Properties

Label 3675.2.a.n
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

\(f(q)\) \(=\) \( q + q^{2} + q^{3} - q^{4} + q^{6} - 3q^{8} + q^{9} + O(q^{10}) \) \( q + q^{2} + q^{3} - q^{4} + q^{6} - 3q^{8} + q^{9} + 4q^{11} - q^{12} - 2q^{13} - q^{16} - 6q^{17} + q^{18} - 4q^{19} + 4q^{22} - 3q^{24} - 2q^{26} + q^{27} - 2q^{29} + 5q^{32} + 4q^{33} - 6q^{34} - q^{36} - 6q^{37} - 4q^{38} - 2q^{39} - 2q^{41} + 4q^{43} - 4q^{44} - q^{48} - 6q^{51} + 2q^{52} - 6q^{53} + q^{54} - 4q^{57} - 2q^{58} - 12q^{59} + 2q^{61} + 7q^{64} + 4q^{66} - 4q^{67} + 6q^{68} - 3q^{72} - 6q^{73} - 6q^{74} + 4q^{76} - 2q^{78} - 16q^{79} + q^{81} - 2q^{82} - 12q^{83} + 4q^{86} - 2q^{87} - 12q^{88} + 14q^{89} + 5q^{96} + 18q^{97} + 4q^{99} + O(q^{100}) \)

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 1.00000 −1.00000 0 1.00000 0 −3.00000 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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.n 1
5.b even 2 1 147.2.a.a 1
7.b odd 2 1 525.2.a.d 1
15.d odd 2 1 441.2.a.f 1
20.d odd 2 1 2352.2.a.v 1
21.c even 2 1 1575.2.a.c 1
28.d even 2 1 8400.2.a.bn 1
35.c odd 2 1 21.2.a.a 1
35.f even 4 2 525.2.d.a 2
35.i odd 6 2 147.2.e.b 2
35.j even 6 2 147.2.e.c 2
40.e odd 2 1 9408.2.a.m 1
40.f even 2 1 9408.2.a.bv 1
60.h even 2 1 7056.2.a.p 1
105.g even 2 1 63.2.a.a 1
105.k odd 4 2 1575.2.d.a 2
105.o odd 6 2 441.2.e.b 2
105.p even 6 2 441.2.e.a 2
140.c even 2 1 336.2.a.a 1
140.p odd 6 2 2352.2.q.e 2
140.s even 6 2 2352.2.q.x 2
280.c odd 2 1 1344.2.a.g 1
280.n even 2 1 1344.2.a.s 1
315.z even 6 2 567.2.f.b 2
315.bg odd 6 2 567.2.f.g 2
385.h even 2 1 2541.2.a.j 1
420.o odd 2 1 1008.2.a.l 1
455.h odd 2 1 3549.2.a.c 1
560.be even 4 2 5376.2.c.l 2
560.bf odd 4 2 5376.2.c.r 2
595.b odd 2 1 6069.2.a.b 1
665.g even 2 1 7581.2.a.d 1
840.b odd 2 1 4032.2.a.k 1
840.u even 2 1 4032.2.a.h 1
1155.e odd 2 1 7623.2.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.2.a.a 1 35.c odd 2 1
63.2.a.a 1 105.g even 2 1
147.2.a.a 1 5.b even 2 1
147.2.e.b 2 35.i odd 6 2
147.2.e.c 2 35.j even 6 2
336.2.a.a 1 140.c even 2 1
441.2.a.f 1 15.d odd 2 1
441.2.e.a 2 105.p even 6 2
441.2.e.b 2 105.o odd 6 2
525.2.a.d 1 7.b odd 2 1
525.2.d.a 2 35.f even 4 2
567.2.f.b 2 315.z even 6 2
567.2.f.g 2 315.bg odd 6 2
1008.2.a.l 1 420.o odd 2 1
1344.2.a.g 1 280.c odd 2 1
1344.2.a.s 1 280.n even 2 1
1575.2.a.c 1 21.c even 2 1
1575.2.d.a 2 105.k odd 4 2
2352.2.a.v 1 20.d odd 2 1
2352.2.q.e 2 140.p odd 6 2
2352.2.q.x 2 140.s even 6 2
2541.2.a.j 1 385.h even 2 1
3549.2.a.c 1 455.h odd 2 1
3675.2.a.n 1 1.a even 1 1 trivial
4032.2.a.h 1 840.u even 2 1
4032.2.a.k 1 840.b odd 2 1
5376.2.c.l 2 560.be even 4 2
5376.2.c.r 2 560.bf odd 4 2
6069.2.a.b 1 595.b odd 2 1
7056.2.a.p 1 60.h even 2 1
7581.2.a.d 1 665.g even 2 1
7623.2.a.g 1 1155.e odd 2 1
8400.2.a.bn 1 28.d even 2 1
9408.2.a.m 1 40.e odd 2 1
9408.2.a.bv 1 40.f even 2 1

Atkin-Lehner signs

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

Hecke kernels

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

\( T_{2} - 1 \)
\( T_{11} - 4 \)
\( T_{13} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + 2 T^{2} \)
$3$ \( 1 - T \)
$5$ 1
$7$ 1
$11$ \( 1 - 4 T + 11 T^{2} \)
$13$ \( 1 + 2 T + 13 T^{2} \)
$17$ \( 1 + 6 T + 17 T^{2} \)
$19$ \( 1 + 4 T + 19 T^{2} \)
$23$ \( 1 + 23 T^{2} \)
$29$ \( 1 + 2 T + 29 T^{2} \)
$31$ \( 1 + 31 T^{2} \)
$37$ \( 1 + 6 T + 37 T^{2} \)
$41$ \( 1 + 2 T + 41 T^{2} \)
$43$ \( 1 - 4 T + 43 T^{2} \)
$47$ \( 1 + 47 T^{2} \)
$53$ \( 1 + 6 T + 53 T^{2} \)
$59$ \( 1 + 12 T + 59 T^{2} \)
$61$ \( 1 - 2 T + 61 T^{2} \)
$67$ \( 1 + 4 T + 67 T^{2} \)
$71$ \( 1 + 71 T^{2} \)
$73$ \( 1 + 6 T + 73 T^{2} \)
$79$ \( 1 + 16 T + 79 T^{2} \)
$83$ \( 1 + 12 T + 83 T^{2} \)
$89$ \( 1 - 14 T + 89 T^{2} \)
$97$ \( 1 - 18 T + 97 T^{2} \)
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