# Properties

 Label 78.2.i.a Level $78$ Weight $2$ Character orbit 78.i Analytic conductor $0.623$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$78 = 2 \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 78.i (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.622833135766$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ 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 primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/78\mathbb{Z}\right)^\times$$.

 $$n$$ $$53$$ $$67$$ $$\chi(n)$$ $$1$$ $$\zeta_{12}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
43.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
−0.866025 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 3.73205i 0.866025 0.500000i 2.36603 1.36603i 1.00000i −0.500000 0.866025i 1.86603 3.23205i
43.2 0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 0.267949i −0.866025 + 0.500000i 0.633975 0.366025i 1.00000i −0.500000 0.866025i 0.133975 0.232051i
49.1 −0.866025 + 0.500000i −0.500000 0.866025i 0.500000 0.866025i 3.73205i 0.866025 + 0.500000i 2.36603 + 1.36603i 1.00000i −0.500000 + 0.866025i 1.86603 + 3.23205i
49.2 0.866025 0.500000i −0.500000 0.866025i 0.500000 0.866025i 0.267949i −0.866025 0.500000i 0.633975 + 0.366025i 1.00000i −0.500000 + 0.866025i 0.133975 + 0.232051i
 $$n$$: 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
13.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 78.2.i.a 4
3.b odd 2 1 234.2.l.c 4
4.b odd 2 1 624.2.bv.e 4
5.b even 2 1 1950.2.bc.d 4
5.c odd 4 1 1950.2.y.b 4
5.c odd 4 1 1950.2.y.g 4
12.b even 2 1 1872.2.by.h 4
13.b even 2 1 1014.2.i.a 4
13.c even 3 1 1014.2.b.e 4
13.c even 3 1 1014.2.i.a 4
13.d odd 4 1 1014.2.e.g 4
13.d odd 4 1 1014.2.e.i 4
13.e even 6 1 inner 78.2.i.a 4
13.e even 6 1 1014.2.b.e 4
13.f odd 12 1 1014.2.a.i 2
13.f odd 12 1 1014.2.a.k 2
13.f odd 12 1 1014.2.e.g 4
13.f odd 12 1 1014.2.e.i 4
39.h odd 6 1 234.2.l.c 4
39.h odd 6 1 3042.2.b.i 4
39.i odd 6 1 3042.2.b.i 4
39.k even 12 1 3042.2.a.p 2
39.k even 12 1 3042.2.a.y 2
52.i odd 6 1 624.2.bv.e 4
52.l even 12 1 8112.2.a.bj 2
52.l even 12 1 8112.2.a.bp 2
65.l even 6 1 1950.2.bc.d 4
65.r odd 12 1 1950.2.y.b 4
65.r odd 12 1 1950.2.y.g 4
156.r even 6 1 1872.2.by.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.i.a 4 1.a even 1 1 trivial
78.2.i.a 4 13.e even 6 1 inner
234.2.l.c 4 3.b odd 2 1
234.2.l.c 4 39.h odd 6 1
624.2.bv.e 4 4.b odd 2 1
624.2.bv.e 4 52.i odd 6 1
1014.2.a.i 2 13.f odd 12 1
1014.2.a.k 2 13.f odd 12 1
1014.2.b.e 4 13.c even 3 1
1014.2.b.e 4 13.e even 6 1
1014.2.e.g 4 13.d odd 4 1
1014.2.e.g 4 13.f odd 12 1
1014.2.e.i 4 13.d odd 4 1
1014.2.e.i 4 13.f odd 12 1
1014.2.i.a 4 13.b even 2 1
1014.2.i.a 4 13.c even 3 1
1872.2.by.h 4 12.b even 2 1
1872.2.by.h 4 156.r even 6 1
1950.2.y.b 4 5.c odd 4 1
1950.2.y.b 4 65.r odd 12 1
1950.2.y.g 4 5.c odd 4 1
1950.2.y.g 4 65.r odd 12 1
1950.2.bc.d 4 5.b even 2 1
1950.2.bc.d 4 65.l even 6 1
3042.2.a.p 2 39.k even 12 1
3042.2.a.y 2 39.k even 12 1
3042.2.b.i 4 39.h odd 6 1
3042.2.b.i 4 39.i odd 6 1
8112.2.a.bj 2 52.l even 12 1
8112.2.a.bp 2 52.l even 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 14 T_{5}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(78, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ $$( 1 + T + T^{2} )^{2}$$
$5$ $$1 + 14 T^{2} + T^{4}$$
$7$ $$4 - 12 T + 14 T^{2} - 6 T^{3} + T^{4}$$
$11$ $$36 - 36 T + 6 T^{2} + 6 T^{3} + T^{4}$$
$13$ $$169 - T^{2} + T^{4}$$
$17$ $$169 + 104 T + 51 T^{2} + 8 T^{3} + T^{4}$$
$19$ $$36 + 36 T + 6 T^{2} - 6 T^{3} + T^{4}$$
$23$ $$676 - 52 T + 30 T^{2} + 2 T^{3} + T^{4}$$
$29$ $$121 + 22 T + 15 T^{2} - 2 T^{3} + T^{4}$$
$31$ $$64 + 32 T^{2} + T^{4}$$
$37$ $$1369 - 444 T + 11 T^{2} + 12 T^{3} + T^{4}$$
$41$ $$11449 - 3852 T + 539 T^{2} - 36 T^{3} + T^{4}$$
$43$ $$5476 - 148 T + 78 T^{2} + 2 T^{3} + T^{4}$$
$47$ $$324 + 72 T^{2} + T^{4}$$
$53$ $$( -3 + 6 T + T^{2} )^{2}$$
$59$ $$4096 - 64 T^{2} + T^{4}$$
$61$ $$121 + 88 T + 75 T^{2} - 8 T^{3} + T^{4}$$
$67$ $$21316 + 6132 T + 734 T^{2} + 42 T^{3} + T^{4}$$
$71$ $$36 + 36 T + 6 T^{2} - 6 T^{3} + T^{4}$$
$73$ $$3721 + 134 T^{2} + T^{4}$$
$79$ $$( 24 + 12 T + T^{2} )^{2}$$
$83$ $$4 + 104 T^{2} + T^{4}$$
$89$ $$576 - 288 T + 24 T^{2} + 12 T^{3} + T^{4}$$
$97$ $$1296 - 36 T^{2} + T^{4}$$