# Properties

 Label 630.2.d.b Level 630 Weight 2 Character orbit 630.d Analytic conductor 5.031 Analytic rank 0 Dimension 4 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$630 = 2 \cdot 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 630.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.03057532734$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{5})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + \beta_{3} q^{5} + ( -\beta_{1} - \beta_{3} ) q^{7} - q^{8} +O(q^{10})$$ $$q - q^{2} + q^{4} + \beta_{3} q^{5} + ( -\beta_{1} - \beta_{3} ) q^{7} - q^{8} -\beta_{3} q^{10} + 4 \beta_{1} q^{11} + 2 \beta_{3} q^{13} + ( \beta_{1} + \beta_{3} ) q^{14} + q^{16} + \beta_{2} q^{17} -\beta_{2} q^{19} + \beta_{3} q^{20} -4 \beta_{1} q^{22} + 4 q^{23} + 5 q^{25} -2 \beta_{3} q^{26} + ( -\beta_{1} - \beta_{3} ) q^{28} + 2 \beta_{1} q^{29} -2 \beta_{2} q^{31} - q^{32} -\beta_{2} q^{34} + ( -5 - \beta_{2} ) q^{35} + 7 \beta_{1} q^{37} + \beta_{2} q^{38} -\beta_{3} q^{40} + 2 \beta_{3} q^{41} + \beta_{1} q^{43} + 4 \beta_{1} q^{44} -4 q^{46} -3 \beta_{2} q^{47} + ( 3 + 2 \beta_{2} ) q^{49} -5 q^{50} + 2 \beta_{3} q^{52} + 4 q^{53} + 4 \beta_{2} q^{55} + ( \beta_{1} + \beta_{3} ) q^{56} -2 \beta_{1} q^{58} + 2 \beta_{3} q^{59} -3 \beta_{2} q^{61} + 2 \beta_{2} q^{62} + q^{64} + 10 q^{65} + 5 \beta_{1} q^{67} + \beta_{2} q^{68} + ( 5 + \beta_{2} ) q^{70} + \beta_{1} q^{71} + 6 \beta_{3} q^{73} -7 \beta_{1} q^{74} -\beta_{2} q^{76} + ( 8 - 4 \beta_{2} ) q^{77} + 6 q^{79} + \beta_{3} q^{80} -2 \beta_{3} q^{82} + 4 \beta_{2} q^{83} + 5 \beta_{1} q^{85} -\beta_{1} q^{86} -4 \beta_{1} q^{88} -2 \beta_{3} q^{89} + ( -10 - 2 \beta_{2} ) q^{91} + 4 q^{92} + 3 \beta_{2} q^{94} -5 \beta_{1} q^{95} + ( -3 - 2 \beta_{2} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{2} + 4q^{4} - 4q^{8} + O(q^{10})$$ $$4q - 4q^{2} + 4q^{4} - 4q^{8} + 4q^{16} + 16q^{23} + 20q^{25} - 4q^{32} - 20q^{35} - 16q^{46} + 12q^{49} - 20q^{50} + 16q^{53} + 4q^{64} + 40q^{65} + 20q^{70} + 32q^{77} + 24q^{79} - 40q^{91} + 16q^{92} - 12q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 6 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 4 \nu$$$$)/2$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 8 \nu$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} + 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} - 3$$ $$\nu^{3}$$ $$=$$ $$-2 \beta_{2} + 4 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/630\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$281$$ $$451$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
629.1
 − 2.28825i 2.28825i 0.874032i − 0.874032i
−1.00000 0 1.00000 −2.23607 0 2.23607 1.41421i −1.00000 0 2.23607
629.2 −1.00000 0 1.00000 −2.23607 0 2.23607 + 1.41421i −1.00000 0 2.23607
629.3 −1.00000 0 1.00000 2.23607 0 −2.23607 1.41421i −1.00000 0 −2.23607
629.4 −1.00000 0 1.00000 2.23607 0 −2.23607 + 1.41421i −1.00000 0 −2.23607
 $$n$$: 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
7.b odd 2 1 inner
15.d odd 2 1 inner
105.g even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 630.2.d.b 4
3.b odd 2 1 630.2.d.c yes 4
4.b odd 2 1 5040.2.k.b 4
5.b even 2 1 630.2.d.c yes 4
5.c odd 4 2 3150.2.b.b 8
7.b odd 2 1 inner 630.2.d.b 4
12.b even 2 1 5040.2.k.c 4
15.d odd 2 1 inner 630.2.d.b 4
15.e even 4 2 3150.2.b.b 8
20.d odd 2 1 5040.2.k.c 4
21.c even 2 1 630.2.d.c yes 4
28.d even 2 1 5040.2.k.b 4
35.c odd 2 1 630.2.d.c yes 4
35.f even 4 2 3150.2.b.b 8
60.h even 2 1 5040.2.k.b 4
84.h odd 2 1 5040.2.k.c 4
105.g even 2 1 inner 630.2.d.b 4
105.k odd 4 2 3150.2.b.b 8
140.c even 2 1 5040.2.k.c 4
420.o odd 2 1 5040.2.k.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.d.b 4 1.a even 1 1 trivial
630.2.d.b 4 7.b odd 2 1 inner
630.2.d.b 4 15.d odd 2 1 inner
630.2.d.b 4 105.g even 2 1 inner
630.2.d.c yes 4 3.b odd 2 1
630.2.d.c yes 4 5.b even 2 1
630.2.d.c yes 4 21.c even 2 1
630.2.d.c yes 4 35.c odd 2 1
3150.2.b.b 8 5.c odd 4 2
3150.2.b.b 8 15.e even 4 2
3150.2.b.b 8 35.f even 4 2
3150.2.b.b 8 105.k odd 4 2
5040.2.k.b 4 4.b odd 2 1
5040.2.k.b 4 28.d even 2 1
5040.2.k.b 4 60.h even 2 1
5040.2.k.b 4 420.o odd 2 1
5040.2.k.c 4 12.b even 2 1
5040.2.k.c 4 20.d odd 2 1
5040.2.k.c 4 84.h odd 2 1
5040.2.k.c 4 140.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(630, [\chi])$$:

 $$T_{11}^{2} + 32$$ $$T_{23} - 4$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{4}$$
$3$ 
$5$ $$( 1 - 5 T^{2} )^{2}$$
$7$ $$1 - 6 T^{2} + 49 T^{4}$$
$11$ $$( 1 + 10 T^{2} + 121 T^{4} )^{2}$$
$13$ $$( 1 + 6 T^{2} + 169 T^{4} )^{2}$$
$17$ $$( 1 - 24 T^{2} + 289 T^{4} )^{2}$$
$19$ $$( 1 - 28 T^{2} + 361 T^{4} )^{2}$$
$23$ $$( 1 - 4 T + 23 T^{2} )^{4}$$
$29$ $$( 1 - 50 T^{2} + 841 T^{4} )^{2}$$
$31$ $$( 1 - 22 T^{2} + 961 T^{4} )^{2}$$
$37$ $$( 1 + 24 T^{2} + 1369 T^{4} )^{2}$$
$41$ $$( 1 + 62 T^{2} + 1681 T^{4} )^{2}$$
$43$ $$( 1 - 84 T^{2} + 1849 T^{4} )^{2}$$
$47$ $$( 1 - 4 T^{2} + 2209 T^{4} )^{2}$$
$53$ $$( 1 - 4 T + 53 T^{2} )^{4}$$
$59$ $$( 1 + 98 T^{2} + 3481 T^{4} )^{2}$$
$61$ $$( 1 - 32 T^{2} + 3721 T^{4} )^{2}$$
$67$ $$( 1 - 84 T^{2} + 4489 T^{4} )^{2}$$
$71$ $$( 1 - 140 T^{2} + 5041 T^{4} )^{2}$$
$73$ $$( 1 - 34 T^{2} + 5329 T^{4} )^{2}$$
$79$ $$( 1 - 6 T + 79 T^{2} )^{4}$$
$83$ $$( 1 - 6 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$( 1 + 158 T^{2} + 7921 T^{4} )^{2}$$
$97$ $$( 1 + 97 T^{2} )^{4}$$