Properties

Label 3185.2.a.j
Level $3185$
Weight $2$
Character orbit 3185.a
Self dual yes
Analytic conductor $25.432$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3185 = 5 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3185.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(25.4323530438\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 65)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta - 1) q^{2} - \beta q^{3} + ( - 2 \beta + 1) q^{4} - q^{5} + (\beta - 2) q^{6} + (\beta - 3) q^{8} - q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\beta - 1) q^{2} - \beta q^{3} + ( - 2 \beta + 1) q^{4} - q^{5} + (\beta - 2) q^{6} + (\beta - 3) q^{8} - q^{9} + ( - \beta + 1) q^{10} + ( - \beta + 2) q^{11} + ( - \beta + 4) q^{12} + q^{13} + \beta q^{15} + 3 q^{16} + (2 \beta + 2) q^{17} + ( - \beta + 1) q^{18} + ( - \beta - 2) q^{19} + (2 \beta - 1) q^{20} + (3 \beta - 4) q^{22} - \beta q^{23} + (3 \beta - 2) q^{24} + q^{25} + (\beta - 1) q^{26} + 4 \beta q^{27} + 4 \beta q^{29} + ( - \beta + 2) q^{30} + ( - 3 \beta - 6) q^{31} + (\beta + 3) q^{32} + ( - 2 \beta + 2) q^{33} + 2 q^{34} + (2 \beta - 1) q^{36} + 6 \beta q^{37} - \beta q^{38} - \beta q^{39} + ( - \beta + 3) q^{40} + (2 \beta + 6) q^{41} + (5 \beta - 4) q^{43} + ( - 5 \beta + 6) q^{44} + q^{45} + (\beta - 2) q^{46} + ( - 2 \beta + 2) q^{47} - 3 \beta q^{48} + (\beta - 1) q^{50} + ( - 2 \beta - 4) q^{51} + ( - 2 \beta + 1) q^{52} + ( - 6 \beta - 6) q^{53} + ( - 4 \beta + 8) q^{54} + (\beta - 2) q^{55} + (2 \beta + 2) q^{57} + ( - 4 \beta + 8) q^{58} + ( - 3 \beta - 6) q^{59} + (\beta - 4) q^{60} + 8 q^{61} - 3 \beta q^{62} + (2 \beta - 7) q^{64} - q^{65} + (4 \beta - 6) q^{66} - 2 q^{67} + ( - 2 \beta - 6) q^{68} + 2 q^{69} + ( - 7 \beta + 2) q^{71} + ( - \beta + 3) q^{72} + 6 \beta q^{73} + ( - 6 \beta + 12) q^{74} - \beta q^{75} + (3 \beta + 2) q^{76} + (\beta - 2) q^{78} + 6 \beta q^{79} - 3 q^{80} - 5 q^{81} + (4 \beta - 2) q^{82} + (2 \beta + 6) q^{83} + ( - 2 \beta - 2) q^{85} + ( - 9 \beta + 14) q^{86} - 8 q^{87} + (5 \beta - 8) q^{88} - 6 q^{89} + (\beta - 1) q^{90} + ( - \beta + 4) q^{92} + (6 \beta + 6) q^{93} + (4 \beta - 6) q^{94} + (\beta + 2) q^{95} + ( - 3 \beta - 2) q^{96} + ( - 4 \beta + 2) q^{97} + (\beta - 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 4 q^{6} - 6 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 4 q^{6} - 6 q^{8} - 2 q^{9} + 2 q^{10} + 4 q^{11} + 8 q^{12} + 2 q^{13} + 6 q^{16} + 4 q^{17} + 2 q^{18} - 4 q^{19} - 2 q^{20} - 8 q^{22} - 4 q^{24} + 2 q^{25} - 2 q^{26} + 4 q^{30} - 12 q^{31} + 6 q^{32} + 4 q^{33} + 4 q^{34} - 2 q^{36} + 6 q^{40} + 12 q^{41} - 8 q^{43} + 12 q^{44} + 2 q^{45} - 4 q^{46} + 4 q^{47} - 2 q^{50} - 8 q^{51} + 2 q^{52} - 12 q^{53} + 16 q^{54} - 4 q^{55} + 4 q^{57} + 16 q^{58} - 12 q^{59} - 8 q^{60} + 16 q^{61} - 14 q^{64} - 2 q^{65} - 12 q^{66} - 4 q^{67} - 12 q^{68} + 4 q^{69} + 4 q^{71} + 6 q^{72} + 24 q^{74} + 4 q^{76} - 4 q^{78} - 6 q^{80} - 10 q^{81} - 4 q^{82} + 12 q^{83} - 4 q^{85} + 28 q^{86} - 16 q^{87} - 16 q^{88} - 12 q^{89} - 2 q^{90} + 8 q^{92} + 12 q^{93} - 12 q^{94} + 4 q^{95} - 4 q^{96} + 4 q^{97} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
−1.41421
1.41421
−2.41421 1.41421 3.82843 −1.00000 −3.41421 0 −4.41421 −1.00000 2.41421
1.2 0.414214 −1.41421 −1.82843 −1.00000 −0.585786 0 −1.58579 −1.00000 −0.414214
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(7\) \(-1\)
\(13\) \(-1\)

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 3185.2.a.j 2
7.b odd 2 1 65.2.a.b 2
21.c even 2 1 585.2.a.m 2
28.d even 2 1 1040.2.a.j 2
35.c odd 2 1 325.2.a.i 2
35.f even 4 2 325.2.b.f 4
56.e even 2 1 4160.2.a.z 2
56.h odd 2 1 4160.2.a.bf 2
77.b even 2 1 7865.2.a.j 2
84.h odd 2 1 9360.2.a.cd 2
91.b odd 2 1 845.2.a.g 2
91.i even 4 2 845.2.c.b 4
91.n odd 6 2 845.2.e.h 4
91.t odd 6 2 845.2.e.c 4
91.bc even 12 4 845.2.m.f 8
105.g even 2 1 2925.2.a.u 2
105.k odd 4 2 2925.2.c.r 4
140.c even 2 1 5200.2.a.bu 2
273.g even 2 1 7605.2.a.x 2
455.h odd 2 1 4225.2.a.r 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.a.b 2 7.b odd 2 1
325.2.a.i 2 35.c odd 2 1
325.2.b.f 4 35.f even 4 2
585.2.a.m 2 21.c even 2 1
845.2.a.g 2 91.b odd 2 1
845.2.c.b 4 91.i even 4 2
845.2.e.c 4 91.t odd 6 2
845.2.e.h 4 91.n odd 6 2
845.2.m.f 8 91.bc even 12 4
1040.2.a.j 2 28.d even 2 1
2925.2.a.u 2 105.g even 2 1
2925.2.c.r 4 105.k odd 4 2
3185.2.a.j 2 1.a even 1 1 trivial
4160.2.a.z 2 56.e even 2 1
4160.2.a.bf 2 56.h odd 2 1
4225.2.a.r 2 455.h odd 2 1
5200.2.a.bu 2 140.c even 2 1
7605.2.a.x 2 273.g even 2 1
7865.2.a.j 2 77.b even 2 1
9360.2.a.cd 2 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3185))\):

\( T_{2}^{2} + 2T_{2} - 1 \) Copy content Toggle raw display
\( T_{3}^{2} - 2 \) Copy content Toggle raw display
\( T_{11}^{2} - 4T_{11} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T - 1 \) Copy content Toggle raw display
$3$ \( T^{2} - 2 \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 4T + 2 \) Copy content Toggle raw display
$13$ \( (T - 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 4T - 4 \) Copy content Toggle raw display
$19$ \( T^{2} + 4T + 2 \) Copy content Toggle raw display
$23$ \( T^{2} - 2 \) Copy content Toggle raw display
$29$ \( T^{2} - 32 \) Copy content Toggle raw display
$31$ \( T^{2} + 12T + 18 \) Copy content Toggle raw display
$37$ \( T^{2} - 72 \) Copy content Toggle raw display
$41$ \( T^{2} - 12T + 28 \) Copy content Toggle raw display
$43$ \( T^{2} + 8T - 34 \) Copy content Toggle raw display
$47$ \( T^{2} - 4T - 4 \) Copy content Toggle raw display
$53$ \( T^{2} + 12T - 36 \) Copy content Toggle raw display
$59$ \( T^{2} + 12T + 18 \) Copy content Toggle raw display
$61$ \( (T - 8)^{2} \) Copy content Toggle raw display
$67$ \( (T + 2)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 4T - 94 \) Copy content Toggle raw display
$73$ \( T^{2} - 72 \) Copy content Toggle raw display
$79$ \( T^{2} - 72 \) Copy content Toggle raw display
$83$ \( T^{2} - 12T + 28 \) Copy content Toggle raw display
$89$ \( (T + 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 4T - 28 \) Copy content Toggle raw display
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