Properties

Label 2601.2.a.n
Level $2601$
Weight $2$
Character orbit 2601.a
Self dual yes
Analytic conductor $20.769$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2601 = 3^{2} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2601.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(20.7690895657\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 867)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta ) q^{2} + ( 1 - 2 \beta ) q^{4} -2 \beta q^{5} + 3 q^{7} + ( -3 + \beta ) q^{8} +O(q^{10})\) \( q + ( -1 + \beta ) q^{2} + ( 1 - 2 \beta ) q^{4} -2 \beta q^{5} + 3 q^{7} + ( -3 + \beta ) q^{8} + ( -4 + 2 \beta ) q^{10} + ( -2 - 2 \beta ) q^{11} + ( 1 - 4 \beta ) q^{13} + ( -3 + 3 \beta ) q^{14} + 3 q^{16} -3 q^{19} + ( 8 - 2 \beta ) q^{20} -2 q^{22} -2 \beta q^{23} + 3 q^{25} + ( -9 + 5 \beta ) q^{26} + ( 3 - 6 \beta ) q^{28} + ( 6 + 2 \beta ) q^{29} + ( 1 + 4 \beta ) q^{31} + ( 3 + \beta ) q^{32} -6 \beta q^{35} + ( 1 - 4 \beta ) q^{37} + ( 3 - 3 \beta ) q^{38} + ( -4 + 6 \beta ) q^{40} + ( -6 + 2 \beta ) q^{41} + 3 q^{43} + ( 6 + 2 \beta ) q^{44} + ( -4 + 2 \beta ) q^{46} + ( 4 - 4 \beta ) q^{47} + 2 q^{49} + ( -3 + 3 \beta ) q^{50} + ( 17 - 6 \beta ) q^{52} + ( -4 - 4 \beta ) q^{53} + ( 8 + 4 \beta ) q^{55} + ( -9 + 3 \beta ) q^{56} + ( -2 + 4 \beta ) q^{58} + ( 6 - 2 \beta ) q^{59} + ( 1 + 4 \beta ) q^{61} + ( 7 - 3 \beta ) q^{62} + ( -7 + 2 \beta ) q^{64} + ( 16 - 2 \beta ) q^{65} + q^{67} + ( -12 + 6 \beta ) q^{70} + ( 8 + 2 \beta ) q^{71} + ( -2 + 4 \beta ) q^{73} + ( -9 + 5 \beta ) q^{74} + ( -3 + 6 \beta ) q^{76} + ( -6 - 6 \beta ) q^{77} + 12 q^{79} -6 \beta q^{80} + ( 10 - 8 \beta ) q^{82} + ( -4 - 2 \beta ) q^{83} + ( -3 + 3 \beta ) q^{86} + ( 2 + 4 \beta ) q^{88} + ( 6 + 8 \beta ) q^{89} + ( 3 - 12 \beta ) q^{91} + ( 8 - 2 \beta ) q^{92} + ( -12 + 8 \beta ) q^{94} + 6 \beta q^{95} + ( -5 + 4 \beta ) q^{97} + ( -2 + 2 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{4} + 6q^{7} - 6q^{8} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{4} + 6q^{7} - 6q^{8} - 8q^{10} - 4q^{11} + 2q^{13} - 6q^{14} + 6q^{16} - 6q^{19} + 16q^{20} - 4q^{22} + 6q^{25} - 18q^{26} + 6q^{28} + 12q^{29} + 2q^{31} + 6q^{32} + 2q^{37} + 6q^{38} - 8q^{40} - 12q^{41} + 6q^{43} + 12q^{44} - 8q^{46} + 8q^{47} + 4q^{49} - 6q^{50} + 34q^{52} - 8q^{53} + 16q^{55} - 18q^{56} - 4q^{58} + 12q^{59} + 2q^{61} + 14q^{62} - 14q^{64} + 32q^{65} + 2q^{67} - 24q^{70} + 16q^{71} - 4q^{73} - 18q^{74} - 6q^{76} - 12q^{77} + 24q^{79} + 20q^{82} - 8q^{83} - 6q^{86} + 4q^{88} + 12q^{89} + 6q^{91} + 16q^{92} - 24q^{94} - 10q^{97} - 4q^{98} + O(q^{100}) \)

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
−1.41421
1.41421
−2.41421 0 3.82843 2.82843 0 3.00000 −4.41421 0 −6.82843
1.2 0.414214 0 −1.82843 −2.82843 0 3.00000 −1.58579 0 −1.17157
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(17\) \(1\)

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 2601.2.a.n 2
3.b odd 2 1 867.2.a.h yes 2
17.b even 2 1 2601.2.a.m 2
51.c odd 2 1 867.2.a.g 2
51.f odd 4 2 867.2.d.b 4
51.g odd 8 2 867.2.e.b 4
51.g odd 8 2 867.2.e.c 4
51.i even 16 4 867.2.h.a 8
51.i even 16 4 867.2.h.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
867.2.a.g 2 51.c odd 2 1
867.2.a.h yes 2 3.b odd 2 1
867.2.d.b 4 51.f odd 4 2
867.2.e.b 4 51.g odd 8 2
867.2.e.c 4 51.g odd 8 2
867.2.h.a 8 51.i even 16 4
867.2.h.h 8 51.i even 16 4
2601.2.a.m 2 17.b even 2 1
2601.2.a.n 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2601))\):

\( T_{2}^{2} + 2 T_{2} - 1 \)
\( T_{5}^{2} - 8 \)
\( T_{7} - 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + 2 T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( -8 + T^{2} \)
$7$ \( ( -3 + T )^{2} \)
$11$ \( -4 + 4 T + T^{2} \)
$13$ \( -31 - 2 T + T^{2} \)
$17$ \( T^{2} \)
$19$ \( ( 3 + T )^{2} \)
$23$ \( -8 + T^{2} \)
$29$ \( 28 - 12 T + T^{2} \)
$31$ \( -31 - 2 T + T^{2} \)
$37$ \( -31 - 2 T + T^{2} \)
$41$ \( 28 + 12 T + T^{2} \)
$43$ \( ( -3 + T )^{2} \)
$47$ \( -16 - 8 T + T^{2} \)
$53$ \( -16 + 8 T + T^{2} \)
$59$ \( 28 - 12 T + T^{2} \)
$61$ \( -31 - 2 T + T^{2} \)
$67$ \( ( -1 + T )^{2} \)
$71$ \( 56 - 16 T + T^{2} \)
$73$ \( -28 + 4 T + T^{2} \)
$79$ \( ( -12 + T )^{2} \)
$83$ \( 8 + 8 T + T^{2} \)
$89$ \( -92 - 12 T + T^{2} \)
$97$ \( -7 + 10 T + T^{2} \)
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