# Properties

 Label 1350.4.c.k Level $1350$ Weight $4$ Character orbit 1350.c Analytic conductor $79.653$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1350 = 2 \cdot 3^{3} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1350.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$79.6525785077$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 270) 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.

 $$f(q)$$ $$=$$ $$q - 2 i q^{2} - 4 q^{4} + 14 i q^{7} + 8 i q^{8} +O(q^{10})$$ q - 2*i * q^2 - 4 * q^4 + 14*i * q^7 + 8*i * q^8 $$q - 2 i q^{2} - 4 q^{4} + 14 i q^{7} + 8 i q^{8} + 3 q^{11} - 47 i q^{13} + 28 q^{14} + 16 q^{16} - 39 i q^{17} - 32 q^{19} - 6 i q^{22} + 99 i q^{23} - 94 q^{26} - 56 i q^{28} - 51 q^{29} + 83 q^{31} - 32 i q^{32} - 78 q^{34} + 314 i q^{37} + 64 i q^{38} - 108 q^{41} - 299 i q^{43} - 12 q^{44} + 198 q^{46} + 531 i q^{47} + 147 q^{49} + 188 i q^{52} - 564 i q^{53} - 112 q^{56} + 102 i q^{58} - 12 q^{59} + 230 q^{61} - 166 i q^{62} - 64 q^{64} - 268 i q^{67} + 156 i q^{68} + 120 q^{71} - 1106 i q^{73} + 628 q^{74} + 128 q^{76} + 42 i q^{77} + 739 q^{79} + 216 i q^{82} - 1086 i q^{83} - 598 q^{86} + 24 i q^{88} + 120 q^{89} + 658 q^{91} - 396 i q^{92} + 1062 q^{94} - 1642 i q^{97} - 294 i q^{98} +O(q^{100})$$ q - 2*i * q^2 - 4 * q^4 + 14*i * q^7 + 8*i * q^8 + 3 * q^11 - 47*i * q^13 + 28 * q^14 + 16 * q^16 - 39*i * q^17 - 32 * q^19 - 6*i * q^22 + 99*i * q^23 - 94 * q^26 - 56*i * q^28 - 51 * q^29 + 83 * q^31 - 32*i * q^32 - 78 * q^34 + 314*i * q^37 + 64*i * q^38 - 108 * q^41 - 299*i * q^43 - 12 * q^44 + 198 * q^46 + 531*i * q^47 + 147 * q^49 + 188*i * q^52 - 564*i * q^53 - 112 * q^56 + 102*i * q^58 - 12 * q^59 + 230 * q^61 - 166*i * q^62 - 64 * q^64 - 268*i * q^67 + 156*i * q^68 + 120 * q^71 - 1106*i * q^73 + 628 * q^74 + 128 * q^76 + 42*i * q^77 + 739 * q^79 + 216*i * q^82 - 1086*i * q^83 - 598 * q^86 + 24*i * q^88 + 120 * q^89 + 658 * q^91 - 396*i * q^92 + 1062 * q^94 - 1642*i * q^97 - 294*i * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 8 q^{4}+O(q^{10})$$ 2 * q - 8 * q^4 $$2 q - 8 q^{4} + 6 q^{11} + 56 q^{14} + 32 q^{16} - 64 q^{19} - 188 q^{26} - 102 q^{29} + 166 q^{31} - 156 q^{34} - 216 q^{41} - 24 q^{44} + 396 q^{46} + 294 q^{49} - 224 q^{56} - 24 q^{59} + 460 q^{61} - 128 q^{64} + 240 q^{71} + 1256 q^{74} + 256 q^{76} + 1478 q^{79} - 1196 q^{86} + 240 q^{89} + 1316 q^{91} + 2124 q^{94}+O(q^{100})$$ 2 * q - 8 * q^4 + 6 * q^11 + 56 * q^14 + 32 * q^16 - 64 * q^19 - 188 * q^26 - 102 * q^29 + 166 * q^31 - 156 * q^34 - 216 * q^41 - 24 * q^44 + 396 * q^46 + 294 * q^49 - 224 * q^56 - 24 * q^59 + 460 * q^61 - 128 * q^64 + 240 * q^71 + 1256 * q^74 + 256 * q^76 + 1478 * q^79 - 1196 * q^86 + 240 * q^89 + 1316 * q^91 + 2124 * q^94

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times$$.

 $$n$$ $$1001$$ $$1027$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
649.1
 1.00000i − 1.00000i
2.00000i 0 −4.00000 0 0 14.0000i 8.00000i 0 0
649.2 2.00000i 0 −4.00000 0 0 14.0000i 8.00000i 0 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
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1350.4.c.k 2
3.b odd 2 1 1350.4.c.j 2
5.b even 2 1 inner 1350.4.c.k 2
5.c odd 4 1 270.4.a.f 1
5.c odd 4 1 1350.4.a.r 1
15.d odd 2 1 1350.4.c.j 2
15.e even 4 1 270.4.a.j yes 1
15.e even 4 1 1350.4.a.e 1
20.e even 4 1 2160.4.a.l 1
45.k odd 12 2 810.4.e.n 2
45.l even 12 2 810.4.e.f 2
60.l odd 4 1 2160.4.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.4.a.f 1 5.c odd 4 1
270.4.a.j yes 1 15.e even 4 1
810.4.e.f 2 45.l even 12 2
810.4.e.n 2 45.k odd 12 2
1350.4.a.e 1 15.e even 4 1
1350.4.a.r 1 5.c odd 4 1
1350.4.c.j 2 3.b odd 2 1
1350.4.c.j 2 15.d odd 2 1
1350.4.c.k 2 1.a even 1 1 trivial
1350.4.c.k 2 5.b even 2 1 inner
2160.4.a.b 1 60.l odd 4 1
2160.4.a.l 1 20.e even 4 1

## Hecke kernels

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

 $$T_{7}^{2} + 196$$ T7^2 + 196 $$T_{11} - 3$$ T11 - 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 4$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 196$$
$11$ $$(T - 3)^{2}$$
$13$ $$T^{2} + 2209$$
$17$ $$T^{2} + 1521$$
$19$ $$(T + 32)^{2}$$
$23$ $$T^{2} + 9801$$
$29$ $$(T + 51)^{2}$$
$31$ $$(T - 83)^{2}$$
$37$ $$T^{2} + 98596$$
$41$ $$(T + 108)^{2}$$
$43$ $$T^{2} + 89401$$
$47$ $$T^{2} + 281961$$
$53$ $$T^{2} + 318096$$
$59$ $$(T + 12)^{2}$$
$61$ $$(T - 230)^{2}$$
$67$ $$T^{2} + 71824$$
$71$ $$(T - 120)^{2}$$
$73$ $$T^{2} + 1223236$$
$79$ $$(T - 739)^{2}$$
$83$ $$T^{2} + 1179396$$
$89$ $$(T - 120)^{2}$$
$97$ $$T^{2} + 2696164$$