Properties

Label 1200.4.f.j
Level 1200
Weight 4
Character orbit 1200.f
Analytic conductor 70.802
Analytic rank 1
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 6)
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.

\(f(q)\) \(=\) \( q -3 i q^{3} + 16 i q^{7} -9 q^{9} +O(q^{10})\) \( q -3 i q^{3} + 16 i q^{7} -9 q^{9} -12 q^{11} -38 i q^{13} -126 i q^{17} + 20 q^{19} + 48 q^{21} + 168 i q^{23} + 27 i q^{27} -30 q^{29} + 88 q^{31} + 36 i q^{33} + 254 i q^{37} -114 q^{39} + 42 q^{41} -52 i q^{43} + 96 i q^{47} + 87 q^{49} -378 q^{51} -198 i q^{53} -60 i q^{57} -660 q^{59} -538 q^{61} -144 i q^{63} -884 i q^{67} + 504 q^{69} -792 q^{71} -218 i q^{73} -192 i q^{77} -520 q^{79} + 81 q^{81} -492 i q^{83} + 90 i q^{87} -810 q^{89} + 608 q^{91} -264 i q^{93} + 1154 i q^{97} + 108 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 18q^{9} + O(q^{10}) \) \( 2q - 18q^{9} - 24q^{11} + 40q^{19} + 96q^{21} - 60q^{29} + 176q^{31} - 228q^{39} + 84q^{41} + 174q^{49} - 756q^{51} - 1320q^{59} - 1076q^{61} + 1008q^{69} - 1584q^{71} - 1040q^{79} + 162q^{81} - 1620q^{89} + 1216q^{91} + 216q^{99} + O(q^{100}) \)

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 16.0000i 0 −9.00000 0
49.2 0 3.00000i 0 0 0 16.0000i 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.j 2
4.b odd 2 1 150.4.c.d 2
5.b even 2 1 inner 1200.4.f.j 2
5.c odd 4 1 48.4.a.c 1
5.c odd 4 1 1200.4.a.b 1
12.b even 2 1 450.4.c.e 2
15.e even 4 1 144.4.a.c 1
20.d odd 2 1 150.4.c.d 2
20.e even 4 1 6.4.a.a 1
20.e even 4 1 150.4.a.i 1
35.f even 4 1 2352.4.a.e 1
40.i odd 4 1 192.4.a.c 1
40.k even 4 1 192.4.a.i 1
60.h even 2 1 450.4.c.e 2
60.l odd 4 1 18.4.a.a 1
60.l odd 4 1 450.4.a.h 1
80.i odd 4 1 768.4.d.c 2
80.j even 4 1 768.4.d.n 2
80.s even 4 1 768.4.d.n 2
80.t odd 4 1 768.4.d.c 2
120.q odd 4 1 576.4.a.q 1
120.w even 4 1 576.4.a.r 1
140.j odd 4 1 294.4.a.e 1
140.w even 12 2 294.4.e.h 2
140.x odd 12 2 294.4.e.g 2
180.v odd 12 2 162.4.c.c 2
180.x even 12 2 162.4.c.f 2
220.i odd 4 1 726.4.a.f 1
260.l odd 4 1 1014.4.b.d 2
260.p even 4 1 1014.4.a.g 1
260.s odd 4 1 1014.4.b.d 2
340.r even 4 1 1734.4.a.d 1
380.j odd 4 1 2166.4.a.i 1
420.w even 4 1 882.4.a.n 1
420.bp odd 12 2 882.4.g.i 2
420.br even 12 2 882.4.g.f 2
660.q even 4 1 2178.4.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.4.a.a 1 20.e even 4 1
18.4.a.a 1 60.l odd 4 1
48.4.a.c 1 5.c odd 4 1
144.4.a.c 1 15.e even 4 1
150.4.a.i 1 20.e even 4 1
150.4.c.d 2 4.b odd 2 1
150.4.c.d 2 20.d odd 2 1
162.4.c.c 2 180.v odd 12 2
162.4.c.f 2 180.x even 12 2
192.4.a.c 1 40.i odd 4 1
192.4.a.i 1 40.k even 4 1
294.4.a.e 1 140.j odd 4 1
294.4.e.g 2 140.x odd 12 2
294.4.e.h 2 140.w even 12 2
450.4.a.h 1 60.l odd 4 1
450.4.c.e 2 12.b even 2 1
450.4.c.e 2 60.h even 2 1
576.4.a.q 1 120.q odd 4 1
576.4.a.r 1 120.w even 4 1
726.4.a.f 1 220.i odd 4 1
768.4.d.c 2 80.i odd 4 1
768.4.d.c 2 80.t odd 4 1
768.4.d.n 2 80.j even 4 1
768.4.d.n 2 80.s even 4 1
882.4.a.n 1 420.w even 4 1
882.4.g.f 2 420.br even 12 2
882.4.g.i 2 420.bp odd 12 2
1014.4.a.g 1 260.p even 4 1
1014.4.b.d 2 260.l odd 4 1
1014.4.b.d 2 260.s odd 4 1
1200.4.a.b 1 5.c odd 4 1
1200.4.f.j 2 1.a even 1 1 trivial
1200.4.f.j 2 5.b even 2 1 inner
1734.4.a.d 1 340.r even 4 1
2166.4.a.i 1 380.j odd 4 1
2178.4.a.e 1 660.q even 4 1
2352.4.a.e 1 35.f even 4 1

Hecke kernels

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

\( T_{7}^{2} + 256 \)
\( T_{11} + 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 9 T^{2} \)
$5$ 1
$7$ \( 1 - 430 T^{2} + 117649 T^{4} \)
$11$ \( ( 1 + 12 T + 1331 T^{2} )^{2} \)
$13$ \( 1 - 2950 T^{2} + 4826809 T^{4} \)
$17$ \( 1 + 6050 T^{2} + 24137569 T^{4} \)
$19$ \( ( 1 - 20 T + 6859 T^{2} )^{2} \)
$23$ \( 1 + 3890 T^{2} + 148035889 T^{4} \)
$29$ \( ( 1 + 30 T + 24389 T^{2} )^{2} \)
$31$ \( ( 1 - 88 T + 29791 T^{2} )^{2} \)
$37$ \( 1 - 36790 T^{2} + 2565726409 T^{4} \)
$41$ \( ( 1 - 42 T + 68921 T^{2} )^{2} \)
$43$ \( 1 - 156310 T^{2} + 6321363049 T^{4} \)
$47$ \( 1 - 198430 T^{2} + 10779215329 T^{4} \)
$53$ \( 1 - 258550 T^{2} + 22164361129 T^{4} \)
$59$ \( ( 1 + 660 T + 205379 T^{2} )^{2} \)
$61$ \( ( 1 + 538 T + 226981 T^{2} )^{2} \)
$67$ \( 1 + 179930 T^{2} + 90458382169 T^{4} \)
$71$ \( ( 1 + 792 T + 357911 T^{2} )^{2} \)
$73$ \( 1 - 730510 T^{2} + 151334226289 T^{4} \)
$79$ \( ( 1 + 520 T + 493039 T^{2} )^{2} \)
$83$ \( 1 - 901510 T^{2} + 326940373369 T^{4} \)
$89$ \( ( 1 + 810 T + 704969 T^{2} )^{2} \)
$97$ \( 1 - 493630 T^{2} + 832972004929 T^{4} \)
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