Properties

Label 1225.2.a.s
Level $1225$
Weight $2$
Character orbit 1225.a
Self dual yes
Analytic conductor $9.782$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 1225 = 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1225.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.78167424761\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + ( -1 + \beta ) q^{3} + ( 2 + \beta ) q^{4} + 4 q^{6} + ( 4 + \beta ) q^{8} + ( 2 - \beta ) q^{9} +O(q^{10})\) \( q + \beta q^{2} + ( -1 + \beta ) q^{3} + ( 2 + \beta ) q^{4} + 4 q^{6} + ( 4 + \beta ) q^{8} + ( 2 - \beta ) q^{9} + ( 1 - \beta ) q^{11} + ( 2 + 2 \beta ) q^{12} + ( 3 - \beta ) q^{13} + 3 \beta q^{16} + ( -3 + \beta ) q^{17} + ( -4 + \beta ) q^{18} + ( 2 + 2 \beta ) q^{19} -4 q^{22} + ( 2 - 2 \beta ) q^{23} + 4 \beta q^{24} + ( -4 + 2 \beta ) q^{26} + ( -3 - \beta ) q^{27} + ( -1 + 3 \beta ) q^{29} + ( 4 + \beta ) q^{32} + ( -5 + \beta ) q^{33} + ( 4 - 2 \beta ) q^{34} -\beta q^{36} -6 q^{37} + ( 8 + 4 \beta ) q^{38} + ( -7 + 3 \beta ) q^{39} -2 \beta q^{41} + ( -6 + 2 \beta ) q^{43} + ( -2 - 2 \beta ) q^{44} -8 q^{46} + ( -1 - 3 \beta ) q^{47} + 12 q^{48} + ( 7 - 3 \beta ) q^{51} + 2 q^{52} + 2 \beta q^{53} + ( -4 - 4 \beta ) q^{54} + ( 6 + 2 \beta ) q^{57} + ( 12 + 2 \beta ) q^{58} + 4 q^{59} -6 \beta q^{61} + ( 4 - \beta ) q^{64} + ( 4 - 4 \beta ) q^{66} -4 \beta q^{67} -2 q^{68} + ( -10 + 2 \beta ) q^{69} + 8 q^{71} + ( 4 - 3 \beta ) q^{72} + ( -2 - 4 \beta ) q^{73} -6 \beta q^{74} + ( 12 + 8 \beta ) q^{76} + ( 12 - 4 \beta ) q^{78} + ( -5 + \beta ) q^{79} -7 q^{81} + ( -8 - 2 \beta ) q^{82} + 4 q^{83} + ( 8 - 4 \beta ) q^{86} + ( 13 - \beta ) q^{87} -4 \beta q^{88} + ( -4 + 2 \beta ) q^{89} + ( -4 - 4 \beta ) q^{92} + ( -12 - 4 \beta ) q^{94} + 4 \beta q^{96} + ( -7 + 5 \beta ) q^{97} + ( 6 - 2 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} - q^{3} + 5q^{4} + 8q^{6} + 9q^{8} + 3q^{9} + O(q^{10}) \) \( 2q + q^{2} - q^{3} + 5q^{4} + 8q^{6} + 9q^{8} + 3q^{9} + q^{11} + 6q^{12} + 5q^{13} + 3q^{16} - 5q^{17} - 7q^{18} + 6q^{19} - 8q^{22} + 2q^{23} + 4q^{24} - 6q^{26} - 7q^{27} + q^{29} + 9q^{32} - 9q^{33} + 6q^{34} - q^{36} - 12q^{37} + 20q^{38} - 11q^{39} - 2q^{41} - 10q^{43} - 6q^{44} - 16q^{46} - 5q^{47} + 24q^{48} + 11q^{51} + 4q^{52} + 2q^{53} - 12q^{54} + 14q^{57} + 26q^{58} + 8q^{59} - 6q^{61} + 7q^{64} + 4q^{66} - 4q^{67} - 4q^{68} - 18q^{69} + 16q^{71} + 5q^{72} - 8q^{73} - 6q^{74} + 32q^{76} + 20q^{78} - 9q^{79} - 14q^{81} - 18q^{82} + 8q^{83} + 12q^{86} + 25q^{87} - 4q^{88} - 6q^{89} - 12q^{92} - 28q^{94} + 4q^{96} - 9q^{97} + 10q^{99} + O(q^{100}) \)

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
−1.56155
2.56155
−1.56155 −2.56155 0.438447 0 4.00000 0 2.43845 3.56155 0
1.2 2.56155 1.56155 4.56155 0 4.00000 0 6.56155 −0.561553 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 1225.2.a.s 2
5.b even 2 1 245.2.a.d 2
5.c odd 4 2 1225.2.b.f 4
7.b odd 2 1 175.2.a.f 2
15.d odd 2 1 2205.2.a.x 2
20.d odd 2 1 3920.2.a.bs 2
21.c even 2 1 1575.2.a.p 2
28.d even 2 1 2800.2.a.bi 2
35.c odd 2 1 35.2.a.b 2
35.f even 4 2 175.2.b.b 4
35.i odd 6 2 245.2.e.i 4
35.j even 6 2 245.2.e.h 4
105.g even 2 1 315.2.a.e 2
105.k odd 4 2 1575.2.d.e 4
140.c even 2 1 560.2.a.i 2
140.j odd 4 2 2800.2.g.t 4
280.c odd 2 1 2240.2.a.bh 2
280.n even 2 1 2240.2.a.bd 2
385.h even 2 1 4235.2.a.m 2
420.o odd 2 1 5040.2.a.bt 2
455.h odd 2 1 5915.2.a.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.a.b 2 35.c odd 2 1
175.2.a.f 2 7.b odd 2 1
175.2.b.b 4 35.f even 4 2
245.2.a.d 2 5.b even 2 1
245.2.e.h 4 35.j even 6 2
245.2.e.i 4 35.i odd 6 2
315.2.a.e 2 105.g even 2 1
560.2.a.i 2 140.c even 2 1
1225.2.a.s 2 1.a even 1 1 trivial
1225.2.b.f 4 5.c odd 4 2
1575.2.a.p 2 21.c even 2 1
1575.2.d.e 4 105.k odd 4 2
2205.2.a.x 2 15.d odd 2 1
2240.2.a.bd 2 280.n even 2 1
2240.2.a.bh 2 280.c odd 2 1
2800.2.a.bi 2 28.d even 2 1
2800.2.g.t 4 140.j odd 4 2
3920.2.a.bs 2 20.d odd 2 1
4235.2.a.m 2 385.h even 2 1
5040.2.a.bt 2 420.o odd 2 1
5915.2.a.l 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1225))\):

\( T_{2}^{2} - T_{2} - 4 \)
\( T_{3}^{2} + T_{3} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -4 - T + T^{2} \)
$3$ \( -4 + T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( -4 - T + T^{2} \)
$13$ \( 2 - 5 T + T^{2} \)
$17$ \( 2 + 5 T + T^{2} \)
$19$ \( -8 - 6 T + T^{2} \)
$23$ \( -16 - 2 T + T^{2} \)
$29$ \( -38 - T + T^{2} \)
$31$ \( T^{2} \)
$37$ \( ( 6 + T )^{2} \)
$41$ \( -16 + 2 T + T^{2} \)
$43$ \( 8 + 10 T + T^{2} \)
$47$ \( -32 + 5 T + T^{2} \)
$53$ \( -16 - 2 T + T^{2} \)
$59$ \( ( -4 + T )^{2} \)
$61$ \( -144 + 6 T + T^{2} \)
$67$ \( -64 + 4 T + T^{2} \)
$71$ \( ( -8 + T )^{2} \)
$73$ \( -52 + 8 T + T^{2} \)
$79$ \( 16 + 9 T + T^{2} \)
$83$ \( ( -4 + T )^{2} \)
$89$ \( -8 + 6 T + T^{2} \)
$97$ \( -86 + 9 T + T^{2} \)
show more
show less