# Properties

 Label 21.3.h.b Level $21$ Weight $3$ Character orbit 21.h Analytic conductor $0.572$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$21 = 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 21.h (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.572208555157$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-5})$$ Defining polynomial: $$x^{4} - 5 x^{2} + 25$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( -\beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{3} + \beta_{2} q^{4} -\beta_{1} q^{5} + ( -5 - 2 \beta_{3} ) q^{6} + 7 \beta_{2} q^{7} -3 \beta_{3} q^{8} + ( 1 + 4 \beta_{1} - \beta_{2} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( -\beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{3} + \beta_{2} q^{4} -\beta_{1} q^{5} + ( -5 - 2 \beta_{3} ) q^{6} + 7 \beta_{2} q^{7} -3 \beta_{3} q^{8} + ( 1 + 4 \beta_{1} - \beta_{2} ) q^{9} -5 \beta_{2} q^{10} + ( -5 \beta_{1} + 5 \beta_{3} ) q^{11} + ( 2 - \beta_{1} - 2 \beta_{2} ) q^{12} -2 q^{13} + 7 \beta_{3} q^{14} + ( 5 + 2 \beta_{3} ) q^{15} + ( 19 - 19 \beta_{2} ) q^{16} + ( 12 \beta_{1} - 12 \beta_{3} ) q^{17} + ( \beta_{1} + 20 \beta_{2} - \beta_{3} ) q^{18} + ( -16 + 16 \beta_{2} ) q^{19} -\beta_{3} q^{20} + ( 14 - 7 \beta_{1} - 14 \beta_{2} ) q^{21} -25 q^{22} -6 \beta_{1} q^{23} + ( -6 \beta_{1} + 15 \beta_{2} + 6 \beta_{3} ) q^{24} -20 \beta_{2} q^{25} -2 \beta_{1} q^{26} + ( -22 - 7 \beta_{3} ) q^{27} + ( -7 + 7 \beta_{2} ) q^{28} + 7 \beta_{3} q^{29} + ( -10 + 5 \beta_{1} + 10 \beta_{2} ) q^{30} + 3 \beta_{2} q^{31} + ( 7 \beta_{1} - 7 \beta_{3} ) q^{32} + ( 25 + 10 \beta_{1} - 25 \beta_{2} ) q^{33} + 60 q^{34} -7 \beta_{3} q^{35} + ( 1 + 4 \beta_{3} ) q^{36} + ( -12 + 12 \beta_{2} ) q^{37} + ( -16 \beta_{1} + 16 \beta_{3} ) q^{38} + ( 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{39} + ( -15 + 15 \beta_{2} ) q^{40} + 14 \beta_{3} q^{41} + ( 14 \beta_{1} - 35 \beta_{2} - 14 \beta_{3} ) q^{42} + 44 q^{43} -5 \beta_{1} q^{44} + ( -\beta_{1} - 20 \beta_{2} + \beta_{3} ) q^{45} -30 \beta_{2} q^{46} + 6 \beta_{1} q^{47} + ( -38 + 19 \beta_{3} ) q^{48} + ( -49 + 49 \beta_{2} ) q^{49} -20 \beta_{3} q^{50} + ( -60 - 24 \beta_{1} + 60 \beta_{2} ) q^{51} -2 \beta_{2} q^{52} + ( 9 \beta_{1} - 9 \beta_{3} ) q^{53} + ( 35 - 22 \beta_{1} - 35 \beta_{2} ) q^{54} + 25 q^{55} + ( 21 \beta_{1} - 21 \beta_{3} ) q^{56} + ( 32 - 16 \beta_{3} ) q^{57} + ( -35 + 35 \beta_{2} ) q^{58} + ( -9 \beta_{1} + 9 \beta_{3} ) q^{59} + ( -2 \beta_{1} + 5 \beta_{2} + 2 \beta_{3} ) q^{60} + ( 26 - 26 \beta_{2} ) q^{61} + 3 \beta_{3} q^{62} + ( 7 + 28 \beta_{3} ) q^{63} -41 q^{64} + 2 \beta_{1} q^{65} + ( 25 \beta_{1} + 50 \beta_{2} - 25 \beta_{3} ) q^{66} -52 \beta_{2} q^{67} + 12 \beta_{1} q^{68} + ( 30 + 12 \beta_{3} ) q^{69} + ( 35 - 35 \beta_{2} ) q^{70} + 42 \beta_{3} q^{71} + ( 60 - 3 \beta_{1} - 60 \beta_{2} ) q^{72} -18 \beta_{2} q^{73} + ( -12 \beta_{1} + 12 \beta_{3} ) q^{74} + ( -40 + 20 \beta_{1} + 40 \beta_{2} ) q^{75} -16 q^{76} -35 \beta_{1} q^{77} + ( 10 + 4 \beta_{3} ) q^{78} + ( 79 - 79 \beta_{2} ) q^{79} + ( -19 \beta_{1} + 19 \beta_{3} ) q^{80} + ( 8 \beta_{1} + 79 \beta_{2} - 8 \beta_{3} ) q^{81} + ( -70 + 70 \beta_{2} ) q^{82} -63 \beta_{3} q^{83} + ( 14 - 7 \beta_{3} ) q^{84} -60 q^{85} + 44 \beta_{1} q^{86} + ( 14 \beta_{1} - 35 \beta_{2} - 14 \beta_{3} ) q^{87} + 75 \beta_{2} q^{88} -22 \beta_{1} q^{89} + ( -5 - 20 \beta_{3} ) q^{90} -14 \beta_{2} q^{91} -6 \beta_{3} q^{92} + ( 6 - 3 \beta_{1} - 6 \beta_{2} ) q^{93} + 30 \beta_{2} q^{94} + ( 16 \beta_{1} - 16 \beta_{3} ) q^{95} + ( -35 - 14 \beta_{1} + 35 \beta_{2} ) q^{96} -93 q^{97} + ( -49 \beta_{1} + 49 \beta_{3} ) q^{98} + ( -100 + 5 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{3} + 2q^{4} - 20q^{6} + 14q^{7} + 2q^{9} + O(q^{10})$$ $$4q - 4q^{3} + 2q^{4} - 20q^{6} + 14q^{7} + 2q^{9} - 10q^{10} + 4q^{12} - 8q^{13} + 20q^{15} + 38q^{16} + 40q^{18} - 32q^{19} + 28q^{21} - 100q^{22} + 30q^{24} - 40q^{25} - 88q^{27} - 14q^{28} - 20q^{30} + 6q^{31} + 50q^{33} + 240q^{34} + 4q^{36} - 24q^{37} + 8q^{39} - 30q^{40} - 70q^{42} + 176q^{43} - 40q^{45} - 60q^{46} - 152q^{48} - 98q^{49} - 120q^{51} - 4q^{52} + 70q^{54} + 100q^{55} + 128q^{57} - 70q^{58} + 10q^{60} + 52q^{61} + 28q^{63} - 164q^{64} + 100q^{66} - 104q^{67} + 120q^{69} + 70q^{70} + 120q^{72} - 36q^{73} - 80q^{75} - 64q^{76} + 40q^{78} + 158q^{79} + 158q^{81} - 140q^{82} + 56q^{84} - 240q^{85} - 70q^{87} + 150q^{88} - 20q^{90} - 28q^{91} + 12q^{93} + 60q^{94} - 70q^{96} - 372q^{97} - 400q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 5 x^{2} + 25$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/5$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$5 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$5 \beta_{3}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/21\mathbb{Z}\right)^\times$$.

 $$n$$ $$8$$ $$10$$ $$\chi(n)$$ $$-1$$ $$-1 + \beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1
 −1.93649 − 1.11803i 1.93649 + 1.11803i −1.93649 + 1.11803i 1.93649 − 1.11803i
−1.93649 1.11803i 0.936492 2.85008i 0.500000 + 0.866025i 1.93649 + 1.11803i −5.00000 + 4.47214i 3.50000 + 6.06218i 6.70820i −7.24597 5.33816i −2.50000 4.33013i
2.2 1.93649 + 1.11803i −2.93649 0.614017i 0.500000 + 0.866025i −1.93649 1.11803i −5.00000 4.47214i 3.50000 + 6.06218i 6.70820i 8.24597 + 3.60611i −2.50000 4.33013i
11.1 −1.93649 + 1.11803i 0.936492 + 2.85008i 0.500000 0.866025i 1.93649 1.11803i −5.00000 4.47214i 3.50000 6.06218i 6.70820i −7.24597 + 5.33816i −2.50000 + 4.33013i
11.2 1.93649 1.11803i −2.93649 + 0.614017i 0.500000 0.866025i −1.93649 + 1.11803i −5.00000 + 4.47214i 3.50000 6.06218i 6.70820i 8.24597 3.60611i −2.50000 + 4.33013i
 $$n$$: e.g. 2-40 or 990-1000 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
7.c even 3 1 inner
21.h odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.3.h.b 4
3.b odd 2 1 inner 21.3.h.b 4
4.b odd 2 1 336.3.bn.f 4
7.b odd 2 1 147.3.h.b 4
7.c even 3 1 inner 21.3.h.b 4
7.c even 3 1 147.3.b.d 2
7.d odd 6 1 147.3.b.c 2
7.d odd 6 1 147.3.h.b 4
12.b even 2 1 336.3.bn.f 4
21.c even 2 1 147.3.h.b 4
21.g even 6 1 147.3.b.c 2
21.g even 6 1 147.3.h.b 4
21.h odd 6 1 inner 21.3.h.b 4
21.h odd 6 1 147.3.b.d 2
28.g odd 6 1 336.3.bn.f 4
84.n even 6 1 336.3.bn.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.3.h.b 4 1.a even 1 1 trivial
21.3.h.b 4 3.b odd 2 1 inner
21.3.h.b 4 7.c even 3 1 inner
21.3.h.b 4 21.h odd 6 1 inner
147.3.b.c 2 7.d odd 6 1
147.3.b.c 2 21.g even 6 1
147.3.b.d 2 7.c even 3 1
147.3.b.d 2 21.h odd 6 1
147.3.h.b 4 7.b odd 2 1
147.3.h.b 4 7.d odd 6 1
147.3.h.b 4 21.c even 2 1
147.3.h.b 4 21.g even 6 1
336.3.bn.f 4 4.b odd 2 1
336.3.bn.f 4 12.b even 2 1
336.3.bn.f 4 28.g odd 6 1
336.3.bn.f 4 84.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 5 T_{2}^{2} + 25$$ acting on $$S_{3}^{\mathrm{new}}(21, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$25 - 5 T^{2} + T^{4}$$
$3$ $$81 + 36 T + 7 T^{2} + 4 T^{3} + T^{4}$$
$5$ $$25 - 5 T^{2} + T^{4}$$
$7$ $$( 49 - 7 T + T^{2} )^{2}$$
$11$ $$15625 - 125 T^{2} + T^{4}$$
$13$ $$( 2 + T )^{4}$$
$17$ $$518400 - 720 T^{2} + T^{4}$$
$19$ $$( 256 + 16 T + T^{2} )^{2}$$
$23$ $$32400 - 180 T^{2} + T^{4}$$
$29$ $$( 245 + T^{2} )^{2}$$
$31$ $$( 9 - 3 T + T^{2} )^{2}$$
$37$ $$( 144 + 12 T + T^{2} )^{2}$$
$41$ $$( 980 + T^{2} )^{2}$$
$43$ $$( -44 + T )^{4}$$
$47$ $$32400 - 180 T^{2} + T^{4}$$
$53$ $$164025 - 405 T^{2} + T^{4}$$
$59$ $$164025 - 405 T^{2} + T^{4}$$
$61$ $$( 676 - 26 T + T^{2} )^{2}$$
$67$ $$( 2704 + 52 T + T^{2} )^{2}$$
$71$ $$( 8820 + T^{2} )^{2}$$
$73$ $$( 324 + 18 T + T^{2} )^{2}$$
$79$ $$( 6241 - 79 T + T^{2} )^{2}$$
$83$ $$( 19845 + T^{2} )^{2}$$
$89$ $$5856400 - 2420 T^{2} + T^{4}$$
$97$ $$( 93 + T )^{4}$$