Properties

Label 9702.2.a.cu
Level $9702$
Weight $2$
Character orbit 9702.a
Self dual yes
Analytic conductor $77.471$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9702 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9702.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(77.4708600410\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 154)
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.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + (\beta + 1) q^{5} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} + (\beta + 1) q^{5} - q^{8} + ( - \beta - 1) q^{10} - q^{11} + ( - \beta + 1) q^{13} + q^{16} + ( - 2 \beta - 2) q^{17} + ( - \beta + 5) q^{19} + (\beta + 1) q^{20} + q^{22} - 4 q^{23} + (2 \beta + 1) q^{25} + (\beta - 1) q^{26} + 2 \beta q^{29} - 2 q^{31} - q^{32} + (2 \beta + 2) q^{34} + ( - 4 \beta - 2) q^{37} + (\beta - 5) q^{38} + ( - \beta - 1) q^{40} + (2 \beta + 2) q^{41} + (2 \beta - 6) q^{43} - q^{44} + 4 q^{46} - 2 q^{47} + ( - 2 \beta - 1) q^{50} + ( - \beta + 1) q^{52} + (2 \beta - 4) q^{53} + ( - \beta - 1) q^{55} - 2 \beta q^{58} + (\beta + 5) q^{59} + (\beta + 3) q^{61} + 2 q^{62} + q^{64} - 4 q^{65} + ( - 6 \beta - 2) q^{67} + ( - 2 \beta - 2) q^{68} + (2 \beta - 2) q^{71} + (4 \beta - 4) q^{73} + (4 \beta + 2) q^{74} + ( - \beta + 5) q^{76} + (\beta + 1) q^{80} + ( - 2 \beta - 2) q^{82} + (5 \beta - 1) q^{83} + ( - 4 \beta - 12) q^{85} + ( - 2 \beta + 6) q^{86} + q^{88} + 10 q^{89} - 4 q^{92} + 2 q^{94} + 4 \beta q^{95} + (2 \beta - 8) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{8} - 2 q^{10} - 2 q^{11} + 2 q^{13} + 2 q^{16} - 4 q^{17} + 10 q^{19} + 2 q^{20} + 2 q^{22} - 8 q^{23} + 2 q^{25} - 2 q^{26} - 4 q^{31} - 2 q^{32} + 4 q^{34} - 4 q^{37} - 10 q^{38} - 2 q^{40} + 4 q^{41} - 12 q^{43} - 2 q^{44} + 8 q^{46} - 4 q^{47} - 2 q^{50} + 2 q^{52} - 8 q^{53} - 2 q^{55} + 10 q^{59} + 6 q^{61} + 4 q^{62} + 2 q^{64} - 8 q^{65} - 4 q^{67} - 4 q^{68} - 4 q^{71} - 8 q^{73} + 4 q^{74} + 10 q^{76} + 2 q^{80} - 4 q^{82} - 2 q^{83} - 24 q^{85} + 12 q^{86} + 2 q^{88} + 20 q^{89} - 8 q^{92} + 4 q^{94} - 16 q^{97}+O(q^{100}) \) 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
−0.618034
1.61803
−1.00000 0 1.00000 −1.23607 0 0 −1.00000 0 1.23607
1.2 −1.00000 0 1.00000 3.23607 0 0 −1.00000 0 −3.23607
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(7\) \(-1\)
\(11\) \(1\)

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 9702.2.a.cu 2
3.b odd 2 1 1078.2.a.w 2
7.b odd 2 1 1386.2.a.m 2
12.b even 2 1 8624.2.a.bf 2
21.c even 2 1 154.2.a.d 2
21.g even 6 2 1078.2.e.q 4
21.h odd 6 2 1078.2.e.n 4
84.h odd 2 1 1232.2.a.p 2
105.g even 2 1 3850.2.a.bj 2
105.k odd 4 2 3850.2.c.q 4
168.e odd 2 1 4928.2.a.bk 2
168.i even 2 1 4928.2.a.bt 2
231.h odd 2 1 1694.2.a.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.a.d 2 21.c even 2 1
1078.2.a.w 2 3.b odd 2 1
1078.2.e.n 4 21.h odd 6 2
1078.2.e.q 4 21.g even 6 2
1232.2.a.p 2 84.h odd 2 1
1386.2.a.m 2 7.b odd 2 1
1694.2.a.l 2 231.h odd 2 1
3850.2.a.bj 2 105.g even 2 1
3850.2.c.q 4 105.k odd 4 2
4928.2.a.bk 2 168.e odd 2 1
4928.2.a.bt 2 168.i even 2 1
8624.2.a.bf 2 12.b even 2 1
9702.2.a.cu 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9702))\):

\( T_{5}^{2} - 2T_{5} - 4 \) Copy content Toggle raw display
\( T_{13}^{2} - 2T_{13} - 4 \) Copy content Toggle raw display
\( T_{17}^{2} + 4T_{17} - 16 \) Copy content Toggle raw display
\( T_{19}^{2} - 10T_{19} + 20 \) Copy content Toggle raw display
\( T_{23} + 4 \) Copy content Toggle raw display
\( T_{29}^{2} - 20 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 2T - 4 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 2T - 4 \) Copy content Toggle raw display
$17$ \( T^{2} + 4T - 16 \) Copy content Toggle raw display
$19$ \( T^{2} - 10T + 20 \) Copy content Toggle raw display
$23$ \( (T + 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 20 \) Copy content Toggle raw display
$31$ \( (T + 2)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 4T - 76 \) Copy content Toggle raw display
$41$ \( T^{2} - 4T - 16 \) Copy content Toggle raw display
$43$ \( T^{2} + 12T + 16 \) Copy content Toggle raw display
$47$ \( (T + 2)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 8T - 4 \) Copy content Toggle raw display
$59$ \( T^{2} - 10T + 20 \) Copy content Toggle raw display
$61$ \( T^{2} - 6T + 4 \) Copy content Toggle raw display
$67$ \( T^{2} + 4T - 176 \) Copy content Toggle raw display
$71$ \( T^{2} + 4T - 16 \) Copy content Toggle raw display
$73$ \( T^{2} + 8T - 64 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 2T - 124 \) Copy content Toggle raw display
$89$ \( (T - 10)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 16T + 44 \) Copy content Toggle raw display
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