# Properties

 Label 325.4.d.a 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: $$x^{2} + 1$$ 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 $$q - 3 q^{2} - i q^{3} + q^{4} + 3 i q^{6} + 15 q^{7} + 21 q^{8} + 26 q^{9} - 48 i q^{11} - i q^{12} + (26 i - 39) q^{13} - 45 q^{14} - 71 q^{16} + 45 i q^{17} - 78 q^{18} + 6 i q^{19} - 15 i q^{21} + 144 i q^{22} + 162 i q^{23} - 21 i q^{24} + ( - 78 i + 117) q^{26} - 53 i q^{27} + 15 q^{28} + 144 q^{29} - 264 i q^{31} + 45 q^{32} - 48 q^{33} - 135 i q^{34} + 26 q^{36} + 303 q^{37} - 18 i q^{38} + (39 i + 26) q^{39} + 192 i q^{41} + 45 i q^{42} - 97 i q^{43} - 48 i q^{44} - 486 i q^{46} + 111 q^{47} + 71 i q^{48} - 118 q^{49} + 45 q^{51} + (26 i - 39) q^{52} - 414 i q^{53} + 159 i q^{54} + 315 q^{56} + 6 q^{57} - 432 q^{58} - 522 i q^{59} + 376 q^{61} + 792 i q^{62} + 390 q^{63} + 433 q^{64} + 144 q^{66} + 36 q^{67} + 45 i q^{68} + 162 q^{69} - 357 i q^{71} + 546 q^{72} + 1098 q^{73} - 909 q^{74} + 6 i q^{76} - 720 i q^{77} + ( - 117 i - 78) q^{78} + 830 q^{79} + 649 q^{81} - 576 i q^{82} - 438 q^{83} - 15 i q^{84} + 291 i q^{86} - 144 i q^{87} - 1008 i q^{88} + 438 i q^{89} + (390 i - 585) q^{91} + 162 i q^{92} - 264 q^{93} - 333 q^{94} - 45 i q^{96} + 852 q^{97} + 354 q^{98} - 1248 i q^{99} +O(q^{100})$$ 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 + (26*i - 39) * 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 + (-78*i + 117) * 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 + (39*i + 26) * 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 + (26*i - 39) * 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 + (-117*i - 78) * 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 + (390*i - 585) * 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 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 6 q^{2} + 2 q^{4} + 30 q^{7} + 42 q^{8} + 52 q^{9}+O(q^{10})$$ 2 * q - 6 * q^2 + 2 * q^4 + 30 * q^7 + 42 * q^8 + 52 * q^9 $$2 q - 6 q^{2} + 2 q^{4} + 30 q^{7} + 42 q^{8} + 52 q^{9} - 78 q^{13} - 90 q^{14} - 142 q^{16} - 156 q^{18} + 234 q^{26} + 30 q^{28} + 288 q^{29} + 90 q^{32} - 96 q^{33} + 52 q^{36} + 606 q^{37} + 52 q^{39} + 222 q^{47} - 236 q^{49} + 90 q^{51} - 78 q^{52} + 630 q^{56} + 12 q^{57} - 864 q^{58} + 752 q^{61} + 780 q^{63} + 866 q^{64} + 288 q^{66} + 72 q^{67} + 324 q^{69} + 1092 q^{72} + 2196 q^{73} - 1818 q^{74} - 156 q^{78} + 1660 q^{79} + 1298 q^{81} - 876 q^{83} - 1170 q^{91} - 528 q^{93} - 666 q^{94} + 1704 q^{97} + 708 q^{98}+O(q^{100})$$ 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

## 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.a 2
5.b even 2 1 325.4.d.b 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.b 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.a 2
65.f even 4 1 169.4.a.c 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.b 1
65.o even 12 2 169.4.c.c 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.b 2
195.j odd 4 1 1521.4.a.i 1
195.s even 4 1 117.4.b.a 2
195.u odd 4 1 1521.4.a.d 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.k even 4 1
169.4.a.c 1 65.f even 4 1
169.4.c.b 2 65.t even 12 2
169.4.c.c 2 65.o 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 1.a even 1 1 trivial
325.4.d.a 2 65.d even 2 1 inner
325.4.d.b 2 5.b even 2 1
325.4.d.b 2 13.b even 2 1
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.u odd 4 1
1521.4.a.i 1 195.j 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

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