Properties

Label 4275.2.a.y
Level $4275$
Weight $2$
Character orbit 4275.a
Self dual yes
Analytic conductor $34.136$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4275 = 3^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4275.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(34.1360468641\)
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 285)
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.

\(f(q)\) \(=\) \( q + ( 1 + \beta ) q^{2} + ( 1 + 2 \beta ) q^{4} + \beta q^{7} + ( 3 + \beta ) q^{8} +O(q^{10})\) \( q + ( 1 + \beta ) q^{2} + ( 1 + 2 \beta ) q^{4} + \beta q^{7} + ( 3 + \beta ) q^{8} + ( -2 + 3 \beta ) q^{11} + ( 2 + \beta ) q^{13} + ( 2 + \beta ) q^{14} + 3 q^{16} + ( 4 - 2 \beta ) q^{17} - q^{19} + ( 4 + \beta ) q^{22} + ( 2 + 4 \beta ) q^{23} + ( 4 + 3 \beta ) q^{26} + ( 4 + \beta ) q^{28} -\beta q^{29} + ( -6 + 2 \beta ) q^{31} + ( -3 + \beta ) q^{32} + 2 \beta q^{34} + ( 2 + \beta ) q^{37} + ( -1 - \beta ) q^{38} + ( -4 + 3 \beta ) q^{41} + ( -8 - 3 \beta ) q^{43} + ( 10 - \beta ) q^{44} + ( 10 + 6 \beta ) q^{46} + ( -2 - 4 \beta ) q^{47} -5 q^{49} + ( 6 + 5 \beta ) q^{52} + 8 q^{53} + ( 2 + 3 \beta ) q^{56} + ( -2 - \beta ) q^{58} + ( -4 - 6 \beta ) q^{59} + ( -4 + 8 \beta ) q^{61} + ( -2 - 4 \beta ) q^{62} + ( -7 - 2 \beta ) q^{64} + ( 4 + 4 \beta ) q^{67} + ( -4 + 6 \beta ) q^{68} + ( 8 + 2 \beta ) q^{71} + ( 2 + 4 \beta ) q^{73} + ( 4 + 3 \beta ) q^{74} + ( -1 - 2 \beta ) q^{76} + ( 6 - 2 \beta ) q^{77} + ( 2 - \beta ) q^{82} + ( 10 + 2 \beta ) q^{83} + ( -14 - 11 \beta ) q^{86} + 7 \beta q^{88} + ( 4 - 7 \beta ) q^{89} + ( 2 + 2 \beta ) q^{91} + ( 18 + 8 \beta ) q^{92} + ( -10 - 6 \beta ) q^{94} + ( 14 - 3 \beta ) q^{97} + ( -5 - 5 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} + 6q^{8} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} + 6q^{8} - 4q^{11} + 4q^{13} + 4q^{14} + 6q^{16} + 8q^{17} - 2q^{19} + 8q^{22} + 4q^{23} + 8q^{26} + 8q^{28} - 12q^{31} - 6q^{32} + 4q^{37} - 2q^{38} - 8q^{41} - 16q^{43} + 20q^{44} + 20q^{46} - 4q^{47} - 10q^{49} + 12q^{52} + 16q^{53} + 4q^{56} - 4q^{58} - 8q^{59} - 8q^{61} - 4q^{62} - 14q^{64} + 8q^{67} - 8q^{68} + 16q^{71} + 4q^{73} + 8q^{74} - 2q^{76} + 12q^{77} + 4q^{82} + 20q^{83} - 28q^{86} + 8q^{89} + 4q^{91} + 36q^{92} - 20q^{94} + 28q^{97} - 10q^{98} + O(q^{100}) \)

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
−0.414214 0 −1.82843 0 0 −1.41421 1.58579 0 0
1.2 2.41421 0 3.82843 0 0 1.41421 4.41421 0 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 4275.2.a.y 2
3.b odd 2 1 1425.2.a.k 2
5.b even 2 1 855.2.a.d 2
15.d odd 2 1 285.2.a.g 2
15.e even 4 2 1425.2.c.l 4
60.h even 2 1 4560.2.a.bf 2
285.b even 2 1 5415.2.a.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.a.g 2 15.d odd 2 1
855.2.a.d 2 5.b even 2 1
1425.2.a.k 2 3.b odd 2 1
1425.2.c.l 4 15.e even 4 2
4275.2.a.y 2 1.a even 1 1 trivial
4560.2.a.bf 2 60.h even 2 1
5415.2.a.n 2 285.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4275))\):

\( T_{2}^{2} - 2 T_{2} - 1 \)
\( T_{7}^{2} - 2 \)
\( T_{11}^{2} + 4 T_{11} - 14 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 - 2 T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( -2 + T^{2} \)
$11$ \( -14 + 4 T + T^{2} \)
$13$ \( 2 - 4 T + T^{2} \)
$17$ \( 8 - 8 T + T^{2} \)
$19$ \( ( 1 + T )^{2} \)
$23$ \( -28 - 4 T + T^{2} \)
$29$ \( -2 + T^{2} \)
$31$ \( 28 + 12 T + T^{2} \)
$37$ \( 2 - 4 T + T^{2} \)
$41$ \( -2 + 8 T + T^{2} \)
$43$ \( 46 + 16 T + T^{2} \)
$47$ \( -28 + 4 T + T^{2} \)
$53$ \( ( -8 + T )^{2} \)
$59$ \( -56 + 8 T + T^{2} \)
$61$ \( -112 + 8 T + T^{2} \)
$67$ \( -16 - 8 T + T^{2} \)
$71$ \( 56 - 16 T + T^{2} \)
$73$ \( -28 - 4 T + T^{2} \)
$79$ \( T^{2} \)
$83$ \( 92 - 20 T + T^{2} \)
$89$ \( -82 - 8 T + T^{2} \)
$97$ \( 178 - 28 T + T^{2} \)
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