# 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 $1$

# Related objects

## 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: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*x + 1 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} - 1) q^{2} + ( - \beta_1 - 1) q^{3} + (\beta_{2} - \beta_1 + 2) q^{4} + (\beta_{2} + 2 \beta_1) q^{6} + (\beta_{2} + \beta_1 - 1) q^{7} + (3 \beta_1 - 4) q^{8} + (\beta_{2} + 3 \beta_1) q^{9}+O(q^{10})$$ q + (-b2 - 1) * q^2 + (-b1 - 1) * q^3 + (b2 - b1 + 2) * q^4 + (b2 + 2*b1) * q^6 + (b2 + b1 - 1) * q^7 + (3*b1 - 4) * q^8 + (b2 + 3*b1) * q^9 $$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})$$ q + (-b2 - 1) * q^2 + (-b1 - 1) * q^3 + (b2 - b1 + 2) * q^4 + (b2 + 2*b1) * q^6 + (b2 + b1 - 1) * q^7 + (3*b1 - 4) * q^8 + (b2 + 3*b1) * q^9 + (-b2 - 2) * q^11 + (-b1 + 1) * q^12 + q^13 + (b2 - b1 - 1) * q^14 + (2*b2 - 4*b1 + 3) * q^16 + (2*b2 - 2) * q^17 - 5*b1 * q^18 - b2 * q^19 + (-2*b2 - 2*b1) * q^21 + (2*b2 - b1 + 5) * q^22 + (b1 - 5) * q^23 + (-3*b2 - 2*b1 - 2) * q^24 + (-b2 - 1) * q^26 + (-4*b2 - 4*b1 - 2) * q^27 + (-b2 + b1 - 1) * q^28 + (-3*b2 - 3*b1 + 3) * q^29 + (b2 + 2*b1 - 4) * q^31 + (-3*b2 + 4*b1 - 5) * q^32 + (b2 + 3*b1 + 1) * q^33 + (2*b2 + 2*b1 - 4) * q^34 + (-2*b2 + 4*b1 - 5) * q^36 + (b2 + 3*b1 - 1) * q^37 + (-b1 + 3) * q^38 + (-b1 - 1) * q^39 + (-2*b2 + 2*b1 - 2) * q^41 + (2*b1 + 4) * q^42 + (2*b2 - 3*b1 - 1) * q^43 + (-3*b2 + 4*b1 - 8) * q^44 + (5*b2 - 2*b1 + 6) * q^46 + (-b2 - b1 - 3) * q^47 + (2*b2 + 3*b1 + 7) * q^48 + (-2*b2 - 3) * q^49 + (-2*b2 + 4) * q^51 + (b2 - b1 + 2) * q^52 + (-4*b2 - 2*b1 - 2) * q^53 + (2*b2 + 4*b1 + 10) * q^54 + (-b2 - b1 + 7) * q^56 + (b2 + b1 - 1) * q^57 + (-3*b2 + 3*b1 + 3) * q^58 + (3*b2 - 2*b1 - 2) * q^59 + (b2 + 3*b1 + 1) * q^61 + (4*b2 - 3*b1 + 3) * q^62 + (b2 + 3*b1 + 5) * q^63 + (b2 - 3*b1 + 12) * q^64 + (-b2 - 5*b1 - 1) * q^66 + (-5*b2 - b1 - 3) * q^67 + (-2*b1 + 4) * q^68 + (-b2 + 3*b1 + 3) * q^69 + (b2 - 6*b1 - 2) * q^71 + (5*b2 + 15) * q^72 + (-b2 - 3*b1 + 9) * q^73 + (b2 - 5*b1 + 1) * q^74 + (-b2 + 2*b1 - 4) * q^76 - 2*b1 * q^77 + (b2 + 2*b1) * q^78 + (4*b2 + 2*b1 - 6) * q^79 + (5*b2 + 5*b1 + 6) * q^81 + (2*b2 - 6*b1 + 10) * q^82 + (b2 - b1 - 7) * q^83 - 2 * q^84 + (b2 + 8*b1 - 8) * q^86 + (6*b2 + 6*b1) * q^87 + (4*b2 - 9*b1 + 11) * q^88 + (4*b2 + 4*b1 + 2) * q^89 + (b2 + b1 - 1) * q^91 + (-6*b2 + 7*b1 - 13) * q^92 + (-3*b2 - b1 + 1) * q^93 + (3*b2 + b1 + 5) * q^94 + (-b2 - 6) * q^96 + (-6*b2 - 4*b1 + 6) * q^97 + (3*b2 - 2*b1 + 9) * q^98 + (-b2 - 8*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{2} - 4 q^{3} + 5 q^{4} + 2 q^{6} - 2 q^{7} - 9 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q - 3 * q^2 - 4 * q^3 + 5 * q^4 + 2 * q^6 - 2 * q^7 - 9 * q^8 + 3 * q^9 $$3 q - 3 q^{2} - 4 q^{3} + 5 q^{4} + 2 q^{6} - 2 q^{7} - 9 q^{8} + 3 q^{9} - 6 q^{11} + 2 q^{12} + 3 q^{13} - 4 q^{14} + 5 q^{16} - 6 q^{17} - 5 q^{18} - 2 q^{21} + 14 q^{22} - 14 q^{23} - 8 q^{24} - 3 q^{26} - 10 q^{27} - 2 q^{28} + 6 q^{29} - 10 q^{31} - 11 q^{32} + 6 q^{33} - 10 q^{34} - 11 q^{36} + 8 q^{38} - 4 q^{39} - 4 q^{41} + 14 q^{42} - 6 q^{43} - 20 q^{44} + 16 q^{46} - 10 q^{47} + 24 q^{48} - 9 q^{49} + 12 q^{51} + 5 q^{52} - 8 q^{53} + 34 q^{54} + 20 q^{56} - 2 q^{57} + 12 q^{58} - 8 q^{59} + 6 q^{61} + 6 q^{62} + 18 q^{63} + 33 q^{64} - 8 q^{66} - 10 q^{67} + 10 q^{68} + 12 q^{69} - 12 q^{71} + 45 q^{72} + 24 q^{73} - 2 q^{74} - 10 q^{76} - 2 q^{77} + 2 q^{78} - 16 q^{79} + 23 q^{81} + 24 q^{82} - 22 q^{83} - 6 q^{84} - 16 q^{86} + 6 q^{87} + 24 q^{88} + 10 q^{89} - 2 q^{91} - 32 q^{92} + 2 q^{93} + 16 q^{94} - 18 q^{96} + 14 q^{97} + 25 q^{98} - 8 q^{99}+O(q^{100})$$ 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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 3x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 2$$ v^2 - v - 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 2$$ b2 + b1 + 2

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(325))$$:

 $$T_{2}^{3} + 3T_{2}^{2} - T_{2} - 5$$ T2^3 + 3*T2^2 - T2 - 5 $$T_{3}^{3} + 4T_{3}^{2} + 2T_{3} - 2$$ T3^3 + 4*T3^2 + 2*T3 - 2

## Hecke characteristic polynomials

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