# Properties

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

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## 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.

 $$f(q)$$ $$=$$ $$q - 2 i q^{2} - 4 q^{4} + 23 i q^{7} + 8 i q^{8} +O(q^{10})$$ q - 2*i * q^2 - 4 * q^4 + 23*i * q^7 + 8*i * q^8 $$q - 2 i q^{2} - 4 q^{4} + 23 i q^{7} + 8 i q^{8} + 30 q^{11} + 34 i q^{13} + 46 q^{14} + 16 q^{16} - 42 i q^{17} + 139 q^{19} - 60 i q^{22} - 192 i q^{23} + 68 q^{26} - 92 i q^{28} - 234 q^{29} - 55 q^{31} - 32 i q^{32} - 84 q^{34} + 191 i q^{37} - 278 i q^{38} + 138 q^{41} - 53 i q^{43} - 120 q^{44} - 384 q^{46} + 366 i q^{47} - 186 q^{49} - 136 i q^{52} + 330 i q^{53} - 184 q^{56} + 468 i q^{58} + 396 q^{59} + 23 q^{61} + 110 i q^{62} - 64 q^{64} + 452 i q^{67} + 168 i q^{68} + 204 q^{71} + 691 i q^{73} + 382 q^{74} - 556 q^{76} + 690 i q^{77} + 709 q^{79} - 276 i q^{82} - 1098 i q^{83} - 106 q^{86} + 240 i q^{88} + 816 q^{89} - 782 q^{91} + 768 i q^{92} + 732 q^{94} + 905 i q^{97} + 372 i q^{98} +O(q^{100})$$ q - 2*i * q^2 - 4 * q^4 + 23*i * q^7 + 8*i * q^8 + 30 * q^11 + 34*i * q^13 + 46 * q^14 + 16 * q^16 - 42*i * q^17 + 139 * q^19 - 60*i * q^22 - 192*i * q^23 + 68 * q^26 - 92*i * q^28 - 234 * q^29 - 55 * q^31 - 32*i * q^32 - 84 * q^34 + 191*i * q^37 - 278*i * q^38 + 138 * q^41 - 53*i * q^43 - 120 * q^44 - 384 * q^46 + 366*i * q^47 - 186 * q^49 - 136*i * q^52 + 330*i * q^53 - 184 * q^56 + 468*i * q^58 + 396 * q^59 + 23 * q^61 + 110*i * q^62 - 64 * q^64 + 452*i * q^67 + 168*i * q^68 + 204 * q^71 + 691*i * q^73 + 382 * q^74 - 556 * q^76 + 690*i * q^77 + 709 * q^79 - 276*i * q^82 - 1098*i * q^83 - 106 * q^86 + 240*i * q^88 + 816 * q^89 - 782 * q^91 + 768*i * q^92 + 732 * q^94 + 905*i * q^97 + 372*i * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 8 q^{4}+O(q^{10})$$ 2 * q - 8 * q^4 $$2 q - 8 q^{4} + 60 q^{11} + 92 q^{14} + 32 q^{16} + 278 q^{19} + 136 q^{26} - 468 q^{29} - 110 q^{31} - 168 q^{34} + 276 q^{41} - 240 q^{44} - 768 q^{46} - 372 q^{49} - 368 q^{56} + 792 q^{59} + 46 q^{61} - 128 q^{64} + 408 q^{71} + 764 q^{74} - 1112 q^{76} + 1418 q^{79} - 212 q^{86} + 1632 q^{89} - 1564 q^{91} + 1464 q^{94}+O(q^{100})$$ 2 * q - 8 * q^4 + 60 * q^11 + 92 * q^14 + 32 * q^16 + 278 * q^19 + 136 * q^26 - 468 * q^29 - 110 * q^31 - 168 * q^34 + 276 * q^41 - 240 * q^44 - 768 * q^46 - 372 * q^49 - 368 * q^56 + 792 * q^59 + 46 * q^61 - 128 * q^64 + 408 * q^71 + 764 * q^74 - 1112 * q^76 + 1418 * q^79 - 212 * q^86 + 1632 * q^89 - 1564 * q^91 + 1464 * 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 23.0000i 8.00000i 0 0
649.2 2.00000i 0 −4.00000 0 0 23.0000i 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.p 2
3.b odd 2 1 1350.4.c.e 2
5.b even 2 1 inner 1350.4.c.p 2
5.c odd 4 1 1350.4.a.m yes 1
5.c odd 4 1 1350.4.a.p yes 1
15.d odd 2 1 1350.4.c.e 2
15.e even 4 1 1350.4.a.b 1
15.e even 4 1 1350.4.a.ba yes 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1350.4.a.b 1 15.e even 4 1
1350.4.a.m yes 1 5.c odd 4 1
1350.4.a.p yes 1 5.c odd 4 1
1350.4.a.ba yes 1 15.e even 4 1
1350.4.c.e 2 3.b odd 2 1
1350.4.c.e 2 15.d odd 2 1
1350.4.c.p 2 1.a even 1 1 trivial
1350.4.c.p 2 5.b even 2 1 inner

## Hecke kernels

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

 $$T_{7}^{2} + 529$$ T7^2 + 529 $$T_{11} - 30$$ T11 - 30

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 4$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 529$$
$11$ $$(T - 30)^{2}$$
$13$ $$T^{2} + 1156$$
$17$ $$T^{2} + 1764$$
$19$ $$(T - 139)^{2}$$
$23$ $$T^{2} + 36864$$
$29$ $$(T + 234)^{2}$$
$31$ $$(T + 55)^{2}$$
$37$ $$T^{2} + 36481$$
$41$ $$(T - 138)^{2}$$
$43$ $$T^{2} + 2809$$
$47$ $$T^{2} + 133956$$
$53$ $$T^{2} + 108900$$
$59$ $$(T - 396)^{2}$$
$61$ $$(T - 23)^{2}$$
$67$ $$T^{2} + 204304$$
$71$ $$(T - 204)^{2}$$
$73$ $$T^{2} + 477481$$
$79$ $$(T - 709)^{2}$$
$83$ $$T^{2} + 1205604$$
$89$ $$(T - 816)^{2}$$
$97$ $$T^{2} + 819025$$
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