# Properties

 Label 630.2.d.d 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: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} -\beta_{3} q^{5} + ( -\beta_{1} - \beta_{2} ) q^{7} + q^{8} +O(q^{10})$$ $$q + q^{2} + q^{4} -\beta_{3} q^{5} + ( -\beta_{1} - \beta_{2} ) q^{7} + q^{8} -\beta_{3} q^{10} -\beta_{2} q^{11} + ( -2 \beta_{1} + \beta_{2} ) q^{13} + ( -\beta_{1} - \beta_{2} ) q^{14} + q^{16} -2 \beta_{3} q^{17} -\beta_{3} q^{20} -\beta_{2} q^{22} + 6 q^{23} -5 q^{25} + ( -2 \beta_{1} + \beta_{2} ) q^{26} + ( -\beta_{1} - \beta_{2} ) q^{28} + 2 \beta_{2} q^{29} + q^{32} -2 \beta_{3} q^{34} + ( -3 \beta_{1} + 4 \beta_{2} ) q^{35} + 3 \beta_{2} q^{37} -\beta_{3} q^{40} + ( 6 \beta_{1} - 3 \beta_{2} ) q^{41} -6 \beta_{2} q^{43} -\beta_{2} q^{44} + 6 q^{46} -2 \beta_{3} q^{47} + ( -2 + 3 \beta_{3} ) q^{49} -5 q^{50} + ( -2 \beta_{1} + \beta_{2} ) q^{52} + 6 q^{53} + ( -2 \beta_{1} + \beta_{2} ) q^{55} + ( -\beta_{1} - \beta_{2} ) q^{56} + 2 \beta_{2} q^{58} + ( 6 \beta_{1} - 3 \beta_{2} ) q^{59} + 6 \beta_{3} q^{61} + q^{64} + 5 \beta_{2} q^{65} -2 \beta_{3} q^{68} + ( -3 \beta_{1} + 4 \beta_{2} ) q^{70} -4 \beta_{2} q^{71} + ( 4 \beta_{1} - 2 \beta_{2} ) q^{73} + 3 \beta_{2} q^{74} + ( -3 + \beta_{3} ) q^{77} -4 q^{79} -\beta_{3} q^{80} + ( 6 \beta_{1} - 3 \beta_{2} ) q^{82} + 4 \beta_{3} q^{83} -10 q^{85} -6 \beta_{2} q^{86} -\beta_{2} q^{88} + ( -6 \beta_{1} + 3 \beta_{2} ) q^{89} + ( 5 + 3 \beta_{3} ) q^{91} + 6 q^{92} -2 \beta_{3} q^{94} + ( -8 \beta_{1} + 4 \beta_{2} ) q^{97} + ( -2 + 3 \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{2} + 4q^{4} + 4q^{8} + O(q^{10})$$ $$4q + 4q^{2} + 4q^{4} + 4q^{8} + 4q^{16} + 24q^{23} - 20q^{25} + 4q^{32} + 24q^{46} - 8q^{49} - 20q^{50} + 24q^{53} + 4q^{64} - 12q^{77} - 16q^{79} - 40q^{85} + 20q^{91} + 24q^{92} - 8q^{98} + O(q^{100})$$

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

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

## 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
 1.58114 + 0.707107i −1.58114 − 0.707107i 1.58114 − 0.707107i −1.58114 + 0.707107i
1.00000 0 1.00000 2.23607i 0 −1.58114 2.12132i 1.00000 0 2.23607i
629.2 1.00000 0 1.00000 2.23607i 0 1.58114 + 2.12132i 1.00000 0 2.23607i
629.3 1.00000 0 1.00000 2.23607i 0 −1.58114 + 2.12132i 1.00000 0 2.23607i
629.4 1.00000 0 1.00000 2.23607i 0 1.58114 2.12132i 1.00000 0 2.23607i
 $$n$$: 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.d yes 4
3.b odd 2 1 630.2.d.a 4
4.b odd 2 1 5040.2.k.a 4
5.b even 2 1 630.2.d.a 4
5.c odd 4 2 3150.2.b.c 8
7.b odd 2 1 inner 630.2.d.d yes 4
12.b even 2 1 5040.2.k.d 4
15.d odd 2 1 inner 630.2.d.d yes 4
15.e even 4 2 3150.2.b.c 8
20.d odd 2 1 5040.2.k.d 4
21.c even 2 1 630.2.d.a 4
28.d even 2 1 5040.2.k.a 4
35.c odd 2 1 630.2.d.a 4
35.f even 4 2 3150.2.b.c 8
60.h even 2 1 5040.2.k.a 4
84.h odd 2 1 5040.2.k.d 4
105.g even 2 1 inner 630.2.d.d yes 4
105.k odd 4 2 3150.2.b.c 8
140.c even 2 1 5040.2.k.d 4
420.o odd 2 1 5040.2.k.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.d.a 4 3.b odd 2 1
630.2.d.a 4 5.b even 2 1
630.2.d.a 4 21.c even 2 1
630.2.d.a 4 35.c odd 2 1
630.2.d.d yes 4 1.a even 1 1 trivial
630.2.d.d yes 4 7.b odd 2 1 inner
630.2.d.d yes 4 15.d odd 2 1 inner
630.2.d.d yes 4 105.g even 2 1 inner
3150.2.b.c 8 5.c odd 4 2
3150.2.b.c 8 15.e even 4 2
3150.2.b.c 8 35.f even 4 2
3150.2.b.c 8 105.k odd 4 2
5040.2.k.a 4 4.b odd 2 1
5040.2.k.a 4 28.d even 2 1
5040.2.k.a 4 60.h even 2 1
5040.2.k.a 4 420.o odd 2 1
5040.2.k.d 4 12.b even 2 1
5040.2.k.d 4 20.d odd 2 1
5040.2.k.d 4 84.h odd 2 1
5040.2.k.d 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 $$S_{2}^{\mathrm{new}}(630, [\chi])$$:

 $$T_{11}^{2} + 2$$ $$T_{23} - 6$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T )^{4}$$
$3$ 
$5$ $$( 1 + 5 T^{2} )^{2}$$
$7$ $$1 + 4 T^{2} + 49 T^{4}$$
$11$ $$( 1 - 20 T^{2} + 121 T^{4} )^{2}$$
$13$ $$( 1 + 16 T^{2} + 169 T^{4} )^{2}$$
$17$ $$( 1 - 14 T^{2} + 289 T^{4} )^{2}$$
$19$ $$( 1 - 19 T^{2} )^{4}$$
$23$ $$( 1 - 6 T + 23 T^{2} )^{4}$$
$29$ $$( 1 - 50 T^{2} + 841 T^{4} )^{2}$$
$31$ $$( 1 - 31 T^{2} )^{4}$$
$37$ $$( 1 - 56 T^{2} + 1369 T^{4} )^{2}$$
$41$ $$( 1 - 8 T^{2} + 1681 T^{4} )^{2}$$
$43$ $$( 1 - 10 T + 43 T^{2} )^{2}( 1 + 10 T + 43 T^{2} )^{2}$$
$47$ $$( 1 - 74 T^{2} + 2209 T^{4} )^{2}$$
$53$ $$( 1 - 6 T + 53 T^{2} )^{4}$$
$59$ $$( 1 + 28 T^{2} + 3481 T^{4} )^{2}$$
$61$ $$( 1 - 8 T + 61 T^{2} )^{2}( 1 + 8 T + 61 T^{2} )^{2}$$
$67$ $$( 1 - 67 T^{2} )^{4}$$
$71$ $$( 1 - 110 T^{2} + 5041 T^{4} )^{2}$$
$73$ $$( 1 + 106 T^{2} + 5329 T^{4} )^{2}$$
$79$ $$( 1 + 4 T + 79 T^{2} )^{4}$$
$83$ $$( 1 - 86 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$( 1 + 88 T^{2} + 7921 T^{4} )^{2}$$
$97$ $$( 1 + 34 T^{2} + 9409 T^{4} )^{2}$$