2352.2.q.w 2 140.s even 6 2
3528.2.a.v 1 105.g even 2 1
3528.2.s.g 2 105.p even 6 2
3528.2.s.w 2 105.o odd 6 2
4032.2.a.bc 1 120.m even 2 1
4032.2.a.bh 1 120.i odd 2 1
4200.2.a.t 1 1.a even 1 1 trivial
4200.2.t.j 2 5.c odd 4 2
5376.2.c.d 2 80.q even 4 2
5376.2.c.bb 2 80.k odd 4 2
7056.2.a.bq 1 420.o odd 2 1
8400.2.a.y 1 4.b odd 2 1
9408.2.a.be 1 280.c odd 2 1
9408.2.a.da 1 280.n even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4200))\):

\( T_{11} \)
\( T_{13} + 6 \)
\( T_{17} - 2 \)
\( T_{19} - 4 \)
\( T_{23} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - T \)
$5$ 1
$7$ \( 1 + T \)
$11$ \( 1 + 11 T^{2} \)
$13$ \( 1 + 6 T + 13 T^{2} \)
$17$ \( 1 - 2 T + 17 T^{2} \)
$19$ \( 1 - 4 T + 19 T^{2} \)
$23$ \( 1 - 4 T + 23 T^{2} \)
$29$ \( 1 + 10 T + 29 T^{2} \)
$31$ \( 1 + 8 T + 31 T^{2} \)
$37$ \( 1 + 6 T + 37 T^{2} \)
$41$ \( 1 + 2 T + 41 T^{2} \)
$43$ \( 1 - 4 T + 43 T^{2} \)
$47$ \( 1 + 8 T + 47 T^{2} \)
$53$ \( 1 - 10 T + 53 T^{2} \)
$59$ \( 1 - 12 T + 59 T^{2} \)
$61$ \( 1 + 2 T + 61 T^{2} \)
$67$ \( 1 + 12 T + 67 T^{2} \)
$71$ \( 1 + 12 T + 71 T^{2} \)
$73$ \( 1 - 14 T + 73 T^{2} \)
$79$ \( 1 + 8 T + 79 T^{2} \)
$83$ \( 1 + 12 T + 83 T^{2} \)
$89$ \( 1 + 2 T + 89 T^{2} \)
$97$ \( 1 + 10 T + 97 T^{2} \)
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