# Properties

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

# Learn more

## 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: $$\Z[a_1, \ldots, a_{17}]$$ 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.

 $$f(q)$$ $$=$$ $$q - 2 i q^{2} - 4 q^{4} + 34 i q^{7} + 8 i q^{8} +O(q^{10})$$ q - 2*i * q^2 - 4 * q^4 + 34*i * q^7 + 8*i * q^8 $$q - 2 i q^{2} - 4 q^{4} + 34 i q^{7} + 8 i q^{8} - 48 q^{11} - 70 i q^{13} + 68 q^{14} + 16 q^{16} + 27 i q^{17} - 119 q^{19} + 96 i q^{22} - 51 i q^{23} - 140 q^{26} - 136 i q^{28} + 30 q^{29} - 133 q^{31} - 32 i q^{32} + 54 q^{34} - 218 i q^{37} + 238 i q^{38} + 156 q^{41} - 88 i q^{43} + 192 q^{44} - 102 q^{46} + 516 i q^{47} - 813 q^{49} + 280 i q^{52} - 639 i q^{53} - 272 q^{56} - 60 i q^{58} + 654 q^{59} + 461 q^{61} + 266 i q^{62} - 64 q^{64} - 182 i q^{67} - 108 i q^{68} + 900 q^{71} + 704 i q^{73} - 436 q^{74} + 476 q^{76} - 1632 i q^{77} + 1375 q^{79} - 312 i q^{82} + 915 i q^{83} - 176 q^{86} - 384 i q^{88} + 1116 q^{89} + 2380 q^{91} + 204 i q^{92} + 1032 q^{94} + 16 i q^{97} + 1626 i q^{98} +O(q^{100})$$ q - 2*i * q^2 - 4 * q^4 + 34*i * q^7 + 8*i * q^8 - 48 * q^11 - 70*i * q^13 + 68 * q^14 + 16 * q^16 + 27*i * q^17 - 119 * q^19 + 96*i * q^22 - 51*i * q^23 - 140 * q^26 - 136*i * q^28 + 30 * q^29 - 133 * q^31 - 32*i * q^32 + 54 * q^34 - 218*i * q^37 + 238*i * q^38 + 156 * q^41 - 88*i * q^43 + 192 * q^44 - 102 * q^46 + 516*i * q^47 - 813 * q^49 + 280*i * q^52 - 639*i * q^53 - 272 * q^56 - 60*i * q^58 + 654 * q^59 + 461 * q^61 + 266*i * q^62 - 64 * q^64 - 182*i * q^67 - 108*i * q^68 + 900 * q^71 + 704*i * q^73 - 436 * q^74 + 476 * q^76 - 1632*i * q^77 + 1375 * q^79 - 312*i * q^82 + 915*i * q^83 - 176 * q^86 - 384*i * q^88 + 1116 * q^89 + 2380 * q^91 + 204*i * q^92 + 1032 * q^94 + 16*i * q^97 + 1626*i * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 8 q^{4}+O(q^{10})$$ 2 * q - 8 * q^4 $$2 q - 8 q^{4} - 96 q^{11} + 136 q^{14} + 32 q^{16} - 238 q^{19} - 280 q^{26} + 60 q^{29} - 266 q^{31} + 108 q^{34} + 312 q^{41} + 384 q^{44} - 204 q^{46} - 1626 q^{49} - 544 q^{56} + 1308 q^{59} + 922 q^{61} - 128 q^{64} + 1800 q^{71} - 872 q^{74} + 952 q^{76} + 2750 q^{79} - 352 q^{86} + 2232 q^{89} + 4760 q^{91} + 2064 q^{94}+O(q^{100})$$ 2 * q - 8 * q^4 - 96 * q^11 + 136 * q^14 + 32 * q^16 - 238 * q^19 - 280 * q^26 + 60 * q^29 - 266 * q^31 + 108 * q^34 + 312 * q^41 + 384 * q^44 - 204 * q^46 - 1626 * q^49 - 544 * q^56 + 1308 * q^59 + 922 * q^61 - 128 * q^64 + 1800 * q^71 - 872 * q^74 + 952 * q^76 + 2750 * q^79 - 352 * q^86 + 2232 * q^89 + 4760 * q^91 + 2064 * 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 34.0000i 8.00000i 0 0
649.2 2.00000i 0 −4.00000 0 0 34.0000i 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.c 2
3.b odd 2 1 1350.4.c.r 2
5.b even 2 1 inner 1350.4.c.c 2
5.c odd 4 1 270.4.a.k yes 1
5.c odd 4 1 1350.4.a.n 1
15.d odd 2 1 1350.4.c.r 2
15.e even 4 1 270.4.a.a 1
15.e even 4 1 1350.4.a.bb 1
20.e even 4 1 2160.4.a.t 1
45.k odd 12 2 810.4.e.d 2
45.l even 12 2 810.4.e.x 2
60.l odd 4 1 2160.4.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.4.a.a 1 15.e even 4 1
270.4.a.k yes 1 5.c odd 4 1
810.4.e.d 2 45.k odd 12 2
810.4.e.x 2 45.l even 12 2
1350.4.a.n 1 5.c odd 4 1
1350.4.a.bb 1 15.e even 4 1
1350.4.c.c 2 1.a even 1 1 trivial
1350.4.c.c 2 5.b even 2 1 inner
1350.4.c.r 2 3.b odd 2 1
1350.4.c.r 2 15.d odd 2 1
2160.4.a.j 1 60.l odd 4 1
2160.4.a.t 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 $$S_{4}^{\mathrm{new}}(1350, [\chi])$$:

 $$T_{7}^{2} + 1156$$ T7^2 + 1156 $$T_{11} + 48$$ T11 + 48

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 4$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 1156$$
$11$ $$(T + 48)^{2}$$
$13$ $$T^{2} + 4900$$
$17$ $$T^{2} + 729$$
$19$ $$(T + 119)^{2}$$
$23$ $$T^{2} + 2601$$
$29$ $$(T - 30)^{2}$$
$31$ $$(T + 133)^{2}$$
$37$ $$T^{2} + 47524$$
$41$ $$(T - 156)^{2}$$
$43$ $$T^{2} + 7744$$
$47$ $$T^{2} + 266256$$
$53$ $$T^{2} + 408321$$
$59$ $$(T - 654)^{2}$$
$61$ $$(T - 461)^{2}$$
$67$ $$T^{2} + 33124$$
$71$ $$(T - 900)^{2}$$
$73$ $$T^{2} + 495616$$
$79$ $$(T - 1375)^{2}$$
$83$ $$T^{2} + 837225$$
$89$ $$(T - 1116)^{2}$$
$97$ $$T^{2} + 256$$
show more
show less