Properties

Label 3675.2.a.c
Level $3675$
Weight $2$
Character orbit 3675.a
Self dual yes
Analytic conductor $29.345$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3675.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(29.3450227428\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2q^{2} + q^{3} + 2q^{4} - 2q^{6} + q^{9} + O(q^{10}) \) \( q - 2q^{2} + q^{3} + 2q^{4} - 2q^{6} + q^{9} - 2q^{11} + 2q^{12} + q^{13} - 4q^{16} - 2q^{18} - q^{19} + 4q^{22} - 2q^{26} + q^{27} + 4q^{29} - 9q^{31} + 8q^{32} - 2q^{33} + 2q^{36} - 3q^{37} + 2q^{38} + q^{39} + 10q^{41} - 5q^{43} - 4q^{44} - 6q^{47} - 4q^{48} + 2q^{52} - 12q^{53} - 2q^{54} - q^{57} - 8q^{58} + 12q^{59} - 10q^{61} + 18q^{62} - 8q^{64} + 4q^{66} + 5q^{67} - 6q^{71} - 3q^{73} + 6q^{74} - 2q^{76} - 2q^{78} - q^{79} + q^{81} - 20q^{82} + 6q^{83} + 10q^{86} + 4q^{87} - 16q^{89} - 9q^{93} + 12q^{94} + 8q^{96} - 6q^{97} - 2q^{99} + O(q^{100}) \)

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 1.00000 2.00000 0 −2.00000 0 0 1.00000 0
\(n\): 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 3675.2.a.c 1
5.b even 2 1 147.2.a.b 1
7.b odd 2 1 3675.2.a.a 1
7.d odd 6 2 525.2.i.e 2
15.d odd 2 1 441.2.a.a 1
20.d odd 2 1 2352.2.a.w 1
35.c odd 2 1 147.2.a.c 1
35.i odd 6 2 21.2.e.a 2
35.j even 6 2 147.2.e.a 2
35.k even 12 4 525.2.r.e 4
40.e odd 2 1 9408.2.a.k 1
40.f even 2 1 9408.2.a.bz 1
60.h even 2 1 7056.2.a.m 1
105.g even 2 1 441.2.a.b 1
105.o odd 6 2 441.2.e.e 2
105.p even 6 2 63.2.e.b 2
140.c even 2 1 2352.2.a.d 1
140.p odd 6 2 2352.2.q.c 2
140.s even 6 2 336.2.q.f 2
280.c odd 2 1 9408.2.a.bg 1
280.n even 2 1 9408.2.a.cv 1
280.ba even 6 2 1344.2.q.c 2
280.bk odd 6 2 1344.2.q.m 2
315.q odd 6 2 567.2.h.f 2
315.u even 6 2 567.2.g.f 2
315.bn odd 6 2 567.2.g.a 2
315.bq even 6 2 567.2.h.a 2
420.o odd 2 1 7056.2.a.bp 1
420.be odd 6 2 1008.2.s.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.2.e.a 2 35.i odd 6 2
63.2.e.b 2 105.p even 6 2
147.2.a.b 1 5.b even 2 1
147.2.a.c 1 35.c odd 2 1
147.2.e.a 2 35.j even 6 2
336.2.q.f 2 140.s even 6 2
441.2.a.a 1 15.d odd 2 1
441.2.a.b 1 105.g even 2 1
441.2.e.e 2 105.o odd 6 2
525.2.i.e 2 7.d odd 6 2
525.2.r.e 4 35.k even 12 4
567.2.g.a 2 315.bn odd 6 2
567.2.g.f 2 315.u even 6 2
567.2.h.a 2 315.bq even 6 2
567.2.h.f 2 315.q odd 6 2
1008.2.s.d 2 420.be odd 6 2
1344.2.q.c 2 280.ba even 6 2
1344.2.q.m 2 280.bk odd 6 2
2352.2.a.d 1 140.c even 2 1
2352.2.a.w 1 20.d odd 2 1
2352.2.q.c 2 140.p odd 6 2
3675.2.a.a 1 7.b odd 2 1
3675.2.a.c 1 1.a even 1 1 trivial
7056.2.a.m 1 60.h even 2 1
7056.2.a.bp 1 420.o odd 2 1
9408.2.a.k 1 40.e odd 2 1
9408.2.a.bg 1 280.c odd 2 1
9408.2.a.bz 1 40.f even 2 1
9408.2.a.cv 1 280.n even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3675))\):

\( T_{2} + 2 \)
\( T_{11} + 2 \)
\( T_{13} - 1 \)