# Properties

 Label 1350.4.c.h 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + 2 i q^{2} - 4 q^{4} + 19 i q^{7} - 8 i q^{8} +O(q^{10})$$ q + 2*i * q^2 - 4 * q^4 + 19*i * q^7 - 8*i * q^8 $$q + 2 i q^{2} - 4 q^{4} + 19 i q^{7} - 8 i q^{8} - 12 q^{11} + 50 i q^{13} - 38 q^{14} + 16 q^{16} - 126 i q^{17} - 29 q^{19} - 24 i q^{22} - 18 i q^{23} - 100 q^{26} - 76 i q^{28} + 102 q^{29} - 265 q^{31} + 32 i q^{32} + 252 q^{34} - 65 i q^{37} - 58 i q^{38} - 240 q^{41} - 367 i q^{43} + 48 q^{44} + 36 q^{46} - 72 i q^{47} - 18 q^{49} - 200 i q^{52} + 636 i q^{53} + 152 q^{56} + 204 i q^{58} + 102 q^{59} - 103 q^{61} - 530 i q^{62} - 64 q^{64} + 52 i q^{67} + 504 i q^{68} + 582 q^{71} + 65 i q^{73} + 130 q^{74} + 116 q^{76} - 228 i q^{77} - 173 q^{79} - 480 i q^{82} - 498 i q^{83} + 734 q^{86} + 96 i q^{88} - 822 q^{89} - 950 q^{91} + 72 i q^{92} + 144 q^{94} - 821 i q^{97} - 36 i q^{98} +O(q^{100})$$ q + 2*i * q^2 - 4 * q^4 + 19*i * q^7 - 8*i * q^8 - 12 * q^11 + 50*i * q^13 - 38 * q^14 + 16 * q^16 - 126*i * q^17 - 29 * q^19 - 24*i * q^22 - 18*i * q^23 - 100 * q^26 - 76*i * q^28 + 102 * q^29 - 265 * q^31 + 32*i * q^32 + 252 * q^34 - 65*i * q^37 - 58*i * q^38 - 240 * q^41 - 367*i * q^43 + 48 * q^44 + 36 * q^46 - 72*i * q^47 - 18 * q^49 - 200*i * q^52 + 636*i * q^53 + 152 * q^56 + 204*i * q^58 + 102 * q^59 - 103 * q^61 - 530*i * q^62 - 64 * q^64 + 52*i * q^67 + 504*i * q^68 + 582 * q^71 + 65*i * q^73 + 130 * q^74 + 116 * q^76 - 228*i * q^77 - 173 * q^79 - 480*i * q^82 - 498*i * q^83 + 734 * q^86 + 96*i * q^88 - 822 * q^89 - 950 * q^91 + 72*i * q^92 + 144 * q^94 - 821*i * q^97 - 36*i * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 8 q^{4}+O(q^{10})$$ 2 * q - 8 * q^4 $$2 q - 8 q^{4} - 24 q^{11} - 76 q^{14} + 32 q^{16} - 58 q^{19} - 200 q^{26} + 204 q^{29} - 530 q^{31} + 504 q^{34} - 480 q^{41} + 96 q^{44} + 72 q^{46} - 36 q^{49} + 304 q^{56} + 204 q^{59} - 206 q^{61} - 128 q^{64} + 1164 q^{71} + 260 q^{74} + 232 q^{76} - 346 q^{79} + 1468 q^{86} - 1644 q^{89} - 1900 q^{91} + 288 q^{94}+O(q^{100})$$ 2 * q - 8 * q^4 - 24 * q^11 - 76 * q^14 + 32 * q^16 - 58 * q^19 - 200 * q^26 + 204 * q^29 - 530 * q^31 + 504 * q^34 - 480 * q^41 + 96 * q^44 + 72 * q^46 - 36 * q^49 + 304 * q^56 + 204 * q^59 - 206 * q^61 - 128 * q^64 + 1164 * q^71 + 260 * q^74 + 232 * q^76 - 346 * q^79 + 1468 * q^86 - 1644 * q^89 - 1900 * q^91 + 288 * 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 19.0000i 8.00000i 0 0
649.2 2.00000i 0 −4.00000 0 0 19.0000i 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.h 2
3.b odd 2 1 1350.4.c.m 2
5.b even 2 1 inner 1350.4.c.h 2
5.c odd 4 1 1350.4.a.c 1
5.c odd 4 1 1350.4.a.y yes 1
15.d odd 2 1 1350.4.c.m 2
15.e even 4 1 1350.4.a.k yes 1
15.e even 4 1 1350.4.a.q yes 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1350.4.a.c 1 5.c odd 4 1
1350.4.a.k yes 1 15.e even 4 1
1350.4.a.q yes 1 15.e even 4 1
1350.4.a.y yes 1 5.c odd 4 1
1350.4.c.h 2 1.a even 1 1 trivial
1350.4.c.h 2 5.b even 2 1 inner
1350.4.c.m 2 3.b odd 2 1
1350.4.c.m 2 15.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 $$S_{4}^{\mathrm{new}}(1350, [\chi])$$:

 $$T_{7}^{2} + 361$$ T7^2 + 361 $$T_{11} + 12$$ T11 + 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 4$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 361$$
$11$ $$(T + 12)^{2}$$
$13$ $$T^{2} + 2500$$
$17$ $$T^{2} + 15876$$
$19$ $$(T + 29)^{2}$$
$23$ $$T^{2} + 324$$
$29$ $$(T - 102)^{2}$$
$31$ $$(T + 265)^{2}$$
$37$ $$T^{2} + 4225$$
$41$ $$(T + 240)^{2}$$
$43$ $$T^{2} + 134689$$
$47$ $$T^{2} + 5184$$
$53$ $$T^{2} + 404496$$
$59$ $$(T - 102)^{2}$$
$61$ $$(T + 103)^{2}$$
$67$ $$T^{2} + 2704$$
$71$ $$(T - 582)^{2}$$
$73$ $$T^{2} + 4225$$
$79$ $$(T + 173)^{2}$$
$83$ $$T^{2} + 248004$$
$89$ $$(T + 822)^{2}$$
$97$ $$T^{2} + 674041$$