Properties

Label 1800.2.k.q
Level $1800$
Weight $2$
Character orbit 1800.k
Analytic conductor $14.373$
Analytic rank $0$
Dimension $8$
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: \(8\)
Coefficient field: 8.0.214798336.3
Defining polynomial: \(x^{8} - 2 x^{7} - 2 x^{5} + 9 x^{4} - 4 x^{3} - 16 x + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 600)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + ( -\beta_{4} - \beta_{5} + \beta_{6} ) q^{4} + ( -1 - \beta_{1} + \beta_{3} + \beta_{6} ) q^{7} + ( 1 + \beta_{2} - \beta_{6} + \beta_{7} ) q^{8} +O(q^{10})\) \( q + \beta_{3} q^{2} + ( -\beta_{4} - \beta_{5} + \beta_{6} ) q^{4} + ( -1 - \beta_{1} + \beta_{3} + \beta_{6} ) q^{7} + ( 1 + \beta_{2} - \beta_{6} + \beta_{7} ) q^{8} + ( 1 + \beta_{3} - \beta_{4} + \beta_{5} ) q^{11} + ( 1 - \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{13} + ( \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{5} - \beta_{6} ) q^{14} + ( -\beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{16} + ( \beta_{1} + \beta_{4} - 2 \beta_{6} - \beta_{7} ) q^{17} + ( -1 - 2 \beta_{1} - 3 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{7} ) q^{19} + ( -2 - \beta_{1} - 3 \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{22} + ( 1 - 2 \beta_{1} + 3 \beta_{3} + 3 \beta_{4} + \beta_{5} - 2 \beta_{7} ) q^{23} + ( -2 \beta_{1} - 4 \beta_{2} + \beta_{3} - 2 \beta_{7} ) q^{26} + ( 2 - \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} ) q^{28} + ( \beta_{1} + 4 \beta_{2} + 2 \beta_{3} - \beta_{4} + 2 \beta_{6} - \beta_{7} ) q^{29} + ( 1 + \beta_{1} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{31} + ( 4 + 2 \beta_{1} + 2 \beta_{7} ) q^{32} + ( 2 - \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} + 3 \beta_{6} ) q^{34} + ( 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} - 2 \beta_{7} ) q^{37} + ( 6 - \beta_{1} - 3 \beta_{2} + 3 \beta_{5} - 3 \beta_{6} ) q^{38} + ( -\beta_{1} + 2 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{41} + ( -1 - \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{43} + ( 2 - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{6} ) q^{44} + ( 2 + 3 \beta_{1} - 3 \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{46} + ( 4 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{47} + ( 4 \beta_{1} - 6 \beta_{3} - 2 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} ) q^{49} + ( 4 - 4 \beta_{2} - 3 \beta_{4} + \beta_{5} - \beta_{6} ) q^{52} + ( 3 \beta_{1} + 2 \beta_{2} + 3 \beta_{4} + 3 \beta_{7} ) q^{53} + ( -3 - 5 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{56} + ( -2 + \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + 4 \beta_{7} ) q^{58} + ( -4 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} ) q^{59} + ( 1 - \beta_{1} - 5 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{61} + ( 4 - 3 \beta_{1} + 3 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} - \beta_{5} + 3 \beta_{6} ) q^{62} + ( -4 + 4 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{64} + ( -3 + 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - 3 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} ) q^{67} + ( -4 + 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{68} + ( 5 - 2 \beta_{1} + 3 \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{71} + ( 2 - 2 \beta_{3} + 6 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{73} + ( -4 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 4 \beta_{7} ) q^{74} + ( -2 - 3 \beta_{1} - 7 \beta_{2} + 6 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} - 4 \beta_{7} ) q^{76} + ( -4 - \beta_{1} - 4 \beta_{2} - 2 \beta_{3} + \beta_{4} - 4 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} ) q^{77} + ( -2 - 2 \beta_{1} + 6 \beta_{3} + 4 \beta_{4} - 2 \beta_{6} - 4 \beta_{7} ) q^{79} + ( 2 - \beta_{1} + \beta_{2} - \beta_{4} - 3 \beta_{5} - \beta_{6} ) q^{82} + ( 2 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} ) q^{83} + ( 2 + \beta_{1} + 7 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} - 4 \beta_{7} ) q^{86} + ( 2 - 4 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} ) q^{88} + ( -2 \beta_{1} + 6 \beta_{4} + 4 \beta_{5} - 2 \beta_{7} ) q^{89} + ( -3 - 5 \beta_{2} - \beta_{3} + \beta_{4} - 3 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{91} + ( -2 + 4 \beta_{2} + 2 \beta_{3} - 6 \beta_{4} + 2 \beta_{6} + 4 \beta_{7} ) q^{92} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{5} + 2 \beta_{6} ) q^{94} + ( 1 - 2 \beta_{1} + 6 \beta_{3} + 4 \beta_{4} - 2 \beta_{6} - 4 \beta_{7} ) q^{97} + ( -4 - 4 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} + 4 q^{4} - 8 q^{7} + 4 q^{8} + O(q^{10}) \) \( 8 q - 2 q^{2} + 4 q^{4} - 8 q^{7} + 4 q^{8} + 6 q^{14} + 8 q^{16} - 12 q^{22} + 8 q^{23} + 2 q^{26} + 4 q^{28} + 8 q^{31} + 28 q^{32} + 12 q^{34} + 30 q^{38} + 12 q^{44} + 20 q^{46} + 20 q^{52} - 8 q^{56} - 12 q^{58} + 30 q^{62} - 32 q^{64} - 28 q^{68} + 40 q^{71} + 16 q^{73} - 8 q^{74} - 20 q^{76} - 16 q^{79} + 24 q^{82} + 18 q^{86} + 8 q^{88} - 36 q^{92} - 4 q^{94} + 8 q^{97} - 48 q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{7} - 2 x^{5} + 9 x^{4} - 4 x^{3} - 16 x + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{7} + 2 \nu^{4} + 3 \nu^{3} - 6 \nu^{2} - 4 \nu \)\()/8\)
\(\beta_{2}\)\(=\)\((\)\( -5 \nu^{7} + 2 \nu^{6} + 4 \nu^{5} + 18 \nu^{4} - 21 \nu^{3} - 12 \nu^{2} - 20 \nu + 56 \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( 5 \nu^{7} - 4 \nu^{6} - 4 \nu^{5} - 18 \nu^{4} + 25 \nu^{3} + 10 \nu^{2} + 24 \nu - 64 \)\()/8\)
\(\beta_{4}\)\(=\)\((\)\( 3 \nu^{7} - 2 \nu^{6} - 2 \nu^{5} - 10 \nu^{4} + 15 \nu^{3} + 8 \nu^{2} + 10 \nu - 36 \)\()/4\)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{7} + 2 \nu^{6} + 4 \nu^{5} + 10 \nu^{4} - 15 \nu^{3} - 12 \nu^{2} - 8 \nu + 36 \)\()/4\)
\(\beta_{6}\)\(=\)\((\)\( -4 \nu^{7} + 3 \nu^{6} + 4 \nu^{5} + 12 \nu^{4} - 18 \nu^{3} - 9 \nu^{2} - 10 \nu + 44 \)\()/4\)
\(\beta_{7}\)\(=\)\((\)\( 7 \nu^{7} - 4 \nu^{6} - 6 \nu^{5} - 22 \nu^{4} + 31 \nu^{3} + 18 \nu^{2} + 26 \nu - 80 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} + \beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{6} - \beta_{5} + \beta_{4} + \beta_{2} - \beta_{1}\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} + \beta_{2} + 2 \beta_{1} + 3\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(4 \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} - 2 \beta_{3} + 5 \beta_{2} + \beta_{1}\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-\beta_{7} + \beta_{6} + 2 \beta_{5} + 7 \beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_{1} - 1\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(4 \beta_{7} + 3 \beta_{6} + \beta_{5} - \beta_{4} - 8 \beta_{3} - 5 \beta_{2} + 5 \beta_{1}\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(7 \beta_{7} - 5 \beta_{6} + 8 \beta_{5} - \beta_{4} - 11 \beta_{3} + 3 \beta_{2} - 2 \beta_{1} + 5\)\()/2\)

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
1.41216 0.0762223i
1.41216 + 0.0762223i
−0.565036 1.29643i
−0.565036 + 1.29643i
−1.08003 0.912978i
−1.08003 + 0.912978i
1.23291 0.692769i
1.23291 + 0.692769i
−1.29150 0.576222i 0 1.33594 + 1.48838i 0 0 1.97676 −0.867721 2.69204i 0 0
901.2 −1.29150 + 0.576222i 0 1.33594 1.48838i 0 0 1.97676 −0.867721 + 2.69204i 0 0
901.3 −1.16863 0.796431i 0 0.731395 + 1.86147i 0 0 −4.72294 0.627801 2.75787i 0 0
901.4 −1.16863 + 0.796431i 0 0.731395 1.86147i 0 0 −4.72294 0.627801 + 2.75787i 0 0
901.5 0.0591148 1.41298i 0 −1.99301 0.167056i 0 0 −1.33411 −0.353863 + 2.80620i 0 0
901.6 0.0591148 + 1.41298i 0 −1.99301 + 0.167056i 0 0 −1.33411 −0.353863 2.80620i 0 0
901.7 1.40101 0.192769i 0 1.92568 0.540143i 0 0 0.0802864 2.59378 1.12796i 0 0
901.8 1.40101 + 0.192769i 0 1.92568 + 0.540143i 0 0 0.0802864 2.59378 + 1.12796i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 901.8
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.q 8
3.b odd 2 1 600.2.k.e yes 8
4.b odd 2 1 7200.2.k.s 8
5.b even 2 1 1800.2.k.t 8
5.c odd 4 1 1800.2.d.s 8
5.c odd 4 1 1800.2.d.t 8
8.b even 2 1 inner 1800.2.k.q 8
8.d odd 2 1 7200.2.k.s 8
12.b even 2 1 2400.2.k.e 8
15.d odd 2 1 600.2.k.d 8
15.e even 4 1 600.2.d.g 8
15.e even 4 1 600.2.d.h 8
20.d odd 2 1 7200.2.k.r 8
20.e even 4 1 7200.2.d.s 8
20.e even 4 1 7200.2.d.t 8
24.f even 2 1 2400.2.k.e 8
24.h odd 2 1 600.2.k.e yes 8
40.e odd 2 1 7200.2.k.r 8
40.f even 2 1 1800.2.k.t 8
40.i odd 4 1 1800.2.d.s 8
40.i odd 4 1 1800.2.d.t 8
40.k even 4 1 7200.2.d.s 8
40.k even 4 1 7200.2.d.t 8
60.h even 2 1 2400.2.k.d 8
60.l odd 4 1 2400.2.d.g 8
60.l odd 4 1 2400.2.d.h 8
120.i odd 2 1 600.2.k.d 8
120.m even 2 1 2400.2.k.d 8
120.q odd 4 1 2400.2.d.g 8
120.q odd 4 1 2400.2.d.h 8
120.w even 4 1 600.2.d.g 8
120.w even 4 1 600.2.d.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.2.d.g 8 15.e even 4 1
600.2.d.g 8 120.w even 4 1
600.2.d.h 8 15.e even 4 1
600.2.d.h 8 120.w even 4 1
600.2.k.d 8 15.d odd 2 1
600.2.k.d 8 120.i odd 2 1
600.2.k.e yes 8 3.b odd 2 1
600.2.k.e yes 8 24.h odd 2 1
1800.2.d.s 8 5.c odd 4 1
1800.2.d.s 8 40.i odd 4 1
1800.2.d.t 8 5.c odd 4 1
1800.2.d.t 8 40.i odd 4 1
1800.2.k.q 8 1.a even 1 1 trivial
1800.2.k.q 8 8.b even 2 1 inner
1800.2.k.t 8 5.b even 2 1
1800.2.k.t 8 40.f even 2 1
2400.2.d.g 8 60.l odd 4 1
2400.2.d.g 8 120.q odd 4 1
2400.2.d.h 8 60.l odd 4 1
2400.2.d.h 8 120.q odd 4 1
2400.2.k.d 8 60.h even 2 1
2400.2.k.d 8 120.m even 2 1
2400.2.k.e 8 12.b even 2 1
2400.2.k.e 8 24.f even 2 1
7200.2.d.s 8 20.e even 4 1
7200.2.d.s 8 40.k even 4 1
7200.2.d.t 8 20.e even 4 1
7200.2.d.t 8 40.k even 4 1
7200.2.k.r 8 20.d odd 2 1
7200.2.k.r 8 40.e odd 2 1
7200.2.k.s 8 4.b odd 2 1
7200.2.k.s 8 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 \(S_{2}^{\mathrm{new}}(1800, [\chi])\):

\( T_{7}^{4} + 4 T_{7}^{3} - 6 T_{7}^{2} - 12 T_{7} + 1 \)
\( T_{11}^{8} + 32 T_{11}^{6} + 336 T_{11}^{4} + 1344 T_{11}^{2} + 1600 \)
\( T_{17}^{4} - 40 T_{17}^{2} + 104 T_{17} - 24 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 16 + 16 T - 8 T^{3} - 6 T^{4} - 4 T^{5} + 2 T^{7} + T^{8} \)
$3$ \( T^{8} \)
$5$ \( T^{8} \)
$7$ \( ( 1 - 12 T - 6 T^{2} + 4 T^{3} + T^{4} )^{2} \)
$11$ \( 1600 + 1344 T^{2} + 336 T^{4} + 32 T^{6} + T^{8} \)
$13$ \( 81 + 1420 T^{2} + 502 T^{4} + 44 T^{6} + T^{8} \)
$17$ \( ( -24 + 104 T - 40 T^{2} + T^{4} )^{2} \)
$19$ \( 380689 + 75156 T^{2} + 4662 T^{4} + 116 T^{6} + T^{8} \)
$23$ \( ( -88 + 152 T - 56 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$29$ \( 627264 + 107584 T^{2} + 6288 T^{4} + 144 T^{6} + T^{8} \)
$31$ \( ( 673 + 204 T - 70 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$37$ \( 4096 + 24576 T^{2} + 4224 T^{4} + 128 T^{6} + T^{8} \)
$41$ \( ( 328 - 56 T - 64 T^{2} + T^{4} )^{2} \)
$43$ \( 4363921 + 508692 T^{2} + 18614 T^{4} + 244 T^{6} + T^{8} \)
$47$ \( ( -176 - 256 T - 72 T^{2} + T^{4} )^{2} \)
$53$ \( 23104 + 425920 T^{2} + 19536 T^{4} + 256 T^{6} + T^{8} \)
$59$ \( 31181056 + 2477824 T^{2} + 57824 T^{4} + 432 T^{6} + T^{8} \)
$61$ \( 3025 + 216396 T^{2} + 14838 T^{4} + 236 T^{6} + T^{8} \)
$67$ \( 25979409 + 1957780 T^{2} + 44598 T^{4} + 372 T^{6} + T^{8} \)
$71$ \( ( -536 + 72 T + 96 T^{2} - 20 T^{3} + T^{4} )^{2} \)
$73$ \( ( -432 + 864 T - 168 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$79$ \( ( 8080 - 864 T - 184 T^{2} + 8 T^{3} + T^{4} )^{2} \)
$83$ \( 3873024 + 1762048 T^{2} + 44256 T^{4} + 368 T^{6} + T^{8} \)
$89$ \( ( 10880 - 64 T - 224 T^{2} + T^{4} )^{2} \)
$97$ \( ( 8881 + 348 T - 202 T^{2} - 4 T^{3} + T^{4} )^{2} \)
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