Properties

Label 1200.4.f.t
Level $1200$
Weight $4$
Character orbit 1200.f
Analytic conductor $70.802$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1200.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(70.8022920069\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 120)
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.

\(f(q)\) \(=\) \( q + 3 i q^{3} + 20 i q^{7} -9 q^{9} +O(q^{10})\) \( q + 3 i q^{3} + 20 i q^{7} -9 q^{9} + 56 q^{11} -86 i q^{13} + 106 i q^{17} + 4 q^{19} -60 q^{21} -136 i q^{23} -27 i q^{27} + 206 q^{29} + 152 q^{31} + 168 i q^{33} -282 i q^{37} + 258 q^{39} -246 q^{41} -412 i q^{43} + 40 i q^{47} -57 q^{49} -318 q^{51} -126 i q^{53} + 12 i q^{57} + 56 q^{59} -2 q^{61} -180 i q^{63} -388 i q^{67} + 408 q^{69} + 672 q^{71} + 1170 i q^{73} + 1120 i q^{77} + 408 q^{79} + 81 q^{81} -668 i q^{83} + 618 i q^{87} -66 q^{89} + 1720 q^{91} + 456 i q^{93} + 926 i q^{97} -504 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 18 q^{9} + O(q^{10}) \) \( 2 q - 18 q^{9} + 112 q^{11} + 8 q^{19} - 120 q^{21} + 412 q^{29} + 304 q^{31} + 516 q^{39} - 492 q^{41} - 114 q^{49} - 636 q^{51} + 112 q^{59} - 4 q^{61} + 816 q^{69} + 1344 q^{71} + 816 q^{79} + 162 q^{81} - 132 q^{89} + 3440 q^{91} - 1008 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(577\) \(751\) \(901\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
1.00000i
1.00000i
0 3.00000i 0 0 0 20.0000i 0 −9.00000 0
49.2 0 3.00000i 0 0 0 20.0000i 0 −9.00000 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 1200.4.f.t 2
4.b odd 2 1 600.4.f.a 2
5.b even 2 1 inner 1200.4.f.t 2
5.c odd 4 1 240.4.a.g 1
5.c odd 4 1 1200.4.a.p 1
12.b even 2 1 1800.4.f.v 2
15.e even 4 1 720.4.a.q 1
20.d odd 2 1 600.4.f.a 2
20.e even 4 1 120.4.a.b 1
20.e even 4 1 600.4.a.i 1
40.i odd 4 1 960.4.a.k 1
40.k even 4 1 960.4.a.bj 1
60.h even 2 1 1800.4.f.v 2
60.l odd 4 1 360.4.a.n 1
60.l odd 4 1 1800.4.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.4.a.b 1 20.e even 4 1
240.4.a.g 1 5.c odd 4 1
360.4.a.n 1 60.l odd 4 1
600.4.a.i 1 20.e even 4 1
600.4.f.a 2 4.b odd 2 1
600.4.f.a 2 20.d odd 2 1
720.4.a.q 1 15.e even 4 1
960.4.a.k 1 40.i odd 4 1
960.4.a.bj 1 40.k even 4 1
1200.4.a.p 1 5.c odd 4 1
1200.4.f.t 2 1.a even 1 1 trivial
1200.4.f.t 2 5.b even 2 1 inner
1800.4.a.f 1 60.l odd 4 1
1800.4.f.v 2 12.b even 2 1
1800.4.f.v 2 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1200, [\chi])\):

\( T_{7}^{2} + 400 \)
\( T_{11} - 56 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 9 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 400 + T^{2} \)
$11$ \( ( -56 + T )^{2} \)
$13$ \( 7396 + T^{2} \)
$17$ \( 11236 + T^{2} \)
$19$ \( ( -4 + T )^{2} \)
$23$ \( 18496 + T^{2} \)
$29$ \( ( -206 + T )^{2} \)
$31$ \( ( -152 + T )^{2} \)
$37$ \( 79524 + T^{2} \)
$41$ \( ( 246 + T )^{2} \)
$43$ \( 169744 + T^{2} \)
$47$ \( 1600 + T^{2} \)
$53$ \( 15876 + T^{2} \)
$59$ \( ( -56 + T )^{2} \)
$61$ \( ( 2 + T )^{2} \)
$67$ \( 150544 + T^{2} \)
$71$ \( ( -672 + T )^{2} \)
$73$ \( 1368900 + T^{2} \)
$79$ \( ( -408 + T )^{2} \)
$83$ \( 446224 + T^{2} \)
$89$ \( ( 66 + T )^{2} \)
$97$ \( 857476 + T^{2} \)
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