Properties

Label 245.4.a.a
Level $245$
Weight $4$
Character orbit 245.a
Self dual yes
Analytic conductor $14.455$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 245.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.4554679514\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 5)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 4q^{2} - 2q^{3} + 8q^{4} + 5q^{5} + 8q^{6} - 23q^{9} + O(q^{10}) \) \( q - 4q^{2} - 2q^{3} + 8q^{4} + 5q^{5} + 8q^{6} - 23q^{9} - 20q^{10} + 32q^{11} - 16q^{12} + 38q^{13} - 10q^{15} - 64q^{16} - 26q^{17} + 92q^{18} - 100q^{19} + 40q^{20} - 128q^{22} - 78q^{23} + 25q^{25} - 152q^{26} + 100q^{27} - 50q^{29} + 40q^{30} + 108q^{31} + 256q^{32} - 64q^{33} + 104q^{34} - 184q^{36} + 266q^{37} + 400q^{38} - 76q^{39} - 22q^{41} + 442q^{43} + 256q^{44} - 115q^{45} + 312q^{46} + 514q^{47} + 128q^{48} - 100q^{50} + 52q^{51} + 304q^{52} + 2q^{53} - 400q^{54} + 160q^{55} + 200q^{57} + 200q^{58} - 500q^{59} - 80q^{60} + 518q^{61} - 432q^{62} - 512q^{64} + 190q^{65} + 256q^{66} + 126q^{67} - 208q^{68} + 156q^{69} + 412q^{71} + 878q^{73} - 1064q^{74} - 50q^{75} - 800q^{76} + 304q^{78} + 600q^{79} - 320q^{80} + 421q^{81} + 88q^{82} - 282q^{83} - 130q^{85} - 1768q^{86} + 100q^{87} + 150q^{89} + 460q^{90} - 624q^{92} - 216q^{93} - 2056q^{94} - 500q^{95} - 512q^{96} - 386q^{97} - 736q^{99} + O(q^{100}) \)

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
−4.00000 −2.00000 8.00000 5.00000 8.00000 0 0 −23.0000 −20.0000
\(n\): 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 245.4.a.a 1
3.b odd 2 1 2205.4.a.q 1
5.b even 2 1 1225.4.a.k 1
7.b odd 2 1 5.4.a.a 1
7.c even 3 2 245.4.e.g 2
7.d odd 6 2 245.4.e.f 2
21.c even 2 1 45.4.a.d 1
28.d even 2 1 80.4.a.d 1
35.c odd 2 1 25.4.a.c 1
35.f even 4 2 25.4.b.a 2
56.e even 2 1 320.4.a.h 1
56.h odd 2 1 320.4.a.g 1
63.l odd 6 2 405.4.e.l 2
63.o even 6 2 405.4.e.c 2
77.b even 2 1 605.4.a.d 1
84.h odd 2 1 720.4.a.u 1
91.b odd 2 1 845.4.a.b 1
105.g even 2 1 225.4.a.b 1
105.k odd 4 2 225.4.b.c 2
112.j even 4 2 1280.4.d.l 2
112.l odd 4 2 1280.4.d.e 2
119.d odd 2 1 1445.4.a.a 1
133.c even 2 1 1805.4.a.h 1
140.c even 2 1 400.4.a.m 1
140.j odd 4 2 400.4.c.k 2
280.c odd 2 1 1600.4.a.bi 1
280.n even 2 1 1600.4.a.s 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.4.a.a 1 7.b odd 2 1
25.4.a.c 1 35.c odd 2 1
25.4.b.a 2 35.f even 4 2
45.4.a.d 1 21.c even 2 1
80.4.a.d 1 28.d even 2 1
225.4.a.b 1 105.g even 2 1
225.4.b.c 2 105.k odd 4 2
245.4.a.a 1 1.a even 1 1 trivial
245.4.e.f 2 7.d odd 6 2
245.4.e.g 2 7.c even 3 2
320.4.a.g 1 56.h odd 2 1
320.4.a.h 1 56.e even 2 1
400.4.a.m 1 140.c even 2 1
400.4.c.k 2 140.j odd 4 2
405.4.e.c 2 63.o even 6 2
405.4.e.l 2 63.l odd 6 2
605.4.a.d 1 77.b even 2 1
720.4.a.u 1 84.h odd 2 1
845.4.a.b 1 91.b odd 2 1
1225.4.a.k 1 5.b even 2 1
1280.4.d.e 2 112.l odd 4 2
1280.4.d.l 2 112.j even 4 2
1445.4.a.a 1 119.d odd 2 1
1600.4.a.s 1 280.n even 2 1
1600.4.a.bi 1 280.c odd 2 1
1805.4.a.h 1 133.c even 2 1
2205.4.a.q 1 3.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(245))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 4 T + 8 T^{2} \)
$3$ \( 1 + 2 T + 27 T^{2} \)
$5$ \( 1 - 5 T \)
$7$ 1
$11$ \( 1 - 32 T + 1331 T^{2} \)
$13$ \( 1 - 38 T + 2197 T^{2} \)
$17$ \( 1 + 26 T + 4913 T^{2} \)
$19$ \( 1 + 100 T + 6859 T^{2} \)
$23$ \( 1 + 78 T + 12167 T^{2} \)
$29$ \( 1 + 50 T + 24389 T^{2} \)
$31$ \( 1 - 108 T + 29791 T^{2} \)
$37$ \( 1 - 266 T + 50653 T^{2} \)
$41$ \( 1 + 22 T + 68921 T^{2} \)
$43$ \( 1 - 442 T + 79507 T^{2} \)
$47$ \( 1 - 514 T + 103823 T^{2} \)
$53$ \( 1 - 2 T + 148877 T^{2} \)
$59$ \( 1 + 500 T + 205379 T^{2} \)
$61$ \( 1 - 518 T + 226981 T^{2} \)
$67$ \( 1 - 126 T + 300763 T^{2} \)
$71$ \( 1 - 412 T + 357911 T^{2} \)
$73$ \( 1 - 878 T + 389017 T^{2} \)
$79$ \( 1 - 600 T + 493039 T^{2} \)
$83$ \( 1 + 282 T + 571787 T^{2} \)
$89$ \( 1 - 150 T + 704969 T^{2} \)
$97$ \( 1 + 386 T + 912673 T^{2} \)
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