Properties

Label 1200.4.f.q
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 150)
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.

\(f(q)\) \(=\) \( q + 3 i q^{3} + 23 i q^{7} -9 q^{9} +O(q^{10})\) \( q + 3 i q^{3} + 23 i q^{7} -9 q^{9} + 30 q^{11} + 29 i q^{13} -78 i q^{17} + 149 q^{19} -69 q^{21} -150 i q^{23} -27 i q^{27} + 234 q^{29} + 217 q^{31} + 90 i q^{33} -146 i q^{37} -87 q^{39} -156 q^{41} + 433 i q^{43} + 30 i q^{47} -186 q^{49} + 234 q^{51} -552 i q^{53} + 447 i q^{57} -270 q^{59} + 275 q^{61} -207 i q^{63} + 803 i q^{67} + 450 q^{69} -660 q^{71} -646 i q^{73} + 690 i q^{77} + 992 q^{79} + 81 q^{81} + 846 i q^{83} + 702 i q^{87} + 1488 q^{89} -667 q^{91} + 651 i q^{93} + 319 i q^{97} -270 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 18 q^{9} + O(q^{10}) \) \( 2 q - 18 q^{9} + 60 q^{11} + 298 q^{19} - 138 q^{21} + 468 q^{29} + 434 q^{31} - 174 q^{39} - 312 q^{41} - 372 q^{49} + 468 q^{51} - 540 q^{59} + 550 q^{61} + 900 q^{69} - 1320 q^{71} + 1984 q^{79} + 162 q^{81} + 2976 q^{89} - 1334 q^{91} - 540 q^{99} + O(q^{100}) \)

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 23.0000i 0 −9.00000 0
49.2 0 3.00000i 0 0 0 23.0000i 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.q 2
4.b odd 2 1 150.4.c.b 2
5.b even 2 1 inner 1200.4.f.q 2
5.c odd 4 1 1200.4.a.r 1
5.c odd 4 1 1200.4.a.v 1
12.b even 2 1 450.4.c.h 2
20.d odd 2 1 150.4.c.b 2
20.e even 4 1 150.4.a.c 1
20.e even 4 1 150.4.a.g yes 1
60.h even 2 1 450.4.c.h 2
60.l odd 4 1 450.4.a.i 1
60.l odd 4 1 450.4.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.4.a.c 1 20.e even 4 1
150.4.a.g yes 1 20.e even 4 1
150.4.c.b 2 4.b odd 2 1
150.4.c.b 2 20.d odd 2 1
450.4.a.i 1 60.l odd 4 1
450.4.a.l 1 60.l odd 4 1
450.4.c.h 2 12.b even 2 1
450.4.c.h 2 60.h even 2 1
1200.4.a.r 1 5.c odd 4 1
1200.4.a.v 1 5.c odd 4 1
1200.4.f.q 2 1.a even 1 1 trivial
1200.4.f.q 2 5.b even 2 1 inner

Hecke kernels

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

\( T_{7}^{2} + 529 \)
\( T_{11} - 30 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 9 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 529 + T^{2} \)
$11$ \( ( -30 + T )^{2} \)
$13$ \( 841 + T^{2} \)
$17$ \( 6084 + T^{2} \)
$19$ \( ( -149 + T )^{2} \)
$23$ \( 22500 + T^{2} \)
$29$ \( ( -234 + T )^{2} \)
$31$ \( ( -217 + T )^{2} \)
$37$ \( 21316 + T^{2} \)
$41$ \( ( 156 + T )^{2} \)
$43$ \( 187489 + T^{2} \)
$47$ \( 900 + T^{2} \)
$53$ \( 304704 + T^{2} \)
$59$ \( ( 270 + T )^{2} \)
$61$ \( ( -275 + T )^{2} \)
$67$ \( 644809 + T^{2} \)
$71$ \( ( 660 + T )^{2} \)
$73$ \( 417316 + T^{2} \)
$79$ \( ( -992 + T )^{2} \)
$83$ \( 715716 + T^{2} \)
$89$ \( ( -1488 + T )^{2} \)
$97$ \( 101761 + T^{2} \)
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