# Properties

 Label 21.34.c.a Level $21$ Weight $34$ Character orbit 21.c Analytic conductor $144.864$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$21 = 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$34$$ Character orbit: $$[\chi]$$ $$=$$ 21.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$144.863940650$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2\cdot 3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -43046721 + 86093442 \zeta_{6} ) q^{3} + 8589934592 q^{4} + ( 49967189809217 - 101524131070974 \zeta_{6} ) q^{7} -5559060566555523 q^{9} +O(q^{10})$$ $$q +(-43046721 + 86093442 \zeta_{6}) q^{3} +8589934592 q^{4} +(49967189809217 - 101524131070974 \zeta_{6}) q^{7} -5559060566555523 q^{9} +(-369768517790072832 + 739537035580145664 \zeta_{6}) q^{12} +(-2738982428485658748 + 5477964856971317496 \zeta_{6}) q^{13} +73786976294838206464 q^{16} +($$$$11\!\cdots\!10$$$$-$$$$23\!\cdots\!20$$$$\zeta_{6}) q^{19} +($$$$65\!\cdots\!51$$$$- 68433587236834121340 \zeta_{6}) q^{21} -$$$$11\!\cdots\!25$$$$q^{25} +($$$$23\!\cdots\!83$$$$-$$$$47\!\cdots\!66$$$$\zeta_{6}) q^{27} +($$$$42\!\cdots\!64$$$$-$$$$87\!\cdots\!08$$$$\zeta_{6}) q^{28} +(-$$$$42\!\cdots\!10$$$$+$$$$84\!\cdots\!20$$$$\zeta_{6}) q^{31} -$$$$47\!\cdots\!16$$$$q^{36} +$$$$62\!\cdots\!30$$$$q^{37} -$$$$35\!\cdots\!24$$$$q^{39} -$$$$59\!\cdots\!40$$$$q^{43} +(-$$$$31\!\cdots\!44$$$$+$$$$63\!\cdots\!88$$$$\zeta_{6}) q^{48} +(-$$$$78\!\cdots\!87$$$$+$$$$16\!\cdots\!60$$$$\zeta_{6}) q^{49} +(-$$$$23\!\cdots\!16$$$$+$$$$47\!\cdots\!32$$$$\zeta_{6}) q^{52} +$$$$14\!\cdots\!30$$$$q^{57} +(-$$$$17\!\cdots\!40$$$$+$$$$34\!\cdots\!80$$$$\zeta_{6}) q^{61} +(-$$$$27\!\cdots\!91$$$$+$$$$56\!\cdots\!02$$$$\zeta_{6}) q^{63} +$$$$63\!\cdots\!88$$$$q^{64} -$$$$19\!\cdots\!60$$$$q^{67} +(-$$$$13\!\cdots\!12$$$$+$$$$26\!\cdots\!24$$$$\zeta_{6}) q^{73} +($$$$50\!\cdots\!25$$$$-$$$$10\!\cdots\!50$$$$\zeta_{6}) q^{75} +($$$$99\!\cdots\!20$$$$-$$$$19\!\cdots\!40$$$$\zeta_{6}) q^{76} -$$$$38\!\cdots\!84$$$$q^{79} +$$$$30\!\cdots\!29$$$$q^{81} +($$$$56\!\cdots\!92$$$$-$$$$58\!\cdots\!80$$$$\zeta_{6}) q^{84} +($$$$41\!\cdots\!88$$$$-$$$$43\!\cdots\!20$$$$\zeta_{6}) q^{91} -$$$$54\!\cdots\!30$$$$q^{93} +(-$$$$40\!\cdots\!44$$$$+$$$$80\!\cdots\!88$$$$\zeta_{6}) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 17179869184q^{4} - 1589751452540q^{7} - 11118121133111046q^{9} + O(q^{10})$$ $$2q + 17179869184q^{4} - 1589751452540q^{7} - 11118121133111046q^{9} +$$$$14\!\cdots\!28$$$$q^{16} +$$$$13\!\cdots\!62$$$$q^{21} -$$$$23\!\cdots\!50$$$$q^{25} -$$$$13\!\cdots\!80$$$$q^{28} -$$$$95\!\cdots\!32$$$$q^{36} +$$$$12\!\cdots\!60$$$$q^{37} -$$$$70\!\cdots\!48$$$$q^{39} -$$$$11\!\cdots\!80$$$$q^{43} -$$$$15\!\cdots\!14$$$$q^{49} +$$$$29\!\cdots\!60$$$$q^{57} +$$$$88\!\cdots\!20$$$$q^{63} +$$$$12\!\cdots\!76$$$$q^{64} -$$$$39\!\cdots\!20$$$$q^{67} -$$$$76\!\cdots\!68$$$$q^{79} +$$$$61\!\cdots\!58$$$$q^{81} +$$$$11\!\cdots\!04$$$$q^{84} +$$$$83\!\cdots\!56$$$$q^{91} -$$$$10\!\cdots\!60$$$$q^{93} + O(q^{100})$$

## Character values

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

 $$n$$ $$8$$ $$10$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
20.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 7.45591e7i 8.58993e9 0 0 −7.94876e11 + 8.79225e13i 0 −5.55906e15 0
20.2 0 7.45591e7i 8.58993e9 0 0 −7.94876e11 8.79225e13i 0 −5.55906e15 0
 $$n$$: 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 CM by $$\Q(\sqrt{-3})$$
7.b odd 2 1 inner
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.34.c.a 2
3.b odd 2 1 CM 21.34.c.a 2
7.b odd 2 1 inner 21.34.c.a 2
21.c even 2 1 inner 21.34.c.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.34.c.a 2 1.a even 1 1 trivial
21.34.c.a 2 3.b odd 2 1 CM
21.34.c.a 2 7.b odd 2 1 inner
21.34.c.a 2 21.c even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{34}^{\mathrm{new}}(21, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$5559060566555523 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$77\!\cdots\!07$$$$+ 1589751452540 T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$22\!\cdots\!12$$$$+ T^{2}$$
$17$ $$T^{2}$$
$19$ $$39\!\cdots\!00$$$$+ T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$53\!\cdots\!00$$$$+ T^{2}$$
$37$ $$( -$$$$62\!\cdots\!30$$$$+ T )^{2}$$
$41$ $$T^{2}$$
$43$ $$($$$$59\!\cdots\!40$$$$+ T )^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$88\!\cdots\!00$$$$+ T^{2}$$
$67$ $$($$$$19\!\cdots\!60$$$$+ T )^{2}$$
$71$ $$T^{2}$$
$73$ $$53\!\cdots\!32$$$$+ T^{2}$$
$79$ $$($$$$38\!\cdots\!84$$$$+ T )^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$48\!\cdots\!08$$$$+ T^{2}$$