Properties

Label 300.2.i.c
Level $300$
Weight $2$
Character orbit 300.i
Analytic conductor $2.396$
Analytic rank $0$
Dimension $4$
CM discriminant -3
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.i (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.39551206064\)
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: $\mathrm{U}(1)[D_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} -3 \beta_{3} q^{7} + 3 \beta_{2} q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} -3 \beta_{3} q^{7} + 3 \beta_{2} q^{9} -\beta_{1} q^{13} + 7 \beta_{2} q^{19} + 9 q^{21} + 3 \beta_{3} q^{27} -11 q^{31} -4 \beta_{3} q^{37} -3 \beta_{2} q^{39} -\beta_{1} q^{43} -20 \beta_{2} q^{49} + 7 \beta_{3} q^{57} - q^{61} + 9 \beta_{1} q^{63} + 7 \beta_{3} q^{67} -8 \beta_{1} q^{73} -4 \beta_{2} q^{79} -9 q^{81} -9 q^{91} -11 \beta_{1} q^{93} -3 \beta_{3} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q + 36q^{21} - 44q^{31} - 4q^{61} - 36q^{81} - 36q^{91} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
257.1
−1.22474 + 1.22474i
1.22474 1.22474i
−1.22474 1.22474i
1.22474 + 1.22474i
0 −1.22474 + 1.22474i 0 0 0 −3.67423 3.67423i 0 3.00000i 0
257.2 0 1.22474 1.22474i 0 0 0 3.67423 + 3.67423i 0 3.00000i 0
293.1 0 −1.22474 1.22474i 0 0 0 −3.67423 + 3.67423i 0 3.00000i 0
293.2 0 1.22474 + 1.22474i 0 0 0 3.67423 3.67423i 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
5.b even 2 1 inner
5.c odd 4 2 inner
15.d odd 2 1 inner
15.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.i.c 4
3.b odd 2 1 CM 300.2.i.c 4
4.b odd 2 1 1200.2.v.h 4
5.b even 2 1 inner 300.2.i.c 4
5.c odd 4 2 inner 300.2.i.c 4
12.b even 2 1 1200.2.v.h 4
15.d odd 2 1 inner 300.2.i.c 4
15.e even 4 2 inner 300.2.i.c 4
20.d odd 2 1 1200.2.v.h 4
20.e even 4 2 1200.2.v.h 4
60.h even 2 1 1200.2.v.h 4
60.l odd 4 2 1200.2.v.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.2.i.c 4 1.a even 1 1 trivial
300.2.i.c 4 3.b odd 2 1 CM
300.2.i.c 4 5.b even 2 1 inner
300.2.i.c 4 5.c odd 4 2 inner
300.2.i.c 4 15.d odd 2 1 inner
300.2.i.c 4 15.e even 4 2 inner
1200.2.v.h 4 4.b odd 2 1
1200.2.v.h 4 12.b even 2 1
1200.2.v.h 4 20.d odd 2 1
1200.2.v.h 4 20.e even 4 2
1200.2.v.h 4 60.h even 2 1
1200.2.v.h 4 60.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 729 \) acting on \(S_{2}^{\mathrm{new}}(300, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 9 + T^{4} \)
$5$ \( T^{4} \)
$7$ \( 729 + T^{4} \)
$11$ \( T^{4} \)
$13$ \( 9 + T^{4} \)
$17$ \( T^{4} \)
$19$ \( ( 49 + T^{2} )^{2} \)
$23$ \( T^{4} \)
$29$ \( T^{4} \)
$31$ \( ( 11 + T )^{4} \)
$37$ \( 2304 + T^{4} \)
$41$ \( T^{4} \)
$43$ \( 9 + T^{4} \)
$47$ \( T^{4} \)
$53$ \( T^{4} \)
$59$ \( T^{4} \)
$61$ \( ( 1 + T )^{4} \)
$67$ \( 21609 + T^{4} \)
$71$ \( T^{4} \)
$73$ \( 36864 + T^{4} \)
$79$ \( ( 16 + T^{2} )^{2} \)
$83$ \( T^{4} \)
$89$ \( T^{4} \)
$97$ \( 729 + T^{4} \)
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