Properties

Label 2275.2.a.j
Level $2275$
Weight $2$
Character orbit 2275.a
Self dual yes
Analytic conductor $18.166$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2275 = 5^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2275.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(18.1659664598\)
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 91)
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.

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

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 1.41421 0 0 −2.00000 −1.00000 2.82843 −1.00000 0
1.2 1.41421 −1.41421 0 0 −2.00000 −1.00000 −2.82843 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(7\) \(1\)
\(13\) \(-1\)

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 2275.2.a.j 2
5.b even 2 1 91.2.a.c 2
15.d odd 2 1 819.2.a.h 2
20.d odd 2 1 1456.2.a.q 2
35.c odd 2 1 637.2.a.g 2
35.i odd 6 2 637.2.e.g 4
35.j even 6 2 637.2.e.f 4
40.e odd 2 1 5824.2.a.bk 2
40.f even 2 1 5824.2.a.bl 2
65.d even 2 1 1183.2.a.d 2
65.g odd 4 2 1183.2.c.d 4
105.g even 2 1 5733.2.a.s 2
455.h odd 2 1 8281.2.a.v 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.a.c 2 5.b even 2 1
637.2.a.g 2 35.c odd 2 1
637.2.e.f 4 35.j even 6 2
637.2.e.g 4 35.i odd 6 2
819.2.a.h 2 15.d odd 2 1
1183.2.a.d 2 65.d even 2 1
1183.2.c.d 4 65.g odd 4 2
1456.2.a.q 2 20.d odd 2 1
2275.2.a.j 2 1.a even 1 1 trivial
5733.2.a.s 2 105.g even 2 1
5824.2.a.bk 2 40.e odd 2 1
5824.2.a.bl 2 40.f even 2 1
8281.2.a.v 2 455.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2275))\):

\( T_{2}^{2} - 2 \)
\( T_{3}^{2} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -2 + T^{2} \)
$3$ \( -2 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( -18 + T^{2} \)
$13$ \( ( -1 + T )^{2} \)
$17$ \( -2 + T^{2} \)
$19$ \( -9 + 6 T + T^{2} \)
$23$ \( 1 - 6 T + T^{2} \)
$29$ \( 1 - 6 T + T^{2} \)
$31$ \( -17 + 2 T + T^{2} \)
$37$ \( -14 - 4 T + T^{2} \)
$41$ \( 28 - 12 T + T^{2} \)
$43$ \( ( -5 + T )^{2} \)
$47$ \( 7 + 6 T + T^{2} \)
$53$ \( 1 - 6 T + T^{2} \)
$59$ \( 4 - 12 T + T^{2} \)
$61$ \( ( -6 + T )^{2} \)
$67$ \( -36 - 12 T + T^{2} \)
$71$ \( -14 + 12 T + T^{2} \)
$73$ \( 7 - 10 T + T^{2} \)
$79$ \( -23 - 14 T + T^{2} \)
$83$ \( 63 + 18 T + T^{2} \)
$89$ \( 7 - 6 T + T^{2} \)
$97$ \( -161 - 2 T + T^{2} \)
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