Properties

Label 1200.4.f.c
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.

\(f(q)\) \(=\) \( q -3 i q^{3} + i q^{7} -9 q^{9} +O(q^{10})\) \( q -3 i q^{3} + i q^{7} -9 q^{9} -42 q^{11} + 67 i q^{13} + 54 i q^{17} -115 q^{19} + 3 q^{21} -162 i q^{23} + 27 i q^{27} + 210 q^{29} + 193 q^{31} + 126 i q^{33} -286 i q^{37} + 201 q^{39} + 12 q^{41} + 263 i q^{43} -414 i q^{47} + 342 q^{49} + 162 q^{51} + 192 i q^{53} + 345 i q^{57} + 690 q^{59} -733 q^{61} -9 i q^{63} -299 i q^{67} -486 q^{69} + 228 q^{71} -938 i q^{73} -42 i q^{77} -160 q^{79} + 81 q^{81} -462 i q^{83} -630 i q^{87} + 240 q^{89} -67 q^{91} -579 i q^{93} -511 i q^{97} + 378 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 18 q^{9} + O(q^{10}) \) \( 2 q - 18 q^{9} - 84 q^{11} - 230 q^{19} + 6 q^{21} + 420 q^{29} + 386 q^{31} + 402 q^{39} + 24 q^{41} + 684 q^{49} + 324 q^{51} + 1380 q^{59} - 1466 q^{61} - 972 q^{69} + 456 q^{71} - 320 q^{79} + 162 q^{81} + 480 q^{89} - 134 q^{91} + 756 q^{99} + O(q^{100}) \)

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 1.00000i 0 −9.00000 0
49.2 0 3.00000i 0 0 0 1.00000i 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.c 2
4.b odd 2 1 150.4.c.e 2
5.b even 2 1 inner 1200.4.f.c 2
5.c odd 4 1 1200.4.a.i 1
5.c odd 4 1 1200.4.a.bb 1
12.b even 2 1 450.4.c.a 2
20.d odd 2 1 150.4.c.e 2
20.e even 4 1 150.4.a.a 1
20.e even 4 1 150.4.a.h yes 1
60.h even 2 1 450.4.c.a 2
60.l odd 4 1 450.4.a.f 1
60.l odd 4 1 450.4.a.o 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.4.a.a 1 20.e even 4 1
150.4.a.h yes 1 20.e even 4 1
150.4.c.e 2 4.b odd 2 1
150.4.c.e 2 20.d odd 2 1
450.4.a.f 1 60.l odd 4 1
450.4.a.o 1 60.l odd 4 1
450.4.c.a 2 12.b even 2 1
450.4.c.a 2 60.h even 2 1
1200.4.a.i 1 5.c odd 4 1
1200.4.a.bb 1 5.c odd 4 1
1200.4.f.c 2 1.a even 1 1 trivial
1200.4.f.c 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 \(S_{4}^{\mathrm{new}}(1200, [\chi])\):

\( T_{7}^{2} + 1 \)
\( T_{11} + 42 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 9 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 1 + T^{2} \)
$11$ \( ( 42 + T )^{2} \)
$13$ \( 4489 + T^{2} \)
$17$ \( 2916 + T^{2} \)
$19$ \( ( 115 + T )^{2} \)
$23$ \( 26244 + T^{2} \)
$29$ \( ( -210 + T )^{2} \)
$31$ \( ( -193 + T )^{2} \)
$37$ \( 81796 + T^{2} \)
$41$ \( ( -12 + T )^{2} \)
$43$ \( 69169 + T^{2} \)
$47$ \( 171396 + T^{2} \)
$53$ \( 36864 + T^{2} \)
$59$ \( ( -690 + T )^{2} \)
$61$ \( ( 733 + T )^{2} \)
$67$ \( 89401 + T^{2} \)
$71$ \( ( -228 + T )^{2} \)
$73$ \( 879844 + T^{2} \)
$79$ \( ( 160 + T )^{2} \)
$83$ \( 213444 + T^{2} \)
$89$ \( ( -240 + T )^{2} \)
$97$ \( 261121 + T^{2} \)
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