# Properties

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

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## 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: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{4}$$ 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.

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

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

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

## 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 −0.437016 + 0.437016i 1.14412 − 1.14412i 0.437016 + 0.437016i −1.14412 − 1.14412i 1.14412 + 1.14412i −0.437016 − 0.437016i
1.00000i 0 −1.00000 0 0 −2.12132 1.58114i 1.00000i 0 0
251.2 1.00000i 0 −1.00000 0 0 −2.12132 + 1.58114i 1.00000i 0 0
251.3 1.00000i 0 −1.00000 0 0 2.12132 1.58114i 1.00000i 0 0
251.4 1.00000i 0 −1.00000 0 0 2.12132 + 1.58114i 1.00000i 0 0
251.5 1.00000i 0 −1.00000 0 0 −2.12132 1.58114i 1.00000i 0 0
251.6 1.00000i 0 −1.00000 0 0 −2.12132 + 1.58114i 1.00000i 0 0
251.7 1.00000i 0 −1.00000 0 0 2.12132 1.58114i 1.00000i 0 0
251.8 1.00000i 0 −1.00000 0 0 2.12132 + 1.58114i 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.c 8
3.b odd 2 1 inner 3150.2.b.c 8
5.b even 2 1 inner 3150.2.b.c 8
5.c odd 4 1 630.2.d.a 4
5.c odd 4 1 630.2.d.d yes 4
7.b odd 2 1 inner 3150.2.b.c 8
15.d odd 2 1 inner 3150.2.b.c 8
15.e even 4 1 630.2.d.a 4
15.e even 4 1 630.2.d.d yes 4
20.e even 4 1 5040.2.k.a 4
20.e even 4 1 5040.2.k.d 4
21.c even 2 1 inner 3150.2.b.c 8
35.c odd 2 1 inner 3150.2.b.c 8
35.f even 4 1 630.2.d.a 4
35.f even 4 1 630.2.d.d yes 4
60.l odd 4 1 5040.2.k.a 4
60.l odd 4 1 5040.2.k.d 4
105.g even 2 1 inner 3150.2.b.c 8
105.k odd 4 1 630.2.d.a 4
105.k odd 4 1 630.2.d.d yes 4
140.j odd 4 1 5040.2.k.a 4
140.j odd 4 1 5040.2.k.d 4
420.w even 4 1 5040.2.k.a 4
420.w even 4 1 5040.2.k.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.d.a 4 5.c odd 4 1
630.2.d.a 4 15.e even 4 1
630.2.d.a 4 35.f even 4 1
630.2.d.a 4 105.k odd 4 1
630.2.d.d yes 4 5.c odd 4 1
630.2.d.d yes 4 15.e even 4 1
630.2.d.d yes 4 35.f even 4 1
630.2.d.d yes 4 105.k odd 4 1
3150.2.b.c 8 1.a even 1 1 trivial
3150.2.b.c 8 3.b odd 2 1 inner
3150.2.b.c 8 5.b even 2 1 inner
3150.2.b.c 8 7.b odd 2 1 inner
3150.2.b.c 8 15.d odd 2 1 inner
3150.2.b.c 8 21.c even 2 1 inner
3150.2.b.c 8 35.c odd 2 1 inner
3150.2.b.c 8 105.g even 2 1 inner
5040.2.k.a 4 20.e even 4 1
5040.2.k.a 4 60.l odd 4 1
5040.2.k.a 4 140.j odd 4 1
5040.2.k.a 4 420.w even 4 1
5040.2.k.d 4 20.e even 4 1
5040.2.k.d 4 60.l odd 4 1
5040.2.k.d 4 140.j odd 4 1
5040.2.k.d 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 $$S_{2}^{\mathrm{new}}(3150, [\chi])$$:

 $$T_{11}^{2} + 2$$ $$T_{13}^{2} + 10$$ $$T_{17}^{2} - 20$$ $$T_{43}^{2} - 72$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{4}$$
$3$ 
$5$ 
$7$ $$( 1 - 4 T^{2} + 49 T^{4} )^{2}$$
$11$ $$( 1 - 20 T^{2} + 121 T^{4} )^{4}$$
$13$ $$( 1 - 16 T^{2} + 169 T^{4} )^{4}$$
$17$ $$( 1 + 14 T^{2} + 289 T^{4} )^{4}$$
$19$ $$( 1 - 19 T^{2} )^{8}$$
$23$ $$( 1 - 10 T^{2} + 529 T^{4} )^{4}$$
$29$ $$( 1 - 50 T^{2} + 841 T^{4} )^{4}$$
$31$ $$( 1 - 31 T^{2} )^{8}$$
$37$ $$( 1 + 56 T^{2} + 1369 T^{4} )^{4}$$
$41$ $$( 1 - 8 T^{2} + 1681 T^{4} )^{4}$$
$43$ $$( 1 + 14 T^{2} + 1849 T^{4} )^{4}$$
$47$ $$( 1 + 74 T^{2} + 2209 T^{4} )^{4}$$
$53$ $$( 1 - 70 T^{2} + 2809 T^{4} )^{4}$$
$59$ $$( 1 + 28 T^{2} + 3481 T^{4} )^{4}$$
$61$ $$( 1 - 8 T + 61 T^{2} )^{4}( 1 + 8 T + 61 T^{2} )^{4}$$
$67$ $$( 1 + 67 T^{2} )^{8}$$
$71$ $$( 1 - 110 T^{2} + 5041 T^{4} )^{4}$$
$73$ $$( 1 - 106 T^{2} + 5329 T^{4} )^{4}$$
$79$ $$( 1 - 4 T + 79 T^{2} )^{8}$$
$83$ $$( 1 + 86 T^{2} + 6889 T^{4} )^{4}$$
$89$ $$( 1 + 88 T^{2} + 7921 T^{4} )^{4}$$
$97$ $$( 1 - 34 T^{2} + 9409 T^{4} )^{4}$$
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