# Properties

 Label 3150.2.bp.f Level 3150 Weight 2 Character orbit 3150.bp Analytic conductor 25.153 Analytic rank 0 Dimension 8 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 3150.bp (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$25.1528766367$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{24})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 630) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{24}^{4} ) q^{2} -\zeta_{24}^{4} q^{4} + ( -3 \zeta_{24}^{3} + 2 \zeta_{24}^{7} ) q^{7} - q^{8} +O(q^{10})$$ $$q + ( 1 - \zeta_{24}^{4} ) q^{2} -\zeta_{24}^{4} q^{4} + ( -3 \zeta_{24}^{3} + 2 \zeta_{24}^{7} ) q^{7} - q^{8} + ( 4 - \zeta_{24}^{3} - 2 \zeta_{24}^{4} + \zeta_{24}^{5} ) q^{11} + ( 2 + \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{7} ) q^{13} + ( -\zeta_{24}^{3} + 3 \zeta_{24}^{7} ) q^{14} + ( -1 + \zeta_{24}^{4} ) q^{16} + ( 4 - 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{4} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{17} + ( -2 \zeta_{24} + \zeta_{24}^{2} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{19} + ( 2 + \zeta_{24} - \zeta_{24}^{3} - 4 \zeta_{24}^{4} + \zeta_{24}^{7} ) q^{22} + ( -2 - \zeta_{24}^{2} + 2 \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{23} + ( 2 + \zeta_{24} - 2 \zeta_{24}^{4} - \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{26} + ( 2 \zeta_{24}^{3} + \zeta_{24}^{7} ) q^{28} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{29} + ( 2 \zeta_{24} - 2 \zeta_{24}^{2} + 2 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{31} + \zeta_{24}^{4} q^{32} + ( 2 - 4 \zeta_{24}^{4} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{34} + ( -\zeta_{24} - 6 \zeta_{24}^{2} - \zeta_{24}^{3} ) q^{37} + ( \zeta_{24}^{2} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - \zeta_{24}^{6} ) q^{38} + ( 4 + 2 \zeta_{24} + 2 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{41} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{43} + ( -2 + \zeta_{24} - 2 \zeta_{24}^{4} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{44} + ( \zeta_{24}^{2} + 2 \zeta_{24}^{4} + \zeta_{24}^{6} ) q^{46} + ( 1 + 4 \zeta_{24} + 2 \zeta_{24}^{2} + \zeta_{24}^{4} - 4 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{47} + ( 8 \zeta_{24}^{2} - 3 \zeta_{24}^{6} ) q^{49} + ( -\zeta_{24}^{3} - 2 \zeta_{24}^{4} - \zeta_{24}^{5} ) q^{52} + ( 2 \zeta_{24} + 2 \zeta_{24}^{2} - 6 \zeta_{24}^{3} + \zeta_{24}^{4} - 6 \zeta_{24}^{5} + 2 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{53} + ( 3 \zeta_{24}^{3} - 2 \zeta_{24}^{7} ) q^{56} + ( -4 \zeta_{24} - 4 \zeta_{24}^{2} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{58} + ( -3 \zeta_{24} - 2 \zeta_{24}^{2} - 2 \zeta_{24}^{3} + 6 \zeta_{24}^{4} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{6} - 3 \zeta_{24}^{7} ) q^{59} + ( -2 \zeta_{24} - 2 \zeta_{24}^{2} - 6 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{61} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{6} ) q^{62} + q^{64} + ( 8 + 6 \zeta_{24} - 6 \zeta_{24}^{2} + 6 \zeta_{24}^{3} - 4 \zeta_{24}^{4} - 6 \zeta_{24}^{5} + 6 \zeta_{24}^{6} - 6 \zeta_{24}^{7} ) q^{67} + ( -2 + 2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{4} ) q^{68} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{6} + 6 \zeta_{24}^{7} ) q^{71} + ( 4 \zeta_{24} - 6 \zeta_{24}^{4} + 4 \zeta_{24}^{7} ) q^{73} + ( -\zeta_{24} - 6 \zeta_{24}^{2} - \zeta_{24}^{3} + \zeta_{24}^{5} + 6 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{74} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{76} + ( 1 + 2 \zeta_{24}^{2} - 8 \zeta_{24}^{3} - 3 \zeta_{24}^{4} + \zeta_{24}^{6} + 10 \zeta_{24}^{7} ) q^{77} + ( 6 + 2 \zeta_{24} + 4 \zeta_{24}^{2} - 6 \zeta_{24}^{3} - 6 \zeta_{24}^{4} + 4 \zeta_{24}^{5} - 8 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{79} + ( 4 - \zeta_{24} + 3 \zeta_{24}^{3} - 4 \zeta_{24}^{4} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{82} + ( -6 \zeta_{24} + 6 \zeta_{24}^{3} + 8 \zeta_{24}^{5} - 2 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{83} + ( -2 \zeta_{24} - 2 \zeta_{24}^{2} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{86} + ( -4 + \zeta_{24}^{3} + 2 \zeta_{24}^{4} - \zeta_{24}^{5} ) q^{88} + ( 4 - 4 \zeta_{24} + 7 \zeta_{24}^{3} - 4 \zeta_{24}^{4} - 3 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{89} + ( -2 - 3 \zeta_{24}^{2} - 6 \zeta_{24}^{3} - \zeta_{24}^{4} + 2 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{91} + ( 2 + 2 \zeta_{24}^{2} - \zeta_{24}^{6} ) q^{92} + ( 2 + 2 \zeta_{24}^{2} + 4 \zeta_{24}^{3} - \zeta_{24}^{4} - 4 \zeta_{24}^{5} - 2 \zeta_{24}^{6} ) q^{94} + ( -6 + 2 \zeta_{24} - 12 \zeta_{24}^{2} + 2 \zeta_{24}^{3} - 6 \zeta_{24}^{5} + 6 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{97} + ( 5 \zeta_{24}^{2} - 8 \zeta_{24}^{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{2} - 4q^{4} - 8q^{8} + O(q^{10})$$ $$8q + 4q^{2} - 4q^{4} - 8q^{8} + 24q^{11} + 16q^{13} - 4q^{16} + 24q^{17} - 8q^{23} + 8q^{26} + 4q^{32} + 32q^{41} - 24q^{44} + 8q^{46} + 12q^{47} - 8q^{52} + 4q^{53} + 24q^{59} + 8q^{64} + 48q^{67} - 24q^{68} - 24q^{73} - 4q^{77} + 24q^{79} + 16q^{82} - 24q^{88} + 16q^{89} - 20q^{91} + 16q^{92} + 12q^{94} - 48q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$451$$ $$2801$$ $$\chi(n)$$ $$-1$$ $$1 - \zeta_{24}^{4}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
899.1
 0.965926 − 0.258819i −0.258819 − 0.965926i 0.258819 + 0.965926i −0.965926 + 0.258819i 0.965926 + 0.258819i −0.258819 + 0.965926i 0.258819 − 0.965926i −0.965926 − 0.258819i
0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 0 −2.63896 + 0.189469i −1.00000 0 0
899.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 0 −0.189469 2.63896i −1.00000 0 0
899.3 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 0 0.189469 + 2.63896i −1.00000 0 0
899.4 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 0 2.63896 0.189469i −1.00000 0 0
1349.1 0.500000 0.866025i 0 −0.500000 0.866025i 0 0 −2.63896 0.189469i −1.00000 0 0
1349.2 0.500000 0.866025i 0 −0.500000 0.866025i 0 0 −0.189469 + 2.63896i −1.00000 0 0
1349.3 0.500000 0.866025i 0 −0.500000 0.866025i 0 0 0.189469 2.63896i −1.00000 0 0
1349.4 0.500000 0.866025i 0 −0.500000 0.866025i 0 0 2.63896 + 0.189469i −1.00000 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1349.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
105.p even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3150.2.bp.f 8
3.b odd 2 1 3150.2.bp.a 8
5.b even 2 1 3150.2.bp.c 8
5.c odd 4 1 630.2.be.b yes 8
5.c odd 4 1 3150.2.bf.c 8
7.d odd 6 1 3150.2.bp.d 8
15.d odd 2 1 3150.2.bp.d 8
15.e even 4 1 630.2.be.a 8
15.e even 4 1 3150.2.bf.b 8
21.g even 6 1 3150.2.bp.c 8
35.i odd 6 1 3150.2.bp.a 8
35.k even 12 1 630.2.be.a 8
35.k even 12 1 3150.2.bf.b 8
35.k even 12 1 4410.2.b.e 8
35.l odd 12 1 4410.2.b.b 8
105.p even 6 1 inner 3150.2.bp.f 8
105.w odd 12 1 630.2.be.b yes 8
105.w odd 12 1 3150.2.bf.c 8
105.w odd 12 1 4410.2.b.b 8
105.x even 12 1 4410.2.b.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.be.a 8 15.e even 4 1
630.2.be.a 8 35.k even 12 1
630.2.be.b yes 8 5.c odd 4 1
630.2.be.b yes 8 105.w odd 12 1
3150.2.bf.b 8 15.e even 4 1
3150.2.bf.b 8 35.k even 12 1
3150.2.bf.c 8 5.c odd 4 1
3150.2.bf.c 8 105.w odd 12 1
3150.2.bp.a 8 3.b odd 2 1
3150.2.bp.a 8 35.i odd 6 1
3150.2.bp.c 8 5.b even 2 1
3150.2.bp.c 8 21.g even 6 1
3150.2.bp.d 8 7.d odd 6 1
3150.2.bp.d 8 15.d odd 2 1
3150.2.bp.f 8 1.a even 1 1 trivial
3150.2.bp.f 8 105.p even 6 1 inner
4410.2.b.b 8 35.l odd 12 1
4410.2.b.b 8 105.w odd 12 1
4410.2.b.e 8 35.k even 12 1
4410.2.b.e 8 105.x even 12 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3150, [\chi])$$:

 $$T_{11}^{8} - \cdots$$ $$T_{13}^{4} - 8 T_{13}^{3} + 20 T_{13}^{2} - 16 T_{13} + 1$$ $$T_{17}^{8} - \cdots$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T + T^{2} )^{4}$$
$3$ 1
$5$ 1
$7$ $$1 - 94 T^{4} + 2401 T^{8}$$
$11$ $$1 - 24 T + 304 T^{2} - 2688 T^{3} + 18481 T^{4} - 104232 T^{5} + 496816 T^{6} - 2036016 T^{7} + 7239280 T^{8} - 22396176 T^{9} + 60114736 T^{10} - 138732792 T^{11} + 270580321 T^{12} - 432905088 T^{13} + 538554544 T^{14} - 467692104 T^{15} + 214358881 T^{16}$$
$13$ $$( 1 - 8 T + 72 T^{2} - 328 T^{3} + 1535 T^{4} - 4264 T^{5} + 12168 T^{6} - 17576 T^{7} + 28561 T^{8} )^{2}$$
$17$ $$1 - 24 T + 316 T^{2} - 2976 T^{3} + 22330 T^{4} - 141240 T^{5} + 776752 T^{6} - 3781944 T^{7} + 16463059 T^{8} - 64293048 T^{9} + 224481328 T^{10} - 693912120 T^{11} + 1865023930 T^{12} - 4225494432 T^{13} + 7627471804 T^{14} - 9848128152 T^{15} + 6975757441 T^{16}$$
$19$ $$1 + 58 T^{2} + 1881 T^{4} + 1872 T^{5} + 44906 T^{6} + 71184 T^{7} + 869156 T^{8} + 1352496 T^{9} + 16211066 T^{10} + 12840048 T^{11} + 245133801 T^{12} + 2728661098 T^{14} + 16983563041 T^{16}$$
$23$ $$( 1 + 4 T - 31 T^{2} + 4 T^{3} + 1312 T^{4} + 92 T^{5} - 16399 T^{6} + 48668 T^{7} + 279841 T^{8} )^{2}$$
$29$ $$( 1 - 36 T^{2} + 470 T^{4} - 30276 T^{6} + 707281 T^{8} )^{2}$$
$31$ $$1 + 100 T^{2} + 5706 T^{4} + 237200 T^{6} + 7904915 T^{8} + 227949200 T^{10} + 5269610826 T^{12} + 88750368100 T^{14} + 852891037441 T^{16}$$
$37$ $$1 + 72 T^{2} + 1441 T^{4} + 2520 T^{5} + 74088 T^{6} + 269496 T^{7} + 5102928 T^{8} + 9971352 T^{9} + 101426472 T^{10} + 127645560 T^{11} + 2700666001 T^{12} + 184732301448 T^{14} + 3512479453921 T^{16}$$
$41$ $$( 1 - 16 T + 232 T^{2} - 2000 T^{3} + 15591 T^{4} - 82000 T^{5} + 389992 T^{6} - 1102736 T^{7} + 2825761 T^{8} )^{2}$$
$43$ $$1 - 296 T^{2} + 39900 T^{4} - 3207640 T^{6} + 168538406 T^{8} - 5930926360 T^{10} + 136410159900 T^{12} - 1871123462504 T^{14} + 11688200277601 T^{16}$$
$47$ $$1 - 12 T + 182 T^{2} - 1608 T^{3} + 14001 T^{4} - 79344 T^{5} + 532582 T^{6} - 2135988 T^{7} + 16324772 T^{8} - 100391436 T^{9} + 1176473638 T^{10} - 8237732112 T^{11} + 68320413681 T^{12} - 368786771256 T^{13} + 1961817189878 T^{14} - 6079477445556 T^{15} + 23811286661761 T^{16}$$
$53$ $$1 - 4 T - 66 T^{2} + 984 T^{3} - 631 T^{4} - 50712 T^{5} + 363998 T^{6} + 1081556 T^{7} - 22439580 T^{8} + 57322468 T^{9} + 1022470382 T^{10} - 7549850424 T^{11} - 4978893511 T^{12} + 411504365112 T^{13} - 1462847834514 T^{14} - 4698844559348 T^{15} + 62259690411361 T^{16}$$
$59$ $$1 - 24 T + 224 T^{2} - 1392 T^{3} + 9358 T^{4} - 16056 T^{5} - 613888 T^{6} + 6806952 T^{7} - 48734957 T^{8} + 401610168 T^{9} - 2136944128 T^{10} - 3297565224 T^{11} + 113394264238 T^{12} - 995174624208 T^{13} + 9448439535584 T^{14} - 59727635635656 T^{15} + 146830437604321 T^{16}$$
$61$ $$1 + 124 T^{2} + 6186 T^{4} + 30240 T^{5} + 293552 T^{6} + 4411200 T^{7} + 18080963 T^{8} + 269083200 T^{9} + 1092306992 T^{10} + 6863905440 T^{11} + 85650372426 T^{12} + 6388526420764 T^{14} + 191707312997281 T^{16}$$
$67$ $$1 - 48 T + 1108 T^{2} - 16320 T^{3} + 166858 T^{4} - 1140048 T^{5} + 3573520 T^{6} + 22726896 T^{7} - 369529661 T^{8} + 1522702032 T^{9} + 16041531280 T^{10} - 342884256624 T^{11} + 3362375747818 T^{12} - 22034041746240 T^{13} + 100227887443252 T^{14} - 290914157055504 T^{15} + 406067677556641 T^{16}$$
$71$ $$1 - 280 T^{2} + 41532 T^{4} - 4371368 T^{6} + 352979654 T^{8} - 22036066088 T^{10} + 1055397935292 T^{12} - 35868079497880 T^{14} + 645753531245761 T^{16}$$
$73$ $$( 1 + 12 T - 6 T^{2} + 48 T^{3} + 6659 T^{4} + 3504 T^{5} - 31974 T^{6} + 4668204 T^{7} + 28398241 T^{8} )^{2}$$
$79$ $$1 - 24 T + 252 T^{2} - 1584 T^{3} + 5050 T^{4} - 17832 T^{5} - 542160 T^{6} + 18801768 T^{7} - 226869549 T^{8} + 1485339672 T^{9} - 3383620560 T^{10} - 8791871448 T^{11} + 196697909050 T^{12} - 4874057336016 T^{13} + 61258038791292 T^{14} - 460893815667816 T^{15} + 1517108809906561 T^{16}$$
$83$ $$1 - 232 T^{2} + 39708 T^{4} - 4492952 T^{6} + 427617638 T^{8} - 30951946328 T^{10} + 1884475010268 T^{12} - 75850166621608 T^{14} + 2252292232139041 T^{16}$$
$89$ $$1 - 16 T - 48 T^{2} + 1824 T^{3} + 2078 T^{4} - 135504 T^{5} + 38912 T^{6} + 7970384 T^{7} - 72309597 T^{8} + 709364176 T^{9} + 308221952 T^{10} - 95526119376 T^{11} + 130378376798 T^{12} + 10185324434976 T^{13} - 23855101966128 T^{14} - 707701358328464 T^{15} + 3936588805702081 T^{16}$$
$97$ $$( 1 + 24 T + 276 T^{2} + 1608 T^{3} + 10022 T^{4} + 155976 T^{5} + 2596884 T^{6} + 21904152 T^{7} + 88529281 T^{8} )^{2}$$