Properties

Label 1200.3.c.k
Level $1200$
Weight $3$
Character orbit 1200.c
Analytic conductor $32.698$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1200.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(32.6976317232\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.40960000.1
Defining polynomial: \(x^{8} + 7 x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 30)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{2} q^{3} + ( -2 \beta_{1} - 2 \beta_{2} + \beta_{7} ) q^{7} + ( 2 - \beta_{5} + \beta_{6} ) q^{9} +O(q^{10})\) \( q -\beta_{2} q^{3} + ( -2 \beta_{1} - 2 \beta_{2} + \beta_{7} ) q^{7} + ( 2 - \beta_{5} + \beta_{6} ) q^{9} + ( -\beta_{4} - \beta_{5} ) q^{11} + 5 \beta_{1} q^{13} + ( 2 \beta_{1} + 4 \beta_{2} + 5 \beta_{3} + \beta_{7} ) q^{17} + ( 8 - 3 \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{19} + ( -13 - 3 \beta_{5} - \beta_{6} ) q^{21} + ( -2 \beta_{1} - 4 \beta_{2} - \beta_{7} ) q^{23} + ( -4 \beta_{1} - \beta_{2} + 6 \beta_{3} + \beta_{7} ) q^{27} + ( 4 \beta_{4} - 4 \beta_{6} ) q^{29} -8 q^{31} + ( -3 \beta_{1} + 3 \beta_{3} + 3 \beta_{7} ) q^{33} + ( -15 \beta_{1} - 8 \beta_{2} + 4 \beta_{7} ) q^{37} + ( -10 + 5 \beta_{6} ) q^{39} + ( 6 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} ) q^{41} + ( 10 \beta_{1} + 6 \beta_{2} - 3 \beta_{7} ) q^{43} + ( 10 \beta_{1} + 20 \beta_{2} + 5 \beta_{7} ) q^{47} + ( -45 - 6 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} ) q^{49} + ( -18 + 15 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} ) q^{51} + ( -2 \beta_{1} - 4 \beta_{2} - 5 \beta_{3} - \beta_{7} ) q^{53} + ( 15 \beta_{1} - 8 \beta_{2} - 9 \beta_{3} + 3 \beta_{7} ) q^{57} + ( -2 \beta_{4} - 6 \beta_{5} - 4 \beta_{6} ) q^{59} + ( -16 + 6 \beta_{4} - 2 \beta_{5} + 4 \beta_{6} ) q^{61} + ( -28 \beta_{1} + 8 \beta_{2} + 6 \beta_{3} + 7 \beta_{7} ) q^{63} + ( 38 \beta_{1} - 6 \beta_{2} + 3 \beta_{7} ) q^{67} + ( 33 - 3 \beta_{5} + 3 \beta_{6} ) q^{69} + ( -\beta_{4} - 5 \beta_{5} - 4 \beta_{6} ) q^{71} + ( -21 \beta_{1} + 8 \beta_{2} - 4 \beta_{7} ) q^{73} + ( 12 \beta_{1} + 24 \beta_{2} + 2 \beta_{3} + 6 \beta_{7} ) q^{77} + ( -28 + 3 \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{79} + ( 7 + 18 \beta_{4} - 2 \beta_{5} - 4 \beta_{6} ) q^{81} + ( -2 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} - \beta_{7} ) q^{83} + ( -36 \beta_{1} - 12 \beta_{2} - 12 \beta_{3} ) q^{87} + ( -8 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} ) q^{89} + ( 20 + 15 \beta_{4} - 5 \beta_{5} + 10 \beta_{6} ) q^{91} + 8 \beta_{2} q^{93} + ( 41 \beta_{1} + 8 \beta_{2} - 4 \beta_{7} ) q^{97} + ( -48 + 9 \beta_{4} - 3 \beta_{5} - 6 \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 16q^{9} + O(q^{10}) \) \( 8q + 16q^{9} + 64q^{19} - 104q^{21} - 64q^{31} - 80q^{39} - 360q^{49} - 144q^{51} - 128q^{61} + 264q^{69} - 224q^{79} + 56q^{81} + 160q^{91} - 384q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 7 x^{4} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -2 \nu^{6} - 16 \nu^{2} \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{7} + \nu^{6} - \nu^{5} - 2 \nu^{4} - 8 \nu^{3} + 8 \nu^{2} - 8 \nu - 7 \)\()/3\)
\(\beta_{3}\)\(=\)\( -2 \nu^{7} + \nu^{5} - 13 \nu^{3} + 5 \nu \)
\(\beta_{4}\)\(=\)\( -2 \nu^{7} + \nu^{6} - 14 \nu^{3} + 6 \nu^{2} + 2 \nu \)
\(\beta_{5}\)\(=\)\( -2 \nu^{7} - \nu^{6} - 2 \nu^{5} - 12 \nu^{3} - 6 \nu^{2} - 12 \nu \)
\(\beta_{6}\)\(=\)\( -2 \nu^{7} - 2 \nu^{6} - 14 \nu^{3} - 12 \nu^{2} + 2 \nu \)
\(\beta_{7}\)\(=\)\((\)\( 10 \nu^{7} + \nu^{5} - 4 \nu^{4} + 71 \nu^{3} + 17 \nu - 14 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} + \beta_{6} + 2 \beta_{4} - \beta_{3} - 2 \beta_{2} - \beta_{1}\)\()/12\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{6} - 2 \beta_{4} - 9 \beta_{1}\)\()/12\)
\(\nu^{3}\)\(=\)\((\)\(2 \beta_{7} - 2 \beta_{6} + 3 \beta_{5} - \beta_{4} + 4 \beta_{3} - 4 \beta_{2} - 2 \beta_{1}\)\()/12\)
\(\nu^{4}\)\(=\)\((\)\(-\beta_{7} - \beta_{3} - 4 \beta_{2} - 2 \beta_{1} - 14\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-5 \beta_{7} - 5 \beta_{6} - 3 \beta_{5} - 13 \beta_{4} + 11 \beta_{3} + 10 \beta_{2} + 5 \beta_{1}\)\()/12\)
\(\nu^{6}\)\(=\)\((\)\(-8 \beta_{6} + 8 \beta_{4} + 27 \beta_{1}\)\()/6\)
\(\nu^{7}\)\(=\)\((\)\(-13 \beta_{7} + 13 \beta_{6} - 21 \beta_{5} + 5 \beta_{4} - 29 \beta_{3} + 26 \beta_{2} + 13 \beta_{1}\)\()/12\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
1.14412 1.14412i
1.14412 + 1.14412i
−1.14412 1.14412i
−1.14412 + 1.14412i
−0.437016 0.437016i
−0.437016 + 0.437016i
0.437016 0.437016i
0.437016 + 0.437016i
0 −2.94317 0.581139i 0 0 0 11.4868i 0 8.32456 + 3.42079i 0
449.2 0 −2.94317 + 0.581139i 0 0 0 11.4868i 0 8.32456 3.42079i 0
449.3 0 −1.52896 2.58114i 0 0 0 7.48683i 0 −4.32456 + 7.89292i 0
449.4 0 −1.52896 + 2.58114i 0 0 0 7.48683i 0 −4.32456 7.89292i 0
449.5 0 1.52896 2.58114i 0 0 0 7.48683i 0 −4.32456 7.89292i 0
449.6 0 1.52896 + 2.58114i 0 0 0 7.48683i 0 −4.32456 + 7.89292i 0
449.7 0 2.94317 0.581139i 0 0 0 11.4868i 0 8.32456 3.42079i 0
449.8 0 2.94317 + 0.581139i 0 0 0 11.4868i 0 8.32456 + 3.42079i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 449.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1200.3.c.k 8
3.b odd 2 1 inner 1200.3.c.k 8
4.b odd 2 1 150.3.b.b 8
5.b even 2 1 inner 1200.3.c.k 8
5.c odd 4 1 240.3.l.c 4
5.c odd 4 1 1200.3.l.u 4
12.b even 2 1 150.3.b.b 8
15.d odd 2 1 inner 1200.3.c.k 8
15.e even 4 1 240.3.l.c 4
15.e even 4 1 1200.3.l.u 4
20.d odd 2 1 150.3.b.b 8
20.e even 4 1 30.3.d.a 4
20.e even 4 1 150.3.d.c 4
40.i odd 4 1 960.3.l.f 4
40.k even 4 1 960.3.l.e 4
60.h even 2 1 150.3.b.b 8
60.l odd 4 1 30.3.d.a 4
60.l odd 4 1 150.3.d.c 4
120.q odd 4 1 960.3.l.e 4
120.w even 4 1 960.3.l.f 4
180.v odd 12 2 810.3.h.a 8
180.x even 12 2 810.3.h.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.3.d.a 4 20.e even 4 1
30.3.d.a 4 60.l odd 4 1
150.3.b.b 8 4.b odd 2 1
150.3.b.b 8 12.b even 2 1
150.3.b.b 8 20.d odd 2 1
150.3.b.b 8 60.h even 2 1
150.3.d.c 4 20.e even 4 1
150.3.d.c 4 60.l odd 4 1
240.3.l.c 4 5.c odd 4 1
240.3.l.c 4 15.e even 4 1
810.3.h.a 8 180.v odd 12 2
810.3.h.a 8 180.x even 12 2
960.3.l.e 4 40.k even 4 1
960.3.l.e 4 120.q odd 4 1
960.3.l.f 4 40.i odd 4 1
960.3.l.f 4 120.w even 4 1
1200.3.c.k 8 1.a even 1 1 trivial
1200.3.c.k 8 3.b odd 2 1 inner
1200.3.c.k 8 5.b even 2 1 inner
1200.3.c.k 8 15.d odd 2 1 inner
1200.3.l.u 4 5.c odd 4 1
1200.3.l.u 4 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1200, [\chi])\):

\( T_{7}^{4} + 188 T_{7}^{2} + 7396 \)
\( T_{11}^{2} + 72 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( 6561 - 648 T^{2} + 18 T^{4} - 8 T^{6} + T^{8} \)
$5$ \( T^{8} \)
$7$ \( ( 7396 + 188 T^{2} + T^{4} )^{2} \)
$11$ \( ( 72 + T^{2} )^{4} \)
$13$ \( ( 100 + T^{2} )^{4} \)
$17$ \( ( 11664 - 936 T^{2} + T^{4} )^{2} \)
$19$ \( ( -296 - 16 T + T^{2} )^{4} \)
$23$ \( ( 26244 - 396 T^{2} + T^{4} )^{2} \)
$29$ \( ( 720 + T^{2} )^{4} \)
$31$ \( ( 8 + T )^{8} \)
$37$ \( ( 913936 + 3848 T^{2} + T^{4} )^{2} \)
$41$ \( ( 944784 + 2664 T^{2} + T^{4} )^{2} \)
$43$ \( ( 376996 + 2012 T^{2} + T^{4} )^{2} \)
$47$ \( ( 16402500 - 9900 T^{2} + T^{4} )^{2} \)
$53$ \( ( 11664 - 936 T^{2} + T^{4} )^{2} \)
$59$ \( ( 3504384 + 6624 T^{2} + T^{4} )^{2} \)
$61$ \( ( -1184 + 32 T + T^{2} )^{4} \)
$67$ \( ( 34975396 + 15068 T^{2} + T^{4} )^{2} \)
$71$ \( ( 1166400 + 5040 T^{2} + T^{4} )^{2} \)
$73$ \( ( 1123600 + 7880 T^{2} + T^{4} )^{2} \)
$79$ \( ( 424 + 56 T + T^{2} )^{4} \)
$83$ \( ( 324 - 684 T^{2} + T^{4} )^{2} \)
$89$ \( ( 186624 + 3744 T^{2} + T^{4} )^{2} \)
$97$ \( ( 16289296 + 13832 T^{2} + T^{4} )^{2} \)
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