Properties

Label 300.3.g.f
Level $300$
Weight $3$
Character orbit 300.g
Analytic conductor $8.174$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 300.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.17440793081\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-35}) \)
Defining polynomial: \(x^{2} - x + 9\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{-35})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{3} -8 q^{7} + ( -9 + \beta ) q^{9} +O(q^{10})\) \( q -\beta q^{3} -8 q^{7} + ( -9 + \beta ) q^{9} + ( -3 + 6 \beta ) q^{11} -2 q^{13} + ( -3 + 6 \beta ) q^{17} + 11 q^{19} + 8 \beta q^{21} + ( 6 - 12 \beta ) q^{23} + ( 9 + 8 \beta ) q^{27} + ( -6 + 12 \beta ) q^{29} -46 q^{31} + ( 54 - 3 \beta ) q^{33} + 16 q^{37} + 2 \beta q^{39} + ( -9 + 18 \beta ) q^{41} -62 q^{43} + ( -6 + 12 \beta ) q^{47} + 15 q^{49} + ( 54 - 3 \beta ) q^{51} + ( -6 + 12 \beta ) q^{53} -11 \beta q^{57} + ( 12 - 24 \beta ) q^{59} -16 q^{61} + ( 72 - 8 \beta ) q^{63} -113 q^{67} + ( -108 + 6 \beta ) q^{69} + ( 18 - 36 \beta ) q^{71} -101 q^{73} + ( 24 - 48 \beta ) q^{77} + 68 q^{79} + ( 72 - 17 \beta ) q^{81} + ( -3 + 6 \beta ) q^{83} + ( 108 - 6 \beta ) q^{87} + ( -9 + 18 \beta ) q^{89} + 16 q^{91} + 46 \beta q^{93} + 22 q^{97} + ( -27 - 51 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{3} - 16q^{7} - 17q^{9} + O(q^{10}) \) \( 2q - q^{3} - 16q^{7} - 17q^{9} - 4q^{13} + 22q^{19} + 8q^{21} + 26q^{27} - 92q^{31} + 105q^{33} + 32q^{37} + 2q^{39} - 124q^{43} + 30q^{49} + 105q^{51} - 11q^{57} - 32q^{61} + 136q^{63} - 226q^{67} - 210q^{69} - 202q^{73} + 136q^{79} + 127q^{81} + 210q^{87} + 32q^{91} + 46q^{93} + 44q^{97} - 105q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1
0.500000 + 2.95804i
0.500000 2.95804i
0 −0.500000 2.95804i 0 0 0 −8.00000 0 −8.50000 + 2.95804i 0
101.2 0 −0.500000 + 2.95804i 0 0 0 −8.00000 0 −8.50000 2.95804i 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 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.3.g.f 2
3.b odd 2 1 inner 300.3.g.f 2
4.b odd 2 1 1200.3.l.m 2
5.b even 2 1 300.3.g.g yes 2
5.c odd 4 2 300.3.b.d 4
12.b even 2 1 1200.3.l.m 2
15.d odd 2 1 300.3.g.g yes 2
15.e even 4 2 300.3.b.d 4
20.d odd 2 1 1200.3.l.k 2
20.e even 4 2 1200.3.c.j 4
60.h even 2 1 1200.3.l.k 2
60.l odd 4 2 1200.3.c.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.3.b.d 4 5.c odd 4 2
300.3.b.d 4 15.e even 4 2
300.3.g.f 2 1.a even 1 1 trivial
300.3.g.f 2 3.b odd 2 1 inner
300.3.g.g yes 2 5.b even 2 1
300.3.g.g yes 2 15.d odd 2 1
1200.3.c.j 4 20.e even 4 2
1200.3.c.j 4 60.l odd 4 2
1200.3.l.k 2 20.d odd 2 1
1200.3.l.k 2 60.h even 2 1
1200.3.l.m 2 4.b odd 2 1
1200.3.l.m 2 12.b even 2 1

Hecke kernels

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

\( T_{7} + 8 \)
\( T_{11}^{2} + 315 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 9 + T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( ( 8 + T )^{2} \)
$11$ \( 315 + T^{2} \)
$13$ \( ( 2 + T )^{2} \)
$17$ \( 315 + T^{2} \)
$19$ \( ( -11 + T )^{2} \)
$23$ \( 1260 + T^{2} \)
$29$ \( 1260 + T^{2} \)
$31$ \( ( 46 + T )^{2} \)
$37$ \( ( -16 + T )^{2} \)
$41$ \( 2835 + T^{2} \)
$43$ \( ( 62 + T )^{2} \)
$47$ \( 1260 + T^{2} \)
$53$ \( 1260 + T^{2} \)
$59$ \( 5040 + T^{2} \)
$61$ \( ( 16 + T )^{2} \)
$67$ \( ( 113 + T )^{2} \)
$71$ \( 11340 + T^{2} \)
$73$ \( ( 101 + T )^{2} \)
$79$ \( ( -68 + T )^{2} \)
$83$ \( 315 + T^{2} \)
$89$ \( 2835 + T^{2} \)
$97$ \( ( -22 + T )^{2} \)
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