# Properties

 Label 252.4.b.b Level $252$ Weight $4$ Character orbit 252.b Analytic conductor $14.868$ Analytic rank $0$ Dimension $4$ CM discriminant -84 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$252 = 2^{2} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 252.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.8684813214$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{7})$$ Defining polynomial: $$x^{4} + 8 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -2 \beta_{1} q^{2} -8 q^{4} + \beta_{2} q^{5} -7 \beta_{3} q^{7} + 16 \beta_{1} q^{8} +O(q^{10})$$ $$q -2 \beta_{1} q^{2} -8 q^{4} + \beta_{2} q^{5} -7 \beta_{3} q^{7} + 16 \beta_{1} q^{8} + 4 \beta_{3} q^{10} -31 \beta_{1} q^{11} + 14 \beta_{2} q^{14} + 64 q^{16} + 37 \beta_{2} q^{17} + 58 \beta_{3} q^{19} -8 \beta_{2} q^{20} -124 q^{22} + 95 \beta_{1} q^{23} + 111 q^{25} + 56 \beta_{3} q^{28} + 76 \beta_{3} q^{31} -128 \beta_{1} q^{32} + 148 \beta_{3} q^{34} -49 \beta_{1} q^{35} -376 q^{37} -116 \beta_{2} q^{38} -32 \beta_{3} q^{40} + 109 \beta_{2} q^{41} + 248 \beta_{1} q^{44} + 380 q^{46} + 343 q^{49} -222 \beta_{1} q^{50} + 62 \beta_{3} q^{55} -112 \beta_{2} q^{56} -152 \beta_{2} q^{62} -512 q^{64} -296 \beta_{2} q^{68} -196 q^{70} -319 \beta_{1} q^{71} + 752 \beta_{1} q^{74} -464 \beta_{3} q^{76} + 217 \beta_{2} q^{77} + 64 \beta_{2} q^{80} + 436 \beta_{3} q^{82} -518 q^{85} + 992 q^{88} + 415 \beta_{2} q^{89} -760 \beta_{1} q^{92} + 406 \beta_{1} q^{95} -686 \beta_{1} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 32q^{4} + O(q^{10})$$ $$4q - 32q^{4} + 256q^{16} - 496q^{22} + 444q^{25} - 1504q^{37} + 1520q^{46} + 1372q^{49} - 2048q^{64} - 784q^{70} - 2072q^{85} + 3968q^{88} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 8 x^{2} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 5 \nu$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 11 \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} + 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} - 4$$ $$\nu^{3}$$ $$=$$ $$($$$$-5 \beta_{2} + 11 \beta_{1}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$73$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
55.1
 − 2.57794i 1.16372i − 1.16372i 2.57794i
2.82843i 0 −8.00000 3.74166i 0 18.5203 22.6274i 0 −10.5830
55.2 2.82843i 0 −8.00000 3.74166i 0 −18.5203 22.6274i 0 10.5830
55.3 2.82843i 0 −8.00000 3.74166i 0 −18.5203 22.6274i 0 10.5830
55.4 2.82843i 0 −8.00000 3.74166i 0 18.5203 22.6274i 0 −10.5830
 $$n$$: 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
84.h odd 2 1 CM by $$\Q(\sqrt{-21})$$
3.b odd 2 1 inner
4.b odd 2 1 inner
7.b odd 2 1 inner
12.b even 2 1 inner
21.c even 2 1 inner
28.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.4.b.b 4
3.b odd 2 1 inner 252.4.b.b 4
4.b odd 2 1 inner 252.4.b.b 4
7.b odd 2 1 inner 252.4.b.b 4
12.b even 2 1 inner 252.4.b.b 4
21.c even 2 1 inner 252.4.b.b 4
28.d even 2 1 inner 252.4.b.b 4
84.h odd 2 1 CM 252.4.b.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.4.b.b 4 1.a even 1 1 trivial
252.4.b.b 4 3.b odd 2 1 inner
252.4.b.b 4 4.b odd 2 1 inner
252.4.b.b 4 7.b odd 2 1 inner
252.4.b.b 4 12.b even 2 1 inner
252.4.b.b 4 21.c even 2 1 inner
252.4.b.b 4 28.d even 2 1 inner
252.4.b.b 4 84.h odd 2 1 CM

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(252, [\chi])$$:

 $$T_{5}^{2} + 14$$ $$T_{11}^{2} + 1922$$ $$T_{19}^{2} - 23548$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 8 + T^{2} )^{2}$$
$3$ $$T^{4}$$
$5$ $$( 14 + T^{2} )^{2}$$
$7$ $$( -343 + T^{2} )^{2}$$
$11$ $$( 1922 + T^{2} )^{2}$$
$13$ $$T^{4}$$
$17$ $$( 19166 + T^{2} )^{2}$$
$19$ $$( -23548 + T^{2} )^{2}$$
$23$ $$( 18050 + T^{2} )^{2}$$
$29$ $$T^{4}$$
$31$ $$( -40432 + T^{2} )^{2}$$
$37$ $$( 376 + T )^{4}$$
$41$ $$( 166334 + T^{2} )^{2}$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$T^{4}$$
$67$ $$T^{4}$$
$71$ $$( 203522 + T^{2} )^{2}$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$( 2411150 + T^{2} )^{2}$$
$97$ $$T^{4}$$