# Properties

 Label 1350.4.c.v 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: $$\Q(i, \sqrt{21})$$ Defining polynomial: $$x^{4} + 11x^{2} + 25$$ x^4 + 11*x^2 + 25 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q + 2 \beta_1 q^{2} - 4 q^{4} + (\beta_{2} - 5 \beta_1) q^{7} - 8 \beta_1 q^{8}+O(q^{10})$$ q + 2*b1 * q^2 - 4 * q^4 + (b2 - 5*b1) * q^7 - 8*b1 * q^8 $$q + 2 \beta_1 q^{2} - 4 q^{4} + (\beta_{2} - 5 \beta_1) q^{7} - 8 \beta_1 q^{8} + ( - 4 \beta_{3} - 15) q^{11} + ( - \beta_{2} + 20 \beta_1) q^{13} + ( - 2 \beta_{3} + 10) q^{14} + 16 q^{16} + (5 \beta_{2} + 15 \beta_1) q^{17} + (5 \beta_{3} - 23) q^{19} + ( - 8 \beta_{2} - 30 \beta_1) q^{22} + ( - 5 \beta_{2} + 12 \beta_1) q^{23} + (2 \beta_{3} - 40) q^{26} + ( - 4 \beta_{2} + 20 \beta_1) q^{28} + ( - 7 \beta_{3} + 45) q^{29} + ( - 20 \beta_{3} - 10) q^{31} + 32 \beta_1 q^{32} + ( - 10 \beta_{3} - 30) q^{34} + (3 \beta_{2} - 20 \beta_1) q^{37} + (10 \beta_{2} - 46 \beta_1) q^{38} + (12 \beta_{3} - 150) q^{41} + (5 \beta_{2} + 305 \beta_1) q^{43} + (16 \beta_{3} + 60) q^{44} + (10 \beta_{3} - 24) q^{46} + ( - 5 \beta_{2} - 312 \beta_1) q^{47} + (10 \beta_{3} + 129) q^{49} + (4 \beta_{2} - 80 \beta_1) q^{52} + (5 \beta_{2} - 123 \beta_1) q^{53} + (8 \beta_{3} - 40) q^{56} + ( - 14 \beta_{2} + 90 \beta_1) q^{58} + (26 \beta_{3} + 45) q^{59} + (45 \beta_{3} + 68) q^{61} + ( - 40 \beta_{2} - 20 \beta_1) q^{62} - 64 q^{64} + ( - 9 \beta_{2} - 695 \beta_1) q^{67} + ( - 20 \beta_{2} - 60 \beta_1) q^{68} + ( - 15 \beta_{3} - 390) q^{71} + (12 \beta_{2} + 290 \beta_1) q^{73} + ( - 6 \beta_{3} + 40) q^{74} + ( - 20 \beta_{3} + 92) q^{76} + (5 \beta_{2} - 681 \beta_1) q^{77} + ( - 15 \beta_{3} + 223) q^{79} + (24 \beta_{2} - 300 \beta_1) q^{82} + (10 \beta_{2} - 954 \beta_1) q^{83} + ( - 10 \beta_{3} - 610) q^{86} + (32 \beta_{2} + 120 \beta_1) q^{88} + ( - 69 \beta_{3} + 285) q^{89} + ( - 25 \beta_{3} + 289) q^{91} + (20 \beta_{2} - 48 \beta_1) q^{92} + (10 \beta_{3} + 624) q^{94} + (26 \beta_{2} - 1175 \beta_1) q^{97} + (20 \beta_{2} + 258 \beta_1) q^{98}+O(q^{100})$$ q + 2*b1 * q^2 - 4 * q^4 + (b2 - 5*b1) * q^7 - 8*b1 * q^8 + (-4*b3 - 15) * q^11 + (-b2 + 20*b1) * q^13 + (-2*b3 + 10) * q^14 + 16 * q^16 + (5*b2 + 15*b1) * q^17 + (5*b3 - 23) * q^19 + (-8*b2 - 30*b1) * q^22 + (-5*b2 + 12*b1) * q^23 + (2*b3 - 40) * q^26 + (-4*b2 + 20*b1) * q^28 + (-7*b3 + 45) * q^29 + (-20*b3 - 10) * q^31 + 32*b1 * q^32 + (-10*b3 - 30) * q^34 + (3*b2 - 20*b1) * q^37 + (10*b2 - 46*b1) * q^38 + (12*b3 - 150) * q^41 + (5*b2 + 305*b1) * q^43 + (16*b3 + 60) * q^44 + (10*b3 - 24) * q^46 + (-5*b2 - 312*b1) * q^47 + (10*b3 + 129) * q^49 + (4*b2 - 80*b1) * q^52 + (5*b2 - 123*b1) * q^53 + (8*b3 - 40) * q^56 + (-14*b2 + 90*b1) * q^58 + (26*b3 + 45) * q^59 + (45*b3 + 68) * q^61 + (-40*b2 - 20*b1) * q^62 - 64 * q^64 + (-9*b2 - 695*b1) * q^67 + (-20*b2 - 60*b1) * q^68 + (-15*b3 - 390) * q^71 + (12*b2 + 290*b1) * q^73 + (-6*b3 + 40) * q^74 + (-20*b3 + 92) * q^76 + (5*b2 - 681*b1) * q^77 + (-15*b3 + 223) * q^79 + (24*b2 - 300*b1) * q^82 + (10*b2 - 954*b1) * q^83 + (-10*b3 - 610) * q^86 + (32*b2 + 120*b1) * q^88 + (-69*b3 + 285) * q^89 + (-25*b3 + 289) * q^91 + (20*b2 - 48*b1) * q^92 + (10*b3 + 624) * q^94 + (26*b2 - 1175*b1) * q^97 + (20*b2 + 258*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 16 q^{4}+O(q^{10})$$ 4 * q - 16 * q^4 $$4 q - 16 q^{4} - 60 q^{11} + 40 q^{14} + 64 q^{16} - 92 q^{19} - 160 q^{26} + 180 q^{29} - 40 q^{31} - 120 q^{34} - 600 q^{41} + 240 q^{44} - 96 q^{46} + 516 q^{49} - 160 q^{56} + 180 q^{59} + 272 q^{61} - 256 q^{64} - 1560 q^{71} + 160 q^{74} + 368 q^{76} + 892 q^{79} - 2440 q^{86} + 1140 q^{89} + 1156 q^{91} + 2496 q^{94}+O(q^{100})$$ 4 * q - 16 * q^4 - 60 * q^11 + 40 * q^14 + 64 * q^16 - 92 * q^19 - 160 * q^26 + 180 * q^29 - 40 * q^31 - 120 * q^34 - 600 * q^41 + 240 * q^44 - 96 * q^46 + 516 * q^49 - 160 * q^56 + 180 * q^59 + 272 * q^61 - 256 * q^64 - 1560 * q^71 + 160 * q^74 + 368 * q^76 + 892 * q^79 - 2440 * q^86 + 1140 * q^89 + 1156 * q^91 + 2496 * q^94

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 11x^{2} + 25$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 6\nu ) / 5$$ (v^3 + 6*v) / 5 $$\beta_{2}$$ $$=$$ $$( 3\nu^{3} + 48\nu ) / 5$$ (3*v^3 + 48*v) / 5 $$\beta_{3}$$ $$=$$ $$6\nu^{2} + 33$$ 6*v^2 + 33
 $$\nu$$ $$=$$ $$( \beta_{2} - 3\beta_1 ) / 6$$ (b2 - 3*b1) / 6 $$\nu^{2}$$ $$=$$ $$( \beta_{3} - 33 ) / 6$$ (b3 - 33) / 6 $$\nu^{3}$$ $$=$$ $$-\beta_{2} + 8\beta_1$$ -b2 + 8*b1

## 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
 − 1.79129i 2.79129i − 2.79129i 1.79129i
2.00000i 0 −4.00000 0 0 8.74773i 8.00000i 0 0
649.2 2.00000i 0 −4.00000 0 0 18.7477i 8.00000i 0 0
649.3 2.00000i 0 −4.00000 0 0 18.7477i 8.00000i 0 0
649.4 2.00000i 0 −4.00000 0 0 8.74773i 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.v 4
3.b odd 2 1 1350.4.c.ba 4
5.b even 2 1 inner 1350.4.c.v 4
5.c odd 4 1 1350.4.a.bg yes 2
5.c odd 4 1 1350.4.a.bl yes 2
15.d odd 2 1 1350.4.c.ba 4
15.e even 4 1 1350.4.a.be 2
15.e even 4 1 1350.4.a.bn yes 2

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

## Hecke kernels

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

 $$T_{7}^{4} + 428T_{7}^{2} + 26896$$ T7^4 + 428*T7^2 + 26896 $$T_{11}^{2} + 30T_{11} - 2799$$ T11^2 + 30*T11 - 2799

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 4)^{2}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 428 T^{2} + 26896$$
$11$ $$(T^{2} + 30 T - 2799)^{2}$$
$13$ $$T^{4} + 1178 T^{2} + 44521$$
$17$ $$T^{4} + 9900 T^{2} + \cdots + 20250000$$
$19$ $$(T^{2} + 46 T - 4196)^{2}$$
$23$ $$T^{4} + 9738 T^{2} + \cdots + 20985561$$
$29$ $$(T^{2} - 90 T - 7236)^{2}$$
$31$ $$(T^{2} + 20 T - 75500)^{2}$$
$37$ $$T^{4} + 4202 T^{2} + \cdots + 1692601$$
$41$ $$(T^{2} + 300 T - 4716)^{2}$$
$43$ $$T^{4} + 195500 T^{2} + \cdots + 7796890000$$
$47$ $$T^{4} + 204138 T^{2} + \cdots + 8578279161$$
$53$ $$T^{4} + 39708 T^{2} + \cdots + 108243216$$
$59$ $$(T^{2} - 90 T - 125739)^{2}$$
$61$ $$(T^{2} - 136 T - 378101)^{2}$$
$67$ $$T^{4} + 996668 T^{2} + \cdots + 218758256656$$
$71$ $$(T^{2} + 780 T + 109575)^{2}$$
$73$ $$T^{4} + 222632 T^{2} + \cdots + 3235789456$$
$79$ $$(T^{2} - 446 T + 7204)^{2}$$
$83$ $$T^{4} + 1858032 T^{2} + \cdots + 794265958656$$
$89$ $$(T^{2} - 570 T - 818604)^{2}$$
$97$ $$T^{4} + 3016778 T^{2} + \cdots + 1569660685321$$