Properties

Label 2205.4.a.q
Level $2205$
Weight $4$
Character orbit 2205.a
Self dual yes
Analytic conductor $130.099$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2205.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(130.099211563\)
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

\(f(q)\) \(=\) \( q + 4q^{2} + 8q^{4} - 5q^{5} + O(q^{10}) \) \( q + 4q^{2} + 8q^{4} - 5q^{5} - 20q^{10} - 32q^{11} + 38q^{13} - 64q^{16} + 26q^{17} - 100q^{19} - 40q^{20} - 128q^{22} + 78q^{23} + 25q^{25} + 152q^{26} + 50q^{29} + 108q^{31} - 256q^{32} + 104q^{34} + 266q^{37} - 400q^{38} + 22q^{41} + 442q^{43} - 256q^{44} + 312q^{46} - 514q^{47} + 100q^{50} + 304q^{52} - 2q^{53} + 160q^{55} + 200q^{58} + 500q^{59} + 518q^{61} + 432q^{62} - 512q^{64} - 190q^{65} + 126q^{67} + 208q^{68} - 412q^{71} + 878q^{73} + 1064q^{74} - 800q^{76} + 600q^{79} + 320q^{80} + 88q^{82} + 282q^{83} - 130q^{85} + 1768q^{86} - 150q^{89} + 624q^{92} - 2056q^{94} + 500q^{95} - 386q^{97} + O(q^{100}) \)

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
0
4.00000 0 8.00000 −5.00000 0 0 0 0 −20.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(7\) \(-1\)

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 2205.4.a.q 1
3.b odd 2 1 245.4.a.a 1
7.b odd 2 1 45.4.a.d 1
15.d odd 2 1 1225.4.a.k 1
21.c even 2 1 5.4.a.a 1
21.g even 6 2 245.4.e.f 2
21.h odd 6 2 245.4.e.g 2
28.d even 2 1 720.4.a.u 1
35.c odd 2 1 225.4.a.b 1
35.f even 4 2 225.4.b.c 2
63.l odd 6 2 405.4.e.c 2
63.o even 6 2 405.4.e.l 2
84.h odd 2 1 80.4.a.d 1
105.g even 2 1 25.4.a.c 1
105.k odd 4 2 25.4.b.a 2
168.e odd 2 1 320.4.a.h 1
168.i even 2 1 320.4.a.g 1
231.h odd 2 1 605.4.a.d 1
273.g even 2 1 845.4.a.b 1
336.v odd 4 2 1280.4.d.l 2
336.y even 4 2 1280.4.d.e 2
357.c even 2 1 1445.4.a.a 1
399.h odd 2 1 1805.4.a.h 1
420.o odd 2 1 400.4.a.m 1
420.w even 4 2 400.4.c.k 2
840.b odd 2 1 1600.4.a.s 1
840.u even 2 1 1600.4.a.bi 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.4.a.a 1 21.c even 2 1
25.4.a.c 1 105.g even 2 1
25.4.b.a 2 105.k odd 4 2
45.4.a.d 1 7.b odd 2 1
80.4.a.d 1 84.h odd 2 1
225.4.a.b 1 35.c odd 2 1
225.4.b.c 2 35.f even 4 2
245.4.a.a 1 3.b odd 2 1
245.4.e.f 2 21.g even 6 2
245.4.e.g 2 21.h odd 6 2
320.4.a.g 1 168.i even 2 1
320.4.a.h 1 168.e odd 2 1
400.4.a.m 1 420.o odd 2 1
400.4.c.k 2 420.w even 4 2
405.4.e.c 2 63.l odd 6 2
405.4.e.l 2 63.o even 6 2
605.4.a.d 1 231.h odd 2 1
720.4.a.u 1 28.d even 2 1
845.4.a.b 1 273.g even 2 1
1225.4.a.k 1 15.d odd 2 1
1280.4.d.e 2 336.y even 4 2
1280.4.d.l 2 336.v odd 4 2
1445.4.a.a 1 357.c even 2 1
1600.4.a.s 1 840.b odd 2 1
1600.4.a.bi 1 840.u even 2 1
1805.4.a.h 1 399.h odd 2 1
2205.4.a.q 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2205))\):

\( T_{2} - 4 \)
\( T_{11} + 32 \)
\( T_{13} - 38 \)
\( T_{17} - 26 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 4 T + 8 T^{2} \)
$3$ 1
$5$ \( 1 + 5 T \)
$7$ 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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