# Properties

 Label 1350.4.c.t 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 54) 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.

 $$f(q)$$ $$=$$ $$q - 2 i q^{2} - 4 q^{4} + 7 i q^{7} + 8 i q^{8} +O(q^{10})$$ q - 2*i * q^2 - 4 * q^4 + 7*i * q^7 + 8*i * q^8 $$q - 2 i q^{2} - 4 q^{4} + 7 i q^{7} + 8 i q^{8} + 60 q^{11} - 79 i q^{13} + 14 q^{14} + 16 q^{16} + 108 i q^{17} - 11 q^{19} - 120 i q^{22} - 132 i q^{23} - 158 q^{26} - 28 i q^{28} - 96 q^{29} + 20 q^{31} - 32 i q^{32} + 216 q^{34} + 169 i q^{37} + 22 i q^{38} + 192 q^{41} + 488 i q^{43} - 240 q^{44} - 264 q^{46} - 204 i q^{47} + 294 q^{49} + 316 i q^{52} + 360 i q^{53} - 56 q^{56} + 192 i q^{58} - 156 q^{59} + 83 q^{61} - 40 i q^{62} - 64 q^{64} - 47 i q^{67} - 432 i q^{68} + 216 q^{71} - 511 i q^{73} + 338 q^{74} + 44 q^{76} + 420 i q^{77} + 529 q^{79} - 384 i q^{82} - 1128 i q^{83} + 976 q^{86} + 480 i q^{88} - 36 q^{89} + 553 q^{91} + 528 i q^{92} - 408 q^{94} - 605 i q^{97} - 588 i q^{98} +O(q^{100})$$ q - 2*i * q^2 - 4 * q^4 + 7*i * q^7 + 8*i * q^8 + 60 * q^11 - 79*i * q^13 + 14 * q^14 + 16 * q^16 + 108*i * q^17 - 11 * q^19 - 120*i * q^22 - 132*i * q^23 - 158 * q^26 - 28*i * q^28 - 96 * q^29 + 20 * q^31 - 32*i * q^32 + 216 * q^34 + 169*i * q^37 + 22*i * q^38 + 192 * q^41 + 488*i * q^43 - 240 * q^44 - 264 * q^46 - 204*i * q^47 + 294 * q^49 + 316*i * q^52 + 360*i * q^53 - 56 * q^56 + 192*i * q^58 - 156 * q^59 + 83 * q^61 - 40*i * q^62 - 64 * q^64 - 47*i * q^67 - 432*i * q^68 + 216 * q^71 - 511*i * q^73 + 338 * q^74 + 44 * q^76 + 420*i * q^77 + 529 * q^79 - 384*i * q^82 - 1128*i * q^83 + 976 * q^86 + 480*i * q^88 - 36 * q^89 + 553 * q^91 + 528*i * q^92 - 408 * q^94 - 605*i * q^97 - 588*i * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 8 q^{4}+O(q^{10})$$ 2 * q - 8 * q^4 $$2 q - 8 q^{4} + 120 q^{11} + 28 q^{14} + 32 q^{16} - 22 q^{19} - 316 q^{26} - 192 q^{29} + 40 q^{31} + 432 q^{34} + 384 q^{41} - 480 q^{44} - 528 q^{46} + 588 q^{49} - 112 q^{56} - 312 q^{59} + 166 q^{61} - 128 q^{64} + 432 q^{71} + 676 q^{74} + 88 q^{76} + 1058 q^{79} + 1952 q^{86} - 72 q^{89} + 1106 q^{91} - 816 q^{94}+O(q^{100})$$ 2 * q - 8 * q^4 + 120 * q^11 + 28 * q^14 + 32 * q^16 - 22 * q^19 - 316 * q^26 - 192 * q^29 + 40 * q^31 + 432 * q^34 + 384 * q^41 - 480 * q^44 - 528 * q^46 + 588 * q^49 - 112 * q^56 - 312 * q^59 + 166 * q^61 - 128 * q^64 + 432 * q^71 + 676 * q^74 + 88 * q^76 + 1058 * q^79 + 1952 * q^86 - 72 * q^89 + 1106 * q^91 - 816 * 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 7.00000i 8.00000i 0 0
649.2 2.00000i 0 −4.00000 0 0 7.00000i 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.t 2
3.b odd 2 1 1350.4.c.a 2
5.b even 2 1 inner 1350.4.c.t 2
5.c odd 4 1 54.4.a.d yes 1
5.c odd 4 1 1350.4.a.h 1
15.d odd 2 1 1350.4.c.a 2
15.e even 4 1 54.4.a.a 1
15.e even 4 1 1350.4.a.v 1
20.e even 4 1 432.4.a.m 1
40.i odd 4 1 1728.4.a.e 1
40.k even 4 1 1728.4.a.f 1
45.k odd 12 2 162.4.c.a 2
45.l even 12 2 162.4.c.h 2
60.l odd 4 1 432.4.a.b 1
120.q odd 4 1 1728.4.a.bb 1
120.w even 4 1 1728.4.a.ba 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.4.a.a 1 15.e even 4 1
54.4.a.d yes 1 5.c odd 4 1
162.4.c.a 2 45.k odd 12 2
162.4.c.h 2 45.l even 12 2
432.4.a.b 1 60.l odd 4 1
432.4.a.m 1 20.e even 4 1
1350.4.a.h 1 5.c odd 4 1
1350.4.a.v 1 15.e even 4 1
1350.4.c.a 2 3.b odd 2 1
1350.4.c.a 2 15.d odd 2 1
1350.4.c.t 2 1.a even 1 1 trivial
1350.4.c.t 2 5.b even 2 1 inner
1728.4.a.e 1 40.i odd 4 1
1728.4.a.f 1 40.k even 4 1
1728.4.a.ba 1 120.w even 4 1
1728.4.a.bb 1 120.q odd 4 1

## Hecke kernels

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

 $$T_{7}^{2} + 49$$ T7^2 + 49 $$T_{11} - 60$$ T11 - 60

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 4$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 49$$
$11$ $$(T - 60)^{2}$$
$13$ $$T^{2} + 6241$$
$17$ $$T^{2} + 11664$$
$19$ $$(T + 11)^{2}$$
$23$ $$T^{2} + 17424$$
$29$ $$(T + 96)^{2}$$
$31$ $$(T - 20)^{2}$$
$37$ $$T^{2} + 28561$$
$41$ $$(T - 192)^{2}$$
$43$ $$T^{2} + 238144$$
$47$ $$T^{2} + 41616$$
$53$ $$T^{2} + 129600$$
$59$ $$(T + 156)^{2}$$
$61$ $$(T - 83)^{2}$$
$67$ $$T^{2} + 2209$$
$71$ $$(T - 216)^{2}$$
$73$ $$T^{2} + 261121$$
$79$ $$(T - 529)^{2}$$
$83$ $$T^{2} + 1272384$$
$89$ $$(T + 36)^{2}$$
$97$ $$T^{2} + 366025$$