Properties

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

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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: \(1\)
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 153)
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.

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

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
−1.41421 0 0 −1.00000 0 −1.41421 2.82843 0 1.41421
1.2 1.41421 0 0 −1.00000 0 1.41421 −2.82843 0 −1.41421
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
51.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2601.2.a.p 2
3.b odd 2 1 2601.2.a.q 2
17.b even 2 1 2601.2.a.q 2
17.d even 8 2 153.2.f.a 4
51.c odd 2 1 inner 2601.2.a.p 2
51.g odd 8 2 153.2.f.a 4
68.g odd 8 2 2448.2.be.r 4
204.p even 8 2 2448.2.be.r 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
153.2.f.a 4 17.d even 8 2
153.2.f.a 4 51.g odd 8 2
2448.2.be.r 4 68.g odd 8 2
2448.2.be.r 4 204.p even 8 2
2601.2.a.p 2 1.a even 1 1 trivial
2601.2.a.p 2 51.c odd 2 1 inner
2601.2.a.q 2 3.b odd 2 1
2601.2.a.q 2 17.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2601))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -2 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( -2 + T^{2} \)
$11$ \( ( 1 + T )^{2} \)
$13$ \( ( -1 + T )^{2} \)
$17$ \( T^{2} \)
$19$ \( ( 3 + T )^{2} \)
$23$ \( ( -5 + T )^{2} \)
$29$ \( ( -8 + T )^{2} \)
$31$ \( -32 + T^{2} \)
$37$ \( -2 + T^{2} \)
$41$ \( ( 11 + T )^{2} \)
$43$ \( ( 9 + T )^{2} \)
$47$ \( -72 + T^{2} \)
$53$ \( -8 + T^{2} \)
$59$ \( -98 + T^{2} \)
$61$ \( -50 + T^{2} \)
$67$ \( ( 10 + T )^{2} \)
$71$ \( ( -10 + T )^{2} \)
$73$ \( -128 + T^{2} \)
$79$ \( -50 + T^{2} \)
$83$ \( -200 + T^{2} \)
$89$ \( -18 + T^{2} \)
$97$ \( -200 + T^{2} \)
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