# Properties

 Label 325.4.a.d Level $325$ Weight $4$ Character orbit 325.a Self dual yes Analytic conductor $19.176$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$325 = 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 325.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$19.1756207519$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 13) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 5 q^{2} + 7 q^{3} + 17 q^{4} + 35 q^{6} + 13 q^{7} + 45 q^{8} + 22 q^{9}+O(q^{10})$$ q + 5 * q^2 + 7 * q^3 + 17 * q^4 + 35 * q^6 + 13 * q^7 + 45 * q^8 + 22 * q^9 $$q + 5 q^{2} + 7 q^{3} + 17 q^{4} + 35 q^{6} + 13 q^{7} + 45 q^{8} + 22 q^{9} - 26 q^{11} + 119 q^{12} - 13 q^{13} + 65 q^{14} + 89 q^{16} - 77 q^{17} + 110 q^{18} - 126 q^{19} + 91 q^{21} - 130 q^{22} + 96 q^{23} + 315 q^{24} - 65 q^{26} - 35 q^{27} + 221 q^{28} - 82 q^{29} + 196 q^{31} + 85 q^{32} - 182 q^{33} - 385 q^{34} + 374 q^{36} + 131 q^{37} - 630 q^{38} - 91 q^{39} + 336 q^{41} + 455 q^{42} + 201 q^{43} - 442 q^{44} + 480 q^{46} + 105 q^{47} + 623 q^{48} - 174 q^{49} - 539 q^{51} - 221 q^{52} + 432 q^{53} - 175 q^{54} + 585 q^{56} - 882 q^{57} - 410 q^{58} - 294 q^{59} - 56 q^{61} + 980 q^{62} + 286 q^{63} - 287 q^{64} - 910 q^{66} - 478 q^{67} - 1309 q^{68} + 672 q^{69} + 9 q^{71} + 990 q^{72} - 98 q^{73} + 655 q^{74} - 2142 q^{76} - 338 q^{77} - 455 q^{78} + 1304 q^{79} - 839 q^{81} + 1680 q^{82} + 308 q^{83} + 1547 q^{84} + 1005 q^{86} - 574 q^{87} - 1170 q^{88} - 1190 q^{89} - 169 q^{91} + 1632 q^{92} + 1372 q^{93} + 525 q^{94} + 595 q^{96} - 70 q^{97} - 870 q^{98} - 572 q^{99}+O(q^{100})$$ q + 5 * q^2 + 7 * q^3 + 17 * q^4 + 35 * q^6 + 13 * q^7 + 45 * q^8 + 22 * q^9 - 26 * q^11 + 119 * q^12 - 13 * q^13 + 65 * q^14 + 89 * q^16 - 77 * q^17 + 110 * q^18 - 126 * q^19 + 91 * q^21 - 130 * q^22 + 96 * q^23 + 315 * q^24 - 65 * q^26 - 35 * q^27 + 221 * q^28 - 82 * q^29 + 196 * q^31 + 85 * q^32 - 182 * q^33 - 385 * q^34 + 374 * q^36 + 131 * q^37 - 630 * q^38 - 91 * q^39 + 336 * q^41 + 455 * q^42 + 201 * q^43 - 442 * q^44 + 480 * q^46 + 105 * q^47 + 623 * q^48 - 174 * q^49 - 539 * q^51 - 221 * q^52 + 432 * q^53 - 175 * q^54 + 585 * q^56 - 882 * q^57 - 410 * q^58 - 294 * q^59 - 56 * q^61 + 980 * q^62 + 286 * q^63 - 287 * q^64 - 910 * q^66 - 478 * q^67 - 1309 * q^68 + 672 * q^69 + 9 * q^71 + 990 * q^72 - 98 * q^73 + 655 * q^74 - 2142 * q^76 - 338 * q^77 - 455 * q^78 + 1304 * q^79 - 839 * q^81 + 1680 * q^82 + 308 * q^83 + 1547 * q^84 + 1005 * q^86 - 574 * q^87 - 1170 * q^88 - 1190 * q^89 - 169 * q^91 + 1632 * q^92 + 1372 * q^93 + 525 * q^94 + 595 * q^96 - 70 * q^97 - 870 * q^98 - 572 * q^99

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
5.00000 7.00000 17.0000 0 35.0000 13.0000 45.0000 22.0000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$1$$
$$13$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 325.4.a.d 1
5.b even 2 1 13.4.a.a 1
5.c odd 4 2 325.4.b.b 2
15.d odd 2 1 117.4.a.b 1
20.d odd 2 1 208.4.a.g 1
35.c odd 2 1 637.4.a.a 1
40.e odd 2 1 832.4.a.a 1
40.f even 2 1 832.4.a.r 1
55.d odd 2 1 1573.4.a.a 1
60.h even 2 1 1872.4.a.k 1
65.d even 2 1 169.4.a.e 1
65.g odd 4 2 169.4.b.a 2
65.l even 6 2 169.4.c.a 2
65.n even 6 2 169.4.c.e 2
65.s odd 12 4 169.4.e.e 4
195.e odd 2 1 1521.4.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.a.a 1 5.b even 2 1
117.4.a.b 1 15.d odd 2 1
169.4.a.e 1 65.d even 2 1
169.4.b.a 2 65.g odd 4 2
169.4.c.a 2 65.l even 6 2
169.4.c.e 2 65.n even 6 2
169.4.e.e 4 65.s odd 12 4
208.4.a.g 1 20.d odd 2 1
325.4.a.d 1 1.a even 1 1 trivial
325.4.b.b 2 5.c odd 4 2
637.4.a.a 1 35.c odd 2 1
832.4.a.a 1 40.e odd 2 1
832.4.a.r 1 40.f even 2 1
1521.4.a.a 1 195.e odd 2 1
1573.4.a.a 1 55.d odd 2 1
1872.4.a.k 1 60.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(325))$$:

 $$T_{2} - 5$$ T2 - 5 $$T_{3} - 7$$ T3 - 7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 5$$
$3$ $$T - 7$$
$5$ $$T$$
$7$ $$T - 13$$
$11$ $$T + 26$$
$13$ $$T + 13$$
$17$ $$T + 77$$
$19$ $$T + 126$$
$23$ $$T - 96$$
$29$ $$T + 82$$
$31$ $$T - 196$$
$37$ $$T - 131$$
$41$ $$T - 336$$
$43$ $$T - 201$$
$47$ $$T - 105$$
$53$ $$T - 432$$
$59$ $$T + 294$$
$61$ $$T + 56$$
$67$ $$T + 478$$
$71$ $$T - 9$$
$73$ $$T + 98$$
$79$ $$T - 1304$$
$83$ $$T - 308$$
$89$ $$T + 1190$$
$97$ $$T + 70$$