Properties

Label 361.4.a.b
Level $361$
Weight $4$
Character orbit 361.a
Self dual yes
Analytic conductor $21.300$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 361 = 19^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 361.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(21.2996895121\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 19)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 3q^{2} + 5q^{3} + q^{4} - 12q^{5} + 15q^{6} + 11q^{7} - 21q^{8} - 2q^{9} + O(q^{10}) \) \( q + 3q^{2} + 5q^{3} + q^{4} - 12q^{5} + 15q^{6} + 11q^{7} - 21q^{8} - 2q^{9} - 36q^{10} - 54q^{11} + 5q^{12} - 11q^{13} + 33q^{14} - 60q^{15} - 71q^{16} - 93q^{17} - 6q^{18} - 12q^{20} + 55q^{21} - 162q^{22} + 183q^{23} - 105q^{24} + 19q^{25} - 33q^{26} - 145q^{27} + 11q^{28} + 249q^{29} - 180q^{30} - 56q^{31} - 45q^{32} - 270q^{33} - 279q^{34} - 132q^{35} - 2q^{36} + 250q^{37} - 55q^{39} + 252q^{40} - 240q^{41} + 165q^{42} - 196q^{43} - 54q^{44} + 24q^{45} + 549q^{46} - 168q^{47} - 355q^{48} - 222q^{49} + 57q^{50} - 465q^{51} - 11q^{52} - 435q^{53} - 435q^{54} + 648q^{55} - 231q^{56} + 747q^{58} - 195q^{59} - 60q^{60} - 358q^{61} - 168q^{62} - 22q^{63} + 433q^{64} + 132q^{65} - 810q^{66} + 961q^{67} - 93q^{68} + 915q^{69} - 396q^{70} + 246q^{71} + 42q^{72} + 353q^{73} + 750q^{74} + 95q^{75} - 594q^{77} - 165q^{78} + 34q^{79} + 852q^{80} - 671q^{81} - 720q^{82} + 234q^{83} + 55q^{84} + 1116q^{85} - 588q^{86} + 1245q^{87} + 1134q^{88} + 168q^{89} + 72q^{90} - 121q^{91} + 183q^{92} - 280q^{93} - 504q^{94} - 225q^{96} - 758q^{97} - 666q^{98} + 108q^{99} + O(q^{100}) \)

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
3.00000 5.00000 1.00000 −12.0000 15.0000 11.0000 −21.0000 −2.00000 −36.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(19\) \(-1\)

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 361.4.a.b 1
19.b odd 2 1 19.4.a.a 1
57.d even 2 1 171.4.a.d 1
76.d even 2 1 304.4.a.b 1
95.d odd 2 1 475.4.a.e 1
95.g even 4 2 475.4.b.c 2
133.c even 2 1 931.4.a.a 1
152.b even 2 1 1216.4.a.a 1
152.g odd 2 1 1216.4.a.f 1
209.d even 2 1 2299.4.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.4.a.a 1 19.b odd 2 1
171.4.a.d 1 57.d even 2 1
304.4.a.b 1 76.d even 2 1
361.4.a.b 1 1.a even 1 1 trivial
475.4.a.e 1 95.d odd 2 1
475.4.b.c 2 95.g even 4 2
931.4.a.a 1 133.c even 2 1
1216.4.a.a 1 152.b even 2 1
1216.4.a.f 1 152.g odd 2 1
2299.4.a.b 1 209.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 3 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(361))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -3 + T \)
$3$ \( -5 + T \)
$5$ \( 12 + T \)
$7$ \( -11 + T \)
$11$ \( 54 + T \)
$13$ \( 11 + T \)
$17$ \( 93 + T \)
$19$ \( T \)
$23$ \( -183 + T \)
$29$ \( -249 + T \)
$31$ \( 56 + T \)
$37$ \( -250 + T \)
$41$ \( 240 + T \)
$43$ \( 196 + T \)
$47$ \( 168 + T \)
$53$ \( 435 + T \)
$59$ \( 195 + T \)
$61$ \( 358 + T \)
$67$ \( -961 + T \)
$71$ \( -246 + T \)
$73$ \( -353 + T \)
$79$ \( -34 + T \)
$83$ \( -234 + T \)
$89$ \( -168 + T \)
$97$ \( 758 + T \)
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