# Properties

 Label 1350.4.c.z Level $1350$ Weight $4$ Character orbit 1350.c Analytic conductor $79.653$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(i, \sqrt{6})$$ Defining polynomial: $$x^{4} + 9$$ x^4 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{3}\cdot 3^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q - 2 \beta_1 q^{2} - 4 q^{4} + ( - \beta_{2} - 10 \beta_1) q^{7} + 8 \beta_1 q^{8}+O(q^{10})$$ q - 2*b1 * q^2 - 4 * q^4 + (-b2 - 10*b1) * q^7 + 8*b1 * q^8 $$q - 2 \beta_1 q^{2} - 4 q^{4} + ( - \beta_{2} - 10 \beta_1) q^{7} + 8 \beta_1 q^{8} + (2 \beta_{3} + 15) q^{11} + ( - 5 \beta_{2} - 5 \beta_1) q^{13} + ( - 2 \beta_{3} - 20) q^{14} + 16 q^{16} + ( - 5 \beta_{2} - 18 \beta_1) q^{17} + (5 \beta_{3} + 16) q^{19} + ( - 4 \beta_{2} - 30 \beta_1) q^{22} + (5 \beta_{2} + 9 \beta_1) q^{23} + ( - 10 \beta_{3} - 10) q^{26} + (4 \beta_{2} + 40 \beta_1) q^{28} + (5 \beta_{3} - 60) q^{29} + (10 \beta_{3} + 122) q^{31} - 32 \beta_1 q^{32} + ( - 10 \beta_{3} - 36) q^{34} + ( - 3 \beta_{2} - 205 \beta_1) q^{37} + ( - 10 \beta_{2} - 32 \beta_1) q^{38} + (6 \beta_{3} + 210) q^{41} + (7 \beta_{2} + 160 \beta_1) q^{43} + ( - 8 \beta_{3} - 60) q^{44} + (10 \beta_{3} + 18) q^{46} + (35 \beta_{2} + 81 \beta_1) q^{47} + ( - 20 \beta_{3} + 27) q^{49} + (20 \beta_{2} + 20 \beta_1) q^{52} + ( - 5 \beta_{2} - 384 \beta_1) q^{53} + (8 \beta_{3} + 80) q^{56} + ( - 10 \beta_{2} + 120 \beta_1) q^{58} + (14 \beta_{3} + 165) q^{59} + ( - 15 \beta_{3} - 229) q^{61} + ( - 20 \beta_{2} - 244 \beta_1) q^{62} - 64 q^{64} + (3 \beta_{2} - 430 \beta_1) q^{67} + (20 \beta_{2} + 72 \beta_1) q^{68} + (15 \beta_{3} + 555) q^{71} + ( - 6 \beta_{2} + 430 \beta_1) q^{73} + ( - 6 \beta_{3} - 410) q^{74} + ( - 20 \beta_{3} - 64) q^{76} + ( - 35 \beta_{2} - 582 \beta_1) q^{77} + (45 \beta_{3} - 290) q^{79} + ( - 12 \beta_{2} - 420 \beta_1) q^{82} + ( - 40 \beta_{2} - 600 \beta_1) q^{83} + (14 \beta_{3} + 320) q^{86} + (16 \beta_{2} + 120 \beta_1) q^{88} + ( - 9 \beta_{3} + 330) q^{89} + ( - 55 \beta_{3} - 1130) q^{91} + ( - 20 \beta_{2} - 36 \beta_1) q^{92} + (70 \beta_{3} + 162) q^{94} + ( - 14 \beta_{2} - 655 \beta_1) q^{97} + (40 \beta_{2} - 54 \beta_1) q^{98}+O(q^{100})$$ q - 2*b1 * q^2 - 4 * q^4 + (-b2 - 10*b1) * q^7 + 8*b1 * q^8 + (2*b3 + 15) * q^11 + (-5*b2 - 5*b1) * q^13 + (-2*b3 - 20) * q^14 + 16 * q^16 + (-5*b2 - 18*b1) * q^17 + (5*b3 + 16) * q^19 + (-4*b2 - 30*b1) * q^22 + (5*b2 + 9*b1) * q^23 + (-10*b3 - 10) * q^26 + (4*b2 + 40*b1) * q^28 + (5*b3 - 60) * q^29 + (10*b3 + 122) * q^31 - 32*b1 * q^32 + (-10*b3 - 36) * q^34 + (-3*b2 - 205*b1) * q^37 + (-10*b2 - 32*b1) * q^38 + (6*b3 + 210) * q^41 + (7*b2 + 160*b1) * q^43 + (-8*b3 - 60) * q^44 + (10*b3 + 18) * q^46 + (35*b2 + 81*b1) * q^47 + (-20*b3 + 27) * q^49 + (20*b2 + 20*b1) * q^52 + (-5*b2 - 384*b1) * q^53 + (8*b3 + 80) * q^56 + (-10*b2 + 120*b1) * q^58 + (14*b3 + 165) * q^59 + (-15*b3 - 229) * q^61 + (-20*b2 - 244*b1) * q^62 - 64 * q^64 + (3*b2 - 430*b1) * q^67 + (20*b2 + 72*b1) * q^68 + (15*b3 + 555) * q^71 + (-6*b2 + 430*b1) * q^73 + (-6*b3 - 410) * q^74 + (-20*b3 - 64) * q^76 + (-35*b2 - 582*b1) * q^77 + (45*b3 - 290) * q^79 + (-12*b2 - 420*b1) * q^82 + (-40*b2 - 600*b1) * q^83 + (14*b3 + 320) * q^86 + (16*b2 + 120*b1) * q^88 + (-9*b3 + 330) * q^89 + (-55*b3 - 1130) * q^91 + (-20*b2 - 36*b1) * q^92 + (70*b3 + 162) * q^94 + (-14*b2 - 655*b1) * q^97 + (40*b2 - 54*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 16 q^{4}+O(q^{10})$$ 4 * q - 16 * q^4 $$4 q - 16 q^{4} + 60 q^{11} - 80 q^{14} + 64 q^{16} + 64 q^{19} - 40 q^{26} - 240 q^{29} + 488 q^{31} - 144 q^{34} + 840 q^{41} - 240 q^{44} + 72 q^{46} + 108 q^{49} + 320 q^{56} + 660 q^{59} - 916 q^{61} - 256 q^{64} + 2220 q^{71} - 1640 q^{74} - 256 q^{76} - 1160 q^{79} + 1280 q^{86} + 1320 q^{89} - 4520 q^{91} + 648 q^{94}+O(q^{100})$$ 4 * q - 16 * q^4 + 60 * q^11 - 80 * q^14 + 64 * q^16 + 64 * q^19 - 40 * q^26 - 240 * q^29 + 488 * q^31 - 144 * q^34 + 840 * q^41 - 240 * q^44 + 72 * q^46 + 108 * q^49 + 320 * q^56 + 660 * q^59 - 916 * q^61 - 256 * q^64 + 2220 * q^71 - 1640 * q^74 - 256 * q^76 - 1160 * q^79 + 1280 * q^86 + 1320 * q^89 - 4520 * q^91 + 648 * q^94

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{2} ) / 3$$ (v^2) / 3 $$\beta_{2}$$ $$=$$ $$2\nu^{3} + 6\nu$$ 2*v^3 + 6*v $$\beta_{3}$$ $$=$$ $$-2\nu^{3} + 6\nu$$ -2*v^3 + 6*v
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 12$$ (b3 + b2) / 12 $$\nu^{2}$$ $$=$$ $$3\beta_1$$ 3*b1 $$\nu^{3}$$ $$=$$ $$( -\beta_{3} + \beta_{2} ) / 4$$ (-b3 + b2) / 4

## 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
 1.22474 + 1.22474i −1.22474 − 1.22474i −1.22474 + 1.22474i 1.22474 − 1.22474i
2.00000i 0 −4.00000 0 0 24.6969i 8.00000i 0 0
649.2 2.00000i 0 −4.00000 0 0 4.69694i 8.00000i 0 0
649.3 2.00000i 0 −4.00000 0 0 4.69694i 8.00000i 0 0
649.4 2.00000i 0 −4.00000 0 0 24.6969i 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.z 4
3.b odd 2 1 1350.4.c.w 4
5.b even 2 1 inner 1350.4.c.z 4
5.c odd 4 1 1350.4.a.bc 2
5.c odd 4 1 1350.4.a.bp yes 2
15.d odd 2 1 1350.4.c.w 4
15.e even 4 1 1350.4.a.bi yes 2
15.e even 4 1 1350.4.a.bj yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1350.4.a.bc 2 5.c odd 4 1
1350.4.a.bi yes 2 15.e even 4 1
1350.4.a.bj yes 2 15.e even 4 1
1350.4.a.bp yes 2 5.c odd 4 1
1350.4.c.w 4 3.b odd 2 1
1350.4.c.w 4 15.d odd 2 1
1350.4.c.z 4 1.a even 1 1 trivial
1350.4.c.z 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 $$S_{4}^{\mathrm{new}}(1350, [\chi])$$:

 $$T_{7}^{4} + 632T_{7}^{2} + 13456$$ T7^4 + 632*T7^2 + 13456 $$T_{11}^{2} - 30T_{11} - 639$$ T11^2 - 30*T11 - 639

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 4)^{2}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 632 T^{2} + 13456$$
$11$ $$(T^{2} - 30 T - 639)^{2}$$
$13$ $$T^{4} + 10850 T^{2} + \cdots + 28890625$$
$17$ $$T^{4} + 11448 T^{2} + \cdots + 25765776$$
$19$ $$(T^{2} - 32 T - 5144)^{2}$$
$23$ $$T^{4} + 10962 T^{2} + \cdots + 28291761$$
$29$ $$(T^{2} + 120 T - 1800)^{2}$$
$31$ $$(T^{2} - 244 T - 6716)^{2}$$
$37$ $$T^{4} + 87938 T^{2} + \cdots + 1606486561$$
$41$ $$(T^{2} - 420 T + 36324)^{2}$$
$43$ $$T^{4} + 72368 T^{2} + \cdots + 225480256$$
$47$ $$T^{4} + 542322 T^{2} + \cdots + 66584125521$$
$53$ $$T^{4} + 305712 T^{2} + \cdots + 20179907136$$
$59$ $$(T^{2} - 330 T - 15111)^{2}$$
$61$ $$(T^{2} + 458 T + 3841)^{2}$$
$67$ $$T^{4} + 373688 T^{2} + \cdots + 33472897936$$
$71$ $$(T^{2} - 1110 T + 259425)^{2}$$
$73$ $$T^{4} + 385352 T^{2} + \cdots + 31372911376$$
$79$ $$(T^{2} + 580 T - 353300)^{2}$$
$83$ $$T^{4} + 1411200 T^{2} + \cdots + 207360000$$
$89$ $$(T^{2} - 660 T + 91404)^{2}$$
$97$ $$T^{4} + 942722 T^{2} + \cdots + 149528382721$$