Properties

Label 1225.4.a.j
Level $1225$
Weight $4$
Character orbit 1225.a
Self dual yes
Analytic conductor $72.277$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 1225 = 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1225.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(72.2773397570\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 7)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} - 2q^{3} - 7q^{4} - 2q^{6} - 15q^{8} - 23q^{9} + O(q^{10}) \) \( q + q^{2} - 2q^{3} - 7q^{4} - 2q^{6} - 15q^{8} - 23q^{9} - 8q^{11} + 14q^{12} + 28q^{13} + 41q^{16} + 54q^{17} - 23q^{18} + 110q^{19} - 8q^{22} - 48q^{23} + 30q^{24} + 28q^{26} + 100q^{27} - 110q^{29} - 12q^{31} + 161q^{32} + 16q^{33} + 54q^{34} + 161q^{36} + 246q^{37} + 110q^{38} - 56q^{39} - 182q^{41} - 128q^{43} + 56q^{44} - 48q^{46} + 324q^{47} - 82q^{48} - 108q^{51} - 196q^{52} + 162q^{53} + 100q^{54} - 220q^{57} - 110q^{58} - 810q^{59} + 488q^{61} - 12q^{62} - 167q^{64} + 16q^{66} - 244q^{67} - 378q^{68} + 96q^{69} - 768q^{71} + 345q^{72} - 702q^{73} + 246q^{74} - 770q^{76} - 56q^{78} + 440q^{79} + 421q^{81} - 182q^{82} - 1302q^{83} - 128q^{86} + 220q^{87} + 120q^{88} - 730q^{89} + 336q^{92} + 24q^{93} + 324q^{94} - 322q^{96} + 294q^{97} + 184q^{99} + O(q^{100}) \)

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
0
1.00000 −2.00000 −7.00000 0 −2.00000 0 −15.0000 −23.0000 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 1225.4.a.j 1
5.b even 2 1 49.4.a.b 1
7.b odd 2 1 175.4.a.b 1
15.d odd 2 1 441.4.a.i 1
20.d odd 2 1 784.4.a.g 1
21.c even 2 1 1575.4.a.e 1
35.c odd 2 1 7.4.a.a 1
35.f even 4 2 175.4.b.b 2
35.i odd 6 2 49.4.c.c 2
35.j even 6 2 49.4.c.b 2
105.g even 2 1 63.4.a.b 1
105.o odd 6 2 441.4.e.e 2
105.p even 6 2 441.4.e.h 2
140.c even 2 1 112.4.a.f 1
280.c odd 2 1 448.4.a.i 1
280.n even 2 1 448.4.a.e 1
385.h even 2 1 847.4.a.b 1
420.o odd 2 1 1008.4.a.c 1
455.h odd 2 1 1183.4.a.b 1
595.b odd 2 1 2023.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.4.a.a 1 35.c odd 2 1
49.4.a.b 1 5.b even 2 1
49.4.c.b 2 35.j even 6 2
49.4.c.c 2 35.i odd 6 2
63.4.a.b 1 105.g even 2 1
112.4.a.f 1 140.c even 2 1
175.4.a.b 1 7.b odd 2 1
175.4.b.b 2 35.f even 4 2
441.4.a.i 1 15.d odd 2 1
441.4.e.e 2 105.o odd 6 2
441.4.e.h 2 105.p even 6 2
448.4.a.e 1 280.n even 2 1
448.4.a.i 1 280.c odd 2 1
784.4.a.g 1 20.d odd 2 1
847.4.a.b 1 385.h even 2 1
1008.4.a.c 1 420.o odd 2 1
1183.4.a.b 1 455.h odd 2 1
1225.4.a.j 1 1.a even 1 1 trivial
1575.4.a.e 1 21.c even 2 1
2023.4.a.a 1 595.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1225))\):

\( T_{2} - 1 \)
\( T_{3} + 2 \)
\( T_{19} - 110 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( 2 + T \)
$5$ \( T \)
$7$ \( T \)
$11$ \( 8 + T \)
$13$ \( -28 + T \)
$17$ \( -54 + T \)
$19$ \( -110 + T \)
$23$ \( 48 + T \)
$29$ \( 110 + T \)
$31$ \( 12 + T \)
$37$ \( -246 + T \)
$41$ \( 182 + T \)
$43$ \( 128 + T \)
$47$ \( -324 + T \)
$53$ \( -162 + T \)
$59$ \( 810 + T \)
$61$ \( -488 + T \)
$67$ \( 244 + T \)
$71$ \( 768 + T \)
$73$ \( 702 + T \)
$79$ \( -440 + T \)
$83$ \( 1302 + T \)
$89$ \( 730 + T \)
$97$ \( -294 + T \)
show more
show less