Properties

Label 3630.2.a.w
Level 3630
Weight 2
Character orbit 3630.a
Self dual yes
Analytic conductor 28.986
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 3630 = 2 \cdot 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3630.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.9856959337\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 30)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + 4q^{7} + q^{8} + q^{9} + O(q^{10}) \) \( q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + 4q^{7} + q^{8} + q^{9} - q^{10} + q^{12} - 2q^{13} + 4q^{14} - q^{15} + q^{16} - 6q^{17} + q^{18} + 4q^{19} - q^{20} + 4q^{21} + q^{24} + q^{25} - 2q^{26} + q^{27} + 4q^{28} + 6q^{29} - q^{30} + 8q^{31} + q^{32} - 6q^{34} - 4q^{35} + q^{36} + 2q^{37} + 4q^{38} - 2q^{39} - q^{40} + 6q^{41} + 4q^{42} + 4q^{43} - q^{45} + q^{48} + 9q^{49} + q^{50} - 6q^{51} - 2q^{52} - 6q^{53} + q^{54} + 4q^{56} + 4q^{57} + 6q^{58} - q^{60} + 10q^{61} + 8q^{62} + 4q^{63} + q^{64} + 2q^{65} - 4q^{67} - 6q^{68} - 4q^{70} + q^{72} - 2q^{73} + 2q^{74} + q^{75} + 4q^{76} - 2q^{78} - 8q^{79} - q^{80} + q^{81} + 6q^{82} - 12q^{83} + 4q^{84} + 6q^{85} + 4q^{86} + 6q^{87} + 18q^{89} - q^{90} - 8q^{91} + 8q^{93} - 4q^{95} + q^{96} + 2q^{97} + 9q^{98} + O(q^{100}) \)

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 −1.00000 1.00000 4.00000 1.00000 1.00000 −1.00000
\(n\): 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 3630.2.a.w 1
11.b odd 2 1 30.2.a.a 1
33.d even 2 1 90.2.a.c 1
44.c even 2 1 240.2.a.b 1
55.d odd 2 1 150.2.a.b 1
55.e even 4 2 150.2.c.a 2
77.b even 2 1 1470.2.a.d 1
77.h odd 6 2 1470.2.i.o 2
77.i even 6 2 1470.2.i.q 2
88.b odd 2 1 960.2.a.e 1
88.g even 2 1 960.2.a.p 1
99.g even 6 2 810.2.e.b 2
99.h odd 6 2 810.2.e.l 2
132.d odd 2 1 720.2.a.j 1
143.d odd 2 1 5070.2.a.w 1
143.g even 4 2 5070.2.b.k 2
165.d even 2 1 450.2.a.d 1
165.l odd 4 2 450.2.c.b 2
176.i even 4 2 3840.2.k.f 2
176.l odd 4 2 3840.2.k.y 2
187.b odd 2 1 8670.2.a.g 1
220.g even 2 1 1200.2.a.k 1
220.i odd 4 2 1200.2.f.e 2
231.h odd 2 1 4410.2.a.z 1
264.m even 2 1 2880.2.a.a 1
264.p odd 2 1 2880.2.a.q 1
385.h even 2 1 7350.2.a.ct 1
440.c even 2 1 4800.2.a.d 1
440.o odd 2 1 4800.2.a.cq 1
440.t even 4 2 4800.2.f.p 2
440.w odd 4 2 4800.2.f.w 2
660.g odd 2 1 3600.2.a.f 1
660.q even 4 2 3600.2.f.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.2.a.a 1 11.b odd 2 1
90.2.a.c 1 33.d even 2 1
150.2.a.b 1 55.d odd 2 1
150.2.c.a 2 55.e even 4 2
240.2.a.b 1 44.c even 2 1
450.2.a.d 1 165.d even 2 1
450.2.c.b 2 165.l odd 4 2
720.2.a.j 1 132.d odd 2 1
810.2.e.b 2 99.g even 6 2
810.2.e.l 2 99.h odd 6 2
960.2.a.e 1 88.b odd 2 1
960.2.a.p 1 88.g even 2 1
1200.2.a.k 1 220.g even 2 1
1200.2.f.e 2 220.i odd 4 2
1470.2.a.d 1 77.b even 2 1
1470.2.i.o 2 77.h odd 6 2
1470.2.i.q 2 77.i even 6 2
2880.2.a.a 1 264.m even 2 1
2880.2.a.q 1 264.p odd 2 1
3600.2.a.f 1 660.g odd 2 1
3600.2.f.i 2 660.q even 4 2
3630.2.a.w 1 1.a even 1 1 trivial
3840.2.k.f 2 176.i even 4 2
3840.2.k.y 2 176.l odd 4 2
4410.2.a.z 1 231.h odd 2 1
4800.2.a.d 1 440.c even 2 1
4800.2.a.cq 1 440.o odd 2 1
4800.2.f.p 2 440.t even 4 2
4800.2.f.w 2 440.w odd 4 2
5070.2.a.w 1 143.d odd 2 1
5070.2.b.k 2 143.g even 4 2
7350.2.a.ct 1 385.h even 2 1
8670.2.a.g 1 187.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(1\)
\(11\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3630))\):

\( T_{7} - 4 \)
\( T_{13} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T \)
$3$ \( 1 - T \)
$5$ \( 1 + T \)
$7$ \( 1 - 4 T + 7 T^{2} \)
$11$ 1
$13$ \( 1 + 2 T + 13 T^{2} \)
$17$ \( 1 + 6 T + 17 T^{2} \)
$19$ \( 1 - 4 T + 19 T^{2} \)
$23$ \( 1 + 23 T^{2} \)
$29$ \( 1 - 6 T + 29 T^{2} \)
$31$ \( 1 - 8 T + 31 T^{2} \)
$37$ \( 1 - 2 T + 37 T^{2} \)
$41$ \( 1 - 6 T + 41 T^{2} \)
$43$ \( 1 - 4 T + 43 T^{2} \)
$47$ \( 1 + 47 T^{2} \)
$53$ \( 1 + 6 T + 53 T^{2} \)
$59$ \( 1 + 59 T^{2} \)
$61$ \( 1 - 10 T + 61 T^{2} \)
$67$ \( 1 + 4 T + 67 T^{2} \)
$71$ \( 1 + 71 T^{2} \)
$73$ \( 1 + 2 T + 73 T^{2} \)
$79$ \( 1 + 8 T + 79 T^{2} \)
$83$ \( 1 + 12 T + 83 T^{2} \)
$89$ \( 1 - 18 T + 89 T^{2} \)
$97$ \( 1 - 2 T + 97 T^{2} \)
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