Properties

Label 3146.2.a.a
Level 3146
Weight 2
Character orbit 3146.a
Self dual yes
Analytic conductor 25.121
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 3146 = 2 \cdot 11^{2} \cdot 13 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 3146.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(25.1209364759\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} - 3q^{3} + q^{4} - q^{5} + 3q^{6} - q^{7} - q^{8} + 6q^{9} + O(q^{10}) \) \( q - q^{2} - 3q^{3} + q^{4} - q^{5} + 3q^{6} - q^{7} - q^{8} + 6q^{9} + q^{10} - 3q^{12} + q^{13} + q^{14} + 3q^{15} + q^{16} + 3q^{17} - 6q^{18} - 6q^{19} - q^{20} + 3q^{21} - 4q^{23} + 3q^{24} - 4q^{25} - q^{26} - 9q^{27} - q^{28} - 2q^{29} - 3q^{30} + 4q^{31} - q^{32} - 3q^{34} + q^{35} + 6q^{36} + 3q^{37} + 6q^{38} - 3q^{39} + q^{40} - 3q^{42} + 5q^{43} - 6q^{45} + 4q^{46} + 13q^{47} - 3q^{48} - 6q^{49} + 4q^{50} - 9q^{51} + q^{52} + 12q^{53} + 9q^{54} + q^{56} + 18q^{57} + 2q^{58} - 10q^{59} + 3q^{60} + 8q^{61} - 4q^{62} - 6q^{63} + q^{64} - q^{65} - 2q^{67} + 3q^{68} + 12q^{69} - q^{70} - 5q^{71} - 6q^{72} + 10q^{73} - 3q^{74} + 12q^{75} - 6q^{76} + 3q^{78} + 4q^{79} - q^{80} + 9q^{81} + 3q^{84} - 3q^{85} - 5q^{86} + 6q^{87} + 6q^{89} + 6q^{90} - q^{91} - 4q^{92} - 12q^{93} - 13q^{94} + 6q^{95} + 3q^{96} + 14q^{97} + 6q^{98} + O(q^{100}) \)

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 −3.00000 1.00000 −1.00000 3.00000 −1.00000 −1.00000 6.00000 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 3146.2.a.a 1
11.b odd 2 1 26.2.a.b 1
33.d even 2 1 234.2.a.b 1
44.c even 2 1 208.2.a.d 1
55.d odd 2 1 650.2.a.g 1
55.e even 4 2 650.2.b.a 2
77.b even 2 1 1274.2.a.o 1
77.h odd 6 2 1274.2.f.l 2
77.i even 6 2 1274.2.f.a 2
88.b odd 2 1 832.2.a.j 1
88.g even 2 1 832.2.a.a 1
99.g even 6 2 2106.2.e.t 2
99.h odd 6 2 2106.2.e.h 2
132.d odd 2 1 1872.2.a.m 1
143.d odd 2 1 338.2.a.a 1
143.g even 4 2 338.2.b.a 2
143.i odd 6 2 338.2.c.g 2
143.k odd 6 2 338.2.c.c 2
143.o even 12 4 338.2.e.d 4
165.d even 2 1 5850.2.a.bn 1
165.l odd 4 2 5850.2.e.v 2
176.i even 4 2 3328.2.b.k 2
176.l odd 4 2 3328.2.b.g 2
187.b odd 2 1 7514.2.a.i 1
209.d even 2 1 9386.2.a.f 1
220.g even 2 1 5200.2.a.c 1
264.m even 2 1 7488.2.a.w 1
264.p odd 2 1 7488.2.a.v 1
429.e even 2 1 3042.2.a.l 1
429.l odd 4 2 3042.2.b.f 2
572.b even 2 1 2704.2.a.n 1
572.k odd 4 2 2704.2.f.j 2
715.c odd 2 1 8450.2.a.y 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.a.b 1 11.b odd 2 1
208.2.a.d 1 44.c even 2 1
234.2.a.b 1 33.d even 2 1
338.2.a.a 1 143.d odd 2 1
338.2.b.a 2 143.g even 4 2
338.2.c.c 2 143.k odd 6 2
338.2.c.g 2 143.i odd 6 2
338.2.e.d 4 143.o even 12 4
650.2.a.g 1 55.d odd 2 1
650.2.b.a 2 55.e even 4 2
832.2.a.a 1 88.g even 2 1
832.2.a.j 1 88.b odd 2 1
1274.2.a.o 1 77.b even 2 1
1274.2.f.a 2 77.i even 6 2
1274.2.f.l 2 77.h odd 6 2
1872.2.a.m 1 132.d odd 2 1
2106.2.e.h 2 99.h odd 6 2
2106.2.e.t 2 99.g even 6 2
2704.2.a.n 1 572.b even 2 1
2704.2.f.j 2 572.k odd 4 2
3042.2.a.l 1 429.e even 2 1
3042.2.b.f 2 429.l odd 4 2
3146.2.a.a 1 1.a even 1 1 trivial
3328.2.b.g 2 176.l odd 4 2
3328.2.b.k 2 176.i even 4 2
5200.2.a.c 1 220.g even 2 1
5850.2.a.bn 1 165.d even 2 1
5850.2.e.v 2 165.l odd 4 2
7488.2.a.v 1 264.p odd 2 1
7488.2.a.w 1 264.m even 2 1
7514.2.a.i 1 187.b odd 2 1
8450.2.a.y 1 715.c odd 2 1
9386.2.a.f 1 209.d even 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(11\) \(-1\)
\(13\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3146))\):

\( T_{3} + 3 \)
\( T_{5} + 1 \)
\( T_{7} + 1 \)
\( T_{19} + 6 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( 1 + 3 T + 3 T^{2} \)
$5$ \( 1 + T + 5 T^{2} \)
$7$ \( 1 + T + 7 T^{2} \)
$11$ \( \)
$13$ \( 1 - T \)
$17$ \( 1 - 3 T + 17 T^{2} \)
$19$ \( 1 + 6 T + 19 T^{2} \)
$23$ \( 1 + 4 T + 23 T^{2} \)
$29$ \( 1 + 2 T + 29 T^{2} \)
$31$ \( 1 - 4 T + 31 T^{2} \)
$37$ \( 1 - 3 T + 37 T^{2} \)
$41$ \( 1 + 41 T^{2} \)
$43$ \( 1 - 5 T + 43 T^{2} \)
$47$ \( 1 - 13 T + 47 T^{2} \)
$53$ \( 1 - 12 T + 53 T^{2} \)
$59$ \( 1 + 10 T + 59 T^{2} \)
$61$ \( 1 - 8 T + 61 T^{2} \)
$67$ \( 1 + 2 T + 67 T^{2} \)
$71$ \( 1 + 5 T + 71 T^{2} \)
$73$ \( 1 - 10 T + 73 T^{2} \)
$79$ \( 1 - 4 T + 79 T^{2} \)
$83$ \( 1 + 83 T^{2} \)
$89$ \( 1 - 6 T + 89 T^{2} \)
$97$ \( 1 - 14 T + 97 T^{2} \)
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