Properties

 Label 3150.3.c.b Level $3150$ Weight $3$ Character orbit 3150.c Analytic conductor $85.831$ 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: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 3150.c (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$85.8312832735$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.157351936.1 Defining polynomial: $$x^{8} + x^{4} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{29}]$$ Coefficient ring index: $$2^{8}$$ Twist minimal: no (minimal twist has level 126) 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.

 $$f(q)$$ $$=$$ $$q -\beta_{3} q^{2} + 2 q^{4} + \beta_{1} q^{7} -2 \beta_{3} q^{8} +O(q^{10})$$ $$q -\beta_{3} q^{2} + 2 q^{4} + \beta_{1} q^{7} -2 \beta_{3} q^{8} + ( -2 \beta_{5} - 4 \beta_{6} ) q^{11} + ( -4 \beta_{1} - 4 \beta_{2} ) q^{13} -\beta_{6} q^{14} + 4 q^{16} + ( -13 \beta_{3} + 2 \beta_{7} ) q^{17} -20 q^{19} + ( 8 \beta_{1} - 2 \beta_{2} ) q^{22} + ( -2 \beta_{3} + 4 \beta_{7} ) q^{23} + ( -8 \beta_{5} + 4 \beta_{6} ) q^{26} + 2 \beta_{1} q^{28} + ( 19 \beta_{5} - 4 \beta_{6} ) q^{29} + ( 4 + 4 \beta_{4} ) q^{31} -4 \beta_{3} q^{32} + ( 26 - 2 \beta_{4} ) q^{34} -19 \beta_{2} q^{37} + 20 \beta_{3} q^{38} + ( 27 \beta_{5} - 6 \beta_{6} ) q^{41} + ( 24 \beta_{1} + 10 \beta_{2} ) q^{43} + ( -4 \beta_{5} - 8 \beta_{6} ) q^{44} + ( 4 - 4 \beta_{4} ) q^{46} -12 \beta_{3} q^{47} -7 q^{49} + ( -8 \beta_{1} - 8 \beta_{2} ) q^{52} + ( -3 \beta_{3} - 24 \beta_{7} ) q^{53} -2 \beta_{6} q^{56} + ( 8 \beta_{1} + 19 \beta_{2} ) q^{58} + ( -20 \beta_{5} + 8 \beta_{6} ) q^{59} + ( 58 - 8 \beta_{4} ) q^{61} + ( -4 \beta_{3} - 8 \beta_{7} ) q^{62} + 8 q^{64} + ( 32 \beta_{1} + 24 \beta_{2} ) q^{67} + ( -26 \beta_{3} + 4 \beta_{7} ) q^{68} + ( -2 \beta_{5} - 4 \beta_{6} ) q^{71} + ( -20 \beta_{1} - 12 \beta_{2} ) q^{73} -38 \beta_{5} q^{74} -40 q^{76} + ( 28 \beta_{3} - 2 \beta_{7} ) q^{77} + ( -76 + 8 \beta_{4} ) q^{79} + ( 12 \beta_{1} + 27 \beta_{2} ) q^{82} + ( 64 \beta_{3} - 8 \beta_{7} ) q^{83} + ( 20 \beta_{5} - 24 \beta_{6} ) q^{86} + ( 16 \beta_{1} - 4 \beta_{2} ) q^{88} + ( -51 \beta_{5} - 18 \beta_{6} ) q^{89} + ( 28 + 4 \beta_{4} ) q^{91} + ( -4 \beta_{3} + 8 \beta_{7} ) q^{92} + 24 q^{94} + ( 44 \beta_{1} + 36 \beta_{2} ) q^{97} + 7 \beta_{3} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 16q^{4} + O(q^{10})$$ $$8q + 16q^{4} + 32q^{16} - 160q^{19} + 32q^{31} + 208q^{34} + 32q^{46} - 56q^{49} + 464q^{61} + 64q^{64} - 320q^{76} - 608q^{79} + 224q^{91} + 192q^{94} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + x^{4} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$2 \nu^{4} + 1$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} + 5 \nu^{2}$$$$)/6$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{7} - 4 \nu^{5} + 7 \nu^{3} + 4 \nu$$$$)/24$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{6} + 3 \nu^{2}$$$$)/2$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{7} - 4 \nu^{5} - 7 \nu^{3} + 4 \nu$$$$)/24$$ $$\beta_{6}$$ $$=$$ $$($$$$5 \nu^{7} + 4 \nu^{5} + 13 \nu^{3} + 44 \nu$$$$)/24$$ $$\beta_{7}$$ $$=$$ $$($$$$-5 \nu^{7} + 4 \nu^{5} - 13 \nu^{3} + 44 \nu$$$$)/24$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} + \beta_{6} + \beta_{5} + \beta_{3}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{4} + 3 \beta_{2}$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{7} + \beta_{6} - 5 \beta_{5} + 5 \beta_{3}$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$3 \beta_{1} - 1$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$\beta_{7} + \beta_{6} - 11 \beta_{5} - 11 \beta_{3}$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$-5 \beta_{4} + 9 \beta_{2}$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$-7 \beta_{7} + 7 \beta_{6} + 13 \beta_{5} - 13 \beta_{3}$$$$)/4$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 1.28897 − 0.581861i −0.581861 − 1.28897i −0.581861 + 1.28897i 1.28897 + 0.581861i 0.581861 + 1.28897i −1.28897 + 0.581861i −1.28897 − 0.581861i 0.581861 − 1.28897i
−1.41421 0 2.00000 0 0 2.64575i −2.82843 0 0
449.2 −1.41421 0 2.00000 0 0 2.64575i −2.82843 0 0
449.3 −1.41421 0 2.00000 0 0 2.64575i −2.82843 0 0
449.4 −1.41421 0 2.00000 0 0 2.64575i −2.82843 0 0
449.5 1.41421 0 2.00000 0 0 2.64575i 2.82843 0 0
449.6 1.41421 0 2.00000 0 0 2.64575i 2.82843 0 0
449.7 1.41421 0 2.00000 0 0 2.64575i 2.82843 0 0
449.8 1.41421 0 2.00000 0 0 2.64575i 2.82843 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 449.8 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
15.d odd 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3150.3.c.b 8
3.b odd 2 1 inner 3150.3.c.b 8
5.b even 2 1 inner 3150.3.c.b 8
5.c odd 4 1 126.3.b.a 4
5.c odd 4 1 3150.3.e.e 4
15.d odd 2 1 inner 3150.3.c.b 8
15.e even 4 1 126.3.b.a 4
15.e even 4 1 3150.3.e.e 4
20.e even 4 1 1008.3.d.a 4
35.f even 4 1 882.3.b.f 4
35.k even 12 2 882.3.s.i 8
35.l odd 12 2 882.3.s.e 8
40.i odd 4 1 4032.3.d.i 4
40.k even 4 1 4032.3.d.j 4
45.k odd 12 2 1134.3.q.c 8
45.l even 12 2 1134.3.q.c 8
60.l odd 4 1 1008.3.d.a 4
105.k odd 4 1 882.3.b.f 4
105.w odd 12 2 882.3.s.i 8
105.x even 12 2 882.3.s.e 8
120.q odd 4 1 4032.3.d.j 4
120.w even 4 1 4032.3.d.i 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.3.b.a 4 5.c odd 4 1
126.3.b.a 4 15.e even 4 1
882.3.b.f 4 35.f even 4 1
882.3.b.f 4 105.k odd 4 1
882.3.s.e 8 35.l odd 12 2
882.3.s.e 8 105.x even 12 2
882.3.s.i 8 35.k even 12 2
882.3.s.i 8 105.w odd 12 2
1008.3.d.a 4 20.e even 4 1
1008.3.d.a 4 60.l odd 4 1
1134.3.q.c 8 45.k odd 12 2
1134.3.q.c 8 45.l even 12 2
3150.3.c.b 8 1.a even 1 1 trivial
3150.3.c.b 8 3.b odd 2 1 inner
3150.3.c.b 8 5.b even 2 1 inner
3150.3.c.b 8 15.d odd 2 1 inner
3150.3.e.e 4 5.c odd 4 1
3150.3.e.e 4 15.e even 4 1
4032.3.d.i 4 40.i odd 4 1
4032.3.d.i 4 120.w even 4 1
4032.3.d.j 4 40.k even 4 1
4032.3.d.j 4 120.q odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{4} + 464 T_{11}^{2} + 46656$$ acting on $$S_{3}^{\mathrm{new}}(3150, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -2 + T^{2} )^{4}$$
$3$ $$T^{8}$$
$5$ $$T^{8}$$
$7$ $$( 7 + T^{2} )^{4}$$
$11$ $$( 46656 + 464 T^{2} + T^{4} )^{2}$$
$13$ $$( 2304 + 352 T^{2} + T^{4} )^{2}$$
$17$ $$( 79524 - 788 T^{2} + T^{4} )^{2}$$
$19$ $$( 20 + T )^{8}$$
$23$ $$( 46656 - 464 T^{2} + T^{4} )^{2}$$
$29$ $$( 248004 + 1892 T^{2} + T^{4} )^{2}$$
$31$ $$( -432 - 8 T + T^{2} )^{4}$$
$37$ $$( 1444 + T^{2} )^{4}$$
$41$ $$( 910116 + 3924 T^{2} + T^{4} )^{2}$$
$43$ $$( 13191424 + 8864 T^{2} + T^{4} )^{2}$$
$47$ $$( -288 + T^{2} )^{4}$$
$53$ $$( 64738116 - 16164 T^{2} + T^{4} )^{2}$$
$59$ $$( 9216 + 3392 T^{2} + T^{4} )^{2}$$
$61$ $$( 1572 - 116 T + T^{2} )^{4}$$
$67$ $$( 23658496 + 18944 T^{2} + T^{4} )^{2}$$
$71$ $$( 46656 + 464 T^{2} + T^{4} )^{2}$$
$73$ $$( 4946176 + 6752 T^{2} + T^{4} )^{2}$$
$79$ $$( 3984 + 152 T + T^{2} )^{4}$$
$83$ $$( 53231616 - 18176 T^{2} + T^{4} )^{2}$$
$89$ $$( 443556 + 19476 T^{2} + T^{4} )^{2}$$
$97$ $$( 70023424 + 37472 T^{2} + T^{4} )^{2}$$