# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ = $$630 = 2 \cdot 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 630.m (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.03057532734$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} + ( 2 + \zeta_{8}^{2} ) q^{5} + \zeta_{8}^{3} q^{7} + \zeta_{8}^{3} q^{8} +O(q^{10})$$ $$q + \zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} + ( 2 + \zeta_{8}^{2} ) q^{5} + \zeta_{8}^{3} q^{7} + \zeta_{8}^{3} q^{8} + ( 2 \zeta_{8} + \zeta_{8}^{3} ) q^{10} + ( \zeta_{8} - 2 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{11} + ( 4 + 4 \zeta_{8}^{2} ) q^{13} - q^{14} - q^{16} + ( -2 + 2 \zeta_{8} - 2 \zeta_{8}^{2} ) q^{17} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{19} + ( -1 + 2 \zeta_{8}^{2} ) q^{20} + ( -1 + \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{22} + ( -2 + 2 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{23} + ( 3 + 4 \zeta_{8}^{2} ) q^{25} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{26} -\zeta_{8} q^{28} + ( 2 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{29} + ( 6 - 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{31} -\zeta_{8} q^{32} + ( -2 \zeta_{8} + 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{34} + ( -\zeta_{8} + 2 \zeta_{8}^{3} ) q^{35} + ( 2 - 2 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{37} + ( 2 - 2 \zeta_{8}^{2} ) q^{38} + ( -\zeta_{8} + 2 \zeta_{8}^{3} ) q^{40} + ( -2 \zeta_{8} + 6 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{41} + ( -1 + 10 \zeta_{8} - \zeta_{8}^{2} ) q^{43} + ( 2 - \zeta_{8} + \zeta_{8}^{3} ) q^{44} + ( -4 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{46} + ( -5 - 6 \zeta_{8} - 5 \zeta_{8}^{2} ) q^{47} -\zeta_{8}^{2} q^{49} + ( 3 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{50} + ( -4 + 4 \zeta_{8}^{2} ) q^{52} + ( 4 - 4 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{53} + ( 2 + \zeta_{8} - 4 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{55} -\zeta_{8}^{2} q^{56} + ( -2 + 2 \zeta_{8} - 2 \zeta_{8}^{2} ) q^{58} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{59} + ( -7 \zeta_{8} + 7 \zeta_{8}^{3} ) q^{61} + ( -3 + 6 \zeta_{8} - 3 \zeta_{8}^{2} ) q^{62} -\zeta_{8}^{2} q^{64} + ( 4 + 12 \zeta_{8}^{2} ) q^{65} + ( 7 - 7 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{67} + ( 2 - 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{68} + ( -2 - \zeta_{8}^{2} ) q^{70} + ( 2 \zeta_{8} - 4 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{71} + 10 \zeta_{8} q^{73} + ( 6 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{74} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{76} + ( -1 + 2 \zeta_{8} - \zeta_{8}^{2} ) q^{77} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{79} + ( -2 - \zeta_{8}^{2} ) q^{80} + ( 2 - 2 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{82} + ( -2 + 2 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{83} + ( -2 + 4 \zeta_{8} - 6 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{85} + ( -\zeta_{8} + 10 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{86} + ( -1 + 2 \zeta_{8} - \zeta_{8}^{2} ) q^{88} + ( -6 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{89} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{91} + ( -2 - 4 \zeta_{8} - 2 \zeta_{8}^{2} ) q^{92} + ( -5 \zeta_{8} - 6 \zeta_{8}^{2} - 5 \zeta_{8}^{3} ) q^{94} + ( -2 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{95} + ( -6 + 6 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{97} -\zeta_{8}^{3} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 8q^{5} + O(q^{10})$$ $$4q + 8q^{5} + 16q^{13} - 4q^{14} - 4q^{16} - 8q^{17} - 4q^{20} - 4q^{22} - 8q^{23} + 12q^{25} + 8q^{29} + 24q^{31} + 8q^{37} + 8q^{38} - 4q^{43} + 8q^{44} - 16q^{46} - 20q^{47} - 16q^{52} + 16q^{53} + 8q^{55} - 8q^{58} - 12q^{62} + 16q^{65} + 28q^{67} + 8q^{68} - 8q^{70} + 24q^{74} - 4q^{77} - 8q^{80} + 8q^{82} - 8q^{83} - 8q^{85} - 4q^{88} - 24q^{89} - 8q^{92} - 24q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$127$$ $$281$$ $$451$$ $$\chi(n)$$ $$-\zeta_{8}^{2}$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
197.1
 −0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 − 0.707107i 0.707107 + 0.707107i
−0.707107 + 0.707107i 0 1.00000i 2.00000 1.00000i 0 0.707107 + 0.707107i 0.707107 + 0.707107i 0 −0.707107 + 2.12132i
197.2 0.707107 0.707107i 0 1.00000i 2.00000 1.00000i 0 −0.707107 0.707107i −0.707107 0.707107i 0 0.707107 2.12132i
323.1 −0.707107 0.707107i 0 1.00000i 2.00000 + 1.00000i 0 0.707107 0.707107i 0.707107 0.707107i 0 −0.707107 2.12132i
323.2 0.707107 + 0.707107i 0 1.00000i 2.00000 + 1.00000i 0 −0.707107 + 0.707107i −0.707107 + 0.707107i 0 0.707107 + 2.12132i
 $$n$$: 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
15.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 630.2.m.b yes 4
3.b odd 2 1 630.2.m.a 4
5.b even 2 1 3150.2.m.a 4
5.c odd 4 1 630.2.m.a 4
5.c odd 4 1 3150.2.m.b 4
15.d odd 2 1 3150.2.m.b 4
15.e even 4 1 inner 630.2.m.b yes 4
15.e even 4 1 3150.2.m.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.m.a 4 3.b odd 2 1
630.2.m.a 4 5.c odd 4 1
630.2.m.b yes 4 1.a even 1 1 trivial
630.2.m.b yes 4 15.e even 4 1 inner
3150.2.m.a 4 5.b even 2 1
3150.2.m.a 4 15.e even 4 1
3150.2.m.b 4 5.c odd 4 1
3150.2.m.b 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 $$S_{2}^{\mathrm{new}}(630, [\chi])$$:

 $$T_{11}^{4} + 12 T_{11}^{2} + 4$$ $$T_{17}^{4} + 8 T_{17}^{3} + 32 T_{17}^{2} + 32 T_{17} + 16$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{4}$$
$3$ 1
$5$ $$( 1 - 4 T + 5 T^{2} )^{2}$$
$7$ $$1 + T^{4}$$
$11$ $$1 - 32 T^{2} + 466 T^{4} - 3872 T^{6} + 14641 T^{8}$$
$13$ $$( 1 - 8 T + 32 T^{2} - 104 T^{3} + 169 T^{4} )^{2}$$
$17$ $$1 + 8 T + 32 T^{2} + 168 T^{3} + 866 T^{4} + 2856 T^{5} + 9248 T^{6} + 39304 T^{7} + 83521 T^{8}$$
$19$ $$( 1 - 30 T^{2} + 361 T^{4} )^{2}$$
$23$ $$1 + 8 T + 32 T^{2} + 120 T^{3} + 386 T^{4} + 2760 T^{5} + 16928 T^{6} + 97336 T^{7} + 279841 T^{8}$$
$29$ $$( 1 - 4 T + 54 T^{2} - 116 T^{3} + 841 T^{4} )^{2}$$
$31$ $$( 1 - 12 T + 80 T^{2} - 372 T^{3} + 961 T^{4} )^{2}$$
$37$ $$1 - 8 T + 32 T^{2} - 72 T^{3} - 622 T^{4} - 2664 T^{5} + 43808 T^{6} - 405224 T^{7} + 1874161 T^{8}$$
$41$ $$1 - 76 T^{2} + 3654 T^{4} - 127756 T^{6} + 2825761 T^{8}$$
$43$ $$1 + 4 T + 8 T^{2} - 220 T^{3} - 3554 T^{4} - 9460 T^{5} + 14792 T^{6} + 318028 T^{7} + 3418801 T^{8}$$
$47$ $$1 + 20 T + 200 T^{2} + 1220 T^{3} + 7246 T^{4} + 57340 T^{5} + 441800 T^{6} + 2076460 T^{7} + 4879681 T^{8}$$
$53$ $$1 - 16 T + 128 T^{2} - 1296 T^{3} + 12338 T^{4} - 68688 T^{5} + 359552 T^{6} - 2382032 T^{7} + 7890481 T^{8}$$
$59$ $$( 1 + 110 T^{2} + 3481 T^{4} )^{2}$$
$61$ $$( 1 + 24 T^{2} + 3721 T^{4} )^{2}$$
$67$ $$1 - 28 T + 392 T^{2} - 4508 T^{3} + 43006 T^{4} - 302036 T^{5} + 1759688 T^{6} - 8421364 T^{7} + 20151121 T^{8}$$
$71$ $$1 - 236 T^{2} + 23494 T^{4} - 1189676 T^{6} + 25411681 T^{8}$$
$73$ $$1 - 8542 T^{4} + 28398241 T^{8}$$
$79$ $$( 1 - 126 T^{2} + 6241 T^{4} )^{2}$$
$83$ $$1 + 8 T + 32 T^{2} + 600 T^{3} + 11186 T^{4} + 49800 T^{5} + 220448 T^{6} + 4574296 T^{7} + 47458321 T^{8}$$
$89$ $$( 1 + 12 T + 206 T^{2} + 1068 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$1 + 24 T + 288 T^{2} + 3960 T^{3} + 49826 T^{4} + 384120 T^{5} + 2709792 T^{6} + 21904152 T^{7} + 88529281 T^{8}$$