Properties

Label 300.3.b.b
Level $300$
Weight $3$
Character orbit 300.b
Analytic conductor $8.174$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(8.17440793081\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + 3 i q^{3} -13 i q^{7} -9 q^{9} +O(q^{10})\) \( q + 3 i q^{3} -13 i q^{7} -9 q^{9} -23 i q^{13} -11 q^{19} + 39 q^{21} -27 i q^{27} + 59 q^{31} + 26 i q^{37} + 69 q^{39} -83 i q^{43} -120 q^{49} -33 i q^{57} -121 q^{61} + 117 i q^{63} -13 i q^{67} + 46 i q^{73} + 142 q^{79} + 81 q^{81} -299 q^{91} + 177 i q^{93} + 167 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 18 q^{9} + O(q^{10}) \) \( 2 q - 18 q^{9} - 22 q^{19} + 78 q^{21} + 118 q^{31} + 138 q^{39} - 240 q^{49} - 242 q^{61} + 284 q^{79} + 162 q^{81} - 598 q^{91} + O(q^{100}) \)

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 3.00000i 0 0 0 13.0000i 0 −9.00000 0
149.2 0 3.00000i 0 0 0 13.0000i 0 −9.00000 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 300.3.b.b 2
3.b odd 2 1 CM 300.3.b.b 2
4.b odd 2 1 1200.3.c.b 2
5.b even 2 1 inner 300.3.b.b 2
5.c odd 4 1 300.3.g.a 1
5.c odd 4 1 300.3.g.c yes 1
12.b even 2 1 1200.3.c.b 2
15.d odd 2 1 inner 300.3.b.b 2
15.e even 4 1 300.3.g.a 1
15.e even 4 1 300.3.g.c yes 1
20.d odd 2 1 1200.3.c.b 2
20.e even 4 1 1200.3.l.a 1
20.e even 4 1 1200.3.l.e 1
60.h even 2 1 1200.3.c.b 2
60.l odd 4 1 1200.3.l.a 1
60.l odd 4 1 1200.3.l.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.3.b.b 2 1.a even 1 1 trivial
300.3.b.b 2 3.b odd 2 1 CM
300.3.b.b 2 5.b even 2 1 inner
300.3.b.b 2 15.d odd 2 1 inner
300.3.g.a 1 5.c odd 4 1
300.3.g.a 1 15.e even 4 1
300.3.g.c yes 1 5.c odd 4 1
300.3.g.c yes 1 15.e even 4 1
1200.3.c.b 2 4.b odd 2 1
1200.3.c.b 2 12.b even 2 1
1200.3.c.b 2 20.d odd 2 1
1200.3.c.b 2 60.h even 2 1
1200.3.l.a 1 20.e even 4 1
1200.3.l.a 1 60.l odd 4 1
1200.3.l.e 1 20.e even 4 1
1200.3.l.e 1 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(300, [\chi])\):

\( T_{7}^{2} + 169 \)
\( T_{11} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 9 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 169 + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 529 + T^{2} \)
$17$ \( T^{2} \)
$19$ \( ( 11 + T )^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( ( -59 + T )^{2} \)
$37$ \( 676 + T^{2} \)
$41$ \( T^{2} \)
$43$ \( 6889 + T^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( ( 121 + T )^{2} \)
$67$ \( 169 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 2116 + T^{2} \)
$79$ \( ( -142 + T )^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( 27889 + T^{2} \)
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