# Properties

 Label 325.2.e.b Level $325$ Weight $2$ Character orbit 325.e Analytic conductor $2.595$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$325 = 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 325.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.59513806569$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{5})$$ Defining polynomial: $$x^{4} - x^{3} + 2 x^{2} + x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 65) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( 1 - 2 \beta_{1} + \beta_{3} ) q^{3} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{4} + ( -\beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{6} + ( 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} ) q^{7} + ( -1 - 2 \beta_{2} ) q^{8} + 2 \beta_{3} q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( 1 - 2 \beta_{1} + \beta_{3} ) q^{3} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{4} + ( -\beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{6} + ( 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} ) q^{7} + ( -1 - 2 \beta_{2} ) q^{8} + 2 \beta_{3} q^{9} + ( -1 - 2 \beta_{1} - \beta_{3} ) q^{11} + ( 3 + \beta_{2} ) q^{12} + ( -1 - 4 \beta_{3} ) q^{13} + ( -2 - \beta_{2} ) q^{14} + 3 \beta_{1} q^{16} + ( -4 \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{17} + 2 \beta_{2} q^{18} + ( -2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{19} + ( 7 + 4 \beta_{2} ) q^{21} + ( -3 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{22} + ( 5 + 2 \beta_{1} + 5 \beta_{3} ) q^{23} + ( -5 - 5 \beta_{3} ) q^{24} + ( -\beta_{1} - 4 \beta_{2} ) q^{26} + ( 1 + 2 \beta_{2} ) q^{27} + ( -5 + 3 \beta_{1} - 5 \beta_{3} ) q^{28} + ( 5 - 4 \beta_{1} + 5 \beta_{3} ) q^{29} + ( -\beta_{1} - \beta_{2} + 5 \beta_{3} ) q^{32} + ( 4 \beta_{1} + 4 \beta_{2} + 3 \beta_{3} ) q^{33} + ( 4 - 3 \beta_{2} ) q^{34} + ( 2 - 2 \beta_{1} + 2 \beta_{3} ) q^{36} + ( 3 + 3 \beta_{3} ) q^{37} + ( 2 + \beta_{2} ) q^{38} + ( 3 + 2 \beta_{1} + 8 \beta_{2} - \beta_{3} ) q^{39} + ( -7 + 8 \beta_{1} - 7 \beta_{3} ) q^{41} + ( -4 + 3 \beta_{1} - 4 \beta_{3} ) q^{42} + ( -2 \beta_{1} - 2 \beta_{2} + 5 \beta_{3} ) q^{43} + ( 1 - \beta_{2} ) q^{44} + ( 7 \beta_{1} + 7 \beta_{2} + 2 \beta_{3} ) q^{46} + 8 \beta_{2} q^{47} + ( -3 \beta_{1} - 3 \beta_{2} - 6 \beta_{3} ) q^{48} + ( -6 + 8 \beta_{1} - 6 \beta_{3} ) q^{49} + ( -9 + 2 \beta_{2} ) q^{51} + ( -4 + 3 \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{52} -6 q^{53} + ( -2 - \beta_{1} - 2 \beta_{3} ) q^{54} + ( -4 \beta_{1} - 4 \beta_{2} + 7 \beta_{3} ) q^{56} + ( -7 - 4 \beta_{2} ) q^{57} + ( \beta_{1} + \beta_{2} - 4 \beta_{3} ) q^{58} + ( 6 \beta_{1} + 6 \beta_{2} + 3 \beta_{3} ) q^{59} + ( -12 \beta_{1} - 12 \beta_{2} + 7 \beta_{3} ) q^{61} + ( 6 - 4 \beta_{1} + 6 \beta_{3} ) q^{63} + ( 1 - 2 \beta_{2} ) q^{64} + ( -4 + 7 \beta_{2} ) q^{66} + ( 1 + 6 \beta_{1} + \beta_{3} ) q^{67} + ( 5 - \beta_{1} + 5 \beta_{3} ) q^{68} + ( -12 \beta_{1} - 12 \beta_{2} + \beta_{3} ) q^{69} + ( 2 \beta_{1} + 2 \beta_{2} - 5 \beta_{3} ) q^{71} + ( 4 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{72} + 6 q^{73} + ( 3 \beta_{1} + 3 \beta_{2} ) q^{74} + ( 5 - 3 \beta_{1} + 5 \beta_{3} ) q^{76} + q^{77} + ( -8 - 3 \beta_{1} + \beta_{2} - 6 \beta_{3} ) q^{78} + ( 11 + 11 \beta_{3} ) q^{81} + ( \beta_{1} + \beta_{2} + 8 \beta_{3} ) q^{82} + ( -4 - 8 \beta_{2} ) q^{83} + ( 7 \beta_{1} + 7 \beta_{2} - 11 \beta_{3} ) q^{84} + ( 2 + 3 \beta_{2} ) q^{86} + ( -6 \beta_{1} - 6 \beta_{2} + 13 \beta_{3} ) q^{87} + ( -3 - 4 \beta_{1} - 3 \beta_{3} ) q^{88} + ( 9 + 9 \beta_{3} ) q^{89} + ( -12 + 6 \beta_{1} - 2 \beta_{2} - 9 \beta_{3} ) q^{91} + ( 3 + 5 \beta_{2} ) q^{92} + ( -8 - 8 \beta_{1} - 8 \beta_{3} ) q^{94} + ( -7 - 9 \beta_{2} ) q^{96} + ( -4 \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{97} + ( 2 \beta_{1} + 2 \beta_{2} + 8 \beta_{3} ) q^{98} + ( 2 - 4 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + q^{2} + q^{4} + 5 q^{6} + 4 q^{7} - 4 q^{9} + O(q^{10})$$ $$4 q + q^{2} + q^{4} + 5 q^{6} + 4 q^{7} - 4 q^{9} - 4 q^{11} + 10 q^{12} + 4 q^{13} - 6 q^{14} + 3 q^{16} + 2 q^{17} - 4 q^{18} - 4 q^{19} + 20 q^{21} + 7 q^{22} + 12 q^{23} - 10 q^{24} + 7 q^{26} - 7 q^{28} + 6 q^{29} - 9 q^{32} - 10 q^{33} + 22 q^{34} + 2 q^{36} + 6 q^{37} + 6 q^{38} - 6 q^{41} - 5 q^{42} - 8 q^{43} + 6 q^{44} - 11 q^{46} - 16 q^{47} + 15 q^{48} - 4 q^{49} - 40 q^{51} - 5 q^{52} - 24 q^{53} - 5 q^{54} - 10 q^{56} - 20 q^{57} + 7 q^{58} - 12 q^{59} - 2 q^{61} + 8 q^{63} + 8 q^{64} - 30 q^{66} + 8 q^{67} + 9 q^{68} + 10 q^{69} + 8 q^{71} + 24 q^{73} - 3 q^{74} + 7 q^{76} + 4 q^{77} - 25 q^{78} + 22 q^{81} - 17 q^{82} + 15 q^{84} + 2 q^{86} - 20 q^{87} - 10 q^{88} + 18 q^{89} - 20 q^{91} + 2 q^{92} - 24 q^{94} - 10 q^{96} + 2 q^{97} - 18 q^{98} + 16 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 2 x^{2} + x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 1$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 2 \nu^{2} - 2 \nu - 1$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{2} - 1$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/325\mathbb{Z}\right)^\times$$.

 $$n$$ $$27$$ $$301$$ $$\chi(n)$$ $$1$$ $$\beta_{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
126.1
 −0.309017 + 0.535233i 0.809017 − 1.40126i −0.309017 − 0.535233i 0.809017 + 1.40126i
−0.309017 + 0.535233i 1.11803 1.93649i 0.809017 + 1.40126i 0 0.690983 + 1.19682i 2.11803 + 3.66854i −2.23607 −1.00000 1.73205i 0
126.2 0.809017 1.40126i −1.11803 + 1.93649i −0.309017 0.535233i 0 1.80902 + 3.13331i −0.118034 0.204441i 2.23607 −1.00000 1.73205i 0
276.1 −0.309017 0.535233i 1.11803 + 1.93649i 0.809017 1.40126i 0 0.690983 1.19682i 2.11803 3.66854i −2.23607 −1.00000 + 1.73205i 0
276.2 0.809017 + 1.40126i −1.11803 1.93649i −0.309017 + 0.535233i 0 1.80902 3.13331i −0.118034 + 0.204441i 2.23607 −1.00000 + 1.73205i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 325.2.e.b 4
5.b even 2 1 65.2.e.a 4
5.c odd 4 2 325.2.o.a 8
13.c even 3 1 inner 325.2.e.b 4
13.c even 3 1 4225.2.a.u 2
13.e even 6 1 4225.2.a.y 2
15.d odd 2 1 585.2.j.e 4
20.d odd 2 1 1040.2.q.n 4
65.d even 2 1 845.2.e.g 4
65.g odd 4 2 845.2.m.e 8
65.l even 6 1 845.2.a.b 2
65.l even 6 1 845.2.e.g 4
65.n even 6 1 65.2.e.a 4
65.n even 6 1 845.2.a.e 2
65.q odd 12 2 325.2.o.a 8
65.s odd 12 2 845.2.c.c 4
65.s odd 12 2 845.2.m.e 8
195.x odd 6 1 585.2.j.e 4
195.x odd 6 1 7605.2.a.ba 2
195.y odd 6 1 7605.2.a.bf 2
260.v odd 6 1 1040.2.q.n 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.e.a 4 5.b even 2 1
65.2.e.a 4 65.n even 6 1
325.2.e.b 4 1.a even 1 1 trivial
325.2.e.b 4 13.c even 3 1 inner
325.2.o.a 8 5.c odd 4 2
325.2.o.a 8 65.q odd 12 2
585.2.j.e 4 15.d odd 2 1
585.2.j.e 4 195.x odd 6 1
845.2.a.b 2 65.l even 6 1
845.2.a.e 2 65.n even 6 1
845.2.c.c 4 65.s odd 12 2
845.2.e.g 4 65.d even 2 1
845.2.e.g 4 65.l even 6 1
845.2.m.e 8 65.g odd 4 2
845.2.m.e 8 65.s odd 12 2
1040.2.q.n 4 20.d odd 2 1
1040.2.q.n 4 260.v odd 6 1
4225.2.a.u 2 13.c even 3 1
4225.2.a.y 2 13.e even 6 1
7605.2.a.ba 2 195.x odd 6 1
7605.2.a.bf 2 195.y odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - T_{2}^{3} + 2 T_{2}^{2} + T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(325, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + 2 T^{2} - T^{3} + T^{4}$$
$3$ $$25 + 5 T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$1 + 4 T + 17 T^{2} - 4 T^{3} + T^{4}$$
$11$ $$1 - 4 T + 17 T^{2} + 4 T^{3} + T^{4}$$
$13$ $$( 13 - 2 T + T^{2} )^{2}$$
$17$ $$361 + 38 T + 23 T^{2} - 2 T^{3} + T^{4}$$
$19$ $$1 - 4 T + 17 T^{2} + 4 T^{3} + T^{4}$$
$23$ $$961 - 372 T + 113 T^{2} - 12 T^{3} + T^{4}$$
$29$ $$121 + 66 T + 47 T^{2} - 6 T^{3} + T^{4}$$
$31$ $$T^{4}$$
$37$ $$( 9 - 3 T + T^{2} )^{2}$$
$41$ $$5041 - 426 T + 107 T^{2} + 6 T^{3} + T^{4}$$
$43$ $$121 + 88 T + 53 T^{2} + 8 T^{3} + T^{4}$$
$47$ $$( -64 + 8 T + T^{2} )^{2}$$
$53$ $$( 6 + T )^{4}$$
$59$ $$81 - 108 T + 153 T^{2} + 12 T^{3} + T^{4}$$
$61$ $$32041 - 358 T + 183 T^{2} + 2 T^{3} + T^{4}$$
$67$ $$841 + 232 T + 93 T^{2} - 8 T^{3} + T^{4}$$
$71$ $$121 - 88 T + 53 T^{2} - 8 T^{3} + T^{4}$$
$73$ $$( -6 + T )^{4}$$
$79$ $$T^{4}$$
$83$ $$( -80 + T^{2} )^{2}$$
$89$ $$( 81 - 9 T + T^{2} )^{2}$$
$97$ $$361 + 38 T + 23 T^{2} - 2 T^{3} + T^{4}$$