Properties

Label 300.4.a.b
Level $300$
Weight $4$
Character orbit 300.a
Self dual yes
Analytic conductor $17.701$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 300.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.7005730017\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 12)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 3q^{3} - 8q^{7} + 9q^{9} + O(q^{10}) \) \( q - 3q^{3} - 8q^{7} + 9q^{9} + 36q^{11} + 10q^{13} - 18q^{17} - 100q^{19} + 24q^{21} - 72q^{23} - 27q^{27} - 234q^{29} - 16q^{31} - 108q^{33} + 226q^{37} - 30q^{39} + 90q^{41} - 452q^{43} - 432q^{47} - 279q^{49} + 54q^{51} - 414q^{53} + 300q^{57} - 684q^{59} + 422q^{61} - 72q^{63} - 332q^{67} + 216q^{69} - 360q^{71} - 26q^{73} - 288q^{77} + 512q^{79} + 81q^{81} + 1188q^{83} + 702q^{87} - 630q^{89} - 80q^{91} + 48q^{93} + 1054q^{97} + 324q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 −3.00000 0 0 0 −8.00000 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.4.a.b 1
3.b odd 2 1 900.4.a.g 1
4.b odd 2 1 1200.4.a.be 1
5.b even 2 1 12.4.a.a 1
5.c odd 4 2 300.4.d.e 2
15.d odd 2 1 36.4.a.a 1
15.e even 4 2 900.4.d.c 2
20.d odd 2 1 48.4.a.a 1
20.e even 4 2 1200.4.f.d 2
35.c odd 2 1 588.4.a.c 1
35.i odd 6 2 588.4.i.e 2
35.j even 6 2 588.4.i.d 2
40.e odd 2 1 192.4.a.l 1
40.f even 2 1 192.4.a.f 1
45.h odd 6 2 324.4.e.a 2
45.j even 6 2 324.4.e.h 2
55.d odd 2 1 1452.4.a.d 1
60.h even 2 1 144.4.a.g 1
65.d even 2 1 2028.4.a.c 1
65.g odd 4 2 2028.4.b.c 2
80.k odd 4 2 768.4.d.j 2
80.q even 4 2 768.4.d.g 2
105.g even 2 1 1764.4.a.b 1
105.o odd 6 2 1764.4.k.b 2
105.p even 6 2 1764.4.k.o 2
120.i odd 2 1 576.4.a.b 1
120.m even 2 1 576.4.a.a 1
140.c even 2 1 2352.4.a.bk 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.4.a.a 1 5.b even 2 1
36.4.a.a 1 15.d odd 2 1
48.4.a.a 1 20.d odd 2 1
144.4.a.g 1 60.h even 2 1
192.4.a.f 1 40.f even 2 1
192.4.a.l 1 40.e odd 2 1
300.4.a.b 1 1.a even 1 1 trivial
300.4.d.e 2 5.c odd 4 2
324.4.e.a 2 45.h odd 6 2
324.4.e.h 2 45.j even 6 2
576.4.a.a 1 120.m even 2 1
576.4.a.b 1 120.i odd 2 1
588.4.a.c 1 35.c odd 2 1
588.4.i.d 2 35.j even 6 2
588.4.i.e 2 35.i odd 6 2
768.4.d.g 2 80.q even 4 2
768.4.d.j 2 80.k odd 4 2
900.4.a.g 1 3.b odd 2 1
900.4.d.c 2 15.e even 4 2
1200.4.a.be 1 4.b odd 2 1
1200.4.f.d 2 20.e even 4 2
1452.4.a.d 1 55.d odd 2 1
1764.4.a.b 1 105.g even 2 1
1764.4.k.b 2 105.o odd 6 2
1764.4.k.o 2 105.p even 6 2
2028.4.a.c 1 65.d even 2 1
2028.4.b.c 2 65.g odd 4 2
2352.4.a.bk 1 140.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} + 8 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(300))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 3 + T \)
$5$ \( T \)
$7$ \( 8 + T \)
$11$ \( -36 + T \)
$13$ \( -10 + T \)
$17$ \( 18 + T \)
$19$ \( 100 + T \)
$23$ \( 72 + T \)
$29$ \( 234 + T \)
$31$ \( 16 + T \)
$37$ \( -226 + T \)
$41$ \( -90 + T \)
$43$ \( 452 + T \)
$47$ \( 432 + T \)
$53$ \( 414 + T \)
$59$ \( 684 + T \)
$61$ \( -422 + T \)
$67$ \( 332 + T \)
$71$ \( 360 + T \)
$73$ \( 26 + T \)
$79$ \( -512 + T \)
$83$ \( -1188 + T \)
$89$ \( 630 + T \)
$97$ \( -1054 + T \)
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