Properties

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

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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

\(f(q)\) \(=\) \( q - 2q^{2} + 7q^{3} - 4q^{4} - 14q^{6} + 24q^{8} + 22q^{9} + O(q^{10}) \) \( q - 2q^{2} + 7q^{3} - 4q^{4} - 14q^{6} + 24q^{8} + 22q^{9} - 5q^{11} - 28q^{12} - 14q^{13} - 16q^{16} - 21q^{17} - 44q^{18} - 49q^{19} + 10q^{22} + 159q^{23} + 168q^{24} + 28q^{26} - 35q^{27} + 58q^{29} - 147q^{31} - 160q^{32} - 35q^{33} + 42q^{34} - 88q^{36} - 219q^{37} + 98q^{38} - 98q^{39} - 350q^{41} + 124q^{43} + 20q^{44} - 318q^{46} + 525q^{47} - 112q^{48} - 147q^{51} + 56q^{52} - 303q^{53} + 70q^{54} - 343q^{57} - 116q^{58} + 105q^{59} + 413q^{61} + 294q^{62} + 448q^{64} + 70q^{66} - 415q^{67} + 84q^{68} + 1113q^{69} - 432q^{71} + 528q^{72} - 1113q^{73} + 438q^{74} + 196q^{76} + 196q^{78} - 103q^{79} - 839q^{81} + 700q^{82} + 1092q^{83} - 248q^{86} + 406q^{87} - 120q^{88} + 329q^{89} - 636q^{92} - 1029q^{93} - 1050q^{94} - 1120q^{96} - 882q^{97} - 110q^{99} + O(q^{100}) \)

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
−2.00000 7.00000 −4.00000 0 −14.0000 0 24.0000 22.0000 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.d 1
5.b even 2 1 49.4.a.c 1
7.b odd 2 1 1225.4.a.c 1
7.d odd 6 2 175.4.e.a 2
15.d odd 2 1 441.4.a.e 1
20.d odd 2 1 784.4.a.r 1
35.c odd 2 1 49.4.a.d 1
35.i odd 6 2 7.4.c.a 2
35.j even 6 2 49.4.c.a 2
35.k even 12 4 175.4.k.a 4
105.g even 2 1 441.4.a.d 1
105.o odd 6 2 441.4.e.k 2
105.p even 6 2 63.4.e.b 2
140.c even 2 1 784.4.a.b 1
140.s even 6 2 112.4.i.c 2
280.ba even 6 2 448.4.i.a 2
280.bk odd 6 2 448.4.i.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.4.c.a 2 35.i odd 6 2
49.4.a.c 1 5.b even 2 1
49.4.a.d 1 35.c odd 2 1
49.4.c.a 2 35.j even 6 2
63.4.e.b 2 105.p even 6 2
112.4.i.c 2 140.s even 6 2
175.4.e.a 2 7.d odd 6 2
175.4.k.a 4 35.k even 12 4
441.4.a.d 1 105.g even 2 1
441.4.a.e 1 15.d odd 2 1
441.4.e.k 2 105.o odd 6 2
448.4.i.a 2 280.ba even 6 2
448.4.i.f 2 280.bk odd 6 2
784.4.a.b 1 140.c even 2 1
784.4.a.r 1 20.d odd 2 1
1225.4.a.c 1 7.b odd 2 1
1225.4.a.d 1 1.a even 1 1 trivial

Hecke kernels

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

\( T_{2} + 2 \)
\( T_{3} - 7 \)
\( T_{19} + 49 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + T \)
$3$ \( -7 + T \)
$5$ \( T \)
$7$ \( T \)
$11$ \( 5 + T \)
$13$ \( 14 + T \)
$17$ \( 21 + T \)
$19$ \( 49 + T \)
$23$ \( -159 + T \)
$29$ \( -58 + T \)
$31$ \( 147 + T \)
$37$ \( 219 + T \)
$41$ \( 350 + T \)
$43$ \( -124 + T \)
$47$ \( -525 + T \)
$53$ \( 303 + T \)
$59$ \( -105 + T \)
$61$ \( -413 + T \)
$67$ \( 415 + T \)
$71$ \( 432 + T \)
$73$ \( 1113 + T \)
$79$ \( 103 + T \)
$83$ \( -1092 + T \)
$89$ \( -329 + T \)
$97$ \( 882 + T \)
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