# Properties

 Label 630.2.m.d Level 630 Weight 2 Character orbit 630.m Analytic conductor 5.031 Analytic rank 0 Dimension 8 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: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.1698758656.6 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} -\beta_{5} q^{4} + ( -\beta_{2} - \beta_{3} - \beta_{5} ) q^{5} -\beta_{1} q^{7} + \beta_{1} q^{8} +O(q^{10})$$ $$q + \beta_{3} q^{2} -\beta_{5} q^{4} + ( -\beta_{2} - \beta_{3} - \beta_{5} ) q^{5} -\beta_{1} q^{7} + \beta_{1} q^{8} -\beta_{4} q^{10} + ( -\beta_{2} - \beta_{4} + 2 \beta_{5} ) q^{11} + ( -\beta_{1} - \beta_{4} - \beta_{7} ) q^{13} + q^{14} - q^{16} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} ) q^{17} + ( -1 + \beta_{7} ) q^{20} + ( -1 - 3 \beta_{1} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{22} + ( -2 - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{6} ) q^{23} + ( -1 + 5 \beta_{1} + \beta_{4} + \beta_{7} ) q^{25} + ( \beta_{6} + \beta_{7} ) q^{26} + \beta_{3} q^{28} + ( 2 - 2 \beta_{2} + 2 \beta_{4} - \beta_{6} + \beta_{7} ) q^{29} + ( -2 + \beta_{1} - \beta_{3} - \beta_{6} + \beta_{7} ) q^{31} -\beta_{3} q^{32} + ( -\beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{34} + ( \beta_{3} + \beta_{6} ) q^{35} + ( -5 \beta_{1} + \beta_{4} - \beta_{7} ) q^{37} + ( -\beta_{3} - \beta_{6} ) q^{40} + ( \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + 6 \beta_{5} ) q^{41} + ( -3 + 2 \beta_{1} + \beta_{2} + 3 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{43} + ( 2 + \beta_{1} - \beta_{3} - \beta_{6} + \beta_{7} ) q^{44} + ( 2 \beta_{2} - 2 \beta_{4} ) q^{46} + ( 3 - 2 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} - 3 \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{47} + \beta_{5} q^{49} + ( -4 - \beta_{3} - \beta_{6} - \beta_{7} ) q^{50} + ( -\beta_{2} - \beta_{3} - \beta_{6} ) q^{52} + ( -4 - 3 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - 4 \beta_{5} + \beta_{6} + \beta_{7} ) q^{53} + ( -2 + 5 \beta_{1} - \beta_{3} - \beta_{6} - 3 \beta_{7} ) q^{55} -\beta_{5} q^{56} + ( 2 - 2 \beta_{1} + \beta_{2} + 3 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{58} + ( 4 + 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{6} - 2 \beta_{7} ) q^{59} + ( -\beta_{2} + \beta_{4} ) q^{61} + ( -1 + \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} ) q^{62} + \beta_{5} q^{64} + ( -4 + 4 \beta_{3} - \beta_{6} - \beta_{7} ) q^{65} + ( 1 - 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{67} + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} ) q^{68} + ( -\beta_{2} - \beta_{3} - \beta_{5} ) q^{70} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{71} + ( 4 + 2 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} - 4 \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{73} + ( 6 + \beta_{6} - \beta_{7} ) q^{74} + ( 1 - \beta_{2} - 3 \beta_{3} - \beta_{5} + \beta_{6} ) q^{77} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{79} + ( \beta_{2} + \beta_{3} + \beta_{5} ) q^{80} + ( -2 - 7 \beta_{1} - \beta_{4} - 2 \beta_{5} + \beta_{7} ) q^{82} + ( -6 - 2 \beta_{1} + 2 \beta_{4} - 6 \beta_{5} - 2 \beta_{7} ) q^{83} + ( 4 + 5 \beta_{1} + \beta_{2} - 3 \beta_{3} - \beta_{4} - 4 \beta_{5} + \beta_{6} + \beta_{7} ) q^{85} + ( -2 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{86} + ( -1 + \beta_{2} + 3 \beta_{3} + \beta_{5} - \beta_{6} ) q^{88} + ( 10 - 4 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} + \beta_{6} - \beta_{7} ) q^{89} + ( \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} ) q^{91} + ( -2 + 2 \beta_{1} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} ) q^{92} + ( 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{94} + ( 2 - 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} ) q^{97} -\beta_{1} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q - 4q^{13} + 8q^{14} - 8q^{16} - 4q^{20} - 4q^{22} - 8q^{23} - 4q^{25} + 24q^{29} - 8q^{31} - 4q^{35} - 4q^{37} + 4q^{40} - 12q^{43} + 24q^{44} + 12q^{47} - 32q^{50} + 4q^{52} - 32q^{53} - 24q^{55} + 12q^{58} + 16q^{59} - 4q^{62} - 32q^{65} + 20q^{67} + 36q^{73} + 40q^{74} + 4q^{77} - 12q^{82} - 56q^{83} + 32q^{85} - 4q^{88} + 72q^{89} - 8q^{92} - 4q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 18 x^{6} + 97 x^{4} + 176 x^{2} + 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{7} + 2 \nu^{6} + 18 \nu^{5} + 28 \nu^{4} + 89 \nu^{3} + 74 \nu^{2} + 104 \nu - 16$$$$)/64$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} + 18 \nu^{5} + 8 \nu^{4} + 105 \nu^{3} + 72 \nu^{2} + 248 \nu + 64$$$$)/64$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} - 2 \nu^{6} + 18 \nu^{5} - 28 \nu^{4} + 89 \nu^{3} - 74 \nu^{2} + 104 \nu + 16$$$$)/64$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{7} + 18 \nu^{5} - 8 \nu^{4} + 105 \nu^{3} - 72 \nu^{2} + 248 \nu - 64$$$$)/64$$ $$\beta_{5}$$ $$=$$ $$($$$$-3 \nu^{7} - 46 \nu^{5} - 179 \nu^{3} - 168 \nu$$$$)/64$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{7} - 6 \nu^{6} + 10 \nu^{5} - 92 \nu^{4} - 15 \nu^{3} - 358 \nu^{2} - 120 \nu - 336$$$$)/64$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{7} + 6 \nu^{6} + 10 \nu^{5} + 92 \nu^{4} - 15 \nu^{3} + 358 \nu^{2} - 120 \nu + 336$$$$)/64$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} + \beta_{6} + 2 \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{7} - \beta_{6} + \beta_{4} + 3 \beta_{3} - \beta_{2} - 3 \beta_{1} - 10$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-9 \beta_{7} - 9 \beta_{6} - 18 \beta_{5} - 5 \beta_{4} - 13 \beta_{3} - 5 \beta_{2} - 13 \beta_{1}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-9 \beta_{7} + 9 \beta_{6} - 17 \beta_{4} - 27 \beta_{3} + 17 \beta_{2} + 27 \beta_{1} + 74$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$81 \beta_{7} + 81 \beta_{6} + 178 \beta_{5} + 37 \beta_{4} + 149 \beta_{3} + 37 \beta_{2} + 149 \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$89 \beta_{7} - 89 \beta_{6} + 201 \beta_{4} + 235 \beta_{3} - 201 \beta_{2} - 235 \beta_{1} - 650$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-761 \beta_{7} - 761 \beta_{6} - 1810 \beta_{5} - 325 \beta_{4} - 1565 \beta_{3} - 325 \beta_{2} - 1565 \beta_{1}$$$$)/2$$

## Character values

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

 $$n$$ $$127$$ $$281$$ $$451$$ $$\chi(n)$$ $$\beta_{5}$$ $$-1$$ $$1$$

## 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
 3.16053i − 2.16053i − 0.692297i 1.69230i − 3.16053i 2.16053i 0.692297i − 1.69230i
−0.707107 + 0.707107i 0 1.00000i −1.52773 1.63280i 0 −0.707107 0.707107i 0.707107 + 0.707107i 0 2.23483 + 0.0743018i
197.2 −0.707107 + 0.707107i 0 1.00000i 2.23483 0.0743018i 0 −0.707107 0.707107i 0.707107 + 0.707107i 0 −1.52773 + 1.63280i
197.3 0.707107 0.707107i 0 1.00000i −1.19663 + 1.88893i 0 0.707107 + 0.707107i −0.707107 0.707107i 0 0.489528 + 2.18183i
197.4 0.707107 0.707107i 0 1.00000i 0.489528 2.18183i 0 0.707107 + 0.707107i −0.707107 0.707107i 0 −1.19663 1.88893i
323.1 −0.707107 0.707107i 0 1.00000i −1.52773 + 1.63280i 0 −0.707107 + 0.707107i 0.707107 0.707107i 0 2.23483 0.0743018i
323.2 −0.707107 0.707107i 0 1.00000i 2.23483 + 0.0743018i 0 −0.707107 + 0.707107i 0.707107 0.707107i 0 −1.52773 1.63280i
323.3 0.707107 + 0.707107i 0 1.00000i −1.19663 1.88893i 0 0.707107 0.707107i −0.707107 + 0.707107i 0 0.489528 2.18183i
323.4 0.707107 + 0.707107i 0 1.00000i 0.489528 + 2.18183i 0 0.707107 0.707107i −0.707107 + 0.707107i 0 −1.19663 + 1.88893i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 323.4 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.d yes 8
3.b odd 2 1 630.2.m.c 8
5.b even 2 1 3150.2.m.j 8
5.c odd 4 1 630.2.m.c 8
5.c odd 4 1 3150.2.m.i 8
15.d odd 2 1 3150.2.m.i 8
15.e even 4 1 inner 630.2.m.d yes 8
15.e even 4 1 3150.2.m.j 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.m.c 8 3.b odd 2 1
630.2.m.c 8 5.c odd 4 1
630.2.m.d yes 8 1.a even 1 1 trivial
630.2.m.d yes 8 15.e even 4 1 inner
3150.2.m.i 8 5.c odd 4 1
3150.2.m.i 8 15.d odd 2 1
3150.2.m.j 8 5.b even 2 1
3150.2.m.j 8 15.e even 4 1

## Hecke kernels

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

 $$T_{11}^{8} + 76 T_{11}^{6} + 1668 T_{11}^{4} + 9728 T_{11}^{2} + 16384$$ $$T_{17}^{8} - 64 T_{17}^{5} + 1040 T_{17}^{4} - 2304 T_{17}^{3} + 2048 T_{17}^{2} + 8192 T_{17} + 16384$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{4} )^{2}$$
$3$ 1
$5$ $$1 + 2 T^{2} - 16 T^{3} + 2 T^{4} - 80 T^{5} + 50 T^{6} + 625 T^{8}$$
$7$ $$( 1 + T^{4} )^{2}$$
$11$ $$1 - 12 T^{2} + 40 T^{4} - 260 T^{6} + 15086 T^{8} - 31460 T^{10} + 585640 T^{12} - 21258732 T^{14} + 214358881 T^{16}$$
$13$ $$1 + 4 T + 8 T^{2} + 28 T^{3} + 80 T^{4} + 372 T^{5} + 1240 T^{6} + 6412 T^{7} + 34174 T^{8} + 83356 T^{9} + 209560 T^{10} + 817284 T^{11} + 2284880 T^{12} + 10396204 T^{13} + 38614472 T^{14} + 250994068 T^{15} + 815730721 T^{16}$$
$17$ $$1 - 64 T^{3} - 252 T^{4} + 960 T^{5} + 2048 T^{6} + 6016 T^{7} - 17786 T^{8} + 102272 T^{9} + 591872 T^{10} + 4716480 T^{11} - 21047292 T^{12} - 90870848 T^{13} + 6975757441 T^{16}$$
$19$ $$( 1 - 19 T^{2} )^{8}$$
$23$ $$1 + 8 T + 32 T^{2} - 8 T^{3} - 924 T^{4} - 5032 T^{5} - 10656 T^{6} + 28456 T^{7} + 420614 T^{8} + 654488 T^{9} - 5637024 T^{10} - 61224344 T^{11} - 258573084 T^{12} - 51490744 T^{13} + 4737148448 T^{14} + 27238603576 T^{15} + 78310985281 T^{16}$$
$29$ $$( 1 - 12 T + 78 T^{2} - 252 T^{3} + 738 T^{4} - 7308 T^{5} + 65598 T^{6} - 292668 T^{7} + 707281 T^{8} )^{2}$$
$31$ $$( 1 + 4 T + 110 T^{2} + 356 T^{3} + 4930 T^{4} + 11036 T^{5} + 105710 T^{6} + 119164 T^{7} + 923521 T^{8} )^{2}$$
$37$ $$1 + 4 T + 8 T^{2} + 220 T^{3} - 48 T^{4} - 8684 T^{5} - 10152 T^{6} - 208404 T^{7} - 3196994 T^{8} - 7710948 T^{9} - 13898088 T^{10} - 439870652 T^{11} - 89959728 T^{12} + 15255670540 T^{13} + 20525811272 T^{14} + 379727508532 T^{15} + 3512479453921 T^{16}$$
$41$ $$1 - 76 T^{2} + 5480 T^{4} - 315940 T^{6} + 12648846 T^{8} - 531095140 T^{10} + 15485170280 T^{12} - 361007922316 T^{14} + 7984925229121 T^{16}$$
$43$ $$1 + 12 T + 72 T^{2} + 300 T^{3} - 1008 T^{4} - 17988 T^{5} - 98280 T^{6} - 509316 T^{7} - 2377282 T^{8} - 21900588 T^{9} - 181719720 T^{10} - 1430171916 T^{11} - 3446151408 T^{12} + 44102532900 T^{13} + 455138139528 T^{14} + 3261823333284 T^{15} + 11688200277601 T^{16}$$
$47$ $$1 - 12 T + 72 T^{2} - 220 T^{3} - 2352 T^{4} + 10116 T^{5} + 72152 T^{6} - 1599596 T^{7} + 18271774 T^{8} - 75181012 T^{9} + 159383768 T^{10} + 1050273468 T^{11} - 11477009712 T^{12} - 50455901540 T^{13} + 776103503688 T^{14} - 6079477445556 T^{15} + 23811286661761 T^{16}$$
$53$ $$1 + 32 T + 512 T^{2} + 5792 T^{3} + 60388 T^{4} + 619872 T^{5} + 5690880 T^{6} + 44817632 T^{7} + 328258854 T^{8} + 2375334496 T^{9} + 15985681920 T^{10} + 92284683744 T^{11} + 476490366628 T^{12} + 2422188295456 T^{13} + 11348152898048 T^{14} + 37590756474784 T^{15} + 62259690411361 T^{16}$$
$59$ $$( 1 - 8 T + 148 T^{2} - 520 T^{3} + 8710 T^{4} - 30680 T^{5} + 515188 T^{6} - 1643032 T^{7} + 12117361 T^{8} )^{2}$$
$61$ $$( 1 + 226 T^{2} + 16 T^{3} + 20162 T^{4} + 976 T^{5} + 840946 T^{6} + 13845841 T^{8} )^{2}$$
$67$ $$1 - 20 T + 200 T^{2} - 1332 T^{3} + 9104 T^{4} - 102596 T^{5} + 1118232 T^{6} - 8938244 T^{7} + 69729150 T^{8} - 598862348 T^{9} + 5019743448 T^{10} - 30857080748 T^{11} + 183455805584 T^{12} - 1798366642524 T^{13} + 18091676433800 T^{14} - 121214232106460 T^{15} + 406067677556641 T^{16}$$
$71$ $$1 - 216 T^{2} + 31388 T^{4} - 3350120 T^{6} + 263319558 T^{8} - 16887954920 T^{10} + 797621843228 T^{12} - 27669661326936 T^{14} + 645753531245761 T^{16}$$
$73$ $$1 - 36 T + 648 T^{2} - 8620 T^{3} + 98192 T^{4} - 980116 T^{5} + 8807960 T^{6} - 73999132 T^{7} + 618591198 T^{8} - 5401936636 T^{9} + 46937618840 T^{10} - 381281785972 T^{11} + 2788480080272 T^{12} - 17869877131660 T^{13} + 98064578635272 T^{14} - 397706346687492 T^{15} + 806460091894081 T^{16}$$
$79$ $$1 - 280 T^{2} + 32732 T^{4} - 1914792 T^{6} + 96372294 T^{8} - 11950216872 T^{10} + 1274914051292 T^{12} - 68064487545880 T^{14} + 1517108809906561 T^{16}$$
$83$ $$1 + 56 T + 1568 T^{2} + 30056 T^{3} + 449924 T^{4} + 5649064 T^{5} + 62548320 T^{6} + 632947128 T^{7} + 5962291750 T^{8} + 52534611624 T^{9} + 430895376480 T^{10} + 3230061357368 T^{11} + 21352637617604 T^{12} + 118391805566008 T^{13} + 512642505442592 T^{14} + 1519618855419112 T^{15} + 2252292232139041 T^{16}$$
$89$ $$( 1 - 36 T + 734 T^{2} - 10004 T^{3} + 106562 T^{4} - 890356 T^{5} + 5814014 T^{6} - 25378884 T^{7} + 62742241 T^{8} )^{2}$$
$97$ $$1 + 4 T + 8 T^{2} + 1852 T^{3} + 7856 T^{4} - 48156 T^{5} + 1459480 T^{6} + 10407580 T^{7} - 90680354 T^{8} + 1009535260 T^{9} + 13732247320 T^{10} - 43950680988 T^{11} + 695486031536 T^{12} + 15903754155964 T^{13} + 6663776039432 T^{14} + 323193137912452 T^{15} + 7837433594376961 T^{16}$$