Properties

Label 1200.4.f.v
Level $1200$
Weight $4$
Character orbit 1200.f
Analytic conductor $70.802$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(i, \sqrt{19})\)
Defining polynomial: \(x^{4} - 9 x^{2} + 25\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 75)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 \beta_{1} q^{3} + ( -13 \beta_{1} + \beta_{3} ) q^{7} -9 q^{9} +O(q^{10})\) \( q + 3 \beta_{1} q^{3} + ( -13 \beta_{1} + \beta_{3} ) q^{7} -9 q^{9} + ( -14 + \beta_{2} ) q^{11} + ( -9 \beta_{1} - 4 \beta_{3} ) q^{13} + ( -34 \beta_{1} + 5 \beta_{3} ) q^{17} + ( 3 + \beta_{2} ) q^{19} + ( 39 - 3 \beta_{2} ) q^{21} + ( 66 \beta_{1} + 3 \beta_{3} ) q^{23} -27 \beta_{1} q^{27} + ( -46 - 7 \beta_{2} ) q^{29} + ( -61 + 7 \beta_{2} ) q^{31} + ( -42 \beta_{1} + 3 \beta_{3} ) q^{33} + ( -142 \beta_{1} - 6 \beta_{3} ) q^{37} + ( 27 + 12 \beta_{2} ) q^{39} + ( 196 + 13 \beta_{2} ) q^{41} + ( 345 \beta_{1} + \beta_{3} ) q^{43} + ( 310 \beta_{1} - 8 \beta_{3} ) q^{47} + ( -130 + 26 \beta_{2} ) q^{49} + ( 102 - 15 \beta_{2} ) q^{51} + ( 424 \beta_{1} + 7 \beta_{3} ) q^{53} + ( 9 \beta_{1} + 3 \beta_{3} ) q^{57} + ( 62 - 16 \beta_{2} ) q^{59} + ( 375 - 14 \beta_{2} ) q^{61} + ( 117 \beta_{1} - 9 \beta_{3} ) q^{63} + ( 179 \beta_{1} + 25 \beta_{3} ) q^{67} + ( -198 - 9 \beta_{2} ) q^{69} + ( -412 + 5 \beta_{2} ) q^{71} + ( 54 \beta_{1} + 2 \beta_{3} ) q^{73} + ( 486 \beta_{1} - 27 \beta_{3} ) q^{77} + ( -440 - 40 \beta_{2} ) q^{79} + 81 q^{81} + ( 78 \beta_{1} - 48 \beta_{3} ) q^{83} + ( -138 \beta_{1} - 21 \beta_{3} ) q^{87} + ( 432 - 36 \beta_{2} ) q^{89} + ( 1099 - 43 \beta_{2} ) q^{91} + ( -183 \beta_{1} + 21 \beta_{3} ) q^{93} + 521 \beta_{1} q^{97} + ( 126 - 9 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 36 q^{9} + O(q^{10}) \) \( 4 q - 36 q^{9} - 56 q^{11} + 12 q^{19} + 156 q^{21} - 184 q^{29} - 244 q^{31} + 108 q^{39} + 784 q^{41} - 520 q^{49} + 408 q^{51} + 248 q^{59} + 1500 q^{61} - 792 q^{69} - 1648 q^{71} - 1760 q^{79} + 324 q^{81} + 1728 q^{89} + 4396 q^{91} + 504 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 9 x^{2} + 25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} - 4 \nu \)\()/5\)
\(\beta_{2}\)\(=\)\((\)\( -4 \nu^{3} + 56 \nu \)\()/5\)
\(\beta_{3}\)\(=\)\( 8 \nu^{2} - 36 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + 4 \beta_{1}\)\()/8\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 36\)\()/8\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{2} + 14 \beta_{1}\)\()/2\)

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
2.17945 0.500000i
−2.17945 0.500000i
−2.17945 + 0.500000i
2.17945 + 0.500000i
0 3.00000i 0 0 0 4.43560i 0 −9.00000 0
49.2 0 3.00000i 0 0 0 30.4356i 0 −9.00000 0
49.3 0 3.00000i 0 0 0 30.4356i 0 −9.00000 0
49.4 0 3.00000i 0 0 0 4.43560i 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.v 4
4.b odd 2 1 75.4.b.c 4
5.b even 2 1 inner 1200.4.f.v 4
5.c odd 4 1 1200.4.a.bl 2
5.c odd 4 1 1200.4.a.bu 2
12.b even 2 1 225.4.b.h 4
20.d odd 2 1 75.4.b.c 4
20.e even 4 1 75.4.a.d 2
20.e even 4 1 75.4.a.e yes 2
60.h even 2 1 225.4.b.h 4
60.l odd 4 1 225.4.a.j 2
60.l odd 4 1 225.4.a.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.4.a.d 2 20.e even 4 1
75.4.a.e yes 2 20.e even 4 1
75.4.b.c 4 4.b odd 2 1
75.4.b.c 4 20.d odd 2 1
225.4.a.j 2 60.l odd 4 1
225.4.a.n 2 60.l odd 4 1
225.4.b.h 4 12.b even 2 1
225.4.b.h 4 60.h even 2 1
1200.4.a.bl 2 5.c odd 4 1
1200.4.a.bu 2 5.c odd 4 1
1200.4.f.v 4 1.a even 1 1 trivial
1200.4.f.v 4 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 \(S_{4}^{\mathrm{new}}(1200, [\chi])\):

\( T_{7}^{4} + 946 T_{7}^{2} + 18225 \)
\( T_{11}^{2} + 28 T_{11} - 108 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 9 + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( 18225 + 946 T^{2} + T^{4} \)
$11$ \( ( -108 + 28 T + T^{2} )^{2} \)
$13$ \( 22877089 + 9890 T^{2} + T^{4} \)
$17$ \( 41525136 + 17512 T^{2} + T^{4} \)
$19$ \( ( -295 - 6 T + T^{2} )^{2} \)
$23$ \( 2624400 + 14184 T^{2} + T^{4} \)
$29$ \( ( -12780 + 92 T + T^{2} )^{2} \)
$31$ \( ( -11175 + 122 T + T^{2} )^{2} \)
$37$ \( 85008400 + 62216 T^{2} + T^{4} \)
$41$ \( ( -12960 - 392 T + T^{2} )^{2} \)
$43$ \( 14094675841 + 238658 T^{2} + T^{4} \)
$47$ \( 5874302736 + 231112 T^{2} + T^{4} \)
$53$ \( 27185414400 + 389344 T^{2} + T^{4} \)
$59$ \( ( -73980 - 124 T + T^{2} )^{2} \)
$61$ \( ( 81041 - 750 T + T^{2} )^{2} \)
$67$ \( 24951045681 + 444082 T^{2} + T^{4} \)
$71$ \( ( 162144 + 824 T + T^{2} )^{2} \)
$73$ \( 2890000 + 8264 T^{2} + T^{4} \)
$79$ \( ( -292800 + 880 T + T^{2} )^{2} \)
$83$ \( 482096926224 + 1413000 T^{2} + T^{4} \)
$89$ \( ( -207360 - 864 T + T^{2} )^{2} \)
$97$ \( ( 271441 + T^{2} )^{2} \)
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