Properties

Label 300.1.b.a
Level $300$
Weight $1$
Character orbit 300.b
Analytic conductor $0.150$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 300.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.149719503790\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.300.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q -i q^{3} -i q^{7} - q^{9} +O(q^{10})\) \( q -i q^{3} -i q^{7} - q^{9} + i q^{13} + q^{19} - q^{21} + i q^{27} - q^{31} + 2 i q^{37} + q^{39} + i q^{43} -i q^{57} - q^{61} + i q^{63} -i q^{67} -2 i q^{73} -2 q^{79} + q^{81} + q^{91} + i q^{93} -i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{9} + 2q^{19} - 2q^{21} - 2q^{31} + 2q^{39} - 2q^{61} - 4q^{79} + 2q^{81} + 2q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/300\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(151\) \(277\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1
1.00000i
1.00000i
0 1.00000i 0 0 0 1.00000i 0 −1.00000 0
149.2 0 1.00000i 0 0 0 1.00000i 0 −1.00000 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
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 300.1.b.a 2
3.b odd 2 1 CM 300.1.b.a 2
4.b odd 2 1 1200.1.c.a 2
5.b even 2 1 inner 300.1.b.a 2
5.c odd 4 1 300.1.g.a 1
5.c odd 4 1 300.1.g.b yes 1
12.b even 2 1 1200.1.c.a 2
15.d odd 2 1 inner 300.1.b.a 2
15.e even 4 1 300.1.g.a 1
15.e even 4 1 300.1.g.b yes 1
20.d odd 2 1 1200.1.c.a 2
20.e even 4 1 1200.1.l.a 1
20.e even 4 1 1200.1.l.b 1
60.h even 2 1 1200.1.c.a 2
60.l odd 4 1 1200.1.l.a 1
60.l odd 4 1 1200.1.l.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.1.b.a 2 1.a even 1 1 trivial
300.1.b.a 2 3.b odd 2 1 CM
300.1.b.a 2 5.b even 2 1 inner
300.1.b.a 2 15.d odd 2 1 inner
300.1.g.a 1 5.c odd 4 1
300.1.g.a 1 15.e even 4 1
300.1.g.b yes 1 5.c odd 4 1
300.1.g.b yes 1 15.e even 4 1
1200.1.c.a 2 4.b odd 2 1
1200.1.c.a 2 12.b even 2 1
1200.1.c.a 2 20.d odd 2 1
1200.1.c.a 2 60.h even 2 1
1200.1.l.a 1 20.e even 4 1
1200.1.l.a 1 60.l odd 4 1
1200.1.l.b 1 20.e even 4 1
1200.1.l.b 1 60.l odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(300, [\chi])\).

Hecke characteristic polynomials

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