# Properties

 Label 630.2.b.b Level 630 Weight 2 Character orbit 630.b 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.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.03057532734$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.7442857984.4 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 26 x^{6} + 205 x^{4} + 540 x^{2} + 324$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{7} + 20 \nu^{5} + 12 \nu^{4} + 73 \nu^{3} + 156 \nu^{2} - 198 \nu + 216$$$$)/144$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{7} - 20 \nu^{5} + 12 \nu^{4} - 73 \nu^{3} + 156 \nu^{2} + 198 \nu + 216$$$$)/144$$ $$\beta_{3}$$ $$=$$ $$($$$$5 \nu^{7} + 112 \nu^{5} + 665 \nu^{3} + 954 \nu$$$$)/432$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{6} - 20 \nu^{4} - 73 \nu^{2} + 54$$$$)/24$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{7} + 6 \nu^{6} + 20 \nu^{5} + 132 \nu^{4} + 73 \nu^{3} + 738 \nu^{2} - 54 \nu + 900$$$$)/144$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{7} - 6 \nu^{6} + 20 \nu^{5} - 132 \nu^{4} + 73 \nu^{3} - 738 \nu^{2} - 54 \nu - 900$$$$)/144$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{7} - 26 \nu^{5} - 187 \nu^{3} - 306 \nu$$$$)/36$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{6} + \beta_{5} + \beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{6} + \beta_{5} + 2 \beta_{4} - \beta_{2} - \beta_{1} - 14$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$2 \beta_{7} - 13 \beta_{6} - 13 \beta_{5} + 12 \beta_{3} - 7 \beta_{2} + 7 \beta_{1}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$13 \beta_{6} - 13 \beta_{5} - 26 \beta_{4} + 25 \beta_{2} + 25 \beta_{1} + 146$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-50 \beta_{7} + 163 \beta_{6} + 163 \beta_{5} - 228 \beta_{3} + 73 \beta_{2} - 73 \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$-187 \beta_{6} + 187 \beta_{5} + 326 \beta_{4} - 427 \beta_{2} - 427 \beta_{1} - 1790$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$854 \beta_{7} - 2113 \beta_{6} - 2113 \beta_{5} + 3684 \beta_{3} - 895 \beta_{2} + 895 \beta_{1}$$$$)/2$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
251.1
 0.916813i 2.73923i − 3.73923i − 1.91681i − 0.916813i − 2.73923i 3.73923i 1.91681i
1.00000i 0 −1.00000 1.00000 0 −2.56510 0.648285i 1.00000i 0 1.00000i
251.2 1.00000i 0 −1.00000 1.00000 0 −1.80230 + 1.93693i 1.00000i 0 1.00000i
251.3 1.00000i 0 −1.00000 1.00000 0 0.0951965 2.64404i 1.00000i 0 1.00000i
251.4 1.00000i 0 −1.00000 1.00000 0 2.27220 + 1.35539i 1.00000i 0 1.00000i
251.5 1.00000i 0 −1.00000 1.00000 0 −2.56510 + 0.648285i 1.00000i 0 1.00000i
251.6 1.00000i 0 −1.00000 1.00000 0 −1.80230 1.93693i 1.00000i 0 1.00000i
251.7 1.00000i 0 −1.00000 1.00000 0 0.0951965 + 2.64404i 1.00000i 0 1.00000i
251.8 1.00000i 0 −1.00000 1.00000 0 2.27220 1.35539i 1.00000i 0 1.00000i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 251.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.c 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.b.b yes 8
3.b odd 2 1 630.2.b.a 8
4.b odd 2 1 5040.2.f.i 8
5.b even 2 1 3150.2.b.f 8
5.c odd 4 1 3150.2.d.a 8
5.c odd 4 1 3150.2.d.f 8
7.b odd 2 1 630.2.b.a 8
12.b even 2 1 5040.2.f.f 8
15.d odd 2 1 3150.2.b.e 8
15.e even 4 1 3150.2.d.c 8
15.e even 4 1 3150.2.d.d 8
21.c even 2 1 inner 630.2.b.b yes 8
28.d even 2 1 5040.2.f.f 8
35.c odd 2 1 3150.2.b.e 8
35.f even 4 1 3150.2.d.c 8
35.f even 4 1 3150.2.d.d 8
84.h odd 2 1 5040.2.f.i 8
105.g even 2 1 3150.2.b.f 8
105.k odd 4 1 3150.2.d.a 8
105.k odd 4 1 3150.2.d.f 8

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{17}^{4} - 50 T_{17}^{2} - 120 T_{17} - 72$$ acting on $$S_{2}^{\mathrm{new}}(630, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{4}$$
$3$ 1
$5$ $$( 1 - T )^{8}$$
$7$ $$1 + 4 T + 6 T^{2} + 4 T^{3} + 2 T^{4} + 28 T^{5} + 294 T^{6} + 1372 T^{7} + 2401 T^{8}$$
$11$ $$1 - 36 T^{2} + 776 T^{4} - 11916 T^{6} + 146094 T^{8} - 1441836 T^{10} + 11361416 T^{12} - 63776196 T^{14} + 214358881 T^{16}$$
$13$ $$1 - 44 T^{2} + 872 T^{4} - 9828 T^{6} + 102190 T^{8} - 1660932 T^{10} + 24905192 T^{12} - 212379596 T^{14} + 815730721 T^{16}$$
$17$ $$( 1 + 18 T^{2} - 120 T^{3} - 38 T^{4} - 2040 T^{5} + 5202 T^{6} + 83521 T^{8} )^{2}$$
$19$ $$1 - 44 T^{2} + 1256 T^{4} - 31140 T^{6} + 702382 T^{8} - 11241540 T^{10} + 163683176 T^{12} - 2070018764 T^{14} + 16983563041 T^{16}$$
$23$ $$1 - 80 T^{2} + 3740 T^{4} - 123312 T^{6} + 3185414 T^{8} - 65232048 T^{10} + 1046605340 T^{12} - 11842871120 T^{14} + 78310985281 T^{16}$$
$29$ $$1 - 128 T^{2} + 8732 T^{4} - 399744 T^{6} + 13409894 T^{8} - 336184704 T^{10} + 6175977692 T^{12} - 76137385088 T^{14} + 500246412961 T^{16}$$
$31$ $$1 - 32 T^{2} + 572 T^{4} - 21984 T^{6} + 1650310 T^{8} - 21126624 T^{10} + 528254012 T^{12} - 28400117792 T^{14} + 852891037441 T^{16}$$
$37$ $$( 1 + 4 T + 84 T^{2} + 532 T^{3} + 3602 T^{4} + 19684 T^{5} + 114996 T^{6} + 202612 T^{7} + 1874161 T^{8} )^{2}$$
$41$ $$( 1 - 4 T + 142 T^{2} - 468 T^{3} + 8354 T^{4} - 19188 T^{5} + 238702 T^{6} - 275684 T^{7} + 2825761 T^{8} )^{2}$$
$43$ $$( 1 + 8 T + 146 T^{2} + 984 T^{3} + 8842 T^{4} + 42312 T^{5} + 269954 T^{6} + 636056 T^{7} + 3418801 T^{8} )^{2}$$
$47$ $$( 1 - 20 T + 166 T^{2} - 396 T^{3} - 838 T^{4} - 18612 T^{5} + 366694 T^{6} - 2076460 T^{7} + 4879681 T^{8} )^{2}$$
$53$ $$1 - 304 T^{2} + 43772 T^{4} - 3946320 T^{6} + 247218022 T^{8} - 11085212880 T^{10} + 345382134332 T^{12} - 6737965783216 T^{14} + 62259690411361 T^{16}$$
$59$ $$( 1 + 42 T^{2} - 312 T^{3} + 3106 T^{4} - 18408 T^{5} + 146202 T^{6} + 12117361 T^{8} )^{2}$$
$61$ $$1 - 236 T^{2} + 31208 T^{4} - 2804772 T^{6} + 194357422 T^{8} - 10436556612 T^{10} + 432101005928 T^{12} - 12158808349196 T^{14} + 191707312997281 T^{16}$$
$67$ $$( 1 - 16 T + 218 T^{2} - 1728 T^{3} + 15338 T^{4} - 115776 T^{5} + 978602 T^{6} - 4812208 T^{7} + 20151121 T^{8} )^{2}$$
$71$ $$1 - 460 T^{2} + 98600 T^{4} - 12828132 T^{6} + 1106047246 T^{8} - 64666613412 T^{10} + 2505591746600 T^{12} - 58926130603660 T^{14} + 645753531245761 T^{16}$$
$73$ $$1 - 232 T^{2} + 30332 T^{4} - 3074136 T^{6} + 255245510 T^{8} - 16382070744 T^{10} + 861375446012 T^{12} - 35109540499048 T^{14} + 806460091894081 T^{16}$$
$79$ $$( 1 - 4 T + 48 T^{2} - 532 T^{3} + 9470 T^{4} - 42028 T^{5} + 299568 T^{6} - 1972156 T^{7} + 38950081 T^{8} )^{2}$$
$83$ $$( 1 + 8 T + 244 T^{2} + 1800 T^{3} + 27878 T^{4} + 149400 T^{5} + 1680916 T^{6} + 4574296 T^{7} + 47458321 T^{8} )^{2}$$
$89$ $$( 1 - 4 T + 118 T^{2} + 1308 T^{3} - 670 T^{4} + 116412 T^{5} + 934678 T^{6} - 2819876 T^{7} + 62742241 T^{8} )^{2}$$
$97$ $$1 - 208 T^{2} + 38300 T^{4} - 4059696 T^{6} + 468613574 T^{8} - 38197679664 T^{10} + 3390671462300 T^{12} - 173258177025232 T^{14} + 7837433594376961 T^{16}$$