Properties

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

Related objects

Downloads

Learn more about

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 12)
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.

\(f(q)\) \(=\) \( q -3 i q^{3} + 8 i q^{7} -9 q^{9} +O(q^{10})\) \( q -3 i q^{3} + 8 i q^{7} -9 q^{9} -36 q^{11} -10 i q^{13} -18 i q^{17} -100 q^{19} + 24 q^{21} -72 i q^{23} + 27 i q^{27} + 234 q^{29} + 16 q^{31} + 108 i q^{33} + 226 i q^{37} -30 q^{39} + 90 q^{41} -452 i q^{43} + 432 i q^{47} + 279 q^{49} -54 q^{51} + 414 i q^{53} + 300 i q^{57} -684 q^{59} + 422 q^{61} -72 i q^{63} + 332 i q^{67} -216 q^{69} + 360 q^{71} + 26 i q^{73} -288 i q^{77} + 512 q^{79} + 81 q^{81} + 1188 i q^{83} -702 i q^{87} + 630 q^{89} + 80 q^{91} -48 i q^{93} + 1054 i q^{97} + 324 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 18q^{9} + O(q^{10}) \) \( 2q - 18q^{9} - 72q^{11} - 200q^{19} + 48q^{21} + 468q^{29} + 32q^{31} - 60q^{39} + 180q^{41} + 558q^{49} - 108q^{51} - 1368q^{59} + 844q^{61} - 432q^{69} + 720q^{71} + 1024q^{79} + 162q^{81} + 1260q^{89} + 160q^{91} + 648q^{99} + O(q^{100}) \)

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 8.00000i 0 −9.00000 0
49.2 0 3.00000i 0 0 0 8.00000i 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.d 2
4.b odd 2 1 300.4.d.e 2
5.b even 2 1 inner 1200.4.f.d 2
5.c odd 4 1 48.4.a.a 1
5.c odd 4 1 1200.4.a.be 1
12.b even 2 1 900.4.d.c 2
15.e even 4 1 144.4.a.g 1
20.d odd 2 1 300.4.d.e 2
20.e even 4 1 12.4.a.a 1
20.e even 4 1 300.4.a.b 1
35.f even 4 1 2352.4.a.bk 1
40.i odd 4 1 192.4.a.l 1
40.k even 4 1 192.4.a.f 1
60.h even 2 1 900.4.d.c 2
60.l odd 4 1 36.4.a.a 1
60.l odd 4 1 900.4.a.g 1
80.i odd 4 1 768.4.d.j 2
80.j even 4 1 768.4.d.g 2
80.s even 4 1 768.4.d.g 2
80.t odd 4 1 768.4.d.j 2
120.q odd 4 1 576.4.a.b 1
120.w even 4 1 576.4.a.a 1
140.j odd 4 1 588.4.a.c 1
140.w even 12 2 588.4.i.d 2
140.x odd 12 2 588.4.i.e 2
180.v odd 12 2 324.4.e.a 2
180.x even 12 2 324.4.e.h 2
220.i odd 4 1 1452.4.a.d 1
260.l odd 4 1 2028.4.b.c 2
260.p even 4 1 2028.4.a.c 1
260.s odd 4 1 2028.4.b.c 2
420.w even 4 1 1764.4.a.b 1
420.bp odd 12 2 1764.4.k.b 2
420.br even 12 2 1764.4.k.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.4.a.a 1 20.e even 4 1
36.4.a.a 1 60.l odd 4 1
48.4.a.a 1 5.c odd 4 1
144.4.a.g 1 15.e even 4 1
192.4.a.f 1 40.k even 4 1
192.4.a.l 1 40.i odd 4 1
300.4.a.b 1 20.e even 4 1
300.4.d.e 2 4.b odd 2 1
300.4.d.e 2 20.d odd 2 1
324.4.e.a 2 180.v odd 12 2
324.4.e.h 2 180.x even 12 2
576.4.a.a 1 120.w even 4 1
576.4.a.b 1 120.q odd 4 1
588.4.a.c 1 140.j odd 4 1
588.4.i.d 2 140.w even 12 2
588.4.i.e 2 140.x odd 12 2
768.4.d.g 2 80.j even 4 1
768.4.d.g 2 80.s even 4 1
768.4.d.j 2 80.i odd 4 1
768.4.d.j 2 80.t odd 4 1
900.4.a.g 1 60.l odd 4 1
900.4.d.c 2 12.b even 2 1
900.4.d.c 2 60.h even 2 1
1200.4.a.be 1 5.c odd 4 1
1200.4.f.d 2 1.a even 1 1 trivial
1200.4.f.d 2 5.b even 2 1 inner
1452.4.a.d 1 220.i odd 4 1
1764.4.a.b 1 420.w even 4 1
1764.4.k.b 2 420.bp odd 12 2
1764.4.k.o 2 420.br even 12 2
2028.4.a.c 1 260.p even 4 1
2028.4.b.c 2 260.l odd 4 1
2028.4.b.c 2 260.s odd 4 1
2352.4.a.bk 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 \(S_{4}^{\mathrm{new}}(1200, [\chi])\):

\( T_{7}^{2} + 64 \)
\( T_{11} + 36 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 9 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 64 + T^{2} \)
$11$ \( ( 36 + T )^{2} \)
$13$ \( 100 + T^{2} \)
$17$ \( 324 + T^{2} \)
$19$ \( ( 100 + T )^{2} \)
$23$ \( 5184 + T^{2} \)
$29$ \( ( -234 + T )^{2} \)
$31$ \( ( -16 + T )^{2} \)
$37$ \( 51076 + T^{2} \)
$41$ \( ( -90 + T )^{2} \)
$43$ \( 204304 + T^{2} \)
$47$ \( 186624 + T^{2} \)
$53$ \( 171396 + T^{2} \)
$59$ \( ( 684 + T )^{2} \)
$61$ \( ( -422 + T )^{2} \)
$67$ \( 110224 + T^{2} \)
$71$ \( ( -360 + T )^{2} \)
$73$ \( 676 + T^{2} \)
$79$ \( ( -512 + T )^{2} \)
$83$ \( 1411344 + T^{2} \)
$89$ \( ( -630 + T )^{2} \)
$97$ \( 1110916 + T^{2} \)
show more
show less