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: \( x^{2} - x - 10 \) Copy content Toggle raw display
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.

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

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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1200))\):

\( T_{7}^{2} - 6T_{7} - 360 \) Copy content Toggle raw display
\( T_{11}^{2} - 42T_{11} + 72 \) Copy content Toggle raw display

Hecke characteristic polynomials

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