# Properties

 Label 1350.4.a.r Level $1350$ Weight $4$ Character orbit 1350.a Self dual yes Analytic conductor $79.653$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1350 = 2 \cdot 3^{3} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1350.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$79.6525785077$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 270) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 2 q^{2} + 4 q^{4} - 14 q^{7} + 8 q^{8}+O(q^{10})$$ q + 2 * q^2 + 4 * q^4 - 14 * q^7 + 8 * q^8 $$q + 2 q^{2} + 4 q^{4} - 14 q^{7} + 8 q^{8} + 3 q^{11} - 47 q^{13} - 28 q^{14} + 16 q^{16} + 39 q^{17} + 32 q^{19} + 6 q^{22} + 99 q^{23} - 94 q^{26} - 56 q^{28} + 51 q^{29} + 83 q^{31} + 32 q^{32} + 78 q^{34} - 314 q^{37} + 64 q^{38} - 108 q^{41} - 299 q^{43} + 12 q^{44} + 198 q^{46} - 531 q^{47} - 147 q^{49} - 188 q^{52} - 564 q^{53} - 112 q^{56} + 102 q^{58} + 12 q^{59} + 230 q^{61} + 166 q^{62} + 64 q^{64} + 268 q^{67} + 156 q^{68} + 120 q^{71} - 1106 q^{73} - 628 q^{74} + 128 q^{76} - 42 q^{77} - 739 q^{79} - 216 q^{82} - 1086 q^{83} - 598 q^{86} + 24 q^{88} - 120 q^{89} + 658 q^{91} + 396 q^{92} - 1062 q^{94} + 1642 q^{97} - 294 q^{98}+O(q^{100})$$ q + 2 * q^2 + 4 * q^4 - 14 * q^7 + 8 * q^8 + 3 * q^11 - 47 * q^13 - 28 * q^14 + 16 * q^16 + 39 * q^17 + 32 * q^19 + 6 * q^22 + 99 * q^23 - 94 * q^26 - 56 * q^28 + 51 * q^29 + 83 * q^31 + 32 * q^32 + 78 * q^34 - 314 * q^37 + 64 * q^38 - 108 * q^41 - 299 * q^43 + 12 * q^44 + 198 * q^46 - 531 * q^47 - 147 * q^49 - 188 * q^52 - 564 * q^53 - 112 * q^56 + 102 * q^58 + 12 * q^59 + 230 * q^61 + 166 * q^62 + 64 * q^64 + 268 * q^67 + 156 * q^68 + 120 * q^71 - 1106 * q^73 - 628 * q^74 + 128 * q^76 - 42 * q^77 - 739 * q^79 - 216 * q^82 - 1086 * q^83 - 598 * q^86 + 24 * q^88 - 120 * q^89 + 658 * q^91 + 396 * q^92 - 1062 * q^94 + 1642 * q^97 - 294 * q^98

## 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
2.00000 0 4.00000 0 0 −14.0000 8.00000 0 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
$$2$$ $$-1$$
$$3$$ $$1$$
$$5$$ $$1$$

## 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 1350.4.a.r 1
3.b odd 2 1 1350.4.a.e 1
5.b even 2 1 270.4.a.f 1
5.c odd 4 2 1350.4.c.k 2
15.d odd 2 1 270.4.a.j yes 1
15.e even 4 2 1350.4.c.j 2
20.d odd 2 1 2160.4.a.l 1
45.h odd 6 2 810.4.e.f 2
45.j even 6 2 810.4.e.n 2
60.h even 2 1 2160.4.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.4.a.f 1 5.b even 2 1
270.4.a.j yes 1 15.d odd 2 1
810.4.e.f 2 45.h odd 6 2
810.4.e.n 2 45.j even 6 2
1350.4.a.e 1 3.b odd 2 1
1350.4.a.r 1 1.a even 1 1 trivial
1350.4.c.j 2 15.e even 4 2
1350.4.c.k 2 5.c odd 4 2
2160.4.a.b 1 60.h even 2 1
2160.4.a.l 1 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1350))$$:

 $$T_{7} + 14$$ T7 + 14 $$T_{11} - 3$$ T11 - 3 $$T_{17} - 39$$ T17 - 39

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 2$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T + 14$$
$11$ $$T - 3$$
$13$ $$T + 47$$
$17$ $$T - 39$$
$19$ $$T - 32$$
$23$ $$T - 99$$
$29$ $$T - 51$$
$31$ $$T - 83$$
$37$ $$T + 314$$
$41$ $$T + 108$$
$43$ $$T + 299$$
$47$ $$T + 531$$
$53$ $$T + 564$$
$59$ $$T - 12$$
$61$ $$T - 230$$
$67$ $$T - 268$$
$71$ $$T - 120$$
$73$ $$T + 1106$$
$79$ $$T + 739$$
$83$ $$T + 1086$$
$89$ $$T + 120$$
$97$ $$T - 1642$$