Properties

Label 1200.4.a.bt
Level $1200$
Weight $4$
Character orbit 1200.a
Self dual yes
Analytic conductor $70.802$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1200.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(70.8022920069\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{41}) \)
Defining polynomial: \(x^{2} - x - 10\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{41}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 q^{3} + ( 3 + 3 \beta ) q^{7} + 9 q^{9} +O(q^{10})\) \( q + 3 q^{3} + ( 3 + 3 \beta ) q^{7} + 9 q^{9} + ( 21 - 3 \beta ) q^{11} + ( -39 - 3 \beta ) q^{13} + ( -51 + 5 \beta ) q^{17} + ( -28 - 12 \beta ) q^{19} + ( 9 + 9 \beta ) q^{21} + ( -24 - 4 \beta ) q^{23} + 27 q^{27} + ( -159 - 21 \beta ) q^{29} + ( -26 - 6 \beta ) q^{31} + ( 63 - 9 \beta ) q^{33} + ( -153 + 27 \beta ) q^{37} + ( -117 - 9 \beta ) q^{39} + ( -204 + 6 \beta ) q^{41} + ( 60 - 48 \beta ) q^{43} + ( 90 + 46 \beta ) q^{47} + ( 35 + 18 \beta ) q^{49} + ( -153 + 15 \beta ) q^{51} + ( -201 - 41 \beta ) q^{53} + ( -84 - 36 \beta ) q^{57} + ( 93 - 3 \beta ) q^{59} + ( 170 - 48 \beta ) q^{61} + ( 27 + 27 \beta ) q^{63} + ( 366 - 30 \beta ) q^{67} + ( -72 - 12 \beta ) q^{69} + ( 18 + 90 \beta ) q^{71} + ( -666 + 54 \beta ) q^{73} + ( -306 + 54 \beta ) q^{77} + ( -190 + 150 \beta ) q^{79} + 81 q^{81} + ( -492 + 104 \beta ) q^{83} + ( -477 - 63 \beta ) q^{87} + ( 558 + 72 \beta ) q^{89} + ( -486 - 126 \beta ) q^{91} + ( -78 - 18 \beta ) q^{93} + ( 384 - 120 \beta ) q^{97} + ( 189 - 27 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{3} + 6 q^{7} + 18 q^{9} + O(q^{10}) \) \( 2 q + 6 q^{3} + 6 q^{7} + 18 q^{9} + 42 q^{11} - 78 q^{13} - 102 q^{17} - 56 q^{19} + 18 q^{21} - 48 q^{23} + 54 q^{27} - 318 q^{29} - 52 q^{31} + 126 q^{33} - 306 q^{37} - 234 q^{39} - 408 q^{41} + 120 q^{43} + 180 q^{47} + 70 q^{49} - 306 q^{51} - 402 q^{53} - 168 q^{57} + 186 q^{59} + 340 q^{61} + 54 q^{63} + 732 q^{67} - 144 q^{69} + 36 q^{71} - 1332 q^{73} - 612 q^{77} - 380 q^{79} + 162 q^{81} - 984 q^{83} - 954 q^{87} + 1116 q^{89} - 972 q^{91} - 156 q^{93} + 768 q^{97} + 378 q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.70156
3.70156
0 3.00000 0 0 0 −16.2094 0 9.00000 0
1.2 0 3.00000 0 0 0 22.2094 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1200.4.a.bt 2
4.b odd 2 1 75.4.a.c 2
5.b even 2 1 1200.4.a.bn 2
5.c odd 4 2 240.4.f.f 4
12.b even 2 1 225.4.a.o 2
15.e even 4 2 720.4.f.j 4
20.d odd 2 1 75.4.a.f 2
20.e even 4 2 15.4.b.a 4
40.i odd 4 2 960.4.f.p 4
40.k even 4 2 960.4.f.q 4
60.h even 2 1 225.4.a.i 2
60.l odd 4 2 45.4.b.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.b.a 4 20.e even 4 2
45.4.b.b 4 60.l odd 4 2
75.4.a.c 2 4.b odd 2 1
75.4.a.f 2 20.d odd 2 1
225.4.a.i 2 60.h even 2 1
225.4.a.o 2 12.b even 2 1
240.4.f.f 4 5.c odd 4 2
720.4.f.j 4 15.e even 4 2
960.4.f.p 4 40.i odd 4 2
960.4.f.q 4 40.k even 4 2
1200.4.a.bn 2 5.b even 2 1
1200.4.a.bt 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1200))\):

\( T_{7}^{2} - 6 T_{7} - 360 \)
\( T_{11}^{2} - 42 T_{11} + 72 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( -3 + T )^{2} \)
$5$ \( T^{2} \)
$7$ \( -360 - 6 T + T^{2} \)
$11$ \( 72 - 42 T + T^{2} \)
$13$ \( 1152 + 78 T + T^{2} \)
$17$ \( 1576 + 102 T + T^{2} \)
$19$ \( -5120 + 56 T + T^{2} \)
$23$ \( -80 + 48 T + T^{2} \)
$29$ \( 7200 + 318 T + T^{2} \)
$31$ \( -800 + 52 T + T^{2} \)
$37$ \( -6480 + 306 T + T^{2} \)
$41$ \( 40140 + 408 T + T^{2} \)
$43$ \( -90864 - 120 T + T^{2} \)
$47$ \( -78656 - 180 T + T^{2} \)
$53$ \( -28520 + 402 T + T^{2} \)
$59$ \( 8280 - 186 T + T^{2} \)
$61$ \( -65564 - 340 T + T^{2} \)
$67$ \( 97056 - 732 T + T^{2} \)
$71$ \( -331776 - 36 T + T^{2} \)
$73$ \( 324000 + 1332 T + T^{2} \)
$79$ \( -886400 + 380 T + T^{2} \)
$83$ \( -201392 + 984 T + T^{2} \)
$89$ \( 98820 - 1116 T + T^{2} \)
$97$ \( -442944 - 768 T + T^{2} \)
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