Properties

Label 4200.2.t.c
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 840)
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.

\(f(q)\) \(=\) \( q -i q^{3} + i q^{7} - q^{9} +O(q^{10})\) \( q -i q^{3} + i q^{7} - q^{9} -4 q^{11} -2 i q^{13} -2 i q^{17} + 4 q^{19} + q^{21} + i q^{27} + 10 q^{29} + 4 i q^{33} -6 i q^{37} -2 q^{39} -6 q^{41} -4 i q^{43} + 8 i q^{47} - q^{49} -2 q^{51} + 6 i q^{53} -4 i q^{57} + 4 q^{59} -10 q^{61} -i q^{63} -4 i q^{67} -16 q^{71} -14 i q^{73} -4 i q^{77} -8 q^{79} + q^{81} -4 i q^{83} -10 i q^{87} -10 q^{89} + 2 q^{91} -10 i q^{97} + 4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{9} - 8q^{11} + 8q^{19} + 2q^{21} + 20q^{29} - 4q^{39} - 12q^{41} - 2q^{49} - 4q^{51} + 8q^{59} - 20q^{61} - 32q^{71} - 16q^{79} + 2q^{81} - 20q^{89} + 4q^{91} + 8q^{99} + O(q^{100}) \)

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.c 2
5.b even 2 1 inner 4200.2.t.c 2
5.c odd 4 1 840.2.a.d 1
5.c odd 4 1 4200.2.a.bb 1
15.e even 4 1 2520.2.a.f 1
20.e even 4 1 1680.2.a.t 1
20.e even 4 1 8400.2.a.l 1
35.f even 4 1 5880.2.a.t 1
40.i odd 4 1 6720.2.a.bl 1
40.k even 4 1 6720.2.a.m 1
60.l odd 4 1 5040.2.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.a.d 1 5.c odd 4 1
1680.2.a.t 1 20.e even 4 1
2520.2.a.f 1 15.e even 4 1
4200.2.a.bb 1 5.c odd 4 1
4200.2.t.c 2 1.a even 1 1 trivial
4200.2.t.c 2 5.b even 2 1 inner
5040.2.a.l 1 60.l odd 4 1
5880.2.a.t 1 35.f even 4 1
6720.2.a.m 1 40.k even 4 1
6720.2.a.bl 1 40.i odd 4 1
8400.2.a.l 1 20.e even 4 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 1 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 1 + T^{2} \)
$11$ \( ( 4 + T )^{2} \)
$13$ \( 4 + T^{2} \)
$17$ \( 4 + T^{2} \)
$19$ \( ( -4 + T )^{2} \)
$23$ \( T^{2} \)
$29$ \( ( -10 + T )^{2} \)
$31$ \( T^{2} \)
$37$ \( 36 + T^{2} \)
$41$ \( ( 6 + T )^{2} \)
$43$ \( 16 + T^{2} \)
$47$ \( 64 + T^{2} \)
$53$ \( 36 + T^{2} \)
$59$ \( ( -4 + T )^{2} \)
$61$ \( ( 10 + T )^{2} \)
$67$ \( 16 + T^{2} \)
$71$ \( ( 16 + T )^{2} \)
$73$ \( 196 + T^{2} \)
$79$ \( ( 8 + T )^{2} \)
$83$ \( 16 + T^{2} \)
$89$ \( ( 10 + T )^{2} \)
$97$ \( 100 + T^{2} \)
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