Properties

Label 300.2.d.a
Level $300$
Weight $2$
Character orbit 300.d
Analytic conductor $2.396$
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: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 300.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.39551206064\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + i q^{3} + i q^{7} - q^{9} +O(q^{10})\) \( q + i q^{3} + i q^{7} - q^{9} + 6 q^{11} + 5 i q^{13} + 6 i q^{17} -5 q^{19} - q^{21} -6 i q^{23} -i q^{27} + 6 q^{29} - q^{31} + 6 i q^{33} -2 i q^{37} -5 q^{39} -i q^{43} -6 i q^{47} + 6 q^{49} -6 q^{51} -12 i q^{53} -5 i q^{57} + 6 q^{59} -13 q^{61} -i q^{63} -11 i q^{67} + 6 q^{69} + 2 i q^{73} + 6 i q^{77} -8 q^{79} + q^{81} -6 i q^{83} + 6 i q^{87} -5 q^{91} -i q^{93} + 7 i q^{97} -6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{9} + 12q^{11} - 10q^{19} - 2q^{21} + 12q^{29} - 2q^{31} - 10q^{39} + 12q^{49} - 12q^{51} + 12q^{59} - 26q^{61} + 12q^{69} - 16q^{79} + 2q^{81} - 10q^{91} - 12q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
1.00000i
1.00000i
0 1.00000i 0 0 0 1.00000i 0 −1.00000 0
49.2 0 1.00000i 0 0 0 1.00000i 0 −1.00000 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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.2.d.a 2
3.b odd 2 1 900.2.d.a 2
4.b odd 2 1 1200.2.f.a 2
5.b even 2 1 inner 300.2.d.a 2
5.c odd 4 1 300.2.a.b 1
5.c odd 4 1 300.2.a.c yes 1
8.b even 2 1 4800.2.f.b 2
8.d odd 2 1 4800.2.f.bi 2
12.b even 2 1 3600.2.f.v 2
15.d odd 2 1 900.2.d.a 2
15.e even 4 1 900.2.a.c 1
15.e even 4 1 900.2.a.e 1
20.d odd 2 1 1200.2.f.a 2
20.e even 4 1 1200.2.a.f 1
20.e even 4 1 1200.2.a.n 1
40.e odd 2 1 4800.2.f.bi 2
40.f even 2 1 4800.2.f.b 2
40.i odd 4 1 4800.2.a.o 1
40.i odd 4 1 4800.2.a.ce 1
40.k even 4 1 4800.2.a.p 1
40.k even 4 1 4800.2.a.cf 1
60.h even 2 1 3600.2.f.v 2
60.l odd 4 1 3600.2.a.s 1
60.l odd 4 1 3600.2.a.z 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.2.a.b 1 5.c odd 4 1
300.2.a.c yes 1 5.c odd 4 1
300.2.d.a 2 1.a even 1 1 trivial
300.2.d.a 2 5.b even 2 1 inner
900.2.a.c 1 15.e even 4 1
900.2.a.e 1 15.e even 4 1
900.2.d.a 2 3.b odd 2 1
900.2.d.a 2 15.d odd 2 1
1200.2.a.f 1 20.e even 4 1
1200.2.a.n 1 20.e even 4 1
1200.2.f.a 2 4.b odd 2 1
1200.2.f.a 2 20.d odd 2 1
3600.2.a.s 1 60.l odd 4 1
3600.2.a.z 1 60.l odd 4 1
3600.2.f.v 2 12.b even 2 1
3600.2.f.v 2 60.h even 2 1
4800.2.a.o 1 40.i odd 4 1
4800.2.a.p 1 40.k even 4 1
4800.2.a.ce 1 40.i odd 4 1
4800.2.a.cf 1 40.k even 4 1
4800.2.f.b 2 8.b even 2 1
4800.2.f.b 2 40.f even 2 1
4800.2.f.bi 2 8.d odd 2 1
4800.2.f.bi 2 40.e odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(300, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 1 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 1 + T^{2} \)
$11$ \( ( -6 + T )^{2} \)
$13$ \( 25 + T^{2} \)
$17$ \( 36 + T^{2} \)
$19$ \( ( 5 + T )^{2} \)
$23$ \( 36 + T^{2} \)
$29$ \( ( -6 + T )^{2} \)
$31$ \( ( 1 + T )^{2} \)
$37$ \( 4 + T^{2} \)
$41$ \( T^{2} \)
$43$ \( 1 + T^{2} \)
$47$ \( 36 + T^{2} \)
$53$ \( 144 + T^{2} \)
$59$ \( ( -6 + T )^{2} \)
$61$ \( ( 13 + T )^{2} \)
$67$ \( 121 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 4 + T^{2} \)
$79$ \( ( 8 + T )^{2} \)
$83$ \( 36 + T^{2} \)
$89$ \( T^{2} \)
$97$ \( 49 + T^{2} \)
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