Properties

Label 361.2.a.g
Level $361$
Weight $2$
Character orbit 361.a
Self dual yes
Analytic conductor $2.883$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(2.88259951297\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
Defining polynomial: \( x^{3} - 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 19)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + (\beta_{2} - 1) q^{3} + (\beta_{2} - 2 \beta_1 + 1) q^{4} + ( - \beta_{2} - 1) q^{5} + ( - \beta_{2} + 2) q^{6} - \beta_1 q^{7} + ( - 3 \beta_{2} + 2 \beta_1 - 2) q^{8} + ( - 3 \beta_{2} + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} + (\beta_{2} - 1) q^{3} + (\beta_{2} - 2 \beta_1 + 1) q^{4} + ( - \beta_{2} - 1) q^{5} + ( - \beta_{2} + 2) q^{6} - \beta_1 q^{7} + ( - 3 \beta_{2} + 2 \beta_1 - 2) q^{8} + ( - 3 \beta_{2} + \beta_1) q^{9} + (\beta_{2} - 2 \beta_1) q^{10} + (\beta_{2} + \beta_1) q^{11} + ( - \beta_{2} + \beta_1 - 1) q^{12} + ( - \beta_{2} - 2 \beta_1) q^{13} + ( - \beta_{2} + \beta_1 - 2) q^{14} + (\beta_{2} - \beta_1 - 1) q^{15} + (3 \beta_{2} - 3 \beta_1 + 1) q^{16} + (\beta_{2} - \beta_1 + 2) q^{17} + (4 \beta_{2} - 4 \beta_1 - 1) q^{18} + ( - \beta_{2} + 3 \beta_1 - 1) q^{20} - q^{21} + 3 q^{22} + (2 \beta_1 - 2) q^{23} + (4 \beta_{2} - 3 \beta_1 - 2) q^{24} + (\beta_{2} + \beta_1 - 2) q^{25} + ( - \beta_{2} + \beta_1 - 5) q^{26} + (3 \beta_{2} - 3 \beta_1 - 2) q^{27} + (2 \beta_{2} - 2 \beta_1 + 3) q^{28} + (\beta_{2} - 5) q^{29} + ( - 2 \beta_{2} + \beta_1) q^{30} + (2 \beta_{2} + \beta_1 - 3) q^{31} + 3 \beta_1 q^{32} + ( - 2 \beta_{2} + \beta_1 + 3) q^{33} + ( - 2 \beta_{2} + 4 \beta_1 - 3) q^{34} + (2 \beta_1 + 1) q^{35} + ( - 2 \beta_{2} + 5 \beta_1 - 3) q^{36} + ( - 3 \beta_{2} + 2 \beta_1) q^{37} + (2 \beta_{2} - \beta_1 - 4) q^{39} + (2 \beta_{2} - \beta_1 + 6) q^{40} + ( - 4 \beta_{2} + 3 \beta_1 - 4) q^{41} + ( - \beta_1 + 1) q^{42} + ( - 3 \beta_{2} + 5 \beta_1) q^{43} + ( - 2 \beta_{2} + \beta_1 - 3) q^{44} + (\beta_1 + 5) q^{45} + (2 \beta_{2} - 4 \beta_1 + 6) q^{46} + (3 \beta_{2} - \beta_1 - 2) q^{47} + ( - 5 \beta_{2} + 3 \beta_1 + 2) q^{48} + (\beta_{2} - 5) q^{49} + ( - 2 \beta_1 + 5) q^{50} + (\beta_1 - 1) q^{51} + (4 \beta_{2} - 3 \beta_1 + 6) q^{52} + (\beta_{2} - 3 \beta_1 - 2) q^{53} + ( - 6 \beta_{2} + 4 \beta_1 - 1) q^{54} + ( - 3 \beta_1 - 3) q^{55} + ( - 2 \beta_{2} + 5 \beta_1 - 1) q^{56} + ( - \beta_{2} - 4 \beta_1 + 6) q^{58} + ( - 2 \beta_1 - 7) q^{59} + (\beta_{2} - \beta_1 + 2) q^{60} + ( - 4 \beta_{2} + 4 \beta_1 + 3) q^{61} + ( - \beta_{2} - 2 \beta_1 + 7) q^{62} + ( - \beta_{2} + 3 \beta_1 + 1) q^{63} + ( - 3 \beta_{2} + 3 \beta_1 + 4) q^{64} + (5 \beta_1 + 4) q^{65} + (3 \beta_{2} - 3) q^{66} + ( - 2 \beta_{2} + 6 \beta_1 + 6) q^{67} + (4 \beta_{2} - 7 \beta_1 + 5) q^{68} + ( - 2 \beta_{2} + 4) q^{69} + (2 \beta_{2} - \beta_1 + 3) q^{70} + ( - 2 \beta_1 - 10) q^{71} + ( - \beta_{2} - 2 \beta_1 + 13) q^{72} - 4 \beta_1 q^{73} + (5 \beta_{2} - 5 \beta_1 + 1) q^{74} + ( - 4 \beta_{2} + \beta_1 + 5) q^{75} + ( - \beta_{2} - \beta_1 - 3) q^{77} + ( - 3 \beta_{2} - \beta_1 + 4) q^{78} + (6 \beta_{2} - 7 \beta_1 - 3) q^{79} + ( - \beta_{2} + 3 \beta_1 - 4) q^{80} + (\beta_{2} + 5) q^{81} + (7 \beta_{2} - 11 \beta_1 + 6) q^{82} + (9 \beta_{2} - 6 \beta_1) q^{83} + ( - \beta_{2} + 2 \beta_1 - 1) q^{84} + ( - 2 \beta_{2} + \beta_1 - 3) q^{85} + (8 \beta_{2} - 8 \beta_1 + 7) q^{86} + ( - 7 \beta_{2} + \beta_1 + 7) q^{87} + (3 \beta_{2} - 6 \beta_1 - 3) q^{88} + (3 \beta_{2} - \beta_1 - 5) q^{89} + (\beta_{2} + 4 \beta_1 - 3) q^{90} + (2 \beta_{2} + \beta_1 + 5) q^{91} + ( - 6 \beta_{2} + 8 \beta_1 - 8) q^{92} + ( - 7 \beta_{2} + 2 \beta_1 + 8) q^{93} + ( - 4 \beta_{2} + 2 \beta_1 + 3) q^{94} + 3 q^{96} + ( - 2 \beta_{2} - 2 \beta_1 + 5) q^{97} + ( - \beta_{2} - 4 \beta_1 + 6) q^{98} + (4 \beta_{2} - 5 \beta_1 - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{5} + 6 q^{6} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{5} + 6 q^{6} - 6 q^{8} - 3 q^{12} - 6 q^{14} - 3 q^{15} + 3 q^{16} + 6 q^{17} - 3 q^{18} - 3 q^{20} - 3 q^{21} + 9 q^{22} - 6 q^{23} - 6 q^{24} - 6 q^{25} - 15 q^{26} - 6 q^{27} + 9 q^{28} - 15 q^{29} - 9 q^{31} + 9 q^{33} - 9 q^{34} + 3 q^{35} - 9 q^{36} - 12 q^{39} + 18 q^{40} - 12 q^{41} + 3 q^{42} - 9 q^{44} + 15 q^{45} + 18 q^{46} - 6 q^{47} + 6 q^{48} - 15 q^{49} + 15 q^{50} - 3 q^{51} + 18 q^{52} - 6 q^{53} - 3 q^{54} - 9 q^{55} - 3 q^{56} + 18 q^{58} - 21 q^{59} + 6 q^{60} + 9 q^{61} + 21 q^{62} + 3 q^{63} + 12 q^{64} + 12 q^{65} - 9 q^{66} + 18 q^{67} + 15 q^{68} + 12 q^{69} + 9 q^{70} - 30 q^{71} + 39 q^{72} + 3 q^{74} + 15 q^{75} - 9 q^{77} + 12 q^{78} - 9 q^{79} - 12 q^{80} + 15 q^{81} + 18 q^{82} - 3 q^{84} - 9 q^{85} + 21 q^{86} + 21 q^{87} - 9 q^{88} - 15 q^{89} - 9 q^{90} + 15 q^{91} - 24 q^{92} + 24 q^{93} + 9 q^{94} + 9 q^{96} + 15 q^{97} + 18 q^{98} - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{18} + \zeta_{18}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display

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
−1.53209
−0.347296
1.87939
−2.53209 −0.652704 4.41147 −1.34730 1.65270 1.53209 −6.10607 −2.57398 3.41147
1.2 −1.34730 −2.87939 −0.184793 0.879385 3.87939 0.347296 2.94356 5.29086 −1.18479
1.3 0.879385 0.532089 −1.22668 −2.53209 0.467911 −1.87939 −2.83750 −2.71688 −2.22668
\(n\): 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 361.2.a.g 3
3.b odd 2 1 3249.2.a.z 3
4.b odd 2 1 5776.2.a.br 3
5.b even 2 1 9025.2.a.bd 3
19.b odd 2 1 361.2.a.h 3
19.c even 3 2 361.2.c.i 6
19.d odd 6 2 361.2.c.h 6
19.e even 9 2 19.2.e.a 6
19.e even 9 2 361.2.e.f 6
19.e even 9 2 361.2.e.g 6
19.f odd 18 2 361.2.e.a 6
19.f odd 18 2 361.2.e.b 6
19.f odd 18 2 361.2.e.h 6
57.d even 2 1 3249.2.a.s 3
57.l odd 18 2 171.2.u.c 6
76.d even 2 1 5776.2.a.bi 3
76.l odd 18 2 304.2.u.b 6
95.d odd 2 1 9025.2.a.x 3
95.p even 18 2 475.2.l.a 6
95.q odd 36 4 475.2.u.a 12
133.u even 9 2 931.2.v.b 6
133.w even 9 2 931.2.x.a 6
133.x odd 18 2 931.2.v.a 6
133.y odd 18 2 931.2.w.a 6
133.z odd 18 2 931.2.x.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.2.e.a 6 19.e even 9 2
171.2.u.c 6 57.l odd 18 2
304.2.u.b 6 76.l odd 18 2
361.2.a.g 3 1.a even 1 1 trivial
361.2.a.h 3 19.b odd 2 1
361.2.c.h 6 19.d odd 6 2
361.2.c.i 6 19.c even 3 2
361.2.e.a 6 19.f odd 18 2
361.2.e.b 6 19.f odd 18 2
361.2.e.f 6 19.e even 9 2
361.2.e.g 6 19.e even 9 2
361.2.e.h 6 19.f odd 18 2
475.2.l.a 6 95.p even 18 2
475.2.u.a 12 95.q odd 36 4
931.2.v.a 6 133.x odd 18 2
931.2.v.b 6 133.u even 9 2
931.2.w.a 6 133.y odd 18 2
931.2.x.a 6 133.w even 9 2
931.2.x.b 6 133.z odd 18 2
3249.2.a.s 3 57.d even 2 1
3249.2.a.z 3 3.b odd 2 1
5776.2.a.bi 3 76.d even 2 1
5776.2.a.br 3 4.b odd 2 1
9025.2.a.x 3 95.d odd 2 1
9025.2.a.bd 3 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(361))\):

\( T_{2}^{3} + 3T_{2}^{2} - 3 \) Copy content Toggle raw display
\( T_{3}^{3} + 3T_{3}^{2} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + 3T^{2} - 3 \) Copy content Toggle raw display
$3$ \( T^{3} + 3T^{2} - 1 \) Copy content Toggle raw display
$5$ \( T^{3} + 3T^{2} - 3 \) Copy content Toggle raw display
$7$ \( T^{3} - 3T + 1 \) Copy content Toggle raw display
$11$ \( T^{3} - 9T - 9 \) Copy content Toggle raw display
$13$ \( T^{3} - 21T + 37 \) Copy content Toggle raw display
$17$ \( T^{3} - 6 T^{2} + 9 T - 3 \) Copy content Toggle raw display
$19$ \( T^{3} \) Copy content Toggle raw display
$23$ \( T^{3} + 6T^{2} - 24 \) Copy content Toggle raw display
$29$ \( T^{3} + 15 T^{2} + 72 T + 111 \) Copy content Toggle raw display
$31$ \( T^{3} + 9 T^{2} + 6 T - 53 \) Copy content Toggle raw display
$37$ \( T^{3} - 21T - 17 \) Copy content Toggle raw display
$41$ \( T^{3} + 12 T^{2} + 9 T - 111 \) Copy content Toggle raw display
$43$ \( T^{3} - 57T + 163 \) Copy content Toggle raw display
$47$ \( T^{3} + 6 T^{2} - 9 T + 3 \) Copy content Toggle raw display
$53$ \( T^{3} + 6 T^{2} - 9 T - 51 \) Copy content Toggle raw display
$59$ \( T^{3} + 21 T^{2} + 135 T + 267 \) Copy content Toggle raw display
$61$ \( T^{3} - 9 T^{2} - 21 T + 181 \) Copy content Toggle raw display
$67$ \( T^{3} - 18 T^{2} + 24 T + 424 \) Copy content Toggle raw display
$71$ \( T^{3} + 30 T^{2} + 288 T + 888 \) Copy content Toggle raw display
$73$ \( T^{3} - 48T + 64 \) Copy content Toggle raw display
$79$ \( T^{3} + 9 T^{2} - 102 T - 809 \) Copy content Toggle raw display
$83$ \( T^{3} - 189T + 459 \) Copy content Toggle raw display
$89$ \( T^{3} + 15 T^{2} + 54 T + 57 \) Copy content Toggle raw display
$97$ \( T^{3} - 15 T^{2} + 39 T + 127 \) Copy content Toggle raw display
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