Properties

Label 1200.4.f.i
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 600)
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.

\(f(q)\) \(=\) \( q -3 i q^{3} + 5 i q^{7} -9 q^{9} +O(q^{10})\) \( q -3 i q^{3} + 5 i q^{7} -9 q^{9} -14 q^{11} + i q^{13} -46 i q^{17} + 19 q^{19} + 15 q^{21} + 46 i q^{23} + 27 i q^{27} -14 q^{29} -133 q^{31} + 42 i q^{33} -258 i q^{37} + 3 q^{39} + 84 q^{41} + 167 i q^{43} + 410 i q^{47} + 318 q^{49} -138 q^{51} + 456 i q^{53} -57 i q^{57} -194 q^{59} -17 q^{61} -45 i q^{63} + 653 i q^{67} + 138 q^{69} -828 q^{71} + 570 i q^{73} -70 i q^{77} -552 q^{79} + 81 q^{81} -142 i q^{83} + 42 i q^{87} + 1104 q^{89} -5 q^{91} + 399 i q^{93} -841 i q^{97} + 126 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 18 q^{9} + O(q^{10}) \) \( 2 q - 18 q^{9} - 28 q^{11} + 38 q^{19} + 30 q^{21} - 28 q^{29} - 266 q^{31} + 6 q^{39} + 168 q^{41} + 636 q^{49} - 276 q^{51} - 388 q^{59} - 34 q^{61} + 276 q^{69} - 1656 q^{71} - 1104 q^{79} + 162 q^{81} + 2208 q^{89} - 10 q^{91} + 252 q^{99} + O(q^{100}) \)

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 5.00000i 0 −9.00000 0
49.2 0 3.00000i 0 0 0 5.00000i 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.i 2
4.b odd 2 1 600.4.f.e 2
5.b even 2 1 inner 1200.4.f.i 2
5.c odd 4 1 1200.4.a.g 1
5.c odd 4 1 1200.4.a.bd 1
12.b even 2 1 1800.4.f.l 2
20.d odd 2 1 600.4.f.e 2
20.e even 4 1 600.4.a.e 1
20.e even 4 1 600.4.a.n yes 1
60.h even 2 1 1800.4.f.l 2
60.l odd 4 1 1800.4.a.m 1
60.l odd 4 1 1800.4.a.v 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.4.a.e 1 20.e even 4 1
600.4.a.n yes 1 20.e even 4 1
600.4.f.e 2 4.b odd 2 1
600.4.f.e 2 20.d odd 2 1
1200.4.a.g 1 5.c odd 4 1
1200.4.a.bd 1 5.c odd 4 1
1200.4.f.i 2 1.a even 1 1 trivial
1200.4.f.i 2 5.b even 2 1 inner
1800.4.a.m 1 60.l odd 4 1
1800.4.a.v 1 60.l odd 4 1
1800.4.f.l 2 12.b even 2 1
1800.4.f.l 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 \(S_{4}^{\mathrm{new}}(1200, [\chi])\):

\( T_{7}^{2} + 25 \)
\( T_{11} + 14 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 9 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 25 + T^{2} \)
$11$ \( ( 14 + T )^{2} \)
$13$ \( 1 + T^{2} \)
$17$ \( 2116 + T^{2} \)
$19$ \( ( -19 + T )^{2} \)
$23$ \( 2116 + T^{2} \)
$29$ \( ( 14 + T )^{2} \)
$31$ \( ( 133 + T )^{2} \)
$37$ \( 66564 + T^{2} \)
$41$ \( ( -84 + T )^{2} \)
$43$ \( 27889 + T^{2} \)
$47$ \( 168100 + T^{2} \)
$53$ \( 207936 + T^{2} \)
$59$ \( ( 194 + T )^{2} \)
$61$ \( ( 17 + T )^{2} \)
$67$ \( 426409 + T^{2} \)
$71$ \( ( 828 + T )^{2} \)
$73$ \( 324900 + T^{2} \)
$79$ \( ( 552 + T )^{2} \)
$83$ \( 20164 + T^{2} \)
$89$ \( ( -1104 + T )^{2} \)
$97$ \( 707281 + T^{2} \)
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