# Properties

 Label 1350.4.c.y Level $1350$ Weight $4$ Character orbit 1350.c Analytic conductor $79.653$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(i, \sqrt{209})$$ Defining polynomial: $$x^{4} + 105x^{2} + 2704$$ x^4 + 105*x^2 + 2704 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 2 \beta_1 q^{2} - 4 q^{4} + ( - \beta_{2} - 7 \beta_1) q^{7} + 8 \beta_1 q^{8}+O(q^{10})$$ q - 2*b1 * q^2 - 4 * q^4 + (-b2 - 7*b1) * q^7 + 8*b1 * q^8 $$q - 2 \beta_1 q^{2} - 4 q^{4} + ( - \beta_{2} - 7 \beta_1) q^{7} + 8 \beta_1 q^{8} + (2 \beta_{3} + 8) q^{11} + (\beta_{2} + 4 \beta_1) q^{13} + ( - 2 \beta_{3} - 12) q^{14} + 16 q^{16} + (2 \beta_{2} - 2 \beta_1) q^{17} + (\beta_{3} - 22) q^{19} + ( - 4 \beta_{2} - 20 \beta_1) q^{22} + (4 \beta_{2} + 68 \beta_1) q^{23} + (2 \beta_{3} + 6) q^{26} + (4 \beta_{2} + 28 \beta_1) q^{28} + (2 \beta_{3} - 16) q^{29} + (2 \beta_{3} + 73) q^{31} - 32 \beta_1 q^{32} + (4 \beta_{3} - 8) q^{34} + (12 \beta_{2} + 121 \beta_1) q^{37} + ( - 2 \beta_{2} + 42 \beta_1) q^{38} + (6 \beta_{3} + 12) q^{41} + ( - 14 \beta_{2} + 7 \beta_1) q^{43} + ( - 8 \beta_{3} - 32) q^{44} + (8 \beta_{3} + 128) q^{46} + (10 \beta_{2} - 154 \beta_1) q^{47} + ( - 13 \beta_{3} - 163) q^{49} + ( - 4 \beta_{2} - 16 \beta_1) q^{52} + (14 \beta_{2} + 430 \beta_1) q^{53} + (8 \beta_{3} + 48) q^{56} + ( - 4 \beta_{2} + 28 \beta_1) q^{58} + (8 \beta_{3} + 8) q^{59} + ( - 27 \beta_{3} + 320) q^{61} + ( - 4 \beta_{2} - 150 \beta_1) q^{62} - 64 q^{64} + (15 \beta_{2} + 622 \beta_1) q^{67} + ( - 8 \beta_{2} + 8 \beta_1) q^{68} + ( - 36 \beta_{3} - 60) q^{71} + (12 \beta_{2} + 701 \beta_1) q^{73} + (24 \beta_{3} + 218) q^{74} + ( - 4 \beta_{3} + 88) q^{76} + ( - 22 \beta_{2} - 1010 \beta_1) q^{77} + 259 q^{79} + ( - 12 \beta_{2} - 36 \beta_1) q^{82} + ( - 62 \beta_{2} + 2 \beta_1) q^{83} + ( - 28 \beta_{3} + 42) q^{86} + (16 \beta_{2} + 80 \beta_1) q^{88} + ( - 24 \beta_{3} + 780) q^{89} + (10 \beta_{3} + 488) q^{91} + ( - 16 \beta_{2} - 272 \beta_1) q^{92} + (20 \beta_{3} - 328) q^{94} + ( - 23 \beta_{2} + 525 \beta_1) q^{97} + (26 \beta_{2} + 352 \beta_1) q^{98}+O(q^{100})$$ q - 2*b1 * q^2 - 4 * q^4 + (-b2 - 7*b1) * q^7 + 8*b1 * q^8 + (2*b3 + 8) * q^11 + (b2 + 4*b1) * q^13 + (-2*b3 - 12) * q^14 + 16 * q^16 + (2*b2 - 2*b1) * q^17 + (b3 - 22) * q^19 + (-4*b2 - 20*b1) * q^22 + (4*b2 + 68*b1) * q^23 + (2*b3 + 6) * q^26 + (4*b2 + 28*b1) * q^28 + (2*b3 - 16) * q^29 + (2*b3 + 73) * q^31 - 32*b1 * q^32 + (4*b3 - 8) * q^34 + (12*b2 + 121*b1) * q^37 + (-2*b2 + 42*b1) * q^38 + (6*b3 + 12) * q^41 + (-14*b2 + 7*b1) * q^43 + (-8*b3 - 32) * q^44 + (8*b3 + 128) * q^46 + (10*b2 - 154*b1) * q^47 + (-13*b3 - 163) * q^49 + (-4*b2 - 16*b1) * q^52 + (14*b2 + 430*b1) * q^53 + (8*b3 + 48) * q^56 + (-4*b2 + 28*b1) * q^58 + (8*b3 + 8) * q^59 + (-27*b3 + 320) * q^61 + (-4*b2 - 150*b1) * q^62 - 64 * q^64 + (15*b2 + 622*b1) * q^67 + (-8*b2 + 8*b1) * q^68 + (-36*b3 - 60) * q^71 + (12*b2 + 701*b1) * q^73 + (24*b3 + 218) * q^74 + (-4*b3 + 88) * q^76 + (-22*b2 - 1010*b1) * q^77 + 259 * q^79 + (-12*b2 - 36*b1) * q^82 + (-62*b2 + 2*b1) * q^83 + (-28*b3 + 42) * q^86 + (16*b2 + 80*b1) * q^88 + (-24*b3 + 780) * q^89 + (10*b3 + 488) * q^91 + (-16*b2 - 272*b1) * q^92 + (20*b3 - 328) * q^94 + (-23*b2 + 525*b1) * q^97 + (26*b2 + 352*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 16 q^{4}+O(q^{10})$$ 4 * q - 16 * q^4 $$4 q - 16 q^{4} + 36 q^{11} - 52 q^{14} + 64 q^{16} - 86 q^{19} + 28 q^{26} - 60 q^{29} + 296 q^{31} - 24 q^{34} + 60 q^{41} - 144 q^{44} + 528 q^{46} - 678 q^{49} + 208 q^{56} + 48 q^{59} + 1226 q^{61} - 256 q^{64} - 312 q^{71} + 920 q^{74} + 344 q^{76} + 1036 q^{79} + 112 q^{86} + 3072 q^{89} + 1972 q^{91} - 1272 q^{94}+O(q^{100})$$ 4 * q - 16 * q^4 + 36 * q^11 - 52 * q^14 + 64 * q^16 - 86 * q^19 + 28 * q^26 - 60 * q^29 + 296 * q^31 - 24 * q^34 + 60 * q^41 - 144 * q^44 + 528 * q^46 - 678 * q^49 + 208 * q^56 + 48 * q^59 + 1226 * q^61 - 256 * q^64 - 312 * q^71 + 920 * q^74 + 344 * q^76 + 1036 * q^79 + 112 * q^86 + 3072 * q^89 + 1972 * q^91 - 1272 * q^94

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 105x^{2} + 2704$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 53\nu ) / 52$$ (v^3 + 53*v) / 52 $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 209\nu ) / 52$$ (v^3 + 209*v) / 52 $$\beta_{3}$$ $$=$$ $$3\nu^{2} + 158$$ 3*v^2 + 158
 $$\nu$$ $$=$$ $$( \beta_{2} - \beta_1 ) / 3$$ (b2 - b1) / 3 $$\nu^{2}$$ $$=$$ $$( \beta_{3} - 158 ) / 3$$ (b3 - 158) / 3 $$\nu^{3}$$ $$=$$ $$( -53\beta_{2} + 209\beta_1 ) / 3$$ (-53*b2 + 209*b1) / 3

## 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
 6.72842i − 7.72842i 7.72842i − 6.72842i
2.00000i 0 −4.00000 0 0 28.1852i 8.00000i 0 0
649.2 2.00000i 0 −4.00000 0 0 15.1852i 8.00000i 0 0
649.3 2.00000i 0 −4.00000 0 0 15.1852i 8.00000i 0 0
649.4 2.00000i 0 −4.00000 0 0 28.1852i 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.y 4
3.b odd 2 1 1350.4.c.x 4
5.b even 2 1 inner 1350.4.c.y 4
5.c odd 4 1 1350.4.a.bd 2
5.c odd 4 1 1350.4.a.bo yes 2
15.d odd 2 1 1350.4.c.x 4
15.e even 4 1 1350.4.a.bh yes 2
15.e even 4 1 1350.4.a.bk yes 2

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

 $$T_{7}^{4} + 1025T_{7}^{2} + 183184$$ T7^4 + 1025*T7^2 + 183184 $$T_{11}^{2} - 18T_{11} - 1800$$ T11^2 - 18*T11 - 1800

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 4)^{2}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 1025 T^{2} + 183184$$
$11$ $$(T^{2} - 18 T - 1800)^{2}$$
$13$ $$T^{4} + 965 T^{2} + 209764$$
$17$ $$T^{4} + 3780 T^{2} + \cdots + 3504384$$
$19$ $$(T^{2} + 43 T - 8)^{2}$$
$23$ $$T^{4} + 23760 T^{2} + \cdots + 10036224$$
$29$ $$(T^{2} + 30 T - 1656)^{2}$$
$31$ $$(T^{2} - 148 T + 3595)^{2}$$
$37$ $$T^{4} + 161882 T^{2} + \cdots + 2969269081$$
$41$ $$(T^{2} - 30 T - 16704)^{2}$$
$43$ $$T^{4} + 184730 T^{2} + \cdots + 8459032729$$
$47$ $$T^{4} + 144612 T^{2} + \cdots + 472801536$$
$53$ $$T^{4} + 542196 T^{2} + \cdots + 7527297600$$
$59$ $$(T^{2} - 24 T - 29952)^{2}$$
$61$ $$(T^{2} - 613 T - 248870)^{2}$$
$67$ $$T^{4} + 966833 T^{2} + \cdots + 73877414416$$
$71$ $$(T^{2} + 156 T - 603360)^{2}$$
$73$ $$T^{4} + 1101482 T^{2} + \cdots + 172481565481$$
$79$ $$(T - 259)^{4}$$
$83$ $$T^{4} + 3617460 T^{2} + \cdots + 3263630128704$$
$89$ $$(T^{2} - 1536 T + 318960)^{2}$$
$97$ $$T^{4} + 1073189 T^{2} + \cdots + 1526464900$$
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