Properties

Label 1800.2.k.p
Level $1800$
Weight $2$
Character orbit 1800.k
Analytic conductor $14.373$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1800.k (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.3730723638\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.399424.1
Defining polynomial: \(x^{6} - 2 x^{5} + 3 x^{4} - 6 x^{3} + 6 x^{2} - 8 x + 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + \beta_{2} q^{4} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} ) q^{7} + ( 2 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{8} +O(q^{10})\) \( q -\beta_{1} q^{2} + \beta_{2} q^{4} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} ) q^{7} + ( 2 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{8} + ( 2 \beta_{1} - \beta_{4} + 2 \beta_{5} ) q^{11} + ( -1 - 3 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} ) q^{13} + ( 4 + 2 \beta_{1} + 2 \beta_{2} ) q^{14} + ( 1 - \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{16} + ( 1 - \beta_{1} - \beta_{2} + 3 \beta_{3} - \beta_{4} + \beta_{5} ) q^{17} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{19} + ( 2 - 2 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} ) q^{22} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} ) q^{23} + ( -4 + 2 \beta_{2} ) q^{26} + ( 4 - 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} ) q^{28} -\beta_{4} q^{29} + ( -3 - \beta_{1} - \beta_{2} + 3 \beta_{3} - \beta_{4} + \beta_{5} ) q^{31} + ( 2 - \beta_{1} + \beta_{2} + 3 \beta_{3} - \beta_{4} ) q^{32} + ( 2 - 2 \beta_{1} - 2 \beta_{4} - 2 \beta_{5} ) q^{34} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} - 3 \beta_{5} ) q^{37} + ( -2 - 2 \beta_{4} + 2 \beta_{5} ) q^{38} + ( 4 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} ) q^{41} + ( 2 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{5} ) q^{43} + ( -2 - 4 \beta_{1} - 2 \beta_{2} - 4 \beta_{4} + 2 \beta_{5} ) q^{44} + ( -4 - 2 \beta_{2} ) q^{46} + ( 3 + \beta_{1} + 3 \beta_{2} - 5 \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{47} + ( 7 + 2 \beta_{1} + 2 \beta_{2} - 6 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{49} + ( 4 + 6 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{52} -\beta_{4} q^{53} + ( -2 - 4 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} + 2 \beta_{5} ) q^{56} -2 \beta_{3} q^{58} + ( -2 \beta_{1} - \beta_{4} - 2 \beta_{5} ) q^{59} + ( -4 \beta_{1} + 2 \beta_{4} - 4 \beta_{5} ) q^{61} + ( 2 + 2 \beta_{1} - 2 \beta_{4} - 2 \beta_{5} ) q^{62} + ( 5 - \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + \beta_{4} - 3 \beta_{5} ) q^{64} -2 \beta_{4} q^{67} + ( -2 - 2 \beta_{1} + 2 \beta_{2} - 6 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{68} + ( -4 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} ) q^{71} + 6 q^{73} + ( 2 \beta_{2} - 4 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} ) q^{74} + ( 2 + 2 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{76} + ( -4 - 4 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} ) q^{77} + ( 5 - \beta_{1} - \beta_{2} + 3 \beta_{3} - \beta_{4} + \beta_{5} ) q^{79} + ( -8 - 6 \beta_{1} - 4 \beta_{2} ) q^{82} + ( 2 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{83} + ( 4 - 4 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} ) q^{86} + ( -2 + 2 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} ) q^{88} + ( 4 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} ) q^{89} + ( 4 + 8 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} ) q^{91} + ( -4 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{92} + ( 2 \beta_{2} + 4 \beta_{4} + 4 \beta_{5} ) q^{94} + ( -4 + 2 \beta_{1} + 2 \beta_{2} - 6 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{97} + ( -4 - 5 \beta_{1} + 4 \beta_{4} + 4 \beta_{5} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{2} - 2 q^{4} - 4 q^{7} + 8 q^{8} + O(q^{10}) \) \( 6 q + 2 q^{2} - 2 q^{4} - 4 q^{7} + 8 q^{8} + 16 q^{14} + 10 q^{16} + 12 q^{17} + 20 q^{22} - 8 q^{23} - 28 q^{26} + 28 q^{28} - 12 q^{31} + 12 q^{32} + 12 q^{34} - 8 q^{38} + 20 q^{41} + 4 q^{44} - 20 q^{46} + 8 q^{47} + 30 q^{49} + 8 q^{52} - 4 q^{56} + 4 q^{62} + 22 q^{64} - 16 q^{68} + 8 q^{71} + 36 q^{73} - 12 q^{74} + 12 q^{76} + 36 q^{79} - 28 q^{82} + 16 q^{86} - 12 q^{88} + 28 q^{89} - 24 q^{92} + 4 q^{94} - 36 q^{97} - 6 q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 2 x^{5} + 3 x^{4} - 6 x^{3} + 6 x^{2} - 8 x + 8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{5} - 2 \nu^{4} + 3 \nu^{3} - 6 \nu^{2} + 6 \nu - 8 \)\()/4\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{5} + \nu^{4} - \nu^{3} + 3 \nu^{2} + 2 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{5} + \nu^{4} - 3 \nu^{3} + 3 \nu^{2} - 2 \nu + 6 \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{5} - 3 \nu^{3} + 4 \nu^{2} - 2 \nu + 8 \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( 3 \nu^{5} - 2 \nu^{4} + 5 \nu^{3} - 6 \nu^{2} + 6 \nu - 12 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5} + \beta_{3} + \beta_{2} + \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - \beta_{1} - 1\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{5} - 3 \beta_{3} + \beta_{2} - \beta_{1} + 3\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(\beta_{5} - 3 \beta_{4} + 3 \beta_{3} + \beta_{2} - \beta_{1} + 3\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(5 \beta_{5} + 3 \beta_{3} - \beta_{2} - 3 \beta_{1} + 1\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1001\) \(1351\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
901.1
−0.671462 + 1.24464i
−0.671462 1.24464i
0.264658 + 1.38923i
0.264658 1.38923i
1.40680 + 0.144584i
1.40680 0.144584i
−0.671462 1.24464i 0 −1.09828 + 1.67146i 0 0 −4.68585 2.81783 + 0.244644i 0 0
901.2 −0.671462 + 1.24464i 0 −1.09828 1.67146i 0 0 −4.68585 2.81783 0.244644i 0 0
901.3 0.264658 1.38923i 0 −1.85991 0.735342i 0 0 −0.941367 −1.51380 + 2.38923i 0 0
901.4 0.264658 + 1.38923i 0 −1.85991 + 0.735342i 0 0 −0.941367 −1.51380 2.38923i 0 0
901.5 1.40680 0.144584i 0 1.95819 0.406803i 0 0 3.62721 2.69597 0.855416i 0 0
901.6 1.40680 + 0.144584i 0 1.95819 + 0.406803i 0 0 3.62721 2.69597 + 0.855416i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 901.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1800.2.k.p 6
3.b odd 2 1 600.2.k.c 6
4.b odd 2 1 7200.2.k.p 6
5.b even 2 1 360.2.k.f 6
5.c odd 4 1 1800.2.d.q 6
5.c odd 4 1 1800.2.d.r 6
8.b even 2 1 inner 1800.2.k.p 6
8.d odd 2 1 7200.2.k.p 6
12.b even 2 1 2400.2.k.c 6
15.d odd 2 1 120.2.k.b 6
15.e even 4 1 600.2.d.e 6
15.e even 4 1 600.2.d.f 6
20.d odd 2 1 1440.2.k.f 6
20.e even 4 1 7200.2.d.q 6
20.e even 4 1 7200.2.d.r 6
24.f even 2 1 2400.2.k.c 6
24.h odd 2 1 600.2.k.c 6
40.e odd 2 1 1440.2.k.f 6
40.f even 2 1 360.2.k.f 6
40.i odd 4 1 1800.2.d.q 6
40.i odd 4 1 1800.2.d.r 6
40.k even 4 1 7200.2.d.q 6
40.k even 4 1 7200.2.d.r 6
60.h even 2 1 480.2.k.b 6
60.l odd 4 1 2400.2.d.e 6
60.l odd 4 1 2400.2.d.f 6
120.i odd 2 1 120.2.k.b 6
120.m even 2 1 480.2.k.b 6
120.q odd 4 1 2400.2.d.e 6
120.q odd 4 1 2400.2.d.f 6
120.w even 4 1 600.2.d.e 6
120.w even 4 1 600.2.d.f 6
240.t even 4 1 3840.2.a.bo 3
240.t even 4 1 3840.2.a.br 3
240.bm odd 4 1 3840.2.a.bp 3
240.bm odd 4 1 3840.2.a.bq 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.k.b 6 15.d odd 2 1
120.2.k.b 6 120.i odd 2 1
360.2.k.f 6 5.b even 2 1
360.2.k.f 6 40.f even 2 1
480.2.k.b 6 60.h even 2 1
480.2.k.b 6 120.m even 2 1
600.2.d.e 6 15.e even 4 1
600.2.d.e 6 120.w even 4 1
600.2.d.f 6 15.e even 4 1
600.2.d.f 6 120.w even 4 1
600.2.k.c 6 3.b odd 2 1
600.2.k.c 6 24.h odd 2 1
1440.2.k.f 6 20.d odd 2 1
1440.2.k.f 6 40.e odd 2 1
1800.2.d.q 6 5.c odd 4 1
1800.2.d.q 6 40.i odd 4 1
1800.2.d.r 6 5.c odd 4 1
1800.2.d.r 6 40.i odd 4 1
1800.2.k.p 6 1.a even 1 1 trivial
1800.2.k.p 6 8.b even 2 1 inner
2400.2.d.e 6 60.l odd 4 1
2400.2.d.e 6 120.q odd 4 1
2400.2.d.f 6 60.l odd 4 1
2400.2.d.f 6 120.q odd 4 1
2400.2.k.c 6 12.b even 2 1
2400.2.k.c 6 24.f even 2 1
3840.2.a.bo 3 240.t even 4 1
3840.2.a.bp 3 240.bm odd 4 1
3840.2.a.bq 3 240.bm odd 4 1
3840.2.a.br 3 240.t even 4 1
7200.2.d.q 6 20.e even 4 1
7200.2.d.q 6 40.k even 4 1
7200.2.d.r 6 20.e even 4 1
7200.2.d.r 6 40.k even 4 1
7200.2.k.p 6 4.b odd 2 1
7200.2.k.p 6 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1800, [\chi])\):

\( T_{7}^{3} + 2 T_{7}^{2} - 16 T_{7} - 16 \)
\( T_{11}^{6} + 64 T_{11}^{4} + 1088 T_{11}^{2} + 4096 \)
\( T_{17}^{3} - 6 T_{17}^{2} - 16 T_{17} + 32 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 8 - 8 T + 6 T^{2} - 6 T^{3} + 3 T^{4} - 2 T^{5} + T^{6} \)
$3$ \( T^{6} \)
$5$ \( T^{6} \)
$7$ \( ( -16 - 16 T + 2 T^{2} + T^{3} )^{2} \)
$11$ \( 4096 + 1088 T^{2} + 64 T^{4} + T^{6} \)
$13$ \( 256 + 784 T^{2} + 56 T^{4} + T^{6} \)
$17$ \( ( 32 - 16 T - 6 T^{2} + T^{3} )^{2} \)
$19$ \( 256 + 272 T^{2} + 40 T^{4} + T^{6} \)
$23$ \( ( -16 - 12 T + 4 T^{2} + T^{3} )^{2} \)
$29$ \( ( 4 + T^{2} )^{3} \)
$31$ \( ( -64 - 16 T + 6 T^{2} + T^{3} )^{2} \)
$37$ \( 65536 + 5648 T^{2} + 136 T^{4} + T^{6} \)
$41$ \( ( 232 - 36 T - 10 T^{2} + T^{3} )^{2} \)
$43$ \( 16384 + 5120 T^{2} + 144 T^{4} + T^{6} \)
$47$ \( ( 496 - 92 T - 4 T^{2} + T^{3} )^{2} \)
$53$ \( ( 4 + T^{2} )^{3} \)
$59$ \( 1024 + 576 T^{2} + 80 T^{4} + T^{6} \)
$61$ \( 262144 + 17408 T^{2} + 256 T^{4} + T^{6} \)
$67$ \( ( 16 + T^{2} )^{3} \)
$71$ \( ( -64 - 112 T - 4 T^{2} + T^{3} )^{2} \)
$73$ \( ( -6 + T )^{6} \)
$79$ \( ( -64 + 80 T - 18 T^{2} + T^{3} )^{2} \)
$83$ \( 65536 + 8448 T^{2} + 224 T^{4} + T^{6} \)
$89$ \( ( 184 - 4 T - 14 T^{2} + T^{3} )^{2} \)
$97$ \( ( -328 - 4 T + 18 T^{2} + T^{3} )^{2} \)
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