Properties

Label 4200.2.t.j
Level $4200$
Weight $2$
Character orbit 4200.t
Analytic conductor $33.537$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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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.t (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(33.5371688489\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -i q^{3} -i q^{7} - q^{9} +O(q^{10})\) \( q -i q^{3} -i q^{7} - q^{9} + 6 i q^{13} + 2 i q^{17} -4 q^{19} - q^{21} -4 i q^{23} + i q^{27} + 10 q^{29} -8 q^{31} -6 i q^{37} + 6 q^{39} -2 q^{41} -4 i q^{43} -8 i q^{47} - q^{49} + 2 q^{51} -10 i q^{53} + 4 i q^{57} -12 q^{59} -2 q^{61} + i q^{63} -12 i q^{67} -4 q^{69} -12 q^{71} -14 i q^{73} + 8 q^{79} + q^{81} + 12 i q^{83} -10 i q^{87} + 2 q^{89} + 6 q^{91} + 8 i q^{93} -10 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{9} - 8q^{19} - 2q^{21} + 20q^{29} - 16q^{31} + 12q^{39} - 4q^{41} - 2q^{49} + 4q^{51} - 24q^{59} - 4q^{61} - 8q^{69} - 24q^{71} + 16q^{79} + 2q^{81} + 4q^{89} + 12q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4200\mathbb{Z}\right)^\times\).

\(n\) \(1177\) \(2101\) \(2801\) \(3151\) \(3601\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1849.1
1.00000i
1.00000i
0 1.00000i 0 0 0 1.00000i 0 −1.00000 0
1849.2 0 1.00000i 0 0 0 1.00000i 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4200.2.t.j 2
5.b even 2 1 inner 4200.2.t.j 2
5.c odd 4 1 168.2.a.a 1
5.c odd 4 1 4200.2.a.t 1
15.e even 4 1 504.2.a.e 1
20.e even 4 1 336.2.a.e 1
20.e even 4 1 8400.2.a.y 1
35.f even 4 1 1176.2.a.f 1
35.k even 12 2 1176.2.q.d 2
35.l odd 12 2 1176.2.q.f 2
40.i odd 4 1 1344.2.a.m 1
40.k even 4 1 1344.2.a.b 1
60.l odd 4 1 1008.2.a.b 1
80.i odd 4 1 5376.2.c.d 2
80.j even 4 1 5376.2.c.bb 2
80.s even 4 1 5376.2.c.bb 2
80.t odd 4 1 5376.2.c.d 2
105.k odd 4 1 3528.2.a.v 1
105.w odd 12 2 3528.2.s.g 2
105.x even 12 2 3528.2.s.w 2
120.q odd 4 1 4032.2.a.bc 1
120.w even 4 1 4032.2.a.bh 1
140.j odd 4 1 2352.2.a.c 1
140.w even 12 2 2352.2.q.d 2
140.x odd 12 2 2352.2.q.w 2
280.s even 4 1 9408.2.a.be 1
280.y odd 4 1 9408.2.a.da 1
420.w even 4 1 7056.2.a.bq 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.a.a 1 5.c odd 4 1
336.2.a.e 1 20.e even 4 1
504.2.a.e 1 15.e even 4 1
1008.2.a.b 1 60.l odd 4 1
1176.2.a.f 1 35.f even 4 1
1176.2.q.d 2 35.k even 12 2
1176.2.q.f 2 35.l odd 12 2
1344.2.a.b 1 40.k even 4 1
1344.2.a.m 1 40.i odd 4 1
2352.2.a.c 1 140.j odd 4 1
2352.2.q.d 2 140.w even 12 2
2352.2.q.w 2 140.x odd 12 2
3528.2.a.v 1 105.k odd 4 1
3528.2.s.g 2 105.w odd 12 2
3528.2.s.w 2 105.x even 12 2
4032.2.a.bc 1 120.q odd 4 1
4032.2.a.bh 1 120.w even 4 1
4200.2.a.t 1 5.c odd 4 1
4200.2.t.j 2 1.a even 1 1 trivial
4200.2.t.j 2 5.b even 2 1 inner
5376.2.c.d 2 80.i odd 4 1
5376.2.c.d 2 80.t odd 4 1
5376.2.c.bb 2 80.j even 4 1
5376.2.c.bb 2 80.s even 4 1
7056.2.a.bq 1 420.w even 4 1
8400.2.a.y 1 20.e even 4 1
9408.2.a.be 1 280.s even 4 1
9408.2.a.da 1 280.y odd 4 1

Hecke kernels

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

\( T_{11} \)
\( T_{13}^{2} + 36 \)
\( T_{17}^{2} + 4 \)
\( T_{19} + 4 \)
\( T_{29} - 10 \)
\( T_{31} + 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + T^{2} \)
$5$ 1
$7$ \( 1 + T^{2} \)
$11$ \( ( 1 + 11 T^{2} )^{2} \)
$13$ \( ( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} ) \)
$17$ \( ( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} ) \)
$19$ \( ( 1 + 4 T + 19 T^{2} )^{2} \)
$23$ \( 1 - 30 T^{2} + 529 T^{4} \)
$29$ \( ( 1 - 10 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 + 8 T + 31 T^{2} )^{2} \)
$37$ \( 1 - 38 T^{2} + 1369 T^{4} \)
$41$ \( ( 1 + 2 T + 41 T^{2} )^{2} \)
$43$ \( 1 - 70 T^{2} + 1849 T^{4} \)
$47$ \( 1 - 30 T^{2} + 2209 T^{4} \)
$53$ \( 1 - 6 T^{2} + 2809 T^{4} \)
$59$ \( ( 1 + 12 T + 59 T^{2} )^{2} \)
$61$ \( ( 1 + 2 T + 61 T^{2} )^{2} \)
$67$ \( 1 + 10 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 + 12 T + 71 T^{2} )^{2} \)
$73$ \( 1 + 50 T^{2} + 5329 T^{4} \)
$79$ \( ( 1 - 8 T + 79 T^{2} )^{2} \)
$83$ \( 1 - 22 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 - 2 T + 89 T^{2} )^{2} \)
$97$ \( 1 - 94 T^{2} + 9409 T^{4} \)
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