Properties

Label 368.2.a.h
Level $368$
Weight $2$
Character orbit 368.a
Self dual yes
Analytic conductor $2.938$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 368 = 2^{4} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 368.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.93849479438\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 23)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{3} + ( -1 - \beta ) q^{5} + ( -1 + \beta ) q^{7} + 2 q^{9} +O(q^{10})\) \( q -\beta q^{3} + ( -1 - \beta ) q^{5} + ( -1 + \beta ) q^{7} + 2 q^{9} + ( 3 - \beta ) q^{11} + 3 q^{13} + ( 5 + \beta ) q^{15} + ( 3 + \beta ) q^{17} + 2 q^{19} + ( -5 + \beta ) q^{21} - q^{23} + ( 1 + 2 \beta ) q^{25} + \beta q^{27} -3 q^{29} + 3 \beta q^{31} + ( 5 - 3 \beta ) q^{33} -4 q^{35} + ( 1 + \beta ) q^{37} -3 \beta q^{39} + ( 1 + 2 \beta ) q^{41} + ( -2 - 2 \beta ) q^{45} -\beta q^{47} + ( -1 - 2 \beta ) q^{49} + ( -5 - 3 \beta ) q^{51} + ( -4 - 2 \beta ) q^{53} + ( 2 - 2 \beta ) q^{55} -2 \beta q^{57} + ( -2 + 2 \beta ) q^{59} + ( 2 + 4 \beta ) q^{61} + ( -2 + 2 \beta ) q^{63} + ( -3 - 3 \beta ) q^{65} + ( 5 + \beta ) q^{67} + \beta q^{69} + ( -10 + \beta ) q^{71} + ( 11 + 2 \beta ) q^{73} + ( -10 - \beta ) q^{75} + ( -8 + 4 \beta ) q^{77} + ( 2 - 4 \beta ) q^{79} -11 q^{81} + ( 11 + \beta ) q^{83} + ( -8 - 4 \beta ) q^{85} + 3 \beta q^{87} + ( -6 + 2 \beta ) q^{89} + ( -3 + 3 \beta ) q^{91} -15 q^{93} + ( -2 - 2 \beta ) q^{95} + ( 11 - 3 \beta ) q^{97} + ( 6 - 2 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{5} - 2q^{7} + 4q^{9} + O(q^{10}) \) \( 2q - 2q^{5} - 2q^{7} + 4q^{9} + 6q^{11} + 6q^{13} + 10q^{15} + 6q^{17} + 4q^{19} - 10q^{21} - 2q^{23} + 2q^{25} - 6q^{29} + 10q^{33} - 8q^{35} + 2q^{37} + 2q^{41} - 4q^{45} - 2q^{49} - 10q^{51} - 8q^{53} + 4q^{55} - 4q^{59} + 4q^{61} - 4q^{63} - 6q^{65} + 10q^{67} - 20q^{71} + 22q^{73} - 20q^{75} - 16q^{77} + 4q^{79} - 22q^{81} + 22q^{83} - 16q^{85} - 12q^{89} - 6q^{91} - 30q^{93} - 4q^{95} + 22q^{97} + 12q^{99} + O(q^{100}) \)

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
1.61803
−0.618034
0 −2.23607 0 −3.23607 0 1.23607 0 2.00000 0
1.2 0 2.23607 0 1.23607 0 −3.23607 0 2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(23\) \(1\)

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 368.2.a.h 2
3.b odd 2 1 3312.2.a.ba 2
4.b odd 2 1 23.2.a.a 2
5.b even 2 1 9200.2.a.bt 2
8.b even 2 1 1472.2.a.s 2
8.d odd 2 1 1472.2.a.t 2
12.b even 2 1 207.2.a.d 2
20.d odd 2 1 575.2.a.f 2
20.e even 4 2 575.2.b.d 4
23.b odd 2 1 8464.2.a.bb 2
28.d even 2 1 1127.2.a.c 2
44.c even 2 1 2783.2.a.c 2
52.b odd 2 1 3887.2.a.i 2
60.h even 2 1 5175.2.a.be 2
68.d odd 2 1 6647.2.a.b 2
76.d even 2 1 8303.2.a.e 2
92.b even 2 1 529.2.a.a 2
92.g odd 22 10 529.2.c.o 20
92.h even 22 10 529.2.c.n 20
276.h odd 2 1 4761.2.a.w 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.2.a.a 2 4.b odd 2 1
207.2.a.d 2 12.b even 2 1
368.2.a.h 2 1.a even 1 1 trivial
529.2.a.a 2 92.b even 2 1
529.2.c.n 20 92.h even 22 10
529.2.c.o 20 92.g odd 22 10
575.2.a.f 2 20.d odd 2 1
575.2.b.d 4 20.e even 4 2
1127.2.a.c 2 28.d even 2 1
1472.2.a.s 2 8.b even 2 1
1472.2.a.t 2 8.d odd 2 1
2783.2.a.c 2 44.c even 2 1
3312.2.a.ba 2 3.b odd 2 1
3887.2.a.i 2 52.b odd 2 1
4761.2.a.w 2 276.h odd 2 1
5175.2.a.be 2 60.h even 2 1
6647.2.a.b 2 68.d odd 2 1
8303.2.a.e 2 76.d even 2 1
8464.2.a.bb 2 23.b odd 2 1
9200.2.a.bt 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(368))\):

\( T_{3}^{2} - 5 \)
\( T_{5}^{2} + 2 T_{5} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -5 + T^{2} \)
$5$ \( -4 + 2 T + T^{2} \)
$7$ \( -4 + 2 T + T^{2} \)
$11$ \( 4 - 6 T + T^{2} \)
$13$ \( ( -3 + T )^{2} \)
$17$ \( 4 - 6 T + T^{2} \)
$19$ \( ( -2 + T )^{2} \)
$23$ \( ( 1 + T )^{2} \)
$29$ \( ( 3 + T )^{2} \)
$31$ \( -45 + T^{2} \)
$37$ \( -4 - 2 T + T^{2} \)
$41$ \( -19 - 2 T + T^{2} \)
$43$ \( T^{2} \)
$47$ \( -5 + T^{2} \)
$53$ \( -4 + 8 T + T^{2} \)
$59$ \( -16 + 4 T + T^{2} \)
$61$ \( -76 - 4 T + T^{2} \)
$67$ \( 20 - 10 T + T^{2} \)
$71$ \( 95 + 20 T + T^{2} \)
$73$ \( 101 - 22 T + T^{2} \)
$79$ \( -76 - 4 T + T^{2} \)
$83$ \( 116 - 22 T + T^{2} \)
$89$ \( 16 + 12 T + T^{2} \)
$97$ \( 76 - 22 T + T^{2} \)
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