# Properties

 Label 1350.4.c.o 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: 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.

 $$f(q)$$ $$=$$ $$q + 2 i q^{2} - 4 q^{4} + 13 i q^{7} - 8 i q^{8} +O(q^{10})$$ q + 2*i * q^2 - 4 * q^4 + 13*i * q^7 - 8*i * q^8 $$q + 2 i q^{2} - 4 q^{4} + 13 i q^{7} - 8 i q^{8} + 30 q^{11} - 61 i q^{13} - 26 q^{14} + 16 q^{16} + 12 i q^{17} + 49 q^{19} + 60 i q^{22} - 18 i q^{23} + 122 q^{26} - 52 i q^{28} - 186 q^{29} - 160 q^{31} + 32 i q^{32} - 24 q^{34} + 91 i q^{37} + 98 i q^{38} - 378 q^{41} - 268 i q^{43} - 120 q^{44} + 36 q^{46} + 144 i q^{47} + 174 q^{49} + 244 i q^{52} - 570 i q^{53} + 104 q^{56} - 372 i q^{58} + 204 q^{59} - 877 q^{61} - 320 i q^{62} - 64 q^{64} + 187 i q^{67} - 48 i q^{68} + 606 q^{71} + 431 i q^{73} - 182 q^{74} - 196 q^{76} + 390 i q^{77} - 1151 q^{79} - 756 i q^{82} - 102 i q^{83} + 536 q^{86} - 240 i q^{88} + 984 q^{89} + 793 q^{91} + 72 i q^{92} - 288 q^{94} + 265 i q^{97} + 348 i q^{98} +O(q^{100})$$ q + 2*i * q^2 - 4 * q^4 + 13*i * q^7 - 8*i * q^8 + 30 * q^11 - 61*i * q^13 - 26 * q^14 + 16 * q^16 + 12*i * q^17 + 49 * q^19 + 60*i * q^22 - 18*i * q^23 + 122 * q^26 - 52*i * q^28 - 186 * q^29 - 160 * q^31 + 32*i * q^32 - 24 * q^34 + 91*i * q^37 + 98*i * q^38 - 378 * q^41 - 268*i * q^43 - 120 * q^44 + 36 * q^46 + 144*i * q^47 + 174 * q^49 + 244*i * q^52 - 570*i * q^53 + 104 * q^56 - 372*i * q^58 + 204 * q^59 - 877 * q^61 - 320*i * q^62 - 64 * q^64 + 187*i * q^67 - 48*i * q^68 + 606 * q^71 + 431*i * q^73 - 182 * q^74 - 196 * q^76 + 390*i * q^77 - 1151 * q^79 - 756*i * q^82 - 102*i * q^83 + 536 * q^86 - 240*i * q^88 + 984 * q^89 + 793 * q^91 + 72*i * q^92 - 288 * q^94 + 265*i * q^97 + 348*i * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 8 q^{4}+O(q^{10})$$ 2 * q - 8 * q^4 $$2 q - 8 q^{4} + 60 q^{11} - 52 q^{14} + 32 q^{16} + 98 q^{19} + 244 q^{26} - 372 q^{29} - 320 q^{31} - 48 q^{34} - 756 q^{41} - 240 q^{44} + 72 q^{46} + 348 q^{49} + 208 q^{56} + 408 q^{59} - 1754 q^{61} - 128 q^{64} + 1212 q^{71} - 364 q^{74} - 392 q^{76} - 2302 q^{79} + 1072 q^{86} + 1968 q^{89} + 1586 q^{91} - 576 q^{94}+O(q^{100})$$ 2 * q - 8 * q^4 + 60 * q^11 - 52 * q^14 + 32 * q^16 + 98 * q^19 + 244 * q^26 - 372 * q^29 - 320 * q^31 - 48 * q^34 - 756 * q^41 - 240 * q^44 + 72 * q^46 + 348 * q^49 + 208 * q^56 + 408 * q^59 - 1754 * q^61 - 128 * q^64 + 1212 * q^71 - 364 * q^74 - 392 * q^76 - 2302 * q^79 + 1072 * q^86 + 1968 * q^89 + 1586 * q^91 - 576 * 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 13.0000i 8.00000i 0 0
649.2 2.00000i 0 −4.00000 0 0 13.0000i 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.o 2
3.b odd 2 1 1350.4.c.f 2
5.b even 2 1 inner 1350.4.c.o 2
5.c odd 4 1 270.4.a.d 1
5.c odd 4 1 1350.4.a.w 1
15.d odd 2 1 1350.4.c.f 2
15.e even 4 1 270.4.a.h yes 1
15.e even 4 1 1350.4.a.i 1
20.e even 4 1 2160.4.a.q 1
45.k odd 12 2 810.4.e.r 2
45.l even 12 2 810.4.e.j 2
60.l odd 4 1 2160.4.a.g 1

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

 $$T_{7}^{2} + 169$$ T7^2 + 169 $$T_{11} - 30$$ T11 - 30

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 4$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 169$$
$11$ $$(T - 30)^{2}$$
$13$ $$T^{2} + 3721$$
$17$ $$T^{2} + 144$$
$19$ $$(T - 49)^{2}$$
$23$ $$T^{2} + 324$$
$29$ $$(T + 186)^{2}$$
$31$ $$(T + 160)^{2}$$
$37$ $$T^{2} + 8281$$
$41$ $$(T + 378)^{2}$$
$43$ $$T^{2} + 71824$$
$47$ $$T^{2} + 20736$$
$53$ $$T^{2} + 324900$$
$59$ $$(T - 204)^{2}$$
$61$ $$(T + 877)^{2}$$
$67$ $$T^{2} + 34969$$
$71$ $$(T - 606)^{2}$$
$73$ $$T^{2} + 185761$$
$79$ $$(T + 1151)^{2}$$
$83$ $$T^{2} + 10404$$
$89$ $$(T - 984)^{2}$$
$97$ $$T^{2} + 70225$$