# Properties

 Label 1960.2.q.x Level $1960$ Weight $2$ Character orbit 1960.q Analytic conductor $15.651$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1960 = 2^{3} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1960.q (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$15.6506787962$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: 8.0.21913473024.16 Defining polynomial: $$x^{8} - 2 x^{7} + 11 x^{6} - 2 x^{5} + 51 x^{4} + 162 x^{2} + 112 x + 196$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{3} -\beta_{3} q^{5} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{3} -\beta_{3} q^{5} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} ) q^{9} + ( -\beta_{1} - \beta_{6} ) q^{11} + ( 3 - \beta_{2} - \beta_{4} + \beta_{7} ) q^{13} + ( 1 - \beta_{2} ) q^{15} + ( -2 + \beta_{1} + 2 \beta_{3} - 2 \beta_{5} ) q^{17} + ( -\beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{19} + ( -2 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{23} + ( -1 + \beta_{3} ) q^{25} + ( 3 + \beta_{2} - \beta_{4} - \beta_{7} ) q^{27} + ( -1 + \beta_{2} + 3 \beta_{4} ) q^{29} + ( 2 + 2 \beta_{1} - 2 \beta_{3} - \beta_{5} - 2 \beta_{6} ) q^{31} + ( -1 + \beta_{1} + \beta_{2} - 4 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{33} + ( 2 \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{37} + ( -2 - 3 \beta_{1} + 2 \beta_{3} - 3 \beta_{6} ) q^{39} + ( 2 + 2 \beta_{2} + \beta_{4} + 2 \beta_{7} ) q^{41} + ( -2 + 2 \beta_{4} - 2 \beta_{7} ) q^{43} + ( -1 - \beta_{1} + \beta_{3} + \beta_{6} ) q^{45} + ( 1 - \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{47} + ( -1 + \beta_{1} + \beta_{2} + 7 \beta_{4} + 2 \beta_{5} - 7 \beta_{6} + 2 \beta_{7} ) q^{51} + ( 2 \beta_{1} + \beta_{5} + 2 \beta_{6} ) q^{53} + ( 1 - \beta_{2} + \beta_{4} ) q^{55} + ( -2 + 4 \beta_{4} + 2 \beta_{7} ) q^{57} + ( -2 + 2 \beta_{3} + 5 \beta_{6} ) q^{59} + ( 2 - 2 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{61} + ( -1 + \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{65} + ( 2 - 2 \beta_{3} - \beta_{5} - 6 \beta_{6} ) q^{67} + ( 6 + 2 \beta_{4} - \beta_{7} ) q^{69} + ( 2 - 2 \beta_{2} + 4 \beta_{7} ) q^{71} + ( -2 - 4 \beta_{1} + 2 \beta_{3} - \beta_{5} - 5 \beta_{6} ) q^{73} + ( -1 + \beta_{1} + \beta_{2} ) q^{75} + ( 1 - \beta_{1} - \beta_{2} - 6 \beta_{3} + 5 \beta_{4} - 5 \beta_{6} ) q^{79} + ( 5 - 5 \beta_{3} - 2 \beta_{5} ) q^{81} + ( 10 - 2 \beta_{2} + \beta_{4} - \beta_{7} ) q^{83} + ( 1 + \beta_{2} - 2 \beta_{7} ) q^{85} + ( 4 + \beta_{1} - 4 \beta_{3} + 3 \beta_{5} - \beta_{6} ) q^{87} + ( -10 \beta_{3} - 3 \beta_{4} + \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{89} + ( 4 - 4 \beta_{1} - 4 \beta_{2} + 6 \beta_{3} + 2 \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{93} + ( \beta_{5} - \beta_{6} ) q^{95} + ( 7 - \beta_{2} + 2 \beta_{4} + 2 \beta_{7} ) q^{97} + ( 4 - 2 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 2q^{3} - 4q^{5} - 6q^{9} + O(q^{10})$$ $$8q - 2q^{3} - 4q^{5} - 6q^{9} - 2q^{11} + 20q^{13} + 4q^{15} - 6q^{17} + 4q^{23} - 4q^{25} + 28q^{27} - 4q^{29} + 12q^{31} - 18q^{33} - 14q^{39} + 24q^{41} - 16q^{43} - 6q^{45} + 2q^{47} - 2q^{51} + 4q^{53} + 4q^{55} - 16q^{57} - 8q^{59} - 20q^{61} - 10q^{65} + 8q^{67} + 48q^{69} + 8q^{71} - 16q^{73} - 2q^{75} - 22q^{79} + 20q^{81} + 72q^{83} + 12q^{85} + 18q^{87} - 40q^{89} + 32q^{93} + 52q^{97} + 24q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2 x^{7} + 11 x^{6} - 2 x^{5} + 51 x^{4} + 162 x^{2} + 112 x + 196$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$329 \nu^{7} + 154 \nu^{6} + 2209 \nu^{5} + 4794 \nu^{4} + 28611 \nu^{3} + 20492 \nu^{2} + 19740 \nu + 169702$$$$)/84134$$ $$\beta_{3}$$ $$=$$ $$($$$$3056 \nu^{7} - 8415 \nu^{6} + 32538 \nu^{5} - 21575 \nu^{4} + 122298 \nu^{3} - 200277 \nu^{2} + 351628 \nu + 204092$$$$)/588938$$ $$\beta_{4}$$ $$=$$ $$($$$$21 \nu^{7} - 68 \nu^{6} + 141 \nu^{5} + 306 \nu^{4} - 353 \nu^{3} + 1308 \nu^{2} + 1260 \nu + 8108$$$$)/3658$$ $$\beta_{5}$$ $$=$$ $$($$$$13 \nu^{7} + 45 \nu^{6} - 174 \nu^{5} + 712 \nu^{4} - 654 \nu^{3} + 1071 \nu^{2} - 2878 \nu + 2058$$$$)/1829$$ $$\beta_{6}$$ $$=$$ $$($$$$-3270 \nu^{7} + 11895 \nu^{6} - 45994 \nu^{5} + 84562 \nu^{4} - 172874 \nu^{3} + 283101 \nu^{2} - 238267 \nu + 543998$$$$)/294469$$ $$\beta_{7}$$ $$=$$ $$($$$$-1064 \nu^{7} + 397 \nu^{6} - 7144 \nu^{5} - 15504 \nu^{4} - 25401 \nu^{3} - 66272 \nu^{2} - 63840 \nu - 138894$$$$)/42067$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{6} + \beta_{4} - 4 \beta_{3} + \beta_{2} + \beta_{1} - 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} + \beta_{4} + 5 \beta_{2} - 9$$ $$\nu^{4}$$ $$=$$ $$9 \beta_{6} - 2 \beta_{5} + 22 \beta_{3} - 9 \beta_{1} - 22$$ $$\nu^{5}$$ $$=$$ $$-11 \beta_{7} + 17 \beta_{6} - 11 \beta_{5} - 17 \beta_{4} + 40 \beta_{3} - 31 \beta_{2} - 31 \beta_{1} + 31$$ $$\nu^{6}$$ $$=$$ $$-28 \beta_{7} - 75 \beta_{4} - 71 \beta_{2} + 217$$ $$\nu^{7}$$ $$=$$ $$-183 \beta_{6} + 103 \beta_{5} - 340 \beta_{3} + 217 \beta_{1} + 340$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1960\mathbb{Z}\right)^\times$$.

 $$n$$ $$981$$ $$1081$$ $$1177$$ $$1471$$ $$\chi(n)$$ $$1$$ $$-\beta_{3}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
361.1
 1.43998 − 2.49412i 1.09199 − 1.89138i −0.591990 + 1.02536i −0.939980 + 1.62809i 1.43998 + 2.49412i 1.09199 + 1.89138i −0.591990 − 1.02536i −0.939980 − 1.62809i
0 −1.43998 + 2.49412i 0 −0.500000 0.866025i 0 0 0 −2.64709 4.58489i 0
361.2 0 −1.09199 + 1.89138i 0 −0.500000 0.866025i 0 0 0 −0.884883 1.53266i 0
361.3 0 0.591990 1.02536i 0 −0.500000 0.866025i 0 0 0 0.799096 + 1.38408i 0
361.4 0 0.939980 1.62809i 0 −0.500000 0.866025i 0 0 0 −0.267126 0.462676i 0
961.1 0 −1.43998 2.49412i 0 −0.500000 + 0.866025i 0 0 0 −2.64709 + 4.58489i 0
961.2 0 −1.09199 1.89138i 0 −0.500000 + 0.866025i 0 0 0 −0.884883 + 1.53266i 0
961.3 0 0.591990 + 1.02536i 0 −0.500000 + 0.866025i 0 0 0 0.799096 1.38408i 0
961.4 0 0.939980 + 1.62809i 0 −0.500000 + 0.866025i 0 0 0 −0.267126 + 0.462676i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 961.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1960.2.q.x 8
7.b odd 2 1 1960.2.q.y 8
7.c even 3 1 1960.2.a.y yes 4
7.c even 3 1 inner 1960.2.q.x 8
7.d odd 6 1 1960.2.a.x 4
7.d odd 6 1 1960.2.q.y 8
28.f even 6 1 3920.2.a.ce 4
28.g odd 6 1 3920.2.a.cd 4
35.i odd 6 1 9800.2.a.cs 4
35.j even 6 1 9800.2.a.cl 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1960.2.a.x 4 7.d odd 6 1
1960.2.a.y yes 4 7.c even 3 1
1960.2.q.x 8 1.a even 1 1 trivial
1960.2.q.x 8 7.c even 3 1 inner
1960.2.q.y 8 7.b odd 2 1
1960.2.q.y 8 7.d odd 6 1
3920.2.a.cd 4 28.g odd 6 1
3920.2.a.ce 4 28.f even 6 1
9800.2.a.cl 4 35.j even 6 1
9800.2.a.cs 4 35.i odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1960, [\chi])$$:

 $$T_{3}^{8} + 2 T_{3}^{7} + 11 T_{3}^{6} + 2 T_{3}^{5} + 51 T_{3}^{4} + 162 T_{3}^{2} - 112 T_{3} + 196$$ $$T_{11}^{8} + \cdots$$ $$T_{13}^{4} - 10 T_{13}^{3} + 19 T_{13}^{2} + 52 T_{13} - 124$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$196 - 112 T + 162 T^{2} + 51 T^{4} + 2 T^{5} + 11 T^{6} + 2 T^{7} + T^{8}$$
$5$ $$( 1 + T + T^{2} )^{4}$$
$7$ $$T^{8}$$
$11$ $$16 + 80 T + 356 T^{2} + 236 T^{3} + 165 T^{4} + 18 T^{5} + 15 T^{6} + 2 T^{7} + T^{8}$$
$13$ $$( -124 + 52 T + 19 T^{2} - 10 T^{3} + T^{4} )^{2}$$
$17$ $$188356 - 105896 T + 81670 T^{2} + 7236 T^{3} + 3631 T^{4} + 182 T^{5} + 87 T^{6} + 6 T^{7} + T^{8}$$
$19$ $$64 - 192 T + 784 T^{2} + 624 T^{3} + 668 T^{4} + 48 T^{5} + 26 T^{6} + T^{8}$$
$23$ $$256 - 768 T + 2912 T^{2} + 1952 T^{3} + 1236 T^{4} + 248 T^{5} + 54 T^{6} - 4 T^{7} + T^{8}$$
$29$ $$( 188 - 20 T - 43 T^{2} + 2 T^{3} + T^{4} )^{2}$$
$31$ $$614656 - 225792 T + 90784 T^{2} - 15936 T^{3} + 4340 T^{4} - 696 T^{5} + 134 T^{6} - 12 T^{7} + T^{8}$$
$37$ $$64 + 320 T + 1936 T^{2} - 1680 T^{3} + 1756 T^{4} - 80 T^{5} + 42 T^{6} + T^{8}$$
$41$ $$( -4228 + 1528 T - 96 T^{2} - 12 T^{3} + T^{4} )^{2}$$
$43$ $$( 752 - 192 T - 48 T^{2} + 8 T^{3} + T^{4} )^{2}$$
$47$ $$3136 + 5376 T + 6920 T^{2} + 3712 T^{3} + 1545 T^{4} + 274 T^{5} + 45 T^{6} - 2 T^{7} + T^{8}$$
$53$ $$322624 - 240832 T + 140016 T^{2} - 34224 T^{3} + 7164 T^{4} - 568 T^{5} + 86 T^{6} - 4 T^{7} + T^{8}$$
$59$ $$( 2116 - 184 T + 62 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$61$ $$28558336 + 5814272 T + 1418880 T^{2} + 165888 T^{3} + 29040 T^{4} + 3056 T^{5} + 356 T^{6} + 20 T^{7} + T^{8}$$
$67$ $$5456896 + 1009152 T + 452928 T^{2} - 11872 T^{3} + 14116 T^{4} + 48 T^{5} + 178 T^{6} - 8 T^{7} + T^{8}$$
$71$ $$( 10976 - 252 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$73$ $$201412864 + 53134848 T + 11604896 T^{2} + 1090624 T^{3} + 102996 T^{4} + 4768 T^{5} + 426 T^{6} + 16 T^{7} + T^{8}$$
$79$ $$3136 + 27328 T + 242232 T^{2} - 33160 T^{3} + 16121 T^{4} + 2582 T^{5} + 411 T^{6} + 22 T^{7} + T^{8}$$
$83$ $$( 256 - 1600 T + 418 T^{2} - 36 T^{3} + T^{4} )^{2}$$
$89$ $$29246464 + 16872960 T + 6716736 T^{2} + 1308320 T^{3} + 181156 T^{4} + 16080 T^{5} + 1042 T^{6} + 40 T^{7} + T^{8}$$
$97$ $$( -1022 - 36 T + 157 T^{2} - 26 T^{3} + T^{4} )^{2}$$