Properties

Label 1127.2.a.c
Level $1127$
Weight $2$
Character orbit 1127.a
Self dual yes
Analytic conductor $8.999$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1127 = 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1127.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.99914030780\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{2} + ( 1 - 2 \beta ) q^{3} + ( -1 + \beta ) q^{4} + 2 \beta q^{5} + ( 2 + \beta ) q^{6} + ( -1 + 2 \beta ) q^{8} + 2 q^{9} +O(q^{10})\) \( q -\beta q^{2} + ( 1 - 2 \beta ) q^{3} + ( -1 + \beta ) q^{4} + 2 \beta q^{5} + ( 2 + \beta ) q^{6} + ( -1 + 2 \beta ) q^{8} + 2 q^{9} + ( -2 - 2 \beta ) q^{10} + ( -4 + 2 \beta ) q^{11} + ( -3 + \beta ) q^{12} -3 q^{13} + ( -4 - 2 \beta ) q^{15} -3 \beta q^{16} + ( -2 - 2 \beta ) q^{17} -2 \beta q^{18} + 2 q^{19} + 2 q^{20} + ( -2 + 2 \beta ) q^{22} + q^{23} -5 q^{24} + ( -1 + 4 \beta ) q^{25} + 3 \beta q^{26} + ( -1 + 2 \beta ) q^{27} -3 q^{29} + ( 2 + 6 \beta ) q^{30} + ( -3 + 6 \beta ) q^{31} + ( 5 - \beta ) q^{32} + ( -8 + 6 \beta ) q^{33} + ( 2 + 4 \beta ) q^{34} + ( -2 + 2 \beta ) q^{36} + 2 \beta q^{37} -2 \beta q^{38} + ( -3 + 6 \beta ) q^{39} + ( 4 + 2 \beta ) q^{40} + ( 1 - 4 \beta ) q^{41} + ( 6 - 4 \beta ) q^{44} + 4 \beta q^{45} -\beta q^{46} + ( 1 - 2 \beta ) q^{47} + ( 6 + 3 \beta ) q^{48} + ( -4 - 3 \beta ) q^{50} + ( 2 + 6 \beta ) q^{51} + ( 3 - 3 \beta ) q^{52} + ( -2 - 4 \beta ) q^{53} + ( -2 - \beta ) q^{54} + ( 4 - 4 \beta ) q^{55} + ( 2 - 4 \beta ) q^{57} + 3 \beta q^{58} + ( -4 + 4 \beta ) q^{59} + ( 2 - 4 \beta ) q^{60} + ( 2 - 8 \beta ) q^{61} + ( -6 - 3 \beta ) q^{62} + ( 1 + 2 \beta ) q^{64} -6 \beta q^{65} + ( -6 + 2 \beta ) q^{66} + ( -4 - 2 \beta ) q^{67} -2 \beta q^{68} + ( 1 - 2 \beta ) q^{69} + ( 11 - 2 \beta ) q^{71} + ( -2 + 4 \beta ) q^{72} + ( -9 - 4 \beta ) q^{73} + ( -2 - 2 \beta ) q^{74} + ( -9 - 2 \beta ) q^{75} + ( -2 + 2 \beta ) q^{76} + ( -6 - 3 \beta ) q^{78} + ( -6 + 8 \beta ) q^{79} + ( -6 - 6 \beta ) q^{80} -11 q^{81} + ( 4 + 3 \beta ) q^{82} + ( 10 + 2 \beta ) q^{83} + ( -4 - 8 \beta ) q^{85} + ( -3 + 6 \beta ) q^{87} + ( 8 - 6 \beta ) q^{88} + ( 8 - 4 \beta ) q^{89} + ( -4 - 4 \beta ) q^{90} + ( -1 + \beta ) q^{92} -15 q^{93} + ( 2 + \beta ) q^{94} + 4 \beta q^{95} + ( 7 - 9 \beta ) q^{96} + ( -14 + 6 \beta ) q^{97} + ( -8 + 4 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} - q^{4} + 2q^{5} + 5q^{6} + 4q^{9} + O(q^{10}) \) \( 2q - q^{2} - q^{4} + 2q^{5} + 5q^{6} + 4q^{9} - 6q^{10} - 6q^{11} - 5q^{12} - 6q^{13} - 10q^{15} - 3q^{16} - 6q^{17} - 2q^{18} + 4q^{19} + 4q^{20} - 2q^{22} + 2q^{23} - 10q^{24} + 2q^{25} + 3q^{26} - 6q^{29} + 10q^{30} + 9q^{32} - 10q^{33} + 8q^{34} - 2q^{36} + 2q^{37} - 2q^{38} + 10q^{40} - 2q^{41} + 8q^{44} + 4q^{45} - q^{46} + 15q^{48} - 11q^{50} + 10q^{51} + 3q^{52} - 8q^{53} - 5q^{54} + 4q^{55} + 3q^{58} - 4q^{59} - 4q^{61} - 15q^{62} + 4q^{64} - 6q^{65} - 10q^{66} - 10q^{67} - 2q^{68} + 20q^{71} - 22q^{73} - 6q^{74} - 20q^{75} - 2q^{76} - 15q^{78} - 4q^{79} - 18q^{80} - 22q^{81} + 11q^{82} + 22q^{83} - 16q^{85} + 10q^{88} + 12q^{89} - 12q^{90} - q^{92} - 30q^{93} + 5q^{94} + 4q^{95} + 5q^{96} - 22q^{97} - 12q^{99} + O(q^{100}) \)

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
−1.61803 −2.23607 0.618034 3.23607 3.61803 0 2.23607 2.00000 −5.23607
1.2 0.618034 2.23607 −1.61803 −1.23607 1.38197 0 −2.23607 2.00000 −0.763932
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)
\(23\) \(-1\)

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 1127.2.a.c 2
7.b odd 2 1 23.2.a.a 2
21.c even 2 1 207.2.a.d 2
28.d even 2 1 368.2.a.h 2
35.c odd 2 1 575.2.a.f 2
35.f even 4 2 575.2.b.d 4
56.e even 2 1 1472.2.a.s 2
56.h odd 2 1 1472.2.a.t 2
77.b even 2 1 2783.2.a.c 2
84.h odd 2 1 3312.2.a.ba 2
91.b odd 2 1 3887.2.a.i 2
105.g even 2 1 5175.2.a.be 2
119.d odd 2 1 6647.2.a.b 2
133.c even 2 1 8303.2.a.e 2
140.c even 2 1 9200.2.a.bt 2
161.c even 2 1 529.2.a.a 2
161.k even 22 10 529.2.c.n 20
161.l odd 22 10 529.2.c.o 20
483.c odd 2 1 4761.2.a.w 2
644.h odd 2 1 8464.2.a.bb 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.2.a.a 2 7.b odd 2 1
207.2.a.d 2 21.c even 2 1
368.2.a.h 2 28.d even 2 1
529.2.a.a 2 161.c even 2 1
529.2.c.n 20 161.k even 22 10
529.2.c.o 20 161.l odd 22 10
575.2.a.f 2 35.c odd 2 1
575.2.b.d 4 35.f even 4 2
1127.2.a.c 2 1.a even 1 1 trivial
1472.2.a.s 2 56.e even 2 1
1472.2.a.t 2 56.h odd 2 1
2783.2.a.c 2 77.b even 2 1
3312.2.a.ba 2 84.h odd 2 1
3887.2.a.i 2 91.b odd 2 1
4761.2.a.w 2 483.c odd 2 1
5175.2.a.be 2 105.g even 2 1
6647.2.a.b 2 119.d odd 2 1
8303.2.a.e 2 133.c even 2 1
8464.2.a.bb 2 644.h odd 2 1
9200.2.a.bt 2 140.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1127))\):

\( T_{2}^{2} + T_{2} - 1 \)
\( T_{3}^{2} - 5 \)

Hecke characteristic polynomials

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