Properties

Label 1425.2.a.p
Level $1425$
Weight $2$
Character orbit 1425.a
Self dual yes
Analytic conductor $11.379$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1425 = 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1425.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.3786822880\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
Defining polynomial: \(x^{2} - 7\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 285)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{7}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + q^{3} + 5 q^{4} + \beta q^{6} + ( 1 - \beta ) q^{7} + 3 \beta q^{8} + q^{9} +O(q^{10})\) \( q + \beta q^{2} + q^{3} + 5 q^{4} + \beta q^{6} + ( 1 - \beta ) q^{7} + 3 \beta q^{8} + q^{9} + ( 3 - \beta ) q^{11} + 5 q^{12} + ( 3 - \beta ) q^{13} + ( -7 + \beta ) q^{14} + 11 q^{16} + 4 q^{17} + \beta q^{18} - q^{19} + ( 1 - \beta ) q^{21} + ( -7 + 3 \beta ) q^{22} + ( -4 - 2 \beta ) q^{23} + 3 \beta q^{24} + ( -7 + 3 \beta ) q^{26} + q^{27} + ( 5 - 5 \beta ) q^{28} + ( 1 + 3 \beta ) q^{29} + 6 q^{31} + 5 \beta q^{32} + ( 3 - \beta ) q^{33} + 4 \beta q^{34} + 5 q^{36} + ( -1 - \beta ) q^{37} -\beta q^{38} + ( 3 - \beta ) q^{39} + ( -7 - \beta ) q^{41} + ( -7 + \beta ) q^{42} + ( -3 - \beta ) q^{43} + ( 15 - 5 \beta ) q^{44} + ( -14 - 4 \beta ) q^{46} + ( -4 + 2 \beta ) q^{47} + 11 q^{48} + ( 1 - 2 \beta ) q^{49} + 4 q^{51} + ( 15 - 5 \beta ) q^{52} + ( -6 - 2 \beta ) q^{53} + \beta q^{54} + ( -21 + 3 \beta ) q^{56} - q^{57} + ( 21 + \beta ) q^{58} + ( 6 + 2 \beta ) q^{59} + ( -6 - 2 \beta ) q^{61} + 6 \beta q^{62} + ( 1 - \beta ) q^{63} + 13 q^{64} + ( -7 + 3 \beta ) q^{66} + ( -4 + 4 \beta ) q^{67} + 20 q^{68} + ( -4 - 2 \beta ) q^{69} + ( 2 + 2 \beta ) q^{71} + 3 \beta q^{72} -10 q^{73} + ( -7 - \beta ) q^{74} -5 q^{76} + ( 10 - 4 \beta ) q^{77} + ( -7 + 3 \beta ) q^{78} + ( -4 + 4 \beta ) q^{79} + q^{81} + ( -7 - 7 \beta ) q^{82} -6 q^{83} + ( 5 - 5 \beta ) q^{84} + ( -7 - 3 \beta ) q^{86} + ( 1 + 3 \beta ) q^{87} + ( -21 + 9 \beta ) q^{88} + ( -9 - 3 \beta ) q^{89} + ( 10 - 4 \beta ) q^{91} + ( -20 - 10 \beta ) q^{92} + 6 q^{93} + ( 14 - 4 \beta ) q^{94} + 5 \beta q^{96} + ( -5 + 3 \beta ) q^{97} + ( -14 + \beta ) q^{98} + ( 3 - \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} + 10q^{4} + 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} + 10q^{4} + 2q^{7} + 2q^{9} + 6q^{11} + 10q^{12} + 6q^{13} - 14q^{14} + 22q^{16} + 8q^{17} - 2q^{19} + 2q^{21} - 14q^{22} - 8q^{23} - 14q^{26} + 2q^{27} + 10q^{28} + 2q^{29} + 12q^{31} + 6q^{33} + 10q^{36} - 2q^{37} + 6q^{39} - 14q^{41} - 14q^{42} - 6q^{43} + 30q^{44} - 28q^{46} - 8q^{47} + 22q^{48} + 2q^{49} + 8q^{51} + 30q^{52} - 12q^{53} - 42q^{56} - 2q^{57} + 42q^{58} + 12q^{59} - 12q^{61} + 2q^{63} + 26q^{64} - 14q^{66} - 8q^{67} + 40q^{68} - 8q^{69} + 4q^{71} - 20q^{73} - 14q^{74} - 10q^{76} + 20q^{77} - 14q^{78} - 8q^{79} + 2q^{81} - 14q^{82} - 12q^{83} + 10q^{84} - 14q^{86} + 2q^{87} - 42q^{88} - 18q^{89} + 20q^{91} - 40q^{92} + 12q^{93} + 28q^{94} - 10q^{97} - 28q^{98} + 6q^{99} + O(q^{100}) \)

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
−2.64575
2.64575
−2.64575 1.00000 5.00000 0 −2.64575 3.64575 −7.93725 1.00000 0
1.2 2.64575 1.00000 5.00000 0 2.64575 −1.64575 7.93725 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(19\) \(1\)

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 1425.2.a.p 2
3.b odd 2 1 4275.2.a.u 2
5.b even 2 1 285.2.a.d 2
5.c odd 4 2 1425.2.c.i 4
15.d odd 2 1 855.2.a.g 2
20.d odd 2 1 4560.2.a.bo 2
95.d odd 2 1 5415.2.a.s 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.a.d 2 5.b even 2 1
855.2.a.g 2 15.d odd 2 1
1425.2.a.p 2 1.a even 1 1 trivial
1425.2.c.i 4 5.c odd 4 2
4275.2.a.u 2 3.b odd 2 1
4560.2.a.bo 2 20.d odd 2 1
5415.2.a.s 2 95.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1425))\):

\( T_{2}^{2} - 7 \)
\( T_{7}^{2} - 2 T_{7} - 6 \)
\( T_{11}^{2} - 6 T_{11} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -7 + T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( T^{2} \)
$7$ \( -6 - 2 T + T^{2} \)
$11$ \( 2 - 6 T + T^{2} \)
$13$ \( 2 - 6 T + T^{2} \)
$17$ \( ( -4 + T )^{2} \)
$19$ \( ( 1 + T )^{2} \)
$23$ \( -12 + 8 T + T^{2} \)
$29$ \( -62 - 2 T + T^{2} \)
$31$ \( ( -6 + T )^{2} \)
$37$ \( -6 + 2 T + T^{2} \)
$41$ \( 42 + 14 T + T^{2} \)
$43$ \( 2 + 6 T + T^{2} \)
$47$ \( -12 + 8 T + T^{2} \)
$53$ \( 8 + 12 T + T^{2} \)
$59$ \( 8 - 12 T + T^{2} \)
$61$ \( 8 + 12 T + T^{2} \)
$67$ \( -96 + 8 T + T^{2} \)
$71$ \( -24 - 4 T + T^{2} \)
$73$ \( ( 10 + T )^{2} \)
$79$ \( -96 + 8 T + T^{2} \)
$83$ \( ( 6 + T )^{2} \)
$89$ \( 18 + 18 T + T^{2} \)
$97$ \( -38 + 10 T + T^{2} \)
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