# Properties

 Label 68.2.e.a Level $68$ Weight $2$ Character orbit 68.e Analytic conductor $0.543$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$68 = 2^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 68.e (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.542982733745$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{13})$$ Defining polynomial: $$x^{4} + 7x^{2} + 9$$ x^4 + 7*x^2 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} - 1) q^{3} + ( - \beta_1 - 1) q^{5} + (\beta_{3} - \beta_1 + 1) q^{7} + ( - \beta_{3} - \beta_{2} + 4 \beta_1) q^{9}+O(q^{10})$$ q + (b2 - 1) * q^3 + (-b1 - 1) * q^5 + (b3 - b1 + 1) * q^7 + (-b3 - b2 + 4*b1) * q^9 $$q + (\beta_{2} - 1) q^{3} + ( - \beta_1 - 1) q^{5} + (\beta_{3} - \beta_1 + 1) q^{7} + ( - \beta_{3} - \beta_{2} + 4 \beta_1) q^{9} + ( - \beta_{3} - \beta_1 + 1) q^{11} + (\beta_{3} - \beta_{2} - \beta_1) q^{13} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{15} + (\beta_{3} - \beta_{2} - 3 \beta_1 + 1) q^{17} + (\beta_{3} + \beta_{2} + 3 \beta_1) q^{19} + ( - \beta_{3} + \beta_{2} + \beta_1 - 8) q^{21} + ( - \beta_{3} + 3 \beta_1 - 3) q^{23} - 3 \beta_1 q^{25} + (2 \beta_{3} - 8 \beta_1 + 8) q^{27} + (2 \beta_{2} + 3 \beta_1 + 1) q^{29} + ( - \beta_{2} + 2 \beta_1 + 3) q^{31} + ( - \beta_{3} + \beta_{2} + \beta_1 + 4) q^{33} + ( - \beta_{3} + \beta_{2} + \beta_1 - 2) q^{35} + (3 \beta_1 + 3) q^{37} + ( - 6 \beta_1 - 6) q^{39} + ( - \beta_1 + 1) q^{41} + (\beta_{3} + \beta_{2} + 7 \beta_1) q^{43} + (2 \beta_{3} - 5 \beta_1 + 5) q^{45} - 4 q^{47} + (\beta_{3} + \beta_{2}) q^{49} + ( - 2 \beta_{3} + \beta_{2} - 4 \beta_1 - 9) q^{51} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{53} + (\beta_{3} - \beta_{2} - \beta_1 - 2) q^{55} + (2 \beta_{3} + 4 \beta_1 - 4) q^{57} + ( - \beta_{3} - \beta_{2} + 5 \beta_1) q^{59} + ( - 4 \beta_{3} + \beta_1 - 1) q^{61} + ( - 5 \beta_{2} + 6 \beta_1 + 11) q^{63} + (2 \beta_{2} + 2 \beta_1) q^{65} + (2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 4) q^{67} + (3 \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 12) q^{69} + ( - 3 \beta_{2} - 4 \beta_1 - 1) q^{71} + ( - 7 \beta_1 - 7) q^{73} + ( - 3 \beta_{3} + 3 \beta_1 - 3) q^{75} + (\beta_{3} + \beta_{2} + 5 \beta_1) q^{77} + ( - \beta_{3} - \beta_1 + 1) q^{79} + ( - 5 \beta_{3} + 5 \beta_{2} + 5 \beta_1 - 13) q^{81} + (3 \beta_{3} + 3 \beta_{2} - 7 \beta_1) q^{83} + (2 \beta_{2} + 3 \beta_1 - 3) q^{85} + (\beta_{3} + \beta_{2} + 11 \beta_1) q^{87} + (3 \beta_{3} - 3 \beta_{2} - 3 \beta_1) q^{89} + ( - 6 \beta_1 + 6) q^{91} + (3 \beta_{3} + 3 \beta_{2} - 9 \beta_1) q^{93} + ( - 2 \beta_{3} - 2 \beta_1 + 2) q^{95} + (2 \beta_{2} - 3 \beta_1 - 5) q^{97} + (\beta_{2} - 1) q^{99}+O(q^{100})$$ q + (b2 - 1) * q^3 + (-b1 - 1) * q^5 + (b3 - b1 + 1) * q^7 + (-b3 - b2 + 4*b1) * q^9 + (-b3 - b1 + 1) * q^11 + (b3 - b2 - b1) * q^13 + (-b3 - b2 + b1) * q^15 + (b3 - b2 - 3*b1 + 1) * q^17 + (b3 + b2 + 3*b1) * q^19 + (-b3 + b2 + b1 - 8) * q^21 + (-b3 + 3*b1 - 3) * q^23 - 3*b1 * q^25 + (2*b3 - 8*b1 + 8) * q^27 + (2*b2 + 3*b1 + 1) * q^29 + (-b2 + 2*b1 + 3) * q^31 + (-b3 + b2 + b1 + 4) * q^33 + (-b3 + b2 + b1 - 2) * q^35 + (3*b1 + 3) * q^37 + (-6*b1 - 6) * q^39 + (-b1 + 1) * q^41 + (b3 + b2 + 7*b1) * q^43 + (2*b3 - 5*b1 + 5) * q^45 - 4 * q^47 + (b3 + b2) * q^49 + (-2*b3 + b2 - 4*b1 - 9) * q^51 + (-2*b3 - 2*b2 - 2*b1) * q^53 + (b3 - b2 - b1 - 2) * q^55 + (2*b3 + 4*b1 - 4) * q^57 + (-b3 - b2 + 5*b1) * q^59 + (-4*b3 + b1 - 1) * q^61 + (-5*b2 + 6*b1 + 11) * q^63 + (2*b2 + 2*b1) * q^65 + (2*b3 - 2*b2 - 2*b1 + 4) * q^67 + (3*b3 - 3*b2 - 3*b1 + 12) * q^69 + (-3*b2 - 4*b1 - 1) * q^71 + (-7*b1 - 7) * q^73 + (-3*b3 + 3*b1 - 3) * q^75 + (b3 + b2 + 5*b1) * q^77 + (-b3 - b1 + 1) * q^79 + (-5*b3 + 5*b2 + 5*b1 - 13) * q^81 + (3*b3 + 3*b2 - 7*b1) * q^83 + (2*b2 + 3*b1 - 3) * q^85 + (b3 + b2 + 11*b1) * q^87 + (3*b3 - 3*b2 - 3*b1) * q^89 + (-6*b1 + 6) * q^91 + (3*b3 + 3*b2 - 9*b1) * q^93 + (-2*b3 - 2*b1 + 2) * q^95 + (2*b2 - 3*b1 - 5) * q^97 + (b2 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{3} - 4 q^{5} + 2 q^{7}+O(q^{10})$$ 4 * q - 2 * q^3 - 4 * q^5 + 2 * q^7 $$4 q - 2 q^{3} - 4 q^{5} + 2 q^{7} + 6 q^{11} - 4 q^{13} - 28 q^{21} - 10 q^{23} + 28 q^{27} + 8 q^{29} + 10 q^{31} + 20 q^{33} - 4 q^{35} + 12 q^{37} - 24 q^{39} + 4 q^{41} + 16 q^{45} - 16 q^{47} - 30 q^{51} - 12 q^{55} - 20 q^{57} + 4 q^{61} + 34 q^{63} + 4 q^{65} + 8 q^{67} + 36 q^{69} - 10 q^{71} - 28 q^{73} - 6 q^{75} + 6 q^{79} - 32 q^{81} - 8 q^{85} - 12 q^{89} + 24 q^{91} + 12 q^{95} - 16 q^{97} - 2 q^{99}+O(q^{100})$$ 4 * q - 2 * q^3 - 4 * q^5 + 2 * q^7 + 6 * q^11 - 4 * q^13 - 28 * q^21 - 10 * q^23 + 28 * q^27 + 8 * q^29 + 10 * q^31 + 20 * q^33 - 4 * q^35 + 12 * q^37 - 24 * q^39 + 4 * q^41 + 16 * q^45 - 16 * q^47 - 30 * q^51 - 12 * q^55 - 20 * q^57 + 4 * q^61 + 34 * q^63 + 4 * q^65 + 8 * q^67 + 36 * q^69 - 10 * q^71 - 28 * q^73 - 6 * q^75 + 6 * q^79 - 32 * q^81 - 8 * q^85 - 12 * q^89 + 24 * q^91 + 12 * q^95 - 16 * q^97 - 2 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 7x^{2} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 4\nu ) / 3$$ (v^3 + 4*v) / 3 $$\beta_{2}$$ $$=$$ $$\nu^{2} + \nu + 4$$ v^2 + v + 4 $$\beta_{3}$$ $$=$$ $$( \nu^{3} - 3\nu^{2} + 7\nu - 12 ) / 3$$ (v^3 - 3*v^2 + 7*v - 12) / 3
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} - \beta_1 ) / 2$$ (b3 + b2 - b1) / 2 $$\nu^{2}$$ $$=$$ $$( -\beta_{3} + \beta_{2} + \beta _1 - 8 ) / 2$$ (-b3 + b2 + b1 - 8) / 2 $$\nu^{3}$$ $$=$$ $$-2\beta_{3} - 2\beta_{2} + 5\beta_1$$ -2*b3 - 2*b2 + 5*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/68\mathbb{Z}\right)^\times$$.

 $$n$$ $$35$$ $$37$$ $$\chi(n)$$ $$1$$ $$\beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1
 − 2.30278i 1.30278i 2.30278i − 1.30278i
0 −2.30278 2.30278i 0 −1.00000 1.00000i 0 2.30278 2.30278i 0 7.60555i 0
13.2 0 1.30278 + 1.30278i 0 −1.00000 1.00000i 0 −1.30278 + 1.30278i 0 0.394449i 0
21.1 0 −2.30278 + 2.30278i 0 −1.00000 + 1.00000i 0 2.30278 + 2.30278i 0 7.60555i 0
21.2 0 1.30278 1.30278i 0 −1.00000 + 1.00000i 0 −1.30278 1.30278i 0 0.394449i 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
17.c even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 68.2.e.a 4
3.b odd 2 1 612.2.k.e 4
4.b odd 2 1 272.2.o.g 4
5.b even 2 1 1700.2.o.c 4
5.c odd 4 1 1700.2.m.a 4
5.c odd 4 1 1700.2.m.b 4
8.b even 2 1 1088.2.o.t 4
8.d odd 2 1 1088.2.o.s 4
12.b even 2 1 2448.2.be.u 4
17.b even 2 1 1156.2.e.c 4
17.c even 4 1 inner 68.2.e.a 4
17.c even 4 1 1156.2.e.c 4
17.d even 8 2 1156.2.a.h 4
17.d even 8 2 1156.2.b.a 4
17.e odd 16 8 1156.2.h.e 16
51.f odd 4 1 612.2.k.e 4
68.f odd 4 1 272.2.o.g 4
68.g odd 8 2 4624.2.a.bq 4
85.f odd 4 1 1700.2.m.b 4
85.i odd 4 1 1700.2.m.a 4
85.j even 4 1 1700.2.o.c 4
136.i even 4 1 1088.2.o.t 4
136.j odd 4 1 1088.2.o.s 4
204.l even 4 1 2448.2.be.u 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.2.e.a 4 1.a even 1 1 trivial
68.2.e.a 4 17.c even 4 1 inner
272.2.o.g 4 4.b odd 2 1
272.2.o.g 4 68.f odd 4 1
612.2.k.e 4 3.b odd 2 1
612.2.k.e 4 51.f odd 4 1
1088.2.o.s 4 8.d odd 2 1
1088.2.o.s 4 136.j odd 4 1
1088.2.o.t 4 8.b even 2 1
1088.2.o.t 4 136.i even 4 1
1156.2.a.h 4 17.d even 8 2
1156.2.b.a 4 17.d even 8 2
1156.2.e.c 4 17.b even 2 1
1156.2.e.c 4 17.c even 4 1
1156.2.h.e 16 17.e odd 16 8
1700.2.m.a 4 5.c odd 4 1
1700.2.m.a 4 85.i odd 4 1
1700.2.m.b 4 5.c odd 4 1
1700.2.m.b 4 85.f odd 4 1
1700.2.o.c 4 5.b even 2 1
1700.2.o.c 4 85.j even 4 1
2448.2.be.u 4 12.b even 2 1
2448.2.be.u 4 204.l even 4 1
4624.2.a.bq 4 68.g odd 8 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(68, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 2 T^{3} + 2 T^{2} - 12 T + 36$$
$5$ $$(T^{2} + 2 T + 2)^{2}$$
$7$ $$T^{4} - 2 T^{3} + 2 T^{2} + 12 T + 36$$
$11$ $$T^{4} - 6 T^{3} + 18 T^{2} + 12 T + 4$$
$13$ $$(T^{2} + 2 T - 12)^{2}$$
$17$ $$T^{4} - 18T^{2} + 289$$
$19$ $$T^{4} + 44T^{2} + 16$$
$23$ $$T^{4} + 10 T^{3} + 50 T^{2} + 60 T + 36$$
$29$ $$T^{4} - 8 T^{3} + 32 T^{2} + 144 T + 324$$
$31$ $$T^{4} - 10 T^{3} + 50 T^{2} - 60 T + 36$$
$37$ $$(T^{2} - 6 T + 18)^{2}$$
$41$ $$(T^{2} - 2 T + 2)^{2}$$
$43$ $$T^{4} + 124T^{2} + 1296$$
$47$ $$(T + 4)^{4}$$
$53$ $$T^{4} + 112T^{2} + 2304$$
$59$ $$T^{4} + 76T^{2} + 144$$
$61$ $$T^{4} - 4 T^{3} + 8 T^{2} + \cdots + 10404$$
$67$ $$(T^{2} - 4 T - 48)^{2}$$
$71$ $$T^{4} + 10 T^{3} + 50 T^{2} + \cdots + 2116$$
$73$ $$(T^{2} + 14 T + 98)^{2}$$
$79$ $$T^{4} - 6 T^{3} + 18 T^{2} + 12 T + 4$$
$83$ $$T^{4} + 332T^{2} + 4624$$
$89$ $$(T^{2} + 6 T - 108)^{2}$$
$97$ $$T^{4} + 16 T^{3} + 128 T^{2} + \cdots + 36$$