Properties

Label 1800.2.f.e
Level $1800$
Weight $2$
Character orbit 1800.f
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.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.3730723638\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{37}]\)
Coefficient ring index: \( 1 \)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 i q^{7} +O(q^{10})\) \( q + 3 i q^{7} -2 q^{11} + 3 i q^{13} -6 i q^{17} + 7 q^{19} + 6 i q^{23} -2 q^{29} -5 q^{31} + 10 i q^{37} -12 q^{41} -3 i q^{43} + 10 i q^{47} -2 q^{49} -6 q^{59} -13 q^{61} + 7 i q^{67} + 4 q^{71} + 6 i q^{73} -6 i q^{77} + 8 q^{79} -6 i q^{83} + 16 q^{89} -9 q^{91} -7 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q - 4q^{11} + 14q^{19} - 4q^{29} - 10q^{31} - 24q^{41} - 4q^{49} - 12q^{59} - 26q^{61} + 8q^{71} + 16q^{79} + 32q^{89} - 18q^{91} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
649.1
1.00000i
1.00000i
0 0 0 0 0 3.00000i 0 0 0
649.2 0 0 0 0 0 3.00000i 0 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
5.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.f.e 2
3.b odd 2 1 600.2.f.d 2
4.b odd 2 1 3600.2.f.o 2
5.b even 2 1 inner 1800.2.f.e 2
5.c odd 4 1 1800.2.a.e 1
5.c odd 4 1 1800.2.a.t 1
12.b even 2 1 1200.2.f.c 2
15.d odd 2 1 600.2.f.d 2
15.e even 4 1 600.2.a.b 1
15.e even 4 1 600.2.a.i yes 1
20.d odd 2 1 3600.2.f.o 2
20.e even 4 1 3600.2.a.i 1
20.e even 4 1 3600.2.a.bl 1
24.f even 2 1 4800.2.f.z 2
24.h odd 2 1 4800.2.f.k 2
60.h even 2 1 1200.2.f.c 2
60.l odd 4 1 1200.2.a.b 1
60.l odd 4 1 1200.2.a.q 1
120.i odd 2 1 4800.2.f.k 2
120.m even 2 1 4800.2.f.z 2
120.q odd 4 1 4800.2.a.bd 1
120.q odd 4 1 4800.2.a.bs 1
120.w even 4 1 4800.2.a.bc 1
120.w even 4 1 4800.2.a.bp 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.2.a.b 1 15.e even 4 1
600.2.a.i yes 1 15.e even 4 1
600.2.f.d 2 3.b odd 2 1
600.2.f.d 2 15.d odd 2 1
1200.2.a.b 1 60.l odd 4 1
1200.2.a.q 1 60.l odd 4 1
1200.2.f.c 2 12.b even 2 1
1200.2.f.c 2 60.h even 2 1
1800.2.a.e 1 5.c odd 4 1
1800.2.a.t 1 5.c odd 4 1
1800.2.f.e 2 1.a even 1 1 trivial
1800.2.f.e 2 5.b even 2 1 inner
3600.2.a.i 1 20.e even 4 1
3600.2.a.bl 1 20.e even 4 1
3600.2.f.o 2 4.b odd 2 1
3600.2.f.o 2 20.d odd 2 1
4800.2.a.bc 1 120.w even 4 1
4800.2.a.bd 1 120.q odd 4 1
4800.2.a.bp 1 120.w even 4 1
4800.2.a.bs 1 120.q odd 4 1
4800.2.f.k 2 24.h odd 2 1
4800.2.f.k 2 120.i odd 2 1
4800.2.f.z 2 24.f even 2 1
4800.2.f.z 2 120.m even 2 1

Hecke kernels

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

\( T_{7}^{2} + 9 \)
\( T_{11} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 9 + T^{2} \)
$11$ \( ( 2 + T )^{2} \)
$13$ \( 9 + T^{2} \)
$17$ \( 36 + T^{2} \)
$19$ \( ( -7 + T )^{2} \)
$23$ \( 36 + T^{2} \)
$29$ \( ( 2 + T )^{2} \)
$31$ \( ( 5 + T )^{2} \)
$37$ \( 100 + T^{2} \)
$41$ \( ( 12 + T )^{2} \)
$43$ \( 9 + T^{2} \)
$47$ \( 100 + T^{2} \)
$53$ \( T^{2} \)
$59$ \( ( 6 + T )^{2} \)
$61$ \( ( 13 + T )^{2} \)
$67$ \( 49 + T^{2} \)
$71$ \( ( -4 + T )^{2} \)
$73$ \( 36 + T^{2} \)
$79$ \( ( -8 + T )^{2} \)
$83$ \( 36 + T^{2} \)
$89$ \( ( -16 + T )^{2} \)
$97$ \( 49 + T^{2} \)
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