Properties

Label 1800.2.k.d
Level $1800$
Weight $2$
Character orbit 1800.k
Analytic conductor $14.373$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-7}) \)
Defining polynomial: \(x^{2} - x + 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 200)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{-7})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta ) q^{2} + ( -1 - \beta ) q^{4} + 4 q^{7} + ( 3 - \beta ) q^{8} +O(q^{10})\) \( q + ( -1 + \beta ) q^{2} + ( -1 - \beta ) q^{4} + 4 q^{7} + ( 3 - \beta ) q^{8} + ( 1 - 2 \beta ) q^{11} + ( -4 + 4 \beta ) q^{14} + ( -1 + 3 \beta ) q^{16} + 3 q^{17} + ( -1 + 2 \beta ) q^{19} + ( 3 + \beta ) q^{22} -4 q^{23} + ( -4 - 4 \beta ) q^{28} + 4 q^{31} + ( -5 - \beta ) q^{32} + ( -3 + 3 \beta ) q^{34} + ( 4 - 8 \beta ) q^{37} + ( -3 - \beta ) q^{38} + 5 q^{41} + ( 2 - 4 \beta ) q^{43} + ( -5 + 3 \beta ) q^{44} + ( 4 - 4 \beta ) q^{46} + 8 q^{47} + 9 q^{49} + ( 4 - 8 \beta ) q^{53} + ( 12 - 4 \beta ) q^{56} + ( 2 - 4 \beta ) q^{59} + ( -4 + 8 \beta ) q^{61} + ( -4 + 4 \beta ) q^{62} + ( 7 - 5 \beta ) q^{64} + ( -3 + 6 \beta ) q^{67} + ( -3 - 3 \beta ) q^{68} -8 q^{71} -7 q^{73} + ( 12 + 4 \beta ) q^{74} + ( 5 - 3 \beta ) q^{76} + ( 4 - 8 \beta ) q^{77} + 4 q^{79} + ( -5 + 5 \beta ) q^{82} + ( 3 - 6 \beta ) q^{83} + ( 6 + 2 \beta ) q^{86} + ( -1 - 5 \beta ) q^{88} + q^{89} + ( 4 + 4 \beta ) q^{92} + ( -8 + 8 \beta ) q^{94} -2 q^{97} + ( -9 + 9 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - 3 q^{4} + 8 q^{7} + 5 q^{8} + O(q^{10}) \) \( 2 q - q^{2} - 3 q^{4} + 8 q^{7} + 5 q^{8} - 4 q^{14} + q^{16} + 6 q^{17} + 7 q^{22} - 8 q^{23} - 12 q^{28} + 8 q^{31} - 11 q^{32} - 3 q^{34} - 7 q^{38} + 10 q^{41} - 7 q^{44} + 4 q^{46} + 16 q^{47} + 18 q^{49} + 20 q^{56} - 4 q^{62} + 9 q^{64} - 9 q^{68} - 16 q^{71} - 14 q^{73} + 28 q^{74} + 7 q^{76} + 8 q^{79} - 5 q^{82} + 14 q^{86} - 7 q^{88} + 2 q^{89} + 12 q^{92} - 8 q^{94} - 4 q^{97} - 9 q^{98} + O(q^{100}) \)

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
0.500000 1.32288i
0.500000 + 1.32288i
−0.500000 1.32288i 0 −1.50000 + 1.32288i 0 0 4.00000 2.50000 + 1.32288i 0 0
901.2 −0.500000 + 1.32288i 0 −1.50000 1.32288i 0 0 4.00000 2.50000 1.32288i 0 0
\(n\): e.g. 2-40 or 990-1000
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.d 2
3.b odd 2 1 200.2.d.c yes 2
4.b odd 2 1 7200.2.k.b 2
5.b even 2 1 1800.2.k.f 2
5.c odd 4 2 1800.2.d.m 4
8.b even 2 1 inner 1800.2.k.d 2
8.d odd 2 1 7200.2.k.b 2
12.b even 2 1 800.2.d.a 2
15.d odd 2 1 200.2.d.b 2
15.e even 4 2 200.2.f.d 4
20.d odd 2 1 7200.2.k.i 2
20.e even 4 2 7200.2.d.m 4
24.f even 2 1 800.2.d.a 2
24.h odd 2 1 200.2.d.c yes 2
40.e odd 2 1 7200.2.k.i 2
40.f even 2 1 1800.2.k.f 2
40.i odd 4 2 1800.2.d.m 4
40.k even 4 2 7200.2.d.m 4
48.i odd 4 2 6400.2.a.bg 2
48.k even 4 2 6400.2.a.cb 2
60.h even 2 1 800.2.d.d 2
60.l odd 4 2 800.2.f.d 4
120.i odd 2 1 200.2.d.b 2
120.m even 2 1 800.2.d.d 2
120.q odd 4 2 800.2.f.d 4
120.w even 4 2 200.2.f.d 4
240.t even 4 2 6400.2.a.bh 2
240.bm odd 4 2 6400.2.a.cc 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.2.d.b 2 15.d odd 2 1
200.2.d.b 2 120.i odd 2 1
200.2.d.c yes 2 3.b odd 2 1
200.2.d.c yes 2 24.h odd 2 1
200.2.f.d 4 15.e even 4 2
200.2.f.d 4 120.w even 4 2
800.2.d.a 2 12.b even 2 1
800.2.d.a 2 24.f even 2 1
800.2.d.d 2 60.h even 2 1
800.2.d.d 2 120.m even 2 1
800.2.f.d 4 60.l odd 4 2
800.2.f.d 4 120.q odd 4 2
1800.2.d.m 4 5.c odd 4 2
1800.2.d.m 4 40.i odd 4 2
1800.2.k.d 2 1.a even 1 1 trivial
1800.2.k.d 2 8.b even 2 1 inner
1800.2.k.f 2 5.b even 2 1
1800.2.k.f 2 40.f even 2 1
6400.2.a.bg 2 48.i odd 4 2
6400.2.a.bh 2 240.t even 4 2
6400.2.a.cb 2 48.k even 4 2
6400.2.a.cc 2 240.bm odd 4 2
7200.2.d.m 4 20.e even 4 2
7200.2.d.m 4 40.k even 4 2
7200.2.k.b 2 4.b odd 2 1
7200.2.k.b 2 8.d odd 2 1
7200.2.k.i 2 20.d odd 2 1
7200.2.k.i 2 40.e odd 2 1

Hecke kernels

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

\( T_{7} - 4 \)
\( T_{11}^{2} + 7 \)
\( T_{17} - 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( ( -4 + T )^{2} \)
$11$ \( 7 + T^{2} \)
$13$ \( T^{2} \)
$17$ \( ( -3 + T )^{2} \)
$19$ \( 7 + T^{2} \)
$23$ \( ( 4 + T )^{2} \)
$29$ \( T^{2} \)
$31$ \( ( -4 + T )^{2} \)
$37$ \( 112 + T^{2} \)
$41$ \( ( -5 + T )^{2} \)
$43$ \( 28 + T^{2} \)
$47$ \( ( -8 + T )^{2} \)
$53$ \( 112 + T^{2} \)
$59$ \( 28 + T^{2} \)
$61$ \( 112 + T^{2} \)
$67$ \( 63 + T^{2} \)
$71$ \( ( 8 + T )^{2} \)
$73$ \( ( 7 + T )^{2} \)
$79$ \( ( -4 + T )^{2} \)
$83$ \( 63 + T^{2} \)
$89$ \( ( -1 + T )^{2} \)
$97$ \( ( 2 + T )^{2} \)
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