# Properties

 Label 3150.2.m.k Level 3150 Weight 2 Character orbit 3150.m 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.m (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$25.1528766367$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{24})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{4}$$ 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 $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$127$$ $$451$$ $$2801$$ $$\chi(n)$$ $$\zeta_{24}^{3}$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1457.1
 −0.965926 − 0.258819i 0.258819 + 0.965926i 0.965926 + 0.258819i −0.258819 − 0.965926i −0.965926 + 0.258819i 0.258819 − 0.965926i 0.965926 − 0.258819i −0.258819 + 0.965926i
−0.707107 + 0.707107i 0 1.00000i 0 0 0.707107 + 0.707107i 0.707107 + 0.707107i 0 0
1457.2 −0.707107 + 0.707107i 0 1.00000i 0 0 0.707107 + 0.707107i 0.707107 + 0.707107i 0 0
1457.3 0.707107 0.707107i 0 1.00000i 0 0 −0.707107 0.707107i −0.707107 0.707107i 0 0
1457.4 0.707107 0.707107i 0 1.00000i 0 0 −0.707107 0.707107i −0.707107 0.707107i 0 0
2843.1 −0.707107 0.707107i 0 1.00000i 0 0 0.707107 0.707107i 0.707107 0.707107i 0 0
2843.2 −0.707107 0.707107i 0 1.00000i 0 0 0.707107 0.707107i 0.707107 0.707107i 0 0
2843.3 0.707107 + 0.707107i 0 1.00000i 0 0 −0.707107 + 0.707107i −0.707107 + 0.707107i 0 0
2843.4 0.707107 + 0.707107i 0 1.00000i 0 0 −0.707107 + 0.707107i −0.707107 + 0.707107i 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2843.4 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 3150.2.m.k yes 8
3.b odd 2 1 3150.2.m.l yes 8
5.b even 2 1 3150.2.m.g 8
5.c odd 4 1 3150.2.m.h yes 8
5.c odd 4 1 3150.2.m.l yes 8
15.d odd 2 1 3150.2.m.h yes 8
15.e even 4 1 3150.2.m.g 8
15.e even 4 1 inner 3150.2.m.k yes 8

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

## Hecke kernels

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

 $$T_{11}^{4} + 12 T_{11}^{2} + 4$$ $$T_{13}^{8} - \cdots$$ $$T_{17}^{8} - 96 T_{17}^{5} + 770 T_{17}^{4} - 2496 T_{17}^{3} + 4608 T_{17}^{2} - 4512 T_{17} + 2209$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{4} )^{2}$$
$3$ 1
$5$ 1
$7$ $$( 1 + T^{4} )^{2}$$
$11$ $$( 1 - 32 T^{2} + 466 T^{4} - 3872 T^{6} + 14641 T^{8} )^{2}$$
$13$ $$1 - 8 T + 32 T^{2} - 128 T^{3} + 574 T^{4} - 2360 T^{5} + 8704 T^{6} - 35112 T^{7} + 139491 T^{8} - 456456 T^{9} + 1470976 T^{10} - 5184920 T^{11} + 16394014 T^{12} - 47525504 T^{13} + 154457888 T^{14} - 501988136 T^{15} + 815730721 T^{16}$$
$17$ $$1 - 96 T^{3} + 158 T^{4} + 2400 T^{5} + 4608 T^{6} + 8544 T^{7} - 207741 T^{8} + 145248 T^{9} + 1331712 T^{10} + 11791200 T^{11} + 13196318 T^{12} - 136306272 T^{13} + 6975757441 T^{16}$$
$19$ $$( 1 - 48 T^{2} + 1202 T^{4} - 17328 T^{6} + 130321 T^{8} )^{2}$$
$23$ $$1 + 16 T + 128 T^{2} + 800 T^{3} + 4670 T^{4} + 26576 T^{5} + 147456 T^{6} + 807760 T^{7} + 4140483 T^{8} + 18578480 T^{9} + 78004224 T^{10} + 323350192 T^{11} + 1306857470 T^{12} + 5149074400 T^{13} + 18948593792 T^{14} + 54477207152 T^{15} + 78310985281 T^{16}$$
$29$ $$( 1 + 58 T^{2} + 96 T^{3} + 2019 T^{4} + 2784 T^{5} + 48778 T^{6} + 707281 T^{8} )^{2}$$
$31$ $$( 1 + 42 T^{2} - 192 T^{3} + 1139 T^{4} - 5952 T^{5} + 40362 T^{6} + 923521 T^{8} )^{2}$$
$37$ $$1 + 16 T + 128 T^{2} + 784 T^{3} + 4324 T^{4} + 27760 T^{5} + 198016 T^{6} + 1322736 T^{7} + 8422566 T^{8} + 48941232 T^{9} + 271083904 T^{10} + 1406127280 T^{11} + 8103872164 T^{12} + 54365662288 T^{13} + 328412980352 T^{14} + 1518910034128 T^{15} + 3512479453921 T^{16}$$
$41$ $$1 - 172 T^{2} + 15786 T^{4} - 1010672 T^{6} + 47929043 T^{8} - 1698939632 T^{10} + 44607463146 T^{12} - 817017929452 T^{14} + 7984925229121 T^{16}$$
$43$ $$1 + 8 T + 32 T^{2} + 368 T^{3} + 1438 T^{4} - 5752 T^{5} - 24320 T^{6} - 270552 T^{7} - 3004701 T^{8} - 11633736 T^{9} - 44967680 T^{10} - 457324264 T^{11} + 4916235838 T^{12} + 54099107024 T^{13} + 202283617568 T^{14} + 2174548888856 T^{15} + 11688200277601 T^{16}$$
$47$ $$1 - 8 T + 32 T^{2} - 520 T^{3} + 2012 T^{4} + 17432 T^{5} - 68640 T^{6} + 1046296 T^{7} - 15494202 T^{8} + 49175912 T^{9} - 151625760 T^{10} + 1809842536 T^{11} + 9817918172 T^{12} - 119259403640 T^{13} + 344934890528 T^{14} - 4052984963704 T^{15} + 23811286661761 T^{16}$$
$53$ $$1 + 32 T + 512 T^{2} + 6592 T^{3} + 79454 T^{4} + 821920 T^{5} + 7348224 T^{6} + 61872608 T^{7} + 480643491 T^{8} + 3279248224 T^{9} + 20641161216 T^{10} + 122364983840 T^{11} + 626930277374 T^{12} + 2756744689856 T^{13} + 11348152898048 T^{14} + 37590756474784 T^{15} + 62259690411361 T^{16}$$
$59$ $$( 1 + 4 T + 170 T^{2} + 856 T^{3} + 13027 T^{4} + 50504 T^{5} + 591770 T^{6} + 821516 T^{7} + 12117361 T^{8} )^{2}$$
$61$ $$( 1 + 16 T + 222 T^{2} + 2480 T^{3} + 20579 T^{4} + 151280 T^{5} + 826062 T^{6} + 3631696 T^{7} + 13845841 T^{8} )^{2}$$
$67$ $$1 - 16 T + 128 T^{2} - 1648 T^{3} + 12196 T^{4} + 9104 T^{5} - 348800 T^{6} + 6630000 T^{7} - 95097114 T^{8} + 444210000 T^{9} - 1565763200 T^{10} + 2738146352 T^{11} + 245763071716 T^{12} - 2225006176336 T^{13} + 11578672917632 T^{14} - 96971385685168 T^{15} + 406067677556641 T^{16}$$
$71$ $$( 1 - 156 T^{2} + 13094 T^{4} - 786396 T^{6} + 25411681 T^{8} )^{2}$$
$73$ $$1 + 2044 T^{4} + 20460870 T^{8} + 58046004604 T^{12} + 806460091894081 T^{16}$$
$79$ $$1 - 320 T^{2} + 58500 T^{4} - 7347136 T^{6} + 669209222 T^{8} - 45853475776 T^{10} + 2278579738500 T^{12} - 77787985766720 T^{14} + 1517108809906561 T^{16}$$
$83$ $$1 - 8 T + 32 T^{2} - 160 T^{3} + 10142 T^{4} - 122968 T^{5} + 672000 T^{6} - 7632056 T^{7} + 75142083 T^{8} - 633460648 T^{9} + 4629408000 T^{10} - 70311503816 T^{11} + 481322291582 T^{12} - 630246502880 T^{13} + 10462091947808 T^{14} - 217088407917016 T^{15} + 2252292232139041 T^{16}$$
$89$ $$( 1 - 8 T + 316 T^{2} - 1912 T^{3} + 40422 T^{4} - 170168 T^{5} + 2503036 T^{6} - 5639752 T^{7} + 62742241 T^{8} )^{2}$$
$97$ $$1 - 16 T + 128 T^{2} - 1136 T^{3} + 10556 T^{4} - 116816 T^{5} + 1163136 T^{6} - 11493168 T^{7} + 113239174 T^{8} - 1114837296 T^{9} + 10943946624 T^{10} - 106614809168 T^{11} + 934515090236 T^{12} - 9755218531952 T^{13} + 106620416630912 T^{14} - 1292772551649808 T^{15} + 7837433594376961 T^{16}$$