# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$325 = 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 325.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

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

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{2} -2 q^{3} + q^{4} -2 i q^{6} + 3 i q^{8} + q^{9} +O(q^{10})$$ $$q + i q^{2} -2 q^{3} + q^{4} -2 i q^{6} + 3 i q^{8} + q^{9} + 2 i q^{11} -2 q^{12} + ( 2 + 3 i ) q^{13} - q^{16} + i q^{18} + 6 i q^{19} -2 q^{22} -6 q^{23} -6 i q^{24} + ( -3 + 2 i ) q^{26} + 4 q^{27} -6 q^{29} + 6 i q^{31} + 5 i q^{32} -4 i q^{33} + q^{36} -6 i q^{37} -6 q^{38} + ( -4 - 6 i ) q^{39} -8 i q^{41} + 6 q^{43} + 2 i q^{44} -6 i q^{46} + 8 i q^{47} + 2 q^{48} + 7 q^{49} + ( 2 + 3 i ) q^{52} + 12 q^{53} + 4 i q^{54} -12 i q^{57} -6 i q^{58} -2 i q^{59} + 6 q^{61} -6 q^{62} -7 q^{64} + 4 q^{66} -12 i q^{67} + 12 q^{69} -2 i q^{71} + 3 i q^{72} -6 i q^{73} + 6 q^{74} + 6 i q^{76} + ( 6 - 4 i ) q^{78} -11 q^{81} + 8 q^{82} -4 i q^{83} + 6 i q^{86} + 12 q^{87} -6 q^{88} + 8 i q^{89} -6 q^{92} -12 i q^{93} -8 q^{94} -10 i q^{96} + 6 i q^{97} + 7 i q^{98} + 2 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{3} + 2q^{4} + 2q^{9} + O(q^{10})$$ $$2q - 4q^{3} + 2q^{4} + 2q^{9} - 4q^{12} + 4q^{13} - 2q^{16} - 4q^{22} - 12q^{23} - 6q^{26} + 8q^{27} - 12q^{29} + 2q^{36} - 12q^{38} - 8q^{39} + 12q^{43} + 4q^{48} + 14q^{49} + 4q^{52} + 24q^{53} + 12q^{61} - 12q^{62} - 14q^{64} + 8q^{66} + 24q^{69} + 12q^{74} + 12q^{78} - 22q^{81} + 16q^{82} + 24q^{87} - 12q^{88} - 12q^{92} - 16q^{94} + O(q^{100})$$

## Character values

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

 $$n$$ $$27$$ $$301$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
51.1
 − 1.00000i 1.00000i
1.00000i −2.00000 1.00000 0 2.00000i 0 3.00000i 1.00000 0
51.2 1.00000i −2.00000 1.00000 0 2.00000i 0 3.00000i 1.00000 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.b even 2 1 inner

## Twists

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

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

## Hecke kernels

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

 $$T_{2}^{2} + 1$$ $$T_{3} + 2$$ $$T_{7}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$( 2 + T )^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$4 + T^{2}$$
$13$ $$13 - 4 T + T^{2}$$
$17$ $$T^{2}$$
$19$ $$36 + T^{2}$$
$23$ $$( 6 + T )^{2}$$
$29$ $$( 6 + T )^{2}$$
$31$ $$36 + T^{2}$$
$37$ $$36 + T^{2}$$
$41$ $$64 + T^{2}$$
$43$ $$( -6 + T )^{2}$$
$47$ $$64 + T^{2}$$
$53$ $$( -12 + T )^{2}$$
$59$ $$4 + T^{2}$$
$61$ $$( -6 + T )^{2}$$
$67$ $$144 + T^{2}$$
$71$ $$4 + T^{2}$$
$73$ $$36 + T^{2}$$
$79$ $$T^{2}$$
$83$ $$16 + T^{2}$$
$89$ $$64 + T^{2}$$
$97$ $$36 + T^{2}$$