# Properties

 Label 3150.2.bf.a Level 3150 Weight 2 Character orbit 3150.bf Analytic conductor 25.153 Analytic rank 0 Dimension 8 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 3150.bf (of order $$6$$, degree $$2$$, 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 126) 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.

 $$f(q)$$ $$=$$ $$q -\zeta_{24}^{2} q^{2} + \zeta_{24}^{4} q^{4} + ( \zeta_{24} + 2 \zeta_{24}^{3} - \zeta_{24}^{4} - 2 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{7} -\zeta_{24}^{6} q^{8} +O(q^{10})$$ $$q -\zeta_{24}^{2} q^{2} + \zeta_{24}^{4} q^{4} + ( \zeta_{24} + 2 \zeta_{24}^{3} - \zeta_{24}^{4} - 2 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{7} -\zeta_{24}^{6} q^{8} + ( -3 \zeta_{24}^{2} + 3 \zeta_{24}^{6} ) q^{11} + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{13} + ( -\zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{14} + ( -1 + \zeta_{24}^{4} ) q^{16} + ( \zeta_{24} + 2 \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{17} + ( -2 + 2 \zeta_{24} + \zeta_{24}^{3} - 2 \zeta_{24}^{4} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{19} + 3 q^{22} + ( 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{23} + ( -2 \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{26} + ( 1 + 2 \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{4} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{28} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 3 \zeta_{24}^{6} ) q^{29} + ( -2 + 3 \zeta_{24} + 6 \zeta_{24}^{3} + \zeta_{24}^{4} - 6 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{31} + ( \zeta_{24}^{2} - \zeta_{24}^{6} ) q^{32} + ( 2 + \zeta_{24} - \zeta_{24}^{3} - 4 \zeta_{24}^{4} + \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{34} + ( 4 + 3 \zeta_{24}^{3} - 4 \zeta_{24}^{4} - 3 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{37} + ( \zeta_{24} + 2 \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{38} + ( -2 \zeta_{24} + 8 \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{41} + ( -4 - 3 \zeta_{24} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} ) q^{43} -3 \zeta_{24}^{2} q^{44} + ( -3 \zeta_{24} - 3 \zeta_{24}^{7} ) q^{46} + ( -2 \zeta_{24} + 2 \zeta_{24}^{2} + \zeta_{24}^{3} + \zeta_{24}^{5} - 4 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{47} + ( 5 - 4 \zeta_{24} - 2 \zeta_{24}^{3} - 5 \zeta_{24}^{4} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{49} + ( \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{52} + ( -3 \zeta_{24} + 3 \zeta_{24}^{2} - 3 \zeta_{24}^{6} + 3 \zeta_{24}^{7} ) q^{53} + ( \zeta_{24} - \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{56} + ( -3 + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{4} - 3 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{58} + ( 4 \zeta_{24} - \zeta_{24}^{2} - 8 \zeta_{24}^{3} - 8 \zeta_{24}^{5} - \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{59} + ( 2 + 2 \zeta_{24} + \zeta_{24}^{3} + 2 \zeta_{24}^{4} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{61} + ( -3 \zeta_{24} + 2 \zeta_{24}^{2} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} - \zeta_{24}^{6} + 6 \zeta_{24}^{7} ) q^{62} - q^{64} -10 \zeta_{24}^{4} q^{67} + ( 2 \zeta_{24} - 2 \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} + 4 \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{68} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} + 6 \zeta_{24}^{6} ) q^{71} + ( -4 - 2 \zeta_{24} - 4 \zeta_{24}^{3} + 2 \zeta_{24}^{4} + 4 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{73} + ( -3 \zeta_{24} - 4 \zeta_{24}^{2} + 4 \zeta_{24}^{6} + 3 \zeta_{24}^{7} ) q^{74} + ( 2 + \zeta_{24} - \zeta_{24}^{3} - 4 \zeta_{24}^{4} + \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{76} + ( -6 \zeta_{24} + 3 \zeta_{24}^{2} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{77} + ( 7 + 3 \zeta_{24}^{3} - 7 \zeta_{24}^{4} - 3 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{79} + ( -4 + 4 \zeta_{24} + 2 \zeta_{24}^{3} - 4 \zeta_{24}^{4} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{82} + ( 2 \zeta_{24} - 2 \zeta_{24}^{2} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{83} + ( 4 \zeta_{24}^{2} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{86} + 3 \zeta_{24}^{4} q^{88} + ( 6 \zeta_{24}^{2} - 12 \zeta_{24}^{6} ) q^{89} + ( -\zeta_{24} - 2 \zeta_{24}^{3} - 6 \zeta_{24}^{4} + 2 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{91} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} ) q^{92} + ( -4 + \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{4} - 2 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{94} + ( 5 + 2 \zeta_{24} - 2 \zeta_{24}^{3} - 10 \zeta_{24}^{4} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{97} + ( -2 \zeta_{24} - 5 \zeta_{24}^{2} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 5 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{4} - 4q^{7} + O(q^{10})$$ $$8q + 4q^{4} - 4q^{7} - 4q^{16} - 24q^{19} + 24q^{22} + 4q^{28} - 12q^{31} + 16q^{37} - 32q^{43} + 20q^{49} - 12q^{58} + 24q^{61} - 8q^{64} - 40q^{67} - 24q^{73} + 28q^{79} - 48q^{82} + 12q^{88} - 24q^{91} - 24q^{94} + O(q^{100})$$

## Character values

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

 $$n$$ $$127$$ $$451$$ $$2801$$ $$\chi(n)$$ $$1$$ $$1 - \zeta_{24}^{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1151.1
 −0.965926 + 0.258819i 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
−0.866025 + 0.500000i 0 0.500000 0.866025i 0 0 −2.62132 0.358719i 1.00000i 0 0
1151.2 −0.866025 + 0.500000i 0 0.500000 0.866025i 0 0 1.62132 + 2.09077i 1.00000i 0 0
1151.3 0.866025 0.500000i 0 0.500000 0.866025i 0 0 −2.62132 0.358719i 1.00000i 0 0
1151.4 0.866025 0.500000i 0 0.500000 0.866025i 0 0 1.62132 + 2.09077i 1.00000i 0 0
1601.1 −0.866025 0.500000i 0 0.500000 + 0.866025i 0 0 −2.62132 + 0.358719i 1.00000i 0 0
1601.2 −0.866025 0.500000i 0 0.500000 + 0.866025i 0 0 1.62132 2.09077i 1.00000i 0 0
1601.3 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0 0 −2.62132 + 0.358719i 1.00000i 0 0
1601.4 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0 0 1.62132 2.09077i 1.00000i 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1601.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g 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.bf.a 8
3.b odd 2 1 inner 3150.2.bf.a 8
5.b even 2 1 126.2.k.a 8
5.c odd 4 1 3150.2.bp.b 8
5.c odd 4 1 3150.2.bp.e 8
7.d odd 6 1 inner 3150.2.bf.a 8
15.d odd 2 1 126.2.k.a 8
15.e even 4 1 3150.2.bp.b 8
15.e even 4 1 3150.2.bp.e 8
20.d odd 2 1 1008.2.bt.c 8
21.g even 6 1 inner 3150.2.bf.a 8
35.c odd 2 1 882.2.k.a 8
35.i odd 6 1 126.2.k.a 8
35.i odd 6 1 882.2.d.a 8
35.j even 6 1 882.2.d.a 8
35.j even 6 1 882.2.k.a 8
35.k even 12 1 3150.2.bp.b 8
35.k even 12 1 3150.2.bp.e 8
45.h odd 6 1 1134.2.l.f 8
45.h odd 6 1 1134.2.t.e 8
45.j even 6 1 1134.2.l.f 8
45.j even 6 1 1134.2.t.e 8
60.h even 2 1 1008.2.bt.c 8
105.g even 2 1 882.2.k.a 8
105.o odd 6 1 882.2.d.a 8
105.o odd 6 1 882.2.k.a 8
105.p even 6 1 126.2.k.a 8
105.p even 6 1 882.2.d.a 8
105.w odd 12 1 3150.2.bp.b 8
105.w odd 12 1 3150.2.bp.e 8
140.p odd 6 1 7056.2.k.f 8
140.s even 6 1 1008.2.bt.c 8
140.s even 6 1 7056.2.k.f 8
315.q odd 6 1 1134.2.t.e 8
315.u even 6 1 1134.2.l.f 8
315.bn odd 6 1 1134.2.l.f 8
315.bq even 6 1 1134.2.t.e 8
420.ba even 6 1 7056.2.k.f 8
420.be odd 6 1 1008.2.bt.c 8
420.be odd 6 1 7056.2.k.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.k.a 8 5.b even 2 1
126.2.k.a 8 15.d odd 2 1
126.2.k.a 8 35.i odd 6 1
126.2.k.a 8 105.p even 6 1
882.2.d.a 8 35.i odd 6 1
882.2.d.a 8 35.j even 6 1
882.2.d.a 8 105.o odd 6 1
882.2.d.a 8 105.p even 6 1
882.2.k.a 8 35.c odd 2 1
882.2.k.a 8 35.j even 6 1
882.2.k.a 8 105.g even 2 1
882.2.k.a 8 105.o odd 6 1
1008.2.bt.c 8 20.d odd 2 1
1008.2.bt.c 8 60.h even 2 1
1008.2.bt.c 8 140.s even 6 1
1008.2.bt.c 8 420.be odd 6 1
1134.2.l.f 8 45.h odd 6 1
1134.2.l.f 8 45.j even 6 1
1134.2.l.f 8 315.u even 6 1
1134.2.l.f 8 315.bn odd 6 1
1134.2.t.e 8 45.h odd 6 1
1134.2.t.e 8 45.j even 6 1
1134.2.t.e 8 315.q odd 6 1
1134.2.t.e 8 315.bq even 6 1
3150.2.bf.a 8 1.a even 1 1 trivial
3150.2.bf.a 8 3.b odd 2 1 inner
3150.2.bf.a 8 7.d odd 6 1 inner
3150.2.bf.a 8 21.g even 6 1 inner
3150.2.bp.b 8 5.c odd 4 1
3150.2.bp.b 8 15.e even 4 1
3150.2.bp.b 8 35.k even 12 1
3150.2.bp.b 8 105.w odd 12 1
3150.2.bp.e 8 5.c odd 4 1
3150.2.bp.e 8 15.e even 4 1
3150.2.bp.e 8 35.k even 12 1
3150.2.bp.e 8 105.w odd 12 1
7056.2.k.f 8 140.p odd 6 1
7056.2.k.f 8 140.s even 6 1
7056.2.k.f 8 420.ba even 6 1
7056.2.k.f 8 420.be odd 6 1

## Hecke kernels

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

 $$T_{11}^{4} - 9 T_{11}^{2} + 81$$ $$T_{37}^{4} - 8 T_{37}^{3} + 66 T_{37}^{2} + 16 T_{37} + 4$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{2} + T^{4} )^{2}$$
$3$ 
$5$ 
$7$ $$( 1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4} )^{2}$$
$11$ $$( 1 + 13 T^{2} + 48 T^{4} + 1573 T^{6} + 14641 T^{8} )^{2}$$
$13$ $$( 1 - 20 T^{2} + 169 T^{4} )^{4}$$
$17$ $$1 - 32 T^{2} + 478 T^{4} + 1024 T^{6} - 81341 T^{8} + 295936 T^{10} + 39923038 T^{12} - 772402208 T^{14} + 6975757441 T^{16}$$
$19$ $$( 1 + 12 T + 92 T^{2} + 528 T^{3} + 2487 T^{4} + 10032 T^{5} + 33212 T^{6} + 82308 T^{7} + 130321 T^{8} )^{2}$$
$23$ $$( 1 + 28 T^{2} + 255 T^{4} + 14812 T^{6} + 279841 T^{8} )^{2}$$
$29$ $$( 1 - 62 T^{2} + 1995 T^{4} - 52142 T^{6} + 707281 T^{8} )^{2}$$
$31$ $$( 1 + 6 T + 23 T^{2} + 66 T^{3} - 468 T^{4} + 2046 T^{5} + 22103 T^{6} + 178746 T^{7} + 923521 T^{8} )^{2}$$
$37$ $$( 1 - 8 T - 8 T^{2} + 16 T^{3} + 1447 T^{4} + 592 T^{5} - 10952 T^{6} - 405224 T^{7} + 1874161 T^{8} )^{2}$$
$41$ $$( 1 + 20 T^{2} - 1146 T^{4} + 33620 T^{6} + 2825761 T^{8} )^{2}$$
$43$ $$( 1 + 8 T + 84 T^{2} + 344 T^{3} + 1849 T^{4} )^{4}$$
$47$ $$1 - 152 T^{2} + 13198 T^{4} - 834176 T^{6} + 42212419 T^{8} - 1842694784 T^{10} + 64402029838 T^{12} - 1638440730008 T^{14} + 23811286661761 T^{16}$$
$53$ $$1 + 158 T^{2} + 13753 T^{4} + 883694 T^{6} + 47672164 T^{8} + 2482296446 T^{10} + 108517785193 T^{12} + 3501969058382 T^{14} + 62259690411361 T^{16}$$
$59$ $$1 - 38 T^{2} - 4727 T^{4} + 30058 T^{6} + 20937316 T^{8} + 104631898 T^{10} - 57278765447 T^{12} - 1602860278358 T^{14} + 146830437604321 T^{16}$$
$61$ $$( 1 - 12 T + 176 T^{2} - 1536 T^{3} + 15591 T^{4} - 93696 T^{5} + 654896 T^{6} - 2723772 T^{7} + 13845841 T^{8} )^{2}$$
$67$ $$( 1 + 10 T + 33 T^{2} + 670 T^{3} + 4489 T^{4} )^{4}$$
$71$ $$( 1 - 176 T^{2} + 15234 T^{4} - 887216 T^{6} + 25411681 T^{8} )^{2}$$
$73$ $$( 1 + 12 T + 182 T^{2} + 1608 T^{3} + 16131 T^{4} + 117384 T^{5} + 969878 T^{6} + 4668204 T^{7} + 28398241 T^{8} )^{2}$$
$79$ $$( 1 - 14 T + 7 T^{2} - 434 T^{3} + 13996 T^{4} - 34286 T^{5} + 43687 T^{6} - 6902546 T^{7} + 38950081 T^{8} )^{2}$$
$83$ $$( 1 + 278 T^{2} + 32811 T^{4} + 1915142 T^{6} + 47458321 T^{8} )^{2}$$
$89$ $$( 1 - 70 T^{2} - 3021 T^{4} - 554470 T^{6} + 62742241 T^{8} )^{2}$$
$97$ $$( 1 - 190 T^{2} + 20643 T^{4} - 1787710 T^{6} + 88529281 T^{8} )^{2}$$