Properties

Label 325.2.a.i
Level $325$
Weight $2$
Character orbit 325.a
Self dual yes
Analytic conductor $2.595$
Analytic rank $0$
Dimension $2$
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: \(0\)
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.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{2} + \beta q^{3} + (2 \beta + 1) q^{4} + (\beta + 2) q^{6} + ( - 2 \beta - 2) q^{7} + (\beta + 3) q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 1) q^{2} + \beta q^{3} + (2 \beta + 1) q^{4} + (\beta + 2) q^{6} + ( - 2 \beta - 2) q^{7} + (\beta + 3) q^{8} - q^{9} + (\beta + 2) q^{11} + (\beta + 4) q^{12} + q^{13} + ( - 4 \beta - 6) q^{14} + 3 q^{16} + ( - 2 \beta + 2) q^{17} + ( - \beta - 1) q^{18} + ( - \beta + 2) q^{19} + ( - 2 \beta - 4) q^{21} + (3 \beta + 4) q^{22} - \beta q^{23} + (3 \beta + 2) q^{24} + (\beta + 1) q^{26} - 4 \beta q^{27} + ( - 6 \beta - 10) q^{28} - 4 \beta q^{29} + ( - 3 \beta + 6) q^{31} + (\beta - 3) q^{32} + (2 \beta + 2) q^{33} - 2 q^{34} + ( - 2 \beta - 1) q^{36} + 6 \beta q^{37} + \beta q^{38} + \beta q^{39} + (2 \beta - 6) q^{41} + ( - 6 \beta - 8) q^{42} + (5 \beta + 4) q^{43} + (5 \beta + 6) q^{44} + ( - \beta - 2) q^{46} + (2 \beta + 2) q^{47} + 3 \beta q^{48} + (8 \beta + 5) q^{49} + (2 \beta - 4) q^{51} + (2 \beta + 1) q^{52} + ( - 6 \beta + 6) q^{53} + ( - 4 \beta - 8) q^{54} + ( - 8 \beta - 10) q^{56} + (2 \beta - 2) q^{57} + ( - 4 \beta - 8) q^{58} + ( - 3 \beta + 6) q^{59} - 8 q^{61} + 3 \beta q^{62} + (2 \beta + 2) q^{63} + ( - 2 \beta - 7) q^{64} + (4 \beta + 6) q^{66} + 2 q^{67} + (2 \beta - 6) q^{68} - 2 q^{69} + (7 \beta + 2) q^{71} + ( - \beta - 3) q^{72} - 6 \beta q^{73} + (6 \beta + 12) q^{74} + (3 \beta - 2) q^{76} + ( - 6 \beta - 8) q^{77} + (\beta + 2) q^{78} - 6 \beta q^{79} - 5 q^{81} + ( - 4 \beta - 2) q^{82} + ( - 2 \beta + 6) q^{83} + ( - 10 \beta - 12) q^{84} + (9 \beta + 14) q^{86} - 8 q^{87} + (5 \beta + 8) q^{88} + 6 q^{89} + ( - 2 \beta - 2) q^{91} + ( - \beta - 4) q^{92} + (6 \beta - 6) q^{93} + (4 \beta + 6) q^{94} + ( - 3 \beta + 2) q^{96} + (4 \beta + 2) q^{97} + (13 \beta + 21) q^{98} + ( - \beta - 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 4 q^{6} - 4 q^{7} + 6 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} + 4 q^{6} - 4 q^{7} + 6 q^{8} - 2 q^{9} + 4 q^{11} + 8 q^{12} + 2 q^{13} - 12 q^{14} + 6 q^{16} + 4 q^{17} - 2 q^{18} + 4 q^{19} - 8 q^{21} + 8 q^{22} + 4 q^{24} + 2 q^{26} - 20 q^{28} + 12 q^{31} - 6 q^{32} + 4 q^{33} - 4 q^{34} - 2 q^{36} - 12 q^{41} - 16 q^{42} + 8 q^{43} + 12 q^{44} - 4 q^{46} + 4 q^{47} + 10 q^{49} - 8 q^{51} + 2 q^{52} + 12 q^{53} - 16 q^{54} - 20 q^{56} - 4 q^{57} - 16 q^{58} + 12 q^{59} - 16 q^{61} + 4 q^{63} - 14 q^{64} + 12 q^{66} + 4 q^{67} - 12 q^{68} - 4 q^{69} + 4 q^{71} - 6 q^{72} + 24 q^{74} - 4 q^{76} - 16 q^{77} + 4 q^{78} - 10 q^{81} - 4 q^{82} + 12 q^{83} - 24 q^{84} + 28 q^{86} - 16 q^{87} + 16 q^{88} + 12 q^{89} - 4 q^{91} - 8 q^{92} - 12 q^{93} + 12 q^{94} + 4 q^{96} + 4 q^{97} + 42 q^{98} - 4 q^{99}+O(q^{100}) \) 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
−0.414214 −1.41421 −1.82843 0 0.585786 0.828427 1.58579 −1.00000 0
1.2 2.41421 1.41421 3.82843 0 3.41421 −4.82843 4.41421 −1.00000 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.i 2
3.b odd 2 1 2925.2.a.u 2
4.b odd 2 1 5200.2.a.bu 2
5.b even 2 1 65.2.a.b 2
5.c odd 4 2 325.2.b.f 4
13.b even 2 1 4225.2.a.r 2
15.d odd 2 1 585.2.a.m 2
15.e even 4 2 2925.2.c.r 4
20.d odd 2 1 1040.2.a.j 2
35.c odd 2 1 3185.2.a.j 2
40.e odd 2 1 4160.2.a.z 2
40.f even 2 1 4160.2.a.bf 2
55.d odd 2 1 7865.2.a.j 2
60.h even 2 1 9360.2.a.cd 2
65.d even 2 1 845.2.a.g 2
65.g odd 4 2 845.2.c.b 4
65.l even 6 2 845.2.e.c 4
65.n even 6 2 845.2.e.h 4
65.s odd 12 4 845.2.m.f 8
195.e odd 2 1 7605.2.a.x 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.a.b 2 5.b even 2 1
325.2.a.i 2 1.a even 1 1 trivial
325.2.b.f 4 5.c odd 4 2
585.2.a.m 2 15.d odd 2 1
845.2.a.g 2 65.d even 2 1
845.2.c.b 4 65.g odd 4 2
845.2.e.c 4 65.l even 6 2
845.2.e.h 4 65.n even 6 2
845.2.m.f 8 65.s odd 12 4
1040.2.a.j 2 20.d odd 2 1
2925.2.a.u 2 3.b odd 2 1
2925.2.c.r 4 15.e even 4 2
3185.2.a.j 2 35.c odd 2 1
4160.2.a.z 2 40.e odd 2 1
4160.2.a.bf 2 40.f even 2 1
4225.2.a.r 2 13.b even 2 1
5200.2.a.bu 2 4.b odd 2 1
7605.2.a.x 2 195.e odd 2 1
7865.2.a.j 2 55.d odd 2 1
9360.2.a.cd 2 60.h even 2 1

Hecke kernels

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

\( T_{2}^{2} - 2T_{2} - 1 \) Copy content Toggle raw display
\( T_{3}^{2} - 2 \) 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
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 4T - 4 \) 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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