Properties

Label 300.3.k.b
Level $300$
Weight $3$
Character orbit 300.k
Analytic conductor $8.174$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 300.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.17440793081\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Defining polynomial: \(x^{4} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + 2 \beta_{3} q^{7} + 3 \beta_{2} q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + 2 \beta_{3} q^{7} + 3 \beta_{2} q^{9} + 6 q^{11} + 10 \beta_{1} q^{13} + 12 \beta_{3} q^{17} + 10 \beta_{2} q^{19} -6 q^{21} + 24 \beta_{1} q^{23} + 3 \beta_{3} q^{27} -48 \beta_{2} q^{29} -26 q^{31} + 6 \beta_{1} q^{33} + 26 \beta_{3} q^{37} + 30 \beta_{2} q^{39} + 30 q^{41} + 24 \beta_{1} q^{43} -12 \beta_{3} q^{47} + 37 \beta_{2} q^{49} -36 q^{51} -12 \beta_{1} q^{53} + 10 \beta_{3} q^{57} -78 \beta_{2} q^{59} + 2 q^{61} -6 \beta_{1} q^{63} -52 \beta_{3} q^{67} + 72 \beta_{2} q^{69} + 120 q^{71} -68 \beta_{1} q^{73} + 12 \beta_{3} q^{77} + 74 \beta_{2} q^{79} -9 q^{81} -36 \beta_{1} q^{83} -48 \beta_{3} q^{87} -150 \beta_{2} q^{89} -60 q^{91} -26 \beta_{1} q^{93} + 4 \beta_{3} q^{97} + 18 \beta_{2} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + O(q^{10}) \) \( 4 q + 24 q^{11} - 24 q^{21} - 104 q^{31} + 120 q^{41} - 144 q^{51} + 8 q^{61} + 480 q^{71} - 36 q^{81} - 240 q^{91} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/3\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(3 \beta_{2}\)
\(\nu^{3}\)\(=\)\(3 \beta_{3}\)

Character values

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

\(n\) \(101\) \(151\) \(277\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
157.1
−1.22474 + 1.22474i
1.22474 1.22474i
−1.22474 1.22474i
1.22474 + 1.22474i
0 −1.22474 + 1.22474i 0 0 0 2.44949 + 2.44949i 0 3.00000i 0
157.2 0 1.22474 1.22474i 0 0 0 −2.44949 2.44949i 0 3.00000i 0
193.1 0 −1.22474 1.22474i 0 0 0 2.44949 2.44949i 0 3.00000i 0
193.2 0 1.22474 + 1.22474i 0 0 0 −2.44949 + 2.44949i 0 3.00000i 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
5.c odd 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.3.k.b 4
3.b odd 2 1 900.3.l.c 4
4.b odd 2 1 1200.3.bg.h 4
5.b even 2 1 inner 300.3.k.b 4
5.c odd 4 2 inner 300.3.k.b 4
15.d odd 2 1 900.3.l.c 4
15.e even 4 2 900.3.l.c 4
20.d odd 2 1 1200.3.bg.h 4
20.e even 4 2 1200.3.bg.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.3.k.b 4 1.a even 1 1 trivial
300.3.k.b 4 5.b even 2 1 inner
300.3.k.b 4 5.c odd 4 2 inner
900.3.l.c 4 3.b odd 2 1
900.3.l.c 4 15.d odd 2 1
900.3.l.c 4 15.e even 4 2
1200.3.bg.h 4 4.b odd 2 1
1200.3.bg.h 4 20.d odd 2 1
1200.3.bg.h 4 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 144 \) acting on \(S_{3}^{\mathrm{new}}(300, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 9 + T^{4} \)
$5$ \( T^{4} \)
$7$ \( 144 + T^{4} \)
$11$ \( ( -6 + T )^{4} \)
$13$ \( 90000 + T^{4} \)
$17$ \( 186624 + T^{4} \)
$19$ \( ( 100 + T^{2} )^{2} \)
$23$ \( 2985984 + T^{4} \)
$29$ \( ( 2304 + T^{2} )^{2} \)
$31$ \( ( 26 + T )^{4} \)
$37$ \( 4112784 + T^{4} \)
$41$ \( ( -30 + T )^{4} \)
$43$ \( 2985984 + T^{4} \)
$47$ \( 186624 + T^{4} \)
$53$ \( 186624 + T^{4} \)
$59$ \( ( 6084 + T^{2} )^{2} \)
$61$ \( ( -2 + T )^{4} \)
$67$ \( 65804544 + T^{4} \)
$71$ \( ( -120 + T )^{4} \)
$73$ \( 192432384 + T^{4} \)
$79$ \( ( 5476 + T^{2} )^{2} \)
$83$ \( 15116544 + T^{4} \)
$89$ \( ( 22500 + T^{2} )^{2} \)
$97$ \( 2304 + T^{4} \)
show more
show less