# Properties

 Label 325.2.n.c Level $325$ Weight $2$ Character orbit 325.n 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.n (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.59513806569$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-7})$$ Defining polynomial: $$x^{4} - x^{3} - x^{2} - 2 x + 4$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 65) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{3} ) q^{2} -\beta_{2} q^{3} + ( \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{4} + ( -1 + \beta_{1} ) q^{6} + ( -1 - \beta_{2} ) q^{7} + ( 1 - 2 \beta_{2} ) q^{8} + ( 2 - 2 \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( 1 - \beta_{3} ) q^{2} -\beta_{2} q^{3} + ( \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{4} + ( -1 + \beta_{1} ) q^{6} + ( -1 - \beta_{2} ) q^{7} + ( 1 - 2 \beta_{2} ) q^{8} + ( 2 - 2 \beta_{2} ) q^{9} + ( \beta_{2} - 2 \beta_{3} ) q^{11} + ( -1 + \beta_{1} + \beta_{3} ) q^{12} + ( 1 - 4 \beta_{2} ) q^{13} + ( -2 + \beta_{1} + \beta_{3} ) q^{14} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{16} + ( 1 - 2 \beta_{1} - 3 \beta_{2} + 4 \beta_{3} ) q^{17} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{18} + ( 1 + \beta_{2} ) q^{19} + ( -1 + 2 \beta_{2} ) q^{21} + ( 3 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{22} + ( 2 - 4 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{23} + ( -2 + \beta_{2} ) q^{24} + ( -3 + 4 \beta_{1} - \beta_{3} ) q^{26} -5 q^{27} + ( -1 - \beta_{2} + 3 \beta_{3} ) q^{28} + ( 2 - 4 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{29} + ( 2 - 6 \beta_{1} - 4 \beta_{2} + 6 \beta_{3} ) q^{31} + ( 4 - \beta_{1} + 3 \beta_{2} ) q^{32} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{33} + ( -2 - 3 \beta_{1} + 4 \beta_{2} + 3 \beta_{3} ) q^{34} + ( -2 + 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{36} + ( 3 \beta_{2} - 6 \beta_{3} ) q^{37} + ( 2 - \beta_{1} - \beta_{3} ) q^{38} + ( -4 + 3 \beta_{2} ) q^{39} + ( -\beta_{2} + 2 \beta_{3} ) q^{41} + ( 1 - 2 \beta_{1} + \beta_{3} ) q^{42} + ( -7 + 2 \beta_{1} + 9 \beta_{2} - 4 \beta_{3} ) q^{43} + ( 3 + \beta_{1} - 6 \beta_{2} - \beta_{3} ) q^{44} + ( 5 - 3 \beta_{1} + 2 \beta_{2} ) q^{46} + ( 4 \beta_{1} - 4 \beta_{3} ) q^{47} + ( -1 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{48} -4 \beta_{2} q^{49} + ( 1 - 2 \beta_{1} - 2 \beta_{3} ) q^{51} + ( -4 + 5 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{52} + ( 2 + 2 \beta_{1} + 2 \beta_{3} ) q^{53} + ( -5 + 5 \beta_{3} ) q^{54} + ( -3 + 3 \beta_{2} ) q^{56} + ( 1 - 2 \beta_{2} ) q^{57} + ( 5 - 3 \beta_{1} + 2 \beta_{2} ) q^{58} + ( 3 + 4 \beta_{1} + 7 \beta_{2} ) q^{59} + ( 5 + 2 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} ) q^{61} + ( 4 - 8 \beta_{1} + 6 \beta_{2} + 4 \beta_{3} ) q^{62} + ( -4 + 2 \beta_{2} ) q^{63} + ( 9 - 2 \beta_{1} - 2 \beta_{3} ) q^{64} + ( -4 + \beta_{1} + \beta_{3} ) q^{66} + ( -8 + \beta_{2} + 6 \beta_{3} ) q^{67} + ( 1 - 2 \beta_{1} + 9 \beta_{2} + \beta_{3} ) q^{68} + ( -1 + 2 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} ) q^{69} + ( -1 + 4 \beta_{1} + 3 \beta_{2} ) q^{71} + ( -2 - 2 \beta_{2} ) q^{72} + ( 9 + 3 \beta_{1} - 6 \beta_{2} - 6 \beta_{3} ) q^{74} + ( 1 + \beta_{2} - 3 \beta_{3} ) q^{76} + ( -1 + 2 \beta_{1} + 2 \beta_{3} ) q^{77} + ( -1 - 3 \beta_{1} + 4 \beta_{3} ) q^{78} -6 q^{79} -\beta_{2} q^{81} + ( -3 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{82} + ( -4 - 2 \beta_{1} + 8 \beta_{2} + 2 \beta_{3} ) q^{83} + ( 2 - 3 \beta_{1} - \beta_{2} ) q^{84} + ( 2 - 3 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} ) q^{86} + ( -1 + 2 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} ) q^{87} + ( -2 + 4 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} ) q^{88} + ( -8 + 5 \beta_{2} - 2 \beta_{3} ) q^{89} + ( -5 + 7 \beta_{2} ) q^{91} + ( 11 - \beta_{1} - \beta_{3} ) q^{92} + ( 2 + 2 \beta_{2} - 6 \beta_{3} ) q^{93} + ( -4 + 8 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} ) q^{94} + ( 3 + \beta_{1} - 6 \beta_{2} - \beta_{3} ) q^{96} + ( 1 - 6 \beta_{1} - 5 \beta_{2} ) q^{97} + ( -4 + 4 \beta_{1} ) q^{98} + ( -2 + 4 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 3q^{2} - 2q^{3} + q^{4} - 3q^{6} - 6q^{7} + 4q^{9} + O(q^{10})$$ $$4q + 3q^{2} - 2q^{3} + q^{4} - 3q^{6} - 6q^{7} + 4q^{9} - 2q^{12} - 4q^{13} - 6q^{14} - q^{16} + 6q^{19} + 7q^{22} - 6q^{24} - 9q^{26} - 20q^{27} - 3q^{28} + 21q^{32} - 2q^{36} + 6q^{38} - 10q^{39} + 3q^{42} - 12q^{43} + 21q^{46} - q^{48} - 8q^{49} - 7q^{52} + 12q^{53} - 15q^{54} - 6q^{56} + 21q^{58} + 30q^{59} + 12q^{61} + 24q^{62} - 12q^{63} + 32q^{64} - 14q^{66} - 24q^{67} + 21q^{68} + 6q^{71} - 12q^{72} + 21q^{74} + 3q^{76} - 3q^{78} - 24q^{79} - 2q^{81} - 7q^{82} + 3q^{84} - 24q^{89} - 6q^{91} + 42q^{92} + 6q^{93} - 20q^{94} - 12q^{97} - 12q^{98} + O(q^{100})$$

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

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

## Character values

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

 $$n$$ $$27$$ $$301$$ $$\chi(n)$$ $$1$$ $$1 - \beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
101.1
 1.39564 + 0.228425i −0.895644 − 1.09445i 1.39564 − 0.228425i −0.895644 + 1.09445i
−0.395644 + 0.228425i −0.500000 0.866025i −0.895644 + 1.55130i 0 0.395644 + 0.228425i −1.50000 0.866025i 1.73205i 1.00000 1.73205i 0
101.2 1.89564 1.09445i −0.500000 0.866025i 1.39564 2.41733i 0 −1.89564 1.09445i −1.50000 0.866025i 1.73205i 1.00000 1.73205i 0
251.1 −0.395644 0.228425i −0.500000 + 0.866025i −0.895644 1.55130i 0 0.395644 0.228425i −1.50000 + 0.866025i 1.73205i 1.00000 + 1.73205i 0
251.2 1.89564 + 1.09445i −0.500000 + 0.866025i 1.39564 + 2.41733i 0 −1.89564 + 1.09445i −1.50000 + 0.866025i 1.73205i 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.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 325.2.n.c 4
5.b even 2 1 325.2.n.b 4
5.c odd 4 2 65.2.l.a 8
13.e even 6 1 inner 325.2.n.c 4
13.f odd 12 2 4225.2.a.bk 4
15.e even 4 2 585.2.bf.a 8
20.e even 4 2 1040.2.df.b 8
65.f even 4 2 845.2.n.c 8
65.h odd 4 2 845.2.l.c 8
65.k even 4 2 845.2.n.d 8
65.l even 6 1 325.2.n.b 4
65.o even 12 2 845.2.b.f 8
65.o even 12 2 845.2.n.c 8
65.q odd 12 2 845.2.d.c 8
65.q odd 12 2 845.2.l.c 8
65.r odd 12 2 65.2.l.a 8
65.r odd 12 2 845.2.d.c 8
65.s odd 12 2 4225.2.a.bj 4
65.t even 12 2 845.2.b.f 8
65.t even 12 2 845.2.n.d 8
195.bf even 12 2 585.2.bf.a 8
260.bg even 12 2 1040.2.df.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.l.a 8 5.c odd 4 2
65.2.l.a 8 65.r odd 12 2
325.2.n.b 4 5.b even 2 1
325.2.n.b 4 65.l even 6 1
325.2.n.c 4 1.a even 1 1 trivial
325.2.n.c 4 13.e even 6 1 inner
585.2.bf.a 8 15.e even 4 2
585.2.bf.a 8 195.bf even 12 2
845.2.b.f 8 65.o even 12 2
845.2.b.f 8 65.t even 12 2
845.2.d.c 8 65.q odd 12 2
845.2.d.c 8 65.r odd 12 2
845.2.l.c 8 65.h odd 4 2
845.2.l.c 8 65.q odd 12 2
845.2.n.c 8 65.f even 4 2
845.2.n.c 8 65.o even 12 2
845.2.n.d 8 65.k even 4 2
845.2.n.d 8 65.t even 12 2
1040.2.df.b 8 20.e even 4 2
1040.2.df.b 8 260.bg even 12 2
4225.2.a.bj 4 65.s odd 12 2
4225.2.a.bk 4 13.f odd 12 2

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 3 T + 2 T^{2} - 3 T^{3} + T^{4}$$
$3$ $$( 1 + T + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$( 3 + 3 T + T^{2} )^{2}$$
$11$ $$49 - 7 T^{2} + T^{4}$$
$13$ $$( 13 + 2 T + T^{2} )^{2}$$
$17$ $$441 + 21 T^{2} + T^{4}$$
$19$ $$( 3 - 3 T + T^{2} )^{2}$$
$23$ $$441 + 21 T^{2} + T^{4}$$
$29$ $$441 + 21 T^{2} + T^{4}$$
$31$ $$3600 + 132 T^{2} + T^{4}$$
$37$ $$3969 - 63 T^{2} + T^{4}$$
$41$ $$49 - 7 T^{2} + T^{4}$$
$43$ $$225 + 180 T + 129 T^{2} + 12 T^{3} + T^{4}$$
$47$ $$256 + 80 T^{2} + T^{4}$$
$53$ $$( -12 - 6 T + T^{2} )^{2}$$
$59$ $$2209 - 1410 T + 347 T^{2} - 30 T^{3} + T^{4}$$
$61$ $$225 - 180 T + 129 T^{2} - 12 T^{3} + T^{4}$$
$67$ $$225 - 360 T + 177 T^{2} + 24 T^{3} + T^{4}$$
$71$ $$625 + 150 T - 13 T^{2} - 6 T^{3} + T^{4}$$
$73$ $$T^{4}$$
$79$ $$( 6 + T )^{4}$$
$83$ $$4624 + 164 T^{2} + T^{4}$$
$89$ $$1681 + 984 T + 233 T^{2} + 24 T^{3} + T^{4}$$
$97$ $$2601 - 612 T - 3 T^{2} + 12 T^{3} + T^{4}$$