# Properties

 Label 420.2.w.a Level $420$ Weight $2$ Character orbit 420.w Analytic conductor $3.354$ Analytic rank $0$ Dimension $8$ CM discriminant -84 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$420 = 2^{2} \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 420.w (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.35371688489$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.12745506816.1 Defining polynomial: $$x^{8} + 23 x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{2} ) q^{2} + \beta_{1} q^{3} -2 \beta_{2} q^{4} + ( -\beta_{1} - \beta_{3} ) q^{5} + ( -\beta_{1} - \beta_{7} ) q^{6} + ( \beta_{1} + \beta_{3} - \beta_{4} - \beta_{7} ) q^{7} + ( 2 + 2 \beta_{2} ) q^{8} + 3 \beta_{2} q^{9} +O(q^{10})$$ $$q + ( -1 + \beta_{2} ) q^{2} + \beta_{1} q^{3} -2 \beta_{2} q^{4} + ( -\beta_{1} - \beta_{3} ) q^{5} + ( -\beta_{1} - \beta_{7} ) q^{6} + ( \beta_{1} + \beta_{3} - \beta_{4} - \beta_{7} ) q^{7} + ( 2 + 2 \beta_{2} ) q^{8} + 3 \beta_{2} q^{9} + ( \beta_{3} + \beta_{4} + \beta_{7} ) q^{10} + ( \beta_{5} - \beta_{6} ) q^{11} + 2 \beta_{7} q^{12} + ( -\beta_{1} - 2 \beta_{3} + \beta_{7} ) q^{14} + ( -1 - \beta_{2} + \beta_{6} ) q^{15} -4 q^{16} + ( \beta_{1} + \beta_{3} - \beta_{4} - 4 \beta_{7} ) q^{17} + ( -3 - 3 \beta_{2} ) q^{18} + ( \beta_{1} - 2 \beta_{3} - \beta_{7} ) q^{19} + ( 2 \beta_{1} - 2 \beta_{4} - 2 \beta_{7} ) q^{20} + ( -1 + \beta_{5} - \beta_{6} ) q^{21} + ( 2 \beta_{2} + 2 \beta_{6} ) q^{22} + ( 1 - \beta_{2} - 2 \beta_{5} ) q^{23} + ( 2 \beta_{1} - 2 \beta_{7} ) q^{24} + ( -2 - \beta_{2} - \beta_{5} - \beta_{6} ) q^{25} -3 \beta_{7} q^{27} + ( 2 \beta_{3} + 2 \beta_{4} ) q^{28} + ( 3 - \beta_{2} - \beta_{5} - \beta_{6} ) q^{30} + ( \beta_{1} - 4 \beta_{4} - 3 \beta_{7} ) q^{31} + ( 4 - 4 \beta_{2} ) q^{32} + ( \beta_{1} - 3 \beta_{3} - 3 \beta_{4} ) q^{33} + ( -4 \beta_{1} - 2 \beta_{3} + 4 \beta_{7} ) q^{34} + ( 4 + 4 \beta_{2} + \beta_{6} ) q^{35} + 6 q^{36} + ( 5 - 3 \beta_{2} + 2 \beta_{6} ) q^{37} + ( -4 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} ) q^{38} + ( -4 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{7} ) q^{40} + ( -\beta_{1} + 2 \beta_{4} + \beta_{7} ) q^{41} + ( 1 + \beta_{2} + 2 \beta_{6} ) q^{42} + ( -4 \beta_{2} - 2 \beta_{5} - 2 \beta_{6} ) q^{44} + ( -3 \beta_{1} + 3 \beta_{4} + 3 \beta_{7} ) q^{45} + ( -2 + 2 \beta_{5} - 2 \beta_{6} ) q^{46} -4 \beta_{1} q^{48} -7 \beta_{2} q^{49} + ( 1 - \beta_{2} + 2 \beta_{5} ) q^{50} + ( -10 + \beta_{5} - \beta_{6} ) q^{51} + ( -3 \beta_{1} + 3 \beta_{7} ) q^{54} + ( \beta_{1} - \beta_{3} + 3 \beta_{4} - 2 \beta_{7} ) q^{55} + ( 2 \beta_{1} - 4 \beta_{4} - 2 \beta_{7} ) q^{56} + ( -5 + 7 \beta_{2} + 2 \beta_{6} ) q^{57} + ( -4 + 4 \beta_{2} + 2 \beta_{5} ) q^{60} + ( -4 \beta_{1} - 4 \beta_{3} + 4 \beta_{4} + 6 \beta_{7} ) q^{62} + ( -3 \beta_{3} - 3 \beta_{4} ) q^{63} + 8 \beta_{2} q^{64} + ( -4 \beta_{1} + 6 \beta_{4} + 2 \beta_{7} ) q^{66} + ( 6 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} ) q^{68} + ( 3 \beta_{1} + 6 \beta_{3} - 3 \beta_{7} ) q^{69} + ( -7 - \beta_{2} - \beta_{5} - \beta_{6} ) q^{70} + ( 10 + \beta_{5} - \beta_{6} ) q^{71} + ( -6 + 6 \beta_{2} ) q^{72} + ( 6 \beta_{2} - 2 \beta_{5} - 2 \beta_{6} ) q^{74} + ( \beta_{1} + 3 \beta_{3} - 3 \beta_{4} - 3 \beta_{7} ) q^{75} + ( 6 \beta_{1} - 4 \beta_{4} + 2 \beta_{7} ) q^{76} + ( \beta_{1} + \beta_{3} - \beta_{4} + 6 \beta_{7} ) q^{77} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{80} -9 q^{81} + ( 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{7} ) q^{82} + ( -2 \beta_{2} - 2 \beta_{5} - 2 \beta_{6} ) q^{84} + ( 10 - 2 \beta_{2} - 3 \beta_{5} + \beta_{6} ) q^{85} + ( 4 \beta_{2} + 4 \beta_{5} ) q^{88} + ( -\beta_{1} + \beta_{7} ) q^{89} + ( 6 \beta_{1} + 3 \beta_{3} - 3 \beta_{4} - 3 \beta_{7} ) q^{90} + ( 2 + 2 \beta_{2} + 4 \beta_{6} ) q^{92} + ( -5 - \beta_{2} + 4 \beta_{5} ) q^{93} + ( -5 - 7 \beta_{2} - 3 \beta_{5} + \beta_{6} ) q^{95} + ( 4 \beta_{1} + 4 \beta_{7} ) q^{96} + ( 7 + 7 \beta_{2} ) q^{98} + ( 6 \beta_{2} + 3 \beta_{5} + 3 \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 8 q^{2} + 16 q^{8} + O(q^{10})$$ $$8 q - 8 q^{2} + 16 q^{8} + 8 q^{11} - 12 q^{15} - 32 q^{16} - 24 q^{18} - 8 q^{22} - 16 q^{25} + 24 q^{30} + 32 q^{32} + 28 q^{35} + 48 q^{36} + 32 q^{37} + 16 q^{50} - 72 q^{51} - 48 q^{57} - 24 q^{60} - 56 q^{70} + 88 q^{71} - 48 q^{72} - 72 q^{81} + 64 q^{85} + 16 q^{88} - 24 q^{93} - 56 q^{95} + 56 q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 23 x^{4} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{5} + 19 \nu$$$$)/5$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} + 24 \nu^{2}$$$$)/5$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} + 24 \nu^{3} + 5 \nu$$$$)/5$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} + \nu^{5} - 24 \nu^{3} + 24 \nu$$$$)/5$$ $$\beta_{5}$$ $$=$$ $$($$$$-3 \nu^{6} - \nu^{4} - 67 \nu^{2} - 9$$$$)/5$$ $$\beta_{6}$$ $$=$$ $$($$$$-3 \nu^{6} + \nu^{4} - 67 \nu^{2} + 9$$$$)/5$$ $$\beta_{7}$$ $$=$$ $$($$$$-4 \nu^{7} - 91 \nu^{3}$$$$)/5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{4} + \beta_{3} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{6} + \beta_{5} + 6 \beta_{2}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} - 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$($$$$5 \beta_{6} - 5 \beta_{5} - 18$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-19 \beta_{4} - 19 \beta_{3} + 29 \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$-12 \beta_{6} - 12 \beta_{5} - 67 \beta_{2}$$ $$\nu^{7}$$ $$=$$ $$($$$$-48 \beta_{7} + 91 \beta_{4} - 91 \beta_{3} - 91 \beta_{1}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/420\mathbb{Z}\right)^\times$$.

 $$n$$ $$211$$ $$241$$ $$281$$ $$337$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$ $$-\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
83.1
 1.54779 + 1.54779i −0.323042 − 0.323042i 0.323042 + 0.323042i −1.54779 − 1.54779i −0.323042 + 0.323042i 1.54779 − 1.54779i −1.54779 + 1.54779i 0.323042 − 0.323042i
−1.00000 + 1.00000i −1.22474 1.22474i 2.00000i 1.22474 1.87083i 2.44949 −1.87083 + 1.87083i 2.00000 + 2.00000i 3.00000i 0.646084 + 3.09557i
83.2 −1.00000 + 1.00000i −1.22474 1.22474i 2.00000i 1.22474 + 1.87083i 2.44949 1.87083 1.87083i 2.00000 + 2.00000i 3.00000i −3.09557 0.646084i
83.3 −1.00000 + 1.00000i 1.22474 + 1.22474i 2.00000i −1.22474 1.87083i −2.44949 −1.87083 + 1.87083i 2.00000 + 2.00000i 3.00000i 3.09557 + 0.646084i
83.4 −1.00000 + 1.00000i 1.22474 + 1.22474i 2.00000i −1.22474 + 1.87083i −2.44949 1.87083 1.87083i 2.00000 + 2.00000i 3.00000i −0.646084 3.09557i
167.1 −1.00000 1.00000i −1.22474 + 1.22474i 2.00000i 1.22474 1.87083i 2.44949 1.87083 + 1.87083i 2.00000 2.00000i 3.00000i −3.09557 + 0.646084i
167.2 −1.00000 1.00000i −1.22474 + 1.22474i 2.00000i 1.22474 + 1.87083i 2.44949 −1.87083 1.87083i 2.00000 2.00000i 3.00000i 0.646084 3.09557i
167.3 −1.00000 1.00000i 1.22474 1.22474i 2.00000i −1.22474 1.87083i −2.44949 1.87083 + 1.87083i 2.00000 2.00000i 3.00000i −0.646084 + 3.09557i
167.4 −1.00000 1.00000i 1.22474 1.22474i 2.00000i −1.22474 + 1.87083i −2.44949 −1.87083 1.87083i 2.00000 2.00000i 3.00000i 3.09557 0.646084i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 167.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
84.h odd 2 1 CM by $$\Q(\sqrt{-21})$$
5.c odd 4 1 inner
7.b odd 2 1 inner
12.b even 2 1 inner
35.f even 4 1 inner
60.l odd 4 1 inner
420.w even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 420.2.w.a 8
3.b odd 2 1 420.2.w.b yes 8
4.b odd 2 1 420.2.w.b yes 8
5.c odd 4 1 inner 420.2.w.a 8
7.b odd 2 1 inner 420.2.w.a 8
12.b even 2 1 inner 420.2.w.a 8
15.e even 4 1 420.2.w.b yes 8
20.e even 4 1 420.2.w.b yes 8
21.c even 2 1 420.2.w.b yes 8
28.d even 2 1 420.2.w.b yes 8
35.f even 4 1 inner 420.2.w.a 8
60.l odd 4 1 inner 420.2.w.a 8
84.h odd 2 1 CM 420.2.w.a 8
105.k odd 4 1 420.2.w.b yes 8
140.j odd 4 1 420.2.w.b yes 8
420.w even 4 1 inner 420.2.w.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.w.a 8 1.a even 1 1 trivial
420.2.w.a 8 5.c odd 4 1 inner
420.2.w.a 8 7.b odd 2 1 inner
420.2.w.a 8 12.b even 2 1 inner
420.2.w.a 8 35.f even 4 1 inner
420.2.w.a 8 60.l odd 4 1 inner
420.2.w.a 8 84.h odd 2 1 CM
420.2.w.a 8 420.w even 4 1 inner
420.2.w.b yes 8 3.b odd 2 1
420.2.w.b yes 8 4.b odd 2 1
420.2.w.b yes 8 15.e even 4 1
420.2.w.b yes 8 20.e even 4 1
420.2.w.b yes 8 21.c even 2 1
420.2.w.b yes 8 28.d even 2 1
420.2.w.b yes 8 105.k odd 4 1
420.2.w.b yes 8 140.j odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{2} - 2 T_{11} - 20$$ acting on $$S_{2}^{\mathrm{new}}(420, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 2 + 2 T + T^{2} )^{4}$$
$3$ $$( 9 + T^{4} )^{2}$$
$5$ $$( 25 + 4 T^{2} + T^{4} )^{2}$$
$7$ $$( 49 + T^{4} )^{2}$$
$11$ $$( -20 - 2 T + T^{2} )^{4}$$
$13$ $$T^{8}$$
$17$ $$160000 + 3824 T^{4} + T^{8}$$
$19$ $$( 100 + 76 T^{2} + T^{4} )^{2}$$
$23$ $$( 1764 + T^{4} )^{2}$$
$29$ $$T^{8}$$
$31$ $$( 2500 - 124 T^{2} + T^{4} )^{2}$$
$37$ $$( 100 + 160 T + 128 T^{2} - 16 T^{3} + T^{4} )^{2}$$
$41$ $$( -14 + T^{2} )^{4}$$
$43$ $$T^{8}$$
$47$ $$T^{8}$$
$53$ $$T^{8}$$
$59$ $$T^{8}$$
$61$ $$T^{8}$$
$67$ $$T^{8}$$
$71$ $$( 100 - 22 T + T^{2} )^{4}$$
$73$ $$T^{8}$$
$79$ $$T^{8}$$
$83$ $$T^{8}$$
$89$ $$( 6 + T^{2} )^{4}$$
$97$ $$T^{8}$$