Properties

Label 325.2.a.j
Level $325$
Weight $2$
Character orbit 325.a
Self dual yes
Analytic conductor $2.595$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 325.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.59513806569\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) 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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - 1) q^{2} + ( - \beta_1 - 1) q^{3} + (\beta_{2} - \beta_1 + 2) q^{4} + (\beta_{2} + 2 \beta_1) q^{6} + (\beta_{2} + \beta_1 - 1) q^{7} + (3 \beta_1 - 4) q^{8} + (\beta_{2} + 3 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - 1) q^{2} + ( - \beta_1 - 1) q^{3} + (\beta_{2} - \beta_1 + 2) q^{4} + (\beta_{2} + 2 \beta_1) q^{6} + (\beta_{2} + \beta_1 - 1) q^{7} + (3 \beta_1 - 4) q^{8} + (\beta_{2} + 3 \beta_1) q^{9} + ( - \beta_{2} - 2) q^{11} + ( - \beta_1 + 1) q^{12} + q^{13} + (\beta_{2} - \beta_1 - 1) q^{14} + (2 \beta_{2} - 4 \beta_1 + 3) q^{16} + (2 \beta_{2} - 2) q^{17} - 5 \beta_1 q^{18} - \beta_{2} q^{19} + ( - 2 \beta_{2} - 2 \beta_1) q^{21} + (2 \beta_{2} - \beta_1 + 5) q^{22} + (\beta_1 - 5) q^{23} + ( - 3 \beta_{2} - 2 \beta_1 - 2) q^{24} + ( - \beta_{2} - 1) q^{26} + ( - 4 \beta_{2} - 4 \beta_1 - 2) q^{27} + ( - \beta_{2} + \beta_1 - 1) q^{28} + ( - 3 \beta_{2} - 3 \beta_1 + 3) q^{29} + (\beta_{2} + 2 \beta_1 - 4) q^{31} + ( - 3 \beta_{2} + 4 \beta_1 - 5) q^{32} + (\beta_{2} + 3 \beta_1 + 1) q^{33} + (2 \beta_{2} + 2 \beta_1 - 4) q^{34} + ( - 2 \beta_{2} + 4 \beta_1 - 5) q^{36} + (\beta_{2} + 3 \beta_1 - 1) q^{37} + ( - \beta_1 + 3) q^{38} + ( - \beta_1 - 1) q^{39} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{41} + (2 \beta_1 + 4) q^{42} + (2 \beta_{2} - 3 \beta_1 - 1) q^{43} + ( - 3 \beta_{2} + 4 \beta_1 - 8) q^{44} + (5 \beta_{2} - 2 \beta_1 + 6) q^{46} + ( - \beta_{2} - \beta_1 - 3) q^{47} + (2 \beta_{2} + 3 \beta_1 + 7) q^{48} + ( - 2 \beta_{2} - 3) q^{49} + ( - 2 \beta_{2} + 4) q^{51} + (\beta_{2} - \beta_1 + 2) q^{52} + ( - 4 \beta_{2} - 2 \beta_1 - 2) q^{53} + (2 \beta_{2} + 4 \beta_1 + 10) q^{54} + ( - \beta_{2} - \beta_1 + 7) q^{56} + (\beta_{2} + \beta_1 - 1) q^{57} + ( - 3 \beta_{2} + 3 \beta_1 + 3) q^{58} + (3 \beta_{2} - 2 \beta_1 - 2) q^{59} + (\beta_{2} + 3 \beta_1 + 1) q^{61} + (4 \beta_{2} - 3 \beta_1 + 3) q^{62} + (\beta_{2} + 3 \beta_1 + 5) q^{63} + (\beta_{2} - 3 \beta_1 + 12) q^{64} + ( - \beta_{2} - 5 \beta_1 - 1) q^{66} + ( - 5 \beta_{2} - \beta_1 - 3) q^{67} + ( - 2 \beta_1 + 4) q^{68} + ( - \beta_{2} + 3 \beta_1 + 3) q^{69} + (\beta_{2} - 6 \beta_1 - 2) q^{71} + (5 \beta_{2} + 15) q^{72} + ( - \beta_{2} - 3 \beta_1 + 9) q^{73} + (\beta_{2} - 5 \beta_1 + 1) q^{74} + ( - \beta_{2} + 2 \beta_1 - 4) q^{76} - 2 \beta_1 q^{77} + (\beta_{2} + 2 \beta_1) q^{78} + (4 \beta_{2} + 2 \beta_1 - 6) q^{79} + (5 \beta_{2} + 5 \beta_1 + 6) q^{81} + (2 \beta_{2} - 6 \beta_1 + 10) q^{82} + (\beta_{2} - \beta_1 - 7) q^{83} - 2 q^{84} + (\beta_{2} + 8 \beta_1 - 8) q^{86} + (6 \beta_{2} + 6 \beta_1) q^{87} + (4 \beta_{2} - 9 \beta_1 + 11) q^{88} + (4 \beta_{2} + 4 \beta_1 + 2) q^{89} + (\beta_{2} + \beta_1 - 1) q^{91} + ( - 6 \beta_{2} + 7 \beta_1 - 13) q^{92} + ( - 3 \beta_{2} - \beta_1 + 1) q^{93} + (3 \beta_{2} + \beta_1 + 5) q^{94} + ( - \beta_{2} - 6) q^{96} + ( - 6 \beta_{2} - 4 \beta_1 + 6) q^{97} + (3 \beta_{2} - 2 \beta_1 + 9) q^{98} + ( - \beta_{2} - 8 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} - 4 q^{3} + 5 q^{4} + 2 q^{6} - 2 q^{7} - 9 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} - 4 q^{3} + 5 q^{4} + 2 q^{6} - 2 q^{7} - 9 q^{8} + 3 q^{9} - 6 q^{11} + 2 q^{12} + 3 q^{13} - 4 q^{14} + 5 q^{16} - 6 q^{17} - 5 q^{18} - 2 q^{21} + 14 q^{22} - 14 q^{23} - 8 q^{24} - 3 q^{26} - 10 q^{27} - 2 q^{28} + 6 q^{29} - 10 q^{31} - 11 q^{32} + 6 q^{33} - 10 q^{34} - 11 q^{36} + 8 q^{38} - 4 q^{39} - 4 q^{41} + 14 q^{42} - 6 q^{43} - 20 q^{44} + 16 q^{46} - 10 q^{47} + 24 q^{48} - 9 q^{49} + 12 q^{51} + 5 q^{52} - 8 q^{53} + 34 q^{54} + 20 q^{56} - 2 q^{57} + 12 q^{58} - 8 q^{59} + 6 q^{61} + 6 q^{62} + 18 q^{63} + 33 q^{64} - 8 q^{66} - 10 q^{67} + 10 q^{68} + 12 q^{69} - 12 q^{71} + 45 q^{72} + 24 q^{73} - 2 q^{74} - 10 q^{76} - 2 q^{77} + 2 q^{78} - 16 q^{79} + 23 q^{81} + 24 q^{82} - 22 q^{83} - 6 q^{84} - 16 q^{86} + 6 q^{87} + 24 q^{88} + 10 q^{89} - 2 q^{91} - 32 q^{92} + 2 q^{93} + 16 q^{94} - 18 q^{96} + 14 q^{97} + 25 q^{98} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 3x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 2 \) 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
−1.48119
2.17009
0.311108
−2.67513 0.481194 5.15633 0 −1.28726 −0.806063 −8.44358 −2.76845 0
1.2 −1.53919 −3.17009 0.369102 0 4.87936 1.70928 2.51026 7.04945 0
1.3 1.21432 −1.31111 −0.525428 0 −1.59210 −2.90321 −3.06668 −1.28100 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 325.2.a.j 3
3.b odd 2 1 2925.2.a.bj 3
4.b odd 2 1 5200.2.a.cj 3
5.b even 2 1 325.2.a.k 3
5.c odd 4 2 65.2.b.a 6
13.b even 2 1 4225.2.a.bh 3
15.d odd 2 1 2925.2.a.bf 3
15.e even 4 2 585.2.c.b 6
20.d odd 2 1 5200.2.a.cb 3
20.e even 4 2 1040.2.d.c 6
65.d even 2 1 4225.2.a.ba 3
65.f even 4 2 845.2.d.b 6
65.h odd 4 2 845.2.b.c 6
65.k even 4 2 845.2.d.a 6
65.o even 12 4 845.2.l.e 12
65.q odd 12 4 845.2.n.f 12
65.r odd 12 4 845.2.n.g 12
65.t even 12 4 845.2.l.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.b.a 6 5.c odd 4 2
325.2.a.j 3 1.a even 1 1 trivial
325.2.a.k 3 5.b even 2 1
585.2.c.b 6 15.e even 4 2
845.2.b.c 6 65.h odd 4 2
845.2.d.a 6 65.k even 4 2
845.2.d.b 6 65.f even 4 2
845.2.l.d 12 65.t even 12 4
845.2.l.e 12 65.o even 12 4
845.2.n.f 12 65.q odd 12 4
845.2.n.g 12 65.r odd 12 4
1040.2.d.c 6 20.e even 4 2
2925.2.a.bf 3 15.d odd 2 1
2925.2.a.bj 3 3.b odd 2 1
4225.2.a.ba 3 65.d even 2 1
4225.2.a.bh 3 13.b even 2 1
5200.2.a.cb 3 20.d odd 2 1
5200.2.a.cj 3 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(325))\):

\( T_{2}^{3} + 3T_{2}^{2} - T_{2} - 5 \) Copy content Toggle raw display
\( T_{3}^{3} + 4T_{3}^{2} + 2T_{3} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + 3T^{2} - T - 5 \) Copy content Toggle raw display
$3$ \( T^{3} + 4 T^{2} + 2 T - 2 \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + 2 T^{2} - 4 T - 4 \) Copy content Toggle raw display
$11$ \( T^{3} + 6 T^{2} + 8 T - 2 \) Copy content Toggle raw display
$13$ \( (T - 1)^{3} \) Copy content Toggle raw display
$17$ \( T^{3} + 6 T^{2} - 4 T - 8 \) Copy content Toggle raw display
$19$ \( T^{3} - 4T - 2 \) Copy content Toggle raw display
$23$ \( T^{3} + 14 T^{2} + 62 T + 86 \) Copy content Toggle raw display
$29$ \( T^{3} - 6 T^{2} - 36 T + 108 \) Copy content Toggle raw display
$31$ \( T^{3} + 10 T^{2} + 20 T - 26 \) Copy content Toggle raw display
$37$ \( T^{3} - 28T - 52 \) Copy content Toggle raw display
$41$ \( T^{3} + 4 T^{2} - 32 T + 32 \) Copy content Toggle raw display
$43$ \( T^{3} + 6 T^{2} - 46 T - 278 \) Copy content Toggle raw display
$47$ \( T^{3} + 10 T^{2} + 28 T + 20 \) Copy content Toggle raw display
$53$ \( T^{3} + 8 T^{2} - 40 T - 304 \) Copy content Toggle raw display
$59$ \( T^{3} + 8 T^{2} - 40 T - 262 \) Copy content Toggle raw display
$61$ \( T^{3} - 6 T^{2} - 16 T - 4 \) Copy content Toggle raw display
$67$ \( T^{3} + 10 T^{2} - 60 T - 604 \) Copy content Toggle raw display
$71$ \( T^{3} + 12 T^{2} - 88 T - 754 \) Copy content Toggle raw display
$73$ \( T^{3} - 24 T^{2} + 164 T - 236 \) Copy content Toggle raw display
$79$ \( T^{3} + 16 T^{2} + 24 T - 16 \) Copy content Toggle raw display
$83$ \( T^{3} + 22 T^{2} + 152 T + 316 \) Copy content Toggle raw display
$89$ \( T^{3} - 10 T^{2} - 52 T + 200 \) Copy content Toggle raw display
$97$ \( T^{3} - 14 T^{2} - 84 T + 200 \) Copy content Toggle raw display
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