# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ = $$3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 3150.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$25.1528766367$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.40960000.1 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 630) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} - q^{4} + ( -\beta_{3} - \beta_{5} ) q^{7} -\beta_{1} q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} - q^{4} + ( -\beta_{3} - \beta_{5} ) q^{7} -\beta_{1} q^{8} + 4 \beta_{4} q^{11} -2 \beta_{5} q^{13} + ( \beta_{2} - \beta_{4} ) q^{14} + q^{16} + \beta_{7} q^{17} + \beta_{6} q^{19} -4 \beta_{3} q^{22} + 4 \beta_{1} q^{23} + 2 \beta_{2} q^{26} + ( \beta_{3} + \beta_{5} ) q^{28} -2 \beta_{4} q^{29} -2 \beta_{6} q^{31} + \beta_{1} q^{32} + \beta_{6} q^{34} + 7 \beta_{3} q^{37} -\beta_{7} q^{38} -2 \beta_{2} q^{41} -\beta_{3} q^{43} -4 \beta_{4} q^{44} -4 q^{46} -3 \beta_{7} q^{47} + ( -3 - 2 \beta_{6} ) q^{49} + 2 \beta_{5} q^{52} + 4 \beta_{1} q^{53} + ( -\beta_{2} + \beta_{4} ) q^{56} + 2 \beta_{3} q^{58} + 2 \beta_{2} q^{59} -3 \beta_{6} q^{61} + 2 \beta_{7} q^{62} - q^{64} + 5 \beta_{3} q^{67} -\beta_{7} q^{68} + \beta_{4} q^{71} -6 \beta_{5} q^{73} + 7 \beta_{4} q^{74} -\beta_{6} q^{76} + ( -8 \beta_{1} - 4 \beta_{7} ) q^{77} -6 q^{79} -2 \beta_{5} q^{82} -4 \beta_{7} q^{83} -\beta_{4} q^{86} + 4 \beta_{3} q^{88} -2 \beta_{2} q^{89} + ( -10 - 2 \beta_{6} ) q^{91} -4 \beta_{1} q^{92} -3 \beta_{6} q^{94} + ( -3 \beta_{1} + 2 \beta_{7} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 8q^{4} + O(q^{10})$$ $$8q - 8q^{4} + 8q^{16} - 32q^{46} - 24q^{49} - 8q^{64} - 48q^{79} - 80q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 7 x^{4} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{6} + 8 \nu^{2}$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$($$$$2 \nu^{4} + 7$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$2 \nu^{7} - \nu^{5} + 13 \nu^{3} - 5 \nu$$$$)/3$$ $$\beta_{4}$$ $$=$$ $$($$$$-2 \nu^{7} - \nu^{5} - 13 \nu^{3} - 5 \nu$$$$)/3$$ $$\beta_{5}$$ $$=$$ $$\nu^{6} + 6 \nu^{2}$$ $$\beta_{6}$$ $$=$$ $$($$$$4 \nu^{7} + \nu^{5} + 29 \nu^{3} + 11 \nu$$$$)/3$$ $$\beta_{7}$$ $$=$$ $$($$$$-4 \nu^{7} + \nu^{5} - 29 \nu^{3} + 11 \nu$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} + \beta_{6} + \beta_{4} + \beta_{3}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{5} + 3 \beta_{1}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{7} + \beta_{6} + 2 \beta_{4} - 2 \beta_{3}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$3 \beta_{2} - 7$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-5 \beta_{7} - 5 \beta_{6} - 11 \beta_{4} - 11 \beta_{3}$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$4 \beta_{5} - 9 \beta_{1}$$ $$\nu^{7}$$ $$=$$ $$($$$$13 \beta_{7} - 13 \beta_{6} - 29 \beta_{4} + 29 \beta_{3}$$$$)/4$$

## Character values

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

 $$n$$ $$127$$ $$451$$ $$2801$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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
 1.14412 − 1.14412i −0.437016 + 0.437016i −1.14412 + 1.14412i 0.437016 − 0.437016i −0.437016 − 0.437016i 1.14412 + 1.14412i 0.437016 + 0.437016i −1.14412 − 1.14412i
1.00000i 0 −1.00000 0 0 −1.41421 2.23607i 1.00000i 0 0
251.2 1.00000i 0 −1.00000 0 0 −1.41421 + 2.23607i 1.00000i 0 0
251.3 1.00000i 0 −1.00000 0 0 1.41421 2.23607i 1.00000i 0 0
251.4 1.00000i 0 −1.00000 0 0 1.41421 + 2.23607i 1.00000i 0 0
251.5 1.00000i 0 −1.00000 0 0 −1.41421 2.23607i 1.00000i 0 0
251.6 1.00000i 0 −1.00000 0 0 −1.41421 + 2.23607i 1.00000i 0 0
251.7 1.00000i 0 −1.00000 0 0 1.41421 2.23607i 1.00000i 0 0
251.8 1.00000i 0 −1.00000 0 0 1.41421 + 2.23607i 1.00000i 0 0
 $$n$$: 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
3.b odd 2 1 inner
5.b even 2 1 inner
7.b odd 2 1 inner
15.d odd 2 1 inner
21.c even 2 1 inner
35.c odd 2 1 inner
105.g even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3150.2.b.b 8
3.b odd 2 1 inner 3150.2.b.b 8
5.b even 2 1 inner 3150.2.b.b 8
5.c odd 4 1 630.2.d.b 4
5.c odd 4 1 630.2.d.c yes 4
7.b odd 2 1 inner 3150.2.b.b 8
15.d odd 2 1 inner 3150.2.b.b 8
15.e even 4 1 630.2.d.b 4
15.e even 4 1 630.2.d.c yes 4
20.e even 4 1 5040.2.k.b 4
20.e even 4 1 5040.2.k.c 4
21.c even 2 1 inner 3150.2.b.b 8
35.c odd 2 1 inner 3150.2.b.b 8
35.f even 4 1 630.2.d.b 4
35.f even 4 1 630.2.d.c yes 4
60.l odd 4 1 5040.2.k.b 4
60.l odd 4 1 5040.2.k.c 4
105.g even 2 1 inner 3150.2.b.b 8
105.k odd 4 1 630.2.d.b 4
105.k odd 4 1 630.2.d.c yes 4
140.j odd 4 1 5040.2.k.b 4
140.j odd 4 1 5040.2.k.c 4
420.w even 4 1 5040.2.k.b 4
420.w even 4 1 5040.2.k.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.d.b 4 5.c odd 4 1
630.2.d.b 4 15.e even 4 1
630.2.d.b 4 35.f even 4 1
630.2.d.b 4 105.k odd 4 1
630.2.d.c yes 4 5.c odd 4 1
630.2.d.c yes 4 15.e even 4 1
630.2.d.c yes 4 35.f even 4 1
630.2.d.c yes 4 105.k odd 4 1
3150.2.b.b 8 1.a even 1 1 trivial
3150.2.b.b 8 3.b odd 2 1 inner
3150.2.b.b 8 5.b even 2 1 inner
3150.2.b.b 8 7.b odd 2 1 inner
3150.2.b.b 8 15.d odd 2 1 inner
3150.2.b.b 8 21.c even 2 1 inner
3150.2.b.b 8 35.c odd 2 1 inner
3150.2.b.b 8 105.g even 2 1 inner
5040.2.k.b 4 20.e even 4 1
5040.2.k.b 4 60.l odd 4 1
5040.2.k.b 4 140.j odd 4 1
5040.2.k.b 4 420.w even 4 1
5040.2.k.c 4 20.e even 4 1
5040.2.k.c 4 60.l odd 4 1
5040.2.k.c 4 140.j odd 4 1
5040.2.k.c 4 420.w even 4 1

## Hecke kernels

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

 $$T_{11}^{2} + 32$$ $$T_{13}^{2} + 20$$ $$T_{17}^{2} - 10$$ $$T_{43}^{2} - 2$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{4}$$
$3$ 
$5$ 
$7$ $$( 1 + 6 T^{2} + 49 T^{4} )^{2}$$
$11$ $$( 1 + 10 T^{2} + 121 T^{4} )^{4}$$
$13$ $$( 1 - 6 T^{2} + 169 T^{4} )^{4}$$
$17$ $$( 1 + 24 T^{2} + 289 T^{4} )^{4}$$
$19$ $$( 1 - 28 T^{2} + 361 T^{4} )^{4}$$
$23$ $$( 1 - 30 T^{2} + 529 T^{4} )^{4}$$
$29$ $$( 1 - 50 T^{2} + 841 T^{4} )^{4}$$
$31$ $$( 1 - 22 T^{2} + 961 T^{4} )^{4}$$
$37$ $$( 1 - 24 T^{2} + 1369 T^{4} )^{4}$$
$41$ $$( 1 + 62 T^{2} + 1681 T^{4} )^{4}$$
$43$ $$( 1 + 84 T^{2} + 1849 T^{4} )^{4}$$
$47$ $$( 1 + 4 T^{2} + 2209 T^{4} )^{4}$$
$53$ $$( 1 - 14 T + 53 T^{2} )^{4}( 1 + 14 T + 53 T^{2} )^{4}$$
$59$ $$( 1 + 98 T^{2} + 3481 T^{4} )^{4}$$
$61$ $$( 1 - 32 T^{2} + 3721 T^{4} )^{4}$$
$67$ $$( 1 + 84 T^{2} + 4489 T^{4} )^{4}$$
$71$ $$( 1 - 140 T^{2} + 5041 T^{4} )^{4}$$
$73$ $$( 1 + 34 T^{2} + 5329 T^{4} )^{4}$$
$79$ $$( 1 + 6 T + 79 T^{2} )^{8}$$
$83$ $$( 1 + 6 T^{2} + 6889 T^{4} )^{4}$$
$89$ $$( 1 + 158 T^{2} + 7921 T^{4} )^{4}$$
$97$ $$( 1 - 97 T^{2} )^{8}$$