# Properties

 Label 325.4.d.b Level $325$ Weight $4$ Character orbit 325.d Analytic conductor $19.176$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$325 = 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 325.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$19.1756207519$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 13) 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.

 $$f(q)$$ $$=$$ $$q + 3 q^{2} -i q^{3} + q^{4} -3 i q^{6} -15 q^{7} -21 q^{8} + 26 q^{9} +O(q^{10})$$ $$q + 3 q^{2} -i q^{3} + q^{4} -3 i q^{6} -15 q^{7} -21 q^{8} + 26 q^{9} + 48 i q^{11} -i q^{12} + ( 39 + 26 i ) q^{13} -45 q^{14} -71 q^{16} + 45 i q^{17} + 78 q^{18} -6 i q^{19} + 15 i q^{21} + 144 i q^{22} + 162 i q^{23} + 21 i q^{24} + ( 117 + 78 i ) q^{26} -53 i q^{27} -15 q^{28} + 144 q^{29} + 264 i q^{31} -45 q^{32} + 48 q^{33} + 135 i q^{34} + 26 q^{36} -303 q^{37} -18 i q^{38} + ( 26 - 39 i ) q^{39} -192 i q^{41} + 45 i q^{42} -97 i q^{43} + 48 i q^{44} + 486 i q^{46} -111 q^{47} + 71 i q^{48} -118 q^{49} + 45 q^{51} + ( 39 + 26 i ) q^{52} -414 i q^{53} -159 i q^{54} + 315 q^{56} -6 q^{57} + 432 q^{58} + 522 i q^{59} + 376 q^{61} + 792 i q^{62} -390 q^{63} + 433 q^{64} + 144 q^{66} -36 q^{67} + 45 i q^{68} + 162 q^{69} + 357 i q^{71} -546 q^{72} -1098 q^{73} -909 q^{74} -6 i q^{76} -720 i q^{77} + ( 78 - 117 i ) q^{78} + 830 q^{79} + 649 q^{81} -576 i q^{82} + 438 q^{83} + 15 i q^{84} -291 i q^{86} -144 i q^{87} -1008 i q^{88} -438 i q^{89} + ( -585 - 390 i ) q^{91} + 162 i q^{92} + 264 q^{93} -333 q^{94} + 45 i q^{96} -852 q^{97} -354 q^{98} + 1248 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{2} + 2 q^{4} - 30 q^{7} - 42 q^{8} + 52 q^{9} + O(q^{10})$$ $$2 q + 6 q^{2} + 2 q^{4} - 30 q^{7} - 42 q^{8} + 52 q^{9} + 78 q^{13} - 90 q^{14} - 142 q^{16} + 156 q^{18} + 234 q^{26} - 30 q^{28} + 288 q^{29} - 90 q^{32} + 96 q^{33} + 52 q^{36} - 606 q^{37} + 52 q^{39} - 222 q^{47} - 236 q^{49} + 90 q^{51} + 78 q^{52} + 630 q^{56} - 12 q^{57} + 864 q^{58} + 752 q^{61} - 780 q^{63} + 866 q^{64} + 288 q^{66} - 72 q^{67} + 324 q^{69} - 1092 q^{72} - 2196 q^{73} - 1818 q^{74} + 156 q^{78} + 1660 q^{79} + 1298 q^{81} + 876 q^{83} - 1170 q^{91} + 528 q^{93} - 666 q^{94} - 1704 q^{97} - 708 q^{98} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
324.1
 1.00000i − 1.00000i
3.00000 1.00000i 1.00000 0 3.00000i −15.0000 −21.0000 26.0000 0
324.2 3.00000 1.00000i 1.00000 0 3.00000i −15.0000 −21.0000 26.0000 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
65.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 325.4.d.b 2
5.b even 2 1 325.4.d.a 2
5.c odd 4 1 13.4.b.a 2
5.c odd 4 1 325.4.c.b 2
13.b even 2 1 325.4.d.a 2
15.e even 4 1 117.4.b.a 2
20.e even 4 1 208.4.f.b 2
40.i odd 4 1 832.4.f.e 2
40.k even 4 1 832.4.f.c 2
65.d even 2 1 inner 325.4.d.b 2
65.f even 4 1 169.4.a.b 1
65.h odd 4 1 13.4.b.a 2
65.h odd 4 1 325.4.c.b 2
65.k even 4 1 169.4.a.c 1
65.o even 12 2 169.4.c.b 2
65.q odd 12 2 169.4.e.d 4
65.r odd 12 2 169.4.e.d 4
65.t even 12 2 169.4.c.c 2
195.j odd 4 1 1521.4.a.d 1
195.s even 4 1 117.4.b.a 2
195.u odd 4 1 1521.4.a.i 1
260.p even 4 1 208.4.f.b 2
520.bc even 4 1 832.4.f.c 2
520.bg odd 4 1 832.4.f.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.b.a 2 5.c odd 4 1
13.4.b.a 2 65.h odd 4 1
117.4.b.a 2 15.e even 4 1
117.4.b.a 2 195.s even 4 1
169.4.a.b 1 65.f even 4 1
169.4.a.c 1 65.k even 4 1
169.4.c.b 2 65.o even 12 2
169.4.c.c 2 65.t even 12 2
169.4.e.d 4 65.q odd 12 2
169.4.e.d 4 65.r odd 12 2
208.4.f.b 2 20.e even 4 1
208.4.f.b 2 260.p even 4 1
325.4.c.b 2 5.c odd 4 1
325.4.c.b 2 65.h odd 4 1
325.4.d.a 2 5.b even 2 1
325.4.d.a 2 13.b even 2 1
325.4.d.b 2 1.a even 1 1 trivial
325.4.d.b 2 65.d even 2 1 inner
832.4.f.c 2 40.k even 4 1
832.4.f.c 2 520.bc even 4 1
832.4.f.e 2 40.i odd 4 1
832.4.f.e 2 520.bg odd 4 1
1521.4.a.d 1 195.j odd 4 1
1521.4.a.i 1 195.u odd 4 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -3 + T )^{2}$$
$3$ $$1 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$( 15 + T )^{2}$$
$11$ $$2304 + T^{2}$$
$13$ $$2197 - 78 T + T^{2}$$
$17$ $$2025 + T^{2}$$
$19$ $$36 + T^{2}$$
$23$ $$26244 + T^{2}$$
$29$ $$( -144 + T )^{2}$$
$31$ $$69696 + T^{2}$$
$37$ $$( 303 + T )^{2}$$
$41$ $$36864 + T^{2}$$
$43$ $$9409 + T^{2}$$
$47$ $$( 111 + T )^{2}$$
$53$ $$171396 + T^{2}$$
$59$ $$272484 + T^{2}$$
$61$ $$( -376 + T )^{2}$$
$67$ $$( 36 + T )^{2}$$
$71$ $$127449 + T^{2}$$
$73$ $$( 1098 + T )^{2}$$
$79$ $$( -830 + T )^{2}$$
$83$ $$( -438 + T )^{2}$$
$89$ $$191844 + T^{2}$$
$97$ $$( 852 + T )^{2}$$