Properties

Label 300.2.j.b
Level $300$
Weight $2$
Character orbit 300.j
Analytic conductor $2.396$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(2.39551206064\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.157351936.1
Defining polynomial: \(x^{8} + x^{4} + 16\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{2} + \beta_{4} ) q^{2} + \beta_{6} q^{3} + ( \beta_{3} + \beta_{5} ) q^{4} + \beta_{1} q^{6} + 2 \beta_{4} q^{7} + ( 2 \beta_{6} + \beta_{7} ) q^{8} + \beta_{3} q^{9} +O(q^{10})\) \( q + ( \beta_{2} + \beta_{4} ) q^{2} + \beta_{6} q^{3} + ( \beta_{3} + \beta_{5} ) q^{4} + \beta_{1} q^{6} + 2 \beta_{4} q^{7} + ( 2 \beta_{6} + \beta_{7} ) q^{8} + \beta_{3} q^{9} + ( 2 - 4 \beta_{1} ) q^{11} + ( \beta_{2} - \beta_{4} ) q^{12} + ( -2 \beta_{6} + 4 \beta_{7} ) q^{13} + ( -2 \beta_{3} + 2 \beta_{5} ) q^{14} + ( -2 + 3 \beta_{1} ) q^{16} + \beta_{7} q^{18} + ( 2 \beta_{3} - 4 \beta_{5} ) q^{19} + 2 q^{21} + ( -2 \beta_{2} + 6 \beta_{4} ) q^{22} + 4 \beta_{6} q^{23} + ( 3 \beta_{3} - \beta_{5} ) q^{24} + ( -8 + 2 \beta_{1} ) q^{26} -\beta_{4} q^{27} + ( 4 \beta_{6} - 2 \beta_{7} ) q^{28} -8 \beta_{3} q^{29} + ( 2 - 4 \beta_{1} ) q^{31} + ( \beta_{2} - 5 \beta_{4} ) q^{32} + ( 2 \beta_{6} - 4 \beta_{7} ) q^{33} + ( -2 + \beta_{1} ) q^{36} + ( 4 \beta_{2} + 2 \beta_{4} ) q^{37} + ( -8 \beta_{6} + 2 \beta_{7} ) q^{38} + ( 2 \beta_{3} - 4 \beta_{5} ) q^{39} -2 q^{41} + ( 2 \beta_{2} + 2 \beta_{4} ) q^{42} -8 \beta_{6} q^{43} + ( -10 \beta_{3} + 6 \beta_{5} ) q^{44} + 4 \beta_{1} q^{46} + ( -2 \beta_{6} + 3 \beta_{7} ) q^{48} + 3 \beta_{3} q^{49} + ( -6 \beta_{2} - 10 \beta_{4} ) q^{52} + ( 4 \beta_{6} - 8 \beta_{7} ) q^{53} + ( \beta_{3} - \beta_{5} ) q^{54} + ( 4 + 2 \beta_{1} ) q^{56} + ( -4 \beta_{2} - 2 \beta_{4} ) q^{57} -8 \beta_{7} q^{58} + ( -2 \beta_{3} + 4 \beta_{5} ) q^{59} + 6 q^{61} + ( -2 \beta_{2} + 6 \beta_{4} ) q^{62} + 2 \beta_{6} q^{63} + ( 7 \beta_{3} - 5 \beta_{5} ) q^{64} + ( 8 - 2 \beta_{1} ) q^{66} -12 \beta_{4} q^{67} + 4 \beta_{3} q^{69} + ( -\beta_{2} - 3 \beta_{4} ) q^{72} + ( -4 \beta_{6} + 8 \beta_{7} ) q^{73} + ( 6 \beta_{3} + 2 \beta_{5} ) q^{74} + ( -4 - 6 \beta_{1} ) q^{76} + ( -8 \beta_{2} - 4 \beta_{4} ) q^{77} + ( -8 \beta_{6} + 2 \beta_{7} ) q^{78} + ( -2 \beta_{3} + 4 \beta_{5} ) q^{79} - q^{81} + ( -2 \beta_{2} - 2 \beta_{4} ) q^{82} -12 \beta_{6} q^{83} + ( 2 \beta_{3} + 2 \beta_{5} ) q^{84} -8 \beta_{1} q^{86} + 8 \beta_{4} q^{87} + ( 12 \beta_{6} - 10 \beta_{7} ) q^{88} + 6 \beta_{3} q^{89} + ( -4 + 8 \beta_{1} ) q^{91} + ( 4 \beta_{2} - 4 \beta_{4} ) q^{92} + ( 2 \beta_{6} - 4 \beta_{7} ) q^{93} + ( -6 + \beta_{1} ) q^{96} + ( 8 \beta_{2} + 4 \beta_{4} ) q^{97} + 3 \beta_{7} q^{98} + ( -2 \beta_{3} + 4 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 4q^{6} + O(q^{10}) \) \( 8q + 4q^{6} - 4q^{16} + 16q^{21} - 56q^{26} - 12q^{36} - 16q^{41} + 16q^{46} + 40q^{56} + 48q^{61} + 56q^{66} - 56q^{76} - 8q^{81} - 32q^{86} - 44q^{96} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + x^{4} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{4} + 2 \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} + 5 \nu \)\()/6\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{6} + 5 \nu^{2} \)\()/12\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{5} + \nu \)\()/6\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{6} + 7 \nu^{2} \)\()/12\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{7} + 7 \nu^{3} \)\()/24\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{7} + 5 \nu^{3} \)\()/12\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{4} + \beta_{2}\)
\(\nu^{2}\)\(=\)\(\beta_{5} + \beta_{3}\)
\(\nu^{3}\)\(=\)\(\beta_{7} + 2 \beta_{6}\)
\(\nu^{4}\)\(=\)\(3 \beta_{1} - 2\)
\(\nu^{5}\)\(=\)\(-5 \beta_{4} + \beta_{2}\)
\(\nu^{6}\)\(=\)\(-5 \beta_{5} + 7 \beta_{3}\)
\(\nu^{7}\)\(=\)\(7 \beta_{7} - 10 \beta_{6}\)

Character values

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

\(n\) \(101\) \(151\) \(277\)
\(\chi(n)\) \(1\) \(-1\) \(\beta_{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
−1.28897 0.581861i
−0.581861 1.28897i
0.581861 + 1.28897i
1.28897 + 0.581861i
−1.28897 + 0.581861i
−0.581861 + 1.28897i
0.581861 1.28897i
1.28897 0.581861i
−1.28897 0.581861i −0.707107 0.707107i 1.32288 + 1.50000i 0 0.500000 + 1.32288i −1.41421 + 1.41421i −0.832353 2.70318i 1.00000i 0
7.2 −0.581861 1.28897i 0.707107 + 0.707107i −1.32288 + 1.50000i 0 0.500000 1.32288i 1.41421 1.41421i 2.70318 + 0.832353i 1.00000i 0
7.3 0.581861 + 1.28897i −0.707107 0.707107i −1.32288 + 1.50000i 0 0.500000 1.32288i −1.41421 + 1.41421i −2.70318 0.832353i 1.00000i 0
7.4 1.28897 + 0.581861i 0.707107 + 0.707107i 1.32288 + 1.50000i 0 0.500000 + 1.32288i 1.41421 1.41421i 0.832353 + 2.70318i 1.00000i 0
43.1 −1.28897 + 0.581861i −0.707107 + 0.707107i 1.32288 1.50000i 0 0.500000 1.32288i −1.41421 1.41421i −0.832353 + 2.70318i 1.00000i 0
43.2 −0.581861 + 1.28897i 0.707107 0.707107i −1.32288 1.50000i 0 0.500000 + 1.32288i 1.41421 + 1.41421i 2.70318 0.832353i 1.00000i 0
43.3 0.581861 1.28897i −0.707107 + 0.707107i −1.32288 1.50000i 0 0.500000 + 1.32288i −1.41421 1.41421i −2.70318 + 0.832353i 1.00000i 0
43.4 1.28897 0.581861i 0.707107 0.707107i 1.32288 1.50000i 0 0.500000 1.32288i 1.41421 + 1.41421i 0.832353 2.70318i 1.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
20.d odd 2 1 inner
20.e even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.2.j.b 8
3.b odd 2 1 900.2.k.h 8
4.b odd 2 1 inner 300.2.j.b 8
5.b even 2 1 inner 300.2.j.b 8
5.c odd 4 2 inner 300.2.j.b 8
12.b even 2 1 900.2.k.h 8
15.d odd 2 1 900.2.k.h 8
15.e even 4 2 900.2.k.h 8
20.d odd 2 1 inner 300.2.j.b 8
20.e even 4 2 inner 300.2.j.b 8
60.h even 2 1 900.2.k.h 8
60.l odd 4 2 900.2.k.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.2.j.b 8 1.a even 1 1 trivial
300.2.j.b 8 4.b odd 2 1 inner
300.2.j.b 8 5.b even 2 1 inner
300.2.j.b 8 5.c odd 4 2 inner
300.2.j.b 8 20.d odd 2 1 inner
300.2.j.b 8 20.e even 4 2 inner
900.2.k.h 8 3.b odd 2 1
900.2.k.h 8 12.b even 2 1
900.2.k.h 8 15.d odd 2 1
900.2.k.h 8 15.e even 4 2
900.2.k.h 8 60.h even 2 1
900.2.k.h 8 60.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(300, [\chi])\):

\( T_{7}^{4} + 16 \)
\( T_{19}^{2} - 28 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 16 + T^{4} + T^{8} \)
$3$ \( ( 1 + T^{4} )^{2} \)
$5$ \( T^{8} \)
$7$ \( ( 16 + T^{4} )^{2} \)
$11$ \( ( 28 + T^{2} )^{4} \)
$13$ \( ( 784 + T^{4} )^{2} \)
$17$ \( T^{8} \)
$19$ \( ( -28 + T^{2} )^{4} \)
$23$ \( ( 256 + T^{4} )^{2} \)
$29$ \( ( 64 + T^{2} )^{4} \)
$31$ \( ( 28 + T^{2} )^{4} \)
$37$ \( ( 784 + T^{4} )^{2} \)
$41$ \( ( 2 + T )^{8} \)
$43$ \( ( 4096 + T^{4} )^{2} \)
$47$ \( T^{8} \)
$53$ \( ( 12544 + T^{4} )^{2} \)
$59$ \( ( -28 + T^{2} )^{4} \)
$61$ \( ( -6 + T )^{8} \)
$67$ \( ( 20736 + T^{4} )^{2} \)
$71$ \( T^{8} \)
$73$ \( ( 12544 + T^{4} )^{2} \)
$79$ \( ( -28 + T^{2} )^{4} \)
$83$ \( ( 20736 + T^{4} )^{2} \)
$89$ \( ( 36 + T^{2} )^{4} \)
$97$ \( ( 12544 + T^{4} )^{2} \)
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