# Properties

 Label 3150.3.c.c Level $3150$ Weight $3$ Character orbit 3150.c Analytic conductor $85.831$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 3150.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$85.8312832735$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.157351936.1 Defining polynomial: $$x^{8} + x^{4} + 16$$ x^8 + x^4 + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{4}$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{6} q^{2} + 2 q^{4} - \beta_1 q^{7} + 2 \beta_{6} q^{8}+O(q^{10})$$ q + b6 * q^2 + 2 * q^4 - b1 * q^7 + 2*b6 * q^8 $$q + \beta_{6} q^{2} + 2 q^{4} - \beta_1 q^{7} + 2 \beta_{6} q^{8} + (7 \beta_{7} - 9 \beta_{3}) q^{11} + (3 \beta_{2} - \beta_1) q^{13} + ( - 2 \beta_{7} + \beta_{3}) q^{14} + 4 q^{16} + ( - 19 \beta_{6} - 4 \beta_{5}) q^{17} + (5 \beta_{4} + 15) q^{19} + ( - 11 \beta_{2} + 7 \beta_1) q^{22} + ( - 7 \beta_{6} - 13 \beta_{5}) q^{23} + ( - 2 \beta_{7} + 4 \beta_{3}) q^{26} - 2 \beta_1 q^{28} + (7 \beta_{7} - 15 \beta_{3}) q^{29} + (8 \beta_{4} + 4) q^{31} + 4 \beta_{6} q^{32} + ( - 4 \beta_{4} - 34) q^{34} + (28 \beta_{2} + 5 \beta_1) q^{37} + (20 \beta_{6} + 10 \beta_{5}) q^{38} + (18 \beta_{7} - 6 \beta_{3}) q^{41} + (55 \beta_{2} + 6 \beta_1) q^{43} + (14 \beta_{7} - 18 \beta_{3}) q^{44} + ( - 13 \beta_{4} - 1) q^{46} + (27 \beta_{6} + 30 \beta_{5}) q^{47} - 7 q^{49} + (6 \beta_{2} - 2 \beta_1) q^{52} + (37 \beta_{6} + 28 \beta_{5}) q^{53} + ( - 4 \beta_{7} + 2 \beta_{3}) q^{56} + ( - 23 \beta_{2} + 7 \beta_1) q^{58} + (36 \beta_{7} - 33 \beta_{3}) q^{59} + (24 \beta_{4} - 22) q^{61} + (12 \beta_{6} + 16 \beta_{5}) q^{62} + 8 q^{64} + ( - 38 \beta_{2} - 7 \beta_1) q^{67} + ( - 38 \beta_{6} - 8 \beta_{5}) q^{68} + ( - 43 \beta_{7} + 6 \beta_{3}) q^{71} + ( - 21 \beta_{2} + 35 \beta_1) q^{73} + (10 \beta_{7} + 23 \beta_{3}) q^{74} + (10 \beta_{4} + 30) q^{76} + (19 \beta_{6} - 11 \beta_{5}) q^{77} + ( - 19 \beta_{4} - 36) q^{79} + (6 \beta_{2} + 18 \beta_1) q^{82} + ( - 96 \beta_{6} + 6 \beta_{5}) q^{83} + (12 \beta_{7} + 49 \beta_{3}) q^{86} + ( - 22 \beta_{2} + 14 \beta_1) q^{88} + (34 \beta_{7} + 39 \beta_{3}) q^{89} + (3 \beta_{4} - 7) q^{91} + ( - 14 \beta_{6} - 26 \beta_{5}) q^{92} + (30 \beta_{4} + 24) q^{94} + (38 \beta_{2} + 36 \beta_1) q^{97} - 7 \beta_{6} q^{98}+O(q^{100})$$ q + b6 * q^2 + 2 * q^4 - b1 * q^7 + 2*b6 * q^8 + (7*b7 - 9*b3) * q^11 + (3*b2 - b1) * q^13 + (-2*b7 + b3) * q^14 + 4 * q^16 + (-19*b6 - 4*b5) * q^17 + (5*b4 + 15) * q^19 + (-11*b2 + 7*b1) * q^22 + (-7*b6 - 13*b5) * q^23 + (-2*b7 + 4*b3) * q^26 - 2*b1 * q^28 + (7*b7 - 15*b3) * q^29 + (8*b4 + 4) * q^31 + 4*b6 * q^32 + (-4*b4 - 34) * q^34 + (28*b2 + 5*b1) * q^37 + (20*b6 + 10*b5) * q^38 + (18*b7 - 6*b3) * q^41 + (55*b2 + 6*b1) * q^43 + (14*b7 - 18*b3) * q^44 + (-13*b4 - 1) * q^46 + (27*b6 + 30*b5) * q^47 - 7 * q^49 + (6*b2 - 2*b1) * q^52 + (37*b6 + 28*b5) * q^53 + (-4*b7 + 2*b3) * q^56 + (-23*b2 + 7*b1) * q^58 + (36*b7 - 33*b3) * q^59 + (24*b4 - 22) * q^61 + (12*b6 + 16*b5) * q^62 + 8 * q^64 + (-38*b2 - 7*b1) * q^67 + (-38*b6 - 8*b5) * q^68 + (-43*b7 + 6*b3) * q^71 + (-21*b2 + 35*b1) * q^73 + (10*b7 + 23*b3) * q^74 + (10*b4 + 30) * q^76 + (19*b6 - 11*b5) * q^77 + (-19*b4 - 36) * q^79 + (6*b2 + 18*b1) * q^82 + (-96*b6 + 6*b5) * q^83 + (12*b7 + 49*b3) * q^86 + (-22*b2 + 14*b1) * q^88 + (34*b7 + 39*b3) * q^89 + (3*b4 - 7) * q^91 + (-14*b6 - 26*b5) * q^92 + (30*b4 + 24) * q^94 + (38*b2 + 36*b1) * q^97 - 7*b6 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 16 q^{4}+O(q^{10})$$ 8 * q + 16 * q^4 $$8 q + 16 q^{4} + 32 q^{16} + 120 q^{19} + 32 q^{31} - 272 q^{34} - 8 q^{46} - 56 q^{49} - 176 q^{61} + 64 q^{64} + 240 q^{76} - 288 q^{79} - 56 q^{91} + 192 q^{94}+O(q^{100})$$ 8 * q + 16 * q^4 + 32 * q^16 + 120 * q^19 + 32 * q^31 - 272 * q^34 - 8 * q^46 - 56 * q^49 - 176 * q^61 + 64 * q^64 + 240 * q^76 - 288 * q^79 - 56 * q^91 + 192 * q^94

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + x^{4} + 16$$ :

 $$\beta_{1}$$ $$=$$ $$( 2\nu^{4} + 1 ) / 3$$ (2*v^4 + 1) / 3 $$\beta_{2}$$ $$=$$ $$( \nu^{6} + 5\nu^{2} ) / 12$$ (v^6 + 5*v^2) / 12 $$\beta_{3}$$ $$=$$ $$( -\nu^{7} + 4\nu^{5} + 7\nu^{3} - 4\nu ) / 24$$ (-v^7 + 4*v^5 + 7*v^3 - 4*v) / 24 $$\beta_{4}$$ $$=$$ $$( -\nu^{6} + 3\nu^{2} ) / 4$$ (-v^6 + 3*v^2) / 4 $$\beta_{5}$$ $$=$$ $$( -\nu^{7} + 2\nu^{5} - 5\nu^{3} + 10\nu ) / 12$$ (-v^7 + 2*v^5 - 5*v^3 + 10*v) / 12 $$\beta_{6}$$ $$=$$ $$( -\nu^{7} - 4\nu^{5} + 7\nu^{3} + 4\nu ) / 24$$ (-v^7 - 4*v^5 + 7*v^3 + 4*v) / 24 $$\beta_{7}$$ $$=$$ $$( \nu^{7} + 2\nu^{5} + 5\nu^{3} + 10\nu ) / 12$$ (v^7 + 2*v^5 + 5*v^3 + 10*v) / 12
 $$\nu$$ $$=$$ $$( \beta_{7} + \beta_{6} + \beta_{5} - \beta_{3} ) / 2$$ (b7 + b6 + b5 - b3) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{4} + 3\beta_{2} ) / 2$$ (b4 + 3*b2) / 2 $$\nu^{3}$$ $$=$$ $$( \beta_{7} + 2\beta_{6} - \beta_{5} + 2\beta_{3} ) / 2$$ (b7 + 2*b6 - b5 + 2*b3) / 2 $$\nu^{4}$$ $$=$$ $$( 3\beta _1 - 1 ) / 2$$ (3*b1 - 1) / 2 $$\nu^{5}$$ $$=$$ $$( \beta_{7} - 5\beta_{6} + \beta_{5} + 5\beta_{3} ) / 2$$ (b7 - 5*b6 + b5 + 5*b3) / 2 $$\nu^{6}$$ $$=$$ $$( -5\beta_{4} + 9\beta_{2} ) / 2$$ (-5*b4 + 9*b2) / 2 $$\nu^{7}$$ $$=$$ $$( 7\beta_{7} - 10\beta_{6} - 7\beta_{5} - 10\beta_{3} ) / 2$$ (7*b7 - 10*b6 - 7*b5 - 10*b3) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$451$$ $$2801$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 0.581861 − 1.28897i −1.28897 − 0.581861i −1.28897 + 0.581861i 0.581861 + 1.28897i 1.28897 + 0.581861i −0.581861 + 1.28897i −0.581861 − 1.28897i 1.28897 − 0.581861i
−1.41421 0 2.00000 0 0 2.64575i −2.82843 0 0
449.2 −1.41421 0 2.00000 0 0 2.64575i −2.82843 0 0
449.3 −1.41421 0 2.00000 0 0 2.64575i −2.82843 0 0
449.4 −1.41421 0 2.00000 0 0 2.64575i −2.82843 0 0
449.5 1.41421 0 2.00000 0 0 2.64575i 2.82843 0 0
449.6 1.41421 0 2.00000 0 0 2.64575i 2.82843 0 0
449.7 1.41421 0 2.00000 0 0 2.64575i 2.82843 0 0
449.8 1.41421 0 2.00000 0 0 2.64575i 2.82843 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 449.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3150.3.c.c 8
3.b odd 2 1 inner 3150.3.c.c 8
5.b even 2 1 inner 3150.3.c.c 8
5.c odd 4 1 3150.3.e.a 4
5.c odd 4 1 3150.3.e.d yes 4
15.d odd 2 1 inner 3150.3.c.c 8
15.e even 4 1 3150.3.e.a 4
15.e even 4 1 3150.3.e.d yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3150.3.c.c 8 1.a even 1 1 trivial
3150.3.c.c 8 3.b odd 2 1 inner
3150.3.c.c 8 5.b even 2 1 inner
3150.3.c.c 8 15.d odd 2 1 inner
3150.3.e.a 4 5.c odd 4 1
3150.3.e.a 4 15.e even 4 1
3150.3.e.d yes 4 5.c odd 4 1
3150.3.e.d yes 4 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{4} + 464T_{11}^{2} + 12321$$ acting on $$S_{3}^{\mathrm{new}}(3150, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 2)^{4}$$
$3$ $$T^{8}$$
$5$ $$T^{8}$$
$7$ $$(T^{2} + 7)^{4}$$
$11$ $$(T^{4} + 464 T^{2} + 12321)^{2}$$
$13$ $$(T^{4} + 32 T^{2} + 4)^{2}$$
$17$ $$(T^{4} - 1268 T^{2} + 272484)^{2}$$
$19$ $$(T^{2} - 30 T + 50)^{4}$$
$23$ $$(T^{4} - 1184 T^{2} + 349281)^{2}$$
$29$ $$(T^{4} + 872 T^{2} + 8649)^{2}$$
$31$ $$(T^{2} - 8 T - 432)^{4}$$
$37$ $$(T^{4} + 1918 T^{2} + 370881)^{2}$$
$41$ $$(T^{4} + 2304 T^{2} + 1245456)^{2}$$
$43$ $$(T^{4} + 6554 T^{2} + 7689529)^{2}$$
$47$ $$(T^{4} - 6876 T^{2} + 8191044)^{2}$$
$53$ $$(T^{4} - 7604 T^{2} + 2842596)^{2}$$
$59$ $$(T^{4} + 9972 T^{2} + 16695396)^{2}$$
$61$ $$(T^{2} + 44 T - 3548)^{4}$$
$67$ $$(T^{4} + 3574 T^{2} + 1212201)^{2}$$
$71$ $$(T^{4} + 13904 T^{2} + 35892081)^{2}$$
$73$ $$(T^{4} + 18032 T^{2} + 66161956)^{2}$$
$79$ $$(T^{2} + 72 T - 1231)^{4}$$
$83$ $$(T^{4} - 39456 T^{2} + \cdots + 379314576)^{2}$$
$89$ $$(T^{4} + 20636 T^{2} + 4955076)^{2}$$
$97$ $$(T^{4} + 21032 T^{2} + 58186384)^{2}$$